Electromagnetic and Optical Pulse Propagation 1: Spectral Representations in Temporally Dispersive Media

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Author: Kurt E. Oughstun

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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. Hänsch, Garching T. Kamiya, Tokyo F. Krausz, Garching B. Monemar, Linköping H. Venghaus, Berlin H. Weber, Berlin H. Weinfurter, München

125

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

Ferenc Krausz Ludwig-Maximilians-Universität München Lehrstuhl für Experimentelle Physik Am Coulombwall I 85748 Garching, Germany and Max-Planck-Institut für Quantenoptik Hans-Kopfermann-Straße I 85748 Garching, Germany E-mail: [email protected]

School of Electrical and Computer Engineering Van Leer Electrical Engineering Building Georgia Institute of Technology 777 Atlantic Drive NW Atlanta, GA 30332-0250 Email:[email protected]

Bo Monemar

Toshimitsu Asakura

Herbert Venghaus

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

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

Heinrich-Hertz-Institut für Nachrichtentechnik Berlin GmbH Einsteinufer 37 10587 Berlin, Germany E-mail: [email protected]

Theodor W. Hänsch

Horst Weber

Max-Planck-Institut für Quantenoptik Hans-Kopfermann-Strasse I 85748 Garching, Germany E-mail: [email protected]

Technische Universität Berlin Optisches Institut Straβe des 17. Juni 135 10623 Berlin, 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]

Harald Weinfurter Ludwig-Maximilians-Universität München Sektion Physik Schellingstraβe 4/III 80799 München, Germany E-mail:[email protected]

Kurt E. Oughstun

Electromagnetic and Optical Pulse Propagation 1 Spectral Representations in Temporally Dispersive Media

Kurt E. Oughstun College of Engineering & Mathematics University of Vermont Burlington, VT 05405 [email protected]

Library of Congress Control Number: 2006926454 ISBN-10: 0-387-34599-X ISBN-13: 978-0387-34599-4

e-ISBN: 0-387-30070-8

Printed on acid-free paper. © 2006 Springer Science+Business Media, LLC 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. 9 8 7 6 5 4 3 2 1 springer.com

The cautious guest who comes to the table speaks sparingly. Listens with ears learns with eyes. Such is the seeker of knowledge. H´avam´al The Sayings of the Vikings Translated by Bj¨orn J´onasson

This volume is dedicated to Professor John B. Bulman Professor George C. Sherman Professor Emil Wolf Professor Kenneth I. Golden Dr. Walter J. Fader Dr. Richard Albanese My teachers, mentors, and valued colleagues.

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, 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. Therein lies both their interest and usefulness. 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 which otherwise have rather general dispersive and absorptive properties. In Volume II, the analysis is specialized to the propagation of 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. 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. Prerequisite material includes a two-semester junioror senior-level undergraduate course sequence in electromagnetic ﬁeld theory as well as a senior-level course in complex variable theory. Challenging problems are given throughout the text. The development presented in Volume I provides a rigorous description of the fundamental time-domain electromagnetics and optics in linear temporally dispersive media. The analysis begins with a detailed review of the classical Maxwell-Lorentz theory and the invariance of the ﬁeld equations in the special theory of relativity. The macroscopic theory is then obtained from the microscopic theory through a well-deﬁned spatial averaging procedure. The analysis then continues with a general description of macroscopic electromagnetics and the role that causality plays in the constitutive (or material) relations. From the conservation laws for the electromagnetic ﬁeld,

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Preface

the concept of electromagnetic energy ﬂow in causally dispersive media is then developed. The angular spectrum of plane waves representation of the pulsed radiation ﬁeld in homogeneous, isotropic, locally linear, temporally dispersive media is derived and expressed in the form of the classical integral representations of Weyl, Sommerfeld, and Ott. The theory is then applied to the general description of pulsed electromagnetic and optical beam ﬁelds in dispersive materials, showing how the eﬀects of temporal dispersion and spatial diﬀraction are coupled. Much of the material presented here may also be found in the earlier Springer-Verlag book Electromagnetic Pulse Propagation in Causal Dielectrics that I coauthored with George Sherman. The detailed theory presented in Volume II provides the necessary mathematical and physical basis to describe and explain in explicit detail the dynamical ﬁeld evolution of a pulse as it travels through a linear temporally dispersive and absorptive medium. This is the subject of a classic theory with origins in the seminal research by Arnold Sommerfeld and Leon Brillouin in the early 1900s for a Lorentz model dielectric and described in modern textbooks on advanced electrodynamics. This classic theory has been carefully reexamined and extended by George Sherman and myself over the time period from 1974 to 1997 using modern asymptotic methods of approximation. In particular, 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. Finally, the persistent controversy regarding the question of superluminal pulse propagation in dispersive media is examined in light of recent results establishing the domain of applicability of the group velocity approximation. My research in this area began in the early 1970s when I was a graduate student at The Institute of Optics of the University of Rochester in Rochester, New York. I am grateful for the ﬁnancial support during that critical period by The Institute of Optics, the Corning Glass Works Foundation, the National Science Foundation, and the Center for Naval Research. This research continued while I was at the United Technologies Research Center, the University of Wisconsin at Madison, and ﬁnally the University of Vermont with an extended sabbatical at the Universitet i Bergen in Norway that was generously supported by the Norwegian Research Council for Sciences and the Humanities (NAVF). The critical, long-term support of this research by Dr. Arje Nachmann at the Physics and Electronics Directorate of the United States Air Force Oﬃce of Scientiﬁc Research is gratefully acknowledged. A diverse variety of textbooks has contributed to my understanding of both electromagnetic and optical ﬁeld and wave theory. Chief among these are J. D. Jackson’s Classical Electrodynamics, M. Born and E. Wolf’s Principle’s of Optics, and J. M. Stone’s Radiation and Optics. Each has had a fundamental inﬂuence on my own research. Critical discussions of speciﬁc as-

Preface

IX

pects of the research presented in this volume with Professors George Sherman, Emil Wolf, Anthony Devaney, Jakob Stamnes, Kenneth Golden, and Natalie Cartwright, and Drs. Richard Albanese and Arthur Yaghjian were indispensable in the preparation of this work.

Burlington, Vermont August 2006

Kurt Edmund Oughstun

Contents

1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 A Critical History of Previous Research . . . . . . . . . . . . . . . . . . . 4 1.3 Organization of the Book . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

2

Microscopic Electromagnetics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 2.1 The Microscopic Maxwell–Lorentz Theory . . . . . . . . . . . . . . . . . 48 2.1.1 Diﬀerential Form of the Microscopic Maxwell Equations 51 2.1.2 Integral Form of the Microscopic Maxwell Equations . . 59 2.2 Invariance of the Maxwell–Lorentz Equations . . . . . . . . . . . . . . 67 2.2.1 Transformation Laws in Special Relativity . . . . . . . . . . . 68 2.2.2 Transformation of Dynamical Quantities . . . . . . . . . . . . 75 2.2.3 Interdependence of Electric and Magnetic Fields . . . . . 81 2.2.4 Transformation Relations for Electric and Magnetic Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 2.2.5 Invariance of Maxwell’s Equations . . . . . . . . . . . . . . . . . . 85 2.3 Conservation Laws for the Microscopic Electromagnetic Field 88 2.3.1 Conservation of Energy and Poynting’s Theorem . . . . . 88 2.3.2 Conservation of Linear Momentum . . . . . . . . . . . . . . . . . 92 2.3.3 Conservation of Angular Momentum . . . . . . . . . . . . . . . . 95 2.4 Uniqueness of Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

3

Microscopic Potentials and Radiation . . . . . . . . . . . . . . . . . . . . . 3.1 The Microscopic Electromagnetic Potentials . . . . . . . . . . . . . . . 3.1.1 The Lorenz Condition and the Lorenz Gauge . . . . . . . . 3.1.2 The Coulomb Gauge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.3 The Retarded Potentials . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 The Hertz Potential and Elemental Dipole Radiation . . . . . . . . 3.2.1 The Hertz Potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.2 Radiation from an Elemental Hertzian Dipole . . . . . . . . 3.3 Li´enard–Wiechert Potentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

109 109 112 113 116 119 120 124 125

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Contents

3.3.1 The Li´enard–Wiechert Potentials . . . . . . . . . . . . . . . . . . . 3.3.2 The Field Produced by a Moving Charged Particle . . . 3.3.3 Radiated Energy from a Moving Charged Particle . . . . 3.4 The Radiation Field Produced by a General Dipole Oscillator 3.4.1 The Field Vectors Produced by a General Dipole Oscillator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.2 The Electric Dipole Approximation . . . . . . . . . . . . . . . . . 3.4.3 The Field Produced by a Monochromatic Dipole Oscillator in the Electric Dipole Approximation . . . . . . 3.5 The Complex Potential and the Scalar Optical Field . . . . . . . . 3.5.1 The Wave Equation for the Complex Potential . . . . . . . 3.5.2 Electromagnetic Energy and Momentum Densities . . . . 3.5.3 A Scalar Representation of the Optical Field . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

125 130 136 137 138 144 147 155 157 157 158 162 162

4

Macroscopic Electromagnetics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165 4.1 Correlation of Microscopic and Macroscopic Electromagnetics 165 4.1.1 Spatial Average of the Microscopic Field Equations . . . 166 4.1.2 Spatial Average of the Charge Density . . . . . . . . . . . . . . 167 4.1.3 Spatial Average of the Current Density . . . . . . . . . . . . . . 172 4.1.4 The Macroscopic Maxwell Equations . . . . . . . . . . . . . . . . 175 4.2 Constitutive Relations in Linear Electromagnetics and Optics 178 4.3 Causality and Dispersion Relations . . . . . . . . . . . . . . . . . . . . . . . 182 4.3.1 The Dielectric Permitivity . . . . . . . . . . . . . . . . . . . . . . . . . 184 4.3.2 The Electric Conductivity . . . . . . . . . . . . . . . . . . . . . . . . . 187 4.3.3 The Magnetic Permeability . . . . . . . . . . . . . . . . . . . . . . . . 189 4.4 Causal Models of the Material Dispersion . . . . . . . . . . . . . . . . . . 193 4.4.1 The Lorentz–Lorenz Relation . . . . . . . . . . . . . . . . . . . . . . 196 4.4.2 The Debye Model of Orientational Polarization . . . . . . . 197 4.4.3 Generalizations of the Debye Model . . . . . . . . . . . . . . . . . 201 4.4.4 The Classical Lorentz Model of Resonance Polarization 207 4.4.5 Composite Model of the Dielectric Permittivity . . . . . . 214 4.4.6 Composite Model of the Magnetic Permeability . . . . . . 215 4.4.7 The Drude Model of Free Electron Metals . . . . . . . . . . . 217 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220

5

Fundamental Field Equations in a Temporally Dispersive Medium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Macroscopic Field Equations in Temporally Dispersive HILL Media . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.1 Temporal Frequency Domain Representation . . . . . . . . . 5.1.2 Complex Time-Harmonic Form of the Field Quantities 5.1.3 The Harmonic Electromagnetic Plane Wave Field . . . .

221 221 223 225 227

Contents

5.2 Electromagnetic Energy and Energy Flow . . . . . . . . . . . . . . . . . 5.2.1 Poynting’s Theorem and the Conservation of Energy . . 5.2.2 The Energy Density and Evolved Heat in a Dispersive and Absorptive Medium . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.3 Complex Time-Harmonic Form of Poynting’s Theorem 5.2.4 Electromagnetic Energy in the Harmonic Plane Wave Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.5 Reversible and Irreversible Electrodynamic Processes in Temporally Dispersive Media . . . . . . . . . . . . . . . . . . . . 5.2.6 Energy Velocity of a Time-Harmonic Field in a Multiple- Resonance Lorentz Model Dielectric . . . . . . . . 5.3 Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.1 Boundary Conditions for Nonconducting Dielectric Media . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.2 Boundary Conditions for Dielectric–Conductor Interfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

7

The Angular Spectrum Representation of the Pulsed Radiation Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 The Fourier–Laplace Integral Representation of the Radiation Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Scalar and Vector Potentials for the Radiation Field . . . . . . . . 6.2.1 The Nonconducting, Nondispersive Medium Case . . . . . 6.2.2 The Spectral Lorenz Condition for Dispersive HILL Media . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 Angular Spectrum of Plane Waves Representation of the Radiation Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4 Polar Coordinate Form of the Angular Spectrum Representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4.1 Transformation to an Arbitrary Polar Axis . . . . . . . . . . 6.4.2 Weyl’s Proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4.3 Weyl’s Integral Representation . . . . . . . . . . . . . . . . . . . . . 6.4.4 Sommerfeld’s Integral Representation . . . . . . . . . . . . . . . 6.4.5 Ott’s Integral Representation . . . . . . . . . . . . . . . . . . . . . . 6.5 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

XIII

234 234 239 242 248 249 256 262 266 267 273 274 275

277 278 284 288 289 291 299 307 310 318 320 323 324 325 326

The Angular Spectrum Representation of Pulsed Electromagnetic and Optical Beam Fields in Temporally Dispersive Media . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 329

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Contents

7.1 The Angular Spectrum Representation of the Freely Propagating Electromagnetic Field . . . . . . . . . . . . . . . . . . . . . . . 7.1.1 Geometric Form of the Angular Spectrum Representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1.2 Angular Spectrum Representation and Huygen’s Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Polarization Properties of the Freely Propagating Electromagnetic Wave Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.1 The Polarization Ellipse for the Complex Field Vectors 7.2.2 Propagation Properties of the Polarization Ellipse . . . . 7.2.3 Relation Between the Electric and Magnetic Polarizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.4 The Uniformly Polarized Wave Field . . . . . . . . . . . . . . . . 7.3 Real Direction Cosine Form of the Angular Spectrum Representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4 Pulsed Electromagnetic Beam Fields and Source-Free Fields . 7.4.1 General Properties of Source-Free Wave Fields . . . . . . . 7.4.2 Separable Pulsed Beam Fields . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

A

Free Fields in Temporally Dispersive Media . . . . . . . . . . . . . . 8.1 Laplace–Fourier Representation of the Free Field . . . . . . . . . . . 8.1.1 Plane Wave Expansion of the Free Field in a Nondispersive Nonconducting Medium . . . . . . . . . . . . . . 8.1.2 Uniqueness of the Plane Wave Expansion of the Initial Value Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2 Transformation to Spherical Coordinates in k-Space . . . . . . . . 8.2.1 Plane Wave Representations and Mode Expansions . . . 8.2.2 Polar Coordinate Axis Along the Direction of Observation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3 Propagation of the Free Electromagnetic Field . . . . . . . . . . . . . 8.3.1 Initial Field Values Conﬁned Within a Sphere of Radius R . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3.2 Initial Field Values Conﬁned Inside a Closed Convex Surface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3.3 Propagation of the Free Electromagnetic Wave Field . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

329 333 341 345 346 351 355 357 361 366 367 381 384 385 387 387 391 396 399 401 403 405 406 410 413 418 418

Helmholtz’ Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 421 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 423

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XV

B

The Dirac Delta Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B.1 The One-Dimensional Dirac Delta Function . . . . . . . . . . . . . . . . B.2 The Dirac Delta Function in Higher Dimensions . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

425 425 432 435

C

The Fourier–Laplace Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . 437 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 440

D

The Eﬀective Local Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 441 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 444

E

Magnetic Field Contribution to the Classical Lorentz Model of Resonance Polarization . . . . . . . . . . . . . . . . . . . . . . . . . 445 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 448

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 449

1 Introduction

1.1 Motivation The dynamical evolution of an electromagnetic pulse as it propagates through a linear, temporally dispersive medium (such as water) or system (such as a dielectric waveguide) is a classical problem in both electromagnetics and optics. With Maxwell’s unifying theory [1, 2] of electromagnetism and optics, Lorentz’s classical model [3] of dielectric dispersion, and Einstein’s special theory of relativity [4], the stage was then set for a long-standing problem of some controversy in classical physics, engineering, and applied mathematics. If the system was nondispersive, an arbitrary plane wave pulse would propagate unaltered in shape at the phase velocity of the wave ﬁeld in the medium. In a dispersive medium, however, the pulse is modiﬁed as it propagates due to two fundamentally interconnected eﬀects. First of all, each monochromatic spectral component of the initial pulse propagates through the dispersive system with its own phase velocity so that the phasal relationship between the various spectral components of the pulse changes with the propagation distance. Secondly, but not of secondary importance, each monochromatic spectral component is absorbed with increasing propagation distance at its own rate so that the relative amplitudes between the spectral components of the pulse also change with the propagation distance. These two simple eﬀects then result in a complicated change in the dynamical structure of the propagated pulsed wave ﬁeld. The rigorous analysis of dispersive pulse propagation phenomena is further complicated by the simple fact that the phasal and absorptive parts of the system response are connected through the physical requirement of causality [5, 6]. For an initial pulse with a suﬃciently rapid rise-time, fall-time, or amplitude change within the body of the pulse, these eﬀects manifest themselves through the formation of well-deﬁned precursor ﬁelds [7–10] whose evolution has been shown [11] to be completely determined by the interrelated dispersive and absorptive properties of the system. The precursor ﬁelds are readily distinguished in the dynamical evolution of the propagated ﬁeld by the fact that the range of their oscillation frequency is typically quite diﬀerent from that of the input ﬁeld and their attenuation is typically much less than that at the carrier frequency of the input ﬁeld. The precursor ﬁelds (or forerunners, as they were originally called) were ﬁrst described by Sommerfeld [8] and Brillouin [9] in their seminal analysis

2

1 Introduction

of optical signal propagation in a locally linear, isotropic, causally dispersive dielectric medium whose frequency dispersion is described by the single resonance Lorentz model [3]. Unfortunately, their analysis errantly concluded that the amplitudes of these precursor ﬁelds were, for the most part, negligible in comparison to the main signal evolution and that the main signal arrival occurred with a sudden rise in amplitude of the ﬁeld. Because of this, it was further concluded that the signal arrival could not be given an unambiguous physical deﬁnition. These misconceptions have, quite unfortunately, settled into some of the standard literature on electromagnetic wave theory [12]. The more recent analysis [13–20] of linear dispersive pulse propagation that is based upon modern asymptotic techniques [21–27] has provided a complete, rigorous description of the dynamical ﬁeld evolution in both single and multiple resonance Lorentz model dielectrics. In particular, this analysis has shown that the precursor ﬁelds that result from an input Heaviside unit step-function modulated signal are a dominant feature of the ﬁeld evolution in the mature dispersion regime. The mature dispersion regime has been shown [28–30] to typically include all propagation distances that are greater than one absorption depth in the medium at the signal frequency of the initial ﬁeld. In addition, this modern asymptotic description [13, 14, 16] has also provided both a precise deﬁnition and physical interpretation [28] of the signal velocity in the dispersive medium. This proper description of the signal velocity is critically dependent upon the correct description and interpretation of the precursor ﬁelds. This signal velocity is shown to be bounded below by zero and above by the speed of light c in vacuum, in complete agreement with the special theory of relativity [4]. The central importance that the precursor ﬁelds hold in both the analysis and interpretation of linear dispersive pulse propagation phenomena is also realized in the study of ultrashort pulse dynamics. The asymptotic theory clearly shows that the resultant pulse distortion due to an input rectangular envelope modulated pulse is primarily due to the precursor ﬁelds that are associated with the leading and trailing edges of the input pulse envelope regardless of the (positive-deﬁnite) initial temporal pulse width [17]. The interference between these two sets of precursor ﬁelds increases with the propagation distance in the dispersive medium and naturally leads to asymmetric pulse distortion. Similar results are obtained for a trapezoidal envelope pulse (of central interest in ultrawideband radar) provided that either the initial rise-time or fall-time is faster than the characteristic relaxation time of the material. The situation is quite diﬀerent for a pulse whose initial envelope function is inﬁnitely smooth, such as that for a Gaussian envelope pulse (of central interest in ultrashort optics). In that case, the entire pulse evolves into a single set of precursor ﬁelds provided that the initial pulse width is shorter than the characteristic relaxation time of the medium [18, 31–33], reinforcing the fundamental role that the precursor ﬁelds play in dispersive pulse dynamics.

1.1 Motivation

3

The subject of electromagnetic pulse propagation in dispersive media has been and continues to be of considerable practical importance in several areas of contemporary optics and engineering electromagnetics. For example, the eﬀects of dispersion are prevalent in all ﬁber optic communication and integrated optics systems [34, 35]. As data rates continue to increase and enter the terabit rate domain, the temporal pulse widths will begin to exceed the characteristic optical relaxation times for typical ﬁber materials and the precursor ﬁelds will then dominate the ﬁeld evolution. In addition, with the current experimental capabilities of producing both ultrashort (femtosecond) optical pulses [36] and digitally coded ultrawideband microwave signals, a new technology is rapidly developing in which the importance of the causal eﬀects of dispersion is greatly magniﬁed. This graduate level text presents a uniﬁed detailed analysis of electromagnetic and optical pulse propagation in linear, causally dispersive media. The theory is an intriguing blend of electromagnetic theory, signal processing, and applied mathematics. The fundamental theory of the electromagnetic ﬁeld in a general temporally dispersive medium is developed in Volume I of this two-volume work. The theoretical development is conducted for the general situation in which the dielectric permittivity, electric conductivity, and magnetic permeability of a homogeneous isotropic medium may be frequency dependent. The rigorous angular spectrum of plane waves representation of the pulsed radiation ﬁeld (in which the spatiotemporal behavior of the current source is speciﬁed) as well as the pulsed electromagnetic beam ﬁeld (in which the spatiotemporal behavior of either the electric or magnetic ﬁeld vector is speciﬁed over a plane) are developed without approximation. The ﬁrst form of the angular spectrum representation is most appropriate for the analysis of antenna systems with a prescribed current distribution and the second form is most appropriate for optical pulse type problems in which the initial ﬁeld is known at some speciﬁed plane. The volume concludes with a detailed description of the free-ﬁeld behavior (the ﬁeld behavior after all of the sources have been turned oﬀ) in a temporally dispersive medium ﬁlling all space. The analysis presented in this ﬁnal chapter is based upon the Advanced Physical Optics lecture notes on free-ﬁelds in homogeneous, isotropic, locally linear (but nondispersive) media given by George Sherman in 1973–1974 at The Institute of Optics of the University of Rochester. Volume II presents a detailed development of the asymptotic description of plane wave pulse propagation in causally dispersive media. This volume begins with a detailed overview of the modern saddle point methods that may be used for the asymptotic evaluation of the temporal frequency contour integral that is contained in the angular spectrum representation, including uniform and transitional asymptotic techniques. A plane wave pulse is considered in order that the analysis may be focused on the temporal pulse evolution alone. Because the asymptotic theory depends upon the speciﬁc saddle point dynamics in the complex frequency plane, and because the saddle point

4

1 Introduction

dynamics depend upon the speciﬁc form of the medium dispersion relation, a particular model of the material dispersion must be chosen. The general properties of the material dispersion, obtained from the Kramers–Kronig relations [37, 38], show that there are two general types of causal models for dispersion in dielectrics [11], the Debye-type dielectric [39] for orientational polarization and the Lorentz-type dielectric [3] for resonance polarization phenomena. Because of this, the classical models of dielectric dispersion considered here are limited to the Rocard–Powles extension [40] of the Debye model and the multiple resonance Lorentz model. The frequency dispersion of the magnetic permeability may also, in certain circumstances, be described by the Rocard–Powles and Lorentz models. In addition, the classical Drude model of conductivity is considered for both pure conductors as well as in composite models for semiconducting materials. The asymptotic description developed here is then used to develop a physical interpretation of dispersive pulse dynamics in a Lorentz model dielectric that may then be extended to more general materials. This physical description of dispersive pulse propagation supplants the previous group velocity description and reduces to it in the limit as the material absorption vanishes. This analysis then establishes the space–time domain in which the group velocity approximation is valid. Finally, the question of superluminal pulse propagation is carefully addressed in view of this rigorous theory.

1.2 A Critical History of Previous Research The history of published research in the area of dispersive wave propagation is extensive and varied in both content and depth of mathematical rigor. This section provides an overview of the most important published literature on the subject. This includes papers that are concerned with such closely related topics as electromagnetic wave propagation in waveguiding systems and ﬁber optics, as well as research on nonelectromagnetic pulse propagation phenomena that may only be peripherally related to the central topic of this work. Much of this research was and continues to be motivated by the fundamental question regarding the velocity of information transfer through wave propagation. A brief description of the shortcomings of much of this previous research is presented throughout this section. Early considerations of the wave theory of light represented the optical wave ﬁeld as a coherent superposition of monochromatic scalar wave disturbances. Dispersive wave propagation was ﬁrst considered in this manner by Sir William R. Hamilton [41] in 1839 where the concept of group velocity was ﬁrst introduced. In that paper, Hamilton compared the phase and group velocities of light, stating that [41] the velocity with which such vibration spreads into those portions of the vibratory medium which were previously undisturbed, is in general diﬀerent

1.2 A Critical History of Previous Research

5

from the velocity of a passage of a given phase from one particle to another within that portion of the medium which is already fully agitated; since we have velocity of transmission of phase = s/k, but velocity of propagation of vibratory motion = ds/dk,

where s denotes the angular frequency and k the wavenumber of the disturbance in Hamilton’s notation. Subsequent to this deﬁnition, Stokes [42] 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 [43] 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, arguing that [44] Unless we can deal with phases, a simple train of waves presents no mark by which its parts can be identiﬁed. The introduction of such a mark necessarily involves a departure from the original simplicity of a single train, and we have to consider how in accordance with Fouriers theorem the new state of things is to be represented. The only case in which we can expect a simple result is when the mark is of such a character that it leaves a considerable number of consecutive waves still sensibly of the given harmonic type, though the wave-length and amplitude may vary within moderate limits at points whose distance amounts to a very large multiple of λ . . . From this we see that . . . the deviations from the simple harmonic type travel with the velocity dn/dk and not with the velocity n/k,

where n denotes the angular frequency and k the wavenumber in Rayleigh’s notation. These early considerations may best be illustrated by the coherent superposition of two time-harmonic waves with equal amplitudes and nearly equal wave numbers (k and k + δk) and angular frequencies (ω and ω + δω, respectively) travelling in the positive z-direction. The linear superposition of these two wave functions then yields the waveform [45, 46] U (z, t) = a cos (kz − ωt) + a cos ((k + δk)z − (ω + δω)t) 1 ¯ −ω ¯ t). = 2a cos (zδk − tδω) cos (kz 2

(1.1)

This superposition then produces an amplitude modulated wave with mean wavenumber k¯ = k + δk/2 and mean angular frequency ω ¯ = ω + δω/2. The surfaces of constant phase propagate with the phase velocity

6

1 Introduction

ω ¯ vp ≡ ¯ , k

(1.2)

while the surfaces of constant amplitude propagate with the group velocity vg ≡

δω . δk

(1.3)

Notice that these results are exact for the waveform given in Eq. (1.1). If the

G,P |

0

z

0

U(z,t)

G P ||

0

0

0

t = ∆t z

G P ||

0

t=0

t = 2∆t z

Fig. 1.1. Evolution of a simple wave group in a temporally dispersive medium with normal frequency dispersion.

medium is nondispersive, then k¯ = ω ¯ /c, δk = δω/c, and the phase and group velocities are equal. However, if the medium exhibits temporal dispersion so that k(ω) = (ω/c)n(ω) where n(ω) is not a constant, then the phase and group velocities will, in general, be diﬀerent. In particular, if n(ω) > 0 increases with increasing ω ≥ 0, then vp ≥ vg > 0 and the phase fronts will advance through the wave group as described by Rayleigh [43]. This elementary phenomenon is illustrated in Figure 1.1 for the simple wave group described in Eq. (1.1). 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 = 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 = 0), these two points become increasingly separated in time, as illustrated in the middle (t = δt) and bottom (t = 2δt) wave patterns, showing that the phase velocity of the wave is greater than the group velocity of the

1.2 A Critical History of Previous Research

7

envelope in this case. This phenomenon is readily observable in nature to the discerning observer. The result given in Eq. (1.1) can be generalized [46] to obtain the Fourier– Laplace integral representation [47] 1 ˜ i(k(ω)z−ωt −iψ ) dω (1.4) ie u ˜(ω − ωc )e U (z, t) = 2π C for plane wave pulse propagation in the +z-direction, where the temporal ˜ (z, ω) of U (z, t) satisﬁes the Helmholtz equation Fourier spectrum U ˜ (z, ω) = 0 ∇2 + k˜2 (ω) U (1.5) with boundary value given by the initial pulse U (0, t) = u(t) sin (ωc t + ψ) at the plane z = 0, where ψ is a phase constant and ω ˜ k(ω) ≡ n(ω) c

(1.6)

˜ describes the complex wavenumber with propagation factor β(ω) ≡ {k(ω)} ˜ and attenuation factor α(ω) ≡ {k(ω)} in the medium with complex index of refraction n(ω). Here C denotes the straight-line contour extending from ia − ∞ to ia + ∞ where a is greater than the abscissa of absolute convergence [12] for the initial envelope function u(t). This exact integral representation forms the basis for much of the research on linear dispersive wave propagation. With Maxwell’s theory [1, 2] of electromagnetic wave propagation ﬁrmly in place at the beginning of the twentieth century, the development of a theory of the dispersive properties of dielectric media was begun in terms of a classical atomistic model that culminated in Lorentz’s classical work [3]. Drude [48] indicates in a footnote that Maxwell (1869) was the ﬁrst to base the theory of anomalous dispersion upon such an atomistic model. In research independent of Maxwell’s, Sellmeier, v. Helmholtz, and Ketteler also used this model as a basis for a theory of material dispersion1 . The distinction between the signal and group velocities originated in the early research by Voigt [51, 52] and Ehrenfest [53] on elementary dispersive waves, and by Laue [54] who ﬁrst considered the problem of dispersive wave propagation in a region of anomalous dispersion where the absorption is both large and strongly dependent upon the frequency. Subsequently, the distinction between the front and signal velocities was considered by Sommerfeld [7, 8] who showed that no signal could travel faster than the vacuum speed of light c and that the signal front progressed with the velocity c in a dispersive medium, as well as by Brillouin [9, 10] who provided a detailed description of the signal evolution in a Lorentz model dielectric. In his 1907 paper, Sommerfeld [7] stated that (as translated by Brillouin [10]): 1

For a more complete discussion of this early work, see E. Mach [49] as well as the well-known undergraduate optics text by Jenkins and White [50].

8

1 Introduction It can be proven that the signal velocity is exactly equal to c, if we assume the observer to be equipped with a detector of inﬁnite sensitivity, and this is true for normal or anomalous dispersion, for isotropic or anisotropic medium, that may or may not contain conduction electrons. The signal velocity has absolutely nothing to do with the phase velocity. There is nothing, in this problem, in the way of Relativity theory.

The “signal velocity” referred to here by Sommerfeld has since become known as the front velocity, the signal velocity being described by Brillouin [9, 10] in terms of the moment of transition from the forerunner evolution to the signal evolution in the dynamical ﬁeld evolution due to an initial Heaviside step function modulated signal. Brillouin’s asymptotic analysis, based upon the then newly developed method of steepest descent due to Debye [55], provided the ﬁrst detailed description of the frequency dispersion of the signal velocity in a single resonance Lorentz model dielectric. Based upon this seminal analysis, Brillouin concluded that [9, 10] The signal velocity does not diﬀer from the group velocity, except in the region of anomalous dispersion. There the group velocity becomes greater than the velocity in vacuum if the reciprocal c/U < 1; it even becomes negative . . . Naturally, the group velocity has a meaning only so long as it agrees with the signal velocity. The negative parts of the group velocity have no physical meaning . . . The signal velocity is always less than or at most equal to the velocity of light in vacuum.

Initial Onset of the Signal U(0,t) t

0

t=0 Arrival of the Signal U(z,t) t

0

t = z/c First Forerunners

t = z/n0c

t = z/vs

Second Forerunners

Signal Evolution

Fig. 1.2. Brillouin’s description of the evolution of a step-function signal in a single resonance Lorentz medium, where n0 = n(0) and vs denotes the signal velocity value at the carrier frequency of the initial signal.

1.2 A Critical History of Previous Research

9

Sommerfeld’s now classic analysis [7, 8] was the ﬁrst to prove that the signal arrival in a dispersive medium (described by the Lorentz model) does not always propagate with the group velocity and that, even though the group velocity may exceed the vacuum speed of light c in a region of anomalous dispersion, the wave ﬁeld arrives with a positive velocity that is always less than or equal to c. In an important extension of Sommerfeld’s results, Brillouin [9] also employed a Fourier integral representation of a Heaviside unit step-function modulated plane wave signal U (0, t) = u(t) sin (ωc t), where u(t) = 0 for t < 0 and u(t) = 1 for t > 0, as it propagates through a semiinﬁnite, single resonance Lorentz medium. This exact integral representation is given by [9]; [see Eq. (1.4)] U (z, t) =

1 2π

ia+∞

ia−∞

e(z/c)φ(ω,θ) dω, ω − ωc

(1.7)

with arbitrary constant a > 0. The phase function appearing in the exponential propagation factor in the integrand of Eq. (1.7) is given by c ˜ − ωt) φ(ω, θ) = i (k(ω)z z = iω(n(ω) − θ),

(1.8)

where θ ≡ ct/z is a dimensionless space–time parameter and n(ω) =

1−

b2 2 ω − ω02 + 2iδω

1/2 (1.9)

is the complex index of refraction of the Lorentz medium with resonance frequency ω0 , plasma frequency b, and damping constant δ. Because φ(ω, θ) is complex-valued, Brillouin [9] applied the then recently developed method of steepest descent [55] in order to obtain the asymptotic behavior of the propagated signal for large propagation distances z 0. In using this asymptotic method, the original contour of integration, which extended from negative to positive inﬁnity in the upper-half of the complex ω-plane, is deformed through the appropriate saddle points ωSPj (θ), j = S, B, of φ(ω, θ) along the path of steepest descent, where φ (ωSPj , θ) = 0 with φ = dφ/dω. The dominant contribution to the asymptotic approximation of the integral representation of the ﬁeld then is due to the behavior in the immediate neighborhood of the dominant saddle point (the saddle point with the least exponential attenuation) together with the pole contribution at ω = ωc as the observation point moves oﬀ to inﬁnity with the ﬁeld. This asymptotic representation due to Brillouin may then be expressed as [13, 14] U (z, t) ∼ US (z, t) + UB (z, t) + Uc (z, t) where

(1.10)

10

1 Introduction

c 1/2 exp zc φ(ωSPj , θ) −1 Uj (z, t) = aj 2πz ωSPj − ωc (−φ (ωSPj , θ))1/2

(1.11)

for j = S, B, where aS = 2 for all θ > 1 and aB = 1 for 1 < θ < θ1 while aB = 2 for θ > θ1 . Here θ1 ≈ θ0 + 2δ 2 b2 /3θ0 ω04 with θ0 ≡ n(0), as originally described by Brillouin [9, 10]. With a ﬁrst-order approximation of the behavior of the phase function φ(ω, θ) in the complex ω-plane, the approximate behavior of the saddle point locations as a function of θ together with their relative importance was deduced, giving rise to the following succession of events illustrated in Figure 1.2. First of all, in complete agreement with the principle of relativistic causality [6], the propagated ﬁeld identically vanishes for all θ = ct/z < 1; this important result is due to the behavior of φ(ω, θ) at |ω| = ∞ and the fact that the integrand appearing in the integral representation of the propagated ﬁeld is analytic in the upper-half of the complex ω-plane. Between θ = ct/z ≥ 1 and larger values of this parameter, two sets of forerunners [US (z, t) and UB (z, t)] are present for a single resonance Lorentz medium. From a physical point of view, these forerunners arise from those Fourier components comprising the initial pulse shape whose velocities of propagation through the dispersive medium are greater than the velocity of propagation of the Fourier component at the applied signal frequency ωc . The main signal at the ﬁnite applied signal frequency ωc will then arrive at some point θ = θs during the evolution of these forerunners. From a purely mathematical point of view, this main signal arrival is dependent upon the crossing of the deformed contour of integration with the simple pole singularity at ω = ωc that appears in the integrand of the integral representation (1.5) of the propagated ﬁeld. At such a crossing, the integral may then be evaluated through use of the residue theorem with result Uc (z, t) = −e−α(ωc )z sin (β(ωc )z − ωc t).

(1.12)

As a consequence, the main signal is found to arrive with a mathematically well-deﬁned (but physically incorrect) signal velocity vs = c/θs which may, in certain special cases, coincide with the group velocity. Thus, in an involved analysis, Brillouin has presented a basic asymptotic description of pulse propagation in a Lorentz medium that is now a classic of electromagnetic theory.2 This research then established the asymptotic theory of pulse propagation in dispersive, absorptive media. An essential feature of this approach is its adherence to relativistic causality through careful treatment of the dispersive properties of both the real and imaginary parts of the complex index of refraction. At approximately the same time, Havelock [57, 58] completed his research on wave propagation in dispersive media based upon Kelvin’s stationary phase method [59]. It appears that Havelock was the ﬁrst to employ the Taylor 2

The role that this research played in Brillouin’s scientiﬁc career may be found in the biographical article by Mosseri [56].

1.2 A Critical History of Previous Research

11

series expansion of the wave number (κ in Havelock’s notation) about a given wavenumber value κ0 that the spectrum of the wave group is clustered about, referring to this approach as the group method. In addition, Havelock stated that [58] “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 κ.” This research then established the group velocity method for dispersive wave propagation. Because the method of stationary phase [60] 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 κ0 about which the Taylor series expansion is taken. In Kelvin’s method, κ0 is the stationary phase point of the wavenumber κ whereas in Havelock’s method κ0 describes the wavenumber value about which the wave group spectrum is peaked. This apparently subtle change in the value of κ0 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 [61] instead of at the vacuum speed of light c. There were then two diﬀerent approaches to the problem of dispersive pulse propagation: the asymptotic approach (based upon Debye’s method [55] of steepest descent) which provided a proper accounting of causality but was considered to be mathematically unwieldy without any simple physical interpretation, and Havelock’s group velocity approximation (based upon Havelock’s reformulation [57, 58] of Kelvin’s asymptotic method [59] of stationary phase) which violates causality but possesses a simple, physically appealing interpretation. It is interesting to note that both methods are based upon an asymptotic expansion technique but with two very diﬀerent approaches, the method of stationary phase relying upon coherent interference and the method of steepest descent relying upon attenuation. The asymptotic approach was revisited by Baerwald [62] in 1930 who reconsidered Brillouin’s original description [9] of the signal velocity in a Lorentz medium. Because of the unnecessary constraint imposed on the deformed contour of integration by the method of steepest descent in Brillouin’s analysis, the signal arrival was deﬁned to occur when the path of steepest descent moved across the simple pole singularity at ω = ωc appearing in the integral representation (1.5). This misconception resulted in a frequency dependence of the signal velocity that erroneously peaks to the vacuum speed of light c near to the medium resonance frequency ω0 and that is incomplete in its description when ω > ω0 . Baerwald [62] was the ﬁrst to show that the signal velocity is at a minimum near the resonance frequency. A comparison of the frequency dependence of the signal velocity in a single resonance Lorentz medium as ﬁrst described by Brillouin [9] and then by Baerwald [62]

12

1 Introduction

1

Brillouin's Signal Velocity

1/n0

vs/c

Baerwald's Signal Velocity 0 0

Ω

Ω

Fig. 1.3. Comparison of Brillouin’s and Baerwald’s description of the relative signal velocity vs /c in a single-resonance Lorentz medium with undamped resonance frequency ω0 , where n0 = n(0) denotes the static value of the index of refraction.

is depicted in Figure 1.3. The asymptotic description was also revisited by Stratton [12] in 1941, who reformulated the problem in terms of the Laplace transform and derived an alternate contour integral representation of the propagated signal. Stratton [12] appears to have ﬁrst referred to the forerunners described by Sommerfeld [8] and Brillouin [9] as precursors. The ﬁrst experimental measurement of the signal velocity was attempted by Shiren [63] in 1962 using pulsed microwave ultrasonic waves within a narrow absorption band. His experimental results [63] were “found to lie within theoretical limits established by calculations of Brillouin and Baerwald.” However, a more detailed analysis of these experimental results by Weber and Trizna [64] indicated that the velocity measured by Shiren was in reality that for the ﬁrst precursor and not the signal. Subsequent research by Handelsman and Bleistein [25] in 1969 provided a uniform asymptotic description of the arrival and initial evolution of the signal front. The ﬁrst experimental measurements of the precursor ﬁelds originally described by Sommerfeld [8] and Brillouin [9] were then published by Pleshko and Pal´ ocz [65]; it is apparent that they were the ﬁrst to refer to the ﬁrst and second precursors as the Sommerfeld and Brillouin precursors, respectively. Although their experiments were conducted in the microwave domain on waveguiding structures with dispersion characteristics that are similar to that described by a single resonance Lorentz model dielectric, the results established the physical propriety of the asymptotic approach.

1.2 A Critical History of Previous Research

13

The signal velocity of sound in superﬂuid 3 He − B has been measured by Avenel, Rouﬀ, Varoquaux and Williams [66, 67] for moderate material damping. Their reported experimental results are in agreement with Brillouin’s original description [9] in which the signal velocity peaks to a maximum value near the material resonance frequency (see Figure 1.3). 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 [67] by Varoquaux, Williams and Avenel in superﬂuid 3 He − B. The group velocity approximation was also reﬁned and extended during this same period, most notably by Eckart [68] 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 [69] and Lighthill [70] on the general mathematical properties of three-dimensional wave propagation and the group velocity for shipwave 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 loss-free media and systems was also established [69–73], thereby providing a physical basis for the group velocity in lossless systems with an inconclusive extension to dissipative media [74, 75]. In addition, the quasimonochromatic or slowly varying envelope approximation was precisely formulated by Born and Wolf [46] in the context of partial coherence theory. This then completed the mathematical and physical basis for the group velocity approximation. This analysis naturally led to the description of propagation in a dispersive medium by means of ray techniques which provides several alternate approaches to the asymptotic approximation of exact integral representations in such media. The ﬁrst is the direct-ray method [76–81] which is applicable to solving partial diﬀerential equations with the appropriately speciﬁed boundary or initial conditions. This is accomplished by assuming an asymptotic series for the solution which is then substituted into the partial diﬀerential equation. Families of rays are introduced along which the functional terms of the assumed series solution then satisfy ordinary diﬀerential equations, which can then be directly solved. Alternatively, there is the space–time ray theory [82–85] which employs the plotting of rays and dispersion surfaces along with the initial or boundary values of the wave-ﬁeld to demonstrate the propagation phenomena and develop the asymptotic representation. Both of these ray techniques are useful in certain applications; however, they are heuristic in origin since each requires additional unnecessary hypotheses about the nature of the solution. Moreover, the applicability of the

14

1 Introduction

results is limited because the theory does not provide error terms for the resultant asymptotic representations. Further evidence of these limitations is noted by Felsen and coworkers [86–88] who have applied space–time ray and dispersion surface techniques to the problems of propagation in dispersive media with applications to isotropic (cold) plasma media. Here Felsen noted the existence of certain transition regions wherein the ray-optic technique fails and one must resort to an exact integral representation and subsequent rigorous asymptotic analysis. This approach has recently been generalized by Heyman [89] and Felsen [90–92] and Melamed [93–96] using complex-source and complex-spectrum representations for pulsed-beam propagation in media that exhibit both spatial inhomogeneity and temporal dispersion. However, the formulation remains restricted to the case when the material attenuation is nondispersive. Consider next the integral representation techniques that have been employed in connection with the one-dimensional boundary value problem for pulse propagation in metallic waveguiding structures. The general method utilized in this type of problem, along with its limitations, is presented in §7.8 of Jackson [97]. Elliott [98], Forrer [99], and Wanselow [100] considered the problem of one-dimensional microwave pulse distortion in a dielectric ﬁlled metallic waveguide with propagation factor 1 2 2 1/2 ˜ k(ω) = ω − ωco , v

(1.13)

where v = ( µ)−1/2 with real-valued dielectric permittivity and magnetic permeability µ for the dielectric ﬁlling the waveguide, and where ωco denotes the cutoﬀ angular frequency for the particular waveguide mode under consideration. Direct application of Fourier transform techniques results in an integral representation for the pulse which is then solved by expanding the propagation factor by Taylor series into a quadratic approximation. Knop and Cohn [101] pointed out that this quadratic approximation applies only in the quasimonochromatic case and that if the initial pulse rise-time is short, this quasimonochromatic assumption no longer applies and the resulting transfer function for the system yields an output before an input is applied, violating the principle of causality. Nevertheless, approximate numerical solutions for this problem with the exact form of the propagation factor indicate that for times t > L/c (L being the propagation length of the waveguide), the approximately degraded waveforms obtained by Elliott (and corrected by Knop and Cohn) are good approximations to the actual output pulse shapes of the waveguide. In order to overcome these diﬃculties, Knop [102] derived an exact solution to the problem of the propagation of a rectangular-pulsemodulated carrier input in a simple plasma medium. By employing transfer function techniques and the Laplace transform, a closed form analytic solution for the output was obtained in the form of a series summation over odd-order Bessel functions. Case and Haskell [103] simpliﬁed the result somewhat by indicating that this series summation could be written as the sum of

1.2 A Critical History of Previous Research

15

two Lommel functions. Finally, the paper by Vogler [104] generalized these results to arbitrary waveforms in ideal (lossless) waveguides. However, these results apply only to idealized situations. A problem of related interest is the transient response as treated in the papers by Wait [105] and Haskell and Case [106]. Wait dealt with approximate methods for determining the distortion of a pulse propagating through a dispersive channel such as a metallic waveguide. Through use of the transfer function approach, the one-dimensional output pulse is obtained as a linear superposition over all frequencies of monochromatic plane waves. The stationary phase method is then applied to evaluate the resultant integral representation for a general phase function ϕ(ω) that describes the system dispersion, the propagation of a pulse in an ideal waveguide being given as an introductory example. The stationary phase method in mode theory is then developed in an application to a more realistic model wherein the group velocity has either a maximum or minimum (the existence of a maximum corresponds to the buildup of the ﬁeld just prior to the arrival of the main signal). The quasimonochromatic pulse is considered next, followed by a discussion of pulse distortion where the leading edge of the envelope of a step-modulated carrier is analyzed. The Taylor series expansion of ϕ(ω) in the asymptotic expansion clearly shows that the even-order derivatives of ϕ(ω) result in symmetrical pulse distortion whereas any asymmetrical pulse distortion is due to odd-order derivatives of ϕ(ω). Finally, a more generalized discussion of the leading edge of the propagated signal was undertaken (i.e., the transient solution in Wait’s terminology) for wave propagation in an idealized waveguide. In a reexamination of Sommerfeld’s problem [7, 8], Haskell and Case [106] presented the complete analysis of the arrival of the main signal in a lossless, isotropic plasma medium with dispersion relation 1/2 1 2 ˜ ω − ωp2 , k(ω) = c

(1.14)

where ωp denotes the plasma angular frequency. By employing the saddle point method of integration, uniform asymptotic solutions for the transient response have been obtained that are in accordance with the principle of relativistic causality. This solution can be divided into three successive regions: a region before the saddle point crosses the pole (the anterior transient), a region when the saddle point is in the neighborhood of the pole (the main signal buildup), and a region after the saddle point has crossed the pole (the posterior transient). Consequently, the one-dimensional boundary value problem of a step-function modulated carrier wave propagating in an isotropic plasma medium has been asymptotically solved for the transient response. A review of the applications of such uniform asymptotic expansion techniques for radiation and diﬀraction problems may be found in the paper by Ludwig [107]. A fundamental extension of this theory is provided by the problem of the reﬂection and transmission of a pulsed electromagnetic beam ﬁeld that is

16

1 Introduction

incident upon the planar interface separating two half-spaces containing different dispersive media. Early treatments of this problem have either focused on time-harmonic beam ﬁelds in lossless media [108, 109] with emphasis on the Goos–H¨anchen eﬀect [110] or on pulsed plane wave ﬁelds when the incident medium is vacuum and the second medium is dispersive [111–113]. The formation of a forerunner upon transmission into a plasma medium across a planar vacuum–plasma interface appears to have ﬁrst been treated by Skrotskaya et al. [111] in 1969. The asymptotic description of the formation of precursors upon transmission into a Lorentz model dielectric was then given by Gitterman and Gitterman [112] in 1976. However, each of these descriptions considers only normal plane wave incidence upon the dispersive half-space. An extension to oblique plane wave incidence upon a dispersive half-space has been given by Blaschak and Franzen [113] using numerical methods. The formulation for the general situation in which both media are dispersive and absorptive and the incident ﬁeld is a pulsed electromagnetic beam ﬁeld has been given by Marozas and Oughstun [114]. The results of this analysis have direct application in integrated and ﬁber optic device technologies, the analysis and design of low-observable surfaces, and the analysis of bioelectromagnetic eﬀects in stratiﬁed tissue. A somewhat diﬀerent application of the problem of pulse distortion in a dispersive medium is presented in the paper by Wait [115] which employs the Laplace transform to obtain an integral representation of a one-dimensional electromagnetic pulse propagating in a geological medium described by a frequency-dependent conductivity. Although no closed-form solutions are obtained and only numerical evaluations of the integral representation are presented, this paper does indicate the applicability of this theory to geological sensing through the analysis of resulting pulse distortions. With use of Laplace transform techniques, Wait [116] also considered the exact (causal) solution for the electromagnetic ﬁeld radiated by a step-function modulated dipole source in a dielectric medium whose complex dielectric permittivity has the form (ω) = K (1 + ia/ω) (1 + ibω), where K, a, and b are constants. These results then have application to pulsed dipole radiation in a simple conducting medium such as a cold plasma. A generalization of these integral representation techniques for the analytic description of pulse distortion in a linear dispersive system is given by Jones [117]. By employing the Fourier integral representation of the pulse and assuming that the system has a quadratic dispersion relation ˜ ˜ c ) + k˜ (ωc )(ω − ωc ) + 1 k˜ (ωc )(ω − ωc )2 , k(ω) = k(ω 2

(1.15)

2 ˜ ˜ and k˜ = ∂ 2 k/∂ω (which is equivalent to the quasiwhere k˜ = ∂ k/∂ω monochromatic approximation and hence yields noncausal results if the initial pulse rise-time is suﬃciently short), Jones showed that the resultant integral can be expressed as

1.2 A Critical History of Previous Research

U (z, t) ≈

1 (2π k˜ (ω

( c )) 1/2)

˜

ei[k(ωc )z−ωc t+3π/4−ψ]

˜ (ωc )z + t − t)2 ( k dt × u(t ) exp −i 2k˜ (ωc )z −∞

∞

17

(1.16)

which shows that the pulse envelope propagates at the group velocity at the input pulse carrier frequency. In particular, the pulse shape is shown to be proportional to the Fresnel transform of the input pulse shape and, after the pulse has propagated suﬃciently far through the dispersive system, the pulse shape becomes proportional to the Fourier transform of the initial pulse envelope shape. Consequently, an analogy can be drawn between the distorted pulse envelope shape and the diﬀraction pattern produced when monochromatic light of uniform intensity passes normally through an aperture in an opaque screen whose aperture function is identical to the initial pulse envelope shape. More recently published treatments concerned with the propagation of wave packets in dispersive and absorptive media [118–121] 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 [34] and nonlinear optics in general [122, 123] with little regard for its accuracy. A phase-space asymptotic description of wave propagation in homogeneous, dispersive, dissipative media has also been introduced by Hoc, Besieris and Sockell [124]. This phase-space approach uses a combined spacewavevector–time domain representation in which wave propagation is realized through the evolution of the Wigner distribution function [125]. However, the dispersive properties of the medium are approximated by an appropriate power series expansion in their analysis, thereby limiting the method to narrowband pulses. A variety of purely numerical techniques has also been developed for the depiction of ultrashort pulse dynamics in temporally dispersive media and systems. In general, there are three computational approaches to time-domain electromagnetics and optics [126]: time-domain, frequency-domain, and hybrid time- and frequency-domain methods. Because of its natural representation in the frequency domain, as exhibited in Eq. (1.4), dispersive pulse propagation phenomena are most amenable to computational methods that are based upon the Fourier–Laplace transform. Discrete Fourier transform methods have been succesfully applied to a variety of problems in dispersive

18

1 Introduction

pulse propagation, most notably by Veghte and Balanis [127] for transient signals on microstrip transmission lines, Moten, Durney and Stockham [128], Albanese, Penn and Medina [129], and Blaschak and Franzen [113] for trapezoidal envelope pulses in Debye and Lorentz model dielectrics with application to bioelectromagnetics [130]. However, because the discrete Fourier transform can only be numerically computed (e.g., using the FFT algorithm with an electronic computer with ﬁnite memory) over a ﬁnite frequency domain [−ωmax , ωmax ], the complete evolution of the Sommerfeld precursor can never be adequately described using this numerical approach alone. For this purpose, an eﬃcient Laplace transform algorithm was developed by Hosono [131–133] and later updated by Wyns, Foty and Oughstun [134]. Comparison of numerical results using Hosono’s Laplace transform algorithm with the uniform asymptotic description of the Sommerfeld precursor due to Handelsman and Bleistein [25] established the accuracy of the modern asymptotic theory for the propagated signal front [134]. A hybrid asymptotic–FFT algorithm has also been developed [135] where the high-frequency structure |ω| > ωmax of the propagated ﬁeld is determined using the uniform asymptotic description. A similar approach has been described by Ziolkowski, Dudley and Dvorak [136] who have introduced an extraction technique that dramatically reduces the required number of sample points for an FFT simulation of pulse propagation through lossy plasma media. The extracted term contains the high-frequency information of the propagated pulse which may then be evaluated analytically. An alternate approach to the numerical inversion of the Laplace transform using the Dubner–Abate algorithm [137] with application to dispersive signal propagation in a Lorentz model dielectric has been given by Barakat [138]. However, the numerical results presented here do not resolve the high-frequency structure that is present at the onset of the Sommerfeld precursor. Time-domain methods, on the other hand, involve the solution of a set of integro-diﬀerential equations that result from the diﬀerential form of Maxwell’s equations taken together with the appropriate constitutive relation for the material dispersion. This is most eﬃciently accomplished with the ﬁnite-diﬀerence time-domain (FTDT) method [139–141] which discretizes space and time into discrete cells with dimensions (∆x, ∆y, ∆z, ∆t). The computational stability of the FDTD method is set by the Courant condition [142] which is based on the minimum time ∆t = ∆z/c required to propagate one cell length ∆z, for example, in the z-direction. The FDTD grid is then causally connected when the Courant condition is satisﬁed. The cell size is determined by the smallest sampled wavelength λmin required by the physical problem, which is related to the maximum angular frequency ωmax in the corresponding frequency-domain representation, where λmin = 2πc/ωmax . Although this method can be used to model very complex propagation geometries, it suﬀers from numerical dispersion [142, 143] that is due to the nonzero space and time grid spacing, which results in a frequency-dependent

1.2 A Critical History of Previous Research

19

phase error. Joseph, Hagness and Taﬂove [140] used the FTDT method to numerically calculate the high-frequency leading edge of the Sommerfeld precursor in a Lorentz model dielectric, comparing it with uniform asymptotic results. When approximate saddle point locations were used in the uniform asymptotic description, they found a maximum error of approximately 10 percent. However, when precise numerical saddle point locations are used in the uniform asymptotic description, the quantitative agreement between the two results is exceptionally good with the rms error between them decreasing with increasing propagation distance.

Ω Ωmax

Ω t

Τ

t

Fig. 1.4. Information diagram partitioned into rectangular Gabor cells of dimension ∆t × ∆ω for a physically realizable function of time measured over a ﬁnite time interval τ by an instrument with maximum angular frequency response ωmax , where ∆ω denotes the indeterminacy in the angular frequency measured in the time interval ∆t.

Hybrid time- and frequency-domain methods are derived from the fact that the angular frequency ω and time t for a wave are conjugate physical variables that satisfy the indeterminacy principle ∆ω∆t ≥

1 , 2

(1.17)

where ∆ω denotes the indeterminacy in the angular frequency measured in the time interval ∆t. Any physically realizable function of time f (t) can only be known over a ﬁnite time interval of duration τ , whereas its frequency structure can only be known up to some maximum angular frequency value ωmax . This function can then be described in terms of the conjugate physical variables (t, ω) in a rectangular region with sides τ and ωmax , respectively, known as an information diagram, illustrated in Figure 1.4. The principle of

20

1 Introduction

indeterminacy then describes elementary cells in the (t, ω) plane with area Sel = ∆ω∆t ∼ 1/2, where an elementary cell’s particular shape is arbitrary. This information diagram can be divided into square elementary cells, called Gabor cells [144], with sides ∆t = 1s and ∆ω = 1r/s. The function f (t) can then be described by its values fj that are speciﬁed in each cell labeled by the index j. Because the values fj are complex-valued (e.g., they have both amplitude and phase) with some speciﬁed reference value, the total number of independent parameters which completely describe the physical process recorded by an instrument with spectral resolving power ωmax in a ﬁnite observation time τ is then given by Nmax = 2τ ωmax + 1. Unit area elementary cells can be formed by dividing the information rectangle into horizontal strips of length τ and width 1/τ , resulting in the Fourier series expansion N max

f (t) =

cj eijω0 t

(1.18)

j=−Nmax

with ω0 ≡ 1/τ . This then corresponds to the temporal frequency domain representation of an optical pulse. Unit area elementary cells can also be formed by dividing the information rectangle into vertical strips of width nikov sampling theτ /Nmax and length ωmax . The Whittaker–Shannon–Kotel´ orem [145–147] then states that the function f (t) is completely deﬁned over the time interval t ∈ [0, τ ] by the set of sampled values fj ≡ f (jτ /Nmax ) if Nmax = 2τ ωmax + 1. This then corresponds to the time-domain representation of an optical pulse. The signal that occupies the minimum area in the information diagram (i.e., a single Gabor cell) is the Gaussian pulse 2

fel (t) ≡ e−(κ

/2)(t−t0 )2 i(ωo t+φ0 )

e

(1.19)

centered at the time t0 with angular frequency ω0 = 1/τ , where κ describes the signal duration and φ0 is a phase constant. The Fourier transform of this elementary signal is √ 2 2 (1.20) f˜el = ( 2π/κ)e−(ω−ω0 ) /(2κ ) ei((ω−ω0 )t0 −φ0 ) . √ The rms √ widths of these two functions are given by ∆t = 1/( 2κ) and ∆ω = κ/ 2, respectively, so that ∆t∆ω = 1/2, which is the minimum allowed by the indeterminacy principle. A given temporal signal can then be decomposed in terms of elementary signals of the form given in Eq. (1.19) in a manner depending upon the value of the parameter κ. In the limit as κ → 0, the elementary signal becomes a pure sinusoid and the decomposition is just the Fourier series expansion of the signal (i.e., a temporal frequency-domain representation is obtained). In the opposite limit as κ → ∞, the elementary signal approaches a Dirac delta function and the decomposition yields the sampling method (i.e., a time-domain representation is obtained). The general series decomposition in terms of the elementary functions fel (t) results in

1.2 A Critical History of Previous Research

21

a hybrid time-frequency-domain representation, which may be ideally suited for speciﬁc problems. One promising approach is the application of wavelets to electromagnetics [148]. Numerical methods have had immediate application to inverse problems in time-domain electromagnetics and optics. Beezley and Krueger [149] used a time-domain technique to derive a nonlinear integro-diﬀerential equation that relates the complex permittivity (through its susceptibility kernel) to the reﬂection operator for a one-dimensional homogeneous slab of an unknown material. The susceptibility kernel can then be determined from measured reﬂection data. An extension of this technique to stratiﬁed media was then given by Kristensson and Krueger [150, 151]. A Green’s function technique developed by Krueger and Ochs [152] for nondispersive inhomogeneous media, which is related to the invariant embedding technique developed by Corones, Davison, Ammicht, Kristensson and Krueger [153–157], has also been applied to both direct- and inverse-scattering problems. The generaliztion of this method by Karlsson, Otterheim, and Stewart [158] has general applicability to one-dimensional dispersive media. An equally important problem in dispersive wave theory, which has also received considerable attention in the open literature, concerns the propagation velocities of light in dispersive and absorptive media. The description of the velocity of energy transport through a causally dispersive medium, originally considered by Brillouin [9] in 1914 and again [159] in 1932, was reinvestigated by Schulz-DuBois [160] in 1969, Askne and Lind [161] and Loudon [162] in 1970, as well as by Anderson and Askne [163] in 1972. General conservation properties for waves in dispersive, dissipative media were also considered by Censor and Brandstatter [164]. By determining the total energy density associated with a time-harmonic, plane wave electromagnetic ﬁeld in the dispersive medium as the sum of the energy density of the wave plus the energy stored in the medium, Loudon [162] arrived at a simple, closed form expression for the energy transport velocity for propagation of a monochromatic wave ﬁeld in a classical single resonance Lorentz medium, given by c , (1.21) vE (ω) = nr (ω) + ni (ω)/δ which is relativistically causal (i.e., is both positive and less than or equal to the vacuum speed of light c for all real angular frequencies ω ≥ 0). This description showed that the energy and group velocities are diﬀerent in a region of anomalous dispersion in causally dispersive dielectrics whereas the energy velocity approaches the group velocity in regions of normal dispersion as the material loss becomes vanishingly small. A generalization of Loudon’s energy velocity to a multiple resonance Lorentz model dielectric was then given by Oughstun and Shen [165] in 1988. The direct generalization of this result to an arbitrary dispersive medium has been shown by Barash and Ginzburg [166] to require a speciﬁc microscopic model of the material dispersion. Neverthe-

22

1 Introduction

less, a general formulation for a material with arbitrary dispersive properties has been proposed [167], but remains to be fully demonstrated. The 1970 review paper by Smith [168] provides a general discussion of the deﬁnitions, physical signiﬁcances, interrelationships, and observabilities of seven diﬀerent velocities of light with an outlook to determine the proper velocity to be used in describing dispersive pulse propagation. Smith states here that “there is no observable physical quantity associated with the phase of a light wave,” and that “a phase velocity cannot be attributed to a wave packet or to any wave except a monochromatic wave.” As a consequence, the concept of the phase velocity is useful only in determining the phase of a monochromatic wave in space and time given the phase at some other position and time. Nevertheless, this elementary concept is indeed an essential ingredient in the mathematical description of dispersive pulse dynamics, as is clearly evident in the Fourier–Laplace integral representation given in Eq. (1.4). Smith also showed that the standard deﬁnition of the group velocity fails to describe the motion of the peak in the envelope of an arbitrary pulse when the pulse frequency is within a region of anomalous dispersion. In order to correct this, a generalized group velocity deﬁned as the velocity of motion of the temporal center of gravity of the amplitude of the wave packet has been proposed. The velocity of energy transport, deﬁned as the ratio of the timeaverage Poynting vector to the time-average total electromagnetic energy density, is incorrectly criticized as not corresponding to any real observable physical quantity, as is the signal velocity introduced by Sommerfeld [7, 8] and Brillouin [9] and reﬁned by Baerwald [62] as well as by Trizna and Weber [169, 170]. For completeness, Smith also brieﬂy considered the relativistic velocity constant and the ratio of units velocity appearing in Maxwell’s equations. Finally, Smith introduced a new deﬁnition for the velocity of light, called the centrovelocity, which is deﬁned as the velocity of motion of the temporal center of gravity of the intensity of the pulse. It is important to note here that both the generalized deﬁnition of the group velocity and the centrovelocity are undeﬁned for a step-function modulated signal. An “eighth” velocity of light has also been proposed by Bloch [171] that is based upon the crosscorrelation of the original and received pulses. Of further interest in this matter is the large number of experimental papers that report measurements of superluminal pulse velocities in apparent violation of the special theory of relativity. Early measurements by Basov et al. [172] and Faxvog et al. [173] report laboratory measurements of superluminal group velocity values in inverted media. Based upon the rate equations for resonant radiative energy transfer in a two-level active medium, Basov et al. [172] point out that as soon as the energy density in the leading edge of the pulse reaches a suﬃcient level, all of the active particles will produce stimulated emission into that leading edge, while the trailing edge of the pulse travels through the medium with either a much lower ampliﬁcation or even attenuation. As a result, the peak of the pulse will undergo an additional

1.2 A Critical History of Previous Research

23

shift forward, thereby resulting in an eﬀective peak velocity that exceeds c. However, this does not contradict the relativistic principle of causality since such a motion of the peak amplitude point of the pulse occurs due to the deformation of the initially weak leading edge of nonzero intensity. That is, only the pulse shape has propagated with a velocity exceeding c and not the energy of the pulse. Later experimental results by Chu and Wong [174] in an absorbing medium are similarly ﬂawed [175] because they do not adequately consider the eﬀects of pulse distortion. The proper (i.e., physically correct) description of these experimental results must then be obtained through a careful analysis of Gaussian pulse dynamics in a dispersive, absorptive medium. Early research by Garrett and McCumber [176] in 1970 on Gaussian pulse propagation in an anomalous dispersion medium employed the complex index of refraction n(ω) = n∞ −

ω0 ωp , ω(ω − ω0 + iγ)

(1.22)

which is valid in the weak dispersion limit when |ωp /γ| n∞ . Here ω0 denotes the resonance frequency, ωp the plasma frequency, and γ describes the atomic linewidth of the medium whose refractive index has the large frequency limit n∞ ≥ 1. Their analysis showed that when a Gaussian pulse is incident upon a thin slab of such a weakly dispersive material, “the power spectrum of the emerging pulse is still substantially Gaussian, and the peak of the pulse emerges at the instant given by the classical group velocity expression, even if that instant is earlier than the instant at which the peak of the input pulse entered the slab.” However, as the slab thickness increases, the pulse spectrum becomes suﬃciently distorted such that the classical group velocity approximation is no longer applicable because [176] “the concept of a dominant central frequency ω ¯ eventually fails.” A more detailed analysis of Gaussian pulse dynamics in a Lorentz model dielectric that is not restricted to the weak dispersion limit was then given by Tanaka, Fujiwara, and Ikegami [177] in 1986. They found 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 become inﬁnite in the same range. 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 Fourier-component 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.

The uniform asymptotic description of Gaussian pulse dynamics in a Lorentz model dielectric was then given by Balictsis and Oughstun [18, 31, 33]. This asymptotic description clearly shows that a Gaussian pulse evolves into a set of precursor ﬁelds that are a characteristic of the dispersive properties

24

1 Introduction

of the medium. In a single-resonance Lorentz model dielectric, an ultrashort Gaussian pulse will then evolve into a pair of pulse components that travel at diﬀerent velocities. The ﬁrst component is a generalized Sommerfeld precursor whose peak amplitude travels near the vacuum speed of light c with near zero attenuation and dominates the total propagated ﬁeld when the input pulse carrier angular frequency ωc is well above the medium resonance frequency ωc . The second component is a generalized Brillouin precursor whose peak amplitude travels near the zero frequency velocity √ c/n(0) and only attenuates algebraically with propagation distance as 1/ z and dominates the total propagated ﬁeld when ωc is either below or near resonance. Furthermore, the peak amplitude point in each pulse component is found [32] to evolve with the propagation distance z ≥ 0 in such a manner that the pulse dynamics evolve from the classical group velocity description to the energy velocity description [178, 179] as the propagation distance increases into the mature dispersion regime.3 The energy velocity description [178, 179] due to Sherman and Oughstun provided the ﬁrst detailed physical explanation of ultrashort pulse dynamics in dispersive, attenuative system. The description is based upon the energy velocity and attenuation of time-harmonic waves in the dispersive system and is derived from the modern asymptotic theory of dispersive pulse propagation [13–18]. This new description reduces to the group velocity description in dispersive systems that are lossless. In particular, Sherman and Oughstun have shown that [178], in the mature dispersion regime, “the ﬁeld is dominated by a single real frequency at each space–time point, That frequency ωE is the frequency of the time-harmonic wave with the least attenuation that has energy velocity equal to z/t.” The modern asymptotic description of dispersive pulse propagation has provided the ﬁrst accurate, analytical description of ultrawideband pulse dynamics in causal systems. This description is based upon Olver’s saddle point method [21] together with those uniform asymptotic techniques [22–25] that are necessary to obtain a continuous evolution of the ﬁeld at some ﬁxed observation distance that is within the mature dispersion regime. Similar asymptotic results were also published at the same time by Vasilev, Kelbert, Sazonov, and Chaban [180] using saddle point techniques. This analysis has shown that the precursor ﬁelds that result from an input Heaviside unit step-function modulated signal are a dominant feature of the ﬁeld evolution in the mature dispersion regime, as illustrated in Figure 1.5 (compare with Fig. 1.2). The appearance of these precursor ﬁelds is critically dependent upon the rise-time of the initial pulse which must be faster than the characteristic relaxation time of the medium [19]. In addition, this modern asymptotic description has also provided both a precise deﬁnition [13, 14, 16] and physical 3

The mature dispersion regime typically includes all propagation distances that are greater than one absorption depth in the medium at some characteristic oscillation frequency of the initial pulse.

1.2 A Critical History of Previous Research

U(0,t)

25

Initial Onset of the Signal t

0

t=0

U(z,t)

Arrival of the Signal

0

t t = z/c t = z/n0c t = z/vs Signal First Second Evolution Precursor Precursor

Fig. 1.5. Modern asymptotic description of the evolution of a step-function signal in a single-resonance Lorentz medium, where n0 = n(0) and vs denotes the signal velocity value at the frequency of the initial signal.

interpretation [28] of the signal velocity in the dispersive medium. This proper description of the signal velocity is critically dependent upon the correct description and interpretation of the precursor ﬁelds. Furthermore, this signal velocity is shown [14] to be bounded below by zero and above by Loudon’s energy transport velocity [162] for monochromatic waves, as illustrated in Figure 1.6 (compare with Fig. 1.3). The analysis of an ultrawideband, double-exponential electromagnetic pulse that is propagating through either a hollow metallic waveguide or a cold plasma medium has been described in detail by Dvorak and Dudley [181]. The dispersion relation is then given either by Eq. (1.13) or (1.14), respectively. Closed-form solutions are obtained in terms of incomplete Lipschitz–Hankel integrals which provide an accurate description of the leading edge of the Sommerfeld precursor that is characteristic of this dispersion relation. The central importance of the precursor ﬁelds in both the analysis and interpretation of linear dispersive pulse propagation phenomena is also realized in the study of ultrashort pulse dynamics. The asymptotic theory clearly shows that the resultant pulse distortion due to an input rectangular envelope modulated pulse is primarily due to the precursor ﬁelds that are associated with the leading and trailing edges of the input pulse envelope regardless of the (positive-deﬁnite) initial temporal pulse width [17]. The interference between these two sets of precursor ﬁelds increases with the propagation distance in the dispersive medium and naturally leads to asymmetric pulse distortion. The situation is quite diﬀerent for a pulse whose initial envelope function is inﬁnitely smooth, such as that for a Gaussian envelope pulse. In that case, the entire pulse evolves into a single set of precursor ﬁelds

26

1 Introduction

1

1/n0

Energy Transport Velocity

v/c

0 0

Ω

Signal Velocity

Ω

Fig. 1.6. Angular frequency dependence of the relative energy transport velocity for a monochromatic signal and the relative signal velocity of a Heaviside step function signal in a single-resonance Lorentz model dielectric with resonance angular frequency ω0 .

provided that the initial pulse width is shorter than the characteristic relaxation time of the medium [18, 31–33], reinforcing the fundamental role that the precursor ﬁelds play in dispersive pulse dynamics. Optical precursors have also been shown by Gagnon [182] to be interpreted in terms of selfsimilar solutions of speciﬁc evolution equations. Predicted new precursors in materials exhibiting spatial dispersion have also been described by Birman and Frankel [183, 184], the eﬀects of spatial dispersion increasing the signal velocity of the pulse. A Sommerfeld “precursor-like” eﬀect has also been described [185] for pulse diﬀraction in vacuum. The diﬀraction of electromagnetic pulses in a Lorentz model dielectric has been described by Solhaug, Stamnes, and Oughstun [186]. It is shown there that the edge diﬀraction process is itself dispersive and adds to the eﬀects of material dispersion. The combined eﬀects of edge diﬀraction and temporal dispersion then result in a singularity in the Brillouin precursor evolution that is due to the geometric focusing of the quasistatic portion of the transient ﬁeld component. Recently published research [29, 30] by Xiao and Oughstun has identiﬁed the space–time domain within which the group velocity approximation is valid. 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 ωc of the initial pulse at which the temporal pulse spectrum is peaked, as originally described by Havelock [57, 58]. The slowly

1.2 A Critical History of Previous Research

27

varying envelope approximation is a hybrid time and frequency domain representation [123] 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 ωc . The envelope function is assumed to be slowly varying on the time scale ∆tc ∼ 1/ωc , which is equivalent [187] to the quasimonochromatic assumption that its spectral bandwidth ∆ω is suﬃciently narrow that the inequality ∆ω/ωc 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 ωc with the unfounded assumption [118, 119, 123] that improved accuracy can always be obtained through the inclusion of higher-order terms. This assumption has been proven incorrect [29, 30] in the ultrashort pulse, ultrawideband signal regime, optimal results being obtained using either the quadratic or the cubic dispersion approximation of the wavenumber. 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 [60] 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 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 → ∞. 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 ωc . For example, zc = ∞ 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 ωc of the input pulse. In an attempt to overcome these critical diﬃculties, Brabec and Krausz [188] 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 [61]. Second, the subsequent imposed Taylor series expan˜ sion of the complex wavenumber k(ω) about ωc approximates the material dispersion by its local behavior about some characteristic angular frequency of the initial pulse. Because this approximation is incapable of correctly de-

28

1 Introduction

scribing the precursor ﬁelds, it is then incapable of correctly describing the dynamical evolution of any ultrashort pulse and its accuracy monotonically decreases [29] 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 [189–196] 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 [190] has argued for more careful analysis of experimental measurements reporting superluminal motions. Diener [191] 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 [7, 8] that the signal arrival cannot exceed c in a causally dispersive medium. Kuzmich, Dogariu, Wang, Milonni, and Chiao [194] 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 information.” In contrast, Nimtz and Haibel [195] 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 frequencyband width. But, amazingly enough, the time span between the cause and eﬀect is reduced by a superluminal signal velocity compared with the time span in the case of light propagation from cause to eﬀect.

In addition, Winful [196] 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. 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 [197, 198], thereby rendering the accuracy of this superluminal tunneling analysis as questionable at best. It is indeed clear that a more physically meaningful pulse velocity measure needs to be considered in order to accurately describe the complicated pulse evolution that occurs in ultrashort dispersive pulse dynamics when the initial

1.3 Organization of the Book

29

pulse envelope is continuous (e.g., Gaussian). One possible velocity measure considers the packet velocity that is described by the temporal moments of the pulse [199]. A related measure would track the temporal centroid of the Poynting vector of the pulse. This pulse centroid velocity of the Poynting vector was ﬁrst introduced by Lisak [200] in 1976. Recent descriptions of its properties by Peatross, Glasgow, and Ware [201, 202] and Cartwright and Oughstun [203, 204] have established its eﬃcacy in describing the evolution of the pulse velocity with propagation distance in a single-resonance Lorentz model dielectric. In particular, it has been shown [204] that the instantaneous centroid velocity of the pulse Poynting vector approaches the classical group velocity in the limit as the propagation distance approaches zero from above when the input pulse carrier frequency lies within the normal dispersion region either above or below the medium absorption band, but in general not in the region of anomalous dispersion. However, the classical group velocity is always obtained at any ﬁxed propagation distance in the limit as the initial pulse width is allowed to increase indeﬁnitely. This limiting behavior reinforces the previously established [29, 30] region of applicability of the group velocity approximation. The experimental observability of optical precursors has been proposed by Aaviksoo, Lippmaa, and Kuhl [205] in 1988 using the transient response of excitonic resonances to picosecond pulse excitation. The experimental observation of the Sommerfeld precursor was then reported [206] in 1991. The experimental observation of the Brillouin precursor in bulk media using an ¨ ultrashort optical pulse has been reported by Choi and Osterberg [207] in 2004, but not without criticism [208] that is itself due criticism. More recent experimental observations [209] 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. Taken together, these experimental results provide an important (albeit partial) veriﬁcation of the modern asymptotic theory in its description of ultrashort dispersive pulse dynamics.

1.3 Organization of the Book The subject of classical electromagnetic pulse radiation and propagation in a linear, temporally dispersive medium is considered here in two major movements. For convenience, this book is then separated into two separate volumes in order to adequately represent these two movements: the fundamental electromagnetic theory and the ensuing analytical description of the propagated ﬁeld through the use of mathematically well-deﬁned asymptotic expansion techniques that are ideally suited for this type of problem. Other major topics, such as inverse methods for the determination of the dispersive material properties from the propagated ﬁeld characteristics, are only brieﬂy considered, while topics such as propagation in temporally dispersive, spatially

30

1 Introduction

inhomogeneous media or in temporally dispersive materials that also exhibit spatial dispersion, are not considered. Nevertheless, the proper detailed solution of the forward problem in a homogeneous, isotropic, locally linear, causal medium that exhibits temporal dispersion contains enough material to ﬁll many volumes, each dedicated to a diﬀerent physical model of the dispersive medium or system. The fundamental electromagnetic theory presented here is developed for a general, linear dispersive medium with both a frequency-dependent dielectric permittivity (ω), a frequency-dependent magnetic permeability µ(ω), and a frequency-dependent conductivity σ(ω). This then provides the underlying theory for pulse propagation in purely dielectric, purely conductive, or semiconducting materials as well as in metamaterials. The asymptotic theory of the propagated pulse dynamics is developed here for the multiple resonance Lorentz model dielectric, the Rocard– Powles–Debye model dielectric, and the Drude model conductor as well as for composite models of semiconducting materials (i.e., a material that is a “good conductor” in one frequency domain and a “good dielectric” in another) and metamaterials (i.e., materials with frequency-dependent dielectric permittivity and magnetic permeability such that the resultant complex index of refraction takes on negative values over a ﬁnite frequency domain). This description then provides the framework for the description of pulse propagation phenomena in other types of linear dispersive media. The fundamental theory developed in Volume I of this book begins with the microscopic Maxwell–Lorentz theory of electromagnetism. This includes the invariance of the Maxwell–Lorentz equations in the special theory of relativity, conservation laws, and potential theory. The Li`enard–Wiechert potentials are then used to derive the radiation ﬁeld that is produced by a general dipole oscillator without any of the usual approximations. Quite naturally, the analysis presented here follows that given by H. A. Lorentz [3], as updated by L. Rosenfeld [210] and J. M. Stone [211]. For convenience, both mksa and Gaussian (cgs) units are used throughout this book through the use of a conversion factor ∗ that appears in double brackets ∗ in each equation (if required). If that factor is included in the aﬀected equation, it is then in cgs units, while if that factor is replaced by unity, it is in mksa units. If no such factor is present, then that equation is correct in either system of units. The derivation of macroscopic electromagnetics from the microscopic Maxwell–Lorentz theory is presented in Chapter 4. General constitutive relations in linear electromagnetics and optics are described in detail together with causality. The speciﬁc causal models of material dispersion that are considered in this book are then described in detail. These include the Lorentz model [3] of dielectric resonance, the Cole–Cole [212] and Rocard–Powles [40] extensions of the Debye model [39] of orientational polarization and the Drude model [48] of free electron metals, as well as composite models for describing semiconducting materials. The fundamental ﬁeld equations and conservation laws in a temporally dispersive medium are then developed in Chapter 5.

1.3 Organization of the Book

31

This includes Poynting’s theorem and the conservation of energy as well as the conservation of linear and angular momentum. The velocity of energy ﬂow in a time-harmonic plane wave ﬁeld is then considered. The chapter concludes with a brief discussion of the boundary conditions for complex media. This fundamental theory is followed in Chapter 6 by a thorough rigorous development of the angular spectrum of plane waves representation of the pulsed radiation ﬁeld in a homogeneous, isotropic, locally linear, temporally dispersive medium. This analysis includes a rigorous exposition of Weyl’s proof and the subsequent derivation of Weyl’s integral representation, as well as both Sommerfeld’s and Ott’s integral representations of a spherical wave in terms of a superposition of plane waves. The angular spectrum representation of pulsed electromagnetic beam ﬁelds is then considered in Chapter 7 along with its relationship to Huygen’s principle through the Rayleigh– Sommerfeld diﬀraction integrals. The general polarization characteristics of the ﬁeld vectors are also considered in detail. A derivation of the speciﬁc integral representation of the pulsed plane wave electromagnetic ﬁeld whose asymptotic description forms the subject of Volume II of the book is then given. The chapter concludes with a detailed analysis of the properties of source-free wave ﬁelds. The volume concludes with a chapter on the properties of free ﬁelds in temporally dispersive media.4 Both source-free wave ﬁelds and free ﬁelds satisfy several uniquely interesting properties that were ﬁrst established by Sherman for scalar wave ﬁelds in vacuum. Volume II begins with a review of the angular spectrum of plane waves representation of pulsed electromagnetic and optical beam wave ﬁelds in temporally dispersive media with application to multipole ﬁelds, localized pulsed beam ﬁelds and so-called electromagnetic “bullets”. This is then followed by a detailed description of those asymptotic methods of analysis that are appropriate for the asymptotic evaluation of the type of integral that appears in this spectral representation. Of central importance to the modern asymptotic theory is Olver’s method which eliminates the central role played by the steepest descent path in previous saddle point methods of analysis. The appropriate uniform asymptotic expansions that are necessary to obtain a continuous description of the dynamical ﬁeld evolution are then described. These include the uniform asymptotic expansion when the saddle point locations are at inﬁnity (necessary for the description of the Sommerfeld precursor front), the uniform asymptotic expansion when two ﬁrst-order saddle points coalesce into a single second-order saddle point for an instant and then separate into two ﬁrst-order saddle points (necessary for the description near the 4

A portion of the material presented in Chapters 2 and 3 has been used as part of the ﬁrst-year graduate course in physical optics, and the material presented in Chapters 6 through 8 has been used as the basis of an advanced graduate-level course in physical optics at The Institute of Optics of the University of Rochester. Both course sequences were taken by the author when he was a graduate student there.

32

1 Introduction

peak amplitude point of the Brillouin precursor), and the uniform asymptotic description when an isolated saddle point passes close to a simple pole singularity of the integrand (necessary for the description of the signal arrival). The chapter then concludes with a description of hybrid asymptotic and numerical methods for the description of dispersive pulse dynamics. The classical group velocity approximation is then considered in detail in Chapter 11. Both the slowly varying envelope approximation, the slowly evolving envelope approximation, and the group velocity approximation are detailed and shown to fail when the initial pulse spectrum becomes ultrawideband, thereby establishing the necessity of an asymptotic description in that situation. Because the asymptotic description requires that the behavior of the complex phase function φ(ω, θ) ≡ iω(n(ω) − θ) that appears in the integral representation of the propagated ﬁeld be known with some detail throughout the complex ω-plane, this asymptotic description must then be carried out with a speciﬁc form for both the complex dielectric permittivity

c (ω) = (ω) + i4πσ(ω)/ω and the magnetic permeability µ(ω). Causality, as embodied in the Kramers–Kronig relations, shows that temporal dispersion in dielectrics separates into two general classes: Lorentz-type dispersion and Debye-type dispersion. Because of this, the speciﬁc model types considered in this book are the multiple-resonance Lorentz model, the Drude model (which is a special case of the Lorentz model), and the multiple relaxation time Rocard–Powles–Debye model. Although speciﬁc to these models, the asymptotic analysis presented here serves as a general guide for other, more exotic types of material dispersion. This detailed analysis begins in Chapter 12 with a thorough study of the behavior of the complex phase function φ(ω, θ) in the complex ω-plane as a function of the real space–time parameter θ = ct/z. Approximate expressions for the saddle point locations for θ ≥ 1 that are accurate over this entire space–time domain are derived for each particular model of the material dispersion considered in this book. Composite models constructed from two or more of the basic dispersion models present a special challenge due to their exceedingly complicated analytic behavior. In that case, numerically determined saddle point locations must be used. Fortunately, this is a straightforward numerical problem. The chapter then concludes with a detailed description of the general procedure for obtaining the asymptotic approximation of the propagated ﬁeld. With these results, the uniform asymptotic behavior of the basic integral representation for the description of dispersive pulse propagation is next considered. This analysis is begun in Chapter 13 where the uniform asymptotic description of the Sommerfeld, Brillouin, and middle precursor ﬁelds is considered with detailed examples given for a variety of initial pulse envelope functions. The ﬁrst forerunner (Sommerfeld’s precursor), which is due to the distant pair of saddle points in a Lorentz model medium, is analyzed ﬁrst

1.3 Organization of the Book

33

using a direct application of Olver’s saddle point method [21]. However, because the distant saddle points are initially located at inﬁnity in the complex ω-plane when θ = 1, Olver’s method is not applicable to obtaining the asymptotic description of the front of the ﬁrst precursor ﬁeld. For that purpose, the uniform asymptotic expansion due to Handelsman and Bleistein [25] is employed. The result is an asymptotic approximation of the ﬁrst (Sommerfeld’s) precursor ﬁeld that is uniformly valid over the entire duration of the ﬁeld. There is no Sommerfeld precursor for a Debye-type dielectric because that type of material dispersion doesn’t result in the appropriate distant saddle points. The second forerunner (Brillouin’s precursor), which is due to the pair of near saddle points located in the region of the complex ω-plane about the origin, is analyzed next using a direct application of Olver’s saddle point method. For a Lorentz-type dielectric, there are two near ﬁrst-order saddle points that are initially located along the imaginary axis in the complex ωplane. These two ﬁrst-order saddle points approach each other along this axis as θ increases from unity, coalesce into a single second-order saddle point at θ = θ1 , where θ1 ≈ θ0 ≡ n(0), after which they separate into a pair of ﬁrstorder saddle points that are symmetrically situated about the imaginary axis in the lower half of the complex ω-plane. As a consequence, Olver’s method yields a discontinuous result at θ = θ1 and is not applicable to obtaining the asymptotic description of the second precursor ﬁeld for small values of |θ −θ1 | with θ = θ1 . For that purpose one must apply the uniform asymptotic expansion due to Chester, Friedman and Ursell [22] for values of θ about the critical value θ1 . The result is an asymptotic approximation of the second precursor ﬁeld that is uniformly valid and continuous over the entire duration of the ﬁeld. For a Debye-type dielectric there is just a single near saddle point that moves down the imaginary as θ increases from unity, crossing the origin at θ = θ0 . Olver’s method then yields a completely continuous description of the Brillouin precursor in this case. The asymptotic description of the middle precursor in a multiple-resonance Lorentz model dielectric is then considered. Because the middle saddle points typically remain isolated, Olver’s method is suﬃcient in obtaining the asymptotic description of the middle precursor, if there is one (a necessary condition for the appearance of the middle precursor is given in Chapter 15 in terms of the energy transport velocity). The eﬀects of conductivity on the Brillouin precursor are then carefully examined. Because conductivity is primarily a low-frequency eﬀect, only the Brillouin precursor is signiﬁcantly aﬀected by the presence of conductivity, although the middle precursor may, to a lesser extent, also be altered. The asymptotic description of the signal is then considered in Chapter 14. This ﬁnal contribution (if present) to the asymptotic approximation of the total propagated ﬁeld is due to the interaction of the deformed contour of integration through the relevant saddle points with any pole singularities appearing in the integrand of the integral representation of the propagated ﬁeld. The analysis begins with the nonuniform asymptotic approximation of

34

1 Introduction

the pole contribution which is based upon a direct application of Cauchy’s residue theorem. This nonuniform asymptotic approximation is valid only if the interacting saddle point and pole remain isolated from each other. If the saddle point and pole come within close proximity of each other, however, this nonuniform approximation becomes invalid and must be replaced by the uniform asymptotic expansion due to Bleistein [23, 24]. This is done for each type of material dispersion considered in this book. With these detailed results, the complete, continuous dynamical evolution of the total propagated ﬁeld is then considered in Chapter 15. The continuous evolution of the total precursor ﬁeld is considered ﬁrst, followed by the interaction of that ﬁeld with the signal contribution. Comparison of each of these ﬁeld components with purely numerical results then establishes the accuracy of this uniform asymptotic description. The signal arrival and its associated signal velocity are then obtained from a careful consideration of the interaction of these two sets of component ﬁelds. In a fundamental modiﬁcation of Brillouin’s approach [9, 10], the signal arrival is shown to occur at the instant when the real exponential value associated with the simple pole singularity appearing in the integrand becomes less (in magnitude) than the real exponential behavior associated with the interacting saddle point [13, 14]. With this new interpretation of the signal arrival, an accurate deﬁnition of the signal velocity is then obtained. Finally, employing approximate analytic expressions for the signal velocity in a Lorentz model dielectric, it is shown that the signal velocity of a pulse as deﬁned here is related to Loudon’s [162] velocity of energy propagation for a time-harmonic signal in that medium. This extremely important result makes possible a better physical understanding of the dynamics of dispersive pulse propagation. This new physical description of dispersive pulse dynamics [178, 179] is then presented in Chapter 16. This description supplants the classical group velocity description that is restricted to nonabsorptive media and reduces to it in the limit of vanishing medium absorption. The ﬁnal chapter considers a variety of applications for ultrashort pulse, ultrawideband signal phenomena. This includes a description of the reﬂection and transmission of an electromagnetic pulse at a planar interface and at a dispersive layer, the latter problem being of some importance in the analysis of superluminal tunneling. Electromagnetic pulse interaction with biological structures is then considered along with the health and safety issues associated with ultrashort pulse radiation. Applications to remote sensing, groundand foliage-penetrating radar, and undersea communications using the Brillouin precursor complete the book.

References

35

References 1. J. C. Maxwell, “A dynamical theory of the electromagnetic ﬁeld,” Phil. Trans. Roy. Soc. (London), vol. 155, pp. 450–521, 1865. 2. J. C. Maxwell, A Treatise on Electricity and Magnetism. Oxford: Oxford University Press, 1873. 3. H. A. Lorentz, The Theory of Electrons. Leipzig: Teubner, 1906. Ch. IV. 4. A. Einstein, “Zur elektrodynamik bewegter k¨orper,” Ann. Phys., vol. 17, pp. 891–921, 1905. 5. J. S. Toll, “Causality and the dispersion relation: Logical foundations,” Phys. Rev., vol. 104, no. 6, pp. 1760–1770, 1956. 6. H. M. Nussenzveig, Causality and Dispersion Relations. New York: Academic, 1972. Ch. 1. 7. A. Sommerfeld, “Ein einwand gegen die relativtheorie der elektrodynamok und seine beseitigung,” Phys. Z., vol. 8, p. 841, 1907. ¨ 8. A. Sommerfeld, “Uber die fortpﬂanzung des lichtes in disperdierenden medien,” Ann. Phys., vol. 44, pp. 177–202, 1914. ¨ 9. L. Brillouin, “Uber die fortpﬂanzung des licht in disperdierenden medien,” Ann. Phys., vol. 44, pp. 204–240, 1914. 10. L. Brillouin, Wave Propagation and Group Velocity. New York: Academic, 1960. 11. 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. 12. J. A. Stratton, Electromagnetic Theory. New York: McGraw-Hill, 1941. 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, “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. 15. S. Shen and K. E. Oughstun, “Dispersive pulse propagation in a doubleresonance Lorentz medium,” J. Opt. Soc. Am. B, vol. 6, pp. 948–963, 1989. 16. 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. 17. K. E. Oughstun and G. C. Sherman, “Uniform asymptotic description of ultrashort rectangular optical pulse propagation in a linear, causally dispersive medium,” Phys. Rev. A, vol. 41, no. 11, pp. 6090–6113, 1990. 18. 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. 19. 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. 20. 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.

36

1 Introduction

21. F. W. J. Olver, “Why steepest descents,” SIAM Rev., vol. 12, no. 2, pp. 228– 247, 1970. 22. 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. 23. N. Bleistein, “Uniform asymptotic expansions of integrals with stationary point near algebraic singularity,” Com. Pure and Appl. Math., vol. XIX, no. 4, pp. 353–370, 1966. 24. 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. 25. 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. 26. L. Felsen and N. Marcuvitz, Radiation and Scattering of Waves. Englewood Cliﬀs, NJ: Prentice-Hall, 1973. 27. N. Bleistein and R. Handelsman, Asymptotic Expansions of Integrals. New York: Dover, 1975. 28. K. E. Oughstun, P. Wyns, and D. P. Foty, “Numerical determination of the signal velocity in dispersive pulse propagation,” J. Opt. Soc. Am. A, vol. 6, no. 9, pp. 1430–1440, 1989. 29. 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. 30. 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. 31. 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 Ultra-Wideband, Short-Pulse Electromagnetics 2 (L. Carin and L. B. Felsen, eds.), pp. 273–283, New York: Plenum, 1994. 32. 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. 33. 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. 34. G. P. Agrawal, Nonlinear Fiber Optics. Academic, 1989. 35. M. N. Islam, Ultrafast Fiber Switching Devices and Systems. Cambridge: Cambridge University Press, 1992. 36. C. Rulli`ere, ed., Femtosecond Laser Pulses: Principles and Experiments. Berlin: Springer-Verlag, 1998. 37. R. de L. Kronig, “On the theory of dispersion of X-Rays,” J. Opt. Soc. Am. & Rev. Sci. Instrum., vol. 12, no. 6, pp. 547–557, 1926. 38. H. A. Kramers, “La diﬀusion de la lumi`ere par les atomes,” in Estratto dagli Atti del Congresso Internazional de Fisici Como, pp. 545–557, Bologna: Nicolo Zonichelli, 1927. 39. P. Debye, Polar Molecules. New York: Dover, 1929. 40. J. McConnel, Rotational Brownian Motion and Dielectric Theory. London: Academic, 1980.

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41. W. R. Hamilton, “Researches respecting vibration, connected with the theory of light,” Proc. Royal Irish Academy, vol. 1, pp. 341–349, 1839. 42. G. G. Stokes, “Smith’s prize examination question no. 11,” in Mathematical and Physical Papers, vol. 5, Cambridge University Press, 1905. pg. 362. 43. L. Rayleigh, “On progressive waves,” Proc. London Math. Soc., vol. IX, pp. 21– 26, 1877. 44. L. Rayleigh, “On the velocity of light,” Nature, vol. XXIV, pp. 52–55, 1881. 45. R. B. Lindsay, Mechanical Radiation. New York: McGraw-Hill, 1960. Ch. 1. 46. M. Born and E. Wolf, Principles of Optics. Cambridge: Cambridge University Press, seventh (expanded) ed., 1999. 47. K. E. Oughstun and G. C. Sherman, Pulse Propagation in Causal Dielectrics. Berlin: Springer-Verlag, 1994. 48. P. Drude, Lehrbuch der Optik. Leipzig: Teubner, 1900. Ch. V. 49. E. Mach, The Principles of Physical Optics: An Historical and Philosophical Treatment. New York: First German edition 1913. English translation 1926, reprinted by Dover, 1953. Ch. VII. 50. F. A. Jenkins and H. E. White, Fundamentals of Optics. New York: McGrawHill, third ed., 1957. Ch. 23. ¨ 51. W. Voigt, “Uber die ¨ anderung der schwingungsform des lichtes beim fortschreiten in einem dispergirenden oder absorbirenden mittel,” Ann. Phys. und Chem. (Leipzig), vol. 68, pp. 598–603, 1899. 52. W. Voigt, “Weiteres zur ¨ anderung der schwingungsform des lichtes beim fortschreiten in einem dispergirenden oder absorbirenden mittel,” Ann. Phys. (Leipzig), vol. 4, pp. 209–214, 1901. 53. P. Ehrenfest, “Mißt der aberrationswinkel in fall einer dispersion des ¨athers die wellengeschwindigkeit?,” Ann. Phys. (Leipzig), vol. 33, p. 1571, 1910. 54. A. Laue, “Die fortpﬂanzung der strahlung in dispergierenden und absorpierenden medien,” Ann. Phys., vol. 18, p. 523, 1905. 55. 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. 56. R. Mosseri, “L´eon Brillouin: une vie ` a la crois´ee des ondes,” Sciences et Vie, vol. numero special “200 ans de sciences 1789-1989”, pp. 256–261, 1989. 57. T. H. Havelock, “The propagation of groups of waves in dispersive media,” Proc. Roy. Soc. A, vol. LXXXI, p. 398, 1908. 58. T. H. Havelock, The Propagation of Disturbances in Dispersive Media. Cambridge: Cambridge University Press, 1914. 59. 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. 60. E. T. Copson, Asymptotic Expansions. London: Cambridge University Press, 1965. 61. P. M. Morse and H. Feshbach, Methods of Theoretical Physics. New York: McGraw-Hill, 1953. Vol. I. ¨ 62. H. Baerwald, “Uber die fortpﬂanzung von signalen in disperdierenden medien,” Ann. Phys., vol. 7, pp. 731–760, 1930. 63. N. S. Shiren, “Measurement of signal velocity in a region of resonant absorption by ultrasonic paramagnetic resonance,” Phys. Rev., vol. 128, pp. 2103– 2112, 1962. 64. T. A. Weber and D. B. Trizna, “Wave propagation in a dispersive and emissive medium,” Phys. Rev., vol. 144, pp. 277–282, 1966.

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Problems 1.1. Derive Eq. (1.1). Hint: Use the trigonometric identity cos α + cos β = 2 cos ((α + β)/2) cos ((α − β)/2). 1.2. Extend the result given in Eq. (1.1) to the linear superposition of three time-harmonic waves with equal amplitudes and nearly equal wavenumbers k, k − δk, and k + δk and associated angular frequencies ω, ω − δω, and ω + δω, respectively. Show that the surfaces of constant phase propagate with the ¯ /k¯ with mean wavenumber k¯ = k and mean angular phase velocity vp = ω frequency ω ¯ = ω, while the surfaces of constant amplitude propagate with the group velocity vg = δω/δk. With ω1 = 2π and δω = ω1 /10, compute the superposition of these three waveforms both directly and using the derived expression when the system dispersion is given by k(ω) = (ω/c)(1.5 + ω/10).

46

1 Introduction

1.3. Extend the result given in Eq. (1.1) to the linear superposition of two time-harmonic waves with unequal amplitudes a1 and a2 and nearly equal wavenumbers k and k − δk and associated angular frequencies ω and ω − δω, respectively. Hint: Express the resulting superposed waveform in the form U (z, t) = a1 cos(ψ − ϕ) + a2 cos(ψ + ϕ), where ϕ = ϕ(δk, δω), and use the angle sum and angle diﬀerence relations for the cosine function. 1.4. Show that the temporal Fourier transform of the Fourier–Laplace integral representation of U (z, t) given in Eq. (1.4) satisﬁes the Helmholtz equa 2 2 ˜ ˜ (z, ω) = 0 with boundary value U (0, t) = u(t) sin(ωc t + tion ∇ + k (ω) U ψ). 1.5. Use the quadratic dispersion relation (1.15) to approximate the dispersion relation given in Eq. (1.14) about the angular frequency ωc = 2ωp . Determine the rms error of this quadratic approximation over the angular frequency domain ω ∈ [ωc − δω, ωc + δω] as δω increases from 0 to ωp . Describe the behavior of the quadratic approximation as ωc decreases to the plasma frequency ωp . 1/2 N 2 /(ω 2 − ωj2 + 2iγj ω) for 1.6. With the expression n(ω) = 1 − j=0 ωpj the complex refractive index of a multiple-resonance Lorentz model dielectric as a starting point, derive the expression given in Eq. (1.22) for the behavior about the resonance frequency ω0 in the weak dispersion limit when |ωp0 /γ0 | n∞ , where n∞ describes the limiting behavior of the index of refraction above ω0 . Explain any additional approximations that may need to be made. 1.7. Show that the temporal Fourier transform of the Gaussian pulse given in Eq. (1.19) is given by Eq. (1.20) and that the respective rms widths ∆t and ∆ω for these two functions satisfy ∆t∆ω = 1/2, the minimum allowed by the indeterminacy principle.

2 Microscopic Electromagnetics

A mathematically rigorous, physically based development of the classical theory of electromagnetism is introduced here through a consideration of the microscopic Maxwell–Lorentz theory [1–4]. Although the Lorentz theory of electrons is a purely classical, heuristic model that is incapable of analyzing many fundamental problems associated with the atomic constituency of matter, it is nevertheless an expedient model in providing the proper source terms for the microscopic Maxwell equations. Indeed, the classical Lorentz theory does yield many results connected with the electromagnetic properties of matter that agree in functional form with that given by the quantum theory. In particular, the Lorentz theory of matter assumes additional forces of just the right nature such that qualitatively correct expressions are obtained and, by empirical adjustment of the parameters appearing in these ad hoc force relations, quantitatively correct predictions may also be obtained. Even though the quantum theory justiﬁes the assumption of these additional forces and shows them to be of electrical origin, the Lorentz theory is incapable of arriving at this fundamental level. The starting point of the theory developed here is the microscopic formulation of Maxwell’s equations. The microscopic ﬁeld vectors are related to the microscopic properties of matter through the elementary source terms, these being the microscopic charge density and the convective current density. The fundamental electromagnetic interaction between the source terms appearing in the microscopic theory and a test particle employed to measure the constituent ﬁeld vectors then occurs in vacuum and is completely speciﬁed by the microscopic Lorentz–Maxwell theory through the Lorentz force relation. Based upon this microscopic ﬁeld formalism, the macroscopic properties of material media in their interaction with an electromagnetic ﬁeld may then be developed in a consistent fashion. An important consequence of this approach is that the relationship of the macroscopic electromagnetic ﬁeld vectors to their microscopic counterparts may be clearly deﬁned with respect to the relationship between the macroscopic and microscopic properties of the particular material medium. Throughout this book the analysis is presented 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

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2 Microscopic Electromagnetics

double brackets ∗ is omitted, whereas the inclusion of this factor yields the appropriate expression in cgs units. If there isn’t any double bracketed quantity present, the expression can be interpreted equally in Gaussian or mksa units unless speciﬁcally noted otherwise.

2.1 The Microscopic Maxwell–Lorentz Theory In accordance with the classical Lorentz theory [3–5] it is assumed that matter is composed of three basic physical entities: mass, positive electric charge, and negative electric charge. The spatial distributions of the two types of charge at any given instant of time are assumed to be speciﬁed at each point of three-dimensional space by two positive-deﬁnite scalar ﬁelds: the density of positive charge ρ+ and the density of negative charge ρ− . These two scalar ﬁelds are formally deﬁned through the use of the following limiting procedure. Let Q+ denote the total positive charge that is contained in the interior of a simply connected closed region of space with volume V and let P be an interior point of the region with position vector r. The ratio Q+ /V then gives the average volume density of positive charge in the region and in the limit as the region shrinks to the point P , the density of positive charge at P is obtained, where Q+ (t) . (2.1) ρ+ (r, t) ≡ lim V →0 V In a similar fashion, the density of negative charge at the point P is speciﬁed by the limit Q− (t) , (2.2) ρ− (r, t) ≡ lim V →0 V where Q− is the total negative charge contained in the region V . By deﬁnition, ρ+ (r, t) is the positive charge per unit volume at any point of space, and ρ− (r, t) is the negative charge per unit volume at any point of space and is taken to be a nonnegative quantity. The charge density ρ(r, t), given by ρ(r, t) = ρ+ (r, t) − ρ− (r, t),

(2.3)

is the net charge per unit volume at any point of space and time. It may also be deﬁned by the limit Q(t) ρ(r, t) ≡ lim , (2.4) V →0 V where Q = Q+ − Q− is the net charge contained in the volume V . Each of these charge densities is assumed to be ﬁnite and to vary continuously from point to point in space. This mathematical technique of deﬁning a ﬁeld through a limiting ratio must be used (and interpreted) with caution in order

2.1 The Microscopic Maxwell–Lorentz Theory

49

to achieve results that correspond in some sense to physical reality. Quite fortunately, it does provide a convenient means for introducing discontinuities into the formalism. In particular, the concept of a point charge may be introduced in a generalized function sense as an appropriate limit of a continuous charge distribution. With this generalized interpretation, all results derived for a continuous distribution remain valid in the discrete case.

Fig. 2.1. Inﬁnitesimal regular surface element da with unit normal vector n ˆ and convective current density j(r, t).

The kinematics of the two types of charge at a given instant are assumed to be speciﬁed at each point of space by two vector ﬁelds, the convective current densities of positive and negative charge. If, at a given point r and instant of time t, the positive charge is moving with velocity v+ (r, t), then the convective current density of positive charge is deﬁned by j+ (r, t) ≡ ρ+ (r, t)v+ (r, t),

(2.5)

and the convective current density of negative charge is deﬁned by j− (r, t) ≡ ρ− (r, t)v− (r, t).

(2.6)

Let da be an inﬁnitesimal element of a regular surface having a unit normal vector n ˆ , as illustrated in Figure 2.1. If j+ is evaluated at an interior point ˆ da is seen to be the rate of the regular surface element da, the quantity j+ · n at which positive charge ﬂows across da into the region of space into which ˆ is then seen to be the rate per unit n ˆ is directed. The scalar quantity j+ · n area of positive charge ﬂow and j+ (r, t) itself can be interpreted as the ﬂow of positive charge per unit area per unit time through a regular surface element oriented perpendicularly to the vector j+ , the ﬂow being in the direction of j+ . In a similar fashion, j− (r, t) can be interpreted as the ﬂow of negative charge per unit area per unit time through a regular surface element oriented perpendicularly to the vector j− , the ﬂow being in the direction of j− which, by the adopted convention, points in the direction in which the negative

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2 Microscopic Electromagnetics

charge is moving. The net convective current density j = j(r, t) is then given by (2.7) j(r, t) = j+ (r, t) − j− (r, t), which is simply called the convective current density. Consequently, through a surface that is oriented perpendicular to the vector j, a net charge j = |j| ﬂows per unit area per unit time, so that across an arbitrarily oriented surface element, the net ﬂow of charge per unit area per unit time is given by J (r, t) = j(r, t) · n ˆ da.

(2.8)

The total convective current that is ﬂowing across a regular surface S into the region of space that the unit normal vector n ˆ to the surface is directed is then given by the surface integral Jtot (r, t) = j(r, t) · n ˆ da. (2.9) S

Suppose now that the surface S appearing in Eq. (2.9) is closed and that, in the standard convention, n ˆ denotes the positive (or outward) unit normal vector to the closed surface. By virtue of the deﬁnition of the convective current as the physical ﬂow of charge across a surface element, it then follows that the surface integral of the normal component of the net convective current density j(r, t) over the closed surface S is a measure of the loss of charge from the region of space that is enclosed by S. By the fundamental principle of conservation of charge, it is then required that d j(r, t) · n ˆ da = − ρ(r, t)d3 r, (2.10) dt V S where V is the volume of the region enclosed by the surface S, and where d3 r is the appropriate diﬀerential element of volume. If the net current ﬂow is out of the region V , then the net charge enclosed by S decreases (dQ/dt < 0) and both sides of Eq. (2.10) are positive, whereas if the net current ﬂow is into the region then the net charge enclosed by S increases (dQ/dt > 0) and both sides of Eq. (2.10) are negative, where ρ(r, t)d3 r (2.11) Q(t) = V

is the total charge contained in V . The ﬂow of charge across the surface S can originate in two diﬀerent ways: the surface S may be ﬁxed in space and the net charge density ρ(r, t) may vary in time, or the net charge density may be invariable with time while the surface S moves in some prescribed manner. Of course, a combination of these two situations may also occur. In either of the latter two cases the integral appearing on the right-hand side of Eq. (2.10) is a function of time by

2.1 The Microscopic Maxwell–Lorentz Theory

51

virtue of its variable limits of integration. If, however, the surface S is ﬁxed in space and the volume integral of the net charge density over the region enclosed by S is convergent for all time t, then the time derivative appearing on the right-hand side of Eq. (2.10) may be brought inside the integration (by Leibniz’ rule) to yield ∂ρ(r, t) 3 d r. j(r, t) · n ˆ da = − (2.12) ∂t S V Application of the divergence theorem to the left-hand side of this equation then results in the expression ∂ρ(r, t) (2.13) ∇ · j(r, t) + d3 r = 0. ∂t V The integrand of Eq. (2.13) is a continuous function of the coordinates so that there must exist small regions of space within which the integrand does not change sign. If the integral is to vanish for arbitrary regions of space V , it is then necessary that the integrand be identically zero. There then results ∇ · j(r, t) +

∂ρ(r, t) = 0, ∂t

(2.14)

which is known as the equation of continuity. This is the diﬀerential form of the conservation of charge. 2.1.1 Diﬀerential Form of the Microscopic Maxwell Equations The microscopic Maxwell–Lorentz theory [1–4] asserts that charges and currents give rise to electric and magnetic ﬁelds, and these ﬁelds in turn exert forces on the charges and inﬂuence their motion that at all times satisfy the equation of continuity (2.14). This interplay of source and ﬁeld is governed by a set of equations which, ﬁrstly, relate the ﬁelds to the charges and currents that produce them, secondly, relate the ﬁelds to the forces that they exert on the charges, and thirdly, specify the law of motion of the charges under the inﬂuence of all forces acting on them, including nonelectromagnetic as well as those forces that are electromagnetic in origin. The elementary source terms that appear in the microscopic Maxwell equations are the net charge density ρ(r, t) = ρ+ (r, t) − ρ− (r, t) and the net current density j(r, t) = j+ (r, t) − j− (r, t). It is important to notice that in the classical theory the source ﬁelds ρ(r, t) and j(r, t) represent the net charge and current distributions in minute detail. For example, even if the net charge of a molecule is zero, the net microscopic charge density ρ(r, t) will not be zero within the region of space occupied by the molecule if there is a separation of positive and negative charges within the molecule. The net charge and current densities give rise to a microscopic electric ﬁeld vector e = e(r, t) and a microscopic magnetic ﬁeld vector b = b(r, t). At

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2 Microscopic Electromagnetics

every point in space and time the microscopic ﬁeld vectors e(r, t) and b(r, t) satisfy the microscopic Maxwell equations 1 ∂b(r, t) , (2.15) ∇ × e(r, t) = − c ∂t 4π 1 ∂e(r, t) , (2.16) ∇ × b(r, t) = µ0 j(r, t) + 0 µ0 c c ∂t where c is the speed of light in vacuum, and where 0 is the dielectric permittivity and µ0 is the magnetic permeability of free space. In Gaussian (or cgs) units, 0 = µ0 = 1. Two additional conditions that are satisﬁed by the electric and magnetic vector ﬁelds e and b may be directly deduced from Maxwell’s equations by noting that the divergence of the curl of any vector vanishes identically. Upon taking the divergence of Eq. (2.15) there results ∇·(∂b/∂t) = ∂(∇·b)/∂t = 0. The commutation of the diﬀerential operators ∇· and ∂/∂t is admissible in the above relation because the ﬁeld vector b(r, t) and all of its space and time derivatives are assumed to be continuous. From this result it is seen that at every point in space, the divergence of b(r, t) is a constant. Experimental results1 have revealed this constant to be zero, so that ∇ · b(r, t) = 0

(2.17)

and the magnetic ﬁeld is solenoidal. In a similar fashion, the divergence of Eq. (2.16) results in ∂(∇·e)/∂t+(4π/ 0 )∇·j = 0, which, with the equation of continuity (2.14), may be rewritten as ∂[∇ · e − (4π/ 0 )ρ]/∂t = 0. The quantity [∇ · e − (4π/ 0 )ρ] is then seen to be a constant which experiment has revealed to be zero, so that ∇ · e(r, t) =

4π ρ(r, t).

0

(2.18)

Consequently, the charges distributed with a net density ρ(r, t) in part constitute the source terms for the microscopic electric ﬁeld vector e(r, t). The divergence relations appearing in Eqs. (2.17) and (2.18) are frequently included as part of Maxwell’s equations. However, they are not independent relations if the conservation of charge is assumed, as has been done here. On the other hand, if Eqs. (2.15)–(2.18) are postulated as Maxwell’s equations, then the conservation of charge as expressed by the equation of continuity (2.14) is a direct consequence of this set of diﬀerential relations. A certain asymmetry is apparent in Eqs. (2.15)–(2.18) that has important physical signiﬁcance. The densities ρ(r, t) and j(r, t) that are electrical in origin appear in Eqs. (2.16) and (2.18), but in the corresponding places in 1

A detailed historical survey of the foundational experiments for both electric and magnetic ﬁeld properties may be found in Chapters 3–4 of Elliott’s Electromagnetics text [6].

2.1 The Microscopic Maxwell–Lorentz Theory

53

Eqs. (2.15) and (2.17) where one would expect similar densities of magnetic charge and current to appear, there are none. The assertion that the Maxwell equations as they stand are entirely adequate to account for all classical electromagnetic phenomena has (so far) withstood the test of time. There has yet to be found any experimental evidence of magnetic charge, and the microscopic Maxwell equations are written in such a way so as to explicitly display this fact. Electric and magnetic ﬁelds are fundamentally ﬁelds of force that ultimately originate from electric charges. Whether such a force ﬁeld may be termed electric, magnetic, or electromagnetic hinges upon the motional state of the electric charges relative to the point at which the ﬁeld observations are made. Electric charges at rest relative to an observation point give rise to an electrostatic ﬁeld at that point. A relative motion of the charges yields a convective current and provides an additional magnetic force ﬁeld at the observation point. This additional ﬁeld is a magnetostatic ﬁeld if the charges are all moving at constant velocities relative to the observation point. Finally, if the charges undergo accelerated motions, both time-varying electric and magnetic ﬁelds are produced that are coupled through the microscopic Maxwell equations (2.15) and (2.16), and are termed electromagnetic ﬁelds. The microscopic Maxwell theory is completed by the equation for the force per unit volume (i.e., the microscopic force density) that is exerted on the charges and convective currents by the electromagnetic ﬁeld. The microscopic force density f+ = f+ (r, t) acting on the positive charge and convective current densities ρ+ (r, t) and j+ (r, t) is given by the Lorentz force 1 (2.19) f+ (r, t) = ρ+ (r, t)e(r, t) + j+ (r, t) × b(r, t), c and the microscopic force density f− = f− (r, t) acting on the negative charge and convective current densities ρ− (r, t) and j− (r, t) is given by 1 f− (r, t) = −ρ− (r, t)e(r, t) − j− (r, t) × b(r, t). (2.20) c The addition of Eqs. (2.19) and (2.20) yields the net microscopic force density f (r, t) = f+ (r, t) + f− (r, t), which is given by the Lorentz force relation [3] 1 (2.21) f (r, t) = ρ(r, t)e(r, t) + j(r, t) × b(r, t), c in terms of the net microscopic charge and current densities. The physical signiﬁcance of the net force density f = f (r, t) may be described in the following manner. Consider an inﬁnitesimal region of space with volume ∆V in which the net charge density is ρ = ρ(r, t) and the net convective current density is j = j(r, t) at some instant of time. At that particular instant, let the microscopic ﬁeld vectors in the region ∆V have the

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2 Microscopic Electromagnetics

spatially averaged values e and b. The Lorentz theory then asserts that the net force exerted on the charged mass in ∆V arising from the action of the electric and magnetic ﬁelds is given by 1 (2.22) ∆F = (ρ∆V )e + (j∆V ) × b. c Let P be an interior point of the region ∆V , as illustrated in Figure 2.2. In the limit as ∆V shrinks to the point P , the average ﬁeld values e and b go over to their respective values e(r, t) and b(r, t) at the point P , and the limiting ratio ∆F (2.23) f (r, t) ≡ lim ∆V →0 ∆V then deﬁnes f = f (r, t) as the net electromagnetic force per unit volume acting at the point r at the instant of time t.

Fig. 2.2. Lorentz force ∆F exerted on an inﬁnitesimal volume element ∆V with net charge density ρ and convective current density j.

Consider now an elementary charged particle of small but nonvanishing spatial extent that is part of some physical system. If it is suﬃciently well removed from any other charges or currents in the system, then the external ﬁelds e0 (r, t) and b0 (r, t) arising from these other charges and currents are essentially constant over the spatial extent of the particle and the net force acting upon the particle is readily obtained by integrating the Lorentz force relation given in Eq. (2.21) over the particle volume, with the result 1 F(r, t) = q e0 (r, t) + v(r, t) × b0 (r, t) . (2.24) c Here q is the total charge of the elementary particle (e.g., q = −qe for an electron) and v(r, t) is the velocity of the particle. The following six comments then apply to this equation [4].

2.1 The Microscopic Maxwell–Lorentz Theory

55

1. The electromagnetic ﬁeld vectors e0 and b0 are the microscopic ﬁelds arising from all charges and convective currents in the system with the exception of those of the charged particle itself. 2. Because of the ﬁniteness of the charge q on the particle, its inﬂuence on neighboring charges and currents cannot always be neglected, and it is assumed that the external ﬁelds e0 and b0 have been determined with this inﬂuence taken into account. 3. If the elementary charged particle is part of or is in collision with another elementary particle or system of particles (such as an atom or molecule), additional forces of a nonelectromagnetic nature must be included.2 4. Equation (2.24) applies to a particle that is moving without rotation so that the velocity v is the same at all points of its volume. It also applies to a rotating particle if v is taken as the velocity of the charge centroid of the elementary particle.3 5. If the motion of the particle involves any acceleration, there is then a small force of radiation reaction4 that needs to be added to Eq. (2.24). 6. If the magnitude of the velocity of the particle is very small in comparison to the vacuum speed of light c (i.e., in the nonrelativistic limit |v| c), then the second term appearing on the right-hand side of Eq. (2.24) is small when compared with the ﬁrst term in those frequent cases in which the electric and magnetic ﬁeld strengths are of the same order of magnitude (in cgs units), and often the eﬀects of this small term can then be neglected. A fundamental property of the microscopic Maxwell–Lorentz equations is their linearity as expressed by the principle of superposition. Let ρ1 (r, t) and j1 (r, t) be any given distribution of microscopic charge and current densities and let the microscopic ﬁeld vectors due to them be denoted by e1 (r, t) and b1 (r, t). Let the ﬁeld vectors e2 (r, t) and b2 (r, t) be due to another distribution of microscopic charge and current densities ρ2 (r, t) and j2 (r, t). Due to the linearity of the microscopic Maxwell equations and the Lorentz force relation, it then follows that the combined charge and current distributions ρ(r, t) = ρ1 (r, t) + ρ2 (r, t) and j(r, t) = j1 (r, t) + j2 (r, t), respectively, produce the ﬁeld vectors e(r, t) = e1 (r, t) + e2 (r, t) and b(r, t) = b1 (r, t) + b2 (r, t). The principle of superposition is of particular importance in the solution of many electromagnetic problems. In its most direct application, it allows one 2

3

4

This occurs, for example, in the collisional broadening of radiation from a Lorentz atom. See Chapter 12 of Stone [4] for a detailed development of collisional broadening eﬀects in a system of Lorentz atoms. The electron spin is a purely quantum-mechanical eﬀect that is not susceptible to a complete analysis in classical physics. For a detailed historical discussion of this point, see chapter VI of Kramers’ classic treatise Quantum Mechanics [7]. For a complete rigorous development of radiation reaction in the Lorentz– Abraham model, see Yaghjian’s treatise Relativistic Dynamics of a Charged Sphere [8].

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2 Microscopic Electromagnetics

to calculate the ﬁeld vectors due to a given ﬁxed distribution of charged particles by ﬁrst determining the separate ﬁelds produced by each particle alone and then adding the individual results to give the total ﬁeld. However, if the electromagnetic ﬁeld is due to an unspeciﬁed motion of some system of interacting charged particles, then the Maxwell–Lorentz equations (with proper account taken for radiation reaction, collisional eﬀects, etc.) must be solved in a self-consistent fashion in which the ﬁelds, in part, determine the motion of each charged particle in the system, and each charged particle is, in turn, a source term for the electromagnetic ﬁeld. In this more complicated (and more realistic) situation, the principle of superposition applies in each separate part of the dynamical computation. An example of this self-consistent procedure may be found in plasma electrodynamics [9–11]. All that remains now is to show how the microscopic electric and magnetic ﬁelds can be measured (at least in principle) and their units deﬁned in terms of the elementary sources of the theory. For that purpose, the classical concept of a charged test particle is employed. Such a particle is deﬁned to consist of a distribution of charge, each element of which is in some way held rigidly ﬁxed with respect to the other elements regardless of any electromagnetic forces that may be acting on it. The part of these electromagnetic forces due to the charges and currents in the test particle itself is assumed to be exactly balanced by internal stabilizing forces of the particle (which remain unspeciﬁed in the classical theory). The remainder of the electromagnetic force that is exerted on the test particle is due to the ﬁelds arising from external charges and other ﬁelds whose origin is not due to the test particle itself, and these are just the ﬁelds that are to be measured by introducing the test particle into the system. Furthermore, in order that the system under consideration be undisturbed by the presence of the test particle when it is introduced, it is assumed that the latter is of inﬁnitesimal size and has inﬁnitesimal charge dq. Moreover, it is assumed that the test particle can be controlled to such a degree that it can be held ﬁxed in space at any point or given a deﬁnite velocity at any point, and in all cases the force exerted upon it due to the interacting ﬁelds under study can always be measured at any instant of time with inﬁnitesimal accuracy. The plausible existence of such an inﬁnitesimal charged test particle along with its complete controllability is possible only within the framework of classical physics. Such is not the case, however, in the physics of quantum ﬁeld theory [12]. In the purely classical framework of the Maxwell–Lorentz theory, the concept of ﬁeld measurements through the use of inﬁnitesimally small test charges is pushed to the limit of physical conceptualization by asserting that the ﬁeld vectors can be measured with microscopic precision even inside the basic entities of matter. It is in this sense that the microscopic ﬁeld vectors e(r, t) and b(r, t) have meaning at all points of space. If the test particle dq moves with the velocity v without acceleration or rotation, it then gives rise to a current density j(r, t) = ρ(r, t)v deﬁned at

2.1 The Microscopic Maxwell–Lorentz Theory

57

each point of space, and Eq. (2.24) gives the total force acting on the test particle as 1 (2.25) dF(r, t) = dq e(r, t) + v × b(r, t) . c Notice that no distinction need be made between the external ﬁeld vectors e0 and b0 employed in Eq. (2.24) and the total ﬁeld vectors e(r, t) and b(r, t) at the ﬁxed observation point P at which the test particle is introduced because dq is inﬁnitesimally small. Let this inﬁnitesimal test charge be at rest at a ﬁxed point in the ﬁeld so that v = 0. The force exerted on the particle is then given by dF(r, t) = e(r, t)dq and e(r, t) =

∆F(r, t) dF(r, t) = lim , ∆q→0 dq ∆q

(2.26)

so that knowing dq and measuring dF(r, t) gives the value of the microscopic ﬁeld vector e(r, t) at that point in space and time. Next, refer space to a ﬁxed Cartesian coordinate system and let the inﬁnitesimal test particle move in the y-coordinate direction with velocity v = ˆ 1y vy and no acceleration. From Eq. (2.25), the force exerted on this test particle is now given by 1 1x vy bz (r, t) − ˆ dF(r, t) = e(r, t)dq + dq ˆ 1z vy bx (r, t) . (2.27) c Subtract out the component of the force that is due to the electric ﬁeld e(r, t), which is now known from the measurement speciﬁed by Eq. (2.26); the x-component of the remaining force is then given by dFx = 1/cvy bz dq, so that 1 ∆Fx (r, t) 1 dFx (r, t) = ||c|| . (2.28) lim bz (r, t) = ||c|| vy dq vy ∆q→0 ∆q Similarly, the components bx (r, t) and by (r, t) can be respectively obtained by ﬁrst taking the test particle velocity in the z-direction and then in the xdirection and measuring the components dFy (r, t) and dFz (r, t), respectively, of the force remaining after that due to the electric ﬁeld has been subtracted out. The results are given by bx (r, t) = ||c||

∆Fy (r, t) 1 dFy (r, t) 1 = ||c|| , lim vz dq vz ∆q→0 ∆q

(2.29)

by (r, t) = ||c||

1 ∆Fz (r, t) 1 dFz (r, t) = ||c|| . lim vx dq vx ∆q→0 ∆q

(2.30)

The limiting expressions appearing in Eqs. (2.26) and (2.28)–(2.30) are not only of a formal mathematical nature but are also connected with the fundamental physical deﬁnition of the electric and magnetic ﬁelds, respectively. At this point it is convenient to introduce the auxiliary ﬁeld vectors that are deﬁned by the pair of relations

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2 Microscopic Electromagnetics

d(r, t) ≡ 0 e(r, t), 1 h(r, t) ≡ b(r, t), µ0

(2.31) (2.32)

where d = d(r, t) is the microscopic electric displacement vector and h = h(r, t) is the microscopic magnetic intensity vector. With these identiﬁcations, the diﬀerential form of the microscopic Maxwell equations becomes 1 ∂b(r, t) ∇ × e(r, t) = − , (2.33) c ∂t 1 ∂d(r, t) 4π , (2.34) ∇ × h(r, t) = j(r, t) + c c ∂t ∇ · d(r, t) = 4πρ(r, t), (2.35) ∇ · b(r, t) = 0. (2.36) Again, notice that in cgs or Gaussian units, 0 = µ0 = 1 so that d(r, t) = e(r, t) and h(r, t) = b(r, t) and there is no real distinction between these microscopic ﬁeld vectors in that system. Gaussian (cgs) Units In Gaussian units (centimeters, grams, seconds), the basic unit of charge is the statcoulomb so that the volume charge density ρ is in units of statcoulombs per cubic centimeter. The fundamental unit of current is then that current which transports charge at the rate of 1 statcoulomb per second and is called the statampere. The unit of current density j is then the statampere per square centimeter. The unit of force in Gaussian units is the dyne so that, from Eq. (2.26), the unit of measure of the microscopic electric ﬁeld intensity is the dyne per statcoulomb, and this is called the esu (electrostatic unit) of electric ﬁeld intensity, viz. 1 esu of electric ﬁeld intensity ≡ 1

dyne . statcoulomb

(2.37)

In terms of energy, 1 esu of electric ﬁeld intensity = 1 statvolt/cm, so that 1 statvolt = 1

erg . statcoulomb

(2.38)

The unit of measure of the microscopic magnetic induction ﬁeld is the gauss, so that from Eqs. (2.28)–(2.30) 1 gauss = 1

dyne . statcoulomb

(2.39)

2.1 The Microscopic Maxwell–Lorentz Theory

59

MKSA (SI) Units The primary relating factor between mksa and cgs units is the vacuum speed of light c, where c = 2.9979×108 m/s. In mksa or SI (Syst`eme Internationale) units (meters, kilograms, seconds, amperes) the basic unit of charge is the coulomb, where 1 coulomb = 2.9979 × 109 statcoulombs. The volume charge density ρ is accordingly in units of coulombs per cubic meter. Furthermore, the basic unit of current is that current which transports charge at the rate of 1 coulomb per second and is called the ampere, where 1 ampere = 2.9979 × 109 statamperes, and the unit of current density j is the ampere per square meter. In the conventional mksa system of electromagnetic units, the electric current is arbitrarily chosen as the fourth fundamental dimension. The unit of force in mksa units is the newton so that, from Eq. (2.26), the unit of measure of the microscopic electric ﬁeld intensity is the newton per coulomb. In terms of energy, 1 volt/m = 1 newton/coulomb, so that 1 volt = 1

joule , coulomb

(2.40)

where (since 1 newton = 1 × 105 dynes), 1 volt = (1/299.79) statvolt. The unit of measure of the microscopic magnetic induction ﬁeld in mksa units is the tesla, so that, from Eqs. (2.28)–(2.30), 1 tesla = 1

newton , coulomb

(2.41)

where [noting the factor c in Eqs. (2.28)–(2.30) when Gaussian units are employed] 1 tesla = 1 × 104 gauss. Finally, the mksa values of the dielectric permittivity and magnetic permeability of free space are given respectively by 0 = (1 × 107 farad · m/s2 )/(4πc2 ) = 8.854 × 10−12 farad/m and µ0 = 4π × 10−7 weber/(ampere · m), where 1 farad = 1 coulomb/volt is the unit of capacitance and 1 weber = 1 volt · s = 1 joule/ampere is the unit of magnetic ﬂux. The displacement vector d is then in units of coulombs/m2 and the magnetic intensity vector h is in units of amperes/m. 2.1.2 Integral Form of the Microscopic Maxwell Equations The properties of an electromagnetic ﬁeld that are described by the set of diﬀerential equations given in Eqs. (2.33)–(2.36) can also be expressed by an equivalent set of integral equations. Consider ﬁrst the microscopic diﬀerential relation given in Eq. (2.33). Upon integrating this equation over any regular open surface S that is bounded by a closed, regular (i.e. nonintersecting) contour C, followed by application of Stokes’ theorem to the surface integral of ∇ × e(r, t), one obtains 1 ∂b(r, t) e(r, t) · dl + ·n ˆ da = 0, (2.42) c ∂t C S

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2 Microscopic Electromagnetics

where the integration around the contour C is traversed in the positive sense (the positive direction about C is clockwise when one views S in the direction of the outward unit normal vector n ˆ to S). Because the electromagnetic ﬁeld vectors and their partial derivatives are assumed to be continuous, if the surface S is ﬁxed in space, then by Leibniz’ rule, the diﬀerential operator ∂/∂t may be brought out from under the integration, with the result 1 d e(r, t) · dl = − b(r, t) · n ˆ da, (2.43) c dt S C which is known as Faraday’s law. The scalar quantity deﬁned by b(r, t) · n ˆ da Φm (R, t) ≡

(2.44)

S

is the ﬂux of the magnetic induction vector b(r, t) through the surfaces S, and is appropriately called the magnetic ﬂux. In cgs units the unit measure of Φm is in gauss · cm2 which is called the maxwell, while in mksa units the unit of measure of Φm is in tesla · m2 , which is called the weber, where 1 weber = 1 × 108 maxwells. Notice that the vector R appearing in the argument of the magnetic ﬂux in Eq. (2.44) depends upon both the position vector r appearing in the argument of the magnetic induction vector as well as upon the shape and orientation of the surface of integration S appearing in that equation. According to Eq. (2.43), the circulation of the microscopic electric ﬁeld vector e(r, t) about any closed, regular contour C that is situated in that ﬁeld is equal to minus 1/c times the time rate of change of the magnetic ﬂux through any open regular surface bounded by that contour. The experiments of Faraday [13] demonstrated that the relation given in Eq. (2.43) holds whatever the cause of the magnetic ﬂux variation. The total time derivative appearing outside the surface integral in Eq. (2.43) was obtained from Eq. (2.42) under the assumption of a variable magnetic ﬂux density threading a ﬁxed contour C, but the total magnetic ﬂux Φm will also change through any deformation or motion of the contour C. It is then clear that this time derivative of the magnetic ﬂux must be interpreted in this more general sense. This more general interpretation of Faraday’s law as given in Eq. (2.43) is in fact a consequence of the diﬀerential ﬁeld equations, but the general proof must be based on the electrodynamics of moving media [14]. It is important to notice that the microscopic electric ﬁeld e(r, t) appearing in Eq. (2.43) is the electric ﬁeld intensity at the diﬀerential path element dl along the contour C in the reference frame in which dl is at rest. If the contour C is moving with a constant velocity V in some direction, as illustrated in Figure 2.3, then the total time derivative appearing in Eq. (2.43) must take this translational motion into account. The magnetic ﬂux Φm linking the circuit then changes because both the ﬂux changes with time at a given point and the translation of the circuit changes the location of the boundary.

2.1 The Microscopic Maxwell–Lorentz Theory

61

V

C

V

Fig. 2.3. Contour C moving with a constant translational velocity V.

If the ordered-triple (x1 , x2 , x3 ) denotes the coordinates of the boundary C, then the rate of change with time of some property of the ﬁeld that is associated with the point (x1 , x2 , x3 ) = (x1 (t), x2 (t), x3 (t)) at the time t is given by 3 3 ∂ ∂xk ∂ ∂ d ∂ = + + = Vk . (2.45) dt ∂t ∂t ∂xk ∂t ∂xk k=1

k=1

Because V = (V1 , V2 , V3 ) is the constant velocity of the contour C, then the total time derivative is seen to be given by the convective derivative ∂ d = + V · ∇, dt ∂t

(2.46)

where ∇ = (∂/∂x1 , ∂/∂x2 , ∂/∂x3 ). The total time derivative of the magnetic ﬂux linking the circuit C is then given by (where the subscript L is used to denote the laboratory frame of reference in which the ﬁeld measurements are made) ∂bL d bL · n ˆ da = [(V · ∇) bL ] · n ˆ da ·n ˆ da + dt S ∂t S S ∂bL ·n ˆ da + = ∇ × (bL × V) · n ˆ da, S ∂t S since ∇×(bL ×V) = (V·∇)bL −V(∇·bL )−(bL ·∇)V+bL (∇·V), where the last three terms appearing on the right-hand side of this identity vanish by virtue of both the divergenceless character of the magnetic induction vector bL and the constant value of the velocity vector V. Application of Stokes’ theorem to the second surface integral on the right-hand side of the above equation then results in the expression

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2 Microscopic Electromagnetics

d dt

bL · n ˆ da =

S

S

∂bL ·n ˆ da + ∂t

(bL × V) · dl.

(2.47)

C

With this result, Faraday’s law (2.43) may then be written in the form 1 ∂bL ·n ˆ da. (2.48) [e − 1/c(V × bL )] · dl = − c C S ∂t Equation (2.48) is an equivalent statement of Faraday’s law applied to a uniformly moving circuit C with constant velocity V. The electric ﬁeld intensity e appearing in Eq. (2.48) is measured with respect to the moving reference frame in which the contour C is at rest. An alternate interpretation of this situation, however, is to view the contour C and surface S as being instantaneously located at a certain position in space in the laboratory frame of reference. Application of Faraday’s law (2.43) to that circuit results in the expression 1 d eL · dl = − bL · n ˆ da, (2.49) c dt S C where eL is the microscopic electric ﬁeld vector in the laboratory reference frame. The assumption of Galilean invariance (i.e., that physical laws should be invariant under a Galilean transformation5 ) then implies that the left-hand sides of Eqs. (2.48) and (2.49) must be equal. This in turn implies that the electric ﬁeld intensity e in the moving coordinate system of the circuit C is given by 1 e(r, t) = eL (r, t) + V × bL (r, t). (2.50) c This is then the Galilean transformation of the electric ﬁeld intensity. A charged test particle q at rest in the moving coordinate frame of the circuit C then experiences the force F = qe. When viewed from the laboratory frame of reference, the charge represents a microscopic current density vector j = qVδ(r−r0 ), where δ(R) is the three-dimensional Dirac delta function and where r0 = r0 (t) denotes the position of the test particle in the laboratory frame; it then experiences the Lorentz force density 1 (2.51) f = qδ(r − r0 ) eL (r, t) + V × bL (r, t) , c in agreement with the result implied by Eq. (2.50) taken together with the force relation f = qeδ(r − r0 ). Galilean invariance is not rigorously valid, however, but holds (to a good degree of approximation) only for velocities 5

The classical concept of Galilean invariance states that physical phenomena are the same when viewed by two observers moving with a constant velocity V relative to each other, provided that the coordinates in space and time are related by the Galilean transformation r = r + Vt, t = t.

2.1 The Microscopic Maxwell–Lorentz Theory

63

that are small in comparison to the vacuum speed of light c. As a consequence, the above considerations are valid only for nonrelativistic velocities and the transformation relation given in Eq. (2.50) for the electric ﬁeld intensity is correct only to ﬁrst order in V /c, but is in error by terms of order V 2 /c2 . Consider now the second curl relation (2.34) of the microscopic Maxwell’s equations. Upon integration of this relation over any given regular open surface S that is bounded by a closed regular contour C, followed by application of Stokes’ theorem to the surface integral of ∇ × h(r, t), there results 4π 1 ∂d(r, t) ·n ˆ da, (2.52) h(r, t) · dl = j(r, t) · n ˆ da + c c ∂t C S S where the integration about the contour C is traversed in the positive sense. The total current J (R, t) linking the contour C is given by [cf. Eq. (2.9)] J (R, t) = j(r, t) · n ˆ da, (2.53) S

where the vector R appearing in the argument of the total current depends upon both the position vector r appearing in the argument of the current density vector j(r, t) as well as upon the shape and orientation of the surface of integration S appearing in Eq. (2.53). As in the preceding analysis, the partial diﬀerential operator ∂/∂t may be brought out from under the surface integral in Eq. (2.52) and replaced by the total time derivative (by Leibniz’ rule), resulting in the ﬁnal expression 4π 1 d h(r, t) · dl = j(r, t) · n ˆ da + d(r, t) · n ˆ da, (2.54) c c dt S C S which is known as Amp`ere’s law [15]. This expression is valid whatever the cause of the electric ﬂux variation. Again, a rigorous proof of the complete generality of this result must be based upon the electrodynamics of moving media [14]. In the steady state, the time derivative of the surface integral appearing on the right-hand side of Amp`ere’s circuital law (2.54) is zero and the total convection current J through any open regular surface S is equal to c/4π times the circulation of the microscopic magnetic intensity vector h about the contour C that forms the boundary of the open surface S. However, if the microscopic electric ﬁeld is variable in time, the vector 1/4π∂d/∂t has associated with it a microscopic magnetic intensity vector h(r, t) exactly equal to that which would be produced by a microscopic current distribution with density j (r, t) = 1/4π∂d(r, t)/∂t; this term is accordingly called Maxwell’s displacement current.6 6

An elementary discussion of the necessity of the displacement current in Amp`ere’s law may be found in §15.1 of Reitz and Milford’s undergraduate text Foundations of Electromagnetic Theory [16].

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2 Microscopic Electromagnetics

The microscopic magnetic intensity vector h appearing in Eq. (2.54) is the magnetic ﬁeld at the path element dl along the contour C in the coordinate system in which dl is at rest. Let the contour C move with a constant velocity V in some ﬁxed direction; the total time derivative appearing in Eq. (2.54) is then given by the convective derivative that is given in Eq. (2.46). The total time derivative of the electric ﬂux linking the contour C is then given by (where the subscript L is again used to denote the laboratory frame of reference in which the ﬁeld measurements are made) ∂dL d ·n ˆ da + dL · n ˆ da = [(V · ∇) dL ] · n ˆ da dt S ∂t S S ∂dL ·n ˆ da + (dL × V) · dl + 4π ρV · n = ˆ da, S ∂t C S where the divergence relation given in Eq. (2.35) has been employed in the ﬁnal surface integral appearing on the right-hand side of the above equation. In addition, the total current J linking the contour C as measured in the moving reference frame is augmented by the convection current with density jV = ρV, so that ρV · n ˆ da, (2.55) J = JL − S

where JL is the current measured in the laboratory frame of reference. With these two results, Eq. (2.54) may then be written in the form 4π 1 1 ∂dL ·n ˆ da. (2.56) h − (dL × V) · dl = JL + c c c C S ∂t This equation is an equivalent statement of the integral representation (2.53) of the microscopic Maxwell equation (2.34) applied to a uniformly moving contour C. The magnetic intensity vector h = h(r, t) is measured here in the moving frame in which the contour C is at rest. An alternate interpretation of this situation, however, is to view the contour C and the associated surface S as being instantaneously located at a certain position in space in the laboratory coordinate system. Application of Eq. (2.52) to that ﬁxed contour then yields the expression 4π 1 ∂dL ·n ˆ da, (2.57) hL · dl = JL + c c C S ∂t where hL is the microscopic magnetic ﬁeld intensity in the laboratory reference frame. The assumption of Galilean invariance implies that the left-hand sides of Eqs. (2.56) and (2.57) must be equal. This in turn implies that the microscopic magnetic intensity vector h in the moving coordinate system of the circuit is related to that in the laboratory frame through the relation

2.1 The Microscopic Maxwell–Lorentz Theory

1 h(r, t) = hL (r, t) − V × dL (r, t). c

65

(2.58)

This is then the Galilean transformation of the magnetic ﬁeld intensity. Just as for Eq. (2.50), this expression is only correct to ﬁrst order in the quantity V /c but, in accordance with the special theory of relativity, is found to be in error by terms of order V 2 /c2 . In order to illustrate the inadequacy of Galilean relativity, consider a convective current density j = ρVc in the moving coordinate frame. In the laboratory frame of reference this corresponds to the convective current density jL = ρ(Vc + V) and the Lorentz force density acting upon it in the laboratory frame is given by 1 (2.59) fL = ρeL + jL × bL . c In the moving coordinate frame the current density experiences the force density 1 (2.60) f = ρe + ρVc × b, c which, with substitution from the Galilean transformation relations given in Eqs. (2.50) and (2.58), becomes 1 1 1 f = ρ eL + V × bL + ρVc × bL − V × eL c c c 1 1 = ρeL + ρ(Vc + V) × bL − 2 ρVc × (V × eL ) c c 1 1 (2.61) = ρeL + jL × bL − 2 ρVc × (V × eL ). c c These two expressions [(2.59) and (2.61)] for the force density diﬀer by an amount that is on the order of Vc V /c2 , which is a direct consequence of the assumption of Galilean invariance. The violation of invariance of the force relation in the Maxwell–Lorentz theory with Galilean relativity is resolved by the special theory of relativity. Classical electrodynamics is indeed consistent with special relativity wherein the Maxwell–Lorentz equations are invariant under a Lorentz transformation. In fact, H. A. Lorentz [17] originally arrived at this transformation by requiring the invariance of the Maxwell–Lorentz equations between any two inertial reference frames. Inconsistencies such as that raised above by calculating the microscopic force density due to a magnetic ﬁeld acting upon a convective current density in two diﬀerent inertial reference frames that are moving with a relative velocity V are resolved by the fact, shown so clearly in the special theory of relativity, that magnetic and electric ﬁelds have no separate meaning; instead, special relativity implies the single uniﬁed concept of an electromagnetic ﬁeld. In general, a ﬁeld that is purely electric, or

66

2 Microscopic Electromagnetics

purely magnetic, in one inertial frame will have both electric and magnetic ﬁeld components in another inertial reference frame. The two remaining diﬀerential ﬁeld relations given in Eqs. (2.35) and (2.36) can easily be expressed in an equivalent integral form with the aid of the divergence theorem. Accordingly, let R be any regular region of space that is bounded by a closed surface S; integration of Eq. (2.36) over the region R and application of the divergence theorem to the resultant volume integral of ∇ · b then results in the expression b(r, t) · n ˆ da = 0, (2.62) S

which is Gauss’ law [18] for the magnetic ﬁeld, where n ˆ denotes the positive (outward directed) unit normal vector to the surface element da. Hence, the total ﬂux of the microscopic magnetic induction vector b(r, t) crossing any closed regular surface is zero. Similarly, upon integration of Eq. (2.35) over the region R and application of the divergence theorem to the resultant volume integral of ∇ · d, one obtains the expression d(r, t) · n ˆ da = 4π ρ(r, t)d3 r = 4πq(t). (2.63) S

R

which is Gauss’ law [18] for the electric ﬁeld. Hence, the total ﬂux of the microscopic electric displacement vector d(r, t) crossing any given closed regular surface S is equal to the total charge q(t) contained within the region R that is bounded by that surface. As a summary of these results, the time-domain integral form of the microscopic Maxwell equations are given by [from Eqs. (2.43), (2.54), (2.62), and (2.63)] 1 d e(r, t) · dl = − b(r, t) · n ˆ da, (2.64) c dt S C 4π 1 d h(r, t) · dl = j(r, t) · n ˆ da + d(r, t) · n ˆ da,(2.65) c c dt S S C d(r, t) · n ˆ da = 4π ρ(r, t)d3 r, (2.66) S R b(r, t) · n ˆ da = 0. (2.67) S

In the ﬁrst pair of these equations, S is any regular open surface that is bounded by a closed regular contour C, while in the second pair of equations R is any regular region of space that is bounded by a closed surface S. Unlike the diﬀerential (or point) form of Maxwell’s equations [given in Eqs. (2.33)– (2.36)] which require that the ﬁeld vectors are piecewise continuous with piecewise continuous ﬁrst partial derivatives, the integral (or space) form of Maxwell’s equations are valid over any region of space. They are then the more general form because they no longer require that the ﬁeld properties be everywhere well behaved.

2.2 Invariance of the Maxwell–Lorentz Equations

67

2.2 Invariance of the Maxwell–Lorentz Equations in the Special Theory of Relativity The failure of the expected invariance of the Maxwell–Lorentz theory under a Galilean transformation in Newtonian relativity eventually led to the development of the special theory of relativity. This special theory describes physical phenomena as measured by diﬀerent observers in reference frames that are moving at a constant velocity relative to each other, appropriately referred to as inertial reference frames. This special theory then did away with the “Newtonian” idea of an absolute frame of reference with respect to which the absolute dynamical state of any physical system could be speciﬁed. This idea of an absolute reference frame resulted in the introduction of a socalled “luminiferous ether” through which electromagnetic waves propagated. Originally introduced by Thomas Young [19] in order to explain Bradley’s discovery of stellar aberration [20] in terms of the wave theory of light, the idea of a “luminiferous ether” was widely accepted at that time and was adopted by Maxwell [1] as an essential part of his theory of electromagnetic waves. Based upon the negative experimental results of Trouton and Noble [21] to detect the eﬀects of the velocity of the Earth through this luminiferous ether, as well as upon the independent mathematical arguments of Lorentz [17] and Poincar´e [22] (who had coined the phrase “principle of relativity”), Einstein [23] was led to the conclusion that “the same laws of electrodynamics and optics will be valid for all frames of reference for which the equations of mechanics hold good.” He then went on to raise this conjecture (the purport of which will hereafter be called the ‘Principle of Relativity’) to the status of a postulate, and also introduce another postulate, which is only apparently irreconcilable with the former, namely, that light is always propagated in empty space with a deﬁnite velocity c which is independent of the state of motion of the emitting body. These two postulates suﬃce for the attainment of a simple and consistent theory. Finally, he concluded that the “introduction of a ‘luminiferous ether’ will prove to be superﬂuous inasmuch as the view here to be developed will not require an absolutely stationary space provided with special properties...” These results are summarized in the following two postulates of special relativity theory: 1. The Fundamental Postulate of Special Relativity: The laws of physics are the same in all inertial reference frames. 2. The Postulate of the Constancy of the Speed of Light: The speed of light in free space has the same value c in all inertial systems. As an immediate consequence of the ﬁrst postulate, no preferred inertial reference system exists. Because of this, it is then physically impossible to

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2 Microscopic Electromagnetics

detect the uniform motion of one inertial frame of reference from observations that are made entirely within that reference frame. With regard to the fundamental postulate of special relativity, the identical formulation of the laws of physics in all inertial reference frames leads to two fundamental concepts that depend upon the manner in which the equations of physics transform from one inertial reference frame to another. Equations that do not change in form under a transformation from one inertial reference frame to another are said to be Lorentz invariant, whereas equations that remain valid because their terms, which are not all invariant, transform according to identical laws are said to be Lorentz covariant. Lorentz covariance then states that, in two diﬀerent inertial reference frames located at the same event in space–time, all nongravitational laws of physics must yield the same results for identical experiments. These two postulates immediately lead to a re-examination of the concept of simultaneity. Because the ﬁrst postulate requires that the concept of a universal time must be abandoned, the second postulate then means that the only way that simultaneity can be deﬁned is in terms of the velocity of light c. This then elevates c to a level of signiﬁcance that is much more fundamental than its property of being the propagation velocity of an electromagnetic wave in free space. It is in fact the maximum velocity of propagation of any interaction and its ﬁnite value then excludes instantaneous action at a distance. The fundamental concept of simultaneity is then deﬁned in the following manner [24]: two instants of time t1 and t2 that are observed at two points P1 and P2 in a particular inertial reference frame are said to be simultaneous if an impulse of light emitted at the geometric midpoint of the line connecting P1 and P2 arrives at the point P1 at the time t1 and at the point P2 at the time t2 . This deﬁnition guarantees that an impulse of light that is emitted from a point source will reach all equidistant points from that source point simultaneously in a given inertial reference frame so that its phase surface is spherical in that reference frame. Notice that the simultaneity of two events at two diﬀerent points in an inertial reference frame has no signiﬁcance independent of the frame. 2.2.1 Transformation Laws in Special Relativity The proper relationship (i.e., one that satisﬁes the postulates of special relativity) between the space–time coordinates (r, t) = (x, y, z, t) of some event as observed in an inertial coordinate frame Σ and the space–time coordinates (r , t ) = (x , y , z , t ) of that same event as observed in another inertial coordinate frame Σ provides the fundamental transformation laws of the special theory of relativity. This relationship is provided by the equations of transformation as formally expressed by the functional relations x = x (x, y, z, t), y = y (x, y, z, t), z = z (x, y, z, t), and t = t (x, y, z, t) which relate the space–time coordinates of an event in the inertial coordinate frame Σ with those of the same event as observed in another inertial coordinate frame Σ .

2.2 Invariance of the Maxwell–Lorentz Equations

69

With the assumption that space and time are both homogeneous (i.e., that all points in space–time are equivalent), these equations of transformation must then be linear. The connection between space-time points in the two inertial reference frames is obtained from the postulate of the constancy of the velocity of light c. Consider an impulse of light that is emitted as a spherical wave from the origin O in the inertial reference frame Σ at the instant t = 0 and assume that the origin O of the inertial reference frame Σ coincides with O at that precise instant of time. The emitted spherical light wave then propagates with the velocity c in all directions in each inertial frame Σ and Σ , so that (2.68) c2 t2 − x2 + y 2 + z 2 = c2 t2 − x2 + y 2 + z 2 . This relationship then provides the connection between the coordinates of any given event as observed in the two inertial refernce frames Σ and Σ . When reference frame Σ is moving with constant velocity V with magnitude V = |V| in some arbitrary (but ﬁxed) direction relative to the inertial reference frame Σ, the most general form of these transformation equations is found to be given by [25] V·r r = r + V (γ − 1) − γt , (2.69) V2 V·r ct = γ ct − , (2.70) c where γ≡

1 1 − β2

(2.71)

with β ≡ V /c denoting the magnitude of the normalized velocity. The inverse coordinate transformation is obtained by interchanging primed and unprimed quantities and by replacing V with −V (the velocity of Σ relative to Σ ), with the result V · r r = r + V , (2.72) (γ − 1) + γt V2 V · r ct = γ ct + . (2.73) c These relations are known as the Poincar´e–Lorentz transformation relations. Notice that the transformation relation given in Eq. (2.69) may be separated into components parallel () and perpendicular (⊥) to the velocity vector V as r = γ(r − Vt), r ⊥ = r⊥ .

(2.74) (2.75)

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2 Microscopic Electromagnetics

Similar expressions may also be written for the inverse transformation given in Eq. (2.72). The Poincar´e–Lorentz transformation relations given in Eqs. (2.69)–(2.70) may be expressed in matrix form as

Σ rΣ = QΣ Σ r ,

(2.76)

where rΣ = (x1 , x2 , x3 , x4 ) ≡ (x, y, z, ct) and rΣ = (x1 , x2 , x3 , x4 ) ≡ (x , y , z , ct ) are the row vector forms of the column vectors appearing in Eq. (2.68) with coordinate transformation matrix ⎞ ⎛ V2 V V 1 + Vx2 (γ − 1) Vx 2y (γ − 1) VVx V2z (γ − 1) − Vcx γ ⎜ V V ⎟ 2 ⎜ y x (γ − 1) 1 + Vy (γ − 1) Vy Vz (γ − 1) − Vy γ ⎟ ⎟. ⎜ V2 V2 V2 2 c (2.77) QΣ = Σ ⎟ ⎜ Vz Vx ⎝ V 2 (γ − 1) VVz V2y (γ − 1) 1 + VVz2 (γ − 1) − Vcz γ ⎠ V −γ cy −γ Vcz γ −γ Vcx The inverse transformation relation is then given by

Σ rΣ = QΣ Σ r ,

(2.78)

Σ −1 where QΣ is the inverse of the transformation matrix appearing Σ = (QΣ ) in Eq. (2.77); that is, Σ Σ Σ (2.79) QΣ Σ QΣ = QΣ QΣ = I,

where

⎛

1 ⎜0 I ≡ (δij ) = ⎜ ⎝0 0

0 1 0 0

0 0 1 0

⎞ 0 0⎟ ⎟ 0⎠ 1

(2.80)

is the identity matrix or idemfactor. The speciﬁc form of the Poincar´e–Lorentz transformation equations is a result of the invariance of the relation given in Eq. (2.68) for the equation of a spherical light wave whose radius increases with time at a rate given by the invariant vacuum speed of light c. As a consequence, the quadratic form ds2 = c2 dt2 − (dx2 + dy 2 + dz 2 )

(2.81)

for the world distance ds between two neighboring events with space–time coordinates (r, t) = (x, y, z, t) and (r+dr, t+dt) = (x+dx, y+dy, z+dz, t+dt) is invariant under the Poincar´e–Lorentz transformation. In fact, the Poincar´e– Lorentz transformation relations are the only nonsingular transformation relations that possess a unique inverse and which maintain the invariance of the quadratic form for the world distance [26]. The world distance s between two events is accordingly said to be either timelike, lightlike, or spacelike depending on whether the squared interval s2 is either positive, zero, or negative, respectively.

2.2 Invariance of the Maxwell–Lorentz Equations

71

A simple test of the physical propriety of the Poincar´e–Lorentz formations is that they should reduce to the approximate Galilean formation equations when V /c 1. Indeed, in the limit as V /c → transformation equation (2.74) becomes V2 r = r − Vt 1 + 2 + · · · → r − Vt, 2c

transtrans0, the

(2.82)

and the transformation equation (2.73), when applied to the motion of the origin O of Σ , for example, which is given by r = Vt, becomes V2 V2 t =t 1− 2 1 + 2 + · · · → t. (2.83) c 2c These are just the classical Galilean transformation equations which are then seen to be approximately valid only in the nonrelativistic limit β 1. The ﬁrst important consequence of the Poincar´e–Lorentz transformations is the phenomenon of time dilation. Consider a clock that is at rest at the point x in the Σ reference frame which is moving with velocity V = ˆ 1x V along the common x − x axis of the Σ and Σ inertial frames. Let two successive events at that point in the Σ frame span the time interval from t to t + ∆t . An observer in the Σ frame observes these two successive events as occurring at the two instances of time given by [cf. Eq. (2.73)] t + (V /c2 )x , 1 − β2 (t + ∆t ) + (V /c2 )x . t2 = 1 − β2

t1 =

A single clock at the ﬁxed position x in the Σ frame is used to measure the time interval from t to t + ∆t while the corresponding time instances t1 and t2 = t1 + ∆t in the Σ frame are measured by two diﬀerent clocks that are synchronized, one stationary Σ-clock that is coincident with the moving Σ -clock at the beginning of the time interval and another stationary Σ-clock that is coincident with the moving Σ -clock at the end of the time interval. Because ∆t = t2 − t1 , the two time intervals are then related by ∆t ∆t = . 1 − β2

(2.84)

As a consequence of this time dilation relation, the duration of a measured time interval is a minimum in the inertial reference frame in which the clock is at rest relative to the observer. When a clock moves with a velocity V relative to the observer, its rate is measured by that observer to have slowed 2 down by the factor 1 − β because any measured time interval is increased 2 by the factor 1/ 1 − β .

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2 Microscopic Electromagnetics

If dτ represents a proper diﬀerential time interval, then the nonproper diﬀerential time interval dt is given by dt =

dτ 1 − β2

= γdτ,

(2.85)

where γ = 1/ 1 − β 2 . For small values of the normalized velocity β = V /c the dilation factor γ is close to unity and dt ≈ dτ , but as β exceeds the approximate value 0.9 the dilation factor begins to rapidly increase, as seen in Figure 2.4.

Dilation Factor Γ

101

100 0

0.5 ΒV/c

1.0

Fig. 2.4. Dependence of the relativistic time dilation factor γ = 1/ 1 − β 2 on the normalized velocity β = V /c. Notice that γ → ∞ in the limit as β → 1− .

The convective or substantial derivative operator D/Dt that is given by [cf. Eq. (2.46)] ∂ D = + V · ∇, (2.86) Dt ∂t relates the rate of change of any given variable in an inertial reference frame Σ that is moving with a ﬁxed velocity V relative to another inertial reference frame Σ. This diﬀerential operator was used in Section 2.1.2 in the derivation of the classical or Galilean transformation relations for the electric and magnetic ﬁeld vectors in the classical Maxwell–Lorentz theory. In the relativistic domain of the Poincar´e–Lorentz transformation this diﬀerential operator is replaced [25] by the partial diﬀerential operator ∂/∂t. From the expression given in Eq. (2.84) for the time dilation that is observed between the two inertial reference frames, it is seen that

2.2 Invariance of the Maxwell–Lorentz Equations

∂ D 1 . = ∂t 1 − β 2 Dt With substitution from Eq. (2.86) the general expression 1 ∂ ∂ + V · ∇ = ∂t 1 − β 2 ∂t

73

(2.87)

(2.88)

results, which reduces directly to the classical expression given in Eq. (2.46) in the nonrelativistic limit β → 0+ . The next important consequence of the Poincar´e–Lorentz transformations is the phenomenon of Lorentz–Fitzgerald contraction. Again, let the Σ reference frame move with velocity V = ˆ 1x V along the common x − x axis of the Σ and Σ inertial frames. Consider a measuring rod that is stationary in the Σ inertial reference frame and is lying along the x -axis of that frame. Let its end points be at the coordinate positions x1 and x2 ≥ x1 so that its length in this proper frame, referred to as its proper length, is given by the diﬀerence ∆x = x2 − x1 . From the Poincar´e–Lorentz transformation given in Eq. (2.74), xj = γ(xj − V tj ) for j = 1, 2 so that their diﬀerence is given by x2 − x1 = γ[(x2 − x1 ) − V (t2 − t1 )]. The length of the rod in the Σ inertial reference frame is given by the distance between the coordinates x2 and x1 of the two end points measured at the same instant of time in that inertial frame. With t2 = t1 , the preceding relation then yields the Lorentz–Fitzgerald contraction (2.89) ∆x = 1 − β 2 (∆x ) . The measured length ∆x of the moving rod is then contracted from its proper length ∆x by the factor 1 − β 2 . This contraction of length was put forth independently by both FitzGerald and Lorentz [27] (who acknowledged FitzGerald’s unpublished speculation on this eﬀect) in order to explain the negative result of the Michelson–Morley experiment to measure the supposed eﬀects of the luminiferous ether on a moving body. For the dimensions of the rod along the y - and z -coordinate directions, which are perpendicular to the relative motion of the two inertial reference frames, the transformation relation given in Eqs. (2.75) immediately shows that the measurement of these dimensions is the same in both frames. As a consequence, a given body’s length is a maximum when it is measured by an observer at rest with respect to it. When the body moves with a velocity V relative to the observer, its measured length is contracted in the direction of its motion by the factor 1 − β 2 , while its dimensions perpendicular to the direction of the motion are unaﬀected. The general relativistic velocity transformation relation of a rigid body moving with velocity u in the Σ inertial reference frame that is moving with ﬁxed velocity V relative to the inertial reference frame Σ is obtained from the Poincar´e–Lorentz transformation relation (2.72) with the inverse of Eq. (2.88) as

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2 Microscopic Electromagnetics

u u=

1 − β 2 + V (u · V/V 2 ) 1 − 1 − β 2 + 1 1 + u · V/c2

.

(2.90)

The corresponding inverse transformation is then obtained by interchanging primed and unprimed quantities and by replacing V with −V in the above relation to obtain u 1 − β 2 + V (u · V/V 2 ) 1 − 1 − β 2 − 1 u = . (2.91) 1 − u · V/c2 The transformation relation given in Eq. (2.90) may be separated into components parallel () and perpendicular (⊥) to the velocity vector V as u =

u + V

, 1 + u · V/c2 1 − β2 u⊥ = u ⊥ , 1 + u · V/c2

(2.92) (2.93)

with similar expressions for the inverse transformation. It is then seen that each perpendicular (or transverse) component of the velocity of an object as measured in the Σ inertial reference frame is related to the corresponding transverse velocity component as well as to the parallel velocity component of the object in the Σ reference frame. The complexity of the resultant transformation relations exhibited in Eqs. (2.90)–(2.91) is a direct consequence of the fact that neither reference frame is a proper one. A case of special interest occurs when an inertial reference frame Σ is chosen such that u = 0. The parallel and transverse velocity transformation equations then become u = V, u⊥ = 1 − β 2 u ⊥ . In this case there is no length contraction involved and the transverse velocity transformation is due to the eﬀects of time-dilation alone. The relativistic acceleration transformation equations are obtained from the velocity transformation relations by direct time diﬀerentiation as (1 − β 2 )3/2 a , (1 + u · V/c2 )3 1 − β2 u ⊥ β/c − a a , a⊥ = ⊥ (1 + u · V/c2 )2 1 + u · V/c2 a =

with inverse

(2.94) (2.95)

2.2 Invariance of the Maxwell–Lorentz Equations

(1 − β 2 )3/2 a , (1 − u · V/c2 )3 1 − β2 u⊥ β/c = a . a⊥ + (1 − u · V/c2 )2 1 − u · V/c2

a = a ⊥

75

(2.96) (2.97)

Unlike the Galilean transformation result which states that the acceleration is independent of the inertial reference frame, the correct relativistic transformation given in Eqs. (2.94)–(2.95) clearly shows that the acceleration of a rigid body depends upon the inertial reference frame in which it is measured. The classical result that a = a is obtained in the nonrelativistic limit as β → 0. 2.2.2 Transformation of Dynamical Quantities Classical Newtonian mechanics is found to be inconsistent with the special theory of relativity because, although its laws are invariant under a Galilean transformation, they are not invariant under a Poincar´e–Lorentz transformation. In addition, Newton’s third law of motion requiring that action and reaction forces must be equal has no meaning in special relativity for actionat-a-distance forces (allowed in the classical theory of Newtonian mechanics) because the simultaneity of spatially separated events depends upon the inertial reference frame from which they are observed. Finally, from Newton’s second law of motion, the action of a force on a particle in Newtonian mechanics can accelerate that particle to an unlimited velocity, in violation of the second fundamental postulate of the special theory of relativity. Consider a body of ﬁxed mass at rest in an inertial reference frame Σ. Because the mass of this body must then be the ordinary mass in nonrelativistic Newtonian mechanics, it is appropriately referred to as the rest mass or proper mass m0 of the body. If this body is now moving with velocity u in the Σ reference frame, conservation of linear momentum requires that its mass, as observed by a stationary observer in Σ, be given by its relativistic mass m0 . (2.98) m= 1 − u2 /c2 Here u is the absolute value of the velocity of the body as observed in the inertial reference frame Σ. Notice that this expression for the relativistic mass preserves the form for momentum while implicitly assuming that the relativistic mass of a body is independent of its acceleration. Notice that m → m0 in the nonrelativistic limit as u → 0, whereas in the relativistic limit as u → c from below, the relativistic mass becomes unbounded. The relativistic form of Newton’s second law of motion is then given by m u d dp 0 . (2.99) F= = dt dt 1 − u2 /c2

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2 Microscopic Electromagnetics

The conservation law of relativistic momentum then states that in the absence of external forces (F = 0), the linear momentum of the system is conserved so that the quantity p = m0 u/ 1 − u2 /c2 is a constant. The kinetic energy Uk of a particle is given by the amount of work that is done by the application of an external force F to increase its velocity from 0 up to some ﬁnal value u, so that u u d (mu) · dr. (2.100) F · dr = Uk = 0 0 dt The integrand appearing in this expression may be rewritten as d/dt(mu) · dr = d(mu) · u = mu · du + u2 dm. From the expression given in Eq. (2.98) for the relativistic mass, one obtains the equation m2 c2 − m2 u · u = m20 c2 with total diﬀerential 2mc2 dm − 2mu2 dm − 2m2 u · du = 0, which may be rewritten as mu · du + u2 dm = c2 dm. With these substitutions, Eq. (2.100) becomes m 2 dm = mc2 − m0 c2 . (2.101) Uk = c m0

With substitution from Eq. (2.98) for the relativistic mass, the ﬁnal expression for the relativistic kinetic energy of a particle with rest mass m0 is found as 1 2 Uk = m0 c −1 . (2.102) 1 − u2 /c2 In the nonrelativistic limit as β = u/c → 0, this expression becomes −1/2 Uk = m0 c2 1 − β 2 −1 3 1 = m0 c2 1 + β 2 + β 4 + · · · − 1 2 8 1 → m0 u 2 , 2 which is just the classical result for the kinetic energy of a body with mass m0 and velocity u. Notice that the omitted terms are O(β 2 ) to the nonrelativistic result.7 In the opposite limit as β → 1− , Uk → ∞ so that an inﬁnite amount of work needs to be expended in order to accelerate a particle with nonzero rest mass up to the vacuum speed of light c. With the total energy of the particle given by E = mc2 = m0 c2 + Uk , 7

(2.103)

The order symbol O is deﬁned as follows. Let f (z) and g(z) be two functions of the complex variable z that possess limits as z → z0 in some domain D. Then f (z) = O(g(z)) as z → z0 iﬀ there exist positive constants K and δ such that |f (z)| ≤ K|g(z)| whenever 0 < |z − z0 | < δ.

2.2 Invariance of the Maxwell–Lorentz Equations

77

so that Uk = (m−m0 )c2 , it is then seen that any change in the kinetic energy of a particle may be related to the change in its inertial mass. The quantity the rest energy of the particle. m0 c2 is accordingly referred to as From the expression p = m0 u/ 1 − u2 /c2 for the relativistic momentum of a particle with rest mass m0 , one ﬁnds that p2 = m20 u2 /(1 − u2 /c2 ) which may be solved for the square of the magnitude of the velocity with the result u2 = p2 /(m20 + p2 /c2 ). Substitution of this result into Eq. (2.102) then yields (Uk + m0 c2 )2 = m20 c4 + p2 c2 which may be written in terms of the total energy E = Uk + m0 c2 as 2 2 E 2 = m0 c2 + (pc) .

(2.104)

Upon expanding the equation (Uk + m0 c2 )2 = m20 c4 + p2 c2 one obtains the quadratic relation Uk2 + 2m0 c2 Uk = p2 c2 for the kinetic energy of the particle. In the nonrelativistic limit when u/c 1, the kinetic energy Uk of the particle will be much less than its rest energy m0 c2 so that the quadratic term Uk2 becomes negligible in comparison to the term linear in Uk , thereby yielding √ the classical result that p → 2mo Uk as u/c → 0. With Eq. (2.104) rewritten as E = c m20 c2 + p2 , the derivative of the total energy with respect to the magnitude of the momentum is found as dE/dp = pc2 /E. With E = mc2 and p2 = m2 u2 , this expression reduces to the elegant result dE =u (2.105) dp so that the magnitude of the velocity of a particle is given by the change in total energy with respect to the momentum of the particle. Consider now obtaining a relativistically correct expression for the acceleration of a particle due to the application of a force, where F=

d du dm (mu) = m +u . dt dt dt

(2.106)

Because m = E/c2 where E = Uk + mo c2 , then dm/dt = (1/c2 )dUk /dt. Furthermore, because dUk = F·dr [cf. Eq. (2.100)], then dUk /dt = F·dr/dt = F · u so that dm/dt = (1/c2 )F · u. Substitution of this expression into Eq. (2.106) then yields F·u du +u 2 . F=m (2.107) dt c The acceleration of the particle is then given by the general expression F·u du 1 = F− a≡ u, (2.108) dt m c2 and the acceleration is, in general, not parallel to the applied force. Two special cases of Eq. (2.108) immediately arise. The ﬁrst is the case in which the applied force F is parallel to the velocity u of the particle. Because

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2 Microscopic Electromagnetics

the vectors a, F, and u are then all parallel, Eq. (2.106) may then be written in the scalar form dm m0 d m0 du du F =m +u = +u 2 2 dt dt dt 1 − u /c dt 1 − u2 /c2 du m0 m0 du u2 /c2 = = . 1+ 2 2 2 2 3/2 2 2 1 − u /c dt dt (1 − u /c ) 1 − u /c Because a = du/dt, the above result may then be written as F =

m0 a , (1 − u2 /c2 )3/2

(2.109)

where the relativistic quantity m0 /(1 − u2 /c2 )3/2 is referred to as the longitudinal mass. The other special case is realized when the applied force F is perpendicular to the velocity u, in which case F · u = 0. Equation (2.108) then yields the expression F⊥ =

m0

a⊥ , (2.110) 1 − u2 /c2 where the relativistic quantity m0 / 1 − u2 /c2 is referred to as the transverse mass. Finally, from the relation following Eq. (2.106) it is seen that dm dUk = c2 , dt dt

(2.111)

which states that a change in the kinetic energy of a body is equal to a proportionate change in its relativistic mass. Einstein’s mass–energy relation E = mc2

(2.112)

is then seen to be an expression of the equivalence of mass and energy. Based upon his derivation of this fundamental expression, Einstein [23] hypothesized that mass and energy form a single invariant quantity that is referred to as the mass–energy. As a consequence of the mass–energy relation, because the rest mass of a body is internal energy, then a body without any mass has no internal energy, its energy being all external. If such a body with zero rest mass moved with a velocity that was less than c in any inertial reference frame, then another inertial reference frame could always be found in which it is at rest. However, if it travels at the velocity c in any inertial reference frame, then it will travel at the velocity c in every inertial reference frame. Thus, a body with zero rest mass must travel at the speed of light and can never be at rest in any inertial reference frame. Any particle moving at the speed of light is called a luxon, an important example of which is the photon. In addition, any particle that

2.2 Invariance of the Maxwell–Lorentz Equations

79

has an initial velocity that is less than c is called a baryon; such a particle can be accelerated to a velocity that approaches c but can never exceed this value. Finally, any particle whose velocity is larger than c during its entire lifetime is referred to as a tachyon. Consider now the transformation relations for the linear momentum and energy of a body with rest mass m0 . These quantities are deﬁned in the inertial reference frame Σ by the pair of relations p=

m0 u 1−

u2 /c2

,

E=

m0 c2 1 − u2 /c2

,

whereas in the inertial reference frame Σ the corresponding quantities are deﬁned by p =

m0 u 1 − u2 /c2

,

m0 c2 E = , 1 − u2 /c2

where the Σ frame is moving with the velocity V with respect to Σ. Application of the velocity transformation relation given in Eq. (2.90) together with the identity 1 1 + u · V/c2 = 1 − u2 /c2 1 − u2 /c2 1 − V 2 /c2 then yields the general transformation relations 1 V p= p +E 2 , c 1 − V 2 /c2 1 (E + V · p ). E= 1 − V 2 /c2

(2.113)

(2.114) (2.115)

The inverse transformation relations are obtained by replacing V with −V and interchanging primed and unprimed quantities, yielding 1 V p = p−E 2 , (2.116) c 1 − V 2 /c2 1 (E − V · p). (2.117) E = 1 − V 2 /c2 The transformation relations for the parallel and perpendicular components of the momentum (relative to the direction of V) are then given by the relations p = (p −EV /c2 )/ 1 − V 2 /c2 and p ⊥ = p⊥ , with similar expressions for the inverse transformation. Notice that the quantities p and E/c2 transform in precisely the same manner as do the space–time coordinates r and t of a particle [cf. Eqs. (2.72)–(2.73)]. On a more fundamental level, notice the

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2 Microscopic Electromagnetics

interdependence of momentum and energy in the special theory of relativity. If the energy and momentum are conserved in an interaction that is observed in one inertial reference frame, then these two quantities must be conserved in every inertial reference frame. In addition, if momentum is conserved, then energy must also be conserved. The transformation relations for mass follow directly from the energy and momentum transformationrelations given in Eqs. (2.114)–(2.117). Because E =mc2 with m = m0 / 1 − u2 /c2 and because E = m c2 with m = m0 / 1 − u2 /c2 , then mc2 = =

1

1 − V 2 /c2 m 1−V

2 /c2

m c2 + V · m u

c2 + u · V .

The transformation relation for the mass is then seen to be given by

with inverse

1 + u · V/c2 m = m , 1 − V 2 /c2

(2.118)

1 − u · V/c2 m = m . 1 − V 2 /c2

(2.119)

The transformation relation for the force F = dp/dt is obtained directly from those for the linear momentum p as 1 − V 2 /c2 F + V 1 − 1 − V 2 /c2 F · V/V 2 − F · u/c2 , F = 1 − u · V/c2 (2.120) where the inverse transformation relation is obtained by replacing V with −V and interchanging primed and unprimed quantities. The transformation relations for the parallel and perpendicular components of the force (relative to the direction of V) are then given by the pair of relations F − F · uV /c2 , 1 − u · V/c2 1 − V 2 /c2 = F⊥ . 1 − u · V/c2

F = F ⊥

Notice that the force F in one inertial reference frame is related to the power F · u that is developed by the force in another inertial reference frame. This then implies that power and force are interrelated in the special theory of relativity. In particular, the transformation relation (2.120) and its inverse are completed by the equation

2.2 Invariance of the Maxwell–Lorentz Equations

u · F =

(u − V) · F . 1 − u · V/c2

81

(2.121)

Notice that these transformation relations for the force reduce to the Newtonian limits F = F and u · F = u · F as V /c → 0. As a case of special interest, consider a body that is at rest at some instant of time t = t0 in the inertial reference frame Σ where it is then subjected to the force F . Because u = 0 at t = t0 in this proper frame, then the inverse of the force transformation relation given in Eq. (2.120) yields F = F and F⊥ = 1 − V 2 /c2 F ⊥ . It is then seen that the force measured in the proper reference frame for the body is greater than the corresponding force measured in any other inertial reference frame. 2.2.3 Interdependence of Electric and Magnetic Fields The simple fact that electric and magnetic ﬁelds have no separate meaning is eminently demonstrated in the special theory of relativity. In particular, a ﬁeld that is purely electric or magnetic in one inertial reference frame will, in general, have both electric and magnetic ﬁeld components in another inertial reference frame. The two “separate” concepts of an electric and a magnetic ﬁeld are then subsumed by the single unifying concept of an electromagnetic ﬁeld. The correct transformation relations for the electromagnetic ﬁeld depend upon the transformation properties of the electric and magnetic ﬁeld vectors as well as upon the transformation properties of their source terms. Consider ﬁrst the electronic charge contained within a macroscopic element of volume that is in the form of a cube whose edges have rest length l0 . If there are N electrons in the cube, then the total charge in the cube is N qe and the volume charge density is given by ρ0 = N qe /l03 . Let the charges be at rest in the inertial reference frame Σ so that their current density in that frame is simply j0 = 0. From an inertial reference frame Σ that is moving with a velocity V relative to Σ along a coordinate direction that is situated along one of the edges of the cube, that edge will have the measured length l0 1 − V 2 /c2 in Σ whereas the measured lengths of the edges along the diof this rections transverse to the motion remain unchanged at l0 . The volume “cube” as observed in the reference frame Σ is then given by l03 1 − V 2 /c2 . However, the number of electrons contained in this “cube” do not change and the charge on each electron remains ﬁxed to an observer in Σ so that the observed charge density is given by ρ = N qe /(l03 1 − V 2 /c2 ), which may be expressed in terms of the rest value ρ0 = N qe /l03 as ρ0 ρ0 = m, ρ= 2 2 m 1 − V /c 0

(2.122)

where m is the relativistic mass that is given in Eq. (2.98). This result is clearly independent of the direction of motion between the two inertial reference frames and so remains valid in the general case. As observed from the

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2 Microscopic Electromagnetics

reference frame Σ, the charges move with velocity V so that there is a measured current density of value j = ρ0 V/ 1 − V 2 /c2 , which may be expressed as ρ0 j= p, (2.123) m0 where p is the relativistic momentum of the moving charged particle. As an immediate consequence of Eqs. (2.122) and (2.123), the charge and current densities ρ and j transform in the same manner as do the mass and momentum m and p, respectively. Notice that the relation between the current density and the charge density is similar to that between space and time coordinates as expressed by the Poincar´e–Lorentz transformation given in Eqs. (2.69)–(2.73) as well as that between momentum and energy as expressed in Eqs. (2.114)–(2.117). As a consequence, just as both of the quantities c2 t2 − |r|2 = c2 τ 2 and c2 m2 − |p|2 = c2 m20 are invariant, the quantity c2 ρ2 − |j|2 = c2 ρ20 is also invariant. Consider now a simpliﬁed model of the convection current ﬂowing in a metal wire whose axis is situated along the x-axis of an inertial reference frame Σ. It is assumed here that the free electrons travel with an average drift velocity u = ˆ 1x u that is directed along the positive x-coordinate direction and that the average number N of free electrons per unit volume is equal to the number of positive ions per unit volume so that the average net charge density is zero throughout the entire volume of the wire. Although the spatial distribution of free electrons in the metal is random on a microscopic scale, because the average charge density is zero, the spatial distribution of free electrons on a macroscopic average scale must be uniform and identical to that of the positive ions. Let Σ denote the inertial reference frame in which the positive ions of the metal are at rest. The average negative charge density is then ρ− = N qe , where qe denotes the magnitude of the electronic charge, the positive charge density is ρ+ = N qe , and the net charge density is then ρ = ρ+ −ρ− = 0. However, the average negative and positive current densities are j− = N qe u and j+ = 0, respectively, so that the net current density is then j = j+ − j− = −N qe u. From the point of view of an observer in another inertial reference frame 1x V relative to Σ, the negative and Σ that is moving with a velocity V = ˆ positive charge densities are, from Eqs. (2.122)–(2.123) with Eqs. (2.119) and (2.116), given by 1 − V u/c2 ρ− , ρ− = 1 − V 2 /c2 1 ρ+ , ρ+ = 1 − V 2 /c2 respectively, where ρ− = ρ+ = N qe . With this substitution, the net charge density observed in Σ is found as

2.2 Invariance of the Maxwell–Lorentz Equations

83

N qe V u/c2 ρ = ρ+ − ρ− = , 1 − V 2 /c2 and the observer in Σ ﬁnds the wire to be positively charged. The origin of this observed net positive charge density in the inertial reference frame Σ is simply due to the relativity of simultaneity [24]; observers in the two diﬀerent inertial reference frames disagree on the simultaneity of measurements made on the end point positions of adjacent like charge pairs, thereby measuring diﬀerent average distances of separation between adjacent pairs of positive ions and adjacent pairs of conduction electrons. This then leads to diﬀerent measured values of the positive and negative charge densities, resulting in a measured net positive charge density in Σ when the net charge density is zero in Σ. In the inertial reference frame Σ where the positive ions are stationary, the net charge density is zero whereas the current density is nonzero. There is then no observed electric ﬁeld whereas the magnetic ﬁeld about the conductor is nonvanishing and the ﬁeld observed in Σ is purely magnetic. In the inertial reference frame Σ , however, there is a positive net charge density together with a net current density so that both an electric and a magnetic ﬁeld are observed. As a consequence, it is seen that whether an electromagnetic ﬁeld is purely magnetic or purely electric, or part electric and magnetic, is dependent upon the inertial reference frame from which the charge and current sources for that electromagnetic ﬁeld are observed. 2.2.4 Transformation Relations for Electric and Magnetic Fields The Lorentz force relation for a point charge q that is moving with velocity u at a particular space-time point at which the microscopic electric ﬁeld is e and the microscopic magnetic ﬁeld is b is given by [cf. Eq. (2.24)] 1 (2.124) F = q e + u × b . c From Eq. (2.120), the set of force transformation relations between an inertial reference frame Σ and another inertial reference frame Σ in which the particle experiencing the force is instantaneously at rest (u = 0) is given by F = F and F⊥ = F ⊥ /γ with γ = 1/ 1 − V 2 /c2 , where Σ moves with velocity V with respect to Σ. Because the charged particle is assumed to be initially at rest in the inertial reference frame Σ , then the observed Lorentz force on the particle is given by F = qe , whereas in the inertial reference frame Σ the corresponding Lorentz force at that instant is given by F = q(e + ||1/c||V × b). Application of the transformation relations for the parallel and perpendicular components of the force then yields the set of transformation relations for the parallel and perpendicular components of the electric ﬁeld vector

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2 Microscopic Electromagnetics

e = e , 1 e ⊥ = γ e⊥ + V × b , c

(2.125)

with inverse e = e , 1 e⊥ = γ e ⊥ − V × b . c

(2.126)

These relativistically correct transformation relations state that the component of the electric ﬁeld vector that is parallel to the relative velocity between the two inertial reference frames is unchanged, whereas the components of the electric ﬁeld vector that are perpendicular to that relative velocity vector are transformed into both electric and magnetic ﬁeld components. Attention is now turned to the transformation properties of the magnetic ﬁeld vector. Assume again that the inertial reference frame Σ moves with velocity V with respect to Σ. Let the point charge q be moving with velocity u in Σ so that the Lorentz force is given by F = q(e +||1/c||u ×b ). In the Σ inertial reference frame the velocity u of the charged particle has components u = (u + V )/(1 + u · V/c2 ) and u⊥ = 1 − V 2 /c2 u ⊥ /(1 + u · V/c2 ) parallel and perpendicular to the relative velocity V between the two inertial reference frames [cf. Eqs. (2.92)–(2.93)]. With the Lorentz force relation F = q(e + ||1/c||u × b) in Σ, substitution of the above results into the general force transformation relation and use of the transformation relations (2.125) for the electric ﬁeld then results in the set of transformation relations for the parallel and perpendicular components of the magnetic ﬁeld vector b = b , 1 b ⊥ = γ b⊥ − ||c|| 2 V × e , c

(2.127)

with inverse b = b , 1 b⊥ = γ b ⊥ + ||c|| 2 V × e . c

(2.128)

These relativistically correct transformation relations state that the component of the magnetic ﬁeld vector that is parallel to the relative velocity between the two inertial reference frames is unchanged, whereas the components of the magnetic ﬁeld vector that are perpendicular to that relative velocity vector are transformed into both electric and magnetic ﬁeld structures. Notice the similarity between the transformation relations (2.125)–(2.126) and (2.127)–(2.128) for the electric and magnetic ﬁeld vectors, respectively;

2.2 Invariance of the Maxwell–Lorentz Equations

85

the only diﬀerences are in the sign change and the additional 1/c2 factor. Finally, because these transformation relations involve all components of both the electric and magnetic ﬁeld vectors, it is once again seen that electric and magnetic ﬁelds are interdependent. In vector form, these transformation relations become [compare with Eqs. (2.50) and (2.58), respectively] 1 e·V e = (1 − γ) (2.129) + γ e + V × b , V c 1 b·V b = (1 − γ) + γ b − ||c|| 2 V × e , (2.130) V c where γ = 1/ 1 − V 2 /c2 , with inverse relations given by interchange of primed and unprimed quantities and replacement of V with −V. 2.2.5 Invariance of Maxwell’s Equations The invariance of the diﬀerential form of the microscopic Maxwell’s equations [cf. Eqs. (2.15)–(2.18)] ||4π|| ρ(r, t),

0 ∇ · b(r, t) = 0, 1 ∂b(r, t) ∇ × e(r, t) = − , c ∂t 4π 1 ∂e(r, t) , ∇ × b(r, t) = µ0 j(r, t) + 0 µ0 c c ∂t ∇ · e(r, t) =

(2.131) (2.132) (2.133) (2.134)

is now ﬁnally considered. These coupled partial diﬀerential equations for the microscopic electric and magnetic ﬁeld vectors hold at any given space-time point (r, t) in the inertial reference frame Σ. The same form of these relations must also hold in an inertial reference frame Σ that is moving at a ﬁxed velocity V relative to Σ so that ||4π|| ρ (r , t ),

0 ∇ · b (r , t ) = 0, 1 ∂b (r , t ) , ∇ × e (r , t ) = − c ∂t 4π 1 ∂e (r , t ) , ∇ × b (r , t ) = µ0 j (r , t ) + 0 µ0 c c ∂t ∇ · e (r , t ) =

(2.135) (2.136) (2.137) (2.138)

where the space–time coordinates (r , t ) ∈ Σ and (r, t) ∈ Σ are related through the Poincar´e–Lorentz transformation, and where ∇ denotes the vector diﬀerential operator in Σ .

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2 Microscopic Electromagnetics

ˆx V relative to Σ For simplicity, let Σ move with ﬁxed velocity V = 1 along their common x − x axes. In this case, the Poincar´e–Lorentz transformation relations become (in component form) x = γ(x − V t), y = y, z = z, 2 and t = γ(t − xV /c ) with γ = 1/ 1 − V 2 /c2 . Application of the chain rule with these transformation relations then yields the set of diﬀerential relations ∂ ∂ ∂x ∂ ∂y ∂ ∂z ∂ ∂t ∂ V ∂ + + + =γ − 2 , = ∂x ∂x ∂x ∂x ∂y ∂x ∂z ∂x ∂t ∂x c ∂t ∂x ∂ ∂ ∂y ∂ ∂z ∂ ∂t ∂ ∂ = + + + = , ∂y ∂y ∂x ∂y ∂y ∂y ∂z ∂y ∂t ∂y ∂ ∂y ∂ ∂z ∂ ∂t ∂ ∂ ∂x ∂ + + + = , = ∂z ∂z ∂x ∂z ∂y ∂z ∂z ∂z ∂t ∂z ∂ ∂ ∂y ∂ ∂z ∂ ∂t ∂ ∂ ∂x ∂ + + + = γ − V . = ∂t ∂t ∂x ∂t ∂y ∂t ∂z ∂t ∂t ∂t ∂x Substitution of these relations into the y-component of Eq. (2.133), viz. 1 ∂by ∂ez ∂ex − = − , ∂z ∂x c ∂t then results in the partial diﬀerential relation 1 ∂by ∂ez ∂ex V ∂ez ∂by −γ − 2 = − γ −V , ∂z ∂x c ∂t c ∂t ∂x which may be rewritten as 1 ∂ 1 ∂ V ∂ex V b − + + ||c|| e γ e = − γ b . z y z c ∂t c y ∂z ∂x c2 Upon comparison of this relation with the y -component of Eq. (2.137), it is seen that these components of the electric and magnetic ﬁeld vectors must then satisfy the transformation relations ex = ex , 1 ez = γ ez + V by , c V by = γ by + ||c|| 2 ez , c in agreement with the corresponding relations given in Eqs. (2.125) and (2.127). Similarly, substitution of the above transformation relations for the partial diﬀerential operators into the z-component of Eq. (2.133), viz. 1 ∂bz ∂ey ∂ex − = − , ∂x ∂y c ∂t

2.2 Invariance of the Maxwell–Lorentz Equations

87

then results in the partial diﬀerential relation 1 ∂bz ∂ey ∂ex V ∂ey ∂bz − 2 − = − γ −V γ , ∂x c ∂t ∂y c ∂t ∂x which may be rewritten as 1 1 ∂ V ∂ex ∂ V b γ e − γ b . − = − − ||c|| e y z y c z c ∂t ∂x ∂y c2 Comparison of this relation with the z -component of Eq. (2.137) then gives the additional transformation relations 1 ey = γ ey − V bz , c V bz = γ bz − ||c|| 2 ey , c in agreement with the corresponding relations given in Eqs. (2.125) and (2.127). Finally, substitution of the above transformation relations for the partial diﬀerential operators into Eq. (2.132) ∂bx ∂by ∂bz + + =0 ∂x ∂y ∂z results in the partial diﬀerential relation ∂bx V ∂bx ∂bz ∂by − + = 0. + γ 2 ∂x c ∂t ∂y ∂z Substitution of the inverses of the above transformation relations for by and bz then gives V ∂bx ∂ V V ∂bx ∂ − 2 + by − ||c|| 2 ez + bz + ||c|| 2 ey = 0, ∂x c ∂t ∂y c ∂z c which may be rewritten as ∂by ∂bx ∂b V + + z − ||c|| 2 ∂x ∂y ∂z c

∂ey 1 ∂bx ∂ez − + = 0. ∂y ∂z c ∂t

Comparison of this expression with that given in Eq. (2.136) then gives the ﬁnal transformation relation bx = bx , in agreement with the corresponding relation given in Eq. (2.127). The same set of transformation relations for the components of the electric and magnetic ﬁeld vectors is obtained using Eqs. (2.131) and (2.134) when the appropriate transformation relations for the source terms are employed. Thus, the microscopic Maxwell equations are invariant in form under a Poinca´re–Lorentz transformation when the ﬁeld vectors are transformed by the relations speciﬁed in Eqs. (2.129)–(2.130) and the charge and current sources are transformed in the manner required by Eqs. (2.122) and (2.123).

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2 Microscopic Electromagnetics

2.3 Conservation Laws for the Microscopic Electromagnetic Field The mathematical formulation of the laws of conservation of energy and momentum for the microscopic electromagnetic ﬁeld are of fundamental importance to the physical interpretation of electromagnetic phenomena. Of particular interest here is the classical interpretation of these microscopic conservation laws for the combined system of charged particles and ﬁelds in the Maxwell–Lorentz theory. 2.3.1 Conservation of Energy and Poynting’s Theorem As described in Section 2.1, matter is assumed to be composed of both positive and negative charged particles that are characterized by their mass m and their (nonnegative) microscopic charge density, ρ+ for a positive charge and ρ− for a negative charge. The kinematics of these two types of charged particles are respectively described by their convective current densities j+ = ρ+ v+ and j− = ρ− v− . The electromagnetic forces that act upon these charges are simply the Lorentz forces with densities at each point of space and time given by 1 f+ (r, t) = ρ+ (r, t)e(r, t) + j+ (r, t) × b(r, t), c 1 f− (r, t) = −ρ− (r, t)e(r, t) − j− (r, t) × b(r, t). c There are also additional unspeciﬁed forces of a nonelectromagnetic nature. Let V be any simply connected region of space containing charges that are speciﬁed by the microscopic densities ρ+ (r, t), ρ− (r, t), j+ (r, t), and j− (r, t). The rate at which the Lorentz forces acting inside the region V do work on the enclosed charged particles is then given by (f+ · v+ + f− · v− ) d3 r = (ρ+ v+ − ρ− v− ) · e d3 r V V = j · e d3 r, V

where j = j+ − j− is the total convective current density. The microscopic magnetic ﬁeld is absent in this expression because terms involving b contain triple products with repeated factors and accordingly vanish. If the particles in the region V are enumerated by the index k, and if the velocity of the kth particle is vk and the nonelectromagnetic force acting on it is Fk , then the rate at which all forces acting in the region V do work (i.e., the power generated by them) is given by

2.3 Conservation Laws for the Microscopic Electromagnetic Field

P =

Fk · vk +

k

j · e d3 r.

89

(2.139)

V

This equation provides a local form of the energy theorem in the sense that it involves the forces acting on each individual particle within a given region V of space. Consider now obtaining an alternate form of the volume integral appearing on the right-hand side of Eq. (2.139) so that it involves only the microscopic electromagnetic ﬁeld vectors within the region V . From the diﬀerential form of Amp´ere’s law given in Eq. (2.34) one obtains the expression c 1 ∂d j · e = e · (∇ × h) − e · , (2.140) 4π 4π ∂t and from the diﬀerential form of Faraday’s law given in Eq. (2.33) there results c 1 ∂b 0 = − h · (∇ × e) − h · . (2.141) 4π 4π ∂t Addition of these two equations then yields the expression c ∂ 1 0 e2 + µ0 h2 j·e=− − ∇ · e × h , ∂t 4π 2 4π

(2.142)

which is simply a consequence of the microscopic ﬁeld equations for the electromagnetic ﬁeld. If one then deﬁnes the electromagnetic energy density by the quantity 1 1 u(r, t) ≡ 0 e2 (r, t) + µ0 h2 (r, t) (2.143) 2 4π so that the total ﬁeld energy in a region V is given by 2 1 1 U (t) ≡

0 e (r, t) + µ0 h2 (r, t) d3 r, u(r, t)d3 r = 2 4π V V and the microscopic Poynting vector as c s(r, t) ≡ e(r, t) × h(r, t), 4π

(2.144)

(2.145)

the relation appearing in Eq. (2.142) takes the form j·e=−

∂u − ∇ · s. ∂t

(2.146)

Substitution of this expression into Eq. (2.139) and application of the divergence theorem to the volume integral of ∇ · s over the regular region V produces the result dU P = Fk · vk − s·n ˆ d2 r, (2.147) − dt S k

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2 Microscopic Electromagnetics

where S is the closed surface bounding the regular region V and where n ˆ is the unit outward normal vector to S. Equation (2.147) is a form of the local energy theorem that is mathematically equivalent to the expression given in Eq. (2.139). The scalar quantity u(r, t) that is deﬁned in Eq. (2.143) can be interpreted as a measure of the density of energy in the microscopic electromagnetic ﬁeld and is, in general, a function of both position and time (i.e., it is a local instantaneous quantity). The scalar quantity U (t) that is deﬁned in Eq. (2.144) may then be interpreted as the total electromagnetic ﬁeld energy in the region V and is a function only of the time for that region. The surface integral appearing in Eq. (2.147) is then interpreted as representing a ﬂow of electromagnetic energy through the regular, closed surface S; the vector quantity s(r, t) that is deﬁned in Eq. (2.145) must then represent the microscopic ﬂux density of the electromagnetic ﬁeld energy. If that surface integral is positive, there must then be a net loss of electromagnetic energy from the region V , whereas if it is negative there must be a net gain of electromagnetic energy within the region V . The vector quantity s(r, t) is called the Poynting vector of the microscopic electromagnetic ﬁeld and is, in general, a function of both position and time at each space–time point at which the electromagnetic ﬁeld is deﬁned. The relation that appears in Eq. (2.147) is commonly referred to as Poynting’s theorem [28] which, with the aid of the local energy theorem as expressed in Eq. (2.139), may be written in the alternate form dU 3 − j · ed r = − s·n ˆ d2 r, (2.148) dt V S where j = ρv is the total convective current density in the region V . Poynting’s theorem8 for the microscopic electromagnetic ﬁeld is a mathematical statement of the conservation of energy of the combined system of charged particles and ﬁelds in the Maxwell–Lorentz theory. The total energy U (t) of a given system is an example of an extensive variable because it refers to the system as a whole. Associated with it is the intensive variable u(r, t) deﬁned in Eq. (2.144) which describes the local (microscopic) behavior. The diﬀerential form of Poynting’s theorem given in Eq. (2.146) is an example of the balance equation for an intensive variable [29]. In general, balance equations of this type naturally lead to a classiﬁcation of physical quantities into two general categories. The ﬁrst category consists of conserved quantities where the source term vanishes. An important example of a conserved quantity is given by the charge density ρ(r, t) which satisﬁes the equation of continuity [cf. Eq. (2.14)] ∂ρ(r, t) = 0, ∂t where the current density j(r, t) describes the ﬂow of the conserved quantity. When j(r, t) = 0 in Eq. (2.146) the diﬀerential form of Poynting’s theorem ∇ · j(r, t) +

8

This result was also derived by Oliver Heaviside in the same year.

2.3 Conservation Laws for the Microscopic Electromagnetic Field

91

becomes ∂u(r, t) = 0, ∂t and the energy density u(r, t) is the conserved quantity whereas the Poynting vector s(r, t) describes the ﬂow of this quantity. The other category consists of nonconserved quantities where the source term does not vanish, as occurs in Eq. (2.146) when j(r, t) = 0. The scalar quantity j(r, t) · e(r, t) appearing in Eq. (2.146) describes the work done by the electromagnetic ﬁeld on the local charge and represents a loss term in that relation. The general interpretation of the Poynting vector s(r, t) is that at any point of observation of an electromagnetic ﬁeld at which s(r, t) is diﬀerent from zero one can assert that electromagnetic energy is ﬂowing in the direction of s(r, t) such that across an elemental plane surface perpendicular to the direction of s(r, t) at that point, the rate of ﬂow of energy in the ﬁeld is 2 2 given by the quantity |s(r, t)| in erg/sec/cm (in cgs units) or joule/sec/m in (mksa units). However, it must be emphasized that the local quantities u(r, t) and s(r, t) merely provide an interpretation (albeit a useful one) of the density and transfer of energy in a given electromagnetic ﬁeld. The physically proper statement of the transfer of electromagnetic energy is embodied in Poynting’s theorem as given in either of Eqs. (2.147) or (2.148). As was ﬁrst pointed out by Thompson,9 even though the total ﬂow of electromagnetic energy through a closed surface S may be correctly represented by the surface integral of the normal component of the Poynting vector, it cannot be deﬁnitely concluded that the time rate of energy ﬂow at any given point is uniquely speciﬁed by the Poynting vector s(r, t) at that point, for one may add to the Poynting vector any solenoidal vector ﬁeld (which then integrates to zero over any closed regular surface) without aﬀecting the statement of conservation of energy that is expressed by Poynting’s theorem (2.148). Hence, there is no strictly valid justiﬁcation for the accepted interpretation of the Poynting vector except that it is useful and seldom leads to erroneous results provided that proper care is taken in its application and interpretation. When using this interpretation, however, one should keep in mind that the energy ﬂow implied by the value of the Poynting vector s(r, t) need not coincide with any intuitive notion of energy ﬂow. In particular, because s(r, t) need not vanish for a static electromagnetic ﬁeld, this interpretation can lead to a nonzero value of the energy ﬂow even in a static ﬁeld. The classical interpretation of Poynting’s theorem as a statement of the conservation of energy is then seen to depend to a considerable degree on hypothesis. Various critiques [30] as well as alternative forms [31–33] have been oﬀered but none has the advantage of greater plausibility so as to supersede the classical Poynting–Heaviside interpretation. Poynting’s theorem as embodied in Eq. (2.148) is a direct mathematical consequence of the Maxwell– Lorentz theory and the associated hypothesis of an energy density and a ﬂow ∇ · s(r, t) +

9

See J. J. Thompson, Recent Researches, pp. 251–387.

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2 Microscopic Electromagnetics

of energy in the classical electromagnetic ﬁeld has proved to be extremely useful. As stated by Stratton [34] in this context, “A theory is not an absolute truth but a self-consistent analytical formulation of the relations governing a group of natural phenomena.” With the acceptance of the validity of the Maxwell–Lorentz theory, Poynting’s theorem is clearly a valid self-consistent relationship. 2.3.2 Conservation of Linear Momentum The conservation of linear momentum in a combined system of charged particles and ﬁelds can be considered in a similar manner in the microscopic Maxwell–Lorentz theory. From the Lorentz force relation given in Eq. (2.21) and Newton’s second law of motion F = dpmech /dt, the time rate of change of the total mechanical momenta pmech of all the charged particles in a speciﬁed volume V is given by (with the assumption that no particles enter or leave the region V ) 1 dpmech = (2.149) ρe + j × b d3 r. dt c V In the same manner as taken in the derivation of Poynting’s theorem, the microscopic Maxwell’s equations given in Eqs. (2.34)–(2.35) are now employed for the purpose of eliminating the microscopic source terms ρ(r, t) and j(r, t) from Eq. (2.149), where 1 ρ = ∇ · d, 4π c 1 ∂d j = ∇ × h − . 4π 4π ∂t With these two substitutions, the integrand appearing in Eq. (2.149) becomes 1 1 1 ∂d ρe + j × b = (∇ · d)e + b × − b × (∇ × h) . c 4π c ∂t Because b × ∂d/∂t = −∂(d × b)/∂t + d × ∂b/∂t and because (∇ · b)h = 0 from Eq. (2.36), the preceding expression becomes 4π ||4π|| ρe + j × b c 1 ∂ 1 ∂b = (∇ · d)e + (∇ · b)h − (d × b) + d × − b × (∇ × h). c ∂t c ∂t Furthermore, because ||1/c||∂b/∂t = −∇ × e from Eq. (2.33), the above expression ﬁnally becomes

2.3 Conservation Laws for the Microscopic Electromagnetic Field

93

4π ||4π|| ρe + j × b c =

1 ∂ (∇ · d)e + (∇ · b)h − d × (∇ × e) − b × (∇ × h) − (d × b). c ∂t

With this result, the expression appearing in Eq. (2.149) can now be written as 1 d dpmech (d × b)d3 r + dt dt V 4πc 1 [(∇ · d)e − d × (∇ × e) + (∇ · b)h − b × (∇ × h)] d3 r. = 4π V (2.150) The volume integral appearing on the left-hand side of this equation can be formally deﬁned as the total electromagnetic momentum pem of the electromagnetic ﬁeld in the region V , so that [35] 1 (e × h)d3 r. (2.151) pem ≡ 0 µ0 4πc V The integrand of this expression, viz. 1 (e(r, t) × h(r, t)) , pem (r, t) ≡ 0 µ0 4πc

(2.152)

can then be interpreted as a density of electromagnetic momentum in the microscopic ﬁeld. Notice that this momentum density is proportional to the energy ﬂux density s(r, t), viz. pem (r, t) =

1 s(r, t), c2

(2.153)

where s(r, t) is the Poynting vector of the microscopic electromagnetic ﬁeld. In order to complete the identiﬁcation of the volume integral of pem (r, t) as the total electromagnetic momentum, and further, to establish Eq. (2.150) as a statement of the conservation of linear momentum, the volume integral appearing on the right-hand side of that expression needs to be converted into a surface integral of the normal component of a linear momentum ﬂow. With the use of the vector diﬀerential identity ∇(U · V) = U · ∇V + V · ∇U + U × (∇ × V) + V × (∇ × U) which yields, with V = U, 1 U × (∇ × U) = −U · ∇U + ∇ U 2 , 2

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2 Microscopic Electromagnetics

the terms involving the microscopic magnetic induction vector b in Eq. (2.150) can be written as 1 b(∇ · b) − b × (∇ × b) = b(∇ · b) + b · ∇b − ∇ b2 2 1 = ∇ · bb − Ib2 , 2 ˆx 1 ˆy 1 ˆz 1 ˆx + 1 ˆy + 1 ˆz is the unit dyadic or idemfactor. Similarly, the where I = 1 terms involving the microscopic electric ﬁeld intensity vector e in Eq. (2.150) can be written as 1 e(∇ · e) − e × (∇ × e) = ∇ · ee − Ie2 . 2 With these substitutions the expression appearing in Eq. (2.150) for the conservation of linear momentum for the combined system of charged particles and ﬁelds in a region V becomes d 3 (pmech + pem ) = ∇·T d r = n ˆ · T d2 r, (2.154) dt V S where S is the closed boundary surface for the regular region V and where n ˆ is the unit outward normal vector to S. The dyadic (second-rank tensor) quantity T deﬁned by 1 1 2 2 T ≡ 0 ee + µ0 hh − I 0 e + µ0 h (2.155) 4π 2 is the microscopic Maxwell stress tensor of the electromagnetic ﬁeld. In terms of the electromagnetic energy density that is deﬁned in Eq. (2.143), the Maxwell stress tensor becomes 1 (2.156) T (r, t) = ( 0 e(r, t)e(r, t) + µ0 h(r, t)h(r, t)) − Iu(r, t). 4π It is clear that T is a symmetric tensor with elements 1 1 2 2

0 e + µ0 h δij , Tij = 0 ei ej + µ0 hi hj − 4π 2

(2.157)

where δij is the Kronecker-delta function (δii = 1, δij = 0 when i = j). If the region V is taken to be all of space and if all of the components of the stress tensor T go to zero with suﬃcient rapidity such that the surface integral of the normal component of T vanishes as the surface S recedes to inﬁnity, then Eq. (2.154) yields d (pmech + pem ) = 0 dt

(2.158)

2.3 Conservation Laws for the Microscopic Electromagnetic Field

95

for all space. Hence, it is the total linear momentum in all of space that does not change with time (and hence is conserved) rather than just the mechanical linear momentum of the system of charged particles in all of space (which then need not be conserved). The quantity n ˆ · T appearing in the integrand of the surface integral of Eq. (2.154) is then seen to represent the normal ﬂow of linear momentum per unit area across the boundary surface S into the enclosed volume V . That is, the quantity n ˆ · T is the force per unit area transmitted across the surface S by the microscopic electromagnetic ﬁeld. Just as was found with regard to the interpretation of both Poynting’s theorem and the Poynting vector, as well as with the electromagnetic energy density, the concept of electromagnetic momentum just presented need not coincide with any intuitive notion of momentum ﬂow in the ﬁeld. For example, in a static electric and magnetic ﬁeld it is possible to have a nonvanishing ˆ · T . In density of electromagnetic momentum p ¯ em and momentum ﬂux n addition, even though the total ﬂow of electromagnetic momentum across a closed surface may be correctly represented by the surface integral appearing on the right-hand side of Eq. (2.154), it cannot be deﬁnitely concluded that the time rate of electromagnetic momentum ﬂux in the direction speciﬁed by the unit vector n ˆ at any given ﬁxed point in space is uniquely speciﬁed by the component n ˆ ·T of Maxwell’s stress tensor at that point. Indeed, one may always add to the vector n ˆ · T any other vector ﬁeld that integrates to zero over a closed surface without aﬀecting the statement of conservation of linear momentum that is expressed in Eq. (2.154). The classical interpretation of Eq. (2.154) as a statement of the conservation of linear momentum in the combined system of charged particles and ﬁelds is then seen to depend to a considerable degree on hypothesis. Nevertheless, Eq. (2.154) is a direct mathematical consequence of the Maxwell–Lorentz theory and, just as for Poynting’s theorem, is a valid self-consistent relationship. 2.3.3 Conservation of Angular Momentum The conservation of angular momentum in a combined system of charged particles and ﬁelds follows directly from the preceding derivation of the conservation of linear momentum. The mechanical angular momentum of a system of point particles (labeled by the index i) of mass mi taken about a ﬁxed point O as illustrated in Figure 2.5 is given by (i) ri × pmech , lmech ≡ R × M v + i

where R is the position vector (relative to the ﬁxedpoint O) of the center of mass of the system of particles of total mass M = i mi , v is the velocity of the center of mass of the system relative to O, and where ri is the position (i) vector and pmech is the linear momentum of the ith particle relative to the center of mass of the system. It is assumed here that the center of mass of

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2 Microscopic Electromagnetics

the system of charged particles is at rest with respect to O so that v = 0 and the total mechanical angular momentum is then given by (i) lmech = ri × pmech . (2.159) i

The total time derivative of this expression then yields dpmech dlmech ri × = . dt dt i (i)

(2.160)

(i)

because (dri /dt) × pmech = vi × (mi vi ) = 0.

O

Ri R

ri

ith Charged Particle

Center of Mass

Fig. 2.5. Center of mass of the system of charged particles and associated position vectors for the determination of the system angular momentum.

From Eq. (2.149), the time rate of change of the sum of the mechanical angular momenta of the total system of charged particles in a given region V is given by (assuming that no particles enter or leave the region V ) 1 dlmech r × ρe + j × b d3 r. (2.161) = dt c V From the relation preceding Eq. (2.150) the integrand appearing on the righthand side of this expression may be written as

2.3 Conservation Laws for the Microscopic Electromagnetic Field

1 ∂pem r × ρe + j × b = r × (∇ · T ) − r × c ∂t ∂ = −∇ · (T × r) − (r × pem ) , ∂t

97

(2.162)

where the fact that dr/dt = 0 (because r now denotes the position vector from the ﬁxed origin O to the diﬀerential volume element d3 r) and the dyadic identity [36] r × (∇ · T ) = −∇ · (T × r) for the microscopic stress tensor have been employed. Substitution of Eq. (2.162) into Eq. (2.161) then yields dlmech d + (r × p ¯ em ) d3 r = − ∇ · (T × r) d3 r dt dt V V =− n ˆ · (T × r) d2 r, (2.163) S

by the divergence theorem, where S is the closed boundary surface of the region V and where n ˆ is the outward unit normal vector to the surface S. The volume integral appearing on the left-hand side of this expression can then be identiﬁed as the total electromagnetic angular momentum lem in the region V , so that (r × pem ) d3 r. (2.164) lem ≡ V

The integrand of this expression, viz. lem ≡ r × pem =

1 r × s, c2

(2.165)

can then be interpreted as the microscopic density of electromagnetic angular momentum. The ﬂux of angular momentum of the microscopic electromagnetic ﬁeld is then described by the tensor M ≡ T × r.

(2.166)

Notice that this quantity may be written as a third-rank tensor as Mijk = Tij rk − Tik rj

(2.167)

where the indices i, j, k take on the integer values 1, 2, 3. Because this tensor quantity is antisymmetric in the indices j and k (i.e., Mijk = −Mikj ), it only has three independent elements. With the inclusion of the index i, the tensor quantity Mijk is then seen to have nine components and can then be written as a pseudotensor of the second rank (i.e., as a pseudodyadic), as has been done in Eq. (2.166). With these identiﬁcations, the expression given in Eq. (2.163) for the conservation of angular momentum for the combined system of charged particles and ﬁelds in a region V of space becomes

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2 Microscopic Electromagnetics

d (lmech + lem ) = − dt

n ˆ · M d2 r.

(2.168)

S

If the region V is taken to be all of space and if the angular momentum ﬂux tensor M goes to zero with suﬃcient rapidity such that the surface integral of the normal component of M vanishes as the boundary surface S recedes to inﬁnity, then Eq. (2.168) for the conservation of angular momentum for the combined system of charged particles and the electromagnetic ﬁeld becomes d (lmech + lem ) = 0 (2.169) dt for all space. Hence, it is the total angular momentum in all of space that is conserved rather than just the total mechanical angular momentum of the entire system of charged particles (which need not be conserved). The quantity n ˆ · M appearing in the surface integral of Eq. (2.168) is then seen to represent the normal ﬂow of electromagnetic angular momentum per unit area out of the region V across the boundary surface S.

2.4 Uniqueness of Solution Consider ﬁnally determining the conditions that must be satisﬁed by the microscopic electromagnetic ﬁeld vectors obtained as solutions of the microscopic Maxwell equations in order that these solutions are unique. Helmholtz’ theorem (see Appendix A) shows that a given vector ﬁeld may be uniquely expressed in a region V in terms of a scalar and vector potential that are determined, respectively, from the divergence and curl of that vector ﬁeld in V together with the normal and tangential components of the vector ﬁeld on the closed surface S bounding the region V . Consider then any well-behaved vector ﬁeld F(r) whose divergence and curl are both speciﬁed throughout a ﬁnite, regular region V of space along with the normal component n ˆ · F(r) of the ﬁeld vectorF on the closed regular surface S that bounds the region V , where n ˆ is the outward directed unit normal vector to S. Furthermore, suppose that another well-behaved vector ﬁeld G(r) exists having exactly the same divergence and curl throughout the region V and satisfying the same boundary condition on S as does the vector ﬁeld F(r). One then has that ∇ · F(r) = ∇ · G(r); r ∈ V, ∇ × F(r) = ∇ × G(r); r ∈ V, n ˆ · F(r) = n ˆ · G(r); r ∈ S, and the vector ﬁeld H(r) deﬁned by their diﬀerence as H(r) ≡ F(r) − G(r) must then satisfy the three conditions ∇ · H(r) = 0; ∇ × H(r) = 0;

r ∈ V, r ∈ V,

n ˆ · H(r) = 0;

r ∈ S.

2.4 Uniqueness of Solution

99

From the second of the above set of relations, H(r) is seen to be an irrotational vector ﬁeld in V so that it may be expressed as the gradient of a scalar function φ(r) throughout the region V ; that is, H(r) = ∇φ(r) for r ∈ V . Substitution of this identiﬁcation into the ﬁrst of the above set of relations then yields ∇2 φ = 0 for r ∈ V so that φ(r) satisﬁes Laplace’s equation throughout the region V . By Green’s ﬁrst integral identity, if both φ(r) and ψ(r) are scalar functions of position deﬁned throughout the region V with continuous partial derivatives through second order, then 2 2 φ∇ ψ + ∇φ · ∇ψ d3 r, φ∇ψ · n ˆd r = S

V

so that, with ψ(r) = φ(r) in V , there results 2 φ∇φ · n ˆd r = (∇φ · ∇φ) d3 r, S

V

because φ(r) satisﬁes Laplace’s equation in V . Furthermore, the last relation in the above set of conditions results in the relation ∇φ · n ˆ = 0 on the surface S and the above result then becomes 2 2 |∇φ| d3 r = 0 ⇒ |H| d3 r = 0. V

V

2

Because |H| is a nonnegative quantity throughout (any) region V , one must then have that H(r) = 0 at each point r ∈ V , and hence F(r) = G(r);

∀r ∈ V.

Thus, the speciﬁcation of both the divergence and curl of a well-behaved vector ﬁeld F(r) throughout a ﬁnite, regular region V of space in addition to the speciﬁcation of a particular boundary condition for F(r) on the closed surface S bounding V (in the above case, the normal component of F(r) on S) results in a unique determination of the ﬁeld F(r) in V . The unique determination of the vector ﬁeld F(r) in a regular region V is also obtained if, instead of specifying the normal component n ˆ · F(r), one speciﬁed the tangential component n ˆ × F(r) of the vector ﬁeld on the closed boundary surface S. The microscopic Maxwell equations ||4π|| ρ(r, t),

0 ∇ · b(r, t) = 0, 1 ∂b(r, t) , ∇ × e(r, t) = − c ∂t 4π 1 ∂e(r, t) ∇ × b(r, t) = µ0 j(r, t) + 0 µ0 , c c ∂t ∇ · e(r, t) =

100

2 Microscopic Electromagnetics

are a set of relations that specify the divergence and curl of both the microscopic electric e(r, t) and magnetic b(r, t) ﬁeld vectors in any region V of space in terms of the fundamental charge ρ(r, t) and current j(r, t) source densities. The only hindrance to the application of the above uniqueness theorem to these microscopic Maxwell equations is presented by the fact that the speciﬁed curl relations depend upon the time rate of change of the other ﬁeld quantity.10 Hence, the uniqueness of each ﬁeld vector is intimately connected to the uniqueness of the other (except in the static case when the ﬁeld vectors are uncoupled) and the above conditions for uniqueness are insuﬃcient as they currently stand. The deﬁciency in the preceding uniqueness theorem is simply due to its inherent neglect in explicitly accounting for the time dependence of the electromagnetic ﬁeld vectors. Because the ﬁeld equations are coupled, the suﬃcient conditions for the uniqueness of solution for the time-dependent ﬁeld vectors e(r, t) and b(r, t) must be obtained simultaneously, and this is most conveniently accomplished through Poynting’s energy theorem. Consider then a ﬁnite region V of space that is bounded by a closed regular surface S. Assume that there are two sets of solutions {e1 (r, t), b1 (r, t)} and {e2 (r, t), b2 (r, t)} to the microscopic Maxwell equations that describe the electromagnetic ﬁeld behavior within the region V and which are equal at some time t = t0 . It is then desired to determine the conditions under which these two sets of solutions remain equal for all time t > t0 . Because the microscopic ﬁeld equations are linear, then the diﬀerence ﬁeld deﬁned by e(r, t) ≡ e1 (r, t) − e2 (r, t), b(r, t) ≡ b1 (r, t) − b2 (r, t),

(2.170)

is also a solution of the ﬁeld equations within the region V . The ﬁeld vectors e(r, t) and b(r, t) therefore satisfy Poynting’s theorem as given in Eq. (2.148), which may be written as 2 3 1 1 d 2 2

d e + µ h r + s · n ˆ d r = − j · e d3 r, (2.171) 0 0 2 4π dt V S V where s = c/4πe×h is the Poynting vector for the diﬀerence ﬁeld and where n ˆ is the unit outward normal vector to the surface S. In order that the surface integral of the Poynting vector appearing in Eq. (2.171) vanish, it is only necessary that either the tangential components of e1 (r, t) and e2 (r, t), or else the tangential components of b1 (r, t) and b2 (r, t) be identical on the surface ˆ ×e = 0 or n ˆ ×b = 0 and s has no normal S for all time t ≥ t0 ; for then either n component over the entire surface S because, by the fundamental property of the scalar triple product, n ˆ · (e × b) = (ˆ n × e) · b = e · (b × n ˆ ). Consequently, 10

For a time-independent or static ﬁeld the ﬁeld equations become uncoupled and the preceding uniqueness theorem is directly applicable.

2.4 Uniqueness of Solution

101

there is no net ﬂow of electromagnetic energy associated with the diﬀerence ﬁeld across the closed surface S and the form (2.171) of Poynting’s theorem becomes 2 3 1 1 d 2

d e + µ h r = − j · e d3 r. (2.172) 0 0 2 4π dt V

V

Here j = ρv is the convective current density in the region V . The volume integral of the quantity j · e over the region V is simply the rate at which the Lorentz forces acting inside the region V do work and hence is a nonnegative quantity for a dissipative system. The right-hand side of Eq. (2.172) is consequently always less than or equal to zero. On the other hand, the electromagnetic ﬁeld energy that is given by the volume integral on the left-hand side of Eq. (2.172) is always greater than or equal to zero and, by construction, vanishes at t = t0 . Because the right-hand side of Eq. (2.172) is always negative or zero, the time derivative of the total electromagnetic ﬁeld energy in the region V must also always be negative or zero and hence does not ever increase with time. Because it vanishes at t = t0 , it must then vanish for all t ≥ t0 . Because each of the two terms appearing in the integrand of the electromagnetic ﬁeld energy integral are nonnegative, Eq. (2.172) can only be satisﬁed if both e(r, t) = e1 (r, t) − e2 (r, t) = 0 and b(r, t) = b1 (r, t) − b2 (r, t) = 0 for all t ≥ t0 . The following uniqueness theorem has therefore been established: Theorem 1. Uniqueness. A microscopic electromagnetic ﬁeld is uniquely determined within a bounded, regular region V for all time t ≥ t0 by both the initial values of the microscopic electric and magnetic ﬁeld vectors throughout the region V and the values of the tangential component of either the electric or magnetic ﬁeld vector over the closed boundary surface S for all t ≥ t0 . Notice that this does not prove the existence of any ﬁeld vectors e(r, t) and b(r, t) that satisfy the imposed conditions; the theorem only states that if such a ﬁeld did exist, it is then the only such ﬁeld. Notice further that for a static ﬁeld the requirement on the initial values of the ﬁeld vectors in the region V is superﬂuous and the previous uniqueness statement is obtained. It has thus been established that the values of the microscopic electric and magnetic ﬁeld vectors are uniquely determined throughout any closed (bounded) regular region V at any given time t by a tangential boundary condition on either of the ﬁeld vectors on the boundary surface S and by the initial values of both ﬁeld vectors everywhere in the region V . If the boundary surface S recedes to inﬁnity, the region V is externally unbounded and one must ensure the vanishing of the surface integral of the Poynting vector over an inﬁnitely remote surface. If the ﬁeld was established in the ﬁnite past, this diﬃculty may be circumvented by the assumption that the boundary surface S lies beyond the spatial zone reached at the time t by a ﬁeld that is propagated with a ﬁnite velocity c; notice that this argument cannot be applied to the strictly static ﬁeld case because the ﬁeld must then be established in the inﬁnite past. The above uniqueness theorem does not take

102

2 Microscopic Electromagnetics

into full account this ﬁniteness of propagation of the electromagnetic ﬁeld. Because the ﬁeld is propagated with a ﬁnite velocity, only those elements of the region V whose distance from the point of observation is less than or equal to the quantity c(t−t0 ) need be accounted for. The classical uniqueness theorem given above has been extended in this sense by Rubinowicz [37].

References

103

References 1. J. C. Maxwell, “A dynamical theory of the electromagnetic ﬁeld,” Phil. Trans. Roy. Soc. (London), vol. 155, pp. 450–521, 1865. 2. J. C. Maxwell, A Treatise on Electricity and Magnetism. Oxford: Oxford University Press, 1873. 3. H. A. Lorentz, The Theory of Electrons. Leipzig: Teubner, 1906. Ch. IV. 4. J. M. Stone, Radiation and Optics, An Introduction to the Classical Theory. New York: McGraw-Hill, 1963. 5. L. Rosenfeld, Theory of Electrons. Amsterdam: North-Holland, 1951. 6. R. S. Elliott, Electromagnetics: History, Theory, and Applications. Piscataway, NJ: IEEE, 1993. 7. H. A. Kramers, Quantum Mechanics. Amsterdam: North-Holland, 1957. 8. A. D. Yaghjian, Relativistic Dynamics of a Charged Sphere. Berlin-Heidelberg: Springer-Verlag, 1992. 9. K. I. Golden and G. Kalman, “Phenomenological electrodynamics of twodimensional Coulomb systems,” Phys. Rev. B, vol. 45, no. 11, pp. 5834–5837, 1992. 10. K. I. Golden and G. Kalman, “Phenomenological electrodynamics of electronic superlattices,” Phys. Rev. B, vol. 52, no. 20, pp. 14719–14727, 1995. 11. K. I. Golden and G. J. Kalman, “Quasilocalized charge approximation in strongly coupled plasma physics,” Physics of Plasmas, vol. 7, no. 1, pp. 14– 32, 2000. 12. N. Bohr and L. Rosenfeld, “Field and charge measurements in quantum electrodynamics,” Phys. Rev., vol. 78, no. 6, pp. 794–798, 1950. 13. M. Faraday, Experimental Researches in Electricity. London: Bernard Quaritch, 1855. 14. P. Penﬁeld and H. A. Haus, Electrodynamics of Moving Media. Cambridge, MA: M.I.T. Press, 1967. 15. A. M. Amp`ere, “Memoir on the mutual action of two electric currents,” Annales de Chimie et Physique, vol. 15, pp. 59–76, 1820. 16. J. R. Reitz and F. J. Milford, Foundations of Electromagnetic Theory. Reading, MA: Addison-Wesley, second ed., 1967. 17. H. A. Lorentz, “Electromagnetic phenomena in a system moving with any velocity less than that of light,” Proc. Acad. Sci. Amsterdam, vol. 6, pp. 809–832, 1904. 18. K. F. Gauss, “Theoria Attractionis Corporum Sphaeroidicorum Ellipticorum Homogeneorum,” in Werke, vol. 5, pp. 1–22, G¨ ottingen: Royal Society of Science, 1870. 19. T. Young, “Experiments and calculations relative to physical optics,” in Miscellaneous Works (G. Peacock, ed.), vol. 1, pp. 179–191, London: John Murray Publishers, 1855. p.188. 20. J. Bradley, “An account of a new discovered motion of the ﬁx’d stars,” Phil. Trans. Roy. Soc. (London), vol. 35, pp. 637–660, 1728. 21. F. T. Trouton and H. R. Noble, “Forces acting on a charged condenser moving through space,” Proc. Roy. Soc. (London), vol. 72, pp. 132–133, 1903. 22. J. H. Poincar´e, “L’etat actuel et l’avenir de la physique math´ematique,” Bull. Sci. Math., vol. 28, pp. 302–324, 1904. English translation in Monist, vol. 15, 1 (1905).

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23. A. Einstein, “Zur elektrodynamik bewegter k¨orper,” Ann. Phys., vol. 17, pp. 891–921, 1905. 24. R. Resnick, Introduction to Special Relativity. New York: John Wiley & Sons, 1968. 25. J. V. Bladel, Relativity and Engineering. Berlin-Heidelberg: Springer-Verlag, 1984. Section 1.5. 26. A. S. Eddington, The Mathematical Theory of Relativity. New York: Chelsea, third ed., 1975. Section 5. 27. H. A. Lorentz, Versuch einer Theorie der elektrischen und optischen Erscheinungen in bewegten K¨ orpern. Leiden: Sections 89-92, 1895. English translation: “Michelson’s Interference Experiment,” in The Principle of Relativity. A Collection of Original Memoirs on the Special and General Theory of Relativity by A. Einstein, H. A. Lorentz, H. Minkowski, and H. Weyl, New York: Dover, 1958. 28. J. H. Poynting, “Transfer of energy in the electromagnetic ﬁeld,” Phil. Trans., vol. 175, pp. 343–xxx, 1884. 29. G. Nicolis, Introduction to Nonlinear Science. Cambridge: Cambridge University Press, 1995. Section 2.2. 30. W. S. Franklin, “Poynting’s theorem and the distribution of electric ﬁeld inside and outside of a conductor carrying electric current,” Phys. Rev., vol. 13, no. 3, pp. 165–181, 1901. 31. MacDonald, Electric Waves. Cambridge: Cambridge University Press, 1902. 32. Livens, The Theory of Electricity. Cambridge: Cambridge University Press, 1926. pp. 238 ﬀ. 33. Mason and Weaver, The Electromagnetic Field. Chicago: University of Chicago Press, 1929. pp. 264 ﬀ. 34. J. A. Stratton, Electromagnetic Theory. New York: McGraw-Hill, 1941. 35. M. Abraham, “Prinzipien der Dynamik der Elektrons,” Ann. Physik, vol. 10, pp. 105–179, 1903. 36. H. B. Phillips, Vector Analysis. New York: John Wiley & Sons, 1933. 37. A. Rubinowicz, “Uniqueness of solution of Maxwell’s equations,” Phys. Zeits., vol. 27, pp. 707–710, 1926.

Problems 2.1. Consider a charged point particle of mass m and charge q that is moving through an externally applied electromagnetic ﬁeld {e, b} with velocity v = dr/dt, where r = r(t) denotes the position vector of the charged particle with respect to a ﬁxed origin of coordinates. From the Lorentz force relation (2.24) and Newton’s second law of motion, the equation of motion for the charged particle is found to be given by 1 d2 r m 2 = qe + qv × b. (2.173) dt c (It should be noted that neither side of this equation is strictly correct: the left-hand side is relativistically incorrect but is a good approximation for

2.4 Problems

105

small particle velocities such that v/c 1, whereas the right-hand side of this equation does not include the radiation damping that is due to the electromagnetic ﬁeld radiated by the charged particle when it undergoes an acceleration, the energy of which must be drawn from the particle’s kinetic energy). (a) Prove that the magnetic ﬁeld vector b has no inﬂuence on the magnitude v = |v| of the velocity of the particle, and hence, on it’s kinetic energy. (b) Consider the motion in a uniform steady magnetic ﬁeld b = ˆ 1z b alone that is directed along the positive z-axis of the chosen system of coordinates. The equations of motion then become m(d2 x/dt2 ) = ||1/c||qb(dy/dt), m(d2 y/dt2 ) = −||1/c||qb(dx/dt), and m(d2 z/dt2 ) = 0 in component form, 1y y + ˆ 1z z. The third equation shows that the component of where r = ˆ 1x x + ˆ motion in the direction of b is unaﬀected by the ﬁeld so that one need only consider the projection of the motion onto the xy-plane (the component of r perpendicular to b). Show that this projection describes an orbit with radius R = ||c||mv/|qb| and that the projected motion has the constant angular velocity ω = ||1/c||qb/m. (c) Write a computer program that computes and plots the three-dimensional trajectory followed by an electron in a uniform, steady magnetic ﬁeld that is directed along the z-axis. " ! ˆ d2 r = V ∇ · Fd3 r), 2.2. Beginning with the divergence theorem ( S F · n where the closed surface S forms the copmplete boundary of the region V with outward unit normal vector n ˆ , show that, for any suﬃciently continuous vector ﬁeld G = G(r) and scalar ﬁeld f = f (r): 2 f (r)ˆ nd r = (∇f (r)) d3 r, (2.174) S V n ˆ × G(r)d2 r = (∇ × G(r))d3 r. (2.175) S

V

! " 2.3. Beginning with Stokes theorem ( C F · dr = S (∇ × F) · n ˆ d2 r), where the closed contour C forms the complete boundary of the surface S with unit normal vector n ˆ taken in the positive direction, show that, for any suﬃciently continuous vector ﬁeld G = G(r) and scalar ﬁeld f = f (r): dr × G(r) = (ˆ n × ∇)G(r)d2 r, (2.176) C S f (r) = (ˆ n × ∇f (r))d2 r. (2.177) C

S

2.4. Show that charge is conserved using only the time-domain integral form of Maxwell’s equations. 2.5. Two events occur at the same spatial point in an inertial reference frame Σ, but not simultaneously. Show that the temporal sequence of these two events remains unchanged in any (all) other inertial reference frame(s) Σ .

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2 Microscopic Electromagnetics

Σ Σ Σ 2.6. Show that QΣ Σ QΣ = QΣ QΣ = I.

2.7. Determine the relativistic motion of a charged particle of charge q and rest mass m0 as it passes through a uniform, time-independent magnetic ﬁeld b = ˆ 1z b. Assume that the initial velocity of the particle is along the x-direction. 2.8. Derive the identity given in Eq. (2.113). 2.9. Derive the general transformation relation for linear momentum given in Eq. (2.114). 2.10. Derive the general transformation relation for energy given in Eq. (2.115). 2.11. Derive the general transformation relation for the force given in Eq. (2.120). 2.12. Verify the expression for the force given in Eq. (2.121). 2.13. Derive the appropriate transformation relations for the microscopic charge density ρ and current density j. 2.14. Derive the transformation relations given in Eq. (2.125) for the rectangular components of the microscopic electric ﬁeld vector e. 2.15. Derive the transformation relations given in Eq. (2.127) for the rectangular components of the microscopic magnetic ﬁeld vector b. 2.16. Show that the microscopic Maxwell equations (2.131) and (2.134) are invariant in form under a Poinca´re–Lorentz transformation when the ﬁeld vectors are transformed by the relations speciﬁed in Eqs. (2.129)–(2.130) and the charge and current sources are transformed by the relations obtained in Problem 2.13. 2.17. Show that the relativistically correct transformation relations given in Eqs. (2.129)–(2.130) for the electric and magnetic ﬁeld vectors, respectively, reduce to the corresponding Galilean transformation relations given in Eqs. (2.50) and (2.58) in the small velocity limit as V /c → 0. Determine the ﬁrst higher-order correction term to these Galilean transformation relations. 2.18. A static distribution of charge and current can set up time-independent electric and magnetic ﬁelds in a common region such that the Poynting vector is nonvanishing, but there is no net power ﬂow. Show that under these conditions s·n ˆ d2 r = 0, S

for any closed surface S in the region.

2.4 Problems

107

2.19. Prove that the speciﬁcation of both the divergence and curl of a wellbehaved vector ﬁeld F(r) throughout a ﬁnite, regular region V of space, in addition to the speciﬁcation of the tangential component n ˆ × F of the vector ﬁeld over the closed boundary surface S of V results in a unique determination of F(r) in V . 2.20. Is uniqueness obtained when the volume integral of the quantity j · e over the region V appearing on the right-hand side of Eq. (2.172) is either zero or negative?

3 Microscopic Potentials and Radiation

The microscopic Maxwell equations consist of a set of coupled ﬁrst-order partial diﬀerential equations relating the various components of the microscopic electric and magnetic ﬁeld vectors that comprise the electromagnetic ﬁeld. The analysis of the properties or the determination of an electromagnetic ﬁeld is often facilitated by the introduction of auxiliary ﬁelds known as potentials. The two homogeneous Maxwell equations ∇·b = 0 and ∇×e = −1/c∂b/∂t indicate that not all of the components of the ﬁeld vectors e and b are entirely independent so that it may be possible to obtain a more compact description of the ﬁeld with fewer components. Such an approach should yield a smaller number of second-order partial diﬀerential equations that may take better advantage in describing the properties of the electromagnetic ﬁeld under certain circumstances.

3.1 The Microscopic Electromagnetic Potentials The vector and scalar potentials1 for an electromagnetic ﬁeld are introduced through the diﬀerential properties of the electric and magnetic ﬁeld vectors as described by the microscopic Maxwell equations given in Eqs. (2.33)–(2.36). Consider a microscopic electromagnetic ﬁeld that is due to a given microscopic distribution of charge density ρ(r, t) and convective current density j(r, t). The spatiotemporal properties of this ﬁeld are then described by 1 ∂b(r, t) = 0, (3.1) ∇ × e(r, t) + c ∂t 1 ∂d(r, t) 4π (3.2) = j(r, t), ∇ × h(r, t) − c ∂t c with ∇·d(r, t) = 4πρ(r, t) and ∇·b(r, t) = 0, where the charge and current densities are related by the equation of continuity ∇ · j(r, t) = −∂ρ(r, t)/∂t. Due to the divergenceless character of the magnetic induction vector b 1

George Green introduced the concept of the potential function into the theory of electricity and magnetism in 1828. Franz Neumann (the father of Karl Neumann) introduced the vector potential in 1845.

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3 Microscopic Potentials and Radiation

throughout all space-time, the magnetic ﬁeld vector b is always solenoidal so that it can always be expressed as the curl of another vector ﬁeld a0 , where b(r, t) = ∇ × a0 (r, t).

(3.3)

However, the subsidiary vector ﬁeld a0 is not uniquely speciﬁed by this equation, for b is also given by the curl of some other vector ﬁeld a, b(r, t) = ∇ × a(r, t),

(3.4)

a(r, t) = a0 (r, t) − ∇ψ(r, t)

(3.5)

where when ψ is any arbitrary scalar function of both position and time. If b is replaced in Eq. (3.1) by either of the expressions given in Eqs. (3.3) or (3.4), there results, respectively 1 ∂a0 (r, t) = 0, ∇ × e(r, t) + c ∂t 1 ∂a(r, t) = 0. ∇ × e(r, t) + c ∂t The vector ﬁelds (e + 1/c∂a0 /∂t) and (e + 1/c∂a/∂t) are therefore everywhere irrotational and each can then be expressed as the gradient of some scalar function, so that 1 ∂a0 (r, t) (3.6) − ∇φ0 (r, t), e(r, t) = − c ∂t 1 ∂a(r, t) − ∇φ(r, t). (3.7) e(r, t) = − c ∂t From Eq. (3.5), the scalar functions φ and φ0 are seen to be related as 1 ∂ψ(r, t) . (3.8) φ(r, t) = φ0 (r, t) + c ∂t Notice that arbitrary scalar constants may always be added to both of the scalar functions ψ and φ without altering the ﬁeld vectors e and b. The vector functions a(r, t) are called the vector potentials and the scalar functions φ(r, t) are called the scalar potentials of the microscopic electromagnetic ﬁeld. The functions a0 and φ0 designate one speciﬁc pair of potentials from which the electromagnetic ﬁeld vectors may be determined through application of Eqs. (3.3) and (3.6). An inﬁnite number of potentials that describe the same electromagnetic ﬁeld can then be constructed from Eqs. (3.5) and (3.8) through choice of the function ψ(r, t). That is, the scalar and vector potentials are not unique for a given situation; the ﬁeld vectors themselves

3.1 The Microscopic Electromagnetic Potentials

111

are, however, uniquely determined (provided that the conditions stated in Theorem 1 of §2.4 are satisﬁed). The homogeneous equations of Maxwell’s ﬁeld equations are naturally satisﬁed by the scalar and vector potentials. The inhomogeneous equations then provide the pair of inhomogeneous partial diﬀerential equations for the potential ﬁelds as 1 ∂ 4π 2 ∇ φ + (∇ · a) = − ρ, (3.9) c ∂t

0 1 4π ∂φ 1 ∂2a ∇2 a − 2 2 − ∇ ∇ · a + 0 µ0 = − µ0 j. (3.10) c ∂t c ∂t c Hence, with the introduction of these two auxiliary functions, the set of four Maxwell’s equations have been reduced to a system of two equations that are, however, still coupled. They may, nevertheless, be uncoupled by exploiting the arbitrariness involved in the deﬁnition of the potential functions through the scalar function ψ(r, t) appearing in Eqs. (3.5) and (3.8). The transformation relation pair a(r, t)

−→

φ(r, t)

−→

a (r, t) = a(r, t) + ∇ψ(r, t), 1 ∂ψ(r, t) , φ (r, t) = φ(r, t) − c ∂t

(3.11) (3.12)

is called a gauge transformation and the invariance of the electromagnetic ﬁeld vectors under such a transformation is called gauge invariance. Finally, the scalar function ψ(r, t) is called the gauge function. The interrelationship between the electric and magnetic ﬁeld vectors and the scalar and vector potential ﬁelds under a gauge transformation is illustrated in Figure 3.1.

Fig. 3.1. Network structure of the gauge transformation for the microscopic vector and scalar potential ﬁelds.

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3 Microscopic Potentials and Radiation

3.1.1 The Lorenz Condition and the Lorenz Gauge The freedom of choice implied by the pair of relations in Eq. (3.11)–(3.12) means that one can always choose a pair of potentials {a(r, t), φ(r, t)} such that they satisfy the diﬀerential relation 1 ∂φ =0 (3.13) ∇ · a + 0 µ0 c ∂t which is known as the Lorenz condition [1].2 In order to show that a pair of potentials can always be found such that they satisfy the Lorenz condition, suppose that the potentials {a(r, t), φ(r, t)} which satisfy the pair of relations given in Eqs. (3.9)–(3.10) do not satisfy the Lorenz condition given in Eq. (3.13). One can then undertake a gauge transformation to a new pair of potential functions {a (r, t), φ (r, t)} and demand that this set does satisfy the Lorenz condition; hence

0 µ0 ∂φ c ∂t

0 µ0 ∂ 1 ∂ψ φ− = ∇ · (a + ∇ψ) + c ∂t c ∂t

0 µ0 ∂φ 1 ∂2ψ 2 = ∇·a+ + ∇ ψ− 2 2 . c ∂t c ∂t

0 = ∇ · a +

Thus, provided that a gauge function ψ(r, t) can be found to satisfy the inhomogeneous wave equation ∇2 ψ −

1 ∂2ψ

0 µ0 ∂φ , = −∇ · a − c2 ∂t2 c ∂t

(3.14)

the new pair of potentials {a (r, t), φ (r, t)} will satisfy the Lorenz condition. For a pair of potentials {a(r, t), φ(r, t)} that satisfy the Lorenz condition the pair of diﬀerential equations given in Eqs. (3.9)–(3.10) for the potentials become uncoupled as 1 ∂2φ 4π =− ρ, c2 ∂t2

0 4π 1 ∂2a 2 ∇ a − 2 2 − = − µ0 j, c ∂t c ∇2 φ −

(3.15) (3.16)

and one is left with two separate inhomogeneous wave equations. 2

Although the erroneous attribution of this gauge condition to H. A. Lorentz was corrected by E. T. Whittaker [2] in 1951, the majority of texts have regrettably continued to attribute it to Lorentz and not to Lorenz who introduced the retarded potentials in a series of three articles beginning in 1867 [1]; see also the letter by J. Van Bladel. [3]

3.1 The Microscopic Electromagnetic Potentials

113

Even for a pair of potentials that satisfy the Lorenz condition (3.13) there still remains a certain degree of arbitrariness in their determination. In particular, the restricted gauge transformation a(r, t)

−→

φ(r, t)

−→

a (r, t) = a(r, t) + ∇ψ(r, t), 1 ∂ψ(r, t) φ (r, t) = φ(r, t) − , c ∂t

(3.17) (3.18)

where the gauge function ψ(r, t) satisﬁes the homogeneous wave equation ∇2 ψ −

1 ∂2ψ = 0, c2 ∂t2

(3.19)

preserves the Lorenz condition provided that the pair of potential functions {a(r, t), φ(r, t)} satisfy it initially. All pairs of potentials in this restricted class are said to belong to the Lorenz gauge. The Lorenz gauge is commonly employed in both the physics and engineering communities; ﬁrst, because it yields the pair of uncoupled inhomogeneous wave equations (3.15)–(3.16) which treat the scalar and vector potentials on an equal level, and second, because it is independent of the particular coordinate system chosen, it ﬁts naturally into the framework of the special theory of relativity. 3.1.2 The Coulomb Gauge In the Coulomb gauge the vector potential is chosen so as to be solenoidal, viz. ∇ · a(r, t) = 0. (3.20) This is also known as either the transverse gauge or as the radiation gauge. In this gauge the scalar potential satisﬁes Poisson’s equation ∇2 φ = − with solution φ(r, t) =

4π 4π 0

4π ρ

0 ρ(r , t) 3 d r, |r − r |

(3.21)

(3.22)

where the integration is taken over all space. Hence, the scalar potential is just the instantaneous Coulomb potential due to the microscopic charge distribution with density ρ(r, t). The vector potential in the Coulomb gauge satisﬁes the inhomogeneous wave equation 4π 1 ∂2a

0 µ0 ∂φ ∇ . (3.23) ∇2 a − 2 2 = − µ0 j + c ∂t c c ∂t The apparent “current” term that appears in Eq. (3.23) can, in principle, be determined from the instantaneous Coulomb potential given in Eq. (3.22).

114

3 Microscopic Potentials and Radiation

With use of the equation of continuity [cf. Eq. (2.14)] in Eq. (3.22), there results ∂φ ∂ρ(r , t)/∂t 3 4π ∇ ∇ = d r ∂t 4π 0 |r − r | 4π ∇ · j(r , t) 3 =− ∇ (3.24) d r. 4π 0 |r − r | Because this current term is irrotational, it may then cancel a corresponding term that is contained in the microscopic current density j(r, t). From Helmholtz’ theorem [4] (see Appendix A), the current density may be expressed as the sum of two terms as j(r, t) = j (r, t) + jt (r, t),

(3.25)

where ∇ × j = 0 and j (r, t) is accordingly referred to as the longitudinal or irrotational current density, whereas ∇ · jt = 0 and jt (r, t) is appropriately referred to as the transverse or solenoidal current density. From Helmholtz’ theorem, these component current densities are uniquely determined from the microscopic current density j(r, t) as ∇ · j(r , t) 3 1 j (r, t) = − ∇ (3.26) d r, 4π |r − r | j(r , t) 3 1 jt (r, t) = ∇×∇× d r. (3.27) 4π |r − r | Comparison of Eqs. (3.24) and (3.26) then yields the identiﬁcation ∇

4π ∂φ(r, t) = j (r, t). ∂t

0

(3.28)

As a consequence, the source term for the inhomogeneous wave equation (3.23) of the vector potential in the Coulomb gauge can be expressed entirely in terms of the transverse current density given in Eq. (3.27), so that 4π 1 ∂2a 2 ∇ a − 2 2 = − µ0 jt . (3.29) c ∂t c This result is the origin of the name “transverse gauge.” The term “radiation gauge” has its origin in the fact that transverse radiation ﬁelds are given by the vector potential alone, the instantaneous Coulomb potential given in Eq. (3.22) contributing only to the near-ﬁeld behavior. This gauge is of particular importance in quantum electrodynamics because a quantum mechanical description of photons would then necessitate quantization of only the vector potential. The Coulomb or transverse gauge is particularly useful when no charge or current sources are present. In that case the instantaneous Coulomb potential

3.1 The Microscopic Electromagnetic Potentials

115

vanishes everywhere (φ(r, t) = 0) and the vector potential a(r, t) satisﬁes the homogeneous wave equation ∇2 a −

1 ∂2a = 0. c2 ∂t2

(3.30)

The microscopic electric and magnetic ﬁeld vectors are then given by the pair of expressions 1 ∂a(r, t) e(r, t) = − , c ∂t b(r, t) = ∇ × a(r, t). (3.31) It is important at this point to note an interesting peculiarity of the Coulomb gauge. It is well known that any electromagnetic disturbance propagates with a ﬁnite velocity c in free space. However, Eq. (3.22) indicates that in the Coulomb gauge the scalar potential for the electromagnetic disturbance is “propagated” instantaneously everywhere in space. The vector potential in the Coulomb gauge, on the other hand, satisﬁes the wave equation given in Eq. (3.23) with its associated ﬁnite velocity of propagation c in free space. Notice further that in the Lorenz gauge, both potential functions satisfy a wave equation with ﬁnite propagation velocity c. The incongruity appearing in the scalar potential of the Coulomb gauge is then seen to be a mere consequence of the gauge choice and hence, has no bearing upon the resultant electromagnetic ﬁeld which is independent of the particular choice of gauge. Indeed, it is not the potentials but the ﬁeld vectors themselves (and the forces they exert on any given test particle) that are the physical quantities of interest in the theory, and these propagate with a ﬁnite velocity c in free space. As a ﬁnal point of interest, return to the Lorenz gauge and consider any region of space wherein the charge density ρ(r, t) is zero. The scalar potential φ(r, t) then satisﬁes the homogeneous wave equation ∇2 φ −

1 ∂2φ =0 c2 ∂t2

(3.32)

and a gauge function ψ(r, t) may then be chosen such that the scalar potential vanishes throughout the region. In particular, from Eqs. (3.12) and (3.19), it is only necessary to take the gauge function to be given by ψ(r, t) = c φ(r, t) dt, (3.33) because then φ(r, t)

−→

1 ∂ψ(r, t) φ (r, t) = φ(r, t) − c ∂t = φ(r, t) − φ(r, t) = 0.

(3.34)

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3 Microscopic Potentials and Radiation

The electromagnetic ﬁeld vectors can then be derived from the vector potential alone, as given by the pair of relations in Eq. (3.31), and the Lorenz condition (3.13) reduces to ∇ · a = 0, which is simply the condition given in Eq. (3.20) for the Coulomb gauge. Hence, in charge-free regions of space the Lorenz and Coulomb gauges become identical when the gauge function ψ(r, t) is chosen such that the scalar potential (in the Lorenz gauge) vanishes throughout the region. 3.1.3 The Retarded Potentials The general solutions of the inhomogeneous wave equations given in Eqs. (3.15)–(3.16) for the scalar and vector potentials in the Lorenz gauge [i.e. that are subject to the Lorenz condition given in Eq. (3.13)] are given by the retarded potentials ρ(r , t − R/c) 3 4π d r, φ(r, t) = (3.35) 4π 0 V R 4π/c j(r , t − R/c) 3 a(r, t) = µ0 d r. (3.36) 4π R V Here R ≡ |r−r | = (x − x )2 + (y − y )2 + (z − z )2 is the distance between the observation point at r and the source point at r at which point the volume element of integration d3 r is located, the integration being carried out throughout the volume V that includes all charge and current sources that produce the electromagnetic ﬁeld under consideration. In the above expressions the argument t−R/c denotes that the pair of potentials {a(r, t), φ(r, t)} in the Lorenz gauge are due to the charge and current distributions at the retarded time t − R/c, where c is the vacuum speed of light. Because of this retardation eﬀect, the pair of potentials {a(r, t), φ(r, t)} are called retarded potentials. Consider now proving3 that these retarded potentials are indeed the special solutions to the inhomogeneous wave equations in the Lorenz gauge as given in Eqs. (3.15)–(3.16). Because of the similarity of the wave equations for the scalar and vector potentials, it is necessary to only consider the proof for the scalar potential that satisﬁes the equation ∇2 φ −

1 ∂2φ 4π =− ρ. c2 ∂t2

0

(3.37)

Let the integration region V appearing in Eq. (3.35) be divided into two distinct regions V1 and V2 such that V = V1 ∪ V2 and V1 ∩ V2 = ∅, where ∅ denotes the empty set, and where V1 is an arbitrarily small volume surrounding the point r at which the potential is to be observed (the observation or ﬁeld point). The scalar potential φ(r, t) is then composed of two parts 3

This proof was ﬁrst given by Georg Friedrich Bernhard Riemann in 1858.

3.1 The Microscopic Electromagnetic Potentials

φ(r, t) = φ1 (r, t) + φ2 (r, t), where

4π φi (r, t) = 4π 0

Vi

ρ(r , t − R/c) 3 d r R

117

(3.38) (3.39)

for i = 1, 2. Let the region V1 containing the ﬁeld point r be suﬃciently small such that retardation eﬀects are negligible for all source points within that region. In the limit as V1 → 0 one then has that ρ(r .t − R/c) → ρ(r , t);

r ∈ V1

and the expression (3.39) for φ1 (r, t) becomes ρ(r , t) 3 4π φ1 (r, t) = d r. 4π 0 V1 R

(3.40)

(3.41)

Because this equation is identical with the instantaneous Coulomb potential [cf. Eq. (3.22)], it then follows that φ1 (r, t) is the solution of Poisson’s equation 4π ∇2 φ1 = − ρ. (3.42)

0 In the region V2 , R > 0 and it is permissible to diﬀerentiate φ2 (r, t) under the integral sign. Because the Laplacian operator ∇2 acts only upon the unprimed coordinates and because the integrand ρ(r , t − R/c)/R depends upon these coordinates only through the radial distance R from the observation point to the source point, one may then use spherical coordinates to obtain the Laplacian of ρ(r , t − R/c)/R as ρ ∂ ρ 1 ∂ ∇2 = 2 R2 R R ∂R ∂r R 2 1 ∂ ρ = , R ∂R2 and because ∂ 2 ρ(t − R/c) 1 ∂ 2 ρ(t − R/c) = 2 , 2 ∂R c ∂t2 one ﬁnally has that ∇

2

ρ(t − R/c) R

=

1 ∂ 2 ρ(t − R/c) . Rc2 ∂t2

With these expressions there then results 1 ∂ 2 φ2 c2 ∂t2 ρ(r , t − R/c) 4π 1 ∂ 2 ρ(r , t − R/c) 3 − = ∇2 d r 4π 0 V2 R Rc2 ∂t2 =0 (3.43)

∇2 φ2 −

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3 Microscopic Potentials and Radiation

Hence, in the limit as V1 → 0, Eqs. (3.38), (3.41), and (3.43) yield ∇2 φ(r, t) = ∇2 (φ1 (r, t) + φ2 (r, t)) 4π 1 ∂ 2 φ(r, t) =− ρ(r, t) + 2 ,

0 c ∂t2 so that ∇2 φ(r, t) −

1 ∂ 2 φ(r, t) 4π =− ρ(r, t), c2 ∂t2

0

(3.44)

and the retarded potential that is given in Eq. (3.35) satisﬁes the inhomogeneous wave equation given in Eq. (3.15) for the scalar potential in the Lorenz gauge. Notice that the term −(4π/ 0 )ρ(r, t) arises from the observation point whereas the term (1/c2 )∂ 2 φ(r, t)/∂t2 arises from everywhere else but the observation point. In a strictly analogous manner, one can show that each Cartesian component of the vector potential appearing in Eq. (3.36) satisﬁes the appropriate component of the inhomogeneous wave equation for the vector potential in the Lorenz gauge as given in Eq. (3.16). Hence, Eq. (3.36) satisﬁes the inhomogeneous wave equation for the vector potential in the Lorenz gauge. Moreover, on account of the equation of continuity given in Eq. (2.14), these solutions also satisfy the Lorenz condition given in Eq. (3.13). The expressions appearing in Eqs. (3.35)–(3.36) show that one may regard the vector and scalar potentials {a(r, t), φ(r, t)} in the Lorenz gauge as arising from contributions from each volume element of space, a typical element d3 r contributing the amounts 4π/c(µ0 /4π)j(r , t − R/c) and (4π/4π 0 )ρ(r , t − R/c) to a(r, t) and φ(r, t), respectively. As stated earlier, the quantity R/c is precisely the time needed for an electromagnetic disturbance to propagate from the source point at r to the observation (or ﬁeld) point at r, so that each contribution has to emanate from the element at such preceding time as to reach the point of observation at the required time t. For this reason, the expressions given in Eqs. (3.35)–(3.36) are called retarded potentials. It is also possible to construct solutions in the form of advanced potentials with the time argument t + R/c appearing in the integrand. Both types of potential satisfy the inhomogeneous wave equations (3.15)–(3.16) in the Lorenz gauge and are mathematically admissible; however, only the retarded potentials that refer to charge and current sources at earlier times correspond to the physical law of causality so that the advanced potentials are not considered. The retarded potentials given in Eqs. (3.35)–(3.36) represent a special solution of the inhomogeneous wave equations (3.15)–(3.16) in the Lorenz gauge, namely that which arises from the given distribution of charge and current sources with densities ρ(r, t) and j(r, t) = ρ(r, t)v(r, t). The general solution is obtained by adding to these the general solutions of the homogeneous wave equations

3.2 The Hertz Potential and Elemental Dipole Radiation

119

1 ∂2φ = 0, (3.45) c2 ∂t2 1 ∂2a ∇2 a − 2 2 = 0, (3.46) c ∂t where φ(r, t) and a(r, t) are again subject to the Lorenz condition. For slowly varying systems the retarded potentials in the Lorenz gauge reduce to the static potentials. The ﬁelds due to a localized charge or current distribution about the origin then fall oﬀ, in the limit as R → ∞, as R−2 or faster, because the potentials are then given by the asymptotic approximations 4π ρ(r , t)d3 r , (3.47) φ(r, t) ∼ 4π 0 R V 4π/c a(r, t) ∼ j(r , t)d3 r , (3.48) µ0 4πR V ∇2 φ −

where the radial distance R from the origin to the observation or ﬁeld point is much larger than the maximum radial extent of the source region V from the origin. On the other hand, if the charge distribution in V varies rapidly with time, then although the vector potential a(r, t) still falls oﬀ as R−1 for large R, the electric ﬁeld vector e(r, t) contains a term proportional to −∂a/∂t that also decays as R−1 . There is a similar term in the magnetic ﬁeld vector h(r, t) as required by the second of Maxwell’s equations [cf. Eq. (3.2)]. Thus, in a system that is oscillating with a period T the dominant asymptotic behavior of the microscopic e and h ﬁelds at suﬃciently large distances R varies as (RT )−1 ; this part of the ﬁeld is responsible for radiation.

3.2 The Hertz Potential and Elemental Dipole Radiation The analysis of the preceding section has shown that the microscopic electromagnetic ﬁeld vectors may be speciﬁed by the scalar and vector potentials. That is, the equations that are required to describe the six components of the combined electric and magnetic ﬁeld vectors {e, b} are over-speciﬁed and one only need consider the four components of the potentials {a, φ} to completely describe the microscopic electromagnetic ﬁeld, thereby reducing the component dimensionality of the theory from six to four components. However, because these potential functions are connected by the Lorenz condition (3.13) when they are expressed in the Lorenz gauge, for example,, it then follows that it is possible to express the electromagnetic ﬁeld vectors in terms of a single vector function, known as the Hertz potential,4 thereby reducing the component dimensionality of the theory to three. 4

The Hertz potential is a generalization of a certain potential function that was introduced for the electromagnetic ﬁeld of an oscillating dipole by H. Hertz [5] in 1889. The vector character of the Hertz potential was ﬁrst noted by A. Righi [6] in 1901.

120

3 Microscopic Potentials and Radiation

3.2.1 The Hertz Potential Consider the vector and scalar potential ﬁelds under the Lorenz condition (3.13), so that 4π 1 ∂2a (3.49) ∇2 a − 2 2 = − µ0 j, c ∂t c 1 ∂2φ 4π ∇2 φ − 2 2 = − ρ, (3.50) c ∂t

0 where the microscopic current and charge densities satisfy the continuity equation ∂ρ ∇·j+ = 0. (3.51) ∂t As an immediate consequence of this relation, it follows that there exists a vector function p(r, t) such that ∂p(r, t) , ∂t ρ(r, t) = −∇ · p(r, t), j(r, t) =

(3.52) (3.53)

in which case Eq. (3.51) is identically satisﬁed. This vector ﬁeld p(r, t) is called the electric moment. In terms of this electric moment, the pair of wave equations in Eqs. (3.49)–(3.50) become 4π ∂p 1 ∂2a 2 ∇ a − 2 2 = − µ0 , (3.54) c ∂t c ∂t 1 ∂2φ 4π ∇2 φ − 2 2 = − ∇ · p. (3.55) c ∂t

0 Assume now that the vector and scalar potentials are given by the respective relations µ0 ∂Π(r, t) , c ∂t 1 φ(r, t) = − ∇ · Π(r, t),

0 a(r, t) =

(3.56) (3.57)

which clearly satisfy the Lorenz condition (3.13). In terms of the vector ﬁeld Π(r, t) the inhomogeneous wave equations given in Eqs. (3.54)–(3.55) become ∂ 1 ∂2Π ∇2 Π − 2 + 4πp = 0, ∂t c ∂t2 1 ∂2Π ∇ · ∇2 Π − 2 + 4πp = 0, c ∂t2 respectively. This pair of equations will clearly be satisﬁed if

3.2 The Hertz Potential and Elemental Dipole Radiation

∇2 Π −

1 ∂2Π = −4πp. c2 ∂t2

121

(3.58)

The vector ﬁeld Π(r, t) is called the Hertz vector or Hertz potential [5] for the electromagnetic ﬁeld. The above analysis shows that the Hertz vector Π(r, t) determines an electromagnetic ﬁeld through its vector and scalar potential functions, but it does not show that a given electromagnetic ﬁeld can be represented in terms of the vector ﬁeld Π(r, t). In order to accomplish this, it must be proven that given the vector and scalar potentials for an electromagnetic ﬁeld, the vector ﬁeld Π(r, t) is “uniquely deﬁned.” To that end, let φ0 (r) ≡ φ(r, 0) be the value of the scalar potential φ(r, t) when t = 0 and let Π0 (r) be any vector function of position that is independent of the time t such that ∇·Π0 = − 0 φ0 . Deﬁne the time-dependent vector ﬁeld Π (r, t) by the relation c Π (r, t) ≡ Π0 (r) + µ0

so that a(r, t) =

t

a(r, t ) dt ,

(3.59)

0

µ0 ∂Π (r, t) c ∂t

(3.60)

by straightforward diﬀerentiation of Eq. (3.59), and c t ∇ · Π (r, t) = ∇ · Π0 (r) + ∇ · a(r, t ) dt µ0 0 t ∂φ(r, t ) = − 0 φ0 (r) − 0 dt ∂t 0 = − 0 φ(r, t),

(3.61)

where the Lorenz condition (3.13) has been invoked. From the pair of relations appearing in Eqs. (3.54)–(3.55) one then obtains, by direct substitution, ∂ ∂p 1 ∂ 2 Π = −4π , ∇2 Π − 2 ∂t c ∂t2 ∂t 2 1 ∂ Π = −4π∇ · p, ∇ · ∇2 Π − 2 c ∂t2 and consequently ∇ 2 Π −

1 ∂ 2 Π + 4πp = ∇ × Ξ, c2 ∂t2

(3.62)

where Ξ(r) is a vector function of position that is independent of the time t. Let (3.63) Π (r, t) = Π(r, t) + ∇ × Ψ(r),

122

3 Microscopic Potentials and Radiation

where Ψ(r) is another vector function of position that is independent of the time t and is such that (3.64) ∇2 Ψ(r) = Ξ(r) as required by Eq. (3.62). It is then clear that the vector ﬁeld Π (r, t) satisﬁes Eq. (3.58), and because ∇ · Π (r, t) = ∇ · Π(r, t), ∂Π (r, t) ∂Π(r, t) = , ∂t ∂t

(3.65) (3.66)

then Eqs. (3.60) and (3.61) show that Π (r, t) satisﬁes the two relations given in Eqs. (3.56) and (3.57). Hence, for a given electromagnetic ﬁeld one can always determine a vector ﬁeld Π(r, t) such that µ0 ∂ 2 Π(r, t) 1 ∇ (∇ · Π(r, t)) − 2 ,

0 c ∂t2 ∂Π(r, t) µ0 ∇× , b(r, t) = c ∂t e(r, t) =

(3.67) (3.68)

where Π(r, t) satisﬁes the inhomogeneous wave equation given in Eq. (3.58) and is the Hertz vector or Hertz potential for the microscopic electromagnetic ﬁeld. This Hertz potential is a superpotential in the sense that the electromagnetic ﬁeld, when described by Π(r, t), is given by a three-component equation, whereas the vector and scalar potential pair {a(r, t), φ(r, t)} amount to a four-component potential. The Hertz potential Π(r, t) is not uniquely determined because the vector and scalar potentials {a(r, t), φ(r, t)} are invariant under the gauge transformation Π(r, t) → Π (r, t) = Π(r, t) + ∇ × Γ(r), (3.69) where Γ(r) is an arbitrary vector function of position that is independent of the time t. An even greater degree of arbitrariness is present here because φ(r, t) and a(r, t) are themselves not uniquely determined in the chosen Lorenz gauge because of the restricted gauge transformation given in Eqs. (3.17)–(3.18). As a consequence, the microscopic electromagnetic ﬁeld vectors are invariant under the more general gauge transformation [7] Π(r, t) → Π (r, t) = Π(r, t) + ∇ × Γ(r) −

1 ∂Λ(r, t) − ∇λ(r, t), c ∂t

(3.70)

where Γ(r), Λ(r, t), and λ(r, t) are arbitrary functions. Under this transformation the vector and scalar potentials given in Eqs. (3.56)–(3.57) undergo the gauge transformation µ0 ∂ 2 Λ(r, t) µ0 ∂λ(r, t) ∇ , − 2 cc ∂t c ∂t ∂Λ(r, t) 1 1 φ(r, t) → φ (r, t) = φ(r, t) + ∇· + ∇2 λ(r, t).

0 c ∂t

0 a(r, t) → a (r, t) = a(r, t) −

(3.71) (3.72)

3.2 The Hertz Potential and Elemental Dipole Radiation

123

Fig. 3.2. Network structure of the restricted gauge transformation for the vector and scalar potentials in the Lorenz gauge and the Hertz vector.

The Lorenz condition (3.13) is satisﬁed by the transformed pair of potentials {a (r, t), φ (r, t)} if it is originally satisﬁed by the pair {a(r, t), φ(r, t)}, because

0 µ0 ∂φ µ0 ∂ 2 Λ ∂ 2 (∇ · Λ 0 µ0 ∂φ − ∇·a = ∇·a+ − ∇· c ∂t c ∂t cc ∂t2 ∂t2 2 ∂λ ∂(∇ λ) µ0 − − ∇2 = 0. (3.73) c ∂t ∂t Upon comparison of this equation with the expressions given in Eqs. (3.71)– (3.72), the gauge function ψ(r, t) appearing in Eqs. (3.11)–(3.12) is seen to satisfy the pair of relations

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3 Microscopic Potentials and Radiation

µ0 1 ∂ 2 Λ ∂λ , + ∇ c c ∂t2 ∂t c 1 ∂Λ ∂ψ =− ∇· + ∇2 λ , ∂t

0 c ∂t

∇ψ = −

(3.74) (3.75)

from which it is readily seen that ψ(r, t) satisﬁes the homogeneous wave equation (3.19). The interrelationship among the electric and magnetic ﬁeld vectors, the scalar and vector potentials, and the Hertz vector under these various gauge transformations within the Lorenz gauge is illustrated in Figure 3.2. Because the gauge transformation for the Hertz potential leaves the scalar and vector potentials in the Lorenz gauge, it then results in a restricted gauge transformation for the scalar and vector potential functions. 3.2.2 Radiation from an Elemental Hertzian Dipole Consider the microscopic electromagnetic ﬁeld produced by an elemental linear electric dipole situated at the ﬁxed point r = r0 and oscillating along a ˆ Such an ideal point dipole is ﬁxed direction speciﬁed by the unit vector d. characterized by the electric moment ˆ p(r, t) = p(t)δ(r − r0 )d,

(3.76)

where δ(r) denotes the Dirac delta function (see Appendix B). The particular solution of the inhomogeneous wave equation (3.58) for the Hertz potential may be expressed in the form given in Eqs. (3.35)–(3.36) for the retarded potentials as 4π p(r , t − R/c) 3 Π(r, t) = d r, (3.77) 4π R so that, with substitution from Eq. (3.76), the Hertz vector for a point dipole is found as 4π ˆ Π(r, t) = p(t − R/c)d, (3.78) 4πR where R = |r − r0 |. The microscopic electric and magnetic ﬁeld vectors are then given by Eqs. (3.67)–(3.68) where [making note of the mathematical equivalence ∇p(t − R/c) = (∂p(t − R/c)/∂t)(−∇R/c) with ∇R = R/R] 1 ˆ = p(t − R/c) ∇ 1 · d ˆ + 1 (∇p(t − R/c)) · d ˆ ∇· p(t − R/c)d R R R p(t − R/c) 1 ∂p(t − R/c) ˆ + R · d =− R3 cR2 ∂t so that

3.3 Li´enard–Wiechert Potentials

4π ∇ (∇ · Π) = 4π

3 1 ∂2p 3 ∂p + 2 3 2 p+ 5 4 R cR ∂t c R ∂t 1 1 ∂p ˆ d, p+ − R3 cR2 ∂t

and 4π ∇×Π= 4π

1 1 ∂p p+ 3 R cR2 ∂t

125

ˆ · R)R (d

ˆ × R), (d

where all quantities on the right-hand side of the above equations are evaluated at the retarded time t−R/c. With these substitutions, Eqs. (3.67)–(3.68) become 3 1 ∂2p ˆ 4π 3 ∂p e(r, t) = p+ + 2 3 2 (d · R)R 4π 0 R5 cR4 ∂t c R ∂t 1 ∂2p ˆ 4πµ0 1 1 ∂p d, (3.79) + 2 p+ − 4πc2 R3 cR2 ∂t c R ∂t2 4π µ0 1 ∂p 1 ∂2p ˆ b(r, t) = + (d × R). (3.80) c 4π R3 ∂t cR2 ∂t2 These are then the electromagnetic ﬁeld vectors for an elemental Hertzian dipole, where p = p(t − R/c). The detailed properties of this radiation ﬁeld are described in Section 3.4 following a more general derivation.

3.3 The Li´ enard–Wiechert Potentials and the Field of a Moving Charged Particle Speciﬁc attention is now turned to the ultimate source of all electromagnetic radiation, that being moving charged particles. It is ultimately desired to be able to calculate the electric and magnetic ﬁeld vectors produced by charged particles that are in arbitrary motion. In order to accomplish this one may employ the retarded potentials that are given in Eqs. (3.35)–(3.36); however, in order to perform the required integrations it is necessary to know the detailed motion of the charged particles because the calculations depend explicitly upon the positions and velocities of the charged particles at the retarded time t − R/c. It is therefore desirable to obtain expressions for the vector and scalar potentials {a(r, t), φ(r, t)} that explicitly exhibit the dependence of these potential functions on the velocity and acceleration of a given charged particle, and this is provided by the Li´enard-Wiechert potentials [8, 9]. 3.3.1 The Li´ enard–Wiechert Potentials Consider a single point particle of charge q whose motion traces out a trajectory that is described by the radius vector rq (t ) relative to some ﬁxed origin

126

3 Microscopic Potentials and Radiation

O in space, as depicted in Figure 3.3. The calculation of the resultant scalar potential function at any given ﬁxed observation point P with position vector r according to the expressions given in Eq. (3.35) requires a retarded time integration over the entire region of space containing charge that contributes to φ(r, t). For the single point charge q one may express the retarded time calculation in terms of a Dirac delta function as ∞ δ(t − t + |r − rq (t )|/c) 4π φ(r, t) = q (3.81) dt . 4π 0 −∞ |r − rq (t )| In order to evaluate this integral one needs to perform a change of variable

Fig. 3.3. Trajectory of a single point charge q that is described by the position vector rq (t ) with ﬁxed origin O. The ﬁeld produced by this charge is observed at the ﬁxed observation point P with position vector r.

so that the variable of integration is the same as the argument of the delta function. The integral is then just the value of the integrand taken at the time allowed by the delta function. This variable change is necessary because the position vector rq (t ) that describes the path of the given charged point particle is an explicit function of the time variable. Deﬁne then the new variable of integration as 1 t ≡ t − t + |r − rq (t )|. c

(3.82)

Upon diﬀerentiation of this expression and taking note of the fact that dt = 0 because t is the ﬁxed time of observation, one obtains

3.3 Li´enard–Wiechert Potentials

dt =

1 d 1+ |r − r (t )| dt . q c dt

127

(3.83)

The scalar quantity |r − rq (t )| is given by the expression

|r − rq (t )| =

3

1/2

2

(xi − xqi (t ))

,

(3.84)

i=1

where xqi = xqi (t ), i = 1, 2, 3 are the coordinates of the source point at the time t and where the coordinates xi , are ﬁxed because they represent the coordinates of the ﬁxed observation point P . Notice that the quantity |r − rq (t )| is only an implicit function of t through each of the coordinate values xqi = xqi (t ) which are themselves explicit functions of t . The derivative of the quantity |r − rq (t )|/c with respect to t is then given by 3

1 d 1 |r − r (t )| = q c dt c i=1 =

dxqi ∂ |r − rq (t )| ∂xqi dt

1 drq (∇q |r − rq (t )|) · , c dt

(3.85)

where the subscript q appearing on the gradient operator in this expression indicates that the partial diﬀerentiation operations are to be taken with respect to the coordinates of the charged particle, so that ∇q |r − rq (t )| = −

R(t ) r − rq (t ) =− . |r − rq (t )| R(t )

(3.86)

Furthermore, the derivative of the position vector rq (t ) with respect to t is just the velocity v(t ) of the charged particle. With the deﬁnition of the normalized velocity as v(t ) Υ(t ) ≡ (3.87) c with magnitude β(t ) ≡ |Υ(t )| = v(t )/c, Eq. (3.85) becomes Υ(t ) · R(t ) 1 d . |r − r (t )| = − q c dt R(t ) Substitution of this expression in Eq. (3.83) then yields Υ(t ) · R(t ) dt dt = 1 − R(t ) so that dt =

R(t )

R(t ) dt . − Υ(t ) · R(t )

(3.88)

(3.89)

(3.90)

128

3 Microscopic Potentials and Radiation

With the change of variable deﬁned in Eq. (3.82), the expression given in Eq. (3.81) for the scalar potential becomes ∞ R(t ) 1 4π φ(r, t) = dt q δ(t ) 4π 0 −∞ R(t ) R(t ) − Υ(t ) · R(t ) q 4π = . 4π 0 R(t ) − Υ(t ) · R(t ) t =0 Because t = 0 implies that t = t − R(t )/c, then φ(r, t) =

q 4π , 4π 0 R(t ) − Υ(t ) · R(t )

(3.91)

where t denotes the retarded time here and throughout the remainder of this section. Because the current density j is just equal to the charge density multiplied by the velocity of the particle, an analogous calculation can be carried out for the vector potential with the result 4π µ0 c qΥ(t ) a(r, t) = . (3.92) c 4π R(t ) − Υ(t ) · R(t ) Equations (3.91)–(3.92) for the scalar and vector potentials, which explicitly exhibit the dependence of the potentials on the velocity of the charged particle, are called the Li´enard-Wiechert potentials.

Fig. 3.4. Spherical “information shell” collapsing to the ﬁxed observation point P with velocity c.

Some physical insight concerning the form of the Li´enard–Wiechert potentials for a moving charge may be gained through a consideration of the

3.3 Li´enard–Wiechert Potentials

129

following construction due to Panofsky and Phillips [10]. Consider the calculation of the scalar potential φ(r, t) at a given ﬁxed observation point P at some speciﬁc instant of time t. Surround the point P with a spherical shell of radius R which is suﬃciently large that it contains all of the charge that contributes to the potential at the time t. At the initial instant of time the shell is allowed to collapse to the center P with the velocity c. As the collapsing shell sweeps through the charge distribution it gathers information regarding the charge density. The spherical shell will then have collapsed to the observation point P at the prescribed instant of time t and will have gathered all of the information necessary to compute the value of the scalar potential φ(r, t) at that particular space–time point, where r is the position vector for the point P . For a distribution of stationary charges this construct just yields the retarded potential given in Eqs. (3.35)–(3.36). However, if the charge distribution has a net outward (inward) velocity, the volume integral of the charge density that is measured by the collapsing sphere will yield a result that is smaller (larger) than the total charge of the system. Consider then a diﬀerential element of charge dq that is distributed uniformly with a charge density ρ throughout a volume element dV , as illustrated in Figure 3.4. Let R be the vector from dV to the observation point P . If the charge dq is stationary, then the amount of charge that the spherical shell will cross as it contracts by an amount dR in a time interval dt is given by ρ(t )dadR = ρ(t )dV . On the other hand, if the charge moves with a velocity v, then the amount of charge that is crossed by the spherical shell in the time interval dt will be altered from its stationary value to the value dq = ρ(t )dV − ρ(t )

v(t ) · R(t ) dadt. R(t )

Clearly, if v is directed inward then v · R is positive and dq is reduced from its stationary value, whereas if v is directed outward then v · R is negative and dq is increased from its stationary value. Because dV = (da)(dR) and dR = cdt, then (da)(dt) = (dV /dR)(dR/c) = (1/c)dV . Substitution of this expression in the previous result then yields Υ(t ) · R(t ) dq = 1 − ρ(t )dV R(t ) ρ(t ) dV, = [R(t ) − Υ(t ) · R(t )] R(t ) so that ρ(t ) dq dV = . R(t ) R(t ) − Υ(t ) · R(t ) Substitution of this expression in Eq. (3.35) for the retarded scalar potential then gives

130

3 Microscopic Potentials and Radiation

φ(r, t) =

4π 4π 0

V

dq . R(t ) − Υ(t ) · R(t )

(3.93)

If the complete charge distribution contains a total charge q that is conﬁned to an inﬁnitesimally small region of space, one can then perform the required integration by neglecting the variation of the denominator over the charge region and Eq. (3.93) reduces to the expression given in Eq. (3.91). A similar argument for the vector potential gives 4π µ0 c Υ(t )dq a(r, t) = , (3.94) c 4π V R(t ) − Υ(t ) · R(t ) which reduces to the expression given in Eq. (3.92) for the case of a single point charge. Notice that R, Υ = v/c, and R = |R| are all functions of the retarded time t = t − R(t )/c. The preceding derivation provides some further insight into the physical interpretation of the concept of a point charge as well as to the physical applicability of the results obtained for such a mathematical idealization. All that is required to obtain the Li´enard–Wiechert potentials (3.91)–(3.92) for the scalar and vector potentials is that the quantities (R(t ) − R(t ) · Υ(t )) and Υ(t ) have a negligible variation over the spatial extent of the charge q. 3.3.2 The Field Produced by a Moving Charged Particle With the expressions given in Eqs. (3.93) and (3.94) for the scalar and vector potentials in the Lorenz gauge for the microscopic electromagnetic ﬁeld that is produced by a point charge undergoing an arbitrary motion in free space, the microscopic ﬁeld vectors themselves are given by 1 ∂a(r, t) , (3.95) e(r, t) = −∇φ(r, t) − c ∂t b(r, t) = ∇ × a(r, t). (3.96) Because the time derivative of the vector potential a(r, t), and hence of the normalized velocity Υ = v/c, is involved, it is then clear that the electric and magnetic ﬁeld vectors will be functions not only of the velocity v, but also of the acceleration ˇ a = dv/dt of the charged particle.5 The electric and magnetic ﬁeld vectors may then each be separated into two components as e(r, t) = ev (r, t) + ea (r, t),

(3.97)

b(r, t) = bv (r, t) + ba (r, t).

(3.98)

Here ea (r, t) and ba (r, t) involve the particle acceleration and are accordingly called the acceleration ﬁelds, both of which go to zero when ˇ a = 0, whereas 5

This holds for the magnetic ﬁeld vector because the spatial derivatives involved in the curl operation are to be applied to the retarded quantities appearing in the Li´enard–Wiechert vector potential.

3.3 Li´enard–Wiechert Potentials

131

ev (r, t) and bv (r, t) involve the particle velocity and are accordingly called the velocity ﬁelds, both of which yield the static ﬁeld for a point charge with velocity v = 0. Let the coordinates of the ﬁeld point P = P (x1 , x2 , x3 ) be denoted by xα , with α = 1, 2, 3, and let the retarded source point coordinates be denoted by xα = xα (t ), where t denotes the retarded time. The ﬁeld and source point variables are connected by the retardation condition 3 1/2 2 R(xα , xα ) ≡ (xα − xα ) = c(t − t ). (3.99) α=1

The components of the vector diﬀerential operator ∇ appearing in Eqs. (3.95)–(3.96) are partial derivatives at a ﬁxed time t, and therefore not at constant retarded time t . Partial diﬀerentiation with respect to the ﬁeld coordinates xα compares the potentials at neighboring points at the same time t, but the signals producing these potentials originated from the source charge at diﬀerent retarded times t . Similarly, the partial derivative with respect to the time t implies constant ﬁeld point coordinates xα , and hence refers to the comparison of the potentials at a given ﬁeld point over an interval of time during which the coordinates xα of the source charge will have changed. Because only the time variation with respect to t is given, it is then necessary to transform the diﬀerential operators ∂/∂t|xα and ∇|t to expressions in terms of the partial diﬀerential operator ∂/∂t |xα in order to compute the ﬁelds. The retarded values of the velocity and acceleration of the point charge q, given by dxα , dt dvα d2 xα a ˇα = = , dt dt2 vα =

(3.100) (3.101)

are assumed known. In terms of the “position” vector R these expressions may be written in vector form as dR , dt dv d2 R ˇ a = = − 2 , dt dt

v=−

(3.102) (3.103)

where the minus sign appears because R is the radius vector from the charge q to the observation point P , and not the reverse. Furthermore, diﬀerentiation of the identity R2 = R · R with respect to t with ﬁxed ﬁeld coordinates xα yields ∂R ∂R = 2R · (3.104) 2R ∂t xα ∂t xα

132

3 Microscopic Potentials and Radiation

so that

∂R ∂t

=− xα

R·v . R

(3.105)

Upon diﬀerentiation of the retardation condition (3.99) with respect to t, one obtains ∂R ∂t = c 1− ∂t ∂t ∂R ∂t R · v ∂t = =− , ∂t ∂t R ∂t so that ∂t 1 1 = = , ∂t 1 − (R · v/(Rc) 1 − (R · Υ)/R or ∂ ∂ R ∂ R = , = ∂t R − R · Υ ∂t s ∂t

(3.106)

which is the desired transformation for the time derivatives, where s ≡ R − R · Υ.

(3.107)

Similarly, diﬀerentiation of the same expression with respect to the spatial coordinates xα gives ∇R = −c∇t = ∇1 R + so that ∇t = −

∂R R R·v − ∇t , ∇t = ∂t R R

R R =− , c(R − R · Υ) sc

(3.108)

and hence, in general ∇ = ∇1 + (∇t )

∂ R ∂ = ∇1 − . ∂t sc ∂t

(3.109)

Here the operator ∇1 implies diﬀerentiation with respect to the ﬁrst argument of the function f (xα , t ) = R(xα , xα (t )) appearing in Eq. (3.99), that is, diﬀerentiation at ﬁxed retarded time t . The expressions appearing in Eqs. (3.106) and ((3.109) constitute the required transformation relations for the diﬀerential operators from the coordinates of the ﬁeld point to those of the moving charged particle. The computation of the electric ﬁeld vector from the Li´enard–Wiechert potentials given in Eqs. (3.91)–(3.92) then becomes

3.3 Li´enard–Wiechert Potentials

1 ∂a e = −∇φ − c ∂t 1 4π 1 ∂ v + 2 q ∇ =− 4π 0 s c ∂t s 4π 1 1 ∂ v = . q ∇s − 4π 0 s2 c2 ∂t s

133

(3.110)

With Eqs. (3.106) and (3.109) one then obtains 1 4π 0 e = 2 ∇1 s − 4πq s 1 = 2 ∇1 s − s 1 = 2 ∇1 s − s

R ∂s R ∂ v − cs3 ∂t c2 s ∂t s a v ∂s R ∂s R ˇ − 2 − 2 cs3 ∂t c s s s ∂t R ∂s R Rv ∂s − 2 2ˇ a + 2 3 . cs3 ∂t c s c s ∂t

(3.111)

Diﬀerentiation of the retardation relation given in Eq. (3.107) yields the pair of expressions 1 ∇1 s = ∇1 R − R · v c R v = − , (3.112) R c ∂s ∂R 1 ∂ = − (R · v) ∂t ∂t c ∂t 1 R · v 1 ∂R ∂v ·v− R· =− − R c ∂t c ∂t R·ˇ a R · v v2 + − , (3.113) =− R c c which, when substituted in Eq. (3.111), results in the expression 1 R v R·v R·ˇ a 4π 0 R v2 e = 2 − − − − 3 4πq s R c cs c R c 2 R R·v R·ˇ a Rv v − 2 2ˇ − − . a+ 2 3 c s c s c R c

(3.114)

The velocity part of the electric ﬁeld vector is then given by ev (r, t) =

4π q (R − RΥ)(1 − β 2 ), 4π 0 s3

(3.115)

and the acceleration part is given by ea (r, t) =

4π q R × ((R − RΥ) × ˇ a) , 4π 0 c2 s3

(3.116)

134

3 Microscopic Potentials and Radiation

where β = |Υ| = v/c. The computation of the magnetic induction vector from the Li´enard– Wiechert potentials follows along similar lines as v 4π 0 c2 b = ∇× 4πcq s 1 1 =∇ ×v+ ∇×v s s 1 1 = − 2 (∇s) × v + ∇ × v s s 1 R ∂s R ∂v 1 = − 2 ∇1 s − ×v+ ∇1 × v − × s sc ∂t s sc ∂t R·ˇ a 1 R v R R · v v2 R×ˇ a − − + − =− 2 − ×v− 2 , s R c sc R c c s c because ∇1 × v = 0, so that

4π 0 c2 R×ˇ a v×R 1 1 R·v R·ˇ a v2 b=− + + + − . 4πcq cs2 s2 R s cR c2 c2

(3.117)

The velocity part of the magnetic induction ﬁeld vector is then given by bv (r, t) =

4πc q (v × R)(1 − β 2 ), 4π 0 c2 s3

(3.118)

and the acceleration part of the ﬁeld is given by ba (r, t) =

4πc q R × {R × [(R − RΥ) × ˇ a]} . 4π 0 c3 s3 R

(3.119)

Taken together, Eqs. (3.115)–(3.116) and (3.118)–(3.119) constitute the set of velocity and acceleration ﬁelds for the electromagnetic ﬁeld produced by a moving charged particle q. The quantities R, R, Υ, and s = R − R · Υ appearing on the right-hand side of each of these expressions are all evaluated at the retarded time t = t−R(t )/c. Upon comparison of Eqs. (3.118)–(3.119) with Eqs. (3.115)–(3.116), it is seen that b(r, t) =

c R × e(r, t). c R

(3.120)

Hence, the magnetic ﬁeld vector is always perpendicular to both the electric ﬁeld vector and the retarded radius vector R. Furthermore, bv is perpendicular to ev , and ba is perpendicular to ea . The retarded radius vector R is perpendicular to ea , bv , and ba , but is not in general perpendicular to ev . It is important to point out that the above set of equations for the electromagnetic ﬁeld vectors is relativistically correct. Indeed, the velocity v = cΥ may simply be considered as the relative velocity between the point charge

3.3 Li´enard–Wiechert Potentials

135

and the observer. However, the Li´enard–Wiechert potentials were derived prior to the advent of the special theory of relativity; it was only after the same results were obtained relativistically that it became apparent that v could indeed be interpreted as the relative velocity. The velocity part of the electric ﬁeld, given in Eq. (3.115), is seen to vary as R−2 for large distances R between the source and ﬁeld point, and is formally identical with the convective ﬁeld produced by a uniformly moving charged particle [10]. With regard to this ﬁeld one may deﬁne the quantity Rv ≡ R − RΥ v = R−R c

(3.121)

as the “virtual present radius vector,” that is, the position vector that would describe the position the point charge would occupy “at present” if it had continued with a uniform velocity from the point xq (t ), as illustrated in Figure 3.5. In terms of this quantity the velocity part of the ﬁeld vector becomes 4π q ev (r, t) = Rv (1 − β 2 ), (3.122) 4π 0 s3 and this is just the electric ﬁeld produced by a charge in uniform motion.

True Present Position Virtual Present Position

R

Retarded Position xq(t')

Rv R

Field Point xΑ(t) Fig. 3.5. Geometric relationship between the retarded position, the true present position, and the virtual present position of a point charge q moving along an arbitrary path in space (indicated by the dotted curve).

136

3 Microscopic Potentials and Radiation

3.3.3 Radiated Energy from a Moving Charged Particle The radiated energy from a moving charged particle q is obtained from the Poynting vector of the electromagnetic ﬁeld. From Eqs. (3.97)–(3.98), the microscopic Poynting vector may be expressed as s(r, t) = sv (r, t) + sa (r, t) + s× (r, t),

(3.123)

where sv (r, t) ≡ c/4πev (r, t) × bv (r, t) is the Poynting vector associated with the velocity part of the ﬁeld, sa (r, t) ≡ c/4πea (r, t) × ba (r, t) is the Poynting vector associated with the acceleration part of the ﬁeld, and s× (r, t) is due to the cross terms from both the velocity and acceleration parts of the ﬁeld. From Eqs. (3.115) and (3.118), the microscopic Poynting vector associated with the velocity part of the ﬁeld is given by 4πc q 2 (1 − β 2 )(R − RΥ) × (Υ × R) (4π)2 0 s6 4πc q 2 (1 − β 2 ) R(R − Υ · R) + (Rβ 2 − Υ · R)R . (3.124) = (4π)2 0 s6

sv =

From Eqs. (3.116) and (3.119), the microscopic Poynting vector associated with the acceleration part of the ﬁeld is given by sa =

4πc q 2 {[R × ((R − RΥ) × ˇ a)] × [R × (R × ((R − RΥ) × ˇ a))]} . (4π)2 0 Rs6

In order to evaluate the complicated cross product appearing in this expression, let Γ ≡ (R − RΥ) × ˇ a = Rv × ˇ a, so that (R × Γ) × (R × (R × Γ)) = (R × Γ) × [(R · Γ)R − R2 Γ] = R2 Γ × (R × Γ) − (R · Γ)R × (R × Γ) = R2 [Γ 2 R − (Γ · R)Γ] − (R · Γ)[(R · Γ)R − R2 Γ] = R2 Γ 2 − (R · Γ)2 R. Hence, sa =

a)2 − (R · (Rv × ˇ a))2 4πc q 2 R2 (Rv × ˇ R, (4π)2 0 s6 R

(3.125)

which is along the direction of the retarded radius vector from the point charge q to the observation point P . The radial dependencies of the Poynting vectors for the velocity and acceleration ﬁelds are therefore given by sv ∝

1 , R4

sa ∝

1 , R2

3.4 The Radiation Field Produced by a General Dipole Oscillator

137

where the magnitude of the contribution s× to the Poynting vector from the cross-terms is proportional to R−3 . In order to determine the energy radiated by a moving charged particle, the normal component of the Poynting vector s(r, t) must be integrated over the surface of a sphere. As the radius of the sphere becomes large, then because the element of surface area involves the quantity R2 , the surface integral involving sv vanishes as 1/R2 , whereas the integral involving sa remains ﬁnite and nonvanishing in general. Hence, a charged particle that moves with a constant velocity, and so has sa = 0, cannot radiate energy; only a charged particle that is undergoing acceleration can produce radiation. The fact that a charged particle moving with a uniform velocity cannot radiate energy is consistent with the relativistic nature of the ﬁeld quantities, for if v is the relative velocity between the charged particle and the observation point, there is a reference frame in which the particle is at rest and the observation point is in uniform motion, and a static charge clearly cannot radiate energy. That is, if it is possible to ﬁnd an inertial reference frame with respect to which the charged particle is at rest, then radiation cannot occur.

3.4 The Radiation Field Produced by a General Dipole Oscillator The radiation ﬁeld that is most fundamental to the microscopic MaxwellLorentz theory is that of an atomic dipole oscillator. Although classical physics fails to provide a correct description of electronic motion in an atomic system, the quantum theory clearly shows that in many cases the atomic radiation ﬁeld is primarily due to the dipole moment of the atomic system so that the radiation ﬁeld can be thought of as being produced by a classical dipole oscillator. Consider then an atomic system in which an electron of charge q1 = −qe moves about a ﬁxed nucleus of charge q2 = +qe (any other nuclear charges are assumed to be eﬀectively screened by the other bound electrons of the electrically neutral system) such that its velocity v is small in comparison to the velocity of light c in free space and its position is always in close proximity to the nucleus. Let d = d(t) denote6 the temporally varying position vector of the electron relative to the nucleus, let r denote the position vector from the nucleus to the ﬁxed observation point P , and let re denote the vector from the electron position to the point P , as illustrated in Figure 3.6. The radiation ﬁeld at any observation point P that is not too near the source atomic site (r d) is then given to a high degree of accuracy by just the ﬁrst few terms in an expansion of the exact ﬁeld in a power series in terms of the dimensionless quantities d/r and v/c. However, instead of 6

The position vector d(t) should not be confused here with the microscopic electric displacement vector d(r, t) ≡ 0 e(r, t).

138

3 Microscopic Potentials and Radiation

-qe d

re

P

r

+qe

Fig. 3.6. Atomic dipole oscillator comprised of a ﬁxed nucleus and a bound electron of equal and opposite charge.

performing this expansion on the exact ﬁeld, it is more appropriate to expand the Li´enard-Wiechert potentials in a power series in terms of the same quantities and then determine the radiation ﬁeld that results from the lowest order terms that appear in this expansion. This latter approach is taken in the analysis presented here which follows that given by Stone [11]. 3.4.1 The Field Vectors Produced by a General Dipole Oscillator The scalar and vector potentials at the stationary ﬁeld observation point P at the time t consist of the superposition of the Li´enard–Wiechert potentials due to the ﬁxed nucleus (ﬁxed relative to P ) and the bound electron. This physical situation then involves the following three interrelated time variables: – t ≡ the time at which both the ﬁelds and potentials at the observation point P are to be determined. – t ≡ t − r/c = the retarded time at the ﬁxed position of the nucleus corresponding to the time t at the observation point P . The ﬁeld contribution at the point P at time t due to the nuclear charge +qe depends upon the presence and position of the nuclear charge at the retarded time t . – te ≡ t − re (te )/c = the retarded time at the variable position of the bound electron corresponding to the time t at the observation point P . The ﬁeld contribution at P at time t due to the bound electronic charge −qe depends upon the electrons position and motional state at the retarded time te . In addition to these speciﬁc time variables, it is convenient to introduce an arbitrary time variable τ that may be assigned any particular value. It is

3.4 The Radiation Field Produced by a General Dipole Oscillator

139

assumed that the position of the bound electron is a known vector function ˙ ) and d(τ ) relative to the ﬁxed position of the nucleus, with velocity d(τ ¨ ). The bound electron’s distance from the nucleus is assumed acceleration d(τ to always be less than some ﬁnite maximum distance dmax , so that |d(τ )| ≤ dmax

(3.126)

for all τ , and the distance r = |r| of the ﬁeld observation point P from the nucleus is assumed to be much greater than that same maximum distance dmax , so that (3.127) r dmax . In addition, the velocity of the bound electron is assumed to always be less than some maximum value vmax , where ˙ (3.128) d(τ ) ≤ vmax c for all τ . These three inequalities constitute the basic approximations under which the analysis of the radiation ﬁeld due to an atomic dipole oscillator is undertaken in this development. From Eqs. (3.91) and (3.92), the Li´enard–Wiechert potentials due to the charge +qe of the (stationary) nucleus are given by 4π qe , 4π 0 r an (r, t) = 0,

φn (r, t) =

and those due to the bound electron of charge −qe are given by −1 d˙ · re 4π qe φe (r, t) = − , 1− 4π 0 re cre −1 d˙ · re 4πµ0 d˙ qe 1− an (r, t) = − . 4πc re cre

(3.129) (3.130)

(3.131)

(3.132)

Because of the inequality given in Eq. (3.128), the preceding pair of expressions for the scalar and vector potentials of the bound electron may be expanded as d˙ · re 4π qe 1+ + ··· , (3.133) φe (r, t) = − 4π 0 re cre d˙ · re 4πµ0 d˙ an (r, t) = − qe + ··· , (3.134) 1+ 4πc re cre respectively, where the sum of the remaining terms in each series is of order O((vmax /c)2 ).

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3 Microscopic Potentials and Radiation

The position vector re from the bound electron position to the observation point P is given by the vector diﬀerence re = r − d so that its magnitude is given by 1/2 re ≡ |re | = r2 − 2r · d + d2 1/2 r · d d2 = r 1−2 2 + 2 , r r and consequently −1/2 1 1 r · d d2 1−2 2 + 2 = re r r r 2 1 r·d d 3(r · d)2 = + 3 − 3+ + ···, r r 2r 2r5

(3.135)

where the sum of the remaining terms in the series is O(d3 /r4 ). The vector r and its magnitude r are to be evaluated at the retarded time t = t − r/c in the Li´enard–Wiechert potentials appearing in Eqs. (3.133)– (3.134) whereas the vector d and its magnitude d are to be evaluated at the retarded time te = t − re (te )/c. It is possible (and quite advantageous) to express d, d, and their various time derivatives in terms of the retarded time t which is simply proportional to the observation time t and is only slightly diﬀerent from te . The relation between these two retarded times is simply te = t +

(te ) , c

(3.136)

where (te ) ≡ r − re (te ). By Lagrange’s theorem [12], if x(τ ) denotes either d(τ ) or one of its derivatives with respect to τ , then x(te ) = x(t ) +

∞ 1 dn−1 n x(t ˙ ) ((t )) n n−1 n!c dt n=1

1 = x(t ) + x(t ˙ )(t ) + · · · . c Because (te ) = r − re (te ) 1/2 r · d(te ) d2 (te ) = r−r 1−2 + r2 r2 2 d(te ) · r d2 (te ) (d(te ) · r) = r−r 1− + − + ··· r2 2r2 2r4 2

=

d(te ) · r d2 (te ) (d(te ) · r) − + + ··· r 2r 2r3

(3.137) (3.138)

3.4 The Radiation Field Produced by a General Dipole Oscillator

141

then, with this result evaluated at t , Eq. (3.138) becomes 2 1 d(t ) · r d2 (t ) (d(t ) · r) x(te ) = x(t )+ x(t + · · · +· · · . (3.139) ˙ ) − + c r 2r 2r3 The relative position vector and velocity of the bound electron at the retarded time te may then be expressed in terms of the retarded time t at the ﬁxed position of the nucleus as 2 d(t ) · r d2 (t ) (d(t ) · r) 1˙ − + + · · · + · · · , (3.140) d(te ) = d(t ) + d(t ) c r 2r 2r3 2 2 1 ˙ e ) = d(t ¨ ) d(t ) · r − d (t ) + (d(t ) · r) + · · · + · · · . (3.141) ˙ ) + d(t d(t c r 2r 2r3 With these expansions the Li´enard–Wiechert potentials due to the bound electron of charge q = −qe may be expressed in terms of quantities that are evaluated at the retarded time t in the following manner. Substitution of Eq. (3.135) into the expansion given in Eq. (3.133) for the scalar potential gives ˙ e ) · re (te ) d(t 4π qe 1 + + ··· φe (r, t) = − 4π 0 re (te ) cre (te ) 4π 1 d(te ) · r d2 (te ) + = − qe − + · · · 4π 0 r r3 2r3 ˙ e ) · re (te ) 1 d(te ) · r d2 (te ) d(t + − + · · · + ··· , × 1+ c r r3 2r3 so that, with the further substitution of the relation re (te ) = r − d(te ), there results 1 d(te ) · r d2 (te ) 4π + φe (r, t) = − qe − + ··· 4π 0 r r3 2r3 ˙ e ) · (r − d(te )) 1 d(te ) · r d2 (te ) d(t − + ··· + ··· × 1+ + c r r3 2r3 1 d(te ) · r d2 (te ) 4π = − qe − + · · · + 4π 0 r r3 2r3 ˙ e ) · r 1 d(te ) · r d2 (te ) d(t × 1+ + − + · · · c r r3 2r3 ˙ e ) · d(te ) 1 d(te ) · r d2 (te ) d(t + − − + ··· + ··· . c r r3 2r3 Substitution of the pair of relations appearing in Eqs. (3.140)–(3.141) into the above expansion of the scalar potential then yields (where all quantities

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3 Microscopic Potentials and Radiation

on the right-hand side of the equal sign are now evaluated at the retarded time t at the stationary position of the nucleus) r (d · r)2 d˙ d · r d2 4π 1 + 3 · d+ − + φe (r, t) = − qe + ··· + ··· 4π 0 r r c r 2r 2r3 d · r d2 (d · r)2 2 1 − + − 3 s2 + d · d˙ + · · · 2r c r 2r 2r3 2 d˙2 d · r d2 (d · r)2 + 2 + ··· + ··· + ··· − + c r 2r 2r3

2 ¨ d · r d2 d (d · r) r − + × 1 + · d˙ + + ··· + ··· c c r 2r 2r3 1 d˙ d · r d2 r (d · r)2 × + 3 · d+ − + + ··· + ··· r r c r 2r 2r3 d · r d2 1 2 (d · r)2 − 3 d2 + d · d˙ − + + · · · 2r c r 2r 2r3 2 (d · r)2 s˙ 2 d · r d2 − + + 2 + ··· + ··· + ··· c r 2r 2r3 ¨ d · r d2 d 1 ˙ (d · r)2 − d+ + ··· + ··· − + c c r 2r 2r3 (d · r)2 d˙ d · r d2 − + · d+ + ··· + ··· c r 2r 2r3 d˙ d · r d2 1 r (d · r)2 × + 3 · d+ − + + · · · + ··· r r c r 2r 2r3 d · r d2 1 (d · r)2 2 2 ˙ − 3 d + d·d + ··· − + 2r c r 2r 2r3

2 d˙2 d · r d2 (d · r)2 + 2 − + + ··· + ··· + ··· + ··· . c r 2r 2r3 (3.142) Upon collecting terms that are ordered according to the number of times the electronic position vector d or any of its time derivatives or combinations thereof appears, one ﬁnally obtains

3.4 The Radiation Field Produced by a General Dipole Oscillator

143

˙ 1 r · d (r · d)(r 4π · d) d2 3 + 3 + φe (r, t) = − qe − 3 + O(d ) 4π 0 r r cr4 2r ˙ ¨ · d) d · d˙ (r · d)(r · d) (r · d)(r r · d˙ 3 + + O(d ) + − × 1+ cr cr3 c2 r2 cr r 1 d·ˆ r d˙ · ˆ d2 4π qe + 2 + − 3 = − 4π 0 r r cr 2r +

¨ ·ˆ r)(d r) 3(d · ˆ r)(d˙ · ˆ r) (d · ˆ + + O(d3 ) , c2 r c2 r

(3.143)

where ˆ r ≡ r/r is the unit vector in the direction of the observation point P . From the pair of relations given in Eqs. (3.133)–(3.134), the vector potential ae (r, t) that is associated with the scalar potential φe (r, t) is seen to be given by

0 µ0 ˙ ae (r, t) = (3.144) d(te )φe (r, t) c so that, with substitution from Eqs. (3.141) and (3.143), there results 4π µ0 r d · ˆ r r d˙ · ˆ 1 d·ˆ ¨ + O(d3 ) qe d˙ + d + 2 + + O(d2 ) ae (r, t) = − c 4π c r r cr 4π µ0 d˙ · ˆ r˙ d·ˆ r d·ˆ r¨ 1˙ qe d + 2 d˙ + d+ d + O(d3 ) . (3.145) = − c 4π r r cr cr The above pair of expressions (3.143) and (3.145) for the scalar and vector Li´enard–Wiechert potentials due to the bound electron of charge q = −qe are both in the form of a series of terms involving the position vector r = rˆ r of the ﬁeld observation point relative to the ﬁxed position of the nucleus, the position vector d(t ) of the bound electron relative to the position of the nucleus, and the derivatives of d(t ) with respect to the retarded time t = t − r/c at the ﬁxed position of the nucleus corresponding to the time t at the stationary observation point P . The terms in each series expansion have been ordered into groups that represent successive approximations to each of the potentials in the following manner: in the zeroth or lowest-order group the electronic position vector d does not appear (the scalar potential alone has this term); in the ﬁrst-order group the electronic position vector d ˙ in the second-order group the appears once in each term (either as d or as d); electronic position vector d appears twice in each term (either as d · d = d2 , ˙ (d ·ˆ ˙ etc.); and so on for higher-order terms. The zerothd · d, r)(d˙ ·ˆ r), (d ·ˆ r)d, order group in the scalar potential represents the monopole contribution due to the electron whose position is taken at the ﬁxed position of the nucleus (this

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3 Microscopic Potentials and Radiation

contribution to the total scalar potential of the atomic dipole oscillator will then be identically canceled by the monopole contribution due to the ﬁxed nucleus of charge q = +qe ). The ﬁrst-order groups in both the scalar and vector potentials represent the electric dipole contribution to the total ﬁeld because only the electric dipole moment p = qe d and its time derivative p˙ = qe d˙ appear. The second-order groups in both the scalar and vector potentials represent both the electric quadrupole and magnetic dipole contributions to the total ﬁeld. In general, the nth-order groups in both the scalar and vector potentials represent both the electric 2n -pole and the magnetic 2n−1 -pole contributions to the total ﬁeld. 3.4.2 The Electric Dipole Approximation It is clear from Eqs. (3.143) and(3.145) that any (n+1)th-order group provides a negligible contribution to the total radiated ﬁeld when compared to the contribution from the nth-order group provided that the pair of inequalities appearing in Eqs. (3.127)–(3.128) are both satisﬁed and that the electric 2n pole and the magnetic 2n−1 -pole contributions are together nonvanishing. As a consequence, an accurate approximation to the ﬁeld produced by an atomic dipole oscillator may be obtained by considering only the terms up through the electric dipole contribution (the ﬁrst group) provided that the above pair of inequalities are both satisﬁed and that the electric dipole moment does not vanish nor is vanishingly small. If the electronic motion is such that at all times the resultant electric dipole moment p = qe d and its time derivative both vanish, then one must consider the electric quadrupole and magnetic dipole contributions or, if these also vanish, the ﬁrst nonvanishing higherorder multipole contribution must then be considered. The ﬁrst-order or electric dipole approximation to the Li´enard–Wiechert potentials due to the motion of a bound electron about the nucleus in an atom with a stationary nuclear site is then given by φ(r, t) = φn (r, t) + φe (r, t)

˙ ) · ˆ 4π q r d(t r 1 d(t ) · ˆ 4π e ∼ − + qe + = 2 4π 0 r 4π 0 r r cr ˙ ) · ˆ 4π r d(t r d(t ) · ˆ =− , qe + 2 4π 0 r cr

a(r, t) = an (r, t) + ae (r, t) ˙ ) 4π µ0 d(t ∼ qe , = − c 4π r

(3.146)

(3.147)

˙ ) are both where ˆ r = r/r is independent of the time and where d(t ) and d(t evaluated at the retarded time t = t − r/c at the stationary position of the nucleus.

3.4 The Radiation Field Produced by a General Dipole Oscillator

145

The electric and magnetic ﬁeld vectors of the radiated ﬁeld are obtained from the scalar and vector Li´enard–Wiechert potentials through use of the diﬀerential relations 1 ∂a(r, t) e(r, t) = −∇φ(r, t) − , (3.148) c ∂t b(r, t) = ∇ × a(r, t). (3.149) Because r is independent of the time, then ∂/∂t = ∂/∂t. However, care must be taken in evaluating the required spatial derivatives because of the presence of r in the retarded time. If one lets the ordered triple (x, y, z) denote the coordinates of the ﬁeld observation point P and (x , y , z ) be the coordinates of the atomic nucleus, then ∂d(t ) dd(t ) ∂t = ∂x dt ∂x ˙ ) ∂ t − r = d(t ∂x c 1/2 1˙ ∂ (x − x )2 + (y − y )2 + (z − z )2 = − d(t ) c ∂x 1 ˙ x − x , = − d(t ) c r

(3.150)

with analogous expressions for the partial derivatives with respect to y and z. With Eqs. (3.146) and (3.147), the electric ﬁeld vector given by Eq. (3.148) becomes ˙ ) · ˆ 4π d(t 1 ) · ˆ r r d(t ¨ ) . e(r, t) ∼ + 2 d(t qe ∇ + = 4π 0 r2 cr c r The vector identity ∇(U·V) = U·∇V +V ·∇U+U×(∇×V)+V ×(∇×U) applied to the two gradient quantities in the above equation then gives ˆ r 4π ˆ r ˆ r ∼ qe d(t ) · ∇ e(r, t) = + 2 · d(t ) + 2 × (∇ × d(t )) 2 4π 0 r r r 1˙ ˆ r ˆ r ˆ r 1 ˙ )+ ˙ )) + ¨ ) , · ∇d(t × (∇ × d(t + d(t ) · ∇ + d(t c r cr cr c2 r (3.151) because ∇ × (ˆ r/rn ) = 0 for n = 0, 1, 2, . . .. The various spatial derivatives appearing in Eq. (3.151) are now evaluated. First of all, for any vector ﬁeld U and integer value n, ˆ r U·ˆ r U U·∇ r, (3.152) = n+1 − (n + 1) n+1 ˆ rn r r

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3 Microscopic Potentials and Radiation

so that

r d(t ) · ˆ d(t ) −3 ˆ r, 3 r r3 ˙ ) · ˆ ˙ ) r d(t d(t r ˙d(t ) · ∇ ˆ − 3 ˆ r. = 2 3 3 r r r

d(t ) · ∇

ˆ r r2

=

(3.153) (3.154)

With the expression given in Eq. (3.150) for ∂d(t )/∂x with analogous expressions for ∂d(t )/∂y and ∂d(t )/∂z, there then results 1˙ ˆ r · ∇d(t ) = − d(t ), c ˙ ) = − 1 d(t ¨ ), ˆ r · ∇d(t c

(3.155) (3.156)

and 1 ˙ ), r × d(t ∇ × d(t ) = − ˆ (3.157) c ˙ ) = − 1 ˆ ¨ ). r × d(t ∇ × d(t (3.158) c Substitution of these expressions into Eq. (3.151) for the electric ﬁeld vector then gives

˙ ) · ˆ d(t 4π r r d(t ) · ˆ d(t ) ∼ ˆ r e(r, t) = qe − 3 +2 4π 0 r3 r3 cr2 ˙ ) d(t ¨ ) d(t . (3.159) −ˆ r× ˆ r× + 2 cr2 c r Because any vector U may always be resolved into components parallel and perpendicular to the unit vector ˆ r through use of the relation U = (U · ˆ r)ˆ r −ˆ r × (ˆ r × U), then the above expression for the electric ﬁeld vector may accordingly be resolved as

˙ ) 4π d(t ) d(t e(r, t) ∼ 2 ·ˆ r ˆ r qe + = 4π 0 r3 cr2 ˙ ) d(t ¨ ) d(t ) d(t . (3.160) +ˆ r× ˆ r× + + 2 r3 cr2 c r For the magnetic ﬁeld vector, Eqs. (3.147) and (3.149) give ˙ ) d(t ¨ ) 4π µ0 d(t ∼ b(r, t) = qeˆ , r× + 2 c 4π r cr

(3.161)

where Eq. (3.158) has been used in obtaining the ﬁnal result. These two expressions constitute the ﬁeld vectors of the radiation ﬁeld produced by an atomic dipole oscillator in the electric dipole approximation, where t = t−r/c is the retarded time at the stationary position of the atomic nucleus.

3.4 The Radiation Field Produced by a General Dipole Oscillator

147

3.4.3 The Field Produced by a Monochromatic Dipole Oscillator in the Electric Dipole Approximation Consider a single atom whose (screened) nucleus with charge +qe is at rest in the laboratory frame of reference, and where a single bound electron of charge −qe oscillates at a constant angular frequency ω in an elliptical path about the nucleus, as illustrated in Figure 3.7. The real-valued position vector

-qe

. d

d

+qe

Ω Fig. 3.7. Elliptical path of motion of a bound electron about a ﬁxed nucleus.

d(t ) extending from the center of the nucleus to the center of the orbiting electron may then be represented as # $ d(t ) = d0 e−iωt # $ = (d0 + id0 ) e−iωt , (3.162) where d0 = d0 +id0 is a ﬁxed complex-valued vector with real and imaginary parts d0 = {d0 } and d0 = {d0 }, respectively, that specify the elliptical orbit. Here t = t−r/c is the retarded time at the stationary nuclear position, so that

e−iωt = e−iω(t−r/c) = ei(kr−ωt) , where

(3.163)

ω 2π ≡ (3.164) c λ is the wavenumber of the time-harmonic wave motion with wavelength λ. This is an extremely satisfying result, stating that the phenomenon of wave propagation is a direct consequence of the retardation condition t = t − r/c. The relevant time derivatives of the electronic position vector d(t ) are then given by k≡

148

3 Microscopic Potentials and Radiation

$ # ˙ ) = −iωd0 e−iωt , d(t # $ ¨ ) = −ω 2 d0 e−iωt . d(t Substitution of these expressions into the pair of relations given in Eqs. (3.160)–(3.161) then yields the following pair of expressions for the electric dipole radiation ﬁeld vectors:

2 2i 4π 3 ∼ qe k − r) ˆ r (d0 · ˆ e(r, t) = − 4π 0 (kr)3 (kr)2

1 i 1 + (ˆ r × (ˆ r × d0 )) ei(kr−ωt) , − − (kr)3 (kr)2 kr

4π µ0 c i 1 qe k 3 b(r, t) ∼ (ˆ r × d0 )ei(kr−ωt) . + = − c 4π (kr)2 kr

(3.165) (3.166)

From the inequalities given in Eqs. (3.127)–(3.128), this pair of expressions for the radiation ﬁeld produced by a time-harmonic electric dipole are good approximations of the actual radiated ﬁeld provided that both of the inequalities dmax r ˙

c d

=⇒ =⇒

max

d0 r 1 c d0 = ω k

(3.167) (3.168)

are satisﬁed, where d0 ≡ |d0 | is the maximum displacement of the bound electron from the nucleus. The pair of relations given in Eqs. (3.165)–(3.166) constitutes the electric dipole approximation of the radiation ﬁeld produced by a monochromatic dipole oscillator. The space–time evolution of the radiation ﬁeld vectors given in Eqs. (3.165)–(3.166) is best illustrated for the special case of a linear oscillation of the bound electron. For that special case, choose a spherical coordinate system with polar axis along the ˆ 1z -direction and let d0 = d0 ˆ 1z . The electric dipole moment is then given by 1z . p = pˆ 1z = q e d 0 ˆ

(3.169)

With this chosen geometry one obtains d0 · ˆ r = d0 cos θ, ˆ r × d0 = −ˆ 1φ d0 sin θ, ˆθ d0 sin θ, ˆ r × (ˆ r × d0 ) = 1

(3.170)

where θ is the angle of declination from the positive z-axis. The polar coordinate representation of the linear dipole radiation ﬁeld vectors is then seen to be given by

3.4 The Radiation Field Produced by a General Dipole Oscillator

e(r, t) = ˆ 1r er (r, θ, t) + ˆ 1θ eθ (r, θ, t), ˆ b(r, t) = 1φ bφ (r, θ, t),

149

(3.171) (3.172)

with

3 2 2i 4π pk cos θ − (3.173) er (r, θ, t) ∼ ei(kr−ωt) , =− 4π 0 (kr)3 (kr)2

1 4π 3 i 1 i(kr−ωt) ∼ pk sin θ eθ (r, θ, t) = − − − , (3.174) e 4π 0 (kr)3 (kr)2 kr

4π µ0 c 3 i 1 i(kr−ωt) pk sin θ bφ (r, θ, t) ∼ . (3.175) e + = c 4π (kr)2 kr The streamlines of the magnetic ﬁeld vector are then comprised of concentric circles about the polar axis and lie in planes perpendicular to the polar axis. The streamlines7 of the electric ﬁeld vector are independent of the azimuthal angle φ and are contained in planes that pass through (or contain) the polar axis. Maps of these streamlines at various instants of time were ﬁrst given by H. Hertz [13] in 1893. The above set of equations also applies to Hertzian dipole antennas with an appropriately speciﬁed dipole moment p [compare with Eqs. (3.179)– (3.80) for an elemental Hertzian dipole]. Because of the complicated radial dependence indicated in Eqs. (3.173)–(3.174), the ﬁeld structure naturally separates into three radial zones (the static zone, the intermediate zone, and the wave zone), each of which is now described in some detail. The Static Zone The static zone of the dipole radiation ﬁeld is deﬁned by the inequality kr 1,

(3.176)

so that r λ/2π. In the static zone, all terms appearing in the expressions given in Eqs. (3.173)–(3.175) for the ﬁeld vectors that contain the factors (kr)−2 and (kr)−1 are negligible in comparison to the term containing the factor (kr)−3 . Furthermore, the approximation eikr ≈ 1 is valid throughout the static zone so that Eqs. (3.173)–(3.175) reduce to the pair of expressions e(r, t) ∼ =− b(r, t) ∼ = 0. 7

4π p ˆ 1r 2 cos θ + ˆ 1θ sin θ cos(ωt), 3 4π 0 r

(3.177) (3.178)

The streamlines of the electric ﬁeld vector are obtained from the vector diﬀerential relation dl = K e, where K is a constant and where dl is a diﬀerential element of length in the chosen coordinate system.

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3 Microscopic Potentials and Radiation

Hence, in the static zone the radiation ﬁeld is essentially the ﬁeld due to an electrostatic dipole of variable moment p cos(ωt). The streamlines of the static zone ﬁeld are illustrated in Figure 3.8 when cos(ωt) = ±mπ where m is an integer. The dashed curves describe the streamlines obtained from the static zone approximation with electric ﬁeld given by Eq. (3.177) and the solid curves describe the same streamlines using the more accurate expressions given in Eqs. (3.173)–(3.174). Because of the inequality d0 r used in the electric dipole approximation, the streamlines depicted in Figure 3.8 are inaccurate near the origin for the idealized case of a point dipole. The static zone approximation breaks down whenever the oscillatory factor

v

p = qed01z

Fig. 3.8. Streamlines of the electric ﬁeld vector in the static zone (kr 1) of a linear electric dipole oscillator. The dashed curves describe the streamlines obtained from the static zone approximation and the solid curves describe the same streamlines in the electric dipole approximation. Both the horizontal and vertical scales of the ﬁgure are equal to λ/2π.

cos(ωt) approaches zero. Instead of the electric streamlines contracting into and disappearing at the origin, they instead pinch oﬀ into closed loops which then propagate outwards from the dipole location, as is illustrated by the ﬁeld streamline sequence in Figure 3.9. The Intermediate Zone The intermediate zone of the dipole radiation ﬁeld is roughly deﬁned by the expression

3.4 The Radiation Field Produced by a General Dipole Oscillator

kr ≈ 1,

151

(3.179)

so that r ≈ λ/2π. The intermediate zone occupies the region of space about the dipole oscillator that is just beyond the static zone (kr 1) but is not so large that the opposite inequality (kr 1) is satisﬁed, where this latter inequality deﬁnes the wave zone of the radiation ﬁeld. The evolution of the electric ﬁeld streamlines in the intermediate zone is illustrated in Figure 3.9 at the successive instants ωt = 0, π/4, π/2, 3π/4. This same ﬁgure sequence with the streamline directions reversed also applies when ωt = π, 5π/4, 3π/2, 7π/4, so that a complete cycle in time is obtained. In this intermediate zone, the radial component er (r, θ, t) is appreciable in comparison to the transverse component eθ (r, θ, t) of the electric ﬁeld vector of the radiation ﬁeld and the electromagnetic ﬁeld has not yet settled into a deﬁnite wavelength. The intermediate zone is distinguished from the static zone by the formation of closed loops in the electric ﬁeld streamlines which begin to occur about r ≈ λ/2.

Λ

Λ

Ωt =

Ωt = Π

Ωt = 3Π

Ωt = Π

Λ

Λ

Fig. 3.9. Streamlines of the electric ﬁeld vector from the static throughout the intermediate zone (kr ≈ 1) of a harmonic linear electric dipole oscillator at successive instants of time.

152

3 Microscopic Potentials and Radiation

The Wave Zone By following a few sets of closed electric ﬁeld streamlines as they propagate outward from the dipole oscillator, they are found to rapidly settle into the simple pattern illustrated in Figure 3.10. Inspection of this ﬁgure shows that the radial component of the electric ﬁeld vector is rapidly becoming negligible in comparison with the θ-component, especially in any small angular region about the θ = π/2 direction. Throughout any such angular region the electric ﬁeld vector is seen to be rapidly approaching (with increasing propagation distance) the condition of being completely transverse to the radial direction from the dipole source location. In addition, the distance between successive zeroes of eθ (r, θ, t) approaches the constant wavelength λ=

c 2π = 2π k ω

(3.180)

that is associated with the angular frequency of oscillation ω of the source dipole. The wave zone of the dipole radiation ﬁeld is deﬁned by the inequality kr 1,

(3.181)

so that r λ/2π. In that case, the ﬁeld vectors given in Eqs. (3.171)–(3.175) simplify to the pair of expressions e(r, t) ∼ 1θ eθ (r, θ, t) =ˆ 4π pk 2 ∼ sin(θ) cos(kr − ωt), 1θ =ˆ 4π 0 r b(r, t) ∼ 1φ bφ (r, θ, t) =ˆ 4π µ0 c pk 2 ∼ 1φ sin(θ) cos(kr − ωt), =ˆ c 4π r

(3.182)

(3.183)

which become increasingly accurate as kr → ∞. Throughout the wave zone, the dipole radiation ﬁeld is essentially transverse to the radial direction from the dipole source and possesses a (near) constant wavelength λ = 2πc/ω that is associated with the angular frequency of oscillation ω of the source dipole oscillator. These general features of the wave zone ﬁeld are clearly evident in the ﬁeld graphs presented in Figures 3.11 and 3.12. The graph in Figure 3.11 describes the radial dependence of the transverse ﬁeld components eθ and bφ (in Gaussian units) in the equatorial plane (θ = π/2) at several successive values of the temporal phase quantity ωt. The radial dependence of these ﬁeld components along a line at any other angle of declination θ may be obtained from these graphs simply by multiplying the ordinate values by sin(θ). The outward propagation of the radiation ﬁeld as the time variable t increases is clearly evident in this ﬁgure. The graph in Figure 3.12 describes the radial dependence of the radial electric ﬁeld component er along the polar

3.4 The Radiation Field Produced by a General Dipole Oscillator

153

z

Λ

V

p = p1z

ΘΠ

Fig. 3.10. Streamlines of the electric ﬁeld vector near the onset of the wave zone (kr 1) for a time-harmonic linear electric dipole oscillator.

axis (θ = 0) at several successive values of the temporal phase quantity ωt. The radial dependence of er along a line at any other angle θ may then be obtained from this graph simply by multiplying the ordinate values by cos(θ). In addition, this pair of graphs clearly shows that the radial component er of the electric ﬁeld vector becomes negligibly small in comparison to eθ in the region between r = 2λ and r = 3λ, and that a deﬁnite ﬁxed wavelength has appeared in the radiated ﬁeld structure at about r = 2λ. As a consequence, the approximate boundary between the intermediate and wave zones is at approximately r ∼ = 2λ.

154

3 Microscopic Potentials and Radiation pk3

eΘbΦ

ΘΠ

Ωt = Π Π Π

r

pk3 Λ ΛΛΛ

Fig. 3.11. Radial dependence of the transverse ﬁeld components eθ and bφ (in gaussian units) in the equatorial plane (θ = π/2) at several successive values of the temporal phase quantity ωt. pk3

e r

Θ

Ωt = Π Π Π

pk3

r

Λ Λ

Fig. 3.12. Radial dependence of the radial ﬁeld components er (in gaussian units) along the positive dipole axis (θ = 0) at several successive values of the temporal phase quantity ωt.

3.5 The Complex Potential and the Scalar Optical Field

155

3.5 The Complex Potential and the Scalar Optical Field In source-free regions of space (i.e., in vacuum) the Lorenz and Coulomb gauges become identical when the gauge function is chosen such that the scalar potential in the Lorenz gauge vanishes throughout the region [cf. Eqs. (3.32)–(3.34)]. In that situation the electromagnetic ﬁeld is completely speciﬁed by the vector potential a(r, t) that satisﬁes the subsidiary condition ∇ · a(r, t) = 0.

(3.184)

Based upon the analysis of Green and Wolf [14] in 1953 and Wolf [15] in 1959, let this vector potential possess the spatial Fourier integral representation 1 C(k, t)eik·r d3 k, (3.185) a(r, t) = (2π)3 K the integration being taken over all of real k-space, as indicated by the symbol K. Application of the subsidiary condition given in Eq. (3.184) to this representation then yields the orthogonality condition k · C(k, t) = 0

(3.186)

that is satisﬁed for all k ∈ K. Furthermore, because a(r, t) is real-valued, the complex conjugate of Eq. (3.185) then gives 1 a(r, t) = C∗ (k, t)e−ik·r d3 k (2π)3 K 1 C∗ (−k, t)eik·r d3 k = (2π)3 K so that the symmetry relation C(−k, t) = C∗ (k, t)

(3.187)

is satisﬁed for all k ∈ K. In order to fully utilize the orthogonality relation given in Eq. (3.186), Wolf [15] introduced a pair of real, mutually orthogonal unit vectors ˆl1 and ˆl2 deﬁned as ˆ ˆl1 (k) ≡ n × k , ˆl2 (k) ≡ k × l1 , (3.188) ˆ |n × k| |k × l1 | where n is an arbitrary but ﬁxed real vector. In terms of these unit vectors, Wolf [15] then deﬁned a set of complex basis vectors as

for kz ≥ 0, with

L(k) ≡ ˆl1 (k) + iˆl2 (k)

(3.189)

L(−k) ≡ L(k).

(3.190)

156

3 Microscopic Potentials and Radiation

The complex potential of the vector potential ﬁeld a(r, t) is then deﬁned as 1 V (r, t) ≡ γ(k, t)eik·r d3 k, (3.191) (2π)3 K where γ(k, t) ≡ L(k) · C(k, t). The Fourier inverse of Eq. (3.185), viz. ∞ a(r, t)e−ik·r d3 r, C(k, t) =

(3.192)

(3.193)

−∞

then gives, with Eq. (3.192), γ(k, t) = L(k) ·

∞

a(r, t)e−ik·r d3 r.

(3.194)

−∞

Substitution of this expression into Eq. (3.191) for the complex potential then yields the result ∞ a(r , t) · M(r − r)d3 r , (3.195) V (r, t) = −∞

where

1 M(r) ≡ (2π)3

L(k)e−ik·r d3 k.

(3.196)

K

This result then shows that the complex potential is obtained from a linear transformation of the real-valued vector potential with transform kernel M given by the Fourier transform of the complex basis vectors L(k). A similar representation in source-free regions has been given by Whittaker [16] in terms of two real scalar wave functions. Because k·C(k, t) = 0, the complex vector C(k, t) lies in the plane formed by the two orthogonal unit vectors ˆl1 (k) and ˆl2 (k). The symmetry relations given in Eqs. (3.187) and (3.190), together with the deﬁnition in Eq. (3.192), then yield the result C(k, t) =

1 [γ(k, t)L∗ (k) + γ ∗ (−k, t)L(k)] . 2

(3.197)

With the Fourier inverse of Eq. (3.191), this relation becomes

∞ ∞ 1 ∗ −ik·r 3 ∗ −ik·r 3 L (k) V (r, t)e d r + L(k) V (r, t)e d r . C(k, t) = 2 −∞ −∞ (3.198) Substitution of this expression into the Fourier integral representation given in Eq. (3.185) then gives

3.5 The Complex Potential and the Scalar Optical Field

a(r, t) =

∞

−∞

V (r , t)M∗ (r − r)d3 r ,

157

(3.199)

where the symbol denotes the real part. This is then the inverse transformation to that given in Eq. (3.195). 3.5.1 The Wave Equation for the Complex Potential Because the vector potential a(r, t) satisﬁes the homogeneous vector wave equation 1 ∂ 2 a(r, t) = 0, (3.200) ∇2 a(r, t) − 2 c ∂t2 in source-free regions [cf. Eq. (3.30)], then each spatial Fourier component C(k, t) of a(r, t) must satisfy the diﬀerential equation ∂ 2 C(k, t) + k 2 c2 C(k, t) = 0. ∂t2 Correspondingly, each spatial Fourier component γ(k, t) of the complex potential V (r, t) must satisfy the diﬀerential equation ∂ 2 γ(k, t) + k 2 c2 γ(k, t) = 0. ∂t2 As a consequence, the complex potential must then satisfy the homogeneous scalar wave equation ∇2 V (r, t) −

1 ∂ 2 V (r, t) = 0, c2 ∂t2

(3.201)

The electromagnetic ﬁeld vectors are then uniquely determined by this single scalar complex potential V (r, t) in source-free regions of space. 3.5.2 Electromagnetic Energy and Momentum Densities The electric and magnetic ﬁeld vectors are given in terms of the vector potential as % % % 1 % ∂a(r, t) % e(r, t) = − % % c % ∂t , b(r, t) = ∇ × a(r, t), where a(r, t) is given by Eq. (3.199) in terms of the complex potential. The total electromagnetic momentum of the ﬁeld is then given by [cf. Eq. (2.151)] ∞

0 µ0 (e(r, t) × h(r, t)) d3 r pem (t) = 4πc −∞ % % ∞ % 1 % 1 ∂a(r, t) % = −% (3.202) × (∇ × a(r, t)) d3 r. % 4π % c2 ∂t −∞

158

3 Microscopic Potentials and Radiation

Substitution from Eq. (3.199) then yields [14, 15] % % ∞ % 1 % 1 ∂V ∗ (r, t) ∂V ( r, t) ∗ % pem (t) = − % (r, t) d3 r, ∇V (r, t) + ∇V % 4π % 2c2 ∂t ∂t −∞ (3.203) and the electromagnetic momentum density is then given as ∂V ∗ (r, t) 1 ∂V (r, t) ∗ ∇V (r, t) + ∇V (r, t) (3.204) pem (r, t) = − 4π2c2 ∂t ∂t ¯ em is then in terms of the complex potential. The Poynting vector s = c2 p given by ∂V ∗ (r, t) ∂V ( r, t) 1 ∇V (r, t) + ∇V ∗ (r, t) . s(r, t) = − (3.205) 24π ∂t ∂t The other relevant quantity appearing in energy conservation and ﬂow in the microscopic electromagnetic ﬁeld is the electromagnetic energy density [cf. Eq. (2.143)] 1 2

0 e (r, t) + µ0 h2 (r, t) 24π 2 1 ∂a(r, t) 1 2 + (∇ × a(r, t)) . = 24πµ0 c2 ∂t

u(r, t) =

(3.206)

Substitution from Eq. (3.199) then yields the expression [14, 15] 1 1 ∂V (r, t) ∂V ∗ (r, t) ∗ u(r, t) = + ∇V (r, t) · ∇V (r, t) . (3.207) 24πµ0 c2 ∂t ∂t From Poynting’s theorem [cf. Eq. (2.146)] in source-free regions, these quantities are found to satisfy the conservation law ∇ · s(r, t) +

∂u(r, t) = 0. ∂t

(3.208)

Furthermore, these expressions for the electromagnetic energy density u(r, t) and Poynting vector s(r, t) in terms of the complex potential V (r, t) for the microscopic electromagnetic ﬁeld in source-free regions may be formally identiﬁed with the respective quantum mechanical expressions for the probability density and current [15, 17]. 3.5.3 A Scalar Representation of the Optical Field Because V (r, t) is in general a complex function, it may be expressed as V (r, t) ≡ V(r, t)eiϕ(r,t) ,

(3.209)

3.5 The Complex Potential and the Scalar Optical Field

159

where V(r, t) is the real amplitude and ϕ(r, t) the phase of the complex potential. Upon substitution of this representation into the wave equation (3.201) for the complex potential and separation of the result into real and imaginary parts then yields the pair of wave equations 2 1 ∂2V ∂ϕ 2 2 −V = 0, (3.210) ∇ V − V(∇ϕ) − 2 2 c ∂t ∂t ∂2ϕ ∂V ∂ϕ 1 2 + V 2 = 0. (3.211) 2∇V · ∇ϕ + V∇ ϕ − 2 2 c ∂t ∂t ∂t The electromagnetic momentum density given in Eq. (3.204) and the electromagnetic energy density given in Eq. (3.207) may be expressed in terms of the real-valued functions V(r, t) and ϕ(r, t) as 1 ˙ t)∇V(r, t) + V 2 (r, t)ϕ(r, V(r, ˙ t)∇ϕ(r, t) , (3.212) 4πc2 2 1 ˙ 1 2 V(r, t) + (V(r, t)ϕ(r, u(r, t) = ˙ t)) 24πµ0 c2

pem (r, t) = −

2

2

+ (∇V(r, t)) + (V(r, t)∇ϕ(r, t))

.

(3.213)

Of particular interest here is the case of a monochromatic (or timeharmonic) wave ﬁeld in which the time enters only through the exponential factor8 e−iωt . The amplitude of the complex potential function is then independent of time (V˙ = 0) and the phase function is of the form ϕ(r, t) = kS(r) − ωt,

(3.214)

where k = ω/c is the wavenumber. With this substitution, the pair of wave equations appearing in Eqs. (3.210)–(3.211) become 1 ∇2 V = 1, k2 V 1 ∇V · ∇S + V∇2 S = 0, 2 2

(∇S) −

(3.215) (3.216)

and Eqs. (3.212)–(3.213) for the electromagnetic momentum and energy densities become 8

A general monochromatic wave must be represented by a complex potential function of the form V (r, t) = V1 (r)e−iωt + V2 (r)eiωt , where V1 and V2 are complex functions of the position. Wolf [15] has shown that the case considered here in which V2 = 0 corresponds to a monochromatic wave of arbitrary shape but with a spatial average circular polarization.

160

3 Microscopic Potentials and Radiation

k2 V 2 (r)∇S(r), (3.217) 4πc 2 1 1 2 V 2 (r) 1 + (∇S(r)) + 2 ∇ ln(V(r)) . (3.218) u(r, t) = 24πµ0 k

pem (r, t) =

Equation (3.217) shows that both the electromagnetic momentum and energy ﬂow are orthogonal to the cophasal surfaces deﬁned by S(r) = constant . In addition, Eq. (3.216) may be written in the form ∇ · V 2 (r)∇S(r) = 0,

(3.219)

(3.220)

so that, from Eq. (3.217), it is found that ∇ · pem (r) = 0.

(3.221)

Hence, the momentum density of a monochromatic electromagnetic wave ﬁeld is solenoidal [this result also follows from the conservation law (3.208) because ∂u/∂t = 0]. Finally, Eq. (3.215) may be written in the form of a generalized free-space eikonal equation 2

(∇S(r)) = N 2 (r) where

N (r) ≡ 1 +

1/2 1 2 ∇ V(r) k 2 V(r)

(3.222)

(3.223)

is a modiﬁed refractive index function that depends upon the amplitude of the complex potential [14]. Consider ﬁnally the implication of Eq. (3.216). With the deﬁnition of the diﬀerential operator [18] ∂ ≡ ∇S · ∇, (3.224) ∂τ so that τ speciﬁes the position along the orthogonal trajectories to the cophasal surfaces S(r) = constant, and hence is along the direction of energy ﬂow, Eq. (3.216) becomes ∂V 1 + V ∇2 S = 0. ∂τ 2

(3.225)

The formal solution to this transport equation for the amplitude of the complex potential along these orthogonal trajectories is then given by 1 τ 2 ∇ S(τ )dτ . (3.226) V(τ ) = V(τ0 ) exp − 2 τ0

3.5 The Complex Potential and the Scalar Optical Field

161

In the inﬁnite frequency limit as k → ∞, Eq. (3.222) reduces to the free-space eikonal equation of geometrical optics [19] (∇S)2 = 1.

(3.227)

In this limiting case the phase function S(r) is completely speciﬁed by its boundary conditions alone. The cophasal surfaces S(r) = constant are then the wavefronts of geometrical optics and their orthogonal trajectories are the familiar rays of geometrical optics.

162

3 Microscopic Potentials and Radiation

References 1. L. Lorenz, “On the identity of the vibrations of light with electrical currents,” Philos. Mag., vol. 34, pp. 287–301, 1867. 2. E. T. Whittaker, A History of the Theories of the Aether and Electricity. London: T. Nelson & Sons, 1951. 3. J. V. Bladel, “Lorenz or Lorentz?,” IEEE Antennas Prop. Mag., vol. 33, p. 69, 1991. 4. P. M. Morse and H. Feshbach, Methods of Theoretical Physics. New York: McGraw-Hill, 1953. Vol. I. 5. H. Hertz, “Die Kr¨ afte electrischer Schwingungen, behandelt nach der Maxwell’schen Theorie,” Ann. Phys., vol. 36, pp. 1–22, 1889. 6. A. Righi, “Electromagnetic ﬁelds,” Nuovo Cimento, vol. 2, pp. 104–121, 1901. 7. A. Nisbet, “Hertzian electromagnetic potentials and associated gauge transformations,” Proc. Roy. Soc. A, vol. 231, pp. 250–263, 1955. 8. A. Li´enard, “Electric and magnetic ﬁeld produced by a moving charged parti´ ´ cle,” L’Eclairage Electrique, vol. 16, pp. 5–14, 53–59, 106–112, 1898. 9. E. Wiechert, “Electrodynamical laws,” Arch. N´eerland., vol. 5, pp. 549–573, 1900. 10. W. K. H. Panofsky and M. Phillips, Classical Electricity and Magnetism. Reading, MA: Addison-Wesley, 1955. Ch. 19–20. 11. J. M. Stone, Radiation and Optics, An Introduction to the Classical Theory. New York: McGraw-Hill, 1963. 12. E. T. Whittaker and G. N. Watson, Modern Analysis. London: Cambridge University Press, fourth ed., 1963. p. 133. 13. H. Hertz, Electric Waves. London: Macmillan, 1893. English translation. 14. H. S. Green and E. Wolf, “A scalar representation of electromagnetic ﬁelds,” Proc. Phys. Soc. A, vol. 66, no. 12, pp. 1129–1137, 1953. 15. E. Wolf, “A scalar representation of electromagnetic ﬁelds: III,” Proc. Phys. Soc., vol. 74, pp. 281–289, 1959. 16. E. T. Whittaker, “On an expression of the electromagnetic ﬁeld due to electrons by means of two scalar potential functions,” Proc. Lond. Math. Soc., vol. 1, pp. 367–372, 1904. 17. P. Roman, “A scalar representation of electromagnetic ﬁelds: II,” Proc. Phys. Soc., vol. 74, pp. 269–280, 1959. 18. A. Nisbet and E. Wolf, “On linearly polarized electromagnetic waves of arbitrary form,” Proc. Camb. Phil. Soc., vol. 50, pp. 614–622, 1954. 19. M. Born and E. Wolf, Principles of Optics. Cambridge: Cambridge University Press, seventh (expanded) ed., 1999.

Problems 3.1. Prove that the instantaneous Coulomb potential given in Eq. (3.22) is the solution to Poisson’s equation ∇2 φ = −(4π/ 0 )ρ. Notice that, in part, this entails proving that 1 2 = −4πδ(r − r ), ∇ |r − r |

3.5 Problems

163

where δ(r) is the three-dimensional Dirac delta function. In order to accomplish this, ﬁrst prove that ∇2 (|r − r |−1 ) = 0 when r = r . When r = r , translate the origin of coordinates to r and consider the quantity ∇2 (1/r) where r = r. Upon integration of the quantity ∇2 (1/r) over an arbitrary volume V that contains the origin, show that 1 d3 r = −4π, ∇2 r V thereby proving the above identity for the three-dimensional delta function. 3.2. Show that the retarded potentials given in Eqs. (3.35)–(3.36) satisfy the Lorenz condition (3.13). Notice that the vector R is the radius vector from the position of the elementary source element ρ(r , t − R/c)d3 r to the ﬁxed observation or ﬁeld point, so that v = −∂R/∂t, where j = ρv. 3.3. From Eqs. (3.62) amd (3.63), prove that ∇2 Ψ = Ξ, Notice that, in part, this entails proving that ∇2 (∇ × Ψ) = ∇ × (∇2 Ψ) for any diﬀerentaible vector ﬁeld Ψ(r). 3.4. Derive Eqs. (3.67) and (3.68). 3.5. Obtain an expression for that part [s× (r, t)] of the Poynting vector that is associated with the cross-terms between the velocity and acceleration parts of the electromagnetic ﬁeld produced by a moving charged point particle. Show that the radial dependence of this term varies as R−3 as R → ∞. 3.6. Through use of the general retarded expressions given in Eqs. (3.115)– (3.116) and (3.118)–(3.119) for the electromagnetic ﬁeld produced by a moving charged particle, determine the ﬁeld vectors and the Poynting vector produced by a point charge q that is moving with a ﬁxed velocity v in terms of the present position vector R(t) of the particle. From these results, describe the ﬁeld behavior in each of the two limiting cases obtained as v → 0 and as v → c. 3.7. Prove the diﬀerential vector identity ∇ × (ˆ r/rn ) = 0 for n = 0, 1, 2, . . .. 3.8. Derive Eq. (3.152). 3.9. Derive Eqs. (3.155)–(3.158). 3.10. Derive the equation for the electric ﬁeld streamlines for the monochromatic dipole radiation ﬁeld given in Eqs. (3.171) and (3.173)–(3.174). Hint: Let er ≡ (1/r sin(θ))∂/∂θ(f (r, θ) sin(θ)) and eθ ≡ −(1/r)∂/∂r(rf (r, θ)).

4 Macroscopic Electromagnetics

In the classical Maxwell–Lorentz theory [1–3], matter is regarded as being composed of point charges (e.g., point electrons and point nuclei) that produce microscopic electric and magnetic ﬁelds. The microscopic equations of electromagnetics given in Eqs. (2.33)–(2.36) together with the Lorentz force relation given in Eq. (2.21) describe the detailed classical behavior of the charged particles and ﬁelds, as presented in Chapters 2 and 3. The macroscopic equations of electromagnetics, in turn, describe the average behavior of the charged particles and ﬁelds. It is then expected that, through a suitable averaging procedure, the macroscopic ﬁeld equations may be derived from the microscopic equations, a viewpoint that was initially developed by H. A. Lorentz [3] in 1906 and has since been extended by J. H. van Vleck [4], R. Russakoﬀ [5], and F. N. H. Robinson [6].

4.1 Correlation of Microscopic and Macroscopic Electromagnetics A macroscopically small volume of matter at rest in the laboratory frame typically contains on the order of between 1018 to 1028 electrons and nuclei that are all in dynamical motion due to, for example, thermal agitation, zero point vibration, or orbital motion. The microscopic electromagnetic ﬁelds that are produced by these fundamental charge sources are then seen to vary with extreme rapidity in both space and time. The spatial variations occur over distances with an upper limit set by intermolecular spacing which is of the order of 10−8 cm or less, and the temporal ﬂuctuations occur with typical average oscillation periods ranging from 10−13 s for nuclear vibrations to 10−17 s for electronic orbital motion. Macroscopic measurement instruments typically average over both space and time intervals that are much larger than these characteristic dimensions so that these microscopic ﬂuctuations are usually averaged out. It is then necessary to obtain a macroscopic electromagnetic theory in order to describe this type of measurement process. The type of averaging that is appropriate to carry the theory from the microscopic domain to the macroscopic domain must ﬁrst be carefully considered. First of all, it is well known that the propagation of light in dielectric materials is adequately described by the Maxwell equations with a continuous

166

4 Macroscopic Electromagnetics

dielectric permittivity, whereas X-ray difraction clearly exhibits the atomistic nature of matter. As a consequence, it is reasonable to take the length 0 ≈ 10−6 cm as the absolute lower limit to the macroscopic domain. The period of scillation associated with electromagnetic radiation of this wavelength is 0 /c ≈ 3 × 10−17 s. In a volume of material of size 30 ≈ 10−18 cm3 there are typically on the order of 106 electrons and nuclei. Hence, in any macroscopically small region of ponderable media with linear dimensions 0 , there are so many microscopic charged particles that the random ﬂuctuations will be completely washed out by a spatial averaging procedure because, in the absence of any special preparation and the establishment of ordering over macroscopic distances, the temporal variations of the microscopic ﬁelds are uncorrelated over distances of order . All that survives from a spatial averaging are the frequency components that correspond (in the classical sense) to oscillations that are driven at the external, applied frequencies. 4.1.1 Spatial Average of the Microscopic Field Equations The spatial average of a microscopic function f (r, t) of position and time is deﬁned as f (r, t) ≡ w(r )f (r − r , t)d3 r , (4.1) where w(r) is a real-valued, positive, suﬃciently well-behaved function that is nonzero only in some nonvanishing region of space surrounding the point r = 0. This “weighting” function is normalized to unity over all of space as ∞ w(r)d3 r = 1, (4.2) −∞

and varies suﬃciently slowly such that the local series approximation 1 w(r + d) ≈ w(r) + (d · ∇)w(r) + (d · ∇)2 w(r) 2

(4.3)

is well satisﬁed for d = |d| of the order of molecular sizes. As shown in the following subsections , this approximation allows one to make any necessary multipole moment expansions of the charge and current densities. It is readily evident that the operations of space and time diﬀerentiation commute with this spatial-averaging operation, because ∂f (r − r , t) 3 ∂ f (r, t) = w(r ) d r ∂xj ∂xj && '' ∂f (r, t) = , (4.4) ∂xj and

4.1 Correlation of Microscopic and Macroscopic Electromagnetics

∂f (r − r , t) 3 w(r ) d r ∂t '' && ∂f (r, t) . = ∂t

∂ f (r, t) = ∂t

167

(4.5)

It is then clear that the spatial diﬀerential operator ∇ also commutes with the spatial-averaging operation in the usual gradient, divergence, curl and Laplacian operations. The macroscopic electric E(r, t) and magnetic B(r, t) ﬁeld vectors are deﬁned as the spatial averages of their respective microscopic ﬁeld vectors e(r, t) and b(r, t) as E(r, t) ≡ e(r, t), B(r, t) ≡ b(r, t).

(4.6) (4.7)

The spatial average of the microscopic ﬁeld equations given in Eqs. (2.33)– (2.36) then gives 4π ρ(r, t),

0 ∇ · B(r, t) = 0, 1 ∂B(r, t) , ∇ × E(r, t) = − c ∂t 4π 1 ∂E(r, t) . ∇ × B(r, t) = µ0 j(r, t) + 0 µ0 c c ∂t ∇ · E(r, t) =

(4.8) (4.9) (4.10) (4.11)

The two averaged homogeneous equations (4.9) and (4.10) then remain the same as their microscopic counterparts. For the averaged inhomogeneous equations the spatial averages ρ(r, t) and j(r, t) remain to be determined, the derived macroscopic ﬁeld vectors D(r, t) and H(r, t) being introduced by the extraction from ρ(r, t) and j(r, t) of certain contributions that can be identiﬁed with the bulk properties of the material medium [7–9]. 4.1.2 Spatial Average of the Charge Density Consider a material medium that is comprised of molecules composed of nuclei and electrons that are bound together and, in addition, contains “free” charges that are not localized around any particular molecule of the material. The microscopic charge density may then be separated into two groups: bound charge ρb (r, t), which belongs to the molecular structure of the material and free charge ρf (r, t), which accounts for the conduction current in the material. The microscopic charge density is then given by ρ(r, t) = ρb (r, t) + ρf (r, t), where

(4.12)

168

4 Macroscopic Electromagnetics

ρf (r, t) =

qj δ (r − rj (t))

(4.13)

j (free)

is the microscopic free charge density, and ρb (r, t) = ρn (r, t)

(4.14)

n (mol)

is the microscopic bound charge density, where qj δ (r − rj (t)) , ρn (r, t) =

(4.15)

j(n)

the summation extending over the bound charge in the nth molecule.

O' n molecule th

rjn

qj

rn rj

O

Fig. 4.1. Position vectors for the nth molecule and the jth electron in that molecule.

Consider ﬁrst taking the spatial average of the charge density in the nth molecule and then summing the contributions from all of the molecules in

4.1 Correlation of Microscopic and Macroscopic Electromagnetics

169

the material. In order to accomplish this, it is appropriate to ﬁrst express the coordinates of the charges in the nth molecule with respect to an origin O that is at rest with respect to that molecule; for example, this is readily accomplished at the center of mass of the molecule. Let the position vector of that ﬁxed point be denoted by rn (t) relative to the ﬁxed origin O and let the position vector of the jth charge in the molecule be denoted by rj (t) relative to O as well as by rjn (t) relative to O , as illustrated in Figure 4.1, where rj (t) = rn (t) + rjn (t).

(4.16)

The spatial average of the charge density in the nth molecule is then given by ρn (r, t) = w(r )ρn (r − r , t)d3 r = qj w(r )δ(r − r − rjn − rn , t)d3 r j(n)

=

qj w(r − rn − rjn ).

(4.17)

j(n)

Because |rjn | is on the order of atomic dimensions, the terms in the above summation have arguments that diﬀer only slightly from the vector r − rn on the spatial scale over which the weighting function w(r) changes appreciably. One may then expand the function w(r − rn − rjn ) about the point r − rn for each term in the summation appearing in Eq. (4.17) so that, with application of Eq. (4.3), there results 1 2 qj w(r − rn ) − (rjn · ∇)w(r − rn ) + (rjn · ∇) w(r − rn ) ρn (r, t) ≈ 2 j(n)

1 = qn w(r − rn ) − pn · ∇w(r − rn ) + Qn · ∇ · ∇w(r − rn ). (4.18) 6 The quantities introduced here, which entail summations over the charges in the molecule, are the molecular multipole moments Molecular Charge (Monopole Moment): qn ≡ qj , (4.19) j(n)

Molecular Dipole Moment Vector: pn ≡

qj rjn ,

(4.20)

j(n)

Molecular Quadrupole Moment Tensor: Qn ≡

qj rjn rjn , (4.21)

j(n)

where the elements of the molecular quadrupole moment tensor are Qn αβ = 3 qj (rjn )α (rjn )β , j(n)

170

4 Macroscopic Electromagnetics

the indices α and β taking on the values 1, 2, 3. When Eq. (4.18) is viewed in light of the deﬁnition of the spatial average given in Eq. (4.1), the ﬁrst term is seen to be the spatial average of a point charge density located at r = rn , the second term as the divergence of the average of a point dipole density at r = rn , and the third term as the second spatial derivative of the average of a point quadrupole density at r = rn . The approximate expression appearing in Eq. (4.18) for the spatial average of the microscopic charge density may then be written as [5, 10] 1 ρn (r, t) ≈ qn δ(r − rn ) − ∇ · pn δ(r − rn ) + ∇ · ∇ · Qn δ(r − rn ). 6 (4.22) Through the process of spatial averaging process, each molecule is viewed as a collection of point multipoles located at a single ﬁxed point O in the molecule (see Fig. 4.1). Although the detailed structure of the molecular charge distribution is important at the microscopic level, for macroscopic phenomena it is replaced by a sum of point multipoles at each molecular site. The total microscopic charge density given in Eq. (4.12) consists of both free and bound charges. Upon taking the summation of Eq. (4.22) over all of the molecules in the material (which may be of diﬀerent species) and combining this result with the spatial average of the free charge density, the spatial average of the (total) microscopic charge density is obtained as ρ(r, t) = ρf (r, t) + ρb (r, t) (( )) (( )) = qj δ(r − rj (t)) + ρn (r, t) n (mol)

j (free)

≈ (r, t) − ∇ · P(r, t) + ∇ · ∇ · Q (r, t),

(4.23)

where ∇ · ∇ · Q (r, t) =

3 3 α=1 β=1

∂2 Q (r, t). ∂xα ∂xβ αβ

Here (r, t) is the macroscopic charge density (r, t) = f (r, t) + b (r, t) that is comprised of the macroscopic free charge density (( )) qj δ(r − rj (t)) f (r, t) ≡ j (free)

plus the macroscopic bound charge density

(4.24)

(4.25)

4.1 Correlation of Microscopic and Macroscopic Electromagnetics

(( b (r, t) ≡

n (mol)

qj δ(r − rn (t))

.

(4.26)

j(n)

In addition, P(r, t) is the macroscopic polarization density (( )) P(r, t) ≡ qj rjn δ(r − rn (t)) , n (mol)

(4.27)

j(n)

and Q (r, t) is the macroscopic quadrupole moment density tensor (( )) 1 Q (r, t) ≡ qj rjn rjn δ(r − rn (t)) 2 n (mol)

with elements Qαβ (r, t)

1 = 2

((

(4.28)

j(n)

n (mol)

171

))

)) qj (rjn )α (rjn )β δ(r − rn (t))

,

(4.29)

j(n)

the indices α and β taking on the values 1, 2, 3. Because not all of the components of the quadrupole moment tensor are independent, a traceless microscopic molecular quadrupole moment density is deﬁned as [8, 10] 2 Qn αβ ≡ Qn αβ − qj (rjn ) δαβ =

j(n)

2 qj 3(rjn )α rjn )α − (rjn ) δαβ .

(4.30)

j(n)

Deﬁne the mean square charge radius rn2 of the molecular charge distribution for the nth molecule as 2 qj (rjn ) , (4.31) qˆ rn2 ≡ j(n)

where qˆ is some convenient unit of (positive) charge such as that of a proton. With this substitution, Eq. (4.30) may be written as Qn αβ = Qn αβ + qˆ rn2 δαβ . (4.32) The elements of the macroscopic quadrupole moment density given in Eq. (4.29) then become (( )) 1 qˆ rn2 δαβ δ(r − rn (t)) . (4.33) Qαβ (r, t) = Qαβ (r, t) + 6 n (mol)

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4 Macroscopic Electromagnetics

The net result is that, in Eq. (4.23) for the spatial average of the microscopic charge density, the traceless macroscopic quadrupole moment density tensor Q(r, t) replaces Q (r, t) and the macroscopic bound charge density b (r, t) is augmented by an additional term so that Eq. (4.26) is replaced by1 (( )) )) (( 1 2 2 b (r, t) = qn δ(r − rn (t)) + ∇ qˆrn δ(r − rn (t)) . 6 n n (mol)

(mol)

(4.34) The trace of the macroscopic quadrupole tensor Q is grouped with the macroscopic charge density because it is an = 0 contribution from the multipole expansion of the molecular charge distribution of the medium [10]. Taken together, the molecular charge and mean square radius terms represent the ﬁrst two terms in an expansion of the = 0 molecular multipole as one goes beyond the inﬁnite wavelength-zero wavenumber limit of electroststics [6]. 4.1.3 Spatial Average of the Current Density The microscopic current density can also be separated into two groups: a “bound” current density jb (r, t) that is due to the motion of the bound charges that belong to the molecular structure of the material, and the free current density jf (r, t) that is due to the motion of the free charge in the material and decribes the conduction current. The microscopic current density is then given by (4.35) j(r, t) = jb (r, t) + jf (r, t), where jf (r, t) =

qj vj δ(r − rj (t))

(4.36)

j (free)

is the microscopic free current density, and jb (r, t) = jn (r, t)

(4.37)

n (mol)

is the microscopic bound current density with jn (r, t) = qj vj δ(r − rj (t))

(4.38)

j(n)

describing the microscopic current density of the nth molecule in the material. Here vj = drj /dt denotes the velocity of the jth charge (bound or free). 1

Notice that the quantity ∇ · P(r, t) is sometimes referred to as the bound charge density. In order to avoid confusion with the bound charge density as described in Eq. (4.34), the quantity ∇ · P(r, t) is referred to here by its proper name, the polarization charge density.

4.1 Correlation of Microscopic and Macroscopic Electromagnetics

173

The spatial average of the current density in the nth molecule is obtained in the same manner as that given in Eq. (4.17) for the charge density. For the current density jn (r, t) = w(r )jn (r − r , t)d3 r = qj w(r )vj δ(r − r − rj (t))d3 r j(n)

=

qj (vjn + vn )

w(r )δ(r − r − rn − rjn )d3 r

j(n)

=

qj (vjn + vn ) w(r − rn − rjn ),

(4.39)

j(n)

where rj (t) = rn (t) + rjn (t) so that (assuming nonrelativistic motion of the charged particles) vj (t) = vn (t) + vjn (t), where vn (t) describes the velocity of the origin O in the nth molecule (see Fig. 4.1) and where vjn denotes the internal relative velocity of the jth charged particle in that molecule. Upon expanding the weighting function w(r − rn − rjn ) about the point r − rn in each term of the above series, there results jn (r, t) ≈ qj (vjn + vn ) j(n)

1 × w(r − rn ) − (rjn · ∇)w(r − rn ) + (rjn · ∇)2 w(r − rn ) 2 dpn = qn vn w(r − rn ) + w(r − rn ) − vn (pn · ∇)w(r − rn ) dt 1 + vn Qn · ∇ · ∇ w(r − rn ) − qj vjn (rjn · ∇)w(r − rn ), 6 j(n)

(4.40) where the last third-order molecular moment has been omitted. Here qn is the molecular charge, pn the molecular dipole moment, and Qn the molecular quadrupole moment, as deﬁned in Eqs. (4.19)–(4.21). The molecular magnetic moment is deﬁned as mn ≡

1 qj (rjn × vjn ). 2c j(n)

In addition, it is seen that d (pn w(r − rn )) + ∇ × (pn × vn w(r − rn )) dt dpn w(r − rn ) − vn (vn · ∇)w(r − rn ), = dt

(4.41)

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4 Macroscopic Electromagnetics

and 1 d Qn · ∇w(r − rn ) 6 dt 1 − ∇ × ∇ · Qn × ∇w(r − rn ) 6 1 = vn Qn · ∇ · ∇ w(r − rn ) − qj vjn (rjn · ∇)w(r − rn ). 6

∇ × (cmn w(r − rn )) −

j(n)

With these substitutions, Eq. (4.40) becomes d (pn w(r − rn )) + ∇ × (pn × vn w(r − rn )) dt 1 d Qn · ∇w(r − rn ) + c∇ × mn w(r − rn )) + 6 dt 1 − ∇ × ∇ · Qn × vn w(r − rn )) 6 d 1 pn δ(r − rn ) − ∇ · Qn δ(r − rn ) = qn vn δ(r − rn ) + dt 6 + c∇ × mn δ(r − rn ) + * 1 + ∇ × pn × vn δ(r − rn ) − ∇ · Qn × vn δ(r − rn ) . 6 (4.42)

jn (r, t) ≈ qn vn w(r − rn ) +

Upon taking the summation of the above expression over all of the molecules in the material and then combining this result with the spatial average of the free current density, one ﬁnally obtains the spatial average of the microscopic current density as j(r, t) = jf (r, t) + jb (r, t) (( )) (( )) = qj vj δ(r − rj (t)) + jn (r, t) n (mol)

j (free)

d ≈ J(r, t) + P(r, t) − ∇ · Q (r, t) + c∇ × M(r, t) dt (( )) +∇ × pn × vn δ(r − rn ) n (mol)

1 − ∇· 6

((

)) Qn

× vn δ(r − rn )

.

n (mol)

(4.43) Here J(r, t) is the macroscopic current density

4.1 Correlation of Microscopic and Macroscopic Electromagnetics

J(r, t) = Jf (r, t) + Jb (r, t)

175

(4.44)

that is comprised of the macroscopic free current density (( )) qj vj δ(r − rj (t)) Jf (r, t) ≡

(4.45)

j (free)

plus the macroscopic bound current density (( )) qn vj δ(r − rj (t)) , Jb (r, t) ≡ n (mol)

(4.46)

j(n)

and M(r, t) is the macroscopic magnetization (( )) M(r, t) ≡ mn δ(r − rn (t)) .

(4.47)

n (mol)

If the free charges also possess intrinsic magnetic moments, then the spatial average of these microscopic moments must also be included in the deﬁning relation (4.47) for the macroscopic magnetization vector. 4.1.4 The Macroscopic Maxwell Equations Substitution of Eq. (4.23) for the spatial average of the microscopic charge density into Eq. (4.8) yields ∇ · E(r, t) ≈

4π (r, t) − ∇ · P(r, t) + ∇ · ∇ · Q (r, t) ,

0

which may be rewritten as ∇ · 0 E(r, t) + 4πP(r, t) − 4π∇ · Q (r, t) ≈ 4π(r, t).

(4.48)

In addition, substitution of Eq. (4.43) for the spatial average of the microscopic current density into Eq. (4.11) yields ∇ × B(r, t) ≈

0 µ0 ∂E(r, t) c ∂t % % % 4π % ∂ % +% % c % µ0 J(r, t) + ∂t P(r, t) − ∇ · Q (r, t) + c∇ × M(r, t) (( )) +∇ × pn × vn δ(r − rn ) n (mol)

1 − ∇· 6

((

n (mol)

)) Qn

× vn δ(r − rn )

,

176

4 Macroscopic Electromagnetics

which may be rewritten as 1 ∇× B(r, t) − 4πM(r, t) µ0 )) % % (( % 4π % % % − % % pn × vn δ(r − rn ) c n (mol)

(( )) % % % 4π % 1 % % Qn × vn δ(r − rn ) +% % ∇ · c 6 n (mol)

% % % % % % % 4π % % J(r, t) + % 1 % ∂ 0 E(r, t) + 4πP(r, t) − 4π∇ · Q (r, t) . ≈% % c % ∂t % c % (4.49) Deﬁne the macroscopic electric displacement vector D(r, t) as the vector ﬁeld whose quadrupole approximation is given by D(r, t) ≈ 0 E(r, t) + 4πP(r, t) − 4π∇ · Q (r, t),

(4.50)

and deﬁne the macroscopic magnetic intensity vector H(r, t) as the vector ﬁeld whose approximation is given by H(r, t) ≈

1 B(r, t) − 4πM(r, t) µ0 )) % % (( % 4π % % % −% % pn × vn δ(r − rn ) c n (mol)

1 + ∇· 6

((

)) Qn

× vn δ(r − rn )

.

(4.51)

n (mol)

Notice that all of the approximation due to neglected higher-order terms has been placed in these two derived ﬁeld vectors. With these deﬁnitions, the macroscopic Maxwell equations assume the familiar form ∇ · D(r, t) = 4π(r, t), ∇ · B(r, t) = 0, % % % 1 % ∂B(r, t) % ∇ × E(r, t) = − % , %c% ∂t % % % % % 1 % ∂D(r, t) % 4π % % % % ∇ × H(r, t) = % . % c % J(r, t) + % c % ∂t

(4.52) (4.53) (4.54) (4.55)

It is clear from this derivation that, because the macroscopic ﬁeld vectors E and B are the respective spatial averages of the microscopic ﬁeld vectors e

4.1 Correlation of Microscopic and Macroscopic Electromagnetics

177

and b, then E and B are of more fundamental signiﬁcance than the derived ﬁeld vectors D and H. The derived ﬁeld vectors D and H should be regarded as a shorthand notation for the expressions appearing on the right-hand sides of Eqs. (4.50) and (4.51), respectively. For the conservation of charge, the spatial average of the microscopic equation of continuity given in Eq. (2.14) directly yields the macroscopic equation of continuity ∇ · J(r, t) +

∂(r, t) = 0. ∂t

(4.56)

This result also follows from the divergence of Eq. (4.55) with substitution from Eq. (4.52). Suppose now that the medium as a whole is moving with the translational velocity v. If one neglects any other motion of the molecules comprising the medium, one then has that vn ≈ v for all n. Equation (4.51) then becomes (neglecting the quadrupole terms) % % % 4π % 1 % (4.57) B(r, t) − H(r, t) ≈ 4πM(r, t) + % % c % P(r, t) × v, µ0 where, from Eq. (4.50) P(r, t) ≈

1 (D(r, t) − 0 E(r, t)) , 4π

(4.58)

so that

% % %1% 1 % B(r, t) − H(r, t) ≈ 4πM(r, t) + % % c % (D(r, t) − 0 E(r, t)) × v. µ0 (4.59)

Hence, the electric polarization P (and quadrupole moment Q, if included) enters the expression for the eﬀective magnetization for a moving medium. Equation (4.59) is the nonrelativistic limit of one of the equations in the Minkowski formulation of the electrodynamics of moving media [11]. Finally, consider the spatial average of the microscopic Lorentz force relation (2.21) % % %1% % (4.60) f (r, t = ρ(r, t)e(r, t) + % % c % j(r, t) × b(r, t). With the macroscopic force density deﬁned by the spatial average F(r, t) ≡ f (r, t), application of the spatial averaging process (4.1) to the above expression yields F(r, t) = w(r )ρ(r − r , t)e(r − r , t)d3 r % % %1% 3 % +% (4.61) % c % w(r )j(r − r , t) × b(r − r , t)d r .

178

4 Macroscopic Electromagnetics

Let e (r, t) ≡ E(r, t) − e(r, t) and b (r, t) ≡ B(r, t) − b(r, t) denote the diﬀerence ﬁelds between the macroscopic and microscopic ﬁeld vectors and assume that E(r − r , t) ≈ E(r, t) and B(r − r , t) ≈ B(r, t) when r varies over the macroscopically small spatial size of the weighting function w(r ). The above expression for the macroscopic force density then becomes % % %1% % F(r, t) ≈ ρ(r, t)E(r, t) + % % c % j(r, t) × B(r, t) % % %1% % −ρ(r, t)e (r, t) − % % c % j(r, t) × b (r, t) % % %1% % ≈ (r, t)E(r, t) + % % c % J(r, t) × B(r, t) −(∇ · P(r, t))E(r, t) + (∇ · ∇ · Q (r, t))E(r, t) % % %1% d % % P(r, t) − ∇ · Q (r, t) × B(r, t) +% % c dt + (∇ × M(r, t)) × B(r, t) % % %1% % −ρ(r, t)e (r, t) − % % c % j(r, t) × b (r, t), (4.62) and it is seen that there is no simple expression of the spatially averaged Lorentz force relation in a ponderable medium. Nevertheless, the microscopic Lorentz force relation (2.21) holds everywhere and so can always be applied to determine the electromagnetic force in microscopic detail.

4.2 Constitutive Relations in Linear Electromagnetics and Optics The macroscopic Maxwell’s equations given in Eqs. (4.52)–(4.55), together with the macroscopic equation of continuity (4.56), are completed by the appropriate constitutive (or material) relations relating the induced electric displacement vector D(r, t), the magnetic intensity vector H(r, t), and the conduction current density Jc (r, t), these three ﬁeld vectors being referred to as the induction ﬁelds, to both the electric ﬁeld intensity E(r, t) and the magnetic induction ﬁeld B(r, t), these latter two ﬁeld vectors being referred to as the primitive ﬁelds. For many physical situations, magnetic ﬁeld eﬀects are entirely negligible in comparison with the eﬀects produced by the electric ﬁeld for both the electric displacement ﬁeld and the conduction current density, and electric ﬁeld eﬀects are likewise negligible in comparison with magnetic ﬁeld eﬀects for the magnetic induction vector.2 With the additional assumption that the ﬁeld strengths considered are suﬃciently small that nonlinear 2

Notable exceptions are bianisotropic and bi-isotropic materials that exhibit chirality [12].

4.2 Constitutive Relations in Linear Electromagnetics and Optics

179

eﬀects are negligible, each material relation may be expressed in the form of the general constitutive relation ∞ t 3 ˆ , t , r, t) · F(r , t ), G(r, t) = d r dt ζ(r (4.63) −∞

−∞

where the upper limit of integration in the time integral is imposed by primiˆ , t , r, t) denotes either the dielectric permittivity tive causality [13]. Here ζ(r response tensor ˆ(r , t , r, t) when F(r, t) = E(r, t) and G(r, t) = D(r, t), the inverse of the magnetic permeability response tensor µ ˆ(r , t , r, t) when F(r, t) = B(r, t) and G(r, t) = H(r, t), or the electric conductivity response tensor σ ˆ (r , t , r, t) when F(r, t) = E(r, t) and G(r, t) = Jc (r, t), where Jc (r, t) denotes the macroscopic conduction current density whose origin is derived from the macroscopic free current density that is deﬁned in Eq. (4.45). That the inverse of the magnetic permeability tensor is used in the constitutive relation is a consequence of the fact that the magnetic intensity vector H(r, t) is the induced ﬁeld whereas the magnetic induction vector B(r, t) is the primitive ﬁeld [14]. The linear properties of the material are then determined by the analytical properties of each of the material tensors. The linearity property expressed in the general constitutive relation given in Eq. (4.63) is due to the independence of the dielectric permittivity, magnetic permeability, and conductivity tensors on the applied ﬁeld strength. For example, this relation states that both D(r, t) and Jc (r, t) are doubled if E(r, t) is doubled for all r and t, while H(r, t) is doubled if B(r, t) is doubled for all r and t. ˆ , t , r, t) = The material is said to be spatially locally linear if and only if ζ(r ˆ t , t)δ(r − r ). With this substitution, the general constitutive relation ζ(r, given in Eq. (4.63) simpliﬁes to t ˆ t , t) · F(r, t )dt , G(r, t) = (4.64) ζ(r, −∞

so that, for example, D(r, t) is doubled at a particular point r = r0 if E(r, t) is doubled at that same point r = r0 for all time t. The mathematical statement of spatial local linearity then corresponds in an intuitive way to the physical property that each molecule in the material is uncoupled from every other molecule in the material. A material is said to be spatially homogeneous if and only if its properties ˆ − r , t , t). If this ˆ , t , r, t) = ζ(r vary with position within the material as ζ(r mathematical property is not satisﬁed, then the material is said to be spatially inhomogeneous. Similarly, a material is said to be temporally homogeneous if ˆ , r, t − t ). If ˆ , t , r, t) = ζ(r and only if its properties vary with time as ζ(r this mathematical property is not satisﬁed, then the material is said to be temporally inhomogeneous. For a spatially and temporally homogeneous material, the general constitutive relation given in Eq. (4.63) becomes

180

4 Macroscopic Electromagnetics

∞

G(r, t) =

3

t

d r −∞

−∞

ˆ − r , t − t ) · F(r , t ). dt ζ(r

(4.65)

The spatiotemporal Fourier transform of this relation then yields, with application of the convolution theorem ˜˜ ˜ ˜ ˜ ˜ G(k, ω) = ζ(k, ω) · F(k, ω). Here ˜ ˜ F(k, ω) =

∞

d3 r

−∞

∞

dt F(r, t)e−i(k·r−ωt)

(4.66)

(4.67)

−∞

is the spatiotemporal Fourier transform of F(r, t), provided that the spatial and temporal integrals exist, with inverse transform ∞ ∞ 1 ˜ 3 ˜ F(r, t) = d k dω F(k, ω)ei(k·r−ωt) . (4.68) (2π)4 −∞ −∞ Hence, a spatially and temporally homogeneous material is linear in Fourier ˜ ˜ ˜ ˜ 0 , ω0 ) space in the sense that if F(k, ω) is doubled at k = k0 , ω = ω0 , then G(k ˜˜ is doubled. If the material tensor ζ(k, ω) depends on the wave vector k, then the medium is said to be spatially dispersive, whereas if it depends upon the temporal angular frequency ω, it is temporally dispersive. Spatial dispersion is typically considered in connection with crystal optics [15], whereas temporal dispersion appears in all ponderable materials. If the material tensor is independent of both k and ω, the medium is said to be nondispersive. In that case, Eq. (4.66) simpliﬁes to the expres˜ ˜ ˜ ˜ ˜ ω), whose inverse transform then yields the result sion G(k, ω) = ζ˜ · F(k, ˜˜ G(k, ω) = ζ · F(k, ω), which is seen to be a special case of Eq. (4.64). Consequently, a linear nondispersive medium is both spatially and temporally locally linear. Notice that the material response of a temporally locally linear medium is instantaneous, which is clearly nonphysical (with the single exception of vacuum) and is not considered any further here. For a material that is locally linear, spatially inhomogeneous, and temˆ , t , r, t) = porally dispersive, the response tensor assumes the form ζ(r ˆ t − t )δ(r − r ). With this substitution, the general constitutive relation ζ(r, given in Eq. (4.63) becomes t ˆ t − t ) · F(r, t )dt , G(r, t) = (4.69) ζ(r, −∞

˜ ω) = ζ(r, ω) · F(r, ˜ ω). The appropriate with temporal Fourier transform G(r, frequency domain constitutive relation for a locally linear, spatially homogeneous, spatially nondispersive, temporally dispersive material is then seen to ˜ ω), where the material properties are now ˜ ω) = ζ(ω) · F(r, be given by G(r, completely independent of position within the material.

4.2 Constitutive Relations in Linear Electromagnetics and Optics

181

Finally, consider the property of isotropy. The material is said to be isotropic if and only if the components of the response tensor satisfy the relaˆ , t , r, t)δij , where δij is the Kronecker delta function, tion ζˆij (r , t , r, t) = ζ(r ˆ , t , r, t) is independent of the indices i, j. With this substituand where ζ(r tion, the general constitutive relation given in Eq. (4.63) becomes

∞

G(r, t) =

3

t

ˆ , t , r, t)F(r , t ). dt ζ(r

d r −∞

(4.70)

−∞

If the material is not isotropic, it is then said to be anisotropic. The material response tensor for a spatially homogeneous, isotropic, locally linear, temporally dispersive material is obtained from the previous two relations with the additional constraint of spatial homogeneity. In that case ˆ − t )δ(r − r ) and the constitutive relation given in Eq. ˆ , t , r, t) = ζ(t ζ(r (4.70) becomes t ˆ − t )F(r, t )dt , (4.71) ζ(t G(r, t) = −∞

˜ ω) = ζ(ω)F(r, ˜ ω). Hence, G(r, ˜ ω) is with temporal Fourier transform G(r, ˜ parallel to F(r, ω); however, this does not necessarily imply that G(r, t) is parallel to F(r, t) for all r and t. For this to occur, the homogeneous, linear, second-order integral equation t ˆ − t )F(r, t )dt (4.72) γF(r, t) = ζ(t −∞

must be satisﬁed. It is of interest to examine in more detail the constitutive relation appearing in Eq. (4.71) for a spatially homogeneous, isotropic, locally linear, temporally dispersive material. Because the ﬁeld G(r, t) depends upon the ˆ ), it is past history of both the ﬁeld F(r, t ) and the material response ζ(t appropriate to express F(r, t ) in a Taylor series expansion about the instant t = t as ∞ 1 ∂ j F(r, t) (t − t)j , (4.73) F(r, t ) = j! ∂t j=1 which is valid provided that F(r, t ) and all of its time derivatives exist at each instant t ≤ t. Substitution of this expression into the constitutive relation given in Eq. (4.71) then gives G(r, t ) = where

∞ 1 ˆ(j) ∂ j F(r, t) ζ (t − t )j , j j! ∂t j=1

(−1)j ζˆ(j) ≡ j!

0

∞

ˆ )τ j dτ ζ(τ

(4.74)

(4.75)

182

4 Macroscopic Electromagnetics

is proportional to the jth moment of the material response function. The temporal Fourier transform of Eq. (4.74) then shows that the temporal frequency spectrum of the material response function is given by ζ(ω) =

∞

ζˆ(j) (−iω)j .

(4.76)

j=0

The real part of ζ(ω) is then seen to be an even function of ω whereas the imaginary part is an odd function of ω. Furthermore, Eq. (4.76) shows that the material expansion coeﬃcients are given by ij ∂ j ζ(ω) (j) ˆ ζ = . (4.77) j! ∂ω j ω=0 Because of uniqueness, Eq. (4.76) is the Maclaurin series expansion of the function ζ(ω).

4.3 Causality and Dispersion Relations Causality is an essential feature of electromagnetic wave theory. At the most fundamental level is the principle of primitive causality [13, 16] which simply states that the eﬀect cannot precede the cause. As trivially obvious as this is, the events of “cause” and “eﬀect” must be carefully deﬁned in order to properly apply this principle [17]. This is modiﬁed in the special theory of relativity to what is referred to as the principle of relativistic causality which states that no signal can propagate with a velocity greater than the speed of light c in vacuum. These two fundamental physical principles have direct bearing on the problem of ultrashort dispersive pulse propagation with regard to modeling the material dispersion. The most direct derivation [18] of the dispersion relations that must be satisﬁed by any linear system begins by expressing the angular frequency spectrum of the output signal fout (t) – the eﬀect – in terms of the spectrum of the input signal fin (t) – the cause – through the linear relationship f˜out (ω) = χ(ω)f˜in (ω),

where

∞

χ(ω) =

iωt χ(t)e ˆ dt

(4.78)

(4.79)

−∞

describes the linear system response at the angular frequency ω. Because of primitive causality, the system response may be expressed as χ(t) ˆ = U (t)Ψˆ (t),

(4.80)

4.3 Causality and Dispersion Relations

183

where U (t) = 0 for t < 0 and U (t) = 1 for t > 0 denotes the Heaviside unit step function (see Appendix C), and where Ψˆ (t) ≡ χ(t) ˆ for all t > 0. The Fourier transform of the Heaviside unit step function is given by ∞ i iωt ˜ e dt = lim U (ω) = ω + i∆ ∆→0+ 0 i =P + πδ(ω), (4.81) ω where P indicates that the Cauchy principal value is to be taken. Equation (4.79) with (4.80) then yields, with application of the convolution theorem and the above result ∞ 1 ˜ (ω − ω )dω χ(ω) = Ψ (ω )U 2π −∞ ∞ Ψ (ω ) 1 i = dω + Ψ (ω). P (4.82) 2π 2 −∞ ω − ω Because χ(t) ˆ = U (t)Ψˆ (t), the behavior of the function Ψˆ (t) for t < 0 may be freely chosen. As the ﬁrst choice, let Ψˆ (−|t|) = Ψˆ (|t|) so that Ψˆ (t) is an even function of t. Its Fourier spectrum Ψ (ω) is then purely real and Eq. (4.82) yields Ψ (ω) = 2{χ(ω)}, so that ∞ {χ(ω )} 1 dω . (4.83) {χ(ω)} = − P π −∞ ω − ω For the second choice, let Ψˆ (−|t|) = −Ψˆ (|t|) so that Ψˆ (t) is an odd function of t. Its Fourier spectrum Ψ (ω) is then purely imaginary and Eq. (4.82) yields Ψ (ω) = 2i{χ(ω)}, so that ∞ {χ(ω )} 1 (4.84) {χ(ω)} = P dω . π −∞ ω − ω The real and imaginary parts of a causal system response function then form a Hilbert transform pair, as expressed in Eqs. (4.83)–(4.84). These integral relations, referred to as either the Plemelj formulae [19] or the dispersion relations, are generalized in Titchmarsh’s theorem [13, 20]: Theorem 2. Titchmarsh’s Theorem. Any square-integrable function χ(ω) with inverse Fourier transform χ(t) ˆ that satisﬁes one of the following four conditions satisﬁes all four of them: 1. χ(t) ˆ = 0 for all t < 0. 2. χ(ω ) = limω →0+ {χ(ω + iω )} for almost all ω , where χ(ω) is holomorphic in the upper-half of the complex ω = ω + iω plane and is square-integrable over any line parallel to the ω -axis in the upper-half plane.

184

4 Macroscopic Electromagnetics

3. {χ(ω)} and {χ(ω)} satisfy the ﬁrst Plemelj formula (4.83). 4. {χ(ω)} and {χ(ω)} satisfy the second Plemelj formula (4.84). The restriction to almost all ω appearing in condition 2 of this theorem refers to the fact that the Fourier transform of a function remains unchanged when that function is changed over a set of measure zero. 4.3.1 The Dielectric Permitivity For a spatially homogeneous, isotropic, locally linear, temporally dispersive dielectric, the constitutive relation (4.71) for the electric displacement vector becomes t D(r, t) =

ˆ(t − t )E(r, t )dt (4.85) −∞

with temporal Fourier transform ˜ ω) = (ω)E(r, ˜ ω), D(r,

where

∞

(ω) =

ˆ(t)eiωt dt

(4.86)

(4.87)

−∞

is the dielectric permittivity. In addition, the constitutive relation for the macroscopic polarization density is given by t P(r, t) = 0 χ ˆe (t − t )E(r, t )dt (4.88) −∞

with temporal Fourier transform ˜ ω) = 0 χe (ω)E(r, ˜ ω), P(r,

where χe (ω) =

∞

−∞

χ ˆe (t)eiωt dt.

(4.89)

(4.90)

From the Fourier transform of Eq. (4.50) for a simple polarizable dielectric 3 ˜ ω) = 0 (1 + 4πχe (ω)) E(r, ˜ ω), D(r,

(4.91)

so that the dielectric permittivity and electric susceptibility are related by

(ω) = 0 (1 + 4πχe (ω)) . 3

(4.92)

A simple polarizable dielectric is deﬁned here as one for which the quadrupole and all higher-order moments of the molecular charge distribution identically vanish.

4.3 Causality and Dispersion Relations

185

Because E(r, t), P(r, t), and D(r, t) are all real-valued vector ﬁelds, then each of their temporal frequency spectra satisﬁes the respective symmetry property ˜ −ω), ˜ ∗ (r, ω) = E(r, E ∗ ˜ (r, ω) = P(r, ˜ −ω), P ∗ ˜ ˜ −ω), D (r, ω) = D(r,

(4.93) (4.94) (4.95)

for real ω. As a consequence, the real part of each spectral vector ﬁeld is an even function of ω whereas the imaginary part is an odd function of ω. Furthermore, Eq. (4.94) with (4.89) and Eq. (4.95) with (4.86) yields the symmetry relations χ∗e (ω) = χe (−ω),

∗ (ω) = (−ω)

(4.96) (4.97)

for real ω. These results also follow respectively from Eqs. (4.87) and (4.90) because the dielectric response functions ˆ(t) and χ(t) ˆ are both real-valued. As a consequence, the real part of (ω) is an even function of ω whereas its imaginary part is an odd function of ω when ω is real-valued. For complex ω = ω + iω with ω ≡ {ω} and ω ≡ {ω} these symmetry relations become χ∗e (ω) = χe (−ω ∗ ),

∗ (ω) = (−ω ∗ ),

(4.98) (4.99)

as seen from Eqs. (4.90) and (4.87), respectively. Hence, along the imaginary axis, χ∗e (iω ) = χe (iω ) and ∗ (iω ) = (iω ) so that both χe (ω) and (ω) are real-valued on the imaginary axis. Because Eq. (4.88) represents the most fundamental cause and eﬀect relation for the dielectric response, the electric susceptibility then satisﬁes Titchmarsh’s theorem. In particular, the Plemelj formulae for the electric susceptibility are given by the pair of Hilbert transform relations ∞ {χe (ω )} 1 dω , (4.100) {χe (ω)} = P π ω − ω −∞ ∞ {χe (ω )} 1 (4.101) {χe (ω)} = − P dω . π ω − ω −∞ Because of the symmetry relation expressed in Eq. (4.96), this transform pair may be expressed over the positive real frequency axis alone as ∞ ω {χe (ω )} 2 {χe (ω)} = P dω , (4.102) π ω 2 − ω 2 0 ∞ {χe (ω )} 2ω {χe (ω)} = − P dω . (4.103) π ω 2 − ω 2 0

186

4 Macroscopic Electromagnetics

With Eq. (4.92), the appropriate dispersion relations for the dielectric permittivity are then seen to be given by ∞ { (ω )} 1 dω , { (ω) − 0 } = P (4.104) π −∞ ω − ω ∞ 1 { (ω ) − 0 } { (ω)} = − P dω , (4.105) π ω − ω −∞ which were ﬁrst derived by Kramers [21] and Kronig [22] in 1927 and 1926, respectively, and are known as the Kramers–Kronig relations. As a consequence of Titchmarsh’s theorem, (ω) is holomorphic in the upper-half of the complex ω−plane. In addition, (ω)/ 0 → 1 in the limit as |ω| → ∞ in the upper-half plane (ω > 0). By continuity, (ω )/ 0 → 1 as ω → ±∞ along the real ω −axis, so that (ω ) → 0 as ω → ±∞. In addition, because (ω ) is an odd function along the real ω −axis, then (0) = 0. Upon returning to the fundamental constitutive relation given in Eq. (4.85) it is seen that at any ﬁxed point in space the displacement vector depends upon the past history of the electric ﬁeld intensity at that point through the dielectric permittivity response function ˆ(t − t ). From a physical point of view, it seems reasonable to expect that the sensitivity of the medium response decreases as the past time t decreases further into the past from the present time t at which the ﬁeld vector D(r, t) is evaluated. It is then seen appropriate to expand the electric ﬁeld intensity E(r, t ) in a Taylor series about the instant t = t as E(r, t ) =

∞ 1 ∂ n E(r, t) (t − t)n , n n! ∂t n=0

(4.106)

provided that E(r, t ) and all of its time derivatives exist at each instant t ≤ t. This Taylor series expansion represents E(r, t ) for all t ≤ t if and only if the remainder RN (t ) after N terms approaches zero as N → ∞. Substitution of this expansion into the constitutive relation (4.85) then gives D(r, t) =

∞

(n)

n=0

with (n)

(−1)n ≡ n!

∞

∂ n E(r, t) ∂tn

(4.107)

ˆ(τ )τ n dτ.

(4.108)

0

Notice that each coeeﬁcient (n) is proportional to the nth-order moment of the dielectric permittivity response function. This form of the constitutive relation explicitly exhibits the local linear dependence of the derived ﬁeld vector D(r, t) on the temporal derivatives of the electric ﬁeld intensity vector E(r, t). For a quasi-static ﬁeld this form of the constitutive relation may approximated as

4.3 Causality and Dispersion Relations

D(r, t) (0) E(r, t) + (1)

187

∂E(r, t) , ∂t

from which it is seen that the zeroth-order moment (0) is just the static dielectric permittivity of the medium; that is, (0) = (0). The temporal Fourier transform of the constitutive relation given in Eq. (4.107) yields, with Eq. (4.86), ∞ ˜ ω) = (ω)E(r, ˜ ω) = ˜ ω), D(r, (−iω)n (n) E(r, n=0

so that the temporal frequency spectrum of the dielectric permittivity is given by ∞

(n) (−iω)n . (4.109)

(ω) = n=0

The real and imaginary parts of this expression then yield the pair of relations

r (ω) ≡ { (ω)} = (0) − (2) ω 2 + (4) ω 4 − (6) ω 6 + · · · , (1)

i (ω) ≡ { (ω)} = −

ω+

(3)

3

(5)

ω −

5

ω + ···.

(4.110) (4.111)

Once again it is seen that the real part of (ω) is an even function of real ω and the imaginary part is an odd function of ω. Furthermore, it is seen that in ∂ n (ω) , (4.112)

(n) = n! ∂ω n ω=0 so that the nth-order moment of the dielectric response function is proportional to the nth-order derivative of the dielectric permittivity at the origin. 4.3.2 The Electric Conductivity For a spatially homogeneous, isotropic, locally linear, temporally dispersive conducting or semiconducting material, the constitutive relation (4.71) for the conduction current vector becomes t Jc (r, t) = σ ˆ (t − t )E(r, t )dt (4.113) −∞

with temporal Fourier transform ˜c (r, ω) = σ(ω)E(r, ˜ ω), J

where

∞

σ(ω) = −∞

σ ˆ (t)eiωt dt

(4.114)

(4.115)

188

4 Macroscopic Electromagnetics

is the electric conductivity. Because both Jc (r, t) and E(r, t) are real-valued vector ﬁelds, the conductivity then satisﬁes the symmetry relation σ ∗ (ω) = σ(−ω ∗ )

(4.116)

for complex ω. The manner in which the electric conductivity enters the dispersive material properties is obtained from the temporal Fourier transform of the macroscopic form (4.55) of Amp´ere’s law in a semiconducting material % % % % %1% % 4π % % % ˜ % ˜ ˜ ∇ × H(r, ω) = % % J(r, ω) − % % c % iω D(r, ω), c ˜ ω) = J ˜ext (r, ω)+ J ˜c (r, ω) with J ˜ext (r, ω) describing any externally where J(r, applied current source. Substitution of the constitutive relations given in Eqs. (4.86) and (4.114) then yields % % % % % % % % ˜ext (r, ω) − % 1 % iω c (ω)E(r, ˜ ω), ˜ ω) = % 4π % J (4.117) ∇ × H(r, %c% % c % where

σ(ω) (4.118) ω is called the complex permittivity of the material. For a nonconducting material, σ(ω) = 0 and c (ω) = (ω), whereas for a purely conducting material,

(ω) = 0 and c (ω) = 0 + i4πσ(ω)/ω. Because of the simple pole singularity at ω = 0 appearing in Eq. (4.118), the ﬁrst part of condition 2 in Titchmarsh’s theorem is not satisﬁed. The Hilbert transform of the complex permittivity then includes a boundary contribution from this pole. The dispersion relations given in Eqs. (4.104)–(4.105) must then be modiﬁed [23, 24] for the complex permittivity as ∞ { c (ω )} 1 (4.119) { c (ω) − 0 } = P dω , π ω − ω −∞ ∞ 1 { c (ω ) − 0 } σ(0) (4.120) { c (ω)} − 4π =− P dω , ω π ω − ω −∞

c (ω) ≡ (ω) + i4π

where the pole contribution at ω = 0 has been subtracted. This pair of relations reduces to the pair of Kramers–Kronig relations given in Eqs. (4.104)– (4.105) in the limit of zero static conductivity. For a purely conducting material they yield the pair of dispersion relations ∞ σ(ω) {σ(ω )/ω } σ(0) 1 − = P dω , (4.121) ω ω π ω − ω −∞ ∞ σ(ω) {σ(ω )/ω } 1 dω , (4.122) =− P ω π ω − ω −∞

4.3 Causality and Dispersion Relations

189

for the electric conductivity. Consider again the fundamental constitutive relation given in Eq. (4.113) for the conduction current. This relation becomes, with substitution of the Taylor series expansion of the electric ﬁeld intensity about the instant t = t, Jc (r, t) =

∞

σ (n)

n=0

∂E(r, t) ∂t

(4.123)

(−1)n ∞ σ ≡ σ ˆ (τ )τ n dτ. (4.124) n! 0 This form of the constitutive relation explicitly displays the local linear dependence of the derived conduction current vector Jc (r, t) on the temporal derivatives of the electric ﬁeld intensity vector E(r, t). Each coeﬃcient σ (n) is then seen to be proportional to the nth-order moment of the electric conductivity response function. In particular, the zeroth-order moment is seen to be identical with the static conductivity of the medium; that is, σ (0) = σ(0). The temporal Fourier transform of the constitutive relation given in Eq. (4.123) yields, with Eq. (4.114) ∞ ˜ ω) = ˜ ω), ˜c (r, ω) = σ(ω)E(r, (−iω)n σ (n) E(r, J

with

(n)

n=0

so that the temporal frequency spectrum of the electric conductivity is given by ∞ σ(ω) = σ (n) (−iω)n . (4.125) n=0

The real and imaginary parts of this expression then yield the pair of relations σr (ω) ≡ {σ(ω)} = σ (0) − σ (2) ω 2 + σ (4) ω 4 − σ (6) ω 6 + · · · , (4.126) σi (ω) ≡ {σ(ω)} = −σ (1) ω + σ (3) ω 3 − σ (5) ω 5 + · · · . (4.127) Once again it is seen that the real part of σ(ω) is an even function of real ω and the imaginary part is an odd function of ω. Furthermore, it is seen that in ∂ n σ(ω) σ (n) = , (4.128) n! ∂ω n ω=0 so that the nth-order moment of the electric conductivity response function is proportional to the nth-order derivative of the conductivity at the origin. 4.3.3 The Magnetic Permeability For a spatially homogeneous, isotropic, locally linear, temporally dispersive, nonferromagnetic material, the constitutive relation (4.71) for the magnetic intensity vector becomes

190

4 Macroscopic Electromagnetics

t

µ ˆ−1 (t − t )B(r, t )dt ,

H(r, t) = −∞

(4.129)

with temporal Fourier transform ˜ ω) = µ−1 (ω)B(r, ˜ ω), H(r, where µ−1 (ω) =

∞

µ ˆ−1 (t)eiωt dt

(4.130)

(4.131)

−∞

is the inverse of magnetic permeability µ(ω). The constitutive relation given in Eq. (4.130) may then be expressed in the more familiar form ˜ ω) = µ(ω)H(r, ˜ ω). B(r,

(4.132)

In addition, the constitutive relation for the macroscopic magnetization vector is given by t 1 M(r, t) = χ ˆb (t − t )B(r, t )dt (4.133) µ0 −∞ with temporal Fourier transform 1 ˜ ˜ ω), M(r, ω) = χb (ω)B(r, µ0

where χb (ω) =

∞

−∞

χ ˆb (t)eiωt dt.

(4.134)

(4.135)

For a simple magnetizable medium, the magnetization vector is given by [cf. Eq.(4.51)]4 1 1 M(r, t) = B(r, t) − H(r, t) . (4.136) 4π µ0 The temporal Fourier transform of this relation then yields, with substitution from Eqs. (4.132) and (4.134) ˜ ω), ˜ ω) = µ0 (1 + 4πχm (ω)) H(r, B(r,

(4.137)

where 1 + 4πχm (ω) = 1/(1 − 4πχb (ω)), so that χm (ω) =

χb (ω) 1 − 4πχb (ω)

(4.138)

is the magnetic susceptibility of the material. Comparison of Eqs. (4.132) and (4.137) then shows that the magnetic permeability and magnetic susceptibility are related by 4

This equation is taken here as the deﬁnition of a simple magnetizable medium.

4.3 Causality and Dispersion Relations

µ(ω) = µ0 (1 + 4πχm (ω)) .

191

(4.139)

This is the familiar textbook relation that is commonly employed. The magnetic susceptibility of a simple magnetizable material is typically much smaller than the electric susceptibility for that material. This is due to the fact that the magnetization of a nonferromagnetic material is a relativistic eﬀect and so is on the order of v 2 /c2 , where v is the velocity of the atomic electrons [23]. For diamagnetic materials, χm (ω) is slightly less than zero (χm ∼ −1 × 10−5 → −1 × 10−8 ) so that µ(ω)/µ0 is slightly smaller than unity, whereas for paramagnetic materials, χm (ω) is slightly larger than zero (χm ∼ 1 × 10−4 → 1 × 10−7 ) so that µ(ω)/µ0 is slightly greater than unity. In either case, |χm (ω)| 1 and Eq. (4.138), which may be rewritten as 1 − 4πχb (ω) = 1/(1 + 4πχm (ω)), shows that χm (ω) ≈ χb (ω)

(4.140)

over the frequency domain of interest. Within the accuracy of this approximate equivalence, the constitutive relation given in Eq. (4.137) may be inverted as t

B(r, t) = −∞

µ ˆ(t − t )H(r, t )dt ,

(4.141)

which, in eﬀect, reverses the proper roles of cause and eﬀect in simple magnetizable materials. Because B(r, t), M(r, t), and H(r, t) are all real-valued vector ﬁelds, then each of their temporal frequency spectra satisﬁes the respective symmetry property ˜ ∗ (r, ω) = B(r, ˜ −ω ∗ ), B ˜ ∗ (r, ω) = M(r, ˜ M −ω ∗ ), ∗ ˜ (r, ω) = H(r, ˜ −ω ∗ ), H

(4.142) (4.143) (4.144)

for complex ω = ω + iω . As a consequence, the real part of each spectral vector ﬁeld is an even function of real ω = ω and the imaginary part is an odd function of ω . Furthermore, Eqs. (4.131), (4.135), and (4.138) yield the symmetry relations χ∗b (ω) = χb (−ω ∗ ),

(4.145)

χ∗m (ω) ∗

∗

(4.146)

µ (ω) = µ(−ω ).

(4.147)

= χm (−ω ), ∗

Hence, along the imaginary axis, χ∗j (iω ) = χj (iω ) for j = m, b and µ∗ (iω ) = µ(iω ) so that both χj (ω) and µ(ω) are real-valued on the imaginary axis. Furthermore, the real part of µ(ω) is an even function and the imaginary part of µ(ω) is an odd function along the real ω −axis.

192

4 Macroscopic Electromagnetics

Because Eq. (4.133) represents the most fundamental cause and eﬀect relation for the magnetic response, the susceptibility χb (ω) then satisﬁes Titchmarsh’s theorem. In particular, the Plemelj formulae for this susceptibility function are given by the pair of Hilbert transform relations ∞ {χj (ω )} 1 (4.148) {χj (ω)} = P dω , π ω − ω −∞ ∞ {χj (ω )} 1 (4.149) {χj (ω)} = − P dω , π ω − ω −∞ with j = b. Because of the approximate equality stated in Eq. (4.140), the above pair of dispersion relations also holds for the magnetic susceptibility χm (ω) when j = m. Because of the symmetry relations expressed in Eqs. (4.145) and (4.146), this transform pair may be expressed over the positive real frequency axis alone as ∞ ω {χj (ω )} 2 dω , (4.150) {χj (ω)} = P π ω 2 − ω 2 0 ∞ {χj (ω )} 2ω {χj (ω)} = − P dω , (4.151) π ω 2 − ω 2 0 for j = m, b. With Eq. (4.139), the appropriate dispersion relations for the magnetic permeability are then seen to be given by ∞ {µ(ω )} 1 {µ(ω) − µ0 } = P dω , (4.152) π −∞ ω − ω ∞ {µ(ω ) − µ0 } 1 dω . (4.153) {µ(ω)} = − P π ω − ω −∞ Notice that these dispersion relations should be used with appropriate caution as the magnetic permeability function µ(ω) loses physical meaning as ω exceeds some angular frequency value ωµ [23]. Consider again the resultant constitutive relation given in Eq. (4.141). Expansion of the magnetic intensity vector H(r, t ) in a Taylor series about the instant t = t results in the expression H(r, t ) =

∞ 1 ∂ n H(r, t) (t − t)n , n n! ∂t n=0

(4.154)

provided that H(r, t ) and all of its time derivatives exist at each instant t ≤ t. Substitution of this expansion in Eq. (4.141) then gives B(r, t) =

∞ n=0

with

µ(n)

∂ n H(r, t) ∂tn

(4.155)

4.4 Causal Models of the Material Dispersion

µ(n) ≡

(−1)n n!

∞

µ ˆ(τ )τ n dτ.

193

(4.156)

0

The zeroth-order moment µ(0) is then seen to be identical with the static permeability of the medium; that is, µ(0) = µ(0). The temporal Fourier transform of the constitutive relation given in Eq. (4.155) yields, with the Fourier transform of Eq. (4.141), ∞ ˜ ω), ˜ ω) = µ(ω)H(r, ˜ ω) = (−iω)n µ(n) H(r, B(r, n=0

so that the temporal frequency spectrum of the magnetic permeability is given by ∞ µ(n) (−iω)n . (4.157) µ(ω) = n=0

The real and imaginary parts of this expression then yield the pair of relations µr (ω) ≡ {µ(ω)} = µ(0) − µ(2) ω 2 + µ(4) ω 4 − µ(6) ω 6 + · · · , (4.158) µi (ω) ≡ {µ(ω)} = −µ(1) ω + µ(3) ω 3 − µ(5) ω 5 + · · · . (4.159) Once again it is seen that the real part of µ(ω) is an even function of real ω and the imaginary part is an odd function of ω. Furthermore, it is seen that in ∂ n µ(ω) (n) µ = , (4.160) n! ∂ω n ω=0 so that the nth-order moment of the eﬀective magnetic permeability response function is proportional to the nth-order derivative of the permeability at the origin.

4.4 Causal Models of the Material Dispersion The theory of temporal frequency dispersion in homogeneous, isotropic, locally linear materials in either the solid, liquid, or gaseous states has its origins in the classical theories due to Drude [25] for free-electron conductors, Lorentz [3] for high-frequency resonant dispersion phenomena in dielectrics, and Debye [26] for low-frequency orientational polarization phenomena in dielectrics. Although these models are phenomenological in their origin, they do provide reasonably accurate expressions for describing the dispersive properties of such media over the appropriate frequency domain of interest. Composite models may then be constructed from appropriate generalizations [9, 27–29] of these individual models in order to more accurately describe the classical frequency dispersion over the entire frequency domain. It is essential that each of these physical models be causal for obvious reasons.

194

4 Macroscopic Electromagnetics 2

10

1

ΕrΩRΕΩ

10

0

10

-1

10

8

10

10

10

12

10

14

10

16

10

18

10

Frequency f (Hz)

Fig. 4.2. Frequency dispersion of the real part of the relative dielectric permittivity data for triply distilled water at 25o C. (Numerical data supplied by the School of Aerospace Medicine, Armstrong Laboratory, Brooks Air Force Base.) 2

10

0

10

-2

ΕiΩIΕΩ

10

-4

10

-6

10

-8

10

-10

10

8

10

10

10

12

10

14

10

16

10

18

10

Frequency f (Hz)

Fig. 4.3. Frequency dispersion of the imaginary part of the relative dielectric permittivity data for triply distilled water at 25o C. (Numerical data supplied by the School of Aerospace Medicine, Armstrong Laboratory, Brooks Air Force Base.)

4.4 Causal Models of the Material Dispersion

195

A dielectric material of central importance to both electromagnetics and optics is water [30–34] as it is so prevalent in our natural environment. Representative data points for the computed frequency dispersion of the real and imaginary parts of the dielectric permittivity of triply distilled water along the positive real frequency axis are presented in Figures 4.2 and 4.3, respectively, where f = ω /2π is the oscillation frequency in Hz. These computed data points have been obtained from representative experimental data points for the real index of refraction nr (ω ) ≡ {n(ω )} and attenuation coeﬃcient α(ω ) ≡ (ω /c)ni (ω ) of triply distilled water at 25o C with )} denoting the imaginary part of the complex index of reni (ω ) ≡ {n(ω fraction n(ω) ≡ (µ/µ0 )( (ω)/ 0 ), where µ/µ0 = 1 for triply-distilled water. At zero frequency, i (0) = 0 and [because (0) = r (0)] ∞

i (ω ) 2 dω , (4.161)

(0) = 0 + P π ω 0 so that the static permittivity is given by the weighted sum of the contributions from all of the absorption mechanisms in the medium. Because of the Kramers–Kronig relations [cf. Eqs. 4.100)–(4.105)], each instance of rapid variation in r (ω ) over some frequency interval is accompanied by a peak in i (ω ) in that same frequency interval. The low frequency dependence of <

r (ω ) depicted in Figure 4.2 for 0 ≤ f ∼ 1013 Hz is characteristic of rotational polarization phenomena in polar dielectrics, whereas the high frequency de> pendence for f ∼ 1013 Hz is more characteristic of resonance polarization phenomena, ﬁrst molecular (due to vibrational modes through the infrared region of the spectrum) and then atomic (due to electronic modes through the ultraviolet region of the spectrum). Additional polarization modes due to atomic core electrons may also appear in the x-ray region of the spectrum as relatively weak variations in the dielectric permittivity [35]. Similar remarks apply to the frequency dependence of the imaginary part of the dielectric permittivity depicted in Figure 4.3. As the frequency increases above zero, a critical frequency value is reached at which the permanent molecular dipoles of the material can no longer follow the oscillations of the driving wave ﬁeld and the real permittivity rapidly de(0) creases from its near static value (0) = (0) to a value V that is given by the low-frequency limit of the combined vibrational and electronic polarization modes of the material. As the frequency f increases through each vibrational polarization mode of the material, the dielectric permittivity exhibits charac(0) teristic resonance responses as r (ω ) approaches a value e that is given by the low-frequency limit of the electronic polarization modes of the material. As f increases further through each electronic polarization mode of the material, the dielectric permittivity exhibits characteristic resonance responses as the dielectric permittivity approaches the high-frequency limits r → 1,

i → 0 as f → ∞.

196

4 Macroscopic Electromagnetics

4.4.1 The Lorentz–Lorenz Relation Because appropriate physical models for the material response originate at the microscopic level, the relationship between the electromagnetic response at the molecular level and its macroscopic equivalent needs to be established. The macroscopic polarization P(r, t) due to a material comprised of molecular species labeled by the index j with number densities Nj is given by P(r, t) = Nj pj (r, t), (4.162) j

where pj (r, t) is the macroscopic spatial average of the microscopic molecular dipole moment, given by the causal linear relation t pj (r, t) = α ˆ j (t − t )Eeff (r, t )dt . (4.163) −∞

Here Eeff (r, t) denotes the eﬀective electric ﬁeld at the local molecular site. The temporal Fourier transform of Eq. (4.163) gives ˜ eff (r, ω), ˜ pj (r, ω) = αj (ω)E

(4.164)

where αj (ω) is the mean polarizability for the j-type molecules at the angular frequency ω. The term “mean” used here implies a spatial average over molecular sites. For a simple polarizable medium, the spatial average of the eﬀective electric ﬁeld is given by (see Appendix D) Eeff (r, t) = E(r, t) +

4π P(r, t), 3 0

(4.165)

where E(r, t) is the external, applied electric ﬁeld. Substitution of Eq. (4.164) with the temporal Fourier transform of Eq. (4.165) into Eq. (4.162) then yields the expression j Nj αj (ω) ˜ ˜ ω) P(r, ω) = E(r, (4.166) 1 − (4π/3 0 ) j Nj αj (ω) relating the macroscopic induced polarization to the external, applied electric ﬁeld. Comparison of this expression with that given in Eq. (4.89) shows that the electric susceptibility of the material is given by j Nj αj (ω) , (4.167) χe (ω) =

0 − (4π/3) j Nj αj (ω) so that the dielectric permittivity (ω) = 0 (1 + 4πχe (ω)) is given by

4.4 Causal Models of the Material Dispersion

(ω) = 0 1 + 4π

197

Nj αj (ω) .

0 − (4π/3) j Nj αj (ω) j

(4.168)

Upon solving this relation for the summation over the molecular polarizabilities of the medium, one ﬁnally obtains the Lorentz–Lorenz formula [36, 37]

Nj αj (ω) =

j

3 0 (ω)/ 0 − 1 , 4π (ω)/ 0 + 2

(4.169)

which is also referred to as the Clausius–Mossotti relation.5 This simple relation provides the required connection between the phenomenological macroscopic Maxwell theory and the microscopic atomic theory of matter [39]. 4.4.2 The Debye Model of Orientational Polarization The microscopic orientational (or dipolar) polarization due to permanent molecular dipole moments may be described by the relaxation equation due to Debye [26] dp(r, t) 1 p(r, t) = aEeff (r, t), (4.170) + dt τm where a is a constant in time and where τm denotes the characteristic exponential relaxation time of the molecular dipole moment in the absence of an externally applied electromagnetic ﬁeld. Rotational Brownian motion theory [27] shows that the dipolar relaxation time is given by τm =

ζ , 2KB T

(4.171)

where KB is Boltzmann’s constant, T is the absolute temperature, and where ζ is a frictional constant describing the resistance to dipolar rotation in the medium. The temporal frequency transform of the spatial average of Eq. (4.170) then yields ˜ p(r, ω) =

aτm ˜ eff (r, ω). E 1 − iωτ

(4.172)

Comparison of this expression with that given in Eq. (4.164) then shows that the molecular polarizability for the Debye model is given by α(ω) =

aτm . 1 − iωτm

(4.173)

With this expression substituted in Eq. (4.167), the electric susceptibility is found to be given by 5

Lorentz [3] attributed Eq. (4.169) with j = 1 to the earlier work by R. Clausius (1879) and O. F. Mossotti (1850); see p. 50 of [38].

198

4 Macroscopic Electromagnetics

χe (ω) =

1 N aτ ,

0 1 − iωτ

where τ ≡ τm /(1 − (4π/3 0 )N aτm ), so that the dielectric permittivity is

(ω) = 0 + 4π

N aτ . 1 − iωτ

Evaluation of this expression at ω = 0 then shows that 1 s − 0 , 4π N τ

a=

(4.174)

where s ≡ (0) denotes the static permittivity of the medium. With this substitution, the single relaxation time Debye model susceptibility and permittivity expressions become 1 s / 0 − 1 , 4π 1 − iωτ

s / 0 − 1

(ω) = 0 1 + , 1 − iωτ

χe (ω) =

where τ≡

s / 0 + 2 τm 3

(4.175) (4.176)

(4.177)

is the eﬀective relaxation time. When considered as a function of complex ω = ω + iω , the Debye model dielectric permittivity given in Eq. (4.176) is found to have a single simple pole singularity at i (4.178) ωp ≡ − τ along the imaginary axis in the lower half of the complex ω−plane. The Debye model dielectric permittivity is then analytic in the upper-half of the complex ω−plane. In addition,

(ω ) = lim

(ω + iω ) ω →0

for all ω . Finally, because 2

| (ω)/ 0 − 1| =

( s / 0 − 1)2 1 + 2τ ω + τ 2 |ω|2

is integrable over any line parallel to the ω −axis for all ω > 0, condition 2 of Titchmarsh’s theorem is then satisﬁed. The Debye model permittivity is then causal. A simple generalization of the Debye model expression given in Eq. (4.176) in order to reﬂect the inﬂuence of higher frequency (ω 1/τ ) polarization mechanisms on the lower frequency behavior is given by

4.4 Causal Models of the Material Dispersion

(ω)/ 0 = ∞ +

sr − ∞ , 1 − iωτ

199

(4.179)

where ∞ ≥ 1 denotes the large frequency limit of the relative dielectric permittivity due to the Debye model alone. The real and imaginary parts of this expression are then given by

sr − ∞ , 1 + τ 2 ω2 τω ,

i (ω)/ 0 = ( sr − ∞ ) 1 + τ 2 ω2

r (ω)/ 0 = ∞ +

so that 2 2 2

r (ω)/ 0 − ( sr + ∞ )/2 + i (ω)/ 0 = ( sr − ∞ )/2 .

(4.180)

The locus of points ( r (ω), i (ω)) then lie on a semicircle with center along the −axis at = ( sr + ∞ )/2 and radius ( sr − ∞ )/2, as illustrated by the solid curve in Figure 4.4 for triply distilled water. The numerical data for triply distilled water presented in Figures 4.2–4.3 is also presented by the + signs in this ﬁgure, where increasing angular frequency values progress from right to left (notice the high-frequency structure near the origin of this plot that is due to resonance polarization contributions). Such a graphical representation is referred to as a Cole–Cole plot [28]. From Eq. (4.180), the maximum value of i (ω)/ 0 occurs when i (ω)/ 0 = ( sr + ∞ )/2 which in turn occurs when ωτ = 1. The relaxation time τ for a single relaxation time Debye model may then be estimated from the experimentally determined angular frequency value at which i (ω) reaches a maximium. The Cole–Cole plot readily shows whether the experimental data points for a given material can be described either by a single relaxation time, multiple relaxation times, or a distribution of relaxation times. In order to account for additional orientational polarization modes as well as for any polarization mechanisms at higher frequencies (due, for example, to resonance polarization eﬀects) the Debye model expression given in Eq. (4.179) may be generalized as aj

(ω)/ 0 = ∞ + , (4.181) 1 − iωτj j where ∞ is the large frequency limit of the relative permittivity due to orientational polarization eﬀects alone, and where aj = sr − ∞ . (4.182) j

with sr ≡ s / 0 denoting the relative static permittivity of the material. The summation appearing in the above two equations is taken over each

200

4 Macroscopic Electromagnetics 50

45

40

ΕiΩIΕΩ

35

30

25

20

15

10

5

0

0

10

20

30

40

50

60

70

ΕrΩRΕΩ

Fig. 4.4. Cole–Cole plot of the single (solid curve) and double (dashed curve) relaxation time Debye models compared with the numerical data (+ signs) for the real and imaginary parts of the relative dielectric permittivity of triply distilled water at 25o C. The single relaxation time Debye model parameters used here are ∞ = 3.1, sr = 79.0, a = 75.9, and τ = 7.88 × 10−12 s, and the double relaxation time parameters are ∞ = 1.7, a1 = 74.7, τ1 = 8.44 × 10−12 s, a2 = 2.8, and τ2 = 4.77 × 10−14 s. The point marked with an × along the abscissa locates the center of the semicircular Cole–Cole plot for the single relaxation time Debye model.

orientational mode present in the material, each mode characterized by its individual relaxation time τj and strength aj . The resultant model is causal. Its Cole–Cole plot for triply distilled water, represented by the dashed curve in Figure 4.4, possesses a better ﬁt to the numerical data at the higher frequency values while maintaining the accuracy at very low frequencies. The frequency dependence of a double relaxation time Debye model that has been numerically ﬁtted to the numerical data for the relative dielectric permittivity of triply distilled water (see Figs. 4.2–4.3) is presented in Figure 4.5 over the frequency domain from f = 100 MHz to f = 1 THz. The frequency behavior illustrated here is typical of orientational polarization phenomena in polar dielectrics. Discrepancies between the model curves and data points may be reduced by employing appropriate extensions of the Debye model.

4.4 Causal Models of the Material Dispersion

201

Real and Imaginary Parts of the Dielectric Permittivity

90

80

70

ΕrΩ

60

50

40

30

ΕiΩ

20

10

0 8 10

9

10

10

10

11

10

12

10

Frequency f (Hz)

Fig. 4.5. Comparison of the double relaxation time Debye model results with ∞ = 1.7, a1 = 74.7, τ1 = 8.44 × 10−12 s, a2 = 2.8, and τ2 = 4.77 × 10−14 s (solid curves) and numerical data (+ signs) for the real and imaginary parts of the relative dielectric permittivity of triply distilled water at 25o C.

4.4.3 Generalizations of the Debye Model A major shortcoming of the Debye model is that it yields a nonvanishing absorption coeﬃcient at suﬃciently high frequencies such that ω 1/τ1 , where it is assumed here that the relaxation times τj are ordered in decreasing value. This is clearly evident in Figure 4.6 which presents a comparison of the double relaxation time Debye model curve for the imaginary part of the dielectric permittivity with the numerical data for water over the entire frequency domain of interest here. Notice that the entire visible window for water has been eﬀectively cut oﬀ by the Debye model curve. This then implies that the early time (t τ ) spontaneous relaxation of orientational molecular polarization cannot be exponential in time. Furthermore, because such an exponential time decay can result only from a macroscopic averaging process taken over the collisions between a polar molecule and the random thermal motion of the surrounding molecules, it then becomes necessary to include the eﬀects of rotational Brownian motion in the dynamical theory [27]. The complex molecular polarizability including, to ﬁrst-order, the eﬀects of rotational Brownian motion on the dipolar molecular relaxation process, is given by [27]

202

4 Macroscopic Electromagnetics

α(ω) =

aτm . (1 − iωτm )(1 − iωτmf )

(4.183)

This ﬁrst-order correction to the Debye model expression appearing in Eq. (4.173) was ﬁrst given by Rocard [40] and Powles [41] through the inclusion of inertial eﬀects in the Debye model. Here τmf =

I ζ

(4.184)

is the associated friction time, where I is the moment of inertia of the polar molecule and ζ is the same frictional constant appearing in Eq. (4.171) for the dipolar relaxation time. With this substitution in Eqs. (4.167)–(4.168), the single relaxation time Rocard–Powles–Debye model susceptibility and permittivity expressions become

s / 0 − 1 1 , 4π (1 − iωτ )(1 − iωτf )

s / 0 − 1 ,

(ω) = 0 1 + (1 − iωτ )(1 − iωτf )

χe (ω) =

(4.185) (4.186)

where the eﬀective relaxation time τ is given in Eq. (4.177), and where τf denotes an eﬀective friction time for the model. When considered as a function of complex ω = ω + iω , the Rocard– Powles–Debye model dielectric permittivity given in Eq. (4.186) is found to have isolated simple pole singularities at i ωp1 ≡ − , τ

ωp2 ≡ −

i , τf

(4.187)

which are situated along the imaginary axis in the lower-half of the complex ω−plane. The Rocard–Powles–Debye model dielectric permittivity is then analytic in the upper-half of the complex ω-plane. In addition,

(ω ) = lim

(ω + iω ) ω →0

for all ω . Finally, because 2

| (ω)/ 0 − 1| =

(1 +

2τ ω

+

( s / 0 − 1)2 2 τ |ω|2 )(1 + 2τ

fω

+ τf2 |ω|2 )

is integrable over any line parallel to the ω -axis for all ω > 0, condition 2 of Titchmarsh’s theorem is then satisﬁed. The Rocard–Powles–Debye model permittivity is then causal. As in Eq. (4.179), the Rocard–Powles model expression (4.186) may be generalized as

sr − ∞

(ω)/ 0 = ∞ + (4.188) (1 − iωτ )(1 − iωτf )

4.4 Causal Models of the Material Dispersion

203

in order to account for the inﬂuence of higher frequency (ω 1/τ ) polarization mechanisms on the lower frequency behavior, where ∞ ≥ 1 denotes the large frequency limit of the relative dielectric permittivity due to the Rocard– Powles–Debye model alone. The real and imaginary parts of this expression are then given by

r (ω)/ 0 = ∞ + ( sr − ∞ )

i (ω)/ 0 = ( sr − ∞ )

(1 − τ τf ω 2 ) , (1 + τ 2 ω 2 )(1 + τf2 ω 2 )

(τ + τf )ω , (1 + τ 2 ω 2 )(1 + τf2 ω 2 )

so that 2 2 2

r (ω)/ 0 − ( sr + ∞ )/2 + i (ω)/ 0 ≤ ( sr − ∞ )/2 ,

(4.189)

where the equality results when τf = 0. The locus of points ( r (ω), i (ω)) for the Rocard–Powles extension then lies within the semicircle of the Cole–Cole plot for the Debye model limit (as τf → 0). In order to account for additional orientational polarization modes as well as for any polarization mechanisms at higher frequencies, the Rocard– Powles–Debye model expression given in Eq. (4.188) may be generalized as aj (4.190)

(ω)/ 0 = ∞ + (1 − iωτ )(1 − iωτf j ) j j with the sum rule given in Eq. (4.182), where ∞ is the large frequency limit of the relative permittivity due to orientational polarization eﬀects alone. The summation appearing here is taken over each orientational mode present in the material, each mode characterized by its individual relaxation time τj , friction time τf j , and strength aj . The frequency dependence of a double relaxation time Rocard–Powles– Debye model that has been numerically ﬁtted to the numerical data for the relative dielectric permittivity of triply distilled water [33] is practically indistinguishable from that presented in Figure 4.5 for the Debye model over the frequency domain from f = 100 MHz to f = 1 THz. However, for sufﬁciently higher frequencies (ω 1/τ1 ) the Rocard–Powles extension of the Debye model results in an expression for the imaginary part of the dielectric permittivity that decreases as 1/ω 3 as compared to 1/ω for the Debye model, as evident in Figure 4.6. Although the Rocard–Powles extension of the Debye model is a signiﬁcant improvement over the classical Debye model over the high-frequency domain ω 1/τ1 , its 1/ω 3 fall-oﬀ is still insuﬃcient to correctly model the visible window in water. Notice that the shallower windows in the infrared are also eﬀectively cut oﬀ by the Rocard–Powles–Debye model, as seen in a closer inspection of Figure 4.6. A further generalization of the Debye model, as well as of the Rocard– Powles extension of the Debye model, is obtained through the introduction of

204

4 Macroscopic Electromagnetics 2

10

0

10

-2

ΕiΩIΕΩ

10

Cole-Cole Extension of the Rocard-PowlesDebye Model

-4

10

Debye Model

-6

10

-8

10

Rocard-PowlesDebye Model

-10

10

8

10

10

10

12

14

10

10

16

10

18

10

Frequency f (Hz)

Fig. 4.6. Comparison of the double relaxation time Debye, Rocard–Powles–Debye, and Cole–Cole extension of the Rocard–Powles–Debye model results (solid curves) with numerical data (+ signs) for the imaginary part of the relative dielectric permittivity of triply distilled water at 25o C. The model parameters used here are ∞ = 2.1, a1 = 74.1, τ1 = 8.44 × 10−12 s, τf 1 = 4.62 × 10−14 s, a2 = 3.0, τ2 = 6.53 × 10−14 s, and τf 2 = 1.43 × 10−15 s with ν1 = 0 and ν2 = −0.5.

a continuous distribution of relaxation times. As introduced by Fr¨ ohlich [29] for the Debye model permittivity and generalized here to the Rocard–Powles– Debye model, ∞ f (τ )

(ω)/ 0 − ∞ = dτ, (4.191) (1 − iωτ )(1 − iωτf ) 0 where τf = γ(τ ) [27]. The function f (τ ) is the distribution function for the relaxation time. Evaluation of this expression at ω = 0 gives ∞

sr − ∞ = f (τ )dτ (4.192) 0

so that the quantity f (τ )dτ measures the relative contribution to the permittivity diﬀerence sr − ∞ from those components of the dielectric response with relaxation times in the interval [τ, τ + dτ ], where sr ≡ (0)/ 0 is the relative static permittivity and ∞ is the high frequency limit of the relative permittivity due to orientational polarization eﬀects alone. Because the

4.4 Causal Models of the Material Dispersion

205

distribution function f (τ ) is nonnegative for all τ ∈ [0, ∞], i (ω) is then comprised of a superposition of absorption curves with varying positions of their corresponding maxima. The half-width of the resulting absorption curve is then broadened from that for a single Rocard–Powles–Debye model. This distribution can also result in a multiplicity of absorption peaks. The inversion of Eq. (4.191) with τf = 0 when (ω)/ 0 is described by an analytic function of ω has been studied by Stieltjes, as ascribed by Titchmarsh [42]. Fr¨ ohlich [43] introduced the distribution function fF (τ ) ≡

sr − ∞ , ln(τ2 /τ1 )τ

τ1 < τ < τ2

(4.193)

that incorporates a ﬁnite range of relaxation times between τ1 and τ2 and is zero elsewhere. Substitution of this expression in Eq. (4.191) with τf independent of τ then results in the expression [44]

τ2 (1 + (ωτ1 )2 )1/2

sr − ∞ ln

(ω)/ 0 = ∞ + ln(τ2 /τ1 )(1 − iωτf ) τ1 (1 + (ωτ2 )2 )1/2

+ i (arctan(ωτ2 ) − arctan(ωτ1 )) . (4.194) Because arctan(ωτ ) → π/2 as ωτ → ∞, the attenuation at high frequencies (ω 1/τ ) for the Fr¨ ohlich distribution function with τf = 0 is due solely to the diﬀerence between two similar curves. Perhaps a more appropriate distribution function is that due to Cole and Cole [28], given by fCC (τ ) ≡ ( sr − ∞ )

(τ /τ1

)1−ν

sin(πν)/(πτ ) , − 2 cos(πν) + (τ1 /τ )1−ν

(4.195)

which is continuous for all τ > 0. Substitution of this expression in Eq. (4.191) with τf independent of τ results in the expression, [44]

(ω)/ 0 = ∞ +

sr − ∞ (1 − iωτ1 )1−ν (1 − iωτf )

(4.196)

with ν < 1. When considered as a function of complex ω = ω + iω , the Cole–Cole extension (4.196) of the Rocard–Powles–Debye model dielectric permittivity with a noninteger value of ν is found to have a branch point singularity ωb and a simple pole singularity ωp at ωb ≡ −

i , τ1

ωp ≡ −

i , τf

(4.197)

respectively, which are situated along the negative imaginary axis in the lower-half of the complex ω-plane. For a negative integer value of ν, the

206

4 Macroscopic Electromagnetics

branch point ωb becomes a pole singularity and the causality analysis following Eq. (4.187) applies. With a branch cut chosen along the negative imaginary axis extending from ωb to −i∞, the Cole–Cole model dielectric permittivity is then analytic in the upper-half of the complex ω-plane. In addition,

(ω + iω )

(ω ) = lim ω →0

for all ω . Finally, because 2

| (ω)/ 0 − ∞ | =

( sr − ∞ )2 (1 + 2τ1 ω + τ12 |ω|2 )1−ν (1 + 2τf ω + τf2 |ω|2 )

is integrable over any line parallel to the ω -axis for all ω > 0 when ν < 1, condition 2 of Titchmarsh’s theorem is then satisﬁed. The Cole–Cole extension of the Rocard–Powles–Debye model permittivity given in Eq. (4.196) is then causal. Notice that the Cole–Cole model is typically stated [44] as being restricted to values of ν satisfying the inequality 0 ≤ ν < 1; however, the lower limit does not appear to be necessary and is consequently omitted here. The generalization of the Cole–Cole expression given in Eq. (4.196) in order to account for additional distributions of orientational polarization modes as well as for any higher frequency polarization mechanisms is given by [cf. Eq. (4.190)] aj (4.198)

(ω)/ 0 = ∞ + 1−ν (1 − iωτj ) j (1 − iωτf j ) j together with the sum rule given in Eq. (4.182). The summation appearing here is taken over each orientational mode distribution present in the material, each mode characterized by a representative relaxation time τj , friction time τf j , strength aj , and distribution parameter νj with νj < 1. The frequency dependence of a double relaxation time Cole–Cole extended Rocard–Powles–Debye model with ν1 = 0 and ν2 = −0.5, viz.

(ω)/ 0 = ∞ +

a2 a1 + , (4.199) (1 − iωτ1 )(1 − iωτf 1 ) (1 − iωτ2 )3/2 (1 − iωτf 2 )

that has been numerically ﬁtted to the numerical data for the relative dielectric permittivity of triply distilled water is practically indistinguishable from that presented in Figure 4.5 for the Debye model for 100 MHz ≤ f ≤ 1 THz. However, for suﬃciently higher frequencies (ω 1/τ1 ), this Cole–Cole model extension results in an expression for the imaginary part of the dielectric permittivity that decreases as 1/ω 4 as compared to 1/ω 3 for the Rocard– Powles–Debye model alone, as evident in Figure 4.6. The visible window in water, as well as several secondary windows at lower frequencies, are now no longer cut oﬀ by lower frequency orientational polarization eﬀects.

4.4 Causal Models of the Material Dispersion

207

4.4.4 The Classical Lorentz Model of Resonance Polarization The classical Lorentz model [3] of resonance polarization phenomena in dielectrics describes the material as a collection of neutral atoms with elastically bound electrons to the nucleus (i.e., as a collection of Lorentz oscillators), where each electron is bound to the nucleus by a Hooke’s law restoring force. Under the action of an applied electromagnetic ﬁeld, the equation of motion of a typical bound electron is given by 2 drj d rj 2 + 2δj (4.200) m + ωj rj = −qe Eeff , dt2 dt where m is the mass of the electron and qe is the magnitude of the electronic charge. The quantity ωj is the undamped resonance angular frequency of the jth oscillator type and δj is the associated phenomenological damping constant of the oscillator. The same dynamical equation of motion also applies to molecular vibration modes when m is the ionic mass and ω0 is the undamped resonance frequency of the transverse vibrational mode of the ionic lattice structure [35]. The ﬁeld Eeff = Eeff (r, t) is the eﬀective local electric ﬁeld intensity that acts on the electron (or ion) as a driving force. The additional force −1/cqe vj × Beff arising from the interaction of the bound electron (or ion) with the eﬀective local magnetic ﬁeld is assumed to be negligible in comparison to the electric ﬁeld interaction (due to the smallness of the velocity of the electron in comparison with the vacuum speed of light c) and is consequently neglected in the classical theory; its eﬀect is considered in Appendix E. Finally, the term 2mδj drj /dt represents a phenomenological damping term for the electronic motion of the jth Lorentz oscillator type. The actual loss mechanism for a free atom is radiation damping [45], but it arises from a variety of scattering mechanisms in both solid and liquids [35]. The temporal frequency transform of Eq. (4.200) directly yields the frequency-domain solution ˜rj (r, ω) =

ω2

qe /m ˜ eff (r, ω), E − ωj2 + 2iδj ω

(4.201)

˜ j = −qe ˜rj is then given by and the local induced dipole moment p ˜ j (r, ω) = p

ω2

−qe2 /m ˜ eff (r, ω). E − ωj2 + 2iδj ω

(4.202)

If there are Nj Lorentz oscillators per unit volume that are characterized by the undamped resonance frequencies ωj and damping constants δj , then the macroscopic polarization induced in the medium is given by the summation over all oscillator types of the spatially averaged locally induced dipole moments as

208

4 Macroscopic Electromagnetics

˜ ω) = P(r,

j

,, =

Nj ˜ pj (r, ω)

-- ˜ eff (r, ω) Nj αj (ω). E

(4.203)

j

Here αj (ω) =

−qe2 /m ω 2 − ωj2 + 2iδj ω

(4.204)

is the atomic polarizability of the Lorentz oscillator type characterized by ωj and δj with number density Nj . In addition, N = j Nj is the total number of electrons per unit volume that interact with the eﬀective local applied electromagnetic ﬁeld. Substitution of this result into the Lorentz– Lorenz formula (4.169) then yields the expression [see Eq. (4.168)] 1 − (2/3) j b2j /(ω 2 − ωj2 + 2iδj ω) (4.205)

(ω)/ 0 = 1 + (1/3) j b2j /(ω 2 − ωj2 + 2iδj ω) for the relative dielectric permittivity, where . 4π Nj qe2 bj ≡

0 m

(4.206)

is the plasma frequency with number density Nj . When the inequality b2j /(6δj ωj ) 1 is satisﬁed, the denominator in Eq. (4.205) may be approximated by the ﬁrst two terms in its power series expansion, so that ⎞⎛ ⎞ ⎛ b2j b2j 1 2 ⎠ ⎝1 − ⎠

(ω)/ 0 ≈ ⎝1 − 3 j ω 2 − ωj2 + 2iδj ω 3 j ω 2 − ωj2 + 2iδj ω ≈ 1−

j

b2j , ω 2 − ωj2 + 2iδj ω

(4.207)

which is the usual expression [3, 39, 46] for the frequency dispersion of a classical Lorentz model dielectric. As an example, consider the single resonance Lorentz√ model material parameters ω0 = 4 × 1016 r/s, δ0 = 0.28 × 1016 r/s, b0 = 20 × 1016 r/s that were chosen by Brillouin [47, 48] in his now classic analysis of dispersive signal propagation. These values correspond to a highly absorptive dielectric. Furthermore, because b20 /(6δ0 ω0 ) = 2.976, the approximation given in Eq. (4.207) does not strictly apply and the Lorentz–Lorenz modiﬁed expression (4.205) must be used for these particular model parameters. The angular frequency dispersion of the relative dielectric permittivity for the Lorentz model alone is illustrated by the dashed curves in Figure 4.7 for

4.4 Causal Models of the Material Dispersion

209

8

b2 = 20X1032r /s2 2

6

ΕrΩRΕΩ

4

2

b2 = 2X1032r /s2 2

0

-2

-4

-6

0

1

2

3

4

5

6

7

8

9

Ω - r/s

10 16

x 10

Fig. 4.7. Angular frequency dependence of the real part of the relative dielectric permittivity for a single-resonance Lorentz model dielectric with (solid curves) and without (dashed curves) the Lorentz–Lorenz formula correction for two diﬀerent values of the material plasma frequency. 12

10

b2 = 20X1032r /s2 2

ΕiΩIΕΩ

8

6

4

2

b2 = 2X1032r2/s2 0

0

1

2

3

4

5

Ω - r/s

6

7

8

9

10 16

x 10

Fig. 4.8. Angular frequency dependence of the imaginary part of the relative dielectric permittivity for a single-resonance Lorentz model dielectric with (solid curves) and without (dashed curves) the Lorentz–Lorenz formula correction for two diﬀerent values of the material plasma frequency.

210

4 Macroscopic Electromagnetics

the real part r (ω)/ 0 = { (ω)/ 0 } and in Figure 4.8 for the imaginary part i (ω)/ 0 = { (ω)/ 0 }. The corresponding solid curves in this pair of ﬁgures describe the resultant frequency dispersion for this Lorentz model dielectric when the Lorentz–Lorenz relation is used, as given by Eq. (4.205). The Lorentz–Lorenz modiﬁed material dispersion is seen to primarily shift the resonance frequency to a lower value while increasing both the absorption and the below resonance real dielectric permittivity. If the plasma frequency √ is decreased to the value b0 = 2 × 1016 r/s so that b20 /(6δ0 ω0 ) = 0.2976, then the modiﬁcation of the Lorentz model by the Lorentz–Lorenz relation is relatively small (and the dispersion is also weak), as exhibited by the second set of curves in Figures 4.7–4.8. The approximate expression given in Eq. (4.207) may then be used in this latter case, but not in the former. Because the primary eﬀect of the Lorentz–Lorenz formula on the Lorentz model is to downshift the eﬀective resonance frequency and increase the static permittivity value, consider then determining the resonance frequency ω∗ appearing in the Lorentz–Lorenz formula for a single-resonance Lorentz model dielectric that will yield the same value for s / 0 ≡ (0)/ 0 as that given by the Lorentz model alone with resonance frequency ω0 . Equations (4.205) and (4.207) then result in the equivalence relation [49, 50] 1 + b20 /ω02 = with solution ω∗ =

1 + (2/3)(b20 /ω∗2 ) 1 − (1/3)(b20 /ω∗2 ) / ω02 + b20 /3.

(4.208)

This equivalence relation then allows the construction of a dielectric model satisfying the Lorentz–Lorenz relation with a resonance frequency ω∗ that is upshifted to match a given Lorentz model resonance frequency ω0 . The branch points of the complex index of refraction n(ω) ≡ (ω)/ 0 for a single-resonance Lorentz model dielctric with relative magnetic permeability µ/µ0 = 1 and dielectric permittivity described by Eq. (4.207) are given by / / ωz± = −iδ0 ± ω02 + b20 − δ02 , ωp± = −iδ0 ± ω02 − δ02 , and the branch points of the complex index of refraction for the Lorentz– Lorenz modiﬁed Lorentz model dielectric with dielectric permittivity described by Eq. (4.205) are given by / / ωp± = −iδ0 ± ω∗2 − b20 /3 − δ02 , ωz± = −iδ0 ± ω∗2 + 2b20 /3 − δ02 . If ω∗ = ω0 , then the branch points ωp± and ωz± of n(ω) for the Lorentz–Lorenz modiﬁed Lorentz model are shifted inward toward the imaginary axis from the respective branch point locations ωp± and ωz± for the Lorentz model alone

4.4 Causal Models of the Material Dispersion

211

3

2.5

nr(Ω)

2

1.5

1

0.5

0

0

1

2

3

4

5

6

7

8

9

Ω - r/s

10 16

x 10

Fig. 4.9. Comparison of the angular frequency dependence of the real part of the complex index of refraction n(ω) = (ω)/ 0 for a single-resonance Lorentz model dielectric alone (solid curve) and for the equivalent Lorentz–Lorenz formula modiﬁed Lorentz model (circles). 2.5

2

ni(Ω)

1.5

1

0.5

0

0

1

2

3

4

5

Ω - r/s

6

7

8

9

10 16

x 10

Fig. 4.10. Comparison of the angular frequency dependence of the imaginary part of the complex index of refraction n(ω) = (ω)/ 0 for a single-resonance Lorentz model dielectric alone (solid curve) and for the equivalent Lorentz–Lorenz formula modiﬁed Lorentz model (circles).

212

4 Macroscopic Electromagnetics

provided that the inequality ω∗2 − b20 /3 − δ02 ≥ 0 is satisﬁed. If the opposite inequality ω∗2 − b20 /3 − δ02 < 0 is satisﬁed, then the branch points ωp± are located along the imaginary axis. However, if ω∗ is given by the equivalence relation (4.208), then the locations of the branch points of the complex index of refraction n(ω) for the Lorentz–Lorenz modiﬁed Lorentz model and the Lorentz model alone are exactly the same. It then follows that the analyticity properties for these two models of the dielectric permittivity are also the same. A comparison of the angular frequency dependence of the real and imaginary parts of the complex index of refraction n(ω) = (ω)/ 0 with dielectric permittivity described by Eqs. (4.205) and (4.207) for a single-resonance Lorentz model dielectric with ω∗ given by the equivalence relation (4.208) is presented in Figures 4.9 and 4.10 for Brillouin’s choice √ of the material parameters (ω0 = 4 × 1016 r/s, δ0 = 0.28 × 1016 r/s, b0 = 20 × 1016 r/s). The rms error between the two sets of data points presnted here is approximately 2.3 × 10−16 for the real part and 2.0 × 10−16 for the imaginary part of the complex index of refraction, with a maximum single point rms error of ≈ 2.5 × 10−16 . The corresponding rms error for the relative dielectric permittivity is ≈ 1.1×10−15 for both the real and imaginary parts with a maximum single point rms error of ≈ 1 × 10−14 . Variation of any of the remaining material parameters in the equivalent Lorentz–Lorenz modiﬁed Lorentz model dielectric, including the value of the plasma frequency from that speciﬁed in Eq. (4.208), only results in an increase in the rms error. This approximate equivalence relation between the Lorentz–Lorenz formula modiﬁed Lorentz model dielectric and the Lorentz model dielectric alone is then seen to provide a “best ﬁt” in the rms sense between the frequency dependence of the two models. Finally, for a multiple-resonance Lorentz model dielectric, the equivalence relation (4.208) becomes / ωj∗ = ωj2 + b2j /3 (4.209) for each resonance line j present in the dielectric. The behavior of the real and imaginary parts of the complex index of refraction along the positive real frequency axis illustrated in Figures 4.9 and 4.10 is typical of a single resonance Lorentz dielectric. The frequency regions wherein the real index of refraction nr (ω) ≡ {n(ω)} increases with increasing ω [i.e., where nr (ω) has a positive slope] are termed normally dispersive, whereas the region where nr (ω) decreases with increasing ω [i.e., where nr (ω) has a negative slope] is said to exhibit anomalous dispersion. Notice that the real index of refraction nr (ω) varies rapidly with ω within the region of anomalous dispersion and that this region essentially coincides with the region of strong absorption for the medium, the angular frequency width of the absorption line increasing with increasing values of the phenomenological damping constant δ0 .

4.4 Causal Models of the Material Dispersion

213

When considered as a function of complex ω = ω + iω , the Lorentz model dielectric permittivity

(ω)/ 0 = 1 −

b20 ω 2 − ω02 + 2iδ0 ω

is found to have isolated simple pole singularities at / ωp± ≡ −iδ0 ± ω02 − δ02 ,

(4.210)

(4.211)

which are symmetrically located about the imaginary axis in the lower halfplane. The Lorentz model dielectric permittivity is then analytic in the upperhalf plane. In addition,

(ω + iω )

(ω ) = lim ω →0

for all ω . Finally, because | (ω)/ 0 − 1| =

(ω 2

+

ω 2 )2

−

2ω02 (ω 2

−

ω 2 )

b40 + ω04 + 4δ0 ω (ω 2 + ω 2 ) + 4δ0 ω02 ω

is integrable over any line parallel to the ω -axis for all ω > 0, condition 2 of Titchmarsh’s theorem is then satisﬁed. The Lorentz model of resonance polarization in dielectrics is then causal. From a quantum mechanical approach [51] it is found that the dielectric response function of a multiple resonance dielectric material is given by

(ω)/ 0 = 1 − 4π

fj N qe2 . m j ω 2 − ωj2 + 2iγj ω

(4.212)

Although there is a formal similarity between this equation and the classical Lorentz model expression given in Eq. (4.207), the physical interpretation of some of the terms is quite diﬀerent. Classically, ωj is the undamped resonance frequency of a bound electron, whereas quantum mechanically it is the transition frequency of a bound electron between two atomic states that are separated in energy by the amount h ¯ ωj , where ¯h ≡ h/2π, h denoting Planck’s constant. The parameter fj , called the oscillator strength, is a measure of the relative probability of a quantum mechanical transition taking place with energy change ¯hωj , which satisﬁes the sum rule [compare with the sum rule N = j Nj following Eq. (4.204)] fj = 1. (4.213) j

Finally, the quantity δj in the classical theory is a phenomenological damping constant that appears in the damping term 2δj r˙ j of the equation of

214

4 Macroscopic Electromagnetics

motion (4.200). This term represents the loss of energy due to electromagnetic radiation, and this is the classical analogue of spontaneous emission in the quantum theory [51]. Consequently, although the physical theory upon which the Lorentz model is based is rather simplistic, the proper functional form of the dielectric response is obtained for dielectric media. The model parameters bj , ωj , and δj cannot be obtained from the classical theory, but they can be obtained by ﬁtting them to experimental measurements of the dispersive dielectric under consideration. 4.4.5 Composite Model of the Dielectric Permittivity Careful examination of Figures 4.2 and 4.3 depicting the frequency dependence of the real and imaginary parts of the dielectric permittivity of triply distilled water shows that there are eight major resonance features above 1 THz. With Eq. (4.199) for the observed rotational polarization phenomena, a composite Cole–Cole–Rocard–Powles–Debye–Lorentz model for the frequency dispersion of the dielectric permittivity of triply distilled water is given by

(ω)/ 0 = 1 +

2 j=1

aj (1 − iωτj )1−νj (1 − iωτf j )

−

8 (j even)

b2j , ω2 − + 2iδj ω ω02

j=0

(4.214) where ν1 = 0 and ν2 = −0.5. A near-optimized rms ﬁt of the parameters appearing in this model results in the Cole–Cole modiﬁed Rocard–Powles– Debye model parameter set given in Table 4.1 and the Lorentz model parameter set given in Table 4.2. The Cole–Cole modiﬁed Rocard–Powles–Debye model parameters listed in Table 4.1 diﬀer from those used in Figure 4.6 because of the inﬂuence of the higher-frequency Lorentz model resonance features that are now included in the composite model given in Eq. (4.214). That only a “near-optimized” rms ﬁt to the numerical data was obtained here is, in part, a consequence of the interplay between the various relaxation and resonance features of the model. In addition, an rms ﬁt to the real part of the relative permittivity data yields results that are diﬀerent from an rms ﬁt to the imaginary part of the relative permittivity data, this being due to measurement errors that are reﬂected in the numerical data. The results presented here then represent a compromise between these two optimizations. A comparison of the frequency dependence of the relative dielectric permittivity described by the composite model given in Eq. (4.214) with the parameters listed in Tables 4.1 and 4.2 and the numerical data for triply distilled water is presented in Figure 4.11 for the real part and in Figure 4.12 for the imaginary part. Notice that the j = 0 resonance feature appears in close proximity to the j = 1 relaxation feature. A major shortcoming of the composite model is its inability to model the full transparency of the

4.4 Causal Models of the Material Dispersion

215

j aj τj τf j 1 74.65 8.30 × 10−12 s 1.09 × 10−13 s 2 2.988 5.91 × 10−14 s 8.34 × 10−15 s Table 4.1. Estimated best rms ﬁt Cole–Cole modiﬁed Rocard–Powles–Debye model parameters for triply distilled water at 25o C. j 0 2 4 6 8

ωj 2.08 × 1013 r/s 1.05 × 1014 r/s 3.27 × 1014 r/s 6.19 × 1014 r/s 2.30 × 1016 r/s

δj 3.56 × 1012 r/s 4.51 × 1013 r/s 3.08 × 1012 r/s 2.86 × 1013 r/s 5.82 × 1015 r/s

bj 4.98 × 1012 r/s 8.48 × 1013 r/s 1.98 × 1013 r/s 1.65 × 1014 r/s 2.01 × 1016 r/s

Table 4.2. Estimated best rms ﬁt Lorentz model parameters for triply distilled water at 25o C.

visible window in water, this being primarily due now to the uppermost resonance feature at ω8 ∼ = 2.30 × 1016 r/s. It is clear that this near-ultraviolet resonance feature is not accurately described by the classical Lorentz model. Nevertheless, the composite model given in Eq. (4.214) with material parameters given in Tables 4.1 and 4.2 provides a reasonably accurate description of both the real and imaginary parts of the dielectric permittivity of triply distilled water at 25o C over the angular frequency domain ω ∈ [0, ωmax ] with ωmax ≈ 1 × 1015 r/s. 4.4.6 Composite Model of the Magnetic Permeability The frequency dependence of the magnetic permeability µ(ω) may also, in some circumstances [52], be described by a composite Rocard–Powles–Debye– Lorentz model, where µ(ω)/µ0 = 1 +

N j=1

2 Ωmj mj − . (1 − iωΥj )(1 − iωΥf j ) ω 2 − Ωj2 + 2iΓj ω (j even)

j=0

(4.215) The ﬁrst summation on the right-hand side of this expression describes the low-frequency orientational response of the material magnetization, where Υj is the macroscopic relaxation time and Υf j is the associated friction time. The second summation describes the high-frequency resonance response of the material magnetization [12], where Ωj denotes the undamped resonance frequency, Ωmj is the magnetic equivalent to the plasma frequency, and Υj is the phenomenological damping constant associated with the jth resonance frequency.

216

4 Macroscopic Electromagnetics 2

ΕrΩRΕΩ

10

1

10

0

10

Ω 10

10

Τ

12

10

Τ

Ω Ω Ω 14

10

16

10

Ω - r/s

Ω

18

10

Fig. 4.11. Comparison of the angular frequency dependence of the real part of the dielectric permittivity given by the composite Cole–Cole modiﬁed Rocard–Powles– Debye model (solid curve) with the numerical data for triply-distilled water at 25o C. 2

10

0

10

-2

ΕiΩIΕΩ

10

-4

10

-6

10

-8

10

Ω

-10

10

10

10

Τ

12

10

Τ

Ω Ω Ω 14

10

Ω 16

10

18

10

Ω - r/s

Fig. 4.12. Comparison of the angular frequency dependence of the real part of the dielectric permittivity given by the composite Cole–Cole modiﬁed Rocard–Powles– Debye model (solid curve) with the numerical data for triply distilled water at 25o C. The dashed curve describes the behavior for the composite model without the near-ultraviolet resonance line.

4.4 Causal Models of the Material Dispersion

217

4.4.7 The Drude Model of Free Electron Metals The classical Drude model [25] treats conduction electrons that are near the Fermi level as essentially free electrons [35]. Their equation of motion under the action of an applied electromagnetic ﬁeld is then given by Eq. (4.200) with the resonance frequency set equal to zero; viz. 2 d r dr = −qe Eeff . m +γ (4.216) dt2 dt The relative dielectric permittivity is then given by

(ω)/ 0 = 1 −

ωp2 , ω(ω + iγ)

(4.217)

where γ = 1/τc is a damping constant given by the inverse of the relaxation time τc associated with the mean-free path for free electrons in the material, and . 4π N qe2 (4.218) ωp ≡

0 m is the plasma frequency. Here N is the number density of free electrons with eﬀective mass m. Because the Lorentz model is causal, the Drude model is then also a causal model.

218

4 Macroscopic Electromagnetics

References 1. J. C. Maxwell, “A dynamical theory of the electromagnetic ﬁeld,” Phil. Trans. Roy. Soc. (London), vol. 155, pp. 450–521, 1865. 2. J. C. Maxwell, A Treatise on Electricity and Magnetism. Oxford: Oxford University Press, 1873. 3. H. A. Lorentz, The Theory of Electrons. Leipzig: Teubner, 1906. Ch. IV. 4. J. H. van Vleck, Electric and Magnetic Susceptibilities. London: Oxford University Press, 1932. 5. R. Russakoﬀ, “A derivation of the macroscopic Maxwell equations,” Am. J. Phys., vol. 38, no. 10, pp. 1188–1195, 1970. 6. F. N. H. Robinson, Macroscopic Electromagnetism. Oxford: Pergamon, 1973. 7. E. M. Purcell, Electricity and Magnetism. New York: McGraw-Hill, 1965. pp. 344–347. 8. C. J. F. B¨ ottcher, Theory of Electric Polarization: Volume I. Dielectrics in Static Fields. Amsterdam: Elsevier, second ed., 1973. 9. C. J. F. B¨ ottcher and P. Bordewijk, Theory of Electric Polarization: Volume II. Dielectrics in Time-Dependent Fields. Amsterdam: Elsevier, second ed., 1978. 10. J. D. Jackson, Classical Electrodynamics. New York: John Wiley & Sons, third ed., 1999. 11. P. Penﬁeld and H. A. Haus, Electrodynamics of Moving Media. Cambridge, MA: M.I.T. Press, 1967. 12. I. V. Lindell, A. H. Sihvola, S. A. Tretyakov, and A. J. Viitanen, Electromagnetic Waves in Chiral and Bi-Isotropic Media. Boston: Artech House, 1994. Ch. 6. 13. H. M. Nussenzveig, Causality and Dispersion Relations. New York: Academic, 1972. Ch. 1. 14. R. W. Whitworth and H. V. Stopes-Roe, “Experimental demonstration that the couple on a bar magnet depends on H, not B,” Nature, vol. 234, pp. 31–33, 1971. 15. V. M. Agranovich and V. L. Ginzburg, Crystal Optics with Spatial Dispersion, and Excitons. Berlin-Heidelberg: Springer-Verlag, second ed., 1984. 16. J. S. Toll, “Causality and the dispersion relation: Logical foundations,” Phys. Rev., vol. 104, no. 6, pp. 1760–1770, 1950. 17. A. Gr¨ unbaum, “Is preacceleration of particles in Dirac’s electrodynamics a case of backward causation? The myth of retrocausation in classical electrodynamics,” Philosophy of Science, vol. 43, pp. 165–201, 1976. 18. B. Y. Hu, “Kramers-Kronig in two lines,” Am. J. Phys., vol. 57, no. 9, p. 821, 1989. 19. J. Plemelj, “Ein Erg¨ anzungssatz zur Cauchyschen Integraldarstellung analytischer Funktionen, Randwerte betreﬀend,” Monatshefte f¨ ur Mathematik und Physik, vol. 19, pp. 205–210, 1908. 20. E. C. Titchmarsh, Introduction to the Theory of Fourier Integrals. London: Oxford University Press, 1939. Ch. I. 21. H. A. Kramers, “La diﬀusion de la lumi`ere par les atomes,” in Estratto dagli Atti del Congresso Internazional de Fisici Como, pp. 545–557, Bologna: Nicolo Zonichelli, 1927. 22. R. de L. Kronig, “On the theory of dispersion of X-Rays,” J. Opt. Soc. Am. & Rev. Sci. Instrum., vol. 12, no. 6, pp. 547–557, 1926.

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4 Macroscopic Electromagnetics

46. J. M. Stone, Radiation and Optics, An Introduction to the Classical Theory. New York: McGraw-Hill, 1963. ¨ 47. L. Brillouin, “Uber die fortpﬂanzung des licht in disperdierenden medien,” Ann. Phys., vol. 44, pp. 204–240, 1914. 48. L. Brillouin, Wave Propagation and Group Velocity. New York: Academic, 1960. 49. K. E. Oughstun and N. A. Cartwright, “On the Lorentz-Lorenz formula and the Lorentz model of dielectric dispersion,” Opt. Exp., vol. 11, no. 13, pp. 1541– 1546, 2003. 50. K. E. Oughstun and N. A. Cartwright, “On the Lorentz-Lorenz formula and the Lorentz model of dielectric dispersion: addendum,” Opt. Exp., vol. 11, no. 21, pp. 2791–2792, 2003. 51. R. Loudon, The Quantum Theory of Light. London: Oxford University Press, 1973. 52. J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, “Magnetism from conductors and enhanced nonlinear phenomena,” IEEE Trans. Microwave Theory & Tech., vol. 47, pp. 2075–2084, 1999.

Problems 4.1. Show that the spatially averaged charge is conserved using the expressions given in Eqs. (4.23) and (4.43) for the spatial averages of the microscopic charge and current densities. "∞ 4.2. Show that F{sgn(t)} = P{2i/ω} where F{f (t)} ≡ −∞ f (t)eiωt dt denotes the Fourier transform of the function f (t) and where sgn(t) ≡ −1 for t < 0 and sgn(t) ≡ 1 for t > 0 is the sign function. From this result, show that F{U (t)} = P{i/ω} + πδ(ω), where U (t) denotes the Heaviside unit step function. "t ˆ−1 (t − t )ˆ µm (t − t )dt = δ(t − t), where µ ˆ−1 (t) is as 4.3. Show that −∞ µ deﬁned in"Eq. (4.129) and where µ ˆm (t) is deﬁned through the linear relation ∞ ˆm (t − t )H(r, t )dt . B(r, t) = −∞ µ 4.4. Explain in physical terms why the eﬀective local ﬁeld given in Eq. (4.165) [see also Appendix D] is larger in magnitude than the macroscopic applied ˜ ω). electric ﬁeld E(r, 4.5. With the molecular polarizability given by Eq. (4.173) for the single relaxation time Debye model, derive Eqs. (4.175)–(4.176) for the electric susceptibility and dielectric permittivity. 4.6. With the molecular polarizability given by Eq. (4.183) for the single relaxation time Rocard–Powles–Debye model, derive Eqs. (4.185)–(4.186) for the electric susceptibility and dielectric permittivity. 4.7. Establish the inequality given in Eq. (4.189) for the Rocard–Powles extension of the Debye model.

5 Fundamental Field Equations in a Temporally Dispersive Medium

The fundamental macroscopic electromagnetic ﬁeld equations and elementary plane wave solutions in homogeneous, isotropic, locally linear, temporally dispersive media are developed in this chapter. The general frequency dependence of the dielectric permittivity, magnetic permeability, and electric conductivity is included in the analysis so that the resultant ﬁeld equations rigorously apply to both perfect and imperfect dielectrics, conductors and semiconducting materials, as well as to metamaterials, over the entire frequency domain. The analysis presented here focuses on the source-free ﬁeld equations and plane wave solutions in such “simple” temporally dispersive materials.

5.1 Macroscopic Field Equations in Temporally Dispersive HILL 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 equations [cf. Eqs. (4.52)–(4.55)] ∇ · D(r, t) = 4πc (r, t), % % % 1 % ∂B(r, t) % , ∇ × E(r, t) = − % %c% ∂t ∇ · B(r, t) = 0, % % % % % 1 % ∂D(r, t) % 4π % % % % +% ∇ × H(r, t) = % % % c % Jc (r, t), c ∂t taken together with the constitutive relations t D(r, t) =

ˆ(t − t )E(r, t )dt , −∞ t

H(r, t) = −∞ t

Jc (r, t) =

−∞

(5.1) (5.2) (5.3) (5.4)

(5.5)

µ ˆ−1 (t − t )B(r, t )dt ,

(5.6)

σ ˆ (t − t )E(r, t )dt ,

(5.7)

222

5 Fundamental Field Equations in a Temporally Dispersive Medium

as described in Eqs. (4.85), (4.141), and (4.113), respectively. Here ˆ(t) denotes the (real-valued) dielectric permittivity, µ ˆ(t) denotes the (real-valued) magnetic permeability, and σ ˆ (t) denotes the (real-valued) electric conductivity response of the simple dispersive medium. By causality, ˆ(t − t ) = 0, ˆ (t − t ) = 0 for t > t, as exhibited in the upper limit of µ ˆ(t − t ) = 0, and σ integration in the above three relations. The dependence of the conduction current density Jc (r, t) on the magnetic ﬁeld through the (J × B) Lorentz force term, known as the Hall eﬀect, is negligible in all but very special materials and is not considered here. The conduction current density Jc (r, t) and charge density c (r, t) are related by the equation of continuity [cf. Eq. (4.56)], ∇ · Jc (r, t) +

∂c (r, t) = 0, ∂t

(5.8)

which also follows from the divergence of Eq. (5.4) with substitution from Eq. (5.1). The temporal Fourier transform (see Appendix C) of this relation yields ˜ c (r, ω) = iω ˜c (r, ω), ∇·J (5.9) and the temporal Fourier transform of the constitutive relation (5.7) yields ˜ ω), with application of the convolution theorem, so that ˜ c (r, ω) = σ(ω)E(r, J ˜c (ω) =

σ(ω) ˜ ω). ∇ · E(r, iω

(5.10)

Furthermore, the temporal Fourier transform of the constitutive relation (5.5) ˜ ω) = (ω)E(r, ˜ ω), with application of the convolution theorem, so yields D(r, that the temporal Fourier transform of Gauss’ law (5.1) gives, with substitution of the above expression for ˜c (ω), ˜ ω) = 4π˜

(ω)∇ · E(r, c (r, ω) σ(ω) ˜ ω). ∇ · E(r, = 4π iω

(5.11)

˜ ω) and (σ(ω)/iω)∇ · In order that this ﬁnal relationship between (ω)∇ · E(r, ˜ ˜ E(r, ω) be satisﬁed for all ω, either ∇ · E(r, ω) = 0 or, if not, then 4πσ(ω)/(ω (ω)) = i, which will not be satisﬁed for general expressions ˜ ω) = 0 must then be satisof σ(ω) and (ω) for all ω. The relation ∇ · E(r, ﬁed, so that ˜c (r, ω) = 0, which in turn implies that c (r, t) = 0. This result is a direct consequence of the continuity equation (5.8), the divergence relation (5.1) for the electric displacement vector, and, most importantly, the two constitutive relations (5.5) and (5.7) for the dielectric permittivity and the electric conductivity of a homogeneous, isotropic, locally linear, temporally dispersive medium. The ﬁnal space–time domain form of the set of Maxwell’s electromagnetic ﬁeld equations to be considered here is then given by

5.1 Macroscopic Field Equations in Temporally Dispersive HILL Media

∇ · D(r, t) = 0,

% % % 1 % ∂B(r, t) % , ∇ × E(r, t) = − % %c% ∂t ∇ · B(r, t) = 0, % % % % % 1 % ∂D(r, t) % 4π % % % % ∇ × H(r, t) = % % +% % c % Jc (r, t), c ∂t

223

(5.12) (5.13) (5.14) (5.15)

with no external charge or current sources, with the constitutive relations given in Eqs. (5.5)–(5.7), and the equation of continuity ∇ · Jc (r, t) = 0.

(5.16)

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. Because the material properties are spatially continuous, it then follows, for example, that ˆ d2 r = 0 Jc (r, t) · n (5.17) S

for any closed surface S, with similar expressions for the ﬁeld vectors. 5.1.1 Temporal Frequency Domain Representation The temporal frequency spectra of the electric ﬁeld intensity and magnetic induction ﬁeld vectors are deﬁned by the Fourier transform integrals ∞ ˜ E(r, t)eiωt dt, (5.18) E(r, ω) = −∞ ∞ ˜ ω) = B(r, t)eiωt dt, (5.19) B(r, −∞

with corresponding inverse transformations ∞ 1 ˜ ω)e−iωt dω, E(r, t) = E(r, 2π −∞ ∞ 1 ˜ ω)e−iωt dω. B(r, t) = B(r, 2π −∞

(5.20) (5.21)

Because both of the electromagnetic ﬁeld vectors E(r, t) and B(r, t) are realvalued vector functions of both position and time, then their corresponding temporal frequency spectra satisfy the symmetry property ˜ ∗ (r, ω) = E(r, ˜ −ω ∗ ), E ˜ −ω ∗ ), ˜ ∗ (r, ω) = B(r, B

(5.22) (5.23)

224

5 Fundamental Field Equations in a Temporally Dispersive Medium

for complex ω. The temporal frequency spectra of the electric displacement vector D(r, t), magnetic intensity vector H(r, t), and conduction current vector Jc (r, t) also satisfy this symmetry property, so that [cf. Eqs. (4.99), (4.116), and (4.147)]

∗ (ω) = (−ω ∗ ), µ∗ (ω) = µ(−ω ∗ ), ∗

(5.24)

∗

σ (ω) = σ(−ω ), for complex ω. Finally, the temporal frequency transform of the source-free Maxwell’s equations (5.12)–(5.15) yields ˜ ω) = 0, ∇ · E(r, % % %1% ˜ % ˜ ∇ × E(r, ω) = % % c % iω B(r, ω), ˜ ω) = 0, ∇ · B(r, % % %1% ˜ ˜ % ∇ × B(r, ω) = − % % c % iωµ(ω) c (ω)E(r, ω),

(5.25) (5.26) (5.27) (5.28)

where

σ(ω) (5.29) ω deﬁnes the complex permittivity c (ω) of the dispersive medium. Upon taking the curl of Eq. (5.26) and substituting Eq. (5.28) there results % % %1% 2 2˜ ˜ % ∇ E(r, ω) + % (5.30) % c2 % ω µ(ω) c (ω)E(r, ω) = 0,

c (ω) ≡ (ω) + i4π

after application of the vector identity ∇×∇× = ∇(∇·)−∇2 for the curl-curl operator and substitution from Eq. (5.25). In a similar fashion, upon taking the curl of Eq. (5.28) and substituting Eq. (5.26) one obtains % % %1% 2 2˜ ˜ % ∇ B(r, ω) + % (5.31) % c2 % ω µ(ω) c (ω)B(r, ω) = 0, where the solenoidal character of the magnetic ﬁeld has been used. Deﬁne the vacuum wavenumber of the ﬁeld as ω (5.32) k0 ≡ , c and the complex index of refraction of the medium as 1/2 µ(ω) c (ω) n(ω) ≡ . µ0 0

(5.33)

Notice that the complex index of refraction combines the frequency dispersion of each individual dispersive material paramter (ω), µ(ω), and σ(ω) into a

5.1 Macroscopic Field Equations in Temporally Dispersive HILL Media

225

single, physically meaningful parameter. With these substitutions, the wave equations given in Eqs. (5.30) and (5.31) become ˜ ω) = 0, ˜ ω) + k 2 n2 (ω)E(r, ∇2 E(r, 0 2˜ 2 2 ˜ ω) = 0. ∇ B(r, ω) + k0 n (ω)B(r,

(5.34) (5.35)

These are the vector Helmholtz equations for the electromagnetic ﬁeld vectors in homogeneous, isotropic, locally linear, temporally dispersive media. 5.1.2 Complex Time-Harmonic Form of the Field Quantities The complex (phasor) form of the electric and magnetic ﬁeld vectors for a strictly time-harmonic ﬁeld with ﬁxed angular frequency ω is obtained from the representation −iωt ˜ Eω (r, t) = E(r)e , −iωt ˜ Bω (r, t) = B(r)e ,

(5.36) (5.37)

where each complex ﬁeld vector is related to its real-valued counterpart by # $ −iωt ˜ E(r, t) = {Eω (r, t)} = E(r)e , (5.38) # $ −iωt ˜ B(r, t) = {Bω (r, t)} = B(r)e . (5.39) With these substitutions, the constitutive relations given in Eqs. (5.5)–(5.7) become # $ −iωt ˜ D(r, t) = (ω)E(r)e , (5.40) # $ −iωt ˜ H(r, t) = µ−1 (ω)B(r)e , (5.41) # $ −iωt ˜ Jc (r, t) = σ(ω)E(r)e . (5.42) The appropriate complex form of these induced ﬁeld vectors is then given by −iωt −iωt ˜ ˜ Dω (r, t) = D(r)e = (ω)E(r)e , −iωt −1 −iωt ˜ ˜ Hω (r, t) = H(r)e = µ (ω)B(r)e , −iωt −iωt ˜ ˜ Jcω (r, t) = Jc (r)e = σ(ω)E(r)e ,

(5.43) (5.44) (5.45)

˜ ˜ ˜ ˜ ˜ c (r) = σ(ω)E(r). ˜ so that D(r) = (ω)E(r), B(r) = µ(ω)H(r), and J Consider now determining the relationship between the complex phasor form of the ﬁeld vectors and the corresponding temporal frequency spectra of the real-valued ﬁeld vectors. Substitution of Eq. (5.38) into Eq. (5.18) for the temporal frequency spectrum of the electric ﬁeld intensity vector results in

226

5 Fundamental Field Equations in a Temporally Dispersive Medium

# $ −iωt ˜ E(r)e eiω t dt −∞ ∞ ∞ 1 ˜ ˜ ∗ (r) = ei(ω −ω)t dt + E ei(ω +ω)t dt , E(r) 2 −∞ −∞

˜ ω ) = E(r,

so that

∞

# $ ˜ ω ) = π E(r)δ(ω ˜ ∗ (r)δ(ω + ω) ˜ − ω) + E E(r,

(5.46)

with similar expressions for the other ﬁeld vectors. Hence, the frequency spectrum of the complex monochromatic ﬁeld given in Eqs. (5.36)–(5.37) with real-valued ﬁeld vectors given in Eqs. (5.38)–(5.39) is comprised of a symmetric pair of spectral lines at ω = ±ω. Substitution of Eqs. (5.38)–(5.42) into the source-free form of Maxwell’s equations given in Eqs. (5.12)–(5.15) then results in the set of equations $ # −iωt ˜ = 0, (5.47) (ω)∇ · E(r)e % # # $ % $ %1% −iωt −iωt ˜ ˜ % ∇ × E(r)e =% , (5.48) % c % iω B(r)e # $ −iωt ˜ ∇ · B(r)e = 0, (5.49) % % # $ $ %1% # −iωt −iωt ˜ ˜ % µ−1 (ω)∇ × B(r)e = −% , (5.50) % c % iω c (ω)E(r)e where c (ω) is the complex permittivity deﬁned in Eq. (5.29). Because it is only the real parts of the above relations that enter into the determination of the real-valued ﬁeld vectors through Eqs. (5.38)–(5.42), one is free to impose the additional condition that the imaginary parts of these quantities also satisfy the same set of relations. One then obtains the complex time-harmonic diﬀerential (phasor) form of Maxwell’s equations ˜ ∇ · E(r) = 0, (5.51) % % %1% ˜ % ˜ (5.52) ∇ × E(r) =% % c % iω B(r), ˜ ∇ · B(r) = 0, (5.53) % % %1% ˜ ˜ % ∇ × B(r) = −% (5.54) % c % iωµ(ω) c (ω)E(r), which are identical with the temporal frequency domain form given in Eqs. (5.25)–(5.28); the diﬀerence between these two sets of equations then lies entirely in the interpretation of the ﬁeld vectors. Furthermore, the complex time-harmonic ﬁeld vectors satisfy the vector Helmholtz equations ˜ ˜ ∇2 E(r) = 0, (5.55) + k 2 n2 (ω)E(r) 0

˜ ˜ ∇2 B(r) = 0, + k02 n2 (ω)B(r)

(5.56)

in temporally dispersive HILL media, where n(ω) is the complex index of refraction deﬁned in Eq. (5.33).

5.1 Macroscopic Field Equations in Temporally Dispersive HILL Media

227

5.1.3 The Harmonic Electromagnetic Plane Wave Field The simplest electromagnetic wave ﬁeld is that of a time-harmonic electromagnetic plane wave for which the complex ﬁeld vectors appearing in Eqs. (5.38)–(5.39) are given by ˜ ˜ E(r) = E0 eik·r , ˜ ˜ B(r) = B0 eik·r ,

(5.57) (5.58)

where E0 and B0 are ﬁxed complex-valued vectors, independent of both time and the coordinate position r. The surfaces of constant phase for this wave˜ · r} = constant which yields a family of parallel ﬁeld are then given by {k ˜ Substitution of plane surfaces with normal vector parallel to the vector {k}. either of these expressions for the complex ﬁeld vectors into the appropriate vector Helmholtz equation given in Eqs. (5.55)–(5.56) then results in the identiﬁcation ˜·k ˜ = k 2 n2 (ω), k 0 where k0 = ω/c is the vacuum wavenumber of the ﬁeld. Let ˜ ≡ k(ω)ˆ ˜ s, k

(5.59)

˜ and where ˆs is a real-valued unit vector in the direction of the wave vector k, where ω ˜ (5.60) k(ω) ≡ k0 n(ω) = n(ω) c is the complex wavenumber of the wave ﬁeld in the temporally dispersive HILL medium. Substitution of each complex ﬁeld vector given in Eqs. (5.57)–(5.58) into its appropriate divergence relation given either in Eq. (5.51) or (5.53) of the complex time-harmonic diﬀerential form of Maxwell’s equations then yields the transversality relation ˆs · E0 = ˆs · B0 = 0.

(5.61)

˜ × E0 = 1/cωB0 , so Furthermore, the curl relation in Eq. (5.52) gives k that n(ω) ˆs × E0 , (5.62) B0 = c c ˜ × B0 = 1/cωµ(ω) c (ω)E ˜ 0 , so and the curl relation in Eq. (5.54) yields k that c 1 ˆs × B0 . E0 = − (5.63) c n(ω) These three relations [Eqs. (5.61)–(5.63)] then show that the ordered triple of vectors {E0 , B0 , ˆs} forms a right-handed orthogonal triad at any point of

228

5 Fundamental Field Equations in a Temporally Dispersive Medium

space. The ﬁeld vectors E0 and B0 then lie in the surfaces of constant phase and their complex amplitudes are related by B0 = c

n(ω) E0 c

(5.64)

at each point of space. The complex intrinsic impedance of the dispersive medium is deﬁned as the ratio of the complex electric ﬁeld intensity to the complex magnetic ﬁeld intensity H0 = B0 /µ(ω), so that, with Eqs. (5.64) and (5.33), 1/2 µ(ω) E0 = . (5.65) η(ω) ≡ H0

c (ω) For free space the intrinsic impedance is given by η0 = µ0 / 0 = 376.73Ω in mksa units whereas it is unity in cgs units. From Eqs. (5.29), (5.33), and (5.60), the complex wavenumber of the harmonic electromagnetic plane wave ﬁeld is found to be given by % % %1% 1/2 ˜ % k(ω) = % % c % ω [µ(ω) c (ω)] % % 1/2 %1% % ω µ(ω) (ω) + i4π σ(ω) =% . (5.66) %c% ω Let ˜ k(ω) ≡ β(ω) + iα(ω),

(5.67) ˜ ˜ where β(ω) ≡ {k(ω)} is the propagation factor and α(ω) ≡ {k(ω)} is the attenuation factor for the harmonic plane wave of angular frequency ω in the dispersive medium. Because (ω) = r (ω) + i i (ω) and σ(ω) = σr (ω) + iσi (ω), then the real and imaginary parts of the complex permittivity are given by σi (ω) , ω σr (ω) ,

c (ω) ≡ { c (ω)} = i (ω) + 4π ω

c (ω) ≡ { c (ω)} = r (ω) − 4π

(5.68) (5.69)

respectively. The real and imaginary parts of the complex wavenumber given in Eq. (5.66) are then determined from the relation % % %1% 2 % β 2 − α2 + 2iαβ = % % c2 % ω (µ + iµ ) ( c + i c ) , which, upon equating real and imaginary parts, yields the pair of relations % % %1% 2 2 2 % β −α = % % c2 % ω (µ c − µ c ) , % % % 1 % 2 % 2αβ = % % c2 % ω (µ c + µ c ) .

5.1 Macroscopic Field Equations in Temporally Dispersive HILL Media

229

Upon solving the second equation for β and substituting in the ﬁrst equation, one obtains the fourth-order polynomial equation % % % % %1% 2 % 1 % 1 4 4 2 2 % % % α + % 2 % ω (µ c − µ c ) α − % % c4 % 4 ω (µ c + µ c ) = 0 c with solution (assuming that µ c − µ c > 0) ⎧ ⎫ % % 1/2 ⎨ ⎬ 2 % % µ c + µ c 1%1 2 ω (µ

− µ

) 1 + − 1 , α2 = % c c % c2 % 2 ⎩ ⎭ µ c − µ c and consequently % % %1% 2 2 % β2 = % % c2 % ω (µ c − µ c ) + α ⎧ ⎫ % % 1/2 ⎨ ⎬ 2 %1%1 2 µ

+ µ

c c % ω (µ c − µ c ) =% 1 + + 1 . % c2 % 2 ⎩ ⎭ µ c − µ c The plane wave propagation and attenuation factors in a temporally dispersive HILL medium are then given by ⎫1/2 ⎧ % % 1/2 ⎨ 1/2 ⎬ 2 %1% 1 µ

+ µ

c c %ω (µ

− µ

) + 1 , (5.70) 1 + β=% c c %c% ⎭ ⎩ 2 µ c − µ c ⎫1/2 ⎧ % % 1/2 ⎨ 1/2 ⎬ 2 %1% 1 µ c + µ c % −1 , (5.71) α=% 1+ % c % ω 2 (µ c − µ c ) ⎭ ⎩ µ c − µ c when µ c − µ c > 0. If µ c − µ c < 0, then the quantity (µ c − µ c ) is replaced by its absolute value and the above pair of expressions for β and α are interchanged. Finally, if µ c − µ c = 0, then % % 1/2 1/2 %1% 1

2 c % % β = α = % %ω , (5.72) 1 + 2 µ c c 2

c where µ /µ = c / c in this special case. These relations may then be used to classify dispersive materials according to the origin of the loss over the frequency domain of interest. For nonmagnetic materials with µ(ω) = µ where µ is real-valued, Eqs. (5.70)–(5.71) simplify to ⎧ ⎫1/2 % % 2 1/2 1/2 ⎨ ⎬ %1% 1

(ω) % β(ω) = % +1 , (5.73) 1 + c % c % ω 2 (µ c (ω)) ⎩ ⎭

c (ω) ⎫1/2 ⎧ % % 1/2 ⎨ 2 1/2 ⎬ %1% 1

(ω) c %ω (ω)) − 1 , (5.74) α(ω) = % 1 + (µ

c %c% ⎭ ⎩ 2

c (ω)

230

5 Fundamental Field Equations in a Temporally Dispersive Medium

when c > 0. For low-loss materials in a given spectral domain, which may be referred to as near-ideal dielectrics,1 the inequality

c (ω)

1

c (ω)

(5.75)

is satisﬁed over the angular frequency domain of interest. The plane wave propagation and attenuation factors may then be approximated as 2 1 c (ω) β(ω) ≈ kn (ω) 1 + , (5.76) 8 c (ω) α(ω) ≈ where

1

(ω) kn (ω) c , 2

c (ω)

(5.77)

% % %1% % kn (ω) ≡ % % c % c µ c (ω)k0 = nr (ω)k0

(5.78)

is the wavenumber in a ﬁctitious lossless dispersive medium (i.e., an “ideal dielectric”) with real-valued refractive index nr (ω). The complex intrinsic impedance of the harmonic plane wave ﬁeld is, from Eq. (5.65), then found as −1/2

c (ω) η(ω) = ηn (ω) 1 + i

c (ω)

2

c (ω) 3 c (ω) , (5.79) ≈ ηn (ω) 1− −i 8 c (ω) 2 c (ω) 6

where ηn (ω) ≡

µ

c (ω)

(5.80)

is the intrinsic impedance of the ﬁctitious lossless medium with refractive index nr (ω). The phase diﬀerence δf between the complex electric and magnetic ﬁeld amplitudes is then given by tan δf ≈

c (ω) , 2 c (ω)

(5.81)

˜ = |ηn |He ˜ −iδf . Finally, the phase velocity of a time-harmonic plane where E wave ﬁeld in a near-ideal dielectric is given by vp (ω) =

ω ≈ β(ω)

c/nr (ω) 2 , (ω) 1 + 18 c (ω)

(5.82)

c

1

Some texts refer to such low-loss materials as being imperfect dielectrics. This terminology is not used here as all dispersive dielectric media have loss over some frequency domain and consequently are imperfect.

5.1 Macroscopic Field Equations in Temporally Dispersive HILL Media

231

where c/nr (ω) is the phase velocity in the ﬁctitious lossless medium. Once again, notice that the above expressions given in Eqs. (5.76)–(5.82) are valid approximations only within those frequency domains where the inequality given in Eq. (5.75) is satisﬁed and the medium behaves as a near-ideal dielectric. For a material such as a semiconductor where the losses are primarily conductive, the complex wavenumber given in Eq. (5.66) for a time-harmonic plane wave ﬁeld may be approximated as % % 1/2 %1% σr (ω) ˜ % % k(ω) ≈ % % ω µ c (ω) + i4π , c ω

(5.83)

where c (ω) = r (ω) − 4πσi (ω)/ω. For ω σr (ω)/ c (ω) the loss term in the above expression is suﬃciently small that the previous approximations apply with c = 4πσr (ω)/ω and the semiconductor behaves like a nearideal dielectric. On the other hand, for ω σr (ω)/ c (ω) the semiconductor material behavior is dominated by the real part σr (ω) of the conductivity and may then be considered as a good conductor. In that low-frequency domain the complex wavenumber given in Eq. (5.83) may be further approximated as 1+i ˜ k(ω) ≈ , (5.84) dp where

% % 1/2 % c % 2 % % dp ≡ % √ % ωµσr (ω) 2 π

(5.85)

is the penetration depth of the conductor. It is then seen from Eq. (5.84) with Eqs. (5.62)–(5.63) that the electric and magnetic ﬁelds are π/4 out of phase in a good conductor. The complex intrinsic impedance of a time-harmonic plane wave ﬁeld with oscillation frequency in the low-frequency domain of a semiconductor material may be approximated as % % 1/2 % 1 % −iπ/4 µω % η(ω) ≈ % √ % e . σr (ω) 2 π%

(5.86)

Because σr (ω) ≈ σ (0) − σ (2) ω 2 as ω → 0, then η(ω) → 0 as ω → 0. Finally, the phase velocity in the low frequency domain of a semiconductor is approximately given by % % 1/2 % c % 2ω % % vp (ω) ≈ ωdp = % √ % , µσr (ω) 2 π

(5.87)

which is very small in a good conductor. An important example of a semiconducting material is provided by seawater. From Eq. (4.217) describing the dielectric permittivity for the Drude

232

5 Fundamental Field Equations in a Temporally Dispersive Medium

model of free electron conductors and Eq. (5.29) for the complex permittivity, the frequency dispersion of the electric conductivity is given by γσ0 σ(ω) = i , (5.88) ω + iγ where σ0 ≡ ( 0 /4π)ωp2 /γ denotes the static conductivity of the material and where τc = 1/γ is the relaxation time associated with the mean free path for free electrons in the material. Estimates of these parameters for seawater are σ0 ≈ 4 mho/m and γ ≈ 1 × 1011 /s. The resultant angular frequency dispersion of the real and imaginary parts of the Drude model conductivity for sea-water is presented in Figure 5.1. Notice that any eﬀects due to the conductivity rapidly dimnish as the frequency increases above γ and that the imaginary part of the conductivity is negligible in comparison to the real part for |ω| 1/γ.

Real and Imaginary Parts of the Complex Conductivity

5

4.5

4

3.5

ΣrΩ

3

2.5

2

1.5

ΣiΩ

1

0.5

0 6 10

7

10

8

10

9

10

10

10

Ω - r/s

11

10

12

10

13

10

14

10

Fig. 5.1. Angular frequency dispersion of the real (solid curve) and imaginary (dashed curve) parts of the complex conductivity described by the Drude model for sea-water with static conductivity σ0 ≈ 4 mhos/m and mean relaxation time τc ≈ 1 × 10−11 s.

The relative complex dielectric permittivity with the Drude model is given by

ω − iγ , (5.89) ω(ω 2 + γ 2 ) where (ω)/ 0 is given in Eq. (4.214) for triply distilled water with parameters given in Tables 4.1 and 4.2. The inﬂuence of the Drude model conductivity

c (ω)/ 0 = (ω)/ 0 − ωp2

5.1 Macroscopic Field Equations in Temporally Dispersive HILL Media

233

3

10

2

Real & Imaginary Parts of ΕcΩ

10

1

10

Εc' Ω 0

10

Εc'' Ω -1

10

-2

10

6

10

8

10

10

10

12

14

10

10

Ω - r/s

16

10

18

10

Fig. 5.2. Angular frequency dispersion of the real and imaginary parts of the complex dielectric permittivity for water both without (solid curves) and with (dashed curves) conductivity appropriate for sea-water. 2

10

1

Real & Imaginary Parts of nΩ

10

nrΩ 0

10

niΩ -1

10

-2

10

-3

10

6

10

8

10

10

10

12

10

Ω - r/s

14

10

16

10

18

10

Fig. 5.3. Angular frequency dispersion of the real and imaginary parts of the complex index of refraction for water both without (solid curves) and with (dashed curves) conductivity appropriate for sea-water.

234

5 Fundamental Field Equations in a Temporally Dispersive Medium

on the imaginary part of the complex permittivity of water is depicted in Figure 5.2 when the numerical value of the coeﬃcient a1 appearing in the Rocard–Powles contribution is changed from a1 = 74.65 to a1 = 79.17 in order to refelect the addition of the contribution −ωp2 /γ 2 to the relative static value from the Drude model. Notice that the imaginary part of the complex permittivity now increases monotonically as the frequency decreases to zero below ω ∼ 1/γ whereas the real part is essentially unaltered from its zero conductivity behavior. The angular frequency dispersion of the real and imaginary parts of the corresponding complex index of refraction are illustrated in Figure 5.3, where µ = µ0 . Notice that the introduction of conductivity has an equal eﬀect on both the real and imaginary parts of n(ω).

5.2 Electromagnetic Energy and Energy Flow in Temporally Dispersive HILL Media Of considerable importance to the physical interpretation of electromagnetic wave propagation in dispersive media are the related concepts of electromagnetic energy and energy ﬂow in the dispersive host medium. The statement of the conservation of energy in the coupled ﬁeld–medium system, as embodied in Poynting’s theorem, is readily obtained as a direct mathematical consequence of the real space–time form of Maxwell’s ﬁeld equations, just as was derived for the microscopic ﬁeld equations in §2.3.1, but now with the appropriate constitutive relations. Although subject to interpretation, this relation provides a mathematically consistent formulation of energy ﬂow in the electromagnetic ﬁeld that is extremely useful in the analysis and physical interpretation of electromagnetic information transmission in temporally dispersive media and systems [1–3]. 5.2.1 Poynting’s Theorem and the Conservation of Energy The analysis begins with the pair of curl relations (5.13) and (5.15) of the real space-time form of Maxwell’s equations with no externally supplied charge or current sources. Upon taking the scalar product of the ﬁrst equation (5.13) with H(r, t) and the second equation (5.15) with E(r, t) and then taking their diﬀerence, one obtains % % % % % % %1% % E · ∂D + H · ∂B + % 4π % Jc · E, E·∇×H−H·∇×E=% % c % %c% ∂t ∂t which may be rewritten as % % % c % % 1 % % % % E · ∂D + H · ∂B . Jc · E = − % % ∇ · (E × H) − % % 4π % 4π ∂t ∂t

(5.90)

5.2 Electromagnetic Energy and Energy Flow

235

The Poynting vector for the macroscopic electromagnetic ﬁeld is deﬁned at each space–time point as [cf. Eq. (2.145)] % c % % % (5.91) S(r, t) ≡ % % E(r, t) × H(r, t), 4π which has the dimensional units of power per unit area. In addition, one may deﬁne the scalar quantities Ue (r, t) and Um (r, t) that are associated with the electric and magnetic ﬁelds, respectively, through the general relations % % 1 % ∂Ue (r, t) % % E(r, t) · ∂D(r, t) , ≡% (5.92) % ∂t 4π % ∂t % % 1 % ∂Um (r, t) % % H(r, t) · ∂B(r, t) , ≡% (5.93) % ∂t 4π % ∂t which have the dimensions of energy per unit volume, with sum U (r, t) = Ue (r, t) + Um (r, t).

(5.94)

With these deﬁnitions, Eq. (5.90) becomes Jc (r, t) · E(r, t) = −∇ · S(r, t) −

∂U (r, t) , ∂t

(5.95)

which is the diﬀerential form of the Heaviside–Poynting theorem for the macroscopic electromagnetic ﬁeld [compare with its microscopic counterpart in Eq. (2.146)]. Integration of this expression over an arbitrary volume V bounded by a simply connected closed surface S followed by application of the divergence theorem then yields d ˆ d2 r − Jc (r, t) · E(r, t)d3 r = − S(r, t) · n U (r, t)d3 r, (5.96) dt V S V ˆ denotes the unit outward normal vector to S. This integral form of where n the Heaviside–Poynting theorem may be taken as a mathematical expression of the conservation of energy in the coupled ﬁeldmedium system. Because of this physical interpretation, the volume integral on the left-hand side of this relation is interpreted as the time rate of work being done on the conduction current in the region V by the electromagnetic ﬁeld, the volume integral on the right-hand side of this relation is interpreted as the time rate of energy loss by the electromagnetic ﬁeld in V , and the surface integral is interpreted as the time rate of electromagnetic energy ﬂow out of V across the boundary surface S. The generally accepted physical interpretation of the scalar quantity U (r, t) is that it represents the total electromagnetic energy density of the coupled ﬁeld–medium system at the space-time point (r, t), measured in 3 3 ergs/cm in Gaussian (cgs) units and joules/m in SI (mksa) units. The sep (r, t) then represent the electric and arate scalar quantities Ue (r, t) and Um magnetic energy densities of the coupled ﬁeld-medium system.

236

5 Fundamental Field Equations in a Temporally Dispersive Medium

If one assumes that as t → −∞, E(−∞) = B(−∞) = 0 and that U (−∞) = 0 everywhere in space, then Eqs. (5.92)–(5.93) yield % % % 1 % ∂D(r, t) % dt, (5.97) Ue (r, t) = % % 4π % E(r, t) · ∂t % % % 1 % ∂B(r, t) % dt. (5.98) Um (r, t) = % % 4π % H(r, t) · ∂t Because the conditions leading to these expressions are not satisﬁed by a strictly monochromatic ﬁeld, these expressions do not have unrestricted applicability. However, they do apply to physically realizable pulsed electromagnetic ﬁelds. The general interpretation of the Poynting vector S(r, t) is that at any point r at which S ≡ |S| is diﬀerent from zero one can assert that electromagnetic energy is ﬂowing in the direction ˆs ≡ S/S such that across an elemental plane surface oriented perpendicular to ˆs at that point the rate of ﬂow of en2 2 ergy in the ﬁeld is S ergs/s/cm (in Gaussian units) or S joules/s/m (in SI units). In the absence of the conductive current density (i.e., in a nonconducting medium), the diﬀerential form (5.95) of Poynting’s theorem becomes ∂U (r, t) = 0, ∂t

∇ · S(r, t) +

(5.99)

which has exactly the same form as the equation of continuity (4.56) for the conservation of charge. By analogy with this equation, the total energy density U (r, t) of the coupled electromagnetic ﬁeld–medium system is the conserved quantity and the Poynting vector S(r, t) then represents the density ﬂow of this conserved quantity. The conservation of electromagnetic energy in the coupled ﬁeld–medium system is now illustrated through a careful consideration of the explicit form that the energy density takes in a temporally dispersive HILL medium. For any general homogeneous, isotropic, locally linear, temporally dispersive medium satisfying the constitutive relations given in Eqs. (5.5) and (5.6), one has the multipole expansions [cf. Eqs. (4.50) and (4.51)] D(r, t) = 0 E(r, t) + 4πP(r, t) − 4π∇ · Q (r, t) + · · · ,

(5.100)

B(r, t) = µ0 H(r, t) + 4πµ0 M(r, t) + · · · ,

(5.101)

where P(r, t) is the macroscopic polarization density vector [see Eq. (4.27)], Q (r, t) is the macroscopic quadrupole moment density tensor [see Eqs. (4.28)–(4.33)], and M(r, t) is the macroscopic magnetization vector [see Eq. (4.47)] of the temporally dispersive medium. If the dielectric response is dominated by the material polarizabilty, then Eqs. (5.5) and (5.67) yield

0 E(r, t) + 4πP(r, t) =

t

−∞

ˆ(t − t )E(r, t )dt ,

(5.102)

5.2 Electromagnetic Energy and Energy Flow

to a high degree of approximation, so that ∞ 1 [ˆ

(t − t ) − 0 δ(t − t )] E(r, t )dt P(r, t) = 4π −∞ ∞ = χ ˆe (t − t )E(r, t )dt ,

237

(5.103)

−∞

where the upper limit of integration in Eq. (5.69) has been extended from t to +∞ because ˆ(t − t ) vanishes when t > t. Here χ ˆe (t) =

1 [ˆ

(t) − 0 δ(t)] 4π

(5.104)

is the electric susceptibility of the polarizable medium. Similarly, if the magnetic response is dominated by the material magnetization, then Eqs. (4.136) and (5.6) yield 1 B(r, t) − 4πM(r, t) = µ0

t

−∞

µ ˆ−1 (t − t )B(r, t )dt ,

(5.105)

to a high degree of approximation, so that ∞ −1 1 µ0 δ(t − t ) − 4πˆ µ−1 (t − t ) B(r, t )dt M(r, t) = 4π −∞ ∞ 1 = χ ˆb (t − t )B(r, t )dt , (5.106) µ0 −∞ where the upper limit of integration in both Eqs. (4.133) and (5.105) has ˆb (t − t ) both vanish been extended from t to +∞ because µ ˆ−1 (t − t ) and χ when t > t. Here χ ˆb (t) =

1 δ(t) − 4πµ0 µ ˆ−1 (t) 4π

(5.107)

is related to the magnetic susceptibility χ ˆm (t) through the frequency domain relation [cf. Eq. (4.138)] (1 − 4πχb (ω)) = 1/(1 + 4πχm (ω)), so that µ ˆ(t) 1 χ ˆm (t) = − δ(t) . (5.108) 4π µ0 The electric and magnetic energy densities of the coupled ﬁeld–medium system are then given by % % % 1 % 0 ∂P(r, t) % % Ue (r, t) = % % E(r, t) · E(r, t) + E(r, t) · dt, (5.109) 4π 2 ∂t % % % 1 % µ0 ∂M(r, t) % Um (r, t) = % dt, (5.110) % 4π % 2 H(r, t) · H(r, t) + µ0 H(r, t) · ∂t

238

5 Fundamental Field Equations in a Temporally Dispersive Medium

respectively. This pair of expressions explicitly states that a portion of the electromagnetic ﬁeld energy resides in the dispersive medium. As for Eqs. (5.97) and (5.98), these two equations do not have unrestricted applicability; however, they are valid if the electromagnetic ﬁeld vanishes as t → −∞. Substitution of Eqs. (5.109) and (5.110) combined with Eq. (5.94) into the diﬀerential form (5.95) of the Heaviside–Poynting theorem yields 1 ∂H(r, t) ∂E(r, t) µ0 H(r, t) · + 0 E(r, t) · ∇ · S(r, t) = − 4π ∂t ∂t ∂P(r, t) ∂M(r, t) + E(r, t) · Jc (r, t) + + µ0 H(r, t) · . ∂t ∂t (5.111) This is the appropriate diﬀerential form of Poynting’s theorem for a homogeneous, isotropic, locally linear, temporally dispersive semiconducting medium whose dielectric response is described by the polarization density vector P(r, t) and whose magnetic response is described by the macroscopic magnetization vector M(r, t). Because its derivation does not rely on any assumption regarding the ﬁeld behavior as t → −∞, this expression then has general applicability. Furthermore, it is straightforward to show that the same expression holds with an external current source if the conduction current density Jc (r, t) is replaced by the total current density J(r, t) = Jc (r, t) + Jext (r, t). It is evident that one may associate the ﬁrst two terms appearing on the right-hand side of Eq. (5.111) with the rate of change of the energy density in the electromagnetic ﬁeld alone, the next two terms with the electric energy density, and the ﬁnal term with the magnetic energy density that is interacting with the dispersive medium. A certain portion of this electric and magnetic interaction energy is reactively stored in the medium and the remainder is absorbed, acting as a source for the evolved heat in the dispersive absorptive medium. The total energy density of the ﬁeld is then given by the sum of the ﬁrst two terms plus the reactively stored portion of the remaining three terms. The open literature is littered with criticisms of the Heaviside–Poynting theorem [4] as well as with revised versions of energy balance and ﬂow in the electromagnetic ﬁeld [5, 6]. An alternate form of the Heaviside–Poynting theorem has been more recently proposed [7] based upon a redeﬁnition of the energy ﬂux vector in terms of the time derivatives of the vector and scalar potentials for the ﬁeld. This alternate deﬁnition of power ﬂow in the electromagnetic ﬁeld would then yield an identically zero value in the static ﬁeld case. However, as shown in Problem 2.18, although the Poynting vector may not vanish in specially constructed static ﬁeld cases, it does indeed yield the physically correct result of zero net energy ﬂow in those exceptional cases [8]. A detailed critique of this alternate form of the energy ﬂux density has been given by F. N. H. Robinson [9], but not without reply [10]. With the support of a wealth of experimental veriﬁcation together with the lack of any deﬁnitive

5.2 Electromagnetic Energy and Energy Flow

239

experiment proving otherwise, the physical validity of the Heaviside–Poynting interpretation of electromagnetic energy ﬂow is maintained throughout this work. 5.2.2 The Energy Density and Evolved Heat in a Dispersive and Absorptive Medium Attention is now given to the precise physical deﬁnition of the expressions for the electric and magnetic energy densities Ue (r, t) and Um (r, t), respectively, as well as to the evolved heat (or dissipation) Q(r, t) in the electrodynamics of a dispersive, absorptive HILL medium. Without the formal deﬁnitions given in Eqs. (5.92) and (5.93), the diﬀerential form (5.95) of the Heaviside– Poynting theorem may be written as 1 ∂B(r, t) ∂D(r, t) −∇ · S(r, t) = + H(r, t) · E(r, t) · + J(r, t) · E(r, t), 4π ∂t ∂t (5.112) where J(r, t) = Jc (r, t) + Jext (r, t) denotes the total current density. In the simplest case of a nonmagnetic and nondispersive HILL medium, the constitutive relations given in Eqs. (5.5) and (5.6) yield D(r, t) = E(r, t) and H(r, t) = µ−1 B(r, t), respectively, where µ and are scalar constants. In that case the quantity ∂B(r, t) 1 H(r, t) · dt Um (r, t) = 4π ∂t ∂ 1 1 (B(r, t) · B(r, t)) dt = 4π 2µ ∂t 1 1 = B(r, t) · H(r, t) (5.113) 4π 2 is immediately identiﬁed with the energy density of the magnetic ﬁeld and is the same as that given by Eq. (5.98), and the quantity ∂D(r, t) 1 E(r, t) · dt Ue (r, t) = 4π ∂t ∂ 1

= (E(r, t) · E(r, t)) dt 4π 2 ∂t 1 1 = E(r, t) · D(r, t) (5.114) 4π 2 is immediately identiﬁed with the energy density of the electric ﬁeld and is the same as that given by Eq. (5.97). The total electromagnetic energy density is then given by the sum of these two quantities as U(r, t) = Ue (r, t) + Um (r, t) 1 1 =

|E(r, t)|2 + µ|H(r, t)|2 , 4π 2

(5.115)

240

5 Fundamental Field Equations in a Temporally Dispersive Medium

which, according to Landau and Lifshitz [11], has the “exact thermodynamic signiﬁcance” of representing “the diﬀerence between the internal energy per unit volume with and without the ﬁeld, the density and entropy remaining unchanged.” The Poynting vector S(r, t) = c/4πE(r, t) × H(r, t) has the physical meaning of representing the total electromagnetic energy ﬂux across a unit area normal to the direction of S(r, t). For a homogeneous, isotropic, locally linear, temporally dispersive medium with frequency-dependent dielectric permittivity (ω) = r (ω) + i i (ω), magnetic permeability µ(ω) = µr (ω) + iµi (ω), and electric conductivity σ(ω) = σr (ω) + iσi (ω), the physical interpretation of the Poynting vector S(r, t) = c/4πE(r, t) × H(r, t) as the total electromagnetic energy ﬂux density remains intact [11]. However, the expressions given in Eqs. (5.113) and (5.114) for the quantities Um (r, t) and Ue (r, t) no longer represent the magnetic and electric energy densities, respectively. Furthermore, the quantities (r, t) given in Eqs. (5.97) and (5.98) may now be interpreted as Ue (r, t) and Um generalized electric and magnetic energy densities, respectively, of the coupled ﬁeld–medium system, as is evident in Eqs. (5.109) and (5.110) for a dispersive medium whose dielectric response is dominated by the medium polarizability and whose magnetic response is dominated by the medium magnetization. A formal separation of these coupled ﬁeld–medium quantities can be made by rewriting Eqs. (5.92) and (5.93) as [12] % % % 1 % ∂D(r, t) ∂Ue (r, t) % + Qe (r, t) = % + Jc (r, t) · E(r, t), (5.116) % 4π % E(r, t) · ∂t ∂t % % % 1 % ∂B(r, t) ∂Um (r, t) % + Qm (r, t) = % , (5.117) % 4π % H(r, t) · ∂t ∂t where Ue (r, t) now represents just that part of the electric energy density residing both in the ﬁeld and reactively stored in the dispersive medium, Um (r, t) now represents just that part of the magnetic energy density residing both in the ﬁeld and reactively stored in the dispersive medium, and where Q(r, t) ≡ Qe (r, t) + Qm (r, t)

(5.118)

represents the evolved heat or dissipation in the medium. Comparison of Eq. (5.116) with Eq. (5.92) and comparison of Eq. (5.117) with Eq. (5.93) shows only in the absence of all loss mechanisms in that Ue = Ue and Um = Um the medium. With Eqs. (5.116) and (5.117), the Heaviside–Poynting theorem (5.112) becomes ∂U(r, t) + Q(r, t) = −∇ · S(r, t) − Jext (r, t) · E(r, t), ∂t

(5.119)

where U(r, t) ≡ Ue (r, t) + Um (r, t). All that remains is to obtain separate expressions for U(r, t) and Q(r, t). The sum of the expressions given in Eqs. (5.116) and (5.117) yields, with substitution of the series expansions of the constitutive relations given in

5.2 Electromagnetic Energy and Energy Flow

241

Eqs. (4.107), (4.123), and (4.155) for homogeneous, isotropic, locally linear, temporally dispersive media, % % % 1 % ∂U % E · ∂D + H · ∂B + Jc · E +Q = % % 4π % ∂t ∂t ∂t % % ∞ ∞ n+1 % 1 % E ∂ n+1 H (n) ∂ % E· = %

+ H · µ(n) n+1 % 4π % n+1 ∂t ∂t n=0

+E ·

∞ n=0

σ (n)

n=0

n

∂ E . ∂tn

Upon collecting terms together with the same order time derivatives and ﬁeld vectors, there results ∂U + Q = σ (0) E 2 ∂t ∂H 2 ∂E 2 1 (0) 1 (0) + +

/4π + σ (1) (µ /4π 2 ∂t 2 ∂t ∂2H 2 2 2 1 (1) 1 (1) (2) ∂ E

/4π + σ (µ /4π + + ∂t2 2 ∂t2 2 2 ∂H 2 2 ∂2E − (1) /4π + σ (2) − (µ(1) /4π ∂t ∂t +···. (5.120) In general, one then has that % % 1 % 1% % α1 E 2 (r, t) + β1 H 2 (r, t) + · · · , U(r, t) = % (5.121) % 2 4π % % % 2 2 1% 1 % % ξ1 ∂E + ζ1 ∂H Q(r, t) = σ (0) E 2 (r, t) + % + · · · , (5.122) 2 % 4π % ∂t ∂t where

α1 + β1 + ξ1 + ζ1 = (0) + µ(0) + 4πσ (1) ,

(5.123)

and so on for higher-order coeﬃcients. Unfortunately, the ﬁrst-order coefﬁcients α1 , β1 , ξ1 , and ζ1 cannot be separately expressed in terms of the zeroth-order dielectric permittivity and magnetic permeability moments (0) and µ(0) [see Eqs. (4.112) and (4.160)] and the ﬁrst-order electric conductivity moment σ (1) [see Eq. (4.128)]. In the simplest situation of a nondispersive medium, in which case = (0) , µ = µ(0) , and σ = σ (0) , the above expression becomes α1 + β1 = + µ, because ξ1 = ζ1 = 0 when a causal medium is nondispersive. It is then seen that the natural choice is to take α1 = and β1 = µ, in which case the expression given in Eq. (5.121) simpliﬁes to that in Eq. (5.115), as required.

242

5 Fundamental Field Equations in a Temporally Dispersive Medium

Such a natural separation is not possible in the general dispersive case. For example, if the electric and magnetic ﬁeld vectors are assumed to vary suﬃciently slowly with time such that |ξ1 |/T σ (0) and |ζ1 |/T σ (0) , where T is the characteristic temporal period describing variations in the electromagnetic ﬁeld, then Eqs. (5.121) and (5.122) may be approximated as % % 1 % 1% % (0) E 2 (r, t) + µ(0) H 2 (r, t) , U(r, t) ∼ = % % % 2 4π Q(r, t) ∼ = σ (0 E 2 (r, t),

(5.124)

when |α1 | |ξ1 | and |β1 | |ζ1 |. However, it is by no means certain that the conditions |α1 | |ξ1 | and |β1 | |ζ1 | will be satisﬁed and it is entirely < < possible that either |α1 | ∼ |ξ1 | or |β1 | ∼ |ζ1 |. Based upon these results, Barash and Ginzburg [12] concluded that “one cannot, in general, express the electromagnetic energy density U(r, t) and dissipation Q(r, t) separately in terms of the dielectric permittivity, magnetic permeability and electric conductivity of a general causal, temporally dispersive medium.”2 Consequently, in order to unambiguously determine these quantities, it may be necessary to employ a speciﬁc physical model of the dispersive medium through, for example, the equation of motion at the microscopic level. 5.2.3 Complex Time-Harmonic Form of Poynting’s Theorem For a completely time-harmonic electromagnetic ﬁeld with ﬁxed, real-valued angular frequency ω, each real-valued ﬁeld quantity may be expressed in the form [cf. Eqs. (5.38)–(5.42)] $ 1 # −iωt −iωt ˜ ∗ (r)eiωt , ˜ ˜ = (5.125) A(r, t) = A(r)e +A A(r)e 2 ˜ where A(r) is in general complex-valued and is related to the positive frequency component of the temporal frequency spectrum of A(r, t), as described in Eq. (5.46). The scalar product of two such real-valued vector ﬁelds with the same oscillation frequency ω is then given by $ 1 #˜ −i2ωt ˜ ˜ ˜ ∗ (r) + A(r) A(r, t) · B(r, t) = A(r) · B(r)e ·B 2 $ 1 #˜∗ −i2ωt ˜ ˜ ˜ , (5.126) = A (r) · B(r) + A(r) · B(r)e 2 2

This result has recently been criticized in an analysis [13] based upon a revised formulation of electromagnetic energy conservation [7]. However, this analysis relies, in part, on the revised constitutive relations [compare with the relations given in Eqs. (5.5) and (5.6)] D = ∗ ∂E/∂t and B = µ ∗ ∂H/∂t, where ∗ denotes the convolution operation. This then results in the real parts of both (ω) and µ(ω) being odd functions of real ω and their imaginary parts are now even, in contrast with experimental results (see Figs. 4.2 and 4.3).

5.2 Electromagnetic Energy and Energy Flow

243

and their vector product is given by $ 1 #˜ −i2ωt ˜ ∗ (r) + A(r) ˜ ˜ ×B × B(r)e A(r, t) × B(r, t) = A(r) 2 $ 1 #˜∗ −i2ωt ˜ ˜ ˜ (r) × B(r) + A(r) × B(r)e = A . (5.127) 2 The time-average of a periodic function A(r, t) is simply deﬁned by the integral 1 T /2 A(r, t) ≡ A(r, t)dt, (5.128) T −T /2 where T > 0 is the fundamental period of oscillation of the function A(r, t); for an aperiodic function one takes the limit as T → ∞ in the above expression. If A(r, t) is a simple harmonic function of time with ﬁxed angular frequency ω, then T = 2π/ω and A(r, t) = 0. The time-average of the scalar and vector products of two such time-harmonic vector ﬁelds with the same oscillation frequency do not vanish, however, as seen from Eqs. (5.126) and (5.127), and are given by $ 1 #˜ ˜ ∗ (r) A(r, t) · B(r, t) = A(r) ·B 2 $ 1 #˜∗ ˜ = A (r) · B(r) , (5.129) 2 $ # 1 ˜ ˜ ∗ (r) A(r, t) × B(r, t) = A(r) ×B 2 $ 1 #˜∗ ˜ = A (r) × B(r) . (5.130) 2 The time-average of the Poynting vector for a time-harmonic ﬁeld is then given by % c % % % ˜ ˜ t) × H(r, t) S(r, t) = % % E(r, 4π % # $ 1% % c % ˜ ˜ ∗ (r) = % % E(r) ×H 2 #4π $ ˜ = S(r) , (5.131) where

% 1% % c %˜ ˜ ∗ (r) ˜ ×H (5.132) S(r) ≡ % % E(r) 2 4π is the complex Poynting vector of the time-harmonic ﬁeld.3 Consider now the complex, time-harmonic diﬀerential form of Maxwell’s equations, as given in Eqs. (5.51)–(5.54), whose curl relations may be written as 3

Notice that some authors choose to leave the factor 12 in Eq. (5.131) when mksa ˜ ˜=E ˜ ×H ˜ ∗ and S = 1 {S}. units are employed, in which case S 2

244

5 Fundamental Field Equations in a Temporally Dispersive Medium

% % %1% ˜ % ˜ ∇ × E(r) =% % c % iω B(r), % % % % % 4π % %1% ˜ ˜ ˜ % iω (ω)E(r) % %J + ∇ × H(r) = −% % c % c (r), %c% in the absence of any external current sources. The complex conjugate of the second curl relation gives % % % c % % 1 % % % ∗ ∗ ∗ ˜ ˜∗ ˜ % Jc (r) = % % ∇ × H (r) − % % 4π % iω (ω)E (r), 4π where ω is assumed here to be real-valued. The scalar product of this expres˜ sion with the complex (phasor) electric ﬁeld vector E(r) yields, with the ﬁrst of the above curl relations, % % c % % % 1 % % %˜ ∗ ˜ ∗ ˜ ·D ˜∗ ˜ ˜ % iω E Jc · E = % % E · ∇ × H − % % 4π 4π % % % c % % % c % % 1 % % % % % ˜∗ ∗ ˜ ˜ ˜ ·D ˜∗ ˜ % iω E = −% %∇ · E × H + % %H · ∇ × E − % % 4π 4π 4π % % % c % % % 1 % % % ∗ ˜ ·H ˜∗ −E ˜ ˜ ˜ ·D ˜∗ . % iω B = −% %∇ · E × H + % % % 4π 4π With the deﬁnition (5.132) of the complex Poynting vector one ﬁnally obtains 1 ˜∗ ˜ ˜ Jc (r) · E(r) = −∇ · S(r) − 2iω (˜ ue (r) − u ˜m (r)) , 2

(5.133)

which is the time-harmonic analogue of the diﬀerential form (5.95) of the Heaviside–Poynting theorem. The scalar quantity % % 1 % 1% ˜ ˜ ∗ (r) % E(r) ·D (5.134) u ˜e (r) ≡ % % 4 4π % is commonly referred to as the harmonic electric energy density, and the quantity % % 1% 1 % ˜ ˜ ∗ (r) % B(r) ·H (5.135) u ˜m (r) ≡ % 4 % 4π % is commonly referred to as the harmonic magnetic energy density. However, ˜m (r) can be rigorously neither of these two deﬁned quantities u ˜e (r) and u related to the respective electric and magnetic energy densities Ue (r, t) and Um (r, t) of the coupled ﬁeld–medium system because Eqs. (5.97) and (5.98) are not strictly applicable for a time-harmonic ﬁeld. Nevertheless, Eqs. (5.92) and (5.93) are applicable in the time-harmonic case and do provide a partial connection of the complex densities deﬁned in Eqs. (5.134) and (5.135) with physically meaningful quantities. The time-average of Eq. (5.92) yields

5.2 Electromagnetic Energy and Energy Flow

&

∂Ue (r, t) ∂t

so that

'

245

% %& ' % 1 % % E(r, t) · ∂D(r, t) =% % 4π % ∂t % % , % 1 %1 ˜ −iωt + iω D ˜ −iωt + E ˜ ∗ eiωt · −iω De ˜ ∗ eiωt % =% % 4π % 4 Ee % % % 1 %1 ˜∗ ˜ ˜ ˜∗ % =% % 4π % 4 iω E · D − iω E · D % % $ % 1 %1 # ˜ ˜∗ % =% ue (r)} , % 4π % 2 iω E(r) · D (r) = 2 {iω˜ 1 {˜ ue (r)} = − 2ω

&

∂Ue (r, t) ∂t

' ,

(5.136)

and the imaginary part of the harmonic electric energy density is related to the time-average of the time-rate of change of the electric energy density of the coupled ﬁeld-medium system. In a similar manner, the time-average of Eq. (5.93) yields & ' % %& ' % 1 % ∂Um (r, t) % H(r, t) · ∂B(r, t) =% % 4π % ∂t ∂t % % % 1 %1 ˜ ˜∗ ˜∗ ˜ % =% % 4π % 4 iω H · B − iω H · B % % $ % 1 %1 # ˜ ˜∗ % =% um (r)} , % 4π % 2 iω H(r) · B (r) = 2 {−iω˜ so that {˜ um (r)} =

1 2ω

&

∂Um (r, t) ∂t

' ,

(5.137)

and the imaginary part of the harmonic magnetic energy density is related to the time-average of the time-rate of change of the magnetic energy density of the coupled ﬁeld–medium system. Unfortunately, such a simple, physically ˜m (r). meaningful relationship does not result for the real parts of u˜e (r) and u Integration of Eq. (5.133) over an arbitrary ﬁxed volume V bounded by a simple closed surface S and application of the divergence theorem yields 1 ˜ ˜ ∗ (r) · E(r) ˜ ˆ d2 r − 2iω d3 r = − S(r) (˜ ue (r) − u ˜m (r)) d3 r, J ·n 2 V S V (5.138) which is the analogue of the integral form (5.96) of the Heaviside–Poynting theorem for time-harmonic ﬁelds. The time-harmonic forms (5.133) and (5.138) of the Heaviside–Poynting theorem are, just as are their general counterparts in Eqs. (5.95)–(5.96), a mathematically rigorous consequence of Maxwell’s equations and are therefore self-consistent relationships within the framework of classical electrodynamics. As in the general case, it is the

246

5 Fundamental Field Equations in a Temporally Dispersive Medium

physical interpretation of the quantities appearing in these relations that depend to a certain degree on hypothesis. The real part of the time-harmonic Heaviside–Poynting theorem given in Eq. (5.138) is, with substitution from Eqs. (5.131), (5.136), and (5.137), # $ 1 ˜ ˜ ∗ (r) · E(r) J d3 r 2 V 2 ˆ d r + 2ω = − S(r, t) · n {˜ ue (r) − u ˜m (r)} d3 r S V ' & ' & ∂Ue (r, t) ∂Um (r, t) 2 ˆd r− = − S(r, t) · n + d3 r, ∂t ∂t S V (5.139) which is just a statement of the conservation of energy in the coupled ﬁeldmedium system. The imaginary part of Eq. (5.138) is # $ 1 ˜ ˜ ∗ (r) · E(r) J d3 r 2 V # $ ˜ ˆ d2 r − 2ω {˜ ue (r) − u ˜m (r)} d3 r. (5.140) = − S(r) ·n S

V

˜ ˜ ∗ (r) · E(r) ˜ ˜ ∗ (r) · E(r), ˜ ˜ c (r) = σ(ω)E(r) so that J = σ(ω)E then the Because J c real and imaginary parts are expressible as # $ ˜ ˜ 2, ˜ ∗ (r) · E(r) J = σr (ω)|E(r| c # $ ˜ ˜ 2. ˜ ∗ (r) · E(r) J = −σi (ω)|E(r| c ˜ 2 = ( r (ω)−i i (ω))|E| ˜ 2 , then [from In addition, because (44π)˜ ue = ∗ (ω)|E| Eq. (5.136)] % % & ' 1 % ∂Ue (r, t) 1% 2 ˜ % ω i (ω)|E(r)| = −2ω {˜ ue (r)} = % , (5.141) % ∂t 2 4π % ˜ 2 = (µr (ω) + iµi (ω))|H| ˜ 2 , then [from Eq. and because (44π)˜ um = µ(ω)|H| (5.137)] % % & ' 1 % ∂Um (r, t) 1% 2 ˜ % ωµi (ω)|H(r)| = 2ω {˜ um (r)} = % . (5.142) ∂t 2 % 4π % Substitution of these expressions into Eq. (5.139) then results in ˆ d2 r − S(r, t) · n S % % 1 % ω% σr (ω) ˜ 2 2 ˜ % % = % %

i (ω) + 4π |E(r)| + µi (ω)|H(r)| d3 r 2 4π V ω % % # $ 1 % ω% 2 2 ˜ ˜ % c (ω)|E(r)| + µ(ω)|H(r)| (5.143) = % d3 r. % % 2 4π V

5.2 Electromagnetic Energy and Energy Flow

247

The left-hand side of this equation is interpreted as the time-average electromagnetic power ﬂow into the region V across the simple closed boundary surface S, so that the right-hand side may then be interpreted as the timeaverage rate of electromagnetic energy that is dissipated in the temporally dispersive HILL medium contained within the region V . In addition, substitution of the above expressions into Eq. (5.140) results in # $ ˜ ˆ d2 r S(r) ·n S % % 1 % ω% σi (ω) ˜ 2 2 ˜ % % =− % % |E(r)| − µr (ω)|H(r)| d3 r

r (ω) − 4π 2 4π V ω % % # $ 1 % ω% 2 2 ˜ ˜ % c (ω)|E(r)| − µ(ω)|H(r)| (5.144) =− % d3 r. % % 2 4π V The right-hand side of this expression may be interpreted as the time-average of the reactive or stored energy within the coupled ﬁeld–medium system enclosed by the surface S. It would seem that these physical interpretations of the various quantities appearing in Eqs. (5.143) and (5.144) provide separate identiﬁcations of both the time-average electromagnetic energy density U(r, t) and dissipation Q(r, t) for a time-harmonic ﬁeld in terms of the dielectric permittivity, magnetic permeability, and electric conductivity of the dispersive medium. However, there is no unambiguous connection between the integrands appearing on the right-hand sides of Eqs. (5.143) and (5.144) and the time-average quantities Ue (r, t), Um (r, t), and Q(r, t). For an attenuative medium, the time-average electromagnetic power ﬂow into any given region V in the medium across the simple closed boundary surface S of V must be positive for any nonzero ﬁeld, so that ˆ d2 r > 0, − S(r, t) · n S

and Eq. (5.143) then yields the inequality # $ 2 2 ˜ ˜ c (ω)|E(r)| + µ(ω)|H(r)| d3 r > 0, ω V

which is valid for all nonstatic ﬁelds4 (ω = 0). Because this inequality holds 2 2 ˜ ˜ + µ(ω)|H(r)| } > 0 and for any region V in the medium, then { c (ω)|E(r)| consequently σr (ω) ˜ 2 ˜ |E(r)|2 + µi (ω)|H(r)| > 0, (5.145)

i (ω) + 4π ω where ω > 0 is the real-valued angular frequency of the (nonstatic) timeharmonic ﬁeld. The ﬁrst term in this inequality is the electric loss and the 4

There is no loss for a strictly static ﬁeld, as is clearly evident from Eq. (5.144).

248

5 Fundamental Field Equations in a Temporally Dispersive Medium

second term is the magnetic loss. From the time-average of Eq. (5.119) with Jext (r, t) = 0 one obtains & ' ∂U(r, t) + Q(r, t) ∂t % % 1 % σr (ω) ˜ ω% 2 2 ˜ % %

i (ω) + 4π = % % |E(r)| + µi (ω)|H(r)| , 2 4π ω (5.146) so that the sum of the time-average time rate of change of the electromagnetic energy density reactively stored in the medium and the time-average dissipation of electromagnetic energy in the medium is determined by the imaginary parts of (ω) and µ(ω) together with the real part of σ(ω). Because of the second law of thermodynamics (the law of increase of entropy), the dissipation of electromagnetic energy must be accompanied by the evolution of heat [11] so that Q(r, t) > 0. However, the sign of the time-average quantity ∂U(r, t)/∂t is left undetermined. The general inequality appearing in Eq. (5.145) is then seen to allow for ampliﬁcation in one or two of the medium parameters provided that it is exceeded by the loss associated with the remaining medium parameter(s). If the dielectric permittivity, magnetic permeability, and electric conductivity of the medium are separately attenuative, then

i (ω) > 0, µi (ω) > 0,

(5.147)

σr (ω) > 0, for all ﬁnite positive ω > 0. However, the signs for the real parts r (ω) of the dielectric permittivity and µr (ω) of the magnetic permeability and the imaginary part σi (ω) of the medium conductivity are not subject to any physical restriction beyond that imposed by causality. 5.2.4 Electromagnetic Energy in the Harmonic Plane Wave Field Consider ﬁnally the time-harmonic energy densities and Poynting vector for a plane wave electromagnetic ﬁeld in a temporally dispersive HILL medium with dielectric permittivity (ω), magnetic permeability µ(ω), and electric conductivity σ(ω). From Eqs. (5.134) and (5.135) with substitution from Eqs. (5.57) and (5.58), respectively, the time-harmonic electric and magnetic energy densities are found as 1

∗ (ω)|E0 |2 e−2α(ω)ˆs·r , 44π 1 u ˜m (r) = µ(ω)|H0 |2 e−2α(ω)ˆs·r . 44π u ˜e (r) =

(5.148) (5.149)

5.2 Electromagnetic Energy and Energy Flow

249

Because H0 = ( c (ω)/µ(ω))1/2 E0 , from Eq. (5.65), where E0 and H0 are complex-valued, then |H0 |2 = | c (ω)/µ(ω)||E0 |2 , so that u ˜m (r) =

1 | c (ω)||E0 |2 eiφµ (ω) e−2α(ω)ˆs·r , 44π

(5.150)

where φµ (ω) ≡ arg{µ(ω)} is the phase angle of µ(ω) = |µ(ω)|eiφµ (ω) . From Eq. (5.132) with substitution from Eqs. (5.57), (5.58), and (5.62) with (5.65), the complex Poynting vector for a time-harmonic plane electromagnetic wave ﬁeld in a temporally dispersive HILL medium is given by % c % 1 % % ˜ |E0 |2ˆse−2α(ω)ˆs·r , (5.151) S(r) =% % ∗ 4π 2η (ω) where η(ω) = [µ(ω)/ c (ω)]1/2 denotes the complex intrinsic impedance of the material. The time-average Poynting vector in the dispersive HILL medium is then % c % η (ω) % % r S(r, t) = % % |E0 |2ˆse−2α(ω)ˆs·r , (5.152) 4π 2|η(ω)|2 where ηr (ω) denotes the real part of the complex intrinsic impedance. 5.2.5 Reversible and Irreversible Electrodynamic Processes in Temporally Dispersive Media An important re-examination of the Barash and Ginzburg result [12] for the electromagnetic energy density and evolved heat in a dispersive dissipative medium (presented in §5.2.2) has been given by Glasgow et al. [14] for a simple dispersive dielectric [µ(ω) = µ0 , σ(ω) = 0]. In that case the total electromagnetic energy density in the coupled ﬁeld–medium system is given by [cf. Eqs. (5.109)–(5.110)] % % t % 1 %1 ∂P(r, t ) 2 2 %

U (r, t) = % E (r, t) + µ H (r, t) + E(r, t ) · dt . 0 0 % 4π % 2 ∂t −∞ (5.153) The ﬁrst two terms on the right-hand side of this equation describe the electromagnetic ﬁeld energy density in the absence of the dispersive medium [cf. Eq. (5.115)] % % % 1 %1 2 2 % (5.154) Uem (r, t) ≡ % % 4π % 2 0 E (r, t) + µ0 H (r, t) , and the last term describes the electromagnetic energy density in the coupled ﬁeld–medium system t ∂P(r, t ) E(r, t ) · dt , (5.155) Uint (r, t) ≡ ∂t −∞

250

5 Fundamental Field Equations in a Temporally Dispersive Medium

which is deﬁned by Glasgow et al. [14] as the interaction energy. By comparison, Barash and Ginzburg [12] separated the interaction energy into two parts: one part which represents the electromagnetic energy that is reactively stored in the dispersive medium, and another part, Q(r, t), which represents the dissipation of electromagnetic energy in the medium. Notice that U(r, t) U (r, t) = + Q(r, t) ∂t ∂t Uem (r, t) Uint (r, t) = + , ∂t ∂t

(5.156)

the ﬁrst form of this separation being due to Barash and Ginzburg [12], and the second form due to Glasgow et al. [14], where U(r, t) = Uem (r, t) + Urev (r, t) and Uint (r, t) = Urev (r, t) + Uirrev (r, t). The Helmholtz free energy (also known as the work function) ψ ≡ U −T S is deﬁned [15] as the diﬀerence between the internal energy U of the system and the heat energy T S, where T is the absolute temperature and S the entropy. Von Helmholtz (1882) called this ψ function the free energy of a system, because [15] “its change in a reversible isothermal process equals the energy that can be ‘freed’ in the process and converted to mechanical work.” The decrease in ψ is then equal to the maximum work done on the system in a reversible isothermal process. Glasgow et al. [14] generalize this Helmholtz free energy to the irreversible, dissipative, nonequilibrium case by deﬁning the dynamical free energy as “the work the system can do on, or return to, an external agent,” which naturally reduces to the Helmholtz free energy in the reversible case. With regard to Eq. (5.156), this dynamical free energy U(r, t) is given by the sum of the ﬁeld energy term Uem (r, t) and the reactive (or reversible) energy term Urev (r, t). The evolved heat energy Q(r, t) then corresponds to the irreversible (or latent) part of the interaction energy. The interaction energy deﬁned in Eq. (5.155) can be expressed in terms of the instantaneous (or causal) spectrum of the electric ﬁeld vector5 , deﬁned as [16] t ˜ t (r, ω) ≡ 1 E(r, t )eiωt dt , (5.157) E 2π −∞ in the following manner due to Peatross, Ware, and Glasgow [17]. From Eq. (4.88), the macroscopic polarization density of a temporally dispersive HILL medium can be expressed as ∞ P(r, t) = E(r, t )G(t − t )dt (5.158) −∞

with Green’s function 5

Notice that this instantaneous spectrum is equal to the Fourier transform of E(r, t )U (t − t ), where U (t) is the Heaviside unit step-function.

5.2 Electromagnetic Energy and Energy Flow

0 2π

G(t) ≡

∞

−∞

χe (ω)e−iωt dω.

251

(5.159)

This Green’s function may be expressed as the sum of two terms, the ﬁrst associated with the real part χe (ω) ≡ {χ(ω)} and the second with the imaginary part χe (ω) ≡ {χ(ω)} of the electric susceptibility, as G(t) = G (t) + G (t), where

(5.160)

0 ∞ χ (ω)e−iωt dω, 2π −∞ e

0 ∞ G (t) ≡ χ (ω)e−iωt dω. 2π −∞ e G (t) ≡

(5.161) (5.162)

Causality is introduced through the fact that the real and imaginary parts of the electric susceptibility satisfy the Plemelj formulae given in Eqs. (4.148)– (4.149), viz. ∞ χe (ω ) 1 χe (ω) = P (5.163) dω , −ω π ω −∞ ∞ χe (ω ) 1 dω . (5.164) χe (ω) = − P −ω π ω −∞ Substitution of Eq. (5.163) into (5.161) then yields ∞ ∞ −iωt e

0 dω. dω χ (ω)P G (t) = e −ω 2π 2 −∞ ω −∞ Because P

∞

−∞

e−iωt dω = ω − ω

iπe−iω t , −iπe−iω t ,

(5.165)

t>0 , t<0

(5.166)

then Eq. (5.165) shows that G (t) = G (t) for t > 0 and G (t) = −G (t) for t < 0, so that

2G (t), t > 0 G(t) = . (5.167) 0, t<0 Substitution of this result in Eq. (5.158) then gives t P(r, t) = 2 E(r, t )G (t − t )dt =i

−∞

0 ∞

π

−∞

dω χe (ω)e−iωt

t

−∞

dt E(r, t )eiωt ,

(5.168)

252

5 Fundamental Field Equations in a Temporally Dispersive Medium

where causality is now explicitly expressed in the upper limit of integration, in agreement with the expression given in Eq. (4.88). The time derivative of this expression is then given by t ∂P(r, t)

0 ∞ dω ωχe (ω)e−iωt dt E(r, t )eiωt = ∂t π −∞ −∞ ∞

0 +i E(r, t) χe (ω)dω. π −∞ Because χe (ω) is an odd function of ω, the ﬁnal integral in the above expression is equal to zero and consequently t

0 ∞ ∂P(r, t) = dω ωχe (ω)e−iωt dt E(r, t )eiωt ∂t π −∞ −∞ ∞ −iωt ˜ t (r, ω)e = 2 0 ωχe (ω)E dω. (5.169) −∞

Substitution of this result into Eq. (5.155) then yields Uint (r, t) = 2 0

∞

−∞

dω ωχe (ω)

t

−∞

˜ t (r, ω)e−iωt dt . E(r, t ) · E

The time derivative of the complex conjugate of Eq. (5.157) results in the identiﬁcation ˜ ∗ (r, ω) 1 ∂E t = E(r, t)e−iωt , ∂t 2π so that the above expression for the interaction energy becomes Uint (r, t) = 4π 0

∞

−∞

dω ωχe (ω)

˜∗ ˜ t (r, ω) · ∂ Et (r, ω) dt . E ∂t −∞ t

Because Uint (r, t) is a real-valued quantity, the above expression may be written as ∞ dω ωχe (ω) Uint (r, t) = 2π 0 −∞ t ˜ ∗ (r, ω) ˜ t (r, ω) ∂E ∂E ∗ t ˜ ˜ + Et (r, ω) · dt . Et (r, ω) · × ∂t ∂t −∞ 2 ∞ t ∂ E ˜ t (r, ω) = 2π 0 dω ωχe (ω) dt ∂t −∞ −∞ ∞ 2 ˜ = 2π 0 ωχe (ω) E (r, ω) (5.170) dω. t −∞

5.2 Electromagnetic Energy and Energy Flow

253

It is then seen that Uint (r, t) ≥ 0

(5.171)

for all time t, where Uint (r, −∞) = 0, in agreement with the Landau–Lifshitz result [11] for the asymptotic heat that Uint (r, +∞) ≥ 0. This generalization [17] of the Landau–Lifshitz result shows that the interaction energy can never run a deﬁcit; that is, at any instant of time t the work that the electromagnetic ﬁeld does on the medium always exceeds the work that the medium does against the ﬁeld. The energy dissipated and consequently lost to the medium from the electromagnetic ﬁeld since time t = −∞ is given by Uint (r, ∞), which may be expressed as Uint (r, ∞) = Uint (r, ∞) − Uint (r, −∞), because Uint (r, −∞) = 0. The quantity Uint (r, ∞)−Uint (r, t) then represents the electromagnetic energy lost to the medium over the future time interval t > t given the present state of the medium at time t that has been established by the electromagnetic ﬁeld over the past time interval t < t. When this energy diﬀerence is negative, its opposite Uint (r, t) − Uint (r, ∞) > 0 is then interpreted as representing a reversible process of energy that is “borrowed” from the electromagnetic ﬁeld and returned to it at a later time as t → ∞. The value of this energy diﬀerence Uint (r, t) − Uint (r, ∞) will be different for diﬀerent future ﬁelds, the physically proper value then being given by an appropriate extremum principle. To that end, the interaction energy given in Eq. (5.170) is denoted as ∞ ˜ t (r, ω)|2 dω, Uint [E](r, t) ≡ 2π 0 ωχe (ω)|E (5.172) −∞

where [cf. Eq. (5.157)] ˜ t (r, ω) ≡ 1 E 2π

t

E(r, t )eiωt dt .

(5.173)

−∞

The reversible and irreversible energies of the electromagnetic ﬁeld in a simple temporally dispersive dielectric [µ = µ0 and σ(ω) = 0] are then given by the following deﬁnitions due to Glasgow et al. [14] Deﬁnition 1. Reversible Electromagnetic Energy. The reversible energy Urev (r, t) = Urev [E](r, t) of the electromagnetic ﬁeld at time t in a simple temporally dispersive dielectric is given by the supremum (least upper bound) of values that the quantity Uint (r, t) − Uint (r, ∞) can attain over all possible alternative future ﬁelds; that is, Urev [E](r, t) ≡ sup {Uint [E](r, t) − Uint [EU (t − t ) + Ef ](r, ∞)} , (5.174) Ef

where U (t) denotes the Heaviside unit step function, and where Ef denotes the possible future electric ﬁelds with Ef (r, t ) = 0 for all t < t.

254

5 Fundamental Field Equations in a Temporally Dispersive Medium

Notice that Uint [EU (t − t ) + Ef ](r, t) = Uint [E](r, t) in the above expression, where t denotes the “past” time variable appearing in the integrand of Eq. (5.173). Deﬁnition 2. Irreversible Electromagnetic Energy.The irreversible energy Uirrev (r, t) = Uirrev [E](r, t) of the electromagnetic ﬁeld at time t in a simple temporally dispersive dielectric is given by the complement of the reversible energy as the inﬁmum (greatest lower bound) of values that the quantity Uint (r, ∞) can attain over all possible alternate future ﬁelds; that is, Uirrev [E](r, t) ≡ inf {Uint [EU (t − t ) + Ef ](r, ∞)} . Ef

(5.175)

This formulation of the irreversible energy is in a form that is now appropriate for the calculus of variations. Because any future ﬁeld solution Ef (r, t ) vanishes for all t < t, as required in Deﬁnition 1 above, then the temporal Fourier transform of the concatenated electric ﬁeld quantity E(r, t )U (t − t ) + Ef (r, t ) is given by t 8 7 1 E(r, t )eiωt dt F E(r, t )U (t − t ) + Ef (r, t ) = 2π −∞ ∞ 1 + Ef (r, t )eiωt dt 2π t ˜ ˜ = Et (r, ω) + Ef (r, ω). (5.176) The variational derivative of the irreversible electromagnetic energy deﬁned in Eq. (5.175) with respect to future possible ﬁelds then gives, with Eq. (5.170), 7 8 δEf Uint [EU (t − t ) + Ef ](r, ∞) ∞ 2 ˜ ˜ = δE˜f ωχe (ω) E t (r, ω) + Ef (r, ω) dω −∞

= 0.

(5.177)

Glasgow et al. [14] then deﬁne the new electric ﬁeld vector functions E+ (r, t ) ≡ Ef (r, t + t),

(5.178)

E− (r, t ) ≡ E(r, t + t)U (−t ),

(5.179)

whose temporal Fourier transforms ∞ ˜ + (r, ω; t) = 1 Ef (r, t + t)eiωt dt E 2π −∞ ∞ 1 ˜ f (r, ω), (5.180) = e−iωt Ef (r, τ )eiωτ dτ = e−iωt E 2π −∞ ∞ ˜ − (r, ω; t) = 1 E(r, t + t)U (−t )eiωt dt E 2π −∞ t ˜ t (r, ω), = e−iωt E(r, τ )eiωτ dτ = e−iωt E (5.181) −∞

5.2 Electromagnetic Energy and Energy Flow

255

are analytic in the upper and lower half-planes, respectively. Substitution of these ﬁeld quantities in Eq. (5.177) with the variation now taken with respect ˜+ then results in the expression to E ∞ ˜ + (r, ω; t) δ E ˜ ∗ (r, ω; t)dω = 0. ˜ − (r, ω; t) + E ωχe (ω) E (5.182) + −∞

˜ + (r, ω; t) is analytic in the upper half of the complex ω-plane, Because E ˜ ∗ (r, ω; t) is then ˜ + (r, ω; t). The variation δ E then so also is its variation δ E + analytic in the lower half plane. The relation given in Eq. (5.182) cannot then be satisﬁed unless the remainder of the integrand is also analytic in the lower half plane. If the integrand in Eq. (5.182) is analytic in the lower halfplane and has magnitude that goes to zero suﬃciently fast as |ω| → ∞ in the lower half-plane, then by completion of the contour with a semicircle in the lower half-plane centered at the origin with radius Ω → ∞, as illustrated in Figure 5.1, application of Cauchy’s theorem results in Eq. (5.182). This result then provides a general variational approach for the determination of the irreversible electromagnetic energy in a simple dispersive dielectric [14]. However, this result does not contradict the Barash and Ginzburg result [12] presented in §5.2.2 because an analytic model for the imaginary part of the electric susceptibility is required. Ω''

Ω'

Ω

Complex ΩPlane

Fig. 5.4. Completion of the contour of integration in the lower half of the complex ω-plane with a semicircular arc of radius |ω| = Ω → ∞.

256

5 Fundamental Field Equations in a Temporally Dispersive Medium

As an illustration, consider a single-resonance Lorentz model dielectric with resonance angular frequency ω0 , phenomenological damping constant δ, and plasma frequency b = (4π/ 0 )N qe2 /m. In their analysis, Glasgow et al. [14] take the electric susceptibility for this causal model medium to be given by the expression χe (ω) = −4π

b2 , (ω − (ω0 − iδ)) (ω + (ω0 + iδ))

(5.183)

with a denominator that diﬀers from the usual expression (ω 2 + 2iδω − ω02 ) by the quantity δ 2 [cf. Eq. (4.204)]. The reversible and irreversible energies are then found as % % # $2 % 1 % π 0 b2 ˜ − (r, ω0 − iδ; t) % − iδ) E (ω Urev [E](r, t) = % 0 % 4π % 2 ω 2 0 $2 # 2 2 ˜ − (r, ω0 − iδ; t) + ω +δ E , (5.184) 0

% % # $2 2 t % 1 % ˜ % 2π 0 δb Uirrev [E](r, t) = % (ω − iδ) E (r, ω − iδ; t ) dt . 0 − 0 % 4π % ω02 −∞ (5.185) ˜ ω) relating the macro˜ ω) = 0 χe (ω)E(r, From the constitutive relation P(r, scopic polarization and electric ﬁeld spectra in a temporally dispersive HILL medium, it is found that [14] $ # ˜ − (r, ω0 − iδ; t) = ω0 P(r, ˙ (ω0 − iδ)E t), (5.186) b2 $ # ˜ − (r, ω0 − iδ; t) = − ω0 P(r, t). (5.187) E b2 With substitution of these two expressions, Eqs. (5.184) and (5.185) become % % 2 2 % 1 % 0 1 2 ˙ (r, t) + ω0 + δ P 2 (r, t) , (5.188) % Urev [E](r, t) = % P % 4π % 2 2b2 2b2 % % t % 1 % δ % P˙ 2 (r, t )dt . Uirrev [E](r, t) = % (5.189) % 4π % 0 b2 −∞ The result given in Eq. (5.188) then shows that the reversible energy for a single Lorentz oscillator type is given by the sum of its kinetic and potential energies. 5.2.6 Energy Velocity of a Time-Harmonic Field in a MultipleResonance Lorentz Model Dielectric A phenomenological point of view that is often useful is to consider the electromagnetic ﬁeld to have an associated electromagnetic energy that ﬂows

5.2 Electromagnetic Energy and Energy Flow

257

through space with an energy velocity vE . The direction and rate of ﬂow of electromagnetic energy per unit area through a surface normal to the direction of ﬂow is taken to be given by the time-average of the Poynting vector S(r, t) = c/4πE(r, t) × H(r, t), where E(r, t) and H(r, t) denote the realvalued electric and magnetic ﬁeld vectors, respectively, of the electromagnetic ﬁeld. The time-average velocity of propagation of electromagnetic energy is then given by S(r, t) , (5.190) vE ≡ Utot (r, t) where Utot (r, t) is the volume density of the total energy associated with the electromagnetic ﬁeld. When the electromagnetic ﬁeld propagates in a nondispersive HILL medium with real-valued dielectric permittivity (r) and propagation is found to magnetic permeability µ(r) , the velocity of energy have magnitude vE = |vE | that is given by c/ (r) µ(r) , where (r) = 1 and µ(r) = 1 in the special case when the medium is free-space (i.e., vacuum). The calculation is much more diﬃcult, however, when the the ﬁeld propagates through a temporally dispersive dielectric medium because it is then impossible to explicitly express the electric energy density and the dissipation separately in terms of the general dielectric permittivity of a general dispersive dielectric, as shown in §§ 5.2.2–5.2.4. Nevertheless, the calculation is possible if a speciﬁc dynamical model for the medium response is given, such as that provided by the Lorentz model [see §4.4.4]. Loudon’s [18, 19] original derivation of the energy velocity for a single-resonance Lorentz model dielectric as well as its generalization [20] to a multiple-resonance dielectric is now presented for a monochromatic electromagnetic ﬁeld. The connection between the resulting energy velocity for a time-harmonic wave and the signal velocity of a pulse in a Lorentz model dielectric is not apparent from the theory presented here. The fact that there is a close connection has been established by the modern asymptotic theory [21, 22] and is carefully developed in Volume II. One of the most spectacular results [23] of this modern asymptotic theory is that the frequency dependence of the energy velocity of a monochromatic wave with angular frequency ωc is shown to be an upper envelope to the frequency dependence of the signal velocity of a pulse with signal frequency ωc . The analysis begins with the diﬀerential form of Poynting’s theorem which may be written as [cf. Eq. (5.111)] % % % 1 % % µ0 H · ∂H + 0 E · ∂E + 4πE · ∂P ∇ · S = −% (5.191) % 4π % ∂t ∂t ∂t for a nonconducting, nonmagnetic, dispersive dielectric with µ = µ0 . First, consider obtaining an expression for the interaction term E · ∂P/∂t for a general, multiple-resonance Lorentz model dielectric. From Eq. (4.200), drj m d2 rj 2 + ωj rj , + 2δj E=− (5.192) qe dt2 dt

258

5 Fundamental Field Equations in a Temporally Dispersive Medium

and from Eqs. (4.201)–(4.203) the macroscopic polarization vector is seen to be given by P=− Nj qe rj , (5.193) j

so that 2 2 drj ∂P 1 d drj 1 2d 2 = Nj m + 2δj + ωj (rj ) . (5.194) E· ∂t 2 dt dt dt 2 dt j The quantity appearing on the right-hand side of Poynting’s theorem (5.191) is then seen to be the sum of a perfect diﬀerential in time and a term in δj which corresponds to the dissipation mechanism. If one then deﬁnes the energy density [cf. Eq. (5.156)] Utot ≡ Uem + Urev , where Uem

(5.195)

% % % 1 %1 2 2 % ≡% % 4π % 2 0 E + µ0 H

(5.196)

is the energy density stored in the electromagnetic ﬁeld alone, and where 2 drj 1 2 2 (5.197) Urev ≡ Nj m + ωj (rj ) 2 j dt is the reversible energy density stored in the multiple resonance Lorentz medium, then the diﬀerential form (5.191) of Poynting’s theorem becomes ∇ · S + 2m

Nj δj

j

drj dt

2 =−

dUtot . dt

(5.198)

Integration of this expression over an arbitrary region V bounded by a simple closed surface S followed by application of the divergence theorem then yields the expression

2

ˆ d r + 2m S·n S

j

Nj δj V

drj dt

2

d d r=− dt 3

Utot d3 r,

(5.199)

V

ˆ is the outward unit normal vector to the surface S. This relation exwhere n presses the conservation of energy in the dispersive Lorentz model dielectric, where the two terms on the left-hand side represent the rate of energy loss in the region V through ﬂow across its surface S and by dissipation in the medium contained in V , respectively, whereas the integral on the right-hand side represents the rate of change of the total electromagnetic energy stored within the region V .

5.2 Electromagnetic Energy and Energy Flow

259

For a time-harmonic ﬁeld with angular frequency ω, the time-average energy density stored in the multiple-resonance Lorentz model dielectric is found from Eq. (5.197) with (4.201) as % % % 1 % 0 2 −2α(ω)ˆs·r b2j (ω 2 + ωj2 ) % Urev = % . % 4π % 4 E0 e (ω 2 − ω 2 )2 + 4δ 2 ω 2 j

j

(5.200)

j

The time-average value of the energy density stored in the monochromatic electromagnetic ﬁeld is obtained from Eq. (5.196) with substitution of the relation H0 /E0 = (ω)/µ0 [cf. Eq. (5.65)] % % % 1 % 0 2 2 −2α(ω)ˆs·r 2 % . (5.201) Uem = % % 4π % 4 nr (ω) + ni (ω) + 1 E0 e With use of the ﬁrst of the pair of relations [cf. Eqs. (4.207) and (5.33)] n2r (ω) − n2i (ω) = 1 −

j

nr (ω)ni (ω) =

j

(ω 2

b2j (ω 2 − ωj2 ) , (ω 2 − ωj2 )2 + 4δj2 ω 2

(5.202)

b2j δj ω , − ωj2 )2 + 4δj2 ω 2

(5.203)

the total time-average electromagnetic energy density stored in both the ﬁeld and the medium is found to be given by Utot = Uem + Urev ⎡ ⎤ % % % 1 % 0 2 −2α(ω)ˆs·r 2 b2j ω 2 % ⎣nr (ω) + ⎦, =% % 4π % 2 E0 e 2 − ω 2 )2 + 4δ 2 ω 2 (ω j j j

(5.204)

where the summation extends over all of the medium resonances. Finally, from Eq. (5.152), the time-average value of the magnitude of the Poynting vector for a time-harmonic plane wave ﬁeld is found to be given by % 2% % c % nr (ω) 2 −2α(ω)ˆs·r % |S(r, t)| = % , (5.205) % 4π % 2µ0 c E0 e where nr (ω) denotes the real part of the complex index of refraction. For a single-resonance Lorentz model dielectric, Eq. (5.200) simpliﬁes to 2 % % 2 2 % 1 % 0 2 −2α(ω)ˆs·r ω b + ω 0 % Urev = % . (5.206) 2 % 4π % 4 E0 e (ω 2 − ω02 ) + 4δ 2 ω 2 Furthermore, the magnitude of the macroscopic polarization vector [given in Eq. (5.193) with substitution from Eq. (4.201)] becomes

260

5 Fundamental Field Equations in a Temporally Dispersive Medium

P = −

b2 (ω 2

−

ω02 )2

˜

+

4δ 2 ω 2

+

4δ 2 ω 2

E0 eik·r

(5.207)

with time derivative P˙ = −

b2 ω (ω 2

−

ω02 )2

˜

E0 eik·r .

(5.208)

With these expressions, the expression given in Eq. (5.206) for the timeaverage energy density stored in a single-resonance Lorentz model dielectric becomes % % 2 % 1 % 0 ω0 2 1 ˙ 2 −2α(ω)ˆs·r % % Urev = % % P + 2P e , (5.209) 4π 4 b2 b in agreement6 with the result given in Eq. (5.188). In addition, Eq. (5.204) becomes, with substitution from Eq. (5.203) % % % 1 % 0 2 −2α(ω)ˆs·r 2 nr (ω)ni (ω) % % Utot = % % E0 e nr (ω) + , (5.210) 4π 2 δ where ni (ω) denotes the imaginary part of the complex index of refraction. The time-average velocity of energy transport in a single-resonance Lorentz model dielectric is then given by the ratio of Eqs. (5.167) and (5.168) as vE ≡

c |S| = , Utot nr (ω) + ωni (ω)/δ

(5.211)

which is Loudon’s now classic result [18]. Brillouin’s derivation [24] of the energy velocity is in error due to the neglect of the electromagnetic energy reactively stored in the Lorentz oscillators. Because nr (ω) + ωni (ω)/δ ≥ 1 for all ω ≥ 0, then the energy transport velocity satisﬁes the inequality 0 ≤ vE ≤ c for all ω ≥ 0, in agreement with the relativistic principle of causality. The angular frequency dependence of the relative time-average energy velocity vE /c in a single-resonance Lorentz model dielectric is illustrated in Figure 5.5 for a highly lossy medium. For comparison, the relative phase and group velocities are also depicted. Although the phase velocity vp = ω/β(ω) yields superluminal values when ω > ω0 , and the group velocity vg = (dβ(ω)/dω)−1 yields both superluminal and negative values in the region of anomalous dispersion that extends from the angular frequency value at which the phase velocity is a minimum to the angular frequency value at which it is a maximum, the energy transport velocity vE is subluminal for all ω ∈ [0, +∞), approaching c from below in the limit as ω → +∞ from below. Notice that the energy velocity attains a minimum value just above the resonance angular frequency ω0 near the point where ni (ω) attains its maximum value and that of anomalous dispersion vE remains relatively small throughout the region that approximately extends from ω0 to ω1 ≡ ω02 + b2 . 6

Notice that the additional δ 2 term appearing in Eq. (5.188) is due to the addition of that term in the expression (5.183) for the model susceptibility.

5.2 Electromagnetic Energy and Energy Flow

261

3

Relative Wave Velocity v/c

2.5

2

1.5

vp /c

1

vg /c

0.5

vE /c 0

0

0.2

Ω

0.6

0.8

1

1.2

1.4

1.6

Ω - r/s

1.8

2 17

x 10

Fig. 5.5. Angular frequency dependence of the relative phase velocity vp /c (dashed curve), group velocity vg /c (dotted curve), and energy transport velocity vE /c (solid curve) for a plane wave ﬁeld in a single-resonance Lorentz model dielectric with √ medium parameters ω0 = 4 × 1016 r/s, b = 20 × 1016 r/s, and δ = 0.28 × 1016 r/s.

For a multiple-resonance Lorentz model dielectric, the ratio of the expression (5.205) for the time-average magnitude of the Poynting vector to the expression given in Eq. (5.204) for the total time-average electromagnetic energy density results in the general expression [20] vE (ω) = nr (ω) +

1 nr (ω)

c

b2j ω 2 j (ω 2 −ωj2 )2 +4δj2 ω 2

(5.212)

for the time-average velocity of energy transport. This result is an important generalization of the single-resonance expression (5.211) due to Loudon [18] and reduces to that result in that special case. Because the denominator appearing in the expression given in Eq. (5.212) is greater than or equal to unity for all ω ∈ [0, +∞), then for all physically realizable values of the angular frequency of oscillation of the monochromatic wave ﬁeld, the energy transport velocity given in Eq. (5.212) yields results that are in agreement with relativistic causality. The angular frequency dependence of the relative time-average energy velocity vE /c in a multiple-resonance Lorentz model of triply distilled water is illustrated in Figure 5.6. For comparison, the relative phase and group velocities are also depicted. Notice that the relative phase

262

5 Fundamental Field Equations in a Temporally Dispersive Medium

Relative Wave Velocity v/c

1.5

vp /c

vg /c 1

vE /c

0.5

0 12 10

13

10

Ω

Ω

Ω Ω1015 Ω - r/s

16

10

Ω

17

10

18

10

Fig. 5.6. Angular frequency dependence of the relative phase velocity vp /c (dashed curve), group velocity vg /c (dotted curve), and energy transport velocity vE /c (solid curve) for a plane wave ﬁeld in a multiple-resonance Lorentz model of triply distilled water with medium parameters given in Table 4.2.

velocity vp /c remains subluminal until the angular frequency of oscillation exceeds the uppermost resonance frequency ω8 . Notice also that the group velocity remains positive throughout the ﬁrst two absorption bands and that the group velocity approaches the energy velocity in each region of normal dispersion. Finally, notice that the energy velocity remains subluminal for all ω ∈ [0, +∞) and that it attains a local minimum near each resonance ωj .

5.3 Boundary Conditions Consider a continuous, smooth surface S that forms the boundary (or interface) between two separate homogeneous, isotropic, locally linear, temporally dispersive materials, across which there occur rapid changes in the values of the constitutive parameters (ω), µ(ω), and σ(ω) at a ﬁxed frequency ω.7 On a macroscopic scale, these changes in the material properties may typically 7

Notice that, because the electric and magnetic material properties are temporally dispersive, speciﬁc angular frequency values may exist such that one or more of the constitutive parameters do not change across S.

5.3 Boundary Conditions

263

be considered to be discontinuous so that the electromagnetic ﬁeld vectors themselves are expected to exhibit corresponding discontinuities across the interface surface S. The discontinuous change in the constitutive parameters across the interface S may be considered as the limiting case of the more physically realistic situation in which S is replaced by an inﬁnitesimally thin transition layer within which the material parameters (ω), µ(ω), and σ(ω) at ﬁxed ω vary rapidly, but continuously, from their (complex) values 1 (ω), µ1 (ω), σ1 (ω) in medium 1 to their (complex) values 2 (ω), µ2 (ω), σ2 (ω) in medium 2, as depicted in Figure 5.7. The thickness of the transition layer is denoted by ∆ and the appropriate limit to the discontinuous change across the interface surface S is obtained as ∆ → 0. The variation in material properties across the transition layer is assumed to be suﬃciently smooth such that the electromagnetic ﬁeld vectors and their ﬁrst derivatives are continuous bounded functions of both position and time.

Medium 2 Ε Μ Σ

v

n

l

v Τ

S

aye r

v

Ν

} TLransition

Medium 1 ΕΜΣ

Fig. 5.7. Transition layer about the interface S separating two temporally disˆ persive HILL media. At each point on the interface surface S, the unit vector n is normal to S and directed from medium 1 into medium 2, and the mutually ˆ ×n ˆ. orthogonal unit vectors τˆ and νˆ are tangent to S, where τˆ = ν

Consider ﬁrst Gauss’ law for the electric and magnetic ﬁelds, given respectively by the pair of integral relations ˜ ω) · ds = 4π ˜(r, ω)d3 r, (5.213) D(r, Σ R ˜ ω) · ds = 0 B(r, (5.214) Σ

264

5 Fundamental Field Equations in a Temporally Dispersive Medium

in the temporal frequency domain, where the simply connected closed surface Σ encloses the region R, and where ds denotes the outward-oriented diﬀerential element of surface area on Σ. Choose this surface Σ to be a vanishingly small right circular cylinder whose generators are normal to the interface S and whose end-caps (each with surface area ∆a) respectively lie in the upper and lower surfaces of the transition layer so that they are separated by the layer thickness ∆. For the magnetic ﬁeld behavior across the interface S, let ˜ 1 (r, ω) denote the magnetic ﬁeld vector at the center of the cylinder base B ˜ 2 (r, ω) denote ˆ 1 , and let B in medium 1 with outward unit normal vector n the magnetic ﬁeld vector at the center of the cylinder base in medium 2 with ˆ 2 . In the limit as both ∆ → 0 and ∆a → 0, the outward unit normal vector n ˜ 2 (r, ω) are evaluated at the same point ˜ 1 (r, ω) and B magnetic ﬁeld vectors B ˆ2 → n ˆ , where ˆ 1 → −ˆ n and n but on opposite sides of the interface S, and n ˆ is normal to S and directed from medium 1 into medium the unit vector n 2, as depicted in Figure 5.7. In that limit, Eq. (5.214) yields the boundary condition ˜ 1 (r, ω) = 0, r ∈ S. ˜ 2 (r, ω) − B ˆ· B (5.215) n Hence, the normal component of the magnetic induction ﬁeld vector is continuous across any surface of discontinuity in the material parameters (ω), µ(ω), and σ(ω). For the behavior of the normal component of the electric displacement vector across the interface S, a similar analysis applied to Eq. (5.213) results in the limiting expression ˜ ˜ ˆ · D2 (r, ω) − D1 (r, ω) = 4π lim lim ˜(r, ω)d3 r. n ∆ →0 ∆a→0

R

In the limit as ∆ → 0, the charge enclosed in the region R approaches the value S (r, ω)∆a, where S (r, ω) denotes the free surface charge density on the interface S, given by S (r, ω) ≡ lim {(r, ω)∆} . ∆ →0

With this identiﬁcation, the above boundary condition becomes ˜ 1 (r, ω) = 4πS (r, ω), r ∈ S, ˜ 2 (r, ω) − D ˆ· D n

(5.216)

(5.217)

provided that the order of the limiting process as ∆ → 0 and the integration over the region R can be interchanged. Thus, the normal component of the electric displacement vector is discontinuous across any surface of discontinuity in the material parameters (ω), µ(ω), and σ(ω), the amount of the discontinuity being proportional to the free surface charge density at that point. Consider next Faraday’s and Amp´ere’s laws, given respectively by the pair of integral relations

5.3 Boundary Conditions

265

˜ ω) · dl = iω ˜ ω) · ν ˆ d2 r, E(r, B(r, (5.218) c Σ C % % % 4π % iω ˜ ˜ ω) · ν ˜ ω) · ν % ˆ d2 r + % ˆ d2 r H(r, ω) · dl = − D(r, J(r, % c % c Σ C Σ (5.219) in the temporal frequency domain, where Σ denotes the surface region enˆ denotes the positive unit normal vector closed by the contour C, and where ν to the surface Σ, the direction of which is determined by the direction of integration about the contour C. Choose the contour C to be an inﬁnitesimally small plane rectangular loop whose sides (each with length ∆) are perpendicular to the interface S and whose top and bottom (each with length ∆s) respectively lie in the upper and lower surfaces of the transition layer about the interface S. Let τˆ 1 and τˆ 2 denote unit vectors in the direction of circulation about the contour C along the lower (medium 1) and upper (medium 2) sides, respectively, of the rectangular contour at the edge of each tran˜ 1 (r, ω) denote the electric ﬁeld value at the sition layer. In addition, let E midpoint of the lower side of C in medium 1 with circulation vector τˆ 1 , and ˜ 2 (r, ω) denote the electric ﬁeld value at the midpoint of the upper side let E of C in medium 2 with circulation vector τˆ 2 . In the limit as both ∆ → 0 ˜ 2 (r, ω) are evaluated at ˜ 1 (r, ω) and E and ∆s → 0, the electric ﬁeld vectors E τ the same point but on opposite sides of the interface S, whereas τˆ 1 → −ˆ and τˆ 2 → τˆ , where (see Fig. 5.7) ˆ ×n ˆ τˆ ≡ ν

(5.220)

deﬁnes the unit tangent vector to the surface S at that point. In this limit, Faraday’s law [Eq. (5.218)] becomes iω ˜ ˜ ˜ ˆ × E2 (r, ω) − E1 (r, ω) − ˆ· n lim B(r, ω)∆ = 0, r ∈ S, ν c ∆ →0 ˜ =ν ˜ Because the orientation of the contour C, ˆ) · E ˆ · (ˆ because (ˆ ν ×n n × E). ˆ , is entirely arbitrary, then and hence the direction of the unit vector ν ˜ 1 (r, ω) = iω lim B(r, ˜ ω)∆ , r ∈ S. ˜ 2 (r, ω) − E ˆ× E n c ∆ →0 Finally, because the ﬁeld vectors and their ﬁrst derivatives are assumed to be bounded, the right-hand side of this relation vanishes with ∆, resulting in the boundary condition ˜ 1 (r, ω) = 0, r ∈ S. ˜ 2 (r, ω) − E ˆ× E (5.221) n Hence, the tangential component of the electric ﬁeld intensity vector is continuous across any surface of discontinuity in the material parameters (ω), µ(ω), and σ(ω).

266

5 Fundamental Field Equations in a Temporally Dispersive Medium

For the behavior of the tangential component of the magnetic intensity vector across the interface S, a similar analysis applied to Amp´ere’s law [Eq. (5.219)] results in the limiting expression % % % 4π % ˜ ˜ ˜ ω)∆ % lim J(r, ˆ × H2 (r, ω) − H1 (r, ω) = % n % c % ∆ →0 iω ˜ ω)∆ , r ∈ S. lim D(r, − c ∆ →0 ˜ ω) and its ﬁrst derivatives are bounded, the second term on the Because D(r, right-hand side of this expression vanishes in the limit as ∆ → 0. However, the ﬁrst term on the right-hand side of this expression does not necessarily ˆ ∆s∆ ﬂowing through vanish as it is proportional to the current J = J(r, t)· ν the rectangular loop C. In the limit as ∆ → 0, the current J approaches ˆ ∆s, where the value JS (r, t) · n # $ ˜ S (r, ω) ≡ lim J(r, ˜ ω)∆ J (5.222) ∆ →0

is the temporal frequency spectrum of the surface current density on the interface S between the two media. With this identiﬁcation, the above boundary condition becomes % % % % ˜ ˜ 1 (r, ω) = % 4π % J ˜ 2 (r, ω) − H ˆ× H (5.223) n % c % S (r, ω) r ∈ S. Hence, the presence of a surface current on the interface S across which the material parameters (ω), µ(ω), and σ(ω) change discontinuously, results in a discontinuous change in the tangential component of the magnetic ﬁeld intensity vector, the amount of the discontinuity being proportional to the surface current density at that point. The surface charge and current densities are not independent, but rather are related by the equation of continuity given in Eq. (4.56), taken in the limit as ∆ → 0, whose temporal Fourier transform yields ˜ S (r, ω) − iω ˜S (r, ω) = 0. ∇·J

(5.224)

Because of this relationship, it is necessary only to apply the boundary conditions given in Eqs. (5.221) and (5.223) on the tangential components of the electric and magnetic ﬁeld vectors, the boundary conditions given in Eqs. (5.215) and (5.217) then being automatically satisﬁed. 5.3.1 Boundary Conditions for Nonconducting Dielectric Media For a purely dielectric material, the conductivity is identically zero at all frequencies; viz., σ(ω) = 0 ∀ω. (5.225)

5.3 Boundary Conditions

267

Such a dielectric cannot then furnish free charge, so that if no excess charge is externally supplied to the interface between two such dielectric materials, then the surface charge density vanishes; viz., ˜S (r, ω) = 0 ∀ω,

r ∈ S.

(5.226)

The equation of continuity (5.224) for the surface charge and current densities then requires that the surface current density be solenoidal. However, because the conductivity for each material is identically zero, then the constitutive relation given in Eq. (4.113) requires that the conduction current density vanish in each medium, so that ˜ S (r, ω) = 0 ∀ω, J

r∈S

(5.227)

on the interface between these two media. The boundary conditions for the electric and magnetic ﬁeld vectors across an interface surface S separating two nonconducting dielectric media with material properties 1 (ω), µ1 (ω) in medium 1 and 2 (ω), µ2 (ω) in medium 2 are then given by the set of relations ˜ 1 (r, ω) = 0, r ∈ S, ˜ 2 (r, ω) − D ˆ· D (5.228) n ˜ 1 (r, ω) = 0, r ∈ S, ˜ 2 (r, ω) − B ˆ· B (5.229) n ˜ 1 (r, ω) = 0, r ∈ S, ˜ 2 (r, ω) − E ˆ× E n (5.230) ˜ 1 (r, ω) = 0, r ∈ S. ˜ 2 (r, ω) − H ˆ× H n (5.231) This set of relations can then be used to determine the bending of the electric and magnetic lines of force across the interface S. The solution in the lossless case is trivial, whereas the solution in the general case when both media are attenuative is suﬃciently interesting to merit inclusion as a problem. 5.3.2 Boundary Conditions for Dielectric–Conductor Interfaces Consider ﬁrst the case in which medium 1 is a perfect conductor, so that ˆ to the interface S be σ(ω) = ∞ for all ω. Let the unit normal vector n directed from this perfectly conducting material into the (nonconducting) dielectric material (medium 2), as depicted in Figure 5.7. In order that the conduction current density in a perfect conductor have, at most, ﬁnite values, the electric ﬁeld inside such an idealized conductor must identically vanish and the boundary condition given in Eq. (5.217) becomes ˜ 2 (r, ω) = 4π˜ ˆ ·D S (r, ω), n

r, ∈ S.

(5.232)

The free charge in a perfect conductor then moves “instantaneously” in response to any applied time-varying electromagnetic ﬁeld, producing the correct surface charge density to identically cancel the externally applied electric ﬁeld within the body of the ideal conductor. In turn, Faraday’s law then

268

5 Fundamental Field Equations in a Temporally Dispersive Medium

requires that the magnetic ﬁeld also identically vanish inside a perfect conductor so that the boundary condition given in Eq. (5.223) becomes % % % 4π % ˜ %˜ ˆ × H2 (r, ω) = % n (5.233) % c % JS (r, ω), r ∈ S. The surface charge on a perfect conductor then moves “instantaneously” in response to the tangential magnetic ﬁeld to identically cancel the externally applied magnetic ﬁeld within the body of the ideal conductor. The remaining two boundary conditions appearing in Eqs. (5.215) and (5.221) for the normal component of the magnetic induction ﬁeld vector and the tangential component of the electric ﬁeld vector, respectively, then become ˜ 2 (r, ω) = 0, ˆ ·B n ˜ 2 (r, ω) = 0, ˆ ×E n

r ∈ S,

(5.234)

r ∈ S.

(5.235)

The boundary conditions stated in Eqs. (5.232)–(5.235) then state that only normal E and tangential H ﬁelds can exist immediately outside the surface of a perfect conductor, the ﬁeld magnitudes dropping discontinuously to zero inside the perfect conductor body. If the conducting body has ﬁnite conductivity, then Ohm’s law [see Eqs. (4.113)–(4.114)] for a homogeneous, isotropic, locally linear conductor applies and ˜ 1 (r, ω) ˜ 1 (r, ω) = σ1 (ω)E (5.236) J in the absence of any externally supplied current source. The divergence of the temporal frequency domain form of Amp´ere’s law [see Eq. (5.28)] yields ˜ ω), ˜ 2 (r, ω) = 0 = − 1 iω c2 (ω)∇ · E(r, ∇· ∇×H c where c2 (ω) = 2 (ω) + i4πσ2 (ω)/ω is the complex permittivity of the conducting material, so that with Gauss’ law ˜ 2 (r, ω) = ˜2 (r, ω) , ∇·E

2 (ω)

(5.237)

one ﬁnds that ˜2 (r, ω) = 0 in the conducting material. This result, taken together with the fact that the nonconducting dielectric medium 1 cannot furnish free charge, then leads to the conclusion that the surface charge density vanishes on S, viz. ˜S (r, ω) = 0,

r ∈ S.

(5.238)

The equation of continuity given in Eq. (5.224) for the surface charge and current then requires that ˜ S (r, ω) = 0, ∇·J

r ∈ S,

(5.239)

5.3 Boundary Conditions

269

and the idealized surface current is solenoidal. Inside the conducting medium the electric and magnetic ﬁelds are attenuated exponentially with the normal distance ξ from the interface as e−ξ/dp , where [see Eq. (5.85)] % % 1/2 % c % 2 % % dp ≈ % √ % ωµ (ω)σ (ω) 2 π 1

(5.240)

1

is the penetration depth or skin depth of the conductor. Here µ1 (ω) ≡ {µ1 (ω)} and σ1 (ω) ≡ {σ1 (ω)} denote the real parts of the magnetic permeability and electric conductivity, respectively, in the conducting medium. Notice that this expression for the penetration depth is valid when the overall material loss is dominated by the conductive loss over the frequency domain of interest. This distance then deﬁnes [25] a physical transition layer between the electromagnetic ﬁeld quantities in medium 2 just outside the surface S and the vanishing ﬁeld quantities suﬃciently deep inside the conductor body (medium 1). As a consequence, the boundary condition appearing in Eq. (5.233) for the tangential component of the magnetic ﬁeld vector is replaced by the boundary condition [cf. Eq. (5.231)] ˜ 1 (r, ω) = 0, r ∈ S. ˜ 2 (r, ω) − H ˆ× H (5.241) n The other tangential boundary condition given in Eq. (5.221) remains valid. ˜ 2 (r, ω) of the magnetic ﬁeld If there exists a tangential component H just outside the surface S of the conducting medium, then the boundary condition given in Eq. (5.241) states that the same tangential component exists just inside the surface S. With this boundary value, the electromagnetic ﬁeld behavior in the conducting material can then be constructed from the appropriate solution to Maxwell’s equations % % %1% ˜ ˜ % (5.242) ∇ × E1 (r, ω) = % % c % iωµ1 (ω)H1 (r, ω), % % % % ˜ 1 (r, ω), ˜ 1 (r, ω) = − % 1 % iω c1 (ω)E (5.243) ∇×H %c% where c1 (ω) = 1 (ω) + i4πσ1 (ω)/ω is the complex permittivity and µ1 (ω) the magnetic permeability of the conducting material. For suﬃciently small real-valued angular frequency values ω such that

1 (ω) 4π

σ1 (ω) , ω

(5.244)

where 1 (ω) ≡ { 1 (ω)}, the conductive loss dominates the dielectric loss, and because the dielectric loss is typically assumed to dominate the magnetic loss [i.e., µ1 (ω) 1 (ω)], the pair of curl relations (the time-harmonic

270

5 Fundamental Field Equations in a Temporally Dispersive Medium

Maxwell’s equations) appearing in Eqs. (5.242)–(5.243) may be approximated as % c % 1 % % ˜ 1 (r, ω), ˜ 1 (r, ω) ∼ ∇×H (5.245) E =% % 4π σ1 (ω) i ˜ 1 (r, ω). ˜ 1 (r, ω) ∼ ∇×E (5.246) H = −c ωµ1 (ω) Let the variable ξ ≥ 0 denote the normal coordinate distance to the interface surface S into the conducting medium, where ξ = 0 on S. Then ∇ ≈ −ˆ n

∂ , ∂ξ

(5.247)

the remaining spatial derivatives of the ﬁeld quantities being assumed negligible within the conducting medium [25]. With this substitution, the pair of curl relations appearing in Eqs. (5.245)–(5.246) becomes % c % 1 ˜ 1 (ξ, ω) ∂H % ˜ 1 (ξ, ω) ≈ − % ˆ× , E n % % 4π σ1 (ω) ∂ξ ˜ 1 (ξ, ω) i ∂E ˜ 1 (ξ, ω) ≈ c ˆ× . n H ωµ1 (ω) ∂ξ

(5.248) (5.249)

This pair of relations immediately yields the (approximate) transversality relation ˜ 1 (ξ, ω) ≈ 0, ˜ 1 (ξ, ω) ≈ n ˆ ·E ˆ ·H n (5.250) ˜ 1 (r, ω) are parallel to the interface surface S. ˜ 1 (r, ω) and H so that both E Substitution of Eq. (5.248) into Eq. (5.249) then yields the expression % 2% %c % i ∂ 2 ˜ ˜ 1 (ξ, ω) n ˜ 1 (ξ, ω) , % ˆ ˆ n · H − H H1 (ξ, ω) ≈ − % % 4π % ωµ (ω)σ (ω) ∂ξ 2 1 1 so that [after applying the approximate transversality relation given in Eq. (5.250)] % % % 4π % ∂2 ˜ ˜ % H1 (ξ, ω) + % % c2 % iωµ1 (ω)σ1 (ω)H1 (ξ, ω) ≈ 0, ∂ξ 2 which may be written in terms of the penetration depth dp as ∂2 ˜ 2i ˜ H1 (ξ, ω) + 2 H 1 (ξ, ω) ≈ 0. ∂ξ 2 dp

(5.251)

The solution of this partial diﬀerential equation that is consistent with the boundary condition given in Eq. (5.241) is then ˜ 2 (r, ω)e−ξ/dp eiξ/dp , ˜ 1 (ξ, ω) ≈ H H

r ∈ S.

(5.252)

5.3 Boundary Conditions

271

From Eq. (5.248), the electric ﬁeld in the conducting material corresponding to this magnetic ﬁeld solution is given by % % % 1 % ωµ1 (ω) 1/2 ˜ 2 (r, ω) e−ξ/dp eiξ/dp e−iπ/4 , ˜ % % ˆ ×H n E1 (ξ, ω) ≈ % √ % σ1 (ω) 2 π

r ∈ S,

(5.253) which is shifted in phase by −π/4 from the magnetic ﬁeld vector. For suﬃciently small angular frequencies such that the inequality ω σ1 (ω)/µ1 (ω) is satisﬁed, the electric ﬁeld strength will be negligible in comparison to the magnetic ﬁeld strength in the conductor, whereas in the opposite limit when ω σ1 (ω)/µ1 (ω), the magnetic ﬁeld strength will be negligible in comparison to the electric ﬁeld strength. From the boundary condition given in Eq. (5.221) for the tangential component of the electric ﬁeld vector, it is seen that there exists a tangential component of the electric ﬁeld just above the interface surface S of the conductor that is given by the expression in Eq. (5.253) evaluated at ξ = 0, so that % % % 1 % ωµ1 (ω) 1/2 ˜ 2 (r, ω) e−iπ/4 , r ∈ S. (5.254) ˜ % % ˆ n × H E2 (r, ω) ≈ % √ % σ1 (ω) 2 π ˜ ˜ ω)] then shows Faraday’s law of induction [H(r, ω) = (c/iωµ(ω)) ∇ × E(r, ˜ that there exists a small normal component H2⊥ (r, ω) of the magnetic ﬁeld vector just above the surface of the conducting medium. With the decomposi˜ (r, ω) of the electric ﬁeld vector into normal and ˜ ω) = E ˜ ⊥ (r, ω) + E tion E(r, ˆ ∂/∂ξ, longitudinal components at the interface and the identity ∇ = ∇⊥ − n Faraday’s law becomes ˜ 2 (r, ω) ∂ E c ˜ 2 (r, ω) − n ˜ 2 (r, ω) = ˜ 2⊥ (r, ω) + E ˆ× ∇⊥ × E . H iωµ2 (ω) ∂ξ (5.255) ˜ 2⊥ (r, ω) and n ˜ 2 (r, ω)/∂ξ terms appearing on the rightˆ × ∂E The ∇⊥ × E hand side of this expression yield the tangential magnetic ﬁeld component H˜2 (r, ω) on the interface surface S, the second of these two terms giving the zeroth-order boundary value for the magnetic intensity vector appearing in Eqs. (5.253)–(5.254) so that the ﬁrst of these two terms provides the ﬁrst˜ 2 (r, ω) term order correction to this prescribed boundary value. The ∇⊥ × E yields the desired expression for the normal component of the magnetic ﬁeld ˜ 2 (r, ω), so ˜ 2⊥ (r, ω) = (c/iωµ2 (ω)) ∇T × E vector on the interface S as H that, with the result given in Eq. (5.254), one obtains the ﬁrst-order correction % % 1/2 % c % µ1 (ω) ˜ ˜ 2⊥ (r, ω) ≈ % √ % ˆ × H (r, ω) eiπ/4 , H n × ∇ T 2 % 2 π % ωµ (ω)σ (ω) 2 1

r∈S (5.256)

272

5 Fundamental Field Equations in a Temporally Dispersive Medium

to the imposed boundary value on S. If required, an iterative procedure may then be used to numerically determine the precise boundary values for a given problem. ˜ 1 (r, ω) in the conduct˜ c (r, ω) ≈ σ (ω)E The conduction current density J 1 ing material is, with substitution from Eq. (5.253), given by % c % √2 % % ˜ ˜ 2 (r, ω) e−iπ/4 e−(1−i)ξ/dp , r ∈ S, (5.257) ˆ ×H n Jc (ξ, ω) ≈ % % 4π dp (ω) for ξ ≥ 0. Because this current density is eﬀectively conﬁned to a thin layer just below the interface surfce S of the conducting medium, it is equivalent to an eﬀective surface current density ∞ ˜ c (ξ, ω)dξ ˜ S (r, ω) ≡ J J ef f 0 % c % % % ˜ 2 (r, ω) , r ∈ S. ˆ ×H ≈% % n (5.258) 4π Thus, a good conductor behaves eﬀectively as a perfect conductor with the idealized surface current replaced by an eﬀective surface current that is exponentially distributed beneath the surface of the conducting material [25]. The existence of nonvanishing tangential components of the electric and magnetic ﬁeld vectors at the interface surface S implies that electromagnetic power is coupled into the conducting medium across S. From the timeharmonic form of Poynting’s theorem (see §5.2.3), the time-average electromagnetic power absorbed per unit area on the surface S is given by the real part of the complex Poynting vector as ' & % # $ dPloss 1% % c % ˜ 1 (r, ω) × H ˜ ∗ (r, ω) ˆ· E = − % % n 1 da 2 4π % # $ 1% % c % ˜ 2 (r, ω) × H ˜ ∗ (r, ω) , ˆ· E ≈ − % % n 2 2 4π for r ∈ S. With substitution from Eq. (5.254), this expression becomes ' % % & 2 % 1 %1 dPloss ˜ 2 (r, ω) , r ∈ S, % ωµ1 (ω)dp (ω) H (5.259) ≈% % 4π % 4 da which should be equal to the time-average ohmic power loss in the conductor [see Eq. (5.138)] per unit surface area on S. This latter quantity may be obtained from the expression & ' dPΩ 1˜ ˜∗ = J c (ξ, ω) · E1 (ξ, ω) dV 2 2 1 ˜ J ≈ (ξ, ω) c 2σ1 (ω) % % 2 % c 2% 1 ˜ −2ξ/dp % H ≈% (r, ω) , (5.260) e 2 % 4π % σ (ω)d2 (ω) p 1

5.4 Discussion

273

for the time-average ohmic power loss per unit volume in the conductor, where r ∈ S. The ohmic power loss in the conducting medium per unit area on the interface surface S is then given by & ' ∞& ' dPΩ dPΩ = dξ da dV %0 % 2 % c 2% 1 ˜ % H ≈% (r, ω) 2 % 4π % σ (ω)dp (ω) 1 % % 2 % 1 %1 ˜ 2 (r, ω) , r ∈ S, % ωµ1 (ω)dp (ω) H =% (5.261) % 4π % 4 which is just that given in Eq. (5.259).

5.4 Discussion The results presented in this chapter provide a detailed description of the fundamental electromagnetic properties that appear in causal, temporally dispersive HILL media. In particular, the general properties of time-harmonic plane wave propagation in a causally dispersive HILL medium have been developed, including energy ﬂow and dissipation. The description is valid for homogeneous, isotropic, locally linear dielectric, conducting, or semiconducting materials. These basic results are fundamental to both the mathematical formulation and physical interpretation of pulsed electromagnetic beam ﬁeld propagation in such media. In addition, the approximate boundary conditions for a dielectric–conductor interface provide the basis for analyzing the electromagnetic mode properties of dielectric-ﬁlled metallic waveguides. The analysis of waveguide modes has been extensively studied for ideal metallic waveguides [26] where mode guidance is achieved by near ideal reﬂection from the highly conducting waveguide walls, as well as for lossless dielectric waveguides [27] where mode guidance is achieved by total internal reﬂection from the lossless dielectric interface.

274

5 Fundamental Field Equations in a Temporally Dispersive Medium

References 1. 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. 2. 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. 3. 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. 4. W. S. Franklin, “Poynting’s theorem and the distribution of electric ﬁeld inside and outside of a conductor carrying electric current,” Phys. Rev., vol. 13, no. 3, pp. 165–181, 1901. 5. J. Neufeld, “Revised formulation of the macroscopic Maxwell theory. I. Fundamentals of the proposed formulation,” Il Nuovo Cimento, vol. LXV B, no. 1, pp. 33–68, 1970. 6. J. Neufeld, “Revised formulation of the macroscopic Maxwell theory. II. Propagation of an electromagnetic disturbance in dispersive media,” Il Nuovo Cimento, vol. LXVI B, no. 1, pp. 51–76, 1970. 7. C. Jeﬀries, “A new conservation law for classical electrodynamics,” SIAM Review, vol. 34, no. 4, pp. 386–405, 1992. 8. H. G. Schantz, “On the localization of electromagnetic energy,” in UltraWideband, Short-Pulse Electromagnetics 5 (P. D. Smith and S. R. Cloude, eds.), pp. 89–96, New York: Kluwer Academic, 2002. 9. F. N. H. Robinson, “Poynting’s vector: Comments on a recent paper by Clark Jeﬀries,” SIAM Review, vol. 36, no. 4, pp. 633–637, 1994. 10. C. Jeﬀries, “Response to a commentary by F. N. H. Robinson,” SIAM Review, vol. 36, no. 4, pp. 638–641, 1994. 11. L. D. Landau and E. M. Lifshitz, Electrodynamics of Continuous Media. Oxford: Pergamon, 1960. Ch. IX. 12. Y. S. Barash and V. L. Ginzburg, “Expressions for the energy density and evolved heat in the electrodynamics of a dispersive and absorptive medium,” Usp. Fiz. Nauk., vol. 118, pp. 523–530, 1976. [English translation: Sov. Phys.Usp. vol. 19, 163–270 (1976)]. 13. J. M. Carcione, “On energy deﬁnition in electromagnetism: An analogy with viscoelasticity,” J. Acoust. Soc. Am., vol. 105, no. 2, pp. 626–632, 1999. 14. C. Broadbent, G. Hovhannisyan, M. Clayton, J. Peatross, and S. A. Glasgow, “Reversible and irreversible processes in dispersive/dissipative optical media: Electro-magnetic free energy and heat production,” in Ultra-Wideband, ShortPulse Electromagnetics 6 (E. L. Mokole, M. Kragalott, and K. R. Gerlach, eds.), pp. 131–142, New York: Kluwer Academic, 2003. 15. F. W. Sears, An Introduction to Thermodynamics, the Kinetic Theory of Gases, and Statistical Mechanics. Reading, MA: Addison-Wesley, second ed., 1953. 16. J. H. Eberly and K. W´ odkiewicz, “The time-dependent physical spectrum of light,” J. Opt. Soc. Am., vol. 67, no. 9, pp. 1252–1261, 1977. 17. J. Peatross, M. Ware, and S. A. Glasgow, “Role of the instantaneous spectrum on pulse propagation in causal linear dielectrics,” J. Opt. Soc. Am. A, vol. 18, no. 7, pp. 1719–1725, 2001.

5.4 Problems

275

18. R. Loudon, “The propagation of electromagnetic energy through an absorbing dielectric,” Phys. A, vol. 3, pp. 233–245, 1970. 19. R. Loudon, The Quantum Theory of Light. London: Oxford University Press, 1973. 20. 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. 21. 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. 22. S. Shen and K. E. Oughstun, “Dispersive pulse propagation in a doubleresonance Lorentz medium,” J. Opt. Soc. Am. B, vol. 6, pp. 948–963, 1989. 23. 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. 24. L. Brillouin, Wave Propagation and Group Velocity. New York: Academic, 1960. 25. J. D. Jackson, Classical Electrodynamics. New York: John Wiley & Sons, third ed., 1999. 26. R. E. Collin, Field Theory of Guided Waves. Piscataway, NJ: IEEE, second ed., 1991. 27. D. Marcuse, Theory of Dielectric Optical Waveguides. New York: Academic, 1974.

Problems 5.1. Derive the diﬀerential equation satisﬁed by the complex charge density cω (r, t) in a temporally dispersive semiconducting material. From the solution to this equation, determine the relaxation time of the charge density in the medium and show that c (r, t) must vanish in the absence of external sources. 5.2. From the symmetry relations given in Eqs. (4.97), (4.116), and (4.147) for real ω, obtain the possible forms of the symmetry relation for the complex index of refraction n(ω) = [µ(ω) c (ω)/(µ0 0 )]1/2 , where c (ω) =

(ω) + i4πσ(ω)/ω is the complex permittivity. 5.3. Consider a homogeneous, bianisotropic, locally linear, nondispersive, nonconducting medium that is characterized by the constitutive relations D(r) = · E(r) + ξ · H(r), B(r) = ζ · E(r) + µ · H(r), where , ξ, ζ, and µ are complex-constant dyadics. Determine the conditions that these material tensors must satisfy if the medium is lossless. 5.4. Obtain appropriate expressions for the time-harmonic plane wave propagation and attenuation factors β(ω) and α(ω), respectively, that are valid when c (ω) < 0 and when c (ω) = 0.

276

5 Fundamental Field Equations in a Temporally Dispersive Medium

5.5. Express the macroscopic Poynting vector deﬁned in Eq. (5.58) in terms of the spatial average of its microscopic counterpart given in Eq. (2.145). 5.6. Derive the approximate expression given in Eq. (5.108) for the magnetic susceptibilty in either diamagnetic or paramagnetic materials. 5.7. Derive the expression given in Eq. (5.152). 5.8. Derive the expression given in Eq. (5.205) for the time-average value of the magnitude of the Poynting vector for a time-harmonic plane wave ﬁeld in a temporally dispersive HILL dielectric. 5.9. Show that the general expression given in Eq. (5.189) for the irreversible electromagnetic energy yields the proper loss term for a single resonance Lorentz model dielectric. 5.10. Show that the energy and group velocities are approximately equal in the normal dispersion regions of a single resonance Lorentz model dielectric. 5.11. Determine how the electric and magnetic streamlines change across an interface S separating two dissimilar lossy dielectric media.

6 The Angular Spectrum Representation of the Pulsed Radiation Field in a Temporally Dispersive Medium

Attention is now directed to the rigorous solution of the electromagnetic ﬁeld that is radiated by a general current source in a homogeneous, isotropic, locally linear, temporally dispersive medium occupying all of space that is characterized by the time-dependent dielectric permittivity response function

ˆ(t), magnetic permeability response function µ ˆ(t), and electric conductivity response function σ ˆ (t). The applied current source J0 (r, t) is assumed here

J0(r,t)

O

z = -Z

z

z=Z

Fig. 6.1. General current source embedded in a dispersive medium.

to be a well-behaved known function of both position r and time t that identically vanishes for |z| ≥ Z, where Z is a positive constant, as illustrated in Figure 6.1. Furthermore, it is assumed that the current source is turned on at time t = 0, so that J0 (r, t) = 0 ,

t ≤ 0.

(6.1)

Because the present analysis is concerned only with the electromagnetic ﬁeld that is radiated by this current source, it is then required that both ﬁeld

278

6 The Angular Spectrum Representation of the Pulsed Radiation Field

vectors E(r, t) and B(r, t) also vanish for t ≤ 0; viz.,

E(r, t) = 0 t ≤ 0. B(r, t) = 0

(6.2)

These requirements on the current source are not restrictive in any physical sense because all real radiation problems may be cast so as to satisfy them.

6.1 The Fourier–Laplace Integral Representation of the Radiation Field From the constitutive relations given in Eqs. (5.5)–(5.6) for a temporally dispersive HILL medium and the conditions given in Eqs. (6.1) and (6.2), the total current density in the medium after the current source has been turned on (i.e., for t > 0) is given by t J(r, t) = J0 (r, t) + σ ˆ (t − t )E(r, t )dt , (6.3) 0

and the electric displacement and magnetic intensity vectors are t

ˆ(t − t )E(r, t )dt , D(r, t) = 0 t µ ˆ−1 (t − t )B(r, t )dt , H(r, t) =

(6.4) (6.5)

0

respectively. The Fourier–Laplace transform of these relations then gives, with application of the convolution theorem given in Eq. (C.15) for the Laplace transform, ˜ ˜ (k, ω) + σ(ω)E(k, ˜ ω) = J ˜˜ ˜ J(k, ω), 0 ˜ ˜ ˜ ˜ D(k, ω) = (ω)E(k, ω),

(6.6)

˜ ˜ ˜ ˜ ω). H(k, ω) = µ−1 (ω)B(k,

(6.8)

(6.7)

The spatiotemporal evolution of the electromagnetic ﬁeld in a temporally dispersive HILL medium occupying all of three-dimensional space with charge and current sources is described by the macroscopic Maxwell’s equations ∇ · D(r, t) = 4π(r, t), % % % 1 % ∂B(r, t) % , ∇ × E(r, t) = − % %c% ∂t ∇ · B(r, t) = 0, % % % % % 1 % ∂D(r, t) % 4π % % % % ∇ × H(r, t) = % % +% % c % J(r, t). c ∂t

(6.9) (6.10) (6.11) (6.12)

6.1 The Fourier–Laplace Integral Representation of the Radiation Field

279

The Fourier–Laplace transform of these relations, with application of Eq. (C.13) and the initial conditions given in Eq. (6.2), then gives the set of vector algebraic equations, using Eqs. (C.19) and (C.20), ˜ ˜ ˜˜(k, ω), ik · D(k, ω) = 4π % % %1% ˜ ˜ ˜ % ˜ ik × E(k, ω) = % % c % iω B(k, ω), ˜ ˜ ik · B(k, ω) = 0,

% % % % %1% ˜ % 4π % ˜ ˜ ˜ ω). ˜ ˜ % iω D(k, % % J(k, ik × H(k, ω) + ω) = − % %c% % c %

(6.13) (6.14) (6.15) (6.16)

With Eq. (6.7), the ﬁrst of the above set of equations may be expressed as (k, ω) ˜ ˜ k · E(k, . ω) = −4πi

(ω) In addition, substitution of Eqs. (6.6)–(6.7) into Eq. (6.16) yields % % % % %1% % 4π % ˜˜ ˜ ˜ ˜ ˜ % % % k × B(k, ω) = − % % ωµ(ω) c (ω)E(k, ω) − i % % c % µ(ω)J(k, ω), c

(6.17)

(6.18)

where c (ω) = (ω) + i4πσ(ω)/ω is the complex permittivity of the dispersive, conducting medium, as deﬁned in Eq. (5.29). The spatiotemporal spectrum of the charge density may be expressed in terms of the spatiotemporal spectrum of the current density through the use of the equation of continuity ∇ · J(r, t) = −

∂(r, t) . ∂t

(6.19)

The Fourier–Laplace transform of this conservation relation then results in ˜ ˜ ω) = iω ˜˜(k, ω), where the fact that the initial value of the equation ik · J(k, the charge density (r, t) vanishes has been used. Substitution of Eq. (6.6) into this equation and solving for the spatiotemporal spectrum of the charge density then yields ˜˜ ˜ ˜ ˜˜(k, ω) = k · J0 (k, ω) + σ(ω)k · E(k, ω) . ω With this result, Eq. (6.17) becomes ˜˜ ˜ ˜ 0 (k, ω) + σ(ω)k · E(k, k·J ω) ˜˜ k · E(k, , ω) = −4π ω (ω) so that

(6.20)

280

6 The Angular Spectrum Representation of the Pulsed Radiation Field

˜ ˜ 0 (k, ω) k·J ˜ ˜ , k · E(k, ω) = −4πi ω c (ω)

(6.21)

where c (ω) is the complex permittivity of the dispersive medium. With Eqs. (6.18) and (6.21), the spatiotemporal form of Maxwell’s equations given in Eqs. (6.13)–(6.16) take the ﬁnal form ˜ ˜ 0 (k, ω) k·J ˜ ˜ k · E(k, , (6.22) ω) = −4πi ω c (ω) % % %1% ˜ ˜ ˜ % ˜ (6.23) k × E(k, ω) = % % c % ω B(k, ω), ˜ ˜ k · B(k, ω) = 0, (6.24) % % % % % % % % 1% ˜˜ ˜ ˜ ˜ ˜ % 4π % k × B(k, ω) = − % % c % ωµ(ω) c (ω)E(k, ω) − % c % iµ(ω)J0 (k, ω). (6.25) The original set of partial diﬀerential relations for the real space-time form of the electric and magnetic ﬁeld vectors have thus been replaced by a set of algebraic equations in which the spatiotemporal spectra of the electric and magnetic ﬁeld vectors are the only remaining unknown quantities. Consider ﬁrst the solution for the spatiotemporal frequency spectrum of the electric ﬁeld vector. The vector product of k with the expression in Eq. (6.23) gives % % % % ˜˜ ˜ ˜ = % 1 % ωk × B. (6.26) k× k×E %c% With the vector identity for the triple vector product, the left-hand side of this equation becomes ˜˜ ˜ ˜ ˜ = k·E ˜ k − k2 E k× k×E = −4πi

˜ ˜0 k·J ˜˜ k − k 2 E, ω c (ω)

with substitution from Eq. (6.22). Furthermore, the right-hand side of Eq. ˜ ˜ (6.26) can be expressed in terms of E(k, ω) with substitution from Eq. (6.25). With these substitutions, Eq. (6.26) becomes

k2 −

% % ˜˜ % % µ(ω) c (ω) 2 ˜ 0 ˜˜ − 4πi k · J ˜ = % 4π % iωµ(ω)J ω k. E 0 % c2 % c2 ω c (ω)

With the deﬁnition of the complex velocity in a temporally dispersive HILL medium with complex permittivity c (ω) and magnetic permeability µ(ω) as the square root of the quantity

6.1 The Fourier–Laplace Integral Representation of the Radiation Field

v 2 (ω) ≡

c2 , µ(ω) c (ω)

281

(6.27)

one ﬁnally obtains the expression ˜ ˜ E(k, ω) = 4πi

%1% ˜ ˜ 0 (k, ω) − % 2 % ωµ(ω)J c

k2 −

˜ ˜ 0 (k,ω) k·J ωc (ω) k

ω2 v 2 (ω)

.

(6.28)

This equation can be somewhat simpliﬁed with the application of the vector ˜ ˜ ˜ ˜ 0 )k = k × (k × J ˜ 0 , with which the numerator in Eq. ˜ 0 ) + k2 J identity (k · J (6.28) becomes % % 2 ˜ ˜ %1% ω 1 ˜ ˜˜ 2 ˜ ˜ 0 − k · J0 k = ˜0 − k × k × J % % ωµ(ω)J − k . J 0 % c2 % ω c (ω) ω c (ω) v 2 (ω) With this substitution, Eq. (6.28) may be rewritten as ⎡ ⎤ ˜ ˜ 0 (k, ω) k× k×J i ˜˜ (k, ω)⎦ . ˜ ˜ ⎣ −J E(k, ω) = 4π 0 ω2 2 ω c (ω) − k 2 v (ω)

(6.29)

The solution for the spatiotemporal frequency spectrum of the magnetic induction ﬁeld vector is obtained by taking the vector product of k with the expression in Eq. (6.25) and applying the relations given in Eqs. (6.23) and (6.24) as follows. Beginning with the vector identity ˜˜ ˜ ˜ ˜ = k·B ˜ k − k2 B k× k×B ˜ ˜ = −k 2 B, substitution from Eq. (6.25) then gives % % % 4π % ωµ(ω) c (ω) ˜˜ ˜ ˜ % k×E−% −k B = − % c % iµ(ω)k × J0 4π % % % 4π % ω2 ˜ ˜˜ ˜ % =− 2 B−% % c % iµ(ω)k × J0 , v (ω) 2˜ ˜

and hence

% % ˜˜ % 4π % ˜ ˜ % iµ(ω) k × J0 (k,2 ω) . B(k, ω) = % % c % k 2 − v2ω(ω)

(6.30)

The inverse Fourier–Laplace transform of the vector solutions appearing in Eqs. (6.28)–(6.30) then yields the following Fourier–Laplace integral representation of the radiation ﬁeld in a homogeneous, isotropic, locally linear, temporally dispersive medium with frequency-dependent dielectric permittivity (ω), magnetic permeability µ(ω), and electric conductivity σ(ω):

282

6 The Angular Spectrum Representation of the Pulsed Radiation Field

4π E(r, t) = i (2π)4

= −i

∞

3

dω

d k

c

k2 −

−∞

C

4π (2π)4

%1% ˜ ˜ 0 (k, ω) − % 2 % ωµ(ω)J

˜ ˜ 0 (k,ω) k·J ωc (ω) k i(k·r−ωt)

ω2 v 2 (ω)

e

(6.31)

dω C

1 ω c (ω) ⎡

˜˜ (k, ω) ⎤ k× k×J 0 ˜ ˜ 0 (k, ω) + ⎦ ei(k·r−ωt) , d3 k ⎣J × 2 − ω2 k −∞ v 2 (ω)

∞

% % ∞ ˜˜ % 4π % 1 3 k × J0 (k, ω) i(k·r−ωt) % B(r, t) = i % dω µ(ω) d k e , 2 % c % (2π)4 k 2 − v2ω(ω) C −∞

(6.32)

(6.33) where C is the Bromwich contour [1] given by ω = ω + ia where ω varies from −∞ to +∞ and where a is the abscissa of absolute convergence for the integrand in each inverse Laplace transform (see Appendix C). In order to apply contour integration techniques for the purpose of simplifying the set of expressions given in Eqs. (6.31)–(6.33), it is necessary to express the ω–integral in a form that involves only nonnegative real values of ω. Consider then the sum I = I+ + I− ,

with I± ≡

∞

dω

˜ ˜ d3 k U(k, ω)ei(k·r−ωt) ,

(6.34)

(6.35)

−∞

C±

where C+ is the contour ω − ω + ia where ω varies from 0 to +∞, and where C− is the contour ω − ω + ia where ω varies from −∞ to 0. If the spectral ˜ ˜ function U(k, ω) has the property that ∗

˜ ˜ ˜ (k, ω + ia), ˜ U(−k, −ω + ia) = U then

(6.36)

∞ ∗ ˜ ˜ (k, ω + ia)e−i(k·r−(ω +ia)∗ t) d(ω + ia)∗ d3 k U 0 −∞ ∞ ∞ ∗ ˜ ˜ (k, ω + ia)e−i(k·r−(ω −ia)t) . = d(ω − ia) d3 k U

I∗+ =

0

∞

−∞

Under the change of variable k → −k, ω → −ω , this integral becomes −∞ ∞ ∗ ˜ ˜ (−k, −ω + ia)e−i(−k·r+(ω +ia)t) d(−ω − ia) d3 k U I∗+ = 0

−∞

6.1 The Fourier–Laplace Integral Representation of the Radiation Field

0

d(ω + ia)

= −∞ ia

=

∞

283

˜ ˜ d3 k U(k, ω + ia)ei(k·r−(ω +ia)t)

−∞ ∞

dω ia−∞

−∞

˜ ˜ d3 k U(k, ω)ei(k·r−ωt) = I− .

(6.37)

Hence, if the symmetry property speciﬁed in Eq. (6.36) is satisﬁed, then I = I+ + I− = I+ + I∗+ = 2 {I+ } ,

(6.38)

where the ω–integral in I+ involves only nonnegative real values of ω. In order to determine if the spectral functions appearing in Eqs. (6.31)– (6.33) satisfy the symmetry property given in Eq. (6.36), consider ﬁrst the ˜ ˜ Fourier–Laplace transform V(k, ω) of a real-valued function V(r, t), where ˜˜ V(k, ω) =

∞

∞

d3 r V(r, t)e−i(k·r−ωt) ,

dt 0

in which case ˜ ˜ V(−k, −ω + ia) =

(6.39)

−∞

∞

∞

dt 0

d3 r V(r, t)e+i(k·r−(ω −ia)t)

−∞

∗ ˜ ˜˜ ∗ (k, ω), ˜ (k, ω + ia) = V =V

(6.40)

ˆ(t), and σ ˆ (t) are all real-valued with ω = ω + ia. Because J0 (r, t), ˆ(t), µ functions, their Fourier–Laplace spectra satisfy the symmetry relations ∗

˜ ˜ ˜ (k, ω), ˜ 0 (−k, −ω + ia) = J J 0

(−ω + ia) = ∗ (ω),

(6.41) (6.42)

µ(−ω + ia) = µ∗ (ω), σ(−ω + ia) = σ ∗ (ω),

(6.43) (6.44)

so that, for the complex permittivity σ(−ω + ia) −ω + ia ∗ σ (ω) = ∗ (ω) − i4π ω − ia ∗ σ(ω) = (ω) + i4π = ∗c (ω). ω

c (−ω + ia) = (−ω + ia) + i4π

(6.45)

As a consequence of this result it is then found that ∗

i (−ω + ia) c (−ω + ia) = (i (ω + ia) c (ω)) ∗

= (iω c (ω)) ,

(6.46)

284

6 The Angular Spectrum Representation of the Pulsed Radiation Field

and hence c2

2

(−ω + ia) 2 = µ(−ω + ia) c (−ω + ia) (−ω + ia) v 2 (−ω + ia) 2

= µ∗ (ω) ∗c (ω) (ω − ia) 2 ∗ = µ(ω) c (ω) (ω + ia) ∗ = ω 2 µ(ω) c (ω) 2 ∗ ω = c2 . v 2 (ω)

(6.47)

It then follows from the relations given in Eqs. (6.41)–(6.47) that the spectral functions appearing in Eqs. (6.31)–(6.33) satisfy the condition speciﬁed in Eq. (6.36). The Fourier–Laplace integral representation of the radiation ﬁeld in a temporally dispersive HILL medium with frequency-dependent dielectric permittivity (ω), magnetic permeability µ(ω), and electric conductivity σ(ω) may then be expressed as ⎧ ⎫ ˜ ˜ 0 (k,ω) ωµ(ω) ˜ k·J ∞ ˜ ⎬ ⎨ (k, ω) − k J 4π c2 0 ωc (ω) 3 i(k·r−ωt) i dω d k e E(r, t) = 2 ⎭ 8π 4 ⎩ C+ k 2 − v2ω(ω) −∞ (6.48)

4π 1 i dω = − 8π 4 ω

(ω) c C+ ⎡

˜˜ (k, ω) ⎤ ∞ k× k×J 0 3 ⎣˜ i(k·r−ωt) ˜ ⎦ × e d k J0 (k, ω) + , 2 k 2 − v2ω(ω) −∞

% % % 4π % 1 % B(r, t) = % % c % 8π 4 i

˜˜ (k, ω) k×J 0 dω µ(ω) d3 k ei(k·r−ωt) 2 2 k − v2ω(ω) C+ −∞ ∞

(6.49)

, (6.50)

for t > 0, where C+ is the contour ω = ω + ia, where ω varies from 0 to +∞ and where a is ﬁxed at a value greater than the abscissa of absolute convergence for the particular radiation problem under consideration. This ﬁnal form of the Fourier–Laplace integral representation of the radiated electric and magnetic ﬁeld vectors explicitly exhibits their real-valued character.

6.2 Scalar and Vector Potentials for the Radiation Field Because the magnetic induction ﬁeld B(r, t) is a transverse (or solenoidal) vector ﬁeld, it can always be expressed as the curl of another vector ﬁeld

6.2 Scalar and Vector Potentials for the Radiation Field

285

A(r, t) as B(r, t) = ∇ × A(r, t),

(6.51)

where A(r, t) is the vector potential (see §3.1). It is then seen from Eqs. (6.33) and (6.50) that such a vector potential for the radiation ﬁeld is given by % % ∞ ˜˜ (k, ω) % 4π % 1 J 0 % A(r, t) = % dω µ(ω) d3 k ei(k·r−ωt) % c % (2π)4 ω2 2 k − v2 (ω) −∞ C (6.52)

% % ∞ ˜ ˜ 0 (k, ω) % 4π % 1 J % =% dω µ(ω) d3 k ei(k·r−ωt) . 2 % c % 8π 4 k 2 − v2ω(ω) C+ −∞ (6.53) Comparison of Eq. (6.52) with (6.31), and (6.53) with (6.48) shows that the electric ﬁeld vector E(r, t) can be expressed as E(r, t) = −i

4π (2π)4

C

% % ˜ ˜ 0 (k, ω) % 1 % ∂A(r, t) k·J i(k·r−ωt) % % ke − 2 %c% ω ∂t k 2 − v2 (ω) −∞

∞ ˜˜ (k, ω) dω k·J 0 d3 k kei(k·r−ωt) 2 ω c (ω) −∞ k 2 − v2ω(ω)

dω ω c (ω)

∞

d3 k

4π i = − 8π 4 C+ % % % 1 % ∂A(r, t) % , −% %c% ∂t

which may be written in the form % % % 1 % ∂A(r, t) % , E(r, t) = −∇ϕ(r, t) − % %c% ∂t

(6.54)

where ϕ(r, t) =

=

4π (2π)4

4π 8π 4

C

dω ω c (ω)

C+

∞

d3 k

−∞

˜˜ (k, ω) k·J 0 ei(k·r−ωt) 2 k 2 − v2ω(ω)

˜˜ (k, ω) ∞ dω k·J 0 d3 k ei(k·r−ωt) 2 ω c (ω) −∞ k 2 − v2ω(ω)

(6.55)

(6.56) is a scalar potential for the radiation ﬁeld. Hence, by determining the vector and scalar potentials A(r, t) and ϕ(r, t), respectively, for a given current source J0 (r, t), the electric and magnetic ﬁeld vectors of the radiation ﬁeld

286

6 The Angular Spectrum Representation of the Pulsed Radiation Field

produced by that current source are directly obtained from Eqs. (6.51) and (6.54). The problem of determining E(r, t) and B(r, t) for a given radiation problem speciﬁed by the current source J0 (r, t) has thus been reduced to determining A(r, t) and ϕ(r, t), which are speciﬁed by somewhat simpler expressions. The gauge transformations described in §3.1 for the microscopic electromagnetic ﬁeld vectors also apply to this macroscopic case. In particular, because the magnetic induction ﬁeld vector B(r, t) is deﬁned through Eq. (6.51) in terms of the curl of the vector potential ﬁeld A(r, t), then this vector potential ﬁeld is arbitrary to the extent that the gradient of some scalar function Λ(r, t) can be added to A(r, t) without aﬀecting B(r, t). That is, the magnetic ﬁeld vector B(r, t) is left unchanged by the transformation [cf. Eq. (3.11)] (6.57) A(r, t) −→ A (r, t) = A(r, t) + ∇Λ(r, t). Under this transformation, the electric ﬁeld vector given in Eq. (6.54) becomes % % % 1 % ∂A (r, t) % E(r, t) = −∇ϕ(r, t) − % %c% ∂t % % % % % 1 % ∂A(r, t) % 1 % ∂Λ(r, t) % % % −% = −∇ϕ(r, t) − % % % c % ∇ ∂t . c ∂t Hence, in order that the electric ﬁeld vector remain unchanged under the vector potential transformation given in Eq. (6.57), the scalar potential ϕ(r, t) must be simultaneously transformed as % % % 1 % ∂Λ(r, t) % (6.58) ϕ(r, t) −→ ϕ (r, t) = ϕ(r, t) − % % c % ∂t . The complete transformation speciﬁed by the pair of relations in Eqs. (6.57)– (6.58) is called a gauge transformation and the invariance of the electric and magnetic ﬁeld vectors under such a transformation is referred to as gauge invariance. The scalar function Λ(r, t) is called the gauge function of the transformation. The freedom of choice that is implied by Eqs. (6.57) and (6.58) means that one can always choose a set of potentials {A , ϕ } such that [cf. Eq. (3.13)] c ∂ϕ (r, t) = 0, (6.59) ∇ · A (r, t) + 2 c ∂t which is the Lorenz condition [2] for the macroscopic potentials. In order to prove that a pair of potentials {A , ϕ } can always be found to satisfy the Lorenz condition, suppose that the original pair of potentials {A, ϕ} does not satisfy Eq. (6.59). One may then undertake a gauge transformation to the new pair of potentials {A , ϕ } and demand that A (r, t) and ϕ (r, t) satisfy the Lorenz condition, so that

6.2 Scalar and Vector Potentials for the Radiation Field

∇2 Λ(r, t) −

1 ∂ 2 Λ(r, t) c ∂ϕ(r, t) = − ∇ · A(r, t) + 2 c2 ∂t2 c ∂t

287

.

(6.60)

Thus, provided that a gauge function Λ(r, t) can be found that satisﬁes Eq. (6.60), the new pair of potentials {A , ϕ } will satisfy the Lorenz condition. Even for a pair of potentials that satisfy the Lorenz condition (6.59) there still remains a certain degree of arbitrariness. This is because the restricted gauge transformation A(r, t) −→ A (r, t) = A(r, t) + ∇ψ(r, t), % % % 1 % ∂ψ(r, t) % ϕ(r, t) −→ ϕ (r, t) = ϕ(r, t) − % % c % ∂t ,

(6.61) (6.62)

with

1 ∂ 2 ψ(r, t) =0 (6.63) c2 ∂t2 preserves the Lorenz condition provided that the potential pair {A, ϕ} satisﬁes it initially, as is readily evident from Eq. (6.60). All pairs of potentials {A, ϕ} in this restricted class belong to the so-called Lorenz gauge. The divergence of the vector potential given in Eq. (6.52) for the radiation ﬁeld is given by % % ∞ ˜˜ 0 (k, ω) % 4π % i k·J % dω µ(ω) d3 k ei(k·r−ωt) , ∇ · A(r, t) = % % c % (2π)4 2 − ω2 k C −∞ v 2 (ω) ∇2 ψ(r, t) −

and the partial time derivative of the scalar potential given in Eq. (6.55) for the radiation ﬁeld is found as ∞ ˜ ˜ 4π dω ∂ϕ(r, t) 3 k · J0 (k, ω) i(k·r−ωt) =− i d k e 2 ∂t (2π)4 C c (ω) −∞ k 2 − v2ω(ω) % % ∞ ˜˜ (k, ω) % 4π % i k·J 0 2 % = −% dω µ(ω)v (ω) d3 k ei(k·r−ωt) , % c2 % (2π)4 2 − ω2 k C −∞ v 2 (ω) where v 2 (ω) is the square of the complex velocity in the temporally dispersive HILL medium, deﬁned in Eq. (6.27). With substitution of these two relations the left-hand side of the Lorenz condition given in Eq. (6.59) is found to be c ∂ϕ(r, t) ∇ · A (r, t) + 2 c ∂t % % ∞ ˜˜ % 4π % i v 2 (ω) 3 k · J0 (k, ω) i(k·r−ωt) % % dω µ(ω) 1 − d k e , = % % 2 c (2π)4 C c2 k 2 − v2ω(ω) −∞ (6.64) which does not, in general, vanish. Hence, the vector and scalar potentials given in Eqs. (6.52) and (6.55) for the radiation ﬁeld produced by the current

288

6 The Angular Spectrum Representation of the Pulsed Radiation Field

source J0 (r, t) in a temporally dispersive HILL medium do not, in general, satisfy the Lorenz condition. Nevertheless, a gauge function Λ(r, t) can always be determined from the solution of Eq. (6.60) so as to deﬁne a gauge transformation to a new pair of potentials {A , ϕ } that do satisfy the Lorenz condition. However, the vector and scalar potentials given in Eqs. (6.52) and (6.55) are clearly the most natural, and hence the simplest forms of the potentials for the given radiation problem. 6.2.1 The Nonconducting, Nondispersive Medium Case It is because of the frequency dependence of the complex velocity appearing in Eq. (6.64) that the quantities ∇ · A and ∂ϕ/∂t for the radiation problem cannot be directly related (without an appropriate gauge transformation) except in certain special cases. To that end, consider an idealized nondispersive, nonconducting (σ = 0) dielectric medium with constant, real-valued dielectric permittivity and magnetic permeability µ, so that c (ω) = and v(ω) = v, where c2 . (6.65) v2 = µ

The vector and scalar potentials given in Eqs. (6.52)–(6.53) and (6.55)–(6.56) for the radiation ﬁeld are then seen to satisfy the generalized Lorenz condition ∇ · A(r, t) +

c ∂ϕ(r, t) = 0, v2 ∂t

(6.66)

so that ϕ(r, t) can be directly determined from A(r, t). Consider now determining the diﬀerential equations that are satisﬁed by the vector and scalar potentials in this generalized Lorenz gauge. From Eq. (6.52) one has that 1 ∂ 2 A(r, t) v2 ∂t2 % % ∞ ˜ ˜ % 4π % µ ω2 3 J0 (k, ω) 2 % % −k + 2 ei(k·r−ωt) =% % dω d k 2 c (2π)4 C v k 2 − ωv2 −∞ % % ∞ % 4π % µ ˜ ˜ 0 (k, ω)ei(k·r−ωt) % = −% dω d3 k J % c % (2π)4 C −∞ % % $ % 4π % −1 −1 # ˜ ˜ 0 (k, ω) , % µF L = −% J % c %

∇2 A(r, t) −

and the vector potential is then seen to satisfy the inhomogeneous wave equation % % % 4π % 1 ∂ 2 A(r, t) % % µJ0 (r, t) = − (6.67) ∇2 A(r, t) − 2 % c % 2 v ∂t

6.2 Scalar and Vector Potentials for the Radiation Field

289

when the medium is nonconducting and nondispersive. Similarly, from Eq. (6.55) one has that 1 ∂ 2 ϕ(r, t) v 2 ∂t2 ˜ ˜ 0 (k, ω) dω ∞ 3 k · J 4π ω2 2 = d k −k + 2 ei(k·r−ωt) 2 (2π)4 C ω −∞ v k 2 − ωv2 ∞ ˜ ˜ 4π 3 k · J0 (k, ω) i(k·r−ωt) =− dω d k . e 4 (2π) C ω −∞

∇2 ϕ(r, t) −

˜˜ (k, ω)/ω, and the ˜˜0 (k, ω) = k · J From Eq. (6.20) with σ = 0 one ﬁnds that 0 scalar potential is then seen to satisfy the inhomogeneous wave equation ∇2 ϕ(r, t) −

1 ∂ 2 ϕ(r, t) 4π = 0 (r, t), 2 2 v ∂t

where 0 (r, t) = F

−1

L

−1

˜˜ (k, ω) k·J 0 ω

(6.68)

(6.69)

is the charge density associated with the externally supplied current source J0 (r, t), as required by the equation of continuity in the nondispersive, nonconducting medium. The relations given in Eqs. (6.66)–(6.68) clearly show that A(r, t) and ϕ(r, t) are the standard macroscopic vector and scalar potentials, respectively, in the Lorenz gauge. 6.2.2 The Spectral Lorenz Condition for Dispersive HILL Media The reason that the general vector and scalar potentials given in Eqs (6.52) and (6.55) do not satisfy a Lorenz condition is due to the fact that the complex velocity is frequency dependent in any temporally dispersive medium. The temporal frequency transforms of these vector and scalar potentials for the radiation ﬁeld in a temporally dispersive HILL medium are found to be % % ˜ ˜ 0 (k, ω) % 4π % µ(ω) ∞ 3 J ˜ % A(r, ω) = % d k eik·r , 2 % c % (2π)3 k 2 − v2ω(ω) −∞ % % ˜˜ (k, ω) % 4π % µ(ω)v 2 (ω) ∞ 3 k · J 0 % ϕ(r, ˜ ω) = % d k eik·r . 2 % c2 % (2π)3 ω k 2 − v2ω(ω) −∞

(6.70)

(6.71)

Consider then the temporal spectral domain form of Eq. (6.66) with a frequency-dependent complex velocity as given by Eq. (6.27). With Eqs. (6.70) and (6.71) one obtains the relation

290

6 The Angular Spectrum Representation of the Pulsed Radiation Field

% % % % ˜ ω) − % 1 % iωµ(ω) c (ω)ϕ(r, ∇ · A(r, ˜ ω) = 0. %c%

(6.72)

It is then seen that the temporal frequency spectra of the vector and scalar potentials for the radiation ﬁeld in a temporally dispersive HILL medium satisfy a generalized spectral Lorenz condition. This condition reduces to the temporal frequency domain form of Eq. (6.66) when the medium is nonconducting and nondispersive. Consider now obtaining the individual wave equations satisﬁed by the temporal frequency spectra of the radiation potentials. From Eq. (6.70) one obtains ω2 ˜ A(r, ω) v 2 (ω) % % ˜ ˜ 0 (k, ω) % 4π % µ(ω) ∞ 3 J ω2 2 % % −k + 2 eik·r d k =% % 2 c (2π)3 −∞ v (ω) k 2 − v2ω(ω) % % % 4π % µ(ω) ∞ 3 ˜ ˜ 0 (k, ω)eik·r % % = −% % d kJ c (2π)3 −∞ % % $ # % 4π % ˜ ˜ 0 (k, ω) , % µ(ω)F−1 J = −% % c %

˜ ω) + ∇2 A(r,

so that the temporal frequency spectrum of the vector potential satisﬁes the inhomogeneous, dispersive wave equation % % % 4π % ω2 ˜ 2˜ ˜ % ∇ A(r, ω) + 2 (6.73) A(r, ω) = − % % c % µ(ω)J0 (r, ω). v (ω) Similarly, from Eq. (6.71) one ﬁnds that % % % 4π % µ(ω)v 2 (ω) ∞ 3 ˜ ω2 2 ˜ 0 (k, ω) eik·r . % % ϕ(r, ˜ ω) = − % 2 % ˜ ω) + 2 d k k · ∇ ϕ(r, J v (ω) c (2π)3 ω −∞ From Eq. (6.20) it is seen that the spatiotemporal frequency spectrum of the charge density may be written as ˜˜(k, ω) = ˜˜0 (k, ω) + ˜˜c (k, ω),

(6.74)

where

˜˜ ˜˜0 (k, ω) ≡ k · J 0 (k, ω) ω is the nonconductive charge density, and where ˜ ˜ ˜˜c (k, ω) ≡ σ(ω) k · E(k, ω) ω

(6.75)

(6.76)

is the conductive charge density. With substitution from Eq. (6.75) the wave equation for the scalar potential becomes

6.3 Angular Spectrum of Plane Waves Representation of the Radiation Field

291

% % % 4π % µ(ω)v 2 (ω) ∞ 3 ω2 % ˜˜0 (k, ω)eik·r ϕ(r, ˜ ω) = − % ∇2 ϕ(r, ˜ ω) + 2 d k % c2 % (2π)3 ω v (ω) −∞ % % % 4π % 2 % = −% 0 (r, ω), % c2 % µ(ω)v (ω)˜ so that the temporal frequency spectrum of the scalar potential satisﬁes the inhomogeneous, dispersive wave equation ˜ ω) + ∇2 ϕ(r,

ω2 v 2 (ω)

ϕ(r, ˜ ω) = −

Here ˜0 (r, ω) = F−1

4π ˜0 (r, ω).

c (ω)

k · J˜˜0 (k, ω) ω

(6.77)

(6.78)

is the temporal frequency spectrum of the nonconductive charge density.

6.3 Angular Spectrum of Plane Waves Representation of the Radiation Field The Fourier–Laplace representations given in Eqs. (6.31)–(6.33) and (6.48)– (6.50) for the electromagnetic ﬁeld vectors and in Eqs. (6.52)–(6.53) and (6.55)–(6.56) for the vector and scalar potentials for the radiation ﬁeld in a temporally dispersive HILL medium occupying all of space are each of the general form i G(r, t) = (2π)4

∞

dω C

−∞

d3 k

˜ ˜ F(k, ω) i(k·r−ωt) e 2 2 k − v2ω(ω)

∞ ˜˜ 1 F(k, ω) = i dω d3 k ei(k·r−ωt) , 2 8π 4 k 2 − v2ω(ω) −∞ C+

(6.79)

(6.80)

where C is the contour ω = ω + ia with ω varying from −∞ to +∞, and C+ is the contour ω = ω + ia with ω varying from 0 to +∞, and where the ˜ ˜ spectral function F(k, ω) satisﬁes the symmetry property [cf. Eq. (6.36)] ∗ ˜ ˜ ˜ (k, ω + ia). ˜ F(−k, −ω + ia) = F

(6.81)

For each equation represented by the generic form given in Eqs. (6.79)–(6.80), ˜ ˜ the spectral function F(k, ω) is given by either the scalar or vector product ˜ ˜ of k with J (k, ω) = FL {J (r, t)}. Because J (r, t) and its ﬁrst-order spatial 0

0

0

derivatives vanish for |z| Z is a positive constant, it then follows # ≥ Z, where $ ˜ ˜ that F(r, t) = F−1 L−1 F(k, ω) also vanishes for |z| ≥ Z, where

292

6 The Angular Spectrum Representation of the Pulsed Radiation Field

˜˜ F(k, ω) =

∞

∞

dt 0

d3 r F(r, t)e−i(k·r−ωt) .

(6.82)

−∞

Let kz = kz + ikz and ω = ω + ia where kz , kz , ω and a are all real-valued variables. With these substitutions Eq. (6.82) becomes ∞ ∞ Z ˜˜ dz e−ikz z ekz z dt dxdy F(r, t)e−i(kx x+ky y−ω t) e−at F(k, ω) = −Z Z

=

−Z

0

−∞

−ikz z

G(kx , ky , ω, z)e

ekz z dz.

(6.83)

Because the only integral involving kz = kz + ikz is taken over a bounded region, and because the integrand in Eq. (6.83) is an analytic function of kz for all kz and is also a continuous function of the variable of integration ˜ ˜ (provided that F(r, t) is continuous), then F(k, ω) is an entire function of the complex variable kz (i.e., it is analytic for all kz ) [3]. As a consequence, each ˜ ˜ component of F(k, ω) is bounded in magnitude as ∞ ∞ Z ˜˜ kz z F ≤ (k, ω) dz e dt dxdy |Fj (r, t)| j −Z 0 −∞ ∞ ∞ ≤ e|kz |Z dt dxdy |Fj (r, t)| 0

≤ Mj e|kz |Z ,

−∞

(6.84)

where Mj is independent of kz . ˜ ˜ Because F(k, ω) is an entire function of complex kz , the only singularities appearing in the integrand of Eq. (6.79) [as well as in Eq. (6.80)] are located where k 2 − ω 2 /v 2 (ω) = 0, and hence, this integrand has two simple pole singularities that are located at 2 1/2 ω 2 − k , (6.85) kz = ± T v 2 (ω) ˜ ˜ where kT2 ≡ kx2 + ky2 . Because |F(k, ω)| is bounded, the kz -integrations in Eqs. (6.79)–(6.80) may then be evaluated through the use of Cauchy’s residue theorem once the locations of the simple pole singularities given in Eq. (6.85) are determined. In order to determine the location of the simple pole singularities satisfying Eq. (6.85) in the complex kz -plane, it is necessary to determine the sign of the imaginary part of the quantity ω 2 /v 2 (ω). From Eqs. (5.29) and (6.27) one has that ω2 σ(ω) = ω 2 µ(ω) c (ω) = ω 2 µ(ω) (ω) + i4π c2 2 v (ω) ω 2 2 = µr ω r − 4πωσi − µi ω i + 4πωσr + i µi ω 2 r − 4πωσi + µr ω 2 i + 4πωσr ,

6.3 Angular Spectrum of Plane Waves Representation of the Radiation Field

293

so that with ω = ω + ia, where ω and a are both real-valued, # ω2 = µr ω 2 − a2 r − 2aω i − 4π (ω σi + aσr ) c2 2 v (ω) $ −µi ω 2 − a2 i + 2aω r + 4π (ω σr − aσi ) # + i µr ω 2 − a2 i + 2aω r + 4π (ω σr − aσi ) $ + µi ω 2 − a2 r − 2aω i − 4π (ω σi + aσr ) , (6.86) with r (ω) ≡ { (ω)}, i (ω) ≡ { (ω)}, µr (ω) ≡ {µ(ω)}, µi (ω) ≡ {µ(ω)}, σr (ω) ≡ {σ(ω)}, and σi (ω) ≡ {σ(ω)}. The only situation to be considered here is the one for which each of the inequalities

i (ω + ia) ≥ 0,

µi (ω + ia) ≥ 0,

σr (ω + ia) ≥ 0,

(6.87)

is satisﬁed for ω ∈ [0, +∞) for all a ≥ 0, so that the medium is absorptive in each of its dielectric, magnetic, and conductive responses. Along the contour C+ in the representation given in Eq. (6.80), ω = 0 → +∞ with a ≥ 0 a ﬁxed constant that is greater than the abscissa of absolute convergence (see Appendix C) for the particular radiation problem under consideration. The inequality 2 ω > 0, (6.88) 2 v (ω) and hence (because kx and ky are both real-valued) 2 ω 2 − k > 0, T v 2 (ω)

(6.89)

will then be satisﬁed along the contour C+ in any of the following cases. 1. If a = 0, then ω i (ω) + 4πσr (ω) +

µi (ω) µi (ω) ω r (ω) > 4π σi (ω). µr (ω) µr (ω)

(6.90)

If µi (ω) = 0, then one must have that i (ω) > 0 at every frequency value where σr (ω) = 0 and σr (ω) > 0 at every frequency value where i (ω) = 0. If µi (ω) = 0, then there are two possibilities: a) If ω r (ω) ≥ 4πσi (ω), then it is suﬃcient to require that i (ω) > 0 at every frequency value where σr (ω) = 0 and σr (ω) > 0 at every frequency value where i (ω) = 0. b) If ω r (ω) ≤ 4πσi (ω) for suﬃciently small nonnegative values of ω , then the general inequality given in Eq. (6.90) must be satisﬁed. This is easily satisﬁed for a simple (nonferromagnetic) magnetizable material for which |µi (ω)/µr (ω)| 1.

294

6 The Angular Spectrum Representation of the Pulsed Radiation Field

2. If σr (ω) = σi (ω) = 0 so that the medium is nonconducting, then

r (ω) > −

(ω 2 − a2 ) − 2aω µµri (ω) (ω) 2aω +

µi (ω) 2 µr (ω) (ω

− a2 )

i (ω).

(6.91)

3. If i (ω) = 0 so that the dielectric permittivity is real-valued, then µi (ω) 2 (ω − a2 ) r (ω) 2aω + µr (ω) µi (ω) (ω σi (ω) + aσr (ω)) > 0. + 4π (ω σr (ω) − aσi (ω)) − µr (ω) (6.92) If a = 0 this inequality requires that 1 µi (ω) σr (ω) > − ω r (ω) − σi (ω) , µr (ω) 4π

(6.93)

and the general inequality given in Eq. (6.92) must be satisﬁed when a > 0. For a nondispersive medium, both r and µr are positive constants and

i = µi = 0. If, in addition, σr = σi = 0 so that the medium is also nonconducting, then case 2 applies with a > 0, and the inequality in Eq. (6.91) becomes r > 0. On the other hand, if the medium is conducting so that σr > 0, then one can take a = 0 and the inequality in Eq. (6.93) of case 3 applies. Hence, for a nondispersive medium, the equivalent inequalities given in Eqs. (6.88)–(6.89) can always be satisﬁed. For a causal, temporally dispersive medium both r (ω) and µr (ω) can be negative over certain frequency intervals 7 but both 8 i (ω) and µi (ω) are always nonnegative on C+ . In that case, ω 2 /v 2 (ω) will be negative for suﬃciently small ω < a, as can be seen from the imaginary part of Eq. (6.86). One can then ensure that the equivalent inequalities given in Eqs. (6.88)– (6.89) are satisﬁed by taking a = 0, as in case 1. If this is not permissible because of a positive abscissa of absolute convergence for the particular radiation problem under consideration, one can always take the lower limit of the ω-integration in Eq. (6.80) to be ω = a and proceed to the limit a → 0 after completion of the integration. In any event, ω will always be considered to be positive when the kz -integral in Eq. (6.80) is evaluated and then the ω-integral possesses a positive lower limit that one then lets pass to zero in a limiting process. In order to proceed with the kz -integration in Eq. (6.80) it is assumed that the equivalent inequalities in Eqs. (6.88)–(6.89) are satisﬁed. The complex quantity 2 1/2 ω 2 − kT , (6.94) γ(ω) ≡ v 2 (ω)

6.3 Angular Spectrum of Plane Waves Representation of the Radiation Field

295

where kT2 ≡ kx2 + ky2 , is taken as the principal branch of the square root that is deﬁned in the following manner: let ω2 v 2 (ω)

− kT2 = Γ eiφΓ ;

0 ≤ φΓ < 2π,

(6.95)

where Γ is the magnitude and φΓ the phase of the complex quantity (ω 2 /v 2 (ω) − kT2 ), and deﬁne √ (6.96) γ(ω) ≡ Γ eiφΓ /2 ; 0 ≤ φΓ < 2π. Then γ(ω) always has a nonnegative imaginary part because 0 ≤ φΓ < π. With reference to Eq. (6.85), the simple pole singularity at kz = γ(ω) is then always in the upper-half of the complex kz -plane and the pole at kz = −γ(ω) is always in the lower-half of the complex kz -plane. kz''

C1 !Γ +

kz ' + Γ

C2

Complex kz - Plane

Fig. 6.2. The semicircular contours C1 and C2 centered at the origin in the complex kz -plane, where C1 is contained entirely in the upper-half plane and C2 is contained entirely in the lower-half plane.

Because of the inequality in Eq. (6.84), the jth component of the integrand in Eq. (6.80) is bounded as ˜˜ ˜˜ (k, ω) F F (k, ω) j j i(k·r−ωt) e−kz z eat e = 2 2 ω 2 k 2 − ω k − 2 v (ω)

v 2 (ω)

Mj e|kz |Z−kz z , ≤ 2 ω k 2 − v2 (ω)

(6.97)

296

6 The Angular Spectrum Representation of the Pulsed Radiation Field

with a → 0. Hence, in the upper-half of the complex kz -plane the integrand of the integral in Eq. (6.80) decays exponentially and goes to zero as |kz | → ∞ if z > Z > 0. The same exponential decay is present in the lower-half of the complex kz -plane if z < −Z. It then follows from Jordan’s lemma [4] that the kz -integral in Eq. (6.80) may be evaluated by application of Cauchy’s residue theorem either by completing the contour along the real axis into a closed circuit using the semicircular contour C1 in the upper-half plane if z > Z, as illustrated in Figure 6.2, or by using the semicircular contour C2 in the lower-half plane if z < −Z, also illustrated in Figure 6.2, each contour centered at the origin, where the radius of each is allowed to go to inﬁnity. If |z| < Z, then the integral in Eq. (6.80) cannot be evaluated by completing the contour in this manner. For the radiation ﬁeld in the positive half-space z > Z one completes the contour in the upper-half of the complex kz -plane and the only contribution to the kz -integral in Eq. (6.80) is due to the simple pole at kz = +γ(ω). For the radiation ﬁeld in the negative half-space z < −Z one completes the contour in the lower-half of the complex kz -plane and the only contribution to the kz -integral in Eq. (6.80) is due to the simple pole at kz = −γ(ω). Because ω2 k2 − 2 = (kz − γ(ω)) (kz + γ(ω)) , (6.98) v (ω) then the residue of the simple pole at kz = γ(ω) is given by Residue (kz = γ(ω))

∞ ˜˜ 1 F(k, ω) = lim (k − γ) dω dk dk ei(k·r−ωt) z x y 4 k →γ 8π z (kz − γ)(kz + γ) −∞ C+ ∞ ˜ ˜ + , ω) ˜ + ˜ k 1 F( (6.99) ei(k ·r−ωt) , = dω dk dk x y 4 8π C+ 2γ(ω) −∞

and the residue of the simple pole at kz = −γ(ω) is given by Residue (kz = −γ(ω)) = = Here

1 8π 4 1 8π 4

lim (kz + γ)

kz →−γ

∞

dω C+

−∞

dkx dky

˜˜ F(k, ω) ei(k·r−ωt) (kz − γ)(kz + γ)

˜ ˜ − , ω) ˜ − ˜ k F( ei(k ·r−ωt) , dω dkx dky −2γ(ω) C+ −∞ ∞

˜ ± (ω) ≡ 1 ˆ x kx + 1 ˆ y ky ± 1 ˆ z γ(ω) k

is the complex wave vector, where

(6.100)

(6.101)

6.3 Angular Spectrum of Plane Waves Representation of the Radiation Field

297

˜ ± (ω) = k 2 + k 2 + γ 2 (ω) ˜ ± (ω) · k k x y =

ω2 = k˜2 (ω), v 2 (ω)

(6.102)

so that 1/2 ˜ ± (ω) ˜ ˜ ± (ω) · k k(ω) = k % % %1% 1/2 ω = % % ω µ(ω) c (ω) = = k0 n(ω) v(ω) % c %

(6.103)

is the complex wavenumber in the temporally dispersive HILL medium [cf. Eq. (5.60) for a time-harmonic plane wave ﬁeld]. Here k0 = ω/c is the vacuum wavenumber and n(ω) is the complex index of refraction of the dispersive medium, deﬁned in Eq. (5.33). Consequently, by the residue theorem [5], Eq. (6.99) is multiplied by 2πi and Eq. (6.100) by −2πi to obtain the ﬁnal expressions for the kz -integration, with the ﬁnal result

∞ ˜˜ k ˜ ± , ω) ˜ ± 1 F( i(k ·r−ωt) i(±2πi) dω dkx dky G(r, t) = e 8π 4 ±2γ(ω) −∞ C+

∞ ˜ ˜ ± , ω) ˜ ± ˜ k 1 F( =− (6.104) ei(k ·r−ωt) dω dkx dky (2π)3 γ(ω) −∞ C+ for the generic integral in Eq. (6.80). This ﬁnal expression is referred to as the angular spectrum of plane waves representation of the ﬁeld G(r, t); the origin of this terminology is made clear in §6.4 when the spatial frequency integral in Eq. (6.104) is transformed to polar coordinates in kx ky -space. For the electric and magnetic ﬁeld vectors for the radiation ﬁeld, given in Eqs. (6.49)–(6.50), the kz -integration of each Fourier–Laplace representation yields the pair of expressions

dω 4π E(r, t) = (2π)3 ω

c (ω) C+

˜˜ (k ∞ ˜ ± , ω) ˜± × k ˜± × J k 0 ˜ ± ·r−ωt) i(k , e dkx dky γ(ω) −∞

B(r, t) = −

4π/c (2π)3

(6.105)

dω µ(ω) C+

˜ (k ˜ ± , ω) ˜ ± ˜± × J ˜ k 0 ei(k ·r−ωt) , dkx dky γ(ω) −∞

∞

(6.106)

298

6 The Angular Spectrum Representation of the Pulsed Radiation Field

˜ + (ω) is used when z > Z > 0 and k ˜ − (ω) is used when for t > 0, where k z < −Z. This pair of expressions is the angular spectrum of plane waves representation of the electromagnetic ﬁeld vectors for the radiation ﬁeld in a temporally dispersive HILL medium occupying all of space. For the vector and scalar potentials given in Eqs. (6.53) and (6.56), respectively, for this radiated electromagnetic wave ﬁeld, the kz -integration of each Fourier–Laplace representation yields the angular spectrum of plane waves representations

∞ ˜˜ (k ˜ ± , ω) ˜ ± J 4π/c 0 ei(k ·r−ωt) , i dω µ(ω) dkx dky A(r, t) = (2π)3 γ(ω) −∞ C+ (6.107)

∞ ˜ ± ± ˜ , ω) ˜ ± ˜ ·J ˜ 0 (k 4π dω k i(k ·r−ωt) e ϕ(r, t) = , i dkx dky (2π)3 γ(ω) C+ ω c (ω) −∞ (6.108) for t > 0. The parameter a for the contour C+ (ω = ω + ia) can be taken to be zero in Eqs. (6.106) and (6.107) because the points at which γ(ω) = 0 are integrable singularities [6], and can also be taken to be zero in Eqs. (6.105) and (6.108) provided that the points at which ω c (ω) = 0 are also integrable singularities. The exponential factor appearing in the angular spectrum of plane waves representations given in Eqs. (6.105)–(6.108) is, with substitution from Eq. (6.101), given by ˜ ± ·r−ωt)

ei(k

= e±iγ(ω)z ei(kx x+ky y−ωt) ,

(6.109)

which represent either homogeneous or inhomogeneous plane waves depend1/2 ing upon whether γ(ω) ≡ ω 2 /v 2 (ω) − kT2 is real, imaginary, or complexvalued. From Eqs. (5.29) and (6.27) one has that % % % % %1% 2 % 4π % ω2 % % % (6.110) = % 2 % ω µ(ω) (ω) + % % c2 % iωµ(ω)σ(ω). 2 v (ω) c Consider ﬁrst the simplest case of a nondispersive nonconducting material where σ = 0 and and µ are both real-valued, so that ω 2 /v 2 = ω 2 µ /c2 . 1/2 With ω taken to be real-valued (so that a = 0), then γ = ω 2 /v 2 − kT2 is either pure real or pure imaginary. If kT2 ≡ kx2 + ky2 ≤ ω 2 /v 2 , then γ is real and Eq. (6.109) represents pure oscillatory plane waves. On the other hand, if kT2 > ω 2 /v 2 , then γ is pure imaginary with γ = i|γ| so that iγ is real and negative and the expression given in Eq. (6.109) possesses exponential decay in the z-direction for |z| > Z, because then e±iγz = e−|γ||z| . Consequently, for kT2 ≤ ω 2 /v 2 , Eq. (6.109) represents homogeneous plane waves for which the surfaces of constant phase coincide with the surfaces of constant amplitude, whereas inhomogeneous plane waves are obtained when kT2 > ω 2 /v 2 , where

6.4 Polar Coordinate Form of the Angular Spectrum Representation

299

the phase is constant in planes parallel to the z-axis (kx x + ky y = κ1 ) and the amplitude is constant in planes perpendicular to the z-axis (γz = κ2 ), where κj , j = 1, 2 are real constants. That the plane wave factors appearing in the angular spectrum of plane waves representations given in Eqs. (6.105)– ((6.108) only decay along the z-direction has no real physical meaning and is simply a consequence of an imposed dependency due to the evaluation of the kz -integral in the Fourier–Laplace representations. It is then clear that, in general, the angular spectrum of plane waves representations for both the electromagnetic ﬁeld vectors, given in Eqs. (6.105)– (6.106), and the vector and scalar potentials, given in Eqs. (6.107)–(6.108), combine both homogeneous and inhomogeneous plane waves in the 2π-solid angle about the positive z-axis for z > Z, as well as in the 2π-solid angle about the negative z-axis for z < −Z. These speciﬁc forms of the plane wave representation are known as Weyl-type expansions [7]; each is valid only in its particular half-space with plane waves propagating only within the 2π-solid angle into the respective half-space. When the source of the ﬁeld ceases to radiate the problem reduces to an initial value problem and one can use a representation that possesses only homogeneous plane waves in the entire 4π-solid angle about the chosen origin. This speciﬁc form of the plane wave representation is known as a Whittaker-type expansion [8] and is deferred to Chapter 8. A detailed comparison of these two expansion types for the optical wave ﬁeld in a nondispersive, nonconducting HILL medium may be found in the published papers by Sherman et al. [9, 10] and Devaney and Sherman [11]. Further discussion of the angular spectrum representation may be found in the research monographs by Clemmow [12], Stamnes [13], Nieto-Vesperinas [14], and Hansen and Yaghjian [15] for a variety of other applications.

6.4 Polar Coordinate Form of the Angular Spectrum Representation Attention is now given to the spatial angular spectrum integral ∞ U (kx , ky ) ik± ·r dkx dky e u(r) = γ −∞

(6.111)

which appears in the angular spectrum representation (6.105) for the electric ˜ ± ± ± ˜ ˜ ˜ ˜ ﬁeld intensity vector with U (kx , ky ) = k × k × J0 (k , ω) , in the angular spectrum representation (6.106) for the magnetic induction ﬁeld vector with ˜ ˜± × J ˜ ± , ω), in the angular spectrum representation (6.107) ˜ 0 (k U (kx , ky ) = k ˜ ± , ω), and in the angular ˜˜ (k for the vector potential ﬁeld with U (k , k ) = J x

y

0

spectrum representation (6.108) for the scalar potential ﬁeld with U (kx , ky ) =

300

6 The Angular Spectrum Representation of the Pulsed Radiation Field

˜ ˜ ± , ω). Here r = 1 ˜± · J ˜ 0 (k ˆx x + 1 ˆy y + 1 ˆ z z is the position vector for the ﬁeld, k and ˜ ± (ω) ≡ 1 ˆ x kx + 1 ˆ y ky ± 1 ˆ z γ(ω) k (6.112) is the complex wave vector with 1/2 γ(ω) ≡ k˜2 (ω) − k 2 − k 2 , x

where ˜ k(ω) =

y

% % %1% ω % ω µ(ω) c (ω) 1/2 =% % % v(ω) c

(6.113)

(6.114)

˜ − for z < −Z, ˜ + for z > Z and k is the complex wave number. One employs k where Z > 0 describes the maximum spatial extent of the applied current source in the z-coordinate direction (see Fig. 6.1). Finally, U (kx , ky ) = f (kx , ky , γ)

(6.115)

where f (kx , ky , kz ) is an entire function of complex kx , ky , and kz [see the discussion in connection with Eq. (6.83)]; however, f (kx , ky , γ) is not necessarily an entire function of complex kx and ky . Consider now the change of variable to polar coordinates that is deﬁned by ˜ sin α cos β, kx = k(ω) ˜ ky = k(ω) sin α sin β,

(6.116) (6.117)

where kx and ky are both real-valued and range over the domain from −∞ to ˜ +∞, k(ω) is the ﬁxed complex number given by Eq. (6.114), and where α and β are, in general, complex-valued angles whose domains must be determined so as to yield the proper ranges for both kx and ky . The Jacobian of this coordinate transformation is ∂(kx , ky ) ∂kx /∂α ∂ky /∂α ˜2 = k (ω) sin α cos α. ≡ (6.118) ∂kx /∂β ∂ky /∂β ∂(α, β) Furthermore, under this coordinate transformation ˜ γ = ±k(ω) cos α,

(6.119)

where the proper sign choice is determined by the domain of integration. The derivation now proceeds with the determination of the integration contours along which the complex angles α and β vary in such a manner that both kx and ky are kept real-valued and varying from −∞ to +∞. The simplest case where k˜ is real-valued is considered ﬁrst, followed by the ˜ general case where k(ω) is complex-valued. Clearly, the ﬁrst case is a limiting situation of the latter and so lends important guidance to the latter general derivation.

6.4 Polar Coordinate Form of the Angular Spectrum Representation

301

Consider then the case where k˜ = k is real-valued. It is clear from Eqs. (6.116)–(6.117) that β must then be real-valued in order that both kx and ky are real-valued. The angle α must then be complex-valued such that sin α is real-valued and ranges from 0 to +∞. The real angle β ranging over 0 to 2π will then give kx and ky a range over −∞ to +∞. As a consequence,

Α''

C'

cosΑ' = 0 (inhomogeneous waves) 0

sinhΑ'' = 0 (homogeneous waves)

Α'

Π

cosΑ' = 0 (inhomogeneous waves) C

Complex Α- Plane

Fig. 6.3. The contours C and C in the complex α-plane for the special case when ˜ = k is real-valued. k

let α = α + iα , where α and α are both real-valued, so that sin α = sin (α + iα ) = sin α cosh α + i cos α sinh α . Because sin α must be realvalued, one then obtains the condition cos α sinh α = 0, which is satisﬁed along either of the following two contours in the complex α-plane, illustrated in Figure 6.3: the contour C where α varies from 0 to π/2 along the α -axis, and then α varies from α = π/2 to π/2 − i∞ along the line α = π/2; the contour C where α varies from 0 to π/2 along the α -axis, and then α varies from α = π/2 to π/2 + i∞ along the line α = π/2. Along the α = {α} axis, sin α = sin α so that 1/2 γ = k 1 − sin2 α = ±k cos α ˜±

and the factor eik ·r appearing in the integrand of Eq. (6.111) describes homogeneous plane waves along this portion of the contour for both C and C . Along the line α = π/2, sin α = cosh α so that

302

6 The Angular Spectrum Representation of the Pulsed Radiation Field

1/2 1/2 γ = k 1 − cosh2 α = k − sinh2 α = ±ik sinh α ˜±

and the factor eik ·r describes inhomogeneous plane waves along this portion of the contour for either C or C . As α varies from 0 to π/2 with α = 0 along both C and C , sin α = sin α varies from 0 to 1, and as α varies from π/2 to π/2−i∞ with α = π/2 along C, sin α = cosh α varies from 1 to +∞, and as α varies from π/2 to π/2+i∞ with α = π/2 along C , sin α = cosh α also varies from 1 to +∞. Along the contour C, cos α varies as follows: π : 2

α = 0,

α = 0 →

π , 2

α = 0 → −∞ :

α =

cos α = cos α = 1 → 0, cos α = −1 sinh α = 0 → +i∞,

so that one must then choose the positive branch for γ when α varies along the contour C, viz. γ = +k cos α; α ∈ C. (6.120) On the other hand, when α varies from 0 to +∞ with α = π/2 along the contour C , cos α = −i sinh α varies from 0 to −i∞ and one must then choose the negative branch for γ; that is, γ = −k cos α when α ∈ C . For simplicity, the positive branch as speciﬁed in Eq. (6.120) is chosen here. ˜ Consider now the case when k(ω) is complex-valued. In that case, let ˜ k(ω) ≡ k(ω)eiκ(ω) ,

(6.121)

˜ ˜ where k(ω) ≡ |k(ω)| is the magnitude and κ(ω) ≡ arg{k(ω)} is the argument (or phase) of the complex wavenumber with 0 ≤ κ < π/2, which will always be satisﬁed for an attenuative medium. Let α = α + iα and β = β + iβ with α ≡ {α}, α ≡ {α}, β ≡ {β}, and β ≡ {β}. With these substitutions, Eqs. (6.116) and (6.117) become kx = eiκ(ω) sin (α + iα ) cos (β + iβ ) k(ω) = (cos κ + i sin κ) (sin α cosh α + i cos α sinh α ) × (cos β cosh β − i sin β sinh β ) = (sin α cosh α cos β cosh β + cos α sinh α sin β sinh β ) cos κ

− (cos α sinh α cos β cosh β − sin α cosh α sin β sinh β ) sin κ +i (sin α cosh α cos β cosh β + cos α sinh α sin β sinh β ) sin κ

− (cos α sinh α cos β cosh β − sin α cosh α sin β sinh β ) cos κ (6.122)

and

6.4 Polar Coordinate Form of the Angular Spectrum Representation

303

ky = eiκ(ω) sin (α + iα ) sin (β + iβ ) k(ω) = (cos κ + i sin κ) (sin α cosh α + i cos α sinh α ) × (sin β cosh β + i cos β sinh β ) = (sin α cosh α sin β cosh β − cos α sinh α cos β sinh β ) cos κ

− (sin α cosh α cos β sinh β + cos α sinh α sin β cosh β ) sin κ +i (sin α cosh α sin β cosh β − cos α sinh α cos β sinh β ) sin κ

+ (sin α cosh α cos β sinh β + cos α sinh α sin β cosh β ) cos κ , (6.123)

respectively. Because the left-hand sides of both of these expressions are realvalued, then the imaginary parts appearing on the right-hand sides must both vanish and one then obtains the pair of expressions (sin α cosh α cos β cosh β + cos α sinh α sin β sinh β ) sin κ − (cos α sinh α cos β cosh β − sin α cosh α sin β sinh β ) cos κ = 0, (sin α cosh α sin β cosh β − cos α sinh α cos β sinh β ) sin κ + (sin α cosh α cos β sinh β + cos α sinh α sin β cosh β ) cos κ = 0. Because sin κ and cos κ are known to exist, this pair of simultaneous linear equations in sin κ and cos κ must have a solution, which in turn requires that the determinant of its coeﬃcients must vanish, which then results in the relation sinh β cosh β sin2 α cosh2 α + cos2 α sinh2 α = 0. Because cosh β = 0 for real β and (sin2 α cosh2 α + cos2 α sinh2 α ) = 0 for all real values of α and α , it then follows that sinh β = 0 so that β = 0

(6.124)

and β = β is real-valued. With this identiﬁcation, the above pair of simultaneous linear equations becomes cos β (sin α cosh α sin κ + cos α sinh α cos κ) = 0, sin β (sin α cosh α sin κ + cos α sinh α cos κ) = 0, and because cos β = cos β and sin β = sin β cannot both be equal to zero, one must then have that sin α tan κ + cos α tanh α = 0;

0≤κ<

π . 2

Furthermore, Eqs. (6.122) and (6.123) become (with β = β )

(6.125)

304

6 The Angular Spectrum Representation of the Pulsed Radiation Field

kx = cos β (sin α cosh α cos κ − cos α sinh α sin κ) , k(ω) ky = sin β (sin α cosh α cos κ − cos α sinh α sin κ) , k(ω)

(6.126) (6.127)

where α , α , and β = β must vary in such a manner that kx and ky both vary continuously over the domain from −∞ to +∞.

Α''

ΚΠ Π

ΚΠ ΚΠ Π

Π Κ Π !Κ Κ

Π

C-1

Π

C0

Π Κ Π

Π

Α'

C1

Complex Α-Plane

Fig. 6.4. Family of contours Cm in the complex α-plane.

Consider now the family of contours in the complex α-plane that are described by Eq. (6.125). In the limit as α → +∞, tanh α → 1 and Eq. (6.125) assumes the limiting form sin α tan κ + cos α = 0, so that

π , tan α = − cot κ = tan κ − 2

and consequently α → κ −

π (mod 2π) 2

as

α → +∞.

(6.128)

On the other hand, in the limit as α → −∞, tanh α → −1 and Eq. (6.125) assumes the limiting form

6.4 Polar Coordinate Form of the Angular Spectrum Representation

305

sin α tan κ − cos α = 0, so that tan α = cot κ = tan

π 2

−κ ,

and consequently α →

π − κ(mod 2π) 2

as

α → −∞.

(6.129)

Furthermore, when α = 0, tanh α = 0 and Eq. (6.125) becomes sin α tan κ = 0, ˜ and because 0 ≤ κ < π/2 for complex k(ω), then sin α = 0 and α = ±mπ;

m = 0, 1, 2, 3, . . .

when

α = 0.

(6.130)

Hence, each member Cm of the family of contours described by Eq. (6.125) begins at α = mπ −π/2+κ, α = +∞, passes across the α -axis at α = mπ, and ends at α = mπ + π/2 − κ, α = −∞ for each postive and negative integer value of the index m, as illustrated in Figure 6.4. The angle of slope as the curve crosses the α -axis (i.e., at α = 0) for each member Cm of this family of contours is obtained by diﬀerentiating Eq. (6.125) to obtain dα sin α tanh α − tan κ cos α = , dα cos α sech2 α so that tan−1

dα dα

α =0

= tan−1 (− tan κ) = −κ,

(6.131)

and the angle of slope is given by minus the phase of the complex wavenumber. ˜ In the special case when κ = 0 and k(ω) is real-valued, Eq. (6.125) reduces to the equation cos α sinh α = 0 that was obtained when the complex wave ˜ number was assumed to be real-valued [i.e., when k(ω) = k with real-valued k], and the lower-half of the contour C0 reduces to the contour C that was obtained in that special case (see Fig. 6.3). Therefore, the general contour C to be employed here is given by the lower-half of C0 and is directed from α = 0 to α = π/2 − κ − i∞, as illustrated in Figure 6.5. Along this contour the real and imaginary parts of α = α + iα vary as

π sin α = 0 → sin (π/2 − κ) = cos κ ≥ 0, α =0→ −κ ⇒ cos α = 1 → cos (π/2 − κ) = sin κ ≥ 0, 2

sinh α = 0 → −∞, α = 0 → −∞ ⇒ cosh α = 1 → +∞,

306

6 The Angular Spectrum Representation of the Pulsed Radiation Field

Α''

Κ

Π Κ

Π

Α'

C Complex Α-Plane

Fig. 6.5. The contour C in the complex α-plane.

so that kx and ky vary along the contour C as kx = A cos β, k(ω)

ky = A sin β, k(ω)

where A varies from 0 to +∞. Thus, with α = α + iα varying along the contour C and with β = β varying between 0 and 2π, both kx and ky vary continuously from −∞ to +∞ and are real-valued, as desired. Furthermore, along the contour C, cos α ≥ 0, sin α ≥ 0, cosh α ≥ 0, and sinh α ≤ 0, so that (6.132) cos α = cos α cosh α + i |sin α sinh α | and the phase of cos α is always between 0 and π. Once again, the positive branch of γ must be taken when α varies along C [see Eq. (6.120)], so that ˜ γ(ω) = +k(ω) cos α;

α ∈ C.

(6.133)

Hence, under the change of variable to polar coordinates given in Eqs. (6.116)–(6.117), the angular spectrum integral given in Eq. (6.111) becomes 2π ˜± ˜ dβ sin αdα U (kx , ky )eik ·r , (6.134) u(r) = k(ω) 0

where

C

˜ ˜ ± (ω) = k(ω) ˆ x sin α cos β + 1 ˆ y sin α sin β ± 1 ˆ y cos α k 1

when the polar axis has been chosen to be along the kz -axis.

(6.135)

6.4 Polar Coordinate Form of the Angular Spectrum Representation

307

6.4.1 Transformation to an Arbitrary Polar Axis It is evident from a physical point of view that the choice of the polar axis to be along the kz -axis is simply a matter of convenience that is unnecessary and the expression in Eq. (6.134) for the angular spectrum integral should be independent of this choice. The rigorous mathematical proof of the simple observation is, however, quite another matter. To that end, consider choosing

kz

Polar Axis Λ Α

∆

k

ky

kx

Fig. 6.6. Relation of the polar axis to the kx , ky , kz coordinate axes when the wave vector k is real-valued.

some other polar axis that is speciﬁed relative to the kx , ky , kz coordinate axes by the (arbitrarily chosen) angles λ and δ, where λ is the angle of declination from the kz -axis and δ is the azimuthal angle of the normal projection of the polar axis onto the kx ky -plane, measured from the kx -axis, as illustrated in Figure 6.6 (the wave vector k depicted here is real-valued for the purpose of illustration). Under this transformation to a new polar axis, the expressions in Eqs. (6.116)–(6.117) become [retaining the complex variables α and β as ˜ the new angular coordinates for the complex wave vector k(ω)] ˜ cos α sin λ cos δ ± sin α cos β cos λ cos δ − sin α sin β sin δ , kx = k(ω) (6.136) ˜ ky = k(ω) cos α sin λ sin δ ± sin α cos β cos λ sin δ + sin α sin β cos δ , (6.137) where the ± sign appearing in these two expressions corresponds to the ± ˜ ± and takes into account the fact that for z < −Z, the sign appearing in k

308

6 The Angular Spectrum Representation of the Pulsed Radiation Field

polar axis is chosen in the opposite direction from that for z > Z, in which case λ → π − λ, as illustrated in Figure 6.7 (remember that the polar axis is no longer ﬁxed as it was in the preceding analysis).

kz

Λ

Polar Axis z>Z

ΠΛ

0

Polar Axis z < -Z Fig. 6.7. Polar axis choices when either z > Z or z < −Z.

For the Jacobian ∂(kx , ky )/∂(α, β) of this transformation, the required partial derivatives are given by 1 ∂kx ˜ k(ω) ∂α 1 ∂ky ˜ k(ω) ∂β 1 ∂ky ˜ k(ω) ∂α 1 ∂kx ˜ k(ω) ∂β so that

= − sin α sin λ cos δ ± cos α cos β cos λ cos δ − cos α sin β sin δ, = ∓ sin α sin β cos λ sin δ + sin α cos β cos δ, = − sin α sin λ sin δ ± cos α cos β cos λ sin δ + cos α sin β cos δ, = ∓ sin α sin β cos λ cos δ − sin α cos β sin δ,

1 ∂(kx , ky ) k˜−1 ∂kx /∂α k˜−1 ∂ky /∂α = ˜−1 k ∂kx /∂β k˜−1 ∂ky /∂β k˜2 (ω) ∂(α, β) = − sin2 α cos β sin λ ± sin α cos α cos λ.

(6.138)

In addition, under the coordinate transformation given in Eqs. (6.136)– (6.137) one has that

6.4 Polar Coordinate Form of the Angular Spectrum Representation

1−

309

ky2 kx2 γ 2 (ω) = + k˜2 (ω) k˜2 (ω) k˜2 (ω) 2 = cos α sin λ cos δ ± sin α cos β cos λ cos δ − sin α sin β sin δ 2 + cos α sin λ sin δ ± sin α cos β cos λ sin δ + sin α sin β cos δ = cos2 α sin2 λ + sin2 α cos2 β cos2 λ + sin2 α sin2 β ± 2 sin α cos α cos β sin λ cos λ = cos2 α − cos2 α cos2 λ + sin2 α cos2 β − sin2 α cos2 β sin2 λ + sin2 α sin2 β ± ±2 sin α cos α cos β sin λ cos λ

so that 1− and hence

2 γ 2 (ω) = 1 − cos α cos λ ∓ sin α cos β sin λ , k˜2 (ω)

˜ γ(ω) = ±k(ω) cos α cos λ ∓ sin α cos β sin λ .

˜ When λ = 0 one must have γ(ω) = +k(ω) cos α in accordance with Eq. (6.133), so that one ﬁnally obtains ˜ γ(ω) = k(ω) cos α cos λ ∓ sin α cos β sin λ (6.139) and Eq. (6.138) then yields ∂(kx , ky ) ˜ = ±k(ω)γ(ω) sin α ∂(α, β)

(6.140)

for the Jacobian of the transformation given in Eqs. (6.136)–(6.137), where ˜±. the sign choice corresponds to the sign choice appearing in k Hence, under the change of variable (6.136)–(6.137) to polar coordinates with an arbitrarily speciﬁed polar axis direction, the angular spectrum integral given in Eq. (6.111) becomes ˜± ˜ u(r) = ±k(ω) U (kx , ky )eik ·r sin α dαdβ, (6.141) D(α,β)

where the polar coordinate variables α = α + iα and β = β + iβ must now both be complex-valued, in general, and where the domain of integration D(α, β) is some two-dimensional surface in the four-dimensional α , α , β , β -space. The determination of this integration domain in the general case is a very diﬃcult problem. Nevertheless, it is shown in §6.4.2 that, given certain restrictions on the analyticity of the spectral function U (kx , ky ), the angular domain of integration D(α, β) can be taken to be the same domain of integration as that appearing in Eq. (6.134). Subject to these conditions on the analyticity of U (kx , ky ), the angular spectrum integral given in Eq. (6.141) becomes

310

6 The Angular Spectrum Representation of the Pulsed Radiation Field

˜ u(r) = ±k(ω)

2π

dβ 0

˜ ± ·r

sin α dα U (kx , ky )eik

,

(6.142)

C

where ± ˜ ˜ ˆ x cos α sin λ cos δ ± sin α cos β cos λ cos δ − sin α sin β sin δ k (ω) = k(ω) 1 ˆ y cos α sin λ sin δ ± sin α cos β cos λ sin δ + sin α sin β cos δ +1 ˆ ± 1z cos α cos λ ∓ sin α cos β sin λ , (6.143) and where the contour C is that branch of the family of contours in the complex α = α + iα plane described by Eq. (6.125) # that $ begins at α = 0 ˜ and extends to α = π/2 − κ − i∞, where κ ≡ arg k(ω) , 0 ≤ κ < π/2, as depicted in Figure 6.5. 6.4.2 Weyl’s Proof The proof that the angular spectrum integral appearing in Eq. (6.141) is independent of the transformation angles λ and δ, and, subject to certain conditions on the analyticity of the spectral function U (kx , ky ), reduces to the expression given in Eq. (6.142) is now given based upon the classic proof by Weyl [7] in 1919. This important proof begins with the spatial angular spectrum integral given in Eq. (6.111) under the general coordinate transformation given in Eqs. (6.136)–(6.137), which may be written in the general form 2π U (kx , ky ) ik˜ ± ·r ∂(kx , ky ) e , (6.144) I(r, λ, δ) ≡ dβ dα γ ∂(α, β) C 0 where C is the contour extending from α = 0 to α = π/2 − κ − i∞ that is described by Eq. (6.125). This particular form of the angular spectrum integral takes into account the polar coordinate form of the domain of integration for kx and ky as expressed in Eq. (6.134) when the polar axis is along kz , which is then modiﬁed by the transformation to an arbitrary polar axis direction speciﬁed by the transformation angles λ and δ through the Jacobian J(kx , ky ; α, β) ≡

∂(kx , ky ) . ∂(α, β)

(6.145)

The proof then entails showing that I(r, λ, δ) is independent of the parameters λ and δ by determining the partial derivatives ∂I/∂λ and ∂I/∂δ and showing that they both vanish. From Eq. (6.115), the spectral amplitude function appearing in Eq. (6.144) is given by

6.4 Polar Coordinate Form of the Angular Spectrum Representation

U (kx , ky ) = f (kx , ky , γ), where

f (kx , ky , kz ) =

∞

311

(6.146)

f (x, y, z)e−i(kx x+ky y+kz z) dxdydz

(6.147)

−∞

is an entire function of complex kx , ky , kz . From the discussion associated with Eq. (6.83), it is seen that the parameters λ and δ, through their speciﬁcation of the polar axis direction, determine the half-space |z| > Z wherein the integral appearing in Eq. (6.147) converges, because the polar axis direction speciﬁes the orientation of the z-axis. Two such possible half-spaces and their associated z-axis directions are depicted in Figure 6.8 for a current source with compact spatial support.

z'

J0(r,t) z

O

Z'

z'> z>Z

Fig. 6.8. Two possible half-spaces and their associated polar axis directions.

Deﬁne the quantity ψ ≡ ψ(kx , ky , kz ) ≡

1 f (kx , ky , kz )eik·r , kz

(6.148)

and let ψ˜ ≡ ψ(kx , ky , γ).

(6.149)

Then the function ψ˜ has an integrable singularity at γ = 0 and ψ˜ vanishes at α = π/2 − κ − i∞. With this substitution, the integral I(r, λ, δ) deﬁned in Eq. (6.144) becomes

312

6 The Angular Spectrum Representation of the Pulsed Radiation Field

I(r, λ, δ) ≡

2π

˜ dα ψJ(k x , ky ; α, β).

dβ 0

(6.150)

C

Consider the dependence of I(r, λ, δ) on the parameter λ. Upon diﬀerentiating Eq. (6.150) with respect to λ and assuming that the orders of diﬀerentiation and integration can be interchanged, one obtains 2π ∂J ∂I ∂ ψ˜ = J + ψ˜ ≡ I1 + I2 , dβ dα (6.151) ∂λ ∂λ ∂λ C 0 where

2π

∂J , ∂λ 0 C 2π ∂ ψ˜ J. I2 ≡ dβ dα ∂λ C 0

I1 ≡

dβ

dα ψ˜

(6.152) (6.153)

Attention is given ﬁrst to the integral I1 . The Jacobian appearing here is given by J(kx , ky ; α, β) = so that ∂J = ∂λ

∂ ∂kx ∂α ∂λ

∂ky ∂kx ∂kx ∂ky − , ∂α ∂β ∂α ∂β

∂ ∂ky ∂kx ∂ky + ∂β ∂β ∂λ ∂α ∂ ∂kx ∂ky ∂ ∂ky ∂kx − , − ∂α ∂λ ∂β ∂β ∂λ ∂α

where it has been assumed that the orders of diﬀerentiation can be interchanged. With this substitution, the integral I1 becomes 2π ∂ ∂ky ∂kx ∂ ∂kx ∂ky + I1 = dβ dα ψ˜ ∂α ∂λ ∂β ∂β ∂λ ∂α C 0 ∂ ∂ky ∂kx ∂ ∂kx ∂ky − − ∂α ∂λ ∂β ∂β ∂λ ∂α

π/2−κ−i∞ 2π ∂ky ∂kx ∂kx ∂ky = − ψ˜ dβ ψ˜ ∂λ ∂β ∂λ ∂β 0 0 ∂kx ∂ ∂ky ∂ky ∂ ∂kx ˜ ˜ ψ − ψ − dα ∂λ ∂α ∂β ∂λ ∂α ∂β C

2π ∂ky ∂kx ∂kx ∂ky + − ψ˜ dα ψ˜ ∂λ ∂α ∂λ ∂α 0 C 2π ∂ky ∂ ∂kx ∂kx ∂ ∂ky ˜ ˜ − dβ ψ − ψ (6.154) ∂λ ∂β ∂α ∂λ ∂β ∂α 0

6.4 Polar Coordinate Form of the Angular Spectrum Representation

313

upon integration by parts. In order to proceed, special consideration must be allowed for the point α = π/2 on the contour C in the special case when ˜ k(ω) is real-valued (i.e., when κ = 0). As illustrated in Figure 6.9, in this

Α''

Ε

Π

Α'

O Ε Π i

8

CΕ

Complex Α-Plane

Fig. 6.9. The contour C in the complex α-plane in the special case when the wave ˜ number k(ω) is real-valued.

special limiting case the contour C must be separated into the disjoint contour C which excludes an inﬁnitesimally small -neighborhood about the point α = π/2 where the slope of the contour is discontinuous. Of course, when ˜ k(ω) is complex-valued (i.e., when 0 < κ < π/2) the slope of the contour C is continuous along its entirety and this separation is unnecessary. In order to include this special case, Eq. (6.154) must be expanded as

π/2− 2π ∂ky ∂kx ∂kx ∂ky ˜ ˜ −ψ dβ ψ I1 = lim →0 0 ∂λ ∂β ∂λ ∂β 0 π/2−i∞ ∂ky ∂kx ∂kx ∂ky + ψ˜ − ψ˜ ∂λ ∂β ∂λ ∂β π/2−i ∂ ∂kx ∂ ∂k ∂k ∂k y y x − dα ψ˜ − ψ˜ ∂λ ∂α ∂β ∂λ ∂α ∂β C

2π ∂kx ∂ky ∂ky ∂kx + lim − ψ˜ dα ψ˜ →0 C ∂λ ∂α ∂λ ∂α 0 2π ∂ky ∂ ∂ ∂k ∂k ∂k x x y ψ˜ − ψ˜ (6.155) − dβ ∂λ ∂β ∂α ∂λ ∂β ∂α 0 when κ = 0.

314

6 The Angular Spectrum Representation of the Pulsed Radiation Field

Consider ﬁrst the “surface terms” appearing in Eq. (6.155). First of all, ˜ kx , and ky are all periodic in β with period 2π, then because the functions ψ, 2π ∂kx ∂ky ∂ky ∂kx ˜ ˜ −ψ ψ = 0, ∂λ ∂α ∂λ ∂α 0

(6.156)

independent of the contour C. Furthermore, from Eq. (6.143) one ﬁnds that ∂kx ˜ = k(ω) sin α ∓ sin β cos λ cos δ − cos β sin δ , ∂β ∂ky ˜ = k(ω) sin α ∓ sin β cos λ sin δ + cos β cos δ , ∂β so that at α = 0, ∂kx ∂ky = = 0, ∂β α=0 ∂β α=0 and because ψ˜ vanishes at α = π/2 − κ − i∞ for 0 ≤ κ < π/2, then when κ = 0 one has that π/2−κ−i∞ ∂ky ∂kx ∂kx ∂ky = 0, − ψ˜ ψ˜ ∂λ ∂β ∂λ ∂β 0

(6.157)

whereas for κ = 0 one obtains

π/2− π/2−i∞ ∂kx ∂ky ∂ky ∂kx ∂ky ∂kx ∂kx ∂ky ˜ ˜ ˜ ˜ lim + ψ ψ −ψ −ψ →0 ∂λ ∂β ∂λ ∂β 0 ∂λ ∂β ∂λ ∂β π/2−i α=π/2− ∂ky ∂kx ∂kx ∂ky − ψ˜ = lim ψ˜ . (6.158) →0 ∂λ ∂β ∂λ ∂β α=π/2−i This “surface term” will also vanish provided that the quantity to be evaluated is continuous at = 0, because then the limit as → 0 can be taken by setting = 0. From Eqs. (6.139) and (6.143) ∂kx ˜ = k(ω) cos α cos λ ∓ sin α cos β sin λ cos δ ∂λ = γ(ω) cos δ, ∂ky ˜ = k(ω) cos α cos λ ∓ sin α cos β sin λ sin δ ∂λ = γ(ω) sin δ, ˜±

and because ψ˜ = (1/γ)f (kx , ky , γ)eik ·r , then the ﬁnal quantity to be evaluated in Eq. (6.158) is indeed continuous at = 0 and this “surface term” vanishes. Thus, all of the “surface terms” in both Eqs. (6.154) and (6.155) vanish, and these expressions can both be written as

6.4 Polar Coordinate Form of the Angular Spectrum Representation

I1 = − lim

→0

2π

dβ 0

dα

C

315

∂ ψ˜ ∂kx ∂ky ∂kx ∂ 2 ky ∂ ψ˜ ∂kx ∂ky + ψ˜ − ∂α ∂λ ∂β ∂λ ∂α∂β ∂α ∂β ∂λ

∂ 2 kx ∂ky ∂ ψ˜ ∂kx ∂ky ∂ 2 kx ∂ky + + ψ˜ ∂α∂β ∂λ ∂β ∂α ∂λ ∂β∂α ∂λ

2 ˜ ∂kx ∂ ky ∂ ψ ∂kx ∂ky − ψ˜ − ∂β ∂λ ∂α ∂λ ∂β∂α

2π ∂kx ∂ky ∂ ψ˜ ∂kx ∂ky = − lim − dβ dα →0 0 ∂α ∂λ ∂β ∂β ∂λ C ∂ ψ˜ ∂kx ∂ky ∂kx ∂ky + − , (6.159) ∂β ∂α ∂λ ∂λ ∂α −ψ˜

where C = C when κ = 0, and where the limiting procedure is ignored when 0 < κ < π/2. Because ψ(kx , ky , kz ) has an integrable singularity at kz = 0, but is otherwise an analytic function of complex kz , and is an entire function of complex kx and ky , so that ψ˜ ≡ ψ(kx , ky , γ) is an analytic function on the contour C with an integrable singularity at γ = 0 (where it also has a branch point), ∂ψ ∂kx ∂ ψ˜ = ∂α ∂kx ∂α ∂ψ ∂kx = ∂kx ∂α ∂ ψ˜ ∂ψ ∂kx = ∂β ∂kx ∂β ∂ψ ∂kx = ∂kx ∂β

∂ψ ∂ky ∂ψ ∂kz + + ∂ky ∂α ∂kz ∂α kz =γ ∂ψ ∂ky ∂ψ ∂γ + + , ∂ky ∂α ∂kz ∂α kz =γ ∂ψ ∂ky ∂ψ ∂kz + + ∂ky ∂β ∂kz ∂β kz =γ ∂ψ ∂ky ∂ψ ∂γ + + , ∂ky ∂β ∂kz ∂β kz =γ

(6.160)

(6.161)

because γ depends upon the parameters α and β in precisely the same manner that kz does along the contour C. With these substitutions, Eq. (6.159) becomes

2π ∂ψ ∂kx ∂kx ∂ky ∂ψ ∂kx ∂ky ∂ky I1 = − lim + dβ dα →0 0 ∂kx ∂α ∂λ ∂β ∂ky ∂λ ∂α ∂β C ∂ψ ∂kx ∂kx ∂ky ∂ψ ∂kx ∂ky ∂γ − ∂kz ∂λ ∂β ∂α ∂kx ∂α ∂β ∂λ ∂ψ ∂kx ∂ky ∂ky ∂ψ ∂kx ∂ky ∂γ − − ∂ky ∂β ∂α ∂λ ∂kz ∂β ∂λ ∂α ∂ψ ∂kx ∂ky ∂ky ∂ψ ∂kx ∂kx ∂ky + + ∂kx ∂β ∂α ∂λ ∂ky ∂α ∂β ∂λ +

316

6 The Angular Spectrum Representation of the Pulsed Radiation Field

+

∂ψ ∂kx ∂ky ∂γ ∂ψ ∂kx ∂kx ∂ky − ∂kz ∂α ∂λ ∂β ∂kx ∂β ∂λ ∂α

∂ψ ∂kx ∂ky ∂γ ∂ψ ∂kx ∂ky ∂ky − − ∂ky ∂λ ∂β ∂α ∂kz ∂λ ∂α ∂β kz =γ

2π ∂ψ ∂kx ∂kx ∂ky ∂kx ∂ky − = − lim dβ dα →0 0 ∂k ∂λ ∂α ∂β ∂β ∂α x C ∂kx ∂ky ∂ψ ∂ky ∂kx ∂ky − + ∂ky ∂λ ∂α ∂β ∂β ∂α ∂ky ∂γ ∂ψ ∂kx ∂ky ∂γ − + ∂kz ∂λ ∂β ∂α ∂α ∂β ∂ky ∂kx ∂γ ∂kx ∂γ − + ∂λ ∂α ∂β ∂β ∂α kz =γ

2π ∂ψ ∂kx ∂ψ ∂ky + = − lim dβ dα J(kx , ky ; α, β) →0 0 ∂k ∂λ ∂k x y ∂λ C ∂ky ∂γ ∂ψ ∂kx ∂ky ∂γ − + ∂kz ∂λ ∂β ∂α ∂α ∂β ∂ky ∂kx ∂γ ∂kx ∂γ − + . ∂λ ∂α ∂β ∂β ∂α kz =γ

(6.162) The last two terms in this expression are evaluated in the following manner. Diﬀerentiation of the expression γ 2 (ω) = k˜2 (ω) − kx2 − ky2 with ω held ﬁxed yields γdγ = −kx dkx − ky dky , so that ∂γ 1 ∂kx ∂ky =− + ky kx , ∂α γ ∂α ∂α 1 ∂γ ∂kx ∂ky =− kx + ky . ∂β γ ∂β ∂β One then has that

∂ky ∂γ ∂ky ∂γ 1 ∂ky ∂kx ∂ky − = − kx + ky ∂β ∂α ∂α ∂β γ ∂β ∂α ∂α 1 ∂ky ∂kx ∂ky + kx + ky γ ∂α ∂β ∂β kx ∂kx ∂ky ∂kx ∂ky = − − γ ∂α ∂β ∂β ∂α kx = − J(kx , ky ; α, β), γ

6.4 Polar Coordinate Form of the Angular Spectrum Representation

317

and ∂kx ∂γ ∂kx ∂ky ∂kx ∂γ 1 ∂kx − = − kx + ky ∂α ∂β ∂β ∂α γ ∂α ∂β ∂β 1 ∂kx ∂kx ∂ky + kx + ky γ ∂β ∂α ∂α ky ∂kx ∂ky ∂kx ∂ky = − − γ ∂α ∂β ∂β ∂α ky = − J(kx , ky ; α, β). γ With these ﬁnal substitutions, Eq. (6.162) becomes

2π ∂ψ ∂ky ∂ψ ∂kx + I1 = − lim dβ dα →0 0 ∂kx ∂λ ∂ky ∂λ C 1 ∂ψ ∂kx ∂ky + ky kx − γ ∂kz λ λ

J(kx , ky ; α, β),

kz =γ

and because −γ

∂kx ∂ky ∂γ = kx + ky , ∂λ ∂λ ∂λ

one ﬁnally obtains the result

2π ∂ψ ∂ky ∂ψ ∂γ ∂ψ ∂kx + + dβ dα J(kx , ky ; α, β) I1 = − lim →0 0 ∂kx ∂λ ∂ky ∂λ ∂kz ∂λ C kz =γ 2π ∂ ψ˜ dβ dα (6.163) = − lim J(kx , ky ; α, β) = −I2 , →0 0 ∂λ C upon comparison with Eq. (6.153). Hence, from Eq. (6.151) one obtains the desired result that I(r, λ, δ) is independent of the angular parameter λ. Similarly, by changing λ to δ everywhere in this proof one also obtains that I(r, λ, δ) is independent of the angular parameter δ. Hence ∂I ∂I = = 0, ∂λ ∂δ

(6.164)

and this completes Weyl’s proof for a dispersive and absorptive medium. As a consequence, Eq. (6.142) is valid for any choice of the angles λ and δ such that the point of observation of the ﬁeld is within the region of convergence of the integral.

318

6 The Angular Spectrum Representation of the Pulsed Radiation Field

6.4.3 Weyl’s Integral Representation If one chooses the polar axis to be along the observation direction that is speciﬁed by the ﬁeld point position vector r = r(r, φ, θ), then the transformation angles δ and λ are chosen equal to the azimuthal angle δ = φ and the angle of declination λ = θ of the observation direction. With this choice the positive sign is used in front of the integral appearing in Eq. (6.142) and ˜ ± (ω) · r = k(ω)r ˜ cos α, k

(6.165)

where α (when it is real-valued) is the angle between the polar axis and the wave vector (when it too is real-valued), as illustrated in Figure 6.10. With

z Polar Axis r ΛΘ Α

k

y

∆Φ

x

Fig. 6.10. Choice of the polar axis in Weyl’s integral.

these substitutions, the angular spectrum integral in Eq. (6.142) becomes 2π ˜ ˜ u(r) = k(ω) dβ sin αdα U (kx , ky )eik(ω)r cos α . (6.166) 0

C

Consider now the special case when the spectral amplitude function U (kx , ky ) appearing in Eq. (6.166) is unity; viz. U (kx , ky ) = 1. With this substitution, Eq. (6.166) becomes [with the change of variable ξ = cos α so that α = 0 corresponds to ξ = 1 and α = π/2 − κ − i∞ corresponds to ξ = (∞)ei(π/2−κ) ]

6.4 Polar Coordinate Form of the Angular Spectrum Representation

319

˜

sin α eik(ω)r cos α dα

˜ u(r) = 2π k(ω) C

˜ = −2π k(ω)

∞ei(π/2−κ)

˜

eik(ω)rξ dξ

1

ξ=∞ei(π/2−κ) ˜ ik(ω)rξ e ˜ = −2π k(ω) ˜ ik(ω)r ξ=1 ˜

eik(ω)r , r where the antiderivative at the upper limit of integration indepen# vanishes $ ˜ dently of κ(ω) ≡ arg k(ω) . Substitution of these two expressions in Eq. (6.111) then yields the general result ∞ ˜ eik(ω)r i 1 ik˜ ± (ω)·r = e dkx dky (6.167) r 2π −∞ γ(ω) = −2πi

known as Weyl’s integral [7], which expresses a spherical wave in terms of a superposition of plane waves in the dispersive medium. Because ˜ ± (ω) · r = kx x + ky y ± γ(ω)z, k ˜ − (ω). ˜ + (ω) and k then on the plane z = 0 there is no distinction between k Notice that the term γ(ω)z provides exponential decay for the convergence of Weyl’s integral. On the plane z = 0 the only decay factor appearing in the integrand of the integral in Eq. (6.167) is due to the term γ −1 (ω), which goes to zero as γ goes oﬀ to inﬁnity, but not with suﬃcient rapidity for absolute convergence requirements to be satisﬁed. Consequently, Weyl’s integral (6.167) converges everywhere except on the plane z = 0, and it is conditionally convergent everywhere on the plane z = 0 except at the origin where it is divergent, as depicted in Figure 6.11 for (a) the general angular spectrum representation given in Eqs. (6.105)–(6.108), and (b) Weyl’s integral given in Eq. (6.167). An alternate representation of Weyl’s integral may be directly obtained from Eq. (6.142) with U (kx , ky ) = 1, so that in place of Eq. (6.167) one obtains π ˜ ˜ eik(ω)r ik(ω) dβ sin αdα ei(kx x+ky y±γ(ω)z) , = r 2π −π C where the change in the integration domain from β = 0 → 2π to β = −π → π is justiﬁed because it results in the same range of values for kx and ky . From Eqs. (6.116)–(6.117) and (6.133), ˜ kx = k(ω) sin α cos β, ˜ ky = k(ω) sin α sin β, ˜ γ(ω) = k(ω) cos α,

320

6 The Angular Spectrum Representation of the Pulsed Radiation Field

~k

J0

~+ k

Angular spectrum representation not valid within the region of space that contains the current source.

(a)

~k

~+ k

Plane z=0 Weyl's integral representation is conditionally convergent everywhere on the plane z = 0 except at the origin where the point source is located. (b)

Fig. 6.11. Regions of validity of the angular spectrum and Weyl’s integral representations.

and the above expression becomes π ˜ ˜ ik(ω) eik(ω)r ˜ = dβ sin αdα eik(ω)(x sin α cos β+y sin α sin β±z cos β) , r 2π −π C (6.168) which is the polar coordinate form of Weyl’s integral representation, where the positive sign appearing in the exponential of the integrand is employed when z > 0 and the negative sign when z < 0. This form of the representation clearly shows that Weyl’s integral expresses a spherical wave as a superposition of inhomogeneous plane waves, where the elements of the ordered triple (sin α cos β, sin α sin β, ± cos β) with α ∈ C and β = −π → π are the complex direction cosines of the elementary plane wave normals. 6.4.4 Sommerfeld’s Integral Representation Consider the change of variable ˜ ζ = k(ω) sin α

(6.169)

in the polar coordinate form (6.168) of Weyl’s integral representation, so that as α = α + iα varies over 0 → π/2 − κ − i∞ on the contour C, the variable −iκ(ω) ˜ (∞) = k · ∞. Deﬁne the quantity ζ varies over 0 → k(ω)e 1/2 Γ (ω) ≡ ζ 2 − k˜2 (ω) ˜ = −ik(ω) cos α = −iγ(ω),

(6.170)

6.4 Polar Coordinate Form of the Angular Spectrum Representation

321

where the branch of the square root is determined such that {Γ (ω)} ≥ 0 as α varies over the contour C as follows: $ # ˜ {Γ (ω)} = −ik(ω) cos (α + iα ) = −k(ω) (cos κ sin α sinh α − sin κ cos α cosh α ) . Because 0 ≤ κ < π/2 so that both cos κ and sin κ are nonnegative, and because, as α varies over the contour C, sin α varies over 0 → cos κ, sinh α varies over 0 → −∞, cos α varies over 1 → sin κ, and cosh α varies over 1 → +∞, then {Γ (ω)} = Ak(ω) where A varies over 0 → +∞ as α varies over C. In addition, along the contour C # $ ˜ {ζ} = k(ω) sin α = k(ω) {(cos κ + i sin κ)(sin α cosh α + i cos α sinh α )} = k(ω) cos κ cosh α (cos α tanh α + tan κ sin α ) = 0 from Eq. (6.125), so that ζ is real-valued as α varies over C. With substitution from Eqs. (6.169) and (6.170), the polar coordinate form (6.168) of Weyl’s integral representation becomes ˜

1 eik(ω)r = r 2π 1 = 2π

∞

π ζdζ dβ eiζ(x cos β+y sin β) e∓Γ z ˜ k(ω) cos α −π π 1 ∓Γ (ω)z e ζdζ dβ eiζ(x cos β+y sin β) , Γ (ω) −π

i 0

∞

0

where the upper sign choice is employed for z > 0 and the lower sign choice for z < 0. If one now transforms the spatial coordinates (x, y, z) in the Cartesian representation to cylindrical coordinates (r, ϕ, z), where x = r cos ϕ, y = r sin ϕ with r = x2 + y 2 the cylindrical coordinate representation of the spherical radius r, then the above expression becomes ˜

eik(ω)r 1 = r 2π

∞

0

1 ∓Γ (ω)z ζdζ e Γ (ω)

π

eiζr cos (β−ϕ) dβ,

−π

and consequently ˜

eik(ω)r = r

0

∞

1 ∓Γ (ω)z e J0 (ζr)ζdζ, Γ (ω)

(6.171)

where J0 (ξ) denotes the Bessel function of the ﬁrst kind of order zero. This result may be written in a more explicit form as ˜

eik(ω)r = r

0

∞

1

1/2

˜ 2 (ω)] ∓[ζ 2 −k

1/2 e ζ 2 − k˜2 (ω)

z

J0 (ζr)ζdζ.

(6.172)

322

6 The Angular Spectrum Representation of the Pulsed Radiation Field

It is then seen that the path of integration extends from the origin along the positive, real ζ-axis with a suitable indentation into the complex ζ-plane ˜ about the branch point at ζ = k when k˜ = k(ω) assumes a real value k. (1) (1) If one now uses the relation [16] J0 (ξ) = H0 (ξ) − H0 (−ξ) /2 which expresses the Bessel function of the ﬁrst kind of order zero in terms of the Hankel function of the ﬁrst kind (or Bessel function of the third kind) of order zero, then the spherical wave representation given in Eq. (6.171) becomes ˜

1 eik(ω)r = r 2 =

1 2

0

0

∞

∞

1 ∓Γ z (1) H0 (−ζr)ζdζ e Γ 0 0 1 ∓Γ z (1) 1 ∓Γ z (1) H0 (ζr)ζdζ + H0 (ζr)ζdζ e e Γ −∞ Γ 1 ∓Γ z (1) H0 (ζr)ζdζ − e Γ

∞

under the transformation ζ → −ζ in the second integral. One then ﬁnally obtains the classical result [17, 18], ˜ eik(ω)r 1 ∞ 1 ∓Γ (ω)z (1) = e H0 (ζr)ζdζ r 2 0 Γ (ω) 1/2 2 ˜2 1 1 ∞ (1) = e∓[ζ −k (ω)] z H0 (ζr)ζdζ, 1/2 2 −∞ 2 ˜2 ζ − k (ω) (6.173) which is known as Sommerfeld’s integral representation, where the upper sign choice appearing in the exponential is selected for z > 0 and the lower sign choice for z < 0. The contour of integration now extends along the real axis from −∞ to +∞ in the complex ζ-plane with the exception of a suitable indentation into the upper half-plane at the origin in order to avoid the logarithmic branch point that the Hankel function exhibits at that point, in addition to any suitable indentation about the branch point at ζ = k when ˜ k(ω) assumes a real value k. Finally, notice that in this representation the polar axis is ﬁxed along the z-axis. It is not necessary that the contour of integration appearing in Eqs. (6.171)–(6.173) lie along the real ζ-axis, but rather that the contour for each representation must only lie within a speciﬁc domain of analyticity deﬁned by each integrand. For the representation given in Eqs. (6.171)–(6.172) it is readily seen that the path of integration must lie within the strip of analyticity # $ # $ ˜ ˜ − k(ω) < {ζ} < k(ω) (6.174) ˜ that is deﬁned by the branch points at complex ζ = ±k(ω). Because of (1) the logarithmic branch point that the Hankel function H0 (ξ) exhibits at the origin, the path of integration for Sommerfeld’s integral representation (6.173) must lie within the strip of analyticity

6.4 Polar Coordinate Form of the Angular Spectrum Representation

# $ ˜ 0 < {ζ} < k(ω) .

323

(6.175)

Of course, there are permissible deformations of the contour of integration outside the strip of analyticity through the use of analytic continuation. Indeed, the path along the real ζ-axis that is described in the discussion following Eq. (6.173) is a trivial example of one such path. 6.4.5 Ott’s Integral Representation The major diﬃculty in both the application and extension of the polar coordinate form (6.168) of Weyl’s integral represenation is the simple fact that there are two integrations to perform. This is somewhat oﬀset by the generality aﬀorded by the arbitrary choice of direction for the polar axis that is not provided by Sommerfeld’s integral representation (6.173) which entails only a single integration. In order to partially overcome these two complications,

Α''

Π

Π Π !Κ

Κ Π Κ

Α'

C0 Complex ΑΑ'!iΑ'' Plane

Fig. 6.12. Contour of integration for Ott’s integral representation.

Ott [19] applied the transformation given in Eq. (6.169) with the deﬁnition in Eq. (6.170) to Sommerfeld’s integral representation (6.173). Under this transformation; viz., ˜ ζ = k(ω) sin α, ˜ dζ = k(ω) cos αdα,

324

6 The Angular Spectrum Representation of the Pulsed Radiation Field

the path of integration along the real ζ-axis is transformed to the contour C0 that is described by Eq. (6.125) and extends from −π/2 + κ + i∞ through the origin to π/2 − κ −#i∞ in$ the complex α-plane, as illustrated in Figure ˜ 6.12, where κ(ω) ≡ arg k(ω) . Under this transformation the representation appearing in Eq. (6.173) becomes ˜

˜ k(ω) eik(ω)r =i r 2

C0

(1)

H0

˜ ˜ k(ω)r sin α e±ik(ω)z cos α sin α dα,

(6.176)

which is known as Ott’s integral representation. This integral representation of a spherical wave forms the basis of Ba˜ nos’ classical research [18] on dipole radiation in the presence of a conducting half-space.

6.5 Applications The rigorous analysis presented in this chapter provides a basis for describing pulsed, ultrawideband electromagnetic radiation from both primary sources (antennas) as well as secondary sources (scatterers) embedded in a temporally dispersive medium. The physical requirement of causality has played a central role throughout this analysis. The latter topic on secondary sources has direct application to inverse scattering [20–24] in biomedical imaging (tumor detection), remote sensing and material identiﬁcation (ground- and foliage-penetrating radar), and the topic on primary sources has application to ultrawideband antenna design [25–28] and undersea communications [29]. These applications are considered in more detail in Volume II of this work.

References

325

References 1. J. A. Stratton, Electromagnetic Theory. New York: McGraw-Hill, 1941. 2. L. Lorenz, “On the identity of the vibrations of light with electrical currents,” Philos. Mag., vol. 34, pp. 287–301, 1867. 3. E. T. Copson, An Introduction to the Theory of Functions of a Complex Variable. London: Oxford University Press, 1935. p. 110. 4. E. T. Whittaker and G. N. Watson, Modern Analysis. London: Cambridge University Press, fourth ed., 1963. Section 6.222. 5. E. T. Copson, An Introduction to the Theory of Functions of a Complex Variable. London: Oxford University Press, 1935. Section 6.1. 6. H. B. Phillips, Vector Analysis. New York: John Wiley & Sons, 1933. 7. H. Weyl, “Ausbreitung elektromagnetischer Wellen u ¨ ber einem ebenen Leiter,” Ann. Physik (Leipzig), vol. 60, pp. 481–500, 1919. 8. E. T. Whittaker, “x,” Math. Ann., vol. 57, pp. 333–355, 1902. 9. G. C. Sherman, A. J. Devaney, and L. Mandel, “Plane-wave expansions of the optical ﬁeld,” Opt. Commun., vol. 6, pp. 115–118, 1972. ´ Lalor, “Contribution of the 10. G. C. Sherman, J. J. Stamnes, A. J. Devaney, and E. inhomogeneous waves in angular-spectrum representations,” Opt. Commun., vol. 8, pp. 271–274, 1973. 11. A. J. Devaney and G. C. Sherman, “Plane-wave representations for scalar wave ﬁelds,” SIAM Rev., vol. 15, pp. 765–786, 1973. 12. P. C. Clemmow, The Plane Wave Spectrum Representation of Electromagnetic Fields. Oxford: Pergamon, 1966. 13. J. J. Stamnes, Waves in Focal Regions: Propagation, Diﬀraction and Focusing of Light, Sound and Water Waves. Bristol, UK: Adam Hilger, 1986. 14. M. Nieto-Vesperinas, Scattering and Diﬀraction in Physical Optics. New York: Wiley-Interscience, 1991. 15. T. B. Hansen and A. D. Yaghjian, Plane-Wave Theory of Time-Domain Fields. New York: IEEE, 1999. 16. G. N. Watson, A Treatise on the Theory of Bessel Functions. Cambridge: Cambridge University Press, second ed., 1958. Sect. 3.62, Eq. (5). ¨ 17. A. Sommerfeld, “Uber die Ausbreitung der Wellen in der drahtlosen Telegraphie,” Ann. Phys. (Leipzig), vol. 28, pp. 665–737, 1909. 18. A. Ba˜ nos, Dipole Radiation in the Presence of a Conducting Half-Space. Oxford: Pergamon, 1966. Sect. 2.12. 19. H. Ott, “Reﬂexion und Brechung von Kugelwellen; Eﬀekte 2. Ordnung,” Ann. Phys., vol. 41, pp. 443–467, 1942. 20. R. S. Beezley and R. J. Krueger, “An electromagnetic inverse problem for dispersive media,” J. Math. Phys., vol. 26, no. 2, pp. 317–325, 1985. 21. G. Kristensson and R. J. Krueger, “Direct and inverse scattering in the time domain for a dissipative wave equation. I. Scattering operators,” J. Math. Phys., vol. 27, no. 6, pp. 1667–1682, 1986. 22. G. Kristensson and R. J. Krueger, “Direct and inverse scattering in the time domain for a dissipative wave equation. II. Simultaneous reconstruction of dissipation and phase velocity proﬁles,” J. Math. Phys., vol. 27, no. 6, pp. 1683– 1693, 1986. 23. J. P. Corones, M. E. Davison, and R. J. Krueger, “Direct and inverse scattering in the time domain via invariant embedding equations,” J. Acoustic Soc. Am., vol. 74, pp. 1535–1541, 1983.

326

6 The Angular Spectrum Representation of the Pulsed Radiation Field

24. G. Kristensson and R. J. Krueger, “Direct and inverse scattering in the time domain for a dissipative wave equation. Part III. Scattering operators in the presence of a phase velocity mismatch,” J. Math. Phys., vol. 28, pp. 360–370, 1987. 25. H. E. Moses and R. T. Prosser, “Initial conditions, sources, and currents for prescribed time-dependent acoustic and electromagnetic ﬁelds in three dimensions,” IEEE Trans. Antennas Prop., vol. 24, no. 2, pp. 188–196, 1986. 26. 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. Roy. Soc. Lond. A, vol. 422, pp. 351–365, 1989. 27. H. E. Moses and R. T. Prosser, “Acoustic and electromagnetic bullets: Derivation of new exact solutions of the acoustic and Maxwell’s equations,” SIAM J. Appl. Math., vol. 50, no. 5, pp. 1325–1340, 1990. 28. H. E. Moses and R. T. Prosser, “The general solution of the time-dependent Maxwell’s equations in an inﬁnite medium with constant conductivity,” Proc. Roy. Soc. Lond. A, vol. 431, pp. 493–507, 1990. 29. 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.

Problems 6.1. Show that the electric displacement vector D(r, t) and magnetic intensity vector H(r, t) are invariant under the gauge transformation given in Eqs. (6.57)–(6.58). 6.2. Evaluate the solid angle integral e±iF·G dΩ(G), 4π

where dΩ(G) is the diﬀerential element of solid angle of the variable vector G about the ﬁxed vector F. 6.3. With the result from the previous problem, determine the Fourier spectrum of a function f (r) that depends only upon the magnitude of the position vector r and not upon the direction. 6.4. Determine the gauge function Λ(r, t) that is required to perform the gauge transformation from the radiation potentials {A, ϕ} given in Eqs. (6.52) and (6.55) for a general temporally dispersive HILL medium with frequency-dependent dielectric permittivity (ω), magnetic permeability µ(ω), and electric conductivity σ(ω), to the potentials {A , ϕ } in the Lorenz gauge, where ∇ · A (r, t) +

c ∂ϕ (r, t) = 0. c2 ∂t

Obtain explicit expressions for the new potentials {A , ϕ } under this gauge transformation and verify that they do indeed satisfy the Lorenz condition.

6.5 Problems

327

6.5. Show that the angular spectrum representations (6.107)–(6.108) of the vector and scalar potentials yield the angular spectrum representations (6.105)–(6.106) of the electromagnetic ﬁeld vectors for the radiation ﬁeld in a temporally dispersive HILL medium. 6.6. Use the angular spectrum representation (6.107) to determine the vector potential ﬁeld A(r, t) in the right half-space region z > Z > 0 for t > 0 when the external current source is a point source conﬁned to the origin, given by ˆ J0 (r, t) = 1δ(r)f (t), where f (t) is a real-valued function of time that vanishes for t < 0 with ˆ is a unit vector in an arbitrary ﬁxed diFourier–Laplace transform F (ω), 1 ˜ rection, and where k(ω) = ω/v, where v is a real, positive-valued constant. Use Weyl’s integral given in Eq. (6.167) to perform the required spatial frequency integration. 6.7. Use the angular spectrum representation (6.107) to determine the vector potential ﬁeld A(r, t) in the right half-space region z > Z > 0 for t > 0 when the external current source is the temporal impulse ˆ (r)δ(t), J0 (r, t) = 1f where f (r) is a real-valued, spherically symmetric function of time that vanˆ is a unit vector in an ishes for r > Z > 0 with Fourier transform F (k), 1 ˜ arbitrary ﬁxed direction, and where k(ω) = ω/v, where v is a real, positivevalued constant. 6.8. Consider a current source that is represented by a pulsed current sheet that begins to radiate at time t = 0 and that is uniformly distributed throughout the entire xy-plane, being completely embedded in a temporally dispersive HILL medium with complex permittivity c (ω), with ˆ x J0 δ (z)F (t), J0 (r, t) = 1 where F (t) = 0 for t < 0, δ (z) is the derivative of the delta function, and J0 is a measure of the current source strength. (a) Determine both the vector and scalar potentials of the radiation ﬁeld produced by this current source. (b) From this pair of potentials, determine both the electric and magnetic ﬁeld vectors for this radiation ﬁeld. 6.9. Derive the angular spectrum of plane waves representation of the vector and scalar potential ﬁelds A(r, t) and ϕ(r, t) that is equivalent to that given in Eqs. (6.107)–(6.108) in a temporally dispersive HILL medium that is valid when the current source J0 (r, t) begins to radiate at time t = t0 (where t0 can be negative) instead of at t = 0. Determine the form that this representation takes when the ω-integration is taken over C (ω = ia − ∞ → ia + ∞) instead of being taken over C+ (ω = ia → ia + ∞).

328

6 The Angular Spectrum Representation of the Pulsed Radiation Field

6.10. Consider the current source distribution

f (r) cos (ω0 t); J0 (r, t) = 0;

t > t0 t < t0

for real-valued ω0 , where f (r) is a continuous, bounded, real-valued function of position such that f (r) = 0;

|r| > R,

so that the current source has compact support in three-dimensional space. Assume that v(ω) is real-valued. (a) Determine the space–time domain (i.e., the values of r and t) for which the ω-integral appearing in the angular spectrum for the vector potential ﬁeld derived in Problem 6.9 can be converted into a contour integral that is closed at |ω| = ∞ in the upper-half of the complex ω-plane. (b) Determine the space–time domain for which the ω-integral appearing in the angular spectrum representation for this vector potential ﬁeld can be converted into a contour integral that is closed at |ω| = ∞ in the lower-half of the complex ω-plane. (c) Show that in a certain speciﬁed space-time domain where the vector potential ﬁeld is nonzero, it can be expressed in the form A(r, t) = A0 (r, t) + At0 (r, t), where A0 (r, t) is independent of t0 and is an expansion of time-harmonic plane waves all with angular frequency ω0 . Derive explicit integral expressions for both A0 (r, t) and At0 (r, t). Finally, discuss the behavior of At0 (r, t) as t0 → −∞. 6.11. Derive the equations of transformation to an arbitrary polar axis given in Eqs.(6.136)–(6.137) that are obtained from the Cartesian coordinate rep˜ through the following succession of rotations resentation of the wave vector k about the origin: (i) the kx and ky coordinate axes are rotated counterclockwise through the angle δ about the kz -axis, forming the (kx , ky , kz )coordinate system; (ii) the kx and kz coordinate axes are rotated counterclockwise through the angle λ about the ky -axis, bringing the kz -axis into alignment with the arbitrarily chosen polar axis along the kˆz -axis and forming the ﬁnal (kˆx , kˆy , kˆz ) coordinate system.

7 The Angular Spectrum Representation of Pulsed Electromagnetic and Optical Beam Fields in Temporally Dispersive Media

A completely general representation of the propagation of a freely propagating electromagnetic wave ﬁeld into the half-space z ≥ z0 > Z of a homogeneous, isotropic, locally linear, temporally dispersive medium is now considered. The term “freely propagating” is used here1 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 {E0 , B0 } of electromagnetic ﬁeld vectors are known functions of time and the transverse ˆx x + 1 ˆ y y in the plane z = z0 , as illustrated in Figure position vector rT = 1 7.1. 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.

7.1 The Angular Spectrum Representation of the Freely Propagating Electromagnetic Field Consider an electromagnetic wave ﬁeld that is propagating into the half-space z ≥ z0 > Z > 0 and let the electric and magnetic ﬁeld vectors on the plane z = z0 , given by the boundary values E(rT , z0 , t) = E0 (rT , t), B(rT , z0 , t) = B0 (rT , t),

(7.1) (7.2)

ˆ x x+ 1 ˆy y be known functions of time and the transverse position vector rT ≡ 1 in the plane z = z0 , as indicated by the 0 subscript, as depicted in Figure 7.1. It is assumed here that the two-dimensional spatial Fourier transform in the 1

A freely propagating ﬁeld is fundamentally diﬀerent from a source-free ﬁeld because the former has an angular spectrum representation that contains both homogeneous and inhomogeneous plane wave components whereas the latter contains only homogeneous plane wave components in lossless media [1–6]. A generalized description of source-free ﬁelds appropriate for dispersive attenuative media is given in the ﬁnal section of this chapter.

330

7 Angular Spectrum Representation of Pulsed Beam Fields

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

v

1z z

Half-Space z z0

Plane z = z0

Fig. 7.1. Geometry of the planar electromagnetic boundary value problem.

transverse coordinates as well as the temporal Fourier–Laplace transform of each ﬁeld vector on the plane z = z0 exists, where ∞ ∞ ˜ ˜ dt dxdy E0 (rT , t)e−i(kT ·rT −ωt) , (7.3) E0 (kT , ω) = −∞ −∞

∞ 1 ˜˜ (k , ω)ei(kT ·rT −ωt) , (7.4) dω dkx dky E E0 (rT , t) = 0 T 4π 3 −∞ C+ and

∞ dt dxdy B0 (rT , t)e−i(kT ·rT −ωt) , (7.5) −∞ −∞

∞ 1 ˜ i(kT ·rT −ωt) ˜ dω dkx dky B0 (kT , ω)e , (7.6) B0 (rT , t) = 4π 3 −∞ C+

˜ ˜ 0 (kT , ω) = B

∞

ˆ x kx + 1 ˆ y ky is the transverse wave vector. If the initial time where kT ≡ 1 dependence of the electromagnetic ﬁeld vectors on the plane z = z0 is such that both the electric E0 (r, t) and magnetic B0 (r, t) ﬁeld vectors vanish for all t < t0 for some ﬁnite value of t0 , then the time–frequency transform pairs appearing in Eqs. (7.3)–(7.6) are Laplace transformations and the contour

7.1 Angular Spectrum Representation of the freely propagating Field

331

C+ is the straight line path ω = ω + ia with a being a constant greater than the abscissa of absolute convergence for the initial time evolution of the wave ﬁeld (see Appendix C); if not, then they are Fourier transformations. From Eqs. (6.105)–(6.106), the electromagnetic ﬁeld vectors for the radiation ﬁeld throughout the positive half-space z > Z > 0 are given by

dω 4π E(r, t) = (2π)3 ω

c (ω) C+

˜˜ (k ∞ ˜ + , ω) ˜+ × k ˜+ × J k 0 ˜ + ·r−ωt) i(k , dkx dky e γ(ω) −∞

B(r, t) = −

4π/c (2π)3

(7.7)

dω µ(ω) C+

˜ ˜ + , ω) ˜ + ˜+ × J ˜ 0 (k k i(k ·r−ωt) , dkx dky e γ(ω) −∞

∞

(7.8) where

˜ + (ω) = 1 ˆ x kx + 1 ˆ y ky + 1 ˆ z γ(ω) k

(7.9)

is the complex wave vector for propagation into the positive half-space z > Z > 0 with the associated complex wavenumber 1/2 ˜ ˜ + (ω) · k ˜ + (ω) k(ω) ≡ k ω 1/2 [µ(ω) c (ω)] , = k0 n(ω) = c

(7.10)

and where γ(ω) is deﬁned as the principal branch of the expression [cf. Eqs. (6.94)–(6.96)] 1/2 , (7.11) γ(ω) = k˜2 (ω) − kT2 ˆ z z0 and the electric with kT2 ≡ kx2 + ky2 . On the plane z = z0 > Z, r = rT + 1 ﬁeld vector given in Eq. (7.7) becomes, with Eq. (7.1),

dω 4π E(rT , t) = (2π)3 ω

c (ω) C+

˜ ∞ ˜ + , ω) ˜+ × k ˜+ × J ˜ 0 (k k iγ(ω)z0 i(kT ·rT −ωt) dkx dky e . e γ(ω) −∞ (7.12)

332

7 Angular Spectrum Representation of Pulsed Beam Fields

Comparison of this expression with that given in Eq. (7.4) shows that the spatiotemporal spectrum of the electric ﬁeld vector on the plane z = z0 is given by ˜ ˜ + , ω) ˜+ × J ˜+ × k ˜ 0 (k k ˜ ˜ 0 (kT , ω) = 4π (7.13) eiγ(ω)z0 . E 2ω c (ω) γ(ω) ˆ z z, and the electric ﬁeld vector Consequently, on any plane z ≥ z0 , r = rT + 1 is given by

∞ 1 ˜ ˜ 0 (kT , ω)eiγ(ω)(z−z0 ) ei(kT ·rT −ωt) , dω dkx dky E E(r, t) = 4π 3 −∞ C+ (7.14) which is the desired angular spectrum representation of the propagated electric ﬁeld vector in terms of the spectrum of its planar boundary value at z = z0 . Similarly, the magnetic induction vector on the plane z = z0 > Z is given by, with substitution from Eq. (7.2),

4π/c dω µ(ω) B(rT , t) = − (2π)3 C+

∞ ˜ ˜ + , ω) ˜+ × J ˜ 0 (k k eiγ(ω)z0 ei(kT ·rT −ωt) . dkx dky γ(ω) −∞ (7.15) Comparison of this expression with that given in Eq. (7.6) shows that the spatiotemporal spectrum of the magnetic induction ﬁeld vector on the plane z = z0 is given by % % ˜ + ˜˜ ˜ + 4π % 1% ˜ ˜ % µ(ω) k × J0 (k , ω) eiγ(ω)z0 . B0 (kT , ω) = − % 2% c % γ(ω)

(7.16)

Consequently, the magnetic induction ﬁeld vector on any plane z ≥ z0 is given by

∞ 1 ˜ ˜ 0 (kT , ω)eiγ(ω)(z−z0 ) ei(kT ·rT −ωt) , dω dkx dky B B(r, t) = 4π 3 C+ −∞ (7.17) which is the desired angular spectrum representation of the propagated magnetic induction ﬁeld vector in terms of the spectrum of its planar boundary value at z = z0 . Taken together, Eqs. (7.14) and (7.17) constitute the angular spectrum representation of the freely propagating electromagnetic wave ﬁeld for all z ≥ z0 > Z > 0.

7.1 Angular Spectrum Representation of the freely propagating Field

333

The spatiotemporal spectra of the electromagnetic ﬁeld vectors at the plane z = z0 cannot be chosen independently of each other because both are ultimately determined from the same radiation source. Indeed, substitution of Eq. (7.16) into (7.13) immediately yields the pair of relations c ˜˜ (k , ω), ˜+ × B k 0 T ωµ(ω) c (ω) ˜ ˜ ˜+ × E ˜ 0 (kT , ω). ˜ 0 (kT , ω) = c k B ω In addition, the transversality condition ˜ ˜ 0 (kT , ω) = − E

˜ ˜ ˜+ · B ˜+ · E ˜ 0 (kT , ω) = k ˜ 0 (kT , ω) = 0 k

(7.18) (7.19)

(7.20)

is satisﬁed. These three equations are precisely the relations that hold between both ﬁeld vectors and the associated wave vector for a time-harmonic electromagnetic plane wave ﬁeld in a temporally dispersive HILL medium [cf. Eqs. (5.61)–(5.63)]. The integrands appearing in the angular spectrum representations given in Eqs. (7.14) and (7.17), given by ˜ ˜ (k , ω)ei[k˜ + (ω)·(1ˆx x+1ˆy y+1ˆz (z−z0 ))−ωt] , ˜ 0 (kT , ω)ei[kT ·rT +γ(ω)(z−z0 )−ωt] = E ˜ E 0 T ˜ ˜ i[k ·r +γ(ω)(z−z )−ωt] 0 ˜ 0 (kT , ω)e T T ˜ 0 (kT , ω)ei[k˜ + (ω)·(1ˆx x+1ˆy y+1ˆz (z−z0 ))−ωt] , B =B are then seen to correspond to a time-harmonic electromagnetic plane wave ﬁeld that is propagating away from the plane z = z0 at each angular freˆ ˆ quency ω and transverse wave vector # kT = 1x kx + 1y ky that $ is present in the ˜ ˜ ˜ ˜ initial spectral amplitude vectors E0 (kT , ω), B0 (kT , ω) at that plane, with but one signiﬁcant diﬀerence: the wave vector components kx and ky are al1/2 ways real-valued and independent of ω whereas γ(ω) = k˜2 (ω) − kx2 − ky2 is, in general, complex-valued. Hence, each spectral plane wave component is attenuated in the z-direction alone; this is just a mathematical consequence of the evaluation of the kz -integral when the angular spectrum representation was derived in §6.3. The change in the amount of attenuation with propagation distance in diﬀerent directions between any two parallel planes z = constant is accounted for solely by the dependence of γ upon kx and ky . 7.1.1 Geometric Form of the Angular Spectrum Representation The plane wave spectral components appearing in the angular spectrum representation given in Eqs. (7.14) and (7.17) of the propagated electromagnetic wave ﬁeld into the positive half-space z ≥ z0 may be cast into a more geometric form by setting ˜ kx = k(ω)p, ˜ ky = k(ω)q, ˜ γ(ω) = k(ω)m,

(7.21) (7.22) (7.23)

334

7 Angular Spectrum Representation of Pulsed Beam Fields

˜ where k(ω) = k0 n(ω) is the complex wave number given in Eq. (7.10), k0 = 1/2 is the ω/c is the wavenumber in vacuum, and n(ω) = (c/c) [µ(ω) c (ω)] complex index of refraction of the temporally dispersive HILL medium [cf. Eq. (5.33)]. The expression given in Eq. (7.11) then requires that 1/2 , m = 1 − p2 − q 2

(7.24)

where the principal branch of the square root expression is to be taken, as deﬁned in Eqs. (6.94)–(6.96). With these substitutions, the spatial phase term appearing in the exponential factors of both Eqs. (7.14) and (7.17) becomes ˜ kT · rT + γ(ω)(z − z0 ) = k(ω) [px + qy + m(z − z0 )] . The ordered triple of complex numbers (p, q, m) is then seen to be the set of ˜+ = 1 ˆ x kx + 1 ˆ y ky + 1 ˆ z γ, complex direction cosines of the complex wave vector k because ˜ + · r = kx x + ky y + γz k kx ky γ(ω) ˜ = k(ω) x+ y+ z ˜ ˜ ˜ k(ω) k(ω) k(ω) ˜ = k(ω) (px + qy + mz) . That these generalized direction cosines are, in general, complex-valued follows directly from the fact that, for a causally dispersive medium the ˜ wavenumber k(ω) = β(ω) + iα(ω) is, in general, complex-valued where β(ω) is the (real-valued) time-harmonic plane wave propagation factor and α(ω) is the (real-valued) time-harmonic plane wave attenuation factor, given in Eqs. (5.70)–(5.71). Because kx and ky must both be real-valued quantities, then the generalized direction cosines p and q must, in general, be complex-valued, so that, with p = p + ip and q = q + iq , ˜ = β(ω) + iα(ω) p + ip kx = k(ω)p (7.25) = β(ω)p − α(ω)p + i β(ω)p + α(ω)p , ˜ ky = k(ω)p = β(ω) + iα(ω) q + iq (7.26) = β(ω)q − α(ω)q + i β(ω)q + α(ω)q . Hence, in order that both kx and ky are real-valued for all values of the angular frequency ω, it is required that α(ω) p, β(ω) α(ω) q. q = − β(ω)

p = −

With these substitutions, Eqs. (7.25) and (7.26) become, respectively,

(7.27) (7.28)

7.1 Angular Spectrum Representation of the freely propagating Field

335

α2 (ω) p , kx = β(ω) 1 + 2 β (ω) α2 (ω) q . ky = β(ω) 1 + 2 β (ω)

(7.29) (7.30)

Hence, in order to determine the manner in which p and q must be required to vary so that kx and ky both vary from −∞ to +∞ for all ω = ω + ia, ω ≥ 0, it is necessary to know the behavior of β(ω) along the contour C+ in the complex ω-plane. Because β(ω) is proportional to the real part of the complex index of refraction which (for the medium models considered in this book, unless otherwise noted; see Problem 7.2) is positive for all ω ∈ C+ , then p and q must both vary from −∞ to +∞. Furthermore, because α(ω) ≥ 0 for all ω ∈ C+ (for the medium models considered in this book), then p varies from +∞ to −∞ as p varies from −∞ to +∞, and q varies from +∞ to −∞ as q varies from −∞ to +∞. Hence, the contour Cw that w = w + iw , w = p, q varies over in the complex w-plane is a straight line path through the origin at the angle − arctan (α(ω)/β(ω)) to the real axis, as illustrated in Figure 7.2. Notice that Cp and Cq both depend upon the angular frequency ω. When the medium is lossless, α vanishes and the contour lies along the real axis; this may occur, for example, at certain frequency values in a dispersive medium or at all frequency values in a nondispersive medium.

p'',q''

O

p',q' -Tan-1(ΑΩ ΒΩ)

Cp ,Cq

Fig. 7.2. Contour of integration in the complex p, q-plane.

With p and q as respectively given in Eqs. (7.27)–(7-28), the expression given in Eq. (7.24) for the complex direction cosine m = (1 − p2 − q 2 )1/2 becomes

336

7 Angular Spectrum Representation of Pulsed Beam Fields

1/2 m(ω) = 1 − (p + ip )2 − (q + iq )2 1/2 α(ω) 2 α2 (ω) 2 2 2 p + q + 2i p +q = 1− 1− 2 . (7.31) β (ω) β(ω) In order to evaluate this expression with the appropriate branch choice as set forth in Eqs. (6.94)–(6.96), let m(ω) ≡ ζ 1/2 (ω)

(7.32)

with ζ = ζ + iζ , where

α2 (ω) 2 p + q 2 , ζ (ω) ≡ {ζ(ω)} = 1 − 1 − 2 β (ω) α(ω) 2 p + q 2 . ζ (ω) ≡ {ζ(ω)} = 2 β(ω)

(7.33) (7.34)

Because α(ω)/β(ω) ≥ 0 for all ω ∈ C+ , then ζ (ω) ≥ 0,

∀ω ∈ C+ .

(7.35)

Furthermore, ζ (ω) > 0, ζ (ω) = 0, ζ (ω) < 0,

1 , 1 − α2 (ω)/β 2 (ω) 1 , when p2 + q 2 = 1 − α2 (ω)/β 2 (ω) 1 . when p2 + q 2 > 2 1 − α (ω)/β 2 (ω) when p2 + q 2 <

(7.36) (7.37) (7.38)

It is then clear that the argument of ζ(ω) satisﬁes the inequality 0 ≤ arg {ζ(ω)} ≤ π,

(7.39)

and hence, that the appropriate branch of the argument of m = ζ 1/2 satisﬁes the inequality π (7.40) 0 ≤ arg {m(ω)} ≤ , 2 for all ω ∈ C+ , as illustrated in the sequence of diagrams given in Figure 7.3. As a consequence, the real and imaginary parts of the complex direction cosine m(ω) satisfy the inequalities m (ω) = {m(ω)} ≥ 0, m (ω) = {m(ω)} ≥ 0,

(7.41) (7.42)

for all ω ∈ C+ . Explicit expressions for both m (ω) and m (ω) are obtained in the following manner. First of all, because p2 + q 2 + m2 = 1, with p = p + ip ,

7.1 Angular Spectrum Representation of the freely propagating Field

Ζ''

Ζ''

O

m = Ζ Ζ'

O

(a) 2

2

p' + q' < 1/(1 - Α Β )

Ζ''

Ζ

Ζ

m = Ζ Π Ζ'

m = Ζ

Ζ

O

2

p' + q' = 1/(1 - Α Β )

Ζ'

(c)

(b) 2

337

2

2

p' + q' > 1/(1 - Α Β )

Fig. 7.3. Proper values of the complex direction cosine m(ω) when (a) p2 + q 2 < 1/(1 − α2 /β 2 ), (b) p2 + q 2 = 1/(1 − α2 /β 2 ), and (c) p2 + q 2 > 1/(1 − α2 /β 2 ).

q = q + iq , m = m + im and substitution from Eqs. (7.27)–(7.28) for p and q , there results α2 2 α 2 2 2 2 2 p + q + m − m − 2i p + q − m m = 1. 1− 2 β β Upon equating real and imaginary parts in the above expression, one obtains the pair of simultaneous equations α2 2 p + q 2 + m2 − m2 = 1, (7.43) 1− 2 β α 2 p + q 2 . (7.44) m m = β Substitution of m from the second relation into the ﬁrst then yields the simple quartic equation 2 α2 2 α2 4 2 p + q − 1 m2 − 2 p2 + q 2 = 0. m + 1− 2 β β The general solution for m2 is then given by

α2 2 1 2 p + q 2 1− 1− 2 m = 2 β 1/2 2 2 α2 α2 2 2 2 2 p +q + 1+ 2 p +q . ± 1−2 1− 2 β β

338

7 Angular Spectrum Representation of Pulsed Beam Fields

Because m is real-valued, then m2 ≥ 0 and only the positive sign choice in the above expression is appropriate. Hence

α2 2 1 2 p +q 1− 1− 2 m = √ β 2 1/2 1/2 2 2 α2 α2 2 2 2 2 p +q + 1+ 2 p +q , ± 1−2 1− 2 β β (7.45) and m is given by

α p2 + q 2 m = , βm

(7.46)

provided that α = 0. When α = 0, Eq. (7.43) becomes m2 − m2 = 1 − p2 + q 2 and Eq. (7.44) states that either m = 0 or m = 0. Then m = 0 and 1/2 m = 1 − p2 + q 2

(7.47)

when p2 + q 2 ≤ 1, whereas m = 0 and m =

1/2 2 p + q 2 − 1

(7.48)

when p2 + q 2 > 1. With these results, the complex phase term appearing in the plane wave ˜+ propagation factor eik ·r may be expressed as ˜ + (ω) · r = kx x + ky y + γ(ω)∆z k α2 (ω) = β(ω) 1 + 2 (p x + q y) + (β(ω)m − α(ω)m ) ∆z β (ω) + i (α(ω)m + β(ω)m ) ∆z, so that ˜ + (ω)·r

eik

= e−(α(ω)m +β(ω)m )∆z 2 2 × eiβ(ω)[(1+α (ω)/β (ω))(p x+q y)+(m −(α(ω)/β(ω))m )∆z] ,

(7.49) ˜+

where ∆z ≡ z − z0 . If p = q = 0, then eik ·r represents the spatial part of a time-harmonic homogeneous plane wave with angular frequency ω because the surfaces of constant amplitude coincide with the surfaces of constant phase, given by ∆z = constant. If α(ω) = 0, then the expression in Eq.

7.1 Angular Spectrum Representation of the freely propagating Field

339

(7.49) represents the spatial part of a time-harmonic homogeneous plane wave when p2 + q 2 ≤ 1 (in which case m = 0), whereas it represents an evanescent wave when p2 + q 2 > 1 (in which case m = 0) because the surfaces of constant amplitude are then orthogonal to the equiphase surfaces. In general, α(ω) = 0 and either p = 0 or q = 0; The expression in Eq. (7.49) then represents the spatial part of a time-harmonic inhomogeneous plane wave with angular frequency ω because the surfaces of constant amplitude ∆z = z − z0 = constant are diﬀerent from the surfaces of constant phase, given by α2 (ω) α(ω) 1+ 2 (p x + q y) + m − m ∆z = constant. (7.50) β (ω) β(ω)

x s

cos-1((1+Α Β )p'/s)

O

cos-1((m'-(Α Β)m'')/s)

Cophasal Surface

cos-1((1+Α Β )q'/s)

z

y

Fig. 7.4. Inhomogeneous plane wave phase front propagating in the direction s.

The inhomogeneous plane wave phase fronts described in Eq. (7.50) propagate in the direction speciﬁed by the real-valued vector α2 (ω) ˆ α(ω) ˆ ˆ p 1x + q 1y + m − m 1z s≡ 1+ 2 (7.51) β (ω) β(ω) with magnitude s=

α2 (ω) 1+ 2 β (ω)

2

2

p +q

2

α(ω) + m − m β(ω)

2 1/2 ,

(7.52)

340

7 Angular Spectrum Representation of Pulsed Beam Fields

which is, in general, not equal to unity. The plane phase fronts or cophasal surfaces of the inhomogeneous plane wave described in Eq. (7.49) are then seen to propagate in the direction speciﬁed by the set of real-valued directions cosines 1 α2 (ω) α2 (ω) α(ω) 1 1 m , (7.53) 1+ 2 p, 1+ 2 q, m − s β (ω) s β (ω) s β(ω) as illustrated in Figure 7.4. These inhomogeneous plane wave phase fronts advance into the positive half-space ∆z > 0 when the inequality m −

α(ω) m >0 β(ω)

(7.54)

is satisﬁed. Substitution of the relation given in Eq. (7.46) into this expression then yields the equivalent inequality α2 (ω) p2 + q 2 > 0. m 1− 2 β (ω) m2 Because m > 0 when α(ω) = 0, then the inequality given in Eq. (7.54) will be satisﬁed when 2 p + q 2 β 2 (ω) , (7.55) < m2 α2 (ω) where the right-hand side of this inequality depends solely upon the dispersive properties of the medium. For a nondispersive medium, α(ω) = 0 for all ω and the inequality given in Eq. (7.54) simply becomes m > 0, which is satisﬁed by the homogeneous plane waves propagating into the positive half-space occupied by a loss-free medium. Consider now obtaining the conditions (if indeed any do exist in the general case) under which the inequality given in Eq. (7.55) is satisﬁed. In order to address this, return to the quartic equation in m following Eq. (7.44); viz. 2 α2 2 α2 p + q 2 − 1 m2 − 2 p2 + q 2 = 0. m4 + 1 − 2 β β Two inequalities may then be obtained from this equation that are dependent upon the sign of the coeﬃcient of m2 , as follows. 1. If this coeﬃcient is negative [i.e., if p2 + q 2 < 1/ 1 − α2 /β 2 ] which occurs when 0 ≤ arg {m(ω)} < π/4, then the quartic equation given above implies that p2 + q 2 β(ω) < . (7.56) m2 α(ω) Hence, if β(ω) ≥ α(ω), then the inequality appearing in Eq. (7.56) implies that the inequality appearing in Eq. (7.55) is satisﬁed. On the other hand, if α(ω) > β(ω), then the inequality appearing in Eq. (7.55) may still be satisﬁed, but there is no guarantee from this method of argument.

7.1 Angular Spectrum Representation of the freely propagating Field

341

2

2. If this coeﬃcient is positive [i.e., if p2 + q 2 > 1/ 1 − α2 /β ] which occurs when π/4 < arg {m(ω)} ≤ π/2, then the quartic equation given above implies that β(ω) p2 + q 2 > . (7.57) m2 α(ω) If β(ω) > α(ω) then the inequality appearing in Eq. (7.55) may be satisﬁed, but there is no guarantee from this method of argument. However, it is deﬁnitely not satisﬁed if α(ω) ≤ β(ω). Notice that when α(ω) = 0, the inequality speciﬁed in case 1 reduces to the inequality p2 + q 2 < 1 which speciﬁes the homogeneous plane wave components in a loss-free medium. On the other hand, when α(ω) = 0, the inequality speciﬁed in case 2 becomes p2 + q 2 > 1 which speciﬁes the evanescent plane wave components in a loss-free medium. The geometric form of the angular spectrum representation of the freely propagating electromagnetic ﬁeld is then given by E(x, y, z, t)

1 ˜ ˜ −iωt ik(ω)(px+qy+m∆z) 2 ˜ ˜ dω e k (ω)dpdq , E0 (p, q, ω)e = 4π 3 C+ Cp Cq (7.58) B(x, y, z, t)

1 ˜ ˜ ˜ 0 (p, q, ω)eik(ω)(px+qy+m∆z) k˜2 (ω)dpdq , = dω e−iωt B 4π 3 C+ Cp Cq (7.59) where p = p + ip varies over the contour Cp and q = q + iq varies over the contour Cq , as illustrated in Figure 7.2. Because the spatiotemporal frequency spectra of the ﬁeld vector boundary values are related [cf. Eqs. (7.18)–(7.19)], then the pair of relations given in Eqs. (7.58)–(7.59) for the propagated ﬁeld vectors in the half-space z ≥ z0 may be expressed in terms of either boundary ˜ ˜˜ (p, q, ω). Similar results ˜ 0 (p, q, ω) or B value alone, that is, in terms of either E 0 for the formal solution of such boundary value problems for time-harmonic (or monochromatic) wave propagation may be found in the published work of Bouwkamp [7] and Goodman [8] for the scalar optical ﬁeld, and Carter [9] and Stamnes [10] for the diﬀraction theory of the electromagnetic ﬁeld. 7.1.2 Angular Spectrum Representation and Huygen’s Principle It is well known that the solution of the planar boundary value problem that is considered in this chapter can also be obtained through a superposition of spherical waves. The solution in terms of spherical waves has its physical origin in Huygen’s principle [11] and has been (and continues to be) a central

342

7 Angular Spectrum Representation of Pulsed Beam Fields

theme in classical diﬀraction theory. It is consequently of some importance to establish the connection between the angular spectrum of plane waves representation and the mathematical embodiment of Huygen’s principle as found in classical diﬀraction theory. It is clear that Weyl’s integral, as given in Eq. (6.167), which expresses a spherical wave in terms of a superposition of plane waves, provides this connection, which is now derived following the treatment due to Sherman [1, 2]. The derivation begins with the angular spectrum representation of the freely propagating electromagnetic wave ﬁeld that is given in Eqs. (7.58)– (7.59) where ∆z = z − z0 ≥ 0 is the normal propagation distance between the observation plane at z ≥ z0 and the input plane at z = z0 . In this spatiotemporal spectral representation ∞ ˜ ˜ ˜ 0 (p, q, ω) = ˜ 0 (x, y, ω)e−ik(ω)(px+qy) dxdy, (7.60) E E −∞ ∞ ˜ ˜ ˜ 0 (x, y, ω)e−ik(ω)(px+qy) ˜ 0 (p, q, ω) = dxdy, (7.61) B B −∞

where

˜ 0 (x, y, ω) = E ˜ 0 (x, y, ω) = B

∞

−∞ ∞ −∞

E0 (x, y, t)eiωt dt,

(7.62)

B0 (x, y, t)eiωt dt,

(7.63)

are the temporal frequency spectra of the initial ﬁeld vectors on the plane z = z0 . Substitution of Eq. (7.60) into Eq. (7.58) results in the expression

∞ ∞ 1 −iωt ˜ 0 (x , y , ω) dω e dx dy E E(x, y, z, t) = 4π 3 C+ −∞ −∞

˜ ik(ω)[p(x−x )+q(y−y )+m∆z] ˜ 2 × e k (ω)dpdq , Cp

Cq

(7.64) and substitution of Eq. (7.61) into Eq. (7.59) yields

∞ ∞ 1 −iωt ˜ 0 (x , y , ω) dω e dx dy B B(x, y, z, t) = 4π 3 −∞ −∞ C+

˜ ik(ω)[p(x−x )+q(y−y )+m∆z] ˜ 2 × e k (ω)dpdq . Cp

Cq

(7.65) The monochromatic spatial impulse response function for the normal propagation distance ∆z = z − z0 in the temporally dispersive HILL medium at the angular frequency ω is deﬁned here as

7.1 Angular Spectrum Representation of the freely propagating Field

h(ξ, η; ∆z, ω) ≡

1 (2π)2

Cp

˜

eik(ω)[pξ+qη+m∆z] k˜2 (ω)dpdq,

343

(7.66)

Cq

where ξ = x−x and η = y−y when applied to Eqs. (7.64)–(7.65). Notice that this impulse response function is space-invariant (or isoplanatic) and that it depends upon the angular frequency ω through the complex wave number ˜ k(ω). With the set of relations given in Eqs. (7.21)–(7.23), this expression may be rewritten as ∞ ∞ 1 eiγ(ω)ζ ei(kx ξ+ky η) dkx dky h(ξ, η; ζ, ω) = (2π)2 −∞ −∞ # $ = F −1 eiγ(ω)ζ , (7.67) 1/2 . The quantity eiγ(ω)ζ is seen to be the where γ(ω) = k˜2 (ω) − kx2 − ky2 spatial transfer function of the linear dispersive system at the angular frequency ω. With the deﬁnition given in Eq. (7.66), the preceding pair of expressions for the propagated ﬁeld vectors becomes E(x, y, z, t)

∞ ∞ 1 ˜ 0 (x , y , ω)h(x − x , y − y ; ∆z, ω)dx dy = dω e−iωt E π −∞ −∞ C+ (7.68) B(x, y, z, t)

∞ ∞ 1 −iωt ˜ 0 (x , y , ω)h(x − x , y − y ; ∆z, ω)dx dy dω e = B π −∞ −∞ C+ (7.69) and the spatial part of each propagated ﬁeld vector is given by the twodimensional convolution of the spatial part of the corresponding initial ﬁeld vector on the plane z = z0 with the spatial impulse response function at each value of the angular frequency ω. Notice that the material dispersion is contained entirely within the spatial impulse response function. Weyl’s integral given in Eq. (6.167) expresses a monochromatic spherical wave in terms of a superposition of monochromatic plane waves as ∞ ∞ ˜ 1 ik˜ ± (ω)·r eik(ω)r i dkx dky , (7.70) = e r 2π −∞ −∞ γ(ω) ˜ ± (ω) = 1 ˆ x kx + 1 ˆ y ky ± 1 ˆ z γ(ω), r = 1 ˆx x + 1 ˆy y + 1 ˆ z z, and where where k 2 2 2 r = |r| = + x + y + z . The positive sign choice is taken in the positive half-space z > 0 whereas the negative sign choice is taken in the negative half-space z < 0. With this result in mind, the relation given in Eq. (7.66) may be expressed as

344

7 Angular Spectrum Representation of Pulsed Beam Fields

∞ ∞ + 1 ˜+ eik (ω)·r dkx dky 2π 2 −∞ −∞ ∞ ∞ 1 = ei(kx x+ky y+γ(ω)∆z) dkx dky 2π 2 −∞ −∞ ∞ ∞ i 1 ik˜ + (ω)·r+ 1 ∂ e =− dkx dky ,(7.71) 2π ∂z 2π −∞ −∞ γ(ω)

h(x, y; ∆z, ω) =

ˆx x + 1 ˆy y + 1 ˆ z ∆z with ∆z = z − z0 > 0. The interchange of the where r+ ≡ 1 order of the diﬀerentiation and integration operations used in the derivation of the ﬁnal expression in Eq. (7.71) is justiﬁed by the following argument due to Lalor [12]. Deﬁne the pair of functions 1 ik(px+qz+mz) ˜ e dpdq, f (z) ≡ Cp Cq m ˜ g(z) ≡ ik˜ eik(px+qz+mz) dpdq, Cp

Cq

where it is desired to show that g(z) = ∂f (z)/∂z or, equivalently, that z g(z )dz = f (z) − f (z0 ). z0

Consider then the integral z g(z )dz = ik˜ z0

z

dz

Cp

z0

˜

eik(px+qy+mz ) dpdq.

Cq

˜ Because the Lebesgue integrability of the function exp ik(px + qy + mz ) is ensured by the existence of the integral z ˜ ik(px+qy+mz ) dz I= e dpdq z0

Cp

Cq

for z > 0, then by Fubini’s theorem [13], the order of integration in the above expression may be interchanged, so that z z 1 ik(px+qy+mz ˜ ) e g(z )dz = dpdq z0 Cp C q m z0 = f (z) − f (z0 ), as was to be shown. With substitution of the identity expressed by Weyl’s integral [Eq. (7.70)] in Eq. (7.71), one ﬁnally obtains the important result that ˜ eik(ω)R 1 ∂ h(x − x , y − y ; ∆z, ω) = − , (7.72) 2π ∂z R

7.2 Polarization Properties of the Electromagnetic Wave Field

345

where R ≡ + (x − x )2 + (y − y )2 + (∆z)2 . This then identiﬁes the spatial impulse response function deﬁned in Eq. (7.66) in terms of the normal derivative of the “free-space” Green’s function ˜

G(R, ω) ≡

eik(ω)R R

(7.73)

that plays a central role in the mathematical embodiment of Huygen’s principle, as given by the integral theorem of Helmholtz and Kirchhoﬀ [11]. Substitution of Eqs. (7.72) and (7.73) into the pair of relations given in Eqs. (7.68)–(7.69) then yields

∞ ∞ −1 ∂G(R, ω) −iωt ˜ E(x, y, z, t) = dx dy , dω e E0 (x , y , ω) 2π 2 ∂z −∞ −∞ C+

B(x, y, z, t) =

−1 2π 2

C+

dω e−iωt

∞

−∞

∞

(7.74)

˜ 0 (x , y , ω) ∂G(R, ω) dx dy B ∂z −∞

.

(7.75) The spatial integrals appearing here are just the ﬁrst Rayleigh–Sommerfeld diﬀraction integrals [14] of classical optics. The solution in terms of the normal derivatives of the initial ﬁeld vectors, which yields the second Rayleigh– Sommerfeld diﬀraction integrals, is left as an exercise.

7.2 Polarization Properties of the Freely Propagating Electromagnetic Wave Field Of considerable interest to the description of the propagation characteristics of a general electromagnetic wave ﬁeld are the polarization properties of its electric and magnetic ﬁeld vectors. The standard treatment [11] of the polarization state of an electromagnetic wave is restricted to the idealized case of a time-harmonic plane wave ﬁeld. This restriction has, in part, been removed by Nisbet and Wolf [15] for the case of a linearly polarized, timeharmonic wave ﬁeld with arbitrary spatial form. A complete extension [16] of this treatment to a general pulsed electromagnetic wave ﬁeld is then directly accomplished through the angular spectrum of plane waves representation for the freely propagating ﬁeld. From the pair of expressions given in Eqs. (7.14) and (7.17), the angular spectrum of plane waves representation of a freely propagating electromagnetic wave ﬁeld may be written as

1 −iωt ˜ dω , (7.76) E(r, ω)e E(r, t) = π C+

346

7 Angular Spectrum Representation of Pulsed Beam Fields

1 B(r, t) = π

˜ ω)e−iωt dω B(r,

,

(7.77)

C+

where the (complex-valued) temporal frequency spectrum of each ﬁeld vector has the angular spectrum representation ∞ ∞ 1 ˜ ˜ ˜ 0 (kT , ω)eik˜ + ·r+ dkx dky E E(r, ω) = (2π)2 −∞ −∞ ≡ pe (r, ω) + iqe (r, ω), (7.78) ∞ ∞ + + 1 ˜ ˜ 0 (kT , ω)eik˜ ·r dkx dky ˜ ω) = B B(r, (2π)2 −∞ −∞ ≡ pm (r, ω) + iqm (r, ω). (7.79) Here pj (r, ω) and qj (r, ω) denote the real and imaginary parts, respectively, of the temporal frequency spectrum domain form of the appropriate ﬁeld vector, as indicated by the subscript j = e, m. With this substitution, the pair of relations given in Eqs. (7.76)–(7.77) becomes

1 ˜ e (r, t; ω)dω , (7.80) E(r, t) = V π C+

1 ˜ B(r, t) = (7.81) Vm (r, t; ω)dω , π C+ where

˜ j (r, t; ω) ≡ pj (r, ω) + iqj (r, ω) e−iωt , V

(7.82)

for j = e, m, are complex-valued vector ﬁelds that describe the spatial properties of each monochromatic ﬁeld component appearing in the propagated ﬁeld representation given in Eqs. (7.80)–(7.81). 7.2.1 The Polarization Ellipse for the Complex Field Vectors With the analysis of Born and Wolf [11] as a guide, consider the behavior of the time-harmonic complex vector ﬁeld ˜ V(r, t) ≡ p(r) + iq(r) e−iωt (7.83) ˜ ˜ e (r, t) or t) represents either V at a ﬁxed point r = r0 in space. Here V(r, ˜ 0 , t) describes a curve ˜ m (r, t). As time varies the end point of the vector V(r V in the plane that is speciﬁed by the pair of (real-valued) vector p(r0 ) and ˜ 0 , t) is periodic in time at any ﬁxed point q(r0 ). Furthermore, because V(r r = r0 in space, this curve must then be closed. Now let p(r0 ) + iq(r0 ) ≡ (a + ib)eiϕ ,

(7.84)

7.2 Polarization Properties of the Electromagnetic Wave Field

347

where ϕ is as yet unspeciﬁed. The real vectors a and b may then be expressed in terms of p(r0 ), q(r0 ), and ϕ as a = p(r0 ) cos ϕ + q(r0 ) sin ϕ, b = −p(r0 ) sin ϕ + q(r0 ) cos ϕ.

(7.85) (7.86)

Notice that a, b, and ϕ are all functions of the (ﬁxed) position vector r0 . Consider now choosing the angle ϕ such that the vectors a and b are orthogonal so that a · b = 0 and tan (2ϕ) =

2p(r0 ) · q(r0 ) . p2 (r0 ) − q2 (r0 )

(7.87)

The parameters that specify the spatial properties of the complex vector ˜ ﬁeld V(r, t) at any ﬁxed point r = r0 may now be considered to be the ﬁve independent components of the orthogonal vectors a and b and the associated phase factor ϕ, instead of the six independent components of the vectors p and q. With substitution from Eq. (7.84), the expression given in Eq. (7.83) ˜ for the monochromatic complex vector ﬁeld V(r, t) at the ﬁxed point r = r0 becomes ˜ 0 , t) ≡ V(r =

˜ (r) (r0 , t) + iV ˜ (i) (r0 , t) V a cos (ωt − ϕ) + b sin (ωt − ϕ) −i a sin (ωt − ϕ) − b cos (ωt − ϕ) .

(7.88)

If a Cartesian coordinate system is now deﬁned with origin at the ﬁxed ﬁeld point r = r0 and with the x- and y-coordinate directions chosen along the direction of the vectors a = a(r0 ) and b = b(r0 ), respectively, then the ˜ 0 , t) with respect to this coordinate system components of the real part of V(r are given by V˜x(r) (r0 , t) = a(r0 ) cos (ωt − ϕ(r0 )), V˜y(r) (r0 , t) = b(r0 ) sin (ωt − ϕ(r0 )), V˜ (r) (r0 , t) = 0,

(7.89)

z

and the components of the imaginary part of V(r0 , t) with respect to this coordinate system are given by V˜x(i) (r0 , t) = a(r0 ) cos (ωt − ϕ(r0 ) + π/2), V˜y(i) (r0 , t) = b(r0 ) sin (ωt − ϕ(r0 ) + π/2), V˜z(i) (r0 , t) = 0,

(7.90)

where a(r0 ) ≡ |a(r0 )| and b(r0 ) ≡ |b(r0 )|. Both of the above two sets of equations describe an ellipse in time, called the polarization ellipse, that is given by

348

7 Angular Spectrum Representation of Pulsed Beam Fields

y

x Γp

Ψ

Fig. 7.5. The polarization ellipse at a ﬁxed point in space.

2 (j) V˜x (r0 , t)

(j) V˜y (r0 , t)

2

+ = 1; j = r, i, (7.91) a2 b2 with semi-axes of lengths a and b along the x- and y-coordinate axes, re˜ 0 , t) are π/2 out spectively. Notice that the real and imaginary parts of V(r of phase as they trace out the polarization ellipse. The real vectors p(r0 ) and q(r0 ) are then seen to form a pair of conjugate semi-diameters of the polarization ellipse, as illustrated in Figure 7.5. The semi-axis lengths a and b of the polarization ellipse described by the complex vector ﬁeld at the ﬁxed point r = r0 are readily obtained from Eqs. (7.85)–(7.87) as 1/2 1 2 2 2 2 2 2 2 + 4 (p · q) , (7.92) a = p +q + p −q 2 1/2 2 1 2 2 . (7.93) b2 = p + q 2 − p2 − q 2 + 4 (p · q) 2 In order to determine the angle ψ between the vectors a and p, depicted in Figure 7.5, one begins by expressing the equation of the polarization ellipse in parametric form as V˜x(j) (r0 , t) = a cos φj , V˜y(j) (r0 , t) = b sin φj ,

(7.94) (7.95)

for j = r, i, where φj = φj (r0 , t) denotes the eccentric angle that is depicted in Figure 7.6 for the case when a ≥ b (this inequality is assumed to hold throughout the remaining analysis). From the geometry of the ﬁgure it is seen that the eccentric angle φj is related to the polar angle of the point (j) (j) V˜x , V˜y on the vibration ellipse by

7.2 Polarization Properties of the Electromagnetic Wave Field

349

y (acosΦj , asinΦj)

~

~

(Vx(j), Vy(j)) = (acosΦj , bsinΦj)

b

Φj Θ x

a

Fig. 7.6. The geometric relation between the eccentric angle φj and the polar angle (j) (j) V˜x , V˜y on the polarization ellipse.

θ of the point

tan θ =

(j) V˜y b = tan φj . (j) a V˜x

(7.96)

Comparison of the pair of expressions in Eq. (7.94)–(7.95) with the expressions given in Eqs. (7.89)–(7.90) shows that for the vibration ellipse of the ˜ (r) (r0 , t) of the complex ﬁeld vector real part V φr (r0 , t) = ωt − ϕ(r0 ),

(7.97)

whereas for the imaginary part φi (r0 , t) = ωt − ϕ(r0 ) + π/2,

(7.98)

at any ﬁxed point r = r0 in the positive half-space. From Eq. (7.83) it is seen ˜ (r) (r, t) = p(r) when t = 0, and V ˜ (i) (r, t) = p(r) when ωt = −π/2, so that V that the eccentric angle of p is −ϕ; from Eq. (7.96), the angle ψ between the vectors p and a (see Fig. 7.5) is then given by tan ψ =

b tan ϕ. a

(7.99)

350

7 Angular Spectrum Representation of Pulsed Beam Fields

Finally, if γp denotes the angle between the vectors p and q, as illustrated in Figure 7.5, and if β is an auxiliary angle deﬁned as tan β ≡

q , p

(7.100)

then Eq. (7.87) becomes tan (2ϕ) =

2pq cos γp = tan (2β) cos γp . − q2

(7.101)

p2

Summarizing these standard results, if the vectors p = p(r0 ) and q = q(r0 ) are given at the ﬁxed point r = r0 , where γp is the angle between these two vectors and β is the auxiliary angle deﬁned in Eq. (7.100) as the inverse tangent of the ratio of the magnitudes of these two vectors, then the principal semi-axes of the vibration ellipse at the point r0 are given by the relations in Eqs. (7.92)–(7.93) and the angle ψ that the major axis makes with the vector p is given by Eq. (7.99) where the phase factor ϕ is found from Eq. (7.101).

Ωt%!Π

Ωt%!Π

b Ωt%!Π

a

Ωt% Ωt%!Π

Ωt%!Π

(a) Left-Handed Polarization Sense

b

a

Ωt%

Ωt%!Π

(b) Right-Handed Polarization Sense

Fig. 7.7. Left-handed (a) and right-handed (b) polarization ellipses in optics. The polarization senses illustrated here are reversed in electrical engineering.

From Eqs. (7.82)–(7.84), each time-harmonic vector ﬁeld component appearing in the propagated ﬁeld representation given in Eqs. (7.80)–(7.81) is of the form ˜ (7.102) V(r, t; ω) = a(r, ω) + ib(r, ω) ei(ϕ(r,ω)−ωt) which is the complex representation of a time-harmonic wave that is propagating in the direction speciﬁed by ∇ϕ(r). There are then two possible senses in which the polarization ellipse is traced out, corresponding to left-handed

7.2 Polarization Properties of the Electromagnetic Wave Field

351

and right-handed polarizations. Curiously enough, there are two opposing deﬁnitions of this polarization sense: one in optics (physics) and one in electrical engineering. Consider ﬁrst the deﬁnition in optics. If the sign of the scalar triple product [a, b, ∇ϕ] = (a × b) · ∇ϕ is positive, then to an observer looking in a direction that is opposite to that in which the wave ﬁeld is propagating, the end point of the ﬁeld vector describes its ellipse in the counterclockwise sense and the polarization is said to be left-handed, as illustrated in part (a) of Figure 7.7. On the other hand (pun intended), if the sign of the scalar triple product [a, b, ∇ϕ] is negative, then the polarization ellipse is described in the clockwise sense and the polarization is said to be right-handed, as illustrated in part (b) of the ﬁgure. In electrical engineering the polarization sense is deﬁned for an observer looking in the direction that the wave ﬁeld is propagating. Left-handed then becomes right-handed and right-handed becomes left-handed when one goes from an optics to an electrical engineering point of view, and vice versa. 7.2.2 Propagation Properties of the Polarization Ellipse Consider now the relation between the initial and propagated polarization properties of the pulsed electromagnetic wave ﬁeld in the positive half-space z ≥ z0 . From the expressions given in Eqs. (7.78)–(7.79) and (7.84), the complex representation of the polarization ellipse for the propagated electric and magnetic ﬁeld vectors is seen to be given by

+ ae (r+ , ω) + ibe (r+ , ω) eiϕe (r ,ω) ∞ ∞ 1 ˜ ˜ 0 (kT , ω)eik+ ·r+ dkx dky , = E 2 (2π) −∞ −∞

+ am (r+ , ω) + ibm (r+ , ω) eiϕm (r ,ω) ∞ ∞ 1 ˜ ˜ 0 (kT , ω)eik+ ·r+ dkx dky , = B 2 (2π) −∞ −∞

(7.103)

(7.104) ˆ x x+1 ˆ y y+1 ˆ z (z−z0 ) is the position vector of the ﬁeld observation where r+ ≡ 1 point in the positive half-space z ≥ z0 . Notice that with the relations given in Eqs. (7.18) and (7.19), these two equations may be expressed solely in terms ˜ ˜ ˜ 0 (kT , ω), if desired, where [cf. Eqs. (7.3)–(7.6)] ˜ 0 (kT , ω) or B of either E ˜ ˜ 0 (kT , ω) = E ˜ ˜ 0 (kT , ω) = B

∞

∞

−∞ ∞

−∞ ∞

−∞

−∞

˜ 0 (rT , ω)e−ikT ·rT dxdy, E

(7.105)

˜ 0 (rT , ω)e−ikT ·rT dxdy, B

(7.106)

352

7 Angular Spectrum Representation of Pulsed Beam Fields

with

˜ 0 (rT , ω) = E ˜ 0 (rT , ω) = B

∞

−∞ ∞ −∞

E0 (rT , t)eiωt dt,

(7.107)

B0 (rT , t)eiωt dt,

(7.108)

ˆx x + 1 ˆ y y and kT = 1 ˆ x kx + 1 ˆ y ky . As in Eqs. (7.80)–(7.81), the with rT = 1 initial ﬁeld vectors on the z = z0 plane may be expressed as

1 ˜ (0) (r, t; ω)dω , E0 (r, t) = (7.109) V e π C+

1 (0) ˜ B0 (r, t) = (7.110) Vm (r, t; ω)dω , π C+ where ˜ e (r, t; ω) = E ˜ 0 (rT , ω)e−iωt V (0) = p(0) (r , ω) + iq (r , ω) e−iωt , T T e e

(7.111)

˜ 0 (rT , ω)e−iωt ˜ m (r, t; ω) = B V (0) −iωt = p(0) , m (rT , ω) + iqm (rT , ω) e

(7.112)

are complex vectors that describe the spatial properties of each monochromatic ﬁeld component that is present in the initial wave ﬁeld. With the complex representation of the polarization ellipse that is given in Eq. (7.84), one then has that ˜ 0 (rT , ω) = p(0) (rT , ω) + iq(0) (rT , ω) E e e (0) (0) iϕe (rT ,ω) = ae (rT , ω) + ib(0) , e (rT , ω) e

(7.113)

˜ 0 (rT , ω) = p(0) (rT , ω) + iq(0) (rT , ω) B m m (0) (0) iϕm (rT ,ω) = am (rT , ω) + ib(0) . m (rT , ω) e

(7.114)

Substitution of Eq. (7.105) with (7.113) into Eq. (7.103) [or (7.106) with (7.114) into (7.104)] then yields the general expression (0) + aj (r+ , ω) + ibj (r+ , ω) eiϕj (r ,ω) ∞ ∞ (0) 1 (0) (0) a = (r , ω) + ib (r , ω) eiϕm (rT ,ω) T T j j (2π)2 −∞ −∞

∞ ∞ + + ˜ × ei(k ·r −kT ·rT ) dkx dky dx dy . −∞

−∞

(7.115)

7.2 Polarization Properties of the Electromagnetic Wave Field

353

The integral appearing in the brackets of this expression is recognized as the monochromatic spatial impulse response function, deﬁned in Eq. (7.66), for the normal propagation distance ∆z = z −z0 in the positive half-space z ≥ z0 of a temporally dispersive HILL medium at the angular frequency ω; viz., ∞ ∞ 1 ˜+ + ei(k ·r −kT ·rT ) dkx dky h(x − x , y − y ; ∆z, ω) = 2 (2π) −∞ −∞ ∞ ∞ 1 = ei[kx (x−x )+ky (y−y )+γ(ω)∆z] dkx dky , 2 (2π) −∞ −∞ (7.116) where γ = γ(ω) is deﬁned in Eq. (7.11) and should not be confused with the angle γp deﬁned in Eq. (7.101) for the polarization ellipse (see Fig. 7.5). With this identiﬁcation, Eq. (7.115) becomes (0) + aj (r+ , ω) + ibj (r+ , ω) eiϕj (r ,ω) ∞ ∞ (0) (0) (0) = aj (rT , ω) + ibj (rT , ω) eiϕj (rT ,ω) −∞

−∞

×h(x − x , y − y ; ∆z, ω)dx dy . (7.117)

Hence, the polarization properties of each monochromatic component present in the propagated wave ﬁeld are given by the convolution of the initial polarization behavior on the plane z = z0 with the spatial impulse response function for the normal propagation distance ∆z = z − z0 in the temporally dispersive HILL medium at the angular frequency ω. As a consequence, if the polarization properties of the initial wave ﬁeld vary from point to point over the plane at z = z0 , then the polarization properties of the propagated wave ﬁeld will also, in general, vary from point to point throughout the positive half-space ∆z ≥ 0. In a strict sense, the state of polarization refers to the electromagnetic ﬁeld vector behavior at a particular point in space and, in general, varies from point to point for each monochromatic component present in the wave ﬁeld. Moreover, the frequency dependence allows for the state of polarization to vary in time at any ﬁxed observation point in the positive half-space when the wave ﬁeld is pulsed in time. For a uniformly polarized ﬁeld vector over the plane at z = z0 it is required that (0) (0) (0) (0) (0) (0) ˆ (0) eiϕˆj , ˆj + ib aj (rT , ω) + ibj (rT , ω) eiϕj (rT ,ω) = Wj (rT , ω) a j (7.118) (0) ˆ (0) are both constant vectors and ϕˆ(0) is a scalar constant. ˆj and b where a j j The only spatial variation in the particular ﬁeld vector (either electric j = e or magnetic j = m) at the plane z = z0 appears in the spectral ﬁeld amplitude (0) (0) function Wj (rT , ω) = Wj (x, y, ω). With this substitution, Eq. (7.123) for the propagated polarization ellipse becomes

354

7 Angular Spectrum Representation of Pulsed Beam Fields

(0) (0) + (0) ˆ (0) eiϕˆj , ˆj + ib aj (r+ , ω) + ibj (r+ , ω) eiϕj (r ,ω) = Wj (r+ , ω) a j (7.119) where ∞ ∞ (0) + Wj (r , ω) = Wj (rT , ω)h(x − x , y − y ; ∆z, ω)dx dy , (7.120) −∞

−∞

and the polarization state for this ﬁeld vector remains unchanged throughout the positive half-space z ≥ z0 . Notice that the necessary condition speciﬁed (0) ˆ (0) and the phase constant ˆj and b in Eq. (7.118) requires that both vectors a j (0)

ϕˆj are independent of the angular frequency ω. If this is not the case, then the polarization state of the wave ﬁeld vector considered will, in general, evolve with time at a given ﬁxed point in space. Two special cases of considerable importance in regard to the polarization state at a ﬁxed point in space are the linearly polarized and circularly polarized wave ﬁelds. For a linearly polarized ﬁeld vector at the point r = r0 , the minor axis of the polarization ellipse vanishes so that bj = 0 and Eq. (7.93) then requires that p2j qj2 = (pj · qj )2 , (7.121) and the angle γp between p and q is either 0 or π (see Fig. 7.5). The complex representation of the ﬁeld vector at this point in space is then given by ˜ j (r0 , t) = aj (r0 )ei(ϕj (r0 −ωt) . V

(7.122)

For a circularly polarized ﬁeld vector at the point r = r0 , the vectors aj and bj are indeterminate and consequently, the angle ϕj is also indeterminate. For this to be the case, Eq. (7.87) then requires that pj · qj = 0

∧

p2j − qj2 = 0,

(7.123)

so that the vectors pj = pj (r0 ) and qj = qj (r0 ) are orthogonal and of equal magnitude. The complex representation of the ﬁeld vector at this point in space is then given by √ ˜ j (r0 , t) = 2pj (r0 )e±π/4 e−iωt , (7.124) V where the sign choice depends upon the polarization sense. From this expression it is seen that any given state of circular polarization may be decomposed into the superposition of two properly phased and orthogonally oriented linearly polarized ﬁelds. Furthermore, any given state of elliptic polarization may be decomposed into the superposition of a left-handed and right-handed circularly polarized ﬁeld with the same angular frequency but with unequal amplitudes.

7.2 Polarization Properties of the Electromagnetic Wave Field

355

7.2.3 Relation Between the Electric and Magnetic Polarizations From Eqs. (7.78)–(7.79) and (7.102), the complex representation of the polarization ellipses for the temporal frequency spectra of the electric and magnetic ﬁeld vectors are respectively given by ˜ ω) = pe (r, ω) + iqe (r, ω) E(r, = ae (r, ω) + ibe (r, ω) eiϕe (r,ω) ≡ ˜e(r, ω)eiϕe (r,ω) , (7.125) ˜ B(r, ω) = pm (r, ω) + iqm (r, ω) ˜ ω)eiϕm (r,ω) , (7.126) = am (r, ω) + ibm (r, ω) eiϕm (r,ω) ≡ b(r, where the complex ﬁeld vectors deﬁned by ˜e(r, ω) ≡ ae (r, ω) + ibe (r, ω), ˜ b(r, ω) ≡ am (r, ω) + ibm (r, ω),

(7.127) (7.128)

have been introduced here for notational convenience and are not to be confused with the microscopic electric and magnetic ﬁeld vectors. Throughout the positive half-space z ≥ z0 , the temporal frequency spectra of the ﬁeld vectors satisfy the ﬁeld equations [cf. Eqs. (5.25)–(5.28)] ˜ ω) = 0, ∇ · E(r, % % %1% ˜ % ˜ ∇ × E(r, ω) = % % c % iω B(r, ω), ˜ ω) = 0, ∇ · B(r, % % %1% ˜ ˜ % ∇ × B(r, ω) = − % % c % iωµ(ω) c (ω)E(r, ω).

(7.129) (7.130) (7.131) (7.132)

Substitution of the complex representations for the electric and magnetic ﬁeld vectors given in Eqs. (7.125) and (7.126), respectively, into the pair of divergence relations given above then yields the pair of relations i˜e(r, ω) · ϕe (r, ω) + ∇ · ˜e(r, ω) = 0, ˜ ω) = 0, ˜ ω) · ϕm (r, ω) + ∇ · b(r, ib(r,

(7.133) (7.134)

and substitution into the pair of curl relations gives ˜ ω)eiϕm (r,ω) ˜ ω) = b(r, B(r, c ∇ϕe (r, ω) × ˜e(r, ω) − i∇ × ˜e(r, ω) eiϕe (r,ω) , = (7.135) ω ˜ ω) = ˜e(r, ω)eiϕe (r,ω) E(r, c ˜ ω) + i∇ × b(r, ˜ ω) eiϕm (r,ω) . − ∇ϕm (r, ω) × b(r, = ωµ(ω) c (ω) (7.136)

356

7 Angular Spectrum Representation of Pulsed Beam Fields

The polarization properties of one ﬁeld vector may then be determined directly from the polarization state of the other ﬁeld vector at each point of space from this ﬁnal pair of relations. For example, if the electric ﬁeld vector is linearly polarized with ˜e(r, ω) = ae (r, ω), where ae is a real-valued vector ﬁeld, then ˜ ω)eiϕm (r,ω) = c ∇ϕe (r, ω) × ae (r, ω) − i∇ × ae (r, ω) eiϕe (r,ω) , b(r, ω and the magnetic ﬁeld vector will, in general, be elliptically polarized provided that ae is spatially dependent; however, if ae is spatially independent so that the electric ﬁeld vector is uniformly linearly polarized throughout space, then ∇ × ae = 0 and the magnetic ﬁeld vector is also linearly polarized, but with an orientation that may vary from point to point in space. The pair of expressions given in Eqs. (7.135)–(7.136) shows that the temporal frequency spectra of the electric and magnetic ﬁeld vectors are not, in general, instantaneously orthogonal to each other, because ˜ ω) · B(r, ˜ ω) = −i c ˜e(r, ω) · ∇ × ˜e(r, ω) ei2ϕ(r,ω) , (7.137) E(r, ω which does not, in general, vanish. As previously noted, the state of polarization strictly refers to the electromagnetic wave ﬁeld behavior at a speciﬁc point in space and, in general, varies from point to point in the ﬁeld. Even in the special case when the electric ﬁeld vector is everywhere linearly polarized in some ﬁxed direction, the magnetic ﬁeld vector will, in general, be elliptically polarized, as was ﬁrst shown by Nisbet and Wolf [15]. As a consequence, in conrast with the orthogonality relation [cf. Eqs. (6.29)–(6.30)] ˜ ˜ ˜ ˜ E(k, ω) · B(k, ω) = 0

(7.138)

for the spatiotemporal spectra of the ﬁeld vectors, the temporal frequency spectra of the electric and magnetic ﬁeld vectors are not, in general, instantaneously orthogonal to each other. However, the long-time average ˜ (r) ˜ (r) ˜ (r) of #the real-valued $ quantity E (r, t; ω)#· B (r, t; ω),$ with E (r, t; ω) ≡ ˜ (r) (r, t; ω) ≡ B(r, ˜ ω)e−iωt and B ˜ ω)e−iωt does indeed vanish, E(r, because [cf. Eq. (5.129)] $ , # ˜ (r) (r, t; ω) = 1 E(r, ˜ (r) (r, t; ω) · B ˜ ω) · B ˜ ∗ (r, ω) E 2 c = ˜e(r, ω) · ∇ϕe (ω) × ˜e∗ (r, ω) 2ω + i˜e(r, ω) · ∇ × ˜e∗ (r, ω) c = i˜e(r, ω) · ∇ × ˜e∗ (r, ω) 2ω = 0. (7.139)

7.2 Polarization Properties of the Electromagnetic Wave Field

357

The fact that the quantity ˜e · (∇ × ˜e∗ ) is real-valued, used in the above proof, directly follows from the vector diﬀerential identity ∇ · (v × w) = w · (∇ × v) − v · (∇ × w). With v = ˜e and w = ˜e∗ as well as the fact that ˜e × ˜e∗ = 0, this identity yields the result ˜e · (∇ × ˜e∗ ) = ˜e∗ · (∇ × ˜e), and consequently 1 ˜e · (∇ × ˜e∗ ) + ˜e∗ · (∇ × ˜e) 2 ∗ 8 17 ˜e · (∇ × ˜e∗ ) + ˜e · (∇ × ˜e∗ ) = 27 8 = ˜e · (∇ × ˜e∗ ) .

˜e · (∇ × ˜e∗ ) =

Thus, the temporal frequency spectra of the electric and magnetic ﬁeld vectors are, on the average, mutually orthogonal. ˜ ˜ ˜ ˜ E(k, ω) · B(k, ω) = 0 Fk ˜ ˜ ω) = 0 E(r, ω) · B(r, + * (r) ˜ (r) (r, t; ω) = 0 ˜ (r, t; ω) · B E Fω *

E(r, t) · B(r, t) = 0 + E(r, t) · B(r, t) = 0.

As a summary of these results, indicated in the above set of relations, ˜ ˜˜ ˜ the spatiotemporal frequency spectra E(k, ω) and B(k, ω) of the electric and magnetic ﬁeld vectors, respectively, are always mutually orthogonal. The tem$ # (r) ˜ ˜ poral frequency spectral wave components E (r, t; ω) ≡ E(r, ω)e−iωt # $ ˜ (r) (r, t; ω) ≡ B(r, ˜ ω)e−iωt of these ﬁeld vectors are not, in general, and B orthogonal, but are mutually orthogonal in the long-time average sense. Finally, the electric E(r, t) and magnetic B(r, t) ﬁeld vectors are not, in general, orthogonal and neither is their long-time average. 7.2.4 The Uniformly Polarized Wave Field For an electromagnetic wave ﬁeld that is uniformly polarized in the electric ﬁeld vector, Eqs. (7.118) and (7.125)–(7.126) require that the initial ﬁeld vectors on the plane at z = z0 are given by ˜ 0 (rT , ω) = E ˜0 (rT , ω)(ae + ibe )eiϕe , (7.140) E iϕm (rT ,ω) ˜ ˜ , (7.141) B0 (rT , ω) = B0 (rT , ω) am (rT , ω) + ibm (rT , ω) e where ae and be are ﬁxed vectors and where ϕe is a scalar constant. These two ﬁeld vectors are not independent and must be oriented such that Eqs. (7.135)–(7.136) are satisﬁed. In particular, Eq. (7.135) requires that

358

7 Angular Spectrum Representation of Pulsed Beam Fields

˜0 (rT , ω) am (rT , ω) + ibm (rT , ω) eiϕm (rT ,ω) B c ˜ ∇E0 (rT , ω) × (ae + ibe )eiϕe , = −i ω (7.142) from which it is seen that the temporal frequency spectra of the electric and magnetic ﬁeld vectors are orthogonal at the plane z = z0 . Although not necessary, the complex vectors (ae + ibe ) and (am + ibm ) are chosen to be normalized, so that 2 (7.143) |aj + ibj | = a2j + b2j = 1 for j = e, m. The spatiotemporal frequency spectra of the initial ﬁeld vectors at the plane z = z0 are then given by ∞ ∞ ˜ ˜ 0 (kT , ω) = (ae + ibe )eiϕe ˜0 (rT , ω)e−ikT ·rT dxdy E E −∞

−∞

˜˜ (k , ω), = (ae + ibe )e E (7.144) 0 T ∞ ∞ ˜ ˜ 0 (kT , ω) = i c (ae + ibe )eiϕe × ˜0 (rT , ω) e−ikT ·rT dxdy ∇E B ω −∞ −∞ c ˜ + iϕe ˜ ˜ 0 (kT , ω), k × (ae + ibe )e E (7.145) = ω iϕe

where ˜˜ (k , ω) = E 0 T

∞

−∞

∞

−∞

˜0 (rT , ω)e−ikT ·rT dxdy. E

(7.146)

In addition, the space–time form of the initial ﬁeld vectors at the plane z = z0 is given by

1 iϕe −iωt ˜ E0 (rT , ω)e dω , (7.147) E0 (rT , t) = (ae + ibe )e π C+

˜0 (rT , ω) ∇E c iϕe −iωt B0 (rT , t) = dω . (7.148) i(ae + ibe )e × e π ω C+ The propagated ﬁeld vectors are found from the angular spectrum representation given in Eqs. (7.14) and (7.17) with substitution from Eqs. (7.144)– (7.145) as

1 (ae + ibe )eiϕe E(r, t) = 4π 3

∞ ∞ + ˜ ˜ i( k ·r−ωt) ˜ 0 (kT , ω)e E × dω dkx dky , C+

−∞

−∞

(7.149)

7.2 Polarization Properties of the Electromagnetic Wave Field

359

B(r, t) = −

c (ae + ibe )eiϕe 4π 3

+ dω ∞ ∞ ˜ + ˜ ˜ ˜ 0 (kT , ω)ei(k ·r−ωt) dkx dky . × k (ω)E −∞ −∞ C+ ω (7.150)

Hence, the propagated ﬁeld vectors are seen to be everywhere orthogonal in the positive half-space z ≥ z0 when the electric ﬁeld vector is uniformly polarized. However, the magnetic ﬁeld vector is not, in general, uniformly polarized because of its directional dependence on the complex wave vector ˜ + (ω) which causes the orientation of the magnetic ﬁeld polarization ellipse k to vary in both space and time. In order that the magnetic ﬁeld vector also be ˜˜ (k , ω) must have a ﬁxed ˜ + (ω)E uniformly polarized the spectral quantity k 0 T direction independent of ω; this in turn implies that the initial ﬁeld amplitude ˜0 (rT , ω) must be independent of one transverse coordinate direction. E For an electromagnetic wave ﬁeld that is uniformly polarized in the magnetic ﬁeld vector the set of relations given in Eqs. (7.140)–(7.141) is replaced by ˜ 0 (rT , ω) = E ˜0 (rT , ω) ae (rT , ω) + ibe (rT , ω) eiϕe (rT ,ω) , (7.151) E ˜ 0 (rT , ω) = B ˜0 (rT , ω)(am + ibm )eiϕm , (7.152) B where am and bm are ﬁxed vectors and where ϕm is a scalar constant. The propagated wave ﬁeld vectors throughout the positive half-space in this case are then found to be given by

c E(r, t) = − 3 (am + ibm )eiϕm 4π

∞ ∞ + dω ˜ ˜ ˜ + (ω)B ˜ 0 (kT , ω)ei(k ·r−ωt) dkx dky . × k C+ ωµ(ω) c (ω) −∞ −∞ (7.153)

B(r, t) =

1 (am + ibm )eiϕm 4π 3 ∞ × dω C+

−∞

∞

−∞

˜ + ·r−ωt) ˜ i(k ˜ B 0 (kT , ω)e dkx dky . (7.154)

The propagated ﬁeld vectors are again seen to be everywhere orthogonal in the positive half-space, however, the electric ﬁeld vector is not, in general, uniformly polarized.

360

7 Angular Spectrum Representation of Pulsed Beam Fields

In order that both the electric and magnetic ﬁeld vectors be uniformly polarized throughout the positive half-space z ≥ z0 , Eqs. (7.150) and (7.154) must both be satisﬁed throughout that region, so that

∞ ∞ ˜+ ˜˜ (k , ω)(a + ib )eiϕm dω dkx dky ei(k ·r−ωt) B 0 T m m C+

−∞

−∞

c ˜˜ ˜ + (ω) + E 0 (kT , ω)(ae + ibe )eiϕe × k ω

= 0,

and consequently ˜ ˜˜ (k , ω)(a + ib )eiϕm = −c k(ω) ˜˜ (k , ω)(a + ib )eiϕe × ˆs, E B 0 T m m 0 T e e ω (7.155) ˆy q + 1 ˆ z m is a real-valued unit vector that is deﬁned by ˆx p + 1 where ˆs = 1 ˜ s. Notice ˜ + (ω) = k(ω)ˆ the direction of the complex-valued wave vector as k that the relation given in Eq. (7.161) also follows from the relation given in ˜˜ (k , ω) = (c/ω)k ˜ ˜˜ (k , ω) given ˜ + (ω)×E ˜ 0 (kT , ω), with E Eq. (7.19); viz., B 0 T 0 T ˜ (k , ω) given by the two-dimensional spatial transform ˜ by Eq. (7.146) and B 0

T

of Eq. (7.152). The orientation of the magnetic polarization ellipse is then seen to depend on the orientation of the electric polarization ellipse through the factor (ae + ibe ) × ˆs which depends upon the direction of the complex ˜ s. Because it is required that both polarization ˜ + (ω) = k(ω)ˆ wave vector k ellipses are ﬁxed in both space and time, the direction of the unit vector ˆs must then be appropriately constrained and this, in turn, constrains the coordinate dependency of the ﬁeld vectors themselves. As an example, consider an electromagnetic wave that is uniformly polarized in the electric ﬁeld vector such that it is linearly polarized along the x-axis in a nondispersive (and hence, nonabsorptive) medium. In that case, ˆ x , be = 0, and ϕe = 0, so that ae = 1 iϕm 2 2 ˆx ˆ ˆ ˆ (am + ibm )e = 1x p + 1y q + 1z 1 − p − q × 1 ˆ y 1 − p2 − q 2 − 1 ˆ z q, =1 from which it is seen that there are two possibilities in order to maintain the requirement that the magnetic ﬁeld is uniformly polarized. Either q = q0 is ﬁxed, in which case p may be allowed to vary such that either p2 ≤ 1 − q02 or p2 > 1 − q02 , or else q is allowed to vary in which case p must vary in such a fashion that m = 0, that is, such that p2 = 1 − q 2 . This latter situation in which m = 0 for all possible allowed values of p and q precludes propagation into the positive half-space ∆z > 0 and so is of no interest here. Hence, q = q0 and the ﬁeld itself must then be independent of the y-coordinate.

7.3 Real Direction Cosine Form of the Angular Spectrum Representation

361

As another example, consider an electromagnetic wave that is uniformly polarized in the electric ﬁeld vector such that it is circularly polarized in the ˆ x , be = 1 ˆ y and xy-plane in a nondispersive medium. In that case, ae = 1 ϕe = 0, so that ˆ x + i1 ˆx p + 1 ˆy q + 1 ˆ z 1 − p2 − q 2 × 1 ˆy (am + ibm )eiϕm = 1 ˆ y 1 − p2 − q 2 − 1 ˆz q − i 1 ˆ x 1 − p2 − q 2 − 1 ˆz p =1 and p and q must both be ﬁxed (i.e., p = p0 and q = q0 ) in order for the magnetic polarization ellipse to be uniform. Because a2m = 1 − p20 and b2m = 1 − q02 the magnetic ﬁeld is elliptically polarized in general and is circularly polarized when p0 = q0 . In either case, the ﬁeld itself is independent of both the x- and y-coordinates. This then establishes the following result: Theorem 3. The requirement that both the electric and magnetic ﬁeld vectors of an electromagnetic wave ﬁeld be uniformly polarized throughout the positive half-space z ≥ z0 requires that the ﬁeld be independent of some coordinate direction.

7.3 Real Direction Cosine Form of the Angular Spectrum Representation The plane wave spectral components appearing in the angular spectrum representation given in Eqs. (7.14) and (7.17) may be cast into a more explicit geometric form [17] by setting2 [compare with the transformation relations given in Eqs. (7.21)–(7.23)] −iψ(ω) ˜ kx = k(ω)e p, −iψ(ω) ˜ ky = k(ω)e q, −iψ(ω) ˜ m, γ(ω) = k(ω)e

where

# $ α(ω) ˜ ψ(ω) ≡ arg k(ω) = arctan β(ω)

(7.156) (7.157) (7.158)

(7.159)

˜ is the phase angle of the complex wavenumber k(ω) = β(ω) + iα(ω). With 1/2 ˜ (7.160) k(ω) ≡ k(ω) = β 2 (ω) + α2 (ω) 2

Notice that the same notation that was used in §7.1.1 is being used here. Care must then be exercised not to mix these results.

362

7 Angular Spectrum Representation of Pulsed Beam Fields

˜ denoting the magnitude of the complex wavenumber k(ω) = k(ω)eiψ(ω) , the set of transformation relations given in Eqs. (7.156)–(7.158) may be expressed as kx = k(ω)p, ky = k(ω)q, γ(ω) = k(ω)m.

(7.161) (7.162) (7.163)

Because both kx and ky are real-valued, then both direction cosines p and 1/2 with kT2 = q are real-valued. In addition, because γ(ω) = k˜2 (ω) − kT2 kx2 + ky2 , then

1/2 m(ω) = ei2ψ(ω) − p2 + q 2 ,

(7.164)

which is, in general, complex-valued. As in Eq. (7.32), let m(ω) ≡ ζ 1/2 (ω) with ζ = ζ + iζ , where (7.165) ζ (ω) ≡ {ζ(ω)} = cos 2ψ(ω) − p2 + q 2 , ζ (ω) ≡ {ζ(ω)} = sin 2ψ(ω) . (7.166) ˜ From the real and imaginary parts of the expression k(ω) = (ω/c)n(ω) one obtains the pair of expressions 1 ω nr (ω) − ω ni (ω) , c 1 α(ω) = ω ni (ω) + ω nr (ω) , c β(ω) =

(7.167) (7.168)

with ω = ω +iω . Along the real angular frequency axis, β(ω ) = (ω /c)nr (ω ) and α(ω ) = (ω /c)ni (ω ), both of which are nonnegative for all ω ≥ 0. If both β(ω) and α(ω) are to be nonnegative for all ω on the contour C+ [see the discussion following Eqs. (7.3)–(7.6)], then the pair of inequalities ω nr (ω) ≥ ω ni (ω),

ω ni (ω) + ω nr (ω) ≥ 0,

(7.169) (7.170)

must be satisﬁed ∀ω ∈ C+ . Substitution of the second inequality into the ﬁrst yields the trivial inequality ω 2 + ω 2 ≥ 0 so that only one of the inequalities appearing in Eqs. (7.169)–(7.170) needs to be assumed, the other then being satisﬁed automatically. Because of its clear necessity, it is assumed here that the plane wave propagation factor β(ω) is nonnegative ∀ω ∈ C+ . It then follows that 0 ≤ ψ(ω) ≤ π/2 ∀ω ∈ C+ . Consequently, sin (2ψ(ω)) ≥ 0 so that ζ (ω) ≥ 0 on C+ . On the other hand, cos (2ψ(ω)) > 0 when 0 ≤ ψ(ω) ≤ π/4, cos (2ψ(ω)) = 0 when ψ(ω) = π/4, and cos (2ψ(ω)) < 0 when π/4 ≤ ψ(ω) ≤ π/2, so that

7.3 Real Direction Cosine Form of the Angular Spectrum Representation

Ζ''

Ζ''

O

Ζ''

Ζ

Ζ

m = Ζ Ζ'

O

363

m = Ζ Π Ζ'

(a)

(b)

p2+ q2 < cos(2ΨΩ)

p2+ q2 = cos(2ΨΩ)

m = Ζ

Ζ

Ζ'

O

(c) p2+ q2 > cos(2ΨΩ)

Fig. 7.8. Inequalities satisﬁed by the direction cosines p and q on the contour C+ .

ζ (ω) ≥ 0 when (p2 + q 2 ) ≤ cos (2ψ(ω)) and ζ (ω) < 0 when (p2 cos (2ψ(ω)) ∀ω ∈ C+ . Consequently, p2 + q 2 < cos 2ψ(ω) ⇒ ζ (ω) > 0 ∧ ζ (ω) ≥ 0 7 8 π ⇒ 0 ≤ arg ζ(ω) < 2 7 8 π ⇒ 0 ≤ arg m(ω) < , 4 p2 + q 2 = cos 2ψ(ω) ⇒ ζ (ω) = 0 ∧ ζ (ω) ≥ 0 7 8 π ⇒ arg ζ(ω) = 2 7 8 π ⇒ arg m(ω) = , 4 p2 + q 2 > cos 2ψ(ω) ⇒ ζ (ω) < 0 ∧ ζ (ω) ≥ 0 7 8 π ⇒ < arg ζ(ω) ≤ π 2 7 8 π π ⇒ < arg m(ω) < , 4 2

+ q2 ) >

(7.171)

(7.172)

(7.173)

for all ω ∈ C+ , as illustrated in Figure 7.8. As a consequence, the real and imaginary parts of the direction cosine m(ω) ≡ m (ω) + im (ω) satisfy the respective inequalities m (ω) ≡ {m(ω)} ≥ 0,

m (ω) ≡ {m(ω)} ≥ 0,

(7.174) (7.175)

for all ω ∈ C+ . Explicit expressions for both m (ω) and m (ω) are obtained from the square of Eq. (7.164) as

364

7 Angular Spectrum Representation of Pulsed Beam Fields

m (ω) =

1 cos (2ψ(ω)) − p2 + q 2 2 1/2 1/2 2 1 1 cos (2ψ(ω)) − p2 + q 2 + , + 2 sin2 (2ψ(ω)) (7.176)

provided that sin (2ψ(ω)) = 0, and m (ω) =

sin (2ψ(ω)) , 2m (ω)

(7.177)

provided that m (ω) = 0. If m (ω) = 0, then sin (2ψ(ω)) = 0 and either ψ(ω) = 0, in which case (p2 + q 2 ) ≥ 1 and 1/2 m (ω) = p2 + q 2 − 1 ,

(7.178)

or ψ(ω) = π/2, in which case 1/2 m (ω) = p2 + q 2 + 1 ,

(7.179)

for all values of p and q. Finally, for the special case of a lossless medium, ψ(ω) = 0 and m(ω) = m (ω) = 1 − (p2 + q 2 ) (7.180) if (p2 + q 2 ) ≤ 1, whereas m(ω) = im (ω) = i (p2 + q 2 ) − 1

(7.181)

if (p2 + q 2 ) > 1. With these results, the plane wave propagation factor appearing in the angular spectrum representation given in Eqs. (7.14) and (7.17) becomes [cf. Eq. (7.49)] ˜ + (ω)·r

eik

= e−k(ω)m

(ω)∆z ik(ω)(px+qy+m (ω)∆z )

e

,

(7.182)

˜ + (ω) · r represents the with ∆z ≡ z − z0 ≥ 0. If p = q = 0, then exp ik spatial part of a homogeneous plane wave because the surfaces of constant amplitude coincide with the surfaces of constant phase given by ∆z =constant. If α(ω) = 0, then Eq. (7.182) represents the spatial part of a homogeneous it repreplane wave when p2 + q 2 ≤ 1 (in which case m = 0), whereas sents the spatial part of an evanescent wave when p2 + q 2 > 1 (in which case m = 0) and the surfaces of constant amplitude are orthogonal to the cophasal surfaces. In general, α(ω) = 0 and if either p = 0 or q = 0, then the expression appearing in Eq. (7.182) represents the spatial part of an inhomogeneous plane wave of angular frequency ω because the surfaces of constant

7.3 Real Direction Cosine Form of the Angular Spectrum Representation

365

x s

cos-1(p/s)

cos-1((m'(Ω)/s)

O cos-1(q/s)

Cophasal Surface px + qy + m'(Ω)z = constant

z

y

Fig. 7.9. Inhomogeneous plane wave phase front propagating in the direction specˆ x p/s + 1 ˆ y q/s + 1 ˆ z m (ω)/s. iﬁed by the unit vector ˆs = 1

amplitude ∆z = constant are diﬀerent from the surfaces of constant phase (px + qy + m (ω)∆z) = constant. These inhomogeneous plane wave phase fronts then propagate in the direction that is speciﬁed by the unit vector [cf. Eq. (7.51)] ˆy q + 1 ˆ z m (ω) , ˆx p + 1 ˆs = 1 (7.183) s s s with s = p2 + q 2 + m2 (ω), (7.184) as depicted in Figure 7.9 [cf. Fig. 7.4)]. Because m (ω) ≥ 0 for all real values of p and q, as well as for all ω ∈ C+ , then those inhomogeneous plane wave components with m (ω) > 0 have phase fronts that advance into the positive half-space ∆z > 0. From Eq. (7.176) it is seen that the inequality < m (ω) (7.185) m (ω) 2 2 2 2 (p +q )>cos (2ψ(ω))

(p +q )
is satisﬁed when sin (2ψ(ω)) = 0, so that m (ω) > m (ω) 2 2 (p +q )>cos (2ψ(ω))

(p2 +q 2 )
,

(7.186)

provided that m (ω) = 0. Consequently, the amplitude attenuation associated with the inhomogeneous plane wave spectral components in the exterior

366

7 Angular Spectrum Representation of Pulsed Beam Fields

7 8 region R> ≡ (p, q)(p2 + q 2 ) > cos (2ψ(ω)) is typically larger than that associated with the inhomogeneous plane wave spectral components in the 7 8 interior region R< ≡ (p, q)(p2 + q 2 ) < cos (2ψ(ω)) . With these results, the angular spectrum of plane waves representation given in Eqs. (7.14) and (7.17) becomes

1 ˜˜ (p, q, ω) dω dpdq k 2 (ω)U U(r, t) = 0 4π 3 C+ R< ∪R>

−k(ω)m (ω)∆z ik(ω)(px+qy+m (ω)∆z ) ×e e (7.187) for ∆z ≥ 0. Here U(r, t) represents either the electric ﬁeld vector E(r, t), in ˜ ˜˜ (p, q, ω) of ˜ 0 (p, q, ω) represents the Fourier–Laplace transform E which case U 0 the boundary value for that ﬁeld vector, or the magnetic ﬁeld vector B(r, t), ˜ ˜˜ (p, q, ω) ˜ 0 (p, q, ω) represents the Fourier–Laplace transform B in which case U 0 of the boundary value for that ﬁeld vector.

7.4 Pulsed Electromagnetic Beam Fields and Source-Free Fields The angular spectrum of plane waves representation of the freely propagating electromagnetic ﬁeld given in Eqs. (7.14) and (7.17) expresses that wave ﬁeld throughout the positive half-space z ≥ z0 as a superposition of both homogeneous and inhomogeneous plane waves. In the idealized limit of a lossless medium, ψ(ω) = 0 and one obtains homogeneous plane wave components 7 8 in the interior circular domain R< ≡ (p, q)(p2 + q 2 )7< 1 and evanescent 8 plane wave components in the exterior domain R> ≡ (p, q)(p2 + q 2 ) > 1 . Because there is no time-average energy ﬂow from the evanescent plane wave components into the half-space z > z0 , an electromagnetic beam ﬁeld in a lossless medium is typically (perhaps casually) deﬁned, in part, by the requirement that its angular spectrum does not contain any evanescent wave components, and consequently that its ﬁeld can be represented by an angular spectrum that contains only homogeneous plane wave components [9]. Because the homogeneous plane wave spectral components do not attenuate when propagating through such a lossless medium, this condition ensures that all of the angular spectrum components of the beam ﬁeld remain present in their initial proportion throughout the positive half-space z ≥ z0 . Wave ﬁelds in lossless media that only contain homogeneous plane wave components in their angular spectrum representation are known as source-free wave ﬁelds [4, 6].

7.4 Pulsed Electromagnetic Beam Fields and Source-Free Fields

367

Because such a distinction cannot be made when the medium is attenuative, Sherman’s deﬁnition [4, 6] of source-free wave ﬁelds needs to be generalized in order to treat that more general case. Because of the inequality given in Eq. (7.186), wave ﬁelds in dispersive attenuative media that contain only inhomogeneous plane wave components with direction cosines that satisfy the inequality (p2 + q 2 ) < cos (2ψ(ω)) for all ω ∈ C+ in their angular spectrum representation are called source-free wave ﬁelds. In the limit as the material loss goes to zero at all frequencies, these inhomogeneous plane wave components become homogeneous plane wave components with direction cosines satisfying the inequality (p2 + q 2 ) < 1. Speciﬁcally, one has the following deﬁnition for a scalar wave ﬁeld that is desribed by the scalar version of the angular spectrum representation given in Eq. (7.187): ˜ (r, ω) is said to be source-free if and Deﬁnition 3. The scalar wave ﬁeld U ˜ ˜ 0 (p, q, ω) ∈ L2 vanishes almost everywhere for (p2 +q 2 ) > cos (2ψ(ω)) only if U for all ω ∈ C+ . An electromagnetic beam ﬁeld in a dispersive lossy medium may then be deﬁned, in part, by the requirement that each of its scalar components is a generalized source-free wave ﬁeld, so that ˜ ˜˜ (p, q, ω) = 0, ˜ 0 (p, q, ω) = B E 0

∀ (p, q) p2 + q 2 > cos (2ψ(ω)).

(7.188)

Each propagated ﬁeld vector is then given by the angular spectrum of plane wave representation appearing in Eq. (7.187) taken only over the interior circular region R< . For reasons of notational simplicity, the properties of source-free wave ﬁelds are presented here for the scalar case. The results then apply directly to the electromagnetic wave-ﬁeld case. 7.4.1 General Properties of Source-Free Wave Fields With the classic analysis of Sherman [6] as a guide, consider the monochro˜ (r, ω) that satisﬁes the scalar Helmholtz equamatic scalar wave function U tion [cf. Eqs. (5.55)–(5.56)] ˜ (r, ω) = 0 ∇2 + k˜2 (ω) U (7.189) throughout the positive half-space z ≥ z0 and which has the boundary value 7 8 ˜ (r, ω) = U ˜0 (rT , ω), (7.190) lim U ∆z→0

where ∆z ≡ z − z0 . This wave function then has the angular spectrum of plane waves representation ∞ ∞ ˜˜ (p, q, ω)H(p, q, ω; x, y, ∆z)k 2 (ω)dpdq ˜ (r, ω) = 1 U U 0 4π 2 −∞ −∞ (7.191)

368

7 Angular Spectrum Representation of Pulsed Beam Fields

throughout the positive half-space ∆z ≥ 0, where ∞ ∞ ˜˜ (p, q, ω) = ˜0 (x, y, ω)e−ik(px+qy) dxdy, U U 0 −∞ −∞ ∞ ∞ ˜˜ (p, q, ω)eik(px+qy) k 2 dpdq, ˜0 (x, y, ω) = 1 U U 0 (2π)2 −∞ −∞

(7.192) (7.193)

form a two-dimensional Fourier transform pair, and where H(p, q, ω; x, y, ∆z) ≡ G(p, q, ω, ∆z)eik(ω)(px+qy)

(7.194)

is the kernel appearing in the angular spectrum integral in Eq. (7.191) [cf. Eq. (7.182)] with G(p, q, ω, ∆z) ≡ eik(ω)m(ω)∆z 1/2

i2ψ(ω) −(p2 +q 2 )] = eik(ω)∆z[e 2 2 1/2 ˜ = eik(ω)∆z[1−(p +q )] .

(7.195)

The complex-valued variables p ≡ pe−iψ and q ≡ qe−iψ have been introduced here in order to simplify the notation.3 The conditions under which the scalar wave ﬁeld given by Eq. (7.191) is the valid representation of the propagated wave ﬁeld throughout the half-space z > z0 that is due to the boundary value given in Eq. (7.190) has been considered in detail by Lalor [12], Montgomery [18], Sherman and Bremermann [5], and Sherman [4, 6] for a lossless nondispersive medium. A sketch of the proof of the validity of this angular spectrum representation of the solution to the Helmholtz equation given in ˜ Eq. (7.189) with complex wave number k(ω) that has the boundary value given in Eq. (7.190) is now given. Validity of the Angular Spectrum Representation The method of proof presented here follows that given by Sherman [6] for a ˜0 (rT , ω) = U ˜0 (x, y, ω) ∈ L2 lossless medium. For all ω ∈ C+ , assume that U ˜ (r, ω) = U ˜ (x, y, z, ω) be given by the angular spectrum representaand let U # $ ˜˜ (p, q, ω) = F U ˜ tion in Eq. (7.191) for all ∆z = z − z0 > 0 with U (x, y, ω) 0 0 being given by the two-dimensional Fourier transform speciﬁed in Eq. (7.192). Because H(p, q, ω; x, y, ∆z) ∈ L2 , then the function formed by the product ˜˜ (p, q, ω)H(p, q, ω; x, y, ∆z) is in L1 , and hence, U ˜ (x, y, z, ω) exists and is U 0 bounded for all ∆z > 0. The spatial partial derivatives of the product func˜˜ (p, q, ω)H(p, q, ω; x, y, ∆z) satisfy the inequalities tion U 0 3

Notice that p and q are identical to the complex direction cosines p and q used in §7.1 [cf. Eqs. (7.21)–(7.23) and Eqs. (7.156)–(7.158)].

7.4 Pulsed Electromagnetic Beam Fields and Source-Free Fields

∂ j+m+n ˜ ˜ 0 (p, q, ω)H(p, q, ω; x, y, ∆z) ≤ M (p, q, δ, ω) U ∂xj ∂y m ∂z n

369

(7.196)

for all ∆z ≥ δ > 0, where δ is an arbitrary positive number, and where j+m+n j m i2ψ(ω) 2 n/2 ˜ 2 ˜ M (p, q, δ, ω) ≡ k U 0 (p, q, ω)G(p, q, ω, δ) p q e − p +q (7.197) is independent of z. Because the right-hand side of Eq. (7.197) is the product of two functions that are in L2 , then it is in L1 . Hence, by Lebesgue’s theorem of bounded convergence [19], the integrand in Eq. (7.191) is suﬃciently well˜ (r, ω) = U ˜ (x, y, z, ω) is continuous with respect to each spatial behaved that U variable and has continuous partial derivatives with respect to each spatial variable, where ∂ j+m+n ˜ U (x, y, z, ω) ∂xj ∂y m ∂z n ∞ ∞ 1 ∂ j+m+n ˜˜ U = (p, q, ω)H(p, q, ω; x, y, ∆z) k 2 (ω)dpdq 0 4π 2 −∞ −∞ ∂xj ∂y m ∂z n (7.198) for all real values of x and y and for all ∆z ≥ δ > 0. The proof that the partial diﬀerentiation operation can be interchanged in its order with the integration operation in obtaining the expression on the right-hand side of Eq. (7.198) may be found in a generalization of a theorem by McShane [20] from a single to a multiple integral; although that theorem is stated only for a single diﬀerentiation, it may be repeatedly applied (j + m + n) times to obtain the result given in Eq. (7.193). If one applies the result given in Eq. (7.198) to determine the Laplacian ˜ (x, y, z, ω), one obtains the result given in Eq. (7.189). Hence, for all of U ˜ (r, ω) = U ˜ (x, y, z, ω) that is given in Eq. (7.191) ω ∈ C+ , the wave function U satisﬁes the scalar Helmholtz equation (7.189) with complex wavenumber ˜ k(ω) = k(ω)eiψ(ω) for all ∆z ≥ δ > 0 with δ arbitrarily small. Consider now the proof of Eq. (7.190) when the limiting procedure is ˜ (r, ω) = U ˜ (x, y, z, ω) ∈ L2 . In that case taken as a limit in the mean for all U it must be shown that 7 8 (7.199) lim I(∆z, ω) = 0, ∆z→0

where

I(∆z, ω) ≡

∞

−∞

∞

−∞

2 ˜ ˜0 (x, y, ω) dxdy U (x, y, ∆z, ω) − U

(7.200)

for ∆z > 0. The two-dimensional spatial Fourier transform of the quantity ˜˜ (p, q, ω) (G(p, q, ω, ∆z) − 1) for ˜ ˜ U (x, y, ∆z, ω) − U0 (x, y, ω) is equal to U 0 ∆z > 0 so that by Parseval’s theorem

370

7 Angular Spectrum Representation of Pulsed Beam Fields

∞

∞

I(∆z, ω) = −∞

−∞

2 ˜˜ U 0 (p, q, ω) G(p, q, ω, ∆z) − 1 dpdq

(7.201)

for ∆z > 0. Because |G(p, q, ω, ∆z) − 1| ≤ 2 for all ∆z > 0, then 2 2 ˜˜ ˜˜ U 0 (p, q, ω) G(p, q, ω, ∆z) − 1 ≤ 4 U 0 (p, q, ω) for ∆z > 0. One can then apply Lebesgue’s theorem of bounded convergence [19] to obtain ∞ ∞ 2 2 7 8 ˜˜ lim U 0 (p, q, ω) G(p, q, ω, ∆z) − 1 lim I(∆z, ω) = ∆z→0

−∞

−∞

∆z→0

× k 2 (ω)dpdq = 0.

(7.202)

This then proves the following generalization of a theorem due to Sherman.4 ˜0 (rT , ω) = Theorem 4. Sherman’s First Theorem. For all ω ∈ C+ , let U ˜ ˜0 (x, y, ω) ∈ L2 with two-dimensional Fourier transform U ˜ 0 (p, q, ω) = U # $ ˜0 (x, y, ω) , and let U ˜ (r, ω) = U ˜ (x, y, z, ω) be given by the angular specF U trum of plane waves representation in Eq. (7.191) for all ∆z = z − z0 > 0. ˜ (r, ω) = U ˜ (x, y, z, ω) is bounded and has continuous Then for all ∆z > 0, U partial derivatives of all orders with respect to each spatial coordinate vari˜ (r, ω) = U ˜ (x, y, z, ω) satisﬁes the scalar Helmholtz equation able. Moreover, U ˜ with complex wave number k(ω) = k(ω)eiψ(ω) for all ∆z > 0 and has the boundary value 7 8 ˜ (x, y, z, ω) = U ˜0 (x, y, ω). lim U (7.203) ∆z→0

From Schwartz’s inequality it follows that if a function F (p, q) ∈ L2 vanishes almost everywhere outside a bounded region in (p, q)−space, then ˜˜ (p, q, ω) ∈ L1 . The fol˜ (r, ω) is source-free, then U F (p, q) ∈ L1 . Hence, if U 0 5 lowing three theorems due to Sherman then remain valid with only minor modiﬁcations (included here) in a dispersive attenuative medium. ˜ (r, ω) Theorem 5. Sherman’s Second Theorem. If the scalar wave ﬁeld U is source-free, then the relation given in Eq. (7.191) can be used to extend ˜ (r, ω) into the region z ≤ z0 to obtain a bounded solution of the Helmholtz U equation (7.189) for all space. ˜ (r, ω) through the use of The extension of the source-free wave ﬁeld U ˜ (r, ω) is the integral representation given in Eq. (7.191) means only that U deﬁned by Eq. (7.191) for all z. This theorem shows that it is not necessary to 4 5

See Theorem 1 of [6]. See Theorems 2 through 4 of [6].

7.4 Pulsed Electromagnetic Beam Fields and Source-Free Fields

371

restrict attention to the positive half-space z ≥ z0 when considering a sourcefree wave ﬁeld. The proof of this theorem is essentially inchanged from that given by Sherman [6] in the lossless case. As a consequence of this theorem, source-free wave ﬁelds may always be considered to extend throughout all space. Because source-free ﬁelds satisfy the homogeneous Helmholtz equation given in Eq. (7.189) throughout all space, there are then no sources of the ﬁeld anywhere in any ﬁnite reach of space; any source of the ﬁeld must then be located inﬁnitely far away. Because a source-free ﬁeld is bounded for all x, y, z, such inﬁnitely removed sources can then have no mathematical signiﬁcance. Because of this property, Sherman [4, 6] regarded source-free ﬁelds to be wave ﬁelds that extended throughout all space with no sources anywhere, as reﬂected in the name “source-free.” This type of ﬁeld is clearly a special case of a freely propagating wave ﬁeld that has no sources in the positive half-space z ≥ z0 . Theorem 6. Sherman’s Third Theorem. For a given boundary value ˜0 (rT , ω) = U ˜0 (x, y, ω) ∈ L2 for all ω ∈ C+ , the ﬁeld U ˜ (r, ω) = U ˜ (x, y, z, ω) U ˜ is source-free if and only if U0 (x, y, ω) is equal almost everywhere to a function f0 (x, y, ω) that can be extended to the entire space of two complex variables X = x + ix , Y = y + iy as an entire function f0 (X, Y, ω) such that

2

|f0 (X, Y, ω)| ≤ Aek (ω)(x

1/2

+y 2 )

(7.204)

for all ω ∈ C+ , where x and y are real variables and A is a positive constant. This theorem, which is a consequence of the Plancherel–P´ olya theorem ˜0 (x, y, ω) [21], allows one to determine directly from the boundary value U ˜ ˜ whether the ﬁeld U (r, ω) = U (x, y, z, ω) that is a result of that boundary value is source-free without having to evaluate the Fourier transform # $ ˜ ˜ ˜ U 0 (p, q, ω) = F U0 (x, y, ω) . Proof. In order to prove the suﬃciency part of theorem 6, let the ﬁeld ˜ (r, ω) = U ˜ (x, y, z, ω) be source-free for all ω ∈ C+ , so that, by deﬁnition, U # $ ˜˜ (p, q, ω) = F U ˜0 (x, y, ω) vanishes almost everywhere for p2 + q 2 > U 0 cos (2ψ(ω)) for all ω ∈ C+ . If one then lets x = r cos ζ, y = r sin ζ, then it follows directly from the necessity portion of the proof of the Plancherel– ˜0 (x, y, ω) is equal almost everywhere ˜0 (rT , ω) = U P´ olya theorem [21] that U to a function f0 (x, y, ω) that can be extended to the entire space of two complex variables X = x + ix , Y = y + iy as an entire function f0 (X, Y, ω) such that |f0 (X, Y, ω)| ≤ AerKf (ζ) ,

(7.205)

where Kf (ζ) is the least upperbound of the quantity k(p cos ζ + q sin ζ with ﬁxed ζ for (p, q) in the region p2 + q 2 ≤ cos (2ψ(ω)) for all ω ∈ C+ , where

372

7 Angular Spectrum Representation of Pulsed Beam Fields

k is the transform parameter appearing in Eqs. (7.192)–(7.193). With p = R cos ξ and q = R sin ξ with R ≤ cos1/2 (2ψ(ω)), then Kf (ζ) is the least upper bound of the quantity kR cos (ζ − ξ), and hence Kf (ζ) = k cos1/2 (2ψ(ω)) ≡ k (ω),

(7.206)

which, when substituted in Eq. (7.205), yields the result given in Eq. (7.204). ˜0 (x, y, ω) = In order to prove the necessity part of the theorem, let U f0 (x, y, ω) almost everywhere, where f0 (x, y, ω) can be extended to an entire function f0 (X, Y, ω) of the two complex variables X = x + ix , Y = y + iy that satisﬁes the inequality given in Eq. (7.204). Consider the function hω (ζ, x, y) that is deﬁned by the limit 1 (7.207) hω (ζ, x, y) ≡ lim ln |f0 (x + ir cos ζ, y + ir sin ζ| , r→∞ r where x = r cos ζ, y = r sin ζ, as before, and let hω (ζ) ≡ sup {hω (ζ, x, y)}

(7.208)

be the least upper bound of hω (ζ, x, y) for all x and y with ﬁxed ζ for olya theorem [21], the Fourier transform all ω ∈ C+ . By the Plancherel–P´ F0 (p, q, ω) = F{f0 (x, y, ω)} vanishes almost everywhere ouside a convex region Df in (p, q)−space, where k(p cos ζ + q sin ζ) ≤ hω (ζ)

(7.209)

for all (p, q) ∈ Df . Because f0 (X, Y, ω) satisﬁes the inequality given in Eq. (7.204), then 1 k r ln Ae = k . (7.210) hω (ζ) ≤ lim r→∞ r With p = R cos ξ and q = R sin ξ, the above two inequalities yield the inequality R cos (ζ − ξ) ≤ cos1/2 (2ψ(ω)),

(7.211)

so that F0 (p, q, ω) vanishes almost everywhere outside the circular region R2 = (p2 + q 2 ) ≤ cos (2ψ(ω)). Finally, because the Fourier transforms of two functions that are equal almost everywhere are themselves equal al˜˜ (p, q, ω) vanishes almost everywhere for (p2 + q 2 ) > most everywhere, then U 0 cos (2ψ(ω)). ! Theorem 7. Sherman’s Fourth Theorem. The angular spectrum repre˜ (r, ω) = sentation given in Eq. (7.191) can be used to extend the wave ﬁeld U ˜ U (x, y, z, ω) to the entire space of three complex variables X = x + ix ,

7.4 Pulsed Electromagnetic Beam Fields and Source-Free Fields

373

˜ (X, Y, Z, ω) such that for Y = y + iy , Z = z + iz as an entire function U constant z = {Z}, 2 2 1/2 ˜ (7.212) U (X, Y, Z, ω) ≤ Bek(x +y ) , where x , y , z are real variables and B is a positive constant that can ˜ (r, ω) is source-free. depend on z , if and only if U Proof. In order to prove the necessity part of this theorem, assume that ˜ (r, ω) = U ˜ (x, y, z, ω) is a source-free wave ﬁeld and consider the represenU tation ˜˜ (p, q, ω)H(p, q, ω; X, Y, Z)k 2 (ω)dpdq, ˜ (X, Y, Z, ω) = 1 U U 0 4π 2 R< (7.213) where X = x + ix , Y = y + iy , and Z = z + iz . From Eqs. (7.194) and (7.195) 1 ˜ ˜˜ −k(ω)m (ω)z −k(ω)m (ω)z 2 U (p, q, ω) e k (ω)dpdq, U (X, Y, Z, ω) ≤ e 0 2 4π R< where m (ω) ≡ {m(ω)} ≥ 0 and m (ω) ≡ {m(ω)} ≥ 0 for all ω ∈ C+ . In any bounded region R of X, Y, Z-space, the product of the two exponentials appearing in the integrand of the above equation is bounded by a positive constant M that depends upon the region R, so that M ˜˜ 2 ˜ U (p, q, ω) k (ω)dpdq = M A. (7.214) U (X, Y, Z, ω) ≤ 0 2 4π R< Hence, the integral deﬁned in Eq. (7.213) exists for all X, Y, Z in any bounded region of X, Y, Z-space. Because the integrand appearing in Eq. (7.213) is an ˜ (X, Y, Z, ω) is an entire function entire function of complex X, Y, Z, then U ˜ ˜ of X, Y, Z, provided that U (r, ω) = U (x, y, z, ω) is source-free [22]. It then ˜ (X, Y, Z, ω) satisﬁes follows from the suﬃciency proof of Theorem 6 that U the inequality given in Eq. (7.212) for constant z ≡ {Z}. In order to prove the suﬃciency part of this theorem, assume that ˜ (r, ω) = U ˜ (x, y, z, ω) may be extended to the entire space of three comU ˜ (X, Y, Z, ω) for all ω ∈ C+ . If plex variables X, Y, Z as an entire function U ˜ one then replaces f0 (x, y, ω) with U (x, y, Z, ω) in the necessity part of the proof of Theorem 6 and keeps z = {Z} constant throughout, it then fol˜˜ (p, q, ω)G(p, q, ω, ∆z) vanishes almost everywhere lows that the quantity U 0 for p2 +q 2 > cos (2ψ(ω)). Because G(p, q, ω, ∆z) does not vanish for any ﬁnite ˜˜ (p, q, ω) vanishes almost everywhere values of p and q [cf. Eq. (7.195)], then U 0 for p2 + q 2 > cos (2ψ(ω)). !

374

7 Angular Spectrum Representation of Pulsed Beam Fields

˜ (r, ω) = As a consequence of this theorem, a source-free wave ﬁeld U ˜ (x, y, z, ω) has all of the properties that are associated with an entire funcU ˜ (r, ω) = U ˜ (x, y, z, ω) can be tion in complex variable theory. In particular, U expanded in each spatial variable in a Taylor series that converges for all values of x, y, and z. In addition, the speciﬁcation of the ﬁeld throughout a ﬁnite region of space is suﬃcient to determine the ﬁeld throughout all space through the process of analytic continuation. This result is of fundamental consequence to diﬀraction and scattering theory [10, 23] as well as to inverse optics [11, 24]. The Sherman Expansion of Source-Free Wave Fields ˜ Because the complex wave number k(ω) is analytic along the contour C+ , then for a source-free wave ﬁeld one may represent the function G(p, q, ω, ∆z) deﬁned in Eq. (7.195) by its Taylor series expansion G(p, q, ω, ∆z) =

∞ ∞ G(2r,2s) (0, 0, ω, ∆z) r=0 s=0

(2r)!(2s)!

p2r q 2s ,

(7.215)

where G(m,n) (p, q, ω, ∆z) ≡

∂ m+n G(p, q, ω, ∆z) . ∂pm ∂q n

(7.216)

Substitution of this Taylor series expansion into the angular spectrum representation given in Eq. (7.191) then yields ˜ (r, ω) = U

∞ ∞ 1 G(2r,2s) (0, 0, ω, ∆z) 4π 2 r=0 s=0 (2r)!(2s)! ∞ ∞ ˜ ˜˜ (p, q, ω)eik(ω) (px+qy) k˜2 (ω)dpdq. × p2r q 2s U 0 −∞

−∞

(7.217) Because m+n ˜ U0 (x, y, ω) ˜ (m,n) (x, y, ω) = ∂ U 0 ∂xm ∂y n ∞ ∞ m+n 1 ˜˜ (p, q, ω) ik(ω) ˜ U = pm q n 0 4π 2 −∞ −∞ ˜ × eik(ω)(px+qy) k˜2 (ω)dpdq,

(7.218) one ﬁnally obtains the Sherman expansion of the source-free wave ﬁeld

7.4 Pulsed Electromagnetic Beam Fields and Source-Free Fields

˜ (r, ω) = U

∞ ∞ r=0 s=0

G(2r,2s) (0, 0, ω, ∆z) ˜ (2r,2s) (x, y, ω). 2(r+s) U0 ˜ (2r)!(2s)! ik(ω)

375

(7.219)

This series expansion explicitly displays the contribution to the transverse spatial behavior of the propagated wave ﬁeld due to each even-ordered spatial partial derivative of the boundary value given in Eq. (7.190). −2(r+s) ˜ appearing in the above series expanNotice that the factor (k(ω)) sion is misleading because the same factor with an opposite-signed exponent is contained in the partial derivative G(2r,2s) (0, 0, ω, ∆z). From Eq. (7.195) ∂ m+n G(p, q, ω, ∆z) ∂pm ∂q n m+n ˜ 2 2 1/2 ˜ = ik(ω)∆z eik(ω)∆z(1−p −q ) ϕ(m,n) (p, q),

G(m,n) (p, q, ω, ∆z) ≡

(7.220) where ϕ(m,n) (p, q) ≡

1/2 ∂ m+n 1 − p2 − q 2 . m n ∂p ∂q

(7.221)

With these substitutions, Sherman’s expansion (7.219) becomes ˜ (r, ω) = U

∞ ∞ (∆z)2(r+s) r=0 s=0

(2r)!(2s)!

˜ ˜ (2r,2s) (x, y, ω)eik(ω)∆z ϕ(2r,2s) (0, 0)U . 0

(7.222) In addition, the quantity ϕ expansion is given by

(2r,2s)

ϕ(2r,2s) (0, 0) =

(0, 0) appearing in this form of the Sherman

∂ 2(r+s) 2 2 1/2 1 − p − q ∂p2r ∂q 2s p=q=0

= θ(2r)θ(2s), where θ(0) = 1 and m1

θ(2m) ≡ (−1)

2

1 −1 2

1 1 (2m)! − 2 ··· −m+1 2 2 m!

(7.223)

(7.224)

for integer values of m > 0. With these results, one ﬁnally obtains the spatial series representation for either the electric or magnetic ﬁeld vector of the pulsed, source-free electromagnetic wave ﬁeld [cf. Eq. (7.187)] as [17] U(r, t) =

∞ ∞ θ(2r)θ(2s)

(∆z)2(r+s) (2r)!(2s)! r=0 s=0

1 ˜ (2r,2s) i(k(ω)∆z−ωt ) ˜ dω , (7.225) × (x, y, ω)e U0 π C+

376

7 Angular Spectrum Representation of Pulsed Beam Fields

where ∞ ∂ 2(r+s) (2r,2s) ˜ (x, y, ω) ≡ U0 (x, y, t)eiωt dt U0 ∂x2r ∂y 2s −∞ ∞ (2r,2s) = U0 (x, y, t)eiωt dt.

(7.226)

−∞

The representation given in Eq. (7.225) explicitly displays the temporal evolution of a pulsed electromagnetic beam ﬁeld through a single contour integral taken over the even-order spatial partial derivatives of the temporal frequency spectrum of the boundary value for the appropriate ﬁeld vector at the plane z = z0 . This representation is exact provided that the wave ﬁeld is sourcefree. For a plane wave ﬁeld propagating in the positive z-direction, only the r = s = 0 term is nonvanishing, in which case Eq. (7.225) reduces to a single contour integral and the pulse evolution is independent of the transverse position in the ﬁeld. For a beam ﬁeld, however, the temporal pulse evolution will, in general, be dependent upon the transverse postion in the ﬁeld through the even-ordered spatial derivatives of the transverse beam ﬁeld proﬁle at the plane z = z0 . Validity of the Sherman Expansion A sketch of the proof of the validity of the Sherman expansion given in Eq. (7.219) for a source-free wave ﬁeld in a dispersive, attenuative medium is now given, following the method of proof given by Sherman [6]. Deﬁnition 4. Let B denote the space of complex-valued functions f (x, y) that can be extended to the entire space of two complex variables X = x + ix , Y = y + iy as an entire function f (X, Y ) such that |f (X, Y )| ≤ Ae(k−δk)(|X|

2

1/2

+|Y |2 )

(7.227)

for some δk > 0, where x, y, x , and y are all real variables and where A is a positive real constant. From the theory of generalized functions [22, 25], every function f (x, y) ∈ B can be considered to be a distribution (a generalized function in the space D ) with Fourier transform F (p, q) that is an ultradistribution (a generalized function in the space Z which, in turn, is deﬁned as the set of all generalized functions in the space Z of slowly increasing entire functions [25]). One can determine F (p, q) by taking the Fourier transform of the two-dimensional Taylor series expansion f (x, y) =

∞ ∞ m=0 n=0

amn xm y n

(7.228)

7.4 Pulsed Electromagnetic Beam Fields and Source-Free Fields

377

with amn = f (m,n) (0, 0)/(m!n!), term by term to obtain ∞ ∞ f (x, y)e−ik(px+qy) dxdy F (p, q) = −∞

=

−∞

2 ∞ ∞ 2π k

amn δ (m) (p)δ (n) (q), m+n (−ik) m=0 n=0

(7.229)

where δ (m) (p) denotes the mth-order derivative of the Dirac delta function [see Appendix B]. If the relation in Eq. (7.229) is now multiplied by a function Ψ (p, q) ∈ Z and the result is integrated over all space, one can integrate the series expansion on the right-hand side term by term to obtain [5]

∞

∞

F (p, q)Ψ (p, q)dpdq = −∞

−∞

2π k

2 ∞ ∞

amn Ψ (m,n) (0, 0). m+n (−ik) m=0 n=0 (7.230)

The convergence of the series appearing on the right-hand side of this expression is guaranteed by generalized function theory for all functions f (x, y) ∈ B and functions Ψ (p, q) ∈ Z. Following the method of proof given by Sherman [6], the series expansion appearing in Eq. (7.230) is now used to extend the class of generalized functions that F (p, q) is deﬁned onto one that is much broader than Z. The method begins with the following two deﬁnitions. Deﬁnition 5. Let Ψ (p, q, X, Y, Z) denote a function of the two real variables p, q and the three complex parameters X, Y, Z that is deﬁned for all p, q, X, Y, Z. The function Ψ (p, q, X, Y, Z) is said to be in C for X, Y, Z in a region R if and only if 1. For ﬁxed values of p and q such that (p2 +q 2 ) < cos θ for real θ ∈ [0, π/2), the quantity |Ψ (p, q, X, Y, Z)| is bounded for all X, Y, Z ∈ R; 2. For ﬁxed values of X, Y, Z ∈ R, the function |Ψ (p, q, X, Y, Z)| can be extended to the space of two complex variables P = p + ip , Q = q + iq as a function that is analytic for |P |2 + |Q|2 < cos θ. Notice that no restrictions are placed on the behavior of the function Ψ (p, q, X, Y, Z) ∈ C outside the circle |P |2 + |Q|2 = cos θ or outside the region R. Deﬁnition 6. The function Ψν (p, q, X, Y, Z) is said to converge to zero as ν → ∞ in the sense of C if and only if, for all X, Y, Z ∈ R, Ψν (p, q, X, Y, Z) → 0 as ν → ∞ uniformly for all p, q such that (p2 + q 2 ) ≤ cos (θ) + δ for some δ > 0. The following theorems due to Sherman [6] then remain valid in the temporally dispersive medium case.

378

7 Angular Spectrum Representation of Pulsed Beam Fields

Theorem 8. Let Ψ (p, q, X, Y, Z) ∈ C for X, Y, Z in a region R, let f (x, y) ∈ B, and let amn be given by amn =

1 (m,n) f (0, 0) m!n!

(7.231)

for all integer values of m and n with a00 = f (0, 0). Then the series S(X, Y, Z) =

2π k

2 ∞ ∞

amn Ψ (m,n) (0, 0, X, Y, Z) (7.232) m+n (−ik) m=0 n=0

converges absolutely and uniformly for all X, Y, Z ∈ R. ˜0 (x, y, ω) ∈ B possess the two-dimensional Taylor series Corollary 1. Let U expansion that is given by ˜0 (x, y, ω) = U

∞ ∞

amn (ω)xm y n

(7.233)

m=0 n=0

with amn (ω) =

1 ˜ (m,n) (0, 0, ω). U m!n! 0

(7.234)

Then the series ∞ ∞

˜mn (x, y, z, ω) ≡ amn (ω)U

m=0 n=0

∞ ∞ amn (ω) (m,n) H (0, 0, ω; x, y, ∆z) (ik)m+n m=0 n=0

(7.235) is absolutely and uniformly convergent for all x, y, ∆z. ˜0 (x, y, ω) ∈ B. Then the series that is obtained by replacTheorem 9. Let U ing x, y, z in the series ˜ (x, y, z, ω) = U

∞ ∞ amn (ω) (m,n) H (0, 0, ω; x, y, ∆z) (ik)m+n m=0 n=0

(7.236)

with X = x + ix , Y = y + iy , Z = z + iz , respectively, converges uni˜ (X, Y, Z, ω) of three complex variables X, Y, Z. formly to an entire function U ˜ (x, y, z, ω) that is given in Eq. (7.236) can be As a consequence, the function U extended to the space of three complex variables X, Y, Z as an entire function ˜ (X, Y, Z, ω). U ˜0 (x, y, ω) ∈ B and let U ˜ (x, y, z, ω) be given by Eq. Corollary 2. Let U ˜ (7.236). Then for all x, y, z, the function U (x, y, z, ω) is continuous and has continuous partial derivatives with respect to x, y, z of all orders that can be obtained simply by diﬀerentiating the series in Eq. (7.236) term by term.

7.4 Pulsed Electromagnetic Beam Fields and Source-Free Fields

379

˜0 (x, y, ω) ∈ B and let U ˜ (x, y, z, ω) be given by Eq. Theorem 10. Let U ˜ (x, y, z, ω) satisﬁes the Helmholtz equation given in Eq. (7.236). Then U (7.189) for all x, y, z and satisﬁes the boundary value given in Eq. (7.190). Proof. Application of the preceding corollary to the expression given in Eq. ˜ (x, y, z, ω), (7.236) yields, for the Laplacian of U 2

˜ (x, y, z, ω) = ∇ U =

∞ ∞ amn (ω) 2 (m,n) ∇ H (0, 0, ω; x, y, ∆z) m+n (ik) m=0 n=0

∞ ∞ amn (ω) ˜ 2 (m,n) (0, 0, ω; x, y, ∆z) ik(ω) H (ik)m+n m=0 n=0

˜ (x, y, z, ω) = −k˜2 (ω)U ˜ (x, y, z, ω) satisﬁes the Helmholtz equation. Furthermore, because and U ˜ U (x, y, z, ω) is continuous, one then has that # $ ˜ (x, y, z0 , ω) = lim U ˜ (x, y, z, ω) U = =

∆z→0 ∞ ∞

amn (ω) (m,n) H (0, 0, ω; x, y, 0) m+n (ik) m=0 n=0 ∞ ∞

˜0 (x, y, ω) amn (ω)xm y n = U

m=0 n=0

˜ (x, y, z, ω), as given by Eq. (7.236), satisﬁes the boundary value given and U in Eq. (7.190). ! ˜0 (x, y, ω) ∈ B as well as Theorem 11. Sherman’s Fifth Theorem. Let U 2 ˜ U0 (x, y, ω) ∈ L . Then the series given in Eq. (7.236) converges to the same ˜ (r, ω) = U ˜0 (x, y, z, ω) that is given by the integral representation function U in Eq. (7.191). ˜0 (x, y, ω) ∈ Proof. Following the method of proof given by Sherman [6], let U 2 ˜ ˜ ˜ B as well as U0 (x, y, ω) ∈ L . Because U0 (x, y, ω) ∈ B, then U0 (X, Y, ω) is an entire function of the complex variables X = x + ix , Y = y + iy , and ˜0 (x, y, ω) is satisﬁes the inequality given in Eq. (7.227). As a consequence, U ˜ continuous for all x, y and U0 (X, Y, ω) satisﬁes the inequality [cf. Eq. (7.227)] 2 2 1/2 ˜ U0 (X, Y, ω) ≤ Ae(k−δk)(|x | +|y | ) for some δk > 0. According to the Plancherel–P´ olya theorem [21], the two˜ ˜ ˜0 (x, y, ω) vanishes aldimensional spatial Fourier transform U 0 (p, q, ω) of U 2 most everywhere for (p2 + q 2 ) > cos1/2 (2ψ(ω)) − δk/k , as shown in Eq. ˜˜ (p, q, ω) ∈ L1 and (7.211). As a consequence, U 0

380

7 Angular Spectrum Representation of Pulsed Beam Fields

˜0 (x, y, ω) = U

1 (2π)2

∞

−∞

∞

−∞

˜˜ (p, q, ω)eik(px+qy) k 2 dpdq U 0

(7.237)

˜0 (x, y, ω) and the right-hand side of Eq. almost everywhere. Because both U (7.237) are continuous for all x, y, then Eq. (7.237) must be valid everywhere. The partial derivatives of the integrand ˜˜ (p, q, ω)eik(px+qy) I(p, q; x, y) ≡ k 2 U 0 appearing in Eq. (7.237) satisfy the inequality m+n ∂ ˜˜ (p, q, ω) , ≤ (ikp)m (ikq)n k 2 U I(p, q; x, y) 0 ∂xm ∂y n where the right-hand side of this inequality is independent of the spatial coordinate variables x, y, z and is in L1 . The orders of diﬀerentiation and integration may then be interchanged when using the Fourier integral representation given in Eq. (7.237) to determine the spatial partial derivatives of ˜0 (x, y, ω), so that U ∞ ∞ ˜˜ (p, q, ω)pm q n eik(px+qy) k 2 dpdq. ˜ (m,n) (x, y, ω) = 1 U (ik)m+n U 0 0 (2π)2 −∞ −∞ ˜ (m,n) (0, 0, ω)/(m!n!) Substitution of this expression into the equation amn = U 0 for the coeﬃcients appearing in the two-dimensional Taylor series expansion ˜0 (x, y, ω) then gives of U amn (ω) =

k 2π

2

∞

−∞

∞

−∞

(ik)m+n ˜˜ U 0 (p, q, ω)pm q n dpdq, m!n!

(7.238)

which, upon substitution into Eq. (7.236), yields ˜ (x, y, z, ω) = U

∞ ∞

1 H (m,n) (0, 0, ω; x, y, ∆z) m!n! m=0 n=0 2 ∞ ∞ k(ω) ˜˜ (p, q, ω)pm q n dpdq. U × 0 2π −∞ −∞

With the interchange of the order of the double summation and integration in this equation, one ﬁnally obtains ∞ ∞ 1 ˜˜ (p, q, ω) ˜ U (x, y, z, ω) = U 0 (2π)2 −∞ −∞ ∞ ∞ pm q n H (m,n) (0, 0, ω; x, y, ∆z) k 2 (ω)dpdq. × m!n! m=0 n=0 (7.239)

7.4 Pulsed Electromagnetic Beam Fields and Source-Free Fields

381

˜˜ (p, q, ω) is nonvanishing only for p2 +q 2 ≤ cos1/2 (2ψ(ω)) − δk/k 2 , Because U 0 the interchange of the order of the operations that led to Eq. (7.238) is justiﬁed if the double series appearing in the integrand is absolutely and uni 2 formly convergent for all x, y, z and p2 + q 2 > cos1/2 (2ψ(ω)) − δk/k . This condition is satisﬁed by this double-series summation because it is just the Taylor series expansion for H(p, q, ω; x, y, ∆z) in the variables p, q and because H(P, Q, ω; X, Y, ∆Z) is an analytic function of the complex variables 2 P = p + ip , Q = q + iq in the region |P |2 + |Q|2 ≤ cos1/2 (2ψ(ω)) − δk/k for all δk > 0 and all X, Y, ∆Z. Hence, the expression given in Eq. (7.239) reduces to the angular spectrum representation given in Eq. (7.191) and the theorem is proved. ! ˜ (r, ω) = U ˜ (x, y, z, ω) as given by Eq. (7.191) be sourceCorollary 3. Let U ˜˜ (p, q, ω) that vanishes almost free with a square-integrable angular spectrum U 0 2 everywhere outside the circle p2 + q 2 = cos1/2 (2ψ(ω)) − δk/k for some ˜ (r, ω) is given by the series in Eq. (7.236). δk > 0. Then U 7.4.2 Separable Pulsed Beam Fields A case of special interest is that in which the spatial and temporal properties of a pulsed electromagnetic beam ﬁeld are separable. Although this separability property is frequently assumed in the open literature, the general conditions under which it is valid are rarely, if ever, addressed. For that special case, assume that the spatial and temporal properties of the initial ﬁeld vectors at the plane z = z0 are separable in the sense that [17] ˆ 0 (x, y)f (t), E0 (rT , t) = E0 (x, y, t) = E ˆ 0 (x, y)g(t), B0 (rT , t) = B0 (x, y, t) = B

(7.240) (7.241)

so that ˜ ˜ ˜ ˜ 0 (p, q, ω) = E ˆ E 0 (p, q, k)f (ω), ˜ ˜ ˜ 0 (p, q, ω) = B ˆ 0 (p, q, k)˜ g (ω). B One now has the spatial Fourier transform pair relations ∞ ∞ ˜ ˆ ˆ 0 (x, y)e−ik(px+qy) dxdy, U0 (p, q, k) = U −∞ −∞ ∞ ∞ 1 ˜ ˆ 0 (p, q, k)eik(px+qy) k 2 dpdq ˆ U U0 (x, y) = (2π)2 −∞ −∞

(7.242) (7.243)

(7.244) (7.245)

ˆ 0 (x, y), and the separate Fourier–Laplace transform ˆ 0 (x, y) and B for both E pair relations

382

7 Angular Spectrum Representation of Pulsed Beam Fields

˜ h(ω) =

∞

h(t)eiωt dt, −∞

1 −iωt ˜ dω h(ω)e h(t) = π C+

(7.246) (7.247)

for both f (t) and g(t). It is seen from Eq. (7.19) that these functions cannot be independemtly chosen because their spectra must satisfy the relation ˜ ˜ˆ 1/2 ˜ ˆ 0 (p, q, k)˜ B g (ω) = (µ(ω) c (ω)) ˆs × E 0 (p, q, k)f (ω),

(7.248)

ˆy q + 1 ˆ z m(ω). The separation of this relation into spatial ˆx p + 1 where ˆs = 1 and temporal parts as ˜ ˜ ˆ 0 (p, q, k) = ˆs × E ˆ 0 (p, q, k), B 1/2 g˜(ω) = µ(ω) c (ω) f˜(ω),

(7.249) (7.250)

does not provide a unique solution of Eq. (7.248) except in special cases. With these results substituted in Eq. (7.225), one obtains the spatial series representation of the separable pulsed, source-free electromagnetic beam ﬁeld as E(r, t) =

B(r, t) =

∞ ∞ θ(2r)θ(2s) ˆ (2r,2s) (x, y) (∆z)2(r+s) E 0 (2r)!(2s)! m=0 n=0

1 ˜ k(ω)∆z−ωt i ) dω , f˜(ω)e ( × π C+ ∞ ∞ θ(2r)θ(2s) ˆ (2r,2s) (x, y) (∆z)2(r+s) B 0 (2r)!(2s)! m=0 n=0

1 ˜ i(k(ω)∆z−ωt ) dω . g˜(ω)e × π C+

(7.251)

(7.252)

This pair of expressions explicitly displays the temporal evolution of the electromagnetic beam ﬁeld through a single contour integral that is independent of the transverse spatial coordinates x and y [cf. Eq. (7.225)]. The temporal ﬁeld evolution is then independent of the transverse position in the pulsed beam ﬁeld throughout the entire half-space z ≥ z0 provided that the initial ﬁeld vectors are separable in the sense speciﬁed in Eqs. (7.240)–(7.241). A suﬃcient condition for the wave-ﬁeld to be separable in this sense is that it be source-free. For the special case of a strictly monochromatic (or time-harmonic) beam ﬁeld, (7.253) f (t) = sin (ωc t + ϕ0 ), where ϕ0 is a phase constant, then

7.4 Pulsed Electromagnetic Beam Fields and Source-Free Fields

7

f˜(ω) = −iπ δ(ω + ωc )eiϕ0

8 − δ(ω − ωc )e−iϕ0 .

Substitution of this expression into Eq. (7.251) then gives # $ ˜ ˆ Eωc (r, t) = E(r) iei(k(ωc )∆z−ωc t) ,

383

(7.254)

(7.255)

where ˆ E(r) ≡

∞ ∞ θ(2r)θ(2s) ˆ (2r,2s) (x, y) (∆z)2(r+s) E 0 (2r)!(2s)! m=0 n=0

(7.256)

describes the spatial variation of the monochromatic source-free electric ﬁeld. For the associated magnetic ﬁeld, Eq. (7.250) gives, with substitution from Eq. (7.254), 8 1/2 7 g˜(ω) = −iπ µ(ω) c (ω) δ(ω + ωc )eiϕ0 − δ(ω − ωc )e−iϕ0 . (7.257) Substitution of this expression into Eq. (7.252) then gives # $ 1/2 i(k(ω ˜ c )∆z−ωc t) ˆ , i µ(ωc ) c (ωc ) e Bωc (r, t) = B(r)

(7.258)

where ˆ B(r) ≡

∞ ∞ θ(2r)θ(2s) ˆ (2r,2s) (x, y) (∆z)2(r+s) B 0 (2r)!(2s)! m=0 n=0

(7.259)

describes the spatial variation of the monochromatic source-free magnetic ﬁeld. Notice that the expressions given in Eqs. (7.256) and (7.259) are both independent of the angular oscillation frequency ωc of the monochromatic source-free wave ﬁeld. The transverse spatial variation of each source-free ﬁeld vector at any plane z > z0 then depends solely upon the transverse spatial variation of the ﬁeld at the initial plane z = z0 through all of the even-order spatial derivatives of the corresponding ﬁeld vector at that plane.6 With the identiﬁcations given in Eqs. (7.256) and (7.259), the expressions given in Eqs. (7.251)–(7.252) for a separable pulsed, source-free electromagnetic beam ﬁeld become

1 ˜ k(ω)∆z−ωt i ) dω , ˆ (7.260) f˜(ω)e ( E(r, t) = E(r) π C+

1 ˜ i(k(ω)∆z−ωt ) ˆ B(r, t) = B(r) dω . (7.261) g˜(ω)e π C+ The propagated source-free pulsed beam ﬁeld is then given as the product of two separate factors, one describing the transverse spatial variation and the other describing the spatiotemporal longitudinal variation. Notice that both factors depend upon the propagation distance ∆z = z − z0 . 6

This remarkable result was ﬁrst derived for the source-free scalar wave ﬁeld by Sherman [4, 6] in 1968.

384

7 Angular Spectrum Representation of Pulsed Beam Fields

References 1. G. C. Sherman, “Application of the convolution theorem to Rayleigh’s integral formulas,” J. Opt. Soc. Am., vol. 57, pp. 546–547, 1967. 2. G. C. Sherman, “Integral-transform formulation of diﬀraction theory,” J. Opt. Soc. Am., vol. 57, pp. 1490–1498, 1967. 3. J. R. Shewell and E. Wolf, “Inverse diﬀraction and a new reciprocity theorem,” J. Opt. Soc. Am., vol. 58, no. 12, pp. 1596–1603, 1968. 4. G. C. Sherman, “Diﬀracted wave ﬁelds expressible by plane-wave expansions containing only homogeneous waves,” Phys. Rev. Lett., vol. 21, no. 11, pp. 761– 764, 1968. 5. G. C. Sherman and H. J. Bremermann, “Generalization of the angular spectrum of plane waves and the diﬀraction transform,” J. Opt. Soc. Am., vol. 59, no. 2, pp. 146–156, 1969. 6. G. C. Sherman, “Diﬀracted wave ﬁelds expressible by plane-wave expansions containing only homogeneous waves,” J. Opt. Soc. Am., vol. 59, pp. 697–711, 1969. 7. C. J. Bouwkamp, “Diﬀraction theory,” Rept. Prog. Phys., vol. 17, pp. 35–100, 1954. 8. J. W. Goodman, Introduction to Fourier Optics. New York: McGraw-Hill, 1968. 9. W. H. Carter, “Electromagnetic beam ﬁelds,” Optica Acta, vol. 21, pp. 871–892, 1974. 10. J. J. Stamnes, Waves in Focal Regions: Propagation, Diﬀraction and Focusing of Light, Sound and Water Waves. Bristol, UK: Adam Hilger, 1986. 11. M. Born and E. Wolf, Principles of Optics. Cambridge: Cambridge University Press, seventh (expanded) ed., 1999. 12. E. Lalor, “Conditions for the validity of the angular spectrum of plane waves,” J. Opt. Soc. Am., vol. 58, pp. 1235–1237, 1968. 13. H. L. Royden, Real Analysis. New York: Macmillan, second ed., 1968. p. 269. 14. A. Sommerfeld, Optics, vol. IV of Lectures in Theoretical Physics. New York: Academic, 1964. paperback edition. 15. A. Nisbet and E. Wolf, “On linearly polarized electromagnetic waves of arbitrary form,” Proc. Camb. Phil. Soc., vol. 50, pp. 614–622, 1954. 16. K. E. Oughstun, “Polarization properties of the freely-propagating electromagnetic ﬁeld of arbitrary spatial and temporal form,” J. Opt. Soc. Am. A, vol. 9, no. 4, pp. 578–584, 1992. 17. 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. 18. W. D. Montgomery, “Algebraic formulation of diﬀraction applied to self imaging,” J. Opt. Soc. Am., vol. 58, no. 8, pp. 1112–1124, 1968. 19. E. C. Titchmarsh, The Theory of Functions. London: Oxford University Press, 1937. Section 10.5. 20. E. J. McShane, Integration. Princeton, NJ: Princeton University Press, 1944. p. 217. 21. W. Kaplan, Introduction to Analytic Functions. Reading, MA: Addison-Wesley, 1966. p. 171. 22. H. Bremermann, Distributions, Complex Variables, and Fourier Transforms. Reading, MA: Addison-Wesley, 1965. Ch. 8.

7.4 Problems

385

23. M. Nieto-Vesperinas, Scattering and Diﬀraction in Physical Optics. New York: Wiley-Interscience, 1991. 24. T. B. Hansen and A. D. Yaghjian, Plane-Wave Theory of Time-Domain Fields. New York: IEEE, 1999. 25. I. M. Gel’fand and G. E. Shilov, Generalized Functions, vol. I. New York: Academic, 1964. Ch. 2.

Problems 7.1. Solve the relation given in Eq. (7.13) for the applied current density ˜ + , ω) of the source. ˜ 0 (k J 7.2. Redo the analysis presented in Eqs. (7.29)–(7.42) for a negative index medium where n(ω) < 0 over some frequency domain. 7.3. Obtain the solution equivalent to that given in Eqs. (7.74)–(7.75) expressed in terms of the normal derivatives of the initial ﬁeld vectors at the plane z = z0 . The spatial integrals appearing in this representation are the second Rayleigh–Sommerfeld diﬀraction integrals of classical optics. 7.4. For a uniformly polarized wave ﬁeld in the electric ﬁeld vector, show that for the magnetic ﬁeld vector to also be uniformly polarized, the spectral ˜˜ (k , ω) must have a ﬁxed direction independent of ω. Show ˜ + (ω)E quantity k 0 T ˜0 (rT , ω) must be that this in turn implies that the initial ﬁeld amplitude E independent of one transverse coordinate direction. 7.5. Show that the temporal frequency transform of the electromagnetic ﬁeld vectors given in Eqs. (7.14) and (7.17) vector Helnmholtz equa satisﬁes the 2 2 2 2 ˜ ˜ ˜ ˜ ω) = 0. tions ∇ + k (ω) E(r, ω) = 0 and ∇ + k (ω) B(r, 7.6. Prove Theorem 5. 7.7. Provide the conditions required for the separation given in Eqs. (7.249)– (7.250) to be a unique solution of Eq. (7.248). ˆ ˆ 7.8. Apply Theorem 11 to derive integral expressions for the E(r) and B(r) ﬁelds whose series representations are respectively given in Eqs. (7.256) and (7.259).

8 Free Fields in Temporally Dispersive Media

If there are no sources of an electromagnetic ﬁeld present anywhere in space during a period of time, then that ﬁeld is said to be a free-ﬁeld during that time. The detailed properties of such free-ﬁelds were ﬁrst studied in detail by Sherman, Devaney and Mandel [1], Sherman, Stamnes, Devaney and Lalor [2], and Devaney and Sherman [3] in the early 1970s. Such ﬁelds are of interest because they form the simplest type of wave phenomena encountered in both electromagnetics and optics. Without any loss of generality, attention is restricted here to the ﬁeld behavior for all nonnegative time t ≥ 0 during which it is assumed that there are no sources for the ﬁeld anywhere in space; viz., (r, t ≥ 0) = 0, J(r, t ≥ 0) = 0.

(8.1) (8.2)

Any sources that produced the ﬁeld were nonvanishing only during negative times t < 0. It is unnecessary to know what these sources were provided that the initial values D(r, 0) = D0 (r), B(r, 0) = B0 (r),

(8.3) (8.4)

for the electric displacement and magnetic induction ﬁeld vectors are known, ˙ 0 (r) and B ˙ 0 (r) where D0 (r) and B0 (r) as well as their ﬁrst time derivatives D are prescribed vector functions of position r ∈ R3 . That the initial values for the magnetic intensity vector H(r, t) and the conduction current density vector Jc (r, t) are not needed is a consequence of the fact that these initial values are not all independent, as is now shown.

8.1 Laplace–Fourier Representation of the Free Field The electromagnetic ﬁeld in a homogeneous, isotropic, locally linear, temporally dispersive medium occupying all of space with no externally supplied charge or current sources is described by the set of source-free Maxwell’s equations

388

8 Free Fields in Temporally Dispersive HILL Media

∇ · E(r, t) = 0,

% % % 1 % ∂B(r, t) % , ∇ × E(r, t) = − % %c% ∂t ∇ · B(r, t) = 0, % % % % % 1 % ∂D(r, t) % 4π % % % % ∇ × H(r, t) = % % +% % c % Jc (r, t), c ∂t with the constitutive (or material) relations t D(r, t) = D0 (r) +

ˆ(t − t )E(r, t )dt , 0 t µ ˆ−1 (t − t )B(r, t )dt , H(r, t) = H0 (r) + 0 t σ ˆ (t − t )E(r, t )dt . Jc (r, t) = Jc0 (r) +

(8.5) (8.6) (8.7) (8.8)

(8.9) (8.10) (8.11)

0

With use of the Laplace transform relation [see Eq. (C.13) in Appendix C] # $ L f˙(r, t) = −f (r, 0) − iωL {f (r, t)} , (8.12) the temporal Laplace transform of the source-free Maxwell’s equations given in Eqs. (8.5)–(8.8) yields the set of temporal frequency domain relations ˜ ω) = 0, ∇ · E(r, % % %1% ˜ ˜ % ∇ × E(r, ω) = % % c % iω B(r, ω) + B0 (r) , ˜ ω) = 0, ∇ · B(r, % % % % %1% % % ˜ ω) + D0 (r) + % 4π % J ˜ ˜ % iω D(r, ∇ × H(r, ω) = − % %c% % c % c (r, ω), where

˜ H(r, ω) =

∞

(8.13) (8.14) (8.15) (8.16)

H(r, t)eiωt dt

0

i ˜ ω), H0 (r) + µ−1 (ω)B(r, ω ∞ ˜ ω) = D(r, t)eiωt dt D(r, =

(8.17)

0

i ˜ ω), = D0 (r) + (ω)E(r, ω ˜ c (r, ω) = J =

∞

0

(8.18)

Jc (r, t)eiωt dt

i ˜ c (r, ω), Jc0 (r) + σ(ω)J ω

(8.19)

8.1 Laplace–Fourier Representation of the Free Field

389

with substitution from Eqs. (8.9)–(8.11), respectively. Substitution of these expressions in Eq. (8.16) then gives % % % 4π % c i ˜ ˜ % Jc0 (r) , ∇ × B(r, ω) = − 2 iω E(r, ω) − µ(ω) ∇ × H0 (r) − % % c % v (ω) ω (8.20) where

c2 µ(ω) c (ω) is the square of the complex velocity [cf. Eq. (6.27)]. Here v 2 (ω) ≡

(8.21)

σ(ω) (8.22) ω is the complex permittivity of the dispersive medium [cf. Eq. (5.29)]. The initial values H0 (r), D0 (r), and Jc0 (r) are related through Amp`ere’s law (8.8) at time t = 0 as % % % % % 4π % %1% % % ˙ % (8.23) ∇ × H0 (r) − % % Jc0 (r) = % % c % D0 (r), c

c (ω) ≡ (ω) + i4π

where

∂D(r, t) ˙ D0 (r) ≡ ∂t t=0 1 ˜ ω)dω. = (−iω) (ω)E(r, 2π C

With these results, Eq. (8.20) becomes i ˜ ω) + ˜ ω) = − c ˙ 0 (r) , iω E(r, ∇ × B(r, D v 2 (ω) ω c (ω)

(8.24)

(8.25)

which directly reduces to the form obtained when the medium is nonconducting and nondispersive.1 The three-dimensional spatial Fourier transform of the set of temporal frequency domain relations given in Eqs. (8.13)–(8.15) and (8.25) results in the set of spatiotemporal frequency domain equations [see Appendix C] ˜ ˜ k · E(k, ω) = 0, % % %1% ˜ ˜˜ ˜ ˜ % k × E(k, ω) = % % c % ω B(k, ω) − iB0 (k) , ˜˜ k · B(k, ω) = 0, c 1 ˜ ˜ ˜ ˜ ˙ ˜ ω E(k, ω) + k × B(k, ω) = − 2 D0 (k) . v (ω) ω c (ω) 1

(8.26) (8.27) (8.28) (8.29)

In the nonconducting, nondispersive medium case, c (ω) = , v = c/nr where √ nr = µ is the real-valued index of refraction of the homogeneous, isotropic, ˙ 0 (r) = −iω E0 (r). ˙ 0 (r) = E locally linear medium, and D

390

8 Free Fields in Temporally Dispersive HILL Media

The set of partial diﬀerential equations given in Eqs. (8.5)–(8.8) with the constitutive relations given in Eqs. (8.9)–(8.11) have thus been replaced by the set of algebraic vector relations given in Eqs. (8.26)–(8.29) which are ˜˜ ˜˜ now directly solved for the Laplace–Fourier spectra E(k, ω) and B(k, ω) of the electromagnetic ﬁeld vectors in terms of the initial values of these ﬁeld vectors and the appropriate medium response. The vector product of gives, after use of k with the relation in Eq. (8.27) ˜˜ ˜ ˜ ˜ ˜ the vector identity k × k × E(k, ω) = k · E(k, ω) k − k 2 E(k, ω) for the ˜˜ vector triple product and the orthogonality relation k · E(k, ω) = 0 given in Eq. (8.26), % % %1% ˜ ˜ ˜ ˜ 0 (k) , ˜ % ωk × B(k, ω) − ik × B −k 2 E(k, ω) = % %c% where k 2 ≡ k · k. With substitution from Eq. (8.29), the above expression becomes i 1 1 ˜˙ ˜ ˜ ˜ 0 (k), ˜ ˜ k×B ω 2 E(k, −k 2 E(k, ω) = − 2 ω) + D0 (k) − v (ω)

c (ω) c with solution ˜˙ (k)/ (ω) − v 2 (ω)/ck × B ˜ 0 (k) iD 0 c ˜ ˜ . E(k, ω) = i 2 2 2 ω − k v (ω)

(8.30)

In an analogous manner for the magnetic induction ﬁeld vector, the vector product of k gives, after use of the vector with the relation in Eq. (8.29) ˜˜ ˜ ˜ ˜ ˜ identity k× k × B(k, ω) for the vector triple ω) = k · B(k, ω) k−k 2 B(k, ˜ ˜ product and the orthogonality relation k · B(k, ω) = 0 given in Eq. (8.28), c 1 ˜ ˜ 2˜ ˜ ˜ ˙ ωk × E(k, ω) + k × D0 (k) . −k B(k, ω) = − 2 v (ω) ω c (ω) With substitution from Eq. (8.27), the above expression becomes ˜ ˜ −k 2 B(k, ω) = −

ω v 2 (ω)

˜ ˜ ˜ 0 (k) − ω B(k, ω) − iB

c ˜˙ (k), k×D 0 ω c (ω)v 2 (ω)

with solution ˜˙ (k) ˜ 0 (k) + i c/ω c (ω) k × D ω B 0 ˜ ˜ . B(k, ω) = i ω 2 − k 2 v 2 (ω)

(8.31)

The inverse Laplace–Fourier transform of the spatiotemporal spectral ﬁeld vectors given in Eqs. (8.30) and (8.31) then yields the Laplace–Fourier integral representation of the free-ﬁeld

8.1 Laplace–Fourier Representation of the Free Field

i E(r, t) = (2π)4

∞

dω

3

d k −∞

C

˜˙ v 2 (ω) i c (ω) D0 (k) − c k × ω 2 − k 2 v 2 (ω)

˜ 0 (k) B

391

ei(k·r−ωt) , (8.32)

B(r, t) =

i (2π)4

∞

dω

d3 k

˜˙ (k) ˜ 0 (k) + i c k × D ωB 0 ωc (ω) ω 2 − k 2 v 2 (ω)

−∞

C

ei(k·r−ωt) , (8.33)

valid for all t ≥ 0. Although it appears that the initial value problem for the source-free ﬁeld has been solved and that Eqs. (8.32)–(8.33) represent that solution, that is not yet the case. Speciﬁcally, it is not yet known as to whether this Laplace–Fourier integral representation forms a solution. All that is known is that if there is a solution with the properties required to ensure that the above analysis is valid (e.g., if the ﬁeld vectors and initial values are squareintegrable), then that solution must then be given by Eqs. (8.32)–(8.33), and hence, there is only one such solution. All that remains to be proven is the existence of the solution. 8.1.1 Plane Wave Expansion of the Free Field in a Nondispersive Nonconducting Medium In order to determine whether the relations given in Eqs. (8.32)–(8.33) represent the solution to the initial value problem stated in Eqs. (8.1)–(8.4) as well as to gain further insight into the properties of such free ﬁelds, the special case of a nondispersive nonconducting medium is considered in order that the ω-integrals in Eqs. (8.32)–(8.33) can be directly evaluated. For a nondispersive nonconducting medium, Eqs. (8.32)–(8.33) become i E(r, t) = (2π)4

B(r, t) =

i (2π)4

∞

dω C

3

d k −∞

v2 c k × ω2 − k2 v2

˜ 0 (k) − ωE

˜ 0 (k) B

ei(k·r−ωt) , (8.34)

∞

˜ 0 (k) ˜ 0 (k) + ck × E ωB dω d3 k ei(k·r−ωt) , ω2 − k2 v2 C −∞ (8.35)

for all t ≥ 0. Attention is then given to integrals of the form αω + β −iωt e dω ICj ≡ 2 − v2 k2 ω Cj

(8.36)

for j = 1, 2, 3, 4 for positive t, where v = c/nr is the real phase velocity with nr denoting the real index of refraction of the nondispersive nonconducting medium. Here α and β are functions that are both independent of ω, k is

392

8 Free Fields in Temporally Dispersive HILL Media

ﬁxed, and Cj are the separate portions of the closed contour C in the complex ω-plane depicted in Figure 8.1: C1 is the straight line contour parallel to the ω -axis and extending from −R + ia to R + ia in the upper-half plane, C2 is the straight line segment from R + ia to R, C3 is the semicircular arc ω = Reiθ with θ varying from 0 to −π in the clockwise direction, and C4 is the straight line segment from −R to −R + ia. The integral about the entire Ω''

C1

-R+ia C4 -R

R+ia C2

O

R

Θ

Ω'

R

C3 Complex Ω-Plane Fig. 8.1. Contour of integration C = C1 ∪ C2 ∪ C3 ∪ C4 in the complex ω-plane.

closed contour C is then given by the sum of the integrals over each of the contours Cj for j = 1, 2, 3, 4, so that IC =

4

ICj ,

(8.37)

j=1

where IC may also be evaluated by application of the residue theorem. If the integrals over C2 , C3 , and C4 vanish in the limit as R → ∞, then IC1 = IC with C1 → C in that limit, where C is the ω integration contour appearing in the integral representation given in Eqs. (8.34)–(8.35). On each contour Cj , the magnitude of the integral ICj is bounded in the manner αω + β −i(ω +iω )t αω + β ω t e ICj ≤ |dω| = |dω| , ω2 − v2 k2 ω2 − v2 k2 e Cj

where

Cj

8.1 Laplace–Fourier Representation of the Free Field

393

2 2 1/2 2 αω + β = |α| |ω| + |β| + 2{αβω} ω2 − v2 k2 |ω 2 |2 + v 4 k 4 − 2v 2 k 2 {ω 2 } 1/2 |α|2 ω 2 + ω 2 + |β|2 + 2 {αβ(ω + iω )} = . ω 4 + ω 4 + 2ω 2 ω 2 + v 4 k 4 − 2v 2 k 2 (ω 2 − ω 2 ) (8.38) Consider ﬁrst the contour integrals over the vertical line segments C2 and C4 . On either of these contours, 0 ≤ ω ≤ a and ω = ±R, where the upper sign choice is taken for C2 and the lower sign choice for C1 , so that eω

t

≤ eat ,

and [from Eq. (8.38)] 1/2 2 2 2 2 αω + β |α| R + |β| + ω + 2 {αβ(R + iω )} ω 2 − v 2 k 2 = R4 + ω 4 + 2R2 ω 2 + v 4 k 4 − 2v 2 k 2 (R2 − ω 2 )

O(R2 ) = O(R4 )

1/2 =O

1 R

as R → ∞. Consequently, as R becomes inﬁnitely large a αω + β ω t IC2,4 ≤ ω 2 − v 2 k 2 e dω 0 M at a M a at e e →0 ≤ dω = R R 0

(8.39)

as R → ∞ for all t ≥ 0. Consider next the integral over the semicircular contour C3 , along which ω = Re−iθ = R(cos θ − i sin θ) with θ = 0 → π, so that dω = −iRe−iθ dθ and |dω| = Rdθ. For 0 ≤ θ ≤ π/2, the inequality 2θ/π ≤ sin θ applies, so that eω

t

= e−Rt sin θ ≤ e−(2/π)Rtθ .

In addition, from Eq. (8.38) 2 2 1/2 2 αω + β = |α| R + |β| + 2 {αβR(cos θ − i sin θ)} ω2 − v2 k2 R4 + v 4 k 4 − 2v 2 k 2 R2 (cos2 θ − sin2 θ) 1/2 O(R2 ) 1 = = O 4 O(R ) R as R → ∞. Consequently, as R becomes inﬁnitely large

394

8 Free Fields in Temporally Dispersive HILL Media

αω + β −Rt sin θ Rdθ ω2 − v2 k2 e 0 M π/2 −Rt sin θ e Rdθ ≤2 R 0 π/2 Mπ 1 − e−Rt → 0 ≤ 2M e−(2/π)Rtθ dθ = Rt 0

π

|IC3 | ≤

as R → ∞ for all t > 0. For t = 0 it is noted that the integral expressions given in Eqs. (8.34)–(8.35) actually represent the quantities U (t)E(r, t) and U (t)B(r, t), where U (t) denotes the Heaviside unit step function. Because these quantities are discontinuous at t = 0, one must take the limit as t → 0+ . Because |IC3 | → 0 as R → ∞ for all t > 0, then |IC3 | → 0 as R → ∞ and t → 0+ . Thus (8.40) |IC3 | → 0 as R → ∞ for all t ≥ 0. With Eq. (8.37), these limiting results then show that αω + β −iωt lim IC1 = lim e dω, R→∞ R→∞ C ω 2 − v 2 k 2

(8.41)

where C = C1 ∪ C2 ∪ C3 ∪ C4 is the entire closed contour depicted in Figure 8.1. This contour integral over C is now evaluated using the residue theorem. The integrand has two simple poles at ω = ±vk with residues Residue αvk + β −ivkt e (ω = +vk) = , 2vk Residue αvk − β ivkt −αvk + β ivkt e e = . (ω = −vk) = −2vk −2vk

(8.42)

For √ ﬁxed k, these are the only poles enclosed by the contour C for all R > v 2 k 2 + δ 2 for arbitrarily small δ > 0. Application of the residue theorem to the right-hand side of Eq. (8.41) then yields β β lim IC1 = −iπ α + (8.43) e−ivkt + α − eivkt R→∞ vk vk

for all t ≥ 0. Application of the result given in Eq. (8.43) to the integral expressions given in Eqs. (8.34)–(8.35) then result in ∞ 1 ˜ 0 (k) − v ˆs × B ˜ 0 (k) ei(k·r−vkt) E(r, t) = E 16π 3 −∞ c

v ˜ 0 (k) + ˜ 0 (k) ei(k·r+vkt) d3 k, ˆs × B + E c (8.44)

8.1 Laplace–Fourier Representation of the Free Field

B(r, t) =

1 16π 3

∞

395

−∞

˜ 0 (k) + c ˆs × E ˜ 0 (k) ei(k·r−vkt) B v

c ˜ 0 (k) ei(k·r+vkt) d3 k, ˜ 0 (k) − ˆs × E + B v (8.45)

where ˆs ≡ k/k. Notice that the exponential factor ei(k·r−vkt) represents a plane wave traveling in the direction ˆs = k/k, while ei(k·r+vkt) represents a plane wave traveling in the opposite direction given by −ˆs. Upon combining similar terms in this pair of expressions for the free ﬁeld, one obtains ∞ v 1 ˜ 0 (k) cos (vkt) + i ˜ 0 (k) sin (vkt) eik·r d3 k, ˆs × B E(r, t) = E (2π)3 −∞ c

B(r, t) =

1 (2π)3

∞

−∞

(8.46)

˜ 0 (k) cos (vkt) − i c ˆs × E ˜ 0 (k) sin (vkt) eik·r d3 k, B v (8.47)

for t > 0. The free ﬁeld representation given in Eqs. (8.44)–(8.45) may also be simpliﬁed by noting that the integrand factors involving eivkt are the complex conjugates of the corresponding integrand factors involving e−ivkt . For example, let ∞ v ˜ ˜ ˆs × B0 (k) ei(k·r±vkt) d3 k, E± (r, t) ≡ E0 (k) ± c −∞ so that E(r, t) =

1 E+ (r, t) + E− (r, t) . 2(2π)3

Because the initial values E0 (r) and B0 (r) for the ﬁeld vectors are both real, ˜ ∗ (k) = then their spatial frequency spectra satisfy the symmetry relations E 0 ∗ ˜ ˜ ˜ E0 (−k) and B0 (k) = B0 (−k) for real-valued k, so that ∞ v ∗ ∗ ∗ ˜ ˜ ˆs × B0 (k) e−i(k·r+vkt) d3 k E+ (r, t) = E0 (k) + c −∞ ∞ v ˜ ˜ ˆs × B0 (−k) e−i(k·r+vkt) d3 k. = E0 (−k) + c −∞ Under the change of variable k = −k so that k = k and ˆs = −ˆs, the above expression becomes

396

8 Free Fields in Temporally Dispersive HILL Media

E∗+ (r, t)

v ˜ ˜ ˆs × B0 (k ) ei(k ·r−vk t) d3 k =− E0 (k ) − c ∞ ∞ v ˜ ˜ ˆs × B0 (k ) ei(k ·r−vk t) d3 k = E0 (k ) − c −∞ = E− (r, t),

−∞

so that 1 ∗ E− (r, t) + E− (r, t) 3 2(2π) 7 8 1 = E− (r, t) , 3 (2π)

E(r, t) = =

with an analogous expression holding for B(r, t). The integral representations appearing in Eqs. (8.44)–(8.45) may then be expressed as ∞ v 1 ˜ ˜ ˆs × B0 (k) ei(k·r−vkt) d3 k, (8.48) E(r, t) = E0 (k) − (2π)3 c −∞ ∞ 1 ˜ 0 (k) ei(k·r−vkt) d3 k, (8.49) ˜ 0 (k) + c ˆs × E B(r, t) = B (2π)3 v −∞ for t > 0. 8.1.2 Uniqueness of the Plane Wave Expansion of the Initial Value Problem ˜ 0 (k) and B ˜ 0 (k) have the necessary properties to ensure that Assume that E the spatial and temporal partial derivatives of the plane wave expansions given in Eqs. (8.48) and (8.49) for the free ﬁeld vectors E(r, t) and B(r, t) can be obtained by diﬀerentiating the integrands in these expressions. The divergence of Eq. (8.48) then gives ∞ v 1 ˜ ˜ ˆs × B0 (k) ei(k·r−vkt) d3 k, ∇ · E(r, t) = ik · E0 (k) − (2π)3 c −∞

∞ 1 i(k·r−vkt) 3 ˜ 0 (k)e i k·E d k . (8.50) = (2π)3 −∞ Hence, ∇ · E(r, t) will vanish in accordance with the Maxwell–Gauss relation ˜ 0 (k) = 0, which is satisﬁed if and (8.5) in a source-free HILL medium if k · E only if (8.51) ∇ · E0 (r) = 0. Therefore, E(r, t) will be divergenceless if E0 (r) is divergenceless. Similarly, the divergence of Eq. (8.49) gives

8.1 Laplace–Fourier Representation of the Free Field

397

∞ 1 ˜ 0 (k) ei(k·r−vkt) d3 k, ˜ 0 (k) − v ˆs × E ∇ · B(r, t) = ik · B (2π)3 c −∞

∞ 1 i(k·r−vkt) 3 ˜ 0 (k)e i k·B d k . (8.52) = (2π)3 −∞

Hence, ∇ · B(r, t) will vanish in accordance with the Maxwell–Gauss relation (8.7) in a source-free nonconducting, nondispersive HILL medium if ˜ 0 (k) = 0, which is satisﬁed if and only if k·B ∇ · B0 (r) = 0.

(8.53)

Therefore, B(r, t) will be divergenceless if B0 (r) is divergenceless. The curl of Eq. (8.48) gives ∞ 1 v ˜ ˜ ˆs × B0 (k) ei(k·r−vkt) d3 k, ik × E0 (k) − ∇ × E(r, t) = (2π)3 c −∞

∞ v 1 ˜ 0 (k) + ˜ 0 (k) ei(k·r−vkt) d3 k , k×E kB i = (2π)3 c −∞ (8.54) ˜ 0 (k) − B ˜ 0 (k) k · ˆs = −k B ˜ 0 (k). In ˜ 0 (k) = ˆs k · B because k × ˆs × B addition, the partial derivative of Eq. (8.49) with respect to time gives ∞ ∂B(r, t) 1 c ˜ ˜ ˆs × E0 (k) ei(k·r−vkt) d3 k (−ivk) B0 (k) + = ∂t (2π)3 v −∞

∞ v 1 i(k·r−vkt) 3 ˜ 0 (k) + ˜ 0 (k) e kB i d k k×E = −c (2π)3 c −∞ = −c∇ × E(r, t),

(8.55)

after comparison with Eq. (8.54). Hence, the Maxwell–Faraday relation (8.6) is satisﬁed if the initial value B0 (r) satisﬁes the relation given in Eq. (8.53). Similarly, the curl of Eq. (8.49) gives ∞ c 1 ˜ ˜ ˆs × E0 (k) ei(k·r−vkt) d3 k ∇ × B(r, t) = ik × B0 (k) + (2π)3 v −∞

∞ c 1 ˜ 0 (k) − ˜ 0 (k) ei(k·r−vkt) d3 k , i k×B = kE (2π)3 v −∞ (8.56) ˜ 0 (k) = ˆs k · E ˜ 0 (k) − E ˜ 0 (k) k · ˆs = −k E ˜ 0 (k). In because k × ˆs × E addition, the partial derivative of Eq. (8.48) with respect to time gives

398

8 Free Fields in Temporally Dispersive HILL Media

˜ 0 (k) ei(k·r−vkt) d3 k ˜ 0 (k) − v ˆs × B (−ivk) E c −∞

∞ c v2 1 i(k·r−vkt) 3 ˜ 0 (k) e ˜ 0 (k) − kE d k k×B = c (2π)3 v −∞

∂E(r, t) 1 = ∂t (2π)3

=

∞

c ∇ × B(r, t)

µ

(8.57)

√ after comparison with Eq. (8.56), where v = c/ µ. Hence, the Maxwell– Amp`ere relation (8.8) in a source-free nonconducting, nondispersive HILL medium is satisﬁed if the initial value E0 (r) satisﬁes the relation given in Eq. (8.51). Maxwell’s equations in a homogeneous, isotropic, locally linear (HILL) nonconducting, nondispersive medium with no charge or current sources are therefore satisﬁed by the plane wave expansions given in Eqs. (8.48)–(8.49), as well as in Eqs. (8.44)–(8.45) and (8.46)–(8.47), provided that the initial values E0 (r) and B0 (r) are solenoidal, satisfying the relations given in Eqs. (8.51) and (8.53), respectively. That is, the expressions given in Eqs. (8.48)– (8.49) for E(r, t) and B(r, t), respectively, represent a solution to the initial value problem if the initial values E0 (r) and B0 (r) are both divergenceless. It now remains to be determined whether these expressions for E(r, t) and B(r, t) satisfy the initial values. Because the integral expressions given in Eqs. (8.48)–(8.49), as well as in Eqs. (8.44)–(8.45) and (8.46)–(8.47), actually represent the vector ﬁeld quantitites U (t)E(r, t) and U (t)B(r, t), where U (t) is the Heaviside unit step function, they are then discontinuous at t = 0. For that reason, one must consider their limiting behavior as t → 0+ . In order to determine the limiting value for the electric ﬁeld vector, construct the diﬀerence vector IE (r, t) ≡ (2π)3 E(r, t) − E0 (r) . ˜ 0 (k) = F {E0 (r)}, one With substitution from Eq. (8.46) and the deﬁnition E ﬁnds that ∞ v ˜ ˜ ˆs × B0 (k) sin (vkt) eik·r d3 k |IE (r, t)| = E0 (k) cos (vkt) + i −∞ c ∞ ˜ 0 (k)eik·r d3 k − E −∞ ∞ v ik·r 3 ˜ ˜ ˆs × B0 (k) sin (vkt) e d k = E0 (k) cos (vkt) − 1 + i −∞ c ∞ v ˜ ˜ 0 (k) |sin (vkt)| d3 k . ≤ E0 (k) |cos (vkt) − 1| + ˆs × B c −∞ The right-hand side of the ﬁnal inequality in the above expression can be made as small as desired by choosing t > 0 suﬃciently small. Hence, IE (r, t)

8.2 Transformation to Spherical Coordinates in k-Space

399

tends to 0 as t → 0+ , and consequently lim E(r, t) = E0 (r)

t→0+

(8.58)

when E(r, t) is given by any of the expressions appearing in Eqs. (8.32), (8.44), (8.46), or (8.48). In a similar fashion, the diﬀerence vector IB (r, t) ≡ (2π)3 B(r, t) − B0 (r) is found to satisfy the inequality ∞ c ˜ ˜ 0 (k) |sin (vkt)| d3 k , |IB (r, t)| ≤ ˆs × E B0 (k) |cos (vkt) − 1| − v −∞ where the right-hand side can be made as small as desired by taking t > 0 suﬃciently small. Hence, IB (r, t) tends to 0 as t → 0+ , and consequently lim B(r, t) = B0 (r)

t→0+

(8.59)

when B(r, t) is given by any of the expressions appearing in Eqs. (8.33), (8.45), (8.47), or (8.49). The following result due to Devaney [4] has thus been established: If the initial ﬁeld values E0 (r) and B0 (r) satisfy the respective conditions given in Eqs. (8.51) and (8.53) of being divergenceless and have the properties required to ensure that the above analysis is valid, then there is one and only one solution for t > 0 to the source-free Maxwell’s equations in a nonconducting, nondispersive HILL medium ﬁlling all of space satisfying the initial conditions E(r, 0) = E0 (r) and B(r, 0) = B0 (r).

8.2 Transformation to Spherical Coordinates in k-Space The expressions given in Eqs. (8.48)–(8.49) for the free ﬁeld vectors E(r, t) and B(r, t) for t > 0 are each in the form of a superposition of monochromatic plane waves and are appropriately called plane wave expansions. As the variables of integration change, both the frequency and the direction of propagation of the plane waves change. In order to explicitly display this, a change of variables to spherical coordinates in k-space is made, as illustrated in Figure 8.2, where ω sin α cos β, c ω ky = sin α sin β, c ω kz = cos α, c

kx =

(8.60) (8.61) (8.62)

400

8 Free Fields in Temporally Dispersive HILL Media kz

k Α Ω/c

O Β

ky

kx

Fig. 8.2. Spherical coordinates in k-space.

with spherical polar coordinate variables that vary over the respective domains 0 ≤ ω < ∞, 0 ≤ α ≤ π, 0 ≤ β ≤ 2π. The Jacobian of this transformation then gives ∂(kx , ky , kz ) dαdβdω ∂(ω, α, β) ω 2 ω = sin α dαdβd , c c

d3 k =

where

(8.63)

ω . (8.64) c Under this transformation, the plane wave expansions given in Eqs. (8.48)– (8.49) become ∞ π 2π 1 2 ω dω sin αdα dβ E(r, t) = (2πc)3 0 0 0 v ˜ ˜ ˆs × B0 (k) ei(k·r−(v/c)ωt) , × E0 (k) − c (8.65) ∞ π 2π 1 ω 2 dω sin αdα dβ B(r, t) = (2πc)3 0 0 0 ˜ 0 (k) + c ˆs × E ˜ 0 (k) ei(k·r−(v/c)ωt) , × B v (8.66) k=

8.2 Transformation to Spherical Coordinates in k-Space

401

for all t > 0. This pair of expressions for the free ﬁeld vectors are plane wave expansions with the integration taken over the angular frequency ω and over the propagation direction speciﬁed by the polar angles α and β explicitly displayed. In terms of the element of solid angle dΩ(k) in the direction of k, where dΩ(k) ≡ dΩ = sin α dαdβ, (8.67) these expressions can be further simpliﬁed as ∞ 1 2 ˜ 0 (k) ei(k·r−(v/c)ωt) , ˜ 0 (k) − v ˆs × B E(r, t) = ω dω dΩ E (2πc)3 c 0 4π (8.68) ∞ c 1 ˜ 0 (k) ei(k·r−(v/c)ωt) , ˜ 0 (k) + ˆs × E ω 2 dω dΩ B B(r, t) = (2πc)3 v 4π 0 (8.69) for all t > 0. Finally, if one introduces the new ﬁeld vector 1 ˜ 0 (k) , ˜ 0 (k) − v ˆs × B ˜ E(k) ≡ E (2π)3 c

(8.70)

in which case c c 1 ˜ ˜ ˜ ˆs × E(k) = ˆs × E0 (k) − ˆs × ˆs × B0 (k) v (2π)3 v 1 c ˜ ˜ ˆs × E0 (k) , = B0 (k) + (2π)3 v

(8.71)

then the expressions given in Eqs. (8.48)–(8.49) and (8.68)–(8.69) become ∞ i(k·r−vkt) 3 ˜ E(r, t) = E(k)e d k −∞

∞ 1 i(k·r−(v/c)ωt) ˜ = 3 ω 2 dω dΩ E(k)e , c 0 4π ∞ c ˜ ˆs × E(k) B(r, t) = ei(k·r−vkt) d3 k v −∞ ∞ c ˜ ω 2 dω dΩ ˆs × E(k) ei(k·r−(v/c)ωt) , = 3 c v 0 4π

(8.72)

(8.73)

for all t > 0. 8.2.1 Plane Wave Representations and Mode Expansions The preceding relations express the solution to the initial value problem in a form that is particularly convenient for a given mode expansion. A mode

402

8 Free Fields in Temporally Dispersive HILL Media

expansion is a superposition of particular solutions (called modes) to the diﬀerential equations governing a physical problem. In the present case the governing equations are Maxwell’s equations in a nonconducting, nondispersive HILL medium with no sources, and the modes are the real parts of the complex, monochromatic electromagnetic plane waves i(k·r−vkt) ˜ t, k) = E(k)e ˜ , E(r, i(k·r−vkt) ˜ t, k) = c ˆs × E(k)e ˜ . B(r, v

These modes are especially convenient to work with because the dynamics of monochromatic plane waves are particularly simple. Mode expansions are convenient to work with in the study of the interaction of an electromagnetic wave ﬁeld with any linear system [5]. The ﬁeld incident on the system can always be represented as a linear superposition of the modes of the system. The manner in which each mode interacts with the system can then be investigated independently of the other modes. The ﬁnal result is then obtained by superimposing the results obtained for each mode separately. One factor that complicates the application of the plane wave mode expansion given in Eqs. (8.72)–(8.73) is that each integrand includes plane waves that are traveling in all directions. As an illustration, let the initial ﬁeld vectors E0 (r) and B0 (r) be conﬁned to the region z < 0 at t = 0, both ﬁeld vectors identically vanishing throughout the half-space z > 0. The wave ﬁeld is then incident upon some linear system (e.g., a telescope) in the region z > 0 for t > 0. In order to obtain an expression for the output from the linear system for t > 0, one can express the incident wave ﬁeld by the plane wave expansion given in Eqs. (8.72)–(8.73), pass each plane wave component through the linear system, and then superimpose the results to obtain the total output from the system. However, in this approach, one must include plane wave components that are traveling through the linear system towards the plane z = 0 (backwards through the telescope) as well as components that are traveling away from that plane (forwards through the telescope). Another feature of the plane wave expansion given in Eqs. (8.72)–(8.73) worth noting is that the free ﬁeld vectors E(r, t) and B(r, t) are expressed as the real parts of the complex wave ﬁelds Ec (r, t) and Bc (r, t), respectively, which are themselves expressed as ∞ ˜ c (r, ω)e−iωt dω, Ec (r, t) = (8.74) E 0 ∞ ˜ c (r, ω)e−iωt dω. (8.75) Bc (r, t) = B 0

˜ c (r, ω) ˜ c (r, ω) and B Functions of this form are called analytic signals [6]. If E are suﬃciently well behaved, then Ec (r, t +it ) and Bc (r, t +it ) are analytic functions of complex t = t + it for t < 0.

8.2 Transformation to Spherical Coordinates in k-Space

403

8.2.2 Polar Coordinate Axis Along the Direction of Observation The preceding plane wave representations of the free ﬁeld are expressed in spherical polar coordinates with the polar axis chosen to be along the z-axis of the original Cartesian coordinate system. Somewhat simpler expressions for the free ﬁeld vectors E(r, t) and B(r, t) can be obtained by expressing the components kx , ky , kz in terms of a new spherical coordinate system whose polar axis is along the direction of the observation point described by the position vector r. The new variables of integration are then given by ω sin α cos β , c ω ky = sin α sin β , c ω kz = cos α , c

kx =

(8.76) (8.77) (8.78)

which vary over the domain 0 ≤ ω < ∞, 0 ≤ α ≤ π, 0 ≤ β ≤ 2π, where kz is the component of k along the ﬁeld observation point position vector r, so that (8.79) kz = k · r = k cos α . The (kx , ky , kz ) coordinate system can be obtained from the (kx , ky , kz ) coordinate system through a sequence of three successive rotations through the Euler angles φ, θ, and ψ as follows [7]. The initial (kx , ky , kz ) coordinate system is ﬁrst rotated through the angle φ counterclockwise about the kz -axis, resulting in the intermediate (ξ, η, ζ) coordinate system. This intermediate coordinate system is then rotated counterclockwise through the angle θ about the the ξ-axis, producing the new intermediate (ξ , η , ζ ) coordinate system. Finally, this intermediate coordinate system is rotated counterclockwise through the angle ψ about the ζ -axis, resulting in the desired (kx , ky , kz ) coordinate system. Because only the kz -axis is speciﬁed by the position vector r, the kx - and ky -axes are completely arbitrary as is the angle ψ. The transformation matrix from the (kx , ky , kz ) coordinate system to the (kx , ky , kz ) system is then given by ⎛ ⎞⎛ ⎞⎛ ⎞ cos ψ sin ψ 0 1 0 0 cos φ sin φ 0 A = ⎝ − sin ψ cos ψ 0 ⎠ ⎝ 0 cos θ sin θ ⎠ ⎝ − sin φ cos φ 0 ⎠ 0 0 1 0 − sin θ cos θ 0 0 1 (cos φ cos ψ − cos θ sin φ sin ψ) (sin φ cos ψ + cos θ cos φ sin ψ) (sin θ sin ψ) = (− cos φ sin ψ − cos θ sin φ cos ψ) (− sin φ sin ψ + cos θ cos φ cos ψ) (sin θ cos ψ) , sin φ sin θ

− cos φ sin θ

cos θ

(8.80) so that

⎞ ⎛ ⎞ kx kx ⎝ ky ⎠ = A ⎝ ky ⎠ , kz kz ⎛

(8.81)

404

8 Free Fields in Temporally Dispersive HILL Media

which may be written in matrix notation as k = Ak. Because the transformation matrix A is orthogonal, the inverse transformation k = A−1 k from the (kx , ky , kz ) coordinate system to the (kx , ky , kz ) system is given by the transpose AT of the transformation matrix A as A−1 = AT (cos φ cos ψ − cos θ sin φ sin ψ) (− cos φ sin ψ − cos θ sin φ cos ψ) (sin φ sin θ) = (sin φ cos ψ + cos θ cos φ sin ψ) (− sin φ sin ψ + cos θ cos φ cos ψ) (− cos φ sin θ) , sin θ sin ψ

sin θ cos ψ

cos θ

(8.82) Upon expressing the components kx , ky , kz in terms of the new spherical polar coordinates ω/c, α , β which have their polar axis along the direction of the position vector r of the ﬁeld observation point, where ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ kx sin α cos β kx ω ⎝ ky ⎠ = A−1 ⎝ ky ⎠ = A−1 ⎝ sin α sin β ⎠ , (8.83) c kz cos α kz it is found that ω kx = sin α cos φ cos (β + ψ) c − sin α cos θ sin φ cos (β + ψ) + cos α sin θ sin φ , (8.84) ω sin α sin φ cos (β + ψ) ky = c + sin α cos θ cos φ sin (β + ψ) − cos α sin θ cos φ , (8.85) ω kz = sin α sin θ sin (β + ψ) + cos θ cos α , (8.86) c where the real angle ψ is completely arbitrary. Under this transformation, the plane wave expansions given in Eqs. (8.65)–(8.66) become 2π ∞ π 1 2 E(r, t) = ω dω sin α dα dβ (2πc)3 0 0 0 v ˜ ˜ ˆs × B0 (k) ei(ω/c)(r cos α −vt) , × E0 (k) − c (8.87) ∞ π 2π 1 ω 2 dω sin α dα dβ B(r, t) = (2πc)3 0 0 0 c ˜ ˜ ˆ × B0 (k) + s × E0 (k) ei(ω/c)(r cos α −vt) , v (8.88) for all t > 0, where the angular dependence of the quantity k · r has been explicitly displayed. This representation of the plane wave expansion of the

8.3 Propagation of the Free Electromagnetic Field

405

free ﬁeld has the same properties as that for the represenation given in Eqs. (8.65)–(8.66). Notice that the solution to the initial value problem can be expressed in other ways that exhibit diﬀerent properties of the free ﬁeld; perhaps the most important form is Poisson’s solution [8] which is equivalent to the plane wave expansions considered here. The extension of this analysis of the free ﬁeld to a dispersive HILL medium remains to be accomplished. Because the residue analysis presented in §8.1.1 requires that the analytic properties of the integrand be known throughout the complex ω-plane, a speciﬁc model for the material dispersion must then be employed. Diﬀerent models will then result in diﬀerent results and it is essential that the model be causal in order that the results have true physical meaning.

8.3 Propagation of the Free Electromagnetic Field The plane wave representation of the free ﬁeld given in Eqs. (8.72)–(8.73) may be rewritten as

∞ 1 i(k·r−(v/c)ωt) ˜ dΩ ω 2 dω E(k)e E(r, t) = 3 2c 4π 0

∞ 2 ∗ −i(k·r−(v/c)ωt) ˜ , (8.89) + dΩ ω dω E (k)e B(r, t) =

c 2c3 v

4π

0

∞

dΩ 0

4π

+

˜ ω 2 dω ˆs × E(k) ei(k·r−(v/c)ωt) ∞

dΩ 0

4π

∗ −i(k·r−(v/c)ωt) ˜ ω dω ˆs × E (k) e , (8.90) 2

for all t > 0. Because the initial values E0 (r) and B0 (r) are real-valued, then their spatial Fourier transforms satisfy the respective symmetry relations ˜ ∗ (k) and B ˜ 0 (−k) = B ˜ ∗ (k), so that E(−k) ˜ 0 (−k) = E ˜ = E˜∗ (k). With this E 0 0 result, Eqs. (8.89)–(8.90) become

∞ 1 i(k·r−(v/c)ωt) ˜ E(r, t) = 3 dΩ ω 2 dω E(k)e 2c 0 4π

∞ 2 −i(k·r−(v/c)ωt) ˜ , + dΩ ω dω E(−k)e B(r, t) =

c 2c3 v

4π

0

∞

dΩ 4π

+

0

dΩ 4π

0

˜ ω 2 dω ˆs × E(k) ei(k·r−(v/c)ωt) ∞

−i(k·r−(v/c)ωt) ˜ ω dω ˆs × E(−k) e . 2

406

8 Free Fields in Temporally Dispersive HILL Media

With the change of variable ω → −ω in the second term of both of these expressions, noting that k → −k under this transformation, one ﬁnally obtains ∞ 1 i(k·r−(v/c)ωt) ˜ E(r, t) = 3 dΩ ω 2 dω E(k)e , (8.91) 2c 4π −∞ ∞ c ˜ dΩ ω 2 dω ˆs × E(k) ei(k·r−(v/c)ωt) , (8.92) B(r, t) = 3 2c v 4π −∞ for all t > 0. The spatiotemporal behavior described by this representation is now considered for several special cases. 8.3.1 Initial Field Values Conﬁned Within a Sphere of Radius R Let the initial values E0 (r) and E0 (r) be conﬁned to the interior of a sphere of radius R > 0 centered at the origin, so that E0 (r) = B0 (r) = 0; and

˜ 0 (k) = E

|r|
˜ 0 (k) = B

|r|
∀r |r| > R,

(8.93)

E0 (r)e−ik·r d3 r,

(8.94)

B0 (r)e−ik·r d3 r.

(8.95)

If E0 (r) and B0 (r) are bounded for all r such that |r| < R, then ˜ 0 (k) ∂E ∂ −ik·r 3 e = d r, E0 (r) ∂ω ∂ω |r|
(8.96) (8.97)

˜ 0 (k) and B ˜ 0 (k) of the initial ﬁeld vectors, and consequently The spectra E ˜ the spectral function E(k), are then entire functions of ω because derivatives ˜ 0 (k) with respect to ω of all orders exist. Consider then the behavior of E ˜ and B0 (k) on the circle |ω| = constant in the complex ω-plane. First, the inequalities ˜ E (k) |E0 (r)| e−i(ω/c)ˆs·r d3 r , 0 ≤ |r|
directly follow from Eqs. (8.94)–(8.95). Because

8.3 Propagation of the Free Electromagnetic Field

407

−i(ω/c)ˆs·r −i((ω +iω )/c)ˆs·r e = e = e(ω

/c)ˆ s·r

,

then these inequalities become ˜ |E0 (r)| e(ω /c)ˆs·r d3 r , E0 (k) ≤ |r|
In addition, one has that max|r|

≤ M1 e|ω |R/c , ˜ |B0 (r)| e|ω |R/c d3 r B0 (k) ≤

(8.98)

|r|
≤ M2 e|ω

|R/c

,

(8.99)

because E0 (r) and B0 (r) are bounded for all r such that |r| < R. Therefore, if the initial values E0 (r) and B0 (r) are bounded for all r such ˜ 0 (k) that |r| < R and identically vanish for all r such that |r| > R, then both E |ω |R/c ˜ , where M = sup(M1 , M2 ).2 and B0 (k) can grow no faster than M e Consequently, 1 ˜ v ˜ ˜ ˆ = E(k) E (k) − (k) s × B 0 0 3 (2π) c v 1 ˜ ˜ + B ≤ (k) (k) E 0 0 (2π)3 c v 1 |ω |R/c ≤ M + M 1 c 2 e (2π)3 ≤ M e|ω

|R/c

,

so that the integrands in Eqs. (8.91)–(8.92) satisfy the inequality 2˜ i(k·r−(v/c)ωt) ˜ ω E(k)ei(k·r−(v/c)ωt) = ω 2ˆs × E(k)e 2 ˜ ei(ω/c)(ˆs·r−vt) = ω E(k) ≤ ω 2 M e|ω |(R/c) e−(ω /c)(ˆs·r−vt) 2

This result is part of the Paley–Wiener theorem.

(8.100)

(8.101)

408

8 Free Fields in Temporally Dispersive HILL Media

in the complex ω-plane. Consequently, in order for the integrands in the plane wave expansions given in Eqs. (8.91)–(8.92) to possess exponential attenuation, the inequality ˆs · r > R + vt (8.102) must be satisﬁed when ω > 0, whereas the inequality ˆs · r < vt − R

(8.103)

must be satisﬁed when ω < 0. The integrands appearing in the plane wave expansions given in Eqs. (8.91)–(8.92) possess exponential decay for those values of ˆs = k/k, r, and t satisfying the inequalities given in Eqs. (8.102) and (8.103). By Jordan’s ˜ ˜ lemma, if E0 (k) and B0 (k) go to zero faster than 1/|ω|2 as ω → ∞, then the integral over ω in Eqs. (8.91) and (8.92) vanishes for these values of ˆs, r, and t. That is, all modes propagating in the direction ˆs appearing in the plane wave expansion for the free ﬁeld vectors E(r, t) and B(r, t) will not contribute to the free ﬁeld at the point P with position vector r if the conditions speciﬁed in Eqs. (8.102) and (8.103) are satisﬁed. A geometrical construction is given in Figures 8.3 and 8.4 that illustrates the directions that the modes either do or do not contribute to the ﬁeld at the point P with position vector r from the origin O as a function of time. Figure 8.3 depicts the construction of three successive cones at the instants t0 = 0, t1 > t0 , and t2 > t1 with generators ˆs0 , ˆs1 , and ˆs2 , respectively, such that the inequality given in Eq. (8.102) is satisﬁed in the interior of a given cone at that instant of time. Any plane wave with direction ˆs lying inside the ﬁrst cone with generator ˆs0 satisﬁes the inequality ˆs · r > R, any plane wave with direction ˆs lying inside the second cone with generator ˆs1 satisﬁes the inequality ˆs · r > R + vt1 , and any plane wave with direction ˆs lying inside the third cone with generator ˆs1 satisﬁes the inequality ˆs · r > R + vt2 . These plane wave modes then do not contribute to the free ﬁeld at P at these instants of time. Figure 8.4 depicts the same type of construction at the same instants of time for the inequality given in Eq. (8.103). Notice that the time instant t1 has been chosen to be given by t1 = R/v when the cone becomes a plane. For earlier times these cones open to the left and for later times they open to the right in the ﬁgure. Taken together, this pair of diagrams shows that, at each instant of time t ≥ t0 = 0, the only plane wave modes that contribute to the free ﬁeld at the point P are those whose propagation directions ˆs lie in the space between the respective cones described by the pair of inequalities in Eqs. (8.102)–(8.103). It is then seen that when t ≥ R/v, only those plane wave modes whose direction ˆs has a positive component along the position vector r to the observation point P (i.e., such that ˆs · r > 0) contribute to the ﬁeld at P , and that when t≥

|r| + R , v

(8.104)

8.3 Propagation of the Free Electromagnetic Field

409

no plane wave modes contribute and the ﬁeld at P vanishes.

V

>R sr sr>R+vt1 s r>R+vt2 V

V

V

V

V

R+vt1

s0 r

P

V

s0 s 2 O R

s2

R+vt2

t1>t0

t2>t1

t0=0 The cone closes up as t increases. Fig. 8.3. Wave vector regions satisfying the inequality in Eq. (8.102).

V

V

V

V

V

V

R-vt2

V

r

V

s1

s1 s 2 O R s0

s2 P

s0

The cone opens up to a plane t >t and then t0=0 t =R/v 2 1 closes up 1 as t increases. Fig. 8.4. Wave vector regions satisfying the inequality in Eq. (8.103).

410

8 Free Fields in Temporally Dispersive HILL Media

8.3.2 Initial Field Values Conﬁned Inside a Closed Convex Surface Let the initial ﬁeld values E0 (r) and B0 (r) identically vanish outside a simply connected region V with convex surface S, so that ˜ 0 (k) = E0 (r)e−ik·r d3 r, (8.105) E V ˜ 0 (k) = B0 (r)e−ik·r d3 r, (8.106) B V

and let E0 (r) and B0 (r) both be continuous in V and have continuous partial derivatives up through the third order in V . Let the surface S be described by the relation S = {(x, y, z)| F (x, y, z) = 0} , (8.107) where the equation F (x, y, z) = 0 can be solved for x, y, and z as Z+ (x, y) z= , Z− (x, y) Y+ (x, z) , y= Y− (x, z) X+ (y, z) , x= X− (y, z)

(8.108) (8.109) (8.110)

with Y+ (x) ≡ maximum value of Y+ (x, z) obtained when z takes on all possible values on S with ﬁxed x on S, Y− (x) ≡ minimum value of Y− (x, z) obtained when z takes on all possible values on S with ﬁxed x on S, X+ ≡ maximum value of X+ (y, z) obtained when y and z take on all possible values on S, and X− ≡ minimum value of X− (y, z) obtained when y and z take on all possible values on S. With these identiﬁcations, Eqs. (8.105)–(8.106) can be written as X+ Y+ (x) Z+ (x,y) ˜ 0 (k) = dx dy dz E0 (r)e−ik·r , (8.111) E Y− (x)

X−

˜ 0 (k) = B

X+

Z− (x,y)

Y+ (x)

dx Y− (x)

X−

Z+ (x,y)

dy Z− (x,y)

dz B0 (r)e−ik·r .

(8.112)

The multiple integration appearing in Eq. (8.111) may be evaluated in a stepwise manner as follows: First, the innermost integral is evaluated as I1 (x, y) =

Z+ (x,y)

Z− (x,y)

i = kz

E0 (r)e−ikz z dz −ikz z

E0 (r)e

Z+ (x,y) Z− (x,y)

−

Z+ (x,y)

Z− (x,y)

∂E0 (r) −ikz z e dz ∂z

,

8.3 Propagation of the Free Electromagnetic Field

411

and the second integral may be evaluated in the same manner as Y+ (x) I1 (x, y)e−iky y dy I2 (x) = Y− (x)

i = ky

−iky y

I1 (x, y)e

Y+ (x) Y− (x)

−

Y+ (x)

Y− (x)

∂I1 (x, y) −iky y e dy ∂y

.

In the same manner, the outermost integral in Eq. (8.111) may be evaluated as X+ ˜ 0 (k) = I2 (x)e−ikx x dx E X−

i = kx = −

−ikx x

I2 (x)e

i kx ky kz

X+ X−

X+

−

X−

z=Z− (x,y)

− −

Y+ (x)

Y− (x)

X−

Z− (x,y)

− −

Y+ (x)

Y− (x)

Z+ (x,y)

∂E0 (r) −ik·r e ∂x

∂E0 (r) −ik·r e dz ∂z

∂E0 (r) −ik·r e ∂y −

X+

Z+ (x,y)

Z− (x,y)

i kx ky kz

z=Z+ (x,y) −ik·r E0 (r)e

+

∂I2 (x) −ikx x e dx ∂x

z=Z+ (x,y)

y=Y− (x)

z=Z− (x,y)

x=X+ ∂ 2 E0 (r) −ik·r dz dy e ∂y∂z

z=Z+ (x,y)

Z+ (x,y)

Z− (x,y)

y=Y+ (x)

z=Z− (x,y)

∂ 2 E0 (r) −ik·r e dz ∂x∂z z=Z+ (x,y)

x=X−

y=Y+ (x) y=Y− (x)

∂ 2 E0 (r) −ik·r e ∂x∂y z=Z− (x,y) Z+ (x,y) 3 ∂ E0 (r) −ik·r − dz dy dx, e Z− (x,y) ∂x∂y∂z (8.113)

after substitution from the preceding two integrations for I2 (x) and I1 (x, y). ˜ 0 (k) of An analogous expression holds for the spatial frequency spectrum B the magnetic ﬁeld. Because kx = (ω/c) sin α cos β, ky = (ω/c) sin α sin β, and kz = (ω/c) cos α, and because ˆs ≡ k/k = (c/ω)k, then the inequality

412

8 Free Fields in Temporally Dispersive HILL Media

−ik·r −i(ω/c)ˆs·r e = e = e(ω /c)ˆs·r ≤ e|ω |R/c

(8.114)

is satisﬁed for all r such that |r| ≤ R. For each complex vector component of ˜ 0 (k), the inequality E 1/2 ˜ ˜0j (k)E ˜ ∗ (k) ≤ E0j (k) = E 0j

Ke|ω |R/c (8.115) 3 (ω/c) sin2 α cos α sin β cos β

is satisﬁed, where j = x, y, z. Then, for α and β both bounded away from both 0 and π/2, one has the inequality e|ω |R/c ˜ , E0j (k) ≤ K1 ω3

(8.116)

and similarly

e|ω |R/c ˜ , (8.117) B0j (k) ≤ K2 ω3 where K1 and K2 are positive constants. However, the integrals appearing in Eqs. (8.91) and (8.92) are both taken over 4π steradians. The apparent “divergence” in the inequality when either α or β is equal to either 0 or π/2 is clearly dependent upon the choice of polar axes. That is, this “divergence” exists for each particular set of coordinate axes so that the overlap produced for two carefully selected coordinate axes eﬀectively eliminates the problem because the solution must be independent of the choice of axes. Consequently, the inequalities appearing in Eqs. (8.116)–(8.117) hold over 4π steradians. It then follows that 1 ˜ v ˜ ˜ ˆs × B0 (k) E0 (k) − E(k) = 3 (2π) c |v| ˜ 1 ˜ ≤ B0 (k) E0 (k) + (2π)3 c 1 1 |ω |R/c |v| ≤ K2 e K1 + 3 (2π) c |ω|3 M |ω |R/c ≤ e , (8.118) |ω|3

where M is another positive constant. The integrands appearing in Eqs. (8.91) and (8.92) then satisfy the inequality 2 ˜ i(k·r−(v/c)ωt) i(k·r−(v/c)ωt) ˜ ω ˆs × E(k)e = ω 2 E(k)e 2 ˜ ei(ω/c)(ˆs·r−vt) = ω E(k) ≤

M |ω |R/c −(ω /c)(ˆs·r−vt) e e , |ω|

(8.119)

8.3 Propagation of the Free Electromagnetic Field

413

in the complex ω-plane. One then obtains the same conditions, given in Eqs. (8.102) and (8.103), for exponential decay in the free ﬁeld. The results of the previous subsection (§8.3.2) then apply here where R now represents the radius of the smallest sphere that completely encloses the simply connected region V . 8.3.3 Propagation of the Free Electromagnetic Wave Field Consider the earlier form of the plane wave expansion of the free ﬁeld given in Eqs. (8.46)–(8.47) as ∞ 1 v ˜ 0 (k) cos (vkt) + i ˜ 0 (k) sin (vkt) eik·r d3 k, ˆs × B E(r, t) = E (2π)3 −∞ c

B(r, t) =

1 (2π)3

∞

−∞

(8.120)

˜ 0 (k) cos (vkt) − i c ˆs × E ˜ 0 (k) sin (vkt) eik·r d3 k, B v (8.121)

for t > 0. From the identity sin (AB) 1 = AB 4π

eiA·B dΩ(A), 4π

where dΩ(A) denotes the diﬀerential element of solid angle about the direction of the vector A, one ﬁnds that sin (vkt) 1 = eik·vt dΩ(v), vkt 4π 4π and d sin (vkt) cos (vkt) = dt vk 1 d ik·vt = t e dΩ(v) . 4π dt 4π Substitution of these results into the plane wave expansions given in Eqs. (8.120)–(8.121) then gives

∞ 1 3 ˜ 0 (k) d teik·(r+vt) dΩ(v) d k E(r, t) = E 32π 4 4π dt −∞

v2 ik·(r+vt) ˜ 0 (k)e +i tk × B c

414

8 Free Fields in Temporally Dispersive HILL Media

d v2 dΩ(v) tE0 (r + vt) + t∇ × B0 (r + vt) dt c 4π

∂ 1 v2 = tE0 (r + vt)dΩ(v) + ∇ × B0 (r + vt)dΩ(v) , t 4π ∂t 4π c 4π 1 = 4π

B(r, t) =

1 32π 4

(8.122)

˜ 0 (k) d teik·(r+vt) dΩ(v) d3 k B dt −∞ 4π ∞

˜ 0 (k)eik·(r+vt) − ictk × E

d tB0 (r + vt) − ct∇ × E0 (r + vt) dΩ(v) dt 4π

∂ 1 = tB0 (r + vt)dΩ(v) − ct ∇ × E0 (r + vt)dΩ(v) , 4π ∂t 4π 4π

1 = 4π

(8.123) for t > 0.

vt

P r' r

Surface & : |r - r'| = vt = constant

O

Fig. 8.5. The spherical surface |r − r | = vt = constant about the ﬁeld observation point P .

Consider now rewriting this pair of integral representations as surface integrals over the spherical surface Σ illustrated in Figure 8.5, given by |r − r | = vt = constant. A diﬀerential element of area on the surface Σ is given by dA = |r − r |2 dΩ(v) = (vt)2 dΩ(v), so that

8.3 Propagation of the Free Electromagnetic Field

dΩ(v) =

dA . vt|r − r |

415

(8.124)

With these identiﬁcations, the relations given in Eqs. (8.122)–(8.123) become

1 ∂ E0 (r ) ∇ × B0 (r ) v2 E(r, t) = dA + dA , (8.125) 4πv ∂t Σ |r − r | c Σ |r − r |

∂ B0 (r ) ∇ × E0 (r ) 1 B(r, t) = dA − c dA , (8.126) 4πv ∂t Σ |r − r | |r − r | Σ for t > 0, where Σ is the surface |r − r | = vt. In the scalar case, this is just Poisson’s solution to the initial value problem.

vt2 P V

vt1 S

&1 &2

Fig. 8.6. Spherical regions about the ﬁeld observation point P that determine the time instants t1 and t2 between which the initial values conﬁned to within V contribute to the free ﬁeld at P .

Let the initial ﬁeld values E0 (r) and B0 (r) as well as their spatial derivatives vanish outside of a simply connected region V with closed outer surface S. Because the initial ﬁeld vectors E0 (r ) and B0 (r ) and their spatial derivatives are then identically zero when the point r = r + vt is outside V , the free ﬁeld vectors E(r, t) and B(r, t) also vanish when this occurs. That is, given an observation point P that is exterior to the initial value region V , the free ﬁeld will be zero at P until the instant of time t1 when the sphere Σ1 of radius vt1 drawn about P as center ﬁrst intersects the region V , as depicted in Figure 8.6. Furthermore, the free ﬁeld at P will return to zero at

416

8 Free Fields in Temporally Dispersive HILL Media

S P (a) t = 0

P i' S S'

vt1 P vt 1

Po'

(b) t = t1 > 0

Pi'' vt2 S'' S P vt2

S''

Po'' (c) t = t2 > t1

Fig. 8.7. Inward and outward propagation of the geometric “wavefront” from the surface S. The shaded regions indicate the space where the free ﬁeld is nonvanishing and the unshaded regions indicate the space where the free ﬁeld identically vanishes.

8.3 Propagation of the Free Electromagnetic Field

417

a later time t2 when the sphere Σ2 of radius vt2 with center at P completely encloses the region V for the ﬁrst time, as also depicted in Figure 8.6. The only time when the free ﬁeld at P is nonzero is when the sphere Σ with radius vt centered at P intersects the surface S that surrounds the region containing the initial ﬁeld values. As a consequence, the surface S of the region V can be considered to be a wavefront that propagates both “inward” and “outward” according to the laws of geometrical optics from the time t = 0 through positive times t, as illustrated in the sequence of diagrams in Figure 8.7. At t = 0 the ﬁeld vectors E(r, t) and B(r, t) are equal to their respective initial values E0 (r) and B0 (r), both of which identically vanish outside the simply connected region V with closed outer surface S, as illustrated by the shaded region in part (a) of Figure 8.7. At a slightly later time t = t1 > 0, the “wavefront” surface S will have propagated both inward and outward through a distance vt1 , where √ v = c/ µ is the constant phase velocity in the nondispersive HILL medium. If the time t1 is short enough that there is no overlap between the inward propagated surface and any other points of the initial surface S, as depicted in part (b) of the ﬁgure, then the free ﬁeld vectors will identically vanish everywhere outside the region V that is enclosed by the outward propagated surface S . At a later time t2 > t1 such that the inward propagating “wavefront” surface overlaps points on the initial surface S, as illustrated in part (c) of the ﬁgure, the free ﬁeld vectors will then vanish everywhere exterior to the annular region V with closed surface S . The outer portion of S ˆ propagates with velocity v in the direction of the outward normal vector n to the initial surface S and consequently retains the shape of that initial surface. The inner portion of S propagates with velocity v in the direction of the inward normal −ˆ n to S so that its shape will be the inversion of S, as illustrated.

418

8 Free Fields in Temporally Dispersive HILL Media

References 1. G. C. Sherman, A. J. Devaney, and L. Mandel, “Plane-wave expansions of the optical ﬁeld,” Opt. Commun., vol. 6, pp. 115–118, 1972. ´ Lalor, “Contribution of 2. G. C. Sherman, J. J. Stamnes, A. J. Devaney, and E. the inhomogeneous waves in angular-spectrum representations,” Opt. Commun., vol. 8, pp. 271–274, 1973. 3. A. J. Devaney and G. C. Sherman, “Plane-wave representations for scalar wave ﬁelds,” SIAM Rev., vol. 15, pp. 765–786, 1973. 4. 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. 5. J. W. Goodman, Introduction to Fourier Optics. New York: McGraw-Hill, 1968. 6. M. Born and E. Wolf, Principles of Optics. Cambridge: Cambridge University Press, seventh (expanded) ed., 1999. 7. H. Goldstein, Classical Mechanics. Reading, MA: Addison-Wesley, 1950. Chapter 4. 8. N. S. Koshlyakov, M. M. Smirnov, and E. B. Gliner, Diﬀerential Equations of Mathematical Physics. Amsterdam: North-Holland, 1964. Ch. VI, §3.

Problems 8.1. Derive the relation given in Eq. (8.25) with Eq. (8.16) as the starting point. 8.2. Derive the identity sin (AB) 1 = AB 4π

eiA·B dΩ(A), 4π

where dΩ(A) denotes the diﬀerential element of solid angle about the direction of the vector A. ˜ 8.3. Determine the electric ﬁeld vector E(r, t) for t > 0 when E(k) ≡ ˜ 0 (k) − (v/c)ˆs × B ˜ 0 (k)] = (1/2π)3 1δ(k ˆ is a ˆ (1/2π)3 [E − ω0 /v), where 1 ﬁxed unit vector in some speciﬁed direction. 8.4. Let the initial values for the electric and magnetic ﬁeld vectors be given ˆ z f (r) and B0 (r) = ∇× 1 ˆ z F (r), where f (r) and F (r) are by E0 (r) = ∇×∇× 1 suﬃciently well-behaved scalar functions of r ≡ |r| alone. Use the plane wave expansion given in Eq. (8.48) to determine the electric ﬁeld E(r, t)" for t > 0. Simplify the solution so that the only integral it contains is u(r) = rF (r)dr. 8.5. Let Π(r, t) be a solution of the scalar wave equation ∇2 Π(r, t) −

1 ¨ Π(r, t) = 0 v2

8.3 Problems

419

f(r)

0 0

R

r

˙ for t > 0 with the initial values Π(r, 0) = f (r) and Π(r, 0) = F (r), where f (r) and F (r) are scalar functions of r ≡ |r| alone. (a) Let F (r) = 0 and let f (r) vary with r as illustrated in the ﬁgure above, where f (r) = 0 for r > R. Sketch the behavior of Π(r, t) as a function of r for ﬁxed t such that t > R/v. (b) Let f (r) = 0 for all r and let F (r) = 1 for r < R and F (r) = 0 for r > R. Determine Π(r, t) for all t > 0. ∂ [rf (r)]. Determine Π(r, t) for all t > 0 in terms of the (c) Let F (r) = − vr ∂r function f (r).

A Helmholtz’ Theorem

Because ∇

2

1 R

= −4πδ(R)

(A.1)

where R = r − r with magnitude R = |R| and where δ(R) = δ(r − r ) = δ(x − x )δ(y − y )δ(z − z ) is the three-dimensional Dirac delta function (see Appendix B), then any suﬃciently well-behaved vector function F(r) = F(x, y, z) can be represented as F(r) = F(r )δ(r − r ) d3 r V 1 1 =− F(r )∇2 d3 r 4π V R F(r ) 3 1 = − ∇2 (A.2) d r, 4π R V the integration extending over any region V that contains the point r. With the identity ∇ × ∇× = ∇∇ · −∇2 , Eq. (A.2) may be written as 1 F(r ) 3 F(r ) 3 1 (A.3) F(r) = ∇×∇× d r − ∇∇ · d r. 4π R 4π R V V Consider ﬁrst the divergence term appearing in this expression. Because the vector diﬀerential operator ∇ does not operate on the primed coordinates, then 1 F(r ) 3 1 1 F(r ) · ∇ (A.4) ∇· d r = d3 r . 4π R 4π R V V Moreover, the integrand appearing in this expression may be expressed as 1 1 = −F(r ) · ∇ F(r ) · ∇ R R ) F(r 1 = −∇ · (A.5) + ∇ · F(r ), R R where the superscript prime on the vector diﬀerential operator ∇ denotes diﬀerentiation with respect to the primed coordinates alone. Substitution of

422

A Helmholtz’ Theorem

Eq. (A.5) into Eq. (A.4) and application of the divergence theorem to the ﬁrst term then yields 1 F(r ) 3 ∇ · F(r ) 3 1 F(r ) 1 3 ∇· d r =− d r ∇ · d r + 4π R 4π V R 4π V R V 1 ∇ · F(r ) 3 1 1 F(r ) · n d r =− ˆ d2 r + 4π S R 4π V R = φ(r), (A.6) which is the desired form of the scalar potential φ(r) for the vector ﬁeld F(r). Here S is the surface that encloses the regular region V and contains the point r. For the curl term appearing in Eq. (A.3) one has that 1 1 F(r ) 3 1 ∇× d r =− d3 r F(r ) × ∇ 4π R 4π V R V 1 1 = F(r ) × ∇ (A.7) d3 r . 4π V R Moreover, the integrand appearing in the ﬁnal form of the integral in Eq. (A.7) may be expressed as 1 F(r ) ∇ × F(r ) − ∇ × F(r ) × ∇ = , (A.8) R R R so that 1 ∇× 4π

V

F(r ) 3 ∇ × F(r ) 3 1 d r = d r − R 4π V R ∇ × F(r ) 3 1 d r + = 4π V R = a(r),

F(r ) 1 ∇ × d3 r 4π V R 1 1 F(r ) × n ˆ d2 r 4π S R (A.9)

which is the desired form of the vector potential. The relations given in Eqs. (A.3), (A.6), and (A.9) then show that F(r) = −∇φ(r) + ∇ × a(r),

(A.10)

where the scalar potential φ(r) is given by Eq. (A.6) and the vector potential a(r) is given by Eq. (A9). This expression may also be written as F(r) = F (r) + Ft (r),

(A.11)

where F (r) = −∇φ(r) ∇ · F(r ) 3 F(r ) 1 1 r + =− ∇ d ∇ ·n ˆ d2 r 4π |r − r | 4π V S |r − r |

(A.12)

References

423

is the longitudinal or irrotational part of the vector ﬁeld (where ∇ × F (r ) = 0), and where Ft (r) = ∇ × a(r) F(r ) 3 1 = ∇×∇× d r 4π V |r − r | ∇ × F(r ) 3 F(r ) 1 1 ∇× d ∇ × ×n ˆ d2 r(A.13) = r + 4π |r − r | 4π V S |r − r | is the transverse or solenoidal part of the vector ﬁeld (where ∇ · F (r ) = 0). If the surface S recedes to inﬁnity and if the vector ﬁeld F(r) is regular at inﬁnity, then the surface integrals appearing in the above expressions and Eqs. (A.12)–(A.13) become F (r) = −∇φ(r) ∇ · F(r ) 3 1 =− ∇ d r, 4π |r − r | V Ft (r) = ∇ × a(r) ∇ × F(r ) 3 1 ∇× d r. = 4π |r − r | V

(A.14)

(A.15)

Taken together, the above results constitute what is known as Helmholtz’ theorem [1]. Theorem 12. Helmholtz’ Theorem. Let F(r) be any continuous vector ﬁeld with continuous ﬁrst partial derivatives. Then F(r) can be uniquely expressed in terms of the negative gradient of a scalar potential φ(r) and the curl of a vector potential a(r), as embodied in Eqs. (A.10) and (A.11).

References 1. H. B. Phillips, Vector Analysis. New York: John Wiley & Sons, 1933.

B The Dirac Delta Function

B.1 The One-Dimensional Dirac Delta Function The Dirac delta function [1] in one-dimensional space may be deﬁned by the pair of equations δ(x) = 0; x = 0, ∞ δ(x) dx = 1.

(B.1) (B.2)

−∞

It is clear from this deﬁnition that δ(x) is not a function in the ordinary mathematical sense, because if a function is zero everywhere except at a single point and the integral of this function over its entire domain of deﬁnition exists, then the value of this integral is necessarily also equal to zero. Because of this, it is more appropriate to regard δ(x) as a functional quantity with a certain well-deﬁned symbolic meaning. For example, one can consider a sequence of functions δ(x, ε) that, with increasing values of the parameter ε, diﬀer appreciably from zero only over a decreasing x-interval about the origin and which are such that ∞ δ(x, ε) dx = 1 (B.3) −∞

for all values of ε. Although it may be tempting to try to interpret the Dirac delta function as the limit of such a sequence of well-deﬁned functions δ(x, ε) as ε → ∞, it must be recognized that this limit need not exist for all values of the independent variable x. However, the limit ∞ δ(x, ε) dx = 1 (B.4) lim ε→∞

−∞

must exist. As a consequence, one may interpret any operation that involves the delta function δ(x) as implying that this operation is to be performed with a function δ(x, ε) of a suitable sequence and that the limit as ε → ∞ is to be taken at the conclusion of the calculation. The particular choice of the sequence of functions δ(x, ε) is immaterial, provided that their oscillations (if any) near the origin x = 0 are not too violent [2]. Each of the following functions forms a sequence with respect to the parameter ε that satisﬁes the required properties.

426

B The Dirac Delta Function 2 2 ε δ(x, ε) = √ e−ε x , π δ(x, ε) = rect1/ε (x), ε δ(x, ε) = sinc(εx), π δ(r, ε) = circ1/ε (r), ε δ(r, ε) = J1 (2πεr), r

where rect1/ε (x) ≡ ε/2 when |x| < 1/ε and is zero otherwise, circ1/ε (r) ≡ ε2 /π when r < 1/ε and is zero otherwise, and sinc(x) ≡ sin (x)/x when x = 0 and is equal to its limiting value of unity when x = 0, where the last two of the above set of functions are appropriate for polar coordinates. Let f (x) be a continuous and suﬃciently well-behaved function of x ∈ (−∞, ∞) and consider the value of the deﬁnite integral ∞ ∞ f (x)δ(x − a)dx = lim f (x)δ(x − a, ε)dx. ε→∞

−∞

−∞

When the parameter ε is large, the value of the integral appearing on the right-hand side of this equation depends essentially on the behavior of f (x) in the immediate neighborhood of the point x = a alone, and the error that results from the replacement of f (x) by f (a) may be made as small as desired by taking ε suﬃciently large. Hence ∞ ∞ lim f (x)δ(x − a, ε)dx = f (a) lim δ(x − a, ε)dx, ε→∞

ε→∞

−∞

so that

∞

−∞

−∞

f (x)δ(x − a)dx = f (a).

(B.5)

This result is referred to as the sifting property of the delta function. That is, the process of multiplying a continuous function by δ(x − a) and integrating over all values of the variable x is equivalent to the process of evaluating the function at the point x = a. Notice that, for this result to hold, the domain of integration need not be extended over all x ∈ (−∞, ∞); it is only necessary that the domain of integration contain the point x = a in its interior, so that

a+∆2

f (x)δ(x − a)dx = f (a),

(B.6)

a−∆1

where ∆1 > 0, ∆2 > 0. It is then seen that f (x) need only be continuous at the point x = a. The above results may be written symbolically as f (x)δ(x − a) = f (a)δ(x − a),

(B.7)

B.1 The One-Dimensional Dirac Delta Function

427

the meaning of such a statement being that the two sides yield the same result when integrated over any domain containing the point x = a. For the special case when f (x) = xk with k > 0 and a = 0, Eq. (B.7) yields xk δ(x) = 0,

∀k > 0.

(B.8)

Theorem 13. Similarity Relationship (Scaling Law). For all a = 0 δ(ax) =

1 δ(x). |a|

(B.9)

Proof. In order to prove this relationship one need only compare the integrals of f (x)δ(ax) and f (x)δ(x)/|a| for any suﬃciently well-behaved continuous function f (x). For the ﬁrst integral one has (for any a = 0) ∞ 1 ∞ 1 f (0), f (x)δ(ax)dx = ± f (y/a)δ(y)dy = a |a| −∞ −∞ where the upper or lower sign choice is taken accordingly as a > 0 or a < 0, respectively, and for the second integral one obtains ∞ 1 1 f (0). f (x) δ(x)dx = |a| |a| −∞ Comparison of these two results then shows that δ(ax) = δ(x)/|a|, as was to be proved. ! For the special case a = −1, Eq. (B.9) yields δ(−x) = δ(x),

(B.10)

so that the delta function is an even function of its argument. Theorem 14. Composite Function Theorem. If y = f (x) is any continuous function of x with simple zeroes at the points xi [i.e., y = 0 at x = xi and f (xi ) = 0] and no other zeroes, then 1 δ(f (x)) = (B.11) δ(x − xi ). |f (xi )| i Proof. In order to prove this theorem, let g(x) be any suﬃciently wellbehaved continuous function and let {xi } denote the set of points at which y = 0. Under the change of variable x = f −1 (y) one has that ∞ 1 dy g(x)δ(f (x))dx = g f −1 (y) δ(y) −1 |f (f (y)) | −∞ R 1 = g f −1 (0) −1 |f (f (0)) | xi 1 , = g(xi ) |f (xi | i

428

B The Dirac Delta Function

where R denotes the range of f (x). In addition,

∞ ∞ 1 1 δ(x − x g(x) ) dx = g(x)δ(x − xi )d i |f (xi )| |f (xi )| −∞ −∞ i i 1 . = g(xi ) |f (xi | i Comparison of these two expressions then proves the theorem.

!

As an example, consider the function f (x) = x2 − a2 which has simple zeroes at x = ±a. Then |f (±a)| = 2|a| so that, for a = 0, δ(x2 − a2 ) =

1 (δ(x − a) + δ(x + a)) . 2|a|

An additional relationship of interest that employs the Dirac delta function is ∞ δ(ξ − x)δ(x − η)dx = δ(ξ − η), (B.12) −∞

which is seen to be an extension of the sifting property to the delta function itself. This equation then implies that if both sides are multiplied by a continuous function of either ξ or η and the result integrated over all values of either ξ or η, respectively, an identity is obtained. That is, because ∞ f (ξ)δ(ξ − η)dξ = f (η), −∞

and

∞

δ(ξ − x)δ(x − η)dx dξ −∞ ∞ ∞ = f (ξ)δ(ξ − x)dξ δ(x − η)dx −∞ −∞ ∞ = f (x)δ(x − η)dx = f (η) ∞

f (ξ) −∞

−∞

then the expression in Eq. (B.12) follows. In a similar manner, because ∞ ∞ g(η) δ(ξ − x)δ(x − η)dx dη −∞ −∞ ∞ ∞ = g(η)δ(x − η)dη δ(ξ − x)dx −∞ −∞ ∞ = g(x)δ(ξ − x)dx = g(ξ) −∞

B.1 The One-Dimensional Dirac Delta Function

and

∞

−∞

429

g(η)δ(ξ − η)dη = g(ξ),

then the expression in Eq. (B.12) is again obtained. Consider next what interpretation may be given to the derivatives of the delta function. This is accomplished through " ∞ use of the function sequence δ(x, ε). Consider then the ordinary integral −∞ f (x)δ (x, ε)dx which may be evaluated by application of the method of integration by parts with u = f (x) and dv = δ (x, ε)dx, so that ∞ ∞ f (x)δ (x, ε)dx = f (∞)δ(∞, ε) − f (−∞)δ(−∞, ε) − f (x)δ(x, ε)dx. −∞

−∞

Upon proceeding to the limit as ε → ∞, the ﬁrst two terms appearing on the right-hand side of this equation both vanish because lim δ(±∞, ε) = 0,

(B.13)

f (x)δ (x)dx = −f (0).

(B.14)

ε→∞

with the result

∞

−∞

Upon repeating this procedure n times for the nth-order derivative of the delta function, one obtains the general result ∞ f (x)δ (n) (x)dx = (−1)n f (n) (0). (B.15) −∞

As a special case of Eq. (B.14), let f (x) = x so that ∞ ∞ xδ (x)dx = −1 = − δ(x)dx, −∞

−∞

and one then has the equivalence xδ (x) = −δ(x).

(B.16)

Because δ(x) is an even function and x is an odd function, it then follows that δ (x) is an odd function of its argument; that is δ (−x) = −δ (x).

(B.17)

The generalization of Eq. (B.16) may be directly obtained from Eq. (B.15) by letting f (x) = xn . In that case, f (n) (x) = n! and this relation gives ∞ ∞ xn δ (x)dx = (−1)n n! = (−1)n n! δ(x)dx, −∞

−∞

430

B The Dirac Delta Function

and one then has the general equivalence xn δ (n) (x) = (−1)n n!δ(x).

(B.18)

This ﬁnal relationship shows that the even-order derivatives of the delta function are even functions and the odd-order derivatives are odd functions of the argument. It is often convenient to express the Dirac delta function in terms of the Heaviside unit step function U (x) that is deﬁned by the relations U (x) = 0 when x < 0, U (x) = 1 when x > 0. Consider the behavior of the derivative of U (x). If, as before, a superscript prime denotes diﬀerentiation with respect to the argument, one obtains formally upon integration by parts (with the limits −x1 < 0 and x2 > 0), x2 x2 x2 f (x)U (x)dx = [f (x)U (x)]−x − f (x)U (x)dx 1 −x1 −x1 x2 = f (x2 ) − f (x)dx 0

= f (x2 ) − [f (x2 ) − f (0)] = f (0), where f (x) is any continuous function. Upon setting x = y − a, and f (x) = f (y − a) = F (y), and then proceeding to the limits as −x1 → −∞ and x2 → +∞, the above result becomes ∞ F (y)U (y − a)dy = F (a), −∞

and the derivative U (x) is seen to satisfy the sifting property given in Eq. (B.5). In particular, with F (y) = 1 and a = 0, this expression becomes ∞ U (y)dy = 1, −∞

and U (x) also satisﬁes the property given in Eq. (B.2) which serves to partially deﬁne the delta function. Moreover, U (x) = 0 for all x = 0 and property (B.1) is also satisﬁed. Hence, one may identify the derivative of the unit step function with the delta function, so that δ(x) =

dU (x) . dx

(B.19)

In addition, it is seen that

x

U (x) =

δ(ξ)dξ, −∞

which follows from Eqs. (B.6) and (B.19).

(B.20)

B.1 The One-Dimensional Dirac Delta Function

431

The Dirac delta function may also be introduced through the use of the Fourier integral theorem [3], which may be written as ∞ ∞ dν dx f (x)ei2πν(x−a) (B.21) f (a) = −∞

−∞

for any suﬃciently well-behaved, continuous function f (x). Deﬁne the function sequence ε K(x − a, ε) ≡ ei2πν(x−a) dν −ε

sin (2π(x − a)ε) = π(x − a)

(B.22)

K(x − a) ≡ lim K(x − a, ε).

(B.23)

with limit ε→∞

Strictly speaking, this limit does not exist in the ordinary sense when x − a = 0; however, the limit does exist and has the value zero when x − a = 0 if it is interpreted in the sense of a Ces´aro limit [4]. Upon inversion of the order of integration, Eq. (B.21) may be formally rewritten as ∞ f (a) = f (x)K(x − a)dx, (B.24) −∞

which should be interpreted as meaning that ∞ f (a) = lim f (x)K(x − a, ε)dx. ε→∞

(B.25)

−∞

Thus, the function K(x − a) satisﬁes the sifting property (B.5) of the delta function. If one sets f (x) = 1 and a = 0 in Eq. (B.24), there results ∞ K(x)dx = 1, −∞

and K(x) satisﬁes the property given in Eq. (B.2) which serves to partially deﬁne the delta function. Because K(x) = limε→∞ K(x, ε) = 0 when x = 0, so that the property given in Eq. (B.1) is also satisﬁed, one then obtains from Eq. (B.23) the relation ∞

δ(x) =

ei2πνx dν.

(B.26)

−∞

That is, the Dirac delta function may be regarded as the Fourier transform of unity. The reciprocal relation follows from Eq. (B.25) upon setting f (x) = exp(i2πνx) and a = 0, so that ∞ 1= δ(x)e−i2πνx dx, (B.27) −∞

which also follows directly from the sifting property given in Eq. (B.5). Notice that this relation by itself is not suﬃcient to imply the validity of Eq. (B.26).

432

B The Dirac Delta Function

B.2 The Dirac Delta Function in Higher Dimensions The deﬁnition of the Dirac delta function may easily be extended to higherdimensional spaces. In particular, consider three-dimensional vector space in which case the deﬁning relations given in Eqs. (B.1)–(B.2) become δ(r) = 0; r = 0, ∞ δ(r)d3 r = 1.

(B.28) (B.29)

−∞

The function δ(r) ≡ δ(x, y, z) ≡ δ(x)δ(y)δ(z),

(B.30)

1y y + ˆ 1z z is the position vector with components (x, y, z) where r = ˆ 1x x + ˆ clearly satisﬁes Eqs. (B.28)–(B.29) and so deﬁnes a three-dimensional Dirac delta function. The sifting property given in Eq. (B.5) then becomes ∞ f (r)δ(r − a)d3 r = f (a), (B.31) −∞

and the similarity relationship or scaling law given in Eq. (B.9) now states that 1 δ(ar) = 3 δ(r), (B.32) |a| where a is a scalar constant. The Fourier transform pair relationship expressed in Eqs. (B.26)–(B.27) becomes ∞ 1 δ(r) = eik·r d3 k, (B.33) (2π)3 −∞ ∞ 1= δ(r)e−ik·r d3 r, (B.34) −∞

1y ky + ˆ 1z kz = 2π(ˆ 1x νx + ˆ 1 y νy + ˆ 1z νz ). where k = ˆ 1x kx + ˆ The generalization of the three-dimensional Dirac delta function to more general coordinate systems requires more careful attention. Suppose that a function ∆(r) is given in Cartesian coordinates as ∆(r) = δ(x)δ(y)δ(z)

(B.35)

and it is desired to express ∆(r) in terms of the orthogonal curvilinear coordinates (u, v, w) that are deﬁned by u = f1 (x, y, z), v = f2 (x, y, z), w = f3 (x, y, z),

(B.36)

B.2 The Dirac Delta Function in Higher Dimensions

433

where f1 , f2 , f3 are continuous, single-valued functions of x, y, z with a unique inverse x = f1−1 (u, v, w), y = f2−1 (u, v, w), z = f3−1 (u, v, w). That is, an expression for ∆(r) is desired in terms of the coordinate variables (u, v, w) that satisﬁes the relation ∞ ∆(r − r )ϕ(u, v, w)dV = ϕ(u , v , w ), (B.37) −∞

where dV is the diﬀerential volume element in u, v, w-space and (u , v , w ) is the point corresponding to (x , y , z ) under the coordinate transformation given in Eq. (B.36). If the point r = (x, y, z) is varied from r to r + δr1 by changing the coordinate variable u to u + δu while keeping v and w ﬁxed, then δr1 =

∂r δu. ∂u

Similarly, if the point r = (x, y, z) is varied from r to r + δr2 by changing the coordinate variable v to v + δv while keeping u and w ﬁxed, then δr2 =

∂r δv. ∂v

The parallelogram with sides δr1 and δr2 then has area δA = |δA| = |δr1 × δr2 | ∂r ∂r δuδv. × = ∂u ∂v

(B.38)

If the point r = (x, y, z) is now varied from r to r + δr3 by changing the coordinate variable w to w + δw while keeping u and v ﬁxed, then δr3 =

∂r δw, ∂w

and the volume of the parallelepiped with edges δr1 , δr2 , and δr3 is then given by δV = |δr3 · (δr1 × δr2 )| ∂r ∂r ∂r δuδvδw. = · × ∂w ∂u ∂v

(B.39)

The quantity J

x, y, z u, v, w

∂(x, y, z) ∂(u, v, w) ∂r ∂r ∂r ≡ · × ∂w ∂u ∂v

≡

(B.40)

434

B The Dirac Delta Function

is recognized as the Jacobian of the coordinate transformation of x, y, z with respect to u, v, w. With this result for the diﬀerential element of volume, Eq. (B.37) becomes ∞ x, y, z dudvdw = ϕ(u , v , w ), ∆(r − r )ϕ(u, v, w) J (B.41) u, v, w −∞ from which it is immediately seen that x, y, z δ(x)δ(y)δ(z). δ(u)δ(v)δ(w) = J u, v, w

(B.42)

Because this transformation is assumed to be single-valued, then δ(u)δ(v)δ(w) δ(x)δ(y)δ(z) = x,y,z J u,v,w u, v, w δ(u)δ(v)δ(w), = J x, y, z

(B.43)

where J(u, v, w/x, y, z) is the Jacobian of the inverse transformation. Consider ﬁnally the description of a function (r) that vanishes everywhere in three-dimensional space except on a surface S and is such that ∞ (r)ϕ(r)d3 r = ς(r)ϕ(r)d2 r, (B.44) −∞

S

where ς(r) is the value of (r) on the surface S, that is, when r ∈ S. Choose orthogonal curvilinear coordinates (u, v, w) such that w = w0 describes the surface S for some constant w0 , in which case ∇w is parallel to the normal to the surface S, and is such that ∇u and ∇v are both perpendicular to the normal to the surface S. The diﬀerential element of area of the surface S is then given by Eq. (B.38). Furthermore, both ∂r/∂w and (∂r/∂u) × (∂r/∂v) are normal to S so that ∂r ∂r x, y, z ∂r . (B.45) J = × u, v, w ∂w ∂u ∂v With this result, Eq. (B.44) may be written as ∞ ∂r ∂r ∂r dudvdw = (r)ϕ(r) ς(r)ϕ(r)d2 r, × ∂u ∂v ∂w S −∞ which, with Eq. (B.38), may be expressed as ∞ ∂r 2 d rdw = (r)ϕ(r) ς(r)ϕ(r)d2 r. ∂w −∞ S From this result it then follows that

(B.46)

References

ς(r) (r) = ∂r δ(w − w0 ),

435

(B.47)

∂w

which is the solution of Eq. (B.44). This result can be simpliﬁed somewhat by noting that when the variable w is varied while u and v are held ﬁxed, then the changes in r and w are related by δw = ∇w · δr, so that ∂r · ∇w = 1. ∂w Moreover, because both ∂r/∂w and ∇w are normal to the surface S described by w = w0 , then ∂r ∂w |∇w| = 1, and, as a result, Eq. (B.47) becomes (r) = |∇w| ς(r)δ(w − w0 )

(B.48)

as the solution to Eq. (B.44).

References 1. P. A. M. Dirac, The Principles of Quantum Mechanics. Oxford: Oxford University Press, 1930. §15. 2. M. J. Lighthill, Introduction to Fourier Analysis and Generalized Functions. London, England: Cambridge University Press, 1970. 3. E. C. Titchmarsh, Introduction to the Theory of Fourier Integrals. London: Oxford University Press, 1939. Ch. I. 4. B. van der Pol and H. Bremmer, Operational Calculus Based on the Two-Sided Laplace Integral. London: Cambridge University Press, 1950. pp. 100–104.

C The Fourier–Laplace Transform

The complex temporal frequency spectrum of a vector function f (r, t) of both position r and time t that vanishes for t < 0 is of central importance to the solution of problems in time-domain electromagnetics and optics and is considered here in some detail following, in part, the treatment by Stratton [1]. The Laplace transform of f (r, t) with respect to the time variable t is deﬁned here as ∞ f (r, t)eiωt dt, (C.1) L{f (r, t)} ≡ 0

which is simply a Fourier transform with complex angular frequaency ω that is taken over only the positive time interval. Let f (r, t) be another vector function of both position and time such that f (r, t) = f (r, t);

t > 0,

(C.2)

but which may not vanish for t ≤ 0. The Laplace transform given in Eq. (C.1) may then be written as ∞ U (t)f (r, t)eiωt dt, (C.3) L{f (r, t)} = −∞

where U (t) = 0 for t < 0 and U (t) = 1 for t > 0 is the Heaviside unit step function. For real ω, the Laplace transform of f (r, t) is then seen to be equal to the Fourier transform of U (t)f (r, t), viz. L{f (r, t)} = Fω {U (t)f (r, t)};

for real ω,

(C.4)

where the subscript ω indicates that it is the Fourier transform variable. The inverse Fourier transform of this equation then gives 7 8 (C.5) U (t)f (r, t) = F−1 L{f (r, t)} for real ω. For complex ω, let ω = ω + iω where ω ≡ {ω} and ω ≡ {ω}. The Laplace transform given in Eq. (C.1) then becomes ∞ L{f (r, t)} = U (t)f (r, t)e−ω t eiω t dt −∞ # $ = Fω U (t)f (r, t)e−ω t . (C.6)

438

C The Fourier–Laplace Transform

The inverse Fourier transform of this expression then yields 7 8 L{f (r, t)} U (t)f (r, t)e−ω t = Fω−1 ∞ 1 = L{f (r, t)}e−iω t dω , 2π −∞ which may be rewritten as U (t)f (r, t) =

1 2π

∞

−∞

L{f (r, t)}e−i(ω +iω

)t

(C.7)

dω

∞+ω 1 = L{f (r, t)}e−iωt dω 2π −∞+iω 1 = L{f (r, t)}e−iωt dω. 2π C

(C.8)

Here C denotes the straight line contour ω = ω + iω with ω ﬁxed and ω varying over the real domain from −∞ to +∞. Because f (r, t) = U (t)f (r, t), Eqs. (C.1) and (C.8) then deﬁne the Laplace transform pair relationship ∞ ˜ f (r, ω) ≡ L {f (r, t)} = f (r, t)eiωt dt, (C.9) 0 # $ 1 ˜ f (r, ω) = f (r, ω)e−iωt dω, (C.10) f (r, t) ≡ L−1 ˜ 2π C where ˜ f (r, ω) is the complex temporal frequency spectrum of f (r, t) with ω = ω + iω . Notice that ω = {ω} plays a passive role in the Laplace transform operation because it remains constant in both the forward and inverse transformations. Nevertheless, its presence can be important because the factor e−ω t appearing in the integrand of the transformation (C.9) may serve as a convergence factor when ω > 0. In particular, ∞ ˜ f (r, ω) = f (r, t)e−ω t eiω t dt (C.11) 0

is just the Fourier transform Fω {f (r, t)e−ω t }. The Fourier transform of f (r, t) alone is ∞ Fω {f (r, t)} = f (r, t)eiω t dt, 0

which exists provided that f (r, t) is absolutely integrable; viz., T |f (r, t)| dt < ∞. lim T →∞

0

If f (r, t) does not vanish properly at inﬁnity, then the above integral fails to converge and the existence of the Fourier transform Fω {f (r, t)} is not guaranteed. However, if there exists a real number γ such that

C The Fourier–Laplace Transform

lim

T →∞

0

T

f (r, t)e−γt dt < ∞,

439

(C.12)

then f (r, t) is transformable for all ω ≥ γ and its temporal frequency spectrum is given by the Laplace transform (C.9). The lower bound γa of all of the values of γ for which the inequality appearing in Eq. (C.12) is satisﬁed is called the abscissa of absolute convergence for the function f (r, t). The Laplace transform of the time derivative ∂f (r, t)/∂t can be related to the Laplace transform of f (r, t) through integration by parts as ∞ ∂f (r, t) iωt ∂f (r, t) e dt = L ∂t ∂t 0 ∞ ∞ = f (r, t)eiωt 0 − iω f (r, t)eiωt dt 0

= −f (r, 0) − iωL {f (r, t)} ,

(C.13)

where the fact that |f (r, t)eiωt | = |f (r, t)|e−ω t must vanish as t → ∞ for all ω ≥ γa has been used in obtaining the ﬁnal form of Eq. (C.13). For the appropriate form of the convolution theorem for the Laplace transform, consider determining the function whose Laplace transform is equal to the product f˜1 (ω)f˜2 (ω) = L{f1 (t)}L{f2 (t)} so that $ # 1 −1 ˜ ˜ f1 (ω)f2 (ω) = L f˜1 (ω)f˜2 (ω)e−iωt dω 2π C ∞ 1 iωτ ˜ = f2 (τ )e dτ e−iωt dω f1 (ω) 2π C 0 ∞ 1 −iω(t−τ ) ˜ = dτ · f2 (τ ) dω , f1 (ω)e 2π C 0 and consequently # $ L−1 f˜1 (ω)f˜2 (ω) =

0

∞

f1 (t − τ )f2 (τ )U (t − τ )dτ,

where the unit step function U (t − τ ) is explicitly included in this expression to emphasize the fact that f1 (t) vanishes for t < 0. Because U (t − τ ) vanishes for negative values of its argument, the upper limit of integration in τ must be t and the above equation becomes # $ t f1 (t − τ )f2 (τ )dτ, (C.14) L−1 f˜1 (ω)f˜2 (ω) = 0

which may be rewritten as ∞ L f1 (t − τ )f2 (τ )dτ = L {f1 (t)} L {f2 (t)} , 0

(C.15)

440

C The Fourier–Laplace Transform

which is the convolution theorem for the Laplace transform. The spatiotemporal Fourier–Laplace transform of a vector function F(r, t) of both position r and time t that vanishes for t < 0 is deﬁned here by the pair of relations ∞ ∞ = = F(k, ω) ≡ FL{F(r, t)} = d3 r dt · F(r, t)e−i(k·r−ωt) , (C.16) −∞ 0 ∞ 1 = = 3 = = d k dω · F(k, F(r, t) ≡ F−1 L−1 {F(k, ω)} = ω)ei(k·r−ωt) , (2π)4 −∞ C (C.17) 1y ky + ˆ 1z kz and r = ˆ 1x x + ˆ 1y y + ˆ 1z z. Because, where k = ˆ 1x kx + ˆ ∞ ∂F(r, t) 1 = = = d3 k dω · ikj F(k, ω)ei(k·r−ωt) , ∂xj (2π)4 −∞ C from Eq. (C.17), then the transforms of the ﬁrst spatial derivatives of F(r, t) are given by ∂F(r, t) = = = ikj F(k, FL ω), (C.18) ∂xj where x1 = x, x2 = y, x3 = z and k1 = kx , k2 = ky , k3 = kz . With this result, the spatiotemporal transform of the divergence of the vector ﬁeld F(r, t) is found to be = = FL {∇ · F(r, t)} = ik · F(k, ω), (C.19) and the spatiotemporal transform of the curl of F(r, t) is given by = = FL {∇ × F(r, t)} = ik × F(k, ω).

(C.20)

The spatiotemporal transforms of higher-order spatial derivatives of F(r, t) may then be obtained through repeated application of the above relations.

References 1. J. A. Stratton, Electromagnetic Theory. New York: McGraw-Hill, 1941.

D The Eﬀective Local Field

The average electric ﬁeld intensity acting on a given molecule in a dielectric is called the local or eﬀective ﬁeld. In a linear isotropic dielectric, the spatially averaged induced molecular charge separation is directly proportional to, and in the same direction as, the local ﬁeld at that molecular site so that the average induced molecular dipole moment is given by ˜ ef f (r, ω) ˜ p(r, ω) = 0 α(ω)E

(D.1)

at the ﬁxed angular frequency ω of the applied time-harmonic ﬁeld [see Eq. (4.164)]. The molecular polarizability α(ω) characterizes the frequencydependent linear response of the molecules comprising the dielectric to the applied electric ﬁeld. The eﬀective local ﬁeld may be determined by removing the molecule under consideration while maintaining all of the remaining molecules in their time-averaged polarized states (time-averaging being used only to remove the eﬀects of thermal ﬂuctuations), the spatially averaged electric ﬁeld intensity then being calculated in the cavity left vacant by that removed molecule [1–3]. Let Vm denote the volume of that single molecular cavity region. The eﬀective local ﬁeld at that molecular site is then given by the diﬀerence 1 1 3 ˜ ˜ e(r , ω)d r − ˜ em (r , ω)d3 r , (D.2) Eef f (r, ω) = Vm Vm Vm Vm where ˜ e(r , ω) is the total microscopic electric ﬁeld at the point r ∈ Vm , and where ˜ em (r , ω) is the electric ﬁeld due to the charge distribution of the molecule under consideration evaluated at the point r ∈ Vm . If the dielectric material is locally homogeneous (i.e., its dielectric properties at any point in the material are essentially constant over a macroscopically small but microscopically large region), then the ﬁrst integral appearing in Eq. (D.2) is essentially just the macroscopic electric ﬁeld as deﬁned in Eq. (4.6). In particular, if the weighting function w(r ) is taken to be given by 1/Vm if r ∈ Vm and 0 otherwise, then Eq. (4.1) gives 1 ˜ ˜ e(r − r , ω)d3 r , (D.3) E(r, ω) ≡ ˜ e(r, ω) = Vm Vm where r ∈ Vm . With the change of variable r = r − r , the ﬁrst integral appearing in Eq. (D.2) is then obtained.

442

D The Eﬀective Local Field

z

d Τ1 q

dΤ2

r' O

R

V

Fig. D.1. Spherical cavity region V of radius R with point charge q situated a distance r < R from the center O.

For the second integral appearing in Eq. (D.2), consider determining the average electric ﬁeld intensity inside a sphere of radius R containing a point charge q that is located a distance r from the center O of the sphere, as illustrated in Figure D.1. The z-axis is chosen to be along the line from the center O of the sphere passing through the point charge q. With this choice, symmetry then shows that the average ﬁeld over the spherical volume must be along the z-axis. The average electric ﬁeld in V is then given by the scalar quantity 1 e˜z d3 r, (D.4) e¯z = V V where V is the volume of the spherical region. It is convenient to separate this volume integral into two parts, one taken over the spherical shell V1 between the radii r and R, and the other taken over the inner sphere V2 of radius r , so that 1 1 3 e˜z d r + e˜z d3 r. (D.5) e¯z = V V1 V V2 The integral over the spherical shell region V1 vanishes because of the equal and opposite contributions arising from the pair of volume elements dτ1 and dτ2 that are intercepted by the element of solid angle dΩ, as depicted in Figure D.1. Because the magnitude of e˜z decreases with the square of the

D The Eﬀective Local Field

443

distance from the point charge q, whereas the volume element dτ = r2 dΩ increases with the square of this distance, their product remains constant. For positive q, e˜z is positive at dτ1 and it is negative at dτ2 , whereas for negative q, e˜z is negative at dτ1 and positive at dτ2 . In either case, the two contributions to the integral over V1 cancel and that integral then vanishes.

z

Θ

q r' r''

O

P V2

Fig. D.2. Spherical polar coordinate system about the point charge q for the integral over the inner spherical region V2 with center at O and radius r .

The integral of e˜z over the inner volume V2 is then equal to the same integral over the entire spherical region V . In order to evaluate this ﬁnal volume integral, choose spherical polar coordinates (r , θ, ϕ) with origin at the point charge q, as illustrated in Figure D.2. At any point P ∈ V2 , ˆz = ˜·1 e˜z = e so that V

4π e˜z d r = q 4π 0 3

4π qr . 3 0

π

dϕ 0

4π =− qr

0 =−

2π

4π q cos(θ), 4π 0 r2

sin(θ) cos(θ) π/2

π

(D.6)

−2r cos(θ)

dr

dθ

0

cos2 (θ) sin(θ)dθ

π/2

(D.7)

The average electric ﬁeld intensity inside the sphere due to the point charge q is then given by

444

D The Eﬀective Local Field

e¯z =

1 (4/3)πR3

e˜z d3 r = −

V

4π qr . 4π 0 R3

(D.8)

The electric dipole moment of the point charge q with reference to the center ˆ z , so that Eq. (D.8) may be ˜ ≡ qr 1 O of the spherical region is given by p written in the general form ¯=− e

4π ˜. p 4π 0 R3

The second integral appearing in Eq. (D.2) is then given by 4π 1 ˜ m (r, ω), ˜ em (r , ω)d3 r = − p 3 Vm Vm 4π 0 rm

(D.9)

(D.10)

˜ m is the dipole moment of the molecule under consideration. Because where p N = 1/Vm is the local volume density of molecules, then with the assumption that all of the local molecules have parallel and equal polarization vectors, the macroscopic polarization is given by ˜ ω) = N p ˜ m (r, ω) P(r, so that the spatially averaged self-ﬁeld of the molecule is given by 1 4π ˜ ˜ em (r , ω)d3 r = − P(r, ω). Vm Vm 3 0

(D.11)

With these substitutions, Eq. (D.2) for the spatially averaged eﬀective ﬁeld becomes ˜ ω), ˜ ω) + 4π P(r, ˜ ef f (r, ω) = E(r, (D.12) E 3 0 and the local ﬁeld is larger than the macroscopic electric ﬁeld. This expression for the eﬀective local ﬁeld was ﬁrst derived by Lorentz [1] who used a somewhat diﬀerent deﬁnition of the local ﬁeld as the ﬁeld value at the center of the molecule rather than that averaged over the molecular volume.

References 1. H. A. Lorentz, The Theory of Electrons. Leipzig: Teubner, 1906. Ch. IV. 2. C. Kittel, Introduction to Solid State Physics. New York: John Wiley & Sons, fourth ed., 1971. Ch. 13. 3. J. D. Jackson, Classical Electrodynamics. New York: John Wiley & Sons, third ed., 1999.

E Magnetic Field Contribution to the Classical Lorentz Model of Resonance Polarization

With the complete Lorentz force relation as the driving force, the equation of motion of a harmonically bound electron is given by drj d2 rj qe 1 drj 2 + ωj rj = − × Beff (r, t) , + 2δj (E.1) Eeff (r, t) + dt2 dt m c dt where Eeff (r, t) is the eﬀective local electric ﬁeld intensity and Beff (r, t) is the eﬀective local magnetic induction ﬁeld at the space–time point (r, t), and where rj = rj (r, t) describes the displacement of the electron from its equilibrium position. Here qe denotes the magnitude of the charge and m the mass of the harmonically bound electron with undamped resonance frequency ωj and phenomenological damping constant δj . The temporal Fourier integral representation of the electric and magnetic ﬁeld vectors of the eﬀective local plane electromagnetic wave is given by ∞ ˜ eff (r, ω) = Eeff (r, t)eiωt dt, (E.2) E −∞ ∞ ˜ eff (r, ω) = Beff (r, t)eiωt dt B −∞

c ˜ eff (r, ω), = k(ω) × E ω

(E.3)

where k(ω) is the wave vector of the plane wave ﬁeld with magnitude given by the wavenumber k(ω) = ω/c because the eﬀective local ﬁeld is essentially a microscopic ﬁeld. With the temporal Fourier integral representation ∞ ˜rj (ω) = rj (t)eiωt dt, (E.4) −∞

the dynamical equation of motion (E.1) becomes 2 qe ˜ ˜ eff ω + 2iδj ω − ωj2 ˜rj = Eeff − i˜rj × k × E m qe ˜ eff − i ˜rj · E ˜ eff k , = (1 + i˜rj · k) E m with formal solution

(E.5)

446

E Magnetic Field Contribution to the Lorentz Model

˜ eff ˜rj · E 1 + i˜rj · k qe ˜ ˜rj = k . Eeff − i 2 m ω 2 − ωj2 + 2iδj ω ω − ωj2 + 2iδj ω

(E.6)

The electron displacement vector may then be expressed as a linear combi˜ eff nation of the orthogonal pair of vectors k and E ˜ eff + bj k, ˜rj = aj E

(E.7)

˜ eff = 0, where, because of the transversality relation k · E aj =

˜ eff ˜rj · E , 2 ˜eff E

bj =

c2 ˜rj · k. ω2

(E.8)

The pair of scalar products appearing in the above expression may be evaluated from Eq. (E.6) as ˜rj · k = −i

˜ eff ˜rj · E qe k2 , 2 m ω 2 − ωj + 2iδj ω

1 + i˜rj · k ˜ eff = qe ˜2 . ˜rj · E E m ω 2 − ωj2 + 2iδj ω eff Substitution of the second relation into the ﬁrst then yields ˜rj · k = −i

(ω 2

−

ωj2

2 2 ˜eff ω (qe /mc)2 E , 2 2 ω2 ˜eff + 2iδj ω) − (qe /mc)2 E

(E.9)

and substitution of this result into the second relation gives ˜ eff = ˜rj · E

qe /m ω 2 − ωj2 + 2iδj ω

2 2 ˜eff ω (qe /mc)2 E 2 ˜eff . (E.10) E × 1+ 2 ω2 ˜eff (ω 2 − ωj2 + 2iδj ω)2 − (qe /mc)2 E The coeﬃcients aj and bj appearing in Eq. (E.7) are then given by aj =

qe /m ω 2 − ωj2 + 2iδj ω

˜ 2 ω2 (qe /mc)2 E eff , × 1+ 2 ω2 ˜eff (ω 2 − ωj2 + 2iδj ω)2 − (qe /mc)2 E bj = −i respectively.

2 ˜eff (qe /m)2 E , ˜ 2 ω2 (ω 2 − ωj2 + 2iδj ω)2 − (qe /mc)2 E eff

(E.11) (E.12)

E Magnetic Field Contribution to the Lorentz Model

447

˜ j ≡ −qe ˜rj for the jth The local (or microscopic) induced dipole moment p Lorentz oscillator type is then given by [compare with Eq. (4.202)] ˜ eff (r, ω) + bj k . ˜ j (r, ω) = −qe aj E (E.13) p If there are Nj Lorentz oscillators per unit volume of the jth type, then the macroscopic polarization induced in the medium is given by the summation over all oscillator types of the spatially averaged locally induced dipole moments as ˜ ω, E ˜eff ) = Nj ˜ pj (r, ω) P(r, j

,, =

-- ˜ eff (r, ω) ˜eff ) + k ˜eff ). Nj αj⊥ (ω, E Nj αj (ω, E E j

j

(E.14) Here (0)

(2)

˜eff ) ≡ α (ω) + α (ω, E ˜eff ) αj⊥ (ω, E j⊥ j⊥ =

−qe2 /m − ωj2 + 2iδj ω

ω2

2 2 ˜eff ω (qe /mc)2 E × 1+ 2 2 ω2 ˜eff (ω 2 − ωj + 2iδj ω)2 − (qe /mc)2 E

(E.15)

is deﬁned here as the perpendicular component of the atomic polarizability, (0) with αj⊥ (ω) ≡ αj⊥ (ω, 0) = αj (ω), where αj (ω) is the classical expression (4.204) for the atomic polarizability when magnetic ﬁeld eﬀects are neglected, and 2 ˜eff (qe3 /m2 )E ˜eff ) ≡ i αj (ω, E (E.16) 2 ω2 ˜eff (ω 2 − ωj2 + 2iδj ω)2 − (qe /mc)2 E is deﬁned here as the parallel component of the atomic polarizability, where αj (ω, 0) = 0. The atomic polarizability is then seen to be nonlinear in the local electric ﬁeld strength when magnetic ﬁeld eﬀects are included. However, numerical calculations [1] show that these nonlinear terms are entirely negligible for eﬀective ﬁeld strengths that are typically less than ∼ 1012 V/m and that they begin to have a signiﬁcant contribution for ﬁeld strengths that are typically greater than ∼ 1015 V/m for a highly absorptive material. The physical origin of the nonlinear term considered here is due to the diamagnetic eﬀect that appears in the analysis of the interaction of an electromagnetic ﬁeld with a charged particle in the quantum theory of electrodynamics [2, 3]. The Hamiltonian for this coupled system is given by {see Eqs. (XIII.71)–(XIII.72) of Messiah [2]}

448

E Magnetic Field Contribution to the Lorentz Model

H = H0 −

Z q2 qe 2 rj⊥ , H · L + e 2 H2 2mc 8mc j=1

in Gaussian units, where H0 is the Hamiltonian of the center of mass system of the isolated atom with Z spinless electrons, H(r) is the magnetic ﬁeld intensity vector with magnitude H ≡ |H|, where r⊥ is the projection of the position vector r on the plane perpendicular to H(r), and where L = Z j=1 (rj × pj ) is the total angular momentum of the Z atomic electrons. The third term in the above expression for the Hamiltonian is the main factor in atomic diamagnetism. The order of magnitude of this factor is given by ∼ (Zqe2 /12mc2 )H 2 r2 , where r2 ∼ 1×10−16 cm2 for a bound electron. The ratio of this quantity to the level distance µB H, where µB ≡ qe ¯h/2mc is the Bohr magneton, is found [2] to be ∼ 10−9 ZH gauss. For a single electron atom (Z = 1), the diamagnetic eﬀect will become signiﬁcant when H ≥ 109 gauss, which corresponds to an electric ﬁeld strength E ≥ 109 esu, or equivalently E ≥ 3 × 1013 V/m, in agreement with the preceeding classical result that the nonlinear eﬀects in the Lorentz model become signiﬁcant for an applied ﬁeld strength between 1012 V/m and 1015 V/m. In addition, nonlinear optical eﬀects are found to dominate the linear response when the local ﬁeld strength becomes comparable to the Coulomb ﬁeld of the atomic nucleus [4]. As an estimate of this ﬁeld strength, if the distance between the nucleus and the bound electron is taken to be given h2 /mqe2 ≈ 5.29 × 10−9 cm, where ¯h ≡ h/2π and h by the Bohr radius a0 ≡ ¯ is Planck’s constant, the electric ﬁeld strength is E ≈ 5.13 × 1011 V/m, in general agreement with the preceding estimates.

References 1. K. E. Oughstun and R. A. Albanese, “Magnetic ﬁeld contribution to the Lorentz model,” J. Opt. Soc. Am. A, vol. 23, no. 7, pp. 1751–1756, 2006. 2. A. Messiah, Quantum Mechanics, vol. II. Amsterdam: North-Holland, 1962. 3. C. Cohen-Tannoudji, J. Dupont-Roc, and G. Grynberg, Photons and Atoms: Introduction to Quantum Electrodynamics. New York: John Wiley & Sons, 1989. Section III.D. 4. R. W. Boyd, Nonlinear Optics. San Diego: Academic, 1992. Ch. 1.

Index

abscissa of absolute convergence, 282, 439 acceleration ﬁeld, 130 advanced potentials, 118 Amp`ere’s law, 63 analytic signals, 402 angular spectrum representation freely propagating wave ﬁeld, 332, 366 geometric form, 341 of radiation ﬁeld, 297 asymptotic heat, 253 atomic polarizability parallel component, 447 perpendicular component, 447 attenuation factor, 7 auxiliary ﬁeld vectors, 57 Barash, Y. S. and Ginzburg, V. L., 242 baryon, 79 Ba˜ nos, A., 324 Boltzmann’s constant KB , 197 Born, M. and Wolf, E., 13, 346 boundary condition normal, 264 tangential, 265, 266 Brillouin precursor, 12 Brillouin, L., 1 Cauchy principal value, 183 causality dispersion relations, 183 Kramers–Kronig relations, 186 Plemelj formulae, 183 primitive, 179, 182 relativistic, 182, 260 charge density macroscoopic, nonconductive, 290

macroscopic (r, t), 170 macroscopic bound b (r, t), 170 macroscopic free f (r, t), 170 macroscopic, conductive, 290 microscopic ρ(r, t), 48 microscopic bound ρb (r, t), 167 microscopic free ρf (r, t), 167 surface, 264 Clausius–Mossotti relation, 197 Cole–Cole extended Rocard–Powles– Debye model triply distilled water, 206 Cole–Cole plot, 199 complex dielectric permittivity, 188 complex index of refraction n(ω), 210, 224, 297 complex permittivity, 224 complex velocity, 280 complex wavenumber, 7 conservation laws electromagnetic energy, 91, 236 electromagnetic momentum angular, 97 linear, 93 energy local form, 89 relativistic momentum, 76 conservation of charge microscopic, 50 constitutive relations, 178 and linearity, 179 general form, 179 induction ﬁelds, 178 primitive ﬁelds, 178 convective derivative, 72 convolution theorem, Laplace transform, 440 cophasal surfaces, 160

450

Index

Courant condition, 18 current density conduction, 172 convective (microscopic) j(r, t), 50 eﬀective, 272 irrotational or longitudinal j (r, t), 114 macroscopic J(r, t), 174 macroscopic bound Jb (r, t), 175 macroscopic free Jf (r, t), 175 microscopic “bound” jb (r, t), 172 microscopic free jf (r, t), 172 solenoidal or transverse jt (r, t), 114 surface, 266 current source J0 (r, t), 277 Debye model and rotational Brownian motion, 197, 201 causality, 198 eﬀective relaxation time, 198 of orientational (dipolar) polarization, 197 permittivity, 198 relaxation equation, 197 Rocard–Powles extension, 201 susceptibility, 198 Debye-type dielectric, 4 Devaney, A. J., 299 diamagnetic, 191 diamagnetic eﬀect, 447 dielectric permittivity complex c (ω), 188 composite model, 214 free space 0 , 52 of sea-water, 232 of triply distilled water, 193 temporally dispersive (ω), 184 dipolar relaxation time τm , 197 dipole Hertzian, elemental, 125 linear electric, 124 moment, electric, 148 dipole oscillator electric dipole contribution, 144 electric monopole contribution, 143 magnetic dipole contribution, 144 dipole radiation intermediate zone, 150

static zone, 149 wave zone, 152 Dirac delta function δ(ξ), 124, 425 composite function theorem, 427 derivatives of, 429 extension to higher dimensions, 432 sifting property, 426 similarity relationship (scaling law), 427 direction cosines, plane wave complex, 334 real, 361 dispersion anomalous, 212 normal, 212 dispersion relations dielectric permittivity, 186 electric conductivity, 188 electric susceptibility, 185 magnetic permeability, 192 magnetic susceptibility, 192 dispersive interface, 16 dissipation, 240 Drude model causality, 217 damping constant γ, 217 dielectric permittivity, 217 electric conductivity σ(ω), 232 static conductivity σ0 , 232 dynamical free energy, 250 eﬀective electric ﬁeld, 196, 445 eﬀective ﬁeld, 441 spatially averaged, 444 eﬀective magnetic ﬁeld, 445 eikonal equation generalized, 160 geometrical optics, 161 Einstein, A., 1 mass–energy relation, 78 special theory of relativity, 67 electric 2n -pole, 144 electric conductivity σ(ω), 188 electric dipole approximation, 144 electric displacement vector macroscopic D(r, t), 176 microscopic d(r, t), 58 electric energy density time-harmonic, 244

Index electric ﬁeld vector macroscopic E(r, t), 167 microscopic e(r, t), 52 electric moment, 120 electric susceptibility χe (ω), 184 electromagnetic angular momentum microscopic density lem , 97 microscopic total lem , 97 electromagnetic beam ﬁeld, 367 separable, 381 electromagnetic energy irreversible, 254 macroscopic density, total U (r, t), 235 microscopic density u(r, t), 90 total U (t), 89 reversible, 253 electromagnetic energy density, 89, 235 electromagnetic ﬁeld, 53 electromagnetic linear momentum microscopic density pem (r, t), 93 microscopic total pem , 93 electromagnetic wave freely propagating, 329 freely propagating boundary conditions, 329 plane, time-harmonic, 227 source-free, 329 source-free initial conditions, 387 electrostatic ﬁeld, 53 energy density electromagnetic ﬁeld Uem , 258 reversible, 258 energy transport velocity, 21, 261 energy velocity description Sherman and Oughstun, 24 equation of continuity conduction current, 222 macroscopic, 177 microscopic, 51 evolved heat (dissipation) Q(r, t), 240 extensive variable, 90 Faraday’s law microscopic, 60 forerunner, 1 Fourier integral theorem, 431 Fourier transform Fω {f (r, t)}, 438

451

free-ﬁeld, 387 Laplace–Fourier integral representation, 390 plane wave expansion, 391–405 freely propagating wave ﬁeld, 329 Fr¨ ohlich distribution function, 205 Gabor cells, 20 Galilean invariance, 62, 64 Galilean transformation electric ﬁeld intensity, 62 magnetic ﬁeld intensity, 65 gauge Coulomb, 113 function, 111, 286 invariance, 111, 286 Lorenz, 113, 287 radiation, 113 transformation, 111, 122, 286 transformation, restricted, 113, 287 transverse, 113 Gauss’ law microscopic electric ﬁeld, 66 magnetic ﬁeld, 66 Gaussian pulse dynamics Balictsis and Oughstun, 23 Garrett and McCumber, 23 Tanaka, Fujiwara, and Ikegami, 23 Gaussian units electrostatic unit (esu) of electric ﬁeld intensity, 58 gauss, 58 maxwell, 60 statampere, 58 statcoulomb, 58 statvolt, 58 Goos–H¨ anchen eﬀect, 16 Green’s function free-space, 345 group method Havelock, T. H., 11 group velocity, 4, 7 Hamilton, Sir W. R., 4 Rayleigh, Lord, 5 Stokes, G. G., 5 group velocity approximation, 11 Eckart, C., 13 Lighthill, M. J., 13

452

Index

Whitham, G. B., 13 group velocity method, 11 half-space, positive and negative, 296 Hall eﬀect, 222 Heaviside unit step function U (ξ), 430 Heaviside–Poynting theorem diﬀerential form, 235 integral form, 235 time-harmonic form, 244 Helmholtz equation, 7, 225, 367 Helmholtz free energy (work function), 250 Helmholtz’ theorem, 423 Hertz potential, 121 vector, 121 Hertz, H., 119, 149 Hilbert transform, 183, 185 homogeneous, isotropic, locally linear (HILL) temporally dispersive media constitutive relations, 221 Huygen’s principle, 345 idemfactor, 70, 94 impulse response function spatial, 342 incomplete Lipschitz–Hankel integrals Dvorak, S. and Dudley, D., 25 indeterminacy principle, 19 inertial reference frames, 67 information diagram, 19 initial ﬁeld values in a closed convex surface, 410 in a sphere, 406 instantaneous (causal) spectrum, 250 intensive variable, 90 interaction energy, 250 intrinsic impedance complex, 228 free space, 228 good conductor, 231 near-ideal dielectric, 230 intrinsic impedance, complex, 228 inverse problems, 21 isoplanatic, see space-invariant Jackson, J. D., 14

Jordan’s lemma, 408 Kramers–Kronig relations, 186 Kronecker-delta function δij , 94 Lagrange’s theorem, 140 Lalor, E., 344 Landau, L. D. and Lifshitz, E. M., 240, 253 Laplace transform L{f (r, t)}, 437 Li´enard–Wiechert potentials, 128 Li´enard-Wiechert potentials and special relativity, 135 local energy theorem, 90 local ﬁeld, see eﬀective ﬁeld Lorentz covariant, 68 Lorentz force relation, 53 Lorentz invariant, 68 Lorentz model atomic polarizability, 208 Lorentz–Lorenz modiﬁed, 210 magnetic ﬁeld contribution, 207 of resonance polarization, 207 oscillator strength, 213 phenomenological damping constant δj , 207 resonance frequency, undamped ωj , 207 sum rule, 213 Lorentz theory, 48 Lorentz, H. A., 1 Lorentz–Lorenz formula, 197 Lorentz-type dielectric, 4 Lorenz condition, 112, 286 generalized, 290 Loudon, R., 21, 260 luminiferous ether, 67 luxon, 78 macroscopic polarization density P(r, t), 171 macroscopic quadrupole moment density tensor Q (r, t), 171 magnetic 2n−1 -pole, 144 magnetic energy density time-harmonic, 244 magnetic ﬁeld vector macroscopic B(r, t), 167 microscopic b(r, t), 52

Index magnetic ﬂux, 60 magnetic intensity vector macroscopic H(r, t), 176 microscopic h(r, t), 58 magnetic permeability composite model, 215 dispersive µ(ω), 190 free space µ0 , 52 magnetic susceptibility χm (ω), 190 magnetization, macroscopic M(r, t), 175 magnetostatic ﬁeld, 53 material relations, see cnstitutive relations178 material response anisotropic, 181 isotropic, 181 nondispersive, 180 spatially dispersive, 180 spatially homogeneous, 179 spatially inhomogeneous, 179 temporally dispersive, 180 temporally homogeneous, 179 temporally inhomogeneous, 179 material response tensors dielectric permittivity ˆ (r , t , r, t), 179 electric conductivity σ ˆ (r , t , r, t), 179 magnetic permeability µ ˆ (r , t , r, t), 179 mature dispersion regime, 2, 24 Maxwell stress tensor microscopic, 94 Maxwell’s displacement current, 63 Maxwell’s equations macroscopic temporal frequency-domain form, 224 time-domain diﬀerential form, 176, 223 microscopic, 52 time-domain diﬀerential form, 58 time-domain integral form, 66 phasor form, 226 source-free, 387 spatial average, 167 spatiotemporal form, 280

453

spatiotemporal frequency domain form, 389 Maxwell, J. C., 1 mean polarizability αj (ω), 196 mean square charge radius, 171 metallic waveguides, 14 microscopic Maxwell–Lorentz theory, 51 Minkowski formulation, 177 mksa units ampere, 59 coulomb, 59 farad, 59 volt, 59 weber, 59, 60 modern asymptotic description, 24 molecular magnetic moment, 173 molecular multipole moments, 169 molecular polarizability α(ω), 441 monochromatic (time-harmonic) wave ﬁeld, 159 near-ideal dielectrics, 230 Newton’s second law of motion relativistic form, 75 Nisbet, A. and Wolf, E., 345, 356 nonlinear optical eﬀects, 448 normalized velocity, 127 number density Nj , 208 numerical techniques discrete Fourier transform method, 17 fast Fourier transform method, 18 ﬁnite-diﬀerence time-domain method, 18 Hosono’s Laplace transform method, 18 ohmic power loss, 273 Olver’s saddle point method, 24 optical wave ﬁeld, 158 order symbol O, 76 Ott’s integral representation, 324 Panofsky, W. and Phillips, M., 129 paramagnetic, 191 penetration depth dp , 231, 269 phase velocity, 5 good conductor, 231

454

Index

near-ideal dielectric, 230 phase-space asymptotic description, 17 phasor representation, 225 plane wave attenuation factor α(ω), 228, 334 evanescent, 339, 364 homogeneous, 298, 339, 364 inhomogeneous, 298, 339, 364 propagation factor β(ω), 228, 334 plane wave expansion and mode expansions, 402 and Poisson’s solution, 405 Devaney, A. J., 399 polar coordinate form, 401, 404 uniqueness, 399 plasma frequency bj , 208 Poincar´e–Lorentz transformation relations, 69–70 coordinate transformation matrix, 70 electric ﬁeld transformation, 84 force transformation, 80 invariance of Maxwell’s equations, 87 magnetic ﬁeld transformation, 84 mass transformation, 80 Poisson’s equation scalar potential, 113 Poisson’s solution, 415 polarization left-handed, 350 right-handed, 351 state, 353 polarization ellipse, 347, 353 polarized ﬁeld circular, 354 linear, 354 uniform, 353, 357 potential functions, electromagnetic complex, 156 Li´enard–Wiechert, 125 macroscopic, 285–291 microscopic, 109–111 retarded, 116 Poynting vector ˜ complex S(r), 243 macroscopic S(r, t), 235 microscopic s(r, t), 89 Poynting’s theorem, 90 Poynting–Heaviside interpretation, 91

precursor, 1 Brillouin, 12 formation on transmission, 16 observation of Aaviksoo, Lippmaa, and Kuhl, 29 ¨ Choi and Osterberg, 29 Jeong, Dawes, and Gauthier, 29 Pleshko and Pal´ ocz, 12 Sommerfeld, 12 principle of superposition, 55 propagation factor, 7 pulse centroid velocity, 29 pulse diﬀraction, 26 pulse distortion, 16 quadratic dispersion relation, 16 radiated energy, 136 radiation electromagnetic, 119 reaction, 55 radiation ﬁeld Fourier–Laplace integral representation, 281, 284 scalar potential, 285 vector potential, 285 ray techniques direct-ray method, 13 Felsen, L. B., 14 Heyman, E., 14 Melamed, T., 14 space–time ray theory, 13 Rayleigh–Sommerfeld diﬀraction integrals, 345 refractive index complex, 210 function, modiﬁed, 160 relativity Newtonian, 67 special theory, 67 relaxation times dipolar, 197 distribution of, 204 Drude model, mean-free path τc , 217 eﬀective, 198, 202 retardation condition, 131 retarded potentials, 118 retarded time, 116 Riemann’s proof, 116

Index Righi, A., 119 Rocard–Powles–Debye model Cole–Cole extension, 205 friction time τmf , 202 of triply distilled water, 203 permittivity, 202 susceptibility, 202 sampling theorem, 20 scalar potential macroscopic, 285 microscopic, 110 semiconductor, 231 Sherman expansion, see source-free wave ﬁeld Sherman, G. C., 3, 31, 299, 342, 367 SI (Syst`eme Internationale) units, see mksa units signal arrival, 9, 12, 15 buildup, 15 velocity, 2 signal velocity, 2, 7 Baerwald, H., 11 Brillouin, L., 7 Ehrenfest, P., 7 Laue, A., 7 Shiren, N. S., 12 Sommerfeld, A., 7 sound, 13 Voigt, W., 7 Weber and Trizna, 12 simple magnetizable medium, 190 simple polarizable dielectric, 184 simultaneity, 68 skin depth, 269 slowly evolving wave approach, 27 slowly varying envelope approximation, 13, 27 Sommerfeld precursor, 12 Sommerfeld’s integral representation, 322 Sommerfeld, A., 1 source-free wave ﬁeld, 366, 371 separability, 382 Sherman expansion, 374 spatial series representation, 382 space-invariant, 343 spatial average

455

of a microscopic function, 166 weighting function, 166 spatial dispersion eﬀects, 26 spatial transfer function, 343 spatially locally linear, 179 spatiotemporal Fourier-Laplace transform, 440 special theory of relativity dilation factor γ, 72 fundamental postulate, 67 longitudinal mass, 78 Lorentz–Fitzgerald contraction, 73 mass–energy, 78 postulate of the constancy of the speed of light, 67 proper diﬀerential time interval, 72 relativistic mass, 75 rest energy, 77 rest mass (proper mass), 75 time dilation, 71 transverse mass, 78 speed of light in vacuum c, 52 stationary phase method Kelvin, L., 10 steepest descent method Debye, P., 8 Olver, 24 Stone, J. M., 138 Stratton, J. A., 12, 91 streamlines electric, 149 magnetic, 149 substantial derivative, 72 superluminal pulse propagation, 28 superluminal pulse velocities, 22 surface of constant phase, 227 symmetry property, 282 tachyon, 79 test particle, 56 time-average of a periodic function, 243 electromagnetic energy velocity, 257 Poynting vector, 243, 249 Titchmarsh’s theorem, 183 transient anterior, 15 posterior, 15 transport equation, 160

456

Index

vacuum wavenumber k0 , 224 vector potential macroscopic, 285 microscopic, 110 velocity complex, 280 energy transport, 21 front, 8 group, 4, 260 phase, 5, 260 pulse centroid, 29 time-average energy transport, 260, 261 velocity ﬁeld, 131 virtual present radius vector, 135

˜ + (ω), 331 complex k ˜ ± (ω), 296 complex k wavelength λ, 147 wavenumber ˜ complex k(ω), 227, 297, 331 vacuum k0 , 297 wavenumber k, 147 wavevector complex part γ(ω), 294 Weyl’s integral polar coordinate form, 320 rectangular coordinate form, 319 Weyl’s proof, 317 Weyl’s proof , 310 Weyl, H., 310 Weyl-type expansion, 299 Whittaker-type expansion, 299 Wolf, E., 155 world distance lightlike, 70 spacelike, 70 timelike, 70

wave vector

Yaghjian, A., 55

transversality relation, 227 uniform asymptotic method Handelsman, R. and Bleistein, N., 12 uniqueness theorem microscopic electromagnetic ﬁeld, 101

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