Electromagnetic and Optical Pulse Propagation 2: Temporal Pulse Dynamics in Dispersive, Attenuative Media (Springer Series in Optical Sciences)

Springer Series in

optical sciences founded by H.K.V. Lotsch Editor-in-Chief: W. T. Rhodes, Atlanta Editorial Board: A. Adibi, Atlanta T. Asakura, Sapporo T. W. H¨ansch, Garching T. Kamiya, Tokyo F. Krausz, Garching B. Monemar, Link¨oping H. Venghaus, Berlin H. Weber, Berlin H. Weinfurter, M¨unchen

For other titles published in this series, go to http://www.springer.com/series/624

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Springer Series in

optical sciences The Springer Series in Optical Sciences, under the leadership of Editor-in-Chief William T. Rhodes, Georgia Institute of Technology, USA, provides an expanding selection of research monographs in all major areas of optics: lasers and quantum optics, ultrafast phenomena, optical spectroscopy techniques, optoelectronics, quantum information, information optics, applied laser technology, industrial applications, and other topics of contemporary interest. With this broad coverage of topics, the series is of use to all research scientists and engineers who need up-to-date reference books. The editors encourage prospective authors to correspond with them in advance of submitting a manuscript. Submission of manuscripts should be made to the Editor-in-Chief or one of the Editors. See also www.springer.com/series/624

Editor-in-Chief William T. Rhodes Georgia Institute of Technology School of Electrical and Computer Engineering Atlanta, GA 30332-0250, USA E-mail: [email protected]

Editorial Board Ali Adibi

Bo Monemar

Georgia Institute of Technology School of Electrical and Computer Engineering Atlanta, GA 30332-0250, USA E-mail: [email protected]

Department of Physics and Measurement Technology Materials Science Division Link¨oping University 58183 Link¨oping, Sweden E-mail: [email protected]

Toshimitsu Asakura Hokkai-Gakuen University Faculty of Engineering 1-1, Minami-26, Nishi 11, Chuo-ku Sapporo, Hokkaido 064-0926, Japan E-mail: [email protected]

Theodor W. H¨ansch Max-Planck-Institut f¨ur Quantenoptik Hans-Kopfermann-Straße 1 85748 Garching, Germany E-mail: [email protected]

Takeshi Kamiya Ministry of Education, Culture, Sports Science and Technology National Institution for Academic Degrees 3-29-1 Otsuka, Bunkyo-ku Tokyo 112-0012, Japan E-mail: [email protected]

Ferenc Krausz Ludwig-Maximilians-Universit¨at M¨unchen Lehrstuhl f¨ur Experimentelle Physik Am Coulombwall 1 85748 Garching, Germany and Max-Planck-Institut f¨ur Quantenoptik Hans-Kopfermann-Straße 1 85748 Garching, Germany E-mail: [email protected]

Herbert Venghaus Fraunhofer Institut f¨ur Nachrichtentechnik Heinrich-Hertz-Institut Einsteinufer 37 10587 Berlin, Germany E-mail: [email protected]

Horst Weber Technische Universit¨at Berlin Optisches Institut Straße des 17. Juni 135 10623 Berlin, Germany E-mail: [email protected]

Harald Weinfurter Ludwig-Maximilians-Universit¨at M¨unchen Sektion Physik Schellingstraße 4/III 80799 M¨unchen, Germany E-mail: [email protected]

Kurt E. Oughstun

Electromagnetic and Optical Pulse Propagation 2 Temporal Pulse Dynamics in Dispersive, Attenuative Media

With 350Figures

ABC

Professor Dr. Kurt E. Oughstun University of Vermont College of Engineering & Mathematical Sciences School of Engineering Burlington, VT 05405-0156 USA [email protected]

ISSN 0342-4111 e-ISSN 1556-1534 ISBN 978-1-4419-0148-4 e-ISBN 978-1-4419-0149-1 DOI 10.1007/978-1-4419-0149-1 Springer Dordrecht Heidelberg London New York Library of Congress Control Number: 2009929777 c Springer Science+Business Media, LLC 2009 ° All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer Science+Business Media, LLC, 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks, and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com)

Chosen by beauty to be a handmaiden of the stars, she passes like a silver brush across the lens of a telescope. She brushes the stars, the galaxies and the light-years into the order that we know them. Rommel Drives on Deep into Egypt Poems by Richard Brautigan This volume is dedicated to my best friend Joyce and to our daughters Marcianna & Kristen In memory of my parents Edmund Waldemar Oughstun & Ruth Kinat Oughstun (New Britain, Connecticut) and my grandparents Julies E. Oughstun & Adeline B. Lehmann (Kalwari, Prussia) Heinrich Kinat & Wanda Bucholz (Sladow, Austria-Poland) and my great-grandfather Karl Øvsttun (Øvsttun, Norway)

Preface

This two volume graduate text presents a systematic theoretical treatment of the radiation and propagation of pulsed electromagnetic and optical fields through temporally dispersive and attenuative media. Although such fields are often referred to as transient, they may be short-lived only in the sense of an observation made at some fixed point in space. In particular, because of their unique properties when the initial pulse spectrum is ultrawideband with respect to the material dispersion, specific features of the propagated pulse are found to persist in time long after the main body of the pulse has become exponentially small. The subject matter divides naturally into two volumes. Volume I presents a detailed development of the fundamental theory of pulsed electromagnetic radiation and wave propagation in causal, linear media that are homogeneous and isotropic but otherwise have rather general dispersive and absorptive properties. In Vol. II, the analysis is specialized to the propagation of plane wave electromagnetic and optical pulses in homogeneous, isotropic, locally linear media whose temporal frequency dispersion is described by a specific causal model. Dielectric, conducting, and semiconducting material models are considered with applications to bioelectromagnetics, remote sensing, ground and foliage penetrating radar, and undersea communications. Taken together, these two volumes present sufficient material to cover a two semester graduate sequence in electromagnetic and optical wave theory in physics and electrical engineering as well as in applied mathematics. Either volume by itself could also be used as the text for a single semester graduate level course. Challenging problems are given throughout the text. The development presented in Vol. I provides a mathematically rigorous description of the fundamental time-domain electromagnetics and optics in linear temporally dispersive media. The analysis begins with a general description of macroscopic electromagnetics and the role that causality plays in the constitutive (or material) relations in linear electromagnetics and optics. The angular spectrum of plane waves representation of the pulsed radiation field in homogeneous, isotropic, locally linear, temporally dispersive media is then derived and applied to the description of pulsed electromagnetic and optical beam fields where the effects of temporal dispersion and spatial diffraction are coupled. Volume II begins with a review of pulsed electromagnetic and optical beam field propagation followed by a concise description of modern asymptotic methods of

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approximation appropriate for the description of pulse propagation in dispersive, attenuative media, including uniform and transitional asymptotic techniques. The detailed theory presented here and in Volume I provides the necessary mathematical and physical basis to describe and explain in explicit detail the dynamical pulse evolution as it propagates through a causally dispersive material. This is the subject of a classic theory with origins in the seminal research by Arnold Sommerfield and Leon Brillouin in the early 1900s for a Lorentz model dielectric and described in modern text-books on advanced electrodynamics. This classic theory has been carefully reexamined and extended by George Sherman and myself, beginning in 1974 when I was a graduate student at The Institute of Optics of the University of Rochester and George Sherman was my research advisor. In addition to improving the accuracy of many of the approximations in the classical theory and applying modern asymptotic methods, we have developed a physical model that provides a simplified quantitative algorithm that not only describes the entire dynamical field evolution in the mature dispersion regime but also explains each feature in the propagated field in simple physical terms. This physical model reduces to the group velocity description in the limit as the material loss approaches zero. More recent analysis has extended these results to more general dispersion models, including the RocardPowles extension of the Debye model of orientational polarization in dielectrics and the Drude model of conductivity. Finally, the controversy regarding the question of superluminal pulse propagation is carefully examined in view of recent results establishing the domain of applicability of the group velocity approximation. I would like to acknowledge the financial support I received during my graduate studies by The Institute of Optics, the Corning Glass Works Foundation, the National Science Foundation, and the Center for Naval Research, as well as the encouragement and support from my thesis committee members: Professors Emil Wolf, Carlos R. Stroud, Brian J. Thompson, John H. Thomas, and George C. Sherman. This research continued while I was at the United Technologies Research Center, the University of Wisconsin at Madison, and the University of Vermont. The critical, long-term support of this research by Dr. Arje Nachmann of the United States Air Force Office of Scientific Research and Dr. Richard Albanese of Armstrong Laboratory, Brooks Air Force Base, is gratefully acknowledged. The majority of results presented here have been published in peer-reviewed journals listed in the references. A good portion of this research has been conducted with several of my former graduate students at the University of Wisconsin (Shioupyn Shen) and at the University of Vermont (Judith Laurens, Constantinos Balictsis, Paul Smith, John Marozas, Hong Xiao, and Natalie Cartwright). Their critical insight has been instrumental in several of the theoretical advances presented here. Burlington, VT January 2009

Kurt Edmund Oughstun

Contents

9

Pulsed Electromagnetic and Optical Beam Wavefields in Temporally Dispersive Media . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1 Angular Spectrum Representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1.1 Real Direction Cosine Form of the Angular Spectrum of Plane Waves Representation . . . . . . . . . . . . . . . . . . . . 9.1.2 Electromagnetic Energy Flow in the Angular Spectrum Representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1.3 Homogeneous and Evanescent Plane Wave Contributions to the Angular Spectrum Representation . . . . . 9.1.4 Paraxial Approximation of the Angular Spectrum of Plane Waves Representation . . . . . . . . . . . . . . . . . . . . 9.2 Angular Spectrum Representation of Multipole Wavefields . . . . . . . . . . 9.2.1 Multipole Expansion of the Scalar Optical Wavefield due to a Localized Source Distribution . . . . . . . . . . . 9.2.2 Multipole Expansion of the Electromagnetic Wavefield Generated by a Localized Charge–Current Distribution in a Dispersive Dielectric Medium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3 Stationary Phase Asymptotic Approximations of the Angular Spectrum Representation in Free-Space. . . . . . . . . . . . . . . 9.3.1 Approximations Valid Over a Hemisphere . . . . . . . . . . . . . . . . . . . 9.3.2 Approximations Valid on the Plane z D z0 . . . . . . . . . . . . . . . . . . . Q E .r; !/ . . . . . Q H .r; !/ and U 9.3.3 Asymptotic Approximations of U 9.3.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.4 Separable Pulsed Beam Wavefields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.4.1 Gaussian Beam Propagation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.4.2 Asymptotic Behavior . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.5 The Inverse Initial Value Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.5.1 The Direct Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.5.2 The Inverse Problem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.6 Summary. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 2 6 9 10 17 23 30

37 50 54 58 66 69 70 75 79 79 83 85 88 88 91

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10 Asymptotic Methods of Analysis using Advanced Saddle Point Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 10.1 Olver’s Saddle Point Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 10.1.1 Peak Value of the Integrand at the Endpoint of Integration. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 10.1.2 Peak Value of the Integrand at an Interior Point of the Path of Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 10.1.3 The Application of Olver’s Saddle Point Method . . . . . . . . . . . . 103 10.2 Uniform Asymptotic Expansion for Two First-Order Saddle Points at Infinity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 10.3 Uniform Asymptotic Expansion for Two First-Order Saddle Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110 10.3.1 The Uniform Asymptotic Expansion for Two Isolated First-Order Saddle Points . . . . . . . . . . . . . . . . . . 111 10.3.2 The Uniform Asymptotic Expansion for Two Neighboring First-Order Saddle Points . . . . . . . . . . . . . 113 10.3.3 The Transitional Asymptotic Approximation for Two Neighboring First-Order Saddle Points . . . . . . . . . . . . . 125 10.4 Uniform Asymptotic Expansion for a First-Order Saddle Point and a Simple Pole Singularity . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 10.4.1 The Complementary Error Function . . . . . . . . . . . . . . . . . . . . . . . . . . 131 10.4.2 Asymptotic Behavior for a Single Interacting Saddle Point. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135 10.4.3 Asymptotic Behavior for Two Isolated Interacting Saddle Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 10.5 Asymptotic Expansions of Multiple Integrals . . . . . . . . . . . . . . . . . . . . . . . . . 141 10.5.1 Absolute Maximum in the Interior of the Closure of D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 10.5.2 Absolute Maximum on the Boundary of the Closure of D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142 10.6 Summary. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146 11 The Group Velocity Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147 11.1 Historical Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147 11.2 The Pulsed Plane Wave Electromagnetic Field . . . . . . . . . . . . . . . . . . . . . . . . 152 11.2.1 The Delta Function Pulse and the Impulse Response of the Medium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160 11.2.2 The Heaviside Unit Step Function Signal . . . . . . . . . . . . . . . . . . . . 161 11.2.3 The Double Exponential Pulse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163 11.2.4 The Rectangular Pulse Envelope Modulated Signal . . . . . . . . . 163 11.2.5 The Trapezoidal Pulse Envelope Modulated Signal . . . . . . . . . 165 11.2.6 The Hyperbolic Tangent Envelope Modulated Signal . . . . . . . 169

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11.2.7 The Van Bladel Envelope Modulated Pulse . . . . . . . . . . . . . . . . . . 174 11.2.8 The Gaussian Envelope Modulated Pulse . . . . . . . . . . . . . . . . . . . . 177 11.3 Wave Equations in a Simple Dispersive Medium and the Slowly Varying Envelope Approximation . . . . . . . . . . . . . . . . . . . . . 178 11.3.1 The Dispersive Wave Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179 11.3.2 The Slowly Varying Envelope Approximation . . . . . . . . . . . . . . . 180 11.3.3 Dispersive Wave Equations for the Slowly Varying Wave Amplitude and Phase . . . . . . . . . . . . . . . . . . . . . . . . . . 185 11.4 The Classical Group Velocity Approximation . . . . . . . . . . . . . . . . . . . . . . . . . 194 11.5 Failure of the Classical Group Velocity Method . . . . . . . . . . . . . . . . . . . . . . . 200 11.5.1 Impulse Response of a Double-Resonance Lorentz Model Dielectric . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209 11.5.2 Heaviside Unit Step Function Signal Evolution. . . . . . . . . . . . . . 212 11.5.3 Rectangular Envelope Pulse Evolution . . . . . . . . . . . . . . . . . . . . . . . 214 11.5.4 Van Bladel Envelope Pulse Evolution . . . . . . . . . . . . . . . . . . . . . . . . 215 11.5.5 Concluding Remarks on the Slowly Varying Envelope and Classical Group Velocity Approximations. . . . 222 11.6 Extensions of the Group Velocity Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223 11.7 Localized Pulsed-Beam Propagation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231 11.7.1 Mathematical Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232 11.7.2 Paraxial Asymptotics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234 11.8 The Necessity of an Asymptotic Description . . . . . . . . . . . . . . . . . . . . . . . . . . 244 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249 12 Analysis of the Phase Function and Its Saddle Points . . . . . . . . . . . . . . . . . . . . . 251 12.1 General Saddle Point Dynamics for Causally Dispersive Dielectrics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253 12.1.1 The Region About the Origin (j!j  !0 ). . . . . . . . . . . . . . . . . . . . 254 12.1.2 The Region About Infinity (j!j  !m ) . . . . . . . . . . . . . . . . . . . . . . 260 12.1.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263 12.2 The Behavior of the Phase in the Complex !-Plane for Causally Dispersive Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 264 12.2.1 Single-Resonance Lorentz Model Dielectrics . . . . . . . . . . . . . . . . 264 12.2.2 Multiple-Resonance Lorentz Model Dielectrics . . . . . . . . . . . . . 279 12.2.3 Rocard–Powles–Debye Model Dielectrics . . . . . . . . . . . . . . . . . . . 293 12.2.4 Drude Model Conductors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 306 12.3 The Location of the Saddle Points and the Approximation of the Phase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 316 12.3.1 Single-Resonance Lorentz Model Dielectrics . . . . . . . . . . . . . . . . 316 12.3.2 Multiple-Resonance Lorentz Model Dielectrics . . . . . . . . . . . . . 353 12.3.3 Rocard–Powles–Debye Model Dielectrics . . . . . . . . . . . . . . . . . . . 366 12.3.4 Drude Model Conductors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 367 12.3.5 Semiconducting Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 370

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12.4

Procedure for the Asymptotic Analysis of the Propagated Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 376 12.5 Synopsis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 386 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 386 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 387

13 Evolution of the Precursor Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 389 13.1 The Field Behavior for  < 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 391 13.2 The Sommerfeld Precursor Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393 13.2.1 The Nonuniform Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 394 13.2.2 The Uniform Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 400 13.2.3 Field Behavior at the Wavefront . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 405 13.2.4 The Instantaneous Oscillation Frequency . . . . . . . . . . . . . . . . . . . . 407 13.2.5 The Delta Function Pulse Sommerfeld Precursor . . . . . . . . . . . . 408 13.2.6 The Heaviside Step Function Pulse Sommerfeld Precursor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 412 13.3 The Brillouin Precursor Field in Lorentz Model Dielectrics . . . . . . . . . . 416 13.3.1 The Nonuniform Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 417 13.3.2 The Uniform Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 430 13.3.3 The Instantaneous Oscillation Frequency . . . . . . . . . . . . . . . . . . . . 437 13.3.4 The Delta Function Pulse Brillouin Precursor . . . . . . . . . . . . . . . 439 13.3.5 The Heaviside Step Function Pulse Brillouin Precursor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 440 13.4 The Brillouin Precursor Field in Debye Model Dielectrics . . . . . . . . . . . 445 13.5 The Middle Precursor Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 448 13.6 Impulse Response of Causally Dispersive Materials . . . . . . . . . . . . . . . . . . 454 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 462 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 464 14 Evolution of the Signal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 467 14.1 The Nonuniform Asymptotic Approximation . . . . . . . . . . . . . . . . . . . . . . . . . 468 14.2 Rocard–Powles–Debye Model Dielectrics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 471 14.3 The Uniform Asymptotic Approximation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 474 14.4 Single Resonance Lorentz Model Dielectrics . . . . . . . . . . . . . . . . . . . . . . . . . . 478 14.4.1 Frequencies below the Absorption Band . . . . . . . . . . . . . . . . . . . . . 479 14.4.2 Frequencies above the Absorption Band . . . . . . . . . . . . . . . . . . . . . 483 14.4.3 Frequencies within the Absorption Band . . . . . . . . . . . . . . . . . . . . . 485 14.4.4 The Heaviside Unit Step Function Signal . . . . . . . . . . . . . . . . . . . . 488 14.5 Multiple Resonance Lorentz Model Dielectrics . . . . . . . . . . . . . . . . . . . . . . . 494 14.6 Drude Model Conductors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 499 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 500 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 501

Contents

xiii

15 Continuous Evolution of the Total Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 503 15.1 The Total Precursor Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 504 15.2 Resonance Peaks of the Precursors and the Signal Contribution . . . . . 507 15.3 The Signal Arrival and the Signal Velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 509 15.3.1 Transition from the Precursor Field to the Signal . . . . . . . . . . . . 509 15.3.2 The Signal Velocity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 515 15.4 Comparison of the Signal Velocity with the Phase, Group, and Energy Velocities. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 526 15.5 The Heaviside Step Function Modulated Signal . . . . . . . . . . . . . . . . . . . . . . . 532 15.5.1 Signal Propagation in a Single Resonance Lorentz Model Dielectric . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 533 15.5.2 Signal Propagation in a Double Resonance Lorentz Model Dielectric . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 554 15.5.3 Signal Propagation in a Drude Model Conductor . . . . . . . . . . . . 561 15.5.4 Signal Propagation in a Rocard–Powles–Debye Model Dielectric . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 566 15.5.5 Signal Propagation along a Dispersive Transmission Line . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 569 15.6 The Rectangular Pulse Envelope Modulated Signal . . . . . . . . . . . . . . . . . . . 572 15.6.1 Rectangular Envelope Pulse Propagation in a Single Resonance Lorentz Model Dielectric . . . . . . . . . . . . 574 15.6.2 Rectangular Envelope Pulse Propagation in a Rocard–Powles–Debye Model Dielectric . . . . . . . . . . . . . . . 591 15.6.3 Rectangular Envelope Pulse Propagation in Triply Distilled Water . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 597 15.6.4 Rectangular Envelope Pulse Propagation in Saltwater. . . . . . . 603 15.7 Noninstantaneous Rise-Time Signals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 605 15.7.1 Hyperbolic Tangent Envelope Signal Propagation in a Single Resonance Lorentz Model Dielectric . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 606 15.7.2 Raised Cosine Envelope Signal Propagation in a Single Resonance Lorentz Model Dielectric . . . . . . . . . . . . 617 15.7.3 Trapezoidal Envelope Pulse Propagation in a Rocard–Powles–Debye Model Dielectric . . . . . . . . . . . . . . . 619 15.8 Infinitely Smooth Envelope Pulses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 621 15.8.1 Gaussian Envelope Pulse Propagation in a Single Resonance Lorentz Model Dielectric . . . . . . . . . . . . . . . . . 621 15.8.2 Van Bladel Envelope Pulse Propagation in a Double Resonance Lorentz Model Dielectric . . . . . . . . . . . 638 15.8.3 Brillouin Pulse Propagation in a Rocard– Powles–Debye Model Dielectric; Optimal Pulse Penetration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 638

xiv

Contents

15.9

The Pulse Centroid Velocity of the Poynting Vector . . . . . . . . . . . . . . . . . . 643 15.9.1 Mathematical Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 643 15.9.2 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 645 15.9.3 The Instantaneous Centroid Velocity . . . . . . . . . . . . . . . . . . . . . . . . . 649 15.10 Dispersive Pulse Propagation in the Singular and Weak Dispersion Limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 651 15.10.1 The Singular Dispersion Limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 652 15.10.2 The Weak Dispersion Limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 653 15.11 Comparison with Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 656 15.12 The Myth of Superluminal Pulse Propagation . . . . . . . . . . . . . . . . . . . . . . . . . 669 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 672 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 676 16 Physical Interpretations of Dispersive Pulse Dynamics . . . . . . . . . . . . . . . . . . . 679 16.1 Energy Velocity Description of Dispersive Pulse Dynamics . . . . . . . . . . 681 16.1.1 Approximations Having a Precise Physical Interpretation . . 683 16.1.2 Physical Model of Dispersive Pulse Dynamics . . . . . . . . . . . . . . 689 16.2 Extension of the Group Velocity Description . . . . . . . . . . . . . . . . . . . . . . . . . . 701 16.3 Signal Model of Dispersive Pulse Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . 702 16.4 Summary and Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 708 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 711 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 712 17 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 713 17.1 On the Use and Application of Precursor Waveforms . . . . . . . . . . . . . . . . . 713 17.2 Electromagnetic Energy Dissipation in Causally Dispersive Media . . 716 17.2.1 General Formulation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 717 17.2.2 Evolved Heat in Lorentz Model Dielectrics . . . . . . . . . . . . . . . . . . 719 17.2.3 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 720 17.3 Reflection and Transmission Phenomena . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 728 17.3.1 Reflection and Transmission at a Dispersive Half-Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 728 17.3.2 The Goos–H¨anchen Shift . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 738 17.3.3 Reflection and Transmission at a Dispersive Layer: The Question of Superluminal Tunneling . . . . . . . . . . . . 751 17.4 Optimal Pulse Penetration through Dispersive Bodies . . . . . . . . . . . . . . . . 751 17.4.1 Ground Penetrating Radar. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 752 17.4.2 Foliage Penetrating Radar . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 756 17.4.3 Undersea Communications using the Brillouin Precursor . . . 759 17.5 Ultrawideband Pulse Propagation through the Ionosphere . . . . . . . . . . . . 761 17.6 Health and Safety Issues Associated with Ultrashort Pulsed Electromagnetic Radiation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 767 17.7 Future Prospects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 771 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 773 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 775

Contents

xv

Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 777 F

Asymptotic Expansion of Single Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 777 F.1 Foundations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 780 F.2 Asymptotic Sequences, Series and Expansions . . . . . . . . . . . . . . . . . . . . . . . . 783 F.3 Integration by Parts. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 790 F.4 The Method of Stationary Phase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 792 F.5 Watson’s Lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 794 F.6 Laplace’s Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 800 F.7 The Method of Steepest Descents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 805 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 810

G Proof of Theorem 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 811 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 818 H The Radon Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 819 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 823 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 825

Chapter 9

Pulsed Electromagnetic and Optical Beam Wavefields in Temporally Dispersive Media

The macroscopic electromagnetic field behavior in a homogeneous, isotropic, locally linear (HILL), temporally dispersive medium with no externally supplied charge or current sources is described by the macroscopic Maxwell’s equations1 [see (5.12)–(5.15) of Vol. 1] r  D.r; t / D 0;

   1  @B.r; t /  r  E.r; t / D    c  @t ; r  B.r; t / D 0;      1  @D.r; t /  4     r  H.r; t / D   C  c  Jc .r; t /; c @t taken together with the constitutive relations Z

t

Z

1 t

Z

1 t

D.r; t / D H.r; t / D Jc .r; t / D

O .t  t 0 /E.r; t 0 /dt0 ; O 1 .t  t 0 /B.r; t 0 /dt0 ; O .t  t 0 /E.r; t 0 /dt0 ;

1

as described in (4.85), (4.141), and (4.113) of Vol. 1, respectively, where the conduction current density satisfies the equation of continuity [see (5.8)–(5.11) of Vol. 1] r  Jc .r; t / D 0: 1

As in Vol. 1, the analysis is presented here in such a manner that it can be interpreted in both cgs (Gaussian) units and mksa units. This is accomplished by writing each equation such that it can be interpreted in mksa units provided that the factor * in the double brackets kk is omitted, while the inclusion of this factor yields the appropriate expression in cgs units. If there is no double bracketed quantity, the expression can be interpreted equally in Gaussian or mksa units unless specifically noted otherwise. K.E. Oughstun, Electromagnetic and Optical Pulse Propagation 2: Temporal Pulse Dynamics in Dispersive, Attenuative Media, Springer Series in Optical Sciences 144, c Springer Science+Business Media, LLC 2009 DOI 10.1007/978-1-4419-0149-1 9, 

1

2

9 Pulsed Beam Wavefields in Temporally Dispersive Media

Hence, the electric displacement vector D.r; t /, the electric field intensity vector E.r; t /, the magnetic induction vector B.r; t /, the magnetic field intensity vector H.r; t /, and the conduction current density vector Jc .r; t / are all solenoidal vector fields in the source-free medium. Here O .t / denotes the (real-valued) dielectric permittivity, .t O / denotes the (real-valued) magnetic permeability, and O .t / denotes the (real-valued) electric conductivity response of the simple dispersive medium. By O  t 0 / D 0, and O .t  t 0 / D 0 for t 0 > t , as exhibited in causality, O .t  t 0 / D 0, .t the upper limit of integration in the above three constitutive relations. In any HILL medium, the angular spectrum of plane waves representation expresses any freely propagating electromagnetic wavefield that propagates into the positive half-space z  z0 as a superposition of both homogeneous and inhomogeneous plane waves [1, 2]. The inhomogeneous plane wave components are unimportant in many instances when the material is lossless because they become evanescent in the propagation distance. Because of this evanescence, an electromagnetic beam field in a lossless medium is typically defined, in part, by the requirement that it can be represented by an angular spectrum that contains only homogeneous plane wave components [3], which then typically leads to the paraxial approximation that is so widely used in Fourier optics [4]. As the homogeneous waves do not attenuate in such a lossless medium, this condition ensures that all of the angular spectrum components of the beam field remain in the original proportions throughout the half-space z  z0 . The situation is much more involved in a temporally dispersive, attenuative medium as the plane wave spectral components no longer conveniently separate into homogeneous and evanescent constituents. The detailed solution of this problem is then of central importance to this work as it forms the basis for the theoretical description of ultrashort pulse dynamics in causally dispersive media, particularly since diffraction is itself a dispersive phenomenon. Associated with this rigorous description is the question of the separability of pulsed beam fields given that the intial field is separated into the product of a temporal and a spatial part. Such space and time separability is trivially obtained for pulsed plane wavefields, but not so in the general case of a pulsed beam wavefield. The related topics of multipole expansions, local pulsed beam solutions, electromagnetic “bullets,” and the inverse initial value problem are also introduced and considered with sufficient detail to provide a sound basis for further research. Although these latter topics are not essential for the topics considered in the remainder of the book, and so may be omitted without any loss of pedagogical continuity, they nevertheless do serve to illustrate the sweeping breadth of this research area.

9.1 Angular Spectrum Representation A completely general representation of the propagation of a freely propagating electromagnetic wavefield into the half-space z  z0 > Z of an HILL temporally dispersive medium is now considered. The phrase “freely-propagating” is used

9.1 Angular Spectrum Representation Fig. 9.1 Geometry of the planar electromagnetic boundary value problem

3 E0(rT ,t) = E0(x,y,t) H0(rT ,t) = H0(x,y,t)

v

1z z

Half-Space z z0

Plane z = z0

here2 to indicate that there are no externally supplied charge or current sources for the field present in this half-space, the field source residing somewhere in the region z  Z. It is unnecessary to know what this source is provided that the pair fE0 ; B0 g of electromagnetic field vectors are known functions of time and the transverse position vector rT D 1O x x C 1O y y in the plane z D z0 , as illustrated in Fig. 9.1. The rigorous, formal solution of this planar boundary value problem for the electromagnetic field in the half-space z  z0 forms the basis of investigation for a wide class of pulsed electromagnetic beam field problems in both optics and electrical engineering. Consider an electromagnetic wavefield that is propagating into the half-space z  z0 > Z > 0 and let the electric and magnetic field vectors on the plane z D z0 , denoted by E.rT ; z0 ; t / D E0 .rT ; t /;

(9.1)

B.rT ; z0 ; t / D B0 .rT ; t /;

(9.2)

be known functions of time and the transverse position vector rT 1O x x C 1O y y in the plane z D z0 , as indicated by the 0 subscript, as depicted in Fig. 9.1. It is 2

A freely propagating field is fundamentally different from a free-field whose externally supplied charge and current sources have all been turned off. The spatiotemporal properties of free-fields are considered in Chap. 8 of Vol. 1.

4

9 Pulsed Beam Wavefields in Temporally Dispersive Media

assumed here that the two-dimensional spatial Fourier transform in the transverse coordinates as well as the temporal Fourier–Laplace transform of each field vector on the plane z D z0 exists, where QQ .k ; !/ D U 0 T

Z

Z

1

1

dt dxdy U0 .rT ; t /e i.kT rT !t/ ; (9.3) 1 1 (Z ) Z 1 1 QQ .k ; !/e i.kT rT !t/ ; (9.4) < d! d kx d ky U U0 .rT ; t / D 0 T 4 3 CC 1

where kT 1O x kx C 1O y ky is the transverse wave vector. Here U0 .rT ; t / represents either the initial electric E0 .rT ; t / or magnetic B0 .rT ; t / field vector at the QQ .k ; !/ then represents the corresponding plane z D z0 ; the spectral field vector U 0 T Q Fourier–Laplace transform EQ 0 .kT ; !/ or BQQ 0 .kT ; !/ of the initial electric or magnetic field vector at that plane. If the initial time dependence of either field vector E0 .rT ; t / or B0 .rT ; t / at the plane z D z0 is such that it vanishes identically for all t < t0 for some finite value of t0 , then the transform relations appearing in (9.3) and (9.4) are Laplace transformations and the contour of integration CC is the straight line path ! D ! 0 C ia, with a greater than the abscissa of absolute convergence for the initial time behavior of the field [5] and with ! 0
(Z

Z

1

d! CC

QQ .k ; !/e i.!/ z e i.kT rT !t/ dkx dky U 0 T

) (9.5)

1

for both field vectors, where z z  z0 . The temporal frequency spectrum Q !/ of this wavefield satisfies the stated boundary value problem for the vecU.r; tor Helmholtz equation   Q !/ D 0: r 2 C kQ 2 .!/ U.r;

(9.6)

In addition, the spatiotemporal spectra of the electric and magnetic field vectors at the plane z D z0 satisfy the transversality relations [see (7.18)–(7.20) of Vol. 1] kck kQ C .!/  BQQ 0 .kT ; !/; !.!/c .!/ kck Q C k .!/  EQQ 0 .kT ; !/; BQQ 0 .kT ; !/ D !

EQQ 0 .kT ; !/ D 

(9.7) (9.8)

9.1 Angular Spectrum Representation

so that

5

kQ C .!/  EQQ 0 .kT ; !/ D kQ C .!/  BQQ 0 .kT ; !/ D 0:

(9.9)

kQ C .!/ 1O x kx C 1O y ky C 1O x .!/

(9.10)

Here is the complex wave vector for propagation into the positive half-space z  z0 , with the associated complex wavenumber  1=2 Q k.!/ kQ C .!/  kQ C .!/ 1=2 !  .!/c .!/ D ; kck and .!/ is defined as the principal branch of the expression 1=2  .!/ kQ 2 .!/  kT2 ;

(9.11)

(9.12)

with kT2 D kx2 C ky2 . This branch choice results in either exponential or zero attenuation with increasing z  0 for each plane wave spectral component. Here c .!/ .!/ C i k4k

 .!/ !

(9.13)

is the complex permittivity c .!/ of the dispersive medium with frequency dependent dielectric permittivity .!/ D r .!/ C ii .!/ and electric conductivity  .!/ D r .!/ C ii .!/, and .!/ D r .!/ C ii .!/ denotes the frequency dependent magnetic permeability, whose corresponding real and imaginary parts are each connected through the appropriate dispersion relation [see Sect. 4.3 of Vol. 1]. The integrand QQ .k ; !/e i .kQ C .!/ r!t / QQ .k ; !/e i.!/ z e i.kT rT !t/ D U U 0 T 0 T

(9.14)

that appears in the angular spectrum representation given in (9.5), where r 1O x x C 1O y y C 1O z z

(9.15)

with z D z  z0  0, corresponds to a specific time-harmonic plane wavefield. This wavefield is propagating away from the initial plane at z D z0 and into the positive half-space z  z0 at each angular frequency ! and transverse O O nwave vector kT 1x kox C 1y ky that is present in the initial spectral field vectors QQ .k ; !/; BQQ .k ; !/ at that plane with one distinction: the wave vector compoE 0 T 0 T nents kx and ky are real-valued and independent of !, whereas .!/ is, in general, complex-valued. Consequently, each plane wave component in the angular spectrum representation given in (9.5) is attenuated in the z-direction alone, and this is merely a mathematical consequence of the choice of the residue evaluation in the Fourier– Laplace representation of the radiated electromagnetic field [see Sect. 6.3 of Vol. 1].

6

9 Pulsed Beam Wavefields in Temporally Dispersive Media

9.1.1 Real Direction Cosine Form of the Angular Spectrum of Plane Waves Representation The plane wave spectral components appearing in the angular spectrum representation given in (9.5) may be expressed in a more revealing geometric form by setting [see Sect. 7.3 of Vol. 1] kx k.!/p;

ky k.!/q;

.!/ k.!/m;

(9.16)

where both p and q are real-valued. Here ˇ ˇ  1=2 ˇQ ˇ k.!/ ˇk.!/ ˇ D ˇ 2 .!/ C ˛ 2 .!/

(9.17)

  n o ˛.!/ Q .!/ arg k.!/ D arctan ˇ.!/

(9.18)

is the magnitude and

is the phase of the complex wavenumber Q k.!/ D ˇ.!/ C i ˛.!/ D k.!/e i

.!/

(9.19)

n o Q defined in (9.11), with plane wave propagation factor ˇ.!/ < k.!/ and attenn o Q Q uation factor ˛.!/ = k.!/ . In addition, notice that k.!/ D k0 n.!/, where k0 !=c is the wavenumber in vacuum and  n.!/

.!/c .!/ 0  0

1=2 (9.20)

is the complex index of refraction of the dispersive medium. With these substitutions, the relation given in (9.12) yields  m.!/ D e i2

.!/

1=2   p2 C q2 ;

(9.21)

whose real and imaginary parts are found to satisfy the respective inequalities [cf. (7.174)–(7.175)] m0 .!/ < fm.!/g  0;

(9.22)

m00 .!/ = fm.!/g  0;

(9.23)

for all ! 2 CC . Explicit expressions for both m0 .!/ and m00 .!/ are obtained from the square of (9.21) as

9.1 Angular Spectrum Representation

( m0 .!/ D

7

  1 cos.2 .!//  p 2 C q 2 2 "    2 C C cos.2 .!//  p 2 C q 2

1 sin2 .2 .!//

#1=2 ) 1=2 ; (9.24)

provided that sin.2 .!// ¤ 0, and m00 .!/ D

sin.2 .!// ; 2m0 .!/

(9.25)

provided that m0 .!/ ¤ 0. If m0 .!/ D 0, then sin.2 .!// D 0 and either in which case .p 2 C q 2 /  1 and  1=2 m00 .!/ D p 2 C q 2  1 ; or

.!/ D 0,

(9.26)

.!/ D =2, in which case  1=2 m00 .!/ D p 2 C q 2 C 1 ;

(9.27)

for all (real) values of p and q. Finally, for the special case of a lossless medium at some angular frequency !, .!/ D 0 and m.!/ D m0 .!/ D

p 1  .p 2 C q 2 /

(9.28)

if .p 2 C q 2 /  1, while m.!/ D m00 .!/ D i

p .p 2 C q 2 /  1

(9.29)

if .p 2 C q 2 / > 1. With these results, the plane wave propagation factor expressed in (9.14) that appears in the angular spectrum representation given in (9.5) becomes Q C .!/r

ei k

D e k.!/m

00 .!/ z

0

e ik.!/.pxCqyCm .!/ z/ ; C

(9.30)

with z z  z0  0. If p D q D 0, then e i kQ .!/r represents the spatial part of a homogeneous plane wave since the surfaces of constant amplitude coincide with C the surfaces of constant phase given by z Dconstant. If ˛.!/ D e i kQ .!/r   20, then represents the spatial part of a homogeneous plane wave when p C q 2  1 (in whichcase m00 D 0), while it represents the spatial part of an evanescent plane wave when p 2 C q 2 > 1 (in which case m0 D 0) and the surfaces of constant amplitude are orthogonal to the cophasal surfaces. In general, ˛.!/ ¤ 0 and if either p ¤ 0

8

9 Pulsed Beam Wavefields in Temporally Dispersive Media C

or q ¤ 0, then e i kQ .!/r represents the spatial part of an inhomogeneous plane wave of angular frequency ! since the surfaces of constant amplitude z D constant are different from the surfaces of constant phase .px C qy C m0 .!/ z/ D constant. These inhomogeneous plane wave phase fronts then propagate in the direction that is specified by the unit vector p q m0 .!/ ; sO D 1O x C 1O y C 1O z s s s with sD

(9.31)

p p 2 C q 2 C m02 .!/;

(9.32)

0

as depicted in Fig. 9.2. Since m .!/  0 for all real values of p and q, as well as for all ! 2 CC , then those inhomogeneous plane wave components with m0 .!/ > 0 have phase fronts that advance into the positive half-space z > 0. From (9.24) it is seen that the inequality ˇ ˇ m0 .!/ˇ

.p 2 Cq 2 />cos .2 .!//

ˇ ˇ < m0 .!/ˇ

(9.33)

.p 2 Cq 2 /
is satisfied when sin .2 .!// ¤ 0, so that ˇ ˇ m00 .!/ˇ

.p 2 Cq 2 />cos .2 .!//

ˇ ˇ > m00 .!/ˇ

.p 2 Cq 2 /
;

(9.34)

provided that m0 .!/ ¤ 0. Consequently, the amplitude attenuation associated with the˚ inhomogeneous plane wave spectral components in the exterior region ˇ R> .p; q/ˇ.p 2 C q 2 / > cos .2 .!// is typically larger than that associated with the plane wave spectral components in the interior region ˇ ˚ inhomogeneous R< .p; q/ˇ.p 2 C q 2 / < cos .2 .!// . x s

cos–1(p/s)

cos–1((m'( )/s)

O cos–1(q/s)

y

Cophasal Surface px + qy + m'( ) Δz = constant

z

Fig. 9.2 Inhomogeneous plane wave phase front propagating in the direction specified by the unit vector sO D 1O x p=s C 1O y q=s C 1O z m0 .!/=s

9.1 Angular Spectrum Representation

9

With these results, the angular spectrum representation given in (9.5) becomes (Z Z 1 QQ .p; q; !/ i!t U.r; t / D < d! e dpdq k 2 .!/U 0 4 3 CC R< [R> ) e k.!/m

00 .!/ z

0

e ik.!/.pxCqyCm .!/ z/ (9.35)

QQ .k =k.!/; k =k.!/; !/. Here U.r; t / repQQ .p; q; !/ D U for z  0, where U 0 0 x y QQ .p; q; !/ represents resents either the electric field vector E.r; t /, in which case U 0 the spatiotemporal Fourier–Laplace transform EQQ 0 .p; q; !/ of the boundary value for QQ .p; q; !/ repthat field vector, or the magnetic field vector B.r; t /, in which case U 0 resents the spatiotemporal Fourier–Laplace transform BQQ 0 .p; q; !/ of the boundary value for that field vector.

9.1.2 Electromagnetic Energy Flow in the Angular Spectrum Representation A quantity of critical interest to this analysis is the time-average flow of energy associated with the individual plane wave components appearing in the angular spectrum representation of the freely–propagating electromagnetic wavefield. This is given by the real part of the complex Poynting vector [see (5.131) and (5.132) of Vol. 1]   QQ k ; !/ 1  c  E.rI QQ  .rI k ; !/; S.rI   QQ kT ; !/  H T T 2 4 where [see (9.7)–(9.8)] QQ k ; !/ D EQQ .k ; !/e i kQ C .!/ r ; E.rI T 0 T QQ .k ; !/e i kQ C .!/ r QQ k ; !/ D H H.rI T 0 T D

kck Q C QC k .!/  EQQ 0 .kT ; !/e i k .!/ r ; .!/!

  with r 1O x x C 1O y y C 1O z z. Since kQ C .!/ D k.!/ 1O x p C 1O y q C 1O z m.!/ ,     so that kQ C .!/ D k.!/ 1O x p C 1O y q C 1O z m .!/ , then the complex Poynting

10

9 Pulsed Beam Wavefields in Temporally Dispersive Media

vector for each individual plane wave spectral component at the real-valued angular frequency ! becomes  2 ˇ2 k.!/ ˇˇ QQ c  ˇ 2k.!/m00 .!/ z QQ k ; !/ D    E .k ; !/ S.rI ˇ ˇ e T  4  2 .!/! 0 T h  i  1O x p C 1O y q C 1O z m0 .!/  i m00 .!/  2 ˇ2 ˇ c  k.!/ ˇ 2k.!/m00 .!/ z ˇ QQ  .k ; !/ E D ˇ e ˇ 0 T  4  2 j.!/j2 ! n    r .!/ 1O x p C 1O y q C 1O z m0 .!/ C i .!/1O z m00 .!/ h   io Ci i .!/ 1O x p C 1O y q C 1O z m0 .!/  r .!/1O z m00 .!/ ; (9.36) where r .!/ < f.!/g and i .!/ = f.!/g, and where ! is taken to be real-valued. The time-average Poynting vector associated with each individual plane wave spectral component at the real-valued angular frequency ! is then obtained by taking the real part of the above expression, with the result  2 ˇ2 ˇ E D   k.!/ ˇ 2k.!/m00 .!/ z ˇ QQ QS! .r; t I p; q/ D  c   4  2 j.!/j2 ! ˇE0 .kT ; !/ˇ e n  

o  r .!/ 1O xp C 1O y q C 1O z r .!/m0 .!/ C i .!/m00 .!/ : (9.37) The direction of the time-average flow of electromagnetic energy for a given plane wave spectral component in the pulsed beam wavefield is then seen to depend upon the sign of the quantity Œr .!/m0 .!/ C i .!/m00 .!/ . If this quantity is positive, then the energy flow at that angular frequency is directed into the positive halfspace z > z0 , while if it is negative the time-average energy flow is directed toward the negative half-space.

9.1.3 Homogeneous and Evanescent Plane Wave Contributions to the Angular Spectrum Representation Consider the monochromatic wavefield ˚ U! .r; t / D < UQ .r; !/e i!t

(9.38)

whose complex (phasor) field amplitude is given by the angular spectrum representation [see (7.191)] 1 UQ .r; !/ D 4 2

Z

1

1

UQQ 0 .kT ; !/e i.!/ z e i .kx xCky y / dkx dky

(9.39)

9.1 Angular Spectrum Representation

11

1=2  with .!/ kQ 2 .!/  kx2  ky2 . This representation is expressed in terms of a superposition of monochromatic plane waves whose amplitude and phase are given by the spectral distribution function UQQ 0 .kT ; !/ D UQQ 0 .kx ; ky ; !/ Z 1 0 D UQ 0 .r0T ; !/e ikT rT dx 0 dy 0 Z1 1 0 0 UQ 0 .x 0 ; y 0 ; !/e i .kx x Cky y / dx 0 dy 0 : D

(9.40)

1

Q In free-space, k.!/ D k D !=c and the spectral components satisfying the inequality kx2 Cky2  k 2 represent homogeneous plane waves while the spectral components satisfying the opposite inequality kx2 C ky2 > k 2 represent evanescent plane waves. Q If the medium is dispersive (and hence absorptive), then k.!/ D ˇ.!/ C i ˛.!/ is complex-valued and this simple separation is not possible (see Sects. 7.1, 7.3, and 9.1). Because of this, the development in the remainder of this section is henceforth restricted to free-space. In free-space, it is found convenient to separate the domain of integration D D R2 in the angular spectrum representation given in (9.39) into the two parts (compare with the regions R< and R> of Sect. 9.1.1) o n ˇ DH .kx ; ky /ˇ kx2 C ky2  k 2 ; o n ˇ DE .kx ; ky /ˇ kx2 C ky2 > k 2 ;

(9.41) (9.42)

where D D DH [ DE . The complex field amplitude then separates into the two parts Z p 2 2 2 1 UQ H .r; !/ UQQ 0 .kx ; ky ; !/e i k kx ky z e i .kx xCky y / dkx dky ; 4 DH Z p 2 2 2 1 UQQ 0 .kx ; ky ; !/e  kx Cky k z e i .kx xCky y / dkx dky ; UQ E .r; !/ 4 DE

(9.43) (9.44)

where UQ .r; !/ D UQ H .r; !/ C UQ E .r; !/:

(9.45)

Here UQ H .r; !/ represents the homogeneous wave contribution and UQ E .r; !/ represents the evanescent wave contribution to the wavefield UQ .r; !/. Substitution of (9.40) into (9.44) gives 1 UQ E .r; !/ D  2

Z

1

UQ 0 .x 0 ; y 0 ; !/I.x  x 0 ; y  y 0 ; z/dx 0 dy 0 ; 1

(9.46)

12

9 Pulsed Beam Wavefields in Temporally Dispersive Media

with I.; ; z/ 

1 2

Z

e

p

kx2 Cky2 k 2 z i .kx Cky /

e

DE

dkx dky ;

(9.47)

which is related to the evanescent part of the monochromatic spatial impulse response function h.; I z; !/ defined in (7.66). The exact evaluation of this integral representation of the evanescent wave contribution forms the main focus of this subsection. The integral appearing in (9.47) may be expressed in a more convenient form through the change of variable kx D cos ˛;

ky D sin ˛;

(9.48)

where D k ! 1 and ˛ D 0 ! 2. With the definition of the angle  through the pair of relations [not to be confused with the complex wavenumber component .!/ defined in (9.12)]  cos  D p ;  2 C 2

 sin  D p ;  2 C 2

(9.49)

the transformed integral expression in (9.47) becomes 1 I.; ; z/  2

Z

1

Z

k

2

e

p

2 k 2 z i

e

p

 2 C2 cos .˛/

d d˛:

0

The integral over ˛ is immediately recognized as the integral representation of the zero-order Bessel function of the first kind, where J0 . / D so that

Z

1 2

Z

1

e

I.; ; z/ 

2

p

2 k 2 z

Under the final change of variable D expression becomes Z

(9.50)

0

k

1

e z J0

I.; ; z/ 

e i cos .˛/ d˛;  p  J0  2 C 2 d :

(9.51)

p 2  k 2 , so that d D d , the above p  p  2 C 2 k 2 C 2 d :

(9.52)

0

With the addition theorem for Bessel functions (see Sect. 8.531(1) of [6]) J0

1 p  X a2 C b 2  2ab cos  D J0 .a/J0 .b/ C 2 Jn .a/Jn .b/ cos .n /; nD1

9.1 Angular Spectrum Representation

13

the Bessel function appearing in (9.52) may be expressed as p  p  2 C 2 k 2 C 2  p   p  D J0 k  2 C  2 J0  2 C  2

J0

C2

1 X

 p   p  Jn k  2 C 2 Jn  2 C 2 cos .n=2/

nD1

 p   p  D J0 k  2 C 2 J0  2 C 2 C2

1  p   p  X .1/n J2n k  2 C 2 J2n  2 C 2 : nD0

Substitution of this result in (9.52) and interchanging the order of integration and summation then results in the expression  p Z I.; ; z/ D J0 k  2 C 2

1

 p  J0  2 C 2 e z d

0

1  p Z X 2 .1/n J2n k  2 C 2

1

 p  J2n  2 C 2 e z d :

0

nD0

(9.53) With the integral identity (see Sect. 6.621(4) of [6]) Z 0

1

 p  J2m x 2 C y 2 e z d 8 "p #2m 9 = x 2 C y 2 C z2  z @ < 1 D p p ; @z : x 2 C y 2 C z2 x2 C y2

the above expression becomes  p  @ I.; ; z/ D J0 k  2 C 2 @z

( p

1

)

 2 C 2 C z2 1  p  X C2 .1/n J2n k  2 C 2 nD0

8 "p #2n 9 =  2 C 2 C z2  z @ < 1  ; p p ; @z :  2 C 2 C z2  2 C 2 (9.54)

14

9 Pulsed Beam Wavefields in Temporally Dispersive Media

which involves an infinite summation of even-order Bessel functions. With the definition of the Lommel functions [7] Um .u; v/ Vm .u; v/

1 X nD0 1 X

.1/n .1/n

nD0

 u mC2n v  v mC2n u

JmC2n .v/;

(9.55)

Jm2n .v/;

(9.56)

which arise naturally in the diffraction theory of light through a circular aperture [8], the expression in (9.54) may be written in the form I.; ; z/

 p   p 9 8 p 2 2 2 2 2 2 2 @ < J0 k  C  C 2U0 k  C  C z  kz; k  C  = ; D p ; @z :  2 C 2 C z2 (9.57)

which is originally due to Bertilone [9]. The evanescent wave contribution to the time-harmonic (phasor) wavefield UQ .r; !/ is then given by [cf. (9.46)] 1 UQ E .r; !/ D  2

Z

1

UQ 0 .x 0 ; y 0 ; !/I.x  x 0 ; y  y 0 ; z/dx 0 dy 0 ;

(9.58)

1

with @ I.x  x ; y  y ; z/ D @z 0

0



2U0 .kR  k z; k/  J0 .k/ ; R

(9.59)

p where R D .x  x 0 /2 C .y  y 0 /2 C . z/2 is the distance between the initial field p point at r0 D .x 0 ; y 0 ; z0 / and the observation point at r D .x; y; z0 /, and where  D .x  x 0 /2 C .y  y 0 /2 . For a pulsed wavefield in vacuum, the spatiotemporal pulse evolution separates into homogeneous and evanescent wave contributions as U.r; t / D UH .r; t / C UE .r; t /;

(9.60)

with Z 1 1 UH .r; t / D < UQ H .r; !/e i!t d!;  0 Z 1 1 UQ E .r; !/e i!t d!: UE .r; t / D <  0

(9.61) (9.62)

9.1 Angular Spectrum Representation

15

Substitution of the expression given in (9.58) with (9.59) into the above Fourier integral representation of the evanescent wave contribution to the pulsed wavefield then yields the expression

Z 1 @ G.x 0 ; y 0 I x; y; z; t / 1 dx 0 dy 0 ; (9.63) UE .r; t / D  2 2 1 @z R where 0

Z

0

1

G.x ; y I x; y; z; t / < 0

UQ 0 .x 0 ; y 0 ; t 0 / ˚ 0  2U0 ..R  z/!=c; !=c/  J0 .!=c/ e i!t d!: (9.64)

Substitution of the Fourier integral relation Z 1 0 UQ 0 .x 0 ; y 0 ; !/ D U0 .x 0 ; y 0 ; t 0 /e i!t dt0 1

into the above expression, followed by an interchange of the integration order, then gives [taking note of the fact that U0 .x 0 ; y 0 ; t 0 / is real-valued] Z

0 0

1

G.x y I x; y; z; t / D c 1

U0 .x 0 ; y 0 ; t 0 / (Z 1

  2U0 .k.R  z/; k/  J0 .k/ 0 )  cos .ckjt 0  t j/d k dt0 ;

(9.65)

where the variable of integration in the inner integral has been changed from ! to k D !=c. Consider first the evaluation of the k-integral in (9.65). Expansion of the zerothorder Lommel function appearing in that integral in terms of an infinite series of integer-order Bessel functions through the defining relation given in (9.55) results in Z 1 U0 .k.R  z/; k/ cos .ckjt 0  t j/d k 0

D

 Z 1  X R  z 2n nD0



1

cos .ckjt 0  t j/J2n .k/dk: 0

From Sects. 6.771(10) and 8.940(1) of [6], the cosine transform of the integer order Bessel function is given by ( Z 1 .1/n p cos .2n arccos .b=a//; 0  b < a a2 b 2 J2n .a / cos .b /d D 0; 0 < a < b; 0

16

9 Pulsed Beam Wavefields in Temporally Dispersive Media

so that, together with the summation (see Sect. 1.447(2) of [6]) 1 X

 n cos .n#/ D

nD0

1   cos # ; 1  2 cos # C  2

jj < 1;

the above integral becomes Z

1

U0 .k.R  z/; k/ cos .ckjt 0  t j/dk 0

( D ( D

p

1 2 c 2 .tt 0 /2

p

1 2 c 2 .tt 0 /2

P1  R z 2n nD0





0; 1 cos # 12 cos #C 2



cos .n#/;

0  cjt 0  t j <  0   < cjt 0  t j

;

0  cjt 0  t j <  0   < cjt 0  t j;

0;

(9.66)

where  R  z 2n ;    0 cjt  t j : # 2 arccos  



(9.67) (9.68)

Finally, the remaining k-integral appearing in (9.65) is given by Z

(

1

0

J0 .k/ cos .ckjt  t j/dk D

1 ; 2 c 2 .t 0 t/2

p

0;

0

0  cjt 0  t j <  0   < cjt 0  t j:

(9.69)

Combination of (9.66) and (9.69) then gives the desired integration Z

1 0

 2U0 .k.R  z/; k/  J0 .k/ cos .ckjt 0  t j/dk h   i ( 1 cos # p 1 2  1 ; 0  cjt 0  t j <  2 12 cos #C 2 c 2 .tt 0 /2 D 0; 0   < cjt 0  t j: (9.70)

Notice that the integration results given in (9.66), (9.69), and (9.70) are improperly defined when  D 0. As a consequence, the final result given in (9.70) has physical meaning here only in the context of the integrand of (9.65) where one may treat the integration in the generalized function sense [9]. Finally, notice that  2 from (9.67)–(9.68).

1   cos # 1  2 cos # C  2

 1D

R z ; R2  c 2 .t 0  t /2

(9.71)

9.1 Angular Spectrum Representation

17

Substitution of the integration result given in (9.70) with (9.71) into (9.65) then yields the following result due to Bertilone [9]: G.x 0 ; y 0 I x; y; z; t / Dc R

Z

tC=c

t=c

U0 .x 0 ; y 0 ; t 0 / p 2  c 2 .t 0  t /2



z 2 R  c 2 .t 0  t /2



dt 0 : (9.72)

The final integral appearing in this expression is nonsingular in the positive halfspace z > 0 for any properly behaved initial field behavior U0 .x; y; t / at the plane z D z0 . Substitution of this result in (9.63) then yields the exact expression UE .r; t / D 

c 2 2

Z

1

Z

1

tC=c

t 0 Dt=c

U0 .x 0 ; y 0 ; t 0 / p 2  c 2 .t 0  t /2

z @ dx 0 dy 0 dt 0  @z R2  c 2 .t 0  t /2 (9.73)

for the evanescent contribution to the propagated pulsed wavefield. In contrast with the total field, the evanescent field is found to be noncausal since UE .r; t / is expressed by (9.73) in terms of the field behavior U0 .x 0 ; y 0 ; t 0 / on the z D z0 plane at the future times from t 0 D t  =c to t 0 D t C =c. Because the total (causal) wavefield is given as the sum of the homogeneous and evanescent parts, the homogeneous part must then possess the precise noncausal behavior to exactly cancel the noncausal contribution from the evanescent part. Such nonphysical behavior was first reported by Sherman, Stamnes, Devaney, and Lalor [10] who concluded that “for many physical problems, it is inappropriate to give physical significance to UH .r; t / and UE .r; t / independently.” Similar remarks would then appear likely to apply to the separate inhomogeneous plane wave contributions from the interior and exterior regions R< and R> , respectively [see (9.33) and (9.34)], when the medium is dispersive; however, a rigorous proof of this extension of Bertilone’s proof to dispersive, attenuative media remains to be given.

9.1.4 Paraxial Approximation of the Angular Spectrum of Plane Waves Representation If the real direction cosines p and q are both sufficiently small in magnitude in comparison to unity, then the complex direction cosine m.!/ may be approximated by the first few terms in its Maclaurin series as    1=2 1  p 2 C q 2 e i2 .!/  1 2 p C q 2 e i .!/ :

e i .!/  2

m.!/ D e i

.!/

(9.74)

18

9 Pulsed Beam Wavefields in Temporally Dispersive Media

With this approximation, the propagation kernel G.p; q; !/ appearing in the angular spectrum representation in (9.35) becomes G.p; q; !/ e ik.!/m.!/ z

(9.75)

Q .!/ z.p 2 Cq 2 /=2

Q

e ik.!/ z e k

;

(9.76)

where the superscript asterisk denotes complex conjugation. The general spatial phase dispersion due to propagation is then replaced by quadratic phase dispersion in this paraxial approximation, so named because of its general validity in the paraxial region about the z-axis described by the pair of inequalities p  1 and q  1. Substitution of the paraxial approximation to the propagation kernel given in (9.76) into the angular spectrum representation given in (9.35) then results in the expression (Z ) 1 Q k.!/ z!t i / d! ; Q !/e . U.r; (9.77) U.r; t / D <  CC where Z 1 Q Q k.!/ Q !/ i k.!/ Q 0 .x 0 ; y 0 ; !/e i 2 z ..xx 0 /2 C.yy 0 /2 / dx 0 dy 0 U.r; U 2 z 1

(9.78)

is one form of the Fresnel–Kirchhoff diffraction integral [4, 8]. The same result is obtained from the first Rayleigh–Sommerfeld diffraction integral representation [see (7.74) and (7.75) of Vol. 1] in the Fresnel approximation wherein the distance R p .x  x 0 /2 C .y  y 0 /2 C z2 between the source and field points is approximated as R zC.xx 0 /2 =.2 z/C.yy 0 /2 =.2 z/. In this approximation, the spherical secondary wavelets embodied in the Huygens–Fresnel principle have been replaced by parabolic secondary wavelets. Since the Fresnel approximation is valid in the vanishingly small diffraction angle limit .x  x 0 /= z  1, .y  y 0 /= z  1 and results in the expression given in (9.39)–(9.40), it is then equivalent to the paraxial approximation. Q Because k.!/ D ˇ.!/ C i˛.!/, the Fresnel–Kirchhoff diffraction integral appearing in (9.40) may be rewritten in the more revealing form Z 1 Q k.!/ Q 0 .x 0 ; y 0 ; !/ Q U U.r; !/ i 2 z 1 ˛.!/

e  2 z ..xx /

0 2 C.yy 0 /2

0 2 0 2 / e i ˇ.!/ 2 z ..xx / C.yy / / dx 0 dy 0 ;

(9.79)

9.1 Angular Spectrum Representation

19

which explicitly displays the manner in which the material attenuation influences the monochromatic beam-field diffraction. Specifically, material attenuation decreases the effects of diffraction relative to the geometrical optics contribution. Nevertheless, care must be exercised when this paraxial representation is applied to pulsed beam problems, particularly when the initial pulse is ultrawideband, since the results may not be strictly causal.

9.1.4.1

Accuracy of the Paraxial Approximation

Limits on the accuracy of the paraxial approximation have been reported for pulsed beam propagation phenomena in lossless dispersive media by Melamed and Felsen [11, 12]. Surprisingly, the situation is simplified when material loss is present [13]. The results are best illustrated through a detailed numerical example. Because of its central role in bioelectromagnetics as well as in foliage and ground penetrating radar, a dispersive material of central interest here is water. The Cole–Cole extension of the Rocard–Powles model for the dielectric permittivity of triply distilled water at 25ı C [see (4.196) of Sect. 4.4.3] is given by the causal function .!/ D 1 C

2 X

aj 1j .1  i !/ .1  i !/ j fj j D1

(9.80)

which accurately describes the complex-valued relative permittivity over the frequency domain 0  !  1  1013 r=s extending from the static into the low infrared region of the electromagnetic spectrum. Here 1 D 2:1 denotes the large frequency limit of the relative dielectric permittivity described by this model, a1 D 74:1 and a2 D 3:0 are nondimensional scalar quantities describing the strength of each Rocard–Powles feature and contributing to the static relative permittivity given by S .0/ D 1 Ca0 Ca1 , where S D 79:1 for water. In addition, 1 D 8:441012 s and 2 D 6:53  1014 s are the macroscopic Debye relaxation times for each feature, f 1 D 4:62  1014 s and f 2 D 1:43  1015 s are the corresponding associated friction times introduced by the Rocard–Powles extension of the Debye model, with Cole–Cole parameters 1 D 0 and 2 D 0:5. The resultant frequency dispersion of Q the real and imaginary parts of the complex wavenumber k.!/ D ˇ.!/ C i˛.!/ is depicted in Fig. 9.3. Notice the peak in the absorption at ! 6  1013 r=s, which is then followed by a monotonic decrease in attenuation for larger frequencies. For angular frequencies greater than ! 11013 r=s, resonance polarization effects begin to appear, and the rotational polarization model leading to (9.80) must be augmented by a sequence of Lorentz lines that add resonance features to the dispersion curves depicted in Fig. 9.3 [see Sect. 4.4]. This additional frequency-dependent structure is unnecessary for the purpose of this discussion and so is not included here; however, its neglect does not limit the accuracy of the results presented here [13].

20

9 Pulsed Beam Wavefields in Temporally Dispersive Media

a

10

~ β(ω)=ℜ{k(ω)} - m−1

10

10 10 10

5

0

−5

−10

10

~ α(ω)=ℑ{k(ω)} - m−1

b

102

10 10 10

10

4

6

10

8

10 ω - r/s

10

10

10

12

10

14

5

0

−5

−10

10

−15

10

102

10

4

6

10

8

10 ω - r/s

10

10

10

12

10

14

Fig. 9.3 Angular frequency dependence of the (a) real and (b) imaginary parts of the complex Q wavenumber k.!/ D ˇ.! C i ˛.!/ for the Cole–Cole extension of the Rocard–Powles–Debye model of triply distilled water at 25ı C. The values marked with a cross () in each diagram indicate the real and imaginary values of the wavenumber at the angular frequency values !c D 2107 r=s in the HF, !c D 2  109 r=s in the UHF, and !c D 2  1011 r=s in the EHF regions of the electromagnetic spectrum that are used in this section

At the lower angular frequency (!c D 2  107 r=s) indicated in Fig. 9.3, the absorption is small [˛.!c / D 4:66  104 m1 ] and cos .2 .!c // D 1:0000 so that the complex direction cosine m.!c / is practically the same as that for an ideal lossless medium. This is evident from the pair of graphs presented in Fig. 9.4 which p depict the behavior of the real and imaginary parts of m.!c / as functions of p 2 C q 2 . The dashed curves in this figure depict the behavior of the paraxial approximation given p in (9.74). The real part of this paraxial approximation is seen to be accurate for p 2 C q 2  0:6, while the imaginaryppart, which is approximately zero for all real values of p and q, is accurate for all p 2 C q 2  1. The above behavior is reflected in Fig. 9.5 which depicts the behavior of the magnitude and phase of the propagation kernel G.p; q; !c /, as well as in Fig. 9.6 which depicts the behavior of the real and imaginary parts of G.p; q; !c / as funcp tions of p 2 C q 2 at the propagation distance z D 0:1 m. These results clearly demonstrate the well-known result that the paraxial approximation is only valid p when p 2 C q 2 < 1 in a lossless medium.

9.1 Angular Spectrum Representation

a

21

1.2 1

ℜ{m(ωc)}

0.8 0.6 0.4 0.2 0

−0.2

0

0.5

2

2 1/2

2

2 1/2

1

1.5

1

1.5

(p +q )

b

1.4 1.2

ℑ{m(ωc)}

1 0.8 0.6 0.4 0.2 0

0

0.5 (p +q )

Fig. 9.4 Behavior of the (a) real and (b) imaginary parts of the complex direction cosine m.!c / D p 1=2 ˚ at !c D 2 107 r=s as functions of p 2 C q 2 . The solid curves exp Œi 2 .!c /  .p 2 C q 2 / depict the exact behavior while the dashed curves describe the behavior obtained with the paraxial approximation

At the intermediate angular frequency (!c D 2  109 r=s) indicated in Fig. 9.3, the absorption is moderate [˛.!c / D 4:65 m1 ] and cos .2 .!c // D 0:9988. The complex direction cosine m.!c / is then a slightly smoothed version of that for an ideal lossless medium, as can be seen in Fig. 9.7. The accuracy of the paraxial approximation is seen to be slightly improved over that for the low-loss case illustrated in Fig. 9.4, and this slight improvement in accuracy is reflected in the paraxial approximation of the propagation kernel G.p; q; !c / whose magnitude and phase are depicted in Fig. 9.8 and whose real and imaginary parts are depicted in Fig. 9.9 for the propagation distance z D 0:1 m. Notice that both the amplitude and pphase of the paraxial approximation of G.p; q; !c / lose accuracy as the quantity p 2 C q 2 approaches unity from below in this moderate-absorption case, as reflected in the nearly identical behavior exhibited by the real and imaginary parts of G.p; q; !c /. At the upper angular frequency (!c D 2  1011 r=s) indicated in Fig. 9.3, the absorption is large [˛.!c / D 4:27  103 m1 ] and cos .2 .!c // D 0:4615. The complex direction cosine m.!c / is then a smoothed version of that for an ideal lossless medium, as can be seen in Fig. 9.10. The accuracy of the paraxial approximation has improved significantly over that for the previous two cases, particularly in the

22

9 Pulsed Beam Wavefields in Temporally Dispersive Media

a

1

|G(p,q,ωc)|

0.95 0.9 0.85 0.8

0

0.5

2

2 1/2

2

2 1/2

1

1.5

1

1.5

(p +q )

b

0.2

arg{G(p,q,ωc)}

0.15 0.1 0.05 0 –0.05

0

0.5 (p +q )

Fig. 9.5 Behavior of the (a) magnitude and (b) phase of the propagation kernel G.p; q; !c / at p !c D 2  107 r=s as functions of p 2 C q 2 at the propagation distance z D 0:1 m. The solid curves depict the exact behavior while the dashed curves describe the behavior obtained with the paraxial approximation

interior region R< where .p 2 C q 2 /  cos .2 .!c /, and remains reasonably accurate out to twice this value. Because of the high value of the absortion at this frequency, the paraxial approximation of the propagation kernel G.p; q; !c /, whose magnitude and phase are depicted in Fig. 9.11 and whose real and imaginary parts are depicted in Fig. 9.12 for the propagation distance z D 0:1 m, is nearly identical to the exact behavior for all values of p and q for which jG.p; q; !c /j is essentially nonzero. Finally, notice that the accuracy of the paraxial approximation of G.p; q; !c / increases as the propagation distance increases. This important property is illustrated through comparison of Figs. 9.9 (for z D 0:1 m), 9.13 (for z D 1:0 m), and 9.14 (for z D 10:0 m) for the intermediate frequency case (!c D 2  109 r=s) where the absorption is moderate [˛.!c / D 4:65 m1 so that zd D 0:215 m]. As the propagation distance increases, both the real and imaginary parts of the paraxial approximation of the propagation kernel are seen to improve in accuracy. At the largest propagation distance considered, the error is almost exclusively due to a small p; q-dependent phase error. These numerical results show that the accuracy of the paraxial approximation improves both as the material attenuation increases as well as when the

9.2 Angular Spectrum Representation of Multipole Wavefields

a

23

1

ℜ{G(p,q,ωc)}

0.95 0.9 0.85 0.8

0

0.5

2

2 1/2

2

2 1/2

1

1.5

1

1.5

(p +q )

b

0.2

ℑ{G(p,q,ωc)}

0.15 0.1 0.05 0 – 0.05

0

0.5 (p +q )

Fig. 9.6 Behavior of the (a) real and p (b) imaginary parts of the propagation kernel G.p; q; !c / at !c D 2  107 r=s as functions of p 2 C q 2 at the propagation distance z D 0:1 m. The solid curves depict the exact behavior while the dashed curves describe the behavior obtained with the paraxial approximation

propagation distance into the attenuative medium increases. This trend indicates that components in the exterior domain ˚ the inhomogeneous plane wave R> D .p; q/j.p 2 C q 2 / > cos .2 .!// become entirely negligible in comparison with the homogeneous and inhomogeneous plane wave components in the ˚ interior domain R< D .p; q/j.p 2 C q 2 / < cos .2 .!// as the propagation distance z typically exceeds a single absorption depth zd ˛ 1 .!c / at some characteristic frequency !c of the initial pulse.

9.2 Angular Spectrum Representation of Multipole Wavefields Multipole expansions of both the scalar optical and vector electromagnetic fields are widely employed in both classical and quantum electrodynamics [14] and optics [15]. The genesis of this type of expansion may be found in the early research of scattering by a spherical particle due to Clebsch [16], Mie [17], Debye [18], and Bromwich [19]. In 1903, E. T. Whittaker [20] presented the first multipole expansion

24

9 Pulsed Beam Wavefields in Temporally Dispersive Media

a

1.2 1

ℜ{m(ωc)}

0.8 0.6 0.4 0.2 0 –0.2

0

0.5 2

2 1/2

2

2 1/2

1

1.5

1

1.5

(p +q )

b

1.4 1.2

ℑ{m(ωc)}

1 0.8 0.6 0.4 0.2 0

0

0.5 (p +q )

Fig. 9.7 Same as in Fig. 9.4 but with !c D 2  109 r=s

a

0.7 0.6

|G(p,q,ωc)|

0.5 0.4 0.3 0.2 0.1 0

0

0.5 2

2 1/2

2

2 1/2

1

1.5

1

1.5

(p +q )

arg{G(p,q,ωc )}

b

4

2

0

−2 −4

0

0.5

(p +q )

Fig. 9.8 Same as in Fig. 9.5 but with !c D 2  109 r=s

9.2 Angular Spectrum Representation of Multipole Wavefields

a

25

0.8

ℜ{G(p,q,ωc )}

0.6 0.4 0.2 0 –0.2 –0.4 –0.6

0

0.5 2

2 1/2

2

2 1/2

1

1.5

1

1.5

(p +q )

b

0.6

ℑ{G(p,q,ωc )}

0.4 0.2 0

–0.2 –0.4 –0.6 –0.8

0

0.5 (p +q )

Fig. 9.9 Same as in Fig. 9.6 but with !c D 2  109 r=s

a

1.2 1

ℜ{m(ωc)}

0.8 0.6 0.4 0.2 0 –0.2

0

0.5

2

2 1/2

2

2 1/2

1

1.5

1

1.5

(p +q )

b

1.4 1.2

ℑ{m(ωc)}

1 0.8 0.6 0.4 0.2 0

0

0.5

(p +q )

Fig. 9.10 Same as in Fig. 9.4 but with !c D 2  1011 r=s

26

9 Pulsed Beam Wavefields in Temporally Dispersive Media

a

x 10

2

–185

|G(p,q,ωc)|

1.5

1

0.5

0

0

0.05

0.1

0.15

0.2

0.25 2

0.3

0.35

0.4

0.45

0.5

0.3

0.35

0.4

0.45

0.5

0.3

0.35

0.4

0.45

0.5

0.3

0.35

0.4

0.45

0.5

2 1/2

(p +q )

arg{G(p,q,ωc)}

b

4

2

0

–2

–4

0

0.05

0.1

0.15

0.2

0.25 2

2 1/2

(p +q )

Fig. 9.11 Same as in Fig. 9.5 but with !c D 2  1011 r=s

a

5

x 10

–186

ℜ{G(p,q,ωc)}

0 −5 −10 −15 −20

0

0.05

0.1

0.15

0.2

0.25 2

2 1/2

(p +q )

b

12

x 10

–186

ℑ{G(p,q,ωc)}

10 8 6 4 2 0 −2

0

0.05

0.1

0.15

0.2

0.25 2

2 1/2

(p +q )

Fig. 9.12 Same as in Fig. 9.6 but with !c D 2  1011 r=s

9.2 Angular Spectrum Representation of Multipole Wavefields

a

27

0.01

ℜ{G(p,q,ωc)}

0.005 0

–0.005 –0.01

0

0.1

0.2

0.3

0.4

0.5 2

0.6

0.7

0.8

0.9

1

0.6

0.7

0.8

0.9

1

2 1/2

(p +q )

b ℑ{G(p,q,ωc)}

0.01 0.005 0

–0.005 –0.01

0

0.1

0.2

0.3

0.4

0.5 2

2 1/2

(p +q )

Fig. 9.13 Same as in Fig. 9.9 but with z D 1:0 m

a

1

x 10

–20

ℜ{G(p,q,ωc)}

0.5 0

–0.5 –1

0

0.05

0.1

0.15

0.2

0.25 2

0.3

0.35

0.4

0.45

0.5

0.3

0.35

0.4

0.45

0.5

2 1/2

(p +q )

b

–21

8

x 10

ℑ{G(p,q,ωc)}

6 4 2 0 –2 –4 –6

0

0.05

0.1

0.15

0.2

0.25 2

2 1/2

(p +q )

Fig. 9.14 Same as in Fig. 9.9 but with z D 10:0 m

28

9 Pulsed Beam Wavefields in Temporally Dispersive Media

of a source-free scalar wavefield. This was accomplished by first expressing the solution of the homogeneous scalar wave equation as a linear superposition of homogeneous plane waves (known as a Whittaker-type expansion [21]). Expansion of the plane wave amplitude functions in a series of spherical harmonics then resulted in Whittaker’s multipole expansion of the free-field. Whittaker’s classic analysis was later extended by A. J. Devaney and E. Wolf [22] to the electromagnetic field produced by a localized charge–current distribution. The analysis presented here closely follows their method of analysis.3 The general description begins with the time-domain form of the macroscopic Maxwell’s equations [see (5.1)–(5.4) of Vol. 1] r  D.r; t / D k4k%ext .r; t /;    1  @B.r; t /  r  E.r; t / D    c  @t ; r  B.r; t / D 0;      1  @D.r; t /  4     C r  H.r; t / D    c  J.r; t /; c @t

(9.81) (9.82) (9.83) (9.84)

in a homogeneous, isotropic, locally linear, temporally dispersive medium (a simple dispersive medium), as described by the constitutive relations [see (5.5)–(5.7) of Vol. 1] Z

t

D.r; t / D 1 Z t

H.r; t / D 1 Z t

Jc .r; t / D

O .t  t 0 /E.r; t 0 /dt 0 ;

(9.85)

O 1 .t  t 0 /B.r; t 0 /dt 0 ;

(9.86)

O .t  t 0 /E.r; t 0 /dt 0 ;

(9.87)

1

where O is the dielectric response, O the magnetic response, and O the conductive response functions of the simple dispersive medium. Here J.r; t / D Jc .r; t / C Jext .r; t / is the total current density given by the sum of the conduction current density Jc .r; t / and the external current density Jext .r; t /. The total current density is related to the charge density through the equation of continuity r  J.r; t / D 

@%.r; t / : @t

(9.88)

Since %c D 0 [see the discussion following (5.11) in Vol. 1], then r Jc .r; t / D 0 and the conduction current density is solenoidal. Furthermore, the externally supplied charge and current sources satisfy the continuity equation 3

See also A. J. Devaney’s Ph.D. thesis [23] in 1971, which became required reading by many later graduate students (including myself) at The Institute of Optics.

9.2 Angular Spectrum Representation of Multipole Wavefields

r  Jext .r; t / D 

@%ext .r; t / : @t

29

(9.89)

The temporal Fourier transform of Maxwell’s equations then results in, with substitution from the Fourier transform of the above constitutive relations, the set of frequency domain field equations Q !/ D k4k %Q ext .r; !/ ; r  E.r; .!/   1 Q !/; Q !/ D   i!.!/H.r; r  E.r; c  Q !/ D 0; r  H.r;         Q !/ D  1  i!c .!/E.r; Q !/ C  4  JQ ext .r; !/; r  H.r; c   c 

(9.90) (9.91) (9.92) (9.93)

where c .!/ .!/ C k4ki

 .!/ !

(9.94)

is the complex permittivity. Here .!/ is the complex-valued dielectric permitQ !/ D tivity and  .!/ is the complex-valued electrical conductivity, where D.r; Q Q Q Q Q .!/E.r; !/, Jc .r; !/ D  .!/E.r; !/, and where B.r; !/ D .!/H.r; !/ with .!/ denoting the complex-valued magnetic permeability of the medium. Upon taking the curl of Faraday’s law (9.91) and using the relations given in Gauss’ law (9.90) and Ampre’s law (9.93), there results     Q !/ D  k4k i !.!/.!/ JQ ext .r; !/  r %Q ext .r; !/ ; r 2 C kQ 2 .!/ E.r; .!/ kc 2 k Q where k.!/ is the complex wavenumber defined in (9.11). Define the complexvalued wavenumber ! ..!/.!//1=2 kQ0 .!/ (9.95) kck Q as the nonconducting limit of the complex wavenumber k.!/. With this identification, the above expression for the temporal frequency transform of the electric field intensity becomes !   Q 2 .!/ k k4k Q !/ D  JQ ext .r; !/  r %Q ext .r; !/ ; r 2 C kQ 2 .!/ E.r; i 0 .!/ !

(9.96)

where %Q ext .r; !/ D r  JQ ext .r; !/=.i !/. In a similar fashion, the curl of Ampre’s law (9.93), together with Faraday’ law (9.91), yields

30

9 Pulsed Beam Wavefields in Temporally Dispersive Media

     4  2 2 Q  Q Q r C k .!/ H.r; !/ D    c  r  Jext .r; !/:

(9.97)

The electric and magnetic field vectors described by Maxwell’s equations (9.81)– (9.84) in a simple dispersive medium [as described by the constitutive relations given in (9.85)–(9.87)] then satisfy this pair of reduced wave equations [(9.96) and (9.97)], and conversely, the solutions of (9.96) and (9.97) which behave as outgoing spherical waves at infinity satisfy Maxwell’s equations (9.81)–(9.84) in all of space [22].

9.2.1 Multipole Expansion of the Scalar Optical Wavefield due to a Localized Source Distribution Consider the particular solution of the reduced scalar wave equation   r 2 C kQ 2 .!/ Q .r; !/ D k4k.r; Q !/

(9.98)

for a given spectral source distribution .r; Q !/ that is assumed to be a continuous function of position and is confined to a finite region about the origin (i.e., has compact support). In particular, let .r; Q !/ 0 for all r jrj > R, where R > 0 is some real constant, independent of the angular frequency !. The well-known particular solution of (9.98) that behaves as an outgoing spherical wave at infinity is given by Z 0 e ikQ.!/jrr j 3 0 Q .r; !/ D k4k d r: .r Q 0 ; !/ (9.99) 4 r 0 R jr  r0 j The spherical wave kernel appearing in this integral transform may be expressed as a superposition of plane waves through Weyl’s integral (see Sect. 6.4.3 of Vol. 1) Z 1 i 1 i kQ ˙ .!/r e ikQ.!/r D e dkx dky r 2 1 .!/ Z Z  Q ik.!/ Q D dˇ sin ˛ d˛ e ik.!/.x sin ˛ cos ˇCy sin ˛ sin ˇ˙z cos ˇ/ ; 2  C (9.100) 1=2  where kQ ˙ .!/  r D kx x C ky y ˙ .!/z with .!/ kQ 2 .!/  kx2  ky2 [see (9.12)], and where the contour C , illustrated in Fig. 9.15, extends from ˛ D 0 to Q ˛ D =2   i 1, where argfk.!/g with 0   =2. The positive sign appearing in the exponential in (9.100) is employed when z > 0 while the negative sign is used when z < 0. The polar coordinate form of Weyl’s integral expresses a spherical wave as a superposition of inhomogeneous plane waves, the ordered triple .sin ˛ cos ˇ; sin ˛ sin ˇ; ˙ cos ˛/ describing the complex direction cosines of the elementary plane wave normals.

9.2 Angular Spectrum Representation of Multipole Wavefields

31

Fig. 9.15 Contour of integration C in the complex ˛ D ˛ 0 C i ˛ 00 plane for Weyl’s integrtal representation

Weyl’s integral representation of the spherical wave appearing in the integrand of (9.99) is Z Z  0 Q Q ik.!/ e ik.!/jrr j 0 Q D dˇ sin ˛ d˛ e ik.!/s.rr / ; jr  r0 j 2  C

(9.101)

where sO 1O x sin ˛ cos ˇ C 1O y sin ˛ sin ˇ ˙ 1O z cos ˛ is a complex-valued vector whose components are the complex direction cosines associated with the inhomogeneous plane wave normals. Substitution of this result in (9.99) and interchanging the order of integration then results in the expression Z Z  Q Q Q .r; !/ D k4k ik.!/ dˇ sin ˛ d˛ O .Os; !/e ik.!/Osr 8 2  C

(9.102)

for the particular solution of the reduced wave equation (9.98), where the spectral amplitude function O .Os; !/ is defined in terms of the temporal frequency spectrum of the source distribution by O .Os; !/ D

Z

Q

0

.r Q 0 ; !/e ik.!/Osr d 3r 0 :

(9.103)

r 0 R

Just as in Whittaker’s representation [20] of the source-free wavefield, the Devaney– Wolf representation [22] expresses the radiated spectral wavefield Q .r; !/ as a

32

9 Pulsed Beam Wavefields in Temporally Dispersive Media

superposition of plane waves; however, while the Whittaker representation contains only homogeneous plane waves, the Devaney–Wolf representation contains both homogeneous and inhomogeneous plane wave components. In the nondispersive limit, the inhomogeneous components become evanescent as the contour C approaches the limiting case comprised of the straight line segment ˛ D 0 ! =2 along the real axis followed by the straight line ˛ D =2 ! =2  i 1 with fixed real part and imaginary part varying from 0 to 1. Integral representations of this general type describe the wavefield as an angular spectrum of plane waves [21, 22]. It is seen from (9.103) that the spectral amplitude function O .Os; !/ is related to the spatiotemporal Fourier transform of the source distribution QQ !/ D .k;

Z

0

.r Q 0 ; !/e ikr d 3 r 0

(9.104)

r 0 R

Q with k D k.!/O s, that is

  Q O .Os; !/ D QQ k.!/O s; ! :

(9.105)

Since the integral appearing on the right side of (9.104) extends over a finite space domain, and since .r; Q !/ has been assumed to be a continuous function of the QQ !/ is the boundary value along the real spatial coordinates, it then follows4 that .k; kx0 ; ky0 ; kz0 axes of an entire function of the complex variables kx D kx0 C ikx00 ; ky D ky0 C iky00 ; kz D kz0 C ikz00 . The relation given in (9.105) is then seen to be valid when sO is complex. Furthermore, O .Os; !/ is the boundary value along the contour C of an entire function of the three complex variables sx D sx0 C i sx00 ; sy D sy0 C isy00 ; sz D sz0 C isz00 . After Whittaker [20], the spectral amplitude function O .Os; !/ is now expanded in a series of spherical harmonics Y`m .; '/ i ` C`m P`m .cos /e im as O .Os; !/ D

1 X ` X

.i /` a`m .!/Y`m .˛; ˇ/;

(9.106)

(9.107)

`D0 mD` coefficients a`m .!/,

otherwise known as the multipole mowhere the expansion ments, are given by the projections of O .Os; !/ onto the spherical harmonics as a`m .!/

4

Di

`

Z

Z





dˇ 

0



d˛ sin .˛/ O .Os; !/Y`m .˛; ˇ/:

(9.108)

As noted by Devaney and Wolf [22], the result that the Fourier transform of a continuous function possessing compact support is a boundary value along the real axis of an entire function follows from a theorem on analytic functions defined by definite integrals that is given in 5:5 of Copson [24]. The multidimensional form of this result is provided by the Plancherel–P´olya theorem (see page 352 of [25]).

9.2 Angular Spectrum Representation of Multipole Wavefields

33

Here

dm P` .u/ (9.109) d um defines the associated Legendre polynomial of degree ` and order m in terms of the Legendre polynomials 1 d` 2 P` .u/ ` .u  1/` ; (9.110) 2 `Š du` this relation being referred to as Rodriques’ formula. Finally, the normalization constants appearing in (9.106) are given by P`m .u/

s C`m

m

.1/ .i /

`

.2` C 1/.`  m/Š : 4.` C m/Š

(9.111)

As noted by Devaney and Wolf [22], although the multipole moments a`m .!/ at a fixed value of the angular frequency !, defined in (9.108), “depend explicitly only on those spectral amplitudes O .Os; !/ which are associated with real sO (i.e., those corresponding to homogeneous plane waves in the angular spectrum representation),” the fact that O .Os; !/ is the boundary value of an entire function implies that the expansion (9.107) is valid for all unit vectors associated with the complex contour C depicted in Fig. 9.15. The multipole moments a`m .!/, given in (9.108), can also be expressed in terms of the spectral source distribution .r; Q !/ through substitution from (9.103). This substitution, followed by an interchange of the order of integration, gives a`m .!/ D

Z

d 3 r 0 .r0 ; !/i ` r 0 R

Z

Z





dˇ 

0

Q



0

d˛ sin .˛/Y`m .˛; ˇ/e ik.!/Osr :

Whittaker [20] showed that the homogeneous plane wave expansion of the spectral multipole field   Q m .r; !/ j (9.112) k.!/r Y`m .; /; ` ` where j` . / denotes the spherical Bessel function of order `, is given by [22] m ` .r; !/

1 D .i / 4 `

Z

Z





dˇ 

0

Q

d˛ sin .˛/Y`m .˛; ˇ/e ik.!/Osr ;

(9.113)

where the ordered-triple .r; ; / denotes the spherical polar coordinates of the field point r with reference to the same coordinate axis system as the complex-valued vector sO. With this identification, the above expression for the multipole moments becomes Z  3 0 .r0 ; !/m (9.114) a`m .!/ D ` .r; !/d r ; r 0 R

which is the desired result.

34

9 Pulsed Beam Wavefields in Temporally Dispersive Media

Substitution of the expansion given in (9.107) for O .Os; !/ into the expression (9.102) for the radiated spectral wavefield Q .r; !/, followed by an interchange of the order of integration and summation, then results in the series expansion [22] Q Q .r; !/ D k.!/

1 X ` X

a`m .!/˘Q `m .r; !/;

(9.115)

`D0 mD`

where k4k ˘Q `m .r; !/ D  .i /`C1 8 2

Z

Z



Q

dˇ 

C

sin ˛ d˛ Y`m .˛; ˇ/e ik.!/Osr :

(9.116)

As first shown by Erd´elyi [26] and later proved by Devaney and Wolf [22], the quantity appearing on the right-hand side of (9.116) is precisely the angular spectrum of plane waves representation of the spectral scalar multipole field of degree ` and order m, given by .C/ Q m .; /; (9.117) ˘Q m .r; !/ D h .k.!/r/Y `

`

`

where the ordered-triple .r; ; / denotes the spherical polar coordinates of the field .C/ point r and where h` . / denotes the spherical Hankel function of the first kind of order `. The expansion given in (9.115) is then seen to be the multipole expansion of the spectral scalar wavefield Q .r; !/ and the coefficients a`m .!/ appearing in that expansion are the corresponding multipole moments. Finally, unlike the angular spectrum of plane waves representation which is valid only in the two half-spaces z > R and z < R, the multipole expansion of the same wavefield is valid for all r > R (that is, for all points outside the source region).

9.2.1.1

Proof of the Validity of the Angular Spectrum Represention of the Scalar Multipole Field

Proof that the spectral multipole field in (9.117) is given by the angular spectrum representation in (9.116) is now presented based upon the proof given by Devaney and Wolf [22]. The proof begins with Erd´elyi’s result [26] that any scalar multipole field ˘Q `m .r; !/ of order m  0 can be generated by a spherical wave .C/ Q Q which is itself proportional to the lowest order scalar mulh0 .k.!/r/ D e ikQr =kr, p .C/ Q m through application of the operator tipole field ˘Q ` .r; !/ D .1= 4/h0 .k.!/r/, relation ˘Q `m .r; !/ D C`m



1 Q ik.!/



@ @ Ci @x @y

m

P`m



1

@ Q ik.!/ @z



e ikQ.!/r Q k.!/r

(9.118)

9.2 Angular Spectrum Representation of Multipole Wavefields

35

for m  0, where P`m



1 @ ikQ @z



ˇ ˇ dm Pz .u/ˇˇ : d um uD.1=ikQ/@=@`

(9.119)

Here P` .u/ denotes the Legendre polynomial of degree ` and the normalization coefficients C`m are as defined in (9.111). The identity ˘Q `jmj .r; ; / D .1/jmj ˘Q `jmj .r; ; /

(9.120)

is used when m < 0, where .r; ; / are the spherical polar coordinates of the position vector r. Substitution of Weyl’s integral representation (9.100) of a spherical scalar wave into (9.118), followed by an interchange of the order of integration and differentiation [justified when jzj > 0 so that the double integral in Weyl’s integral is uniformly convergent], then results in the expression ˘Q `m .r; !/ D

Z

Z



Q

d˛ sin .˛/ F .˛; ˇ/e ik.!/Osr ;

dˇ 

(9.121)

C

where Q

F .˛; ˇ/e ik.!/Osr m     @ 1 1 @ @ i m Q m C` Ci e ik.!/Osr : D P` Q Q 2 @y ik.!/ @x ik.!/ @z Since sO  r D x sin ˛ cos ˇ C y sin ˛ sin ˇ C z cos ˛, the above expression yields F .˛; ˇ/ D

i m m imˇ .m/ C sin ˛ e P` .cos ˛/: 2 `

(9.122)

.m/

Since sinm ˛ P` .cos ˛/ D P`m .cos ˛/, then i m m C P .cos ˛/e imˇ 2 ` ` i .i /` Y`m .˛; ˇ/; D 2

F .˛; ˇ/ D

(9.123)

where Y`m .˛; ˇ/ D i ` C`m P`m .cos ˛/e imˇ is the spherical harmonic of degree ` and order m [see (9.106)]. Substitution of this result in (9.121) then yields the desired result Z Z  i Q m ` Q .i / dˇ d˛ sin .˛/ Y`m .˛; ˇ/e ik.!/Osr ; (9.124) ˘` .r; !/ D 2 C 

36

9 Pulsed Beam Wavefields in Temporally Dispersive Media

valid when jzj > 0 and m  0. The result is also valid when z D 0, except at the origin, in the limiting sense [22] ˇ ˇ ˘Q `m .r; !/ˇ

zD0

i .i /` lim D 2 jzj!0

Z

Z



dˇ 

d˛ C

Q sin .˛/ Y`m .˛; ˇ/e ik.!/Osr

:

(9.125) For the angular spectrum representation of ˘Q `m .r; !/ when m < 0, the analysis begins with the expression sOr D r.sin  sin ˛ cos .ˇ C / C cos  cos ˛/ for the dot product between the complex vector sO along the normals to the inhomogeneous plane wave phase fronts and the position vector r at the field point. With this substitution, (9.124) becomes i .i /` ˘Q `jmj .r; !/ D 2

Z

Z



dˇ 

e

d˛ sin .˛/ Y`jmj .˛; ˇ/ C ikQ.!/rŒsin  sin ˛ cos .ˇC/Ccos  cos ˛

:

Substitution of this expression into the identity given in (9.120) then yields i ˘Q `jmj .r; !/ D .1/jmj .i /` 2

Z

Z



d˛ sin .˛/ Y`jmj .˛; ˇ/  C ikQ.!/rŒsin  sin ˛ cos .ˇ/Ccos  cos ˛

dˇ

e

:

With change of the variable of integration from ˇ to ˇ in this expression and use of the relation Y`jmj .˛; ˇ/ D .1/jmj Y`jmj .˛; ˇ/ there results i .i /` ˘Q `jmj .r; !/ D 2

Z

Z



Q

dˇ 

C

d˛ sin .˛/ Y`jmj .˛; ˇ/e ik.!/r sOr :

With the final identification that jmj D m with m < 0, the above result becomes formally identical to that given in (9.122), establishing the result that (9.124) is valid for `  m  `, which then completes the proof.

9.2.1.2

Far-Zone Behavior of the Multipole Expansion

An important feature of the angular spectrum representation of a multipole field is that it directly yields the far-zone behavior of the radiated wavefield, and hence, the radiation pattern, defined as the angular distribution of the radiated far-zone relative field strength at a fixed distance r from the source. Consider then the behavior of Q .r; !/ as k.!/r ! 1 along a fixed direction that is specified by the unit vector Q O1r r=r, where k.!/ jk.!/j, so that ˇ ˇ ˇr  r0 ˇ  r  r0  1O r :

(9.126)

9.2 Angular Spectrum Representation of Multipole Wavefields

37

Substitution of this asymptotic approximation into (9.99) then yields  e ikQ.!/r  Q Q .1O r r; !/  k4k QQ 1O r k.!/; ! 4 r

(9.127)

QQ !/ is the spatiotemporal Fourier transform of the source as k.!/r ! 1, where .k;   Q charge distribution .r; t /. However, from (9.105), QQ 1O r k.!/; ! is equal to the   spectral amplitude function O 1O r ; ! , so that the above asymptotic approximation may be written as  ikQ.!/r  Q .1O r r; !/  k4k O 1O r ; ! e ; (9.128) 4 r as k.!/r ! 1. This important result shows that the radiation patternof the scalar wavefield is exactly given by the spectral amplitude function O 1O r ; ! , where 1O r is a real unit vector specifying the observation direction. The radiation pattern of the scalar wavefield in any given direction 1O r from the source is then seen to be given by the complex amplitude of the plane wave component in the angular spectrum representation that is propagating in the direction 1O r . In particular, with substitution from (9.107), this asymptotic approximation becomes 1 X ` ikQ.!/r X Q .1O r r; !/  k4k e .i /` a`m .!/Y`m .; / 4 r

(9.129)

`D0 mD`

as k.!/r ! 1, where the ordered pair .; / denotes the spherical polar coordinates of the unit vector 1O r along the direction of observation. Finally, as pointed out by Devaney and Wolf [22], it is useful to stress the following points: The angular spectrum representation (9.102), and the multipole expansion (9.113), are mode expansions in the sense that they express the field Q .r; !/ in terms of certain elementary fields (plane wavefields and multipole fields, respectively), each of which satisfies Q the same  equation as does .r; !/ outside the source region, namely the Helmholtz equa2 2 Q Q tion r C k .!/ .r; !/ D 0. The range of validity of each of the two expansions is different. The angular spectrum expansion represents Q .r; !/ outside the strip jzj  R, the multipole expansion represents it outside the sphere r  R. The expansion coefficients in the two representations are related by (9.105) and (9.106).

9.2.2 Multipole Expansion of the Electromagnetic Wavefield Generated by a Localized Charge–Current Distribution in a Dispersive Dielectric Medium The temporal frequency spectra of the electric and magnetic field vectors in a simple nonconducting dispersive dielectric medium are found to satisfy the reduced wave

38

9 Pulsed Beam Wavefields in Temporally Dispersive Media

equations [see (9.96) and (9.97) when  .!/ D 0] !   Q k.!/ 1 Q !/ D k4k i.!/ JQ ext .r; !/  r %Q ext .r; !/ ; r 2 C kQ 2 .!/ E.r; kck .!/       Q !/ D   4  r  JQ ext .r; !/; r 2 C kQ 2 .!/ H.r;  c 

(9.130) (9.131)

Q with %Q ext .r; !/ D r  JQ ext .r; !/=.i !/, where k.!/ .!=kck/..!/.!//1=2 is the complex wavenumber for a nonconducting dispersive medium (i.e., for a dispersive dielectric), and where .!/ ..!/=.!//1=2 is the complex intrinsic impedance of the medium. As in the scalar case, it is assumed that the external charge %Q ext .r; !/ and current JQ ext .r; !/ sources are both continuous and continuously differentiable functions of position and that both identically vanish when r > R (i.e., have compact ˚ support), where R  0 is some real constant. The electroQ !/; H.r; Q !/ generated by the spectral charge–current magnetic field vectors E.r; ˚ distribution %Q ext .r; !/; JQ ext .r; !/ are accordingly identified with those particular solutions of the reduced wave equations (9.130)–(9.131) that behave as outgoing spherical waves at infinity. Equations (9.130)–(9.131) show that each individual Cartesian component of the temporal frequency ˚ domain electric and magnetic field vectors of the electromagQ !/; H.r; Q !/ satisfy inhomogeneous Helmholtz equations of netic wavefield E.r; the form given in (9.98). The results obtained in the preceeding subsection for the multipole expansion of the scalar wavefield then apply to each of these Cartesian components. Specifically, (9.102) and (9.103) show that the electric and magnetic field vectors have the angular spectrum of plane waves representations Z  Z Q O s; !/e ikQ.!/Osr ; Q !/ D k4k ik.!/ dˇ sin ˛ d˛ E.O E.r; 8 2  C Z Z  Q i k.!/ O s; !/e ikQ.!/Osr ; Q !/ D k4k dˇ sin ˛ d˛ H.O H.r; 8 2  C

(9.132) (9.133)

n o O s; !/; H.O O s; !/ are where the electric and magnetic spectral amplitude vectors E.O given by   Q k.!/ 1 0 Q JQ ext .r; !/  r %Q ext .r; !/ e ik.!/Osr d 3 r 0 ; i.!/ 0 kck .!/ r R (9.134) Z   ikQ.!/Osr0 3 0 1 O s; !/ D r  JQ ext .r; !/ e d r; (9.135) H.O kck r 0 R O s; !/ D E.O

Z

9.2 Angular Spectrum Representation of Multipole Wavefields

39

respectively. These two spectral amplitude functions may be expressed in a simpler form in terms of the spatiotemporal Fourier transforms of the charge and current densities, given by %QQ ext .k; !/ D JQQ ext .k; !/ D

Z r 0 R

Z

r 0 R

0

%Q ext .r0 ; !/e ikr d 3 r 0 ;

(9.136)

0 JQ ext .r0 ; !/e ikr d 3 r 0 ;

(9.137)

respectively, with the result   Q Q O s; !/ D  i .!/k.!/O s  sO  JQQ ext .k.!/O E.O s; !/ ; kck i Q Q O s; !/ D H.O s; !/: k.!/O s  JQQ ext .k.!/O kck

(9.138) (9.139)

The spectral amplitude field vectors are then seen to satisfy the orthogonality relations O s; !/ D .!/Os  H.O O s; !/; E.O O s; !/ D 1 sO  E.O O s; !/; H.O .!/ O s; !/ D sO  H.O O s; !/ D 0: sO  E.O

(9.140) (9.141) (9.142)

Consequently, at each angular frequency ! and direction sO, the spectral amplitude vector fields appearing in the integrands of (9.132)–(9.133) are monochromatic plane waves satisfying the source-free Maxwell’s equations throughout all of space. The angular spectrum of plane waves representation given in (9.132)–(9.133) is then seen to be a mode expansion of the electromagnetic field, valid throughout its domain of validity given by jzj > R. To obtain the multipole expansions of the monochromatic electromagnetic field ˚ Q !/; H.r; Q !/ in the spectral frequency domain, one first expands the vectors E.r; n o O s; !/; H.O O s; !/ in terms of the vector spherical spectral amplitude field vectors E.O harmonic functions Ym ` .˛; ˇ/ , defined in terms of the ordinary spherical harmonic functions Y`m .˛; ˇ/ by the relation [22] m Ym ` .˛; ˇ/ Ls Y` .˛; ˇ/;

(9.143)

where Ls is the orbital angular momentun operator   1 @ @  uO ˛ ; Ls i uO ˇ @˛ sin ˛ @ˇ

(9.144)

40

9 Pulsed Beam Wavefields in Temporally Dispersive Media

with uO ˛ and uO ˇ denoting unit vectors in the positive ˛ and ˇ directions, respectively. The vector spherical harmonics can be shown [27] to be everywhere tangent to the unit sphere, viz., sO  Ym ` .˛; ˇ/ D 0;

(9.145)

form an orthogonal set in the sense that Z

Z



dˇ 

0





0

m d˛ sin ˛Ym ` .˛; ˇ/  Y`0 .˛; ˇ/ D `.` C 1/ı`;`0 ım;m0 ;

(9.146)

and, taken together with the associated vector spherical harmonic functions sO  Ym ` .˛; ˇ/, form a complete orthogonal basis [22] for all sufficiently well-behaved vector functions F.Os/ defined on and tangential to the unit sphere jOsj D 1, so that sO  F.Os/ D 0. In particular, the spectral amplitude field vectors defined by (9.138) and (9.139) in terms of the spatiotemporal spectrum of the current source, may be expanded as [cf. (9.107)] O s; !/ D E.O

1 X ` X

 m m .i /` a`m .!/Os  Ym ` .˛; ˇ/ C b` .!/Y` .˛; ˇ/ ;

(9.147)

`D1 mD`

O s; !/ D H.O

1 `

 1 X X m .i /`  a`m .!/Ym s  Ym ` .˛; ˇ/ C b` .!/O ` .˛; ˇ/ ; .!/ `D1 mD`

(9.148) where the first expansion has been written in a form that is analogous to the scalar multipole expansion given in (9.107), the second expansion then following from the orthogonality relations given in (9.140)–(9.142). Notice that the summations over ` now begin with ` D 1 rather than with ` D 0 since there is no vector spherical harmonic of degree zero. Application of the orthogonality relation in (9.146) to the multipole expansions given in (9.147) and (9.148) then yields the pair of expressions Z  Z  i`  O s; !/  Ym dˇ d˛ sin .˛/ H.O ` .˛; ˇ/; (9.149) `.` C 1/  0 Z Z   i`  O s; !/  Ym b`m .!/ D dˇ d˛ sin .˛/ E.O (9.150) ` .˛; ˇ/; `.` C 1/  0

a`m .!/ D .!/

for the multipole moments as projections of the spectral ampolitude field vectors onto the vector spherical harmonics. One may also express a`m .!/ and b`m .!/ in terms of the spatiotemporal transform of the source current density by substituting the expressions given in (9.138)–(9.139) into the above relations with the result [22]

9.2 Angular Spectrum Representation of Multipole Wavefields

Q i `C1 .!/k.!/ kck `.` C 1/

Z

Q i `C1 .!/k.!/ b`m .!/ D  kck `.` C 1/

Z

a`m .!/ D 

Z





dˇ 



41

0

Z

d˛ sin .˛/ i h  Q s; !/  Ym  sO  JQQ ext .k.!/O ` .˛; ˇ/; (9.151) 

dˇ d˛ sin .˛/ 0 io n h  Q s; !/  Ym  sO  sO  JQQ ext .k.!/O ` .˛; ˇ/:



(9.152) Since sO  Ym ` .˛; ˇ/ D 0 [see (9.145)], (9.151) can be rewritten as [22] a`m .!/

Q i `C1 .!/k.!/ D kck `.` C 1/

Z



Z



dˇ d˛ sin .˛/ io h i n  h 0  Q s; !/  sO  Ym .˛; ˇ/ :  sO  sO  JQQ ext .k.!/O ` (9.153)

h i QQ !/ =jkj2 is the spatiotemporal Fourier transform JQQ .k; !/ of Since k  k  J.k; T the transverse part JT .r; t / of the current distribution, then (9.152) and (9.153) show that [22] all of the multipole moments a`m .!/ and b`m .!/, and consequently the electromagnetic field vectors external to the source region, depend only on those Fourier components of the transverse part of the source curQ rent distribution for which k.!/ D .!=kck/n.!/ [see Sect. 6.3 of Vol. 1], where n.!/ .c .!/.!//1=2 is the complex index of refraction of the medium. Substitution of the expansions in (9.147) and (9.148) for the spectral amplitude O s; !/ and H.O O s; !/ into the angular spectrum of plane waves represenvector fields E.O tation of the temporal frequency spectra of the electric and magnetic field vectors given in (9.132) and (9.133), followed by an interchange in the order of integration and summation, results in the series expansions [22] Q !/ D E.r;

1 X ` X

m  a` .!/EQ e`;m .r; !/ C b`m .!/EQ h`;m .r; !/ ;

(9.154)

`D1 mD`

Q !/ D H.r;

1 X ` X

m  Q e`;m .r; !/ C b`m .!/H Q h`;m .r; !/ ; a` .!/H `D1 mD`

(9.155)

42

9 Pulsed Beam Wavefields in Temporally Dispersive Media

where Q h`;m .r; !/ EQ e`;m .r; !/ D .!/H Z Z  Q

 ikQ.!/Osr ik.!/ k4k.i /` dˇ d˛ sin ˛ sO  Ym ; ` .˛; ˇ/ e 2 8 C  (9.156) h e Q `;m .r; !/ EQ `;m .r; !/ D .!/H Z Z  Q ik.!/ ikQ.!/Osr k4k.i /` dˇ d˛ sin ˛ Ym : ` .˛; ˇ/e 8 2  C (9.157) Devaney and Wolf [22] have shown that the integrals appearing in (9.156) and (9.157) are precisely the electromagnetic multipole fields ˚

 Q h`;m .r; !/ D r  r  r˘`m .r; !/ ; EQ e`;m .r; !/ D .!/H

 Q Q e`;m .r; !/ D ik.!/r EQ h`;m .r; !/ D .!/H  r˘`m .r; !/ ;

(9.158) (9.159)

field defined in (9.116) and (9.117). The where ˘`m .r; !/ is the scalar multipole o n e e Q Q component vector field quantities E .r; !/; H .r; !/ are then seen to be the `;m

`;m

temporal frequency spectra of the electric and magnetic fields n generated by an elec-o Q h .r; !/; H Q h .r; !/ tric multipole while the component vector field quantities E `;m `;m are the corresponding fields generated by a magnetic multipole, each of degree ` and order m. The pair of expressions in (9.154) and (9.155) are then the multipole expansions of the temporal frequency domain electric and magnetic field vectors generated by a spatially localized charge–current source distribution (i.e., a charge–current source with compact spatial support), the expansion coefficients a`m .!/ and b`m .!/ given in (9.151) and (9.152) then being the electric and magnetic multipole moments, respectively. The temporal frequency spectrum of any electromagnetic multipole wavefield then satisfies the homogeneous Maxwell equations for all r > 0, valid for all r > R (i.e., valid everywhere external to the source region). Substitution of the expressions in (9.136) and (9.137) for the spatiotemporal Fourier transforms of the charge and current densities into the expressions in (9.151) and (9.152) for the multipole moments results in a`m .!/

Z Q i `C1 .!/k.!/ D d 3 r 0 JQ ext .r0 ; !/ kck `.` C 1/ r 0 R Z  Z   ikQ.!/Osr0 dˇ d˛ sin .˛/ sO  Ym ;  ` .˛; ˇ/e

(9.160)

Z Q i .!/k.!/ d 3 r 0 JQ ext .r0 ; !/ kck `.` C 1/ r 0 R Z  Z   ikQ.!/Osr0 dˇ d˛ sin .˛/ Ym ;  ` .˛; ˇ/e

(9.161)

b`m .!/ D



0 `C1



0

9.2 Angular Spectrum Representation of Multipole Wavefields

43

m where the relation sO  .Os  Ym ` / D Y` , a consequence of the fact that the vector spherical harmonic functions are everywhere tangential to the unit sphere jsj D 1, has been used. Application of the two identities



` k ikr  rm ` .r/ D .i / 4

Z

Z





dˇ 

0

ikOsr d˛ sin .˛/Ym ; ` .˛; ˇ/e

(9.162) ˚

 k r  r  rm D .i /` ` .r/ 4

Z

Z





dˇ 

0

ikOsr d˛ sin .˛/Os  Ym ; ` .˛; ˇ/e

(9.163) m where m ` .r/ j` .kr/Y` .; / is the source-free multipole field, the above pair of expressions for the multipole moments becomes [22]

a`m .!/

Z n h  io Q 4 i.!/k.!/  D .r0 / d 3r 0 JQ ext .r0 ; !/  r  r  r0 m ` kck`.` C 1/kQ  .!/ r 0 R (9.164)

b`m .!/ D

Q 4 i.!/k.!/ kck`.` C 1/

Z r 0 R

h  i  0 JQ ext .r0 ; !/  r  r0 m .r / d 3r 0: `

(9.165)

p Q Notice that .!/k.!/ D .!=kck/.!/, where .!/ .!/=.!/ is the complex intrinsic impedance of the nonconducting medium [see (5.65) of Vol. 1], and Q that k.!/= kQ  .!/ D e i2 .!/ , where .!/ is the phase of the complex wavenumber Q D 1 , the given in (9.18). In free space [.!/ D 1 (in cgs units) and kQ  .!/=k.!/ above pair of expressions for the multipole moments simplify to the well-known expressions given in graduate-level electromagnetics texts (see, for example, Chap. 9 of [28]). The pair of expressions given in (9.164) and (9.165) then provide the generalization of the multipole moments to the dispersive, attenuative, nonconducting medium case. The generalization of these expressions to the dispersive, conducting case remains to be given.

9.2.2.1

Proof of the Validity of the Angular Spectrum Represention of the Electromagnetic Multipole Field

Proof that the spectral electromagnetic multipole field given in (9.158) and (9.159) is given by the angular spectrum representation in (9.156) and (9.157) is now presented based on the proof by Devaney and Wolf [22]. Consider first the vector quantity  ˚ m ˚

Q Q ˘Q ` .r; !/r  r  r  r ˘Q `m .r; !/ ik.!/ r  r˘Q `m .r; !/ D ik.!/ Q Qm D k.!/L r ˘` .r; !/;

(9.166)

44

9 Pulsed Beam Wavefields in Temporally Dispersive Media

because r  r D 0, and where Lr i r  r   1 O @ @  D i 1O  1 @ sin  @ is the “orbital angular momentum operator” in r-space [cf. (9.144)]. Substitution of the definition given in (9.117) for the spectral scalar multipole field ˘Q `m .r; !/ into (9.166) then gives   ˚

.C/  Q Q Q k.!/r Ym ik.!/ r  r˘Q `m .r; !/ D k.!/h ` .; /; `

(9.167)

where m Ym ` .; / Lr Y` .; /

(9.168)

is the vector spherical harmonic function of degree ` and order m [cf. (9.143)]. The vector spherical harmonic functions Ym ` .; / may be expressed as a linear combination of the ordinary spherical harmonic functions Y`m .; / as [27] mC1 .; / C eC aC Y`m1 .; / C 1O z mY`m .; /; Ym ` .; / D e a Y`

(9.169) where p .`  m/.` C m C 1/; p aC .` C m/.`  m C 1/;  1 O e 1x  i 1O y ; 2  1 O eC 1x C i 1O y : 2 a

(9.170) (9.171) (9.172) (9.173)

Substitution of (9.169) into (9.167) then gives  ˚

Q ik.!/ r  r˘Q `m .r; !/ 

Q D k.!/ e a ˘Q mC1 .r; !/ C eC aC ˘Q `m1 .r; !/ C m˘Q `m .r; !/ : `

(9.174) Each of the spectral scalar multipole fields appearing on the right-hand side of the above equation are now expressed in terms of the angular spectrum representation given in (9.116) with the result  ˚

Q ik.!/ r  r˘Q `m .r; !/ Z Z  k4k Q Q `C1 dˇ sin ˛d˛ G.˛; ˇ/e ik.!/Osr ; k.!/.i / D 8 2  C (9.175)

9.2 Angular Spectrum Representation of Multipole Wavefields

45

where G.˛; ˇ/ D e a Y`mC1 .˛; ˇ/ C eC aC Y`m1 .˛; ˇ/ C 1O z mY`m .˛; ˇ/: (9.176) Comparison of this expression with that given in (9.169) immediately shows that G.˛; ˇ/ D Y.˛; ˇ/, so that  ˚

Q ik.!/ r  r˘Q `m .r; !/ Z Z  k4k Q `C1 ikQ.!/Osr dˇ sin ˛d˛ Ym ; k.!/.i / D ` .˛; ˇ/e 8 2 C  (9.177) which is precisely the angular spectrum representation (9.157) of the pair of electromagnetic multipole fields given in (9.159). Finally, application of the curl operator to the expression in (9.177), noting that the curl operator may be taken inside both integrals on the right-hand side since that double integral converges uniformly for jzj > 0, results in  ˚

r  r  r˘Q `m .r; !/ Z Z  k4k Q `C1 ikQ.!/Osr dˇ sin ˛d˛ sO  Ym ; k.!/.i / D ` .˛; ˇ/e 2 8  C (9.178) which is precisely the angular spectrum representation (9.156) of the pair of electromagnetic multipole fields given in (9.158). This then completes the proof.

9.2.2.2

Far-Zone Behavior of the Electromagnetic Multipole Field

As in the scalar wave case, the angular spectrum representation of an electromagnetic multipole field directly yields the far-zone behavior of the radiated wavefield. Application of the asymptotic approximation given in (9.128) to each of the Q !/ and H.r; Q !/ then yields Cartesian components of each spectral field vector E.r; ikQ.!/r O 1O r ; !/ e Q 1O r r; !/  k4k E. ; E. 4 r

(9.179)

ikQ.!/r O 1O r ; !/ e Q 1O r r; !/  k4k H. H. ; 4 r

(9.180)

46

9 Pulsed Beam Wavefields in Temporally Dispersive Media

as k.!/r ! 1 along the fixed direction specified by the unit vector 1O r D r=r. The radiation pattern of the radiated electromagnetic wavefield is thus seen to be O 1O r ; !/, where 1O r O 1O r ; !/ and H. described by the spectral amplitude field vectors E. specifies the observation direction relative to the source coordinate system. From (9.138) and (9.139), this far-zone asymptotic approximation of the radiated wavefield may be expressed in terms of the spatiotemporal Fourier–Laplace transform of the source current distribution as   h  i ikQ.!/r  i  Q  4  .!/k.!/ O r  1O r  JQQ ext .1O r ; !/ e O Q 1 E.1r r; !/   ; 4  c  r

(9.181)

    ikQ.!/r  i  Q  4  k.!/ O r  JQQ ext .1O r ; !/ e O Q 1 ; H.1r r; !/  4  c  r

(9.182)

as k.!/r ! 1. Alternatively, the far-zone asymptotic behavior of the radiated electromagnetic wavefield may be expressed in terms of the multipole moments a`m .!/ and b`m .!/ through (9.147) and (9.148) as ikQ.!/r Q 1O r ; !/  k4k e E. 4 r 1 X ` X

 m m  .i /` a`m .!/1O r  Ym ` .; / C b` .!/Y` .; / ; `D1 mD`

(9.183) ikQ.!/r O 1O r ; !/ D k4k e H. 4.!/ r



1 X ` X

 m m O .i /`  a`m .!/Ym ` .; / C b` .!/1r  Y` .; / ;

`D1 mD`

(9.184) as k.!/r ! 1, where .; / are the spherical polar coordinates of the unit vector 1O r . As noted by Devaney and Wolf [22], since the radiation pattern of the electroO 1O r ; !/ or magnetic field is given by either of the spectral amplitude field vectors E. O O O 1r ; !/ for all real values of the unit direction vector 1r , then (9.149) and (9.150), H. taken together with the orthogonality relations given in (9.140)–(9.142), may be interpreted as specifying the multipole moments a`m .!/ and b`m .!/ in terms of the radiation pattern of the radiated electromagnetic wavefield. Hence [22], “all the multipole moments and: : : the electromagnetic field at all points outside the sphere r > R, are completely specified by the radiation pattern.”

9.2 Angular Spectrum Representation of Multipole Wavefields

9.2.2.3

47

Radiated Power and the Angular Distribution of Multipole Radiation

Consider first the time-averaged power radiated by the localized charge–current source. From (5.132) of Vol. 1 [see also Sect. 9.1.2], the time-average Poynting vector of the radiated time-harmonic field component with angular frequency ! is given by the real part of the complex Poynting vector  c  Q Q !/ D 1  Q  .r; !/: !/  H S.r;   E.r; 2 4

(9.185)

Substitution from (9.179) and (9.180) into this expression results in the far-zone behavior 2˛.!/r Q !/  k4ck E. O 1O r ; !/  H O  .1O r ; !/ e (9.186) S.r; 32 2 r2 n o Q as k.!/r ! 1 along the 1O r -direction, where ˛.!/ = k.!/ is the plane wave attenuation factor in the dispersive medium [see (9.19)]. The time-average power radiated by the localized charge–current source is then given by the surface integral of the radial component of the time-average Poynting vector over a sphere ˙ of radius r as k.!/r ! 1, so that

Q !/  1O r r 2 d˝ S.r; ˙

Z  Z  h i k4ck 2˛.!/r O 1O r ; !/  H O  .1O r ; !/  1O r :  e < d sin  d E. 32 2  0 Z

hP i D <

Application of the orthogonality relation given in (9.141) to the integrand in the above expression then results in Z  Z  h i k4ck 2˛.!/r 1  O O O O e <  d sin  d E.1r ; !/  E .1r ; !/ : hP i  32 2  .!/  0 (9.187) The time-average radiated power may also be expressed in terms of the magnetic field intensity vector as hP i 

Z  Z  h i k4ck 2˛.!/r O  .1O r ; !/ : O 1O r ; !/  H e < .!/ d sin  d H. 32 2  0 (9.188)

O 1O r ; !/ into (9.187) [or, equivSubstitution of the multipole expansion (9.147) for E. O 1O r ; !/ into (9.188)] alently, substitution of the multipole expansion (9.148) for H.

48

9 Pulsed Beam Wavefields in Temporally Dispersive Media

with application of the orthogonality relation (9.146) for the vector spherical harmonics then yields the well-known result [22]

1 X ` i h X 1 k4ck 2 2 2˛.!/r m m e < `.` C 1/ .!/j C .!/j jb hP i  ja ` ` 32 2  .!/ `D1 mD` (9.189) expressing the time-average radiated power in terms of the electric and magnetic multipole moments of the spectral amplitude field vectors as k.!/r ! 1. This result then shows that, if both the electric and magnetic multipole moments a`m .!/ and b`m .!/ of a source comprised of a set of incoherently superimposed multipoles of fixed order ` are all independent of m, independent of the angular frequency !, then the far-field angular distribution of the radiated power by that source will be isotropic. The time-averaged far-field power radiated per unit solid angle is given by [28] n o dhP i Q !/  1O r r 2 ; D < S.r; d˝

(9.190)

so that, with substitution from (9.179) and (9.180) followed by application of the orthogonality relation in (9.141) and then substitution of the multipole expansion in (9.147), one obtains ( ` ˇ X dhP i k4ck 2˛.!/r 1 `ˇ m  e < .i / ˇa` .!/1O r  Ym ` .; / d˝ 32 2  .!/ mD`

ˇ2

ˇ Cb`m .!/Ym ` .; /ˇ

)

(9.191) as k.!/r ! 1. Notice that the electric [a`m .!/] and magnetic [b`m .!/] multipoles of any fixed order .`; m/ possess the same angular dependence at any far-field distance r but have orthogonal polarization states. Notice also that the relative angular distribution of the multipole radiation due to a pulsed source can change with observation distance r in a dispersive medium due to the frequency dependent attenuation factor e 2˛.!/r appearing in this expression. For a pure electric multipole of order .`; m/ the above expression (9.191) for the time-averaged far-field power radiated per unit solid angle becomes

dhP i k4ck 1 2 2˛.!/r  <  ja`m .!/j2 jYm ` .; /j e d˝ 32 2  .!/

(9.192)

as k.!/r ! 1, with a similar expression for the magnetic case. By comparison, the far-zone electric field strength for this pure multipole is given by [from (9.183)]

9.2 Angular Spectrum Representation of Multipole Wavefields

49

˛.!/r Q 1O r ; !/  k4k ja`m .!/j jYm E. ` .; /j e 4

(9.193)

as k.!/r ! 1, with a similar expression for the magnetic field strength. A relative angular plot of either of these two expressions [(9.192) or (9.193)] for given values of ` and m provides a radiation or antenna pattern5 for that particular order multipole, the latter (9.193) being a field strength pattern and the former (9.192) a power pattern. Since both involve relative values, the power pattern for a pure multipole is just the square of the corresponding field pattern. The power patterns for dipole radiation (` D 1, m D 0; ˙1) are depicted in Fig. 9.16, the dipole distribution for m D 0 being a linear dipole oscillating along the z-axis at the origin and the dipole distributions for m D ˙1 being comprised of a pair of linear dipoles in the equatorial plane =2 out of phase with each other, one at the origin along the x-axis and the other at the origin along the y-axis. The power patterns for quadrupole radiation (` D 2, m D 0; ˙1; ˙2) are depicted in Fig. 9.17.

z

|Y1±1(q,f)|2 Fig. 9.16 Dipole radiation ˇ ˇ2 power patterns ˇY01 .; /ˇ D 3 sin2  (solid curve) 8 ˇ ˇ2 ˇ ˇ and ˇY˙1 D 1 .; /ˇ   3 2 1 C cos  (dashed 16 curve). Since each radiation pattern depicted here is independent of the azimuthal angle , the threedimensional pattern is given by the corresponding surface of revolution about the polar z-axis

|Y10(q,f)|2

5 From the IEEE Standard Definitions of Terms for Antennas (IEEE Std 145-1983), a radiation pattern or antenna pattern is defined as “a mathematical function or a graphical representation of the radiation properties of the antenna as a function of space coordinates. In most cases, the radiation pattern is determined in the far-field region and is represented as a function of the directional coordinates. Radiation properties include power flux density, radiation intensity, field strength, directivity phase or polarization.”

50

9 Pulsed Beam Wavefields in Temporally Dispersive Media

Fig. 9.17 Quadrupole radiation power ˇ ˇ2 patterns ˇY02 .; /ˇ D 15 sin2  cos2  (solid 8 ˇ ˇ2 ˇ ˇ curve), ˇY˙1 2 .; /ˇ D   5 1  3 cos2  C 4 cos4  16

z

|Y2±1(q,f)|2 |Y20(q,f)|2

(dashed curve), and ˇ ˇ2 ˇ ˙2 ˇ ˇY2 .; /ˇ D   5 1  cos4  (dot16 ted curve). Since each radiation pattern depicted here is independent of the azimuthal angle , the threedimensional pattern is given by the corresponding surface of revolution about the polar z-axis

|Y2±2(q,f)|2

9.3 Stationary Phase Asymptotic Approximations of the Angular Spectrum Representation in Free-Space Q !/ of a pulsed wavefield U.r; t / may be The temporal frequency spectrum U.r; expressed in the positive half-space z > 0 of a nonabsorptive (and hence, nondispersive) medium by the real direction cosine form of the angular spectrum of plane waves representation as [see Sect. 9.1.1] Q E .r; !/ Q !/ D U Q H .r; !/ C U U.r; with Q J .r; !/ D U

Z

QQ U.p; q; !/e ik.pxCqyCmz/ dpdq

(9.194)

(9.195)

DJ

for z > 0 with J D H; E. Here k D !=c is real-valued and  1=2 ˚ m D C 1  p2  q2 ; .p; q/ 2 DH D .p; q/j 0  p 2 C q 2  1 ; (9.196) 1=2 ˚  ; .p; q/ 2 DE D .p; q/j p 2 C q 2 > 1 : (9.197) m D Ci p 2 C q 2  1 QQ For a sufficiently well-behaved spatiotemporal frequency spectrum U.p; q; !/, the Q !/ is found [29, 30] to satisfy the homogeneous Helmholtz spectral wavefield U.r; equation [see Sect. 7.1.1 of Vol. 1]   2 Q !/ D 0; r C k 2 U.r;

(9.198)

9.3 Stationary Phase Asymptotic Approximations

51

and the Sommerfeld radiation condition [31, page 250, Eq. (1d)] lim r

r!1

! Q !/ @U.r; Q !/ D 0  ikU.r; @r

(9.199)

in the half-space z > 0, which guarantees that there are only outgoing waves at Q H .r; !/ is expressed in (9.195) as a suinfinity. The spectral wavefield component U perposition of homogeneous plane waves that are propagating in directions specified by the real-valued direction cosines .p; q; m/, as illustrated in Fig. 9.2, whereas the Q E .r; !/ is expressed in (9.195) as a superposition spectral wavefield component U of evanescent plane waves that are propagating in directions perpendicular to the z-axis as specified by the real-valued direction cosines.p; q; 0/ and exponentially decaying at different rates in the z-direction. Because the behavior of the evanescent plane wave components is so vastly different from that for the homogeneous plane Q E .r; !/ Q H .r; !/ and U wave components, the two component spectral wavefields U are usually treated separately, as done here. In view of the fact that the integral appearing in the angular spectrum representation in (9.195) cannot be evaluated analytically (except in special cases), meaningful approximations are then required. Numerical methods may always be used, of course, but they yield little real understanding of the phenomena involved unless they are coupled with approximate analytic results. Typically, the most appropriate approximations in the majority of applications are those that are valid as the distance p (9.200) R D .x  x0 /2 C .y  y0 /2 C .z  z0 /2 from the fixed point .x0 ; y0 ; z0 / becomes large in comparison with the wavelength  D 2=k D 2c=! of the spectral amplitude wavefield. This approximation Q E .r; !/ as Q !/, U Q H .r; !/, and U is provided by the asymptotic expansion of U.r; kR ! 1 with fixed k and fixed observation direction specified by the (fixed) direction cosines .1 ; 2 ; 3 / defined by 1

x  x0 ; R

2

y  y0 ; R

3

z  z0 : R

(9.201)

The analysis presented here follows that given by Sherman, Stamnes, and Lalor [32] Q !/ as the point of observation in 1975 who consider the asymptotic behavior of U.r; .x; y; z/ recedes to infinity in a fixed direction .1 ; 2 ; 3 / with positive z-component 3 > 0 through the point .x0 ; y0 ; z0 /, as well as when the point of observation recedes to infinity in a fixed direction .1 ; 2 ; 0/ perpendicular to the z-axis through a fixed point .x0 ; y0 ; z0 / in the positive half-space z > 0. The generalized asymptotic approximation presented here for the angular spectrum integral appearing in (9.195) is based upon an earlier extension of the method of stationary phase (see Sect. 4 of Appendix F) to multiple integrals of the form Z

g.p; q/e ikRf .p;q/ dpdq;

I.kR/ D D

(9.202)

52

9 Pulsed Beam Wavefields in Temporally Dispersive Media

as kR ! 1. In this extended method of stationary phase, originally due to Focke [33], Braun [34], Jones and Kline [35], and Chako [36], the functions f .p; q/ and g.p; q/ are taken to be independent of kR and to be sufficiently smooth in the domain D and the quantity kRf .p; q/ is required to be real-valued for all .p; q/ 2 D. Heuristically, as kR becomes large, small variations in .p; q/ result in rapid oscillations of the exponential factor appearing in the integrand of (9.202) and this, in turn, results in destructive interference in the integration about that point in the integration domain D so that most of the domain D provides only a negligible contribution to the integral I.kR/. Significant contributions to I.kR/ come from the respective neighborhoods of specific types of critical points that are ordered according to their relative asymptotic order as follows: Critical Point of the First Kind:

An interior point .ps ; qs / 2 D where

ˇ ˇ @f .p; q/ ˇˇ @f .p; q/ ˇˇ D D 0; @p ˇ.ps ;qs / @q ˇ.ps ;qs /

(9.203)

at which point the phase function f .p; q/ is stationary, called an interior stationary phase point because the exponential factor in the integrand of (9.202) does not oscillate at that point, is a critical point of the first kind because it results in an ˚asymptotic contribution to the integral appearing in (9.202) that is of order O .kR/1 as kR ! 1. Critical Point of the Second Kind: A boundary point .pb ; qb / 2 @D on the curve that forms the boundary of the integration domain D at which @f [email protected] ;qb / D 0, where ds describes a differential element of arc length along the boundary curve, is ˚a critical point of the second kind because it yields a contribution of order O .kR/3=2 as kR ! 1. Critical Point of the Third Kind: A corner point .pc ; qc / 2 @D on the curve that forms the boundary of the integration domain D is a critical point of the third kind because the slope ˚of the boundary curve is discontinuous, resulting in a contribution of order O .kR/2 as kR ! 1. Additional critical points can arise at any singularities of the function g.p; q/ appearing in the integrand of (9.202). Unfortunately, this heuristic description does not lead to a rigorous asymptotic description of integrals of the form given in (9.202). Although these rigorous extensions of the stationary phase method have little resemblance to the heuristic stationary phase argument presented above, they do provide justification of that heuristic argument provided that certain restrictions (as specified above) are placed on both f .p; q/ and g.p; q/. Unfortunately, these restrictions are not satisfied by the angular spectrum integrals given in (9.195). In particular, the restriction that the functions f .p; q/ and g.p; q/ must possess a finite number of continuous partial derivatives in D except possibly at a finite number of isolated integrable singularities of g.p; q/ is not satisfied by the angular spectrum integral (9.195) on the unit circle p 2 C q 2 D 1 that forms the boundary between DH and DE . In addition, the requirement that kRf .p; q/ is real-valued in D cannot be Q !/ or U Q E .r; !/ when 3 > 0 since m is imaginary in DE . applied to either U.r;

9.3 Stationary Phase Asymptotic Approximations

53

Q E .r; !/ is indeed negligible in comNevertheless, if the evanescent contribution U Q H .r; !/ then, at the very least, the leadparison to the homogeneous contribution U Q !/ can be obtained from the asymptotic ing term in the asymptotic expansion of U.r; Q H .r; !/. This is not always the case, however, as shown by the folexpansion of U lowing example due to Sherman et al. [32]. A time-harmonic spherical wave e ikr UQ s .r; !/ D r 2 2 2 1=2 e ik.x Cy Cz / D 2 .x C y 2 C z2 /1=2 radiating away from the origin has spectral amplitude [37] UQQ s .p; q; !/ D D

ik 2 m ik : 2.1  p 2  q 2 /1=2

When the point of observation lies along the z-axis, p D q D 0 and the integrals in (9.195) may be directly evaluated to obtain e ikz 1  ; UQ sH .r; !/ D z z 1 UQ sE .r; !/ D : z It is then seen that the homogeneous and evanescent wave contributions are of the same order in 1=z, so that, in this special case, the evanescent contribution to the wavefield cannot be neglected in comparison to the homogeneous contribution. This result is shown to be a direct consequence of the singular behavior of UQQ s .p; q; !/ D ik=2 m on the unit circle p 2 C q 2 D 1. In the generalized asymptotic method due to Sherman, Stamnes, and Lalor [32] that is considered here, specific restrictions are placed on the spectral amplitude QQ QQ function U.p; q; !/. In particular, U.p; q; !/ 2 TN , where TN is defined for QQ positive, even integer N as the set of all spectral functions U.p; q; !/ that are independent of the spatial coordinates x; y; z and that satisfy the following conditions for all !: QQ 1. U.p; q; !/ can be expressed in the form (with the dependence on the angular frequency ! suppressed) V.p; q; m/ QQ ; U.p; q; !/ D m

(9.204)

where m is as defined in (9.196) and (9.197), and where V .p; q; m/ jV.p; q; m/j is bounded for all p; q.

54

9 Pulsed Beam Wavefields in Temporally Dispersive Media

2. V.p; q; s/ is a complex, continuous vector function of the three independent variables p; q; s that are defined (a) for all real p and q, (b) for real s 2 Œ0; 1 , and (c) for pure imaginary s D i with  > 0. 3. V.p; q; s/ possesses continuous, bounded partial derivatives up to the order N with respect to p; q; s for all p; q; s within its domain of definition. Notice that condition 1 ensures the convergence of the angular spectrum integrals appearing in (9.195) for all z > 0, while conditions 2 and 3 are required for the derivation of the asymptotic approximations of these integrals.

9.3.1 Approximations Valid Over a Hemisphere Q !/ D U.x; Q The asymptotic approximation of U.r; y; z; !/ that is valid as the point of observation r D .x; y; z/ recedes to infinity in a fixed direction with z > 0 through the point r0 D .x0 ; y0 ; z0 / is considered first. This is done in two steps. First, the method of stationary phase is extended using the technique due to Focke [33] and QQ Chako [36] in order to show that when U.p; q; !/ 2 TN has an interior stationary phase point at .ps ; qs / D .1 ; 2 /, only an arbitrarily small neighborhood of that stationary phase point contributes to the asymptotic behavior as R ! 1 up to terms of order Of.kR/N g. A standard application of the method of stationary phase is then used to obtain the asymptotic contribution due to this interior stationary phase point.

9.3.1.1

Extension of the Method of Stationary Phase

The method of approach used by Focke [33] and Chako [36] employs a neutralizer function .p; q/ to isolate the interior stationary point. Notice that since the point of observation r D .x; y; z/ recedes to infinity in a fixed direction with z > 0 through the point r0 D .x0 ; y0 ; z0 /, then the direction cosine values 1 ; 2 ; 3 are constant with 3 > 0, and consequently, the stationary phase point .ps ; qs / D .1 ; 2 / is fixed within the interior of the region DH . Define ˝1 and ˝2 to be two arbitrarily small neighborhoods of the point .ps ; qs /, both of which are completely contained within the interior of DH , where ˝1 ˝2 is a proper subset of ˝2 . Let .p; q/ be a real, continuous function of the independent variables p and q with continuous partial derivatives of all orders for all real values of p and q that satisfies the property 0  .p; q/  1;

(9.205)

with .p; q/ D

1; 0;

when .p; q/ 2 ˝1 : when .p; q/ … ˝1

(9.206)

9.3 Stationary Phase Asymptotic Approximations

55

The existence of such a continuous function for arbitrary regions ˝1 and ˝2 with ˝1 ˝2 has been given by Bremmerman [38]. Additional details about the explicit form of this neutralizer function .p; q/ are unnecessary. The principal result of this section is expressed in the following theorem due to Sherman, Stamnes, and Lalor [32]. QQ Theorem 1. Let U.p; q; !/ 2 TN for some positive even integer N . Then for z > 0, Q !/ D U.x; Q the spectral wavefield U.r; y; z; !/ given in (9.194)–(9.195) with k a positive real constant satisfies Q Q 0 .x; y; z/ C R.x; y; z/; U.x; y; z; !/ D U

(9.207)

where6 Q 0 .x; y; z/ D U

Z

QQ .p; q/U.p; q; !/e ik.pxCqyCmz/ dpdq;

(9.208)

DH

˚ and where R.x; y; z/ D O .kR/N as kR ! 1 uniformly with respect to 1 and q 2 for all real 1 ; 2 such that ı < 1  12  22  1 for any positive constant ı < 1. Notice that the dependence on the angular frequency ! has been suppressed in ˚ the above expressions. In addition, the statement that R.x; y; z/ D O .kR/N as kR ! 1 uniformly with respect to 1 and 2 means that R.x; y; z/ is bounded as jR.x; y; z/j  M.kR/N as kR ! 1 with M a positive constant independent of 1 and 2 . This theorem then directly leads to the result that the asymptotic beQ !/ of order lower than .kR/N as kR ! 1 is havior of the spectral wavefield U.r; completely determined by an arbitrarily small neighborhood of the point .ps ; qs /. QQ In particular, if U.p; q; !/ 2 TN when N is arbitrarily large, then the asymptotic Q Q 0 .x; y; z/. The proof expansion of U.r; !/ is equal to the asymptotic expansion of U of this theorem [32] is given in Appendix G based upon a modification of the proof of Theorem 1 by Chako [36] or Theorem 3 by Focke [33].

9.3.1.2

Q Asymptotic Approximation of U.r; !/

Theorem 1 establishes that any terms of order lower than .kR/N in the asymptotic Q !/ as kR ! 1 with fixed k > 0 must be contributed by the term behavior of U.r; [see (9.208)] Z QQ Q .p; q/U.p; q; !/e ik.pxCqyCmz/ dpdq: (9.209) U0 .x; y; z/ D DH

6

Notice the typographical error in (2.5) of Ref. [32] where the spectral amplitude function QQ U.p; q; !/ was inadvertently omitted from that equation.

56

9 Pulsed Beam Wavefields in Temporally Dispersive Media

This integral representation can be expressed in the form of the integral I.kR/ in (9.202) with phase function f .p; q/ D 1 p C 2 q C 3 m

(9.210)

and amplitude function (now a vector) QQ g.p; q/ D .p; q/U.p; q; !/e ik.px0 Cqy0 Cmz0 / :

(9.211)

Because .p; q/ D 0 when .p; q/ … ˝2 , the domain of integration DH appearing in (9.209) can be replaced by a domain D that is taken as any region within the interior of the region DH that contains the subregion ˝2 within its interior (see Fig. 9.18). With that choice, the phase function f .p; q/ defined in (9.210) is realvalued and infinitely differentiable in D, and the amplitude function g.p; q/ has continuous, bounded partial derivatives up to order N in D. The integral in (9.209) then satisfies all of the conditions required for the direct application of the stationary phase method due to Braun [34]. Because there is only the single critical point [the interior stationary phase point .p; q/ D .1 ; 2 /] in D, and because the integrand and its derivatives all vanish on the boundary of the domain D, application of the method of stationary phase due to Braun [34] then gives7 N

2 1 o n Bn .#; '/ e ikR X  N2 Q ; U.x; y; z/ D C O .kR/ kR nD0 .kR/n

(9.212)

as kR ! 1 with fixed k > 0, positive z, and fixed direction cosines 1 ; 2 , and QQ with U.p; q; !/ 2 TN , where the coefficients Bn .#; '/ are independent of R. Notice that, as required by the definition of the set TN of spectral functions introduced just prior to (9.204), N is a positive, even integer. Finally, notice that the order of the remainder term in (9.212) is a refinement of Braun’s estimate as reported by Sherman, Stamnes, and Lalor [32]. Application of the result given in Sect. 5.1 of the paper by Jones and Kline [35] yields the zeroth-order term B0 .#; '/ D 2 i V.1 ; 2 ; 3 /e ik.1 x0 C2 y0 C3 z0 / ; H1

(9.213)

E1

p 2+ q 2−1 ... H2

E2

Fig. 9.18 Illustration of the integration regions DJ1 and DJ 2 , J D H; E, appearing in (9.224) and (9.225) 7

Although the stationary phase result given by Braun [34] is for a scalar field, the result may be directly generalized to a vector field by seperately applying it to each of the scalar components of the vector field.

9.3 Stationary Phase Asymptotic Approximations

57

p QQ where [cf. (9.204)] V.p; q; m/ mU.p; q; !/ with m D C 1  p 2  q 2 , and where [see (9.200) and (9.201)] 1 D .x  x0 /=R, 2 D .y  y0 /=R, p 3 D .z  z0 /=R with R D .x  x0 /2 C .y  y0 /2 C .z  z0 /2 are related to the angles # and ' through the relations 1 D sin # cos '; 2 D sin # sin ';

(9.214)

3 D cos #; with 0  # < =2 and 0  ' < 2. Q !/ satisfies the By making use of the fact that the spectral wavefield U.r; Helmholtz equation (9.198) for z > 0, Sherman [39] has shown that if a solution Q !/ of the Helmholtz equation has an asymptotic expansion of the form given U.r; in (9.212) for arbitrarily large (even) values of N , and if the asymptotic expansion Q !/ with respect to the spherical coordinate of each of the partial derivatives of U.r; variables R; #; ' up to order 2 can be obtained by term-by-term differentiation of the asymptotic expansion given in (9.212), then the coefficients Bn .#; '/ satisfy the recursion formula [see Problem 9.9] .n C 1/BnC1 .#; '/ D

 i 2 L  n.n C 1/ Bn .#; '/; 2

(9.215)

where L2 is the differential operator defined by L2 

1 @ sin # @#

 sin #

@ @#

 

1 @2 : 2 sin # @' 2

(9.216)

Q !/ given in (9.194)– Since the partial derivatives of the spectral wavefield U.r; (9.195) satisfy these requirements, then the coefficients BnC1 .#; '/ appearing in the asymptotic expansion (9.212) of this wavefield satisfy the recursion formula given QQ in (9.215) provided that U.p; q; !/ 2 T1 . Since the functional dependence of the QQ coefficients BnC1 .#; '/ on the spatiotemporal spectrum U.p; q; !/ of the wavefield Q Q is independent of N , then (9.215) also holds for U.p; q; !/ 2 TN with N finite. Sherman’s recursion formula (9.215) then provides a straightforward method for obtaining higher-order terms in the asymptotic approximation of the specQ !/ from the first term in the expansion (9.212). For example, tral wavefield U.r; QQ q; !/ 2 TN with N  6, B1 .#; '/ D .i=2/L2 B0 .#; '/, so that if U.p;  ikR  i Q !/ D 2 i e 1C L2 V.1 ; 2 ; 3 /e ik.1 x0 C2 y0 C3 z0 / U.r; kR 2kR ˚ (9.217) CO .kR/3 as kR ! 1 with fixed k > 0 and fixed 1 ; 2 with z > 0.

58

9 Pulsed Beam Wavefields in Temporally Dispersive Media

9.3.2 Approximations Valid on the Plane z D z0 Q !/ D U.x; Q The asymptotic approximation of U.r; y; z; !/ that is valid as the point of observation r D .x; y; z/ recedes to infinity in a fixed direction perpendicular to the z-axis with z > 0 through the point r0 D .x0 ; y0 ; z0 / is now considered. This case is excluded in Sect. 9.3.1 because the method of analysis used there cannot be extended to include the special case when 3 D 0. As in Theorem 1, it is again QQ assumed that U.p; q; !/ 2 TN for some positive even integer N and that z D z0 > 0. Upon setting z D z0 in (9.195) an integral of the form given in (9.202) is obtained, viz., Z g.p; q/e ikRf .p;q/ dpdq; (9.218) I.kR/ D D

with f .p; q/ D 1 p C 2 q; QQ g.p; q/ D U.p; q; !/e ik.px0 Cqy0 Cmz0 / ;

(9.219) (9.220)

and D D DJ , J D H; E. The phase function f .p; q/ is then real and analytic in both regions DH and DE . However, the amplitude function g.p; q/ need not have any of its partial derivatives with respect to p and q exist on the unit circle p 2 C q 2 D 1. To circumvent this problem, Sherman, Stamnes, and Lalor [32] isolate that unit circle using a neutralizer function as follows. Let 0 .p; q/ be a real, continuous function with continuous partial derivatives of all orders for all p; q that satisfies the inequality 0  0 .p; q/  1;

(9.221)

with 0 .p; q/ D

1 for 1  "  p 2 C q 2  1 C "; (9.222) 2 2 0 for either p C q  1  2" or p 2 C q 2  1 C 2";

where " > 0 is a positive constant with " < 1=3. The integral in (9.195) can then be written in the form Q J1 .x; y; z0 ; !/ C U Q J 2 .x; y; z0 ; !/ Q J .x; y; z0 ; !/ D U U

(9.223)

with J D H; E, where Q J1 .x; y; z0 ; !/ D U Q J 2 .x; y; z0 ; !/ D U

Z Z

DJ1

DJ 2

QQ 0 .p; q/U.p; q; !/e ik.pxCqyCmz0 / dpdq;

(9.224)

QQ Œ1  0 .p; q/ U.p; q; !/e ik.pxCqyCmz0 / dpdq: (9.225)

9.3 Stationary Phase Asymptotic Approximations

59

The various regions of integration in the p; q-plane appearing in these integral expressions are defined as ˚ DH1 .p; q/ j 1  3"  p 2 C q 2  1 ; ˚ DH 2 .p; q/ j 0  p 2 C q 2  1  "=2 ; ˚ DE1 .p; q/ j 1  p 2 C q 2  1 C 3" ; ˚ DE2 .p; q/ j p 2 C q 2  1 C "=2 ; and are depicted in Fig. 9.18. The asymptotic approximation of each of the integrals appearing in (9.224) and (9.225) is now separately considered.

9.3.2.1

The Region DH2

Q H2 .x; y; z0 ; !/ is found [32] The integrand appearing in the expression (9.225) for U to satisfy all of the conditions required in Theorem 1 of Chako [36]. Because the region DH2 does not contain any critical points of that integrand, and because the QQ amplitude function Œ1  0 .p; q/ U.p; q; !/ and all N of its partial derivatives with respect to p and q vanish on the boundary of DH2 , it then follows that ˚ Q H 2 .x; y; z0 ; !/ D O .kR/N U

(9.226)

as kR ! 1 with fixed k > 0.

9.3.2.2

The Region DE2

Q E2 .x; y; z0 ; !/ is also found The integrand appearing in the expression (9.225) for U [32] to satisfy the conditions in Theorem 1 of Chako [36], but the integration domain DE2 appearing in (9.225) extends to infinity whereas Chako’s proof is given only for a finite domain of integration. Its extension to the case of an infinite domain has been given by Sherman, Stamnes, and Lalor [32] who consider the same integral but with the region of integration now given by 1 C "=2  p 2 C q 2  K, where K > 1 C "=2 is an arbitrary constant that is allowed to go to infinity. This change then introduces two complications into Chako’s proof. For the first, the amplitude QQ function U.p; q; !/ and its partial derivatives with respect to p and q do not, in general, vanish on the boundary p 2 Cq 2 D K for finite K, but they do vanish in the limit as Kp ! 1 because each of these boundary terms contains the exponential factor e z0 K1 with z0 > 0.8 The second complication occurs in the remainder integral 8

Notice that the special case when z0 D 0 can be treated only if additional restrictions are placed QQ on the behavior of the spectral amplitude function U.p; q; !/ and its partial derivatives with respect 2 2 to p and q in the limit as p C q ! 1.

60

9 Pulsed Beam Wavefields in Temporally Dispersive Media

after integration by parts N times, which is now taken over an infinite domain. However, all that is required in the proof is that this integral p is convergent and this is z0 p 2 Cq 2 1 guaranteed by the presence of the exponential factor e in the integrand. Hence, the result of the theorem due to Chako applies in this case, so that ˚ Q E2 .x; y; z0 ; !/ D O .kR/N U

(9.227)

as kR ! 1 with fixed k > 0.

9.3.2.3

The Region DH1

Attention is now turned to obtaining the asymptotic approximation of Q H1 .x; y; z0 ; !/ as kR ! 1 with fixed k > 0. Under the change of variable U p D sin ˛ cos ˇ, q D sin ˛ sin ˇ the integral representation appearing in (9.224) becomes Q H1 .x; y; z0 ; !/ D U

Z

ˇ0 C2 ˇ0

Z

=2

˛10

A0 .˛; ˇ/e ikR sin ˛ cos .ˇ'/ d˛dˇ;

(9.228)

with fixed direction cosines 1 D sin # cos ', 2 D sin # sin ' [see (9.201)] with 0  #p < =2 and 0 p' < 2. Here ˇ0 is an arbitrary real constant, ˛10 arcsin 1  3" D arccos 3", and A0 .˛; ˇ/ 0 .p; q/V .p; q; m/e ik.px0 Cqy0 Cmz0 / sin ˛;

(9.229)

with m D .1  p 2  q 2 /1=2 D cos ˛ and V .p; q; m/ is as defined in (9.204). Since ˇ0 is an arbitrary constant, it is chosen for later convenience to be given by ˇ0 D '  =4. The integral appearing in (9.228) is now in the form of the integral given in (9.202) with phase and amplitude functions f .˛; ˇ/ D sin ˛ cos .ˇ  '/; g.˛; ˇ/ D A0 .˛; ˇ/;

(9.230) (9.231)

respectively. Because both of these functions satisfy all of the requirements necessary for application of the method of stationary phase, this method may now be directly applied. The critical points of the integral in (9.228) occur at the stationary phase points that are defined by the condition [cf. (9.203)] ˇ ˇ @f ˇˇ @f ˇˇ D D 0; @˛ ˇ.˛s ;ˇs / @ˇ ˇ.˛s ;ˇs /

(9.232)

9.3 Stationary Phase Asymptotic Approximations

61

so that both cos ˛ cos .ˇ  '/ D 0 ) ˛ D =2 _ ˇ D ' C =2; ' C 3=2 and sin ˛ sin .ˇ  '/ D 0 ) ˇ D '; ' C . The stationary phase points are then Point a: Point b:

.˛s ; ˇs / D .=2; '/; .˛s ; ˇs / D .=2; ' C /;

both of which occur on the boundary of the integration region DH1 , as illustrated in Fig. 9.19. Additional critical points occur at the two corners .=2; ˇ0 / and .=2; ˇ0 C 2/. Because of the above chioice that ˇ0 D '  =4, neither of the stationary phase points coincide with a boundary corner of the integration domain, as depicted in Fig. 9.19. Consider first the contribution from the two corner critical points which are just an artifact of the change of variables of integration. Because their location along the line ˛ D =2 is completely determined by the choice of the constant ˇ0 , it is expected that, taken together, they do not contribute to the asymptotic behavior of

f/ '' f/

''

b

''

f/

a

'' ''

f/

f/

'

Fig. 9.19 Illustration of the integration region DH1 appearing in (9.228) and the location of the critical points appearing in that integral

62

9 Pulsed Beam Wavefields in Temporally Dispersive Media

Q H1 .x; y; z0 ; !/ as kR ! 1. The verification of this expected result is given by U the following argument [32]. Construct a periodic neutralizer function  00 .ˇ/ with period 2 that is a real, continuous function of ˇ with continuous derivatives of all orders, and is such that  00 .ˇ/ D 1;

when

'  =8  ˇ  ' C 9=8;

and  00 .ˇ/ D 0 in some neighborhood of ˇ0 , as depicted on the left side of Fig. 9.19. Q H1 .x; y; z0 ; !/ may then be exThe integral representation (9.228) of the field U pressed as Q H1 .x; y; z0 ; !/ D U

Z Z

ˇ0 C2 ˇ0 ˇ0 C2

C ˇ0

Z

=2

˛10

Z

=2

˛10

 00 .ˇ/A0 .˛; ˇ/e ikR sin ˛ cos .ˇ'/ d˛dˇ   1   00 .ˇ/ A0 .˛; ˇ/e ikR sin ˛ cos .ˇ'/ d˛dˇ:

Because the integrand vanishes (due to the neutralizer function) at the corner points, the only critical points in the first integral are the stationary phase points labeled a and b in Fig. 9.19. Moreover, because  00 .ˇ/ D 1 in a separate neighborhood of each of the stationary phase points, the asymptotic behavior of this first integral is identical to the stationary phase point contributions to the asymptotic behavior of Q H1 .x; y; z0 ; !/ as kR ! 1. On the other hand, because the quantity .1   00 .ˇ// U vanishes at both of the stationary phase points, the only critical points of importance in the second integral are the corner points. Because the integrand in this second integral is a periodic function of ˇ with period 2, the integrand can be made to vanish at the corner points simply by changing the region of integration so that it extends from '9=8 to 'C7=8. It then from Theorem 1 of the paper [36] ˚ follows by Chako that the second integral is O .kR/N as kR ! 1. Q H1 .x; y; z0 ; !/ may then be expressed as The field U ˚ Q .a/ .x; y; z0 ; !/ C U Q .b/ .x; y; z0 ; !/ C O .kR/N ; (9.233) Q H1 .x; y; z0 ; !/ D U U H1 H1 .a/

.b/

Q .x; y; z0 ; !/ denote the separate contributions to Q .x; y; z0 ; !/ and U where U H1 H1 Q H1 .x; y; z0 ; !/ from the stationary phase points a the asymptotic behavior of U and b, respectively. To obtain the separate asymptotic approximations of both Q .b/ .x; y; z0 ; !/, the asymptotic expansion due to a boundQ .a/ .x; y; z0 ; !/ and U U H1 H1 ary stationary phase point of the hyperbolic type must be used [32]. This has been shown to be of the same form as that for a boundary stationary phase point of the elliptic type, which has been given in Sects. 3.3–3.4 of Bremmerman [38]. Thus Q .j1 / .x; y; z0 ; !/ C U Q .j2 / .x; y; z0 ; !/; Q .j / .x; y; z0 ; !/ D U U H1 H1 H1

(9.234)

9.3 Stationary Phase Asymptotic Approximations

63

with N

Q .j1 / .x; y; z0 ; !/ U H1

.j / 2 ˚ BHn1 .'/ e ˙ikR X D C o .kr/N=4 ; n kR nD0 .kR/

(9.235)

N

.j / 2 1 ˚ BHn2 .'/ e ˙ikR X .j2 / Q UH1 .x; y; z0 ; !/ D C o .kr/N=4 ; 3=2 n .kR/ .kR/ nD0 .j /

(9.236)

.j /

as kR ! 1 with fixed k > 0, where BHn1 .'/ and BHn2 .'/ are both independent of R. The upper sign in the exponential appearing in (9.235)–(9.236) is used when j D a and the lower sign is used when j D b. Explicit expressions for the coeffi.j / .j / cients BH1n .'/ and BH2n .'/ are given in Appendix II of the paper [32] by Sherman, Stamnes, and Lalor.

9.3.2.4

The Region DE 1

Q E1 .x; y; z0 ; !/ as Consider finally obtaining the asymptotic approximation of U kR ! 1 with fixed k > 0. With the change of variable defined by p D cosh  cos ; q D cosh  sin ;

(9.237) (9.238)

m D Ci sinh ;

(9.239)

so that with Jacobian J.p; q=; / D sinh  cosh , the integral appearing in (9.224) becomes Q E1 .x; y; z0 ; !/ D U

Z

ˇ0 C2 ˇ0

where 1 sinh1

Z

1

A00 .; /e ikR cosh ./ cos .'/ dd;

(9.240)

0

p  p  3 D cosh1 1 C 3 and

A00 .; / i 0 .p; q/V .p; q; m/e ik.px0 Cqy0 Cmz0 / cosh ;

(9.241)

with p and q as given in (9.237) and (9.238). The integral appearing in (9.240) is now in the form of the integral in (9.202) with amplitude and phase functions g.; / D A00 .; /; f .; / D cosh ./ cos .  '/; respectively.

(9.242) (9.243)

64

9 Pulsed Beam Wavefields in Temporally Dispersive Media

As was found for the integral in (9.228) for the spectral field component Q H1 .x; y; z0 ; !/, all of the critical points lie on the boundary @DE1 of the inteU gration domain. Just as in that case, the critical points on the boundary  D 1 and at the corners .; / D .0; 0/ and .; / D .0; 2/ do not contribute to the asymptotic behavior of the integral in (9.240). The asymptotic behavior of Q E1 .x; y; z0 ; !/ is then completely determined by the the spectral field component U contributions from the stationary phase points (a) at .s ; s / D .0; '/ and (b) at Q E1 .x; y; z0 ; !/ is expressed as .s ; s / D .0; ' C /. As in (9.233), the field U ˚ Q .a/ .x; y; z0 ; !/ C U Q .b/ .x; y; z0 ; !/ C O .kR/N ; (9.244) Q E1 .x; y; z0 ; !/ D U U E1 E1 .a/

.b/

Q .x; y; z0 ; !/ denote the separate contributions to the Q .x; y; z0 ; !/ and U where U E1 E1 Q H1 .x; y; z0 ; !/ from the stationary phase points (a) and (b), asymptotic behavior of U respectively. An analysis similar to that following (9.233) then yields [32] Q .j1 / .x; y; z0 ; !/ C U Q .j2 / .x; y; z0 ; !/; Q .j / .x; y; z0 ; !/ D U U E1 E1 E1

(9.245)

with N

Q .j1 / .x; y; z0 ; !/ U E1

.j / 2 ˚ BEn1 .'/ e ˙ikR X D C o .kr/N=4 ; n kR nD0 .kR/

(9.246)

N

.j / 2 1 ˚ BEn2 .'/ e ˙ikR X .j2 / Q UE1 .x; y; z0 ; !/ D C o .kr/N=4 ; 3=2 n .kR/ .kR/ nD0 .j /

(9.247)

.j /

as kR ! 1 with fixed k > 0, where BEn1 .'/ and BEn2 .'/ are both independent of R. The upper sign in the exponential appearing in (9.246) and (9.247) is used when j D a and the lower sign is used when j D b.

9.3.2.5

.j /

.j /

Relationship between the Coefficients BH 1n .'/, BH 2n .'/, .j / .j / and BEn1 .'/, BEn2 .'/ .j /

.j /

The various relationships between the coefficients BH1n .'/ and BH2n .'/ in Q H1 .x; y; z0 ; !/ and the asymptotic expansion given in (9.235) and (9.236) for U .j1 / .j2 / the coefficients BEn .'/ and BEn .'/ in the asymptotic expansion given in (9.246) Q E1 .x; y; z0 ; !/ are given by Jones and Kline [35], as modified by and (9.247) for U Sherman, Stamnes, and Lalor [32], as9 9

Explicit expressions for all of these coefficients are unnecessary for the final asymptotic expansion and so are not given here. The interested reader should consult Appendices I–III in the 1976 paper [32] by Sherman, Stamnes, and Lalor.

9.3 Stationary Phase Asymptotic Approximations .a /

65

BEn1 .'/ D CBH 1n .'/;

.a /

(9.248)

.a / BEn2 .'/ .b / BEn1 .'/ .b / BEn2 .'/

.a / BH 2n .'/; .b / BH1n .'/; .b / BH2n .'/:

(9.249)

D D D

(9.250) (9.251)

The pair of relations in (9.250) and (9.251) then show that the contribution of Q E1 .x; y; z0 ; !/ is equal in magnitude but opposite in sign with the point b to U Q H1 .x; y; z0 ; !/. Consequently, the point b does the contribution of the point b to U Q H1 .x; y; z0 ; !/ C not contribute to the asymptotic behavior ˚of the field quantity U Q E1 .x; y; z0 ; !/ with order lower than O .kR/N=4 that arises from the sum of U the second terms on the right hand side of (9.233) and (9.244). In a similar manner, (9.249) implies that the contribution of the point a to the series involving inverse half Q E1 .x; y; z0 ; !/ is equal in magnitude but oppopowers of .kR/ in the expansion of U Q H1 .x; y; z0 ; !/. Taken together site in sign to the same contribution of the point a to U with the previous result, this implies that the asymptotic expansion of the field quanQ E1 .x; y; z0 ; !/ does not include terms with half powers of Q H1 .x; y; z0 ; !/ C U tity U ˚ .kR/ of order lower than O .kR/N=4 . Finally, the relation given in (9.248) implies Q E1 .x; y; z0 ; !/ contribute equally Q H1 .x; y; z0 ; !/ and U that the field components U Q to the remaining terms in the asymptotic expansion of the total field U.x; y; z0 ; !/ involving only inverse integral powers of .kR/. It then follows from this result together with (9.194), (9.223), (9.227), and (9.235) that N

.a / 2 ˚ BH 1n .'/ e ikR X Q C o .kr/N=4 ; U.x; y; z0 ; !/ D 2 n kR nD0 .kR/

(9.252)

as kR ! 1 with fixed k > 0 and z D z0 . .a / The coefficients BH 1n .'/ appearing in the asymptotic expansion (9.252) are found [32] to be given by the limiting expression .a /

BH 1n .'/ D

1 lim Bn .#; '/ 2 ! 2

(9.253)

of the coefficients Bn .#; '/ appearing in the asymptotic expansion (9.212). It then follows that the results of this section can be combined with those of Sect. 9.3.1 to yield the asymptotic expansion N

2 ˚ Bn .#; '/ e ikR X Q C o .kr/N=4 ; U.x; y; z0 ; !/ D kR nD0 .kR/n

(9.254)

as kR ! 1 with fixed k > 0 and z0 > 0 that is uniformly valid with respect to the angles # and ' in their respective domains 0  #  =2 and 0  '  2. The zeroth-order coefficient B0 .#; '/ is given in (9.213) and the first-order coefficient is given by B1 .#; '/ D .i=2/L2 B0 .#; '/, where the differential operator L2 is defined in (9.216).

66

9 Pulsed Beam Wavefields in Temporally Dispersive Media

Q H .r; !/ and U Q E .r; !/ 9.3.3 Asymptotic Approximations of U The respective dominant terms in the asymptotic expansion of the homogeneous Q H .r; !/ and in the asymptotic expansion of the evanescent field field component U Q component UE .r; !/ are now derived for each of the three direction cosine regions 0 < 3 < 1, 3 D 0, and 3 D 1. 9.3.3.1

Approximations Valid over the Hemisphere 0 < 3 < 1

To obtain the asymptotic approximation of the homogeneous wave contribution Q H .r; !/ to the total spectral wavefield, the notation and definitions used in (9.221)– U (9.225) are applied with z0 replaced by z. The only critical point of the integral for Q H 2 .r; !/, where the integrand is nonzero, is the interior stationary phase point U Q H 2 .r; !/ is then seen to be iden.ps ; qs / D .1 ; 2 /. The asymptotic expansion of U Q !/ given in (9.212) and (9.217). tical with the asymptotic expansion of U.r; It then follows from the above result together with (9.194) that the asymptotic Q E .r; !/ to the total wavefield is expansion of the evanescent wave contribution U Q H1 .r; !/. The asymptotic expansion of the integral expression identical to that of U Q H1 .x; y; z; !/ is treated in the same manner as was employed for Q H1 .r; !/ D U for U Q H1 .x; y; z0 ; !/ in Sect. 9.3.2. The change of integration variables given by p D U sin ˛ cos ˇ, q D sin ˛ sin ˇ that was used in the proof of Theorem 1 is first made, resulting in the integral [cf. (9.228)] Z

ˇ0 C2

Z

=2

A0 .˛; ˇ/e ikRf .˛;ˇ/ d˛dˇ;

(9.255)

f .˛; ˇ/ D sin # sin ˛ cos .ˇ  '/ C cos # cos ˛:

(9.256)

Q H1 .x; y; z; !/ D U

˛10

ˇ0

with phase function [cf. (9.230)]

The location of the critical points for this phase function are the same as those depicted in Fig. 9.19. As in that case, the points a and b are the only critical points that contribute to the asymptotic behavior of the integral under consideration. However, unlike that case, the saddle points here are not ordinary stationary phase points where both @f =@˛ and @f =@ˇ vanish, but rather they are boundary stationary phase points where the isotimic contours f .˛; ˇ/ D constant are tangent to the boundary of the integration domain; in the present case they are both points on the boundary line ˛ D =2 where @f =@ˇ D 0. The resultant asymptotic expansion is then found to be [32, 34, 35] p i=4 h 0 0 Q H1 .x; y; z; !/ D 2= sin #e V.10 ; 20 ; 0/e ik.x0 1 Cy0 2 / e ikR sin # U 3=2 .kR/ cos # i ˚ 0 0 Ci V. 0 ;  0 ; 0/e ik.x0 1 Cy0 2 / e ikR sin # C O .kR/2 1

2

(9.257)

9.3 Stationary Phase Asymptotic Approximations

67

as kR ! 1 with fixed k > 0 and N  8, where 10 D 1 = sin # D cos '; 20 D 2 = sin # D sin ':

(9.258) (9.259)

Q E .r; !/ are then given by Q H .r; !/ and U The resultant asymptotic expansions for U (for N  8) ikR

Q H .x; y; z; !/ D 2 i e V.1 ; 2 ; 3 /e ik.1 x0 C2 y0 C3 z0 / U kR ˚ Q H1 .x; y; z; !/ C O .kR/2 ; CU ˚ Q H1 .x; y; z; !/ C O .kR/2 ; Q E .x; y; z; !/ D U U

(9.260) (9.261)

as kR ! 1 with fixed 1 , 2 , and k > 0. Q E .r; !/ is of These results then show that the evanescent wave contribution U 1 Q higher order in .kR/ than the homogeneous wave contribution UH .r; !/. This then Q H .r; !/ Q E .r; !/ in comparison to U provides a rigorous justification for neglecting U as kR ! 1 with fixed k > 0 when 0 < 3 < 1. In addition, the evanescent Q E .r; !/ is of even higher order in .kR/1 in comparison to the wave contribution U Q H .r; !/ as kR ! 1 in the special case when homogeneous wave contribution U V.p; q; 0/ D 0.

9.3.3.2

Approximations Valid on the Plane z D z0

The asymptotic behavior of the spectral wavefield on the plane z D z0 (or, equivalently, when 3 D 0) is directly obtained from the analysis presented in Sect. 9.3.2, the present analysis focusing on obtaining explicit expressions for the dominant terms in the separate homogeneous and evanescent wave contributions. The asympQ H .x; y; z0 ; !/ is totic expansion of the homogeneous spectral wave contribution U obtained from (9.223), (9.226), and (9.233)–(9.236), taken together with the sej1 j2 ries expressions [32] for the coefficients BH n .'/ and BH n .'/, with the result (for N  8) h Q H .x; y; z0 ; !/ D  i V.1 ; 2 ; 0/e ikR e ik.x0 1 Cy0 2 / U kR

i ˚ V.1 ; 2 ; 0/e ikR e ik.x0 1 Cy0 2 / C O .kR/3=2 (9.262)

as kR ! 1 with fixed 1 , 2 , and k > 0. Similarly, the asymptotic expansion of Q E .x; y; z0 ; !/ is obtained from (9.223), the evanescent spectral wave contribution U (9.227), (9.244), (9.248), and (9.250) with the result (for N  8)

68

9 Pulsed Beam Wavefields in Temporally Dispersive Media

h Q E .x; y; z0 ; !/ D  i V.1 ; 2 ; 0/e ikR e ik.x0 1 Cy0 2 / U kR

i ˚ CV.1 ; 2 ; 0/e ikR e ik.x0 1 Cy0 2 / C O .kR/3=2 (9.263)

as kR ! 1 with fixed 1 , 2 , and k > 0. These two expressions then show that the homogeneous and evanescent wave contributions are of the same order in .kR/1 as Q E .x; y; z0 ; !/ cankR ! 1. As a consequence, the evanescent wave contribution U not, in general, be neglected in comparison to the homogeneous wave contribution Q H .x; y; z0 ; !/ on the plane z D z0 . U

9.3.3.3

Approximations Valid on the Line x D x0 , y D y0

Along the line x D x0 , y D y0 , one has that 3 D 0 and the homogeneous spectral wave contribution can be written as Q H .x0 ; y0 ; z; !/ D U

Z

Z

2

=2

d˛ A.˛; ˇ/e ikz cos ˛ ;

dˇ 0

(9.264)

0

where A.˛; ˇ/ D V.p; q; m/e ik.px0 Cqy0 Cmz0 / sin ˛; (9.265) p 2 2 with p D sin ˛ cos ˇ, q D sin ˛ sin ˇ, and m D 1  p  q D cos ˛ [see the proof of Theorem 1]. Because the phase function in (9.264) does not involve the integration variable ˇ, the asymptotic behavior of the integral can be obtained from the single integral Q H .x0 ; y0 ; z; !/ D U

Z

=2

A.˛/e ikR cos ˛ d˛;

(9.266)

0

where R D z  z0 , with A.˛/ e ikz0 cos ˛

Z

2

A.˛; ˇ/dˇ:

(9.267)

0

Application of the method of stationary phase for single integrals in Appendix F shows that the only contributions to the asymptotic behavior of the integral in (9.266) arise from the endpoints of the integral at ˛ D 0 and ˛ D =2 with the result [see (F.50)] ikR ˚ Q H .x0 ; y0 ; z; !/ D 2 i e V.0; 0; 1/e ikz0 C i A.=2/CO .kR/3=2 (9.268) U kR kR

as kR ! 1 with k > 0 and for N  4.

9.3 Stationary Phase Asymptotic Approximations

69

Because the first term in (9.268) is the dominant term in the asymptotic approxQ 0 ; y0 ; z; !/, (9.194) then shows that the asymptotic behavior of the imation of U.x evanescent wave contribution is given by ˚ Q E .x0 ; y0 ; z; !/ D  i A.=2/ C O .kR/3=2 U kR

(9.269)

as kR ! 1 with k > 0. The homogeneous and evanescent wave contributions are again seen to be of the same order in .kR/1 as kR ! 1 so that, in general, Q H .x0 ; y0 ; z; !/ for large Q E .x0 ; y0 ; z; !/ cannot be neglected in comparison to U U kR ! 1.

9.3.4 Summary The stationary phase asymptotic expansions presented in this section are all valid as kR ! 1 with fixed wavenumber k > 0 and fixed direction cosines 1 D .x  x0 /=R, 2 D .y  y0 /=R with N  12 in a nonabsorptive (and hence, strictly speaking, nondispersive) medium. Less restrictive conditions on N result in special cases. Q !/ is the same for all The asymptotic behavior of the spectral wavefield U.r; 3 D .z  z0 /=R such that 0  3  1 and is given by [cf. (9.254)]   i e ikR 2 Q 1C L V.1 ; 2 ; 3 /e ik.1 x0 C2 y0 C3 z0 / U.r; !/ D 2 i kR 2kR ˚ (9.270) CO .kR/3 as kR ! 1 with fixed k > 0, where L2 is the differential operator defined in (9.216). The asymptotic behavior of the separate homogeneous and evanescent comQ J .r; !/, J D H; E, however, depends on the value of 3 , ponent wavefields U separating into the three cases (a) 0 < 3 < 1, (b) 3 D 0, and (c) 3 D 1, as described in Sect. 9.3.3. The results show that the evanescent wave contriQ E .r; !/ is negligible in comparison to the homogeneous wave contribution U Q H .r; !/ for large kR ! 1 in case (a), but not necessarily in cases bution U (b) and (c). In some important applications of the angular spectrum representation in electromagnetic wave theory, integral representations of the form given in (9.194) and QQ (9.195) are obtained with U.p; q; !/ … TN because of the presence of isolated singularities in the integrand. This occurs, for example, in the analysis of the refelection and refraction of a nonplanar wavefield (e.g., an electromagnetic beam field) at a planar interface separating two different media [37, 40]. A neutralizer function can QQ then be used to isolate each singularity. Because U.p; q; !/ 2 TN , the asymptotic approximation of the resultant integral that does not contain any of the singularities

70

9 Pulsed Beam Wavefields in Temporally Dispersive Media

can then be obtained using the two-dimensional stationary phase results of Sherman, Stamnes, and Lalor [32] presented here. Each of the remaining integrals contains QQ one of the isolated singularities, U.p; q; !/ … TN and so its asymptotic approximation must be obtained using some other technique. In some case, as in the reflection and refraction problem [40], a change of integration variable can result in a transformed integral that is amenable to the stationary phase method presented here. If that is not the case, uniform asymptotic methods [41–43] may then need to be employed.

9.4 Separable Pulsed Beam Wavefields A problem of special interest in both optics and electromagnetics is that in which the spatial and temporal properties of a pulsed electromagnetic (or optical) beam wavefield are separable. The general conditions under which this simplifying assumption is valid are presented in Sect. 7.4.2 of Vol. 1. In this case, the spatial and temporal properties of the initial field vectors at the input plane z D z0 are assumed to be separable in the sense that [44] E0 .rT ; t / D E0 .x; y; t / D EO 0 .x; y/f .t /; B0 .rT ; t / D B0 .x; y; t / D BO 0 .x; y/g.t /;

(9.271) (9.272)

in which case EQQ 0 .p; q; !/ D EQO 0 .p; q; k/fQ.!/; BQQ 0 .p; q; !/ D BQO 0 .p; q; k/g.!/: Q

(9.273) (9.274)

One now has the spatial Fourier transform pair relations QO .p; q; k/ D U 0

Z

1

Z

1

O 0 .x; y/ D U

1 .2/2

1

O 0 .x; y/e ik.pxCqy/ dxdy; U

(9.275)

1

Z

Z

1

1

1

QO .p; q; k/e ik.pxCqy/ k 2 dpdq U 0

(9.276)

1

for both EO 0 .x; y/ and BO 0 .x; y/, and the separate Fourier–Laplace transform pair relations Z 1 Q h.!/ D h.t /e i!t dt; (9.277) 1

1 h.t / D < 

(Z

) i!t Q d! h.!/e CC

(9.278)

9.4 Separable Pulsed Beam Wavefields

71

for both f .t / and g.t /. It is seen from (9.8) that these functions cannot be independently chosen because their spectra must satisfy the relation BQO 0 .p; q; k/g.!/ Q D ..!/c .!//1=2 sO  EQO 0 .p; q; k/fQ.!/;

(9.279)

where sO D 1O x p C 1O y q C 1O z m.!/. The separation of this relation into spatial and temporal frequency components as OQ 0 .p; q; k/; c BQO 0 .p; q; k/ D sO  E 1=2  g.!/ Q D .!/c .!/ fQ.!/;

(9.280) (9.281)

does not provide a unique solution of (9.279) except in special cases. Consider now the propagation of such a separable pulsed wavefield in a (fictitious) dispersive medium that is lossless. Substitution of the generic form of the separibility condition given in (9.273) and (9.274) into the angular spectrum representation given in (9.35) then yields (Z Z 1 i!t Q < d! h.!/e k 2 .!/dpdq U.r; t / D 4 3 CC R< [R>

) Q O 0 .p; q; k/G.p; q; !; z/e ik.!/.pxCqy/ ; U (9.282)

˚ q/j p 2 C q 2 < 1 defines the homogeneous plane wave domain, where R ˚ < D .p; R> D .p; q/j p 2 C q 2 > 1 defines the evanescent plane wave domain, and where [see (7.195)] G.p; q; !; z/ e ik.!/m z D e ik.!/ z.1p

2 q 2 1=2

/

(9.283)

is the propagation factor for a plane wave progressing in the direction specified by the direction cosines .p; q; m/. As described in Sect. 7.4 of Vol. 1, because the evanescent plane wave components in the angular spectrum representation do not provide any time-average energy flow into the positive half-space z > z0 [see (9.37)], an electromagnetic beam field is defined, in part, by the requirement that its angular spectrum does not contain any evanescent plane wave components [3], and consequently can be represenated by an angular spectrum that contains only homogeneous plane wave components. Because the homogeneous plane wave spectral components do not attenuate with propagation distance in a lossless medium, this condition ensures that all of the angular spectrum components of the beam field are maintained in their initial proportion throughout the positive half-space z  z0 . Such wavefields are known as

72

9 Pulsed Beam Wavefields in Temporally Dispersive Media

source-free wavefields [45, 46] and, as such, possess several rather unique properties [see Sect. 7.4 of Vol. 1]. QO .p; q; k/ of the sepAs a consequence, if the initial spatial frequency spectrum U 0 arable pulsed beam wavefield contains only homogeneous plane wave components, then the inner integration domain R< can be extended to all of p; q-space and the propagation factor G.p; q; !; z/ may then be represented by its Maclaurin’s series expansion as [see (7.215–7.216)] G.p; q; !; z/ D

1 X 1 X G .2r;2s/ .0; 0; !; z/ rD0 sD0

.2r/Š.2s/Š

p 2r q 2s ;

(9.284)

where G .m;n/ .p; q; !; /

@mCn G.p; q; !; / : @p m @q n

(9.285)

Substitution of this expansion in (9.282) and interchanging the order of integration and summation then yields ( 1 1 Z XX G .2r;2s/ .0; 0; !; z/ Q 1 < d! U.r; t / D h.!/e i!t 3 4 .2r/Š.2s/Š C C rD0 sD0 ) Z 1Z 1 Q 2r 2s ik.!/.pxCqy/ 2 O U0 .p; q; k/p q e  k .!/dpdq : 1

1

(9.286) Because [see (7.218)] mCn O U0 .x; y/ O .m;n/ .x; y/ @ U 0 @x m @y n Z 1Z 1 1 QO .p; q; k/ .ik.!//mCn p m q n U D 0 2 2 1 1 e ik.!/.pxCqy/ k 2 .!/dpdq;

(9.287) one then obtains ( 1 1 Z XX 1 i!t Q U.r; t / D < d! h.!/e  C C rD0 sD0

) G .2r;2s/ .0; 0; !; z/ O .2r;2s/ U  .x; y/ ; .2r/Š.2s/Š.ik.!//2.rCs/ 0 (9.288)

9.4 Separable Pulsed Beam Wavefields

73

for all z  0. The spatial series expansion appearing in the above expression is due to G. C. Sherman [45, 46] and is correspondingly referred to as Sherman’s expansion. Notice that the factor .ik.!//2.rCs/ appearing in the above series expansion is misleading because the same factor with an opposite-signed exponent is contained in the partial derivative G .2r;2s/ .0; 0; !; z/. It is seen from (9.283) that G .m;n/ .p; q; !; / D .ik.!/ /mCn e ik.!/ .1p where ' .m;n/ .p; q/

2 q 2 1=2

/ ' .m;n/ .p; q/;

1=2 @mCn  1  p2  q2 : m n @p @q

(9.289)

(9.290)

With these substitutions, the above expression for the propagated, separable wavefield becomes U.r; t / D

1 X 1 X . z/2.rCs/ rD0 sD0

' .2r;2s/ .0; 0/ .2r/Š.2s/Š ) (Z 1 .2r;2s/ i.k.!/ z!t/ Q O  < U .x; y/h.!/e d! ; 0  CC (9.291)

for all z  0. The transverse spatial variation of both the source-free electric and magnetic field vectors at any plane z > z0 is thus seen to depend solely upon the transverse spatial variation of the field at the initial plane at z D z0 through all of the even-order spatial derivatives of the corresponding field vector at that plane. At first impression, it would appear that this transverse spatial variation is independent of the wavenumber k.!/, and hence of the angular frequency !, so that the function O .2r;2s/ .x; y/ could be taken out from under the integral in (9.291). This clearly U 0 deserves a more careful examination. O .2r;2s/ .x; y/ is indeed independent of the wavenumber k.!/, then the propaIf U 0 gated source-free pulsed beam field is given by the product of two separate factors [see (7.260) and (7.261) in Vol. 1], one describing the transverse spatial variation and the other describing the longitudinal spatiotemporal variation, each factor dependent upon the propagation distance z D z  z0 . This apparent wavenumber (or frequency) independence is a direct consequence of the two-dimensional ikm z , with Maclaurin p series expansion of the propagation kernel G.p; q; z/ e m D 1  p 2  q 2 , whose explicit dependence upon k.!/ is then transferred to O 0 .x; y/. With this in mind, (9.291) may be the transverse spatial derivatives of U rewritten as (Z ) 1 i.k.!/ z!t/ Q O d! ; (9.292) h.!/e U.r; t / D U.x; y/ <  CC

74

9 Pulsed Beam Wavefields in Temporally Dispersive Media

with [see (7.222) in Vol. 1] O U.x; y/

1 X 1 X . z/2.rCs/ rD0 sD0

.2r/Š.2s/Š

.2r;2s/

O ' .2r;2s/ .0; 0/U 0

.x; y/:

(9.293)

Unfortunately, except in trivial cases (such as for a plane wave), this series representation converges extremely slowly so that an exceedingly large (perhaps even infinite) number of terms is required. The integral representation corresponding to this series expansion of the transverse field distribution is found to be given by (see Problem 9.10) 1 O U.x; y/ D 4 2

Z

1 1

Z

1

QO . ;  /e i .x xCy yC. / z/ d d ; U 0 x y x y

(9.294)

1

1=2  with where   2  x2  y2 QO . ;  / D U 0 x y

Z

1

1

Z

1

O 0 .x; y/e i.x xCy y/ dxdy: U

(9.295)

1

Notice that this integral representation of the source-free transverse field vector O U.x; y/ that is defined by the infinite summation given in (9.293) is independent of the value of the spatial frequency  appearing in the propagation factor in the integrand of (9.294) only in the limit as  ! 1. The reason that this is so is found in the infinite double summation of (9.293). The transverse field variation is exO 0 .x; y/, and pressed there in terms of all of the even-order spatial derivatives of U this, in turn, requires that the transverse spatial variation of this initial field structure be known all the way down to an infinitesimal scale, that is, as 1= ! 0. This, of course, is just the geometrical optics limit. This rather curious result deserves further explanation. First of all, notice that the field must be source-free. This means that the wavefield does not contain any evanescent field components in a lossless medium. This, in turn, means that the wavenumber component [see (9.12)] 1=2  .!/ D k 2 .!/  kT2 QO .k ; k / is nonzero, where k 2 D is real-valued for all points .kx ; ky / at which U 0 x y T 2 2 kx C ky . In general, for each value of !, the value of the wavenumber k.!/ defines the transition circle kT2 D k 2 .!/ in kx ky -space between homogeneous and evanescent plane wave components. This wavenumber value then appears as the upper limit of integration for the homogeneous plane wave contribution to the angular spectrum representation and this, in turn, sets an upper limit on the spatial frequency scale (or equivalently, a lower limit on the spatial scale) for the transverse spatial structure of the wavefield. However, for a source-free wavefield, this upper limit is replaced by infinity as the homogeneous wave propagation factor

9.4 Separable Pulsed Beam Wavefields

75

G.kx ; ky ; !; z/ is replaced by its Maclaurin series expansion [see (9.284)]. The O result is an expression for the transverse field variation U.x; y/ that is indeed independent of the wavenumber. The results presented here then show that, except in special cases (such as for a plane wave), if the initial wavefield is separable in the sense defined in (9.271) and (9.272), then the propagated wavefield will not, in general, remain separable unless it is strictly source-free. In that idealized case, the wavefield remains separable throughout its propagation and its transverse spatial variation is independent of the wavenumber (or frequency). This then demonstrates how a seemingly innocent assumption (in this case, the assumption that the wavefield contains only homogeneous wave components) can lead to extreme results when they are taken to their logical limit. If the initial pulse is strictly quasimonochromatic (in which case [47] !=!c  1, where ! is the bandwidth of the pulse spectrum centered at !c ), then the propagated field is, to some degree of approximation, separable with the spatial frequency  being taken as the wavenumber k.!c / evaluated at this characteristic angular frequency of the pulse. Except in special cases, this approximation does not hold in the ultrawideband case. In that case, the expression (9.292) for the propagated wavefield should be written as (Z ) 1 i.k.!/ z!t/ Q O d! ; U.x; y/h.!/e U.r; t / D <  CC

(9.296)

O with U.x; y/ given by (9.294) with  D k.!/.

9.4.1 Gaussian Beam Propagation An example of central importance in both optics and microwave theory considers the spatial properties of a gaussian beam wavefield. Let the initial transverse field behavior at the plane z D z0 be described as 2 2 2 2 UO 0 .x; y/ D A0 e .x x Cy y / ;

(9.297)

with fixed amplitude A0 and constants x and y that set the initial beam widths in the x- and y-directions, respectively. With use of the integral identity Z

1

2

e ˛ e ˙iˇ d D

1

  1=2 ˛

e ˇ

2 =4˛

;

˛ > 0;

(9.298)

the initial field spectrum [see (9.295)] is found to be given by  .x2 =4x2 Cy2 =4y2 / UQO 0 .x ; y / D A e : x y

(9.299)

76

9 Pulsed Beam Wavefields in Temporally Dispersive Media

The propagated field distribution is then obtained from (9.294) as UO .x; y/ D

A 4x y

Z

1

1

Z

1

e .x =4x Cy =4y / e i .x xCy yC. / z/ dx dy : 2

2

2

2

1

(9.300) With the paraxial approximation    D 

  1=2 1  x2 = 2  y2 = 2 1



x2 C y2

(9.301)

2

of the propagation factor, valid when .x2 C y2 /= 2  1, the propagated field given in (9.300) may be separated into the product of a pair of two-dimensional gaussian beam fields as UO .x; y/ AUO .x/UO .y/; (9.302) with UO .x/

1 2 1=2 x

Z

1

e .x =4x Ci z=2 / e ix x dx 2

2

1

1 x 2 =.1=x2 C2i z= / D  ; 1=2 e 1 C 2i x2 z=

(9.303)

with an exactly analogous expression for UO .y/. Notice that this result depends explicitly on the spatial frequency value ; this is due to the paraxial approximation given in (9.301) used in obtaining (9.303). Furthermore, in the limit as  ! 1, 2 2 UO .x/ ! e x x , which is just the geometrical optics limit. The argument of the exponential factor appearing in (9.303) can be separated into real and imaginary parts as 

x2 2 z= x2 D  x2 C i x2 1=x2 C 2i z= 1 C 4x4 . z/2 = 2 1=x4 C 4. z/2 = 2 D

1  x2 C i x2; w2x . z/ 2Rx . z/

where

 wx . z/ wx0 1 C

2ız w2x0 

(9.304)

2 !1=2 (9.305)

is the beam radius (or “spot size”) at the e 1 amplitude point in the x-direction with beam waist

9.4 Separable Pulsed Beam Wavefields

77

wx0 wx .0/ D and where

 Rx . z/ z 1 C

1 ; x w2x0  2 z

(9.306) 2 ! (9.307)

is the radius of curvature of the phase front10 in the xz-plane in the paraxial approximation. Finally, the square root factor appearing in (9.303) may be written as 1

1

 1=2 1 C 2i x2 z=

e i 2 arctan .2 z=.wx0 // D  1=4 1 C 4. z/2 =.w4x0  2 / r wx0 i x . z/=2 D ; e wx . z/

where

 x . z/ arctan

2

2 z w2x0 

(9.308)

 (9.309)

describes the phase shift with propagation distance z away from the beam waist at z D z0 . With these identifications, the two-dimensional gaussian beam field given in (9.303) becomes r wx0 .x=wx . z//2 .i=2/Œ.=Rx . z//x 2  x . z/

e e UO .x/

; (9.310) wx . z/ with an analogous expression for UO .y/. The gaussian beam wavefront is a plane wave at the beam waist [Rx .0/ D Ry .0/ D 1], and for large j zj it approaches a spherical wavefront with center at the point r0 D .0; 0; z0 / and with radius equal to z, and in between these two limits it is, in general, an astigmatic wavefront. The divergence angle x of the gaussian beam in the xz-plane is obtained from the limiting behavior of the expression tan x D wx . z/=Rx . z/ as j zj ! 1 with the result 2 : (9.311) tan x D wx0 Finally, the collimated beam length or Rayleigh range 2zp R is defined as the distance 2 of its value at the beam over which the beam radius remains less than or equal to p waist, so that w.zR / 2w0 , with solution zR D

1 2 w : 2 0

(9.312)

p   Notice that z D R2  .x 2 C y 2 /  R  x 2 C y 2 =.2R/ in the parabolic approximation of a spherical wavefront with radius R, as appears in the imaginary part of (9.304) due to the paraxial approximation made in (9.301).

10

78

9 Pulsed Beam Wavefields in Temporally Dispersive Media

The gaussian beam approximates a collimated beam over the Rayleigh range j zj  zR on either side of the beam waist, whereas outside of this range it behaves more like a converging spherical (or more generally astigmatic) wave when z < zR and a diverging spherical (or more generally astigmatic) wave when z > zR . All of the paraxial beam parameters for a gaussian beam are explicitly dependendent upon the spatial frequency parameter  which, in effect, sets an upper limit to the spatial frequency scale. With  D k D 2=, the above gaussian beam parameters assume their usual form in the paraxial approximation as [48, 49]  !1=2 z 2 ; wx . z/ D wx0 1 C zR  2 ! zR Rx . z/ D z 1 C ; z   z ; . z/ D arctan x zR 

(9.313)

(9.314) (9.315)

tan x D

 ; wx0

(9.316)

zR D

 2 w :  x0

(9.317)

The physical relation of these parameters to the gaussian beam propagation pattern they describe is illustrated in the contour plot depicted in Fig. 9.20 for the case when w0 = D 20. The symmetric pair of upper and lower dashed curves describe the beam half-width values ˙w. z/ which asymptotically approach the pair of dashed lines at ˙ as z ! ˙1. Notice that if the spot size parameter w0 is decreased from the value used in computing this field pattern, the Rayleigh range will decrease and the angular divergence will increase, resulting in a more divergent field behavior, whereas if it is increased, the Rayleigh range will be lengthened and the divergence angle narrowed, resulting in a more collimated beam behavior.

2w0

Fig. 9.20 Contour plot of the gaussian beam amplitude with propagation distance z about the beam waist when w0 D 20, illustrating the relationship of the beam waist 2w0 , Rayleigh range 2zR , and angular divergence 2 parameters with the main features of the propagation pattern

2q

2zR

9.5 The Inverse Initial Value Problem

79

9.4.2 Asymptotic Behavior Finally, consider the asymptotic behavior as R ! 1 of the integral representation O of the transverse field U.x; y/ defined in (9.294). Under the change of variable x D p, y D q, this representation may be expressed as  2 i z O U.x; y/ D e 4 2

Z

1 1

Z

1

QO .p; q/e i.pxCqyCm z/ dpdq; U 0

(9.318)

1

1=2  . With x0 D y0 D 0 and fixed direction cosines 1 D where m D 1  p 2  q 2 QO .p; q/ [see (9.204)], the x=R, 2 D y=r, 3 D z=R, and with V.p; q; m/ D mU 0 O asymptotic approximation of U.x; y/ is obtained from (9.270) as  z OQ O U0 .x=R; y=R/e iR e i z.z0 =R1/ ; U.x; y/  i 2R2

(9.319)

as R ! 1. With  D k D 2=, the above expression becomes z OQ O U0 .x=R; y=R/e i2R= e i2. z=/.z0 =R1/ ; U.x; y/  i R2

(9.320)

as R= ! 1. As an example, for the gaussian beam wavefield given in (9.297) with initial beam widths (or beam waists) wx0 D 1=x and wy0 D 1=y , the initial field spectrum given in (9.299) becomes QO .x=R; y=R/ D Aw w e . 2 =4R2 /.w2x0 x 2 Cw2y0 y 2 / : U 0 x0 y0 Substitution of this expression in (9.320) then yields wx0 wy0 .=R/2 .w2x0 x 2 Cw2y0 y 2 / i2R= i2. z=/.z0 =R1/ O e U.x; y/  i A 2 e e ; R = z as R= ! 1. An analogous expression is obtained from (9.302) and (9.310) in the limit as z R ! 1 (see Problem 9.11).

9.5 The Inverse Initial Value Problem This chapter concludes with a concise description of the solution to the following time-dependent inverse source problem: Determine the charge sources %.r; t / and currents J.r; t / that are zero everywhere except over the time interval T < t < T

80

9 Pulsed Beam Wavefields in Temporally Dispersive Media

such that for t > T they produce prescribed solutions of the source-free (or homogeneous) Maxwell’s equations    1  @H.r; t /  r  E.r; t / D    c  0 @t ;    1  @E.r; t /  r  H.r; t / D   c  0 @t ; r  E.r; t / D r  H.r; t / D 0;

(9.321) (9.322) (9.323)

in free-space. Although the solution to the inverse problem has been shown to be nonunique [50, 51], Moses and Prosser [52] have shown that by specifying a partial time-dependence in the wave-zone of the radiation field, a unique solution can be found for each such specification. The early analysis of Moses and Prosser [52] begins with the Bateman– Cunningham form of Maxwell’s equations, which is obtained in the following manner. First of all, notice that the pair of curl relations in (9.321) and (9.322) may be expressed as 1 @H.r; t / ; r  E.r; t / D  0 c @t 1 @E.r; t / 0 r  H.r; t / D ; c @t

(9.324) (9.325)

p where 0 0 =0 is the intrinsic impedance of free space. If one then defines the complex vector field .r; t / as .r; t / E.r; t /  i0 H.r; t /;

(9.326)

Maxwell’s equations may be expressed in complex form as r

.r; t / D i

r

.r; t / D 0;

1 @ .r; t / ; c @t

(9.327) (9.328)

which is the Bateman–Cunningham form of Maxwell’s equations when t is replaced by ct . The general solution of the these equations may then be expressed in terms of the eigenfunctions of the curl operator [53]. In a subsequent refinement of their work that is based upon the Radon transform [54], Moses and Prosser [55] have shown that there always exists a vector function of position G.r/ such that for any causal, finite energy solution fE.r; t /; H.r; t /g of Maxwell’s equations, lim .rE.r; t // D G.r  ct; ; /;

(9.329)

lim .rH.r; t // D 0 rO  G.r  ct; ; /;

(9.330)

r!1 r!1

9.5 The Inverse Initial Value Problem

81

with rO  G.r/ D 0;

(9.331)

where rO r=r is the unit vector in the direction of the position vector r D .r; ; / with spherical polar coordinates .r; ; / such that x D r sin  cos , y D r sin  sin , z D r cos  with 0     and 0   < 2. Causality then requires that the radiated field vectors E.r; t / and H.r; t / are obtained from their respective initial temporal field behaviors. The pair of asymptotic limits given in (9.329) and (9.330) then show that, with the exception of the 1=r factor, all causal, finite energy solutions of Maxwell’s equations propagate outward to infinity like one-dimensional electromagnetic waves along rays specified by the polar angles  and . Furthermore, the exact electromagnetic field vectors are given in terms of the wave zone vector field G.r/ as [55] Z 2 Z  @ 1 0 0 sin  d d 0 G.r  uO  ct;  0 ;  0 /; (9.332) E.r; t / D 2 0 @p 0 Z 2 Z  @ 1 G.r  uO  ct;  0 ;  0 /; (9.333) H.r; t / D sin  0 d 0 d 0 uO  2 0 @p 0 where p D r  uO  ct and uO D .sin  0 cos  0 ; sin  0 sin  0 ; cos  0 /. So-called electromagnetic bullets may then be constructed by requiring that the vector function G.r  ct; ; / D G.r  ct; / describing the far-zone behavior be independent of the azimuthal angle  and that it identically vanishes both outside of the cone 0   < c as well as outisde of the radial space–time region a C ct < r < b C ct , where a < b, as depicted in Fig. 9.21. The solution of the inverse source problem then provides the source distribution required to produce this electromagnetic bullet. Unfortunately, the preceeding analysis of the inverse source problem due to Moses and Prosser [52, 55], while of considerable historical importance, is only

z

Fig. 9.21 Space–time domain (shaded region) of an electromagnetic bullet that is defined in the interior region of the cone 0   < c and radial space–time region a C ct < r < b C ct

c

O

82

9 Pulsed Beam Wavefields in Temporally Dispersive Media

applicable to sources with a specific separable space–time dependence. In addition, these solutions have been shown, in general, not to be minimum-energy solutions. These limitations have been addressed by Marengo, Devaney, and Ziolkowski [56] who reconsider the time-dependent inverse source problem with far-field data based upon a limited-view Radon transform, a problem analogous to a limited-view computed tomography reconstruction. Their analysis considers the inverse source problem for the inhomogeneous scalar wave equation 

r2 

1 @2 c 2 @t 2

 U.r; t / D k4kQ.r; t /;

(9.334)

where the radiating source Q.r; t /, which is assumed to be localized within a simply–connected region D 2 <3 , is to be determined from measurements of the radiated field [see (3.81) of Vol. 1] U.r; t / D

k4k 4

Z

1

dt0

Z

1

1

1

d 3r 0

ˇ ˇ  Q.r0 ; t 0 /  0 ı t  t C ˇr  r0 ˇ =c 0 jr  r j

(9.335)

for all r … D and all t 2 <. Because of the existence of nonradiating sources within the spatial support D of the source [50, 51], whose fields identically vanish for all r … D, this inverse source problem does not admit a unique solution unless certain a priori constraints are applied, as done by Moses and Prosser [52, 55], but not without introducing unnecessary limitations on the solution. The asymptotic behavior of the radiated field given in (9.335) is given by (see Problem 9.13) k4k F .Os; / (9.336) U.r; t /  4 r as r ! 1, where sO r=r is a unit vector that specifies the direction of observation of the radiated wavefield,  t  r=c is the retarded time, and where Z

1

F .Os; / D 1

dt0

Z

1

d 3 r 0 Q.r0 ; t 0 /ı.t 0    r0  sO=c/:

(9.337)

1

If the time-domain radiation pattern F .Os;  / is known for all times  2 < over all observation directions sO 2 S 2 , where S 2 2 <3 is the unit sphere, the radiated wavefield U.r; t / can be then determined [57] everywhere outside the source domain D; however, a unique determination of U.r; t / for r … D is not possible if the radiation pattern F .Os; / is known only over a discrete set of directions sO. The analysis of the inverse source problem with far-field data taken over a discrete set of directions has been given by Marengo, Devaney, and Ziolkowski [56] using a limited-view Radon transform inversion. Their motivation in solving this important problem was (a) the direct problem of synthesizing the minimum energy (minimum L2 norm) source Q.r; t / with prescribed spatial support D that produces a given far-field radiation pattern F .Os;  /, as for an electromagnetic bullet, and (b)

9.5 The Inverse Initial Value Problem

83

the inverse problem of reconstructing an unknown source using far-field data, as for source (target) identification and interrogation. These two interrelated direct and inverse problems are now treated in the following subsections based upon their analysis.

9.5.1 The Direct Problem Define the source function %./ as %./ Q.r; t /;

(9.338)

and define where  .0 ; 1 ; 2 ; 3 / with 0 D ct , 1 D x, p 2 D y, 3 D z, p O . ;  ;  ;  / with  D 1= 2,  D s = 2, s2 D the unit vector  s s0 s1 x p ps0 s1 s2 s3 sy = 2, s3 D sz = 2, where sx , sy , sz are the components of the unit vector sO D .sx ; sy ; sz /. In addition, define the masking function M./ as ( M./

1; if  2 D ; 0; if  … D

(9.339)

where D denotes the space–time region containing the source. With these definitions, the expression (9.337) for the radiation pattern becomes 1 F .Os; / D p 2

Z

1

  p M./%./ı c= 2    O s d 4 :

(9.340)

1

This result may then be expressed in terms of the fourfold Radon transform (see Appendix H) Z 1 O 4 O / D %./ı.    /d (9.341) RŒ% .; 1

O of the function %./ over the set of hyperplanes   D 0, where O is a unit vector defining the orientation of the hyperplane in the four-dimensional Radon domain and where the real-valued parameter  defines the distance of the hyperplane to the origin. The expression given in (9.340) for the time-domain radiation pattern may then be expressed as p 1 F .Os; / D p RŒM% .O s ; c= 2/: 2

(9.342)

Hence, the radiation pattern F .Os;  / along a fixed direction sO is given by the projection of the source function %./ onto the line taken along the direction of the unit vector O s , depicted in part (a) of Fig. 9.22. It then follows from (9.342) that, for a prescribed radiation pattern F .Os;  /, the data necessary to reconstruct the source

84

9 Pulsed Beam Wavefields in Temporally Dispersive Media

Fig. 9.22 (a) Threedimensional depiction of the Radon transform representation of the radiation pattern along the fixed direction sO due to the source distribution %./ in a “viewing” direction specified by the unit vector sO. (b) The relation of the unit vector sO to the light cone in three-dimensional space– time. (After Fig. 1 in [56])

a 0

= ct

](

v

R[ s

v

/4

)

v

0

s

v

s

2

1

b 0

= ct

v s

2 1

function %./ consist of the Radon projections along radial lines that are tangent to the generalized four-dimensional light cone with apex at the origin, as depicted in Part (b) of Fig. 9.22 in the three-dimensional case. In this manner, Marengo, Devaney, and Ziolkowski have shown that [56] “Radon projections onto directions outside the light cone cannot be inferred from F .Os; /, making the inversion nonunique. The inverse source problem with far-field data reduces to finding source functions consistent with radon projections provided only for directions that lie tangentially on the surface of the light cone (nonradiating sources are those whose Radon projections onto those directions vanish . . . ).” The frequency-domain radiation pattern FQ .Os; !/ D

Z

1

F .Os;  /e i! d

(9.343)

1

is, with substitution from (9.340), found to be given by 1 FQ .Os; !/ D c

Z

1

p

M./%./e i.

2!=c/O s 

d 4 ;

(9.344)

1

which is just the spatiotemporal Fourier transform of the masked source function M./%./ evaluated on the surface of a generalized cone centered at the origin

9.5 The Inverse Initial Value Problem

85

in the four-dimensional .!=c; k/ Fourier domain. It is then seen that [56] “while F .Os; / provides information about Radon projections of %./ only along lines tangent to the light cone . . . , FQ .Os; !/ provides information about the fourfold Fourier transform of %./ only over the surface of an analogous generalized cone in the [four-dimensional] Fourier domain. Thus FQ .Os; !/ alone is insufficient to uniquely determine %./ through Fourier inversion, confirming the nonunique nature of the inverse source problem.”

9.5.2 The Inverse Problem The development of the minimum energy solution to the inverse source problem from far-field data is now given. By “minimum energy” solution is meant the minimum L2 norm s j%./j2 d 4  solution %Emi n ./ QEmi n .r; t /. Only the general formulation of the solution is given here. More specific details with applications to discrete samples may be found in the paper by Marengo, Devaney, and Ziolkowski [56] as well as in related work cited there. The general solution begins by expressing (9.340) for the radiation pattern in the form of a linear mapping as [56] F D P%;

(9.345)

˚ where %./ 2 X with X L2 <4 denoting the Hilbert space of L2 functions of the variable  2 <4 , F .Os;  / 2 Y with Y D T  S denoting the Hilbert space 2 2 that is formed from the direct product of the˚ space T 2 L f
For example, the inner product of two functions %1 ./ 2 X and %2 ./ 2 X is defined as Z 4 h%1 ; %2 iX % 1 ./%2 ./d ;

where the superscript asterisk (*) denotes the complex conjugate.

86

9 Pulsed Beam Wavefields in Temporally Dispersive Media

which is associated with the operation of backprojection (see [56] and the references contained therein). The appearance of the masking function in this expression guar  antees that P  maps Y ! X, so that  PP maps Y ! Y and P P maps X ! X. The final form of (9.346) shows that P  F ./ is given by the product of the masking function M./ with a linear superposition of time-dependent plane waves  with amplitudes given by the far-field pattern F .Os;  /. In particular, the quantity P  F ./ is seen to be a free-field (see Sect. 7.4.1 of Vol. 1) that is defined by a plane wave expansion truncated within the space-time domain D containing the source [see (9.339)]. It then follows from Sherman’s first theorem (see Theorem 4 of Vol. 1)  that P  F ./ satisfies the homogeneous wave equation    1 @2  r 2  2 2 P  F ./ D 0; c @t

(9.347)

for all  2 D excluding the boundary @D of D. The unique, minimum L2 norm solution to the inverse source problem expressed in (9.345) is found to be [59, 60]   %Emi n ./ D P  FN ./;

(9.348)

where FN .Os; / denotes the filtered time-domain radiation pattern that is defined by the relation   PP  FN .Os;  / D F .Os;  /: (9.349) If the far-field data measurements are noise-free, then (9.348) and (9.349) provide the normal solution to the inverse source problem. However, in the inevitable presence of noise in the data, some regularization procedure [59] (such as Tikhonov– Phillips regularization) may be used to generate approximate minimum energy solutions to the inverse source problem [61]. The noise-free solution is only considered here. Taken together, (9.347) and (9.348) give   1 @2 2 r  2 2 %Emi n ./ D 0 c @t

(9.350)

for all  2 D with  … @D. Notice that the frequency domain analog of this result, which is implicitly contained in the inverse source problem solutions due to Bleistein and Cohen [51] and Devaney and Porter [62, 63] and others, is the usual form employed in this methodology. For the case of a time-independent masking function M./ D M.r/, (9.340), (9.346), and (9.349) yield

9.5 The Inverse Initial Value Problem

  1 PP  FN .Os; / D c

Z

87

1

dt0

1

1 c

Z

d 2s0

S2

Z

Z

1 c

1

dt00 FN .Os0 ; t 00 /

1 1

 D

d 3 r 0 M.r0 /ı.  t 0 C r0  sO=c/

d 2 s 0 FN .Os0 ; t 0  r0  sO0 =c/

S2

Z

1 1

 D

Z

  d 3 r 0 M.r0 /ı t 00    .Os  sO0 /  r0 =c

1

Z

d 2 s 0 FN .Os0 ;  / ˝ h.Os  sO0 ;  /;

(9.351)

S2

where the symbol ˝ here denotes the temporal convolution operation Z

1

f ./ ˝ g. / D

f .  t /g.t /dt;

(9.352)

  d 3 r 0 M.r0 /ı  C .Os  sO0 /  r0 =c :

(9.353)

1

and where Z

0

1

h.Os  sO ; / 1

Taken together, (9.349) and (9.351) show that F .Os; / D

1 c

Z

d 2 s 0 FN .Os0 ;  / ˝ h.Os  sO0 ;  /:

(9.354)

S2

Upon comparison of (9.353) with (9.340), the quantity h.Os  sO0 ; / is identified as the time-domain radiation pattern due to the space–time separable source Q.r; t / D M.r/ı.t  sO0  r=c/;

(9.355)

which is uniformly distributed over the spatial support M.r/ and is impulsively excited with a progressive time delay t 0 D sO0  r=c. With this understanding, the integral of the convolution given in (9.354) is seen to state that [56] “the unfiltered data F .Os; / are equal to the sum over all available directions sO0 of the time-domain radiation pattern of a source that consists of a uniform distribution of point radiators (within the spatial support of the sought-after source) all of which are excited, with a progressive time delay r  sO0 =c, by the same time signature, the latter being precisely – for a given sO0 – the time signature of the filtered data FN .Os0 ; /.” That is, along each fixed direction sO0 , the filtered data FN .Os0 ;  / may be interpreted as the excitation that must be applied at each point of the uniform source distribution Q.r; t / D M.r/ı.t  sO0  r=c/ in order to obtain the original, unfiltered data F .Os; / through a linear superposition of the time-domain radiation patterns due to each of these synthetic sources.

88

9 Pulsed Beam Wavefields in Temporally Dispersive Media

To compute the minimum energy source from (9.346) and (9.348), the relation appearing in (9.354) must be inverted. Upon taking the temporal Fourier transform of (9.354), one obtains Z 1 Q s  sO0 ; !/; Q d 2 s 0 FQN .Os0 ; !/h.O (9.356) F .Os; !/ D c S2 where [from the temporal Fourier transform of (9.353)] Q s  sO0 ; !/ D h.O

Z

1

0

0

M.r0 /e i.!=c/r .OsOs / d 3 r 0 :

(9.357)

1

The minimum energy solution is then obtained in the following manner. First, numerically (or, if possible, analytically) solve the filtering operation specified in (9.356) to determine FQN .Os; !/ along each specified direction sO. Then, upon taking the inverse temporal Fourier transform of FQN .Os; !/, recover the filtered data FN .Os;  /. Finally, use the first form of (9.346) to backproject this filtered data FN .Os; /, as required by the reconstruction formula given in (9.348). Further details may be found in the paper by Marengo, Devaney, and Ziolkowski [56] and the references contained therein.

9.6 Summary This rather lengthy chapter has provided both a review of the fundamental material necessary for the study of dispersive pulse propagation as well as extending this theory into several related topics of interest. Of significant interest here is the question concerning separable pulsed beam fields and the full implications of the commonly used paraxial approximation. In particular, it was shown here that, with the exception of trivial cases (such as a pulsed, plane wavefield), space-time separability for a pulsed, beam wavefield is strictly valid only in the geometrical optics limit as ! ! 1.

References 1. K. E. Oughstun and G. C. Sherman, Pulse Propagation in Causal Dielectrics. Berlin: SpringerVerlag, 1994. 2. K. E. Oughstun, Electromagnetic & Optical Pulse Propagation Volume I: Spectral representations in Temporally Dispersive Media. Berlin: Springer-Verlag, 2006. 3. W. H. Carter, “Electromagnetic beam fields,” Optica Acta, vol. 21, pp. 871–892, 1974. 4. J. W. Goodman, Introduction to Fourier Optics. New York: McGraw-Hill, 1968. 5. J. A. Stratton, Electromagnetic Theory. New York: McGraw-Hill, 1941. 6. I. S. Gradshteyn and I. M. Ryzhik, Tables of Integrals, Series, and Products. New York: Academic Press, 1980.

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7. G. N. Watson, A Treatise on the Theory of Bessel Functions. Cambridge: Cambridge University Press, 1922. 8. M. Born and E. Wolf, Principles of Optics. Cambridge: Cambridge University Press, seventh (expanded) ed., 1999. 9. D. C. Bertilone, “The contribution of homogeneous and evanescent plane waves to the scalar optical field: exact diffraction formulae,” J. Mod. Opt., vol. 38, no. 5, pp. 865–875, 1991. ´ Lalor, “Contribution of the inhomo10. G. C. Sherman, J. J. Stamnes, A. J. Devaney, and E. geneous waves in angular-spectrum representations,” Opt. Commun., vol. 8, pp. 271–274, 1973. 11. T. Melamed and L. B. Felsen, “Pulsed-beam propagation in lossless dispersive media. I. Theory,” J. Opt. Soc. Am. A, vol. 15, pp. 1268–1276, 1998. 12. T. Melamed and L. B. Felsen, “Pulsed-beam propagation in lossless dispersive media. II. A numerical example,” J. Opt. Soc. Am. A, vol. 15, pp. 1277–1284, 1998. 13. K. E. Oughstun, “Asymptotic description of pulsed ultrawideband electromagnetic beam field propagation in dispersive, attenuative media,” J. Opt. Soc. Am. A, vol. 18, no. 7, pp. 1704–1713, 2001. 14. W. Heitler, The Quantum Theory of Radiation. Oxford: Clarendon Press, 1954. 15. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics. Cambridge: Cambridge University Press, 1995. ¨ 16. R. F. A. Clebsch, “Uber die Reflexionan einer Kugelfl¨ache,” J. Reine Angew. Math., vol. 61, pp. 195–262, 1863. 17. G. Mie, “Beitr¨age zur Optik truber Medien, speziell kollaidaler Metallosungen,” Ann. Phys. (Leipzig), vol. 25, pp. 377–452, 1908. 18. P. Debye, “Der lichtdruck auf Kugeln von beliegigem Material,” Ann. Phys. (Leipzig), vol. 30, pp. 57–136, 1909. 19. T. J. I. Bromwich, Phil. Trans. R. Soc. Lond., vol. 220, p. 175, 1920. 20. E. T. Whittaker, “On the partial differential equations of mathematical physics,” Math. Ann., vol. 57, pp. 333–355, 1903. 21. A. J. Devaney and G. C. Sherman, “Plane-wave representations for scalar wave fields,” SIAM Rev., vol. 15, pp. 765–786, 1973. 22. A. J. Devaney and E. Wolf, “Multipole expansions and plane wave representations of the electromagnetic field,” J. Math. Phys., vol. 15, pp. 234–244, 1974. 23. A. J. Devaney, A New Theory of the Debye Representation of Classical and Quantized Electromagnetic Fields. PhD thesis, The Institute of Optics, University of Rochester, 1971. 24. E. T. Copson, An Introduction to the Theory of Functions of a Complex Variable. London: Oxford University Press, 1935. p. 110. 25. B. A. Fuks, Introduction to the Theory of Analytic Functions of Several Complex Variables. Providence: American Mathematical Society, 1963. 26. A. Erd´elyi, “Zur Theorie der Kugelwellen,” Physica (The Hague), vol. 4, pp. 107–120, 1937. 27. E. L. Hill, “The theory of vector spherical harmonics,” Am. J. Phys., vol. 22, pp. 211–214, 1954. 28. J. D. Jackson, Classical Electrodynamics. New York: John Wiley & Sons, Inc., third ed., 1999. 29. C. J. Bouwkamp, “Diffraction theory,” Rept. Prog. Phys., vol. 17, pp. 35–100, 1954. 30. E. Lalor, “Conditions for the validity of the angular spectrum of plane waves,” J. Opt. Soc. Am., vol. 58, pp. 1235–1237, 1968. 31. A. Sommerfeld, Optics, vol. IV of Lectures in Theoretical Physics. New York: Academic Press, 1964. paperback edition. ´ Lalor, “Asymptotic approximations to angular-spectrum 32. G. C. Sherman, J. J. Stamnes, and E. representations,” J. Math. Phys., vol. 17, no. 5, pp. 760–776, 1976. 33. J. Focke, “Asymptotische Entwicklungen mittels der Methode der station¨aren phase,” Ber. Verh. Saechs. Akad. Wiss. Leipzig, vol. 101, no. 3, pp. 1–48, 1954. 34. G. Braun, “Zur Methode der station¨aren Phase,” Acta Phys. Austriaca, vol. 10, pp. 8–33, 1956. 35. D. S. Jones and M. Kline, “Asymptotic expansion of multiple integrals and the method of stationary phase,” J. Math. Phys., vol. 37, pp. 1–28, 1958.

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36. N. Chako, “Asymptotic expansions of double and multiple integrals occurring in diffraction theory,” J. Inst. Math. Appl., vol. 1, no. 4, pp. 372–422, 1965. 37. A. Ba˜nos, Dipole Radiation in the Presence of a Conducting Half-Space. Oxford: Pergamon Press, 1966. Sect. 2.12. 38. H. Bremermann, Distributions, Complex Variables, and Fourier Transforms. Reading: Addison-Wesley, 1965. Chap. 8. 39. G. C. Sherman, “Recursion relations for coefficients in asymptotic expansions of wavefields,” Radio Sci., vol. 8, pp. 811–812, 1973. 40. J. Gasper, “Reflection and refraction of an arbitrary wave incident on a planar interface,” M.S. dissertation, The Institute of Optics, University of Rochester, Rochester, NY, 1972. 41. N. Bleistein, “Uniform asymptotic expansions of integrals with stationary point near algebraic singularity,” Com. Pure Appl. Math., vol. XIX, no. 4, pp. 353–370, 1966. 42. N. Bleistein, “Uniform asymptotic expansions of integrals with many nearby stationary points and algebraic singularities,” J. Math. Mech., vol. 17, no. 6, pp. 533–559, 1967. 43. N. Bleistein and R. Handelsman, Asymptotic Expansions of Integrals. New York: Holt, Rinehart and Winston, 1975. 44. K. E. Oughstun, “The angular spectrum representation and the Sherman expansion of pulsed electromagnetic beam fields in dispersive, attenuative media,” Pure Appl. Opt., vol. 7, no. 5, pp. 1059–1078, 1998. 45. G. C. Sherman, “Diffracted wavefields expressible by plane-wave expansions containing only homogeneous waves,” Phys. Rev. Lett., vol. 21, no. 11, pp. 761–764, 1968. 46. G. C. Sherman, “Diffracted wavefields expressible by plane-wave expansions containing only homogeneous waves,” J. Opt. Soc. Am., vol. 59, pp. 697–711, 1969. 47. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics. Cambridge: Cambridge University Press, 1995. Chap. 3. 48. H. Kogelnik, “Imaging of optical modes – Resonators with internal lenses,” Bell Syst. Tech. J., vol. 44, pp. 455–494, 1965. 49. H. Kogelnik and T. Li, “Laser beams and resonators,” Proc. IEEE, vol. 54, no. 10, pp. 1312– 1329, 1966. 50. A. J. Devaney and E. Wolf, “Radiating and nonradiating classical current distributions and the fields they generate,” Phys. Rev. D, vol. 8, pp. 1044–1047, 1973. 51. N. Bleistein and J. K. Cohen, “Nonuniqueness in the inverse source problem in acoustics and electromagnetics,” J. Math. Phys., vol. 18, pp. 194–201, 1977. 52. H. E. Moses and R. T. Prosser, “Initial conditions, sources, and currents for prescribed time-dependent acoustic and electromagnetic fields in three dimensions, Part I: Acoustic and electromagnetic “bullets”, expanding waves, and imploding waves,” IEEE Trans. Antennas Prop., vol. 34, no. 2, pp. 188–196, 1986. 53. H. E. Moses, “Eigenfunctions of the curl operator, rotationally invariant Helmholtz theorem, and applications to electromagnetic theory and fluid mechanics,” SIAM J. Appl. Math., vol. 21, pp. 114–144, 1971. 54. H. E. Moses and R. T. Prosser, “A refinement of the Radon transform and its inverse,” Proc. R. Soc. Lond. A, vol. 422, pp. 343–349, 1989. 55. H. E. Moses and R. T. Prosser, “Exact solutions of the three-dimensional scalar wave equation and Maxwell’s equations from the approximate solutions in the wave zone through the use of the Radon transform,” Proc. R. Soc. Lond. A, vol. 422, pp. 351–365, 1989. 56. E. A. Marengo, A. J. Devaney, and R. W. Ziolkowski, “New aspects of the inverse source problem with far-field data,” J. Opt. Soc. Am. A, vol. 16, pp. 1612–1622, 1999. 57. E. Heyman and A. J. Devaney, “Time-dependent multipoles and their application for radiation from volume source distributions,” J. Math. Phys., vol. 37, pp. 682–692, 1996. 58. T. F. Jordan, Linear Operators for Quantum Mechanics. New York: John Wiley & Sons, 1969. 59. M. Bertero, “Linear inverse and ill-posed problems,” in Advances in Electronics and Electron Physics (P. W. Hawkes, ed.), pp. 1–120, New York: Academic Press, 1989. 60. D. N. G. Roy, Methods of Inverse Problems in Physics. Boca Raton, Florida CRC Press, second ed., 1991.

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91

61. R. W. Deming and A. J. Devaney, “A filtered backpropagation algorithm for GPR,” J. Env. Eng. Geo., vol. 0, pp. 113–123, 1996. 62. R. P. Porter and A. J. Devaney, “Generalized holography and computational solutions to inverse source problems,” J. Opt. Soc. Am., vol. 72, pp. 1707–1713, 1982. 63. A. J. Devaney and R. P. Porter, “Holography and the inverse source problem. Part II: inhomogeneous media,” J. Opt. Soc. Am. A, vol. 2, pp. 2006–2011, 1985.

Problems 9.1. Derive the expression Q  x 0 ; y  y 0 ; z/ D z .ikR  1/e ikR K.x R3 Q for the monochromatic impulse response function K.x; y; z/ D 2 UQ .r; !/ [i.e., the scalar wave-field produced in the positive half-space z > z0 when UQ 0 .x 0 ; y 0 ; !/pD ı.x 0 ; y 0 /] for the planar boundary value problem in free-space, where R D .x  x 0 /2 C .y  y 0 /2 C .z  z0 /2 . This function is also referred to as the scalar dipole field. Using the relation 2

   v 2 v @U0 .u; v/ D  1 U1 .u; v/  J1 .v/; @u u u

show that the evanescent wave contribution to the scalar dipole field is given by  z 0 0 Q KE .x  x ; y  y ; z/ D 3 J0 .k/  2U0 .kR  k z; k/ R  kR J1 .k/  2kRU1 .kR  k z; k/ : C z The homogeneous plane wave contribution to the scalar dipole field is then given by Q  x 0 ; y  y 0 ; z/  KQ E .x  x 0 ; y  y 0 ; z/: KQ H .x  x 0 ; y  y 0 ; z/ D K.x Show that, along the axis of the dipole Q 0; z/ D K.0;



k2 i k2  e ikz ; kz .kz/2

k2 ; KQ E .0; 0; z/ D  .kz/2

i k 2 ikz=2 sin .kz=2/ ikz=2 e e KQ H .0; 0; z/ D  : kz kz=2

92

9 Pulsed Beam Wavefields in Temporally Dispersive Media

Notice that the homogeneous wave contribution along the dipole axis is negligible in comparison to the evanescent contribution when kz  1, while the opposite is true when kz  1. 9.2. Show that the solutions of (9.96)–(9.97) which behave as outgoing spherical waves at infinity satisfy Maxwell’s equations (9.81)–(9.84) in all of space. 9.3. Show that the particular solution given in (9.99) satisfies the reduced scalar wave equation given in (9.98). 9.4. Derive the pair of relations given for the electric and n in (9.138)–(9.139) o O s; !/; H.O O s; !/ given in (9.134)–(9.135), magnetic spectral amplitude vectors E.O making use of the inverse of the spatial Fourier transform relations given in (9.136)– (9.137) as well as the spatiotemporal Fourier transform of the equation of continuity. 9.5. Derive (9.189) for the time-average radiated power given by (9.187). 9.6. Use the expansion of the vector spherical harmonics Ym ` .; / in terms of the ordinary spherical harmonics Y`m .; / that is given in (9.169), together with the relations given in (9.170)–(9.173), to show that the expression given in (9.192) for the time-averaged far-field power radiated per unit solid angle by a pure multipole of order .`; m/ may be expressed as ˇ2

ˇˇ m a` .!/ˇ 2˛.!/r k4ck dhP i 1  e <  d˝ 32 2  .!/ 2`.` C 1/ ˇ ˇ2  .`  m/.` C m C 1/ ˇY`mC1 .; /ˇ ˇ ˇ2 C.` C m/.`  m C 1/ ˇY`m1 .; /ˇ C 2m2 Y`m .; /

as k.!/r ! 1. 9.7. Show that a scalar plane wave has the multipole expansion e

ikr

D 4

1 X `D0

`

i j` .kr/

` X

 Y`m .; /Y`m . 0 ;  0 /;

mD`

where r D .r; ; / and k D .k;  0 ;  0 / are the spherical coordinate representations of these two vectors. 9.8. Determine the spectral amplitude UQQ s .p; q; !/ of the time-harmonic spherical wave UQ s .r; !/ D e ikr =r radiating away from the origin. With this result, use (9.195)–(9.197) to obtain explicit expressions for the homogeneous and evanescent components of this spherical wavefield along the z-axis (i.e., when p D q D 0). 9.9. Derive explicit expression for the endpoints ˛1 and ˛4 of the contour C depicted in Fig. 9.19.

Problems

93

9.10. Let UQ .r; !/ D UQ .r; #; '; !/ be a solution of the homogeneous scalar Helmholtz equation   2 r C k 2 UQ .r; !/ D 0; and let UQ .r; #; '; !/ have an asymptotic expansion of the form N 1 ˚ e ikr X Bn .#; '; !/ UQ .r; #; '; !/ D C O .kr/N n kr nD0 .kr/

for arbitrarily large N as kr ! 1 with fixed k > 0 and fixed direction cosines 1 D sin # cos ', 2 D sin # sin ', 3 D cos #. With the assumption that the asymptotic expansions of the partial derivatives of UQ .r; !/ D UQ .r; #; '; !/ with respect to r; #; ' up to the second order can be obtained by differentiating the above asymptotic expansion term by term, derive the general recursion relation between the coefficients BnC1 .#; '; !/ and Bn .#; '; !/. In particular, show that B1 .#; '; !/ D

i 2 L B0 .#; '; !/; 2

where L2 is the differential operator defined by L2 

1 @ sin # @#

  @ 1 @2 sin #  : 2 @# sin # @' 2

9.11. Show that the integral representation of the source-free transverse field vector O U.x; y/ given in (9.294) has the series expansion given in (9.293). 9.12. Obtain the limiting behavior as z R ! 1 of the gaussian beam field given in (9.302) and (9.310). 9.13. Derive the asymptotic approximation given in (9.336) and (9.337) of the farfield behavior of the radiation field given in (9.335).   9.14. Show that the expression given in (9.346) for the quantity P  F ./ follows from the definition of the adjoint P  of the linear operator P and (9.340) for the radiation pattern F .Os; / with  t  r=c.

Chapter 10

Asymptotic Methods of Analysis using Advanced Saddle Point Techniques

The integral representation developed in Vol. 1 and reviewed in Chap. 9 of this volume provides an exact, formal solution to the problem of electromagnetic pulse propagation in homogeneous, isotropic, locally linear, temporally dispersive media either filling all of space or filling the half-space z > z0 . However, an exact, analytic evaluation of the resultant contour integral is typically not possible for a rather broad class of realistic initial pulse shapes. Consequently, a well-defined approximate evaluation of the integral representation for a given initial pulse is necessary in order to determine the behavior of the temporal phenomena of primary interest here. This includes the spatiotemporal properties of the precursor fields, the arrival of the signal, the signal velocity, and the spatiotemporal evolution of the pulse. To have complete confidence in the results, it is essential that this approxiamate evaluation procedure possess a useful, well-defined error bound. There are two possible approaches to accomplish this approximate evaluation. The first is a direct numerical evaluation of the integral representation. With the continued development of faster electronic computers with larger memory (particularly RAM), such an accurate numerical evaluation is now possible to be routinely carried out on a desktop computer.1 However, this can be done for only one initial pulse type with one particular set of characteristics (e.g., pulse width, pulse riseand fall-times, and carrier frequency) and one type of dispersive material (e.g., with Debye, Drude, or Lorentz model medium dispersion) with one particular set of material parameters at a time. Since the dependence of the propagation characteristics on these parameters is complicated, such an approach would require a vast number of individual cases in order to arrive at a general understanding of dispersive pulse propagation phenomena. The alternate approach is to conduct an asymptotic analysis of the integral representation for any given initial pulse shape in a specific type of dispersive material. This approach is also difficult to accomplish, but it results in analytic approximations for the propagated wave field that display clearly all of the basic features of the propagation phenomena as a function of both the input pulse properties and the dispersive medium properties. Such an asymptotic 1

This was not the case when George Sherman and I completed our earlier research on dispersive pulse propagation, published in the now retired Springer Series on Wave Phenomena as Electromagnetic Pulse Propagation in Causal Dielectrics in 1994 with a corrected edition in 1997. K.E. Oughstun, Electromagnetic and Optical Pulse Propagation 2: Temporal Pulse Dynamics in Dispersive, Attenuative Media, Springer Series in Optical Sciences 144, c Springer Science+Business Media, LLC 2009 DOI 10.1007/978-1-4419-0149-1 10, 

95

96

10 Asymptotic Methods of Analysis

approach yields a very accurate approximation to the actual wave field behavior in the regions of primary interest, and the analysis provides fundamental insight into the resultant dynamical behavior not given by the numerical approach. Nevertheless, numerical results can and do serve as an independent numerical “experiment” that the asymptotic analysis is compared to for both guidance and verification. In addition, numerical methods can also be combined with asymptotic results to produce a hybrid solution methodology. Consequently, the basic approach taken throughout the remainder of this book relies upon well-founded asymptotic methods of analysis. The generic form of the basic single contour integral representation which remains to be evaluated using asymptotic techniques is [cf. (9.292)] Z

q.!/e zp.!/ d!;

I.z/ D C

where z is the asymptotic parameter. There exists two standard approaches for obtaining the asymptotic expansion of such integrals as z ! 1. The simplest is the method of stationary phase (see Appendix F.4). However, that approach is applicable to only those integrals in which the argument of the exponential term appearing in the integrand is purely imaginary. Since dispersive media must necessarily be absorptive, the method of stationary phase is then not strictly applicable to this problem. The other approach is the method of steepest descent (see Appendix F.7), which relies upon a factor that decays exponentially with the parameter z in the integrand. Consequently, that approach is applicable to the analysis of the asymptotic behavior of the type of single contour integrals appearing in dispersive pulse propagation. However, a strightforward application of the method of steepest descent leads to discontinuous results in the asymptotic approximation of the integral if, for example, the order of the relevant saddle point changes abruptly at some critical time or if another contribution to the asymptotic behavior of the integral suddenly becomes dominant over the saddle point contribution, as occurs in the classical analysis of dispersive pulse propagation due to Brillouin [1, 2]. Such seemingly discontinuous changes can be more accurately described as a rapid but continuous transition using a specifically designed uniform asymptotic expansion method that is based upon an appropriate extension of the method of steepest descent. A summary description of the basic results of the modern theory of asymptotic analysis that are necessary for a complete description of dispersive pulse dymaics is presented in this chapter. For purposes of conciseness, the results are stated in theorem form without proof (which may naturally be found in the cited literature). Each theorem is followed by a detailed discussion and analysis regarding its application to the asymptotic approximation of the integral of interest. The methods presented are sufficient to obtain the complete uniform asymptotic of electromagnetic and optical pulses in a variety of causally dispersive media. A detailed knowledge of the basic theory of asymptotic expansions, presented in Appendix F, is assumed throughout the analysis presented in this chapter. The starting point of the theory reviewed here has its origin in the results developed by F. W. J. Olver in 1970 [3]. Olver’s method is an alternative to the method

10 Asymptotic Methods of Analysis

97

of steepest descents (see Sect. 7 of Appendix F) that is less stringent in its requirements on the deformation of the contour of integration C through the saddle point (or isolated saddle points) of the function p.!/ appearing in the integral representation of I.z/. Olver’s method2 is used throughout the modern asymptotic theory of dispersive pulse propagation in order to obtain the basic asymptotic nature of the propagated wave field in those space–time regions that are removed from certain critical transition points at which the method breaks down. Olver’s method is also used to obtain the transition in the asymptotic behavior of the integral I.z/ when there are two relevant isolated saddle points through which the contour of integration C may be deformed and the relative dominance of these two saddle points changes with time. That analysis is necessary to describe the transition from the first to the second precursor field in a Lorentz model dielectric [4, 5]. The uniform asymptotic expansion that is valid when the saddle point is at infinity is presented in Sect. 10.2, based upon the method developed by Handelsman and Bleistein [6]. Such a uniform expansion is necessary to describe the initial arrival and evolution of the first prescursor field [7]. As the saddle point moves into the finite complex plane, this uniform asymptotic expansion reduces to the expansion given by Olver’s method. The uniform asymptotic expansion due to two nearby first-order saddle points is next presented in Sect. 10.3 based upon the analysis of both Chester, Friedman, and Ursell [8] and Felsen and Marcuvitz [9, Sect. 4.5]. When two saddle points approach one another and coalesce, a single saddle point of higher-order results; such is the case during the evolution of the second precursor field in a Lorentz model dielectric [1,2,4,5]. The change in form of the asymptotic expansion (obtained using either the method of steepest descent or Olver’s saddle point method) due to the abrupt change in order of the relevant saddle points makes it necessary to have an asymptotic expansion that is valid uniformly in a neighborhood of the exceptional value of the parameter describing the order of the saddle point. That expansion is necessary to obtain a complete, continuous evolution of the second precursor field [7, 10]. The uniform asymptotic expansion that is valid when the saddle point is near a simple pole singularity of the spectral function q.!/ appearing in the integrand of I.z/ is considered next in Sect. 10.4 based upon the analysis of both Bleistein [11,12] and Felsen and Marcuvitz [9, Sect. 4.4], as well as upon its recent extension by Cartwright [13, 14]. In this case, the uniform asymptotic expansion provides a continuous transition as the deformed contour of integration crosses the simple pole singularity and a residue is either contributed to or subtracted from the integral. Such a uniform asymptotic analysis is necessary to properly describe the continuous transition from the precursor field to the main signal [7, 14]. Finally, the extension of Laplace’s method (see Appendix F.6) to multiple integrals is considered in Sect. 10.5 based upon the analysis of Fulks and Sather [15] 2

Olver’s 1970 SIAM Review paper “Why Steepest Descents?” played a critical role in my dissertation research on dispersive pulse propagation that began at the Institute of Optics in 1974.

98

10 Asymptotic Methods of Analysis

which, in turn, is based on a series of papers by L. C. Hsu3 on the asymptotic behavior of multiple integrals that first appeared in 1948 and continued through 1956. This analysis then provides the basis for an extension of the saddle point method for single contour integrals to multiple integrals.

10.1 Olver’s Saddle Point Method The saddle point method due to Olver [3] provides an important alternative to the method of steepest descent that was originally developed by B. Riemman [16] in 1876 and P. Debye [17] in 1909, described in Appendix F.7. Olver’s saddle point method, or just Olver’s method (as it is referred to here), relaxes the stringent requirements placed upon the deformed countour of integration, and therein lies its central importance in the asymptotic theory of dispersive pulse propagation. In his reliance upon the method of steepest descent, Brillouin [1,2] imparted a special significance to the path of steepest descent which resulted in an incorrect definition of the signal velocity. Olver’s method removes this requirement and that, in turn, results in a correct description of the signal velocity in a causally dispersive medium.

10.1.1 Peak Value of the Integrand at the Endpoint of Integration The analysis begins with the generic contour integral Z I.z/ D q.!/e zp.!/ d!

(10.1)

P

that is taken along some specified simply connected contour P extending from !1 to !2 in the complex !-plane. Both p.!/ and q.!/ are assumed to be holomorphic (regular analytic) fuctions of the complex variable ! in a domain D containing the contour P . The purpose of this consideration is to obtain the asymptotic expansion of the contour integral I.z/ for large absolute values of the real or complex parameter z that is uniformly valid with respect to the phase of z. The case considered in this subsection is the one in which the real part of the exponential argument zp.!/ appearing in the integrand of (10.1) attains its maximum value along the contour P at the starting point !1 . The angle of slope of the contour P at !1 is given by ˛N lim arg f!  !1 g; !!!1

(10.2)

P

where the limit is taken along the contour of integration P , as indicated. 3

The interested reader should consult the list of references given in the paper [15] by Fulks and Sather.

10.1 Olver’s Saddle Point Method

99

The assumptions made in Olver’s analysis [3] are as follows: 1. The functions p.!/ and q.!/ are both independent of the parameter z, and are both single-valued and holomorphic in an open domain D in the complex !-plane. 2. The contour of integration P is independent of the parameter z, the starting endpoint magnitude j!1 j is finite whereas j!2 j may be either finite or infinite, and the entire path P extending from !1 to !2 lies in the domain of holomorphicity D with the possible exception of the endpoints which may be boundary points of D. 3. In a neighborhood of the endpoint !1 , the functions p.!/ and q.!/ can be expanded in convergent series of the form p.!/ D p.!1 / C

1 X

ps .!  !1 /sC ;

(10.3)

sD0

q.!/ D

1 X

qs .!  !1 /sC1 ;

(10.4)

sD0

where p0 ¤ 0, the quantity  is real and positive, and 0. When  and  are not integers (which can occur only when !1 is a boundary point of the domain D), the appropriate branch choices of the quantities .!  !1 / and .!  !1 / are then determined by the limiting forms .!  !1 / D j!  !1 j e i˛N ; .!  !1 / D j!  !1 j e i˛N ;

(10.5) (10.6)

as ! ! !1 along P , and by continuity elsewhere on the contour P . 4. The parameter z ranges either along a ray or over an angular subsector 1    2 in the complex z-plane with jzj  Z > 0, where  argfzg and 2  1 < . Furthermore, it is assumed that the integral I.z/ converges at the endpoint !2 both absolutely and uniformly ˚ with respect to z. 5. Finally, it is assumed that the quantity < e i Œp.!1 /  p.!/ is positive for all values of ! along the contour P , except at the starting endpoint !1 of P , and that this quantity is bounded away from zero uniformly with respect to  as ! ! !2 along P . This then implies that the real part of the exponential argument zp.!/ appearing in the integrand of (10.1) attains its maximum value along the contour P at the starting endpoint !1 . Notice that neither ˛N nor  need be constrained to their principal range .;  , provided that consistency is maintained throughout their usage. Finally, the following convention is introduced for the phase of a quantity which appears in each term of the asymptotic expansion of I.z/: the value of the angle ˛N 0 argfp0 g is not necessarily its principal value, but rather is chosen so as to satisfy the inequality N  j˛N 0 C  C ˛j

 : 2

(10.7)

100

10 Asymptotic Methods of Analysis

This branch choice for argfp0 g is then to be used in constructing all of the fractional powers of .p0 / which appear in the asymptotic expansion. Since  is restricted to lie within an angular interval that is less than , the value of ˛N 0 which satisfies the above inequality is independent of  D argfzg. Subject to these assumptions and conventions, the asymptotic expansion of the contour integral given in (10.1) is then given by the following theorem due to Olver [3]. Theorem 2. Olver’s Theorem. Subject to the conditions 1 through 5, the contour integral I.z/ has the asymptotic expansion Z q.!/e

zp.!/

d!  e

zp.!1 /

P

1 X

 

sD0

sC 



as ; z.sC/=

(10.8)

as jzj ! 1 uniformly with respect to  D argfzg for 1    2 . Here  . / denotes the gamma˚ function. The branch of z.sC/= to be employed in this expansion has phase arg z.sC/= D .s C /=, and the first three coefficients as are given by q0 ; .p0 /=   1 q1 . C 1/p1 q0 a1 D  ; 2   p0 .p0 /.C1/= ( q2 . C 2/p1 q1  a2 D  2 p 0 a0 D



C . C  C

2/p12

 2p0 p2

(10.9) (10.10)

 . C 2/q0

)

23 p02

1 : .p0 /.C2/= (10.11)

By Definition 5 in Appendix F, the asymptotic expansion given in (10.8) states that for an arbitrary positive integer N , Z

q.!/e zp.!/ d! D e zp.!1 / P

where

"N 1 X sD0

 

sC 



as z.sC/=

˚ RN D O z.N C/=

# C RN ;

(10.12)

(10.13)

as jzj ! 1 is the remainder after N terms. The statement that the asymptotic expansion in (10.8) is satisfied uniformly with respect to  for 1    2 means that the order relation given in (10.13) for the remainder RN is satisfied uniformly with respect to  for all  2 Œ1 ; 2 . The finite sum appearing in (10.12) is referred

10.1 Olver’s Saddle Point Method

101

to as an asymptotic approximation of the integral, and the first term of that series is called the dominant term of the asymptotic expansion. If the complex phase function p.!/ and the contour of integration P appearing in (10.8) are both continuous functions of some parameter  that varies continuously over a specified domain D, so that p.!/ D p.!;  / and P D P . /, then the asymptotic behavior of the integral I.z/ D I.z;  / can change discontinuously as  varies over D even when all of the conditions of Olver’s theorem are satisfied for all  2 D. This can occur, for example, when either of the parameters  or  changes discontinuously as  changes continuously, as seen in (10.3) and (10.4). However, if for all  2 D, conditions 1 through 5 are satisfied with both  and  independent of  , and if the path P moves in the complex !-plane in a continuous fashion as  varies continuously, then the asymptotic expansion given in (10.8) is uniform with respect to the parameter  for all  2 D and the asymptotic behavior of the integral I.z; / varies continuously with .

10.1.2 Peak Value of the Integrand at an Interior Point of the Path of Integration Consider now the contour integral I.z/ given in (10.1) with the lower limit !1 replaced by !0 . Both p.!/ and q.!/ are taken to be regular analytic functions in an open domain D containing the contour of integration .!0 ; !2 /P , and  argfzg is either fixed or ranges over a closed interval Œ1 ; 2 such that 2  1 < . Unlike in the subsection, suppose now that the maximum value of the ˚ previous quantity < e i p.!/ occurs at some point !1 that is interior to the path P , so that !1 2 .!0 ; !2 /P and is independent of . Accordingly, the contour P may be partitioned at !1 so that Z q.!/e

I.z/ D

zp.!/

PC

Z

q.!/e zp.!/ d!;

d! 

(10.14)

P

where P C is that portion of the original contour P extending from !1 to !2 and P  is that portion of P extending from !1 to !0 . Notice that P C is traversed in the same sense as P , whereas P  is traversed in the opposite sense that P is traversed. The results of Olver’s theorem (Theorem 2) then apply to each of the above two contour integrals for large jzj ! 1 subject to the conditions 1 through 5 given in the previous subsection. Since the functions p.!/ and q.!/ are now both analytic about the point !1 2 D, the parameter  appearing in (10.3) is an integer and the expansion appearing in (10.3) is then the Taylor series expansion of p.!/ about the point !1 with coefficients p .sC/ .!1 / ps D ; (10.15) .s C /Š where p .n/ .!1 / @n p.!/=@! n j!D!1 .

102

10 Asymptotic Methods of Analysis

.1/ Consider first the˚ case when p .!1 / ¤ 0, so that  D 1. The fifth condition that the quantity <˚ e i p.!/ be a maximum at ! D !1 then gives [upon taking the derivative of < e i p.!/ along the contour P and setting the result to zero at ! D !1 ] N D 0; p .1/ .!1 / ¤ 0; (10.16) cos .˛N 0 C  C ˛/

where ˛N 0 argfp0 g D argfp .1/ .!1 /g and where ˛N is the angle of slope of the contour P at !1 along which the derivative is evaluated. Since the two possible N is equal to =2 for one integral values of ˛N differ by , the quantity .˛N 0 C  C ˛/ and =2 for the other. As a result, the value of ˛N 0 and hence, the coefficients as , s D 0; 1; 2; 3; : : : , are exactly the same for the two contour integrals appearing in (10.14), one taken over P C and the other over P  . Consequently, the asymptotic expansions of these two contour integrals are the same, and all that ˚ remains after substitution of (10.12) into (10.14) is an error term R0N D O z.N C/ e zp.!1 / , where N is an arbitrary positive integer. Hence, the method does not provide an asymptotic expansion of the integral I.z/ in this case. However, if !1 is a saddle point of p.!/ so that p .1/ .!1 / D 0, then by condition 3 and the expansion given in (10.3) with Taylor series coefficients ps given in (10.15), it is seen that the parameter  is an integer such that   2. Notice that the quantity .  1/ specifies the order of the saddle point. Thus, ˛N differs by  for the two contour integrals appearing in (10.14), causing the values of ˛N 0 which satisfy the inequality in (10.7) to differ by either  of .  1/, according as to whether  is even or odd, respectively. Consequently, different branches are used for the quantity .p0 /1= in constructing the coefficients as , and the asymptotic expansions of these two contour integrals no longer cancel upon substitution into (10.14). In this case then, an asymptotic expansion of the contour integral I.z/ is obtained with the application of Olver’s saddle point method. As an example, consider the case in which the contour of integration P passes through a single, isolated, first-order saddle point, as illustrated in Fig. 10.1. The shaded area in this ˚ diagram indicates the local region about the saddle point wherin the quantity < e i Œp.!1 /  p.!/ is positive; i.e., the region of the complex !-plane within which the contour of integration P must lie in order that condition 5 of Olver’s theorem is satisfied. From (10.14), the integral I.z/ taken over the

P

P+

Fig. 10.1 Local geometry about an interior first-order saddle point at !1 . The paths P C and P  both descend away from the saddle point along the contour P

1

P−

10.1 Olver’s Saddle Point Method

103

contour P may be expressed as the difference between two contour integrals I C .z/ and I  .z/ taken over the contours P C and P  , respectively, both of which start at opposite sides of the saddle point !1 and progress away from it, P C being taken in the same sense as the original contour P and P  being taken in the opposite sense, as illustrated. Since  D 2, then ˛N 0C and ˛N 0 differ by 2 and the coefficients as for the asymptotic expansion of these two contour integrals are related by as D .1/sC1 asC ;

s D 0; 1; 2; 3; : : : :

(10.17)

Consequently, the even-order coefficients add whereas the odd-order coefficients cancel each other when the asymptotic expansion of I.z/ is constructed from the sum of I C .z/ and I  .z/. From (10.8), (10.9), and (10.11) of Olver’s theorem, the first two terms of the asymptotic expansion of the contour integral defined in (10.1) are then given by ) (     ˚ .2C=2/ a2C   a0C zp.!1 / ; C 1C CO z  I.z/ D 2e 2 z=2 2 z1C=2 (10.18) as jzj ! 1, where specific expressions for the coefficients a0C and a0 are readily obtained from (10.9) and (10.11). With  D 1, this result reduces directly to the well-known result obtained using the method of steepest descent [see (F.115) of Appendix F].

10.1.3 The Application of Olver’s Saddle Point Method The type of integral that needs to be evaluated in the analysis of dispersive pulse propagation has the same form as the contour integral appearing in (10.1), but the path of integration does not necessarily pass through the saddle point of the phase function in the integrand. As a consequence, Olver’s saddle point method cannot be applied directly to obtain the asymptotic expansion of the desired contour integral. The first step in the asymptotic analysis is to apply Cauchy’s residue theorem to change the path of integration in such a manner that Olver’s saddle point method can be applied to the resulting deformed contour integral. When the contour of integration P passes through a saddle point !1 of the phase function p.!/ in such a way that the integral I.z/ can be expressed in the form given in (10.14) with each of the component integrals over the contours P C and P  satisfying conditions 1 through 5 of Olver’s theorem, then P is defined here as an Olver-type path with respect to the saddle point !1 . Furthermore, if Cauchy’s residue theorem can be applied to express the integral appearing in (10.1) taken over the contour P 0 as the sum of the same integral taken over a contour P plus the contributions of any pole singularities of the function q.!/ appearing in the

104

10 Asymptotic Methods of Analysis

integrand, then the contour P 0 is said to be deformable to the contour P , and vice versa. In particular, the contour P 0 is not deformable to the contour P if the difference between the two contour integrals includes nonvanishing contributions due to any integral along arcs at j!j D 1 or along branch cuts of either of the functions p.!/ or q.!/ appearing in the integrand. To apply Olver’s method to obtain an asymptotic expansion of a given integral I.z/ taken over a contour P 0 as jzj ! 1 in some specified sector of argfzg, the first step is to try to find an Olver-type path P with respect to a saddle point !1 of the integrand to which the contour P 0 may be deformed. Even when such a path exists, the task of finding it can be formidable when the function p.!/ appearing in the exponential of the integrand is complicated. Nevertheless, that task is usually much simpler than the one of determining a path of steepest descent through the saddle point to which P 0 may be deformed, as is required in order to apply the method of steepest descent. The essential feature of an Olver-type path is that the real part of the quantity zp.!/ is larger at the saddle point than at any other point along the contour P . This condition is much more general than the constraint on the path of steepest descent through the saddle point. There is always a finite domain in the complex !-plane that has the property that any path in this domain that passes through the saddle point !1 is an Olver-type path with respect to !1 . Since the path of steepest descent is one of these paths, the method of steepest descent is then seen to be a special case of Olver’s method. The only real significance of the steepest descent path is that it permits the determination of the smallest upper bound on the estimate of the magnitude of the remainder term that results when the asymptotic series is terminated after a finite number of terms [3]. Additional complications can arise when the complex phase function p.!/ D p.!;  / is also a function of a parameter  that varies over some domain of interest R, as is the case for the type of integrals arising in dispersive pulse propagation [1, 2, 18, 19]. Suppose that p.!;  / is a continuous function of  2 R and that for each such value of  , there is an Olver-type path P D P ./, to which the original contour of integration P 0 may be deformed, that moves in a continuous fashion in the complex !-plane as  varies continuously over R. Although the integral I.z/ D I.z; / itself will then vary continuously with  2 R, its asymptotic approximation that is obtained by applying Olver’s saddle point method for each value of  2 R may change discontinuously as  varies continuously. These discontinuities can arise from a variety of complications. In particular, Brillouin’s [1, 2] asymptotic approximations of signal propagation in a Lorentz medium based upon Debye’s method of steepest descent [17], exhibit three discontinuities with  that arise from three different sources. The same discontinuities result when Olver’s saddle point method is applied in place of the method of steepest descent, but with the former method, the sources of these discontinuities are made more transparent. The discontinuous nature of Brillouin’s results are an artifact of the asymptotic analysis known as “Stokes’ phenomena” [20]. For fixed values of z, the integrals being evaluated are actually continuous functions of the parameter . To obtain asymptotic approximations of these integrals that provide their true functional behavior as  varies over the region of interest R, it is necessary to apply uniform

10.2 Uniform Expansion for Two Saddle Points at Infinity

105

asymptotic expansion methods. Three different methods are required in order to deal with the three different sources of discontinuous behavior encountered in the analysis of dispersive pulse propagation phenomena. A review of the required methods is presented in the following three sections of this chapter.

10.2 Uniform Asymptotic Expansion for Two First-Order Saddle Points at Infinity The asymptotic expansion of certain integrals I.z/ D I.z;  / of the form given in (10.1) that is valid uniformly as the relevant saddle points tend towards infinity in the complex !-plane as the parameter  approaches some critical value. In particular, let the complex phase function p.!/ D p.!;  / appearing in (10.1) have two first-order saddle points !˙ . / with equal imaginary parts and with real parts that approach ˙1, respectively, as  approaches unity from above. Let the contour of integration P 0 be deformable to a continuous path P ./ D P C ./ C P  . / where P ˙ . / is the path of steepest descent through !˙ ./ with one endpoint satisfying
(10.19)

have a Laurent series expansion of the form .!;  / D .1  /! C

1 X

an . /! n

(10.20)

nD0

for all ! such that j!j  R1 and for all  2 Œ1;  0 , where R1 is a finite positive constant and  0 > 1 is a positive constant. All other saddle points of p.!;  /, if any, are assumed to be finite in number and confined to some bounded region of the complex !-plane such that j!j  R2 < R1 for all  2 Œ1;  0 . Moreover, the amplitude function q.!/ may be written in the form Q q.!/ D ! .1C/ q.!/;

(10.21)

for large j!j with real  > 0, where the function q.!/ Q has a Laurent series expansion that is convergent for j!j  R1 and is such that Q ¤ 0: lim q.!/

j!j!1

(10.22)

If  < 0, then the uniform asymptotic approximation presented in Theorem 3 following is still applicable for all values of  2 Œ1;  0 provided that its limiting value as  approaches unity is finite [6].

106

10 Asymptotic Methods of Analysis

Since the two saddle points !˙ ./ are located at infinity when  D 1, condition 2 of Theorem 2 (Olver’s theorem) is not satisfied and Olver’s saddle point method becomes inapplicable in this limit as  ! 1C . An asymptotic approximation of the integral I.z; / is then desired that is uniform in the parameter  as  approaches the critical value of unity from above. For simplicity, only the dominant term in this uniform asymptotic expansion is considered here. This uniform asymptotic approximation is given by the following theorem due to Handelsman and Bleistein [6]. Theorem 3. (Handelsman and Bleistein) In the integrand of the contour integral Z I.z;  / D

q.!/e zp.!;/ d!

(10.23)

P ./

with real z, let the function p.!;  / possess a pair of first-order saddle points !˙ . / with equal imaginary parts and whose real parts approach ˙1, respectively, as the parameter  approaches unity from above. Let the contour P ./ D P C ./CP  . / be a continuous function of , where P ˙ ./ is the path of steepest descent through !˙ . / with one endpoint satisfying
  0 J .˛. /z/ C 2˛./e i=2 1 JC1 .˛. /z/ C R.z;  /; (10.24) where (with K a positive real constant independent of both  and z) the remainder term satisfies the inequality jR.z; /j  K

 j2˛./jC1

jJC1 .˛./z/j C jJC2 .˛. /z/j z

(10.25)

for z  Z > 0 and  2 Œ1;  0 . Notice that this error term is small for large z independent of ˛. /. The coefficients appearing here are given by  1 .!C ;  /  .! ;  / ; 2  1 .!C ;  / C .! ;  / ; ˇ./  (2 1=2  4˛ 3 . / 1 q.!C /  0 . / 00 .! ;  / 2 .2˛.//C1 C 1=2 )  4˛ 3 . / q.! / C ; 00 .! ;  / .2˛.//C1  ˛. / 

(10.26) (10.27)

(10.28)

10.2 Uniform Expansion for Two Saddle Points at Infinity

107

(

1 . /

1=2  4˛ 3 ./ 1 q.!C /  00 .! ;  / 4˛. / .2˛.//C1 C 1=2 )  4˛ 3 ./ q.! /  : 00 .! ;  / .2˛.//C1 

(10.29)

The argument of the quantity Œ 00 .!˙ ;  / is chosen so as to satisfy the inequality given in (10.7) with  D argfi zg D =2 for z real and positive and ˛N C being the angle of slope of the contour P C ./ leading away from the saddle point !C ./ in the limit as  ! 1C . For values of  close to unity, these coefficients reduce to np o p ˛. / D 2 a1 ./.  1/ C o  1 ; np o ˇ./ D a0 ./ C o  1 ;

(10.30) (10.31)

where a0 and a1 are the first two coefficients in the Laurent series expansion of .!;  / D ip.!;  // given in (10.20), and where the coefficients 0 ./ and 1 . / are both O.1/ for  2 Œ1;  0 . Finally, the proper branch of the quantity ˛ 1=2 ./ is chosen so as to satisfy the relation

4˛ 3 . /=

00

1=2

.!˙ ; /

D

p

h np oi 2a1 ./ 1 C O  1

(10.32)

as  ! 1C for all  2 Œ1;  0 . As an illustration of the application of Theorem 3 to obtaining the uniform asymptotic expansion of the integral I.z;  / given in (10.23), consider the case in which  D 0 and assume that the complex-valued phase function p.!;  / X.!; / C iY.!;  / satisfies the symmetry property p  .!  ;  / D p.!;  /;

(10.33)

so that, with ! D ! 0 C i ! 00 , X.! 0 C i ! 00 ;  / D X.! 0 C i ! 00 ;  /; 0

00

0

00

Y.! C i ! ;  / D Y.! C i ! ;  /:

(10.34) (10.35)

The saddle points of p.!;  / are then symmetrically located in the complex !-plane about the imaginary axis. In particular, let p.!;  / possess a pair of first-order saddle points !˙ . / which approach ˙1 C i !a00 as  approaches unity from above; that is lim!1 !˙ . / D ˙1 C i !a00 , where !a00 is a constant. The phase function .!;  / ip.!;  / is then given by .!;  / D Y .!; /  iX.!; /;

(10.36)

108

10 Asymptotic Methods of Analysis

and the coefficients appearing in the uniform asymptotic expansion given in (10.24) are then given by ˛. / D Y.!C ;  /;

(10.37)

(10.38) ˇ. / D iX.!C ;  /; ( ) 1 q.!C / q.! / 0 . / D Y 1=2 .!C ; /  C ; (10.39) 2 Œip 00 .!C ;  / 1=2 Œip 00 .! ;  / 1=2 ) ( 1 q.! / q.!C /  : (10.40) 1 . / D  4Y 1=2 .!C ; / Œip 00 .!C ;  / 1=2 Œip 00 .! ;  / 1=2 With these substitutions, the uniform expansion given in (10.24) becomes I.z; / D 2 i e zX.!C ;/ ( " 1 1=2 q.!C /  Y .!C ; / 

1=2 2 ip 00 .!C ;  / C

 "

q.! /

1=2 ip 00 .! ;  /

#

  J0  Y.!C ;  /z

q.!C / 1 C e i=2 Y 1=2 .! ;  / 

1=2 2 ip 00 .!C ;  / ) #   q.! / 

1=2 J1  Y.!C ;  /z  ip 00 .! ;  / CR1 .z; /;

(10.41)

as z ! 1, which is uniformly valid in  as  ! 1C . At  D 1, (10.30) and (10.37) show that Y.!˙ ; 1/ D 0 and the first two terms in the uniform expansion (10.41) for I.z; / vanish, leaving just the remainder term R1 .z;  /. Notice that the asymptotic behavior of the integral I.z;  / in a neighborhood  2 Œ1; 1 C of the critical point  D 1 strongly depends on the value of the parameter   for the amplitude function q.!/ [see (10.21)]. For  D 0 the peak of the J  Y.!C ; /z Bessel function occurs at  D 1, as illustrated in Fig. 10.2, but as  increases, the location of this peak shifts to larger values of , as illustrated in Figs. 10.2 and 10.3. For values of  bounded away from unity, the magnitude of the argument jY.!C ; /jz of the Bessel functions appearing in the uniform expansion (10.41)

10.2 Uniform Expansion for Two Saddle Points at Infinity

109

Bessel Functions J0(ζ) & J1(ζ)

1

J0(ζ)

0.5

J1(ζ)

0

−0.5 0

10

20

ζ

30

40

50

Fig. 10.2 Bessel functions J0 . / and J1 . / for real 0.6

Bessel Functions J1(ζ) & J2(ζ)

J1(ζ) 0.4

J2(ζ)

0.2

0

– 0.2

– 0.4

0

10

20

ζ

30

40

50

Fig. 10.3 Bessel functions J1 . / and J2 . / for real

becomes large so that each Bessel function may be replaced by its asymptotic approximation s J . / D

  2 cos .  =2  =4/ C O 1 e j=f gj 

110

10 Asymptotic Methods of Analysis

as j j ! 1 with j arg . /j < . Hence, for values of  bounded away from unity, the uniform asymptotic expansion given in (10.41) becomes r 2 zX.!C ;/ I.z; / D e z (" #  q.!C /e i=4 q.! /e i=4  

1=2 C

1=2 cos Y.!C ;  /z C 4  p 00 .!C ; /  p 00 .! ;  / " # )  q.!C /e i=4 q.! /e i=4  Ci

1=2 

1=2 sin Y.!C ;  /z C 4  p 00 .!C ; /  p 00 .! ;  / ˚ CO ŒY.!C ; /z 1 1=2  2 e zp.!C ;/ D q.!C /  00 zp .!C ;  / 1=2  ˚ 2 Cq.! /  00 e zp.! ;/ C O ŒY.!C ;  /z 1 zp .! ;  / (10.42) as jY.!C ; /zj ! 1 with j arg fY.!C ;  /zgj < . This is the same result as that obtained by direct application of Olver’s saddle point method except that the dependence of the remainder term on the saddle point location !C is explicitly displayed in (10.42) through the factor Y.!C ;  /. Since Y.!C ;  / ! 0 as  ! 1C , the estimate of the remainder term in (10.42) is not useful when
10.3 Uniform Asymptotic Expansion for Two First-Order Saddle Points Consider now the situation when the phase function p.!/ D p.!;  / has two saddle points that evolve in the finite complex !-plane as the parameter  varies over some domain R. There are then three possibilities. If the two saddle points remain isolated from each other as  varies over R and if one of the saddle points is dominant over the other during that entire variation, then Olver’s saddle point method directly applies and this case need not be considered any further. However, if the two saddle points remain isolated from each other as  varies over R, and if one of the saddle points is dominant over the other for  < s while the other is dominant over the first for  > s with s 2 R, then a direct application of Olver’s saddle point method will result in a discontinuous change in behavior at  D s . Finally, if the two saddle points approach each other as  approaches some critical value s and coalesce into a single higher-order saddle point at  D s , then a direct application of Olver’s saddle point method will again result in a discontinuous change in behavior at  D s . These latter two cases are now separately treated in some detail so as to obtain an asymptotic expansion of I.z;  / in each case that is uniformly valid in  for all  2 R.

10.3 Uniform Expansion for Two First-Order Saddle Points

111

10.3.1 The Uniform Asymptotic Expansion for Two Isolated First-Order Saddle Points Consider a contour integral I.z;  / of the form given in (10.1) taken over a path of integration P 0 which extends from j!j D 1 through the finite complex !-plane and back to j!j D 1 without forming a closed contour. Let the complex phase function p.!;  / be a continuous function of a real parameter  that varies over a domain R. In addition, let !1 . / and !2 ./ denote two isolated4 first-order saddle points of p.!;  / such that the inequality s , and where s . Let the original contour of integration P 0 be deformable to a path P ./ that, for all  2 R, passes through both of the saddle points !1 . / and !2 . / and has the following properties. 1. For all  2 R, the contour P ./ changes continuously in the complex !-plane as  varies over R continuously. 2. The contour P . / can be divided into two parts P1 ./ and P2 . / such that P . / D P1 . / C P2 . /, where, for i D 1; 2, Pi . / passes through the saddle point !i . / and is an Olver-type path with respect to !i ./. The integral I.z; / taken over the contour P ./ can then be expressed as I.z;  / D I1 .z;  / C I2 .z;  /; P'

(10.43)

P'

P'

1 2

1

2

1

P

a

P

P

b s

2

c s

s

Fig. 10.4 Interaction of two isolated first-order saddle points !1 and !2 . The C45o hatched area indicates the region of the complex !-plane wherein the inequality
4

The term “isolated”, as used here, means that the distance between the two points is bounded away from zero for all  2 R.

112

10 Asymptotic Methods of Analysis

where

Z

q.!/e zp.!;/ d!;

Ii .z;  / D

(10.44)

Pi ./

for i D 1; 2. It then follows from the conditions imposed on the component contours P1 . / and P2 . / that P . / is an Olver-type path for the integral I.z;  / with respect to the saddle point !1 . / when  < s and with respect to the saddle point !2 . / when  > s . Hence, according to Theorem 2 (Olver’s theorem) and the results of Sect. 10.1.2, the asymptotic expansion of I.z;  / as jzj ! 1 is given by I.z; /  2e

zp.!i ;/

1 X

.i/

 .j C =2/

j D0

a2j

zj C=2

;

(10.45)

where !i D !1 for  < s and !i D !2 for  > s , and where the (first three) .i/ coefficients a2j are calculated with respect to the dominant saddle point !i using (10.9)–(10.11). The discontinuous nature at  D s of this asymptotic approximation of I.z; / as a function of  for fixed z is obvious. In addition, at  D s Olver’s saddle point method cannot be applied to obtain an asymptotic expansion of the integral I.z; / because condition 5 of Theorem 3 is not satisfied. This discontinuity can be avoided and an asymptotic expansion at  D s can be obtained, however, by applying Olver’s method to each component contour integral Ii .z; / for i D 1; 2 instead of just applying it to the full contour integral I.z;  /. The asymptotic expansion of each component contour integral Ii .z;  / as jzj ! 1 that is uniformly valid with respect to  for all  2 R is given by the right-hand side of (10.45). Substitution of each expansion into (10.43) then yields the desired uniform asymptotic expansion, as expressed by the following corollary [4, 7] to Theorem 2. Corollary 1. In the contour integral Z I.z;  / D

q.!/e zp.!;/ d!

(10.46)

P ./

for  2 R, let !1 . / and !2 ./ denote two isolated first-order saddle points of p.!;  / such that s , and
8 1 <M X :

.1/

 .j C =2/

j D0

C2e zp.!2 ;/

8 1
j D0

a2j

zj C=2

9  M C=2 = CO z ; .2/

 .j C =2/

a2j

zj C=2

9 =  C O zN C=2 ; ; 

(10.47)

10.3 Uniform Expansion for Two First-Order Saddle Points

113

as jzj ! 1 uniformly with respect to  for all  2 R, where M and N are arbitrary positive integers. For sufficiently large values of jzj and for fixed  2 R with  ¤ s , the second term in (10.46) is asymptotically negligible in comparison to the first term when  < s , while the first term is asymptotically negligible in comparison to the second term when  > s . As a result, the asymptotic expansion given in (10.47) is equivalent to that given in (10.45) under either of these two conditions. As the quantity j  s j tends to zero, however, jzj must be increased without bound in order for the expansion given in (10.45) to provide a good (i.e., accurate) asymptotic approximation of I.z; / with a finite fixed number of terms N . The uniform asymptotic expansion given in (10.47) does not suffer from this difficulty. For sufficiently large fixed values of jzj, (10.47) can be used with fixed values of M and N to obtain an asymptotic approximation of I.z;  / that is uniformly valid for all  2 R. The result is a continuous function of  for all  2 R. The difficulty with the nonuniform expansion given in (10.45) arises from the error term associated with that asymptotic expansion truncated after N terms [see (10.12) and (10.13)]. The magnitude of this error term satisfies the inequality jRN j  Ajzj.N C=2/ C A0 e
(10.48)

where !i denotes the dominant saddle point and !j denotes the other saddle point. Since !i gives the dominant contribution, the second term on the right in (10.48) is negligible in comparison to the first for sufficiently large ˚jzj, so that  jRN j D O z.N C=2/ . However, the closer  is to s , the closer < p.!j ;  / is to < fp.!i ; /g and the larger jzj must be made in order for the second term to be negligible in comparison to the first. By comparison, the error term in the uniform asymptotic expansion given in (10.47) does not have this additional second term because the contour Pi . / for the integral Ii .z;  / does not pass through the saddle point !j . /.

10.3.2 The Uniform Asymptotic Expansion for Two Neighboring First-Order Saddle Points Consider again a contour integral I.z;  / of the form given in (10.1) taken over a path of integration P 0 which extends from j!j D 1 through the finite complex !plane and back to j!j D 1 without forming a closed contour. Let the complex phase function p.!;  / be a continuous function of a real parameter  that varies over a domain R. Furthermore, let !1 ./ and !2 . / denote two first-order saddle points of p.!;  / which coalesce into a single saddle point !s of second order when  D s 2 R. Let the original contour of integration P 0 be deformable to a path P . / that, for all  2 R, passes either through one of the saddle points or through both of them. When the deformed contour P ./ passes through just one of the saddle

114

10 Asymptotic Methods of Analysis P'

P'

P

P

P'

P

1

1 2

s

2

a

b s

c s

s

Fig. 10.5 Interaction of two neighboring first-order saddle points !1 . / and !2 . / that coalesce into a single second-order saddle point !s when  D s . The shaded area indicates the region of the complex !-plane wherein the inequality
points, it is assumed to be an Olver-type path with respect to that saddle point. When it passes through both saddle points, it may be expressed as P . / D P1 ./ C P2 ./, where Pi . / is an Olver-type path with respect to the saddle point !i . / for i D 1; 2. Furthermore, it is assumed that, for all  2 R, the contour P . / changes continuously in the complex !-plane as  varies continuously over the domain R. As an illustration, the situation that is encountered in the asymptotic description of dispersive pulse propagation in a Lorentz model dielectric is depicted in Fig. 10.5. In this case, the path P 0 is not deformable to an Olver-type path with respect to the saddle point !2 . / when  < s and the deformed contour P ./ passes through the saddle point !1 . / only. At  D s the deformed contour P .s / passes through the second-order saddle point !1 .s / D !2 .s / !s , and for  > s , the deformed contour P . / passes through both saddle points !1 . / and !2 . /. Although Olver’s saddle point method can be applied to obtain an asymptotic approximation of the integral that is valid for sufficiently large jzj for each value of  2 R, the result is a discontinuous function of  at  D s . The source of the discontinuity is the discontinuous change in the parameter  (which describes the order of the saddle points) from  D 2 when  ¤ s to  D 3 when  D s . The resulting asymptotic approximation obtained for  ¤ s remains useful as  approaches the critical value s only if jzj increases without bound as j  s j becomes arbitrarily small. To obtain an asymptotic approximation of the integral I.z;  / for large fixed jzj that is a continuous function of  for all  2 R, it is necessary to apply a uniform asymptotic expansion technique. The key is to introduce a transformation of variable that accounts for the change in saddle point order. In this case the change in order is from  D 2 to  D 3 and the appropriate change of variable is of the form [cf. (F.76)] 1 (10.49) p.!;  / D ˛0 ./ C ˛1 ./v  v3 : 3

10.3 Uniform Expansion for Two First-Order Saddle Points

115

The required result is stated most simply if the asymptotic parameter z is taken to be real and positive and if the deformed contour of integration P . /, or Pi . / for i D 1; 2 when P . / D P1 ./ C P2 ./, is taken along the path of steepest descent with respect to the relevant saddle point !i ./. Extensions of the result to include other appropriate paths P ./ as well as complex z are discussed following the theorem. For simplicity, only the dominant term in the asymptotic expansion is considered here. The result, due to Chester, Friedman, and Ursell [8, 9] in 1957, as later updated by Felsen and Marcuvitz [9] in 1973, can be stated as follows. Theorem 4. (Chester, Friedman, and Ursell) In the integrand of the contour integral Z q.!/e zp.!;/ d!; (10.50) I.z; / D P ./

let the functions p.!;  / and q.!/ both be holomorphic in a domain D containing the two first-order saddle points !1 ./ and !2 ./ of p.!;  /, both of which vary in position in the complex !-plane as the parameter  varies over a domain R in such a way that as  approaches some critical value s 2 R, the two saddle points coalesce into a single saddle point of second order; specifically, p 0 .!1 ;  / D p 0 .!2 ; / D 0 with p 00 .!1 ; / ¤ 0 and p 00 .!2 ;  / ¤ 0 for  ¤ s , while at  D s , !1 .s / D !2 .s / !s with p 0 .!s ; s / D p 00 .!s ; s / D 0 and p 000 .!s ; s / ¤ 0. The path P . /, or Pi . / for i D 1; 2 when P ./ D P1 ./ C P2 . /, is the path of steepest descent with respect to the relevant saddle point, and the parameter z is real and positive. Both p.!;  / and P ./ are taken to be continuous functions of  for all  2 R. Then the asymptotic behavior of the integral I.z;  / that is uniformly valid for all  2 R is given by (

I.z; / D e

   q.!1 /h1 ./ C q.!2 /h2 ./ 2 i  2=3 1 C O.z / C ˛1 ./z z1=3 2 " #)  q.!1 /h1 ./  q.!2 /h2 . / 2 i 0  2=3 1 C O.z / C 2=3 C ˛1 ./z 1=2 z 2˛ . /

˛0 ./z

1

(10.51) as z ! 1, where the function C. / is defined by the contour integral C. /

1 2 i

Z

3 =3

e vv

dv

(10.52)

L

and C 0 . / denotes its first derivative. The path of integration L appearing in (10.52) is the path that is mapped into the complex v-plane as ! traverses the contour P ./ by the root of the cubic equation given in (10.49) that satisfies the relation ˇ d v ˇˇ 1 D ˇ d! !D!s hs .s /

(10.53)

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10 Asymptotic Methods of Analysis

when  D s . The coefficients appearing here are defined as  1

p.!1 ;  / C p.!2 ;  / ; 2

 1=3 3

1=2 p.!1 ;  /  p.!2 ;  / ˛1 . / ; 4 !1=2 1=2 j 2˛1 ./ hj . / .1/ 00 I j D 1; 2 ; p .!j ;  / ˛0 . /

(10.54) (10.55) (10.56)

for  ¤ s . At the critical value s of  when the two first-order saddle points coalesce into a single second-order saddle point, the coefficients in the uniform expansion take on the limiting values 

2 lim hj . / D  000 !s p .!s / lim

!s

lim

1=3 hs .s /I j D 1; 2 ;

q.!1 /h1 . / C q.!2 /h2 ./ D q.!s /hs .s /; 2 q.!1 /h1 . /  q.!2 /h2 ./ 1=2

!s

2˛1 ./

D h2s .s /q 0 .!s /;

(10.57) (10.58) (10.59)

and where ˛0 .s / D p.!s ; s /. The cube root in (10.57) is made single-valued by the requirement that arg fhs .s /g D ˛N s ;

(10.60)

where ˛N s is the angle of slope of the path P .s / as it leaves the second-order saddle point at ! D !s when  D s . The cube root in (10.55) is made single-valued by the requirement that 1=2

lim

!s

˛1 ./ !1 ./  !2 ./

! D

1 : 2hs .s /

(10.61)

Finally, the square roots in (10.56) are made single-valued by the requirement that lim hj ./ D hs .s /I j D 1; 2 ;

!s

(10.62)

with the argument of hs .s / specified by (10.60). The requirement stated in (10.60) on the argument of hs .s / is the same as that obtained when the more general condition used in Olver’s theorem [see (10.7)] is applied to the case when z is real and positive and P . / is taken along the path of steepest descent through the second-order saddle point when  D s , in which case  D 3. Theorem 4 can then be extended to complex z and arbitrary Olver-type

10.3 Uniform Expansion for Two First-Order Saddle Points

117

paths with respect to the saddle points by applying (10.7) instead of (10.60) here. The theorem is sufficient as stated, however, for the type of problems encountered in the asymptotic description of dispersive pulse propagation that is treated in this volume. 1=2 Consider now the argument of the coefficient ˛1 ./ that is defined in (10.55). If the saddle point !2 encircles the saddle point !1 once as  varies over R, then 3=2 1=2 the argument of ˛1 . / varies over a range of 6 so that the argument of ˛1 . / varies over a range of 2. Hence, the cube root in (10.55) is not confined to a 3=2 single branch of the cube root of ˛1 ./ as would be obtained by using a branch 3=2 cut to restrict the argument of ˛1 ./ to a range that is less than 2. To determine 1=2 the argument of ˛1 . / as implied by (10.61), it is useful to apply the following geometrical construction. Let ˛N 12 be the angle of slope of the vector from !2 to !1 in the complex !-plane. Then, according to (10.60) and (10.61), n o 1=2 lim arg ˛1 ./ D ˛N 12  ˛N s C 2 n;

(10.63)

!s

where n is an arbitrary integer. Hence, as !1 approaches !2 along a straight line, 1=2 the argument of ˛1 . / approaches 2 n plus the angle that the line makes with the vector tangent to the path of steepest descent as it leaves the second-order saddle point !s at  D s . If desired, the integer n can be chosen so that the argument of 1=2 ˛1 . / lies within the principle range .; / for all  2 R. As found later, however, it is most convenient to choose n D 0. With this choice in the example depicted in 1=2 Fig. 10.5, the argument of ˛1 ./ is approximately equal to =3 for small, positive .s   / and is approximately equal to =6 for small, positive .  s /. Consider now the argument of the coefficient hj ./, j D 1; 2 as specified in Theorem 4. Since [9] lim

!s



p 00 .!1 ; / !1 . /  !2 . /

D lim

!s

p 00 .!2 ;  / !1 ./  !2 ./

1 D  p 000 .!s ; s /; 2 (10.64)

it then follows from (10.57) and (10.60) that lim

!s



!1 . /  !2 ./ p 00 .!1 ; /

D lim

!s

!1 . /  !2 . / p 00 .!2 ;  /

D 3˛N s :

(10.65)

Application of (10.56) and (10.63) with the above result then gives n o lim arg h2j ./ D 2˛N s C 2 nI

!s

j D 1; 2 ;

(10.66)

where n is the same integer appearing in (10.63). It is thus seen to be most convenient to choose n D 0 because it then follows from the requirement that

118

10 Asymptotic Methods of Analysis

lim!s hj . / D hs .s /, j D 1; 2 given in (10.57), together with (10.60) and (10.66), that the argument of the coefficient hj ./ is given by n o ˚ 1 arg hj . / D arg h2j ./ I 2

j D 1; 2 :

(10.67)

Application of the theorem is completed by the determination of the contour L onto which the path P . / is mapped under the cubic coordinate transformation given in (10.49). This transformation maps the path P ./ into three contours in the complex v-plane. In particular, it follows from (10.60) and (10.64) that the path of integration L appearing in (10.52) is the one contour of the three that satisfies the relation ˇ d v ˇˇ D ˛N s (10.68) d! ˇ !D!s

when  D s . Since the integrand in (10.52) is an entire function of complex v, the only features of the path L that effect the value of C. / are the endpoints of the path. For the type of problem considered in this volume, the endpoints are at infinity since jp.!;  /j ! 1 as j!j ! 1 in each direction along P . /. Consequently, it follows from (10.49) that the arguments of the endpoints of L can lie only in the following regions: Region 1: Region 2: Region 3:

  < argfvg < ; 6 6  5 < argfvg < ; 2 6  5 < argfvg <  ;  6 2



(10.69) (10.70) (10.71)

as illustrated in Fig. 10.6. Any contour that originates in Region i and terminates in Region j is labeled Lij . Three such contours are depicted in Fig. 10.6. If L is taken to be an Lij contour, then the value of C. / is determined completely by specifying i and j independent of any other details of the path Lij . In particular, it is found that [3, 9, 21] 8 9 Ai . /I for i D 3; j D 2 > ˆ ˆ > < i2=3 = Ai . e i2=3 /I for i D 2; j D 1 e ; (10.72) C. / D e i=3 Ai . e i2=3 /I for i D 3; j D 1 > ˆ ˆ > : ; 0I for i D j where Ai . / denotes the Airy function [22] Ai . / D

31=3 

Z 0

1

  cos t 3 C 31=3 t dt;

(10.73)

10.3 Uniform Expansion for Two First-Order Saddle Points

119

ℑ{v}

Region 2 32

21

ℜ{v}

31

Region 1

Region 3 Fig. 10.6 Possible contours of integration Lij in the complex v-plane onto which the path P . / may be mapped under the cubic transformation given in (10.49) 1

Airy Function Aι(ζ) & Its First Derivative A'ι(ζ)

A'ι(ζ) Aι(ζ)

0

−1 −10

−8

−6

−4

−2

0

2

4

6

8

10

ζ Fig. 10.7 Airy function Ai . / and its first derivative A0i . / for real

which is historically referred to as Airy’s ‘celebrated rainbow integral’ as it was introduced by Airy to describe the observed intensity distribution in the rainbow [23]. The behavior of the Airy function Ai . / and its first derivative A0i . / is illustrated in Fig. 10.7.

120

10 Asymptotic Methods of Analysis

If the transformation given in (10.49), taken together with the requirement given in (10.53), maps the contour P ./ into an Lij path for one value of  2 R, then that transformation maps P . / into that same Lij path (that is, the indices i and j are fixed) for all  2 R. Hence, in order to determine the particular values of the indices i and j for all  2 R, it suffices to determine the arguments of the endpoints of L for just one value of  2 R. For that purpose, it is most convenient to choose the value  D s since P .s / is mapped into two straight lines, one extending from infinity in Region i to the origin and the other extending from the origin to infinity in Region j . The values of i and j can then be determined by examining the slopes of the path L at the origin when  D s , where [from (10.68)] argf vg D ˛N s C argf !g;

(10.74)

at v D 0 when  D s . Here ! denotes the change in ! along the contour P .s / taken in the direction leading away from !s [i.e., ! D !  !s with ! on P .s /] and v denotes the corresponding change in v along the path L taken in the direction leading away from v D 0. This relationship is also approximately valid for values of  ¤ s such that the quantity j  s j is sufficiently small. As an illustration of the determination of the path L, consider the situation illustrated in Fig. 10.5. From part (b) of the figure, the angle of slope of the path P .s / as it leaves the second-order saddle point at !.s / D !s is seen to be ˛N s D =6. If the change ! appearing in (10.74) is taken to lie along the portion of the path P .s / that approaches the saddle point !s from the left in Fig. 10.5b, then it is seen that argf !g D 5=6. Equation (10.74) then states that the argument of v that lies along the corresponding portion of the transformed contour is given by argf vg D 2=3, showing that the transformed contour originates in Region 2. If the change ! is taken to lie along the portion of the path P .s / that leaves the saddle point !s toward the right in Fig. 10.5b, then it is seen that argf !g D =6. Equation (10.74) then states that the argument of v that lies along the corresponding portion of the transformed contour is given by argf vg D 0, showing that the transformed contour terminates in Region 1. Consequently, the coordinate transformation given in (10.49), taken together with the condition stated in (10.68), transforms the contour P ./ into an L21 path so that the function C. / appearing in the uniform asymptotic approximation given in (10.51) is given by the second form listed in (10.72), viz. C. / D e i2=3 Ai . e i2=3 /. As an aside, it is of peripheral interest to examine the behavior of the contour P . / depicted in Fig. 10.5 when it is transformed under the cubic change of variable defined in (10.49). Upon application of this cubic transformation, the two saddle 1=2 points !1 . / and !2 . / map into the pair of points v1;2 ˙˛1 ./ for all  2 R. The following sequence of events is then obtained.  For values of  such that .s  / > 0 is sufficiently small, argf !g D  along

the contour P . / approaching the saddle point !1 in Fig. 10.5a so that, since (10.74) remains approximately valid for  < s , argf vg Š 5=6 along the 1=2 transformed contour approaching the transformed saddle point v1 D ˛1 ./, and argf !g D 0 along the contour P ./ leaving the saddle point !1 so that

10.3 Uniform Expansion for Two First-Order Saddle Points

121

ℑ{v} Tangent line to the path Pv at the saddle point v1 =

x

Pv

ℜ{v}

x

a ℑ{v}

s

The two first-order saddle points v1 & v2 have coalesced into a single second-order saddle point located at the origin.

Pv

ℜ{v}

b ℑ{v}

s

Tangent line to the path Pv at the saddle point v1 = –

x

Pv x

c

ℜ{v} Tangent line to the path Pv at the saddle point v1 =

s

Fig. 10.8 Depiction of the continuous evolution with  of the transformed contour of integration Pv . /. Notice that this is the image of the contour of integration P . / illustrated in Fig. 10.5 for the same values of 

argf vg Š =6 along the transformed contour leaving the saddle point v1 D 1=2 ˛1 . /, as depicted in Fig. 10.8a.  For  D s , argf !g D 5=6 along the contour P .s / approaching the saddle point !s in Fig. 10.5b so that argf vg D 2=3 along the transformed contour approaching the transformed saddle point v D 0, and argf !g D =6 along the contour P .s / leaving !s so that argf vg D 0 along the transformed contour leaving v D 0, as depicted in Fig. 10.8b.  For values of  such that .  s / > 0 is sufficiently small, argf !g D 3=4 along the contour P ./ approaching the saddle point !2 in Fig. 10.5c so that, since (10.74) remains approximately valid for  > s , argf vg Š 7=12

122

10 Asymptotic Methods of Analysis

along the transformed contour approaching the transformed saddle point v2 D 1=2 ˛1 . /, argf !g D =4 along the contour P . / leaving the saddle point !2 so that argf vg Š 5=12 along the transformed contour leaving the saddle 1=2 point v2 D ˛1 . /, argf !g D 3=4 along the contour P . / approaching the saddle point !1 in Fig. 10.5c so that argf vg Š 11=12 along the 1=2 transformed contour approaching the transformed saddle point v1 D ˛1 ./, and argf !g D =4 along the contour P ./ leaving the saddle point !1 so that argf vg Š =12 along the transformed contour leaving the saddle point 1=2 vD ˛1 . /, as depicted in Fig. 10.8c. This analytical description then indicates that the approaching and departing slopes 1=2 of the contour Pv . / at the single saddle point v1 D ˛1 . / when  < s and at the 1=2 pair of saddle points v1;2 D ˙˛1 ./ when  > s are discontinuous at  D s . This discontinuous behavior, however, is just an anomally of the coalescence of the two first-order saddle points into a single second-order saddle point when  D s . Indeed, as  approaches closer to s , the path of steepest descent Pv ./ exterior to a small neighborhood of the saddle point passes through deviates more sharply from the tangent line to the appropriate slope at the relevant saddle point in such a way that at  D s , the path Pv ./ has evolved in a continuous manner into the path Pv .s /, as depicted in Fig. 10.8. Consequently, the transformed contour of integration Pv . / and its preimage P ./ evolve in a continuous manner with  for all  2 R, as required in Theorem 4. For sufficiently large but fixed finite values of z, (10.51) provides an asymptotic approximation for the integral I.z;  / that is a continuous function of  for all  2 R, a feature that does not result when Olver’s saddle point method is applied directly. In return for this continuous behavior, however, (10.51) is more complicated than the nonuniform results of Olver’s method due to the presence of the Airy function and its first derivative. To obtain the simpler results that follow from Olver’s method, the uniform asymptotic approximation given in (10.51) is now reduced for the three cases of the example depicted in Fig. 10.5. Case I:  < s and j˛. /jz2=3  1. For values of  < s , the angle of slope of the vector directed from !2 to !1 is given by ˛N 12 D =2, as obtained from Fig. 10.5a. Then, according to (10.63) with n D 0 1=2 and ˛N s D =6, argf˛1 . /g Š =3 when  Š s , and, according to (10.57) and (10.60), argfhi . /g Š =6 when  Š s with i D 1; 2. Although these results for 1=2 argf˛1 . /g and argfhi . /g have been obtained only in the limit as  approaches s from below, they serve to fix the proper phase of these two functions for all  < s such that  2 R. For the situation that is encountered in the asymptotic description of dispersive pulse propagation, treated in Chap. 13, these limiting values are found [7] to be valid for all  < s . In this case, the argument of the Airy function in the expression [see (10.72)] C. / D e i2=3 Ai . e i2=3 / appearing in the uniform expansion given in (10.51) is given by ˛1 . /z2=3 e i2=3 D j˛1 ./jz2=3 ;

10.3 Uniform Expansion for Two First-Order Saddle Points

123

which is real and positive for z > 0. For j˛1 ./jz2=3  1, the large argument asymptotic expansion of the Airy function and its first derivative with a positive real argument may be employed in the uniform asymptotic expansion given in (10.51), where (see Problem 10.4)   3=2   1 e .2=3/j˛1 ./j z ; 1 C O Ai j˛1 . /jz2=3 D p j˛1 ./j3=2 z 2 j˛1 ./j1=4 z1=6 A0i

    1 j˛1 . /j1=4 z1=6 .2=3/j˛1 ./j3=2 z 2=3 ; 1CO j˛1 . /jz D e p j˛1 ./j3=2 z 2 

as j˛1 . /jz2=3 ! 1. Substitution of these two asymptotic expansions into (10.51) then gives ( I.z; / D e zp.!1 ;/

 q.!1 / 

2 zp 00 .!1 ;  /

1=2

h

C O .˛1 . /z/

3=2

) i

;

(10.75)

as j˛1 . /jz2=3 ! 1. This result is the same as that obtained when Olver’s saddle point method is directly applied to the case of a single isolated saddle point at ! D !1 . /, except that the dependence of the remainder term on the separation between the two saddle points !1 . / and !2 ./ is displayed explicitly through the factor ˛1 . / appearing in (10.75). Case II:  D s . At the critical value  D s when the two first-order saddle points coalesce into a single second-order saddle point !1 .s / D !2 .s / D !s and ˛1 .s / identically vanishes, the limiting expressions given in (10.57)–(10.59) must be employed in the unifoirm asymptotic expansion given in (10.51). In particular, the Airy function and its first derivative at this critical value are given by 31=6  .1=3/; 2 31=6 A0i .0/ D   .2=3/: 2 Ai .0/ D

Substitution of these results into (10.51) then yields ) ( 1=3    2i  .1=3/ e i=6  000 q.!s / C O z4=3 ; I.z; s / D e zp.!s ;s / 31=6 zp .!s ; s / (10.76) as z ! 1. The same result is obtained when Olver’s method is directly applied to the case of a single isolated second-order saddle point at ! D !s .

124

10 Asymptotic Methods of Analysis

Case III:  > s and j˛. /jz2=3  1. For values of  > s , the angle of slope of the vector directed from !2 to !1 is given by ˛N 12 D 0, as obtained from Fig. 10.5c. Then, according to (10.63) with 1=2 n D 0 and ˛N s D =6, argf˛1 ./g Š =6 when  Š s , and, according to (10.57) and (10.60), argfhi . /g Š =6 when  Š s with i D 1; 2. Although 1=2 these results for argf˛1 . /g and argfhi ./g have been obtained only in the limit as  approaches s from above, they serve to fix the proper phase of these two functions for all  > s such that  2 R. For the situation that is encountered in the asymptotic description of dispersive pulse propagation, treated in Chap. 13, these limiting values are found [7] to be valid for all  > s . In this case, the argument of the Airy function appearing in the uniform expansion given in (10.51) is given by ˛1 . /z2=3 e i2=3 D j˛1 . /jz2=3 ; which is real and negative for z > 0. For j˛1 ./jz2=3  1, the large argument asymptotic expansion of the Airy function and its first derivative with a negative real argument may be employed in the uniform asymptotic expansion given in (10.51), where (see Problem 10.4)       sin .2=3/j˛1 ./j3=2 z C =4 1 2=3 ; 1 C O D Ai j˛1 . /jz p j˛1 ./j3=2 z j˛1 ./j1=4 z1=6     j˛1 ./j1=4 z1=6 A0i j˛1 . /jz2=3 D  cos .2=3/j˛1 ./j3=2 z C =4 p    1 ;  1CO j˛1 . /j3=2 z as j˛1 . /jz2=3 ! 1. Substitution of these two asymptotic expansions into (10.51) then gives (

) 1=2 i h 2 3=2 I.z; / D e C O .˛1 . /z/ q.!1 /  00 zp .!1 ;  / ( ) 1=2  i h 2 3=2 zp.!2 ;/ C O .˛1 . /z/ q.!2 /  00 Ce ; zp .!2 ;  / zp.!1 ;/



(10.77) as j˛1 . /jz2=3 ! 1. This expression is the same result as that obtained when Olver’s saddle point method is directly applied to the case of a pair of isolated first-order saddle points at ! D !1 . / and ! D !2 ./ that are of equal dominance, except that the dependence of the remainder term on the separation between the two saddle points is displayed explicitly through the factor ˛1 . / appearing in (10.77).

10.4 Uniform Expansion for a Saddle Point and Nearby Singularity

125

10.3.3 The Transitional Asymptotic Approximation for Two Neighboring First-Order Saddle Points Although the uniform asymptotic expansion for two neighboring first-order saddle points given in Theorem 4 provides an asymptotic approximation of the integral I.z; / that is uniformly valid in  over a domain R that contains the critical point s when the two saddle points coalesce into a single second-order saddle point at ! D !s , the coefficients K1 . / q.!1 /h1 ./ C q.!2 /h2 . /;

(10.78)

q.!1 /h1 ./  q.!2 /h2 . / ; j˛1 ./j1=2

(10.79)

K2 . /

appearing in that expansion [see (10.51)] may become indeterminate (due, for example, to a lack of numerical precision in the computation) when  approaches s . To avoid this situation, a transitional asymptotic approximation [24] may be used to bridge the transitional region where these coefficients become numerically unstable. In particular, the transitional asymptotic approximation of the integral I.z;  / considered in Theorem 4 is given by [10] ˇ ˇ 2 IT .z; /  2 ˇˇ 000 zp .!

s

ˇ1=3 ˇ ˇ q.!s /Ai ./e zp.!s ;/ ; /ˇ

(10.80)

as z ! 1 with s  ı1 <  < s C ı2 , where  z2=3

p 0 .!s ;  /sgn.p 000 .!s ;  // : j2p 000 .!s ;  /j1=3

(10.81)

The incremental  -values ıj , j D 1; 2 are chosen in such a way that this expression has a continuous match with the uniform asymptotic expansion on either side of the critical value s .

10.4 Uniform Asymptotic Expansion for a First-Order Saddle Point and a Simple Pole Singularity Consider again a contour integral I.z;  / of the form given in (10.1) taken over a path of integration P 0 which extends from j!j D 1 through the finite complex !-plane and back to j!j D 1 without forming a closed contour. Let the complex phase function p.!;  / be a continuous function of a real parameter  that varies over a domain R. In addition, let the complex amplitude function q.!/ and the contour P 0 both be independent of . Let P ./ be an Olver-type path with respect to a saddle point !sp . / of p.!;  / and let the original contour P 0 be deformable to

126

10 Asymptotic Methods of Analysis

the path P . / which evolves in a continuous manner in the complex !-plane as  varies continuously over R. If no pole singularities of the amplitude function q.!/ are crossed when P 0 is deformed to P ./, then the integral I.z;  / is equal to the integral Isp .z; / that is defined by Z

q.!/e zp.!;/ d!:

Isp .z;  / D

(10.82)

P ./

Moreover, if any pole singularities of q.!/ are crossed when P 0 is deformed to P . /, then the value of the original integral I.z;  / can be obtained from the value of Isp .z; / by adding the appropriate residue contributions of the pole singularities according to Cauchy’s residue theorem [25]. In particular, suppose that for all values of  2 R less than some critical value s 2 R, the path P 0 is deformable to P ./ without crossing any poles of q.!/, and that for all values of  2 R such that  > s , a single, simple pole singularity of q.!/ located at ! D !c is crossed when P 0 is deformed to P . / such that the pole is encircled in the clockwise sense, as depicted in the sequence of diagrams in Fig. 10.9 for the three cases when (a)  < s , (b) s <  < c , and (c)  > s . In each part of the figure, the shaded area indicates the region of the complex !-plane ˚ wherein the inequality < fp.!;  /g < < p.!sp ;  / is satisfied, that is, the region within which the deformed contour P ./ must lie in order for it to be an Olver-type path with respect to the saddle point !sp ./. The integral I.z;  / is then given in terms of the integral Isp .z; / by 8 <

9 Isp .z;  /I for  < s = I.z; / D Isp .z;  /   i e zp.!c ;s / I for  D s ; : ; Isp .z;  /  2 i e zp.!c ;/ I for  > s

P'

(10.83)

P'

P

P'

P xc

x c

SP

P c x

SP

a

SP

b s

s

<

c c

c

Fig. 10.9 Interaction of a first-order saddle point with a fixed simple pole singularity at !c . The dotted line in each part of the figure indicates the tangent line to the path of steepest descent through the saddle point at ! D !SP . /

10.4 Uniform Expansion for a Saddle Point and Nearby Singularity

where

  D lim .!  !c /q.!/ !!!c

127

(10.84)

is the residue of the simple pole singularity at ! D !c . If Olver’s saddle point method is applied to obtain an asymptotic approximation of the “saddle point integral” Isp .z;  /, the result is a continuous function of  for fixed, finite values of z of sufficiently large magnitude. In particular, this result is continuous at  D s . However, the resulting asymptotic approximation for the integral I.z; /, obtained from (10.83), is a discontinuous function of  at  D s . The discontinuity is of little or no consequence for fixed values of jzj larger than some positive constant Z, however, because the saddle point integral Isp .z;  / varies exponentially as e zp.!sp ;/ which dominates the exponential behavior of the residue contribution appearing in the second and third parts of (10.83) so long as !c lies within the shaded region of Fig. 10.9 [i.e., provided that c > s , as depicted in part (c) of Fig. 10.9 and c . Hence, the asymptotic behavior of I.z; / changes abruptly as  crosses from the region  < c to the region  > c . For fixed, finite values of jzj > Z, however, the asymptotic approximation of I.z;  / obtained by substituting an asymptotic approximation of Isp .z;  / in (10.83) is a continuous function of  for all  > s as long as the residue contribution is retained for all  > s . An additional complication arises when the saddle point !sp . / approaches close to the pole singularity at ! D !c when  D s . As the distance j!sp .s /ˇ  !c j between these two ˇpoints becomes small, then so does the difference ˇp.!sp .s /; !s /  p.!c ; s /ˇ and Z accordingly increases. For values of z such that jzj < Z, the discontinuity displayed in (10.83) can then be significant. To avoid this discontinuous behavior when the saddle point is near the pole, it is necessary to apply a technique known as “subtraction of the pole.” This technique yields an asymptotic expansion of the integral Isp .z;  / as jzj ! 1 that is uniformly valid for all  2 R. The resulting asymptotic approximation of the integral I.z;  / is then a continuous function of  as  varies continuously in R for fixed, finite values of z. The asymptotic behavior of the saddle point integral Isp .z;  / can be stated most simply if the asymptotic parameter z is taken to be real and positive and if the

128

10 Asymptotic Methods of Analysis

Olver-type path P . / through the saddle point is taken as the path of steepest descent defined by the equation =fp.!;  /g D =fp.!sp ;  /g. The result for more general Olver-type paths is discussed following the theorem. For simplicity, only the dominant term in the asymptotic expansion is considered here. The result, originally developed by Felsen and Marcuvitz [26] in 1959 and later generalized by Bleistein [11, 12] in 1966 and 1967, can be stated [9] as follows. Theorem 5. (Felsen, Marcuvitz, and Bleistein) In the integral Z

q.!/e zp.!;/ d!;

Isp .z;  / D

(10.85)

P ./

let the contour of integration P ./ be the path of steepest descent through the first-order saddle point !sp . / of p.!;  / that is isolated from any other saddle points of the function p.!;  /, and let z be real and positive. It is further assumed that all of the conditions required in order for P . / to be an Olver-type path with respect to !sp . / are satisfied for all  2 R except that the function q.!/ exhibits a single first-order pole singularity at ! D !c with !c 2 D, where D denotes the domain of analyticity of p.!;  / in the complex !-plane. In addition, the complex phase function p.!;  / is assumed to be a continuous function of  for all  2 R, whereas both q.!/ and z are independent of  . Under these conditions, the asymptotic behavior of the saddle point integral Isp .z;  / is given by  Isp .z; / D q.!sp / 

2 zp 00 .!sp ;  /

1=2

e zp.!sp ;/



 p C ˙i erfc i . / z e zp.!c ;/ C CR1 e zp.!sp ;/ I

r

when =f . /g ¤ 0;

 e zp.!sp ;/ z . /



(10.86)

˚ where the ˚ upper sign choice is used when = . / > 0 and the lower sign choice when = ./ < 0, and  Isp .z; s / D q.!sp / 

2 00 zp .!sp ; s /

1=2

e zp.!sp ;s /

r    p  zp.!c ;s /  e zp.!sp ;s / C i erfc i .s / z e C z .s / i e zp.!c ;s / C R1 e zp.!sp ;s / I when =f .s /g D 0; .s / ¤ 0;

(10.87)

10.4 Uniform Expansion for a Saddle Point and Nearby Singularity

Isp .z; s / D

129

1=2 2 e zp.!sp ;s / zp 00 .!sp ; s /   p 000 .!sp ; s /   q.!sp /    00 !sp .s /  !c 6p .!sp ; s /

 

CR1 e zp.!sp ;s / I

when .s / D 0;

(10.88)

where R1 D O.z3=2 / as z ! 1 uniformly with respect to  for all  2 R, where  is the residue of the simple pole singularity at ! D !c , defined in (10.84), and

1=2 ./ p.!sp ./; /  p.!c ;  / :

(10.89)

1=2 The argument of the quantity  zp 00 .!sp ./; / is defined to be equal to arg.d!/!sp , where d! is a differential element of path length along the path of steepest descent through the saddle point !sp ./, and the argument of . / is defined such that  

   p 00 .!sp . /;  / 1=2 ./ D !c  !sp ./  : !c !!sp ./ 2 lim

(10.90)

p R1 2 Finally, the function erfc. / .2= / e  d  is the complementary error function. The asymptotic behavior of the saddle point integral Isp .z;  / is given by (10.86) with the upper sign choice when the contour P . / lies on one side of the pole (with respect to the original path P 0 ) such that =f . /g > 0, and with the lower sign choice when P . / lies on the other side of the pole (with respect to P 0 ) such that =f ./g < 0. When  D s the pole lies on the contour P .s /, by definition, then =f .s /g D 0 and the asymptotic behavior of the saddle point integral Isp .z; s / is given by (10.87) if .s / ¤ 0, and it is given by (10.88) if .s / D 0, in which case the saddle point coalesces with the pole. Since the order relation R1 D O.z3=2 / for the error term as z ! 1 is satisfied uniformly with respect to  for all  2 R, the apparent discontinuities in the asymptotic behavior of Isp .z;  / exhibited in (10.86)–(10.88) are real. In particular, when the path P . / passes from one side of the pole (again, with respect to the original path P 0 ) to the other, the discontinuous jump in Isp .z; / due to the change in sign of =f . /g in (10.86) is equal to 2 i e zp.!c ;s / . This discontinuity in Isp .z;  / exactly cancels the discontinuity in I.z; / introduced by the contribution of the simple pole singularity when Cauchy’s residue theorem is applied to deform the original contour P 0 to the path of steepest descent P . / through the saddle point, as exhibited in the set of relations given in (10.83). As a result, the asymptotic behavior of I.z;  / is a continuous function of  for all  2 R for fixed, finite values of z. If P . / is an Olver-type path other than the path of steepest descent through the saddle point at ! D !sp . /, then Theorem 5 remains valid provided that P . / is deformable to the path of steepest descent without crossing the pole singularity.

130

10 Asymptotic Methods of Analysis

If the pole is crossed when P . / is deformed to the steepest descent path through the saddle point, then the set of relations given in (10.86)–(10.88) are changed [27] by the addition or subtraction of the term 2 i e zp.!c ;/ . Since the change in the expression for the saddle point integral Isp .z;  / is equal but with opposite sign to the change introduced between I.z;  / and Isp .z;  / when Cauchy’s residue theorem is applied to change the contour of integration from the steepest descent path to the new Olver-type path P . /, the resulting asymptotic expression for I.z;  / remains unchanged. Hence, the uniform asymptotic approximation obtained for I.z;  / is independent of the particular Olver-type path chosen. Nevertheless, in order to apply Theorem 5 to obtain the uniform asymptotic approximation of I.z;  /, it is still necessary to determine the path of steepest descent relative to the position of the pole in order to determine whether or not the residue contribution due to the pole should be added to the right-hand side of the appropriate expression in (10.86)–(10.88). Consider now the determination of the proper argument of the complex quantity ./ defined in (10.89). If the pole encircles the saddle point once as  varies over R, the argument of 2 . / varies over a range of 4 so that the argument of . / varies over a range of 2. Hence, . / is not confined to a single branch of the square root of 2 . / as would be obtained by using a branch cut to restrict the argument of 2 . / to a range of less than 2. To determine the argument of . / that is implied by (10.90), it is useful to apply the following geometrical construction. Let ˛N c denote the angle of slope of the vector from the saddle point !sp. / to the pole !c in the complex !-plane. Equation (10.90) then yields lim

!c !!sp ./

n

 1=2 o C 2 n; arg ./ D ˛N c C arg  p 00 .!sp ./; /

(10.91)

where the limit is taken along the straight line with slope ˛N c and where n is an arbitrary integer. Since arg

n

 p 00 .!sp ./; /

1=2 o

D ˛N sd ;

(10.92)

as required by Theorem 5, where ˛N sd is the angle of slope of a vector tangent to the path of steepest descent at the saddle point, as defined in (10.2) with P taken as the steepest descent path and !1 D !sp ./, then (10.91) becomes lim

!c !!sp ./

 arg . / D ˛N c  ˛N sd C 2 n:

(10.93)

Hence, as the pole approaches the saddle point along a straight line, the argument of ./ approaches 2 n plus the angle that the line makes with the vector tangent to the steepest descent path at the saddle point !sp . /. The integer n can be chosen so that the argument of ı. / lies within the principal range .; 

for all  2 R. For example, in the situation depicted in Fig. 10.9, the limit of arg . / is a small negative angle in part (a) and a small positive angle in parts (b) and (c).

10.4 Uniform Expansion for a Saddle Point and Nearby Singularity

131

10.4.1 The Complementary Error Function To obtain a better understanding of the uniform asymptotic nature of the saddle point integral Isp .z; / that is given in (10.86)–(10.88) of Theorem 5, it is necessary to understand the analytic properties of the complementary error function erfc. / 2 for complex . Since e  is an entire function, both the error function erf. /, which p 2 is defined as the integral of the Gaussian distribution function g./ D .2= /e  from 0 to , viz., Z 2 2 erf. / p e  d ; (10.94)  0 and the complementary error function erfc. /, which is defined as the integral of the Gaussian distribution function from to 1, viz., 2 erfc. / p 

Z

1

2

e  d ;

(10.95)

are entire functions of complex , where erfc. / D 1  erf. /. Along the real axis, erf.1/ D 0, erf.0/ D 1=2, and erf.1/ D 1, whereas erfc.1/ D 2, erfc.0/ D 1, and erfc.1/ D 0, as illustrated in Fig. 10.10. The behavior of the complementary error function erfc. / in the complex -plane is more complicated, as illustrated in Fig. 10.11. Part (a) of the figure shows the real part and part (b) the imaginary part. Notice that the real part is even symmetric about the imaginary axis whereas the imaginary part is odd symmetric, that is < ferfc.  /g D < ferfc. /g and = ferfc.  /g D = ferfc. /g.

Error & Complementary Error Functions

2

erfc(ζ) = 1 - erf(ζ) 1.5

1

0.5

0

erf(ζ)

−0.5

−1 −3

−2

−1

0

1

2

3

ζ

Fig. 10.10 The error function erf. / and the complementary error function erfc. / for real

132

10 Asymptotic Methods of Analysis

ℜ{erfc( )}

a 20 10 0

−10 −20

2

1 ℑ{␨}

b

0

−1

−1

−2 −2

0

1

2

ℜ{␨}

ℑ{erfc( )}

20 10 0

−10 −20

2 1 ℑ{␨}

0

−1

−1

−2 −2

0

1

2

ℜ{␨}

Fig. 10.11 Real (a) and imaginary (b) parts of the complementary error function erfc. / for complex

Along the imaginary axis where D i 00 D e ˙i=2 with   0, the integral expression for the complementary error function becomes Z    2 2 erfc e ˙i=2 D 1 i p e t dt; (10.96)  0 which is related to Dawson’s integral FD ./ D e 

2

Z



2

e t dt:

(10.97)

0

  Consequently, erfc e ˙i=2 ! 1 i 1 as  ! 1, as indicated in Fig. 10.12.

10.4 Uniform Expansion for a Saddle Point and Nearby Singularity

133

20 15 10 5 0 −5 −10 −15 −20 1 1

0.5 0.5

0 ( )

0

−0.5

−0.5 −1

( )

−1

Fig. 10.12 Cosine C ./ and sine S ./ Fresnel integrals with real argument . Notice that C .1/ D S .1/ D 1=2, C .0/ D S .0/ D 0, and C .1/ D S .1/ D 1=2

Along the diagonal axes where D e ˙i=4 with real-valued , the complementary error function is given by h p  p i p   2= i S 2= ; erfc e ˙i=4 D 1  2e ˙i=4 C

(10.98)

where C./ and S./ are the cosine and sine Fresnel integrals, respectively, defined by C./

Z



  cos .=2/t 2 dt;

(10.99)

  sin .=2/t 2 dt:

(10.100)

0

S./

Z



0

The cosine and sine Fresnel integrals are plotted against each other in Fig. 10.12 with the real argument  plotted along the vertical axis. The projection of this curve onto the horizontal plane then yields the well-known Cornu spiral in  illustrated  Fig. 10.13. Since C.˙1/ D S.˙1/ D ˙1=2, it is seen that erfc e ˙i=4 ! 0   as  ! 1 and that erfc e ˙i=4 ! 2 as  ! 1. The asymptotic expansion of the complementary error function is given by [21] 1

X 1  .n C 1=2/ 2 erfc. / D p e  .1/n  .1=2/ 2nC1  nD0

(10.101)

134

10 Asymptotic Methods of Analysis 0.8 0.6 0.4 0.2

( )

0

−0.2 −0.4 −0.6 −0.8 −0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

( ) Fig. 10.13 The Cornu spiral, plotting the cosine Fresnel integral C ./ vs. the sine Fresnel integral S ./ at the same value of the real argument  which varies from negative to positive infinity. Notice that C .1/ D S .1/ D 1=2, C .0/ D S .0/ D 0, and C .1/ D S .1/ D 1=2

as j j ! 1 uniformly in j arg. /j < =2. With use of the identity erfc. / C erfc. / D 2;

(10.102)

the asymptotic behavior of erfc. / in the half-plane
X  .n C 1=2/ 1 2 .1/n erfc. / D 1 C p e   .1=2/ 2nC1  nD0

(10.103)

as j j ! 1 with arg. / D ˙=2. The asymptotic expansions given in (10.101) and (10.103) are equivalent for arg. / D ˙=2 because the first term on the righthand side of (10.103) is asymptotically negligible in comparison to the second term as j j ! 1. Both equations give the correct asymptotic behavior of erfc. / for arg. / D ˙ =2. The expression given in (10.103) has the advantage, however, in that it gives the correct asymptotic behavior of the real and imaginary parts of erfc. / separately, as can be seen by comparison with (10.96) and noting that the second term on the right-hand side of (10.103) is purely imaginary. Finally, it is

10.4 Uniform Expansion for a Saddle Point and Nearby Singularity

135

seen from (10.101)–(10.103) that the asymptotic behavior of the complementary error function erfc. / as j j ! 1 is exponentially attenuated in the angular sectors =4 < arg. / < =4 and 3=4 < arg. / < 5=4, whereas it is exponentially amplified for =4 < arg. / < 3=4 and 3=4 < arg. / < =4.

10.4.2 Asymptotic Behavior for a Single Interacting Saddle Point When the saddle point !sp . / is sufficiently distant from the pole at ! D !c such p that j ./j z  1, then the dominant term in the corresponding asymptotic expansion of the complementary error function with large argument can be substituted into either (10.86) or (10.87). If the integrer n appearing in (10.93) is chosen so that arg. . // lies within the principal range .;  , and if the pole does not lie on the path of steepest descent through the saddle point, then the phase of the argument of the complementary phase function appearing in (10.86) satisfies the inequality ˇ  ˇ ˇarg i . /pz ˇ <  I 2

˚ = . / ¤ 0:

(10.104)

Hence, the asymptotic expansion of the complementary error function given in (10.101) is applicable in (10.86). From (10.101), the dominant term in the asymptotic behavior of the complementary error function appearing in (10.86) is given by  p erfc i . / z D

n

p 3 o 1 p e zŒp.!sp ;/p.!c ;/ C O . / z i . / z

p as j ./j z ! 1 uniformly with respect to arg. .// with =2 < p arg i . / z < =2. Substitution of this expression into (10.86) then results in the asymptotic expression  Isp .z; / D q.!sp / 

2 zp 00 .!sp ;  /

1=2

e zp.!sp ;/ C O

n

p 3 o ; . / z (10.105)

p which is valid as j ./j z ! 1 uniformly with respect to arg. .// with 0 < arg . .// < . Notice that this result is the same as that obtained with a direct application of Olver’s saddle point method except that the dependence of the remainder term on the separation between the pole and the saddle point is displayed in (10.105) through the function . /. If the pole lies on the path of steepest descent (in which case  D s ) but remains far from the saddle point, then the phase of the argument of the complementary error function appearing in (10.87) is given by  p  arg i .s / z D  ; 2

(10.106)

136

10 Asymptotic Methods of Analysis

since =f .s /g D 0 with .s / ¤ 0. It is then seen that the asymptotic expansion of the complementary error function given in (10.103) is applicable in this case, so that  p erfc i .s / z D 1 

n

p 3 o 1 p e zŒp.!sp ;s /p.!c ;s / C O .s / z i .s / z

p as j .s /j z ! 1 with arg. .s // D 0. Substitution of this expression into (10.87) then results in the asymptotic expression 

2 Isp .z; s / D q.!sp /  00 zp .!sp ; s /

1=2

e zp.!sp ;s / C O

n

p 3 o ; .s / z (10.107)

p which is valid as j .s /j z ! 1 with arg. .s // D 0. This result is the same as that obtained with direct application of Olver’s saddle point method except that the dependence of the remainder term on the separation between the pole and the saddle point is displayed in (10.107) through .s /. The example depicted in Fig. 10.9 is now continued in order to illustrate the application of Theorem 5. Substitution of (10.86) and (10.87) into (10.83) yields the sequence of asymptotic expressions 1=2 2 I.z; / D q.!sp /  00 e zp.!sp ;/ zp .!sp ;  / r

 p  e zp.!sp ;/ C i erfc i . / z e zp.!c ;/ C z . / 

CR1 e zp.!sp ;/ I  < s ; (10.108) 1=2  2 e zp.!sp ;s / I.z; s / D q.!sp /  00 zp .!sp ; s / r

 p  e zp.!sp ;s / C i erfc i .s / z e zp.!c ;s / C z .s / 2i e zp.!c ;s / C R1 e zp.!sp ;s / I  D s ; 1=2  2 I.z; / D q.!sp /  00 e zp.!sp ;/ zp .!sp ;  / r

 p  zp.!c ;/  e zp.!sp ;/ C i erfc i . / z e C z . / 2i e zp.!c ;/ C R1 e zp.!sp ;/ I where R1 D O all  2 R.

 > s ;

(10.109)

(10.110)

n

p p 3 o as j . /j z ! 1 uniformly with respect to  for ./ z

10.4 Uniform Expansion for a Saddle Point and Nearby Singularity

137

 p At  D s , arg i .s / z D =2 for real z > 0. In that case, the complementary error function appearing in (10.109) is given by (10.96), so that 1=2 2 e zp.!sp ;s / zp 00 .!sp ; s / r

 p p  zp.!c ;s /  e zp.!sp ;s / C 2 FD .s / z e C z .s /

 I.z; s / D q.!sp / 

i e zp.!c ;s / C R1 e zp.!sp ;s / I

 D s ;

(10.111)

p 2 R 2 as j ./j z ! 1, where FD ./ D e  0 e t dt is Dawson’s integral, defined in (10.97). The numerically determined behavior of Dawson’s integral as a function of real   0 is illustrated in Fig. 10.14. The dashed curve in the figure depicts the behavior of the first two (dominant) terms in the asymptotic expansion of FD ./, given by N 1   1 X  .n C 1=2/ (10.112) C O .2N C1/ FD ./ D 2nC1 2 nD0  .1=2/ as  ! 1. Notice that this two term asymptotic approximation provides an accurate estimate of Dawson’s integral when  > 2.

0.6 FD( r) ~ (1/2){1/r + 1/(2 r3)}

Dawson's Integral FD( r)

0.5

0.4 FD(r) 0.3

0.2

0.1

0 0

1

2

3

4

5

6

7

8

9

10

Fig. 10.14 The functional dependence of Dawson’s integral FD ./ for real  0. The numerically determined behavior is represented by the solid curve and the behavior of the first two (dominant) terms in the asymptotic expansion of FD ./ as  ! 1 is represented by the dashed curve

138

10 Asymptotic Methods of Analysis

 p At  D c , arg i .c / z D =4 for real z > 0. In that case, the complementary error function appearing in (10.110) is given by (10.98), so that 1=2  2 I.z; c / D q.!sp /  00 e zp.!sp ;c / zp .!sp ; c / (  p i p h p C i 2 C 2z=j .c /j  S 2z=j .c /j e zp.!c ;c / ) r  e zp.!sp ;c / C  3i e zp.!c ;c / C R1 e zp.!sp ;c / I  D c ; z .c / (10.113) p as j ./j z ! 1, where C. / and S. / are the cosine and sine Fresnel integrals defined in (10.99) and (10.100). For computational purposes, these two functions may be written as     1 C f . / sin

2  g. / cos

2 ; 2 2 2     1

2  g. / sin

2 ; S. / D  f . / cos 2 2 2 C. / D

(10.114) (10.115)

where the functions f . / and g. / may be accurately computed using the rational approximations [22] 1 C 0:926 C ". /; 2 C 1:792 C 3:104 2 1 C ". /; g. / D 2 C 4:142 C 3:492 2 C 6:670 3

f . / D

(10.116) (10.117)

for all  0, where j". /j  2  103 . The Fresnel integrals graphed in Figs. 10.12 and 10.13 were computed using these rational approximations. The asymptotic expressions given in (10.111) and (10.113) describe the behavior of the integral I.z; / at the two critical values at  D s and  D c , respectively, given as a function of the real variable z in terms of well-known real-valued functions. When the saddle point is far enough away from the pole that (10.105) and (10.107) can be applied in (10.83), then (10.108)–(10.110) become  I.z; / D q.!sp / 

2 zp 00 .!sp ;  /

CR1 e zp.!sp ;/ I

1=2

e zp.!sp ;/

 < s ;

(10.118)

10.4 Uniform Expansion for a Saddle Point and Nearby Singularity

 I.z; s / D q.!sp / 

2 00 zp .!sp ; s /

1=2

139

e zp.!sp ;s /  i e zp.!c ;s /

CR1 e zp.!sp ;s / I  D s ; (10.119) 1=2  2 I.z; / D q.!sp /  00 e zp.!sp ;/  2i e zp.!c ;/ zp .!sp ;  / CR1 e zp.!sp ;/ I

 > s ;

(10.120)

n

p p 3 o , as j . /j z ! 1 uniformly with respect to  for where R1 D O ./ z all  2 R. These results are the same as those obtained with a direct application of Olver’s saddle point method, except that the dependence of the remainder term on the separation between the saddle point and the pole is displayed explicitly through the factor ./ in (10.118)–(10.120). This set of equations provides a continuous asymptotic approximation of the integral I.z;  / for fixed large values of z as  varies continuously over R provided that the quantity zjp.!sp ; s /  p.!c ; s /j is large enough so that the discontinuity at  D s when the steepest descent path through the saddle point crosses the pole is negligible. When this quantity is small enough that this discontinuity is significant, then (10.108)–(10.110) must be employed.

10.4.3 Asymptotic Behavior for Two Isolated Interacting Saddle Points Finally, consider the case in which there are two relevant saddle points !1 ./ and !2 . / of the complex phase function p.!;  / which remain isolated from each other over the entire range R of values of  and interact with the simple pole singularity at ! D !c of the amplitude function q.!/ appearing in the path integral I.z;  / of the form given in (10.1). As in Sect. 10.3.1, the path P can then be deformed into an Olver-type path P . / that is composed of two parts P1 . / and P2 ./ such that P . / D P1 . / C P2 . /, where Pj ./, j D 1; 2, passes through the saddle point !j . / and is an Olver-type path with respect to that saddle point. The integral Isp .z; / taken over the contour P ./ can then be expressed as the sum of the two integrals Isp1 .z; / and Isp2 .z;  / taken over the respective paths P1 ./ and P2 ./. The following Corollary to Theorem 5 due to Cartwright [13] in 2004 then applies. Corollary 2. (Cartwright) In the integral Z

q.!/e zp.!;/ d!

Isp .z;  / D

(10.121)

P ./

considered in Theorem 5 for real z > 0, let all of the conditions stated there hold with the exception that the complex-valued phase function p.!;  / possesses two

140

10 Asymptotic Methods of Analysis

first-order saddle points !1 . / and !2 ./ that are isolated from each other as well as from any other saddle points of p.!;  / for all  2 R. The positions of these two saddle points are assumed to move in a vicinity of the isolated simple pole singularity of the amplitude function q.!/ that is located at ! D !c such that !j . / ¤ !c , j D 1; 2, for all  2 R. In addition, let the contour of integration P . / be composed of the two Olver-type paths P1 . / and P2 . / such that P ./ D P1 . / C P2 . /, where Pj . /, j D 1; 2, is the path of steepest descent through the corresponding saddle point !j ./. Define the functions

1=2 j . / p.!j ;  /  p.!c ;  / ;

j D 1; 2:

(10.122)

It is assumed that only the steepest descent path emanating from ˚ the saddle point !1 . / may cross the simple pole at !c , in which case = 2 ./ ¤ 0 for all  2 R. The appropriate argument of 1 ./ is then determined by (10.90). Under these conditions, the asymptotic behavior of the saddle point integral Isp .z;  / is given by 1=2 2 Isp .z; / D q.!1 /  00 e zp.!1 ;/ zp .!1 ;  / 1=2  2 Cq.!2 /  00 e zp.!2 ;/ zp .!2 ;  / r    p  e zp.!1 ;/ C ˙i erfc i 1 ./ z e zp.!c ;/ C z 1 ./ r    p  zp.!c ;/  e zp.!2 ;/ C ˙i erfc i 2 ./ z e C z 2 ./  zp.!1 ;/  ˚ zp.!2 ;/ CK e I when = j . / ¤ 0; Ce (10.123) ˚ where the ˚ upper sign choice is used when = j ./ > 0 and the lower sign choice when = j . / < 0, for j D 1; 2, and 

1=2 2 Isp .z; / D q.!1 /  00 e zp.!1 ;/ zp .!1 ;  / 1=2  2 Cq.!2 /  00 e zp.!2 ;/ zp .!2 ;  / r    p  zp.!c ;/  e zp.!1 ;/ C i erfc i 1 ./ z e C z 1 ./ r    p  zp.!c ;/  e zp.!2 ;/ C ˙i erfc i 2 ./ z e C z 2 ./   i e zp.!c ;/ C K e zp.!1 ;/ C e zp.!2 ;/ I ˚ ˚ when = 1 ./ D 0; ./ ¤ 0; = 2 . / ¤ 0; 

(10.124)

10.5 Asymptotic Expansions of Multiple Integrals

141

˚ where the upper sign choice is used when = 2 ./ > 0 and the lower sign choice  ˚  when = 2 . / < 0, where K D O z3=2 as z ! 1 uniformly with respect to  for all  2 R.

10.5 Asymptotic Expansions of Multiple Integrals The extension of Laplace’s method (see Appendix F.6) to the two-dimensional case is now considered. This extension has its origin in the analysis due to Hsu [28] in 1948 that was later extended by Fulks and Sather [15] in 1961. The description presented here is based upon the detailed analysis presented by Bleistein and Handelsman [29] in 1975. Consider then the asymptotic behavior as z ! 1 of the function I.z/ defined by the double integral [cf. (F.73)] Z I.z/ D

g./e zh./ d 2 ;

(10.125)

D

where  D .1 ; 2 / with 1 and 2 real, and where the finite, simply connected integration domain D is bounded by a smooth curve  . In particular, the closed contour  is defined by the parametric equation ˚  D .1 ; 2 /j 1 D 1 .s/; 2 D 2 .s/; 0  s  L ;

(10.126)

where s is the arc-length along the curve such that  is traced out in the counterclockwise sense as s increases, and where both of the functions 1 .s/ and 2 .s/ are continuously differentiable on Œ0; L . Finally, it is assumed that the functions g./ D g.1 ; 2 / and h./ D h.1 ; 2 / are both continuous with continuous partial derivatives through at least the second order. The asymptotic behavior of the integral I.z/ as z ! 1 depends upon whether the function h./ possesses an absolute maximum value either in the interior of the closure DN of the integration domain D or on the boundary curve  of DN . Each of these two cases is now considered separately.

10.5.1 Absolute Maximum in the Interior of the Closure of D Assume that the function h./ possesses a single absolute maximum in the interior of the closure DN of the domain D at the point  0 D .10 ; 20 /, so that rh. 0 / D 0;

(10.127)

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10 Asymptotic Methods of Analysis

with rh./ ¤ 0 at all other points of DN , and "

@2 h./ @12



  2 2 # @2 h./ @ h./  > 0; @1 @2 @22 D 0   2 @ h./ < 0: @22 D 0

(10.128) (10.129)

If several maxima occur in the interior of DN , then the integration domain can be partitioned into a set of subdomains, each containing a single absolute maximum. It is then expected that the dominant contribution to the integral I.z/ arises from the local behavior of h./ about the critical point  D  0 , as described by the Taylor series expansion 1 h./ D h. 0 / C h1 1 . 0 /.1  10 /2 C h1 2 . 0 /.1  10 /.2  20 / 2 1 (10.130) C h2 2 . 0 /.2  20 /2 C    ; 2 where hi j ./ @2 h./=@i @j , i D 1; 2, j D 1; 2. With this in mind, the asymptotic approximation of the integral I.z/ in (10.125) is found to be given by [29] 2g. 0 /e zh. 0 / I.z/  h  2 i1=2 ; z h1 1 . 0 /h2 2 . 0 /  h1 2 . 0 /

(10.131)

as z ! 1.

10.5.2 Absolute Maximum on the Boundary of the Closure of D Consider now the case when rh. 0 / ¤ 0 in DN , so that the absolute maximum of h./ in DN occurs on the boundary  of D . In particular, assume that this maximum occurs at the point   D .1 ; 2 / 2  and that this maximum is unique. For simplicity, let this point occur when s D 0, so that [from (10.126)] 1 D 1 .0/;

2 D 2 .0/:

(10.132)

The first directional derivative of h./ along the unit tangent vector T to the boundary curve  must then vanish at  D   , in which case ˇ rh./  TˇsD0 D h 1 .1 /10 .0/ C h 2 .2 /20 .0/ D 0;

(10.133)

10.5 Asymptotic Expansions of Multiple Integrals

143

where the prime denotes differentiation with respect to the arc length s along the curve  . Equation (10.133) then states that the vector rh./ is orthogonal to the boundary curve  at the point  D   . With the expression   O n.s/ 20 .s/; 10 .s/

(10.134)

for the unit outward normal vector to  , it is seen that ˇ ˇ O rh.  / D ˇrh.  /ˇn.0/;

(10.135)

as rh./ is in the direction of decreasing h./ and the absolute maximum of h./ occurs on the boundary of DN and not in its interior. Define the vector field H./ g./

rh./ ; jrh./j2

(10.136)

so that g./e zh./ D 

  1 e zh./ r  H./ C r  H./e zh./ : z z

(10.137)

Substitution of this result in the integrand of (10.125) followed by application of the divergence theorem (in two dimensions) then results in 1 I.z/ D z

I O H.s/  n.s/e 

zh.s/

1 ds  z

Z

.r  H.// e zh./ d 2 :

(10.138)

D

Because the second integral in this expression is of the same form as the original surface integral over D and because it has a 1=z multiplicative factor, it is typically of lower order than I.z/. As a consequence, the dominant term in the asymptotic expansion of I.z/ as z ! 1 comes from the boundary integral in (10.138). The asymptotic approximation of this term then gives [29] r

I.z/ 

2 g.  /e zh.  / z3  ˇ  ˇh1 1 .  /h22 .  /  2h1 2 .  /h1 .  /h2 .  / ˇ ˇ3 ˇ Ch2 2 .  /h21 .  / .  / ˇrh.  /ˇ ˇ

1=2 ; (10.139)

as z ! 1. Here .  / is the curvature of the contour  at the point  D   , where the upper sign in the above equation is used when  is convex and the lower sign when  is concave at   .

144

10 Asymptotic Methods of Analysis

Comparison of this result with that given in (10.131) shows that when the absolute maximum of the function h./ occurs at an interior point  D  0 of the integration domain [where rh. 0 / D 0] ˚ I.z/e zh. 0 / D O z1 ; whereas

˚ I.z/e zh.  / D O z3=2 ;

as z ! 1;

as z ! 1;

(10.140)

(10.141)

when the absolute maximum of the function h./ occurs at a boundary point  D   of the integration domain and rh.  / ¤ 0. However, when the absolute maximum at  D   occurs at a boundary point of the domain and rh.  / D 0, the asymptotic behavior of the integral I.z/ is found [29] to be given by 1=2 of the expression given in (10.131) for an interior maximum with  0 replaced by   .

10.6 Summary The inherent complexity of the asymptotic method of analysis for a given problem is offset by its ability to accurately describe a complicated physical process with a compact mathematical expression that explicitly displays the dependence on each factor appearing in the physical model used. This is to be contrasted with a purely numerical method of analysis which may be easier to implement and provides quicker results, but is much less capable of providing a deeper understanding of the physical phenomena involved. The philosophical approach to dispersive pulse propagation used in this book is to fully develop the asymptotic description and to then illustrate the predicted results from this description through the use of accurate numerical results. When warranted, the asymptotic theory may be augmented by numerical techniques, resulting in a hybrid asymptotic-numerical procedure with tremendous practical applicability.

References ¨ 1. L. Brillouin, “Uber die fortpflanzung des licht in disperdierenden medien,” Ann. Phys., vol. 44, pp. 204–240, 1914. 2. L. Brillouin, Wave Propagation and Group Velocity. New York: Academic, 1960. 3. F. W. J. Olver, “Why steepest descents?,” SIAM Rev., vol. 12, no. 2, pp. 228–247, 1970. 4. K. E. Oughstun, Propagation of Optical Pulses in Dispersive Media. PhD thesis, The Institute of Optics, University of Rochester, 1978. 5. K. E. Oughstun and G. C. Sherman, “Propagation of electromagnetic pulses in a linear dispersive medium with absorption (the Lorentz medium),” J. Opt. Soc. Am. B, vol. 5, no. 4, pp. 817–849, 1988.

References

145

6. R. A. Handelsman and N. Bleistein, “Uniform asymptotic expansions of integrals that arise in the analysis of precursors,” Arch. Rat. Mech. Anal., vol. 35, pp. 267–283, 1969. 7. K. E. Oughstun and G. C. Sherman, “Uniform asymptotic description of electromagnetic pulse propagation in a linear dispersive medium with absorption (the Lorentz medium),” J. Opt. Soc. Am. A, vol. 6, no. 9, pp. 1394–1420, 1989. 8. C. Chester, B. Friedman, and F. Ursell, “An extension of the method of steepest descents,” Proc. Cambridge Phil. Soc., vol. 53, pp. 599–611, 1957. 9. L. Felsen and N. Marcuvitz, Radiation and Scattering of Waves. Englewood Cliffs, NJ: Prentice-Hall, Inc., 1973. 10. J. A. Solhaug, K. E. Oughstun, J. J. Stamnes, and P. Smith, “Uniform asymptotic description of the Brillouin precursor in a single-resonance Lorentz model dielectric,” Pure Appl. Opt., vol. 7, no. 3, pp. 575–602, 1998. 11. N. Bleistein, “Uniform asymptotic expansions of integrals with stationary point near algebraic singularity,” Com. Pure Appl. Math., vol. XIX, no. 4, pp. 353–370, 1966. 12. N. Bleistein, “Uniform asymptotic expansions of integrals with many nearby stationary points and algebraic singularities,” J. Math. Mech, vol. 17, no. 6, pp. 533–559, 1967. 13. N. A. Cartwright, Uniform Asymptotic Description of the Unit Step Function Modulated Sinusoidal Signal. PhD thesis, College of Engineering & Mathematical Sciences, University of Vermont, 2004. 14. N. A. Cartwright and K. E. Oughstun, “Uniform asymptotics applied to ultrawideband pulse propagation,” SIAM Rev., vol. 49, no. 4, pp. 628–648, 2007. 15. W. Fulks and J. O. Sather, “Asymptotics II: Laplace’s method for multiple integrals,” Pacific J. of Math., vol. 11, pp. 185–192, 1961. 16. B. Riemann, Gesammelte Mathematische Werke. Leipzig: Teubner, 1876. 17. P. Debye, “N¨aherungsformeln f¨ur die zylinderfunktionen f¨ur grosse werte des arguments und unbeschr¨ankt verander liche werte des index,” Math Ann., vol. 67, pp. 535–558, 1909. ¨ 18. A. Sommerfeld, “Uber die fortpflanzung des lichtes in disperdierenden medien,” Ann. Phys., vol. 44, pp. 177–202, 1914. 19. K. E. Oughstun and G. C. Sherman, Pulse Propagation in Causal Dielectrics. Berlin: SpringerVerlag, 1994. 20. P. M. Morse and H. Feshbach, Methods of Theoretical Physics. New York: McGraw-Hill, 1953. Vol. I. 21. E. T. Copson, Asymptotic Expansions. London: Cambridge University Press, 1965. 22. M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, vol. 55 of Applied Mathematics Series. Washington, D.C.: National Bureau of Standards, 1964. 23. H. M. Nussenzveig, Diffraction Effects in Semiclassical Scattering. Cambridge: Cambridge University Press, 1992. 24. J. J. Stamnes, Waves in Focal Regions: Propagation, Diffraction and Focusing of Light, Sound and Water Waves. Bristol, England: Adam Hilger, 1986. 25. E. T. Copson, An Introduction to the Theory of Functions of a Complex Variable. London: Oxford University Press, 1935. p. 110. 26. L. B. Felsen and N. Marcuvitz, “Modal analysis and synthesis of electromagnetic fields,” Polytechnic Inst. Brooklyn, Microwave Res. Inst. Rep., 1959. 27. A. Ba˜nos, Dipole Radiation in the Presence of a Conducting Half-Space. Oxford: Pergamon Press, 1966. Sect. 3.3. 28. L. C. Hsu, “On the asymptotic evaluation of a class of multiple integrals involving a parameter,” Duke Math. J., vol. 15, pp. 625–634, 1948. 29. N. Bleistein and R. Handelsman, Asymptotic Expansions of Integrals. New York: Holt, Rinehart and Winston, 1975. 30. F. W. J. Olver, Asymptotics and Special Functions. Natick: A K Peters, 1997.

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Problems 10.1. Beginning with the contour integral definition of the Bessel function J . / due to Schl¨afli, given by J . /

1 2 i

Z

1Ci

e sinh   d ;

1i

for argf g < =2, where the integration contour extends from 1  i  to 1 C i  in the domain i    00  i , 0   0  C1, use Olver’s theorem to derive the asymptotic expansion of J . / as j j ! 1 with argf g < =2. See pp. 130–133 of Olver [30]. 10.2. Prove the limiting result given in (10.64). 10.3. Show that C. / D Ai . / along the contour L32 depicted in Fig. 10.6. 10.4. Using the integral representation of the Airy function Ai . / given in (10.73), determine both the Maclaurin series approximation and the asymptotic expansion of the Airy function and its first derivative for real . Estimate the number of terms required in each expansion to provide a matched expansion representation of both the Airy function and its first derivative that is valid for all real , as illustrated in Fig. 10.7. 10.5. Show that the real part of the complementary error function erfc. / is even symmetric about the 00 -axis whereas the imaginary part is odd symmetric, where

D 0 C i 00 ; that is, show that < ferfc.  /g D < ferfc. /g and = ferfc.  /g D = ferfc. /g. 10.6. Use the rational approxiamations of the cosine and sine Fresnal integrals given in (10.114)–(10.117) to generate the graphs of the Cornu spiral given in Figs. 10.12 and 10.13. 10.7. Use the integral definition of the error function given in (10.94) to determine the asymptotic expansion of the complementary error function erfc. / given in (10.101) as j j ! 1 with j arg. /j < =2. 10.8. Determine the asymptotic expansion of Dawson’s integral FD ./, given in (10.97), for real  > 0 as  ! 1.

Chapter 11

The Group Velocity Approximation

Because of its mathematical simplicity and direct physical interpretation, the group velocity approximation has gained widespread use in the physics, engineering, and mathematical science communitites. However, the fundamental assumptions that are used to obtain this description are violated when either the loss component of the material dispersion cannot be neglected or the pulse spectrum becomes ultrawideband, which is taken here to mean that the bandwidth of the pulse spectrum spans at least one critical feature in the material dispersion. This inconsistency then results in intellectual mayhem over such topics as superluminal pulse velocites and superluminal tunneling in the ultrashort pulse dispersion regime. Because of this, it is essential to fully understand this approximate theory so that a better appreciation of the necessity of an asymptotic theory may be gained.

11.1 Historical Introduction Dispersive wave propagation was first considered in terms of a coherent superposition of monochromatic scalar wave disturbances by Sir William R. Hamilton [1] in 1839 where the concept of group velocity was first introduced. In that paper, Hamilton compared the phase and group velocities of light, showing that the phase velocity of a wave is given by the ratio !=k while the velocity of the wave group is given by d!=d k, where ! denotes the angular frequency and k the wavenumber of the disturbance. Subsequent to this definition, Stokes [2] posed the concept of group velocity as a “Smiths Prize examination” question in 1876. Lord Rayleigh then mistakenly attributed the original definition of the group velocity to Stokes, stating that [3] when a group of waves advances into still water, the velocity of the group is less than that of the individual waves of which it is composed; the waves appear to advance through the group, dying away as they approach its anterior limit. This phenomenon was, I believe, first explained by Stokes, who regarded the group as formed by the superposition of two infinite trains of waves, of equal amplitudes and of nearly equal wave-lengths, advancing in the same direction.

Rayleigh then applied these results to explain the difference between the phase and group velocities of light with respect to their observability [4]. K.E. Oughstun, Electromagnetic and Optical Pulse Propagation 2: Temporal Pulse Dynamics in Dispersive, Attenuative Media, Springer Series in Optical Sciences 144, c Springer Science+Business Media, LLC 2009 DOI 10.1007/978-1-4419-0149-1 11, 

147

148

11 The Group Velocity Approximation

These early considerations are best illustrated by the coherent superposition of two time-harmonic waves with equal amplitudes and nearly equal wave numbers (k and k C ık) and angular frequencies (! and ! C ı!, respectively) traveling in the positive z-direction. The linear superposition of these two monochromatic wave functions then yields the polychromatic waveform [5, 6] U.z; t / D a cos .kz  !t / C a cos ..k C ık/z  .! C ı!/t/   1 N  !t N /; D 2a cos .zık  t ı!/ cos .kz 2

(11.1)

which is an amplitude modulated wave with mean wavenumber kN D k C ık=2 and mean angular frequency !N D ! C ı!=2. The surfaces of constant phase propagate with the phase velocity !N (11.2) vp ; kN while the surfaces of constant amplitude propagate with the group velocity vg

ı! : ık

(11.3)

Notice that these results are exact for the waveform given in (11.1). If the medium is nondispersive, then kN D !=c, N ık D ı!=c, and the phase and group velocities are equal. However, if the medium exhibits temporal dispersion so that k.!/ D .!=c/n.!/ where n.!/ is not a constant, then the phase and group velocities will, in general, be different. In particular, if n.!/ > 0 increases with increasing !  0, then vp  vg > 0 and the phase fronts will advance through the wave group as described by Rayleigh [3]. This elementary phenomenon is illustrated in Fig. 11.1 for the simple wave group described in (11.1). Each wave pattern illustrated in this figure describes a “snapshot” of the wave group at a fixed instant of time. In the upper wave pattern, the coincidence at z D 0 of a particular peak amplitude point in the envelope (marked with a G) with a peak amplitude point in the waveform (marked with a P ) is indicated. As time increases from this initial instant of time (t D 0), these two points become increasingly separated in time, as illustrated in the middle (t D ıt ) and bottom (t D 2ıt ) wave patterns, showing that the phase velocity of the wave is greater than the group velocity of the envelope in this case. The group velocity approximation was precisely formulated by Havelock [7,8] in 1914 based upon Kelvin’s stationary phase method [9]. It is apparent that Havelock was the first to employ the Taylor series expansion of the wavenumber k.!/ about a given wavenumber value k0 that the spectrum of the wave group is clustered about, referring to this approach as the group method. In addition, Havelock stated that [8] “The range of integration is supposed to be small and the amplitude, phase and velocity of the members of the group are assumed to be continuous, slowly varying, functions” of the wavenumber k.!/. This research then established the group velocity method for dispersive wave propagation. Because the method of

11.1 Historical Introduction

149 G,P |

0

z

0

U(z,t)

GP || 0

t = dt

0

z

G P || 0

t=0

0

t = 2dt z

Fig. 11.1 Evolution of a simple wave group in a temporally dispersive medium with normal frequency dispersion (i.e., when d n.!/=d! > 0). In this case, the phase front point P , which is coincident with the wave group amplitude point G at t D 0 in the upper part of the figure, advances through the wave group as t increases

stationary phase [10] requires that the wavenumber be real-valued, this method cannot properly treat causally dispersive, attenuative media. Furthermore, notice that Havelock’s group velocity method is a significant departure from Kelvin’s stationary phase method with regard to the wavenumber value k0 about which the Taylor series expansion is taken. In Kelvin’s method, k0 is the stationary phase point of the wavenumber k.!/ whereas in Havelock’s method k0 describes the wavenumber value about which the wave group spectrum is peaked. This apparently subtle change in the value of k0 results in significant consequences for the accuracy of the resulting group velocity description. Finally, notice that the fundamental hyperbolic character of the underlying wave equation is approximated as parabolic in this formulation, the characteristics then propagating instantaneously [11] instead of at the vacuum speed of light c. The group velocity approximation was later refined and extended during the period from 1950 through 1970, most notably by Eckart [12] who considered the close relationship between the method of stationary phase and Hamilton–Jacobi ray theory in dispersive but nonabsorptive media. Of equal importance are the papers by Whitham [13] and Lighthill [14] on the general mathematical properties of threedimensional wave propagation and the group velocity for ship-wave patterns and magnetohydrodynamic waves. The appropriate boundary value problem is solved in both papers through application of the method of stationary phase to a plane-wave expansion representation. Their approach, however, is useful only for nonabsorbing media, thereby limiting the types of dispersion relations that may be considered. The equivalence between the group velocity and the energy-transport velocity in lossfree media and systems was also established [13–17], thereby providing a physical

150

11 The Group Velocity Approximation

basis for the group velocity in lossless systems with an inconclusive extension to dissipative media [18, 19]. The mathematical basis for the group velocity approximation was completed when the quasimonochromatic or slowly varying envelope approximation was precisely formulated by Born and Wolf [6] in the context of partial coherence theory. More recently published treatments concerned with the propagation of wave packets in dispersive and absorptive media [20–23] have employed Havelock’s technique of expanding the phase function appearing in the integral representation of the field in a Taylor series about some fixed characteristic frequency of the initial pulse. This approach may also be coupled with a recursive technique in order to obtain purported correction terms of arbitrary dispersive and absorptive orders for the resultant envelope function. This analysis again relies upon the quasimonochromatic approximation, and hence, can only be applied to study the evolution of pulses with slowly varying envelope functions in weakly dispersive systems. This approximate approach has since been adopted as the standard in both fiber optics [24] and nonlinear optics in general [25,26] with little regard for either its accuracy or its domain of applicability. In summary, this group velocity description of dispersive pulse propagation is based on both the slowly varying envelope approximation and the Taylor series approximation of the complex wavenumber about some characteristic angular frequency !c of the initial pulse at which the temporal pulse spectrum is peaked, as originally described by Havelock [7, 8]. The slowly varying envelope approximation is a hybrid time and frequency domain representation [26] in which the temporal field behavior is separated into the product of a slowly varying temporal envelope function and an exponential phase term whose angular frequency is centered about !c . The envelope function is assumed to be slowly varying on the time scale tc  1=!c , which is equivalent [27] to the quasimonochromatic assumption that its spectral bandwidth ! is sufficiently narrow that the inequality !=!c  1 is satisfied. Under these approximations, the frequency dependence of the wavenumber may then be approximated by the first few terms of its Taylor series expansion about the characteristic pulse frequency !c with the unfounded assumption [20, 21, 26] that improved accuracy can always be obtained through the inclusion of higher-order terms. This assumption has been proven incorrect [28, 29] in the ultrashort pulse, ultrawideband signal regime, optimal results being obtained using either the quadratic or the cubic dispersion approximation of the wavenumber. Recently published research [28, 29] by Xiao and Oughstun has identified the space–time domain within which the group velocity approximation is valid. Because of the slowly varying envelope approximation together with the neglect of the frequency dispersion of the material attenuation, the group velocity approximation is invalid in the ultrashort pulse regime in a causally dispersive material or system, its accuracy decreasing as the propagation distance z  0 increases. This is in contrast with the modern asymptotic description whose accuracy increases in the sense of Poincar´e [10] as the propagation distance increases. There is then a critical propagation distance zc > 0 such that the group velocity description using either the quadratic or cubic dispersion approximation provides an accurate description of the

11.1 Historical Introduction

151

pulse dynamics when 0  z  zc , the accuracy increasing as z ! 0, while the modern asymptotic theory provides an accurate description when z > zc , the accuracy increasing as z ! 1. This critical distance zc depends upon both the dispersive material and the input pulse characteristics including the pulse shape, temporal width, and characteristic angular frequency !c . For example, zc D 1 for the trivial case of vacuum for all pulse shapes, whereas zc  zd for an ultrashort, ultrawideband pulse in a causally dispersive dielectric with e 1 penetration depth zd at the characteristic oscillation frequency !c of the input pulse. In an attempt to overcome these critical difficulties, Brabec and Krausz [30] have proposed to replace the slowly varying envelope approximation with a slowly evolving wave approach that is supposed to be “applicable to the single-cycle regime of nonlinear optics.” As with the slowly varying envelope approximation, the difficulty with the slowly evolving wave approach is twofold. First, the fundamental hyperbolic character of the underlying wave equation is approximated as parabolic. The characteristics then propagate instantaneously [11]. Second, the subsequent Q imposed Taylor series expansion of the complex wavenumber k.!/ about !c approximates the material dispersion by its local behavior about some characteristic angular frequency of the initial pulse. Because this approximation is incapable of correctly describing the precursor fields, it is then incapable of correctly describing the dynamical evolution of any ultrashort pulse and its accuracy monotonically decreases [28] as z exceeds a single absorption depth zd in the dispersive medium. Recent research has been focused on the contentious topic of superluminal pulse propagation [31–38] in both linear and nonlinear optics. Again, the origin of this controversy may be found in the group velocity approximation which is typically favored by experimentalists. In response, Landauer [32] has argued for more careful analysis of experimental measurements reporting superluminal motions. Diener [33] then showed that “the group velocity cannot be interpreted as a velocity of information transfer” in those situations in which it exceeds the vacuum speed of light c. This analysis is in fact based upon an extension of Sommerfeld’s now classic proof [39, 40] that the signal arrival cannot exceed c in a causally dispersive medium. Kuzmich, Dogariu, Wang, Milonni, and Chiao [36] defined a signal velocity that is operationally based upon the optical signal-to-noise ratio and showed that, in those cases when the group velocity is negative, “quantum fluctuations limit the signal velocity to values less than c.” In addition, they argue that a more general definition of the “signal” velocity of a light pulse must satisfy two fundamental criteria:  “First, it must be directly related to a known and practical way of detecting a

signal.”  “Second, it should refer to the fastest practical way of communicating infor-

mation.” In contrast, Nimtz and Haibel [37] argue regarding superluminal tunneling phenomena that “the principle of causality has not been violated by superluminal signals as a result of the finite signal duration and the corresponding narrow frequency-band

152

11 The Group Velocity Approximation

width.” In addition, Winful [38] argues that “distortionless tunneling of electromagnetic pulses through a barrier is a quasistatic process in which the slowly varying envelope of the incident pulse modulates the amplitude of a standing wave.” In particular, “for pulses longer than the barrier width, the barrier acts as a lumped element with respect to the pulse envelope. The envelopes of the transmitted and reflected fields can adiabatically follow the incident pulse with only a small delay that originates from energy storage.” Unfortunately, each of these arguments neglects the frequency-dependent attenuation of the material comprising the barrier. When material attenuation is properly included, the possibility of evanescent waves is replaced by inhomogeneous waves [41,42], thereby rendering the accuracy of this superluminal tunneling analysis as questionable at best. This fundamental question of superluminal pulse propagation and tunneling provides the impetus for obtaining a deeper and physically correct understanding of the dispersive pulse phenomena that are involved.

11.2 The Pulsed Plane Wave Electromagnetic Field An important class of wave fields that arises in many practical situations is that in which either of the field vectors are transverse to some specified propagation direction. These are the transvers electric (TE) and transverse magnetic (TM) mode fields whose importance arises, for example, in the analysis of reflection and transmission phenomena at a dielectric interface. Common to both of these mode solutions is the plane wave field which also holds a position of fundamental importance in the angular spectrum of plane waves representation of time-domain electromagnetic wave-fields [see Vol. 1]. Because the analysis of plane wave pulse propagation through a dispersive medium yields the fundamental dynamics of pulse dispersion that is unencumbered by diffraction effects, this field type is of central importance to the theoretical development presented here. For a transverse electromagnetic wave field with respect to the z-axis, it is required that both Ez .r; t / and Bz .r; t / vanish for all z  z0 . The appropriate solution may then be obtained from the angular spectrum of plane waves representation given in (9.5). One may, without any loss of generality, choose the plane wave field to be linearly polarized along some convenient direction that is orthogonal to the z-axis. Any other state of polarization may then be obtained through an appropriate linear superposition of such linearly polarized wave fieldes with suitable orientations of the vibration plane [see Sect. 7.2 of Vol. 1]. Let the electric field vector be linearly polarized along the y-axis so that E.r; t / D 1O y Ey .z; t /;

(11.4)

B.r; t / D 1O x Bx .z; t /;

(11.5)

11.2 The Pulsed Plane Wave Electromagnetic Field

153

where 1 Ey .z; t / D < 

(Z CC

kck < Bx .z; t / D  

Q EQ y.0/ .!/e i .k.!/ z!t / d!

(Z CC

) ;

) Q k.!/ Q EQ y.0/ .!/e i .k.!/ z!t / d! ; !

(11.6) (11.7)

with z D z  z0 denoting the propagation distance into the positive half-space z  z0 from the input plane at z D z0 . Let the initial time behavior of the plane wave electric field vector at the plane z D z0 be specified by the dimensionless function f .t / of the time t as (11.8) Ey.0/ .t / D E0 f .t /; .0/ with temporal frequency spectrum EQ y .!/ D E0 fQ.!/, where

fQ.!/ D

Z

1

f .t /e i!t dt

(11.9)

1

is the Fourier–Laplace transform of f .t /. With this substitution, the pair of relations appearing in (11.6) and (11.7) become Z iaC1

1 Q E0 < fQ.!/e i .k.!/ z!t / d! ;  ia Z iaC1

kck Q i .k.!/ z!t / Q n.!/f .!/e d! ; E0 < Bx .z; t / D  c ia

Ey .z; t / D

(11.10) (11.11)

Q with use of the expression k.!/ D .!=c/n.!/. For convenience, this pair of expressions may be rewritten as Z iaC1

1 E0 < fQ.!/e . z=c/.!;/ d! ;  ia Z iaC1

kck Bx .z; t / D  E0 < n.!/fQ.!/e . z=c/.!;/ d! ; c ia

Ey .z; t / D

(11.12) (11.13)

for z D z  z0  0. The complex phase function .!;  / appearing in these equations is defined as [43, 44]   .!; / i ! n.!/   ;

(11.14)

Q where n.!/ D .c=!/k.!/ is the complex index of refraction of the dispersive medium, and where ct (11.15)  z

154

11 The Group Velocity Approximation

is a nondimensional space–time parameter defined for all z > 0. Notice that both the electric and magnetic field vectors given in (11.4) and (11.5) and (11.12)–(11.13) may be obtained from the single vector potential A.z; t / D 1O y A.z; t / kck D 1O y E0 < 

(Z

iaC1 ia

fQ.!/ i .k.!/ z!t Q / d! e !

) (11.16)

as 1 @A.z; t / 1 @A.z; t / D 1O y ; kck @t kck @t @A.z; t / B.z; t / D r  A.z; t / D 1O x ; @z

E.z; t / D 

(11.17) (11.18)

in agreement with the results of Sect. 6.2 of Vol. 1. On the other hand, if the initial time behavior of the plane wave magnetic induction field vector at the plane z D z0 is specified by the dimensionless function g.t / so that (11.19) Bx.0/ .z; t / D B0 g.t /; .0/

with temporal frequency spectrum BQ x .!/ D B0 g.!/, Q where Z

1

g.!/ Q D

g.t /e i!t dt

(11.20)

1

is the Fourier–Laplace transform of g.t /, then, from (9.8) [see also (9.279)] B0 g.!/ Q D kck

Q k.!/ kck E0 fQ.!/ D n.!/E0 fQ.!/; ! c

(11.21)

and the pair of relations given in (11.12) and (11.13) become Z iaC1

g.!/ Q c B0 < e . z=c/.!;/ d! ; kck n.!/ ia Z iaC1

1 . z=c/.!;/ Bx .z; t / D  B0 < g.!/e Q d! ;  ia

Ey .z; t / D

(11.22) (11.23)

for the propagated plane wave field for z D z  z0  0. The corresponding vector potential is then given by A.z; t / D 1O y A.z; t / with A.z; t / D

Z iaC1

g.!/ Q 1 B0 < i e . z=c/.!;/ d! ; Q  k.!/ ia

(11.24)

where the electric and magnetic field vectors are as given in (11.17) and (11.18), respectively.

11.2 The Pulsed Plane Wave Electromagnetic Field

155

Because of the requirement of causality, admissable models for describing the behavior of the complex index of refraction n.!/ in the complex !-plane must satisfy the symmetry relation [see Sect. 4.3 and Problem 5.2 of Vol. 1] n.!/ D n .!  /;

(11.25)

  .!; / D i ! n.!/    

D   .!  ;  /; D i !  n.!  /  

(11.26)

and consequently

so that the complex phase function .!;  / satisfies the same symmetry relation in the complex !-plane. Furthermore, for any real-valued initial pulse function f .t / one has that fQ.!/ D

Z

1

f .t /e i!t dt

1

Z

1



f .t /e i! t dt

D



D fQ .!  /:

(11.27)

1

Since g.t / is real-valued, its spectrum also satisfies this symmetry relation [i.e., g.!/ Q D gQ  .!  /]. It is then seen that the symmetry relation given in (11.25) Q D for the complex index of refraction directly follows from the relation B0 g.!/ .kck=c/ n.!/E0 fQ.!/. Because of these fundamental symmetry relations, the integral expression given in (11.12) may be rewritten as E0 Ey .z; t / D 2

(Z

iaC1

fQ.!/e . z=c/.!;/ d!

ia

Z

iaC1

C

)  fQ .!/e . z=c/ .!;/ d! 

ia

D

E0 2

(Z

iaC1

fQ.!/e . z=c/.!;/ d!

ia

Z

iaC1

C

) fQ.  ! 0 C i ! 00 /e

. z=c/.! 0 Ci! 00 ;/

d.! 0 C i ! 00 / ;

ia

where ! 0
156

11 The Group Velocity Approximation

second integral of the above expression there results E0 Ey .z; t / D 2 E0 D 2

(Z

iaC1

fQ.!/e . z=c/.!;/ d! C

ia

Z

Z

ia

) fQ.!/e . z=c/.!;/ d!

ia1

iaC1

fQ.!/e . z=c/.!;/ d!;

ia1

for z  0. A precisely analogous result holds for the magnetic induction field so that the pair of expressions appearing in (11.12) and (11.13) then become Ey .z; t / D

1 E0 2

Z

iaC1

fQ.!/e . z=c/.!;/ d!;

(11.28)

ia1

kck E0 Bx .z; t / D  2c

Z

iaC1

n.!/fQ.!/e . z=c/.!;/ d!;

(11.29)

ia1

for z D zz0  0. On the other hand, if the initial time behavior of the plane wave magnetic induction field vector is specified at the plane z D z0 , as in (11.19), then the propagated plane wave field components are, from (11.22) and (11.23), found as Z iaC1 g.!/ Q c B0 e . z=c/.!;/ d!; 2kck n.!/ ia1 Z iaC1 1 . z=c/.!;/ g.!/e Q d!; Bx .z; t / D  B0 2 ia1

Ey .z; t / D

(11.30) (11.31)

for z D zz0  0. Both field vectors may be obtained through (11.17) and (11.18) from the single vector potential A.z; t / D 1O y A.z; t / with Z iaC1 Q kck f .!/ . z=c/.!;/ A.z; t / D i E0 e d! 2 ! ia1 Z iaC1 g.!/ Q 1 B0 Di e . z=c/.!;/ d!; Q 2 ia1 k.!/

(11.32) (11.33)

where the first form is appropriate for the representation given in (11.18) and (11.19), whereas the second form is appropriate for the representation given in (11.30) and (11.31). This final set of expressions for the propagated linearly polarized plane wave electromagnetic field components in the positive half-space z  z0 that is occupied by a homogeneous, isotropic, locally linear, temporally dispersive medium is of the same form as that treated in classical descriptions of dispersive pulse propagation [8, 43, 44] provided that either the electric or magnetic field component alone is considered. The classical treatment may also be taken to apply to the single vector potential A.z; t / D 1O y A.z; t / given in (11.32) with an input pulse spectrum that is given by fQA .!/ D i kckE0 fQ.!/=!.

11.2 The Pulsed Plane Wave Electromagnetic Field

157

A case of special interest in both communications and radar is that of an input pulse-modulated sinusoidal wave with constant applied angular signal frequency !c D 2fc , given by (11.34) f .t / D u.t / sin .!c t C /; where D 0 for a sine wave carrier and D =2 for a cosine wave carrier. Here u.t / is the real-valued initial envelope function of the input pulse with temporal frequency spectrum Z 1

u.t /e i!t dt:

uQ .!/ D

(11.35)

1

The temporal frequency spectrum of the initial signal given in (11.34) is given by  1 i e uQ .! C !c /  e i uQ .!  !c / ; fQ.!/ D 2i

(11.36)

so that for a sine wave carrier, D 0 and fQ.!/ D ŒQu.! C !c /  uQ .!  !c / =.2i /, and for a cosine wave carrier, D =2 and fQ.!/ D ŒQu.! C !c / C uQ .!  !c / =.2/. Substitution of (11.36) into (11.28) and (11.29) yields the pair of expressions ( Z iaC1 E0 i Ey .z; t / D  uQ .! C !c /e . z=c/.!;/ d! ie 4 ia1 ) Z iaC1 i . z=c/.!;/ i e uQ .!  !c /e d! ;

(11.37)

ia1

( Z iaC1 kckE0 i Bx .z; t / D n.!/Qu.! C !c /e . z=c/.!;/ d! ie 4c ia1 ) Z iaC1 i . z=c/.!;/ i e n.!/Qu.!  !c /e d! ; (11.38) ia1

for z  0, whereas substitution into (11.12) and (11.13) yields the equivalent pair of expressions ( Z iaC1 E0 uQ .! C !c /e . z=c/.!;/ d! Ey .z; t / D  < i e i 2 ia ) Z iaC1 i e i uQ .!  !c /e . z=c/.!;/ d! ; ia

(11.39)

( Z iaC1 kckE0 < i ei Bx .z; t / D n.!/Qu.! C !c /e . z=c/.!;/ d! 2c ia ) Z iaC1 i e i n.!/Qu.!  !c /e . z=c/.!;/ d! ; (11.40) ia

158

11 The Group Velocity Approximation

for z  0. Under the change of variable ! ! !, the first integral appearing in (11.40) becomes Z

iaC1

n.!/Qu.! C !c /e . z=c/.!;/ d! Z ia1   n .!  /Qu .!   !c /e . z=c/ .! ;/ d!; D

ia

ia

which, under the further change of variable ! ! !  , becomes Z

iaC1

n.!/Qu.! C !c /e . z=c/.!;/ d! Z ia  n .!/Qu .!  !c /e . z=c/ .!;/ d!  : D

ia

ia1

Clearly, the same result holds with n.!/ D 1. As a consequence, (11.39) may be rewritten as (Z iaC1 E0 < i e i uQ .!  !c /e . z=c/.!;/ d! Ey .z; t / D 2 ia ) Z ia

i  . z=c/.!;/ C i e uQ .!  !c /e d! D

E0 4

(Z

ia1 iaC1

i e i uQ .!  !c /e . z=c/.!;/ d!

ia

Z

iaC1

C

i  i e uQ .!  !c /e . z=c/.!;/ d!

ia

Z

C

ia

i  i e uQ .!  !c /e . z=c/.!;/ d! ia1 ) Z ia

i e i uQ .!  !c /e . z=c/.!;/ d!

C E0 D 4

(Z

ia1 iaC1

i e i uQ .!  !c /e . z=c/.!;/ d!

ia1

Z

iaC1

C (

E0 < i e i D 2

i  i e uQ .!  !c /e . z=c/.!;/ d!

ia1

Z

)

iaC1

uQ .!  !c /e ia1

with an analogous result for (11.40).

. z=c/.!;/

d! ;

)

11.2 The Pulsed Plane Wave Electromagnetic Field

159

The final expressions for the propagated linearly polarized plane wave field vectors due to the input pulse-modulated sinusoidal carrier wave given in (11.34) are then given by ( ) Z iaC1 E 0 < i e i E.z; t / D 1O y uQ .!  !c /e . z=c/.!;/ d! ; (11.41) 2 ia1 ( ) Z iaC1 kckE 0 i . z=c/.!;/ < ie n.!/Qu.!  !c /e d! ; B.z; t / D 1O x 2c ia1 (11.42) for z  0. An alternate representation in which the path of integration is contained in the right half-space ! 0  0 is obtained from (11.39) and (11.40) as ( Z iaC1 h E 0 < i uQ .! C !c /e i E.z; t / D 1O y 2 ia ) i i . z=c/.!;/ e Qu.!  !c /e d! ; (11.43) ( Z h iaC1 kckE 0 < i B.z; t / D 1O x n.!/ uQ .! C !c /e i 2c ia Qu.!  !c /e

i

i

) e

. z=c/.!;/

d! ; (11.44)

for z  0. Through a comparison of (11.12) and (11.13) with (11.22) and (11.23), one may directly express the above two representations in terms of the initial magnetic induction field behavior [with g.t / expressed in the same envelope modulated signal form given in (11.34) for f .t /] at the input plane at z D z0 . The dynamical evolution of either field vector alone may then be analyzed through a study of the scalar plane wave field whose integral representation in the positive half-space z  0 (where, for simplicity, z0 is now chosen to be at z0 D 0) is given by Z 1 A.z; t / D (11.45) fQ.!/e .z=c/.!;/ d!; 2 C where fQ.!/ D

Z

1

f .t /e i!t dt

(11.46)

1

is the temporal Fourier spectrum of the initial pulse f .t / D A.0; t / at the plane z D 0. Here A.z; t / represents either the scalar potential or any scalar component of the electric field, magnetic field, Hertz vector, or vector potential field whose Q !/ satisfies the Helmholtz equation spectral amplitude A.z;   Q !/ D 0: r 2 C kQ 2 .!/ A.z; (11.47)

160

11 The Group Velocity Approximation

For an input envelope modulated sinusoidal carrier wave with constant angular signal frequency !c and envelope function u.t /, as given in (11.34), the expression (11.45) for the propagated wave field becomes

Z 1 i .z=c/.!;/ uQ .!  !c /e d! ; A.z; t / D < ie 2 C

(11.48)

for z  0. Furthermore, notice that the Fourier–Laplace integral appearing in the plane wave field representation given in (11.45) appears in a similar form in (9.296) for a pulsed electromagnetic beam wave field. Again, the separability of the spatial and temporal parts of the representation given in (9.296) into the form given in (9.292) is, except in special cases such as the plane wave field, strictly valid only in the geometrical optics limit. Of central interest in the subsequent analysis of dispersive pulse propagation presented in the remainder of this book is the attainment of accurate analytical approximations of the spatiotemporal dynamics of the scalar plane wave field A.z; t / due to several canonical pulse shapes that are of particular interest in communication (both optical and cellular) and radar systems. Each canonical problem has been chosen because it illustrates some fundamental feature of dispersive pulse propagation phenomena. These canonical pulse problems are ordered here from the most sharply defined and discontinuous in time (the Dirac delta function) to the smoothest (the gaussian envelope pulse). Pulses with a sharply defined, discontinuous leading edge (such as the Heaviside step function modulated sinusoidal signal) are ideally suited for consideration of the signal velocity in a dispersive medium whereas smooth rise-time pulses (such as the trapezoidal envelope and hyperbolic tangent envelope modulated signals) provide a useful measure of the rise-time necessary to observe the temporally transient (but spatially persistent) wave phenomena associated with discontinuous changes in either the initial pulse amplitude or phase.

11.2.1 The Delta Function Pulse and the Impulse Response of the Medium The delta function pulse is of central importance to any linear system as it provides the impulse response that is a characteristic of that system. With the initial pulse wave form (11.49) fı .t / ı.t  t0 /; which corresponds to an input Dirac delta function located at the instant of time t D t0 > 0 whose Fourier–Laplace transform is fQı .!/ D

Z

1

ı.t  t0 /e i!t dt D e i!t0 ; 0

(11.50)

11.2 The Pulsed Plane Wave Electromagnetic Field

161

the propagated scalar wave disturbance is then given by the integral representation 1 < Aı .z; t / D 2

Z e

.z=c/t0 .!;t0 /

d! ;

(11.51)

C

where t0 .!; t0 / i ! .n.!/  t0 /

(11.52)

is the retarded complex phase function with retarded space–time parameter t0

c .t  t0 /: z

(11.53)

The propagated wave field given by (11.51) is called the impulse response of the dispersive medium. Its central importance in dispersive pulse propagation lies in the fact that the propagated wave field due to any other input pulse f .t / is given by the convolution operation A.z; t / D f .t / ˝ Aı .z; t /

(11.54)

for all z  0.

11.2.2 The Heaviside Unit Step Function Signal For a unit step function modulated signal, the initial pulse envelope is given by the Heaviside unit step function 0 I for t < 0 uH .t / I (11.55) 1 I for t > 0 that is, the external current source for the field abruptly begins to radiate harmonically in time at the instant t D 0 and continues indefinitely for all t > 0 with a constant amplitude and frequency. The Fourier–Laplace transform of this initial envelope function is then given by Z 1 i e i!t dt D uQ H .!/ D (11.56) ! 0 for =f!g > 0. The Fourier–Laplace integral representation of the propagated plane wave signal is then given by 1 AH .z; t / D  < 2

Z C

1 e .z=c/.!;/ d! !  !c

(11.57)

for t > 0 and is zero for t < 0, where z  0. This canonical wave field is precisely the signal considered by Sommerfeld [40] and Brillouin [43, 45] in 1914, Baerwald

162

11 The Group Velocity Approximation 2

10

1

10

0

10

−1

c

10

−2

10

−3

10

−4

10

−5

10

−6

10

0

10

1

10

2

10

3

10 (r/s)

4

10

5

10

6

10

Fig. 11.2 Angular frequency dependence of the magnitude of the spectrum for the Heaviside unit step function signal with angular carrier frequency !c D 1  103 r=s

[46] in 1930, and Oughstun and Sherman [47–49] in 1975 in order to give a precise definition of the signal velocity in a dispersive medium. As such, it is one of the most fundamental canonical problems to be considered in this area of research. The angular frequency dependence of the magnitude of the spectrum uQ H .!/ D i=.! !c / for the Heaviside unit step function signal with angular carrier frequency !c D 1  103 r=s is presented in Fig. 11.2. The frequency behavior depicted here illustrates the basic features of an ultrawideband signal, the most important feature being the ! 1 fall-off in spectral amplitude as ! ! 1. Notice that the signal does not have to be ultrashort in order for it to be ultrawideband, as the temporal duration of the Heaviside step function modulated signal is infinite. Notice further that an ultrawideband pulse does not need to be an envelope modulated sinusoidal signal, as the delta function pulse given in (11.49) is certainly both ultrashort and ultrawideband.1 1

In its 2002 report (Tech. Rep. FCC 02-48), the Federal Communications Commission defined an “ultra-wideband” device as any device for which the fractional bandwidth FB 2

fH  fL fH C fL

is greater than 0:25 or otherwise occupies 1:5 GHz or more of spectrum when the center frequency is greater than 6 GHz. Here fH denotes the 10 dB upper limit and fL the 10dB lower limit of the energy bandwidth. The center frequency of the waveform is defined there as the average of the

11.2 The Pulsed Plane Wave Electromagnetic Field

163

11.2.3 The Double Exponential Pulse A pulse shape that is similar in temporal structure to the delta function pulse but that possesses a nonvanishing temporal width is the double exponential pulse   fde .t / a e ˛1 t  e ˛2 t uH .t / (11.58) with ˛j > 0 for j D 1; 2, and where the constant a is chosen such that the peak amplitude of the pulse is unity. The peak amplitude point of the pulse occurs at the instant of time tm > 0 when dfde .t /=dt D 0, so that ˛1 e ˛1 tm  ˛2 e ˛2 tm D 0; with solution tm D

ln ˛1 =˛2 : ˛1  ˛2

(11.59)

Because ude .tm / 1, substitution of (11.59) in (11.58) then gives 1  : a D e ˛1 tm  e ˛2 tm

(11.60)

A measure of the temporal width of the pulse is given by the temporal difference between the e 1 points of the leading and trailing edge exponential functions in (11.58), so that t D j˛1  ˛2 j=.˛1 ˛2 /. Finally, with the result given in (11.56), the spectrum of the double exponential pulse is found to be given by fQde .!/ D a



1 1  ! C i ˛1 ! C i ˛2

 ;

(11.61)

which is clearly ultrawideband.

11.2.4 The Rectangular Pulse Envelope Modulated Signal For a unit amplitude, rectangular pulse envelope modulated signal, the initial pulse envelope is given by the rectangle function uT .t /

n0I 1I

for either t < 0 or t > T I for 0 < t < T

(11.62)

upper and lower 10 dB points, so that fc

1 .fH C fL /: 2

In turn, the energy bandwidth was defined in 1990 by the OSD/DARPA Ultra-Wideband Radar Review Panel (Tech. Rep. Contract No. DAAH01-88-C-0131, ARPA Order 6049) as the frequency range within which some specified fraction of the total signal energy resides.

164

11 The Group Velocity Approximation

AT (0,t)

1

uT (t)

0

−1

-uT (t) 0

T/2 t (s)

T

Fig. 11.3 Temporal field structure of a 10-cycle, unit amplitude, rectangular envelope modulated sinusoidal signal. The dashed curves describe the envelope function ˙uT .t /

that is, the external current source for the wave field abruptly begins to radiate harmonically in time at time t D 0 and continues with a constant amplitude and frequency up to the time T > 0 at which it abruptly ceases to radiate, as illustrated in Fig. 11.3 for a 10-cycle pulse. Notice that this rectangular envelope function can be represented by the difference between two Heaviside step function envelopes displaced in time by the pulse width T . The Fourier–Laplace transform of the rectangular envelope function uT .t / defined in (11.62) is given by Z

T

e i!t dt D

uQ T .!/ D 0

 1  i!T e 1 : i!

(11.63)

The Fourier–Laplace integral representation of the propagated plane wave pulse then becomes (Z 1 1 AT .z; t / D  < e .z=c/.!;/ d! 2 C !  !c ) Z 1 i!c T .z=c/T .!;T / e e d! ; (11.64) C !  !c for t > 0 and is zero for t < 0, where z  0. Here   T .!; T / i ! n.!/  T

(11.65)

11.2 The Pulsed Plane Wave Electromagnetic Field

165

is the generalized complex phase function [cf. (11.52)] with retarded space–time parameter cT c : (11.66) T .t  T / D   z z Notice that the first integral in (11.64) is exactly the same as that given in the integral representation (11.57) for the unit step function modulated signal and that the second integral appearing in (11.64) is of the exact same form except that the phase function is retarded in time by the initial pulse width T . Finally, because the simple pole singularity at ! D !c appearing in each of the integrands in (11.64) can be removed by simply combining these two integrals as ( )   Z sin .!  !c /T =2 .z=c/T =2 .!;T =2 / 1 i!c T =2 AT .z; t / D < i e e :  !  !c C

(11.67)

Because   sin .!  !c /T =2 T lim D ; !!!c !  !c 2 the spectrum uQ T .!/ of the rectangular envelope pulse is actually analytic for all complex values of ! (i.e., it is an entire function of complex !), as it must be because the envelope function uT .t / has compact temporal support (i.e., it identically vanishes ouside of a finite time domain). The ultrawideband character of the spectrum uQ T .!/ of the rectangular envelope pulse is clearly evident in Figs. 11.4 and 11.5. The solid curve in each figure displays the magnitude of the initial pulse spectrum, Fig. 11.4 for a single cycle pulse with carrier frequency !c D 1  103 r=s and Fig. 11.5 for a 10-cylce pulse. The dashed curve in each figure displays the j!  !c j1 frequency dependence of the step function signal for comparison. Notice the manner in which the step function spectrum is approached by the rectangular envelope spectrum as the initial pulse width T is increased. The spectrum of the rectangular envelope pulse is then seen to be ultrawideband for all T > 0.

11.2.5 The Trapezoidal Pulse Envelope Modulated Signal A canonical pulse envelope shape of central importance to both radar and cellular communication systems is the trapezoidal pulse envelope modulated signal of initial duration T > 0 with envelope rise-time Tr > 0 and fall-time Tf > 0. Such a pulse may be described by the time delayed difference between a pair of trapezoidal envelope signals with equal angular carrier frequencies !c and trapezoidal envelope functions given by ( ut rapj .t /

0; .t  Tj 0 /=Tj ; 1;

for t  Tj 0 ; for Tj 0  t  Tj 0 C Tj ; for  Tj 0 C Tj  t;

(11.68)

166

11 The Group Velocity Approximation 2

10

1

10

0

10

~

|uT ( )|

–1

10

c –2

10

–3

10

–4

10

–5

10

–6

10

10

0

10

1

10

2

3

4

10

10

5

10

6

10

(r/s)

Fig. 11.4 Log–log plot of the angular frequency dependence of the magnitude of the spectrum for the rectangular pulse envelope modulated signal for a single cycle pulse with carrier frequency !c D 1  103 r=s and initial pulse duration T D 2=!c D 6:28 ms. The dashed curve displays the j!  !c j1 angular frequency dependence of the step function signal at the same carrier frequency

10

2

10

1

10

0 c

–1

~

|uT ( )|

10

–2

10

–3

10

–4

10

–5

10

–6

10

0

10

1

10

2

10

3

10 (r/s)

10

4

10

5

10

6

Fig. 11.5 Log–log plot of the angular frequency dependence of the magnitude of the spectrum for the rectangular pulse envelope modulated signal for a 10-cycle pulse with carrier frequency !c D 1  103 r=s and initial pulse duration T D 2=!c D 62:8 ms. The dashed curve displays the j!  !c j1 angular frequency dependence of the step function signal at the same carrier frequency

11.2 The Pulsed Plane Wave Electromagnetic Field

167 uT (t)

1

AT (0,t)

T+(Tr+Tf ) /2

0

–uT (t)

–1 –0.04 –0.02

0

0.02

0.04 t (s)

0.06

0.08

0.1

0.12

Fig. 11.6 Temporal field structure of a 10-cycle (between the half-amplitude points in the envelope function), unit amplitude, trapezoidal envelope modulated sinusoidal signal with !c D 1  103 r=s and equal rise- and fall-times Tr D Tf D 2=fc . The dashed curves describe the envelope function ˙uT .t /

for j D r; f . The total initial pulse duration is then given by T C Tr C Tf and the half-amplitude pulse width is T C .Tr C Tf /=2, as illustrated in Fig. 11.6 with Tr0 D 0s. The temporal angular frequency spectrum of this trapezoidal envelope function is then given by Z uQ t rapj .!/ D

1 1

ut rapj .t /e i!t dt

Z 1 Z Tj 0 CTj 1 i!t .t  Tj 0 /e dt C e i!t dt D Tj Tj 0 Tj 0 CTj   i D sinc !Tj =2 e i!.Tj 0 CTj =2/ ; !

(11.69)

where sinc. /  sin . /= . Notice that in the infinite rise-time (or fall-time) limit as  Tj ! 1, sinc !Tj =2 ! ı.!/ and the initial trapezoidal envelope signal spectrum becomes lim uQ t rapj .!/ D

Tj !1

i ı.!/e i!Tj 0 ; !

168

11 The Group Velocity Approximation

and a monochromatic,   time-harmonic signal is obtained. In the opposite limit as Tj ! 0, sinc !Tj =2 ! 1 and the initial trapezoidal envelope signal spectrum becomes lim uQ t rapj .!/ D

Tj !0

i i!Tj 0 ; e !

which is precisely the ultrawideband spectrum for a step function envelope signal [cf. (11.56)]. Notice that the trapezoidal envelope function is continuous with a discontinuous first derivative at both t D Tj 0 and t D Tj , whereas the Heaviside step function envelope is discontinuous in both its value and its first derivative at t D Tj 0 . The trapezoidal envelope function then retains just the latter feature of the step function envelope, albeit displaced in time by the initial rise-time Tr . In general, the envelope spectrum uQ t rapj .!  !c / described by (11.69) for a trapezoidal envelope signal with fixed angular carrier frequency !c > 0 will be ul> trawideband provided that the inequality 2=Tj  !c is satisfied. In that case, the 1 spectral factor .!  !c / that is characteristic of an ultrawideband signal will remain essentially unchanged over the positive angular frequency domain Œ0; !c , as illustrated in Fig. 11.7. This inequality is equivalent to the inequality <

Tj  T c ;

(11.70)

for j D r; f , where Tc 1=fc D 2=!c is the oscillation period of the timeharmonic carrier wave that is modulated by the trapezoidal envelope function given

2

Tr / Tc = 0.1

1 Tr / T c = 1 Tr / Tc = 10

~

0

−1

−2

0

1

2

3

4

Fig. 11.7 Comparison of the angular frequency dependence of the sine-function component of the trapezoidal envelope signal (solid curves) with that for the Heaviside step function signal (dashed curve) for several values of the initial signal rise-time Tr relative to the period Tc D 1=fc of the carrier wave

11.2 The Pulsed Plane Wave Electromagnetic Field

169

103

c

101

|~utrapr (

–

c)

|

102

100

10–1 Tr / Tc = 0.1 T r / Tc = 1

10–2

10–3

Tr / Tc = 10 0

1

2

3

4

c

Fig. 11.8 Comparison of the angular frequency dependence of the magnitude of the trapezoidal envelope signal spectrum (solid curves) with that for the Heaviside step function signal (dashed curve) for several values of the initial trapezoidal signal rise-time Tr relative to the period Tc D 1=fc of the carrier wave

in (11.68). A comparison of the angular frequency dependence of the magnitude of the trapezoidal envelope signal spectrum with that for the Heaviside step function signal is illustrated in Fig. 11.8. Notice that the transition of the trapezoidal envelope spectrum to an ultrawideband character occurs when the initial rise-time decreases below the value Tr Tc , in agreement with the inequality given in (11.70).

11.2.6 The Hyperbolic Tangent Envelope Modulated Signal To more fully investigate the rise-time effects of a finite turn-on time signal, consideration is now given to the hyperbolic tangent envelope modulated signal with envelope function uht .t /

 1 1 1 C tanh .ˇT t / D ; 2 1 C e 2ˇT t

(11.71)

where the real-valued parameter ˇT > 0 is indicative of the rapidity of turn-on of the signal. In contrast with the trapezoidal envelope function, this envelope function is continuous in time with continuous derivatives for all finite, positive values of ˇT . The corresponding rise-time is then seen to be inversely proportional to ˇT , so that Tr  1=ˇT , as seen in Fig. 11.9. In the limit as ˇT ! 1, uht .t / ! uH .t /

170

11 The Group Velocity Approximation

1

b T = 10 b T =1

0.8

uht (t)

b T = 0.1 0.6

0.4

0.2

0 −10

−8

−6

−4

−2

0

2

4

6

8

10

t

Fig. 11.9 Time dependence of the hyperbolic tangent envelope function. For whatever units the time scale t is in, the scale of the inverse rise-time parameter ˇT is given by its inverse so that the product ˇT t remains dimensionless

and the Heaviside unit step function envelope is obtained. In the opposite limit as ˇT ! 0, uht .t / ! 12 which results in a time-harmonic signal with amplitude of 1=2. An example of a hyperbolic tangent envelope modulated signal Aht .0; t / D uht .t / sin .!c t / is illustrated in Fig. 11.10 when ˇT D !c =10. In that case the initial signal rise-time occurs in approximately ten oscillations of the carrier wave. The Fourier transform of the trapezoidal envelope function is given by the integral Z

1

1 e i!t dt 2ˇT t 1 C e 1 Z 1 1 1 x i!=ˇT dx; D 2 ˇT 0 x .x C 1/

uQ ht .!/ D

under the change of variable x D e ˇT t . For convergence, the variable ! is complex-valued and lies in the upper-half of the complex !-plane; that is, ! 00 =f!g > 0. To evaluate this definite integral, consider the following contour integral in the complex z-plane (with z D x C iy) I I.˛/

zi˛ d z; z.z2 C 1/

11.2 The Pulsed Plane Wave Electromagnetic Field

171

1

Aht(0,t)

uht(t)

0

−uht(t) −1 −10

0 t

10

Fig. 11.10 Initial temporal field behavior Aht .0; t / of a hyperbolic tangent envelope modulated signal with inverse rise-time parameter ˇT D !c =10, where !c is the angular carrier frequency of the signal. The dashed curves describe the hyperbolic tangent envelope functions ˙uht .t /

for complex ˛ D ˛ 0 C i ˛ 00 with ˛ 00 > 0, where the integrand has simple poles at z D 0; ˙i and a branch point at z = 0. With the positive real axis chosen as the branch cut, the contour of integration to be used in evaluating this integral is as illustrated in Fig. 11.11. Along the contour C1 , z D x so that, in both the limit as R ! 1 and the limit as  ! 0 (see Fig. 11.11), Z C1

zi˛ dz D z.z2 C 1/

Z

1

0

x i˛ dx; x.x 2 C 1/

whereas along the contour C2 , z D xe i2 so that in the same limits as R ! 1  ! 0, Z C2

zi˛ d z D e 2 ˛ z.z2 C 1/

Z

1 0

x i˛ dx; x.x 2 C 1/

Along the outer circular contour  (see Fig. 11.1), z D Re i , and the following inequality is obtained ˇ Z 2 ˇZ 00 ˇ ˇ zi˛ R˛ 0 ˇ ˇ d z e ˛  d; ˇ z.z2 C 1/ ˇ 2 R C1  0

172

11 The Group Velocity Approximation iy

+i

R C1 C2

Branch Cut

x

–i

Complex z-Plane

Fig. 11.11 Contour of integration used in the evaluation  of the Fourier transform of the hyperbolic tangent envelope function uht .t / D 12 1 C tanh .ˇT t / . The circular contour  has radius R and extends over the angular domain  2 .0; 2/ in the counterclockwise sense, and the circular contour  has radius  and extends over the angular domain  2 .2; 0/ in the clockwise sense. The straight line contours C1 and C2 connect these two circular contours on either side of the branch cut taken along the positive x-axis, C1 extending from  to R in the upper-half plane and C2 extending from R to  in the lower-half plane

where the integral on the right-hand side of this inequality goes to zero as R ! 1 for ˛ 00 < 2. In addition, along the inner circular contour  , z D e i , and the following inequality is obtained ˇ Z 2 ˇZ 00 ˇ ˇ zi˛ ˛ 0 ˇ ˇ e ˛  d; ˇ z.z2 C 1/ d zˇ  2C1  0  where the integral on the right-hand side of this inequality goes to zero as  ! 0 for ˛ 00 > 0. Finally, by application of the residue theorem to the evaluation of the integral I.˛/, taking note that only the simple pole singularities at z D ˙i are enclosed by the contour, there results I

   zi˛ zi˛ Res C zDi zDCi z.z C i /.z  i / z.z C i /.z  i /    3 ˛ ˛ D i  e 2 C e 2 :

zi˛ d z D 2 i z.z2 C 1/



Res



Taken together, these integral evaluations then yield the result Z 1 =2 x i˛ dx D i I 0 < =f˛g < 2: 2 C 1/ .˛=2/ x.x sinh 0

11.2 The Pulsed Plane Wave Electromagnetic Field

173

The temporal frequency spectrum of the hyperbolic tangent envelope function defined in (11.71) is then given by uQ ht .!/ D i

=.2ˇT / I sinh .!=.2ˇT //

0 < =f!g < 2ˇT :

(11.72)

Since sinh .z/ D 0 at z D ˙n i for n an integer, the right-hand side of (11.72) possesses simple pole singularities at ! D !˙n where !˙n ˙2nˇT i I

n D 0; 1; 2; 3; : : : ;

(11.73)

so that the spectrum of the hyperbolic tangent envelope function possesses an infinite number of simple pole singularities evenly spaced along the imaginary ! 00 -axis with spacing 2ˇT . The inequality 0 < =f!g < 2ˇT appearing in (11.72) requires that the contour of integration appearing in the integral representation (11.48) of the propagated pulse wave field lies in the upper half of the complex !-plane between the real axis and the line parallel to the real axis passing through the first (n D 1) simple pole singularity at ! D !1 D 2ˇT i . Notice that in the limit as ˇT ! 1, the spectrum for the hyperbolic tangent envelope function given in (11.72) approaches the limit lim uQ ht .!/ D

ˇT !1

i !

(11.74)

which is precisely the expression for the temporal frequency spectrum of the Heaviside unit step function envelope [cf. (11.56)]. A comparison of the relative angular frequency dependence !=!c of the magnitude of the hyperbolic tangent envelope signal spectrum with that for the Heaviside step function signal for several values of the relative inverse rise-time parameter ˇT =!c is presented in Fig. 11.12. As for the trapezoidal envelope modulated signal, the hyperbolic tangent envelope modulated signal is seen to become ultrawideband when the approximate inequality >

ˇT =!c  1;

(11.75)

is satisfied, where !c is the angular frequency of the carrier wave, and becomes increasingly ultrawideband as ˇT ! 1, as seen in Fig. 11.12. Effectively, this inequality means that the initial envelope rise-time Tr  1=ˇT of the signal Aht .0; t / D uht .t / sin .!c t / occurs in approximately a single period of oscillation of the carrier wave or faster. Notice from (11.73) that the simple pole singularities !˙n D !c ˙ 2nˇT i move away from the real axis toward !c ˙ 1i as ˇT =!c increases above unity and the initial pulse envelope becomes increasingly ultrawideband, leaving just the single simple pole singularity at ! D !c along the positive real axis when ˇT D 1 and the hyperbolic tangent envelope signal has attained its Heaviside step function envelope signal limit.

174

11 The Group Velocity Approximation

102 100 10−2 10−4 10−6 10−8 10−10

0

1

2

3

4

Fig. 11.12 Comparison of the relative angular frequency dependence of the magnitude of the hyperbolic tangent envelope signal spectrum (solid curves) with that for the Heaviside step function signal (dashed curve) for several values of the inverse rise-time parameter ˇT relative to the angular carrier frequency !c

11.2.7 The Van Bladel Envelope Modulated Pulse An example of an infinitely smooth envelope function with compact temporal support (i.e., one that identically vanishes outside of a finite time domain) is given by the unit amplitude Van Bladel envelope function [50] ( uvb .t /



e

2

1C 4t.t  /

0I



I

when 0 < t <  ; when either t  0 or t  

(11.76)

p with temporal duration  > 0 and full pulse width = 2 at the e 1 amplitude points in the envelope function, as illustrated in Fig. 11.3 for a 2-cycle pulse ( D 2Tc ) and in Fig. 11.14 for a 10-cycle pulse ( D 10Tc ), with Tc 1=fc D 2=!c for a cosine carrier wave. This canonical pulse envelope function is of some importance to ultrashort optical pulse dynamics because its properties of infinite smoothness and temporal compactness are common to all experimental pulses. Notice that, although the Van Bladel envelope function equals unity at its midpoint when t D =2, the resultant modulated carrier wave will not unless one of its peak amplitude points coincides with the midpoint of the envelope function, as it does for the examples presented here with a cosine carrier wave.

11.2 The Pulsed Plane Wave Electromagnetic Field

175

1 uvb(t)

Avb(0,t)

_

0

–1 −0.01

–uvb(t) 0

0.01

t - t /2 (s) Fig. 11.13 Temporal field structure of a 2-cycle Van Bladel envelope modulated pulse with angular carrier frequency !c D 1103 r=s and temporal duration  D 2Tc . The dashed curves describe the envelope function ˙uvb .t /

1

Avb(0,t)

uvb(t)

0

–uvb(t) −1 −0.05

0

0.05

t - t /2 (s)

Fig. 11.14 Temporal field structure of a 10-cycle Van Bladel envelope modulated pulse with angular carrier frequency !c D 1  103 r=s and temporal duration  D 10Tc . The dashed curves describe the envelope function ˙uvb .t /

176

11 The Group Velocity Approximation 100 10−1 10−2 10−3

1

−4

10

10−5

/ Tc = 2

−6

10

10−7 /Tc = 10

10−8 10−9 10−10 1 10

102

103

104

105

(r/s)

Fig. 11.15 Comparison of the relative angular frequency dependence of the magnitude of the Van Bladel envelope pulse spectrum (solid curves) with that for the Heaviside step function signal (dashed curve) for several values of the initial temporal pulse duration 

Because the Van Bladel envelope function defined in (11.76) possesses compact temporal support, its Fourier transform uQ vb .!/ is an entire function [51] of complex !, where Z    2 1C 4t.t  / i!t e e dt: (11.77) uQ vb .!/ D 0

An accurate numerical evaluation of this Fourier transform integral may be accomplished using the fast Fourier transform (FFT) algorithm, provided that due care is given to both sampling and aliasing [52]. The results are presented in Fig. 11.15 for the spectral magnitude jQuvb .! !c /j with angular carrier frequency !c D 1103 r=s for both the two ( D 2Tc ) and 10-cycle ( D 10Tc ) Van Bladel envelope modulated pulses illustrated in Figs. 11.13 and 11.14, respectively. For comparison, the dashed curve in the figure describes the j!  !c j1 ultrawideband frequency behavior of the Heaviside step function envelope signal. Because of its infinitely smooth character, the Van Bladel envelope pulse is not as ultrawideband as its corresponding rectangular envelope modulated pulse is, as seen from a comparison of the two spectral amplitude curves in Fig. 11.15 with their rectangular envelope counterparts in Figs. 11.4 (for a single cycle rectangular envelope pulse) and 11.5 (for a 10-cycle rectangular envelope pulse).

11.2 The Pulsed Plane Wave Electromagnetic Field

177

11.2.8 The Gaussian Envelope Modulated Pulse An example of an infinitely smooth pulse envelope function that does not possess compact temporal support is given by the unit amplitude gaussian envelope function 2 =T 2

ug .t / D e .tt0 /

(11.78)

that is centered at the time t D t0 with initial pulse width t D 2T measured at the e 1 amplitude points, as illustrated in Fig. 11.16 when t0 D 0 and D =2 [which then corresponds to a cosine carrier wave, as described in (11.34)]. Notice that the choice of a cosine carrier wave places a peak at the peak amplitude point, whereas a sine carrier wave places a null at the peak amplitude point. The temporal frequency spectrum of the gaussian envelope function given in (11.78) is given by Z

1

2 =T 2

e .tt0 /

uQ g .!/ D

e i!t dt

1

D

p

1

T e  4 T

2!2

e i!t0 ;

(11.79)

which is another gaussian with angular frequency width ! D 4=T at the e 1 amplitude points. One then has the uncertainty product f t D 2=. For comparison,

1 ug(t)

Ag(0,t)

2T

0

−ug(t) −1 −0.01

0

0.01

t (s) Fig. 11.16 Temporal field structure of a 2-cycle gaussian envelope modulated pulse centered at t0 D 0 with angular carrier frequency !c D 1  103 r=s and temporal duration 2T D 2Tc . The dashed curves describe the envelope function ˙ug .t /

178

11 The Group Velocity Approximation 100

c

T = Tc 10

−5

T = 2Tc

T = 3Tc 10

−10

10

0

10

1

10

2

10

3

10

4

(r/s)

Fig. 11.17 Comparison of the relative angular frequency dependence of the gaussian envelope pulse spectrum (solid curves) with that for the Heaviside step function signal (dashed curve) for several values of the initial temporal pulse width 2T

the dashed curve in the figure describes the j!  !c j1 ultrawideband frequency behavior of the Heaviside step function envelope signal. Because of its infinitely smooth character, the gaussian envelope pulse spectrum is similar to that for the Van Bladel envelope pulse, as seen from a comparison of Figs. 11.15 and 11.17.

11.3 Wave Equations in a Simple Dispersive Medium and the Slowly Varying Envelope Approximation Because of its direct generalizability to pulse propagation in nonlinear media, the wave equations describing electromagnetic pulse propagation in a dispersive medium are of central importance to nonlinear optics in particular [24, 26, 53] and nonlinear wave phenomena in general [54, 55]. This approach parallels the Fourier integral formulation in the linear dispersion regime but, most importantly, it best illustrates the simplifying assumptions that are made in typical treatments of dispersive pulse propagation phenomena.

11.3 Dispersive Wave Equations

179

The analysis is presented here for the case of a simple polarizable dielectric2 for which the electric displacement vector is given by [see (4.50) and (4.91) of Vol. 1] D.r; t / D 0 E.r; t / C k4kP.r; t /;

(11.80)

where P.r; t / is the macroscopic polarization density [see (4.27) of Vol. 1], and where both B.r; t / D 0 H.r; t / and Jc .r; t / D 0 are satisfied. The temporal Fourier transform of the macroscopic polarization density is related to the macroscopic electric field intensity through the electric susceptibility e .!/ of the material as Q !/ Q !/ D 0 e .!/E.r; P.r;

(11.81)

where e .!/ (a macroscopic quantity) is related to the set of molecular polarizabilities ˛j .!/ of the material (microscopic quantities) through the expression P

e .!/ D

Nj ˛j .!/ P 0  .k4k=3/ j Nj ˛j .!/ j

(11.82)

which follows from the Clausius–Mossotti relation [see (4.169) in Vol. 1] X

Nj ˛j .!/ D

j

30 .!/=0  1 ; k4k .!/=0 C 2

(11.83)

which is also referred to as the Lorentz–Lorenz formula. Here Nj is the number density of molecular species with microscopic polarizability ˛j .!/, the latter quantity determined by a dynamical model for the molecular response to an appled timeharmonic electromagnetic wave field.

11.3.1 The Dispersive Wave Equations The source-free, time-domain form of Maxwell’s equations in a simple polarizable dielectric are given by 0 @H.r; t / ; kck @t   4  0 @E.r; t /   @P.r; t / ; C r  H.r; t / D  kck @t c  @t r  E.r; t / D 

2

(11.84) (11.85)

A simple polarizable dielectric is defined here as one for which the quadrupole and all higherorder moments of the molecular charge distribution identically vanish. Equivalently, it is one for which the electric displacement vector is described by (11.80) without approximation

180

11 The Group Velocity Approximation

Rt with r  D.r; t / D r  B.r; t / D 0. Since D.r; t / D 1 .t  t 0 /E.r; t 0 /dt 0 , then Rt r  D.r; t / D 1 .t  t 0 /r  E.r; t 0 /dt 0 D 0, so that r  E.r; t / D 0. The curl of (11.84) then gives, with substitution from (11.85) and the divergenceless character of the electric field intensity vector, r 2 E.r; t / 

   4  @2 P.r; t / 1 @2 E.r; t /   0 D :  c2  c 2 @t 2 @t 2

(11.86)

Similarly, the curl of (11.85) gives, with substitution from (11.84) and the divergenceless character of the magnetic intensity vector, r 2 H.r; t / 

   4  @   1 @2 H.r; t /   r  P.r; t / : D    2 2 c @t c @t

(11.87)

This pair of expressions are the inhomogeneous vector wave equations in a simple dispersive medium. With emphasis typically placed on the electric field component of the electromagnetic wave field, the electric field vector is then taken to be linearly polarized along the 1O x -direction with the direction of propagation in the positive z-direction. In that case E.r; t / D 1O x E.r; t / so that P.r; t / D 1O x P .r; t / and (11.86) becomes    4  @2 P .r; t / 1 @2 E.r; t /   0 r E.r; t /  2 D ;  c2  c @t 2 @t 2 2

(11.88)

which is the inhomogeneous scalar wave equation in a simple dispersive medium. The proper (i.e., without unnecessary simplifying assumptions and approximations) solution of either (11.86) and (11.87) for the electromagnetic wave field or (11.88) for the scalar optical wave field is entirely equivalent to the proper solution of the Fourier–Laplace integral representation for the electromagnetic wave field vectors given, for example, in (11.43) and (11.44), or to the Fourier–Laplace integral representation for the scalar wave field in (11.48). Finally, the central importance of this partial differential equation approach to dispersive wave propagation phenomena is fully realized when the macroscopic polarization density P .r; t / is extended to include nonlinear effects [56–58]. In that case, the polarization is written as P .r; t / D PL .r; t / C PNL .r; t / where PL .r; t / describes the linear response and PNL .r; t / describes the nonlinear response which vanishes as E ! 0.

11.3.2 The Slowly Varying Envelope Approximation The precise definition of the slowly varying envelope (SVE) or quasimonochromatic approximation has its origin in the classical theory of coherence developed by M. Born and E. Wolf [6]. Consider first the complex representation of a real

11.3 Dispersive Wave Equations

181

polychromatic scalar wave w.r; t / that exists for all time t 2 .1; 1/ and is square-integrable, viz., Z 1 w2 .r; t /dt < 1 (11.89) 1

at each point r 2 R . By the Fourier integral theorem [59], the wave field w.r; t / may then be represented by the Fourier integral expression 3

w.r; t / D

1 2

where

Z

1

w.r; Q !/e i!t d!;

(11.90)

w.r; t /e i!t dt:

(11.91)

1

Z

1

w.r; Q !/ D 1

Since w.r; t / is real, it then follows from (11.91) that w Q  .r; !/ D w.r; !/ for real ! and the real part of the wave spectrum is even-symmetric and the imaginary part is odd-symmetric. Because no additional information is then contained in the frequency spectrum w.r; Q !/ for ! < 0, Gabor [60] introduced the complex analytic signal Z 1 1 w.r; Q !/e i!t d! (11.92) v.r; t / 2 0 associated with the real signal w.r; t /. Notice that the complex analytic signal v.r; t / is obtained from its associated wave field w.r; t / at each point r of space simply by suppressing the amplitudes of all of the negative frequency components in the Fourier integral representation of w.r; t / given in (11.90). Because of this, v.r; t / is also referred to as the complex half-range function associated with the real scalar wave field w.r; t /. Let the angular frequency spectrum w.r; Q !/ of the scalar wave field u.r; t / be expressed in the form 1 (11.93) w.r; Q !/ D a.r; !/e i'.r;!/ ; 2 where both a.r; !/ and '.r; !/ are real-valued functions. Substitution of this representation into (11.90) then gives Z 1

Z 1 1 i!t  i!t w.r; t / D w.r; Q !/e d! C wQ .r; !/e d! 2 0 0   Z 1    1 1 i '.r;!/!t i '.r;!/!t Ce d! D a.r; !/ e 2 0 2 Z 1   1 D a.r; !/ cos '.r; !/  !t d!; (11.94) 2 0 whereas substitution of (11.93) in (11.92) gives v.r; t / D

1 4

Z 0

1

a.r; !/e i



'.r;!/!t



d!:

(11.95)

182

11 The Group Velocity Approximation

Comparison of these two expressions then shows that ˚ w.r; t / D 2< v.r; t / :

(11.96)

Application of the Plancherel–Parseval theorem [51] then shows that Z

Z 1 1 w .r; t /dt D Q !/j2 d! jw.r; .2/2 1 1 Z 1 Z 1 1 2 D Q !/j d! D 2 jw.r; jv.r; t /j2 dt 2 2 0 0 Z 1 Z 1 1 1 2 a .r; !/d! D a2 .r; !/d!: D 8 2 0 .4/2 1 1

2

(11.97)

These general results are now applied to the description of the properties of a quasimonochromatic wave field. Let the Fourier spectrum w.r; Q !/ of the real scalar wave field w.r; t / be centered about the angular frequencies C!0 and !0 with effective width !, as illustrated in Fig. 11.18. The wave field is then said to be quasimonochromatic if the effective width ! is small in comparison with the angular frequency !0 ; that is, provided that !  1: !0

(11.98)

~ wrel (r, )

1

0.5

0

–

0

(r/s)

0

Fig. 11.18 Angular frequency dependence of the normalized magnitude of the frequency spectrum of a quasimonochromatic wave centered about the angular frequency !0 with spectral width !

11.3 Dispersive Wave Equations

183

A polychromatic wave field is then defined as a superposition of mutually incoherent quasimonochromatics extending over a finite range of frequencies. As an illustration, consider the case of a strictly monochromatic wave field, as described by the real wave function w.r; t / D A0 .r/ cos .'0 .r/  !0 t / D 0 .r/e i!0 t C 0 .r/ e i!0 t

(11.99)

with 0 .r/ D

1 A0 .r/e i'0 .r/ : 2

(11.100)

The temporal frequency spectrum of this monochromatic scalar wave field is then given by the Fourier transform of (11.99) as 

w.r; Q !/ D 2 0 .r/ı.!  !0 / C 0 .r/ı.! C !0 / :

(11.101)

The associated complex analytic signal to this strictly monochromatic wave field is then given by Z

1

v.r; t / D 0

 0 .r/ı.!  !0 / C 0 .r/ı.! C !0 / e i!t d!

D 0 e i!0 t D

1 A0 .r/e i.'0 .r/!0 t/ : 2

(11.102)

The quasimonochromatic generalization of the scalar wave feld given in (11.99) may be synthesized from it by including temporal dependency in both the amplitude and phase functions as w.r; t / D A.r; t / cos .'.r; t /  !0 t /:

(11.103)

By analogy with (11.102), let the complex analytic signal corresponding to the scalar wave field in (11.103) be of the form v.r; t / D

1 A.r; t /e i.'.r;t/!0 t/ ; 2

(11.104)

v.r; t / D

 1 w.r; t / C i u.r; t / ; 2

(11.105)

which may be written as

where u.r; t / D A.r; t / sin .'.r; t /  !0 t /;

(11.106)

184

11 The Group Velocity Approximation

from (11.103) and (11.104). With (11.103) and (11.106) expressed as w.r; t / D A.r; t / cos ..r; t // and u.r; t / D A.r; t / sin ..r; t //, respectively, the amplitude and phase functions are found to be given by

1=2 A.r; t / D 2jv.r; t /j D w2 .r; t / C u2 .r; t / ; '.r; t / D !0 t C .r; t /; mod.2/;

(11.107) (11.108)

where cos ..r; t // D

w.r; t / ; 2jv.r; t /j

sin ..r; t // D

u.r; t / ; 2jv.r; t /j

(11.109)

at each point r 2 R3 . From (11.92) and (11.104) one has that A.r; t /e i'.r;t/ D

1 

Z

1

w.r; Q !/e i.!!0 /t d!;

(11.110)

0

at each point r 2 R3 . Under the change of variable !N D !  !0 , this transform relation becomes Z 1 1 Q i'.r;t/ N A.r; t /e D d !; N (11.111)

.r; !/e N i !t  !0 Q !/ where .r; N D w.r; Q ! C !0 / for !N  !0 describes the Fourier spectrum of a field which is essentially contained in the low frequency domain .!0 ; !0 /. Because the original wave field was assumed to be quasimonochromatic, the spectral width Q !/ ! of .r; N satisfies the inequality !  !0 and this then implies that the quantity A.r; t /e i'.r;t/ essentially contains only low frequency components. The quantity A.r; t /e i'.r;t/ , which is referred to as the complex envelope of the wave field, is then essentially constant over a time interval t satisfying the inequality t !  1, so that with !  !0 , the complex envelope is seen to be essentially constant over the time interval T0 D 2=!0 . Hence, the complex envelope of a quasimonochromatic wave field changes by only a negligible amount over a few oscillations of the carrier wave, so that the two inequalities ˇ ˇ ˇ ˇ 2 ˇ ˇ ˇ @ A.r; t / ˇ ˇ  !0 ˇ @A.r; t / ˇ ; ˇ ˇ @t ˇ ˇ @t 2 ˇ ˇ 2 ˇ ˇ ˇ ˇ @ '.r; t / ˇ ˇ ˇ ˇ ˇ  !0 ˇ @'.r; t / ˇ ; ˇ @t 2 ˇ ˇ @t ˇ are both satisfied.

(11.112) (11.113)

11.3 Dispersive Wave Equations

185

11.3.3 Dispersive Wave Equations for the Slowly Varying Wave Amplitude and Phase Let the quasimonochromatic scalar wave field E.r; t / have the complex phasor representation (see Sect. 5.1.2 of Vol. 1) E!c .r; t / A.r; t /e i.'.r;t/!c t/ ;

(11.114)

˚ where E.r; t / D < E!c .r; t / with A.r; t / and '.r; t / both being real-valued functions of position r and time t . The phasor representation of the induced polarization density [see (11.81)] is then given by P!c .r; t / 0 e .!c /E!c .r; t / D 0 e .!c /A.r; t /e i.'.r;t/!c t/ ;

(11.115)

˚ where P .r; t / D < P!c .r; t / . With these two substitutions, the scalar wave equation given in (11.88) becomes r 2 E!c .r; t / 

1 @2 E!c .r; t / .1 C k4k .! // D 0; e c c2 @t 2

which may be written as r 2 E!c .r; t / 

n2 .!c / @2 E!c .r; t / D 0; c2 @t 2

(11.116)

where n.!/ D .1 C k4ke .!//1=2 is the complex index of refraction of the dispersive medium [see (4.92) of Vol. 1]. This phasor form of the scalar wave equation is then seen to be characterized by the complex phase velocity vp .!/

c : n.!/

(11.117)

Finally, notice that (11.116) has been obtained without any approximation. This then makes it an ideal starting point for any approximate description of dispersive scalar wave propagation phenomena. Consider now the approximation of (11.116) in the SVE approximation as specified by the two inequalities in (11.112) and (11.113). Because @E!c D @t



 @A @' C iA  i !c A e i.'!c t/ ; @t @t

186

11 The Group Velocity Approximation

then @2 E!c D @t 2

"

 2 @2 ' @2 A @A @' @' C iA C 2i  A 2 2 @t @t @t @t @t

# @A @' 2i !c C 2!c A  !c2 A e i.'!c t/ @t @t "  ! #  2  @' @A @' @' 2  !c C 2!c A 

2i C !c A e i.'!c t/ : @t @t @t @t

The SVE approximation of the scalar wave equation given in (11.116) is then found to be given by   r 2 C kQ 2 .!c / A.r; t /e i'.r;t/ "   @A.r; t / n2 .!c / @'.r; t /  2i  !c c2 @t @t #   @'.r; t / i'.r;t/ @'.r; t /  2!c A.r; t /

0; C e @t @t (11.118) Q where k.!/ D .!=c/n.!/ is the complex wavenumber in the dispersive medium with complex index of refraction n.!/. This is then the general form of the scalar wave equation in the SVE approximation. Because   r 2 A.r; t /e i'.r;t/ D r  r A.r; t /e i'.r;t/   D r  .rA/e i' C iA.r'/e i'   D r 2 A C 2i.rA/  .r'/ C iAr 2 '  A.r'/2 e i' ; the SVE wave equation given in (11.118) becomes 

 r 2 C kQ 2 .!c / A.r; t / C 2i.rA.r; t //  .r'.r; t // CiA.r; t /r 2 '.r; t /  A.r; t /.r'.r; t //2 "   @A.r; t / n2 .!c / @'.r; t /  !  2i c c2 @t @t #   @'.r; t / @'.r; t /  2!c A.r; t / C

0; @t @t (11.119)

11.3 Dispersive Wave Equations

187

after the common factor e i'.r;t/ has been cancelled. As no additional approximations have been made, this form of the SVE wave equation is equivalent to that given in (11.118). Because A.r; t / and '.r; t / are both real-valued functions, then together with the facts that n2 .!c / D .nr .!c / C i ni .!c //2 D n2r .!c /  n2i .!c / C 2i nr .!c /ni .!c / with nr .!/
0; C4 c2 @t @t (11.120)

n2 .!c /  n2i .!c /  r c2



1 A.r; t /r 2 '.r; t / C .rA.r; t //  .r'.r; t // 2

˛.!c /ˇ.!c /A.r; t /   n2 .!c /  n2i .!c / @'.r; t / @A.r; t / C r  ! : c 2 c @t @t   @'.r; t / nr .!c /ni .!c / @'.r; t /  2! : C c A.r; t / 2 c @t @t (11.121) This pair of equations explicitly displays the coupling between the slowly varying amplitude and phase functions A.r; t / and '.r; t /, respectively. As pointed out by Butcher and Cotter [26], the SVE approximation is a hybrid time and frequency domain representation in which the temporal pulse behavior is separated into the product of a slowly varying temporal envelope function and an exponential phase term whose angular frequency is centered about some characteristic angular frequency !c of the pulse, as has been done in (11.114). The fundamental difficulty with the SVE approximation occurs when the inequalities given in (11.112) and (11.113) are applied. Because of these rather innocent looking approximations, the fundamental hyperbolic character of the wave equation has been approximated as parabolic. The characteristics of the SVE wave equation then propagate instantaneously [11] in violation of relativistic causality. The appearance of superluminal pulse velocities in such an approximate theory should then be of no surprise and, more importantly, of little or no consequence.

188

11 The Group Velocity Approximation

If the propagation direction is primarily in the positive z-direction, the amplitude and phase functions may then be expressed as A.r; t / D a.r; t /e ˛.!c /z ;

(11.122)

'.r; t / D ˇ.!c /z C .r; t /;

(11.123)

where the amplitude and phase functions a.r; t / and .r; t /, respectively, are both slowly varying functions of z, satisfying the respective inequalities ˇ ˇ ˇ ˇ 2 ˇ ˇ ˇ @ a.r; t / ˇ ˇ  k.!c / ˇ @a.r; t / ˇ ; ˇ ˇ @z ˇ ˇ @z2 ˇ ˇ 2 ˇ ˇ ˇ ˇ @ .r; t / ˇ ˇ ˇ ˇ ˇ  k.!c / ˇ @.r; t / ˇ ; ˇ @z2 ˇ ˇ @z ˇ

(11.124) (11.125)

p Q where k.!/ jk.!/j D ˇ 2 .!/ C ˛ 2 .!/. Notice that @A=@t D @a=@t and @'=@t D @=@t . In addition,   @ rA.r; t / D rT C 1O z a.r; t /e ˛.!c /z @z    @a.r; t /  ˛.!c /a.r; t / e ˛.!c /z ; D rT a.r; t / C 1O z @z where rT 1O x @=@x C 1O y @=@y, so that      @a @ O O  rT a C 1z  ˛.!c /a e ˛.!c /z r A.r; t / D rT C 1z @z @z   @2 a @a 2 2 C ˛ .!c /a e ˛.!c /z D rT a C 2  2˛.!c / @z @z   @a.r; t / 2 2 C ˛ .!c /a.r; t / e ˛.!c /z ;

rT a.r; t /  2˛.!c / @z 2

which could also have been obtained from the identity r 2 D rT2 C @2 =@z2 , and r'.r; t / D r.r; t / C 1O z ˇ.!c / so that r 2 '.r; t / D r 2 .r; t /: With these additional approximations, the SVE wave equation given in (11.119) becomes

11.3 Dispersive Wave Equations

189





@a @ C 2i ˛.!c /ˇ.!c /a  .rT /2 C 2ˇ.!c / a rT2 a  2˛.!c / @z @z     1 2 @a @ C2i ar  C .rT a/  .rT / C  ˛.!c /a C ˇ.!c / 2 T @z @z "  #    @ @a @ @ n2 .!c /  ! C  2! 2i 

0: c c a 2 c @t @t @t @t (11.126) To obtain a better physical understanding of this equation, as well as of the SVE wave equation given in (11.119), two special cases are now considered. Consider first the special case of a time-harmonic wave field. In that case, both A.r; t / and '.r; t / are independent of the time (viz. @A=@t D @'=@t D 0) and (11.119) reduces to   r 2 C kQ 2 .!c / A.r/ C 2i.rA.r/  .r'.r// CiA.r/r 2 '.r/  A.r/.r'.r//2 0: (11.127) This equation then describes the diffraction of a monochromatic scalar wave field with fixed angular carrier frequency !c in the dispersive medium. Separation of this SVE wave equation into real and imaginary parts gives 

2  2 r C ˇ .!c /  ˛ 2 .!c /  .r'.r//2 A.r/ 0; 1 A.r/r 2 '.r/ C .rA.r//  .r'.r// C ˛.!c /ˇ.!c /A.r/ 0: 2

(11.128) (11.129)

These two equations show how the amplitude and phase of a monochromatic wave field are diffractively coupled, the spatial evolution of the amplitude A.r/ dependent upon the spatial variation of the phase and the spatial evolution of the phase '.r/ dependent upon the spatial variation of the amplitude. Similar remarks hold for the monochromatic limit of the SVE wave equation given in (11.126). Consider next the case of a pulsed plane wave field traveling in the positive z-direction. In that case rT a D rT  D 0 and (11.126) becomes @a.z; t / @.z; t / C ˇ.!c /a.z; t /  i ˛.!c /ˇ.!c /a.z; t / @z @z    @.z; t / @a.z; t / 2i  ˛.!c /a.z; t / C ˇ.!c / @z @z "   @a.z; t / @.z; t / n2 .!c /  !c 2i C 2c 2 @t @t #   @.z; t / @.z; t / C  2!c a.z; t /

0: @t @t

˛.!c /

(11.130)

190

11 The Group Velocity Approximation

This two-dimensional wave equation then describes dispersive pulse evolution in the SVE approximation. Separation into real and imaginary parts yields @a.z; t / @.z; t / C ˇ.!c /a.z; t / @z @z "    1  2 @.z; t / @.z; t / C 2 nr .!c /  n2i .!c /  2!c a.z; t / 2c @t @t #   @a.z; t / @.z; t / 2nr .!c /ni .!c /  !c

0; @t @t

˛.!c /



(11.131)   @a.z; t / @.z; t /  ˛.!c /a.z; t / C ˇ.!c / C ˛.!c /ˇ.!c /a.z; t / @z @z "    @.z; t / @a.z; t / 1  2  !c  2 nr .!c /  n2i .!c / c @t @t #   @.z; t / @.z; t / Cnr .!c /ni .!c /  2!c a.z; t /

0: @t @t (11.132)

These two equations show the complicated manner in which the slowly varying amplitude a.z; t / and phase .z; t / functions are coupled for a plane wave pulse. A more physically appealing approach to the description of dispersive wave propagation in the SVE approximation is based upon that given by Akhmanov, Vysloukh, and Chirkin [23, 25]. Instead of dealing with the macroscopic polarization density P .r; t /, the inhomogeneous scalar wave equation given in (11.88) is written in terms of the electric displacement D.r; t / D 0 E.r; t / C k4kP .r; t / as r 2 E.r; t / 

1 @2 D.r; t / D 0; @t 2 0

(11.133)

O .t  t 0 /E.r; t 0 /dt 0

(11.134)

c2

where [see (4.85) of Vol. 1] Z

t

D.r; t / D 1

is the causal constitutive relation between the electric field E.r; t / and the electric displacement in a homogeneous, isotropic, locally linear (HILL), temporally dispersive dielectric. Let the quasimonochromatic wave field E.r; t / have the complex phasor representation Q t /e i!c t ; (11.135) E!c .r; t / A.r; Q t / D A.r; t /e i'.r;t/ . The constitutive relation given where [cf. (11.114)] A.r; in (11.134) states that at any fixed point r in the simple dielectric, the electric

11.3 Dispersive Wave Equations

191

displacement D.r; t / depends upon the past history of the electric field intensity E.r; t / at that point through the dielectric permittivity response function O .t  t 0 /. It is then reasonable to expect that the sensitivity of the medium response decreases as the past time t 0 decreases further into the past from the present time t at which the field D.r; t / is evaluated (see Sect. 4.3 of Vol. 1). This sensitivity to prior behavior Q t 0 / of the may be captured by expanding the complex phasor representation A.r; electric field intensity in a Taylor series about the instant t 0 D t as Q t 0/ D A.r;

1 X Q t/ 1 @m A.r; .t 0  t /m ; m mŠ @t mD0

(11.136)

Q t 0 / and all of its time derivatives exist at each which is valid provided that A.r; 0 instant t  t . Substitution of this expansion, together with (11.135), into the constitutive relation (11.134) then yields the phasor representation 1 X Q t/ Z t 1 @m A.r; 0 O .t  t 0 /.t 0  t /m e i!c t dt 0 m mŠ @t 1 mD0 Z 1 1 X Q t/ .1/m @m A.r; i!c t D e O ./ m e i!c  d ; m mŠ @t 0 mD0

D!c .r; t / D

which may be expressed as D!c .r; t / D

1 X

 .m/ .!c /

mD0

where  .m/ .!c /

.1/m mŠ

Z

Q t/ @m A.r; e i!c t ; @t m

(11.137)

1

O . / m e i!c  d 

(11.138)

1

is proportional to the mth-order moment of the dielectric permittivity response function about !c . Since Z 1 .!/ D O .t /e i!t dt; 1

so that @m .!/ D im @! m

Z

1

O .t /t m e i!t dt;

1

it is then seen that the dielectric moments about the angular carrier frequency !c , defined in (11.138), may be expressed in terms of the derivatives of the dielectric permittivity as ˇ i m @m .!/ ˇˇ : (11.139)  .m/ .!c / D mŠ @! m ˇ!c

192

11 The Group Velocity Approximation

Notice that  .0/ .!c / D .!c / is just the value of the complex-valued dielectric permittivity at the angular carrier frequency !c . Because " ! # ! m Q @2 @mC1 AQ @m AQ i!c t @mC2 AQ 2@ A  2i !c mC1  !c m e i!c t ; e D @t 2 @t m @t mC2 @t @t the complex phasor form of the scalar wave equation given in (11.133) is found as, with substitution from (11.135) and (11.137), 2 Q Q Q t / C k .!c / A.r; Q t / C 2i @A.r; t /  1 @ A.r; t / r A.r; 2 !c @t !c @t 2 2

!

Q2

1 !c2 X  .m/ .!c / @m C 2 c mD1 0 @t m

2 Q Q Q t / C 2i @A.r; t /  1 @ A.r; t / A.r; !c @t !c2 @t 2

! D 0; (11.140)

Q where k.!/ D .!=c/n.!/ is the complex wavenumber in the dispersive dielectric with complex index of refraction n.!/ D Œ.!/=0 1=2 . This is then the general wave Q t / of the quasimonochroequation for the complex phasor wave “amplitude” A.r; i!c t Q . Notice that the SVE approximatic electric wave field E!c .r; t / D A.r; t /e mation has yet to be applied in this approach. Consider now the particular case of plane wave pulse propagation in the positive z-direction. In that case, let [cf. (11.122)–(11.123)] Q Q t / D a.z; A.r; Q t /e i k.!c /z ;

(11.141)

so that Q t/ D r 2 A.r;



 Q t/ @2 a.z; Q t / Q2 Q Q c / @a.z; C 2i k.! .! / a.z; Q t / e i k.!c /z :  k c @z2 @z

With these substitutions, the phasor wave equation in (11.140) becomes "

!# @2 1 @2  a.z; Q t/ @z2 v2p .!c / @t 2   1 X 1 @2 a.z; 2i @a.z; !c2  .m/ .!c / @m Q t/ Q t/  2 a.z; Q t/ C D 0; i Q c /c 2 0 @t m !c @t !c @t 2 2k.! mD1 @ 1 @ i C  Q @z vp .!c / @t 2k.!c /

(11.142) Q where vp .!/ c=n.!/ D !=k.!/ is the complex phase velocity [see (11.117)]. 2 2 Q2 Because .!/=0 D .c =! /k .!/, then, from the dielectric permittivity moment expression given in (11.139),

11.3 Dispersive Wave Equations

 .m/ .!/ i m @m .!/ i m @m D 0 mŠ @! m mŠ @! m

193





c2 Q2 k .!/ : !2

(11.143)

In particular,  2  c Q2  .1/ .!/ @ Di .!/ k 0 @! ! 2  c2  Q kQ 0 .!/  kQ 2 .!/ ; D 2i 3 ! k.!/ ! and  2   .2/ .!/ c Q2 1 @2 D k .!/ 0 2 @! 2 ! 2    2 c2 Q Q kQ 0 .!/ C 3kQ 2 .!/ : kQ 00 .!/ C ! 2 kQ 0 .!/  4! k.!/ D  4 ! 2 k.!/ ! With substitution from (11.143), the phasor wave equation given in (11.142) becomes !# " 1 @ i @2 1 @2 @ C   a.z; Q t/ Q c / @z2 v2p .!c / @t 2 @z vp .!c / @t 2k.! !ˇ 1 !c2 X i m @m kQ 2 .!/ ˇˇ i ˇ Q c/ ˇ mŠ @! m !2 2k.! mD1 !c   1 @2 a.z; 2i @a.z; Q t/ Q t/ @m  2 Q t/ C D 0:  m a.z; @t !c @t !c @t 2 (11.144) Because the m D 1 and m D 2 terms appearing in the summation of this wave equation possess time derivatives of the slowly varying phasor amplitude function a.z; Q t / that are of the same order as those appearing in the first line of this equation, they may be combined with these terms to yield the approximate expression "

!# 1 @ i Q 00 @2 @ @2 i 1 @2 C C k .!c / 2   a.z; Q t/ Q c / @z2 v2g .!c / @t 2 @z vg .!c / @t 2 @t 2k.! !ˇ 1 !c2 X i m @m kQ 2 .!/ ˇˇ i ˇ Q c/ ˇ mŠ @! m !2 2k.! mD1 !c   1 @2 a.z; 2i @a.z; Q t/ Q t/ @m  2 Q t/ C

0;  m a.z; @t !c @t !c @t 2 (11.145)

where vg .!/ 1=kQ 00 .!/ is the complex group velocity.

194

11 The Group Velocity Approximation

Provided that @2 a.z; Q t/ Q t/ 1 @2 a.z;  D 0; @z2 v2g .!c / @t 2

(11.146)

the SVE phasor wave equation given in (11.145) becomes "

# @ @ @2 1 i Q 00 Q t/ C C k .!c / 2 a.z; @z vg .!c / @t 2 @t 1 !c2 X i m @m i Q c/ mŠ @! m 2k.! mD1

kQ 2 .!/ !2

!ˇ ˇ ˇ ˇ ˇ

!c

  2i @a.z; Q t/ Q t/ @ 1 @2 a.z; Q t/ C

0;  m a.z;  2 @t !c @t !c @t 2 (11.147) m

which is precisely that given by Akhmanov, Vysloukh, and Chirkin [23, 25], who state that this equation “is exact in the sense that it takes into account the dispersive properties of a linear medium.” However, notice that the infinite series summation Q over the derivatives of the square of the complex wavenumber k.!/ D .!=c/n.!/ is valid only within its domain of convergence which is determined by the analyticity properties of the complex index of refraction n.!/ D ..!/=0 /1=2 of the dispersive medium. In particular, the nearest resonance feature !r in n.!/ to the pulse angular carrier frequency !c will restrict the radius of convergence of this series summation to a value set by the distance between these two points. As a consequence, the SVE phasor wave equation given in (11.146) is only locally valid about the angular frequency !c when the material dispersion contains any resonance feature.

11.4 The Classical Group Velocity Approximation The parabolic wave equation used in the classical group velocity approximation is obtained from the SVE phasor wave equation given in (11.147) by neglecting all of the higher-order (m  3) dispersion terms appearing in the infinite summation of that equation with the result 1 @a.z; Q t / i Q 00 Q t/ @2 a.z; @a.z; Q t/

  k .!c / : @z vg .!c / @t 2 @t 2

(11.148)

This wave equation provides the starting point in most of the popular formulations of dispersive pulse propagation. In this formulation, the pulse envelope propagates at the group velocity vg .!c / with distortion described by the so-called group velocity dispersion (GVD) term kQ 00 .!c /. However, because the fundamental hyperbolic

11.4 The Classical Group Velocity Approximation

195

character of the underlying wave equation has been approximated as parabolic in this formultion, the characteristics then propagate instantaneously [11] through the dispersive medium. A convenient description of linear dispersive pulse propagation phenomena in the group velocity approximation as described by the SVE wave equation (11.148) is now given based upon the Fourier–Laplace integral representation of the propagated plane wave pulse given in (11.48) as 1 < E.z; t / D 2

Z

1

uQ .!  !c /e

Q i .k.!/z!t /

d! ;

(11.149)

1

where E.0; tn/ D u.t / cos .!c t /. Ino the phasor notation of (11.135) and (11.141), let Q E.z; t / D < a.z; Q t /e i .k.!c /z!c t / so that Q

a.z; Q t/ D

e i .k.!c /z!c t / 2

Z

1

Q

uQ .!  !c /e i .k.!/z!t / d!;

(11.150)

1

with a.0; Q t / D u.t /. If the initial pulse envelope function u.t / is slowly varying such that the initial pulse spectrum uQ .!  !c / is sharply peaked about the angular carrier frequency !c , then it may be argued that the dominant contribution to the integral representation (11.150) arises from those values of ! which lie in a small Q neighborhood about ! D !c . As a consequence, the complex wavenumber k.!/ may be expanded in a Taylor series about the applied angular carrier frequency !c as Q Q c / C kQ 0 .!c /.!  !c / C 1 kQ 00 .!c /.!  !c /2 C    : k.!/ D k.! 2

(11.151)

Notice that the coefficient in the first term in this expansion is associated with the Q c /, the coefficient in the second term with complex phase velocity vp .!c / D !c =k.! the complex group velocity vg .!c / D 1=kQ 0 .!c /, and the coefficient in the third term with the GVD. Consider first the case of a linear dispersion relation in which the complex wavenumber is approximated by the first two terms of its Taylor series expansion as Q k.!/

kQ .1/ .!/ where Q c / C kQ 0 .!c /.!  !c /: kQ .1/ .!/ k.!

(11.152)

With this substitution in the integral representation (11.150) of the propagated complex envelope, the resultant integral may be directly evaluated to yield   a.z; Q t / u t  kQ 0 .!c /z

(11.153)

in the linear dispersion approximation. It is then seen that, to this first-order of approximation, the oscillatory nature of the signal propagates with the phase velocity

196

11 The Group Velocity Approximation

vp .!c / evaluated at the applied angular carrier frequency whereas the pulse envelope itself propagates undistorted in shape with the group velocity vg .!c /. Notice that the complex envelope function given in (11.153) satisfies the SVE wave equation (see Problem 11.7) 1 @a.z; Q t/ @a.z; Q t/ C

0 @z vg .!c / @t

(11.154)

in the linear dispersion approximation, in agreement with the SVE wave equation given in (11.148) when the GVD term is neglected. Finally, notice that the condition given in (11.146) is indeed satisfied in the linear dispersion approximation. Consider next the more general case of a quadratic dispersion relation in which the complex wavenumber is approximated by the first three terms of its Taylor series Q expansion as k.!/

kQ .2/ .!/ where Q c / C kQ 0 .!c /.!  !c / C 1 kQ 00 .!c /.!  !c /2 : kQ .2/ .!/ k.! 2

(11.155)

With this substitution, the integral representation for the propagated complex envelope function becomes a.z; Q t/ 

e i=4 2 kQ 00 .!c /z

Z 1=2

1

Q0

u.t 0 /e i .k .!c /zCt

0 t 2 =

/ .2kQ00 .!c /z/ dt 0

(11.156)

1

in the quadratic dispersion approximation, where u.t 0 / is the initial pulse envelope function. The complex envelope is then seen to propagate in the dispersive medium at the complex group velocity vg .!c / D 1=kQ 0 .!c / evaluated at the input pulse angular carrier frequency !c . The complex envelope function described in (11.156) is found to satisfy the SVE wave equation (see Problem 11.8) 1 @a.z; Q t/ i Q t/ @a.z; Q t/ @2 a.z; C C kQ 00 .!c /

0; @z vg .!c / @t 2 @t 2

(11.157)

in the quadratic dispersion approximation, which is precisely the SVE wave equation given in (11.148). Notice, however, that the condition given in (11.146) is not satisfied in the quadratic dispersion approximation. In particular, it is found that @2 a.z; Q t/ Q t/ 1 @2 a.z;  @z2 v2g .!c / @t 2   Z 1 1 1

 uQ ./kQ 00 .!c / 2 kQ 0 .!c / C kQ 00 .!c / 2 2 1 4 i Œ.kQ0 .!c /zt /C 12 kQ00 .!c / 2

e d ; which, it may be argued, are of higher-order than each of the terms appearing in (11.157).

11.4 The Classical Group Velocity Approximation

197

Based upon the analysis due to J. Jones [61], the behavior of the remaining integral appearing in (11.156) may be easily understood through a comparison with the two-dimensional Fresnel–Kirchhoff diffraction integral [see (9.79)] 1 a.x; z/ D .i z/1=2

Z

1



0 2

s.x 0 /e i z .xx / dx 0 :

1

This integral describes the diffracted wave amplitude and phase along the x-axis due to the passage of a normally incident plane wave field through a transmitting screen along the x 0 -axis with transmission function s.x 0 /, where the separation of these two parallel axes is z and the wavelength of the incident plane wave field is . As described by the Fresnel wave zone theory, the behavior of the diffracted wave field a.x; z/ at a fixed distance z > 0 is determined by the scale of variation of the transmission function s.x 0 / and the size of the principal Fresnel zone, which is given by [6] r0

p z:

If the spatial extent of s.x 0 / is much larger than r0 , then the geometrical optics approximation applies to the wave field, whereas if the spatial extent of s.x 0 / is much less than r0 , then the Fraunhofer diffraction approximation is appropriate. On the basis of this analogy between the integral description of dispersive pulse propagation in the quadratic dispersion approximation and scalar diffraction theory in the Fresnel approximation, Jones defined, in analogy to the principal Fresnel zone, the complex Fresnel parameter i1=2 h : FQ .z/ 2 kQ 00 .!c /z

(11.158)

With this definition, the integral representation (11.156) for the propagated complex envelope function in the quadratic dispersion approximation becomes e i=4 a.z; Q t/

FQ .z/

Z

1

Q0

u.t 0 /e i .k .!c /zCt

0 t 2 =FQ 2 .z/

/

dt 0 :

(11.159)

1

The quantity jFQ .z/j then sets the scale for dispersive pulse spreading in the quadratic dispersion approximation. As in the geometrical optics approximation of the Fresnel–Kirchhoff diffraction integral, if the initial width T of the pulse envelope is much larger than jFQ .z/j over a given range of values of z  0, then the pulse does not broaden significantly over that range. However, for a sufficiently large distance z, the quantity jFQ .z/j must always become larger than the initial pulse width T , whereupon the pulse begins to spread in exactly the same manner as light passing through an aperture diverges in the Fraunhofer region. These two limiting cases for the integral expression given in (11.159) are now treated separately.

198

11 The Group Velocity Approximation

 Case 1 (T  jFQ .z/j): When z is sufficiently small that T  jFQ .z/j, the scale of

variation of the initial pulse envelope u.t 0 / is much larger than jFQ .z/j so that over the time interval t 0 D jFQ .z/j, the envelope function u.t 0 / does not appreciably change. In that case, over the range of significant values of t 0 contributing to the integral in (11.159), namely t  kQ 0 .!c /z  jFQ .z/j < t 0 < t  kQ 0 .!c /z C jFQ .z/j;

(11.160)

the pulse envelope function u.t 0 / does not vary appreciably. The function u.t 0 / may then be regarded as being essentially constant over this region so that the integral representation (11.159) may be evaluated as   a.z; Q t / u t  kQ 0 .!c /z ; (11.161) and the complex pulsed envelope propagates undistorted in shape at the group velocity [cf. (11.153)].  Case 2 (T  jFQ .z/j): When z is sufficiently large that jFQ .z/j  T , the contributions from the quadratic phase term in the exponential of the integrand in (11.159) are negligible due to the fact that the relative narrowness of the original pulse width T effectively confines the integration interval to small values of t 0 . Consequently, for space–time points .z; t / that satisfy the inequality ˇ ˇ ˇ ˇ0 ˇt  t C kQ 0 .!c /zˇ  jFQ .z/j;

(11.162)

the approximate integral representation for the complex envelope given in (11.159) may be simplified to read a.z; Q t/

1 i e FQ

h

2 =4 .kQ0 .!c /zt / =FQ 2

i

Z

1

Q0

u.t 0 /e i2 .k .!c /zt /t

0 =FQ 2

dt 0 : (11.163)

1

The propagated pulse envelope function is then approximately proportional to the Fourier transform of the initial pulse envelope u.t / and travels at the group velocity through the dispersive medium. Because both the phase and group velocities appearing in this group velocity description are complex-valued, it is convenient nto rewrite this approximate foro Q mulation in terms of the propagation ˇ.!/ < k.!/ and attenuation ˛.!/ n o Q = k.!/ factors, where Q k.!/ D ˇ.!/ C i ˛.!/: (11.164) With this substitution, the integral representation (11.150) of the complex envelope function becomes Z Q e i .k.!c /z!c t / 1 uQ .!  !c /e ˛.!/z e i.ˇ.!/z!t/ d!: (11.165) a.z; Q t/ D 2 1

11.4 The Classical Group Velocity Approximation

199

With the neglect of the dispersion of the attenuation coefficient, so that ˛.!/ ˛.!c /

(11.166)

the integral representation (11.165) becomes a.z; Q t/ D where

e i.ˇ.!c /z!c t/ 2

Z

1

uQ .!  !c /e i.ˇ.!/z!t/ d!;

(11.167)

1

n o E.z; t / D < a.z; Q t /e i.ˇ.!c /z!c t/ e ˛.!c /z :

(11.168)

The quadratic dispersion approximation is now taken as ˇ.!/ ˇ2 .!/ where [cf. (11.155)] 1 ˇ2 .!/ ˇ.!c / C ˇ 0 .!c /.!  !c / C ˇ 00 .!c /.!  !c /2 : 2

(11.169)

With this substitution, the integral representation for the propagated complex envelope function becomes

a.z; Q t/

Z

e i=4 .2ˇ 00 .!c /z/1=2

1

0

u.t 0 /e i.ˇ .!c /zCt

0 t/=.2ˇ 00 .!

c /z/

dt 0

(11.170)

1

in the quadratic dispersion approximation. The complex envelope then propagates through the dispersive medium at the real-valued group velocity vg .!c / D 1=ˇ 0 .!c / with distortion described by the real-valued group velocity dispersion p (GVD) coefficient ˇ 00 .!c / through the real-valued Fresnel parameter F .z/ D 2jˇ 00 .!c /jz, where the absolute value of ˇ 00 .!c / is taken since the GVD may be negative. As an illustration, consider the evolution of a unit amplitude gaussian envelope pulse centered at t D 0 with initial half-width T0 > 0, viz., ug .t / D e t

2 =T 2 0

:

Substitution of this initial envelope function in (11.170) then yields the expression (see Problem 11.10) p T0 e i=4

aQ g .z; t /  1=2 e F 2 .z/ C i T02



T02 CiF 2 .z/ F 4 .z/C 2 T04

.ˇ 0 .!c /zt/2

where ˚ Eg .z; t / D < aQ g .z; t /e i.ˇ.!c /z!c t/ e ˛.!c /z

;

200

11 The Group Velocity Approximation

describes the electric field component of the propagated gaussian pulse wave field. Notice that the peak amplitude point of the propagated gaussian pulse envelope occurs at the point t D ˇ 0 .!c /z and consequently travels at the group velocity vg .!c / D 1=ˇ 0 .!c /. The propagated pulse half-width is seen to be given by s

F 4 .z/ T .z/ D T0 1 C 2 4 D T0  T0 s  2 z D T0 1 C `D

s 1C



2ˇ 00 .!c /z T02

2

where `D

T02 ; 2jˇ 00 .!c /j

is the dispersion length. The dispersion length `D sets the maximum propagation distance over which the gaussian pulse envelope experiences minimal dispersive pulse spreading. In particular, when z  `D with z  0, the propagated gaussian pulse half-width may be approximated by the expression T .z/ T0 C 2.ˇ 00 .!c /z/2 =.T0 /3 as the propagated gaussian pulse half-width increases quadratip cally with the propagation distance, approaching the critical value T .`D / D 2T0 , beyond which T .z/ 2jˇ 00 .!c /jz=T0 when z  `D and the propagated pulse half-width increases linearly with the propagation distance. Notice the close analogy between this group velocity approximation of gaussian pulse propagation and the scalar diffraction theory of gaussian beam propagation presented in Sect. 9.4.1. The space–time evolution of the real and imaginary parts of the complex envelope for an input 52:36 fs gaussian envelope pulse (T0 D 26:18 fs) in fused silica with !c D 2:4  1015 r=s, ˇ.!c / D 1:147  107 r=m, ˇ 0 .!c / D 4:882 fs=¯m, and ˇ 00 .!c / D 3:853  102 fs2 =¯m is illustrated in Figs. 11.19 and 11.20, respectively, where the dispersion length is `D D 8:9  103 m. Notice that the temporal oscillations that appear in both the real and imaginary parts of the propagated pulse envelope function are solely due to the nonvanishing group velocity dispersion term ˇ 00 .!c /. However, the magnitude of the pulse envelope remains gaussian as the propagation distance increases, as illustrated in Fig. 11.21.

11.5 Failure of the Classical Group Velocity Method Recent advances in ultrashort pulse generation techniques, which have led to the creation of both near- and sub-10 fs optical pulses [62–69] call into question the accuracy of the slowly varying envelope approximation. Because the necessary condition [see (11.98)] that !=!c  1 for the applicability of the SVE approximation is not satisfied by such ultrashort pulses, greater care must then be exercised in modeling their dynamic evolution in dispersive lossy media. Moreover, because of the

11.5 Failure of the Classical Group Velocity Method

201

1 0.8 [ag(z,t)]

0.6 0.4 0.2 0 −0.2 −0.4 0 2

20

4

10

6

z /lD

8 10 −20

−10

0 (t - vg)/ T0

Fig. 11.19 Space–time evolution of the real part of the complex envelope for an input 52:36 fs gaussian envelope pulse in fused silica with !c D 2:4  1015 r=s, ˇ.!c / D 1:147  107 r=m, ˇ 0 .!c / D 4:882 fs=¯m, and ˇ 00 .!c / D 3:853  102 fs2 =¯m

0.4 0.3 [ag(z,t)]

0.2 0.1 0 −0.1 −0.2 −0.3 0 2

20 4

10 6

z/lD

0 –10

8 10 –20

(t - vg)/T0

Fig. 11.20 Space–time evolution of the imaginary part of the complex envelope for an input 52:36 fs gaussian envelope pulse in fused silica

202

11 The Group Velocity Approximation

1 0.8

|ag (z,t)|

0.6 0.4 0.2 0 0 2

20 4

10 6

z /lD

8 10

−20

−10

0 (t - vg)/ T0

Fig. 11.21 Space–time evolution of the magnitude of the complex envelope for an input 52:36 fs gaussian envelope pulse in fused silica

interrelated layers of approximation in this theory, each deserves to be carefully addressed in increasing order of complexity. The first approximation in the classical group velocity method is the neglect of the second-order partial derivatives of the complex envelope with respect to time in comparison with its zeroth- and first-order time derivatives [see (11.112) and (11.113)]. The fundamental hyperbolic character of the underlying wave equation is then approximated as being hyperbolic. The characteristics (i.e., the wavefront of any sharply defined signal) then propagate instantaneously [11] through the dispersive material, in violation of the relativistic principle of causality [70]. Similar remarks hold for the quadratic dispersion approximation of the exact integral representation of the propagated plane wave pulse that is given in (11.156), although the slowly varying envelope approximation was not explicitly invoked in its derivation. This is so because the complex envelope given (11.156) satisfies the SVE wave equation given in (11.157). Furthermore, notice that the quadratic dispersion relation given in (11.155) results in an approximate effective transfer function for plane wave pulse propagation that is given by Q

Q0

1 Q00 .!

.!/ e i Œk.!c /Ck .!c /.!!c /C 2 k

2 c /.!!c /

;

with the output of this hypothetical linear system being given by (11.156). Unfortunately, this transfer function is not physically realizable due to its violation of the principle of causality. That is, with the above system transfer function, an output can exist prior to the application of an input, as can easily be seen through a consideration of the case of an input delta function pulse. This nonphysical result is due

11.5 Failure of the Classical Group Velocity Method

203

to the quadratic approximation of the dispersion relation which tacitly assumes that the contribution of the superimposed monochromatic spectral components comprising the input pulse is appreciable only for those components confined to a specific narrow frequency neighborhood about the input pulse carrier frequency !c . Those angular frequency components that lie outside this small neighborhood are assumed to have negligible spectral amplitudes. As a consequence, this group velocity description of dispersive pulse propagation applies only in the quasimonochromatic case. The expression given in (11.156), as well as the solution of the SVE wave equation given in either (11.148) or (11.157), will then, to some extent, approximately represent the propagated pulse behavior for input pulse envelope functions that do not vary too abruptly, such as, for example, the slowly varying gaussian and hyperbolic tangent envelope waveforms which are present for all time t . It may then be argued that, for input pulse envelope functions that do turn on abruptly at time t D t0 at the plane z D 0, the approximate description provided by (11.156) may then be applied for t > t0 C z=c for z > 0 with some unspecified degree of accuracy. This then leads to the second layer of approximation in the group velocity description, that being the accuracy of the quadratic and higher-order dispersion approximations. It is widely believed that the accuracy of the Taylor series approximation of the Q complex wavenumber k.!/ D .!=c/n.!/, and consequently, that of the complex index of refraction n.!/, increases with the inclusion of each higher-order term from the Taylor series expansion of that material dispersion. For example, in the abstract of their paper, Anderson, Askne, and Lisak [21] state that “the evolution of slowly varying wave pulses in strongly dispersive and absorptive media is studied by a recursive method. It is shown that the resulting envelope function may be obtained by including correction terms of arbitrary dispersive and absorptive orders.” In Sect. 7.1.6 of their book, Butcher and Cotter [26] state that “to describe pulse propagation in dispersive media in general we must retain the second-order dispersion, and for ultrashort pulses or those with a wide frequency spectrum it may sometimes be necessary to also include higher-order terms.” This popular sentiment is continued in Sect. 1.3 of the book by Akhmanov, Vysloukh, and Chirkin [25] who state that “one can analyze how the dispersion of a medium affects a propagating pulse for any higher-order approximation of the dispersion theory. Naturally, the higher-order approximations make the quantitative picture of dispersive spreading more precise although its basic features obtained for the second- and third-order approximations remain unchanged.” The inherent error that results from this appealing but unfounded assumption was finally detailed by Xiao and Oughstun [28, 29] in 1997 for the case of a double-resonance Lorentz model dielectric [see Sect. 4.4.4 of Vol. 1] with complex index of refraction 

b22 b02  2 n.!/ D 1  2 2 2 !  !0 C 2i ı0 ! !  !2 C 2i ı2 !

1=2 ;

(11.171)

where !j is the undamped angular p resonance frequency, ıj the phenomenological damping constant, and bj .k4k=0 /Nj qe2 =me the plasma frequency for the

204

11 The Group Velocity Approximation

j th resonance line (j D 0; 2) with number density Nj (number of j -type Lorentz oscillators – atomic or molecular – per unit volume). This causal model [70] provides an accurate description [71] of both the normal and the anomalous dispersion phenomena observed in homogeneous, isotropic, locally linear optical materials when the carrier frequency of the input wave field is situated either in the normal dispersion region inside the passband q .!1 ; !2 / between the q two absorption bands

Œ!0 ; !1 and Œ!2 ; !3 , where !1 !02 C b02 and !3 !22 C b22 , or within the anamolous dispersion region of either of these two absorption bands; i.e., for all angular frequencies in the frequency domain !0  !  !3 . If !2 is the largest angular resonance frequency for the dispersive dielectric material, then (11.171) privides an accurate description of the material dispersion for all !  !0 . Consider first the Taylor series expansion of the complex index of refraction given in (11.171) for a double-resonance Lorentz model of a fluoride-type glass with infrared (!0 D 1:74  1014 r=s, b0 D 1:22  1014 r=s, ı0 D 4:96  1013 r=s) and near-ultraviolet (!2 D 9:145  1015 r=s, b2 D 6:72  1015 r=s, ı2 D 1:434  1015 r=s) resonance lines [28, 29]. The angular frequency dispersion of the real and imaginary parts of the complex index of refraction for the full double-resonance Lorentz model of this glass is described by the solid curves in parts (a) and (b) of Fig. 11.22, respectively. The Taylor series expansion of n.!/ D nr .!/ C i ni .!/ is taken about the angular frequency !c D !min Š 1:615  1015 r=s, which occurs at the inflection point !min 2 .!1 ; !2 / in the real index of refraction nr .!/ where the dispersion is a minimum. The dashed-dotted curves in the figure describe the angular frequency behavior of the real and imaginary parts of the three-term (or second-order) Taylor series approximation n.!/ n.2/ .!/ with 1 n.2/ .!/ n.!c / C n0 .!c /.!  !c / C n00 .!c /.!  !c /2 ; 2 the dashed curves describe the four-term (or third-order) Taylor series approximation n.!/ n.3/ .!/ with 1 1 n.3/ .!/ n.!c / C n0 .!c /.!  !c / C n00 .!c /.!  !c /2 C n000 .!c /.!  !c /3 ; 2 3Š

and the dotted curves describe the ten-term (or ninth-order) Taylor series approximation n.!/ n.9/ .!/. Notice that this large increase in the number of terms results in only a slight improvement in the local accuracy of the Taylor series approximation of n.!/ about !c , whereas the accuracy outside of the passband .!1 ; !2 / is greatly decreased. This is a result of the finite radius of convergence of the Taylor series that is determined, in part, by the branch point singularities of (11.171), situated at q .j / !˙ ˙ !j2  ıj2  i ıj ; in the lower-half of the complex !-plane.

j D 1; 2;

(11.172)

11.5 Failure of the Classical Group Velocity Method

205

nr ( )

a

nr(3)( ) nr(2)( ) nr(9)( )

(r/s)

ni ( )

b

ni(2)( )

ni(9)( )

ni(3)( )

(r/s) Fig. 11.22 Angular frequency dependence of (a) the real and (b) the imaginary parts of the doubleresonance Lorentz model of the complex index of refraction of a fluoride-type glass with infrared .!0 ; b0 ; ı0 / and near-ultraviolet .!2 ; b2 ; ı2 / resonance lines (solid curves). The three-, four-, and ten-term Taylor series approximations about the minimum dispersion point !c D !min between the two resonance lines are depicted by the dashed-dotted, dashed, and dotted curves, respectively

The decrease in accuracy of the Taylor series approximation of the real part of the complex index of refraction for the double-resonance Lorentz model of this fluoride-type glass as the number of terms is increased is presented in Fig. 11.23 for several different angular frequency domains centered about !c . For each set of data presented, the rms error was numerically determined over the angular frequency domain Œ!c  !; !c C ! . For the data described by the lowest curve in the figure, 2 ! was just less than 25% of the available bandwidth .!2  !1 / of the material passband; in this case the rms error of the Taylor series approximation decreases monotonically as the number of terms M increases (i.e., as the order of approximation increases). However, when 2 ! is increased to just over 33% of the available bandwidth, the rms error is found to reach a minimum at M D6 and then increases monotonically as additional terms are included in the approximation. When 2 ! is increased to nearly 36% of the available bandwidth, the minimum in the rms error occurs at M D 2. Finally, when 2 ! is increased to just less than 45% of the available bandwidth, the rms error of the Taylor series approximation remains essentially unchanged as M increases from 1 to 2 and then increases monotonically as the number of terms increases, as illustrated in Fig. 11.23. As a consequence, the assumptions of the group velocity approximation are seen to be valid provided that

206

11 The Group Velocity Approximation 100

rms error

10−1

10−2

10−3

10−4 1

2

3

4

5 6 7 Number of Terms M

8

9

10

Fig. 11.23 The rms error over the angular frequency interval Œ!c  !; !c C ! of the Taylor series approximation of the real part nr .!/ of the complex index of refraction for the double-resonance Lorentz model of n.!/ of a fluoride-type glass with infrared .!0 ; b0 ; ı0 / and near-ultraviolet .!2 ; b2 ; ı2 / resonance lines about the minimum dispersion point !c D !min in the normal dispersion region between the two resonance lines as a function of the number M of terms

the pulse spectrum is strictly band-limited about the input pulse carrier frequency. For the example considered here, this is satisfied when . !/p  .!2 !1 /=3, where . !/p is the bandwidth of the pulse. Q The angular frequency dependence of the complex wavenumber k.!/ .!=c/n.!/ for the full double-resonance Lorentz model for this fluroide-type glass is described by the solid curves in Fig. 11.24; detailed views of the frequency behavior about the infrared resonance line at !0 are presented in Fig. 11.25. The three-, four-, and ten-term Taylor series approximations about the minimum dispersion point !c D !min between the two resonance lines are depicted in these two figures by the dashed-dotted, dashed, and dotted curves, respectively. Just as was found for the complex index of refraction n.!/ for this lossy dielectric, a large increase in the number of terms results in only a slight improvement in the local Q accuracy of the Taylor series approximation of k.!/ about !c , whereas the accuracy outside of the passband .!1 ; !2 / containing !c is greatly decreased. In addition, the complete accuracy of each of these Taylor series approximations kQ kQ .M / .!/ decreases as !c is moved into either absorption band because of the decreased Q radius of convergence of the Taylor series expansion of k.!/, where Q .M /

k

ˇ M ˇ Q X 1 @m k.!/ ˇ .!/ ˇ .!  !c /m ; m ˇ mŠ @! mD0 !c

(11.173)

11.5 Failure of the Classical Group Velocity Method

207

a kr(3) ( ) kr(2) ( )

kr ( )

kr(9) ( )

c

0

2

(r/s)

b

ki ( )

ki(3) ( )

ki(2) ( ) c 0

ki(9) ( )

2

(r/s) Fig. 11.24 Angular frequency dependence of (a) the real and (b) the imaginary parts (solid curves) Q of the complex wavenumber k.!/ along the positive real frequency axis for the double-resonance Lorentz model with complex index of refraction illustrated in Fig. 11.22. The three-, four-, and ten-term Taylor series approximations about the minimum dispersion point !c D !min between the two resonance lines are depicted by the dashed-dotted, dashed, and dotted curves, respectively

denotes the .M C1/-term Taylor series approximation of the complex wavenumber.3 The accuracy of each order of approximation n.M / .!/ of the complex index of refraction also does not improve as the phenomenological damping constants ı0 and ı2 of the double-resonance Lorentz model dielectric are decreased to zero, as illustrated in Fig. 11.26 for a reduced-loss fluoride-type glass with infrared .!0 ; b0 ; ı0 =10/ and near-ultraviolet .!2 ; b2 ; ı2 =10/ resonance lines. Similar results hold for the complex wavenumber. Similar results are also obtained when the phenomenological damping constants are further reduced to the values ı0 =100 and ı2 =100, which correspond to a nearly loss-free dielectric outside of the two absorption bands Œ!0 ; !1 and Œ!2 ; !3 . These detailed numerical results then establish the validity of the following general result [28, 29]: With the exception of a small neighborhood about some characteristic frequency !c of the initial pulse, the inclusion of higher-order terms in the Taylor series approximation of

Notice that, in certain situations, the notation f .j / ./ denotes the j th partial derivative of the function f ./ with respect to the variable . The context in which it appears should always specify just what is meant by this notation.

3

208

11 The Group Velocity Approximation

a

kr ( )

kr(2) ( )

kr

(3)

( )

kr(9) ( ) 0

(r/s)

ki ( )

b

ki(9) ( )

ki(2) ( )

ki(3) ( ) 0

(r/s)

Fig. 11.25 Detail of the angular frequency dependence of (a) the real and (b) the imaginary parts (solid curves) of the complex wavenumber illustrated in Fig. 11.24

Q either the complex index of refraction n.!/ or the complex wavenumber k.!/ in a causally dispersive, attenuative medium beyond the quadratic dispersion approximation is practically meaningless from both the physical and mathematical points of view.

As a consequence, optimal results in the global sense are obtained for the group Q velocity method with either the quadratic dispersion approximation k.!/

kQ .2/ .!/ .3/ Q Q or the cubic dispersion approximation k.!/ k .!/. With this understanding of the inherent limitations of the group velocity method, several numerical examples of dispersive pulse dynamics are now given in which the group velocity description using either the quadratic or cubic dispersion approximations is compared with numerical results using the full (i.e., without approximation) material dispersion. In each case presented here, the material dispersion is described by the double-resonance Lorentz model of a fluoride-type glass with infrared (!0 D 1:74  1014 r=s, b0 D 1:22  1014 r=s, ı0 D 4:96  1013 r=s) and nearultraviolet (!2 D 9:145  1015 r=s, b2 D 6:72  1015 r=s, ı2 D 1:434  1015 r=s) resonance lines, whose frequency dependence is illustrated in Fig. 11.22 (together with both its quadratic and cubic dispersion approximations about the minimum dispersion point !min ) and repeated here in Fig. 11.27. Also illustrated in Fig. 11.27 are the spectral magnitudes of a single-cycle, five-cycle, and ten-cycle Van Bladel envelope pulse [see (11.76)] with angular carrier frequency at the minimum dispersion

11.5 Failure of the Classical Group Velocity Method

209

a nr(3) ( )

nr ( )

nr(9) ( )

nr(2) ( )

c

0

2

(r/s)

ni ( )

b

ni(2) ( ) c 0

ni(9) ( )

2

ni(3) ( )

(r/s) Fig. 11.26 Angular frequency dependence of (a) the real and (b) the imaginary parts (solid curves) of the complex index of refraction n.!/ along the positive real frequency axis for the doubleresonance Lorentz model of the reduced-loss fluoride-type glass with infrared .!0 ; b0 ; ı0 =10/ and near-ultraviolet .!2 ; b2 ; ı2 =10/ resonance lines (solid curves). The three-, four-, and ten-term Taylor series approximations about the minimum dispersion point !c D !min between the two resonance lines are depicted by the dashed-dotted, dashed, and dotted curves, respectively

point in the .!1 ; !2 / passband of the dielectric material (!c D !min ). Notice that the five- and ten-cycle pulse spectra are both quasimonochromatic with spectra essentially contained within the passband of the material dispersion whereas the single-cycle pulse is not.

11.5.1 Impulse Response of a Double-Resonance Lorentz Model Dielectric The numerically determined impulse response of the double-resonance Lorentz model dielectric whose frequency-dependent complex refractive index is illustrated in Figs. 11.22 and 11.27 is presented in Fig. 11.28 at the fixed propagation distance z D 3:24zd , where zd ˛ 1 .!min / is the e 1 absorption depth at the minimum dispersion point !min D 1:615  1015 r=s in the .!1 ; !2 / passband of the material.

210

11 The Group Velocity Approximation

nr ( )

1 cycle pulse

5 cycle pulse

10 cycle pulse

ni ( )

c

(r/s)

Fig. 11.27 Relative magnitudes (drawn to an arbitrary scale) of the input pulse spectrum for single-cycle, five-cycle, and ten-cycle van Bladel envelope pulses with carrier frequency !c D !min at the minimum dispersion point in the passband of the double-resonance Lorentz model of a fluoride-type glass with infrared and a near-ultraviolet resonance lines. For comparison, the angular frequency dependence of the real and imaginary parts of the double-resonance Lorentz model of the complex index of refraction of this dielectric material are described by the solid curves in the figure

For comparison, this propagation distance in the medium is also equal to one hundred absorption depths ˛ 1 .!2 / at the upper resonance frequency !2 , which is the more absorptive of the two resonance lines in the double-resonance Lorentz model dielectric considered here. The impulse response presented in Fig. 11.28 is comprised entirely of the precursor fields that are characteristic of the dispersive medium. The propagated field behavior is described here as a function of the dimensionless space–time parameter  D ct =z which, for fixed z > 0, represents a dimensionless time parameter [see (11.15)]. The propagated wave field at any fixed z > 0 identically vanishes for all  < 1, in agreement with the relativistic principle of causality. The temporal field evolution then begins at  D 1 with the Sommerfeld precursor evolution ES .z; t / which, in effect, describes the high-frequency q medium response j!j  !3 above !22 C b22 . As such, it cannot be accuQ rately described by the Taylor series approximation of k.!/ about !c D !min . For  > mb , where mb > sm for any naturally occurring dielectric other than vacuum, the temporal field evolution is dominated by the Brillouin precursor evolution Eb .z; t / which, in effect, describes the low-frequency medium response j!j  !0 below the lower absorption band. As such, it too cannot be accurately described by the upper absorption band, where !3

211

E(z,t)

11.5 Failure of the Classical Group Velocity Method

sm

mb

0

ct /z

Fig. 11.28 Impulse response of the double-resonance Lorentz model of a fluoride-type glass at the fixed propagation distance z D 3:24zd , where zd ˛ 1 .!min / is the e 1 absorption depth at the minimum dispersion point !min D 1:615  1015 r=s in the passband of the material. The field evolution is described as a function of the dimensionless space–time parameter  D ct =z which, for fixed z > 0, represents a dimensionless time parameter

Q the Taylor series approximation of k.!/ about !c D !min . For sm <  < mb the temporal field evolution is dominated by a middle precursor field Em .z; t / which, in effect, describes the intermediate frequency response !0 < j!j < !3 of the medium. As such, it can be described by the group velocity method to some unspecified degree of accuracy. Notice that the middle precursor describes the wave field transition from the high-frequency Sommerfeld precursor to the low-frequency Brillouin precursor. The total propagated wave field for the delta function pulse E.0; t / D ı.t / may then be expressed as E.z; t / D Es .z; t / C Em .z; t / C Eb .z; t /

(11.174)

for all t  z=c with z > 0. In as much as the propagated wave field structure of any input pulse can be obtained through a convolution with the impulse response of the linear dispersive medium, it is then seen that the group velocity method fails to accurately describe the full effects of linear dispersion in ultrashort, ultrawideband pulse propagation. The remaining subsections begin to address precisely what its domain of applicability is.

212

11 The Group Velocity Approximation

11.5.2 Heaviside Unit Step Function Signal Evolution The inaccuracy of the group velocity method is clearly evident in Fig. 11.29 which illustrates the numerically determined dynamical wave field evolution due to an input Heaviside unit step function envelope signal [see (11.34) and (11.55)] with angular carrier frequency !c D !min Š 1:615  1015 r=s at the minimum dispersion point at three absorption depths z D 3zd with zd ˛ 1 .!c / in the double-resonance Lorentz model of a fluoride-type glass whose temporal frequency dispersion is illustrated in Figs. 11.22, 11.24 and 11.25. The solid curves in parts (a) and (b) of Fig. 11.29 describe the numerically determined field evolution using the full dispersion relation of the double-resonance Lorentz model with complex index of refraction given in (11.171). The dotted curves in the figure describe the numerically determined field evolution using (a) the quadratic dispersion approximation Q k.!/

kQ .2/ .!/ of the complex wavenumber, resulting in the quadratic dispersion approximation E.z; t / E .2/ .z; t / of the propagated wave field, and (b) the Q cubic dispersion approximation k.!/

kQ .3/ .!/ of the complex wavenumber, resulting in the cubic dispersion approximation E.z; t / E .3/ .z; t / of the propagated wave field. The interpretation of the Sommerfeld Es .z; t /, Brillouin Eb .z; t /, and middle Em .z; t / precursor field components in the exact field behavior illustrated in Fig. 11.29 is the same as that for the delta function pulse illustrated in Fig. 11.28. The final contribution to the propagated wave field is the signal contribution Ec .z; t / whose spectral content arises primarily from a small neighborhood of the input pulse spectrum about the input angular carrier frequency !c . The total propagated wave field for the Heaviside step function envelope signal E.0; t / D uH .t / sin .!c t / may then be expressed as E.z; t / D Es .z; t / C Em .z; t / C Eb .z; t / C Ec .z; t /

(11.175)

for all t  z=c with z > 0. Because of the comparatively small spectral amplitude [see Fig. 11.2)] above the upper absorption band when !c < !2 , the amplitude of the Sommerfeld precursor is negligibly small in comparison with the other contribution to the propagated wave field and so is not evident on the scale depicted in Fig. 11.29; a magnified vertical scale would indeed reveal its presence just following the speed of light point t D z=c provided that a sufficient amount of the frequency structure above the upper absorption band was adequately sampled [72, 73]. Notice that the precursor fields are primarily responsible for the observed distortion of the leading edge of the signal while the main body of the signal arises from the signal contribution Ec .z; t / that is primarily due to the residue contribution arising from the simple pole singularity in the spectrum at ! D !c [see (11.56) and (11.57)]. The signal contribution Ec .z; t / approximately corresponds to the entire propagated wave field E .1/ .z; t / obtained using the group velocity method with the linear dispersion approximation ˇ.!/ ˇ .1/ .!/ of the real part of the complex wave number, where ˇ .1/ .!/ D ˇ.!c /Cˇ 0 .!c /.!!c /. With this substitution in (11.167) one obtains

11.5 Failure of the Classical Group Velocity Method

213

z /zd = 3

a

E(z,t)

E(2)(z,t)

'0

sm

mb

0

ct /z

E(z,t)

b

E(3)(z,t) '0

sm

mb

0

ct /z Fig. 11.29 Dynamical wave field evolution due to an input Heaviside step function envelope signal with angular carrier frequency !c D !min at three absorption depths z D 3zd in a double-resonance Lorentz model of a fluoride-type glass. The solid curves in parts (a) and (b) describe the numerically determined field evolution using the full dispersion relation and the dotted curves describe the numerically determined field evolution using (a) the quadratic dispersion approximation and (b) the cubic dispersion approximation

Z 1 1 0 uQ .!  !c /e i.tˇ .!c /z/.!!c / d! 2 1   D u t  ˇ 0 .!c /z

aQ .1/ .z; t / D

and (11.168), with appropriate alteration to account for a sine wave carrier, gives   E .1/ .z; t / D u t  ˇ 0 .!c /z e ˛.!c /z sin .ˇ.!c /z  !c t /:

(11.176)

214

11 The Group Velocity Approximation

In this first order of approximation, the step-function signal propagates undistorted in shape at the group velocity vg D 1=ˇ 0 .!c /. The pulse distortion in the group velocity description is then seen to be a result of the quadratic and cubic dispersion terms in the Taylor series approximation of the wavenumber ˇ.!/ about the angular carrier frequency !c . This distortion is illustrated by the dotted curve in Fig. 11.29a using the quadratic dispersion approximation and by the dotted curve in Fig. 11.29b using the cubic dispersion approximation. The actual distortion of the leading edge of the propagated signal (described by the identical solid curves in both parts of the figure) is primarily due to the middle precursor followed in time by the Brillouin precursor. Both the quadratic and cubic group velocity approximations provide a rough estimate of the behavior of the combined middle and Brillouin precursors with a predicted peak amplitude point at  D 00 1:258 in the quadratic dispersion approximation. The actual peak amplitude point of the Brillouin precursor occurs at  D 0 , where 0 D n.0/ Š 1:424; this peak amplitude point emerges from the remaining field structure as the propagation distance is further increased above the absorption depth zd D ˛ 1 .!c /.

11.5.3 Rectangular Envelope Pulse Evolution Similar results are obtained for the propagated wavefield evolution due to an input rectangular envelope pulse with initial pulse width T D 38:9 fs at the same angular carrier frequency !c D !min at the minimum dispersion point of a doubleresonance Lorentz model of a fluoride-type glass, whose dynamical field evolution is illustrated in Fig. 11.30 at three absorption depths [i.e., z D 3˛ 1 .!c /]. As in Fig. 11.29, the solid curve in both parts (a) and (b) of Fig. 11.30 describes the numerically determined field evolution using the full dispersion relation with complex index of refraction given in (11.171). The dotted curve in Fig. 11.30a describes the numerically determined field evolution using the quadratic dispersion approximation and that in Fig. 11.30b describes the numerically determined field evolution using the cubic dispersion approximation of the wavenumber ˇ.!/. Because the propagated wave field due to an input rectangular envelope pulse may be represented as the difference between the propagated wave fields that result from two unit step function modulated signals that are separated in time by the initial pulse width T [see (11.64)], the observed pulse distortion is then due to the leading- and trailing-edge precursor fields as well as to the interference between them [74]. This pulse distortion is seen to be misrepresented by both the quadratic and cubic dispersion approximations. A comparison of the dotted curves in Parts (a) and (b) of Fig. 11.30 shows that the inclusion of the cubic dispersion term in the approximate dispersion relation actually degrades the overall accuracy of the group velocity description for such an ultrawideband, ultrashort pulse.

11.5 Failure of the Classical Group Velocity Method

215

z/zd = 3

a

E(z,t)

E(2)(z,t)

sm

'0

mb

0

ct /z

E(z,t)

b

E(3)(z,t) sm

'0

mb

0

ct /z Fig. 11.30 Dynamical wave field evolution due to an input rectangular envelope signal with initial pulse width T D38:9 fs and angular carrier frequency !c D!min at three absorption depths z D 3zd in a double-resonance Lorentz model of a fluoride-type glass. The solid curves in parts (a) and (b) describe the numerically determined field evolution using the full dispersion relation and the dotted curves describe the numerically determined field evolution using (a) the quadratic dispersion approximation and (b) the cubic dispersion approximation

11.5.4 Van Bladel Envelope Pulse Evolution Consider finally an input unit amplitude Van Bladel envelope modulated pulse with  1C

2

4t.t  / E.0; t / D uvb .t / sin .!c t /, where the initial envelope function uvb .t / D e for 0 < t <  and zero otherwise is infinitely smooth with compact temporal support and full temporal width  > 0 [see (11.76) and Figs. 11.13 and 11.14]. The temporal form of this canonical pulse envelope function is of some importance to

216

11 The Group Velocity Approximation

ultrashort pulse physics because its properties of infinite smoothness and temporal compactness are common to all experimental pulses. The dynamical field evolution of a Van Bladel envelope pulse with specific initial pulse duration  > 0 and fixed angular carrier frequency !c D !min Š 1:615  1015 r=s at the minimum dispersion point of a double-resonance Lorentz model of a fluoride-type glass is illustrated in Figs. 11.31–11.35. Notice that this causal model of the dielectric

a E(z,t)

z/zd = 0.001

b E(z,t)

z/zd = 0.01

c E(z,t)

z/zd = 0.1

sm

mb

0

'0

d E(z,t)

z/zd = 0.5

sm

mb 0

'0 Fig. 11.31 Numerically determined propagated wave field evolution due to an input unit amplitude, single-cycle Van Bladel envelope pulse (D3:89 fs) with angular carrier frequency !c D !min at the minimum dispersion point using the exact dispersion relation (solid curves) and the cubic dispersion approximation (dotted curves) of the complex wavenumber of a double-resonance Lorentz model of a fluoride-type glass as a function of the dimensionless space–time parameter  D ct =z at several fixed positive values z < zd of the propagation distance into the dispersive, attenuative dielectric, where zd ˛ 1 .!c /

11.5 Failure of the Classical Group Velocity Method

217

E(z,t)

a

z/z d = 1 mb

sm

0

b E(z,t)

z/zd = 5

sm

0 mb

c E(z,t)

z/zd = 10

sm

0 mb

d E(z,t)

z/zd = 20

sm

0 mb

Fig. 11.32 Continuation of the numerically determined propagated wave field evolution presented in Fig. 11.31 of an input unit amplitude, single-cycle Van Bladel envelope pulse ( D 3:89 fs) with angular carrier frequency !c D !min at the minimum dispersion point using the exact dispersion relation (solid curves) and the cubic dispersion approximation (dotted curves) of the complex wave number of a double-resonance Lorentz model of a fluoride-type glass as a function of the dimensionless space–time parameter  D ct =z at several fixed values z zd of the propagation distance into the dispersive, attenuative dielectric, where zd ˛ 1 .!c /

frequency dispersion is characterized by an infrared and near-ultraviolet resonance line with associated relaxation times (see Fig. 11.22)

respectively.

r0 

2 D 127 fs; ı0

r2 

2 D 4:38 fs; ı2

218

11 The Group Velocity Approximation

E(z,t)

a

z/zd = 1

sm

0 mb

'0

E(z,t)

b

z/zd = 5

sm

0 mb

'0

c E(z,t)

z/zd = 10

sm

0 mb

'0

d E(z,t)

z/zd = 20

sm

0 mb

'0 Fig. 11.33 Numerically determined propagated wave field evolution due to an input unit amplitude, 5-cycle Van Bladel envelope pulse ( D 19:45 fs) with angular carrier frequency !c D !min at the minimum dispersion point using the exact dispersion relation (solid curves) and the cubic dispersion approximation (dotted curves) of the complex wavenumber of a double-resonance Lorentz model of a fluoride-type glass as a function of the dimensionless space–time parameter  D ct =z at several fixed positive values z zd of the propagation distance into the dispersive, attenuative dielectric, where zd ˛ 1 .!c /

Each graph in a given figure sequence describes the temporal wave field evolution as a function of the dimensionless space–time parameter  D ct =z at a fixed propagation distance z > 0 relative to the e 1 absorption depth zd D ˛ 1 .!c / in the dispersive medium at the input angular carrier frequency !c D !min at the minimum dispersion point in the passband between the two absorption bands, where ˛ 1 .!min / Š 14:4 ¯m. The solid curve in each graph describes the numerically

11.5 Failure of the Classical Group Velocity Method

219

E(z,t)

a

z/zd = 1

sm mb

0

'0

b E(z,t)

z/zd = 5

mb

0

'0

c E(z,t)

z/z d = 10

sm

0 mb

'0

d E(z,t)

z/zd = 20

sm

0 mb

'0 Fig. 11.34 Numerically determined propagated wave field evolution due to an input unit amplitude, 10-cycle Van Bladel envelope pulse ( D 38:9 fs) with angular carrier frequency !c D !min at the minimum dispersion point using the exact dispersion relation (solid curves) and the cubic dispersion approximation (dotted curves) of the complex wavenumber of a double-resonance Lorentz model of a fluoride-type glass as a function of the dimensionless space–time parameter  D ct =z at several fixed positive values z zd of the propagation distance into the dispersive, attenuative dielectric, where zd ˛ 1 .!c /

determined propagated wave field at the indicated propagation distance when the full Lorentz model of the material frequency dispersion is used, and the dotted curve in each graph depicts the numerically determined propagated wave field using the Q cubic dispersion approximation k.!/

kQ .3/ .!/ of the complex wavenumber about the carrier frequency !c . The cubic dispersion approximation was selected over the quadratic dispersion approximation because it includes the group velocity dispersion (GVD) term as well as providing a description of pulse asymmetry effects in

220

11 The Group Velocity Approximation

a E(z,t)

z/zd = 1

sm mb

0

'0

b E(z,t)

z/zd = 5

sm mb

'0

0

E(z,t)

c

z/zd = 10

sm mb

0

'0

E(z,t)

d

z/zd = 20

sm

0 mb

'0 Fig. 11.35 Numerically determined propagated wave field evolution due to an input unit amplitude, 20-cycle Van Bladel envelope pulse ( D 77:8 fs) with angular carrier frequency !c D !min at the minimum dispersion point using the exact dispersion relation (solid curves) and the cubic dispersion approximation (dotted curves) of the complex wave number of a double-resonance Lorentz model of a fluoride-type glass as a function of the dimensionless space–time parameter  D ct =z at several fixed positive values z zd of the propagation distance into the dispersive, attenuative dielectric, where zd ˛ 1 .!c /

dispersive pulse propagation [24, 25]. Because all of the propagation distances are scaled by the absorption depth zd ˛ 1 .!c / in the dispersive medium at the input pulse carrier frequency, the numerical results presented here are representative of the results obtained for both weakly and strongly absorptive dispersive media. Because the initial pulse envelope function for the Van Bladel envelope pulse possesses compact temporal support, its Fourier transform uQ vb .!/ is an entire

11.5 Failure of the Classical Group Velocity Method

221

function of complex ! [see (11.76) and (11.77)]. As a consequence, the propagated wave field may be expressed as the superposition of the precursor fields that are a characteristic of the dispersive medium for that particular pulse spectrum [see (11.174) for the impulse response], so that E.z; t / D Es .z; t / C Em .z; t / C Eb .z; t /

(11.177)

for all t  z=c with z > 0, the propagated wave field identically vanishing for all t < z=c. Because the middle precursor field Em .z; t / is due to the below resonance spectral region j!j  !2 that contains the spectral domain .!1 ; !2 / described by the group velocity approximation, this wave field component corresponds closest to the GVD that results from either the quadratic or cubic dispersion approximations of the complex wavenumber. For a single-cycle Van Bladel envelope pulse the input full pulse width is given by  D 2=!c Š 3:89 fs with equal rise- and fall-times r;f D =2 Š 1:95 fs, so that r;f < r2  r0 and both the initial rise- and fall-times are less than the relaxation times for both the upper and lower resonance lines of the dispersive medium. In this ultrashort, ultrawideband pulse case, !=!c 5:0 with .!2 !1 /=!c 5:6, so that !=.!2  !1 / 0:893, and 99:83% of the initial pulse pectral energy is contained in the medium passband, as shown in Fig. 11.27. Although r;f < r2 in this case, the Sommerfeld precursor is negligible in comparison with both the middle and Brillouin precursor fields and so its evolution has not been included in the plots of the propagated field evolution presented in Figs. 11.31 and 11.32. The dynamical single-cycle pulse evolution for propagation distances less than a single absorption depth into the medium (0 < z < zd ), illustrated in Fig. 11.31, shows that the group velocity description provides an accurate description of the observed pulse behavior over this small (jzj=zd < 1) relative propagation distance range. However, when the propagation distance exceeds a single absorption depth into the medium (z > zd ), as illustrated in Fig. 11.32, the observed error in the GVD of the propagated pulse evolution increases monotonically with increasing propagation distance. Indeed, the temporal pulse structure depicted in Fig. 11.32a shows that the GVD has already begun to break down at a single absorption depth into the dispersive medium. At five absorption depths the GVD has completely broken down, as is evident in Fig. 11.32b. This breakdown of the GVD is a direct consequence of its inability to correctly model the precursor field components that are a characteristic of the dispersive medium, as is evident in parts (c) and (d) of Fig. 11.32 for 10 and 20 absorption depths, respectively. The value 0 n.0/ Š 1:42 indicated in Figs. 11.31–11.34 marks the space– time point at which the peak amplitude in the Brillouin precursor occurs in the full dispersion model, and the value 00 Š 1:26 marks the corresponding space– time point that is obtained in the cubic dispersion approximation. The amplitude of the Brillouin precursor decays with propagation distance z > 0 only as z1=2 at this critical space–time point, making it a unique feature in the dynamical field evolution. In addition, the space–time point sm Š 1:02 marks the transition from the Sommerfeld to the middle precursor field evolution that is a characteristic of this

222

11 The Group Velocity Approximation

double-resonance Lorentz model dielectric, and the space–time point mb Š 1:29 marks the transition from the middle precursor to the Brillouin precursor, which then dominates the dynamical field evolution of such an ultrashort Van Bladel envelope pulse for all  > mb as z becomes large in comparison to zd . For an input 5-cycle pulse with angular carrier frequency !c D!min at the minimum dispersion point in the .!1 ; !2 /-passband of the double resonance Lorentz model of a fluoride-type glass, the full-pulse width is given by  D 10=!c Š 19:45 fs with equal rise- and fall-times r;f D =2 Š 9:73 fs, so that r2 < r;f  r0 . In this intermediate case, !=!c 0:62 with !=.!2  !1 / 0:111 and 99:98% of the input pulse spectral energy is contained in the medium passband, as shown in Fig. 11.27. The dynamical pulse evolution illustrated in Fig. 11.33 shows that the GVD has already begun to break down at five absorption depths into the medium. As the propagation distance is increased from this point, the propagated wave field becomes increasingly dominated by the middle and Brillouin precursor fields as the GVD becomes increasingly inaccurate. For an input 10-cycle pulse with !c D !min , the full-pulse width is given by  D 20=!c Š 38:9 fs with equal rise- and fall-times r;f D =2 Š 19:45 fs, so that r2 < r;f < r0 . In this nearly quasimonochromatic case, !=!c 0:25 with !=.!2  !1 / 0:045 and 99:999% of the input pulse spectral energy is contained in the medium passband, as shown in Fig. 11.27. The dynamical pulse evolution illustrated in Fig. 11.34 shows that the GVD has already begun to break down at five absorption depths into the medium. As the propagation distance is increased from this point, the propagated wave field becomes increasingly dominated by the middle and Brillouin precursor fields as the GVD becomes increasingly inaccurate. For an input 20-cycle pulse with !c D !min , the full-pulse width is given by  D 40=!c Š 77:8 fs with equal rise- and fall-times r;f D =2 Š 38:9 fs, so that r2  r;f < r0 . In this quasimonochromatic case, !=!c 0:17 with !=.!2  !1 / 0:030 and the percentage of the input pulse’s spectral energy that is contained in the medium passband differs from 100% in the 15th decimal place. The dynamical pulse evolution illustrated in Fig. 11.35 shows that the GVD has begun to break down at five absorption depths into the medium and has completely broken down at ten absorption depths. As the propagation distance is increased from this point, the propagated wave field is again found to become increasingly dominated by the middle and Brillouin precursor fields as the GVD becomes increasingly inaccurate.

11.5.5 Concluding Remarks on the Slowly Varying Envelope and Classical Group Velocity Approximations The numerical results presented here have shown that the GVD (in the SVE approximation) of dispersive pulse propagation in a double-resonance (as well as in a multiple-resonance) Lorentz model dielectric is valid only for small propagation

11.6 Extensions of the Group Velocity Method

223

distances in the dispersive, absorptive medium. Analogous results are obtained for a single-resonance Lorentz model dielectric. Similar results [75–77] are also obtained for gaussian envelope pulses in the ultrashort limit. In particular, the body of results presented here has shown that [28, 29]:  The SVE or quasimonochromatic approximation of linear dispersive pulse prop-

agation in either a single-resonance or multiple-resonance Lorentz model dielectric is valid provided that each of the following inequalities (listed in decreasing order of importance) min > max .r;j ; r;j C2 /;

(11.178)

. !/p  1; !j C2  !j C1

(11.179)

. !/p  1; !c

(11.180)

are strictly satisfied, where . !/p is the spectral width of the input pulse with envelope rise-time r and fall-time f , with min min .r ; f /. Here r;j  2=ıj denotes the relaxation time associated with the lower angular resonance frequency !j and r;j C2  2=ıj C2 denotes the relaxation time associated with the upper angular resonance frequency !j C2 .  For either an ultrashort or an ultrawideband pulse with either an initial rise- or fall-time less than the maximum of the two medium relaxation times r;j and r;j C2 , the accuracy of the GVD using either the quadratic or cubic dispersion approximations for the complex wavenumber decreases monotonically as the propagation distance exceeds one absorption depth zd ˛ 1 .!c / at the input angular carrier frequency !c of the pulse.  The inclusion of higher-order terms in the Taylor series approximation of the complex wavenumber beyond the quadratic dispersion approximation does not improve the accuracy of the GVD for an ultrawideband pulse, and the inclusion of higher-order terms beyond the cubic dispersion approximation decreases the accuracy of the GVD.

11.6 Extensions of the Group Velocity Method An important extension of the GVD has recently been developed by Brabec and Krausz [30]. Their analysis begins with the exact linear dispersive wave equation given in (11.133), which may be generalized to the nonlinear case as r 2 E.r; t / 

1 @2 D.r; t / k4k @2 Pnl .r; t / D 2 ; 2 @t c 0 @t 2 0

c2

(11.181)

224

11 The Group Velocity Approximation

where the displacement field D.r; t / is given by the linear constitutive relation in (11.134), and where Pnl .r; t / denotes the induced nonlinear medium polarization. As in (11.114), (11.122) and (11.123), let E.r; t / have the complex phasor representation N !tC N E!N .r; t / A.r; t /e i.ˇ.!/z

/

(11.182)

with E.r; t / D
/

(11.183)

with Pnl .r; t / D
(11.185)

at the input plane z D 0. As pointed out by Brabec and Krausz [30], this method of defining the complex envelope representation of the pulse is physically meaningful provided that the envelope function A.r; t / is invariant under a change in the phase value of , which is equivalent to the requirement that a phase shift of the phasor electric field E!N .r; t / ! E!0N 0 .r; t / D E!N .r; t /e i does not result in a change in the value of the mean angular carrier frequency given by (11.185), so N Numerical results [30] indicate that this approximate result is highly that !N 0 D !. accurate for pulse durations p down to a single cycle T0 2=!c of the input carrier frequency, as illustrated in Fig. 11.36 for a gaussian envelope pulse, similar results holding for both hyperbolic secant and Lorentzian envelope pulses. The pulse width p is defined there as the full temporal width at the half-amplitude points of the quantity jA.0; t /j2 . For a unit amplitude gaussian envelope pulse with 2 2 2 2t˙ =T 2 ug .t / D e t =T this p condition becomes e p D 1=4 with p tC  t with solution t˙ D ˙T ln .4/=2 so that p D T 2 ln .4/ 1:665T . As can be seen from the numerical data presented in Fig. 11.36, the approximation !N !c applies to a high degree of accuracy (less than a 0:01% error) for gaussian pulse durations satisfying p  T0 ; that is, down to one cycle of oscillation at !c . Returning now to the nonlinear dispersive wave equation given in (11.181), the complex phasor representation of the electric and polarization field quantities given

11.6 Extensions of the Group Velocity Method

225

0.6 0.5 0.4 0.3 0.2 0.1 0 −0.1 0

0.2

0.4

t p / T0

0.6

0.8

1

Fig. 11.36 Variation of the relative difference !=!c between mean !N and the carrier !c angular frequency values for a gaussian envelope pulse with respect to the relative full pulse width p =T0 where T0 2=!c denotes a single oscillation period of the carrier

in (11.182) and (11.183), respectively, are now substituted. From the derivation preceeding (11.119) one obtains 2



r E.r; t / !

 @2 C 2 A.r; t /e i.ˇ.!c /z!c t/ @z   @2 A @A 2 2  ˇ .!c /A e i.ˇ.!c /z!c t/ ; D rT A C 2 C 2iˇ.!c / @z @z

rT2

where the variation of the phase quantity .r/ with the propagation distance z is obtained separately using (11.129). In addition, @2 @2 Pnl .r; t / ! B.r; t; A/e i.ˇ.!c /z!c t/ @t 2 @t 2   i @ 2 2 i.ˇ.!c /z!c t/ 1C D !c e B.r; t; A/: !c @t Finally, from (11.136)–(11.139),    1 X 1 @2 D.r; t /  .m/ .!c / @m @2 A @A 2  ! !  2i ! A e i.ˇ.!c /z!c t/ ; c c m 2 0 @t 2  @t @t @t 0 mD0

226

11 The Group Velocity Approximation

where [see (11.143)]  .m/ .!/ i m @m 0 mŠ @! m



 c2 Q2 .!/ : k !2

In particular  .0/ .!/ c2 D 2 kQ 2 .!/; 0 !   .1/ .!/ c2  Q kQ 0 .!/  kQ 2 .!/ ; D 2i 3 ! k.!/ 0 ! Q and so on for higher-order terms, where k.!/ D ˇ.!/Ci ˛.!/. With these substitutions, the linear electric displacement term in the nonlinear wave equation (11.181) becomes ( ! Q 2 .!c / @ 1 @2 D.r; t / k 2 0 Q c /kQ .!c /   2 ! kQ .!c / C 2i k.! c 0 @t 2 !c @t ! ) 1 X i m @m @m kQ 2 .!/ 2 C!c mŠ @! m !2 @t m mD2 !c   1 @2 A 2i @A  2 2 AC !c @t ! @t ( ! c 2 Q Q c /kQ 0 .!c /  k .!c / @

kQ 2 .!c / C 2i k.! !c @t ! ) 1 X i m @m @m kQ 2 .!/ 2 A.r; t /; C!c mŠ @! m !2 @t m mD2 !c

(11.186) where the final approximation made here is valid in the SVE approximation [see (11.112)]. In addition, the variation of the phase quantity .r/ with the propagation distance z is obtained from (11.129), which is expressed here as @A @ C ˛.!c /ˇ.!c / 0 @z @z

(11.187)

in the SVE approximation. Brabec and Krausz [30] approximate the right-hand side of (11.86) as 2  @ 1 @2 D!N .r; t / 0 Q Q b

k.!c / C i k .!c / C Dt A.r; t /;  2 c 0 @t 2 @t

(11.188)

11.6 Extensions of the Group Velocity Method

227

b t is defined here4 as where the differential operator D 1 X im b Dt mŠ mD2

Q @m k.!/ @! m

! !c

@m : @t m

(11.189)

Comparison of (11.188) with the expression given in (11.186) then shows that this approximation is valid provided that ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇQ ˇk.!c /ˇ  !c ˇkQ 0 .!c /ˇ () !c2 ˇn0 .!c /ˇ  0

(11.190)

which should be readily satisfied in the optical region of the electromagnetic spectrum, and provided that ˇ ˇ ˇ ˇ ˇ @m kQ 2 .!/ ˇ m  ˇ 2 2 ˇ @ ˇ ˇ 2 ˇ! kQ .!/ ! ˇˇ  ˇ ˇ ˇ m ˇ ˇ @! m @! !c

(11.191) !c

for all m  2, which may be more difficult to satisfy in general. Notice further that additional, more subtle approximations have been applied in obtaining (11.188) from (11.186). With these various approximations, the nonlinear wave equation given in (11.181) becomes   @2 @ 2 2 ˇ .!c / C 2iˇ.!c / C 2 C rT A.r; t / @z @z 2  @ @ b t A.r; t / C ˇ.!c / C i ˛.!c / C iˇ 0 .!c /  ˛ 0 .!c / C D @t @t   i @ 2 k4k !c2 1C B.r; t; A/:

 0 c 2 !c @t (11.192) Under the change of coordinates  D t  ˇ 0 .!c /z,  D z to a moving coordinate system traveling at the real group velocity vg .!c / D 1=ˇ 0 .!c / of the pulse in the dispersive medium, the approximate wave equation given in (11.192) becomes (with further approximation) 4

Notice that this definition differs from that given by Brabec and Krausz [30] through the noninclusion of the real part of the term i kQ0 .!c /@=@t . Note also that the attenuation factor ˛ used there is for the intensity and not for the wave amplitude as used here.

228

11 The Group Velocity Approximation

   @ i @ b C ˛.!c /  i D A 1C !c @ @    k4k !c2 1 i @ C B C r2 A 1C 2 0 c !c @t 2iˇ.!c / T     @ ˇ.!c /  !c ˇ 0 .!c / i @ b

C ˛.!c /  i D A ˇ.!c / !c @ @  2  1 @ @ 2 2 0 b  A; C D  ˛ .!c /  ˛.!c /ˇ .!c / 2iˇ.!c / @ 2 @ (11.193) where

1 X im b D D mŠ mD2

Q @m k.!/ @! m

! !c

@m : @ m

(11.194)

The magnitudes of the two terms appearing on the right-hand side of (11.193) are small in comparison with the magnitudes of the terms appearing on the left-hand side if both ˇ ˇ ˇ @A ˇ ˇ ˇ  ˇ.!c / jAj (11.195) ˇ @ ˇ and either

or

ˇ ˇ ˇ @A ˇ ˇ ˇ  !c jAj ˇ @ ˇ

(11.196)

ˇ ˇ ˇ ˇ.!c /  !c ˇ 0 .!c / ˇ ˇ  1: ˇ ˇ ˇ ˇ.! /

(11.197)

c

Hence, if either the two inequalities given in (11.195) and (11.196) or the two inequalities given in (11.195) and (11.197) are both satisfied, then the nonlinear, SVE wave equation appearing in (11.193) reduces to @A.rT ; ; / b  A.rT ; ; /

˛.!c /A.rT ; ;  / C i D @   i @ 1 2 i 1C C rT A.rT ; ; / 2ˇ.!c / !c @   k4k !c2 i @ Ci B.rT ; ; ; A/: 1C 0 c 2 !c @t (11.198) Brabec and Krausz [30] refer to this expression as a “generic nonlinear envelope equation (NEE) first-order in the propagation coordinate ,” where the differential operator .1C.i=!c /@=@/1 is to be evaluated in the frequency domain. In addition, under the same change of variable to the moving coordinate system, (11.187) for the phase factor becomes

11.6 Extensions of the Group Velocity Method

229

@

ˇ.!c /  !c ˇ 0 .!c /; @ with solution ./

0

  C ˇ.!c /  !c ˇ 0 .!c / :

(11.199)

(11.200)

With this result, the electric field E.r; t / D
˛.!c /A.;  / C i D @   i @ k4k !c2 B.; ; A/: 1C Ci 0 c 2 !c @t (11.201) b  .ˇ 00 .!c /=2/@2 =@ 2 , With the further approximations that ˛.!/ 0 and D (11.201) simplifies to the nonlinear Schr¨odinger equation i @A.; / @2 A.; / k4k !c2

 ˇ 00 .!c / C i @ 2 @ 2 0 c 2

  i @ B.; ; A/; 1C !c @t (11.202)

which is commonly used [24] to describe soliton evolution in Kerr-type media where B.; ; A/ / jA.; /j2 A.; /. Notice that in the linear dispersion limit, this wave equation reduces to the SVE wave equation given in (11.148). Notice that the SVE condition ˇ ˇ ˇ ˇ ˇ @A ˇ ˇ @A @A ˇˇ 0 ˇ ˇDˇ  ˇ.!c / jAj  ˇ .! / (11.203) c ˇ @z ˇ ˇ @ @ ˇ is satisfied if the inequality given in (11.197) is satisfied. As a consequence of this, the nonlocal requirements of the SVE approximation may then be dropped in favor of the simpler inequality given in (11.197), which may be rewritten as ˇ ˇ ˇ ˇ ˇ1  vp .!c / ˇ  1; ˇ vg .!c / ˇ

(11.204)

which is satisfied if the ratio of the phase velocity vp .!c / D !c =ˇ.!c / to the group velocity vg D 1=ˇ 0 .!c / is very nearly unity. This then implies that the GVD,

230

11 The Group Velocity Approximation

whether in the SVE approximation or in the slowly-evolving-wave approximation (SEWA), as defined by Brabec and Krausz [30], is only valid in the weak dispersion limit. It is seen from (11.199) that the condition given in (11.197) may be expressed as j@ =@j  ˇ.!c /. With this result, it is seen that the two conditions given in (11.195) and (11.197) can be merged into the single requirement that ˇ ˇ ˇ @E ˇ ˇ ˇ  ˇ.!c /jEj: ˇ @ ˇ

(11.205)

Based upon these results, Brabec and Krausz [30] conclude that “the SEWA requires more from the propagation medium than the SVEA; not only the envelope A but also the relative carrier phase must not significantly vary as the pulse covers a distance equal to the wavelength c D 2=!c . In return, it does not explicitly impose a limitation on the pulse width. Therefore, in the frame of the SEWA the nonlinear envelope equation accurately describes light pulse propagation down to the single cycle regime.” The SEWA will provide a reasonably accurate description of both linear and nonlinear dispersive pulse propagation phenomena provided that the inequality ˇ.!c /Lc  1

(11.206)

is satisfied, where Lc is the minimum of the various characteristic propagation distances involved in the particular problem under consideration. These include the phase change length 1 ˇ ˇ L (11.207) ˇ 2 d n=d ˇc over which

changes by 1 rad, the mth-order absorption lengths pm ˇ; ˇ L.m/ ˛ ˇ˛ .m/ .!c /ˇ

(11.208)

where ˛ .m/ .!/ @m ˛=@! m , the mth-order dispersion lengths pm .m/ ˇ; Lˇ ˇ .m/ ˇˇ .!c /ˇ

(11.209)

where ˇ .m/ .!/ @m ˇ=@! m , the diffraction length LT ˇ.!c /w20 ;

(11.210)

and the characteristic nonlinear propagation length Lnl

ˇ ˇ 2 ˇ A ˇ n .!c / : ˇˇ ˇˇ r B 2ˇ.! / c

(11.211)

11.7 Localized Pulsed-Beam Propagation

231

Here p denotes the temporal pulse duration and w0 the beam radius at the beam .0/ waist. Notice that the zeroth-order distance L˛ D ˛ 1 .!c / defined in (11.208) is the same as that given in the second remark of Sect. 11.5.5. When combined with (11.206), the inequality nr .!c /  ni .!c /

(11.212)

is obtained, restricting the applicability of the GVD, whether in the SVE or SEWA, to the weak dispersion limit. As such, the SEWA fails in the same manner as that presented in Sect. 11.5 for the classical GVD.

11.7 Localized Pulsed-Beam Propagation The final extension of the GVD presented here considers the analysis of pulsed electromagnetic beam wave field propagation in temporally dispersive media in terms of localized pulsed-beam wave packet solutions. These have been defined in lossless dispersive media by Melamed and Felsen [78, 79] as analytic signals in complex space–time with the following properties: 1. Completeness. The set of pulsed-beam solutions form a complete basis for both the decomposition and synthesis of physically realizable space–time dependent pulsed wave fields [80, 81]. 2. Space–time localization. The pulsed-beam solutions are highly localized propagating wave fields whose dynamical evolution in both the configuration (space– time) and spectrum (wavenumber–frequency) domains can be traced analytically [81–84]. 3. Initial conditions. The initial conditions for the pulsed-beam solutions are chosen to be described by the space–time Gaussian window functions with temporal frequency domain form Q 0 .r; !/ D Qf.!/e  12 k.!/rT2 = ; U

(11.213)

where rT2 rT rT , the frequency-dependent vector function Qf.!/ is independent of r, and where  D r C ii is a frequency-independent parameter with r > 0 for ! > 0. Finally, k.!/ D .!=c/n.!/ is the frequency-dependent wavenumber in the (fictitious) lossless medium. 4. Isodiffracting propagation. The pulsed-beam solutions are all isodiffracting; that is, all of their temporal frequency components have the same collimation length and phase front curvature. The aim of this analysis is to use asymptotic methods to extract the effects of pure phase dispersion on the pulsed paraxial beam propagators.

232

11 The Group Velocity Approximation

11.7.1 Mathematical Preliminaries Given the configuration (space–time) domain field U.r; t /, the corresponding temporal frequency domain field is specified by the Fourier transform pair relationship Z

Q !/ D U.r;

1

U.r; t /e i!t dt; Z 1 1 Q !/e i!t d!: U.r; U.r; t / D 2 1

(11.214)

1

(11.215)

The analytic field UC .r; t / corresponding to the temporal frequency domain field Q !/ is then defined by the single-sided inverse Fourier transform U.r; UC .r; t /

1 

Z

1

Q !/e i!t d!; U.r;

(11.216)

0

Q !/ is as defined in (11.214). The real provided that = ft g  0 for real !, where U.r; space–time domain field is then given by the real part of the analytic field, viz., ˚ U.r; t / D < UC .r; t / :

(11.217)

The initial time-domain field distribution U0 .rT ; t / for the localized pulsed-beam solutions on the plane z D z0 is synthesized from the inversion of the correspondQ T ; !/, where rT D 1O x x C 1O y y. ing temporal frequency domain distribution U.r The wavenumber–frequency (spectrum) domain distribution on the initial surface is given by [cf. (9.3)] QQ .s ; !/ D U 0 T

Z

1

Q 0 .rT ; !/e ik.!/sT rT d 2 rT ; U

(11.218)

1

where rT D .x1 ; x2 / is the transverse position vector in the z D z0 plane with d 2 rT D dx1 dx2 , k.!/ D .!=c/n.!/ is the wavenumber in the (fictitious) lossless dispersive medium with index of refraction n.!/ that is real-valued along the real !-axis, and where sT D .s1 ; s2 / denotes the transverse part of the normalized spatial wave vector that is the conjugate transform variable to the position vector r D rT C 1O z z [compare with the real direction cosine form of the angular spectrum representation given in Sect. 9.1.1]. The reconstruction of the temporal frequency domain form of the initial localized pulsed-beam field is then given by [cf. (9.4)] 2 Q 0 .r; !/ D k .!/ U 4 2

Z

1

QQ .s ; !/e ik.!/sT rT d 2 s : U 0 T T

(11.219)

1

The plane wave spectrum representation of the temporal frequency domain wave field in the positive half-space z  z0 is then given by

11.7 Localized Pulsed-Beam Propagation 2 Q !/ D k .!/ U.r; 4 2

Z

1

233

QQ .s ; !/e ik.!/.sT rT Cm z/ d 2 s U 0 T T

(11.220)

1

with z D z  z0 , where mD

q 1  sT2

(11.221)

with = fmg  0 and sT2 D sT  sT . Finally, substitution of this expression into (11.216) results in the plane wave spectrum representation of the propagated analytic field [cf. (9.5)] UC .r; t / D

k 2 .!/ 4 3

Z

1

1

Z

1

QQ .s ; !/e iŒk.!/.sT rT Cm z/i!t d 2 s d!; (11.222) U 0 T T

1

for z  0. The initial spatial field distribution at the plane z D z0 is now chosen in such a manner that it produces a pulsed beam wave packet in the dispersive medium [78]. Such isodiffracting initial field distributions have been proposed by Heyman and Melamed [85] to have the temporal frequency domain form Q 0 .rT ; !/ D Qf.!/e  12 k.!/rT2 =% ; U

(11.223)

where rT2 D rT  rT and where % D %r C i%i is independent of the frequency with %r > 0 for real ! > 0. Substitution of this expression into (11.218) then yields the corresponding initial plane wave spectrum QQ .s ; !/ D 2 % Qf.!/e  12 k.!/%sT2 : U 0 T k.!/

(11.224)

Finally, the initial time domain field distribution U0 .r; t / is given by the real part of the initial analytic field UC 0 .r; t / D

1 

Z

1

Qf.!/e Π12 k.!/rT2 =%Ci!t d!

(11.225)

0

which intimately depends upon the temporal frequency response of the dispersive medium as well as upon the temporal frequency spectrum of the input pulse. The evaluation of the integral representation appearing in (11.222) can be approached in two different ways. The first is to interchange the order of integration and evaluate the !-integral first, resulting in a transient plane wave spectral representation. The other way is to evaluate the spatial-frequency domain integrals first and to then substitute this result into (11.215) to obtain the time-domain form of the propagated wave field. The latter approach is typically taken, as done in Sect. 9.1.3 using the paraxial approximation and continued here using asymptotic analysis based upon the extension of Laplace’s method to multiple integrals (see Sect. 10.5).

234

11 The Group Velocity Approximation

11.7.2 Paraxial Asymptotics Substitution of the initial isodiffracting plane wave spectrum given in (11.224) into the plane wave spectrum representation (11.220) results in the integral expression Q !/ D % k.!/Qf.!/ U.r; 2

Z

1

i

e ik.!/. 2 %sT CsT rT Cm z/ d 2 sT ; 2

(11.226)

1

for z  0. This result may be expressed in the form of (9.202) as Q !/ D % k.!/Qf.!/ U.r; 2

Z

1

e ik0 n.!/˚.p;q/ dpdq

(11.227)

1

with p s1 , q s2 , where k0 !=c is the wavenumber in vacuum, n.!/ is the (fictitious) real-valued index of refraction of the dispersive medium, and i ˚.p; q/ rT  sT C %sT2 C m z 2

(11.228)

is the phase function. As the method of stationary phase can only be applied when the phase function is real-valued, a saddle point technique must then be applied to obtain the asymptotic behavior of the integral in (11.227) as k0 ! 1. Based upon the extension of Laplace’s method to multiple integrals presented in Sect. 10.5, the analysis begins with the paraxial approximation of the phase function defined in (11.228) as 1 ˚.p; q/ rT  sT  . z  i%/sT2 C z: 2

(11.229)

The saddle points are then determined from the condition r˚.ps ; qs / 0, with the result .sT /s D .ps ; qs /, where ps

x ; z  i%

qs

y z  i%

(11.230)

in the paraxial approximation, with @2 ˚=@p 2 D @2 ˚=@q 2 D . z  i%/ and @2 ˚=@p@q D 0. In addition, ˚.ps ; qs / '.r/ z C

rT2 2. z  i%/

(11.231)

in the paraxial approximation. With these results, the asymptotic behavior of (11.227) is obtained from (10.131) as Q !/  U.r;

i% Q f.!/e ik.!/'.r/ z  i%

as k0 ! 1, where '.r/ is the normalized phase.

(11.232)

11.7 Localized Pulsed-Beam Propagation

235

In analogy with gaussian beam theory (see Sect. 9.4.1), Melamed and Felsen [78] rewrite this phase function as rT2 =2 z  z0  iF   r2 1 i C ; D z C T 2 R I

'.r/ D z C

(11.233)

where z0 %i and F %r with z D z  z0 , and where F2 ; z  z0

(11.234)

.z  z0 /2 C F 2 k.!/D 2 ; F

(11.235)

R .z  z0 / C I with

s DD

 1=2 .z  z0 /2 F 1C : k.!/ F2

(11.236)

With these substitutions, (11.232) for the asymptotic field behavior becomes r2 Q !/  z0  iF Qf.!/e ik.!/ z e ik.!/ 2T . R1 C Ii / U.r; z  z0  iF

(11.237)

1 as k0 ! 1. The phase front radius p of curvature is then given by R and the e beam width is given by w. z/ D 2 2D. The gaussian beam waist is then located at z D z0 and its Rayleigh range is given by 2zR D 2F . Notice that the beam waist location z0 , the Rayleigh range 2F , the phase front radius of curvature R, and the quantity I , and hence the normalized phase '.r/ as a whole are frequency independent, whereas the beam width w. z/ / k 1=2 .!/ is frequency dependent. The time-domain representation of the pulsed beam wave field corresponding to the paraxial, asymptotic frequency-domain wave field given in (11.232) is obtained from the single-sided inverse Fourier transform relation in (11.216) as

UC .r; t / 

i% .z  i%/

Z

1

Qf.!/e i.k.!/'.r/!t/ d!

(11.238)

0

as k0 ! 1. The evaluation of this integral in the nondispersive and dispersive cases is now treated separately in the following two subsections.

236

11 The Group Velocity Approximation

11.7.2.1

Pulsed Beam Evolution in the Nondispersive Case

In the nondispersive case, k.!/ D !=c and (11.238) becomes Z 1 i% Qf.!/e i!.t'.r/=c/ d! U .r; t /  .z  i%/ 0 i% C  f .t  '.r/=c/ z  i% C

(11.239)

as k0 ! 1. The wave form is then seen to propagate undistorted in shape with phase velocity given by c ; (11.240) vp .r/ D 1O p jr'.r/j where 1O p r'.r/=jr'.r/j is a unit vector along the direction of the normal to the phase front '.r/ D constant. From (11.233), it is found that  jr'.r/j D 1 C

.x 2 C y 2 /4 4.z  z0  iF /4

1=2

1C

.x 2 C y 2 /4 ; 8.z  z0  iF /4

(11.241)

so that the phase velocity is nearly equal to the vacuum speed of light c in the paraxial region about the z-axis, decreasing from c as the transverse distance from the z-axis increases. As defined by Melamed and Felsen [85], a “conventional waveform” can be obtained from (11.239) with substitution of the spectrum Qf.!/ D Ae !T =2

(11.242)

with T > 0 and fixed, real-valued vector A. This then corresponds to the analytic wave form Z 1 1Q fC .t / D f.!/e i!t d!  0 Z 1 1 e i!.tiT =2/ d! D Aı C .t  iT =2/; (11.243) D A  0 where 1 ı .t /  C

Z

1

e i!t d! ˇ 1 i!t ˇˇ1 1 e I D D ˇ it it 0 0

=ft g < 0;

(11.244)

is the analytic delta function defined in the lower-half of the complex t -plane. The real wave form is then obtained by taking the real part of (11.243) with the result

11.7 Localized Pulsed-Beam Propagation

237

1

fd +(t )/(2/p T )

0.8

0.6 T 0.4

0.2

0 −2

−1

0

1

t (s)

2 −10

x 10

Fig. 11.37 Amplitude normalized temporal structure of the initial analytic delta function pulse wave form fıC .t / described by (11.245) with T D 0:005=c

fıC .t / D

T =2 1 ; 2  t C .T =2/2

(11.245)

where f.t / D AfıC .t /. The full pulse width at the half-amplitude points of this wave form is then equal to T , as indicated in Fig. 11.37 which illustrates the amplitude normalized temporal structure of the initial analytic delta function pulse wave form fıC .t / that is described by (11.245) with T D 0:005=c. Notice that the unnormalized peak amplitude of this pulse is given by 2=T , which occurs at t D 0, and that the maximum angular frequency of the spectrum for this pulse may be estimated from (11.242) as (11.246) !max T 1 : Such a band-limited pulse may then be taken as a model for a real (i.e., measured) sampled signal. The propagated pulsed-beam wave field resulting from this analytic delta function pulse wave form is obtained from (11.238) and (11.245) as i% Af C .t  '.r/=c/ z  i% ı i%  Aı C .t  iT =2  '.r/=c/ z  i%

UC .r; t / 

(11.247)

as k0 ! 1. The spatial structure of this pulsed beam wave field at the instant of time given by ct D 2 m is depicted in the pair of graphs given Figs. 11.38 and

238

11 The Group Velocity Approximation

10

x 10 3.5 3

U(r,t)

2.5 2 1.5 1 0.5 0 −0.5 0.5 0.4 0.3

rT

0.2 0.1 0

1.94

1.96

1.98

2

2.02

2.04

2.06

z

Fig. 11.38 Three-dimensional plot of the propagated nondispersive pulsed beam field due to an input analytic delta function pulse with cT D 0:005 m, % D 5, and ct D 2 m, which results in the beam waist D0 ' 0:316 m and collimation length 2zR D 5 m with angular divergence D ' 0:0632. At the wave front where z D ct D 2, this set of pulsed beam parameters results in the spatial beam width D.z/ ' 0:341 m, a wave front radius of curvature R D 14:5 m, and a temporal, on-axis pulse width Tp .0/ D T ' 1:67  1011 s

11.39. This pulsed beam wave field propagates in the positive direction along the z-axis with wave front radius of curvature R.z/ given by (11.234) with temporal pulse width r2 (11.248) Tp .r/ D T C T ; cI.z/ where I.z/ is defined in (11.235). The temporal pulse width is then shortest along the beam axis where the peak pulse amplitude is largest and increases while the pulse amplitude decreases as the transverse distance rT increases away from that symmetry axis, as evident in both Figs. 11.38 and 11.39. Along the wave front, defined by [see (11.233) and (11.247)] ct D z  z0 C

rT2 ; 2R.z/

(11.249)

the field amplitude is proportional to the inverse Tp1 .r/ of the temporal pulse width. The half-amplitude beam width in the transverse direction is then specified by the requirement that Tp .rT / D 2Tp .0/, with solution p D.z/ D 2 cT I.z/: (11.250) The Rayleigh range (or collimation length) is given by 2zR D 2F and the beam waist is located atpz D z0 where the beam width is a minimum and is given by D0 D.z0 / D 2 cTF. It is then seen from (11.250) and (11.235) that the profile

11.7 Localized Pulsed-Beam Propagation

239

0.5

rT

0

0

−0.5 1.95

2

2.05

z Fig. 11.39 Contour plot of the propagated nondispersive pulsed beam field depicted in Fig. 11.38. The 0-level contour is labeled in the figure and the remaining contours successively increase in value from zero in units of 0:5  1010 to a maximum value of 3:0  1010

of the pulsed beam wave field remains essentially unchanged over the Rayleigh range domain jz  z0 j < F , whereas outside this collimation zone when jzj > z0 this spatial profile broadens as it approaches the asymptotic angular limit diffraction angle given by p D D 2 cT=F : (11.251) Hence, as the collimation length 2F increases, the beam width D0 at the beam waist decreses while the far-field angular divergence D increases. Numerical values of these pulsed beam parameters for the example depicted in Figs. 11.38 and 11.39 are given in the figure caption of Fig. 11.38.

11.7.2.2

Pulsed Beam Evolution in the Lossless Dispersive Case

In general, the integral appearing in (11.238) cannot be evaluated in closed form when the medium is dispersive. In that case, asymptotic methods of analysis may be applied to obtain approximate solutions that explicitly display the essential physics of pulsed beam wave propagation phenomena in dispersive media. The method of analysis taken here follows that developed by L. B. Felsen and coworkers [78, 82, 86–91] which relies upon the construction of dispersion surfaces and space–time rays and which typically assumes that the medium is lossless. In the most recent version of this formulation, Melamed and Felsen [78] consider a lossless, dispersive medium with the aim of obtaining a meaningful parameterization

240

11 The Group Velocity Approximation

of the effects of temporal dispersion on a given paraxial, pulsed beam wave field. Their analysis separates into two separate cases, dependent upon whether or not the spectral function Qf.!/ appearing in (11.238) has a phase term that is !-dependent. Case 1: Qf.!/ with frequency-independent phase With Qf.!/ having a phase that is !-independent, the critical points of the integrand in (11.238) are given by the stationary points of the phase function (in the paraxial approximation) .r; !/ D k.!/'.r/  !t with respect to !, defined by the relation d=d! 0, with the result ˇ t d k.!/ ˇˇ ; (11.252) D '.r/ d! ˇ!s where '.r/ is given in (11.233). Although k.!/ is taken here to be real-valued along the real !-axis, '.r/ is complex-valued in general so that solutions of (11.252) are also complex-valued in general (i.e., they are saddle points). However, instead of directly determining the solution of the saddle point equation given in (11.252), Melamed and Felsen [78] employ the geometrical method of construction using dispersion surfaces and space–time rays developed by Felsen and Marcuvitz [92, Sect. 1.6] for lossless dispersive media. This method considers the real dispersion surface defined by the dispersion relation5 k D k.!/ which may be expressed in implicit form as f .kx ; ky ; kz ; !/ D 0:

(11.253)

In general, f .k; !/ is a hypersurface in four-dimensional .k; !/-space that simplifies when the dispersion relation possesses specific types of symmetry. For a spatially isotropic medium (as considered here), the dispersion relation becomes kDk.!/, where kDjkj is the magnitude of the wave vector k, and (11.253) simplifies to f .k; !/ D 0: (11.254) The total differential of this implicit relation then yields df D

@f @f dk C d! D 0 @k @!

(11.255)

in the isotropic medium case, which may be expressed in two-dimensional vector form as   @f @f  .d!; d k/ D 0: (11.256) ; @! @k 5

Notice that many authors, including Felsen and Markuvitz [92], prefer to express the dispersion relation in the equivalent “inverse” form ! D !.k/ which is advantageous in problems that are dominated by spatial effects such as anisotropy and spatial dispersion.

11.7 Localized Pulsed-Beam Propagation Fig. 11.40 Graphical description of (a) a typical dispersion surface for a simple, lossless dispersive medium and (b) the space– time ray to the on-axis observation point at .z; ct / corresponding to the saddle point at !N s

241

a

ck ( ) Tan −1(ct /z) ck (

_

s)

_ p

s

b

z z

ct

ct

Because .!; k/ and .! C d!; k C dk/ describe two neighboring points on the dispersion surface f .k; !/ D 0, then the vector .d!; dk/ is tangential to the dispersion surface at the point .!; k.!//, as illustrated in Fig. 11.40. The orthogonality relation given in (11.256) then states that the vector .@f =@!; @f =@k/ is normal to the dispersion surface at that point. If the !; k-coordinate axes are chosen such that they are parallel to the t; z-coordinate axes, respectively, then the saddle points of the phase function .z; t; !/ k.!/z  !t are located at those particular points .!s ; k.!s // of the dispersion surface where the normal vector .@f =@!; @f =@k/ is parallel to the space–time vector .1; z=t /. That is, each saddle point !s describes a point of constant spatial frequency k.!/ at which the normal vector satisfies the relation .@f =@!; @f =@k/ D .1; z=t /. Along the z-axis, rT D 0 so that ' D z and the time-domain pulsed beam wave field described by (11.238) becomes UC .r; t / 

i% .z  i%/

Z 0

1

Qf.!/e i.k.!/z!t/ d!

(11.257)

242

11 The Group Velocity Approximation

as k0 ! 1. The on-axis pulsed beam wave field is then seen to propagate like a plane wave pulse. The (real-valued) on-axis stationary phase point is given by the solution of the equation z (11.258) D vg .!N s /; t where vg .!/ .dk=d!/1 is the real-valued group velocity in the lossless, dispersive medium. Figure 11.40 provides a geometric interpretation of this relation. Notice that as the time t increases at a fixed observation point along the z-axis (the beam axis), the point .!s ; ck.!s // moves along the dispersion surface and approaches the point .!p ; 0/ situated on the !-axis. For off-axis observation points that are near to the beam axis (i.e., paraxial observation points), the phase function .r; !/ D k.!/'.r/  !t may be approximated by the first three terms in its Taylor series expansion about the on-axis stationary point !N s as .r; !/ 0 .r; t / C 1 .r; t /.!  !N s / C .2 .r; t /=2/.!  !N s /2 ;

(11.259)

where 0 .r; t / .r; !N s / D k.!N s /'.r/  !N s t; ˇ   @ ˇˇ '.r/ 0 1 ; 1 .r; t / D k .!N s /'.r/  t D t @! ˇ!N s z ˇ @2  ˇˇ 2 .r; t / D k 00 .!N s /'.r/; @! 2 ˇ!N s

(11.260) (11.261) (11.262)

the final form of the expression in (11.261) resulting from substitution of the relation given in (11.258) for the on-axis stationary point. With the quadratic dispersion approximation given in (11.259), the saddle point equation  0 .r; !s / D 0 results in the expression 1 .r; t / : (11.263) !s !N s  2 .r; t / Substitution of this result in (11.259) then results in the approximate expression .r; !s / 0 .r; t /  12 .r; t /=22 .r; t /. The asymptotic behavior of the integral in (11.238) is then obtained by a direct application of the asymptotic result in either (10.18) with  D 1 or (F.115) of Appendix F with the result  1=2 2 i% Qf.!N s /e iŒ0 .r;t/12 .r;t/=22 .r;t/ ; U .r; t /  00 z  i% i k .!N s /'.r/ C

(11.264)

as k0 z ! 1. Notice that the rT dependence in the approximate relation given in (11.263) is not restricted to O.rT2 / so that its accuracy goes beyond that of the paraxial approximation. Finally, notice that the amplitude functions appearing in (11.264) have been evaluated at the on-axis stationary point !N s instead of at the

11.7 Localized Pulsed-Beam Propagation

243

saddle point !s given by (11.263) based upon the assumption that the difference between these two points may be safely neglected in the amplitude dependence of the pulse beam wave field, but not in its phase behavior. Case 2: Qf.!/ with frequency-dependent phase The asymptotic approximation as k0 z ! 1 of the propagated pulsed beam wave field given in (11.264) is invalid when the spectral function Qf.!/ contains a frequency-dependent phase term, this being due to the k0 ! 1 asymptotic behavior. This can be circumvented by including this frequency-dependent phase term in the propagation phase factor. Because the resultant phase function then depends explicitly upon the specific form of the input pulse spectrum, its effect on the asymptotic behavior must be determined in a case by case manner. As an example, for the analytic delta function pulse spectrum given in (11.242) the phase function appearing in (11.238) must be modified to read ıC .r; !/ D k.!/'.r/  !.t  iT =2/:

(11.265)

The on-axis stationary phase point is given by the solution of (11.258). However, in place of (11.252), the off-axis stationary point is now found to satisfy the relation ˇ dk.!/ ˇˇ t  iT =2 D ; '.r/ d! ˇ!s

(11.266)

whose solution requires analytic continuation of the real-valued dispersion surface k.!/ into the complex !-plane for both off-axis and on-axis observation points. Following the method of analysis given in (11.259)–(11.263), the approximate solution for the off-axis saddle points is found to be given by   rT2 t 1 C iT : !s !N s  00 2k .!N s /'.r/ z.z  i%/

(11.267)

Substitution of this approximation in (11.265) for the phase function then results in the approximate expression   rT2 t .t  iT =2/ ıC .r; !s / k.!N s /'.r/  !N s .t  iT =2/ C 00 C iT 2k .!N s /'.r/ z.z  i%/

k.!N s /'.r/  !N s .t  iT =2/ ; (11.268) where the final approximation is valid provided that the magnitude of the neglected quantity from the previous line is much less than 2. Specific conditions under which this paraxial approximation is valid have been shown [78] to depend upon the ratio rT =z, the on-axis propagation distance z, and the proximity of the wave front curvature to the local curvature of the dispersion surface at the stationary point. The

244

11 The Group Velocity Approximation

asymptotic behavior of the propagated pulsed beam wave field for the initial analytic delta function waveform f.t / D AfıC .t / is then given by UC .r; t /  ıC

 1=2 2 i% Ae iŒk.!N s /'.r/!N s .tiT =2/ ; z  i% ik00 .!N s /'.r/

(11.269)

as k0 z ! 1. Further analysis of pulsed beam properties in lossless dispersive media may be found in the pair of papers by Melamed and Felsen [78, 79] upon which the present analysis has been based.

11.8 The Necessity of an Asymptotic Description The fact that the GVD of dispersive pulse propagation, whether it be in the SVE or SEWA, is approximately valid only in the weak dispersion limit shouldn’t be of any surprise based upon the multiple layers of approximation used in deriving the evolution equations used in these various formulations. In this regard, it is meaningful to return to the original formulation of the problem. Subject to the approximation that, at the very most, the frequency-dependence of the absorption may be neglected in a weakly dispersive medium, the resultant integral representation of the propagated field may be properly evaluated using Kelvin’s method of stationary phase [9]. The subsequent approximations made by Havelock [8] of replacing the stationary phase point by the wavenumber value k0 that the initial pulse spectrum was peaked about and then expanding the dispersion relation about that point then restrict the group velocity method to the normal dispersion regions of weakly dispersive media. Notice that in Kelvin’s method, k0 is the stationary phase point of the dispersion relation for the real-valued wavenumber k.!/, whereas in Havelock’s approximation, k0 describes the fixed wavenumber value about which the initial pulse spectrum is peaked. This subtle change in the value of k0 leads to significant consequences regarding the accuracy of the GVD. What is lacking in each of these GVDs is a consideration of the total medium response to the input pulsed wave field, not just the medium response about the fixed carrier frequency of the pulse. To obtain a description of dispersive pulse propagation that is not limited to the normal dispersion regions of weakly dispersive media, each of the unnecessary approximations that the GVD is based on must not be used; in particular, the dispersive properties of the medium must be accurately accounted for over the entire frequency spectrum in the ultrashort pulse case. The uniform asymptotic description based upon the saddle point method provides precisely this description. The remainder of this volume is devoted to its development and application to ultrashort pulse, ultrawideband signal dynamics in causally dispersive media and systems. As a final note, it is important to recognize that the localized pulsed-beam approximation developed over the years by L. B. Felsen and co-workers is only restricted by the lossless dispersive medium assumption. Because of this, its

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6

Leo Felsen and I had a long series of conversations regarding this extension, beginning at the 1989 URSI International Symposium on Electromagnetic Theory in Stockholm, Sweden and continuing through to the 2000 Progress In Electromagnetics Research Symposium in Cambridge, Massachusetts. At our last meeting we agreed that we should be able to extend his localized pulsedbeam theory to a causally dispersive medium using the the paraxial approximation of the angular spectrum representation presented in Sect. 9.1. This extension remains to be done.

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Problems

249

Problems 11.1. Derive the sequence of expressions given in (11.59)–(11.61) for the double exponential pulse, obtaining approximate expressions for these results when ˛2 D ˛1 C ˛ with ˛=˛1  1. In addition, compare the pulse width measure t D j˛2 ˛1 j=˛1 ˛2 to the numerically determined pulse width for the double exponential pulse in the three cases ˛1  ˛2 ˛1 ˛2 , and ˛1  ˛2 . Finally, compute the spectrum fQde .!/ and plot its magnitude in these three cases. 11.2. Derive the series of expressions given in (11.97). 11.3. Beginning with the scalar wave equation given in (11.88), derive the phasor form of this wave equation as given in (11.116). 11.4. Derive the SVE wave equation given in (11.118). 11.5. Derive the SVE wave equation given in (11.126), as well as its separation into real and imaginary parts. 11.6. Beginning with the phasor wave equation given in (11.140), derive the SVE phasor wave equations given in (11.145) and (11.147). 11.7. Show that the complex envelope function a.z; Q t / given by (11.153) in the linear dispersion approximation satisfies the SVE wave equation given in (11.154). 11.8. Show that the complex envelope function a.z; Q t / given by (11.156) in the quadratic dispersion approximation satisfies the (SVE) wave equation given in (11.157). Hint: Substitute the quadratic dispersion relation given in (11.155) into the integral representation of the complex phasor envelope function given in (11.150) and work with that result. 11.9. Beginning with the integral representation (11.150) describing the propagation of the complex envelope function through a dispersive medium, derive the approximate integral representation given in (11.156) when the complex wavenumQ ber k.!/ is represented by its quadratic dispersion approximation as given in (11.155). 11.10. Using the integral representation (11.170) for the propagated complex envelope function, derive the propagation properties for the initial unit amplitude gaus2 2 sian envelope pulse ug .t / D e t =T0 centered at t D 0 with initial pulse half-width > 0. T0 p In particular, show that the pulse half-width is given by T .z/ D 1 C .z=`D /2 for all z  0, where `D is the dispersion length of the pulse in the dispersive medium. 11.11. Under the change of variable  D t  ˇ 0 .!c /z,  D z to a moving coordinate system, show that (11.191) is transformed into (11.192). 11.12. Beginning with (11187), derive the differential equation for the phase given in (11.199) in the moving coordinate frame.

./

Chapter 12

Analysis of the Phase Function and Its Saddle Points

In preparation for the asymptotic analysis of the exact Fourier–Laplace integral representation given either in (11.45) as A.z; t / D

1 2

Z

fQ.!/e .z=c/.!;/ d!

(12.1)

C

R1 with fQ.!/ D 1 f .t /e i!t dt denoting the temporal Fourier spectrum of the initial pulse f .t / D A.0; t /, or in (11.48) as

Z 1 i .z=c/.!;/ < ie uQ .!  !c /e d! A.z; t / D 2 C

(12.2)

when f .t / D u.t / sin .!c t C / with real-valued envelope function u.t / and fixed angular carrier frequency !c , for the propagated plane wave pulse with z  0 in a (causal) temporally dispersive medium, it is necessary to first determine the topography of the real part  .!;  /
(12.3)

Q Here k.!/ D .c=!/n.!/ is the complex wavenumber in the simple (i.e., homogeneous, isotropic, locally linear), causal dispersive medium with complex index of refraction n.!/ D Œ.!/.!/=.0 0 / 1=2 , and where 

ct z

(12.4)

is a dimensionless space–time parameter defined for z > 0. In particular, the location of the saddle points of .!;  /, the value of .!; / at these saddle points, K.E. Oughstun, Electromagnetic and Optical Pulse Propagation 2: Temporal Pulse Dynamics in Dispersive, Attenuative Media, Springer Series in Optical Sciences 144, c Springer Science+Business Media, LLC 2009 DOI 10.1007/978-1-4419-0149-1 12, 

251

252

12 Analysis of the Phase Function and Its Saddle Points

and the regions of the complex !-plane wherein  .!;  / is less than the value of  .!;  / at the dominant saddle point for a given value of  are all required as  varies continuously over some specified space–time domain. This chapter is devoted to that purpose. The analysis presented here is involved because the behavior of the complex-valued phase function is complicated. For example, for a singleresonance Lorentz model dielectric, .!; / possesses four branch points and four saddle points, and its topography evolves with  in a complicated manner. By examining the structure of .!; / in special regions of the complex !-plane where its behavior is relatively simple (e.g., along the real and imaginary axes, in the vicinities of the branch points and the origin, and for large values of j!j), following Sommerfeld [1], Brillouin [2, 3] was able to determine a rough picture of the topography of  .!;  / D
12.1 General Saddle Point Dynamics for Causally Dispersive Dielectrics

253

12.1 General Saddle Point Dynamics for Causally Dispersive Dielectrics The physically correct analysis of the entire dynamical field evolution in dispersive pulse propagation is critically dependent upon the model of the frequency dependence of the linear medium response. To maintain strict adherence to the fundamental physical principle of causality, it is essential that any model chosen for the medium response must be causal. Because of the analyticity properties of the dielectric permittivity .!/ as expressed by Titchmarsh’s theorem [8], the angular frequency dependence of the dielectric permittivity is required to satisfy the dispersion relation [see Sect. 4.3 of Vol. 1] .!/  0 D

1 P i

Z

1 1

. /  0 d ;

!

(12.5)

where the principal value of the integral is to be taken, as indicated. In terms of the relative dielectric permittivity ".!/ .!/=0 this relation becomes 1 P ".!/  1 D i

Z

1 1

". /  1 d :

!

(12.6)

The real and imaginary parts of this relation then yield the pair of Kramers–Kronig relations Z 1 "i . / 1 "r .!/ D 1 C P d ; (12.7) 

1  ! Z 1 "r . /  1 1 d ; (12.8) "i .!/ D  P  1  ! where "r .!/ 0 for finite ! 0 D 0, (12.7) will yield a function "r .!/ that is consistent with all physical requirements, i.e., one which is in principle possible. This makes it possible to use (12.7) even when the function "i .!/ is approximate. On the other hand, (12.8) does not yield a physically possible function "i .!/ for an arbitrary choice of "r .!/, since the condition "i .! 0 / > 0 for finite ! 0 > 0 is not necessarily fulfilled.

Hence, in any serious attempt at obtaining the approximate behavior of the causally interrelated real and imaginary parts of the dielectric permittivity in some specified region of the complex !-plane, special care must be given to the mathematical form of the dispersion relation pair given in (12.7) and (12.8) in order that physically meaningful results are obtained.

254

12 Analysis of the Phase Function and Its Saddle Points

For a nonconducting medium as considered here, the material absorption identically vanishes at zero frequency [most simply because i .! 0 / is an odd function of real ! 0 ] so that, from (12.8), one immediately obtains the sum rule Z 1 "r .! 0 /  1 0 P d! D 0: (12.9) !0 1 The material absorption also identically vanishes at infinite real frequency, as can be seen from the limiting behavior of the dispersion relations given in (12.7) and (12.8) lim "r .! 0 / D 1;

(12.10)

lim "i .! 0 / D 0:

(12.11)

! 0 !˙1

! 0 !˙1

Hence, with little or no loss in generality, it is safe to assume that the angular frequency dependence of "i .! 0 / along the positive ! 0 -axis is such that the loss is significant only within a finite angular frequency domain Œ!0 ; !m with 0  !0 < !m  1:

(12.12)

For all nonnegative values of ! 0 outside of this frequency domain, the material absorption is then negligible by comparison. Attention is now focused on the two special regions of the complex !-plane wherein the dielectric permittivity is reasonably well behaved, these being the circular region about the origin where j!j  !0 and the annular region about infinity where j!j  !m . The analysis presented here follows that given by the author [10] in 1994.

12.1.1 The Region About the Origin (j!j  !0 ) As "i .!/ identically vanishes at the origin ! D 0, one may expand the denominator in the integrand of (12.7) for small j!j in a Maclaurin series as Z 1 "i . / 1 .1  != /1 d "r .!/ D 1 C P 

1 Z 1 1 X 1 "i . / Š 1C !j P d : (12.13) j C1  1 j D0 The validity of this expansion relies upon the property that when j j  j!j and the Maclaurin series expansion of the factor .1  != /1 in the integrand breaks down, "i . / is very close to zero and serves to neutralize the effect. Due to the odd symmetry of "i . / [i.e., "i . / D "i .  /], one then obtains the expansion "r .!/ Š 1 C

1 X j D0

ˇ2j ! 2j

(12.14)

12.1 General Saddle Point Dynamics for Causally Dispersive Dielectrics

with coefficients ˇ2j

1 P 

Z

1 1

"i . / d ;

j C1

255

(12.15)

which is valid for j!j  !0 . Since "r . / does not vanish when D 0, the same expansion technique cannot be used to obtain a low frequency expansion of the dispersion relation given in (12.8). However, because of the even symmetry of "r . / [i.e., "r . / D "r .  /], that equation may be rewritten as "i .!/ D !

2 P 

Z

1 0

"r . /  1 d ;

2  !2

(12.16)

which explicitly displays the property that "i .0/ D 0. In addition, for an attenuative medium one must have that "i .! 0 /  0 for all ! 0  0. Based upon these two results, one may then take the approximation "i .!/ 2ı1 !

(12.17)

for j!j  !0 , where ı1 is a nonnegative real number. Hence, for sufficiently small values of complex ! such that j!j  !0 , the complex relative dielectric permittivity ".!/ .!/=0 may be approximated as ".!/ "s C 2i ı1 ! C ˇ2 ! 2 ; where "s 1 C ˇ0 D 1 C

1 P 

Z

1 1

"i . / d

(12.18)

(12.19)

is the static relative dielectric permittivity of the dispersive material. Notice that, because of the odd symmetry of "i . /, the static relative dielectric permittivity satisfies the inequality "s  1 where the equality holds [i.e., where "s D 1] only in the case of the ideal vacuum [i.e., when "i .!/ D 0 for all !]. The complex index of refraction in the small frequency region about the origin is then given by  1=2 ı1 1 n.!/ D ".!/

0 C i ! C 0 20

  ı12 ˇ2 C 2 ! 2 ; 0

(12.20)

where D n.0/ 0 "1=2 s

(12.21)

is the static index of refraction of the dispersive dielectric material. The saddle points of the complex phase function .!;  / D i !.n.!/ / defined in (12.3) are determined by the condition  0 .!; / D 0, where the prime denotes differentiation with respect to !. The saddle points are then given by the roots of the saddle point equation (12.22) n.!/ C !n0 .!/ D ;

256

12 Analysis of the Phase Function and Its Saddle Points

which explicitly depend upon the space–time parameter  D ct =z. With the approximation given in (12.20) for the complex index of refraction in the region about the origin, the saddle point equation becomes !2 C i

4ı1 20 ! .  0 / 0; 3˛1 3˛1

(12.23)

where ˛1 ˇ2 Cı12 =02 . The roots of (12.23) then yield the approximate near saddle point locations 2ı1 ˙ ./ ˙ ./  i ; (12.24) !SP n 3˛1 with

 1=2 ı12 1 0 6 .  0 /  4 2 . / : 3 ˛1 ˛1

(12.25)

This approximate result is precisely in the form of the first approximation for the location of the near saddle points in a single-resonance Lorentz model dielectric [2, 3, 6, 7] as well as in multiple-resonance Lorentz model dielectrics [11, 12]. The saddle point dynamics (i.e., the evolution of the saddle point locations with  ) are thus seen to depend upon the sign of the quantity ˛1 ˇ2 C ı12 =02 . For a Lorentz model dielectric, ˇ2 is typically positive so that ˛1 > 0; such a medium is classified here as a Lorentz-type dielectric. On the other hand, for a Debye model dielectric (as well as for the Rocard–Powles extension of the Debye model), ˇ2 is typically negative so that ˛1 < 0; such a medium is classified here as a Debye-type dielectric. Of course, it may just happen that ˛1 D 0; in this highly unusual event, a so-called transition-type dielectric is obtained. The dynamical evolution of the near saddle points are now separately treated for these three cases.

12.1.1.1

Case 1: The Lorentz-Type Dielectric (˛1 > 0)

For Lorentz-type dielectrics, the saddle point solution separates into two subluminal (i.e.,  > 1) space–time domains separated by the critical space–time value 1 0 C

2ı12 ; 3˛1 0

(12.26)

where 1 > 0 for nonvanishing ı1 . Application [1–3, 6, 7] of the method of proof of Jordan’s lemma [13, Sect. 6.222] shows that if the initial time behavior A.0; t / of the plane wave pulse at the input plane z D 0 is zero for all time t < 0, then the propagated wavefield in a Lorentz-type dielectric is zero for all superluminal space–time points  < 1 with z > 0. The saddle point dynamics in a Lorentz-model dielectric then need to be considered for such finite duration pulsed signals over just the subluminal space–time domain  > 1.

12.1 General Saddle Point Dynamics for Causally Dispersive Dielectrics

257

For initial values of the space–time parameter  2 .1; 1 , the near saddle point locations are given by ˙ ./ !SP n

  2ı1 :

i ˙ j ./j  3˛1

(12.27)

The two near, first-order saddle points SP˙ n are then seen to initially lie along the imaginary angular frequency axis, symmetrically situated about the point ˙ .1 / D .2ı1 =3˛1 /i , approaching each other along the imaginary axis as !SP n  increases from unity to 1 . Detailed analysis (see Sect. 12.3.1) shows that only the upper near saddle point SPC n is relevant to the subsequent asymptotic analysis of the integral representations given in (12.1) and (12.2) over this -domain [6, 7]. The approximate behavior of the complex phase at the upper near saddle point is then obtained from (12.3) with the substitution of the approximate expressions from (12.20) and (12.27) as ( 2ı 1 C ; /   j . /j   0 .!SP n 3˛1      ) ı12 2ı1 0 2ı1 ı1   j ./j  ˇ2 C 2  j . /j 2 3˛1 ˛1 3˛1 0 

(12.28) for 1 <  < 1 . At the critical space–time point  D 1 , .1 / D 0 and the two near, first-order saddle points SP˙ n have coalesced into a single second-order saddle point SPn at !SPn .1 / 

2ı1 i: 3˛1

(12.29)

The approximate value of the complex phase at this critical point is given by .!SPn ; 1 / 

4ı13 9˛13 0

  ı2 ˇ2 C 12 ; 0

(12.30)

which directly follows from (12.28) with substitution of (12.26). Notice that .!SPn ; 1 / vanishes as ı1 ! 0. Finally, for  > 1 the two near, first-order saddle points SP˙ n have moved off of the imaginary axis and are now symmetrically situated about the imaginary axis in the lower-half of the complex !-plane. The near saddle point locations are now given by 2ı1 ˙ ./ ˙ ./  i; (12.31) !SP n 3˛1

258

12 Analysis of the Phase Function and Its Saddle Points

where . / is real-valued over this space–time domain. The approximate behavior of the complex phase at these two near saddle points is then found to be given by (   ı12 3 2ı 1 1 ˙ 2 ˛1  ˇ2  2 .!SPn ; /  ./   0 C 3˛1 0 2 0   ) 2ı12 ı12 C 2 ˇ2  3˛1 C 2 9˛1 0 ) (    2 2 8ı ı12 8ı 1 2 ˇ2 C 2 ./  12 C 1 ˙i ./ 0   C 20 9˛1 0 9˛1 (12.32) for  > 1 . Hence, the complex phase function .!;  / at the relevant near saddle points is nonoscillatory for 1 <   1 while it has an oscillatory component for all  > 1 . Notice that the accuracy of these approximate solutions for the near saddle ˇ ˙ pointˇ . /ˇ dynamics rapidly diminishes as the quantity j  0 j increases, because ˇ!SP n will then no longer be small in comparison to !0 . An accurate description of the near saddle point dynamics for a Lorentz-type dielectric that is valid for all  > 1 can only be constructed once the behavior of the complex index of refraction n.!/ is explicitly known in the region of the complex !-plane about the first absorption peak at !0 , as has been done [6, 7] for a single-resonance Lorentz model dielectric. These results remain valid in the special case when ı1 D 0. In that case, the !-dependence of "i .!/ varies as ! 3 or higher about the origin. The approximate saddle point equation given in (12.23) is then still correct to order O.! 2 / and, for values of ! about the origin such that the inequality j!j  !0 is well satisfied, the approximate saddle point locations are now given by ˙ .!/ !SP n



20

˙ .  0 / 3ˇ2

1=2 :

(12.33)

The same dynamical behavior then results, but with the two near, first-order saddle points SP˙ n , which are now symmetrically situated about the origin, coalescing into a single and second-order saddle point SPn at the origin when  D 0 . Clearly, (12.28), (12.30), and (12.32) for the complex phase behavior at the near saddle points remain valid in this case with ı1 D 0. This is the only special situation that can arise for a Lorentz-type dielectric since neither ˇ0 nor ˇ2 can vanish for a causally dispersive dielectric, the trivial case of a vacuum being excluded. 12.1.1.2

Case 2: The Debye-Type Dielectric (˛1 < 0)

For a Debye-type dielectric, ˛1 < 0 so that 1 D 0 

2ı12 3j˛1 j0

(12.34)

12.1 General Saddle Point Dynamics for Causally Dispersive Dielectrics

259

and 1 < 0 . Application [12, 14] of the method of proof of Jordan’s lemma [13, Sect. 6.222] shows that if the initial time behavior A.0; t / of the plane wave pulse at the input plane z D 0 is zero for all time t < 0, then the propagated wavefield in a Debye-type dielectric is zero for all space–time points  < 1 with z > 0; a detailed derivation is given in Sect. 13.1. This due to the fact that the absorption does not go to zero as ! ! 1 for a Debye-type dielectric. As a consequence, the saddle point dynamics in a Debye-model dielectric need to be considered for such finite duration pulsed signals over just the space–time domain  > 1 > 1. From (12.24)–(12.25) the two near, first-order saddle point locations are seen to be given by   2ı1 ˙ (12.35) !SPn . / i ˙j ./j C 3j˛1 j for  > 1 . These two saddle points are symmetrically situated about the fixed point ! D i 2ı1 =3j˛1 j and move away from each other along the imaginary axis as  increases above 1 . Because the upper saddle point moves away from the origin as  increases (and consequently increasingly violates the condition j!j  !), only the lower near saddle point SP n is relevant to the asymptotic analysis over this small angular frequency domain about the origin. The saddle point SP n then moves down the imaginary axis and crosses the origin at  D 0 . The complex phase behavior at this first-order near saddle point is then given by (  2ı1    0 .!SPn ; /  j . /j  3j˛1 j   )    ı12 2ı1 2ı1 1 2ı1  ˇ2 C 2 j . /j  j . /j   20 3j˛1 j 3j˛1 j 0 (12.36) for  > 1 . Hence, the complex phase behavior at the near saddle point SP n of a Debye-model dielectric is nonoscillatory for all  > 1 .

12.1.1.3

Case 3: The Transition-Type Dielectric (˛1 D 0)

In the highly unusual event that ˛1 D 0, in which case ˇ2 D ı12 =02 , a so-called transition-type dielectric is described. The approximation given in (12.20) for the complex index of refraction then becomes n.!/ 0 C i

ı1 ! C O.! 3 /; 0

(12.37)

which then results in a single first-order near saddle point that moves down the imaginary axis linearly with increasing  as !SPn ./ i

0 .  0 /: 2ı1

(12.38)

260

12 Analysis of the Phase Function and Its Saddle Points

A more accurate description of the near saddle point dynamics in this transitional case requires that the expansion of the complex index of refraction given in (12.37) be extended to include the ! 3 term.

12.1.2 The Region About Infinity (j!j  !m ) Since "i . / vanishes as ! ˙1, the denominator in the integrand of (12.7) may be expanded for large j!j in a Laurent series so that 1 "r .!/ D 1  P 

Š 1

Z

1

"i . / .1  =!/1 d !

1

1 X

1

1 j C1 !  j D0

Z

1

"i . / j d :

(12.39)

1

The validity of this expansion relies on the fact that when j j  j!j and the expansion of the quantity .1  =!/1 that appears in the integrand of the above principal value integral breaks down, the quantity "i .!/ is vanishingly small and serves to neutralize this behavior. Because of the odd symmetry of "i . /, the relation given in (12.39) can be rewritten as "r .!/ Š 1 

1 X a2j ; ! 2j j D1

(12.40)

which is valid for j!j  !m , with coefficients a2j

Z

1 

1

"i . / 2j 1 d :

(12.41)

1

Notice that the first coefficient, a2 , is nonvanishing for any lossy dielectric, namely, 1 a2 

Z

1

"i . / d > 0:

(12.42)

1

Because the quantity ."r . /  1/ also vanishes as ! ˙1, the same expansion procedure can be applied in the integrand of (12.8), which can be rewritten as "i .!/ D !

2 P 

Z 0

1

"r . /  1 d ;

2  !2

(12.43)

12.1 General Saddle Point Dynamics for Causally Dispersive Dielectrics

261

to yield 2 "i .!/ D P  Š

1 X j D0

Z

1 0

1 "r . /  1  1  2 =! 2 d !

bj ; 2j ! C1

(12.44)

which is valid for j!j  !m , with coefficients 2 bj 

Z

1

."r . /  1/ 2j d :

(12.45)

0

There are then two distinct classes of dielectrics that are distinguished by the value of the zeroth-order coefficient Z 2 1 ."r . /  1/ d : b0 (12.46)  0 For the first class, b0 ¤ 0, which is characteristic of a Debye-type dielectric. For the second class, b0 D 0, which is characteristic of a Lorentz-type dielectric These two cases are now treated separately.

12.1.2.1

Case 1: The Debye-Type Dielectric (b0 ¤ 0)

For a Debye-type dielectric, b0 ¤ 0, and the complex-valued relative dielectric permittivity in the annular region j!j  !m about infinity is given approximately by ".!/ 1 C i

a2 b0  2; ! !

(12.47)

and the associated complex index of refraction n.!/ D .".!//1=2 is then given approximately by a2  b02 =4 b0  : (12.48) n.!/ 1 C i 2! 2! 2 With this substitution in (12.22), the saddle point equation becomes  1

a2  b02 =4

0: 2! 2

(12.49)

The location of these distant saddle point solutions in the complex !-plane is then seen to depend critically upon the sign of the quantity .a2  b02 =4/.

262

12 Analysis of the Phase Function and Its Saddle Points

For a simple Debye-model dielectric with frequency-dependent relative permittivity [see (4.179) of Vol. 1] ".!/ D "1 C ."s  "1 /=.1  i ! / with relaxation time  , static relative permittivity "s , and high-frequency limit "1 D 1, the coefficients appearing in (12.47) are found to be given by b0 D ."s  1/= and a2 D .1  "s /= 2 , in which case it is found that .a2  b02 =4/ < 0 and is equal to zero only when "s D 1 (i.e., only in the trivial of a vacuum). The approximate distant saddle point locations are then given by ˙ ./ ˙i ; (12.50) !SP d .  1/1=2 q where .b02 =4  a2 /=2. These distant saddle point solutions are symmetrically situated about the origin along the imaginary axis and move in toward the origin as  increases from unity, evolving into the near saddle points for a Debye-type dielectric [see (12.35)]. They are then of no further interest in the asymptotic theory.

12.1.2.2

Case 2: The Lorentz-Type Dielectric (b0 D 0)

A Lorentz-type dielectric is further distinguished by the fact that the expansion coefficient b0 appearing in the approximate description of the angular frequency dependence of the relative dielectric permittivity in the region about infinity is identically zero, so that the sum rule [15] Z

1

  "r . /  1 d D 0

(12.51)

0

is satisfied. The complex-valued relative dielectric permittivity in the region j!j  !m is then given approximately by [16] b2 a2 Ci 3 !2 ! a2 ;

1 !.! C ib2 =a2 /

".!/ 1 

(12.52)

and the associated complex index of refraction is then given by n.!/ 1 

a2 : 2!.! C ib2 =a2 /

(12.53)

With this substitution in (12.22), the saddle point equation becomes 1 C

a2

0: 2!.! C ib2 =a2 /2

(12.54)

The roots of this equation then yield the approximate expression for the distant saddle point locations as

12.1 General Saddle Point Dynamics for Causally Dispersive Dielectrics ˙ !SP . / ˙ d



b2 a2  22 2.  1/ 4a2

1=2 i

b2 : 2a2

263

(12.55)

This is precisely the form of the first approximate expressions for the distant saddle point locations in both single-resonance [2, 3, 6, 7] and multiple-resonance [11,12] Lorentz model dielectrics. As can be seen from (12.55), these distant saddle points are symmetrically situated about the imaginary axis and lie along the line ! D ! 0  i b2 =2a2 in the lower-half of the complex !-plane. As  increases from unity, they move in from infinity along this line. Notice that the accuracy of this approximation for the distant saddle point locations diminishes as  increases away ˙ . /j will then no longer be large in comparison with !m . from unity because j!SP d In that case, higher-order approximations that are valid for all   1 need to be obtained. However, such an accurate description of the distant saddle point dynamics for a Lorentz-type dielectric that is valid for all  > 1 can only be constructed once the behavior of the complex index of refraction n.!/ is explicitly known in the region of the complex !-plane about the upper edge of the uppermost absorption band at !m , as has been done [6, 7] for a single-resonance Lorentz model dielectric. If this is not possible, then accurate numerical methods need to be used. With substitution of the approximate expression given in (12.55) into (12.3), together with the approximation of the complex index of refraction given in (12.53), the approximate complex phase behavior at the distant saddle point locations in a Lorentz-type dielectric is found to be given by ˙ ; /  .!SP d

   1=2 b2 b2 1 C 23 .  1/ : .  1/ i 2a2 .  1/ a2 a2

(12.56)

˚ ˙ ˙ ; / < .!SP ;  / vanishes at the speed of light point Notice that  .!SP d d  D 1 and then decreases linearly with  as  increases above unity.

12.1.3 Summary This general, but very approximate, description of the saddle point dynamics of the complex phase function .!;  / in a causally dispersive dielectric has shown that, as far as linear dispersive pulse propagation is concerned, only Lorentz-type and Debye-type dielectrics need be considered. Because of its highly unusual likelihood, the analysis of the transition-type dielectric is left as an exercise (see Prob. 12.1). The necessary detail of the saddle point dynamics in both Lorentz-model and Debye-model dielectrics, as well as in Drude model metals, then forms the subject matter of the remainder of this chapter.

264

12 Analysis of the Phase Function and Its Saddle Points

12.2 The Behavior of the Phase in the Complex !-Plane for Causally Dispersive Materials   The behavior of the complex phase function .!;  / i ! n.!/   in the complex !-plane at any fixed real-value of the space–time parameter   1 is dictated by the analytic form of the complex index of refraction n.!/. Because causality is an essential, physical feature of dispersive pulse dynamics, only causal, physical models of the complex index of refraction are considered here. The analysis begins with a detailed review and extension of Brillouin’s classical analysis [2, 3] of this problem for a single-resonance Lorentz-model dielectric, followed by that for a multiple-resonance Lorentz-model dielectric, a Rocard Powles Debye model dielectric, and finally for a Drude model metal.

12.2.1 Single-Resonance Lorentz Model Dielectrics The complex index of refraction for a single-resonance Lorentz model dielectric is given by (see Sect. 4.4.4 of Vol. 1)  n.!/ D 1 

b2 ! 2  !02 C 2iı!

1=2 ;

(12.57)

p where b D .k4k=0 /N qe2 =m is the plasma frequency of the dispersive medium with number density N of harmonically bound electrons of charge magnitude qe and mass m with angular resonance frequency !0 , and where ı is the associated phenomenological damping constant of the bound electron system. The values of these material parameters as chosen by Brillouin in his analysis [2, 3] for a medium possessing a single near-ultraviolet resonance frequency are !0 D 4:0  1016 r/s; p b D 20  1016 r/s; 16

ı D 0:28  10

(12.58)

r/s:

Although this choice of the medium parameters corresponds to an extremely dispersive, absorptive dielectric medium, it is used in many of the numerical examples presented here to facilitate a comparison with Brillouin’s results. To facilitate comparison with experimental results in the optical region of the spectrum which primarily are conducted in highly transparent glasses, the weak dispersion limit as N ! 1 is considered throughout the analysis as deemed necessary. The singular dispersion limit, obtained as ı ! 0, is also considered. Finally, this analysis is equally applicable to dispersive dielectrics that possess resonance frequencies from the infrared through the ultraviolet regions of the electromagnetic spectrum.

12.2 The Behavior of the Phase in the Complex !-Plane

265

Attention is now turned to the description of the analytic structure of the complex phase function .!;  / in the complex !-plane. This analysis is simplified by the general symmetry relation [see (11.26)] .!; / D   .!  ;  /;

(12.59)

so that the real part  .!;  / D
(12.60)

$ .! 0 C i ! 00 ;  / D $ .! 0 C i ! 00 ;  /;

(12.61)

where ! 0
! 2  !12 C 2iı! ! 2  !02 C 2iı!

1=2

 D

0 /.!  !0 / .!  !C .!  !C /.!  ! /

q !1 C !02 C b 2 :

1=2 ;

(12.62)

(12.63)

The branch point locations for a single-resonance Lorentz model dielectric are then given by q !˙ D ˙ !02  ı 2  i ı; q 0 !˙ D ˙ !12  ı 2  i ı;

(12.64) (12.65)

and are symmetrically located about the imaginary axis, lying along the line ! 00 D ı, provided that !0 > ı, which is assumed to be the case throughout this analysis. 0 , as depicted The branch cuts are then taken as the line segments !0 ! and !C !C in Fig. 12.1. The complex index of refraction n.!/ and the complex phase function .!;  / are then analytic throughout the complex !-plane with the exception of the 0 and !˙ . branch points !˙ The complex index of refraction may be separated into real and imaginary parts as (12.66) n.!/ D nr .!/ C ini .!/;

266

12 Analysis of the Phase Function and Its Saddle Points

''

' ' cut

cut

'

0 0 0 Fig. 12.1 Location of the branch points !˙ and !˙ and the branch cuts ! ! and !C !C in the complex !-plane for a single-resonance Lorentz model dielectric with undamped resonance q

frequency !0 , damping constant ı, and plasma frequency b, where !1 D

!02 C b 2

where nr .!/
00   

   D  ! nr .!/   C ! 0 ni .!/ C i ! 0 nr .!/    ! 00 ni .!/ ; (12.67) so that

    .!;  / D  ! 00 nr .!/   C ! 0 ni .!/ ;   $ .!;  / D ! 0 nr .!/    ! 00 ni .!/:

(12.68) (12.69)

Explicit expressions for the real and imaginary parts of the complex index of refraction (12.57) for a single-resonance Lorentz model dielectric now need to be obtained. It follows from (12.62) that n2 .!/ D

! 2  !12 C 2iı! ! 2  !02 C 2iı! 

D

! 02  ! 002  !02  2ı! 00

2

   b 2 ! 02  ! 002  !02  2ı! 00 C 4! 02 .! 00 C ı/2 C 2i! 0 .! 00 C ı/ b 2 :   2 ! 02  ! 002  !02  2ı! 00 C 4! 02 .! 00 C ı/2 (12.70)

12.2 The Behavior of the Phase in the Complex !-Plane

267

The magnitude and phase of this complex-valued expression are then given by ˇ ˇ 2 ˇn .!/ˇ2 D 1 C

. / ; (12.71) . /

.!/ arg Œn2 .!/

  2! 0 .! 00 Cı/b 2 ; D arctan 2 .! 02 ! 002 !02 2ı! 00 / b 2 .! 02 ! 002 !02 2ı! 00 /C4! 02 .! 00 Cı/2 (12.72) b 4 2b 2 ! 02 ! 002 !02 2ı! 00 2 02 ! ! 002 !02 2ı! 00 C4! 02 .! 00 Cı/2

where the principal branch is to be chosen in this last expression, so that  

< . As a consequence, the complex index of refraction for a single-resonance Lorentz model dielectric may be rewritten in phasor form as #1=4   b 4  2b 2 ! 02  ! 002  !02  2ı! 00 e i =2 : n.!/ D 1 C  2 ! 02  ! 002  !02  2ı! 00 C 4! 02 .! 00 C ı/2 "

(12.73)

With these exact expressions, the analytic structure of  .!;  / may now be examined in several specific regions of the complex !-plane, as was originally done by Brillouin [2, 3].

12.2.1.1

Behavior Along the Real !0 -Axis

Along the real axis, ! 00 D 0 and the real and imaginary parts of the complex index of refraction are directly obtained from (12.73) as "

  #1=4   b 4 C 2b 2 !02  ! 02 cos 12 .! 0 / ; nr .! / D 1 C   2 ! 02  !02 C 4ı 2 ! 02 "  2  #1=4 4 2 02   ! C 2b  ! b 0 ni .! 0 / D 1 C  sin 12 .! 0 / ;  2 2 02 2 02 !  !0 C 4ı ! 0

(12.74)

(12.75)

respectively. The spectral regions along the real ! 0 -axis where the real index of refraction nr .! 0 / increases with ! 0 [i.e., where d nr .! 0 /=d! 0 > 0] are termed normally dispersive whereas the region wherein nr .! 0 / decreases with increasing ! 0 [i.e., where dnr .! 0 /=d! 0 < 0] is said to exhibit anomalous dispersion. As seen in Fig. 12.2, the real index of refraction nr .! 0 / varies rapidly with ! 0 within the region of anomalous dispersion, where this spectral region essentially coincides with the region of strongest absorption of the medium. From (12.68), the behavior of  .!/ along the real ! 0 -axis is given by  .! 0 / D ! 0 ni .! 0 /:

(12.76)

268

12 Analysis of the Phase Function and Its Saddle Points 3

Real & Imaginary Parts of the Complex Index of Refraction

nr ( ' )

2

ni( ' )

1

0

0

5

' (x1016r/s)

10

15

Fig. 12.2 Frequency dependence of the real (solid curve) and imaginary (dashed curve) parts of the complex index of refraction n.! 0 / D nr .! 0 / C i ni .! 0 / along the positive real angular frequency axis forpa single-resonance Lorentz model dielectric with medium parameters !0 D 4  1016 r/s, b D 20  1016 r/s, ı D 0:218  1016 r/s. The dotted line indicates the nondispersive limit of the vacuum [n.!/ D 1]

Since ni .! 0 /  0 for all ! 0  0 and ni .! 0 /  0 for all ! 0  0, it then follows that  .! 0 /  0 for all ! 0 2 Œ1; 1 . Notice that  .! 0 / is independent of the space– time parameter, so that its behavior along the real ! 0 -axis is fixed in space–time, in contrast to its behavior off of this axis. Finally, notice that the magnitude of  .! 0 / is largest in the region of anomalous dispersion about the medium resonance frequency !0 , is vanishingly small for very small [j! 0 j  !0 ] or very large [j! 0 j  !0 ] absolute frequencies j! 0 j, and vanishes identically at both the origin and infinity. Consider now determining the approximate real angular frequency value !min along the positive real frequency axis at which  .! 0 / attains its minimum value. As shown in Sect. 15.3, the signal velocity associated with the propagation of a Heaviside step-function signal in a single-resonance Lorentz model dielectric attains a minimum value when its carrier frequency is at this angular frequency value. Because  .! 0 / is an even function along the real ! 0 -axis, one needs to only consider its behavior along the positive real axis. For values of the phenomenological damping constant ı small in comparison to both the angular resonance frequency !0 and the medium plasma frequency b, it is seen that the minimum in  .! 0 / is attained very near to !0 , as can be inferred from the behavior of ni .! 0 / illustrated in Fig. 12.2. Consequently, for values of ! 0 close to !0 , the approximation

.! 0 / Š arctan



b2 2ı! 0

 D

 2ı! 0 8ı! 03  2 C   2 b 3b 6

(12.77)

12.2 The Behavior of the Phase in the Complex !-Plane

269

may be employed, provided that ı! 0 =b 2  1, from which it is seen that .! 0 / varies slowly in the region about ! 0 D !0 in comparison to the variation in the magnitude of  .! 0 / about this point. The approximate behavior of  .! 0 / in the vicinity of !0 may then be expressed as "  .! 0 / Š ! 0 1 C

b 4  4b 2 !0 .! 0  !0 / 4!02

.! 0

2

 !0 / C

4ı 2 !02

#1=4 sin

1 2



.!0 / :

(12.78)

Upon differentiating this approximate expression with respect to ! 0 and setting the result equal to zero, there results i2 h  2 16 !02 ! 0  !0 C ı 2 !02 i

  h  2 C4b 2 b 2  4!0 ! 0  !0 !02 ! 0  !0 C ı 2 !02 h i  2   Cb 2 ! 0 4!03 ! 0  !0  2b 2 !02 ! 0  !0  4ı 2 !03 D 0

(12.79)

at ! 0 D !min . To solve this quartic equation in ! 0 for the appropriate root at which  .! 0 / attains its minimum value, let ! 0 D !mi n D !0 C $ , where $ is assumed small in comparison to !0 . Then, retaining only terms of order $ and lower, the approximate solution    !2 2ı 2 (12.80) !min Š !0 1 C 2 1  20 b !0 is obtained. Notice that !min ! !0 as ı ! 0. However, when the plasma frequency b becomes small, the approximation given in (12.77) for .! 0 / is no longer valid. In that case, (12.77) must be replaced by the approximate expression  2  b2 b 0

; (12.81)

.! / Š arctan 2ı! 0 2ı! 0 provided that b 2 =ı! 0  1, so that sin . 21 .! 0 // b 2 =4ı! 0 . The approximate behavior of  .! 0 / D ! 0 ni .! 0 / in the vicinity of !0 is then given by ! b 2 .! 02  !02 / b2 0  :  .! /  (12.82) 1  02 4ı 2 .!  !02 /2 C 4ı 2 ! 02 Upon differentiating this approximate expression with respect to ! 0 and setting the result equal to zero, there immediately results s 2ı !mi n !0 1 C ; (12.83) !0 which is independent of the plasma frequency b, and is valid provided that b 2 =ı! 0  1. Notice that !mi n ! !0 as ı ! 0, just as for (12.80).

270

12.2.1.2

12 Analysis of the Phase Function and Its Saddle Points

Limiting Behavior as j!j ! 1

Consider now the behavior of  .!;  / in the limit as j!j approaches infinity in any given direction. It is readily seen from (12.74) and (12.75) that lim n.!/ D 1;

(12.84)

j!j!1

so that limj!j!1 nr .!/ D 1 and limj!j!1 ni .!/ D 0. With these limiting results, (12.68) then gives (12.85) lim  .!;  / D ! 00 .  1/; j!j!1

and the following limiting behavior for  .!;  / at j!j D 1 is obtained:  For  < 1,  .!;  / is equal to 1 in the upper-half of the complex !-plane,

zero at the real ! 0 -axis [i.e.,  .! 0 ;  / D 0 a ! 0 D ˙1], and is equal to C1 in the lower-half of the complex !-plane.  For  D 1,  .!; 1/ D 0 everywhere at j!j D 1.  For  > 1,  .!;  / is equal to C1 in the upper-half of the complex !-plane, zero at the real ! 0 -axis [i.e.,  .! 0 ;  / D 0 at ! 0 D ˙1], and is equal to 1 in the lower-half of the complex !-plane.

12.2.1.3

Behavior Along the Line !00 D ı

Consider next the behavior of  .!;  / along the straight line ! 00 D ı, along which 0 [where n.!/ D 0] and !˙ [where n.!/ becomes infinite]. lie the branch points !˙ With the substitution ! D ! 0  i ı, (12.57) for the complex index of refraction becomes 1=2  b2 0 n.!  i ı/ D 1 C 2 : (12.86) !0  ı 2  ! 02 0 Consequently, n.! 0 qi ı/ is real and positive q q along the line !qD !  i ı when either ! 0   !12  ı 2 ,  !02  ı 2  ! 0  !02  ı 2 , or ! 0  !12  ı 2 , whereas it is 0 purely imaginary along the two branch cuts !0 ! and !C !C . In addition,  .!;  / along this line is given by

   .! 0  i ı/ D ı nr .! 0  i ı/    ! 0 ni .! 0  i ı/;

(12.87)

so that, with (12.86), the following behavior along this line is obtained: ˚ 0 0 < ! 0 , then  When either ! 0 < < f! g, < f! g < ! 0 < < f!C g, or < !C 0

 .!  i ı/ D ı

"

b2 1C 2 !0  ı 2  ! 02

1=2

#  :

(12.88)

12.2 The Behavior of the Phase in the Complex !-Plane

271

˚



0 0  When either < f! , then g  ! 0  < f! g or < f!C g  ! 0  < !C

 .! 0  i ı/ D ı  ! 0



1=2 b2  1 ! 02 C ı 2  !02

(12.89)

0 along either of the two branch cuts !0 ! and !C !C .

12.2.1.4

Behavior in the Vicinity of the Branch Points

Consider finally the limiting behavior in the immediate vicinity of the two branch 0 , the behavior about the respective branch points ! and !0 being points !C and !C 0 , the complex given by symmetry. In the region about the upper branch point !C angular frequency ! may be written as q (12.90) ! D !12  ı 2  i ı C re i' ; 0 where the ordered-pair .r; '/ denotes the polar coordinates about the point !C D q 2 2 !1  ı  i ı. The square of the complex index of refraction given in (12.62) can be expressed as 0 /.!  !0 / .!  !C .!  !C /.!  ! /  q    q  2 2 2 2 ! !1  ı  i ı !   !1  ı  i ı q    q  ; D  2 2 2 2 ! !0  ı  i ı !   !0  ı  i ı

n2 .!/ D

so that  q  re i' 2 !12  ı 2 C re i'  q  n2 .r; '/ D q q q 2 2 2 2 2 2 i' 2 2 i' !1  ı  !0  ı C re !1  ı C !0  ı C re q !12  ı 2 re i' ; Š2 b2 where the final approximation here is valid in the limit as r ! 0. Consequently, the complex index of refraction in a small neighborhood of the complex !-plane about 0 is given by the upper branch point !C p 1=4 1=2 i'=2 2 2 !1  ı 2 r e (12.91) n.r; '/ Š b as r ! 0.

272

12 Analysis of the Phase Function and Its Saddle Points

Similarly, in the region about the lower branch point !C , the complex angular frequency ! may be written as q ! D !02  ı 2  i ı C Re i ;

(12.92)

where the ordered-pair .R; / denotes the polar coordinates about the point !C D q 2 !0  ı 2 i ı. The square of the complex index of refraction can then be expressed as q  q  q q 2 2 2 2 i i 2 2 2 2 !0  ı  !1  ı C Re !0  ı C !1  ı C Re  q  n2 .R; / D 2 !02  ı 2 C Re i Re i b 2 b2 Š q e i D q e i. / ; 2 2 2 2 2 !0  ı R 2 !0  ı R where the final approximation here is valid in the limit as R ! 0. Consequently, the complex index of refraction in a small neighborhood of the complex !-plane about the lower branch point !C is given by b e i. n.R; / Š p  1=4 2 2 !0  ı 2 R1=2

/=2

(12.93)

as R ! 0. The limiting behavior of the complex index of refraction about each of the two 0 and !C , as described by (12.91) and (12.93), respectively, is illusbranch points !C trated in part (a) of Fig. 12.3. Analogous results hold for the behavior of the complex index of refraction about the branch points !0 and ! , respectively, in the left-half of the complex !-plane. From these results, the limiting behavior of  .!;  / about each of the two branch 0 and !C is readily determined from (12.87). Thus, in a small neighborhood points !C 0 , one obtains the limiting of the complex !-plane about the upper branch point !C behavior described by "p  .r; ';  / Š ı

1=4 1=2 2 2 !1  ı 2 r cos .'=2/   b p 3=4 1=2 2 2  !1  ı 2 r sin .'=2/ b

#

(12.94)

as r ! 0, whereas in a small neighborhood of the complex !-plane about the lower branch point !C , one obtains the limiting behavior

12.2 The Behavior of the Phase in the Complex !-Plane

a

273

''

' (1+i) +i −i (1– i)

b

(1+i)0 +i0 −i0

'

0

(1–i)0

''

' − − +

+

'

+

Fig. 12.3 Limiting behavior of (a) the complex index of refraction n.!/ and (b) the real part 0  .!;  / of the complex phase function .!;  / about the branch points !C and !C in the righthalf of the complex !-plane for a single-resonance Lorentz model dielectric. The dashed curve describes the approximate behavior of the isotimic contour  .!;  / D 0 for  > 0

"

b

 .R; ;  / Š ı p  cos ..  /=2/   1=4 2 !02  ı 2 R1=2 1=4  b !02  ı 2  p sin ..  /=2/ 2R1=2

#

(12.95)

as R ! 0. Hence,  .!;  / is negative on both sides of the branch cut near the 0 upper branch point !C for  > 0, is zero at  D 0, and is positive for  < 0. Near the lower branch point !C , however,  .!;  / is negative on the upper side of the branch cut and positive on the lower side for all  , as depicted in part (b) of Fig. 12.3. From the behavior of  .!;  / in the region of the complex !-plane about the lower branch point !C , it is seen that the zero isotimic1 contour  .!;  / D 0 must pass through the branch point !C from above [since  .!;  / ! C1 as ! ! !C along 1

From the Greek isotimos, of equal worth.

274

12 Analysis of the Phase Function and Its Saddle Points

the line ! D ! 0  i ı from below] and then continues on from the lower side of 0 for  > 0, as described by the dashed curve the branch cut between !C and !C in Fig.12.3b. For  < 0, the zero isotimic contour  .!;  / D 0 continues on from 0 , and for  D 0, this contour the upper side of the branch cut between !C and !C 0 . continues on from the upper branch point !C 12.2.1.5

Numerical Results

This simple sketch of the behavior of  .!;  /
1 C b 2 =!02 D 1:5 for Brillouin’s choice of the medium

12.2 The Behavior of the Phase in the Complex !-Plane

275

'' 16

(x10 r/s) SPn+ (−1.2 x1016)

'

SPn−

' 16 (x10 r/s)

(0)

(+1.2 x1016)

Fig. 12.4 Isotimic contours of the real part  .!;  / of the complex phase function .!;  / in the right-half of the complex !-plane at the fixed space–time point  D 1. The shaded area indicates the region of the complex !-plane below the dominant distant saddle point pair where ˙ .!; 1/ < .!SP ; 1/, and the darker shaded area indicates the region below the upper near d C C ˙ saddle point where .!; 1/ <  .!SP ; 1/, where  .!SP ; 1/ <  .!SP ; 1/. Notice that the distant n n d ˙ saddle points SPd are symmetrically located at ˙1  2ı1 at this luminal space–time point

''

(x1016r/s)

SPn+

(+1.0 x1016)

(−1.0 x1016) (0) '

SPn−

'

(x1016r/s)

SPd+

(+1.0 x1016)

16

(−1.0 x10 )

Fig. 12.5 Isotimic contours of the real part  .!;  / of the complex phase function .!;  / in the right-half of the complex !-plane at the fixed space–time point  D 1:25. The shaded area indicates the region of the complex !-plane below the dominant distant saddle point pair where ˙  .!; 1:25/ <  .!SP ; 1:25/, and the darker shaded area indicates the region below the upper d C C ˙ ; 1:25/, where  .!SP ; 1:25/ <  .!SP ; 1:25/ near saddle point where  .!; 1:25/ <  .!SP n n d

276

12 Analysis of the Phase Function and Its Saddle Points

''

SB

(x1016r/s)

(+1.0 x1016)

SPn+

16

(−1.0 x10 ) (0) SPd+

'

SPn−

'

(x1016r/s)

(+1.0 x1016)

16

(−1.0 x10 )

Fig. 12.6 Isotimic contours of the real part  .!;  / of the complex phase function .!;  / in the right-half of the complex !-plane at the fixed space–time point  D SB  1:33425 when C the upper near and distant saddle points are of equal dominance, i.e., such that  .!SP ; SB / D n ˙ .!SPd ; SB /. The shaded area indicates the region of the complex !-plane below the equally ˙ dominant upper near saddle point SPC n and distant saddle point pair SPd

'' 1

(x1016 r/s)

(+1.0 x1016) (–1.0x1016) SPn-

(0)

SPn+ '

(0)

(+1.0x1016)

' (x1016r/s)

SPd+

(–1.0x1016)

Fig. 12.7 Isotimic contours of the real part  .!;  / of the complex phase function .!;  / in the right-half of the complex !-plane at the fixed space–time point  D 1:501 just prior to the coalescence of the two first-order near saddle points SP˙ n into a single second-order saddle point when  D 1 . The shaded area indicates the region of the complex !-plane below the C dominant near saddle point where  .!; 1:501/ <  .!SP ; 1:501/, and the darker shaded area n ˙ ; 1:501/, indicates the region below the distant saddle point pair where  .!; 1:501/ <  .!SP d C ˙ where .!SPn ; 1:501/ >  .!SPd ; 1:501/

12.2 The Behavior of the Phase in the Complex !-Plane

277

'' (x1016r/s)

(+1.0x1016) (−1.0x1016) SPn+

'

(+1.0x1016)

(0)

' (x1016r/s)

SPd+

(−1.0x1016)

Fig. 12.8 Isotimic contours of the real part  .!;  / of the complex phase function .!;  / in the right-half of the complex !-plane at the fixed space–time point  D 1:65 after the near saddle point pair has moved off of the imaginary axis into the lower-half of the complex !-plane. The shaded area indicates the region of the complex !-plane below the dominant near saddle point pair where ˙  .!; 1:65/ <  .!SP ; 1:65/, and the darker shaded area indicates the region below the distant n ˙ ˙ ˙ ; 1:65/, where  .!SP ; 1:65/ <  .!SP ; 1:65/ saddle point pair where  .!; 1:65/ <  .!SP n d d

parameters, and then coalescing into a single second-order saddle point at  D 1 , as approximately described by (12.27) and (12.29) for a general Lorentz-type dielectric, where the value of this critical space–time point for Brillouin’s choice of the medium parameters is just slightly larger than the space–time value  D 1:501 illustrated in Fig. 12.7. As  increases above 1 , the two near first-order saddle points are seen to separate from each other, symmetrically situated about the imaginary axis, approaching the inner branch points !˙ , respectively, as  ! 1. The two distant saddle points SPd˙ , on the other hand, are located in the lowerhalf of the complex !-plane for all   1 and are located at ˙1  i ı at the luminal space–time point  D 1, as approximately described by (12.55) for a general Lorentz-type dielectric. As  increases from unity, these two saddle points symmetrically move in from infinity and approach the respective outer branch points 0 as  increases to infinity, as evident in Figs. 12.8 and 12.9. !˙ Initially, the distant saddle points SPd˙ have less exponential decay associated with them than does the upper near saddle point SPC n in Figs. 12.4 and 12.5, that is C ˙ ; / >  .!SP ; /  .!SP d n

when 1   < SB :

(12.96)

278

12 Analysis of the Phase Function and Its Saddle Points

''

(x1016r/s)

(2.0 x1016)

(−3.4x1016)

(0)

'

(x1016r/s)

'

SPn+

SPd+

(2.0x1016)

(−3.4x1016)

Fig. 12.9 Isotimic contours of the real part  .!;  / of the complex phase function .!;  / in the right-half of the complex !-plane at the fixed space–time point  D 5:0. The shaded area indicates the region of the complex !-plane below the dominant near saddle point pair where ˙ .!; 5/ < .!SP ; 5/, and the darker shaded area indicates the region below the distant saddle n ˙ ˙ ˙ ; 5/, where  .!SP ; 5/ < .!SP ; 5/. Notice the approach point pair where .!; 5/ <  .!SP n d d C of the near saddle point SPn to the lower branch point !C and the approach of the distant saddle 0 point SPC d to the upper branch point !C

Because the original contour of integration C appearing in the integral representation of the propagated plane wave pulse given in either (12.1) or (12.2) is not deformable into an Olver-type path through the lower near saddle point SP n over the initial space–time domain 1   < 1 , that saddle point is irrelevant for the present analysis for  below and bounded away from 1 . At the critical space–time point  D SB Š 1:33425, illustrated in Fig. 12.6, the upper near saddle point SPC n has precisely the same exponential decay associated with it as do the two distant saddle points, that is C ˙ ; SB / D  .!SP ; SB /  .!SP d n

when  D SB :

(12.97)

 Consequently, at the space–time point  D SB , those three saddle points (SPC d , SPd , C and SPn ) are of equal importance (or dominance) in the asymptotic description of the propagated wavefield. The remaining figures show that for values of  2 .SB ; 1 / the upper near saddle point SPC n is dominant over the two distant saddle points SP˙ d , so that C ˙ ; / >  .!SP ; /  .!SP d n

when SB   < 1 :

(12.98)

12.2 The Behavior of the Phase in the Complex !-Plane

279

At  D 1 , the two near first-order saddle points SP˙ n have coalesced into a single second-order saddle point SPn which is dominant over the distant saddle point pair, so that ˙ ; 1 /  .!SPn ; 1 / >  .!SP d

when  D 1 :

(12.99)

Finally, for all  > 1 , the near saddle points SP˙ n are dominant over the distant saddle points SPd˙ , so that ˙ ˙ ; / >  .!SP ; /  .!SP n d

when  > 1 ;

(12.100)

as evident in Figs. 12.8–12.9. Notice the change in scale of the real and imaginary coordinate axes in Fig. 12.9, demonstrating how the topography of  .!;  / 0 as  becomes increasingly concentrated about the branch cuts !0 ! and !C !C increases above the critical space–time value 1 with the near saddle points SP˙ n approaching the lower branch points !˙ , respectively, and the distant saddle points 0 . SPd˙ approaching the upper branch points !˙ These detailed numerical results demonstrate the necessity of obtaining approximate analytic expressions for both the near and distant saddle point locations that accurately describe their dynamical evolution in the complex !-plane for all   1 as well as the complex phase behavior at them. In addition, accurate analytic expressions are needed for each of the critical space–time points encountered here. These include the critical value SB at which the upper near and distant saddle points are of equal importance (see Fig. 12.6) and the critical value 1 at which the two near firstorder saddle points coalesce into a single second-order saddle point (see Fig. 12.7). Moreover, approximate (if an exact solution is unattainable) analytic expressions for each of these quantities need to be obtained that are accurate over the entire space–time domain of interest for both the strong, intermediate, and weak dispersion limits, the former being of central interest to bioelectromagnetics and the latter being of central interest to optics.

12.2.2 Multiple-Resonance Lorentz Model Dielectrics For a double-resonance Lorentz model dielectric with two isolated resonance frequencies, the complex index of refraction is given by (see Sect. 4.4.4 of Vol. 1)  n.!/ D 1 

b02 b22  2 2 ! 2  !0 C 2i ı0 ! ! 2  !2 C 2i ı2 !

1=2 ;

(12.101)

where !j denotes the undamped angular p resonance frequency, ıj is the phenomenological damping constant, and bj D .k4k=0 /Nj qe2 =m is the plasma frequency with number density Nj of Lorentz oscillators denoted by the indices j D 0; 2.

280

12 Analysis of the Phase Function and Its Saddle Points

The branch points of this double-resonance complex index of refraction function, and consequently of the complex phase function .!;  /, may be determined by rewriting the expression given in (12.101) as     31=2 .3/  !  !.1/ !  !C !  !.3/   n.!/ D 4   5 : .0/  .2/  !  !C !  !.0/ !  !C !  !.2/ 2

.1/

!  !C

.0/

(12.102)

.2/

The branch point singularities !˙ and !˙ appearing in this expression are given by the zeros of the denominator in (12.102), so that q .0/ !˙ D ˙ !02  ı02  i ı0 ; q .2/ !˙ D ˙ !22  ı22  i ı2 : .1/

(12.103) (12.104)

.3/

Unfortunately, the branch point zeros !˙ and !˙ appearing in (12.102) are much more difficult to determine, even in approximate form. These four zeros are given by the roots of the quartic equation ! 4 C 2i.ı0 C ı2 /! 3  .!12 C !32 C 4ı0 ı2 /! 2 2i.ı2 !12 C ı0 !32 /! C !02 !22 C b02 !22 C b22 !02 D 0; (12.105) where q !1 C !02 C b02 ; q !3 C !22 C b22 :

(12.106) (12.107)

Approximate analytic solutions to this quartic equation may be obtained in the following manner. Because the zeros must appear in symmetric pairs about the imaginary axis, let .1/

!˙ D ˙1  i ı0 ; .3/

!˙ D ˙3  i ı2 ; .0/

.1/

where it has been assumed that the branch points !˙ and !˙ lie along the line .2/ .3/ ! 00 D ı0 and that the branch points !˙ and !˙ lie along the line ! 00 D ı2 . In that case, the quartic equation given in (12.105) must then be factorizable as .!  .1  i ı0 //.!  .1  i ı0 //.!  .3  i ı2 //.!  .3  i ı2 // D 0;

12.2 The Behavior of the Phase in the Complex !-Plane

281

which then results in ! 4 C 2i.ı0 C ı2 /! 3  .12 C ı02 C 32 C ı22 C 4ı0 ı2 /! 2   2i ı0 .32 C ı22 / C ı2 .12 C ı02 / ! C .12 C ı02 /.32 C ı22 / D 0: Comparison of the terms in this quartic equation with the corresponding terms in (12.105) then yields the set of relations 12 C 32 D !12  ı02 C !33  ı22 ; ı0 .32 C ı22 / C ı2 .12 C ı02 / D ı0 !32 C ı2 !12 ; .12 C ı02 /.32 C ı22 / D !02 !22 C b02 !22 C b22 !02 ; which are overdetermined. This overdetermination of the solution implies that the .1/ .3/ zeros !˙ and !˙ do not, in general, lie along the lines ! 00 D ı0 and ! 00 D ı2 , respectively, as was assumed in constructing this solution. Nevertheless, if ı0 ı2 , then they approximately lie along these lines and the first pair of the above set of equations are nearly identical, resulting in the reduced system of equations 12 C 32 D !12  ı02 C !33  ı22 ; .12 C ı02 /.32 C ı22 / D !02 !22 C b02 !22 C b22 !02 : The solutions to this pair of equations then yields the approximate branch point locations2 s ! 4 C 2.!12  ı02 /!32  b02 b22 C ı03 .1/ !˙ ˙ !12  ı02 C 3  i ı0 ; (12.108) !12 C !32  2ı02 s ! 4 C 2.!32  ı22 /!32  b02 b22 C ı23 .3/ !˙ ˙ !32  ı22 C 1  i ı2 : (12.109) !12 C !32  2ı22 The branch cuts chosen here are the straight line segments !.3/ !.2/ and !.1/ !.0/ .0/ .1/ .2/ .3/ in the left-half plane, and !C !C and !C !C in the right-half plane, as depicted in Fig.12.10. It is typically assumed that 0  !0 < !1  !2 < !3 so that the outer and inner branch cuts do not overlap each other. Finally, the complex index of refraction n.!/ and the complex phase function .!; / are analytic throughout the .0/ .1/ .2/ .3/ complex !-plane with the exception of the branch points !˙ , !˙ , !˙ , and !˙ . The limiting behavior of the complex index of refraction n.!/ and the real part  .!;  / of the complex phase function .!; / for a double-resonance Lorentz model dielectric is quite similar to that of a single-resonance Lorentz model dielectric, especially when the branch cuts are sufficiently separated from each other, and 2

Notice that these approximate expressions for the branch point zero locations are different from those given in an earlier paper [11].

282

12 Analysis of the Phase Function and Its Saddle Points

''

' (1)

(0)

branch

branch

cut

cut

(3) cut

(1)

(0)

(2) cut

(2)

(3)

.j /

.3/ .2/ Fig. 12.10 Location of the branch points !˙ , j D 0; 1; 2; 3 and the branch cuts ! ! and .0/ .1/ !C !C

.2/ .3/ !C !C

.1/ .0/ ! ! in the left-half and and in the right-half of the complex !-plane for a double-resonance Lorentz model dielectric with undamped resonance frequency !0 , damping q constant ı0 , and plasma frequency b0 with !1 D !02 C b02 for the lower resonance line, and with undamped resonance frequency !2 , damping constant ı2 , and plasma frequency b2 with !3 D q

!22 C b22 for the upper resonance line

even more so if one of the resonance lines is much stronger than the other [e.g., when either b2  b0 or b0  b2 ]. In particular, the limiting behavior as j!j ! 1 described in (12.84) and (12.85) remains valid in the multiple-resonance case, as .1/ does the behavior in the vicinity of the branch points given in (12.91) for !˙ and .3/ .0/ .2/ !˙ and in (12.93) for !˙ and !˙ , provided again that the outer and inner branch cuts are sufficiently separated from each other. The real angular frequency dependence of the real and imaginary parts of the complex index of refraction for a double-resonance Lorentz model dielectric is depicted in Fig. 12.11 for a highly absorptive material with visible and near-ultraviolet resonance lines with parameters !0 D 1:0  1016 r=s; p b0 D 0:6  1016 r=s; ı0 D 0:1  1016 r=s;

!2 D 7:0  1016 r=s; p b2 D 12:0  1016 r=s; ı2 D 0:28  1016 r=s:

Notice that the real index of refraction nr .! 0 / varies rapidly with ! 0 within each region of anomalous dispersion and that these two regions essentially coincide with each region Œ!0 ; !1 and Œ!2 ; !3 where the imaginary part ni .! 0 / of the complex index of refraction peaks to a local maximum and the absorption is strongest. The cross along the graph of the function nr .! 0 / in the upper graph of Fig. 12.11 indicates the (numerically determined) inflection point in the real refractive index in the pass-band between the two resonance frequencies where the second derivative @2 nr .! 0 /=@! 02 changes sign from negative to positive values as ! 0 increases, and the cross along the graph of the function ni .! 0 / in the lower graph of Fig. 12.11

12.2 The Behavior of the Phase in the Complex !-Plane

283

nr ( )

2

1

0 1015

1016

(r/s)

1017

1018

1017

1018

102

ni ( )

100 10–2 10–4 10–6 15 10

1016

(r/s)

Fig. 12.11 Frequency dependence of the real (upper graph) and imaginary (lower graph) parts of the complex index of refraction n.! 0 / D nr .! 0 / C i ni .! 0 / along the positive real angular frequency axis for apdouble-resonance Lorentz model dielectric with medium parameters !0 D 1  1016 r=s, b0 D p 0:6  1016 r=s, ı0 D 0:1  1016 r=s for the lower resonance line and !2 D 7  1016 r=s, b2 D 12  1016 r=s, ı2 D 0:28  1016 r=s for the upper resonance line. The dotted line indicates the nondispersive limit of the vacuum [n.!/ D 1]

indicates the point at which the imaginary part of the index of refraction is a minimum in that pass-band. Notice that these two points occur, in general, at different frequency values. The inclusion of this single additional resonance feature in the Lorentz model results in the appearance of four additional saddle points of the complex phase function .!; / in the complex !-plane. In addition to the distant saddle point pair SP˙ d and the near saddle point pair SP˙ n , there are now four additional first-order saddle ˙ points SP˙ m1 and SPm2 , symmetrically situated about the imaginary axis, that evolve with   1 in the intermediate frequency domain between the lower and upper resonance frequencies, i.e., such that <

<

˙ ./j  !2 ; !0  j!mj

j D 1; 2:

(12.110)

Because of this feature, they are referred to here as middle saddle points, where   C  . / D  !mj ./ ; !mj

j D 1; 2;

(12.111)

284

12 Analysis of the Phase Function and Its Saddle Points 1 0.8

+

+

0.6

SPn

+

0.2 0

(−1.0x1016)

+ SPm1

(0)

(2)

+ SPm2

+

'' (x1017r/s)

0.4

−0.2

(+1.0x1016)

−0.4

+

−0.6

SPn−

−0.8 −1 0

5

' (x1016r/s)

10

15

Fig. 12.12 Isotimic contours of the real part  .!;  / of the complex phase function .!;  / for a double-resonance Lorentz model dielectric in the right-half of the complex !-plane at the fixed space–time point  D 1:0. At this luminal space–time value the distant saddle points SPd˙ at ˙1  i ı2 are dominant over both the upper near saddle point SPC n and the upper middle saddle ˙ points SPm1

for all   1. These middle saddle points are clearly evident in the sequence of isotimic contour plots presented in Figs. 12.12–12.19 for the same double-resonance Lorentz model dielectric whose temporal frequency dispersion along the positive real frequency axis is presented in Fig. 12.11. For each additional resonance feature added to the Lorentz model description of resonance frequency dispersion in a dielectric material, an additional set of four firstorder saddle points will appear in the intermediate region of the complex !-plane, (approximately) between the lowest and highest resonance frequency. Although the fundamental structure of the Sommerfeld and Brillouin precursor fields is primarily described by the distant and near saddle points, these additional, intermediate saddle points (or middle saddle points in the case of a double-resonance medium) can have a significant impact on their dynamical evolution [17], even when they never become the dominate saddle points over the entire space–time domain of interest. If any do become the dominant saddle points over some subluminal space–time domain, they will then produce an additional precursor field structure that will appear in the total propagated wavefield structure, provided that the input pulse spectral energy is nonvanishing in that intermediate frequency domain.

12.2 The Behavior of the Phase in the Complex !-Plane

285

5 4 3

(–1.0x1016)

+ SPm1

+

+

1 SPn

+

'' (x1016r/s)

2

0 (0)

+

−1

SPn

16

(+1.0x10 )

+

SPm2

+

−2

+ + SPd

(2)

–

−3 −4 −5

0

5

' (x1016r/s)

10

15

Fig. 12.13 Isotimic contours of the real part  .!;  / of the complex phase function .!;  / for a double-resonance Lorentz model dielectric in the right-half of the complex !-plane at the fixed space–time point  D 1:1. At this space–time value the distant saddle points SPd˙ are dominant ˙ over both the upper near saddle point SPC n and the upper middle saddle points SPm1

Just as in the single-resonance medium case, the two distant saddle points SPd˙ are situated in the lower-half of the complex !-plane for all   1 and are located at ˙1  i.ı0 C ı2 / at the luminal space–time point  D 1, as approximately described by (12.55) for a general Lorentz-type dielectric. As  increases away from unity, these two saddle points symmetrically move in from infinity, approaching the outer .3/ branch points !˙ , respectively, as  ! 1, as evident in Figs. 12.13–12.14. Notice that the change in scale in Figs. 12.15–12.19, made to focus attention on the near and middle saddle points, eliminates the view of the distant saddle point evolution over this latter space–time domain. The two near first-order saddle points SP˙ n are seen in Figs.12.12–12.18 to lie along the imaginary axis over the initial space–time domain  2 Œ1; 1 , approaching each other as  increases,q the upper near saddle point SPC n crossing the origin at  D 0 , where 0 n.0/ D 1 C b02 =!02 C b22 =!22 with 0 Š 1:358 for the particular double-resonant Lorentz model dielectric considered here (see Fig. 12.18), and then coalescing into a single second-order saddle point at  D 1 , just as for a single resonance medium, where 1 is just slightly larger than 0 . As  increases above 1 , the two near first-order saddle points SP˙ n are seen to separate from each other, symmetrically situated about the imaginary axis, approaching the inner branch points

286

12 Analysis of the Phase Function and Its Saddle Points 5 4 3

+

1

+

+

(−1.0x1016)

SPm1

SPn+

0 −1 SPn

(0)

−2

+ + SPd

(2)

+

'' (x1016r/s)

2

(+1.0x1016)

+

+

SPm2

−3 −4 −5

0

5

' (x1016r/s)

10

15

Fig. 12.14 Isotimic contours of the real part  .!;  / of the complex phase function .!;  / for a double-resonance Lorentz model dielectric in the right-half of the complex !-plane at the fixed ˙ are dominant over the upper space–time point  D 1:2 when the upper middle saddle points SPm1 near and distant saddle points

.0/

!˙ , respectively, as  ! 1, as evident in Fig. 12.19. The near saddle point behavior for a double-resonance dielectric is then seen to be entirely analogous to that found for a single-resonance Lorentz model dielectric. The middle saddle point dynamics in the complex !-plane exhibit a complicated evolution with  that imitates several characteristic features of both the near and distant saddle point dynamics as  ! 1. For initial values of   1 the middle C is located in the upper-half of the complex !-plane above the saddle point SPm1 .0/ .1/ C is located in the lower-half !C !C branch cut and the middle saddle point SPm2 of the complex !-plane below that branch cut, as seen in Fig. 12.12. As  increases away from unity, these middle saddle points (with a symmetric set in the left-half plane) move into the intermediate frequency domain between the two branch cuts .0/ .1/ .2/ .3/ !C !C and !C !C , as illustrated in Figs. 12.13–12.15. At the critical space–time point  D N1 Š 1:25, where 1 < N1 < 0 < 1 ;

(12.112)

˙ ˙ and SPm2 come into the two symmetric sets of first-order middle saddle points SPm1 closest proximity with each other in the left- and right-half planes, respectively, as

12.2 The Behavior of the Phase in the Complex !-Plane

287

3

2

(0) (−1.0x1016)

+

SPn

+

+

+

SPm1

0 (0)

+

'' (x1016r/s)

1

−1

(1)

(0)

−

SPn

+

+

SPm2

(+1.0x1016) (0)

−2

−3

(2)

0

1

2

3

4

' (x1016r/s)

5

6

7

Fig. 12.15 Magnified view of the isotimic contours of the real part  .!;  / of the complex phase function .!;  / for a double-resonance Lorentz model dielectric in the right-half of the complex ˙ !-plane at the fixed space–time point  D 1:2 when the upper middle saddle points SPm1 are dominant

illustrated in Fig. 12.16. The more detailed analysis presented in Sect. 12.3.2 shows ˙ are dominant over both the distant saddle points that the middle saddle points SPm1 ˙ SPd and the upper near saddle point SPC n in a small space–time domain about this critical value. At  D 1:3 and all larger values of  , the middle saddle points are dominated by the upper near saddle point SPC n until  D 1 and then by the for  >  , as depicted in Figs. 12.17–12.19. From the near saddle points SP˙ 1 n perspective of the lower resonance line at !0 , the dynamical evolution of the first˙ over the space–time domain  > N1 imitates order middle saddle point pair SPm1 that of the distant saddle point pair in a single-resonance dielectric as they approach .1/ the outer branch points !˙ , respectively, as  ! 1. Similarly, from the perspective of the upper resonance line at !2 , the dynamical evolution of the first-order middle ˙ for  > N1 imitates that of the near saddle point pair for saddle point pair SPm2  > 1 in a single-resonance dielectric as they approach the inner branch points .2/ !˙ , respectively, as  ! 1. Attention is now turned to the determination of the space–time sequence of saddle point dominance over the subluminal space–time domain   1. Because the original contour of integration C appearing in the integral representation of the

288

12 Analysis of the Phase Function and Its Saddle Points 3

2

(0)

(−1.0x1016)

+

+

SPn

(0)

0

+

'' (x1016r/s)

1

−

+ (0)

SPn

(1)

(0)

−1

+ SPm1 +

+ SPm2

(0)

(2)

(+1.0x1016)

−2

−3

0

1

2

3

4

5

6

7

' (x1016r/s) Fig. 12.16 Isotimic contours of the real part  .!;  / of the complex phase function .!;  / for a double-resonance Lorentz model dielectric in the right-half of the complex !-plane at the fixed ˙ space–time point  D N1  1:25 when the two pairs of first-order middle saddle points SPm1 and ˙ SPm2 come into closest proximity with each other in the left- and right-half planes and dominate ˙ both the upper near saddle point SPC n and the distant saddle points SPd

propagated plane wave pulse given in either (12.1) or (12.2) is not deformable into an Olver-type path through the lower near saddle point SP n over the initial space– time domain 1   < 1 , that saddle point is irrelevant for the present analysis for  below and bounded away from 1 . Similarly, because the integration path C is not ˙ deformable into an Olver-type path through the lower middle saddle point pair SPm2 over the space–time domain 1   < N1 , those two saddle points are irrelevant for values of  below and bounded away from N1 . However, for  > N1 the contour C can be deformed into an Olver-type path through this lower middle saddle point pair ˙ , but they are then dominated by (i.e., possess greater exponential attenuation SPm2 ˙ over this entire space–time domain. than) the upper middle saddle point pair SPm1 Finally notice that the simple fact that a saddle point does not become the dominant saddle point does not necessarily mean that it does not influence the asymptotic field behavior (e.g., see Sect. 10.3.2). ˙ do not necessarily become the dominant The upper middle saddle points SPm1 saddle points in all cases. The necessary condition [11] for whether or not they do become the dominant saddle points over some subluminal space–time domain is

12.2 The Behavior of the Phase in the Complex !-Plane

289

3

2

+ +

''

SP n

+

0

SP n−

(−1.0x1016)

(0)

+

16

(x10 r/s)

1

(0)

(1)

+SP m1

+ + SP m2

(0)

−1

(2)

(0)

(+1.0x1016)

−2

−3

0

1

2

3

'

4 (x1016r/s)

5

6

7

Fig. 12.17 Isotimic contours of the real part  .!;  / of the complex phase function .!;  / for a double-resonance Lorentz model dielectric in the right-half of the complex !-plane at the fixed space–time point  D 1:3 when the upper near saddle point SPC n is dominant over both the middle and distant saddle points

obtained from a consideration of the energy transport velocity vE .!/ D

c E .!/

(12.113)

for a monochromatic electromagnetic plane wave in a double-resonance Lorentz model dielectric, where [see (5.212) of Vol. 1] 1 E .!/ D nr .!/ C nr .!/

"

# b02 ! 2 b22 ! 2 C :  2 2 ! 2  !02 C 4ı02 ! 2 ! 2  !22 C 4ı22 ! 2 (12.114)

Let p 2 .1; 0 / denote the space–time value at which  .!SP ˙ ;  / at the upper m1

˙ middle saddle point pair SPm1 has a local maximum. If the upper middle saddle ˙ point SPm1 has less exponential decay associated with it at the space–time point  D p than does the upper near saddle point SPC n , that is, if

 .!SP ˙ ; p / >  .!SPnC ; p /; m1

(12.115)

290

12 Analysis of the Phase Function and Its Saddle Points 3

2

(−1.0x1016) ++

0

''

(x1016r/s)

1

(0)

SP n+ SP n−

+

(0)

(1)

+SP m1

+ + SP m2

(0)

(2)

(0)

−1

(1.0x1016)

−2

−3

0

1

2

3

'

4 (x1016r/s)

5

6

7

Fig. 12.18 Isotimic contours of the real part  .!;  / of the complex phase function .!;  / for a double-resonance Lorentz model dielectric in the right-half of the complex !-plane at the fixed space–time point  D 0 ' 1:3583, just prior to the coalescence of the two first-order near saddle points SP˙ n into a single second-order saddle point SPn at  D 1 . At this space–time value 0 the upper near saddle point SPC n is dominant over both the middle and distant saddle points

˙ then the upper middle saddle point pair SPm1 will be the dominant saddle points in a small  -neighborhood about that space–time point. As shown in Sect. 15.3.2, this condition is equivalent to the condition that the maximum value of the energy transport velocity in that intermediate frequency interval between the upper and lower absorption bands be greater than the value of the energy velocity at zero frequency. Let !p denote the real angular frequency value at which this maximum value occurs, in which case

p D E .!p /:

(12.116)

Because vE .0/ D c=n.0/ D c=0 , then the condition vE .!p / > vE .0/ for the ˙ over the upper near saddle dominance of the upper middle saddle point pair SPm1 over a small space–time interval about  becomes point SPC p n p < 0 :

(12.117)

12.2 The Behavior of the Phase in the Complex !-Plane

291

2

1

16

(x10 r/s)

(−1.0x1016)

(0)

+

SPn

0

''

+

(0)

(0)

(1)

SP + + m1

+

+ SPm2

(2)

( 0) −1

−2

(1.0x1016)

0

1

2

3

'

4 (x1016r/s)

5

6

7

Fig. 12.19 Isotimic contours of the real part  .!;  / of the complex phase function .!;  / for a double-resonance Lorentz model dielectric in the right-half of the complex !-plane at the fixed space–time point  D 1:5 when the near saddle points SP˙ n are dominant over both the middle and distant saddle points

This is then the necessary condition for the dominance of the upper middle saddle point pair. If this condition is not satisfied, the middle saddle points never become the dominant saddle points for all   1.

Case 1: p < 0

12.2.2.1

Initially, the distant saddle points SPd˙ have less exponential decay associated with them than does either the upper near saddle point SPC n or the upper middle saddle ˙ , as seen in Figs. 12.12 and 12.13, and this remains the case until  points SPm1 reaches the space–time point SM , that is  .!SP ˙ ; / >  .!SPnC ; / >  .!SP ˙ ;  /

when 1   < BM ;

 .!SP ˙ ; / >  .!SPnC ; / D  .!SP ˙ ;  /

when  D BM ;

 .!SP ˙ ; / >  .!SP ˙ ; / >  .!SPnC ;  /

when BM <  < SM ;

m1

d

m1

d

d

m1

(12.118)

292

12 Analysis of the Phase Function and Its Saddle Points

where SM is defined by the condition ˙ ˙ ; SM / D  .!SP ; SM /:  .!SP m1 d

(12.119)

Notice that the upper near saddle point SPC n is initially dominant over the middle ˙ , this dominance switching at the space–time point  D BM . saddle point pair SPm1 ˙ is then dominant over both the upper near SPC The middle saddle point pair SPm1 n ˙ and distant SPd saddle points over the space–time domain SM <  < MB , as seen in Figs. 12.14–12.16, where  .!SP ˙ ; / >  .!SP ˙ ; / >  .!SPnC ;  /

when SM   < SB ;

 .!SP ˙ ; / >  .!SPnC ; / D  .!SP ˙ ;  /

when  D SB ;

 .!SP ˙ ; / >  .!SPnC ; / >  .!SP ˙ ;  /

when SB <  < MB ;

m1

d

m1

d

m1

d

(12.120)

where MB is defined by the condition  .!SP ˙ ; MB / D  .!SPnC ; MB /: m1

(12.121)

The upper near saddle point SPC n is then the dominant saddle point over the space– time domain MB <   1 , as seen in Figs. 12.17 and 12.18, where  .!SPnC ; / >  .!SP ˙ ; / >  .!SP ˙ ;  / m1

d

when MB <   1 ;

(12.122)

the two near first-order saddle points coalescing into a single second-order saddle point at  D 1 . The near saddle point pair SP˙ n is then dominant over both the ˙ and distant SPd˙ saddle point pairs over the final space–time domain middle SPm1  > 1 , as seen in Fig. 12.19, where  .!SP ˙ ; / >  .!SP ˙ ;  / >  .!SP ˙ ;  / n

m1

d

when  > 1 :

(12.123)

Notice that the critical space–time points encountered in this saddle point evolution are ordered such that 1 < SM < SB < N1 < MB < 0 < 1

(12.124)

when both ıj > 0 and Nj > 0 are bounded away from zero.

12.2.2.2

Case 2: p > 0

When the inequality given in (12.117) is not satisfied, then the upper middle saddle ˙ never become the dominant saddle points over the entire space–time points SPm1 domain   1. In that case, the saddle point dominance sequence is the same as

12.2 The Behavior of the Phase in the Complex !-Plane

293

that for a single-resonance Lorentz model dielectric. However, this does not mean that these middle saddle points can never influence the dynamical field evolution. Indeed, it can happen that  .!SP ˙ ;  / at the upper middle saddle point pair is m1 just below the value  .!SP ˙ ;  / at the near saddle point pair and interferes with n that contribution over the space–time domain  > 1 , as occurs for triply distilled water [17]. Because of this, the middle saddle points play an important role in the dynamical field evolution and their detailed behavior must be determined even when they are never the dominant saddle points.

12.2.3 Rocard–Powles–Debye Model Dielectrics The complex index of refraction for a single relaxation time Rocard–Powles–Debye model dielectric is given by (see Sect. 4.4.3 of Vol. 1)  n.!/ D 1 C

a0 .1  i !0 /.1  i !f 0 /

1=2 ;

(12.125)

where 1  1 denotes the large frequency limit of the relative dielectric permittivity due to the Rocard–Powles–Debye model alone (typically when ! exceeds  1  1012 r/s). Here 0 denotes the effective relaxation time [see (4.177) of Vol. 1] with the associated friction time f 0 [see (4.184) of Vol. 1] introduced in the Rocard– Powles extension [18] of the Debye model [19], where a0 s  1 with s .0/ denoting the relative static dielectric permittivity of the material. Values of these model coefficients for triply distilled water at 25ı C are given by 1 D 2:1; a0 D 74:1; 0 D 8:44  1012 s; 14

f 0 D 4:62  10

(12.126)

s;

as determined by an rms fit to the numerical data presented in Figs. 4.2 and 4.3 of Vol. 1. Although a double relaxation time model provides a near-optimal fit to the numerical data (see Sect. 4.4.5 of Vol. 1), the complications introduced by the inclusion of a second (and comparatively weaker) relaxation mode does not justify its inclusion in the analysis presented here. Nevertheless, this secondary feature is included in numerical simulations when necessary. Attention is now turned to the description of the analytic structure of both the complex index of refraction n.!/ and the complex phase function .!;  / D i !.n.!/   / in the complex !-plane for the Rocard–Powles–Debye model [20]. Note first that the general symmetry relations given in (12.59)–(12.61) hold here, so that only the right-half of the complex !-plane needs to be considered here. The branch points of n.!/, and consequently of .!;  /, can be directly determined

294

12 Analysis of the Phase Function and Its Saddle Points

by rewriting the expression given in (12.125) for the frequency-dependence of the complex index of refraction as  n.!/ D  D

1 0 f 0 ! 2 C i 1 .0 C f 0 /!  s 0 f 0 ! 2 C i.0 C f 0 /!  1 .!  !z1 /.!  !z2 / .!  !p1 /.!  !p2 /

1=2

1=2 :

(12.127)

The branch point singularities !pj , j D 1; 2 are given by the two zeros of the denominator of the above expression as !p1 

i ; 0

!p2 

i f 0

;

(12.128)

which are both situated along the negative imaginary axis, and the branch point zeros !zj , j D 1; 2 are given by the two zeros of the numerator of the above expression as !zj

0 C f 0 20 f 0

(

) 1=2 0 f 0 s 1 i ; ˙ 4 1 .0 C f 0 /2 

(12.129)

where the upper sign choice is used for j D 1 and the lower sign choice for j D 2. Notice that these two branch point zeros are symmetrically situated about the point !z i

0 C f 0 ; 20 f 0

(12.130)

which also happens to be the midpoint of the two branch point singularities !p1 and !p2 . There are then three possibilities for the location of the branch point zeros, dependent upon the sign of the quantity appearing in the square root of the above expression, as follows:  If 40 f 0 =.0 C f 0 /2 < 1 =s , then

!zj

0 C f 0 D i 20 f 0

(

s 1

0 f 0 s 14 1 .0 C f 0 /2

) ;

(12.131)

and the two branch point zeros are located along the imaginary axis, symmetrically situated about the point !z , as depicted in Fig. 12.20a.  If 40 f 0 =.0 C f 0 /2 D 1 =s , then !zj D !z , j D 1; 2, and there is just a single branch point zero, located along the negative imaginary axis, as depicted in Fig. 12.20b.

12.2 The Behavior of the Phase in the Complex !-Plane

295

''

a

'

= −i/

cut

p1

z1

= −i (

f

)/(2

f

)

cut

z z2

= −i/

p2

b

f

'' ' = −i/

cut

p1

=

z

= −i/

f

= −i(

f

)/(2

)

f

cut

z1,2

p2

c

''

cut

'

branch branch

z2

p1

= −i/

cut z1

z p2

= −i/

f

Fig. 12.20 Location of the branch point singularities !pj and branch point zeros !zj , j D 1; 2, in the complex !-plane for a single relaxation time Rocard–Powles–Debye model dielectric with relaxation time 0 and associated friction time f 0 . The branch cuts are chosen as the line segments (a) !p1 !z1 and !z2 !p2 when 40 f 0 =.0 C f 0 /2 < 1 =s , (b) !p1 !z and !z !p2 when 40 f 0 =.0 C f 0 /2 D 1 =s , and (c) !p1 !p2 and !z1 !z2 when 40 f 0 =.0 C f 0 /2 > 1 =s . Notice that the classical Debye model is obtained in the limit as f 0 ! 0, in which case part (a) of the figure applies. In that limiting case there are just two branch points located along the negative imaginary axis at !p D i=0 and !z D i.s =1 /=0

 If 40 f 0 =.0 C f 0 /2 > 1 =s , then

!zj

0 C f 0 D 20 f 0

) 0 f 0 s 1i ; ˙ 4 1 .0 C f 0 /2

( s

(12.132)

296

12 Analysis of the Phase Function and Its Saddle Points

and the two branch point zeros are located in the lower-half of the complex !-plane, symmetrically situated about the imaginary axis along the line ! 00 D .0 C f 0 /=.20 f 0 /, as depicted in Fig. 12.20c. Notice that in the limit as f 0 ! 0, the Rocard–Powles–Debye model reduces to the classical Debye model. In that limiting case one obtains the pair of branch points !p 

i ; 0

!z i

s =1 ; 0

(12.133)

where !z is not to be confused with the symmetry point defined in (12.130) for just the Rocard–Powles extension of the Debye model. Because s =1 > 1, the branch cut extends along the line segment !p !z down the negative imaginary axis. Because the inequality 40 f 0 =.0 C f 0 /2 < 1 =s is satisfied for the set of Rocard–Powles–Debye model parameters given in (12.126), the branch points for that case are as depicted in part (a) of Fig. 12.20. Because this case also represents the limiting behavior of the classical Debye model, it is the focus of the remaining analysis for the Rocard–Powles–Debye model presented here. If the conditions require it, either of the other two cases may be treated in a similar manner.

12.2.3.1

Behavior Along the Real !0 -Axis

The complex index of refraction given in (12.125) for a single relaxation time Rocard–Powles–Debye model dielectric along the real angular frequency axis may be expressed in phasor form as

n.! 0 / D

h

1=4   i2 2 1 .1C02 ! 02 / 1Cf2 0 ! 02 Ca0 .10 f 0 ! 02 / Ca02 .0 Cf 0 / ! 02 p 2 q 2 e i =2 ; 1C0 ! 02 1Cf 0 ! 02

where

"

D arctan

a . C /! 0  0 0 f 0 1 .1C02 ! 02 / 1Cf2 0 ! 02 Ca0 .10 f 0 ! 02 /

(12.134)

# (12.135)

is the phase angle of n2 .! 0 /. The real and imaginary parts of the complex index of refraction along the real ! 0 -axis are then given by nr .! 0 / D 0

ni .! / D

ˇ ˇ ˇn.! 0 /ˇ cos . 1 .! 0 //; 2 ˇ ˇ ˇn.! 0 /ˇ sin . 1 .! 0 //; 2

(12.136) (12.137)

where ˇ ˇ ˇn.! 0 /ˇ D

h

1=4   i2 2 1 .1C02 ! 02 / 1Cf2 0 ! 02 Ca0 .10 f 0 ! 02 / Ca02 .0 Cf 0 / ! 02 p 2 q 2 : 1C0 ! 02 1Cf 0 ! 02

(12.138)

12.2 The Behavior of the Phase in the Complex !-Plane

297

9

Real & Imaginary Parts of the Complex Index of Refraction

8 7 nr ( ')

6 5 4

ni ( ')

3 2 1 0 108

109

1010

x 1011

1012

1013

x

f 1014

' (r/s)

Fig. 12.21 Frequency dependence of the real (solid curve) and imaginary (dashed curve) parts of the complex index of refraction n.! 0 / D nr .! 0 / C i ni .! 0 / along the positive real angular frequency axis for a single relaxation time Rocard–Powles–Debye model of triply-distilled H2O with medium parameters 1 D 2:1, a0 D 74:1, 0 D 8:44  1012 s, f 0 D 4:62  1014 s

As seen in Fig. 12.21, the real index of refraction is nearly constant over both the low frequency domain j! 0 j  1=0 and the high frequency domain j! 0 j  1=f 0 . In the intermediate frequency domain 0 < ! 0 < f 0 , the real index of refraction rapidly p decreases from its near static value n.0/ D s to its high-frequency limit n1 D p 1 . The imaginary part of the complex index of refraction peaks to its maximum value near the lower end of this intermediate frequency domain 0 < ! 0 < f 0 , whereas it is comparatively small in both the low frequency domain j! 0 j  1=0 and the high frequency domain j! 0 j  1=f 0 , approaching zero as ! 0 ! 0 and as ! 0 ! 1.

12.2.3.2

Limiting Behavior as j!j ! 1

Consider now the behavior of  .!;  /
j!j!1

p 1 ;

(12.139)

298

12 Analysis of the Phase Function and Its Saddle Points

where 1  1, so that limj!j!1 nr .!/ D these limiting results, (12.68) then gives

p 1 and limj!j!1 ni .!/ D 0. With

 p  lim  .!;  / D ! 00   1 ;

(12.140)

j!j!1

and the following limiting behavior for  .!;  / at j!j D 1 is obtained: p  For  < 1 ,  .!;  / is equal to 1 in the upper-half of the complex !plane, zero at the real ! 0 -axis [i.e.,  .! 0 ;  / D 0 a ! 0 D ˙1], and is equal to C1 in the lower-half of the complex !-plane. p p  For  D 1 ,  .!; 1 / D 0 everywhere at j!j D 1. p  For  > 1 ,  .!;  / is equal to C1 in the upper-half of the complex !plane, zero at the real ! 0 -axis [i.e.,  .! 0 ;  / D 0 at ! 0 D ˙1], and is equal to 1 in the lower-half of the complex !-plane.

12.2.3.3

Behavior Along the Imaginary Axis

The behavior of the complex index of refraction for a single relaxation time Rocard– Powles–Debye model dielectric along the imaginary axis is obtained from (12.125) with the substitution ! D i ! 00 , with the result  n.! 00 / D 1 C

a0 00 .1 C 0 ! /.1 C f 0 ! 00 /

1=2 :

(12.141)

It is then seen that n.! 00 / is real-valued everywhere along the imaginary axis excluding the branch cuts !p1 !z1 and !z2 !p2 [see part (a) of Fig. 12.20)]; that is, when either ! 00 > 1=0 , ! 00 < 1=f 0 , or j!z2 j < ! 00 < j!z1 j. The index of refraction n.! 00 / is also positive-valued over each of these intervals. In particular, as ! 00 increases over the upper interval ! 00 > 1=0 , n.! 00 / monotonically decreases from its positive infinite branch point singularity !p1 to its zero frep value at the p quency value n.0/ D 1 C a0 D s , approaching its infinite frequency limit p n.i 1/ D 1 as ! 00 ! 1, as illustrated in Fig. 12.22. Similarly, as ! 00 decreases over the lower interval ! 00 < 1=f 0 , nr .! 00 / monotonically decreases from its positive infinite value at the branch point singularity !p2 , approaching its infinite p frequency limit n.i 1/ D 1 as ! 00 ! 1. Finally, in the intermediate interval j!z2 j < ! 00 < j!z1 j between the two branch cuts, nr .! 00 / increases from zero to a local maximum and then decreases back to zero, as illustrated in Fig. 12.22. Notice that ni .! 00 / D 0 in those regions where nr .! 00 / is nonvanishing and that nr .! 00 / D 0 in those regions where ni .! 00 / is nonvanishing. Since .! 00 ; / D ! 00 .n.! 00 /  / along the imaginary axis, then  .! 00 ;  / D ! 00 .nr .! 00 /   /:

(12.142)

12.2 The Behavior of the Phase in the Complex !-Plane

299

Real & Imaginary Parts of the Complex Index of Refraction

5

4

3 ni ( '' ) 2

ni ( '' )

nr( '')

1

0 −5

nr( '' )

nr ( '' ) f z2

z1

5

0 13

'' (x10 r/s)

Fig. 12.22 Frequency dependence of the real (solid curve) and imaginary (dashed curve) parts of the complex index of refraction n.! 00 / D nr .! 00 / C i ni .! 00 / along the imaginary angular frequency axis for a single relaxation time Rocard–Powles–Debye model of triply distilled H2 O with medium parameters 1 D 2:1, a0 D 74:1, 0 D 8:441012 s, f 0 D 4:621014 s. Notice that nr .! 00 / D 0 and ni .! 00 / > 0 when either 1=f 0 < ! 00 < j!z2 j or j!z1 j < ! 00 < 1=0 , and that ni .! 00 / D 0 and nr .! 00 / > 0 when either ! 00 < 1=f 0 , j!z2 j < ! 00 < j!z1 j, or 1=0 < ! 00

As nr .! 00 / D 0 along the two branch cuts !p1 !z1 and !z2 !p2 , then  .! 00 ;  / D ! 00  . Notice that ! 00 < 0 along the two branch cuts so that  .! 00 ;  / < 0 for all   1 when either 1=f 0 < ! 00 < j!z2 j or j!z1 j < ! 00 < 1=0 . 12.2.3.4

Behavior in the Vicinity of the Branch Points

Consider finally the limiting behavior of both n.!/ and  .!;  / in the immediate vicinity of each of the four branch points depicted in part (a) of Fig. 12.20. In the region about the upper branch point pole !p1 , the complex angular frequency may be written as (12.143) ! D !p1 C r1 e i'1 ; where the ordered-pair .r1 ; '1 / denotes the polar coordinates about the point !p1 D i=0 . Substitution of this expression into (12.127) results in the limiting behavior   .!p1  !z1 /.!p1  !z2 / 1=2 n.r1 ; '1 / D .!p1  !p2 /r1 e i'1 s s =1  1 1 i.=4'1 =2/ e (12.144) Š 0  f 0 r 1=2 1

300

12 Analysis of the Phase Function and Its Saddle Points

as r1 ! 0 about the upper branch point !p1 . Similarly, in the region about the lower branch point pole !p2 , the complex angular frequency may be written as ! D !p2 C r2 e i'2 ;

(12.145)

where the ordered-pair .r2 ; '2 / denotes the polar coordinates about the point !p2 D i=f 0 . Substitution of this expression into (12.127) results in the limiting behavior  .!p2  !z1 /.!p2  !z2 / 1=2 n.r2 ; '2 / D .!p2  !p1 /r2 e i'2 s s =1  1 1 i.=4C'2 =2/ Š e 0  f 0 r 1=2 

(12.146)

2

as r2 ! 0 about the upper branch point !p2 . In the region about the upper branch point zero !z1 , the complex angular frequency may be expressed as ! D !z1 C R1 e i 1 ;

(12.147)

where the ordered-pair .R1 ; 1 / denotes the polar coordinates about the point !z1 given in (12.131) with the upper sign choice. Substitution of this expression into (12.127) results in the limiting behavior 1=2 .!z1  !z2 /R1 e i 1 .!z1  !p1 /.!z1  !p2 / v s u u 0 C f 0 0 f 0 s 1=2 t Š 14 R e i. s =1  1 1 .0 C f 0 /2 1 

n.R1 ;

1/ D

1 =2C=4/

(12.148)

as R1 ! 0 about the upper branch point !z1 . Similarly, in the region about the lower branch point zero !z2 , the complex angular frequency may be expressed as ! D !z2 C R2 e i 2 ;

(12.149)

where the ordered-pair .R2 ; 2 / denotes the polar coordinates about the point !z2 given in (12.131) with the lower sign choice. Substitution of this expression into (12.127) results in the limiting behavior 1=2 .!z2  !z1 /R2 e i 2 .!z2  !p1 /.!z2  !p2 / v s u u 0 C f 0 0 f 0 s 1=2 t Š 14 R e i. s =1  1 1 .0 C f 0 /2 2 

n.R2 ;

2/ D

as R2 ! 0 about the upper branch point !z2 .

2 =2=4/

(12.150)

12.2 The Behavior of the Phase in the Complex !-Plane

301

Fig. 12.23 Limiting behavior of the complex index of refraction n.!/ about the branch point poles !pj and branch point zeros !zj , j D 1; 2, for a single relaxation time Rocard–Powles–Debye model dielectric

''

'

p1

(1– i)

(1+i) −i

+i

−i0 +i0 (1+i)0

(1– i)0 z1

−0

+0 (1+ i)0

z2

+i0 −i0

+i (1+ i)

−i

(1− i)0

(1− i) p2

The limiting behavior of the complex index of refraction for a single relaxation time Rocard–Powles–Debye model dielectric about each of the branch point poles !pj and branch point zeros !zj , j D 1; 2, as described by (12.144), (12.146), (12.148), and (12.150), is illustrated in Fig. 12.23. Because the near saddle point for a Debye-type dielectric moves down the imaginary axis as the space–time parameter  increases [see (12.35) and the discussion following], crossing the origin p at the space–time point  D 0 , where [see (12.21)] 0 s , it primarily interacts with the upper branch point pole !p1 , which also happens to be the only branch point in the finite !-plane in the Debye model limit. From these results, the limiting behavior of the real part  .!;  / D
"s

  ' s =1  1 1 1    cos 0  f 0 r 1=2 4 2

# (12.151)

1

as r1 ! 0. This limiting behavior is depicted in Fig. 12.24. Because  .r1 ; '1 / varies from positive infinity over the angular region 0  '1   to =0 over both of the angular regions =2 < '1 < 0 and  < '1 < 3=2, the zero isotimic contour

302

12 Analysis of the Phase Function and Its Saddle Points

Fig. 12.24 Limiting behavior of the real part  .!;  / of the complex phase function .!;  / about the upper branch point pole !p1 for a single relaxation time Rocard–Powles–Debye model dielectric. The dashed curve describes the approximate behavior of the isotimic contour  .!;  / D 0 for p 1   < 0

''

'

+ +

p1

+

Branch Cut

−θ/τ0 −θ/τ0

z1

 .!;  / D 0 from the origin must pass through the branch point !p1 in both of those angular sectors when 1    0 , where 1

p

1 ;

(12.152)

forming a cardioid-like contour, as illustrated in Fig. 12.24. When  > 0 , the zero isotimic contour through the origin passes into the upper-half of the complex !-plane.

12.2.3.5

Numerical Results

This sketch of the behavior of the real phase function  .!;  /
12.2 The Behavior of the Phase in the Complex !-Plane

303

1 0.8 0.6 SPn +

0.2

13

''' (x10 r/s)

0.4

0

+

p1

−0.2 −0.4 −0.6 −0.8 −1 −1

z1

+

+

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

13

' (x10 r/s)

Fig. 12.25 Isotimic contours of the real part  .!;  / of the complex phase function .!;  / for a single relaxation time Rocard–Powles–Debye model dielectric at the fixed space–time point  D 1 ' 1:4491. At this initial space–time value the upper near saddle point SPn , located along the imaginary axis in the upper-half of the complex !-plane, is the dominant saddle point

1.5

1

11

'' (x10 r/s)

0.5

SP n

+

0

–0.5

–1 + p1

–1.5 −1.5

−1

−0.5

0

0.5

1

1.5

11

' (x10 r/s)

Fig. 12.26 Isotimic contours of the real part  .!;  / of the complex phase function .!;  / for a single relaxation time Rocard–Powles–Debye model dielectric at the fixed space–time point  D 0 ' 8:7293. At this space–time value the dominant upper near saddle point SPn is located at the origin of the complex !-plane

304

12 Analysis of the Phase Function and Its Saddle Points 1.5

1

11

'' (x10 r/s)

0.5

0

−0.5

+

SPn

−1 + −1.5 −1.5

−1

−0.5

p1

0

0.5

1

1.5

11

' (x10 r/s)

Fig. 12.27 Isotimic contours of the real part  .!;  / of the complex phase function .!;  / for a single relaxation time Rocard–Powles–Debye model dielectric at the fixed space–time point  D 20 . At this space–time value the dominant upper near saddle point SPn is located along the negative imaginary axis

through this saddle point without encircling this branch cut. One then needs only consider the upper near saddle point SPn . As  increases from 1 to 0 , the near saddle point SPn moves down the positive imaginary axis to the origin. At the critical space–time point  D 0 , the near saddle point SPn is located at the origin, as illustrated in Fig. 12.26, where  .!SPn ; 0 / D 0. Finally, as  increases above 0 , this saddle point moves down the negative imaginary axis, as illustrated in Fig. 12.27, approaching the upper branch point !p1 D i=0 as  ! 1. For a double relaxation time Rocard–Powles–Debye model dielectric, the appearance of an additional set of relatively “weaker” branch cuts (a0  a1 ) further removed from the origin (typically 0  1 ) results in the appearance of an additional pair of near saddle points SP˙ n that evolve in the intermediate frequency domain j!j > 1=0 in the lower-half of the complex !-plane, as illustrated in Figs. 12.28–12.30. This sequence of figures illustrates the  .!;  / D constant isotimic contours for a double relaxation time Rocard–Powles–Debye model with parameters 1 D 2:1, a0 D 74:1, 0 D 8:441012 s, f 0 D 4:931014 s, a1 D 2:90, 1 D 6:051014 s, and f 1 D 8:59  1015 s that are representative of triply distilled water; notice that a1 =a0 Š 0:039 and 1 =0 Š 0:0072. At the initial space–time point  D 1 Š 1:4491 the two saddle points SP˙ n are symmetrically situated about the imaginary

12.2 The Behavior of the Phase in the Complex !-Plane

305

5 4 3

1

12

'' (x10 r/s)

2

0

+

p1

SP N−

−1

+

SP + N

+ +

−2

z1

−3 −4 −5 −5

−4

−3

−2

−1

0

1

2

3

4

5

12

' (x10 r/s)

Fig. 12.28 Isotimic contours of the real part  .!;  / of the complex phase function .!;  / for a double relaxation time Rocard–Powles–Debye model dielectric at the fixed space–time point  D 1 ' 1:4491. At this initial space–time value the upper near saddle point SPn , located along the imaginary axis in the upper-half of the complex !-plane (not visible in the figure), is the dominant saddle point 5 4 3

+

SPn

1 0

+

12

'' (x10 r/s)

2

p1

−1 +

−2

z1

+− SPN

−3

+ + SPN

−4 −5 −5

−4

−3

−2

−1

0

1

2

3

4

5

12

' (x10 r/s)

Fig. 12.29 Isotimic contours of the real part  .!;  / of the complex phase function .!;  / for a double relaxation time Rocard–Powles–Debye model dielectric at the fixed space–time point  D 2:5. At this space–time value the upper near saddle point SPn is the dominant saddle point

306

12 Analysis of the Phase Function and Its Saddle Points 5 4 3

1

SPn +

12

'' (x10 r/s)

2

0

+

p1

−1 +

−2

z1

−3

+

−4 −5 −5

−4

−3

−2

−1

SPN

0

1

2

3

4

5

12

' (x10 r/s)

Fig. 12.30 Isotimic contours of the real part  .!;  / of the complex phase function .!;  / for a double relaxation time Rocard–Powles–Debye model dielectric at the fixed space–time point  D 2:8443. At this space–time value the upper near saddle point SPn is the dominant saddle point

axis in the lower-half plane such that j!p1 j < j!SP˙ j < j!z1 j, as seen in Fig. 12.28. n Notice that the dominant upper near saddle point SPn , which is situated along the positive imaginary axis at this space–time point, is off the scale in the figure. As  increases to 0 , the two first-order saddle points SP˙ n approach each other and then coalesce into a single second-order saddle point along the negative imaginary axis between the two branch cuts !p1 !z1 and !z2 !p2 , as illustrated in Figs. 12.29– 12.30. They then move in opposite directions along the negative imaginary axis as  increases further, approaching the respective branch points !z1 and !z2 as  ! 1. Notice that the upper near saddle point SPn remains the dominant saddle point over this entire space–time domain.

12.2.4 Drude Model Conductors The angular frequency dependence of the complex index of refraction for a Drude model conductor is given by [see (4.217) of Vol. 1]

n.!/ D 1 

!p2 !.! C i  /

!1=2 ;

(12.153)

12.2 The Behavior of the Phase in the Complex !-Plane

307

where  1=c is a damping constant given by the inverse of the relaxation time c associated with p the mean-free path for electrons in the conducting material. Here !p .k4k=0 /N qe2 =m is the plasma frequency associated with the conduction electrons with number density N , charge magnitude qe , and effective mass m. Notice that this expression is just that for a single-resonance Lorentz model dielectric [cf. (12.57)] with the resonance frequency set equal to zero. When expressed in terms of the relative complex permittivity c .!/=0 .!/=0 C i.k4k=0 / .!/=!, the Drude model describes a conducting material with unity relative dielectric permittivity [viz. .!/=0 D 1] and conductivity [see (5.88) of Vol. 1] 0 ; (12.154)  .!/ D i ! C i where 0 .0 =k4k/!p2 = denotes the static conductivity of the material. Estimates of these parameters for sea-water are 0 4 mho m1 ;

 1  1011 r/s;

(12.155)

with corresponding plasma frequency !p 2:125  1011 r/s. For comparison, approximate values of these Drude model medium parameters for the E-layer of the ionosphere are given by [21] !p   107 r/s and    105 r/s. The branch points of n.!/, and consequently of .!;  / i !.n.!/  /, can be determined by rewriting the expression given in (12.153) for the angular frequency dependence of the complex index of refraction as

n.!/ D

! 2 C i !  !p2 !.! C i  /

where

!1=2

 D

.!  !zC /.!  !z / !.! C i  /

r !z˙

˙ !p2 

  2 2

1=2 ;

 i : 2

(12.156)

(12.157)

The branch point zeros are then located at ! D !z˙ and the branch point poles are at ! D !p˙ with !pC D 0 and !p D i  . The branch cuts are then taken as the horizontal line segment !z !zC and the vertical line segment !p !pC along the negative imaginary axis, as depicted in Fig. 12.31. Because the Drude model is a special case of the Lorentz model with zero resonance frequency, the analysis developed in Sect. 12.2.2 is applicable here with !0 set equal to zero and ı replaced by =2. In particular, (12.72)–(12.73) for the complex phasor form of the complex index of refraction become " n.!/ D 1 C

  !p4  2!p2 ! 02  ! 002  ! 00 .! 02  ! 002  ! 00 /2 C 4! 02 .! 00 C =2/2

#1=4 e i =2 ; (12.158)

308

12 Analysis of the Phase Function and Its Saddle Points

cut

''

branch branch

−

z

'

+ =0 p

cut + z

−

p

= −i

Fig. 12.31 Location of the branch point singularities !p˙ and branch point zeros !z˙ in the complex !-plane for a Drude model conductor with damping constant . The branch cuts are taken as the horizontal line segment !z !zC and the vertical line segment !p !pC along the negative imaginary axis



.!/ D arctan

2! 0 .! 00 C=2/!p2

.! 02 ! 002 ! 00 /

2

!p2 .! 02 ! 002 ! 00 /C4! 02 .! 00 C=2/

 ; 2

(12.159)

where ! 0 D
12.2.4.1

Behavior along the Real !0 -Axis

Along the real axis, ! 00 D 0 and the real and imaginary parts of the complex index of refraction are directly obtained from (12.158) as   31=4 !p2 !p2  2! 02   5 cos 1 .! 0 / ; nr .! 0 / D 41 C 02 02 2 2 ! .! C  / 2

  31=4 !p2 !p2  2! 02   5 sin 1 .! 0 / ; ni .! 0 / D 41 C 02 02 2 2 ! .! C  /

(12.160)

2

(12.161)

respectively, with " 0

.! / D arctan

!p2 ! 0 ! 04  !p2 ! 02 C  2 ! 02

# :

(12.162)

12.2 The Behavior of the Phase in the Complex !-Plane

309

Real & Imaginary Parts of the Complex Conductivity

4

3

r(

')

2

1

0 105

i(

')

1010 ' (r/s)

1015

Fig. 12.32 Frequency dependence of the real (solid curve) and imaginary (dashed curve) parts of the complex conductivity .! 0 / D r .! 0 /Ci i .! 0 / along the positive real angular frequency axis for a Drude model of sea-water with static conductivity 0  4 mho m1 and damping constant   1  1011 r/s

As seen in Fig. 12.31, the real part of the complex conductivity, given by [from (12.154)] r .! 0 / D 0 =.1 C ! 02 = 2 /, rapidly decreases from its static value 0 to zero as ! 0 increases past the damping constant  . At the same time, the imaginary part of the Drude model conductivity, given by i .! 0 / D .0 = /! 0 =.1 C ! 02 = 2 /, peaks to its maximum value near the angular frequency value ! 0 D  . A Drude model material is then considered to be (relatively) conducting when j! 0 j   whereas it is considered to be (relatively) nonconducting when ! 0   . This frequency-dependent behavior in the electric conductivity  .! 0 / is reflected in the angular frequency dependence of both the relative complex dielectric permittivity "c .! 0 / D 1 C i  .! 0 /=! 0 !p2 !p2 D 1 2 C i ! C 2 ! .! 2 C  2 /

(12.163)

and the complex index of refraction n.! 0 / D Œ"c .! 0 / 1=2 , with =0 D 1, whose real and imaginary parts are illustrated in Figs. 12.33 and 12.34, respectively, for the Drude model of sea-water. The cut-off frequency !co for the purely conducting material is defined as the positive, real angular frequency value at which "r .!co / D 0, so that q (12.164) !co !p2   2 :

310

12 Analysis of the Phase Function and Its Saddle Points 4

10

3

Real & Imaginary Parts of the Relative Dielectric Permittivity

10

102 101

''c( ' )

'c( ' )

'c( ' )

100 −1

10

−2

10

−3

10−4 5 10

_

10

co

1010 ' (r/s)

1015

Fig. 12.33 Frequency dependence of the real (solid curve) and imaginary (dashed curve) parts of the relative complex dielectric permittivity "c .! 0 / D 1 C i.! 0 /=! 0 with "c .! 0 / D "0c .! 0 / C i "00c .! 0 / along the positive real angular frequency axis for a Drude model of sea-water with static conductivity 0  4 mho m1 and damping constant   1  1011 r/s. Notice that "r .! 0 / becomes negative-valued below the angular cut-off frequency !co so that its absolute value j"0c .! 0 /j is graphed when ! 0 < !co

102

101 ni( ' ) nr ( ' )

100

10−1 105

co

_

Real & Imaginary Parts of the Complex Index of Refraction

103

1010

1015

' (r/s)

Fig. 12.34 Frequency dependence of the real (solid curve) and imaginary (dashed curve) parts of the complex index of refraction n.! 0 / D nr .! 0 / C i ni .! 0 / along the positive real angular frequency axis for a Drude model of sea-water with static conductivity 0  4 mho m1 and damping constant   1  1011 r/s

12.2 The Behavior of the Phase in the Complex !-Plane

311

When j! 0 j < !co , the real part of the complex dielectric permittivity is negative with zero frequency limit "0c .0/ D 1  !p2 = 2 . Notice that this behavior is modified if the material is not a pure conductor; in that case the zero frequency limit becomes "0c .0/ D ".0/  !p2 = 2 , where ".0/ is the static dielectric permittivity of the semiconducting material. If ".0/ > !p2 = 2 , then there is no cut-off frequency. Notice that the imaginary part "00c .! 0 / of the complex dielectric permittivity along the positive real angular frequency axis decreases below unity as ! 0 increases above the angular cut-off frequency, approaching zero as ! 0 ! 1. On the other hand, "00c .! 0 / increases as ! 0 decreases below !co , approaching infinity as ! 0 ! 0. Because of this behavior, both the real and imaginary parts of the complex index of refraction approach infinity as ! 0 ! 0, as seen in Fig. 12.34.

12.2.4.2

Limiting Behavior as j!j ! 1

As the Drude model is a special case of the single-resonance Lorentz model, the limiting behavior of  .!;  /
j!j!1

(12.165)

so that limj!j!1 nr .!/ D 1 and limj!j!1 ni .!/ D 0, and consequently lim  .!;  / D ! 00 .  1/:

j!j!1

(12.166)

The following limiting behavior at j!j D 1 is then obtained:  For  < 1,  .!;  / is equal to 1 in the upper-half of the complex !-plane,

zero at the real ! 0 -axis [i.e.,  .! 0 ;  / D 0 a ! 0 D ˙1], and is equal to C1 in the lower-half of the complex !-plane.  For  D 1,  .!; 1/ D 0 everywhere at j!j D 1.  For  > 1,  .!;  / is equal to C1 in the upper-half of the complex !-plane, zero at the real ! 0 -axis [i.e.,  .! 0 ;  / D 0 at ! 0 D ˙1], and is equal to 1 in the lower-half of the complex !-plane.

12.2.4.3

Behavior in the Vicinity of the Branch Points

Consider now the limiting behavior of the complex index of refraction n.!/ and the real part  .!;  / of the complex phase function .!; / i !.n.!/   / about the branch points !p˙ and !z˙ for a Drude model conductor. Only the results are presented here, their derivation being left as an exercise. In the region about the upper branch point pole !pC D 0, the complex angular frequency may be written as (12.167) ! D r1 e i'1 :

312

12 Analysis of the Phase Function and Its Saddle Points

With this substitution in (12.156), the limiting behavior of the complex index of refraction in a small region of the complex !-plane about the upper branch point pole !pC at the origin is found to be given by !p

n.r1 ; '1 / Š

1=2  1=2 r1

'1  ei . 4  2 /

(12.168)

as r1 ! 0. This limiting behavior is depicted in part (a) of Fig. 12.35. The resultant limiting behavior of the real part of the complex phase function about this point is then found to be given by ' !p 1=2  1 C r1 sin 1=2  2 4

 .r1 ; '1 ;  / Š 

(12.169)

as r1 ! 0. This limiting behavior is depicted in part (b) of Fig. 12.35. Notice that n.!/ is singular at this branch point whereas  .!;  / vanishes there.

a

'' + + p

(1−i)

0

− z

'

(1+i)

(1−i)0

−i

−i0

(1+i)0

+i +i0 −i

+i (1+i)

0

−i0

+i0 (1+i)0

+ z

(1−i)0 − p

(1−i)

+

b

'' −0 −0

+ p

'

−0 − z

−0

−0

+0

+0

+ +

− p

+ z

+

Fig. 12.35 Limiting behavior of (a) the complex index of refraction n.!/ and (b) the real part  .!;  / of the complex phase function .!;  / about the branch points !p˙ and !z˙ in the complex !-plane for a Drude model conductor

12.2 The Behavior of the Phase in the Complex !-Plane

313

In the region about the lower branch point pole !p D i  , the complex angular frequency may be written as ! D i  C r2 e i'2 :

(12.170)

With this substitution in (12.156), the limiting behavior of the complex index of refraction in a small region of the complex !-plane about the lower branch point pole !p is found to be given by n.r2 ; '2 / Š

!p

'2



1=2  1=2 r2

e i . 4 C 2 /

(12.171)

as r2 ! 0. This limiting behavior is depicted in part (a) of Fig. 12.35. The resultant limiting behavior of the real part of the complex phase function about this point is then found to be given by  .r2 ; '2 ;  / Š

!p  1=2 1=2

r2

cos

'

2

2

C

 4

(12.172)

as r2 ! 0. This limiting behavior is depicted in part (b) of Fig. 12.35. Notice that both n.!/ and  .!;  / are singular at this branch point. In the region about the right branch point zero !zC , the complex angular frequency may be written as !D

q !p2  .=2/2  i =2 C R1 e i 1 :

(12.173)

With this substitution in (12.156), the limiting behavior of the complex index of refraction in a small region of the complex !-plane about the branch point zero !zC is found to be given by q n.R1 ;

1/

Š

4!p2   2 !p

1=2

R1 e i

1 2

(12.174)

as R1 ! 0. The resultant limiting behavior of the real part of the complex phase function about this point is then found to be given by  .R1 ;

1;  /

 Š  2

(12.175)

as R1 ! 0. Because of the even symmetry of  .!;  / about the imaginary axis, the behavior about the left branch point zero !z is the mirror image of that about !zC , as illustrated in Fig. 12.35.

314

12 Analysis of the Phase Function and Its Saddle Points

12.2.4.4

Numerical Results

This sketch of the behavior of the real phase function  .!;  /
5 4 3

SPn

1

+

11

'' (x10 r/s)

2

0

− z

−1

+ p +

SP −

− p

+ z

+ SP+

−2 −3 −4 −5 −5

−4

−3

−2

−1

0

1

2

3

4

5

11

' (x10 r/s)

Fig. 12.36 Isotimic contours of the real part  .!;  / of the complex phase function .!;  / for a Drude model conductor at the fixed, luminal space–time point  D 1:0. At this space–time value the two distant saddle points SPd˙ are situated at ˙1 in the lower-half plane and are the dominant saddle points

12.2 The Behavior of the Phase in the Complex !-Plane

315

5 4 3

1

SPn +

11

'' (x10 r/s)

2

0 − SPd +

−1

− z

+ p

− p

−2

+ z

+SP +

+

SP−

+

+ SPd

−3 −4 −5 −5

−4

−3

−2

−1

0

1

2

3

4

5

11

' (x10 r/s)

Fig. 12.37 Isotimic contours of the real part  .!;  / of the complex phase function .!;  / for a Drude model conductor at the fixed space–time point  D 1:5. At this space–time value the two distant saddle points SPd˙ are the dominant saddle points 5 4 3

1 SPn

11

'' (x10 r/s)

2

+

0 −

SPd +

−1

+SP + d

+SP +

SP−+

−2 −3 −4 −5 −5

−4

−3

−2

−1

0

1

2

3

4

5

11

' (x10 r/s)

Fig. 12.38 Isotimic contours of the real part  .!;  / of the complex phase function .!;  / for a Drude model conductor at the fixed space–time point  D 2. At this space–time value the two distant saddle points SPd˙ are (approximately) equally dominant with the near saddle point SPn

316

12 Analysis of the Phase Function and Its Saddle Points

symmetrically move in towards the branch point zeros !z˙ , respectively, approaching each in the limit as  ! 1, as seen in Figs. 12.37–12.38. They are the dominant saddle points over the space–time domain illustrated here. However, the near saddle point SPn , which is situated along the positive imaginary axis and approaches the branch point pole !pC at the origin as  ! 1, as seen in Figs. 12.26–12.28, is nearly of equal dominance with the distant saddle points at the largest  -value considered here. It eventually does become the dominant saddle point at some larger value SB of  defined by [cf. (12.97)]  .!SP ˙ ; SB / D  .!SPn ; SB /;

(12.176)

d

and remains so for all larger space–time values.

12.3 The Location of the Saddle Points and the Approximation of the Phase To obtain the asymptotic expansion of the propagated wavefield A.z; t / for large values of the propagation distance z > 0, where A.z; t / is given by the Fourier– Laplace integral representation in either (12.1) or (12.2), the saddle points of the complex phase function .!; / must be located in the complex !-plane and the behavior of .!;  / at these points determined. The condition that .!;  / be stationary at a saddle point is  0 .!;  / D i .n.!/  / C i!n0 .!/ 0; where the prime denotes differentiation with respect to !. The saddle point equation is then given by n.!/ C !n0 .!/ D : (12.177) Approximate analytic expressions for A.z; t / then require approximate analytic solutions of (12.177) for the dynamical evolution of the saddle point locations with  as well as for the complex phase behavior at each of these saddle points. In those exceptional cases when reasonably accurate, approximate analytic expressions are unavailable, numerical results alone will have to suffice.

12.3.1 Single-Resonance Lorentz Model Dielectrics An exact analytic expression for the location of the saddle points of .!;  / for a single-resonance Lorentz model dielectric is considered first. With the complex index of refraction given by (12.57), namely,  n.!/ D 1 

b2 ! 2  !02 C 2i ı!

1=2 ;

12.3 The Location of the Saddle Points and the Approximation of the Phase

317

with the first derivative b 2 .! C i ı/

0

n .!/ D  ! 2  !02 C 2i ı!

 2 1 

b2 ! 2  !02 C 2i ı!

1=2 ;

the saddle point equation (12.177) becomes  1

b2 ! 2 !02 C2iı!

1=2

C



b 2 !.!Ciı/ 2

.! 2 !02 C2iı! /

1

b2 ! 2 !02 C2iı!

1=2

D : (12.178)

To eliminate the square root factors appearing in this expression, it may be rewritten as ! 2  !12 C 2i ı! b 2 !.! C i ı/ C  2 D  2 ! 2  !0 C 2i ı! ! 2  !02 C 2i ı!



! 2  !12 C 2i ı! ! 2  !02 C 2i ı!

1=2 ;

where !12 D !02 C b 2 . Squaring both sides of this equation then yields  ! 2  !12 C 2i ı! C

2 b 2 !.! C i ı/ ! 2  !02 C 2i ı!    D  2 ! 2  !12 C 2i ı! ! 2  !02 C 2i ı! : (12.179)

Since both of the expressions given in (12.178)–(12.179) are complicated functions of the complex variable !, it is difficult (if not indeed impossible) to determine exact analytic expressions for the saddle point locations as a function of  for all   1. However, from the computer-generated contour plots illustrating the topography of  .!;  / in the complex !-plane given in Figs. 12.4–12.9, it is found that there are in general a pair of saddle points that evolve with   1 in the region j!j  !0 about the origin and a pair of saddle points that evolve with   1 in the region j!j  !1 removed from the origin. These two regions may then be considered separately in develop approximate analytic expressions for the respective saddle point locations. These approximate solutions may then be used as initial values in a numerical solution of the exact saddle point equation to test both their accuracy and numerically determine more accurately the roots of that equation. Before proceeding with this approximate analysis, notice that two exact roots of the saddle point equation given in (12.174) are readily obtained in the limit as  ! 1. In that limit, either ! 2  !02 C 2i ı! D 0, yielding the roots q ! ! !˙ D ˙ !02  ı 2  i ı;

as

 ! 1;

(12.180)

318

12 Analysis of the Phase Function and Its Saddle Points

or n.!/ D 0, yielding the roots !!

0 !˙

q D ˙ !12  ı 2  i ı;

as

 ! 1:

(12.181)

Thus, in the limit as  ! 1, the saddle points move into the branch points !C and 0 in the right-half plane and ! and !0 in the left-half plane. !C Moreover, an exact polynomial equation describing the location of the saddle points can be obtained as follows: (12.178) is rewritten to eliminate the square roots as  

! 2  !12 C 2i ı! ! 2  !02 C 2i ı!

1=2

 2 2 !  !02 C 2i ı!

    D ! 4 C 4i ı! 3  2 !02 C 2ı 2 ! 2  i ı 4!02 C 3b 2 ! C !12 !02 :

Upon squaring both sides of this equation, there results   3  2 ! 2  !12 C 2i ı! ! 2  !02 C 2i ı!   2

  D ! 4 C 4i ı! 3  2 !02 C 2ı 2 ! 2  i ı 4!02 C 3b 2 ! C !12 !02 : After a bit of algebraic manipulation, one finally obtains the following exact polynomial equation for the saddle point locations in a single-resonance Lorentz model dielectric [4, 7]:     

   2   1 ! 8 C 8i ı  2  1 ! 7  4 !02 C 6ı 2  2  1 C b 2  2 ! 6    2i ı 12!02 C 3b 2 C 16ı 2  2  1 ! 5 

  C 6!04 C .48ı 2 C 2b 2 /!02 C 12b 2 ı 2 C 16ı 4  2  1   Cb 2 !02  2  12ı 2 ! 4 

  C4i ı 6!04 C 4b 2 !02 C 8ı 2 !02 C 4ı 2 b 2  2  1    C !02 C 2ı 2 2!02  2  b 2 ! 3 

   !02 4!04 C 3b 2 !02 C 24ı 2 !02 C 12ı 2 b 2  2  1   b 2 !04 C 20ı 2 !02 C 9ı 2 b 2 ! 2      2i ı!02 4!02 C 3b 2 !02  2  1  b 2 !      C!04 !02 C b 2 !02  2  1  b 2 D 0: (12.182) Because this eighth-order polynomial is extremely formidable as well as difficult to approximate, the approximate solution of the saddle point equation as given in either (12.178) or (12.179) is now developed for both the distant and near saddle points that are a characteristic of a single-resonance Lorentz model dielectric.

12.3 The Location of the Saddle Points and the Approximation of the Phase

12.3.1.1

319

The Region Removed from the Origin (j!j  !1 )

The First Approximation To permit comparison with the classical asymptotic theory due to Brillouin [2, 3], a critical review of this first approximation is now considered. For sufficiently large values of j!j the quantity !02 can be neglected in comparison to the quantity ! 2 in the expression (12.57) for the complex index of refraction of a single-resonance Lorentz model dielectric, so that n2 .!/ 1 

b2 ; !.! C 2iı/

(12.183)

provided that j!j2  !02 . Notice that this approximation simplifies the highfrequency response of a Lorentz model dielectric by the exact behavior of a Drude model conductor [cf. (12.153)]. Because the magnitude of the second term in the above expression for n2 .!/ is small in comparison to unity for j!j2  b 2 , the complex index of refraction may then be further approximated as n.!/ 1 

b2 : 2!.! C 2iı/

(12.184)

Notice that the accuracy of these approximations only improves in the weak dispersion limit as b ! 0 (more fundamentally, in the limit as N ! 0, where N is the number density of Lorentz oscillators). Differentiation of the approximation given in (12.184) with respect to ! yields n0 .!/ b 2

! C iı .! 2

C 2iı!/2

:

(12.185)

Substitution of these approximate expressions into the saddle point equation given in (12.177) then results in the approximate saddle point equation 1

b 2 .! C iı/ b2 C

 2!.! C 2iı/ ! .! C 2iı/2

(12.186)

in the region removed from the origin, with solutions b !SP ˙ ./ ˙ p  2iı; d 2.  1/

(12.187)

for   1. This result, first obtained by Brillouin [2, 3], is referred to as the first approximation of the distant saddle point locations [cf. (12.55)]. The distant saddle points are then seen to be symmetrically located about the imaginary axis, lying

320

12 Analysis of the Phase Function and Its Saddle Points

along the line ! D 2iı. At the luminal space–time point  D 1 these two saddle points are at !SP ˙ .1/ D ˙1  2iı, and as  increases away from unity they d move in toward the imaginary axis that is
!0 and b (i.e., when the distant saddle points SPd˙ enter into the vicinity of the 0 , respectively), the above approximation loses its validity. outer branch points !˙ The first approximation is then seen to be valid only in a small space–time domain  2 Œ1; 1 C / where > 0 increases as the number density N decreases. A more accurate approximation that is valid over the entire space–time domain  2 Œ1; 1/ is then seen to be desirable.

The Second Approximation To obtain a more accurate description of the distant saddle point locations, particularly for large values of  , the exact saddle point equation given in (12.179) is first rewritten as 1  2 .! 2 !02 C2iı! /

 ! 2  !12 C 2iı! C

b 2 !.!Ciı/ ! 2 !02 C2iı!

2

D ! 2  !12 C 2iı!: (12.188)

This particular form of the saddle point equation explicitly displays the desired limiting behavior as  ! 1, because in that limit, the right-hand side of this equation must approach zero, so that q lim !SP ˙ ./ D ˙ !12  ı 2  iı:

!1

(12.189)

d

With this limiting behavior in mind, the rational function appearing in the squared term of (12.188) may be approximated as    2  ı b 2 !.! C iı/ 2 1 C iı=! 2

b C O ! 1  i ; (12.190)

b 1 C 2iı=! ! ! 2  !02 C 2iı! provided that j!j  !0 and j!j  ı . As a first approximation [for the purpose of comparison with Brillouin’s first approximation given in (12.187)], let the above expression be approximated by the first term on the right-hand side of (12.190). In that case, the saddle point equation (12.188) becomes 2  1   ! 2  !02 C 2iı! ! 2  !12 C 2iı!;  2 ! 2  !02 C 2iı!

12.3 The Location of the Saddle Points and the Approximation of the Phase

with solution

321

r

b2 2  iı; (12.191) d 2  1 for   1, which is to be compared with the expression given in (12.187). Although the distant saddle points SPd˙ now lie along the line ! D iı in this first-order approximation, residing at ˙1  iı at  D 1, they do approach the respective outer 0 as  ! 1, in agreement with the exact result [see (12.181)]. branch points !˙ Consider now obtaining the second-order approximation of the distant saddle point locations, in which case (12.190) is approximated as !SP ˙ . / ˙ !02  ı 2 C

ıb 2 b 2 !.! C iı/ 2 :

b  i ! ! 2  !02 C 2iı! With this substitution, the saddle point equation given in (12.179) becomes  2 2  ı2b4 ıb 2  2 !  !02 C 2iı!  2i !  !02 C 2iı!  2 !   !  D  2 ! 2  !02 C 2iı! ! 2  !12 C 2iı! : The term ı 2 b 4 =! 2 may be neglected in comparison to the other two terms on the left-hand side of this equation with the result   ıb 2 b2 2 ! C 2i 2

0: ! 3 C 2iı! 2  !02 C 2  1  1

(12.192)

The zeros of this cubic equation can be obtained by first determining the form of its reduced equation as follows: Define the coefficients a1 2iı;   b2 2 ; b1  !02 C 2  1 ıb 2 c1 2i 2 :  1 Then, under the change of variable ! 

2 a1 D   iı; 3 3

(12.193)

the cubic equation given in (12.192) is reduced to the form  3 C a2  C b2 0;

(12.194)

322

12 Analysis of the Phase Function and Its Saddle Points

where 1 b2 2 4 a2 b1  a12 D !02  2 C ı2; 3  1 3   2 C 3 2 3 1 2 8 a1  a1 b1 C c1 D iı !02  ı 2 C b 2 2 : b2 27 3 3 9  1 To construct the solutions to this reduced cubic equation, let 0

b2 A˙ @ ˙ 2

s

11=3 b22 a23 A ; C 4 27

where s "  2  2 2  2  2 2 2  2  a23 b22 i 4 2 2 3!0  2ı !0  C 2ı 9!0  8ı C D p !0 !0  ı C b 4 27 2  1 3 3 #1=2  2  4  2  2 2 b6 6 4 3!0  ı  C 9ı 2 C 3 Cb C : . 2  1/2 . 2  1/3 Notice that the algebraic expression appearing under the square root operation in the above expression is positive for all   1. If one then defines the real valued quantities   2 ı 8 2 2 2 C 3 !0  ı C b 2 ; (12.195) ˇ1 3 9  1 "  2  2 2  2  2 2 2  2  1 4 2 2 3!0  2ı !0  C 2ı 9!0  8ı ˇ2 p !0 !0  ı C b 2  1 3 3 #1=2  2  4  2  2 2 6 6 3!  2  ı C 9ı C 3  b 0 Cb 4 C ; . 2  1/2 . 2  1/3 (12.196) one finds that A˙ D i .ˇ1 ˇ2 /1=3 ;

(12.197)

where the principal branch 0  arg .A˙ / < 2 has been chosen. The three solutions of the reduced cubic equation given p in (12.194) are then given by  D AC C A and ˙ D .AC CA /=2˙i.AC A / 3=2. Because there are only two distant saddle points which are symmetrically situated about the imaginary axis, it is seen that the last two solutions are of the proper form. Thus, the two sought-after solutions are given by

12.3 The Location of the Saddle Points and the Approximation of the Phase

323

p ˙ D ˙

 3

.ˇ1 C ˇ2 /1=3  .ˇ1  ˇ2 /1=3 2  i

 .ˇ1 C ˇ2 /1=3 C .ˇ1  ˇ2 /1=3 : 2

(12.198)

Substitution of this solution into (12.193) then results in the approximate expression for the distant saddle points locations p  3

!SP ˙ . / ˙ .ˇ1 C ˇ2 /1=3  .ˇ1  ˇ2 /1=3 d 2    2 1 1=3 1=3 .ˇ1 C ˇ2 / C .ˇ1  ˇ2 / : (12.199) i ıC 3 2 This complicated expression is rather formidable to work with, however, and a more simplified expression that possesses greater accuracy than either of the two first-order approximations given in (12.187) and (12.191) is desired. Because ˇ2  ˇ1 for all   1, the following two approximations can then be made:   ˇ1 1=3 ˇ1 1=3 1C

ˇ2 C 2=3 ; ˇ2 3ˇ2  1=3 ˇ1 1=3 ˇ1 1=3 D ˇ2 1

ˇ2 C 2=3 : ˇ2 3ˇ 1=3

.ˇ1 C ˇ2 /1=3 D ˇ2 .ˇ1  ˇ2 /1=3

2

With substitution of these two approximations, (12.199) for the distant saddle point locations becomes ! p 1=3 ˇ1 2 ı C 2=3 : !SP ˙ . / ˙ 3ˇ2  i (12.200) d 3 3ˇ 2

1=3

Finally, the quantity ˇ2 p

may be approximated as

#1=2  2  4   ! ! b2 2 1 2 0 C !02  ı 2 C !02  ı 2

C 04 !02  ı 2 2  1 3 3 b2 3b r b2 2 ;

!02  ı 2 C 2  1 "

1=3 3ˇ2

to a fair degree of approximation [cf. (12.191)]. In addition, with this result one finds that # " ˇ1 ı 3b 2 

: 1C  2 2=3 3 !0  ı 2 . 2  1/ C b 2  2 3ˇ2

324

12 Analysis of the Phase Function and Its Saddle Points

The distant saddle point locations may then be expressed as   !SP ˙ . / D ˙./  iı 1 C . /

(12.201)

d

with the second approximate expressions3 r

b2 2 ; 2  1 b2  ./  2 : !0  ı 2 . 2  1/ C b 2  2 ./

!02  ı 2 C

(12.202) (12.203)

  Notice that ./ D b 2 = . 2  1/ 2 ./ to this order of approximation. The expressions given in (12.201)–(12.203) then comprise the second approximation of the distant saddle point locations. For values of  close to unity, the above expressions simplify to ./ ! p

b 2.  1/

;

. / ! 1; so that the second approximation reduces to the first approximation [cf. (12.187)] in this limit. In particular, in the limit as  approaches unity from above lim !SP ˙ ./ D ˙1  2iı:

!1C

(12.204)

d

On the other hand, for sufficiently large values of  the second approximate expressions given in (12.202) and (12.203) become q ./ ! !12  ı 2 ; . / ! 0; so that in the limit as  approaches infinity q 0 lim !SP ˙ ./ D ˙ !12  ı 2  iı D !˙ ; !1

(12.205)

d

and the distant saddle points SPd˙ respectively approach the outer branch points 0 . The second approximation to the distant saddle point locations then captures !˙ the exact limiting behavior in the two opposite extremes at either  D 1 or  D 1. A sketch of the respective paths followed by these two distant saddle points in the complex !-plane is presented in Fig. 12.39. 3

Notice that the second approximate expression for . / that is used here is slightly modified from that given in earlier publications [4, 6, 7].

12.3 The Location of the Saddle Points and the Approximation of the Phase

325

''

' ' cut −

' cut +

SPd

SPd

Fig. 12.39 Depiction of the behavior of the distant saddle points SPd˙ in the complex !-plane for a single-resonance Lorentz model dielectric. The dotted curves indicate the respective directed paths that these saddle points follow as  increases to infinity. The dashed lines through each saddle point indicate the local behavior of the isotimic contour  .!;  / D  .!SP ˙ ;  / through d that saddle point, the shaded region indicating the local region about each saddle point where the inequality  .!;  / <  .!SP ˙ ;  / is satisfied, the vectors indicating the local direction of ascent d along the lines of steepest descent and ascent through each saddle point

An analytic approximation of the complex phase behavior of the phase function .!;  / that is valid in the region of the complex !-plane traversed by the distant saddle points as  varies from unity to infinity is now considered. For this analysis, the complex index of refraction that is given by the first approximate expression in (12.184) is sufficiently accurate so that the complex phase behavior in the region j!j  !1 of the complex !-plane that is removed from the origin may be approximated as b2 : (12.206) .!;  / i !.1  /  i 2.! C 2iı/ To obtain the approximate behavior of  .!;  /
  .r; '; / .1   / i./ C ı 1 C . / C irei'   b2 ./iı 1./ Cre i'     : i 2  2 ./Cı2 1./ C2./r cos 'C2ı 1./ r sin 'Cr 2 (12.207)

326

12 Analysis of the Phase Function and Its Saddle Points

The real part of this equation then gives

    .r; ';  / .1   / ı 1 C . /  r sin '   b2 ı 1./ Cr sin '      ; 2  2 ./Cı2 1./ C2./r cos 'C2ı 1./ r sin 'Cr 2 (12.208) from which it is seen that  .r; ';  / attains its maximum variation with r about the distant saddle point SPdC when ' D =4; 3=4; 5=4; 7=4. Consequently, in the right-half plane the lines of steepest descent through the distant saddle point SPdC are at the angles ' D 3=4 and ' D 7=4, and the lines of steepest ascent are at ' D =4 and ' D 5=4. Because of the even symmetry of  .!;  / about the imaginary axis, the reverse holds true for the distant saddle point SPd in the left-half of the complex !-plane. This local behavior about the distant saddle points SPd˙ is depicted in Fig. 12.39, where the vectors indicate the direction of ascent along the lines of steepest descent and ascent through each respective distant saddle point, which are at angles of 45ı to the coordinate axes, and the shaded areas indicate the local regions about each saddle point wherein the inequality  .!;  / <  .!SP ˙ ;  / is d satisfied and in which the path of steepest descent from the respective saddle point lies. These general results are in complete agreement with the numerical results presented in Figs. 12.4–12.9. 12.3.1.2

The Region About the Origin (j!j  !0 )

The First Approximation Again, to permit comparison with the classical asymptotic theory due to Brillouin [2, 3], a critical review of this first approximation is considered first. To determine the saddle point locations in the region of the complex !-plane about the origin, the complex index of refraction n.!/ is first expanded in an ascending power series in !. Let the complex index of refraction given in (12.57) be expressed in the form  n.!/ D

! 2  !12 C 2iı! ! 2  !02 C 2iı!

1=2

 D

!12  " !02  "

1=2 ;

(12.209)

where " ! 2 C 2iı!. Because j"j is small in comparison to both !02 and !12 D !02 C b 2 , one then has  1=2 !1 1  "=!12 n.!/ D !0 1  "=!02  2 1=2 b2 !1 2 b 1C" 2 2 C" 2 4

!0 !1 !0 !1 !0  2 ! 2 2 2 !1 b 2 b 4!1  b

1C" 2 2 C" : !0 2!1 !0 8!12 !04

12.3 The Location of the Saddle Points and the Approximation of the Phase

327

Substitution of the expression " ! 2 C 2iı! then gives the result   ı 2 b 2 4!12  b 2 2 !1 b2 n.!/

C !.! C 2iı/  ! ; !0 2!1 !03 2!13 !05

(12.210)

which is correct to O.! 2 /, the neglected terms being of order O.! 3 /. Notice that this result is also correct to O.b 2 /, the neglected terms being of order O.b 4 /. Differentiation of this approximate expression with respect to ! then gives   ı 2 b 2 4!12  b 2 b2 .! C iı/  !: n .!/

!1 !03 !13 !05 0

Therefore, within the constraints of the above approximations, the locations of the near saddle points are given by the zeros of the approximate saddle point equation [cf. (12.23)] 2!03 !1 4ı .  0 / 0; (12.211) !2 C i ! C 3˛ 3˛b 2 where the defined parameters [cf. (12.21)] !1 0 n.0/ D D !0 ˛ 1  ı2

s 1C

b2 ; !02

4!12  b 2 ; !02 !12

(12.212) (12.213)

depend only upon the Lorentz model medium parameters. The roots of this approximate quadratic saddle point equation then give the first approximation of the near saddle point locations as [cf. (12.24)–(12.25)] 1 !SP˙ . / ˙ n 3

s 6

0 !04 ı2 2ı .   /  4 i ; 0 2 2 ˛b ˛ 3˛

(12.214)

for   1. This expression reduces to Brillouin’s result [2, 3] when ˛ is approximated by unity. To analyze the behavior of the near saddle point dynamics as given by the above first approximation, it is necessary first to determine the algebraic sign of the argument of the square root appearing in (12.214). The value of  D 1 at which this argument vanishes is given by [cf. (12.26)] 1 0 C

2ı 2 b 2 : 3˛0 !04

(12.215)

Although this result is exact for the expression given in (12.214), it is only a firstorder approximation of the result that would be obtained from the exact saddle

328

12 Analysis of the Phase Function and Its Saddle Points

point equation given in (12.182). According to this first approximation as given by (12.214), the two near saddle points SP˙ n lie along the imaginary axis, symmetrically situated about the point ! D i 2ı=.3˛/ for  2 Œ1; 1 /, approaching each other as  increases. These two first-order saddle points then coalesce into a single second-order saddle point SPn at this symmetry point ! D i 2ı=.3˛/ when  D 1 . Finally, as  increases above 1 , the two first-order near saddle points SP˙ n move off of the imaginary axis and into the lower-half of the complex !-plane along the line ! 00 D 2ı=.3˛/, remaining symmetrically situated about the imaginary axis for all  > 1 , approaching ˙1  i 2ı=.3˛/ as  ! 1. However, when  > 1 becomes sufficiently large such that the inequality j"j  !02 is no longer satisfied, then this first approximation loses its validity. An estimate of the space–time value at which this occurs is obtained from (12.214) as  0 C 3b 2 =.20 !02 /, which is not too different from 0 for small values of the number density N . A more accurate description that is valid over the entire space–time domain  2 Œ1; 1/ is then seen to be desirable.

The Second Approximation In order to obtain a more accurate description of the near saddle point locations, particularly for  > 1 , the exact saddle point equation given in (12.179) is again employed. This saddle point equation may be rewritten in the form    2 ! 2  !02 C 2iı! D ! 2  !12 C 2iı! C 2b 2

!.! C iı/ ! 2  !02 C 2iı!

! 2 .! C iı/2 Cb 4   2 : ! 2  !12 C 2iı! ! 2  !02 C 2iı! (12.216) For j!j small in comparison to !0 , the two expansions !.! C iı/ 1 ! 2 C iı! D  2 ! 2  !0 C 2iı! !02 1  i 2ı!2  ! 22 !0 !0       ı2 ı2 1 ı

 2 iı! C 1  2 2 ! 2 C i 2 3  4 2 ! 3 ; !0 !0 !0 !0 and   ! 2 ! 2 C 2iı!  ı 2 ! 2 .! C iı/2     2

2iı!02 2!12 C !02 !  !12 !04 ! 2  !12 C 2iı! ! 2  !02 C 2iı!    !2 2ı 3 

2 4 ı 2  2iı! C i 2 2 2!12 C !02 ! ; !1 !0 !1 !0

12.3 The Location of the Saddle Points and the Approximation of the Phase

329

are useful. With substitution of these approximations, the exact saddle point equation given in (12.216) assumes the approximate polynomial form      b2 ıb 2 ı2 ı2b2   2 ! 2  !02 C 2iı! 2i 4 3 C 2  4 2  4 2 2!12 C !02 ! 3 !0 !1 !0 !1 !0   2  2 ı b ı2b2 !2 C 1 2 24 2  2 2 !0 !0 !1 !0   b2 C2iı 1  2 !  !02  b 2 : !0 Because the coefficient of the cubic term in ! is small in comparison to the other terms appearing in this polynomial expression, it may be neglected. The approximate saddle point equation for the near saddle point locations then becomes 2

2

! C 2iı

 2  02 C 2 !b 2 0

!

2

 2  02 C 3˛ !b 2 0

  !02  2  02 2

 2  02 C 3˛ !b 2

0;

(12.217)

0

where 0 is as defined in (12.212) and where the parameter ˛ has been redefined in this second-order approximation as [cf. (12.213)] ˛ 1

 ı2  2 4!1 C b 2 : 2 2 3!0 !1

(12.218)

Notice that for values of  very close to 0 , the coefficients appearing in the second approximation of the saddle point equation given in (12.217) reduce to those appearing in the first approximation given in (12.211). This then shows that the first approximation of the near saddle point locations is valid only in the immediate space–time region about the value 0 . The near saddle point locations may then be expressed as 2 !SP˙ ./ D ˙ ./  iı ./ n 3

(12.219)

with the second approximate expressions 2 6 . / 4

!02 2



 2    02 02

C

2 3˛ !b 2 0

0  ı2 @

2

 2  02 C 2 !b 2 0

2



02

C

2 3˛ !b 2 0

12 31=2 A 7 5 ;

(12.220)

2

b 2 2 3   0 C 2 !02 :

. /

2  2  02 C 3˛ b 22

(12.221)

!0

The expressions given in (12.219)–(12.221) then comprise the second approximation of the near saddle point locations.

330

12 Analysis of the Phase Function and Its Saddle Points

For values of  close to 0 , the above expressions simplify to s 0 ! 4 1 ı2 . / ! 6 20  4 2 ; 3 ˛b ˛ 1

. / ! ; ˛ so that the second approximation reduces to the first approximation [cf. (12.214)]. On the other hand, in the limit as  approaches infinity q lim !SP˙ ./ D ˙ !02  ı 2  iı D !˙

!1

n

(12.222)

and the near saddle points SP˙ n , respectively, approach the inner branch points !˙ , in agreement with the numerical results presented in Sect. 12.2.1. To analyze the behavior of the near saddle points as described by this second approximation, it is again necessary to first determine the algebraic sign of the argument of the square root in (12.220). This amounts to determining a more accurate value of the critical space–time point  D 1 at which the two near first-order saddle points coalesce into a single second-order saddle point, where 0

  !02 12  02 2

12  02 C 3˛ !b 2 0

 ı2 @

2

12  02 C 2 !b 2 0

2

12  02 C 3˛ !b 2

12 A 0;

0

in this second-order approximation, which simplifies to    2  2  ı2  ı2b4 !0  ı 2 12  02 C b 2 3˛  4 2 12  02  4 4 0: !0 !0 Because 1 is greater then 0 for positive-definite values of the phenomenological damping constant ı [see (12.26)], the appropriate solution of this binomial equation gives v 3 2v u   u u 2 2 ! 2  ı2 2 u ı 3˛!  4ı u 2 0  4t1 C 16  1 t0 C b 2 2  0 2 2  15; 2!0 !0  ı 2 3˛!02  4ı 2

(12.223)

where the positive values of both square roots appearing in this expression are to be taken. To compare this second approximation to the value of the critical space–time point 1 with that given in the first approximation by (12.215), the square of the above expression may be approximated as

12.3 The Location of the Saddle Points and the Approximation of the Phase

331

#   2 2 ı 2 !02  ı 2 2 2 2 3˛!0  4ı   1 C 8 1 0 C b 2  1 2!02 !02  ı 2 3˛!02  4ı 2 "

D 02 C

4ı 2 b 2  ; !02 3˛!02  4ı 2

(12.224)

so that 1 0 C

 2

!0

2ı 2 b 2 ; 3˛!02  4ı 2

(12.225)

which reduces to the first approximate expression given in (12.215) through neglect of the term 4ı 2 in comparison to 3˛!02 in the denominator. Because of its simplicity, the approximate expression given in (12.224) for 12 is used in subsequent calculations concerning the behavior of the near saddle points at that critical space–time value. An analytic approximation of the complex phase behavior .!;  / that is valid in the region of the complex !-plane traversed by the near saddle points as  varies from unity to infinity is now considered. For this analysis, the complex index of refraction that is given by the first approximate expression in (12.210) is sufficiently accurate. With (12.213), this approximate expression may be written as n.!/ 0 C

b2 !.˛! C 2iı/; 20 !04

(12.226)

so that the complex phase behavior in the region j!j  !0 of the complex !-plane may be approximated as .!;  / i ! .0  / C

b2 ! 2 .i˛!  2ı/: 20 !04

(12.227)

the dynamical behavior of the near saddle points and the local complex phase behavior about them, as described by this second approximation, is now considered for the three separate cases 1   < 1 ,  D 1 , and  > 1 . Case 1 (1   < 1 ) Over this initial space–time domain the near saddle point locations are given by  !SP˙ ./ D i ˙ n

2 o ./  ı ./ 3

 (12.228)

332

12 Analysis of the Phase Function and Its Saddle Points

with the second approximate expressions 31=2  2  2   0 7 6 2@ 0 A  ; (12.229) o . / 4ı 2 5 b2 2 2 2   0 C 3˛ ! 2   02 C 3˛ !b 2 2

0

2

 2  02 C 2 !b 2

12

!02

0

2

. /

02

0

2 2 !b 2 0

3   C ; 2  2  02 C 3˛ b 22

(12.230)

!0

that are appropriate over this domain. As depicted in Fig. 12.40, the near saddle points SP˙ n are located along the imaginary axis, symmetrically situated about the point ! 00 D  23 ı ./, where . / varies slowly over this space–time domain. To obtain the approximate local behavior of  .!;  /
D i ! 00 ./ C re i' ;

''

+

SPn

'

−

SPn

Fig. 12.40 Depiction of the behavior about the near saddle points SP˙ n situated along the imaginary axis of the complex !-plane for a single resonance Lorentz model dielectric over the initial space–time domain 1   < 1 . The dashed lines through each saddle point indicate the local behavior of the isotimic contour  .!;  / D  .!SP˙ ;  / through that saddle point, the shaded region n indicating the local region about each saddle point where the inequality  .!;  / <  .!SP˙ ;  / is n satisfied, the vectors indicating the local direction of ascent along the lines of steepest descent and ascent through each saddle point

12.3 The Location of the Saddle Points and the Approximation of the Phase

333

where 2 ! 00 D ˙ ./  ı ./: 3 With this substitution in (12.227), the approximate phase behavior in the vicinity of the near saddle points is found to be given by   .r; '; / ! 00 C i re i' .0  /   b 2 003 C ˛! C 2ı! 002  i 3˛! 002 C 4ı! 00 re i' 20 !04

 .3˛! 00 C 2ı/r 2 e i2' C i ˛r 3 e i3' : (12.231)

The real part of this equation then yields    .r; ';  / ! 00 C r sin ' .  0 /   b 2 003 C ˛! C 2ı! 002 C 3˛! 002 C 4ı! 00 r sin ' 4 20 !0

 .3˛! 00 C 2ı/r 2 cos 2'  ˛r 3 sin 3' ; (12.232)

from which it is seen that  .r; ';  / attains its maximum variation about each saddle point when ' D 0; =2; ; 3=2. The lines of steepest descent and ascent through the near saddle points are then parallel to the coordinate axes, as depicted in Fig. 12.40, where the vectors indicate the direction of ascent along these lines. The dashed lines through each saddle point indicate the local behavior of the isotimic contour  .!;  / D  .!SP˙ ;  / through that saddle point, and the n shaded region indicates the local region about each saddle point where the inequality  .!;  / <  .!SP˙ ; / is satisfied. Finally, a further consideration of (12.232) n

shows that the paths of steepest descent through the upper near saddle point SPC n are at the angles ' D 0; , whereas the paths of steepest descent through the lower near saddle point SP n are at the angles ' D =2; 2=2, as indicated in the figure. At the space–time point  D 0 (where 1 < 0 < 1 ), the near saddle points are located at !SPC .0 / D 0; n !SP .0 / i n

(12.233) 4ı ; 3˛

(12.234)

where the solution for the upper near saddle point at this special  -value is exact, as indicated by the equal sign in (12.233). Furthermore .!SPC .0 /; 0 / D  0 .!SPC .0 /; 0 / D 0 n n

(12.235)

334

12 Analysis of the Phase Function and Its Saddle Points

exactly. It is this latter property that makes this saddle point so important in the subsequent asymptotic field behavior. Case 2 ( D 1 ) At the critical space–time point  D 1 , (12.219)–(12.221) yield a single near saddle point located along the negative imaginary axis at 2 2ı !SPn .1 / D  iı .1 /  i; 3 3˛

(12.236)

as depicted in Fig. 12.41. Both the first and second derivatives of the complex phase function vanish at this saddle point, namely  0 .!SPn .1 /; 1 / D  00 .!SPn .1 /; 1 / D 0;

(12.237)

and the two first-order near saddle points have coalesced into a single second-order saddle point. From (12.232) with ! 00 D 2ı=.3˛/, the local behavior of  .!; 1 / about this second-order near saddle point is found to be given by

''

'

SP n

SP n

q

i d

a

Fig. 12.41 Depiction of the behavior about the near saddle point SPn in the complex !-plane for a single-resonance Lorentz model dielectric at the critical space–time point  D 1 when the two first-order saddle points SP˙ n have coalesced into a single second-order saddle point. The dashed lines indicate the local behavior of the isotimic contour  .!;  / D  .!SPn ;  / through this second-order saddle point, the shaded area indicating the local region about the saddle point where the inequality  .!;  / <  .!SPn ;  / is satisfied, the vectors indicating the local direction of ascent along the lines of steepest descent and ascent through the saddle point

12.3 The Location of the Saddle Points and the Approximation of the Phase

 .r; '; 1 / 

b2 0 !04



335



4ı 3 ˛ C r 3 sin 3' : 2 27˛ 2

(12.238)

Hence,  .r; '; 1 / attains its maximum variation about this second-order near saddle point when ' D =6; =2; 5=6; 7=6; 3=2; 11=6. The lines of steepest descent from this saddle point are at ' D =6; 5=6; 3=2, and the lines of steepest ascent are at ' D =2; 7=6; 11=6, as depicted in Fig. 12.41. Case 3 ( > 1 ) Over this final space–time domain the near saddle point locations are given by 2 !SP˙ ./ D ˙ ./  iı . /; n 3

(12.239)

where . / and . / are both real-valued and are given by (12.220) and (12.221), respectively, in the second approximation. Thus, as  increases away from 1 , the two near first-order saddle points move off of the imaginary axis into the lowerhalf of the complex !-plane, symmetrically situated about the imaginary axis, as depicted in Fig. 12.42. In the limit as  ! 1, these two near saddle points SP˙ n approach the inner branch points !˙ , respectively.

''

'

_ − SPn

'

cut

SPn+ cut

'

Fig. 12.42 Depiction of the behavior of the near saddle points SP˙ n in the complex !-plane for a single-resonance Lorentz model dielectric over the final space–time domain  > 1 . The dotted curves indicate the respective directed paths that these first-order saddle points follow as  increases to infinity. The dashed lines through each saddle point indicate the local behavior of the isotimic contour  .!;  / D  .!SP˙ ;  / through that saddle point, the shaded region indicating n the local region about each saddle point where the inequality  .!;  / <  .!SP˙ ;  / is satisfied, n the vectors indicating the local direction of ascent along the lines of steepest descent and ascent through each saddle point

336

12 Analysis of the Phase Function and Its Saddle Points

To obtain the approximate behavior of the real phase function  .!;  / in the local vicinity of these near saddle points, ! is again expressed in polar coordinates .r; '/ about the specific saddle point. Because  .!;  / is symmetric about the imaginary axis, only the behavior about the near saddle point SPC n in the right-half of the complex !-plane needs to be considered. Hence, let ./ C re i' ! D !SPC n D

2 ./  i ı ./ C re i' : 3

With substitution of this expression into (12.227), the approximate complex phase behavior about the near saddle point SPC n for  > 1 is found to be given by   2 .r; '; /

ı ./ C i ./ C irei' .0   / 3 (   2   ˛ 8 3 2 b

. / 1  C ı

. / C 2ı 2 ./ ˛ . /  1 4 9 3 20 !0     4 2 3 ı . / . / 2  ˛ . / C ˛ ./ Ci 3        4 2 2 C 4ı . / ˛ . /  1 C i 3˛ ./ C ı . / 2  ˛ . / re i' 3 )

   2 i2' 3 i3' C 2ı ˛ . /  1 C 3i ˛ ./ r e C i ˛r e : (12.240) The real part of this equation then yields the result   2  .r; ';  / r sin '  ı ./ .0  / 3 (     ˛ 8 3 2 b2 ı . / 1  . / C 2ı 2 ./ ˛ . /  1 C 4 9 3 20 !0       4 C4ı . / ˛ . /  1 r cos '  3˛ 2 ./ C ı 2 . / 2  ˛ . / r sin ' 3 )   2 2 3 C2ı ˛ . /  1 r cos 2'  3˛ ./r sin 2'  ˛r sin 3' ; (12.241) from which it is seen that  .r; ';  / attains its maximum variation about the near saddle point SPC n when ' D =4; 3=4; 5=4; 7=4. The lines of steepest descent through this saddle point are at ' D =4; 7=4 and the lines of steepest ascent are at ' D 3=4; 7=4. Because of the even symmetry of  .!;  / about the imaginary axis, the lines of steepest ascent and descent through the near saddle point SP n are reversed, as illustrated in Fig. 12.42.

12.3 The Location of the Saddle Points and the Approximation of the Phase

12.3.1.3

337

Determination of the Dominant Saddle Points

The asymptotic description of dispersive pulse dynamics in a given medium relies upon the determination of the saddle point (or points) that give the least exponential decay as the propagation distance z ! 1. Such a saddle point SP at which  .!SP ; /
where the second approximate expressions of the functions ./ and . / are given in (12.202) and (12.203), respectively. From (12.208), the real part of the complex phase behavior at this saddle point is described by the approximate expression    ı 1  . / b2  .!SP ˙ ; / ı 1 C . / .  1/    ; d 2  2 . / C ı 2 1  . / 2 

(12.243)

for all   1. Notice that this expression is valid at both the distant saddle points SPd˙ , as indicated. For completeness, the imaginary part of the complex phase behavior at the distant saddle points SPd˙ is given by "

# b 2 =2 $ .!SP ˙ ; / ./   1 C  2 ; d  2 ./ C ı 2 1  . /

(12.244)

for all   1. It is then seen that at the luminal space–time point  D 1,  .!SP ˙ ; 1/ D 0 (exactly), and that as  increases away from unity,  .!SP ˙ ;  / d d is negative with monotonically increasing magnitude, where lim!1  .!SP ˙ ;  / d

0 D 1 as the distant saddle points SPd˙ approach the outer branch points !˙ , respectively [see Figs. 12.3(b) and 12.39]. Consider next the near saddle point behavior in the right-half plane, given by [from (12.228) and (12.239)]

 !SP˙ . / D i ˙ n

!SPC . / D n

o ./

 2  ı ./ I 3

2 ./  i ı ./I 3

1    1 ;

  1 ;

(12.245) (12.246)

338

12 Analysis of the Phase Function and Its Saddle Points

where the second approximate expressions of the functions . / and . / are given in (12.220) and (12.221), respectively, and where o . / is given in (12.229). From (12.232), (12.238), and (12.241), the real part of the complex phase behavior at these saddle points is described by the set of approximate expressions   .!SP˙ ; / ˙ n

 2 o . /  ı ./ .  0 / 3 2  

 2 2 2 b ı ./ ı . / C 2ı I ˙ ˛ ˙ ./  . /  C o o 3 3 20 !04 1    1 ; (12.247)

 .!SP˙ ; 1 /  n

4ı 3 b 2 I 27˛ 2 0 !04

 D 1 ;

(12.248)

2  .!SP˙ ; /  ı . /.  0 / n 3

i h

 8 3 2 ˛ b2 2 ı ./ 1  . / C 2ı ./ ˛ . /  1 I C 3 20 !04 9 (12.249)   1 : Notice that (12.249) is valid at the both near saddle points SP˙ n , as indicated. For completeness, the imaginary part of the complex phase behavior at the near saddle points SP˙ n is seen to identically vanish over the space–time domain 1    1 , and is given by (

b2 $ .!SP˙ ; / . /   0  n 20 !04



  4 2 ı . / 2  ˛ . / C ˛ 3

2

) . / (12.250)

for   1 . These expressions then show that at the upper near saddle point SPC n , ; / is initially negative over the space–time domain  2 Œ1;  /, increasing  .!SPC 0 n to zero as  increases to 0 , identically vanishes at  D 0 , and then grows negative monotonically over the short space–time interval  2 .0 ; 1 . At the lower  near saddle point SP n ,  .!SPn ;  / is initially positive and monotonically decreases to the approximate value 4ı 3 b 2 =.27˛ 2 0 !04 / as  increases to 1 . Because the original contour of integration is not deformable into an Olver-type path through this lower near saddle point over this space–time domain (see Fig. 12.40), this saddle point is not dominant and may then be ignored in the present analysis. Finally, for   1 , during which the two near first-order saddle points first emerge from their coalescence as a single second-order saddle point at  D 1 and then move off symmetrically into the complex !-plane, the quantity  .!SP˙ ;  / continues to n decrease monotonically with increasing  , where lim!1  .!SP˙ ;  / D 1 as n

1 0

339

SB

-

SP

12.3 The Location of the Saddle Points and the Approximation of the Phase

+ SPn−

+ SPd−

+ SPn

Fig. 12.43 Sketch of the behavior of the real part  .!;  / of the complex phase function .!;  / at the relevant saddle points for a single-resonance Lorentz model dielectric

the near saddle points SP˙ n approach the inner branch points !˙ , respectively [see Figs. 12.3(b) and 12.42]. It is then seen that the distant saddle points SPd˙ are at first dominant over the upper near saddle point SPC n , but that for some critical space–time value  D SB between unity and 0 the upper near saddle point SPC n becomes dominant and remains dominant for all later values of  , as depicted in Fig. 12.43. This critical space–time value is then defined by the expression ; SB /;  .!SP ˙ ; SB /  .!SPC n

(12.251)

d

where 1 < SB < 0 . If the second approximate expressions for  .!SP ;  / given by (12.243) for the distant saddle points SPd˙ and (12.247) for the upper near saddle point SPC n are substituted in (12.251), the resulting algebraic expression for SB is intractable to solve. To obtain a manageable expression for SB , the corresponding first approximate expressions are used. Comparison with numerical results shows this to be a sufficiently accurate approximation, at least for Brillouin’s choice of the medium parameters. If a more accurate result is desired, a numerical solution may always be employed. From (12.242), the first approximate real phase behavior at the distant saddle points is given by  .!SP ˙ ;  / 2ı.  1/ d

(12.252)

340

12 Analysis of the Phase Function and Its Saddle Points

for  2 Œ1; 0 /. From (12.247), the first approximate behavior at the upper near saddle point SPC n is given by (with ˛ 1) 2 ; /   .!SPC n 27

(



s

0 !04 .0   / b2 ) ı3b2 9ı.0   /  4 ; (12.253) 0 !04

2ı 2 b 2 C 3.0  / 0 !04

4ı 2 C 6

for  2 Œ1; 0 /. These two first-order approximate expressions are then equated at  D SB , with the result [4, 7] 

2ı 2 b 2 C 3.0  SB / 0 !04



s

0 !04 .0  SB / b2

18ı.0  SB / C 27ı.0  1/;

4ı 2 C 6

where the last term on the right in (12.252) has been neglected because it is small in comparison to the other terms. Upon squaring both sides of this equation and neglecting terms of order Ofı 4 g, the cubic equation .0  SB /3  4

ı2b2 .0  SB /2 0 !04

C18

ı2b2 ı2b2 .  1/.0  SB /  27 .0  1/2 0 4 0 0 !0 20 !04

in the quantity .0  SB / is obtained. Under the change of variable SB D 0 

4ı 2 b 2  ; 30 !04

(12.254)

one obtains the reduced cubic equation  3 C a C c D 0

(12.255)

with coefficients a 18

ı2b2 .0  1/; 0 !04

c 27

ı2b2 .0  1/2 ; 20 !04

where terms of order Ofı 4 g have been neglected. To obtain the solution of this reduced cubic equation, let

12.3 The Location of the Saddle Points and the Approximation of the Phase

341

"

#1=3 r a3 c2 c C A  C 2 4 27 s " !# 2b2 ı2b2 64ı

27 .0  1/2 1C C1 ; 40 !04 270 .0  1/!04 " #1=3 r a3 c2 c C B   2 4 27 s " !# ı2b2 64ı 2 b 2 2

 27 .0  1/ 1C 1 ; 40 !04 270 .0  1/!04 where the branch of each cube root is chosen so that both A and B are real. The real-valued solution of the reduced cubic equation given in (12.255) is then given by  D A C B, and hence, from (12.254), the critical space–time value SB is given by the rather complicated expression SB 0 

4ı 2 b 2 30 !04



ı2b2 3 .0  1/2 40 !04

1=3 " s 1C

!1=3 64ı 2 b 2 C1 270 .0  1/!04 s !1=3 # 64ı 2 b 2  1C 1 ; 270 .0  1/!04 (12.256)

which may be simplified further to the form SB 0 

1=3  2 2 4ı 2 b 2 ı b 2  3 .  1/ ; 0 30 !04 20 !04

(12.257)

2  provided that ı 2 b 2 =!04  1. A useful, simple estimate of the value of SB is provided by the first two terms on the right-hand side of (12.257). q A related quantity of interest is the value of the real angular frequency !SB  !12  ı 2 that is defined by the relation

 .!SP C ; SB /  .!SB /;

(12.258)

d

where  .! 0 / D ! 0 ni .! 0 / along the real axis [see (12.76)]. The importance of this value is realized in the subsequent asymptotic description of dispersive pulse dynamics. According to the definition given in (12.258), the angular frequency !SB is the real coordinate value at which the isotimic contour  .!;  / D  .!SP C ;  / d

342

12 Analysis of the Phase Function and Its Saddle Points

through the distant saddle point SPdC crosses the positive ! 0 -axis when  D SB . Unfortunately, the solution of (12.258) for !SB is an extremely formidable, if not impossible, task. However. because the isotimic contour through SPdC at the angle =2 to the ! 0 -axis remains at essentially this angle when it intersects the ! 0 -axis, as seen in Fig. 12.6, an approximate expression for !SB is given by the real coordinate value of the distant saddle point SPdC at  D SB . Accordingly, substitution of the leading two-term approximation of (12.257) into (12.202) yields s !SB Š .SB / D s

!0

2C

!02  ı 2 C b 2

2 SB 2 SB 1

b2 5ı 2 C ; 2 !0 3!02

(12.259) (12.260)

where terms in ı 4 =!04 and higher in the radical have p been neglected in obtaining the final approximate expression. Notice that !SB 2!0 C b 2 =.4!0 / when !0 > b  ı, and that !SB b C !02 =b when b > !0  ı. Hence, when considered as a function of the number density N p of Lorentz oscillators comprising the material, !SB is seen to be bounded below by 2!0 in the weak dispersion limit and bounded above by the plasma frequency b.

12.3.1.4

Comparison with Numerical Results

A numerical determination of the exact saddle point locations and the exact behavior of the real and imaginary parts of the complex phase function .!;  / D  .!;  / C i $ .!; / at these saddle points is now presented. These results are then compared to both the first and second approximations for the distant and near saddle points developed in the preceding sections. This comparison is done over a reasonable representation of the entire range of values of the space–time parameter   1 of importance in order that the range of values of  over which a given approximation closely describes the exact, numerically determined behavior may be ascertained. Because of its historical importance, Brillouin’s choice of thepsingle-resonance Lorentz medium parameters (namely, !0 D 4  1016 r=s, b D 20  1016 r=s, and ı D 0:28  1016 r=s) are used in the first part of this comparison. Because this choice corresponds to a highly absorptive medium, the second part of this numerical comparison investigates the behavior in the weak dispersion limit as b ! 0 (i.e., as N ! 0). With the complex index of refraction given by (12.57), the exact locations of the saddle points of .!;  / are given by the roots of (12.178), which may be written more simply (and more suitably for the purposes of numerical analysis) as b 2 !.! C iı/   0: F .!;  / D n.!/ C  2 ! 2  !02 C 2iı! n.!/

(12.261)

12.3 The Location of the Saddle Points and the Approximation of the Phase

343

The numerical solution of this equation can then be accomplished using Newton’s method at each fixed value of  with either the second approximate solution at that same  -value as an initial guess or the exact, numerical solution at a neighboring  value as the initial guess. Suppose then that ! is an approximate solution of (12.261) at some fixed  -value and let ! be a small correction that is to be determined such that F .! C !;  / D 0. If this equation is expanded in a Taylor series about the approximate solution ! and then truncated after the first-order term in !, there results F .!;  / ; (12.262) ! D  @F .!; /=@! from which the next approximation is obtained and the process repeated, if necessary. That is, let !1 denote either the second approximate solution of the desired saddle point location at a given fixed  -value or the exact, numerically determined solution at the previous -value, and let !1 , as determined from (12.262), denote the associated correction factor. The second trial solution of (12.261) is then given by !2 D !1 C !1 , and after j such iterations, the .j C 1/th solution is given by !j C1 D !j 

F .!j ;  / : @F .!; /=@!j!D!j

(12.263)

This numerical iteration procedure is terminated at a value !k at which jF .!k ;  /j < ", where " > 0 sets the accuracy of the numerical procedure. For the numerical results presented here, " D 11011 . Finally, the exact expression for @F .!; /=@! appearing in (12.263), given by @F .!;  / b2 D  2 @! ! 2  !02 C 2iı! n.!/ 2

0

13 b2 4n.!/ C ! 2 ! 2 C2iı! . /n.!/ A5 0   43! C 2iı  !.! C iı/2 @  2 ; !  !02 C 2iı! n.!/ (12.264)

is employed in the calculation of the exact saddle point equations. Once the numerically determined exact saddle point location at a given value of  is obtained, the exact values of both  .!;  / and $ .!; / at that space–time point are calculated using the exact expressions given in (12.68) and (12.69), respectively. Notice that Newton’s method fails when j@F .!;  /=@!j becomes exceedingly small, as may occur near the critical space–time point  D 1 as the two near firstorder saddle points approach each other and coalesce into a single second-order saddle point at  D 1 and then separate and move apart. In that case, the method of bisection may be used to numerically determine the saddle point location along the imaginary axis. The numerically determined saddle point locations as a function of  are illustrated in Figs. 12.44–12.46 along with the first and second approximate results, as

344

12 Analysis of the Phase Function and Its Saddle Points 20

30

5.0 4.0 3.0 2.0 2.0

3.0

s

1.009

1.010 1.009

1.010

1.020

1.025

1.030

1.050

1.040

1.10

1.15

1.50 1.40 1.30 1.25 1.20

at io n First Approximation

1.015

im

1.015

ox

on

1.020

pr

ati

1.02 5

nd

1.1 0

co

Ap

Loc

1.03 0

Se

0 1.2

0

0 1.5 40 1. 30 E 1. .25 x a ct 1

1.0 50

1.50 1.40 0 1.3

−0.5

−0.6

25

branch cut

1.04

−0.4

ω ‘ (x1016r/s)

15

1.1 0

ω“ (x1016r/s)

−0.3

10

1. 20 1. 15

−0.2

5

Fig. 12.44 Comparison of the exact (numerically determined) distant saddle point locations C !SP . / in the right-half of the complex !-plane as a function of  > 1 with that given by the d first and second approximations for a single-resonance Lorentz model dielectric with Brillouin’s choice of the medium parameters

4 3 + SPn

'' (x1016r/s)

2 1 0 −1

1.1

1.2

1.3

1.4

1.5

−2 −3

SPn−

−4 ˙ Fig. 12.45 Comparison of the exact (solid curves) near saddle point locations !SP . / along the n imaginary axis as a function of  2 Œ1; 1 with that given by the first (short dashed curves) and second (long dashed curves) approximations for a single-resonance Lorentz model dielectric with Brillouin’s choice of the medium parameters

12.3 The Location of the Saddle Points and the Approximation of the Phase

345

ω' (x1016r/s) 4.0

3.0

2.5

2.0

1.90

1.80

1.70

80 1.

1. 90

2. 0

1. 90

1.

2. 0

p p ro at

2.5

m io

3.0

xi

ca

ω” (x1016r/s)

A

2.

Lo

on

4.0

n

ti s

−0.28

6

nd

5

t

−0.26

5

co

ac

−0.24

4

First Approximation

Ex

−0.22

3

Se

1.7 0 80

1.60

2

1.7 0

1.51 1.52 1.5 1.5 3 1.5 4 5 1.6 0

−0.20

1

1.51

−0.18

0

3.0

5.0 4.0 10

5.0 20 50 10 0 2

ω+

b r a n c h c u t

ω‘+

C !SP . / n

Fig. 12.46 Comparison of the exact (solid curve) near saddle point locations in the righthalf of the complex !-plane as a function of  1 with that given by the first (short dashed line) and second (long dashed curve) approximations for a single-resonance Lorentz model dielectric with Brillouin’s choice of the medium parameters

indicated in each figure. Consider first the  -evolution of the distant saddle point locations !SP C in the right-half of the complex !-plane that is depicted in Fig. 12.44. d The  -values corresponding to the saddle point locations marked by the crosses along the respective path are indicated along each curve in the figure. The second approximate locations of this distant saddle point are seen to be in close agreement with the exact locations over the entire range of values of  considered. Furthermore, it is seen that these second approximate locations move at approximately the same rate with respect to  as do the exact locations. As regards, this rate of motion of the distant saddle points with increasing , both the exact and the second approximations show that for  very close to unity, a small increase in the value of  produces a large change in the distant saddle point location, and that as  increases away from unity, the change in location of the saddle point with  steadily diminishes, so that 0 , as the distant saddle points SPd˙ approach ever closer to the outer branch points !˙ respectively, a large increase in the value of  produces only a small change in the saddle point location. Finally, the second approximate distant saddle point locations are seen to provide a marked improvement in accuracy over the first approximate lo˙ . / approaches near to cations, which are seen to fail rapidly as the real part of !SP n 0 the real part of the branch point !˙ , which, for Brillouin’s choice of the medium parameters, corresponds to values of  greater than approximately 1:05. Consequently, the second approximation of the distant saddle point behavior accurately describes

346

12 Analysis of the Phase Function and Its Saddle Points

the exact behavior over the entire -domain of interest [i.e.,  2 Œ1; 1/], whereas the first approximation is valid only for values of  close to unity. Consider next the  -evolution of the near saddle point locations in the right-half of the complex !-plane. For  2 Œ1; 1 the near saddle points SP˙ n are situated along the imaginary axis, approximately symmetric about the value ! 00 D 2ı=.3˛/, as illustrated in Fig. 12.45. Notice that the second approximation provides only a slightly more accurate description of the near saddle point locations over this initial -domain than does the first approximation. The exact and approximate locations are seen to be in excellent agreement for values of  in the range 1:3    1 . The critical space–time value  D 1 when the two near first-order saddle points SP˙ n coalesce into a single second-order saddle point SPn is found numerically to lie in the range 1:50275 < 1 < 1:50300; where (12.225) yields the first approximate value 1 1:50414 and (12.223) yields the second approximate value 1 1:50275, in very close agreement with the numerically determined lower bound given above. At  D 1:502752 the numerically determined saddle point locations are found to be separated by a very small distance along the imaginary axis. For  > 1 the near saddle points move off symmetrically from the imaginary axis and into the lower-half of the complex !-plane, as illustrated in Fig. 12.46 for the near saddle point SPC n in the right-half plane. Notice that the first approximation of the near saddle point behavior rapidly fails as  increases away from 1 because . /j quickly becomes comparable to !0 . The second approximate locations, j!SPC n on the other hand, closely follow along with the exact near saddle point locations as  increases away from 1 , SP˙ n approaching the inner branch points !˙ , respectively, as  ! 1. Notice that the path traced out by the second approximation lies closely adjacent to the path traced out by the exact near saddle point locations for all  > 1 , but that the positions predicted by the second approximation lie slightly ahead of the actual positions. Furthermore, the second approximate locations plotted in Fig. 12.46 are seen to move with increasing  at approximately the same rate as do the exact locations over the entire range of  -values depicted. As in the case for the distant saddle points, this rate of motion is rapid at first, but as the near saddle point locations !SP˙ ./ approach closer to the inner branch points n !˙ , respectively, their rate of motion with  rapidly decreases. Consequently, taken together with the previous results over the initial space–time domain  2 Œ1; 1 , the second approximation of the near saddle point behavior accurately describes the exact behavior over the entire  -domain of interest (namely,  > 1), whereas the first approximation is valid only for values of  within the limited space–time interval 1:3    1 . Finally, consider the behavior of the complex phase function .!;  / at these distant and near saddle points as a function of , as illustrated in Figs. 12.47–12.49. Figures 12.47 and 12.48 describe the real part  .!;  / of the complex phase behavior at the saddle points and Fig. 12.50 describes the imaginary part $ .!; / of

12.3 The Location of the Saddle Points and the Approximation of the Phase

347

1.2 1.0

0.8 0.6 SPn−

0.4

SP

(x1016 /s)

0.2 0

SB

1.2

1.5

+ SPn−

−0.2 −0.4 −0.6

2.0

+ SPd− SPn+

−0.8 −1.0

Fig. 12.47 Comparison of the exact behavior (solid curves) of the real part  .!;  / of the complex phase function .!;  / at the near and distant saddle points of a single-resonance Lorentz model dielectric (with Brillouin’s choice of the medium parameters) as a function of  for 1    2:2 with that given by the first (dotted curves) and second (dashed curves) approximations

the complex phase behavior at the saddle points. The real and imaginary parts of the complex phase behavior described by the first approximation is indicated by the dotted curves, that by the second approximation by the dashed curves, and the exact, numerically determined behavior by the solid curves in each figure. Consider first the  -dependence of the real phase function  .!;  / at the saddle points, as depicted in Figs. 12.47 and 12.48. At the two first-order distant saddle points SPd˙ ,  .!SP ˙ ; / is seen to identically vanish at  D 1, and then to decrease d monotonically as  increases away from unity, where lim!1  .!SP ˙ ;  / D 1. d The first approximation to  .!SP ˙ ;  / is seen to rapidly diverge away from the d exact behavior as  increases above 0 , whereas the second approximation closely follows the exact behavior over the entire range of values of  considered. At the upper near saddle point SPC n , both the first and second approximations to ; / are very close to each other and are in fair agreement with the exact  .!SPC n

348

12 Analysis of the Phase Function and Its Saddle Points

0

2

4

6

8

10

12

14

−1

−2 + SPd−

+ SPn−

SP

(x1016/s)

−3

−4

−5

−6

−7

−8

Fig. 12.48 Comparison of the exact behavior (solid curves) of the real part  .!;  / of the complex phase function .!;  / at the near and distant saddle points of a single-resonance Lorentz model dielectric (with Brillouin’s choice of the medium parameters) as a function of  for 2    14 with that given by the first (dotted curves) and second (dashed curves) approximations

behavior for 1   < SB the agreement becoming excellent over the space–time interval SB    1 . The numerically determined space–time value SB at which ; SB / is found to be given by  .!SP ˙ ; SB / D  .!SPC n d

SB Š 1:334; where the first two terms of (12.257) result in the rough estimate SB 1:495, all three terms in (12.257) yielding the first-order approximate value SB 1:255, and (12.256) yielding the second approximate value SB 1:295, which is in error by only 2:9%. The related angular frequency value !SB defined in (12.258) is found to be given by !SB Š 8:70  1016 r/s; where the simplified expression given in (12.260) yields the estimate !SB

7:22  1016 r/s and (12.259) gives !SB 8:41  1016 r/s, which is in error by only

12.3 The Location of the Saddle Points and the Approximation of the Phase

349

3:3%. For all  > SB , the near saddle point SPC n , and then both near saddle points for    >  are dominant over the distant saddle points, the second apSP˙ 1 SB n ; / for  2 Œ ; 

and  .! ;  / for   1 providing proximation of  .!SPC SB 1 SP˙ n n an accurate description of the exact, numerically determined behavior. For values of  in a small neighborhood about 0 , the first approximation is also seen to accurately describe the exact behavior at the near saddle points SP˙ n , as was expected, but as  increases further and further away from 0 , the accuracy of the first approximation is seen to steadily diminish. Finally, notice from Fig. 12.48 that for values of  sufficiently greater than 1 ,  .!SP ;  / at both the near and distant saddle points decreases steadily in a nearly linear relationship to , with  .!SP˙ ;  / >  .!SP ˙ ;  /. n d Consider next the -dependence of the imaginary part $ .!; / of the complex phase behavior at the saddle points in the right-half of the complex !-plane, illustrated in Fig. 12.49. Because $ .! 0 C i ! 00 ;  / D $ .! 0 C i ! 00 ;  /, the negative

1

2

3

4

5

0

−2

−4

SP

(x1016/s)

−6

−8

−10

+ SPd

SPn+

−12

−14

−16

Fig. 12.49 Comparison of the exact behavior (solid curves) of the imaginary part $ .!;  / of the complex phase function .!;  / at the near and distant saddle points in the right-half of the complex !-plane for a single-resonance Lorentz model dielectric (with Brillouin’s choice of the medium parameters) as a function of  for 1    5 with that given by the first (dotted curves) and second (dashed curves) approximations

350

12 Analysis of the Phase Function and Its Saddle Points

of this behavior is exhibited in the left-half plane. As is evident from the figure, the first approximate behavior at the distant saddle point SPdC rapidly diverges away from the exact behavior as  increases away from unity, and the first approximate behavior at the near saddle point SPC n rapidly diverges away from the exact behavior as  increases away from 1 , where $ .!SP˙ ;  / D 0 for  2 Œ1; 1 . Notice n that the first approximate curves for the near and distant saddle points cross each other at  4:3, a behavior that is not exhibited by the exact solutions at these saddle points. The second approximate behavior for both the distant and near saddle points, however, is seen to provide a significant improvement over the respective first approximate behavior over the entire  -domain illustrated. Consider finally, the saddle point behavior in the weak dispersion limit as the number density N of Lorentz oscillators decreases, resulting in a decrease in the p plasma frequency b D .k4=0 /N qe2 =m of the dispersive medium which approaches vacuum in the vanishing dispersion limit as N ! 0. The change in the frequency dependence of the complex index of refraction along the positive real angular frequency axis as the number density N is decreased is illustrated in Fig. 12.50, where the initial case is for Brillouin’s choice of the medium parameters

3

nr ( )

2.5 2

N

1.5 N/10 1

N/100

0.5 0

10

ni ( )

10

0

5

' ( x1016r/s)

10

15

10

15

1

0

N –1

10

N/10

–2

10

N/100

–3

10 0

5

' ( x1016r/s)

Fig. 12.50 Frequency dependence of the real (upper graph) and imaginary (lower graph) parts of the complex index of refraction n.! 0 / D nr .! 0 / C i ni .! 0 / along the positive real angular frequency axis for a single-resonance Lorentz p model dielectric with medium parameters !0 D 4  1016 r/s, ı D 0:218  1016 r/s, and b D 20  1016 r/s with number density N (solid curves). The dashed curves indicate the behavior when the number density is reduced by 10 and the dotted curves when it is reduced by 100.

12.3 The Location of the Saddle Points and the Approximation of the Phase

351

[N D .0 =k4k/.m=qe2 /b 2 6:275  1029 ] and the other two cases correspond to the number densities N=10 6:275  1028 and N=100 6:275  1027 . Notice that the zero frequency value n.0/ D 1:5 in the initial case is reduced to n.0/ 1:0607 in the N=10 case and is further reduced to n.0/ 1:0062 in the N=100 case. Although the real frequency dispersion nr .!/ may be considered weak when the number density has been reduced by 100, the material absorption ˛.!/ D .!=c/ni .!/ is still significant about the medium resonance frequency !0 . As regard to the behavior of n.!/ in the complex !-plane, notice that [see (12.64) and (12.65)] 0 lim !˙ D !˙ I (12.265) N !0

0 that is, the outer branch points !˙ move in toward the inner branch points !˙ as the number density decreases to zero, cancelling each other out at N D 0 when the vacuum is obtained. The  -dependence of the real and imaginary parts of the distant saddle point locations, as described by the second approximate expressions given in (12.219)– (12.221) for   1 is illustrated in Fig. 12.51 for the N , N=10, and N=100 cases.

18

' (x1016r/s)

10

N

17

10

N/10 N/100 16

10

1

1.1

1.2

1.3

1.4

1.5

1.6

1.7

1.8

1.9

2

1.3

1.4

1.5

1.6

1.7

1.8

1.9

2

6

'' (x1016r/s)

5.5 5

N

4.5 4

N/10

3.5

N/100

3 2.5

1

1.1

1.2

Fig. 12.51 space–time  -dependence of the real (upper graph) and imaginary (lower graph) parts of the distant saddle point SPdC evolution for a single-resonance Lorentz model dielectric with p medium parameters !0 D 4  1016 r/s, ı D 0:218  1016 r/s, and b D 20  1016 r/s with number density N (solid curves). The dashed curves indicate the behavior when the number density is reduced by 10 and the dotted curves when it is reduced by 100

352

12 Analysis of the Phase Function and Its Saddle Points

Notice that, as the number density decreases, the large  limiting behavior is attained at earlier values of  . This is reflected in the numerical value of the critical space– time parameter SB which decreases from the initial (second approximate) value SB 1:2949 to the value SB 1:0347 at N=10 and then to the value SB

1:00298 at N=100. Accompanying this change, the angular frequency value !SB decreases from its initial (second approximate) value !SB 8:41  1016 r/s to the  1016 r/s value !SB 7:18  1016 r/s at N=10 and then to the value !SB 7:05q

at N=100, while at the same time the angular frequency value !1 !02 C b 2 that sets the scale for the outer branch points decreases from its initial value !1 D 6  1016 r/s to the value !1 4:24  1016 r/s at N=10 and then to the value !1

4:025  1016 r/s at N=100, where !1 ! !0 as N ! 0. Finally, the  -dependence of the real and imaginary parts of the near saddle point locations, as described by the second approximate expressions given in (12.239) and (12.220)–(12.221) for   1 is illustrated in Fig. 12.52 for the N , N=10, and N=100 cases. Notice that, just as for the distant saddle point behavior, the large  limiting behavior is attained at earlier values of  as the number density decreases. This is q reflected in the numerical values of the critical space–time values 0 1 C b 2 =!02 and 1 , whose second approximation is given in (12.223). Initially, these critical

' (x1016r/s)

4

N/100 N/10

3

N 2 1 0

1

2

3

4

5

6

7

8

9

10

4

5

6

7

8

9

10

0

'' (x1016r/s)

−0.5 −1 −1.5

N

−2

N/10 N/100

−2.5 −3

1

2

3

Fig. 12.52 space–time  -dependence of the real (upper graph) and imaginary (lower graph) parts of the near saddle point SPC dielectric n evolution for  1 for a single-resonance Lorentz model p with medium parameters !0 D 4  1016 r/s, ı D 0:218  1016 r/s, and b D 20  1016 r/s with number density N (solid curves). The dashed curves indicate the behavior when the number density is reduced by 10 and the dotted curves when it is reduced by 100. Notice that the value of 1 decreases to unity as the number density is decreased

12.3 The Location of the Saddle Points and the Approximation of the Phase

353

 -values are given by 0 D 1:500 and 1 1:50275 for Brillouin’s choice of the medium parameters. Their values are reduced to 0 1:06066 and 1 1:06105 when the number density is decreased to N=10, and are further reduced to 0

1:00623 and 1 1:00627 when the number density is decreased to N=100. Hence, in the weak dispersion limit as N ! 0, the near and distant saddle point dynamics become increasingly compressed about the luminal space–time point  D 1. Because of this, it is seen best to describe dispersive pulse dynamics in terms of the critical space–time parameters that characterize a specific dispersive medium model, such as SB , 0 , and 1 for a single-resonance Lorentz model dielectric. For a given propagation distance z into the dispersive medium, these space–time parameters then set the appropriate time-scale in which the pulse dynamic may be best described.

12.3.2 Multiple-Resonance Lorentz Model Dielectrics For a double-resonance Lorentz model dielectric with two isolated resonance frequencies, the complex index of refraction is given by [cf. (12.101)]  n.!/ D 1 

b22 b02  2 2 ! 2  !0 C 2iı0 ! ! 2  !2 C 2iı2 ! .1/

1=2 ;

(12.266)

.2/

where it is assumed here that j!˙ j  j!˙ j; see (12.103), (12.104) and (12.108), (12.109) for the branch point locations. Approximate expressions for the distant saddle point SPd˙ dynamics, the near saddle point SP˙ n dynamics, and the middle ˙ , j D 1; 2, dynamics may then be obtained by approximating saddle point SPmj the complex index of refraction in a manner that captures the essential frequency behavior in the appropriate region of the complex !-plane.

12.3.2.1

The Region Above the Upper Resonance Line (j!j  !3 )

q For sufficiently large values of j!j  !3  !1 , where !3 !22 C b22 and !1 q !02 C b02 , the complex index of refraction given in (12.266) may be approximated as [cf. (12.184)] n.!/ 1 

b22 b02  ; 2!.! C 2iı0 / 2!.! C 2iı2 /

(12.267)

with derivative n0 .!/

b02 .! C iı0 / b22 .! C iı2 / C : ! 2 .! C 2iı0 /2 ! 2 .! C 2iı2 /2

(12.268)

354

12 Analysis of the Phase Function and Its Saddle Points

The first-order approximation to the saddle point equation (12.177) is then given by 2.1   /.! C 2iı0 /2 .! C 2iı2 /2 C b02 .! C 2iı2 /2 C b22 .! C 2iı0 /2 0; which may be approximated as 2 2 N 2  b0 C b2 0; .! C 2iı/ 2.  1/

(12.269)

provided that ı0 ı2 , where ıN .ı0 C ı2 /=2. The solution of this equation then gives the first approximate distant saddle point locations [11] s !SP ˙ ./ ˙ d

b02 C b22 N  2iı; 2.  1/

(12.270)

for   1. Based upon the analogy of this result with (12.187) and its corresponding second approximate expression given in (12.201)–(12.203), the second approximation of the distant saddle point locations for a double-resonance Lorentz model dielectric is found to be given by   !SP ˙ . / D ˙./  2iıN 1 C . / ;

(12.271)

d

with s ./

./

.b 2 C b 2 / 2 ; !22  ıN2 C 0 2 2  1

(12.272)

b02 C b22 b02 C b22 D    ; (12.273)  . 2  1/  2 ./ !22  ıN2 . 2  1/ C b02 C b22  2

for all   1. For values of  close to but not less than unity, these two second approximate expressions for ./ and . / may be further approximated as [noting that  2  1 D . C 1/.  1/ 2.  1/] s ./

b02 C b22 ; 2.  1/

. / 1; and this second approximation reduces to the first approximation given in (12.270), with limiting value N lim !SP ˙ ./ D ˙1  2iı: (12.274) !1C

d

12.3 The Location of the Saddle Points and the Approximation of the Phase

355

On the other hand, for sufficiently large values of  ! 1, q ./ ! !32  ıN2 C b02 ; . / ! 0; so that

q .3/ lim !SP ˙ . / D ˙ !32  ıN2 C b02  iıN !˙ ;

!1

(12.275)

d

.3/

where !˙ denotes the outer branch point locations in the left- and right-half planes given in (12.109). Comparison of this second approximate expression for the distant saddle point SPdC locations in the right-half of the complex !-plane with the exact, numerically determined locations [as determined from repeated application of Newton’s method, as described in (12.262) and (12.263)] is provided in Fig. 12.53. Notice that both the real and imaginary parts of the distant saddle point locations are described by this second approximation, as given in (12.271)–(12.273), with sufficient accuracy over the entire space–time domain of interest. In particular, the limiting values described in (12.274) and (12.275) are both realized by the exact solution.

6

' (x1017r/s)

5 4 3 2

Exact Solution

1 0

Second Approximation 1

1.1

1.2

1.3

1.4

1.5

1.4

1.5

'' (x1015r/s)

−2.5 −3

Exact Solution

−3.5

Second Approximation

−4 −4.5 −5 −5.5

1

1.1

1.2

1.3

Fig. 12.53 Comparison of the exact and second approximate  -dependences of the real (upper graph) and imaginary (lower graph) parts of the distant saddle point SPdC evolution for a doublep resonance Lorentz model dielectric with medium parametersp!0 D 1:0  1016 r/s, b0 D 0:6  16 16 16 16 10 r/s, ı0 D 0:1010 r/s, and !2 D 7:010 r/s, b2 D 12:010 r/s, ı2 D 0:281016 r/s

356

12 Analysis of the Phase Function and Its Saddle Points

12.3.2.2

The Region Below the Lower Resonance Line (j!j  !0 )

For sufficiently small values of j!j  !0  !2 , the complex index of refraction given in (12.266) for a double-resonance Lorentz model dielectric may be approximated as    2  b0 ı2 b22 b22 1 i ı0 b02 C 4 !C C 4 !2; (12.276) n.!/ 0 C 0 !04 20 !04 !2 !2 with derivative n0 .!/

i 0



ı0 b02 ı2 b 2 C 42 4 !0 !2

 !C

1 0



b02 b2 C 24 4 !0 !2

 !;

(12.277)

q where 0 n.0/ D 1 C b02 =!02 C b22 =!22 . The first-order approximation of the saddle point equation (12.177) is then found to be given by !04 !24 4 ı0 b02 !24 C ı2 b22 !04 2  !2 C i C .0  / 0; 0 3 b02 !24 C b22 !04 3 b02 !24 C b22 !04

(12.278)

with solution4 "

20 ! 4 ! 4 4  2 4 0 22 4  .  0 /  !SP˙ . / ˙ n 9 3 b0 !2 C b2 !0   2 4 2 ı0 b0 !2 C ı2 b22 !04  :  i 3 b02 !24 C b22 !04



ı0 b02 !24 C ı2 b22 !04 b02 !24 C b22 !04

2 #1=2

(12.279)

This then constitutes the first approximation of the near saddle point locations for a double-resonance Lorentz model dielectric. The critical space–time point  D 1 at which the argument of the square root expression appearing in (12.279) vanishes is given by 2  2 ı0 b02 !24 C ı2 b22 !04  : 1 0 C 30 !04 !24 b02 !24 C b22 !04

(12.280)

axis, symmetrically The two near saddle points SP˙ n then      lie along the imaginary situated about the point ! D 2i ı0 b02 !24 C ı2 b22 !04 = 3 b02 !24 C b22 !04 for  2 Œ1; 1 /, approaching each other as  increases. These two first-order saddle points then coalesce into a single second-order saddle point SPn at the critical space–time point  D 1 , where 4

Notice that this result is somewhat different from (and more accurate than) that given in [11].

12.3 The Location of the Saddle Points and the Approximation of the Phase

!SPn .1 / i

357

 4



2 ı0 b02 !24 C ı2 b22 !0  :  3 b02 !24 C b22 !04

(12.281)

Finally, as  increases above 1 , the two near saddle points SP˙ n separate as they move off of the imaginary axis and into the lower-half of the complex !-plane, ap     proaching ˙1  2i ı0 b02 !24 C ı2 b22 !04 = 3 b02 !24 C b22 !04 as  ! 1. However, when  > 1 becomes sufficiently large such that the inequality j!SP˙ j  !0 is no n longer satisfied, then this first-order approximation is no longer valid. Based upon the analogy of this result with (12.214) and its corresponding second approximate expression given in (12.219)–(12.221), the second approximation of the near saddle point locations for a double-resonance Lorentz model dielectric is found to be given by 2 !SP˙ ./ D ˙ ./  iı0 . / (12.282) n 3 for   1, where 2

 2    02 6  2 . / 4 b  2  02 C 3 !02 C

31=2

!02

0

b22 !02 !24

4 7   ı02 2 . /5 9

; (12.283)

with   b02 ı2 b22 !04 2 2 1 C    C 2 0 3 !02 ı0 b02 !24   :

. /

2 b 2  2   2 C 3 0 1 C b22 !04 2 2 4 0 ! b ! 0

(12.284)

0 2

By construction, this second approximation of the near saddle point locations reduces to the first approximation given in (12.279) for values of  close to 0 . On the other hand, for sufficiently large values of  ! 1, ./ !

. / ! so that

q !02  ı02 ; 3 ; 2

q .0/ lim !SP˙ ./ D ˙ !02  ı02  iı0 D !˙ ;

!1 .0/

n

(12.285)

where !˙ denotes the inner branch point locations in the left- and right-half planes [see (12.103)]. Notice also that this second approximation of the near saddle point

358

12 Analysis of the Phase Function and Its Saddle Points 8

6

'' (x1016r/s)

4

2

Exact Solution

0

Second Approximation

−2

Exact Solution

−4 −6 −8

1

1.1

1.2

1.3

Fig. 12.54 Comparison of the exact (solid curves) and second approximate (dashed curves)  -dependences of the imaginary part of the near saddle point SP˙ n evolution over the space–time domain  2 Œ1; 1 for a double-resonance Lorentz model dielectric with medium parameters p !0 Dp1:0  1016 r/s, b0 D 0:6  1016 r/s, ı0 D 0:10  1016 r/s, and !2 D 7:0  1016 r/s, b2 D 12:0  1016 r/s, ı2 D 0:28  1016 r/s

locations for a double-resonance Lorentz model dielectric reduces to that given in (12.219)–(12.221) for a single-resonance Lorentz model dielectric when b2 D 0. Comparison of this second approximate expression for the near saddle point SP˙ n locations with the exact, numerically determined locations [as determined from repeated application of Newton’s method, described in (12.262)–(12.263)] is presented in Fig. 12.54 for  2 Œ1; 1 when the near saddle points SP˙ n are situated along the imaginary axis, and in Fig. 12.55 for the near saddle point SPC n in the righthalf of the complex !-plane when   1 . Notice that both the real and imaginary parts of the near saddle point locations are described with sufficient accuracy by this second approximation over the entire space–time domain of interest. In particular, the limiting values described in (12.285) is indeed realized by the exact solution.

12.3.2.3

The Region Between the Upper and Lower Resonance Lines (!0 < j!j < !3 )

To analyze the saddle point behavior in the angular frequency region !0 < j!j < !3 between the upper and lower resonance lines in the right-half of the complex !-plane (the behavior in the left-half plane being given by symmetry), consider

12.3 The Location of the Saddle Points and the Approximation of the Phase

359

10

' (x1015 r/s)

Second Approximation 8

Exact Solution 6 4 2 0

1.5

2

2.5

3

3.5

4

4.5

5

'' (x1014 r/s)

−6 −7 −8

Exact Solution −9

Second Approximation −10

1.5

2

2.5

3

3.5

4

4.5

5

Fig. 12.55 Comparison of the exact and second approximate  -dependences of the real (upper graph) and imaginary (lower graph) parts of the near saddle point SPC n evolution over the space– time domain  1 for a double-resonance Lorentz model dielectric with medium parameters p !0 Dp1:0  1016 r/s, b0 D 0:6  1016 r/s, ı0 D 0:10  1016 r/s, and !2 D 7:0  1016 r/s, b2 D 12:0  1016 r/s, ı2 D 0:28  1016 r/s

the change of variable ! D !N s C !; N

(12.286)

where !N s

1 .!0 C !2 / 2

(12.287)

is the mean angular resonance frequency of the double-resonance Lorentz model dielectric, where !N s2  !02 > 0 and !N s2  !22 < 0. With this change of variable, the two resonance terms appearing in the complex index of refraction become ! 2  !02 C 2iı0 ! D !N 2 C 2.!N s C iı0 /!N C !˛2 ; ! 2  !22 C 2iı2 ! D !N 2 C 2.!N s C iı2 /!N C !ˇ2 ; where the two complex-valued quantities !˛2 !N s2  !02 C 2iı0 !N s ;

(12.288)

!ˇ2

(12.289)



!N s2



!22

C 2iı2 !N s :

360

12 Analysis of the Phase Function and Its Saddle Points

have been introduced for notational convenience. With this substitution, the square of the complex index of refraction given in (12.266) for a double-resonance Lorentz model dielectric becomes n2 .!/ N D 1

b02 =!˛2  1 C 2.!N s C iı0 /=!˛2 !N C !N 2 =!˛2 

b22 =!ˇ2   1 C 2.!N s C iı2 /=!ˇ2 !N C !N 2 =!ˇ2 ! # " b02 .!N s C iı0 /2 1 !N s C iı0

1 2 12 !N C 2 4  1 !N 2 !˛ !˛2 !˛ !˛2 ! # " b22 .!N s C iı2 /2 1 !N s C iı2  2 12 !N C 2 4  1 !N 2 !ˇ !ˇ2 !ˇ !ˇ2 ( " # N s C iı0 N s C iı2 2 2 2! 2! D nN s 1 C 2 b0 C b2 !N nN s !˛4 !ˇ4 ! !# ) " b22 .!N s C iı0 /2 .!N s C iı2 /2 1 b02 1 C 4 4  1 !N 2 ; 4  2 nN s !˛4 !˛2 !ˇ !ˇ2 

N < j!ˇ j, where provided that j!j N < j!˛ j and j!j b2 b2 nN s n.!N s / D 1  02  22 !˛ !ˇ

!1=2 :

(12.290)

Notice that nN s is, in general, complex-valued. The complex index of refraction in the intermediate frequency domain between the two resonance lines may then be approximated as " # N s C iı0 N s C iı2 1 2! 2! C b2 b !N n.!/ N nN s C nN s 0 !˛4 !ˇ4 ! ! " b22 .!N s C iı0 /2 .!N s C iı2 /2 1 b02 1 C 4 4 1  4 2nN s !˛4 !˛2 !ˇ !ˇ2 !2 # 1 N s C iı0 N s C iı2 2! 2! C C b2 b0 !N 2 ; nN s !˛4 !ˇ4 (12.291)  3 which is correct to O !N . For notational convenience, this approximate expression may be expressed as ˝1 ˝2 2 n.!/ N nN s C !N  !N ; (12.292) nN s 2nN s

12.3 The Location of the Saddle Points and the Approximation of the Phase

361

with derivative [noting that dn.!/=d! D dn.!/=d N !] N n0 .!/ N

˝1 ˝2 !N  !; N nN s nN s

(12.293)

where !N s C iı0 !N s C iı2 C b22 ; !˛4 !ˇ4 ! b02 b22 .!N s C iı0 /2  1 C ˝2 4 4 !˛ !˛2 !ˇ4

˝1 b02

(12.294) ! ˝12 .!N s C iı2 /2  1 C : 4 nN s !ˇ2 (12.295)

N 0 .!/ N D  for the middle The transformed saddle point equation n.!/ N C .!N s C !/n 5 saddle points in the right-half plane then becomes !N 2 

 2 .2˝1  ˝2 !N s / 2nN s    N0 0; !N C 3˝2 3˝2

where

˝1 N0 nN s C !N s nN s

(12.296)

(12.297)

is a complex-valued space–time value. Notice that, like the critical space–time point 0 D n.0/ for the near saddle points, the complex space–time point is, in part, given by the value of the complex index of refraction at the mean angular frequency !N s where !N D 0. The solution of this transformed saddle point equation, together with the change of variable given in (12.286), then gives the first approximate middle saddle point locations in the right-half of the complex !-plane as 2 2˝1 C .1/j !SP C . / !N s C mj 3 3˝2

"

 .2˝1  ˝2 !N s /2 2nN s    N0  2 3˝2 9˝2

#1=2 ; (12.298)

for j D 1; 2. The middle saddle point locations in the left-half plane are then given  . / D !  by !SPmj C . /. The critical (but complex-valued) space–time value SPmj

 D N1 at which the argument of the square root expression appearing in (12.298) vanishes is given by .2˝1  ˝2 !N s /2 : (12.299) N1 N0 C 6˝2 nN s 5

This result is an extension of that given in [11].

362

12 Analysis of the Phase Function and Its Saddle Points

C C Because N1 is complex-valued in general, the middle saddle points SPm1 and SPm2 do not coalesce into a single second-order saddle point, but rather come into close proximity with each other at the space–time point  D
2

Exact Solution + for SPm1

+

SPm1

'' (x1016r/s)

1

(1)

+

0 + SPm2

−1

Exact Solution + for SPm2

−2 0

1

2

3

' (x1016r/s)

4

5

Fig. 12.56 Comparison of the first approximate solutions for the middle saddle point locations !SP C . /, j D 1; 2, in the right-half of the complex !-plane with the exact, numerically determj

mined middle saddle point locations for  -values increasing from  D 1 to  D 1:5 in  D 0:02 16 steps for p a double-resonance Lorentz model dielectric with parameters !0 D p 1:0  10 r/s, 16 16 16 b0 D 0:6  10 r/s, ı0 D 0:10  10 r/s, and !2 D 7:0  10 r/s, b2 D 12:0  1016 r/s, .1/ ı2 D 0:28  1016 r/s. The outer branch point !C of the lower resonance is indicated by the plus sign

12.3 The Location of the Saddle Points and the Approximation of the Phase

363

to the exact, numerically determined saddle point behavior only over the initial space–time domain immediately following the luminal space–time point  D 1. C comAs  continues to increase, this approximate saddle point solution for SPm1 pletely breaks down as it yields a solution that errantly progresses into the left-half plane. On the other hand, the first approximate solution for the lower middle saddle point location !SP C . / does provide a reasonably accurate description of the m2 exact, numerically determined saddle point behavior over the approximate space– time interval 1:2 <  < 1:5. Although a better approximation for the middle saddle point behavior is desirable, it is not necessary as numerically determined saddle point locations may always be used. If desired, a better analytic approach might be to first determine the minimum dispersion point between the two absorption bands and expand about that point in place of the estimate given in (12.287). The similarity in limiting behavior as  ! 1 between the upper middle saddle C and the distant saddle point SPdC , as well as between the lower middle point SPm1 C and the near saddle point SPC saddle point SPm2 n , is simply a manifestation of the additional branch cut introduced by the second resonance feature in the complex refractive index of the model medium [11]. For example, if the number density N2 is allowed to vanish, these two middle saddle points would combine with the distant saddle point evolution, whereas if the number density N1 is allowed to vanish, these two middle saddle points would combine with the near saddle point evolution. On the other hand, if the upper resonance frequency !2 is allowed to increase to infinity, then these two middle saddle points would combine to form the distant saddle point for a single-resonance Lorentz model dielectric.

12.3.2.4

Determination of the Dominant Saddle Points

The sequence of space–time intervals over which the various saddle points become the dominant saddle point is determined from the space–time dependence of the real part  .!;  / of the complex phase function .!;  / at the distant, near, and middle saddle points, as given in Figs. 12.57 and 12.58 for a double-resonance Lorentz p 1016 r/s, b0 D 0:6  1016 r/s, ı0 D model dielectric with parameters !0 D 1:0 p 0:101016 r/s, and !2 D 7:01016 r/s, b2 D 12:01016 r/s, ı2 D 0:281016 r/s. Notice that the inequality p < 0 is satisfied in this case, where p denotes the ˙ ;  / occurs [see (12.117)]; space–time point at which the peak in the curve  .!SP m1 ˙ this then implies that the middle saddle point pair SPm1 do become the dominant saddle points over the finite space–time domain  2 .SM ; MB /. In this case,the distant saddle points SPd˙ are the dominant saddle points over the initial space– time domain  2 Œ1; SM /, where the space–time value SM is defined in (12.119), as indicated in Figs. 12.57 and 12.58, followed by the upper middle saddle points ˙ over the space–time domain  2 .SM ; MB /, where the space–time value SPm1 MB is defined in (12.121), as indicated in Figs. 12.57 and 12.58. The upper near saddle point SPC n is then the dominant saddle point over the space–time domain  2 .MB ; 1 , the two near first-order saddle point coalescing into a single secondorder saddle point at  D 1 (where 1 is slightly larger than 0 ) which then separate

364

12 Analysis of the Phase Function and Its Saddle Points 5 4

−

SP n 3

+ − SPm2

SP

(x1015/s)

2 1

MB

SM

0

pI

I

I

I

+ SPn−

−1

+− SPm1

+ −

−2

SPm1

−3

+ − SPm2

+ SPn

+ SPd−

−4 −5

1.8

1.6

1.4

1.2

1

2

Fig. 12.57 space–time dependence of the real part  .!;  / of the complex phase function .!;  / at the (numerically determined) distant, near, and middle saddle points of a double-resonance Lorentz model dielectric when the inequality p < 0 is satisfied. In that case, the middle sad˙ become the dominant saddle points over the space–time interval  2 .SM ; MB / dle points SPm1 2

1 +

−

SP

(x1015/s)

− SPm2

SPn

0

I

SM

I

p MB

I

I

+ − SPm1

+

− SPm2

−1 + SPn +

− SPm1

−2

1

1.1

+

SPd− 1.2

1.3

1.4

Fig. 12.58 Magnified view of the real phase behavior depicted in Fig. 12.57

1.5

12.3 The Location of the Saddle Points and the Approximation of the Phase

365

into a pair of first-order saddle points SP˙ n that are the dominant saddle points for all   1 , as seen in Figs. 12.57 and 12.58. Notice that the lower middle saddle ˙ are never the dominant saddle point even though they may have the points SPm2 least exponential decay over some finite space–time domain because the original contour of integration C [see (12.1) and (12.2)] cannot be properly deformed into an Olver-type path (see Sect. 10.1.3) through both them and the other (near, distant, and middle) saddle points (see Figs. 12.12–12.19). Consider next, the situation when the opposite inequality p > 0 is satisfied, ˙ never become the dominant saddle in which case the middle saddle point pair SPm1 points. This situation is illustrated in Fig. 12.59 which depicts the space–time dependence of the real part  .!;  / of the complex phase function .!;  / at the distant, near, and middle saddle points for a double-resonance Lorentz model dielectric with p 16 r/s, b0 D 0:6  1016 r/s, ı0 D 0:10  1016 r/s, and parameters !0 D 1:0  10p !2 D 5:0  1016 r/s, b2 D 12:0  1016 r/s, ı2 D 0:28  1016 r/s, the same as in the previous example with the single exception that the value of !2 has been reduced. The dominant saddle point sequence is now the same as that for a single-resonance Lorentz model dielectric where the distant saddle points SPd˙ are dominant over the initial space–time domain  2 Œ1; SB /, followed by the upper near saddle point ˙ SPC n for  2 .SB ; 1 /, and then the near saddle points SPn for all  > 1 , as seen in Fig. 12.59.

5 4 3 2

+ − SPm2 SPn−

SP

(x1015/s)

1 0

I

SB

I

I

+ SPn−

p

−1 −2 + − SPm1

−3 −4 −5

+ SP d−

+ SPn 1

1.2

1.4

1.6

1.8

2

Fig. 12.59 space–time dependence of the real part  .!;  / of the complex phase function .!;  / at the (numerically determined) distant, near, and middle saddle points of a double-resonance Lorentz model dielectric when the inequality p > 0 is satisfied. In that case, the middle sad˙ never become the dominant saddle points dle points SPm1

366

12 Analysis of the Phase Function and Its Saddle Points

12.3.3 Rocard–Powles–Debye Model Dielectrics The complex index of refraction for a single relaxation time Rocard–Powles–Debye model dielectric is given by [cf. (12.125)]  n.!/ D 1 C

a0 .1  i !0 /.1  i !f 0 /

1=2 ;

(12.300)

where a0 s 1 with s .0/ denoting the static (zero frequency) permittivity and 1 denoting the high-frequency limit of this relaxational model. For a Debyetype dielectric, the saddle point equation yields just a near saddle point solution in the low-frequency domain about the origin (see Sect. 12.1). For j!j  j!p2 j, where !p1 D i=0 is the upper branch point along the negative imaginary axis (see Fig. 12.20), the behavior of the complex index of refraction about the origin may be approximated by the quadratic expression a0 m2 n.!/ 0  20

"

#

p2 4s m2

.1 C 3s /  1 ! 2 C i

where 0 n.0/ D

a0 p !; 20

p s ;

(12.301)

(12.302)

and p 0 C f 0 ; p m 0 f 0 :

(12.303) (12.304)

With this substitution, the saddle point equation (12.177) yields the approximate near saddle point solution [14] # " r 3 !SPn . / i 1  1 C 2 .  0 3 for   0  2 =3 , with and



a0 m2 20

a0 p ; 20

  1 C 3s p2  1 : 4s m2

(12.305)

(12.306)

(12.307)

A comparison of this approximate near saddle point solution with numerically determined near saddle point locations for a single relaxation time Rocard–Powles– Debye model of triply-distilled water at 25ı C is given in Fig. 12.60. The Rocard– Powles–Debye model material parameters used here are given in (12.126). The

12.3 The Location of the Saddle Points and the Approximation of the Phase

367

1 0.8 0.6

'' (x1011r/s)

0.4 0.2 0 −0.2

Exact Solution

−0.4 −0.6 −0.8 −1

Approximate Solution 2

4

6

8

10

12

14

16

18

20

Fig. 12.60 Comparison of the exact (solid curve) and approximate (dashed curve)  -dependences of the near saddle point evolution along the imaginary axis for a single relaxation time Rocard– Powles–Debye model of triply distilled water at 25ı C

agreement between the approximate and exact results is remarkably good about the critical space–time point  D 0 , the accuracy of the approximate solution decreasing monotonically as  increases away from this point. These results then show that the near saddle point SPn for a Rocard–Powles–Debye model dielectric moves p down the imaginary axis as  increases from the value 1 1 , crossing the origin at  D 0 and then approaching the branch point singularity !p1 D i=0 as  ! 1. As this is the only accessible saddle point for a Rocard–Powles–Debye model dielectric, it is the dominant saddle point for all   1 .

12.3.4 Drude Model Conductors The angular frequency dispersion of the complex index of refraction of a Drude model conductor [22] is given by [cf. (12.153)]

n.!/ D 1 

!p2 !.! C i  /

!1=2 ;

(12.308)

368

12 Analysis of the Phase Function and Its Saddle Points

with damping constant  > 0 and angular plasma frequency !p . Although this is a special case of the single-resonance Lorentz model, special care must be given to the region about the origin due to the branch cut structure there (see Fig. 12.31). The isotimic contour plots depicted in Figs. 12.36–12.38 show that, in addition to the distant saddle points SPd˙ that behave very much like those for a single-resonance Lorentz model dielectric, there is near saddle point SPn that moves down the positive imaginary axis and approaches the branch point !pC at the origin as  ! 1, and two inaccessible saddle points SP˙ that evolve with  below the branch cut !z !zC . One then only needs to consider the distant and near saddle points [23].

12.3.4.1

The Region Removed from the Origin (j!j  j!˙ z j)

From the second approximate solution of the distant saddle point locations given in (12.201)–(12.203), the distant saddle point dynamics in a Drude model conductor are seen to be given by !SP ˙ ./ D ˙./  i d

  1 C . / 2

(12.309)

for   1, with the second approximate expressions s ./

!p2  2 2  1

!p2  2 1

. /

C



2 ; 4

2 .27/.4/

./

:

(12.310)

(12.311)

Notice that in the limit as  goes to unity from above lim !SP ˙ ./ D ˙1  i;

!1C

(12.312)

d

whereas in the limit as  goes to infinity lim !SP ˙ ./ D !z˙ ;

!1

(12.313)

d

respectively. The symmetric pair of distant saddle points SPd˙ then begins at infinity when  D 1 and move into the respective branch point zeros !z˙ as  increases to infinity, as depicted in Figs. 12.36–12.38.

12.3 The Location of the Saddle Points and the Approximation of the Phase

12.3.4.2

369

The Region About the Origin (j!j  j!˙ z j)

To obtain an approximate expression for the motion of the near saddle point SPn that is situated along the positive imaginary axis, let ! D i . With this change of variable, the complex index of refraction becomes

n. / D 1 C

!p2

. C  /

!1=2 ;

(12.314)

from which it is seen that n.!/ is real-valued along the positive imaginary axis. The square of this expression for the complex index of refraction may then be approximated for small as !p2 1 C  =!p2 C 2 =!p2 

. C  /  1 C = !  2 !p



1 1C 2  !p  ! 2 2 2 !p !p  

; 1  !p2

n2 . / D

2 C  C !p2

D

which is valid provided that both of the inequalities  !p and   are satisfied. The first approximate behavior of the complex index of refraction in the region about the origin along the positive imaginary axis is then given by n. /

!p2   2 1=2 !p 

;  1=2 1=2 2 3=2 !p

(12.315)

with derivative n0 . / 

!p2   2 !p  : 2 1=2 3=2 4 3=2 !p 1=2

Because . ; / D  .n. /  /, the transformed saddle point equation is given by n. / C n0 . / D  , so that with substitution from the above first-order approximation one obtains the approximate near saddle point equation  1 0 2 2 3 2 3 !   ! 4 2 !p4 16 p p 2 A C 

2   2 @ C 2 0: 4 2 9 !p2   2 9 !p2   2

(12.316)

370

12 Analysis of the Phase Function and Its Saddle Points

The appropriate near saddle point solution is then given by the root of this equation that vanishes as  ! 1. The first approximation of the near saddle point location in a Drude model conductor is then given by 3

!p2

8 !SPn . / i  9 !p2  

1  3 !p2   2 2 A 2 @ C 4 2 2 0

8 2 31=2 9  2 ˆ > ˆ > 2 2 < 9 !p   6 7 = 6 7    1  41  ˆ > 3.!p2  2 / 5 ˆ > : ; 16 4  2 C 4 2

(12.317)

for all   1. Notice that this approximate solution has the limiting behavior lim !SPn ./ D 0;

!1

(12.318)

so that the near saddle point SPn asymptotically approaches the branch point singularity !pC at the origin. A comparison of this approximate near saddle point solution with numerically determined near saddle point locations for a Drude model of sea-water [see (12.155)] with angular plasma frequency !p 2:125  1011 r/s and damping constant  11011 r/s is given in Fig. 12.61. The agreement between the approximate and exact results is remarkably good for all  > 2 and only improves with increasing  as  increases to infinity. Because the distant saddle points SPd˙ are dominant over the near saddle point SPn for  2 Œ1; SB / and the near saddle point SPn is the dominant saddle point for all  > SB , as depicted in Fig. 12.62, where the critical space–time point SB is defined in (12.176), a more accurate description of the near saddle point behavior is unnecessary. Notice that SB Š 1:697 for this simple Drude model of sea-water. By comparison, SB Š 60:07 for the Drude model of the E-layer of the ionosphere. Notice further that lim  .!SPn ;  / D 0;

!1

(12.319)

while  .!SP ˙ ; / decreases monotonically to 1 with increasing . d

12.3.5 Semiconducting Materials The saddle point dynamics become increasingly complicated when several causal dispersion models are required to model the observed material dispersion over a

12.3 The Location of the Saddle Points and the Approximation of the Phase

371

11

'' (r/s)

10

Exact Solution 10

10

Approximate Solution

9

10

1

2

3

4

5

6

7

8

9

10

Fig. 12.61 Comparison of the exact (solid curve) and approximate (dashed curve)  -dependences of the near saddle point evolution along the positive imaginary axis for a Drude model of sea-water with static conductivity 0  4 mho m1 and damping constant   1  1011 r/s 0

−2

(x1010/s)

SP

−4

SPn

−6 +

SPd−

−8

−10

−12

−14

I

1

1.2

1.4

1.6

SB

1.8

2

2.2

2.4

2.6

2.8

3

Fig. 12.62 space–time dependence of the real part  .!;  / of the complex phase function .!;  / at the (numerically determined) distant and near saddle points of the Drude model of sea-water

372

12 Analysis of the Phase Function and Its Saddle Points

given frequency domain (which is set in a given application by the initial pulse spectrum), just as occurred for the double-resonance Lorentz model dielectric presented in Sect. 12.3.2. Another important example is provided by a causal model of the frequency dispersion of a semiconducting material (see Sect. 5.1.3 of Vol. 1). The simplest causal model of a semiconducting material is given by the single relaxation-time Debye model with static conductivity which provides a reasonably accurate description of the frequency dispersion of sea-water for j!j < 1010 r/s (see Fig. 5.1 of Vol. 1). The complex index of refraction of a single relaxation-time Debye model with static conductivity 0 is given by  n.!/ D 1 C

a0 N 0 Ci 1  i !0 !

1=2 ;

(12.320)

where N 0 k4k0 =0 (see Sect. 5.1.1 of Vol. 1). This expression may be rewritten as 1=2  i 1 0 ! 2 C .s C N 0 0 /! C i N 0 n.!/ D ; (12.321) !.1  i !0 / where a0 D s  1 with s .0/ denoting the (relative) static dielectric permittivity of the material. There are then two branch point singularities located at !p1 D 0; !p2 D 

(12.322) i ; 0

(12.323)

and two branch point zeros located at

!zj

s " # s C N 0 0 41 0 N 0 D i 1˙ 1 ; 21 0 .s C N 0 0 /2

(12.324)

for j D 1; 2, where the plus sign is used for j D 1 and the minus sign for j D 2. The branch cuts are then taken as the straight line segments !p1 !z1 and !z2 !p2 along the negative imaginary axis. Computed isotimic contour plots of the real part  .!;  / of the complex phase function .!;  / D i !.n.!/  / for several fixed, subluminal values of the space– p time parameter  > 1 , depicted in Figs. 12.63–12.65, reveal that there are four first-order saddle points for this simple model of a semiconducting material [24]. p One near saddle point SPn lies along the positive imaginary axis for all  > 1 , asymptotically approaching the branch point pole !p1 at the origin as  ! 1.

12.3 The Location of the Saddle Points and the Approximation of the Phase

373

2

SPn

SP

−

z1

n

+

p1

0

+

'' (x1010r/s)

+

1

SP

+

n

−1

−2 −2

−1

0

1

2

10

' (x10 r/s)

Fig. 12.63 Isotimic contours of the real part  .!;  / of the complex phase function .!;  / for a single relaxation time Debye model of sea-water with static conductivity 0 D 4 mho at the fixed p space–time point  D s ' 8:73. The upper near saddle point SPn , located along the imaginary axis in the upper-half of the complex !-plane, is the dominant saddle point and remains so for all larger space–time points 2

+

'' (x1010r/s)

1

SPn p1

0

z1

SP −

−2 −2

+

+

−1

n

−1

0

SP + n

1

2

' (x1010r/s)

Fig. 12.64 Isotimic contours of the real part  .!;  / of the complex phase function .!;  / for a single relaxation time Debye p model of sea-water with static conductivity 0 D 4 mho at the fixed space–time point  D 1:1 s ' 9:60

374

12 Analysis of the Phase Function and Its Saddle Points 2

SPn

+

'' (x1010r/s)

1

p1

0

z1

+

−1

SPn+

+

SPn− −2 −2

−1

0

1

2

10 ' (x10 r/s)

Fig. 12.65 Isotimic contours of the real part  .!;  / of the complex phase function .!;  / for a single relaxation time Debye p model of sea-water with static conductivity 0 D 4 mho at the fixed space–time point  D 1:1 s ' 9:60

Another saddle point lies along the negative imaginary axis below the branch point !z2 (which is off the scale depicted in Figs. 12.63–12.65). The other two saddle points SPn˙ evolve in the low-frequency region 0 < j!j < !p2 in the lower-half of the complex !-plane. Because the original contour of integration C is situated in the upper-half of the complex !-plane, only the three saddle points SPn and SPn˙ are accessible. With the derivative of the complex index of refraction in (12.320) given by   a0 0 i N 0 ; n .!/ D  2n.!/ .1  i !0 /2 ! 2 0

the saddle point equation  0 .!; / D 0 becomes 

i 1 0 ! 2 C .s C N 0 0 /! C i N 0 !.1  i !0 /

1=2 Ci

  a0 0 ! N 0 D ;  2n.!/ .1  i !0 /2 ! 2

12.3 The Location of the Saddle Points and the Approximation of the Phase

375

which may be rewritten as

2 i 1 0 ! 2 C .s C N 0 0 /! C i N 0    .a0 C N 0 0 /0 ! 2 C 2i N 0 0 !  N 0

i 1 0 ! 2 C .s C N 0 0 /! C i N 0 Ci 1  i !0 2  .a0 C N 0 0 /0 ! 2 C 2i N 0 0 !  N 0  2.1  i !0 / 

2 (12.325) D  !.1  i !0 / i 1 0 ! 2 C .s C N 0 0 /! C i N 0 ; without any approximation. Because the saddle points SPn and SPn˙ evolve in the low frequency domain j!j < 1=0 , the quantity 1=.1  i !0 / appearing in the above saddle point equation may be expanded as   1 D 1 C i 0 !  02 ! 2  i 03 ! 3 C 04 ! 4 C O ! 5 : 1  i !0 Substitution of terms up through the cubic term in ! into (12.325) then results in the approximate saddle point equation

  i 0 s .s  31 / C .s C 1 / 2 C N 0 0  2  1 ! 3       N 2  2 31 C s C s s   2 C N 0 0  2 2 C 0 0 ! 2 2 4   2 N N 0 0 !  0 0: (12.326) Ci N 0 s   2 C 2 4 If one now divides through by the quantity D. / s .s  31 / C .s C 1 / 2

(12.327)

and retains terms of order 1=0 , the above saddle point equation simplifies as !3  i

    s s   2 2 N 0 s   2 N 02 ! C !Ci

0; 0 D. / 0 D./ 40 D. /

(12.328)

which is valid provided that the inequality j!j < 1=0 is satisfied. To construct the approximate solutions to this cubic equation, notice first that when  D 0 , where p 0 s [see (12.302)], this equation reduces to ! 3 i

N 02 : 40 D.0 /

(12.329)

376

12 Analysis of the Phase Function and Its Saddle Points

The approximate solutions of the approximate saddle point equation given in (12.328) then separate about this critical space–time point in the following manner:  8  9 p s .s  2 /  N ˆ > 0 ˆ > i ; if C  <  <  1 0 ˆ > 0 D./ s ˆ > ˆ > < =   1=3 N 02 ; !SPn . /

i 40 D.0 / ; if  D 0 ˆ >   ˆ > ˆ > ˆ > N 0 ˆ > p : i 2 ; 1 ; if  > 0 2 s

(12.330)

 s

!SPC . /

n

8 ˆ ˆ ˆ ˆ ˆ <

N 0 2s



 p  i ; s  2   1=3 2 N

0 e i=6 40 D. 0/ ˆ  ˆ ˆ ˆ s .s  2 / ˆ i=6 :e C  D./ 0

N 0 s

if ;  ;

9 > 1 <  < 0 > > > > = ; if  D 0 > > > > > ; if  > 0

p

  9 p >  p  2  i ; if 1 <  < 0 > > > s  > =  1=3 2  N i7=6 0 !SP : . /

e ; if  D  0 n ˆ >  40 D.0 /  ˆ > ˆ > 2 ˆ >  s  / ˆ > : e i7=6 s. D./ ; if  > 0 C Ns0 ; 0 8 ˆ ˆ ˆ ˆ ˆ <

(12.331)

N 0 2s

(12.332)

p Consequently, as  increases above 1 , the near saddle point SPn moves down the positive imaginary axis, asymptotically approaching the branch point singularity !p1 D 0 at the origin from above as  ! 1. Notice that this first-order saddle point p SPn is the dominant saddle point for all  > 1 (see Figs. 12.63–12.65), where p all three saddle points are of equal dominance when  D 1 .

12.4 Procedure for the Asymptotic Analysis of the Propagated Field Equipped with the knowledge of the topography of the complex phase function .!;  / and the space–time dynamics of its saddle points for both Lorentz and Debye-type dielectrics as well as for both conducting and semiconducting media, it is now possible to formulate a procedure to perform the asymptotic analysis of the propagated wavefield A.z; t / as given by either (12.1) or (12.2). This final section describes such a procedure based on the modern asymptotic methods described in Chap. 10. Because of its central importance to the theory, the Lorentz-type dielectric is considered first in greatest detail. The results of this analysis are then extended to Debye-type dielectrics as well as to conducting and semiconducting media.

12.4 Procedure for the Asymptotic Analysis of the Propagated Field

377

The first step in the asymptotic analysis of the wavefield A.z; t / is to express the Fourier–Laplace integral representation of A.z; t / in terms of an integral I.z;  / with the same integrand but with a new contour of integration P . / to which the original contour of integration C may be deformed. In the present application, it is found that any poles of either the spectral function fQ.!/ for (12.1) or uQ .!  !c / for (12.2) that are crossed when the original contour C is deformed to P . / are encircled in the process in the clockwise sense.6 Hence, according to Cauchy’s residue theorem [25], the integral representation of A.z; t / and the integral I.z;  / are related by A.z; t / D I.z;  /  < f2 i./g ; where . / D

X p

Res ! D !p



i uQ .!  !c /e .z=c/.!;/ 2

(12.333)

(12.334)

is the sum of the residues of the poles that were crossed7 and where I.z;  / is defined as Z

1 < i uQ .!  !c /e .z=c/.!;/ d! (12.335) I.z; / 2 P ./ for all z  0. Analogous expressions hold for (12.1) in terms of the spectral function fQ.!/. Attention is now focused on Lorentz-type dielectrics and the single-resonance Lorentz model dielectric in particular. Because the distant saddle points SPd˙ are dominant for some values of  and either the upper near saddle point SPC n or both are dominant for other values of  (as well as possibly of the near saddle points SP˙ n ˙ for a double resonance medium), there the pair of upper middle saddle points SPm1 is no single path P . / that is an Olver-type path (see Sect. 10.1.1) with respect to a single saddle point and that evolves continuously with  for all   1. Consequently, the method of analysis presented in Sect. 10.3.1 is required to obtain an asymptotic representation that remains uniformly valid for all   1. To apply that method, the contour P . / must evolve continuously for all   1 and, in the vicinity of the space–time point  D SB when the saddle point dominance changes (or at each of the space–time points  D SM and  D MB for a double resonance medium satisfying the inequality p < 0 ), the path must pass through both the dominant and nondominant saddle points involved in the dominance change. Moreover, the path P . / must be divisible into a sum of subpaths Pj ./, each of which is an Olver-type path with respect to one of the saddle points. Recall that it is assumed here that each of the spectral functions fQ.!/ and uQ .! !c / is an analytic function of the complex variable !, regular in the entire complex !-plane except at a countable number of isolated points where that function may exhibit poles. 7 Notice that . / changes discontinuously with the space–time parameter  as the path P . / crosses over each pole. However, each of these discontinuities is cancelled by a corresponding discontinuous change in I.z;  /. 6

378

12 Analysis of the Phase Function and Its Saddle Points

For space–time values in the range  2 Œ1; 1 / during which the two near saddle 00 points SP˙ n are situated along the imaginary ! -axis, the lower near saddle point is dominant over all of the other saddle points [see Fig. 12.47 for the single SP n resonance case and Figs. 12.57 and 12.58 for the double resonance case]. That saddle point is not useful, however, because the Olver-type paths with respect to it are not deformable to the original contour C (and vice versa) due to the presence of the branch cuts. For this reason, the lower near saddle point SP n is not included in the subsequent discussion about which saddle point is dominant over the space–time domain  2 Œ1; 1 /. There are many paths having the required properties that pass through both the ˙ upper near saddle point SPC n and the distant saddle points SPd (as well as the pair ˙ of upper middle saddle points SPm1 for a double resonance medium) for space–time points in the domain  2 Œ1; 1 /. There are also many paths having the required properties that pass through both of the near saddle points SP˙ n and the distant saddle ˙ for a double resonance points SPd˙ (as well as the upper middle saddle points SPm1 medium) for  > 1 . Finally, there are many paths having the required properties that pass through the single second-order near saddle point SPn and both of the ˙ for a distant saddle points SPd˙ (as well as the upper middle saddle points SPm1 double resonance medium) when  D 1 . As a consequence, the contour P ./ can always be chosen so that it passes through the upper near saddle point SPC n and the ˙ for a distant saddle points SPd˙ (as well as the upper middle saddle points SPm1 double resonance medium) for  2 Œ1; 1 and through all four (or six) saddle points for  > 1 so that it evolves in a continuous manner as  varies over the entire subluminal space–time domain   1 and can be divided into the desired subpaths with respect to each relevant saddle point. An example of such a path P ./ and its component subpaths Pj . / is illustrated in Fig. 12.66 for a single-resonance Lorentz model dielectric. For values of  in the range  2 Œ1; 1 , the component subpaths (from left to right in the figure) are Pd . /, PnC . /, and PdC ./, and for  > 1 the component subpaths are Pd . /, Pn . /, PnC . /, and PdC ./. The subpaths Pd˙ ./ and Pn˙ ./ are Olvertype paths with respect to the saddle points SPd˙ and SP˙ n , respectively. For a double-resonance Lorentz model dielectric, the subpaths Pm˙ ./ that are Olver-type paths through the respective middle saddle points SPm˙ must also be included. Provided that the path P . / and its component subpaths Pj . / satisfy the above constraints, it is unimportant which particular paths are used; by Olver’s theorem (Theorem 2 of Chap. 10), the asymptotic results are independent of the choice. Some particular choices may be more convenient, however, in that they reduce the computation required. In the analysis that follows, the paths are sometimes taken to follow the path of steepest descent in the vicinity of each saddle point to simplify the determination of the appropriate values of the multivalued functions appearing in the asymptotic expressions. The deformed contour of integration employed by Brillouin [2,3] followed along the entire paths of steepest descent through the distant saddle points SPd˙ and the entire steepest descent path through the upper near saddle point SPC n for 1    1 and through both near saddle points SP˙ n for  > 1 , the various individual paths

12.4 Procedure for the Asymptotic Analysis of the Propagated Field

379

'' C SPn Pn

Pd

'

'

' SPn SPd

SPd

Pd

SB

'' Contour

Original

Integration

of

C

SPn

' Pn

Pd

'

'

SPn

SPd

SPd

SB

''

Pd

1

C

Pd

SPn

' SPd

Pn

SPn

'

Pn

' SPd Pd

1

Fig. 12.66 Olver-type paths through the relevant saddle points of a single-resonance Lorentz model dielectric. The shaded area indicates the region of the complex !-plane wherein the inequality  .!;  / <  .!SP> ;  / is satisfied, where SP> denotes the dominant saddle point (or points) and the darker shaded area indicates the region of the complex !-plane wherein the inequality  .!;  / <  .!SP< ;  / is satisfied, where SP< denotes the nondominant saddle point (or points) over the indicated space–time interval

380

12 Analysis of the Phase Function and Its Saddle Points

being connected along the branch cuts. Although this is a perfectly valid deformed contour of integration, it is unnecessarily complicated and places unnecessary importance to the steepest descent path in the resultant asymptotic description. Because of this, it is not used in this analysis, Olver’s method being used instead. In accordance with the method of analysis described in Sect. 10.3.1, the integral I.z; / is expressed as the sum of integrals with the same integrand over the various subpaths, so that for a single-resonance Lorentz model dielectric I.z; / D Id .z; / C InC .z;  / C IdC .z;  /I I.z; / D

Id .z; /

C

In .z;  /

C

InC .z;  /

C

for 1    1 ; IdC .z;  /I

(12.336)

for  > 1 ; (12.337)

where Id˙ .z; / and In˙ .z; / denote the contour integrals taken over the Olver-type paths Pd˙ and Pn˙ , respectively. The same set of relations hold for a doubleresonance Lorentz model dielectric when the inequality p > 0 is satisfied (see Fig. 12.59). However, when the opposite inequality p < 0 is satisfied, then the upper middle saddle points do become the dominant saddle points (see Figs. 12.57 and 12.58) and the above set of relations is modified to read I.z; / D Id .z; / C Im .z;  / C InC .z;  / C ImC .z;  /IdC .z;  /

(12.338)

for 1    1 , and I.z; / D Id .z; / C Im .z; / C In .z;  / C InC .z;  / C ImC .z;  /IdC .z;  / (12.339) for  > 1 , where Im˙ .z; / denote the contour integrals taken over the Olver-type paths Pm˙ , respectively. To obtain an asymptotic approximation of the integral representation of the propagated wavefield A.z; t / in a Lorentz model dielectric, it now only remains to obtain asymptotic approximations of the various contour integrals appearing on the right-hand sides of either (12.336) and (12.337) for a single resonance medium or (12.338) and (12.339) for a double resonance medium. If the distant saddle points SPd˙ do not pass too near to any poles of either the spectral function fQ.!/ for (12.1) or the spectral function uQ .!  !c / for (12.2), then the results of Sect. 10.2 can be applied to obtain an asymptotic approximation of the quantity Id .z; / C IdC .z; / in the form Id .z; / C IdC .z;  / D As .z; t / C Rd .z;  /;

(12.340)

where As .z; t / is obtained from (10.24) and an estimate of the remainder Rd .z;  / as z ! 1 is given by (10.25). The expression given in (12.340) is uniformly valid for all   1 so long as both of the distant saddle points remain isolated from any poles of either fQ.!/ or uQ .!  !c /. For values of  bounded away from unity from above, (12.340) reduces to the result obtained by application of Olver’s theorem directly to both Id .z; / and IdC .z;  / and adding the results.

12.4 Procedure for the Asymptotic Analysis of the Propagated Field

381

˙ If the near saddle points SPC n for 1 <   1 and SPn for  > 1 do not pass too close to any poles of either the spectral function fQ.!/ for (12.1) or the spectral function uQ .! !c / for (12.2), then the results of Sect. 10.3.2 can be applied to obtain asymptotic approximations of InC .z;  / for 1 <   1 and In .z;  / C InC .z;  / for  > 1 in the form

In .z; /

C

InC .z; / D Ab .z; t / C Rn .z;  /I

for 1    1 ; (12.341)

InC .z; /

for  > 1 ;

D Ab .z; t / C Rn .z;  /I

(12.342)

where the expression for Ab .z; t / and an estimate of the remainder term Rn .z;  / as z ! 1 are obtained from (10.51). Taken together, (12.341) and (12.342) yield an asymptotic approximation of the quantity I.z;  /  Id .z;  /  IdC .z;  / for a single-resonance Lorentz model dielectric that is valid uniformly for all   1 as Q long as the near saddle points SP˙ n remain isolated from any poles of either f .!/ or uQ .!  !c /. For values of  bounded away from 1 from below, the expression in (12.341) reduces to the result that would be obtained by applying Olver’s method directly to obtain the asymptotic approximation of InC .z;  /. Similarly, for values of  bounded away from 1 from above, the expression in (12.342) reduces to the result that would be obtained by applying Olver’s method directly to obtain the asymptotic approximations of In .z; / and InC .z;  / and summing the results. C , j D 1; 2, of a double-resonance Lorentz model If the middle saddle points SPmj dielectric do not pass too close to any poles of either the spectral function fQ.!/ for (12.1) or the spectral function uQ .!  !c / for (12.2), then the uniform asymptotic method of Sect. 10.3.1 can be directly applied to both Im .z;  / and ImC .z;  / and the results summed to obtain an asymptotic approximation of Im .z;  / C ImC .z;  / in the form (12.343) Im .z; / C ImC .z;  / D Am .z; t / C Rm .z;  /; where  Im .z;  / D Im1 .z;  /;

(12.344)

C ImC .z;  / D Im1 .z;  /;

(12.345)

  Im .z;  / D Im1 .z;  / C Im2 .z;  /;

(12.346)

C C ImC .z;  / D Im1 .z;  / C Im2 .z;  /;

(12.347)

for 1   < p , and

˙ for   p . Notice that each component contour integral Imj .z;  / may be obtained from a direct application of Olver’s method. An estimate of the remainder term Rm .z; / as z ! 1 may be obtained from (10.12). Consider now the situation when either one of the distant saddle points SPd˙ approaches (as  varies) a pole of either the spectral function fQ.!/ for (12.1) or the spectral function uQ .!  !c / for (12.2) that is located in a region of the complex

382

12 Analysis of the Phase Function and Its Saddle Points

!-plane bounded away from the limiting values at plus or minus infinity in the lower-half of the complex !-plane approached by SPd˙ as  ! 1C [see (12.204) and (12.274)]. The methods of analysis described in Sects. 10.2 and 10.4 can then be applied to obtain an asymptotic approximation of the quantity Id .z;  / C IdC .z;  / in the form Id .z; / C IdC .z; / D As .z; t / C Cd˙ .z; t / C Rd .z;  /;

(12.348)

where As .z; t / is obtained from (10.24), just as for (12.340). Because the pole is bounded away from the infinite limiting values of !SP ˙ . / as  ! 1C , the sadd dle point and pole do not interact for values of  near unity. Hence, the asymptotic approximation of Id .z; /CIdC .z;  / is determined by applying the uniform asymptotic methods of Sect. 10.2, Cd˙ .z; t / is asymptotically negligible, and the expression given in (12.348) reduces to that in (12.340) for values of  near unity. For values of  bounded away from unity from above, the results of Sect. 10.4 are applicable and the right-hand side of (12.348) is obtained from (10.86) of Theorem 5 with As .z; t / being given by the first term and Cd˙ .z; t / being given by the second term. The resulting expression for As .z; t / is then the same as before. Hence, As .z; t / in (12.348) is given by the same expression as is As .z; t / in (12.340) for all   1. Finally, notice that the quantity Cd˙ .z; t / appearing in (12.348) is asymptotically negligible if either of the distant saddle points SPd˙ does not approach a pole of either fQ.!/ or uQ .!  !c /, in which case (12.348) reduces to (12.340). In a similar manner, consider the situation when either of the near saddle points Q SP˙ n approaches (as  varies) a pole of either the spectral function f .!/ for (12.1) or the spectral function uQ .!  !c / for (12.2) with location bounded away from the critical point where the two near first-order saddle points coalesce to form a single second-order saddle point SPn when  D 1 . The methods of analysis presented in Sects. 10.3–10.4 can then be applied to obtain a asymptotic approximations of InC .z; / for 1 <  < 1 and In .z;  / C InC .z;  / for  > 1 in the form InC .z; / D Ab .z; t / C CnC .z; t / C Rn .z;  /;

(12.349)

for 1   < 1 , and In .z; / C InC .z; / D Ab .z; t / C Cn˙ .z; t / C Rn .z;  /;

(12.350)

for  > 1 . In both cases, the expression for Ab .z; t / is the same as that in (12.341) and (12.342). The quantity CnC .z; t / or Cn˙ .z; t / is asymptotically negligible if the corresponding saddle point does not approach a pole; in that case, the expressions in (12.349) and (12.350) reduce to the corresponding expressions in (12.341) and (12.342). ˙ , j D 1; 2, Consider next the situation when any of the middle saddle points SPmj Q approaches (as  varies) a pole of either the spectral function f .!/ for (12.1) or the spectral function uQ .!!c / for (12.2). The method of analysis presented in Sect. 10.4

12.4 Procedure for the Asymptotic Analysis of the Propagated Field

383

can then be applied to obtain an asymptotic approximation of Im .z;  / C ImC .z;  / in the form Im .z; / C ImC .z; / D Am .z; t / C Cm˙ .z; t / C Rm .z;  /;

(12.351)

where Im˙ .z; / is as given in (12.344)–(12.347). The expression for Am .z; t / is the same as that in (12.343). The quantity Cm˙ .z; t / is asymptotically negligible if both C C and SPm2 or in the saddle points in either the pair of middle saddle points SPm1   pair SPm1 and SPm2 do not approach a pole; in that case, the expression in (12.351) reduces to that in (12.343). Combination of (12.333), (12.336)–(12.343), and (12.348)–(12.351) results in the general expression A.z; t / D As .z; t / C Am .z; t / C Ab .z; t / C Ac .z; t / C R.z;  /

(12.352)

for the asymptotic approximation of the integral representation of the propagated wavefield A.z; t / as z ! 1 in a Lorentz model dielectric. This approximation is uniformly valid for all subluminal space–time points   1 provided that all of the poles of either the spectral function fQ.!/ for (12.1) or the spectral function uQ .!  !c / for (12.2) are bounded away from the limiting locations taken by the distant saddle points SPd˙ as  ! 1C and by SPC n as  ! 1 from below. For a single resonance Lorentz model dielectric, the field term Am .z; t / is set equal to zero in (12.352). The contribution Ac .z; t / appearing in (12.352) is obtained by adding all of the terms that involve the poles, namely Ac .z; t / D < f2 i./g C Cd .z; t / C CdC .z; t / CCm .z; t / C CmC .z; t / C CnC .z; t /;

(12.353)

for 1    1 , and Ac .z; t / D < f2 i./g C Cd .z; t / C CdC .z; t / CCm .z; t / C CmC .z; t / C Cn .z; t / C CnC .z; t /;

(12.354)

for  > 1 . For a single-resonance Lorentz model dielectric, the terms Cm˙ .z; t / are all set equal to zero in these two expressions. An estimate of the remainder term R.z; / as z ! 1 is obtained by taking the largest estimate of the remainder terms appearing in (12.348)–(12.351). An important feature of the general expression given in (12.352) is that the asymptotic behavior of the propagated wavefield A.z; t / in a Lorentz model medium is expressed as the sum of three to four terms which are essentially uncoupled so that they can be treated independently of one another. Each term is determined both by the dynamical behavior of specific saddle points that are a characteristic of the dispersive medium as well as by the analytic behavior of the input pulse spectrum, as described in the paragraphs to follow.

384

12 Analysis of the Phase Function and Its Saddle Points

The dynamic behavior of As .z; t / is determined by the dynamical evolution of the distant saddle points SPd˙ and the value of the input pulse spectrum at these saddle points. Because the distant saddle points are dominant over the initial space–time domain feither  2 Œ1; SB / for a single resonance medium or  2 Œ1; SM / for a double-resonance medium that satisfies p < 0 g, the propagated wavefield component As .z; t / describes the dynamical space–time behavior of the first or Sommerfeld precursor field. This first precursor field is asymptotically negligible during most of the remaining field evolution. The dynamic behavior of Ab .z; t / is determined by the dynamical evolution of the near saddle points SP˙ n and the value of the input pulse spectrum at these saddle points. Because the near saddle points are dominant immediately following the distant saddle point dominance in a single resonance medium, the propagated wavefield component Ab .z; t / describes the dynamical space–time behavior of the second or Brillouin precursor field. This second precursor field is asymptotically negligible during most of the first precursor and remaining field evolution. The dynamic behavior of Am .z; t / in a double-resonance Lorentz model dielec˙ and the tric is determined by the dynamic evolution of the middle saddle points SPm1 value of the input pulse spectrum at these saddle points. Because the middle saddle points are dominant (provided that the inequality p < 0 is satisfied) in the space– time domain  2 .SM ; MB / between the first and second precursor dominance, the propagated wavefield component Am .z; t / describes the dynamical space–time behavior of the middle precursor field. This middle precursor field is asymptotically negligible during most of the first and second precursor evolution. If the opposite inequality p > 0 is satisfied, then the middle precursor is asymptotically negligible during the entire field evolution. The dynamic behavior of Ac .z; t / is determined by the poles of either the spectral function fQ.!/ for (12.1) or the spectral function uQ .!!c / for (12.2) and the dynamics of the saddle points that interact with them. The wavefield component Ac .z; t / is nonzero only if fQ.!/ or uQ .!  !c / has poles. If the envelope function u.t /, defined in (11.34), of the initial plane wavefield A.0; t / at the plane z D 0 is bounded for all time t , then its spectrum uQ .!  !c / can have poles only if u.t / does not tend to zero too fast as t ! 1.8 Hence, the implication of nonzero Ac .z; t / is that the wavefield A.z; t / oscillates with angular frequency !c for positive time t on the plane z D 0 and will tend to do the same at larger values of z for sufficiently large time t . As a result, the propagated wavefield component Ac .z; t / describes the dynamic behavior of the signal contribution oscillating with angular frequency !c . This contribution to the total field evolution is negligible during most of the precursor field evolution. For most values of , only one of the terms As .z; t /, Ab .z; t /, Am .z; t /, and Ac .z; t / appearing in (12.352) is important at a time. There are short space–time intervals in , however, during which two or more of these terms are significant for fixed values of z. These space–time intervals mark the transition periods when the wavefield is changing its character from one form to another and the presence 8

If u.t / is bounded and tends to zero rapidly enough such that the Fourier transform of u.t / converges uniformly for all real !, then uQ .!  !c / is an entire function of complex !.

12.4 Procedure for the Asymptotic Analysis of the Propagated Field

385

of both terms in the expression leads to a continuous transition in the space–time behavior of the propagated wavefield. As a result, (12.352) displays the entire evolution of the field through its various forms in a continuous manner. Analogous results are obtained for Debye model dielectrics and Drude model conductors, as well as for composite models describing semiconducting materials. In particular, (12.355) A.z; t / D Ab .z; t / C Ac .z; t / C R.z;  / for the asymptotic approximation of the integral representation of the propagated wavefield A.z; t / as z ! 1 in a Rocard–Powles–Debye model dielectric. This approximation is uniformly valid for all subluminal space–time points   1 provided that all of the poles of either the spectral function fQ.!/ for (12.1) or the spectral function uQ .!  !c / for (12.2) are bounded away from the origin. The dynamic behavior of Ab .z; t / is determined by the dynamical evolution of the near saddle point SPn (see Figs. 12.25–12.30) and the value of the input pulse spectrum at this saddle point. Because this near saddle point evolution is analogous to the upper near saddle point evolution along the imaginary axis for a Lorentz model dielectric, the propagated wavefield component Ab .z; t / describes the dynamical space–time behavior of the Brillouin precursor field. This precursor field is asymptotically negligible during most of the signal evolution Ac .z; t / which is determined by the poles of either spectral function fQ.!/ or uQ .!  !c / and the dynamics of the saddle point SPn that interacts with them. As before, if present, Ac .z; t / describes the dynamic behavior of the signal contribution oscillating with angular frequency !c . This contribution to the total field evolution is negligible during most of the precursor field evolution. Notice that this same general description also applies to the simple model of a semiconducting medium given by the Debye model with static conductivity (see Figs. 12.63–12.65). The asymptotic approximation of the integral representation of the propagated wavefield A.z; t / in a Drude model conductor is given by A.z; t / D As .z; t / C Ab .z; t / C Ac .z; t / C R.z;  /;

(12.356)

as z ! 1. Notice that this general expression is the same as that for a singleresonance Lorentz model dielectric [cf. (12.352)]. This approximation is uniformly valid for all subluminal space–time points   1 provided that all of the poles of either the spectral function fQ.!/ for (12.1) or the spectral function uQ .!  !c / for (12.2) are bounded away from the origin. The dynamic behavior of As .z; t / is determined by the dynamical evolution of the distant saddle points SPd˙ (see Figs. 12.36–12.38) and the value of the input pulse spectrum at these saddle points. Because the distant saddle points are dominant over the initial space–time domain  2 Œ1; SB /, the propagated wavefield component As .z; t / describes the dynamical space–time behavior of the first or Sommerfeld precursor field. This first precursor field is asymptotically negligible during most of the remaining field evolution. The dynamic behavior of Ab .z; t / is determined by the dynamical evolution of the near saddle point SPn and the value of the input pulse spectrum at it. Because this near

386

12 Analysis of the Phase Function and Its Saddle Points

saddle point is dominant immediately following the distant saddle point dominance (see Fig. 12.62), the propagated wavefield component Ab .z; t / describes the dynamical space–time behavior of the second or Brillouin precursor field. This second precursor field is asymptotically negligible during most of the first precursor and remaining field evolution. The signal contribution Ac .z; t / is determined by the poles of either spectral function fQ.!/ or uQ .!  !c / and the dynamics of the saddle points that interacts with them. If present, Ac .z; t / describes the dynamic behavior of the signal contribution oscillating with angular frequency !c . This contribution to the total field evolution is negligible during most of the precursor field evolution.

12.5 Synopsis The results presented in this rather lengthy chapter are critical to the full understanding of dispersive pulse dynamics. As the propagation distance increases, the observed pulse dynamics are increasingly determined by the dynamics of the saddle points that are a characteristic of the dispersive medium as well as by the analytical behavior of the initial pulse spectrum at them. Each feature in the observed pulse distortion can then be traced back to a specific saddle point behavior, much in the same way as each feature in a scattering process can be traced back to some specific feature in the scattering object. Because of this, specific pulse types can be designed to either strongly interact with a particular set of saddle points (for selective heating, imaging, and remote sensing applications) or to weakly interact with an obscuring medium (for communication and imaging through barriers). On the other hand, specific materials can be designed that weakly interact with specific radar pulses for low-observable (stealth) applications.

References ¨ 1. A. Sommerfeld, “Uber die fortpflanzung des lichtes in disperdierenden medien,” Ann. Phys., vol. 44, pp. 177–202, 1914. ¨ 2. L. Brillouin, “Uber die fortpflanzung des licht in disperdierenden medien,” Ann. Phys., vol. 44, pp. 204–240, 1914. 3. L. Brillouin, Wave Propagation and Group Velocity. New York: Academic, 1960. 4. K. E. Oughstun, Propagation of Optical Pulses in Dispersive Media. PhD thesis, The Institute of Optics, University of Rochester, 1978. 5. K. E. Oughstun and G. C. Sherman, “Uniform asymptotic description of dispersive pulse propagation,” J. Opt. Soc. Am. A, vol. 69, no. 10, p. 1448A, 1979. 6. K. E. Oughstun and G. C. Sherman, “Propagation of electromagnetic pulses in a linear dispersive medium with absorption (the Lorentz medium),” J. Opt. Soc. Am. B, vol. 5, no. 4, pp. 817–849, 1988. 7. K. E. Oughstun and G. C. Sherman, Pulse Propagation in Causal Dielectrics. Berlin: Springer, 1994. 8. H. M. Nussenzveig, Causality and Dispersion Relations. New York: Academic, 1972. Chap. 1.

Problems

387

9. L. D. Landau and E. M. Lifshitz, Electrodynamics of Continuous Media. Oxford: Pergamon, 1960. Ch. IX. 10. K. E. Oughstun, “Dynamical evolution of the precursor fields in linear dispersive pulse propagation in lossy dielectrics,” in Ultra-Wideband, Short-Pulse Electromagnetics 2 (L. Carin and L. B. Felsen, eds.), pp. 257–272, New York: Plenum, 1994. 11. S. Shen and K. E. Oughstun, “Dispersive pulse propagation in a double-resonance Lorentz medium,” J. Opt. Soc. Am. B, vol. 6, pp. 948–963, 1989. 12. J. E. K. Laurens and K. E. Oughstun, “Electromagnetic impulse response of triply-distilled water,” in Ultra-Wideband, Short-Pulse Electromagnetics 4 (E. Heyman, B. Mandelbaum, and J. Shiloh, eds.), pp. 243–264, New York: Plenum, 1999. 13. E. T. Whittaker and G. N. Watson, Modern Analysis. London: Cambridge University Press, fourth ed., 1963. p. 133. 14. K. E. Oughstun, “Dynamical evolution of the Brillouin precursor in Rocard-Powles-Debye model dielectrics,” IEEE Trans. Ant. Prop., vol. 53, no. 5, pp. 1582–1590, 2005. 15. M. Altarelli, D. L. Dexter, H. M. Nussenzveig, and D. Y. Smith, “Superconvergence and sum rules for the optical constants,” Phys. Rev. B, vol. 6, pp. 4502–4509, 1972. 16. H. Xiao and K. E. Oughstun, “Hybrid numerical-asymptotic code for dispersive pulse propagation calculations,” J. Opt. Soc. Am. A, vol. 15, no. 5, pp. 1256–1267, 1998. 17. K. E. Oughstun, J. E. K. Laurens, and C. M. Balictsis, “Asymptotic description of electromagnetic pulse propagation in a linear dispersive medium,” in Ultra-Wideband, Short-Pulse Electromagnetics (H. L. Bertoni, L. B. Felsen, and L. Carin, eds.), pp. 223–240, New York: Plenum, 1992. 18. J. McConnel, Rotational Brownian Motion and Dielectric Theory. London: Academic, 1980. 19. P. Debye, Polar Molecules. New York: Dover, 1929. 20. J. E. K. Laurens, Plane Wave Pulse Propagation in a Linear, Causally Dispersive Polar Medium. PhD thesis, University of Vermont, 1993. Reprinted in UVM Research Report CSEE/93/05-02 (May 1, 1993). 21. M. A. Messier, “A standard ionosphere for the study of electromagnetic pulse propagation,” Tech. Rep. Note 117, Air Force Weapons Laboratory, Albuquerque, NM, 1971. 22. P. Drude, Lehrbuch der Optik. Leipzig: Teubner, 1900. Chap. V. 23. N. A. Cartwright and K. E. Oughstun, “Ultrawideband pulse penetration in an isotropic collisionless plasma,” in 2007 CNC/USNC North American Radio Science Meeting, 2007. 24. N. A. Cartwright and K. E. Oughstun, “Ultrawideband pulse penetration in a Debye medium with static conductivity,” in Fourth IASTED International Conference on Antennas, Radar, and Propagation, 2007. 25. E. T. Copson, An Introduction to the Theory of Functions of a Complex Variable. London: Oxford University Press, 1935. Section 6.1.

Problems 12.1. If ı1 D 0, then the imaginary part of the relative dielectric permittivity varies as "i .!/ ı3 ! 3 about the origin, where ı3 is a nonnegative real number. With this term included in (12.18) with ı1 set equal to zero, determine the near saddle point dynamics. 12.2. Derive the approximate expressions given in (12.80) and (12.83) for !mi n in a single-resonance Lorentz model dielectric. 12.3. Derive the approximate expressions given in (12.108) and (12.109) for the .1/ .3/ branch point locations !˙ and !˙ in a double-resonance Lorentz model dielectric.

388

12 Analysis of the Phase Function and Its Saddle Points

12.4. Derive (12.144), (12.146), (12.148), and (12.150). 12.5. Derive (12.168), (12.171), and (12.174) describing the limiting behavior of the complex index of refraction about the branch points !p˙ and !z˙ for a Drude model conductor. Use these expressions to reproduce the limiting behavior depicted in part (a) of Fig. 12.35. 12.6. Derive (12.169), (12.172), and (12.175) describing the limiting behavior of the real part  .!;  /
than that given in (12.298). The correct solution to this problem, together with the uniform asymptotic approximation of the resultant middle precursor field, could result in an important publication.

Chapter 13

Evolution of the Precursor Fields

On the basis of the foundational analysis just completed, the asymptotic description of dispersive pulse propagation in both Lorentz-type and Debye-type dielectrics as well as in conducting and semiconducting media may now be fully developed. The analysis presented in this chapter begins with an examination of the exact propagated wavefield behavior for superluminal times t such that  D ct =z < 1 for a fixed propagation distance z > 0. By applying the method Sommerfeld [1,2] used to examine the wavefront evolution of a step-function modulated signal in a causally dispersive medium (the Lorentz medium in particular), it is shown here [3,4] that for wavefields with an initial pulse function f .t / that identically vanishes for all times t < 0, the propagated wavefield is identically zero for all superluminal space–time points  < 1, in complete agreement with the relativistic principle of causality [5]. The remainder of the chapter is devoted to the determination of the evolutionary properties of the precursor fields that, because of their intimate connection to the evolutionary properties of the saddle points, are a charcteristic of the dispersive medium. The analysis follows the now classic approach pioneered by Brillouin [6,7] in his treatment of the Heaviside step-function modulated signal with fixed angular carrier frequency !c > 0 propagating in a single resonance Lorentz medium. That analysis was based upon the then recently developed method of steepest descent (see Sect. F.7 of Appendix F) due to Debye [8]. The analysis presented here is based upon the advanced saddle point methods described in Chap.10. When combined with the more accurate approximations of the saddle point locations and the complex phase behavior at them developed in Chap. 12 for both Lorentz- and Debye-type dielectrics as well as for Drude model conductors and semiconducting materials, accurate asymptotic approximations of the associated precursor fields result that are uniformly valid over the entire space–time domain of interest. If necessary, greatly improved accuracy can always be obtained by using numerically determined saddle point locations in the asymptotic expressions. As a result of this detailed analysis, each feature appearing in the propagated wavefield sequence illustrated in Fig. 13.1 may be traced back to the dynamical behavior of a particular saddle point (or points) together with their interaction with any pole singularity in the initial pulse envelope spectrum. The numerically determined propagated wavefield sequence presented in this figure is due to an initial Heaviside unit step function modulated signal with below resonance carrier frequency !c D !0 =2 at 0, 1, 2, and 3 absorption depths K.E. Oughstun, Electromagnetic and Optical Pulse Propagation 2: Temporal Pulse Dynamics in Dispersive, Attenuative Media, Springer Series in Optical Sciences 144, c Springer Science+Business Media, LLC 2009 DOI 10.1007/978-1-4419-0149-1 13, 

389

390

13 Evolution of the Precursor Fields

0

z=0

0

A(z,t)

z = zd

0

z = 2z d

0

z = 3z d 0

2

4

6

8

t (fs) Fig. 13.1 Numerically determined propagated wavefield evolution due to an initial Heaviside unit step function modulated signal f .t / D uH .t / sin .!c t / with below resonance carrier frequency !c D !0 =2 at 0, 1, 2, and 3 absorption depths in a single resonance Lorentz model dielectric. Notice that the vertical (wave amplitude) scale of the initial wavefield structure (z D 0) is in units of 1. For the remaining wavefield plots, the vertical scale is in units of 0:5

[zd ˛ 1 .!c /] in a single resonance Lorentz model dielectric. Notice that the steady-state wave structure oscillating at the input angular carrier frequency !c at each propagation distance z has amplitude given by the attenuation factor e z=zd . The complicated field structure preceding this steady-state behavior is then due to the saddle points and is referred to as the first and second precursor fields. Of particular interest here is the observation that the peak amplitude of the second precursor field attenuates with increasing propagation distance z at a significantly smaller rate than does the remainder of the propagated wavefield. This unique feature may then be exploited in both imaging and communications systems. In addition, its impact on health and safety issues concerning exposure to ultrawideband electromagnetic radiation may have far-reaching implications, particularly in regard to digital cellular telephony. In the asymptotic analysis that follows in both this chapter and later chapters, the expression f .z; t /  g.z;  / is used to mean that g.z;  / is an approximation of the dominant term in the asymptotic expansion of f .z; t / as z ! 1 with fixed  ct =z. The reason that g.z;  / is not equal to the dominant term exactly is that approximations of the saddle point locations and the complex phase behavior at them have been used in determining g.z;  /. When the exact (numerically

13.1 The Field Behavior for  < 1

391

determined1 ) saddle point locations and phase behavior are employed so that g.z;  / is equal to the dominant term exactly, then the asymptotic relation is written ˚ f .z; t / D g.z; / C O z3=2 as z ! 1.

13.1 The Field Behavior for  < 1 If the initial pulse function f .t / of the plane wavefield on the plane z D 0 is zero for all time t < 0, then the propagated wavefield A.z; t / can be zero for space–time values  ct =z < 1 only if the wavefront propagates with a velocity greater than the speed of light c in vacuum, in direct violation of the special theory of relativity (or the principle of relativistic causality) [5]. In his 1914 paper, Sommerfeld [2] proved that for a Heaviside step-function envelope signal f .t / D uH .t / sin .!c t / where [see (11.55)] uH .t / D 0 for t < 0 and uH .t / D 1 for t > 0, the propagated wavefield in a Lorentz model dielectric is identically zero for all superluminal space–time points  < 1. The extension of this proof to a broad class of pulse functions f .t / that vanish for all time t < 0 on the plane z D 0 in a linear, causally dispersive medium is now presented [3, 4]. The proof begins with the exact integral representation of the propagated plane wavefield given in (12.1), viz., A.z; t / D

1 2

Z

fQ.!/e .z=c/.!;/ d!;

(13.1)

C

where A.0; t / D f .t /. Here C denotes the Bromwich contour ! D ! 0 C i a where ! 0
Z

1

f .t /e i!t dt:

(13.2)

0

If the initial pulse function f .t / is bounded for all t , then it immediately follows from direct differentiation of (13.2) that its spectrum fQ.!/ is an analytic function of complex ! D ! 0 C i ! 00 for ! 00 > 0. In addition, if its derivative df .t /=dt is bounded for all t , then integration of the integral in (13.2) by parts shows that the magnitude of uQ .!/ tends to zero uniformly with respect to the angle arg .˝/ for 0    as j˝j ! 1, where ˝ D !  i a with a > 0. 1

Because exact analytic solutions for the saddle point locations are rarely, if ever, available, precise numerical solutions have to suffice.

392

13 Evolution of the Precursor Fields

Because the spectral function fQ.!/ satisfies the above conditions, it is now possible to express A.z; t / by the integral representation given in (13.1) with the change that the integration is now taken over the closed contour that encircles the region ! 00 > a > 0 of the complex !-plane. All that is required is to show that I.z; ; j˝j/ ! 0 uniformly with respect to both z and  for z  Z and   1  " as j˝j ! 1 for arbitrary Z > 0 and " such that 0 < " < 1, where Z

fQ.!/e .z=c/.!;/ d!;

I.z; ; j˝j/

(13.3)

C˝

  with C˝ denoting the semicircular contour !  i a D j˝je i with 0  for fixed j˝j. The proof makes use of the proof of Jordan’s lemma [9]. It directly follows from (13.3) that the inequality Z

ˇ ˇ ˇfQ.!/ˇe .z=c/.!;/ d j!j

jI.z; ; j˝j/j 

(13.4)

C˝

is satisfied, where  .!;  /
(13.5)

with nr .!/ 0 for j˝j > ˝0 . Hence, for j˝j > ˝0 and 0   ,  .!;  /  ! 00 .nr .!/  / :

(13.6)

Furthermore, for any lossy medium there exists a positive constant ˝1 such that nr .!/  1 for j˝j > ˝1 ; notice that for Debye-model dielectrics, this inequality p may be refined to nr .!/  1  1 [see the discussion following (12.34)]. Henceforth, ˝0 is chosen to be larger than ˝1 . It then follows from (13.6) that  .!;  /  ! 00 .1  /

(13.7)

 . Consequently, for   1  " with 0 < " < 1, for j˝j > ˝0 with 0  combination of the inequalities in (13.4) and (13.7) yields Z jI.z; ; j˝j/j 

ˇ ˇ ˇfQ.!/ˇe .z=c/"! 00 d j!j

(13.8)

C˝

for j˝j > ˝0 . From here on, the proof follows exactly the proof of Jordan’s lemma as given by Whittaker and Watson [10]. Hence, I.z; ; j˝j/ ! 0 uniformly with respect to both z and  for z  Z and   1  " as j˝j ! 1 for arbitrary Z > 0 and " such that 0 < " < 1. Consequently, that semicircular contour integral can be

13.2 The Sommerfeld Precursor Field

393

added to the integral in (13.1) in order to express A.z; t / as an integral over a closed contour that encircles the region ! 00 > a of the complex !-plane, viz., 1 A.z; t / D 2

I

fQ.!/e .z=c/.!;/ d!;

(13.9)

C CC˝

for z  Z > 0 and   1  ". Because the integrand in this integral is a regular analytic function of complex ! for ! 00 > a > 0, it then follows from Cauchy’s residue theorem [11,12] that this integral is identically zero for z  Z and   1  " as j˝j ! 1 for arbitrary Z > 0 and arbitrarily small " > 0. This then proves the following generalized form [3, 4] of the theorem due to Sommerfeld [2]: Theorem 6. Sommerfeld’s Relativistic Causality Theorem. If the initial time behavior A.0; t / D f .t / of the plane wavefield at the plane at z D 0 is zero for all time t < 0 and if the model of the linear material dispersion is causal, then the propa1=2 gated wavefield identically vanishes for all t < "1 z=c with z > 0, where "1  1 denotes the high-frequency limit of the relative complex dielectric permittivity of the material. This fundamental theorem can then be applied to any portion of a pulse in order to prove that neither energy nor information can move forward in the pulse body at a superluminal rate, as has essentially been done by Landauer [13] and Diener [14]. In spite of this, the debate concerning superluminal pulse velocities persists (see Sect. 15.12).

13.2 The Sommerfeld Precursor Field The symmetric contributions of the two distant saddle points to the asymptotic behavior of the propagated wavefield A.z; t / for sufficiently large values of the propagation distance z > 0 yield the dynamical space–time evolution of the first or Sommerfeld precursor field. This contribution to the asymptotic behavior of the total wavefield A.z; t / is denoted by As .z; t / and is dominant over the second or Brillouin precursor field in single resonance Lorentz model dielectrics (as well as in double resonance Lorentz model dielectrics that satisfy the inequality p > 0 ) and Drude model conductors over the initial space–time domain  2 Œ1; SB /, whereas it is dominant over both the middle and Brillouin precursors over the space–time domain  2 Œ1; SM / in double resonance Lorentz model dielectrics when the inequality p < 0 is satisfied. Because the pair of first-order distant saddle points SPd˙ remain isolated from each other and do not change their order throughout their evolution, a straightforward application of Olver’s method (see Sect. 10.1) is applied in Sect. 13.2.1 to obtain the asymptotic behavior of the Sommerfeld precursor evolution for  > 1. However, as the space–time parameter is allowed to approach unity from above, these two distant saddle points approach infinity, condition 2 of Olver’s theorem [15] (Theorem 2) is no longer satisfied and Olver’s

394

13 Evolution of the Precursor Fields

method breaks down. To obtain a valid description of the initial behavior of the first precursor field (the wavefront) for values of  in a neighborhood of unity, the uniform asymptotic expansion due to Handelsman and Bleistein [16] given in Theorem 3 is then applied in Sect. 13.2.2. This uniform expansion is valid for all   1 and reduces to the nonuniform result obtained using Olver’s method for all  > 1 bounded away from unity.

13.2.1 The Nonuniform Approximation The asymptotic behavior of the first or Sommerfeld precursor field As .z; t / as z ! 1 for a given initial pulse envelope function u.t / is obtained from the asymptotic expansion of the integral representation of the propagated wavefield [see (11.48)]

Z 1 < i e i uQ .!  !c /e .z=c/.!;/ d! (13.10) A.z; t / D 2 C about the two distant saddle points SPd˙ . The more general integral representation given in (11.45) for the propagated wavefield due to the intial pulse A.0; t / D f .t / at the plane˚z D 0 may be directly obtained from (13.10) with the identification that fQ.!/ D < i e i uQ .!  !c / , the integral representation given in (13.10) resulting when the initial pulse function is given by f .t / D u.t / sin .!c t C /. Because most of the pulse types considered here are expressed in this envelope-modulated carrier wave form, this form of the integral representation is explicitly considered here, the other (more general) case then being obtained through the above identification. The distant saddle point locations may be expressed as !SP ˙ . / D ˙./  i ı .1 C . //

(13.11)

d

for both Lorentz model dielectrics and Drude model conductors. The second approximations for the functions ./ and . / are respectively given by (12.202) and (12.203) for a single resonance medium and by (12.272) and (12.273) for a double resonance medium with ı D 2ıN D ı0 C ı2 . For a Drude model conductor, ./ is given by (12.310) and ./ is given by (12.311) with ı D =2. For each of the examples considered in this text, both the spectral function uQ .!  !c / and the complex phase function .!;  / appearing in the integrand of (13.10) are analytic about these two distant saddle points for all   1. For a single resonance Lorentz model dielectric, as well as for a double resonance Lorentz model dielectric that satisfies the condition p > 0 , the conditions of Olver’s theorem are satisfied at the two distant saddle points SPd˙ when the original contour of integration C is deformed to the path P ./ D Pd . / C PnC ./ C PdC . / for  2 .1; 1 and then to the path P ./ D Pd . / C Pn . / C PnC . / C PdC . / for  > 1 , where Pd . / is an Olver-type path with respect to the distant saddle

13.2 The Sommerfeld Precursor Field

395

point SPd and PdC . / is an Olver-type path with respect to the distant saddle point SPdC (see Fig. 12.66). For a double resonance Lorentz model dielectric satisfying the condition p < 0 , the conditions of Olver’s theorem are satisfied at the two distant saddle points SPd˙ when the original contour of integration C is deformed to the path P . / D Pd . / C PnC ./ C PdC ./ for  2 .1; SM , to the path C  . / C PnC ./ C Pm1 ./ C PdC ./ for  2 ŒSM ; p /, to the P . / D Pd . / C Pm1 C C   . / C Pm2 ./ C PnC ./ C Pm1 ./ C Pm2 ./ C PdC . / path P . / D Pd . / C Pm1   . / C Pm2 . / C for  2 .p ; 1 , and then to the path P ./ D Pd ./ C Pm1 C C C   C Pn . / C Pn . / C Pm1 . / C Pm2 ./ C Pd ./ for  > 1 , where Pd . / is an Olver-type path with respect to the distant saddle point SPd and PdC . / is an Olver-type path with respect to the distant saddle point SPdC . For a Drude model conductor, the conditions of Olver’s theorem are satisfied at the two distant saddle points SPd˙ when the original contour of integration C is deformed to the path P . / D Pd . / C Pn . / C PdC ./ for all  > 1, where Pd ./ is an Olver-type path with respect to the distant saddle point SPd and PdC ./ is an Olver-type path with respect to the distant saddle point SPdC . In each case, (10.18) applies for each of the two distant saddle points with (10.3) and (10.4) taken as Taylor series expansions about these saddle points. Because SPd˙ are first-order saddle points,  D 2, and because uQ .!  !c / is regular at these two saddle points,  D 1. Hence, from Olver’s theorem (Theorem 2) and the results of Sect. 12.4, the contour integral in (13.10) taken over the two Olver-type paths Pd˙ ./ yields the first or Sommerfeld precursor field As .z; t /, which is given by [17–19]  As .z; t / D

c z

(

1=2 < ie

i

a0 .!SP C /e

.z=c/.!

C ;/ SPd

d

Ca0 .!SPd /e

  1 C O z1

.z=c/.!SP  ;/ d

  1 C O z1

)

(13.12) as z ! 1 uniformly for all   1 C " with arbitrarily small " > 0. To evaluate the pair of coefficients a0 .!SP ˙ / D a0 .!SP ˙ . // appearing in d d (13.12), the first two coefficients in the Taylor series expansion (10.3) of the complex phase function .!;  /, as well as the first coefficient in the Taylor series expansion (10.4) of the initial envelope spectrum uQ .!  !c / about the distant saddle points SPd˙ must first be determined. The latter quantity is given by   q0 .!SP ˙ .// D uQ !SP ˙ ./  !c ; d

(13.13)

d

the specific form of which depends upon the particular initial pulse envelope function u.t /. With  D 2 and  D 1 and the observation that the second coefficient in the Taylor series expansion (10.3) of the complex phase function is given by

396

13 Evolution of the Precursor Fields

p0 .!SP ˙ ; / D  00 .!SP ˙ ; /=2Š, the coefficients a0 .!SP ˙ ;  / appearing in the d d d asymptotic expansion (13.12) are found to be given by [see (10.9)]   uQ !SP ˙ ./  !c d a0 .!SP ˙ ;  / D h i1=2 : d 2 00 .!SP ˙ ;  /

(13.14)

d

With this substitution, the nonuniform asymptotic expansion (13.12) of the Sommerfeld precursor becomes ("

As .z; t /  < i e

i

#1=2  .z=c/.!SP C ;/  c=z d uQ !SP C . /  !c e  00 d 2 .!SP C ;  / d " ) #1=2  .z=c/.!SP  ;/  c=z d C  uQ !SPd ./  !c e 2 00 .!SPd ;  / (13.15)

as z ! 1 uniformly for all   1C" with arbitrarily small " > 0. Although numerically determined distant saddle point positions (as a function of  ) may always be used in the exact expressions for the complex pahse function and its second derivative appearing in this equation in order to obtain the precise asymptotic behavior of the Sommerfeld precursor for a given input pulse, approximate analytic expressions of these quantities are useful in their own right. This approximate analysis of the complex phase behavior at the distant saddle points is now treated separately for the single and double resonance cases.

13.2.1.1

The Single Resonance Case

From (12.184) for the approximate behavior of the complex index of refraction in a single resonance Lorentz model dielectric, the approximate behavior of the complex phase function .!;  / i !.n.!/  / in the region j!j > !1 of the complex !-plane above the absorption band is given by .!;  / i !.1  /  i

b2 : 2.! C 2i ı/

The same approximate expression applies to a Drude model conductor [see (12.308)] with b D !p , and ı D =2. Differentiation of this approximate expression twice with respect to ! then yields

13.2 The Sommerfeld Precursor Field

397

 0 .!/ i.1  / C i  00 .!/ i

b2 ; 2.! C 2i ı/2

b2 : .! C 2i ı/3

Thus, at ! D !SP ˙ . / with !SP ˙ ./ given by (13.11), one obtains d

d

"   .!SP ˙ ; / ı 1 C . / .  1/ C

 2   # b =2 1  . /  2  2 . / C ı 2 1  . / " # b 2 =2

i./   1 C  2 ; (13.16)  2 ./ C ı 2 1  . /

d

and

b2  00 .!SP ˙ ; / i

 3 : d ˙./ C i ı 1  . /

(13.17)

The second coefficient in the Taylor series expansion (10.3) of the complex phase function .!;  / is then given by p0 .!SP ˙ ; / d

 00 .!SP ˙ ;  / d

2Š

b 2 =2

i

 3 : ˙./ C i ı 1  . /

(13.18)

  The proper value of the quantity ˛N 0˙ arg  p0 .!SP ˙ ;  / must now be d determined according to the convention defined in Olver’s method [see (10.7)]. C For simplicity, the Olver-type path Pd ./ through the distant saddle point SPdC in the right-half of the complex !-plane is taken to locally lie along the path of steepest descent through that saddle point, as illustrated in Fig. 13.2. From (12.208),

'

+

SPd

_

+

Pd ( )

Fig. 13.2 Illustration of the steepest descent choice of the Olver-type path PdC . / through the distant saddle point SPdC . The shaded area in the figure indicates the local region of the complex !-plane about the saddle point where the inequality  .!;  / <  .!SP C ;  / is satisfied d

398

13 Evolution of the Precursor Fields

the angle of slope of the contour at the saddle point is then given by ˛N C D =4. Therefore, because  arg .z/ D 0, the proper value of ˛N 0C , as determined by the inequality in (10.7), is ˛N0C ' =2. Through a similar argument, the proper value of ˛N 0 arg p0 .!SPd ; / is ˛N 0 ' =2. With these approximate results, the coefficients a0 .!SP ˙ ;  / given in (13.14) for d the asymptotic expansion (13.12) are found to be given by

 3 1=2 1

˙ ./ C i ı 1  . / d 2i b 2    i  3=2  1=2 1  3  4  . / ˙ ıi 1  . /  ./ ;

p uQ !SP ˙ . /  !c e d 2 2b (13.19)

  a0 .!SP ˙ ; / uQ !SP ˙ . /  !c d



where the facts that 0 < 1  . / < 1 and ./  ı for all  2 .1; 1 have been employed in the expansion of the square root expression. With these substitutions in the asymptotic expansion (13.12), one obtains the nonuniform asymptotic approximation of the Sommerfeld precursor field in a single resonance Lorentz model dielectric as [3, 4, 17] s

   z  .b 2 =2/.1  . // c./ exp ı 1 C . / .  1/ C 2 2z c  . / C ı 2 .1  . //2 (      3  < i e i uQ !SP C ./  !c ./ C ıi 1  . / d 2  

 z b 2 =2   exp i ./   1 C 2 C c  ./ C ı 2 .1  . //2 4       3 CQu !SPd . /  !c ./  ıi 1  . / 2 

)   b 2 =2  z ./   1 C 2 C  exp i ; c  ./ C ı 2 .1  . //2 4

1 As .z; t /  b

(13.20) as z ! 1 uniformly for all   1 C " with arbitrarily small " > 0. This expression also describes the asymptotic behavior of the Sommerfeld precursor in a Drude model conductor with b D !p and ı D =2.

13.2.1.2

The Double Resonance Case

From (12.267) for the approximate behavior of the complex index of refraction of a double resonance Lorentz model dielectric, the approximate behavior of the

13.2 The Sommerfeld Precursor Field

399

complex phase function .!; / in the region j!j > !3 of the complex !-plane above the upper absorption band is given by .!;  / i !.1  /  i

b22 b02 i : 2.! C 2i ı0 / 2.! C 2i ı2 /

Differentiation of this expression twice with respect to ! then yields b02 b22 C i ; 2.! C 2i ı0 /2 2.! C 2i ı2 /2 b02 b22  00 .!/ i  i : .! C 2i ı0 /3 .! C 2i ı2 /3  0 .!/ i.1   / C i

Thus, at ! D !SP ˙ . / with !SP ˙ ./ given by (13.11) with ı D ı0 Cı2 , one obtains d

d

"   .!SP ˙ ; / .ı0 C ı2 / 1 C . / .  1/ 

1 2

 2  b0 C b22 . /

#

 2 ./ C .ı0 C ı2 /2 2 . / #   1 b02 C b22 2

i. /   1 C ; (13.21)  2 ./ C .ı0 C ı2 /2 2 ./

d

"

and  00 .!SP ˙ ; / i d

b02 C b22 Œ˙./  i.ı0 C ı2 /. / 3

:

(13.22)

The second coefficient in the Taylor series expansion (10.3) of the complex phase function .!;  / is then given by p0 .!SP ˙ ; /

 00 .!SP ˙ ;  / d

2Š

d

i

1 2

 2  b0 C b22

Œ˙./  i.ı0 C ı2 /. / 3

:

(13.23)

  As in the single resonance case, the proper value of ˛N 0˙ arg p0 .!SP ˙ ;  / , as d

determined by the inequality in (10.7), is ˛N 0˙ ' ˙=2. With these approximate results, the coefficients a0 .!SP ˙ ;  / given in (13.14) for d the asymptotic expansion (13.12) are found to be given by  

uQ !SP ˙ . /  !c 1

3 1=2 d ˙ ./  i.ı0 C ı2 /. / a0 .!SP ˙ ; /

q d 2i b02 C b22     uQ !SP ˙ . /  !c 3 i 4 3=2 1=2 d  ./ i.ı0 C ı2 /. / ./ :

q e 2 2.b 2 C b 2 / 0

2

(13.24)

400

13 Evolution of the Precursor Fields

With these substitutions in the asymptotic expansion (13.12), one obtains the nonuniform asymptotic approximation of the Sommerfeld precursor field in a double resonance Lorentz model dielectric as s c./ As .z; t /  2.b02 C b22 /z ( " #) 1 2 2  .b C b /./ z  0 2  exp .ı0 C ı2 / 1 C . / .  1/  2 2 c  ./ C .ı0 C ı2 /2 2 . / (     3  < i e i uQ !SP C ./  !c ./  i.ı0 C ı2 /. / d 2 ( " !# ) 1 2 2 .b C b / z  0 2 2  exp i ./   1 C 2 C c  ./ C .ı0 C ı2 /2 2 . / 4     3 CQu !SPd . /  !c ./ C i.ı0 C ı2 /. / 2 ( " !# )) 1 2 .b C b22 / z  2 0 ./   1 C 2  exp i C ; c  ./ C .ı0 C ı2 /2 2 . / 4 (13.25) as z ! 1 uniformly for all   1 C " with arbitrarily small " > 0.

13.2.2 The Uniform Approximation As the space–time parameter  ct=z is allowed to approach unity from above, the asymptotic approximations given in (13.20) and (13.25) lose their validity and each must be replaced by the uniform asymptotic representation presented in Theorem 3 of Sect. 10.2. In their 1969 paper [16], Handelsman and Bleistein performed the required analysis for the step-function envelope signal using the first approximation of the distant saddle point locations, obtaining a result derived by Sommerfeld [2] in 1914. As Handelsman and Bleistein pointed out in this paper, their result is not truly uniform because the first approximation of the distant saddle point locations is useful only for very small, positive values of the quantity   1 so that their asymptotic approximation of the Sommerfeld precursor As .z; t / is valid only for space–time values  in the vicinity of  D 1. As shown by Oughstun and Sherman [4, 20] in 1989, when the second approximation of the distant saddle point locations is used, a uniform asymptotic approximation of the first precursor field As .z; t / is obtained that is valid uniformly for all   1. The results of this modern asymptotic theory are now presented. From Theorem 3 (due to Handelsman and Bleistein) and the results of Sect. 12.4, the uniform asymptotic expansion of the contour integral appearing in the integral

13.2 The Sommerfeld Precursor Field

401

representation (13.10) taken over the two Olver-type paths PdC ./ and Pd . / through the distant saddle points SPd and SPdC , respectively, results in a uniform asymptotic expansion of the first or Sommerfeld precursor field that is given by [17, 20] ( z

As .z; t / D < e i c ˇ./ e i h

 0 J

z c

h i   2˛./e i 2 

˛. / C 2˛./e

i 2

1 JC1

z c

i ˛. /

) C R1 .z;  /; (13.26)

as z ! 1 for all   1. The remainder term R1 .z;  / is bounded by the inequality given in (10.25) of Theorem 3 for all   1 with the constant K independent of  , (13.26) then providing the dominant term in the asymptotic expansion of the first precursor field that is uniformly valid for all   1. The real parameter  which sets the order of the Bessel functions J . / and JC1 . / appearing in the uniform expansion (13.26) is defined by (10.21). Finally, the coefficients appearing in (13.26) are defined as [from (10.26)–(10.29)]  i .!SP C ;  /  .!SPd ;  / ; d 2  i ˇ./ .!SP C ;  / C .!SPd ;  / ; d 2 ˛. /

(13.27) (13.28)

and "

#1=2 ˛ 3 ./ i  00 .!SP C ;  / d " #1=2 uQ .!SPd  !c / ˛ 3 . / C

; .1C/  00 i  .!SP C ;  /  2˛./ d " ( #1=2 3 u Q .! C  !c / 1 ˛ ./ SPd 1 . /

.1C/ 00 2˛./ i  .!SP C ;  / 2˛./ d " #1=2 ) uQ .!SPd  !c / ˛ 3 ./ 

; .1C/  00 i  .!SP C ;  /  2˛./

uQ .!SP C  !c / d 0 . /

.1C/ 2˛./

(13.29)

(13.30)

d

h i1=2 where the branch of the square root expression ˙˛ 3 . /=i  00 .!SP ˙ ;  / appeard ing in (13.29) and (13.30) is determined by the limiting relation given in (10.32) of Theorem 3. Explicit expressions for these coefficients in the single and double resonance cases are now given.

402

13 Evolution of the Precursor Fields

13.2.2.1

The Single Resonance Case

From the second approximate expressions given in (13.16) and (13.17) for the complex phase behavior at the distant saddle points in a single resonance Lorentz model dielectric (as well as for a Drude model conductor), one obtains the approximate expressions [20]   b 2 =2 ; ˛. / ./   1 C  2 ./ C ı 2 .1  . //2 "  2   #   b =2 1  . / ˇ./ i ı 1 C . / .  1/ C  2 ;  2 ./ C ı 2 1  . /

(13.31) (13.32)

and   . 12  / b 2 =2  1=2 . / 0 . / .1C/ ./   1 C 2 b  2 ./ C ı 2 .1  . //2 (   3  uQ .!SP C  !c / ./ C i ı.1  . // d 2 )  3 C.1/.1C/ uQ .!SPd  !c / ./  i ı.1  . // ; 2   . 12 C / b 2 =2  1=2 . / 1 . / .2C/ ./   1 C 2 b  2 ./ C ı 2 .1  . //2 (   3  uQ .!SP C  !c / ./ C i ı.1  . // d 2 )  3 .1C/ .1/ uQ .!SPd  !c / ./  i ı.1  . // : 2

(13.33)

(13.34)

h i1=2 The branch of the square root expression ˙˛ 3 ./=i  00 .!SP ˙ ;  / appearing in d (13.29) and (13.30) has been determined by the limiting relation given in (10.32) in the following manner. In accordance with Theorem 3, the argument of the quantity ˙i  00 .!SP ˙ ; / b 2 = Œ./ ˙ i ı.1  . // 3 as determined by the inequality d

given in (10.7) with  arg .i z/ D =2 and ˛N C D =4 is approximately zero [because ./  ı.1  . // for all   1]. Then, according to (10.32), arg .˛. // D 23 arg .a1 / in the limit as  approaches unity from above, where [from the Laurent series expansion given in (10.20) applied to (12.184)], a1 D b 2 =2, which is real and positive. Hence, arg .a1 / D 0 so that lim!1C farg .˛. //g D 0. By continuity, this result is approximately valid for all   1. This branch

13.2 The Sommerfeld Precursor Field

403

requirement, together with the inequality ./  ı.1  . // for all   1, has been used in obtaining the approximations given in (13.33) and (13.34) for the coefficients 0 . / and 1 . /, respectively. Substitution of these second approximate expressions for the coefficients ˛. /, ˇ./, 0 . /, and 1 . / into the uniform asymptotic expansion of the Sommerfeld precursor given in (13.26) then yields [4, 17, 20]  1=2 b 2 =2 ./  1C As .z; t /  2b  2 ./ C ı 2 .1  . //2 ( "  #)  2   b =2 1  . / z   exp ı 1 C . / .  1/ C  2 c  2 . / C ı 2 1  . / ((   3 i . 2 C / <e uQ .!SP C  !c / ./ C i ı.1  . // d 2 )  3 C.1/.1C/ uQ .!SPd  !c / ./  i ı.1  . // 2    b 2 =2 z ./   1 C J c  2 . / C ı 2 .1  . //2 (   3 i 2 uQ .!SP C  !c / ./ C i ı.1  . // Ce d 2 )  3 .1C/ uQ .!SPd  !c / ./  i ı.1  . // .1/ 2   )  z b 2 =2 JC1 ./   1 C c  2 ./ C ı 2 .1  . //2 (13.35) as z ! 1 uniformly for all   1. This expression constitutes the second-order approximation of the uniform asymptotic approximation of the first or Sommerfeld precursor field in single resonance Lorentz model dielectrics (as well as in Drude model conductors with b D !p and ı D =2). As shown at the end of Sect. 10.2, this result reduces to the nonuniform asymptotic approximation given in (13.20) for values of  > 1 bounded away from unity.

13.2.2.2

The Double Resonance Case

From the second approximate expressions given in (13.21) and (13.22) for the complex phase behavior at the distant saddle points in a double resonance Lorentz model dielectric, one obtains the approximate expressions

404

13 Evolution of the Precursor Fields

" ˛. / ./   1 C "

1 2 .b 2 0

 2 ./

#

C b22 / ; C .ı0 C ı2 /2 2 ./

  ˇ./ i.ı0 C ı2 / 1 C . / .  1/ 

(13.36)

# C b22 /./ ; (13.37)  2 ./ C .ı0 C ı2 /2 2 . / 1 2 .b 2 0

and " !#. 12  / 1 2 .b C b22 /  1=2 . / 2 0 ./   1 C 2 0 . /

q  ./ C .ı0 C ı2 /2 2 . / 2.1C/ b02 C b22 (   3  uQ .!SP C  !c / ./  i.ı0 C ı2 /./ d 2  ) 3 C.1/.1C/ uQ .!SPd  !c / ./ C i.ı0 C ı2 /./ ; (13.38) 2 !#. 12 C / " 1 2 .b C b22 /  1=2 . / 2 0 1 . /

./   1 C 2 q  ./ C .ı02 C ı22 /2 ./ 2.2C/ b02 C b22 (   3  uQ .!SP C  !c / ./  i.ı0 C ı2 /./ d 2  ) 3 .1C/ .1/ uQ .!SPd  !c / ./ C i.ı0 C ı2 /. / ; (13.39) 2 where all branch choices are determined in the same manner as in the single resonance case. With these substitutions, the uniform asymptotic expansion of the Sommerfeld precursor given in (13.26) yields " #1=2 1 2 .b C b22 / ./ 2 0  1C 2 As .z; t /  q  ./ C .ı0 C ı2 /2 2 ./ 2 b02 C b22 ( " #) 1 2  .b C b22 /./ z  2 0  exp ı 1 C . / .  1/ C 2 c  ./ C .ı0 C ı2 /2 2 ./ ((   3  <e i . 2 C / uQ .!SP C  !c / ./  i.ı0 C ı2 /. / d 2  ) 3 .1C/  C.1/ uQ .!SPd  !c / ./ C i.ı0 C ı2 /. / 2 " !# 1 2 .b C b22 /./ z 2 0 ./   1 C 2 J c  . / C .ı0 C ı2 /2 2 ./

13.2 The Sommerfeld Precursor Field

405

(

  3  Ce i 2 uQ .!SP C  !c / ./  i.ı0 C ı2 /./ d 2  ) 3 .1C/ .1/ uQ .!SPd  !c / ./ C i.ı0 C ı2 /. / 2 " !# ) 1 2 .b C b22 /./ z 2 0 ./   1 C 2 JC1 c  ./ C .ı0 C ı2 /2 2 ./ (13.40) as z ! 1 uniformly for all   1. This expression constitutes the second-order approximation of the uniform asymptotic approximation of the first or Sommerfeld precursor field in a double resonance Lorentz model dielectric. This result reduces to the nonuniform asymptotic approximation given in (13.25) for values of  > 1 bounded away from unity.

13.2.3 Field Behavior at the Wavefront By Sommerfeld’s relativistic causality theorem (Theorem 6 of Sect. 13.1), the propagated plane wavefield due to any initial pulse f .t / at the plane z D 0 that identically vanishes for all t < 0 will identically vanish for all superluminal space– time points  ct =z < 1 for all z > 0. If the initial pulse f .t / is then abruptly turned on at time t D 0, the propagated wavefront arrival then occurs at the luminal space–time point  D 1. To investigate the propagated wavefield behavior at this point, attention is now given to the limiting behavior of the uniform asymptotic approximation [as given in either (13.35) for the single resonance case or in (13.40) for the double resonance case] of the Sommerfeld precursor field As .z; t / as  approaches unity from above. In this limit, the functions ./ and . / attain the limiting forms [see (12.202) and (12.203) for the single resonance case and (12.272) and (12.273) for the double resonance case] b lim ./ D p ; C !1 2.  1/ lim . / D 1;

!1C

q where b D b02 C b22 for the double resonance case. In this limit, the argument of the Bessel functions J . / and JC1 . / appearing in the uniform asymptotic expansion of the Sommerfeld precursor becomes  lim

!1C

  b 2 =2 z z p ./   1 C D b 2.  1/: 2 c c  2 ./ C ı 2 .1  . //

406

13 Evolution of the Precursor Fields

Consequently, for subluminal space–time values  very close to unity, the argument of the Bessel functions appearing in the uniform asymptotic expansion of the Sommerfeld precursor is sufficiently small that the small argument limiting form of these Bessel functions may be employed, where (for integer values of the order ) [21] J . / 

 1 

2 ;  . C 1/

as ! 0 with  fixed and nonnegative. For negative values of the order , the relation J . / D .1/ J . / may be employed in conjunction with the above asymptotic expression to obtain  1 

2 ; J . /  .1/  . C 1/ 

as ! 0 with  fixed and nonnegative. For integer   0, substitution of the above results into either (13.25) for the single resonance case or (13.40) for the double resonance case yields the limiting behavior lim As .z; t / 

!1C

z b e 2ı c .1/ p 4  1 ( h i  < e i . 2 C / uQ .!SP C  !c / C .1/1C/ uQ .!SPd  !c / d

 p  bz 1 2.  1/   . C 1/ 2c h i C uQ .!SP C  !c /  .1/1C/ uQ .!SPd  !c / d  p C1 ) i 2 bz e 2.  1/ (13.41)   . C 1/ 2c as z ! 1. Because the initial envelope function u.t / is real-valued, its spectrum satisfies the symmetry property uQ .!/ D uQ  .!  /, and because 0, this expression is, in general, nonzero. This then establishes the following first part of the result for the wavefront velocity:  For integer   0, the front of the first (or Sommerfeld) precursor field in either

a Lorentz model dielectric or a Drude model conductor travels with the velocity of light c in vacuum. For  D 1, the limiting behavior of the first precursor field for space–time values  close to but greater than or equal to unity becomes

13.2 The Sommerfeld Precursor Field

lim As .z; t / 

!1C

407

(

i bz h uQ .!SP C  !c / C uQ .!SPd  !c / d 2c ) i h 1 p uQ .!SP C  !c /  uQ .!SPd  !c / d 2.  1/ (13.42)

b 2ı z .1/ i <e p e c 2 2

i

as z ! 1. Notice that this expression diverges as .  1/1=2 as  ! 1C but is finite for any  D 1 C " with arbitrarily small " > 0. This then establishes the following second part of the result for the wavefront velocity:  For negative integer  D 1, the front of the first (or Sommerfeld) precursor field

in either a Lorentz model dielectric or a Drude model conductor is singular and this point travels with the velocity of light c in vacuum. For smaller negative integer values of  (viz., for  D 2; 3; 4; : : : ), the initial envelope spectrum uQ .!/ does not remain finite in the limit as j!j ! 1, so that the preceding analysis does not apply. Such initial pulse envelope functions u.t / are excluded from this analysis.

13.2.4 The Instantaneous Oscillation Frequency The instantaneous angular frequency of oscillation of the Sommerfeld precursor field is defined [6, 7] as the time derivative of the oscillatory phase. Notice that the oscillatory phase terms appearing in the uniform and nonuniform asymptotic approximations of the Sommerfeld precursor are identical. Because d=dt D c=z and cb 2  d . / D  ; 2 dt z./ .  1/2 2b 2 c ../  1/ d ./ D  ; dt z . 2  1/2  2 . / then the instantaneous angular frequency of oscillation of the first precursor field in a single resonance Lorentz medium is given by i d hz ˛. / dt c    b 2 =2 d z ./   1 C ' dt c  2 ./ C ı 2 .1  . //2 ( ) 2 2 2 b2 2  ./  5ı .1  . // D ./ C b  2.  1/; Œ 2 ./ C ı 2 .1  . //2 2 2./ . 2  1/2 (13.43)

!s . /

408

13 Evolution of the Precursor Fields

which may be approximated as n o !s . / < !SP C ./ D ./:

(13.44)

d

An analogous derivation shows that the approximation given in (13.44) also holds in the double resonance case. Although an approximation, the identification of the instantaneous angular frequency of oscillation of the Sommerfeld precursor as being given by the real part of the distant saddle point location !SP C ./ in the d right half of the complex !-plane is intuitively pleasing. Furthermore, notice that this notion of an instantaneous angular frequency of oscillation is, strictly speaking, only a heuristic mathematical identification which, in certain circumstances, may yield completely erroneous or misleading results [22]. This is not the case for the Sommerfeld precursor whose instantaneous oscillation frequency monotonically decreases with increasing  from q an initial infinite value at  D 1, approaching !12  ı 2 for a single resonance Lorentz model q medium or the limiting value .1/ D !32 C b02  ıN2 for a double resonance Lorentz model medium as  ! 1, where ıN .ı0 C ı2 /=2. either the limiting value .1/ D

13.2.5 The Delta Function Pulse Sommerfeld Precursor Because uQ .!  !c / D i for a delta function pulse [compare (11.51) and (13.10) with D 0], then  D 1 [see (10.21)–(10.22)]. As was pointed out in Sect. 10.2, the uniform asymptotic expansion stated in Theorem 3 remains applicable for all values of   1 when  < 0 provided that its limiting behavior as  ! 1C remains finite [16]. With these substitutions, the uniform asymptotic approximation given in (13.26) becomes [4, 17, 20] Aıs .z; t / 

 1=2 b 2 =2 ./  1C 2b  2 ./ C ı 2 .1  . //2 ( "  #)  2   b =2 1  . / z   exp ı 1 C . / .  1/ C  2 c  2 ./ C ı 2 1  . / (    z b 2 =2   2. /J1 ./   1 C c  2 ./ C ı 2 .1  . //2    )   b 2 =2 z ./   1 C C3ı 1  ./ J0 c  2 . / C ı 2 .1  . //2 (13.45)

13.2 The Sommerfeld Precursor Field

409

as z ! 1 either in a single resonance Lorentz model dielectric or in a Drude model conductor (with b D !p and ı D =2). A similar expression is obtained for a double resonance Lorentz model dielectric. Because this expression diverges as  ! 1C , then this uniform asymptotic approximation of the Sommerfeld precursor for a delta function pulse is valid for all  > 1 as z ! 1. The wavefront behavior of the propagated field due to a delta function pulse certainly merits further investigation. Substitution of the limiting forms taken by ./ and ./ as well as by the argument of the Bessel functions appearing in (13.45), as given at the beginning of Sect. 13.2.3, into (13.45) results in the asymptotic approximation Aıs .z; t /  b

z J1 .b/ ı z .1/ e c ; c 

(13.46)

p where  .z=c/ 2.  1/. For sufficiently small values of b one may substitute the first term in the small argument expansion of the Bessel function J1 .b/ with the result b 2 z 2ı z .1/ e c ; (13.47) Aıs .z; t /   2c as z ! 1 and  ! 1C . The field behavior right on the wavefront at  D 1 is then seen to be given by b2z (13.48) Aıs .z; t /   2c for large but finite propagation distances z. If this result is valid, then the magnitude of the wavefront would appear to grow with increasing propagation distance in the dispersive medium! To investigate this result in more detail, return to the exact integral representation of the propagated wavefield for the delta function pulse given in (11.51) with t0 D 0, which may be rewritten as2 Aı .z; t / D

1 < 2

Z

1

z

e c .!;/ d!;

(13.49)

1

the integration being taken along the real frequency axis. Because .!;  / ' i ! .1   / for sufficiently large j!j, then Z ˝ z 1 e c .!;/ d! < 2 j!j ˝ ˝ Z 1 Z ˝h i z z z 1 1 < < e c .!;/  e c i!.1/ d!; e c i!.1/ d! C D 2 2 1 ˝

Aı .z; t / '

2

1 < 2

Z

z

e c i!.1/ d! C

This derivation is based upon an analysis due to George C. Sherman as conveyed to me in a private communication in 2007.

410

13 Evolution of the Precursor Fields

which then yields Aı .z; t / ' ı.t  z=c/ C G.z; t /;

(13.50)

where the function 1 < G.z; t / 2

Z

˝

e

z c .!;/

˝

1 sin d!  

 z c z c

  t ˝ t

(13.51)

is bounded for all finite z. Consider now the remaining integral in this expression, given by Z ˝ z 1 g.z; t / e c .!;/ d!; (13.52) 2 ˝ where ˝ is taken to be larger than the distance j!SP ˙ . /j of the distant saddle d point from the origin. To evaluate this integral asymptotically, the original contour of integration from ˝ to C˝ along the real ! 0 -axis is deformed to the contour CI [ CII [ CIII illustrated in Fig. 13.3. The contour CII is an Olver-type path through the distant SPd˙ and near SPnC saddle points (as well as through the upper middle saddle points in a double resonance medium), the contour CI extending vertically downward from the point ˝ until it intersects CII at the point !I , and the contour CIII extending vertically upward from the point !III on the contour  . Because .!; / ' i !.1   / along both paths CI CII to ˝, where !I D !III and CIII , the integrals along them may be directly evaluted as Z

z z e i . c t /!I  e i . c t /˝   e d! D ; 2 i cz  t ˝ z z Z ˝ z 1 e i . c t /˝  e i . c t /!III z  e c i!.1/ d! D gIII .z; t / D ; 2 !III 2 i c  t

1 gI .z; t / D 2

!I

z c i!.1/

'' SP n CII CI

SP d

' CIII

SP d CII

I

III

Fig. 13.3 Deformed contour of integration CI [ CII [ CIII for the asymptotic evaluation of the function g.z; t /. The contour CII is an Olver-type path through the distant SPd˙ and near SPnC saddle points

13.2 The Sommerfeld Precursor Field

411

so that g.z; t / D

 z

     t ˝ sin cz  t !I0 !I00 . cz t /     Ce C gII .z; t /;  cz  t  cz  t

sin

c

(13.53) where !I D !I0 C i !I00 . Substitution of this result into (13.51) then gives G.z; t / D e

!I00 . cz t /

sin

  ˚  t !I0 z  C < gII .z; t / :  c t  z

c

(13.54)

The first term on the right in this equation is asymptotically negligible in comparison to the contribution from the distant saddle points because !I lies on an Olver-type path through those saddle points. In addition, the uniform asymptotic expansion of the second term in this equation yields the same result that was obtained for the Sommerfeld precursor, so that G.z; t /  Aıs .z; t /;

(13.55)

as z ! 1 for small   1  0. The propagated wavefront behavior due to a delta function pulse is then given by [from (13.50) and (13.55)] Aı .z; t /  ı.t  z=c/ C Aıs .z; t /

(13.56)

as z ! 1, where Aıs .z; t / denotes the Sommerfeld precursor for the delta function pulse with uniform asymptotic approximation given in (13.45). The same result holds for the double resonance case. A similar result was obtained by He and Str¨om [23] in 1996 using a time-domain wave-splitting technique and by Karlsson and Rikte [24] in 1998 using a time-domain method based on dispersive wave splitting. With this result, (13.46)–(13.48) for the propagated field behavior near the wavefront due to a delta function pulse become z J1 .b/ ı z .1/ e c c  b 2 z 2ı z .1/ e c  ı.t  z=c/  2c b2z  ı.t  z=c/  2c

Aı .z; t /  ı.t  z=c/  b

(13.57) (13.58) (13.59)

as z ! 1 for small   1  0, each succeeding expression valid for smaller values of the quantity   1, the final expression only holding right on the wavefront at  D 1. This result then shows that the function of the term b 2 z=2c is to decrease the contribution of the delta function at the wavefront as the propagation distance z increases, the wavefront propagating at the vacuum speed of light c.

412

13 Evolution of the Precursor Fields

13.2.6 The Heaviside Step Function Pulse Sommerfeld Precursor For a Heaviside unit step function modulated signal with D 0, the spectrum of the initial envelope function uH .t / is given by (11.56), so that uQ H .!SP ˙  !c / D d

i !SP ˙ ./  !c d

D

i   ˙./  !c  i ı 1 C . /

(13.60)

for both Lorentz model dielectrics and Drude model conductors. It then follows that  D 0 in the uniform asymptotic approximation given in (13.35) and that this asymptotic approximation is uniformy valid for all   1. The uniform asymptotic approximation of the Sommerfeld precursor for the Heaviside unit step function modulated signal in either a single resonance Lorentz model dielectric or a Drude model conductor is then given by [4, 17, 20] AH s .z; t / 

 1=2 b 2 =2 ./  1C 2b  2 ./ C ı 2 .1  . //2 ( "  #)  2   b =2 1  . / z   exp ı 1 C . / .  1/ C  2 c  2 ./ C ı 2 1  . / ( "     ı ./ 5  . / C 3!c 1  . /   2  2 2 ./ C !c C ı 2 1 C . /  #   ./ 5  . /  3!c 1  . /   2  2 ./  !c C ı 2 1 C . /    z b 2 =2 J0 ./   1 C c  2 ./ C ı 2 .1  . //2 "  3 2   ./ ./  !c  2 ı 1  2 . / C  2  2 ./  !c C ı 2 1 C . /    # ./ ./ C !c  32 ı 2 1  2 ./   2  2 ./ C !c C ı 2 1 C . /    ) b 2 =2 z ./   1 C J1 c  2 . / C ı 2 .1  . //2 (13.61)

as z ! 1 uniformly for all   1. An analogous expression holds for a double resonance Lorentz model dielectric (see Problem 13.2). It is evident from this result that the Sommerfeld precursor AH s .z; t /, and consequently the total propagated

13.2 The Sommerfeld Precursor Field

413

wavefield AH .z; t / due to an input Heaviside step function signal, vanishes on the wavefront at  D 1, but is nonzero for  D 1 C " where " > 0 is arbitrarily small. Consequently, the wavefront travels at the vacuum speed of light c. For space–time values  > 1 bounded away from unity, the two Bessel functions appearing in (13.61) may be replaced by their own large argument asymptotic approximations given by [see Sect. 10.2] s J . / D

     2 cos    C O 1 e j=. /j  2 4

as ! 1 with j arg . /j < . With this substitution in (13.61), the nonuniform asymptotic approximation of the first precursor results, given by [3, 4, 17] s ( "   #)  z  1 c./ .b 2 =2/ 1./ exp ı AH s .z; t /   1 C . / .  1/ C  2 b 2z c  2 ./Cı 2 1./ ("     ./ ./  !c  32 ı 2 1  2 ./   2  2 ./  !c C ı 2 1 C . /    # ./ ./ C !c  32 ı 2 1  2 ./   2  2 ./ C !c C ı 2 1 C . /     b 2 =2  z C  cos ./   1 C c 4  2 ./ C ı 2 .1  . //2 "     ı ./ 5  . / C 3!c 1  . / C  2  2 2 ./ C !c C ı 2 1 C . /  #   ./ 5  . /  3!c 1  . /   2  2 ./  !c C ı 2 1 C . /  )   b 2 =2 z   sin ./   1 C C c 4  2 . / C ı 2 .1  . //2 (13.62) as z ! 1 with   1 C " with " > 0. The same result is obtained from the nonuniform asymptotic expression in (13.20) with substitution from (13.60). The dynamical evolution with  D ct =z of the first, or Sommerfeld, precursor field AH s .z; t / for the Heaviside unit step function envelope modulated signal with below resonance angular carrier frequency !c D 1  1016 r=s is illustrated in Figs. 13.4 and 13.5 at a fixed observation distance z D zd of one absorption depth, where zd ˛ 1 .!c /, the field evolution illustrated in Fig. 13.5 being a close-up view of the wave form immediately following the wavefront. This temporal field evolution was computed using the uniform asymptotic approximation given in (13.61) for a single resonance Lorentz model dielectric with numerically

414

13 Evolution of the Precursor Fields 0.003

0.002

AHs(z,t)

0.001

0

−0.001

−0.002

−0.003

1

1.1

1.2

Fig. 13.4 Temporal evoution of the Sommerfeld precursor field AH s .z; t / at one absorption depth z D zd ˛ 1 .!c / in a single resonance Lorentz p model dielectric with Brillouin’s choice of the medium parameters (!0 D 4  1016 r=s, b D 20  1016 r=s, ı D 0:28  1016 r=s) for a Heaviside unit step function modulated signal with below resonance angular carrier frequency !c D 1  1016 r=s 0.003

0.002

AHs(z,t)

0.001

0

−0.001

−0.002

−0.003

1

1.001

1.002

Fig. 13.5 Close-up of the leading edge of the Sommerfeld precursor field AH s .z; t / depicted in Fig. 13.4. The solid curve results from numerically determined saddle point locations and the dashed curve from the second approximate expressions

13.2 The Sommerfeld Precursor Field

415

determined distant saddle point locations using Brillouin’s choice of the medium parameters [see (12.58)]. The accuracy of these results is remarkably good when compared with purely numerical results and only improve as the propagation distance z increases. The rapid amplitude build-up of the first precursor field envelope from its zero value on the wavefront at  D 1 to a maximum value is clearly evident in both figures. For larger values of  , the amplitude damps out exponentially with increasing  . The instantaneous angular frequency of oscillation is also seen to rapidly decrease as  increases away from unity, this decrease becoming less rapid as  continues to increase, as seen in Fig. 13.4. Finally, as the propagation distance z increases away from the initial plane at z D 0, the peak amplitude of the Sommerfeld precursor diminishes and shifts to earlier space–time points, approaching the wavefront at  D 1 as z ! 1. Finally, notice that the relatively small peak amplitude of the Sommerfeld precursor field depicted in Figs. 13.4 and 13.5, as well as in Fig. 13.1, is due to the fact that the input carrier frequency !c of the signal is below the medium resonance frequency !0 . As !c is increased above !0 , the spectral amplitude uQ .!  !c / increases there and so the peak amplitude of the Sommerfeld precursor also increases relative to the remainder of the propagated wavefield. For example, at one absorption depth in the same medium with above resonance carrier frequency !c D 7  1016 r=s, the peak amplitude of the Sommerfeld precursor is found to be ŒAH s .z; t / peak 0:24. As pointed out earlier, the nonuniform asymptotic expression given in (13.62) is not a valid asymptotic approximation of the first precursor field in the limit as  ! 1C for fixed values of the propagation distance z > 0. To establish connection with the now classical result obtained by Brillouin [6, 7] for the first precursor field, however, the behavior of this expression in that limit is now examined. For space– time values  approximately equal to but greater than unity, (13.62) simplifies to s

3=4 2ı z .1/ bc  2.  1/ e c 2z (" b .2.  1//1=2  !c   2 b .2.  1//1=2  !c C 4ı 2

AH s .z; t /  

#  zp  b .2.  1//1=2 C !c 2.  1/ C  cos b  2 c 4 b .2.  1//1=2 C !c C 4ı 2 " 1 C2ı  2 1=2 b .2.  1//  !c C 4ı 2 # )  zp 1   2.  1/ C sin b 2 c 4 b .2.  1//1=2 C !c C 4ı 2 (13.63)

as z ! 1 with subluminal  1. This same result would be obtained using the first approximate expressions of the distant saddle point locations [see (12.187)]

416

13 Evolution of the Precursor Fields

in the nonuniform asymptotic approximation given in (13.20). This expression can be further simplified by noting that for space–time values  very close to unity, any finite angular carrier frequency !c > 0 will be negligible in comparison to the p quantity b= 2.  1/. Hence, in the limit as  ! 1C , the nonuniform asymptotic approximation given in (13.63) simplifies to s

 1=4  zp 2bc !c 2.  1/  2ı cz .1/ cos b 2.  1/ C AH s .z; t /  e z b 2 C 8ı 2 .  1/ c 4 (13.64) as z ! 1 and  ! 1C . This expression is precisely Brillouin’s result for the first forerunner [6]; see also page 73 of Ref. [7]. Hence, Brillouin’s asymptotic approximation of the first precursor field is an approximation, valid for  near 1, of an expression [(13.62)] that is not valid for  near 1.

13.3 The Brillouin Precursor Field in Lorentz Model Dielectrics The contributions of the near saddle points to the asymptotic behavior of the propagated wavefield A.z; t / for sufficiently large values of the propagation distance z > 0 yield the dynamical space–time evolution of the second or Brillouin precursor field. This contribution to the asymptotic behavior of the total wavefield A.z; t / is denoted by Ab .z; t / and is dominant over the first or Sommerfeld precursor field in single resonance Lorentz model dielectrics, as well as in double resonance Lorentz model dielectrics that satisfy the inequality p > 0 ), for all  > SB , whereas it is dominant over both the Sommerfeld and middle precursor fields for all  > MB in double resonance Lorentz model dielectrics when the inequality p < 0 is satisfied. For Debye model dielectrics, it is the only precursor field and, unlike that for Lorentz model dielectrics, it is due to a single saddle point that moves down the imaginary axis. These two cases must then be treated separately, the more complicated Lorentz model case being treated here first because of its central importance to the classical theory due to Brillouin [6, 7]. From the results presented in Sect. 12.3, the two first-order near saddle points SPn˙ , which are initially isolated from each other at the luminal space–time point  D 1, SPnC situated along the positive imaginary axis and SPn situated along the negative imaginary axis, approach each other along the imaginary axis as  increases to the critical value 1 and coalesce into a single second-order saddle point SPn along the negative imaginary axis when  D 1 , after which they separate into two first-order saddle points and symmetrically move away from each other in the lower half of the complex !-plane as  increases above 1 , approaching in the inner branch points !˙ as  ! 1. A straightforward application of Olver’s theorem is first presented in Sect. 13.3.1 to determine the asymptotic behavior of the Brillouin precursor field in each of the separate space–time domains 1 <  < 1 ,  D 1 , and  > 1 . Because these results are nonuniform in a neighborhood of the critical space–time point  D 1 when the two near first-order saddle points coalesce into a single second-order saddle point and the saddle point order abruptly changes, one

13.3 The Brillouin Precursor Field in Lorentz Model Dielectrics

417

must then resort to the uniform asymptotic expansion due to Chester, Friedman, and Ursell [25] described by Theorem 4 in Sect. 10.3.2 in order to obtain an asymptotic approximation of the Brillouin precursor that is continuous for all  > 1. This is done in Sect. 13.3.2.

13.3.1 The Nonuniform Approximation The asymptotic behavior of the second or Brillouin precursor field Ab .z; t / for a particular initial pulse envelope function u.t / is derived from the asymptotic expansion of the integral representation of the propagated wavefield given in (13.10) taken about the near saddle points. The near saddle point locations may be expressed as [see (12.245), and (12.246), and (12.282)] (

 i ˙ 0 ./  23 ı ./ ; !SP ˙ . / D n ˙ ./  i 23 ı ./;

1    1 ;   1

(13.65)

where [see (12.220) and (12.221)] 2 . / 4

  !02  2  02 2

 2  02 C 3˛ !b 2

31=2 4 2 2 5  ı . / ; 9

(13.66)

0

2

b 2 2 3   0 C 2 !02

. /

; 2  2  02 C 3˛ b 22

(13.67)

!0

for a single resonance Lorentz model dielectric with ˛ given by (12.218), and where [see (12.283) and (12.284)] 31=2  2  2   0 4 7 6   ı02 2 . /5 ;  2 . / 4 2 2 b b ! 9 2 0 2 0 2   0 C 3 ! 2 C ! 4 0 2   2 b ı b22 !04 2 2 2 0 1 C    C 2 2 2 4 0 3 !0 ı0 b0 !2   ;

. /

2  2   2 C 3 b02 1 C b22 !04 0 !2 b2 ! 4 2

!02

0

(13.68)

(13.69)

0 2

for a double resonance Lorentz model dielectric with ı D ı0 in (13.65). In either case, 02 . / D  2 . /. For each of the examples considered in this text, both the spectral function uQ .!  !c / and the complex phase function .!;  / appearing in the integrand of (13.10) are analytic about the two near saddle points for all   1. Because the near saddle point behavior separates into the three separate space–time domains 1 <  < 1 ,  D 1 , and  > 1 , each case is now separately examined.

418

13.3.1.1

13 Evolution of the Precursor Fields

Case 1: 1 <  < 1

For space–time values in the subluminal space–time domain  2 .1; 1 /, the conditions of Olver’s theorem are satisfied at the upper near saddle point SPnC when the original contour of integration C is deformed to the path P ./ D Pd ./CPnC . /C PdC . /, where PnC . / is an Olver-type path with respect to the upper near saddle point SPnC , as illustrated in Fig. 12.66. For a double resonance Lorentz model dielectric that satisfies the inequality p < 0 , the conditions of Olver’s theorem are satisfied at the upper near saddle point SPnC when the original contour of integration is deformed to the path P . / D Pd ./ C PnC ./ C PdC . / for  2 .1; SM , to the C  path P . / D Pd . / C Pm1 . / C PnC ./ C Pm1 ./ C PdC ./ for  2 ŒSM ; p /, C    . / C PnC . / C Pm1 ./ C and then to the path P . / D Pd ./ C Pm1 ./ C Pm2 C C Pm2 . / C Pd . / for  2 .p ; 1 /. In either case, (10.18) applies for the upper near saddle point SPnC with (10.3) and (10.4) taken as Taylor series expansions about this first-order saddle point, in which case  D 2. Furthermore, because uQ .!  !c / is analytic at this saddle point, then  D 1. Hence, from Olver’s theorem (Theorem 2) and the results of Sect. 12.4, the contour integral in (13.10) taken over the Olvertype path PnC . / yields the asymptotic expansion of the initial space–time behavior of the second or Brillouin precursor field Ab .z; t / given by [17–19]  Ab .z; t / D

c z

1=2

< i e i

n  o .z=c/.! C ;/

SPn a0 .!SPnC /e 1 C O z1

(13.70)

as z ! 1 uniformly for all 1 <   1  " with arbitrarily small " > 0. To evaluate the coefficient a0 .!SPnC / appearing in (13.70), the first two coefficients in the Taylor series expansion (10.3) of the complex phase function .!;  / as well as the first coefficient in the Taylor series expansion (10.4) of the initial envelope spectrum uQ .!  !c / about the upper near saddle point SPnC must first be determined. The latter quantity is given by q0 .!SPnC .// D uQ .!SPnC ./  !c /;

(13.71)

the specific form of which depends upon the particular pulse envelope function u.t /. Because the second coefficient in the Taylor series expansion (10.3) of the complex phase function is given by p0 .!SPnC ;  / D  00 .!SPnC ;  /=2Š, the coefficient a0 .!SPnC ; / is found to be given by [see (10.9)] uQ .!SPnC ./  !c / a0 .!SPnC ;  / D h i1=2 : 2 00 .!SPnC ;  /

(13.72)

With this substitution, the nonuniform asymptotic expansion (13.70) of the Brillouin precursor becomes

13.3 The Brillouin Precursor Field in Lorentz Model Dielectrics Fig. 13.6 Illustration of the steepest descent choice of the Olver-type path PnC . / through the upper near saddle point SPnC for 1 <  < 0 . This first-order saddle point crosses the origin at  D 0 n.0/. The shaded area indicates the local region of the complex !-plane about the saddle point where the inequality  .!;  / < .!SPnC ;  / is satisfied

Ab .z; t /  < i e i

419 ''

Pn SPn '

8" 9 #1=2 < = c=z .z=c/.! C ;/ SPn uQ .!SPnC ./  !c /e  00 : 2 .!SP C ;  / ; n

(13.73) small " > 0. as z ! 1 uniformly for all 1 <   1  " with arbitrarily  The proper value of ˛N 0 arg   00 .!SPnC ;  / must now be determined according to the convention defined in Olver’s method [see (10.7)]. For simplicity, the Olver-type path PnC . / through the upper near first-order saddle point SPnC is taken to locally lie along the path of steepest descent through that saddle point, as illustrated in Fig. 13.6. From (12.232), the angle of slope of the contour at the saddle point is given by ˛N D 0. Because  arg .z/ D 0, the proper value of ˛N 0 , as determined by the inequality in (10.7), is ˛N 0 D 0.

The Single Resonance Lorentz Model Dielectric From (12.227), the approximate behavior q of the complex phase function .!;  / in the below resonance region j!j < !02  ı 2 of the complex !-plane about the origin is given by .!;  / i !.0  / C

b2 ! 2 .i ˛!  2ı/: 20 !04

Differentiation of this expression twice with respect to ! then yields  0 .!;  / i.0  / C  00 .!;  /

b2 .3i ˛! 2  4ı!/; 20 !04

b2 .3i ˛!  2ı/; 0 !04

420

13 Evolution of the Precursor Fields

so that at ! D !SPnC . / D i .!SPnC ; /



 2 0 ./  3 ı ./ , there results

 1 2ı . /  3 0 ./ .0  / 3  2  b2  2ı ./  3 0 ./ 2ı 3  ˛ . / C 3˛ C 4 540 !0

0 ./

 ;

(13.74)  b2   00 .!SPnC ; /  2ı 1  ˛ . / C 3˛ 4 0 !0

 0 ./ :

(13.75)

Substitution of these approximate expressions into (13.73) then gives the asymptotic approximation of the second or Brillouin precursor over the initial subluminal space–time domain  2 .1; 1 / as [3, 4, 17] #1=2 " n o !02 0 c=.z/   Ab .z; t /  < i e i uQ .!SPnC  !c / b 4ı 1  ˛ . / C 6˛ 0 ./ z .2ı ./  3 0 .//  exp 3c i h 2 0 ./

(13.76)  0   C b .2ı ./3 0 .//Œ2ı.3˛ .//C3˛ 18 ! 4 0 0

as z ! 1 uniformly for 1 <   1  " with arbitrarily small " > 0. The dynamical evolution of the wavefield described by (13.76) depends on the algebraic sign of the exponential argument appearing in that expression for z > 0. Because the quantity .2ı ./  3 0 .// is negative for 1 <  < 0 , vanishes at  D 0 , and is positive for 0 <  < 1 , and because the quantity Œ2ı.3  ˛ . // C 3˛ 0 . / is positive for 1 <  < 1 , and because the inequality j0  j 

b 2 j2ı . /  3

0 ./j Œ2ı.3

 ˛ .// C 3˛ 180 !04

0 ./

is satisfied for all  2 .1; 1 /, with the equality holding only at  D 0 when both sides of this equation vanish, the argument of this exponential function is negative for 1 <  < 0 , vanishes identically at  D 0 , and is again negative for   0 <  < 1 . Consequently, the second precursor field Ab .z; t / first grows with increasing  as the exponential argument decreases with increasing  2 .1; 0 /, becoming exponentially dominant over the first precursor field when  > SB , where 1 < SB < 0 . At  D 0 the exponential argument identically vanishes (because the approximate expressions for both the upper near saddle point and the complex phase behavior at this saddle point become exact when it crosses the origin) and the wavefield given in (13.76) varies only as z1=2 for ı > 0,3 making the wavefield behavior at this 3

The wavefield behavior in the special case when ı D 0 is examined in Case 2. In that case, 1 D 0 and the two near saddle points SPn˙ coalesce into a single second-order saddle point SPn at the origin.

13.3 The Brillouin Precursor Field in Lorentz Model Dielectrics

421

space–time point strikingly different from the wavefield at any other space–time point in the dynamical field evolution. Finally, for increasing values of  2 .0 ; 1 /, the exponential argument decreases and the field is again exponentially attenuated with propagation distance z > 0. Although the functional form of the second precursor gives it the appearance of being nonoscillatory over this initial space–time domain, it does have an effective oscillation frequency with half period given by the temporal width at the e 1 amplitude point and the peak amplitude point at  D 0 .

The Double Resonance Lorentz Model Dielectric From (12.276), the approximate behavior q of the complex phase function .!;  / in the below resonance region j!j < !02  ı02 of the complex !-plane about the origin is given by .!;  / i!.0   / 

1 0



ı0 b02 ı2 b 2 C 42 4 !0 !2



!2 C

i 20



b02 b2 C 24 4 !0 !2



!3:

Differentiation of this expression twice with respect to ! then yields   2  ı0 b02 b0 ı2 b22 b22 3i ! C !2; C C 20 !04 !04 !24 !24     ı2 b22 b22 3i b02 2 ı0 b02 00  .!;  /  C 4 C C 4 !; 0 !04 0 !04 !2 !2  0 .!;  / i.0   / 

so that at ! D !SPnC . / D i .!SPnC ; /



2 0



0 ./

  23 ı0 . / , there results

 1 2ı0 . /  3 0 ./ 3(    ı0 b02 ı2 b22 1  2ı0 . /  3 0 . /  0   C C 4 30 !04 !2  2  )  b22  1 b0  C 4 2ı0 . /  3 0 ./ ; 6 !04 !2 (13.77)

2  .!SPnC ; /  0 00



ı0 b02 ı2 b22 C !04 !24



1  0



b02 b22 C !04 !24





3

0 ./

  2ı0 . / : (13.78)

Substitution of these approximate expressions into (13.73) then gives the asymptotic approximation of the Brillouin precursor over the initial space–time domain  2 .1; 1 / in a double resonance Lorentz model dielectric as

422

13 Evolution of the Precursor Fields

s

  2   2    1=2 ı0 b 0 b0 ı2 b22 b22  0 c 2 Ab .z; t /  C 4 C C 4 3 0 . /  2ı0 . / 2z !04 !2 !04 !2 n o i < i e uQ .!SPnC  !c /  z .2ı . /  3 0 .// 0    exp 3c h 2   2  i  2 ı0 b0 ı2 b22 b0 b22 .2ı ./3 0 .// 0 ./ 3 ı0 ./ C C !4 C !4 C !4 30 2 !4 0

2

0

2

(13.79) as z ! 1 uniformly for 1 <   1  " with arbitrarily small " > 0. The dynamical behavior described by this equation is the same as that described in the single resonance case.

13.3.1.2

Case 2:  D 1

At the space–time point  D 1 D ct1 =z, the conditions of Olver’s theorem are satisfied at the second-order near saddle point SPn when the original contour of integration C is deformed to the path P .1 / D Pd .1 / C PnC .1 / C PdC .1 /, where PnC .1 / is an Olver-type path with respect to the near saddle point SPn , located at [see (12.236)] 2ı (13.80) !SPn .1 / i ; 3˛ with ˛ D 1  ı 2 .4!12 C b 2 /=.3!02 !12 / 1. For a double resonance Lorentz model dielectric that satisfies the inequality p < 0 , the conditions of Olver’s theorem are satisfied at the second-order near saddle point SPn when the original contour of in  .1 /CPm2 .1 /CPnC .1 /C tegration is deformed to the path P .1 / D Pd .1 /CPm1 C C C Pm1 .1 / C Pm2 .1 / C Pd .1 /, where PnC .1 / is an Olver-type path with respect to the second-order near saddle point SPn whose location is also given by (13.80) with   1 ı D ı0 [cf. (12.281)] and with ˛ D 1 C b22 !04 =b02 !24 1 C .ı2 =ı0 /.b22 !04 =b02 !24 / . Because both of the functions uQ .!  !c / and .!;  / appearing in the integrand of (13.10) are analytic about this second-order near saddle point, then (10.3) and (10.4) are Taylor series expansions about that point with  D 3 and  D 1. Because  D 3, the argument presented in Sect. 10.1.2 leading up to (10.18), which is valid only for the case when  D 2, must now be appropriately modified. The second coefficient in the Taylor series expansion in (10.3) of the complex phase function .!; 1 / about this second-order near saddle point is given by p0 .!SPn ; 1 / D

1 000  .!SPn ; 1 / 3Š

(13.81)

and the first coefficient in the Taylor series expansion in (10.4) of the initial pulse envelope spectrum is given by

13.3 The Brillouin Precursor Field in Lorentz Model Dielectrics

423

    q0 .!SPn 1 / D uQ !SPn .1 /  !c ;

(13.82)

the specific form of which depends upon the particular initial pulse envelope function u.t / under consideration. Notice that for both the single and double resonance cases,  000 .!SPn ; 1 / ' i j 000 .!SPn ;1 /j.  The proper value of ˛N 0 arg   000 .!SPn ; 1 / ' arg .i / must now be determined according to the convention defined in Olver’s method [see (10.7)]. For simplicity, the Olver-type path Pn .1 / through the second-order near saddle point SPn is taken to locally lie along the path of steepest descent through that saddle point, as illustrated in Fig. 13.7. The contour integral for the wavefield A.z; t / along this path may be written as A.z; t1 / D Ab .z; t1 / D I C  I  ; where I ˙ denote the contour integrals taken in opposite directions leading away from the saddle point along that corresponding half of the Olver-type path Pn .1 /, as indicated in Fig. 13.7. The angle of slope of the two steepest descent paths leaving the saddle point are given by [see (12.238) and Fig. 12.41] ˛N C D =6 and ˛N  D 5=6, as indicated in the figure. Consequently, because  arg .z/ D 0, the proper values of ˛N 0˙ for these two paths, as determined by the inequality given in (10.7), is ˛N 0C D =2 for I C and ˛N 0 D 5=2 for I  . The phase difference between the two coefficients a0˙ .!SPn ; 1 / then results in the factor C

e i ˛N 0

=3

 =3

 e i ˛N 0

D e i=6  e i5=6 D 2 cos

  6

D

p

3;

where a0C .!SPn ; 1 /

1=3   3Š 1  D uQ !SPn .1 /  !c e i=6 : 3 j 000 .!SPn ; 1 /j

(13.83)

'' I−

_

I+

Pn

' _ SPn

Fig. 13.7 Illustration of the steepest descent choice of the Olver-type path Pn .1 / through the second-order near saddle point SPn at  D 1 . The shaded area indicates the local region of the complex !-plane about the saddle point where the inequality  .!; 1 / <  .!SPn ; 1 / is satisfied

424

13 Evolution of the Precursor Fields

The asymptotic expansion of the second or Brillouin precursor field Ab .z; t / at the fixed space–time point  D 1 D ct1 =z is then given by [17–19] Ab .z; t / D

1=3   . 13 / 6c p 2 3 j 000 .!SPn ; 1 /jz n  z

 o  < i e i uQ !SPn .1 /  !c e c .!SPn ;1 / 1 C O z1=3 (13.84)

as z ! 1.

The Single Resonance Lorentz Model Dielectric From the set of relations preceeding (13.74), one finds that 2ı .!SPn ; 1 /

3˛  000 .!SPn ; 1 / 3i



4ı 2 b 2 0  1 C 9˛0 !04

˛b 2 : 0 !04

 ;

(13.85) (13.86)

Substitution of these results in (13.84) then yields the asymptotic approximation Ab .z; t / 

  20 !0 c 1=3 ˚ i  < i e uQ !SPn .1 /  !c 2 ˛b z    2ız 4ı 2 b 2 0  1 C  exp (13.87) 3˛c 9˛0 !04

 . 13 / p !0 2 3



as z ! 1 at the fixed space–time point  D 1 D ct1 =z. From (12.225), it is seen that 0  1 C

4ı 2 b 2 2ı 2 b 2

 ; 4 9˛0 !0 9˛0 !04

and hence, as was expected, the second precursor field attenuates exponentially with increasing propagation distance z > 0 at the fixed space–time point  D 1 . In the special (limiting) case when ı D 0, it then follows that 1 D 0 exactly and the two first-order near saddle points coalesce into a single second-order saddle point at the origin. The peak amplitude in the Brillouin precursor then decays with the propagation distance z > 0 only as z1=3 [as described by (13.87)] instead of the usual z1=2 behavior [see (13.76)] observed when ı > 0 [26].

13.3 The Brillouin Precursor Field in Lorentz Model Dielectrics

425

The Double Resonance Lorentz Model Dielectric From (12.280) and (12.281) for 1 and !SPn .1 / in a double resonance Lorentz model dielectric and the set of relations preceeding (13.77), one finds that 3  4 ı0 b02 !24 C ı2 b22 !04 .!SPn ; 1 /   2 ; 270 !04 !24 b02 !24 C b22 !04  000 .!SPn ; 1 / 3i

b02 !24 C b22 !04 : 0 !04 !24

(13.88) (13.89)

Substitution of these approximate expressions into (13.84) then yields the asymptotic approximation !1=3 ˚ 20 !04 !24 c   2 4 < i e i uQ .!SPn .1 /  !c / Ab .z; t1 /  p 2 4 2 3 z b0 !2 C b2 !0 " 3 #  4z ı0 b02 !24 C ı2 b22 !04  exp  (13.90) 2  270 !04 !24 c b02 !24 C b22 !04 

1 3

as z ! 1 at the fixed space–time point  D 1 D ct1 =z.

13.3.1.3

Case 3:  > 1

For space–time values  in the domain  > 1 , the conditions of Olver’s theorem are satisfied at the symmetric pair of first-order near saddle points SPn˙ when the original contour of integration C is deformed to the path P . / D Pd ./CPn . /C PnC . /CPdC . /, where Pn . / is an Olver-type path with respect to the near saddle point SPn and PnC . / is an Olver-type path with respect to the near saddle point SPnC , as illustrated in Fig. 12.66. For a double resonance Lorentz model dielectric that satisfies the inequality p < 0 , the conditions of Olver’s theorem are satisfied at the pair of near saddle points SPn˙ when the original contour of integration is   ./ C Pm2 ./ C Pn ./ C PnC ./ C deformed to the path P . / D Pd ./ C Pm1 C C C Pm1 . / C Pm2 . / C Pd . /. In either case, (10.18) applies for each of the two near saddle points SPn˙ with (10.3) and (10.4) taken as Taylor series expansions about this first-order saddle point, in which case  D 2. Furthermore, because uQ .!  !c / is analytic at each of these saddle points, then  D 1. Hence, from Olver’s theorem (Theorem 2) and the results of Sect. 12.4, the contour integral in (13.10) taken over the two Olver-type paths Pn . / and PnC ./ yields the asymptotic expansion of the conclusion of the space–time behavior of the second or Brillouin precursor field Ab .z; t / given by [17–19]

426

13 Evolution of the Precursor Fields

 Ab .z; t / D

c z

1=2

( < i e i

a0 .!SPnC /e

.z=c/.!

Ca0 .!SPn /e

C ;/ SPn



  1 C O z1

.z=c/.!SPn ;/

  1 C O z1

)

(13.91) as z ! 1 uniformly for all   1 C " with arbitrarily small " > 0. To evaluate the pair of coefficients a0 .!SP ˙ / appearing in (13.91), the first two n coefficients in the Taylor series expansion (10.3) of the complex phase function .!;  / as well as the first coefficient in the Taylor series expansion (10.4) of the initial envelope spectrum uQ .!  !c / about each saddle point SPn˙ must now be determined. The latter quantity is given by q0 .!SP ˙ .// D uQ .!SP ˙ ./  !c /; n

n

(13.92)

the specific form of which depends upon the particular pulse envelope function u.t /. Because the second coefficient in the Taylor series expansion (10.3) of the complex phase function is given by p0 .!SP ˙ ;  / D  00 .!SP ˙ ;  /=2Š, the coefficient n n a0 .!SP ˙ ; / is found to be given by [see (10.9)] n

uQ .!SP ˙ ./  !c / n a0 .!SP ˙ ;  / D h i1=2 : n 00 2 .!SP ˙ ;  /

(13.93)

n

With this substitution, the nonuniform asymptotic expansion (13.91) of the Brillouin precursor becomes ("

Ab .z; t /  < i e

i

#1=2 c=z .z=c/.! C ;/ SPn uQ .!SPnC ./  !c /e  00 2 .!SPnC ;  / ) 1=2  c=z  C  uQ .!SPn ./  !c /e .z=c/.!SPn ;/ 2 00 .!SPn ;  / (13.94)

as z ! 1 uniformly for all   1 C " with arbitrarilysmall " > 0. The proper value of ˛N 0˙ arg   00 .!SP ˙ ;  / must now be determined n according to the convention defined in Olver’s method [see (10.7)]. Consider first C the value for the near saddle point SPn in the right half of the complex !-plane. For simplicity, the Olver-type path PnC ./ through this first-order saddle point is taken to locally lie along the path of steepest descent through that saddle point, as illustrated in Fig. 13.8. From (12.241), the angle of slope of the contour at the sadin the diagram. Because  arg .z/ D 0, dle point is ˛N C D =4, as indicated   the proper value of ˛N 0C arg   00 .!SPnC ;  / , as determined by the inequality in

13.3 The Brillouin Precursor Field in Lorentz Model Dielectrics

Pn

_

427

'

SPn

Fig. 13.8 Illustration of the steepest descent choice of the Olver-type path PnC . / through the first-order near saddle point SPnC in the right half of the complex !-plane for  > 1 . The shaded area indicates the local region of the complex !-plane about the saddle point where the inequality  .!;  / < .!SPnC ;  / is satisfied

(10.7), is ˛N 0C ' =4. By a similar argument for the near saddle SPn in the  point  00 left half of the complex !-plane, the proper value of ˛N 0 arg   .!SPn ;  / is ˛N 0 ' =4. The Single Resonance Lorentz Model Dielectric From the set of relations preceeding (13.74), the complex phase function and its second derivative at the two first-order near saddle point locations ! D !SP ˙ . / D n ˙ . /  i 23 ı ./ are found to be given by (

  2 b2  1  ˛ . / 2 . /

. /.  0 / C 4 3 0 !0  )  4 2 2 1 ˛ . /  1 C ı . / 9 3 (  )   4 2 b2 2 ı . / 2  ˛ . / C ˛ . / ; ˙i ./ 0   C 20 !04 3

.!SP ˙ ; / ı n

(13.95)  00 .!SP ˙ ; /

n

  b2  2ı ˛ . /  1 ˙ 3i ˛ . / : 4 0 !0

(13.96)

One then has that #1=2 " h i1=2 2 !  0   00 .!SP ˙ ; /

0   n b 2ı ˛ . /  1 ˙ 3i ˛ . / s !02 0 e ˙i=4 ;

b 3˛ ./

(13.97)

428

13 Evolution of the Precursor Fields

where the finalapproximation is valid for all  > 1 such that the inequality  3˛ ./  2ı ˛ . /  1 is satisfied. Notice that this inequality will be satisfied provided that  is not too close to 1 , a requirement that isn’t overly restrictive in the nonuniform asymptotic description (when  approaches 1 from above, the nonuniform description must be replaced by the uniform asymptotic description). Substitution of these approximate expressions in (13.94) then gives the asymptotic approximation of the second or Brillouin precursor field as [3, 4, 17] s

(  z 2 0 c exp  ı

./.  0 / 6 ˛ ./z c 3 )    ˛ b2 4 2 2 2

. /  1 .1  ˛ .// ./ C ı ./ C 9 3 0 !04 " ( ( z i < i e . / 0   uQ .!SPnC ./  !c / exp i c )  #   4 2 b2  2 C ı . / 2  ˛ . / C ˛ . / Ci 4 20 !04 3 " ( z . / 0   CQu.!SPn ./  !c / exp  i c ))  #   4 2 b2  2 ı . / 2  ˛ . / C ˛ ./ C i 4 20 !04 3

!2 Ab .z; t /  0 b

(13.98) as z ! 1 for  > 1 . The second precursor field is then seen to be oscillatory and increasingly attenuated with the propagation distance z > 0 as  increases above 1 , the attenuation factor increasing with increasing  . Notice that the quantity is a fixed phase factor associated with the initial pulse carrier wave [where D 0 corresponds to a sine-wave carrier and D =2 corresponds to a cosine-wave carrier, as described in (11.34)], whereas the function . / describes the space– time dependence of the real part of the near saddle point location for  > 1 , as described by (13.66).

The Double Resonance Lorentz Model Dielectric With the substitution ! D !SP ˙ ./ D ˙ ./  i 23 ı0 . / in the set of relations n preceeding (13.77), one obtains

13.3 The Brillouin Precursor Field in Lorentz Model Dielectrics

2 62 .!SP ˙ ; / ı0 4 . /.  0 / C n 3

 b02 1 C

429

 4

ı2 b22 !0 ı0 b02 !24 0 !04



3  4 7 2 ./  ı02 2 . / 5 9

    ı2 b22 !04 4ı 2 b 2

. / ; ˙i ./ 0   C 0 40 1 C 0 !0 ı0 b02 !24

(13.99)

and     b22 !04 ı2 b22 !04 1 C 2 4 . /  2 1 C b0 !2 ı0 b02 !24   b2!4 3b02 1 C 22 04 ./ ˙i 4 20 !0 b0 !2   3b02 b22 !04

˙i 1C 2 4 ./; (13.100) 20 !04 b0 !2

ı0 b02  .!SP ˙ ; /

n 0 !04 00

where the final approximation is valid for all  > 1 such that the inequality 3 ./  2ı0 . /  .b02 !24 C .ı2 =ı0 /b22 !04 /=.b02 !24 C b22 !04 / is satisfied. When this inequality is not satisfied, the nonuniform expansion is becoming invalid and must then be replaced by the uniform expansion. One then has that h i1=2 !2  00 .!SP ˙ ; /

0 n b0

s

20 b02 !24  e ˙i=4 : 3 b02 !24 C b22 !04 . / 

(13.101)

Substitution of these approximate expressions in (13.94) then gives the asymptotic approximation of the Brillouin precursor in a double resonance Lorentz model dielectric as s 0 b 2 ! 4 c !02  2 4 0 22 4  Ab .z; t /  b0 6 b0 !2 C b2 !0 ./z   z 2 .ı0 b02 !24 Cı2 b22 !04 /. 2 ./ 49 ı02 2 .//

./.   / C  exp ı0 0 0 ı0 !04 !24 c 3 ( < i e i

uQ .!SPnC ./  !c / (

zh  exp i ./ 0   C c CQu.!SPn . /  !c / (  exp

4ı02 b02 0 !04

zh  i ./ 0   C c



1C

4ı02 b02 0 !04

ı2 b22 !04 ı0 b02 !24

 1C



)

i

. / C i

ı2 b22 !04 ı0 b02 !24



i

 4

))

. /  i

 4

(13.102)

430

13 Evolution of the Precursor Fields

as z ! 1 for  > 1 . As in the single resonance case, the Brillouin precursor field is seen to be oscillatory and increasingly attenuated with the propagation distance z > 0 as  increases above 1 , the attenuation factor increasing with increasing  .

13.3.2 The Uniform Approximation The set of relations given in (13.76), (13.84), and (13.98) for the single resonance Lorentz model and in (13.79), (13.90), and (13.102) for the double resonance Lorentz model represent the nonuniform asymptotic approximation of the second or Brillouin precursor field for space–time values  ct =z in the successive domains 1 <  < 1 ,  D 1 , and  > 1 , respectively. The results are discontinuous at the critical transition point at  D 1 at which the two first-order near saddle points have coalesced into a single second-order saddle point. To obtain a continuous transition in the behavior of the Brillouin precursor field as  is allowed to vary across the critical space–time point  D 1 , the uniform asymptotic expansion due to Chester, Friedman, and Ursell [25] is employed (see Theorem 4 of Sect. 10.3). The required uniform asymptotic approximation is obtained by direct application of Theorem 4. Because the behavior of the near saddle points and the path of integration in the present case is the same as in the example treated in Sect. 10.3.2, it follows from the discussion in that example that the path of integration L appearing in (10.52) is an L21 contour (see Fig. 10.6) so that the function C. / appearing in (10.51)  is given  by (10.72) with indices i D 2; j D 1, so that C. / D e i2=3 Ai e i2=3 , where Ai ./ is the Airy function. Although Theorem 4 is directly applicable to the present problem over the entire space–time domain  > 1, it is still necessary to treat the two cases 1 <   1 and   1 separately because the approximate expressions for the near saddle point locations differ in the two cases. Nonetheless, the results for these two cases combined are continuous at the critical space–time point at  D 1 and therefore constitute an asymptotic approximation of the Brillouin precursor field Ab .z; t / that is uniformly valid for all  > 1. Consider first the uniform asymptotic behavior of the Brillouin precursor field over the initial space–time domain 1 <  < 1 . In that case, the two near saddle point locations are given by (13.65) as  !SP ˙ . / D i ˙ n

 2 ./  ı . / ; 0 3

(13.103)

where 02 . / D  2 . / with ./ and . / given in (13.66) and (13.67) for a single resonance Lorentz model dielectric and by (13.68) and (13.69) for the double resonance case. The analysis presented here will focus on the single resonance case, the double resonance case being left as an exercise. Application of Theorem 4 yields the asymptotic expansion [4, 17, 20, 27]

13.3 The Brillouin Precursor Field in Lorentz Model Dielectrics

431

 1=3 z c < e i e c ˛0 ./ z h  i  uQ .!SPnC  !c /h1 ./ C uQ .!SPn  !c /h2 ./ C O z1

1 Ab .z; t / D  2 (

e i C

2 3

  2 Ai ˛1 ./e i 3 .z=c/2=3

.c=z/1=3 h uQ .!SPnC  !c /h1 . /  uQ .!SPn  !c /h2 . / 1=2 ˛1 . / )   1  i i 4 0  2 i 2=3 e 3 Ai ˛1 ./e 3 .z=c/ CO z (13.104)

as z ! 1 uniformly for all  2 .1; 1 . From (10.54)–(10.56) with substitution from (13.74) and (13.75), the coefficients appearing in this uniform expansion are found to be given by  1 .!SPnC ; / C .!SPn ;  / 2 2

 ı . /.  0 / 3      4 2 2 1 ıb 2 2 ˛ . /  1 ;  0 . / ˛ . /  1 C ı ./ 9 3 0 !04 (13.105)  1=3  3 1=2 ˛1 . / .!SPnC ; /  .!SPn ;  / 4 "   #1=3 3 3 b2 1=3 2 2 2 2

0 . / .  0 / C ; ˛ . / C ˛ı ./  2ı . / 2 0 !04 4 0 ˛0 . /

(13.106) and "

#1=2 1=2 2˛1 . / h1;2 . / 00  .!SP ˙ ; / n " #1=2 2 !0 1=6 20  

. / b 0 3˛ 0 ./ ˙ 2ı 1  ˛ . / "   #1=6 3 3 b2 2 2 2 2 ˛ ./ C ˛ı . /  2ı . /  .  0 / C ; 2 0 !04 4 0 (13.107)

432

13 Evolution of the Precursor Fields

for  2 .1; 1 . Notice that the upper sign choice in (13.107) corresponds to h1 . / and the lower sign choice to h2 ./. In the limit as  approaches the critical value 1 from below, this expression reduces to [see (10.57)]  h.1 / lim h1;2 ./ D  !1

1=3  20 !04

 ; 3i ˛b 2

2

1=3

 000 .!SPn ; 1 / (13.108)

where the final approximation is obtained by substitution from (13.86). Analogous expressions are obtained for the double resonance case. The proper values of the multivalued functions appearing in (13.106)–(13.108) are determined by the conditions presented in Sect. 10.3.2. In particular, the phase of h1;2 . / is specified by (10.60) as (in the notation of the present chapter)   lim arg h1;2 ./ D ˛N C ;

!1

(13.109)

where ˛N C is the angle of slope of the steepest descent path leaving the secondorder saddle point SPn at  D 1 . From Fig. 13.7 it is seen that ˛N C D =6. Hence, (13.109) shows that the argument of h.1 / is =6. Moreover, because the sixth power of the quantity appearing on the right-hand side of (13.107) is real and negative for all  2 .1; 1 , the argument  of h1;2 . / is independent of  over that space–time domain. Hence arg h1;2 ./ D =6 and (13.107) may be rewritten as ˇ1=2 ˇ ˇ ˇ1=6 ˇˇ !02 ˇˇ 2 ˇ 0 ˇ ˇ   ˇ e i=6 h1;2 . /

0 . / ˇ 3˛ 0 ./ ˙ 2ı 1  ˛ . / ˇ b ˇ  ˇ1=6 ˇ3 ˇ 3 b2 2 2 2 2 ˇ ˛ 0 ./ C ˛ı . /  2ı . / ˇˇ ;  ˇ .  0 / C 4 2 0 !0 4 (13.110) for all  2 .1; 1 . 1=2 The proper value of the phase of the quantity ˛1 ./ is determined from (10.63) with n D 0, so that (in the notation of the present chapter)   1=2 lim arg ˛1 ./ D ˛N 12  ˛N C ;

!1

(13.111)

where ˛N 12 is the angle of slope of the vector from the saddle point SPn to the saddle point SPnC . Because ˛N 12 D =2 (see Fig. 12.40) and ˛N C D =6 , then    1=2 lim arg ˛1 ./ D : !1 3

(13.112)

13.3 The Brillouin Precursor Field in Lorentz Model Dielectrics

433

Moreover, because the cube of the quantity appearing on the right-hand side of 1=2 (13.106) is real and negative for all  2 .1; 1 , the argument of ˛1 ./ is indepen 1=2  dent of  over that space–time domain. Hence, arg ˛1 . / D =3 and (13.106) may be rewritten as ˇ 1=2 ˛1 . / ˇ

0 . /

ˇ1=3 i=3 ˇ e

ˇ  ˇ3 3 b2 ˇ ˛ ˇ .  0 / C ˇ2 0 !04 4

 ˇˇ1=3 ˇ 2 2 2 2 ; 0 ./ C ˛ı ./  2ı . / ˇ ˇ (13.113)

for all  2 .1; 1 .  Because arg ˛1 . / D 2=3, the argument of the Airy function Ai . / and its first derivative A0i . / appearing in the uniform asymptotic expansion in (13.104) is real and nonnegative for all  2 .1; 1 . With the above results for the arguments of the quantitites h1;2 . / and ˛1 ./, one finally obtains the uniform asymptotic approximation of the Brillouin precursor [4, 17, 20, 27] Ab .z; t / 

 1=3 z c e c ˛0 ./ z ( h ˇ ˇ ˇ ˇi i uQ .!SPnC  !c /ˇh1 ./ˇ C uQ .!SPn  !c /ˇh2 . /ˇ < i e 1 2

ˇ ˇ  Ai ˇ˛1 ./ˇ.z=c/2=3 ˇ ˇ ˇ ˇi .c=z/1=3 h ˇ ˇ1=2 uQ .!SPnC  !c /ˇh1 . /ˇ  uQ .!SPn  !c /ˇh2 . /ˇ ˇ˛1 . /ˇ ) ˇ ˇ  0 ˇ 2=3 ˇ A ˛1 ./ .z=c/ (13.114) i

as z ! 1 uniformly for all  2 .1; 1 . As  approaches the critical value 1 from below, the argument of the Airy function and its first derivative in (13.114) tends to zero as the amplitude coefficients in that equation tend to indeterminate forms. The determinate form of (13.114) in this limit is found from the limiting forms given in (10.57)–(10.59) and (13.108) as Ab .z; t1 / 

  20 !0 c 1=3 ˚ i  < i e uQ !SPn .1 /  !c 2 ˛b z    2ız 4ı 2 b 2 0  1 C (13.115)  exp 3˛c 9˛0 !04

 . 13 / p !0 2 3



as z ! 1 with  D 1 D ct1 =z. This result is identical with that given in (13.87) using Olver’s saddle point method.

434

13 Evolution of the Precursor Fields

Consider now the uniform asymptotic behavior of the Brillouin precursor field for   1 . In that case the near saddle points form a symmetric pair with locations given by (13.65) as 2 (13.116) !SP ˙ ./ D ˙ ./  i ı ./; n 3 the approximate complex phase behavioir at these points being given by (13.95) and (13.96) in the single resonance case, the double resonance case being left as an exercise. Application of Theorem 4 then yields the uniform asymptotic expansion [4, 17, 20, 27]  1=3 z c < e i e c ˛0 ./ z h  i  uQ .!SPnC  !c /h1 ./ C uQ .!SPn  !c /h2 ./ C O z1

1 Ab .z; t / D  2 (

e i C

2 3

  2 Ai ˛1 ./e i 3 .z=c/2=3

.c=z/1=3 h uQ .!SPnC  !c /h1 . /  uQ .!SPn  !c /h2 . / 1=2 ˛1 . / )   1  i i 4 0  2 i 2=3 e 3 Ai ˛1 ./e 3 .z=c/ CO z (13.117)

as z ! 1 uniformly for all   1 . From (10.54)–(10.56) with substitution from (13.95) and (13.96), the coefficients appearing in this expression are found to be given by  1 .!SPnC ; / C .!SPn ;  / 2 ( 2

ı . /.  0 / 3   b2  1  ˛ . / C 4 0 !0

˛0 . /

1=2



˛1 . / (

2

)  4 2 2 1 ./ C ı . / ˛ . /  1 9 3 (13.118)

  1=3 3 .!SPnC ; /  .!SPn ;  / 4

 i 32

h . /   0 

b2 20 !04

4

  ı 2 . / 2  ˛ . / C ˛ 3

2

./

i

) 1=3 ;

(13.119)

13.3 The Brillouin Precursor Field in Lorentz Model Dielectrics

435

and " h˙ . /

1=2

2˛ . /

00 1  .!SP ˙ ;  /

#1=2

n

#1=2  1=6 " 3 20    i ./

i b 2 3˛ ./ 2i ı ˛ . /  1 "   #1=6   4 2 b2 2 ı . / 2  ˛ . / C ˛ ./    0  ; 20 !04 3  1=6  1=2 !2 3 20 i

0  i ./ b 2 3˛ ./ "   #1=6   4 2 b2 2    0  ; ı . / 2  ˛ . / C ˛ ./ 20 !04 3 !02

(13.120)   where the final approximation is valid provided that 3˛ ./  2ı ˛ . /  1 , which is found to be satisfied for all   1 . In the limit as  approaches 1 from above, this expression is replaced by its limiting form [see (10.57)]  h.1 / lim h˙ ./ D  !1C

2

1=3

 000 .!SPn ; 1 /

1=3  20 !04

 ; 3i ˛b 2

(13.121)

which is the same as that given in (13.108). The proper values of the multivalued functions appearing in (13.119)–(13.121) are determined by the conditions presented in Sect. 10.3.2. In particular, the phase of h˙ . / is specified by (10.60) as   lim arg h˙ ./ D ˛N C ;

!1C

(13.122)

where ˛N C is the angle of slope of the steepest descent path leaving the second-order saddle point SPn at  D 1 . From Fig. 13.7 it is seen that ˛N C D =6. Hence, (13.122) shows that the argument of h.1 / is =6. Moreover, because the sixth power of the approximate quantity appearing on the right-hand side of (13.120) is real and negative for all   1 , the argument of h˙ . / is approximately independent of  over that space–time domain. Hence arg h˙ ./ =6 and (13.120) may be rewritten as

436

h˙ . /

13 Evolution of the Precursor Fields

ˇ ˇ ˇ1=6 ˇ ˇ ˇ 20 ˇ1=2 i=6 !02 ˇˇ 3 ˇ ˇ ˇ . / ˇ ˇ 3˛ ./ ˇ e b ˇ2 ˇ  ˇ   4 2 b2 ˇ ˇ  0  ı . / 2  ˛ . / C ˛ 4 ˇ 20 !0 3

2

 ˇˇ1=6 ˇ ./ ˇ ˇ

(13.123)

  for all   1 , the accuracy of the approximation arg h˙ . / =6 decreasing as  increases above 1 . 1=2 The proper value of the phase of the quantity ˛1 ./ is determined from (10.63) with n D 0, so that   1=2 lim arg ˛1 ./ D ˛N 12  ˛N C ; (13.124) !1

where ˛N 12 is the angle of slope of the vector from the saddle point SPn to the saddle point SPnC . Because ˛N 12 D 0 (see Fig. 12.42) and ˛N C D =6 , then    1=2 (13.125) lim arg ˛1 ./ D  : C 6 !1 Moreover, because the cube of the quantity appearing on the right-hand side of 1=2 (13.119) is negative imaginary for all   1 , the argument of ˛1 ./ is indepen 1=2  dent of  over that space–time domain. Hence, arg ˛1 . / D =6 and (13.119) may be rewritten as ˇ ˇ ˇ h i ˇ1=3    2 ˇ3 ˇ 1=2 ˛1 . / ˇ 2 . /   0  2b ! 4 43 ı 2 . / 2  ˛ . / C ˛ 2 . / ˇ e i=6 0 0 ˇ ˇ (13.126) for all   1 . Because arg .˛1 . / D =3, the argument of the Airy function Ai . / and its first derivative A0i . / appearing in (13.117) is real and nonpositive for all   1 . With these results for the arguments of the quantities h˙ ./ and ˛1 . /, one finally obtains the uniform asymptotic approximation of the Brillouin precursor [4, 17, 20, 27]   1 c 1=3 z ˛0 ./ ec Ab .z; t /  2 z ( h ˇ ˇ ˇ ˇi i uQ .! C  !c /ˇhC ./ˇ C uQ .!SP   !c /ˇh . /ˇ < i e SPn

n

ˇ  ˇ  Ai ˇ˛1 ./ˇ.z=c/2=3 ˇ C ˇ ˇ  ˇi .c=z/1=3 h ˇ ˇ1=2 uQ .!SPnC  !c /ˇh ./ˇ  uQ .!SPn  !c /ˇh . /ˇ ˇ˛1 . /ˇ ) ˇ  ˇ  0 2=3 A ˇ˛1 ./ˇ.z=c/ (13.127) i

as z ! 1 uniformly for all   1 .

13.3 The Brillouin Precursor Field in Lorentz Model Dielectrics

437

As  approaches the critical value 1 from above, the argument of the Airy function and its first derivative in (13.127) tends to zero as the amplitude coefficients in that equation tend to indeterminate forms. The determinate form of (13.127) in this limit is found from the limiting forms given in (10.57)–(10.59) and (13.121) to be precisely that given in (13.87) and (13.115). Taken together, the set of expressions given in (13.114), (13.115), and (13.127) constitute the uniform asymptotic approximation of the Brillouin precursor field in a single resonance Lorentz model dielectric that is uniformly valid for all  > 1. Because the argument of the Airy function and its first derivative in (13.127) is real and negative for all  > 1 , the Brillouin precursor is oscillatory over this space–time domain. In a similar manner, because the argument of the Airy function and its first derivative in (13.114) is real and positive for  2 .1; 1 , one may conclude that the Brillouin precursor is nonoscillatory over this initial space–time domain; however, this does not mean that the Brillouin precursor begins as a static field as it rapidly builds to its peak amplitude point near  D 0 over this initial space–time domain. Finally, for values of  bounded away from unity, the uniform asymptotic approximation in (13.114) simplifies to the nonuniform approximation given in (13.76), and the uniform asymptotic approximation given in (13.127) simplifies to the nonuniform approximation given in (13.94) as z ! 1.

13.3.3 The Instantaneous Oscillation Frequency The instantaneous angular frequency of oscillation of the Brillouin precursor field Ab .z; t / is defined [6, 7] as minus the time derivative of the oscillatory phase, where the minus sign is included in order to obtain a positive-valued angular frequency. Notice that the oscillatory phase terms appearing in both the nonuniform (13.94) and uniform (13.127) asymptotic approximations of this second precursor field are identical for  > 1 . Because d=dt D c=z for all z > 0 and (with ˛ 1) 0 2 2 d . / 3c d @   0 C D dt 2z d  2   2 C 0 2 !02

2b 2 !02 3b 2 !02

 2    02

1 A

3b 2 c  !02 z  2  02 C

0

 2  02 C

2b 2 !02

3b 2 !02

2 ;

12 31=2 A 7 5

d . / c d 6 D  ı2 @ 4 2 dt z d  2   2 C 3b22  2  02 C 3b 0 !0 !02    3b 2 ı2 2 2  2  2  02 C 3    C 2 2 2 0 cb  !0 !0

  2 3 z . /  2  02 C 3b 2 !

2b 2 !02

 ;

0

then the instantaneous angular frequency of oscillation of the second or Brillouin precursor field is given by

438

13 Evolution of the Precursor Fields

i d hz j˛. /j dt c 

    z d 4 2 b2  2 . /   0  ı '

. / 2 

. / C ./ C c dt 4 20 !04 3     2 2 2 " 3b ı 2b 2 2 2 2 3b 4  3   0 C !02  2 !02   0 C !02

. / 1    2 3 20 !04  2  02 C 3b 2 !0 # 2 4 1  . / 4ı b     2 2 0 !06  2  02 C 3b 2 !0     3b 2 ı2 2b 2 2 2 2 2 b 2  3   0 C !02  2 !02   0 C !02 C   2 3 . /  2  02 C 3b !02   2 2   2ı b    0 

. / 2  . / : 30 !04

!b . / 

  2ı 2 b 2 From (12.225) it is seen that   0  3 4 . / 2  . /   1 for all  > 1 . 0 !0 With this substitution and a bit of algebra, the above expression simplifies to 2

!b . /



b4 . / 41  0 !04

9 8ı 2 2 C !2 0

   3 2 2 2  2 02 C 3b2  15ı2  2 02 C 2b2 !0 !0 !0 5  3 3b 2  2 02 C 2 !0

   3b 2 ı2 2 2  2  2  02 C   C 3  2 2 0 b  !0 !0 C .  1 /   2 3 . /  2  02 C 3b 2 ! 2

2b 2 !02



0

(13.128) for  > 1 , which may be approximated as ˚ !b . / < !SPnC ./ D

. /:

(13.129)

The approximation given in (13.129) also holds in the double resonance case. Although an approximation, the identification of the instantaneous angular frequency of oscillation of the Brillouin prescursor as being given by the real part of the near saddle point location !SPnC ./ in the right half of the complex !-plane for  > 1 is intuitively pleasing and complements the analogous result given in (13.44) for the Sommerfeld precursor. As in that case, the notion of an instantaneous oscillation frequency is only a heuristic mathematical identififcation which, in certain circumstances, may yield completely erroneous or misleading results [22]. Although this is not the case for the Brillouin precursor whose instantaneous oscillation frequency monotonically increases with increasing  > 1 , approaching either the

13.3 The Brillouin Precursor Field in Lorentz Model Dielectrics

439

q limiting value .1/ D !02  ı 2 for a single resonance Lorentz model dielectric q or the limiting value .1/ D !02  ı02 for a double resonance Lorentz model dielectric as  ! 1, the zero oscillation frequency it predicts for the Brillouin precursor over the initial space–time domain  2 .1; 1 is misleading, leading some to erroneously conclude that the Brillouin precursor is a static field over this space– time domain. It is not for all finite z > 0. A physically meaningful effective oscillation frequency may be defined over the initial space–time domain  2 .1; 1 by the temporal width of the build-up to the peak amplitude point that occurs between the space–time points  D SB and  D 0 . From (12.257),   between these two space–time points is given  the difference by  D 0  SB 4ı 2 b 2 = 30 !04 . This difference corresponds to the effective half-period Teff D 2=!eff of the field over this space–time domain through the relation  D .c=z/.Teff =2/, so that !eff .0 /

30 !04 c : 4ı 2 b 2 z

(13.130)

Notice that this effective angular oscillation frequency of the Brillouin precursor at  D 0 asymptotically approaches zero as z ! 1, in agreement with the limiting behavior as  ! 1C of the asymptotic result given in (13.129). The efficacy of this description is considered in Sect. 13.3.5 for the Heaviside step function signal.

13.3.4 The Delta Function Pulse Brillouin Precursor For an input delta function pulse at time t D 0 the corresponding initial envelope D 0. For space–time values  2 spectrum is given by uQ .!  !c / D i with .1; 1 , the uniform asymptotic approximation given in (13.114) becomes [4, 17, 20]

Aıb .z; t / 

  !02 c 1=3 z ˛0 ./ ec 2b z ( " 

3˛

" 

3˛

20

0 ./C2ı



1˛ ./

1=2

 C

3˛

20

0 ./2ı



1˛ ./

ˇ1=4 ˇ ˇ ˇ  ˇ˛1 ./ˇ Ai ˇ˛1 ./ˇ. cz /1=3 1=2  20 20  

0 ./C2ı

1˛ ./

3˛

0 ./2ı

1=2 #



1=2 #

1˛ ./

ˇ z 1=3  .c=z/1=3 0 ˇˇ ˇ ˇ1=4 Ai ˛1 . /ˇ. c / ˇ˛1 ./ˇ

)

(13.131)

440

13 Evolution of the Precursor Fields

as z ! 1 for all  2 .1; 1 . Accurate approximations of the functions ˛0 ./ and ˛1 . / appearing in this expression are given in (13.106) and (13.113), respectively. At the critical space–time point  D 1 the asymptotic field value is obtained from (13.115) as Aıb .z; t1 / 

 . 13 / p !0 2 3



20 !0 c ˛b 2 z

1=3

 exp

2ız 3˛c

  4ı 2 b 2 0  1 C (13.132) 9˛0 !04

as z ! 1 with  D 1 D ct1 =z. Finally, for space–time values   1 , the uniform asymptotic approximation given in (13.127) becomes [4, 17, 20] 1=2  1=3 ˇ ˇ ˇ  ˇ  c ˇ˛1 ./ˇ1=4 e cz ˛0 ./ Ai ˇ˛1 ./ˇ. z /2=3 c z (13.133) as z ! 1 uniformly for all   1 . A transitional asymptotic approximation (see Sect. 10.3.3) of the Brillouin precursor for the delta function pulse has also been given [28] in order to numerically bridge the small  -interval about the critical space–time point at  D 1 where the the coefficient ˛1 . / may become numerically indeterminate due to a lack of numerical accuracy. This problem is now completely eliminated through the use of accurate numerically determined saddle point locations [27]. Nevertheless, the transitional expansion is useful when analytic approximations for the saddle point locations and the complex phase behavior at them are used (see Problem 13.6). !2 Aıb .z; t /  0 b



20 3˛ ./

13.3.5 The Heaviside Step Function Pulse Brillouin Precursor For a Heaviside unit step function modulated signal with fixed angular carrier frequency !c > 0, the spectrum of the envelope function is given by (11.56) so that 

uQ H !SP ˙ . /  !c n



  ˙ 13 3 0 ./ 2ı . /  i !c i D D  2 (13.134) !SP ˙ ./  !c !c2 C 19 3 0 . / 2ı . / n

for 1 <   1 , and   uQ H !SP ˙ . /  !c D n

   23 ı . / C i ˙ . /  !c i (13.135) D 2 !SP ˙ ./  !c ˙ . /  !c C 49 ı 2 2 . / n

for   1 . At  D 1 , both of these equations simplify to the approximate expression 2ı    i !c uQ H !SPn .1 /  !c '  3˛ : (13.136) 4ı 2 !c2 C 9˛ 2 Analogous expressions hold in the double resonance Lorentz model case.

13.3 The Brillouin Precursor Field in Lorentz Model Dielectrics

441

For space–time values  2 .1; 1 , the uniform asymptotic approximation given in (13.114) becomes [4, 17, 20] AH b .z; t / 

!02 !c 2b (" 

 1=3 z c e c ˛0 ./ z

1=2 20   3˛ 0 ./ C 2ı 1  ˛ . / 0 ./  2ı ./ 1=2 #  20 1   C  2 3˛ 0 . /  2ı 1  ˛ . / !c2 C 19 3 0 ./ C 2ı ./ ˇ ˇ1=4 ˇ ˇ  ˇ˛1 ./ˇ Ai ˇ˛1 ./ˇ. cz /1=3 " 1=2  20 1     2 3˛ 0 ./ C 2ı 1  ˛ . / !c2 C 19 3 0 ./  2ı ./  1=2 # 20 1     2 3˛ 0 ./  2ı 1  ˛ . / !c2 C 19 3 0 ./ C 2ı ./ ) ˇ z 1=3  .c=z/1=3 0 ˇˇ ˇ ˇ A ˛1 ./ˇ. c / ˇ˛1 ./ˇ1=4 i (13.137)  !c2 C 19 3

1



2

as z ! 1 for all  2 .1; 1 . Accurate approximations of the functions ˛0 ./ and ˛1 . / appearing in this expression are given in (13.106) and (13.113), respectively. At the critical space–time point  D 1 the asymptotic field value is obtained from (13.115) as   4ı 2 b 2 0  1 C 9˛0 !04 (13.138) as z ! 1 with  D 1 D ct1 =z. Finally, for space–time values   1 , the uniform asymptotic approximation in (13.127) becomes [4, 17, 20] AH b .z; t1 / 

 . 13 / !0 !c p 4ı 2 2 3 !c2 C 9˛ 2



20 !0 c ˛b 2 z

1=3



exp

2ız 3˛c

 1=2  1=3 z !2 0 c AH b .z; t /   0 e c ˛0 ./ b 6˛ . / z ( " ˇ1=4 ˇ 2 1  ı ./ˇ˛1 ./ˇ  2 4 3 ./ C !c C 9 ı 2 2 ./ # ˇ  ˇ  1 Ai ˇ˛1 ./ˇ. cz /1=3  2 4 . /  !c C 9 ı 2 2 ./

442

13 Evolution of the Precursor Fields

"

.c=z/1=3 ./  !c ˇ ˇ  2 ˇ˛1 . /ˇ1=4 ./  !c C 49 ı 2 2 ./ ) # ˇ z 1=3   ˇ ./ C !c 0 ˇ ˇ  Ai  ˛1 ./ . c / 2 . / C !c C 49 ı 2 2 ./ (13.139) as z ! 1 uniformly for all   1 . Accurate approximations of the functions ˛0 . / and ˛1 . / appearing in this expression are given in (13.118) and (13.126), respectively. Taken together, (13.137)–(13.139) constitute the uniform asymptotic approximation of the Brillouin precursor for an input Heaviside unit step function modulated signal. For space–time values  2 .1; 1 / with  bounded away from 1 , substitution of the expression given in (13.134) for the input signal envelope spectrum evaluated at the near saddle point locations into the nonuniform asymptotic approximation given in (13.76) results in [3]

AH b .z; t / 

!02 !c

"

0 c=.z/   4ı 1  ˛ . / C 6˛

h  2 i b !c2 C 19 3 0 ./  2ı ./ (   z  2ı ./  3 0 ./ 0    exp 3c b2  C 2ı ./  3 180 !04

 h  0 ./ 2ı 3  ˛ . /  3˛

#1=2 0 ./

0 ./

) i

(13.140) as z ! 1. The same nonuniform result is obtained from the uniform asymptotic approximation given in (13.137) with substitution of the dominant term in the large argument asymptotic expansion of both the Airy function and its first derivative [see the pair of expressions preceeding (10.75)]. For  0 , this nonuniform asymptotic approximation simplifies somewhat to AH b .z; t / 

 1=2 !02 !c 0 c i h   2ı 2 6 ˛./z b !c2 C ./  3˛ (   z 2ı  ./ 0    exp c 3˛    ) 2ı b2 4 C  ./ ˛./ C ı (13.141) 3 20 !04 3˛

13.3 The Brillouin Precursor Field in Lorentz Model Dielectrics

443

as z ! 1, where  . /

20 !04 4ı 2 C .0   / 9˛ 2 3˛b 2

1=2 :

The nonuniform asymptotic approximation of the second precursor field for the unit step function modulated signal given in (13.141) is the classical result for the second forerunner obtained by Brillouin [6] for  2 .1; 1 /; see also page 66 of [7]. That result is then seen to be accurate only for values of  near 0 , becoming invalid as  approaches 1 . At the space–time point  D 0 D ct0 =z, at which 0 .0 / D .0 / D 2ı=3˛, both (13.140) and (13.141) simplify to the result !2 AH b .z; t0 /  0 b!c



0 c 4ız

1=2 ;

(13.142)

at which point the Brillouin precursor varies with the propagation distance z > 0 only as z1=2 as z ! 1, making this space–time point in the field evolution entirely unique. The same result is obtained for the uniform asymptotic approximation given in (13.139) for sufficiently large values of z > 0. For space–time values  > 1 with  bounded away from 1 , substitution of the expression given in (13.135) for the input signal envelope spectrum evaluated at the near saddle point locations into the nonuniform asymptotic approximation given in (13.98) results in [3]

AH b .z; t / 

q

20 c 3 ˛ ./z

i h i 2 2 . /  !c C 49 ı 2 2 ./ ./ C !c C 49 ı 2 2 ./ ( " z 2 .  0 / ./  exp  ı c 3 b

h

!02 !c

 b 2  C 1  ˛ . / 4 0 !0

2

 ˛ 4

. /  1 ./ C ı 2 2 ./ 9 3

(  4 2 2 2 2  !c C ı ./  ./ 9   h 4 ./.2˛ .// ı2 b2    C C  cos ./z 0 4 c 3 2 ! 0 0

4 C ı . / . / 3 h   sin ./z 0   C c

ı2 b2 20 !04



4 ./.2˛ .// 3

C

˛

2 ./

˛



ı2

2 ./

ı2



#)

C 4

C 4

i

) i

(13.143)

444

13 Evolution of the Precursor Fields

as z ! 1. This nonuniform asymptotic approximation ˇ ˇreduces to the classical result given by Brillouin [6, 7] if ./ is replaced by ˇ. /ˇ and . / is replaced by ˛ 1 ' 1. However, these two replacements are valid only for space–time points not too distant from 0 . Hence, Brillouin’s classical expression for the second precursor field over the space–time domain  > 1 is an approximation, valid for  near 1 , of an expression that becomes invalid as  approaches 1 from above. As a result, Brillouin’s expression for the asymptotic behavior of the second precursor field (or second forerunner) for the unit step function modulated signal over the space–time domain  > 1 is not applicable. The temporal evolution of the second or Brillouin precursor field AH b .z; t / for a unit step function modulated signal with below resonance angular carrier frequency !c D 1  1016 r=s at one absorption depth in a single resonance Lorentz model dielectric with Brillouin’s choice of the model parameters is represented by the solid curve in Fig. 13.9 when numerical saddle point locations are used in the uniform asymptotic approximation given in (13.137)–(13.139). In that case, 0 D 1:5 and 1 1:50275. As evident in Fig. 13.9, the Brillouin precursor field amplitude builds up rapidly as  increases to 0 and then decays with increasing  > 0 . The instantaneous oscillation frequency !b ./ of the Brillouin precursor is also seen to monotonically increase with increasing  > 1. Comparison of this

0.4

0.3

AHb(z,t)

0.2

0.1

0 −0.1 −0.2

1.4

1.5

1.6

1.7

1.8

Fig. 13.9 Temporal evoution of the Brillouin precursor field AH b .z; t / at one absorption depth model dielectric with Brillouin’s choice of z D zd ˛ 1 .!c / in a single resonance Lorentz p the medium parameters (!0 D 4  1016 r=s, b D 20  1016 r=s, ı D 0:28  1016 r=s) for a Heaviside unit step function modulated signal with below resonance angular carrier frequency !c D 1  1016 r=s. The solid curve results from numerically determined saddle point locations and the dashed curve from the second approximate expressions

13.4 The Brillouin Precursor Field in Debye Model Dielectrics

445

behavior with that depicted in Figs. 13.4 and 13.5 for the Sommerfeld precursor at the same propagation distance in the same medium shows that the peak amplitude of the Brillouin precursor is approximately two orders of magnitude larger than that for the Sommerfeld precursor in this below resonance frequency case. As !c is increased above the angular resonance frequency !0 , the peak amplitude of the Brillouin precursor will diminish and the peak amplitude of the Sommerfeld precursor will increase at any fixed propagation distance z > 0. However, because of its unique z1=2 peak amplitude decay, the Brillouin precursor will eventually dominate the Sommerfeld precursor for sufficiently large observation distances provided that the input pulse spectral energy is nonvanishing in the spectral domain j!j < !0 below resonance. Previously published research [4, 20] exhibited a discontinuity in the uniform asymptotic behavior of the Brillouin precursor about the critical space–time point  D 1 in a Lorentz medium, originally thought to be due to numerical instabilities in the limiting behavior of the coefficients ˛11 ./, h1;2 . /, and h˙ . / for  near 1 .4 To bridge this small space–time neighborhood about  D 1 , a transitional asymptotic approximation (see Sect. 10.3.3) was then used [28] with complete success for the delta function pulse Brillouin precursor (see Problem 13.6) but only with limited success for the Heaviside step function signal case. However, as was pointed out by Cartwright [27, 29], this instability is actually due to the use of an unnecessary approximation of the lower near saddle point location !SPn . / for  2 .1; 1 /. The dashed curve in Fig. 13.9 describes the transitional asymptotic behavior of the Brillouin precursor when the second approximate expressions (13.65)–(13.67) for the near saddle point locations are used, resulting in a discontinuity in the field behavior just prior to the critical space–time point at  D 1 . This discontinuous behavior in the Brillouin precursor evolution is completely eliminated when accurate, numerical saddle point locations are used in the uniform asymptotic expressions without any further approximation, as seen in Fig. 13.9.

13.4 The Brillouin Precursor Field in Debye Model Dielectrics For a single relaxation time Rocard–Powles–Debye model dielectric with complex index of refraction given in (12.125), there is a single near saddle point with location given approximately in (12.305) as [30] # " r 3 !SPn . / i 1  1 C 2 .  0 3 4

(13.144)

At that time (circa 1975), numerical computations of the precursor fields using the second approximate saddle point locations with the appropriate, approximate expressions of the complex phase .!;  / behavior at them were performed in FORTRAN IV using double precision complex arithmetic on the University of Rochester’s IBM 370 computer. Any observed discontinuous behavior in the computed field evolution was then thought to be due to numerical instabilities caused by this limited numerical accuracy.

446

13 Evolution of the Precursor Fields

p for all   0  2 =3 with 0 n.0/ D s , where [cf. (12.306) and

 (12.307)] a0 p =.20 / and .a0 m2 =.20 // p2 .1 C 3s /=.4s m2 /  1 , with p 0 C f 0 and m2 0 f 0 . Numerical results presented in Figs. 12.25–12.30 show that this saddle point moves down the imaginary axis as  increases from the value p 1 1 , crossing the origin at  D 0 and then approaching the upper branch point singularity !p2 D i=0 as  ! 1. The accuracy of this approximation is illustrated in Fig. 12.60. A direct application of Olver’s theorem to the contour integral in (13.10) taken over the contour that results when C is deformed to an Olver-type path through the single near first-order saddle point SPn results in the asymptotic description of the Brillouin precursor in a Debye-type dielectric given by [30] r Ab .z; t / 

) ( c Q .!SPn ./  !c / z .!SPn ;/ i u c e < e 2z . 00 .!SPn ;  //1=2

(13.145)

p as z ! 1 with  > 1 . Unlike that for a Lorentz model dielectric where there are two neighboring near saddle points that coalesce into a single second-order saddle point at some critical space–time point, thereby requiring the application of a uniform asymptotic expansion technique, the asymptotic expression given in (13.145) p is uniformly valid for all finite space–time points  > 1 provided that any pole singularities of the spectral envelope function uQ .!SPn . /  !c / are sufficiently well removed from the near saddle point location lying along the imaginary axis. The dynamical structure of the Brillouin precursor in a single relaxation time Rocard–Powles–Debye model dielectric when the input pulse is a Heaviside unitstep-function signal with fc D 1 GHz carrier frequency at three absorption depths (z D 3zd ) into the simple Rocard–Powles–Debye model of triply distilled water whose frequency dispersion is depicted in Fig. 12.21, where zd ˛ 1 .!c /, is illustrated in Fig. 13.10. The solid curve describes the asymptotic solution given in (13.145) with numerically determined near saddle point locations and the dashed curve describes the asymptotic solution with the approximate saddle point location given in (13.144). Notice that this Brillouin precursor, which is characteristic of Debye-type dielectrics (see Case 2 in Sect. 12.1.1), appears as a single positive pulse with peak amplitude occurring at the space–time point  D 0 D n.0/. This peak amplitude point then propagates with the velocity v0 D c=0 D c=n.0/ through the dispersive material [31]. Since n.0/ > nr .!/ for all real ! > 0, the peak amplitude velocity is the minimum phase velocity for a pulse in the dispersive dielectric. Since !SPn .0 / D 0 and .!SPn .0 /; 0 / D .0; 0 / D 0, (13.145) then shows that [30, 31] ( AB .z; t0 /  < e

i



c uQ .!c / i 4 n0 .0/z

1=2 ) ;

(13.146)

13.4 The Brillouin Precursor Field in Debye Model Dielectrics

447

0.16 0.14

AHb(z,t)

0.12 0.1 0.08 0.06 0.04 0.02 0

8

8.2

8.4

8.6

8.8

9

9.2

9.4

9.6

9.8

Fig. 13.10 Temporal evoution of the Brillouin precursor field AH b .z; t / at three absorption depths z D 3zd with zd ˛ 1 .!c / in the simple Rocard–Powles–Debye model of triply distilled water for a Heaviside unit step function modulated signal with fc D 1 GHz carrier frequency. The solid curve describes the temporal behavior when numerically determined saddle point locations are used in the asymptotic approximation and the dashed curve when the approximate near saddle point solution is used

as z ! 1 with t0 D 0 z=c, and the peak amplitude point in the Brillouin precursor p only decays algebraically as 1= z . The instantaneous oscillation frequency of the Brillouin precursor at the peak amplitude point is identically zero; however, this does not mean that the Brillouin precursor is a static field. In fact, the instantaneous oscillation frequency can be quite misleading [22] so that the effective oscillation frequency of the Brillouin precursor needs to be carefully examined. One frequency measure that is physically meaningful is determined by the e 1 points of the exponential function expŒ.z=c/.!SPn . /;  / when !c  0, which are given by the solutions of the equation .!N ./; / D c= z:

(13.147)

Since these points occur about the origin where the peak value in the Brillouin precursor occurs, the complex phase function may be approximated by the first few terms in its Maclaurin series expansion as 1 .!;  / Š .0; / C  0 .0; /! C  00 .0;  /! 2 ; 2

(13.148)

448

13 Evolution of the Precursor Fields

where .0; / D 0,  0 .0; / D i.  0 /, and  00 .0;  / D 2i n0 .0/. As a first approximation, .!;  / i. 0 /!, in which case (13.145) with (13.144) yields the solution pair   a0 .0 C f 0 /c 1=2 ˙ 0 ˙ (13.149) 0 z for z > 0. The temporal width of the Brillouin precursor is then given by TB

  a0 .0 C f 0 / 1=2 z .C   / 2 z ; c 0 c

(13.150)

as z ! 1. This then corresponds to the effective oscillation frequency [30] fB

1 1

2 TB 4



0 c a0 .0 C f 0 /z

1=2 (13.151)

of the Brillouin precursor as z ! 1. These two results then show that the temporal width and oscillation frequency of the Brillouin precursor are set by the material parameters independent of the input pulse for sufficiently large propagation distances z > 0. Notice that the effective oscillation frequency of the Brillouin precursor approaches zero as the propagation distance increases to infinity, in which limit the Brillouin precursor becomes a static field (with zero amplitude), but is nonzero for finite propagation distances.

13.5 The Middle Precursor Field The asymptotic behavior of the middle precursor field in a double resonance Lorentz model dielectric is determined by the phase behavior about the middle saddle points ˙ , j D 1; 2, whose dynamical evolution in the right half of the complex !-plane SPmj C (i.e., the -evolution of SPmj ), illustrated in the sequence of graphs in Figs. 12.12– 12.19, is summarized here in Fig. 13.11. Because of the inherent symmetry of the problem about the imaginary axis, as expressed in the equivalent field representations [from (11.12) and (11.13), and (11.28) and (11.29) with (11.45)] Z iaC1 z 1 fQ.!/e c .!;/ d! A.z; t / D <  ia Z iaC1 z 1 D fQ.!/e c .!;/ d! 2 ia1

(13.152) (13.153)

C for all z  0, just the middle saddle point pair SPmj , j D 1; 2, in the right half plane needs to be considered. The asymptotic description of the middle precursor field Am .z; t / is then obtained from the uniform asymptotic expansion of the contour

13.5 The Middle Precursor Field

449

''

a

+ SPm1

+

SPn

' (0)

SPd+

(1) (2)

(3)

+ SPm2

P( )

''

b

+ SPm1

+

SPn

(0)

' + SPm2

(1)

SPd+ (2)

(3)

P( )

c

''

' SPn+

+ SPm1 (0)

(1)

+

SPm2

SPd+ (2)

(3)

P( )

Fig. 13.11 Illustration of the saddle point evolution in a double resonance Lorentz model dielectric for (a) 1 <  < N1 , (b)  D N1 , and (c)  > N1 in the right half of the complex !-plane. The hatched areas in each plot indicate the local region about each saddle point where the real part  .!;  / of the complex phase function .!;  / is less than that at that saddle point. The shaded region in part b illustrates the local behavior about the effective second-order middle saddle point at C C  D N1 when the two first-order middle saddle points SPm1 and SPm2 come into closest proximity to each other

450

13 Evolution of the Precursor Fields

C integral appearing in (13.152) about the middle saddle point pair SPmj for all  > 1. Because the space–time evolution of this middle saddle point pair is analogous to that for the near saddle point pair, coming into closest proximity to each other at  D N1 , the analysis of their asymptotic contribution is analogous to that given in Sect. 13.3.2 for the Brillouin precursor. Although this middle saddle point pair remains separated at this critical space–time point, they do approach close enough that an effective second-order saddle point is produced at  D N1 , as indicated in part (b) of Fig. 13.11 (also see Fig. 12.16), reinforcing this analogy with the near saddle point behavior. The angle of slope of the steepest descent path P .Ns / through the upper midC as it leaves the effective second-order saddle point is seen dle saddle point SPm1 in Fig. 13.11b to be ˛N s D =6. In addition, with the change ! appearing in (10.74) taken to lie along the portion of the path P .Ns / approaching the effective saddle point from the left in Fig. 13.11b, then arg.f !g/ D 5=6, and (10.74) then states that the argument of v lying along the corresponding portion of the transformed contour [under the cubic change of variable defined in (10.49)] is given by arg.f vg/ D =6 C 5=6 D 2=3, showing that the transformed contour originates in Region 2 of Fig. 10.6. If the change ! is taken to lie along the portion of the path P .Ns / leaving this effective saddle point toward the right in Fig. 13.11b, then arg.f !g/ D =6 so that arg.f vg/ D 0, showing that the transformed contour terminates in Region 1 of Fig. 10.6. Consequently, the contour of integration P . / is transformed into an L21 path so that the function C. / appearing in the uniform expansion given in (10.51) of Theorem 4 is given by

  C. / D e i2=3 Ai e i2=3 where Ai . / denotes the Airy function. For early space–time values  2 .1; N1 , the deformed contour of integration is taken along a set of Olver-type paths passing through the upper near saddle point C , and the distant saddle point SPdC in the SPnC , the upper middle saddle point SPm1 right-half plane, as depicted in Fig. 13.11a. For later space–time values  > N1 , the deformed contour of integration is taken along a set of Olver-type paths passing C C and SPm2 , through the near saddle point SPnC , the pair of middle saddle point SPm1 C and the distant saddle point SPd in the right-half plane, as depicted in Fig. 13.11c. At the critical space–time value  D N1 when the two first-order middle saddle points are in closest proximity to each other, the deformed contour passing through the C is chosen to pass through the effective secondupper middle saddle point SPm1 order saddle point residing between the middle saddle point pair, as depicted in Fig. 13.11b. The analysis leading to (13.104) and (13.117) then applies, so that, beginning ˚ with (13.152) with fQ.!/ D < iei uQ .!  !c / , one obtains

13.5 The Middle Precursor Field

451

 1=3 z c Am .z; t / D  < e i e c ˛N 0 ./ z ( h      i  uQ !SP C  !c hN 1 ./ C uQ !SP C  !c hN 2 ./ C O z1 m1

m2

  e i2=3 Ai ˛N 1 ./e i2=3 .z=c/2=3    .c=z/1=3 h  uQ !SP C  !c hN 1 . /  uQ !SP C  !c hN 2 ./ C 1=2 m1 m2 ˛N 1 . / )   1 i i4=3 0  i2=3 2=3 e Ai ˛N 1 ./e .z=c/ CO z (13.154) as z ! 1 for  > 1, where    1   !SP C C  !SP C ; m1 m2 2        1=3 3 1=2  !SP C   !SP C ˛N 1 . / ; m1 m2 4 2 31=2 1=2 ./ 2 ˛ N 5 ; hN 1;2 . / 4 00  1  !SP C ;  ˛N 0 . /

(13.155) (13.156)

(13.157)

m1;2

the upper sign corresponding to hN 1 ./ and the lower sign to hN 2 . /. The proper values of the multivalued functions appearing in (13.156) and (13.157) are determined by the conditions presented in Sect. 10.3.2. Consider first obtaining these values when  2 .1; N1 /. The phase of hN 1;2 ./ is determined by the limiting behavior given in (10.60) as   lim arg hN 1;2 ./ D ˛N C ;

!N1

(13.158)

where ˛N C D =6 is the angle of slope of the steepest descent path leaving the effective second-order saddle point at  D N1 . However, notice that different values are obtained at the actual middle saddle point locations, as depicted in Fig. 13.12. In particular, the angle of slope of the steepest descent path leaving the upper middle saddle point at  D N1 is equal to =6, as seen in Fig. 13.11b. This then translates into the effective value ˛N C D =6 for the steepest descent path leaving the effective second-order saddle point depicted in Fig. 13.11b, as stated above. The phase of the 1=2 quantity ˛N 1 . / is then obtained from (10.63) with n D 0 as  1=2  lim arg ˛N 1 ./ D ˛N 12  ˛N C ;

!N1

(13.159)

452

13 Evolution of the Precursor Fields

-

a

-

-

arg {h1,2 ( )}

b

-

arg {h1}

-

-

arg {h2}

arg { 1 ( )}

c

Fig. 13.12 Depiction of the  -dependence of (a) the angle of slope ˛N 12 of the vector from the 1=2 C C middle saddle point SPm2 to SPm1 and the arguments of the complex quantities (b) ˛N 1 . /, and N (c) h1;2 . / for the middle saddle point pair in a double resonance Lorentz model dielectric

where ˛N 12 is the angle of slope of the vector from the lower middle saddle point C C to the upper middle saddle point SPm1 . Numerical calculations show that ˛N 12 SPm2 increases from  =2 at  D 1 to   as  increases above N1 , passing through the value  3=4 at  D N1 , as depicted in Fig. 13.12a. The limiting relation stated C then shows that the in (13.159) as applied to the upper middle saddle point SPm1 1=2 proper branch of ˛N 1 . / passes through the value 11=12, increasing from  2=3 at  D 1 to  5=6 as  increases above N1 , as depicted in Fig. 13.12c. However, for the effective middle saddle point, ˛N 12 D =2 and ˛N C D =6 so that  1=2   lim arg ˛N 1 ./ eff D :  N 3 !1

(13.160)

13.5 The Middle Precursor Field

453

The dotted curves in each part of Fig. 13.12 describe this effective behavior. With these results, (13.154) yields the uniform asymptotic approximation of the middle precursor field as Am .z; t / 

 1=3 z c < e i e c ˛N 0 ./ z ( h  ˇ ˇi  ˇ ˇ  uQ !SP C  !c ˇhN 1 ./ˇ C uQ !SP C  !c ˇhN 2 . /ˇ m1

m2

ˇ ˇ  Ai ˇ˛N 1 ./ˇ.z=c/2=3 ˇ ˇi  ˇ ˇ .c=z/1=3 h  ˇN ˇ Q ! C  !c ˇhN 2 . /ˇ ˇ C  !c h1 . /  u ˇ1=2 uQ !SPm1 SPm2 ˇ˛N 1 . /ˇ ) ˇ ˇ  0 ˇ 2=3 ˇ A ˛N 1 ./ .z=c/ ; (13.161) i

as z ! 1 uniformly for all  2 .1; N1 /. In a similar manner, the effective limiting behavior as  ! N1C that is depicted in Fig. 13.12 results in  1=2  5 : lim arg ˛N 1 ./ eff D C 6 !N1

(13.162)

The uniform asymptotic approximation of the middle precursor is then obtained from (13.154) as  1=3 z c Am .z; t /  < e i e c ˛N 0 ./ z ( h  ˇ ˇi  ˇ ˇ  uQ !SP C  !c ˇhN 1 ./ˇ C uQ !SP C  !c ˇhN 2 . /ˇ m1

m2

ˇ  ˇ  Ai ˇ˛N 1 ./ˇ.z=c/2=3 ˇ ˇi  ˇ ˇ .c=z/1=3 h  ˇN ˇ Q ! C  !c ˇhN 2 . /ˇ ˇ C  !c h1 . /  u ˇ1=2 uQ !SPm1 SPm2 ˇ˛N 1 . /ˇ ) ˇ  ˇ  A0 ˇ˛N 1 ./ˇ.z=c/2=3 ; (13.163) i

as z ! 1 uniformly for all   N1 . The temporal evolution of the middle precursor field AH m .z; t / for a Heaviside unit step function modulated signal with angular carrier frequency !c D 2  1016 r=s at five absorption depths in the passband between the two absorption bands of a double resonance Lorentz model dielectric with medium parameters !0 D 1  1016 r=s, p 0:6  1016 r=s, ı0 D 0:1  1016 r=s for the lower resonance line and b0 D

454

13 Evolution of the Precursor Fields 0.015

0.01

AHm(z,t)

0.005 _

0

−0.005

−0.01

−0.015 1.15

1.2

1.25

1.3

1.35

1.4

1.45

Fig. 13.13 Temporal evoution of the middle precursor field Am .z; t / at five absorption depths z D 5zd inp a double resonance Lorentz model dielectric with medium parameters !0 D 1  1016 r=s, b0 D p0:6  1016 r=s, ı0 D 0:1  1016 r=s for the lower resonance line and !2 D 7  1016 r=s, b2 D 12  1016 r=s, ı2 D 0:1  1016 r=s for the upper resonance line for a Heaviside unit step function modulated signal with angular carrier frequency !c D 2  1016 r=s

p !2 D 7  1016 r=s, b2 D 12  1016 r=s, ı2 D 0:1  1016 r=s for the upper resonance line is illustrated in Fig. 13.13. Notice that this middle precursor reaches its peak amplitude (which is nearly twice the value e 5 0:0067 for the signal at the applied carrier frequency) near the critical space–time point  D N1 . In addition, notice that the instantaneous oscillation frequency chirps up as  increases to N1 and then chirps down as  increases above this critical critical space–time point with angular frequency that is essentially contained within the passband. Finally, notice the interference between the two middle saddle points for  > N1 .

13.6 Impulse Response of Causally Dispersive Materials A canonical pulse type of fundamental mathematical interest in the description of dispersive pulse propagation phenomena is the Dirac delta function pulse (see Sect. 11.2.1) (13.164) Aı .0; t / D ı.t / whose dynamical evolution yields the impulse response of the dispersive medium. Because the temporal frequency spectrum of this initial pulse function is unity for all !, the propagated plane wave pulse field, given by

13.6 Impulse Response of Causally Dispersive Materials

Aı .z; t / D

1 2

Z

455

e .z=c/.!;/ d!

(13.165)

C

for all z  0, exhibits the pure asymptotic contributions from the saddle points of the complex phase function .!; / at sufficiently large z > 0. The impulse response is then comprised of the precursor fields that are a characteristic of the material dispersion. For a single resonance Lorentz model dielectric the asymptotic behavior of the impulse response is given by [3] Aı .z; t /  Aıs .z; t / C Aıb .z; t /

(13.166)

as z ! 1 for all   1, the propagated wavefield identically vanishing over the entire superluminal space–time domain  < 1. As illustrated in Fig. 13.14, the Sommerfeld precursor Aıs .z; t / arrives with infinite frequency at the speed of light point  D 1 (see Sect. 13.2.5). As  increases above unity, the distant sad0 in from ˙1, approaching the outer branch points !˙ as dle points SPd˙ move ˚  ! 1, so that < ˙ !SP ˙ . / monotonically decreases toward the limiting value d q ˚ !12  ı 2 . At the same time, the attenuative part  .!SP ˙ ;  / < .!SP ˙ ;  / d d of the complex phase function at the distant saddle points monotonically decreases from zero as  increases above unity, resulting in an increase in the wave amplitude attenuation as  increases and the wavefield evolves, as evident in Fig. 13.14. At

2

x 104

Ad (z,t)

1

0

+ SB

−1

−2

1

1.2

1.4

1.6

1.8

2

Fig. 13.14 Numerically determined impulse response of a single resonance Lorentz p model dielectric with Brillouin’s choice of the medium parameters (!0 D 4  1016 r=s, b D 20  1016 r=s, ı D 0:28  1016 r=s) at z D 1 m

456

13 Evolution of the Precursor Fields

the space–time point  D SB , the saddle point dominance changes from the distant to the near saddle points and the Brillouin precursor Aıb .z; t / then dominates the impulse response for all  > SB . As  increases over the space–time interval .SB ; 0 /, the value of  .!SPnC ;  / monotonically increases to zero, vanishes at  D 0 , and then monotonically decreases as  increases above 0 . The field amplitude then experiences zero exponential decay at  D 0 , the amplitude varying with propagation distance z > 0 only as z1=2 at this space–time point, the effective angular oscillation frequency being given by (13.130) over this space–time domain. The instantaneous angular frequency of oscillation of the Brillouin precursor then monotonically increases with increasingq > 1 from the effective value !eff .0 / at

 D 0 , approaching the limiting value !02  ı 2 as  ! 1, the amplitude attenuation also increasing monotonically with increasing  > 0 , as seen in Fig. 13.14. Similar behavior is obtained for a double resonance Lorentz model dielectric if the resonance frequencies !0 and !2 are sufficiently close that the middle saddle ˙ , j D 1; 2, are never the dominant saddle points because p > 0 [see points SPmj (12.116) and (12.117)]. This is illustrated in Fig. 13.15 which depicts the impulse response of a double-resonance medium with model parameters 16 !0 D 1 p 10 r=s; b0 D 0:6  1016 r=s; ı0 D 0:1  1016 r=s;

!2 D 4p 1016 r=s; b2 D 12  1016 r=s; ı2 D 0:1  1016 r=s;

Ad (z,t)

5000

0

−5000 1

SB

1.2

1.4

1.6

1.8

2

Fig. 13.15 Numerically determined impulse response at z D 5 m in a double resonance Lorentz model dielectric when the inequality p > 0 is satisfied

13.6 Impulse Response of Causally Dispersive Materials

457

at the propagation distance z D 5 m. In that case, the asymptotic behavior is described by (13.166) with the transition between the Sommerfeld precursor field dominance and the Brillouin precursor field dominance occuring at the space–time point  D SB ' 1:370. This impulse response is seen to be essentially indistinguishable from that of an equivalent single resonance medium. If the upper resonance frequency in this double resonance medium example is increased to !2 D 7  1016 r=s, then p < 0 and the middle saddle points become the dominant saddle points over the space–time interval  2 .SM ; MB / following the distant saddle point dominance and preceeding the near saddle point dominance, where SM ' 1:201 and MB ' 1:279 for this set of model material parameters. In that case the asymptotic behavior of the impulse response is given by [32] Aı .z; t /  Aıs .z; t / C Aım .z; t / C Aıb .z; t /

(13.167)

as z ! 1 for all   1, the propagated wavefield identically vanishing over the entire superluminal space–time domain  < 1. The impulse response of the medium then contains a middle precursor the evolves between the Sommerfeld and Brillouin precursors, as illustrated in Fig. 13.16. The instantaneous angular frequency of oscillation ofqthis middle precursor first increases as  increases to N1 and then decreases to ! 2  ı 2 as  increases above N1 , whereas its rate of exponential at1

0

tenuation with propagation distance z > 0 first decreases and then increases with increasing  .

Ad (z,t)

5000

0 SM

−5000 1

1.2

MB

1.4

1.6

1.8

2

Fig. 13.16 Numerically determined impulse response at z D 5 m in a double resonance Lorentz model dielectric when the inequality p < 0 is satisfied

458

13 Evolution of the Precursor Fields

A dispersive material of central interest to current research in ultrawideband electromagnetics, particularly to bioelectromagnetics, is water, as this substance is pervasive. The dielectric permittivity of triply distilled water may be expressed as (see Sect. 4.4.5 of Vol. 1) .!/ D or .!/ C res .!/  1;

(13.168)

where or .!/ describes the frequency dependence due to orientational polarization phenomena as described, for example, by the Rocard–Powles extension [33] of the Debye model [34], and where res .!/ describes the frequency dependence due to resonance polarization phenomena as described, for example, by the Lorentz model [35]. The 1 term appearing in this equation is introduced to compensate for the fact that each component model includes a C1 term to describe the vacuum. The multiple resonance Lorentz model description of the dominant resonance features appearing in the measured frequency dispersion of triply distilled water (see Figs. 4.2 and 4.3 of Vol. 1) is given by [see (4.214)] res .!/=0 D 1 

8 X

bj2

.jeve n/

! 2  !j2 C 2i ıj !

(13.169)

j D0

with (rms best fit) parameter values (see Table 4.2) !0 !2 !4 !6 !8

D 2:08  1013 r=s; D 1:05  1014 r=s; D 3:27  1014 r=s; D 6:19  1014 r=s; D 2:30  1016 r=s;

b0 b2 b4 b6 b8

D 4:98  1012 r=s; D 8:48  1013 r=s; D 1:98  1013 r=s; D 1:65  1014 r=s; D 2:01  1016 r=s;

ı0 ı2 ı4 ı6 ı8

D 3:56  1012 r=s; D 4:51  1013 r=s; D 3:08  1012 r=s; D 2:86  1013 r=s; D 5:82  1015 r=s:

The angular frequency dispersion described by this multiple resonance Lorentz model of triply distilled water is given in Fig. 13.17, where the solid curve describes the real part  0 .!/=0 and the dashed curve the imaginary part  00 .!/=0 of the relative dielectric permittivity. The numerically determined impulse response at z D 1 m in this multiple resonance Lorentz model description of the dielectric frequency dispersion of triply distilled water is illustrated in Fig. 13.18, where 0 D

s X 1C .bj =!j /2 ;

(13.170)

j

in this pure resonance case. Notice that this impulse response is almost entirely comprised of a high-frequency Sommerfeld precursor field component followed by a low-frequency Brillouin precursor field component, as described by (13.166). Although the middle precursor never completely appears as the dominant field behavior over some space–time domain in the dynamical field evolution in water, the

13.6 Impulse Response of Causally Dispersive Materials

459

101

Real & Imaginary Parts of

0

'

0

0

10

''

0

−1

10

10−2 1012

0

1013

4

1014

6

8

1015

1016

-r/s

1017

1018

Fig. 13.17 Angular frequency dependence of the real  0 .!/=0 (solid curve) and imaginary  00 .!/=0 (dashed curve) parts of the relative dielectric permittivity as described by the dominant Lorentz model resonance features of triply distilled H2 O

5

x 105

4 3 2

Ad (z,t)

1 0 −1 −2 −3 −4 −5

1

1.2

1.4

1.6

1.8

2

Fig. 13.18 Numerically determined impulse response at z D 1 m in the multiple resonance Lorentz model description of the dielectric frequency dispersion of triply distilled water depicted in Fig. 13.17. Notice that the horizontal axis for the wavefield amplitude is in arbitrary units

460

13 Evolution of the Precursor Fields

middle saddle points do show their influence on the Brillouin precursor by quenching the oscillatory relaxation of this field component, as seen through a comparison of the field evolution illustrated in Fig. 13.18 with that in Fig. 13.14. This is due to interference between the asymptotic contribution from the near saddle points SPn˙ ˙ and the upper middle saddle points SPm1 for  > 1 whose exponential attenuation is only slightly greater than that for the near saddle points for all  > 1 [36]. As the propagation distance z > 0 increases and the dispersive behavior matures,5 the space–time point at which the peak amplitude in the Brillouin precursor occurs shifts upward to the critical space–time point at  D 0 1:5963 where the wavefield amplitude A.z; t0 /, with t0 D .c=z/0 , only decays with the propagation distance as z1=2 . This is illustrated in Fig. 13.19 which gives the impulse response following the Sommerfeld precursor evolution at the propagation distance z D 1 mm in the multiple resonance Lorentz model description of the frequency dispersion of triply distilled water. The Cole–Cole extension of the multiple relaxation time Rocard–Powles–Debye model of the orientational polarization component in the dielectric permittivity of triply distilled water is given by [see (4.214)]

140 120 100

Ad (z,t)

80 60 40 20 0 −20 1.4

1.6

1.8

2

Fig. 13.19 Numerically determined impulse response at z D 1 mm in the multiple resonance Lorentz model description of the dielectric frequency dispersion of triply distilled water depicted in Fig. 13.17. Notice that the horizontal axis for the wavefield amplitude is in the same units as that in Fig. 13.18

5

In the mature dispersion regime [37], “the field is dominated by a single real frequency at each space–time point. That frequency !E is the frequency of the time-harmonic wave with the least attenuation that has energy velocity equal to z=t .” A more detailed description is given in Chap.16.

13.6 Impulse Response of Causally Dispersive Materials

or .!/=0 D 1 C

461

2 X

aj  1j  : 1  i !fj j D1 1  i !j

(13.171)

With 1 D 0 and 2 D 1=2, the rms best fit parameter values appearing in this model are given by (see Table 4.1) a1 D 74:65; a2 D 2:988;

1 D 8:30  1012 s; 2 D 5:91  1014 s;

f 1 D 1:09  1013 s; f 2 D 8:34  1015 s:

The angular frequency dispersion of the relative dielectric permittivity described by the full Cole–Cole extension of the composite Rocard–Powles–Debye–Lorentz model of triply distilled water, obtained by combining (13.169) and (13.171) in (13.168), is illustrated in Fig. 13.20. Comparison of this behavior with that presented in Fig. 13.17 reveals the influence of the low-frequency orientational polarization response on the high-frequency resonance polarization response of the material dispersion. The numerically determined impulse response at z D 100 m in this composite dispersion model of the dielectric frequency dispersion of triply distilled water is illustrated in Fig. 13.21 over a space–time domain immediately following the Sommerfeld precursor evolution. In this case

2

Real & Imaginary Parts of

0

10

101 '

0

100 ''

0

10−1

−2

10

1010

1012

1014

1016

1018

- r/s Fig. 13.20 Angular frequency dependence of the real  0 .!/=0 (solid curve) and imaginary  00 .!/=0 (dashed curve) parts of the relative dielectric permittivity as described by the Cole–Cole extension of the composite Rocard–Powles–Debye–Lorentz model of triply distilled H2 O

462

13 Evolution of the Precursor Fields 4

1

x 10

0.8 0.6 0.4

Ad (z,t)

0.2 0

−0.2 −0.4 −0.6 −0.8 −1 1.5

2.5

3.5

4.5

5.5

Fig. 13.21 Numerically determined impulse response at z D 100 m in the Cole–Cole extension of the Rocard–Powles–Debye model description of the dielectric frequency dispersion of triply distilled water depicted in Fig. 13.20. Notice that the horizontal axis for the wavefield amplitude is in arbitrary units

0 D

s X X 1C aj C .bj =!j /2 ; j

(13.172)

j

resulting in the approximate value 0 8:9547. The detailed temporal field evolution illustrated in Fig. 13.21 displays the interference between the middle and Brillouin precursor field contributions to the total wavefield evolution. As the propagation distance z > 0 increases and the dispersive wavefield behavior matures, the space–time point at which the peak amplitude in the Brillouin precursor occurs shifts upward to the critical space–time point at  D 0 where the wavefield amplitude A.z; t0 /, with t0 D .c=z/0 , only decays with the propagation distance as z1=2 . The dynamical evolution of the impulse response Aı .z; t / becomes increasingly dominated by the Sommerfeld and Brillouin precursor fields as the propagation distance increases to infinity (i.e., as z ! 1) through the mature dispersion regime.

References ¨ 1. A. Sommerfeld, “Uber die Ausbreitung der Wellen in der drahtlosen Telegraphie,” Ann. Phys. (Leipzig), vol. 28, pp. 665–737, 1909. ¨ 2. A. Sommerfeld, “Uber die fortpflanzung des lichtes in disperdierenden medien,” Ann. Phys., vol. 44, pp. 177–202, 1914.

References

463

3. K. E. Oughstun and G. C. Sherman, “Propagation of electromagnetic pulses in a linear dispersive medium with absorption (the Lorentz medium),” J. Opt. Soc. Am. B, vol. 5, no. 4, pp. 817–849, 1988. 4. K. E. Oughstun and G. C. Sherman, Pulse Propagation in Causal Dielectrics. Berlin: SpringerVerlag, 1994. 5. H. M. Nussenzveig, Causality and Dispersion Relations. New York: Academic, 1972. Chap. 1. ¨ 6. L. Brillouin, “Uber die fortpflanzung des licht in disperdierenden medien,” Ann. Phys., vol. 44, pp. 204–240, 1914. 7. L. Brillouin, Wave Propagation and Group Velocity. New York: Academic, 1960. 8. P. Debye, “N¨aherungsformeln f¨ur die zylinderfunktionen f¨ur grosse werte des arguments und unbeschr¨ankt verander liche werte des index,” Math. Ann., vol. 67, pp. 535–558, 1909. 9. E. T. Copson, An Introduction to the Theory of Functions of a Complex Variable. London: Oxford University Press, 1935. Sect. 6.52. 10. E. T. Whittaker and G. N. Watson, A Course of Modern Analysis. New York: MacMillan, 1943. Sect. 6.222. 11. E. T. Copson, An Introduction to the Theory of Functions of a Complex Variable. London: Oxford University Press, 1935. Sect. 6.1. 12. E. T. Whittaker and G. N. Watson, A Course of Modern Analysis. New York: MacMillan, 1943. Sect. 5.2. 13. R. Landauer, “Light faster than light?,” Nature, vol. 365, pp. 692–693, 1993. 14. G. Diener, “Superluminal group velocities and information transfer,” Phys. Lett. A, vol. 223, pp. 327–331, 1996. 15. F. W. J. Olver, “Why steepest descents?,” SIAM Rev., vol. 12, no. 2, pp. 228–247, 1970. 16. R. A. Handelsman and N. Bleistein, “Uniform asymptotic expansions of integrals that arise in the analysis of precursors,” Arch. Rat. Mech. Anal., vol. 35, pp. 267–283, 1969. 17. K. E. Oughstun, Propagation of Optical Pulses in Dispersive Media. PhD thesis, The Institute of Optics, University of Rochester, 1978. 18. K. E. Oughstun and G. C. Sherman, “Uniform asymptotic theory of pulse propagation in Lorentz media,” in Proceedings of the URSI Symposium on Electromagnetic Wave Theory, (Stanford University), pp. 34–36, 1977. 19. K. E. Oughstun and G. C. Sherman, “Uniform asymptotic description of dispersive pulse propagation,” J. Opt. Soc. Am. A, vol. 69, no. 10, p. 1448A, 1979. 20. K. E. Oughstun and G. C. Sherman, “Uniform asymptotic description of electromagnetic pulse propagation in a linear dispersive medium with absorption (the Lorentz medium),” J. Opt. Soc. Am. A, vol. 6, no. 9, pp. 1394–1420, 1989. 21. M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, vol. 55 of Applied Mathematics Series. Washington, D.C.: National Bureau of Standards, 1964. 22. L. Mandel, “Interpretation of instantaneous frequencies,” Am. J. Phys., vol. 42, no. 10, pp. 840– 846, 1974. 23. S. He and S. Str¨om, “Time-domain wave splitting and propagation in dispersive media,” J. Opt. Soc. Am. A, vol. 13, no. 11, pp. 2200–2207, 1996. 24. A. Karlsson and S. Rikte, “Time-domain theory of forerunners,” J. Opt. Soc. Am. A, vol. 15, no. 2, pp. 487–502, 1998. 25. C. Chester, B. Friedman, and F. Ursell, “An extension of the method of steepest descents,” Proc. Cambridge Phil. Soc., vol. 53, pp. 599–611, 1957. 26. K. E. Oughstun and N. A. Cartwright, “Ultrashort electromagnetic pulse dynamics in the singular and weak dispersion limits,” in Progress in Electromagnetics Research Symposium, (Prague, Czech Republic), 2007. 27. N. A. Cartwright and K. E. Oughstun, “Uniform asymptotics applied to ultrawideband pulse propagation,” SIAM Rev., vol. 49, no. 4, pp. 628–648, 2007. 28. J. A. Solhaug, K. E. Oughstun, J. J. Stamnes, and P. Smith, “Uniform asymptotic description of the Brillouin precursor in a single-resonance Lorentz model dielectric,” Pure Appl. Opt., vol. 7, no. 3, pp. 575–602, 1998.

464

13 Evolution of the Precursor Fields

29. N. A. Cartwright, Uniform Asymptotic Description of the Unit Step Function Modulated Sinusoidal Signal. PhD thesis, College of Engineering & Mathematical Sciences, University of Vermont, 2004. 30. K. E. Oughstun, “Dynamical evolution of the Brillouin precursor in Rocard-Powles-Debye model dielectrics,” IEEE Trans. Ant. Prop., vol. 53, no. 5, pp. 1582–1590, 2005. 31. K. E. Oughstun, “Dynamical evolution of the precursor fields in linear dispersive pulse propagation in lossy dielectrics,” in Ultra-Wideband, Short-Pulse Electromagnetics 2 (L. Carin and L. B. Felsen, eds.), pp. 257–272, New York: Plenum Press, 1994. 32. S. Shen and K. E. Oughstun, “Dispersive pulse propagation in a double-resonance Lorentz medium,” J. Opt. Soc. Am. B, vol. 6, pp. 948–963, 1989. 33. J. McConnel, Rotational Brownian Motion and Dielectric Theory. London: Academic Press, 1980. 34. P. Debye, Polar Molecules. New York: Dover Publications, 1929. 35. H. A. Lorentz, The Theory of Electrons. Leipzig: Teubner, 1906. Chap. IV. 36. J. E. K. Laurens and K. E. Oughstun, “Electromagnetic impulse response of triply-distilled water,” in Ultra-Wideband, Short-Pulse Electromagnetics 4 (E. Heyman, B. Mandelbaum, and J. Shiloh, eds.), pp. 243–264, New York: Plenum Press, 1999. 37. G. C. Sherman and K. E. Oughstun, “Description of pulse dynamics in Lorentz media in terms of the energy velocity and attenuation of time-harmonic waves,” Phys. Rev. Lett., vol. 47, pp. 1451–1454, 1981.

Problems 13.1. Derive the second order approximations given in (13.21) and (13.22) of the complex phase behavior at the distant saddle points of a double resonance Lorentz model dielectric. 13.2. Derive the uniform asymptotic approximation of the Sommerfeld precursor for a Heaviside unit step function signal in a double resonance Lorentz model dielectric using the second approximate expressions for the distant saddle point locations. 13.3. Derive the general expression for the uniform asymptotic expansion of the Sommerfeld precursor in a single resonance Lorentz model dielectric when the input pulse is given by A.0; t / D f .t / [see (12.1)]. 13.4. Derive the asymptotic approximation given in (13.45) of the delta function pulse Sommerfeld precursor Aıs .z; t / in either a single or double resonance Lorentz model dielectric. 13.5. Derive (13.88) and (13.89) for the approximate phase behavior at the secondorder near saddle point in a double resonance Lorentz model dielectric when  D 1 . 13.6. Obtain the uniform asymptotic approximation for the Brillouin precursor in a double resonance Lorentz model dielectric in the separate space–time domains  2 .1; 1 and   1 . Show that each of these uniformly valid expressions reduces to the asymptotic approximation given in (13.90) at the critical space–time point  D 1 when the two first-order near saddle points have coalesced into a single second-order saddle point.

Problems

465

13.7. Apply the asymptotic method presented in Sect. 10.3.3 to obtain the transitional asymptotic approximation of the delta function pulse Brillouin precursor Aıb .z; t / in a single resonance Lorentz model dielectric. 13.8. Derive the asymptotic description given in (13.145) of the Brillouin precursor field Ab .z; t / in a single relaxation time Rocard-Powles-Debye model dielectric.

Chapter 14

Evolution of the Signal

The contribution Ac .z; t / to the asymptotic behavior of the propagated plane wavefield A.z; t / that is due to the presence of any simple pole singularities of the spectral function uQ .!  !c /, where A.0; t / D u.t / sin .!c t C / with fixed angular carrier frequency !c  0, is now condidered in some detail, with primary attention given to the oscillatory case when !c > 0. As discussed in Sect. 12.4, the field component Ac .z; t / is associated with any long-term signal that is being propagated through the dispersive material. The velocity of propagation of the signal and the transition from the total precursor field to the signal field are determined by the relative asymptotic dominance of the component fields As .z; t /, Ab .z; t /, Am .z; t /, and Ac .z; t /. Consequently, discussion of these topics is deferred to Chap. 15 where the asymptotic description of the dynamical evolution of the total propagated wavefield A.z; t / is considered by combining the results of this chapter with those of Chap. 13. The first section of this chapter presents the nonuniform asymptotic analysis based on the direct application of Olver’s method [1] and the Cauchy residue theorem [2, 3]. Even though the nonuniform expression exhibits a discontinuous change in behavior as the space–time parameter  ct =z varies, it is a useful approximation for the pole contribution Ac .z; t / in the final expression for the total wavefield A.z; t / for all  > 1 provided that the dominant saddle point remains isolated from the pole, because in that case, Ac .z; t / is asymptotically negligible in comparison to the precursor field at the space–time point when the discontinuity occurs. This is precisely the situation for Debye-type dielectrics as the only saddle point in that case is the near saddle point SPn that moves down the imaginary axis as  increases, crossing the origin at  D 0 n.0/. The nonuniform approximation may then be applied in this case provided that the pole is sufficiently removed from the origin. When applied to the unit step function modulated signal in a single-resonance Lorentz model dielectric, the nonuniform expression for the pole contribution Ac .z; t / is the same as that obtained by Brillouin [4, 5] except for the space–time value s at which the discontinuous change occurs. For those cases in which the nonuniform result is useful, the difference in values of the space–time point at which the discontinuity occurs is of no consequence in the final expression for the total wavefield A.z; t / because Ac .z; t / is asymptotically negligible during a space–time interval that includes both values. Although Brillouin attached physical K.E. Oughstun, Electromagnetic and Optical Pulse Propagation 2: Temporal Pulse Dynamics in Dispersive, Attenuative Media, Springer Series in Optical Sciences 144, c Springer Science+Business Media, LLC 2009 DOI 10.1007/978-1-4419-0149-1 14, 

467

468

14 Evolution of the Signal

significance to the space–time value  D s at which this discontinuous change occurs, calling ts D .z=c/s the time of arrival of the signal, it is shown in Chap. 15 that the time of this discontinuous change has no physical significance and that the signal arrives at a later time. The nonuniform expression for the pole contribution Ac .z; t / is not useful in those cases in which the dominant saddle point passes near the pole singularity because Ac .z; t / is then not negligible at the space–time point s at which the discontinuity occurs. For such cases, it is necessary to apply the uniform asymptotic expansion technique due to Bleistein [6, 7] and Felsen and Marcuvitz [8], as summarized in Theorem 5 of Sect. 10.4, in order to obtain an asymptotic expression for the pole contribution Ac .z; t / that is uniformly valid in  . That analysis, as extended by Cartwright [9, 10], is presented in Sect. 14.3. The remaining sections apply this uniform asymptotic description to the analysis of the simple pole contribution for a Heaviside unit step function modulated signal in both Lorentz-type dielectrics and conducting media.

14.1 The Nonuniform Asymptotic Approximation This section obtains the nonuniform asymptotic approximation of the field component Ac .z; t / due to the contribution of any pole singularities appearing in the integrand of the Fourier–Laplace integral representation of the propagated plane wavefield [see (11.48)] A.z; t / D



Z 1 < i e i uQ .!  !c /e .z=c/.!;/ d! 2 C

(14.1)

as z ! 1. For that purpose, let !p denote a simple pole singularity of the spectral function uQ .!  !c /e i with residue p given by

 p D lim .!  !p /Qu.!  !c /e i :

(14.2)

!!!p

In accordance with the asymptotic procedure presented in Sect. 12.4, each of the functions Cd˙ .z; t /, Cm˙ .z; t /, and Cn˙ .z; t / appearing in (12.353) and (12.354) is zero in the nonuniform asymptotic approximation, so that ˚ Ac .z; t / D < 2 i./ ;

(14.3)

where . / D

X p

Res ! D !p



i uQ .!  !c /e i e .z=c/.!;/ 2

X i p e .z=c/.!p ;/ D 2 p

(14.4)

14.1 The Nonuniform Asymptotic Approximation

469

is the sum of the residues of the poles that are crossed when the original contour of integration C is deformed to the path P ./ that is specified in Sect. 12.4 (see Fig. 12.66). For reasons of simplicity, it is assumed here that the deformed contour of integration P . / is near only one pole at a time, attention then being restricted to obtaining the nonuniform asymptotic contribution due to that single pole alone. The results obtained are easily generalized to account for multiple pole contributions. The contribution of the simple pole singularity at ! D !p occurs when the original contour of integration C , extending along the straight line path from i a1 to i a C 1 in the upper half of the complex !-plane, is deformed across the pole to P . /. More specifically, assume that there is only one pole and let the original contour C and the deformed contour P ./ lie on the same side of the pole for  < s and on opposite sides for  > s . Then, from (14.4), one has that . / D 0; z i .s / D p e c .!p ;s / ; 4 z i . / D p e c .!p ;s / ; 2

for  < s ; for  D s ;

(14.5)

for  > s :

Upon substitution of this set of results into (14.3), one immediately obtains the nonuniform asymptotic approximation for the simple pole contribution at ! D !p as

Ac .z; t /  0;  < s ; h    i 1 z Ac .z; t /  e c .!p ;s / p0 cos cz $ .!p ; s /  p00 sin cz $ .!p ; s / ;  D s ; 2 h    i z Ac .z; t /  e c .!p ;/ p0 cos cz $ .!p ;  /  p00 sin cz $ .!p ;  / ;  > s ; (14.6) as z ! 1, where p0
(14.7) (14.8)

Consequently, the amplitude attenuation coefficient at the real angular frequency !p appearing in (14.6) is given by !p 1 ni .!p /: ˛.!p /   .!p / D c c

(14.9)

Furthermore, the oscillatory phase term appearing in the nonuniform approximation in (14.6) may, in this case, be written as

470

14 Evolution of the Signal

 z z  $ .!p ; / D !p nr .!p /   D ˇ.!p /z  !p t; c c

(14.10)

where the propagation factor ˇ.!p / at the real angular frequency !p is given by ˇ.!p /

!p nr .!p /: c

(14.11)

With these results, the simple pole contribution given in (14.6) becomes Ac .z; t /  0;  < s ; h  i   1 Ac .z; t /  e z˛.!p / p0 cos ˇ.!p /z  !p ts  p00 sin ˇ.!p /z  !p ts ; 2 Ac .z; t /  e

z˛.!p /

 D s ; h    i 0 00 p cos ˇ.!p /z  !p t  p sin ˇ.!p /z  !p t ;  > s ; (14.12)

as z ! 1 for real-valued !p > 0. It is then seen that, in this special case, the simple pole contribution results in a field contribution that is oscillating with fixed angular frequency !p and with an amplitude that is attenuated with propagation distance z > 0 at a constant, time-independent rate given by the attenuation coefficient ˛.!p / [compare this result with the general expression given in (14.6)]. The exact integral representation of the propagated plane wavefield given in (14.1) is a continuous function of the space–time parameter  ct =z for all z  0, and, in particular, is continuous at the space–time point  D s . However, the resulting asymptotic approximation of A.z; t / is a discontinuous function of  at  D s when the pole contribution Ac .z; t / is nonvanishing (i.e., when p ¤ 0) and is given by the nonuniform asymptotic approximation in (14.6).1 The discontinuity is of no consequence for fixed values of the propagation distance z larger than some positive constant Z, however, because the contribution to the propagated wavefield from the dominant saddle point at !SP varies exponentially as e .z=c/.!SP ;/ which dominates2 the exponential behavior of the pole contribution given in the second part of (14.6) at  D s . Hence, at the space–time point  D s when the discontinuity in the asymptotic behavior of the propagated wavefield occurs, the pole contribution is asymptotically negligible in comparison to the saddle point contribution, and, as a consequence, the discontinuous behavior is itself asymptotically negligible. For that reason, the particular value of s at which the pole crossing occurs is of little or no importance to the asymptotic behavior of the total propagated wavefield A.z; t /. 1

The value of s depends upon which Olver-type path is chosen for P . /. If that path is taken to lie along the path of steepest descent that passes through the saddle point nearest the pole, then the value of s is specified by the relation $ .!SP ; s / D $ .!p ; s /. 2 The asymptotic dominance of the saddle point contribution over the pole contribution at  D s is guaranteed by the fact that P .s / is an Olver-type path.

14.2 Rocard–Powles–Debye Model Dielectrics

471

14.2 Rocard–Powles–Debye Model Dielectrics Because the Debye-type dielectric has just the single near saddle point SPn that p moves down the imaginary axis as  increases from 1 D 1 , crossing the origin p at  D 0 s and then approaching the branch point !p1 D i=0 as  ! 1, as depicted in Fig. 14.1 (see also Figs. 12.25–12.30), the pole contribution at the real angular signal frequency !c > 0 will remain isolated from this saddle point for all   1 provided that !c is not too small. In that case, the pole contribution is given by (14.12) with s determined by the steepest descent path P . / passing through the near saddle point SPn into the right half of the complex !-plane. Because this steepest descent path is parallel to the real ! 0 -axis as it leaves the saddle point, s ' 0 ;

(14.13)

the accuracy of this approximation improving as the value of !c > 0 decreases. Notice further that (14.14) s  0 ; as evident from the sequence of illustrations in Fig. 14.1. For a Heaviside unit step function modulated signal with fixed angular carrier frequency !c > 0, the initial pulse spectrum is given by [from (11.56)] uQ H .!  !c / D i=.!  !c / which possesses a simple pole singularity at !p D !c with residue   i D i: (14.15)  D lim .!  !c / !!!c .!  !c / With this substitution, (14.12) becomes AHc .z; t /  0;

 < s ;

  1 AHc .z; t /   e z˛.!c / sin ˇ.!c /z  !c ts ;  D s ; 2   AHc .z; t /  e z˛.!c / sin ˇ.!c /z  !c t ;  > s ;

(14.16)

as z ! 1. The accuracy of this asymptotic approximation of the pole contribution may be assessed by first numerically computing the propagated wavefield at a fixed distance z > 0 into a particular Rocard–Powles–Debye model dielectric and then subtracting the asymptotic behavior of the Brillouin precursor [using (13.145) with numerically determined near saddle point locations] at that same propagation distance in the same dielectric, resulting in the numerical estimate of the pole contribution AnHc .z; t / AH .z; t /  AH b .z; t /;

(14.17)

which can then be compared with that described by the sequence of expressions in (14.16). An example of this calculation is presented in Figs. 14.2 and 14.3 for a fc D 1 GHz Heaviside unit step function signal at one absorption depth [z=zd D 1 with zd ˛ 1 .!c /] in the single relaxation time Rocard–Powles–Debye model of

472

14 Evolution of the Signal

a

ω“

SP

ω‘ P(θ)

branch cut

ωa

b

ω“

SP

ω‘

branch cut

P(θ)

c

ω“

SP

ω‘

branch cut

P(θ)

Fig. 14.1 Evolution of the steepest descent path P . / through the near saddle point in a single relaxation-time Debye-model dielectric. In part (a) 1 <  < 0 , (b)  D 0 , and (c)  > 0 . The short dashed curves are isotimic contours of  .!;  / below the value  .!SPn ;  / at the saddle point and the alternating long and short dashed curves are isotimic contours above that value. The shaded area in each part indicates the region of the complex !-plane where the inequality  .!;  / < .!SPn ;  / is satisfied

14.2 Rocard–Powles–Debye Model Dielectrics

473

0.5 AH(z,t)

0 8

A(z,t)

AHb(z,t)

−0.5

RP

2

4

6

8

10

12

14

16

18

20

Fig. 14.2 Numerically determined Heaviside step function AH .z; t / evolution with 1GH z carrier frequency (solid curve) and the asymptotic behavior of the associated Brillouin precursor AH b .z; t / (dashed curve) at one absorption depth in H2 O

0

−0.5

8

AHc(z,t)

0.5

2

RP

4

6

8

10

12

14

16

18

20

Fig. 14.3 Estimated signal contribution AHc .z; t / D AH .z; t /  AH b .z; t / for a 1GH z Heaviside step function signal at one absorption depth in H2 O

474

14 Evolution of the Signal

triply distilled water described by (12.300). Both the numerically determined wavefield A.z; t / and the asymptotic description of the Brillouin precursor field Ab .z; t / are presented in Fig. 14.2, the former by the solid curve and the latter by the dashed curve. Notice that the leading edge peak in the total wavefield A.z; t / is primarily due to the Brillouin precursor field.3 The resultant estimation of the pole contribution, given by the difference between these two wavefields as expressed by (14.17), is presented in Fig. 14.3. This result is in keeping with the nonuniform asymptotic approximation of the signal contribution Ac .z; t / given in (14.16). In particular, the pole contribution is seen to occur at the space–time point  D s ' 0 , as stated in (14.13). Similar results are obtained as the propagation distance increases, as illustrated in Fig. 14.4 when z D 3zd and in Fig. 14.5 when z D 5zd . Notice that the relative peak amplitude of the leading edge of the estimated signal (or pole) contribution AHc .z; t / increases with increasing propagation distance. Unfortunately, this numerically observed phenomenon is not described by the nonuniform asymptotic description given in (14.16). Nevertheless, the accuracy of the asymptotic description of the total wavefield evolution increases as z ! 1.

14.3 The Uniform Asymptotic Approximation If the saddle point at !sp . / approaches close to the pole singularity at !p when  D s so that the quantity j!sp .s /  !p j becomes small, then the quantity j.!sp .s /; s /  .!p ; s /j also becomes small. In that case the positive constant Z introduced at the end of Sect. 14.1 becomes increasingly large. As a result, it becomes impractical to take z > Z in order to make the pole contribution Ac .z; t / asymptotically negligible at  D s . To avoid the discontinuous behavior of Ac .z; t / at the space–time point  D s when the saddle point is near the pole, the uniform asymptotic approximation stated in Theorem 5 (see Sect. 10.4) due to Felsen and Marcuvitz [8, 11] and Bleistein [6, 7, 12], and later extended by Cartwright [9, 10] must be applied. That asymptotic technique is employed in this section in order to obtain the uniform asymptotic approximation of the pole contribution Ac .z; t / in both Lorentz model dielectrics and Drude model conductors. From (12.353) and (12.354) in Sect. 12.4, the pole contribution in a double resonance Lorentz model dielectric is given by Ac .z; t / D < f2 i./g C Cd .z; t / C CdC .z; t / CCm .z; t / C CmC .z; t / C CnC .z; t /;

3

(14.18)

The numerically determined wavefield A.z; t / in Fig. 14.2, as well as that used in obtaining the results presented in Figs. 14.3–14.5, has been slightly shifted in space–time by a fixed amount (  D 0:060) that is determined by the requirement that the peak amplitude point in A.z; t / for a sufficiently large propagation distance occurs at the same space–time point ( D 0 ) as that described by the asymptotic description of the Brillouin precursor.

14.3 The Uniform Asymptotic Approximation

475

AHc(z,t)

0.1

0

−0.1

8

9

10

11

12

Fig. 14.4 Estimated signal contribution AHc .z; t / D AH .z; t /  AH b .z; t / for a 1GH z Heaviside step function signal at three absorption depths (z=zd D 3) in a single relaxation-time Debye model of H2 O

0.02

AHc(z,t)

0.01

0

−0.01

8

9

10

11

12

Fig. 14.5 Estimated signal contribution AHc .z; t / D AH .z; t /  AH b .z; t / for a 1GH z Heaviside step function signal at five absorption depths (z=zd D 5) in a single relaxation-time Debye model of H2 O

476

14 Evolution of the Signal

for 1    1 , and Ac .z; t / D < f2 i./g C Cd .z; t / C CdC .z; t / CCm .z; t / C CmC .z; t / C Cn .z; t / C CnC .z; t /;

(14.19)

for  > 1 , where . /, which is given by (14.4), is the sum of the residues of the poles that are crossed when the original contour of integration C is deformed to the path P . /. For either a single resonance Lorentz model dielectric or a Drude model conductor, the terms Cm˙ .z; t / are set equal to zero in these two expressions. For reasons of simplicity, it is again assumed that the deformed contour of integration P . / is near only one pole at a time (i.e., that the poles are isolated), and attention is restricted to obtaining the uniform asymptotic contribution due to that single pole alone. The results obtained are easily generalized to account for several individual pole contributions. By Theorem 5 in Sect. 10.4, the C -functions appearing in (14.18) and (14.19) are given by ( "  p  z 1 < i  ˙ i erfc i . / z=c e c .!p ;/ C.z; t / D 2 #) p c=z z .!sp ;/ ec C ; . /

˚ > = . / < 0;

(14.20)   p z z 1 < i  i erfc i . / z=c e c .!p ;/  i e c .!p ;/ C.z; t / D 2 #) p ˚ c=z z .!sp ;/ C ; = . / D 0; ./ ¤ 0; ec . / (

"

(14.21)  1=2 2c 1 C.z; t / D  < i   00 2 z .!sp ;  / )    000 .!sp ;  / z 1 .! ;/ sp ec  C 00 ; ./ D 0; !sp  !c 6 .!sp ;  / (

(14.22) as z ! 1, where !sp denotes the location of the interacting saddle point. Here erfc. / D 1  erf. / denotes the complementary error function defined in Sect. 10.4.1. The particular form of the C -function to be employed depends upon the sign of the imaginary part of the quantity . /, which is defined by [cf. (10.89)]

1=2 ./ .!sp ;  /  .!p ;  / :

(14.23)

14.3 The Uniform Asymptotic Approximation

477

The proper argument of this square root expression is determined by the limiting relation given in (10.93) [see also (10.90)] as lim

!p !!sp ./

 arg . / D ˛N c  ˛N sd C 2n;

(14.24)

where ˛N c is the angle of slope of the vector from !sp to !p in the complex !-plane, ˛N sd is the angle of slope of the tangent vector to the path of steepest descent at the interacting saddle point, and where n is an integer value which is chosen such that the argument of ı. / lies within the principal domain .;  for all   1. Consider first the case in which either one of the distant saddle points SPd˙ approaches the pole singularity at ! D !p that is located in a region of the complex !-plane bounded away from the limiting values ˙1  2ıi approached by !SP ˙ . / d

as  ! 1C , respectively. Specifically, let SPdC approach the point ! D !p which is assumed here to be the only pole singularity of the spectral function uQ .! !c /. Then CdC .z; t / is given by (14.20)–(14.22) with !SP C ./ substituted for !sp throughout, d and the remaining C -functions appearing in (14.18) and (14.19) are asymptotically negligible in comparison to both CdC .z; t / and the residue contribution. Similarly, if the distant saddle point SPd approaches the simple pole singularity at ! D !p , then Cd .z; t / is given by (14.20)–(14.22) with !SPd ./ substituted for !sp throughout, and the remaining C -functions appearing in (14.18) and (14.19) are asymptotically negligible in comparison to both Cd .z; t / and the residue contribution. Consider next the case in which one of the near saddle points SPnC for  2 Œ1; 1

or either SPn˙ for  > 1 approaches the pole singularity at ! D !p . Specifically, let SPnC approach the point ! D !p , which is again assumed to be the only pole singularity of the spectral function uQ .!  !c /. Then CnC .z; t / is given by (14.20)–(14.22) with !SPnC . / substituted for !sp throughout, and the remaining C -functions appearing in (14.18) are asymptotically negligible in comparison to both CnC .z; t / and the residue contribution. If the near saddle point SPnC approaches the simple pole singularity at ! D !p when  > 1 , then CnC .z; t / is given by (14.20)–(14.22) with !SPnC . / substituted for !sp throughout, and the remaining C -functions appearing in (14.19) are asymptotically negligible in comparison to both CnC .z; t / and the residue contribution. Similarly, if the near saddle point SPn approaches the simple pole singularity at ! D !p when  > 1 , then Cn .z; t / is given by (14.20)–(14.22) with !SPnC . / substituted for !sp throughout, and the remaining C -functions appearing in (14.19) are asymptotically negligible in comparison to both Cn .z; t / and the residue contribution. ˙ approaches Consider now the case when one of the middle saddle points SPmj C the pole singularity at ! D !p . Specifically, let SPm1 approach the point ! D !p , which is again assumed to be the only pole singularity of the spectral function uQ .!  !c /. Then CmC .z; t / is given by (14.20)–(14.22) with !SP C . / substituted m1 for !sp throughout, and the remaining C -functions appearing in (14.18) are asymptotically negligible in comparison to both CmC .z; t / and the residue contribution. If C approaches the point ! D !p , then CmC .z; t / is given the middle saddle point SPm2

478

14 Evolution of the Signal

by (14.20)–(14.22) with !SP C ./ substituted for !sp throughout, and the remainm2 ing C -functions appearing in (14.18) are asymptotically negligible in comparison to both CmC .z; t / and the residue contribution. Analogous results hold for the middle  in the left-half plane. saddle points SPmj Finally, for the case in which none of the saddle points approaches close to the pole singularity at ! D !p , then the two nearest saddle points to the pole through which the deformed contour of integration passes must be considered through application of Corollary 2 in Sect. 10.4.3. This situation may occur, for example, when the simple pole at ! D !p is situated near a branch cut of the complex phase 0 function .!;  /. Specifically, let !p be situated just above the branch cut !C !C associated with the resonance frequency !0 in a single resonance Lorentz model dielectric. The deformed contour of integration P ./ passing through both the distant saddle point SPdC and the near saddle point SPnC then interacts with this pole with both of these saddle points remaining isolated from it for all   1. In that case, both CdC .z; t / and CnC .z; t / are given by (14.20)–(14.22) with !SP C ./ and !SPnC . / d substituted for !sp throughout, respectively, the remaining C -functions appearing in (14.18) being asymptotically negligible in comparison to both CdC .z; t /, CnC .z; t /, and the residue contribution. Analogous results hold for Drude model conductors as well as for each branch cut in a multiple resonance Lorentz model dielectric. A case of special interest is that for which the pole singularity !p of the spectral function uQ .!  !c / is real and positive. The complex phase behavior .!;  /  i ! n.!/   at the simple pole singularity at ! D !p is then given by   .!p ; / D !p ni .!p / C i !p nr .!p /   ;

(14.25)

˚  .!p / < .!p ;  / D !p ni .!p /;  ˚  $ .!p ; / = .!p ;  / D !p nr .!p /   ;

(14.26) (14.27)

so that

where nr .!p / and ni .!p / denote the real and imaginary parts of the complex index of refraction, respectively [see, for example, (12.74) and (12.75)]. The saddle points in the left half of the complex !-plane then do not interact with the pole at ! D !p , and hence, their contributions Cj , j D d; m; n, are negligible. Attention is now focused on the pole contribution in a single resonance Lorentz model dielectric.

14.4 Single Resonance Lorentz Model Dielectrics Because the real coordinate location of the near saddle point SPnC in the right q half of the complex !-plane lies within the below resonance domain 0  ! 0 

!02  ı 2

and the real coordinate location of the distant saddle point SPdC in the right-half

14.4 Single Resonance Lorentz Model Dielectrics

479

q plane lies within the above absorption band domain ! 0  !12  ı 2 for all   1, the uniform asymptotic description of the pole contribution in a single resonance Lorentz model dielectric separates naturally into three cases. For real-valued q angular frequency values !p in the below absorption band domain 0  !p 

!02  ı 2 ,

SPnC

the near saddle point will interact with the simple pole singularity at ! D !p . In that case, the function CnC .z; t / appearing in (14.18) and (14.19) may be significant, the remaining C -functions being asymptotically negligible q by comparison. For values of !p in the above absorption band domain !p 

!12  ı 2 , the dis-

tant saddle point SPdC interacts with the simple pole singularity at ! D !p . In that case, the function CdC .z; t / appearing in (14.18) and (14.19) may be significant, the remaining C -functions being asymptotically negligible by comparison. q q Finally, for values of !p within the absorption band, so that !02  ı 2 < !p < !12  ı 2 , neither the near nor distant saddle points paases in close proximity to the simple pole singularity at ! D !p . In that case, both of the functions CdC .z; t / and CnC .z; t / appearing in (14.19) may be significant, the remaining C -functions being asymptotically negligible by comparison. The uniform asymptotic description of the pole contribution in each of these three cases is now treated in detail.

14.4.1 Frequencies below the Absorption Band For real-valued q angular frequency values !p in the below absorption band domain

0  !p  !02  ı 2 , it is the near saddle point SPnC in the right half of the complex !-plane that interacts with the simple pole singularity at ! D !p . The set of uniform asymptotic expressions given in (14.20)–(14.22) then apply to CnC .z; t / with !sp denoting the near saddle point location !SPnC ./ for all  > 1. Furthermore, the quantity ./ is given by (14.23) with either numerically determined near saddle point locations or with the second approximate expressions [from (13.74), (13.85), and (13.95)]  1 2ı . /  3 0 ./ .0  / 3  2  b2  2ı ./  3 0 ./ 2ı 3  ˛ . / C 3˛ C 4 540 !0 1 <  < 1 ;  2 2  2ı 4ı b 0  1 C ;  D 1 ; .!SPn ; 1 /

3˛ 9˛0 !04

.!SPnC ; /

0 ./

 ;

(14.28) (14.29)

480

14 Evolution of the Signal

(

  2 b2 

. /.  0 / C 1  ˛ . / 2 . / 4 3 0 !0  )  4 2 2 1 ˛ . /  1 C ı ./ 9 3 (  )   4 2 b2 2 Ci ./ 0   C ı . / 2  ˛ . / C ˛ . / ; 20 !04 3

.!SPnC ; / ı

 > 1 :

(14.30)

The argument of ./ must now be determined by the limiting expression given in (14.24), taken in the limit as !p approaches the saddle point location, with the integer n chosen such that this argument lies within the principal domain .; /. A sequential depiction of the near saddle point SPnC interaction with the simple pole singularity at ! D !p with !p bounded away from the origin along the positive real ! 0 -axis is given in Fig. 14.6. It is then seen that =2  ˛N c <  for all positive, real values of !p for all  > 1, where ˛N c increases monotonically with increasing  > 1. Furthermore, as described in the derivation of the uniform asymptotic description of the Brillouin precursor in Sect. 13.3.2, the angle of slope ˛N sd of the path of steepest

a

b

''

sd

SPn

c

''

P

=

P

P

c

'

p

''

SPn

sd = =0 c

p

'

'

p c

sd =

SPn

d

e

''

f

''

P

P s

c

SPn

p

'

p sd

=

P

'

p

c= sd =

SPn s

''

s

SPnC

' c

sd

=

SPn s

Fig. 14.6 Interaction of the near saddle point with the simple pole singularity at ! D !p located along the real ! 0 -axis when !p is near the upper end of the below absorption band domain q 0  !p  !02  ı 2 . The shaded area in each diagram of this  -sequence indicates the region of the complex !-plane where the inequality  .!SPnC ;  / > .!;  / is satisfied

14.4 Single Resonance Lorentz Model Dielectrics

481

descent through the near saddle point SPnC is equal to 0 as  increases from unity to 1 , ˛N sd D =6 at  D 1 , and ˛N sd D =4 as  increases above 1 . Notice that, although the value of ˛N sd changes abruptly at the critical space–time point  D 1 , the path of steepest descent through this near saddle point varies in a continuous  fashion  with  for all  > 1. Substitution of these results in (14.24) then yields arg . / D     =2 for 1 <  < s , arg .s / D 0, and arg . / D 3=4 for  > s . Notice   q that s > 0 for all !p 2 0; !02  ı 2 and that s D 0 for !p D 0. Upon application of (14.18)–(14.21), the uniform  contribution of the  qasymptotic simple pole singularity at ! D !p with !p 2 0; For  < s , =f ./g < 0 and (14.20) gives

!02  ı 2 is given as follows.

q #) c q   z z z 1 .! C ;/ c .! ;/ z SPn Ac .z; t /  ; < i   i erfc i . / c e c p C e 2 . / (

"

 < s ;

(14.31)

as z ! 1. At the space–time point  D s D cts =z, =f . /g D 0 and (14.21) gives q #) c q   z z z 1 .! ; / s C Ac .z; ts /  < i  i erfc i .s / cz e c .!p ;s / C e c SPn 2 .s / n z o C< e c .!p ;s / ;  D s ; (14.32) (

"

as z ! 1 with !p ¤ 0. Because the argument of the complementary error function is pure imaginary at  D s ,R this equation can be expressed in terms of Dawson’s  integral FD ./ exp .2 / 0 exp . 2 /d [see (10.97)] as q #) ( " c q   z p z 1 .! ; / C s c z SP n Ac .z; ts /  < i e  2 FD j .s /j c C 2 .s / o n z 1  D s ; (14.33) C < e c .!p ;s / ; 2 as z ! 1 with !p ¤ 0. For  > s , =f . /g > 0 and (14.20) gives q #) c q  z z z 1 c .!SP C ;/ .!p ;/ z c n Ac .z; t /  C ; < i  i erfc i . / c e e 2 . / o n z  > s ; (14.34) C< e c .!p ;/ ; (

as z ! 1.

"



482

14 Evolution of the Signal

For the special case when !p D 0, the upper near saddle point SPnC crosses the simple pole singularity at ! D !p when  D 0 , so that s D 0 and .0 / D 0. For  < 0 , =f ./g < 0 and the uniform asymptotic behavior of the pole contribution is given by (14.31). For  > 0 , =f . /g > 0 and the uniform asymptotic behavior of the pole contribution is given by (14.34). At  D s D 0 , (14.22) applies for CnC .z; t / appearing in (14.18). Because the path P .0 / crosses over the pole at !p D 0, then z 1 1 i i e c .0;0 / D ; (14.35) .0 / D 2 2 4 where .0; 0 / D 0. Hence, in this special case the simple pole contribution at  D s D 0 is given by 8 " #1=2 " #9  000 .!SPnC ; 0 / = 2c 1 < 1 Ac .z; t0 /  < i C 00 2 : z 00 .!SPnC ; 0 / !SPnC .0 / 6 .!SPnC ; 0 / ; 1 C
(14.36)

as z ! 1. With the approximations [see the development leading to (13.74) and (13.75)]  00 .!SPnC ; 0 /   000 .!SPnC ; 0 / i

2ıb 2 ; 0 !04

3˛b 2 ; 0 !04

where ˛ 1 is defined in (12.218), the above expression becomes !2 Ac .z; t0 /  0 2b

s

( " #) i˛ 1 0 c 1 < i  C
(14.37)

as z ! 1 with fixed 0 D ct0 =z. Notice that even though the term 1=!SPnC .0 / is singular because !SPnC .0 / D 0, this expression for the pole contribution Ac .z; t0 /, when combined with the asymptotic approximation of the Brillouin precursor field, yields a uniform asymptotic asymptotic approximation of the total wavefield A.z; t / that is well behaved at  D 0 [13, 14]. Taken together, (14.31)–(14.34) constitute the uniform asymptotic approximation of the pole q contribution Ac .z; t / due to the simple pole at ! D !p with

0 < !p  !02  ı 2 , whereas (14.31), (14.36), and (14.34) constitute the uniform asymptotic approximation of the pole contribution Ac .z; t / when !p D 0. For space–time values  < p s and sufficiently large propagation distances z > 0 term in such that the quantity j ./j z=c  1 is sufficiently large, the dominant  p  the asymptotic expansion of the complementary error function erfc i . / z=c

14.4 Single Resonance Lorentz Model Dielectrics

483

[see (10.101)] may be substituted into (14.31) with the result that the first and second terms in that equation identically cancel. Hence, for space–time values  > 1 that are sufficiently less than s , there is no contribution to the asymptotic behavior of the total wavefield A.z; t / from the simple pole singularity. On the other hand, for space–time values  > p s and sufficiently large propagation distances z > 0 term in such that the quantity j ./j z=c  1 is sufficiently large, the dominant  p  the asymptotic expansion of the complementary error function erfc i . / z=c [see (10.101) and (10.102)] may be substituted into (14.34) with the result o n z Ac .z; t /  < e c .!p ;/

     e ˛.!p /z  0 cos ˇ.!p /z  !p t   00 sin ˇ.!p /z  !p t

(14.38)

as z ! 1 with  > s bounded away from s . These two limiting results are in agreement with the nonuniform asymptotic approximation given in the first and third parts of (14.12), where the amplitude attenuation coefficient ˛.!p / is given in (14.9) and the phase propagation factor ˇ.!p / is given in (14.11).

14.4.2 Frequencies above the Absorption Band For finite, real-valued angular frequency values !p in the above absorption band q domain !p  !12  ı 2 , it is the distant saddle point SPdC in the right-half of the complex !-plane that interacts with the simple pole singularity at ! D !p . The set of uniform asymptotic expressions given in (14.20) and (14.21) then apply to CdC .z; t / with !sp denoting the distant saddle point location !SP C ./ for all d   1. Furthermore, the quantity . / is given by (14.23) with either numerically determined distant saddle point locations or with the second approximate expression [from (13.16)] (

  ) .b 2 =2/ 1  . /  2  2 . / C ı 2 1  . / ( ) .b 2 =2/ i./   1 C (14.39)  2  2 ./ C ı 2 1  . /

  .!SP C ; / ı 1 C . / .  1/ C d

for all   1. The argument of . / must now be determined by the limiting expression given in (14.24) with the integer n chosen such that it lies within the principal domain .; /. A sequential depiction of the distant saddle point SPdC 0 interaction with q the simple pole singularity at ! D !p along the positive real ! -axis when !p >

!12  ı 2 is given in Fig. 14.7. The angle of slope of the tangent vector

to the path of steepest descent leaving the distant saddle point SPdC is ˛N sd D =4

484

14 Evolution of the Signal

a

b

p

'

c

p

c

SPd

sd

SPd

' c

SPd

sd

sd

P

P

P s

p

' c

s

s

Fig. 14.7 Interaction of the distant saddle point SPC at ! D !p d with the simple pole singularity q 0 located along the real ! -axis when !p is in the above absorption band domain !p !02  ı 2 . The shaded area in each diagram of this  -sequence indicates the region of the complex !-plane where the inequality  .!SP C ;  / >  .!;  / is satisfied d

for all  > 1. In addition, the angle of slope ˛N c of the vector from !SPC ./ to !p d decreases monotonically from  to 0 as  increases. The quantity ˛N c  ˛N sd then varies from 5=4 to =4 as  increases from unity, which does not lie within the principal domain. The correct behavior4 of arg Œ ./ is given by Cartwright [9] as follows: as  D 1 ! s , arg Œ ./ D 3=4 ! ; at  D s , arg Œ . / jumps from  to C; and as  increases above s , arg Œ ./ D  ! =4. Upon application of (14.18)–(14.21), the uniform q asymptotic approximation of the simple pole singularity at ! D !p with !p   < s , =f ./g < 0 and (14.20) gives

!12  ı 2 is given as follows. For

q #) c q   z .! C ;/ z z 1 c .! ;/ z SPd < i   i erfc i . / c e c p C e Ac .z; t /  ; 2 . / (

"

 < s ;

(14.40)

as z ! 1. At the space–time point  D s D cts =z, =f . /g D 0 and (14.21) gives q #) c q   z .! C ;s / z z 1 c SPd < i  i erfc i .s / cz e c .!p ;s / C e Ac .z; ts /  2 .s / n z o C< e c .!p ;s / ;  D s ; (14.41) (

"

as z ! 1. Because the argument of the complementary error function is pure imaginary at  D s , this equation can be expressed in terms of Dawson’s integral as 4

The original derivation given in [14] and [15] is off by a factor of  due to an error made by taking the angle ˛N sd to be the angle of the steepest descent path leading into the distant saddle point SPdC instead of leading away from it.

14.4 Single Resonance Lorentz Model Dielectrics

485

q

" ( c #) q   z p z 1 c .!SP C ;s / d Ac .z; ts /   2 FD j .s /j cz C < i e 2 .s / o 1 n z  D s ; (14.42) C < e c .!p ;s / ; 2 as z ! 1. Finally, for  > s , =f . /g > 0 and (14.20) gives q #) c q  z .! C ;/ z z 1 c .! ;/ z SP d < i  i erfc i . / c e c p C e Ac .z; t /  ; 2 . / o n z  > s ; (14.43) C< e c .!p ;/ ; (

"



as z ! 1. Taken together, (14.40)–(14.43) constitute the uniform asymptotic approximation of q the pole contribution Ac .z; t / due to the simple pole at ! D !p with

!12  ı 2 . For space–time values  < s and sufficiently large propagap tion distances z > 0 such that the quantity j . /j z=c  1 is sufficiently large, the dominant term in the asymptotic expansion of the complementary error function  p  erfc i . / z=c may be substituted into (14.40) with the result that the first and second terms in that equation identically cancel. Hence, for space–time values  > 1 that are sufficiently less than s , there is no contribution to the asymptotic behavior of the total wavefield A.z; t / from the simple pole singularity. On the other hand, for space–time values  > p s and sufficiently large propagation distances z > 0 term in such that the quantity j ./j z=c  1 is sufficiently large, the dominant  p  the asymptotic expansion of the complementary error function erfc i . / z=c may be substituted into (14.43) with the result !p 

o n z Ac .z; t /  < e c .!p ;/

     e ˛.!p /z  0 cos ˇ.!p /z  !p t   00 sin ˇ.!p /z  !p t

(14.44)

as z ! 1 with  > s bounded away from s . These two limiting results are in agreement with the nonuniform asymptotic approximation given in the first and third parts of (14.12), where the amplitude attenuation coefficient ˛.!p / is given in (14.9) and the phase propagation factor ˇ.!p / is given in (14.11).

14.4.3 Frequencies within the Absorption Band For real-valued angular frequency q values !p within q the absorption band of the 2 2 Lorentz model dielectric, so that !0  ı < !p < !12  ı 2 , both the near SPnC

486

14 Evolution of the Signal

and distant SPdC saddle points interact with the simple pole singularity at ! D !p . It is then important to determine which saddle point’s steepest descent path crosses the pole and thereby determines the space–time value  D s > 1 at which this crossing occurs. Because the steepest descent path through a given saddle point is detemined by the imaginary part $ .!;  / =f.!; /g of the complex phase function, the space–time point s at which this crossing occurs must satisfy the relation $ .!sp ; s / D $ .!p ; s /;

(14.45)

where !sp denotes the relevant saddle point. The relevant saddle point whose steepest descent path sets the value of s has been shown [9] to be determined by the value of $ .!p ; / at the initial space–time point  D 1. This can be seen by considering of the imaginary part of the complex phase function .!; /   the behavior  i ! n.!/   along the positive real ! 0 -axis, where $ .! 0 ;  / D nr .! 0 /   ! 0 . For a single resonance Lorentz model dielectric, there exists a real angular frequency value !$ within the absorption band such that nr .! 0 / > 1 for 0  ! 0 < !$ and nr .! 0 / < 1 for finite ! 0 > !$ , where5 nr .!$ / 1;

q q !02  ı 2 < !$ < !12  ı 2 :

(14.46)

It is then seen that $ .!p ; 1/ > 0 for 0  ! 0 < !$ and that $ .!p ; 1/ < 0 for finite ! 0 > !$ . Because of this behavior, if $ .!p ; 1/ > 0, then the steepest descent path passing through the near saddle point SPnC and crossing the positive real ! 0 -axis determines the value of s through (14.45). If $ .!p ; 1/ < 0, then the steepest descent path passing through the distant saddle point SPdC and crossing the positive real ! 0 -axis determines the value of s through (14.45). Finally, if $ .!p ; 1/ D 0, then s D 1. The resultant uniform asymptotic approximation of the pole contribution is then determined through a direct application of Corollary 2 due to Cartwright [9,10]; see Sect. 10.4.3. The uniform asymptotic description of the pole contribution at ! D !p when $ .!p ; 1/  0 is then given by 2

3 q c z .! ;/   p z z 1 C c 6 7 SPd < i  4i erfc i d ./ cz e c .!p ;/ C e Ac .z; t /  5 2 d ./ (

2  p  z 6 Ci  4i erfc i n ./ cz e c .!p ;/ C

q

c z

n . /

3 e

z c .!SP C ;/ n

 < s ; 5

7 5

)

(14.47)

As an example, consider the real part of the complex index of refraction for a single resonance Lorentz model dielectric illustrated in Fig. 12.2 for Brillouin’s choice of the medium parameters. Equation (14.46) is then found to be satisfied when !$ ' 4:2925  1016 r=s.

14.4 Single Resonance Lorentz Model Dielectrics

487

2

3

q

( c z  p z  z .!p ;/ z 1 c .!SP C ;/ 7 6 c d Ac .z; t /  C < i  4i erfc i d ./ c e e 5 2 d . / 2  p  z 6 Ci  4i erfc i n ./ cz e c .!p ;/ C

q

n . /

n z o C< e c .!p ;/ as z ! 1 with

3

c z

e

z c .!SP C ;/ n

  s ;

7 5

)

(14.48)

q !02  ı 2 < !p  !$ . Here

1=2 j . / .!SP C ;  /  .!p ;  /

(14.49)

j

for j D d; n. The uniform asymptotic description of the pole contribution at ! D !p when $ .!p ; 1/  0 is given by 2

(

Ac .z; t / 

q

3

c z

 p  z 1 6 < i  4i erfc i d ./ cz e c .!p ;/ C e 2 d ./ 2 

6 Ci  4i erfc i n ./

pz c

q e

z c .!p ;/

C

z c .!SP C ;/ d

3

c z

n . /

e

z c .!SP C ;/ n

 < s ; 2

(

Ac .z; t / 

q

c z

 p  z 1 6 < i  4i erfc i d ./ cz e c .!p ;/ C e 2 d . / 2  p  z 6 Ci  4i erfc i n ./ cz e c .!p ;/ C o n z C< e c .!p ;/

q

c z

n . /

)

3 z c .!SP C ;/ d

z c .!SP C ;/ n

  s ;

7 5

(14.50)

3 e

7 5

7 5

7 5

)

(14.51)

q as z ! 1 with !$  !p < !12  ı 2 , where j . / for j D d; n is given by (14.49). The set of expressions given in (14.47) and (14.48), and (14.50) and (14.51) constitute the uniform asymptotic approximation of the pole contribution Ac .z; t / due

488

14 Evolution of the Signal

to the simple pole singularity at ! D !p when !p is situated within the absorption band of a single resonance Lorentz model dielectric. These uniform asymptotic expressions reduce to the nonuniform result given in either (14.38) or (14.39) for zero for  < s , for sufficiently large propagaspace–time values  > s , and yield p tion distances z > 0 such that j . /j z=c  1.

14.4.4 The Heaviside Unit Step Function Signal For a Heaviside unit step function signal f .t / D uH .t / sin .!c t /, the initial pulse envelope spectrum [see (11.56)] uQ H .!  !c / D i=.!  !c / possesses a simple pole singularity at the applied signal (or carrier) angular frequency !c  0, with residue   D lim

!!!c

i .!  !c / !  !c

 D i:

(14.52)

Substitution of this result into (14.31)–(14.34) with !p set equal to !c then yields the uniform asymptotic descriptionqof the pole contribution for below resonance 

angular signal frequencies !c 2 0; !02  ı 2 as 9 q c > = q  z z z 1 .! C ;/ AHc .z; t /  ; < i erfc i . / cz e c .!c ;/  e c SPn > 2 ˆ . / ; : 8 ˆ <



q

(

 < s ; (14.53) )

c z

q  z  z 1 .! C ;s / <  i erfc i .s / cz e c .!c ;s /  e c SPn 2 .s /   ˛.!c /z sin ˇ.!c /z  !c t ;  D s ; (14.54) e 8 9 q c ˆ > < = q   z z z 1 .! C ;/ c .! ;/ z c SP n < i erfc i . / c e c e ;  AHc .z; t /  > 2 ˆ . / : ;

AHc .z; ts / 

  e ˛.!c /z sin ˇ.!c /z  !c t ;

 > s ;

(14.55)

1=2 as z ! 1 with !c ¤ 0, where . / .!SPnC ;  /  .!p ;  / . If !c D 0, in which case s D 0 , (14.54) must be replaced by !2 AHc .z; t0 /  0 2b

s

1 0 c ; ız !SPnC .0 /

 D s D 0 ;

(14.56)

14.4 Single Resonance Lorentz Model Dielectrics

489

0.4

AHc(z,t)

0.2

0 s

c

−0.2

−0.4 1.4

1.6

1.8

2

2.2

Fig. 14.8 Temporal evoution of the pole contribution AHc .z; t / at one absorption depth z D zd ˛ 1 .!c / in a single resonance Lorentz p model dielectric with Brillouin’s choice of the medium parameters (!0 D 4  1016 r=s, b D 20  1016 r=s, ı D 0:28  1016 r=s) for a Heaviside unit step function modulated signal with below resonance angular carrier frequency !c D 1  1016 r=s

as z ! 1 with fixed 0 D ct0 =z. Again, notice that even though the term 1=!SPnC .0 / is singular because !SPnC .0 / D 0, this expression for the pole contribution Ac .z; t0 /, when combined with the asymptotic approximation of the Brillouin precursor field (which is also singular at  D 0 , but with the opposite sign to the singularity appearing in the above expression), yields a uniform asymptotic asymptotic approximation of the total wavefield A.z; t / that is well behaved at  D 0 . The dynamical evolution of the pole contribution AHc .z; t / as described by (14.53)–(14.55) for a Heaviside unit step function signal is illustrated in Fig. 14.8 at one absorption depth z D zd ˛ 1 .!c / in a single resonance Lorentz model dielectric choice of the medium parameters (!0 D 4  1016 r=s, p with Brillouin’s 16 b D 20  10 r=s, ı D 0:28  1016 r=s) with below resonance angular carrier frequency !c D 1  1016 r=s. This field componment was calculated using numerically determined near saddle point locations. Comparison of this field component with that given in Figs. 13.4 and 13.9 shows that, in this particular case, the peak amplitude in the pole contribution is two orders of magnitude greater than that for the Sommerefeld precursor field and is the same order of magnitude as that for the Brillouin precursor field. Both precursor fields diminish in amplitude for smaller propagation distances as the pole contribution increases to that of the initial wavefield A.0; t / D uH .t / sin .!c t /. However, as the propagation distance increases above a single absorption depth, the peak amplitudes of both precursor fields will decrease at a slower rate with z than that for the pole contribution so that either one or both of them dominate the propagated field evolution.

490

14 Evolution of the Signal

The uniform asymptotic description of the pole contribution for a Heaviside unit step function signal with finite, above absorption band angular carrier frequency q !c  !12  ı 2 is obtained from (14.40)–(14.43) with substitution from (14.52) as 8 ˆ <

9 q c > = q  z z z 1 c .!SP C ;/ .! ;/ z c d < i erfc i . / c e c e AHc .z; t /  C ; > 2 ˆ . / : ; 8 ˆ <



q

 < s ; c

(14.57) 9 > =

q  z  z z 1 c .!SP C ;s / d AHc .z; ts /  < i erfc i .s / cz e c .!c ;s / C e > 2 ˆ .s / : ;

  e ˛.!c /z sin ˇ.!c /z  !c t ;  D s ; (14.58) 8 9 q c ˆ > < = q   z z z 1 c .!SP C ;/ d < i erfc i . / cz e c .!c ;/ C e ; AHc .z; t /  > 2 ˆ . / : ;   e ˛.!c /z sin ˇ.!c /z  !c t ;  > s ; (14.59) as z ! 1. The dynamical evolution of the pole contribution AHc .z; t / as described by (14.57)–(14.59) for a Heaviside unit step function signal is illustrated in Fig. 14.9 at three absorption depths z D 3zd in a single resonance Lorentz model dielectric choice of the medium parameters (!0 D 4  1016 r=s, p with Brillouin’s 16 b D 20  10 r=s, ı D 0:28  1016 r=s) with above absorption band angular carrier frequency !c D 1  1017 r=s. This propagated wavefield componment was calculated using numerically determined near saddle point locations. Because the distant saddle point SPdC passes within close proximity to the simple pole singularity at ! D !c when  D s in this above absorption band case, the magnitude of the quantity ./ becomes small about this space–time point, resulting in the appearance of resonance peak in the pole contribution, as seen in Fig. 14.9. A similar resonance peak also appears in the Sommerfeld precursor field for this case, illustrated in Fig. 14.10, but with the opposite sign. When added together to construct the total wavefield, these two resonance peaks destructively interfere and cancel each other out, as described in Sect. 15.2. Consider finally the case when the carrier frequency !c of the Heaviside unit step within the absorption function signal A.0; t / D uH .t / sin .!c t / is situated q q band of

the single resonance Lorentz model dielectric, so that !02  ı 2 < !c < !12  ı 2 . The uniform asymptotic description then separates into two cases that are dependent upon the sign of the quantity $ .!c ; 1/. From (14.47) and (14.48) with the substitution  D i , the uniform asymptotic description of the pole contribution at ! D !c when $ .!c ; 1/  0 is given by [9, 10]

14.4 Single Resonance Lorentz Model Dielectrics

491

0.06

0.04

AHc(z,t)

0.02

0 s

c2

c1

−0.02

−0.04

−0.06 1

1.1

1.2

1.3

1.4

Fig. 14.9 Temporal evoution of the pole contribution AHc .z; t / at z D 3zd in a single resonance Lorentz model dielectric for a Heaviside unit step function modulated signal with above absorption band angular carrier frequency !c D 1  1017 r=s

0.06

0.04

AHS(z,t)

0.02

0

−0.02

−0.04

−0.06 1

1.1

1.2

1.3

1.4

Fig. 14.10 Temporal evoution of the Sommerfeld precursor AHS .z; t / at z D 3zd in a single resonance Lorentz model dielectric for a Heaviside unit step function modulated signal with above absorption band carrier frequency !c D 1  1017 r=s

492

14 Evolution of the Signal

q

( c z  p z  z .!c ;/ z 1 c .!SP C ;/ c d AHc .z; t /  C < i erfc i d ./ c e e 2 d . / q ) c z  p z  z .!c ;/ z .! C ;/ c SPn e Ci erfc i n ./ c e c C n . /  < s ; (14.60) q

(

c

z  p  z z 1 c .!SP C ;/ d AHc .z; t /  < i erfc i d ./ cz e c .!c ;/ C e 2 d . / q ) c z  p z  z .!c ;/ z .! C ;/ c SPn erfc i n ./ c e c C e n ./   e ˛.!c /z sin ˇ.!c /z  !c t ;   s ; (14.61)

as z ! 1 with

q !02  ı 2 < !c  !$ . Here

1=2 j . / .!SP C ;  /  .!c ;  /

(14.62)

j

for j D d; n. The uniform asymptotic description of the pole contribution at ! D !c when $ .!c ; 1/  0 is obtained from (14.50) and (14.51) as [9, 10] (

q

c

z  p  z z 1 c .!SP C ;/ d < i erfc i d ./ cz e c .!c ;/ C e AHc .z; t /  2 d . / q ) c z  p z  z .!c ;/ z .! C ;/ c SPn e i erfc i n ./ c e c C n . /

(

AHc .z; t / 

q

 < s ; (14.63) c z

z  p  z 1 c .!SP C ;/ d <  i erfc i d ./ cz e c .!c ;/ C e 2 d ./ q ) c z  p z  z .!c ;/ z .! C ;/ c SP n e i erfc i n ./ c e c C n . /   e ˛.!c /z sin ˇ.!c /z  !c t ;   s ; (14.64)

as z ! 1 with !$  !c < (14.62).

q !12  ı 2 , where j ./ for j D d; n is given by

14.4 Single Resonance Lorentz Model Dielectrics

493

The accuracy of this uniform asymptotic description of the pole contribution has been thoroughly investigated beginning with the earlier work of Smith et al. [16,17] and culminating in the recent work by Cartwright et al. [9, 10]. An accurate numerical estmate of the pole contribution can be determined by first computing the total field evolution AH .z; t / at a fixed propagation distance z > 0 and then subtracting the uniform asymptotic approximations of the Sommerfeld and Brillouin precursor fields from it, resulting in the numerical estimate AnHc .z; t / D AH .z; t /  AH s .z; t /  AH b .z; t /. For greatest accuracy, numerically determined saddle point locations are used in each of the uniform asymptotic field descriptions. The above and below absorption band cases all yield expected results, the rms error between the asymptotic approximation and numerical estimate of the pole contribution decreasing monotonically with increasing propagation distance z  zd , all in keeping with the asymptotic sense of Poincar´e’s definition (see Definition 5 of Appendix F). For the intra-absorption band case, consider first the on-resonance carrier frequency !c D !0 example, which satisfies the condition that $ .!c ; 1/ > 0. A comparison of the asymptotic (dashed curve) and numerical estimate (solid curve) of the pole contribution at the fixed propagation distance z D 21:3zd in a single resonance Lorentz model dielectric p with Brillouin’s choice of the medium parameters (!0 D 4  1016 r=s, b D 20  1016 r=s, ı D 0:28  1016 r=s) is presented in Fig. 14.11, where the open circle in the figure indicates the critical space–time point  D s when the steepest descent path P ./ through the near saddle point SPnC

6

x 10

−3

4

AHc(z,t)

2

0

−2

−4

−6

1

2

3

4

5

6

7

8

Fig. 14.11 Comparison of the numerical (solid curve) and uniform asymptotic (dashed curve) pole contributions for a Heaviside step function signal with resonant angular carrier frequency !c D !0 at z  21:3zd (from Fig. 7.9 of Cartwright [9]). In this case, $ .!c ; 1/ > 0

494

14 Evolution of the Signal −4

10

x 10

8

RMS error

6

4

2

0 0

2

4 z (m)

6

8 −8

x 10

Fig. 14.12 RMS error between the uniform asymptotic and numerical pole contributions for a Heaviside step function signal with resonant angular carrier frequency !c D !0 as a function of the propagation distance z (from Fig. 7.10 of Cartwright [9])

crosses the pole at ! D !c . Notice that the high-frequency ripple in the asymptotic result (the dashed curve in the figure) is due to the contribution from the distant saddle point SPdC in (14.60) and (14.61), as can easily be ascertained by eliminating this contribution from this uniform asymptotic description [9]. As seen in Fig. 14.12, the rms error between this asymptotic approximation and the numerical estimate of the pole contribution decreases monotonically with increasing propagation distance z  zd . Similar remarks apply to the intra-absorption band example !c D 1:25!0 illustrated in Fig. 14.13, which satisfies the condition $ .!c ; 1/ < 0. As seen in Fig. 14.14, the rms error between this asymptotic approximation and the numerical estimate of the pole contribution decreases monotonically with increasing propagation distance z  zd .

14.5 Multiple Resonance Lorentz Model Dielectrics The asymptotic analysis of the pole contribution at ! D !p in a double resonance Lorentz model dielectric naturally separates into five separate frequency domains the low-frequency, normally dispersive below resowhen !p  0 is real-valued: 

q 2 nance domain !p 2 0; !0  ı02 , the high-frequency, normally dispersive above q resonance domain !p > !32  ı22 , the intermediate frequency, normally dispersive

14.5 Multiple Resonance Lorentz Model Dielectrics

495

0.04

AHc(z,t)

0.02

0

−0.02

−0.04

1

2

3

4

5

q

6

7

8

9

10

Fig. 14.13 Comparison of the numerical (solid curve) and uniform asymptotic (dashed curve) pole contributions for a Heaviside step function signal with above resonance angular carrier frequency !c D 1:25!0 at z  21:3zd (from Fig. 7.11 of Cartwright [9]). In this case, $ .!c ; 1/ < 0

10

x 10−4

RMS error

8

6

4

2

0 0

0.2

0.4

0.6 z (m)

0.8

1

1.2 x 10−7

Fig. 14.14 RMS error between the uniform asymptotic and numerical pole contributions for a Heaviside step function signal with above resonance angular carrier frequency !c D 1:25!0 as a function of the propagation distance z (from Fig. 7.12 of Cartwright [9])

496

14 Evolution of the Signal

q  q 2 passband !p 2 !1  ı02 ; !22  ı22 , and the two anomalously dispersive abq q  

q 2

q 2 !0  ı02 ; !12  ı02 and !p 2 !2  ı22 ; !32  ı22 . sorption bands !p 2 The uniform asymptotic approximation ofqthe simple pole contribution Ac .z; t /   in the below resonance domain !p 2 0; !02  ı02 is given by (14.31)–(14.34) with (14.36) substituted for (14.33) in the special case when !p D 0. The uniform asymptotic q approximation of the pole contribution in the above resonance domain

!p > !32  ı22 is given by (14.40)–(14.43). In the lower absorption band !p 2 q 

q 2 !0  ı02 ; !12  ı02 , the uniform asymptotic approximation of the pole contribution is given by either (14.47)–(14.48) when $ .!p ; 1/ > 0 or (14.50) and (14.51) when $ .!p ; 1/ < 0 with the distant saddle point SPdC replaced by the middle saddle q 

q 2 C point SPm1 throughout. In the upper absorption band !p 2 !2  ı22 ; !32  ı22 , the uniform asymptotic approximation of the pole contribution is given by either (14.47)–(14.48) when $ .!p ; 1/ > 0 or (14.50) and (15.51) when $ .!p ; 1/ < 0 C throughwith the near saddle point SPnC replaced by the middle saddle point SPm2 out. q q   !12  ı02 ; !22  ı22 between the Attention is now focused on the passband

C two absorption bands. Because the middle saddle point SPm1 crosses the real ! 0 axis at the angular frequency value ! D !˛mi n where the absorption is a minimum within that passband, this crossing occuring at the space–time point  D ˛mi n , the uniform asymptotic description of the pole contribution at ! D !p includes the case where .˛mi n / D 0 when !p D !˛mi n . As an illustration, for the double resonance Lorentz model dielectric example considered in Sect. 12.3.2, it is seen in Fig. 12.56 that !˛mi n 2:6  1016 r=s, which is in good agreement with the numerically determined angular frequency value !˛mi n ' 2:6283  1016 r=s. C C and SPm2 in the right-half plane then interact with Both middle saddle points SPm1 the simple pole at ! D !p . The resultant uniform asymptotic description of this pole contribution is then determined through a direct application of Corollary 2 (see  q 2 Sect. 10.4.3) to each of the three cases !p 2 !1  ı02 ; !˛mi n , !p D !˛mi n , and q   !p 2 !˛mi n ; !22  ı22 . Theq uniform asymptotic description of the pole contribution at ! D !p when   !p 2 !12  ı02 ; !˛mi n is given by

14.5 Multiple Resonance Lorentz Model Dielectrics

497

2

q

2

q

3

( c z  p z  z .!p ;/ z 1 6 c .!SP C ;/ 7 c m2 Ac .z; t /  C < i  4i erfc i 2 ./ c e e 5 2 2 . /  p  z 6 Ci  4i erfc i 1 ./ cz e c .!p ;/ C

c z

1 ./

3 e

z c .!SP C ;/ m1

 < s ; 2

(

Ac .z; t / 

q

7 5

)

(14.65) 3

c z

z  p  z 1 6 c .!SP C ;/ 7 m2 < i  4i erfc i 2 ./ cz e c .!p ;/ C e 5 2 2 . /

2  p  z 6 Ci  4i erfc i 1 ./ cz e c .!p ;/ C

q

c z

1 ./

o n z C< e c .!p ;/

3 e

z c .!SP C ;/ m1

  s ;

7 5

)

(14.66)

as z ! 1, where

1=2 j . / .!SP C ;  /  .!p ;  / mj

(14.67)

for j D 1; 2. The uniform asymptotic description of the pole contribution at ! D !p when !p D !˛mi n is given by (14.65) and (14.66) when  ¤ s . When  D s , where s D ˛mi n , the pole contribution is obtained from (14.22) as 2

3 q c z   .! ;/ p z z 1 C 6 7 c SPm2 e Ac .z; ts / D  < i  4i erfc i 2 ./ cz e c .!p ;/  5 2 2 ./ (

"

#1=2 2c Ci   00 z .!SP C ; s / m1 " ) # z  000 .!SP C ; s / .! C ;s / 1 c m1 SPm1  C 00 ; e !SP C  !˛mi n 6 .!SP C ; s / m1 m1 o 1 n z (14.68) C < e c .!˛mi n ;s / 2

498

14 Evolution of the Signal

as z ! 1 with fixed s D ˛mi n D cts =z. Notice that even though the term .!SP C  m1

!˛mi n /1 is singular at  D s , this expression for the pole contribution Ac .z; ts /, when combined with the asymptotic approximation of the middle precursor field, yields a uniform asymptotic approximation of the total wavefield A.z; t / that is well behaved at  D s . The uniform description of the pole contribution at ! D !p when q asymptotic   !p 2 !˛mi n ; !22  ı22 is given by 2

(

Ac .z; t / 

q

3

c z

 p  z 1 6 < i  4i erfc i 2 ./ cz e c .!p ;/ C e 2 2 . / 2  p  z 6 Ci  4i erfc i 1 ./ cz e c .!p ;/ C

q

z c .!SP C ;/ m2

3

c z

1 ./

e

z c .!SP C ;/ m1

 < s ; 2

(

Ac .z; t / 

q

c z

 p  z 6 Ci  4i erfc i 1 ./ cz e c .!p ;/ C o n z C< e c .!p ;/

q

c z

1 ./

)

(14.69)

z c .!SP C ;/ m2

3 e

7 5

3

 p  z 1 6 < i  4i erfc i 2 ./ cz e c .!p ;/ C e 2 2 . / 2

7 5

z c .!SP C ;/ m1

  s ;

7 5

7 5

)

(14.70)

as z ! 1, where j . / for j D 1; 2 is given by (14.67). The set of expressions given in (14.65), (14.66), and (14.68)–(14.70) constitute the uniform asymptotic approximation of the pole contribution Ac .z; t / due to the simple pole singularity at ! D !p when !p is situated within the passband of a double resonance Lorentz model dielectric. These uniform asymptotic expressions reduce to the nonuniform result given in either (14.38) or (14.39) for space–time values  > s , and yield zero p for  < s , for sufficiently large propagation distances z > 0 such that j ./j z=c  1. A similar analysis holds in each additional passband of a multiple resonance Lorentz model dielectric. Numerical illustrations are given in the next chapter where the uniform asymptotic description of the total propagated wavefield behavior is constructed. With reference to the middle saddle point dynamics depicted in Figs. 12.56 and 13.11, as well as to (14.23) and (14.24), the argument of j ./ is determined in the following manner. The angle of slope ˛N sd1 of the tangent vector to the path of C is found to vary over the domain steepest descent at the middle saddle point SPm1

14.6 Drude Model Conductors

499

˛N sd1  ! =6 as  D 1 ! N1 and then over the domain ˛N sd1 =6 ! =4 as  D N1 ! 1. Because the angle of slope ˛N c1 of the vector from !SP C to m1 !p is seen to vary over the domain ˛N c1 =2 ! =2 as  D 1 ! 1, it then follows from (14.23) with n D 0 that argŒ 1 ./ D 3=2 ! 3=4 as  D 1 ! 1. On the other hand, the angle of slope ˛N sd2 of the tangent vector to the path of C is found to vary over the domain steepest descent at the middle saddle point SPm2 ˛N sd2 =2 ! =3 as  D 1 ! N1 and then over the domain ˛N sd2 =3 ! =4 as  D N1 ! 1. Because the angle of slope ˛N c2 of the vector from !SP C to !p is m2 seen to vary over the domain ˛N c2 =2 !  as  D 1 ! 1, it then follows from (14.23) with n D 0 that argŒ 2 ./ D 0 ! 3=4 as  D 1 ! 1. Finally, notice that when !p D !˛mi n , lim!˛mi n ˛N c1 D =2 whereas lim!˛C ˛N c1 D C=2. mi n

14.6 Drude Model Conductors   The complex phase function .!; / D i ! n.!/   for a Drude model conductor with complex index of refraction [see (12.153)] !1=2 !p2 (14.71) n.!/ D 1  !.! C i  / possesses a pair of distant saddle points SPd˙ given by [see (12.309)–(12.311)] !SP ˙ ./ D ˙./  i d

  1 C . / 2

(14.72)

that beginqat ˙1  i  at  D 1 and move into the respective branch point zeros !z˙ D ˙ !p2  .=2/2  i =2 as  ! 1 and a single near saddle point SPn that moves down the positive imaginary axis, approaching the branch point singularity !pC D 0 as  ! 1, as described by the approximate expression given in (12.317). The Sommerfeld precursor in a Drude model conductor is then very similar to that for a Lorentz model dielectric whereas the Brillouin precursor is nonoscillatory and hence, more like that in a Debye model dielectric but with a long exponential tail due to the asymptotic approach of the near saddle point to the origin as  ! 1 [18,19]. Let the deformed contour of integration P ./ through the near and distant saddle points be composed of the set of Olver-type paths with respect to each saddle point such that, within a neighborhood about each saddle point, the Olver-type path is taken along the path of steepest descent through that saddle point. The space–time point  D s when the contour P ./ crosses the pole at ! D !p along the positive real ! 0 -axis is then defined by the equation $ .!sp ; s / D $ .!p ; s /;

(14.73)

500

14 Evolution of the Signal

where $ .!;  / =f.!;  /g, which may then be used to determine which saddle point interacts with the pole. First of all, because the near saddle point SPn is situated along the imaginary axis for all   1, then $ .!SPn ;  / D 0 for all   1. At the distant saddle point SPdC in the right-half plane, $ .!SP C ;  /  0 is equal d to zero at  D 1 and then decreases monotonically with increasing  > 1. Because  $ .!p ; / D !p nr .!p /   when !p is real-valued [cf. (14.27)], it follows that the near saddle point SPn interacts with the pole when !p 2 .0; !$ / whereas the distant saddle point SPdC interacts with the pole when !p  !$ , where the finite, real-valued angular frequency value !$ is defined by the relation (see Fig. 12.34) nr .!$ / D 1:

(14.74)

The uniform asymptotic description of the pole contribution at ! D !p is then given by q #) c q   z .! C ;/ z z 1 c .! ;/ z SP d < i  ˙ i erfc ˙i . / c e c p C e Ac .z; t /  ; 2 . / (

"

(

"

q

 < s ;

(14.75) #)

c q  z  z z 1 c .!SP C ;s / d < i  i erfc i .s / cz e c .!p ;s / C e 2 .s / n z o C< e c .!p ;s / ;  D s D cts =z; (14.76) q ( " #) c q  z  z .! C ;/ z 1 c .! ;/ SPd < i  i erfc i . / cz e c p C e ; Ac .z; t /  2 . / o n z  > s ; (14.77) C< e c .!p ;/ ;

Ac .z; ts / 

as z ! 1, where

1=2 ./ .!sp ;  /  .!c ;  //

(14.78)

with !sp D !SPn . / when 0 < !p < !$ and !sp D !SP C . / when !p  !$ . d Numerical illustrations are given in the next chapter where the uniform asymptotic behavior of the total propagated wavefield is constructed.

References 1. F. W. J. Olver, “Why steepest descents?,” SIAM Rev., vol. 12, no. 2, pp. 228–247, 1970. 2. E. T. Copson, An Introduction to the Theory of Functions of a Complex Variable. London: Oxford University Press, 1935. p. 110. 3. E. C. Titchmarsh, The Theory of Functions. London: Oxford University Press, 1937. Sect. 10.5.

Problems

501

¨ 4. L. Brillouin, “Uber die fortpflanzung des licht in disperdierenden medien,” Ann. Phys., vol. 44, pp. 204–240, 1914. 5. L. Brillouin, Wave Propagation and Group Velocity. New York: Academic, 1960. 6. N. Bleistein, “Uniform asymptotic expansions of integrals with stationary point near algebraic singularity,” Com. Pure Appl. Math., vol. XIX, no. 4, pp. 353–370, 1966. 7. N. Bleistein, “Uniform asymptotic expansions of integrals with many nearby stationary points and algebraic singularities,” J. Math. Mech., vol. 17, no. 6, pp. 533–559, 1967. 8. L. Felsen and N. Marcuvitz, Radiation and Scattering of Waves. Englewood Cliffs, NJ: Prentice-Hall, Inc., 1973. 9. N. A. Cartwright, Uniform Asymptotic Description of the Unit Step Function Modulated Sinusoidal Signal. PhD thesis, College of Engineering & Mathematical Sciences, University of Vermont, 2004. 10. N. A. Cartwright and K. E. Oughstun, “Uniform asymptotics applied to ultrawideband pulse propagation,” SIAM Review., vol. 49, no. 4, pp. 628–648, 2007. 11. L. B. Felsen and N. Marcuvitz, “Modal analysis and synthesis of electromagnetic fields,” Polytechnic Inst. Brooklyn, Microwave Res. Inst. Rep., 1959. 12. N. Bleistein and R. Handelsman, Asymptotic Expansions of Integrals. New York: Holt, Rinehart and Winston, 1975. 13. K. E. Oughstun, Propagation of Optical Pulses in Dispersive Media. PhD thesis, The Institute of Optics, University of Rochester, 1978. 14. K. E. Oughstun and G. C. Sherman, “Uniform asymptotic description of electromagnetic pulse propagation in a linear dispersive medium with absorption (the Lorentz medium),” J. Opt. Soc. Am. A, vol. 6, no. 9, pp. 1394–1420, 1989. 15. K. E. Oughstun and G. C. Sherman, Pulse Propagation in Causal Dielectrics. Berlin: SpringerVerlag, 1994. 16. P. D. Smith, Energy Dissipation of Pulsed Electromagnetic Fields in Causally Dispersive Dielectrics. PhD thesis, University of Vermont, 1995. Reprinted in UVM Research Report CSEE/95/07-02 (July 18, 1995). 17. K. E. Oughstun and P. D. Smith, “On the accuracy of asymptotic approximations in ultrawideband signal, short pulse, time-domain electromagnetics,” in Proceedings of the 2000 IEEE International Symposium on Antennas and Propagation, (Salt Lake City), pp. 685–688, 2000. 18. S. Dvorak and D. Dudley, “Propagation of ultra-wide-band electromagnetic pulses through dispersive media,” IEEE Trans. Elec. Comp., vol. 37, no. 2, pp. 192–200, 1995. 19. N. A. Cartwright and K. E. Oughstun, “Ultrawideband pulse penetration in an isotropic collisionless plasma,” in 2007 CNC/USNC North American Radio Science Meeting, 2007.

Problems 14.1. Show that the uniform asymptotic expressions given in (14.31)–(14.34) of the pole contribution Ac .z; t / due to the simple pole singularity at ! D !p when !p is real-valued and situated below the absorption band of a single resonance Lorentz model dielectric reduce to the nonuniform result p given in (14.12) for sufficiently large propagation distances z > 0 such that j . /j z=c  1. 14.2. Show that the uniform asymptotic expressions given in (14.40)–(14.43) of the pole contribution Ac .z; t / due to the simple pole singularity at ! D !p when !p is real-valued and situated above the absorption band of a single resonance Lorentz model dielectric reduce to the nonuniform result p given in (14.12) for sufficiently large propagation distances z > 0 such that j . /j z=c  1.

502

Problems

14.3. Show that the uniform asymptotic expressions given in (14.47) and (14.48), and (14.50) and (14.51) of the pole contribution Ac .z; t / due to the simple pole singularity at ! D !p when !p is real-valued and situated within the absorption band of a single resonance Lorentz model dielectric reduce to the nonuniform result given p in (14.12) for sufficiently large propagation distances z > 0 such that j ./j z=c  1. 14.4. Show that the uniform asymptotic expressions given in (14.65) and (14.66), and (14.69) and (14.70) of the pole contribution Ac .z; t / due to the simple pole singularity atq! D !p when !p is real-valued and situated within the passband q 2  !1  ı02 ; !22  ı22 between the two absorption bands of a double resonance Lorentz model dielectric reduce to the nonuniform result given p in (14.12) for sufficiently large propagation distances z > 0 such that j . /j z=c  1. 14.5. Derive an approximate expression for (a) the real angular frequency value q 

q 2 2 2 2 !$ 0 2 !0  ı0 ; !1  ı0 at which $ .!$ 0 ; 1/ D 0, and (b) the real angular q 

q 2 !2  ı22 ; !32  ı22 at which $ .!$ 2 ; 1/ D 0 in a double frequency value !$ 2 2 resonance Lorentz model dielectric. 14.6. Derive an approximate expression for the finite, real-valued angular frequency value !$ which satisfies (14.74) for a Drude model conductor.

Chapter 15

Continuous Evolution of the Total Field

This chapter combines the results of the preceding two chapters in order to obtain the uniform asymptotic description of the total pulsed wavefield evolution in a given causally dispersive material. From the discussion given in Sect. 12.4, the propagated plane wavefield in either a single resonance Lorentz model dielectric [see (12.352)] or a Drude model conductor [see (12.356)] may be expressed either in the form A.z; t / D As .z; t / C Ab .z; t / C Ac .z; t /

(15.1)

for all subluminal space–time points  D ct=z  1, or as a linear superposition of fields that are each expressible in this form. The field components Ab .z; t / and Ac .z; t / are both negligible when the first (or Sommerfeld) precursor field As .z; t / is predominant, As .z; t / and Ac .z; t / are both negligible when the second (or Brillouin) precursor field Ab .z; t / is predominant, and the field components Ab .z; t / and Ac .z; t / are both negligible when the pole contribution Ac .z; t / is predominant. Two or three field components become important at the same time during transition periods, giving a continuous asymptotic description of the space–time evolution of the total wavefield A.z; t / for sufficiently large z > 0 for all   1. Analogous results hold for the asymptotic description of the propagated plane wavefield in a double resonance Lorentz model dielectric, which may be expressed either in the form A.z; t / D As .z; t / C Ab .z; t / C Am .z; t / C Ac .z; t /

(15.2)

for all  D ct =z  1, or as a linear superposition of fields that are each expressible in this form, where the dominance of the field component Am .z; t / describing the middle precursor over a finite space–time interval is dependent upon whether or not the necessary condition that p < 0 is satisfied for that particular medium [see (12.116) and (12.117)]. Each additional resonance feature appearing in the material dispersion then introduces the possibility of an additional middle precursor appearing in the dynamical field evolution. Finally, the asymptotic description of the propagated plane wavefield in a Rocard–Powles–Debye model dielectric may be expressed either in the form A.z; t / D Ab .z; t / C Ac .z; t / K.E. Oughstun, Electromagnetic and Optical Pulse Propagation 2: Temporal Pulse Dynamics in Dispersive, Attenuative Media, Springer Series in Optical Sciences 144, c Springer Science+Business Media, LLC 2009 DOI 10.1007/978-1-4419-0149-1 15, 

(15.3) 503

504

15 Continuous Evolution of the Total Field

for all  D ct =z  1, or as a linear superposition of fields that are each expressible in this form. Composite models then naturally combine the features of each separate model. If the initial pulse A.0; t / at the plane z D 0 identically vanishes for all t < 0, then application of Cauchy’s residue theorem showed that A.z; t / D 0 for all superluminal space–time points <1. If this is not the case, as it so happens for a gaussian envelope pulse, then the asymptotic description must be extended into this  < 1 space–time domain in order to obtain a complete description of the propagated pulse dynamics for all  2 .1; C1/. Because the pulsed wavefield amplitude is now nonvanishing for all finite space–time points, the question of superluminal pulse velocities must then be considered with due caution. This chapter begins with the analysis of the total precursor field Ap .z; t / in a given dispersive medium. The results are based upon the analysis of the individual precursor fields presented in Chap. 13. The possible occurrence of a resonance peak in the precursor field whose saddle point interacts with the pole and its complete cancellation by a similar, but oppositely phased, resonance peak in the pole contribution when the two fields are combined to construct the total field is considered next in Sect. 15.2. Because this resonance phenomena does not appear in the total wavefield A.z; t /, it is then seen to simply be an artifact of the asymptotic analysis and so is completely nonphysical. The uniform transition from the total precursor field to the signal (the pole contribution) and the interaction of the signal with each of the precursor fields is then considered in Sect. 15.3. Based upon these results, refined expressions for the signal velocity replacing those described by Brillouin [1,2] are then derived. The signal velocity for pulse propagation in a single resonance Lorentz model dielectric is then compared to the phase velocity, group velocity, and velocity of energy propagation in such a dispersive medium, with generalization to multiple resonance Lorentz model dielectrics. The chapter then concludes with a detailed description of the dispersive pulse dynamics of specific pulse types in both Lorentz-type and Debye-type dielectrics as well as in Drude model conductors and semiconducting materials.

15.1 The Total Precursor Field The total precursor field Ap .z; t / is the combined contribution to the asymptotic behvior of the propagated wavefield from all of the relevent saddle points, so that Ap .z; t / D As .z; t / C Ab .z; t /;

(15.4)

for a single resonance Lorentz model dielectric. For space–time points   1 bounded away from SB , it follows from the results of Sect. 12.3.1 that for sufficiently large propagation distances z > 0, the second precursor field Ab .z; t / is asymptotically negligible in comparison to the first precursor field As .z; t / when 1   < SB , and the first precursor field As .z; t / is asymptotically negligible in

15.1 The Total Precursor Field

505

comparison to the second precursor field Ab .z; t / when  > SB . Both field components As .z; t / and Ab .z; t / are important in the transition region between the first and second precursors that lies in a small neighborhood of the space–time point  D SB . When asymptotic approximations of As .z; t / and Ab .z; t /, each uniformly valid for  in a specific space–time domain, are applied to (15.4), it follows from Corollary 1 of Sect. 10.3.1 that the result is an asymptotic approximation of Ap .z; t / that is uniformly valid over the same space–time domain. Hence, application in (15.4) of the uniform asymptotic approximations of As .z; t / and Ab .z; t / obtained in Sect. 13.2.2 and Sect. 13.3.2, respectively, provides an asymptotic approximation of the total precursor field Ap .z; t / in a single resonance Lorentz model dielectric that is valid uniformly for all   1. Based upon these results and the results of Chap. 13, the general dynamic behavior of the total precursor field Ap .z; t / in a single resonance Lorentz model dielectric is as follows.1 At  D 1 the front of the first (or Sommerfeld) precursor arrives, and this first precursor is dominant for all  2 Œ1; SB /. For space–time values  soon after the luminal space–time point  D 1, the peak amplitude in the first precursor occurs, and for all later space–time values the amplitude of the field envelope decays exponentially. Furthermore, the instantaneous angular frequency of oscillation !s . / of the first precursor field, which is initially infinite at  D 1, monotonically q

decreases with increasing  and approaches the limiting value !s ./ ! !12  ı 2 from above as  ! 1. For space–time values close to SB , the transition from the first to the second precursor field takes place. Because 1 < SB < 1 and because the second precursor is nonoscillatory (but not static!) for all  2 Œ1; SB , the total precursor field behavior for space–time values about the point  D SB is that of the superposition of an exponentially decaying, down-chirped oscillatory wavefield with an increasing nonoscillatory field. Finally, for all  > SB , the second (or Brillouin) precursor is dominant. The magnitude of the second precursor amplitude increases as  approaches 0 from below. At the space–time point  D 0 , the second precursor experiences zero exponential decay and decays with the propagation distance z > 0 only as z1=2 . The field behavior at this critical space–time point is then unique in all of dispersive pulse propagation phenomena with far-reaching implications (from a biological perspective) and applications (from an imaging perspective). Finally, for  > 0 , the exponential decay increases with increasing  and, for  > 1 , the second precursor becomes oscillatory with instantaneous angular frequency !b . / monotonically increasing q from zero with increasing   1 , approaching the limiting value !b ./ ! !12  ı 2 from below as  ! 1. An illustration of this total precursor field evolution for a delta function pulse is given in Fig. 13.14. Similar behavior is obtained for the total precursor field in a Drude model conductor, except that the Brillouin precursor remains nonoscillatory for all  > 1.

1

This description does not include the nonphysical phenomena of a so-called resonance peak which, if it appears, is exactly cancelled by the pole contribution.

506

15 Continuous Evolution of the Total Field

The total precursor field for a double resonance Lorentz model dielectric is given by (15.5) Ap .z; t / D As .z; t / C Am .z; t / C Ab .z; t / when the inequality p < 0 is satisfied [see (12.116) and (12.117)]; if the opposite inequality is statisfied (i.e., if p  0 ), then the total precursor field is given by (15.4) and the preceding asymptotic description for a single resonance Lorentz model dielectric applies. For space–time points   1 bounded away from both SM and MB , it follows from the results of Sect. 12.3.2 that for sufficiently large propagation distances z > 0, both the middle precursor field Am .z; t / and second precursor field Ab .z; t / are asymptotically negligible in comparison to the first precursor field As .z; t / when 1   < SM , both the first precursor field As .z; t / and second precursor field Ab .z; t / are asymptotically negligible in comparison to the middle precursor field As .z; t / when SM   < MB , and both the first precursor field As .z; t / and middle precursor Am .z; t / are asymptotically negligible in comparison to the second precursor field Ab .z; t / when  > MB . Both field components As .z; t / and Am .z; t / are important in the transition region between the first and middle precursors that lies in a small neighborhood of the space–time point  D SM , and both field components Am .z; t / and Ab .z; t / are important in the transition region between the middle and second precursors that lies in a small neighborhood of the space–time point  D MB . When asymptotic approximations of As .z; t /, Am .z; t /, and Ab .z; t /, each uniformly valid for  in a specific space–time domain, are applied to (15.5), it follows from Corollary 1 of Sect. 10.3.1 that the result is an asymptotic approximation of Ap .z; t / that is uniformly valid over the same space– time domain. Hence, application in (15.5) of the uniform asymptotic approximations of As .z; t /, Ab .z; t / and Am .z; t / obtained in Sects. 13.2.2, 13.3.2 and 13.4, respectively, provides an asymptotic approximation of the total precursor field Ap .z; t / in a double resonance Lorentz model dielectric that is valid uniformly for all   1. For each additional resonance feature included in a multiple resonance Lorentz model dielectric, the possibility of an additional middle precursor field is introduced subject to the condition given in (12.117). The general dynamic behavior of the total precursor field in a double resonance Lorentz model dielectric is the same as that in a single resonance medium when p > 0 . However, when p < 0 , the middle precursor becomes the dominant precursor field over the space–time domain  2 .SM ; MB / between the Sommerfeld precursor evolution and the Brillouin precursor evolution. For space–time values close to SM , the transition from the first to the middle precursor field takes place, and for space–time values close to MB , the transition from the middle to the second precursor field takes place. An illustration of this total precursor field evolution for a delta function pulse is given in Fig. 13.15. The total precursor field for a Rocard–Powles–Debye model dielectric is given by Ap .z; t / D Ab .z; t /

(15.6)

and is comprised of just the Brillouin precursor whose dynamical evolution is illustrated in Fig. 13.10. Although this precursor is not oscillatory in the usual

15.2 Resonance Peaks of the Precursors and the Signal Contribution

507

time-harmonic sense, its effective frequency of oscillation has been shown in Sect. 13.4 [see (13.151)] to depend upon the material parameters alone. When combined with either the Lorentz or Drude models, the resulatant composite model of the material dispersion yields a total precursor field that possesses the salient features of each individual model.

15.2 Resonance Peaks of the Precursors and the Signal Contribution Examination of the asymptotic expressions for the Sommerfeld, Brillouin, and middle precursor fields shows that each of them exhibits a resonance peak as  varies if the relevant saddle point passes near a first-order pole of the input pulse (or pulse envelope) spectrum [3]. Indeed, it is apparent from the general expression given in (10.18) for the asymptotic contribution of a first-order saddle point, viz., Z I.z; t / D

q.!/e P

zp.!;/



2 d!  q.!sp /  00 zp .!sp ;  /

1=2

e zp.!sp ;/

(15.7)

as z ! 1, that I.z; t / becomes large if the saddle point !sp . / approaches a pole !p of the spectral function q.!/ as  varies. It is the sole purpose of this short section to show that such a resonance peak is not exhibited by the total wavefield A.z; t /. That resonance peak is cancelled by an identical resonance peak with opposite sign in the term C.z; t / appearing in the uniform asymptotic expression for the pole contribution Ac .z; t /. That is to say, these resonance peaks are an artifact of the separation of the asymptotic behavior of the integral representation of the pulse into the various component wavefield contributions of precursor and pole contribution. Under the conditions that lead to the appearance of a resonance peak in the saddle point contribution to the wavefield behavior, the saddle point passes near a firstorder pole. As a result, the uniform asymptotic expression for the pole contribution Ac .z; t / must be included in the asymptotic approximation of the total wavefield A.z; t /. The result can be written as  A.z; t /  q.!sp / 

2 zp 00 .!sp ;  /

1=2

e zp.!sp ;/ C Ac .z; t /

(15.8)

as z ! 1. From (14.20) and (14.21), if the simple pole and saddle point do not coalesce, the uniform asymptotic approximation of Ac .z; t / can be written as Ac .z; t / 

 . /

r

 zp.!sp ;// e C f0 .!p / z

(15.9)

508

15 Continuous Evolution of the Total Field

as z ! 1, where f0 .!p / is an analytic function of complex !p and where  is the residue of the pole of the spectral function q.!/. Because !sp is a first-order saddle point of the complex phase function p.!;  /, this phase function evaluted at ! D !p can be expanded in a Taylor series about !sp in the form  2 1 p.!p ; / D p.!sp ; / C p 00 .!sp ;  / !p  !sp . / C    : 2

(15.10)

As a consequence [see (14.23)],

1=2 ./ p.!sp ;  /  p.!p ;  /

1=2   !p  !sp . / C f1 .!p / D  12 p 00 .!sp ;  /

(15.11)

for !p sufficiently close to !p , where f1 .!p / is an analytic function of !p that goes to zero as !sp . / ! !p . As a result, (15.9) can be written as  Ac .z; t / 

   1=2  12 p 00 .!sp ; / !p  !sp ./

r

 zp.!sp ;// e C f2 .!p / z

(15.12)

as z ! 1, where f2 .!p / is an analytic function of !p . Similarly, because q.!/ has a first-order pole at ! D !p with residue  , the first term on the right-hand side of (15.8) can be written as  q.!sp / 

2 00 zp .!sp ; /

1=2 e zp.!sp ;/ D

1=2  2   00 e zp.!sp ;/ !sp ./  !p zp .!sp ;  / Cf3 .!p /; (15.13)

where f3 .!p / is an analytic function of !p . Substitution of (15.12) and (15.13) into (15.8) then yields A.z; t /  f2 .!p / C f3 .!p /:

(15.14)

Hence, A.z; t / is an analytic function of !p in a neighborhood of the saddle point !sp . /, and therefore cannot have a singularity at !p D !sp . Consequently, the resonance peak appearing in the precursor field is exactly cancelled by an identical resonance peak appearing in the pole contribution. The total propagated wavefield A.z; t / then does not exhibit the resonance peaks exhibited by its component subfields Ap .z; t / and Ac .z; t /.

15.3 The Signal Arrival and the Signal Velocity

509

15.3 The Signal Arrival and the Signal Velocity Attention is now given to to the detailed description of the arrival of the signal due to the contribution of any simple pole singularity appearing in the spectral function uQ .!  !c / in the integrand of the propagated plane wavefield given in (14.1), viz.,

Z 1 i .z=c/.!;/ A.z; t / D uQ .!  !c /e d! < ie 2 C

(15.15)

for z  0. The transition of the total propagated wavefield A.z; t / from the precursor field Ap .z; t / to the signal Ac .z; t / then defines the signal velocity of the pulse in the dispersive medium. As in Chap. 14, the pole !p is taken to lie along the positive real ! 0 -axis of the complex !-plane. Because of its historical significance, the analysis focuses on the signal velocity in a single resonance Lorentz medium. The extension of these results to more complicated dispersive model media is also included.

15.3.1 Transition from the Precursor Field to the Signal From the results of Chap. 14, the contribution of the simple pole singularity at ! D !p occurs when the original contour of integration C , which extends along the straight line from i a1 to i aC1 in the upper half of the complex !-plane, lies on the opposite side of the pole singularity than does the Olver-type path P . / through the accessible saddle points. That is, P ./ and the original integration contour C lie on the same side of the pole when  < s and lie on opposite sides when  > s [see (14.5)]. Consequently, for  < s the pole is not crossed when the original contour is deformed to P . / and there is no residue contribution, whereas for  > s the pole is crossed in deforming the contour C to P ./ and there is a residue contribution to the asymptotic behavior of the propagated wavefield. The value of s depends upon which Olver-type path is chosen for P ./. If that path is taken to lie along the path of steepest descent through the saddle point nearest the pole, then, because $ .!;  / =f.!;  /g is constant along the path of steepest descent, it follows that the value of s is defined by the expression [1, 2] $ .!sp ; s / D $ .!p ; s /;

(15.16)

where !sp D !sp . / denotes the saddle point which interacts with the pole singularity. At  D s , however, the pole contribution is asymptotically negligible in comparison to the saddle point contribution to A.z; t / because P . / is an Olvertype path with respect to that saddle point. Consequently, the particular value of s at which the pole contribution occurs is of little or no significance to the asymptotic behavior of the propagated wavefield A.z; t /. An example of such an Olver-type path at a fixed space–time point  > 1 when the two near saddle points SP˙ n

510

15 Continuous Evolution of the Total Field

''

c1

c2

c6

c4

' −

SPn

+

SPn

'

c3

c5

'

SPd-

+

SPd

P(q)

Fig. 15.1 A deformed contour of integration P . / passing through both the near and distant saddle points for a fixed space–time value  > 0 . This contour is an Olver-type path with respect to the near saddle point SPC n in the right half of the complex !-plane, and is an Olver-type path with respect to the near saddle point SP n in the left-half of the complex !-plane. The lighter shaded area indicates the region of the complex !-plane wherein the inequality  .!;  / <  .!SP˙ ;  / n is satisfied and the darker shaded area indicates the region of the complex !-plane wherein the inequality  .!;  / <  .!SP˙ ;  / is satisfied d

are dominant over the two distant saddle points SP˙ d in a single resonance Lorentz model dielectric is depicted in Fig. 15.1. The path P . / through the pair of near C saddle points SP n and SPn can lie anywhere within the shaded region of the figure. With the path P . / shown in this figure,  < s if the pole !p lies in the angular frequency interval !c2 < !p < !c4 ,  > s if !p lies within either of the angular frequency intervals 0  !p < !c2 or !p > !c4 , and  D s if either !p D !c2 or !p D !c4 . If the path P . / was chosen to be completely in the lower half of the complex !-plane (such an Olver-type path is possible for the situation illustrated in Fig. 15.1), then  > s for all !p  0. The pole contribution at ! D !p is the dominant contribution to the asymptotic behavior of the propagated wavefield when  > c > s , where c is defined as the space–time value that satisfies the relation [3–6]  .!sp ; c / D  .!p /;

(15.17)

15.3 The Signal Arrival and the Signal Velocity

511

where !sp D !sp .c / denotes the dominant saddle point at  D c . Notice that  .!p / is independent of the value of  when !p is real-valued, as is assumed here. For space–time values  < c such that the inequality  .!sp ;  / >  .!p / is satisfied, the saddle point is the dominant contribution to the asymptotic behavior of the propagated wavefield and the pole contribution is asymptotically negligible by comparison. For space–time values  > c such that the inequality  .!sp ;  / <  .!p / is satisfied, however, the pole contribution is the dominant contribution to the asymptotic behavior of the propagated wavefield and the saddle point contribution is asymptotically negligible by comparison. For example, for the space–time point depicted in Fig. 15.1, (15.17) is satisfied if either !p D !c1 or !p D !c6 and the value of  is then c for either of these two pole locations. Furthermore, for the space–time ;  / >  .!p / is value  depicted in Fig. 15.1,  < c and the inequality  .!SPC n ;  / <  .!p / is satisfied if !c1 < !p < !c6 , and  > c and the inequality  .!SPC n satisfied if either 0  !p < !c1 or !p > !c6 . Consequently, the pole contribution is asymptotically negligible in comparison to the saddle point contribution to the propagated wavefield at the space–time value depicted in Fig. 15.1 if !c1 < !p < !c6 , whereas the pole contribution is the dominant contribution to the asymptotic behavior of the propagated wavefield and the saddle point contribution is asymptotically negligible in comparison to it if either 0  !p < !c1 or !p > !c6 . Based upon these results, it is seen that the signal arrival for a fixed value of !p occurs at the space–time point  D c satisfying (15.17). Notice that this definition of the signal arrival yields a signal velocity [3–6] vc .!p /

c c .!p /

(15.18)

that is independent of the initial pulse envelope function and the propagation distance z, depending only upon the dispersive medium properties and the value of !p . Notice that this pulse velocity measure always satisfies relativistic causality; that is, vc .!p /  c 8 !p . A general overview of the arrival of the signal and its interaction with the Sommerfeld and Brillouin precursor fields is now presented for the case of a single resonance Lorentz model dielectric. This description is based upon (15.17) and the numerical results describing the topography of  .!;  / in the complex !-plane as a function of  presented in Figs. 12.4–12.9 of Sect. 12.2 for Brillouin’s choice of the medium parameters. A similar description may be given for the other models of the material dispersion considered here. Consider first the topography of  .!;  / in the right half of the complex !-plane over the space–time domain  2 Œ1; SB in which the distant saddle points SP˙ d are and are of equal dominance initially dominant over the upper near saddle point SPC n at  D SB , as illustrated in the sequence of illustrations given in Figs. 12.4–12.6. It is seen from these three diagrams that for high angular frequency values !p  !SB , where !SB is defined in (12.258), the signal arrival will occur during the evolution of the first (or Sommerfeld) precursor field, as determined by the space–time point when the isotimic contour  .!;  / D  .!SPC ;  / through the distant saddle point d

512

15 Continuous Evolution of the Total Field

0 SPC d crosses the pole at ! D !p . At the luminal space–time point  D 1 (Fig. 12.4), the symmetric pair of distant saddle points SP˙ d are located at ˙1  2i ı [see (12.204)] and no signal can have arrived (except for the nonphysical signal with infinite angular frequency !p ). At  D 1:25 (Fig. 12.5) it is seen that for values of !p  9:4  1016 r=s the signal has already arrived and is the dominant contribution to the asymptotic behavior of the propagated wavefield A.z; t /. Finally, at  D SB ' 1:33425 (Fig. 12.6), the pair of distant saddle points SP˙ d and the upper near saddle point SPC n are of equal exponential importance in their individual contributions to the asymptotic behavior of the total wavefield A.z; t /. In that case, for values of !p > !SB , where !SB ' 8:6  1016 r=s, the signal has already arrived and is the dominant contribution to the total wavefield A.z; t /. For values !p < !SB , the signal (or pole contribution) has yet to arrive at  D SB and the precursor field Ap .z; t / is the dominant contribution to the total wavefield A.z; t /. For all space–time points  > SB , first the upper near saddle point SPC n for SB <   1 and then the two near saddle points SP˙ n for all  > 1 are dominant over the distant saddle point pair. The asymptotic contribution of these near saddle points to the field A.z; t / yields the second (or Brillouin) precursor. For space–time values  2 .SB ; 0 , during which the upper near saddle point SPC n , lying along the positive ! 00 -axis, is the dominant saddle point, a careful consideration of the iso;  / (inclined at a positive angle of =4 radians timic contour  .!;  / D  .!SPC n to the positive ! 0 -axis) through that saddle point, reveals that the signal due to any pole singularity at !p > !SB loses its asymptotic dominance in the total wavefield evolution because of the decreasing exponential decay of the evolving Brillouin precursor. That is, as  increases over the space–time domain .SB ; 0 , the isotimic ;  / at the angle =4 through the dominant upper near contour  .!;  / D  .!SPC n saddle point recrosses any pole singularity at !p > !SB that had previously been crossed by the isotimic contour  .!;  / D  .!SPC ;  / through the distant saddle d

point SPC d when  varied over the initial space–time interval .1; SB /. Note, however, that the pole contribution at !p > !SB has not been cancelled or negated by this occurrence, but rather has only become less dominant than the evolving sec; 0 / D 0 and the isotimic contour ond precursor field [4]. At  D 0 ,  .!SPC n 0  .!;  / D  .!SPC ;  / intersects the real ! -axis at the origin (where the near 0 n C saddle point SPn happens to be) and at infinity, and remains above the ! 0 -axis for all other positive values of ! 0 . Consequently, at the space–time point  D 0 , the second precursor field (which experiences zero exponential attenuation at this point) is exponentially dominant over all other contributions to the asymptotic behavior of the propagated wavefield A.z; t /. Consider finally the remaining three plots depicting the topography of  .!;  / in the right half of the complex !-plane when  > 0 . At the space–time point  ' 1 ' 1:501 (Fig. 12.7), it is seen that the pole contribution at ! 0 D !p is the dominant contribution to the asymptotic behavior of the propagated wavefield A.z; t / in either the low-frequency domain 0  !p < 0:005  1016 r=s or in the highfrequency domain !p > 20  1016 r=s, whereas for values in the interval 0:005  1016 r=s < !p < 20  1016 r=s, the second precursor is the dominant contribution.

15.3 The Signal Arrival and the Signal Velocity

513

At  D 1:65 (Fig. 12.8), it is seen that the pole contribution at ! 0 D !p is the dominant contribution in either the low-frequency domain 0  !p < 1:27  1016 r=s or in the high-frequency domain !p > 15:6  1016 r=s, whereas for values in the interval 1:27  1016 r=s < !p < 15:6  1016 r=s, the second precursor is the dominant contribution. At  D 5:0 (Fig. 12.9), it is seen that the pole contribution at ! 0 D !p is the dominant contribution in either the low-frequency domain 0  !p < 3:2  1016 r=s or in the high-frequency domain !p > 6:35  1016 r=s, whereas for values in the interval 3:2  1016 r=s < !p < 6:35  1016 r=s, the second precursor is the dominant contribution. Because the near saddle points are dominant over the distant saddle points for all  > SB , and because  .!sp / at the near saddle points monotonically decreases with increasing  > 0 , a critical space–time point  D m will finally be reached at which the relation ; m / D  .!min /  .!SPC n

(15.19)

is satisfied. Here !min is that value of ! 0 along the positive real axis at which  .! 0 / attains its minimum value [see (12.80) and (12.83)]. At this space–time point, the ; m / lies entirely in the lower half of the complex isotimic contour  .!/ D  .!SPC n !-plane with the exception of the two points ! 0 D ˙!min where it just touches the real ! 0 -axis, as illustrated in Fig. 15.2. Consequently, if !p D !min , the signal arrival is at c D m which is larger than any other value of c . That is, the signal velocity vc .!min / D c=m is the absolute minimum signal velocity in any given single resonance Lorentz model dielectric. For all later space–time points  > m , the signal contribution at any real frequency value !p has already arrived and is the dominant contribution to the propagated wavefield A.z; t /. In summary, the signal arrival in a single resonance Lorentz model dielectric separates naturally into two distinct cases dependent upon the value of the real angular frequency !p of the pole in comparison to the critical angular frequency value !SB defined in (12.258). For values of !p in the angular frequency interval 0  !p  !SB , the signal arrival is due to the crossing of the isotimic contour  .!/ D  .!sp / with the simple pole singularity at ! D !p , where !sp denotes the location of the upper near saddle point SPC n for 1 <  < 1 , the second-order near saddle point SPn at  D 1 , and the near saddle point SPC n for all  > 1 . For such values of !p , the signal due to the pole contribution at ! D !p is preceded by the first and second precursor fields, and the signal evolves essentially undisturbed as  increases above c . For pole values !p > !SB , however, the signal arrival first occurs due to the crossing of the isotimic contour  .!/ D  .!SPC ;  / through the d

distant saddle point SPC d with the simple pole singularity. This first arrival occurs at some space–time point  2 .1; SB / for finite !p . At some later space–time point  2 .SB ; 0 /, this pole is again crossed, but in the opposite direction, by the isotimic ; / through the upper near saddle point SPC contour  .!/ D  .!SPC n , rendering n the pole contribution asymptotically less dominant than the second precursor field. Finally, for some still later space–time point  > 0 , the pole is again recrossed ;  / through the in the original direction by the isotimic contour  .!/ D  .!SPC n

514

15 Continuous Evolution of the Total Field

''

O r i g i n a l

C o n t o u r

o f

I n t e g r a t i o n

min

SPd−

Branch Cut

min

SPn−

SPn+

Branch Cut

'

SPd+

P(qm )

Fig. 15.2 A deformed contour of integration P .m / passing through both the near and distant saddle points at the fixed space–time point  D m when the condition  .!SP˙ ; m / D  .!min / n

is satisfied. This contour is an Olver-type path with respect to the near saddle point SPC n in the right half of the complex !-plane, and is an Olver-type path with respect to the near saddle point SP n in the left half of the complex !-plane. The lighter shaded area indicates the region of the complex !-plane wherein the inequality  .!; m / < .!SP˙ ; m / is satisfied and the darker shaded area n indicates the region of the complex !-plane wherein the inequality  .!; m / <  .!SP˙ ; m / is d satisfied

near saddle point SPC n so that it finally becomes asymptotically dominant over all other contributions to the propagated wavefield A.z; t / for all remaining space–time values. Consequently, for pulsed sources with !p > !SB there is the existence of a so-called prepulse [3] due to the interuption of the signal evolution by the second precursor field which becomes dominant over the pole contribution for some short space–time interval. This prepulse formation is seen to be an integral part of the dynamic evolution of the second precursor field superimposed upon the evolution of the signal contribution. The space–time evolution of the signal contribution when !p > !SB may then be considered to be separated into three parts: the so-called prepulse which is preceeded by the first precursor and then followed by the second precursor field superimposed upon the signal contribution, which is then finally followed by the signal which remains dominant for all later space–time points. It is important to keep in mind that the prepulse is not independent of the signal evolution. Indeed, the prepulse formation is simply a consequence of the superposition of the signal (or pole) contribution with the second precursor field which becomes dominant over the signal for a finite space–time interval. As a final point regarding the signal arrival, it is of interest to notice that the uniform asymptotic approximation of the pole contribution at ! D !p takes on a

15.3 The Signal Arrival and the Signal Velocity

515

particularly useful form for numerical calculations at the critical q space–time point  D c . For example, for the below resonance case 0 < !p  !02  ı 2 of a single resonance Lorentz model dielectric, the uniform asymptotic approximation of the pole contribution at the space–time point  D c > s is given by (14.34), where ./ and . / is given by (14.23). At !sp denotes the near saddle point location !SPC h p i n  D c , ˛N sd D =4, arg i .c / z=c D =4, and the complementary error function appearing in (14.34) may be replaced by the right-hand side of (10.98) [see also (10.113)], with the result [3, 7, 8] (     q  q   p 1 i 4 2z 2z < i  i 2e C j .c /j c C i S j .c /j c Ac .z; tc /  2 q ) c o z z z 3 n z .! C ;c / c .! ; / SPn e c p c C C < e c .!p ;c / e .c / 2 (15.20) as z ! 1 with fixed  D c D ctc =z. Here C. / and S. / are the cosine and sine Fresnel integrals, respectively [see (10.99) and (10.100)]. This expression is reminiscent of the special form the Lommel function expression for the diffracted wavefield takes at the geometric shadow boundary (see of Born and Wolf [9]). Analogous results hold for multiple resonance Lorentz model dielectrics as well as for Drude model conductors.

15.3.2 The Signal Velocity The analysis presented in Chaps. 13 and 14 furnishes a complete uniform asymptotic description of the propagated ultrawideband pulsed wavefield in both single and multiple Lorentz model dielectrics, Debye model dielectrics, and Drude model conductors. The signal velocity in each of these causal medium models is now considered in some detail.

15.3.2.1

Signal Velocity in Single Resonance Lorentz Model Dielectrics

The first (or Sommerfeld) precursor field As .z; t / in a single resonance Lorentz model dielectric arrives at the luminal space–time point  D 1, rapidly building to a peak amplitude value immediately following its arrival, the amplitude then decreasing2 as it experiences increasing exponential attenuation as  continues to increase 2

The possible appearance of any resonance peak in the individual precursor evolution is not considered here as it does not appear in the total wavefield evolution, as discussed in Sect. 15.2.

516

15 Continuous Evolution of the Total Field

above unity. At the space–time point  D SB > 1, the second (or Brillouin) precursor field Ab .z; t / becomes dominant over the first precursor field and remains so for all  > SB . The amplitude of this field component rapidly builds up to a peak amplitude value around the space–time point  D 0 > SB , after which its amplitude experiences increasing exponential attenuation as  increases above 0 . If the spectral envelope function uQ .!  !c / for the initial pulse has a pole singularity at ! D !p , then the final contribution to the propagated wavefield A.z; t / arises from the contribution due to that pole. It is assumed here that !p  0 is real-valued. From the results of Sect. 14.4, this pole contribution, when it is the dominant contribution to the asymptotic behavior of the propagated wavefield A.z; t /, is given by [see, for example, (14.38) and (14.44)]

    Ac .z; t /  e ˛.!p /z  0 cos ˇ.!p /z  !p t   00 sin ˇ.!p /z  !p t

(15.21)

as z ! 1 with  > s bounded away from s , where  0
c  0 ;

(15.22)

. / denotes the location of the upper near saddle point SPC where !SPC n for c 2 n Œ0 ; 1 /, the second-order near saddle point SPn at c D 1 , and the near saddle point SPC n for all c > 1 . For all space–time points  > c , the pole contribution given in (15.21) is the dominant contribution to the asymptotic behavior of the propagated wavefield A.z; t / for all !p  0. For angular frequency values !p > !SB , the pole contribution given in (15.21) is the dominant contribution to the asymptotic behavior of the propagated wavefield A.z; t / for space–time points in the interval  2 .c1 ; c2 /, where 1 < c1 < SB and where SB < c2 < c for finite !p . The space–time point  D c1 is defined by the relation (15.23)  .!SPC ; c1 / D  .!p /I 1 < c1 < SB ; !p > !SB ; d

at which point the pole contribution is of equal dominance with the first precursor field, this pole contribution remaining dominant over the first precursor for all  > c1 . However, at the space–time point  D c2 defined by the relation ; c2 / D  .!p /I  .!SPC n

SB < c2 < 0 ; !p > !SB ;

(15.24)

15.3 The Signal Arrival and the Signal Velocity

517

the second precursor is of equal dominance with the pole contribution, and over the subsequent space–time interval  2 .c2 ; c /, the second precursor field is dominant over the pole contribution. Finally, at the space–time point  D c defined in (15.22), these two contributions are again of equal dominance, and for all later space–time points  > c , the pole contribution remains as the dominant contribution to the asymptotic behavior of the propagated wavefield A.z; t /. Physically, the first (or Sommerfeld) precursor field is due to the high-frequency (above absorption band) energy present in the frequency spectrum of the initial pulse as filtered by the material dispersion, whereas the second (or Brillouin) precursor field is due to the low-frequency (below resonance) energy present in the initial pulse spectrum as filtered by the material dispersion. The pole contribution given in (15.21) is physically due to the frequency component in the initial pulse spectrum at the angular frequency !p , as can readily be seen from that equation. For the majority of canonical pulse types considered in this book, the pole occurs at !p D !c , the applied carrier or signal (radian) frequency of the initial plane wave pulse at z D 0. A well-defined signal velocity for these canonical pulse types is now given. The main signal arrival is defined to occur at that space–time value  D c satisfying (15.22) at which the pole contribution given in (15.21) becomes the dominant contribution to the asymptotic behavior of the propagated wavefield A.z; t /. The velocity at which this space–time point in the wavefield propagates through the dispersive medium is defined as the main signal velocity, given by [3–6] vc .!c /

c ; c .!c /

(15.25)

where c is the vacuum speed of light. Furthermore, for angular frequency values !c > !SB , there is the appearance of a so-called pre-pulse whose front arrives at the space–time point  D c1 satisfying (15.23) when the pole contribution given in (15.21) becomes the dominant contribution to the asymptotic behavior of the propagated wavefield, and whose back arrives at the space–time point  D c2 satisfying (15.24) when the second precursor field becomes the dominant contribution to the asymptotic behavior of the propagated wavefield. The velocity at which the front of this prepulse propagates through the dispersive medium is called the anterior pre-signal velocity, given by vc1 .!c /

c I c1 .!c /

!c > !SB ;

(15.26)

and the velocity at which the back of this prepulse propagates through the dispersive medium is called the posterior pre-signal velocity, given by vc2 .!c /

c I c2 .!c /

!c > !SB :

(15.27)

518

15 Continuous Evolution of the Total Field

From the inequalities given in (15.22)–(15.24), these three pulse velocities are seen to satisfy the inequality vc .!c / 

c c < vc2 .!c / < < vc1 .!c / < c: 0 SB

(15.28)

Notice that the main signal,anterior pre-signal, and posterior pre-signal velocities depend only upon the value of !c and the medium parameters. The angular frequency dependence of these three signal velocities is presented in Fig. 15.3 for a single-resonance Lorentz model dielectric characterized p by Brillouin’s choice of the medium parameters (!0 D 4  1016 r=s, !p D 20  1016 r=s, ı D 0:28  1016 r=s). The numerical values for these signal velocity graphs are obtained [3] using (15.22)–(15.24) to determine accurate numerical estimates of the values of c .!c /, c1 .!c /, and c2 .!c / through a comparison of the numerically determined behavior of  .!;  / at either the near or distant saddle points with the value of  .!/ at the angular frequency value !c . As evident from these numerical results presented in Fig. 15.3, the main signal velocity vc .!c / attains a minimum value near the resonance frequency of the medium. The actual minimum occurs at

1

vc 1 / c 0.8

vc 2 / c 0.6

v/c

vE /c vc / c

0.4

0.2

0

0

0

5 c

(r/s)

SB

10

15 x 1016

Fig. 15.3 Angular frequency dependence of the relative main signal velocity vc .!c /=c D 1=c .!c /, relative anterior pre-signal velocity vc1 .!c /=c D 1=c1 .!c /, and relative posterior pre-signal velocity vc2 .!c /=c D 1=c2 .!c / in a single-resonance Lorentz model dielectric characterized by Brillouin’s choice of the medium parameters. The dashed curve describes the frequency dependence of the relative energy transport velocity vE .!c /=c in the dispersive medium

15.3 The Signal Arrival and the Signal Velocity

519

p the value !c D !min !0 1 C 2ı=!0 where  .!c / attains its minimum value along the positive real axis [see (12.83)]. Consequently, the signal velocity does not peak to the vacuum speed of light c near resonance, as indicated by Brillouin [1, 2], but rather attains a minimum value near resonance.3 Approximate expressions for the critical space–time points c .!c /, c1 .!c /, and c2 .!c /, whose inverses give the relative main signal, anterior preisignal, and posterior pre-signal velocities, respectively, may be obtained from the defining expressions given in (15.22)–(15.24), repsectively, through the application of appropriate approximations for  .!;  /. For the main signal velocity one obtains the approximate relationship !2

 .!c / ' ı.c  0 /

c2  02 C 2 !p2 0

c2



02

!p2

(15.29)

C 3 !2 0

for c  0 . For values of c close to 0 (i.e., when either 0  !c  !0 or !c  !1 ), this relation yields the solution c .!c / ' 0 

3 .!c / ; 2ı

(15.30)

whereas for large values of c (i.e., when !0  !c  !1 ), this relation yields the solution  .!c / : (15.31) c .!c / ' 0  ı Substitution of these expressions into (15.25) then yields the desired analytic approximation for the main signal velocity. For the anterior pre-signal velocity, one finds the approximation  .!c / (15.32) c1 .!c / ' 1  2ı for 1  c1 < SB when !c > !SB , from which the approximate behavior of the anterior pre-signal velocity vc1 .c / may be determined. Finally, for the posterior pre-signal velocity, one obtains the approximation  .!c / C

0 !04 02 !0 2 .   /  .0  c2 /3 ' 0 0 c2 4ı!p2 16ı 3 !p4

(15.33)

for SB < c2 < 0 when !c > !SB . The proper solution to this cubic equation corresponds to the one that has the limiting values c2 .1/ D 0 and c2 .!SB / D !SB . 3

Brillouin [1, 2] incorrectly interpreted the signal arrival to occur when the simple pole singularity at the signal frequency was crossed in deforming the original contour of integration to the path of steepest descent through the relevant near and distant saddle points. This result was partially corrected by Baerwald [10] in 1930.

520

15 Continuous Evolution of the Total Field

15.3.2.2

Signal Velocity in Multiple Resonance Lorentz Model Dielectrics

The main signal arrival in a double resonance Lorentz model dielectric is defined to occur at the space–time point  D c satisfying the relation [cf. (15.22)] ; c / D  .!c /;  .!SPC n

c  0

(15.34)

at which the pole contribution becomes the dominant contribution to the asymptotic behavior of A.z; t / and remains so for all  > c . The main signal velocity describes the rate at which this space–time point travels through the dispersive medium and so is given by [cf. (15.25)] c vc .!c / D : (15.35) c .!c / Because of the double resonance character of the dispersive medium, this velocity possesses a local minimum near each resonance frequency as well as attaining a local maximum c=p at some frequency value !c D !p in the passband between the two resonance frequencies, as illustrated in the graph presented in Fig. 15.4. This

1

vc1 /c 0.8

vc2 /c

1/q 0 0.6

vE /c

v/c

vE /c vc /c

vc /c

0.4

0.2

0

0

0

2

6 c (r/s)

SB

8

10 x 1016

Fig. 15.4 Angular frequency dependence of the relative main signal velocity vc .!c /=c D 1=c .!c / and the relative pre-signal velocities vc1 .!c /=c D 1=c1 .!c / and vc2 .!c /=c D 1=c2 .!c / 16 in a double p 1  10 r=s, p resonance Lorentz model dielectric with medium parameters !0 D 16 16 16 b0 D 0:6  10 r=s, ı0 D 0:1  10 r=s and !2 D 4  10 r=s, b2 D 12  1016 r=s, ı2 D 0:1  1016 r=s. The dashed curve describes the frequency dependence of the relative energy transport velocity vE .!c /=c in the dispersive medium, and the dotted line indicates the zero frequency velocity value v.0/=c D 1=0

15.3 The Signal Arrival and the Signal Velocity

521

local maximum value cannot exceed the value c=0 D c=n.0/, which the main signal velocity reaches at both zero and infinite frequency values, as seen in the figure. Notice that this local maximum is significantly less than the peak value attained by the energy velocity vE .!c / in this passband, as described by the dashed curve in the figure, where (see Sect. 5.2.6 of Vol. 1) vE .!/ D c=E .!/ with 1 E .!/ D nr .!/ C nr .!/

"

# b02 ! 2 b22 ! 2 C ;  2 2 ! 2  !02 C 4ı02 ! 2 ! 2  !22 C 4ı22 ! 2 (15.36)

from (5.212) of Vol. 1. It is from this result that the necessary condition given in (12.117) for the appearance of the middle precursor in a double resonaqnce Lorentz model dielectric was obtained [11]. Separate attention must then be given to these two individual cases. If p > 0 , then the middle saddle points never become the dominant saddle points (see Fig. 12.59) and the middle precursor doesn’t appear in the dynamical field evolution. The signal arrival and velocity is then similar to that for a single resonance Lorentz model dielectric, being described by a main signal velocity and anterior and posterior pre-signal velocities for !c > !SB , the only additional feature being the appearance of a local maximum in the main signal velocity in the passband between the two absorption bands, as illustrated in Fig. 15.4. The main signal arrival then occurs at the space–time point  D c satisfying (15.22) and, for angular frequency values !c > !SB , the anterior pre-signal arrival and posterior pre-signal departure occur at the successive space–time points  D cj , j D 1; 2, satisfying (15.23) and (15.24), respectively. This particular branching character of the signal velocity dispersion is a direct consequence of the asymptotic dominance of the Brillouin precursor over the space–time domain  2 .c2 ; c / between the main signal and posterior presignal velocities when !c > !SB and is the same as that obtained for a single resonance Lorentz model dielectric (see Fig. 15.3). The impulse response for this double resonance Lorentz model medium is given in Fig. 13.15. This branching character of the signal velocity is complicated further when the middle saddle points SP˙ m1 become the dominant saddle points over some nonzero space–time interval. In this case the prepulse signal velocity when !c > !SM > !SB is interrupted by the dominance of the upper middle saddle point pair SP˙ m1 over the space–time interval  2 .SM ; MB /, as seen in Figs. 12.57–12.58. The impulse response for this double-resonance Lorentz model medium is given in Fig. 13.16. The space–time description of the prepulse arrival and departure points then separates into two angular frequency domains (see Fig. 12.57). For !c > !MB , where the real-valued angular frequency value !MB is defined by the relation  .!SPC ; MB / D  .!MB /; m1

(15.37)

with  .!SPC ; MB / D  .!SPC ; MB / [see (12.121)] the prepulse arrival and deparn m1 ture is described by the pair of relations [cf. (15.23) and (15.24) with SB replaced by SM in (15.23) and with SB replaced by MB in (15.24)]

522

15 Continuous Evolution of the Total Field

 .!SPC ; c1 / D  .!c /I

1 < c1 < SM ; !c > !MB ;

(15.38)

 .!SPC ; c2 / D  .!c /I n

MB < c2 < 0 ; !c > !MB :

(15.39)

d

For !c 2 .!SM ; !MB /, where the real-valued angular frequency value !SM is defined by the relation (15.40)  .!SPC ; SM / D  .!SM /; m1

with  .!SPC ; SM / D  .!SPC ; SM / [see (12.119)], the set of expressions in m1 d (15.38) and (15.39) is augmented by the prepulse arrival–departure branch  .!SPC ; cm / D  .!c /I

SM < cm < MB :

m1

(15.41)

The corresponding signal velocity branches vc1 .!c / c=c1 .!c /, vcm .!c / c=cm .!c /, vc2 .!c / c=c2 .!c /, and vc2 .!c / c=c2 .!c /, illustrated in Fig. 15.5, are then seen to satisfty the inequality

1

vc1 /c vcm /c

0.8

vcm /c vE /c

vc2 /c

1/q 0

vE /c

vc /c

vc /c

v/c

0.6

vc2 /c

0.4

0.2

0

0

1

0

c

(r/s)

2 x 1017

Fig. 15.5 Angular frequency dependence of the relative main signal velocity vc .!c /=c D 1=c .!c / and the relative pre-signal velocities vc1 .!c /=c D 1=c1 .!c /, vc2 .!c /=c D 1=c2 .!c /, and vcm .!c /=c D 1=cm .!c / in a double-resonance Lorentz model dielectric with medium pap 16 16 16 16 rameters p !0 D 1  10 r=s, b0 D 0:6  10 r=s, ı0 D 0:1  10 r=s and !2 D 7  10 r=s, b2 D 12  1016 r=s, ı2 D 0:1  1016 r=s. The dashed curve describes the frequency dependence of the relative energy transport velocity vE .!c /=c in the dispersive medium, and the dotted line indicates the zero frequency velocity value v.0/=c D 1=0

15.3 The Signal Arrival and the Signal Velocity

0 < vc .!c / <

523

c c c < vc2 .!c / < < vcm .!c / < < vc1 .!c / < c: 0 MB SM (15.42)

In addition, a prepulse also exists for signal frequency values in the angular frequency interval [see (12.108) for a more accurate estimate of the lower limit] q

!12  ı02 < !c <

q

!22  ı22

(15.43)

that lies in the passband between the two absorption bands of the double resonance Lorentz model dielectric. The space–time value cm .!c / defining the anterior pre-signal velocity vcm .!c / D c=cm .!c / of the front of this prepulse satisfies (15.41), and the space–time value c2 .!c / defining the posterior pre-signal velocity vc2 .!c / D c=c2 .!c / of the back of this prepulse satisfies (15.39). These prepulse velocities also satisfy the inequalties given in (15.42). Finally, notice that the energy transport velocity vE .!c /, described by the dashed curves in Figs. 15.4 and 15.5, forms an upper envelope for the signal velocity curves.

15.3.2.3

Signal Velocity in Drude Model Conductors

Because the real phase behavior  .!;  /
q !p2   2 :

(15.45)

Here !p is the angular plasma frequency and  is the damping constant of the conducting medium. Approximate values of these material parameters for sea-water are !p 2:125  1011 r=s and  1  1011 r=s, which corresponds to a static conductivity value of 0 D .0 =k4k/!p2 = 4 mho=m. With !0 D 0, b0 D !p ,  D 2ı0 , and b2 D 0, the expression given in (12.113) and (12.114) for the energy velocity vE .!/ D c=E .!/ becomes E .!/ D nr .!/ C

!p2 nr .!/ .! 2 C  2 /

;

(15.46)

524

15 Continuous Evolution of the Total Field

where nr .!/ is given by the real part of the square root of the expression given in (15.44). The angular frequency dependence of this energy velocity expression (relative to c) is described by the dashed curve in Fig. 15.6. The space–time point  D c at which the signal arrival occurs is described by the equation  .!SPn ; c / D  .!c /;

!c > !co

(15.47)

with associated signal velocity vc .!c / D

c : c .!c /

(15.48)

Numerical solutions to (15.47) for sea-water result in the signal velocity values indicated by the open circles in Fig. 15.6. Notice that these signal velocity values follow closely along the energy velocity curve for all !c  !SB , where !SB ' 2:37013  1011 r=s. Because !co D 1:875  1011 r=s, each of these solutions is above cut-off. 1

0.8

v/c

0.6

0.4

vE /c 0.2

0

p

0

co

SB

4

c (r/s)

6

8

10 x 1011

Fig. 15.6 Angular frequency dependence of the relative signal velocity (open circles) vc .!c /=c D 1=c .!c / in a Drude model conductor with angular plasma frequency !p  2:125  1011 r=s and damping constant   1  1011 r=s. The dashed curve describes the frequency dependence of the relative energy transport velocity vE .!c /=c in the dispersive conducting medium

15.3 The Signal Arrival and the Signal Velocity

15.3.2.4

525

Signal Velocity in Rocard–Powles–Debye Model Dielectrics

Because the Rocard–Powles–Debye model of orientational polarization phenomena in dielectrics is characterized by a single near saddle point SPn that moves down the imaginary axis, crossing the origin at  D 0 and moving into the lower-half plane for all  > 0 , where 0 n.0/ is given in (12.302), the space–time point  D c at which the signal arrival occurs satisfies the equation  .!SPn ; c / D  .!c /;

c  0

(15.49)

with associated signal velocity vc .!c / D

c ; c .!c /

(15.50)

where vc .0/ D c=n.0/. Numerical solutions to (15.49) for the double relaxation time Rocard–Powles–Debye model of triply distilled water result in the signal velocity curve presented in Fig. 15.7. For comparison, the frequency dispersion of the absorption coefficient ˛.!/ D .!=c/ni .!/ is described by the dashed curve in the figure in arbitrary units [notice that  .!/ D c˛.!/ along the real frequency axis]. The signal velocity is then seen to be a minimum where the absorption coefficient

0.12 q0 0.1 a

c)

0.08

vc /c

vc / c 0.06

0.04

0.02

0 108

1010

0

1012

2

c

1014

1016

1018

(r/s)

Fig. 15.7 Angular frequency dependence of the relative signal velocity (solid curve) vc .!c /=c D 1=c .!c / in a double relaxation time Rocard–Powles–Debye model of triply distilled H2 O. The dashed curve describes the frequency dependence of the absorption coefficient ˛.!c / D .!c =c/ni .!c / in arbitrary units

526

15 Continuous Evolution of the Total Field

is a maximum, and, where the absorption is minimal, the signal velocity approaches the zero frequency value c=0 from below.

15.4 Comparison of the Signal Velocity with the Phase, Group, and Energy Velocities The velocity of propagation of the signal in a single resonance Lorentz model dielectric is now compared to the phase, group, and energy velocities in that medium [12]. For a monochromatic plane ˚ wave with fixed angular frequency !  0, the propaQ gation factor ˇ.!/ < k.!/ is related to the real part nr .!/
Q

e c .!;/ D e i .k.!/z!t / z

D e ˛.!/z e i.ˇ.!/z!t/ ;

(15.53)

the first term on the right-hand side describing attenuation and the second term describing the phase change on propagation. The real-valued phase velocity, obtained from the real phase factor appearing in the second term on the right-hand side of (15.36) as vp .!/ D

c ! D ; ˇ.!/ nr .!/

(15.54)

describes the rate at which the phase fronts appearing in the Fourier–Laplace integral representation given in (15.15) propagate through the dispersive medium. The angular frequency dependence of both the relative phase velocity vp .!c / and the phase delay c=vp .c / D p .!c / D nr .!c / in a single resonance Lorentz model dielectric characterized by Brillouin’s choice of the medium parameters (!0 D 4  1016 r=s, p !p D 20  1016 r=s, ı D 0:28  1016 r=s) is illustrated in Fig. 15.8 by the solid and dashed curves, respectively. Notice that the phase velocity becomes superluminal when the angular frequency moves above the medium resonance frequency !0 . This is not a difficulty because the phase velocity is not a directly observable physical quantity, but rather is a mathematical construct from which the phase of a

15.4 Comparison of the Signal, Phase, Group, and Energy Velocities

527

Relative Phase Velocity & Phase Delay

4

3 vp /c 2 q0 1 q0

0

c / vp

0

0

5 c

(r/s)

10

15 x 1016

Fig. 15.8 Angular frequency dependence of the relative phase velocity vp .!c /=c (solid curve) and phase delay c=vp .c / D p .!c / (dashed curve) in a single resonance Lorentz model dielectric with Brillouin’s choice of the medium parameters

monochromatic plane wave at some point in space–time may be determined if the phase is known at some other space–time point. Because a strictly monochromatic plane wave exists for all time t , there is no violation of the relativistic principle of causality by the fact that the phase velocity may exceed the vacuum speed of light c because no real physical information is associated with the phase velocity [13]. Nevertheless, the phase velocity is a useful mathematical construct because any physical wavefield may be constructed by a suitable unique Fourier representation in terms of monochromatic plane waves. Indeed, such a superposition of monochromatic plane waves, each traveling with a different phase velocity with amplitude being attenuated at a different rate with increasing propagation distance, forms the mathematical basis for understanding the underlying physics of pulse propagation in temporally dispersive absorptive media. In addition, it is seen from (15.37) that the phase velocity for a monochromatic plane wavefield serves to define the real index of refraction of the medium. Notice that the phase velocity does describe the pulse velocity in the simplest nonphysical situation when the index of refraction of the medium is nondispersive (excluding the trivial, physically realizable case when the medium is the ideal vacuum). In that special case, n.!/ n.!c /, where !c is some characteristic oscillation frequency of the initial pulse, and the propagation factor is given by

528

15 Continuous Evolution of the Total Field

ˇ.!/

! nr .!c /: c

(15.55)

With the additional usual approximation that the attenuation factor is nondispersive so that ˛.!/ ˛.!c /, the propagated wavefield described by the integral representation in (13.1) may be directly evaluated as   A.z; t / f t  ˇ.!c /z=!c e ˛.!c /z : (15.56) This result then constitutes the phase velocity approximation of dispersive pulse propagation in which the pulse propagates undistorted in shape, but attenuated in amplitude, at the phase velocity vp .!c / D !c =ˇ.!c /. Consider next the real-valued group velocity, defined by ˇ ˇ 1 c ˇ D vg .!c / ; (15.57) @ˇ.!/=@! ˇ!D!c nr .!c / C !c n0r .!c / where n0r .!/ D @nr .!/=@!. The angular frequency dependence of this real group velocity in a single resonance Lorentz model dielectric is presented in Fig. 15.9 for Brillouin’s choice of the medium parameters, the dashed curve in the figure describing the group delay c=vg .!c / D nr .!c / C !c n0r .!c /. Notice that the group velocity varies between negative and positive infinity as !c increases through the region of

10

Relative Group Velocity & Group Delay

8 6

vg /c 4

c/vg c/vg

2 q0 q0

vg /c

0

0

−2 −4 −6 −8 −10

0

5

10 c

(r /s)

15

x 1016

Fig. 15.9 Angular frequency dependence of the relative group velocity vg .!c /=c (solid curve) and group delay c=vg .c / D g .!c / (dashed curve) in a single resonance Lorentz model dielectric with Brillouin’s choice of the medium parameters

15.4 Comparison of the Signal, Phase, Group, and Energy Velocities

529

anomalous dispersion, but that it is well behaved and causal in both normal dispersion regions above and below the absorption band. The group velocity is another mathematical construct intended to describe the propagation of the peak in the pulse envelope under the ad hoc assumption of the quasimonochromatic approximation. Although useful in the normal dispersion region where the absorption is small and the group velocity is well behaved and causal, its accuracy in any anomalous dispersion region where the absorption is large and strongly frequency-dependent is untenable. A slightly more accurate description of the material dispersion (in a normally dispersive region) about the pulse carrier frequency than that given by the simple linear relation given in (15.51) is provided by the linear dispersion approximation [cf. (11.152)] (15.58) ˇ .1/ .!/ D ˇ.!c / C ˇ 0 .!c /.!  !c / with ˛.!/ ˛.!c /, where ˇ 0 .!/ D @ˇ.!/=@! is the group delay. With this substitution for ˛.!/ and ˇ.!/ in the integral representation of the propagated wavefield given in (12.1), there results   A.z; t / f t  ˇ 0 .!c /z e ˛.!c /z

(15.59)

for z  0. In contrast with the overly simplified result given in (15.56), the pulse now propagates undistorted in shape, but attenuated in amplitude, at the group velocity vg .!c / D 1=ˇ 0 .!c /. A somewhat more accurate description of the normal material dispersion is provided by the widely used quadratic dispersion approximation [cf. (11.155)] 1 ˇ .2/ .!/ D ˇ.!c / C ˇ 0 .!c /.!  !c / C ˇ 00 .!c /.!  !c /2 2

(15.60)

with ˛.!/ ˛.!c /, where ˇ 00 .!/ D @2 ˇ.!/=@! 2 describes the so-called group velocity dispersion (GVD). With this substitution for ˛.!/ and ˇ.!/ in (12.1), there results

Z 1 ˇ 0 .!c /zCt 0 t e ˛.!c /z i.ˇ.!c /z!c tC 3 0 i 2ˇ 00 .!c /z 0 4 / A.z; t / ˇ < e f .t /e dt ˇ ˇ2ˇ 00 .!c /zˇ1=2 1 (15.61) for z  0. The pulse phase then propagates with the phase velocity vp .!c / while the pulse itself propagates with the group velocity vg .!c /, the propagated pulse shape being proportional to the Fresnel transform of the initial pulse shape A.0; t / D f .t /. The propagated pulse structure is then seen to be dependent upon the time-scale parameter [14] ˇ ˇ1=2 TF ˇ2ˇ 00 .!c /zˇ (15.62) which depends upon the value of the GVD coefficient and corresponds to the principal Fresnel zone in the analogous scalar wave diffraction problem through a slit aperture. If T > 0 denotes the initial temporal pulse width, then for sufficiently small

530

15 Continuous Evolution of the Total Field

propagation distances z  0 such that the inequality T > TF is satisfied, the pulse shape evolves in the same fashion as the diffracted wavefield in the near-field of the diffracting slit aperture, whereas for sufficiently large propagation distances such that the opposite inequality T
c hjSji D ; hUtot i nr .!c / C !ıc ni .!c /

(15.63)

where the magnitude of the Poynting vector S.r; t / and the total energy density Utot .r; t / have been averaged over one oscillation cycle of the wavefield. The angular frequency dependence of this energy transport velocity is described by the dashed curve in Fig. 15.3 for Brillouin’s choice of the medium parameters. Notice that ve .!c /  c for all real-valued !c  0 with ve .!c / ! c from below as !c ! 1. In addition, the energy velocity is seen to attain a minimum value just above the medium resonance frequency !0 near to the angular frequency value where ni .!/ attains its maximum value, and remains small throughout the anamolous dispersion region. This behavior occurs over each absorption band in a multiple resonance Lorentz model dielectric, as seen in Figs. 15.4 and 15.5. Each frequency interval where vE .!c / is a local minimum corresponds to the angular frequency region where the time-average electromagnetic energy density Urev .r; t / stored in the Lorentz oscillators is near its local maximum value [see (5.200) of Vol. 1]. This linear electromagnetics result complements the nonlinear optics result of a minimal propagation velocity observed in self-induced transparency [16, 17].

15.4 Comparison of the Signal, Phase, Group, and Energy Velocities

531

In the limit as the phenomenological damping constant ı goes to zero, it can be shown that (see Problem 15.2)  ni .!/ D n0r .!/; lim ı!0 ı 

(15.64)

where n0r .!/ D @nr .!/=@!. Hence, in that limit the velocity of energy transport given in (15.63) reduces to the real-valued group velocity given in (15.57) in a single resonance Lorentz model dielectric; that is lim vE .!c / D vg .!c /:

ı!0

(15.65)

Consider now showing that the energy velocity vE .!c / is (to at least a very good approximation) equal to the main signal velocity vc .!c / in the angular frequency domain !c 2 Œ0; !1 and that it is equal to the anterior pre-signal velocity vc1 .!c / when !c  !SB , as evident in Fig. 15.3. This is accomplished by considering the approximate functional form of the quantity E .!c / D

c !c D nr .!c / C ni .!c / vE .!c / ı

(15.66)

in each of these frequency domains. For below resonance angular frequency values, one finds that 3b 2 2 3 .!c / ! ' 0  E .!c / ' 0 C (15.67) 2ı 20 !04 c when 0  !c  !0 , which is precisely the result given in (15.30) for the main signal delay c .!c / in this below resonance frequency domain. For !c 2 Œ!0 ; !1 , one finds that b2  .!c / ; (15.68) E .!c / ' 1 C 2 ' 1  4ı ı which is precisely the result given in (15.31) for the main signal delay c .!c /. Finally, for !c > !SB one finds that E .!c / ' 1 C

b2  .!c / ; '1 2 2!c 2ı

(15.69)

which is precisely the result given in (15.32) for the anterior pre-signal delay c1 .!c /. These results have shown that, to at least a first approximation, the energy transport velocity vE .!c / and the main signal velocity vc .!c / are equal for angular signal frequencies !c 2 Œ0; !1 , where the approximate equality between vE .!c / and vc .!c / begins to fail as !c increases above !0 in such a manner that vE .!c / > vc .!c /, and that the energy transport velocity vE .!c / and the anterior pre-signal velocity vc1 .!c / are equal for !c > !SB . The energy transport velocity

532

15 Continuous Evolution of the Total Field

does not, however, describe the signal velocity behavior when !c 2 .!1 ; !SB /, nor does it describe the break-up of the propagated wavefield into a prepulse and main signal when !c > !SB . This is due simply to the fact that the energy transport velocity vE .!c / was derived strictly for the case of a monochromatic (or time-harmonic) plane wave signal of fixed angular frequency !c , and hence, does not take into account any of the precursor phenomena associated with ultrawideband pulse propagation in a causally dispersive medium. The signal velocity, on the other hand, takes fully into account the precursor fields associated with an ultrawideband signal as it propagates through a temporally dispersive absorptive medium. This transient field structure, comprised of the first and second precursor fields in a single resonance Lorentz model dielectric, is intimately related to both the signal arrival and the prepulse formation for a Heaviside step-function modulated signal. Indeed, the prepulse formation is due entirely to the superposition of the second precursor field with the pole contribution, as described by the uniform asymptotic theory. The applicability of these various pulse velocity measures to specific canonical pulse types is presented in the concluding sections of this chapter. When necessary, additional pulse velocity measures that have been proposed in the open literature are discussed with reference to a particular pulse type. A general overview of these pulse velocity measures in dispersive media may be found in the papers by Smith [18] and Cartwright and Oughstun [12].

15.5 The Heaviside Step Function Modulated Signal A canonical problem of central importance to dispersive pulse propagation, not just for historical reasons, but also for the central role it plays in understanding the underlying space–time structure observed in dispersive pulse dynamics, is provided by the Heaviside step function envelope signal introduced in Sect. 11.2.2. Connected with all of this, as a matter of course, is the persistent question of superluminal pulse propagation and information transmission. As first proven by Sommerfeld in 1914 [19] (see Theorem 6 in Sect. 13.1), the propagated wavefield due to a unit step function modulated signal identically vanishes or all superluminal space–time points  < 1. As a direct consequence of this result, the question of superluminal pulse propagation should be readily answered as not being physically possible, yet the question persists. The remaining evolution of the propagated wavefield for   1 is now described in some detail for Lorentz model dielectrics. The uniform asymptotic behavior of the precursor fields have primarily been illustrated for this canonical pulse in Chap. 13 as has also been done for the pole contribution in Chap. 14. These results are now combined in order to illustrate their linear interaction in the construction of the total wavefield evolution AH .z; t /, given by AH .z; t / D AHs .z; t / C AHb .z; t / C AHc .z; t /

(15.70)

15.5 The Heaviside Step Function Modulated Signal

533

in a single resonance Lorentz model dielectric, and AH .z; t / D AHs .z; t / C AHm .z; t / C AHb .z; t / C AHc .z; t /

(15.71)

in a double resonance Lorentz model dielectric. The single resonance case is considered first.

15.5.1 Signal Propagation in a Single Resonance Lorentz Model Dielectric Consider first the spatio temporal field structure of the first (or Sommerfeld) precursor field AHs .z; t / whose dynamical evolution is due to the  -evolution of the pair of distant saddle points SP˙ d . The front of the Sommerfeld precursor arrives at the luminal space–time point  D 1 with zero amplitude, the amplitude rapidly increasing to a peak value as  initially increases above unity, and then is exponentially damped out with increasing values of  > 1, as illustrated in Fig. 15.10. However, due to the presence of the resonance term .!SPC . /  !c /1 as a factor in d the asymptotic approximation of the first precursor field [see (13.61)], a resonance peak in the wavefield may occur when the distant saddle point SPC d approaches close to the applied signal frequency !c (see Sect. 15.2), as illustrated in Fig. 15.11.

1.5

x 10−3

1

AHs(z,t)

0.5

0

−0.5 −1 −1.5 1

1.02

1.04

1.06

1.08

1.1

q Fig. 15.10 Sommerfeld precursor field evolution when either q the Lorentz model dielectric is highly absorptive or when the applied signal frequency !c  !12  ı 2

534

15 Continuous Evolution of the Total Field

AHs(z,t)

0.05

0

qr −0.05

1

1.1

1.3

1.2

1.4

1.5

q

Fig. q 15.11 Sommerfeld precursor field evolution when the applied signal frequency !c > !12  ı 2 and the absorption is not too large

The amplitude of the first precursor field for space–time values about  ' r is approximately lorentzian in shape with a resonance peak at the space–time point  D r at which the equation .r / D !c

(15.72)

is qsatisfied; an approximate expression for ./ is given in (12.202). Because ./ > !12  ı 2 for finite values of   1, such a resonance peak may only occur when q !c > !12  ı 2 . However, even for such a high applied signal frequency !c above the medium absorption band, this resonance peak may not appear for the case of a highly absorptive medium (i.e., large ı) because of the larger distance between the distant saddle point SPC d and !c at  D r . The dynamical evolution of the first (or Sommerfeld) precursor field is illustrated in Figs. 15.10 and 15.11 for these two different possibilities. The first precursor wave evolution illustrated in Fig. 15.10 is typical of that observed in either a highly absorptive Lorentz model dielectric or when the applied signal frequency !c  q !12  ı 2 ; the resonance peak is absent in either of these two situations. The first precursor wave evolution illustrated in Fig. 15.11, on the q other hand, is typical of that

observed in a weakly absorbing medium when !c > !12  ı 2 ; the resonance peak is present in this situation. As shown in Sect. 15.2, a resonance peak of similar form but opposite sign also appears at  D r in the pole contribution to the propagated wavefield such that this resonance phenomena does not appear in the total wavefield evolution, as illustrated in Fig. 15.12.

15.5 The Heaviside Step Function Modulated Signal

535

AHs(z,t)

0.05 0 −0.05 qr

AHs(z,t) + AHc(z,t)

AHc(z,t)

0.05 0 −0.05 0.05 0 −0.05 1

1.1

1.2

1.3

1.4

1.5

q

Fig. 15.12 Superposition of the Sommerfeld precursor field AHs .z; t / and the pole contribution AHc .z; t / resulting in the cancellation of the resonance peaks in each component field

The instantaneous frequency of oscillation of the Sommerfeld precursor, given in (13.43), is approximately equal to the real part of the distant saddle point location in the right half of the complex !-plane, so that [see (13.44)] n o !s . / < !SPC ./ D ./:

(15.73)

d

This instantaneous angular frequency is at first infinite (when  D 1 and the field amplitude vanishes), and then rapidly decreases as  initially increases away from unity, the rate of this decrease decreasing as  continues to increase and the distant 0 saddle point SPC d asymptotically approaches the outer branch point !C . Consider next the spatiotemporal field structure of the second (or Brillouin) precursor field AHb .z; t / whose dynamical evolution is due to the -evolution of first the upper near saddle point SPC n for 1 <  < 1 and then the pair of near saddle points SP˙ n for   1 . As  increases away from unity and approaches 0 from below, the amplitude of the Brillouin precursor steadily increases as the attenuation monotonically decreases, the attenuation vanishing at  D 0 . Then, as  increases above 0 , the attenuation increases monotonically and the amplitude steadily decreases, but remains larger than the corresponding amplitude of the first precursor field which, for all  > SB , possesses a larger exponential decay rate with propagation distance z > 0 than does the second precursor field. However, due to the presence of the . /  !c /1 as a factor in the asymptotic approximation of resonance term .!SPC n the second precursor field [see (13.139)], a resonance peak in the wavefield may now occur when the near saddle point SPC n approaches close to the applied signal

536

15 Continuous Evolution of the Total Field 0.12 0.1 0.08 0.06 AHb(z,t)

0.04 0.02 0 −0.02

qSB

q0

−0.04 qr

−0.06 −0.08 1.4

1.5

1.6

1.7

Fig. 15.13 Brillouin precursor field evolution when the applied signal frequency !c satisfies 0  q !c <

!02  ı 2 and the absorption is not too large

frequency !c (see Sect. 15.2), as illustrated in Fig. 15.13. As in the first precursor field, the amplitude of the second precursor field for space–time values about  ' r is approximately lorentzian in shape with a resonance peak at the space–time point  D r at which the equation (15.74) .r / D !c is satisfied; q an approximate expression for ./ is given in (12.220). Because . / < !02  ı 2 for finite values of   1, such a resonance peak may only occur q when 0  !c < !02  ı 2 . However, even for such an applied signal frequency !c in the spectral domain below the medium absorption band, this resonance peak may not appear for the case of a highly absorptive medium (i.e. large ı) because of the larger distance between the near saddle point SPC n and !c at  D r . The dynamical evolution of the second (or Brillouin) precursor field is illustrated in Figs. 15.13 and 15.14 for these two different possibilities. The second precursor wave evolution illustrated in Fig. 15.14 is typical of that observed in either a highlyqabsorptive Lorentz model dielectric or when the applied signal frequency

!c  !02  ı 2 ; the resonance peak is absent in either of these two situations. The second precursor wave evolution illustrated in Fig. 15.13, on the otherqhand, is typical of that observed in a weakly absorbing medium when 0  !c < !02  ı 2 ; the resonance peak is present in this situation. As shown in Sect. 15.2, a resonance peak of similar form but opposite sign also appears at  D r in the pole contribution to

15.5 The Heaviside Step Function Modulated Signal

537

0.02

AHb(z,t)

0.01

0

qSB

q0

−0.01

1.2

1.4

1.6

1.8

2

2.2

Fig. 15.14 Brillouin precursor field evolution when either q the Lorentz model dielectric is highly absorptive or when the applied signal frequency !c

!02  ı 2

the propagated wavefield such that this resonance phenomena does not appear in the total wavefield evolution. Finally, the instantaneous frequency of oscillation of the Brillouin precursor, given in (13.129) and (13.130), is given by the effective oscillation frequency !eff .0 /

30 !04 c 4ı 2 b 2 z

(15.75)

over the space–time domain 1 <   1 , and is approximately equal to the real part of the near saddle point location in the right half of the complex !-plane for  > 1 , so that [see (13.129)] n o !b . / < !SPC ./ D n

./:

(15.76)

This instantaneous angular frequency is at first (relatively) small but nonzero for finite z > 0, and then rapidly increases as  initially increases away from 1 , the rate of this decrease decreasing as  continues to increase and the near saddle point SPC n asymptotically approaches the inner branch point !C . The construction of the complete, asymptotic space–time evolution of the propagated Heaviside unit step function signal AH .z; t / at a fixed propagation distance z  0 in a single resonance Lorentz model dielectric is now considered. Because of the dependence of the precursor field amplitudes on the applied signal frequency !c , the total wavefield behavior will differ considerably in each of the

538

15 Continuous Evolution of the Total Field

q q q angular frequency domains 0 < !c  !02  ı 2 , !02  ı 2 < !c < !12  ı 2 , q !12  ı 2 < !c < !SB , and !c > !SB , as illustrated in Figs. 15.15–15.18, respectively. The first (or uppermost) graph in each figure describes the dynamic evolution of the Sommerfeld precursor field AHs .z; t /, the second graph describes the dynamic evolution of the Brillouin precursor field AHb .z; t /, and the third graph describes the dynamic evolution of the pole contribution AHc .z; t /, each at the same propagation distance z  0. The fourth (or lowermost) graph in each figure describes the dynamic evolution of the total propagated Heaviside step function signal AH .z; t / D AHs .z; t / C AHb .z; t / C AHc .z; t /, obtained from the linear superposition of the upper three graphs. Because the resonance phenomena which may occur in either the Sommerfeld or Brillouin precursor field, dependent upon whether !c is above or below the absorption band, respectively, is identically cancelled by the corresponding resonance peak in the pole contribution, this artifact of the asymptotic theory is not specifically considered any further here. Consider first the asymptotic construction of the complete space–time behavior of the propagated Heaviside unit step function signal q when the carrier frequency is

in the below absorption band domain 0 < !c  !02  ı 2 . This construction from the superposition of the component sub-fields is illustrated in Fig. 15.15 at three

AHs(z,t)

0.1 X 50

0

AHb(z,t)

0.1 0

AHc(z,t)

0.1 0 Second Precursor Evolution

AH(z,t)

0.1 0

− 0.1

Main Signal Evolution

First Precursor Evolution

1

1.2

+

+

+

qSB 1.4

q0 q

qc

1.8

2

Fig. 15.15 Superposition of the Sommerfeld precursor AHs .z; t /, Brillouin precursor AHb .z; t / and the pole contribution AHc .z; t / to produce the totalqpropagated wavefield AH .z; t / for a below absorption band applied signal frequency 0 < !c < !02  ı 2 . Notice that the amplitude of the Sommerfeld precursor field AHs .z; t / has been magnified 50 times in the uppermost graph

15.5 The Heaviside Step Function Modulated Signal

539

AHs(z,t)

0.1 0

AHb(z,t)

0.1 0

AHc(z,t)

0.1 X 100 0

AH(z,t)

0.1

First Precursor Evolution

Second Precursor Evolution

0

−0.1

1

+

+

2 q

q0

qSB

3

Fig. 15.16 Superposition of the Sommerfeld precursor AHs .z; t /, Brillouin precursor AHb .z; t / and the pole contribution AHc .z; t / to produce q the total propagated q wavefield AH .z; t / for an intra-

absorption band applied signal frequency !02  ı 2 < !c < !12  ı 2 . Notice that the amplitude of the pole contribution AHc .z; t / has been magnified 100 times in the second graph from the bottom

AHs(z,t)

0

AHb(z,t)

0

AHc(z,t)

0.1

0

First Precursor Evolution

AH(z,t)

0.1

Second Precursor Evolution

Main Signal Evolution

0

−0.1

1

+

+

qSB

q0

2 q

+

q c 2.5

Fig. 15.17 Superposition of the Sommerfeld precursor AHs .z; t /, Brillouin precursor AHb .z; t / and the pole contribution AHc .z; t / to produce q the total propagated wavefield AH .z; t / for an above absorption band applied signal frequency

!12  ı 2 < !c < !SB

540

15 Continuous Evolution of the Total Field

AHs(z,t)

0

AHb(z,t)

0.1

0

AHc(z,t)

0.1

0

AH(z,t)

First Precursor Evolution

Second Precursor Evolution

Prepulse Evolution

Main Signal Evolution

0 qc1

−0.1 1

+ 1.2

+

qSB

qc

qc2

+ 1.4

q0 q

1.6

1.8

+ 2

Fig. 15.18 Superposition of the Sommerfeld precursor AHs .z; t /, Brillouin precursor AHb .z; t / and the pole contribution AHc .z; t / to produce the total propagated wavefield AH .z; t / for an above absorption band applied signal frequency !c > !SB . Notice the prepulse formation between  D c1 and  D c1

absorption depths z D 3zd in a single resonance Lorentz medium when !c D !0 =4, where zd D ˛ 1 .!c /. As can be seen, the first precursor (magnified 50 times in the uppermost graph) evolves essentially undisturbed over its dominant space–time domain  2 Œ1; SB /, and is essentially isolated from the remainder of the signal evolution in this below absorption band case. The transition from the first to the second precursor field then occurs as  increases through SB . The second precursor then evolves essentially undisturbed over the space–time domain  2 .SB ; 1 /, interfering with the build-up to the pole contribution over the space–time domain  2 .1 ; c /. The main signal (due to the pole contribution), oscillating with fixed angular frequency !c and attenuated in ampltude by e z=zd then evolves essentially undisturbed for all  > c . Notice the interference effects between the second precursor and the transient phenomena associated with the main signal arrival. These leading-edge transient interference effects primarily serve to distort the main signal behavior about the space–time point  D c . As  increases above c , these transient effects, together with the interference from the trailing edge of the second precursor, rapidly attenuate and the propagated wavefield settles down to its steady-state behavior given by the residue contribition alone as [from (15.21) with  D i ]   AHss .z; t / D e z˛.!c / sin ˇ.!c /z  !c t :

(15.77)

15.5 The Heaviside Step Function Modulated Signal

541

Finally, notice that as  approaches c from below, the instantaneous angular oscillation frequency !b . / . / of the second precursor field approaches the applied signal frequency !c from below and is approximately equal to !c in the space–time interval about  D c during the transition in the total wavefield behavior from the second precursor to the main signal. For an input Heaviside step function modulated signal with !c D 0, the main signal arrival occurs at the space–time point  D c D 0 when the upper near saddle point SPC n coalesces with the simple pole singularity at the origin. The Sommerfeld precursor field then evolves essentially undisturbed over the space–time domain  2 .1; SB /, followed by a transition about the space–time point  D SB to the Brillouin precursor field. This second precursor then evolves essentially undisturbed over the space–time domain  2 .SB ; 0 /, during which the amplitude of this precursor grows with increasing  as the attenuation decreases to zero at  D 0 , at which point the main signal arrival occurs. This zero frequency main signal component then evolves essentially undisturbed for all  > 0 . Because all of the other spectral frequency components present in the propagated field experience some attenuation with propagation distance, by proceeding to a sufficiently large value of the propagation distance z  0, they can be made exponentially negligible in comparison to the zero frequency main signal evolution that is superimposed with the second precursor evolution about  D 0 , as this point in the second precursor also experiences zero exponentially attenuation. Consider next the behavior of q the propagated wavefield AH .z; t / for applied sigq

nal frequencies !c 2 . !02  ı 2 ; !12  ı 2 / in the absorption band of the dispersive medium. This construction from the superposition of the component sub-fields is illustrated in Fig. 15.16 at 50 absorption depths4 z D 50zd in a single resonance Lorentz medium when !c D 5!0 =4. Notice that the amplitude of the main signal comonent AHc .z; t / has been magnified 100 times in the figure. Because the amplitude of the pole contribution is negligibly small in comparison to either of the precursor fields, and because the signal arrival occurs at such a large space–time point (c reaching its maximum value m in the absorption band at !c D !min , as depicted in Fig. 15.2), the propagated signal behavior is mainly comprised of the interacting first and second precursor fields. Notice further that the instantaneous angular oscillation frequency of either of these two precursor fields cannot reach the value of the applied signal frequency !c of the main signal component. The instantaneous angular q oscillation frequency of the first precursor asymptotically ap-

proaches the value !12  ı 2 / from above as  ! 1 and the instantaneous angular oscillation frequency of the second precursor asymptotically approaches the value q

!02  ı 2 / from below as  ! 1. Because of this, there is a discontinuous change

4

Because a single absorption depth at any frequency in the absorption is so much smaller than the typical absorption depth in the normal dispersion region outside the absorption band, a larger number of absorption depths is needed to propagate a sufficiently large distance in the dispersive medium (i.e., for the dispersion to become mature) for the asymptotic description to be valid.

542

15 Continuous Evolution of the Total Field

in the instantaneous angular frequency of oscillation of the total wavefield in the space–time region about the main signal arrival. Consider next the behavior q of the propagated wavefield AH .z; t / for applied

signal frequencies !c 2 . !12  ı 2 ; !SB / above the absorption band of the dispersive medium but below the high-frequency domain where prepulse formation occurs. The construction of the total wavefield from the superposition of the component sub-fields is illustrated in Fig. 15.17 at three absorption depths z D 3zd in a single resonance Lorentz medium when !c D 2!0 . Because of its increased spectral amplitude when !c is above the absorption band, the Sommerfeld precursor evolution now overlaps with the the Brillouin precursor evolution, resulting in the superposition of a high-frequency wavefield with a low frequency wave. In this intermediate frequency domain, s is determined by the steepest descent path approaching the distant saddle point SPC d and so occurs at a space–time point preceding SB while c is determined by the attenuation at the near saddle point SPC n and so occurs at a space–time point succeeding 0 , as illustrated in the figure. Finally, notice that a weak resonance peak appears in the first precursor evolution near  D 0 for the example presented here. Finally, consider the behavior of the propagated wavefield AH .z; t / for applied signal frequencies in the high-frequency domain !c > !SB . The construction of the total wavefield from the superposition of the component sub-fields is illustrated in Fig. 15.18 at three absorption depths z D 3zd in a single resonance Lorentz medium when !c D 5!0 /2. The Sommerfeld precursor now evolves essentially undisturbed over the initial space–time domain  2 Œ1; c1 /, where c1 < SB . At  D c1 , the prepulse signal arrives and evolves along with the decaying first precursor field up through the space–time point  D SB , at which point there is a transition to the second precursor field which also evolves along with the prepulse. Notice the resonance peak in the Sommerfeld precursor about the space–time point  D c1 , as well as its cancellation by the pole contribution. At  D c2 , the second precursor field becomes dominant over the pole contribution. The second precursor then evolves along with the pole contribution over the space–time domain  2 .c2 ; c /, with the second precursor being the dominant component of the two interfering wavefields. The decaying tail of the second precursor finally becomes negligible in comparison to the pole contribution at  D c , so that for all later space–time values  > c the main signal evolves essentially undisturbed, as evidenced in Fig. 15.18. Consequently, interference exists between the pole contribution and the first precursor field over the space–time interval  2 Œc1 ; SB , which primarily serves to distort the front of the prepulse, and interference exists between the pole contribution and the second precursor field over the space–time interval  2 ŒSB ; c . Because the second precursor experiences virtually zero attenuation in a small space–time interval about the point  D 0 (where the attenuation identically vanishes), the total propagated signal behavior about the space–time point  D 0 is described by a high-frequency ripple due to the pole contribution superimposed upon the slowly varying, large amplitude second precursor for a sufficiently large propagation distance z  0. As the propagation distance increases, the break-up of the pole contribution into a prepulse and the main signal becomes more pronounced than that depicted in Fig. 15.18. Finally,

15.5 The Heaviside Step Function Modulated Signal

543

notice that as  approaches c1 from below, the instantaneous angular oscillation frequency !s . / ' ./ of the first precursor field approaches the applied signal frequency !c from above and is approximately equal to !c when  c1 during the transition from the first precursor to the prepulse evolution. The remarkable accuracy of this asymptotic description is readily evident in Figs. 15.19–15.28 which present the numerically determined dynamical field evolution due to an input Heaviside unit step function signal in a single resonance Lorentz model dielectric with Brillouin’s choice of the medium parameters (!0 D 4:01016 r=s, ı D 0:281016 r=s, and b 2 D 321032 r=s), in which case SB Š 1:334, !SB Š 8:70  1016 r=s, 0 D 1:50, and 1 Š 1:50275. Each figure presents three successive propagated signal waveforms at three successive propagation distances at a fixed angular carrier frequency !c . These numerical results were computed using a 222 -point fast fouries transform (FFT) simulation of the Fourier–Laplace integral representation of the propagated plane-wave pulse given in (12.1) with !max D   1019 r=s. The results are displayed most conveniently as a function of the dimensionless space–time parameter  D ct =z with fixed z > 0, because critical aspects of the wavefield evolution (e.g., the critical space–time points SB , 0 , c , and cj , j D 1; 2) are then independent of the propagation distance z. The dynamical signal evolution depicted in Figs. 15.19–15.21 illustrates the q behavior in the below absorption band domain !c 2 .1; !02  ı 2 /. In each case, the dynamical field evolution begins with the Sommerfeld (or first) precursor over the space–time domain  2 .1; SB /, followed by the Brillouin (or second) precursor

0.05 0 z = zd

AH(z,t)

−0.05

0.1 z = 3zd 0

0.05

z = 5zd

0 1

qSB

2

q0 qc

2.5

q

Fig. 15.19 Dynamical signal evolution with below resonance angular carrier frequency !c D 1  1016 r=s at one, three, and five absorption depths

544

15 Continuous Evolution of the Total Field

0.1 0

z = zd

AH(z,t)

−0.1

0.2 z = 3zd 0

0.1

z = 5zd

0 qc 1

qSB q 0

2

3 q

4

5

Fig. 15.20 Dynamical signal evolution with below resonance angular carrier frequency !c D 2  1016 r=s at one, three, and five absorption depths

0.1 0 z = zd

AH(z,t)

−0.1

0.2 z = 3zd 0

0.1

z = 5zd

0 −0.1

1

qSB q 0

2

3 q

qc 4

5

Fig. 15.21 Dynamical signal evolution with below resonance angular carrier frequency !c D 3  1016 r=s at one, three, and five absorption depths

15.5 The Heaviside Step Function Modulated Signal x 1016

4

Instantaneous Oscillation Frequency (r/s)

545

c

2

1

0

q0

1

2

3 q

qc 4

5

Fig. 15.22 Dynamical evolution of the instantaneous angular frequency of oscillation of the !c D 3  1016 r=s propagated signal wavefields in Fig. 15.21

1

0.5

AH(z,t)

0

z = zd

0.5 z = 3zd 0

0.25

z = 5zd

0 −0.25

q0 0 qSB

10

20

q

30 q c

40

50

Fig. 15.23 Dynamical signal evolution with on resonance angular carrier frequency !c D 4  1016 r=s at one, three, and five absorption depths

546

15 Continuous Evolution of the Total Field

0.1 0 z = 10zd

AH(z,t)

−0.1

0.2 z = 30zd 0

0.1

z = 50zd

0 q0

−0.1

1 qSB

3

5 q

7

9

Fig. 15.24 Dynamical signal evolution with on resonance angular carrier frequency !c D 4  1016 r=s at 10, 30, and 50 absorption depths

1

z = zd

0.5

AH(z,t)

0

0.5 z = 3zd 0

0.25

z = 5zd

0 q0 0 qSB

10

20 q c

30

40

50

q

Fig. 15.25 Dynamical signal evolution with intra-absorption band angular carrier frequency !c D 5  1016 r=s at one, three, and five absorption depths

15.5 The Heaviside Step Function Modulated Signal

547

0.2 z = 10zd

0.1

AH(z,t)

0

0.1 z = 30zd 0 −0.1 0.1 z = 50zd 0 −0.1

q0 1q SB

3

5

7

9

q

Fig. 15.26 Dynamical signal evolution with intra-absorption band angular carrier frequency !c D 5  1016 r=s at 10, 30, and 50 absorption depths

Instantaneous Oscillation Frequency (r/s)

10

x 1016

8

6 c

4

2

0

5

10

q

15

20

qc

25

Fig. 15.27 Dynamical evolution of the instantaneous angular frequency of oscillation of the !c D 51016 r=s propagated signal wavefield at 5 (illustrated in Fig. 15.25), 10 (illustrated in Fig. 15.26), and 15 absorption depths

548

15 Continuous Evolution of the Total Field 0.6 0.4 0.2

AH(z,t)

0

z = zd

0.2

z = 3zd

0 0.2 z = 5zd

0 q0 1 q SB

3

5

q

7

qc

9

Fig. 15.28 Dynamical signal evolution with (barely) above absorption band angular carrier frequency !c D 6  1016 r=s at one, three, and five absorption depths

field, which evolves over the space–time domain  2 .SB ; c /, followed by the main signal evolution for all  > c . Because of the low carrier frequency in Fig. 15.19, the Sommerfeld precursor is barely visible in comparison to both the Brillouin precursor and main signal. However, as the carrier frequency !c is increased, the amplitude j!  !c j1 of the initial pulse spectrum increases in the high-frequency domain above the absorption band, resulting in an increased amplitude of the Sommerfeld precursor, as seen in Figs. 15.20 and 15.21. Finally, notice that both the first and second precursor fields are well defined at one absorption depth (z=zd D 1) in both the !c D 1  1016 r=s (Fig. 15.19) and !c D 2  1016 r=s (Fig. 15.20) cases, but not so in the !c D 3  1016 r=s (Fig. 15.21). This is due to the fact that the absorption depth decreases as the carrier frequency !c approaches the absorption band and the propagation distance must be further increased to reach the so-called mature dispersion regime where the precursor fields are well defined and described by the asymptotic theory. The numerically determined instantaneous angular frequency of oscillation of each of the propapagated signal structures presented in Fig. 15.21 is presented in Fig. 15.22, where the  symbols are for the single absorption depth wavefield, the C symbols are for the three absorption depth wavefield, and the  symbols are for the five absorption depth wavefield. Each numerically determined sampled value of the instantaneous angular oscillation frequency is computed from the expression c ; !j .Nj / D z j

(15.78)

15.5 The Heaviside Step Function Modulated Signal

549

where j D j C1  j represents the absolute difference between the space–time values j at successive zero crossings in the computed wavefield evolution at a fixed value of the propagation distance z > 0. The space–time point at which the numerically determined angular frequency value !j is then assigned to the midpoint value Nj of the space–time interval .j ; j C1 / between two adjacent zeros. Because the near saddle point interacts with the below absorption band pole at ! D !c and the total field evolution is dominated by the Brillouin precursor after the Sommerfeld precursor dies out and before the main signal arrival, the instantaneous oscillation frequency of the total field in this below absorption case is found to increase monotonically to the carrier frequency !c as  increases above SB , and is approximately equal to !c at  D c and remains so for all larger  > c , as seen in Fig. 15.22. The dynamical signal and instantaneous oscillation frequency evolution depicted q in Figs. 15.23–15.27 illustrates the behavior in the absorption band domain q

!c 2 Œ !02  ı 2 ; !12  ı 2 . For applied signal frequencies at the lower end of the absorption band (!c !0 ) the dominant field structure is dominated by the Brillouin precursor, as seen in Figs. 15.23 and 15.24 for the on-resonance angular carrier frequency case (!c D !0 4  1016 r=s), but as !c increases up through the absorption band, the Sommerfeld precursor becomes increasingly important in the total propagated field structure, as seen in Figs. 15.25 and 15.26 for the !c D 5  1016 r=s angular carrier frequency case (which is approximately midway through the absorption band). Because the absorption reaches a maximum so that the absorption depth zd D ˛ 1 .!c / reaches a minimum in the absorption band, it can take several absorption distances to reach the mature dispersion regime in the absorption band. The propagated signal field structures presented in Figs. 15.23 and 15.25 at one, three, and five absorption depths are seen to be in the immature dispersion regime, whereas those presented in Figs. 15.24 and 15.26 are seen to be in the mature dispersion regime, the transition from immature to mature dispersion occuring at approximately ten absorption depths (this is to be contrasted with a single absorption depth in the below absorption band frequency domain). Notice that, although the main signal at !c has largely attenuated away when z > 10zd , the peak amplitudes in the Sommerfeld and Brillouin precursors have not (for example, at ten absorption depths, the main signal amplitude has been attenuated from unity to the value e 10 ' 0:0000454, while the peak amplitude of the interfering Sommerfeld and Brillouin precursors is approximately 0:2, over three orders of magnitude larger). The dynamical evolution of the instantaneous angular frequency of oscillation of the !c D 5  1016 r=s propagated signal wavefield at 5 ( symbols), 10 (C symbols), and 15 ( symbols) absorption depths is illustrated in Fig. 15.27. The effects of interference between the high-frequency Sommerfeld precursor and the low-frequency Brillouin precursor are evident as the oscillation frequency approaches the signal carrier frequency !c from below as  approaches c from below. This dynamical field behavior continues just above the upper edge of the absorption band, as illustrated in Figs. 15.28 and 15.29 when !c D 6  1016 r=s. It is evident that an experimental measurement of the propagated field structure presented in either Figs. 15.24, 15.26, or 15.29 would detect only the interfering

550

15 Continuous Evolution of the Total Field

0.1 z = 10zd

AH(z,t)

0 0.1 z = 30zd 0

0.1 z = 50zd 0

1

q SB q 0

2

3

4

q Fig. 15.29 Dynamical signal evolution with (barely) above absorption band angular carrier frequency !c D 6  1016 r=s at 10, 30, and 50 absorption depths

precursor field structure as the main signal has attenuated away, and consequently would measure a “signal velocity” that is associated with the pronounced peak in the precursor field. This measured velocity value would then be close to the vacuum speed of light c, which is much greater than the actual signal velocity (see, for example, the signal velocity measurement reported in [20] and its criticism in [21]). As the applied angular signal frequency !c is increased above the medium absorption band, the Sommerfeld (or first) precursor field becomes more pronounced in the total field evolution, as is readily evident in Fig. 15.30 when !c D 7  1016 r=s and Fig. 15.31 when !c D 8  1016 r=s. In both cases, the angular carrier frequency is sufficiently small that !c < !SB so that there is no prepulse formation. Notice that, in both cases, the propagated field structure at a single absorption distance (z D zd ) is in the immature dispersion regime wherein the precursor field structure is not yet sufficiently well defined, whereas at five absorption depths (z D 5zd ) the propagation distance is large enough to be in the mature dispersion regime wherein the precursor field structure is well defined. The transition between these two dispersion regimes occurs at approximately three absorption depths for both the !c D 7  1016 r=s (Fig. 15.30) and !c D 8  1016 r=s (Fig. 15.31) angular carrier frequency cases. The interference between the trailing tail of the Sommerfeld precursor with the entire Brillouin precursor evolution is clearly evident in these two field structures. Notice that as the carrier frequency !c increases above the medium absorption band and the material attenuation ˛.!c / decreases monotonically, it is still much larger than that over much of the precursor field evolution so that, for the

15.5 The Heaviside Step Function Modulated Signal

551

0.5 z = zd

0.1 0

AH(z,t)

−0.1 0.2 z = 3zd 0 0.1

z = 5zd

0

q0 1 qSB

3 qc

5

7

9

q Fig. 15.30 Dynamical signal evolution with above absorption band angular carrier frequency !c D 7  1016 r=s at 10, 30, and 50 absorption depths

0.5 z = zd

0.1

AH(z,t)

0

0.2 z = 3zd 0 0.1

z = 5zd

0 1

qSB q 0

2

qc

3

q Fig. 15.31 Dynamical signal evolution with above absorption band angular carrier frequency !c D 8  1016 r=s at 10, 30, and 50 absorption depths

552

15 Continuous Evolution of the Total Field 0.4 z = zd 0.2

AH(z,t)

0 −0.2 0.2 z = 3zd 0 z = 5zd 0

q c1 q c2 1

qSB

q0

2

q

qc

2.5

Fig. 15.32 Dynamical signal evolution with above absorption band angular carrier frequency !c D 9  1016 r=s at 10, 30, and 50 absorption depths

largest propagation distance considered (z D 5zd ) in these two figures, the amplitude of the main signal evolution is becoming negligible in comparison to the peak amplitude of either precursor field. The dynamical field evolution of the propagated signal wavefield when !c > !SB is illustrated in Figs. 15.32 (for !c D 9  1016 r=s) and 15.33 (for !c D 10  1016 r=s) at one, three, and five absorption depths. The corresponding numerically determined values of the instantaneous angular oscillation frequency for the three signal evolutions described in Fig. 15.33 is given in Fig. 15.34, where the symbols are for the single absorption depth wavefield, the C symbols are for the three absorption depth wavefield, and the  symbols are for the five absorption depth wavefield. This instantaneous angular oscillation frequency is then seen to first reach the signal carrier frequency !c from above at  D c1 , remains equal to !c over the space–time interval c1 <  < c2 , then scatters about !c as  increases above c2 (because of interference with the Brillouin precursor field evolution), and finally stabilizes at !c at  D c and remains at that value for all  > c . The prepulse formation predicted by the asymptotic theory is therefore clearly obtained when !c > !SB . The numerically determined values of c1 , c2 , and c at each selected value of !c may then be used to calculate the relative signal velocity values vc1 =c D 1=c1 , vc2 =c D 1=c2 , and vc =c D 1=c , which may then be compared to the signal velocity values predicted by the asymptotic theory for the same Lorentz model dielectric. This has been done by Oughstun, Wyns, and Foty [22] for a single resonance Lorentz model dielectric with model parameters similar to that used by Brillouin except that the value of the phenomenological damping constant ı has been halved. The results are presented in Fig. 15.35. As can be seen, excellent agreement between the

15.5 The Heaviside Step Function Modulated Signal 0.4

553

z = zd

0.2

AH(z,t)

0 −0.2 0.2 z = 3zd 0 z = 5zd 0

qSB 1

q c1

q c2

q0

q

qc 2

2.5

Fig. 15.33 Dynamical signal evolution with above absorption band angular carrier frequency !c D 10  1016 r=s at 10, 30, and 50 absorption depths

Instantaneous Oscillation Frequency (r/s)

2

x 1017

1.8 1.6 1.4 1.2 wc 0.8 0.6 0.4 0.2 0

1

q c1

q c2 1.5

q

qc 2

2.5

Fig. 15.34 Dynamical evolution of the instantaneous angular frequency of oscillation of the !c D 10  1016 r=s propagated signal wavefields in Fig. 15.33

554

15 Continuous Evolution of the Total Field 1.0

v c 1 /c 0.8

v c2 /c

1/q0

vE/c

0.6

v /c

v c /c vc/c

0.4

0.2

0 0

2

0

6

8

SB

10

12

14

(X 1016 r/s)

Fig. 15.35 Angular frequency dependence of the relative main signal velocity vc .!c /=c D 1=c .!c /, relative anterior pre-signal velocity vc1 .!c /=c D 1=c1 .!c /, and relative posterior pre-signal velocity vc2 .!c /=c D 1=c2 .!c / in a single-resonance Lorentz model dielectric characp terized by the medium parameters !0 D 4  1016 r=s, b D 20  1016 r=s, and ı D 0:14  1016 r=s as described by the asymptotic theory (solid curves) and the numerically measured results (data points with error bars) of Ref. [22]. The dashed curve describes the frequency dependence of the relative energy transport velocity vE .!c /=c in the dispersive medium

numerical experimental results (indicated by the data points with error bars) and the description afforded by the modern asymptotic theory is maintained over the entire angular frequency domain considered. In addition to demonstrating the accuracy of the modern asymptotic theory in describing the complete evolution of a Heaviside step function modulated signal in a single resonance Lorentz model dielectric, these results also provide a physical measure of the signal velocity that is based solely on the measurable instantaneous angular frequency of oscillation of the propagated field in the mature dispersion regime.

15.5.2 Signal Propagation in a Double Resonance Lorentz Model Dielectric Because the distant saddle points in a double resonance Lorentz model dielectric evolve in the angular frequency domain above the upper absorption band, the dynamical evolution of the Sommerfeld precursor is similar to that in an equivalent singleq resonance Lorentz model dielectric with angular resonance frequency !N2 D !2 .b02 C b22 /=.b22 C b02 !22 =!02 /, plasma frequency b 2 b02 C b22 , and

15.5 The Heaviside Step Function Modulated Signal

555

damping constant ı .ı0 C ı2 /=2 with ı0 ı2 [see (13.11) and compare (12.202) and (12.203) with (12.272) and (12.273)]. The description of the Heaviside step function Sommerfeld precursor AHs .z; t / presented in Sect. 15.5.1 for a single resonance Lorentz model dielectric then applies here, including the description of the resonance phenomena and its cancellation by the pole contribution in the construction of the total propagated signal. In like fashion, because the near saddle points in a double resonance Lorentz model dielectric evolve in the angular frequency domain below the lower absorption band, the dynamical evolution of the Brillouin precursor is similar to that for an equivalent singleq resonance Lorentz model dielectric with angular resonance frequency !N 0 D !0 .b02 C b22 /=.b02 C b22 !02 =!22 /,

plasma frequency b 2 b02 .1 C b22 !04 =.b02 !24 //, and damping constant ı ı0 ı2 [see (13.65) and compare (13.66) and (13.67) with (13.68) and (13.69)]. The description of the Heaviside step function Brillouin precursor AHb .z; t / presented in Sect. 15.5.1 for a single resonance Lorentz model dielectric then applies here, including the description of the resonance phenomena and its cancellation by the pole contribution in the construction of the asymptotic behavior of the total propagated signal. As a consequence, the analysis presented here q can focus on q the propagated   2 2 !1  ı0 ; !22  ı22 in the signal behavior for angular carrier frequencies !c 2 passband between the two absorption bands of the double resonance Lorentz model dielectric. The uniform asymptotic behavior of the individual Sommerfeld, middle, and Brillouin precursor fields at five absorption depths (z D 5zd ) in a double resonance Lorentz model dielectric with p < 0 is illustrated in Fig. 15.36 when the angular q carrier frequency !c is near the lower end of the medium pass band q 2  2 !1  ı0 ; !22  ı22 . The superposition of these three precursor fields then produces the resulatant total precursor field evolution illustrated in Fig. 15.37. At this propagation distance the main signal contribution has been reduced in amplitude to the value e z=zd D e 5 ' 0:006738, which is almost an order of magnitude smaller than the peak amlitude values of both the middle and Brillouin precursor fields. Similar results are obtained when the angular carrier frequency !c is shifted upward toward the upper end of the medium pass band, as illustrated in Figs. 15.38 and 15.39. Notice the increase in the relative amplitude of the Sommerfeld precursor AHs .z; t / and the decrease in the relative amplitudes of both the middle AHm .z; t / and Brillouin AHb .z; t / precursor fields as the carrier frequency is increased through the pass band. As a result, the main signal amplitude is now only about a third of the peak amplitude in the precursor field evolution. Nevertheless, as the propagation distance increases further, the main signal amplitude will decrease at a faster rate than the precursor fields, resulting in a propagated wavefield structure that is completely dominated by the total precursor field AHp .z; t / D AHs .z; t / C AHm .z; t / C AHb .z; t /;

(15.79)

556

15 Continuous Evolution of the Total Field

AHs(z,t)

0.02 0

−0.02

AHm(z,t)

0.02 0

−0.02

AHb(z,t)

0.02 0 _

−0.02

1

q1 1.2

qSB

q1 1.4 q0

1.6

1.8

2

q

Fig. 15.36 Superposition of the Sommerfeld AHs .z; t /, middle AHm .z; t /, and Brillouin AHb .z; t / precursors to produce the total precursor field for the propagated Heaviside step function signal wavefield AH .z; t / with angular carrier frequency !c near the lower end of the pass band of a double resonance Lorentz model dielectric Middle Precursor Evolution

0.04

AHp(z,t)

0.02 Second Precursor Evolution

First Precursor Evolution

0

q1

qSB

q0

−0.02

−0.04

_ q1 1

1.2

1.4

q

1.6

1.8

2

Fig. 15.37 Total precursor field for the propagated Heaviside step function signal wavefield AH .z; t / with angular carrier frequency !c near the lower end of the pass band of a double resonance Lorentz model dielectric

for z > 0 when the inequality p > 0 is satisfied. When the opposite inequality is satisfied, so that p < 0 , the total precursor field is given by AHp .z; t / D AHs .z; t / C AHb .z; t /; for z > 0, the middle precursor field now being absent.

(15.80)

15.5 The Heaviside Step Function Modulated Signal

557

AHs(z,t)

0.01

0

AHm(z,t)

0.01

0

AHb(z,t)

0.01

0

−0.01

1

_ q1 1.2 qSB

q1 q 1.4 0

1.6

1.8

2

q

Fig. 15.38 Superposition of the Sommerfeld AHs .z; t /, middle AHm .z; t /, and Brillouin AHb .z; t / precursors to produce the total precursor field for the propagated Heaviside step function signal wavefield AH .z; t / with angular carrier frequency !c near the upper end of the pass band of a double resonance Lorentz model dielectric 0.02

0.01

AHp(z,t)

Middle Precursor Evolution First Precursor Evolution

Second Precursor Evolution

q1

0 qsb

q0 _ q1

−0.01

1

1.2

1.4

1.6

1.8

2

q Fig. 15.39 Total precursor field for the propagated Heaviside step function signal wavefield AH .z; t / with angular carrier frequency !c near the upper end of the pass band of a double resonance Lorentz model dielectric

The accuracy of this uniform asymptotic description is clearly evident in the associated figure pairs presented in Figs. 15.40 and 15.41, and Figs. 15.42 and 15.43. The first figure in each figure pair presents three successive propagated signal waveforms

558

15 Continuous Evolution of the Total Field 0.4 0.2

z = zd

AH(z,t)

0 −0.2 0.2 z = 3zd 0 z = 5zd

0 |

1

|

qSB q0

|

q

2 qc

3

Fig. 15.40 Dynamical signal evolution with intra-passband angular carrier frequency !c 2 .!1 ; !2 / at one, three, and five absorption depths when p > 0

Instantaneous Oscillation Frequency (r/s)

3

x 1016

2.5

c

1.5

1

0.5

0

1

2

3

q

Fig. 15.41 Dynamical evolution of the instantaneous angular frequency of oscillation of the propagated signal wavefields in Fig. 15.40

at three successive propagation distances at the same fixed angular carrier frequency !c that is chosen to be in the medium pass band between the two absorption bands. These numerical results were computed using a 222 -point FFT simulation of the exact Fourier–Laplace integral representation of the propagated plane-wave pulse given in (12.1) with !max D   1019 r=s. The results are displayed most conveniently as a function of the dimensionless space–time parameter  D ct=z with

15.5 The Heaviside Step Function Modulated Signal

559

0.4 0.2

z = zd

AH(z,t)

0 −0.2 0.2 z = 3zd 0 z = 5zd

0

|

1

1.2

|

q 01.4

q c1.6

1.8

q Fig. 15.42 Dynamical signal evolution with intra-passband angular carrier frequency !c 2 .!1 ; !2 / at one, three, and five absorption depths when p < 0

Instantaneous Oscillation Frequency (r/s)

5

x 1016

4.5 4 3.5 ωc 2.5 2 1.5 1 0.5 0

|

1.2

qc 1.6

1.4

1.8

q Fig. 15.43 Dynamical evolution of the instantaneous angular frequency of oscillation of the propagated signal wavefields in Fig. 15.42

fixed z > 0, because critical aspects of the wavefield evolution (e.g., the critical space–time points SB , 0 , c , cm , and cj , j D 1; 2) are then independent of the propagation distance z. The sequence of numerically determined wavefield plots presented in Fig. 15.40 describe the dynamical evolution of a Heaviside unit step function signal AH .z; t /

560

15 Continuous Evolution of the Total Field

with fixed angular carrier frequency !c D 2  1016 r=s in the pass band of a douD 1  1016 r=s, ble resonance Lorentz model dielectric with medium parameters !0 p p b0 D 0:61016 r=s, ı0 D 0:11016 r=s, and !2 D 41016 r=s, b2 D 0:21016 r=s, ı2 D 0:1  1016 r=s. In that case, p > 0 , the middle saddle points never become the dominant saddle points and the middle precursor doesn’t appear in both the total precursor field evolution [see (15.80)] and the dynamical field evolution (see Fig. 13.15 for the impulse response of this particular medium and Fig. 15.4 for the frequency dispersion of the signal velocity). The corresponding numerically determined instantaneous angular oscillation frequency for each signal pattern illustrated in Fig. 15.40 is presented in Fig. 15.41, where the  symbols are for the single absorption depth wavefield, the C symbols are for the three absorption depth wavefield, and the  symbols are for the five absorption depth wavefield. Notice that in this case when p > 0 , the instantaneous angular oscillation frequency settles to the signal frequency !c after the complete evolution of the precursor fields. The sequence of numerically determined wavefield plots presented in Fig. 15.42 describe the dynamical evolution of a Heaviside unit step function signal AH .z; t / with fixed angular carrier frequency !c D 3  1016 r=s in the pass band of a douD 1  1016 r=s, ble resonance Lorentz model dielectric with medium parameters !0 p p 16 16 16 b0 D 0:610 r=s, ı0 D 0:110 r=s, and !2 D 710 r=s, b2 D 0:21016 r=s, ı2 D 0:1  1016 r=s. In that case p < 0 , the middle saddle points become the dominant saddle points over the short space–time interval  2 .SM ; MB /, and the middle precursor appears between the Sommerfeld and Brillouin precursors in both the total precursor field evolution [see (15.79)] and the dynamical field evolution (see Fig. 13.16 for the impulse response of this particular medium and Fig. 15.5 for the frequency dispersion of the signal velocity). The corresponding numerically determined instantaneous angular oscillation frequency for each signal pattern illustrated in Fig. 15.42 is presented in Fig. 15.43, where the symbols are for the single absorption depth wavefield, the C symbols are for the three absorption depth wavefield, and the  symbols are for the five absorption depth wavefield. Notice that in this case when p < 0 , the instantaneous angular oscillation frequency first settles to the signal frequency !c at the space–time point  D cm after the evolution of the Sommerfeld and middle precursor fields, but then diverges away from this value over the space–time interval  2 .c2 ; c / because of interference with the Brillouin precursor evolution, after which it resettles to !c and remains at that value for all  > c . These numerical results then confirm the prepulse formation in the pass band of a double resonance Lorentz model dielectric when p < 0 . Comparison of these results with the description afforded by the group velocity approximation in Sect. 15.5.2 reveals the importance of the precursor field evolution in correctly describing the observed signal distortion (see Fig. 11.29). This middle precursor evolution in a double resonance Lorentz model dielectric has also been observed by Karlsson and Rikte [23] using the dispersive wave splitting approach introduced by He and Str˘om [24] in 1996.

15.5 The Heaviside Step Function Modulated Signal

561

15.5.3 Signal Propagation in a Drude Model Conductor The complex index of refraction of a Drude model conductor [25] is given by [see (12.153)] !1=2 !p2 n.!/ D 1  ; (15.81) !.! C i  / where !p is the plasma frequency and  is the damping constant which, for a plasma, is given by the effective collision frequency. Estimates of these model parameters for the E-layer of the ionosphere, given by [26] !p   107 r=s;    105 r=s; are used in this section to describe the transient wavefield phenomena associated with ionospheric signal propagation. The results presented here are based on the asymptotic results presented by Cartwright et al. [27, 28]. From the analysis presented in Sect. 12.3.4, the Drude model possesses a pair of distant saddle points [cf. (12.309)–(12.311)] !SP˙ . / D ˙./  i d

  1 C . / 2

(15.82)

for   1, with the second approximate expressions s ./

. /

!p2  2 2  1

!p2  2 1

C



2 ; 4

2 .27/.4/

./

;

(15.83)

(15.84)

which begin at !SP˙ .1/ D ˙1  i  and move into the outer branch point zeros d q !z˙ D ˙ !p2  .=2/2  i =2 (see 12.31) as  ! 1, and a single near saddle point [cf. (12.317)] 8 3 !p2

1  3 !p2   2 2 A 2 @ C 4 2 2 0

!SPn . / i  9 !p2   8 2 31=2 9  2 ˆ > ˆ > 2 2 < 9 !p   6 7 = 7   1   16 4 ˆ > 3.!p2  2 / 5 ˆ > : ; 16 4  2 C 4 2

(15.85)

562

15 Continuous Evolution of the Total Field

that moves down the positive imaginary axis as   1 increases, asymptotically approaching the origin as  ! 1 (see Figs. 12.36–12.38). The uniform asymptotic description of the Sommerfeld precursor AHs .z; t / for the unit Heaviside step function signal in a Drude model conductor is then given by (13.61) for all   1. The resultant first precursor field evolution of a Heaviside unit step function modulated signal with VLF angular carrier frequency !c D 1  105 r=s at five absorption depths (z D 5zd ) into the Drude model of the E-layer of the ionosphere is presented in Fig. 15.44. This transient field structure is characteristic of the Sommerfeld precursor in Lorentz-type dielectrics, with amplitude that begins at zero at the luminal space–time point  D 1, rapidly building to its peak value soon after this point, and then decreasing monotonically to zero as  ! 1. In addition, the instantaneous angular frequency of oscillation !s . / of this Sommerfeld precursor (see Sect. 13.2.4) begins at infinity (when the field amplitude vanishes) and then chirps down qwith increasing  > 1, asymptotically approaching the limiting value C 1, with the result [27] (

c=z  2 00 .!SPn ;  /

AHb .z; t /  <

1=2

z

e c .!SPn ;/ !SPn . /  !c

) (15.86)

as z ! 1. The resultant second precursor field evolution of a Heaviside unit step function modulated signal with VLF angular carrier frequency !c D 1  105 r=s at

AHs(z,t)

0.0005

0

−0.0005

2

4

q

6

8

10

Fig. 15.44 Dynamical evolution of the Sommerfeld precursor AHs .z; t / due to a Heaviside unit step function modulated signal with VLF angular carrier frequency !c D 1  105 r=s at five absorption depths (z D 5zd ) into the E-layer of the ionosphere (from Cartwright and Oughstun [27])

15.5 The Heaviside Step Function Modulated Signal

563

AHb(z,t)

0.02

0.01

0 0

1000

2000 q

3000

4000

Fig. 15.45 Dynamical evolution of the Brillouin precursor AHb .z; t / due to a Heaviside unit step function modulated signal with VLF angular carrier frequency !c D 1  105 r=s at five absorption depths (z D 5zd ) into the E-layer of the ionosphere (from Cartwright and Oughstun [27])

five absorption depths (z D 5zd ) into the Drude model of the E-layer of the ionosphere is presented in Fig. 15.45. Although this behavior of the Brillouin precursor is similar to that in a Debye-model dielectric, the amplitude increasing from zero at  D 1 and then steadily building to its peak value, after which it decreases monotonically back to zero as  ! 1, there are several important differences. First, the space–time point at which the peak amplitude appears increases with increasing propagation distance z > 0. Most noticeable, however, is the very long tail exhibited by the Brillouin precursor in a conducting medium. This characteristic is not observed in a dispersive medium with zero conductivity and may have important health and safety implications. The uniform asymptotic description of the signal contribution AHc .z; t / due to the simple pole singularity at ! D !c is given in (14.75)–(14.78) and is illustrated in Fig. 15.46 with VLF angular carrier frequency !c D 1  105 r=s at five absorption depths into the Drude model of the E-layer of the ionosphere. The resultant total signal evolution, given by the superposition of the Sommerfeld precursor field, the Brillouin precursor field, and the signal contribution as AH .z; t / D AHs .z; t / C AHb .z; t / C AHc .z; t /;

(15.87)

is then given by the sum of the field plots in Figs. 15.44–15.46, with result given by the solid curve in Fig. 15.47. For comparison, the numerically determined Heaviside unit step function signal wavefield with the same carrier frequency (!c D 1105 r=s) propagated the same distance (z D 5zd ) in the same Drude model medium using a 222 -point FFT simulation of the dispersive propagation problem with maximum

564

15 Continuous Evolution of the Total Field 0.01

AHc(z,t)

Fig. 15.46 Dynamical evolution of the signal contribution AHc .z; t / due to a Heaviside unit step function modulated signal with VLF angular carrier frequency !c D 1  105 r=s at five absorption depths (z D 5zd ) into the E-layer of the ionosphere (from Cartwright and Oughstun [27])

0

−0.01

0

1000

3000

2000

4000

0.03

AH(z,t)

0.02

0.01

0

−0.01 0

1000

2000 q

3000

4000

Fig. 15.47 Dynamical evolution of the total propagated signal wavefield AH .z; t / due to a Heaviside unit step function modulated signal with VLF angular carrier frequency !c D 1105 r=s at five absorption depths (z D 5zd ) into the E-layer of the ionosphere (from Cartwright and Oughstun [27]). The solid curve describes the uniform asymptotic behavior and the dashed curve the numerically determined behavior

sampling frequency fmax D 5  108 r=s sampled at the Nyquist rate is described by the dashed curve in Fig. 15.47. Although this is not sufficiently large to adequately describe the Sommerfeld precursor component,5 it is more than sufficient to provide an accurate description of the Brillouin precursor and signal contributions to the total field evolution, as evident in Fig. 15.47. The result is a high-frequency ripple riding on the low-frequency Brillouin precursor, as illustrated in Fig. 15.47 at 5 5

Complementary hybrid numerical-asymptotic techniques have been developed by both Dvorak et al. [29,30] and Hong et al. [31] to properly handle this problem by extracting the high-frequency behavior in order to treat it analytically, leaving the lower frequency behavior to be dealt with numerically.

15.5 The Heaviside Step Function Modulated Signal

565

0.006

AH(z,t)

0.004

0.002

0

0

2000

4000 q

6000

8000

Fig. 15.48 Dynamical evolution of the total propagated signal wavefield AH .z; t / due to a Heaviside unit step function modulated signal with VLF angular carrier frequency !c D 1  105 r=s at ten absorption depths (z D 10zd ) into the E-layer of the ionosphere (from Cartwright and Oughstun [27]). The solid curve describes the uniform asymptotic behavior and the dashed curve the numerically determined behavior

AH(z,t)

0.00005

0

−0.00005

0

10000

20000

30000

40000

50000

q

Fig. 15.49 Dynamical evolution of the total propagated signal wavefield AH .z; t / due to a Heaviside unit step function modulated signal with VLF angular carrier frequency !c D 1  105 r=s at 200 absorption depths (z D 200zd ) into the E-layer of the ionosphere (from Cartwright and Oughstun [27]). The solid curve describes the uniform asymptotic behavior and the dashed curve the numerically determined behavior

absortpion depths, Fig. 15.48 at ten absorption depths, and in Fig. 15.49 at 200 absorption depths. At this final propagation distance (z D 200zd ), the high-frequency Sommerfeld precursor is the dominant contribution to the total propagated wavefield structure. However, because of its (comparatively) very short space–time lifetime

566

15 Continuous Evolution of the Total Field

AH(z,t)

0.00005

0

−0.00005

0

1.5

2.0

2.5

Fig. 15.50 Early space–time evolution of the total propagated signal wavefield AH .z; t / due to a Heaviside unit step function modulated signal with VLF angular carrier frequency !c D 1105 r=s at 200 absorption depths (z D 200zd ) into the E-layer of the ionosphere (from Cartwright and Oughstun [27]). The solid curve describes the uniform asymptotic behavior and the dashed curve the numerically determined behavior

at the front of the signal, it is not discernable in Fig. 15.49, and so is depicted in Fig. 15.50 with the same amplitude scale used in Fig. 15.49. When a Heaviside step function signal propagates through a pure (i.e., zero conductivity) dispersive dielectric, such as that described by either the Lorentz or Debye models, the Brillouin precursor is found to decay only algebraically as z1=2 due to the fact that the near saddle point crosses the origin at  D 0 n.0/. In a purely conducting medium such as that described by the Drude model, however, the complex dielectric permittivity c .!/ D .!/ C k4ki  .!/=! possesses a simple pole at the origin which prevents the near saddle point from reaching that critical point for finite  . As a consequence, the Brillouin precursor in a conductor will decay at a faster rate than the algebraic z1=2 rate found in a pure dielectric. Nevertheless, detailed numerical studies [28] show that the algebraic decay rate of the peak amplitude point of the Sommerfeld precursor approaches z3=4 as z ! 1, while that of the Brillouin precursor approaches z2 as z ! 1.

15.5.4 Signal Propagation in a Rocard–Powles–Debye Model Dielectric The asymptotic description of dispersive signal propagation in a Debye-type dielectric results in the representation [see (15.3)] AH .z; t / D AHb .z; t / C AHc .z; t /;

(15.88)

15.5 The Heaviside Step Function Modulated Signal

567

where the asymptotic behavior of the Brillouin precursor is described in Sect. 13.4 and the asymptotic behavior of the signal contribution is described in Sect. 14.2. The numerically determined signal propagation in a double relaxation time Rocard-Powles-Debye model of triply-distilled water, whose angular frequency dispersion is illustrated in Fig. 15.51, is depicted in Fig. 15.52 at one (z D zd ), three (z D 3zd ), and five (z D 5zd ) absorption depths zd ˛ 1 .!c / when the angular carrier frequency is !c D 2fc with fc D 1 GHz. This carrier frequency is indicated by the plus sign in Fig. 15.51 on both the curves for the real and imaginary parts of .!/. The numerical results presented in Fig. 15.52 were obtained using a 221 -point FFT simulation of the dispersive propagation problem with maximum sampling frequency fmax D 1  1012 s sampled at the Nyquist rate, where the minimum, nonzero sampled frequency value is then given by fmin D fmax =.2N / with N D 221 ; the radian equivalent values of both of these frequency values are indicated by the -symbol on both of the curves in Fig. 15.51. A similar set of calculations is presented in Figs. 15.53 and 15.54 when fc D 200 GHz with N D 221 and fmax D 11013 s. These numerical results show that the observed signal distortion is primarily due to the Brillouin precursor, as described by the asymptotic theory [32], whose peak amplitude at  D 0 decays algebraically with propagation distance z > 0 as z1=2 . Comparison of these two sets of numerical results shows that, as the carrier frequency increases through the absorption peak at ! 2=0 , where 0 D 8:30  1012 s for triply distilled water, the relative time scale between the Brillouin precursor and the signal contribution changes as the latter becomes more of a ripple riding on the (relatively) slowly decaying tail of the Brillouin precursor.

10

r ( c)

Real and Imaginary Parts of the Relative Dielectric Permittivity

8 r( )

6

4

2 i( ) i ( c)

0

105

1010 (r/s)

1015

Fig. 15.51 Frequency dispersion of the real and imaginary parts of the relative dielectric permittivity of triply distilled H2 O. The values of r .!c / and i .!c / at !c D 2fc with fc D 1 GHz are marked on each curve

568

15 Continuous Evolution of the Total Field z = zd 0.4 0.2 0

AH(z,t)

−0.2 0.2 z = 3zd 0 0.2 z = 5zd 0 q0 5

7

9

11

q

13

15

Fig. 15.52 Dynamical signal evolution with a 1 GHz carrier frequency at one, three, and five absorption depths in triply distilled H2 O

10

r( )

Real and Imaginary Parts of the Relative Dielectric Permittivity

8

6

4 r ( c)

i( )

2

i ( c)

0 105

1010 (r/s)

1015

Fig. 15.53 Frequency dispersion of the real and imaginary parts of the relative dielectric permittivity of triply distilled H2 O. The values of r .!c / and i .!c / at !c D 2fc with fc D 100 GHz are marked on each curve

15.5 The Heaviside Step Function Modulated Signal

569

0.6 z = zd

0.4 0.2

AH(z,t)

0 −0.2 0.2 z = 3zd 0 z = 5zd 0

q0 0

10

20

30

40

50

q

Fig. 15.54 Dynamical signal evolution with a 100 GHz carrier frequency at one, three, and five absorption depths in triply distilled H2 O

15.5.5 Signal Propagation along a Dispersive Transmission Line Although it was not identified as such at the time, the Brillouin precursor has also been observed by Veghte and Balanis [33] in their analysis of transient signal propagation along a dispersive microstrip transmission line. The cross-sectional geometry of the microstrip waveguide is depicted in Fig. 15.55, the space between the metallic conductors being filled with a dispersive dielectric material with relative permittivity .!/ and constant magnetic permeability  D 0 . The voltage signal along the dispersive line is given by the Fourier–Laplace integral representation [33] 1 V .z; t / D 2

Z

1

Q VQ .z0 ; !/e i .k.!/ z!t / d!

(15.89)

1

for all z z  z0  0, where VQ .z0 ; !/ D

Z

1

V .z0 ; t /e i!t dt

(15.90)

1

is the Fourier spectrum of the initial voltage pulse at z D z0 . Here ! Q k.!/ ˇ.!/ C i ˛.!/ D  1=2 .!/ c

(15.91)

570

15 Continuous Evolution of the Total Field

Fig. 15.55 Cross-sectional geometry of a microstrip transmission line

w t sub

h

is the complex wavenumber that is a characteristic of the transmission line properties. Veghte and Balanis [33] considered a lossless, dispersive transmission line using the effective microstrip dispersion model developed by Pramanick and Bhartia [34], with real relative dielectric permittivity given by 0 .f / D sub  eff

sub  eff .0/   ;  .0/ f 2 1 C eff sub ft

(15.92)

Z0 ; 20 h

(15.93)

0 where eff .0/ D eff .0/, with

ft

where sub is the relative dielectric permittivity of the substrate (itself a dispersive medium) of thickness h, and Z0 is the characteristic impedance of the microstrip line. Typical values for these parameters for a microstrip line operating over the frequency domain from 1 GHz to 1  104 GHz are sub D 10:2, eff .0/ D 6:76, h D 0:0635 cm, and Z0 D 53:6 (, with width to height ratio w= h < 1 (see Fig. 15.55), so that ft D 33:585 GHz. Comparison of this dispersion model with the real part of the Debye model [see (4.179) in Vol. 1] shows that (15.92) can readily be generalized to a causal model as eff .!/ D sub C

eff .0/  sub ; 1 C i eff !

(15.94)

where the plus sign is now used in the denominator in order to make the model attenuative. The effective relaxation time in this model is defined here as p eff .0/=sub eff (15.95) !t with !t 2ft . The corresponding parameter values for the above microstrip line example are then eff D 3:8578  1012 s with !t D 2:1102  1011 r=s. The resultant angular frequency dispersion is illustrated in Fig. 15.56 (note that the imaginary part of the relative effective dielectric permittivity has been magnified ten times in the figure). Notice that, unlike the Debye model, the real part of the (effective) dielectric permittivity now increases as ! increases above 2=eff . The numerically determined voltage signal propagation along this dispersive microstrip transmission line using an N D 220 FFT simulation of the Fourier integral representation in

15.5 The Heaviside Step Function Modulated Signal

571 r(

)

3

Real and Imaginary Parts of the Relative Dielectric Permittivity

r ( c)

2 10 i (

c)

10 i ( )

1

0 105

1010 (r/s)

1015

Fig. 15.56 Frequency dispersion of the real and (ten times the) imaginary parts of the relative effective dielectric permittivity of a microstrip transmission line with sub D 10:2, eff .0/ D 6:76, Z0 D 53:6 (, and h D 0:0635 cm. The values of r .!c / and i .!c / at !c D 2fc with fc D 10 GHz are marked on each curve 0.6 z = zd

0.4

VH(z,t) - volts

0.2 0

− 0.2 0.2 z = 3zd

0 0.2 z = 5zd

0 1

2

q0

3 q

4

5

Fig. 15.57 Dynamical signal evolution with a 10 GHz carrier frequency at one, three, and five absorption depths along the microstrip transmission line

(15.89) with fmax D 1  1013 Hz sampled at the Nyquist rate (the radian equivalent values of both fmax and fmin D fmax =2N are indicated by the  symbols on the graphs in Fig. 15.56) is illustrated by the sequence of voltage plots in Fig. 15.57 at

572

15 Continuous Evolution of the Total Field

one, three, and five absorption depths when the input signal is a 1 V heaviside step function signal with fc D 10 GHz carrier frequency. These results clearly show the development of a Brillouin precursor at the leading edge of the propagated voltage signal that is strikingly similar to that for a Rocard–Powles–Debye model dielectric [cf. Fig. 15.52].

15.6 The Rectangular Pulse Envelope Modulated Signal From the results of Sect. 11.2.4, the propagated wavefield due to an initial rectangular envelope modulated sinusoidal wave with fixed angular frequency !c and initial time duration T > 0 (assumed to be an integral number of periods Tc D 2=!c of the carrier wave) may be represented as the difference between two Heaviside unit step function modulated signals separated in time by the initial pulse width T , so that [see (11.64)] AT .z; t / D AH .z; t; 0/  AH .z; t; T / Z

Z z z 1 1 1 z c 0 .!;/ d!  e i!c T z c T .!;T / d! D < 2 C !  !c C !  !c (15.96) for all z  0, where AH .z; t; T / denotes the propagated signal wavefield due to an input Heaviside unit step function signal that turns on (i.e., jumps discontinuously from zero to one) at time t D T , with   0 .!;  / .!; / D i ! n.!/   ; ct D ; z

(15.97) (15.98)

and   T .!; / i ! n.!/  T ; c T .t  T /: z

(15.99) (15.100)

Consequently, the results of Sect. 15.5 may be directly applied to obtain a detailed understanding of the asymptotic behavior of any given rectangular envelope pulse as z ! 1. Specifically, the uniform asymptotic behavior of the front AH .z; t; 0/ of the pulse is completely described by the results presented in Sect. 15.5, and the uniform asymptotic behavior of the back AH .z; t; T / of the pulse is also described by the results in Sect. 15.5 with  replaced by the retarded space–time parameter T and the constant phase factor e i!c T multiplying the final result. The total uniform asymptotic behavior of the propagated rectangular envelope modulated pulse is then given by the difference between these two component wavefields.

15.6 The Rectangular Pulse Envelope Modulated Signal

573

Consider first the asymptotic behavior of the rectangular pulse envelope wavefield AT .z; T / given in (15.96) for space–time values   T c , where T c is that value of T at which the pole contribution to AH .z; t; T / becomes the dominant contribution to that field component. Because T c > c for all T > 0 and finite z > 0, so that the pole contribution to AH .z; t; 0/ is also the dominant contribution to that field component, the asymptotic behavior of the total wavefield is then given by   !c nr .!c /    

 Ce ˛.!c /z sin cz !c nr .!c /  T  !c T

AT .z; t /  e ˛.!c /z sin

z

c

D0

(15.101)

as z ! 1 with   T c . That is, the pole contribution from the back of the pulse identically cancels the pole contribution from the front of the pulse for all   T c . Because T D   .c=z/T , it is then seen that the space–time evolution of the back of the pulse is retarded in  by the amount .c=z/T from the corresponding  -evolution of the front of the pulse. The critical space–time value  D T c at which the pole contribution from the back of the pulse arrives is then given by T c D c C .c=z/T . Hence, the -duration of the propagated rectangular envelope pulse AT .z; t / between the front and back pole contribution arrivals is c D

c T; z

(15.102)

and the corresponding temporal duration of the propagated pulse between these two space–time points is z (15.103) tc D c D T: c Consequently, any pulse broadening and envelope degradation due to dispersion is caused primarily by the precursor fields associated with the front and back of the pulse. If the precursor fields are neglected and only the pole contributions are considered by themselves (i.e., without any interference from the interacting saddle points), then the propagated wavefield becomes AT .z; t /  e ˛.!c /z sin

z c

  !c nr .!c /  

(15.104)

for all  2 Œc ; T c , and is zero for all other values of . Consequently, by neglecting the precursor phenomena associated with the front and back edges of the pulse, as well as any interference from the interacting saddle points with the pole contributions, the propagated pulse retains its rectangular envelope shape and temporal width T , and is simply attenuated with the propagation distance z > 0 by the amplitude factor e ˛.!c /z .

574

15 Continuous Evolution of the Total Field

15.6.1 Rectangular Envelope Pulse Propagation in a Single Resonance Lorentz Model Dielectric To obtain a complete understanding of the propagation of a rectangular envelope modulated signal through a causally dispersive medium, the detailed interaction of the precursor fields associated with the front and back edges of the pulse with both the pole contributions and each other must be fully taken into account, and that requires a specific model for the material dispersion. Attention is first given here to the single resonance Lorentz model dielectric. As described in Sect. 15.5.1, the propagation characteristics of both the front and back edges of the pulse depend upon the value of the applied angular carrier frequency !c , the asymptotic description separating into the normally dispersive below absorption band do

q 2 main !c 2 0; !0  ı 2 , the normally dispersive above absorption band domain

q 2  !1  ı 2 ; 1 , and the anomalously dispersive absorption band domain !c 2 q  q 2 !0  ı 2 ; !12  ı 2 . As a further complication, the propagated wavefield !c 2 behavior also depends strongly upon both the initial pulse duration T > 0 and the propagation distance z > 0 at which the dynamical field behavior is observed, as characterized by the following sequence of space–time domains:  Minimal Distortion Domain: For a sufficiently small propagation distance z > 0

and=or a sufficiently long initial pulse duration T such that the inequality c  1 < .c=z/T is satisfied, the first and second precursor fields associated with the leading edge of the pulse will evolve undisturbed by the precursor field contributions associated with the trailing edge of the pulse, and the first and second precursor fields associated with the back of the pulse will only interfere with the pole contribution associated with the front of the pulse.  Intermediate Distortion Domain: For either larger propagation distances z or shorter initial pulse durations T such that SB  1 < .c=z/T < c  1, the first precursor field associated with the leading edge of the pulse will still evolve undisturbed, but the leading edge second precursor field will overlap and interfere with the trailing edge first precursor field, and the trailing edge second precursor field will interfere with the pole contribution associated with the leading edge of the pulse.  Maximal Distortion Domain: Finally, for even larger propagation distances z or even shorter initial pulse widths T such that the inequality 0 < .c=z/T < SB  1 is satisfied, the first and second precursor fields associated with either the leading or trailing edges of the pulse will overlap and interfere with each other. Consequently, for any given initial pulse duration T > 0, its  -value .c=z/T progressively falls into each one of the above space–time domains as the propagation distance z increases, so that for a sufficiently large value of z, the inequality 1 < .c=z/T < SB is eventually satisfied. For positive values of z and T in the initial, minimal distortion domain c  1 < .c=z/T , the envelope degradation and temporal spread of the propagated pulse is

15.6 The Rectangular Pulse Envelope Modulated Signal

575

minimal. As the propagation distance z increases, the envelope degradation and temporal width of the pulse increase as the precursor fields associated with the leading and trailing edges of the pulse increasingly interfere with each other and the two pole contributions come closer together in -space. This transitional behavior marks the intermediate distortion domain. For values of z and T in the final, maximal distortion domain where 0 < .c=z/T < SB  1, the pulse envelope degradation and temporal spread of the propagated pulse are severe. Pulses with short initial pulse widths T then degrade much faster with propagation distance than those with longer initial pulse widths. For two initial rectangular envelope pulses with the same angular carrier frequency !c but with different initial pulse widths T1 and T2 D mT1 , m > 0, the propagation distances in the same dispersive medium at which their propagated field structure in the -domain is identical are related by z2 D mz1 . Before full consideration can be given to a detailed description of the dynamical wavefield evolution due to an input rectangular envelope modulated pulse with constant angular carrier frequency !c , the signal arrival and associated signal velocity must first be carefully described. This description is afforded by the detailed asymptotic analysis presented in [3, 8, 35] which forms the basis of the presentation given here. For an input rectangular envelope modulated signal AT .z; t / D AH .z; t; 0/  AH .z; t; T / of initial time duration T > 0 and fixed angular carrier frequency !c 2 Œ0; !SB /, the signal arrival occurs at the space–time point  D c and the propagated wavefield ceases to oscillate at !c when T   cT=z D c . Both of these transition points propagate with the signal velocity vc .!c / D c=c .!c /. The main body of the propagated pulse that is oscillating at ! D !c then evolves over the space–time interval from  D c to  D c C cT=z with -width c D cT=z and corresponding temporal width tc D

z c D T: c

(15.105)

Consequently, any temporal pulse broadening and envelope degradation of the initial rectangular envelope pulse when !c 2 Œ0; !SB / is due primarily to the precursor field structure of the propagated wavefield that arises from the leading and trailing edges of the pulse. This interference of the signal contribution with the precursor field structure tends to shorten the space–time domain over which the propagated wavefield oscillates predominantly at the input angular carrier frequency !c . Strictly speaking, the temporal width of the rectangular envelope pulse signal is then found to decrease with increasing propagation distance z  0. The commonly observed phenomenon [14, 36–44] of dispersive pulse spreading is obtained only when the propagated signal is redefined to include a range of frequencies about !c , and this, in turn, implies the incorporation of some portion of the leading and trailing edge precursor fields in the definition of the main body of the pulse. For !c > !SB , the signal arrival first occurs at  D c1 when the simple pole at ! D !c is first crossed, and the propagated wavefield finally ceases to oscillate predominantly at !c when T   cT=z D c1 and the pole contribution is subtracted out. Both of these transition points propagate with the presignal velocity vc1 .!c / D c=c1 .!c /. The main body of the propagated pulse that is oscillating at

576

15 Continuous Evolution of the Total Field

! D !c then evolves over the space–time interval from  D c1 to  D c1 C cT=z with -width c D cT=z and corresponding temporal width tc D

z c D T: c

(15.106)

Between these two space–time points that define the main body of the pulse there are, at most, two other distinct transition points at  D c2 and  D c at which the propagated wavefield either ceases to oscillate predominantly at !c or begins again to oscillate predominantly at ! D !c due to the asymptotic dominance of the leading edge Brillouin precursor between these two space–time points. Because of this, the propagated wavefield due to such an input rectangular envelope pulse separates into, at most, two subpulses provided that cT=z > c  c1 , which eventually reduces to a single pulse at a sufficiently large propagation distance z such that cT=z < c  c1 . Apart from this pulse breakup, the only other source of envelope degradation and pulse broadening of the rectangular envelope pulse with carrier frequency !c > !SB is from the leading and trailing edge precursor fields. The resultant angular frequency dependence of the signal velocity for a finite duration rectangular envelope pulse when the inequality cT=z < c  c1 is satisfied is illustrated in Fig. 15.58. When the propagation distance is sufficiently small such that opposite inequality cT=z > c  c1 is satisfied, then the signal velocity is described by Fig. 15.3. As in that figure, the dashed curve describes the energy

1.0 vc 1 /c 0.8 1/q0 0.6

v/c

vE /c

0.4 vc /c 0.2

0 0

2

0

6

8

SB

10

12

14

(X 1016 r/s)

Fig. 15.58 Angular frequency dependence of the relative signal velocity for a rectangular envelope pulse with finite initial duration T > 0 and propagation distance z > 0 such that cT=zp< c c1 in a single resonance Lorentz model dielectric with parameters !0 D 41016 r=s, b D 201016 r=s, and ı D 0:14  1016 r=s. The long dashed curve describes the angular frequency dispersion of the energy transport velocity for a strictly monochromatic wavefield in the medium

15.6 The Rectangular Pulse Envelope Modulated Signal

577

transport velocity of a strictly monochromatic wavefield in the dispersive medium. The discontinuous jump in the signal velocity at ! D !SB is fundamentally due to the change in dominance of the precursor fields at  D SB . Below !SB the signal arrival occurs following the evolution of the leading edge Sommerfeld and Brillouin precursors, whereas above !SB the signal arrival occurs during the evolution of the leading edge Sommerfeld precursor field. When opposite inequality cT=z > c  c1 is satisfied (with z sufficiently large to be in the mature dispersion regime), the propagated wavefield is found to be separated into two pulse whose velocities are described in Fig. 15.3. The first is a prepulse with front velocity vc1 .!c / D c=c1 .!c / and back velocity vc2 .!c / D c=c2 .!c /, followed by a second subpulse with front velocity vc .!c / D c=c .!c / and back velocity vc1 .!c / D c=c1 .!c /. Because vc1 .!c / > vc .!c / [see (15.28)], the back of this second subpulse eventually catches up with the front and cancels it out, this occuring when cT=z D c  c1 . This then marks the transition from the signal velocity being described by Fig. 15.3 to that described in Fig. 15.58 when !c > !SB . From (15.96) and the uniform asymptotic representation of the Heaviside unit step function envelope signal given in (15.70), the uniform asymptotic description of the propagated wavefield due to an input rectangular envelope modulated signal with initial temporal duration T > 0 is given by [3, 8, 35] AT .z; t /  AHs .z; t; 0/ C AHb .z; t; 0/ C AHc .z; t; 0/ AHs .z; t; T /  AHb .z; t; T /  AHc .z; t; T /;

(15.107)

which is simply the difference between the propagated wavefields due to an input Heaviside unit step function modulated signal AH .z; t; 0/  AHs .z; t; 0/ C AHb .z; t; 0/ C AHc .z; t; 0/ that begins to oscillate harmonically at !c at time t D 0 in the z D 0 plane and the Heaviside unit step function modulated signal AH .z; t; T /  AHs .z; t; T / C AHb .z; t; T / C AHc .z; t; T / that begins to oscillate harmonically at !c at time t D T in the z D 0 plane. It is now shown that this asymptotic representation of the propagated rectangular envelope modulated wavefield provides a complete, accurate description of the dynamical pulse evolution in the mature dispersion regime for all !c 2 Œ0; 1/. Consider first the dynamical pulse evolution in the below absorption band domain   q 2 !c 2 0; !0  ı 2 . In this case, the peak amplitude of the Sommerfeld precursor is typically several orders of magnitude less than the peak amplitude of the Brillouin precursor so that the entire propagated wavefield structure in the mature dispersion regime is dominated by the Brillouin precursor and the signal contibution. In the minimal distortion domain cT=z > c  1, the precursor fields associated with the leading edge AH .z; t; 0/ of the pulse will completely evolve prior to the arrival of the precursor fields associated with the trailing edge AH .z; t; T /. Indeed, the trailing edge precursors will arrive only after the leading edge signal component has arrived (at  D c ) and is evolving. Hence, when this condition prevails the interference between the leading and trailing edge precursor fields is minimal and the resultant pulse distortion is also minimal, as illustrated in Fig. 15.59. In the intermediate

578

15 Continuous Evolution of the Total Field

AH(z,t,0)

0

q 1

qSB

qc

AH(z,t,T) 0

q 1+cT/z qSB + cT/z

1

qc+cT/z

AT(z,t,) 0

q 1

qSB

qc

qc+cT/z

cT/z

Fig. 15.59 Construction of the dynamical space–time structure of the propagated wavefield AT .z; t / D AH .z; t; 0/  AH .z; t; T / due to an input rectangular envelope pulse with temporal duration q T in the normally dispersive, below absorption band angular signal frequency domain   !c 2 0; !02  ı 2 with propagation distance z in the minimal distortion domain cT=z > c  1. When this situation prevails, the interference between the leading and trailing edge precursor fields is minimal and the resultant pulse distortion is minimal

distortion domain c  1 > cT=z > SB  1, the leading edge first precursor field will still evolve undisturbed, but during the evolution of the leading edge Brillouin precursor, the arrival and evolution of the trailing edge precursors occurs. Hence, when this condition prevails there will be interference between the leading edge Brillouin precursor and the trailing edge Sommerfeld precursor, the trailing edge Brillouin precursor appearing soon after the signal arrival at  D c , as illustrated in Fig. 15.60, so that the resultant pulse distortion is found to be moderate. Finally, in the maximal distortion domain cT=z < SB  1 there will occur a nearly complete overlap of the leading and trailing edge precursors, as illustrated in Fig. 15.61, and the resultant pulse distortion is severe.

15.6 The Rectangular Pulse Envelope Modulated Signal

579

AH(z,t,0) q

0 1

qSB

qc

AH(z,t,T) q

0

1

1+cT/z

qSB + cT/z

qc+cT/z

AT(z,t,) q

0

1

qSB

qc

qc+cT/z

cT/z

Fig. 15.60 Construction of the dynamical space–time structure of the propagated wavefield AT .z; t / D AH .z; t; 0/  AH .z; t; T / due to an input rectangular envelope pulse with temporal duration T q in the normally dispersive, below absorption band angular signal frequency   domain !c 2 0; !02  ı 2 with propagation distance z in the intermediate distortion domain c  1cT=z > SB  1. When this situation prevails, the interference between the leading and trailing edge precursor fields is moderate and the resultant pulse distortion is also moderate

In each below absorption band case, the pole contribution to the total wavefield evolution occurs at the space–time point  D c and is then subtracted out at  D c C cT=z so that the overall temporal width of the propagated signal contribution is equal to the initial pulse width T , as stated in (15.105). However, because of the asymptotic dominance of the trailing edge Brillouin precursor AHb .z; t; T /, this signal contribution is the dominant contribution to the total wavefield evolution only over the space–time domain from  D c to  ' 0 C cT=z. The corresponding temporal width of the propagated signal is then given by z tc ' T  .c  0 / c

(15.108)

580

15 Continuous Evolution of the Total Field

AH(z,t,0) 0

q 1

qSB

qc

AH(z,t,T) 0

q

qSB+cT/z qc+cT/z

1 1+cT/z

AT(z,t,) 0

q 1

qSB

qc

qc+cT/z

cT/z

Fig. 15.61 Construction of the dynamical space–time structure of the propagated wavefield AT .z; t / D AH .z; t; 0/  AH .z; t; T / due to an input rectangular envelope pulse with temporal duration q T in the normally dispersive, below absorption band angular signal frequency domain   !c 2 0; !02  ı 2 with propagation distance z in the maximal distortion domain cT=z < SB  1. When this situation prevails, the interference between the leading and trailing edge precursor fields is nearly complete and the resultant pulse distortion is severe

provided that 0 C cT=z > c , which is satisfied up through most of the intermediate distortion domain. When the opposite inequality 0 C cT=z < c is satisfied, the pulse distortion is severe and the total propagated wavefield is dominated by the leading and trailing edge precursor fields. Similar results are obtained in the intermediate signal frequency domain q !c 2

q 2 

q 2 2 2 !0  ı ; !SB which contains the medium absorption band !0 ı ; !12 ı 2 where the medium dispersion is anomalous and extends up to the critical angular frequency value !SB at which the signal velocity of the Heaviside step function modulated signal bifurcates into three branches (see Fig. 15.3). The major difference

15.6 The Rectangular Pulse Envelope Modulated Signal

581

from the below absorption band case is that the leading and trailing edge Sommerfeld precursor fields become more pronounced in the total wavefield evolution as the angular signal frequency moves up through this intermediate frequency domain. The construction of the dynamical space–time structure of the propagated rectandispersive, intra-absorption gular envelope wavefield AT .z; t / in the anomalously q 

q 2 2 band angular signal frequency domain !c 2 !0  ı 2 ; !1  ı 2 is illustrated in Fig. 15.62 when the propagation distance z is in the maximal distortion domain cT=z < SB  1. For values of the initial pulse width T > 0 and propagation distance z > 0 satisfying this inequality there is a nearly complete overlap of the two sets of precursor fields so that the propagated wavefield structure AT .z; t / is dominated by a pair of interfering leading and trailing edge Sommerfeld precursors, followed by a pair of interfering leading and trailing edge Brillouin precursors, which is then followed by the signal oscillating at ! D !c that evolves over the space–time interval from  D c to  D c C cT=z, as illustrated. Just prior to the signal arrival at  D c , the propagated wavefield is dominated by the interfering pair of Brillouin precursors whose instantaneous oscillation frequency is less than !q c.

 Near the lower end of the angular signal frequency domain !02  ı 2 ; !SB , the Sommerfeld precursor field is relatively insignificant in comparison to both the Brillouin precursor field and the signal contribution so that the temporal width of the propagated pulse is given by (15.108). On the other hand, near the upper end of this frequency domain the Sommerfeld precursor field is a dominant feature in the total propagated wavefield over the two space–time domains  2 .1; SB / and  2 .1CcT=z; SB CcT=z/. The propagated signal contribution, which arises from the pole contribution that evolves over the space–time domain extending from  D c to  D c C cT=z when cT=z > c  1, with temporal width z tc D T  .c  1/; c

(15.109)

and is then again the dominant contribution over a small space–time interval about the point  D SB C cT=z provided that cT=z > c  SB . When the inequality cT=z > c  1 is satisfied, the signal contribution is separated into two pulses, each oscillating at the input angular carrier frequency ! D !c . As the propagation distance increases such that the inequality c 1 > cT=z > c SB is satisfied, these two signal pulses reduce to a single pulse oscillating at ! D !c . Finally, when the propagation distance increases such that the inequality cT=z < c  SB is satisfied, the pulse distortion is severe and the total propagated wavefield is dominated by the leading and trailing edge precursor fields over the entire subluminal space–time domain   1. Consider finally the construction of the propagated wavefield in the high angular frequency domain !c > !SB where the propagated Heaviside step function signal separates into a prepulse that evolves over the space–time domain  2 .c1 ; c2 / and a main signal that evolves over the space–time domain  > c , these two signal

582

15 Continuous Evolution of the Total Field

AH(z,t,0) 0

q

1

qSB

qc

AH(z,t,T) 0

q

1

1+cT/z

qc+cT/z

AT(z,t,) q

0

1 1+cT/z qSB

qc

qc+cT/z

cT/z

Fig. 15.62 Construction of the dynamical space–time structure of the propagated wavefield AT .z; t / D AH .z; t; 0/  AH .z; t; T / due to an input rectangular envelope pulse with temporal duration T in the anomalously dispersive, intra-absorption band angular signal frequency doq

q 2  2 2 !0  ı ; !1  ı 2 with propagation distance z in the maximal distortion domain main !c 2 cT=z < SB  1. When this situation prevails, the interference between the leading and trailing edge precursor fields is nearly complete and the resultant pulse distortion is severe

components being separated by the Brillouin precursor field which is the asymptotically dominant field component over the space–time domain  2 .c2 ; c /, as described in Sect. 15.51. In this high-frequency domain the Sommerfeld precursor field is a dominant feature in the total wavefield evolution, evoling essentially undisturbed over the initial space–time domain  2 .1; c1 /. For a sufficiently long initial pulse width T and=or a sufficiently small propagation distance z such that cT=z > c  1, the precursor fields and prepulse associated with the leading edge AH .z; t; 0/ of the rectangular envelope pulse will completely evolve prior to the arrival and evolution of the precursor fields and prepulse associated with the railing edge AH .z; t; T / of the pulse, so that their interference is

15.6 The Rectangular Pulse Envelope Modulated Signal

583

minimal. When this condition is satisfied, the total propagated wavefield evolves in the following sequential manner: 1   < c1 c1    c2 c2 <  < c c    1 C cz T 1 C cz T   < c1 C cz T c1 C cz T   < SB C cz T SB C cz T < 

AT .z; t /  AHs .z; t; 0/ AT .z; t /  AHc .z; t; 0/ AT .z; t /  AHb .z; t; 0/ C AHc .z; t; 0/ AT .z; t /  AHc .z; t; 0/ AT .z; t /  AHs .z; t; T / C AHc .z; t; 0/ AT .z; t /  AHs .z; t; T / AT .z; t /  AHb .z; t; T /

where the leading term in each asymptotic expression given here indicates that it is asymptotically dominant over any additional term included over that particular space–time domain. The propagated rectangular envelope wavefield is thus separated into a prepulse that evolves over the space–time domain  2 Œc1 ; c2 with temporal width z tp D .c2  c1 / (15.110) c that increases linearly with the propagation distance z  0, and a main pulse that evolves over the space–time domain  2 Œc ; 1 C cT=z with temporal width z tc D T  .c  1/ c

(15.111)

that decreases to zero linearly with the propagation distance z over the propagation domain cT=z > c  1. The front and back of the prepulse then propagate with the anterior and posterior pre-signal velocities vc1 .!c / D c=c1 .!c / and vc2 .!c / D c=c2 .!c /, respectively, and the front of the main pulse propagates with the signal velocity vc .!c / D c=c .!c /, just as for the Heaviside step function envelope signal. A moments reflection on the limiting behavior of this asymptotic pulse structure as the propagation distance z becomes small (ignoring momentarily that these results are derived from asymptotic theory as z ! 1) shows that the prepulse width tp given in (15.110) approaches zero while the main pulse width tc given in (15.111) approaches the initial pulse width T as z ! 0. The main pulse is then clearly associated with the initial rectangular envelope pulse. As the propagation distance z increases so that the inequality cT=z < c  1 is satisfied, the main pulse vanishes from the propagated field structure and all that remains is the prepulse and the leading and trailing edge precursor fields. The prepulse remains intact, evolving essentially undisturbed over the space–time interval  2 Œc1 ; c2 until the inequality cT=z < c2  1 is satisfied. When this condition prevails, the prepulse becomes distorted as the trailing edge Sommerfeld precursor begins to evolve over this space–time interval. The construction of the propagated wavefield when c1  1 < cT=z < c2  1 is illustrated in Fig. 15.63. When this

15 Continuous Evolution of the Total Field

AH(z,t,0)

584

0

q

AH(z,t,T)

qc1

qc2

q

0 qc1+cT/z

1

AT(z,t,)

qc

qc2+cT/z

q

0

qc1 1

qc2

qc1+cT/z

qc2+cT/z

qc

1+cT/z

Fig. 15.63 Construction of the dynamical space–time structure of the propagated wavefield AT .z; t / D AH .z; t; 0/  AH .z; t; T / due to an input rectangular envelope pulse with temporal duration T in the normally dispersive, high angular signal frequency domain !c > !SB with propagation distance z in the space–time domain c1 1  cT=z < c2 1. When this situation prevails, the interference between the leading and trailing edge precursor fields is moderate to severe and the resultant pulse distortion is becoming severe

condition prevails, the total propagated wavefield evolves in the following sequential manner: 1   < c1 AT .z; t /  AHs .z; t; 0/ AT .z; t /  AHc .z; t; 0/ c1   < 1 C cz T AT .z; t /  AHs .z; t; T / C AHc .z; t; 0/ 1 C cT=z   < c2 AT .z; t /  AHb .z; t; 0/  AHs .z; t; T / C AHc .z; t; 0/ c2 <  < c1 C cz T AT .z; t /  AHb .z; t; 0/ c1 C cz T <  < c2 C cz T AT .z; t /  AHb .z; t; T / C AHb .z; t; 0/ c2 C cz T <  where the leading term in each asymptotic expression given here indicates that it is asymptotically dominant over any additional term included over that particular space–time domain. The propagated wavefield structure is then seen to be dominated by the leading and trailing edge precursor fields over all but the space–time interval  2 Œc1 ; 1 C cT=z . The temporal width of the prepulse is now given by z tp D T  .c1  1/; c

(15.112)

15.6 The Rectangular Pulse Envelope Modulated Signal

585

which decreases from its maximum value .z=c/.c2  c1 / when cT=z D c2  1 to zero when cT=z D c1  1 as the propagation distance z increases from z D cT =.c2  1/ to z D cT =.c2  c1 /. The validity of this uniform asymptotic description of rectangular envelope pulse propagation in a single resonance Lorentz model dielectric is completely borne out by comparison with detailed numerical calculations of the dynamical pulse evolution [8, 35]. The calculations presented here are for a strongly absorptive medium with Brillouin’s choice of the medium parameters [1, 2], viz., !0 D 4:0  1016 r=s; p b D 20  1016 r=s; ı D 0:28  1016 r=s: The dynamical evolution of the propagated rectangular envelope wavefield at several increasing values of the relative propagation distance z=zd is illustrated in Figs. 15.64–15.67 for the below resonance angular carrier frequency !c D 1:0  1016 r=s. The e 1 penetration depth at this signal frequency is given by zd .!c / ˛ 1 .!c / D 1:82  104 cm. The time origin in each sequence of propagated waveforms has been shifted by the amount tc D c z=c so that the signal arrival at time t D tc and the signal departure at time t D tc C T are aligned at each propagation distance; these time instances are indicated by the vertical dotted lines in each figure. The initial rectangular envelope pulse width in Fig. 15.64 is T D 0:6283 fs D 6:283  1016 s and corresponds to a single period of oscillation of the signal. In this case, the pulse distortion becomes severe (cT=z < SB  1) after only 1=3 of an absorption depth zd into the medium, after which the propagated pulse is dominated by the interfering leading and trailing edge Brillouin precursors. The initial pulse width is doubled to T D 1:257 fs in Fig. 15.65, corresponding to two periods of oscillation of the input signal. In this case the pulse distortion is minimal when z=zd D 0:055, moderate when z=zd D 0:55, and severe when z=zd  2:75. Each of these cases corresponds qualitively to the asymptotic constructions depicted in Figs. 15.59–15.61, respectively. The initial rectangular envelope pulse width is again doubled to T D 2:513 fs in Fig. 15.66. In this case, the pulse distortion is minimal when z=zd  0:7 and becomes severe when z=zd D 1:24, after which the propagated waveform is dominated by the leading and trailing edge Brillouin precursors. Finally, the initial pulse width is doubled once more to T D 5:026 fs in Fig. 15.67 which corresponds to eight oscillation periods of the initial signal. In this case the transition from minimal to moderate pulse distortion occurs when z=zd D 1:41 and the transition to severe pulse distortion occurs when z=zd D 2:48. By comparison, the transition to the severe pulse distortion regime for a picosecond pulse occurs when z=zd 500. Again, in the severe pulse distortion regime the propagated waveform is dominated by the interfering leading and trailing edge Brillouin precursors. Similar results for a single resonance Lorentz model dielectric have been obtained numerically by Barakat [45].

586

15 Continuous Evolution of the Total Field

Fig. 15.64 Dynamical wavefield evolution due to an input rectangular envelope pulse with below resonance angular carrier frequency !c D 1:0  1016 r=s and initial pulse width T D 0:6283 fs in a single resonance Lorentz model dielectric at several increasing values of the relative propagation distance z=zd

T = 0.6283 fs 1 z=0 0

−1 0.5 z / zd = 0.055 0 −0.5

AT (z,t)

0.5 z/ zd = 0.55 0 −0.5 0.2 z/ zd = 2.75

0 −0.2 0.1

z / zd = 5.5

0 −0.1

−4

−2

0

2

t − qc z/c (fs)

Careful inspection of Figs. 15.64–15.67 shows that the propagated rectangular envelope pulse width given in (15.108) correctly describes the time duration over which the propagated waveform is dominated by the signal component oscillating at the input angular signal frequency !c . In particular, this pulse-signal width tc is seen to decrease with increasing propagation distance z from its initial value T to zero at the transition point to the maximal distortion domain. Nevertheless, the overall temporal width of the entire propagated pulse is seen to increase with the propagation distance z. Up into the maximal distortion domain, the propagated rectangular envelope pulse waveform is seen to be defined between the two space–time points  D 0 and  D c C cT=z, with corresponding temporal width z t D T C .c  0 /: c

(15.113)

15.6 The Rectangular Pulse Envelope Modulated Signal Fig. 15.65 Dynamical wavefield evolution due to an input rectangular envelope pulse with below resonance angular carrier frequency !c D 1:0  1016 r=s and initial pulse width T D 1:257 fs in a single resonance Lorentz model dielectric at several increasing values of the relative propagation distance z=zd

587

T = 1.257 fs 1 z=0 0

−1 0.5

z/ zd = 0.055

0 −0.5

AT (z,t)

1 z/ zd = 0.55

0

−1 0.1

z/ zd = 2.75

0 −0.1 0.1

z / zd = 5.5

0 −0.1 −4

−2

0

2

t – qc z/c (fs)

Once into the maximal distortion domain, the propagated rectangular envelope wavefield structure becomes completely dominated by the leading and trailing edge Brillouin precursors whose peak amplitude points occur at the space–time points  D 0 and  D 0 C cT=z, and are thus separated in time by the initial pulse width T . Because these two points in the wavefield evolution experience no exponential decay, but rather decrease in amplitude with the propagation distance z > 0 only as z1=2 , they will remain the prominent feature in the propagated wavefield structure long after the signal contribution has attenuated away. This behavior applies throughout q the normally dispersive below absorption band angular frequency 

domain !c 2 0; !02  ı 2 and remains applicable up through most of the anomaq 

q 2 !0  ı 2 ; !12  ı 2 . Near the upper lously dispersive absorption band !c 2 q end of the absorption band and for signal frequencies !c > !12  ı 2 above the

588

15 Continuous Evolution of the Total Field

Fig. 15.66 Dynamical wavefield evolution due to an input rectangular envelope pulse with below resonance angular carrier frequency !c D 1:0  1016 r=s and initial pulse width T D 2:513 fs in a single resonance Lorentz model dielectric at several increasing values of the relative propagation distance z=zd

T = 2.513 fs 1 z=0 0

−1 0.5

z / zd = 0.055

0 −0.5

AT (z,t)

0.5

z/ zd = 0.55

0 −0.5 0.2 z/ zd = 2.75

0 −0.2 0.1

z / zd = 5.5 0 −0.1

−6

−4

−2

0

2

4

t – qc z/c (fs)

absorption band, the leading and trailing edge Sommerfeld precursor fields become a dominant feature in the propagated wavefield AT .z; t / and must then be included in any description of its overall temporal width, as is now done. The dynamic evolution of the propagated rectangular envelope pulse wavefield AT .z; t / at several increasing values of the propagation distnace z is illustrated in Fig. 15.68 for the above absorption band signal frequency !c D 1:0  1017 r=s, where !c > !SB . The e 1 penetration depth at this signal frequency is zd ˛ 1 .!c / D 2:68  105 cm. The initial temporal pulse width in this example is T D 0:6283 fs D 6:283  1016 s, which corresponds to ten oscillation periods of the signal frequency. At the smallest propagation distance presented in Fig. 15.68, z=zd D 0:037 and at the intermediate propagation distance illustrated, z=zd D 0:37, so that both of these propagated pulse waveforms are in the immature dispersion regime. In both cases the inequality cT=z > c  1 is satisfied so that the pulse

15.6 The Rectangular Pulse Envelope Modulated Signal Fig. 15.67 Dynamical wavefield evolution due to an input rectangular envelope pulse with below resonance angular carrier frequency !c D 1:0  1016 r=s and initial pulse width T D 5:026 fs in a single resonance Lorentz model dielectric at several increasing values of the relative propagation distance z=zd

589

T = 5.026 fs 1 z=0 0

−1 0.5

z / zd = 0.055

0 −0.5

AT(z,t)

1 z / zd = 0.55 0

−1 0.1

z / zd = 2.75

0 −0.1

0.1

z / zd = 5.5

0 −0.1

−6

−4

−2

0

2

4

6

t – qc z/c (fs)

distortion is minimal. At the largest propagation distance illustrated in the figure, z=zd D 3:73 so that the propagated rectangular envelope waveform is in the mature dispersion regime. In this last case, cT=z c1  1, so that the prepulse is almost fully distorted due to interference with the trailing edge Sommerfeld precursor and the main pulse has almost completely disappeared, being replaced by the interfering leading and trailing edge Brillouin precursors. Notice that the time origin for each propagated waveform illustrated in Fig. 15.68 has been shifted by the amount c z=c, the vertical dotted lines in the figure depicting the location of the front and back of the initial, undistorted rectangular envelope pulse, both propagating at the main signal velocity vc .!c / D c=c .!c /.

590

15 Continuous Evolution of the Total Field T = 0.6283 fs 1.08

z/ zd = 0.037

0

−1.00

AT (z,t)

0.84

z/ zd = 0.373

0

−0.90 0.085

z/ zd = 3.73

0

−0.082 −3

−2

−1 t – qc z / c (fs)

0

1

Fig. 15.68 Dynamical wavefield evolution due to an input rectangular envelope pulse with above absorption band angular carrier frequency !c D 1:0  1017 r=s and initial pulse width T D 0:6283 fs in a single resonance Lorentz model dielectric at several increasing values of the relative propagation distance z=zd

The temporal width tc of the main pulse is seen to decrease from its initial value T to zero as the propagation distance increases from zero, as described by (15.111). In addition, the temporal width tp of the prepulse is seen to first increase with increasing propagation distance z  0, as described by (15.110), and then decrease with increasing propagation distance as the pulse distortion becomes severe, as described by (15.112). Nevertheless, the overall temporal width of the entire propagated waveform is seen to increase with the propagation distance z > 0. If just the high-frequency structure in the mature dispersion regime, which evolves

15.6 The Rectangular Pulse Envelope Modulated Signal

591

over the space–time domain from  D 1 to  0 , is included, the overall temporal pulse width becomes z t 0 ' .0  1/; (15.114) c provided that cT=z < c 1, whereas if both the low- and high-frequency structure is included, which evolves over the space–time domain from  D 1 to  ' 0 C cT=z, the overall temporal width is found to be given by z t ' T C .0  1/; c

(15.115)

again provided that cT=z < c  1. The relation given in (15.114) is the appropriate measure of the overall propagated rectangular envelope pulse width in the mature dispersion regime if only the high-frequency content of the wavefield is detected, whereas the relation given in (15.115) is the appropriate measure if all significant frequency components are included (i.e., detected).

15.6.2 Rectangular Envelope Pulse Propagation in a Rocard–Powles–Debye Model Dielectric Because the asymptotic description of a Heaviside step function signal in either a simple Debye model or a more accurate Rocard–Powles–Debye model dielectric is described by the superposition of the Debye-type Brillouin precursor (see Case 2 of Sect. 12.1.1) and the signal contribution, as described in (15.88), the dynamical evolution of a rectangular envelope pulse is then seen to be dominated by a pair of leading and trailing edge Brillouin precursors as the propagation distance z > 0 exceeds a single absorption depth zd D ˛ 1 .!c / at the initial pulse carrier frequency !c , as described in detail in [32]. The pulse evolution to this asymptotic behavior in the mature dispersion regime is illustrated in the sequence of graphs presented in Fig. 15.69 for an input ten-cycle rectangular envelope pulse with fc D 1 GHz carrier frequency and T D 10=fc D 10 ns initial pulse width at one, three, five, seven, and nine absorption depths in triply distilled water (see Fig. 15.53). Notice that the leading and trailing edge Brillouin precursors persist long after the 1 GHz signal has been significantly attenuated by the dispersive absorptive medium. Although these two Brillouin precursors penetrate very far into the material, they only carry a small fraction of the initial pulse energy in the particular case under consideration. The leading edge Brillouin precursor is essentially a remnant of the first half-cycle of the initial pulse and the trailing edge Brillouin precursor is a remnant of the last halfcycle. The input pulse energy available to the leading and trailing edge Brillouin precursors is then limited to that contained in a single cycle of the input rectangular envelope pulse. For a ten-cycle pulse as considered here and illustrated in Fig. 15.69, this means that, at most, only 10% of the input pulse energy is available to this precursor pair [46–48].

592

15 Continuous Evolution of the Total Field z/zd = 1

0.4 0.2 0

A(z,t)

−0.2 0.2 z/zd = 3

0 z/zd = 5

0.1 0 0.1 0 0.1 0 −0.1

z/zd = 7 z/zd = 9

1.24

1.26

1.28

1.3

1.32

t – q0z/c (s)

1.34

1.36

1.38 x 10−7

Fig. 15.69 Dynamical wavefield evolution of an input unit amplitude, ten-cycle rectangular envelope pulse with fc D 1:0 GHz carrier frequency at one, three, five, seven, and nine absorption depths in the simple Rocard–Powles–Debye model of triply distilled water

A more efficient way to generate a Brillouin precursor pair in a dispersive material is with a single cycle pulse because the input pulse energy available to this precursor pair then approaches 100% [32]. The pulse sequence presented in Fig. 15.70 illustrates the dynamical pulse evolution as a unit amplitude, rectangular envelope single-cycle pulse with fc D 1 GHz and T D 1=fc D 1 ns penetrates into triply distilled water at the successive pentration depths z=zd D 0; 1; 2; 3; 4; 5; 6; 7; 8; 9; 10. The evolution of the pulse into a pair of leading and trailing edge Brillouin precursors is clearly evident as the propagation distance exceeds a single absorption depth (z=zd > 1) and the peak amplitude attenuation transitions from exponential e z=zd to the z1=2 algebraic decay described in (13.146). This transition to nonexponential, algebraic decay is illustrated in Fig. 15.71 which presents a semilogarithmic graph of the numerically determined peak amplitude decay as a function of the relative propagation distance z=zd . The solid line in the figure describes the pure exponential decay function e z=zd . The numerically determined peak amplitude decay for three different initial single-cycle pulses with carrier frequency values fc D 0:1 GHz, fc D 1:0 GHz, and fc D 10:0 GHz is presented in this figure by the , ı, and C symbols, respectively, each data set connected by a cubic spline fit. The temporal width of the leading edge Brillouin precursor as a function of the propagation distance z  0 is illustrated in Fig. 15.72. The ordinate in part (a) of the

15.6 The Rectangular Pulse Envelope Modulated Signal 1

593

z/ zd = 0

0 .8 0 .6 1

0 .4

2 3

A ( z,t)

0 .2

4

5

6

7

8

9

10

0 − 0 .2 − 0 .4 − 0 .6 − 0 .8 −1 1 .2

1 .3

1 .4

1.5

1 .6

1. 7

1 .8

1.9

2

x 10–7

t (s )

Fig. 15.70 Propagated pulse sequence of a unit amplitude, single-cycle rectangular envelope pulse with fc D 1:0 GHz carrier frequency in the simple Rocard–Powles–Debye model of triply distilled water 100

Peak Amplitude

10−1

10−2

10−3

10−4

10−5

0

2

4

6

8

10

z / zd Fig. 15.71 Peak amplitude attenuation as a function of the relative propagation distance for input unit amplitude single-cycle rectangular envelope pulses with carrier frequencies fc D 0:1 GHz ( symbols), fc D 1:0 GHz (ı symbols), and fc D 10 GHz (C symbols) in the simple Rocard– Powles–Debye model of triply distilled water. The solid curve describes the pure exponential attenuation experienced by the signal component oscillating at fc , given by e z=zd

594

15 Continuous Evolution of the Total Field

Temporal Width (s)

a 10−8

fc = 0.1GHz fc = 1GHz

10−10

fc = 10GHz

10−12

Temporal Width (s)

b

10−4

10−3

10−2

10−1 z(m)

100

101

102

10−6

10−8 fc = 0.1GHz

10−10

fc = 1GHz fc = 10GHz

10−12 −2 10

10−1

z / zd

100

101

Fig. 15.72 Temporal width of the leading-edge Brillouin precursor (in seconds) as a function of (a) the propagation distance z (in meters), and (b) the relative propagation distance z=zd D ˛.!c /z for a single-cycle rectangular envelope pulse with fc D 0:1 GHz, fc D 1:0 GHz, and fc D 10 GHz. The solid curves describe the limiting asymptotic behavior as z ! 1

figure is the absolute propagation distance z in meters and the ordinate in part (b) is the relative propagation distance z=zd . The dependence of the absorption depth zd ˛ 1 .!c / on the angular carrier frequency !c of the pulse is reflected in the individual curves appearing in Fig. 15.72b. The solid curves in the figure describe the asymptotic result given in (13.150), viz., TB

  a0 .0 C f 0 / 1=2 z .C   / 2 z ; c 0 c

(15.116)

as z ! 1, where ˙ describe the e 1 amplitude points in the Brillouin precursor [see (13.149)], and the three sets of data points represent numerical results obtained for the fc D 0:1 GHz, fc D 1:0 GHz, and fc D 10:0 GHz single-cycle pulse cases, each data set connected by a cubic spline fit. Notice that TB ! 3=.8fc / as z ! 0 for each case and that the numerical results approach the asymptotic behavior described by (15.116) as z ! 1. The temporal width of either the leading or trailing edge Brillouin precursor is then seen to increase monotonically with increasing propagation distance z  0 from the value 3=.8fc / at z D 0, asymptotically approaching the curve described by (15.116) as z ! 1, the transition to the asymptotic behavior occurring when z=zd  1.

15.6 The Rectangular Pulse Envelope Modulated Signal

595

a 1011 1010 fc = 10GHz

Effective Oscillation Frequency (Hz)

109 fc = 1GHz

108 fc = 0.1GHz

107

10−4

10−3

10−2

10−1

100

101

102

z (m)

b 1012

1010

fc = 10GHz fc = 1GHz

108

fc = 0.1GHz

106 10−2

10−1

z/zd

100

101

Fig. 15.73 Effective oscillation frequency (in Hz) of a single cycle rectangular envelope pulse as a function of (a) the propagation distance z (in meters), and (b) the relative propagation distance z=zd D ˛.!c /z for a single-cycle rectangular envelope pulse with fc D 0:1 GHz, fc D 1:0 GHz, and fc D 10 GHz. The solid curves describe the limiting asymptotic behavior as z ! 1

The effective oscillation frequency feff of the single-cycle pulse as a function of the propagation distance z  0 is illustrated in Fig. 15.73. The solid curve describes the asymptotic result [cf. (13.151)]  1=2 0 c 1 1

(15.117) feff ! fB 2 TB 4 a0 .0 C f 0 /z as z ! 1, and the three sets of data points present numerical results for the fc D 0:1 GHz, fc D 1:0 GHz, and fc D 10:0 GHz single-cycle pulse cases, each data set connected by a cubic spline fit. Each measured value of the effective oscillation frequency was determined from the temporal distance between the peak amplitude points in the leading and trailing edge half-cycles of the pulse as it propagated through the dispersive model of water given in (12.300). Notice that feff ! fc as z ! 0 for each case and that the numerical results for the effective oscillation frequency approach the asymptotic behavior given in (15.117) as z ! 1. The effective oscillation frequency of the Brillouin precursor in a Debye model dielectric is then seen to decrease monotonically with increasing propagation distance z  0 from the initial pulse carrier frequency fc at z D 0, asymptotically approaching the curve described by (15.117) as z ! 1, the transition to the asymptotic behavior occuring when z=zd  1.

596

15 Continuous Evolution of the Total Field z /zd = 0 100

80

|Ã(z, )|

1 60

2 3 40

10

20

0 105

106

107

108

109

1010

f (Hz) Fig. 15.74 Magnitude of the pulse spectra for the single-cycle rectangular envelope pulse sequence presented in Fig. 15.70

The magnitude of the spectra for the single-cycle rectangular envelope pulse sequence illustrated in Fig. 15.70 is presented in Fig. 15.74. Notice that the peak value of the spectrum for the input single-cycle pulse is slightly downshifted from the input pulse carrier frequency value fc D 1:0 GHz to the value fp ' 0:83 GHz. This peak amplitude point in the pulse spectrum then shifts to lower frequency values as the propagation distance z  0 increases, shifting from the value fp ' 0:54 GHz at z=zd D 1 to the value fp ' 0:21 GHz at z=zd D 10, in general agreement with the results presented in Fig. 15.73 describing the decrease in the effective pulse frequency with propagation distance. Notice further that the initial and propagated pulse spectra depicted in Fig. 15.74 are effectively contained above 1 MHz over the range of propagation distances considered here. When an ideal high-pass filter with cutoff frequency fmin D 1 MHz is applied to the initial pulse spectrum when computing the propagated pulse wavefield using an adequately sampled FFT simulation of the integral representation of the propagated plane wave pulse given in (12.1), the results presented here for the single-cycle pulse with fc D 1:0 GHz remain essentially unchanged from zero through at least 20 absorption depths [32]. In particular, the algebraic, nonexponential peak amplitude decay of the leading and trailing edge Brillouin precursors [see (13.146)] presented in Fig. 15.71 remains essentially unaltered by application of this ideal high-pass filter operation. It is then clear that this unique, distinguishing behavior of the Brillouin precursor is not a zero frequency phenomenon for finite propagation distances.

15.6 The Rectangular Pulse Envelope Modulated Signal

597

15.6.3 Rectangular Envelope Pulse Propagation in Triply Distilled Water The general dynamical characteristics for rectangular envelope pulse propagation in both Debye-model and Lorentz-model dielectrics presented in the previous subsections are now examined in greater detail for triply distilled water. These results are based on the published research by P. D. Smith [46] et al. [49]. The angular frequency dispersion of the relative dielectric permittivity for this complicated medium is described here by [cf. (4.214) of Vol. 1 as well as (13.168), (13.169), and (13.171)] .!/=0 D 1 C

2 X

X bj2 aj  ; .1  i !j /.1  i !fj / j D0;2;4;6 ! 2  !j2 C 2i ıj ! j D1 (15.118)

with parameter values (compare with those given in Tables 4.1 and 4.2 of Vol. 1) a0 D 74:1, 0 D 8:44  1012 s, f 0 D 4:93  1014 s and a2 D 2:90, 2 D 6:05  1014 s, f 2 D 8:59  1015 s for the orientational polarization part of the model (describing the angular frequency dispersion up through the microwave region of the spectrum), and with !0 =2 D 1:81013 Hz, b0 =2 D 1:21013 Hz, ı0 =2 D 4:3 1012 Hz, !2 =2 D 4:9  1013 Hz, b2 =2 D 6:8  1012 Hz, ı2 =2 D 8:4  1011 Hz, and !4 =2 D 1:0  1014 Hz, b4 =2 D 2:0  1013 Hz, ı4 =2 D 2:8  1012 Hz for the resonance polarization part of the model describing the angular frequency dispersion in the infrared region of the spectrum, and with !6 =2 D 3:7  1015 Hz, b6 =2 D 3:2  1015 Hz, ı6 =2 D 8:0  1014 Hz describing the angular frequency dispersion in the ultraviolet region of the electromagnetic spectrum. Although these model parameters are slightly different from the values given in Tables 4.1 and 4.2 of Vol. 1 with one less resonance line, the resulting frequency dispersion is remarkably similar over the spectral domain from zero through the infrared where this numerical study is focused. The numerically determined dynamical field evolution for each of the five frequency cases depicted in Fig. 15.75 is presented in Figs. 15.76–15.80 [46, 49]. For each frequency case the temporal location of the peak amplitude point in the leading edge Brillouin precursor is denoted by t0eff D

z 0 ; c eff

(15.119)

where 0eff denotes the space–time point whose value is given by the effective zero frequency limit of the index of refraction that is “seen” by the initial pulse spectrum. The temporal location of the peak amplitude point in the trailing edge Brillouin precursor is then given by t0eff C T . As the pulse carrier frequency fc moves up through the various spectral domains that are dominated by different aspects of the dispersion model, the value of 0eff changes as does the character of the leading and trailing edge Brillouin precurors.

598

15 Continuous Evolution of the Total Field c1

1

c2

c3

c4

c5

nr ( )

10

0

10

8

10

10

10

10

12

14

16

10

(r/s)

18

10

10

Fig. 15.75 Angular frequency dependence of the real part nr .!/ D
a

t0 0.6

c

t0 +T

eff

eff

t0

t0 +T

eff

0.15

eff

z = 5zd

0

−0.6

b

ET (z,t) (V/m)

ET (z,t) (V/m)

z = zd

0

10 t (ns) t0

0.2

−0.15

20

d

t0 +T eff

eff

0

0

10 t (ns) t0

0.1

t0 +T eff

eff

z = 10zd

ET (z,t) (V/m)

ET (z,t) (V/m)

z = 3zd

0

−0.2

0

10 t (ns)

20

20

0

−0.1

0

10 t (ns)

20

Fig. 15.76 Dynamical wavefield evolution of an input 1 V=m, 10 ns rectangular envelope modulated sinusoidal carrier wave with fc1 D 1:0 GHz UHF carrier frequency in a composite R–P–D–L model of the dielectric frequency dispersion of triply distilled water at (a) z D zd , (b) z D 3zd , (c) z D 5zd , and (d) z D 10zd , where zd ˛ 1 .!c1 / D 5:83  103 m

15.6 The Rectangular Pulse Envelope Modulated Signal t0

t0 +T

−0.6 0

0.2

t0

b

0.4

0.6 0.8 t (ns)

1.0

1.2

z = 3zd

d

1.0 t (ns) t0

2.0

t0 +T

eff

0.06

eff

z = 10zd

ET (z,t) (V/m)

0

0

−0.08 0

1.4

eff

0.1

eff

z = 5zd

t0 +T

eff

t0 +T

eff

0.08

z = zd

0

ET (z,t) (V/m)

t0

c

eff

eff

ET (z,t) (V/m)

ET (z,t) (V/m)

a 0.6

599

0

−0.1 −0.06

0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 t (ns)

0

0.5

1.0 1.5 t (ns)

2.0

2.5

Fig. 15.77 Dynamical wavefield evolution of an input 1 V=m, 1 ns rectangular envelope modulated sinusoidal carrier wave with fc2 D 10 GHz SHF carrier frequency in a composite R–P–D–L model of the dielectric frequency dispersion of triply–distilled water at (a) z D zd , (b) z D 3zd , (c) z D 5zd , and (d) z D 10zd , where zd ˛ 1 .!c2 / D 3:05  104 m

0.1

ET (z,t) (V/m)

t0 +T eff

eff

40

t0

80 t (ps)

120

d

eff

z = 3zd

50

100 t (ps)

eff

z = 5zd

150

0.01 0 −0.01 0

160

0

−0.1 0

t0 +T

eff

0.02

t0 +T

eff

t0

c

z = zd

0

−0.6 0

b

t0

ET (z,t) (V/m)

ET (z,t) (V/m)

0.6

ET (z,t) (V/m)

a

100

t0

0.008

200 300 t (ps)

eff

400

500

t0 +T eff

z = 10zd

0.004

0

−0.004

0

200

400 t (ps)

600

800

Fig. 15.78 Dynamical wavefield evolution of an input 1 V=m, 100 ps rectangular envelope modulated sinusoidal carrier wave with fc3 D 100 GHz EHF carrier frequency in a composite R–P–D–L model of the dielectric frequency dispersion of triply distilled water at (a) z D zd , (b) z D 3zd , (c) z D 5zd , and (d) z D 10zd , where zd ˛ 1 .!c3 / D 1:17  104 m

600

15 Continuous Evolution of the Total Field

a

t0

t0 +T

0

b

4

t0

eff

z = 5zd

8 t (ps)

0

5

d

t0 +T

eff

0.04

−0.04 0

12

eff

10 15 t (ps) t0

z = 3zd

20

ET (z,t) (V/m)

0

25

t0 +T

eff

eff

z = 10zd

0.01

0.1

ET (z,t) (V/m)

t0 +T

eff

z = zd

0

−0.6

t0

c

eff

eff

ET (z,t) (V/m)

ET (z,t) (V/m)

0.6

0

−0.01 −0.1

0

4

8 t (ps)

12

14

0

10

20 t (ps)

30

40

Fig. 15.79 Dynamical wavefield evolution of an input 1 V=m, 10 ps rectangular envelope modulated sinusoidal carrier wave with fc4 D 1 THz low-IR carrier frequency in a composite R–P–D–L model of the dielectric frequency dispersion of triply distilled water at (a) z D zd , (b) z D 3zd , (c) z D 5zd , and (d) z D 10zd , where zd ˛ 1 .!c4 / D 1:98  105 m

a

t0

c

t0 +T eff

eff

0

−0.6

t0 8

z = zd ET (z,t) (mV/m)

ET (z,t) (V/m)

0.6

2

3

4

0

10

11

t (ps)

b

t0

d

t0 +T eff

eff

t0

z = 3zd

6

7 t (ps)

8

eff

0.06

13

t0 +T eff

z = 10zd

0

−0.06

12 t (ps)

ET (z,t) (mV/m)

ET (z,t) (V/m)

0.06

eff

z = 5zd

−8

5

t0 +T

eff

9

0

−0.06 20

21

22 t (ps)

23

24

Fig. 15.80 Dynamical wavefield evolution of an input 1 V=m, 1 ps rectangular envelope modulated sinusoidal carrier wave with fc5 D 10 THz IR carrier frequency in a composite R–P–D–L model of the dielectric frequency dispersion of triply distilled water at (a) z D zd , (b) z D 3zd , (c) z D 5zd , and (d) z D 10zd , where zd ˛ 1 .!c5 / D 7:02  105 m. Notice the change in abscissa scale to mV=m in parts (c) and (d)

15.6 The Rectangular Pulse Envelope Modulated Signal

601

For the fc1 D 1 GHz UHF radio frequency band case, illustrated in Fig. 15.76, the effective zero frequency limit of the index of refraction is given by 0eff '

q 1 C a1 C a2 C b02 =!02 C b22 =!22 C b42 =!42 D 8:92;

(15.120)

which is dominated by the Rocard–Powles–Debye component of the frequency dispersion model. The dynamical wavefield evolution depicted in Fig. 15.76 clearly shows the exponential decay of the signal component while the leading and trailing edge Brillouin precursors, whose peak amplitudes only decay algebraically as z1=2 , dominate the propagated wavefield structure for all z  zd . Furthermore, the predicted values of t0eff and t0eff C T are seen to accurately describe the peak amplitude points in the leading and trailing edge Brillouin precursors, which are then seen to be a characteristic of the rotational polarization contribution to the frequency dispersion of the dielectric permittivity of water at this input carrier frequency fc1 . For the fc2 D 10 GHz SHF radio frequency band case, illustrated in Fig. 15.77, the effective zero frequency limit of the index of refraction is, to a good approximation, given by (15.120). The dynamical wavefield evolution depicted here again exhibits the exponential decay of the signal component of the propagated rectangular envelope pulse while the leading and trailing edge Brillouin precursors, whose peak amplitudes only decay as z1=2 , dominate the propagated wavefield structure for all z  zd . Notice that the accuracy of the predicted values of t0eff D .z=c/0eff and t0eff C T in describing the peak amplitude points in the leading and trailing edge Brillouin precursors of the propagated wavefield has decreased from that obtained in the previous case, indicating that these precursors are still primarily a characteristic of the rotational polarization phenomena in the frequency dispersion of the dielectric permittivity of water. For the fc3 D 100 GHz EHF radio frequency band case, illustrated in Fig. 15.78, the effective zero frequency limit of the index of refraction is bounded below by the value 0eff '

q 1 C a2 C b02 =!02 C b22 =!22 C b42 =!42 D 2:55;

(15.121)

which is essentially unaffected of the first Rocard–Powles–Debye model component in (15.118). The dynamical pulse evolution depicted here again shows the exponential decay of the signal component while the leading and trailing edge Brillouin precursors dominate the propagated wavefield structure for all z  zd . Notice that only the upper bounds to the time values t0eff D .z=c/0eff and t0eff C T are indicated in Fig. 15.78. In this EHF radio frequency band case, the leading and trailing edge Brillouin precursors are characteristic of both the orientational and resonance polarization components of the dispersion model. The slow temporal decay of both the leading and trailing edge Brillouin precursors displayed in Fig. 15.78 is due to the slow relaxation of the orientational polarization part of the dielectric medium response.

602

15 Continuous Evolution of the Total Field

For the fc4 D 1 THz low-infrared (IR) frequency band case, illustrated in Fig. 15.79, the effective zero frequency limit of the index of refraction is approximately given by (15.121). The dynamical pulse evolution depicted here once again shows the exponential decay of the signal component while the leading and trailing edge Brillouin precursors dominate the propagated wavefield structure for all z  zd . In this low-infrared frequency band case, the leading and trailing edge Brillouin precursors are now primarily a characteristic of the infrared resonance polarization component of the dispersion model (see Fig. 15.75). The slow temporal decay of both the leading and trailing edge Brillouin precursors displayed in Fig. 15.79 is again due to the slow relaxation of the orientational polarization part of the dielectric medium response. Finally, notice the appearance of small amplitude leading and trailing edge Sommerfeld precursors in the dynamical wavefield structure, particularly at the larger propagation distances illustrated in the figure. Finally, for the fc5 D 10 THz case, illustrated in Fig. 15.80, the infrared (IR) carrier frequency is now completely removed from the Rocard–Powles–Debye model structure and the interacting material dispersion is nearly completely determined by the resonance polarization response described here by the Lorentz model component in (15.118). The effective zero frequency limit of the index of refraction is then given by q (15.122) 0eff ' 1 C b02 =!02 C b22 =!22 C b42 =!42 C b62 =!62 D 1:50: The dynamical pulse evolution depicted in Fig. 15.80 shows that the pulse distortion is no longer primarily due to the leading and trailing edge Brillouin precursors, as it was in each of the previous, lower frequency cases, but is now due to leading and trailing edge middle precursors that are a characteristic of the infrared resonance lines in the dielectric frequency response of triply distilled water (see Figs. 13.20 and 13.21). The slowly decaying trailing edge of the propagated wavefield that is observed in parts (a)–(d) of Fig. 15.80 is due to the trailing edge Brillouin precursor that is primarily a characteristic of the effective zero frequency limiting behavior of the infrared Lorentz lines in the composite Rocard–Powles–Debye–Lorentz model given in (15.118), with effective limiting refractive index neff .0/ D 0eff given by (15.122). These results then show that the observed dynamical wavefield evolution due to an input ultrawideband electromagnetic pulse as it propagates through a complex dispersive medium is primarily due to the causal model components of the dielectric permittivity that is spanned by the bandwidth of the initial pulse spectrum. The various dispersive components of a given dielectric material may then be individually probed by a properly designed ultrawideband pulse so that, by varying the initial pulse carrier frequency, different features may be examined and their associated model parameter values may possibly be extracted. An ideal pulse for this inverse problem for material identification [50] is the rectangular envelope modulated signal because it is always ultrawideband even though it may not be ultrashort. The initial pulse width may then be tailored so that its spectrum may sample any desired feature in the dielectric dispersion of the chosen material, and this can always be done remotely.

15.6 The Rectangular Pulse Envelope Modulated Signal

603

15.6.4 Rectangular Envelope Pulse Propagation in Saltwater The effects of conductivity on the unique evolutionary properties of the Brillouin precursor are now considered based on the earlier analyses of Fuller and Wait [51], who consider the effects of dispersion on dipole radiation in geological media, King and Wu [52], who consider the propagation of an ultrawideband radar pulse generated by an electric dipole in sea-water, and of Cartwright and Oughstun [53], who consider ultrawideband pulse propagation in a Debye medium with static conductivity. Consider then a dispersive half-space occupied by a single relaxation time Rocard–Powles–Debye-type material with static conductivity 0 whose relative complex dielectric permittivity is given by [cf. (12.320)] c .!/=0 D 1 C

 .!/ a0 Ci ; .1  i !0 /.1  i !f 0 / !

(15.123)

with  .!/ given by the Drude model as [see (5.88) of Vol. 1]  .!/ D i

0 ; ! C i

(15.124)

where 0 .0 =k4k/!p2 = , where  D 1=c is a damping constant given by the inverse of the relaxation time c associated with the mean-free path for free electrons in the material, and !p is the plasma frequency (see Sect. 4.4.7 of Vol. 1). Estimates of these parameters for sea-water are 0 4 mho=m and  11011 s. In addition, the appropriate dielectric parameters for a single relaxation time model of water are 1 D 2:1, a0 D 74:1, 0 D 8:44  1012 s, and f 0 D 4:62  1014 s. The resultant frequency dependence of the real and imaginary parts of the compex index  1=2 of refraction n.!/ D c .!/=0 with complex dielectric permittivity described by (15.123) and (15.124) with these parameter estimates for saltwater is presented in Fig. 15.81. Because of the nonvanishing conductivity (which introduces a simple pole at the origin), the near saddle point SPn cannot pass through the origin as it moves down the imaginary axis with increasing , instead asymptotically approaching the origin as  ! 1, as described in Sect. 12.3.5 [see (12.330)]. There are also two additional near saddle points SP˙ n that evolve about the origin in the lower half of the complex !-plane, symmetrically situated about the imaginary ! 00 -axis [see (12.331)–(12.332)]. These saddle points then describe the dynamical evolution of the Brillouin precursor [53]; however, the peak amplitude no longer experiences just algebraic decay, as it now also experiences some exponential attenuation due to the presence of conductivity. Nevertheless, the Brillouin precursor persists, dominating the propagated wavefield evolution up to a certain amount of conductivity, as seen in Fig. 15.82 for a fc D 1 GHz, ten-cycle rectangular envelope pulse. A detail of the five absorption depth case given in Fig. 15.83 reveals the characteristic trailing tail of the Brillouin precursor that is due to the material conductivity. Analogous results have been presented by King and Wu [52] for a pulsed dipole in sea-water, showing

604

15 Continuous Evolution of the Total Field

Real & Imaginary Parts of the Complex Index of refraction

104

102 nr ( ni (

0

10

c)

nr ( )

c)

ni( ) −2

10

10−4

10−6 5 10

1010 (r/s)

1015

Fig. 15.81 Angular frequency dispersion of the real (solid curve) and imaginary (dashed curve) parts of the complex index of refraction for saltwater with static conductivity 0 D 4 mhos/m

0.3

0.1 z = zd

AT (z,t)

0 −0.1

0.2 z = 3zd

0

z = 5zd

0 2.5

2.6

2.7 t (s)

2.8 x 10−7

Fig. 15.82 Rectangular envelope pulse evolution with fc D 1 GHz carrier frequency and T D 10=fc D 10 ns pulse width in saltwater with static conductivity 0 D 4 mhos=m

15.7 Noninstantaneous Rise-Time Signals

605

0.02

AT (z,t)

0.01

z = 5zd 0

−0.01 2.4

2.6

3

2.8 t (s)

3.2

3.4 x 10−7

Fig. 15.83 Detail of the five absorption depth waveform in Fig. 15.82

that undersea radar communication may indeed be feasible with use of the Brillouin precursor. The effect of conductivity on the decay rate of the Brillouin precursor is considered in Sect. 15.8.4.

15.7 Noninstantaneous Rise-Time Signals Although the Heaviside unit step function and rectangular envelope pulse signals examined in Sects. 15.5 and 15.6 form a fundamentally important class of canonical pulse types in the theory of dispersive pulse propagation, they are (as my experimental colleagues like to remind me) overly idealized. This is a valid criticism in the same sense as a perfectly sharp edge is an idealization in diffraction theory. Nevertheless, the mathematically precise solution to these idealized canonical problems does indeed provide tremendous insight into the physical processes involved. In particular, although the Dirac delta function pulse is an extreme idealization, the asymptotic description of its propagated wavefield provides the impulse response of the model medium in terms of which any other propagated pulse behavior may be obtained through convolution. Because this impulse response is comprised solely of the precursor fields that are a characteristic of the dispersive medium, the central role that they play in describing pulse distortion is undeniable. However, in order to establish a closer connection to experimental results, the effects of a finite rise-time on the precursor behavior is now considered in some detail.

606

15 Continuous Evolution of the Total Field

15.7.1 Hyperbolic Tangent Envelope Signal Propagation in a Single Resonance Lorentz Model Dielectric For a hyperbolic tangent modulated signal, the envelope function is given by (11.71) as  1 1 ; (15.125) uht .t / D 1 C tanh.ˇT t / D 2 1 C e 2ˇT t with rise-time Tr  1=ˇT , where it is assumed here that ˇT > 0. The temporal frequency spectrum of this envelope function is then given by [see the derivation of (11.72)] uQ ht .!/ D i

=.2ˇT / ; sinh.!=.2ˇT //

0 < =f!g < 2ˇT :

(15.126)

Because sinh.z/ D 0 at z D ˙n i , n D 0; 1; 2; : : : , then uQ ht .!  !c / possesses an infinite number of simple pole singularities at !˙n D !c ˙ i 2nˇT :

(15.127)

The values of the residues of the spectrum in (15.126) at these simple poles singularities are given by n˙ D lim

!!!˙n

.!  !˙n /

i =.2ˇT / sinh Œ.!  !˙n /=.2ˇT /

Di

(15.128)

and hence are independent of the particular pole singularity. The inequality 0 < =f!g < 2ˇT appearing in (15.126) requires that the contour of integration C appearing in the integral representation given in (12.2) lie in the upper half of the complex !-plane between the real axis and the straight line parallel to the real axis that passes through the n D 1 pole singularity at !C1 D !c C i 2ˇT . In the limit as ˇT approaches infinity, the spectrum (15.126) for the hyperbolic tangent envelope function approaches the limiting behavior uQ 1 .!/ lim uQ ht .!/ D ˇT !1

i ; !

(15.129)

which is precisely the spectrum for the Heaviside step function signal. In the opposite limit as ˇT approaches zero from above, the parametric family of functions ˚ 1 4ˇT sinh Œ.!  !c /=.2ˇT /

approaches a Dirac delta function at ! D !c , so that for small, but positive values of ˇT approaching zero, a monochromatic wavefield is approached. This latter limiting case is of little interest to the remaining analysis which focuses on the large ˇT behavior of the propagated wavefield. The ratio of the finite rise-time pulse envelope spectrum uQ ht .!/ to its instantaneous limit uQ 1 .!/ is given by

15.7 Noninstantaneous Rise-Time Signals

607

.!  !c /=.2ˇT / uQ ht .!  !c / D : uQ 1 .!  !c / sinh Œ.!  !c /=.2ˇT /

(15.130)

For large values of the envelope rise-time parameter ˇT and finite values of the quantity .!  !c /, the hyperbolic sine function appearing in the denominator of (15.130) may be approximated by the first two terms of its Taylor series expansion, with the result 2 uQ ht .!  !c /

1 .!  !c /2 : (15.131) uQ 1 .!  !c / 24ˇT2 Because the closest that either the near or distant saddle point can approach the point ! D !c along the positive ! 0 -axis is bounded below by the phenomological damping constant ı, then the value of the hyperbolic tangent envelope spectrum about the point ! D !c is essentially unaltered from its instantaneous rise-time value if the inequality ˇT > ı (15.132) is satisfied. This na¨ıve argument [54] would then lead one to suspect that the precursor fields will persist essentially unchanged for values of the rise-time parameter ˇT of the order of ı or greater. Because ı  1=Tı , where Tı is the characteristic relaxation time of that resonance feature in the frequency response of the dispersive medium, then the inequality given in (15.132) requires that the signal rise-time Tr satisfy the inequality 1 Tr < Tı  (15.133) ı in order for the precursor fields to persist essentially unchanged from their limiting instantaneous rise-time forms. A more detailed analysis [3, 8] that considers the interaction of the near and distant saddle points with the set of simple pole singularities !˙n given in (15.127) for the hyperbolic tangent envelope spectrum validates the conclusions obtained through this simple argument. A brief description of this analysis is now given. In the instantaneous rise-time limit ˇT D 1 of a Heaviside step function signal, the only pole singularity in the finite complex !-plane is situated along the positive ! 0 -axis at ! D !˙0 D !c . As the rise-time parameter ˇT is decreased, the other pole singularities !˙n D !c ˙ i 2nˇT , n ¤ 0, will interact with the deformed contour of integration P . / passing through the saddle points, and their contributions must be taken into account in the construction of the propagated wavefield component Ac .z; t /. Let  D r be defined as the space–time point at which the real part of the interacting saddle point location !SPC ./, j D d; n, is equal to the real part j

of the simple pole singularity location, given by
(15.134)

608

15 Continuous Evolution of the Total Field

The particular pole singularities !˙n that interact with the relevant saddle point then depend upon the values of the quantity ˙2nˇT in comparison to the imaginary part of the saddle point location at  D r . The analysis then separates into the above, below, and intra-absorption band cases. Case 1. Above Absorption Band Domain !c 

q !12  ı 2 .

In the normally dispersive above absorption band domain, the distant saddle point SPC d is the interacting saddle point and the value of r is defined by .r / D !c ;

(15.135)

where the second-order approximation for ./, valid for all   1, is given in (12.202). In addition, the imaginary part of this distant saddle point location is given by [see (12.201)] o n   (15.136) = !SPC ./ D ı 1 C . / ; d

where the second-order approximation for . /, valid for all   1, is given in (12.203). Because the original contour of integration C is constrained to lie in the upperhalf plane between the real ! 0 -axis and the parallel straight line that passes through the n D 1 pole singularity at !C1 D !c C i 2ˇT , only the pole singularities on and below the real axis necessarily interact with the deformed contour of integration P . / that passes through the distant saddle point SPC d , as illustrated in each plot presented in Fig. 15.84. The approximate behavior of the complex phase function .!;  / at these simple pole singularities is given by nˇT  ı C 4.nˇT  ı/2   b 2 =2 i !c   1 C 2 !c C 4.nˇT  ı/2

.!n ; / 2nˇT .  1/ C b 2

!c2

(15.137)

for n D 0; 1; 2; 3; : : : , where !0 D !c . By comparison, the approximate behavior of the complex phase function at the distant saddle point SPC d is given by [see (13.16)] "

  # .b 2 =2/ 1  . /  2  2 ./ C ı 2 1  . / " # b 2 =2 i./   1 C (15.138)  2 ;  2 ./ C ı 2 1  . /

  .!SPC ; / ı 1 C . / .  1/ C d

for all   1.

15.7 Noninstantaneous Rise-Time Signals

609

a C

+1

C

+1

' c

'

' c

+ SPd

SPd+

P

C

+1

c +

SPd

−1

−1

P

r

−1

P

r T

r r

b +2

+2 +1

+1

C '

c

SPd+

−1

P

+2

c

C

c

'

'

+

SPd

SPd+

−1

−2

P

−2 r

−1

P

−2

r T

c

+1

C

r r

+5

+5

+5

+4

+4

+4

+3 +2 +1 c −1 −2 −3

C

+3

+3

+2

+2

C

+1

'

'

c −1

SPd+ P

C

+1

' −2 −3

SPd+

−4

−4

−5

−5

r

P

−3 −4

P

−5 r

T

c −1

r r

Fig. 15.84 Interaction of the deformed contour of integration P . / through the distant saddle point SPC d in the right half of the complex !-plane with the simple pole singularities !˙n D !c ˙ 2nˇT i of the initial envelope q function for the hyperbolic tangent modulated signal with angular carrier frequency !c > !12  ı 2 in a single resonance Lorentz model dielectric. The contour C denotes the original contour of integration that extends along the horizontal line from i a  1 to i a C 1 with 0 < a < 2ˇT . The shaded area in each figure plot indicates the region of the complex !-plane wherein the inequality  .!;  / <  .!SPC ;  / is satisfied d

610

15 Continuous Evolution of the Total Field

Consider first the case where the inequality ˇT >

 ı 1 C .r / 2

(15.139)

is satisfied. The geometry of this situation in the complex !-plane is illustrated in part (a) of Fig. 15.84. In this case, the simple pole singularities in the lower half of the complex !-plane at !n D !c  i 2nˇT , n D 1; 2; 3; : : : , all lie below the distant point SPC d for all   1. Nevertheless, for a sufficiently small value of ˇT satisfying the inequality in (15.137), the simple pole singularity !1 D !c  i 2ˇT , together with the pole singularity !˙0 D !c , will interact with the path P . / that passes through the saddle point SPC d , and so will contribute to the asymptotic behavior of the signal contribution to Aht .z; t /. The approximate expression given in (15.137) for the complex phase behavior at the simple pole singularities shows that the behavior at the poles with n  2 possess a larger exponential decay than that at both the n D 0 and n D 1 poles, and hence are asymptotically negligible in this case. For smaller values of ˇT , such as for the two cases depicted in parts (b) and (c) of Fig. 15.84, the original contour of integration C will be deformed across at least one of the simple pole singularities !Cn in the upper-half plane. However, any such pole contribution to the asymptotic approximation of Aht .z; t / occurs only during a space–time interval when the exponential attenuation associated with it is much greater than that at the distant saddle point SPC d . In the space–time domain when the value of  .!Cn ; / at any of the pole singularities with n D 1; 2; 3; : : : is either comparable with or less than that at the distant saddle point SPC d , the original contour of integration C does not cross that pole singularity when it is deformed to the path P . / through SPC d . Consequently, for all values of the rise-time parameter ˇT on the order of ı or greater, the simple pole singularities in the upper-half of the complex !-plane do not significantly contribute to the asymptotic behavior of the q

propagated wavefield A.z; t / for all !c  !12  ı 2 . The propagated wavefield is then essentially unchanged from the limiting instantaneous rise-time case, the Sommerfeld and Brillouin precursor fields dominating the wavefield structure as z ! 1, in agreement with the simple argument given in connection with (15.130)–(15.133). For even smaller values of ˇT such that Tr  1=ˇT  1=ı, the pole singularities located at !Cn D !c C i 2nˇT , n D 1; 2; 3; : : : , approach close to the real axis and the interaction of the deformed contour of integration with them becomes increasingly important. In that case, several of their contributions to the asymptotic behavior of the propagated wavefield Aht .z; t / are no longer negligible in comparison with the contribution that is due to the simple pole singularity at !˙0 D !c as well as the contributions that are due to several of the simple pole singularities at !n D !c  i 2nˇT , n D 1; 2; 3; : : : , in the lower-half plane. Each of these pole contributions oscillates at the angular signal frequency !c , so that the precursor fields, whose spectral amplitudes decrease as ˇT decreases, become negligible in comparison with the total signal contribution oscillating at !c . The hyperbolic tangent envelope wavefield is then quasimonochromatic.

15.7 Noninstantaneous Rise-Time Signals Fig. 15.85 Numerically determined dynamical wavefield evolution of a hyperbolic tangent envelope modulated signal with above absorption band carrier frequency !c D 2:5!0 at the fixed propagation distance z D 5:84zd in a single resonance Lorentz model dielectric for several values of the initial rise-time parameter ˇT

611

0.002

0

b T = 0.5d

−0.002

0.002

Aht(z,t)

0

bT = d

−0.002

0.002 b T = 5d

0 −0.002

0.002 b T = 10d

0 −0.002

1.0

qc1

qc2

1.2

1.4 q

1.6

1.8

The numerically determined [54] dynamical wavefield evolution of the hyperbolic q tangent envelope signal in the very high-frequency domain !c > !SB > !12  ı 2 of a single resonance Lorentz model dielectric is illustrated in Fig. 15.85 for several values of the rise-time parameter ˇT when !c D 2:5!0 at the fixed propagation distance z D 5:84zd , where zd ˛ 1 .!c /. The propagated wavefield is seen to be quasimonochromatic for ˇT < ı. As the rise-time parameter ˇT is increased above the medium damping constant ı, the Sommerfeld precursor begins to dominate the early time wavefield evolution, and for ˇT  ı the Brillouin precursor appears in a short space–time interval about  D 0 , thereby bifurcating the steady-state signal evolution into a prepulse that evolves over the space–time interval c1    c2 and the main signal which oscillates undisturbed for all  > c . Finally, notice that the ratio of the peak amplitude of the Sommerfeld precursor to the steady-state amplitude of the signal monotonically increases as ˇT increases above ı.

612

15 Continuous Evolution of the Total Field

Case 2. Below Absorption Band Domain !c 

q !02  ı 2 .

In the normally dispersive below absorption band domain, the near saddle point SPC n is the interacting saddle point and the value of r is defined by .r / D !c ;

(15.140)

where the second-order approximation for ./, valid for all  > 1 , is given in (12.220). In addition, the imaginary part of this near saddle point location is given by [see (12.219)] n o 2 = !SPC ./ D  ı ./; (15.141) n 3 where the second-order approximation for . /, valid for all   1 , is given in (12.221). As in the above absorption band case, only the pole singularities on and below the real ! 0 -axis necessarily interact with the deformed contour of integration P . / that passes through the near saddle point SPC n for  > 1 . The approximate behavior of the complex phase function at these simple pole singularities is given by    b2

.n˛ˇT  ı/ !c2  4n2 ˇT2 C 2n˛ˇT !c2 4 0 !0

  b2  2 2 2 i !c   0  ˛ !c  4n ˇT  8nˇT .n˛ˇT  ı 20 !04 (15.142)

.!n ; / 2nˇT .  0 / C

for n D 0; 1; 2; 3; : : : . By comparison, the approximate behavior of .!;  / at the near saddle point SPC n for  > 1 is given by [see (13.95)] ( .!SPC ; / ı n

2

. /. 3

 0 /

)  2 1  b 2  4 2 2 1  ˛ . / C ./ C 9 ı . / 3 ˛ . /  1 0 !04

   b2 4 2 2 i ./   0  ı . / 2  ˛ . / C ˛ . / ; 20 !04 3 (15.143) where the frequency-independent factor ˛ 1 [not to be confused with the frequency-dependent absorption coefficient ˛.!/] is given by (12.218). Consider first the case when the inequality ˇT >

ı

.r / 3

(15.144)

15.7 Noninstantaneous Rise-Time Signals

613

0.004 0

b T = 0.2d

–0.004 0.004 b T = 0.5d

0 –0.004 0.01 Aht(z,t)

0

bT = d

–0.01 0.02 b T = 1.5d

0 –0.02 0.02

b T = 2d

0 –0.02 0.01

b T = oo

0 –0.01 1.4

qc 1.6

1.8

2.0

q

Fig. 15.86 Numerically determined dynamical wavefield evolution of a hyperbolic tangent envelope modulated signal with below absorption band carrier frequency !c D 0:25!0 at the fixed propagation distance z D 5:495zd in a single resonance Lorentz model dielectric for several values of the initial rise-time parameter ˇT

is satisfied. The deformed integration contour P . / through the near saddle point SPC n for  > 1 then interacts with the simple pole singularities at !0 D !c and !1 D !c  i 2ˇT . The other pole singularities either do not enter into the asymptotic description of Aht .z; t / or are asymptotically negligible in comparison with these contributions. For increasingly large values of the rise-time parameter such that ˇT  ı .r /=3, the exponential decay associated with the contribution from the !1 pole singularity becomes much greater than that at !0 D !c and so is asymptotically negligible by comparison. In that limiting case the asymptotic behavior of the propagated wavefield Aht .z; t / approaches that for a Heaviside step function modulated signal, as seen in Fig. 15.86. In the opposite sense, as the risetime parameter ˇT approaches close to the value ı .r /=3, the contribution of the

614

15 Continuous Evolution of the Total Field

simple pole singularity at !1 D !c  i 2ˇT to the asymptotic behavior of the propagated wavefield Aht .z; t / must be taken into account. Furthermore, for values ˇT  ı .r /=3, the original contour of integration C will be deformed across at least one of the simple pole singularities at !Cn D !c C i 2nˇT , n D 1; 2; 3; : : : , in the upper half of the complex !-plane. However, such a pole contribution occurs only during a space–time domain when the exponential attenuation associated with it is much greater than that associated with the near saddle point SPC n . Furthermore, ;  /, the original contour for those space–time values  when  .!Cn ;  /   .!SPC n of integration C will not cross that pole singularity when it is deformed to the path P . / that lies along the steepest descent path through the saddle point SPC n . Consequently, for all values of the rise-time parameter ˇT on the order of ı or greater, the simple pole singularities in the upper half of the complex !-plane do not contribute significantly to the asymptotic behavior of the propagated hyperbolic tangent envefrequency lope modulated wavefield Aht .z; t / for all values of the angular carrierq

in the normally dispersive, below absorption band domain 0 < !c  !02  ı 2 . In the quasimonochromatic limit of small ˇT  ı, however, the pole singularities !Cn approach close to the real axis and the interaction of the deformed contour of integration with them becomes important, as seen in Fig. 15.86. Similar results are obtained as the carrier frequency is shifted up to the medium resonance frequency, as illustrated in Fig. 15.87. Case 3. Intra-Absorption Band Domain

q q !02  ı 2 < !c < !12  ı 2 .

For applied signal frequencies in the intermediate angular frequency domain  q angularq 2 2 !c 2 !0  ı 2 ; !1  ı 2 , which contains the anomalously dispersive absorption band of the dielectric, neither the near nor distant saddle points come within close proximity of any of the simple pole singularities at !˙n D !c ˙ i 2nˇT , 0 n D 0; 1; 2; 3; : : : . Furthermore, because of the presence of the branch cut !C !C just below the region of anomalous dispersion in the lower half of the complex !plane (see Fig. 12.1), the pole singularities at !n D !c ˙i 2nˇT , n D 0; 1; 2; 3; : : : , do not contribute to the asymptotic behavior of the propagated hyperbolic tangent envelope signal Aht .z; t / unless ˇT < ı=2n. As in both the above and below absorption band cases, the contributions to the propagated wavefield that are due to the simple pole singularities located in the upper half of the complex !-plane at !Cn D !c C i 2nˇT , n D 1; 2; 3; : : : , are all asymptotically negligible by comparison unless ˇT  ı, in which case the quasimonochromatic limit is attained. The numerically determined dynamical wavefield evolution of a hyperbolic tangent envelope modulated signal Aht .z; t / with intra-absorption band carrier frequency is presented in Fig. 15.87 when !c D !0 (near the lower end of the absorption band) and in Fig. 15.88 when !c D 1:4375!0 (near the upper end of the absorption band) in a single resonance Lorentz model dielectric p with Brillouin’s choice of the medium parameters (!0 D 4  1016 r=s, b D 20  1016 r=s, and ı D 0:28  1016 r=s). The propagated wavefield structure in the on-resonance case

15.7 Noninstantaneous Rise-Time Signals Fig. 15.87 Numerically determined dynamical wavefield evolution of a hyperbolic tangent envelope modulated signal with on-resonance carrier frequency !c D !0 at the fixed propagation distance z D 2:67zd in a single resonance Lorentz model dielectric for several values of the initial rise-time parameter ˇT

615 0.08

b T = 0.2d

0

– 0.08 0.04 b T = 0.5d

0 – 0.04

Aht(z,t)

0.08 0

bT = d

– 0.08 0.4

b T = 5d

0 0.4

b T = 10d

0

– 40

0

q 0 40 q

80

120

depicted in Fig. 15.87 is similar to that in the below absorption band case illustrated in Fig. 15.86, the field being quasimonochromatic for ˇT < ı, whereas the Brillouin precursor dominates the field evolution when ˇT > ı. The dynamical wavefield evolution becomes more complicated when the carrier frequency is near the upper end of the absorption band, as illustrated in Fig. 15.88 when !c is just below !1 . This particular value of the carrier frequency corresponds to the angular frequency value at which the real-valued group velocity vg .!/ D .@ˇ.!/=@!/1 is equal to the speed of light in vacuum, viz., vg .!c / ' c. The propagated wavefield is clearly quasimonochromatic for ˇT  ı. As ˇT approaches ı from below, the wavefield begins to lose its quasimonochromatic character, as seen in part (a) of Fig. 15.88. As ˇT is increased through and above ı, the leading edge of the wavefield steepens and becomes increasing complicated as both the Sommerfeld and Brillouin precursors increase in relative amplitude. Finally, for ˇT  ı, these two precursor fields dominate the entire wavefield evolution, as seen in part (b) of Fig. 15.88.

616

15 Continuous Evolution of the Total Field

a

b T = 0.2d

b

.0002

0

0.02

−.0002

0

.0005

b T = 0.5d

−0.02 0.04

Aht (z,t)

0 Aht (z,t)

b T = 3d

b T = 5d

0

0.002 0

−0.04 0.1

−0.002

0

bT = d

b T = 10d

0.01 b T = 2d

0

−0.1 0

−0.01 0

1 q 02

4

q

6

8

1q 0 2

4

q

6

8

10

10

Fig. 15.88 Numerically determined dynamical wavefield evolution of a hyperbolic tangent envelope modulated signal with intra-absorption band carrier frequency !c D 1:4375!0 at the fixed propagation distance z D 8:26zd in a single resonance Lorentz model dielectric. (a) Small to moderate values for several values of the initial rise-time parameter ˇT and (b) large values of ˇT

In summary, the analysis presented here has shown that the Sommerfeld and Brillouin precursor fields that are a characteristic of the dynamical field evolution in a single Lorentz model dielectric will persist nearly unchanged from their ideal behavior for a Heaviside step function envelope signal for values of the rise-time parameter ˇT for a hyperbolic tangent envelope signal that are on the order of ı or greater, where ı is the characteristic damping constant of the Lorentz model dielectric, or, equivalently, for values of the rise-time Tr  1=ˇT that are less than or equal to the characteristic relaxation time Tı  1=ı of the dispersive medium. Notice that this result can also be related to the slope of the envelope function which is equal to ˇT =2 at the 1=2 amplitude point. This inequality requires that the maximum initial rise-time in a Lorentz model dielectric with Brillouin’s choice of the medium parameters is given by Tı  1=ı D 0:357 fs.

15.7 Noninstantaneous Rise-Time Signals

617

15.7.2 Raised Cosine Envelope Signal Propagation in a Single Resonance Lorentz Model Dielectric Similar results are obtained [54] for the raised cosine envelope signal with envelope function 8 9 0; t 0 <  =  (15.145) urc .t / 12 1  cos .ˇr t / ; 0  t  Tr : ; 1; Tr  t with rise-time parameter ˇr D =Tr , where Tr is the signal rise-time. Unlike the hyperbolic tangent envelope signal with envelope function given in (15.125), which is nonzero for finite times t  0 when ˇT is finite, the canonical envelope function defined above in (15.145) is identically zero for all t  0 and, moreover, fully attains its steady-state amplitude in the finite rise-time Tr . The temporal frequency spectrum of this envelope function is found to be given by    !  2 ! 1 i i 2ˇ 1 r  : cos 2ˇr  uQ rc .!/ D e 2 ! ! C ˇr !  ˇr

(15.146)

In the limit as ˇr ! 1, the initial signal rise-time goes to zero (Tr ! 0) and the envelope function urc .t / defined in (15.145) goes over to the Heaviside unit step function, viz., (15.147) lim urc .t / D uH .t /; ˇr !1

its spectrum having the appropriate limit given by lim uQ rc .!/ D

ˇr !1

i : !

(15.148)

The opposite limit as ˇr ! 0, however, is of no interest for the raised cosine envelope signal, as the entire wavefield then vanishes. The temporal frequency spectrum of the initial raised cosine envelope signal f .t / D urc .t / sin .!c t / with fixed angular carrier frequency !c is given by uQ rc .!  !c / D

  i i 2ˇ .!!c / e r cos 2ˇ r .!  !c / 2  1 1 2 ;    !  !c !  !c C ˇr !  !c  ˇr (15.149)

which  points at ! D !c and ! D !c ˙ ˇr . Because the factor  has critical cos 2ˇ r .!  !c / appearing in (15.149) vanishes at ! D !c ˙ˇr , these two critical points are removable singularities. Hence, the only pole contribution to the asymptotic behavior of the propagated wavefield Arc .z; t / that is due to an input raised

618

15 Continuous Evolution of the Total Field

cosine envelope signal is from the simple pole singularity at ! D !c . This contribution, when it is the dominant contribution to the asymptotic wavefield behavior, yields the steady-state signal evolution that oscillates harmonically in time with the input angular signal frequency !c . The ratio of the finite rise-time envelope spectrum uQ rc .! !c / to its instantaneous limit uQ H .!  !c / is given by    uQ rc .!  !c / D e i 2ˇr .!!c / cos 2ˇ r .!  !c / uQ H .!  !c /   !  !c !  !c  1  !  !c C ˇr !  !c  ˇr     .!  !c /2 i 2ˇr .!!c / 1 ; .!  !c / 1 C

e 2ˇr ˇr2

(15.150)

where the final approxiamtion is valid for sufficiently large values of the rise-time parameter ˇr and finite values of the quantity !  !c . Because the closest that either the near or distant saddle point can approach the point ! D !c > ı is given by ı, the behavior of the spectrum uQ rc .! !c / about ! D !c is essentially unaltered from its instantaneous rise-time behavior if the inequality ˇr > ı

(15.151)

is satisfied [54]. This is the same approximate inequality obtained for the hyperbolic tangent envelope signal [see (15.132)]. In terms of the characteristic relaxation time Tı  1=ı of the single resonance Lorentz model medium and the initial rise-time Tr  1=ˇr of the raised cosine envelope signal, this inequality becomes Tr < Tı :

(15.152)

Consequently, for finite angular carrier frequencies !c > 0, the precursor fields that are a characteristic of the dynamical field evolution in a single Lorentz model dielectric will persist nearly unchanged from their ideal behavior for a Heaviside step function envelope signal for values of the rise-time parameter ˇr for a raised cosine envelope signal that are on the order of ı or greater, where ı is the characteristic damping constant of the Lorentz model dielectric, or, equivalently, for values of the rise-time Tr  1=ˇr that are less than or equal to the characteristic relaxation time Tı  1=ı of the dispersive medium. Notice that this result can also be related to the slope of the envelope function which is equal to ˇr =2 at the 1=2 amplitude point. These results are completely borne out by precise numerical calculations of the propagated wavefield behavior in a single resonance Lorentz model dielectric [54].

15.7 Noninstantaneous Rise-Time Signals

619

15.7.3 Trapezoidal Envelope Pulse Propagation in a Rocard–Powles–Debye Model Dielectric A canonical pulse envelope shape of central importance to both radar and communications systems is the trapezoidal envelope pulse with envelope function [see (11.68)] 9 8 0; t  T0 > ˆ > ˆ > ˆ tT0 > ˆ > ˆ ; T  t  T C T 0 0 r > ˆ Tr > ˆ < 1; T0 C Tr  t  T0 C Tr C T = (15.153) utrap .t / r CT / ; T0 C Tr C T  t 1  t.T0 CT > ˆ > ˆ Tf > ˆ > ˆ ˆ  T0 C T r C T C Tf > > ˆ > ˆ ; : 0; T0 C Tr C T C Tf  t with initial rise-time Tr > 0, fall-time Tf > 0, and peak amplitude pulse duration T > 0. The total initial pulse duration is then given by T C Tr C Tf and the half-amplitude pulse width is T C .Tr C Tf /=2, as illustrated in Fig. 11.6. This initial pulse type may be described by the time-delayed difference between a pair of trapezoidal envelope signals with envelope functions given by 8 < 0;

9 t  Tj 0 = tTj 0 ; T  t  T C T utrapj .t / j0 j0 j : Tj ; 1; Tj 0 C Tj  t

(15.154)

for j D r; f . The temporal angular frequency spectrum of this trapezoidal signal envelope function is then given by [see (11.69)] Z

1

uQ j .!/ D

uj .t /e i!t dt

1

  i sinc !Tj =2 e i!.Tj 0 CTj =2/ ; (15.155) !   where sinc. / sin . /= . Notice that sinc !Tj =2 ! ı.!/ in the limit as Tj ! 1, in which case the initial signal envelope spectrum becomes uQ trapj .!/ ! .i=!/ı.!/e .i!.Tj 0 CTj =2// , where ı. / denotes the Dirac delta function. A monochroD

matic, time-harmonic signal isthen obtained in this limiting case. In the opposite  limit as Tj ! 0, sinc !Tj =2 ! 1 and the initial signal envelope spectrum becomes uQ trapj .!/ ! .i=!/e .i!Tj 0 / , which describes the ultrawideband spectrum for a step function envelope signal. In general, the spectrum uQ j .!  !c / described by (15.155) for a trapezoidal envelope signal with carrier frequency !c > 0 will be ultrawideband if the in> equality 2=Tj  !c is satisfied. In that case, the ultrawideband spectral factor 1 .!  !c / will remain essentially unchanged over the positive angular frequency

620

15 Continuous Evolution of the Total Field

domain Œ0; 2!c , as depicted in Fig. 11.7 and illustrated in Fig. 11.8. This inequality is equivalent to the inequality < (15.156) Tj  Tc ; for j D r; f , where Tc 1=fc D 2=!c is the oscillation period of the carrier signal that is modulated by the trapezoidal envelope function given in (15.154). The inequality given in (15.156) provides a necessary condition for the dominant appearance of the Brillouin precursor in the propagated field structure in a Rocard–Powles–Debye model dielectric. Notice that this condition is independent of the dispersive material, unlike that for either a hyperbolic-tangent modulated signal [54], a gaussian envelope pulse [55–58], or a van Bladel envelope pulse [59,60]. The reason for this is quite simple: the hyperbolic tangent, gaussian, and van Bladel envelope functions are each continuous with continuous partial derivatives while the trapezoidal envelope function is continuous with a discontinuous first derivative at two points in time (t D Tj 0 and t D Tj ). These two discontinuities in the envelope slope will always yield a precursor contribution in a dispersive material, but that precursor contribution will only be significant in comparison to the pole contribution when the above inequality in (15.156) is satisfied. Notice further that the Heaviside step function signal is discontinuous in both its value and its first derivative at time t D Tj 0 , so that the trapezoidal envelope signal retains just the latter feature, albeit displaced in time by the initial rise-time Tr . These results are completely borne out by detailed numerical calculations, as illustrated in Fig. 15.89 showing the propagated symmetric (i.e., equal rise- and

0.1 0.05 Tr = Tf = Tc / 2

Atrap (z,t)

0 0.05

Tr = Tf = Tc

0 0.05

Tr = Tf = 2Tc

0

2.6

2.65 t (s)

2.7 x 10

–7

Fig. 15.89 Propagated symmetric (Tr D Tf ) trapezoidal envelope pulse structure with fc D 3 GHz carrier frequency at five absorption depths (z D 5zd ) in triply distilled water for Tj < Tc , Tj D Tc , and Tj > Tc , j D r; f

15.8 Infinitely Smooth Envelope Pulses

621

fall-times Tr D Tf ) trapezoidal envelope pulse structure with fc D 3 GHz carrier frequency at five absorption depths [z D 5zd with zd ˛ 1 .!c /] in triply distilled water for the three cases Tj < Tc , Tj D Tc , and Tj > Tc , j D r; f . Because the carrier wave period is given by Tc D fc1 D 3:33  1010 s and the characteristic relaxation time for water is given by 1 D 8:3  1012 s, each trapezoidal envelope case depicted in Fig. 15.89 satisfies the inequality Tj  1 , j D r; f , demonstrating that the Brillouin precursor fidelity is indeed independent of this dispersive material factor, in spite of the fact that the material dispersion is an essential ingredient in the appearance of the Brillouin precursor; notice that the Brillouin precursor is still present in the Tr D Tf D 2Tc case depicted in Fig. 15.89, accounting for the leading- and trailing-edge pulse distortion, but stretched out in time with an amplitude approximately equal to the signal amplitude. Rather, the fidelity of the Brillouin precursor in the dispersive material is governed by the inequality given in (15.156). Because the trapezoidal envelope pulse is the canonical pulse type upon which both pulsed radar and digital wireless telecommunication systems are based [61, 62], this result poses a special challenge regarding public health and safety. To avoid the formation of the Brillouin precursor upon penetration into the human body, the rise and fall times (as well as any other rapid amplitude changes) of both radar and wireless digital communication systems (as well as any other pulsed electromagnetic radiation emitters) should strictly satisfy the inequality Tj > 1=fc ;

j D r; f

(15.157)

where fc is the characteristic oscillation frequency of the radiated pulse.

15.8 Infinitely Smooth Envelope Pulses The final canonical pulse type of interest here is the infinitely smooth envelope pulse. Because of this smoothness property, this pulse type, and in particular the gaussian envelope pulse, is a favorite among the group velocity adherents. In spite of this, the asymptotic theory has much to explain about its ultrashort behavior in a causal, linear dispersive system that the group velocity approximation fails to provide.

15.8.1 Gaussian Envelope Pulse Propagation in a Single Resonance Lorentz Model Dielectric Because of its central importance in optics as well as the central role it plays in the group velocity description, the correct description of gaussian pulse propagation in a dispersive medium provides a unique challenge to both theoreticians and experimentalists alike. An accurate description of gaussian pulse propagation in a

622

15 Continuous Evolution of the Total Field

dispersive medium begins with the 1970 analysis of Garrett and McCumber [63] who considered pulse propagation in the anomalous dispersion regime. Their main result showed that the peak amplitude point can, under certain conditions, propagate at the classical group velocity even when it exceeds the vacuum speed of light c. Crisp [64] then argued that the observed superluminal group velocity was due to asymmetric absorption of energy from the light pulse. More energy is absorbed from the trailing half of the pulse than from the front half, causing the center of gravity of the pulse to move at a velocity greater than the phase velocity of light.

A decade later, Sherman and Oughstun [65] provided a detailed physical description of dispersive pulse dynamics based on Loudon’s [15] time-harmonic electromagnetic energy transport velocity in a Lorentz medium. Not only did this description explain each feature observed in the propagated pulse structure, it was also in complete keeping with relativistic causality. Shortly thereafter, Chu and Wong [66] presented experimental results for picosecond laser pulses propagating through thin samples of a linear dispersive dielectric whose peak absorption never exceeded six absorption depths, showing that the peak amplitude point of a gaussian light pulse travels in such a medium with the group velocity vg .!c / at the optical frequency even when vg .!c / > c, purporting to disprove the enrgy velocity description while verifying the group velocity description. An asymptotic description of the propagation of a gaussian wave packet in a Lorentz medium was then given by Tanaka, Fujiwara, and Ikegami [55] in 1986. They showed that the velocity of the wave packet, defined as the traveling distance of the peak amplitude divided by its flight time, decreases in the absorption range of frequency, although the group velocity becomes infinite in the same range

in agreement with the Sherman–Oughstun energy velocity description [65, 67]. In addition, Tanaka et al. concluded that [55] fast pulse propagation, which was observed by Chu and Wong and is characterized by a packet velocity faster than the light velocity, turns out to be a characteristic in the early stage of the flight and is understood in terms of packet distortion due to damping of Fouriercomponent waves in an anomalous dispersion region. It also turns out that slow pulse propagation characterized by a packet velocity less than the light velocity appears for long travelling distance.

Balictsis and Oughstun [55–58] then showed that the fast pulse component is nothing more than the Sommerfeld precursor and that the slow pulse component is just the Brillouin precursor. These final results provided a complete explanation of the apparent discrepancy between the energy and group velocity results for dispersive gaussian pulse propagation. Consider then an input gaussian envelope modulated harmonic wave f .t / D ug .t / sin .!c t C / with fixed angular carrier frequency !c > 0 and initial full pulse width 2T > 0 that is centered about the instant of time t0 at the plane z D 0, where [see (11.78)] 2 2 ug .t / D e .tt0 / =T ; (15.158)

15.8 Infinitely Smooth Envelope Pulses

623

which is propagating in the positive z-direction through a single resonance Lorentz model dielectric. The constant phase factor is used to adjust the location of the carrier wave with respect to the peak amplitude point in the gaussian envelope. Typically D =2 for an ultrashort pulse so that the carrier maximum coincides with the envelope maximum.

15.8.1.1

Classical Asymptotic Description

The exact, classical integral representation of the propagated gaussian pulse wavefield is given by [56] 1 < Ag .z; t / D 2

Z uQ g .!  !c /e

z 0 c .!; /

d!

(15.159)

C

for all z  0, with initial pulse spectrum uQ g .!/ D  1=2 T e T

2 ! 2 =4

e i.!c t0 C / ;

(15.160)

where

c (15.161)  0   t0 z is the shifted space–time parameter relative to the initial peak amplitude point of the gaussian envelope pulse. The contour of integration C is taken here as any contour in the complex !-plane that is homotopic to the real frequency axis extending from 1 to C1. Because this pulse envelope spectrum is an entire function of complex !, the propagated gaussian pulse wavefield has the representation [see (15.1)] Ag .z; t / D Ags .z; t / C Agb .z; t /

(15.162)

for all z  0, where the asymptotic behavior of the two component wavefields is given by [56, 57, 68]  Agj .z; t /  aj

c 2z

1=2

(

uQ .!SPj . 0 /  !c / z .!SP ; 0 / j < i

1=2 e c  00 .!SPj ;  0 /

) (15.163)

as z ! 1 for j D s; b. Here as D 2 and !SPs D !SPC . 0 / denotes the distant firstd

0 order saddle point SPC d of the complex phase function .!;  / in the right-half of 0 the complex !-plane for all  > 1, whereas ab D 1 for 1 <  0 < 1 and ab D 2 . 0 / denotes the near first-order saddle point SPC for  0 > 1 where !SPb D !SPC n n of the complex phase function .!;  0 / in the right-half of the complex !-plane. The nonuniform behavior exhibited in (15.163) in any small neighborhood of either the space–time point  0 D 1 or of the space–time point  0 D 1 may be corrected using the appropriate uniform asymptotic expansion procedure described in either

624

15 Continuous Evolution of the Total Field

Sect. 13.2.2 or Sect. 13.3.2, respectively. The gaussian pulse wavefield component Ags .z; t / is referred to as a gaussian Sommerfeld precursor field and the component Agb .z; t / is referred to as a gaussian Brillouin precursor field. Because of the form of the initial gaussian envelope spectrum uQ g .!/ given in (15.160), the asymptotic description of each gaussian pulse component Agj .z; t /, j D s; b, contains a gaussian amplitude factor of the form exp .T =2/2  2 i <.!SPj /  !c . In addition, each pulse component contains an exponential attenuation factor that is given by the product of the propagation distance z > 0 with the ˚material attenuation that is given by the real phase behavior  .!SPj ;  0 / D < .!SPj ;  0 / at the relevant saddle point, and the instantaneous angular oscillation frequency of each component in the mature dispersion ˚ pulse regime is approximately given by < !SPj in the ultrashort pulse limit as T ! 0.   q 2 2 Consequently, for a below absorption band carrier frequency !c 2 0; !0  ı , the instantaneous angular oscillation frequency ofqthe gaussian Brillouin precursor

Agb .z; t / crosses !c as it chirps upward toward !02  ı 2 , whereas for an above q  resonance carrier frequency !c 2 !12  ı 2 ; 1 , the instantaneous angular oscillation frequency of the gaussian Sommerfeld precursor Ags .z; t / crosses !c as q 2 it chirps downward toward !1  ı 2 , in each case the gaussian amplitude factor ˚ peaking to unity when < !SPj D !c . For an intra-absorption band angular car q q !02  ı 2 ; !12  ı 2 the carrier frequency value is never rier frequency !c 2 attained by either pulse component [56]. If the input angular signal frequency !c is in  the medium absorption  band q q 2 2 2 2 !0  ı ; !1  ı , then where the dispersion is anomalous, so that !c 2 both gaussian pulse components Ags .z; t / and Agb .z; t / will be present in the propagated waveform in roughly equal proportion. The Brillouin precursor component q

Agb .z; t / becomes more pronounced as !c is decreased from !12  ı 2 !1 to q !02  ı 2 !0 and dominates the propagated gaussian pulse evolution as !c is decreased into the normally dispersive region below the medium resonance frequency, whereas the Sommerfeld precursor q q Ags .z; t / becomes more pronounced as

!c is increased from !02  ı 2 !0 to !12  ı 2 !1 and dominates the propagated gaussian pulse evolution as !c is increased into the normally dispersive region above the medium absorption band. As an illustration, the numerically determined dynamical wavefield evolution due to an input ultrashort gaussian envelope pulse with initial pulse width 2T D 0:2 fs and carrier frequency !c D 5:75  1016 r=s that is near the upper end of the absorption band of a single resonance Lorentz model dielectric with Brillouin’s medium parameters is illustrated in Fig. 15.90 for several values of the relative propagation distance z=zd D ˛.!c /z. This case is of particular interest because the group velocity vg .!/ D .@ˇ.!/=@!/1 at this carrier frequency

15.8 Infinitely Smooth Envelope Pulses X 10

−2

X 10−3 z /zd = 20.66

0

z/zd = 61.98

1 Ag(z,t)

Ag(z,t)

2.5

625

0 −1

−2.5 1

3

4

5

1 q

X 10−3

6

z /zd = 41.32

0

−5 1

X 10−4

3

q

4

5

z/zd = 82.64

0

−6

q0 2

2.8

2

q

Ag(z,t)

Ag(z,t)

5

q0 2

q0

1

q0 q

2

2.8

Fig. 15.90 Numerically determined dynamical wavefield evolution due to a 0:2 fs gaussian envelope pulse with intra-absorption band carrier frequency !c 2 .!0 ; !1 / satisfying vg .!c / ' c in a single resonance Lorentz model dielectric

in this medium is very nearly equal to the speed of light c in vacuum. The gaussian Sommerfeld precursor component is seen to first emerge from the propagated pulse structure as the propagation distance increases into the mature dispersion regime, its peak amplitude point traveling with a velocity just below c, as seen in the 21 and 41 absorption depth cases in the figure. As the propagation distance continues to increase, the gaussian Brillouin precursor component of the pulse emerges, its peak amplitude point traveling with a velocity that approaches the value c=0 D c=n.0/ from above as z ! 1. The propagated wavefield Ag .z; t / due to an ultrashort gaussian envelope pulse then separates (or bifurcates) into two distinct pulse components that propagate with different peak velocities, the faster pulse component being the high-frequency gaussian Sommerfeld precursor Ags .z; t / with instantaneous angular oscillation frequency !s . 0 / that chirps downward toward !1 as the space–time parameter  0 increases, followed by the slower, low-frequency gaussian Brillouin precursor Agb .z; t / with instantaneous angular oscillation frequency !b . 0 / that chirps upward toward !0 as  0 increases, in complete agreement with the asymptotic results presented by Tanaka, Fujiwara, and Ikegami [55]. Notice that this gaussian pulse bifurcation is a linear phenomenom and that, for a multiple resonance Lorentz model dielectric, the pulse can separate into as many subpulses as there are precursor fields; for a double resonance Lorentz model dielectric, the gaussian pulse can separate into three subpulses (a Sommerfeld, middle, and Brillouin

626

15 Continuous Evolution of the Total Field

precursor pulse) when the inequality given in (12.117) is satisfied. Each feature of this dynamical pulse evolution is properly described by the Sherman–Oughstun energy velocity description [65, 67]. As the initial pulse width 2T is increased, the asymptotic approximation of the Sommerfeld and Brillouin pulse components given in (15.163) for the propagated gaussian pulse wavefield representation given in (15.162) remains qualitatively correct while its quantitative accuracy decreases at any fixed, finite propagation distance z > 0. This nonuniform asymptotic description (as well as its uniform counterpart) will remain quantitatively accurate as the initial pulse width is increased provided that the propagation distance is also allowed to increase, in keeping with the definition of an asymptotic expansion in Poincar´e’s sense as z ! 1 (see Definition 5 of Appendix F). However, because the medium is attenuative, the usefulness of this description also decreases as 2T increases as the important features of the dynamical pulse evolution (particularly when compared to experimental observations) are typically measured at some fixed observation distance in the medium.

15.8.1.2

Modified Asymptotic Description

The classical integral representation of gaussian pulse propagation given in (15.159)–(15.161) may be rearranged so as to yield the modified integral representation [55, 58] Ag .z; t / D

Z z 1 0 e c ˚m .!; / d! < i UQ m 2 C

(15.164)

for all z  0, where UQ m  1=2 T e i.!c t0 C

/

(15.165)

is independent of the angular frequency !, and where ˚m .!;  0 / .!;  0 / 

cT 2 .!  !c /2 4z

(15.166)

is the modified complex phase function which depends not only on the dispersive properties of the host medium, but also on the initial pulse width 2T and carrier frequency !c as well as upon the propagation distance z > 0. In the ultrashort pulse limit as 2T ! 0, the modified phase function ˚m .!;  0 / reduces to the classical phase function .!;  0 / D i !.n.!/   0 / so that the asymptotic behavior of the modified integral representation given in (15.164) is determined by the behavior of the integrand about the saddle points of .!;  0 /, as in (15.162) and (15.163). This then establishes the following scaling law for gaussian pulse propagation [57]: if the classical asymptotic description given in (15.162) and (15.163) is valid (to some specifis degree of accuracy) for some given input pulse width 2T at a given propagation

15.8 Infinitely Smooth Envelope Pulses

627

distance z, then this description will remain equally valid (to that same degree of accuracy) as the initial pulse width is increased provided that z is also increased in such a manner that the ratio T 2 =z remains fixed.

The saddle point dynamics of the modified phase function ˚m .!;  0 / are now considered based on the analyses of Tanaka et al. [55] and Balictsis et al. [58, 69, 70]. Although the complex phase function .!;  0 / satisfies the symmetry relation   .!  ;  0 / D .!;  0 /, the modified phase function does not, viz., ˚m .!  ;  0 / ¤ ˚m .!;  0 /:

(15.167)

As a consequence, the modified complex phase behavior, as well as its saddle points, are not symmetric about the imaginary axis. Nevertheless, the branch cuts remain determined by the complex index of refraction and so are still the symmetric line 0 segments !0 ! and !C !C about the imaginary axis (see Fig. 12.1). Numerical results show that there are five first-order saddle points SPmk , k D 1; 2; : : : ; 5, for a single resonance Lorentz model dielectric with respective locations !SPmk . 0 / that remain isolated from each other over the entire space–time domain  0 2 .1; 1/. The dynamical evolution of these saddle points in the complex !-plane is illustrated in Fig. 15.91 as a function of the space–time parameter  0 for a 2T D 0:2 fs gaussian envelope pulse with D =2 and !c D 5:75  1016 r=s angular carrier frequency at 83:92 absorption depths of a single resonance Lorentz model dielectric with Brillouin’s choice of the medium parameters.6 The space–time points rcmj at which these saddle points may cross the real ! 0 -axis are defined by the condition !SPmj .rcmj / D !rcmj ;

j D 1; 2; : : : ; 5;

(15.168)

−

0 −

−30 –15

SPm1

SPm2

SPm5

16

'' (10 r/s)

30

+ +

SPm4

SPm3

rcm5

rcm1

rcm2 c

0

15

16

' (10 r/s)

Fig. 15.91 Dynamical evolution of the five first-order saddle points SPmj , j D 1; 2; : : : ; 5, of the modified complex phase function ˚m .!;  0 / in the complex !-plane as a function of the space– time parameter  0 for a 2T D 0:2 fs gaussian envelope pulse with D =2 and !c D 5:75  1016 r=s angular carrier frequency at 83:92 p absorption depths of a single resonance Lorentz model dielectric with !0 D 4  1016 r=s, b D 20  1016 r=s, and ı D 0:28  1016 r=s 6

Notice that, for reasons of consistency, the j D 1 and j D 2 saddle points SPmj are interchanged from that used in [58, 69, 70].

628

15 Continuous Evolution of the Total Field

where the angular frequency value !rcmj is real-valued. As seen in Fig. 15.91, only the saddle points SPm1 , SPm2 , and SPm5 satisfy this condition, where [58] !rcm1 ' C9:0261  1016 r=s; !rcm2 ' C1:4220  1016 r=s; !rcm5 ' 7:3953  1016 r=s;

rcm1 ' 1:2831 rcm2 ' 1:6871 rcm5 ' 1:7036:

The integration contour C is then deformed into a new path P . 0 / that passes through all of the accessible saddle points of the modified phase function at any given space–time value  0 in such a manner that it may be partitioned into a continuous chain of component subpaths Pmj . 0 /, each an Olver-type path with respect to its corresponding saddle point SPmj . Under this transformation, the modified integral representation (15.164), of the propagated gaussian envelope pulse takes the form X Agj .z; t / (15.169) Ag .z; t / D j

with

( ) Z z 1 0/ ˚ .!; m Q < i Um ec d! Agj .z; t / D 2 Pmj . 0 /

(15.170)

for all z  0. Because the modified complex phase function ˚m .!;  0 / now explicitly depends upon the propagation distance z, the first condition of Olver’s theorem (see Sect. 10.1.1) is not satisfied. This condition serves to ensure that the phase function does not vanish as z ! 1. However, because lim ˚m .!;  0 / D .!;  0 /;

(15.171)

z!1

where the classical complex phase function .!;  0 / strictly satisfies all of the conditions of Olver’s theorem, then the first condition of Olver’s theorem may be relaxed for the modified complex phase function case considered here. Application of Olver’s theorem to each integral in (15.170) then gives  Agj .z; t / 

c 2z

(

1=2 <



i UQ m ˚m00 .!SPmj ;  0 /

1=2 e

z c ˚m .!SPmj

; 0 /

) (15.172)

as z ! 1. Detailed numerical results [58, 69, 70] show that only the two saddle points SPm1 and SPm2 , whose dynamical evolution lies in the right-half of the complex !-plane, become a dominant saddle point during the entire space–time domain of interest, each in its respective space–time interval mj , j D 1; 2. The asymptotic representation of the propagated gaussian envelope pulse is then given by Ag .z; t / D Ag1 .z; t / C Ag2 .z; t /;

(15.173)

15.8 Infinitely Smooth Envelope Pulses

629

where lim Ag1 .z; t / D Ags .z; t /;

(15.174)

lim Ag2 .z; t / D Agb .z; t /I

(15.175)

z!1 z!1

that is, the pulse component Ag1 .z; t / corresponds to the gaussian Sommerfeld precursor Ags .z; t / and the pulse component Ag2 .z; t / corresponds to the gaussian Brillouin precursor Agb .z; t /. Each pulse component Agj .z; t / contains a gaussian amplitude factor that is contained in the modified complex phase function ˚m .!SPmj ;  0 / appearing in the exponential factor of (15.172). To explicitly display this, (15.166) may be expressed as ˚m .!;  0 / D .!;  0 /  g .!/;

(15.176)

where

cT 2 .!  !c /2 (15.177) 4z accounts for the gaussian amplitude factor in each pulse component. The angular frequency dependence of the real part m .! 0 / D
0

'' (1016r/s)

g m

' '

'

m

'

−5

rcm5

rcm1

rcm2

−10 –15

0

c

15

' (1016r/s)

Fig. 15.92 Frequency dependence of the real part m .! 0 / D
630

15 Continuous Evolution of the Total Field

83:92 absorption depths of a single resonance Lorentz model dielectric. The relative maxima of m .! 0 / are found to occur at the real ! 0 -axis crossing points !rcmj , j D 1; 2; 5 that are defined in (15.168), so that ˇ @ .! 0 / ˇˇ @! 0 ˇ! 0 D!rc

D 0;

(15.178)

mj

for j D 1; 2; 5, as indicated in the figure. The asymptotic behavior of each of the two pulse components Ag1 .z; t / D Ags .z; t / and Ag2 .z; t / D Agb .z; t /, given in (15.172), is the same as that obtained from the modified integral representation for Ag .z; t / given in (15.164), taken along the real ! 0 -axis, when the modified phase function is replaced by its quadratic approximation about each of the two angular frequency values !rcm1 and !rcm2 along the positive real frequency axis, as shown by Balictsis [69]. Each of these angular frequency values is, in general, different from the angular carrier frequency !c of the initial gaussian envelope pulse at z D 0. If !c is above the absorption band, then !rcm1 approaches !c in the limit as 2T ! 1 at fixed z > 0, viz., q lim !rcm1 D !c >

2T !1

!12  ı 2 :

(15.179)

In that case, the propagated optical wavefield given in (15.173) becomes dominated by the gaussian Sommerfeld pulse component as the initial pulse width increases, so that Ag .z; t /  Ags .z; t /. On the other hand, if !c is below the medium absorption band, then !rcm2 approaches !c in the limit as 2T ! 1 at fixed z > 0, viz., lim !rcm2 D !c <

2T !1

q !02  ı 2 :

(15.180)

In that case, the propagated optical wavefield given in (15.173) becomes dominated by the gaussian Brillouin pulse component as the initial pulse width increases, so that Ag .z; t /  Agb .z; t /. In this manner, the classical group velocity result is obtained in this large initial gaussian pulse width limit when the carrier frequency is in either of the normally dispersive regions above or below the anomalously dispersive medium absorption band. The transition from the ultrashort to the quasimonochromatic pulse regime is more involved when the carrier frequency lies in the anomalous dispersion region of the medium absorption band. Because the initial pulse spectrum is now located between the high-frequency domain that gives rise to the Sommerfeld precursor and the low-frequency domain that gives rise to the Brillouin precursor, the critical initial pulse width 2T at which the propagated pulse separates into two subpulses is minimized. As an illustration, consider the propagation of a 2T D 0:2 fs gaussian envelope pulse centered at t0 D 15 T with D =2 and !c D 5:75  1016 r=s angular carrier frequency that is near the upper end of the absorption band p of a single resonance Lorentz model dielectric with !0 D 4  1016 r=s, b D 20  1016 r=s, and ı D 0:28  1016 r=s, where vg .!c / ' c. The propagated pulse wavefield at z D 83:92zd , computed using the modified asymptotic description described in

15.8 Infinitely Smooth Envelope Pulses

631

0.001 Ags(z,t)

Agb(z,t)

qpm1

qpm2

Ag(z,t)

Apm1 0

Apm2 −0.001 1

1.5

'

2

2.5

3

Fig. 15.93 Dynamical space–time evolution of the propagated optical wavefield due to a 2T D 0:2 fs gaussian envelope pulse with D =2 and !c D 5:75  1016 r=s angular carrier frequency 16 at 83:92 p absorption depths in a single resonance Lorentz model dielectric with !0 D 4  10 r=s, 16 16 b D 20  10 r=s, and ı D 0:28  10 r=s

15.172 and 15.173 with numerically determined saddle point locations, is illustrated in Fig. 15.93. The resultant waveform compares very well with the FFT-based numerical calculation of the propagated gaussian pulse waveform at z D 82:64zd that is presented in the final graph of Fig. 15.90. The gaussian Sommerfeld precursor pulse component Ag1 .z; t / D Ags .z; t / is found to be the dominant pulse component over the initial space–time domain  0 < 1:455 when the saddle point SPm1 is dominant. The peak amplitude point in the envelope of this pulse component occurs at the space–time point  D ps ' 1:256, so that ps ' rcm1 ;

(15.181)

at which point the propagated pulse wavefield is oscillating at the instantaneous angular frequency !s D !ps ' 9:2669  1016 r=s, so that [see (15.168)] !ps ' !rcm1 D !SPm1 .rcm1 /:

(15.182)

That is, the peak amplitude in the envelope of the gaussian Sommerfeld pulse component approximately occurs at the space–time point when the saddle point SPm1 crosses the real ! 0 -axis and the pulse wavefield at that point is oscillating with an instantaneous angular frequency that is approximately equal to the real coordinate value at which this saddle point crosses the ! 0 -axis. The gaussian Brillouin precursor pulse component Ag2 .z; t / D Agb .z; t / is found to be the dominant pulse component over the final space–time domain  0 > 1:455 when the saddle point SPm2 is dominant. The peak amplitude point in the envelope of this pulse component occurs at the space–time point  D pb ' 1:6724, so that pb ' rcm2 ;

(15.183)

632

15 Continuous Evolution of the Total Field

at which point the propagated pulse wavefield is oscillating at the instantaneous angular frequency !b D !pb ' 1:3534  1016 r=s, so that !pb ' !rcm2 D !SPm2 .rcm2 /:

(15.184)

That is, the peak amplitude in the envelope of the gaussian Brillouin pulse component approximately occurs at the space–time point when the saddle point SPm2 crosses the real ! 0 -axis and the pulse wavefield at that point is oscillating with an instantaneous angular frequency that is approximately equal to the real coordinate value at which this saddle point crosses the ! 0 -axis. The numerically determined space–time evolution of the instantaneous angular oscillation frequency of the propagated gaussian pulse wavefield illustrated in Fig. 15.93 is described by the C signs in Fig. 15.94. The two solid curves in the figure depict the respective  0 -evolution of the real parts of the locations of the two saddle points SPm1 and SPm2 . It is then seen that the instantaneous oscillation frequency for each pulse component is given by the real part of the respective saddle point location describing that pulse component over its respective space–time domain, so that ˚ !s . 0 / D < !SPm1 . 0 / ; ˚ !b . 0 / D < !SPm2 . 0 / :

(15.185) (15.186)

Because the angular carrier frequency !c of the initial gaussian pulse is situated in the medium absorption band and because < f!SPm1 . 0 /g approaches the upper end of the absorption band from above as  0 ! 1 (see Fig. 15.91), then !s . 0 / > !c . Similarly, because < f!SPm2 . 0 /g approaches the lower end of the absorption band from below as  0 ! 1, then !b . 0 / < !c .

'SP ( ' ) m1

16

(X10 r/s)

15

1/2

(

)

(

)

c 1/2

'SP ( ' ) m2

0 –10

–5

0

5

10

'

Fig. 15.94 space–time evolution of the instantaneous angular frequency of oscillation of the propagated optical wavefield illustrated in Fig. 15.93

15.8 Infinitely Smooth Envelope Pulses

15.8.1.3

633

Comparison of the Group and Peak Amplitude Velocities

Attention is now turned to the propagation velocity of a gaussian envelope pulse in a single resonance Lorentz model dielectric, particularly in the region of anomalous dispersion where the group velocity vg .!/ D .@ˇ.!/=@!/1 can exceed c and even become negative. The angular frequency dependence of the inverse of the relative group velocity (i.e., the relative group delay) is depicted by the solid curve in Fig. 15.95. The data values indicated in Fig. 15.95 describe the numerically determined relative peak amplitude velocity values for the following gaussian envelope pulse cases [58, 69]: 1: 2: 3: 4: 5: 6: 7: 8: 9: 10: 11: 12: 13: 14:

!c !c !c !c !c !c !c !c !c !c !c !c !c !c

D 5:75  1016 r=s; D 5:75  1016 r=s; D 5:75  1016 r=s; D 1:00  1016 r=s; D 3:14  1016 r=s; D 4:00  1016 r=s; D 10:0  1016 r=s; D 5:75  1016 r=s; D 5:75  1016 r=s; D 5:625  1016 r=s; D 5:625  1016 r=s; D 5:25  1016 r=s; D 5:25  1016 r=s; D 5:00  1016 r=s;

2T 2T 2T 2T 2T 2T 2T 2T 2T 2T 2T 2T 2T 2T

D 0:20 fs; D 2:00 fs; D 20:0 fs; D 2:00 fs; D 2:00 fs; D 2:00 fs; D 2:00 fs; D 2:00 fs; D 2:00 fs; D 5:00 fs; D 5:00 fs; D 10:0 fs; D 10:0 fs; D 20:0 fs;

z D 83:92 zd D 1 m z D 83:92 zd D 1 m z D 83:92 zd D 1 m z D 0:543 zd D 1 m z D 22:50 zd D 1 m z D 266:6 zd D 1m z D 3:020 zd D 1 m z D 8:392 zd D 0:1 m z D 839:2 zd D 10:0 m z D 19:96 zd D 0:2 m z D 49:91 zd D 0:5 m z D 58:05 zd D 0:4 m z D 145:13 zd D 1:0 m z D 176:13zd D 1:0 m

The numerical value of the space–time point pj D c=vpj , j D 1; 2, at which the peak amplitude in the envelope of each pulse component of the propagated gaussian pulse wavefield Ag .z; t / is plotted in Fig. 15.94 at the corresponding value of the instantaneous angular oscillation frequency !pj at that space–time point for each of these cases. The data points indicated by the open squares in the figure describe purely numerical results whereas the data points indicated by the C signs describe results obtained from the modified asymptotic theory. Both sets of results lie exactly along the group delay curve c=vg .!/, showing that the group velocity does indeed describe the velocity of the peak amplitude point for each pulse component of the propagated gaussian envelope pulse. The sequence of red squares (cases 12 and 13) illustrate the dynamical evolution of the peak amplitude velocity for a 10 fs gaussian envelope pulse with initial angular carrier frequency !c D 5:25  1016 r=s where the group velocity is negative with magnitude less than c. The sequence of green squares (cases 10 and 11) illustrate the dynamical evolution of the peak amplitude velocity for a 5 fs gaussian envelope pulse with initial angular carrier frequency !c D 5:625  1016 r=s where the group velocity is superluminal. The sequence of blue squares (cases 8, 2, and 9) illustrate the dynamical evolution of the peak amplitude velocity for a 2 fs gaussian envelope pulse with initial angular carrier frequency !c D 5:751016 r=s where the group velocity is approximately equal to c.

634

15 Continuous Evolution of the Total Field 8

c/vg

c/v

1 0

c/vg

−8 0

7.5

15

16 pj ( 10 r/s)

Fig. 15.95 Inverse of the peak amplitude velocity for gaussian p pulse propagation in a single resonance Lorentz model dielectric with !0 D 41016 r=s, b D 201016 r=s, and ı D 0:281016 r=s. The solid curve describes the relative group phase delay c=vg .!/ in the dispersive medium. (From Balictsis and Oughstun [58])

The dispersive action of the single resonance Lorentz model dielectric on a 5 fs gaussian envelope pulse with carrier frequency !c D 5:625  1016 r=s just below the upper end of the medium absorption band, results in a superluminal velocity of the peak in the envelope of the propagated pulse at a sufficiently small propagation distance, as illustrated in the middle field plot of Fig. 15.96 corresponding to the superluminal velocity case 10 in Fig. 15.95. The envelope peak in the propagated wavefield at this propagation distance (approximately 20 absorption depths) has the associated instantaneous angular oscillation frequency !ps ' 5:71  1016 r=s > !c and travels at the classical group velocity vps D vg .!ps / ' 1:16c. This peak in the pulse envelope then slows down to a subluminal velocity (case 11 in Fig. 15.95) as the propagation distance increases because the instantaneous oscillation frequency at this peak amplitude point increases as the propagation distance increases. The peak amplitude point in the envelope of the propagated gaussian pulse at this larger propagation distance (approximately 50 absorption depths), illustrated in the bottom field plot of Fig. 15.96, has now shifted to the higher instantaneous angular oscillation frequency !ps ' 5:83  1016 r=s and now travels at the classical group velocity value vps D vg .!ps / ' 0:65c. Thus, as illustrated by the green squares in Fig. 15.95, the instantaneous oscillation frequency at the peak amplitude point in the pulse envelope evolves out of the material absorption band and the pulse dynamics evolve toward the energy velocity description as the propagation distance increases.

15.8 Infinitely Smooth Envelope Pulses

635

Ag(z,t)

1

z=0

0

−1 −10fs

Ag(z,t)

8 X 10

−5fs

0 t' = (t − t 0)

5fs

−9

10fs

z = 19.96zd = 0.2mm

0

−9

– 8 X 10 −10fs

t ps = 0.5767fs −5fs

z/c

0 t' = (t − t 0)

−19

Ag(z,t)

2X10

5fs

10fs

z = 49.91zd = 0.5mm

0

z /c

−19

t ps = 2.5655fs

−2X10

−10fs

−5fs

0

5fs

10fs

t' = (t − t 0)

Fig. 15.96 Numerically determined dynamical pulse evolution for a 2T D 5:0 fs, !c D 5:625  1016 r=s gaussian envelope pulse at (from top to bottom) z D 0, z D 19:96zdp, and z D 49:91zd in a single resonance Lorentz model dielectric with !0 D 4  1016 r=s, b D 20  1016 r=s, and ı D 0:28  1016 r=s, where zd ˛ 1 .!c /. The solid vertical line in the bottom two wavefield plots marks the retarded instant of time t 0 D z=c when the peak pulse amplitude would have arrived at that propagation distance if it had traveled with the speed of light c in vacuum, and the vertical dashed line marks the actual retarded instant of time t 0 D tps when the peak pulse amplitude point actually arrives at that propagation distance. The middle field plot corresponds to the superluminal velocity case 10 and the bottom diagram to the subluminal velocity case 11 in Fig. 15.95. (From Balictsis and Oughstun [58])

636

15 Continuous Evolution of the Total Field

Negative velocity movement of the peak amplitude point are obtained for the case of a 10 fs gaussian envelope pulse with intra-absorption band carrier frequency !c D 5:25  1016 r=s, as indicated by the red squares (cases 12 and 13) in Fig. 15.95. At the smallest (nonzero) propagation distance considered for this case (approximately 58 absorption depths), illustrated in the middle field plot of Fig. 15.97, the peak in the envelope of the propagated pulse has the associated instantaneous oscillation frequency !ps ' 5:291016 r=s > !c and travels at the classical group velocity vps D vg .!ps / ' 2:86c. As the propagation distance is increased to approximately 145 absorption depths, illustrated in the bottom field plot of Fig. 15.97, the instantaneous oscillation frequency at the space–time point where the peak in the pulse envelope occurs shifts to the higher angular frequency value !ps ' 5:35  1016 r=s and now travels at the classical group velocity vps D vg .!ps / ' 4:45c. As the propagation distance continues to increase, the low-frequency components that are present in the initial pulse spectrum are attenuated at a larger rate than are the highfrequency components, so that the propagated pulse spectrum becomes dominated by an increasingly higher frequency component, the peak in the envelope of the propagated pulse propagating with the classical group velocity at this frequency value. Again, as the propagation distance increases into the mature dispersion regime, the pulse dynamics evolve toward that described by the energy velocity description; however, the overall field amplitude also rapidly attenuates to zero in this case. Similar results are obtained for each of the subluminal group velocity cases depicted in Fig. 15.94. For example, the sequence of blue squares (cases 8, 2, and 9) show this shift away from the absorption band for a 2 fs gaussian envelope pulse with intra-absorption band carrier frequency !c D 5:751016 r=s where vg .!c / ' c. The dependence of the peak amplitude velocity on the initial gaussian envelope pulse width 2T for fixed carrier frequency !c D 5:75  1016 r=s and propagation distance z D 83:92zd D 1m is displayed by cases 1, 2, and 3 in Fig. 15.95, illustrating the manner in which the pulse dyamics change as the pulse becomes ultrawideband. Because of the small propagation distance of at most six absorption depths in their laboratory arrangement, the experimental results of Chu and Wong [66] are restricted to the small propagation distance limit below the mature dispersion regime. The modified asymptotic description introduced by Tanaka et al. [55] in 1986 and then fully developed by Balictsis et al. [58, 70] in the 1990s bridges the gap between these two regimes, being in agreement with the experimental results [66] at small propagation distances (i.e., in the immature dispersion regime) and reducing to the classical asymptotic description at sufficiently large propagation distances (i.e., in the mature dispersion regime) in the dispersive, attenuative medium [57]. Moreover, the modified asymptotic description provides a mathematically rigorous derivation of the correct group velocity description of gaussian pulse propagation in a dispersive, attenuative medium and clearly shows how that description evolves into the Sherman–Oughstun energy velocity description [65, 67] as the propagation distance increases into the mature dispersion regime.

15.8 Infinitely Smooth Envelope Pulses

637

Ag(z,t)

1

0

−1 −10fs 2X10

z=0

−5fs

0 t' = (t − t 0)

5fs

10fs

−25

Ag(z,t)

z = 58.05zd = 0.2mm

–2X10

0

−25

−10fs 35X10

tps = −0.4662fs −5fs

0 t' = (t − t 0)

z/c 5fs

10fs

−62

Ag(z,t)

z = 145.13zd = 0.2mm

–35X10

−62

−10fs

tps = −0.7505fs

z/c

−5fs

5fs

0 t' = (t − t 0)

10fs

Fig. 15.97 Numerically determined dynamical pulse evolution for a 2T D 10:0 fs, !c D 5:25  1016 r=s gaussian envelope pulse at (from top to bottom) z D 0, z D 58:05zdp , and z D 145:13zd in a single resonance Lorentz model dielectric with !0 D 4  1016 r=s, b D 20  1016 r=s, and ı D 0:28  1016 r=s, where zd ˛ 1 .!c /. The solid vertical line in the bottom two wavefield plots marks the retarded instant of time t 0 D z=c when the peak pulse amplitude would have arrived at that propagation distance if it had traveled with the speed of light c in vacuum, and the vertical dashed line marks the actual retarded instant of time t 0 D tps when the peak pulse amplitude point actually arrives at that propagation distance. The middle field plot corresponds to the negative velocity case 12 and the bottom diagram to the negative velocity case 13 in Fig. 15.95. (From Balictsis and Oughstun [58])

638

15 Continuous Evolution of the Total Field

15.8.2 Van Bladel Envelope Pulse Propagation in a Double Resonance Lorentz Model Dielectric The major difficulty with the gaussian pulse is that its envelope function is strictly nonzero for all finite time except in the vanishing pulse width limit 2T ! 0 when the initial pulse amplitude is inversely proportional to 2T . An important example of an infinitely smooth pulse envelope with compact temporal support is provided by the Van Bladel envelope pulse Avb .0; t / D uvb .t / sin .!c t C / with envelope function [see (11.76)] ( uvb .t /



e

2

1C 4t.t  /

0I



I

when 0 < t <  ; when either t  0 or t  

(15.187)

p with temporal duration  > 0 and full pulse width = 2 at the e 1 amplitude points in the envelope function, as illustrated in Fig. 11.3 for a two-cycle pulse ( D 2Tc ) and in Fig. 11.14 for a ten-cycle pulse ( D 10Tc ), with Tc 1=fc D 2=!c for a cosine carrier wave ( D =2). Because the envelope function uvb .t / vanishes identically outsde of the finite time interval .0;  /, its Fourier transform uQ vb .!/ is an entire function of complex !. Its resultant propagated wavefield in a double resonance Lorentz model dielectric is then given by Avb .z; t / D Avbs .z; t / C Avbm .z; t / C Avbb .z; t /

(15.188)

for all t  z=c with z > 0, the propagated wavefield identically vanishing for all t < z=c. The Sommerfeld precursor pulse component Avbs .z; t / is due to the distant saddle points SP˙ d , the middle precursor pulse component Avbm .z; t / is due to the middle saddle points SP˙ m1 , and the Brillouin precursor pulse component Avbb .z; t / is due to the near saddle points SP˙ n . The dynamical field evolution when the angular carrier frequency !c is set equal to the value !min at the minimum dispersion point in the passband between the two absorption bands of resonance Lorentz model dielectric is presented in Figs. 11.31–11.35.

15.8.3 Brillouin Pulse Propagation in a Rocard–Powles–Debye Model Dielectric; Optimal Pulse Penetration A problem of particular practical importance is the determination of the structural form of the input pulse that will best penetrate a finite distance into a given dispersive dielectric. The results presented in Figs. 15.70 and 15.71 indicate that the pulse that will provide near-optimal, if not indeed optimal, penetration is comprised of a pair of Brillouin precursor structures with the second precursor delayed in time and  phase shifted from the first. This so-called Brillouin pulse is obtained from

15.8 Infinitely Smooth Envelope Pulses

639

(13.145) with z D zd D ˛ 1=2 .!c / in the exponential, the other factors not appearing in the exponential set equal to unity, and is given by [32]    .!N .T /; T / .!N ./; /  exp ; fBP .t / D exp !c ni .!c / !c ni .!c / 

(15.189)

where T   cT=zd with T > 0 describing the fixed time delay between the leading and trailing-edge Brillouin precursors. If T is chosen too small then there will be significant destructive interference between the leading and trailing-edges and the pulse will be rapidly extinguished. For practical reasons, 2T should be chosen near to the inverse of the operating frequency fc of the antenna used to radiate this Brillouin pulse. With T D 1=.2fc / the input Brillouin pulse is approximately a single-cycle pulse with effective oscillation frequency equal to fc . The input Brillouin pulse when fc D 1 GHz is depicted in Fig. 15.98; part (a) of the figure shows the separate leading and trailing-edge Brillouin precursor structures and part (b) shows the final pulse obtained from the superposition of these two parts, as described by (15.189). The initial rise and fall time for this pulse is  0:6 ns. The dynamical evolution of this input Brillouin pulse in triply distilled water is illustrated by the pulse sequence given in Fig. 15.99 with z=zd D 0; 1; 2; : : : ; 10. Comparison with the pulse sequence 1

A1 & −A2

0.5 0 −0.5 −1 5

5.5

6

6.5

7

7.5

8

8.5 −9 x 10

7

7.5

8

8.5 x 10−9

t (s) 1

ABP1(z,t)

0.5 0 −0.5 −1

5

5.5

6

6.5

t (s)

Fig. 15.98 Temporal structure of the Brillouin pulse BP1 with time delay T D 1=.2fc / for fc D 1 GHz. The separate leading and trailing-edge precursor components are illustrated in (a) and their superposition is given in (b)

640

15 Continuous Evolution of the Total Field 1

z/zd = 0

0.8

1 2

0.6

3

4

0.4

5

6

7

8

9

ABP1(z,t)

0.2

10

0 −0.2 −0.4 −0.6 −0.8 −1 0

1

2

3

4 t (s)

5

6

7 −8 x 10

Fig. 15.99 Propagated pulse sequence for the Brillouin pulse BP1 with delay time T D 1=.2fc / for fc D 1 GHz in the simple Rocard-Powles-Debye model of triply-distilled water

depicted in Fig. 15.70 for a 1 GHz single-cycle rectangular envelope pulse shows that the Brillouin pulse decays much slower with propagation distance. Improved results are obtained when the delay is doubled to the value T D 1=fc . In this case there is a noticeable “dead-time” between the leading and trailing-edge Brillouin precursor structures which decreases the effects of destructive interference between these two components of the Brillouin pulse, resulting in improved penetration into the dispersive, absorptive material. However, this destructive interference can never be completely eliminated for all propagation distances as the time delay between the peak amplitude points for the leading and trailing-edge Brillouin precursors decreases with the inverse of the propagation distance (see Sect. 15.6.1). Nevertheless, it can be effectively eliminated over a given finite propagation distance by choosing the time delay T sufficiently large. The tradeoff in doing this is to decrease the effective oscillation frequency of the radiated pulse. The numerically determined peak amplitude decay with relative propagation distance z=zd is presented in Fig. 15.100. The lower solid curve depicts the exponential attenuation described by the function exp. z=zd /, and the lower dashed curve describes the peak amplitude decay for a single-cycle rectangular envelope pulse with fc D 1 GHz. Notice that the departure from pure exponential attenuation occurs when z=zd 0:5 as the leading and trailing-edge Brillouin precursors begin to emerge from the pulse. The dashed curve labeled BP1 describes the peak amplitude decay for the Brillouin pulse with T D 1=.2fc /, BP2 describes that for the Brillouin pulse with T D 1=fc , and BP3 describes that for T D 3=.2fc /. There

15.8 Infinitely Smooth Envelope Pulses

641

1 0.9 0.8

Peak Amplitude

0.7 0.6 0.5 0.4 BP3

0.3

BP2 BP1

0.2

Single Cycle Pulse exp(−Δ z /zd)

0.1 0

0

1

2

3

4

5

6

7

8

9

10

Δz/zd Fig. 15.100 Peak amplitude as a function of the relative propagation distance z=zd for the input unit amplitude single-cycle rectangular envelope pulse and the Brillouin pulses BP1 , BP2 , and BP3 with fc D 1 GHz. The solid curve describes the pure exponential decay given by exp. z=zd /

isn’t any noticeable improvement in the peak amplitude decay as the delay time T is increased beyond 3=.2fc / over the illustrated range of propagation distances. Notice that at ten absorption depths, exp. z=zd / D exp.10/ Š 4:54  105 , the peak amplitude of the single-cycle pulse is 0:0718, the peak amplitude of the Brillouin pulse BP1 is 0:2123, the peak amplitude of the Brillouin pulse BP2 is 0:2943, and the peak amplitude of the Brillouin pulse BP3 is 0:3015, over three orders of magnitude larger than that expected from simple exponential attenuation. The power associated with the observed peak amplitude decay presented in Fig. 15.100 may be accurately determined by plotting the base ten logarithm of the peak amplitude data vs. the base ten logarithm of the relative propagation distance. If the algebraic relationship between these two quantities is of the form Apeak D B. z=zd /p where B is a constant, then the value of the power p is given by the slope of the relation log.Apeak / D log.B/ C plog. z=zd /. The numerically determined average slope of the base ten logarithm of the numerical data presented in Fig. 15.100 is given in Fig. 15.101. The power factor p for the single-cycle pulse rapidly decreases to the value 1 as the propagation distance increases, this being due to destructive interference between the leading and trailing-edge Brillouin precursors. This destructive interference is somewhat reduced for the Brillouin pulse BP1 whose power factor p varies between 0 and 0:5 when the propagation distance increases up to 3 absorption depths. As the propagation distance increases further, the effects of destructive interference increase and p slowly decreases toward 1. This destructive interference is practically eliminated for the Brillouin

642

15 Continuous Evolution of the Total Field 0 – 0.1 – 0.2

Average Slope

– 0.3 – 0.4

BP3

– 0.5

BP2

– 0.6 BP1

– 0.7 – 0.8

Single Cycle Pulse – 0.9 –1

0

1

2

3

4

5

6

7

8

9

10

Δz/zd Fig. 15.101 Average slope of the base ten logarithm of the numerical data presented in Fig. 15.100

pulse BP2 for propagation distances up to 4 absorption depths. Near optimal (if not indeed optimal) results are obtained for the Brillouin pulse BP3 which experiences negligible destructive interference for propagation distances through at least 20 absorption depths. At z=zd D20 the peak amplitude of this Brillouin pulse is 0:2154, eight orders of magnitude larger than that expected from exponential attenuation. At this penetration depth the power factor p has decreased to the value 0:485 as destructive interference between the leading and trailing-edge Brillouin precursors begins to take effect. The nonexponential, . z/1=2 algebraic decay of the Brillouin precursor makes it the ideal field structure for penetrating attenuative dielectric materials as well as for underwater communications. The fact that its temporal width and effective oscillation frequency depend upon the material parameters makes the Brillouin precursor ideally suited for remote sensing with direct application to foliage and ground penetrating radar as well as to biomedical imaging. However, this also means that the current IEEE/ANSI safety standards may need to be carefully examined for such ultrawideband pulses. Finally, if not indeed optimal, near optimal material penetration is obtained with the Brillouin pulse described by (15.189). If the initial pulse field is perturbed from that given in (15.189), the peak amplitude evolution is decreased from that described by BP1 , BP2 , or BP3 with the period T held fixed. By adjusting the time delay between the leading and trailing-edge Brillouin precursors in the initial pulse, near optimal (if not indeed optimal) pulse penetration can be obtained over a given finite propagation distance. However, as this delay time is increased, the effective oscillation frequency of the initial pulse is decreased.

15.9 The Pulse Centroid Velocity of the Poynting Vector

643

15.9 The Pulse Centroid Velocity of the Poynting Vector Many different definitions have been introduced for the sole purpose of describing the velocity of an electromagnetic pulse in a dispersive medium, the most prevalent of these being phase, group, and energy velocities. Although these different velocity measures provide comparable results in those frequency regions of the material dispersion where the loss is small, they disagree wherever the material loss is large. In fact, some definitions of the pulse velocity (e.g., the phase and group velocities) yield seemingly nonphysical results (superluminal or negative velocities), while others (e.g., the group velocity) apply only to certain pulse characteristics. In 1970, Smith [18] introduced the definition of the pulse centrovelocity in the hope of introducing a measurable pulse velocity which overcomes these shortcomings. This pulse centrovelocity is defined by the quantity ˇ Z ˇ ˇr ˇ

1

tE 2 .r; t /dt

.Z

1

1 1

ˇ1 ˇ E 2 .r; t /dt ˇˇ

(15.190)

where E.r; t / is the real-valued electric field intensity vector. Notice that this ratio is analogous to a center of mass calculation since it tracks the temporal center of gravity of the intensity of the pulse. Recently, Peatross et al. [71, 72] introduced a variant of Smith’s centrovelocity which tracks the temporal center of gravity of the real-valued Poynting vector  c    S.r; t / D   E.r; t /  H.r; t / 4

(15.191)

rather than that of the pulse intensity. Peatross and coworkers [71, 72] then proved that the delay between the initial and propagated temporal center of gravities of the Poynting vector can be expressed as the sum of two terms calculated in the frequency domain, which they call the group delay and the reshaping delay.

15.9.1 Mathematical Formulation For a plane-wave electromagnetic pulse traveling in the positive z-direction through a temporally dispersive HILL medium occupying the positive half-space z > 0 with initial field value specified at z D 0, the average centrovelocity vcv is defined by the equation z vcv (15.192) htz i  ht0 i where

R1 1O z  1 t S.z; t / dt htz i R1 S.z; t / dt 1O z  1

(15.193)

644

15 Continuous Evolution of the Total Field

is the arrival time of the temporal centroid of the Poynting vector at the plane z  0. Notice that calculation of the centrovelocity requires knowledge of the Poynting vector S.z; t / for the propagated wavefield which, in turn, requires expressions for E.z; t / and B.z; t / D 0 H.z; t /. As shown by Peatross et al. [71, 72], the difference between the propagated and initial temporal centers of gravity of the pulse Poynting vector can be expressed as the sum of two terms as (15.194) htz i  ht0 i D Gr C Rr0 ; where the centroid group delay Gr is defined by the expression R1 Gr

Q

@<.k/ z S.z; !/d! R 1@! 1 S.z; !/d!

1

(15.195)

and the reshaping delay Rr0 is defined by R1 Rr 0 i

@ 1 @!

h i e =fkQ gz E.0; !/  e =fkQ gz H .0; !/d! R1 2=fkQ gz S.0; !/d! 1 e

R1 i

@ 1 @!

ŒE.0; !/  H .0; !/d! R1 ; 1 S.0; !/d!

(15.196)

where =fıg denotes the imaginary part of the quantity ı appearing in the brackets. The group delay of a pulse is then seen to be a spectral average of the group delay of individual frequencies which is calculated at the output plane with propagation distance z > 0, whereas the reshaping delay “represents a delay which arises solely from a reshaping of the spectrum through absorption (or amplification)” [71] and is calculated on the initial plane z D 0. In a typical experimental arrangement, the pulse (taken here to be traveling in the positive z-direction) is normally incident upon the material interface at the plane z D 0. The transmitted pulse is then calculated in the spectral domain through application of the normal incidence Fresnel transmission coefficients [9] EQ t .!/ 2n1 .!/ D n1 .!/ C n2 .!/ EQ i .!/ Q 2n2 .!/ Bt .!/ B .!/ D Q n .!/ C n2 .!/ Bi .!/ 1

E .!/

(15.197) (15.198)

where n1 .!/ D 1 for the case of vacuum in the negative half-space z < 0, n2 .!/ is the complex index of refraction of the dispersive material occupying the positive half-space z > 0, EQ t .!/ and EQ i .!/ are the transmitted and incident electric field spectra, and BQ t .!/ and BQ i .!/ are the transmitted and incident magnetic field spectra at the interface plane z D 0, respectively.

15.9 The Pulse Centroid Velocity of the Poynting Vector

645

15.9.2 Numerical Results The numerical method used to calculate the Poynting vector at any plane z  0 utilizes the fast Fourier transform (FFT) algorithm to determine the propagated plane wave electric and magnetic field components. For reasons of definiteness, MKS units are now used. To numerically determine the propagated electric field vector E.z; t / D 1O y E.z; t /, this code computes the FFT of the temporal evolution of the initial electric field vector E.0; t / D 1O y E.0; t /, propagates each monochroQ matic component by multiplication with the propagation factor expŒi k.!/z/ , where Q k.!/ D .!=c/n.!/, and then computes the inverse FFT of the propagated field spectrum, thereby constructing the temporal structure of the propagated electric field Z

1

E.z; t / D

Q Q !/e i.k.!/z!t/ E.0; d!

(15.199)

1

Q !/ is the Fourier spectrum of E.0; t /. Likewise, the propagated magwhere E.0; netic field vector B.z; t / D 1O x B.z; t / is numerically determine by performing the FFT of the initial electric field vector E.0; t /, multiplying the resulting spectrum by .1=c/n.!/, propagating each monochromatic component using the same propQ agation factor expŒik.!/z , and then computing the inverse FFT of the propagated spectrum, with the result B.z; t / D

1 c

Z

1

Q

Q !/e i.k.!/z!t/ d!: n.!/E.0;

(15.200)

1

The Poynting vector for the plane wave pulse is then directly calculated as S.z; t / D

1 E.z; t /B.z; t / 1O z 0

(15.201)

for any z  0. The accuracy of this numerical approach depends directly upon the highest frequency sampled at the Nyquist rate, the highest frequency necessary to accurately describe the initial pulse spectrum, and the highest frequency necessary to accurately describe the material dispersion. For a single resonance Lorentz model dielectric with Brillouin’s choice of the medium parameters, the maximum frequency sampled in the published calculations by Cartwright and Oughstun [73] is at least 2  1017 rad/s with at least 217 points sampled; a higher maximum frequency with more sample points is used when the applied signal frequency !c is set above the material absorption band. Because of its experimental importance, let the initial electric field vector of a plane wave pulse normally incident upon the dielectric interface at z D 0 be described by a single cycle gaussian envelope modulated cosine wave ( D =2) with fixed angular carrier frequency !c > 0. The transmitted electric and magnetic field vectors at z D 0C , and hence the transmitted Poynting vector, will then

646

15 Continuous Evolution of the Total Field

experience a small frequency chirp caused by the frequency dependence of the material refractive index appearing in the Fresnel transmission coefficients E .!/ and B .!/. An estimate of the average dispersive transmission induced frequency chirp in the transmitted Poynting vector is found [73] to be less than 5% of the doubled frequency value 2!c at the center of the Poynting vector for all cases considered.

15.9.2.1

Carrier Frequency Below the Absorption Band

When the carrier frequency of the pulse lies in the normal dispersion region below the region of anomalous dispersion, the amplitude of the gaussian Sommerfeld precursor pulse component Ags .z; t / is negligible compared to that of the gaussian Brillouin precursor pulse component Agb .z; t /. Hence, the centrovelocity will rapidly approach the limit vc D c=0 as z ! 1, which is the rate at which the peak amplitude point in the Brillouin precursor travels through the dispersive material. The average centrovelocity of the single cycle gaussian envelope modulated pulse was numerically calculated for each of the below resonance carrier frequency cases !c D 0:25!0 , !c D 0:5!0 , and !c D 0:75!0 over propagation distances from 0:1zd to 100zd , where zd D ˛ 1 .!c /. The results are presented in Fig. 15.102, where the circles, asterisks, and plus signs denote the data points for the !c D 0:25!0 , !c D 0:5!0 , and !c D 0:75!0 cases, respectively, and the solid curves are cubic 0.7

vcv /c

0.6

0.5

c = 0.25

0

c = 0.5

0

0.4 c = 0.75

0

0.3 10−1

100

z/zd

101

102

Fig. 15.102 Relative average centroid velocity for gaussian pulse propagation in a Lorentz model dielectric as a function of the relative propagation distance z=zd for several values of the initial carrier angular frequency !c in the normal dispersion region below the medium absorption band. (From Cartwright and Oughstun [73])

15.9 The Pulse Centroid Velocity of the Poynting Vector

647

spline fits through these data points. The approach to the limiting value .vc /=c D 1=n.0/ D 1=0 D 2=3 as z ! 1 is clearly evident in the figure, in agreement with the asymptotic theory. Similar behavior is obtained for a rectangular envelope pulse with below resonance signal frequency. In terms of the group and reshaping delays, the group delay begins and remains dominant over the reshaping delay for all propagation distances, in agreement with numerical results [73] obtained for a rectangular envelope modulated pulse with carrier frequency below the region of anomalous dispersion.

15.9.2.2

Carrier Frequency in the Absorption Band

The numerically determined average centrovelocity in the absorption band where the dispersion is anomalous is presented in Fig. 15.103 where the circles, asterisks and plus signs represent the data points for the !c D !0 , !c D 1:25!0 , and !c D 1:5!0 cases, respectively. As evident in the figure, the centrovelocity rapidly approaches the limiting value vc D c=0 D .2=3/c as z ! 1. Because the gaussian envelope pulse considered here effectively contains only one oscillation, it does not experience the strong phase delay effects that, for example, a ten-oscillation rectangular envelope pulse undergoes when the carrier frequency lies within the absorption band, as illustrated in Fig. 15.104. Because of this, the centrovelocity is expected to quickly ascend to the limit set by the peak

0.7

0.6

vcv /c

0.5

0.4

0.3

0.2 10−1

c = 1.5 0 c = 1.25 0

100

c= 0

z/zd

101

102

Fig. 15.103 Relative average centroid velocity for gaussian pulse propagation in a Lorentz model dielectric as a function of the relative propagation distance z=zd for several values of the initial carrier angular frequency !c in the anomalous dispersion region in the medium absorption band. (From Cartwright and Oughstun [73])

648

15 Continuous Evolution of the Total Field 1 c =1.25 0

0.8 0.6

c=

c =1.5 0

0

0.4

vcv /c

0.2

c =1.5 0

0

−0.2 −0.4

c =1.25 0

−0.6 −0.8 −1 10−1

100

z/zd

101

102

Fig. 15.104 Relative average centroid velocity for rectangular envelope pulse propagation in a Lorentz model dielectric as a function of the relative propagation distance z=zd for several values of the initial carrier angular frequency !c in the anomalous dispersion region in the medium absorption band. (From Cartwright and Oughstun [73])

amplitude point of the Brillouin precursor. However, when one considers a gaussian envelope pulse with five oscillations or more, the phase delay effects become increasingly significant for sufficiently small propagation distances and negative and superluminal centrovelocities are indeed observed. This extreme behavior is found [73] to be due primarily to pulse reshaping rather than to motion of the pulse itself. As in the case of a rectangular envelope modulated pulse with carrier frequency in the region of anomalous dispersion, the group and reshaping terms are of the same order of magnitude for small propagation distances, and hence, the reshaping term cannot then be ignored. 15.9.2.3

Carrier Frequency Above the Absorption Band

For carrier frequencies !c that lie in the normal dispersion region above the absorption band of the material, both the gaussian Sommerfeld and gaussian Brillouin precursor pulse components are evident during the pulse propagation. The large relative amount of spectral energy situated in the high-frequency domain above the absorption band implies that the gaussian Sommerfeld precursor pulse component is a significant contribution to (15.162) even for large propagation distances. The peak amplitude point of the gaussian Sommerfeld precursor component travels at a velocity just below c while the peak amplitude point of the gaussian Brillouin precursor

15.9 The Pulse Centroid Velocity of the Poynting Vector

649

0.9

c = 2.5 0

vcv /c

0.8

0.7

c=

2

0

0.6 10−1

100

z/zd

101

102

Fig. 15.105 Relative average centroid velocity for gaussian pulse propagation in a Lorentz model dielectric as a function of the relative propagation distance z=zd for several values of the initial carrier angular frequency !c in the normal dispersion region above the medium absorption band. (From Cartwright and Oughstun [73])

travels at the velocity 0 c D .2=3/c. Thus, the value of the pulse centrovelocity will start above .2=3/c and slowly descend to this limit as the gaussian Sommerfeld precursor gradually decays in amplitude with increasing propagation distance. The relative average centrovelocity values for the gaussian envelope modulated pulse cases with angular carrier frequencies !c D 2!0 and !c D 2:5!0 , which correspond to frequencies in the normal dispersion region above the absorption band, are presented in Fig. 15.105 over propagation distances from 0:1zd to 100zd . The graph clearly shows that the centrovelocity starts above and then descends to the limiting value of .vc /=c D 2=3 in agreement with the asymptotic theory. As found for the corresponding rectangular envelope pulse case [73], the group delay is dominant over the reshaping delay when the propagation distance is greater than 101 absorption depths.

15.9.3 The Instantaneous Centroid Velocity The preceding numerical results are for an average centroid velocity of the Poynting vector that is determined by the initial and final centroid locations within the dispersive medium. This velocity measure, as it was originally introduced by Peatross et al. [71, 72], is appropriate for experimental measurements as one typically measures the input and ouput pulse shapes through a slab of dielectric material with

650

15 Continuous Evolution of the Total Field

given thickness. A more localized centroid velocity measure is given by the instantaneous centroid velocity of the Poynting vector that is defined as [73] vci .zj /

lim

zj C1 !zj

zj C1  zj ; htj C1 i  htj i

(15.202)

where htj i is the centroid of the Poynting vector of the pulse at the propagation distance zj . An accurate numerical estimate of this limiting expression may be obtained by selecting neighboring points .zj ; zj C1 / from the average centroid velocity data sets that are sufficiently close to each other. With the exception of the below resonance case for the rectangular envelope case presented in [73], the instantaneous centroid velocity results are are found to be qualitatively similar to the average centroid velocity results. In particular, the limiting value c (15.203) lim vci .z/ D z!1 0 is always obtained. For the case of a rectangular envelope modulated signal with below resonance carrier frequency, the instantaneous centroid velocity is found to peak to a maximum value at a propagation distance between z=zd D 2 and z=zd D 3, after which it approaches the limiting value c=0 from above as z ! 1, as seen in Fig. 15.106. 1.2

1

vci /c

0.8 c = 0.25

0.6

0.4

0.2 10−1

0

c = 0.50

0

c = 0.75

0

100 z/zd

101

Fig. 15.106 Relative instantaneous centroid velocity for rectangular envelope pulse propagation in a Lorentz model dielectric as a function of the relative propagation distance z=zd for several values of the initial carrier angular frequency !c in the normal dispersion region below the medium absorption band. (From Cartwright and Oughstun [73])

15.10 Dispersive Pulse Propagation in the Singular and Weak Dispersion Limits

651

This peak value in the instantaneous centroid velocity increases as the initial pulse carrier frequency increases through the below resonance frequency domain and just becomes superluminal for angular signal frequency values !c 0:75!0 . The numerical results presented in Fig. 15.106 show that the pulse energy centroid initially “accelerates” until its instantaneous velocity reaches a peak value between two and three absorption depths, after which it “decelerates” toward the asymptotic value c=0 set by the velocity of the peak amplitude point of the Brillouin precursor in the dispersive medium.

15.10 Dispersive Pulse Propagation in the Singular and Weak Dispersion Limits The asymptotic description of the dynamical evolution of an ultrawideband electromagnetic pulse in a dispersive medium has established that the temporal pulse structure evolves into a set of precursor fields that are characteristic of the dispersive medium. Of particular interest is the evolution of the Brillouin precursor whose peak amplitude experiences zero exponential decay with propagation distance z > 0, decreasing algebraically as z1=2 in a dispersive, absorptive medium. The limiting behavior of this algebraic peak amplitude decay in both the zero damping limit as well as the zero density limit is now considered for a Lorentz model dielectric in order to establish whether or not this rather unique behavior persists in these two different limits. The weak dispersion limit is of particular interest as many optical systems are designed to possess minimal loss over the pulse bandwidth. The causal complex index of refraction is given here by  n.!/ D 1 

b2 ! 2  !02 C 2i ı!

1=2 ;

(15.204)

for a single resonance Lorentz model dielectric with undamped angular resonance frequency !0 and phenomenological damping constant ı > 0 with b 2 D k4=0 kN qe2 =m the square of the plasma frequency, where N denotes the number density of Lorentz oscillators in the medium. The material absorption then decreases when either ı ! 0 or when N ! 0. In the first limiting case, the material dispersion becomes increasingly localized about the resonance frequency as ı ! 0 and so is referred to here as the singular dispersion limit. In the second limiting case, the material absorption vanishes while the material dispersion approaches unity at all frequencies as N ! 0 and so is referred to here as the weak dispersion limit. These two limiting cases are fundamentally different in their effects upon ultrashort pulse propagation and are thus treated separately in the following two sections.

652

15 Continuous Evolution of the Total Field

15.10.1 The Singular Dispersion Limit ˙ The asymptotic theory shows that the two first-order near saddle points !SP ./ of n the complex phase function .!; / for a single resonance Lorentz model dielectric coalesce into a single second-order saddle point at [see (12.236)]

!SPn .1 / Š 

2ı i; 3˛

(15.205)

where [from (12.225)] 1 0 C

2ı 2 !p2 3˛0 !04

;

(15.206)

with 0 D n.0/ and ˛ 1. At the space–time point  D 0 D n.0/, the dominant . / crosses the origin [!SPC .0 / D 0] so that its contribution near saddle point !SPC n n to the asymptotic behavior of the propagated wavefield experiences zero exponential attenuation, viz., .0 /; 0 / D 0; (15.207) .!SPC n the peak amplitude point decaying only as z1=2 as z ! 1, while at the space–time point  D 1 this contribution to the asymptotic wavefield experiences a small (but nonzero) amount of exponential attenuation as well as a z1=3 algebraic decay as z ! 1, provided that ı > 0. In the singular dispersion limit as ı ! 0, however, the two near saddle points !SP˙ . / coalesce into a single second-order saddle point at the origin, resulting n in an asymptotic behavior whose peak amplitude experiences zero attenuation, the amplitude now decaying only as z1=3 . Notice that this limiting behavior is entirely consistent with the modern asymptotic theory. The numerically determined peak amplitude decay with relative propagation distance z=zd is presented in Fig. 15.107 for an input Heaviside unit step function modulated signal A.0; t / D f .t / D UH .t / sin .!c t / with below resonance carrier frequency !c D 3:0  1014 r=s in a single resonance Lorentz model dielectric with p angular resonance frequency !0 D 3:9  1014 r=s and plasma frequency b D 9:29  1014 r=s for several decreasing values of the phenomenological dampdepth in ing constant ı. Here zd ˛ 1 .!c / denotes the e 1 amplitude penetration n o Q the dispersive dielectric at the angular frequency !c , where ˛.!/ = k.!/ is the attenuation coefficient. The dashed line in the figure describes the pure exponential attenuation described by the function e z=zd . The peak amplitude used here is given by the measured amplitude of the first maximum in the temporal evolution of the propagated pulse at a fixed observation distance z  0. Notice that this “leadingedge” peak amplitude point initially attenuates more rapidly than that of the signal at ! D !c , but that as the mature dispersion regime is reached and the Brillouin precursor emerges, a transition is made from exponential attenuation to algebraic decay. Notice further that this transition occurs at a larger relative propagation distance z=zd as the phenomenological damping constant ı decreases and the medium

15.10 Dispersive Pulse Propagation in the Singular and Weak Dispersion Limits

653

dispersion becomes increasingly localized about the medium resonance frequency !0 , and hence, more singular. As the material dispersion becomes more singular (i.e., as ı decreases), the number of sample points required to accurately model the material dispersion and resultant propagated field structure increases. At the smallest value of ı considered here, a 223 point FFT was required. The algebraic power associated with the measured peak amplitude decay presented in Fig. 15.107 may be accurately determined by plotting the base ten logarithm of the peak amplitude data vs. the base ten logarithm of the relative propagation distance z=zd , as described in Sect. 15.8.3. If the algebraic relationship between these two quantities is of the form Apeak D B.z=zd /p where B is a constant,then the value of the power p is given by the slope of the relation log .Apeak / D log .B/ C p log .z=zd /. The numerically determined average slope of the base ten logarithm of the data presented in Fig. 15.107 is given in Fig. 15.108 for each value of ı considered. These numerical results show that the power p increases from a value approaching 1=2 as z ! 1 to a value approaching 1=3 as z ! 1 when ı is decreased such that ı=!0  1, in complete agreement with the asymptotic theory. An example of the numerically computed dynamical field evolution in the singular dispersion limit is presented in Fig. 15.109. The initial wavefield at z D 0 is a Heaviside unit step function signal with below resonance angular carrier frequency !c D 3:0  1014 r=s. The propagated wavefield illustrated here was calculated at ten absorption depths into a single resonance Lorentz modelpdielectric with resonance frequency !0 D 3:9  1014 r=s, plasma frequency b D 9:29  1014 r=s, and phenomenological damping constant ı D 3:02  1010 r=s. Because ı=!0 D 7:74  105 , this case is well within the singular dispersion regime.

15.10.2 The Weak Dispersion Limit In the weak dispersion limit as N ! 0, the material dispersion approaches that for vacuum at all frequencies, i.e., n.!/ ! 1. This then introduces a rather curious difficulty into the numerical FFT simulation of pulse propagation in this weak dispersion limit as the number of sample points required to accurately model the propagated pulse behavior rapidly increases as the number density N decreases to zero. To circumvent this problem, an approximate equivalence relation may be used that allows one to compute the propagated field behavior in an equivalent dispersive medium that is strongly dispersive. This approximate equivalence relation, which becomes exact in the limit as N ! 0, directly follows from the integral representation of the propagated wavefield, given by (12.1) as Z 1 (15.208) fQ.!/e .z=c/.!;/ d!; A.z; t / D 2 C for z  0.

654

15 Continuous Evolution of the Total Field

Peak Amplitude of the Brillouin Precursor

10"0

10−1

X

r/s

10−2

10

−3

0

20

40

z/zd

60

80

100

Fig. 15.107 Numerically determined peak amplitude decay due to an input unit step function modulated signal with below resonance carrier frequency !c D p 3:0  1014 r=s in a single resonance Lorentz model dielectric with !0 D 3:9  1014 r=s and b D 9:29  1014 r=s as a function of the relative propagation distance z=zd for decreasing values of the phenomenological damping constant ı

Average Slope of the Logarithm of the Peak Amplitude Data

0

−1

r/s

X

−0.5

0

20

40

z/zd

60

80

100

Fig. 15.108 Average slope of the base ten logarithm of the numerical data presented in Fig. 15.107

15.10 Dispersive Pulse Propagation in the Singular and Weak Dispersion Limits

655

0.02 z/zd = 10

0.01

AH (z,t)

Brillouin Precursor

0 Sommerfeld Precursor

−0.01

−0.02 2

3

4 t (x10

5

6

–10

r/s)

Fig. 15.109 Propagated wavefield at ten absorption depths (z D 10zd ) due to an input Heaviside unit step function modulated signal with below resonance angular carrier frequency !c p D 3:0  1014 r=s in a single resonance Lorentz model dielectric with !0 D 3:9  1014 r=s, b D 9:29  1014 r=s, and ı D 3:02  1010 r=s

Two different propagation problems for the same input pulse A.0; t / D f .t / are identical provided that the relation kQ1 .!/z1  !t1 D kQ2 .!/z2  !t2

(15.209)

is satisfied for all !. Upon equating real and imaginary parts, there results the pair of relations ˇ1 .!/z1  !t1 D ˇ2 .!/z2  !t2 ; ˛1 .!/z1 D ˛2 .!/z2 ;

(15.210) (15.211)

n o Q both of which must be satisfied for all !, where ˇ.!/ < k.!/ and ˛.!/ n o Q = k.!/ . For the absorptive part, one obtains the equivalence relation z2 D

˛1 .!/ z1 ; ˛2 .!/

8 !:

(15.212)

If the two media differ only through p their densities, then because ˛.!/ D .!=c/ni .!/ for real ! and n.!/ D 1 C Ng.!/ ! 1 C 12 Ng.!/ as N ! 0, so that ni .!/ 12 Ng.!/, the above equivalence relation becomes z2

N1 z1 : N2

(15.213)

656

15 Continuous Evolution of the Total Field

The corresponding equivalence relation for the phase part then becomes ! c

    1 1 ! 1 C N1 g.!/ z1  !t1

1 C N2 g.!/ z2  !t2 2 c 2   1 ! N1 1 C N2 g.!/

z1  !t2 ; c 2 N2 (15.214)

so that

 t2 t1 C

 N1 z1 1 ; N2 c

(15.215)

which is the second part of the desired equivalence relation. For example, if N1 =N2 D 1  102 , then z2 D z1  102 and t2 t1  .0:33  108 s=m/z1 . In that case, the propagated wavefield structure illustrated in Fig. 15.109 also applies to the case when the plasma frequency b is reduced by the factor 10 and the propagation distance z is increased by the factor 100 provided that the time scale is adjusted according to the relation given in (15.215).

15.11 Comparison with Experimental Results The first experimental measurements of the precursor fields originally described by Sommerfeld [19] and Brillouin [1] in a single resonance Lorentz medium were published by Pleshko and Pal´ocz [74, 75] in 1969; it is apparent that they were the first to refer to the first and second precursors as the Sommerfeld and Brillouin precursors, respectively. As reported in Pleshko’s Ph.D. thesis [74], the transient responses for three different types of waveguiding structures were investigated: an air-filled rectangular cross-section metallic waveguide, a surface-waveguide, and a coaxial line that is filled with a longitudinally magnetized ferromagnetic material. Two types of baseband pulse generators were used in these experiments, each producing phased locked signals. The first is a so-called Bouncing Ball Pulse Generator (BBPG) [76] which produces a gaussian-type pulse in the X-band7 with 150200 ps pulse widths at the half-amplitude points, and the second was an HP 1105 A pulse generator with a tunnel diode mount which produced a 20 ps rise-time, 3 ¯s width pulse that could then be passed through shaping circuits in order to produce a variety of pulse waveforms, including a single-cycle pulse with ˙2:0 GHz bandwidth centered about a 4:0 GHz carrier frequency. The pulse waveforms were then measured with a 12:4 GHz sampling oscilloscope with 16 GHz maximum frequency response [74, pages 4–7]. Their principal experimental results are now briefly described. 7

The X-band denotes the microwave region of the electromagnetic spectrum extending from 7 to 12:5 GHz.

15.11 Comparison with Experimental Results

657

The dispersion relation for the fundamental mode of an air-filled rectangular waveguide oriented along the z-axis is given by ! kz D c

 1=2 !c2 1 2 ; !

(15.216)

where !c is the angular cutoff frequency. This dispersion relation then approximates the optic mode in a single resonance Lorentz model dielectric, obtained from (12.57) for j!j  !0 with !c replaced by the plasma frequency b. The transient wavefield is then comprised of just a Sommerfeld precursor with zero exponential attenuation, decaying algebraically as z1=2 as z ! 1, as described in (13.62) with ı D 0. This result was first verified experimentally by Pleshko [74] using three lengths of an air-filled rectangular cross-section waveguide with a BBPG source pulse waveform. The measured output waveforms are illustrated in Fig. 15.110 and the measured relative peak amplitude points (indicated by the arrows in Fig. 15.110) are compared in Fig. 15.111 with the theoretical z1=2 amplitude decay. The experimentally measured temporal evolution of the Sommerfeld precursor structure produced by a 20 ps rise-time step function is presented in Fig. 15.112. As then concluded by P. Pleshko [74, page 45], “it may be concluded that the agreement

Fig. 15.110 BBPG source pulse waveforms at increasing propagation distances in an air-filled rectangular waveguide. (Figure 15.17 from P. Pleshko [74])

658

15 Continuous Evolution of the Total Field 1

Relative Peak Amplitude

0.9

0.8

0.7

1/2

1/zrel

0.6

0.5

1

2

zrel

3

4

Fig. 15.111 Measured peak amplitude decay with relative propagation distance (open circles) of the Sommerfeld precursor dominated propagated waveforms in an air-filled rectangular waveguide presented in Fig. 15.109. The solid curve describes the z1=2 behavior predicted by the asymptotic theory. (Data from Fig. 15.18 of P. Pleshko [74])

Fig. 15.112 Step function response of an air-filled rectangular waveguide. (Figure 15.19 from P. Pleshko [74])

15.11 Comparison with Experimental Results

659

between the transient response calculated by the stationary phase method and transient response obtained experimentally is exceptionally good, with an accuracy well within expected measurement error.” Consider next the experimental results for a coaxial line filled with a longitudinally (i.e., along the waveguide axis) magnetized ferrimagnetic material.8 The dispersion relation may be approximated [78] by that given by Suhl and Walker [79] for the dominant TEM mode in a parallel plate waveguide filled with a lossless ferrite in the limit of small plate separation a as kz

 1=2 .!0 C !M /2  ! 2 !p r ; c !0 .!0 C !M /  ! 2

(15.217)

p provided that .!=c/ r a  1. Here r denotes the relative (real-valued) dielectric permittivity of the ferrite, and !0 D g Hi ; !M D k4kg Ms ;

(15.218) (15.219)

where g denotes the gyromagnetic ratio, Hi is the externally applied magnetic field intensity in the ferrite, and k4kMs is the saturation magnetization of the ferrite. The dispersion relation givenp in (15.217) has a low-frequency branch ! 2 Œ0; !R

p with asymptote k D . r =c/! 1 C .!M =!0 / as ! ! 0, where !R

p !0 .!0 C !M /;

(15.220)

p and a high-frequency branch !  !C with asymptote k D . r =c/! as ! ! 1, where (15.221) !C !0 C !M : As the externally applied magnetic field strength is increased, !0 increases and both !R and !C increase and approach each other as the two asymptotes merge, resulting p in a single, nondispersive dielectric mode with dispersion relation k D . r =c/!. This is illustrated in Fig. 15.113 which shows the measured line response to a narrow pulse excitation for increasing values of the applied magnetic field strength. As this is approximately the impulse response of the dispersive line, the resultant waveforms are dominated by the high-frequency branch Sommerfeld precursor followed by the low-frequency branch Brillouin precursor, as clearly evident in the figure. The experimentally measured Heaviside step function envelope modulated sine wave response of this garnet-filled coaxial line is presented in Fig. 15.114 for increasing values of the of the applied magnetic field strength. The input waveform had a 1 ns rise-time with a 625 MHz carrier frequency. as described by Pleshko [74]: Initially, with zero applied field, the sine wave pulse propagates on essentially a low dispersion portion of the high frequency mode. at 20 gauss, an initial high-frequency oscillation 8

A detailed description of ferromagnetic, antiferromagnetic, and ferrimagnetic materials may be found in Chap. 1 of [77].

660

15 Continuous Evolution of the Total Field

Fig. 15.113 Narrow pulse response of a garnet-filled coaxial line. (Figure 15.32 from P. Pleshko [74])

due to the high frequency components of the pulse is seen to be the first signal arriving, which is a Sommerfeld type of precursor, and since the cutoff frequency fC is low, it has appreciable amplitude. At this point also, the Brillouin type of precursor is also seen. There is no main signal at this field strength because the carrier frequency of the pulse lies in the stop band of the system. As the field strength is increased, the Sommerfeld type of precursor is no longer visible to the eye (masked by the noise of the oscilloscope) but the Brillouin precursor still has appreciable amplitude. The waveform obtained at an external field strength of 150 gauss . . . the carrier frequency is close to !R and thus the main body of the signal has very large dispersion and low amplitude. Thus, olnly the low frequency portion of the spectrum comes through with large amplitudes. With a field strength of 200 gauss, the Brillouin type precursor, and the main signal are seen with large amplitudes as Brillouin stated in his book (p. 128) but the Sommerfeld type precursor is not visible due to the fact that the noise of the oscilloscope is greater than the amplitude of the signal comprising the Sommerfeld type precursor.

Although their experiments were conducted in the microwave domain on waveguiding structures with dispersion characteristics that are similar to that described by either a Drude model conductor [compare (15.216) and (12.153)] or a single resonance Lorentz model dielectric [compare the low and high-frequency branches of (15.217) with the below resonance and above absorption band approximations of (12.57)], the results established the physical property of the now classical asymptotic theory developed by Sommerfeld and Brillouin. In particular, through several rather clever experimental arrangements, Pleshko [74] and Pal´ocz [75] were able

15.11 Comparison with Experimental Results

661

Fig. 15.114 Heaviside step function envelope modulated sine wave response of a garnet-filled coaxial line. (Figure 15.39 from P. Pleshko [74])

to isolate the dynamical evolution of the individual pulse components comprising the propagated signal representation given in (15.1). This “proof of principle” done, experimental verification in bulk (e.g., non-waveguiding) media then remained to be given. In an extension of this early experimental work, D. D. Stancil [80] measured magnetostatic precursory wave motion in thin ferrite films.9 These results showed the existence of three types of Brillouin-type precursors in an Yttrium iron garnet (YIG) film, with experimental observations for forward volume waves, backward volume waves, and surface waves. A complete, detailed description of this observed precursor-type phenomena in magnetostatic wave motion remains to be given. Precursor-type phenomena is also observed in fluids and acoustics. The signal velocity of sound in superfluid 3 He–B was measured by Avenel, Rouff, Varoquaux and Williams [82, 83] for moderate material damping. Their reported experimental results are in agreement with Brillouin’s original description [1] where the deformed contour of integration was constrained to entirely lie along the union of steepest descent paths through the distant and near saddle points, resulting in a signal 9

Magnetostatic waves (MSW), also called magnetic polarons or magnons, refer to oscillations in the magnetostatic properties of a magnetic material such as a ferrite. See, for example, the book on magnetostatic waves by Stancil [81].

662

15 Continuous Evolution of the Total Field

velocity that peaks to a maximum value near the material resonance frequency. However, their experiment did not use a step function modulated signal for which the signal velocity has been defined. Rather, they used a continuous envelope pulse for which the signal velocity is undefined. The observation of a “precursory” motion that is similar to the Sommerfeld (or first) precursor was later observed [83] by Varoquaux, Williams, and Avenel in superfluid 3 He–B. Detailed experiments measuring precursor phenomena on fluid surfaces have been presented by Falcon, Laroche, and Fauve [84]. The dispersion relation for the angular frequency !.k/ in terms of the wavenumber k (neglecting dissipation) for such surface waves is given by s   3 gk C k tanh .kh/; !.k/ D 

(15.222)

where g is the acceleration due to gravity,  is the fluid mass density,  is the surface tension, and h is the p depth of the liquid body. Associated with this fluid is the capillary length `c =.g/ and Bond number B0 .`c = h/2 . The wavenumber dispersion of the phase velocity vp .k/ !.k/=k is described by the dotted curve in Fig. 15.115 for mercury ( D 13:5  103 kg=m3 ,  D 1:5  103 Ns=m2 ,  D 0:4 N=m) with depth h D 3:7 mm. The solid curve in the figure describes the (numerically determined) exact wavenumber dependence of the group velocity vg .k/ @!.k/=@k. In either the long wavelength approximation or shallow fluid limit given by kh  1, the dispersion relation in (15.222) may be expanded and differentiated with respect to k to yield the approximate expression [84] vg .k/

i p h @!.k/ a4

gh 1  a2 .kh/2 C .kh/4 ; @k 4

(15.223)

where a2 13  B0 and a4 19  12 B0  13 B02 . The wavenumber dependence 90 of this approximation is described by the dashed curve in Fig. 15.115. Although this approximation is only valid for k  1= h  270, it does properly show that a minimum in the group velocity exists only when the Bond number satisfies the inequality 0  B0 < 13 , the value B0 D 13 corresponding to the critical depth p hc 3`c . A typical surface wave pattern generated by an impulsional excitation is illustrated in Fig. 15.116. The source is located to the right of the figure with the surface wave propagating to the left, led by a high-frequency Sommerfeld precursor SH followed by a low-frequency Brillouin precursor SL (referred to as a low-frequency Sommerfeld precursor in [84]). The observed pulse evolution can be described by the group velocity dispersion curve presented in Fig. 15.115 by first noting that as time increases at a fixed propagation distance z > 0, the horizontal line at vg .k/ D z=t moves p down the group velocity curve. For early times t < vg .0/, where vg .0/ D gh, there is just a single wavenumber solution ks to this equation [viz., vg .ks / z=t ] which decreases as t increases. For all values of the Bond

15.11 Comparison with Experimental Results

663

wave velocity (m/s)

0.22

0.2

vp(k) 0.18 vg(k)

0.16 0

100

200

300 400 wavenumber (r/m)

500

600

700

Fig. 15.115 Wavenumber dependence of the phase velocity vp .k/ D !=k (dotted curve) and the group velocity velocity vg .k/ D @!=@k (solid curve) for mercury with fluid depth h D 3:7 mm and Bond number B0 D 0:22. The dashed curve describes the approximate group velocity dispersion described by (15.223)

Fig. 15.116 Photograph of typical surface wave precursors for mercury with fluid depth h D 3:7 mm and Bond number B0 D 0:22. The full vertical scale corresponds to a 7 cm canal width. (Figure 15.1 from Falcon, Laroche, and Fauve [84])

number B0  13 , so that h  hc , there is just a single solution to this equation as the group velocity monotonically decreases to its static value. In that case, the propagated impulsive response is comprised of just a Sommerfeld-type precursor whose instantaneous oscillation frequency monotonically decreases to zero. On the other hand, when 0  B0 < 13 , so that h > hc , there are two solutions to (15.222) when

664

15 Continuous Evolution of the Total Field

p t > z= gh. In that case, the propagated impulsive response is comprised of two types of precursors: the fastest is the high-frequency Sommerfeld-type precursor SH (from the capillary branch k > kmin of the dispersion curve) characterised by an instantaneous oscillation that decreases with time, that is followed by the slower, low-frequency Brillouin-type precursor SL (from the gravity branch k < kmin of the dispersion curve) that is characterised by an instantaneous oscillation frequency that increases with time. Measurements of these precursor-type wave motions were performed by Falcon et al. [84] for a mercury fluid layer with heights h varying from 2:12 to 13:75 mm, so that 0:02  B0  0:67, the critical value B0 D 13 of the Bond number occuring at the critical fluid height h D hc 3 mm. The source was a horizontal impulsion and the resultant free-surface wave profiles were recorded at a fixed distance z from the source. Their results, using both optical (a and b) and inductive (c) measurement techniques, are presented in Fig. 15.117; the insets in parts (b) and (c) of the figure present a comparison of the two measurement techniques. In part (a) of the figure, h < hc and the dynamical wave evolution at z D 0:2 m is dominated by a Sommerfeld-type precursor whose oscillation frequency monotonically decreases to zero. In part (b) of the figure, h > hc and the high-frequency Sommerfeld-type precursor precedes the low-frequency Brillouin-type precursor at z D 0:2 m. The observation distance in (b) is increased to z D 0:6 m in part (c) of the figure. At this larger propagation distance the high-frequency Sommerfeld-type precursor has disappeared due to attenuation (from viscous dissipation) and only the low-frequency contribution from the gravity branch (the Brillouin-type precursor) is observed. The measured oscillation period of the high-frequency Sommerfeld-type and lowfrequency Brillouin-type precursors are presented p in Fig. 15.118 as a function of the dimensionless space–time parameter .z=t /= gh for a variety of experimental conditions. The solid curves in the figure describe the theoretical behavior obtained from the dispersion relation given in (15.222). Although the description provided by the stationary phase approximation with this real-valued dispersion relation is adequate in some respect, it fails to completely describe the dynamical evolution of the impulsive surface wave phenomena illustrated in Fig. 15.117. A more accurate description that properly (i.e., causally) includes the attenuative part of the dispersion relation remains to be given. The experimental observability of optical precursors has been proposed by Aaviksoo, Lippmaa, and Kuhl [85] in 1988 using the transient response of excitonic resonances to picosecond pulse excitation. The experimental observation of the Sommerfeld and Brillouin precursors was then reported by Aaviksoo, Kuhl, and Ploog [86] in 1991. In their experimental arrangement, an approximate double ex1 =˛2

400 fs ponential pulse [see (11.58)–(11.60)] with a steep rise-time of tr  ln˛1˛˛ 2 1 and a slow decay-time of td  ˛2 6:2 ps was transmitted through a 0:2 ¯m thick layer of GaAs crystal near the exciton resonance. The transmitted pulse was then measured through its cross-correlation with the incident pulse. The results agreed qualitatively (the amplitudes were off by a factor between 2 and 3) with theoretical cross-correlation results based on dispersive pulse propagation in a single resonance Lorentz model medium. Because these theoretical results are described

15.11 Comparison with Experimental Results

665

Fig. 15.117 Measured free-surface profiles of the impulsive wave response at the surface of merp cury as a function of the dimensionless time t =t0 , where t0 D z= gh using optical [(a) and (b)] and inductive [(c)] techiniques. The insets in (b) and (c) show a comparison of measurements made with both techniques. In (a) h < hc so that B0 > 1=3 and just the high-frequency Sommerfeld-type precursor SH is present and in (b) and (c) h > hc so that B0 < 1=3 and both the high-frequency Sommerfeld-type and low-frequency Brillouin-type precursors are present. (Figure 15.3 from Falcon, Laroche, and Fauve [84])

666

15 Continuous Evolution of the Total Field 300

Period (ms)

Low-Frequency Brillouin Precursors

200 High-Frequency Sommerfeld Precursors

100 Tmin

0

0.5

1

1.5

2

(z/t)/(gh)1/2

Fig. 15.118 Measured oscillation period of the high-frequency Sommerfeld and low-frequency p Brillouin precursors as a function of the dimensionless space–time parameter .z=t /= gh with 0:2 m  z  0:8 m for the fluid depth cases h D 2:12 mm (r) for depression pulses, h D 2:12 mm (4) for elevation pulses, h D 3:4 mm (), h D 5:6 mm (u t), h D 7:2 mm (ı), h D 10:4 mm (˘), and h D 13:75 mm (). (From Fig. 15.4 of Falcon, Laroche, and Fauve [84])

by overlapping Sommerfeld and Brillouin precursors (see, for example, Fig. 15.88), together with an exciton precursor that has been described by Birman and Frankel [87,88] their experimental results provide indirect evidence of these precursor fields. As stated by the authors in the conclusion of their paper [86]: we have experimentally demonstrated the existence of precursors for transient electromagnetic wave propagation through dispersive media in the optical frequency range. These precursors or forerunners appear in the transmitted optical pulse if long narrow-bandwidth pulses with steep fronts propagate near material resonances. Experiments on thin GaAs crystals with nearly exponential type pulses and a frequency tuned close to the free-exciton resonance have confirmed this prediction in a good agreement with corresponding theoretical calculations. The observation of separate Sommerfeld, Brillouin, and exciton precursors in the optical regime remains a challenge for future work.

Indeed, the observation of the separate Sommerfeld and Brillouin precursors in the optical domain has posed a special challenge to experimentalists. The experimental observation of the Brillouin precursor in bulk media using an ¨ ultrashort optical pulse has been reported by Choi and Osterberg [89] in 2004, but not without criticism [90] that is itself due criticism. In their reported obser¨ vation of a Brillouin precursor in deionized water, Choi and Osterberg state [89] that they “observe pulse breakup in a linear regime for 540 fs long pulses with a bandwidth of 60 nm. . . propagating through 700 mm of deionized water.” They “attribute the pulse breakup to the formation of optical precursors.” Their conclusions are “further supported by subexponential attenuation with distance for the new peak p as well as 1= z attenuation at distances exceeding 3:5 m.” Their experimental measurements of the peak amplitude decay as a function of propagation distance are

15.11 Comparison with Experimental Results

667

Peak Amplitude

100

e −z/zd

Tr f = T/10 Tr f = T

10−1

0

1

z/zd

2

3

Fig. 15.119 Experimentally measured peak amplitude decay (ı symbols) of an ultrashort optical pulse in deionized water. The dotted line describes pure exponential decay and the dashed curves describe the numerically determined peak amplitude decay with rise/fall time equal to the period of oscillation T D 1=f of the carrier wave (lower dashed curve) and to one-tenth of that period ¨ (upper dashed curve). (Experimental data provided by Choi and Osterberg [89])

presented in Fig. 15.119 by the open circles connected by solid line segments from a linear spline interpolation. Notice that the initial pulse spectrum in their experiments is centered about 700 nm, corresponding to a pulse frequency f ' 0:428 PHz with bandwidth f ' 0:0367 PHz. Because f =f ' 0:0857  1, this pulse is quasimonochromatic. ¨ In their published critique of the Choi and Osterberg results, Alfano et al. [90] state that “Sommerfeld precursors arise from the higher frequencies in the pulse while Brillouin precursors arise from low frequencies far away from resonance frequencies.” However, this statement is not entirely correct. What is necessary is that the pulse spectrum have sufficient energy either above resonance (for the Sommerfeld precursor) or below resonance (for the Brillouin precursor). Alfano et al. [90] further state that “the dispersion of water is almost flat in the region about 700 nm. No asymptotic behavior (saddle point) exists.” Although the real part of the dielectric permittivity of water exhibits relatively weak normal dispersion over the pulse bandwidth, the imaginary part does not (see Figs. 4.2 and 4.3 in Vol. 1). This does not eliminate the possibility of saddle point evolution and hence, the appearance of a precursor. Finally, they attribute the observed pulse breakup as being due to a vibrational overtone absorption band in water that is centered at 760 nm (0:394 PHz). This can also help explain the appearance of a Brillouin

668

15 Continuous Evolution of the Total Field

precursor in their measured attenuation data because small nonlinear effects have been shown [91] to enhance the Brillouin precursor in the dynamical field evolution. ¨ Choi and Osterberg’s experimental results [89], reproduced in Fig. 15.119, clearly exhibit nonexponential decay. The question as to whether or not this data p exhibit the 1= z amplitude attenuation characteristic of the Brillouin precursor is best addressed by determining the algebraic power law described by these values. If the relation between the peak amplitude Apeak and the relative propagation distance z=zd , where zd D ˛ 1 .!c / is the e 1 penetration depth at the pulse frequency, is given by Apeak D B.z=zd /p , where B is a constant, then the power p of the peak amplitude decay may be determined [32] from the slope of the base ten logarithm of the data. The results, presented in Fig. 15.120. show that the averaged experimental data varies between that for the the numerically determined peak amplitude decay for an ultrawideband pulse with rise/fall time equal to the period of oscillation T D 1=f of the carrier wave (lower dashed curve) and to one-tenth of that period (upper dashed curve) in a double resonance Lorentz model of the optical frequency dispersion of triply distilled water over the intermediate propagation distances between 0:25 and 1:75 absorption depths. The average slope values are presented here in order to help smooth the inherent variability in Choi and ¨ Osterberg’s experimental results, indicated by the C signs in Fig. 15.120. These averaged results are indicative of the appearance of a Brillouin precursor over this propagation domain. Further experimental results involving ultrashort optical pulse propagation in water have since been reported by Okawachi et al. [92]. However,

0 −0.2 −0.4 Averaged Experimental Results

Average Slope

−0.6 −0.8 −1

Trf = T/10

−1.2 −1.4

Trf = T

−1.6 −1.8

0

1

z/zd

2

3

Fig. 15.120 Averaged valuse (ı symbols) of the slope of the base ten logarithm of the experimental data (C symbols) obtained from Fig. 15.119 compared with that obtained from the two theoretical (dashed) curves in Fig. 15.119 for the T and T =10 rise/fall time cases

15.12 The Myth of Superluminal Pulse Propagation

669

their experiments were performed using a 540 fs gaussian envelope pulse whose spectrum in the ultraviolet region of the optical domain (using wavelengths of 800 and 1,530 nm) is clearly not ultrawideband, as reflected in their experimental results. Thier conclusion that [92] “we observe strictly monoexponential decay, confirming that propagation of femtosecond pulses in water obeys the Beer-lambert law” has no bearing in the ultrawideband domain. The asymptotic and numerical results p presented in this chapter show that the characteristic 1= z peak amplitude decay of the Brillouin precursor will not be observed in their experimental arrangement. More recent experimental observations [93] of both the Sommerfeld and Brillouin precursors in the optical domain when the input ultrashort pulse is in the region of anomalous dispersion has been reported by Jeong, Dawes, and Gauthier. In their experiment, a step function modulated signal with a 1:7 ns rise-time, which corresponds to a 206 MHz bandwidth with !c ' 3:9  1014 r=s, was transmitted through a dilute gas of potassium (39 K) atoms. The pulse frequency and bandwidth then essentially interact with a single resonance frequency at !0 ' 3:9  1014 r=s with frequency dispersion described by a single resonance Lorentz model dielectric with plasma frequency b ' 3:05  109 r=s and damping constant ı ' 3:02  107 r=s. This then corresponds to both the singular and weak dispersion limits described in Sect. 15.10. Because the pulse carrier frequency is in the anomalous dispersion region of this resonance and because the dispersion is both weak and singular, both the Sommerfeld and Brillouin precursors are observed superimposed on each other [see Fig. 15.88b]. Taken together, these experimental results provide an important (albeit partial) verification of the modern asymptotic theory in its description of ultrashort dispersive pulse dynamics. Additional experimental measurements are clearly needed, not just to verify the theoretical predictions, but also to apply the unique features of precursors to a variety of practical applications, including medical imaging, remote sensing, and communications in adverse environments. With the use of a temporal coherence synthesization scheme proposed by Park et al. [94], transform-limited pulses with fairly arbitrary envelope functions may be constructed in the optical domain.

15.12 The Myth of Superluminal Pulse Propagation There was a young chap named Devaney whose arguments went faster than electromagnetic energy. He published a paper in May, in an extremely noncausal way, with errata published the previous February! Clever limericks aside, the allure of superluminal pulse propagation in classical physics is practically irresistable (see the episode “The Lure of Light” from the 1953–1954 Flash Gordon television series). It certainly is more newsworthy than the fundamental restriction imposed by the special theory of relativity, as evidenced

670

15 Continuous Evolution of the Total Field

by the May 30, 2000 New York Times article “Faster Than Light, Maybe, But Not Back to the Future” as well as by the May 16, 2006 New York Times article “Impressive New Tricks of Light, All Within the Laws of Physics.” Such experiments are typically conducted with a gaussian envelope pulse as that particular pulse shape is ideally suited for the group velocity approximation. Therein lies the entire difficulty with these reported observations of superluminal pulse propagation. Sommerfeld’s relativistic causality theorem (see Theorem 6 in Sect. 13.1), first given in 1914, rigorously proves that information cannot be transmitted through a causal medium faster than the speed of light c in vacuum. In particular, notice that any initial plane wave pulse A.0; t / D f .t / at the plane z D 0 and propagating in the positive z-direction can be formally separated into two distinct parts as A.0; t / D f .t /uH .t0  t / C f .t /uH .t  t0 /;

(15.224)

where uH .t / denotes the Heaviside unit step function [defined here as uH .t / D 0 for t < 0, uH .t / D 12 for t D 0, and uH .t / D 1 for t > 0], and where t0 2 .1; C1/ is any finite, fixed instant of time. Sommerfeld’s theorem then rigorously shows that no part (i.e., information, energy, etc.) of the initial wavefield component A> .0; t / f .t /uH .t  t0 / can appear ahead of the luminal space–time point c.t  t0 /=z D 1 in the propagated wavefield for all z > 0; that is, A> .z; t / D 0

(15.225)

for all .z; t / with z > 0 such that cz .t  t0 / < 1. Notice that sufficiently slow parts of the initial wavefield component A< .0; t / f .t /uH .t0  t / can appear behind the luminal space–time point c.t  t0 /=z D 1 in the propagated wavefield. The observation of both superluminal and negative group velocities for gaussian pulse propagation in any causally dispersive system is then due to pulse reshaping. Because peak amplitude points are not causally related [95,96], there is no violation of special relativity. This simple fact was beautifully demonstrated with a simple electronic circuit by Prof. Kitano of Kyoto University at the 2002 Quantum Optics Workshop on Slow and Fast Light at the Kavli Institute of Theoretical Physics (see Fig. 15.121 for a photograph of some of the participants at this workshop). All of the talks for this workshop can be found at the KITP Web site. The subtle effects of pulse reshaping on the group velocity of a gaussian envelope pulse in the anomalous dispersion region of a Lorentz model dielectric are illustrated in Figs. 15.95–15.97. Generalizations of the group velocity do not fare any better. In particular, the generalization of the group velocity to the pulse centroid velocity does not remove this fundamental difficulty. The detailed numerical study of the evolution of the pulse Poynting vector centrovelocity for both ultrawideband rectangular envelope and gaussian envelope plane-wave pulses traveling through a single resonance Lorentz model dielectric, presented here in Sect. 15.9 for the gaussian envelope pulse with more detailed results in [73] for both gaussian and rectangular envelope pulses, leads to the following set of conclusions:

15.12 The Myth of Superluminal Pulse Propagation

671

Fig. 15.121 Participants in the 2002 Quantum Optics Workshop on Slow and Fast Light at the Kavli Institute of Theoretical Physics, University of California at Santa Barbara. From left to right: First Row: Raymond Chiao, Daniel Gauthier, Lijun Wang, Aephraim Steinberg, Justin Peatross; Second Row: Masao Kitano, Michael Fleischhauer, Scott Galsgow, Peter Milonni, Ulf Leonhardt, A. Zee, Herbert Winful, Guenter Nimtz; Third Row: Curtis Broadbent, Joseph Eberly, Kurt Oughstun

 As z ! 1, both the average and instantaneous centrovelocity of an ultrawide-

band pulse tends toward the rate at which the peak of the Brillouin precursor travels through the medium, independent of the applied pulse frequency. This is precisely the result obtained from the asymptotic theory because the Brillouin precursor dominates the propagated ultrawideband pulse wavefield for sufficiently large propagation distances (typically greater than one absorption depth at the applied pulse frequency).  The reshaping delay [see (15.196)] may become significant [i.e., of the same order of magnitude as the group delay given in (15.195)] for small propagation distances when the carrier frequency of the pulse lies within the region of anomalous dispersion. In general, the relative significance of the reshaping delay is highly dependent on the dispersive material properties.  A phase delay between the electric and magnetic field vectors occurs when the carrier frequency of the input pulse lies within the region of anomalous dispersion of the medium. This phase delay primarily effects the trailing edge of a rectangular envelope pulse for small propagation distances and rapidly diminishes with increasing propagation distance. Because of this phase delay effect, the centroid of the propagated Poynting vector for a rectangular envelope pulse rapidly shifts to earlier times with small increases in the propagation distance, resulting in centrovelocity values that are initially negative, then become negatively infinite, jump discontinuously to positive infinity and then finally become subluminal for sufficiently large z, as illustrated in Fig. 15.104.

672

15 Continuous Evolution of the Total Field

 The effect of the phase delay on the centrovelocity is dependent upon the initial

time duration of the pulse. For example, for an input signal of one oscillation at a carrier frequency within the anamolous dispersion regime, both the rectangular and guassian modulated pulses are found to have centrovelocity values which are subluminal for all propagation distances z > 0 considered. However, when the input pulse consists of ten oscillations at a carrier frequency within the anamolous dispersion regime, both the rectangular and guassian modulated pulses experience superluminal centrovelocity values.  The centrovelocity may not accurately describe the pulse velocity through a given Lorentz model dielectric with regard to energy transport. For example, when the Sommerfeld precursor amplitude is of the same order as the Brillouin precursor amplitude, the centrovelocity will fall between the two precursors at a point where a negligible amount of pulse energy is located. It may then be concluded that superluminal pulse propagation is just an illusion. Neither electromagnetic energy nor information encoded in the electromagnetic field can travel faster than the speed of light in vacuum, all in keeping with Einstein’s [97] special theory of relativity.

References ¨ 1. L. Brillouin, “Uber die fortpflanzung des licht in disperdierenden medien,” Ann. Phys., vol. 44, pp. 204–240, 1914. 2. L. Brillouin, Wave Propagation and Group Velocity. New York: Academic, 1960. 3. K. E. Oughstun, Propagation of Optical Pulses in Dispersive Media. PhD thesis, The Institute of Optics, University of Rochester, 1978. 4. K. E. Oughstun and G. C. Sherman, “Optical pulse propagation in temporally dispersive Lorentz media,” J. Opt. Soc. Am., vol. 65, no. 10, p. 1224A, 1975. 5. K. E. Oughstun and G. C. Sherman, “Comparison of the signal velocity of a pulse with the energy velocity of a time-harmonic field in Lorentz media,” in Proceedings of the URSI Symposium on Electromagnetic Wave Theory, (M¨unchen), pp. C1–C5, 1980. 6. K. E. Oughstun and G. C. Sherman, “Propagation of electromagnetic pulses in a linear dispersive medium with absorption (the Lorentz medium),” J. Opt. Soc. Am. B, vol. 5, no. 4, pp. 817–849, 1988. 7. K. E. Oughstun and G. C. Sherman, “Uniform asymptotic description of electromagnetic pulse propagation in a linear dispersive medium with absorption (the Lorentz medium),” J. Opt. Soc. Am. A, vol. 6, no. 9, pp. 1394–1420, 1989. 8. K. E. Oughstun and G. C. Sherman, Pulse Propagation in Causal Dielectrics. Berlin: SpringerVerlag, 1994. 9. M. Born and E. Wolf, Principles of Optics. Cambridge: Cambridge University Press, seventh (expanded) ed., 1999. ¨ 10. H. Baerwald, “Uber die fortpflanzung von signalen in disperdierenden medien,” Ann. Phys., vol. 7, pp. 731–760, 1930. 11. S. Shen and K. E. Oughstun, “Dispersive pulse propagation in a double-resonance Lorentz medium,” J. Opt. Soc. Am. B, vol. 6, pp. 948–963, 1989. 12. K. E. Oughstun and N. A. Cartwright, “Dispersive pulse dynamics and associated pulse velocity measures,” Pure Appl. Opt., vol. 4, no. 5, pp. S125–S134, 2002. 13. H. M. Nussenzveig, Causality and Dispersion Relations. New York: Academic, 1972. Chap. 1.

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42. R. E. Haskell and C. T. Case, “Transient signal propagation in lossless, isotropic plasmas,” IEEE Trans. Antennas Prop., vol. 15, pp. 458–464, 1967. 43. L. E. Vogler, “An exact solution for waveform distortion of arbitrary signals in ideal wave guides,” Radio Sci., vol. 5, pp. 1469–1474, 1970. 44. J. R. Wait, “Electromagnetic-pulse propagation in a simple dispersive medium,” Elect. Lett., vol. 7, pp. 285–286, 1971. 45. R. Barakat, “Ultrashort optical pulse propagation in a dispersive medium,” J. Opt. Soc. Am. B, vol. 3, no. 11, pp. 1602–1604, 1986. 46. P. D. Smith, Energy Dissipation of Pulsed Electromagnetic Fields in Causally Dispersive Dielectrics. PhD thesis, University of Vermont, 1995. Reprinted in UVM Research Report CSEE/95/07-02 (July 18, 1995). 47. P. D. Smith and K. E. Oughstun, “Electromagnetic energy dissipation of ultra-wideband plane wave pulses in a causal, dispersive dielectric,” in Ultra-Wideband, Short-Pulse Electromagnetics 2 (L. Carin and L. B. Felsen, eds.), pp. 285–295, New York: Plenum, 1995. 48. P. D. Smith and K. E. Oughstun, “Electromagnetic energy dissipation and propagation of an ultrawideband plane wave pulse in a causally dispersive dielectric,” Radio Sci., vol. 33, no. 6, pp. 1489–1504, 1998. 49. P. D. Smith and K. E. Oughstun, “Ultrawideband electromagnetic pulse propagation in triplydistilled water,” in Ultra-Wideband, Short-Pulse Electromagnetics 4 (E. Heyman, B. Mandelbaum, and J. Shiloh, eds.), pp. 265–276, New York: Plenum, 1999. 50. H. T. Banks, M. W. Buksas, and T. Lin, Electromagnetic Material Interrogation Using Conductive Interfaces and Acoustic Wavefronts. Frontiers in Applied Mathematics, Philadelphia: Society for Industrial and Applied Mathematics, 2000. 51. J. A. Fuller and J. R. Wait, “A pulsed dipole in the earth,” in Transient Electromagnetic Fields (L. B. Felsen, ed.), pp. 237–269, New York: Springer-Verlag, 1976. 52. R. W. P. King and T. T. Wu, “The propagation of a radar pulse in sea water,” J. Appl. Phys., vol. 73, no. 4, pp. 1581–1590, 1993. 53. N. A. Cartwright and K. E. Oughstun, “Ultrawideband pulse penetration in a Debye medium with static conductivity,” in Fourth IASTED International Conference on Antennas, Radar, and Propagation, 2007. 54. K. E. Oughstun, “Noninstantaneous, finite rise-time effects on the precursor field formation in linear dispersive pulse propagation,” J. Opt. Soc. Am. A, vol. 12, pp. 1715–1729, 1995. 55. M. Tanaka, M. Fujiwara, and H. Ikegami, “Propagation of a Gaussian wave packet in an absorbing medium,” Phys. Rev. A, vol. 34, pp. 4851–4858, 1986. 56. C. M. Balictsis and K. E. Oughstun, “Uniform asymptotic description of ultrashort Gaussian pulse propagation in a causal, dispersive dielectric,” Phys. Rev. E, vol. 47, no. 5, pp. 3645–3669, 1993. 57. K. E. Oughstun and C. M. Balictsis, “Gaussian pulse propagation in a dispersive, absorbing dielectric,” Phys. Rev. Lett., vol. 77, no. 11, pp. 2210–2213, 1996. 58. C. M. Balictsis and K. E. Oughstun, “Generalized asymptotic description of the propagated field dynamics in Gaussian pulse propagation in a linear, causally dispersive medium,” Phys. Rev. E, vol. 55, no. 2, pp. 1910–1921, 1997. 59. K. E. Oughstun and H. Xiao, “Failure of the quasimonochromatic approximation for ultrashort pulse propagation in a dispersive, attenuative medium,” Phys. Rev. Lett., vol. 78, no. 4, pp. 642– 645, 1997. 60. H. Xiao and K. E. Oughstun, “Failure of the group velocity description for ultrawideband pulse propagation in a double resonance Lorentz model dielectric,” J. Opt. Soc. Am. B, vol. 16, no. 10, pp. 1773–1785, 1999. 61. S. P. Sira, A. Papandreou-Suppappola, and D. Morrell, “Dynamic configuration of time-varying waveforms for agile sensing and tracking in clutter,” IEEE Trans. Signal Proc., vol. 55, no. 7, pp. 3207–3217, 2007. 62. V. Mitlin, Performance Optimization of Digital Communications Systems. Boca-Raton: Auerbach, 2006. Sect. 4.9. 63. C. G. B. Garrett and D. E. McCumber, “Propagation of a Gaussian light pulse through an anomalous dispersion medium,” Phys. Rev. A, vol. 1, pp. 305–313, 1970.

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64. M. D. Crisp, “Concept of group velocity in resonant pulse propagation,” Phys. Rev. A, vol. 4, no. 5, pp. 2104–2108, 1971. 65. G. C. Sherman and K. E. Oughstun, “Description of pulse dynamics in Lorentz media in terms of the energy velocity and attenuation of time-harmonic waves,” Phys. Rev. Lett., vol. 47, pp. 1451–1454, 1981. 66. S. Chu and S. Wong, “Linear pulse propagation in an absorbing medium,” Phys. Rev. Lett., vol. 48, pp. 738–741, 1982. 67. G. C. Sherman and K. E. Oughstun, “Energy velocity description of pulse propagation in absorbing, dispersive dielectrics,” J. Opt. Soc. Am. B, vol. 12, pp. 229–247, 1995. 68. K. E. Oughstun and J. E. Laurens, “Asymptotic description of ultrashort electromagnetic pulse propagation in a linear, causally dispersive medium,” Radio Sci., vol. 26, no. 1, pp. 245–258, 1991. 69. C. M. Balictsis, Gaussian Pulse Propagation in a Causal, Dispersive Dielectric. PhD thesis, University of Vermont, 1993. Reprinted in UVM Research Report CSEE/93/12-06 (December 31, 1993). 70. C. M. Balictsis and K. E. Oughstun, “Uniform asymptotic description of Gaussian pulse propagation of arbitrary initial pulse width in a linear, causally dispersive medium,” in UltraWideband, Short-Pulse Electromagnetics 2 (L. Carin and L. B. Felsen, eds.), pp. 273–283, New York: Plenum, 1994. 71. J. Peatross, S. A. Glasgow, and M. Ware, “Average energy flow of optical pulses in dispersive media,” Phys. Rev. Lett., vol. 84, no. 11, pp. 2370–2373, 2000. 72. M. Ware, S. A. Glasgow, and J. Peatross, “Role of group velocity in tracking field energy in linear dielectrics,” Opt. Exp., vol. 9, no. 10, pp. 506–518, 2001. 73. N. A. Cartwright and K. E. Oughstun, “Pulse centroid velocity of the Poynting vector,” J. Opt. Soc. Am. A, vol. 21, no. 3, pp. 439–450, 2004. 74. P. Pleshko, Transients in Guiding Structures. PhD thesis, New York University, 1969. 75. P. Pleshko and I. Pal´ocz, “Experimental observation of Sommerfeld and Brillouin precursors in the microwave domain,” Phys. Rev. Lett., vol. 22, pp. 1201–1204, 1969. 76. J. B. Gunn, “Bouncing ball pulse generator,” Electron. Lett., vol. 2, no. 5, pp. 172–173, 1966. 77. E. D. Torre, Magnetic Hysteresis. New York: IEEE Press, 1999. 78. M. E. Brodwin and D. A. Miller, “Propagation of the quasi-TEM mode in ferrite-filled coaxial line,” IEEE Trans. Microwave Theory Tech., vol. 12, no. 9, pp. 496–503, 1964. 79. H. Suhl and L. R. Walker, “Topics in guided wave propagation through gyromagnetic media,” Bell Syst. Tech. J., vol. 33, no. 9, pp. 1133–1194, 1954. 80. D. D. Stancil, “Magnetostatic wave precursors in thin ferrite films,” J. Appl. Phys., vol. 53, no. 3, p. 2658, 1982. 81. D. D. Stancil, Theory of Magnetostatic Waves. New York: Springer, 1993. 82. O. Avenel, M. Rouff, E. Varoquaux, and G. A. Williams, “Resonant pulse propagation of sound in superfluid 3 He–B,” Phys. Rev. Lett., vol. 50, no. 20, pp. 1591–1594, 1983. 83. E. Varoquaux, G. A. Williams, and O. Avenel, “Pulse propagation in a resonant medium: Application to sound waves in superfluid 3 He  B,” Phys. Rev. B, vol. 34, no. 11, pp. 7617–7640, 1986. ´ Falcon, C. Laroche, and S. Fauve, “Observation of Sommerfeld precursors on a fluid sur84. E. face,” Phys. Rev. Lett., vol. 91, no. 6, pp. 064502–1–064502–4, 2003. 85. J. Aaviksoo, J. Lippmaa, and J. Kuhl, “Observability of optical precursors,” J. Opt. Soc. Am. B, vol. 5, no. 8, pp. 1631–1635, 1988. 86. J. Aaviksoo, J. Kuhl, and K. Ploog, “Observation of optical precursors at pulse propagation in GaAs,” Phys. Rev. A, vol. 44, no. 9, pp. 5353–5356, 1991. 87. J. L. Birman and M. J. Frankel, “Predicted new electromagnetic precursors and altered signal velocity in dispersive media,” Opt. Comm., vol. 13, no. 3, pp. 303–306, 1975. 88. M. J. Frankel and J. L. Birman, “Transient optical response of a spatially dispersive medium,” Phys. Rev. A, vol. 15, no. 5, pp. 2000–2008, 1977. ¨ 89. S.-H. Choi and U. Osterberg, “Observation of optical precursors in water,” Phys. Rev. Lett., vol. 92, no. 19, pp. 1939031–1939033, 2004.

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90. R. R. Alfano, J. L. Birman, X. Ni, M. Alrubaiee, and B. B. Das, “Comment on ‘Observation of optical precursors in water’,” Phys. Rev. Lett., vol. 94, no. 23, p. 239401, 2005. 91. R. Albanese, J. Penn, and R. Medina, “Ultrashort pulse response in nonlinear dispersive media,” in Ultra-Wideband, Short-Pulse Electromagnetics (H. L. Bertoni, L. B. Felsen, and L. Carin, eds.), pp. 259–265, New York: Plenum, 1992. 92. Y. Okawachi, A. D. Slepkov, I. H. Agha, D. F. Geraghty, and A. L. Gaeta, “Absorption of ultrashort optical pulses in water,” J. Opt. Soc. Am. A, vol. 24, no. 10, pp. 3343–3347, 2007. 93. H. Jeong, A. M. C. Dawes, and D. J. Gauthier, “Direct observation of optical precursors in a region of anomalous dispersion,” Phys. Rev. Lett., vol. 96, no. 14, p. 143901, 2006. 94. Y. Park, M. H. Asghari, T. J. Ahn, and J. Aza˜na, “Transform-limited picosecond pulse shaping based on temporal coherence synthesization,” Opt. Express, vol. 15, no. 15, pp. 9584–9599, 2007. 95. R. Landauer, “Light faster than light?,” Nature, vol. 365, pp. 692–693, 1993. 96. G. Diener, “Superluminal group velocities and information transfer,” Phys. Lett. A, vol. 223, pp. 327–331, 1996. 97. A. Einstein, “Zur elektrodynamik bewegter k¨orper,” Ann. Phys., vol. 17, pp. 891–921, 1905.

Problems 15.1. Derive the approximate expressions given in (15.29)–(15.33) for the main signal, anterior pre-signal, and posterior pre-signal space–time points c .!c /, c1 .!c /, and c2 .!c /, respectively. 15.2. Prove that the energy velocity vE .!/ reduces to the group velocity vg .!/ in the limit as ı ! 0 in a single resonance Lorentz model dielectric. 15.3. Derive the approximate expressions given in (15.67)–(15.69) for the energy velocity in the below resonance, intra-absorption band, and high-frequency domains. 15.4. Prove that the dispersion model for the effective dielectric permittivity of a microstrip transmission line given in (15.94) is causal. 15.5. (a) Derive the approximate expression for .!n ;  / given in (15.137) for n D 0; 1; 2; 3; : : : and compare the real part with that at the distant saddle point SPC d . (b) Derive the approximate expression for .!n ;  / given in (15.142) for n D 0; 1; 2; 3; : : : and compare the real part with that at the near saddle point SPC n for  > 1 . 15.6. Obtain the uniform asymptotic descriptions of the generalized Sommerfeld Ags .z; t / and Brillouin Agb .z; t / precursor fields for the gaussian envelope pulse wavefield Ag .z; t / described by (15.159). 15.7. Prove (15.178), showing that the relative maxima of the real part m .! 0 / of the modified complex phase function defined in (15.166) for the gaussian envelope pulse occur at the real ! 0 -axis crossing points !rcmj , j D 1; 2; 5 that are defined in (15.168) for the saddle points SPmj of ˚m .!;  0 /.

Problems

677

15.8. Use the method of stationary phase to derive the asymptotic expression for the Sommerfeld precursor in an air-filled rectangular waveguide with dispersion relation given by (15.216). Compare this result with the limiting behavior obtained from the uniform asymptotic approximation of the Sommerfeld precursor in a single resonance Lorentz model dielectric given in (13.35) as ı ! 0. 15.9. Derive the approximate expression for the group velocity of surface waves given in (15.223), valid when kh  1.

Chapter 16

Physical Interpretations of Dispersive Pulse Dynamics

The causally interrelated effects of phase dispersion and absorption on the evolution of an electromagnetic pulse as it propagates through a homogeneous linear dielectric, particularly when the pulse is ultrawideband, developed originally by Sommerfeld [1] and Brillouin [2–4] in 1914 in support of Einstein’s 1905 special theory of relativity [5], Brillouin’s signal velocity description partially corrected by Baerwald [6] in 1930, and the theory finally completed in the 1970–1980’s by Oughstun [7] and Sherman [8–12] in a series of papers that forms the basis of the modern asymptotic theory have been described in detail in Chaps. 12–15 of this volume. The results show that after the pulse has propagated sufficiently far in the medium, its spatiotemporal dynamics settle into a relatively simple regime, known as the mature dispersion regime, for the remainder of the propagation. In this regime, the wavefield becomes locally quasimonochromatic with fixed local frequency and wavenumber in small regions of space–time which move with their own characteristic constant velocity. The theory provides accurate but approximate analytic expressions for the local wave properties at any given space–time point in the mature dispersion regime. The expressions are complicated, however, as is their derivation from a well-defined asymptotic theory (presented in Chap. 10), and neither do the results nor their derivations provide complete insight into the physical reasons for the wavefield having the particular local space–time properties it does have in the various subregions of space moving with specific velocities. The mature dispersion regime is well known in the theory of propagation of rather general linear waves in homogeneous dispersive media in which there is no absorption or gain. It is exhibited by all waves whose monochromatic spectral components are described by the Helmholtz equation with real propagation factor; examples include electromagnetic, acoustic, elastic, and gravity waves in lossless, gainless linear systems. Furthermore, a physical explanation is available for the local properties of all of these waves that is based on the concept of the group velocity of time-harmonic waves [13–15]. When either (frequency-dependent) absorption or gain is present in the medium, however, the group velocity description breaks down. As stated by L. B. Felsen in his 1976 review paper [16]: The concept of group velocity vg D .dk=d!/1 for describing the energy propagation characteristics of a wave packet with small frequency spread becomes obscured in a lossy medium since vg is now complex. Furthermore, different propagation speeds may K.E. Oughstun, Electromagnetic and Optical Pulse Propagation 2: Temporal Pulse Dynamics in Dispersive, Attenuative Media, Springer Series in Optical Sciences 144, c Springer Science+Business Media, LLC 2009 DOI 10.1007/978-1-4419-0149-1 16, 

679

680

16 Physical Interpretations of Dispersive Pulse Dynamics

be associated with different features of the pulse (for example, the spatial and temporal maxima). Although attempts have been made to extend the notion of group velocity to wave packets in dissipative media by examining such quantities as Œ
This is a severe limitation of a basic kind because a lossless, gainless dispersive system is noncausal [17]. Hence, the widely accepted physical understanding of the details of dispersive pulse propagation in terms of the well-known (and all too often abused) group velocity description is strictly confined to the special case of no absorption or gain which is itself fundamentally unphysical. Curiously enough, the complex group velocity is real-valued at the saddle points. This can easily be seen by differentiating the first form of the expression for the complex phase function in Eq. (12.3) with respect to ! and equating the result to zero at the saddle points. In spite of the fact that this uniform asymptotic theory provides an accurate, detailed, causal description of the entire pulse evolution in the mature dispersion regime, all too many in the engineering and applied science community refuse to accept the theory because of its mathematical complexity, preferring instead to rely upon the ill-founded group velocity description which, by its very nature, is inapplicable to dispersive attenuative media, and hence, to causally dispersive media and systems. The numerical analysis presented by Xiao and Oughstun [18, 19] (see Sect. 11.5) clearly shows that the group velocity description is restricted to the immature dispersion regime z 2 Œ0; zc / that describes the initial pulse evolution, the asymptotic theory holding when z > zc , where zc D Ofzd g is some critical propagation distance that depends both on the material dispersion and the pulse type. For example, for a Heaviside step function signal with fixed carrier frequency !c 2 Œ0; 1/, zc D zd ˛ 1 .!c /. Beginning in 1981, Sherman and Oughstun published a series of papers [20–22] that provided the first physical explanation of the local wave properties of electromagnetic pulses propagating in causal, dispersive absorptive dielectrics in the mature dispersion regime. The explanation is similar to that given by the group velocity description for lossless, gainless energy in time-harmonic waves, but is fundamentally different from it because the dispersive attenuation of the wave energy is included. This is the only such physical description of the details of pulse dynamics that is known for any dispersive system that includes absorption (or gain) or that is causal and that remains valid in the ultrawideband signal, ultrashort pulse limits.1 In addition to explaining the details of the local space–time behavior of the pulse in simple, physical terms, this energy velocity description also provides a simple mathematical algorithm for predicting those details quantitatively. A detailed derivation of this energy velocity description is given in the first section of this chapter. This 1

Remember that an ultrashort pulse is also ultrawideband, but that an ultrawideband signal need not be ultrashort.

16.1 Energy Velocity Description of Dispersive Pulse Dynamics

681

is followed by a reformulation of the classical group velocity description through a direct, unaltered application of the asymptotic method of stationary phase, followed by a signal analysis of dispersive pulse propagation due to Stratton [23] that separates the propagated pulse dynamics into steady state and transient responses, as has recently been updated by H. T. Banks [24].

16.1 Energy Velocity Description of Dispersive Pulse Dynamics Consider a pulsed, plane electromagnetic wavefield with real-valued (electric or magnetic) wavefield A.z; t / linearly polarized in the xy-plane and travelling in the positive z-direction through a dispersive dielectric which occupies the half-space z > 0. The wavefield is taken to be zero for all t < 0 and therefore can expressed in a Fourier–Laplace representation as Z

iaC1

A.z; t / D

Q !/e i!t d!; A.z;

(16.1)

ia1

where a is a positive constant for the Bromwich contour extending along the straight Q !/ satisfies the Helmholtz from ia  1 to ia C 1. The spectral wave function A.z; equation   Q !/ D 0 r 2 C kQ 2 .!/ A.z; (16.2) throughout the half-space z > 0. The angular frequency dispersion of the comQ plex wavenumber k.!/ .!=c/n.!/ is specified by the frequency dispersion of  1=2 the complex index of refraction n.!/ ..!/=0 /.c .!/=0 of the medium, where c .!/ D .!/Ci k4k .!/=!. The analysis presented here considers a simple dielectric, in which case .!/ D 0 ,  .!/ D 0, and c .!/ D .!/, which is taken here to be described by the single resonance Lorentz model, so that  n.!/ D 1 

b2 2 !  !02 C 2i ı!

1=2 ;

(16.3)

where !0 is the undamped angular resonance frequency, b the plasma frequency, and ı  0 the phenomenological damping constant of the medium. As in the classical asymptotic theory, it is assumed here that the pulsed, plane wavefield satisfies the boundary value A.0; t / D f .t /;

(16.4)

where f .t / is a real-valued function that identically vanishes for all negative time [i.e., f .t / D 0 for all t < 0]. Because of its central importance in linear system theory, the analysis focuses on the impulse response of the dispersive medium, in which case f .t / D A0 fı .t / (see Sect. 11.2.1) with fı .t / D ı.t /, where A0 is a constant and ı.t / the Dirac delta function, the impulse response then being given

682

16 Physical Interpretations of Dispersive Pulse Dynamics

by A.t /=A0 . The exact integral solution to this boundary value problem can be expressed in the form [cf. (12.1)] A.z; t / D

1 2

Z

iaC1

fQ.!/e .z=c/.!;/ d!;

(16.5)

ia1

where a is a real constant greater than the abscissa of absolute convergence [see (C.12) of Appendix C in Vol. 1] for the function f .t /, fQ.!/ D

Z

1

f .t /e i!t dt;

(16.6)

  .!; / D i! n.!/   ;

(16.7)

1

and where with

ct (16.8) z being a dimensionless space–time parameter that, for any fixed value of  , travels with the wavefield at the fixed velocity z=t D c=. Sommerfeld’s relativistic causality theorem [1] (see theorem 6, Sect. 13.1) proves that it directly follows from the exact integral solution in (16.5) that the propagated wavefield A.z; t / identically vanishes for all superluminal space–time points  < 1, that is D

A.z; t / D 0;

8 t < z=c;

(16.9)

in keeping with Einstein’s special theory of relativity [5]. The physical description presented here is developed for initial pulse functions f .t /DA.0; t / that possess temporal Fourier spectra fQ.!/ that are entire functions of the complex angular frequency variable !. In that case the propagated pulse wavefield may be expressed in the form A.z; t / D As .z; t / C Ab .z; t /;

8 t  z=c:

(16.10)

The asymptotic behavior as z ! 1 of these two field components is given by [cf. (13.15)] ( ) 1=2 z 2c .! ;/ s Q As .z; t /  2< (16.11) f .!s /e c z 00 .!s / for  > 1, where  0 .!s / D 0 and  00 .!s / ¤ 0, and by [cf. 13.73) and (13.94), respectively] ( ) 1=2 z 2c Ab .z; t /  < fQ.!b /e c .!b ;/ ; 1 <  < 1 ; (16.12) z 00 .!b / ( ) 1=2 z 2c .! ;/ b Q Ab .z; t /  2< (16.13) ;  > 1 ; f .!b /e c z 00 .!b /

16.1 Energy Velocity Description of Dispersive Pulse Dynamics

683

where  0 .!b / D 0 and  00 .!b / ¤ 0. Here !s !SP C . / denotes the first-order d distant saddle point in the right half of the complex !-plane [see (12.201)] and !b !SPC . / denotes the first-order near saddle point along the imaginary axis d for 1 <  < 1 [see (12.228)] and then in the right half of the complex !-plane for  > 1 [see (12.239)]. The asymptotic expressions given in (16.11)–(16.13) are referred to as nonuniform asymptotic results because they break down at certain critical space–time points. In particular, the right-hand sides of (16.12) and (16.13) become infinite at  D 1 and give discontinuous asymptotic behaviors on opposite sides of that space– time point because  00 .!b ; 1 / D 0. Although uniform asymptotic results have been derived in Sects. 13.2.2 and 13.3.2, they are not necessary in the initial analysis as the nonuniform expressions are much simpler to work with. The uniform results are employed in the final analysis, however, in order to provide an energy velocity description that is valid for all   1 .

16.1.1 Approximations Having a Precise Physical Interpretation Approximations of (16.11)–(16.13) are now obtained which have a precise physical interpretation and result in a simple physical model of dispersive pulse dynamics in the mature dispersion regime. These approximations are valid provided that the phenomenological damping constant ı is much smaller than both the angular resonance frequency !0 and plasma frequency b of the Lorentz model dielectric, viz., ı  !0

&

ı  b:

(16.14)

This requires that the medium not be too highly absorbing. This requirement is not overly restrictive as it is satisfied by Brillouin’s choice of the medium parameters [see (12.58)] for which ı=!0 D 0:07 and ı=b ' 0:0626, where this medium is so absorbing that it would be considered to be opaque at nearly all nonzero, finite frequency values. To obtain the desired simplifications, Sherman and Oughstun [21,22,25] replaced the saddle points appearing in the asymptotic expressions by other frequencies which yield approximately the same results but which have clearer physical interpretations. In particular, the saddle points !s !SPC . / and !b !SPC ./ appearing d d in (16.11) and (16.13) are replaced by specific real angular frequencies leading to quasi-time-harmonic (quasimonochromatic) waves with local frequency, phase, and amplitude which are easily understood in physical terms. Similarly, the saddle point !b !SP C . / appearing in (16.12) is replaced with a specific purely imaginary fred quency leading to a nonoscillatory field with local amplitude and growth rate which is also easily understood in physical terms. The analysis presented here is based entirely upon the earlier published analysis of Sherman and Oughstun [21, 22, 25].

684

16.1.1.1

16 Physical Interpretations of Dispersive Pulse Dynamics

The Quasimonochromatic Contribution

To identify the real frequencies of interest, attention is focused on the attenuation of the wavefield with increasing propagation distance z > 0. It is important that the resultant approximation have the correct attenuation because the theory is centered on the properties of an exponentially decaying wavefield after it has propagated a large distance (relative to some characteristic absorption depth) in the dispersive medium. Hence, it is desired to determine those time-harmonic waves (with real frequencies) that are attenuated in the dispersive medium at the same rate as the wavefield components given in (16.11) and (16.13). To that end, first define the notation2 that gr and gi represent, respectively, the real, and imaginary parts of the arbitrary complex quantity g. Then, for any fixed space–time point   1, the attenuation with increasing propagation distance z > 0 of a wave of the form exp f.z=c/.!; /g for complex ! is determined by r .!; /. For any given space–time value   1, define !Ej as the real frequency value nearest the saddle point !j that satisfies r .!Ej .// D r .!j ;  /

(16.15)

with j D s; b. A time-harmonic plane wave with real angular frequency !Ej then has the same attenuation as the pulsed wave described by (16.11) and (16.13) in the mature dispersion regime. The locations of these real angular frequency values !Es D !Es . / and !Eb D !Eb . / in the complex !-plane are indicated in Fig. 16.1 for some fixed value  > 1 as the corresponding intersections with the real ! 0 -axis of the isotimic contours of r .!; / D  r .!j ;  / that pass through the saddle points !j , j D s; b. ''

r

Eb

'

branch cut

r

Es

b

branch cut

'

b

'

s

r

r

s

Fig. 16.1 Location of the real angular frequencies !Eb D !Eb . / and !Es D !Es . / relative to the locations of the near and distant saddle points !b . / and !s . / in the complex !-plane for a fixed space–time value  > 1 . The dashed curves describe the isotimic contours of constant r .!;  / D
Notice that the variable notation ! 0 D
16.1 Energy Velocity Description of Dispersive Pulse Dynamics

685

The angular frequencies !Ej so defined are intimately connected to the physics of the propagation of time-harmonic (monochromatic) waves in the dispersive medium. It is shown in Sect. 16.1.2 that to a good approximation, they satisfy z vE .!Ej / D ; t

(16.16)

provided that the inequalities in (16.14) are satisfied, where vE .!/ is the velocity of energy transport (or energy transport velocity) for a time-harmonic wave with real-valued angular frequency ! that is defined by (see Sect. 5.2.6 of Vol. 1) vE .!c /

hjSji ; hUtot i

(16.17)

where the magnitude of the Poynting vector S.r; t / and the total energy density Utot .r; t / have been averaged over one oscillation cycle of the wavefield. For a single resonance Lorentz model dielectric, the energy velocity is given by [see (15.63)] vE .!c / D

c ; nr .!c / C !ıc ni .!c /

(16.18)

as derived by Loudon [26]. This result then establishes a fundamental connection between the attenuation of the pulse and the physics of the problem. In particular, it is seen that in the mature dispersion regime, the attenuation of As .z; t / as the observation point moves with fixed velocity (i.e., for fixed  ) is approximately the same as the attenuation of a time-harmonic wave with real frequency that has energy velocity equal to the velocity of that point of observation. The same is true of Ab .z; t / for all  > 1 . Although this result doesn’t say anything about the phase of the wavefield or the amplitude of the exponential term, it does provide a description of the main dynamics of the energy of the pulse in simple physical terms. Moreover, it provides a simpler mathematical algorithm for calculating those dynamics than has been available previously. This result is now extended to include the rest of the pulse dynamics. First, consider the amplitude of the exponential term. If the damping constant ı is not too large [as required by the two inequalities in (16.14)], the slowly varying functions of ! appearing in (16.11) and (16.13) can be approximated by replacing !j by !Ej with j D s; b. This can be seen by applying as follows some general properties of the saddle point locations for a single resonance Lorentz model medium (see 0 Sect. 12.3.1). The distant saddle q point !s is less than 2ı below the real ! -axis and its real part is greater than !12  ı 2 , where !12 !02 C b 2 . Even for the highly absorbing medium considered by Brillouin in his classic analysis [2, 4], the former value is much smaller than the latter. Hence, the imaginary part of the distant saddle point !s can be neglected in comparison to its real part. The near saddle point 0 !b is less than ı below the real q ! -axis and its real part starts at zero for  D 1

and increases rapidly toward

!02  ı 2 as  increases above 1 . Again, the values

686

16 Physical Interpretations of Dispersive Pulse Dynamics

are such that even for Brillouin’s choice of the medium parameters, the imaginary part of !b can be neglected compared to its real part in slowly varying functions for sufficiently large values of  . Next, notice that the relevant isotimic contour of constant r .!/ passing through each saddle point is vertical in the vicinity of that saddle point, as depiced in Fig. 16.1. This means that the real part of the saddle point location !j can be approximated by !Ej for values of ı satsifying the inequalities in (16.14). Again, this approximation is found to be valid for the highly absorbing case represented by Brillouin’s choice of the medium parameters. Combination of these two approximations for each saddle point, it is concluded that the slowly varying functions of ! appearing in (16.11) and (16.13) can be approximated by replacing !j by !Ej , provided that  > 1 is bounded away from 1 in (16.13). Hence, the asymptotic expressions given in (16.11) and (16.13) may be, respectively, approximated as ( As .z; t /  2<

2c z 00 .!Es /

1=2

) fQ.!Es /e

z c .!s ;/

fQ.!Eb /e

z c .!b ;/

(16.19)

as z ! 1 with  > 1, and ( Ab .z; t /  2<

2c z 00 .!Eb /

1=2

) (16.20)

as z ! 1 with  > 1 bounded away from 1 . Greater care must be taken in approximating the exponential function in the above expressions because it is a more rapidly varying function of !. The attenuation has already been expressed in terms of !Ej in (16.15). That result can be expressed in terms of the complex index of refraction n.!/ D nr .!/ C ini .!/ by noting that r .! 0 / D ! 0 ni .! 0 / for real ! 0 . Equation (16.15) can therefore be written as (16.21) r .!j / D !Ej ni .!Ej / D c kQi .!Ej /: The oscillatory portion of the exponential phase is determined by the expression h i z z z i i .!/ D i ! 0 t C ! 0 nr .!/  ! 00 ni .!/ c c c

(16.22)

evaluated at the relevant saddle point ! D !j . If !j is replaced by !Ej in the index of refraction terms and if !j0 is replaced by !Ej elsewhere in this expression, it becomes h i z z (16.23) i i .!j / ' i !Ej t C zkQr .!Ej /  !j00 ni .!Ej / : c c Combination of the expressions given in (16.21) and (16.22), the quantity appearing in the exponential of the integrands of (16.19) and (16.20) can then be approximated by h i z Q Ej /  i z ! 00 ni .!Ej / : (16.24) i .!j / ' i !Ej t C zk.! j c c

16.1 Energy Velocity Description of Dispersive Pulse Dynamics

687

In the case when j D b, (16.24) applies when  > 1 and is accurate only when  is not too close to 1 . The last term appearing in (16.24) includes the location of the saddle point !j00 . That term is not very important, however, because it is negligible except when !Ej is in the absorption band and it contributes only a small phase shift to the field even then. Hence, that term is ignored in this physical model of dispersive pulse dynamics with the understanding that the phase of the wavefield that is obtained using this model may be slightly shifted when !Ej is in the absorption band. In the same spirit, the approximations given in (16.20) and (16.24) with j D b are applied for all  > 1 with the understanding that they become inaccurate as  approaches the critical value 1 from above. The small space–time interval where this problem exists decreases as ı decreases relative to !0 . The numerical calculations of the dynamical wavefield using this physical model that are presented later in this section display the effects of these simplifications in the highy absorbing case. 16.1.1.2

The Nonoscillatory Contribution

Attention is now turned to the contribution to the propagated wavefield that is given in (16.12). This contribution is nonoscillatory and is important only for space–time points .z; t / in the region about the space–time point given by z D vE .0/: t

(16.25)

The group velocity or phase velocity for time-harmonic waves with zero frequency could be applied in (16.25) equally well because all three velocities are equal for zero frequency. Hence, this field is essentially quasistatic. Consider then the nonoscillatory electromagnetic fields of the form h i QQ !/z w.z; t; !/ Q D exp !t Q  k. Q

(16.26)

where the growth rate !Q is a real-valued constant. This wavefield is a solution to QQ !/ Maxwell’s equations in a single resonance Lorentz medium if k. Q is given by !Q QQ !/ Q !/ k. Q D ik.i Q D c

s 1C

!Q 2

b2 ; C !02 C 2ı !Q

(16.27)

which is also real-valued. Because !b is purely imaginary over the initial space–time domain 1    1 , the exponential term appearing in the integrand of (16.12) is a wavefield of the form given in (16.26) and (16.27) except that the growth rate is a function of position and time. Equation (16.16) for the near saddle point !b can be written in the form !b D i !Q b ;

(16.28)

688

16 Physical Interpretations of Dispersive Pulse Dynamics

where !Q b is the real-valued solution to the equation vQ G .!Q b / D

z t

with vQ G .!/ Q defined as Q vQ G .!/

QQ !/ dk. Q d!Q

(16.29) !1 ;

(16.30)

which can be taken as the group velocity of the nonoscillatory waves given in (16.26). This identification does not provide much physical insight, however, because the group velocity of a nonoscillatory wave is more a mathematical object rather than a physical one. To connect it with a more physical quantity, consideration is now given to the velocity of energy transport in a nonoscillatory wave. Take the velocity of energy flow in fields of the form given in (16.26) to be given by (16.17) with the change that the Poynting vector and energy density are not to be time-averaged because the field is nonoscillatory. With this definition, it is shown in [22] that an electromagnetic field of the form given in (16.26) and (16.27) has Q given by energy velocity vQ E .!/ vQ E .!/ Q D

Q cn.i!/Q Q 2 .!/ ; 2 2 n .i!/Q Q .!/ Q  b 2 ı !Q

(16.31)

where Q Q.!/ Q D !Q 2 C !02 C 2ı !; s b2 n.i!/ Q D 1C : Q.!/ Q

(16.32) (16.33)

It is also shown in [22] that the group velocity of the nonoscillatory waves is given by Q cn.i !/Q Q 2 .!/ Q D 2 : (16.34) vQ G .!/ 2 2 n .i!/Q Q .!/ Q  b ı !Q  b 2 !Q 2 Comparison of (16.31) and (16.34) shows that the two velocities differ only by the additional term b 2 !Q 2 appearing in the denominator of (16.34). This term is negligible for all growth rates that satisfy the inequality !Q  !0 I

(16.35)

notice that this is a sufficient condition but that it is far from necessary. Because the attenuation with propagation distance z of the nonoscillatory waves increases with increasing growth rate, it is clear that for z sufficiently large, the nonoscillatory waves which do not satisfy the inequality given in (16.35) will be negligible compared to those that do. In particular, it is shown in [22] that if z satisfies the inequality

16.1 Energy Velocity Description of Dispersive Pulse Dynamics

z>

4Kc !1 .!0 C ı/2 ; r2 ı!0 b 2

689

(16.36)

then w.z; t; !/=w.z; Q t; 0/ < e K for !Q  r!0 , where K and r are arbitrary positive constants which can be chosen to give as good an approximation as desired when neglecting the nonoscillatory waves that do not satisfy the inequality given in Q by vQ E .!/. Q For Brillouin’s choice of the (16.35) well enough to approximate vQ G .!/ medium parameters [see (12.58)] and with K D 2 and r D 0:1, the inequality given in (16.36) gives z > 0:012 cm. It is shown in the next subsection through a numerical example that this approximation is useful even for smaller values of the propagation distance z > 0. Hence, for sufficiently large values of the propagation distance z > 0, a good approximation of the near saddle point location over the initial space–time domain 1    1 can be obtained by taking !Q b as the solution to the equation vQ .!/ Q D

z t

(16.37)

that is closest to the saddle point. One more approximation is useful for the formulation of the physical model. It follows from (12.225) that the critical space–time point 1 can be approximated by the space–time value 0 defined by 0 D

c c D D n.0/; vE .0/ vQ E .0/

(16.38)

which is simply the index of refraction for a static field. The validity of the approximation decreases with increasing ı but is still very good even for the parameter values used by Brillouin [2, 4] (in which case 0 D 1:500 and 1 1:503). This result implies that the second precursor field Ab .z; t / changes from the nonoscillatory form with zero growth rate to the time-harmonic form with zero oscillation frequency at the observation point that is traveling with velocity vE .0/ D vQ E .0/.

16.1.2 Physical Model of Dispersive Pulse Dynamics The physical model of dispersive pulse dynamics is now presented based on the previous results. The simpler nonuniform model is first developed followed by the more complicated, but more accurate, uniform model [22]. 16.1.2.1

The Nonuniform Physical Model

As the propagation distance z tends to infinity with fixed  D ct=z, the wavefield Q t / which, for ı A.z; t / can be expressed as the real part of a complex wavefield A.z; much smaller than both !0 and b, can be approximated as Q t / AQTH .z; t / C AQQS .z; t /; A.z;

(16.39)

690

16 Physical Interpretations of Dispersive Pulse Dynamics

where 1=2 2c Q fQ.!Ej /e i .k.!Ej /z!Ej t / ; z 00 .!Ej / j Ds;b 1=2    QQ 2c !Q E tk. !Q E /z Q Q : AQS .z; t / D f .i !Q E /e z 00 .i !Q E /

AQTH .z; t / D 2

X 

(16.40)

(16.41)

Here, the angular frequency values !Ej are defined to be the nonnegative real-valued solutions of the equation z c (16.42) vE .!Ej / D ;  t and !Q E is defined to be the positive real-valued solution to vQ E .!E /

z c D :  t

(16.43)

The field quantity AQTH .z; t / given in (16.40) is the time-harmonic component discussed in the preceding subsection. It is shown in the following that the sum in (16.40) includes only one term, the Sommerfeld precursor As .z; t /, over the initial space–time domain 1 <  < 0 , and includes two terms, the Sommerfeld precursor As .z; t / and the Brillouin precursor Ab .z; t /, for   0 . The field quantity AQQS .z; t / given in (16.40) is the nonoscillatory component discussed in the preceding subsection. These results constitute a physical model [21,22] of dispersive pulse propagation because they can be used to describe the local dynamics of the pulse in physical terms. They are similar to the mathematical results that lead to the group velocity description that is valid for lossless, gainless systems but are different in three respects: 1. The nonoscillatory contribution is included in addition to the time-harmonic contribution, 2. The energy velocity is used to determine the pulse dynamics in place of the group velocity, 3. The pulse dynamics are strongly affected by the relative attenuation of the various time-harmomic and nonoscillatory contributions. It follows from (16.39)–(16.43) that the principal quantities that determine the local dynamics of the dispersive pulse evolution are the energy velocity vE and attenuation ˛ of time-harmonic waves as functions of frequency and the energy velocity vQ E and attenuation coefficient ˛Q of nonoscillatory waves as a function of the growth rate !. Q Graphs of these functions for a Lorentz model medium with Brillouin’s parameter values [see (12.58)] are given, respectively, in Figs. 16.2–16.5. The energy velocities for the oscillatory and nonoscillatory waves were computed using (16.18) and (16.31), respectively. The attenuation coefficients of the waves are the rates of exponential decay of the wave amplitudes as the propagation distance z > 0 increases with constant  . For time-harmonic waves with real frequencies,

16.1 Energy Velocity Description of Dispersive Pulse Dynamics

691

1.0

0.8

vE /c

0.6

0.4

0.2

0 0

5

10

15

(X1016 r/s)

Fig. 16.2 Angular frequency dispersion of the normalized energy velocity vE .!/=c of a monochromatic electromagnetic wave with angular frequency ! in a single resonance Lorentz model medium

(X108/m)

3.0

2.0

1.0

0 0

5

10

15

(X1016 r/s)

Fig. 16.3 Angular frequency dispersion of the attenuation coefficient ˛.!/ of a monochromatic electromagnetic wave with angular frequency ! in a single resonance Lorentz model medium

692

16 Physical Interpretations of Dispersive Pulse Dynamics 0.70

0.69

v~E /c

0.68

0.67

0.66

0.65 0

0.2

0.4 0.6 ~ (X1016 r/s)

0.8

1.0

Fig. 16.4 Normalized energy velocity vQE .!/=c Q of the nonoscillatory wave components in a single resonance Lorentz model medium

5.0

~(X105/m)

4.0

3.0

2.0

1.0

0 0

0.2

0.4

0.6 ~ (X1016 r/s)

0.8

1.0

Fig. 16.5 Attenuation coefficient ˛. Q !/ Q of the nonoscillatory wave components in a single resonance Lorentz model medium

16.1 Energy Velocity Description of Dispersive Pulse Dynamics

693

the attenuation coefficient ˛.!/ is given by the imaginary part of the complex Q wavenumber k.!/ D ˇ.!/Ci˛.!/ D .!=c/n.!/ with complex index of refraction n.!/ given here by (16.3). For the nonoscillatory waves the attenuation coefficient ˛. Q !/ Q is given by [22] QQ !Q /   !Q ˛. Q !Q E / k. E E c QQ !Q /  !Q E ; D k. E vQ E .!Q E /

(16.44) (16.45)

QQ !/ where k. Q is given by (16.31). Q is given by (16.27) and vQ E .!/ A qualitative description of all of the main features of dispersive pulse dynamics can be obtained through a careful consideration of these four figures (Figs. 16.2– 16.5). Notice first from Fig. 16.2 that there is only one solution to (16.42) over the initial space–time domain 1 <  < 0 , where 0 D 1:5 for Brilllouin’s choice of the medium parameters, so that the summation appearing In (16.40) for the timeharmonic contribution includes only one term. The angular frequency !E of this term is large for  near 1 and decreases monotonically with increasing  . From Fig. 16.3, it is seen that as the frequency decreases from a large value, the attenuation increases. This means that this high-frequency, quasi-time-harmonic term decreases in frequency and amplitude as  increases, in agreement with the asymptotic description of the Sommerfeld precursor (see Sect. 13.2). Continuing on with the physical description, it is noticed in Fig. 16.4 that there is one positive solution !Q E of (16.43) over the initial space–time domain 1 <  < 0 . The growth rate !Q E decreases with increasing , tending toward 0 as  approaches 0 from below. From Fig. 16.5, it is seen that the attenuation with propagation distnce z > 0 of the nonoscillatory contribution is large for large growth rate, but gradually decreases to 0 as !Q E approaches 0. Hence, the nonoscillatory contribuQ t / given in (16.39) is negligible in tion AQQS .z; t / to the complex wavefield A.z; comparison to the Sommerfeld precursor contribution for small  > 1, but gradually increases in amplitude until it dominates the Sommerfeld precursor field as  approaches 0 from below. This marks the arrival of the Brillouin precursor. Finally, notice from Fig. 16.4 that there is no positive solution to (16.43) for   0 . Hence, the nonoscillatory contribution no longer contributes to the complex waveQ t / when   0 . Notice that the nonoscillatory contribution with !Q E D 0 field A.z; at  D 0 has been disallowed by including only positive solutions to (16.43). This has been done so as to avoid the inclusion of the zero frequency solution twice, as it is included in the time-harmonic contribution [because all nonnegative solutions of (16.42) have been included]. Returning to Fig. 16.2, notice that for   0 , there are now two nonnegative solutions of (16.42). The first is a high-frequency solution which is the continuation of the Sommerfeld precursor. The second solution is a low-frequency solution with angular frequency which begins at zero for  D 0 and then increases with increasing  . Consideration of Fig. 16.3 shows that the attenuation of this wave is much less than that for the high-frequency solution. Hence, this low-frequency

694

16 Physical Interpretations of Dispersive Pulse Dynamics

wave contribution dominates the sommerfeld precursor as z increases. Consideration of Fig. 16.3 shows also that the attenuation of this wave increases with increasing !, causing the wave to decrease in amplitude with increasing  . Hence, this contribution has increasing frequency and decreasing amplitude with increasing  , in agreement with the asymptotic description of the Brillouin precursor (see Sect. 13.3). In addition to providing a description of the qualitative pulse behavior in physical terms, the physical model gives approximate analytical expressions which predict the propagated pulse dynamics quantitatively without requiring the evaluation of the saddle point locations in the complex !-plane. Of course, the model does require the solution of (16.42) and (16.43), which are transcendental equations, but these equations are simpler to deal with than the saddle point equations because they involve only real quantities. To investigate the accuracy of the nonuniform physical theory, these expressions have been evaluated numerically for the case of the delta-function pulse A.0; t / D ı.t / in a single resonance Lorentz model medium using Brillouin’s choice of the material parameters with a propagation distance of z D 1 )m. Equations (16.42) and (16.43) were then solved numerically using Mueller’s method [22]. The numerical results of the physical model are described by the solid curve in Fig. 16.6. For comparison, the nonuniform asymptotic result given by (16.10)–(16.13) for the

Ad (z,t)

1.0X1016

0

−1.0X1016 1.0

1.2

1.4

1.6

1.8

2.0

q = ct/z

Fig. 16.6 Nonuniform results for the propagated wavefield due to an input delta function pulse in a single resonance Lorentz model medium. The solid curve is the result of the physical model and the dashed curve is the result of the nonuniform asymptotic theory

16.1 Energy Velocity Description of Dispersive Pulse Dynamics

695

same parameter values is described by the dotted curve in the figure. These latter numerical results used numerically determined saddle point locations. It is apparent from the graph in Fig. 16.6 that the accuracy of the physical model is very good. The main discrepancy between the physical model and asymptotic description is a minor shift in the phase of the Brillouin precursor that has been discussed in connection with the approximations made leading to (16.24). The scale in Fig. 16.6 was chosen so that the transition between the two precursors was clearly displayed. Because the field amplitudes are off this scale for small space–time values  near unity in that figure, the same numerically determined wavefield values are replotted in Fig. 16.7 for small  near the luminal space–time point  D 1 with an appropriate vertical scale. The results of the physical model are almost indistinguishable from the nonuniform asymptotic results. The large discontinuous peak that occurs in both the physical model and asymptotic results presented in Fig. 16.6 for space–time values  near 0 is a consequence of the nonuniform nature of the results as discussed in Sect. 16.1 following (16.10)– (16.13). This behavior is an artifact of the nonuniform asymptotic analysis which makes the results invalid in that space–time region. To obtain results that are valid there, it is necessary to employ the uniform asymptotic description of the propagated wavefield.

Ad (z,t)

5.0X10

17

0

−5.0X1017 1.0

1.01

1.02

1.03 q = ct/z

1.04

1.05

Fig. 16.7 Expanded view of the nonuniform results for the propagated wavefield due to an input delta function pulse in a single resonance Lorentz model medium for space–time values  near 1. The solid curve is the result of the physical model and the dashed curve is the result of the nonuniform asymptotic theory

696

16.1.2.2

16 Physical Interpretations of Dispersive Pulse Dynamics

The Uniform Physical Model

The asymptotic results (and the resulting physical model derived from them) that have been used so far are nonuniform in the vicinity of two critical space–time points: 1.  D 1, which corresponds to the luminal arrival of the Sommerfeld precursor field, and 2.  D 1 > 1, which occurs during the arrival of the Brillouin precursor. This means that in order for the results to provide useful approximations for large z > 0, the propagation distance must be taken larger and larger as  approaches one of these critical space–time values. Furthermore, the functional form of the results are different for space–time values  on opposite sides of a critical value. These difficulties may be removed from the physical model by using the appropriate uniform asymptotic approximations described in Chap. 10 (see Sects. 10.2 and 10.3) and applied in Sects. 13.2.2 and 13.3.2 of Chap. 13. The propagated wavefield is then expressed in terms of special functions (e.g., the Bessel and Airy functions) which are more complicated than the exponential functions occurring in the nonuniform expressions. The arguments of these functions involve the same saddle points as applied in the nonuniform analysis. As the value of  tends away from either one of the critical values, the uniform results tend asymptotically to the same expressions given in the nonuniform description. To employ the uniform asymptotic results of Chap. 13 in the physical model, the saddle points occurring in the uniform asymptotic expressions must be replaced with the approximations used in the nonuniform physical model, these being given by the real solutions of (16.42) and (16.43). Because the asymptotic expressions involve Bessel and Airy functions instead of quasi-time-harmonic waves and nonoscillatory exponentially growing waves, the physical interpretation of the resulting description is not as apparent as that described by the nonuniform description. Nevertheless, such functions frequently arise in the theory of wave propagation and can certainly be considered as representing physical waves. Moreover, in the uniform physical model, the arguments of these special functions are real-valued, involving real frequencies and growth rates which are clearly connected with the physics of both time-harmonic and nonoscillatory waves in the dispersive medium through (16.42) and (16.43). For the case of the delta function pulse A.0; t / D ı.t /, one really doesn’t have to take the trouble to make the physical model uniform in the vicinity of the luminal space–time point  D 1 because the nonuniform model yields reasonably accurate results for values of  quite close to 1. This is demonstrated in Fig. 16.8 which displays the results of a numerical evaluation3 of the exact integral representation of the propagated wavefield given in (16.5) with fQ.!/ D 1 for the same 3

The numerical algorithm used to evaluate the integral representation of the propagated pulse wavefield for this and subsequent examples in this section is described in [22]. As discussed in that reference, the algorithm begins to produce numerical artifacts in the results as  approaches 1

16.1 Energy Velocity Description of Dispersive Pulse Dynamics

697

17

Ad (z,t)

5.0X10

0

−5.0X1017 1.0

1.01

1.02

1.03

1.04

1.05

q = ct/z

Fig. 16.8 Propagated wavefield evolution due to an input delta function pulse A.0; t / D ı.t / for space–time values near the speed of light point at  D 1. The solid curve describes the result of a numerical integration of the exact integral representation, and the dotted curve describes the result given by the nonuniform physical model

parameter values used in Fig. 16.7. Comparison of Fig. 16.8 with the wavefield plot in Fig. 16.7 shows that the latter result has the same form but with a high-frequency ripple superimposed. It has been verified in [22] that this high-frequency ripple is an artifact of the numerical algorithm by showing that its frequency changed when the numerical value of the initial sum index k was changed in the implementation of this inverse Laplace transform algorithm [27], whereas the rest of the curve remained unchanged. To compare the results of the physical model with the numerical integration results, the results of the nonuniform physical model are presented in Fig. 16.8 by the dotted curve. That curve is barely visible following along the centerline of the rippled curve in Fig. 16.8. This then demonstrates that the nonuniform physical model gives valid results for these small space–time values when   start where start D 1:00055 marks the space–time point where the calculation began. For a smaller starting value, one would need to make the physical model uniform in the vicinity of the point  D 1. The results are not as critical for other initial pulse shapes whose spectra vanish as j!j ! 1. from above for the delta function pulse. This is a consequence of the fact that the integral itself is ill behaved at  D 1 (see Sect. 13.2.5).

698

16 Physical Interpretations of Dispersive Pulse Dynamics

The physical model is now modified in order to make it uniform in the vicinity of the critical space–time point  D 0 . The analysis begins with (16.10)–(16.13) with the exception that the expressions in (16.12) and (16.13) are now replaced by the uniform asymptotic expressions given in Sect. 13.3.2. The asymptotic approximation is made uniform in the vicinity of the critical space–time point  D 1 0 by replacing the nonuniform expressions given in (16.12) and (16.13) for Ab .z; t / with i z AQb .z; t / D e c ˛0 ./ 2

(  i c 1=3 i 2  h Q e 3 f .!C /hC ./ C fQ.! /h . / z   z 2=3  Ai j˛1 ./j c  2=3 h i c 4 C e i 3  fQ.!C /hC ./  fQ.! /h . / z )   z 2=3  1=2 0 ˛1 ./Ai j˛1 . /j ; (16.46) c

where Ab .z; t / D
1=3 3 1=2 Œ.!C ;  /  .! ;  /

˛1 . / D ; 4 " #1=2 1=2 2˛1 ./ h˙ . / D 00 I  ¤ 1 :  .!˙ ;  / ˛0 . / D

(16.47) (16.48) (16.49)

At the critical space–time value 1 at which the two near first-order saddle points coalesce into a single second-order saddle point at ! D !1 , where  0 .!1 ; 1 / D  00 .!1 ; 1 / D 0, the limiting behavior [see (10.57)–(10.59)]  h1 lim h˙ ./  !1

2  000 .!1 ; 1 /

1=3 (16.50)

16.1 Energy Velocity Description of Dispersive Pulse Dynamics

699

is obtained with h i lim fQ.!C /hC ./ C fQ.! /h ./ D 2fQ.!1 /h1 ;

!1

lim

!1

fQ.!C /hC ./  fQ.! /h ./ 1=2

˛1 ./

D 2fQ0 .!1 /h21 :

(16.51) (16.52)

The specification of the branch choices used to make the multivalued functions appearing in the above equations single valued is given in Sect. 10.3.2 (see also Sect. 13.3.2). The uniform physical model is obtained from this uniform asymptotic approximation by making the same approximations that were used in the nonuniform asymptotic approximation to obtain the nonuniform physical model. The critical space–time value 1 is approximated by 0 and the saddle points are approximated by using the appropriate solutions to the energy velocity relations given in (16.42) and (16.43). In particular, for 1 <   0 , !C is approximated by i!Q C and ! is approximated by i !Q  , where !Q ˙ are the real-valued solutions of (16.43) with !Q C  !Q  . Similarly, for   0 , !D is approximated by the low-frequency real solution of (16.42) and ! is approximated by the negative of that value. As in the nonuniform physical model, !s is approximated in (16.19) by the high-frequency real solution of (16.42). As  moves away from 0 in either direction, the uniform physical model approaches the nonuniform physical model asymptotically. Hence, one can apply the nonuniform model for all values of  > 1 except those in the vicinity of 0 where the uniform physical model must be applied. To examine the validity of the uniform physical model, the computations that were used to obtain the results presented in Fig. 16.6 were repeated with the change that the uniform physical model was applied in the space–time interval 1:43 <   1:55, where 0 D 1:5. Because the expressions for some of the coefficients in (16.46) become indeterminate when  D 0 , the limiting expressions given in (16.50)–(16.52) were used when 1:43 <   1:55. The results are described by the solid curve in Fig. 16.9 that are superimposed on the results of the numerical integration of the exact integral solution as described by the dotted curve in that figure. The discontinuity in the solid curve at  D 0 and its departure from the dotted curve for values of  larger than but near 0 is a result of the fact that the approximations of the saddle point location by a real solution to the energy velocity relation given in (16.42) is not very good for  values in this space–time region about 0 , as discussed in Sect. 16.1.1. The small shift in phase between the two curves for  > 0 is the same as that exhibited in Fig. 16.6 and discussed there. Apart from these rather minor discrepencies, the results of the uniform physical model are in excellent agreement with the exact numerical solution for the propagated wavefield for all  > 1. As has already been mentioned several times, the Lorentz medium parameter values chosen for these computations correspond to a dispersive dielectric with very high absorption, much too high for the material to be considered to be transparent. Because the approximations improve as the overall material absorption decreases,

700

16 Physical Interpretations of Dispersive Pulse Dynamics

Ad (z,t)

1.0X1016

0

−1.0X1016 1.0

1.2

1.4

1.6

1.8

2.0

q = ct/z

Fig. 16.9 Comparison of the uniform physical model (solid curve) with the numerical integration of the exact integral representation (dotted curve) for the propagated plane wavefield due to an input delta function pulse in a highly absorbing single resonance Lorentz model medium

these numerical results can be considered as a worst case test of the validity of the physical model for describing pulse propagation in dispersive absorbing dielectrics of interest. To verify the utility of the physical model in the opposite extreme of a highly transparent medium, the same computations have been repeated with the same parameter values with the exception that the phenomenological damping constant ı is now taken as ı D 1:0 r=s, which corresponds to the singular dispersion limit described in Sect. 15.10.1. The results for both the uniform physical model and the numerical evaluation of the exact integral solution are presented by the solid and dotted curves in Fig. 16.10, respectively. The agreement is so good that the two results are practically indistinguishable except at a few isolated points. One might expect the group velocity description to be applicable in the singular dispersion limit case presented in Fig. 16.10 because that case is nearly lossless almost everywhere. Indeed, it can be shown that the energy velocity given in (16.18) approaches the group velocity for all frequencies for which the medium is lossless as ı approaches zero (see Problem 5.10 of Vol. 1). The medium is not lossless for all frequencies, however, even when ı is identically zero, because the complex index of refraction given in (16.3) is purely imaginary for values of ! 2 slightly larger than !02 . As a result, the group velocity description is not strictly valid the the Lorentz model medium even when ı is identically zero. Nevertheless, the group velocity description may still be applied to that case in order to see what it would yield. The

16.2 Extension of the Group Velocity Description

701

Ad (z,t)

1.0X1016

0

−1.0X1016 1.0

1.2

1.4

1.6

1.8

2.0

q = ct /z

Fig. 16.10 Comparison of the uniform physical model (solid curve) with the numerical integration of the exact integral representation (dotted curve) for the propagated plane wavefield due to an input delta function pulse in the singular dispersion limit of a single resonance Lorentz model medium

results were found [22] to be identical to those presented in Fig. 16.10 except for in the space–time interval SB    0 where the propagated wavefield failed to rise as it does in Fig. 16.10. It is apparent that all that is missing in the group velocity description in this singular dispersion limit is the nonoscillatory contribution.

16.2 Extension of the Group Velocity Description Based upon these results, the question then naturally arises as to whether or not the group velocity description can be extended to include ultrawideband pulse propagation in dispersive absorptive systems. The fact that it is incapable of completely describing the singular dispersion case presented in Fig. 16.10 suggests not. In addition, the detailed analyses of Xiao and Oughstun [18, 19, 28] has shown that the validity of the classical group velocity description increases as z ! 0 and breaks down as the propagation distance approaches and exceeds a critical propagation distance zc , whereas the validity of the asymptotic and energy velocity descriptions increases as z exceeds this critical propagation distance zc , but fails for propagation distances below it. The critical propagation distance zc is typically set by the absorption depth zd ˛ 1 .!c / of the dispersive attenuative medium evaluated at the

702

16 Physical Interpretations of Dispersive Pulse Dynamics

characteristic oscillation frequency of the pulse. As they stand, no one description is valid for all propagation distances z  0. Nevertheless, the results of the generalized asymptotic description of gaussian envelope pulse propagation developed independently by Tanaka, Fujiwara, and Ikegami [29] and Balictsis and Oughstun [30–32], presented in Sect. 15.8.1, suggests otherwise. In that case, a detailed asymptotic analysis lead to an extension of the group velocity description for gaussian pulse propagation that is not only valid for all z  0, but that is also valid for all initial pulse widths 2T > 0. This generalized asymptotic description quantitatively describes the transition of the pulse evolution from the group velocity description (valid in the immature dispersion region z < zc ) to the energy velocity description (valid in the mature dispersion region z > zc ) as the propagation distance increases. Although this generalized asymptotic description [30–32] is restricted to the gaussian envelope pulse, it does provide direction as to how the classical group velocity description can easily be generalized without altering the simple formulation that group velocity adherents so dearly desire. In particular, the parabolic wave equation given in (11.148) that provides the starting point in many descriptions of dispersive pulse propagation in the group velocity approximation should be changed to read @aQ j .z; t / @2 aQ j .z; t / 1 @aQ j .z; t / i Q 00 (16.53)

  k .!j / @z vg .!j / @t 2 @t 2 where a.z; Q t/ D

X

aQ j .z; t /

(16.54)

j

describes the complex envelope of the plane wave pulse [see (11.135) and (11.141)]. The angular frequency values !j are determined by the maxima in the propagated pulse spectrum. Because of dispersion, these maxima evolve with increasing propagation distance z, beginning at (or near to) the input pulse carrier frequency !c . In a single resonance Lorentz model medium, the pulse spectrum at z D 0 evolves into a low- and a high-frequency peak as the propagation distance increases and the set of wave equations given by (16.53) describes the local behavior about each maxima. The slower low-frequency peak then describes the generalized Brillouin precursor component of the pulse that follows the faster high-frequency peak that describes the generalized Sommerfeld precursor component in this approximation.

16.3 Signal Model of Dispersive Pulse Dynamics It is appropriate to conclude this chapter with an alternate, “subtractive formulation” of dispersive signal propagation as has been described by Stratton4 [23] in 1941. The formulation is based on the propagation of a Heaviside step function signal f .t / D 4

Notice that Stratton’s analysis is based on the Laplace transform with integration variable s D i!.

16.3 Signal Model of Dispersive Pulse Dynamics

703

uH .t / sin .!c t / in a single Lorentz model dielectric and provides a representation of the propagated signal in terms of separate steady-state and transient responses [24]. Here uH .t / is the Heaviside step function defined in (11.55). The analysis begins with the exact Fourier–Laplace integral representation of the propagated plane wave signal, given by [see (11.57)] 1 AH .z; t / D  < 2

Z C

z 1 e c .!;/ d! !  !c

(16.55)

for all z  0, where C is the horizontal straight-line contour ! D ! 0 C ia extending from ! 0 D 1 to ! 0 D C1 in the upper half of the complex !-plane (a > 0). Q Here .!;  / i!Œn.!/   D i.c=z/Œk.!/z  !t is the complex phase function with  ct =z, as described in Sect. 11.2; for z D 0, the limiting form Q .z=c/.!;  / ! i !t is used in (16.55). Here k.!/ .!=c/n.!/ is the complex wavenumber in the dispersive Lorentz medium with complex index of refraction  n.!/ D 1 

b2 ! 2  !02 C 2i ı!

1=2 ;

(16.56)

which is analytic everywhere in the complex !-plane except at the branch cuts 0 [see Fig. 12.1 and (12.64) and (12.65)], where !0 ! and !C !C q !˙ ˙ !02  ı 2  iı; q 0 ˙ !12  ı 2  iı; !˙

(16.57) (16.58)

q where !1 !02 C b 2 . Application of Sommerfeld’s relativistic causality theorem (Theorem 6 in Sect. 13.1) shows that z (16.59) AH .z; t / D 0; 8 t < ; c for all z  0, in keeping with relativistic causality. Application of the derivation given in Sect. 13.1 to the space–time domain  > 1 shows that the original contour C can then be closed in the lower half of the complex !-plane with the contribution from the semicircular path at j!j D 1 identically vanishing. As illustrated in Fig. 16.11, this completion of the contour for   1 results in the enclosure of the simple pole singularity at ! D !c as well as about the two branch cuts !0 ! and 0 . The propagated wavefield may then be expressed as !C !C AH .z; t / D AHss .z; t / C AHtr .z; t /;

8t

z ; c

(16.60)

704

16 Physical Interpretations of Dispersive Pulse Dynamics

''

C

' c

'

'

C−

C+

C

Fig. 16.11 Deformed contour of integration for  > 1 encircling the simple pole singularity at 0 0 ! D !c and encircling the two branch cuts ! ! and !C !C . The contour integral along the semicircular arc C˝ in the lower half-plane vanishes as its radius increases to infinity

for all z  0. The wavefield component AHss .z; t / describes the steady-state response given by the residue contribution from the simple pole singularity at ! D !c , where AHss .0; t / D AH .0; t / D uH .t / sin .!c t /;

8 t 2 .1; C1/;

(16.61)

and the wavefield component AH t r .z; t / describes the transient response given by the contour integration about both branch cuts, where AHtr .0; t / D 0;

8 t 2 .1; C1/:

(16.62)

The steady-state response for all z > 0 is given by the residue contribution from the simple pole singularity at ! D !c as [see (15.77)] AHss .z; t / D e ˛.!c /z sin .ˇ.!c /z  !c t /;

8t>

z : c

(16.63)

The difference between this result and the pole contribution appearing in the signal contribution Ac .z; t / of the modern asymptotic theory is that the latter is nonzero for t > c z=c when !c 2 .0; !SB / and is nonzero for t > c1 z=c when !  !SB , as described in detail in Sect. 15.3. The transient response for all z > 0 is given by the sum of the contour integrals about each of the branch cuts as

16.3 Signal Model of Dispersive Pulse Dynamics

AHtr .z; t / D 

8 <X I

1 < 2 :

705

9 =

j D˙ Cj

z 1 e c .!;/ d! ; ; !  !c

8t>

z ; c

(16.64)

where C is the closed contour encircling the branch cut !0 ! and CC is the closed 0 , both taken in the counterclockwide sense, contour encircling the branch cut !C !C as illustrated in Fig. 16.11. Because of the symmetry relations given in (11.25)– (11.27), the contour integral about the left branch cut is the complex conjugate of the contour integral over the right branch cut, so that 1 AH t r .z; t / D  < 

(I CC

) z 1 .!;/ ec d! ; !  !c

8t>

z ; c

(16.65)

for all z > 0. The resultant wave motion at an point within the dispersive medium has thus been represented in (16.60) by the sum of two terms. As stated by Stratton [23]: Physically these two components may be interpreted as forced and free vibrations of the charges that constitute the medium. The forced vibrations, defined by AH ss .z; t /, are undamped in time and have the same frequency as the impinging wave train. The free vibrations AHtr .z; t / are damped in time as a result of the damping forces acting on the oscillating ions and their frequency is determined by the elastic binding forces. The course of tghe propagation into the medium can be traced as follows: Up to the instant t D z=c, all is quiet. Even when the phase velocity v is greater than c, no wave reaches z earlier thyan t D z=c. At t D z=c the integral AHtr .z; t / first exhibits a value other then zero, indicating that the ions have been set into oscillation. If by the term “wavefront” we understand the very first arrival of the disturbance, then the wavefront velocity is always equal to C , no matter what the medium. It may be shown, however, that at this first instant t D z=c the forced or steady-state term AHss .z; t / just cancels the free or transient term AHtr .z; t /, so that the process starts always from zero amplitude. The steady state is then gradually built up as the transient dies out, quite in the same way that the sudden application of an alternating e.m.f. to an electrical network results in a transient surge which is eventually replaced by a harmonic oscillation.

Notice that Stratton’s description of dispersive pulse dynamics is subtractive in the time domain whereas the asymptotic description is additive. Comparison of (16.60) with the asymtotic description of the propagated signal given by (15.1) shows that the transient response is given by AHtr .z; t / D AHs .z; t / C AHb .z; t / C AHc .z; t /  AHss .z; t /;

8t>

z ; (16.66) c

for all z  0, so that the transient response is given by the superposition of the Sommerfeld and Brillouin precursors plus the difference between the pole contribution (which includes the interaction with the relevant saddle point in the uniform asymptotic theory) and the time-harmonic steady-state response. From (16.60) and (16.61), the transient response at any fixed propagation distance z  0 can be

706

16 Physical Interpretations of Dispersive Pulse Dynamics 0.3 0.2

AHtr (z,t)

0.1 0

−0.1 −0.2 −0.3 1

2

3

4

5

6

t (fs)

Fig. 16.12 Transient response AHtr .z; t / for the below resonance angular carrier frequency !c D !0 =2 at two absorption depths z D 2zd

numerically determined from the difference between the propagated signal AH .z; t / and the steady-state response AH ss .z; t / as AHtr .z; t / D AH .z; t /  AHss .z; t /:

(16.67)

The result of such a computation is presented in Fig. 16.12 for a Heaviside step function signal AH .z; t / with below resonance angular carrier frequency !c D Lorentz medium with 2  1016 r=s at two absorption depths in a single resonance p 20  1016 r=s, ı D Brillouin’s medium parameters (!c D 4  1016 r=s, b D 16 0:28  10 r=s). The corresponding reconstruction of the total signal wavefield AH .z; t / D AHtr .z; t / C AHss .z; t / is illustrated in Fig. 16.13. The corresponding set of calculations for a Heaviside step function signal AH .z; t / with above resonance angular carrier frequency !c D 10  1016 r=s > !SB at two absorption depths in the same single resonance Lorentz medium is presented in Figs. 16.14 and 16.15. The below resonance transient response presented in Fig. 16.12 illustrates the complicated superposition of the Sommerfeld and Brillouin precursor fields with the difference between the pole and the steady-state response that is described by (16.66). The complicating influence of this difference AHc .z; t /  AHss .z; t / on the transient response is somewhat lessened in the above resonance case illustrated in Fig. 16.14. This is because the pole contibution occurs during the Sommerfeld precursor evolution so that the complicating influence of the difference AHc .z; t /  AHss .z; t / only appears at the front of the Sommerfeld precursor, whereas for the below resonance case, the pole contribution occurs during the evolution of the Brillouin precursor so that the complicating influence of the difference

16.3 Signal Model of Dispersive Pulse Dynamics

707

AHtr (z,t)

0.2 0

AHss(z,t)

0.2 0

AH(z,t)

0.2 0

1

2

3

4

5

6

7

8

9

10

t (fs)

Fig. 16.13 Superposition of the transient response AHtr .z; t / and the steady-state response AHss .z; t / to produce the propagated signal AH .z; t / for the below resonance angular carrier frequency !c D !0 =2 at two absorption depths z D 2zd

0.2

AHtr (z,t)

0.1

0

−0.1

−0.2 1

2

3

4

5

6

t (fs)

Fig. 16.14 Transient response AHtr .z; t / for the above resonance angular carrier frequency !c D 2:5!0 at two absorption depths z D 2zd

AHc .z; t /  AHss .z; t / occurs during most of the Sommerfeld and Brillouin precursor evolution. Hence, the full precursor field structure will be revealed in the transient response AHtr .z; t / as the carrier frequency !c is increased to infinity.

708

16 Physical Interpretations of Dispersive Pulse Dynamics

AHtr (z,t)

0.2

0

AHss(z,t)

0.2 0

AH(z,t)

0.2

0

1

2

4

3

5

6

t (fs)

Fig. 16.15 Superposition of the transient response AHtr .z; t / and the steady-state response AHss .z; t / to produce the propagated signal AH .z; t / for the above resonance angular carrier frequency !c D 2:5!0 at two absorption depths z D 2zd

16.4 Summary and Conclusions Several different views of dispersive pulse propagation have been given in this chapter, the most fundamental being the energy velocity description due to Sherman and Oughstun [21, 22, 25]. When combined with the extended group velocity description presented in Sect. 16.2, which is valid in the immature dispersion regime, a complete, physical description of dispersive pulse dynamics is obtained that is valid for all propagation distances. The resultant physical model, however, is (so far) restricted to initial pulse functions whose temporal frequency spectra are entire functions of complex !. If this condition is not satisfied, then one must turn either to the complete asymptotic description, numerical results, or hybrid results which optimally combine both asymptotic and numerical results [33,34]. The signal model of dispersive pulse dynamics, presented in Sect. 16.3, provides a subtractive model in the time domain that is complementary to that provided by the modern asymptotic theory, but not much else beyond that. Although the energy velocity and signal models have been developed for a Lorentz model medium, the results can be extended to other dispersive models, preferably causal. According to the energy velocity model presented in Sect. 16.1, once the plane wave pulse has propagated far enough to be in the mature dispersion regime of a single resonance Lorentz medium, it separates into two distinct components at any given subluminal space–time point .z; t / where t > z=c. Each component is either a quasimonochromatic wave of the form Q

e i .k.!/!t /

16.4 Summary and Conclusions

709

with real angular frequency ! D !./ that is a slowly varying function of position and time, or a nonoscillatory wave of the form 

e

 QQ !/z !t Q k. Q

with real growth rate !Q D !. Q / that is a slowly varying function of position and time. The propagation factor appearing in the quasimonochromatic wave form is just Q the complex wavenumber k.!/ D .!=c/n.!/, where the complex index of refracQQ !/ tion is given in (16.3), and the quantity k. Q appearing in the nonoscillatory wave form in given in (16.27). The angular frequency values of the quasimonochromatic wave components satisfy z (16.68) vE .!/ D ; t where vE .!/ is the velocity of electromagnetic energy transport in a monochromatic wave with real angular frequency !. The growth rate of the nonoscillatory components satisfy z (16.69) vQ E .!/ Q D ; t where vQ E .!/ Q is the velocity of energy transport in a nonoscillatory wave with growth rate !. Q The easiest way to describe the propagated pulse components is to consider the point of observation to be moving with a fixed velocity v D z=t . Then according to (16.67) and (16.68), the components that contribute at a that specific point of observation are the ones with energy velocities equal to the velocity of that point. If v D z=t satisfies the inequality c ; (16.70) v> n.0/ where n.0/ is the index of refraction at zero frequency, then there is only one real solution to each of (16.67) and (16.68). Hence, in the space–time region where (16.69) is satisfied, the pulse has one quasimonochromatic component (the Sommerfeld precursor) and one nonoscillatory component (the initial rise of the Brillouin precursor). If v D z=t satisfies the inequality v<

c ; n.0/

(16.71)

then there is no real solution to (16.68) and there are two real solutions to (16.67). Hence, in that space–time region the pulse has two quasimonochromatic components (the Sommerfeld and Brillouin precursors) and no nonoscillatory component. In the transition region where c v

; (16.72) n.0/ the functional form of the wavefield is more complicated but its dynamics are still controlled by the solutions to (16.67) and (16.68).

710

16 Physical Interpretations of Dispersive Pulse Dynamics

The variations of the precursor fields with changes in the observation point velocity v D z=t are determined by the variation of the energy velocities and attenuation rates of monochromatic waves and nonoscillatory waves as functions of angular frequency ! and growth rate !, Q respectively. These latter variations are depicted by the simple smooth curves presented in Figs. 16.2–16.5. They are determined by the physics of energy flow and absorption in the dispersive medium through the interaction of the electromagnetic field with the medium molecules according to the microscopic model of the material (see Chap. 4 of Vol. 1). With this description of the dispersive wavefield evolution in hand, consideration is now given to its description from a different point of view. If the pulse is observed in a small region of space at a fixed time t > z=c in the mature dispersion regime, it will be found to be made up of the superposition of a quasimonochromatic component of some high real angular frequency !E !s and another component which is either a quasimonochromatic wave with a lower real frequency !E !b or a nonoscillatory wave with real growth rate !. Q These two component will have the same energy velocity. Moreover, if these wave components are then followed through space as time progresses, they move together with that velocity and each one’s amplitude will decay exponentially as it propagates with the attenuation coefficient corresponding to that wave component in the dispersive medium. This will be true throughout the pulse evolution with the wave components having higher energy velocities being ahead of those with lower energy velocities and the separation between these high and low velocity components increasing with time. As a consequence of this wave component spreading, the energy in any given frequency (or growth) interval will spread out over an increasing space–time region as the propagation progresses, causing the wave component amplitudes to decrease (in addition p to the exponential attenuation) by the inverse square root factor 1= z appearing in (16.40) and (16.41). The accuracy of the physical model is illustrated in Fig. 16.9 for a highly absorbing dispersive medium and in Fig. 16.10 in the singular dispersion limit through a superposition of the field behavior predicted by the model with that obtained from a numerical evaluation of the exact integral representation of the propagated wavefield. The agreement between the model and the “exact” numerical results is remarkably good in both cases. The agreement is especially striking in Fig. 16.10 where the wavefield evolution exhibits a very complicated behavior. Without the physical model, a complete explanation of this complicated wave form is impossible. The physical model, however, reveals the source of this behavior. Because the attenuation of the monochromatic waves is so small in this case, the high-frequency component does not rapidly decay like it does for the case illustrated in Fig. 16.9. Hence, it strongly interferes, first with the nonoscillatory component and then with the low-frequency quasimonochromatic wave component, to produce the observed complicated evolution. The physical model presented here is intuitively appealing from a physical point of view. It appears to be a natural extension of the group velocity description known to be valid for lossless, gainless dispersive media. In fact, the classical group velocity description can be considered to be an energy velocity description because, under

References

711

very general conditions, it has been shown that the energy and group velocities are identical in lossless, gainless media [35, 36]. It is then seen that the classical group velocity is only valid for small propagation distances z  zd in a dispersive lossy medium. The extended group velocity description presented in Sect. 16.2 extends this propagation domain up to the mature dispersion region z > zd where the energy velocity description applies, thereby completing the theory.

References ¨ 1. A. Sommerfeld, “Uber die fortpflanzung des lichtes in disperdierenden medien,” Ann. Phys., vol. 44, pp. 177–202, 1914. ¨ 2. L. Brillouin, “Uber die fortpflanzung des licht in disperdierenden medien,” Ann. Phys., vol. 44, pp. 204–240, 1914. 3. L. Brillouin, “Propagation of electromagnetic waves in material media,” in Congr`es International d’Electricit´e, vol. 2, pp. 739–788, Paris: Gauthier-Villars, 1933. 4. L. Brillouin, Wave Propagation and Group Velocity. New York: Academic, 1960. 5. A. Einstein, “Zur elektrodynamik bewegter k¨orper,” Ann. Phys., vol. 17, pp. 891–921, 1905. ¨ 6. H. Baerwald, “Uber die fortpflanzung von signalen in disperdierenden medien,” Ann. Phys., vol. 7, pp. 731–760, 1930. 7. K. E. Oughstun, Propagation of Optical Pulses in Dispersive Media. PhD thesis, The Institute of Optics, University of Rochester, 1978. 8. K. E. Oughstun and G. C. Sherman, “Optical pulse propagation in temporally dispersive Lorentz media,” J. Opt. Soc. Am., vol. 65, no. 10, p. 1224A, 1975. 9. K. E. Oughstun and G. C. Sherman, “Uniform asymptotic theory of pulse propagation in Lorentz media,” in Proceedings of the URSI Symposium on Electromagnetic Wave Theory, (Stanford University), pp. 34–36, 1977. 10. K. E. Oughstun and G. C. Sherman, “Uniform asymptotic description of dispersive pulse propagation,” J. Opt. Soc. Am. A, vol. 69, no. 10, p. 1448A, 1979. 11. K. E. Oughstun and S. Shen, “Velocity of energy transport for a time-harmonic field in a multiple-resonance Lorentz medium,” J. Opt. Soc. Am. B, vol. 5, no. 11, pp. 2395–2398, 1988. 12. K. E. Oughstun and G. C. Sherman, “Uniform asymptotic description of electromagnetic pulse propagation in a linear dispersive medium with absorption (the Lorentz medium),” J. Opt. Soc. Am. A, vol. 6, no. 9, pp. 1394–1420, 1989. 13. I. Tolstoy, Wave Propagation. New York: McGraw-Hill, 1973. Chaps. 1 and 2. 14. L. A. Segel and G. H. Handelsman, Mathematics Applied to Continuum Mechanics. New York: Macmillan, 1977. Chap. 9. 15. B. R. Baldock and T. Bridgeman, Mathematical Theory of Wave Motion. New York: Halsted, 1981. Chap. 5. 16. L. B. Felsen, “Propagation and diffraction of transient fields in non-dispersive and dispersive media,” in Transient Electromagnetic Fields (L. B. Felsen, ed.), pp. 1–72, New York: SpringerVerlag, 1976. p. 65. 17. H. M. Nussenzveig, Causality and Dispersion Relations. New York: Academic, 1972. Chap. 1. 18. K. E. Oughstun and H. Xiao, “Failure of the quasimonochromatic approximation for ultrashort pulse propagation in a dispersive, attenuative medium,” Phys. Rev. Lett., vol. 78, no. 4, pp. 642– 645, 1997. 19. H. Xiao and K. E. Oughstun, “Failure of the group velocity description for ultrawideband pulse propagation in a double resonance Lorentz model dielectric,” J. Opt. Soc. Am. B, vol. 16, no. 10, pp. 1773–1785, 1999. 20. K. E. Oughstun and G. C. Sherman, “Comparison of the signal velocity of a pulse with the energy velocity of a time-harmonic field in Lorentz media,” in Proceedings of the URSI Symposium on Electromagnetic Wave Theory, (M¨unchen), pp. C1–C5, 1980.

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16 Physical Interpretations of Dispersive Pulse Dynamics

21. G. C. Sherman and K. E. Oughstun, “Description of pulse dynamics in Lorentz media in terms of the energy velocity and attenuation of time-harmonic waves,” Phys. Rev. Lett., vol. 47, pp. 1451–1454, 1981. 22. G. C. Sherman and K. E. Oughstun, “Energy velocity description of pulse propagation in absorbing, dispersive dielectrics,” J. Opt. Soc. Am. B, vol. 12, pp. 229–247, 1995. 23. J. A. Stratton, Electromagnetic Theory. New York: McGraw-Hill, 1941. 24. H. T. Banks, M. W. Buksas, and T. Lin, Electromagnetic Material Interrogation Using Conductive Interfaces and Acoustic Wavefronts. Frontiers in Applied Mathematics, Philadelphia: Society for Industrial and Applied Mathematics, 2000. 25. K. E. Oughstun and G. C. Sherman, Pulse Propagation in Causal Dielectrics. Berlin: SpringerVerlag, 1994. 26. R. Loudon, “The propagation of electromagnetic energy through an absorbing dielectric,” Phys. A, vol. 3, pp. 233–245, 1970. 27. P. Wyns, D. P. Foty, and K. E. Oughstun, “Numerical analysis of the precursor fields in dispersive pulse propagation,” J. Opt. Soc. Am. A, vol. 6, no. 9, pp. 1421–1429, 1989. 28. H. Xiao, Ultrawideband Pulse Propagation in Complex Dispersive Media. PhD thesis, University of Vermont, 1998. Reprinted in UVM Research Report CSEE/98/03-01 (March 10, 1998). 29. M. Tanaka, M. Fujiwara, and H. Ikegami, “Propagation of a Gaussian wave packet in an absorbing medium,” Phys. Rev. A, vol. 34, pp. 4851–4858, 1986. 30. C. M. Balictsis, Gaussian Pulse Propagation in a Causal, Dispersive Dielectric. PhD thesis, University of Vermont, 1993. Reprinted in UVM Research Report CSEE/93/12-06 (December 31, 1993). 31. K. E. Oughstun and C. M. Balictsis, “Gaussian pulse propagation in a dispersive, absorbing dielectric,” Phys. Rev. Lett., vol. 77, no. 11, pp. 2210–2213, 1996. 32. C. M. Balictsis and K. E. Oughstun, “Generalized asymptotic description of the propagated field dynamics in Gaussian pulse propagation in a linear, causally dispersive medium,” Phys. Rev. E, vol. 55, no. 2, pp. 1910–1921, 1997. 33. S. He and S. Str¨om, “Time-domain wave splitting and propagation in dispersive media,” J. Opt. Soc. Am. A, vol. 13, no. 11, pp. 2200–2207, 1996. 34. H. Xiao and K. E. Oughstun, “Hybrid numerical-asymptotic code for dispersive pulse propagation calculations,” J. Opt. Soc. Am. A, vol. 15, no. 5, pp. 1256–1267, 1998. 35. M. A. Biot, “General theorems on the equivalence of group velocity and energy velocity,” Phys. Rev., vol. 105, pp. 1129–1137, 1957. 36. M. J. Lighthill, “Group velocity,” J. Inst. Math. Applics., vol. 1, pp. 1–28, 1964.

Problems 16.1. Derive (16.31) for the energy velocity vQ E .!/ Q of nonoscillatory waves in a single resonance Lorentz model dielectric. 16.2. Derive (16.34) for the group velocity vQ G .!/ Q of nonoscillatory waves in a single resonance Lorentz model dielectric. 16.3. Show that the attenuation coefficient ˛. Q !/ Q for the nonoscillatory waves is given by (16.44) and (16.45). 16.4. Prove that the contour integral appearing in (16.55) taken along the semicircular contour C˝ in the lower half of the complex !-plane, illustrated in Fig. 16.11, vanishes as its radius increases to infinity for all  > 1. Explain what happens when  D 1.

Chapter 17

Applications

Although the complete mathematical description of ultrawideband dispersive pulse propagation can be rather involved, its physical interpretation is really rather straightforward. Simply put, the input pulse spectrum is like a block of granite to a sculptor, the dispersive attenuative medium being the sculptor. Just as the sculptor never adds material to the block of granite, the material never adds spectral content to the pulse. Rather, it chips away at the spectral content, gradually shaping the pulse down to the precursor field structures that are a characteristic of the material dispersion (i.e., the temporal material response). The precursor fields are then already contained in the initial pulse. The more ultrawideband the pulse, the more they are completely present. Because the precursor fields are a characteristic of the dispersive material, they are precisely tuned to travel through that medium with minimal distortion and, most importantly, with minimal loss. This property makes them ideally suited for a variety of communication and imaging problems. The analysis presented in this concluding chapter describes several potential applications of electromagnetic and optical precursor fields. This discussion was formally initiated by the Defense Advanced Research Projects Agency (DARPA) in the 1980’s and continues to the present. I spite of initial errors in this potential assessment by DARPA, the unique physical features of precursor waveforms are finally being realized in a variety of areas. Nevertheless, it is still important to review these early discussions in order to better understand and appreciate some of the challenges that confronted this research.

17.1 On the Use and Application of Precursor Waveforms Because of its presumed potential as a counter-stealth technology, the Defense Advanced Research Projects Agency (DARPA), together with the Office of the Secretary of Defence (OSD), commissioned a study [1] in the late 1980s to be performed by Battelle Memorial Laboratory to assess the realisable capabilities of ultrawideband (UWB) or impulse radar, with peripheral consideration to be given to UWB communications and electronic warfare. The executive summary of the report was then published in the November 1990 issue of the IEEE Aerospace and Electronic Systems Society (AESS) Magazine. K.E. Oughstun, Electromagnetic and Optical Pulse Propagation 2: Temporal Pulse Dynamics in Dispersive, Attenuative Media, Springer Series in Optical Sciences 144, c Springer Science+Business Media, LLC 2009 DOI 10.1007/978-1-4419-0149-1 17, 

713

714

17 Applications

Ultrawideband radar systems are obviously characterized by an ultrawideband spectrum, which is taken by some to mean an ! 1 fall-off in spectral amplitude as ! ! 1, a simple, physically appealing definition that has been tempered by the more practical FCC definition given in the footnote on page 165. Because of this characteristic, they have the desirable property of fine range resolution. An impulsive-type implementation of UWB radar, or impulse radar, is then defined as a radar system that radiates a single cycle sine wave, which is clearly ultrawideband. Other types of UWB radar systems are then referred to as nonimpulse radars. In their executive summary [1], the UWB Review Panel stated that an “impulse radar can have substantial low frequency content and typically has high peak power and short pulse length. These properties are the basis for claims of unusual capabilities. In examining the subject, the Panel found it useful to separate such claimed capabilities into two categories: 1) those involving phenomena which are unique to impulsive radars and 2) those in which impulse radar may offer one or more advantages in implementation.” Most claims of unique performance were nonlinear in origin, such as self-induced transparency, with no real bearing to radar. Furthermore [1]: Other claims for unique capabilities were examined and found to be in error. Specifically, “precursors,” which have figured prominently in some discussions, are linear transients in distributed media and not unique to impulse systems. Further, the Panel saw no practical radar application of this phenomenon.

Ironically, whether or not they knowingly take advantage of them, rectangular envelope pulses do engender precursors upon passage through dispersive attenuative media, such as foliage or ground, as described in detail in Sect. 15.6. The three proposed capabilities of impulse radar that received the greatest attention by the UWB Review Panel “centered on claims involving counter-stealth capabilities, Low Probability of Intercept (LPI), and detection of relocatable targets (in camouflage and foliage).” The panel’s conclusions regarding these proposed capabilities were [1]:  Counter-Stealth. The Panel concluded that impulse radar is not “inherently antistealth.”. . . There are no effects in radar absorbing material (RAM) that are unique to impulse radar. . . . All observed effects are due to “out-of-band” operation (with respect to the RAM) and predictions to the contrary are due to a misunderstanding of electromagnetics.  Detectability of the Radar (LPI). . . . The Panel concluded that the impulse radar, which typically has less processing gain, has no special LPI characteristics and is readily detectable by an appropriately designed intercept receiver.  Detection of Relocatable Targets. A capability of interest to both strategic and tactical forces is the detection of military targets when shielded or obscured by trees. Consequently, there has been interest in developing a foliage-penetration imaging radar with sufficient resolution to detect targets of interest with an acceptable false alarm rate. A radar with a resolution on the order of a few feet and operating at frequencies low enough to have tolerable attenuation through foliage might provide a useful capability. The Panel suggests that an impulse radar with a center frequency of a few hundred Megahertz may well be the best way to implement such a system. . . .

It is clear that the Brillouin pulse described in Sect. 15.8.3 provides an optimal solution to the last capability.

17.1 On the Use and Application of Precursor Waveforms

715

Not surprisingly, this report raised a good deal of controversey in certain quarters of the aerospace industry. In response to this controversy, a second panel was formed to, among other things, review the earlier panel’s findings. In this controversy, outlandish claims were made concerning the inability of linear electromagnetics to describe precursor phenomena. Although these claims were shown to be completely invalid, they nonetheless had a negative impact on legitimate research on precursor phenomena. The title of the panel’s published report [2], “The UWB (impulse caper) or ‘punishment of the innocent”’ that appeared in the December 1992 issue of the IEEE AESS clearly expresses this sentiment. A more recent report on the “use and application of precursor waveforms” has since been published in 2004, stating that there is good evidence, both theoretical and experimental indicating that precursor waveforms can be created that have less than exponential decay properties. However, here we (sic) more concerned with the utility of such waveforms for remote sensing applications where penetration of otherwise opaque media is an advantage.. . . it is quite clear that there are immense challenges in designing a system that has useful range for remote sensing applications due to the very small power levels that exist inherently in short pulses.

A detailed analysis of electromagnetic energy flow and power loss in ultrawideband pulse propagation through dispersive attenuative media has been given by Paul Smith et al. [4–6]. These results, presented here in Sect. 17.2, show that the precursor fields associated with ultrawideband pulse propagation in dispersive attenuative media carry a significant fraction of the initial electromagnetic energy much deeper into or through the medium, thereby delivering more power on any hidden target. An upper estimate of the fractional input pulse energy that goes into the Brillouin precursor for a rectangular envelope pulse in a Debye-type dielectric is given by the inverse of the number of complete oscillation periods in the initial pulse. For example, for a single-cycle pulse, 100% of the input pulse energy is available to the leading- and trailing-edge Brillouin precursors, whereas for a ten-cycle pulse, only 10% is available. Finally, for a pulse with initial temporal shape comprised of a pair of Brillouin precursors, the second inverted with respect to the first, optimal pulse penetration with maximum energy transfer can be achieved, as described in Sect. 15.8.3. A coded sequence of such Brillouin pulses will then possess optimal detectability characteristics. The engineering technology assessment presented in [3] continued with the following critical concern: However, as the main advantage of precursors is their potential for reduced attenuation (sic) then their propagation properties through the various media of interest (e.g. foliage etc) will be a key factor in determining utility. This seems to be the area of greatest uncertainty. The theoretical work has used the Debye and Lorentz models for the propagation medium, and it is not clear whether these models may accurately represent the behavior of foliage canopies in practice, nor how this may depend on parameters such as foliage density or moisture content. . . . any consideration of the practical usage of precursors should be compared directly with this form of system, i.e. any attenuation advantage of precursors should be offset against the levels of average power that can be more easily generated with conventional wideband SAR.

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17 Applications

The advantages of using precursor waveforms for ground penetrating radar and imaging through obstructing walls and barriers are clear. The problem is more difficult, however, for foliage penetrating radar (FOLPEN) because of the complicated dispersive scattering associated with leafy foliage. A detailed numerical investigation of this important problem is given in [7]. The resulting algorithm, which is O.N / with respect to the number N of faces in the mesh, will permit one to efficiently model the dispersive properties of a large number of individual leaves and branches in a tree canopy. Finally, in the conclusion of this engineering technology assessment [3], it is stated that: In this paper we have introduced the phenomenon and theory of precursors. There is no doubt that the phenomenon exists, but the theory illustrates that there is a delicate balance between the specification of the waveform and the properties of the medium through which it must propagate. This has led to much debate as to the range of validity and the utility to which precursors can be put. For example, proponents of the idea, particularly K. E. Oughstun and colleagues [8], have had details of their analyses and predictions seriously challenged . . .

All of these challenges and criticisms of the modern asymptotic description of ultrawideband dispersive pulse propagation have been appropriately addressed, both here and elsewhere [9, 10], for both Debye- and Lorentz-type dielectrics. This modern asymptotic theory is derived from the macroscopic form of Maxwell’s equation with the appropriate causal constitutive relations for a temporally dispersive medium using mathematically well-defined asymptotic techniques without any unnecessary approximations and so must be correct. Arguments based on the contradictory assumption that both the temporal pulse and its spectrum have compact support are invalid, as are any conclusions drawn from them. The conclusion that there is no practical radar application of precursor-type wave forms is clearly incorrect.

17.2 Electromagnetic Energy Dissipation in Causally Dispersive Media The mathematical formulation and interpretation of Poynting’s theorem [11] as a statement of the conservation of energy in the coupled electromagnetic fieldmedium system are widely accepted [12–14]. However, its interpretation with respect to the thermal energy that is generated by an electromagnetic field that is propagating through a causally dispersive, attenuative medium must be carefully treated [15], particularly if the electromagnetic field is ultrawideband (see Sect. 5.2 of Vol. 1). The natural resonance structure of the material dispersion results in the emergence of precursor fields in the propagated wavefield as the propagation distance typically exceeds one absorption depth zd ˛ 1 .!c / in the material. Because the peak amplitude points in these precursor field components possess lower loss than does the main body of the pulse for any nonzero, finite angular pulse frequency !c , they will penetrate much farther into the dispersive medium, thereby generating

17.2 Electromagnetic Energy Dissipation in Causally Dispersive Media

717

heat at propagation depths where virtually none is generated by the main body of the pulse. Therein lies their usefulness for both imaging, detection, and communication, as well as their potential harmfulness to biological systems, most importantly with regard to human exposure.

17.2.1 General Formulation The differential representation of the Heaviside–Poynting theorem for the macroscopic electromagnetic field [see (5.95) in Vol. 1] r  S.r; t / D

@U 0 .r; t / C Jc .r; t /  E.r; t / @t

(17.1)

is a direct consequence of the macroscopic form of Maxwell’s equations (see the introduction to Chap. 9), where  c    S.r; t /   E.r; t /  H.r; t / 4

(17.2)

is the Poynting vector, and where U 0 .r; t / D Ue0 .r; t / C Um0 .r; t /

(17.3)

is the sum of the electric and magnetic energy densities that are defined by the differential relations  c  @D.r; t / @Ue0 .r; t /     E.r; t /  ; @t 4 @t  c  @B.r; t / @Um0 .r; t /     H.r; t /  ; @t 4 @t

(17.4) (17.5)

respectively. Here E.r; t / is the electric field intensity vector, D.r; t / is the electric displacement vector in the dispersive medium, B.r; t / is the magnetic induction vector in the dispersive medium, H.r; t / is the magnetic field intensity vector, and Jc .r; t / is the conduction current density vector. For an explicit physical representation, the sum of the electric and magnetic energy densities of the coupled field-medium system given in (17.3) must be expressed instead as a sum of two physically distinct quantities, one representing the energy density that is dissipated in the medium in the form of heat and one that represents the sum of the electromagnetic energy density stored in the field and the reactively stored energy density in the medium. Let Q.r; t / denote the evolved heat (a power density) in the medium. In addition, let the sums of the field energy densities and

718

17 Applications

the reactively stored energy densities for the electric field, the magnetic field, and the electromagnetic field be denoted by the unprimed quantities Ue .r; t /, Um .r; t /, and U.r; t /, respectively, where U.r; t / D Ue .r; t / C Um .r; t /:

(17.6)

The formal separation of the right-hand side of the Heaviside–Poynting theorem as stated in (17.1) into lossy and reactively stored components is then expressed by the relation @Ue .r; t / @Um .r; t / @Ue0 .r; t / @Um0 .r; t / C CJc .r; t /E.r; t / D C CQ.r; t /: (17.7) @t @t @t @t Because explicit expressions for Ue .r; t /, Um .r; t /, and Q.r; t / cannot be obtained in general [15] (see Sect. 5.2.2 in Vol. 1), a specific model of the material response is now required. Consider a simple polarizable dielectric described by the locally linear constitutive relations (see Sect. 4.3.1 of Vol. 1) D.r; t / D 0 E.r; t / C k4kP.r; t /; 1 H.r; t / D B.r; t /; 

(17.8) (17.9)

where  is the magnetic permeability of the material and where P.r; t / is the induced macroscopic polarization density of the material. In this case, the problem of separating the power densities of the coupled field-medium system into its lossy and reactive parts is contained entirely in the electric part of (17.7), which may be expressed as  1 @  0 2 @P.r; t / @Ue .r; t / E .r; t / C E.r; t /  D C Q.r; t /: k4k @t 2 @t @t

(17.10)

Because the first term appearing on the left-hand side of this equation is independent of the medium properties, all of the power dissipation must then be accounted for in the term E  @P=@t . The macroscopic polarization density vector is given by the spatial average1 [see (4.27) and (4.162) of Vol. 1] P.r; t / D

X

Nj hpj .r; t /i;

(17.11)

j

where pj .r; t / D qe rj .r; t / 1

(17.12)

The angle brackets hi denote a spatial average of the quantity  over a macroscopically small but microscopically large region region of space (see Sect. 4.1.1).

17.2 Electromagnetic Energy Dissipation in Causally Dispersive Media

719

is the induced microscopic polarization vector moment of the j th molecular type (see Fig. 4.1 of Vol. 1) with number density Nj , where qe denotes the charge magnitude of the displaced charge with displacement vector rj .r; t / relative to its mean equilibrium position. With these substitutions the time rate of change of the macroscopic polarization density is found to be given by X @P.r; t / @ D qe Nj hrj .r; t /i: @t @t j

(17.13)

An explicit expression for the microscopic displacement vector rj .r; t / is now required in order to determine P.r; t /, and that necessitates that a specific dynamical model of its equation of motion be specified under the action of the applied electromagnetic field.

17.2.2 Evolved Heat in Lorentz Model Dielectrics For a Lorentz model dielectric [16–19] the microscopic electric field vector e.r; t / is directly related to the bound electron displacement vector through the Lorentz force relation, as described by the classical equation of motion (see Sect. 4.4.4 of Vol. 1) 

@2 rj .r; t / @rj .r; t / qe C !j2 rj .r; t /; e.r; t / D C 2ıj 2 me @t @t

(17.14)

where me is the electronic mass. Here !j is the undamped angular resonance frequency and ıj the phenomenological damping constant of the j th resonance structure of the Lorentz model dielectric. The spatial average of this microscopic relation then yields the macroscopic equation of motion 

@ @2 qe E.r; t / D 2 hrj .r; t /i C 2ıj hrj .r; t /i C !j2 hrj .r; t /i; me @t @t

(17.15)

The lossy component of the quantity E  @P=@t may now be unambiguously identified as the scalar product of the right-hand side of (17.13) with qe =me times the middle term appearing on the right-hand side of the above macroscopic equation of motion. The evolved heat is then given by [15] 20 Q.r; t / D k4k



me qe

2 X j

ıj bj2

ˇ ˇ2 ˇ@ ˇ ˇ hrj .r; t /iˇ ; ˇ @t ˇ

(17.16)

where bj2 D .k4k=0 /Nj qe2 =me is the square of the plasma frequency for the j th resonance.

720

17 Applications

Notice that this expression for the evolved heat Q.r; t / may be directly obtained .j / from the damping force fı .r; t / D 2me ıj .@rj .r; t /=@t / appearing in the microscopic equation of motion in (17.14). The thermal power developed by this damping force is given by its scalar product with the induced charge velocity @rj .r; t /=@t , so .j / that wı .r; t / D 2me ıj j@rj .r; t /=@t j2 . The evolved heat is then given by the total P .j / dissipated power density through the summation Q.r; t / D j Nj wı .r; t /, which then results in the expression given in (17.16). To complete the description, the time derivative of the electronic displacement vector rj .r; t / must be expressed in terms of the local macroscopic electric field vector E.r; t /. The Fourier integral representation of the solution of (17.14) is readily obtained as rj .r; t / D

1 2

Z

1

!2

1

qe =me eQ .r; !/e i!t d!;  !j2 C 2i ıj !

where

Z

(17.17)

1

e.r; t /e i!t dt

eQ .r; !/ D

(17.18)

1

is the temporal Fourier transform of the microscopic electric field vector. The spatial average of (17.17) then yields 1 hrj .r; t /i D 2

Z

1 1

!2

qe =me Q !/e i!t d!; E.r;  !j2 C 2i ıj !

(17.19)

Q !/ D hQe.r; !/i is the temporal Fourier transform of the macroscopic where E.r; electric field vector E.r; t /. The induced bound electron velocity is then given by @hrj .r; t /i i qe D @t 2 me

Z

1

1

!2



! Q !/e i!t d!: E.r; C 2i ıj !

!j2

(17.20)

Taken together, (17.16) and (17.20) provide an explicit set of relations for determining the evolved heat in a multiple resonance Lorentz model dielectric.

17.2.3 Numerical Results The results of several numerical calculations by P. D. Smith et al. [4–6] of the propagated electric field, magnetic field, Poynting vector magnitude, and evolved heat density for an input rectangular envelope pulse modulated signal with below resonance angular signal frequency !c in a single resonance Lorentz model dielectric are presented in Fig. 17.1 at three absorption depths (z D 3zd ) with details of the evolved heat density Q.z; t / presented in Fig. 17.2 at one, three, five, and ten

17.2 Electromagnetic Energy Dissipation in Causally Dispersive Media

721

cT/z

c

E(z,t) (V/m)

0.2

0

H(z,t) (X 10 −4A/m)

−0.2 1.4

1.5

1.6

1.7

1.8

1.9

2.0

1.5

1.6

1.7

1.8

1.9

2.0

1.5

1.6

1.7

1.8

1.9

2.0

1.5

1.6

1.7

1.8

1.9

2.0

5 0 −5 1.4

S(z,t) (W/m 2)

15 10 5 0 1.4

(z,t) (W/m 3)

60 40 20 0 1.4

Fig. 17.1 Uniform asymptotic (solid curves) and numerically determined (dashed curves) evolution of the electric E.z; t / and magnetic H.z; t / fields, Poynting vector magnitude S.z; t /, and evolved heat density Q.z; t / due to an input 1 V=m electric field strength, 10-cycle rectangular envelope modulated signal with below resonance angular signal frequency !c D 1  1016 r/s at three absorption depths (z D 3zd ) in a single resonance Lorentz model dielectric. (From Smith and Oughstun [6])

absorption depths. For comparison, the evolution of the evolved when  q heat density q 2 2 2 !0  ı ; !1  ı 2 the applied signal frequency !c is in the absorption band and when it as above the critical angular frequency value !SB [see (12.58)–(12.60)] is presented in Figs. 17.3 and 17.4. The solid curves in each figure were computed using the uniform asymptotic description available at that time using numerically determined saddle point locations and the dashed curves describe numerical results. Brillouin’s choice ofpthe model medium parameters (!0 D 4  1016 r/s, ı D 0:28  1016 r/s, b D 20  1016 r/s) were used in all of these calculations. Although these values of the model medium parameters are representative of a

722

17 Applications

a

c

cT/z

100

cT/z 8 z = 5z

d

d

6 (z,t)

(z,t)

z=z

4 2

0 1.4

1.6

1.8

2.0

2.2

2.4

b

2.6

0 1.4

2.6

1.5

d

cT/z

1.6

1.7

1.8

cT/z

60

40

d

z = 3zd (z,t)

(z,t)

z = 10z

1

20

0 1.4

1.5

1.6

1.7

1.8

1.9

2.0

0

1.5

1.6

Fig. 17.2 Uniform asymptotic (solid curves) and numerically determined (dashed curves) evolution of the evolved heat density Q.z; t / due to an input 1 V=m electric field strength, 10-cycle rectangular envelope pulse with below resonance angular signal frequency !c D !0 =4 at (a) 1, (b) 3, (c) 5, and (d) 10 absorption depths in a single resonance Lorentz model dielectric. (From Smith and Oughstun [6])

highly lossy material, the results are presented in terms of the absorption depth zd ˛ 1 .!c / at the pulse carrier frequency and plotted as a function of the dimensionless space–time parameter  D ct =z so that they can be interpreted for lower loss media. For each case, notice the persistence of the evolved heat density due to the leading- and trailing-edge precursor fields long after that due to the carrier frequency has been nearly completely dissipated. In the below resonance case with !c D !0 =4 D 1:0  1016 r/s illustrated in Fig. 17.2, the evolved heat density becomes increasingly due to the leading- and trailing-edge Brillouin precursors, with little effect from the leading- and trailing-edge Sommerfeld precursors, as the propagation distance increases. As the input pulse carrier frequency !c is increased into the medium absorption band, the leading- and trailing-edge Sommerfeld precursors begin to have a more significant contribution to the evolved heat density, as seen in Fig. 17.3 when !c D 5:75  1016 r/s, which is near the upper end of the absorption band.2 This trend continues as the input pulse carrier frequency !c is increased above the medium absorption band, as illustrated in Fig. 17.4 when !c D 2:5!0 D 10:0  1016 r/s. In that case, !c > !SB and the pulse body oscillating at !c is comprised of the prepulse evolution which evolves over the space–time 2

The group velocity vg .!c / at this angular frequency value is, to a very good approximation, equal to the speed of light c (see Fig. 15.9), whereas both the energy velocity vE .!c / and signal velocity vc .!c / are both near to their respective minimum values (see Fig. 15.3).

17.2 Electromagnetic Energy Dissipation in Causally Dispersive Media 10

qSB

723

q0 + cT/z

q0

z = 75zd (z,t) (W/m3)

8 6 4 2 0 1.0

1.5

qSB

6

q q0

2.0

2.5

q0 + cT/z

(z,t) (W/m3)

z = 100zd 4

2

0 1.0

1.2

1.4

1.6

1.8

2.0

2.2

q Fig. 17.3 Uniform asymptotic (solid curves) and numerically determined (dashed curves) evolution of the evolved heat density Q.z; t / due to an input 1 V=m electric field strength, 10-cycle rectangular envelope pulse with intra-absorption band angular signal frequency !c D 1:4375!0 at 75 (upper graph) and 100 (lower graph) absorption depths in a single resonance Lorentz model dielectric. (From Smith and Oughstun [6])

interval .c1 ; c1 C cT =z/, as illustrated in Fig. 15.63. As the propagation distance z increases, this space–time interval shrinks to a single space–time point at  D c1 . If the temporal duration of some given electromagnetic pulse at some fixed penetration distance z > 0 into the dispersive material is significantly less than the thermal diffusion time constant of the material, then approximately all of the evolved heat at that propagation distance is generated by the electromagnetic pulse as it passes that point before any significant diffusive heat transfer can take place in the material. In this case, the physical quantity of interest is the net heat density that is generated by the plane wave pulse as a function of the penetration distance z in the dispersive absorptive material, given by W3D .z/

Z 0

1

Q.z; t /dt:

(17.21)

724

17 Applications

qc1

qSB qc2

q0 z = 5zd

(z,t) (W/m3)

4 3 2 1 0 1.0

0.6

1.2

qc1

1.4

qSB qc2

q

1.6

1.8

2.0

q0

(z,t) (W/m3)

z = 10zd 0.4

0.2

0 1.0

1.2

1.4

q

1.6

1.8

2.0

Fig. 17.4 Uniform asymptotic (solid curves) and numerically determined (dashed curves) evolution of the evolved heat density Q.z; t / due to an input 1 V=m electric field strength, 10-cycle rectangular envelope pulse with above absorption band angular signal frequency !c D 2:5!0 at 5 (upper graph) and 10 (lower graph) absorption depths in a single resonance Lorentz model dielectric. (From Smith and Oughstun [6])

The z-dependence of this function then yields the thermal density profile of the net heat generated by the passage of the ultrashort electromagnetic pulse prior to diffusion. The numerically determined thermal density profiles W3D .z/ produced by an input 1 V=m electric field strength, 10-cycle rectangular envelope pulse with below resonance, intra-absorption band, and above resonance angular carrier frequencies are presented in Figs. 17.4–17.6, repsectively. The open circles in each figure describe the decibel equivalent numerical data points3 with the solid curve through each set describing the cubic spline fit to them. The dashed curve in each figure indicates the comparative z1 amplitude decay (in decibels) that is characteristic of the therrmal density profile of the Brillouin precursor. The dotted line in Fig. 17.5 describes the pure exponential decay e 2z=zd (in decibels) associated with 3

The decibel equivalent net heat density is given by 10 log10 .W3D .z// with units of dBJ=m3 .

17.2 Electromagnetic Energy Dissipation in Causally Dispersive Media

725

−115

Thermal Density (dBJ/m3)

−120 −125 −130 −135 e−2z/zd −140 ~(zd /z)

−145 −150

0

1

2

3

4

5 z /zd

6

7

8

9

10

Fig. 17.5 Numerically determined thermal density profile W3D .z/ produced by an input 1 V=m electric field strength, 10-cycle rectangular envelope pulse with below resonance angular carrier frequency !c D !0 =4

Thermal Density (dBJ/m3)

−130 −135 −140 −145 −150 ~(zd /z)

−155 −160

0

100

200 z/z d

300

400

Fig. 17.6 Numerically determined thermal density profile W3D .z/ produced by an input 1 V=m electric field strength, 10-cycle rectangular envelope pulse with intra-absorption band carrier frequency !c D 1:4375!0

726

17 Applications −115

Thermal Density (dBJ/m3)

−120 −125 −130 −135 −140 −145 −150 −155

~(zd /z)

0

10

20

30 z/zd

40

50

60

Fig. 17.7 Numerically determined thermal density profile W3D .z/ produced by an input 1 V=m electric field strength, 10-cycle rectangular envelope pulse with above absorption band carrier frequency !c D 2:5!0

the main body of the signal. Notice that the thermal density profile produced by the below-resonance pulse in Fig. 17.5 starts out along this exponential line but then departs from it as the dispersion matures and approaches the algebraic z1 decay due to the emerging dominance of the Brillouin precursor in the total field evolution. Because the pulse duration is on the order of the temporal width of the initial pulse, which is given by T D 2  1015 s for the below resonance ten-cycle pulse case illustrated in Figs. 17.1 and 17.2, the thermal power density profile is then approximately 150 dB higher than the thermal density profile illustrated in Fig. 17.5. A sequence of 6 fs rectangular envelope pulses separated in time by 4 fs would then be able to deliver significant power deep into the dispersive attenuative material. Similar results are obtained in the intra-absorption band and above absorption band cases presented in Figs. 17.6 and 17.7, the transformation from an exponential to an algebraic thermal density profile W3D .z/ now being due to the emerging dominance of both the Sommerfeld and Brillouin precursors. Notice that, because the characteristic e 1 penetration depth zd ˛ 1 .!c / is significantly smaller when the angular carrier frequency !c is in the absorption band than when it is in either of the normal dispersion regions above or below the absorption band, then the absolute propagation distance scale in Fig. 17.6 is on the same order of magnitude as that in Figs. 17.5 and 17.7. These results demonstrate the relatively deep heating produced by the precursor fields as compared with the heat produced by the signal component of the pulse alone. Because of this, large errors will result in predicting the heat generated by

17.2 Electromagnetic Energy Dissipation in Causally Dispersive Media

727

60 50 Thermal Density Ratio (dB)

z /zd = 10 40 30 20 10

z /zd = 5 z /zd = 1

0 −10

0

5

10

15 20 25 Number of Oscillations

30

35

Fig. 17.8 Ratio of the dispersive to nondispersive thermal densities as a function of the (integral) number of oscillations in the initial rectangular envelope pulse with below resonance angular carrier frequency !c D !0 =4 at 1, 5, and 10 absorption depths into a single resonance Lorentz model dielectric

an ultrawideband pulse as it penetrates into a dispersive attenuative body if the precursor fields are neglected, particularly when the penetration distance exceeds just a few absorption depths at the input pulse carrier frequency. For example, even though it may be argued that the field strength is negligible at eight absorption depths, at which distance the energy density in the carrier wave is reduced by e 16 ' 1:125  107 or  70 dB from its initial level, the evolved heat produced by the precursor fields at this depth is  32 dB above that produced by the carrier portion (or main body) of the pulse alone. The ratio of the dispersive to nondispersive thermal densities as a function of the (integral) number of oscillations in the initial rectangular envelope pulse with below resonance angular carrier frequency !c D 1:0  1016 r/s is illustrated in Fig. 17.8 at several different values of the relative penetration distance z=zd into the dispersive attenuative medium. At a single absorption depth (z=zd D 1) this ratio is very nearly unity for all pulse widths above a single oscillation. However, as the penetration distance into the material increases above just a few absorption depths and the precursor fields begin to dominate the propagated pulse structure, this is no longer true unless the initial pulse width is exceedingly large. For example, at z=zd D 5, this thermal density ratio for a 30-cycle pulse (T D 18:85 fs) is  6:9 dB, so that there is approximately 4:9 times greater thermal energy density generated at five absorption depths than that described by the nondispersive approximation. Such results as this can have a profound impact on the establishment of safe exposure levels for ultrawideband radiation.

728

17 Applications

17.3 Reflection and Transmission Phenomena A fundamental extension of the modern asymptotic theory of dispersive attenuative pulse propagation is provided by the problem of the reflection and transmission of a pulsed electromagnetic beam field that is incident upon the planar interface separating two half-spaces containing different causally dispersive media. Previous treatments of this problem have either focused on time-harmonic beam fields in lossless media [20, 21] with application to integrated and fiber optics or on pulsed plane wavefields when the incident medium is vacuum and the second medium is a dispersive attenuative dielectric [22,23] with application to ground and foliage penetrating radar as well as to medical imaging. The paper by Gitterman and Gitterman [22] is noteworthy not only in its analysis of the vacuum-dispersive medium problem, but also in the fact that they first noted the appearance of an additional precursor field in a double resonance Lorentz medium (see Sect. 13.5). The analysis presented here begins with the description of the single interface problem, based on that given by Marozas [24, 25] et al., as extended to include electric conductivity, followed by its application to describe the Goos–H¨anchen effect that is fundamental to both integrated and fiber optics. The results of this analysis are then used to describe the basic features observed in the reflection and transmission of an ultrawideband pulse at a dispersive attenuative layer as well as to address the question of superluminal barrier penetration.

17.3.1 Reflection and Transmission at a Dispersive Half-Space Let the homogeneous, isotropic, locally linear medium in which the incident and reflected wavefields reside be described by the frequency-dependent complex valued dielectric permittivity 1 .!/, electric conductivity 1 .!/, and constant magnetic permeability 1 , and let the homogeneous, isotropic, locally linear medium in which the transmitted (or refracted) wavefield reside be described by the frequencydependent complex valued dielectric permittivity 2 .!/, electric conductivity 2 .!/, and constant magnetic permeability 2 . Take the incident electromagnetic beam field to be propagating along the direction specified by the unit vector 1O w which is at the angle i with respect to the normal N to the interface S, take the reflected wavefield to be propagating along the direction specified by the unit vector 1O 0w which is at the angle r with respect to the surface normal N , and take the transmitted electromagnetic beam field to be propagating along the direction specified by the N , as unit vector 1O 00w which is at the angle t with respect to the surface normal   O O depicted in Fig. 17.9. The right-handed rectangular coordinate systems 1u ; 1v ; 1O w ,     1O 0u ; 1O 0v ; 1O 0w , and 1O 00u ; 1O 00v ; 1O 00w are then defined along each of these wave directions

>

' 1w

>

1v

1u'

>

>

1u

>

729

>

17.3 Reflection and Transmission Phenomena

1v'

1w w

i

r

w'

Medium 1

Medium 2 t

>

w''

>

>

1''u

1''v

'' 1w

      Fig. 17.9 Incident 1O u ; 1O v ; 1O w , reflected 1O 0u ; 1O 0v ; 1O 0w , and transmitted 1O 00u ; 1O 00v ; 1O 00w coordinate systems at a planar interface S with normal N separating two dielectric half-spaces. The upper half-space is occupied by the dispersive attenuative semiconducting medium 1 with dielectric permittivity 1 .!/, electric conductivity 1 .!/, and constant magnetic permeability 1 , and the lower half-space is occupied by the dispersive attenuative semiconducting medium 2 with dielectric permittivity 2 .!/, electric conductivity 2 .!/, and constant magnetic permeability 2

such that the unit vectors 1O v , 1O 0v , and 1O 00v are each directed out of the plane of incidence that is defined by the unit vector 1O w of the incident wave and the normal N to the interface, as indicated in Fig. 17.9. If the unit vector 1O w is along the normal N to the interface, then the plane of incidence is not uniquely defined; in that case, any plane containing the normal N may be chosen as the plane of incidence.

17.3.1.1

Angular Spectrum Representation of the Incident, Reflected, and Transmitted Pulsed Beam WaveFields

Let the incident electric and magnetic field vectors be specified on the plane w D w0 that is at a distance w0 > 0 from the interface along the 1O w -direction as [cf. (9.1) and (9.2)] E.rT ; w0 ; t / D E0 .rT ; t /;

(17.22)

B.rT ; w0 ; t / D B0 .rT ; t /;

(17.23)

where E0 .rT ; t / D E0 .u; v; t / and B0 .rT ; t / D B0 .u; v; t / are known functions of time and the transverse position vector rT D 1O u u C 1O v v (see Fig. 9.1). It is assumed here that the two-dimensional spatial Fourier transform in the transverse coordinates

730

17 Applications

.u; v/ and temporal Fourier–Laplace transform of each field vector exists, where [see (9.3)] EQQ 0 .kT ; !/ D

Z

Z

1 1

BQQ 0 .kT ; !/ D

1

Z

1

dt Z

1

Z

1

1

Z

1

dt 1

dudvE0 .rT ; t /e i.kT rT !t/ ;

(17.24)

dudvB0 .rT ; t /e i.kT rT !t/ ;

(17.25)

dku dkv EQQ 0 .kT ; !/e i.kT rT !t/ ;

(17.26)

dku dkv BQQ 0 .kT ; !/e i.kT rT !t/ ;

(17.27)

1

1

1

with inverse transforms E0 .rT ; t / D

1 .2/3

1 B0 .rT ; t / D .2/3

Z

Z

1

Z

1

d! 1

C

Z

Z

1

1

Z

1

d! 1

C

1

where kT D 1O u ku C 1O v kv is the transverse wave vector. If the initial time-dependence of the field vectors E0 .rT ; t / and B0 .rT ; t / at the plane w D w0 is such that both field vectors vanish for all t < t0 for some finite value of t0 , then the corresponding time– frequency transform pairs appearing in (17.24)–(17.27) are both Laplace transforms with integration contour C as a straight line path ! D ! 0 C i a with ! 0 D w0 or the negative half-space w < w0 are then given by the angular spectrum representation [see (7.14) and (7.17) of Vol. 1] E.r; t / D B.r; t / D

1 .2/3 1 .2/3

Z

Z

1

Z

1

d! 1

C

Z

Z

1

1

(17.28)

dku dkv BQQ 0 .kT ; !/e i.k

(17.29)

1

Z

1

d! C

Q˙ dku dkv EQQ 0 .kT ; !/e i.k r!t/ ;

Q ˙ r!t/

:

1

Here [see (6.101) of Vol. 1] kQ ˙ .!/ 1O u ku C 1O v kv ˙ 1O w .!/

(17.30)

is the complex wave vector, where kQ C .!/ is used for propagation into the positive half-space w > w0 and kQ  .!/ is used for propagation into the negative half-space w < w0 . Here .!/ is defined [see (6.94) of Vol. 1] as the principal branch of the expression i1=2 h ; (17.31) .!/ D kQ 2 .!/  kT2

17.3 Reflection and Transmission Phenomena

731

with kT2 D ku2 C kv2 , where [see (6.103) of Vol. 1]

1=2 ! Q k.!/ D kQ ˙ .!/  kQ ˙ .!/ D n.!/ c

(17.32)

is the complex wavenumber in the dispersive medium with complex index of refraction n.!/ D Œc .!/=.0 0 / 1=2 , where c .!/ .!/ C i k4k .!/=! is the complex permittivity [see (9.13)]. Finally, the spatiotemporal spectra of the electromagnetic field vectors at the plane at w D w0 are related by the transversality relations [see (7.18)–(7.20) of Vol. 1] EQQ 0 .kT ; !/ D 

kck Q ˙ k .!/  BQQ 0 .kT ; !/; !c .!/

(17.33)

kck Q ˙ k .!/  EQQ 0 .kT ; !/; BQQ 0 .kT ; !/ D !

(17.34)

kQ ˙ .!/  EQQ 0 .kT ; !/ D kQ ˙ .!/  BQQ 0 .kT ; !/ D 0:

(17.35)

The electromagnetic field vectors incident upon the interface S are obtained from (17.28) and (17.29) as E.i/ .r; t / D

1 .2/3

Z

Z

1

Z

1

d! 1

C

QQ .k ; !/e i.ku uCkv vC1 .!/w!t/ ; dku dkv E 0 T

1

(17.36) B.i/ .r; t / D

1 .2/3

Z

Z

1

Z

1

d! C

1

QQ .k ; !/e i.ku uCkv vC1 .!/w!t/ ; dku dkv B 0 T

1

(17.37) where 12 .!/ D kQ12 .!/  kT2 with kQ1 .!/ D .!=c/n1 .!/. The propagated plane wave spectra of the incident pulsed beam wavefield vectors at the interface S are QQ .k ; !/e i1 .!/w , with w denoting then seen to be given by EQQ 0 .kT ; !/e i1 .!/w and B 0 T the propagation distance along the direction 1O w to S. The corresponding reflected plane wave spectra at the interface are then given by rN .kT ; !/EQQ 0 .kT ; !/e i1 .!/w and rN .kT ; !/BQQ 0 .kT ; !/e i1 .!/w , and the corresponding transmitted plane wave spectra at the interface are given by Nt.kT ; !/EQQ 0 .kT ; !/e i1 .!/w and Nt.kT ; !/BQQ 0 .kT ; !/e i1 .!/w . Here rN .kT ; !/ D rN .ku ; kv ; ! denotes the amplitude reflection matrix and Nt.kT ; !/ D Nt.ku ; kv ; !/ denotes the amplitude transmission matrix for monochromatic (or timeharmonic) plane wave reflection and transmission at the planar interface S. With reference to Fig. 17.9, the reflected electromagnetic field vectors at the u0 v0 -plane of

732

17 Applications

the reflected coordinate system located a distance w0 from the interface along the 1O 0w -direction are then given by Z 1Z 1 Z 1 d! dku dkv rN .ku ; kv ; !/EQQ 0 .kT ; !/ E.r/ .r; t / D .2/3 C 1 1 0

e i.ku uCkv vC1 .!/.wCw /!t/ ; Z

Z

1 B.r/ .r; t / D .2/3

1

Z

(17.38) 1

d! 1

C

dku dkv rN .ku ; kv ; !/BQQ 0 .kT ; !/

1 0

e i.ku uCkv vC1 .!/.wCw /!t/ : (17.39) The transmitted electromagnetic field vectors are given by Z 1Z 1 Z 1 .t/ E .r; t / D d! dku dkv Nt.ku ; kv ; !/EQQ 0 .kT ; !/ .2/3 C 1 1 e i.ku uCkv vC1 .!/wC2 .!/w 1 B.t/ .r; t / D .2/3

Z

Z

1

Z

1

;

(17.40) 1

d! C

00 !t/

dku dkv Nt.ku ; kv ; !/BQQ 0 .kT ; !/

1

e i.ku uCkv vC1 .!/wC2 .!/w

00 !t/

; (17.41)

at the u00 v00 -plane of the transmitted coordinate system located a distance w00 from the interface along the 1O 00w -direction, where 22 .!/ D kQ22 .!/  kT2 with kQ2 .!/ D .!=c/n2 .!/. 17.3.1.2

Generalized Fresnel Equations

Application of the electromagnetic field boundary conditions given in (5.215) and (5.217) with %Q S .r; !/ D 0 and (5.221) and (5.223) with JQ S .r; !/ D 0 (i.e., with zero surface charge and current density on the interface S) at the planar interface separating semiconducting medium 1 with dielectric permittivity 1 .!/, electric conductivity 1 .!/, and constant magnetic permeability 1 , from the dispersive attenuative semiconducting medium 2 with dielectric permittivity 2 .!/, electric conductivity 2 .!/, and constant magnetic permeability 2 , results in both the generalized laws of reflection and refraction as well as in the generalized Fresnel equations for the amplitude reflection and transmission coefficients. Consider then the inhomogeneous plane wave spectral components of the incident electromagnetic field vectors given in (17.36) and (17.37), each with the complex wave vector O O O kQ C 1 .!/ D 1u ku C 1v kv ˙ 1w 1 .!/

(17.42)

17.3 Reflection and Transmission Phenomena

733

which has the complex direction cosine representation (see Sect. 7.1.1 of Vol. 1)   Q1 .!/ 1O u p C 1O v q C 1O w m1 .!/ D k kQ C 1

(17.43)

with

! kQ1 .!/ ˇ1 .!/ C i ˛1 .!/ D n1 .!/; (17.44) c where p ku =kQ1 .!/, q kv =kQ1 .!/, and m1 1 .!/=kQ1 .!/. Because ku and kv must both be real-valued quantities, then p D p 0 C ip 00 and q D q 0 C i q 00 must both be complex-valued with p 00 D 

˛1 .!/ 0 p ˇ1 .!/

&

q 00 D 

˛1 .!/ 0 q: ˇ1 .!/

(17.45)

Under this transformation to the complex direction cosine representation, the ku ; kv spatial frequency integrals appearing in (17.36) and (17.37) become p; q direction cosine integrals taken over the respective straight-line contours Cp and Cq which pass through the origin and are both at the angle  arctan .˛1 .!/=ˇ1 .!// with respect to the p 0 ; q 0 -axis, extending from negative to positive infinity (see Fig. 7.2 of Vol. 1). The complex direction cosine m1 D m01 C i m001 is then given by 1=2  m1 D 1  p 2  q 2       1=2 ˛ 2 .!/  02 ˛1 .!/  02 p C q 02 p C q 02 C 2i D 1  1  12 ; (17.46) ˇ1 .!/ ˇ1 .!/ where the principal branch of the square root expression is to be taken. With these results, the spatial part of the exponential phase term appearing in the angular spectrum representation in (17.36) and (17.37) is found to be given by kQ C 1 .!/  r D ku u C kv v C 1 .!/w    ˛ 2 .!/  0 D ˇ1 .!/ 1 C 12 p u C q0v ˇ1 .!/     0 C ˇ1 .!/m1  ˛1 .!/m001 w C i ˛1 .!/m01 C ˇ1 .!/m001 w: (17.47) Analogous expressions hold for both the reflected and transmitted electromagnetic beam fields. This then represents the spatial part of an inhomogeneous plane wave of angular frequency ! whose surfaces of constant amplitude w constant are, in general, different from the surfaces of constant phase (or cophasal surfaces)      ˛ 2 .!/  0 ˛1 .!/ 00 m1 w constant: p u C q 0 v C m01  1 C 12 ˇ1 .!/ ˇ1 .!/

(17.48)

734

17 Applications

The attenuative part of the complex wave vector for each inhomogeneous plane wave spectral component is then directed along the 1O w -direction and each inhomogeneous plane wave phase front (cophasal surface) propagates in the direction specified by the real-valued vector      ˛ 2 .!/  0 O ˛1 .!/ 00 O s 1 C 12 p 1u C q 0 1O v C m01  m1 1w ˇ1 .!/ ˇ1 .!/

(17.49)

with magnitude "  #1=2  2  ˛12 .!/  02 ˛1 .!/ 00 2 02 0 m p C q C m1  sD 1C 2 ˇ1 .!/ 1 ˇ1 .!/

(17.50)

This cophasal surface propagation direction is then completely specified by the set of real-valued direction cosines        ˛ 2 .!/ ˛ 2 .!/ 1 1 ˛1 .!/ 00 1 1 C 12 1 C 12 m01  ; p0; q0; .; ; / D m1 s s s ˇ1 .!/ ˇ1 .!/ ˇ1 .!/ (17.51) as illustrated in Fig. 17.10. These expressions then yield both the angle and plane of incidence of each inhomogeneous plane wave spectral component that is incident upon the interface S. Let the normal to the phase front of the incident inhomogeneous plane wave spectral component be at the angle i with respect to the normal N to the interface S, as specified by its direction cosines relative to the positive w-axis which itself is at the angle i with respect to N , as depicted in Fig. 17.9. The generalized law of reflection [24] then states that the reflected coordinate axis in the direction described

u s cos−1((

O cos−1((

v

)p'/s)

cos−1((m'- m''/ )/s)

Cophasal Surface

)q'/s)

w

Fig. 17.10 Inhomogeneous plane wave phase front propagating in the direction specified by the vector s. The attenuative part of the complex wave vector is directed along the positive w-axis

17.3 Reflection and Transmission Phenomena

735

by the unit vector 1O 0w lies in the ˛-plane of incidence that is defined by the incident unit vector 1O w and the normal N to the interface S and is at the angle r D i

(17.52)

with respect to N , while the reflected inhomogeneous plane wave spectral components’ phase front normal lies in the ˇ-plane of incidence that is defined by the vector s and the interface normal N and is at the angle r D i

(17.53)

with respect to the positive w0 -axis of the reflected coordinate system (see Fig. 17.9). In addition, the generalized law of refraction [24] states that the refracted coordinate axis in the direction described by the unit vector 1O 00w lies in the ˛-plane of incidence and is at the angle t with respect to N given by ˛t sin t D ˛i sin i ;

(17.54)

while the refracted inhomogeneous plane wave spectral components’ phase front normal lies in the ˇ-plane of incidence and is at the angle t with respect to the positive w00 -axis of the refracted coordinate system (see Fig. 17.9) given by ˇt sin t D ˇi sin i :

(17.55)

The normal component ˇtn of ˇt is given by the real part and the normal component ˛tn of ˛t is given by the imaginary part of the expression h i1=2  2 ˇtn .!/ C i ˛tn .!/ D n22 .!/k02  ˇi .!/ C i ˛i .!/ C ˛i2 .!/

(17.56)

with k0 D !=c, where the principal branch of the square root expression is to be taken. Here the subscript  denotes the tangential component defined by the plane of incidence and the interface plane, and the subscript  denotes denotes the component that lies in the interface plane but is normal to the plane of incidence such that the unit normal vector to the interface that is directed from medium 1 into medium 2 is given by (17.57) nO D 1O   1O  ; where 1O  is the unit vector in the tangential direction to the interface S defined by the plane of incidence and the interface plane and 1O  is the unit vector normal to the plane of incidence lying in the interface plane. The tangential components of the reflected and transmitted inhomogeneous plane wave spectral component electric field vectors are related to those of the incident electromagnetic beam field by the generalized Fresnel equations [24] N  .!/EQQ ./ EQQ ./ r .!/ D r i .!/; ./ ./ Q Q EQ t .!/ D tN .!/EQ i .!/;

(17.58) (17.59)

736

17 Applications

where the generalized Fresnel reflection matrix is given by



 N i .!/  Y N i .!/ C Y N t .!/ 1 Y N t .!/ ; rN  .!/ Y

(17.60)

and where the generalized Fresnel transmission matrix is given by



 N i .!/ : N i .!/ C Y N t .!/ 1 2Y Nt .!/ Y

(17.61)

Notice that the amplitude reflection and transmission matrices rN .ku ; kv ; !/ and Nt.ku ; kv ; !/ appearing in (17.38)–(17.41) are linearly related to the generalized Fresnel reflection and transmission matrices given in (17.60)and (17.61), respectively. The incident and transmitted complex admittance matrices appearing here are given by N i .!/ D Y

1 0 k0 kQin .!/

N t .!/ D Y

1 0 k0 kQtn .!/

i ˛i .!/kQi .!/ kQi2 .!/  n21 .!/k02 n21 .!/k02 C ˛i2 .!/ i ˛i .!/kQi .!/ 

i ˛t .!/kQt .!/ kQt2 .!/  n22 .!/k02 n22 .!/k02 C ˛t2 .!/ i ˛t .!/kQt .!/

! ; (17.62)  ;

(17.63)

p respectively, where 0 D 0 =0 D 376:73 ( is the intrinsic impedance of free space in mksa units. The angular frequency dispersion of the angular dependence of the real part of the Fresnel reflection coefficient for a TM-mode plane wavefield incident upon the planar interface separating two double resonance Lorentz model dielectrics whose frequency dispersion is described by the relative dielectric permittivity j .!/ D j C

X

bj2`

`D0;2

! 2  !j2` C 2i ıj ` !

;

j D 1; 2

(17.64)

is illustrated in Fig. 17.11 for the special case when the ˛- and ˇ-planes of incidence coincide. For simplicity, the two Lorentz model dielectrics considered here differ only in their limiting high-frequency values4 1 D 2:9938 and 2 D 1:9938, where !j 0 D 1:74121014 r/s, ıj 0 D 0:495551014 r/s, bj 0 D 1:21551014 r/s for the infrared resonance lines and !j 2 D 9:1448  1015 r/s, ıj 2 D 1:4241  1015 r/s, bj 2 D 6:7198  1015 r/s for the visible resonance lines in both dielectrics. Because 1 > 2 so that 1 .!/ > 2 .!/ for all real-valued !, incidence is from the optically rarer medium for all frequencies. Notice that the frequency dependence of the 4

Because the limiting behavior of j .!/ as ! ! 1 must be unity, the fact that j is not unity in both cases means that additional resonance lines are present (but not modeled) at higher frequencies in each medium.

17.3 Reflection and Transmission Phenomena

737

1.5 1 0.5

{rTM}

0

−0.5 −1 5 4 3 log10( ) in THz)

60

2 1 0

20 0

80

100

40 i

Fig. 17.11 Real part of the Fresnel reflection coefficient rTM .!/ as a function of both the angular frequency ! (in THz) and the angle of incidence i (in degrees) of a TM-mode plane wavefield incident upon the planar interface separating two similar double resonance Lorentz model dielectrics that differ only in their limiting high-frequency relative values 1 and 2 with 1 > 2 so that 1 .!/ > 2 .!/ for all real-valued !. (From Marozas and Oughstun [25])

Fresnel reflection coefficient is greatest near the critical angle i D c .!/ and that it becomes negligibly small as the angle of incidence i approaches grazing incidence (i.e., as i ! 90ı ).

17.3.1.3

Asymptotic Description of the Dynamical Transmitted Signal Evolution

Let the time-dependence of the intial pulsed beam wavefield at the input plane at w D w0 be such that its temporal Fourier–Laplace spectrum is ultrawideband, and (for reasons of specificity) let the initial angular carrier frequency !c of the pulse lie in the normally dispersive passband of medium 1 between the two q absorption bands

2 2 of the double resonance Lorentz model of that dielectric, so that !11  ı10 < !c < q 2 2 2 2 2 !20  ı20 , where !11 !10 C b10 . If the propagation distance w to the interface S

is large in comparison to the absorption depth zd1 ˛11 .!c / in medium 1 evaluated at the pulse carrier frequency, then each component of either the electric or magnetic field vector that is incident upon the interface has an asymptotic representation that may be expressed either in the form [see (15.2)] Ai .r; t / D As .r; t / C Ab .r; t / C Am .r; t / C Ac .r; t /;

(17.65)

738

17 Applications

or in a somewhat more complicated form that is given by a linear superposition of wavefields that are themselves expressed in the form given in (17.65). The reflected and transmitted wavefields will also have the same form for their asymptotic representations as either w0 ! 1 or w00 ! 1, respectively. If medium 1 is the vacuum and medium 2 is a double resonance Lorentz model dielectric, then this representation applies just to the transmitted wavefield. In that case, the passband for the incident pulse is set by the normal dispersion region between the two absorption bands of Lorentz medium 2. Clearly, other situations may occur and they are best treated on a case-by-case basis. Several uniquely interesting phenomena appear in the dynamical space–time evolution of the transmitted wavefield. Because the instantaneous angular frequency of oscillation !s . / of the Sommerfeld precursor wavefield [see (13.44)] decreases monotonically from infinity as it evolves, the real part of the complex index of refraction presented to this transient wavefield component will decrease monotonically and its angle of refraction will consequently change dynamically as the wavefield evolves such that the Sommerfeld precursor will spatially fan out from the initial refraction angle equal to the angle of incidence at the infinite frequency front of the Sommerfeld precursor to smaller angles, approaching the angle of refraction set by the angular frequency value at the upper edge of the upper absorption band in the dispersive medium 2, as depicted in Fig. 17.12. A similar space–time effect will occur for the transmitted Brillouin precursor wavefield, but in the opposite direction and beginning at the quasistatic angle of refraction set by the static indices of refraction and increasing to the limiting angle of refraction set by the angular frequency value at the lower edge of the lower absorption band in the dispersive medium 2, also depicted in Fig. 17.12. A smaller space–time evolution in the transmitted middle precursor field will also be observed about the steady-state angle of refraction t set by the main body of the pulse (if there is one5 ). This spatiotemporal coupling of the transmitted pulse is due to the combined effects of angular dispersion at the interface and temporal dispersion in the transmission medium as well as (but typically to a lesser extent) temporal dispersion in the incident medium. Notice that this space–time effect allows one to spatially separate the individual precursor fields from both each other as well as from the main body of the pulse in a well-designed experiment.

17.3.2 The Goos–H¨anchen Shift In the special, idealized case of lossless media, the incident, reflected and transmitted plane waves are either all homogeneous plane waves or, in the case of total internal reflection, the transmitted wave is evanescent (see Sect. 9.1.3). The ˛- and 5

There isn’t any pole contribution for a gaussian envelope pulse whose dynamical evolution is described completely by the precursor fields.

17.3 Reflection and Transmission Phenomena

739

Θi

Medium 1 n1 Medium 2 n2 s

Θ(0)

∞

As(r,t) Θt

Θi Ab(r,t)

i

c

b

Fig. 17.12 Graphical depiction of the dynamical space–time evolution of the refracted signal wavefield due to an incident Heaviside step-function modulated plane wavefield with angular carrier frequency !c . The steady-state angle of refraction t is for the main signal at ! D !c , as specified Snell’s law of refraction nr1 .!c / sin i D nr2 .!c / sin t . The transmitted Sommerfeld precursor front is at the angle of incidence i and, as the Sommerfeld precursor evolves in time, its angle of refraction sweeps to smaller angles as its instantaneous angular oscillation frequency !s chirps downward, as indicated. The transmitted Brillouin precursor front is refracted at the quasistatic angle of refraction .0/ satisfying nr1 .0/ sin i D nr2 .0/ sin .0/ and, as the Brillouin precursor evolves in time, its angle of refraction sweeps to larger angles as its instantaneous angular oscillation frequency !b chirps upward, as indicated. (From Marozas and Oughstun [25])

ˇ-planes then become coincident, the law of reflection is then just given by (17.52) and the law of refraction (Snell’s law) is obtained from (17.55) as n1 .!/ sin 1 D n2 .!/ sin 2 ;

(17.66)

where both (idealized) indices of refraction n1 .!/ and n2 .!/ are real-valued. In addition, the generalized Fresnel reflection and transmission matrices given in (17.60) and (17.61), respectively, become diagonal matrices. For TE-mode or s-polarization with the electric field vector linearly polarized along the v-direction (i.e., perpendicular, or “senkrecht” in German, to the plane of incidence), the Fresnel reflection and transmission coefficients are found to be given by [26] rE .!/ D

n1 .!/ cos 1  n2 .!/ cos 2 ; n1 .!/ cos 1 C n2 .!/ cos 2

(17.67)

tE .!/ D

2n1 .!/ cos 1 ; n1 .!/ cos 1 C n2 .!/ cos 2

(17.68)

740

17 Applications

respectively. Notice that tE .!/ D 1 C rE .!/; this does not violate conservation of energy because the Fresnel coefficients relate field vector amplitudes and not electromagnetic energy. If n1 .!/ < n2 .!/ at the incident wave frequency, so that the plane wavefield is incident upon the optically denser medium, then the angle of refraction 2 has a real-valued solution for all angles of incidence in the fundamental domain 1 2 Œ0; =2 . Furthermore, the amplitude reflection coefficient is negative-valued for all 1 2 Œ0; =2 , so that the reflected TE-wavefield is  out of phase with the incident field, as illustrated in Fig. 17.13 when n1 D 1:5 and n2 D 2:0. Notice that rE .0/ D .n1  n2 /=.n1 C n2 / and tE .0/ D 2n1 =.n1 C n2 / and that rE .=2/ D 1 and tE .=2/ D 0. If n1 .!/ > n2 .!/ at the incident wave frequency, so that the plane wavefield is incident upon the optically rarer medium from the optically denser medium, then the angle of refraction 2 has a real-valued solution for angles of incidence in the restricted domain 1 2 Œ0; c , where   n2 .!/ (17.69) c .!/ arcsin n1 .!/ is the critical angle for the two media at the angular frequency of oscillation ! of the wave. As illustrated in Fig. 17.14, the reflected wavefield is in phase with the incident plane wavefield in the restricted domain of incidence 1 2 Œ0; c . At the critical angle of incidence 1 D c , rE D 1, and the incident wave is totally reflected, and at supercritical angles of incidence 1 2 Œc ; =2 , jrE j D 1, and the well-known phenomenon of total internal reflection occurs, where rE D e iE ; with tan .E =2/ D

1 2 Œc ; =2 ; q sin2 1  .n2 =n1 /2 cos 1

(17.70)

:

(17.71)

Notice that the real part of the transmission coefficient tE does not vanish at either the critical angle (where tE D 2) or at all supercritical angles of incidence, but rather decreases monotonically to zero as 1 increases to grazing incidence, as illustrated in Fig. 17.14. Because the angle of refraction 2 is then pure imaginary, the transmitted wavefield is evanescent (see Sect. 9.1.3). The angular dependence of the internally reflected TE-mode phase angle E at supercritical angles of incidence is illustrated in Fig. 17.15. For TM-mode or p-polarization with the electric field vector linearly polarized along the u-direction (i.e., parallel to or in the plane of incidence), the Fresnel reflection and transmission coefficients are found to be given by [26] rH .!/ D

n1 .!/ cos 2  n2 .!/ cos 1 ; n1 .!/ cos 2 C n2 .!/ cos 1

(17.72)

tH .!/ D

2n1 .!/ cos 2 ; n1 .!/ cos 2 C n2 .!/ cos 1

(17.73)

respectively. Notice that tH .!/ D 1 C rH .!/.

17.3 Reflection and Transmission Phenomena

741

1 2n1 n2 + n1

tE

Fresnel Coefficients TE-mode (s-polarization)

0.8

0.6

0.4 −rE 0.2 n2 – n1 n2 + n1 0

0

10

20

30

40

50

60

70

80

90

Θ1 (degrees)

Fig. 17.13 Fresnel reflection rE and transmission tE coefficients for s-polarization when the wavefield is incident upon the optically denser medium [n1 .!/ < n2 .!/]. Notice that rE is plotted

Fresnel Coefficients TE-mode (s-polarization)

2

2n1 n1 + n2 1

ℜ{tE}

n1 – n2

ℜ{rE}

n1 + n2 0

–1

ℑ{rE} = ℑ{tE}

Θc 0

10

20

30

40 50 60 Θ1 (degrees)

70

80

90

Fig. 17.14 Real and imaginary parts of the Fresnel reflection rE and transmission tE coefficients for s-polarization when the wavefield is incident upon the optically rarer medium [n1 .!/ > n2 .!/]. Notice that the imaginary parts of the reflection and transmission coefficients are equal

742

17 Applications p 3

fE (radians)

2

1

0 qc 50

60

70 Θ1 (degrees)

80

90

Fig. 17.15 Phase change E on total internal reflection for TE-mode or s-polarization as a function of the supercritical angle of incidence 1 c when n1 .!/ > n2 .!/

If n1 .!/ < n2 .!/ at the incident wave frequency, so that the plane wavefield is incident upon the optically denser medium, then the angle of refraction 2 has a real-valued solution for all angles of incidence in the fundamental domain 1 2 Œ0; =2 . At 1 D p the reflection coefficient rH vanishes. This special angle of incidence is called the polarizing angle or Brewster’s angle and is defined by the condition [from (17.72)] n2 cos p D n1 cos 2 . Combination of this expression with Snell’s law (17.66) then yields the well-known result tan p D

n2 .!/ n1 .!/

(17.74)

 ; 2

(17.75)

Furthermore, at the polarizing angle p C 2 D

so that the reflected and transmitted wave vectors are at right angles to each other. For angles of incidence 1 2 Œ0; p / the Fresnel reflection coefficient rH is negative-valued so that the reflected wavefield is  out of phase with the incident wavefield, and for 1 2 .p ; =2 the Fresnel reflection coefficient is positivevalued so that the reflected wavefield is in phase with the incident wavefield, as illustrated in Fig. 17.16.

17.3 Reflection and Transmission Phenomena

743

2

Fresnel Coefficients TM-mode (p-polarization)

1.5 tH 1 2n1 n1 + n2 0.5 rH 0 n1 – n2

Θp

n1 + n2 −0.5 0

10

20

30

40

50

60

70

80

90

Θ1 (degrees)

Fig. 17.16 Fresnel reflection rH and transmission tH coefficients for p-polarization when the wavefield is incident upon the optically denser medium [n1 .!/ < n2 .!/] 2

Fresnel Coefficients TE-mode (s-polarization)

ℜ{tE} 2n1 n1 + n2 1

ℜ{tE}

n1 – n2

ℜ{rE}

n1 + n2 0

−1 0

ℜ{rE}

ℑ{rE} = ℑ{tE}

ℑ{rE} = ℑ{tE}

10

20

30

Θp

40 60 Θc Θ1 (degrees)

70

80

90

Fig. 17.17 Real and imaginary parts of the Fresnel reflection rH and transmission tH coefficients for p-polarization when the wavefield is incident upon the optically rarer medium [n1 .!/ > n2 .!/]. Notice that the imaginary parts of the reflection and transmission coefficients are equal

744

17 Applications 2p

fH (radians)

6

5

4

p qp 40

qc 50

70 60 Θ1 (degrees)

80

90

Fig. 17.18 Phase change H on total internal reflection for TM-mode or p-polarization as a function of the supercritical angle of incidence 1 c when n1 .!/ > n2 .!/

If n1 .!/ > n2 .!/ at the incident wave frequency, so that the plane wavefield is incident upon the optically rarer medium from the opticall denser medium, then one again finds a polarizing or Brewster’s angle p given by (17.74) at which angle of incidence the reflection coefficient rH vanishes, and a critical angle c given by (17.69) beyond which the angle of refraction 2 becomes pure imaginary and the transmitted field becomes evanescent. For incidence angles in the restricted domain 1 2 Œ0; p / the reflected wavefield is in phase with the incident wave, and in the domain 1 2 .p ; c the refracted wavefield is  out of phase with the incident wavefield, as illustarted in Fig. 17.17. At the critical angle of incidence 1 D c , rH D 1 and the wavefield is totally reflected  out of phase with the incident plane wavefield. At supercritical angles of incidence 1 2 Œc ; =2 , jrH j D 1 and total internal reflection occurs, where rH D e iH ;

1 2 Œc ; =2 ;

(17.76)

with .n2 =n1 /2 cos 1 tan .H =2/ D  q : sin2 1  .n2 =n1 /2

(17.77)

The phase change H of the reflected wavefield (relative to the incident plane wavefield) then varies from  at 1 D c to 2 at grazing incidence (1 D =2), as illustrated in Fig. 17.18.

17.3 Reflection and Transmission Phenomena

745

Hence, when a time-harmonic electromagnetic plane wavefield with angular frequency ! propagating in an idealized lossless dielectric medium 1 with index of refraction n1 .!/ is incident upon a planar dielectric interface separating it from another idealized dielectric medium with smaller index of refraction n2 .!/ < n1 .!/ at a supercritical angle of incidence 1 > c , the transmitted field is evanescent and the reflected plane wavefield has amplitude equal to that of the incident wave but is shifted in phase by an amount dependent upon the supercritical angle of incidence. The reflection coefficient for supercritical angles of incidence r D e i is given by (17.70) and (17.71) for TE-mode or s-polarization and by (17.76) and (17.77) for TM-mode or p-polarization. For either polarization type, the phase shift  is seen to be a strong function of the supercritical angle of incidence, as illustrated in Figs. 17.15 and 17.18. If one then considers the supercritical incidence of an electromagnetic beam field which may be represented as a superposition of homogeneous plane waves, each propagating in a slightly different direction about some mean propagation direction 1O w , then each plane wave component will be incident on the planar dielectric interface S at a slightly different angle and hence, each will undergo a slightly different phase change upon reflection, assuming that each is incident at a supercritical angle. The reflected beam field will then be comprised of a superposition of these phase-shifted plane wave components which (from the shift theorem in Fourier analysis) then primarily results in a spatial displacement of the reflected beam along the interface, an effect first described by F. Goos and H. H¨anchen [27] in 1947 and of fundamental importance in dielectric waveguide phenomena. Consider then an electromagnetic beam wavefield that is propagating along the 1O w -direction in an idealized lossless dielectric medium 1 with (real-valued) refractive index n1 .!/ and is incident upon a planar interface S separating medium 1 from another idealized lossless dielectric medium 2 with (real-valued) refractive index n2 .!/ where the inequality n1 .!/ > n2 .!/ is satisfied over the bandwidth of the pulse. Let the w-axis be at the angle 1 to the normal N to the interface S so that 1 is the  angle ofincidence of the beam field, as depicted in Fig. 17.9. The unit vectors 1O u ; 1O v ; 1O w are then chosen such that they form a right-hand orthog  onal triad. In addition, let the 1O 0u ; 1O 0v ; 1O 0w coordinate system be introduced for the reflected beam field such that the w0 -axis intersects the interface S at the same point as the w-axis and is in the plane of incidence formed by the w-axis and the normal N to S and, from the law of reflection given in (17.52), is at theangle 1 from the normal N , as depicted in Fig. 17.9. Although the unit vectors 1O 0u ; 1O 0v ; 1O 0w for the reflected coordinate system also form a right-hand orthogonal triad, the transverse unit vecors 1O 0u and 1O 0v are taken here (as in [28]) to be the negative of those depicted in Fig. 17.9; this change affects the sign of the reflection coefficient which is determined by the orientation of the electric field intensity vector of the reflected wave (either with respect to the v0 -axis for s-polarization or with respect to the u0 -axis for p-polarization).

746

17 Applications

If a plane wavefield of the form QQ .k ; k ; !/e i.ku uCkv vCkw w!t/ A 0 u v is incident upon the interface S, it produces a reflected plane wavefield of the form QQ .k 0 ; k 0 ; !/e i.ku0 u0 Ckv0 v0 Ckw0 w0 !t/ ; A 0 u v where .ku0 ; kv0 ; kw0 / are the components of the wave vector for the reflected wavefield in the .u0 ; v0 ; w0 /-coordinate system. These components are determined by the law of reflection in terms of the wave vector components for the incident wavefield as (17.78) ku0 D ku ; kv0 D kv ; kw0 D kw : In addition, the reflected wavefield vector is given by QQ .k ; k ; !/ D r.k ; k ; !/A QQ .k ; k ; !/e ikw w0 ; A r u v u v 0 u v

(17.79)

where w0 denotes the distance from the plane w D 0 along the w-axis to the interface S and then along the w0 -axis to the plane w0 D 0. The factor r.ku ; kv ; !/ is the appropriate Fresnel reflection coefficient for the polarization state considered (i.e., TE- or TM-mode polarization). Let the angular spectrum of the incident electromagnetic beam field at the plane w D 0 be denoted by EQQ 0 .ku ; kv ; !/. The angular spectrum of the reflected wavefield at the plane w0 D 0 is then given by (17.79) and the temporal frequency spectrum of the electric field vector of the reflected electromagnetic beam wavefield is given by EQ r .u0 ; v0 ; w0 ; !/ D

1 4 2

Z

1 1

Z

1

r.ku ; kv ; !/EQQ 0 .ku ; kv ; !/

1 0

0

e i.ku0 u Ckv0 v Ckw0 w0 / dku dkv :

(17.80)

The precise behavior of the reflected beam wavefield is then determined by the form of the Fresnel reflection coefficient r.ku ; kv ; !/. Consider first the case of ideal reflection when the reflection coefficient [in the reflected .u0 ; v0 ; w0 /-coordinate system] is unity. The reflected beam wavefield given in (17.80) then becomes Z 1Z 1 1 QQ .k ; k ; !/e i.ku0 u0 Ckv0 v0 Ckw0 w0 / dk dk 0 0 EQ .g/ .u ; v ; w ; !/ D E 0 0 u v u v r 4 2 1 1 D EQ i .u0 ; v0 ; w0 ; !/: (17.81) The ideal reflected beam field is then identical to the incident beam field after propagation through the distance w0 in medium 1. This result is just that described by geometrical optics [the reason for the superscript .g/].

17.3 Reflection and Transmission Phenomena

747

In reality, the reflection coefficient r.ku ; kv / appearing in (17.80) is given by the polarization appropriate Fresnel coefficient for the reflection of a plane wave at a planar interface separating two dielectric media. The situation of primary interest here is when the beam field is incident upon the dilectric interface S from the optically denser medium (n1 .!/ > n2 .!/) when all of the angular spectrum components of the incident wavefield are at supercritical angles of incidence. Let .ku /max denote the maximum value of ku at which the u-component of the spectrum EQQ 0 .ku ; kv ; !/ has a non-negligible amplitude. The quantity 

.ku /max kw

(17.82)

then describes the angular extent of the beam field about the w-axis (the propagation direction of the electromagnetic beam) in the plane of incidence. In order that the entire beam field is incident at supercritical angles, it is required that the angle of incidence of the beam field satisfy the inequality 1 > c C :

(17.83)

Because of this, the immediate neighborhood about the critical angle is excluded in this description. In addition, it is required that all significant angular spectrum components of the incident beam wavefield are incident at angles less than grazing incidence, so that  (17.84) 1 <  : 2 If both inequalities given in (17.83) and (17.84) are strictly satisfied, then each plane wave component of the incident electromagnetic beam wavefield will undergo total internal reflection at the interface S. The Fresnel reflection coefficient for each plane wave spectral component is then given by [cf. (17.70) and (17.76)] r.ku ; kv ; !/ D e i.ku ;kv ;!/ ;

(17.85)

where .ku ; kv ; !/ denotes the phase change upon total internal reflection, given by (17.71) for a TE-mode field (s-polarized) and by (17.77) for a TM-mode field (p-polarized). With this substitution, the angular spectrum representation given in (17.80) becomes EQ r .u0 ; v0 ; w0 ; !/ D

1 4 2

Z

1

1

Z

1

EQQ 0 .ku ; kv ; !/e i.ku ;kv ;!/

1 0

0

e i.ku u Ckv v Ckw0 w0 / dku dkv : for the total internally reflected beam wavefield.

(17.86)

748

17 Applications

Because of the pair of inequalities given in (17.83) and (17.84), the phase change .ku ; kv ; !/ is continuous over the domain of interest. It may then be expanded in a Taylor series about ku D 0 as [28]  .ku ; kv ; !/ D .0; kv ; !/ C ku D  0 C ku D C

@ @ku

 C ku D0

ku2 2



@2  @ku2



ku2 F C  ; 2k

C  ku D0

(17.87)

where 0 .0; kv ; !/ .0; 0; !/;

(17.88)

ˇ @.ku ; kv ; !/ ˇˇ ; D ˇ @ku ku D0

(17.89)

ˇ @2 .ku ; kv ; !/ ˇˇ F k : ˇ @ku2 ku D0

(17.90)

Substitution of the first two terms of this Taylor series into (17.86) for the reflected beam wavefield then gives e i0 EQ r .u0 ; v0 ; w0 ; !/ D 4 2

Z

1

1

Z

1

0 0 EQQ 0 .ku ; kv ; !/e iŒku .u D/Ckv v Ckw0 w0 dku dkv

1

which may be expressed in terms of the ideal (geometrical optics) reflected beam given in (17.81) as 0 0 EQ r .u0 ; v0 ; w0 ; !/ ' e i0 EQ .g/ r .u  D; v ; w0 ; !/:

(17.91)

Thus, aside from an overall phase shift 0 , the effect of the first-order term in the Taylor series expansion in (17.87) is to displace the center of the reflected beam a distance D along the u0 -direction, leaving the shape of the beam undistorted, as depicted in Fig. 17.19. This lateral displacement from the geometrical optics path is known as the Goos–H¨anchen shift. Inclusion of the second-order term in the Taylor series expansion in (17.87) results in a focal shift of the reflected beam wavefield, as described by McGuirk and Carniglia [28]. Consider now obtaining an explicit analytic expression for the lateral displacement D in terms of the supercritical angle of incidence 1 > c of the electromagnetic beam wavefield. The angle of incidence u of the ku -component of the angular spectrum for the incident beam wavefield is given by u D 1  arcsin .ku =k/ so that 1 1 u D  arcsin .ku =k/ and ku D k sin . 1 /, where ku D 0 corresponds to u D 1 . One then has that d ku D k cos .1  u /d1 ! kd1 at u D 1 . With this result, (17.89) becomes

17.3 Reflection and Transmission Phenomena

749

Medium 2 n2 n1

>

Medium 1 n D

> 1w

1 w'

1

1

Fig. 17.19 Depiction of the Goos–H¨anchen shift for an electromagnetic beam wavefield incident at a supercritical angle of incidence on the interface S

ˇ ˇ 1 @ ˇˇ  @ ˇˇ DD D ; k @ ˇD1 2 @ ˇD1

(17.92)

where the final expression is appropriate for a monochromatic beam with wavelength  in medium 1. From (17.71), the Goos–H¨anchen shift for s-polarization (TE-mode) is found to be given by DE D

sin 1  ; q  sin2   n2 =n2 1

2

(17.93)

1

and from (17.77), the Goos–H¨anchen shift for p-polarization (TM-mode) is found to be given by DE ; (17.94) DH D 2 2 .1 C n1 =n2 / sin2 1  1 for 1 > c C . Notice that both of these expressions for the Goos–H¨anchen shift are singular at the critical angle 1 D c ; this critical value can be approached from above by making the angular spread of the incident beam wavefield  very small. The angular dependence of the Goos–H¨anchen shift for both s- and p-polarization is illustrated in Fig. 17.20 when n1 D 2:0 and n2 D 1:5. This lateral Goos–H¨anchen shift of the reflected beam upon total internal reflection is equivalent to perfect (i.e., geometrical optics) reflection from a hypothetical interface a distance dj into medium 2. From the simple geometry indicated in Fig. 17.19 it is seen that Dj ; (17.95) dj D 2 sin 1

750

17 Applications 2

q D/ λ 1

DE / λ DH / λ

0 Θc

60

70 Θ1 (degrees)

80

90

Fig. 17.20 Supercritical angular depenedence of the Goos–H¨anchen shift for both s-polarization (solid curve) and p-polarization (dashed curve), where q D DE =DH

for j D E; H . Hence, dH D dE =q and DH D DE =q, where  qD

 n21 n2 C 1 sin2 1  1 D 12 sin2 1  cos2 1 ; 2 n2 n2

(17.96)

for 1 2 .c ; =2/. The supercritical angular dependence of this factor is indicated by the dotted curve in Fig. 17.20. Notice that q D q.1 / has a well-defined limiting value at the critical angle given by q.c / D n22 =n21 . A more detailed analysis of the Goos–H¨anchen shift at and near the critical angle has been given by Chan and Tamir [21]. Generalizations of these results to dispersive absorptive media remains to be fully addressed. The case when the reflecting medium is absorptive has been given by Wild and Giles [29] where it was shown that, under certain conditions, the Goos– H¨anchen shift can become negative. Because this effect is an essential part of the guided mode condition in dielectric optical waveguides [30], particularly in integrated optics [31], its rigorous solution when the frequency dispersion of both the core and cladding materials are properly described by causal models is of fundamental importance, particularly at terabit per second (Tbit=s) transmission rates [32]. For example, a 1 Tbit=s data rate requires that the rectangular envelope pulse bit duration Tb be on the order of Tb  5  1013 s D 500 fs so that a 100 Tbit=s data rate requires that Tb  5 fs where precursor effects become critical.

17.4 Optimal Pulse Penetration through Dispersive Bodies

751

17.3.3 Reflection and Transmission at a Dispersive Layer: The Question of Superluminal Tunneling The problem of the reflection and transmission of an ultrawideband electromagnetic pulse from and through a dispersive absorptive material layer is abundant in physical wave phenomena with a wide variety of practical applications. For example, although the design methodology for antireflection coatings for continuous wave applications is well established, its extension to ultrawideband pulses is not as clearly defined because of the nonzero timing delay between the reflected pulse sequence. A quarter-wave dielectric layer will not extinguish the reflected wavefield when the incident wavefield is a single cycle pulse, and it may even enhance the reflected pulse under certain conditions. At a more fundamental level, the question of superluminal tunneling through a dispersive dielectric layer is of considerable interest [33]. Because the Fresnel transmission coefficients for s- and p-polarization are causal when the material dispersion models are causal, then Sommerfeld’s relativistic causality theorem (Theorem 6 in Sect. 13.1) applies to the transmtted pulse which then identically vanishes for all superluminal transit times when the input pulse vanishes for all t < 0. Of course, for gaussian envelope pulses, pulse reshaping effects can give the appearance of superluminal tunneling, as described in Sects. 15.9 and 15.12.

17.4 Optimal Pulse Penetration through Dispersive Bodies The analysis presented in Sect. 15.8.3 resulted in the identification of a so-called Brillouin pulse that is comprised of a pair of Brillouin precursor structures with the trailing precursor delayed in time and  phase shifted from the leading precursor, given by (15.189) as [9]    .!N .T /; T / .!N ./; /  exp : fBP .t / D exp !c ni .!c / !c ni .!c / 

(17.97)

Here T   cT =zd where T > 0 describes the fixed time delay between the leading and trailing Brillouin precursors. If T is chosen too small then there will be significant destructive interference between the leading and trailing components and the pulse will be rapidly extinguished. For practical reasons, 2T should be chosen near to the inverse of the operating frequency fc of the antenna used to radiate this Brillouin pulse. Because of its dependence on the complex index of refraction along the imaginary axis, the Brillouin pulse shape fBP .t / is dependent on the dispersive properties of the medium, so that each Brillouin pulse is uniquely matched to the dispersive material it is to penetrate.With the normally incident pulse in vacuum,

752

17 Applications

the transmitted plane wave pulse in the dispersive medium with complex index of refraction n.!/ is given by the Fourier integral representation 1 ABP .z; t / D 

Z

1 1

n.!/ Q Q fBP .!/e i .k.!/z!t / d! 1 C n.!/

(17.98)

for all z > 0, where the planar interface is situated at z D 0. Here fQBP .!/ denotes the Fourier transform of the dispersion-matched Brillouin pulse. Notice that the transmitted pulse will be distorted by the frequency-dependent behavior of the transmission coefficient so that the initial pulse just inside the dispersive material is no longer optimal. If necessary, this effect can be corrected by pre-distorting the incident pulse spectrum by the inverse of the transmission coefficient, so that fQBP .!/ ! Œ1 C n.!/=2n.!/ fQBP .!/ in (17.98).

17.4.1 Ground Penetrating Radar With the experimentally measured material dispersion data given by Tinga and Nelson [34], the relative complex dielectric permittivity of loamy soil may be accurately described by a three term Rocard–Powles–Debye model (see Sect. 4.4.3 of Vol. 1) 3 X aj 0 =0 Ci (17.99) c .!/ D 1 C .1  i !j /.1  i !fj / ! j D1 augmented by a static conductivity factor to account for both an ambient material conductivity and that resulting from the moisture content in the soil. The rms best-fit parameters for loamy soil at 25ı C are given in Tables 17.1 and 17.2 for 0% and 2:2% moisture contents, respectively. A comparison of the resultant angular frequency dispersion described by (17.99) with the experimental data for each case is presented in Fig. 17.21. The open circles describe the experimental data for 0% moisture content and the diamonds describe the experimental data for 2:2% moisture content. For comparison, the frequency dependence described by (17.99) for the 0% moisture content case with 0 D 0 is described by the dotted curves in Fig. 17.21. The only discernible difference is in the low-frequency behavior of

Table 17.1 Estimated rms “best-fit” Rocard–Powles–Debye model parameters for loamy soil at 25ı C with 0% moisture content, where 1 D 2:444 and 0 D 6:7  109 mho=m. Here rmsf<Œc .!/ g D 0:0044 and rmsf=Œc .!/ g D 1:158 between the model values and the experimental data j aj j fj 1 2 3

0.167 0.057 0.025

1:032  105 .s/ 6:54  108 .s/ 1:514  109 .s/

2:6  1013 .s/ 7:74  108 .s/ 5:0  1014 .s/

17.4 Optimal Pulse Penetration through Dispersive Bodies

753

Table 17.2 Estimated rms “best-fit” Rocard–Powles–Debye model parameters for loamy soil at 25ı C with 2:2% moisture content, where 1 D 3:305 and 0 D 2:5  106 mho=m. Here rmsf<Œc .!/ g D 0:065 and rmsf=Œc .!/ g D 0:822 between the model values and the experimental data j aj j fj 1 2 3

13.63 0.330 0.293

1:82  106 .s/ 1:97  108 .s/ 2:36  1010 .s/

8:78  1011 .s/ 1:12  1010 .s/ 1:0  1015 .s/

2.2%

10

ℜ{

c

}

20

0% 0 102

104

106

(r/s)

108

1010

1012

105

ℑ{

c

}

2.2% 0

10

0%

s0 = 0 10–5 2 10

104

108

106

1010

1012

(r/s)

Fig. 17.21 Angular frequency dispersion of the real (upper graph) and imaginary (lower graph) parts of the complex dielectric permittivity c .!/ for loamy soil at 0% (solid curves) and 2:2% (dashed curves) moisture contents. (Experimental data points from Tinga and Nelson [34])

the imaginary part of the complex dielectric permittivity. Notice that both the real and imaginary parts of the complex dielectric permittivity increase as the moisture content increases. Experimental data [34] up through a 13:77% moisture content shows that both 1 and each of the coefficients aj increase as the moisture content increases, whereas each of the material relaxation times j decrease as the moisture content increases. The Brillouin pulse for this more complicated medium model can be determined numerically by computing the propagated pulsed wavefield due to an initial singlecycle pulse (the mother pulse) at the desired carrier frequency fc at a sufficiently large propagation distance; typically z  10zd is sufficient as the amplitude of the signal contribution has been attenuated by at least the factor e 10 ' 0:0000454

754

17 Applications 1 Ab(0 -,t)

Ab(0,t)

0.5

0

Ab(0 +,t) −0.5 0

0.1

0.2 t (ms)

0.3

0.4

0.5

Fig. 17.22 Incident (solid curve) and transmitted (dashed curve) Brillouin pulses Ab .0˙ ; t / with fp  0:1 MHz effective oscillation frequency at the vacuum-soil interface at z D 0. The mother pulse for the numerically determined incident Brillouin pulse Ab .0 ; t / was a single-cycle rectangular envelope pulse with fc D 0:1 MHz angular carrier frequency

while the peak amplitude of the leading edge Brillouin precursor is approximately 1500 times larger (see Fig. 15.71). The result of such a calculation in loamy soil with zero conductivity when the initial single cycle pulse frequency is fp D 0:1 MHz is described by the solid curve in Fig. 17.22 after the peak amplitude has been renormalized to unity. The transmitted Brillouin pulse Ab .0C ; t / just inside the medium at z D 0C , given by the inverse Fourier transform of the product of the Fresnel transmission coefficient with the Fourier transform of the incident Brillouin pulse Ab .0 ; t / is described by the dashed curve in Fig. 17.22. The transmitted pulse energy is reduced to 57:6% of the incident pulse energy, as compared to 59:5% energy transmitted for a rectangular envelope pulse with the same frequency. Notice that minimal pulse distortion occurs in the transmitted pulse in this case. A comparison of the numerically determined peak amplitude decay and energy decay of the dispersion-matched, unit amplitude Brillouin pulse as a function of the penetration distance into loamy soil is presented in Figs. 17.23 and 17.24, respectively, for the 0% and 2:2% moisture content cases, with the zero conductivity case serving as a comparitive baseline. The mother pulse in each case is a singlecycle, fc D 0:1 MHz rectangular envelope pulse. For the purpose of comparison, the peak amplitude decay and energy decay when the incident pulse is the mother pulse is described by the dotted curves in both figures. While the deleterious effects of material conductivity on the evolved pulse amplitude and energy are significant, resulting in a peak amplitude decay that is increased from the ideal z1=2 behavior for a pure dielectric, the unique penetrating capabilities of the Brillouin pulse are

17.4 Optimal Pulse Penetration through Dispersive Bodies

755

Relative Peak Amplitude

0.8

0.6

0.4 mho/m

X

0.2 mho/m

X

0 0

2

4

z/zd

6

8

10

Fig. 17.23 Peak amplitude decay of the dispersion-matched, fp D 0:1 MHz Brillouin pulse (solid curves) in loamy soil with 0 D 0, 0 D 6:7  109 mho=m (0% moisture content), and 0 D 2:5  106 mho=m (2:2% moisture content) for a unit amplitude incident pulse on the vacuum-soil interface. For comparison, the peak amplitude decay for the rectangular envelope mother pulse is described by the dotted curves in each case

Relative Pulse Energy

0.6

0.4

0.2 X

mho/m X

0

0

2

4

z /zd

6

mho/m 8

10

Fig. 17.24 Relative pulse energy decay of the dispersion-matched, fp D 0:1 MHz Brillouin pulse (solid curves) in loamy soil with 0 D 0, 0 D 6:7109 mho=m (0% moisture content), and 0 D 2:5  106 mho=m (2:2% moisture content) for a unit amplitude incident pulse on the vacuum-soil interface. For comparison, the relative energy decay for the rectangular envelope mother pulse is described by the dotted curves in each case

756

17 Applications

preserved up through the largest level of conductivity considered here. In each case, the dispersion-matched Brillouin pulse provides near optimal (if not indeed optimal) peak amplitude and energy content for all z > 0. Applications of this unique ground penetrating capability makes this pulse type optimal for such diverse military and civil engineering applications as mine and underground bunker detection, and underground void detection for both highways and airport runways, respectively.

17.4.2 Foliage Penetrating Radar The propagation of an electromagnetic wave through foliage is a complicated problem because it is fundamentally a multiple scattering problem through a spatially random array of dispersive scatterers. Such a detailed, numerically intensive approach is currently being developed [7]. However, if the wavelength of the electromagnetic wave is larger than  4 cm, in which case the frequency is smaller than  8 GHz, then the dispersive effects of foliage can be approximated by a continuous model. In its simplest form, this continuous model of the complex dielectric permittivity of leafy foliage is represented by a fractional mixing model [35] that combines the complex permittivity of leafy vegetation [36] augmented by the conductivity of wood from the associated tree limbs combined with the surrounding air, as described by the causal expression  c .!/ D .1  w1 / C w1 1 C

 0 =0 a C i w2 .1  i !/.1  i !f / !

(17.100)

for the relative complex permittivity, where 0  wj  1 for j D 1; 2. Here 1 D 15:24 is the large frequency limit of the relative permittivity due to rotational polarization in leafy foliage, a1 D 7:86 is the appropriate coefficient in the nonconducting case that yields a static relative permittivity value of .0/ D 1 C a1 D 23:10,  D 1:16  1010 s is the rotational relaxation time, and f D 3:9  1012 s is the associated frictional relaxation time. An estimate of the value of the static conductivity that is appropriate for Douglas fir [37] is given by 0 1  1010 mho=m, so that 0 =0 11:29 r/s. The weighting coefficients w1 and w2 may be considered as fractional volumes of the constituent materials present in the foliage canopy. When w1 D w2 D 0, c .!/ D 1 and the foliage canopy is absent. When w1 D 1 and w2 D 0, the dielectric model for leafy vegetation is obtained. When w1 D 0 and w2 D 1, on the other hand, the simple conductivity model of Douglas fir is obtained. When ! D 0, ˚ (17.101) < c .0/ D 1 C w1 .1 C a  1/: In the opposite extreme of large frequencies that are sufficiently far above the angular frequency value 2= that is characteristic of the rotational polarization of leafy vegetation, one obtains

17.4 Optimal Pulse Penetration through Dispersive Bodies

757

1.01

ℜ{

c

}

1.014

1.006 102

106

104

108

1010

1012

1014

1010

1012

1014

(r/s)

10−5

ℑ{

c

}

100

10−10 10−15 102

104

106

108

(r/s) Fig. 17.25 Angular frequency dispersion of the real (upper graph) and imaginary (lower graph) parts of the complex dielectric permittivity c .!/ for leafy foliage. The dashed curves describe the frequency dispersion for zero conductivity

lim

! 2=

˚ c .!/ D 1 C w1 .1  1/:

(17.102)

Taken together, (17.101) and (17.102) provide an estimate of the values of the fractional volume coefficients w1 and w2 from measured values of the dielectric permittivity of the foliage canopy and the previously determined values of the Rocard–Powles–Debye model of leafy vegetation. Typically, w1  0:01 and w2  0:001 for leafy foliage. The material dispersion described by the causal composite model given in (17.100) of the complex permittivity for leafy foliage is illustrated in Fig. 17.25. Because of the composite material conductivity, a window exists in the low frequency (LF) band around 0:1 MHz. This then makes the ideal frequency at which to launch the single-cycle mother pulse in order to generate the dispersion-matched Brillouin pulse. The numerically determined peak amplitude and pulse energy decays of both the incident mother pulse (dashed curves) and the resultant dispersion matched Brillouin pulse (solid curves) are described in Figs. 17.26 and 17.27, respectively. Notice that the peak amplitude of both the mother pulse and the Brillouin pulse are reduced to 34:45% when transmitted across the vacuum-foliage interface while the transmitted pulse energy is reduced to 11:87% of the incident pulse energy. The transmitted single-cycle rectangular envelope mother pulse initially experiences exponential decay, but as the pulse evolves with increasing propagation

758

17 Applications

Relative Peak Amplitude

0.3

0.2

0.1

0 0

2

4

z /zd

6

8

10

Fig. 17.26 Peak amplitude decay of the dispersion-matched, fp D 0:1 MHz Brillouin pulse (solid curve) in leafy foliage for a unit amplitude incident pulse on the vacuum-foliage interface. For comparison, the peak amplitude decay for the rectangular envelope mother pulse is described by the dotted curve 0.12

Relative Pulse Energy

0.1

0.08

0.06

0.04

0.02

0

0

2

4

z/zd

6

8

10

Fig. 17.27 Relative pulse energy decay of the dispersion-matched, fp D 0:1 MHz Brillouin pulse (solid curve) in leafy foliage for a unit amplitude incident pulse on the vacuum-foliage interface. For comparison, the relative energy decay for the rectangular envelope mother pulse is described by the dotted curve

17.4 Optimal Pulse Penetration through Dispersive Bodies

759

distance, this changes to z1=2 algebraic decay, the transition occurring between one and two absorption depths zd ˛ 1 .!c /. The unique penetrating capabilities of this dispersion-matched Brillouin pulse are clearly evident. Slightly improved results would be obtained if the incident Brillouin pulse spectrum was pre-distorted through division by the transmission coefficient. Analogous results are found for other mixing models of complex dispersive systems, such as that for sand, clouds, and fog. Again, the results obtained using these fractional mixing models are approximately valid provided that the pulse wavelengths are larger than the individual particle sizes. Detailed numerical calculations that include multiple scattering effects are needed to complete these approximate results.

17.4.3 Undersea Communications using the Brillouin Precursor The causal Rocard–Powles–Debye–Drude model of the complex dielectric permittivity of saltwater is given by [see (15.123) and (15.124)] c .!/=0 D 1 C

!  i a0  !p2 ; .1  i !0 /.1  i !f 0 / !.! 2 C  2 /

(17.103)

with 1 D 2:1, a1 D 74:1, 0 D 8:44  1012 r/s, f 0 D 4:62  1014 s for the Rocard–Powles model component, and  1  1011 r/s, 0 4 mho=m, so that p !p D 0 =0 2:13  1011 r/s for the Drude model component. The resultant angular frequency dependence for the complex index of refraction for sea-water is illustrated in Fig. 15.81. The numerically determined dispersion-matched Brillouin pulse derived from a single-cycle rectangular envelope mother pulse with fc D 1:0 GHz carrier frequency is illustrated by the dashed curve in Fig. 17.28. As this incident pulse experiences significant distortion on transmission across the vacuum-salt water interface, its spectrum is pre-distorted through division by the transmission coefficient. The resultant incident pulse (with normalized peak amplitude) is decribed by the solid curve in Fig. 17.28. Notice that the peak amplitude of the transmitted pulse is reduced by 94:4% upon transmission across the vacuum–saltwater interface. By comparison, the dispersion-matched Brillouin pulse amplitude is reduced by 95:4% upon transmission across the vacuum–saltwater interface when its spectrum is not pre-distorted by the transmission coefficient. The numerically determined peak amplitude decay of the pre-distorted, dispersion-matched Brillouin pulse is described by the solid curve in Fig. 17.29 as a function of the relative penetration distance z=zd , the dotted curve describing the peak amplitude decay when the incident Brillouin pulse is not pre-distorted. Notice that both curves describe an algebraic, nonexponential decay.

760

17 Applications 1 0.8

Ab(0 −,t)

Ab(0,t)

0.6 0.4 0.2 Ab(0 +,t)

0 −0.2 −0.4

0

0.1

0.2

t (ms)

0.3

0.4

0.5

Fig. 17.28 Pre-distorted Brillouin pulse Ab .0 ; t / with fp  1:0 GHz effective frequency incident on the vacuum–saltwater interface (solid curve). The dashed curve describes the transmitted, dispersion-matched Brillouin pulse Ab .0C ; t /

Relative Peak Amplitude

0.06

0.04

Predistorted Brillouin Pulse

0.02 Transmitted Brillouin Pulse 0 0

2

4

z /zd

6

8

10

Fig. 17.29 Peak amplitude decay of the pre-distorted, dispersion-matched, fp  1:0 GHz Brillouin pulse (solid curve) in saltwater for a unit amplitude incident pulse on the vacuum–saltwater interface. The dotted curve describes the peak amplitude decay when the inicdent Brillouin pulse is not pre-distorted by the inverse of the transmission coefficient

17.5 Ultrawideband Pulse Propagation through the Ionosphere

761

17.5 Ultrawideband Pulse Propagation through the Ionosphere A closed form solution of the propagation of a double-exponential pulse through a cold plasma (e.g., the ionosphere) has been given by Dvorak and Dudley [38] in terms of the incomplete Lipschitz–Hankel integrals which may then be efficiently computed using both convergent and asymptotic series expansions. The analysis presented in this section is based on their analysis. Because of the long, slowly decaying tails associated with ultrawideband pulse propagation in conducting media, this representation avoids the inordinately large number of sample points that are required using a straightforward FFT simulation of the problem. In addition, the asymptotic behavior of the incomplete Lipschitz–Hankel integrals may then be used to obtain a relatively simple description of the late-time behavior of the pulsed field evolution. The double exponential pulse (see Sect. 11.2.3)   fde .t / a e ˛1 t  e ˛2 t uH .t /

(17.104)

with ˛j > 0 for j D 1; 2, is similar in temporal structure to the delta function pulse but with a nonvanishing temporal width, the constant a chosen such that the peak amplitude is unity. The peak amplitude point of the pulse occurs at the instant of time tm > 0 when dfde .t /=dt D 0, so that tm D

ln ˛1 =˛2 : ˛1  ˛2

(17.105)

Because ude .tm / 1, substitution of (15.190) in (15.189) then gives  1 a D e ˛1 tm  e ˛2 tm :

(17.106)

A measure of the temporal width of the pulse is given by the temporal difference between the e 1 points of the leading and trailing edge exponential functions in (15.189), so that t D j˛1  ˛2 j=.˛1 ˛2 /. Finally, with the result given in (11.56), the spectrum of the double exponential pulse is found to be given by fQde .!/ D a



1 1  ! C i ˛1 ! C i ˛2

 ;

(17.107)

which is clearly ultrawideband. The propagated plane wavefield due to this doubleqexponential pulse in a cold Q plasma described by the dispersion relation k.!/ D .!=c/2  kp2 , where kp D !p =c, is obtained by substituting (17.107) into (11.45) with the result

 Ade .z; t / D a e.˛1 /  e.˛2 / ;

(17.108)

762

17 Applications

for all z  0, where i e.˛/ 2

Z

1

e

h i p i .z=c/ ! 2 !p2 !t

! C i˛

1

d!:

(17.109)

With the linearly polarized electric field vector given by E.z; t / D 1O y Ey .z; t / with Ey .z; t / D E0 Ade .z; t /;

(17.110)

the corresponding magnetic field vector is given by H.z; t / D 1O x Hx .z; t / with (see Sect. 11.2) ( !p2

 E0 Hx .z; t / D f .˛1 / C f .0/ C ˛1 f .˛1 / a 0 ˛1 

!p2

˛2

 f .˛2 / C f .0/  ˛2 f .˛2 /

) (17.111)

for all z  0, where [38] 1 f .˛/ 2

Z

1 1

e

h i p i .z=c/ ! 2 !p2 !t

d!: q .! C i ˛/ ! 2  !p2

(17.112)

p Here 0 0 =0 denotes the intrinsic impedance of free space. The inverse Fourier transform appearing in (17.109) has been evaluated by Dvorak [39] using contour integration as well as by Knop [40], Wait [41], and Case and Haskell [42] using asymptotic methods. Based upon the approach taken by Wu, Olsen, and Plate [43] for the evaluation of the Sommerfeld integrals, Dvorak and Dudley [38] began by determining the differential equation that serves to define both e.˛/ and f .˛/. The change of variables t D z=c D with

D !p

1

cosh ; !p

(17.113)

1

sin ; !p

(17.114)

p t 2  .z=c/2

(17.115)

17.5 Ultrawideband Pulse Propagation through the Ionosphere

763

results in the pair of expressions

e.˛/ D

i 2

1 f .˛/ D 2

Z

1

e

i

! C i˛

1

Z

1

1

hp i .!=!p /2 1 sinh .!=!p / cosh

e

i

(17.116)

d!:

(17.117)

hp i .!=!p /2 1 sinh .!=!p / cosh

q .! C i ˛/ ! 2  !p2

Because z 2 Œ0; 1/ and t 2 .0; 1/, the variables and valued. At any fixed value of z  0, t D 0 ! z=c  ı t D z=c  ı ! z=c C ı t D z=c C ı ! 1

d!;

D i !p z=c ! i "

D i" ! 

D"!1

are, in general, complex-

D i =2 ! 1  i =2; D 1  i =2 ! 1; D 1 ! 0;

where ı and " are arbitrarily small real numbers. From the integral representation (see Abramowitz and Stegun [44]) h i p Z 1 i .z=c/ ! 2 !p2 !t  p  i e d!; (17.118) uH .t  z=c/J0 !p t 2  .z=c/2 D q 2 1 ! 2  !p2

for all z  0, it follows that the function e.˛/ may be determined from f .˛/ through the relation

   @f .˛/

1 !p C ˛f .˛/ C uH . =!p /e  J0 . / cosh e.˛/ D : sinh @ (17.119) In addition, it follows that f .˛/ satisfies the second-order, inhomogeneous, ordinary differential equation (ODE) !  2 ˛ d2 ˛ d C C 2 cosh . /  sinh2 . / f .˛/ d 2 !p d !p   @ @ 1 C˛ D  2 cosh .2 / C sinh .2 / !p @t @.z=c/  p  uH .t  z=c/J0 !p t 2  .z=c/2    1 D  2 e 2 ı . =!p /e  !p      CuH . =!p /e ˛J0 . /  !p cosh . /J1 . / : (17.120)

764

17 Applications

With the homogeneous solutions fh .˛/ D e a˙ ;

(17.121)

where q i 1 h ˛ cosh ˙ ˛ 2 C !p2 sinh !p q ˛t ˙ .z=c/ ˛ 2 C !p2 ; D p !p t 2  .z=c/2

a˙ D

(17.122)

the general solution of the ODE (17.120) is found from the method of variation of parameters to be given by   uH . =!p /e  f .˛/ D  q 2 ˛ 2 C !p2 sinh "

(  e

aC

Z



CC C 0

" e a C C

Z

#  ˛ aC  J0 ./ C cosh . /J1 ./ e d !p

#)  ˛ J0 ./ C cosh . /J1 ./ e a  d  ; !p



0

which becomes, after integration by parts,   uH . =!p /e  f .˛/ D  q 2 ˛ 2 C !p2 sinh h n i    e aC CC C aC cosh C ˛=!p Je0 .aC ; / h io   e a C C a cosh C ˛=!p Je0 .a ; / :

(17.123)

Here Jen .a; / denotes the incomplete Lipschitz–Hankel integral (ILHI) of the first kind of integer order n, defined by the integral [45] Z Jen .a; / 0

e a  n Jn ./d :

(17.124)

17.5 Ultrawideband Pulse Propagation through the Ionosphere

765

A brief description of the properties of the ILHI and its applicability to problems in electromagnetics has been given by Dvorak [46]. Finally, application of the pair of initial conditions that lim !0 f . / D 0 and lim !0 @f . /=@ D 0 results in C˙ D  sinh :

(17.125)

With this result, (17.123) becomes [with (17.113)–(17.115)] ( q   uH .t  z=c/ ˛t sinh .z=c/ ˛ 2 C !p2 f .˛/ D q e ˛ 2 C !p2 " q  z  1 ˛  t ˛ 2 C !p2 e aC Je0 .aC ; / C p c 2!p t 2  .z=c/2 #) q   z a 2 2  ˛ C t ˛ C !p e Je0 .a ; / : c (17.126) In addition, substitution of (17.123) with (17.125) into (17.119) results in [with (17.113)–(17.115)] (

q   e.˛/ D uH .t  z=c/ e ˛t cosh .z=c/ ˛ 2 C !p2 C

1

"

p 2!p t 2  .z=c/2

q  z  ˛  t ˛ 2 C !p2 e aC Je0 .aC ; / c

q   z C ˛ C t ˛ 2 C !p2 e a Je0 .a ; / c

#) :

(17.127) These two expressions are valid for all z  0 and finite t > 0. For numerical computations, the hyperbolic functions appearing in (17.126) and (17.127) are found to result in unacceptable round-off errors when the argument becomes large. Because of this, Dvorak and Dudley [38] revised these solutions in terms of the complementary incomplete Lipschitz and Hankel integral (CILHI) of the first kind of integer order n J en .a; ı; /

Z

ı

e a  n Jn ./d ;

(17.128)

766

17 Applications

where ı D 1 when a  0 and ı D 1 when a < 0. The resulting pair of expressions ( uH .t  z=c/ f .˛/ D q uH .aC /e aC 2 2 ˛ C !p " q  z  1 ˛  t ˛ 2 C !p2 e aC J e0 .aC ; ıC ; / C p c 2!p t 2  .z=c/2 #) q   z  ˛ C t ˛ 2 C !p2 e a J e0 .a ; ı ; / ; c (17.129)

( e.˛/ D uH .t  z=c/ uH .aC /e aC C

1

"

q  z  ˛  t ˛ 2 C !p2 e aC J e0 .aC ; ıC ; / c #) q   z a 2 2 C ˛ C t ˛ C !p e J e0 .a ; ı ; / ; c

p 2!p t 2  .z=c/2

(17.130) avoid these numerical rounding errors. These two expressions, when used together with the electric and magnetic wavefield expressions given in (17.110) and (17.111), and (17.108), represent the exact, closed-form, Dvorak–Dudley representation [38] of the transient response in a cold plasma. Because the cold plasma dispersion relation used by Dvorak and Dudley approximates the high-frequency behavior of the Drude model as 1=2 q  .!p =c/2 Q D .!=c/2  kp2 ; D lim .!=c/2  lim k.!/ !  !  1 C i =!

(17.131)

the transient field response described here provides a closed-form solution of the Sommerfeld precursor (see Sect. 15.5.3) in this limiting case. Applications of these results to communication through the ionosphere show that the Sommerfeld precursor may be ideally suited for this purpose. However, in order to obtain the optimal solution to this problem, the effects of spatial inhomogeneity must be included.

17.6 Health and Safety Issues

767

17.6 Health and Safety Issues Associated with Ultrashort Pulsed Electromagnetic Radiation It is well established that electromagnetic radiation has two fundamentally different effects on biological systems: heating (or thermal) effects and what may be called electromagnetic interaction effects (or athermal effects) that include all effects that are not thermal in origin. Established health and safety standards for electromagnetic radiation exposure have been instituted in the United States by the American National Standards Institute (ANSI) based on the assumption that adverse effects are all thermal in origin. The resultant ANSI Standard C95.1 has then been adopted by the Occupational Safety and Health Administration (OSHA). This standard is based solely on the specific absorption rate (SAR) as the appropriate exposure metric, defined as the electromagnetic power per unit mass (in watts per kilogram – W/kg) absorbed by the target body, viz.,6 SAR

eff jErms j2 ; %m

(17.132)

where eff describes the effective loss coefficient (both conductive and dielectric) in mhos=m that is obtained from the imaginary part of the complex permittivity c .!/ D .!/ C i  .!/=!, %m is the mass density of the object in kg=m3 , and where Erms denotes the rms electric field strength (in V/m) at the absorption point within the body. Notice that this exposure metric represents a whole-body average of the absorbed power [47]. This is done because the safety level for human exposure is determined by the basal metabolic rate (BMR) which describes the rate of energy that is expended by a body while it is at rest in a thermally neutral environment, another whole body metric. The rate of release of energy from the body in this relaxed state is then sufficient for the normal functioning of the vital body organs. This then serves to set the maximum permissible exposure (MPE). Because the effective material loss coefficient is frequency dependent, the maximum permissible exposure is also frequency dependent, as illustrated in Fig. 17.30 for an uncontrolled environment. Two of the solid curves describe the MPE for the electric field strength E and the magnetic field strength H over the frequency domain from 3 kHz to 300 MHz, and the other solid curve describes the MPE power density over the frequency domain from 100 MHz to 300; 000 MHz. For reference, the dashed curves in the figure describe the frequency dependence of the imaginary part  00 .!/ of the relative dielectric permittivity of triply distilled water and the imaginary part c00 .!/ of the relative complex permittivity c .!/ D .!/Ci  .!/=!/ of salt-water with 0 D 4 mho=m static conductivity. In addition, the dotted curves describe the frequency dependence of the penetration depth zd ˛ 1 .!/ in meters for both the noncoducting and semiconducting cases. Notice that the MPE for both 6

Notice that all equations and units of measurement in this section are in the MKSA system of units.

768

17 Applications 103 ''c

E (V/m)

102 101

H (A /m) 2 Pd (mW/cm )

100

zd (m)

10−1 10−2

''

zd (m)

10−3 10−4 10−2

10−1

100

101

102 f (MHz)

103

104

105

106

Fig. 17.30 Frequency dependence (in MHz) of the maximum permissible exposure (MPE – solid curves) in an uncontrolled environment for the electric field strength E (in V=m), the magnetic field strength H (in A=m), and the power density Pd (in mW=cm3 ). The dashed curves describe the frequency dependence of the relative dielectric permittivity .!/ of distilled water and the relative complex permittivity c .!/ D .!/ C i .!/=! of saltwater with 0 D 4 mho=m. The dotted curves describe the frequency dependence of the penetration depth zd ˛ 1 .!/ for both the semiconducting (0 D 4 mho=m) and nonconducting (0 D 0) cases

E and H decreases with decreasing c00 .!/ as the frequency f D !=2 increases over the frequency domain extending  from 3 kHzto 300 MHz, and that the (timeaverage) power density Pd  .!=2/ E 2 C H 2 increases with increasing  00 .!/ as the frequency f increases over the frequency domain extending from 100 MHz to 300;000 MHz. (or penetration) The frequency dependence of the e 1 ˚ amplitude absorption depth zd ˛ 1 .!/, where ˛.!/ = .!=c/Œc .!/ 1=2 is the absorption coefficient of the dispersive material at the angular frequency !, is described by the dotted curves in Fig. 17.30 for both the nonconducting (0 D 0) and semiconducting (0 D 4 mho=m) cases. This measure describes the distance that the transmitted electromagnetic wave penetrates into the body for a time-harmonic wave with frequency f D !=2, the electroamgnetic energy penetration, having been reduced by the factor e 2 at this penetration depth, typically considered to be negligible beyond this distance. For an ultrawideband pulse, however, the peak amplitude of the Brillouin precursor now only decays algebraically as z1=2 with associated electromagnetic energy decaying as z1 as the propagation distance increases above zd , thereby carrying significant electromagnetic energy deeper into the body. Because

17.6 Health and Safety Issues

769

the effect has not been taken into account in setting the current IEEE/ANSI safety standards, these standards may not be sufficient for ultrawideband communication system usage [9]. Athermal effects are more subtle than simple heating and may take further research before safety standards reflect their influence on biological systems. As an electromagnetic pulse travels through biological tissue, the transient response presents a distribution of electromagnetic energy that is continuously transferred to mechanical modes of molecular motion [48, 49]. Because this transient response is comprised of a continuum of frequencies, multiple modes of motion may then be stimulated, thereby driving molecular motion, a matter of concern raised by Ravitz [50] in 1962. More recently, Kotnik and Miklavcic have reported that [51]: Exposure of a biological cell to electric field can lead to a variety of biochemical and physiological responses. If the field is sufficiently strong, the exposure can cause a significant increase in the electric conductivity and permeability of the cell plasma membrane [52]. Provided that the exposure is neither too strong nor too long, this phenomenon (referred to as electroporation or electropermeabilization) is reversible. Using electroporation, many molecules to which the cell plasma membrane is otherwise impermeable can be introduced into the cells or inserted into their plasma membrane. Due to its efficiency, this method is rapidly b ecoming an established approach for treatment of solid cutaneous and subcutaneous tumors, and it also holds great promise for gene therapy.

Because the Brillouin precursor carries a low-frequency field component deep into biological tissue that persists long after the main body of the pulse has been attenuated away, electroporation effects may then occur much deeper than previously assumed. If the precursor induced membrane perturbation occurs over repeated pulses with a high degree of temporal coherence, as occurs, for example, in a wireless digital communication system, changes in the dielectric properties of the material may then result. The biomedical effect of a small transient membrane depolarization is unknown at this time. Chemical reaction rates are also known to be influenced by an externally applied electromagnetic field. If K denotes the chemical equilibrium constant in the absence of an externally applied electromagnetic field and Ke denotes the chemical equilibrium constant in the presence of a timeharmonic (or steady state) electromagnetic wave field with angular frequency !, then these two equilibrium constants are found [53] from both experiment and theory to be related by the simple expression ln ŒKe .!/=K D w1 jE.!/j2 C w2 jH.!/j2 :

(17.133)

Here w1 and w2 are weighting coefficients that depend inversely on the absolute temperature T and directly on the dielectric permittivities and magnetic suceptibilities of both the chemical reactants and products. As stated by Albanese et al. [54]: Exposure of a chemical reacting system to a single sharp electromagnetic pulse or a rapid sequence of such pulses does not represent an equilibrium exposure: therefore, one should not expect that the above equilibrium thermodynamic equation should strictly hold. Were the equation to hold, an electric field intensity of approximately 1;000 V =m could engender a 1% change in equilibrium constant K. Research on transient electromagnetic field-induced chemical reaction rate changes might well be very useful and instructive.

770

17 Applications

Finally, as described earlier, electroporation occurs when a high peak field strength pulse opens a small channel through a cell membrane. Electroporation has been reported [58, 59] to occur for peak field strengths between 400 kV =m and 600 kV =m. The transmembrane potential of a living cell was also reported to be modified by short pulses prior to the occurrence of electroporation. As stated by Albanese et al. [54]: “From the point of view of establishing health and safety guidelines and from the point of view of biophysical curiosity, it appears important to determine the threshold of this phenomenon and the fundamental mechanisms by which it occurs.” All of the concerns raised here are echoed in the more recent 2007 report “Evidence for Brain Tumors and Acoustic Neuromas” by Drs. Hardell, Mild, and Kundi that was prepared for the BioInitiative Working Group. They state in their conclusion that:  Only few studies of long-term exposure to low levels of RF fields and brain tumors exist, all of which have methodological shortcomings including lack of quantitative exposure assessment. Given the crude exposure categories and the likelihood of a bias towards the null hypothesis of no association the body of evidence is consistent with a moderately elevated risk.  Although the population attributable risk is low (likely below 4%), still more than 1,000 cases per year in the US can be attributed to RF exposure at workplaces alone. Due to the lack of conclusive studies of environmental RF exposure and brain tumors the potential of these exposures to increase the risk cannot be estimated.  Epidemiological studies as reviewed in the IEEE C95.1 revision (2006) are deficient to the extent that the entire analysis is professionally unsupportable. IEEEs dismissal of epidemiological studies that link RF exposure to cancer endpoints should be disregarded, as well as any IEEE conclusions drawn from this flawed analysis of epidemiological studies.

Various critiques of the IEEE/ANSI Standard C95.1 have been published over the years, most notably the paper [55] “Science and Standards: The Case of ANSI C95.1 - 1982” by N. H. Steneck. In this paper, Steneck argues that the scientific foundation of C95.1 suffers from more than incompleteness. . . the science used in setting the standard is biased in very pronounced ways. Thermal thinking still plays a major role in this field. Effects that can be traced to heating are not taken as seriously as effects that may arise apart from heating, even though the distinction may make no difference in fact.

These other types of thermal effects are described below. Steneck’s conclusion in 1984 that research on RF bioeffects [55] “has yet to produce a full understanding of the way in which RF energy interacts with living tissue” remains valid to this date. Complicating an already complicated subject is the social health issue surrounding proof of safety. The paper [56] “Why Proof of Safety is Much More Difficult than Proof of Hazard” by I. D. Bross shows that the quantity of data required for a valid assurance of safety is of the order of 30 times greater than that required for a valid proof of hazard. Indeed, the size of the sample needed so far exceeds what is ordinarily attainable in biostatistical-epidemiological studies that official assurances supposedly given on the basis of such studies can have no scientific validity.

That is [56], “although many assurances of safety ‘in the name of science’ have been issued by government agencies and others, few if any of these assurances are statistically valid.”

17.7 Future Prospects

771

The 1994 paper by Albanese, Blaschak, Medina, and Penn [54] provides a brief description of four potential tissue damage mechanisms due to ultrawideband electromagnetic pulse exposure. These are thermal damage, molecular conformation changes, chemical reaction rate alterations, and membrane effects (e.g., electroporation). The thermal damage that Albanese et al. refer to is an electromagnetic field driven event that causes charged particles to collide with biological structures which then increases the total kinetic energy, and thus the localized temperature, of the structure. This is a matter of concern because, as they state it [54], “it is not known how much time the collision process takes to offload the electromagnetic energy a molecule has absorbed into one or more modes of motion. If the rate of energy removal from a target molecule is slower than its absorption from the electromagnetic field, in some sense significant heating of the individual molecule could occur during a radiation exposure with little gross change in the overall medium temperature causing highly localized damage.” Molecular conformation changes describe any physical changes in molecular structure caused by an electromagnetic wave field. As described in [54, 57], the electric potential of a nerve membrane can be shifted from approximately 70 mV to C50 mV through its transient interaction with an action potential. With a mem˚ D 1  108 m, this change in voltage brane thickness of approximately 100A corresponds to an electric field displacement from approximately 7;000 kV =m to C5;000 kV =m across the nerve membrane. Based on this result, Neumann and Katchalsky [57] considered the possibility that such a high peak voltage ultrashort pulsed electric field “could alter the conformation of large macromolecules and thus, possibly provide a mechanism for memory in the central nervous system.” [54]

17.7 Future Prospects The analysis presented in this volume is by no means complete and several important topics of current interest have not been included. Nevertheless, the analysis presented here provides a framework with which each of these research topics may be thoughtfully pursued. Numerical studies of dispersive pulse propagation phenomena require sufficient care in order to yield physically correct results. Indeed, it makes no sense whatsoever to employ unnecessary approximations of either the material dispersion or the pulse behavior in a numerical code, as is commonly done in connection with the group velocity description, nor does it make physical sense to report superluminal pulse propagation for gaussian envelope pulses (the ultimate magician’s smoke and mirror arrangement) that have no finite beginning or end. To this end, several hybrid analytical-numerical approaches have been developed [60, 61] to which the reader is referred. Recent numerical results describe, for example, the joint time-frequency structure [62] of the Sommerfeld and Brillouin precursor fields in a Lorentz model medium. A time-domain theory of forerunners has also been presented by Karlsson and Rikte [63].

772

17 Applications

The asymptotic theory has also been applied to the diffraction of focused electromagnetic pulses in a dispersive medium [64]. These results illustrate the fundamental dispersive coupling between edge diffraction and material dispersion, as described in 9.1 and 9.4. Extensions of dispersive pulse propagation phenomena in the classical Lorentz model to more complicated dispersive media have also been presented in the open literature. Notable here is the work by Zablocky and Engheta [65,66] on signal propagation in temporally dispersive chiral media as well as by Egorov and Rikte [67] on forerunners in bi-gyrotropic materials. The extension to an active Lorentzian medium has been given by Safian, Mojahedi, and Sarris [68]. Here too they show that the signal front travels through the dispersive gain medium at the speed of light c, in agreement with relativistic causality. Numerical analysis of the ultrashort pulse response in a nonlinear dispersive medium given by Albanese, Penn, and Medina [69] shows that the precursor fields become increasingly dominant with the inclusion of a nonlinear response in the causally dispersive model. An analysis of the propagation of thermal light through a dispersive medium has been given by Wang, Magill, and Mandel [70] who show that all of the multipoint correlation functions of any order for a stationary thermal pulse remain unchanged under propagation through a lossless dispersive medium. The invariance properties of random pulses in dispersive attenuative media has been presented by Wang and Wolf [71]. Their analysis shows that random plane wave wave fields comprised of identical but randomly distributed optical pulses “are necessarily stationary and that their power spectra and their longitudinal coherence properties do not change on propagation” through a linear dispersive medium. Application of precursor waveforms to a variety of engineering systems holds great potential using current technology. For example, remote sensing applications include estimating the water content in concrete to determine new construction activity in prohibited areas, measuring effluents in rivers to determine factory production for pollution control, detecting military equipment hidden beneath foliage canopies, and imaging through walls for security. This can be directly accomplished using a repeated sequence of Brillouin pulses that are designed at a frequency fB for the particular material dispersion being interrogated. For optimal detectability of the return pulse signal, the radiated sequence can be appropriately coded. Precursor field effects have even been proposed [72] to monitor human respiration for the prevention of sudden infant death syndrome (SIDS) as well as for the treatment of hypothermia. It is my belief that deeper research into the biomedical effects of ultrawideband electromagnetic pulses, both medically useful as well as potentially injurious, is needed. In particular, it would be of great interest to determine whether electromagnetic transients have a role in the health and safety issues associated with cellular communication systems. Such an analysis would require a detailed understanding of the interaction and interference of radiated pulses from multiple towers with biological systems at the cellular level.

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Problems

775

52. Neumann, Sowers, and Jordan, Electroporation and Electrofusion in Cell Biology. New York: Plenum, 1989. 53. C. Chen, R. J. Heinsohn, and L. N. Mulay, “Effect of electrical and magnetic fields on chemical equilibrium,” J. Phys. Soc. Japan, vol. 25, pp. 319–322, 1968. 54. R. Albanese, J. Blaschak, R. Medina, and J. Penn, “Ultrashort electromagnetic signals: Biophysical questions, safety issues, and medical opportunities,” Aviation. Space and Environmental Medicine, vol. 65, no. 5, pp. 116–120, 1994. 55. N. H. Steneck, “Science and standards: The case of ANSI C95.1 - 1982,” J. Microwave Power, vol. 19, no. 3, pp. 153–158, 1984. 56. I. D. Bross, “Why proof of safety is much more difficult than proof of hazard,” Biometrics, vol. 41, pp. 785–793, 1985. 57. E. Neumann and A. Katchalsky, “Long-lived conformation changes induced by electric impulses in bipolymers,” in Proc. National Academy Sci. USA, vol. 69, pp. 993–997, 1972. 58. T. Y. Tsong, “Electroporation of cell membranes,” Biophys. J., vol. 60, pp. 297–306, 1991. 59. D. C. Chang, B. M. Chassy, J. A. Saunders, and A. E. Sowers, Guide to Electroporation and Electrofusion. New York: Academic Press, 1992. 60. S. L. Dvorak, R. W. Ziolkowski, and L. B. Felsen, “Hybrid analytical-numerical approach for modeling transient wave propagation in Lorentz media,” J. Opt. Soc. Am. A, vol. 15, no. 5, pp. 1241–1255, 1998. 61. H. Xiao and K. E. Oughstun, “Hybrid numerical-asymptotic code for dispersive pulse propagation calculations,” J. Opt. Soc. Am. A, vol. 15, no. 5, pp. 1256–1267, 1998. 62. R. Safian, C. D. Sarris, and M. Mojahedi, “Joint time-frequency and finite-difference timedomain analysis of precursor fields in dispersive media,” Phys. Rev. E, vol. 73, pp. 0666021– 0666029, 2006. 63. A. Karlsson and S. Rikte, “Time-domain theory of forerunners,” J. Opt. Soc. Am. A, vol. 15, no. 2, pp. 487–502, 1998. 64. J. A. Solhaug, J. J. Stamnes, and K. E. Oughstun, “Diffraction of electromagnetic pulses in a single-resonance Lorentz model dielectric,” Pure Appl. Opt., vol. 7, no. 5, pp. 1079–1101, 1998. 65. P. G. Zablocky and N. Engheta, “Transients in chiral media with single-resonance dispersion,” J. Opt. Soc. Am. A, vol. 10, pp. 740–758, 1993. 66. K. E. Oughstun, “Transients in chiral media with single resonance dispersion: comments,” J. Opt. Soc. Am. A, vol. 12, no. 3, pp. 626–628, 1995. 67. I. Egorov and S. Rikte, “Forerunners in bigyrotropic materials,” J. Opt. Soc. Am. A, vol. 15, no. 9, pp. 2391–2403, 1998. 68. R. Safian, M. Mojahedi, and C. D. Sarris, “Asymptotic description of wave propagation in an active Lorentzian medium,” Phys. Rev. E, vol. 75, pp. 66611–1–66611–8, 2007. 69. R. Albanese, J. Penn, and R. Medina, “Ultrashort pulse response in nonlinear dispersive media,” in Ultra-Wideband, Short-Pulse Electromagnetics (H. L. Bertoni, L. B. Felsen, and L. Carin, eds.), pp. 259–265, New York: Plenum Press, 1992. 70. L. J. Wang, B. E. Magill, and L. Mandel, “Propagation of thermal light through a dispersive medium,” J. Opt. Soc. Am. A, vol. 6, no. 5, pp. 964–966, 1989. 71. W. Wang and E. Wolf, “Invariance properties of random pulses and of other random fields in dispersive media,” Phys. Rev. E, vol. 52, no. 5, pp. 5532–5539, 1995. 72. K. G. Ong, W. R. Dreschel, and C. A. Grimes, “Detection of human respiration using squarewave modulated electromagnetic impulses,” Microwave and Optical Tech. Lett., vol. 36, no. 5, pp. 339–343, 2003.

Problems 17.1. Derive the generalized laws of reflection and refraction given in (17.52) and (17.53) and (17.54)–(17.56), respectively.

776

17 Applications

17.2. Derive the generalized laws of reflection and refraction given in (17.52) and (17.53) and (17.54)–(17.56), respectively. 17.3. Show that the generalized Fresnel equations given in (17.58)–(17.63) reduce to the expressions given in (17.67) and (17.68) and (17.72) and (17.73) for the Fresnel reflection and transmission coefficients when there is no material loss. 17.4. Derive (17.71) for the phase angle E of the TE-mode reflection coefficient at supercritical angles of incidence. 17.5. Derive (17.74) for the polarizing or Brewster angle p and show that the reflected and transmitted wave vectors are at right angles to each other when the angle of incidence is equal to this polarizing angle. 17.6. Derive (17.93) and (17.94) for the Goos–H¨anchen shift for s- and p-polarizations, respectively. 17.7. Derive (17.111) for the magnetic field intensity of a double exponential pulse in the cold plasma model of the ionosphere. 17.8. Prove the validity of (17.119). 17.9. With the initial conditions lim !0 f . / D 0 and lim !0 @f . /=@ D 0 for (17.123), show that C˙ D  sinh .

Appendix F

Asymptotic Expansion of Single Integrals

Asymptotic analysis is a powerful analytical approach to obtaining elegantly simple analytic approximations to problems that contain either a parameter or a variable whose magnitude becomes either large or small in comparison to some value that is characteristic of the problem. Its elegance lies in the fact that the results may usually be expressed in a single dominant term that contains all of the essential physics of the problem instead of through the subtle interaction of a large (perhaps infinite) number of terms in a summation. The basic idea behind this general approximating method of analysis may be illustrated by the evaluation of the real exponential integral [1] Z

1

E1 .x/ x

1 t e dt; t

(F.1)

where x > 0 is real-valued. This function possesses the convergent series expansion [2] 1 X xn E1 .x/ D  C ln.x/ C (F.2) nnŠ nD1 P1 1  for x > 0, where  limn!n kD1 k  ln.n C 1/ D 0:57721 : : : is Euler’s constant. Although this series converges for all positive values of x, it becomes computationally useless for x  1. To obtain a useful expression for the value of this function for large values of its argument, repeated integration by parts results in [1] ˇ Z 1 1 t e t ˇˇ1  e dt ˇ t x t2 x ˇ Z 1 e t ˇ1 e x 1 t C 2 ˇˇ C 2 D e dt 3 x t x t x   1Š 2Š nŠ 3Š e x 1  C 2  3 C    C .1/n n C RnC1 .x/; D  D x x x x x

E1 .x/ D 

(F.3)

777

778

F Asymptotic Expansion of Single Integrals

where the remainder after n terms is given by nC1

RnC1 .x/ D .1/

Z

1

.n C 1/Š x

e t dt: t nC2

(F.4)

Since the integral appearing in this remainder term is bounded by e x =x nC2 for x > 0, then the magnitude of this remainder term is bounded as jRnC1 .x/j <

.n C 1/Š x e : x nC2

(F.5)

The remainder after n terms is then seen to be bounded in magnitude by the magnitude of the first term neglected in the series summation Sn .x/ D

  1Š 2Š nŠ e x 3Š 1  C 2  3 C    C .1/n n : x x x x x

(F.6)

However, if one considers the expansion given in (F.6) as an infinite series, the result is divergent. Nevertheless, for sufficiently large values of x > 0, the series summation given in (F.6) is rapidly convergent for a finite number of terms n. An estimate of the optimum number of terms to be used in this expansion for a given value of x may be obtained from the ratio of successive terms as jun .x/=unC1 .x/j D n=x  1, so that the optimum number of terms to be used in the summation Sn .x/ for an estimate of E1 .x/ for a given large value of x is approximately given by the greatest integer in x, as illustrated in Fig. F.1 when x D 5:7. In that case the optimum number of terms is given by n D 5 where 5e 5 Sn .5/ D 0:8704, which is in good agreement with the actual value of 5e 5 E1 .5/ D 0:8663. Inclusion of additional terms in the summation only results in a decrease in accuracy. Most importantly, since the remainder after the first (or dominant) term becomes exponentially small as x ! 1, the approximation E1 .x/ S1 .x/ becomes increasingly accurate as x increases. This example then leads to the following distinction between an asymptotic expansion and a power series expansion of some function: For the power series expansion f .x/ Š

N X

un .x/

nD0

of a given function f .x/, the approximation to the value of f at some fixed value of x improves in some well-defined sense as N ! 1, while for an asymptotic expansion f .x/ D Sn .x/ C RnC1 .x/ of that function, the approximation to f .x/ by the series summation Sn .x/ improves in some (as yet undefined) sense for fixed n as x ! 1. The first (or dominant) term

F Asymptotic Expansion of Single Integrals

779

1

0.95

xexSn(x)

0.9

0.85

0.8

0.75

0.7

2

3

4

5

6 n

7

8

9

10

Fig. F.1 Dependence of the series summation Sn .x/ approximation of the exponential integral E1 .x/ for x D 5:7 as a function of the number n of terms in the summation. The open circles connected by the solid line segments describe the values of the quantity xe x Sn .x/, while the dashed curves describe upper and lower envelopes to these values. Notice that the approximate values oscillate about the actual value of 5e 5 E1 .5/ D 0:8663

Fig. F.2 Statue of the Norwegian mathematician Niels Henrik Abel (1802–1829) wrestling with a sea serpent on the royal palace grounds in Oslo, Norway. (Photograph by K. E. Oughstun)

in the asymptotic expansion of of f .x/ represents the asymptotic approximation of that function as x ! 1. However, care must always be taken to ensure that the given asymptotic expansion is not only well defined mathematically but is also properly applied and interpreted. If not, critical errors may result. Such was the motivation for Abel (see Fig. F.2) to lament in 1828 that

780

F Asymptotic Expansion of Single Integrals

Divergent series are the invention of the devil, and it is a shame to base on them any demonstration whatsoever. By using them, one may draw any conclusion he pleases and that is why these series have produced so many fallacies and so many paradoxes. . .

A brief outline of the essential theory is presented here. A more detailed development may be found in the texts by E. T. Copson [3], N. Bleistein and R. Handelsman [4], and J. D. Murray [5].

F.1 Foundations Definition 1. Spherical Neighborhood. A spherical neighborhood of a point z0 is given by the set of points in the open disc jz  z0 j < ı

(F.7)

if z0 is at a finite distance from the origin (i.e., if jz0 j < 1), while it is given by the open region jzj > ı (F.8) if z0 is the point at infinity. Definition 2. O-order. Let f .z/ and g.z/ be two functions of the complex variable z that possess definite limits as z ! z0 in some domain D. Then f .z/ D O.g.z//

as z ! z0

(F.9)

if there exist positive constants K and ı such that jf .z/j  Kjg.z/j whenever 0 < jz  z0 j < ı. If jf .z/j  Kjg.z/j for all z 2 D, then f .z/ D O.g.z// in D. Definition 3. o-order. Let f .z/ and g.z/ be two functions of the complex variable z that possess definite limits as z ! z0 in some domain D. Then f .z/ D o.g.z//

as z ! z0

(F.10)

if there exists a positive constant ı such that jf .z/j  jg.z/j for any  > 0 whenever 0 < jz  z0 j < ı. Thus, as long as g.z/ is not zero in a neighborhood of z0 , except possibly at the point z0 itself, then f .z/ ! 0 as z ! z0 ; g.z/ ˇ ˇ ˇ f .z/ ˇ ˇ  K as z ! z0 ; ˇ f .z/ D O.g.z// H) ˇ g.z/ ˇ f .z/ D o.g.z// H)

(F.11) (F.12)

where K is some positive constant. The O-order is clearly more important than the o-order in asymptotic analysis since it provides more specific information about the

F.1 Foundations

781

behavior of the function at the point under consideration. For example, if f .z/ ! 0 as z ! z0 , then the O-order states how rapidly f .z/ approaches zero at that point whereas the o-order merely confirms that f .z/ approaches zero at that point. In particular f .z/ D o.g.z// as z ! z0 H) f .z/ D O.g.z// as z ! z0 ;

(F.13)

while the opposite is not necessarily true. As an example, consider the exponential function f .z/ D e z which is an entire function of complex z D x C iy. With z 2 D1 W 0 < jzj < 1; jarg.z/j < =2, then x > 0 and f .z/ D e x e iy D o.x n /

8 n > 0 as jzj ! 1 in D1 :

Any function of the complex variable z that is o.zn / for all n > 0 as jzj ! 1 demonstrates the exponential character of the function. However, for the domain D2 W 0 < jzj < 1;  < arg.z/  , all that can be said is that f .z/ D O.e z /. Theorem 7. Let fn .z/ D Ofgn .z/g in some domain D for n D 1; 2; : : : ; N . Then N X

( an fn .z/ D O

nD1

N X

) jan jjgn .z/j

(F.14)

nD1

for all z 2 D, where the coefficients an , n D 1; 2; : : : ; N are complex constants. Proof. Since fn D Ofgn g in D, then there exists a set of real positive constants Kn such that jfn j  Kn jgn j in D. Let K D maxfKn g, so that ˇN ˇ N N ˇX ˇ X X ˇ ˇ an fn ˇ  jan jjfn j  K jan jjgn j; ˇ ˇ ˇ nD1

nD1

nD1

which is precisely the statement of the above order relation.

t u

Theorem 8. Let f .z/ D Ofg.z/g for all z 2 D. Then for any ˛ > 0, jf .z/j˛ D O fjg.z/j˛ g

(F.15)

for all z 2 D. Proof. Since f D Ofgg in D, then jf j  Kjgj 8z 2 D for some positive constant K. For any ˛ > 0, jf j˛  jKgj˛  K ˛ jgj˛ D Kjgj˛ ; which is precisely the statement of the above order relation.

t u

782

F Asymptotic Expansion of Single Integrals

Theorem 9. Let fi .z/ D Ofgi .z/g in some domain D for i D 1; 2; : : : ; n, and let jgi .z/j  jg.z/j for all z 2 D for each i D 1; 2; : : : ; n. Then n X

ai fi .z/ D Ofg.z/g

(F.16)

iD1

for all z 2 D, where the coefficients ai , i D 1; 2; : : : ; n, are all constants. Proof. Since fi D Ofgi g for i D 1; 2; : : : ; n, then there exist positive constants Ki , i D 1; 2; : : : ; n, such that jfi j  Ki jgi j for all z 2 D. Let K D maxfKi g and let ai , i D 1; 2; : : : ; n be arbitrary constants. Then ˇ ˇ n n n n ˇ X ˇX X X ˇ ˇ ai fi ˇ  jai jjfi j  K jai jjgi j  Kjgj jai j D AKjgj; ˇ ˇ ˇ iD1

iD1

iD1

iD1

t u

which is precisely the statement of the above order relation. Theorem 10. Let fi .z/ D Ofgi .z/g in some domain D for i D 1; 2; : : : ; n. Then n Y

( fi .z/ D O

iD1

n Y

) gi .z/

(F.17)

iD1

for all z 2 D. Proof. Since fi D Ofgi g for i D 1; 2; : : : ; n, then there exists a set of positive constants Ki , i D 1; 2; : : : ; n, such that jfi j  Ki jgi j for all z 2 D. Let K D maxfKi g, i D 1; 2; : : : ; n. Then ˇ n ˇ n n n ˇY ˇ Y Y Y ˇ ˇ jfi j  Ki jgi j  K n jgi j; ˇ fi ˇ  ˇ ˇ iD1

iD1

iD1

iD1

which is precisely the statement of the above order relation.

t u

Notice that these four theorems remain valid if the order symbol O is changed to the (weaker) order symbol o. Order relations can be integrated (a smoothing operation), but they cannot, in general, be differentiated with respect to some other variable. If a function f .z; / is a function of the two variables z and  and if f .z; / D Ofg.z; /g as z ! z0 for all  2 D , then for two values 1 ; 2 2 D with 2 > 1 , ˇZ ˇ ˇ ˇ

2

1

ˇ Z ˇ f .z; /dˇˇ 

Z

2

1

2

jf .z; /jd  K

jg.z; /jd; 1

F.2 Asymptotic Sequences, Series and Expansions

so that

Z

2

f .z; /d D O

1

783

Z

2

g.z; /d

(F.18)

1 ‹

as z ! z0 . However, it is not always true that @f .z; /=@ D [email protected]; /=@g as z ! z0 ; order relations under differentiation must then be considered on a case by case basis.

F.2 Asymptotic Sequences, Series and Expansions Definition 4. Asymptotic Sequence. A finite or infinite sequence of functions fn .z/g is an asymptotic sequence as z ! z0 if there exists a spherical neighborhood of z0 within which none of the functions n .z/ vanish (except possibly at the point z0 ) and (F.19) nC1 .z/ D ofn .z/g as z ! z0 for all n. That is lim

z!z0

nC1 .z/ D0 n .z/

for all n. As an example, the sequence of functions f.z  z0 /n g, n D 0; 1; 2; : : : is an asymptotic sequence as z ! z0 with finite jz0 j since .z  z0 /nC1 nC1 .z/ D D .z  z0 / ! 0 as z ! z0 : n .z/ .z  z0 /n The sequence of functions fzn g, n D 0; 1; 2; : : : is an asymptotic sequence as z ! 1 since z.nC1/ nC1 .z/ D D z1 ! 0 as z ! 1: n .z/ zn Finally, the sequence of functions fe z zan g with real-valued an satisying anC1 > an is an asymptotic sequence as z ! 1 since nC1 .z/ e z zanC1 D z a D z.anC1 an / ! 0 n .z/ ez n

as z ! 1:

Because of the order relation operations given in (F.14)–(F.18), new asymptotic sequences may be formed by the appropriate combination of existing asymptotic sequences. In particular, integration of an asymptotic sequence over some variable other than the asymptotic variable [cf. (F.18)] results in another asymptotic sequence, while differentiation does not, in general.

784

F Asymptotic Expansion of Single Integrals

Definition 5. Asymptotic Expansion in the Sense of Poincar´eP (1886). If fn .z/g is an asymptotic sequence of functions as z ! 1, then the series 1 nD1 an n .z/, not necessarily convergent, where the coefficients an are independent of the variable z, is said to be an asymptotic expansion of a function f .z/ in the sense of Poincar´e with respect to the asymptotic sequence fn .z/g if for every value of N , f .z/ D

N X

an n .z/ C ofN .z/g

(F.20)

nD1

as z ! z0 . If the asymptotic expansion given in (F.20) of a function f .z/ exists for a given asymptotic sequence fn .z/g, it is unique and the expansion coefficients are uniquely determined as f .z/ ; 1 .z/ f .z/  a1 1 .z/ a2 D lim ; z!z0 2 .z/ :: : P 1 f .z/  N nD1 an n .z/ : aN D lim z!z0 N .z/ a1 D lim

z!z0

(F.21) (F.22)

(F.23)

If a function f .z/ possesses an asymptotic expansion in this sense, one then writes f .z/ 

1 X

an n .z/ as z ! z0 :

(F.24)

nD1

A partial sum of this series is called an asymptotic approximation to the function f .z/. Since f .z/ 

N 1 X

an n .z/ D aN N .z/ C ofN .z/g

nD1

as z ! z0 , then f .z/ D

N 1 X

an n .z/ C OfN .z/g

(F.25)

nD1

is an asymptotic approximation to f .z/ as z ! z0 with an error of order OfN .z/g, which is of the same order of magnitude as the first term omitted in the series. The first nonzero term in the asymptotic expansion is called the leading or dominant term in the expansion. If a1 ¤ 0, then f .z/  a1 1 .z/ as z ! z0 in the appropriate domain, meaning that f .z/=1 .z/ ! a1 as z ! z0 . In many practical cases, the

F.2 Asymptotic Sequences, Series and Expansions

785

asymptotic sequence may not be known and the leading term in the expansion is all that is or can be determined, and frequently is all that is required. The asymptotic expansion of a given function f .z/ depends upon the specific sequence of functions fn .z/g that is chosen, so that a function may possess several asymptotic expansions. This interesting property may be quite useful when expansions are required in different domains. However, care must always be taken not to mistakenly use an asymptotic expansion outside of its domain of validity, regardless of how tempting the result may appear, as the consequences may lead to Abel’s lament. The most common asymptotic sequence is the power sequence f.z  z0 /n g as z ! z0 in some domain D containing the point z0 . Without any loss in generality, the point z0 may either be taken as the point at infinity through the change of variable  D 1=.z  z0 /, or else as the origin with the change of variable  D z  z0 . If z0 is the point at infinity, then a typical power series expansion of the function f .z/ about that point is of the form f .z/  g.z/

1 X an nD1

as z ! 1;

zn

(F.26)

where g.z/ is some function, valid in some domain of the variable z. Asymptotic expansions that are based on asymptotic power sequences are called asymptotic power series. For example, the asymptotic expansion of the exponential integral given in (F.3) is an asymptotic power series in terms of the sequence f.1/nC1 .n1/Šzn g as z ! 1 with g.z/ D e z in the domain jarg.z/j < =2. As another example, consider the function f .z/ D 1=.z  1/ for jzj > 1 which possesses the power series expansion 1

X 1 1  z  1 nD1 zn

as z ! 1;

in terms of the power sequence fzn g. Since 1

X 1 1  2 z  1 nD1 z2n

as z ! 1;

then one also has the power series expansion 1 X 1 1  .z C 1/ 2n z1 z nD1

as z ! 1;

in terms of the power sequence fz2n g. Not only can a function possess more than one asymptotic expansion in a given domain, a given asymptotic expansion may also be the expansion for more than one function. As an illustration, consider the asymptotic expansion of the exponential

786

F Asymptotic Expansion of Single Integrals

function e z in terms of the power sequence fzn g as z ! 1 in the domain jarg.z/j < =2, given by e z 

1 X ˛n nD0

as z ! 1; jarg.z/j <

zn

 : 2

whose coefficients are determined from the limiting procedure given in (F.21)– (F.23) as e z D 0; z!1 zn

˛n D lim

for all n D 0; 1; 2; : : : . Thus, if a given function f .z/ possesses the asymptotic power series expansion f .z/  g.z/

1 X an nD1

zn

as z ! 1; jarg.z/j <

 ; 2

then also f .z/ C e z  g.z/

1 X an nD1

zn

as z ! 1; jarg.z/j <

 : 2

Theorem 11. Linear Superposition of Asymptotic Expansions. If f .z/ and g.z/ possess the asymptotic expansions f .z/  g.z/ 

1 X nD1 1 X

an n .z/; bn n .z/;

nD1

with respect to the asymptotic sequence fn .z/g as z ! 1 in a common domain D, then 1 X .˛an C ˇbn /n .z/; (F.27) ˛f .z/ C ˇg.z/  nD1

as z ! 1 in D, where ˛ and ˇ are complex constants. Proof. It follows from from (F.25) that ˇ ˇ N 1 ˇ ˇ X ˇ ˇ f .z/  an n .z/ C OfN .z/g H) ˇf .z/  an n .z/ˇ  K1 jN .z/j; ˇ ˇ nD1 nD1 ˇ ˇ N 1 N 1 ˇ ˇ X X ˇ ˇ bn n .z/ C OfN .z/g H) ˇg.z/  bn n .z/ˇ  K2 jN .z/j; g.z/  ˇ ˇ N 1 X

nD1

nD1

F.2 Asymptotic Sequences, Series and Expansions

787

so that ˇ ˇ N 1 ˇ ˇ X ˇ ˇ .˛an C ˇbn /n .z/ˇ ˇ˛f .z/ C ˇg.z/  ˇ ˇ nD1 ˇ " # " #ˇ N 1 N 1 ˇ ˇ X X ˇ ˇ D ˇ˛ f .z/  an n .z/ C ˇ g.z/  bn n .z/ ˇ ˇ ˇ nD1 nD1 ˇ ˇ ˇ ˇ N 1 N 1 ˇ ˇ ˇ ˇ X X ˇ ˇ ˇ ˇ  j˛j ˇf .z/  an n .z/ˇ C jˇj ˇg.z/  bn n .z/ˇ ˇ ˇ ˇ ˇ nD1

nD1

 j˛jK1 jN .z/j C jˇjK2 jN .z/j D KjN .z/j; where K D j˛jK1 C jˇjK2 .

t u

Theorem 12. Product of Asymptotic Power Series. If f .z/ and g.z/ possess the asymptotic power series expansions f .z/ D

N X an nD1

g.z/ D

zn

N X bn nD1

zn

˚ C O z.N C1/ ; ˚ C O z.N C1/ ;

as z ! 1 in a common domain D, then f .z/g.z/ D

N X cn nD1

zn

˚ C O z.N C1/

(F.28)

as z ! 1 in D, where cn D a0 bn C a1 bn1 C    C an1 b1 C an b0 . The proof of this theorem follows directly from the product of the two asymptotic power series. It then follows from the result given in (F.28) that Œf .z/ m , where m is a positive integer, possesses an asymptotic power series as z ! 1 in D, and hence that any polynomial of f .z/ and any rational function of f .z/, where the denominator has no zeros in the domain D, also possess asymptotic power series as z ! 1 in D. In particular, if the coefficient a0 ¤ 0 in the asymptotic power series expansion P a =zn as z ! 1 in D, then f .z/  1 n nD1 1

X dn 1 1  C f .z/ a0 nD1 zn

(F.29)

as z ! 1 in D. To establish this result and determine explicit expressions for the coefficients dn , notice first that 1=f .z/ tends to the finite limit 1=a0 d0 as z ! 1.

788

F Asymptotic Expansion of Single Integrals

One then has that

1 1 1=f .z/  1=a0 Dz  1=z f .z/ a0

1 1  Dz a0 C a1 =z C O.1=z2 / a0 a0  .a0 C a1 =z C O.1=z2 // Dz a0 Œa0 C a1 =z C O.1=z2 /

a1 a1 C O.1=z/ !  2 d1 ; D a0 Œa0 C a1 =z C O.1=z2 /

a0 1=f .z/  1=a0 C a1 =a02 z a2  a0 a2 1=f .z/  1=.a0  a1 =z/ D ! 1 3 d2 ; 1=z 1=z a0 and so on for higher order coefficients. Theorem 13. Integration of Asymptotic Power Series. If f .z/ has the asymptotic power series expansion f .z/ D

N X an nD1

zn

˚ C O z.N C1/

as z ! 1 in a domain D, then Z

1

f ./  a0 

z

N X anC1 a1 d   nzn nD1

(F.30)

as z ! 1 in D for all simply connected paths of integration in D. ˚ .N C1/ P n Proof. Since f .z/  a0  a1 =z D N nD2 an =z C r.z/, where r.z/ D O z as z ! 1 in D, then for any simply connected path of integration in D, Z z

1



Z 1 Z 1 N X 1 a1 d D f ./  a0  an d C r./d  n z z nD2 Z 1 N X an C r./d: D .n  1/zn1 z nD2

ˇ ˇ Since jr.z/j  K ˇz.N C1/ ˇ, if the path of integration is taken along a path in D with fixed argument of the complex variable z, then ˇZ ˇ ˇ ˇ

z

1

ˇ Z ˇ ˇ r./d  ˇ  K

1 jzj

jj.N C1/ d jj D

K N jzj ; N

F.2 Asymptotic Sequences, Series and Expansions

789

so that Z

1

f ./  a0 

z

N X   an a1 d D t C o zN C1 :u n1  .n  1/z nD2

Term by term differentiation of an asymptotic expansion, on the other hand, does not necessarily result in an asymptotic expansion except in the case of a function whose derivative possesses an asymptotic power series, as stated in the following: Theorem 14. Differentiation of Asymptotic Power Series. If the analytic function f .z/ and its first derivative f 0 .z/ both possess asymptotic power series expansions as z ! 1 in some domain D, then f .z/ 

1 X an nD0

zn

f 0 .z/ 

H)

1 X nan nC1 z nD1

(F.31)

as z ! 1 in D. Proof. Since f 0 .z/ possesses an asymptotic power series expansion as z ! 1 in D, then f 0 .z/ 

1 X bn nD0

zn

:

Since f 0 .z/ is continuous in D, then Z

z

f .z/  f .z0 / D

f 0 ./d 

z0



D b0 .z  z0 / C b1 ln

z z0



Z z C

f 0 ./  b0 

z0

b1 d: 

0 2 Since f .z/ ! an0 as z ! 1 in D, o and since f .z/  b0  b1 =z D Of1=z g so that Rz b the integral z0 f 0 ./  b0  1 d is convergent as z ! 1, then b0 D b1 D 0, so that the above expression becomes

Z

1

a0  f .z/ D

f 0 ./d 

z

1 X bnC1 nD1

nzn

as z ! 1:

However, a0  f .z/  

1 X an nD1

zn

as z ! 1:

Since an asymptotic power series expansion is unique in a given domain, then t u bnC1 D nan .

790

F Asymptotic Expansion of Single Integrals

If the function f .z/ is both single-valued and analytic for all jzj > R, then it possesses a convergent power series expansion that is valid for all arg.z/ in the domain jzj > R. The uniqueness property of asymptotic power series then states that this convergent power series must be the asymptotic power series for f .z/ as z ! 1 based on the asymptotic sequence fzn g. The preceding two theorems then state that this asymptotic series expansion is both differentiable and integrable for all jzj > R. If the function f .z/ is not analytic everywhere in the region jzj > R, then it cannot have a single asymptotic power series expansion in the sequence fzn g that is valid for all arg .z/. Different expansions will then be necessary for different argument domains and caution must be exercised not to use the wrong expansion in a given subdomain.

F.3 Integration by Parts In those cases where it is possible, a straightforward method of obtaining an asymptotic expansion of the integral representation of a given function is to use the method of integration by parts, as illustrated in the following example: Consider the integral representation of the normalized incomplete gamma function [2] Z x e t t a1 dt; (F.32) .a; x/ D 0

where 0. The gamma function is then defined as Z 1  .a/ .a; 1/ D e t t a1 dt;

(F.33)

0

for 0, from which it follows that  .a C 1/ D a .a/: Since  .1/ D

R1 0

(F.34)

e t dt D 1, one then has that  .n C 1/ D nŠ

(F.35)

for n D 0; 1; 2; : : : , so that 0Š D  .1/ D 1. A power series expansion of the normalized incomplete gamma function may be obtained by expanding the integrand in (F.32) in a power series in the variable t and integrating term by term as ! Z x X Z 1 1 X .1/n n a1 .1/n x nCa1 t t dt D .a; x/ D t dt nŠ nŠ 0 0 nD0 nD0 Dx

a

1 X nD0

.1/n xn; .a C n/nŠ

(F.36)

F.3 Integration by Parts

791

which converges for all x. However, this power series representation is only useful for small values of x. For large values of x, an asymptotic power series will clearly be of more use. This divergent series representation is obtained from the integral representation (F.32), which may be rewritten as Z

1

t a1

e t

.a; x/ D

Z

1

dt 

0

e t t a1 dt

x

D  .a/   .a; x/ where

Z

1

 .a; x/ D

(F.37)

e t t a1 dt;

(F.38)

x

is the incomplete gamma function [2]. Successive integration by parts of this integral representation of the incomplete Gamma function then gives Z

1

let u D t a1 ; d v D e t dt Z 1 e t t a2 dt let u D t a2 ; d v D e t dt D e x x a1 C .a  1/

 .a; x/ D

e t t a1 dt

x

x

:: :

˚ D e x x a1 C .a  1/x a2 C    C .a  1/    .a  N C 1/x aN Z 1 C.a  1/.a  2/    .a  N / e t t aN 1 dt: (F.39) x

For any fixed value of N > a  1, ˇZ ˇ ˇ ˇ

1

t aN 1

e t x

ˇ Z ˇ dtˇˇ < x aN 1

1

˚ e t dt D o x aN e x ;

as x ! 1;

x

and so the above expansion for the incomplete gamma function  .a; x/ is asymptotic as x ! 1; that is Z

1

e t t a1 dt x

.a  1/.a  2/ a1 x a1 C e x 1C C    ; x x2

 .a; x/ D

(F.40)

as x ! 1. The asymptotic expansion of the normalized incomplete gamma function is then given by .a; x/   .a; x/  e

x a1

x



.a  1/.a  2/ a1 C 1C C  ; x x2 (F.41)

792

F Asymptotic Expansion of Single Integrals

as x ! 1. An estimate of the optimal number of terms to be used for a given value of x in either of the two asymptotic expansions given in (F.40) and (F.41) may be obtained by taking the ratio of the .N C 1/th term to the N th and setting the result equal to 1, with the result that ja  N j x. Finally, notice that if a is an integer, then both of the series in (F.40) and (F.41) terminate and the right-hand sides of these equations become exact, rather than asymptotic, representations of the incomplete gamma functions  .a; x/ and .a; x/, respectively.

F.4 The Method of Stationary Phase The method of stationary phase was originally developed by G. G. Stokes [6] in 1857 and Lord Kelvin [7] in 1887 for the asymptotic approximation of Fourier transform type integrals of the form Z

b

g.t /e ih.t/ dt

f ./ D

(F.42)

a

as  ! 1, where a, b, g.t /, h.t /,  and t are all real-valued. The functions h. / and g. / are assumed to be analytic functions of the complex variable D t C i  in some domain containing the closed interval Œa; b along the real axis [3]. The exponential kernel e ih.t/ appearing in the integrand is then purely oscillatory so that as  becomes large, its oscillations become very dense and destructive interference occurs almost everywhere. The exceptions occur at any stationary phase point tj defined by (F.43) h0 .tj / 0; since the phase term h.t / is nearly constant in a neighborhood of each such point, as well as from the lower and upper end points t D a and t D b of the integration domain since the effects of destructive interference will be incomplete there. Assume that there is a single interior stationary phase point at t D t0 where a < t0 < b, and is such that h00 .t0 / ¤ 0. In a neighborhood about this stationary phase point, h.t / may be represented by its Taylor series expansion 1 h.t / D h.t0 / C h00 .t0 /.t  t0 /2 C    (F.44) 2 ˚ so that h.t /  h.t0 / D O .t  t0 /2 , whereas in a neighborhood of any other point  2 Œa; b , h.t /  h./ D O f.t   /g. The expansion given in (F.44) then suggests the change of variable (F.45) h.t /  h.t0 / ˙s 2 ; where Cs 2 is used when h00 .t0 / > 0 while s 2 is used when h00 .t0 / < 0. For values of t in a neighborhood of t0 , (F.45) may be inverted with use of the Taylor series expansion given in (F.44) as

F.4 The Method of Stationary Phase

793

 t  t0 D

2 00 jh .t0 /j

1=2

˚ s C O s2 ;

(F.46)

valid when either h00 .t0 / > 0 or h00 .t0 / < 0. With the Taylor series expansion g.t / D g.t0 / C g 0 .t0 /.t  t0 / C    ;

(F.47)

the integral in (F.42) becomes Z

b

f ./ D

g.t /e ih.t/ dt

a

1=2 Z s2 2 2 ih.t0 / g.t0 /e e ˙is Œ1 C O fsg ds  jh00 .t0 /j s1 1=2  Z 1

Z 1 2 ih.t0 / ˙is 2 2 ˙is 2  g.t /e e ds C O s e ds 0 jh00 .t0 /j 1 1 

as  ! 1, so that Z

b

g.t /e ih.t/ dt 

a



2  jh00 .t0 /j

1=2

˚ g.t0 /e ih.t0 / e ˙i=4 C O 1

(F.48)

as  ! 1, where C=4 corresponds to the case when h00 .t0 / > 0 while =4 corresponds to h00 .t0 / < 0. The end point contributions to the asymptotic behavior of this integral are obtained through a straightforward integration by parts as Z

b

g.t /e

ih.t/

a

ˇ  Z b  g.t / d g.t / ˇˇb e ih.t/ dt dt D  i h0 .t / ˇa i h0 .t / a dt   ˚ 1 g.b/ ih.b/ g.a/ ih.a/ e C O 2 ; (F.49)  0 e  0 i  h .b/ h .a/

as  ! 1. Combination of (F.48) and (F.49) then results in the asymptotic approximation Z

b

g.t /e a

ih.t/

1=2 2 dt  g.t0 /e ih.t0 / e ˙i=4  jh00 .t0 /j   ˚ 1 g.b/ ih.b/ g.a/ ih.a/ e C O 2 C  0 e i  h0 .b/ h .a/ 

(F.50) as  ! 1. As an example, consider the asymptotic behavior of the Fourier integral of a continuous function f .t / that identically vanishes outside of the finite domain Œ˛; ˇ ;

794

F Asymptotic Expansion of Single Integrals

i.e., has compact support. Since h.t / D t does not possess any stationary phase points, the asymptotic behavior is due solely to the endpoints. Repeated integration by parts N times then results in the expression Z

ˇ

f .t /e ˛

i!t

dt D

N 1  X nD0

i !

nC1

˚ .n/ f .˛/e i!˛  f .n/ .ˇ/e i!ˇ C RN .!/;

where  RN .!/

i !

N Z

ˇ

f .N / .t /e i!t dt:

˛

By the Riemann–Lebesgue lemma [8], if f .N / .t / is continuous in the interval Œ˛; ˇ , then the integral appearing in the remainder term RN .!/ tends to zero in the limit as ! ! 1, so that RN .!/ D Of! N g. The above result is then the asymptotic series as ! ! 1 of the Fourier transform of the N th-order continuous function f .t / with compact support Œ˛; ˇ .

F.5 Watson’s Lemma An important class of integrals that is amenable to asymptotic analysis is the class of Laplace integrals Z 1

.t /e xt dt;

f .x/ D

(F.51)

0

where f.t /g is integrable over every finite interval Œ0; T . Watson’s lemma provides an asymptotic expansion for such Laplace transform integrals for the fairly wide class of functions of the form .t / D t  g.t /;

(F.52)

where g.t / possesses a Taylor series expansion about t D 0 with g.0/ ¤ 0, and where  is real-valued with  > 1 in order to ensure convergence of the integral appearing in (F.51) at t D 0. If g.t / possesses a zero of order r at t D 0, then the quantity t r is combined with the factor t  to create a new function g.t / that does not vanish at t D 0. With the method of analysis given by Murray [5] as a guide, consider the asymptotic expansion of the function f .x/ given by the Laplace integral representation Z f .x/ D 0

T

t  g.t /e xt dt

(F.53)

F.5 Watson’s Lemma

795

for real-valued x > 0 as x ! 1 and for any finite or infinite value of T > 0, where g.t / possesses a Taylor series expansion about t D 0 with g.0/ ¤ 0, and where  > 1. In order that the integral appearing in (F.53) converges as T ! 1, it is required that the inequality jg.t /j < Ke ct ;

0t T

is satisfied for some constants K and c < x, since then Z

T

jf .x/j < K

t  e .xc/t dt;

0

where t  e .xc/t ! 0 as T ! 1. Since g.t / possesses a Taylor series expansion about the point t D 0 (more specifically, a Maclaurin series), then g.t / D

1 X g .n/ .0/

nŠ

nD0

tn D

N X

an t n C rn .t /;

(F.54)

nD0

where an D g .n/ .0/=nŠ, n D 0; 1; 2; : : : , and where jrn .t /j < Lt N C1 ;

when jt j < R

(F.55)

for some finite radius of convergence R and finite constant L. Consider first the case when T < R. Substitution of (F.54) into (F.53) then gives f .x/ D

N X nD0

where

ˇZ ˇ ˇ ˇ

0

T

Z

T

an

t

Cn xt

e

Z dt C

0

T

t  e xt rN .t /dt;

(F.56)

0

ˇ Z ˇ t  e xt rN .t /dtˇˇ < L

T

t CN C1 e xt dt:

(F.57)

0

With the change of variable  D xt , the integral appearing on the right-hand side of (F.57) becomes Z

T

t 0

CN C1 xt

e

dt D x

.CN C2/

Z

xT

 CN C1 e  d 

0

Z

Z 1  CN C1 e  d    CN C1 e  d  xT 0

Z 1 .CN C2/  . C N C 2/  Dx  CN C1 e  d  : D x .CN C2/

1

xT

(F.58)

796

F Asymptotic Expansion of Single Integrals

Consider the final change of variable  D xT .1 C u/ in the last integral above, which then becomes x .CN C2/

Z

1

 CN C1 e  d  D T CN C2 e xT

Z

xT

1

e xT u .1 C u/CN C1 d u:

0

Because .1 C u/a < e au for any a > 0 and u > 0, then x .CN C2/

Z

1

 CN C1 e  d  < T CN C2 e xT

Z

xT

1

e ŒxT .CN C1/ u d u 0

1 xT  . C N C 1/   CN C1 CN C2 xT 1 1C C  DT e xT xT e xT ;  T CN C1 x D T CN C2 e xT

as x ! 1. Substitution of this result in (F.58) then gives Z

T

˚ t CN C1 e xt dt D x .CN C2/  . C N C 2/ C o x .CN C2/ ;

(F.59)

0

so that (F.57) yields Z

T

˚ e xt t  RN .t /dt D O x .CN C2/

(F.60)

0

as x ! 1. Furthermore, with the estimate given in (F.59), the summation appearing in (F.56) becomes N X nD0

Z an

T

t

Cn xt

e

dt D

0

N X nD0

D

N X

Z

Z

1

an

t 0

Cn xt

e

dt 

1

t

Cn xt

e

dt

T

˚ an  . C n C 1/x .CnC1/ C o x .CN C1/

nD0

(F.61) as x ! 1. Substitution of (F.60) and (F.61) in (F.56) then gives f .x/ D

N X nD0

˚ an  . C n C 1/x .CnC1/ C O x .CN C2/

F.5 Watson’s Lemma

so that

797

Z

T

t  g.t /e xt dt 

0

1 X  . C n C 1/g .n/ .0/ nD0

 .n C 1/x CnC1

(F.62)

as x ! 1. This result is known as Watson’s lemma (see Ex. 2 of [1]). Notice that all of the contributions to the above asymptotic expansion as x ! 1 arise from the region about the point t D 0 irrespective of the order of the zero of t  g.t /. Also notice that the upper limit T does not appear in the asymptotic expansion given in (F.62). When T > R, separate the integral into the sum of two integrals, the first from 0 to T1 < R and the second from T1 to T as Z

T1

f .x/ D

Z

t  g.t /e xt dt C

0

T

t  g.t /e xt dt:

(F.63)

T1

Since jg.t /j < Ke ct , for 0  t  T , then ˇZ ˇ ˇ ˇ

T 

t g.t /e

xt

T1

ˇ Z ˇ ˇ dtˇ < K

1

t  e .xc/t dt:

T1

Under the change of variable t D T1 .1Cu/, the integral appearing on the right-hand side of the above inequality becomes Z

1

T1

Z

t  e .xc/t dt D T1C1 e .xc/T1 <

T1C1 e .xc/T1

D

T1C1 e .xc/T1

 T1

1

.1 C u/ e .xc/T1 u d u

0

Z

1

e Œ.xc/T1  u d u

0

1 .x  c/T1  

e .xc/T1 ; xc

since .1 C u/ < e u , so that the second integral appearing on the right-hand side of (F.63) is asymptotically negligible in comparison to the first as x ! 1. This then establishes the form (F.62) of Watson’s lemma for the general case when t is real-valued. The general form of Watson’s lemma for complex t is given by (see page 49 of [3]). Lemma 1. Watson’s lemma. Let .t / be an analytic function of the complex variable t , apart from a possible branch point at t D 0, when jt j  R C ı, jarg.t /j  < , where R, ı, and are positive constants, and let .t / D

1 X mD1

am t

m r 1

;

jt j < R;

(F.64)

798

F Asymptotic Expansion of Single Integrals

where r is a positive constant. In addition, let j.t /j < Ke bt , where k and b are both positive numbers independent of t when t  R is real and positive. Then Z

1

.t /e zt dt 

0

1 X

m

am 

r

mD1

as jzj ! 1 in the sector jarg.z/j 

 2

zm=r

(F.65)

 . 2

 <

Proof. For any fixed positive integer M , there exists a constant C such that ˇ ˇ ˇ 1 ˇ M 1 ˇ ˇ ˇX ˇ X m m ˇ ˇ ˇ ˇ 1 1 am t r ˇ D ˇ am t r ˇ ˇ.t /  ˇ ˇ ˇ ˇ mD1 mDM ˇ1 ˇ ˇX ˇ M m1 ˇ ˇ 1 Dtr ˇ aM Cm1 t r ˇ ˇ ˇ mD1

t

M r

1

 Ct

M r

K 0 j.t /j

1 bt

e ;

for real t  0. Then Z

1

.t /e zt dt D

0

M 1 X

Z

t

D

m r 1

e zt dt C RM

0

mD1 M 1 X

1

am

am 

m r

mD1

zm=r C RM ;

where the remainder after M  1 terms is given by Z

1

(

RM D 0

Z

1

(

D

1 X

) am t

m r 1

e zt dt

mDM

.t /  0

M 1 X

) am t

m r 1

e zt dt:

mD1

With x
1

C e bt t

jRM j  0

M r

1 xt

e

dt D

C  .x  b/M=r



M r

 ;

when x > b. Since jarg.z/j  2  , then x  jzj sin , and so x > b when jzj > b csc . Consequently, if jarg.z/  2   and jzj > b csc , then

F.5 Watson’s Lemma

799

ˇ M=r ˇ ˇ z RM ˇ 



C jzjM=r  .jzj sin   b/M=r

M r

 ;

(F.66)

˚ which is bounded as jzj ! 1. Hence, RM D O zM=r .

t u

As an extension of Watson’s lemma as expressed in (F.62), consider the asymptotic behavior of the class of real-valued integrals of the form Z

ˇ

f .x/ D

2

.t /e xt dt;

(F.67)

˛

as x ! 1, where ˛ and ˇ are positive constants and where .t / possesses the Taylor series expansion 1 X

.t / D

an t n ;

jt j < R;

(F.68)

nD0

about t D 0 with .0/ ¤ 0. Let t D  1=2 I t D 

1=2

I

0  t  ˇ; ˛  t  0;

so that 1 f .x/ D 2

Z

ˇ2



1=2

.

1=2

/e

x

0

1 d C 2

Z

˛2

 1=2 . 1=2 /e x d :

(F.69)

0

Assuming for the moment that R > max.˛; ˇ/, application of Watson’s lemma to each of the integrals appearing in (F.68) gives 1

1X f .x/ D an 2 nD0 Z 1 X  a2n nD0

(Z

ˇ2



.n1/=2 x

e

0 T

d  C .1/

n

Z

)

˛2



.n1/=2 x

e

d

0

 n1=2 e x d ;

(F.70)

0

where T is any positive number. The integral appearing in this expression is given by Z 0

T

Z 1  .2n1/=2 e x d    .2n1/=2 e x d  0 T xT Z 1 e 1

.2n1/=2 e  d C O D .2nC1/=2 x x 0 xT  ..2n C 1/=2/ e D CO x .2nC1/=2 x

 .2n1/=2 e x d  D

Z

1

800

F Asymptotic Expansion of Single Integrals

˚ as x ! 1, where the O e xT =x term results from the second integral after repeated integration by parts. Substitution of this result in (F.70) then gives Z

ˇ

2

.t /e xt dt 

˛

1 X

a2n  ..2n C 1/=2/x .2nC1/=2

(F.71)

nD0

p as x ! 1. Since  .1=2/ D  and  ..2nC1/=2/ D ..2n1/=2/ ..2n1/=2/, then the asymptotic series in (F.71) may be expressed as Z

ˇ

.t /e

xt 2

˛

   1=2  3a4 a2 C 2 C  a0 C dt  x 2x 4x

(F.72)

as x ! 1. As before, all of the contributions to the asymptotic expansion arise from the neighborhood about the point t D 0.

F.6 Laplace’s Method Laplace’s method [9] considers the asymptotic behavior of integrals of the type Z

ˇ

g.t /e xh.t/ dt

f .x/ D

(F.73)

˛

as x ! 1, where x is real and positive, g.t / is a real-valued, continuous function on the interval ˛  t  ˇ, and where h.t /, together with its first two derivatives h0 .t / and h00 .t /, are real-valued and continuous on the interval ˛  t  ˇ with ˛ and ˇ both real. The essence of Laplace’s method is that the dominant contributions to the asymptotic behavior of the integral appearing in (F.73) as x ! 1 arise from each neighborhood of the points in the interval ˛  t  ˇ where h.t / attains relative maxima. In general, the function h.t / will possess several relative maxima in the interval Œ˛; ˇ , including possibly the points at either endpoint, as depicted in Fig. F.3. To accomodate this possibility, the integral in (F.73) is then separated into the sum of several integrals over each of the subintervals where h.t / attains a single maximum value in each subinterval. For the example illustrated in Fig. F.3, this separation is given by Z

t1

g.t /e

f .x/ D ˛

xh.t/

Z

t2

dt C

g.t /e t1

xh.t/

Z

ˇ

g.t /e xh.t/ dt;

dt C t2

where the dominant term in the first integral arises from the right neighborhood of the endpoint at t D ˛, the dominant term in the second integral from the maximum at t D p1 , and the dominant term in the third integral from the maximum at t D p2 .

801

h(t)

F.6 Laplace’s Method

a

t1

p1

t2

p2

t

b

Fig. F.3 Example of a continuous function h.t / with several relative maxima in the interval ˛  t  ˇ. The left endpoint at t D ˛ is a relative maximum while the right endpoint at t D ˇ is not

Consider the case when h.t / has a single maximium at the interior point t D a, where ˛ < a < ˇ and h0 .a/ ¤ 0. One then has that Z f .x/ D

g.t /e Z

Z

a xh.t/

ˇ

g.t /e xh.t/ dt

dt C

˛

a a˛

g1 ./e xh1 ./ d  C

D 0

Z

ˇa

g2 ./e xh2 . / d ;

(F.74)

0

where the first integral in the above expression results from the change of variable t D a   while the second results from t D a C  . Here g1 . / g.a   /; g2 . / g.a C  /;

h1 ./ h.a   /; h2 ./ h.a C /:

Each integral appearing in the expression (F.74) is now of the form where the maximum value of the exponential function in the integrand occurs at the lower endpoint  D 0. Without any loss of generality, attention is now focused on the asymptotic behavior of integrals of the form Z

T

g.t /e xh.t/ dt;

f .x/ D 0

(F.75)

802

F Asymptotic Expansion of Single Integrals

where h.0/ is the maximum value of h.t / in the interval 0  t  T with T > 0. The function h.t / may then either possess a genuine maximum at the origin with h0 .0/ D 0 and h00 .0/ < 0, or it may not, in which case h0 .0/ < 0. These two cases must then be treated separately. Case 1: Let h0 .0/ D 0, h00 .0/ < 0, and h.0/ > h.t / for all t 2 .0; T /. Assume that g.t / and h00 .t / are both real continuous functions on 0  t  T . Since h00 .t / is continuous and h.0/ is the maximum value of h.t / on this interval, then there exists a ı-neighborhood of the point t D 0 such that h00 .t / < 0 for 0  t  ı < T . Then, by the mean value theorem, there exists a point  2 Œ0; ı such that h.t /  h.0/ D

1 00 h ./t 2 ; 2

when 0  t  ı, where h00 ./ < 0. Define the variable s by the relation h.t /  h.0/ s 2 ;

(F.76)

so that 2

e xh.t/ D e xh.0/ e xs ;

(F.77)

and the extension of Watson’s lemma given in (F.72) applies. Assume that g.t / possesses the Taylor series expansion 1 g.t / D g.0/ C g 0 .0/t C g 00 .0/t 2 C    2

(F.78)

in a neighborhood of the origin that is valid for some finite radius of convergence. Furthermore, substitution of the Taylor series expansion of h.t / about t D 0 in (F.76) results in the expression 1 1 00 h .0/t 2 C h000 .0/t 3 C    D s 2 ; 2 3Š so that

 tD 

2 h00 .0/

1=2

˚ s C O s2 :

(F.79)

Substitution of this expression in (F.78) then gives  g.t / D g.0/ C g 0 .0/ 

2 h00 .0/

1=2

˚ s C O s2

(F.80)

F.6 Laplace’s Method

803

The change of variable from t to s is now made in (F.75) with the above substitutions as h i 1=2 Z  h00 .0/ 1=2 T 2 2 2 f .x/   00 e xh.0/ e xs Œg.0/ C Ofsg ds h .0/ 0 1=2 Z 1

 Z 1 2 xh.0/ xs 2 h.0/x xs 2 e e ds C e O se ds ;  g.0/  00 h .0/ 0 0 1=2  ˚  e h.0/x C e h.0/x O x 1 ;  g.0/  00 2h .0/x



which then gives the result Z

T

g.t /e

xh.t/

0



 dt  g.0/  00 2h .0/x

1=2

˚ e h.0/x C e h.0/x O x 1 ;

(F.81)

as x ! 1. With the above result in hand, attention is now turned to the asymptotic behavior of the integral Z T2 f .x/ D g.t /e xh.t/ dt; T1

where T1 and T2 are both positive numbers and where h.0/ is the maximum value of h.t / in the interval T1  t  T2 with h0 .0/ D 0 and h00 .0/ < 0. The preceding R1 2 derivation then applies, leading to the definite integrals 1 e xs ds D .=x/1=2 R1 2 and 1 se xs ds D 0. Because of the of this second integral, the sec ˚ vanishing h.0/x 1 as it is O x ond term in˚this result is not e ˚ ˚ in (F.81). As a consequence, terms of O s 2 and O s 3 in (F.79) and O s 2 in (F.80) are required. The first R1 2 nonzero contribution to this second term arises from the integral 1 s 2 e xs ds D .1=2/.=x 3 /1=2 , so that Z

T2

g.t /e T1

xh.t/



2 dt  g.0/  00 h .0/x

1=2

˚ e h.0/x C e h.0/x O x 3=2 ;

(F.82)

as x ! 1. For the general integral given in (F.73), where the maximum value of the function h.t / in the interval Œ˛; ˇ occurs at t D a, with ˛ < a < ˇ, the asymptotic behavior is obtained from the result given in (F.82) by simply replacing the point at t D 0 with t D a with the result 1=2  Z ˇ ˚ 2 (F.83) g.t /e xh.t/ dt  g.a/  00 e h.a/x C e h.a/x O x 3=2 ; h .a/x ˛ as x ! 1 with h0 .a/ D 0 and h00 .a/ < 0.

804

F Asymptotic Expansion of Single Integrals

Case 2: Let h.0/ > h.t / for all t 2 .0; T / with h0 .0/ < 0. Then as x ! 1, the dominant contribution to the integral Z T g.t /e xh.t/ dt f .x/ D 0

comes from a ı-neighborhood of the point at t D 0. The mean value theorem then states that there exists a value  with 0 <  < ı < T such that h.t /  h.0/ D h0 ./t; when 0  t  ı, where h0 ./ < 0. Consider then the change of variable h.t /  h.0/ D s

(F.84)

so that, from the Taylor series expansion of h.t / about t D 0, one obtains t D

˚ 2 1 s C O s : h0 .0/

(F.85)

Under this change of variable, the above integral becomes     xh.0/ xs 1 f .x/  ds  0 g.0/ C O fsg e e h .0/ 0 Z 1

Z 1 g.0/ xh.0/ xs xh.0/ xs  0 e e ds C e O se ds ; h .0/ 0 0 Z

so that

Z

h0 .0/T

T

g.t /e xh.t/ dt   0

˚ g.0/ xh.0/ e C e xh.0/ O x 2 0 h .0/x

(F.86)

as x ! 1. For the general case where the maximum value of h.t / in the interval ˛  t  ˇ occurs at the lower endpoint t D ˛ with h0 .˛/ < 0, the above result becomes Z

ˇ

g.t /e xh.t/ dt  

˛

˚ g.˛/ xh.˛/ e C e xh.˛/ O x 2 0 h .˛/x

(F.87)

as x ! 1. On the other hand, when the maximum value of h.t / in the interval ˛  t  ˇ occurs at the upper endpoint t D ˇ with h0 .ˇ/ > 0, one obtains Z ˛

as x ! 1.

ˇ

g.t /e xh.t/ dt 

˚ g.ˇ/ xh.ˇ/ e C e xh.ˇ/ O x 2 0 h .ˇ/x

(F.88)

F.7 The Method of Steepest Descents

805

As an illustration of Laplace’s method, consider obtaining an asymptotic approximation of the gamma function Z

Z

1 x t

1

e tCx ln t dt

t e dt D

 .x C 1/ D 0

0

for real values of x as x ! 1. With the change of variable t D x, this integral becomes Z 1 xC1  .x C 1/ D x e x.ln   / d ; 0

which is the same form as the integral in (F.74) with g./ D 1 and h./ D ln    with h0 .1/ D 0 and h00 .1/ < 0. Direct application of the result given in (F.83) then yields the asymptotic approximation  .x C 1/  x x .2x/1=2 e x

(F.89)

as x ! 1. If x D n is an integer, then  .n C 1/ D nŠ and the above result yields Stirling’s formula p (F.90) nŠ  nn 2 ne n as n ! 1.

F.7 The Method of Steepest Descents Originated by B. Riemann [10] in 1876 and then fully developed by P. Debye [11] in 1909, the method of steepest descents provides a much needed generalization of Laplace’s method to integrals in the complex plane. The method is applicable to the specific class of integrals of the form Z f ./ D g.z/e h.z/ d z; (F.91) C

where C is some piecewise continuous contour in the complex z-plane, g.z/ and h.z/ are both analytic functions of the complex variable z in some domain D which contains the contour C , both independent of  with  a real positive number. Since h.z/ D .x; y/ C i .x; y/ is analytic in some domain D, it then follows from the Cauchy–Riemann conditions that h.z/ cannot have either maxima or minima in that domain, only saddle points. Assume that the point z0 D x0 C iy0 is a relative maximum of .x; y/
@.x; y/ @ .x; y/ D @y @x

(F.92)

806

F Asymptotic Expansion of Single Integrals

are satisfied at z0 , then r h0 .z/ D

D 0 at that point also. Consequently

@ .x; y/ @.x; y/ @ .x; y/ @.x; y/ Ci D Ci D0 @x @x @y @y

(F.93)

at z D z0 . Since .x; y/ and .x; y/ are then both potential functions, each satisfying Laplace’s equation r 2  D 0, r 2 D 0, then by the maximum modulus theorem, both .x; y/ and .x; y/ cannot have either a maximum or a minimum in the domain of analyticity D of h.z/. The point z0 is then a saddle point of both .x; y/ and .x; y/, and hence of h.z/. A first-order saddle point (a saddle point of order one) of h.z/ at the point z D z0 satisfies the relations h0 .z0 / D 0;

h00 .z0 / ¤ 0;

(F.94)

while an nth-order saddle point (a saddle point of order n) at the point z D z0 satisfies the relations h.1/ .z0 / D h.2/ .z0 / D    D h.n/ .z0 / D 0;

h.nC1/ .z0 / ¤ 0:

(F.95)

Since r  r

D

@ @ @ @ C D0 @x @x @y @y

(F.96)

by virtue of the Cauchy–Riemann conditions, then the family of isotimic1 contours .x; y/ D const ant are everywhere orthogonal to the family of isotimic contours .x; y/ D const ant in the domain of analyticity D of h.z/. The contour lines along the direction of r, where .x; y/ changes most rapidly, are then the contours D const ant . A path of steepest descent through the saddle point z0 D x0 C iy0 is then defined by the contour .x; y/ D .x0 ; y0 /. Two comments regarding the path of steepest descent are in order here. First, .x; y/ =fh.z/g generally produces an oscillatory contribution e i .x;y/ in the integrand of (F.91) with oscillation frequency that increases with the asymptotic parameter ; however, this oscillation identically vanishes along any path of steepest descent through the saddle point, reinforcing the central importance of this specific path of integration in the asymptotic expansion procedure for contour integrals of the form given in (F.91). Second, because of Cauchy’s theorem, the original contour of integration C appearing in (F.91) can always be deformed into the path of steepest descent through a given saddle point provided that the saddle point is in the domain of analyticity D of the function h.z/, and provided that appropriate care is given to any singularities of the function g.z/ that may be crossed in that deformation. In a neighborhood of an isolated first-order saddle point at z D z0 , z0 2 D, the function h.z/ possesses the Taylor series expansion 1

From the Greek: iso (of equal) timos (worth).

F.7 The Method of Steepest Descents

807

˚ 1 h.z/ D h.z0 / C h00 .z0 /.z  z0 /2 C O .z  z0 /3 : 2

(F.97)

With the identifications h00 .z0 / ae i˛ ; z  z0 re i ;

a D jh00 .z0 /j > 0; r D jz  z0 j  0;

(F.98) (F.99)

the above expansion becomes .x; y/ C i .x; y/ D 0 C i

0

˚ 1 C ar 2 e i.2C˛/ C O r 3 ; 2

(F.100)

where 0 .x0 ; y0 / and 0 .x0 ; y0 /. Upon equating real and imaginary parts, one obtains the pair of expressions ˚ 1 .x; y/ D 0 C ar 2 cos .2 C ˛/ C O r 3 ; 2 ˚ 1 .x; y/ D 0 C ar 2 sin .2 C ˛/ C O r 3 : 2

(F.101) (F.102)

There are then two isotimic contours .x; y/ D 0 which, for sufficiently small radial distances r from the saddle point z0 , are tangent to the two orthogonal lines that are given by the solutions of cos .2 C ˛/ D 0, so that  1  ˛ 2 2  1  C˛  D 2 2 D

and its continuation and its continuation

 1  ˛ ; 2 2  1   D C˛ : 2 2

 DC

(F.103) (F.104)

The local valley regions below the saddle point where .x; y/ < 0 are then given by  ˛ 3 ˛  < <  ; 4 2 4 2

&

5 ˛ 7 ˛  < <  : 4 2 4 2

(F.105)

The two isotimic contours .x; y/ D 0 along which .x; y/ changes most rapidly as one moves away from the saddle point z0 , i.e., the paths of steepest descent and ascent from z0 , are tangent at z D z0 to the two orthogonal lines that are given by the solutions of sin .2 C ˛/ D 0, so that ˛ 2  ˛ D  2 2  D

and its continuation and its continuation

˛ ; 2 3 ˛ D  : 2 2  D

(F.106) (F.107)

808

F Asymptotic Expansion of Single Integrals

Fig. F.4 Local geometric structure in the complex z-plane of .x; y/
(x,y)

(x,y)

0

0

(x,y)

0

z0

(x,y) (x,y)

0

0

(x,y)

0

The geometric structure of this local behavior about an isolated first-order saddle point is illustrated in Fig. F.4. With the behavior about the saddle point determined, the next step in the method is to deform the contour of integration C , assuming that each of the original endpoints lie on opposite sides of the saddle point in the region satisfying .x; y/ < 0 , so that it lies along the path of steepest descent through the saddle point, as depicted by the contour .x; y/ D 0 in the shaded region of Fig. F.4. Along this steepest descent path .z/  0 D h.z/  h.z0 / 1 D h00 .z0 /.z  z0 /2 ; 2

with h00 .z0 / < 0;

(F.108)

which is real-valued since the imaginary parts of h.z/ and h.z0 / cancel along the path of steepest descent. From Laplace’s method, define the real variable  by the relation (F.109) h.z/  h.z0 /  2 when z is along the steepest descent path through the saddle point, which then determines z D z. / as a function of  . Under this change of variable the contour integral appearing in (F.91) becomes Z

g.z/e h.z/ d z Z b 2 dz d ; D e h.z/ g.z.//e  d a

f ./ D

C

(F.110)

F.7 The Method of Steepest Descents

809

where a > 0 and b > 0 correspond to the endpoints of the deformed contour through the saddle point under the coordinate transformation given in (F.109). Consequently, Z 1 2 dz h.z/ d (F.111) g.z.//e  f ./  e d 1 as  ! 1. Upon expanding the left-hand side of (F.109) in a Taylor series about the saddle point at z D z0 , one obtains ˚ 1 00 h .z0 /.z  z0 /2 C O .z  z0 /3 D  2 ; 2 so that



2 z  z0 D  00 h .z0 /

1=2 :

(F.112)

Since h00 .z0 / D ae i˛ is complex-valued, one must choose the appropriate branch of the quantity Œ1= h00 .z0 / 1=2 when z lies along the path of steepest descent through that saddle point. This branch choice is determined by the sense of direction that the path makes through the branch point at z D z0 . Suppose that when the original contour C is deformed to lie along the steepest descent path through z0 , it progresses from the region where .x; y/ < 0 with  ˛2 to the region with arg.z  z0 / D 2  ˛2 , as depicted in arg.z  z0 / D 3 2 Fig. F.4, where ˛ argfh00 .z0 /g. The appropriate choice in (F.112) must then be the one such that argf1= h00 .z0 /g gives  > 0 when z is in the final region arg.zz0 / D   ˛2 , so that 2 ( arg

1  00 h .z0 /

1=2 ) D

 ˛  2 2

H)

ˇ ˇ h00 .z0 / D ˇh00 .z0 /ˇ e i˛ ;

and 1=2 ˚ 2  ˛ ei . 2  2 / C O  2 z  z0 D 00 jh .z0 /j 1=2  ˚ 2  C O 2 : D i 00 h .z0 / 

(F.113)

If the direction of integration were reversed, then the above result would be replaced by the expression 

2 z  z0 D i 00 h .z0 /

1=2

˚  C O 2 I

however, it is advised that each case be treated individually.

810

F Asymptotic Expansion of Single Integrals

The asymptotic expansion of the integral in (F.111) is now completed with substitution of the Taylor series expansion 1 g.z. // D g.z0 / C g 0 .z0 /.z  z0 / C g 00 .z0 /.z  z0 /2 C    2 1=2  ˚ 2 D g.z0 / C g 0 .z0 /  00  C O 2 ; h .z0 /

(F.114)

so that  f ./  g.z0 / 

2 h00 .z0 /

1=2

e h.z0 /

Z

1

2

e  d  C    ;

1

p 2 as  ! 1. Since 1 e  d  D =, one finally obtains the general result R1

 f ./  g.z0 / 

2 h00 .z0 /

1=2 e

h.z0 /

Ce

h.z0 /

1 ; O 

(F.115)

as  ! 1, where the specific branch of the quantity Œ1= h00 .z0 / 1=2 must be chosen so as to be consistent with the direction of the deformed contour of integration through the saddle point at z D z0 . For the cases given in (F.113), the above result becomes r 2 g.z0 / f ./  e h.z0 / e i.˛/=2 ; p  jh00 .z0 /j as  ! 1.

References 1. E. T. Whittaker and G. N. Watson, A Course of Modern Analysis. New York: MacMillan, 1943. Chap. VIII. 2. M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, vol. 55 of Applied Mathematics Series. Washington, D.C.: National Bureau of Standards, 1964. 3. E. T. Copson, Asymptotic Expansions. London: Cambridge University Press, 1965. 4. N. Bleistein and R. Handelsman, Asymptotic Expansions of Integrals. New York: Dover, 1975. 5. J. D. Murray, Asymptotic Analysis. New York: Springer-Verlag, 1984. 6. G. G. Stokes, “On the discontinuity of arbitrary constants which appear in divergent developments,” Trans. Camb. Phil. Soc., vol. X, pp. 106–128, 1857. 7. L. Kelvin, “On the waves produced by a single impulse in water of any depth, or in a dispersive medium,” Proc. Roy. Soc., vol. XLII, p. 80, 1887. 8. E. T. Whittaker and G. N. Watson, A Course of Modern Analysis. New York: MacMillan, 1943. Sect. 9.41. 9. P. S. de Laplace, Th´eorie Analytique des Probabiliti´es. Paris: V. Courcier, 1820. 10. B. Riemann, Gesammelte Mathematische Werke. Leipzig: Teubner, 1876. 11. P. Debye, “N¨aherungsformeln f¨ur die zylinderfunktionen f¨ur grosse werte des arguments und unbeschr¨ankt verander liche werte des index,” Math. Ann., vol. 67, pp. 535–558, 1909.

Appendix G

Proof of Theorem 1

A proof of Theorem 1 in Sect. 9.3.1, due to Sherman, Stamnes, and Lalor [1], which states that QQ if U.p; q; !/ 2 TN for some positive even integer N , then for z > 0, the spectral wave field Q Q U.r; !/ D U.x; y; z; !/ given in (9.194)–(9.195) with k a positive real constant satisfies Q Q 0 .x; y; z/ C R.x; y; z/; U.x; y; z; !/ D U where Q 0 .x; y; z/ D U

Z DH

QQ .p; q/U.p; q; !/e i k.pxCqyCmz/ dpdq;

(G.1)

(G.2)

˚ and where R.x; y; z/ D O .kR/N as kR ! 1 uniformly with respect to 1 and 2 for q all real 1 ; 2 such that ı < 1  12  22  1 for any positive constant ı < 1,

is given here based upon a modification of the proof of Theorem 1 by Chako [2] or Theorem 3 by Focke [3]. Proof. Construct three neutralizer functions j .p; q/, j D 1; 2; 3, each of which is a real continuous function of p and q with continuous partial derivatives of all orders for all p; q and which satisfy 0  j .p; q/  1 with 1 .p; q/ C 2 .p; q/ C 3 .p; q/ D 1: Choose positive constants C1 ; C2 ; C3 ; C4 such that q 12 C 22 < C1 < C2 < 1 < C3 < C4 ; ˝2 -neighborhood of the point .ps ; qs / is comwith C1 sufficiently large that the p pletely contained within the region p 2 C q 2 < C1 , as illustrated in Fig. G.1. Since 12 C 22 D 1  32 with 3 > ı > 0, the constants C1 ; C2 ; C3 ; C4 clearly exist.

811

812

G Proof of Theorem 1

q

D1

D2 C1 C2 1

D3 C3 C4

p

Fig. G.1 Graphical illustration p of the notation used in the proof of Theorem 1. The shaded area 1  p 2  q 2 is real-valued, and the unshaded area indicates the indicates the region where m D p 2 2 region where m D i p C q  1 is pure imaginary

With these four constants specified, the neutralizer functions j .p; q/, j D 1; 2; 3 are now required to satisfy the conditions 1 .p; q/ D 2 .p; q/ D 3 .p; q/ D

1; 0; 1; 0; 1; 0;

p for pp 2 C q 2  C1 ; for p 2 C q 2  C2 p 2 2 p for C2  p Cpq  C3 ; 2 2 for p C q  C1 & p 2 C q 2  C4 p for pp 2 C q 2  C4 : for p 2 C q 2  C3

G Proof of Theorem 1

813

Finally, three overlapping regions D1 ; D2 ; D3 in the p; q-plane are defined as n o ˇp D1 .p; q/ˇ p 2 C q 2  C2 C " ; n o p ˇ D2 .p; q/ˇ C1  "  p 2 C q 2  C4 C " ; o n p ˇ D3 .p; q/ˇ C3  "  p 2 C q 2 ; where " > 0 is a positive constant that satisfies the three inequalities " < 1  C2 ; " < C3  1; q " < C1  12 C 22 : Each of these three regions is illustrated in Fig. G.1. Features of note that are important for this proof are the following:  Each of the neutralizer functions j .p; q/, j D 1; 2; 3, vanishes both for points

.p; q/ … Dj and for points .p; q/ in some neighborhood of the boundary @Dj of Dj  The stationary phase point .ps ; qs / lies within the interior of D1 and is exterior to both regions D2 and D3 p  The direction cosine m D C 1  p 2  q 2 is real-valued for all points .p; q/ 2 D1 p  And m D Ci p 2 C q 2  1 is pure imaginary for all points .p; q/ 2 D3 . Q !/ that is given in (9.194) and Separate the angular spectrum integral for U.r; (9.195) into three parts as Q 2 .r; !/ C U Q 3 .r; !/; Q !/ D U Q 1 .r; !/ C U U.r; where Q j .r; !/ U

Z

QQ j .p; q/U.p; q; !/e ik.pxCqyCmz/ dpdq Dj

for j D 1; 2; 3. Each of these integrals is now separately treated. Consider first the asymptotic approximation of the integral representation of Q 1 .r; !/ which is in a form that is appropriate for the method of stationary phase. U Since the stationary phase point .ps ; qs / is the only critical point of significance in the domain D1 , and since the critical points of the phase function k  r D k.px C qy C mz/ on the boundary @D1 of D1 do not contribute to the integral because of the neutralizer function 1 .p; q/, and since the region ˝2 which contains

814

G Proof of Theorem 1

the neighborhood ˝1 of the stationary phase point .ps ; qs / lies entirely within D1 , then it follows from the proof of Theorem 1 in Chako [2] that Q 0 .x; y; z/ C R.x; y; z/; Q 1 .r; !/ D U U ˚ where R.x; y; z/ D O .kR/N as kR ! 1 with fixed k > 0, uniformly with q respect to 1 ; 2 for all real 1 and 2 such that ı < 1  12  22  1 for any positive constant ı with 0 < ı < 1. Consider next the asymptotic approximation p of the integral representation of Q 3 .r; !/. In this case, the quantity m D Ci p 2 C q 2  1 is pure imaginary with U p im  p  .C3  "/2  1 < 0 for all .p; q/ 2 D3 . For a sufficiently large value of R D .x  x0 /2 C .y  y0 /2 C .z  z0 /2 , a positive constant a then exists such that z > a. In that case Z ˇ ˇ ˇ ˇ ˇ QQ ˇ ˇU Q 3 .r; !/ˇ  ˇU.p; q; !/ˇ e ikmz dpdq D3 Z ˇ ˇ p ˇ ˇ QQ k.za/ .C3 "/2 1 e ˇU.p; q; !/ˇ e ikma dpdq: D3

QQ Since U.p; q; !/ 2 TN , it then follows from the first condition [see (9.204)] in the definition of TN that the final integral appearing in the above inequality converges to a finite, nonnegative constant M3 that is independent of the direction cosines 1 ; 2 ; 3 . Since z D z0 C 3 R [see (9.201)] and since 3 > 0, then the above inequality becomes p ˇ ˇ ˇU Q 3 .r; !/ˇ  M3 e kŒ.z0 a/CRı .C3 "/2 1 ; ˇ ˇ ˚ Q 3 .r; !/ˇ D O .kR/N uniformly with respect to independent of 1 ; 2 . Hence, ˇU 1 ; 2 as kR ! 1 for all positive even integer values N . Consider finally the asymptotic approximation of the integral representation of Q 2 .r; !/, which turns out to be much more involved than the previous two cases. U The analysis begins with the change of variables p D sin ˛ cos ˇ; q D sin ˛ sin ˇ; with Jacobian J .p; q=˛; ˇ/ D sin ˛ cos ˇ. In addition, 1=2  D cos ˛; m D 1  p2  q2 where ˛ is, in general, complex as the region D2 extends from inside the inner region where m is real-valued into the outer region where m is pure imaginary, as illustrated in Fig. G.1. In addition, let the angles #; ' describe the intersection point

G Proof of Theorem 1

815

on the unit sphere with center at the origin r D 0 in coordinate space with the line through the origin that is parallel to the direction of observation r D .x; y; z/ as it recedes to infinity through the fixed point r0 D .x0 ; y0 ; z0 / with z > 0, where 1 D sin # cos '; 2 D sin # sin '; 3 D cos #; with 0  # < =2 and 0  ' < 2. Under these two changes of variables, the Q 2 .r; !/ becomes integral for U Q 2 .r; !/ D U

Z

2

0

Z

A.˛; ˇ/e ikRŒsin # sin ˛ cos .ˇ'/Ccos # cos ˛ d˛dˇ;

C

where A.˛; ˇ/ 2 .p; q/V .p; q; m/e ik.px0 Cqy0 Cmz0 / sin ˛; with V .p; q; m/ defined in (9.204). The contour ofintegration C in the complex p ˛-plane, which extends from ˛1 D arcsin C1  " along the real ˛ 0 -axis to =2 p  C4 C " , depicted in Fig. G.2, and then to the endpoint at ˛4 D =2  i cosh1 then results in a complete, single covering of the original integration domain D2 . Notice that, in contrast with the original form of the integral, the phase ˚ sin # sin ˛ cos .ˇ  '/ C cos # cos ˛

''

'

C

Fig. G.2 Contour of integration C for the ˛-integral in Q 2 .r; !/ U

816

G Proof of Theorem 1

Q 2 .r; !/ is analytic and the ampliappearing in the above transformed integral for U tude function A.˛; ˇ/ is continuous with continuous partial derivatives with respect to ˛ and ˇ up to order N over the entire integration domain D2 . However, the contour of integration C is now complex, as illustrated in Fig. G.2, and this results in the phase ˚ being complex-valued when ˛ D ˛ 0 C i ˛ 00 varies from =2 to ˛4 along C . As ˛ varies over the contour C with the angle ˇ held fixed, the phase function ˚ varies over a simple curve C.ˇ/ of finite length in the complex ˛-plane. Since the partial derivative 

@˚ @˛

 D sin # cos ˛ cos .ˇ  '/  cos # sin ˛ ˇ

is an entire function of complex ˛ for all ˇ, then 

@˛ @˚

"

 D ˇ

@˚ @˛

 #1 ˇ

  is an analytic function of complex ˛ for all ˛; ˇ provided that @˚ ¤ 0. Since @˛ ˇ  @˚  cos # D 3 > ı, then @˛ ˇ ¤ 0 when cos ˛ D 0. It then follows that the zeros of  @˚  occur for those values of ˛ that satisfy the relation @˛ ˇ tan ˛ D tan # cos .ˇ  '/: On the portion of the contour C over which ˛ is real, ˛1  ˛  =2, and since cos # D 3  C1  " (see Fig. G.1), then #  ˛  =2 and one obtains the inequality tan ˛ > tan # cos .ˇ  '/. On the portion of the contour C over which ˛ is complex,
Z

2 0

Z

  A ˛.˚/; ˇ C.ˇ/



@˛ @˚



e ikR˚ d˚dˇ:

ˇ

As a consequence of the prescribed properties for both the function V.p; q; m/ D Q Q mU.p; q; !/ [cf. (9.204)] and the neutralizer function 2 .p; q/, the function A.˛; ˇ/ possesses N continuous partial derivatives with respect to the variable ˛ 2 C for all

G Proof of Theorem 1

817

 @˛ 

fixed ˇ 2 Œ0; 2/.1 Because the quantity @˚ ˇ is an analytic function of ˛ 2 C , the  @˛  also possesses N continuous partial derivatives taken with product A.˛; ˇ/ @˚ ˇ respect to ˛ along the contour C with fixed ˇ 2 Œ0; 2/. Notice that differentiation with respect to ˚ along the contour C.ˇ/ with fixed ˇ is equivalent to differentiation along quantity  respect to ˛ @˛  C with fixed ˇ followed by multiplication by the @˛  with @˛ . Because is analytic with respect to ˛, the quantity A.˛; ˇ/ has @˚ ˇ @˚ ˇ @˚ ˇ N continuous partial derivatives with respect to ˚ 2 C.ˇ/ for all fixed ˇ 2 Œ0; 2/. The ˚-integral Z I.kR; ˇ/ D

  A ˛.˚/; ˇ

C.ˇ/



@˛ @˚



e ikR˚ d ˚

ˇ

Q 2 .r; !/ is now integrated by parts N times appearing in the integral expression for U ikR˚ each time and differentiating the remainby integrating the exponential factor e ing factor, with the result [4] I.kR; ˇ/ D LN .˚4 /  LN .˚1 / C RN .kR; ˇ/; where LN .˚j / D

N 1 X

( i

n1

nD0

"  #)    @˛ @n A ˛.˚/; ˇ @˚ n @˚ ˇ

ˇ ˇ ˇ ˇ ˇ

ˇ ˚D˚j

e ikR˚j ; .kR/nC1

with ˚j D sin # sin ˛j cos .ˇ  '/ C cos # cos ˛j for j D 1; 4, where ˛1 and ˛4 denote the endpoints of the contour C (see Fig. G.2), and where N

RN .kR; ˇ/ D .i kR/

Z C.ˇ/

(

@N @˚ N

"

  A ˛.˚/; ˇ



@˛ @˚

 #) ˇ

e ikR˚ d ˚: ˇ

As a consequence of the prescribed properties for the neutralizer function 2 .p; q/, n , n D 0; 1; : : : ; N 1, the function A.˛; ˇ/ and all N of its partial derivatives @ A.˛;ˇ/ n @˛ and ˚ of the contour C . Each taken along the contour C , vanish at the endpoints ˚ 1 4 n taken along the contour C.ˇ/ then also vanish of the N partial derivatives @ A.˛;ˇ/ n @˚ at the endpoints ˚1 and ˚4 , so that LN .˚1 / D LN .˚4 / D 0: Furthermore, because the integrand in the above expression for the remainder term RN .kR; ˇ/ is continuous along the contour C.ˇ/ and as this contour is of finite 1

Notice that the derivatives with respect to ˛ must be taken along the contour C since both V .p; q; m/ and 2 .p; q/ are defined only for real-valued p and q. That is, variation of ˛ along the contour C corresponds to p; q varying along the real axis.

818

G Proof of Theorem 1

length for all #, ', and ˇ, then that integral is bounded by some positive constant M2 independent of #, ', and ˇ, so that jRN .kR; ˇ/j  M2 .kR/N : Consequently, ˇ ˇ ˇU Q 2 .r; !/ˇ  2M2 .kR/N ; ˚ Q 2 .r; !/ D O .kR/N uniformly with respect to 1 ; 2 as kR ! 1 with so that U fixed k > 0. In summary, following the method of proof given by Sherman, Stamnes, and Q j .r; !/, j D 1; 2; 3, whose Lalor [1], it has been established that the three terms U Q sum gives U.r; !/, satisfy the order relations ˚ Q 0 .x; y; z/ C O .kR/N ; Q 1 .r; !/ D U U ˚ Q 2 .r; !/ D O .kR/N ; U ˚ Q 3 .r; !/ D O .kR/N ; U uniformly with respect to 1 ; 2 as kR ! 1 with fixed k > 0. This then completes the proof of the theorem.

References ´ Lalor, “Asymptotic approximations to angular-spectrum 1. G. C. Sherman, J. J. Stamnes, and E. representations,” J. Math. Phys., vol. 17, no. 5, pp. 760–776, 1976. 2. N. Chako, “Asymptotic expansions of double and multiple integrals occurring in diffraction theory,” J. Inst. Math. Appl., vol. 1, no. 4, pp. 372–422, 1965. 3. J. Focke, “Asymptotische Entwicklungen mittels der Methode der station¨aren phase,” Ber. Verh. Saechs. Akad. Wiss. Leipzig, vol. 101, no. 3, pp. 1–48, 1954. 4. A. Erd´elyi, “Asymptotic representations of Fourier integrals and the method of stationary phase,” SIAM J. Appl. Math., vol. 3, pp. 17–27, 1955.

Appendix H

The Radon Transform

The Radon transform, named after the Czech mathematician Johann Radon (1887– 1956) has its origin in his 1919 paper [1] (which then led to the Radon-Nikod´ym theorem). For the purpose of this brief development, it is best introduced in connection with the projection-slice theorem in Fourier analysis in the following manner. Consider a two-dimensional scalar (object) function O.r/ D O.x; y/, which may be expressed as Z O.r/ı.R  r/d 2 r;

O.R/ D

(H.1)

D

where D is any region containing the point R, and where ı.r/ is the two-dimensional Dirac delta function which has the Fourier integral representation 1 ı.R  r/ D 4 2

Z

1

e ik.Rr/ d 2 k:

(H.2)

1

O With this Let nO be the unit vector in the dirction of the vector k so that k D k n. substitution, (H.2) becomes ı.R  r/ D

1 4 2

Z

1

d 2 k e ikR e ikrOn ;

1

and since e ikrOn D

Z

1

O ikx ; dx ı.x  r  n/e

1

one then obtains the expression 1 ı.R  r/ D 4 2

Z

1 2

d ke

ikR

1

Z

1

O ikx : dx ı.x  r  n/e

(H.3)

O x/e ikx ; dx P .nI

(H.4)

1

Substitution of this result into (H.1) then gives O.R/ D

1 4 2

Z

1

d 2 k e ikR 1

Z

1 1

819

820

H The Radon Transform

where

Z

O 2r O.r/ı.x  r  n/d

O x/ P .nI

(H.5)

D

denotes the projection of the object function O.r/ onto the direction defined by the O The variable x appearing in this expression is now interpreted as the unit vector n. O real variable defining the location in the n-direction at which the rectilinear line integral through the object function O.r/ is taken. As an illustration, the vertical line integral through the object function depicted in Fig. H.1 is given by Z

O.x; y 0 /dy 0 D

Z

O.x 0 ; y 0 /ı.x  x 0 /dx 0 dy 0

ZD

`

D

O 2 r; O.r/ı.x  r  n/d

D

which is precisely the expression defined in (H.5). The two-dimensional Radon transform of a (sufficiently well behaved) function O.r/ D O.x; y/ is then defined as [2] O x/ RŒO .nI

Z

O 2 r: O.r/ı.x  r  n/d

(H.6)

D

y'

x

v

O (r)

n

r x'

Fig. H.1 Geometry of the vertical line integral through the object function O.r/

H The Radon Transform

821

With use of the notation (see Appendix C in Vol. 1) Fk ff .x/g D n

Fx1

Z

1

f .x/e ikx dx;

(H.7)

1

Z 1 o 1 Q fQ.k/e ikx d k; f .k/ D 2 1

(H.8)

for the Fourier transform and its inverse, respectively, where x and k are referred to as Fourier conjugate variables, then as a special case of the Fourier integral theorem [3], Fx1 ı Fk D 1, that is, the subsequent application of the forward and inverse Fourier transforms (on a sufficiently well-behaved class of functions) results in the identity operator. (H.4) may then be expressed as O x/g ; O.R/ D Fk1 ı Fk fP .nI

(H.9)

O x/g : Fk fO.r/g D Fk fP .nI

(H.10)

so that This important result is known as the projection slice theorem (or central slice theorem) [4]. In its N -dimensional generalization, this theorem relates the .N 1/dimensional Fourier transforms of the projections of an object function onto hyperplanes to the N -dimensional Fourier transform of that original function. Simply stated, this theorem states that the .N 1/-dimensional Fourier transform of a given projection of the function is equal to a “slice” through the N -dimensional Fourier transform of that function. The Fourier integral representation of the delta function given in (H.2) involves a vector k which may be expressed in polar coordinate form as being the vector with magnitude k 2 Œ0; 1/ and with direction specified by a unit vector nO which is at an angle  2 Œ0; 2/ with respect to some fixed reference direction; one then has that Z

1 2

Z

d kD 1

Z

2

1

d 0

kdk: 0

However, it is equally valid to regard the integration variable in (H.2) as being a vector  with “magnitude” 2 .1; 1/ and direction specified by the unit vector NO which is at an angle  2 Œ0; / with respect to the reference direction; in this case one has that Z 1 Z  Z 1 2 d kD d j jd ; 1

0

1

where the absolute value of must be employed since the differential element of O one also has that  D NO . area d 2 k must always be nonnegative. Just as k D k n, There is then a one-to-one correspondence between the points in k-space and the points in -space. In addition, NO may be treated as a conventional unit vector with

822

H The Radon Transform

its range of directions (with respect to some fixed reference direction) appropriately restricted. The identity appearing in (H.3) may then be expressed in -space as Z

1 ı.R  r/ D 4 2

Z



Z

1

d

1

d 1

0

O dx j je i.RN x/ ı.x  R  NO /:

(H.11)

1

Substitution of this expression into (H.1) and using the definition of the projection of the object function onto the direction specified by the unit vector NO given in (H.5) then results in the representation of the object function as O.R/ D

Z

1 4 2

Z



1

d 1

0

n o O d j je i RN Fk P .NO I x/ :

(H.12)

This result then yields the well-known filtered backprojection algorithm for object reconstruction [5]. It is said to be “filtered” because of the quantity j j appearing in the integrand of (H.12). The inverse of the two-dimensional Radon transformation given in (H.6) may be obtained in the following manner [2]. Since jkj D k sgn.k/, where sgn.k/ D C1 if k > 0, sgn.k/ D 0 if k D 0, and sgn.k/ D 1 if k < 0 is the signum function , then the representation of the delta function given in (H.11) may be rewritten as 1 ı.R  r/ D 4 2

Z

Z



Z

1

d 1

0

1

d

O dx sgn. /e i.RN x/ ı.x  R  NO /: (H.13)

1

Since [see (B.14) of Vol. 1] Z Z f .x/ı 0 .x  a/dx D  f 0 .x/ı.x  a/dx; where the prime denotes differentiation with respect to the variable x, then with O O f 0 .x/ D sgn. /e i.RN x/ so that f .x/ D i sgn. /e i.RN x/ , the final integral in (H.13) becomes Z O sgn. /e i.RN x/ ı.x  R  NO /dx Z O D i sgn. /ı 0 .x  R  NO /e i.RN x/ dx; and hence i ı.R  r/ D  2 4

Z

Z



d 0

Z

1

1

d 1

1

O dx sgn. /ı 0 .x  R  NO /e i.RN x/ : (H.14)

References

823

With the identity [2] P

Z

Z

i f .x/ dx D  x 2

Z

1

d k sgn.k/e ikx f .x/;

dx

(H.15)

1

where P indicates that the Cauchy principal value of the integral is to be taken, (H.14) becomes 2 ı.R  r/ D  2 P 

Z

Z



1

d

dx 1

0

ı 0 .x  R  NO / : R  NO  x

(H.16)

P 0 .NO I x/ ; R  NO  x

(H.17)

Substitution of this expression in (H.1) then yields O.r/ D

1 P 2 2

Z

Z



1

d 0

dx 1

which is known as the inverse Radon transform relationship, where P 0 .NO I x/ D

Z

O.r/ı 0 .x  R  NO /d 2 r

(H.18)

D

is the derivative of the projection with respect to the variable x.

References ¨ 1. J. Radon, “Uber lineare Funktionaltransformationen und Funktionalgleichungen,” Acad. Wiss. Wien, vol. 128, pp. 1083–1121, 1919. 2. H. H. Barrett, “The Radon transform and its applications,” in Progress in Optics (E. Wolf, ed.), vol. XXI, pp. 217–286, Amsterdam: North-Holland, 1984. 3. E. C. Titchmarsh, Introduction to the Theory of Fourier Integrals. London: Oxford University Press, 1939. Ch. I. 4. R. M. Mersereau and A. V. Oppenheim, “Digital reconstruction of multidimensional signals from their projections,” Proc. IEEE, vol. 62, pp. 1319–1338, 1974. 5. A. C. Kak, “Computerized tomography with X-ray, emission, and ultrasound sources,” Proc. IEEE, vol. 67, pp. 1245–1272, 1979.

Index

Abel’s lament, 779, 780, 785 Abel, N. H., 779 absorption depth zd , 23 Airy function Ai . /, 118 American National Standards Institute (ANSI), 767 analytic delta function ı C .t /, 236 analytic field, 232 definition, 232 plane wave spectrum representation, 233 angular spectrum of plane waves representation, 4, 32 angular spectrum representation freely-propagating wavefield, 9 radiated wavefield, 32 anomalous dispersion, 267 ANSI Standard C95.1, 767 antenna pattern IEEE definition, 49 anterior pre-signal velocity, 517 associated Legendre polynomial P`m .u/, 33 asymptotic approximation, 101, 784 expansion, 784 dominant term, 101, 784 power series, 785 sequence, 783 asymptotic expansions integration by parts, 790 linear superposition of, 786 method of stationary phase, 792 method of steepest descent, 805 sense of Poincar´e, 626 asymptotic power series derivative of, 789 integration of, 788 product of, 787

attenuation coefficient ˛.!/, 469 attenuation coefficient ˛. Q !/, Q nonoscillatory waves, 693

backprojection, 86 bandwidth energy, 163 fractional, 162 basal metabolic rate (BMR), 767 Bertilone, D. C., 14, 17 Bessel functions J . /, 109 Born, M. and Wolf, E., 150, 180 branch points Lorentz model double resonance, 280 single resonance, 265 Rocard–Powles–Debye model, 294 Brewster’s angle, 742 Brillouin precursor, 211, 384–386, 656 algebraic decay, 596 asymptotic approximation, 420, 428 asymptotic expansion, 418, 424, 425 instantaneous oscillation frequency, 437 uniform asymptotic approximation, 433, 436 Brillouin precursor, Debye-type dielectric asymptotic approximation, 446 effective oscillation frequency, 448, 595 temporal width, 448, 592 Brillouin precursor, dispersive transmission line, 572 Brillouin precursor, Drude model conductor, 562 Brillouin precursor, gaussian, 624 Brillouin pulse, 639, 751 coded sequence of, 715 Brillouin, L., 104

825

826 Cauchy-Riemann conditions, 805 causality, 155 relativistic, 391 centroid group delay Gr , 644 centrovelocity, Poynting vector average, 643 instantaneous, 650 chemical reaction rate changes, 769 Clausius-Mossotti relation, 179 cold plasma transient response Dvorak-Dudley representation, 766 compact temporal support, 174 complementary error function erfc. /, 131 complementary incomplete Lipschitz-Hankel integral J en .a; /, 765 complex analytic signal, 181 complex envelope, 184 complex half-range function, 181 complex index of refraction, 6 complex intrinsic impedance .!/, 38 complex permittivity, 5 complex permittivity c .!/, 29 complex phase function retarded, 161 complex phase function .!;  /, 153, 251 complex phase function, modified, 626 complex wavenumber, 6 conductivity electric .!/, 29, 307 response function, 28 static 0 , 307 configuration domain, 231 constitutive relations simple dispersive medium, 28 convolution temporal, 87 cophasal surfaces, 733 Cornu spiral, 133 counter-stealth, 714 critical ! values !MB , 521 !SM , 522 !SB , 513 !SB , 341 !co , 309, 523 critical  values 0 , 255, 277, 285, 301, 304, 327, 366 1 , 256, 258, 277, 285, 327, 330, 356 1 , 302, 367 c , 510 m , 513 s , 509 0eff , 597 BM , 292

Index MB , 292 SM , 292 SB , 278, 316, 339, 341 N0 , 361 N1 , 286, 361 critical angle c , 740 cut-off frequency !co , 309, 523 Dawson’s integral, 132, 137 Debye model dielectric asymptotic field behavior in, 385 effective, transmission line, 570 Debye, P., 805 deformable contour of integration, 104 delta function pulse, 160, 454 Brillouin precursor, 439 Sommerfeld precursor, 411 detectability of radar, 714 detection of relocatable targets, 714 Devaney, A. J., 17, 28 Devaney-Wolf representation, 31 dielectric Debye-type, 256, 258, 261 Lorentz-type, 256, 261, 262 permittivity .!/, 29 response function, 28 transition-type, 256, 259 dielectric permittivity mth-order moment, 191 fractional mixing model, 756 relative, 253 response function, 191 dispersion anomalous, 267 normal, 267 dispersion relation, 240, 253 linear, 195 dispersion surface, 240 dispersive wave equation nonlinear, 223 distant saddle points, 262, 277, 285 first approximation, 319, 354 second approximation, 324, 354, 368 dominant saddle point, 337 double exponential pulse, 163, 761 Drude model metal, 306 Dvorak-Dudley representation, 766 effective 0 value, 597 effective oscillation frequency single cycle pulse, 595 Einstein, A. special theory of relativity, 672, 682

Index electric energy density, 717 electric susceptibility e .!/, 179 electromagnetic beam field separable, 70 electromagnetic bullets, 81 electropermeabilization, 769 electroporation, 769, 770 energy bandwidth, 163 energy transport velocity, 289, 530, 685 energy velocity nonoscillatory waves, 688 energy velocity description nonuniform model, 689 uniform model, 696, 699 envelope function, 157 equation of continuity, 28 Erd´elyi, A., 34 error function erf. /, 131 Euler’s constant , 777 evolved heat Q, 719 exciton precursor, 666 Birman and Frankel, 666 experimental measurements Aaviksoo, Kuhl, and Ploog, 666 Avenel, Rouff, Varoquaux and Williams, 662 ¨ Choi and Osterberg, 666 D. D. Stancil, 661 Falcon, Laroche, and Fauve, 662 Jeong, Dawes, and Gauthier, 669 Pleshko and Pal´ocz, 656 exponential integral E1 .x/, 777

Felsen, L. B., 239, 244, 679 filtered backprojection algorithm, 822 first forerunner Brillouin’s result, 416 first precursor, 384, 385 asymptotic expansion, 395 Fourier conjugate variables, 821 Fourier integral asymptotic approximation, 794 Fourier transform, 821 fractional bandwidth, 162 fractional mixing model, 756 Fresnel approximation, 18 Fresnel coefficients p-polarization, 740 s-polarization, 739 Fresnel equations generalized, 735 Fresnel integral cosine C ./, 133

827 rational approximations, 138 sine S ./, 133 Fresnel parameter complex FQ , 197 real F , 199 Fresnel reflection matrix, 736 Fresnel transmission matrix, 736 Fresnel-Kirchhoff diffraction integral, 18

G. G. Stokes, 792 Gamma function  .a/, 790 asymptotic approximation, 805 gaussian beam, 75 beam waist, 76 divergence angle, 77 Rayleigh range, 77 spot size, 76 gaussian envelope function, 177 gaussian envelope pulse dispersion length `D , 200 group velocity, 633 gaussian pulse propagation asymptotic description, 623 experimental results, 636 group velocity approximation, 199 proper group velocity description, 636 pulse separation, 625 scaling law, 626 transition to the group velocity description, 630 geometrical optics limit, 74 Gitterman, E. and M., 728 Goos-H¨anchen shift, 748 group delay, 528 group method Havelock, T. H., 148 group velocity, 147, 528 complex, 193, 195 gaussian envelope pulse, 633 Hamilton, Sir W. R., 147 nonoscillatory waves, 688 Rayleigh, Lord, 147 real, 199 Stokes, G. G., 147 group velocity approximation Eckart, C., 149 Lighthill, M. J., 149 Whitham, G. B., 149 group velocity dispersion, 529 group velocity dispersion (GVD), 194, 195, 199 group velocity method, 149 extended, 702

828 health and safety issues, 621 heat density evolved, 719 net, 723 Heaviside step function signal Brillouin precursor, 442 Sommerfeld precursor, 412 steady-state behavior, 540 Heaviside unit step function, 161 Heaviside-Poynting theorem, 717 Helmholtz equation, 4, 159 homogeneous, isotropic, locally linear (HILL) temporally dispersive media constitutive relations, 1 Huygens-Fresnel principle, 18 hyperbolic tangent envelope function, 169 hyperbolic tangent envelope spectrum, 173

IEEE/ANSI safety standards, 642 immature dispersion regime, 680 impulse radar, 713 impulse response, 161, 454, 682 impulse response function spatial, 12 incomplete gamma function asymptotic expansion, 791 normalized .a; x/, 790 incomplete Gamma function  .a; x/, 791 asymptotic expansion, 791 incomplete Lipschitz–Hankel integral Jen .a; /, 764 inhomogeneous wave, 733 instantaneous oscillation frequency Brillouin precursor, 437 Sommerfeld precursor, 407 intermediate distortion domain, 574 intrinsic impedance complex .!/, 43 of free space 0 , 80 inverse problem material identification, 602 inverse source problem nonuniqueness, 85 normal solution, 86 time-dependent, 79 isodiffracting wave field, 233 isotimic contour, 274

Index Laplace’s method, 800 Legendre polynomials P` .u/, 33 Rodriques’ formula, 33 linear dispersion approximation, 195, 529 Lommel functions, 14 Lord Kelvin, 792 Lorentz model dielectric asymptotic field behavior in, 383 double-resonance, 279 single-resonance, 264 Lorentz–Lorenz formula, 179 Low Probability of Intercept (LPI), 714 magnetic permeability .!/, 29 response function, 28 magnetic energy density, 717 magnetostatic waves, 661 main signal arrival, 517, 520 main signal velocity, 517, 520, 583 masking function, 83 mature dispersion regime, 460, 548, 625, 679, 680 maximal distortion domain, 574 maximum permissible exposure (MPE), 767 Maxwell’s equations Bateman–Cunningham form, 80 frequency-domain form, 29 homogeneous, 80 time-domain form, 28 mean angular frequency of a pulse, 224 mean angular resonance frequency !N s , 359 middle precursor, 211, 384 uniform asymptotic approximation, 453 middle saddle point dominance necessary condition for, 289 middle saddle points, 283, 286 first approximation, 361 mine detection, 756 minimal distortion domain, 574 modern asymptotic theory, 679 modified complex phase function ˚m .!;  0 /, 626 molecular conformation changes, 771 mother pulse, 753 multipole expansion electromagnetic wavefield, 42 scalar wavefield, 34 multipole moments, 32, 34, 42

Kramers–Kronig relations, 253 ´ 17 Lalor, E., Laplace integral, 794

near saddle point, 256, 277, 285, 366 first approximation, 327 second approximation, 329

Index near saddle points first approximation, 356, 370 second approximation, 357 neighborhood spherical, 780 net heat density W3D .z/, 723 neutralizer function .p; q/, 54 non-impulse radar, 714 nonlinear envelope equation (NEE), 228 nonlinear response, 180 nonlinear Schr¨odinger equation, 229 nonoscillatory waves, 687 nonradiating sources, 84 normal dispersion, 267

Occupational Safety and Health Administration (OSHA), 767 Olver’s saddle point method, 98 Olver’s theorem, 100 Olver-type path, 103 optical precursor observation of, 664 orbital angular momentun operator Ls , 39 order O, 780 orthogonality relations spectral amplitude vectors, 39

paraxial approximation, 76 quadratic phase dispersion, 18 phase delay, 526 phase velocity, 148 complex, 192, 195 phase velocity vp .!/, 185, 526 phase velocity approximation, 528 plane wave attenuation factor ˛.!/, 6 evanescent, 7, 11 homogeneous, 7, 11 inhomogeneous, 8 propagation factor ˇ.!/, 6 transversality relations, 5 Poincar´e, H., 784 polarization density, 718 polarizing angle, 742 pole contribution Drude model conductor, 500 Heaviside spep function signal below absorption band case, 488 Heaviside step function signal above absorption band case, 490 intra-absorption band case, 490

829 multiple resonance Lorentz model dielectric, 496, 498 nonuniform asymptotic description, 469 Rocard–Powles–Debye model dielectrics, 471 single resonance Lorentz model dielectric above absorption band case, 485 below absorption band case, 481 intra-absorption band case, 486, 487 zero frequency case, 482 uniform asymptotic description, 476 accuracy, 492 polychromatic definition, 183 posterior pre-signal velocity, 517 Poynting vector, 717 complex, 9 time-average, 10 pre-signal velocity anterior, 517, 583 posterior, 517, 583 precursor Brillouin, 656 observability of Aaviksoo, Lippmaa, and Kuhl, 666 Alfano, Birman, Ni, Alrubaiee, and Das, 667 Okawachi, Slepkov, Agha, Geraghty, and Gaeta, 669 observation of Aaviksoo, Kuhl, and Ploog, 666 ¨ Choi and Osterberg, 666 D. D. Stancil, 661 Falcon, Laroche, and Fauve, 662 Jeong, Dawes, and Gauthier, 669 Pleshko and Pal´ocz, 656 Varoquaux, Williams and Avenel, 662 Sommerfeld, 656 space–time measurement, 738 precursor field Brillouin, 211, 384–386 middle, 211, 384 Sommerfeld, 210, 384, 385 total, 504, 506, 556 prepulse, 514, 552 projection slice theorem, 821 propagation factor ˇ.!/, 470 propagation kernel, 18 pulse spreading rectangular envelope, 575 pulse synthesization, 669

830 quadratic dispersion approximation, 196, 199, 529 quadratic dispersion relation, 196 Quantum Optics Workshop on Slow and Fast Light, 670 quasi-static field, 687 quasimonochromatic, 184 definition, 182 limit, 610, 614

radiation pattern, 36, 46 filtered, 86 frequency-domain, 84 IEEE definition, 49 scalar wavefield, 37 time-domain, 82 Radon transform, 83, 819, 820 inverse, 823 raised cosine envelope signal, 617 Rayleigh range, 238 rectangle function, 163 reflection generalized law, 735 refraction generalized law, 735 relativistic causality, 391, 682 reshaping delay Rr0 , 644 residue simple pole, 127 resonance peak, 490, 507 Brillouin precursor, 536 Sommerfeld precursor, 534 Riemann, B, 805 Rocard–Powles–Debye model dielectric, 293

saddle point, 806 order, 102 saddle point dynamics, 256 saddle point equation, 255, 316 saddle point method, 98 saddle points distant, 262, 277, 285, 368 energy velocity equivalent !Ej , 684 isolated, 111 middle, 283, 286 near, 256, 277, 285, 370 scalar dipole field, 91 second forerunner Brillouin’s result, 442, 443 second precursor, 384, 386 asymptotic expansion, 418, 424, 425 self-induced transparency, 530

Index semiconducting material asymptotic field behavior in, 385 Sherman’s expansion, 73 Sherman’s recursion formula, 57 Sherman, G. C., 17 signal arrival, 511 signal contribution, 384–386 signal frequency !c , 157 signal velocity, 511 comparison of numerical and asymptotic results, 552 comparison with energy velocity, 531 main, 517, 520, 522, 583 measurement, 550, 554 prepulse, 518, 522, 583 rectangular envelope pulse, 577 sound, 661 signum function, 822 simple polarizable dielectric, 179 single-sided Fourier transform, 232 singular dispersion limit, 264, 651, 700 slowly evolving wave approach, 151 slowly varying envelope approximation, 150 slowly-evolving-wave approximation (SEWA), 230 Snell’s law, 739 soliton evolution wave equation for, 229 Sommerfeld precursor, 210, 384, 385, 656 asymptotic approximation, 398, 400 asymptotic expansion, 395 instantaneous oscillation frequency, 407 uniform asymptotic approximation, 403, 404 uniform asymptotic expansion, 401 Sommerfeld precursor, Drude model conductor, 562 Sommerfeld precursor, gaussian, 624 Sommerfeld radiation condition, 51 Sommerfeld’s Relativistic Causality Theorem, 393 source function, 83 space–time parameter retarded, 161 space–time parameter  , 154 spatiotemporal Fourier-Laplace transform, 9 special theory of relativity relativistic causality, 682 specific absorption rate (SAR), 767 spectral amplitude vectors orthogonality relations, 39 spectral distribution function, 11 spectrum domain, 231 spherical Bessel function j` . /, 33

Index

831 .C/

spherical Hankel function h` . /, 34 spherical harmonic functions Y`m .; '/, 32 Stamnes, J. J., 17 stationary phase method Kelvin, L., 148 stationary phase point, 792 interior, 52 steady-state response, 704 steepest descent path, 806 Stirling’s formula, 805 Stokes’ phenomena, 104 Stratton, J. A., 705 strictly monochromatic field, 183 subtraction of the pole technique, 127 sum rule, 254, 262 superluminal propagation Flash Gordon, 670 M. Kitano, 670 superluminal pulse propagation, 151 thermal damage localized, 771 total internal reflection, 740, 744 total precursor field, 504, 506 transient response, 704 transmission line effective dispersion model, 570 transversality relations, 731 trapezoidal envelope, 165 trapezoidal envelope function, 165 trapezoidal envelope spectrum, 167 ultrawideband definition of, 162 FCC definition, 162 UWB radar, 713 undersea radar communication feasibility using the Brillouin precursor, 605 uniform asymptotic expansion, 96

Van Bladel envelope function, 174, 638 vector potential plane wave field, 154 vector spherical harmonic functions Ym ` .˛; ˇ/, 39 velocity group, 147 phase, 148 void detection, 756

Watson’s lemma complex argument, 797 real argument, 797 wave equation reduced, 30 slowly-varying envelope approximation, 186 slowly-varying-envelope (SVE), 194 wave vector complex, 5, 730 wavefield freely-propagating, 2 nonoscillatory, 690 time-harmonic, 690 waveform monochromatic, 148 nonoscillatory, 687 polychromatic, 148 wavefront velocity, 406, 407 wavenumber complex, 5, 186, 731 weak dispersion limit, 264, 319, 342, 350, 651 absorptive equivalence relation, 655 phasal equivalence relation, 656 Weyl’s integral, 30 Whittaker’s representation, 31 Whittaker, E. T., 23 Wolf, E., 28