Progress in Optics, Volume 37

PROGRESS IN OPTICS VOLUME XXXVII

EDITORIAL ADVISORY BOARD G. S. AGARWAL,

Ahmedabad, India

T. ASAKURA,

Sapporo, Japan

M.V. BERRY,

Bristol, England

C. COHEN-TANNOUDJI, Paris, France V. L. GINZBURG,

Moscow,Russia

F. GORI,

Rome, Italy

A. KUJAWSKI,

Warsaw, Poland

J.

Olomouc, Czech Republic

&&A,

R. M. SILLITTO,

Edinburgh, Scotland

H. WALTHER,

Garching, Germany

PROGRESS IN OPTICS VOLUME XXXVII

EDITED BY

E. WOLF University of Rochester, N.Y. US.A

Contributors G.P. AGRAWAL, T. ASAKURA, R.Y. CHIAO, D. DRAGOMAN, R.-J. ESSIAMBRE, 1.L. FABELINSKII, 0. KELLER, K.-E. PEIPONEN, A.M. STEINBERG, E.M. VARTIAINEN

1997

ELSEVIER AMSTERDAM LAUSANNE . NEW YORK . OXFORD. SHANNON. SINGAPORE. TOKYO

ELSEVIER SCIENCE B.V SARA BURGERHARTSTRAAT 25 P.O. BOX 21 1 1000 AE AMSTERDAM THE NETHERLANDS

Library of Congress Catalog Card Number: 6 1- I9297 ISBN Volume XXXVII: 0 444 82796 X

0 1997

ELSEVIER SCIENCE B.V.

All rights reserved

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PREFACE This volume presents six articles describing theoretical and experimental research of current interest in optics. The first article, by D. Dragoman, reviews some of the important applications of the Wigner distribution function in optics and optoelectronics, which is useful in the characterization of optical systems and beams, in the coupling between light sources and waveguides, and in nonlinear and ultrafast optics. Although the article focuses on the theoretical aspects, numerical simulations and experiments are also discussed. The usefulness of the Wigner distribution function approach in such a large area of research suggests that its applications are far from being exhausted. In the second article, K.-E. Peiponen, E.M. Vartiainen and T. Asakura review the mathematical foundations and the applicability of Kramers-Kronig relations to data inversion in linear and nonlinear optical spectroscopy. Also, the subject of phase retrieval by use of the maximum entropy model is discussed in connection with reflection spectroscopy and nonlinear optical processes. Sum rules for linear and nonlinear optical constants are presented. The next article, contributed by I.L. Fabelinskii, discusses experimental investigations and theoretical studies of spectra of molecular scattering of light arising from the temporal changes of optical inhomogeneities induced by pressure, entropy, concentration and anisotropy fluctuations. The investigations of such spectra make it possible to obtain much information about various equilibrium and nonequilibrium phenomena, such as the absorption and the velocity of hypersound (frequencies greater than lo9 Hz) and their temperature dependence in solids, liquids and solutions, and the velocity of first and second sound in liquid He 11. The fourth article, by R.-J. Essiambre and G.P. Agrawal, reviews the field of fiber-optical soliton communication systems. It starts by discussing the fundamental properties of fiber-optic solitons. The effects of fiber loss, dispersion, polarization-mode dispersion, and the Raman phenomenon on such solitons are described. The topics include periodic amplification, timing jitter and its control, time-division multiplexing, dispersion management, wavelength-division multiplexing, polarization-division multiplexing, and dark solitons. The article includes a discussion of the system design aspects and experimental results. V

VI

PREFACE

In the next article, contributed by 0. Keller, theoretical aspects of the local field electrodynamics in mesoscopic media are reviewed. These media are of particular importance because the so-called quantum size effects link the local field theory of bulk media to self-field (Lamb shift) and radiation-reaction (spontaneous emission) effects in atoms and molecules. After describing an electromagnetic propagator approach and a microscopic linear and nonlocal many-body response theory, the author goes on to treat the so-called loop equation for the transverse part of the local field. Various topics, including the short- and long-range electrodynamics of mesoscopic media with strongly localized electron orbitals are reviewed, and it is shown how the theory accounts for the linear and nonlinear local field electrodynamics of quantum wells and thin films. The review concludes with a brief description of optical near-field phase conjugation of the field from a mesoscopic particle, and an explanation of why understanding local-field effects may be helpful in the study of strong (subwavelength) spatial localization of matter-attached optical fields. In the concluding article, by R.Y. Chiao and A.M. Steinberger, recent experiments and theories are reviewed concerning the time it takes for a photon or an electromagnetic wave packet to tunnel across a barrier. Two controversial questions about the tunneling time are examined: the first concerns conflicting theoretical predictions of this time, and the second concerns the question whether the observed superluminal group velocities in tunneling of single-photon wave packets, femtosecond laser pulses and microwave pulses violate causality. In connection with the second controversy, the meaning of the superluminal group velocity which is predicted to occur in media with atomic population inversion is discussed. The authors conclude that Einstein causality is, in fact, not violated in any of these phenomena. Emil Wolf Department of Physics and Astronomy University of Rochester Rochester, New York 14627, USA May 1997

CONTENTS I . THE WIGNER DISTRIBUTION FUNCTION IN OPTICS AND OPTOELECTRONICS by D. DRAGOMAN (BUCHAREST. ROMANIA) $ 1 . INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . $ 2. LIGHTPROPAGATION IN PHASE SPACE . . . . . . . . . . . . . . . . . . . . $ 3. WIGNER DISTRIBUTION FUNCTION . . . . . . . . . . . . . . . . . . . . . . 3.1. Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2. Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3. Transformation laws . . . . . . . . . . . . . . . . . . . . . . . . 3.4. Optical generation . . . . . . . . . . . . . . . . . . . . . . . . . OPTICAL SYSTEMS WITH THE WlGNER $ 4 . LIGHTBEAMCHARACTERIZATION IN FIRST-ORDER DISTRIBUTION FUNCTION . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1, Transformation laws for the Wigner distribution function moments . . . . . 4.2. Propagation invariants . . . . . . . . . . . . . . . . . . . . . . . 4.3. Wigner distribution function transformation law and the beam energy variation 4.4. Comparison with other methods ofbeam characterization . . . . . . . . . $ 5. OPTICAL SYSTEM CHARACTERIZATION WITH THE WIGNER DISTRIBUTION FUNCTION . . . 5 . 1. Characterization of aberrated optical systems . . . . . . . . . . . . . . 5 6 . WIGNER DISTRIBUTION FUNCTION REPRESENTATION OF THE COUPLING E ~ ~ C I E N C Y . . . 6.1. Completely coherent single mode sources and waveguides . . . . . . . . . 6.2. Multimode, completely coherent sources and waveguides . . . . . . . . . 6.3. Partially coherent single mode sources and waveguides . . . . . . . . . . $ 7. THEFRACTIONAL WIGNERDISTRIBUTION FUNCTION . . . . . . . . . . . . . . . 7.1. Properties of the fractional Wigner distribution function . . . . . . . . . . 7.2. Optical beam characterization in the near-field diffraction regime . . . . . . 7.3. Optical production of fractional Wigner distribution hnction . . . . . . . . $ 8. OPTICAL BEAMCHARACTERIZATION IN NONLINEAR OPTICAL SYSTEMS . . . . . . . . 8.1. Soliton solution of the NLS equation . . . . . . . . . . . . . . . . . $ 9. COMPLEX FIELDRECONSTRUCTION FROM THE WlGNER DISTRIBUTION FUNCTION . . . . $ 10. WlGNER DISTRIBUTION FUNCTION IN QUANTUM OPTICS . . . . . . . . . . . . . . $ ~ ~ . C O N C L U ~ I. O. N. S. . . . . . . . . . . . . . . . . . . . . . . . . . . REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

vii

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I1. DISPERSION RELATIONS AND PHASE RETRIEVAL IN OPTICAL SPECTROSCOPY by KAI-ERIKPEIPONEN (JOENSUU. FINLAND). ERIKM . VARTIAINEN ASAKURA (SAPPORO. JAPAN) (LAPPEENRANTA. FINLAND) AND TOSHIMITSU

INTRODUCTION

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$ 2 . KRAMERS-KRONIG RELATIONS. . . . . . . . . . . . . . . 2.1. Hilbert transforms . . . . . . . . . . . . . . . . . 2.2. Kramers-Kronig relations in linear absorption spectroscopy 2.3, Kramers-Kronig relations in reflection spectroscopy . . . . . . . . . 2.4. Dispersion relations in nonlinear optics $ 3 . PHASERETRIEVAL IN OPTICAL SPECTROSCOPY . . . . . . . . 3.1. Phase retrieval using maximum entropy model . . . . 3.1 . I . Maximum entropy model . . . . . . . . . . . 3.1.2. Phase retrieval procedure . . . . . . . . . . . 3.2. Phase retrieval in practice: examples . . . . . . . . . 3.2.1. Reflection spectroscopy . . . . . . . . . . . . 3.2.2. Nonlinear optical spectroscopy . . . . . . . .

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$ 4. SWRULES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1. Sum rules in linear optics . . . . . . . . . . . . . . . . . . . . . . . 4.1.1. Complex refractive index . . . . . . . . . . . . . . . . . . . . 4.1.2. Complex reflectance . . . . . . . . . . . . . . . . . . . . . . 4.2. Sum rules in nonlinear optics . . . . . . . . . . . . . . . . . . . . . p 5 . CONCLUSIONS. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ACKNOWLEDGEMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

59 62 62 64 67 69 73 73 73 76 80 80 83 86 86 86 89 90 91 92 92

111. SPECTRA OF MOLECULAR SCATTERING OF LIGHT by I.L. FABELINSKII (Moscow, RUSSIAN FEDERATION)

INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 5 2 . THEFLUCTUATIONS OF THERMODYNAMIC QUANTITIES . . . . . . . . . . . . . . . 99 $ 3 . SPECTRA OF MOLECULAR LIGHTSCAITERING ARISING FROMPRESSURE FLUCTUATIONS &(P) . EQUILIBRIUM PHENOMENA . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1. Elastic thermal waves and emitter-generated acoustic waves . . . . . . . . . 3.2. Hypersound velocity and absorption . . . . . . . . . . . . . . . . . . . 3.3. Molecular light scattering spectrum and its intensity at low temperatures . . . . 3.4. Molecular light scattering spectra of gases . . . . . . . . . . . . . . . . 3.5. Molecular light scattering spectra in viscous liquids and glasses . . . . . . . 3.6. Light scattering by a two media interface . . . . . . . . . . . . . . . . . $ 4. SPECTRA OF MOLECULAR LIGHTSCATTERING ARISING FROMPRESSURE FLUCTUATIONS A E ( P .) SOMENONEQUILIBRIUM PHENOMENA . . . . . . . . . . . . . . . . . . . . . . 4.1. The influence of a steady temperature gradient on the light-scattering spectra . . 4.2. Phonon "bottleneck" in acoustic paramagnetic resonance . . . . . . . . . . 4.3. Amplification of hypersound waves in piezosemiconductors subjected to an external static electric field . . . . . . . . . . . . . . . . . . . . . . . . . .

102 110 112 114 116 119 122 124 124 129 I30

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8 5. SPECTRA OF MOLECULAR LIGHTSCATTERING ARISING FROM ISOBARIC ENTROPY FLUCTUATIONS A&(S)AND FROM CONCENTRATION FLUCTUATIONS A&(C) . . . . . . . . . . . . . 132 5.1. Central peak: thermal diffusivity and diffusion . . . . . . . . . . . . . . . 132 5.2. Rayleigh line and concentration fluctuations correlation radii in a critical region . 140 5.3. The investigation of acoustic peculiarities in the region of critical points of the 142 guaiacol-glycerol solution . . . . . . . . . . . . . . . . . . . . . . . 5.4. The Landau-Placzek relation . . . . . . . . . . . . . . . . . . . . . . 148 6. SPECTRA OF MOLECULAR LIGHT SCATTERING ARISING FROM ANISOTROPY FLUCTUATIONS A~('ik) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151 . . . . . . . . . . . . . 15 1 6.1, Spectrum of depolarized light scattered in liquids 6.2. Detection of the doublet structure of the spectrum . . . . . . . . . . . . . 157 6.3. General and simplified equations describing the spectra of light scattered in liquids consisting of anisotropic molecules . . . . . . . . . . . . . . . . . . . 169 177 5 7. ABOWSOMEPROBLEMS . . . . . . . . . . . . . . . . . . . . . . . . . . REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178

IV. SOLITON COMMUNICATION SYSTEMS by RENB-JEAN EsslAMBRE AND GOVIND P. AGRAWAL (ROCHESTER, NY. USA)

9: 1 . INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189

5 2.

p 3.

5 4.

0 5. $ 6. p 7.

OPTICAL SOLITONS IN FIBERS . . . . . . . 2.1. Nonlinear Schriidinger equation . . . . . . . . . . . . . 2.2. Soliton properties 2.3. Adiabatic perturbation theory . . . . SOLITON-BASED COMMUNICATION SYSTEMS. . 3.1. Information transmission with solitons . 3.2. Loss compensation . . . . . . . . . 3.3. Amplifier noise . . . . . . . . . . AVERAGE-SOLITON REGIME . . . . . . . . . . . 4.1, Evolution of the average soliton . . . . . . . . . . . 4.2. Timing jitter 4.2.1, Gordon-Haus jitter . . . . . . 4.2.2. Polarization-mode dispersion jitter 4.2.3. Acoustic jitter . . . . . . . . 4.3. Soliton interaction . . . . . . . . . 4.4. Soliton control . . . . . . . . . . 4.4.1, Optical bandpass filters . . . . 4.4.2. Synchronous modulators . . . . 4.4.3. Other techniques of soliton control 4.5. Experimental progress . . . . . . . QUASI-ADIABATIC REGIME . . . . . . . . DISTRrBUTED AMPLIFICATION. . . . . . . DISPERSION-DECREASING FIBERS . . . . . . 7. 1 . Basic idea . . . . . . . . . . . .

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190 190 192 195 196 196 198 200 201 201 203 203 204 205 206 209 209 211 213 214 217 221 223 223

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7.2. Timing jitter . . . . . . . . . . . . 7.3. Optical phase conjugation . . . . . . . $ 8. DISPERSION MANAGEMENT. . . . . . . . . 8.1, Dispersion compensation . . . . . . . 8.2. Dispersion profiling . . . . . . . . . . . . . . . . . . . $ 9. CHANNEL MULTIPLEXING 9.1. Wavelength-division multiplexing . . . . 9.1.1. Collision-induced frequency shifts . 9.1.2. Limitations on WDM channels . . 9.1.3. Timing jitter . . . . . . . . . . 9.1.4. Dispersion management . . . . . 9.2. Polarization multiplexing . . . . . . . $ 10. DARK-SOLITON COMMUNICATION SYSTEMS . . . 10.1. Dark-soliton characteristics . . . . . . 10.2. Dark-soliton advantages . . . . . . . . LISTOF SYMBOLS. . . . . . . . . . . . . . . LISTOF ACRONYMS. . . . . . . . . . . . . . REFERENCES . . . . . . . . . . . . . . . . .

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V. LOCAL FIELDS IN LINEAR AND NONLINEAR OPTICS OF MESOSCOPIC SYSTEMS by OLE KELLEK(AALBORG. DENMARK)

INTRODUCTION

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$ 2. LOCALFIELDS AND NONLOCAL OPTICS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1. Electromagnetic propagator approach 2.2. Electrodynamics within the framework of linear response theory . . . . . . . 2.3. Local-field calculations in mesoscopic media . . . . . . . . . . . . . . . . . . 2.4. Retarded local-field interaction at distance: space and time-like couplings 2.5. Non-retarded self-field interaction . . . . . . . . . . . . . . . . . . . . 5 3. LOCALFIELDSIN MESOSCOPIC MEDIAWITH STRONGLY LOCALIZED ELECTRON ORBITALS . 3 . 1 . Short- and long-range interactions in electronically decoupled molecular systems 3.2. Optically dilute molecular systems; point-particle model . . . . . . . . . . . 3.3. Mesoscopic media dominated by short-range interactions . . . . . . . . . . 3.4. Linear short-range interactions in two-level hydrogen-like (Is ++ 2p, ) systems . . 5 4 . LOCALFIELDELECTRODYNAMICS IN QUANTUM WELLS AND THINFILMS . . . . . . . . 4.1. Single quantum wells with 2D Bloch and free-electron dynamics . . . . . . . 4.2. Local-field resonances and eigenmodes . . . . . . . . . . . . . . . . . . 4.3. Non-retarded dynamics: self-field and scalar theories . . . . . . . . . . . . 5 5 . 2D SPATIAL CONFINEMENT OF LIGHTBY OPTICAL PHASE CONJUGATION . . . . . . . . . . . . . 5.1. Source field of a mesoscopic particle: attached and de-attached parts 5.2. Confinement by means of an ideal phase conjugator . . . . . . . . . . . . ACKNOWLEDGEMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

259 265 265 269 274 281 287 292 292 295 299 302 305 305 310 317 324 324 331 337 337

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VI . TUNNELING TIMES AND SUPERLUMINALITY by RAYMOND Y. CHIAO(BERKELEY. CA) AND AEPHRAIM M. STEINBERG (TORONTO. CANADA) Q 1. $ 2. $ 3. $ 4.

INTRODUCTION. . . . . . . . . . . . . . . A BRIEFHISTORY OF Tb"EL1NG TIMES . . . . . TUNNELING AND ITS OPTICAL ANALOGS . . . . . OPTICAL EXPERIMENTS ON TUNNELING TIMES . . . 4 . 1. Carniglia and Mandel's FTIR experiment . . 4.2. Absorptive media with anomalous dispersion 4.3. The Milwaukee group . . . . . . . . . 4.4. The Florence group, part I . . . . . . . . 4.5. The Cologne group, part I . . . . . . . . 4.6. The Berkeley group . . . . . . . . . . 4.7. The Florence group, part I1 . . . . . . . . . . . . . . 4.8. The Cologne group, part I1 4.9. The Vienna group . . . . . . . . . . . 4.10. Deutsch and Golub's Larmor-clock experiment 4.1 1. Balcou and Dutriaux's FTIR experiment . . 5 5 . NEWTHEORETICAL PROGRESS . . . . . . . . . $ 6 . TUNNELING IN DE BROGLIE OPTICS . . . . . . . $ 7 . SUPERLUMR'IALITY AND INVERTED ATOMS . . . . . $ 8. WHY Is EINSTEIN CAUSALITY NOTVIOLATED? . . Q 9. CONCLUSION . . . . . . . . . . . . . . . . ACKNOWLEDGMENTS . . . . . . . . . . . . . . . NOTEADDED IN PROOF . . . . . . . . . . . . . . REFERENCES. . . . . . . . . . . . . . . . . .

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AUTHOR INDEX . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . SUBJECT INDEX . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . CONTENTS OF PREVIOUS VOLUMES . . . . . . . . . . . . . . . . . . . . . . . . CUMULATIVE INDEX . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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E. WOLF, PROGRESS IN OPTICS XXXVII 0 1997 ELSEVIER SCIENCE B.V. ALL RIGHTS RESERVED

I

THE WIGNER DISTRIBUTION FUNCTION IN OPTICS AND OPTOELECTRONICS BY

D. DRAGOMAN University of Bucharest, Physics Department, PO.Box MG-63. Bucharest, Romania

1

CONTENTS

PAGE

INTRODUCTION. . . . . . . . . . . . . . . . . . .

3

LIGHT PROPAGATION IN PHASE SPACE . . . . . . . .

4

WIGNER DISTRIBUTION FUNCTION . . . . . . . . .

6

LIGHT BEAM CHARACTERIZATION IN FIRST-ORDER OPTICAL SYSTEMS WITH THE WIGNER DISTRIBUTION FUNCTION . . . . . . . . . . . . . . . . . . . . .

13

OPTICAL SYSTEM CHARACTERIZATION WITH THE WIGNER DISTRIBUTION FUNCTION . . . . . . . . . .

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WIGNER DISTRIBUTION FUNCTION REPRESENTATION OF THE COUPLING EFFICIENCY . . . . . . . . . . . . . 26 THE FRACTIONAL WIGNER DISTRIBUTION FUNCTION.

32

OPTICAL BEAM CHARACTERIZATION IN NONLINEAR OPTICAL SYSTEMS . . . . . . . . . . . . . . . . .

38

COMPLEX FIELD RECONSTRUCTION FROM THE WIGNER DISTRIBUTION FUNCTION . . . . . . . . . . . . . .

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WIGNER DISTRIBUTION FUNCTION IN QUANTUM OPTICS 49 CONCLUSIONS. . . . . . . . . . . . . . . . . . . REFERENCES.. . . . . . . . . . . . . . . . . . . . . .

2

53 53

5

1. Introduction

The Wigner distribution function (WDF) was introduced by Wigner [ 19321 in quantum mechanics as a mathematical tool that correctly yields the expectation values of any function of the coordinates or the momenta. Since its introduction, the WDF has been applied in many branches of physics, including statistical mechanics, nuclear physics, condensed matter, acoustics, and optical signal processing. The excellent book by Kim and Noz [1991] about WDF applications in quantum mechanics includes a comprehensive list of applications in these branches. Other WDF applications in speech analysis, instantaneous frequency measurement, binary detection problems, coherence, and others with corresponding references are listed in Cohen [1989]. Gase [I9901 calculated the WDF of multilayer systems. In optics the WDF was introduced by Walther [1968] to relate partial coherence to radiometry. The success of the WDF formalism is based on the fact that it provides a simultaneous description of optical phenomena in phase space (spatial and angular coordinates for time harmonic fields or in both spatiotemporal and angular-frequency coordinates for light pulses), and is constant along the geometrical ray path at propagation through first-order optical systems. This property suggests that the WDF is an intermediate description of the field distribution between wave and geometrical optics that can be regarded as a local spatial spectrum for time harmonic field distributions or a local and momentary spectrum for light pulses. The purpose of this review is to present WDF applications to the characterization of light fields and optical systems and to the problem of coupling optimization between sources and waveguides. This brief review is supported by a comprehensive and more detailed bibliography.

3

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THE WIGNER DISTRlBUTION FUNCTION IN OPTICS AND OPTOELECTRONICS

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[I, 8 2

2. Light Propagation in Phase Space

The phase space approach to light propagation was introduced in geometrical optics by Hamilton many years ago. The Hamiltonian equations of motion of a light ray, dr dz

-

dH

dp

8p’

dz

-

dH dr’

can be derived from Fermat’s principle of extremal optical path (Luneburg [1966]). In eq. (2.1) rT=(x,y) is the ray position vector in a z=const. plane of an orthogonal system of coordinates Oxyz and pT= (px,p,,) is its canonical conjugate variable - an angular variable, sometimes called spatial frequency. In geometrical optics z and p correspond to the time t and to the momentum in classic mechanics. The geometrical optical Hamiltonian H(r, p ) = where n(r) is the local refractive index of the medium, reduces in the paraxial approximation (IpI << n) to H(r, p ) = p2/2n(r)- n(r). When H(r,p) is a quadratic function of its variables as, for example, in graded index media where n(r) = no + n2(lrI2) with In21 << no, the ray evolution along the optical axis consists of a succession of linear canonical transformations that can be put in a matrix form:

- d w ,

The matrix S relates the ray vectors (rTpT)in the output (z,, =const.) and input planes (zi= const. =z, - Az) of the optical systems denoted by the subscripts o and i, respectively. S is symplectic (Guillemin and Sternberg [1984]); that is, it satisfies the condition STJS= J , where J

=

(I ) 0 -I

(2.3) with 0 and I as the two-dimensional (2D) null and unit

matrix, respectively. Thus in the Hamiltonian formalism the ray evolution can be described in a 4D space spanned by r andp, which is called phase space (PS). In geometrical optics the 4 x 4 matrix S with unit determinant and real elements takes particular simple forms for optical systems with two principal axes, when the submatrices A to D are diagonal, and for rotationally symmetric optical systems, when A to D are multiples of the 2 x 2 identity matrix. In these cases

1,

5 21

5

LIGHT PROPAGATION IN PHASE SPACE

the ray evolution can be studied independently in two 2D PS (x,px)and (y,py) with corresponding 2 x 2 S matrices. The A , B, C, and D elements are then numbers, and the symplectic condition reduces to det S = 1. To avoid complicated notations, the elements A , B, C , and D will denote both 2 x 2 matrices or numbers; the distinction between these two cases is clearly indicated in the text. S matrices with complex elements were introduced from a wave optical point of view (see, e.g., Yura and Hanson [1987] and Siegman [1986]). The optical systems that can be completely characterized by a symplectic matrix S are called first-order optical systems. A ray that at a z = const. plane has a position vector r and a direction, in the paraxial approximation, o f p is represented in the PS plane by a point. An optical light source formed of a bundle of independent rays is then represented in the PS as a closed area. Under a canonical transformation the shape of the PS area generally changes, but its magnitude remains constant. This result is known as the Liouville theorem, and is not restricted to the paraxial approximation. Examples of ray tracing in PS coordinates can be found in Goethals [ 19891 and Lichtenberg [ 19691. Phenomena such as diffraction, interference, coherence, or polarization cannot be managed in the framework of geometrical optics but only within wave optics, where the light field is characterized by a vectorial distribution A(r, z, t ) that satisfies the Helmholtz equation. A PS analysis of light propagation must now be performed in a more general 6D space of coordinates (r,p, t , w), where w denotes the light frequency (see, e.g., Kostenbauder [ 19901). Generally, not all PS coordinates are implied in the field characterization: for time harmonic fields only the spatial PS coordinates (r,p) are necessary, whereas for light pulses only the temporal PS coordinates ( t ,w ) are used. Moreover, for rotationally symmetric or 1D fields the spatial PS coordinates are reduced to (x,px). Although, in general, this chapter examines the spatial PS, references are also made throughout to the temporal PS. We consider here only scalar wave fields, and if the formulas become excessively complicated or for graphic reasons, discussion is restricted to 1D field distributions. A spatial PS description of a scalar coherent and time harmonic field rp(r,z) can be achieved with either one member, of a class of PS distribution functions that at a z=const. plane are written as 1 c V ( r , p ) = 4n2

J exp [ i k ( q T r - g ~ p - q ~ U ) ]Q(q, g ) v ( u + f )

v*

(-4)

dudgdq,

(2.4) where @(q,5) is the kernel of the integral representation and k =2;1d/L By changing r with t and kp with w, a class of temporal PS distributions is obtained

6

THE WGNER DISTRIBUTION FUNCTION IN OPTICS AND OPTOELECTRONICS

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that describes light pulses. Different members of the class are individualized by a different choice of the kernel. A systematic discussion of the constraints to be imposed on the kernel to obtain a PS distribution function with required properties can be found in Claasen and Meklenbrauker [198Oc] and Cohen [1989]. Any two members of the class of PS distributions C,, and C,,, with corresponding kernels @ I and @2, are related as

By using this relation new results for one distribution can be simply derived if the corresponding results for another distribution are known. Among the members of the class of PS distribution functions, we note the Wigner distribution function (@(q, f) = l), the Margenau-Hill distribution (@(q, 5) = cos(kqTf/2)), the Rihaczek distribution (@(q, f) = exp(ikqTf/2)), the Page distribution (@(q, f) = exp(ikqT lljl/2)), the Choi-Williams distribution (@(q, f) = exp(-k2q2g2/a)), and the spectrogram @(q, 5) =

/

h*(u- f/2) exp(ikqTu/2) h(u + f/2) du.

Although any of these distribution functions can be used for a PS description, the WDF is preferred in a large number of applications for its properties and for its very simple transformation law through first-order optical systems.

0

3. Wigner Distribution Function

The WDF of a scalar, time harmonic, and coherent field distribution p)(r,z) can be defined at a z=const. plane in terms of either the field distribution or its Fourier transform G(p, z): (3.la)

s

where @(p) = q(r) exp(ikrTp)dr. The z parameter was dropped from the formulas because all the integrals are performed over a z=const. plane. The definition of the WDF in eq. (3.la) was chosen such that p retains its meaning

1,

I 31

WlGNER DISTRIBUTION FUNCTION

I

as in geometrical optics. Since other definitions may appear in the literature, care must be taken when comparing the results. The WDF is a bilinear function of the field distribution, a fact that can be harmful, for example, in pattern recognition because the WDF of a linear combination of fields contains all kinds of cross-WDF terms; that is,

where the cross-WDF is defined as:

Wq,,,, (r, p ) =

J qI (r +

g)

q; ( r -

g)

exp(ikrtTp)dr'.

The quadratic behavior of the WDF can be desirable in other applications where the field enters the theory quadratically. For example, it allows the extension of the WDF formalism to stochastic instead of deterministic fields without increasing the dimensionality of the formulas (Bastiaans [ 19861). The WDF of a temporally stationary stochastic field distribution (partially coherent light) is defined by

W r (r, p ) =

1r

(r

r'

+ 7 ,r -

(3.3)

where T(rl, r2) is the correlation function defined as the ensemble average of q(r1)q*(r2).For a vectorial field distribution the WDF is a tensor, as discussed in Bugnolo and Bremmer [1983]. 3.1. PROPERTIES

The most important properties of the WDF of a coherent field are listed here. A more detailed discussion and demonstration of WDF properties can be found in Claasen and Meklenbrauker [ 1980al. PI : The WDF of any field distribution cp is a real but not non-negative function. For real field distributions the WDF is an even function of the angular variable; that is, W,(r,p) = W,(r, -p). P2: A shift in the spatial coordinate of the field distribution shifts the WDF with the same amount; that is, the WDF of cp'(r) = q ( r + ro) is W,(r + ro, p ) . P3: A modulation of the field with exp(ikrTpo) shifts the WDF in the angular coordinate; that is, the WDF of cp'(r) = p ( r )exp(ikrTpo) is W,(r, p +PO). Pq: If the field distribution cp(r) is limited to a certain space interval, the WDF is limited to the same interval. By analogy, if the Fourier transform of the field q ( p ) is limited to an angular interval, the WDF is limited to the same interval.

8

THE WGNER DISTRIBUTION FUNCTION IN OPTICS AND OPTOELECTRONICS

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5

3

Ps: The integral of the WDF over the angular variable has a clear physical meaning. Namely it is proportional to the intensity of the field distribution (3.4) P g : The integral of the WDF over the spatial variable equals the intensity of the Fourier transform of the field distribution (the field spectrum):

The WDF integrals in P5 and P6 are called the marginals of the WDF. P7: The field distribution and its Fourier transform can be reconstructed from the WDF, up to a constant phase factor, by using the inversion formulas:

W I ) CP*(r2)= 4n2 k2

/

W q ( 2E , p ) exp(-ik(rl -r2)Tp) dp,

(3.6a) (3.6b)

Pg: Another property that will be used later was formulated by Moyal, and refers to the relationship between the WDF of two field distributions and the fields themselves:

(3.7)

It states that the average of the product of two WDF’s is always non-negative. Equations (3.6a) and (3.6b) can be interpreted as the conditions that a function must satisfy in order to be a WDF: a function of r and p is a WDF only if the integrals in the right-hand side of eqs. (3.6a) and (3.6b) are separable in the form of the left-hand side. De Groot and Suttorp [I9721 demonstrated that the necessary and sufficient condition for a function of two variables to be a WDF is (3.8) 4x2

for any a and 6 .

1,

9: 31

WIGNER DISTRIBUTION FUNCTION

9

Properties PI to P4 hold also for partially coherent field distributions. Properties Ps and P6 are valid in a slightly modified form: the integral of the WDF of a partially coherent field over the angular variable equals (4x2/k2)r(r, r), where T(r,r) is the positional intensity of the light field, and the WDF integral over the spatial variable equals the directional intensity of the field r ( p ,p ) . [ r ( p ,p ) is the Fourier transform of T(r, r).] The Moyal’s relation for partially coherent light reads as (Bastiaans [ 19861):

-

k2 4x2

1i.l

(PI,

p2) f;(pl,p2) dpl dp2.

(3.9) The definition and properties of the discrete WDF are extensively presented in Claasen and Meklenbrauker [1980b]. 3.2. EXAMPLES

1. A point source located at ro is described by q(r) = 6(r - ro), and its WDF is Ww(r,p ) = 6(r - ro). This shows that only the point r = ro contributes to the WDF. All the frequencies are present at this point. 2. The plane wave can be represented by a harmonic field distribution q(r) = exp(ikr*po) or equivalently by G ( p )= (4n2/k2)S(p +PO). The WDF is WV(r,p ) = (4n2/k2)6 ( p +PO).The angular variable has a unique value p =-PO at all points. 3. A 1D Gaussian beam q(x) = exp(-x2/xi - ikx2/2R) with real xo and R, also has a Gaussian WDF: WV(x,p x ) = fixoexp(-2x2/x~ - ( p x- ~ / R ) ~ k ~ x ; / 2 ) . The Gaussian beam is the only one for which the WDF is positive everywhere. 4. A 1D Gauss-Schell beam that successfully models a class of partially coherent field distributions is described by a correlation function

The WDF is Wr(x, p x ) = (&/m)exp(-4q2x2

-

( p - ~/R)~k’/4rn’).

3.3. TRANSFORMATION LAWS

The transformation law for the WDF follows from that of the field distribution. For coherent time harmonic fields we distinguish among three situations:

10

THE WGNER DISTRIBUTION FUNCTION IN OPTICS AND OPTOELECTRONICS

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5

3

Case A. The field evolution through an optical system is described by an integral equation qo(ro)=

1

h (roj ri>qi (ri) dri,

(3.10)

where h is the point-spread function of the system and ro, r, are the transverse ray positions at the output and input planes, respectively, of the optical system. The integral equation can relate not only the output and input field distributions but also their Fourier transforms or even the field at one plane to the Fourier transform at the other. All these situations are detailed in Bastiaans [1979a]. The final propagation law for the WDF is the same in all cases, and therefore we refer here, for clarity, only to the case described by eq. (3.10) The transport equation for the WDF reads wcp0(ro, Po) =

(5) 1

K (ro, Po,

rir

pi) wq,(ri, Pi>dridpit

(3.11)

where the kernel of the integral equation is given by K(ro, po, ri, pi) = x exp(ik (r:Tp,- rbTpo))drb drl.

Bastiaans [1979a] showed that if the modulus and phase of the point-spread function h(ro, ri) = Ih(ro, ri)I exp(ikV(r,, ri)) are slowly varying hnctions of their variables; that is, for geometrical optical systems, the kernel of the integral equation simplifies to

For real optical systems, V(ro,r,) is identical to the Hamilton's point characteristic that can be expressed in the paraxial approximation in terms of the elements of the geometrical optical matrix S (Luneburg [ 19661): V(ro, ri) = Vo +

ir? B-' A r i + lr;fDB-'

ro - r: B-' ro.

Thus the propagation law of the WDF through first-order optical systems characterized by a real matrix S becomes Wqo(r, p) = WV,(DTr- BTp, -CTr + ATp) .

(3.13)

Equation (3.13) implies that the WDF is constant along the geometrical ray paths. The propagation law of the WDF of ID fields through first-order optical systems

1,

P

31

11

WGNER DISTRIBUTION FUNCTION

characterized by a complex matrix S was derived by Dragoman [1995a,b]. The general expression simplifies for Gaussian field distributions to a form similar to that in eq. (3.13) with elements A , B , C, and D that also depend on the Gaussian field parameters. The transformation law of the spatiotemporal WDF through first-order optical systems was used by Paye and Migus [ 19951 to analyze a pulse shaper. Case B. The evolution of the field distribution is described by a linear transformation that can be put in a matrix form: =

(

T(A-4

V(ri)) o,z(rJ

’

(3.14) where o,,=ao,/dz. This case is represented by a stratified medium (Born and Wolf [ 19641). For example, the electric field component o, of a TE wave and its z derivative at two planes situated at a distance Az apart in a homogeneous layer are related by eq. (3.14) with tl(Az) = t 4 ( b ) = cos(kAz), t>(Az) = sin(kAz)/k, t3(Az) = -ksin(kAz). The vector of WDF, WT = (Wp, Wp,+,=, Wpz.p, Wpz), also satisfies a linear transformation law between the input and output planes. Namely,

w,= ( T

T * )wi

@I

=IWi,

(3.15)

where c31 denotes the direct product of matrices (Barnett [1990]). Case C. The evolution of the field is described by a generally nonlinear differential equation: (3.16) If H = d k 2 ( r ) + d2/dr2, eq. (3.16) is the Helmholtz equation in a transverse inhomogeneous medium. If H =-(k + ( 1/2k)d2/dr2), (3.16) describes the free space evolution of a field in the Fresnel approximation. In the framework of quantum mechanics, eq. (3.16) represents (with z replaced by t ) the timedependent Schrodinger equation. The transport equation for the WDF is (Bastiaans [ 1979b1, Dragoman [ I996a1)

(3.17)

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THE WIGNER DlSTRlBUTlON FUNCTION IN OPTICS AND OPTOELECTRONICS

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53

Here the parametric dependence of the WDF on z is shown. The last term in the argument list of the operators H and H' in eq. (3.17) disappears for linear operators. The transport equations for the WDF in cases A, B and C given, respectively, by eqs. (3.13), (3.15) and (3.17) hold also for partially coherent field distributions (Bastiaans [1986], Simon, Sudarshan and Mukunda [1985]). Case C for inhomogeneous and dispersive media was examined by Bastiaans [ 1979~1.The WDF transformation law of a Gaussian-Schell field through optical systems characterized by complex matrices satisfies in certain cases, a relation similar to eq. (3.13) but with matrix elements which depend on the field distribution (Dragoman [ 1995b1). 3.4. OPTICAL GENERATION

Many devices have been proposed to implement the WDF of 1D or 2D field distributions. One approach to the WDF realization of real 1D fields is by a one-way light passage through a Fourier transform setup (Bartelt, Brenner and Lohmann [1980]). The input consists of a transparency that represents the field distribution, multiplied by a moving Gaussian window. At the output plane that moves synchronously with the window the local spectrum of the WDF of the field convolved with the Gaussian slit transmission is recorded and the WDF is then determined. For real time applications another setup is developed with a fixed but rotated Gaussian window (Bartelt, Brenner and Lohmann [1980]). To avoid the influence of the window, two-way light passage setups were developed for the WDF realization of ID real and arbitrary complex fields (Brenner and Lohmann [1982]). The object consists of a transparency for real fields and a hologram for complex light distributions. The coordinate inversion needed for WDF generation (see eq. (3.la)) is realized with a roof top prism. Another oneway light passage setup was proposed instead of two-way light passage devices (Eichmann and Dong [1982]), but it uses two transparencies rotated relative to each other. If the incident field distribution is 2D, its WDF is 4D, and is therefore impossible to record. To obtain information about the WDF of 2D field distributions however, several setups have been proposed that generate sections of the 4D WDF (Bamler and Glunder [1983], Conner and Li [1985], Iwai, Gupta and Asakura [1986]. These sections can then be displayed in parallel or as temporal sequences. Generating the WDF for digital processing is difficult because of the large number of operations needed to compute the WDF from sampled data. For

I,

5 41

13

WDF LIGHT BEAM CHARACTERIZATION IN FIRST-ORDER OPTICAL SYSTEMS

input plane

.

f

fo

Slit 2

Slit 1

4

output plane

Quadrupole

Lens fo

4

.

Fig. 1. Setup for phase space (PS) characterization of a field distribution

applications where a discrete WDF is required, it is computed from other PS distributions functions easier to implement as the Radon transform (Easton, Ticknor and Barrett [ 19841) or Hartley transform (Berriel-Valdos, Gonzalo and Bescos [ 19881). A final setup that has been used not only for the PS representation of field distributions but also for the measurement of the beam quality factor is shown in fig. 1. This device was analyzed from both a geometrical and a wave optical point of view by Weber [1992]. It consists of a rotated quadrupole of focal length f placed in the focal plane of a spherical lens of focal lengthfo and of two slits that eliminate the incident field structure along the y direction and collimate the beam. If d , and d2 are chosen such that I l d l + l/(d2 +YO)= l/fo, a PS map of the incident field distribution along the x axis can be observed at the output plane; namely, x, = -(d2/f0)xi, yo = -(fod2/f)px,.Since the WDF evolves along optical ray paths, this PS representation can also be regarded as a WDF generation. A time-frequency analog of the setup represented in fig. 1 will be described in 9 9. It generates a temporal PS representation of the incident field distribution.

5 4. Light Beam Characterization in First-Order Optical Systems with the Wigner Distribution Function An important application of the WDF is the characterization of light beams by means of the WDF moments. The moment of order i + j + k + 1 of the WDF of a field distribution is defined as k I = XY PxPy

( x -fIi

0,- j Y

@x

s

-mk(Py - P y ) '

W&, Y , P x , P y ) h d Y dPx dPy W , k Y >P x , P y ) dY dPx dPy

(4.1)

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THE WGNER DISTRIBUTION FUNCTION IN OPTICS AND OPTOELECTRONICS

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54

where

and i to 1 are non-negative integers. The same definition applies for partially coherent fields with W, replaced by Wr. One of the areas in which the WDF moments have important applications is that of pattern recognition. Teague [1980a] demonstrated that an image can be reconstructed if all its spatial moments are known. The first-order spatial moments locate the center of the considered field distribution, the second-order spatial moments characterize the size and orientation of the image, and higherorder moments determine the details of the field distribution. In practice the first 20 order spatial moments are usually sufficient for an accurate image reconstruction. The optical calculation of the spatial moments of an arbitrary order was analyzed by Teague [198Ob]. For the overall characterization of completely or partially coherent field distribution as for the definition of the optical axis, the principal axes, beam width, divergence, and other characteristics, the exact shape of the light field is not important. The optical axis is defined by the condition that the first-order moments of the WDF are zero. These moments can always be made equal to zero by appropriately shifting or rotating the system of coordinates. The principal axes of an arbitrary field can be defined by two conditions. One set of principal axes is determined by X y = 0, and another set is defined by the requirement that pxpy=O. Unlike the first set of principal axes, the second is independent of z. The z dependent rotation that should be imposed on the first set of principal axes, to superimpose them on the second set of principal (or absolute) axes provides a classification scheme of partially or completely coherent beams (Serna, Mejias and Martinez-Herrero [ 19921). The second-order spatial moments x' and are a measure of the beam width along the x and y directions. Analogously, and characterize the angular divergence of _ the _ field distribution. Moreover, the beam waist is defined as the _ _ plane where (x2 + y 2 ) is minimum. The equation a(x2 +y2)ldz = 0 is equivalent to xp,+yp,=O, both of which are invariant to rotations of the axes. _ reference _ The condition for the beam waist always has a solution unless p: = p j = 0, that is unless the field is a plane wave (Serna, Martinez-Herrero and Mejias [1991]). In applications where a small focus diameter and/or a large Rayleigh range are important, it is often desirable to have a criterion to compare two field

7

2

3

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WDF LIGHT BEAM CHARACTERJZATION IN FIRST-ORDER OPTICAL SYSTEMS

15

distributions from the point of view of the just mentioned requirements. Such a criterion is offered by the beam quality factor Q, defined as

Q takes its minimum value for a Gaussian beam. The ratio between the Q value of an arbitrary beam and that of a Gaussian, usually denoted by M 2 , is a measure of the relative far-field spreading of an arbitrary beam relative to an ideal Gaussian beam with the same waist. It is also a measure of the number of transverse modes of a given field distribution (Lavi, Prochaska and Keren [ 19881). A discussion of the beam quality factor for annular lasers can be found in Gase [1992]. A definition of WDF moments and of the beam quality factor in the general spatiotemporal PS coordinates can be found in Lin, Wang, Alda and Bernabeu [ 19931. The sharpness or flatness of a light field is characterized by the higherorder moments of the WDF. For 1D field distributions the kurtosis parameter

z/

K = (7)zprovides a quantitative measure for beam classification according to their sharpness. A beam is leptokurtic, mesokurtic, or platykurtic if K is higher, equal to, or lower than 3, which is the value for a pure 1D Gaussian (Piquero, Mejias and Martinez-Herrero [ 19941). Martinez-Herrero, Mejias, Sanchez and Neira [ 19921 showed that x'accounts for the symmetry of the 1D field distribution, .'px and x'px are, respectively, measure of the spatial -a -range of the beam's symmetry and sharpness, and x2p:/(x2p:) is a measure of the similarity of an arbitrary beam with a quasihomogeneous field distribution. Finally, we _-note that for any 1D completely or partially coherent field distribution x2p: >, 1/2k2 (Papoulis [1968]); this inequality reminds us of the uncertainty principle in quantum mechanics if k is replaced by h. The analogy is only formal: in quantum mechanics it refers to the probabilistic aspect of measuring, whereas in optics it is only a relation satisfied by WDF moments and is not concerned with the measurement of the WDF. The equality sign holds for the Gaussian field distribution. Apart from the moments defined by eq. (4.1) that are called global moments, the first-order local moments have also a clear physical meaning. Thus P(r) = JpW(r, p ) d p / s W(r, p ) dp can be interpreted as the average angle at position r , and R ( p ) = rW(r, p)dr/J W ( r ,p)dr is the average ray position at the anglep. In time-frequency coordinates, T ( w ) = J tW(t, O ) dt/J W ( t , w)dt is the group delay of the optical pulse and Q ( t ) = wW(t, w)do/J W ( t , W ) dw represents the instantaneous frequency (Claasen and Meklenbrauker [ 1980al).

s

s

16

THE WIGNER DISTRIBUTION FUNCTION IN OPTICS AND OPTOELECTRONICS

[I, 0 4

The beam quality factor can be measured with the device in fig. 1 (Hodgson, Haase, Kostka and Weber [ 1992j), arbitrary-order spatial moments can be measured as suggested by Teague [ 1980bl and all the second-order moments of a 2D field distribution can be measured as proposed by Nemes and Siegman [1994]. 4.1. TRANSFORMATION LAWS FOR THE WIGNER DISTRIBUTION FUNCTION

MOMENTS

In the previous section we presented some applications of the WDF moments for light beam characterization. As for the WDF itself, the moments are calculated at a z=const. plane. The transformation laws of the WDF moments from one z = const. plane to another allow the prediction of beam parameters’ modification at propagation. The transformation laws of the WDF moments follow from those of the WDF itself and were derived only for cases A and B in 9: 3.3. Case A. Transformation laws through first-order optical systems. To obtain a compact form of the transformation laws of WDF moments through real firstorder optical systems, the moments of a given orderj are arranged in a matrix Mi defined as

j times

where again @ denotes the direct product of matrices and the bar over the matrix in eq. (4.3) indicates that each matrix element is averaged in the sense of eq. (4.1). Dragoman [ 19941 showed that the moment matrix of orderj propagates through a first-order optical system characterized by a real matrix S according to the following relation:

--

Mjo = (S @ s @ . . .) Mji (S.@ [j/2] times

s @ . . .y .

(4.4)

( j - [.;/2]) times

In particular, one has MI, = SMl,, Mzo = SM2,ST, M30 = SM3,(S @J S)T,and M~o=(S@S)M~~(S@S)~. The transformation law of WDF moments through complex matrix optical systems has a form similar to eq. (4.4) only in a few cases: for Gaussian beams and for Gaussian-Schell beams that propagate through ripple systems and systems characterized by a matrix with elements A = a(l + i)/&, B = 6,

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WDF LIGHT BEAM CHARACTERIZATION IN FIRST-ORDER OPTICAL SYSTEMS

17

C = c, and D = d( 1 - i ) / a with real a to d and ad - bc = 1. In these cases the symplectic matrix S in eq. (4.4) should be replaced by another symplectic matrix that depends on both the optical system and the field distribution (Dragoman [ 1995b1). Case B. Transformation laws through a stratified medium. In a stratified medium where the electric field q and its z derivative at two different planes are related by a matrix T and the WDF vector transforms according to eq. (3.15), the moments of arbitrary order satisfy the same transformation law as the WDF itself. By assuming that Wq(r, p ) dr dp = Wqz(r,p ) dr dp = 1 and defining a moment vector of order i + j + k + Z with respect to the optical axis as

s

s

where

it follows that the transformation law of the moment vector is x'yJp:p;,

=

7xiyJp:p;i.

(4.5)

4.2. PROPAGATION INVARIANTS

The existence of propagation invariants of a field distribution can help to compare or even to classify different fields according to a required criterion. Up to now invariants have been searched only for the propagation through firstorder optical systems. The first invariant that was found is the determinant of the second-order moment matrix, det M2 (Bastiaans [ 19891). The invariance of det M 2 follows from the transformation law of M2 and the symplectic properties of the real or complex S matrix (see (2.3), (4.4) and the discussion that follows). det M2 = Q2, a result that shows that the beam quality factor is constant at propagation through first-order optical systems. This result holds also for the definition of the beam quality factor in the general spatiotemporal PS (Lin, Wang, Alda and Bernabeu [ 19931). Unlike Q, the kurtosis parameter is not invariant at propagation. Its behavior at free propagation was used to classify the light beams (Martinez-Herrero, Piquero and Mejias [ 19951). Another class of invariants at propagation through real or complex matrix systems for which eq. (3.13) holds was introduced by Onciul [ 1993al and Dragoman

18

THE WGNER DISTRIBUTION FUNCTION IN OPTICS AND OPTOELECTRONICS

[I,

54

[ 1995b] in terms of the Lagrange ray invariant a(r, p , r’, p ’ ) = (rTpT)J($) = a(p, p’), where p=(r,p), p’= (r‘,p’) are two points in the PS. The elements of this class are defined as 121 =

J a2‘(r, p , r’, p ’ ) W ( r ,p ) W(r’, p’) dr dp dr’ dp’ J W(r, p ) W(r’,p’) d r dp dr’ dp’

9

where 1 is an integer; 1212 is again identical to Q. The determinant and trace of some combinations of moment matrices up to the fourth order were found to be invariant (Martinez-Herrero, Mejias, Sanchez and Neira [1992]). Although physical interpretations have been given to these invariants, it is not clear why these combinations were chosen or if these are the only invariants. A systematic approach to find beam invariants, first published by Bastiaans [1991], showed that the product between the 4 x 4 second-order moment matrix M2 and the antisymmetric matrix J satisfies a similarity transformation at propagation through first-order optical systems. The eigenvalues of M2J and any combination of them are then invariant at propagation. Several properties of the eigenvalues of M2J have been derived such as (i) the eigenvalues are is an eigenvalue, is also an eigenvalue; and (iii) if M2 is real; (ii) if proportional to a symplectic matrix with a positive proportionality factor m, the two positive eigenvalues of MzJ are both equal to +m and the two negative eigenvalues are equal to -m. Similarity transformations at propagation through first-order optical systems have also been derived for combinations of higherorder moment matrices and the J matrix. Properties (i) and (iii) were found to hold for the combinations of even order moment matrices, whereas (ii) holds for odd order moment matrices and for even order moment matrices M2k with k an odd integer (Dragoman [1994]). A similar approach for finding beam invariants from the second-order moment matrix was taken by Anan’ev and Bekshaev [ 19941. WDF moments’ propagation laws and propagation invariants through misaligned optical systems were found by Onciul [ 1993bl.

x

-x

4.3. WIGNER DISTRIBUTION FUNCTION TRANSFORMATION LAW AND THE BEAM ENERGY VARIATION

The integral of the WDF over its variables is proportional to the total energy of the beam E: W ( r ,p ) d r dp = ( 4 n 2 / k 2 ) E .This relation follows from eq. (3.4). The propagation law for the WDF through first-order optical systems determines the relation between the beam energy at the output plane and that at the input

1,

5 41

WDF LIGHT BEAM CHARACTERIZATION IN FIRST-ORDER OPTICAL SYSTEMS

19

plane of the optical system. For 1D field distributions such a relation was derived by Dragoman [ 1996bl for arbitrary light distributions: E,=y/Wi(x,p,)exp

Re-Im--Re-lmA C B B B

”> B

where k2exp(4kIm VO)

4n36(Bl2 ’ and V, is the constant part of the Hamilton’s point characteristic function. If IB ( = 0, other expressions in terms of the angular or mixed characteristic functions can be used instead of eq. (4.8), as explained in Dragoman [1996b]. When all elements of the optical system matrix as well as VO are real, E, = (k/2n) Wi(x,p x )dx dp, = Ei as expected, otherwise the beam loses or gains energy. 4.4. COMPARISON WITH OTHER METHODS OF BEAM CHARACTERIZATION

This section compares the method of beam width or beam quality factor definition with the WDF moments with other methods. Among these other methods Sasnett [I9891 mentioned the measurement of lie2 of the highest peak or of the outermost peak and the measurement at 86.5% of total power, but neither method gives good results for all field distributions. For example, the use of the technique of measurement at lie2 of the highest peak intensity inadequately characterizes the higher-order modes when the outer peaks fall below this level. The measurement of lie2 of the outermost peak does not solve this problem because for mixtures of two or more modes with outer peaks eliminated, the determined beam width is incorrect. The method of beam width definition with the second-order moment of the WDF has some advantages over other methods because for pure modes the values fall midway in the range of values provided by other methods, and for mixed modes the beam width is equal to the power-weighted sum of M 2 values for the pure modes (Sasnett [1989]). The results obtained from all these methods agree only for the TEMoo mode. Siegman [1993a] described other methods of beam width definition for an arbitrary field distribution as the knife-edge

20

THE WIGNER DISTRIBUTION FUNCTION IN OPTICS AND OPTOELECTRONICS

[I,

55

method, the scanning slit or pinholes method, and the Gauss-fit method. He again pointed out that for each measurement method certain beam profiles can be found for which the method gives wrong results. For example, any nearfield pattern with a step discontinuity produces a farfield pattern whose second-order spatial moment diverges to infinity in the paraxial approximation The Gauss-fit method fails to provide a reasonable value for the beam width of a field that consists of a Gaussian central lobe plus a Gaussian pedestal. In conclusion, the problem of finding a universal method for the definition of beam width, or other beam characteristics, is not completely solved. In most cases, however, the definition based on the WDF moments provides reasonable results.

@ 5. Optical System Characterization

with the Wigner Distribution Function

The previous section presented some applications of the WDF to light field characterization. In this section we show that the WDF can also be used to characterize the optical systems through which the field distribution passes. For reasons of graphic representation this section will be restricted to 1D field distributions and rotationally symmetric optical systems for which eq. (3.13) becomes W,(x, p , ) = W,(Dx- Bp,, -Cx + Ap,). To characterize a first-order optical system, the WDF must be measured at its input and output planes and then the A , B , C, and D elements of the optical system matrix determined according to the WDF transformation law. Although this procedure can be carried out for arbitrary field distributions, it would be helpful if the light field had a WDF with a simple form. Let us suppose that the input plane of the optical system is located at the waist plane of a Gaussian field. Its WDF is then (see the example in 3.2 with I/R=O):

For a better comparison of the WDF at the input and output planes of the optical system, we will examine the transformation of the phase space acceptance (PSA). The PSA represents the intersection of the WDF with a plane parallel to the ( x , p x )plane at a height equal to l/e2 of the maximum height of

I,

5

51

OPTICAL SYSTEM CHARACTERIZATION WITH THE WIGNER DISTRIBUTION FUNCTION

21

the WDF. For a Gaussian light distribution with the waist at the input plane, the PSA in the normalized coordinates X = X / X OP, = kpxxo is an ellipse given by X 2 + iP’

=

(5.1)

1.

The PSA at the output plane is obtained by inserting the WDF transformation law into eq. (5.1). The result in the X , P coordinates is

(54 The transformation of the PSA between the input and output planes of a firstorder optical system is identical to the transformation of an elliptical PS area through a first-order optical system obtained by ray tracing in geometrical optics (Goethals [1989]). The area of the PSA at both the input and output planes has the same value: 2nlk = ilin the original x , p x coordinates or 2 n in the X,P coordinates, a result that follows from Liouville’s theorem. In figs. 2a-d a dotted line represents the PSA ellipse at the input plane chosen at the waist of the beam, and a solid line represents the PSA after passing through a length of free space (fig. 2a) a thin lens (fig. 2b), a magnifier (fig. 2c), and a Gaussian aperture (fig. 2d). Both the eccentricity and the inclination of the input PSA in the normalized coordinates may change after passing through a first-order optical system, their variation being related to the elements of the optical system matrix. For example, after passing through the free space of length d characterized by an optical matrix with A = D = 1, C = 0, and B = d , the angle between the PSA ellipse and the X axis changes from zero to

and the eccentricity changes with

[

A E = E- E . =

’

2J(3k2x: - 4d2)2 + 64 (dkxi)2 5k2x4, + 4d2 + J(3k2xi

-

1/2

4d2)2 + 64 (dkx;)2

After a thin lens characterized by A = D = 1 , C = -l/f and B = 0, where f is the focal length, the PSA ellipse rotates with respect to the X axis with an angle

e = iarctan

(

3;:$x4,)

’

22

THE WIGNER DISTIUBUTION FUNCTION IN OPTICS AND OPTOELECTRONICS

[I,

5

1

:

F40

-1

.

i

-2

..__ .

.,f

a

'

-2

-1

0

1

-3

2

-1 -0.5

X

0 0.5

1

X

2

1

nlo -1 -2-2

-1

0

X

1

2

-1 -0.5 0

0.5

1

X

Fig. 2. The phase space acceptance (PSA) of a Gaussian beam in the waist (dashed line) and after (a) a free space of length d = k x i ; (b) a thin lens of focal lengthf=-O.S/kxi; (c) a magnifier with m = 1.5; (d) a Gaussian aperture with g=-O.S/kx$.

and the eccentricity varies with 1/2

Note that the rotation angle of the PSA ellipse depends on the lens type; that is, it is positive for convergent lenses c f > O ) and negative for divergent lenses

cf. The elements of the optical matrix that describe a magnifier are A = rn, D = l/m, and B = C = 0. After passing through a magnifier, only the eccentricity of the PSA ellipse changes. The variation of the eccentricity,

is, unlike for the previous optical systems, independent on the wavenumber of the incident light and the beam parameter X O .

I,

5 51

OPTICAL SYSTEM CHARACTERIZATION WITH THE WIGNER DISTRIBUTION FUNCTION

23

The Gaussian aperture is characterized by a matrix with complex elements A = 1, B=O, C=-i/g, and D = 1. As demonstrated by Dragoman [1995b] at propagation of a Gaussian field distribution through a complex optical system, the transformation law of the WDF is the same as for real optical systems but with A, B, C, and D replaced by elements that depend also on the incident field: 1, 0, -kxi/2gR (= 0 at waist) and l-kxi/2g, respectively. The determinant of this equivalent matrix is not equal to 1. Moreover, the area of the PSA ellipse in the normalized coordinates is 2161I l-kxi/2g I at the output of the aperture, whereas at the input it is equal to 216. This result agrees with the fact that Liouville's theorem holds only for lossless or gainless optical systems. Again, only the eccentricity of the PSA ellipse changes:

Although the matrices of a lens and a Gaussian aperture are similar, their effect on the field distribution (and on the WDF) is totally different. The Gaussian aperture acts as an absorber, thus modifying the PSA area and the extent on the X direction, whereas the lens modifies the divergence of the beam, that is, the extent on the P direction, without changing the PSA area. 5.1. CHARACTERIZATION OF ABERRATED OPTICAL SYSTEMS

The applications of the WDF also include the characterization of aberrated optical systems. With respect to the WDF, Lancis, Sicre and Pons [1995] discussed the propagation of polychromatic spatially coherent light through an optical system developed for achromatizing Fresnel diffraction patterns. The selected diffraction pattern to be achromatized is located at a distance Ro from the input plane and is followed by an achromatic lens and a zone plate. The WDF transformation law through first-order optical systems has been used to derive relations among the various parameters of the system in order to minimize the chromatic aberration. Experimental verification was performed to illustrate the theoretical results. Lohmann, Ojeda-Castaneda and Streibl [ 19831 took a different approach, calculating the WDF of an aberrated optical system and observing that the influence of aberrations on the WDF can be expressed by means of a differential operator of exponential type. The first term of the operator predicts a coordinate transformation of the WDF identical to the WDF transformation through first-order (unaberrated) optical systems, whereas the second term of

24

THE WIGNER DISTRIBUTION FUNCTION IN OPTICS AND OPTOELECTRONICS

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5

5

the operator is responsible for the shape distortion of the WDF. The analysis was done for the 1D WDF. Dragoman [1996c] discussed in detail the influence of the different aberration types on a 2D WDF. The WDF of an aberrated optical system has been derived as in 93.3 (case A) with the input and output planes chosen as the object and exit pupil planes of the optical system. The unaberrated part, Vunab, of the point characteristic function V(xo,yo, xi, Yi)

=

[A(x? +Y:) + D(xi +v:)- 2(xixo + ~ i v o ) /2B ]

+ VO

Vab(Xo, y o ,

= Vunab(Xo,

xi, Vi)

yo, xi, yi) + Vab(Xo, yo, xi, ui)

was assumed to describe a rotationally symmetric optical system. The aberrated part (Born and Wolf [ 1964]), Vab(Xo, yo,

xi, yi) = -

2 2 (XO +yo) - b(xoxi

2

+yoyi)

+ e (xi + Y;) (xoxi + yoyi)

5

contains the contributions of all types of aberration: spherical aberration (through a), astigmatism (through b), field curvature (through c), distortion (through d), and coma (through e ) . As an example, the influence of these aberration types on an incident 2D Gaussian beam q(xi,yi)=exp[-(x? +y?)/xi] was studied. The WDF at the exit pupil plane was calculated in the normalized coordinates X = xi/xo, Y = yi/xo, P, = kxopxi, and Py = kxopyi, and the PSA of the projection of the WDF on the Y = Py = 0 plane, that is, of the W ( X ,0, P,, 0), was represented for an unaberrated or an aberrated passage through a free space. A free space is understood here as an optical system characterized by a matrix with A = D = 1, C=O, and B = d that can be realized not only by a free space of length d but also by a combination of lenses and free spaces (Sudarshan, Mukunda and Simon [ 19851). At least in the latter case the source of aberrations can be easily identified. The results are shown in fig. 3. After an unaberrated free space the PSA is a rotated ellipse, in agreement with fig. 2a. If spherical aberration is present, the PSA has an S-shaped form, as observed previously by Goethals [ 19891 and Lichtenberg [1969]. For other aberration types the PSA has different shapes, except for astigmatism and field curvature, for which the PSA’s of the WDF projection on the Y = Py = 0 plane are identical. To distinguish between them, one should be able to plot the PSA of the WDF projection on another plane, for example, on the Y =0.5, Py = 0 plane as discussed by Dragoman

1,

5

51

25

OPTICAL SYSTEM CHARACTERIZATION WITH THE WIGNER DISTRIBUTION FUNCTION

-2 -1 0

1 2

-2 -1

0

1

2

z ; p i

-1

-2

g

-1

1

0

4

p

2

1

-2

-2 -1 0

X

1

2

-3-15 0 15 3

X

-4

-2

0

2

4

X

Fig. 3. The phase space acceptance (PSA) of a Gaussian beam in the waist after an (a) unaberrated free space of length d = k x i and in the presence of (b) spherical aberrations with a=0.2/kx;; (c) astigmatism with b=O.l/kx: or field curvature with c=OS/kx;; (d) distortion with d =-0.2/kx; (barrel type); (e) distortion with d =0.2/kx; (pin-cushion type); and (f) coma with e=O.l/kx;.

[1996c]. The WDF projection on the Y = Py = 0 or the Y = 0.5, Py = 0 plane can be realized with the experimental device in fig. 1. In the first case the input beam has to be centered in the input plane, whereas for the experimental realization of the second case a downward shift of the optical beam at the input plane is required. Figure 3 suggest that different types of aberrations can be separately identified by observing their effects on the PSA of a given incident beam. Both the WDF and the WDF moments are modified in the presence of aberrations. Siegman [ 1993bl showed that the spherical aberration degrades the beam quality factor. Piquero, Mejias and Martinez-Herrero [ 19941 investigated the influence of the spherical aberration on the kurtosis parameter K . For an incident 1D Gaussian beam, the kurtosis parameter is equal to 3 and remains constant at propagation through first-order optical systems. After an aberrated lens, however, the output beam is no longer Gaussian and the value of K depends on the plane where it is determined. The experimental results showed that the beam becomes strongly platykurtic before the paraxial focal point of the aberrated lens and strongly leptokurtic after the focal point. Significant kurtosis changes can arise from small spherical aberrations. At distances much larger than

26

THE WGNER DISTRIBUTION FUNCTION IN OPTICS AND OPTOELECTRONICS

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6

the focal length the beam is only slightly leptokurtic. Thus a Gaussian beam can be made sharper or flatter by means of a single spherically aberrated lens.

8 6. Wigner Distribution Function Representation of the Coupling Efficiency Another problem for which the WDF approach has proved useful is that of coupling efficiency optimization between a light source and a waveguide. A series of papers by Onciul [1993c, 1994a,b] and Dragoman [ 1995c,d] showed that a definition of the coupling efficiency in terms of the WDF of the source and waveguide offered some advantages compared with the usual coupling efficiency definition in terms of the source and waveguide field distributions. One advantage is that it takes explicitly into account through the angular variable of the WDF the effect of the receiver’s finite angular aperture on the coupling efficiency. Moreover, Dragoman [ 1995dl defined an effective numerical aperture for single-mode or few-mode waveguides, based on the WDF representation of the waveguide modes that is similar to the geometric definition of the numerical aperture valid for multimode waveguides. This effective numerical aperture, defined for a ID waveguide mode with a WDF W,(x,p,) as

was shown to depend not only on the core and cladding refractive indices but also on the core diameter. This does not contradict the geometric optical definition of the numerical aperture in terms of the critical reflection angle at the boundary between core and cladding, because the latter gives the numerical aperture of the light that excites all bound modes in highly multimode waveguides, whereas the effective numerical aperture is related to a particular mode described by a WDF Wrn(x,px). Another advantage of the coupling efficiency definition in terms of the WDF is that it allows a simpler approach to the coupling efficiency optimization. As one might expect coupling efficiency between a source and a waveguide is higher if the fields (and the WDF) of the source and waveguide at the illumination plane are as similar as possible. Since the WDF transforms through a first-order optical system in an very simple manner (see eq. 3.13), it is possible to find a first-order optical system to be inserted between the source and the waveguide such that at the illumination plane the WDF of the source is as similar as possible to that

WIGNER DISTRIBUTION FUNCTION REPRESENTATION OF THE COUPLING EFFICIENCY

I, $ 61

27

of the waveguide. For sources with unknown field distributions the WDF can be measured with one of the devices described in sect. 3.4. 6.1. COMPLETELY COHERENT SINGLE MODE SOURCES AND WAVEGUIDES

Under the assumption that the surface of illumination is planar, the coupling efficiency between a source and a waveguide can be defined in terms of the field distributions of the source and waveguide, qs(r) and cpw(r),respectively,

or in terms of the corresponding WDF, Ws(r,p), and Ww(r,p)

'

=

J Ws(r, P) Ww(r, P) dr dP k2 J Ws(r, p ) dr dp s Ww(r,p>d r dp' 4n2

where the integrals extend over the illumination plane. The expressions of the coupling efficiency given by eq. (6.2) and eq. (6.3) are identical, as can be demonstrated by using eq. (3.4) and eq. (3.7). Apart from the advantages of the coupling efficiency definition (6.3) discussed above, by a simple inspection, the graphic representation of the WDF of the source and waveguide allows an a priori estimation of the coupling efficiency. In particular from the WDF inspection of different light sources an estimation can be made regarding the value of the coupling efficiency to a given waveguide mode. To illustrate this statement, we study the coupling efficiency between a Gaussian light source and the first four modes of a waveguide. For reasons of graphic representation we will now restrict our discussion to ID field distributions. The WDF of the Gaussian light source q S ( x )= exp(-x2/x:) is

whereas the WDF of thejth mode of the waveguide in the Gaussian approximation qj(x) = (x/xw)Jexp(-x2/xi) is

j-k

C

22W-i

(-1);

i!c; c2; 2i 2 ( j - k ) (kpXxW)-2',

i=O

where C; =j!l[k!(j-k)!]. The normalized WDF [i.e., W(x,p,)lW(O, O)] of the first four modes of the waveguide ( j = O , 1, 2, 3) are represented in fig. 4 in

28

THE WlGNER DISTRIBUTION FUNCTION IN OPTICS AND OPTOELECTRONICS

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56

Fig. 4.The normalized Wigner distribution function (WDF) c";.(X, P)/ ltt;.(O, 0)I of (a) the fundamental mode in a waveguide ( j = O ) ; (b) the first odd mode ( j = 1); (c) the second even mode ( j = 2 ) ; and (d) the second odd mode (J' = 3).

the normalized coordinates X = x / x w , P = kp,x,. Figure 4a also represents the normalized WDF of the light source in the coordinates xlx,, kpxx,. The source couples the most amount of light in the zero-th (fundamental) mode, because the two field distributions have similar WDF's. Moreover, the maximum coupling efficiency ( q = 1) occurs when x , =xw, that is, when the field distributions and the WDF are identical. This result can be obtained with either eq. (6.2) or eq. (6.3). The advantage of using the definition eq. (6.3) becomes obvious when one is interested in the coupling efficiency estimation of the light source in the misaligned waveguide modes. In fig. 5 the coupling eficiency between the light source and the first four waveguide modes is plotted as a function of the normalized lateral displacement XO= xolx, for xs = x , . As expected, for XO= 0 (aligned light source and waveguide modes) the coupling efficiency into odd modes ( j = 1 and j = 3 in our example) is zero, due to the different symmetry of the field distributions of the source and waveguide modes. The symmetry of a field distribution can be determined from its WDF. From the WDF definition it follows that for either odd or even modes W(-x,p,) = W ( x ,-px), but the value of the WDF is positive at the phase space origin (i.e., at x = p x = 0) for even modes and negative for odd modes. This feature can be also observed in fig. 4. The

I,

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WIGNER DISTRIBUTION FUNCTION REPRESENTATION OF THE COUPLING EFFICIENCY

29

xo -3

-1

1

3

Fig. 5 . The dependence on the lateral displacement of the coupling efficiency between a Gaussian source and the fundamental waveguide mode (solid line), the first odd mode (dashed line), the second even mode (dash-dotted line) and the second odd mode (long dashes).

coupling efficiency of the light source into aligned even modes decreases with j because the corresponding WDF’s become more dissimilar when j increases. The displacement of the source and waveguide modes is described in the PS as a lateral displacement along the X axis. By a lateral displacement the coupling efficiency between the light source and the fundamental waveguide mode decreases exponentially because the overlap of the two WDF decreases with the distance between their centers. By analogy, the overlap between the WDF of the source and that of the first odd waveguide mode should have one maximum with the displacement along the positive X axis and another, symmetric along the negative X axis because the WDF are symmetric. This fact is confirmed by fig. 5. The dependence on XOof the coupling efficiency between the source and the mode with j = 3 can be interpreted in the same manner. From figs. 4a and 4c it appears that the coupling efficiency between the source and the even mode with j = 2 should have three maxima: one at XO= 0 and the other two symmetrically with respect to it. The central maximum at XO= 0 does not appear, however, because all the extrema of the WDF with respect to X,along the P = 0 line, are positive. In this case the number of maxima in the coupling efficiency as a function of X is smaller (with the central peak missing) than the number of the relative maxima in the WDF of the waveguide mode. A similar interpretation can be given to the coupling efficiency as function of the tilt angle. Figure 6 shows the dependence of the coupling efficiency

30

THE WlGNER DISTRIBUTION FUNCTION IN OPTICS AND OPTOELECTRONICS

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56

PO -3

-6

3

6

Fig. 6. Same as in fig. 5 as a function of the tilt.

between the Gaussian light source and the first four waveguide modes on the tilt Po=kp,ox,. In the PS the tilt between the source and the waveguide mode is represented as a displacement along the P axis of their WDF. The overlap between the WDF of the source and that of a tilted waveguide mode determines the form of the coupling efficiency dependence on Po. The only case that needs a separate comment is the coupling efficiency between the source and the mode with j = 2. The three maxima in the coupling efficiency in fig. 6 correspond to what is expected from an inspection of figs. 4a and 4c because the extrema of W, with respect to P for X = O have alternative positive and negative values. In this case the number of maxima in the coupling efficiency as a function of the parameter of misalignment is equal to the number of relative maxima in the WDF of the waveguide mode with respect to the misalignment parameter. Other examples of the connection between the form of the WDF graphic representation and the amount of coupled light, including a light modulator, can be found in Dragoman [1995c]. 6.2. MULTIMODE, COMPLETELY COHERENT SOURCES AND WAVEGUIDES

Let us suppose that the field q p , ( r )in the waveguide can be expressed as a superposition of modal fields rpc(r),

m

I,

5 61

WIGNER DISTMBUTION FUNCTION REPRESENTATION OF THE COUPLING EFFICIENCY

31

with complex coefficients a,. The q{ are not necessarily orthonormal and are not necessarily the eigenmodes of the total field. If qw(r) is the bound field in a waveguide, q{ can be chosen as the fundamental and higher-order bound modes. The field of the light source can also be multimoded:

The total coupling efficiency between the light source and the waveguide is then given by

where

are the components of the cross-coupling tensor, with Wfk(r,p ) =

/

qf(r

+

g)

qf*(r -

g)

exp(ikrtTp)dr',

i = s, w.

(6.8) represents the coupling efficiency between the nth source mode and the Ith waveguide mode. The realness of the WDF implies that rlnnl'

p d k =

sw

mnkl = nmkl = mnlk rlsw rlsw rlsw '

Further conditions can be imposed on the components of the cross-coupling tensor, depending on the choice of the modal fields qc,q:. 6.3. PARTIALLY COHERENT SINGLE MODE SOURCES AND WAVEGUIDES

Onciul [ 1994bl showed that the definition (6.3) is not valid for partially coherent light distributions, although the WDF that appears in this definition can be

32

THE WIGNER DISTRIBUTION FUNCTION IN OPTICS AND OPTOELECTRONICS

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7

defined for partially coherent light. It was demonstrated that for partially coherent light the coupling efficiency is

in terms of the field distributions, where the angle brackets denote the ensemble average and (6.10)

W,,(r,p)= =

I(.. 1

( r + g ) c p i ( r - g ) ) exp(ikrlTp)dr'

Tsw(r

r'

(6.1 1)

+ T ,r -

For completely coherent field distributions rsw( rl, r2) = cp,(rl) qk(r2), and eq. (6.3) and eq. (6.10) coincide. If the mutual coherence function is expanded in its eigenmodes (Wolf [ 1982]), that is, (6.12) where p is a scaling factor and

/

~ ( r~ )( rd r) = d m n ,

(6.13)

the coupling efficiency can be written as (6.14)

5

7. The Fractional Wigner Distribution Function

The WDF is an appropriate mathematical tool for beam characterization. Its definition in terms of the Fourier transform of the field means that it adequately describes the light distribution in the far-field diffraction region.

1,

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THE FRACTIONAL WIGNER DISTRIBUTION FUNCTION

33

The corresponding mathematical tool that completely characterizes the field distribution in the near-field regime is the fractional Wigner Distribution Function (FWDF). It was introduced by Dragoman [1996d], starting from the definition of the fractional Fourier transform. This transform was introduced by Namias [ 19801 in quantum mechanics to solve several types of Schrodinger equation, and its properties were investigated by Namias [1980] and McBride and Kerr [1987]. The definition of the fractional Fourier transform of order a of the 1D function v(x> is

where C = sgn(sin a), 1. 1 denotes the modulus of the argument, and &(pX) = q(kp,), s * , ( p x ) = q(-kpx). When a = nI2, the fractional Fourier transform coincides with the Fourier transform. This mathematical definition supports a physical interpretation based on the light propagation properties in a graded index optical fiber. Ozaktas and Mendlovic [1993a] defined the fractional Fourier transform of order a as what happens to a light distribution that propagates through a graded index fiber after a length al;, where L is the propagation length, after which a graded index fiber generates a Fourier transform of the incident light distribution. The fractional Fourier transform was successfully applied in optics for spatial filtering (Ozaktas and Mendlovic [ 1993b], Ozaktas, Barshan, Mendlovic and Onural [ 19941, Granieri, Trabocchi and Sicre [ 1995]), for analyzing the beam propagation in the Fresnel approximation and spherical mirror resonators (Ozaktas and Mendlovic [1994, 1995]), and for shift-variant image detection based on the concept of fractional correlation (Mendlovic, Ozaktas and Lohmann [ 19951). The search for new applications of the fractional Fourier transform continues. Recently Abe and Sheridan [1994a,b, 19951 showed that the fractional Fourier transform is a member of the class of special affine Fourier transforms that represent the most general lossless inhomogeneous linear mapping in PS. The two preceding definitions of the fractional Fourier transform are identical to a third, indirect definition as the transform performed on a function that leads to a rotation with an angle a=an/2 of the associated WDF (Lohmann [1993],

34

THE WGNER DISTRIBUTION FUNCTION IN OPTICS AND OPTOELECTRONICS

[I, $ 7

Mendlovic, Ozaktas and Lohmann [1994]). This means that by applying the WDF to a fractional Fourier transformed distribution, one obtains

J 43, ( x + ); 43; ( x -

g)

exp(ikx'p,) dx'

cos a - kp, sin a , x-

sina + p x cosa k

The FWDF of order a is defined, by analogy with (3.lb), as exp(-ikxp')dp'.

(7.3)

The relation between the FWDF and WDF is simple: Wt'2 (x,p,)= W,(x,px), and for arbitrary a, W,*(x, p,)

=

W,(kpx cosa + x sina, p x sina - x cos a/k);

(7.4)

that is, Wg represents a rotation of the WDF with an angle a - n/2 in the plane (x, kp,) (Dragoman [1996d]). Although the FWDF and WDF are so closely related, their physical meaning is different. This follows from the relationship between the fractional Fourier transform and the Fresnel transform derived by Gori, Santarsiero and Bagini [ 19941:

(7.5) where the Fresnel transform F,(px) is defined as (Papoulis [1968]) (7.6) From eqs. (7.5) and (7.6) it follows that the FWDF can be directly defined in terms of the Fresnel transform, so that it replaces the WDF in the near-field diffraction regime as a mathematical tool for beam characterization. 7.1. PROPERTIES OF THE FRACTIONAL WIGNER DISTRIBUTION FUNCTION

The following properties of the FWDF were derived by Dragoman [ 1996dl:

1, B 71

35

THE FRACTIONAL WIGNER DISTRIBUTION FUNCTION

Pi: A space shift in q ( x ) yields the same shift in the spatial coordinates of W;(x sin a - kp,cos a, x cos a/k +p,sina). Pi: A frequency shift in Ta(p1) (or T ( p , ) ) leads to the same shift in the frequency coordinate of W;(x,p,) (or Wg(xsin a - kp, cos a, x cos alk + pxsin a)). Pi: If q ( x ) is limited to a certain spatial interval, W;(x sin a - kp, cos a, x cos alk + pxsin a ) is limited to the same interval. By analogy, if T a ( p x )(or q ( p x ) ) is limited to a certain frequency interval, W;(x,p,) (or W;(xsina kp,cos a, x cos alk + pxsin a ) ) is limited to the same interval.

XI + x 2

.

sin a - kp, cos a,

~

+X2

2k

cos a + p x sin a

(7.10a)

(7.10b) -

rp,

( m )5 :( m )=

1wg

+P x 2 7) exp [ikn (p,l -pX2)1 h.

Pxl (X,

(7.10c)

These properties of the FWDF were numbered such that properties P', to P{ correspond to properties PI to P7 of the WDF. 7.2. OPTICAL BEAM CHARACTERIZATION IN THE NEAR-FIELD DIFFRACTION REGIME

The WDF moments are used to characterize a field distribution and its propagation through first-order optical systems. Due to the close connection

36

THE WGNER DISTRIBUTION FUNCTION IN OPTICS AND OPTOELECTRONICS

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between the FWDF and WDF, the whole moment formalism of the WDF can be translated to the FWDF in order to characterize field distributions in the nearfield region (Dragoman [1996d]). The moments of the 1D FWDF are defined as (7.1 1)

5"

where = J EWG(x, px) dx dpx/J W,"(x, px) dx dp, with 5 =x,px and i, j integers. By analogy with the transformation law of the WDF through first-order optical systems, the transformation law for the FWDF is W&(X, px) = W ; I P

- Bpx, -CX

where A B

sina

(7.12)

+ &x),

-kcosa)

(:

i3)

(

sin a -cosa/k

k cos a

sina (7.13) is an equivalent optical matrix different from the symplectic S matrix with elements A , B, C, and D that characterizes the first-order optical system; it takes into account the near-field diffraction effects through a. Of course, when a = nI2, one has S = S. The moment matrix of order j is defined as

MY= (xpx)@

(L) -.

@..-',

(7.14)

j times

where the overbar and the superscript a denote the average of all matrix elements in the sense of eq. (7.1 1). The transformation law of moment matrices is similar to eq.(4.4): (7.15) The symplectic property of S implies the invariance of det M f j , in particular the invariance of the beam quality parameter in the near-field diffraction regime defined as Qa =

?aza

- ( G O )

2

=

det M:.

(7.16)

1,

9: 71

37

THE FRACTIONAL WIGNER DISTRIBUTION FUNCTION

The class (4.6) of invariants defined in terms of the WDF simply transforms in the class of invariants

p 2j

=

J a2jW,a(x,Px) W,a(x/,P:) dx dP.x dx’ dP: J tV@, Px) W p ’ , P:) dx dPx dx/dP:

-

(7.17)

’

where a(x,px,x’,pc) is the Lagrange invariant. Combinations of moment matrices of even and odd order and the antisymmetric J matrix can be found that satisfy similarity transformations at propagation through first-order optical systems. All properties of the eigenvalues of these combinations are identical to the corresponding properties in the far-field diffraction regime, with S replaced by S. All combinations of WDF moments that have a physical meaning retain their significance if calculated in terms of FWDF, but in the near-field diffraction regime. 7.3. OPTICAL PRODUCTION OF FRACTIONAL WIGNER DISTRIBUTION FUNCTION

Several setups have been suggested to implement the fractional Fourier transform. Lohmann [ 19931 suggested two setups for the fractional Fourier transform generation that correspond to two different ways of synthesizing a WDF rotation with an angle a. These setups are shown in fig. 7, where the parameters R and Q determine the fractional order a, andfo is an arbitrary focal length. R = tan(a/2) and Q = sin a for the setup in fig. 7a, and R = sin a and Q = tan(a/2) for that in fig. 7b. Granieri, Trabocchi and Sicre [ 19951 proposed a free space Fresnel diffraction configuration equivalent to the setup in fig. 7a, and Lohmann [ 19951 described alternative realizations of the setups in figs. 7a,b with fake zoom lenses. The optical implementation of the FWDF can be realized by using any of the setups that generate the WDF, the input of which must be the output of a setup output

input plane

plane

f = f,/Q

input plane

output plane f

f = f,lQ

-~

d = Rf,

Fig. 7. Two setups for the generation of the fractional Fourier transform.

38

THE WGNER DISTRIBUTION FUNCTION IN OPTICS AND OPTOELECTRONICS

[I,

58

that generates the fractional Fourier transform. Of course, the result will be given in ( p , x ) coordinates instead of ( x , p ) ,and the Fourier transform part of the setup that generates the WDF must have inverted spatial coordinates.

6

8. Optical Beam Characterization in Nonlinear Optical Systems

The analysis of light propagation through nonlinear media is a subject of great practical importance due to the numerous applications of nonlinear media. Since the literature on this subject is vast, we will restrict discussion to publications that are closely connected with the WDF approach to light characterization in nonlinear optical systems. Martinez-Herrero and Mejias [ 199 11 made the first attempt to characterize the beam propagation through nonlinear media by using the WDF. They studied the propagation of a 1D stochastic field distribution through an active medium characterized by a small signal gain and a saturation intensity with an approximation procedure that restricts both the field diffraction along the amplifier and the variation of the medium parameters. Some general propagation characteristics but no clear solution for the WDF in the nonlinear medium were established. For example, it was demonstrated that the WDF does not propagate in active media along straight lines, contrary to what happens in passive media. The radiant intensity of the field varied at propagation, and the transformation laws of the first- and second-order moments were determined. X and px, as well as the beam quality parameter, varied with z, and it was demonstrated that even if 0 at z=O, these moments can take nonzero values at other z values. Carter [1995] also developed a detailed WDF treatment of quantum pulse propagation in nonlinear fibers. The WDF approach allowed the introduction of the excess thermal noise sources in the analysis. He pointed out that the conditions for the reliability of the WDF method are those for which the linearization of the Heisenberg equation is valid; that is, it involves an approximation formulation of the problem. Hirlimann and Morhange [1992] used the temporal WDF to describe the propagation of ultrashort light pulses under the separate effects of linear dispersion and self-phase modulation. Graphic representations of the WDF for propagation in the linear dispersion regime and the self-phase modulation nonlinear regime were derived, starting from the known solutions of the field distributions in these regimes. The time broadening induced by the linear dispersion of the refractive index and the frequency changes induced by selfphase modulation are accounted for in the WDF approach, and the numerical

X=z=

1,

5 81

OPTICAL BEAM CHARACTERIZATION IN NONLINEAR OPTICAL SYSTEMS

39

results are in excellent agreement with those deduced from a classical Fourier analysis. Marcuvitz [1980] simulated the WDF of a Gaussian pulse propagating through a dispersive nonlinear medium. A WDF representation of self-phase modulation and soliton propagation in the time-frequency domain was given by Paye [ 19921. Dragoman [ 1996al formulated a complete treatment of 1D light propagation through an inhomogeneous Kerr medium. This treatment allows the calculation of the propagation law through a nonlinear medium of any quantity that can be expressed in terms of the WDF. We reproduce here the basic results obtained by Dragoman [1996a]. The propagation of a ID monochromatic, paraxial beam through an inhomogeneous Kerr-type medium is governed by the nonlinear Schrodinger (NLS) equa2 tion, which can be written as in eq. (3.16) with H ( x , id/dx, 1q(x)l ) = -yx2+ ad2/dx2 + @ I ~ P ( X > and ~ ~ real y , a, and p. The differential equation for the WDF is then

3 --2akp,-- aw9 + 2yxdW9 az

-

dx

k

dp,

Equation (8.1) is not easier to solve than the original differential equation for the field distribution. This is not necessary, however, if one is only interested in deriving propagation laws for WDF moments of arbitrary order or other quantities that can be expressed in terms of the WDF and are of practical importance. For example, by using eq. (8.1) it was demonstrated that the total beam energy E ( z ) = ( k / 2 n ) J W ( x ,p x ; z ) dx dp, was constant at propagation, and transformation laws were derived for the radiant intensity of the field distribution J ( p , ; z ) = W(x, p x ; z ) dx and for the radiant emittance R(x; 2 ) = j” W ( x ,P,; z ) dp,:

s

40

THE W C N E R DISTRIBUTION FUNCTION IN OPTICS AND OPTOELECTRONICS

[I, 5 8

Similar to the results of Martinez-Herrero and Mejias [1991], the radiant intensity J depends on z because of both the nonlinearity and inhomogeneity of the medium. The nonlinearity also contributes to the variation of R at propagation. The first-order WDF moments satisfy a set of coupled differential equations

ax az

aE az

2yX -- -~

- = 2akpx,

k ’

whereas the second-order WDF moments evolve according to

where I=-k

2n

f

W x , pX;z> Wx, q; z ) b d p , d q -

f

W ( x ,px; Z>d.dPX

f

I c p k z)14

f

I c p k 412&’

(8.6)

If initially T(z=O)=O=px(z=O), then, according to eq. (8.4), X = p x = O at arbitrary z values. The opposite conclusion reported by Martinez-Herrero and Mejias [1991] is a result of a different calculation method and a different nonlinear medium. Transformation laws for second-order moments were also found in Porras, Alda and Bernabeu [ 19931 by an inspection of the differential equation for the field distribution (their results are equivalent to eq. 8.5), and in Pare and Belanger [1992]. A moment theory of the field propagation through a nonlinear medium was also derived by Suydam [1975]. Kamp [1987] found the transport equation for the WDF for field distributions that satisfy the NLS equation, as well as other nonlineartype equations such as the Korteweg-de Vries and Burgers equations. For these nonlinear equations an infinite hierarchy of balance equations was constructed by taking the local moments with respect to

1,

P 81

OPTICAL BEAM CHARACTERIZATION IN NONLINEAR OPTICAL SYSTEMS

41

the angular variable of the WDF transport equation. The balance equations lead to an infinite class of conservation laws equation for the NLS. The beam quality factor and the kurtosis parameter of the field distribution satisfy the following propagation laws, respectively:

-__

--

x2 x3px - x4 xp, _ aK - 8ak

aZ

Equation (8.7) differs from the propagation law for the beam quality factor in Porras, Alda and Bernabeu [1993] because of the different number of transverse dimensions considered (one here, two in the reference just quoted). Park and BClanger [ 19921proposed various definitions of the beam quality factor, and discussed the influence of the number of transverse dimensions on the propagation law of second-order WDF moments. Dragoman [1996a] showed that if in the last term of eq. (8.1), the terms with n 3 1 can be neglected as, for example, for field distributions that vary slowly with x , the propagation law of the WDF simplifies to that through a first-order optical system (eq. 3.13), where for ID fields the matrix elements are A

=

D

=

cos(a(x)z),

B

with a(x) = J2a(2y+@

pa I qI2/ax)/(ka(x)).

=

K~(x)sin(a(x)z), C

=

K~(x)sin(a(x)z),

Iq12/ax), Kl(x) = -2aWu(x),

(8.9)

and K ~ ( x = ) (2y+

The matrix that characterizes the WDF propagation through an inhomogeneous Kerr-type medium depends on the field distribution and has unit determinant; that is, it is symplectic. Magni, Cerullo and De Silvestri [1993] also derived a unit determinant matrix that characterizes the propagation through Kerr media by studying the passage of a cylindrically symmetric Gaussian beam through such a medium. In their derivation the nonlinear term was approximated by a parabola, an approximation that can cause the discrepancies between the theory and experimental results of Nemoto [ 19951. The nonlinear propagation including soliton propagation can also be treated by complex ray tracing (Nasalski [ 19951). This method describes the nonlinear propagation by a scale transformation applied to the linear propagation such that in the scaled space the complex ray follows a straight trajectory as a usual ray in a linear medium. The transformation of an incident Gaussian beam through such

42

THE WlGNER DISTRIBUTION FUNCTION IN OPTICS AND OPTOELECTRONICS

[I,

5

8

a scaled space leads to the determination of a matrix for each of the thin sections of the nonlinear medium, with elements that depend on the beam parameter in the considered sections. Although not directly related to the WDF treatment, two interesting recent papers provide a PS description of nonlinear directional couplers. Snyder, Mitchell, Poladian, Rowland and Chen [1991] and Artigas and Dios [1994] studied the behavior of the nonlinear directional couplers in terms of the initial power by plotting the trajectories of motion in the PS for an initial excitation point and different powers.

8.1. SOLITON SOLUTION OF THE NLS EQUATION

Both the evolution of the WDF (and implicitly that of the field distribution through a nonlinear medium) and also the stationary solution of the NLS equation can be described by the formalism presented in the last section. The NLS equation satisfied by the complex amplitude q(z,t) of an optical signal that propagates along the z direction in a frame of reference that moves with the group velocity ug can be written as

.aq

p2 a2q

1----+yJql

az

2

at2

2

q=O.

(8.10)

In eq. (8.10), z=t-z/ug is the time coordinate in the reference frame, p2 is the dispersion parameter, and y is the nonlinear coefficient. Equation (8.10) is an evolution equation of the field distribution that can be put in the form of eq. (3.16) in the temporal PS, for which H ( z , idldt, Iq(t)12)=(P2/2)d2/dt2y l q ( t ) 1 2 . The soliton solution of eq. (8.10) is shape invariant with respect to z; that is, it has the form q(z, z ) = u ( t ) exp(iu(z, t)), where u and u are the amplitude and phase of the envelope, respectively (Du, Chan and Chui [ 19951). If sgn(B2).sgn(y) < 1, the bright soliton solution of the NLS equation is obtained: q(z, t)= A d m s e c h ( A t ) exp(-iA2zB2/2>,

(8.1 1)

where A is an arbitrary constant. It can be shown that if du/dt = 0 as in our case, WP(t,w)= WU(t,w). Moreover, if du/dz=P=const., aWqlaz=-2(ImP)W,; that is, the WDF is constant with respect to the z variable in a lossless or gainless medium.

4

§ 81

OPTICAL BEAM CHARACTERIZATION IN NONLINEAR OPTICAL SYSTEMS

43

$1r, 1

w

-1

4

-2

-1

0

1

2

T Fig. 8. The normalized Wigner distribution function (WDF) of the bright soliton (left) and its PSA (right).

The temporal WDF of the bright soliton has an analytic form (Konno and Lomdahl [ 19941): Wp(z, W)

II

= 4nA -

sin(2 zw) . sinh(2Az) sinh(nw/A)'

(8.12)

The normalized WDF of the bright soliton W ( T ,w)= W,(T, w)/W,(O, 0 ) and its PSA is represented in fig. 8 in the normalized coordinates T=Az, w=w/A. The PSA has a diamond shape, and it can be used for the determination of the ratio between the 0 2 and y . The maximum T and w values of the PSA are found by numerically solving the equations 2T = exp(-2) sinh(2T) and wn= exp(-2) sinh(wz), respectively. The result is T, = 4.1/24 and w, = 4.1AIn . From either of these two values one can determine the constant A , and then from W,(O, 0) = 4A IPz/yl the ratio between the dispersion parameter and the nonlinear coefficient is determined. T,, wmaX= 2.67 = const., independent of A, which is a characteristic of the WDF of a bright soliton. Moreover, for a pulse with a known peak input power POand halfwidth TO the determined ratio of the dispersion and nonlinear coefficients serves as a test for identifying it with a pure soliton. [ 02/y( = LNLPoTilLD, where LD= T2,/(02[ is the dispersion length and LNL= 1/( [ yIP0) is the nonlinear length. For a pure bright soliton, LNL/LDmust be equal to 1. At propagation through a gainless or lossless medium, all WDF moments as well as the total soliton energy E=2A[Pzlyl are constant. In the frame of reference that moves with the group velocity T=w=O, G = O , p=n2/12,

44

THE WlGNER DISTRIBUTION FUNCTION IN OPTICS AND OPTOELECTRONICS

[I,

89

-

w2 = 1/3, Q=n2/36. The bright soliton is a self-Fourier function (Dragoman [ 1996e1) for which the higher-order moments are related by

(8.13)

All these results are valid if the soliton field is the stationary solution of the NLS equation. If at the input of the medium a soliton is launched that is not the stationary solution of the NLS equation, its WDF evolves according to eq. (8.9). Since dl p7I2/dt has different signs for t < 0 and t > 0, it follows that the leading and trailing edge of the pulse propagate in a different manner. In this case eq. (8.9) describes the evolution of the WDF of the total field, that is, of the part of the pulse that evolves to the stationary solution of the NLS equation in the medium and of the radiated part. A WDF representation of the two-soliton solution of the NLS equation as well as of solitons in open systems was derived by Konno and Lomdahl [1994].

0

9. Complex Field Reconstruction

from the Wigner Distribution Function Equation (3.6a) for r l = r , r2 = O is the basis of time harmonic field reconstruction from the WDF. For ultrashort light pulses the field distribution q(t) can be reconstructed from the temporal WDF W(t,w)by a formula similar to eq. (3.6a), with r replaced by t and kp replaced by w. In a similar manner the field distribution can be reconstructed from the spatiotemporal PS. This chapter examines the field reconstruction problem of ultrashort light pulses from their WDF. To determine the field distribution, one must determine both amplitude and phase of the light pulse. Of several methods that have been proposed, one of the most common is the measurement of the intensity autocorrelation obtained by second harmonic generation in a nonlinear crystal. It can provide some information about the pulse amplitude only for intensities symmetric with time, but yields no information about the phase. Similarly, the measurement of the spectrum can provide information about the amplitude of the pulse in the frequency domain, but for the phase measurement separate methods must be used, such as interferometric measurements and iterative phase-retrieval

I,

5 91

COMPLEX FIELD RECONSTRUCTION FROM THE WGNER DISTRIBUTION FUNCTION

45

procedures. Other methods for phase measurements are mentioned in Paye [1994]. An interesting method of amplitude and phase determination from the measurement of only one quantity - the spectrally resolved autocorrelation of the pulse - is described by Paye, Ramaswamy, Fujimoto and Ippen [1993]. The spectrally resolved autocorrelation is the spectrum of the second harmonic pulse obtained by nonlinear mixing in a noncollinear geometry of the pulse to be measured and a delayed replica of itself. The amplitude and phase of the pulse are determined from the measured data by an iterative algorithm. Although the temporal WDF was recognized long ago as a time frequency distribution that provides the same information about light pulses as the field distribution. the first attempt using the WDF to derive the amplitude and phase of a pulse appeared recently (Beck, Raymer, Walmsley and Wong [ 19931). This approach does not involve nonlinear correlation processes and can therefore also be applied to pulses with low peak intensities. The method, called chronocyclic tomography, consists of the temporal WDF recovering from its rotated projections that can be measured by a setup formed from a temporal free space and a temporal lens. The complex field distribution (or the correlation function for stochastic fields) are then obtained from the WDF inversion formula. The temporal free space and temporal lens are devices that act in the temporal PS in the same way on a field distribution as the free space and lens in the spatial PS. The analogy between fields modulated in time (pulses) and space (finite spatial beams) was first pointed out by Akhmanov, Chirkin, Drabovich, Kovrigin, Khokhlov and Sukhorukov [ 19681, whereas the analogy between the spatial problem of Fresnel diffraction and the temporal problem of first-order dispersion was first realized by Treacy [ 19691. The duality between the problems of paraxial diffraction and narrow-band dispersion originates in two different approximations to the wave equation. The paraxial diffraction assumes a monochromatic wave, and allows the propagation to take place mostly in one direction (z) such that the total 1D electric field takes the form E(x,z, t)=E(x,z) exp[i(wot-k(wo)z)], where E(x,z) is a slowly varying envelope function. The narrow-band dispersion assumes a plane wave spatial profile of the field distribution, and limits the frequency spectrum of the pulse to a suitable range such that the propagation of any spectral component can be accounted for by a Taylor series expansion of the propagation constant p(w) to second order in w. The total electric field can be represented in this approximation as E(x,z, t ) =A(z, t ) exp[i(wot-B(wo)z)] with A(z, t ) a slowly varying envelope function. The analog equations and quantities in the space and

46

THE WIGNER DISTRIBUTION FUNCTION IN OPTICS AND OPTOELECTRONICS

[I,

8

9

Table 1 Space-time analogies in optics a Space domain

Time domain

Governing partial differential equation

d E / d z = (i/2k)d2E/dx2

dA/dE = (i/32/2)d2A/at2

Propagation variable

Z

E=z-zo

Profile variable

X

t = r - t o -(z -zo)/ug

Propagation constant

k0

WO

Frequency

spatial: k,

temporal: w - 00

Transmission function

Fresnel propagation

first-order dispersion

exp(-izkzl(2ko)) lens:

time lens:

exp(-ikox2/(2f)) Imaging law

Ildl +I/d2=lIf

Here zo and r o are arbitrary references, length of the dispersive element.

a

exp(i(w - ~ 0 ) ~ / 3 2 1 / 2 )

82 = d2/3/do2 is the

exp(-iwor2/(2fT)) 1/@2l[l)+ 1/@2212)=WO/fT dispersion coefficient, and I is the

time domain are presented in Table 1 (Kolner [1994a], Godil, Auld and Bloom [ 19941). The temporal analog of the free space can be realized with either a dispersive fiber or a pair of diffraction gratings. In the first case the length d of the free space is equivalent to -021, where 1 is the fiber length, whereas in the second case d is equivalent to 1/p - the group delay dispersion of the grating pair. Since for a fiber 02 can be either positive or negative, it is possible to implement temporal analogs to positive or negative lengths of free space. Moreover, with a birefringent fiber that has the linear eigenpolarization vectors parallel to the orthogonal x and y axes, it is possible to implement anisotropic 2D free spaces (Dragoman and Dragoman [ 19961). The ID time lens introduced by Kolner and Nazarathy [1989] can be implemented by applying a voltage waveform quadratic in time to an electrooptic modulator. The phase acquired by a field that passes through the modulator is qj(t)=Acos(w,t), with w, as the modulation frequency and A as a function of the electric field, the electrooptic coefficient of the crystal and of the relative orientations of the incident field, optical axis of the crystal and of the applied field. In the quadratic approximation around t = 0, qj(t)=A( 1 - okt2/2), from which the focal length f~ is determined as f ~w ~=/ ( A w k(Table ) 1). A general 2D time lens with different focal lengths along the x and y directions can be implemented in

I,

5 91

COMPLEX FIELD RECONSTRUCTION FROM THE WICNER DISTRIBUTION FUNCTION

47

the same manner, but with an incident linear polarized field with components along both x and y directions. The 2D time lens can be spherical, cylindrical, or a quadrupole according to the symmetry of the electrooptic crystal and its excitation (Dragoman and Dragoman [ 19961). A detailed discussion of 1D time lens implementation and characterization can be found in Kolner [1994b] and Godil, Auld and Bloom [1994], and examples of time lenses’ applications are described in Kauffman, Godil, Auld, Banyai and Bloom [1993], Godil, Auld and Bloom [ 19931 and Kauffman, Banyai, Godil and Bloom [ 19941. Implementation of other ID optical temporal processors are discussed in Yang [1995], and the temporal PS characterization of a general birefringent optical fiber is derived from Dragoman and Dragoman [1996]. The setup of Beck, Raymer, Walmsley and Wong [1993] consists of a dispersive element characterized by the dispersion 1Ip = - d2@/dw2, calculated at the central frequency 00 followed by a time lens with a focal distancefT such that d2@ldt2= - w o / f ~ ,where @ is the phase through the respective elements. The action of this setup on the input field spectrum q ( w ) is described by an integral transform similar to the fractional Fourier transform of order 6,:

(e)

qout (we) = J

F e x p 2 n w sin 8 2sin 6 x exp (-i (we- w‘ - -cot 0’2 sin 13 2

/

= P(00)/

q

e

x

p

6,))

(w’) dw’

(9.1)

(2)

with

Now, IP(w0)12=

wq(te cos 8 + we sine, -to sin

e + we cos 0) dte,

(9.3)

where Wp(t, w ) is the temporal WDF and 0 0 =

wcose

+ t sin 8,

tt,

=

-wsin 8

+ t cos 8.

Equation (9.3) resembles eq. (7.9) and can be inverted to obtain Wp(t,w) by the use of the inverse Radon transform (Hermann [1980], Lohmann and Soffer

48

THE WlGNER DISTRIBUTION FUNCTION IN OPTlCS AND OPTOELECTRONICS

input

[I, 6 9

Quadratic phase modulator Birefringent and dispersive fiber

Dispersive fiber

4

fT

dT,,

Fig 9 Setup for the generation of a temporal phase space (PS) representation of a field distnbution

[ 19941). The procedure is as follows: (1) measure Ii&ut(w~)12for different values of the rotation angle 8, that is, for different values of l/p + 00/fT; (2) rescale the measurement to obtain IP(o0)l2;(3) perform the inverse Radon transform to obtain Wq(t, 0 ) ;and (4) obtain the field distribution cp(t), up to an unimportant phase constant $0, from

exp(i$o) u?(t>=

~

fi

Wq(t/2, w ) exp(-itw) dw

Jm

(9.4)

Equation (9.4) follows from the temporal form of eqs. (3.4) and (3.6a). For the stochastic field by Fourier inverting the WDF definition, one obtains the correlation function. The method proposed by Beck, Raymer, Walmsley and Wong [I9931 determines the WDF indirectly, by measuring its rotated projections. A direct method of WDF measurement by Dragoman and Dragoman [ 19961 consists of the transposition in the temporal domain of the setup in fig. 1. The result is the setup in fig. 9. As discussed before, the free space of length dl is replaced by a dispersive fiber of temporal length d T = - b 2 1 1 , , and the spherical lens of focal length f 0 is replaced by a quadratic phase modulator of focal length f T . Since the spatial rotation has no exact analog in the time domain, the remaining part of the device, (i.e., the free space of length fo, the rotated quadrupole, and the free space of length d 2 ) was replaced by a single birefringent dispersive fiber characterized by a set of parameters d,,, where i,j =x,y , depending on the cross polarization dispersion coefficients of the fiber 0 2 , and the fiber length 1 2 . To perform a temporal PS representation of the incident field distribution along the x axis, the parameters of the setup must be chosen such that d T + d T x x - d T d T x x 1f T = 0 and d T y x =fT. The first condition determines the length of the first fiber 11 that must be chosen such that dT= - 8 2 1 11 = d T x x f T /(dTxx -fT). The second condition can be used to determine the length of the birefringent fiber 1 2 , if f T is fixed, or the peak value of the applied electric field that determines the value of f T , if 12 is given. i the With these conditions a temporal PS map: txo= -dTxxt,i / d T , tyo= d ~ y ~ O , of

1, § 101

WIGNER DISTRIBUTION FUNCTION IN QUANTUM OPTICS

49

incident field distribution is obtained at the output plane, where tx and ty are the times associated with the x and y components of the polarized field and 0, and uy their corresponding frequencies measured from W O . For ultrashort optical pulses the time length of the output pulse can be increased by appropriately choosing d T x x and f T . If d T x x = NfT with N > 2, the length of the pulse at the output plane is increased by ( N - 1). This setup can be used for the amplitude and phase reconstruction of the incident field in the same manner as the previous device.

0

10. Wigner Distribution Function in Quantum Optics

Wigner [ 19321 introduced the WDF to calculate the quantum corrections for thermal equilibrium that become important at low temperatures. The idea was to extend the PS treatment of thermodynamics from the temperature region, where the classic mechanics is valid and where all the calculations are based on the classical PS density, to lower temperature regions not too far away from the validity of classical physics. A new quasiprobability function of the position and momentum variables, the WDF, (10.1)

was introduced in terms of the density matrix 6 that represents the state of a spinless system. Soon it became evident that the WDF was the basis of a quantum-mechanical formulation alternative to the Heisenberg or Schrodinger representations. In the PS formulation of quantum mechanics both the position and momentum variables are c-numbers, so that the quantum theory can be treated in a manner similar to classical statistical mechanics. The main aspects of this formulation are (Ruggeri [1971]): (a) The states of the system are described by means of a distribution function F(r,p;t), defined on the classical PS, that depends parametrically on time. This distribution function should be as close as possible to a true probability distribution. (b) A c-number function A(r,p) defined on the PS is associated with a quantum operator such that the quantum expectation value can be calculated classically by means of a PS integration rather than through the operator formalism:

A^

A (->

=

J

A(r, p ) F(r, p ; t) d r dp.

(10.2)

A large number of papers (e.g. Aganval and Wolf [ 1970a-c], Cahil and Glauber [1969a,b], Srinivas and Wolf [1975], Cohen [1966, 19761, Ruggeri [1971])

50

THE WIGNER DISTRIBUTION FUNCTION IN OPTICS AND OPTOELECTRONICS

[I, 9: 10

discuss in detail the problem of associating a c-number function with a quantum operator in such a way that eq. (10.2) holds (and, implicitly, the problem of defining the corresponding PS distribution function). As in classical theory, a class of PS distribution functions correctly satisfies the marginals, the members of which are derived by applying different ordering rules between the quantum mechanical operators and the associated classical quantities. In particular in quantum optics, three such PS distributions are known: the WDF, the P function and the Q function. For each, eq. (10.2) holds if the following ordering rule is obeyed: normal ordering for the P function, antinormal ordering for the Q function, and symmetric (or Weyl) ordering for the Wigner function. Among the PS distribution functions, a class of non-negative distributions is associated with the antinormal ordering, which also includes the Q function. However, Wigner [ 197I] demonstrated that no non-negative distribution function exists that is a joint probability distribution for the noncommuting operators F and p^. This theorem can also be stated in the form: the quantum mechanics cannot be formulated as a classical stochastic theory, which would require that at least a distribution function exists such that if A(r,p) is the PS function used to calculate the expectation value of the operator A^(F, ?), then K ( A ( r , p ) )will give the expectation value of K(A^(P,p^)) for any arbitrary function K (Cohen [ 19661). The WDF is a joint probability distribution for i and p^, so that it must take negative values in some PS regions. Narcowich and O’Connell [1986] showed that the necessary and sufficient condition for a quantum PS function F to be a WDF is to be of h positive type; that is, for arbitrary al to a, the m x m matrix with elements exp(iha(ak, a,)/2)F(a, - a k ) is non-negative, where F ( a ) = F ( x ) exp(ia(a, x)) dx is the symplectic Fourier transform of F with o(al,a2) - the Lagrange invariant defined on two PS points a1 and a2. The properties of the quantum WDF generally parallel those in the classic theory; see, for example, Wigner [1932], Aganval [I9871 and Leonhardt and Paul [1995]. The WDF has been used in quantum optics to study the coherence properties of light (Glauber [1963], Mehta and Wolf [1964], Lax and Louise11 [1967], Bialynicki-Birula and Bialynicki-Birula [ 1973]), to describe quantum noise in interferometers (Aganval [ 1987]), to study the propagation through a parametric amplifier (Mollow and Glauber [ 19671) or in connection with the quantum theory of radiative transfer (Sudarshan [1981]). Moreover, the WDF is the simplest scientific language for the study of coherent and squeezed states (see Kim and Noz [1991], ch. 6). Finally, the measurement of the WDF of a quantum system allows the determination of the complex Schrodinger wave function (or the density matrix) that carries the maximum information about the quantum system. For an excellent review about this subject see Leonhardt and Paul [1995].

WIGNER DISTRIBUTION FUNCTION IN QUANTUM OPTICS

51

vacuum

sig

reference beam Fig. 10. Homodyne detection scheme for quantum Wigner distribution function (WDF) measurement. The dashed beam splitter is fictitious; it models the imperfect photon counting.

All these applications require the knowledge of the WDF of a quantum system. Some information about the WDF (about the quantum state) can be inferred from the process of preparing the quantum system, but a measurement method of the WDF is desirable for the characterization of unknown quantum systems. Since the position and momentum operators do not commute, the Heisenberg’s uncertainty principle forbids the simultaneous measurement of the position and momentum and therefore it seems that it is not possible to measure the quantum WDF. Recently, however, both theoretically (Vogel and Risken [ 19891) and experimentally (Beck, Smithey and Raymer [ 19931, Beck, Smithey, Cooper and Raymer [ 19931, Smithey, Beck, Cooper and Raymer [ 1993a,b], Smithey, Beck, Raymer and Faridani [1993]) it was shown that the WDF can be reconstructed from a set of distribution functions wg(rg) by a technique known as optical homodyne tomography: wg(rg) =

J ~ ( r cos g s -Po sin S, r g sin s +Po cos S) dpe.

(10.3)

Note the similarity between eqs. (9.3) and (10.3). The distribution function W g ( r 0 ) is measured with a balanced homodyne detector (fig. 10) that consists of a 5050 beam splitter, two ideal photodetectors, and a reference beam with a well-defined phase 8 with respect to the signal. If the reference beam is coherent and intense with respect to the signal, the difference between the measured photocurrents is proportional to the expectation value wg(r0) of

52

THE WGNER DISTRIBUTION FUNCTION IN OPTICS AND OPTOELECTRONICS

[I, (i 10

i cos 0 + 5 sin 0, with i a n d a the incident operators. Thus by changing the phase between the reference beam and the signal, different combinations of the position and momentum variables can be measured. The WDF is then determined from the (eventually discrete) set of data by using the inversion formula:

Fo

=

(10.4) Leonhardt [I 9953 proposed a tomographic scheme to infer the quantum states of finite dimensional systems for which he developed a discrete WDF formalism. The influence of the imperfect photon counting is taken into account by inserting a fictitious beam splitter (dashed line in fig. 10) in front of the ideal homodyne detection. The transmissivity t of the fictitious beam splitter is given by the efficiency r] of the photodetector: t = q 2 . In this case the ID measured distribution is a smoothed version of the ideal one obtained from the latter by an averaging procedure with a real Gaussian function. To improve the accuracy of the homodyne detection, an amplifier (“anti-squeezer”) is placed in front of the whole setup. If the gain of the amplifier G is large, the measured distribution becomes a scaled version of the ideal distribution: W ~ ( X O= )(r]G)-’/2wt((~ G ) - ’ / ~ x o thus ) , allowing perfect measurement of distributions with imperfect detectors. The optical homodyne tomography is a complicated procedure to determine the WDF. Useful information about the WDF can be gained by means of simpler techniques that do not require a reference signal. Leonhardt and Paul [1994a] proposed the measurement of the square of the WDF by optically mixing the wave function and a phase conjugate replica of it in a beam splitter. A similar measurement can be performed using a parametric amplifier. For an important class of quantum states, the squeezed states, even the WDF itself can be measured directly, because in this case the WDF is a product of r and p distributions that can be measured separately, each on one half of the statistical ensemble. Information about the WDF can also be gained from measurements of the Q function. The Q function is non-negative and can be interpreted as a smoothed WDF. This smoothing process, which causes negative values of the WDF to disappear, also leads to a loss of information where the finer details of the WDF are wiped out. The Q function, which is related to the WDF by Q(r, p) =

1

W(r’, p’)exp

(- [(r

-

r’)” + (p

-’.,I

) dr’ dp’,

(10.4)

I1

REFERENCES

53

is a member of the so-called non-negative Wigner-type distributions (Cartwright [1976], Mourgues, Feix, Andrieux and Bertrand [1985]), and can be measured, for example, in an eight-port homodyne detector scheme, with a parametric amplifier or with a heterodyning scheme. For a discussion of measurement techniques for the Q function, see Leonhardt and Paul [1994b, 19951, and the references therein, Leonhardt, Bohmer and Paul [ 19951.

5

11. Conclusions

The applications of the WDF presented in this review, together with other applications mentioned in the introduction, support the assertion that the WDF is a valuable theoretical and experimental tool in optics and optoelectronics. New applications will certainly be proposed in the future. Further expansion of WDF applications will probably result from the recent extension of the WDF definition as a quantum quasiprobability distribution of number and phase (Vaccaro [1995]) and as a wide-band distribution hnction in signal processing (Shenoy and Parks [ 19951).

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E. WOLF, PROGRESS IN OPTICS XXXVII 0 1997 ELSEVIER SCIENCE B.V ALL RIGHTS RESERVED

I1

DISPERSION RELATIONS AND PHASE RETRIEVAL IN OPTICAL SPECTROSCOPY BY

KAI-ERIKPEIPONEN Viiisalii Laboratoy, Department of Physics, Universig of Joensuu, PO. Box I l l . FIN-80101, Joensuu, Finland

ERIKM. VARTIAINEN Department of Physics, Lappeenranta Universig of Technologv. PO. Box 20, FIN-53851, Lappeenranta, Finland

AND

TOSHIMITSU ASAKURA Research Institute for Electronic Science, Hokkaido University, Sapporo, Hokkaido 060, Japan

57

CONTENTS

PAGE

5 1. 8 2. 9 3.

INTRODUCTION . . . . . . . . . . . . . . . . . . .

59

KRAMERS-KRONIG RELATIONS . . . . . . . . . . . .

62

PHASE RETRIEVAL IN OPTICAL SPECTROSCOPY . . . .

73

9 4 . SUMRULES . . . . . . . . . . . . . . . . . . . . .

86

3 5. CONCLUSIONS . . . . . . . . . . . . . . . . . . . .

91

ACKNOWLEDGEMENTS . . . . . . . . . . . . . . . . . .

92

REFERENCES . . . . . . . . . . . . . . . . . . . . . . .

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58

6

1. Introduction

In materials science various types of spectroscopies, exploiting for instance radioactive particles or electromagnetic radiation as a probe, have been wellestablished in order to gain information about the physical properties of the media. Such information has provided scientists with a better understanding of the electronic and nuclear systems of elements and compounds. Furthermore, the knowledge of the spectroscopic properties of materials has had an impact on many technical applications, which include, for example, different detector devices. An electromagnetic field that is incident on a medium will have interactions with electrons or protons. The strength of the interaction depends upon the energy of the incident photons. In linear optical spectroscopy, probably the bestknown and most often exploited measurement techniques involve the detection of wavelength-dependent light transmission of transparent materials or the normal reflection of opaque materials. During this century the spectral investigation of the intrinsic optical properties of insulators, metals and semiconductors has drawn much attention. Theoretical models dealing with absorption and dispersion were formulated rather long ago. A relatively simple classical description of the permittivity of insulators, which is a complex valued function, is due to Lorentz whereas that of metals is due to Drude (see, e.g., Born and Wolf [1980], Wooten [1972]). The permittivity of semiconductors can be described by coupling the models of Lorentz and Drude together. The advantage of these models lies in the fact that one can explicitly resolve mathematical formulae for absorption and dispersion. However, the classical description usually gives a qualitative, but not a quantitative, picture of the optical properties of the gaseous, liquid or condensed matter. An improvement in dispersion models is to take into account quantum effects as devised by Kramers and Heisenberg. Nevertheless, dispersion formulae of general validity have not yet been formulated. The measurement of either the optical density or reflectance of materials is relatively easy, but the measurement of the wavelength-dependent change of the real refractive index and the phase of the reflected wave may be problematic. An important step toward resolution of this problem was taken at the beginning 59

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DISPERSION RELATIONS AND PHASE RETRIEVAL IN OPTICAL SPECTROSCOPY

[II,

9

1

of this century independently by Kramers [1927] and Kronig [1926], who were the first to derive the dispersion relations later known as the KramersKronig (KK) relations. These relations are valid for linear optical constants and also for nonlinear susceptibilities, excluding degenerate four-wave mixing processes for which they do not hold. The idea of the KK relations is that the measured wavelength-dependent data is inverted by numerical calculations to yield the appropriate quantity that is needed. This inversion procedure, involving measured and computed data, has provided intrinsic optical properties and has also affected the quantum mechanical description of electronic systems of media. As an example we mention here the seminal paper devoted to the properties of aluminum written by Ehrenreich, Philipp and Segall [1963]. KK relations have been applied not only in optical physics but also in high energy physics, geophysics, chemistry, etc. (Bohren and Huffman [ 19831). It took a considerably long time before the KK relations could be applied in practice. The main obstacle was the lack of means for numerical computations which were tedious a few decades ago. The obstacle disappeared when computers were developed and became widely available for use by scientists. Despite general agreement that the KK relations are valid, it is a well-known fact that for the data inversion the measured data must be extrapolated beyond the measured range. This procedure has brought up criticism (see, e.g., Aspnes [ 19851) about the reliability of the absolute values of the intrinsic linear optical constants calculated using the KK relations. Attempts to avoid usage of the KK relations have been presented in the literature. Among them is a simple method introduced by King [1977, 19781. He proposed that linear spectral data could be inverted with the aid of conjugate Fourier series. This was tested by Ketolainen, Peiponen and Karttunen [ 19911, who applied King’s method to the calculation of the refractive index change of mixed alkali-halide crystals containing F color centers. It was observed that the method of Fourier series also needs data extrapolation beyond the measured range, and the inverted extinction coefficient should have relatively low values at the wings of the spectrum for successful data inversion. The data extrapolation is usually the critical part of KK calculations. This is the case when optical constants are derived, in particular, from reflectance measurement. Therefore, it is a considerable advantage if only the measured spectrum of finite wavelength range is needed for calculating the optical constants in that range. This is possible in reflection spectroscopy with the aid of a maximum entropy procedure that has been introduced for that purpose by the authors of this review article (Vartiainen, Asakura and Peiponen [ 19931). Nonlinear spectroscopies have become important tools in material sciences.

11, § 11

INTRODUCTION

61

Therefore the question of the validity of the Kramers-Kronig type relations coupling the real and imaginary parts of nonlinear susceptibilities has been investigated, starting in the early sixties by Kogan [1963], Price [1963], and Caspers [1964] and a decade later by Ridener and Good [1974, 19751, Smet and Smet [1974], Smet and van Groenendael 119791 and recently by Peiponen [ 1987a, 19881, Bassani and Scandolo [ 19911, Hutchings, Sheik-Bahae, Hagan and van Stryland [1992], and Kircheva and Hadjichristov [1994]. The consistency between the KK relations and experiments in the case of nonlinear wave interaction with media was shown by Kishida, Hasegawa, Iwasa, Koda and Tolcura 119931, who performed rather tedious experiments with the thirdharmonic wave generation in polysilanes. Inversion of optical data, related to the phase retrieval from the measured modulus of nonlinear susceptibility, without any data extrapolations, has been dealt with by Vartiainen [ 19921, who investigated coherent anti-Stokes Raman spectra using maximum entropy procedure. The same procedure was applied for solving the nonlinear optical constants of polysilanes by Vartiainen, Peiponen, Kishida and Koda [1996]. It has been pointed out by many authors (Peiponen [ 19881, Hutchings, Sheik-Bahae, Hagan and van Stryland [1992], Kircheva and Hadjichristov [1994]) that the KK relations do not hold in the case of degenerate four-wave mixing spectroscopy. However, the maximum entropy model can be applied even in the description of such pathological meromorphic nonlinear susceptibilities as shown by Vartiainen and Peiponen [ 19941. Martin [ 19671 and Mezincescu [ 19851 have examined the cases of meromorphic linear susceptibilities to provide modifications in the KK relations and related sum rules. Since the days of the invention of the KK relations, it has been realized that they can be used to provide deep information about the physical parameters of the investigated system. For instance, one can derive with the aid of a particular KK relation a sum rule that is known as thef-sum rule (see, e.g., Smith and Dexter [ 19721). The f-sum rule is the optical counterpart of the quantum mechanical Thomas-Reiche-Kuhn sum rule. The message of the optical f-sum rule is that by measurement of absorption one can get information about the electron density and the oscillator strengths of the electronic transitions. Another feature of many sum rules for optical constants is that they can be exploited to test the consistency of the theoretical models and, importantly, also the success of the data inversions. Altarelli, Dexter, Nussenzveig and Smith [ 19721 were pioneers in deriving a set of new and important sum rules to characterize linear optical constants of materials. Altarelli and Smith [I9741 continued the work and could show by

62

DISPERSION RELATIONS AND PHASE RETRIEVAL IN OPTICAL SPECTROSCOPY

[n,5 2

using the novel sum rules that, at that time, corrections were needed to describe precisely the magnitudes of the optical constants of some materials obtained by data inversions. Thereafter, several authors (Villani and Zimerman [ 1973a,b], Furya, Villani and Zimerman [1977], Smith [1976a,b, 19801, King [1976,1979], Peiponen [ 19851) continued to realize more general or sophisticated sum rules for linear optical constants. Sum rules for nonlinear susceptibilities have been introduced by Peiponen [1987a,b, 19881, and by Peiponen, Vartiainen and Tsuboi [ 19901, Peiponen, Vartiainen and Asakura [1992], Vartiainen, Peiponen and Asakura [ 1993a1, Bassani and Scandolo [1991, 1992a,b], and Scandolo and Bassani [ 19921. In this review article we deal with the Kramers-Kronig relations appearing in linear and nonlinear optical spectroscopy. In addition, we consider the phase retrieval in linear reflection spectroscopy and in nonlinear optics with the aid of the maximum entropy procedure. Finally, sum rules for linear and nonlinear optical constants are presented.

Q 2. Kramers-Kronig Relations 2.1. HILBERT TRANSFORMS

The mathematical theory of complex analysis has provided good means to deal with complex valued optical constants. The reason for this is due to the holomorphicity (we prefer here the terminology where the property “holomorphicity” of a function is used instead of the analogous concept “analyticity”) of the optical constants. From the mathematical standpoint, the holomomorphicity of a function means that a complex valued function of a complex variable is derivable, the derivative is finite and obeys Cauchy-Riemann conditions. In physics, holomorphicity has a profound interpretation and it is a result of causality. In other words, the response is always later than the cause of the response. A practical example illustrating the principle of causality is as follows: if one is kicking a football towards a wall the response, rebounding of the ball from the wall, will be at a later instant than the kicking of the ball. The principle of causality and its relation to holomorphicity and the existence of dispersion relations have been described in the elegant paper written by Toll [ 19561. A comprehensive treatment of general dispersion theory can be found in the book of Nussenzveig [1972]. For a complex valued physical quantity, f(z) = u(z) + iu(z), that is a holomorphic function of the complex variable, z = x + iy, and has appropriate

11,

9 21

KRAMERS-KRONIG RELATIONS

63

asymptotic properties, one can derive the so-called Hilbert transforms with the aid of complex contour integration (Morse and Feshbach [ 19531). The function can in principle be holomorphic in the whole complex plane. However, in spectroscopy, f is holomorphic in a half-plane and it has poles in the other half-plane. The poles can be simple (linear optics) or of higher order (nonlinear optics) and the set of the poles may be countably infinite (like in the case of the Kramers-Heisenberg model for permittivity). The poles represent the resonance points of the system. In the case of measured data they express information about the resonance frequencies and line widths of the spectra and can depend on the temperature of the medium. In the quantum mechanical description they yield information about the electronic transition energies and life times of the related exited states. Hilbert transforms are as follows:

The integrands in eqs. (2.1) are singular for the value x = x’. However, the integrals are finite provided that the Cauchy principal value, which is denoted by the symbol P, exists. Taking the Cauchy principal value means that during the integration the nonessential singular point x = x’ is approached symmetrically from the left and the right side. In numerical integrations it means that we cannot allow the integrals of eqs. (2.1) to have precisely the singular value. Reliable estimates are obtained for u(x’) or u(x’) when the values of the integrals are, practically speaking, not changing while the symmetric approach of the point x’ becomes closer and closer. The advantage of Hilbert transforms is that if we can measure, e.g., the function u then we may calculate the hnction u and vice uersa. The shortcoming of the Hilbert transforms is that one integration, performed for the data inversion, will yield the desired physical parameter at only one point. This means that the point x’ must be scanned over the infinite real x-axis in order to resolve the wanted quantity. It is noteworthy that the Hilbert transform of a constant function yields a zero value; i.e., the Hilbert transforms of u(x) and u(x) + C , where C is a constant, are the same. Therefore, we obtain the same value, e.g., for an integral where the integrand u(x)/(x - x’) is replaced by the integrand (u(x) - u(x’))/(x - x’). Obviously, the latter form will approach the derivative of u when x is approaching x’. It is thus evident that by the Hilbert transforms we may calculate the change in the physical parameter. The additive constant C that may be needed is usually known from the physics of the system, and it has a unique value. Actually, this is a necessary property in the derivation of the Hilbert transforms. That is to say,

64

DISPERSION RELATIONS AND PHASE RETRIEVAL IN OPTICAL SPECTROSCOPY

[II,

5

2

we must choose C, in accordance with physics, to guarantee the vanishing of a complex valued integral along an arc of a semi circle while the radius tends to infinity. This property is known as Jordan’s lemma. The mathematical assumption of the asymptotic behavior of the physical quantity, needed for the validity of eqs. (2. l), is rather weak: f ( z ) IZI-~, 6 > 0, as IzI + 00. For linear and also nonlinear optical constants, the asymptotic fall-off is always stronger. The universal property for linear permittivity (Smith [ 19851) states that the fall-off of the permittivity is inversely proportional to the second power of the angular frequency for high frequency values. For nonlinear susceptibilities, an even stronger fall-off can usually be found. In the case of optical physics, x can be the time variable. Then Hilbert transforms characterize an analytic signal, which is of great importance in the description of coherence properties of optical fields as devised in the seminal paper of Mandel and Wolf [1965]. In optical spectroscopy, x is often replaced by the wavelength of the incident light or more commonly by the corresponding angular frequency. The contour integration using complex angular frequency variable for dispersion relations of the permittivity and the permeability is presented, e.g., in the book of Landau and Lifshitz [1960]. Detailed mathematical procedures of the contour integrations and the estimations which are needed can be found, e.g., from the paper of Hutchings, Sheik-Bahae, Hagan and van Stryland [ 19921.

-

2.2. KRAMERS-KRONIG RELATIONS IN LINEAR ABSORPTION SPECTROSCOPY

Inspection of the Hilbert transforms described above reveals that at first sight they would not be of much importance in practical optical data inversion since integration is also needed in the domain of negative frequencies. Negative angular frequencies are not related to the physical reality, which allows only positive values. However, optical constants have symmetry properties which will reduce the integration to positive values. The symmetry properties will result from the fact that a real valued electromagnetic field (input) must result to a real valued polarization (response) of the electric charges. The symmetry relations or crossing relations, as they are also called, can be written for complex permittivity and permeability and hold for all known systems. The message of the symmetry relations is simple, and it states the even and odd parities of the optical constants. In the case of the angular frequencydependent, complex refractive index, . N ( w ) = a ( w ) + iK(w), where4 is the real

11,

5 21

KRAMERS-KRONIG RELATIONS

65

refractive index and K is the extinction coefficient, one can write the symmetry relations as follows:

In the event that complex angular frequencies, B = Re& + i ImB, are permitted the symmetry relation for complex refractive index is N * ( B ) = N ( - B * ) , where * denotes the complex conjugate. Note that in the case of circularly polarized light and in magneto-optics the crossing relations are asymmetric and therefore dispersion relations of modified Kramers-Kronig form (Smith [ 1976a,b]) are valid for the corresponding complex refractive indices. The best-known form of the KK relations often dealt with in optics are given for the complex refractive index (related to the the linearly polarized light modes in isotropic media) as follows:

n.(w) - 1 (32

- (3‘2

do.

These relations express the principle of causality in the frequency space. Accordingly, the linear response of the system to a light field is defined completely by m(w) or K ( o ) . Similar relations, but of more general validity (not limited by the isotropy of the material) which take into account the tensoric nature of the permittivity (Smith [1980]) can be written for the complex permittivity of insulators, metals and also semiconductors. However, the measurement of the permittivity as a function of angular frequency is problematic and therefore the corresponding Kramers-Kronig relations have importance mainly in theoretical considerations. Usually it is easier to measure and calculate the complex refractive index and thereafter resolve the complex permittivity, t, using the well known identity Efi = N 2 ,where fi is the complex permeability. For insulators, as well as for metals at optical frequencies, the permeability is equal to unity. In fig. 1 are shown the measured extinction coefficients (in a region of strong absorption) and corresponding refractive index changes, calculated from eq. (2.3), for a KBr crystal that contains defects which are F , M and R color centers. The optical constants change as a function of the crystal temperature. Precautions must be taken in order to meet the assumptions imposed on the use of the KK relations. Firstly, we must regard the symmetry properties in theoretical lineshape models. If a line model that does not fulfill the particular

66

DISPERSION RELATIONS AND PHASE RETRIEVAL IN OPTICAL SPECTROSCOPY

I

[II,

5

2

a)T=120K

F

Fig. I . (a) The extinction data of F , M and R color centers of a KBr crystal at various temperatures, and (b) corresponding refractive index changes (Peiponen and Vaittinen [ 19841).

symmetry model is employed in the calculations, an error in the result is unavoidable. This was shown by Peiponen and Vartiainen [1991] with the aid of a gaussian line model. The moral of that study was that if we invert a function that does not obey symmetry relations, the correct values are obtained from the Hilbert transform but not from the Kramers-Kronig form. This observation

11, § 21

KRAMERS-KRONIG RELATIONS

67

stimulated Lee and Sindoni [1992] in their studies of the susceptibility of a semiclassical gas. Inserting w' = 0 in the upper part of eq. (2.3) will provide us the information about the static refractive index. Such information related to the static term or low frequency value of the real part of the permittivity of insulators is important in electric power applications like those of the insulating materials of capacitors. A method to generalize the Kramers-Kronig relations to hold for finite frequency intervals was proposed by Hulthtn [ 19821. His method enables one to compute a complex refractive index, N ( w ) , for u < W I and w > w2, if N ( w ) is known for 01 w w2. So far, this remarkable method has not attracted much interest - probably because usually only either the real or imaginary part of N ( w ) is measured within some interval 01 < w < w2, and Hulthtn's method requires both of them to be known.

< <

2.3. KRAMERS-KRONIG RELATIONS IN REFLECTION SPECTROSCOPY

Kramers-Kronig relations in reflection spectroscopy have drawn much attention. The validity of the Kramers-Kronig relations in reflection spectroscopy was treated by Jahoda [1957] and Velicky [1961]. Criticism about their validity has been presented by Goedecke [1975a,b] (see also Agudin, Palumbo and Platzeck [1986]) on the basis of the proposed violation of the micro causality due to radiation reaction. Smith [ 19771 has provided a simplified proof of the validity of the dispersion relations for complex reflectivity. The KK relations in reflection spectroscopy are important when dealing with strongly absorbing materials like metals and semiconductors for which the measurement of the optical density is cumbersome. The advantage of the measurement of reflectance is due to its simplicity and the wide spectral range of probe wavelengths that can be used. Complex refractive index can be determined solely by ellipsometric measurement, but that procedure is more dependent on instrumental factors and a relatively narrow spectral range can usually be covered. The complex electric field normal reflectance, r, can be given in the polar form r(w) = Ir(w)l exp[iO(w)]. By measurement, we obtain information about the amplitude reflectance R = lrI2.The desire is to resolve the optical constants n and k . The normal reflectance and the complex refractive index are coupled by the Fresnel formula,

r ( u )=

1- N(0) 1+ N ( w ) '

68

DISPERSION RELATIONS AND PHASE RETRIEVAL IN OPTICAL SPECTROSCOPY

[I],

5

2

The remarkable property of eq. (2.4) is that it provides a means, together with the proper KK-relation, to resolve the absolute value of n instead of its change, which is the case when absorption data are inverted. The information about the optical constants of n and k of an opaque medium can be calculated when we notice that the logarithm of the reflectance and the corresponding phase of the reflected wave can be resolved by taking the logarithm Inr = In Ir( + i0. This function is holomorphic on the real axis and in the upper-half of the complex angular frequency plane. However, the asymptotic behaviour of r, i.e., Ir(w)l 0 and therefore lnr(w) + -00 as w + 00, (no interaction with material for infinite energy values of the incident photon) has brought up much confusion in the context of the related Kramers-Kronig relation. Smith [ 19771 has formulated dispersion relations for reflectances where the pathological behavior as decribed above is shown to be nonessential. Following Smith’s argument, we can write the KK relations for the phase retrieval problem in the form: ---f

The constant 60 has the value zero for insulators and metals. However, in some special cases it may have a nonzero value; this has been observed recently by Nash, Bell and Alexander [1995]. Lee [1995] has considered the phase recovery and reconstruction of the Raman amplitude from the Raman exitation profile. He generalized the KK relations to a case where the complex function may possess, in addition to the poles, also zeros in the complex plane. It should be stressed that the latter relation in eq. (2.5) is not a difference of two KJS relations since both integrals are individually divergent, whereas the difference is convergent. The above definition of the reflectance in the polar form means that there is an indeterminacy in r, just as in the case of Hilbert transforms related to an additive constant considered in $2.1. This can be realized when we investigate a product C Irl, where C is a real constant so that C Irl < 1. Then we can write In C Irl = In C + In Irl. When calculating the KK relation, the additive constant term is filtered out. This property with the polar reflectance has importance in the deduction of possible sum rules for Irl. Nevertheless, as already stated in the context of absorption data inversion, the constant C must be consistent with physics. Therefore, the only choice is C = 1 in order to fulfill the demand of Jordan’s lemma. The numerical integrations to calculate the phase angle are very critical to the width of the spectral range of the measured data and also to data

11,

P 21

69

KRAMERS-KRONIG RELATIONS

extrapolations beyond the measured range. Aspnes [ 19851 has investigated the published values for optical constants obtained by KK analysis and found substantial errors in them. He ascertained that these errors are caused mainly by the data extrapolations, and stated that "A Kramers-Kronig analysis should be attempted only if accurate reflectance data are available over a very, very wide energy range". Nevertheless, various fitting procedures have been presented in the literature (Rasigni and Rasigni [1977], Jezierski [1984]). The failure of KK relations in reflectance spectroscopy in the case of layered materials was dealt with by Young [1972]. 2.4. DISPERSION RELATIONS IN NONLINEAR OPTICS

The history of KK type dispersion relations for nonlinear response dates back to the early sixties. Kogan [ 19631, Price [ 19631, Caspers [ 19641, and Smet and Smet [ 19741 were dealing with second order nonlinear responses, whereas Ridener and Good [1974, 19751 derived dispersion relations for a third and an arbitrary order nonlinear response of the system. Smet and van Groenendael [1979] showed that no dispersion relation can be established between different nonlinear phenomena. All of the systems mentioned above were treated in such a way that only one angular frequency of the incident electric field was permitted to change whereas other pump fields were assumed to have a fixed wavelength. Then the dispersion relations are of the KK form and the derivation of them is based on the complex contour integration of one complex variable. The corresponding dispersion relation for the nth order nonlinear susceptibility can be then written as follows: p ( u I , .

. . , w;,.

l

. . , 0,)= 7-P

W X ( " ) ( W l , . . . ,uj,. .. '

s,

0. - w! J

dwj.

(2.6)

J

The above dispersion relation was generalized by Peiponen [1987a, 19881 to allow the changing of the angular frequency of all incident fields. With the aid of the theory of several complex variables and by applying the complex contour integration separately for each variable, one can write the generalized form to read:

x'"'( w;,w;, . . . ,0,)

However, only the odd orders will yield dispersive-dissipative effects to be written by KK type dispersion relations. The symmetry relation relating

70

DISPERSION RELATIONS AND PHASE RETRIEVAL IN OPTICAL SPECTROSCOPY

[II,

92

to the nth order nonlinear susceptibility and involving real frequencies is x(")*(ul,. . . 0,) = x(")(-ul7 . . . -u,). If all frequencies are allowed to vary and the symmetry property is used, the integrations in eq. (2.7) can be performed from zero frequency to infinity. Bassani and Scandolo [1991] have derived dispersion relations, by considering the nonlinear process of all orders, for the refractive index that is a sum of the linear ( h ~and ) nonlinear (R,NL) angular frequency dependent terms, R = n ' +~~ , N L ,and extinction coefficient, K = KL + KNL, in the regime of a probe and one pump beam (with fixed wavelength). These relations are as follows:

where u]is the angular frequency of the probe beam to be scanned, 0 2 is the fixed frequency of the pump beam and E2 is the electric field of the pump beam. From eq. (2.8) we can observe that in the absence of the probe beam there will be a nonzero "static" contribution, RNL(O, u2,Ez), to the refractive index. This kind of procedure with probe and pump beams is quite common in many experiments like those involving coherent anti-Stokes Raman scattering (see, e.g., Shen [1984]). Unfortunately, most of the experiments with the nonlinear optical processes will provide information only about the modulus, Ix(")l,of the nonlinear susceptibility. Then the problem of phase retrieval will again arise, and problems similar to those in linear reflectance spectroscopy are present. The maximum entropy method will provide us with a way to solve the phase retrieval problem and to avoid the data extrapolation in resolving the real and imaginary parts of the nonlinear susceptibility from the wavelenght-dependent modulus data, as will be described later. Nonlinear susceptibilities can also be described by series expansions (Peiponen [1988]) and by making use of the concept of conformal mapping by a similar way as King [ 19781 did in linear optics to avoid KK relations. However, complicated measurements and data extrapolations are then needed. The generalized dispersion relations of several angular frequency variables of eq. (2.7) as well as those of one frequency are valid only for nondegenerate cases. The degenerate case means that a frequency pair like (u,, -uj)appears in the functional dependence of the nonlinear nth order susceptibility. This in turn means that there appear simultaneously poles in the upper and lower half planes. Therefore dispersion relations of the KK type are no longer valid. A function

11, § 21

KRAMERS-KRONIG RELATIONS

71

with poles in both the lower and upper half planes is called meromorphic (Peiponen [ 19881). Shore and Chan [ 19901 and Hutchings, Sheik-Bahae, Hagan and van Stryland [ 19921 have discussed the dispersion relations for meromorphic susceptibilities like ~ ( ~ ) w, ( w-w), , which are closely related, for example, to semiconductors (see Hopf and Stegeman [ 19861). For the possible dispersion relations one must know the resonance points in the upper half plane. This may not, however, be possible with the aid of measured data due to, e.g., the overlapping of adjacent lines belonging to different electronic transitions. Using the complex contour integration we can write a dispersion relation for the meromorphic nonlinear susceptibility. In the case of one angular frequency variable (the others being fixed), a dispersion relation for the nth order nonlinear susceptibility can be given as (Vartiainen and Peiponen [ 19941): X'"'(w;, . . . , wj, -oj,. . . ,w;)

(9- y,!)

do,

=

inx(")(wi,.. . , w,!, -u,!, . . . ,w,,)

(2.9) The latter term on the right-hand side of eq. (2.9) is the contribution of the residues, which takes into account the poles in the upper half plane. Apparently the KK type dispersion relation is now modified by adding a term due to the poles, but as mentioned above, we are not usually provided with the information of resonance frequencies of such poles. An example is shown in fig. 2 (Vartiainen and Peiponen [1994]). Typical theoretical dispersion curves of real and imaginary parts of degenerate susceptibility x ( ~ ) ( w, o ,-w) are plotted along with corresponding curves computed using the dispersion integral of eq. (2.9), but omitting the residues. Clearly, the contribution of residues is significant; otherwise these curves would coincide with Re ~ ( and ~ Im 1 ~ ( ~ 1 . It should be emphasized that there is no causality breaking with meromorphic nonlinear susceptibilities. Merely causality is necessary, but not a sufficient condition for the existence of the conventional KK relations, as pointed out by Kircheva and Hadjichristov [1994]. Kishida, Hasegawa, Iwasa, Koda and Tokura [ 19931 were the first to show the consistency between the KK relations and experiments in nonlinear optics. They made laborious experiments with the third harmonic generation from polysilanes. Inverting the modulus of the third order susceptibility, they calculated the corresponding phase and thereafter the intrinsic real and imaginary parts of

72

DISPERSION RELATIONS AND PHASE RETRIEVAL IN OPTICAL SPECTROSCOPY .

,

.

,

.

,

,

.

,

.

,

.

,

.

,

.

,

[II, $ 2

_

0.2 3

?

0.1 -

/-*'.

.

..mc.-".'

0.0

-0.1 -0.2 -0.3 -

d

-0,4 -0.5

-0.6

-8 .Fi --j a

0.2

j

0.0

m

"

"

"

"

"

"

"

"

"

"

1.2

1.4

1.6

1.8

2.0

"

"

"

"

0.1

-0.1

-;- 0 2 -0.3 -0.4' 1.0

'

"

2.2

"

2.4

"

2.6

"

2.8

'

3.0

Energy (eV) Fig. 2. Theoretical curves (a) R e ~ ( ~ ) ( w , w , - oand ) (b) I m ~ ( ~ ) ( w , w , - w (solid ) lines) and the corresponding curves given by the dispersion integral of eq. (2.9) (dots) (Vartiainen and Peiponen [ 19941).

the third order susceptibility related to the third-harmonic wave generation. The consistency between the theory and experiments was verified by ellipsometric measurements and exploiting various laser lines to generate the third harmonic wave. Finally we remark that care should be taken while employing KK relations in time-resolved spectroscopy as devised by Tokunaga, Terasaki and Kobayashi [ 1993, 19951. They applied femtosecond time-resolved spectroscopy to study the pumpprobe processes in CS2 liquid. According to their experimental and theoretical results the causality condition is not always satisfied when a pump pulse causes changes in the state of the material before the probe field is applied. Basically the situation corresponds the meromorphism of the nonlinear susceptibility.

11,

P 31

PHASE RETRIEVAL IN OPTICAL SPECTROSCOPY

0

73

3. Phase Retrieval in Optical Spectroscopy

Practical experiments in optics usually provide only partial measurements of the electric field E(t). For example, in optical spectroscopy only an intensity spectrum, IE(w)I2 0; lf(w)I2,is measured while the entire complex response function,f(w) = If(w)l exp[iO(o)], is needed for obtaining the desired material properties. This leads to the question of phase retrieval: With some extra obtainable information, is it possible to compute the phase function, O(w), from a measurement of the amplitude If(w)l? Recently, significant progress has been made in solving this problem in spectroscopic applications. The progress is due mainly to the idea of applying a certain type of the maximum entropy model in phase retrieval. This approach was first shown to be applicable in coherent anti-Stokes Raman spectrum (CARS) analysis (Vartiainen [ 19921) and in reflection spectroscopy (Vartiainen, Peiponen and Asakura [ 1992, 1993b1). Later, an improved version of the procedure, applicable to any case of spectrum analysis wheref(w) is to be computed from If(o ) l, was given by Vartiainen, Asakura and Peiponen [1993]. In particular, this method was shown to be valid even in the case where the alternative Kramers-Kronig method is not: whenf(c;)) is a meromorphic complex function, such as the degenerate third-order nonlinear susceptibility x ( ~ ) (61, &-61) , (Vartiainen and Peiponen [ 19941). In this section we give a short review of the phase retrieval in optical spectroscopy using the maximum entropy model: its theorical background, why it works and how it is applied. 3.1. PHASE RETRIEVAL USING MAXIMUM ENTROPY MODEL

3.1.1. Maximum entropy model

It has been conjectured in information theory (Shannon [ 19481, Jaynes [ 1957a,b]) that entropy defines a measure on the space of probability distributions, such that those of high entropy are favored over others. Accordingly, the maximum entropy (ME) principle states that any inferences made from incomplete information should be based on the probability distribution that has the maximum entropy permitted by the available data. Most notably this principle has been utilized with iterative ME algorithms in various image-restoration problems arising in astronomy (Gull and Daniel [1978], Bryan and Skilling [1986]), X-ray crystallography (Collins [ 1982]), medical tomography (Kemp [ 19811) and electron holography (Matsumoto, Tanji and Tonomura [ 19941).

14

DISPERSION RELATIONS AND PHASE RETRIEVAL IN OPTICAL SPECTROSCOPY

[II,

5

3

There exists another type of maximum entropy method that was originally developed by Burg [1967]. His method is an algorithm for computing the power spectrum, S(w), when the measured data consist of sampled signal,

x,

= x(t,),

0

<m <M,

whose autocorrelation values can be given by:

where & denotes the expectation operator. The autocorrelation function, C(m), and the power spectrum, S(w), form a Fourier transform pair, 00

C(m)exp(-imwht),

S(w) = At

,=-a

in

C ( m )= - L " " S (

w ) exp(imwAt) dw,

where At is the time between equispaced samples. In the following it is more convenient to use a normalized frequency v = (2n)-'wAt (i.e., Y is normalized within the interval 0 v 1). Equations (3.3) and (3.4) can now be written as

< <

oc,

S(V) =

C

~ ( mexp(-i2nmv), )

m=-m

S(v) exp(i2nmv) dv, where S( v) = S( w)/At. The conventional method of obtaining a power spectrum estimate from known autocorrelation values, C(m), lml < M , is to assume that C(m) = 0 for )m1 > M , and to take a Fourier transform of w(m)C(m), where w(m)is a window function. The windowing is used to reduce the errors due to the truncation of the autocorrelation function. A shortcoming of this kind of estimation is the fact that there is no knowledge about the window function that would give the best result. In Burg's maximum entropy method, the idea is to choose the spectrum that corresponds to the most random or the most unpredictable time series whose autocorrelation function agrees with a set of known values. This leads to an extrapolation of the autocorrelation function by maximizing the entropy (a

11,

5

31

PHASE RETRIEVAL IN OPTICAL SPECTROSCOPY

75

measure of the average information content) of the process x ( t ) ; hence, the name of the method. As a consequence, Burg's procedure leads to a certain maximum entropy model for S(w). On the one hand, this model is the final result of Burg's approach, and on the other hand, it is the starting-point in the phase retrieval procedure described in sect. 3.2.2. We emphasize that this model for S(w) is the only feature of the phase retrieval method common with Burg's maximum entropy method. For any stationary time series, {xn}, it can be shown that the entropy rate of the time series is (see, e.g., Haykin and Kesler [1983]):

1 I

h

0:

logS(v)dv,

(3.7)

where it is assumed that the time series is limited to the angular frequency band 0 < w < w, and u, = 2 n / A t . Suppose that we know the first 2M + 1 values of the autocorrelation function, namely:

I'

C(m)=

S ( v ) exp(i2nmv) dv,

Iml

< M.

(3.8)

Then the most reasonable choice for the unknown values of C(m), Iml > M , is the one that adds no information or entropy to the process. Therefore, the required maximum entropy power spectrum is the fimction k ( v ) that maximizes eq. (3.7) under the constraint of eq. (3.8). That is,

This leads to the following maximum entropy model (MEM) for S(v) (Haykin and Kesler [ 19831): 3(v)=

1

1+

M

IN2

C a k exp(-i2;zkv) k=l

1'

where the unknown MEM coefficients ak and Toeplitz system, C(0) C(1)

C(-I) C(0)

...

C(-W

. ' . C(1-M)

C ( M ) C ( M - 1) . . '

C(0)

(3.10)

IPl2

can be obtained from a

16

DISPERSION RELATIONS AND PHASE RETRIEVAL IN OPTICAL SPECTROSCOPY

[II, $ 3

Van den Bos [ 197I] has shown that the MEM, as defined in eqs. (3.10) and (3.1 l), is equivalent to the autoregressive (AR) model of a stationary stochastic process x ( t ) . A time process is called autoregressive if an observation x , can be given by a linear combination of the A4 preceding observations: M

(3.12) k= 1

where em is a corresponding error at m (with E[e(n)e(m)]= 0 for n f m), and the ak's are the same coefficients as in eqs. (3.10) and (3.11). Here we are interested in using the model of eq. (3.10) for the phase retrieval. Therefore, in addition to the model for S ( Y ) = lf(v)I2, we must have an ME model forf(v) = If(v)I exp[i@(v)]. We can derive this by taking a z-transform (see, e.g., Haykin [1986]) from both sides of eq. (3.12) and using the convolution theorem to obtain:

+ a2zp2 + . . . + aMz-M>X(z)+ ~ ( z ) ,

X(Z) = -(alz-'

(3.13)

where X(z) and E(z) are z-transforms of the sequences { x m } and { e m } , respectively. Thus, we get the expression for X(z): X(z) = 1+

E(z) M akz-k

c

(3.14)

k=l

Finally, if we replace the signal x ( t ) with the time response function g(t) and set z = exp(i2nv), the z-transforms can be recognized as the discrete-time Fourier transforms. Hence, we can write the maximum entropy estimate for the complex spectral responsef(v) = .F{g(t)} as a function of the normalized frequency v as (3.15) I

+

ak exp(-i2nkv) k= 1

3.1.2. Phase retrieval procedure

Suppose that a power spectrum, S(w) = lf(o)I2, is measured within the frequency range of 01 < w < 0 2 . Now, defining a normalized frequency as: v = (0- 0 1 ) / ( 0 2 - 01),

(3.16)

11,

P 31

PHASE RETRIEVAL IN OPTICAL SPECTROSCOPY

I7

S ( W ) can be fitted by eqs. (3.6), (3.10) and (3.11). However, instead of using

the continuous Fourier transform of eq. (3.6), the autocorrelations, C(rn), are in practice computed by using the discrete Fourier transform: N-l

C(m) = N-'

S,, exp(i2nmn/N),

(3.17)

n=O

where N is the number of S(W,,) = S,, samples. In this case there exists an upper limit to the number of C(m) and, therefore, to the MEM parameter M in eqs. (3.10) and (3.11); i.e., A4 < N/2. Contrary to Burg's maximum entropy method, here there is no ambiguity in selecting the order of M . Namely, if the object is to get the MEM estimate S(W) as close as possible to the original one, then using the maximum value for M = M,, gives the best result. However, if the original spectrum, S ( W ) is noisy, it is possible to reduce the noise in $0) by computing it with the lower order of M < Mmm. This is illustrated in fig. 3 (Vartiainen, Asakura and Peiponen [ 19931). Three simulated spectra, with the same spectral features but different amounts of noise, are shown in fig. 3a. The number of samples was selected to be N = 1001 and, consequently, Mmax= 500. Figure 3b shows how the order of M affects the root-mean-square (RMS) errors in the MEM estimates of the spectra in fig. 3a. The RMS error is defined here as: (3.18) where i E ( v ) , i = A , B, C, is an MEM estimate computed from spectrum i in fig. 3a, and So(v) is the original noiseless spectrum (curve A in fig. 3a). In the noise-free case of curve A, the minimum error (ERMSM is obtained with the maximum value M = 500. The error increases with a decrease of M , although the increase is only nominal over a wide range of M . At the low order of M < 70, however, the error increases very strongly. The latter observation is also true for the noisy spectra of curves B and C. For noisy spectra, the smallest error (compared with the noiseless spectrum) is obtained using the optimum value M = Moptthat is a compromise between the high orders which also reconstruct the noise (see fig. 3c) and the low orders which do not result in much noise but result in distortions instead because the higher-order MEM is required. Therefore, in practice, when Mop,cannot be obtained we should clearly use a high value of M > Mop, rather than a low one of M < Mop,to avoid the distortions at the expense of some noise to be reconstructed.

78

DISPERSION RELATIONS AND PHASE RETRIEVAL IN OPTICAL SPECTROSCOPY

,"t 9

[II,

5

3

(b)

I

N

W

L

200

400

MEM parameter M

Fig. 3. (a) Simulated power spectra S(w) with the noise-to-signal (N/S) ratios (defined as RMS values) of 0 (curve A), 5% (curve 8) and 10% (curve C). (b) RMS errors versus the order of M in MEM estimates of spectra in (a). (c) Power spectrum S ( o ) with N/S=5% (curve A) and its MEM ) M = 500 (curve B) and M = 60 (curve C) (Vartiainen, Asakura and Peiponen fits $ ~ ( owith [ 19931).

The next step in the procedure is to find a way to compute

](Y)

from

$(Y ) = lf(v)I2,assuming that some additional information onf(v) is available. Now, on the presumption that the complex spectral response function f(v) is given by eq. (3.16) and its modulus by eq. (3. lo), we can write the error spectrum as:

where # ( Y ) is the phase of the error spectrum. Thus the error spectrum has a constant amplitude and only its phase can have a frequency dependence. The error phase @ ( Y ) is now the only quantity in eq. (3.16) that cannot be obtained from the measured modulus If(v)I. Consequently, using the MEM in phase retrieval reduces the problem of finding the phase off(v) = If(v)I exp[iO(v)] to the problem of finding the corresponding error phase. In order to find a reasonable estimate for @ ( Y ) , we must have some additional information onf(v). For example, if we know the value of the real (or imaginary) part off(v) or the value of the phase O(v) at L + 1 discrete frequency values Y/ inside the measurement range, we can compute the corresponding error phase values qj(v,) and estimate the error phase by a polynomial interpolation as L

$ ( Y ) = B~

+ B , Y + . ..+ B

~ =Y

CB/Y/, ~

l=O

(3.20)

11, § 31

PHASE RETRIEVAL IN OPTICAL SPECTROSCOPY

19

where the Bl’s satisfy a Vandermonde system (see, e.g., Press, Teukolsky, Vetterling and Flannery [1992]) as:

(3.21)

The idea of using the MEM in resolvingf(v) from its modulus is that the error phase is usually a much smoother function than the actual phase and therefore, that it can be estimated with a low order polynomial (i.e., the number of required known values of $ ( Y [ ) is low). Clearly, the ideal situation would be if the error phase could be given by a linear estimation; i.e., L < 1 in eq. (3.20). In fact, it is possible to reduce the optimum degree, Lopt,of the polynomial in eq. (3.20) by a simple “squeezing” procedure. Namely, instead of using the measured power spectrum S ( Y )as such in eq. (3.9), we should use a modified spectrum Ssqgiven by : (3.22) where ZK(W) =

(2K + 1)-’ w2

- w1

(3.23)

and

Now, choosing the squeezing parameter K = 0, eq. (3.22) restores the original spectrum in Ssqas a function of the normalized frequency Y . However, when K > 0, the original is transformed (squeezed) into a narrower range. For example, setting K = 2 the original spectrum S ( Y ) is squeezed from the interval Y E [0, I] into interval Y E [0.4,0.6]. The rest of the squeezed spectrum is obtained by adding constant wings having the same values as the original spectrum at its end points Y = 0 and Y = 1. Quite commonly, this procedure reduces Lopt enough that a linear approximation ( L = 1) gives a reasonably good estimation for the error phase.

80

DISPERSION RELATIONS AND PHASE RETRIEVAL IN OPTICAL SPECTROSCOPY 0 . 7 - ,

,

O.'

(a)

Q)

0

,

I

I

,

,

,

I

I

I

,

"

"

I

,

,

.

[II,

03

-

c (d

c

0

-a,

0.5

-

c

2

0.3

"

1.5

"

2.0

'

'

"

2.5

'

'

J

3.0

3.5

4.0

4.5

"

5.0

5.5

6.0

Energy (eV)

4

0.8

0.7

-

0.5

-

1.5

1

2.0

2.5

3.0

3.5

4.0

4.5

5.0

5.5

8.0

Energy (eV) Fig. 4.(a) Reflectance of GaAs as a funtion of energy. (b) Phase 0 and the error phases obtained with (&) and without (9) the squeezing procedure. Exact (dots) and computed (solid lines); (c) real refractive index and (d) extinction coefficient. The arrows in (c) and (d) point out the energies at which the phase values were assumed to be known a priori; e.g., determined by ellipsometry (Vartiainen, Peiponen and Asakura [ 19961).

3.2. PHASE RETRIEVAL IN PRACTICE: EXAMPLES

3.2.1. Rejection spectroscopy An example of the utility of using the maximum entropy model in phase retrieval in reflection spectroscopy is illustrated in fig. 4 (Vartiainen, Peiponen and Asakura [1996]). In fig. 4a the reflectance spectrum, R ( o ) = lr(o)I2,of GaAs is shown within the energy range from 1.5 eV to 6.0eV This spectrum (dots) was obtained using complex refractive index data from the literature (Palik [1985a]). The corresponding phase function, @w), and the error phase functions are shown in fig. 4b. Clearly, both the error phase obtained with

11,

9 31

81

PHASE RETRIEVAL IN OPTICAL SPECTROSCOPY

4

5.5

:

:

: 2.0

-

1 .o 1.5

1.5

2.0

2.5

3.0

3.5

4.0

4.5

5.0

5.5

6.0

5.0

5.5

6.0

Energy (eV)

1.5

2.0

2.5

3.0

3.5

4.0

4.5

Energy (eV)

($sq(w)) and without ($(w))the squeezing procedure (defined in eqs. 3.22-3.24) are “slowly varying” functions compared with O(w). Moreover, gSq(w)can be well described by a linear approximation. Consequently, the complex refractive index N ( w ) = n(w) + ik(w) = [l + r(w)]/[I- r(w)] can be retrieved, if in addition to R ( w ) the value of r(w) is known at two frequencies inside the measurement range of R(w). In the present example this additional information was assumed to be obtained by ellipsometric measurements at wavelengths of a red (A = 633 nm) and a green (A = 543 nm) light emitting He-Ne lasers. The resultant real (refractive index, ~ ( w )and ) imaginary (extinction coefficient, ~ ( w )parts ) of JV(0) are shown in figs. 4c and 4d (solid lines), respectively. The exact values (dots) are also shown for comparison. Another example, concerning the reflectance spectrum of KC1, is shown in fig. 5 (Vartiainen, Peiponen and Asakura [ 19961). The spectrum shown in fig. 5a extends over the energies from the visible (2.0 eV) to the far ultraviolet (34 eV)

82

DISPERSION ELATIONS AND PHASE RETRIEVAL IN OPTICAL SPECTROSCOPY

[I], 4 3

o.3}

a

0 C c 0

0.2 -

-a .c

a

a

-

0.1

0.0

i .

3.0

20

10

0

,

.

,

I

.

.

,

30

,

2.5 h

u

2

v

2.0 -

1.5 -

v)

nr

1.0

-

0.5

-

0.0

"

"'

"

"

'

"

"

"

'

-

Fig. 5. (a) Reflectance of a KCI crystal. (b) Phase 6 and the corresponding error phase @ obtained with the squeezing procedure. Exact (dots) and computed (solid lines); (c) real refractive index and (d) extinction coefficient. The arrows in (c) and (d) point out the energies in the visible range where the extinction coefficient k x 0; these two points were used to give a linear estimate for the error phase (Vartiainen, Peiponen and Asakura [1996]).

region. This spectrum was also obtained using complex refractive index data from the literature (Palik [1985b]). In fig. 5b, the idea of the MEM procedure becomes evident: although the phase O(w) has a very complex line shape, the corresponding error phase $sq(o) can still be given rather well by a linear approximation. Furthermore, in this case no extra measurement for recovering O(w) is needed, because KCl is transparent in the visible range and thus we have the constraint ~ ( wM) 0 when w E [2.0 eV, 3.1 eV]. Using this constraint, the error phase was estimated and the complex electric field reflectance was obtained. The resulting real refractive index and extinction coefficient are shown in figs. 5c and 5d (solid lines), respectively.

83

PHASE RETRIEVAL IN OPTICAL SPECTROSCOPY

11, § 31

3.0 X

Q,

-0

1

'

'

'..

'

'

'

'

'

'

'

'

'

.

'

'

'

1

2.5

.-K

a,

> .c. 0 E .c

= Q,

2.0

1.5

:I

1.0

0.5

0

c

.-

.-

.-0

20

10

30

Energy (eV)

c

0

.E c

0.5

X

UI 0.0 0

10

20

30

Energy (eV)

3.2.2. Nonlinear optical spectroscopy

In nonlinear optical spectroscopy the most typical measurement is a power 2 spectrum measurement: the squared modulus of nth-order susceptibility = lx(")Iexp(i0) is measured within some frequency interval. Since both the modulus and the phase 0 provide crucial information on the multiphoton processes of a medium, the phase retrieval problem arises. It has been shown recently (Kishida, Hasegawa, Iwasa, Koda and Tokura [ 19931) that the phase of third-order susceptibility x ( ~w, ) (w, w ) of poly(dihexylsi1ane) can be computed from its modulus by Kramers-Kronig integration. In that calculation the extrapolation problem was avoided by a procedure similar to the Roesler method (Roesler [1965]). In that method, the integrals outside of the measurement region were replaced by two experimental phase data points. In fig. 6 these results are shown together with the corresponding maximum entropy

Ix(")I

x(")

Ix(")I

84

DISPERSION RELATIONS AND PHASE RETNEVAL IN OPTICAL SPECTROSCOPY

L

0.5

1 .o

1.5

Energy (eV)

2.0

0.5

[]I,

8

3

.

1 .o

1.5

2.0

Energy (eV)

Fig. 6 . Experimental values of (a) real and (b) imaginary parts (dots and open circles) of the , o)of PDHS, and the corresponding curves obtained by the KK analysis susceptibility x ( ~ ) ( oo, (dotted lines) and by the MEM procedure (solid lines). The additional information used for the MEM estimates with L = 1 are the two phase values indicated by the arrows (Vartiainen, Peiponen, Kishida and Koda [1996]).

calculations [the real and imaginary parts of x ( ~ ) ( ou, , o)] (Vartiainen, Peiponen, Kishida and Koda [1996]). The MEM curves were computed by estimating the error phase with first-order ( L = 1) and third-order ( L = 3) polynomials. It is observed that these MEM curves are identical, although the curves with L = 1 were obtained using only two measured phase values as additional information, whereas all the measured phase values (about 60 values) were used in the cases of the curves with L = 3. Moreover, the KK and the MEM estimates are very similar, excluding the fact that the KK computation could not reproduce the two-photon resonance peak at 2.1 eV Accordingly, both the KK and the MEM approaches for phase retrieval required phase data at two frequency points as additional information. However, the significant difference between the KK and the MEM analysis is the fact that in the KK calculation the additional information is needed outside, whereas in the case of MEM it is needed inside the measurement range of ( x ( ~ ) ( . Another third-order susceptibility having much practical interest is the frequency degenerate susceptibility x ( ~w, ) (w, -w), which determines the degenerate four-wave mixing processes. Furthermore, it links together the Kerr coefficient, y(w) 0: Re~(’)(w,u,-u), and the two-photon absorption coefficient, B(w) K Im ~ ( ~ ) w (w , -0). , Unfortunately, the phase of ~ ( ~ ’w( ,0 -w) , cannot be

11,

5 31

85

PHASE RETRIEVAL IN OPTICAL SPECTROSCOPY

0.2

-d

-

7

-H3

01

OD -01 -02

-03

X

2

-04 -0 5 -0.6 I 1.5

'

1.0

'

'

'

1.7

i n

I

'

1.0

2.0

2.1

2.2

2.3

2.1

2.5

Energy (eV)

i

0.45

--

0.85

7

0.25 0.1 5 0.05 -0.06

X

5

-0.15

-0.25 -0.35 1.5

1.8

1.1

ia

1.a

2.0

2.1

1.2

2.5

1.4

2.5

Energy (eV)

Fig. 7. (a) The theoretical amplitude ~ x ( ~ ) ( w , wThe , - ~KK (open squares) and MEM (dots) estimates of (b)

I I

and (c) ImX(3) computed with the aid of x ( ~. )The corresponding solid lines are the actual curves (Vartiainen and Peiponen [1994]).

obtained by the KK integration. This is because of the fact that the dispersion relation for ~ ( ~ )o, (-w) o , (given in eq. 2.9) includes in addition to the ordinary KK dispersion relation a term due to contribution of residues. This term can be neither computed nor omitted (see Vartiainen and Peiponen [ 19941). However, it is possible to use the MEM approach also in this case for computing the phase. This was demonstrated recently (Vartiainen and Peiponen [ 19941) by using a theoretical model for ~ ( ~ ) (w,w-o) , (see fig. 7). Figures 7b and c show its real and imaginary parts computed using the amplitude spectrum I,-((3)(o,w,--0)1 (fig. 7a) by the MEM procedure and by the KK

86

DISPERSION RELATIONS AND PHASE RETRIEVAL IN OPTICAL SPECTROSCOPY

[II,

5

4

integrations (the residue-term is omitted). The MEM curves were obtained by assuming that, beside the amplitude spectrum, the phase is known at the resonant frequency m. The same assumption was used for KK integrations apart from the fact that was perfectly “extrapolated” beyond the “measurement” range. The subtractive KK procedure (Ahrenkiel [1971]) was used, when use could be made of the fact that the phase is known a priori at a single frequency m. This offers a greater convergence than does the conventional KK method. Nevertheless, although this procedure forces the phase to have a correct value at m, a good estimate for its dispersion could not be obtained due to significant contribution of unknown residues. By contrast, the MEM procedure yields a good result.

I

9

4. Sum Rules

Sum rules are important in quantum mechanics where they yield information about the electronic transitions. For that purpose we usually must solve the Schrodinger equations to have the wave functions of the states and thereafter to calculate, for instance, the dipole matrix elements to obtain the oscillator strengths. After some commutator algebra, we can then derive the ThomasReiche-Kuhn sum rule for the oscillator strengths (Wooten [1972]). It is interesting to note that the KK relations were first derived for the purpose of optics but later adopted in high energy physics. On the contrary, sum rules have played an important role in high energy physics for a considerably long time. Superconvergence sum rules for particle physics as described by De Alfaro, Fubini, Rosetti and Furlan [I9661 probably stimulated the study of novel sum rules related to optical constants.

4. I . SUM RULES IN LINEAR OPTICS

4.1. I . Complex refractive index

The sum rule that probably has the longest history in optical spectroscopy is the so-called f-sum rule for the extinction coefficient (or the imaginary part of the permittivity), which is analogous to the quantum mechanical ThomasReiche-Kuhn sum rule (Smith and Dexter [ 19721). The f -sum rule is obtained

11, I 41

87

SUM RULES

by recognizing the asymptotic behavior of the complex refractive index for high energies as follows

where = Ne2/m,Eois the square of the plasma frequency, wp, defined by the well-known constants and the electron number density N . Now applying the result of eq. (4.1) to the first KK relation of eq. (2.3) we can obtain thef-sum rule,

which will provide, via absorption measurement, information about the number density of electrons participating in the absorption. The sum rule of eq. (4.2) is related to the generalized Smakula’s equation (Smith and Dexter [ 19721, Smith [1985]), which makes it possible to estimate the oscillator strengths of defects and impurities in a host material provided that the density of the defects is much less than the density of the host atoms. Altarelli, Dexter, Nussenzveig and Smith [ 19721 were the pioneers in optical spectroscopy to derive novel sum rules for the complex refractive index. They made use of the asymptotic properties of the complex refractive index for high angular frequency values. They also employed sophisticated results of complex analysis to introduce the superconvergence of the dispersion-absorption integrals related to the optical constants. Using the superconvergence property they derived the elegant sum rules for isotropic materials as follows:

i

00

[&(w)- 13 d w = 0,

[n( w )- 112dw =

~ ~ dw. ( 0 )

(4.3) The first sum rule of eq. (4.3) states that the average of the real refractive index over all frequencies is unity. It holds for insulators and metals, and is a restatement of causality. The asymptotic property to yield the first sum rule is directly related to Newtonian laws of particle dynamics in the classical description. In quantum mechanical description, it is related to the time dependent Schrodinger equation. The second sum rule, which is valid only for insulators, can be employed for data analysis in order to check the success of the Kramers-Kronig data inversion. Altarelli and Smith [1974] gave further a set of new sum rules by observing that the power of the complex refractive index, (JV- 1)” (m is a positive integer), and also the products w‘(JV - 1>”

88

DISPERSION RELATIONS AND PHASE RETRIEVAL IN OPTKAL SPECTROSCOPY

[I],

54

(r and s are positive integers), are holomorphic functions in the upper half of complex angular frequency plane. Applying the superconvergence theorem they gave sum rules for the complex refractive index and the permittivity of insulators, and analogous sum rules (taking into account the DC-conductivity) for metals. For instance, the following equations hold for the complex refractive index of insulators: lmRe{[N(w)-l]m} dw=O; (4.4)

From the first sum rule of eq. (4.4) we can resolve the sum rules of eq. (4.3). Using different weighting functions, Villani and Zimerman [ 1973a,b], Furya, Zimerman and Villani [1976a,b] and Furya, Villani and Zimerman [I9771 derived a set of new sum rules for the optical constants. King [ 19761 considered the Kramers-Kronig relations and avoided the use of the superconvergence theorem in his derivation of sum rules that was based on the properties of the Dirac’s delta function. Indeed, by such a procedure he could give, e.g., the sum rules of eq. (4.3). Furthermore, King considered the imaginary angular frequencies and the zeros of the real refractive index to obtain other sum rules. Griindler [1983] derived sum rules by using the superconvergence property and Cauchy theorem, Peiponen [1985] by making use of the subtraction of Kramers-Kronig relations, and Peiponen [ 1987~1by using the Cauchy theory and less restrictive assumptions on the optical constants. It is evident that sum rules are holonomic constraints which involve integrals. The sum rule integrals which yield a zero value are important since, e.g., the sum rule for the real refractive index in eq. (4.3) holds separately for the real refractive indices of the host materials and the impurities, as devised by Smith [1974]. Other types of sum rules can give a definite value for the integral and therefore are related directly to physical parameters like plasma frequency. The third class of sum rules involve integrals on both sides of an equation. These are usually applied to check the consistency of theories and more importantly to check the validity of Kramers-Kronig calculations and sometimes the uncertainties in the measurements of spectra. The dependence of the sum rules on physical parameters can be calculated using some simple line model like that of Lorentz but such a result may not hold in general cases of any spectra. The shortcoming with sum rules is that integration, like in the case of Kramers-Kronig relations, is needed to cover the whole spectral range from zero to infinity. Fortunately, the use of the powers of the optical constants usually provide a very fast asymptotic

11, P 41

SUM RULES

89

fall-off of the optical constants and therefore an integration reduced to a finite range is usually enough. Sum rules for the complex refractive index have been given for materials with natural and magneto-optical activity by Smith [ 1976b], who has been the key person not only in realization of many sum rules but also in explaining the physics involved therein. 4.1.2. Complex reflectance

Sum rules in the context of reflectivities were first considered by Smith [ 1976~1 who dealt with magnetoreflectivity. Later, sum rules for complex reflectance were derived by King [1979], who exploited Kramers-Kronig relations and the technique based on the Dirac's delta function. He observed that the following relation holds for insulators and conductors:

I

00

~ - ' R ( O ) ' ' sin ~ O(w)dw = inR(0)"2,

(4.5)

where R is the amplitude reflectance. For metals, R(0) = 1, whereas for insulators, R(0) < 1. King derived many other sum rules, using a weighting factor approach, for the amplitude reflectance and the phase angle. It became evident that dealing with the complex reflectance was not as straightforward as dealing with the complex refractive index. The main problem was that it was not possible to derive a sum rule that would contain only the amplitude reflectance or the phase angle. Indeed sum rules, as can be seen from eq. (4.5) and the following (4.6): Ir(w)l" cos [mO(w)]dw = 0 w Ir(w)l" sin [ m O ( o ) ] d o =

( m = 1,2,3,. . .),

$nwi ( m = I),

(4.6)

involve mixing of the modulus and the phase in the integrands. The results of eq. (4.6) were given by Smith and Manogue [1981], who used the superconvergence theorem for the powers of the complex reflectivity, r = Irl exp(iO) of insulators and conductors. Smith and Manogue argued that due to the indeterminacy that the same phase is associated with the electric field reflectance IrI and with C IrI, where C is a constant and C Irl < 1, we cannot derive the conventional type f-sum rules for the phase and the modulus. King [ 19791

90

DISPERSION RELATIONS AND PHASE RETRIEVAL IN OPTICAL SPECTROSCOPY

[II,

44

considered imaginary angular frequencies and could resolve a sum rule that is related only to R = lrI2. It can be written in the following form: O0

lnR(w)

3t

d o = -lnR(io’). 2o/

(4.7)

Unfortunately, this sum rule has not much use in practical data analysis since an imaginary frequency is involved. Therefore, IOng proposed that it can be applied in testing the consistency of theoretical models for the reflectance of insulators and conductors. The sum rule of eq. (4.7) can also be obtained by using the Poisson’s formulae for the upper-half of the complex angular frequency plane (see, e.g., Morse and Feshbach [1953]). 4.2. SUM RULES IN NONLINEAR OPTICS

Sum rules for nonlinear susceptibilities were introduced by Peiponen [ I987a,b,c, 19881, who considered the model of an anharmonic oscillator and derived sum rules for the cases of sum and difference frequency generation. With the aid of the theory of several complex angular frequency variables and the powers related to the nonlinear susceptibility, we can obtain for instance: (4.8)

r=l,2,...

m = 1 , 2 ,...

r<m,

provided that the nonlinear susceptibility is not meromorphic. After the application of the symmetry property related to the nonlinear susceptibility, we may write, e.g., for the susceptibility of the third harmonic wave:

Furthermore, sum rules for meromorphic nonlinear susceptibilities can be obtained if the integration is not performed using the angular frequency variable that produces simultaneously poles in both half-planes. Then we can write, for instance: (4.10)

r = 1 , 2 ,...

m = 1 , 2) . . .

r<m.

Obviously, sum rules for nonlinear susceptibilities which are analogous to those for linear optical constants can be found.

11,

I 51

91

CONCLUSIONS

Sum rules for the modulus of the nonlinear Raman susceptibility were considered by making use of the Poisson’s formulae for a disk (Peiponen, Vartiainen and Tsuboi [1990]), but there was the difficulty with the imaginary frequency just as in the case of linear reflectance observed by King [ 19791. Sum rules which hold for the nonlinear parts of the complex refractive index were given by Bassani and Scandolo [1991], who observed that:

1

M

~ I K N L ( W I , % , & )=o, ~~I

loc)fl~~(m~,

m2,&) dwi

=

0. (4.11)

Furthermore they predicted that

(4.12)

where the quantity C depends on the nonlinear properties of the medium as well as the strength of the pump beam, and can be calculated if some theoretical line model is employed. Scandolo and Bassani [ 19921 studied nonlinear absorption by analyzing the three-level model of atomic transitions. By using the nonlinear sum rules they explained the anomalous asymmetry in the nonlinear absorption line shape. Bassani and Scandolo [1992a,b] continued their work by studying nonlinear sum rules for atomic hydrogen and thereby to predict the nonlinear absorption spectra of atomic hydrogen. Vartiainen, Peiponen and Asakura [ 1993al applied sum rules for the third order nonlinear susceptibility of coherent anti-Stokes Raman spectrum of the nitrogen Q-branch. Sum rules were exploited in the context of the maximum entropy procedure. According to the sum rules it was possible to estimate the value of the background susceptibility, which is important in CARS spectroscopy. Sum rules for nonlinear reflectances, which may be interesting for instance in the case of the second harmonic wave generation from a metal surface (Bloembergen and Pershan [ 19621, Sonnenberg and Heffner [ 196811, and those for nonlinear magnetoreflectances have not been, as far as we know, described in the literature.

8

5. Conclusions

Mathematical methods for optical spectrum analysis were reviewed. KramersKronig (KK) relations have their origin in the principle of causality and are

92

DISPERSION RELATIONS AND PHASE RETRIEVAL IN OPTICAL SPECTROSCOPY

[I]

therefore connected to fundamental physics. They have been used tradionally in data inversion of linear optical constants. The KK relations also hold in most cases of nonlinear optical spectra. The shortcoming of KK analysis is the need for data extrapolation. The maximum entropy model is interesting, in particular, for phase retrieval problems arising in both linear and nonlinear optical spectroscopies.There is no need for data extrapolation. Sum rules provide information about the microscopic properties of systems. Sum rules of general validity can be found for linear and nonlinear optical constants.

Acknowledgements

The authors wish to express their gratitude to Professor Pertti Ketolainen for his careful reading of the manuscript. One of us (K.-E. P) is grateful to the Academy of Finland for financial support.

References Agudin, J.L., L.J. Palumbo and A.M. Platzeck, 1986, J. Opt. SOC.Am. 3, 715. Ahrenkiel, R.K., 1971, J. Opt. SOC.Am. 61, 1651. Altarelli, M., D.L. Dexter, H.M. Nussenzveig and D.Y. Smith, 1972, Phys. Rev. B 6, 4502. Altarelli, M., and D.Y. Smith, 1974, Phys. Rev. B 9, 1290. Aspnes, D.E., 1985, The accurate determination of optical properties by ellipsometry, in: Handbook of Optical Constants of Solids, ed. E.D. Palik (Academic Press, Orlando, FL) p. 89. Bassani, F., and S. Scandolo, 1991, Phys. Rev. B 44, 8446. Bassani, F., and S. Scandolo, 1992% Phys. Status Solidi B 173, 263. Bassani, F., and S. Scandolo, 1992b, Nuovo Cimento 14D, 873. Bloembergen, N., and P.S. Pershan, 1962, Phys. Rev. 128, 606. Bohren, C.F., and D.R. Huffman, 1983, Absorption and Scattering of Light by Small Particles (Wiley, New York). Born, M., and E. Wolf, 1980, Principles of Optics, 6th Ed. (Pergamon Press, New York). Bryan, R.K., and J. Skilling, 1986, Opt. Acta 33, 287. Burg, J.P., 1967, Maximum entropy spectral analysis. Paper delivered at 37th Ann. Int. Meeting SOC. Explor. Geophysics, Oklahoma City, OK. Caspers, W.L., 1964, Phys. Rev. 133, A1249. Collins, D.M., 1982, Nature 298, 49. De Alfaro, V., G. Fubini, G. Rosetti and G. Furlan, 1966, Phys. Lett. 21, 576. Ehrenreich, H., H.R. Philipp and B. Segall, 1963, Phys. Rev. 132, 1918. Furya, K., A. Villani and A.H. Zimerman, 1977, J. Phys. C 10, 3189. Furya, K., A.H. Zimerman and A. Villani, 1976a, Phys. Rev. B 13, 1357. Furya, K., A.H. Zimerman and A. Villani, 1976b, J. Phys. C 9, 4329.

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Goedecke, G.H., 1975a, J. Opt. SOC.Am. 65, 146. Goedecke, G.H., 1975b, J. Opt. SOC.Am. 65, 1075. Griindler, R., 1983, Phys. Status Solidi B 115, K147. Gull, S.F., and G.J. Daniel, 1978, Nature 272, 686. Haykin, S., 1986, Adaptive Filter Theory (Prentice-Hall, Englewood Cliffs, NJ). Haykin, S., and S. Kesler, 1983, Prediction-error filtering and maximum-entropy spectral estimation, in: Nonlinear Methods of Spectral Analysis, ed. S. Haykin (Springer, Berlin) ch. 2. Hopf, F.A., and (3.1. Stegeman, 1986, Applied Classical Electrodynamics, Nonlinear Optics (Wiley, New York). Hulthen, R., 1982, J. Opt. SOC.Am. 72, 794. Hutchings, D.C., M. Sheik-Bahae, D.J. Hagan and E.W. van Stryland, 1992, Opt. Quantum Electron. 24, I . Jahoda, F.C., 1957, Thesis (Cornell University) unpublished. Jaynes, E.T., 1957a, Phys. Rev. 106, 620. Jaynes, E.T., 1957b, Phys. Rev. 108, 171. Jezierski, K., 1984, J. Phys. C 17, 475. Kemp, M.C., I98 1, Maximum entropy reconstruction in emission tomography, in: Medical Radionucleaic Imaging 1980, Publ. IAEA-SM 247, Vol. 1 (International Atomic Energy Agency, Vienna) p. 128. Ketolainen, I?, K.-E. Peiponen and K. Karttunen, 1991, Phys. Rev. B 43, 4492. King, F.W., 1976, J. Math. Phys. 17, 4726. King, F.W., 1977, J. Phys. C 10, 4726. King, F.W., 1978, J. Opt. SOC.Am. 68, 994. King, F.W., 1979, J. Chem. Phys. 71,4726. Kircheva, PI?, and G.B. Hadjichristov, 1994, J. Phys. B 27, 3781. Kishida, H., T. Hasegawa, Y. Iwasa, T. Koda and Y. Tokura, 1993, Phys. Rev. Lett. 70, 3724. Kogan, S.M., 1963, Sov. Phys. JETP 16, 217. Kramers, H.A., 1927, Estratto dagli Atti del Congress0 Internazionale de Fisici Como, Bologna, 545. Kronig, R., 1926, J. Opt. SOC.Am. 12, 247. Landau, L.D., and E.M. Lifshitz, 1960, Electrodynamics of Continuous Media ( h g a m o n Press, New York). Lee, M.H., and 0.1. Sindoni, 1992, Phys. Rev. A 46, 3028. Lee, S.-Y., 1995, Chem. Phys. Lett. 245, 620. Mandel, L., and E. Wolf, 1965, Rev. Mod. Phys. 37, 231. Martin, P.C., 1967, Phys. Rev. 161, 143. Matsumoto, T., T. Tanji and A. Tonomura, 1994, Optik 97, 169. Mezincescu, G.A., 1985, Phys. Lett. A 108, 477. Morse, P.M., and H. Feshbach, 1953, Methods for Theoretical Physics (McGraw-Hill, New York). Nash, P.L., R.J. Bell and R. Alexander, 1995, J. Mod. Opt. 9, 1837. Nussenzveig, H.M., 1972, Causality and Dispersion Relations (Academic Press, New York). Palik, E.D., 1985a, Gallium arsenide, in: Handbook of Optical Constants of Solids, ed. E.D. Palik (Academic Press, New York) p. 429. Palik, E.D., 1985b, Potassium chloride, in: Handbook of Optical Constants of Solids, ed. E.D. Palik (Academic Press, New York) p. 703. Peiponen, K.-E., 1985, Lett. Nuovo Cimento 44, 445. Peiponen, K.-E., 1987%Phys. Rev. B 35,4116. Peiponen, K.-E., 1987b, J. Phys. C 20, L285. Peiponen, K.-E., 1987c, J. Phys. C 20, 2785.

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Peiponen, K.-E., 1988, Phys. Rev. B 37, 6463. Peiponen, K.-E., and A. Vaittinen, 1984, Phys. Lett. 103A, 209. Peiponen, K.-E., and E.M. Vartiainen, 1991, Phys. Rev. B 44, 8301. Peiponen, K.-E., E.M. Vartiainen and T. Asakura, 1992, J. Phys.: Condens. Matter 4, L299. Peiponen, K.-E., E.M. Vartiainen and T. Tsuboi, 1990, Phys. Rev. A 41, 527. Press, W.H., S.A. Teukolsky, W.T. Vetterling and B.P. Flannery, 1992, Numerical Recipes, 2nd. Ed. (Cambridge University Press, New York) pp. 113-1 16. Price, P.J., 1963, Phys. Rev. 130, 1792. Rasigni, M., and G. Rasigni, 1977, J. Opt. SOC.Am. 67, 54. Ridener, EL., and R.H. Good Jr, 1974, Phys. Rev. B 10, 4980. Ridener, EL., and R.H. Good Jr, 1975, Phys. Rev. B 11, 2768. Roesler, D.M., 1965, Br. J. Appl. Phys. 16, 1 1 19. Scandolo, S., and E Bassani, 1992, Phys. Rev. B 45, 13257. Shannon, C.E., 1948, Bell Syst. Tech. J. 27, 623. Shen, Y.R., 1984, The Principles of Nonlinear Optics (Wiley, New York). Shore, K.A., and D.A.S. Chan, 1990, Electron. Lett. 26, 1207. Smet, F., and P. Smet, 1974, Nuovo Cimento 20B, 273. Smet, E, and A. van Groenendael, 1979, Phys. Rev. A 19, 334. Smith, D.Y., 1974, Conference of Colour Centres in Ionic Crystals, paper 118, Sendai, Japan. Smith, D.Y., 1976a, J. Opt. SOC.Am. 66, 454. Smith, D.Y., 1976b, Phys. Rev. B 13, 5303. Smith, D.Y., 1976c, J. Opt. SOC.Am. 66, 547. Smith, D.Y., 1977, J. Opt. SOC.Am. 67,570. Smith, D.Y., 1980, Dispersion theory and moments relations in magneto-optics, in: Theoretical Aspects and New Developments in Magneto-Optics, ed. J.T. Deversee (Plenum, New York). Smith, D.Y., 1985, Dispersion theory, sum rules, and their application to the analysis of optical data, in: Handbook of Optical Constants of Solids, ed. E.D. Palik (Academic Press, New York) p. 35. Smith, D.Y., and D.L. Dexter, 1972, Optical absorption strength of defects in insulators, in: Progress in Optics, Vol. X, ed. E. Wolf (North-Holland, Amsterdam) ch. V Smith, D.Y., and C.A. Manogue, 1981, J. Opt. SOC.Am. 71, 935. Sonnenberg, H., and H. Heffner, 1968, J. Opt. SOC.Am. 58, 209. Tokunaga, E., A. Terasaki and T. Kobayashi, 1993, Phys. Rev. A 47, R4581. Tokunaga, E., A. Terasaki and T. Kobayashi, 1995, J. Opt. SOC.Am. B 12, 753. Toll, J.S., 1956, Phys. Rev. 104, 1760. Van den Bos, A,, 1971, IEEE Trans. Inf. Theory IT-17, 493. Vartiainen, E.M., 1992, J. Opt. SOC.Am. B 9, 1209. Vartiainen, E.M., T. Asakura and K.-E. Peiponen, 1993, Opt. Commun. 104, 149. Vartiainen, E.M., and K.-E. Peiponen, 1994, Phys. Rev. B 50, 1941. Vartiainen, E.M., K.-E. Peiponen and T. Asakura, 1992, Opt. Commun. 89, 37. Vartiainen, E.M., K.-E. Peiponen and T. Asakura, 1993a, J. Phys.: Condens. Matter 5, LI 13. Vartiainen, E.M., K.-E. Peiponen and T. Asakura, 1993b, Appl. Opt. 32, 1126. Vartiainen, E.M., K.-E. Peiponen and T. Asakura, 1996, Appl. Spectrosc. 50, 1283. Vartiainen, E.M., K.-E. Peiponen, H. Kishida and T. Koda, 1996, J. Opt. SOC.Am. B 13, 2106. Velicky, B., 1961, Czech. J. Phys. B11, 541. Villani, A,, and A.H. Zimerman, 1973a, Phys. Lett. 44A, 295. Villani, A,, and A.H. Zimerman, 1973b, Phys. Rev. B 8, 3914. Wooten, E, 1972, Optical Properties of Solids (Academic Press, New York). Young, R.H., 1972, J. Opt. SOC.Am. 67, 520.

E. WOLF, PROGRESS IN OPTICS XXXVII 0 1997 ELSEVIER SCIENCE B.V. ALL RIGHTS RESERVED

SPECTRA OF MOLECULAR SCATTERING OF LIGHT

I.L. FABELINSKII PN. Lebedeu Physical Institute of the Academy of Sciences of Russian Federation, Leninsky Prosp.. 53. I I7924 Moscow, Russian Federation

95

CONTENTS

PAGE

INTRODUCTION . . . . . . . . . . . . . . . . . . .

97

THE FLUCTUATIONS OF THERMODYNAMIC QUANTITIES

99

SPECTRA OF MOLECULAR LIGHT SCATTERING ARISING FROM PRESSURE FLUCTUATIONS Ae(P). EQUILIBRIUM PHENOMENA . . . . . . . . . . . . . . . . . . . .

102

SPECTRA OF MOLECULAR LIGHT SCATTERING ARISING FROM PRESSURE FLUCTUATIONS A&(P).SOME NONEQUILIBRrUM PHENOMENA . . . . . . . . . . .

124

SPECTRA OF MOLECULAR LIGHT SCATTERING ARISING FROM ISOBARIC ENTROPY FLUCTUATIONS A&(S) AND FROM CONCENTRATION FLUCTUATIONS A&(C) . . . .

132

SPECTRA OF MOLECULAR LIGHT SCATTERING ARISING FROM ANISOTROPY FLUCTUATIONS A E ( ~ / K ). . . . . .

151

ABOUT SOME PROBLEMS. . . . . . . . . . . . . . .

177

REFERENCES . . . . . . . . . . . . . . . . . . . .

96

178

8

1. Introduction

Molecular (Rayleigh) light scattering spectra have long been known as an efficient instrument of both theoretical and experimental studies of various phenomena and their dynamics in physics, chemistry, and biology. They led to the discovery of several new phenomena and appeared to be usefbl for the solution of many applied problems. Light scattering can appear only in the case when the light flux encounters some optical inhomogeneities on its way. Optical inhomogeneities can be of diverse origin. They can be represented by water droplets forming clouds and fog, dust and soot particles in a haze or smog, colloidal particles in a liquid, or single molecules in a rarefied gas when the mean free path of a molecule is greater than the wavelength of light, A. Reviews of the investigations in this field are given in the books of Shifrin [1951], van de Hulst [1957], Kerker [1969], and others. In a condensed medium, free of any foreign admixtures (i.e., an optically “empty” medium), optical inhomogeneities arise from the statistical fluctuations of those physical quantities which lead to the fluctuations of the optical permittivity, A&.Light scattering due to such fluctuations is called “molecular”, or Rayleigh light scattering to underscore its difference from light scattering due to the reasons mentioned above or to combinational (Raman) scattering. Several books and review articles devoted to the investigations of molecular light scattering have already been published. Here we mention only a few that appeared earlier than the others - Fabelinskii [1968] (in Russian [1965]), Griffin [ 1968b], Fleury [ 19701, Fleury and Boon [ 19731, Chu [ 19741, Lallemand [ 19741, Crosignani, DiPorto and Bertolotti [ 19751, Cardona [ 19751, Cardona and Guntherodt [1982a,b, 19841, Berne and Pecora [1976], Dil [1982], BorovikRomanov and Kreines [1982], and Cottam and Lockwood [1986]. The foundations of modern ideas about the light scattered by optical inhomogeneities of the fluctuative origin, were laid down at the very beginning of the twentieth century and were further developed simultaneously with the improvements of the theory of a solid state specific heat - both, it appeared, were bound tightly together. The development of the molecular light scattering theory has proved to be so closely related to the theory of the specific heat of 91

98

SPECTRA OF MOLECULAR SCATTERING OF LIGHT

[IIL § 1

solids that without much error, one could assume that both phenomena have a common origin. Since the history of the theory of the specific heat is of interest, I shall now briefly recount the principal stages in the studies which have been carried out. As early as the beginning of the previous century, I? Dulong and A. Petit (18 19) established an empirical law that under normal conditions the specific heat of any solid is C = 6 cal mol-’ K-’ , explained on the hypothesis of a uniform distribution of energy among the degrees of freedom. However, a discrepancy between the predictions of the above law and the experimental results for certain substances, especially at low temperatures, required an improvement of the theory. In 1907, in his theory of the specific heat of a solid, Einstein replaced the uniform distribution of energy among the degrees of freedom by a Planck’s distribution of energy among the frequencies of elastic vibrations. Einstein took here a qualitatively new step by applying Planck’s law, previously used solely in emission theory problems, to the theory of elastic vibrations. However, Einstein considered only the average effective frequency of the particles constituting the solid. Einstein [ 19071 did not discuss the problems of light scattering or of the diffraction of light by the elastic vibrations of a solid. A striking phenomenon, involving a marked increase in the light scattering in a narrow temperature and pressure range at a phase transition and called the critical opalescence, has long been known, it was finally explained correctly in 1908 by Smoluchovski as being the result of a strong increase in the density fluctuations in pure liquids and of the concentration fluctuations in solutions. In 1910, Einstein employed Smoluchovski’s fruitful idea of fluctuations in a quantitative calculation of the intensities of light in liquids and solutions. In this calculation, Einstein expanded the density fluctuations as a three-dimensional Fourier series. The determination of the amplitudes of these series makes it possible to calculate the scattered-light intensity. Einstein not only obtained his famous formula for the scattered-light intensity but also devised the modern method for the calculation of fluctuations of any thermodynamic quantity using thermodynamic relations together with Boltzmann’s statistical principle relating entropy to the probability of a closed system state. In his lengthy paper, Einstein says nothing about the specific heat of a solid. The component of the Fourier series are referred to as static “waves”. In 1913, Mandelstam superimposed surface fluctuation irregularities onto planar ‘‘diffraction gratings”; in this case, the surface “grating waves” are static. The next significant step was taken in 1912 by Debye, who tackled the still unsolved problem of the specific heat of a solid, especially at low temperatures,

111,

5 21

THE FLUCTUATIONS OF THERMODYNAMIC QUANTITIES

99

and developed in essence Einstein’s theory already mentioned above. In Debye’s theory, a solid is regarded as a continuous medium, but one having a finite number of normal vibrations equal to 3N (where N is the number of atoms or molecules in a sample) and with the minimal wavelength of the elastic waves, which is determined by the interparticle distances, while the maximum frequency is correspondingly equal to the velocity of sound divided by the interparticle distance. Debye says nothing about the scattering or diffraction of light by these dynamic elastic waves. It was not until later that it became clear the Einstein’s and Debye’s studies of the theory heat of a solid, and Einstein’s and Mandelstam’s studies of the theory of molecular light scattering deal with waves of the same nature. One can even say that the Einstein and Mandelstam Fourier components and the Debye thermal elastic waves are the same thing. It was not easy to understand this at the time the theory was proposed, but the realization of this identity has stimulated the development of optics and of molecular acoustics, and has led to the discovery of new phenomena and the appearance of new fields of experimental and theoretical research (Leontovich [ 19311, Fabelinskii [ 19681, Ginzburg [ 19721). The present review is devoted to some problems of the molecular Iight scattering spectroscopy in liquids and solutions which possess two critical points or a double critical point; to phase transitions; to nonequilibrium phenomena in acoustic paramagnetic resonance in the presence of a temperature gradient; to the hypersound amplification in piezosemiconductors subjected to a static electric field and also to the molecular light scattering spectra arising from the anisotropy fluctuations in liquids consisting of anisotropic molecules.

5

2. The Fluctuations Of Thermodynamic Quantities

In a continuous medium (7 << A, with 1the mean free path and A the wavelength of light), optical inhomogeneities arise as a consequence of the statistical nature of the motion of the particles constituting the medium. Different quantities characterizing the states of the substance may fluctuate. Only the fluctuations of physical quantities which lead to the appearance of an optical inhomogeneity in the investigated medium are significant for light scattering. These include density and temperature fluctuations, and - in their turn -the density fluctuations depend on the pressure and entropy fluctuations. Pressure fluctuations arise when particles with momenta somewhat smaller or greater than the volume-average momentum accumulate in a definite place and at a definite time while a fluctuation of temperature, (AT), or entropy, (AS), means

100

[W li 2

SPECTRA OF MOLECULAR SCATTERING OF LIGHT

that particles with a kinetic energy greater or smaller than the value averaged over the sample have gathered in a certain small region at a particular time. Pressure fluctuations (AP)are quite independent of temperature fluctuations (AT) or entropy fluctuations (AS). This means that the statistical average products of the pressure and entropy fluctuations are zero. Considering the entire picture, we can assume that the scattering as a consequence of pressure fluctuations at a particular point in a sample and at a particular time is unrelated to the same fluctuations at another point in the sample at the same time. This also applies to entropy fluctuations. Consequently, in order to calculate the intensity of light scattered as a consequence of fluctuations of different origin, it is sufficient to calculate the scattering by one fluctuation and to multiply the result by the number of such fluctuations in the scattering volume. The integral intensity of the molecular light scattering is proportional to the mean square of the fluctuation, A&: I

- (A&’).

It is worthwhile to assume that A&is a function of a set of independent variables of pressure P , of entropy S and of concentration C (although we might consider A&to be a function of any other set of variables). In the selected case:

(ap)‘

(A&’ ( P ,S, C)) = (AP’)+ ap s,c

Er

(g):,c(AS’) + (ac

(AC’) ,

P,S

(2) because due to the mutual independence of S and C, APAS = 0; ASAC = 0 and APAC = 0. The evaluation of the mean square of the fluctuations of the thermodynamic quantities, that appear in eq. (2), being fulfilled with the help of a method first proposed by Einstein [1910] (see also Landau and Lifshitz [ 19581, Fabelinskii [1968]) leads to the following: \

kT

(AS’) - k p C p . V’ ’

(AC’)

=

CkT

v*ap,/ac’

I

(3)

Here, V’, ,6s, PI, T , and k are, respectively, the effective volume of the fluctuations, the adiabatic compressibility, the osmotic pressure, the absolute temperature, and the Boltzmann constant. The angular brackets denote averaging over the ensemble. Let us consider the case in which the effective dimensions of a fluctuation are much smaller than the wavelength A of the scattering light. One can make the approximating assumption that ( V*)1’3= ler < U 2 5 .

111,

5 21

THE FLUCTUATIONS OF THERMODYNAMIC QUANTITIES

101

The intensity of the molecular light scattering due to fluctuations described by eq. (3) is given by the following expression (Fabelinskii [1968]) for natural exciting light:

where I0 is the exciting-light intensity and L, p and 8 are, respectively, the distance between the scattering volume and the point of observation, the density, and the scattering angle;

I . ~ E2A4L2 ( L a r )u' do Tz k T 2Cpp (l+cos28), where u and Cp are, respectively, the coefficient of volume expansion and the heat capacity at constant pressure; 2

Ion2V ckT ( 1 + cos2 8) , Ic = 2A4L2 &)p,s dpl/dc where C is the concentration and PIthe osmotic pressure. Here, l a d and liS,are the intensities of light scattered by the adiabatic and isobaric density fluctuations respectively, and I , is the intensity of light scattered by the fluctuations of concentration. The overall intensity of the light scattered by the density fluctuations in a liquid is:

(

If one assumes that

(P$)L

=

(;g):

2

=

(P$)

T

9

and takes account of the familiar thermodynamic relationships

(the subscript S denotes throughout the adiabatic value of a quantity and the subscript T corresponds to isothermal values), then the familiar Einstein formula is obtained from eq. (7) (Einstein [1910], Fabelinskii [1968]):

I emphasize that the light scattered as a consequence of the density fluctuations and Ijs) and the concentration fluctuations is in all cases linearly polarized

(lad

I02

SPECTRA OF MOLECULAR SCATTERING OF LIGHT

"9

3

for observations at the scattering angle f3 = 90" relative to the electric field vector of the light wave (this vector is in a plane perpendicular to the scattering plane). If the exciting light is linearly polarized in such a way that its electric field vector lies in the scattering plane, there is no scattered light ( I and I,) for observations at f3 = 90". The light scattered by, for example, the anisotropy fluctuations is depolarized, but this is due to nonthermodynamic fluctuations which are discussed below. However, it is possible to determine the contribution of the intensity of light scattered by the anisotropy fluctuations to the total light flux in the way it was done by Cabannes [1929], who used the depolarization coefficients of a total light flux A and the depolarization coefficient of light, scattered only by the anisotropy fluctuations p, both being easily measured experimentally. Cabannes' correction factor is:

In harmony with the theory and the experiment, the depolarization factor p = 6 / 7 if the exciting light is polarized arbitrarily, and p = 3 / 4 if it is polarized linearly (Placzek [ 19351). Consequently, the intensity of light, scattered by the fluctuations of pressure, entropy and anisotropy in a single liquid, and being observed at an angle of 90", according to eqs. (7) and (10), will be:

As was shown earlier (see e.g. Fabelinskii [1968, 1994]), the ratio of the integral intensity i of light scattered by the anisotropy fluctuations to the intensity I (eq. 7) for 0=90" is:

B = -i = A ( l + p ) I p-A As follows from the measurements of A, keeping p=6/7, the values of B are approximately 2 for benzene, 5 for CS2, and 13 for both salol and benzophenone, while they are 0.56 for acetone and 0.2 for ethyl ether.

5

3. Spectra Of Molecular Light Scattering Arising From Pressure Fluctuations A&(P).Equilibrium phenomena

In all these cases the optical inhomogeneities of fluctuation origin have been assumed to be static ("frozen") and independent of time, so that eqs. (4)-(7) and

111,

5

31

SPECTRA FROM A&(P).EQUILIBRIUM PHENOMENA

103

(9) yield the integral intensity. Nevertheless, the fluctuations defined by eq. (1) vary as a function of time, like statistical fluctuations in general, leading to fluctuations of the permittivity, A&, or of the refractive index, An ( E = n’). They appear and are suppressed, appear again and are again suppressed, and so on. The mean square of a fluctuation, (A&’), or its root-mean-square, (A&’)”’, is thus a function of time. The exciting light (with a certain frequency q )incident on such a fluctuation is scattered sideways and the field of the scattered light wave is also a function of time. In other words, the time dependence of an optical inhomogeneity leads to modulation of the scattered light. The modulation phenomenon occurs in a wide variety of fields in physics, radiophysics and engineering. A theory of this phenomenon has been developed satisfactorily by Rytov [1940], Mandelstam [ 19501, and Fabelinskii [ 19681. In the case of light scattering, the modulation and the Doppler effect are the physical causes of a scattered-light spectrum differing from the exciting-light spectrum. Light scattering always occurs under the influence of exciting light, and is therefore a stimulated process. The theory of time dependence of the concentration fluctuations in a solution and of the density fluctuations in a liquid was developed in detail by Leontovich [1931]. Its results have been applied to the problem of the scattered-light spectrum and a formula has been obtained for the frequency distribution of the scattered-light intensity. Our ultimate aim is, in fact, to do this for all possible fluctuations, but for the moment we shall deal with the pressure and entropy fluctuations or the temperature and the concentration fluctuations. Naturally, the time dependencies of different fluctuations are different, in other words, the modulation functions @(t)are different. We shall designate by d i p ( t ) , di,(t), dic(t), etc., the root-mean-square values of the permittivity fluctuations determined by AP, AS, and AC. Suppose that the monochromatic light exciting the scattering is given by:

E

= Eo exp [i (mot

-

k . r)]

The electric field of the scattered light can then be described by:

E’ ( t )= E ( t )di ( t ) .

(13)

It is necessary to find the law governing the variation of @(t)in order to obtain a specific form of E’(t). The function @(t)or the average fluctuations, (AE’)”’, induced by the pressure fluctuations, are described by the phenomenological equations which apply to and have been formulated for the average values. For

104

SPECTRA OF MOLECULAR SCATTERING OF LIGHT

[IN

§ 3

the pressure fluctuations in an ideal medium, that is free of losses and dispersion, this is a wave equation of the type:

& - U2V2@= 0.

(14)

Since any function of a known argument t - ( d o ) can be a solution of the wave equation (14), we shall proceed in the usual way and seek its solution in the form: @ ( t )=

cos (Qot - q . r - q),

(15)

where @o, Qo,q and r are, respectively, the maximum value of @(t),the angular frequency, the wave vector, and the coordinate, while u is the velocity of sound. Thus, in this case the modulation function @(t) varies in accordance with the cosine law and E’(t) can be readily determined from eqs. (13) and (15):

It follows from eq. (16) that there are two satellites in the scattered-light spectrum: a Stokes satellite with a frequency 00 - QO and an anti-Stokes satellite with a frequency wo+SZo; according to eqs. (14) and (15), the elastic wave frequency is 90= uq. The satellites arising in the course of the modulation of the scattered light are known in the Western literature as the Brillouin components. However, they were predicted independently by Mandelstam [ 19261 and Brillouin [ 19221. Mandelstam also carried out an experimental search for the phenomenon which he predicted. On Landsberg’s and Mandelstam’s suggestion, Gross [ 1930al joined this investigation and detected the Mandelstam-Brillouin (MB) components in a quartz single crystal with the aid of improved apparatus. Gross also observed the same phenomenon in liquids. It follows from the law of conservation of energy and momentum that:

where w, is the scattered-light frequency, 520 the frequency of the elastic (sound) wave, and k , and q are the wave vectors of the scattered light and the elastic wave, respectively (fig. 1).

SPECTRA FROM A&(P).EQUILIBRlUM PHENOMENA

105

k

Fig. 1. Scattering (diffraction) of light by an elastic acoustic wave representing the Fourier component of a pressure fluctuation. The wave vectors represent the exciting light (ko), the scattered light (k&, and the elastic wave (4).

Since Q/WOis small

one can put lksl M Ikol, while

Here, A is the wavelength of the elastic waves and 8 the scattering angle. The relationship between Qo, q, and u follows from eq. (17): 4nn . 8 u SZ,=uq= - n n - = 2 n - ~ s i n - . A 2 c

8 2

(19)

Hence, the shift of the frequency SZ relative to wo is proportional to the ratio of the velocity of sound u to the velocity of light c, as well as to the refractive index and the sine of half the scattering angle:

A _ u_ W

u c

8 2

- f 2 n - sin -,

(20)

where Am is the frequency shift of a ME3 component. It follows from eqs. (19) and (20) that the investigated sound frequency QOcan be varied between wide limits by varying the angle of observation of the scattered light from Q O = O for 8=0 to Q0=2n(u/c)uo for 0=180". For wo = 4 x 1015rad s-l, we obtain 520 M 5 x 10" rad s-' or f M 10" Hz for water, where u = 1.5x lo5 cm s-l and the green-light refractive index is n = 1.33. For diamond, SZO is an order of magnitude higher than for water.

106

SPECTRA OF MOLECULAR SCATTEMNG OF LIGHT

[“I,

I3

It is thus possible to investigate sound waves of frequencies which may be varied by several orders of magnitude by varying the scattering angle. Studies at hypersonic frequencies QO>109-10’0 Hz are especially effective. The details will be given later. Elastic or sound and hypersound waves are the Debye thermal waves and their number is 3 N , where N is the number of molecules or species within a volume of a sample. These extremely numerous sound waves propagate along very diverse directions within a bulk sample and, at first sight, it appears that there is no hope whatsoever of investigating the properties of matter at any one frequency 520 with a wave vector q. However, this would be a premature conclusion. Figure 1 helps one to understand how this can be achieved experimentally. If a plane light wave with a wave vector ko is incident on a sample and observations are made in the direction of the wave vector k,, then in the same direction k , one can observe light diffracted by a grating formed by a standing wave with wave vectors &q which satisfy eq. (1 7) or Bragg’s condition. This condition can be formulated as follows:

0

2An sin - = A, 2

(21)

where A is the period of this grating or the wavelength of the sound, and A is the wavelength of light in vacuum. One may speak of the formation of a standing grating because among the many sound waves there are always some having wave vectors identical in magnitude but with opposite directions. An expression for the field of the scattered light (eq. 16) is obtained by taking account of the modulation of the scattered light. Using eq. ( 1 6), we obtain an expression for the time variation of the intensity: E ’ ( t ) E ’ * ( t ) = I ( t ) = I o [ l+cos(2Q0t-2q.r)], where I0 = ;(E;@O)~(see eqs. 13 and 16). Equation (22) shows that the expression for the intensity of the MB components contains a time-independent constituent equal to I 0 and, against the background of the constant intensity, there is an intensity which varies at a frequency 2520 FZ 2 x 10” Hz. At the maximum, the variable constituent is I o , while at the minimum it is determined by the constant constituent also equal to 1 0 . Thus, the overall maximum intensity is 210. Hence, in our idealized model, with no losses or dispersion of the velocity of sound in matter, the scattered-light spectrum of a liquid phase exhibits two MB scattering components arranged symmetrically around WO.In this model the

111,

5 31

SPECTRA FROM At.(P). EQUILIBRIUM PHENOMENA

I07

MB components do not have a finite width; they are represented by a &function. For an isotropic body, such as a glass, one observes two MI3 components due to longitudinal sound and two components due to transverse sound. In the case of an anisotropic crystal, the formula for the frequency of sound QO or for the optical shift in the spectrum Am is (Chandrasekharan [ 195 11):

Here, ni and n, are the refractive indices for the incident (exciting) and scattered light, respectively. One quasilongitudinal and two quasitransverse waves (“fast” and “slow”) can propagate in any direction in a crystal, and the position of the MB components differ in the two polarization states of the exciting light and in the two polarizations of the scattered light, so that the total number of components may be 24, but - for certain polarizations and orientations of the crystal - it is possible to observe six components of the spectrum simultaneously. Figure 2 shows six MB scattering components. More information can be deduced from the scattered light spectrum if the function @(t)is found for a more realistic medium in which there are losses but the dispersion is neglected. It is then necessary to describe @(t)by the phenomenological hydrodynamic equation for the average values; i.e., the Stokes equation (Rayleigh [ 1894/96], Leontovich [ 193 11, Lamb [ 19321)

iii-u2u2@-ru2&=o.

(24)

Here,

r=-

K

-q+q/+-(y-i) P [43 CP

1

,

where q is the shear and q’ the bulk viscosity, K the thermal conductivity, y = Cp/C,, and C p and Cy are the specific heats at constant pressure and constant volume, respectively. The solution of eq. (24) will be sought in the form of an expansion representing a three-dimensional Fourier series of the type

Substitution of eq. (26) in eq. (24) gives the following expression for the components of the sum (26):

& + rq2& + u2g2@ = 0.

(27)

108

SPECTRA OF MOLECULAR SCATTERING OF LIGHT I

I

I

T2

Fig. 2. Spectrum of light, scattered in a quartz (SiOz) single crystal. 1'7 and Tz are the MB components corresponding to the slow and the fast transverse elastic wave; and L is that corresponding to the longitudinal wave. (Laboratory of LA. Yakovlev, Moscow University.)

The Laplace transform,

1

m

@(PI =

@ ( t ) exp (-pt> dt,

applied to eq. (27) yields the following expression for @ ( p ) :

where 1

6 = - r q 2 , Q; 2

= u2$.

111, § 31

109

SPECTRA FROM A&(P).EQUILIBRIUM PHENOMENA

The time dependence in the modulating function @(t)is obtained from eq. (29) by the inverse Fourier transformation: @ ( t ) = @ (0) exp (-6t) [cos

(Qi- a2) t + sin (0: b2) -

t] .

(3 1)

When p is replaced by iw in the Laplace transform, the latter becomes a Fourier integral and then, if eq. (28) is multiplied by @(O) and it is assumed that (@ (0)& (0)) = 0, the result is the following expression for the intensity distribution:

Integration of eq. (32) over all the frequencies should give the intensity fad of the light scattered by the adiabatic density fluctuations (eq. 4). Hence, the normalization conditions yield

it therefore follows that

If 6 << szo

(35 )

then

It follows from eq. (36) that the overall width of a MB scattering component at half the maximum intensity is (6w),,

=

26.

(37)

I10

SPECTRA OF MOLECULAR SCATTERING OF LIGHT

UK 8 3

Equation (34) was obtained in a different way by Ginzburg [1972, 19791. The factor representing decay with time can be expressed simply in terms of the amplitude absorption coefficient a, namely,

S = ao,

(38)

where o is the velocity of sound. On the other hand, the hydrodynamic theory yields the following expression for a:

a = -q2r , 20

where gives

(39)

r is defined by eq. (25). Substitution of eqs. (39) and (38) into eq. (37)

Consequently, if the molecular scattering spectrum is suitably recorded, it is possible to find the phase velocity of sound from the position of the MB components, while the sound or hypersound decay factor can be found from the halfwidth of these components. Thus, a new field of optical and acoustic research has arisen. The dispersion of the velocity of sound (Fabelinskii and Shustin [ 19531, Molchanov and Fabelinskii [ 19551) has been observed by means of such procedures, and hypersound relaxation and absorption process in liquids have been investigated quantitatively. Many phenomena also affect the intensity of MB components; but this will be discussed in detail below. 3. I . ELASTIC THERMAL WAVES AND EMITTER-GENERATED ACOUSTIC WAVES

It must be emphasized that the elastic thermal or Debye waves differ significantly from the elastic acoustic waves, which are generated by an emitter and are then injected into the medium being investigated. A plane radiation wave then generated is

A

= A0

exp [i (Qat - Qx)] ,

(41)

where Ao, Qa, and Q are the amplitude, the frequency and the wave number of the sound wave, respectively. If A = A0 at time t = 0 and at a point in space x = 0, then this wave is described by eq. (41) as time progresses and the wave propagates in the medium.

111,

P

31

SPECTRA FROM Ae(P). EQUILIBRIUM PHENOMENA

Ill

If the medium suffers acoustic losses under these conditions, the implication is that the frequencies Q, or Q are complex variables. For example, we may assume that Qa =

QI + i s .

Substitution of eq. (42) in eq. (41) yields A

= A0

exp (-st) exp [i (521 t - Qx)]

(43 )

It follows from the last expression that the amplitude of the sound wave may decay as a function of time and, when account is taken of eq. (38), it can also be attenuated in space. It is easy to show that b represents the same quantity in eqs. (43) and (38) [Rayleigh [ 1894/96], Isakovich [ 19731, Landau and Lifshitz [ 19871). In the case of the elastic or Debye waves, the situation is completely different. According to the Debye [1912] theory an average energy hQ [exp (hQ/kT) - I]-' corresponds to each normal vibration and, bearing in mind that h W k T << 1, an average energy kT corresponds to each normal vibration in the MB spectrum. Hence it follows that at a temperature T the elastic thermal or Debye waves retain a constant amplitude A and therefore this amplitude (eq. 43) does not decay in the usually accepted sense. At a constant temperature T , the amplitude of the thermal wave is constant at any point within a sample and at any time, and it can be described in terms of a combination of constants, the velocity of sound, and the frequency interval. The natural conclusion from the foregoing considerations is that, having the MB spectrum at one's disposal and knowing the intensity distribution in the MB components, it is possible to find the decay factor of the sound wave, generally speaking in the range from Q = 0 to 52 = 2n(u/c)oo. However, it is worthwhile to emphasize once again that this possibility arises not because the thermal elastic wave decays, but because the modulating function for the time variation of @(t) includes as parameters quantities such as the viscosity, which determines the elastic perturbation losses, the frequency, and the elastic-vibration wave vector. Because of that, in some of the cases, one can identify an elastic thermal wave with a customary acoustic wave, and this is what we shall be doing, bearing in mind, what has been noted above. It is of interest to find the value of pressure fluctuations and the value of the amplitude of an elastic thermal wave that leads to the appearance of the MB components.

112

SPECTRA OF MOLECULAR SCATTERING OF LIGHT

“11, § 3

The calculation of the value of [(AP2)]’’2can be done only approximately and it is based on eq. (3) where the value of the volume of the fluctuation V* stays undefined. This value can only be estimated on the basis of the following considerations. Because the media studied in this work are supposed to be far from a critical point, the scattering indicatrix has a Rayleigh outlook, and this means that a typical linear dimension of a fluctuation is much less than the scattered light wavelength 12. Let us suppose that a fluctuation has a spherical shape with a radius Y = 12/25 (Fabelinskii [1968]). Then for the value of cm3. For the temperature T = 300 K, cm we find that V’ = 3 x A =5x the hypersound velocity u = 1.5x lo5 cm s-’ and if p = 1 g ~ m - we ~ , can estimate: [ (AP’)] = 5 x lo6 dyne cm-2, which is equal to 5 atm. The internal or “molecular” pressure in liquids, which originates from intermolecular interactions, reaches enormous values lying between 1O3 and 1O5 atm. Supposing that the internal pressure is lo4 atm, one can see that the pressure fluctuations locally change the pressure in a liquid by a 5 x 10-4-th part of its average value. A rough estimate of the amplitude of an elastic thermal wave can be done on the basis of Planck’s formula. In our case, when hP/kT << 1, Planck’s formula transforms into the Rayleigh-Jeans formula: I/ (52, d P ) = (x2u3)-’ . 52’kT. d P = p P 2 A & ,

(44)

where p P 2 A & is the energy density of an acoustic wave. From eq. (44) it follows that:

A,R

=

[(px2u3)-’. kT d P ]

1/2

(45)

Equation (45) allows one to see that A,ff is proportional to the square root of the temperature and is a function of the velocity of sound u. For the numerical calculations, it is necessary to know the value of d P , which is somewhat undefined. The author supposes that for the calculation of the value of A,ff one can assume that the value of d52 equals the true value of a halfwidth of a MB component. For benzene, if we assume that u = 1.5x lo5 cm s-’ , T = 300 K and dSZ M 1 x lo9 rad s-I, from eq. (45) it follows that A,ff M 4 x lo-” cm. In the course of such estimates we can discuss only the orders of magnitude. 3.2. HYPERSOUND VELOCITY AND ABSORPTION

As already mentioned, the first effective application of MB spectroscopy was made by Fabelinskii and Shustin [1953] and Molchanov and Fabelinskii [1955]

111,

P

31

SPECTRA FROM A E ( P ) .EQUILIBRIUM PHENOMENA

I13

in connection with the observation of the dispersion of the velocity of sound in benzene, carbon tetrachloride, chloroform, carbon disulfide, and then also in a series of other liquids'. The dispersion of the velocity of sound Auh is 0.01 1 for C6H6, 0.09 for CS2, 0.12 for CC14, and 0.135 for quinoline. To date, huh has been measured for many liquids. We measured the hypersound absorption coefficient a only after laser light sources became available in our laboratory (Mash, Starunov, Tiganov and Fabelinskii [ 19641). We found that a = 4.5 x 1O3 cm-' for benzene and a = 1.6x 1O4 cm-' for carbon tetrachloride. The technique used by us in the measurement of the dispersion of the velocity of sound and of the absorption coefficient made it possible to carry out a direct experimental test of a formula from the relaxation theory, developed by Mandelstam and Leontovich [ 19371, and to determine the relaxation time z of the bulk viscosity. For example, z = 2.7 x lo-'' s for benzene and z = 28.3 x lo-'' s for carbon disulphide. The relaxation time z has also been found for other liquids. One can say that the experimental studies described here have laid the foundations of a new quantitative method for investigating the optical and acoustic properties of matter in a wide variety of states, for example in the course of phase transitions in the critical region, in paramagnetic materials under the phonon "bottleneck" conditions, in magnetic materials, in piezoelectric semiconductors subjected to an external static electric field, in transparent and opaque insulators, in semiconductors and met?ls, and in viscous liquids and glasses. Naturally, this list does not exhaust the numerous applications of this method. The first reported spectrum of the molecular scattering of He-Ne laser radiation in benzene, will be considered as an example. This spectrum can be used to determine the amplitude absorption coefficient of hypersound and its velocity with the aid of eqs. (36) and (20) (fig. 3). The MB spectra also make it possible to investigate the propagation of hypersound in the region of the liquid-vapor phase transition as well as in liquid solutions. The temperature dependence of the velocity of sound near the phase transition in carbon dioxide (CO2) was determined above and below the critical temperature from the positions of the MB components (Cummins [1971]).

'

It is noteworthy that attempts to use the MB components for the determination of the velocity of hypersound had been undertaken much earlier (Gross and Khvostikov [1932], Rao [1937], Venkatesvaran [1942a], Sunanda Bai [1943]), but were unsuccessful; a positive dispersion of the velocity of sound, expected for a series of liquids, was not observed.

I14

"IL 5 3

SPECTRA OF MOLECULAR SCATTERING OF LIGHT

Av, cm-'

Fig. 3 . Spectrum of light scattered in liquid benzene (Fabelinskii [1968])].

3.3. MOLECULAR LIGHT SCATTERING SPECTRUM AND ITS INTENSITY AT LOW TEMPERATURES

As a result of the investigations of Tamm [1930] and Ginzburg [1943], the formulae for the intensity of each of the two MB components have been derived:

From eq. (44), the overall intensity at 8= 90" is I (w- W )+ I (w + W ) = l a d and, hence:

The quantum-mechanical eq. (47) is valid for any temperature of the scattering medium. In the case when hQ/kT << 1, eq. (47) turns to the classical formula (4) for 8 = 90". If hQ/kT M 1, the classical formula (4) lacks sufficient precision and one must use eq. (47). The temperature limit for the validity of eq. (4) can be estimated to be T = hQ/k M 0.1 K , which means that the formulae of the theory of molecular

“1,

9 31

SPECTRA FROM Ae(P). EQUILIBRIUM PHENOMENA

115

scattering of light obtained with the help of a classical approach remain valid ~ K. We also must note that as far down to extremely low temperatures 7 ‘ 0.1 as for liquid He y = 1.008, the isothermal values differ very slightly from the adiabatic ones. It follows from these examples that molecular light-scattering spectroscopy is effective in the study of various substances and various phenomena under wide variety of conditions. The method is universal. It affords optical and acoustic information, but one should not imagine that molecular light scattering spectroscopy constitutes a simple and easy way of obtaining fundamental results. This is by no means so. The method requires experimental skill and experience. Major and varied difficulties, which are far from easy to overcome, await the physicist in this field. Perhaps the most significant experimental difficulty is the low intensity of scattered light. Equations (4) and (5) make it possible to estimate the proportion of the exciting light scattered by fluctuations of various origins. It has been found that the proportion of the blue exciting light scattered in air, 4 x in molecular hydrogen, in quartz is as follows: 2 x crystal, and in benzene. Thus, the scattered light intensity, integrated - ~the exciting light with respect to frequency, lies in the range from l ~ l O of for liquids such as benzene. The difficulty for various substances to 1 x increases even further when this weak light flux has to be “spread out” between various frequencies and the spectrum must be recorded reliably. Another serious difficulty is the need to investigate lines with very different intensities over a narrow spectral range of -1 cm-I. There are other difficulties, but these will not be discussed here. More than 50 years ago Ginzburg [I9431 pointed out an interesting MB spectrum of He 11, where (apart from the usual longitudinal sound) yet another type of sound is propagated, so that four ME3 components should be observed in the spectrum: two due to the longitudinal sound, and two due to the second sound. The frequency shift of these components in helium I1 is A d w x 1 0-6, for the second sound we have A m l o x and the overall intensity is lo-* of the intensity of the exciting light. In 1968 I wrote (Fabelinskii [1968]): “However, the displacement of the components of the second doublet, called “second sound” is so small and its intensity so negligible that there is no hope of finding these components experimentally”. Lasers had only just begun to be used in the laboratory at the time, and in 1978, Vinen and Hurd [I9781 in their fine spectroscopic investigations obtained excellent records of the MB components for helium I1 at 20 bar pressure (fig. 4). Thus, my general “prophecy” has proved ill-founded. However for a normal pressure the MB components have

1 I6

SPECTRA OF MOLECULAR SCATTERING OF LIGHT

AT=-10.86 rnK

. AT=-3.05 rnK

(c) Fig. 4. MB spectrum of liquid helium below the A point (a, b) at a pressure of 20 bar. The inner narrow doublet is due to second sound. The outer components are due to the first sound (c), The latter spectrum was obtained by reducing the pressure to 17bar. Only the components due to the first sound can be seen. AT is the temperature drop below the A point. The spectra were recorded by a Fabry-Perot interferometer with plane mirrors separated by 80cm. (Vinen and Hurd [1978], Vinen [1980].)

been obtained only for the usual sound. The results of Vinen and Hurd [1978] (see also Vinen [1980]) must be regarded as a record achievement. 3.4. MOLECULAR LIGHT SCATTERING SPECTRA OF GASES

The study of the molecular light scattering spectra of gases belongs to the class

111, § 31

117

SPECTRA FROM Ae(P). EQUILIBRIUM PHENOMENA

of interesting and important problems. The study has an unusual history, which began mote than 50 years ago (1942) with the experimental search for the MB components in H2 at 100 atm, in N2 and 0 2 at 80 atm, and in C02 at 50 atm (Venkatesvaran [ 1942b1). This search was unsuccessful. The components were not recorded and it was actually claimed that the MI3 components should not exist. I did not agree with this claim and demonstrated, on the basis of elementary and qualitative considerations, that discrete MI3 components should be observed for hydrogen, nitrogen, oxygen, and carbon dioxide at the pressures used and for all the investigated gases (Fabelinskii [ 1955, 19591). If a medium can be regarded as continuous or, in other words, if the mean free path 1 of a particle of the medium is shorter or, better still, much shorter, than the wavelength A (eq. 21), then all the results referring to liquids also apply to gases, subject only to the condition that the attenuation of hypersound at a given wavelength is much less than 2n. If we assume that discrete MB components cannot be observed when the halfwidth of a component 60= au is equal to the separation between the maxima of the components and the central line,

60

=

V 8 A o = 2n-00 sin -, C 2

we can obtain the required criterion on the basis of eqs. (19), (20) and (48):

aA = 2n.

(49)

Thus, if aA << 23t, discrete MB components should be observed. The quantity a A can be estimated from elementary gas-kinetic data:

-

1 aA = A - , A

where a rough estimate yields A M 25 and this makes it possible to find aA under experimental conditions when the gas pressure is 50-100 bar (Fabelinskii [1955, 19681). If scattered light is observed for 8 = 90" at atmospheric pressure (7 M cm) and if a A > 1 , discrete MB components cannot be observed, but since aA < 1 at just 20 bar discrete components can be observed. This has been confirmed experimentally, although admittedly the first evidence for the existence of the fine structure in gases was obtained under the conditions of stimulated MB scattering (STBS) (Mash, Morozov, Starunov and Fabelinskii [1965]), but the

I18

SPECTRA OF MOLECULAR SCATTERING OF LIGHT

"§

3

MB components were subsequently also detected in gases due to the scattering by thermal fluctuations (Eastman, Wiggins and Rank [ 19661, Greytak and Benedek [ 19661). Gases are suitable systems for experimental and theoretical studies because different regimes can be realized. For rarefied gases, when 1 >> A, the gaseous medium cannot be regarded as continuous - each particle scatters the exciting light independently of other scattering centers. The scattering spectrum then reproduces that of the exciting light, but the former is somewhat broader as a consequence of the Doppler shift of the frequency of the scattering centers migrating at thermal velocities. The Maxwellian velocity distribution is assumed. The scattered line half-width is then

where rn is the mass of a scattering center (molecule). The experimental results can thus be simply described in the two extreme cases. In the intermediate range, when 1x A, it is not difficult to obtain experimental results, but a quantitative theoretical description is no longer quite so simple. Here it is necessary to solve the Boltzmann kinetic equation. If the changes in the frequency distribution of the intensity I ( o ) are studied at increasing pressures, it can be seen how the Gaussian distribution becomes gradually distorted and, when 7 << A, is converted into a distinct triplet in which there are two MB components and a central (Rayleigh) line (see below). The first experimental study of the spectra of the molecular scattering of light by thermal fluctuations in gaseous Ar, Xe, N2, C02, and CH4 was carried out by Greytak and Benedek [1966] and was followed by other experimental and theoretical investigations (Hara, May and Knaap [1971], Boley and Yip [ 19721, Cazabat-Longequeue and Lallemand [ 1972]), in which the changes in the frequency distribution of the intensity as function of the gas pressure could be readily traced. Theoretical studies by Andreeva and Malyugin [1986, 19871 are of special interest. These authors predicted that a new picture of the molecular scattering of light would emerge when all three components can be seen clearly in the scattering spectrum (1 << A). When account is taken of the interaction of the translational and rotational degrees of freedom and the dispersion of the kinetic coefficients, a new section appears in the molecular light-scattering spectrum: it represents a narrow band with a maximum at an unshifted frequency resulting from a redistribution of the

111, 9: 31

SPECTRA FROM A&(P).EQUILIBRIUM PHENOMENA

I19

intensity in the spectrum. At higher pressures, the new band is converted into the MB components. The redistribution of light in the spectrum is than complete. An extensive review of the work on the subject is given in the paper by Andreeva [1993]. Theoretical investigation of a Rayleigh scattering in a dense spin-polarized gas was performed by Andreeva, Rubin and Yukov [1995].

3.5. MOLECULAR LIGHT SCATTERING SPECTRA IN VISCOUS LIQUIDS AND GLASSES

The history of the spectra of light scattered in viscous liquids and glasses has been very similar to the history of such investigations of gases, in the sense that initially the MB components were sought in viscous liquids and glasses, but could not be detected. It was then decided that they should not exist, whereupon they were in fact observed and such studies have continued and have yielded interesting results. The first investigations were carried out by Gross [1930b], who observed the MB components in a quartz single crystal, but he failed to find these components in the spectrum of light scattered by a glass. The next unsuccessful attempt to observe the MB components in glasses was made by Raman and Rao [1937]. Other attempts to detect these components in the spectrum of light scattered by glasses and viscous liquids were likewise unsuccessful (Raman and Rao [ 1937, 19381, Venkatesvaran [1942a], Velichkina [1953, 19581). The studies were of a fundamental nature and the results seemed to be so reliable that it became desirable to explain this. Such an explanation was formulated (Rank and Douglas [1948]). Briefly, the explanation is as follows: it is well known that the halfwidth 6w of the MB components is determined by the absorption coefficient of the hypersound (eqs. 37,38) and this coefficient is proportional to the viscosity: ac( Q (eq. 39). Next it is claimed that the viscosity of highly viscous liquids P. The viscosity measured under static and glasses is enormous and reaches conditions is indeed as high as this and, accordingly to Rank and Douglas [1948], its enormous value broadens the MB components so much that they cannot be observed as discrete lines. If a viscosity of P determines the absorption of sound in glasses, then even a thin glass layer should not transmit sound waves. However, it is well known that this is not the case and that even high frequency sound passes through glass readily. The important point is that, even at low sound frequencies, the relaxation process renders the absorption of sound incomparably weaker than might have been expected on the basis of the viscosity at the zero frequency. The relaxation process also leads to the dispersion of the velocity of sound (Mandelstam and Leontovich [ 19371).

I20

SPECTRA OF MOLECULAR SCATTERING OF LIGHT

“11,

P

3

This lack of knowledge or misunderstanding of the role of relaxation has been largely responsible for the long-lasting failure to observe the MB components in viscous liquids and glasses. All the above-mentioned experimental searches for these components in viscous liquids and glasses were carried out before laser light sources became available in laboratories, and for this reason the exposures lasted many hours and even many tens of hours. For a long time it has been unclear why, for example, the MB components are clearly seen in glycerol at 5O-6O0C, while in the region of the glass-transition temperature T, of the same glycerol it has not been possible to find these components. If one bears in mind that the relaxation process results in the dispersion of the velocity of sound and that at high viscosities and high hypersonic frequencies it can reach loo%, the velocity of sound is doubled. This means that the adiabatic compressibility, Ps, decreases by a factor of 4 and the integral MB component intensity (eq. 4) also decreases by the same factor. Consequently, the exposure must also be increased by a factor of 4. Under such conditions, exposures lasting many hundreds of hours are needed to observe these components. When this became understood, Pesin and Fabelinskii [ 1959, 19601 increased the aperture ratio of the experimental apparatus by an order of magnitude and discovered the MB components for a number of viscous media. Krishnan [ 19501 was the first to observe these components in fused quartz at 300°C at the A = 263.6 nm line in apparatus with a mercury vapour resonance filter. Flubacher, Leadbetter, Morrison and Stoicheff [ 19591 also detected the MB components related to longitudinal and transverse hypersound in fused quartz. They used a lamp operating on the basis of the mercury-198 isotope. These problems were examined in greater detail in my book (Fabelinskii [ 19681). Further experiments have shown distinctly that for viscous and extremely viscous liquids, the hypersound propagation velocity o and the amplitude absorption coefficient, a, depend on the hypersound frequency, Q, in quite a different way than they do for the liquids with a low viscosity (centipoise range). Below the relaxation region, for small r], a Q2, while for high viscosity, aInside the relaxation region the frequency dependencies of v and a are also different for low- and high-viscosity liquids. Above this region both parameters experience practically no frequency dependence while their value becomes extremely low or even reaches zero. A relaxation process is a process in which a medium approaches the equilibrium after it has been withdrawn from it. This statement is valid for any physical parameter, in our case - viscosity. The relaxation time, z, is the time needed for a parameter under consideration to become e times closer to

-

111,

5

31

SPECTRA FROM AE(P).EQUILIBRIUM PHENOMENA

121

its equilibrium value. For low-viscosity liquids (e.g., benzene), the theories of Mandelstam and Leontovich [I9371 and Isakovich [I9391 are valid (see also Herzfeld and Litovitz [ 19591). But it became clear that for viscous media, the single relaxation time model is not applicable. Many authors unwilling to leave the equations of local relaxation theory introduced a continuous spectrum of relaxation times; i.e., f ( t ) . It is clear that having at one’s disposal a continuous function of a set of some parameters, one can describe the result of any experiment, but it appeared that f(t) was not a universal function and separate ones had to be introduced for longitudinal and transverse sound. Different functions were also needed for different substances. From here it becomes quite evident that this manifold of continuous spectra of z does not bear any physical sense. Still, experimental investigations of viscous liquids carried out at ultra- and hypersonic frequencies can be described by a single parameter. However, one must formulate the requirements (Krivokhizha and Fabelinskii [ 19661) for the desired theory. A version of such a theory was developed by Isakovich and Chaban [1966]. This theory was not a local one; it described the experimental results quite satisfactorily and yielded an a 12”2 dependence. The theory dealt with a single, and not an arbitrary parameter:

-

where p = (a/a,)Q,el, r] is the shear viscosity, and u, and 00 are the velocities of sound at extremely high and extremely low frequency limits, respectively. The variables a and a , are the absorption coefficients; the former stands for the total absorption and the latter for the absorption which is due only to the shear viscosity. At the basis of this theory lies the hypothesis of a micro-nonuniform structure of a viscous liquid that is believed to be composed of two components which differ by some internal parameter; one of these components is supposed to form a “drop” with distinct borders inside the other. The equations given by this theory describe the experimental results quite well, but the basic hypothesis seems less than completely convincing. Also, it appeared to be impossible to expand the results of this theory on the theory of molecular light scattering spectra. The attempts to build up a more comprehensive theory are difficult to understand, but there remains some hope for future progress. Before a complete and adequate theory of the amorphous state will be created, every step will require enormous efforts.

I22

SPECTRA OF MOLECULAR SCATTERING OF LIGHT

"II,

5

3

There now exist many papers devoted both to the theory and to the experiments in the field and in particular to physico-chemical properties of silicate glasses and other viscous and glassy substances. Most are investigations of polymers and liquid crystals. Here we shall mention only a few papers devoted to the discussed problem: Gotze [ 19871 and Dembovskii and Chechetkina [ 19901. 3.6. LIGHT SCATTERING BY A TWO MEDIA INTERFACE

Light scattering at the interface between two media was predicted by Smoluchovski [I9081 and observed by Mandelstam [1913], who also devised a theory of this phenomenon, which was subsequently developed further by Andronov and Leontovich [1926]. These first studies involved the use of light that was not decomposed spectrally, and it was not until 1968 that the first experimental study of light scattered inelastically by the surface of liquid methane was carried out by Kayte and Ingrad [1968]. The development of spectroscopic studies of light scattered by a surface required devices permitting a marked increase in contrast within the spectrum. A high contrast in the spectrum is necessary in all spectroscopic studies, and this was well understood a long time ago. Thus Dufour [ 195 I ] was apparently the first to propose the use of one instead of two consecutively arranged interferometers (Houston [1927]), making sure that light passes through it twice. Although the advantages of this scheme have been demonstrated experimentally (Hariharan and Sen [ 196 l]), the low luminosity of the apparatus has prevented wide-scale employment of multipass Fabry-Perot interferometers. The situation changed radically when laser light sources and improved, weakly absorbing dielectric mirrors appeared in laboratories. Such apparatus has become available in numerous laboratories throughout the world largely owing to the excellent experimental studies by Sandercock [1970, 1972, 19821, who constructed numerous Fabry-Perot interferometers with a corner-cube (triple) prism for the turnback of the light beam and who used his apparatus in a wide variety of investigations. Multipass Fabry-Perot interferometers are nowadays manufactured in unlimited numbers by companies such as Burleigh and others. This has opened extensive opportunities for research requiring a high contrast of -109-10'2 and in particular that involving the determination of the spectra of scattered light modulated by surface waves of various origins. The study of elastic surface and volume waves, propagated in opaque media such as metals and semiconductors, has yielded particularly striking results. In such cases the volume scattering in one direction (into the bulk of a sample) is determined by the depth of a

111, I 31

SPECTRA FROM A E ( P ) .EQUILIBRIUM PHENOMENA

Intensity, arb. units 500

5

10

15

R M '

I

2

3

4

5

v, h i s Fig. 5 . Mandelstam-Brillouin spectrum of light scattered by a sample of GaAs with a ( I I I ) free surface. The surface wave vector q is parallel to the ( 1 10) direction, a=60", 1 = 5 1 4 S n m MBC corresponding to Rayleigh wave, pseudo-surface mode and longitudinal resonance are marked by RM, PSM and LR respectively (Aleksandrov, Saphonov, Velasco, Yakovlev and Martynenko [ 19941).

skin layer cm at the optical wavelength of 5 x cm) and by the crosssectional area of a light beam on the surface of a sample mm2). Thus the scattering volume is just 1 0-l' cm3. Furthermore, one must bear in mind that the intensity of the scattered light is maximal in the direction of the reflected beam, but one must also not forget that the intensity of the parasitic light, preventing determination of the scattered-light spectrum when the contrast is low, is maximal in the same direction. Examples of high-precision molecular light-scattering spectra are presented in figs. 5, 6 and 7, taken from the papers of Aleksandrov, Velichkina, Vodolazskaya, Voronkova, Yakovlev and Yanovskii [ 19901, Aleksandrov, Velichkina, Mozhaev, Potapova, Khmelev and Yakovlev [ 19921, and Aleksandrov, Saphonov, Velasco, Yakovlev and Martynenko [ 19941, respectively. A multipass Fabry-Perot interferometer is suitable for various experiments and is naturally widely used to solve various problems. Figure 5 presents the spectrum of light scattered by a GaAs (1 11) surface when a Rayleigh wave (mode) (RM), a pseudosurface wave (PSM), and a longitudinal acoustic resonance mode (LM) propagate along it. The vector q is parallel to

-

SPECTRA OF MOLECULAR SCATTERING OF LIGHT

[III,

94

0 -20

-12

4

12

20

Frequency shift, GHz Fig. 6 . MB spectrum due to a surface Rayleigh wave in a Ge single crystal (Aleksandrov, Velichkina, Mozhaev, Potapova, Khmelev and Yakovlev [ 19921).

(1 lo), the light incidence angle a = 60°, and the scattering angle is 180”.Figure 6 presents the light-scattering spectrum of TmBa2Cu307, and fig. 7 the spectrum, originating from a Rayleigh mode of Ge.

0 4. Spectra Of Molecular Light Scattering Arising From Pressure Fluctuations A&(P).Some nonequilibrium phenomena 4.1. THE INFLUENCE OF A STEADY TEMPERATURE GRADIENT ON THE LIGHT-

SCATTERING SPECTRA

In all previous cases, it has been assumed that the scattering medium is at an equilibrium temperature identical at all points within the medium. However, about 60 years ago, Mandelstam [1934] stated that if there is a temperature gradient, the intensity of light scattered at a particular point within a body depends on the temperature at other points of the body. The distribution of the scattered-light intensity is then more uniform than the temperature distribution. Mandelstam predicted that this may be observed in solids and in liquids, while according to Andreeva and Malyugin [1987, 19901 this phenomenon with its interesting features should also be observed in gases at suitable pressures. In the presence of a steady temperature gradient, the scattering at a particular “point” in a sample depends also on the temperature of the neighboring regions.

SPECTRA FROM AE(P).SOME NONEQUILIBRIUM PHENOMENA

125

twin-free TmBa,Cu,O,~, Intensity, arb. units 140

k*

a1

'0011

120 100

anti-Stokes

80 60 40

a = 51.9" 20

I U U J ~ ~ L U ~ JVL ~ U ~-L~ Gu

jawr.

L I ~ ~ L L G I I I I ~EC) U L U C L ~ J

10 ~ C D C U L C U I U

uic

uppi

p i , UI LLLC

~ r g w c iicir, .

n is the normal to the sample surface, k , , k , and k~ are the wave vectors of incident and scattered

light and of an elastic surface wave respectively; a is the incidence angle, and E is the electric field vector of the incident light wave. The scattering angle is 180" and the excitation wavelength is 514.5 nm. (Aleksandrov, Velichkina, Vodolazskaya, Voronkova, Yakovlev and Yanovskii [1990].)

The effect depends on the shape and size of a sample, as pointed out by Mandelstam in his very first communication. Mandelstam [ 19341 also mentioned that the distribution of the scattered'light intensity does not reflect exactly the temperature distribution because the energy of the Debye waves, responsible for light scattering constitutes a very small proportion to the entire Debye spectrum. The physical nature of the phenomenon can be readily understood if one takes into account the fact that the scattering of light may be regarded as dimaction by a Fourier component; e.g., by an adiabatic fluctuation or an elastic thermal wave with a wave vector fq. A maximum of the scattered (diffracted) light is observed for exciting light with a wavelength A at an angle 6 when condition (21) is fulfilled (fig. 1). Fluctuations at the most diverse points within a given body generate an elastic wave with a wave vector q. Therefore, if a body is heated nonunifonnly, elastic waves or Fourier components with different amplitudes

I26

SPECTRA OF MOLECULAR SCATTERING OF LIGHT

[HI, $ 4

arrive at the point of observation of the scattered light. For this reason, if the wave with -q differs in amplitude from the wave with +q, the intensity of the Stokes MB scattering component also differs from that of the anti-Stokes component. Following Mandelstam’s proposal, the problem of the temperature dependence of the scattered-light intensity in a medium with a steady temperature gradient has been solved by Leontovich [1935] for a specially derived model. For the scattered-light intensity he obtained a formula where a temperature gradient is taken into account. A more general case of light scattered in a nonuniformly heated body was also examined by Leontovich [ 19391. This problem concerns the energy distribution of elastic lattice vibrations and is solved in a similar way to the problem of the light intensity distribution in an emitting and absorbing medium. Here elastic heat waves, having frequencies in the range responsible for light scattering, are treated as noncoherent beams of various directions. The investigated case is the case of a crystalline plate infinite along the xy-plane and finite in the z-direction. The temperature changes only in the zdirection so that T ( z )= TO+ Cz, where C = ( T I- T0)il is the temperature ( T ) gradient and To corresponds to the temperature on the lower (z=O) face of a crystal and T I corresponds to the temperature on the upper face. For the functions A(z,b), which are proportional to the energy of elastic waves that propagate in opposite directions and determine the scattered-light intensity, Leontovich’s [ 19391 theory yields: ~c - . i , , a . z (A

[ l - (1 + r ) exp (-2az/cos 1 + r exp (-2al/cos

a) a)

for 0 < 8 < nJ2, and A(Z,a)=c(T-G${

1 - (I

+ r ) exp [2a (I z)/cos a] 1 + r exp (2a1/cos a) -

for n / 2 < z9 < n.Here, a is the amplitude absorption coefficient, determined by eq. (39), 0 is the angle between the direction of the temperature gradient and the wave vector of an elastic wave q, and r is the reflection coefficient. If for the directions of the two elastic waves we assume the angles to be and t9‘8, the scattered-light intensity will be:

111,

I 41

SPECTRA FROM A & ( P ) SOME NONEQUlLlBRlUM PHENOMENA

127

where D is a factor that depends on the scattering angle, but not on the temperature distribution. Equation (54) yields the scattered-light intensity dependence on temperature and temperature gradient. If 1998 > 0, then

In 1943, Vladimirskii suggested the use of difference in the intensity of the Stokes and the anti-Stokes MB component for the determination of the hypersound decay coefficient a. In fact, supposing z and I - z to be infinite in eqs. (52) and (53) and taking into account eq. (55), one obtains:

For the uniformly heated body, I , = Ias.One can find from eqs. (56) and (57): Is-Ias I

-

GCOSfi~- ( q V T ) 2aT 2aT '

where q is the unit vector in the direction of a wave vector of an elastic wave, and I = I s + Ia,.Taking into account eq. (39) instead of eq. (58), one obtains

Taking into consideration the above mentioned properties of the elastic heat waves, one can assume that the calculations of Leontovich [1939], carried out for crystals, can also be applied to liquids and to gases at proper pressures. The first attempt to investigate experimentally the scattering of light in a nonuniformly heated body was made by Landsberg and Shubin [1939], who investigated the scattering in a nonuniformly heated quartz single crystal illuminated with white light. Within the limits of an experimental error of about I%, Landsberg and Shubin found no deviation of the intensity distribution from that of a uniformly heated body, which enabled them to conclude that the absorption coefficient is a > 0.75 cm-' for the elastic wave responsible for the light scattering. On the other hand, since the discrete MB components are observed in a quartz crystal, one can postulate that (r < 5 x 1O4 cm-' . In any case, the first quantitative theory and the first experimental attempts to investigate the

128

SPECTRA OF MOLECULAR SCATTERING OF LIGHT

Th scattered

iLzzTC

Fig. 8. Schematic diagram of the apparatus for determining the MB spectrum of a medium with a constant temperature gradient. PM is a photomultiplier (Kiefte, Clouter and Penney [1984]).

light scattering in a medium with a steady temperature gradient were produced before 1940. During the next 40 years, there appeared only the theoretical note by Griffin [1968a,b] who was aware of the investigations of his predecessors. The problem did not attract other scientists' attention. Only in the late 1970s several independent groups of physicists published (nearly simultaneously) a number of theoretical papers devoted to the influence of the temperature gradient on light-scattering spectra in liquids and solutions. Among them are the papers by Procaccia, Ronis and Oppenheim [ 1979a,b], Ronis, Procaccia and Oppenheim [ 1979a,b,c], Kirkpatrick, Cohen and Dorfman [ 1979, 1980, 1982a,b,c], Ronis, Procaccia and Machta [1980], van der Zwan and Mazur [1980], Tremblay, Arai and Siggia [1981], Tremblay and Tremblay [1982], and Tremblay [1984]. Of course, this list does not cover all the good papers where the influence of a steady temperature gradient on the ratio of the intensities of the MB components has been studied. All these papers differed in the approach to the problem, but before the paper of Tremblay, Arai and Siggia [ 19811 was published (and even after that) the authors were not aware of the papers of Mandelstam [1934], Leontovich [1935, 19391, and Vladimirskii [I9431 and did not cite them. These papers were discussed in detail in my book, published in Russia in 1965 (Fabelinskii [1965]) and in the USA in English in 1968 (Fabelinskii [ 19681). The latter was known to Griffin [1968a]. Tremblay, Arai and Siggia [1981] included a short history of the problem in their publication. The theoretically predicted asymmetry of the MB components (see eqs. 5 8 and 59) was confirmed experimentally in a most convincing manner by Kiefte, Clouter and Penney [ 19841 who used water and fused silica as the samples. The schematic of their setup is presented in fig. 8 and the results in fig. 9. Study of the influence of a steady temperature gradient in liquids and solutions

111, P 41

SPECTRA FROM A.E(P). SOME NONEQUlLlBRlUM PHENOMENA

129

(@VT)/q*(lO‘K.cm)

Fig. 9. Dependence of the asymmetry of the MB components on V T I q 2 :(a) Various data for water, V T = 1 7 4 0 Kcm-’ and T = 288 K (solid circles), 298 K (open squares), and 303 K (solid squares); V T = 28.3 K cm-I and T = 296 K. (b) “Best” results for water with V T = 45 Kcm-’ and T = 307 K; the outer and inner arrows in the upper part of the figure correspond, respectively, to q = 3000 cm-l and q = 4000 cm-’ . (c) Data obtained for fused quartz, V T = 80 K cm-I and T = 3 15 K; the arrow at the top corresponds to q=4000cm-’. (Kiefte, Clouter and Penney [1984].)

on molecular light-scattering spectra including an unshified (Rayleigh) line was continued in a number of publications. We shall cite some of them without discussing the results: van der Zwan and Mazur [1980, 19811, Law, Segre, Gammon and Sengers [ 19901, Nieuwoudta and Law [ 19901, Pagonabarraga, Rubi and Torner [1991], Segre, Gammon, Sengers and Law [1992], and Li, Segre, Sengers and Gammon [1994]. Discussion of these papers is beyond the scope of this publication. A physical idea proposed by Mandelstam [1934] more than 60 years ago and further developed by Leontovich [1935, 19391 and Vladimirskii [1943] did not attract any attention for many years. But now it became an established branch of experimental and theoretical research. This branch deserves a special review, and it would be of interest to treat in detail all the aspects of this problem. I hope that further improvements of experimental techniques will allow the measurements of the hypersound absorption coefficient, and perhaps of some other characteristics of the media, to be performed. 4.2. PHONON “BOTTLENECK IN ACOUSTIC PARAMAGNETIC RESONANCE

The application of MB-spectroscopy to the investigation of the acoustic paramagnetic resonance was first proposed by Al’tshuler and Kochelaev [ 19651 (see also Al’tshuler and Kozyrev [ 19721).

130

SPECTRA OF MOLECULAR SCATTERING OF LIGHT

Ҥ 4

Of the most interest is the phenomenon that received the name of a phonon “bottleneck”. The essence is the following. In a paramagnetic crystal, the levels of incorporated ions (for example, Ni2+ or Ce3+) are split in a static magnetic field, and in the simplest case, two levels with an energy difference AE are formed. If such a system is acted upon by a high-frequency electromagnetic field with hw = AE, the usual ESR line is observed. The absorbed energy is consumed in resonant “excitation of spins”. If the spin excitation energy flux to the crystal lattice is greater than the energy flux from the crystal lattice to the thermostat (liquid helium), the number of phonons with an energy hw M h 0 exceeds the equilibrium value. Consequently, in the range of frequencies not exceeding the paramagnetic resonance line width, the nonequilibrium temperature T,, exceeds the equilibrium temperature Te. The quantity U = T,,/T,, which may be calculated, is referred to as the phonon “bottleneck” factor. If the MB components are now observed in light scattered in a crystal exhibiting the phonon “bottleneck” effect in such a way that the components are induced by the resonance phonons hQ, the intensity of these components is greater than for a crystal at the temperature of the heat reservoir surrounding it. Such experiments have been carried out by Brya, Geschwind and Devlin [ 19681 on a MgO crystal with Ni2+ ions and by Al’tshuler, Valishev and Khasanov [ 19691 on double cerium nitrate with Ce3+ ions. In the case of MgO crystals, the temperature of the reservoir was 2 K and the intensity of the MB components corresponding to 60K or u=30. In the study of double cerium nitrate, the temperature of the reservoir was 1.5 K and the intensity of the MB components corresponded to 100 K and hence to u = 70. There are also data obtained by both groups which give u values exceeding 1000. The calculations given by Brya, Geschwind and Devlin [ 19681 yield: U=

n0 t p h

TIP (0) 6u ’

where no is the thermal-equilibrium population difference between spin levels, x p h is the phonon lifetime, T I is the intrinsic spin-lattice relaxation time in the absence of a “bottleneck”, p(v) is the density of phonon states, and dv is the phonon bandwidth. Since T I u 5 , where u is the velocity of sound, p(v) u - ~ , and ( 5 - 8 . The given equation allows one to select a proper acoustic mode in an experimental investigation of the phenomenon.

-

-

4.3. AMPLIFICATION OF HYPERSOUND WAVES IN PIEZOSEMICONDUCTORS

SUBJECTED TO AN EXTERNAL STATIC ELECTRIC FIELD

A striking and impressive phenomenon is displayed in the spectrum of light

111, 9; 41

SPECTRA FROM A t ( S and A&(C)

131

scattered in piezoelectric semiconductors in a static electric field. Little more than 30 years ago it was established that, under certain conditions, an ultrasonic wave propagating in a piezoelectric semiconductor can actually be enhanced rather than, as usual, attenuated (White [ 19621, McFee [ 19661). These conditions include the application of an external static electric field of magnitude E capable of accelerating charge carriers to velocities equal to or greater than the phase velocity of sound. ZnO and CdS are examples of such substances. Calculations have shown that the sound damping factor is:

r, = r, +

x"

-

( E / E c ) ]Qrq

( 1 + q 2 R 2 ) 2 +[ I - ( E / E , ) ] 2 R 2 r 2 '

where To is the absorption by the crystal lattice; t=cl4na; E is the permittivity; a is the conductivity; is the electromechanical coupling constant; SZ is the sound frequency; q is the wave number; R is the Debye radius; and E, is the critical field, (i.e., the field corresponding to the case when the speed of the charge carriers reaches the phase velocity of sound). If the second term of eq. (60) becomes negative for EIE, > 1 and r E is negative, which can be readily realized, the ultrasonic wave, defined by eq. (43) does not decay, but is amplified. Naturally, the amplification takes place within a certain band of frequencies, and the gain maximum in this band corresponds to the frequency SZ,,, = uIR.Here R is the Debye radius

x

and no is the number of carriers of a charge e in 1 cm3. If the frequency of the elastic thermal wave responsible for the appearance of a MB component lies within the band of the amplified elastic wave frequencies of a semiconductor, the molecular light scattering spectrum should change. If the elastic wave that is amplified has the frequency, polarization, and direction of propagation such that it generates a Stokes (or anti-Stokes) MB component, then this component should increase appreciably compared with the other component. This phenomenon was first observed in Wetling's [ 19671 experiments. There followed a large number of investigations devoted to this outstanding phenomenon. We shall mention some of them. The work was continued by Smith [1970], Wakita, Umeno, Hamada and Miki [1973], and this phenomenon was studied in special detail by Velichkina, Dyakonov, Vasileva, Aleksandrov and Yakovlev [ 19821, and Velichkina, Dyakonov, Aleksandrov and Yakovlev [ 19861

132

SPECTRA OF MOLECULAR SCATTERING OF LIGHT

x 1400

I

l-

-2

-1

0

1

i

Fig. 10. MB spectrum of piezoelectric semiconductor ZnO in an external static electric field (Velichkina, Dyakonov, Aleksandrov and Yakovlev [ 19861).

(fig. 10). See also the references given in this last publication. Gurevich [1964] developed the theory with noteworthy thoroughness.

0

5. Spectra of Molecular Light Scattering Arising from Isobaric Entropy Fluctuations A&(S) and from Concentration Fluctuations A&(C) 5.1. CENTRAL PEAK: THERMAL DIFFUSIVITY AND DIFFUSION

A central, or Rayleigh line of molecular light scattering spectra in single liquids and other media is determined by the entropy or temperature fluctuations and, additionally, by the concentration fluctuations in solutions and mixtures. These fluctuations themselves are governed by the thermal diffusivity and, in the case of solutions also by diffusion. Because of that the measurements of the width and of the intensity of a Rayleigh line can provide a lot of various data. These can be especially interesting and valuable when phase transitions and critical phenomena are being investigated. For example, on the basis of this kind of measurements, one can determine the correlation radius of the concentration

111,

I 51

133

SPECTRA FROM AE(S and AE(C)

fluctuations and its critical index; the temperature dependence of the thermal diffusivity and of the diffusion coefficients, and critical opalescence and other phenomena can be investigated. Fluctuations of the entropy in liquids and solutions will now be considered, but the conclusions reached are fully applicable to all types of media within the hydrodynamic approximation. As in the case of adiabatic fluctuations of the density, the spectral singularities of the light scattered as a result of the entropy fluctuations are governed by the law of modulation of the light scattered by the time-dependent fluctuations of the entropy, temperature or concentration (if a solution is considered). The entropy or temperature fluctuations appear and disperse at a rate governed by the thermal diffusivity of a given medium or by the diffusion coefficient D if the medium is a solution. The nature of the time dependence of the rms fluctuations of the entropy (temperature) is described by the well-known Fourier equation or by the heat conduction equation. If we assume the previously discussed time dependence of an optical inhomogeneity described by (An’) (AS’) I”,

x

”’,

(AC2)’12,and if we postulate that (An’)”’ is described by @(t),we can write the Fourier equation in the following form:

We shall seek the solution of eq. (61) in the usual form O(t)= 0 0 exp [ i (mt - q . r ) ]

Substitution of this solution into eq. (61) gives @(t)= @0 exp (-&t) exp (-iq

. r ),

where

6s

= 4’x

and

x=-.K

CPP

Here, K is the thermal conductivity and Cp is the specific heat at constant pressure.

I34

"ij

SPECTRA OF MOLECULAR SCATTERING OF LIGHT

5

1/2

It follows from eq. (62) that the resultant fluctuations (AS2) and (AT')"' disperse in accordance with an exponential law and the rate of dispersal or the spreading time, t,= 1/6,, is governed by the constant and by the wave number q. By way of example, it is assumed that A = 5 x cm, n = 1.5 and cm2 SK'; then at H=90" (eq. 18), the spreading time Z, = lo-* s. The scattered-light field, modulated by eq. (62), is then

x

x=

E ( t ) = &@o exp (-d,t) exp (iwt),

(65)

which leads to the following time dependence of the scattered-light intensity:

I

( t ) = I.

exp (-264.

(66)

It follows from eq. (66) that the light scattered by the entropy fluctuations decays cxponentially without oscillations of the kind observed for the adiabatic fluctuations described by eq. (22). 1 /2 Since the concentration fluctuations (AC2) disperse in accordance with the same law as (AS2)"',

it follows that the nature of the solution and the

x

subsequent expressions should be the same as for (AS2)'12 if is replaced with the diffusion coefficient D, so that the function modulating the scattered light 1s: F ( t ) = @o exp (-act) exp (-iq . r ) ,

(67)

where

6,= q2D.

(68)

In one of his early papers, Mandelstam [ 19261 considered the fluctuations of the temperature and concentration and derived eq. (66). He discussed the process leading to the appearance of an unshifted line in the scattered-light spectrum. The distribution of the intensity on the frequency scale of such a line can be obtained by expanding E ( t ) as a Fourier integral and then calculating the scattered-light intensity as a function of the frequency. This is exactly the procedure adopted above in eqs. (30)-(35), and the final result is:

The total intensity of the central component Ii, is: 00

1, where

/is

I, ( w )d o = I,,, is given by eq. ( 5 ) .

111,

P 51

135

SPECTRA FROM A t ( S and Ar(C)

The half-width of this line at half-maximum l,(w)/2 is then:

6,= 6w, = q2x.

(71)

Here, q is given by eq. (18). In this same way we can obtain the intensity I c ( w ) of the light scattered as a result of the concentration fluctuations:

where I , is described by eq. (6) and the line half-width at half-maximum /,(w)/2 is:

An estimate of the time needed to establish the spectrum of light scattered by the concentration fluctuations, obtained on the assumption that D = 1 0-5 cm2 s-' and s. that the conditions are the same as in the preceding example, gives t = The Einstein expression for the mutual diffusion coefficient D is:

where r] is the shear viscosity and r is, strictly speaking, the radius of a macroscopic particle. Einstein also applied eq. (74) to molecular systems but this application is not well enough justified physically. The half-width of the central (Rayleigh) line for the scattering in a onecomponent medium or in a solution can be described by a general expression of the following type:

L d w = -,

(75)

X'

x*

is the generalized where L are the Onsager kinetic coefficients and susceptibility. In the critical range the value of L is practically independent on temperature, increases strongly because of critical opalescence (Oxtoby and whereas Gelbart [ 19741, Lakoza and Chalyii [ 19831, Zubkov and Romanov [ 1989]), and, therefore, the central line half-width tends to zero. Therefore, a reduction in the half-width of the Rayleigh scattered-light line and an increase in its intensity

x*

136

SPECTRA OF MOLECULAR SCATTERING OF LIGHT

UII,

§ 5

are due to the same cause, which is an increase in the correlation radius of the concentration fluctuations in a solution or of the density fluctuations in a onecomponent liquid because of approach to the critical point. Expressions (7) and (9) derived for the intensity of the scattered light are inapplicable to the case when the temperature of a liquid or a solution is close to the critical temperature. Expressions (7) and (9) are derived on condition that the fluctuations are independent of one another. In the region of the critical temperature this condition is not satisfied. In the critical region the fluctuations influence one another at effective distances r,, which can be called the correlation radii. This important point was allowed for by Ornstein and Zernike [ 1918, 19261 who generalized expressions (7) and (9) to the critical range of temperatures. This generalization by Ornstein and Zernike leads to the following expression for the intensity of the scattered light IOZ:

where I is given by eqs. (7) and (9). In the limiting case of long-wavelength harmonics (q + 0), we find that (76) reduces to the initial expressions (7) and (9). The dependence of 16; on q2 is a straight line, the slope of which makes it possible to find r,'. In this method, the determination of r, requires finding first the dependence of the relative intensity on the scattering angle, or, equivalently, on q. This has been done on many occasions (see, e.g., Komarov and Fisher [ 19621, Fisher [ 19651, Cummins [1971], Swinney [ 19741, and the literature cited there). Another method for determination of the correlation radius of the entropy, temperature, or concentration fluctuations is purely spectroscopic. It basically involves determination of the width of the Rayleigh line in the scattered-light spectrum. The first determination of the Rayleigh line width in the critical region was reported by Alpert, Yeh and Lipsworth [1965], but since then there have been extensive and multifaceted theoretical investigations. A brief account will be given of some results of the studies of critical solutions which have a double critical point. Systems of this kind have been largely neglected, although they are of considerable interest for the understanding of the physical features of phase transitions in general (Chaikov [ 19911). As is well known, second order (or close to them) phase transitions and accompanying critical phenomena, wherever they take place, possess similar or nearly similar features. Because of that, the properties and peculiarities observed for some particular case can be expanded on other transitions of a similar type. This significant conclusion, if it is valid, justifies the search for an investigation

137

SPECTRA FROM A t ( S and A&(C)

t'

a

'

c ,

b

T

wc.I'g:

d

C . )

Fig. 1 1. Typical phase diagrams of different solutions. The open circles denote critical points. In the shaded area the solution is exfoliated. (a) Upper critical point; (b) lower critical point; (c) closed-loop exfoliation region; (d) increase of exfoliation region due to increasing amounts of a third component ( H 2 0 or CC14). (Chaikov [1991].)

object, which, being convenient for an experimentalist, would provide new and van ous know 1edge. Perhaps binary critical mixtures (solutions which possess a critical point) are the systems that suit the above requirements better than any other object. Phase diagrams (fig. 11a-d) represent solutions with different critical points. There are many solutions with the upper critical point (fig. 1 la); as examples, we can mention nitrobenzene-hexane, aniline-cyclohexane and nitroethane~ CL trimethylpentane. A crater-like phase diagram with its extremum at T c and has the lower critical point (fig. 1 lb). Examples include trimethylamine-water, gamma-collidine-water and lutidine-water. There are fewer solutions with the lower critical point. There are also certain systems which, described in terms of the coordinates T and C, form closed regions or loops inside which the components of the binary solution become separated and the positions of the separate phases are stable. There also exists a special group of solutions which are homogeneous across the total phase plane, but the admixture of a small amount of a third component leads to the appearance of a closed-loop exfoliation (separation) region (CRE) on their phase plane (fig. 1 Ic). The size of the CRE depends on the amount of the third component. Since the nature of the phase diagram depends strongly on the amount of the third component, a three-dimensional phase diagram can be constructed, as is done for a guaiacol-glycerol solution in fig. 12. In this figure, the vertical axis gives C, which is the concentration of water. The other two coordinates are the temperature and the concentration of guaiacol in glycerol. The phase diagram is then a crater-like surface and the upper and lower critical points are transformed into lines of such points. The double critical point is where the lines of critical points meet at the extremum. As examples of such solutions we can mention nicotine-water, glycerol-benzylamine, glycerol-m-toluidine,

138

SPECTRA OF MOLECULAR SCATTERING OF LIGHT

tLX

T

Fig. 12. Phase diagram with a crater-like phase separation surface (C, is the concentration of the third component): ( I ) line of lower critical points; (2) line of upper critical points. The surface minimum corresponds to the double critical point (which is the point where lines 1 and 2 meet). (Chaikov [1991].)

glycerol-guaiacol. Special attention will be paid to the latter, because the results obtained can apparently be expanded on the other systems of the same kind. The studies of a guaiacol-glycerol solution revealed from their very beginning some amazing properties. It is enough to add one water molecule per 23 molecules of this solution (which is homogeneous at ambient conditions), in order to create a CRE with a temperature width of AT = 2 K. Addition of one CC14 molecule per 170 molecules of the same solution also leads to the origination of a CRE with A T = 2 K . It seems improbable that a negligible amount of the third component would lead to such a dramatic change in a solution like origination of a CRE! At present, it is difficult to name a physical reason leading to the observed phenomenon. It seems that the origination of a CRE is not a direct consequence of the addition of the third component but rather the result of some process triggered by such an addition. In their detailed theoretical treatment, devoted to the modern methods of evaluation of phase diagrams of the solutions with a CRE, Walker and Vause [ 19831 built up the following concept of a formation of a homogeneous solution

111,

9: 51

SPECTRA FROM AE(S and Ae(L‘)

139

and origination of a CRE.Their opinion is that the hydrogen bonds in a guaiacolglycerol solution are so strong that they do not allow the components to separate and the solution remains homogeneous all over the phase plane. If water is added to this solution, part of the guaiacol hydrogen bonds are “locked” to the water bonds, the interaction between the “main” components becomes weaker, and the solution separates within some region of the phase diagram. Thus, according to Walker and Vause, the main role in a solution separation is played by hydrogen bonds. If it is true that the hydrogen bonds of the third component lead to the solution separation, one can think that adding some fourth component “rich” in hydrogen bonds (e.g. ethanol), would enlarge the CRE already existing in a solution with preliminary added water. But the reality is more rich than one’s imagination, and it appeared that the addition of a very small amount of ethanol (smaller than the amount of preadmixed water) completely took away the CRE.The solution again became homogeneous all over the phase plane. These results suggest that the hydrogen bonds are probably not as significant as it seemed. An attempt was made to find out whether a substance without any hydrogen bonds (e.g., CC14), if added to the solution, would lead to the formation of a CRE.CC14 is a molecule that contains neither hydrogen nor oxygen. As already mentioned, a negligible amount of CC14 added to a homogeneous guaiacol-glycerol solution leads to the formation of a CRE.Therefore the hydrogen bonds either do not play any role in the formation of a CRE, or this role is different from the role ascribed. Further observations of the behavior of this solution led to the discovery that the substance of the third component leads to the formation of a CRE in homogeneous solutions only if this substance itself can be dissolved in one of the “main” components of the solution and cannot be dissolved in the other (water can be dissolved in glycerol but not in guaiacol, CC14 can be dissolved in guaiacol, but not in glycerol). This is not the only pair of investigated substances. There were others, but the results were the same. If the substance of the third component can be dissolved in both of the “main” components (alcohols), this substance does not lead to the formation of a CRE, and, moreover, destroys it if it already existed in a solution. Up to now no exceptions from this rule are known. Other results of the investigations of this extraordinary solution will be presented below. We shall start with the results of the measurements of the temperature dependence of the concentration fluctuations correlation radii in the vicinity of the upper, lower, and double critical points of the solutions with various sizes of a CRE.

140

SPECTRA OF MOLECULAR SCATTERING OF LIGHT

[ a

05

5.2. RAYLEIGH LINE AND CONCENTRATION FLUCTUATIONS CORRELATION RADII IN A CRITICAL REGION

As pointed out earlier, the half-width of the central (Rayleigh) line in the

scattered-light spectrum is equal to the product of the square of the wave number and the mutual diffusion coefficient, as given by eq. (73). It follows from eq. (74) that the diffusion coefficient is itself a function of temperature, as well as of the shear viscosity and of the size r which in the critical region should differ greatly from the size of a molecule and which represents the correlation radius of the concentration fluctuations (Cummins and Swinney [ 19701). Extensive experimental and theoretical investigations have been made of phase transitions and critical phenomena in various media under diverse conditions. See, for example, Fisher [1965], McIntyre and Sengers [1968], Cummins [1971], Stanley [1971], Fleury and Boon [1973], Scott [1974], Swinney [1974], Ma [ 19761, Hohenberg and Halperin [ 19771, Potashinskii and Pokrovskii [1982], Ginzburg, Levanyuk and Sobyanin [ 19831, and Anisimov [ 19871. It is impossible to list all the papers devoted to this field, and the references given must be treated as examples. There have also been many attempts to develop a theory of phase transitions and critical phenomena of solutions which exhibit phase separation. Here, I mention primarily the papers which have led to the development of a theory of interacting modes: Kawasaki [1966, 19711 and Kadanoff and Swift [1968]. This theory accounts well for the experimental results and describes the half-width of the central line in the spectrum representing the concentration fluctuations by the expression:

r = r, + r, = ~~q~ (1 +x2) +D,Rq2K (x).

(77)

The subscripts B and c indicate that the quantity is divided into the background (B) and critical (c) parts, q is the wave number of the scattered light, x = qrc, where r, is the correlation radius of the concentration fluctuations, R = 1.027, and K ( x ) is a function of the following type (Burstyn and Sengers [1982], Chen, Lai, Rouch and Tartaglia [1983]) 5

K (x) = -Y2 [ 1 + x2 + (x' - x-') arctan x] . (78) 4 The background D B and critical D, values predicted by this theory (Kawasaki [ 19661, Kadanoff and Swift [ 19681, and Bhattacharjee, Ferrell, Basu and Sengers [1981]) are:

D,

=

kT

-,

6JWC

(79)

SPECTRA FROM AE(S and A&(C)

141

Here, r] is the shear viscosity and its background component is T]B. The value of qc can be found from (Bhattacharjee, Ferrell, Basu and Sengers [1981])

and Qo are quantities with the dimensions of the wave number. Qo can be obtained from the temperature dependence of the shear viscosity (Bhattacharjee, Ferrell, Basu and Sengers [1981]):

77

- = (Qorc)x~. r]E

is a finite number, Here, x,, is the critical exponent of the viscosity and but such that the theory based on the approximation of a continuous medium is applicable. It therefore follows from expressions (77)-(82) predicted by this theory, that measurements of all the quantities in these expressions (with the exception of r,) can be used to find rc and its temperature dependence, as was done for the guaiacol-glycerol solution (Chaikov, Fabelinskii, Krivokhizha, Lugovaya, Citrovsky and Jany [ 19941). If rc is written in terms of dimensionless temperature (Davidovich and Shinder [ 1989]), the following expression can be used

where

cL T -

El =

-,

T,"

T,'

&2=-.

-T

T,"

Here, CL and T," are the lower and upper critical temperatures of the guaiacolglycerol mixture, ro is a constant, and Y is the critical exponent. This investigation has established that if A T = T u - T," > 1.5"C, the critical exponent is 0.63 150.094, in agreement with Krivokhizha, Fabelinskii and Chaikov [ 19931 and our earlier measurements (Johnston, Clark, Wiltzins and Cannell [ 19851, Larsen and Sorensen [ 19851, Sorensen and Larsen [ 19851, Krivokhizha, Fabelinskii and Chaikov [ 19871, and Chaikov, Fabelinskii, Krivokhizha, Lugovaya, Citrovsky and Jany [1994]). This result fits within the framework of the fluctuation theory, but if the phase separation region

I42

SPECTRA OF MOLECULAR SCATTERING OF LIGHT

[IIL ii

5

corresponds to AT < 1.5"C, the value of Y decreases, tending to Y = O S as predicted by the Landau theory. This result is somewhat unexpected. In the immediate vicinity of the double critical point the experimental value of the critical exponent is Y = 1.02kO.O1, although one would expect Y = I .26 (for details see Chaikov, Fabelinskii, Krivokhizha, Lugovaya, Citrovsky and Jany [ 19941). An extensive theoretical investigation of some of the systems with the doublc critical point was reported by Walker and Vause [ 19831. This purely theoretical paper describes phase diagrams and gives a physical explanation of the existence of the upper and lower critical points on the basis of the strength of the hydrogenbonding force. However, it seems that the physical origin of the lower critical point requires additional study and convincing arguments to ensure the correct understanding of the nature of the phenomena involved.

5.3. THE INVESTIGATION OF ACOUSTIC PECULIARITIES IN THE REGION OF CRITICAL POINTS OF THE GUAIACOL-GLYCEROL SOLUTION

The investigation of the MB components allows the acquisition of extensive and sometimes unique data about the hypersound propagation in the vicinity of critical and double critical points of solutions (Krivokhizha, Fabelinskii and Chaikov [ 19931, Kovalenko, Krivokhizha, Fabelinskii and Chaikov [ 19961). A solution with a CRE created by addition of a third component was examined in special detail. Thc hypersound velocity was derived from the shift of the MB components with the help of eq. (2O), and the absorption coefficient derived according to eqs. (39) and (40). In a solution in which there is a temperature difference AT = 7.28 K between the upper and lower critical points, the hypersound velocity in the homogeneous phase is described by one of two straight lines, which differ in slope, depending on the temperature (fig. 13). The velocity temperature coefficient duidT for the upper critical point is -6.5 ms-' K-', while that for the lower one is - 1 1.6 m s-' K-' . The velocity temperature coefficient near the lower critical point is SG large that it is greater than the corresponding coefficients for the individual liquids. The velocity temperature coefficients of the guaiacol-glycerol solutions with different exfoliation regions, from AT = 39.52 K to AT = 0.062 K in our experiments, were the same for the upper and lower critical temperatures, respcctively. The absolute values of the hypersound vclocities near the upper and lower critical points are quite accurately the same for all the solutions studied.

111,

9 51

SPECTRA FROM AE(S and AE(C)

143

v.io-’,rn.s-1

30

40

50

60

70

80

T,OC

Fig. 13. Temperature dependence of the velocity of hypersound near the upper and lower critical points of a guaiacol-glycerol solution. (Kovalenko, Krivokhizha, Fabelinskii and Chaikov [ 19931.)

In the solutions with AT = 0.062 K, we find duldT = 0 in a narrow temperature interval in the vicinity of the double critical point. In a “dry” solution, in which there is no exfoliation region, the temperature dependence of the hypersound velocity is approximately linear, with a temperature coefficient duldT = -1 1.6m s-’ K-’ . This is the same as the temperature coefficient near the lower critical point. From the width of the MB components for the solution with AT = 7.28 K we can find the hypersound absorption as a function of temperature. Figure 14 shows the width of these components versus the temperature. Near the critical points we clearly see a sharp increase in the width (i.e., in the absorption) on a curve with a maximum of the same type as on curve 1, for the “dry” solution without an exfoliation region. This increase in width is due to a phase transition from a homogeneous solution to a region in which the components of the solution are separated. Figure 15 shows the same result, but for a solution with AT=O.O62K, i.e., essentially for a solution with a double critical point. The nature of the absorption feature near the double critical point is similar to that of the curve in fig. 14. It is seen against a strong background curve which essentially coincides with the temperature dependence of the width of the MB components of the “dry” solution. For the guaiacol-glycerol solution where the CRE was formed by adding CC14

144

-0.1

14

24

34

44

54

64

74

84 T,OC

Fig. 14. Temperature dependence of the absorption coefficient of hypersound near the upper and lower critical points of a guaiacol-glycerol solution. Curve 1 (circles): “dry” solution; curve 2 (crosses): solution with water, AT = 7.28 K. (Kovalenko, Krivokhizha, Fabelinskii and Chaikov [1993].)

GHz

Fig. 15. Temperature dependence of the absorption coefficient of hypersound near the double critical points of a guaiacol-glycerol solution. (Kovalenko, Krivokhizha, Fabelinskii and Chaikov [I 9931.)

the phase diagrams were obtained and the hypersound velocity was measured as in the case of the water-containing solution. The results of these measurements

SPECTRA FROM Ae(S and A&(C)

111, § 51

145

2.142.01.91.8-

1.7-

1.61.51.4L 20

t

30

40

50

60

70

80

90 T, “C

Fig. 16. Temperature dependence of the hypersound velocity in a guaiacol-glycerol solution with added water, CRE A T = 7 . 2 8 K (A, squares), and with added CCl4, CRE A T = 1 K (B, asterisks). (Kovalenko, Krivokhizha, Fabelinskii and Chaikov [ 19961.)

are presented in fig. 16 (lines B). The results of the hypersound velocity measurements for the water-induced CRE solution are presented for comparison on the same figure (lines A). The temperature dependence of the hypersound velocity is similar for both systems. Lines B correspond to the case AT = 1 K. The velocity temperature coefficient is du/dT =-4.8 m s-’ K-’ above the upper critical point, and du/dT =-11 m s-’ K-’ below. Thus, in both cases, there is not only a qualitative coincidence in the temperature dependencies of the hypersound velocity, but the values of the velocity temperature coefficients are also quite similar. In both cases the value of duldT above the upper critical point is nearly two-fold smaller than this value below the lower critical point. It is of interest to find out first how the hypersound velocity temperature dependence behaves in a “dry” solution (in other words, when the solution is homogeneous and possesses no CRE), and also in a solution with a double critical point. The results of these experiments are presented in fig. 17 (lines A). For a “dry” solution, the temperature dependence is nearly a straight line with duldT = -1 1 m s-’ K-’ . Straight lines B correspond to the situation where water is first added to a solution and a CRE with AT = 2.4 K is formed, and the CRE is subsequently destroyed by adding some ethanol. In this case the solution

146

[Ill, 9: 5

SPECTRA OF MOLECULAR SCATTERING OF LIGHT

v.io-’, m.sl

1.31 15

f

25

35

45

55

65

75

85

95

T,“C Fig. 17. Temperature dependencies of the hypersound velocity in a guaiacol-glycerol solution: “dry” solution (A, crosses); solution with added water, CRE AT=2.4K, that is later “collapsed” with the help of ethanol (B, asterisks); solution with added water with a double critical point (C, squares). (Kovalenko, Krivokhizha, Fabelinskii and Chaikov [ 19961.)

remains homogeneous all over the phase plane, but for such a procedure the entire temperature interval can be described by the two lines with the slopes d d d T = -4.8 m s-I K-’ and duldT = - 1 1.5 m s-’ K-’ for the higher and the lower temperatures, respectively. The solution somehow “remembers” its past. Curve C in fig. 17 corresponds to the case when the lower and upper critical points have merged, forming a double critical point. However, in the immediate vicinity of the critical point, duldT = 0. Over a small temperature interval, the hypersound velocity does not depend on temperature. A nearly twofold difference in the values of the velocity temperature coefficients below the lower and above the upper critical points is a result which is very significant and not yet completely understood. The velocity of sound is, indeed, a well-known quantity, u = (ap/ap);’’, that is, a derivative of pressure with respect to density taken at constant entropy (adiabatic value). As far as u is determined by the state equation, it means that the same solution is described by different state equations above the upper and below the lower critical points, while the temperature difference between these two regions is of the order of a fraction of a degree, or might be even very close

IK

D 51

SPECTRA FROM AE(S and AE(C)

147

to zero. This result is amazing, but it has been confirmed sufficiently to dispel doubt in its reliability. We must emphasize that the difference in the velocity temperature coefficients reaches two-fold!! If in a guaiacol-glycerol solution with a CRE one moves along the temperature axis T across the critical points from the higher temperatures to the lower (fig. 1 Id), phase conversion can be observed: the homogeneous solution after passing the upper critical point becomes heterogeneous (exfoliated into two components), and subsequently, after passing the lower critical point, it becomes homogeneous again. At a critical point the difference between the phases disappears, and the difference between the free energies disappears simultaneously - this difference becomes equal to zero. This statement is valid both for the upper and the lower critical points. However, in the vicinity of the double critical point, the temperature difference becomes negligible ( A T z 0.02 K) or even equal to zero, while at the same time doldT differs twice on both sides of the double critical point (see above). It seems unlikely that this can be explained by the difference in action of the hydrogen bonds (Walker and Vause [1983]), and it is necessary to look for the physical reason that leads to the minimization of the free energy at the lower critical point. We can only suppose that below the lower critical point, the clusters are formed that differ from those (if any) which are formed above the upper critical point. The total energy of a cluster is less than the sum of the energies of the molecules associated in it. This mechanism (if it exists) would have led to the decrease of the total energy E, and that would lead directly to the minimization of the free energy. If we continue to speculate in this direction, we can imagine that the clusters might appear to be acoustical inhomogeneities and the propagation conditions for the hypersound would be different from those for sound with much greater wavelength (ultrasound) in the same medium. This difference in the propagation conditions might be responsible for the difference in the velocity temperature coefficients du/dT. The proper experiment has recently been carried out by Krivokhizha, Fabelinskii, Chaikov and Shubin [1996] using a set-up with a previously developed (Velichkina and Fabelinskii [ 19501) sonic interferometer. The results of the experiment are presented in fig. 18a. The upper curves correspond to the hypersound, and the lower curves to the ultrasound. The hypersound frequency is -12 GHz while that of the ultrasound is 2.6 MHz; the wavelength AHSM 2x cm, and A u =0.076cm. ~ The ultrasound velocity temperature coefficient is nearly constant in the whole range of temperatures and its value is close to -4m SS' K-' . Thus, the result of this experiment is consistent with

148

SPECTRA OF MOLECULAR SCATTERING OF LIGHT

v, m/s

20

30

40

50

60

70

80

90~,oc

Fig. 18a. Sound propagation velocity in guaiacol-glycerol solutions with separation regions of different size A T . Curves A: circles for AT =39.65 K, crosses for AT =7.28 K. Curves B, ultrasound: asterisks for AT = 39.2 K, squares for 4 T = 7.2 K. (Krivokhizha, Fabelinskii, Chaikov and Shubin [ 19961.)

the hypothesis that says that from the point of view of energy balance it is advantageous to form the clusters (above the upper critical point) and that the solution possesses some sort of “memory” about the temperature at which such a formation was the most favorable. But it is necessary to consider the argument only as a probable version of the explanation of the lower critical point. Further investigations of the situation are definitely necessary. From the results of the experiments presented in fig. 18a it is clearly seen that there exists a dispersion of the velocity of sound, with its value depending linearly on temperature (fig. 18b). Above the upper critical point the dispersion of the velocity of sound lies in the range from 1.7 to 4%, which is quite usual, while below the lower critical point a more pronounced temperature dependence of the dispersion of the velocity of sound is observed and its value lies in the range from 16 to 22%, which seems to be rather unusual. 5.4. THE LANDAU-PLACZEK RELATION

In the spectrum of molecular light scattering the central (or Rayleigh) component is due to the scattering by isobaric fluctuations and is described by eq. (5).

IR

I49

SPECTRA FROM AE(S and AE(C')

Ty"C 20

30

40

80

90

Fig. I8b. Temperature dependence of the dispersion of the sound velocity in guaiacol-glycerol solutions: curve A, on the side of the lower critical temperature of separation; curve B, on the side of the upper critical temperature of separation (Krivokhizha, Fabelinskii, Chaikov and Shubin [ 19961).

Both MB components 2 / M B are due to adiabatic fluctuations and are given by eq. (4).The ratio of these intensities is:

where

If the thermodynamic relationship described by eq. (8) is taken into account and it is assumed that C = 1, the familiar Landau-Placzek [1934] formula is obtained from eq. (84)

All the relationships given above are purely thermodynamic and apply to a dispersion-free medium. In reality, there are strictly speaking no such media, and

I50

SPECTRA OF MOLECULAR SCATTERING OF LIGHT

“5 5

therefore the Landau-Placzek formula gives estimates of the quantities which are in quantitative disagreement with experiments. The MB components are due to high-frequency (- 10” Hz) thermal waves and, therefore, the dispersion must be taken into account in a quantitative description of the ratio / R / 2 / M B . This was precisely the calculation that Fabelinskii [ 19561 carried out and which yielded a nonthermodynamic expression for /ad. The quantity /is is related to long wavelengths, and therefore we can use eq. (5) and retain the relationship (84), bearing in mind that = [pu’ (a)] -I. Therefore, expression (84) allows partly for the dispersion via the dispersion of the velocity of sound. In view of this, the value of /ad becomes smaller and eq. (84) better describes the experimental results. Nevertheless, a better agreement between theory and experiment still needs to be achieved. The dispersion of the velocity of sound, which originates from relaxation of the bulk viscosity, reduces the value of /ad, but - as demonstrated by Rytov [1957] - such relaxation creates a new region of the spectrum of scattered light which is called the compression wing. This wing is superimposed on the central component, and consequently, if the experimental compression wing is sufficiently narrow, the value of / R I 2 / M B increases and approaches the experimental results. Therefore, the dispersion of a medium must result in some redistribution of the intensity in the scattered-light spectrum in such a way that the contribution subtracted from 2 1 M B is added to I R . In Mountain’s [ 19661 theory of the distribution of the intensity in the spectrum of light scattered in a dispersive medium as a result of the density fluctuations, the following expression describes the ratio of the intensity at any frequency: (y- l)/y+A/C

B/C where

In the high-frequency limit, u q t >> 1, it is found from eq. (86) that

111, § 61

SPECTRA FROM A€(€,*)

151

and apart from the factor C,the formula (86) coincides with that derived earlier by Rytov [1957]. Here uo and u, are the velocities at zero and extremely high frequencies, respectively, and other notations are as defined earlier. In Mountain's formula (87) the factor C was absent. The author of the present review introduced this factor in order to obtain the correct result (eq. 87) in the limit I2 4 0. The value of L depends on the substance. For water it is most probably the highest and equal to 1.7; for other substances, it varies from 1 to 1.5.

0 6. Spectra of Molecular Light Scattering Arising from Anisotropy Fluctuations A&(Eik) Anisotropy fluctuations in a condensed medium consisting of anisotropic molecules originate under the influence of thermal motion. This motion can lead to the formation at some particular moment of small regions inside of which have gathered the molecules that have a polarizability in one arbitrary direction greater (or smaller) than the average one. Chaotic motion aligns the anisotropic molecules for a very short time, approximately in the same way as an electric field aligns the molecules for a long time (Kerr effect). The formation and dispersion of anisotropy fluctuations is closely related to the orientation of the molecules due to Kerr effect. The orientation and disorientation time, or, in another words, anisotropy relaxation time, is the same in both cases. The only difference is that in the Kerr effect the molecules are aligned by an electric field and misaligned by thermal motion, while in the case of anisotropy fluctuations, the alignment and misalignment are both due to the thermal molecular motion. 6.1. SPECTRUM OF DEPOLARIZED LIGHT SCATTERED IN LIQUIDS

The depolarized light scattered by the anisotropy fluctuations has been discovered in the spectrum of scattered light in the form of a fairly wide band with a maximum that coincides with the position of the frequency of the exciting light and which falls on either side, but extends by 100-150cm-' or further and is usually called the wing of the Rayleigh line. This effect was discovered simultaneously by French and Indian physicists in 1928. Attempts to account for the physical origin of the depolarized light in the spectrum had been made by French, Indian and Russian physicists, but the explanations proposed up to 1934 had been incorrect and will not be discussed here (see Fabelinskii [1968] and the literature cited there).

152

SPECTRA OF MOLECULAR SCATTERING OF LIGHT

W, P

6

The origin of the Rayleigh line wing was explained correctly by Landau and Placzek [ 19341 in a paper dealing with a completely different topic. In this paper the Rayleigh line wing is referred to only once as follows: “in the case of liquids the Debye relaxation time creates a definite role in the structure of this part”. This first indication of the relaxation origin of the Rayleigh line wing has played its role and has been significant. The first quantitative relaxation theory of the spectrum of light scattered in a viscous liquid, based on the Maxwellian theory of viscosity, was proposed by Leontovich [1941]. This simplified, but still quite complex theory, yields expressions describing the distribution of the intensity in the MB components due to the longitudinal and transverse thermal waves, as well as the distribution of the intensity in the Rayleigh line wing. Leontovich’s calculations also give the degree of depolarization of the scattered light as a function of the frequency. One of the simplifications in Leontovich’s theory is that it postulates the existence of just one anisotropy relaxation time t. There are also other simplifications in this theory, but they will not be discussed, because use will be made of the results of more general theories. It should, however, be pointed out that the advantage of Leontovich’s theory is its physical clarity in the description of the nature of the phenomenon and a demonstration of how this phenomenon is linked to other physical effects. Leontovich’s expressions for the distribution of the intensity in the Rayleigh line wing in the case of excitation of the scattering by natural light are as follows: 7A2kT 2 t rv ( w ) = 12p 1+w222’

where I v ( w ) and I H ( W ) are the intensities of the scattered light (considered as a function of frequency measured from the frequency of the exciting light) with the vertical and horizontal polarizations, respectively, A is a quantity related to the Maxwellian constant A4 by M = A t ; p is the shear modulus. It follows from eqs. (88) and (89) that the depolarization coefficient is p = I H / l v = 6/7, in agreement with the value given above. Figure 19 shows the distribution of the intensity of the far part of the Rayleigh line wing (RLW) of salol obtained at three different temperatures (Fabelinskii [ 19681). The theoretical expressions (88) and (89) provide rational ways of comparing theory with experiment. One such way, proposed by Fabelinskii [1945], is as

153

SPECTRA FROM AE(5,t)

300

250200-

7

150 100

\

50

%, 0

1 AV

cm-'

Fig. 19. Distribution of the intensity in the far-frequency part of the Rayleigh line wing obtained for the scattering in salol at various temperatures: ( I ) 120°C; (2) 20°C; (3) 0°C (Fabelinskii [1955]).

follows. The reciprocal of the intensity f-'(o)is plotted along the ordinate and u2along the abscissa. Then the equation for the straight line can be written as:

I-'

(0) =c

+co2t2.

This straight line readily yields

'={

d[I-' (o)] cdo2 '

]

where l/c is I at o = 0 . It is therefore possible to easily determine the relaxation time of the anisotropy fluctuations if the simplified theory is not in conflict with the experimental results. Figure 20 shows schematically the results usually obtained experimentally for the RLW of various liquids over a wide frequency range. This typical graph shows immediately that there are two regions with linear dependence: AB and BC. It follows from this experimental result that the RLW

154

SPECTRA OF MOLECULAR SCATTERING OF LIGHT

I“

[III, 9: 6

t

Fig. 20. Schematic dependence of I-’(o) on w2 in the Rayleigh line wing in typical cases. ExplanatIons are provided in the text.

has at least two anisotropy relaxation times: tl (corresponding to AB) and z2 (BC), where zl > z2. The region CD cannot be described by a straight line and very different explanations of this region have been put forward. Fisher [1981] assumes, like many others, that this far part of the RLW is described by an exponential frequency dependence of the intensity. Starunov [ 1963, 19651 and others assume that the region CD can be described by a power function. The question cannot be regarded as finally resolved, because no reliable experimental investigations of this part of the spectrum have yet been made. In the region CD, the intensity is very low, and therefore within the limits of considerable error of such measurements, both points of view can be “confirmed”. The anisotropy relaxation times for the region AB and BC can be determined quite reliably by the above method as reported by Fabelinskii [1945]. It should be stressed once again that the anisotropy relaxation times are determined along the spectrum on the assumption that only one physical process, which is the anisotropy relaxation, is responsible for the RLW. However, in general, there maybe some other processes which contribute to the spectrum of the depolarized light scattered in liquids. Determination of the anisotropy relaxation time from the spectral line width is an indirect method and so far there have been no direct measurements which would have raised doubts about the values of zl and ~2 obtained from the spectrum of depolarized scattered light. Nonetheless, many such doubts

111,

I 61

155

SPECTRA FROM A E ( & )

Table 1 Room-temperature anisotropy relaxation times, tl and rz, determined indirectly and directly Deduced from spectra

Liquid

10'2t' (s)

Carbon disulfide Nitrobenzene

Toluene Benzene Chlorobenzene

2.4

10'27, (s)

1 0 ~ ~ 5(s) 2

0.21

44.6-3 9

4.I

Direct measurements

3.6; 1

2

f

0.5

47.4

10'*72 (S)

0.24 3 1A f 2 . 0

0.17

5f1

-

3.3

0.24

4f0.5

-

~

~

-

6.3f0.3

have been expressed and the results obtained have even been rejected outright. However, there is no need to consider these arguments here, because direct methods of determination of z became available a quarter of a century ago, and have produced results which are in surprisingly accurate agreement with the values found from the spectra of depolarized light scattered in liquids. The optical methods for the direct determination of the anisotropy relaxation times began with the work of Duguay and Hansen [ 19691 and are now conducted on a wide scale. Reviews of this topic were written a long time ago (Pesin and Fabelinskii [1976]) and books on the subject are available. The optical methods for the direct determination of the anisotropy relaxation time are in most cases based on the optical Kerr effect observed for the first time by Mayer and Gires [I9641 and also by Maker, Terhune and Savage [1964]. These can be described as follows: a short intense light pulse induces birefringence in a liquid of anisotropic molecules, in a manner similar to that induced by a static electric field (Kerr effect). The optical Kerr effect can therefore be used to construct an optical shutter that acts during an intense light puise. The use of intense picosecond light pulses has made possible the direct determination of the anisotropy relaxation time, called 51 above, determined from the region AB of the spectrum (fig. 20). Relatively recently it has become possible to measure t2 with the aid of femtosecond light pulses, but so far the measurements are limited to the liquids listed in table 1. The results given in this table are taken from reviews and original papers (Lotshaw, McMorrow, Kalpouzoz and Kenney-Wallace [ 19871). The direct and indirect measurements of 51 are in good agreement. There is also a good agreement for z2 in the case of carbon disulfide studied on different occasions, but a strong divergence between the results of direct and indirect measurements of 52 of nitrobenzene (Starunov

156

UII,

SPECTRA OF MOLECULAR SCATTERING OF LIGHT

Q

6

[ 19631) and those obtained with femtosecond pulses (Lotshaw, McMorrow, Kalpouzoz and Kenney-Wallace [ 19871). Further work will show whether this reflects a physical process or a difference resulting from imperfections of the measurements. The good agreement between the direct and indirect measurements of t I makes it possible to use the spectrum of depolarized light scattered in various media to study relaxation processes and to find T I . Development of femtosecond pulse methods should perform a similar task in the case of t2. Fast processes can be studied by optical switches right down to times T I governing the response time of such switches. Ordinary organic liquid can be used in the range of times defined by t land t2. The best liquid for short times is carbon disulfide for which the permissible and 2 x time lies between 2 . 4 ~ s. The response of a femtosecond pulse in the region CD (fig. 20) is not yet clear, because the physical origin of this s this is a spectrum has not yet been identified. For pulses longer than relatively simple task; namely one should select liquids such as nitrobenzene ( T IM 5 x lo-’ I s) or even more viscous media for which tl can be 10-6-1 0-5 s. Selection of the optical switch material for times shorter than s is more difficult, but it can be done. In the latter case, one must rely on the electronic relaxation times of the switch, s, and these are the times expected for glasses and some other materials in which the Kerr (Voigt) effect is governed by the electronic polarization but not by the orientational effect. A natural and important question arises as to the microscopic nature of the various parts of the RLW shown schematically in fig. 20. Since the whole spectrum of depolarized light scattered in various media is under intensive investigation, new contributions will be made to the understanding of the nature of the wing, but at this stage the most satisfactory and physically clear is the explanation proposed by Starunov [1965], which can be described as follows. Any anisotropic molecule in a liquid is in a potential well created by the environment of its neighboring molecules. This molecule in a potential well exhibits at least two types of thermal motion, one of which is due to rotational diffusion with jumps to other potential wells. Such motion of anisotropic molecules is relatively slow and modulates the scattered light which contributes to the region AB (fig. 20). The relaxation time z I determines the rotational diffusion time. In the interval between two diffusion rotations, the molecule in a potential well executes, under the influence of thermal effects, librations (vibrations in a potential well), which lead to “fast” modulation of the scattered light and make their own contributions to the region BC in the spectrum (fig. 20), so that t2 is a characteristic (“period”) of the librational motion.

-

111, 9: 61

SPECTRA FROM A E ( € , ~ )

I57

The scattering in liquids is not on single molecules, but on anisotropy fluctuations, so that tl and t2 should be regarded as the effective measures of the processes described here. 6.2. DETECTION OF THE DOUBLET STRUCTURE OF THE SPECTRUM

Experimental investigations of the frequency distribution of the intensity of light scattered on the anisotropy fluctuations, carried out with the aid of spectroscopic apparatus with low resolution and with excitation sources which have wide spectral lines, have provided only a general picture of the phenomenon. It would be pointless to use such instruments as the Fabry-Perot interferometer to study the RLW of liquids such as benzene, toluene, and many others with short relaxation times T I and t2 and, consequently, wider spectral regions. But in the case of liquids with a narrow region AB (fig. 20), a Fabry-Perot interferometer could be used to find tl and its dependence on the viscosity (temperature). A Fabry-Perot interferometer is used with the free spectral range A v z I cm-' in investigations of this kind to observe dearly the MB scattering components. In the case of benzene, toluene, etc., there was a continuous background to I H and in case of Iv an MB component was observed against the continuous background. The spectra of light scattered in quinoline, salol, benzophenone, etc. were quite different. In these liquids the spectrum of / H had a weak background and a very strong line, whose width was of the order of the spectral width of the exciting light. For 1" there was also a strong line lying within the MB doublet. This strong line was due to the region AB (fig. 20) of RLW. The anisotropy relaxation time of such liquids, found from the region AB (fig. 20) would be one or even two orders of magnitude longer than t2 found from the region BC. According to the initial relaxation theory of the spectrum of light scattered on anisotropy fluctuations, proposed by Leontovich [ 194I], the anisotropy relaxation time is proportional to the shear viscosity and inversely proportional to the absolute temperature T : t=

rl v--, kT

where V is a constant (according to Leontovich, it is the volume of a molecule) and k is the Boltzmann constant. The first experimental determination of t (Fabelinskii [1945]) and of its temperature dependence has shown that there is a qualitative agreement with eq. (91), but - for example in the case of salol - there is no quantitative

I58

SPECTRA OF MOLECULAR SCATTERING OF LIGHT

W, I

6

agreement. Between 20 and 170°C the shear viscosity varies by a factor of 80 and t calculated from eq. (91) varies by a factor of 120, whereas the experiments show that in this temperature range z changes by a factor of just -4. In calculation of z from eq. (91) it is hardly valid to substitute the shear viscosity Q measured in a static experiment, as is usually done, because the formation of the TUW is related to high frequencies at which the shear viscosity may change considerably because of relaxation. Further experimental investigations of RLW have been hindered by the considerable width of the exciting line from the mercury spectrum generated in high-pressure lamps. The width of the exciting line is greater than the width of the emitted (scattered) line. Nonetheless, investigations of the RLW have been continuing. Reviews of the results obtained can be found in books by Fabelinskii [1968] and by Vuks [1977], but new data have been obtained only when satisfactory light sources have been used, such as the mercury lamp with the I9’Hg isotope (Flubacher, Leadbetter, Morrison and Stoicheff [ 19591). The traditional light sources have been replaced with lasers emitting intense lines with a degree of monochromaticity that could not have been even dreamed of by the earlier experimentalists. The situation in the case of the narrow part of the RLW has also changed drastically. Our first interferometric experimental investigations with a laser (Mash, Starunov, Tiganov and Fabelinskii [ 1964]), which immediately gave new results on the width of the MB component, have revealed a very narrow part of the RLW of liquid nitrobenzene (Starunov, Tiganov and Fabelinskii [1966]). A more detailed experimental study of this part of the wing in nitrobenzene and quinoline made it possible to observe a new effect in the spectrum of depolarized light scattered in these liquids. In our first paper on this topic (Starunov, Tiganov and Fabelinskii [1967]) we stated right at the beginning: “the spectrum of thermal depolarized scattering of light (Rayleigh line wing) revealed a new phenomenon in which the x component, IvH(w), in this spectrum is split into two. The separation between the components of this doublet is considerably less than the separation between the MB components.” Our first explanation of this new phenomenon was that light scattered by the anisotropy fluctuations caused by shear strains is modulated by the Fourier component of these strains or, in other words, by strongly damped transverse acoustic waves. In our first experiments the detector was a photographic plate. We subsequently acquired electronic means for recording the spectra. Figure 2 1 shows a record of the fine structure of TUW in salol (Kovalenko, Krivokhizha and Fabelinskii [1993]) and fig. 22 gives the spectra of the molecular scattering of light in liquid aniline, recorded for different polarizations of the scattered light

159

SPECTRA FROM A&(&,)

Lp

,

,

-0.20

,

.

-0.10

,

,

0

,

,

0.10

,

I

,

c

0.20 cm.’

Fig. 21. Fine structure (doublet) of Rayleigh line wing of salol at 76.7”C (Kovalenko, Krivokhizha and Fabelinskii [1993]).

and showing all the singularities reported previously (Fabelinskii, Kolesnikov and Starunov [ 19771, Kolesnikov [ 19771). Here, IVV represents the MB components, IVHis the fine structure of the Rayleigh line wing, and finally IHH shows some singularity at the frequency of the MB components associated with the interaction between the longitudinal acoustic modes and the orientational motion in liquids (Stegeman and Stoicheff [1968, 19731). The observed doublet structure of the wing had been unexpected and even gave rise to a misunderstanding, primarily because the presence of a structure (two broad lines) in the RLW should mean that the scattered light is modulated by a process with nonmonotonic time dependence. This process can be the propagation of strongly damped transverse sound, as mentioned above. However, since the doublet structure of the RLW is observed in liquids with viscosities in the range 2-4 CP, it is clear that the hypothesis of the propagating of transverse sound is in conflict with classical hydrodynamics, which predicts that at the acoustic wavelength A the absorption coefficient a of transverse sound in such a medium is 2n and, consequently, such sound cannot propagate. According to hydrodynamics, such a medium can support only Newtonian “viscous waves” or, in other words, exponentially decaying sound for which the arguments of the exponential function aA is 2 n (Landau and Lifshitz [1987], Frenkel [1946]).

160

SPECTRA OF MOLECULAR SCATTERING OF LIGHT

0.3 0.2 0.1

0.1 0.2 0.3

cm-1

Fig. 22. Spectrum of the molecular scattering of light in aniline, obtained for various polarizations. Explanations are provided in the text. (Kolesnikov [ 19771.)

Such a modulating function cannot give rise to a doublet in the spectrum and can only alter the width of the spectral band or line. Note that the name “viscous wave” is quite arbitrary because the amplitude of such a “wave” decreases by a factor of 535 in one wavelength and one cannot speak of wave-like motion. Continuation of the experimental investigations definitely confirmed the existence of a doublet, an essentially new and unusual phenomenon. Over a year after our first paper (Starunov, Tiganov and Fabelinskii [ 1967]), Stegeman and Stoicheff [ 19681 confirmed our observations and explanation of the nature of the phenomenon in the case of nitrobenzene, quinoline, aniline and m-nitrotoluene. This new phenomenon has attracted the interest of experimentalists and theoreticians working in a number of laboratories and countries. We shall give only some of the results of the experimental investigations of the fine structure

SPECTRA FROM Ae(€,t)

b

I

15

10

I

I

L

I

0 5 10 FREQUENCY, GHz

5

t

15

Fig. 23. Fine structure of the Rayleigh line wing in quinoline for scattering angles from 8=45" to H = 176O (Stegeman and Stoicheff [1973]).

of RLW for liquids consisting of anisotropic molecules with viscosities which are in approximately the same range. Figure 23 shows the fine structure of quinoline as a function of the scattering angle 8 (Stegeman and Stoicheff [1973]) and we can see that the pronounced doublet disappears at high scattering angles. The fine structure of the wing has also been studied in benzyl alcohol (Gross, Romanov, Solov'ev and Chernyshova [ 1969]), in a-bromonaphtalene and triphenylphosphate at 70.2"C for the scattering angle 8=90" and A= 514.5 nm; the fine structure disappears at 41.6"C and 8=90" (Tsay and Kivelson [1975]). The temperature dependence of the fine structure has also been revealed clearly by an experimental study in the specific case of anisaldehyde (Alms, Bauer, Brauman and Pecora [ 19731) between 79 and 6°C (fig. 24). If the fine structure does indeed appear as a result of modulation of the scattered light by a damped sound wave at 79"C, it would seem that it should be even more readily visible at the lower temperature of 6"C, whereas the experimental evidence shows that the reverse is true. This result is in conflict with hydrodynamics and cannot be reconciled in any way with the Maxwellian scheme of viscosity. A clear fine structure of the RLW has also been observed in acetophenone at 15°C (fig. 25), in salol at 81°C (curve 1 in fig. 26), and in many other cases. There is no space to give all the evidence and there is no need for this. In the case of a substance consisting of anisotropic molecules, such as carbon

162

SPECTRA OF MOLECULAR SCATTERING OF LIGHT

I

-3

I

I

0

3

GHz

[111, 9: 6

Fig. 24. Temperature dependence of the fine structure of Rayleigh line wing of anisaldehyde obtained in the interval from 79°C to 6°C (Alms, Bauer, Brauman and Pecora [1973]).

Frequency, GHz Fig. 25. Fine structure of the Rayleigh line wing of acetophenone (Sixon, Bezot and Searby (1975)l.

SPECTRA FROM A&(€,k)

163

M

Fig. 26. Dependencies I I / H ( w )obtained for salol at various temperatures (for various viscosities): ( I ) 81°C; (2) 36OC; (3) 15°C; (4) 20°C. (Kovaienko, Krivokhizha and Fabelinskii [1993].)

disulfide, there is no fine structure at room temperature and the intensity falls monotonically on either side of the maximum located at the unshifted frequency. Enright and Stoicheff [ 19741 carried out a careful and detailed investigation of the RLW IVH(W)at nine different temperatures between 162 and 192 K, and did the same for I H H ( W at ) six different temperatures. In the case of IVH(O)the fine structure became less clear as the width of the depolarized spectrum increased. Figure 27 shows the spectrum of CS;! obtained at two different temperatures at which the fine structure could be observed. This result is very important because it shows that a liquid consisting of anisotropic molecules can be in a state in which its viscosity lies within an interval of -( 1-2)x P, when this scatteredlight spectrum exhibits the fine structure of the RLW. This is evidence that the observed fine structure is of general nature. Enright and Stoicheff assume, with

164

SPECTRA OF MOLECULAR SCATTERING OF LIGHT

I

I

30

20

I

I

I

I

10 0 10 20 FREQUENCY, GHz

Fig. 27. Fine structure of the Rayleigh line wing of CS2 at two temperatures (Enright and Stoicheff [1974]).

good grounds, that the fine structure may also be observed in such simple liquids as nitrogen and oxygen. At the beginning of the experimental investigations the very first observation of the fine structure of the RLW provided the correct but purely qualitative explanation of the physical nature of the observed effect (Fabelinskii and Starunov [ 19671, Fabelinskii, Sabirov and Starunov [ 19691). A quantitative description of this fine structure is now needed. It is not permissible to calculate the anisotropy fluctuations in the same way as in the case of the density, entropy, and concentration fluctuations because the anisotropy is not a thermodynamic parameter but an internal one. The anisotropy fluctuations and the spectrum of light scattered on such fluctuations must be calculated by a completely different procedure. The first quantitative, simplified, but still very complex calculation of the spectrum of the light scattered on the anisotropy and pressure fluctuations was published by Leontovich [1941], as mentioned above. He obtained formulas for I ( o ) in the region near the exciting line. We shall not repeat the whole complex procedure of obtaining the results in question (they can be found in the book by Fabelinskii [1968]). The final result will be given. If the vector E of the incident linearly polarized light lies in a plane perpendicular to the scattering plane, then:

111,

ii 61

SPECTRA FROM A t ( € , r )

I65

If the same vector E lies in the scattering plane, then:

Here, QL = 2 q u ~ ,QT = 2 q u ~ Q , ; = Q: + (4Q;/3), A = M l z , where M is the Maxwellian constant, p is the shear modulus, the frequency w is measured from the frequency of the exciting light, U L = (/?sp)-’l2,,& is the adiabatic / ~ . Maxwellian scheme yields the following compressibility, and UT = ( p / ~ ) ’ The relationship between the viscosity and elasticity:

There are two terms in eq. (92): the first represents the discrete components at , the second is superimposed on the shifted components and frequencies + d 2 ~and has its maximum at the unshifted frequency (o= 0). Under certain conditions (at a specific temperature) the predicted profile can resemble the doublet structure presented in figs. 21-27. This is the reason why we assumed that the doublet structure observed by us in the near part of the wing can be explained by Leontovich’s theory [1941] and expression (92), which follows from it. This opinion was shared by other physicists (Stegeman [ 19691, Rozhdestvenskaya and Zubkov [1970]). The same misunderstanding can be found in the book by Berne and Pecora [1976]. This interpretation of the experimental results can be confirmed or rejected by further experiments and, in particular, by determination of the temperature dependence of the maxima positions of the doublet components. According to Leontovich’s formula (92) cooling and increase in the shear viscosity (when the relaxation process “converts” the viscosity into elasticity) should increase the high-frequency shear modulus p, velocity of sound uT, and, therefore, QT. Consequently, an increase in the velocity should increase the separation between the doublet maxima and finally at very high viscosities and in glasses there should be a clear triplet with two narrow shifted components and the central one. This spectrum should superficially resemble the spectrum of light scattered by the pressure and entropy fluctuations, but the origin of this spectrum is completely different and represents depolarized light. These predictions follow from Leontovich’s [ 19411 formula (92). However, the first experimental results obtained in our studies (Fabelinskii, Sabirov and Starunov [ 19691, Sabirov [ 19701, Fabelinskii and Starunov [ 19721, Sabirov, Starunov and Fabelinskii [ 19711, Starunov and Fabelinskii [ 19741) produced

I66

SPECTRA OF MOLECULAR SCATTERING OF LIGHT

[ I K ii 6

nothing like this, and subsequent experiments simply confirmed the earlier results. In an experimental study of the temperature dependence of the frequency positions of the maxima of the doublet components of the doublet observed in the RLW (Starunov, Tiganov and Fabelinskii [1967]) it is necessary to select a liquid that consists of molecules as anisotropic as possible and with a shear viscosity that can be varied between the widest possible limits by altering the temperature of the liquid. On the basis of my previous experience, I selected salol for this purpose. It shows clearly the doublet structure of the Rayleigh wing and its viscosity (after suitable purification) can vary from a very low value to that of the glassy state. If the separation between the maxima of the doublet components (lines) is denoted by ~Av,,,, cooling (increase in the velocity) should make it possible to follow the changes in ~Av,,,. Where Leontovich's theory predicts an increase in 2Avv,,, when the velocity rises, our experiments gave a directly opposite result. An increase in the velocity reduced somewhat ~Av,,, (Fabelinskii, Sabirov and Starunov [1969]). The same effect was reported by Stegeman [1969]. This was an unexpected and surprising result. Even more surprising was the behavior of the investigated spectrum at still lower temperatures. Between 120 and 46°C a doublet was observed but at +45"C the doublet structure disappeared and there was no RLW structure of salol between -2 and +45"C; the profile was a smooth dome. This behavior of the spectrum was not predicted by Leontovich's theory or by any other theory available at the time. In our experiments we continued to observe the spectrum when the temperature was reduced even further in a continuous manner. At -2.5"C, a nonmonotonic dependence appeared on the sides of the smooth dome and then clear lines appeared; the value of ~Av,,, increased as the viscosity became higher and finally a clear triplet was observed. The appearance of the side components followed exactly the behavior predicted by Leontovich's theory, at least in the qualitative sense. Figure 26 (curves 3 and 4) shows how the theoretically predicted and the experimentally observed triplet behaves in the case of the spectrum of light scattered in salol at -20°C (Kovalenko, Krivokhizha and Fabelinskii [ 19931). The narrow shifted components are evidence of weak damping of a transverse acoustic wave in salol at -20°C and the narrow central line indicates that the anisotropy relaxation time z is long compared with the value of T at a temperature which is higher, but still such that a triplet is observed. The spectrum of depolarized light had also been studied for the scattering and ] this was done over a wide range of shear in benzophenone [ ( C ~ H S ) ~ C O

I67

SPECTRA FROM A€(&)

111, § 61

viscosities. It was found that the changes in the spectrum with temperature were exactly the same as in the case of salol. Initially, between 1 15 and 15"C, the fine structure of RLW was clearly visible, but between + I 5 and -2.5"C it was no longer detectable. At -2.5"C a triplet appeared and persisted at temperatures as low as -55"C, exactly as described earlier for the case of salol. In these experiments on benzophenone the shear viscosity varied by 12 orders to lo9 P (Sabirov [1970]). of magnitude from When the experimental results are replotted so that the ordinate gives ~Av,,, and the abscissa represents temperature, the results are as shown in fig. 28a for salol and fig. 28b for benzophenone. We can see clearly two branches of

~ A v , , cm-' 0.3 0.2 0.1

-,

T, "C

T,"C Fig. 28. Temperature dependencies of the positions of the maxima (2AvmaX)of a doublet and triplet in the spectrum of Light scattered in (a) salol and (b) henzophenone (Fahelinskii, Sabirov and Starunov [ 19691).

I68

SPECTRA OF MOLECULAR SCATTERING OF LlGHl

15

10

5

0

5

FREQUENCY, GHz Fig. 29. Spectra of light scattered in quinoline. Curve C is the difference between curves A and B (multiplied by 5 ) and reflects the interaction between the acoustic and orientational modes of motion. (Stegeman and Stoicheff [ 19681.)

each spectrum which are practically identical for both substances. At higher temperatures, the points correspond to the value of ~AY,,, for the doublet spectrum and this is quite arbitrarily called the “high-temperature’’ branch, whereas the points lying at lower temperatures represent 2Av,,, corresponding to the positions of the displaced components in the triplet spectrum. This is called the “low-temperature” branch. The continuous curves represent the optimal representation of the experimental points. One further nontrivial observation in the spectrum of molecular scattering of light should be mentioned: it was first reported by Stegeman and Stoicheff (Stegeman and Stoicheff [ 1968, 19731, Stegeman [ 19691). This observation is related to I H H ( w ) . Figure 29 shows its nonmonotonic behavior in the range of frequencies corresponding to the superposition of the MB components. A more detailed experimental investigation of this region showed that, in accordance with the predictions of Leontovich’s theory (eq. 93), the longitudinal acoustic and orientational motion modes interact: the curve is similar to the anomalous dispersion curve in fig. 29.

111, $ 61

SPECTRA FROM A t ( € , n )

169

This is the experimental behavior of the spectrum of molecular scattering of light by the anisotropy fluctuations, specifically the behavior of the RLW, and these are the qualitative and simplified explanations of this behavior.

6.3. GENERAL AND SIMPLIFIED EQUATIONS DESCRIBING THE SPECTRA OF LIGHT SCATTERED IN LIQUIDS CONSISTING OF ANISOTROPIC MOLECULES

As already mentioned above, the first theory published in 1941 was developed by Leontovich. At the time the author of the present chapter had accumulated experimental results that were compared with Leontovich’s theory but were not published. World War I1 had begun in Russia, in Europe it had begun even earlier and physicists had to leave their beloved jobs. The first general theory of the molecular light scattering was developed by Rytov and published in 1957. Rytov’s theory was based on the application of the fluctuation-dissipation theorem and predicted new spectral features named by Rytov as a “compression wing” (later discovered experimentally) and a “shear wing”. However, Romanov, Solov’ev and Filatova [ 19701 pointed out the imperfections of this theory; the criticism was taken into account by Rytov and in 1970 he published the new general theory. The discovery of the fine structure of the Rayleigh wing (FSRW) by Starunov, Tiganov and Fabelinskii [ 19671 attracted attention to this field. In 197 1, on the threshold of a Paris conference on light scattering spectroscopy in liquids, Keyes and Kivelson [1972], who themselves made a significant contribution to the field, pointed out that eight theories explaining the FSRW had been proposed: Leontovich [ 19411, Mountain [ 19661, Volterra [ 19691, Rytov [ 1970a,b], Andersen and Pecora [ 19711, Ben-Reuven and Gershon [ 19711, Keyes and Kivelson [ 19711, and Zamir and Ben-Reuven [1972]. The authors of these theories, and authors of others who proposed different theories of the fine structure, have since developed their proposals and suggested different variants. It is not my aim to provide any complete critical account of these theories. These theories can be divided arbitrarily into two groups: purely phenomenological theories and statistical (if the term can be used) theories. The phenomenological theories are based on the equations of motion and the fluctuation-dissipation theorem, whereas the statistical theories rely on calculations of the methods developed by Keyes and Kivelson [1971] and others cited above. Theoretical and experimental investigations have continued (Bezot, Hesse-Bezot, Ostrowsky and Quentrec [ 19801, Chapell, Allan, Hallem and Kivelson [ 198 I]).

170

[IK9

SPECTRA OF MOLECULAR SCATTERING OF LIGHT

6

The most general formulas describing the molecular light scattering spectra for various polarizations are those which follow from Rytov’s [ 1970al phenomenological theories and will be given here: - 2YC

+ 1ZKa)

1

+ Y 2Cq2 - 2 YZK aq2 - Z2 ( A + Bq2) +Xf+ m2- E 2- - c . nc: . i j 3 k A k

(95)

(96) cos 8

1

+ W (YC - ZKa) cos 0

.

(97)

Here, A

=

pq2-pow2,

B = K + -iu, 3

c = -L TO (poCv -

g)

,

where p is the shear m o d u s , po is the L,nsity, K is the thermal con.xtivity, K is the bulk modulus, and A = ( A + Bq2) C + K2aq2. In Rytov’s theory a fluctuation of the optical permittivity is described by:

Here, uup is the strain tensor, and m; and

nk

are real constants such that:

The above formulas provide a full description of the spectra of light scattered by the density and anisotropy fluctuations in a medium, which is on the whole

111,

I 61

SPECTRA FROM A E ( € , ~ )

171

isotropic. These formulas take account of the dispersion and any number of relaxation times of scalar and of tensor parameters. This is the most general solution of the problem, but the formulas given in eqs. (93497) are difficult to apply to experimental results. They contain many parameters representing quantities which, in principle, can be derived from independent experiments. These theoretical formulas adniit the possibility of any number of relaxation times of different origin. The Ivv(w) spectra have been discussed quite thoroughly above and the quantities which can be deduced from such spectra have been identified. Let us now consider in detail IVH(w,q) given by eq. (96). The experimental results can be judged and compared with the theoretical formulas if eq. (96) for IVH(0,q) is simplified taking into account the experimental results, namely that only two relaxation times tl and z2 are observed in a clear manner (fig. 20). Equation (96) can then be written as follows:

Here, p and k are the complex values of the shear modulus and of the magnetooptic coefficient, respectively:

The constants N and n, which occur in eqs. (98)-(loo), can be expressed in terms of physical quantities such as the static viscosity and the maximum shear modulus p = pu;, where or is the velocity of transverse sound, X can be expressed in terms of the Maxwellian constant M , and D = n:/ni is the ratio of the integral intensities of the spectral components with half-widths t;' and 'z; . If we know the six parameters needed for calculation on the basis of eq. (98) viz., q, 51,r2,pm( t ~ )D, , and A4 - then calculations of this kind can be carried out on a computer. In the case of salol and benzophenone this calculation was undertaken by Starunov and Fabelinskii [1974]. The results of this calculation will be discussed below. The fine structure of the Rayleigh line wing, but not of the whole spectrum, can be described by making further simplifications which are not in conflict with

I72

SPECTRA OF MOLECULAR SCATTERING OF LIGHT

[Wij

6

experiments: we can assume that n2 = 0 and also that T I >> r2 and w2tl z2 << 1 . Assumptions of this kind, which agree with experiments, modify eq. (98) for /VH(w,q) to

Here,

r = ti', where t M is the axwellian relaxation time (see eq. 94). Equation (10 ) is a simplified formula from the general theory of Rytov [ 1970al and is practically identical to the formulas derived from the statistical theories. An analysis of the situation as a whole led Starunov [1971, 19871 to the conclusion that the fine structure is the result of the interaction of orientational motion with shear strains, which gives rise to a scattered-light doublet and can be described by eq. (101). Although eq. (101) contains explicitly only the relaxation time r-' = T I = r, in fact the other times t2 and z M occur in this formula via the coupling parameter R. Equation (101) and its equivalents describe well the doublet structure of the spectrum, which can be seen quite clearly from figs. 2325 (the calculated results are represented by continuous lines). A similar good agreement is obtained in other cases, which are not cited here, but can be found in the literature (Berne and Pecora [ 19761). It should be stressed particularly that so far we have considered the scatteredlight spectra of liquids with viscosities such that the fine structure of the RLW is clearly visible in the form of a doublet or, in other words, there is a narrow dip at the exciting line frequency to=O. The very first experiments have shown that the fine structure disappears when the temperature is lowered and the viscosity increases; for example, the fine structure of anisaldehyde is no longer observed at 8 CP (Alms, Bauer, Brauman and Pecora [ 19731). When the experimental investigations have provided sufficiently diverse data on the dynamics of the spectrum of light scattered in materials, each of which has a wide range of viscosities from to 10" P and in various substances it has been found that the two branches in fig. 28 can be described by a single theory which is of a phenomenological nature and, therefore, does not pretend to

SPECTRA FROM A ~ ( t , k )

I73

~Av,,,,cm-'

+

0.31

01

b

-50

50

0

100

T,"C

Fig. 30. Temperature dependencies of the separation ~Av,,, between the fine-structure components

of the Rayleigh line wing of salol. Circles represent the experimental results. Curve I is plotted with the influence of instrumental function neglected and curve 2 is calculated on the basis of the Lorentzian instrumental function with a half-width of 2 x I O-* cm-' , (Starunov and Fabelinskii [ 19741.)

provide a description of the spectrum throughout the full range of the viscosities (Rytov [1970a]). Since the formulas deduced from the general theory contain six parameters, Rytov [1970b] found that these formulas can fully describe the phenomenon, or in other words, can account for the high- and low-temperature branches simultaneously, if these parameters are regarded as the fitting quantities. However, the parameters of Rytov's phenomenological theory have a physical meaning and can be found by independent experiments, so they cease to be the fitting parameters. The necessary six quantities for salol and benzophenone have made it possible to compare the experimental results with the theoretical formulas. It has been found that the situation is not as simple as one would like. All the necessary parameters and all the required details of the calculations can be found in the paper by Sabirov [1970]. Here we shall give graphically the results of the calculations and represent the experimental results by circles or other symbols. Figure 30 gives the data obtained for salol and fig. 31 gives the corresponding data for benzophenone. It is clear from these figures that the phenomenological and statistical theories both describe qualitatively and in a satisfactory manner the fine structure of the RLW.

174

[HI,

SPECTRA OF MOLECULAR SCATTERING OF LIGHT

2Av,,

56

cm-'

1, -50

2 0

T

50

0 100

T, "c

Fig. 3 I . Dependencies of the separation ~Av,,, between the maxima of the fine structure of Rayleigh line wing of benzophenone. Circles represent the experimental results. Curve I is calculated for U T =920ms-' and curve 2 for U T = 1200ms-'. The positions of the points are not very sensitive to the value of U T . (Starunov and Fabelinskii [1974].)

The situation is quite different for the low-temperature branch. In this case there is not even a qualitative agreement between eq. (98), predicted by Rytov's theory, and experimental results. This discrepancy between the theory and experiment is evidently of the same origin as the discrepancies which result from the use of relaxation theories with local derivatives to describe the propagation of sound in a viscous medium. It is well known that a relaxation theory with local derivatives does not predict the correct frequency dependencies of the velocity and absorption of sound. This difficulty was overcome by lsakovich and Chaban [I9661 in their theory of propagation of sound in viscous and highly viscous media. Their theory provides a satisfactory description of the temperature dependencies of the velocity and absorption of sound, but neither this nor similar theories have been generalized to describe the spectrum of the molecular light scattering. Some attempts to develop a theory of the spectrum of light scattered in a viscous liquid have been made, but they represent only a very early stage. Much work on the fine structure of the RLW has been done on low-viscosity liquids when a relaxation theory based on local derivatives can be applied (see, e.g., lsakovich and Chaban [ 19661, Barlow, Erginsav and Zamb [ 19721, Gershon

111,

0 61

SPECTRA FROM AE(5,k)

175

and Oppenheim [ 19741, Hynne [ 19801, Quentrec and Bezot [ 19801, Wang, Ma, Fytas and Dorfmiiller [ 19831, Wang and Zhang [ 19861, Tao, Li and Cummins [ 19921). As demonstrated above, the details of this phenomenon are much “richer” than predicted by hydrodynamics. The spectrum of molecular scattering of light includes doublets and triplets, depending on the state of an isotropic medium. The doublet structure of the spectrum is described well both by the phenomenological and statistical theories which contain such quantities as the velocity and the absorption of a transverse or shear elastic wave, so that the theoretical expression together with the experimental data make it possible to determine the principal characteristics of transverse sound, which are its velocity and absorption. A detailed analysis of this situation was made by Starunov [1971, 19871 (see also Starunov and Fabelinskii [1974]), who considered the velocity u and the absorption coefficient a at the wavelength A and the frequency QT, and thus derived expressions on the assumption that there are only two anisotropy relaxation times, TI and z2, and one Maxwellian T M , such that 51 > ZM > t2:

The above formulas make it possible to plot the dependence of aAl2n on Q ~ = 2 x f T , which is done in fig. 32. The arrow in this figure identifies the ordinate at which the doublet structure is observed. The dashed horizontal line identifies the value below which the modulating function is oscillatory, and consequently, discrete lines appear in the spectrum. If a N 2 n = 1, it follows - as shown above (see eqs. 48 and 49) - the discrete spectral lines cannot be observed. Figure 33 gives the temperature dependence of aAMn. The dashed line still gives the value of aA/2n below which the modulating function is oscillatory. In the region where the fine structure of the RLW or the doublet structure exists in the spectrum, the modulating function ceases to be exponential, but

I76

SPECTRA OF MOLECULAR SCATTERING OF LIGHT

1.5

0

3 .O

Fig. 32. Dependencies of a A I 2 n on the frequency s 2 =~2 n f T representing observations of light scattered at H = 90". An arrow identifies the value of a.4/2n at which the fine structure ofthe Rayleigh wing appeared. The horizontal dashed line is the level below which the modulation (correlation) function @(r, q) is oscillatory. (Stamnov and Fabelinskii [ 19741.)

0.251

25

50

I5

loo

T,"C

Fig. 33. Temperature dependence of a N 2 n for salol. Curve I corresponds to 0.9GHz and curve 2 corresponds to 3 GHz. (Starunov and Fabelinskii [1974].)

111,

I 71

I77

ABOUT SOME PROBLEMS

0, I~

i

~

~

~

.

~

-

l

U , 10” cm-’

T, “C Fig. 34. Temperature dependencies of ( I ) the velocity u and (2) the absorption coefficient a of hypersound. The continuous curves are results calculated on the basis of the theory (Isakovich and Chaban [1966]). Symbols connected by dashed lines are the experimental results (Kovalenko, Krivokhizha and Fabelinskii [ 19931).

does not yet become regularly oscillatory, and the region where a well-resolved triplet is observed corresponds to a regularly oscillatory modulating function. Figure 26 shows the /VH(U) spectrum of salol for different viscosities. An analysis of a triplet spectrum makes it possible to use the half-widths of the shifted components in order to determine the attenuation coefficient of hypersound in salol as a function of temperature. The shifts of the discrete lines from the exciting line can be used to deduce the temperature dependence of the velocity of hypersound. The results of such an analysis are presented in fig. 34. The continuous curves in this figure give the results of calculations carried out on the basis of the formulas given by Isakovich and Chaban [ 19661. The qualitative agreement between the experimental and theoretical results can be regarded as satisfactory, but there is no quantitative match and the task of developing a satisfactory theory of the effect remains to be done.

9

7. About Some Problems

The present review is devoted to the problems which have been investigated experimentally and theoretically, but which have not been reviewed in books

178

[III

SPECTRA OF MOLECULAR SCATTERING OF LIGHT

or papers comprehensively enough. The significant problems which have not been investigated theoretically or experimentally (or problems which have been predicted theoretically, but have not yet been examined experimentally) will be pointed out below. In a critical region in the process of a vapour-liquid or a homogeneousheterogeneous solution phase transitions the angular motion of anisotropic molecules remains practically unknown. In the light-scattering spectroscopy this is a problem of the influence of a critical state on the depolarized scattering spectra or on the Rayleigh line wing. Andreev [ 19741 has undertaken calculations which yield that in a critical region, if light is scattered by the symmetrical part of

-(

the anisotropy tensor I ( A E ~ ) ~ the ) , depolarized light must undergo critical opalescence. The equation defining the extinction coefficient and its spectral dependence has been derived. The predicted phenomenon has not yet been investigated experimentally. In some solutions, several authors have reported the observation of a narrowing of the Rayleigh line wing or of the depolarized Raman line, while other authors have not observed similar phenomena, or, in another words, a critical region did not affect depolarized light scattering (see the review by Fabelinskii [ 19941). In the theory of Chaban [ 1975, 19781 the narrowing of a Rayleigh line wing must obey the following law: 6w = a + bE", where E = ( T - T,)/T, and o is the critical index 0=0.8. The theory of Wilson [1974] also predicts the narrowing of a Rayleigh line wing, but it must obey the equation 60 = 6 0 , ( T - T,)". Here, D, = kT/8?cgr3.According to Wilson, in a critical region r must be substituted by r , E-0.63 and then 60 E" where 0 FZ 2. In the paper by Wilson, o = 312. In fact, only Wilson's paper [ 19741 contains a direct indication of the influence of the critical region of the solution exfoliation on the angular motion of anisotropic molecules in a solution, but this point of view did not experience any further development. The available experimental results can be described by the theories of Chaban and Wilson only in a purely qualitative way.

-

-

-

References Aleksandrov, V.V., M.V. Saphonov, V.R. Velasco, N.L. Yakovlev and L.Ph. Martynenko, 1994, J. Phys.: Condens. Matter 6, 3347. Aleksandrov, V.V., T.S. Velichkina, V.G. Mozhaev, Yu.B. Potapova, A.K. Khmelev and I.A. Yakovlev, 1992, Phys. Lett. A162, 418. Aleksandrov, V.V., T.S. Velichkina, I.V. Vodolazskaya, V.I. Voronkova, I.A. Yakovlev and VK. Yanovskii. 1990, Solid State Commun. 74, 749. Alms, G.R., D.R. Bauer, J.I. Brauman and R.J. Pecora, 1973, J. Chem. Phys. 59, 5304.

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[111

Lakoza, E.L., and A.V. Chalyii, 1983, Usp. Fiz. Nauk 140, 393. In Russian. Lallemand, P., 1974, in: Photon Correlation, and Light Beating Spectroscopy, Proc. NATO Adv. Study Inst. Ser. B Phys., Vol. 111, eds H.Z. C u m i n s and E.R. Pike (Plenum Press, New York). Lamb, H., 1932, Hydrodynamics (Cambridge University Press, Cambridge). Landau, L.D., and E.M. Lifshitz, 1958, Statistical Physics, 6th Ed. (Addison-Wesley, Reading, MA). Landau, L.D., and E.M. Lifshitz, 1987, Fluid Mechanics, 2nd Ed. (Pergamon Press, Oxford). Landau, L.D., and G. Placzek, 1934, Phys. Z. Sowjetunion 5, 172. Landsberg, G.S., and A.A. Shubin, 1939, Zh. Eksp. Teor. Fiz. 9, 1309. Larsen, G.A., and C.M. Sorensen, 1985, Phys. Rev. Lett. 34, 345. Law, B.M., P.N. Segre, R.W. Gammon and J.V. Sengers, 1990, Phys. Rev. A 41, 816. Leontovich, M.A., 1931, Z. Phys. 72, 247. Leontovich, M.A., 1935, Dokl. Akad. Nauk USSR 1, 97. In Russian; 1985, Selected Works (Nauka, Moscow). In Russian. Leontovich, M.A., 1939, Zh. Eksp. Teor. Fiz. 9, 1314. In Russian. Leontovich, M.A., 1941, J. Phys. USSR, 4, 499. Li, W.B., P.N. Segre, J.V. Sengers and R.W. Gammon, 1994, J. Phys.: Condens. Matter 6, A1 19. Lotshaw, W.T., D. McMorrow, G.A. Kalpouzoz and G.A. Kenney-Wallace, 1987, Chem. Phys. Lett. 136, 323. Ma, Sh.K., 1976, Modern Theory of Critical Phenomena (Benjamin, New York). Maker, P.D., R.W. Terhune and C.M. Savage, 1964, Phys. Rev. Lett. 12, 507. Mandelstam, L.I., 1913, Ann. Phys. (Leipzig) 41, 609. Mandelstam, L.I., 1926, Zh. Russ. Fiz. Khim. Ova. 58, 381. In Russian. Mandelstam, L.I., 1934, Dokl. Akad. Nauk USSR, 2, 219. In Russian. Mandelstam, L.I., 1950, Polnoe Sobranie Trudov [Complete Works], Vols. 1-5 (Izd. AN USSR, Moscow). In Russian. Mandelstam, L.I., and M.A. Leontovich, 1937, Zh. Eksp. Teor. Fiz. 7, 438. In Russian. Mash, D.I., V.V. Morozov, V.S. Starunov and I.L. Fabelinskii, 1965, JETP Lett. 2, 349. Mash, D.I., V.S. Starunov, E.V. Tiganov and I.L. Fabelinskii, 1964, Zh. Eksp. Teor. Fiz. 47, 783. In Russian; 1965, Sov. Phys. JETP 20, 523. Mayer, G., and G. Gires, 1964, C. R. Acad. Sci. 258, 2039. McFee, J.H., 1966, in: Physical Acoustics: Principles and Methods, Vol. 14, Part A, ed. W.B. Mason (Academic Press, New York). Mclntyre, D., and J.V. Sengers, 1968, in: Physics of Simple Liquids, eds H.N.V. Temperley, J.S. Rolinson and C.S. Rushbrooke (North-Holland, Amsterdam). Molchanov, V.A., and I.L. Fabelinskii, 1955, Dokl. Akad. Nauk USSR 105, 248. In Russian. Mountain, R.D., 1966, J. Res. Nat. Bur. Stand. Sect. A, 70, 207. Nieuwoudta, J.C., and B.M. Law, 1990, Phys. Rev. A 42, 2003. Ornstein, L.S., and F, Zernike, 1918, Phys. Z. 19, 134. Ornstein, L.S., and F. Zernike, 1926, Phys. Z. 27, 761. Oxtoby, D.W., and M. W. Gelbart, 1974, J. Chem. Phys. 61, 2957. Pagonabarraga, I., J.M. Rubi and L. Torner, 1991, Physica A 173, 1 1 1. Pesin, M.S., and I.L. Fabelinskii, 1959, Dokl. Akad. Nauk USSR 129, 299. In Russian; 1960, Sov. Phys. Dokl. 4, 1264. Pesin, M.S., and I.L. Fabelinskii, 1960, Dokl. Akad. Nauk USSR 135, 1114. In Russian; 1961, Sov. Phys. Dokl. 5, 1290. Pesin, M.S., and 1.L. Fabelinskii, 1976, Sov. Phys. Usp. 19, 844. Placzek, G., 1935, Rayleigh Scattering and Raman Effect (GNTIU, Kharkov). In Russian; 1934, Handbuch der Radiologie, Band IV, Teil 11 (Academische Verlagsgesellschaft, Leipzig).

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E. WOLF, PROGRESS IN OPTICS XXXVIl 0 1997 ELSEVIER SCIENCE B.V ALL RIGHTS RESERVED

IV SOLITON COMMUNICATION SYSTEMS BY

R E N ~ J E AESSIAMBRE N

' AND GOVINDP. AGRAWAL *

The Insfitute of Optics, Wniversiv of Rochester; Rochester NY 14627, USA

'

Now with Bell Laboratories, Crawford Hill Laboratory, Room HOH L-129, P.O. Box 400, 791 Holmdel-Keyport Road, Holmdel, NJ 07733-0400. email: [email protected] * Corresponding author. 185

CONTENTS

PAGE

4 1. INTRODUCTION . . . . . . . . . . . . . . . . . . .

187

OPTICAL SOLITONS IN FIBERS . . . . . . . . . . . .

188

SOLITON-BASED COMMUNICATION SYSTEMS . . . . .

194

4 4 . AVERAGE-SOLITON REGIME . . . . . . . . . . . . . .

199

4 5 . QUASI-ADIABATIC REGIME . . . . . . . . .

. . . . .

215

9 6. DISTRIBUTED AMPLIFICATION . . . . . . . 9: 7 . DISPERSI0N.DECREASINGFIBERS . . . . . .

. . . . .

219

. . . . .

221

9: 8. DISPERSION MANAGEMENT . . . . . . . . . . . . . .

227

4 9 . CHANNEL MULTIPLEXING . . . . . . . . . . . . . . 9: 10. DARK-SOLITON COMMUNICATION SYSTEMS . . . . . .

233

LIST OF SYMBOLS . . . . . . . . . . . . . . . . . . . .

247

LIST OF ACRONYMS . . . . . . . . . . . . . . . . . . .

251

9 2. 4 3.

r,

CFERENCES . . . . . . . . . . . . . . . . . . . . . .

I86

244

252

tj 1. Introduction

Solitons have been discovered in many branches of physics, including plasma physics, fluid dynamics, particle physics, solid-state physics, and optics. The word soliton was coined (Zabusky and Kruskal [1965]) in 1965 to describe the particle-like properties of pulse envelopes in dispersive nonlinear media. The existence of solitons in optical fibers was suggested (Hasegawa and Tappert [ 1973a1) in 1973, and such solitons were observed experimentally (Mollenauer, Stolen and Gordon [1980]) by 1980. The potential of solitons for long-haul optical communications was demonstrated in 1988 in an experiment in which fiber loss was compensated by using the technique of Raman amplification (Mollenauer and Smith [ 19881). Remarkable progress made during the decade of 1990s has converted fiber solitons into a practical candidate for the next generation of lightwave communication systems. Several terrestrial field trials making use of solitons are planned for 1998, and there is a possibility that solitons may be used for the next trans-Pacific cable (TPC-6) planned to operate at lOOGb/s by the year 2000. Fiber solitons and their applications have been covered in several review articles and book chapters (Kodama and Hasegawa [ 1992b], Agrawal [ 19921, Kodama, Maruta and Hasegawa [ 19941, Agrawal [ 1995a1, Hasegawa and Kodama [ 19951, Haus and Wong [ 19961). This review provides an up-to-date account of soliton communication systems with emphasis on the physics and the design of such systems. The basic concepts behind fiber solitons are introduced in 4 2 which also discusses the properties of such solitons. Section 3 describes how fiber solitons can be used for optical communications and 9 4 discusses the issues involved in the design of such systems when solitons are propagated over thousands of kilometers by using optical amplifiers. Sections 5-8 consider various techniques which can be used to improve the performance of single-channel soliton systems, whereas 9 9 considers channel multiplexing. The final section is devoted to the use of dark solitons for long-haul transmission, a topic still in its infancy.

187

188

SOLITON COMMUNICATION SYSTEMS

Q 2. Optical Solitons in Fibers

The peculiar importance taken by the nonlinear effects in optical fibers (made of silica glass that is only weakly nonlinear) can be attributed to its waveguiding geometry. Optical fibers (especially single-mode fibers) are able to strongly confine an electromagnetic field in the plane transverse to the fiber axis over a 6-8 pm core region. High optical intensities (-10MW/cm2 for pulse energies 10 pJ) can be maintained over tens of kilometers because of an extraordinarily low loss of silica fibers. Since most nonlinear optical effects scale upward with both the optical intensity and the distance of propagation, they become quite important in optical fibers, especially in long-haul fiber links that can stretch to thousands of kilometers.

-

2. I . NONLINEAR SCHRODINGER EQUATION

The wave equation governing the evolution of the optical field in silica fibers is derived from the Maxwell equations. The waveguiding nature of an optical fiber and its weak nonlinearity allow considerable simplifications while solving the Maxwell equations. In particular, a weak nonlinearity allows one to neglect the change in the transverse mode profile with the nonlinear change of refractive index. Consequently, the transverse profile of the field is determined by the radial variation of the linear part of the refractive index and remains fixed along the fiber axis. Only fibers supporting the fundamental HE1 1 transverse mode (also called the LPol mode because of its linear polarization) are considered, since only single-mode fibers are used for soliton transmission. To study soliton evolution along the fiber axis, we make use of the slowlyvarying-envelope approximation (SVEA). Such an approximation is justified when changes in the optical intensity occur on a time scale much longer than an optical cycle, a condition that is equivalent to requiring that the pulse envelope contains many optical cycles. At 1.55 pm, the wavelength commonly used for soliton transmission, an optical cycle lasts about 5 fs. The SVEA is thus expected to be quite accurate for pulse widths > lOOfs. The SVEA is also equivalent to the quasi-monochromatic approximation that is valid when the pulse spectrum is much narrower than the carrier frequency ~0 associated with the optical field. Assuming that the SVEA holds and the optical field propagates along the z axis coinciding with the fiber axis, the electric field appearing in the Maxwell equations can be written as: E ( X , ~ , Z t, ) =

i ( F ( X , ~ ) A ( Zt ), exp[i(hz

-

wot>l+c.c.} ,

Iv,

5 21

189

OPTICAL SOLITONS IN FIBERS

where .i is the polarization unit vector, F ( x , y ) is the fiber-mode profile, Q is the carrier frequency, = n ( ~ ) u o / isc the propagation constant, n ( 9 ) is the linear refractive index at the carrier frequency Q, and A ( z , t ) governs the temporal evolution of the pulse envelope along the fiber length. For simplicity, we ignore the polarization effects for the moment, but will consider them later. After some algebra (see Agrawal [1995a]), one can obtain the evolution equation for the pulse envelope A ( z , t ) , d A +B1-d A

dz

dt

+ -/$i d2A - -p3--iyIAI 1 d3A 2

at2

6

dt3

2A

+ -A a0 2

=

0,

where the last term accounts for the fiber loss through the parameter ao. The coefficients p,,, = [d"fl/du"],, with m = 1,2,3 result from an expansion of the propagation constant p(u) in a Taylor series (Agrawal [1995a]). Physically, PI is simply the inverse of the group velocity ug, pZ accounts for group-velocity dispersion (GVD), and p3 is called the third-order dispersion (TOD) parameter. The effects of become important only when IpZ I < 0.1 ps2/km or for ultrashort solitons of widths below 2ps. We neglect p3 initially and consider its effect in later sections. The parameter y is responsible for the nonlinear phenomenon known as self-phase modulation (SPM) and is defined as:

a

where n2 is the nonlinear refractive index, c is the speed of light in vacuum and A,E is known as the effective mode area (typically A e ~= 50pm2). At the 1.55-pm wavelength, where most soliton communication systems operate, typical values of the parameters in eq. (2.2) are j3, = 5 ns/m, = -1 ps2/km, y = 2 W-'/km, and Q = 0.046 km-' (0.2 dB/km). In the context of soliton propagation, it is useful to transform eq. (2.2) into a normalized form by applying the transformations:

a

The dispersion length, L D = Ti/IpZl,governs the distance scale at which dispersion starts to play a role for a pulse of width TO.The soliton peak power is defined as P, = ( y LD)-I and, as will be seen later, corresponds to the peak power that a secant hyperbolic pulse should have to become a fundamental soliton.

I90

[IV,

SOLITON COMMUNICATION SYSTEMS

Using the transformations in eq. (2.4) and neglecting the can be written in a normalized form:

/33

9: 3

term, eq. (2.2)

.au - sgn(h)-1 (3% + Iu(’u = -I-. a u, aE 2 at2 2

I-

where sgn(h) equals ‘+1’ or ‘-1’ depending on whether p2 > 0 or p2 < 0, and a is a dimensionless parameter representing fiber loss over each dispersion length and defined as

When fiber loss is neglected (ao= 0), eq. (2.5) becomes the standard nonlinear Schrodinger equation (NSE). It is one of the few nonlinear partial differential equations which can be solved analytically by using the inverse scattering theory (IST, Zakharov and Shabat [1971]). The NSE solutions are called solitons to convey their particle-like properties such as robustness against collisions with other solitons and an exceptional stability under small perturbations. One can divide the solutions of the NSE into two classes. When & < 0 (anomalous GVD), solutions of the NSE are called “bright” solitons. When > 0 (normal GVD), solutions of the NSE are referred to as “dark” solitons since they are characterized by an intensity dip in a constant-amplitude background of infinite extent. Dark solitons and their application to fiber-optic communications are discussed separately in the last section. Bright solitons represent localized pulses and are used almost exclusively for the design of soliton communication systems. In what follows, the word soliton refers to bright solitons unless stated otherwise.

a

2.2. SOLITON PROPERTIES

Optical solitons in fibers constitute a whole family of pulses with different peak powers for a given pulse width. The members of the family are characterized by a parameter N, called the soliton order. The Nth-order soliton has the peak power PN = N’P,, where P, is the peak power of the first-order (called fundamental) soliton and N is a positive integer. The IST shows that the different-order solitons represent different “bound states” of the NSE, similar to the energy states of the linear Schrodinger equation in quantum mechanics, with the nonlinear term playing the role of potential energy responsible for the formation of bound states. An input pulse propagating in the fiber can excite one or more of these nonlinear bound states together with a continuum of states which correspond to linear

Iv, I 2 1

OPTICAL SOLITONS IN FIBERS

191

dispersive waves (Gordon [ 19921). If the input-pulse shape, width, and power are such that they correspond closely to a specific bound state, no dispersive waves will be generated, and the pulse will propagate as a soliton whose order is determined by the input-pulse peak power. Figures la, l b and l c show the evolution of first-, second- and third-order solitons, respectively. In all cases, solitons recover their shape after propagating over a distance known as the soliton period, E, = ~ / 2 .In physical units, the soliton period L, = (3t/2)L~, indicating that the dispersion length sets the scale for this behavior. The firstorder soliton has the property that neither its shape nor its phase (see fig. Id) changes during propagation. Such stability allows a first-order soliton to conserve the minimum time-bandwidth product (see eq. 2.9) all along propagation. In contrast, higher-order solitons compress periodically, resulting in soliton chirping. This chirp, developed by higher-order solitons, leads to a broadening of the soliton spectrum (figs. l h and li for N = 2 and N = 3 respectively) and increases the time-bandwidth product. Moreover, the use of higher-order solitons would require more energy per bit of information, a feature not desirable for communication systems. Although higher-order solitons can propagate over large distances (Hamaide, Brun, Audouin and Biotteau [ 19941) under certain conditions, the current soliton communication systems use exclusively first-order solitons because of their shape-preserving nature. The IST provides a simple analytic solution for the first-order soliton. In its most general form, it can be written as (Kodama and Hasegawa [1992b]):

u,(E, z)

= uo sech[uo(t

+ O&

-

zd)] exp[-iw,t

+ -21 (ug - o,’>E + i@],

(2.7)

where the parameters uo, zd, o,,and @ represent the amplitude, position, frequency and phase of the input pulse at E = 0, respectively. The most important property of the soliton is that its amplitude and width are coupled in eq. (2.7) in such a way that a reduction in the soliton width requires an increase in its amplitude UO. The full width at half maximum (FWHM) of the soliton of eq. (2.7) is T, = 2 In(1 + fi)To/uo = 1.763 TO&. The simplest form of eq. (2.7) corresponds to the choice of input-pulse parameters uo = 1, Zd = 0, 0,= 0, and @ = 0. In that case, the soliton evolves as us(E,Z) = sech(t)exp(iE/2). The soliton shape does not change while its phase changes linearly with the distance of propagation. It is important to note that the soliton phase is time independent; i.e., the optical phase is uniform across the entire pulse. The energy of a soliton (in normalized units) can be calculated by using ( ~ , ( ~ =d t2uo = 3 . 5 2 6 / ~ , ,where Z, = T,/To is the soliton FWHM normalized to TO.An important implication for soliton communication systems

s’-

192

SOLITON COMMUNICATION SYSTEMS N = l

N=2

[IV,

s2

.I' = 3

Fig. 1. Temporal, chirp and spectral evolution of N = 1, N = 2 and N = 3 solitons over one soliton period. Although all solitons periodically recover their shape, only the fundamental soliton preserves its shape and does not develop chirp during propagation.

is that the bit energy (and hence the average launch power) increases as the soliton width decreases. The spectral amplitude vs of the soliton of eq. (2.7) is defined as vs(E,o)s .F{us(E, r ) } , where 3 represents the Fourier-Transform operation, and is given by:

From eqs. (2.7) and (2.8), one can easily find the soliton bandwidth Av, (FWHM) and show that the time-bandwidth product for a soliton is given by: Av, t, = [2 In( I

+ v5)/x]*rv 0.3149.

(2.9)

The practical implementation of any soliton communication system is not likely to provide ideal conditions under which the soliton evolution is governed by

Iv, 9 21

OPTICAL SOLITONS IN FIBERS

193

the standard NSE. Indeed, in practice, solitons suffer from many perturbations related to the input pulse shape, chirp and power, fiber loss, and amplifiers noise, to mention a few among them. Because of their particle-like nature, it turns out that solitons remain stable under most perturbations. As an example of soliton robustness, the peak power necessary to generate a fundamental soliton lies in a range as broad as 0.25Ps < Pi, < 2.25Ps. Within this range of peak power, an input pulse will become a fundamental soliton after propagating over a few dispersion lengths. In general, a soliton responds to perturbations by reshaping itself into another soliton of different peak power P,, and thus of different characteristic width T,, since the two are related by P, = Ij3l/(yTi). During this adaptation phase, the perturbed soliton generates dispersive waves, which correspond to continuous-wave (CW) radiation of low power that is not part of the soliton, and hence spread out as any linear wave should do in a dispersive medium. For soliton communication, one must try to minimize the energy lost to dispersive waves because their interaction with the remaining solitons (Midrio, Romagnoli, Wabnitz and Franco [1996]) degrades the bit stream and results in a high bit-error rate during the detection process. In this review, the word soliton is used in a general sense as it is applied even to a perturbed soliton. 2.3. ADIABATIC PERTURBATION THEORY

Since solitons are invariably perturbed in a realistic communication system, the four parameters UO, zd, w,, and 9 used to describe a soliton in eq. (2.7) evolve with soliton propagation along the fiber length, rather than remaining constant. Perturbations can also be imposed externally by using components such as optical filters, modulators, etc., in order to control the four soliton parameters. In general, one must resort to numerical simulations to evaluate the effects of these perturbations on the soliton evolution. However, when the perturbations are sufficiently small, solitons can adapt to perturbations in an adiabatic manner. The evolution of the four soliton parameters can then be studied analytically by using an adiabatic perturbation theory (APT). If €(us)represents the small perturbation and us is the soliton field of eq. (2.7) at the fiber input ( E = 0), the evolution equations are (Karpman and Maslov [ 19771, Georges [1995]): (2.10) (2.11) (2.12)

194

SOLITON COMMUNICATION SYSTEMS

[IV,

9: 3

(2.13) where Re and Im stand for real and imaginary parts, respectively. The four functions over which the perturbation is projected are given by: (2.14) (2.15) (2.16) (2.17) Equations (2.10H2.17) are used extensively in the theory of soliton communication systems. The usefulness of the APT will become clear in the later sections. A comprehensive list of the effects of various perturbations on the four solitons parameters, studied by using eqs. (2.10)-(2.17) of the APT, has been compiled by Hasegawa [ 19951.

tj

3. Soliton-Based Communication Systems

Long-distance transmission of information using optical fibers is hampered by three types of signal degradation which are intrinsic to the fiber - loss, dispersion, and nonlinearity. Fiber loss can be minimized by operating near A = 1.55 pm. However, even with fiber loss as low as 0.2 dB/km, the signal power is reduced by 20 dB (a factor of 100) after transmission over l O O k m of fiber. The loss problem can still be solved by periodically using in-line optical amplifiers to restore the signal power to its original level. Fiber dispersion then becomes the most limiting factor for long-haul systems. The use of a dispersion-compensation scheme can solve the dispersion problem to some extent, but the system performance is then limited by the fiber nonlinearity. Solitons provide an ideal solution to this problem since they use fiber dispersion and the nonlinearity to their advantage in such a way that the two “harmful” effects become useful. This is the main reason behind the enormous interest in soliton communication systems. 3.1. INFORMATION TRANSMISSION WITH SOLITONS

Most fiber-optic communication systems code the information by using a binary digital system in which the presence or absence of an optical pulse represent “1” or “0” bits, a technique referred to as on-off keying. Two methods can be

195

SOLITON-BASED COMMUNICATION SYSTEMS

IV, § 31

T

(b)

Time

Fig. 2. Schematic representation of the two formats used for transmission of digital optical signals: (a) return-to-zero (RZ) format, and (b) non-return-to-zero (NU)format.

used to generate such a bit stream as shown schematically in fig. 2. When the signal returns to the zero level in each bit, the coding is referred to as the returnto-zero (RZ) format. In the second format, called the non-return-to-zero (NU) format, the optical pulse occupies the entire bit slot, and the signal does not return to the zero level between neighboring “1” bits. The N U format is used almost universally because its signal bandwidth is about 50% smaller compared with the RZ format for a given bit rate. The NRZ format cannot be used when solitons are used as information bits. In fact, the soliton width must be a small fraction of the bit slot in order to ensure that the neighboring solitons are well separated, since solitons propagate undistorted only when they are well isolated. This requirement can be used to relate the soliton width TOto the bit rate B as:

where T, is the duration of the bit slot and 240 = TB/Tois the separation between neighboring solitons in normalized units. The input pulse shape and amplitude needed to excite the fundamental soliton can be obtained by setting 5 = 0 in eq. (2.7). It is common to choose uo = 1, 0,= 0, r d = 0, and # = 0. In physical units, the amplitude of the pulse is then given by:

A(0, t ) = &sech(t/To),

(3 4

where the input peak power P, = I/&$(yTi). As a simple example, TO = lops for a IO-Gb/s soliton system if we choose qo = 5 . Of course, the FWHM of the

196

SOLITON COMMUNICATION SYSTEMS

[Iv,

P3

soliton is about 1 7 . 6 ~ when s To = lops, as discussed in 5 2.2. The peak power of the input pulse should be 5 mW for typical values k and y for dispersionshifted fibers. This value corresponds to a pulse energy of only 0.1 pJ and an average power level of 1 mW. An optical source capable of generating picosecond pulses at a repetition rate equal to the bit rate B is needed for a soliton communication system. Several such sources are available. A mode-locked fiber or semiconductor laser is ideal since it produces pulses whose shape is close to the “sech” shape, especially if a passive mode-locking technique is used. Gain-switched semiconductor lasers have also been used because of the relative simplicity with which such lasers can produce picosecond optical pulses. However, since this technique produces chirped pulses, the chirp must be removed before they can be used. Spectral filtering together with pulse compression in a optical fiber can produce nearly chirp-free pulses. An external modulator is used to code the information on the periodic pulse train. The modulator simply blocks the pulse corresponding to every “0” bit. An optical amplifier often follows the modulator to boost the pulse peak power to the level given by eq. (3.2). Although the NSE describes the two important properties of fibers (GVD and nonlinearity), several other physical effects must be included to properly describe the propagation of electromagnetic waves in optical fibers. These includes fiber loss, the Raman effect, TOD, fiber birefringence, and variations of the GVD over the fiber length (Agrawal [1995a]). In addition, in a communication link transmitting a train of solitons over multiple fiber sections, the influence of multiple amplifiers as well as of soliton interactions should be included. Even though a soliton is quite a robust pulse, the management of all these perturbing effects which solitons undergo on their way to the receiver is the key to the design of soliton communication systems. Starting with the very first 1985 experiment (Mollenauer, Stolen and Islam [ 19851, Mollenauer, Gordon and Islam [ 1986]), compensation of fiber loss was recognized as one of the important issues for stable soliton transmission. The following sections are divided according to the schemes used to compensate fiber loss, with separate sections on dispersion management, multiplexing, and dark soliton communication systems. Various physical mechanisms deleterious to error-free transmission and their control are discussed separately in each section. 3.2. LOSS COMPENSATION

To compensate for the fiber loss, it is necessary to amplify the soliton train periodically along the transmission line. Figure 3 illustrates two schemes used to

IV, 9: 31

SOLITON-BASED COMMUNICATION SYSTEMS

197

Tx

(b) Fig. 3. Schematics of the two loss-compensation techniques for solitons: (a) lumped amplification and (b) distributed amplification.

compensate for the fiber loss. One scheme (fig. 3a) makes use of in-line optical amplifiers (typically erbium-doped fiber amplifiers) to restore the soliton energy. Since amplification over a very short distance (-l0m) compensates for the loss occurring over 40-50 km, this scheme is referred to as lumped amplification. It is used almost universally for current soliton systems because of its implementation simplicity, even though its use limits the bit rate as discussed in 0 4. A different approach to loss compensation uses the gain in the transmission fiber itself. Its use requires periodic injection of the pump power into the transmission fiber (fig. 3b). Since the gain is now distributed over the entire fiber link, one refers to such a scheme as distributed amplification. Stimulated Raman scattering can be used to provide distributed gain if the pump frequency is higher than the signal frequency by the Raman shift (-13THz). In fact, the Raman-gain technique was used until 1988; i.e., before erbium-doped fiber amplifiers became available. Distributed amplification can also be achieved by lightly doping the transmission fiber with erbium (or another rare-earth element). Ideally, injection of pump power at a wavelength suitable for population inversion (0.98 or 1.48 pm radiation) provides low but just large enough gain to compensate the fiber loss all along the fiber. In practice, gain variations due to pump absorption make it difficult to compensate the loss exactly at every point. The distributedamplification technique is discussed in Q 6.

198

SOLITON COMMUNICATION SYSTEMS

"v, I 3

3.3. AMPLIFIER NOISE

The use of in-line optical amplifiers affects the soliton evolution in several ways. An optical amplifier, inserted periodically to compensate for the fiber loss, not only restores the soliton energy but also adds noise originating from amplified spontaneous emission (ASE). The spectral density of ASE depends on the amplifier gain, G, and can be written as

where nsp is the spontaneous emission factor whose value depends on the degree of population inversion (typically nsp M 2) and h o 0 is the photon energy for solitons of carrier frequency 00. The effect of ASE noise is to change randomly the values of the four soliton parameters U O , t d , o,,and @ in eq. (2.7) at the output of each amplifier. The variances of such fluctuations for the four soliton parameters can be calculated by using the APT of 9 2.3 with the perturbation €(us) = i n(E, t)exp(-io,t

+ @),

(3.4)

where n(5, t) is the random-noise term, which is assumed to be complex to include both the amplitude and phase fluctuations. It vanishes on average and its variance is related to the spectral noise density SASE. By using eqs. (2.10k (2.13), the (normalized) variances of fluctuations in U O , y,r d , and 4 are given by (Haus and Lai [ 19901, Georges and Favre [1993]):

where F(G) = (G - l)'/(GIn G), and N, is the number of photons in the soliton. N , = 2 P, TO/ ( h YO),where pS is computed by using the average dispersion of the fiber section following the amplifier of gain G. Amplitude fluctuations, as one might expect, lead to a degradation of the signal-to-noise ratio of the soliton bit stream. Although not immediately obvious, it will be seen later that amplifier noise induces timing jitter of solitons that affects the performance of soliton communication systems and limits the total transmission distance. The timing-jitter issue is covered in $4.2.

IY P 41

199

AVERAGE-SOLITON REGIME

§ 4. Average-Soliton Regime

The way a soliton reacts to the energy loss caused by fiber losses depends strongly on the loss per dispersion length, a,and the length LA over which the energy loss occurs. If a << 1 and LA >> LD, the soliton reshapes itself to preserve its nature. This regime is referred to as the quasi-adiabatic regime and is discussed in 5 5. On the other hand, if the amplifier spacing LA is much smaller than the dispersion length LD (ga = LA/LD<< l), the soliton shape is not distorted significantly by fiber loss between successive amplifications. In such a system, a soliton can be amplified hundreds of time while preserving its shape. The soliton properties are then given by the soliton energy (or peak power) averaged over one amplifier spacing. For this reason, this mode of operation is referred to as the average-soliton regime. This section discusses the issues that need attention when the average-soliton regime is used for the design of soliton communication systems. 4.1. EVOLUTION OF THE AVERAGE SOLITON

The periodic lumped amplification can be accounted for by adding a “gain” term to eq. (2.5). By using < 0, it can be written as (Hasegawa and Kodama [ 19911):

al + -1 -+ d2U

i-

ag

2

a t 2

I U ~ = ~ -iU

a 2

N

u+iC6(g-rnEa)(&-

I)U,

n=l

where N is the total number of amplifiers spaced apart by Ea, and G = exp(aga) is the amplification factor by which the soliton peak power is boosted at E = Ea. The factor @ - 1 represents the change in the soliton amplitude occurring during lumped amplification. Because of the rapid amplitude variations introduced by the lumped-amplification scheme, it is useful to make the transformation

where a(E) contains rapid amplitude variations and u(E, z) is a slowly varying function of E. By substituting eq. (4.2) in eq. (4.1), u(5, t) satisfies

.au + -I a2u + a (g)lu12u = 0,

1-

aE

2

a t 2

(4.3)

200

SOLITON COMMUNICATION SYSTEMS

where u(E) is the solution of

By noting that the last term in eq. (4.4) contributes only at &' = mEa, u(E) is found to be periodic. In each period, u(E) decreases exponentially with a jump to its initial value at the end of the period. The concept of the average soliton makes use of the fact that u2(E)in eq. (4.3) varies rapidly with a period Ea << 1. Since solitons evolve little over a short distance Ea, one can replace u2(c) by its average value. This approximation can be justified mathematically by assuming a solution of eq. (4.3) in the form u = 0 + 60, where U is the average soliton satisfying the standard NSE,

.au + -I a2u + (u2(E))lo12a= 0,

1-

a5

2

ax2

(4.5)

and 60 is a perturbation. The practical importance of the average-soliton concept results from the fact that the perturbation 60 turns out to be relatively small when E << 1, since the leading-order correction varies as Ei rather than Ea. As a result, the average-soliton description is quite accurate even for E' = 0.2, corresponding to the case LA = L d 5 . The input peak power Pi, of the average soliton is chosen such that (u2(E))= 1 in eq. (4.5). By using G = exp(a$), it is given by: P,,

GInG G-1

= -ps 9

where 4 is the peak power in lossless fibers. As an example, G = 10 and Pi, zz 2 . 5 6 4 for 50-km amplifier spacing, if we assume a fiber loss of 0.2 dB/km. The condition Ea << 1 required to operate within the average-soliton regime can be related to the soliton width TOand the bit rate B by using eq. (3.1) together with LD = T:/lhl, with the result

By choosing the typical values k = -0.5ps2/km, L A = 50km, and qo = 5, we obtain To >> 5 ps and B << 20 GHz. Clearly, the use of the average-soliton regime imposes a severe limitation on the bit rate of soliton communication systems.

rv, 6 41

AVERAGE-SOLITON REGIME

20 1

4.2. TIMING JITTER

If optical amplifiers compensate for the fiber loss, one may ask what limits the total transmission distance of the soliton link. The answer is provided by the timing jitter. A soliton communication system can operate reliably only if all solitons arrive at the receiver within their assigned bit slot. Several physical mechanisms induce deviations in the soliton position from its original location at the bit center. This section considers mechanisms which limit the performance of systems operating in the average-soliton regime. Section 7.2 considers other mechanisms which become important for ultrashort solitons. 4.2.1. Gordon-Haus jitter

As discussed in (i 3.3, optical amplifiers add both the amplitude and phase noise to the amplified soliton. Since a time-dependent phase shift is equivalent to a change in the soliton central frequency, us fluctuates around its initial value because of amplifier noise. Since the soliton velocity depends only on the soliton frequency in the absence of higher-order effects in fibers, soliton transit time through the fiber link also becomes random following the frequency fluctuations. Fluctuations in the arrival time of a soliton at the receiver (when only frequency fluctuations are taken into account) are referred to as the Gordon-Haus timing jitter (Gordon and Haus [ 19861, Marcuse [ 19921). It is relatively easy to calculate the variance of the timing jitter by using the APT of (i 2.3. In the absence of other perturbations within the fiber, we set €(us) = 0 in eqs. (2.10)-(2.13). Equation (2.12) governing the soliton position is integrated easily to yield t d ( E ) = -w&. At the end of one amplifier spacing, the delay becomes t d = -o&. Since the random frequency shifts accumulate from amplifier to amplifier, the total timing jitter for a series of N amplifiers is obtained by adding all contributions and becomes N

P

p=l i=l

where w, is the frequency shift induced by the ith amplifier. In the limit of large N so that the summations in eq. (4.8) can be replaced by an integral, the (normalized) variance is found to be

&c where

=

N3Eio;J3,

C J is ~

given by eq. (3.6).

(4.9)

2 02

SOLITON COMMUNICATION SYSTEMS

[IV,

9: 4

Since a soliton must arrive within its allocated bit slot, the arrival-time jitter should be a small fraction of the bit slot. By using L = N E a L ~B, = ( 2 qo To)-l, N, = 2 P, T o / ( h vo) and P, = ( ~ L D ) -in ' eq. (4.9), the total bit-rate-distance product B L (in physical units) is found to be limited by (4.10) where D = -2ncp2/A2 and f is the fraction of the bit slot by which a soliton can move without adversely affecting the system performance. The limit set by eq. (4.10) is often referred to as the Gordon-Haus limit. The tolerable value of f depends on the acceptable BER and on details of the receiver design; typically f 0.1. To see how ASE noise limits the total transmission distance, consider a specific soliton communication system operating at 1.55 pm with qo = 6, a. = 0.046km-' (0.2 dB/km), y = 3 W-likm, D = 0.5ps/(km-nm), nsp = 2 and LA = 50 km. For such parameters, eq. (4.10) requires that BL stay below 60000 (Gb/s)-km. For a 15-Gb/s system, the distance is limited to at most 4000km. 4.2.2. Polarization-mode dispersion jitter A soliton train usually possesses a definite state of linear polarization at the input of a fiber link. However, as it is amplified periodically, the state of polarization of individual solitons fluctuates since the ASE added at every amplifier is of random polarization. Such polarization fluctuations induce jitter in the transit time of individual solitons through fiber birefringence. Since the fiber link used for soliton transmission is not made of polarization-preserving fibers, the polarization state of a soliton also changes randomly inside the fiber because of changes in the fiber birefringence resulting from local variations in quantities which affect the modal index (e.g., stress, temperature, core diameter, etc.). This phenomenon is known as the polarization-mode dispersion (PMD), and its effects are quantified through a parameter D,, called the PMD parameter. The timing jitter introduced by the combination of ASE and PMD is found to be (Mollenauer and Gordon [ 19941)

n F ( G )n,, D; L2

02

=_--

pol

16

Ns

LA

(4.1 1) '

Note that u,,l increases linearly with both the transmission distance L and the PMD parameter D,. As an estimate, apol = 0 . 3 8 ~ sfor a long-haul communication system having a. = 0.046kn-' (0.2 dB/km), LA = 50km,

IV, 9 41

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203

nSp = 2, N , = 500000 , D, = 0.1 ps/& and L = I O O O O k m . Such a low value of ap,l is unlikely to affect a 10-Gb/s soliton communication systems for which the bit slot is loops wide. However, for fibers having larger values of the PMD-induced timing jitter becomes the PMD parameter (Dp > 1 ps/&), important enough that its impact should be considered together with the GordonHaus timing jitter.

4.2.3. Acoustic jitter There exists another timing-jitter mechanism that would limit the total transmission distance even if optical amplifiers were noise-free. It originates from the simple phenomenon of acoustic-wave generation (Dianov, Luchnikov, Pilipetskii and Prokhorov [1991, 19921). Confinement of the optical field within the fiber core creates a field gradient in the radial direction of the fiber. This gradient of electric field leads to the generation of acoustic waves through electrostriction, a phenomenon that creates density variations in response to variations in the electric field. Since the refractive index of fused silica is related to the material density, one can associate a change in refractive index (and hence in the soliton group velocity) with the generation of acoustic waves. The electrostrictive mechanism is thus bringing an additional contribution to the intensity-dependent refractive index on top of the Kerr nonlinearity (Buckland and Boyd [1996]). However, the electrostriction-induced index change is not as fast as the Kerr nonlinearity as it occurs on a time scale of roughly the time required by the acoustic wave to traverse the fiber core. Since typically solitons follow one another on a much shorter time scale (loops or less for B 3 10Gb/s), the acoustic wave generated by a single soliton affects tens or even hundreds of the following solitons. Such an acoustic-wave-assisted interaction among solitons (Dianov, Luchnikov, Pilipetskii and Starodumov [1990]) is referred to as longrange interaction and has been observed experimentally (Smith and Mollenauer [ 19891). If a bit stream were composed of only “1” bits such that a soliton occupied each bit slot, all solitons would be shifted in time by the same amount by the emission of acoustic waves (ignoring the boundaries of the bit pattern), creating a uniform shift of the soliton train with no impact on the timing jitter. However, since an information-coded bit stream is generally closer to a random string of “1”s and “0”s bits, the change in the group velocity of a given soliton depends on the presence or absence of solitons in the preceding tens of bit slots; i.e., of the preceding bit pattern. As a result, different solitons acquire slightly different velocities, resulting in timing jitter. For this reason, acoustic jitter is

2 04

SOLITON COMMUNICATION SYSTEMS

"I 4

sometimes said to be of deterministic origin, in contrast with the Gordon-Haus and PMD timing jitters, both of which are stochastic in nature. The deterministic nature of acoustic jitter makes it possible to reduce its impact in practice by moving the detection window at the receiver through an automatic tracking circuit (Mollenauer [ 19961). Attempts have been made to evaluate the extent of timing jitter by using simple analytic models (Dianov, Luchnikov, Pilipetskii and Prokhorov [ 1991, 19921). For bit rates above 5 Gb/s, acoustic jitter can be approximated by (Mollenauer, Mamyshev and Neubelt [ 19941): (4.12) where u,,,, is expressed in ps, Aeff in pm2, D in ps/(km-nm), T, in ps, L in thousands of km, and B in Gb/s. The parameter r corresponds to the intensity reflection coefficient of the sound from the cladding-coating interface. As an example, by choosing the parameters, A,ff = 50 pm2, D = 0.5ps/(km-nm), T, = 15ps, L = 10000 km, B = 10 Gb/s and r = 0.25, one obtains u,,,,= 22ps. Although generally smaller than the Gordon-Haus jitter, the acoustic jitter can contribute to the total timing jitter of a soliton communication system. However, as for the Gordon-Haus jitter, its value can be reduced considerably (Mollenauer, Mamyshev and Neubelt [ 19941) by using sliding-frequency filters (described in ij 4.4.1). 4.3. SOLITON INTERACTION

The soliton of eq. (2.7) is an exact solution of NSE only when it occupies the entire time window such that u,(E, t) -+ 0 as ( T I + 03. In a realistic communication system, as discussed in 5 3, a soliton is confined to the bit slot T, = 1/B, determined by the bit rate B , and accompanies other solitons in the neighboring bit slots. As two solitons separated in time by a few soliton widths propagate together inside an optical fiber, the overlap and interference of their tails become a source of interaction. One can study the interaction between two solitons by solving the NSE with an input field of the form: u(0, t) = sech( t - 40) + r sech[ r (z + qo)] exp(i O),

(4.13)

where 240 = TdTo is the (normalized) separation between the two solitons, is the ratio of their amplitudes (and hence widths), and O is their relative

r

IV,

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AVERAGE-SOLITON REGIME

205

e=o,r=i

Fig. 4. Interaction between two solitons: (a) I9 = 0 and r = I ; (h) I9 = n/4 and r = 1; (c) H = n/2 and r = I; (d) 0 = 0 and r = 1.1. In each case, the soliton pair is propagated over 90 dispersion lengths.

phase difference. Numerical simulations show that the soliton half-separation, qs, changes from its initial value, 40, because of soliton interaction and may increase or decrease depending on the values of r and 8. Figure 4 shows the evolution of a pair of solitons for several combinations of r and 0 by choosing qo = 3.5. Evolution of qs(E)can be studied analytically by using the IST (Karpman and Solovev [1981], Gordon [1983]). In the specific case r = 1 and 19 = 0, qS(Q at any distance E is obtained by solving the following transcendental equation (Gordon [1983]):

Clearly, qs(E) varies periodically along the fiber with the period:

206

SOLITON COMMUNICATION SYSTEMS

[IV, 9; 4

The APT can be used for qo >> 1 and yields the same result (Karpman and Solovev [1981]). A more accurate expression, valid for arbitrary values of qo, is given by (Desem and Chu [1987a,b]):

n sinh(2q0) cosh(qo)

"=

240 + sinh(2qo)

(4.16) '

Equation (4.15) is quite accurate for qo >> 3, as also found numerically. Its prediction is in agreement with fig. 4, drawn for the case qo = 3.5. It can be used for the system design as follows. If EpLD is much greater than the total transmission distance L, soliton interaction can be neglected since soliton spacing would deviate little from its initial value. For qo = 5, Ep = 233. Since the dispersion length typically exceeds 100 km in the average-soliton regime, L << g p L ~can be realized even when L is as large as 10000km. By using LD = 7',2/]&1 and To = (2 Bq&I from eq. (3.1), the condition L << Ep LD can be written in the form of the following design criterion: (4.17) As an illustration, B2L << 5 (Tb/s)*-km if we use p2 = -0.5ps2/km, and qo = 5 as representative values. For a lO-Gb/s system, the total distance is then limited to L << 50000km. A relatively large soliton spacing (typically qo 3 5), necessary to avoid soliton interaction, limits the bit rate of soliton communication systems. The tolerable spacing can be reduced by using unequal amplitudes for the neighboring solitons. The numerical results (Desem and Chu [1987c]) show that for two in-phase solitons, the separation does not change by more than 10% for an initial soliton spacing as small as qo = 4 if their initial amplitudes differ by 10% ( Y = 1.1). Since small changes in the peak power are not detrimental for the soliton nature of pulse propagation, this scheme is feasible in practice and can be useful for increasing the system capacity. The design of such systems, however, requires attention to many details. Soliton interaction can also be modified by other factors such as an initial frequency chirp. It is also influenced by the periodic amplification of solitons (Desem and Chu [ 1987a,b]). In deriving eq. (4.10) for the timing jitter, solitons were assumed to be sufficiently far apart so that their interaction is negligible. However, to maximize the bit rate, solitons are sometimes closely packed together. Since the interaction force between two solitons is strongly dependent on their separation and relative phase, both of which fluctuate due to amplifier noise (see eq. 3 3 , soliton

IV,

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207

interaction may modify considerably the timing jitter. By considering noiseinduced fluctuations of the relative phase of neighboring solitons, timing jitter of interacting solitons is generally found to be enhanced by amplifier noise (Georges and Favre [ 19911). However, for a large input phase difference close to n between neighboring solitons, phase randomization leads to a reduction of the timing jitter. 4.4. SOLITON CONTROL

It should be clear from the preceding discussion that the timing jitter, resulting from various phenomena such as amplifier noise, soliton interaction, PMD and soliton collisions for multichannel systems (see 4 9.1) ultimately limits the performance of soliton communication systems. It is therefore essential to find a solution to the timing-jitter problem before the use of solitons become practical. Several techniques have indeed been developed during the 1990s for controlling the timing jitter (Georges [1995], Wabnitz, Kodama and Aceves [ 19951, Smith and Doran [ 1995a1). This section reviews briefly the main techniques for soliton control. 4.4.I . Optical bandpass jilters

Periodic insertion of optical filters (typically a Fabry-Perot ttalon) along a communication link has been used (Mollenauer, Neubelt, Haner, Lichtman, Evangelides and Nyman [ 19911) since 1991 to realize soliton transmission beyond the Gordon-Haus limit (Mecozzi, Moores, Haus and Lai [ 19911, Kodama and Hasegawa [ 1992al). This approach makes use of the fact that the ASE occurs over the entire amplifier bandwidth, whereas the soliton spectrum is a small fraction of it. The bandwidth of optical filters is chosen such that they let the soliton pass but block most of the ASE. Unfortunately, the transmission distance shows only a modest improvement since the timing jitter is reduced by less than 50% when all in-line optical filters have the same center frequency. The filter technique can be improved dramatically by allowing the center frequency of the successive filters to increase (or decrease) along the link. This method, known as sliding-frequency filters (Mollenauer, Gordon and Evangelides [ 19921, Mollenauer, Lichtman, Neubelt and Harvey [ 1993]), avoids the accumulation of ASE occurring at the central frequency when fixedfrequency filters are used. Sliding-frequency filters also reduce the growth of dispersive waves (Kodama and Wabnitz [ 19941, Romagnoli, Wabnitz and Midrio [ 19941) which are generated by strongly perturbed solitons. There still remains

208

SOLITON COMMUNICATION SYSTEMS

[IV, 5 4

the choice of ‘up-sliding’ or ‘down-sliding’, depending on whether the center frequency of the filters increases or decreases along the link; up-sliding is found to provide better performance (Mollenauer, Gordon and Evangelides [ 19921, Golovchenko, Pilipetskii, Menyuk, Gordon and Mollenauer [ 19951). The reason can be understood by considering the TOD associated with filters which reduces the effective GVD experienced by solitons. The use of sliding-frequency filters is difficult in practice because of the need to maintain a precise frequency control. In an alternative approach, significant reduction of the timing jitter has been realized (Aubin, Montalant, Moulu, Nortier, Pirio and Thomine [ 19951, Toda, Yamagishi and Hasegawa [ 19951) by periodically sliding the signal frequency (instead of the central frequency of filters) while using fixed-frequency filters. APT can be used to study how optical filters benefit a soliton communication system. The effect of a bandpass filter is to modify the soliton spectrum such that:

where vs(&, w ) is the soliton spectral amplitude given by eq. (2.8) and Tf(w) is the amplitude transmissivity of the optical filter located at a distance &. If periodic filtering is modeled by a distributed filtering over the distance & and the filter spectrum is approximated by a parabola over a range covering the soliton spectrum, the perturbation term for APT becomes (4.19) where Sg is the excess gain required to compensate for the loss introduced by each filter and Cf is related to the curvature of the filter transmissivity Tf(w). For a Fabry-Perot filter, Cf is given by (Georges [1995]):

Cf

=

1 R 27‘2A$& ( 1 - R)’ ’

(4.20)

where Avf is the free-spectral range and R is the mirror reflectivity of the FabryPerot filter. For a sliding-frequency filter, the center frequency wf becomes dependent and should be written as of = w;E, where wi is the linear sliding rate of the center frequency of filters. The use of €(us)in eqs. (2.10)-(2.13) then predicts how the four soliton parameters evolve in the presence of optical filters. The results show that the soliton frequency slides with the filters, keeping the soliton train intact, but the ASE accumulated over multiple amplifiers is filtered

IV,

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209

out later when the soliton spectrum has shifted by more than its own width. As a result, the timing jitter is reduced considerably by the use of sliding-frequency filters. Optical filters benefit a soliton communication system in several ways. Their use reduces not only timing jitter but also soliton interaction (Kodama and Wabnitz [ 199 11, Afanasjev [ 1993]), making it possible to pack solitons closer. The physical mechanism behind the reduction of soliton interaction is related to the change in the soliton phase at each filter. A rapid variation of the relative phase between neighboring solitons, occurring as a result of filtering, averages out the soliton interaction by alternating the nature of the interaction force from attractive to repulsive. Such phase changes also reduce the importance of fluctuations in the soliton phase induced by amplifier noise (Menyuk [1995]). The reduction of soliton interaction is even more effective if the filter frequency is alternatively up-shifted and down-shifted in a zig-zag pattern (Dung, Chi and Wen [1995]). For ultrashort solitons for which the TOD and the Raman effect become important, soliton interaction is reduced considerably with optical filtering (Aceves, De Angelis, Nalesso and Santagiustina [ 19941). Although most experiments use Fabry-Perot filters, other types of filters have been also used (Suzuki, Edagawa, Taga, Tanaka, Yamamoto and Akiba [1994]). In particular, filters having top-hat-like transmission are of special interest since they minimize energy loss. In one study (Mecozzi [ 1995b1) on Buttenvorth filters of fixed frequency, it was found that such filters can reduce significantly the accumulation of ASE that invariably occurs when Fabry-Perot filters of fixed frequency are used. Under certain conditions, such reduction in the accumulated ASE may eliminate the need of sliding-frequency filters. Moreover, Buttenvorth filters may also be quite efficient in reducing soliton-soliton interaction (Wabnitz and Westin [ 19961). 4.4.2. Synchronous modulators

Solitons can also be controlled in the time domain. An experiment demonstrated in 1991 (Nakazawa, Yamada, Kubota and Suzuki [1991]) the transmission of a soliton train over 1 million kilometers using the technique of synchronous intensity modulation (often implemented using a LiNbO3 modulator). The technique works by introducing additional losses for solitons which have shifted from their original position (center of the bit slot). The modulator forces solitons to move toward its transmission peak where the loss is minimum. Mathematically, the action of modulator is to change the soliton amplitude as: (4.21)

210

SOLITON COMMUNICATION SYSTEMS

“v, 9: 4

where T,( t) is the time-dependent amplitude transmissivity of the modulator located at 5 = Ern and peaking at t = t,, the center of the bit slot. For the case of sinusoidal modulation commonly used in practice, T m ( t ) can again be approximated by a parabola in the vicinity of t = .,z The APT can again be used in a manner similar to the case of optical filters. The results show that synchronous modulators work by forcing the soliton to move toward their transmission peak, and such forcing reduces the timing jitter considerably. Synchronous modulation can be combined with optical filters to simultaneously control solitons in both the time and frequency domains. A numerical study (Nakazawa, Kubota, Yamada and Suzuki [ 1992]), followed by an experimental realization (Nakazawa, Suzuki, Yamada, Kubota, Kimura and Takaya [ 1993]), indicated the possibility of achieving arbitrarily large transmission distances by this combination. Synchronous intensity modulation not only allows large transmission distances but also permits relatively large amplifier spacings (Kubota and Nakazawa [1993]) because it can simultaneously reshape the soliton and damp the dispersive waves. This property of modulators has been exploited to transmit a 20-Gbis soliton train over 150000km with an amplifier spacing of 105 km (Aubin, Jeanny, Montalant, Moulu, Pirio, Thomine and Devaux [ 19951). In another experiment, a single synchronous modulator, inserted just after the transmitter, allowed transmission of a 20 Gb/s signal over 3000 km (Nakazawa, Suzuki, Kubota, Yamada and Kimura [ 1994]), well beyond the 2300-km GordonHaus limit of the system without modulation. In this experiment, the clock signal used to generate the soliton train was used to drive the modulator. In contrast, when in-line synchronous modulators are used, the signal clock must be recovered periodically. Synchronous modulation can also use a phase modulator (Wabnitz [1993], Smith, Firth, Blow and Smith [1994]). One can understand the effect of periodic phase modulation by recalling that a frequency shift dw = -a#(t)/at is associated with any phase variation # ( t ) . Since a change in soliton frequency is equivalent to a change in the group velocity, phase modulation induces a temporal displacement. Synchronous phase modulation is implemented in such a way that the soliton experiences a frequency shift only if it moves away from the center of the bit slot, thereby confining it to its original position in spite of the timing jitter induced by ASE and other sources. Intensity and phase modulations can be combined together (Bigo, Audouin and Desurvire [ 19951) to further improve the system performance. Similar to the case of guiding-center optical filters, synchronous modulators help a soliton communication system in several other ways. Among other things, they reduce soliton interaction, clamp

IV,

9 41

AVERAGE-SOLITON REGIME

21 1

the level of amplifier noise, and inhibit the growth of dispersive waves (Smith, Blow, Firth and Smith [1993]). 4.4.3. Other techniques of soliton control Numerous ways through which solitons interact with each other and with other optical fields lend themselves to many diverse techniques for soliton control. This subsection describes some of these techniques. In one technique, helpful in reducing soliton interaction, the amplitude of neighboring solitons is alternated between two values differing typically by about 10%. Such a difference in the amplitude uo results in different rates of phase accumulation with increasing distance 6 see eq. 2.7) for the two types of solitons. As a result, the phase difference 8 (-n< 8 5 n)between neighboring solitons changes with propagation, resulting in an averaging of soliton interaction since the interaction force depends on the phase difference between the neighboring solitons and changes from attractive to repulsive with changes in 8 (see 9 4.3). Such a technique has been used successfully (Suzuki, Edagawa, Taga, Tanaka, Yamamoto and Akiba [1994]) to transmit solitons at a bit rate of 20Gb/s over 11 500km, a distance larger than the distance over which two solitons would collide in the absence of amplitude alternation. Another approach to soliton control consists of periodically inserting along the fiber link a device that acts like a fast saturable absorber (FSA). A FSA absorbs low-intensity optical fields but becomes transparent to high-intensity fields provided its response is much faster than the soliton width. It is difficult to find an absorber that can respond at femtosecond time scales. However, nonlinear phase effects in fibers can be used to make an interferometer that acts like an FSA (Agrawal [ 1995al). For example, a nonlinear optical-loop mirror (NOLM; Doran and Wood [ 19881) or a nonlinear amplifying-loop mirror (NALM; Fermann, Haberl, Hofer and Hochreiter [ 19901) can act as an FSA and reduce considerably the timing jitter of solitons (Matsumoto, Ikeda and Hasegawa [1994], Yamada and Nakazawa [ 19941). The same device can also simultaneously stabilize the soliton amplitude (Takada and Imajuku [ 19961). Retiming of a soliton train can also be realized (Widdowson, Malyon, Ellis, Smith and Blow [1994b]) by taking advantage of the nonlinear phenomenon of cross-phase modulation (XPM) in optical fibers (Agrawal [ 1995a1). The technique overlaps the soliton data stream and another pulse train composed of “1” bits (generated through clock recovery, for example) in a few-kilometer fiber where XPM induces a phase shift on the soliton data stream whose magnitude can be controlled. Such a phase modulation of the soliton translates into a net

212

SOLITON COMMUNICATION SYSTEMS

[IV,

P4

frequency shift only when the soliton does not lie in the middle of the bit slot. Similar to the case of synchronous phase modulation, the direction of the frequency shift is such that the soliton is confined to the center of the bit slot. Other nonlinear effects occurring in optical fibers (Agrawal [ 1995a1) can also be exploited to control the soliton parameters. In one study (Kumar and Hasegawa [ 1995]), stimulated Raman scattering was used for this purpose. If a pump beam, modulated at the signal bit rate and upshifted in frequency by the Raman shift (about 13THz) is copropagated with the soliton bit stream, it simultaneously provides gain (through Raman amplification) and phase modulation (through XPM) to each soliton. Such a technique results in both phase and intensity modulations of the soliton stream and can reduce the timing jitter. Another approach (Grigoryan, Hasegawa and Maruta [ 19951) makes use of four-wave mixing for realizing soliton reshaping and the control of soliton parameters. Timing jitter can also be reduced through dispersion compensation. This mechanism is discussed in 4 8. I . 4.5. EXPERIMENTAL PROGRESS

The first experiment (Mollenauer and Smith [ 19881) that demonstrated the possibility of soliton transmission over transoceanic distances was performed in 1988 by using a recirculating fiber loop whose loss was compensated through the Raman-amplification scheme. The main drawback from a practical standpoint was that the experiment used two color-center lasers for soliton generation and amplification. The availability of diode-pumped EDFAs by 1989 provided an opportunity to use them for amplification of picosecond pulses emitted by modelocked or gain-switched semiconductor lasers so that the peak power is high enough for launching of fundamental solitons in optical fibers. Furthermore, EDFAs could also be used as in-line amplifiers to compensate for fiber loss. Many experimental demonstrations of soliton communications were carried out worldwide starting in 1990. The soliton communication experiments can be divided into two categories, depending on whether a linear fiber link or a recirculating fiber loop is used in the experiment. Although the experiments using a linear fiber link are more realistic, most soliton experiments use a recirculating fiber loop because of cost considerations. In an early experiment (Mollenauer, Nyman, Neubelt, Raybon and Evangelides [ 1991]), 2.5-Gb/s solitons were transmitted over 12000km by using a 75-km fiber loop containing three EDFAs spaced 25km apart. In this experiment, the bit-rate-distance product of BL = 30 (Tb/s)-km was limited mainly by the ASE-induced timing jitter as discussed in

~ v 9:, 41

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213

$4.2. The timing-jitter problem was reduced in several experiments by using optical filters. In a 1991 experiment (Mollenauer, Neubelt, Haner, Lichtman, Evangelides and Nyman [ 1991]), the 2.5-Gb/s signal could be transmitted over 14000 km when fixed-frequency filters were used inside the recirculating fiber loop. Soon afterward, the use of sliding-frequency guiding filters (Mollenauer, Gordon and Evangelides [ 19921) resulted in transmission over 15 000 km of a 5-Gb/s signal (Mollenauer, Lichtman, Harvey, Neubelt and Nyman [ 19921). Moreover, when the bit rate was doubled using wavelength-division multiplexing (discussed in 9), the resulting 10-Gb/s signal could still be transmitted over 11 000 km. In another experiment, the use of sliding-frequency guiding filters provided error-free soliton transmission over 20 000 km at 10 Gb/s in a single channel and over 13 000 km at 20 Gb/s in a two-channel experiment (Mollenauer, Lichtman, Neubelt and Harvey [ 19931). Further improvements in the singlechannel transmission experiments have resulted in soliton transmission over 35 000 km at 10 Gb/s and over 24 000 km at 15 Gb/s (Mollenauer, Mamyshev and Neubelt [ 19941) The BL product reaches 360 (Tb/s)-km for these experiments. In the time-domain approach to soliton control, the timing-jitter problem can be solved by using synchronous modulators within the fiber loop. In a 1991 experiment, solitons at 10 Gb/s could be maintained over one million kilometers when a LiNb03 modulator was used within the 510-km loop incorporating EDFAs with 50-km spacing (Nakazawa, Yamada, Kubota and Suzuki [1991]). In a later experiment, solitons were controlled in both the time and frequency domains by using both modulators and optical filters inside the fiber loop. No performance degradation was observed even after one million kilometers, suggesting that such an approach can maintain solitons over unlimited distances (Nakazawa, Suzuki, Yamada, Kubota, Kimura and Takaya [ 19931). Various successful experimental demonstrations of soliton transmission over distances in excess of 10000 km have prompted the use of solitons for undersea lightwave systems. It is possible that the next transpacific cable (TPC-6) will make use of soliton-like propagation by using the RZ format. Such a system is in the planning stage in 1996 and is scheduled to begin operation in the year 2000. Recently, terrestrial use of soliton communication systems, operating in the 1.3-pm wavelength region with transmission distances -1000 km or less, has been considered. The motivation behind the development of such systems stems from the need to update the existing terrestrial fiber links to 10GbIs and beyond while making use of the 40-50 million kilometers of pre-installed fiber. This worldwide network of standard telecommunication fibers has a relatively high dispersion at the 1.55-pm wavelength [p2 M -21 ps2/km], resulting in a dispersion length LD FZ 10 km if we use TO= 15 ps for a 10-Gb/s soliton system.

214

SOLITON COMMUNICATION SYSTEMS

[IY I 4

Since the required amplifier spacing is - 3 0 4 0 km, it is not possible to satisfy the condition LA << LD needed to operate in the average-soliton regime. However, if the operating wavelength is chosen to be 1.3 pm, LD exceeds 400 km, making it easy to satisfy the condition LA << LD. The penalty to be paid is that the fiber loss is higher at 1.3 pm, and thus a larger amplifier gain is needed to compensate for it. Since practical fiber amplifiers operating at 1.3 pm are not readily available, semiconductor laser amplifiers (Agrawal [ 1995b1) provide an alternative, especially with the recent progress realized in reducing their polarization sensitivity. The main drawback of semiconductor laser amplifiers stems from the fact that the amplified pulse is heavily chirped because of the dynamic index changes which occur with gain saturation (Agrawal and Olsson [ 19891). Another drawback is that the carrier density does not recover fully after the amplification of a single soliton, leaving less gain for the following solitons and producing pattern-dependent power fluctuations. Since soliton amplification is accompanied by pulse-energy-dependent frequency shifts, chirping, and unequal gains, it results in variations in the solitons amplitude and frequencies which degrade the system performance considerably. Sliding-frequency filters (Mecozzi [ 1995a1) or a combination of fixed-bandpass filters and acousto-optic modulators (Wabnitz [1995b]) can solve some of these problems. The design of 1.3-pm soliton communication systems is under extensive study and several European field trials are expected to take place by 1998. Several attempts have been made to increase the bit rate of soliton communication systems beyond the 20-Gb/s limit imposed by the average-soliton regime. In a 1992 experiment (Andrekson, Olsson, Haner, Simpson, TanbunEk, Logan, Coblentz, Presby and Wecht [ 1992]), the 32-Gb/s solitons were transmitted over 90km by using 16-ps pulses obtained from a mode-locked semiconductor laser. Since the bit slot is only about 3 1 ps wide at the bit rate of 32Gb/s, neighboring 16-ps solitons were so close to each other [the parameter qo = 1 from eq. (3. I)] that soliton interaction limited the transmission distance to only 90 km. Nonetheless, in a 1993 experiment (Iwatsuki, Suzuki, Nishi and Saruwatari [ 1993]), the bit rate could be doubled to 80 Gb/s by using polarization-division multiplexing (discussed in Q 9) such that the neighboring bit slots carried orthogonally polarized soliton pulses. The 80-Gbh signal was transmitted over 80 km by this technique. The same technique was later used to extend the bit rate to 160 Gb/s (Nakazawa, Suzuki, Yoshida, Yamada, Kitoh and Kawachi [ 19951) by multiplexing two 80-Gb/s channels. At such high bit rates, the bit slot is so small (lops wide at lOOGb/s) that typically the soliton width is 3 ps or less. For such ultrashort solitons several higher-order nonlinear effects

~ vI, 51

QUASI-ADIABATIC REGIME

215

become quite important. The next few sections are devoted to the design of such high-speed soliton communication systems.

0 5. Quasi-Adiabatic Regime In 4 4, we discussed a regime in which the amplifier spacing is short compared with the dispersion length (LA << LD). In this average-soliton regime, large loss-induced energy variations may occur over one amplifier spacing with little distortion of the soliton. As the soliton width is decreased in order to increase the bit rate, the condition LA << L, no longer holds, resulting in shape distortion and considerable emission of dispersive waves when LA M LD. A further decrease of the soliton width leads to a new regime in which LD << LA and considerable dynamic evolution of the soliton occurs over one amplifier spacing. The fate of a soliton in such a regime depends strongly on the loss per dispersion length and is governed by the dimensionless parameter a introduced in eq. (2.6). If a >> I, solitons are perturbed strongly and cannot survive over long lengths. On the other hand, if a << I , each soliton can adapt adiabatically to losses by increasing its width and decreasing its peak power while preserving its soliton nature. For this reason, this regime is referred to as the quasi-adiabatic regime. The condition a << 1 can be related to the bit rate by using a = = aoTi/IpZl and eq. (3.1), and becomes

For Q = 0.2 dB/km, qo = 5, and k = -1 ps2/km, B >> 21 Gb/s or the bit rate should exceed 40 Gb/s. Clearly, the quasi-adiabatic regime is most suitable for high-speed soliton communication systems. The APT theory can be used to study how the soliton width increases in the quasi-adiabatic regime because of fiber losses. From eq. (2.5), the perturbation ~ ( u=) -ia/2. The use of this perturbation in eqs. (2.10)-(2.13) then shows that the soliton amplitude decreases as exp(-aE). Since soliton width varies inversely with the soliton amplitude, it increases exponentially as:

Similar results can also be obtained by using the moment method (Belanger and Belanger [ 19951). For periodically amplified solitons with an amplifier spacing Ea = LA/LD, the width increases by a factor of exp(a$) when it reaches the

216

SOLITON COMMUNICATION SYSTEMS

tw § 5

amplifier. Since each soliton sheds a part of its energy in the form of dispersive waves, whose emission should be minimized as much as possible, clearly the soliton width should not increase by too large a factor. A compromise consists of choosing LA such that a& < 1 so that the soliton width does not increase by more than a factor of 2.7. For 00 = 0.2 dB/km, the amplifier spacing is then limited to about 22 km. The main point to note is that the soliton character of the optical pulse is maintained in spite of an increase in the soliton width because of the adiabatic nature of soliton evolution. According to eq. (5.1), operation in the adiabatic regime requires the use of ultrashort solitons (Ts < 5ps). As the soliton width decreases, higher-order dispersive and nonlinear effects become important. These effects are included by adding three terms to the right side of eq. (2.5) resulting in a generalized NSE (Agrawal [1995a]),

where the three dimensionless parameters are defined as (5.4) The terms involving 6, s and ZR take into account the effects of TOD, self-steepening, and the Raman effect, respectively. Self-steepening becomes important only for solitons shorter than lOOfs and can be safely neglected by setting s = 0. The Raman effect leads to a continuous downshift of the soliton carrier frequency, an effect known as the soliton self-frequency shift (SSFS, Gordon [1986]). SSFS is negligible for T, > 10 ps, but becomes of considerable importance for short solitons (T, < 5ps) since it scales as T,". Since TR FZ 5 fs, the Raman term can be treated accurately as a small perturbation, even for T, = 1 ps. The TOD term becomes important when IpZl is below 0.1 ps2/km. As an example of the smallness of the three higher-order terms, 6 x 0.01, s FZ 0.001, and t~ x 0.005 when To = 1 ps for typical values of fiber parameters. The operation of a soliton communication system in the quasi-adiabatic regime requires removal of dispersive waves. Numerical simulations based on eq. (5.3) show that dispersive waves not only accumulate over multiple amplification stages but also eventually destroy the integrity of the soliton because of its interaction with dispersive waves. Figure 5 shows the destruction of a 4.5-ps soliton after only seven amplifiers because of this interaction. An

IV,

0 51

QUASI-ADIABATIC REGIME

217

Fig. 5. Stabilization of a 4.5-ps soliton by FSAs in the quasi-adiabatic regime with 20-km amplifier spacing. For comparison, soliton evolution without FSAs is shown on the right. The soliton is destroyed after only 8 amplification stages when FSAs are not used.

FSA inserted before or after each amplifier can solve the interaction problem since it blocks the low-power dispersive waves while letting the soliton pass through it. Several kinds of FSAs have been considered for this purpose, including multiquantum-well structures (Atkinson, Loh, Afanasjev, Grudinin, Seeds and Payne [ 1994]), twin-core fibers (VallCe and Essiambre [ 1994]), nonlinear amplifying-loop mirrors (Smith and Doran [ 1995b1, Matsumoto, Ikeda, Uda and Hasegawa [1995]), and devices based on the nonlinear polarization evolution (Knox, Harper, Kean, Doran and Bennion [1995]). Figure 5 shows how a 4.5-ps soliton can propagate over 50 amplifiers when an FSA is inserted before each amplifier. The use of FSAs stabilizes the soliton propagation not only by reducing the dispersive waves but also by inhibiting formation of secondary solitons (Tai, Hasegawa and Bekki [ 19881). Numerical simulations show that the amplifier spacing is limited to below -25 km because of the onset of secondary solitons, in agreement with the qualitative estimate of 22 km obtained by using the criterion a& = 1. SSFS becomes the most limiting factor in the quasi-adiabatic regime when FSA are used to control the growth of dispersive waves. To control the accumulation of the SSFS, insertion of optical filters (bandwidth -3 times the soliton spectral width) has been considered (Malomed [ 19941, Essiambre and Agrawal [1995b]). Because the SSFS must always stay a small fraction (typically 10%) of the filter bandwidth, the width of the soliton is limited to values above 1 ps. By considering simultaneously the effects of lumped gain with filtering, the SSFS, and energy absorption through the FSA, a simple semi-analytic model (Essiambre and Agrawal [ 1995b1) can predict the width (just before amplification where the soliton is the widest) of the steady-state soliton. The results show that the amplifier spacing needed for stable transmission of solitons drops from 20 km to 5 km as the soliton width is reduced from 5 to 2 ps G(32 = -1 ps2/km). Since

218

"§ 5

SOLITON COMMUNICATION SYSTEMS

-75

-50

-25

0

Time (ps)

25

50

-75

-50

-25

0

25

50

Time (ps)

Fig. 6 . Destruction of a IO-Gb/s soliton train in the quasi-adiabatic regime because of dispersivewavessoliton interaction (left) and its restoration with the use of FSAs (right). In these constantintensity contours, a tilt of the soliton's trajectory is due to TOD. Accumulation of dispersive waves is also evident when FSAs are not used.

the bit slot is only l o p s at lOOGb/s, one must use short solitons of 2 p s width to avoid soliton interaction. However, such a system requires 5-km amplifier spacing, a value considered to be too small to be practical. The transmission of ultrashort solitons in the quasi-adiabatic regime has been reported (Nakazawa, Yoshida, Yamada, Suzuki, Kitoh and Kawachi [ 19941) by using 2.7-3.0~s solitons with LA = 25 km and 3-nm bandwidth filters in dispersion-shifted fibers having pZ = -0.25 ps2/km and fn, = 0.1 ps3/km. A close packing of solitons allowed transmission at a bit rate of 80Gb/s, but the transmission distance was limited to 500km. Since FSAs were not used in this experiment, accumulation of dispersive waves is expected to occur. Indeed, numerical simulations performed by using the experimental parameters show that the transmission distance is limited by the soliton-dispersive waves and solitonsoliton interactions (Essiambre and Agrawal [ 1995al). In fact, the transmission distance can be increased by inserting periodically either saturable absorbers or synchronous modulators, both of which control the growth of dispersive waves. Figure 6 shows the results of numerical simulations of the 80-Gb/s experiment of Nakazawa together with the improvement occurring with the insertion of FSAs before every amplifier. In an effort to increase the bit rate, polarization multiplexing (see $9.2) has been used (Nakazawa, Suzuki, Yoshida, Yamada, Kitoh and Kawachi [1995]) with a set-up similar to the one used for the 8O-Gb/s experiment. The reduced soliton-soliton interaction and shorter solitons ( 1.6ps width) allowed transmission at 160 Gb/s over 200 km. The transmission distance is again limited

IV, § 61

DISTRIBUTED AMPLIFICATION

219

by the interaction of dispersive waves with the soliton bit stream. The use of FSAs or synchronous modulators should help. Dispersion management (see 5 8) may also play an important role in increasing the transmission distance as well as the amplifier spacing for such systems.

Q 6. Distributed Amplification As discussed earlier, the distributed-amplification scheme presents inherent advantages over lumped amplification since its use provides a virtually “lossfree” fiber by locally compensating the loss at every point along the fiber link. In fact, this scheme was used as early as 1985 to demonstrate soliton transmission over a 10-km-long fiber (Mollenauer, Stolen and Islam [ 1985]), and than later extended in 1988 to a 4000-km length by using a recirculating fiber loop (Mollenauer and Smith [1988]). In these experiments, the gain was provided through Raman amplification by pumping the transmission fiber at a wavelength of about 1.46 pm from a color-center laser. When erbium-doped fiber amplifiers became available at the end of the 1980s, lumped amplification became more popular because of the lower pump-power requirements of such amplifiers. Their advent led to the notion that one can dope the transmission fiber itself and pump it periodically to provide distributed amplification to the soliton. Several experiments have demonstrated that solitons can be propagated in such “active” fibers. Ideally, distributed amplification requires a constant gain gs per unit of length to match the constant loss rate 00 of fibers. However, since not only the signal but also the pump suffer from fiber loss, the effective gain coefficient decreases with increasing pump depletion. Variations in the soliton energy can be studied by solving (Mollenauer, Gordon and Islam [1986]):

The gain coefficient gs(z) should be calculated by solving the rate equations appropriate for the pumping scheme used (Agrawal [ 1995a1). Assuming that the fiber is pumped equally on both ends to reduce the effects of gain variations and neglecting the gain saturation, the gain coefficiht gs(z) can be approximated as:

where ap is the fiber loss at the pump wavelength and L is the length of the active fiber. The gain constant go is related to the pump power injected at both

220

[N P

SOLITON COMMUNICATION SYSTEMS

6

ends and can be calculated easily for achieving transparency (the condition for which the output soliton has the same energy as the input soliton). The variation of energy along the fiber length is obtained by integrating eq. (6.2) and follows:

where E,, is the soliton energy at the fiber input. The range of soliton energy variations (called the energy or power excursion) increases with the fiber length as a consequence of a more severe depletion of the pump beams. Nevertheless, distributed amplification can significantly reduce the extent of power excursion when compared to the lumped-amplification case. The effect of power excursion on solitons propagating in a chain of active fibers strongly depends on the ratio = LA/LD, where LA now stands for the distributed-amplifier spacing or the spacing between the pump stations. When L A << LD, little soliton reshaping occurs even in presence of large power excursion, and stable transmission over long distance can be achieved. This regime is similar to the average-soliton regime for passive fibers. For LA >> L D , the soliton will adiabatically reshape itself following the power excursion with limited emission of dispersive waves. For intermediate values of a more complicated behavior occurs. In particular, dispersive waves and solitons are amplified resonantly when 2ja 4n. The periodic amplification of the soliton and dispersive waves and their interference lead to unstable and chaotic behavior (Mollenauer, Gordon and Islam [1986]). Although the qualitative description of the distributed and lumped amplification seems similar, two important differences should be pointed out. First, the power excursion in active fibers can be smaller than for passive fibers. This feature permits one to push the average-soliton limit to LA M 0 . 7 L ~(Spirit, Marshall, Constantine, Williams, Davey and Ainslie [ 19911, Rottwitt, Povlsen, Gundersen and Bjarklev [ 19931, Lester, Bertilsson, Rottwitt, Andrekson, Newhouse and Antos [ 19951). Second, for an active fiber pumped to transparency, the energy of the output soliton in each section of a chain of active fibers is identical to its value at the input. Consequently, in the adiabatic regime of distributed amplification, the width of the output soliton should be nearly the same as at the input end. This is in contrast to the adiabatic regime in passive fibers where considerable soliton broadening occurs, resulting in a continuous increase of the local dispersion length LD along the fiber length. Since LD increases with z, the condition LA << LD is more stringent for systems using passive fibers. Modeling of soliton communication systems making use of distributed amplification requires the addition of the gain term to the generalized NSE

ca

-

ca,

IV,

I 71

22 1

DISPERSION-DECREASING FIBERS

given by eq.(5.3). However, it is important to include the effects of finite gain bandwidth. In a simple approach, the equation describing distributed amplification can be written as: .&

1-

1d2U ia + -+ 1111224 = ---u+ agr 2~ 2

ig, 2

-

d +i6-d 3 U -is-(lul at3 dt

2

u)+

dluI2 dt ’

~Ru-

where g, represents the distributed gain. The parameter t 2 = T2/T0,where T2 is the relaxation time of the gain medium and is inversely related to the gain bandwidth Avg. Numerical simulations based on eq. (6.4) show some of the benefits of the distributed-amplification scheme. For solitons < 5ps, the SSFS leads to considerable changes in the soliton evolution: the SSFS changes the effective gain (due to a finite gain bandwidth) and GVD (due to TOD) experienced by the solitons. However, the finite gain bandwidth can reduce the effects of SSFS, stabilizing the soliton frequency at a value slightly below the frequency of the gain peak (Nakazawa, Kubota, Kurokawa and Yamada [ 19911). Under certain conditions, the SSFS can become so large that it cannot be compensated. In that situation, the soliton just moves out of the gain window and loses all of its energy. It is worth pointing out that the gain of a distributed erbium-doped fiber amplifier typically exhibits a double-peak structure due to inhomogeneous broadening. If homogeneous broadening dominates, it can be modeled accurately by a Lorentzian profile in place of the parabolic approximation used in the derivation of eq. (6.4). For a Lorentzian profile, the control of the SSFS becomes less efficient (Afanasjev, Serkin and Vysloukh [ 1992]), consistent with the experimental observation of subpicosecond soliton propagation in 18 km of distributed erbium-doped amplifiers (Nakazawa and Kurokawa [ 19911, Kurokawa and Nakazawa [ 19921).

Q 7. Dispersion-Decreasing Fibers 7.1. BASIC IDEA

An interesting scheme for stable propagation of solitons restores the balance between GVD and SPM in a lossy fiber by changing the dispersion properties of the transmission fiber (Tajima [1987]). Such fibers are called dispersiondecreasing fibers (DDFs) because their GVD must decrease in such a way

222

SOLITON COMMUNICATION SYSTEMS

[IV, § 7

that it compensates for the reduced SPM experienced by the soliton weakened from fiber loss. Since the soliton peak power decreases exponentially, the balance between GVD and SPM can be maintained if the GVD also decreases exponentially as

For such a dispersion profile, the soliton keeps a constant width even in a lossy fiber. However, for solitons shorter than 2ps, the dispersion profile should be modified (Essiambre and Agrawal [1996a]) to take into account the combined action of the Raman effect and TOD. DDFs are also beneficial for WDM application. This topic is treated separately in Q 9.1. Fibers with an exponential GVD profile have been fabricated. A practical technique for fabricating DDFs consists of reducing the core diameter along the fiber length in a controlled manner during the computer-controlled fiber-drawing process. Variations in the fiber diameter change the waveguide contribution to k and reduce its magnitude. Typically GVD varies by a factor of 10 over a length of 30-50km. The accuracy realized by the use of this technique is estimated to be better than 0.1 ps2/km (Bogatyrjov, Bubnov, Dianov and Sysoliatin [ 19951, Richardson, Chamberlin, Dong and Payne [ 19951). Since fibers with a continuously varying dispersion are not always readily available, a simple approach consists of approximating the exponential profile with a “staircase” by splicing together several constant-dispersion fibers with different In, values (Ren and Hsu [1988], Chi and Lin [1991]). This method is discussed in Q 8.2. Note that the concept of dispersion-varying fiber can also be applied to the distributed amplification scheme (van Tartwijk, Essiambre and Agrawal [ 19961) to alleviate the condition LA < LD. Propagation of short solitons in DDFs has been demonstrated in two experiments by using a 40-km DDF and soliton widths down to 2ps (Stentz, Boyd and Evans [1995], Richardson, Chamberlin, Dong and Payne [1995]). In both cases, the 40-km DDF resulted from splicing two 20-km sections of DDFs having their GVD matched at the splice. The soliton preserved its width in spite of an energy loss of more than 8 dB. Data transmission in DDFs has also been attempted. In a recirculating-loop experiment (Richardson, Dong, Chamberlin, Ellis, Widdowson and Pender [1996]), a 6.5-ps soliton train at 10Gb/s could be transmitted nearly error-free (BER < over a distance of 300km. The acoustic timing jitter is believed to be limiting factor because of a relatively high value of the average dispersion for the DDF used in the experiment. The effects of SSFS and TOD would also limit the performance of such systems, especially

Iv, B 71

DISPERSION-DECREASINGFIBERS

223

through the timing jitter for a DDF having a low average dispersion, an issue discussed next. 7.2. TIMING JITTER

The propagation of ultrashort solitons through DDFs can be studied by using eq (5.3) after including the axial variation of GVD. The resulting equation becomes

where p ( Q = lk(E)/h(O)l is the normalized GVD at and the shock term has been neglected by setting s = 0. The distance 5 is normalized to the dispersion length L D = Ti/IpZ(O)l,defined by using the GVD value at the fiber input. Because of the E dependence of the second term, eq. (7.2) is no longer a standard NSE. However, it can be transformed into a perturbed NSE by using the transformations:

These transformations represent renormalization of the soliton amplitude and the distance scale to the local GVD. In terms of u and q, eq. (7.2) becomes

If the GVD profile is chosen such that dp/dr] = -a, or p(E) = exp(-aE), fiber loss has no effect on soliton propagation. This is the same profile as in eq. (7.1). Since the left side of eq. (7.4) corresponds to the standard NSE, one can apply the APT of 4 2.3 to study the effects of SSFS and TOD on the timing jitter. By using eq. (2.10)-(2.12) with the last two terms of eq. (7.4) acting as perturbation, one can obtain the following expression for the soliton displacement at the end of a single DDF of length Ea (Essiambre and Agrawal [1996b]):

where two higher-order cross terms resulting from the combined effect of SSFS and TOD have been neglected since they are proportional to the product of 6 and

224

SOLITON COMMUNICATION SYSTEMS

[IV,

I7

The parameters q G H and q R govern the soliton jitter induced by the GordonHaus and Raman effects, respectively, and are defined as: ZR.

qcH

=

qR

=

-[ 1 - exp(-a$)]/a, 1 5 - - exp(-aEa) 15a2 2

[

(7.6)

1

+exp(-2a~ .

(7.7)

By considering amplitude and frequency fluctuations induced by the amplifier noise and adding the contribution of N amplifiers in a long amplifier chain ( N >> l), timing jitter can be calculated by following a procedure similar to that of 94.2. The variance of timing jitter is found to be (Essiambre and Agrawal [ 1997a1)

+(N3/3)q&ais

+

Variances of soliton parameters are given in eqs. (3.5)-(3.8). In eq. (7.8), the term ( N 3 / 3 )q& a& represents the Gordon-Haus jitter for a soliton link making use of DDFs. The term NO,”, comes from the direct effect of the ASE on soliton position. The terms involving are related to the Raman-induced and to TOD-induced timing jitters. The Raman-induced timing jitter originates from the SSFS experienced by a soliton and depends on its amplitude. Fluctuations in the soliton amplitude introduced by amplifier noise result in fluctuations in the soliton frequency through the Raman effect, which are translated in position fluctuations by the GVD. The first Raman term proportional to N 5 generally dominates for ultrashort solitons. The TOD contribution to timing jitter becomes of importance if the magnitude of the minimum dispersion I/$’”[ of DDFs falls below 0.1 ps2/km or if the Raman jitter is compensated by some optical technique. Figure 7 shows the individual contributions of amplitude, frequency, and position fluctuations together with the total timing jitter as a function of transmission distance for soliton widths in the range 1-40ps by choosing @’” = -0.1 ps2/km and L A = 80 km. For 40ps-wide solitons, timing jitter originates mostly from frequency fluctuations (Gordon-Haus jitter), except at distances below 800 km for which position fluctuations a,, dominate over frequency fluctuations. For 20 ps-wide solitons, a typical width in the averagesoliton regime, only Gordon-Haus jitter contributes significantly since the contributions of Raman and TOD effects are small for such relatively broad

IV,

P 71

-2 m 14 n

cn

.c E

1

(a)

T, = 40 PS

---

Amplitude Frequency

Position

I '

8 _______c--------

0

0

2500

(b)

g

225

DISPERSION-DECREASING FIBERS

5000

7500

J

10000

3 2

_..' A -

01

0

T, = 20 ps

_.A

250

(d) T,=

500

750

1000

1 PS

20

m c

'E t o i= 0

-- --0

2500

5000

7500

Distance (km)

____----10000

0

100

200

300

__-400

500

Distance (krn)

Fig. 7. Relative contributions of amplifier-induced frequency, amplitude and position fluctuations to the timing jitter as a hnction of propagation distance in dispersion-decreasing fibers for several soliton widths. Total timing jitter is also shown by a solid line.

solitons. When shorter solitons are used, the contribution of higher-order effects, especially the Raman effect, increases rapidly with transmission distance. For 3-ps or shorter solitons, the contribution of amplitude fluctuations to the timing jitter (mediated through the Raman effect) becomes so important that the total transmission distance is limited to only a few hundred kilometers in the absence of a soliton-control mechanism. Since the effect of amplitude fluctuations on the timing jitter increases more rapidly than that of frequency fluctuations [ N s versus N 3 dependences in eq. (7.8)], the former will dominate for long distances. Amplitude fluctuations start to dominate after 8OOkm for 3-ps solitons. For a transoceanic distance of 10000 km, amplitude fluctuations dominate for soliton widths below 7 ps. For 1-ps solitons, amplitude fluctuations totally dominate the timing jitter at all distances. 7.3. OPTICAL PHASE CONJUGATION

The increase in the timing jitter brought by the Raman and TOD effects and a shorter bit slot at higher bit rates (only lops wide at 100 Gb/s) make the control of timing jitter essential before such systems become practical. The technique of optical phase conjugation (OPC) is known to be very effective in reducing the timing jitter even for average solitons. The same technique turns out to be quite beneficial to DDF-based systems. The implementation of OPC requires

226

SOLITON COMMUNICATION SYSTEMS

4

I

.,'

'

I

',.

.

.

, . . . .,,

.

.

. . ,

h

g3

v

P, = 0 ps3/km P, = 0.002ps3/krn - _ - P, = 0.01ps3/km - Ps = 0.05ps3/krn P, = - 0.0002ps3/krn

6

z

..___

:=. 2

.-!=

0)

E l i= 0

1000

3000

5000

7000

9000

Distance (krn) Fig. 8. Effect of third-order dispersion (p3) on timing jitter in a periodically amplified DDF-based soliton communication system making use of phase conjugation. The thick solid curve shows the Gordon-Haus jitter.

insertion of an optical element before each amplifier that changes the soliton field from us to u: while preserving all other features of the bit stream. Such a change is equivalent to inverting the soliton spectrum around the wavelength of the pump laser used for the four-wave-mixing process. The timing jitter changes considerably because of OPC. We refer the interested reader to Essiambre and Agrawal [1997b] for details, and give only the result here. The variance of the timing jitter now becomes

(o:")~= ( 8 N q i u ; + 4N2qR&ut + 4- N 3 d 2 & 4 ; )

ui,

3

+ (N/2)q&& + NO:*.

(7.9)

This equation should be compared with eq. (7.8) obtained without OPC. The N dependence of the Gordon-Haus contribution changes from N 3 to N while the Raman-induced jitter is reduced even more dramatically - from N 5 to N . These reductions come from the OPC-induced spectral inversion that provides compensation for the effects both the GVD and SSFS. However, OPC does not compensate for the effects of TOD. As a result, the term involving $ N 3 b 2 & 4 g is the same while the cross-term is reduced, as it involves the Raman effect. The effects of TOD are shown in fig. 8 by plotting the timing jitter of 2-ps solitons undergoing periodic OPC. The dashed horizontal line represents the tolerable timing jitter. For comparison, the timing jitter obtained by considering only the Gordon-Haus term in eq. (7.9) is shown by the thick solid line. Other curves correspond to different values of the TOD parameter /J-J.For = 0.05ps3/km, a typical value for dispersion-shifted silica fibers,

IV,

5 81

DISPERSION MANAGEMENT

221

the transmission distance is limited by TOD to below 1500km. However, considerable improvement occurs when is reduced. Transmission over 7500km is possible for = 0 and the distance can be further increased for slightly negative values of It should be stressed that a complete description of timing jitter requires inclusion of other effects such as dispersive waves, acoustic waves, and PMD. Dispersive waves are likely to become important for low average dispersion while the acoustic effect may take over for large values of average dispersion.

a

a.

0

8. Dispersion Management

Generally speaking, operation at a wavelength where the GVD is low should improve the overall system performance as it lowers the required average power, reduces the timing jitter, and lowers the magnitude of the SSFS for short solitons. However, as the operating wavelength approaches the zero-dispersion wavelength, several effects prevent reliable operation at a low value of Ip21 (Ellis, Widdowson and Shan [ 19961). For single-channel systems, the effects of TOD can induce severe pulse distortion and emission of dispersive waves when lp2 I < 0.1 ps2/km. In WDM systems, another limitation arises because of the four-wave mixing process which becomes nearly phased-matched when Ip2 1 is low. This section discusses techniques which can be used to counteract such effects. 8. I , DISPERSION COMPENSATION

To overcome the difficulties encountered for solitons propagating near the zerodispersion wavelength, a simple approach consists of concatenating fibers with different dispersion characteristics (Kubota and Nakazawa [ 19921). Each fiber section is chosen to have a moderate value (IpZl > 0.3ps2/km) of GVD that could be negative or positive such that the average dispersion over the entire link remains low (IpyI < 0.1 ps2/km). Under certain conditions, such a dispersion compensation is referred to as “partial soliton communication” (Kubota and Nakazawa [1992]). It can also be realized by assembling a collection of fiber segments with a distribution of zero-dispersion wavelengths, resulting in a technique known as dispersion allocation (Nakazawa and Kubota [ 19951). In both methods, dispersion compensation allows a reduction of the average dispersion while avoiding the impairments related to TOD and four-wave mixing.

228

SOLITON COMMUNICATION SYSTEMS

0 DISTA NCE Fig. 9. Two dispersion-management techniques. Thick lines correspond to the dispersion profile resulting when a short section of dispersion-compensating fiber & N +80ps2/km) is inserted just before each amplifier. Thin lines show the case of two fiber segments of equal lengths but opposite dispersions. In both cases the average dispersion is slightly anomalous.

In its simplest form, dispersion compensation is realized by adding a relatively short fiber segment of large positive dispersion to the long transmission fiber of negative dispersion (see fig. 9, thick lines). The short section is referred to as the dispersion-compensating fiber (DCF) since its role is to compensate for the GVD in the transmission fiber. If the long and short sections have lengths Ll and L2, respectively, with /3:” and @’ being the corresponding GVDs, the average dispersion experienced by a soliton is given by:

A condition similar to the average-soliton criterion LA << LD applies for dispersion-compensated systems, but it is much less restrictive as higher values of the ratio Ea = LA/LD are allowed. In the absence of fiber nonlinearity, the effect of GVD in a fiber segment of dispersion pi’’ and length L1 can be described in terms of the cumulative dispersion #)L1. In such a linearly dispersive system, the effect of dispersion can be canceled exactly by joining a fiber of dispersion and length L2 such that the cumulative dispersion cancel each other (flf’L2 = -/3$”L,). However, optimum dispersion compensation does not correspond to exact cancelation of cumulative dispersion in operating systems because of the presence of nonlinear effects. In fact, when the effects of SPM are included, optimum dispersion compensation occurs when the total dispersion of the DCF inserted just before each amplifier is chosen to be 85-95% of the cumulative dispersion of the

of’

IV, § 81

DISPERSION MANAGEMENT

229

transmitting fiber (Suzuki, Morita, Edagawa, Yamamoto, Taga and Akiba [ 19951, Naka, Matsuda and Saito [1996], Wabnitz, Uzunov and Lederer [1996]). In the case of dynamic soliton communication, a similar condition exists (Kubota and Nakazawa [ 1992]), but the cumulative dispersion of the transmission fiber is now given by fii’)(LI - L-), where LC is the length over which soliton recovers its width when launched at the fiber input. In dispersion-compensated systems, soliton propagation is influenced not only by the average GVD value, but also by the length and the number of fiber segments as well as the order in which they are spliced together. For example, placing the DCF before instead of after (fig. 9, thick lines) the transmitting fiber degrades the system performance by introducing an excess of SPM (Knox, Forysiak and Doran [1995]). Increasing the number of fiber segments allows for a longer amplifier spacing (Nakazawa and Kubota [1995]) and improves transmission fidelity (Ohhira, Hasegawa and Kodama [ 19951). When two fiber segments of equal lengths but of nearly opposite GVD values are used (see fig. 9, thin lines), the power required to maintain the soliton increases with the GVD difference, Ap2 3 - fi;”]. This feature suggests that the description of soliton dynamics in terms of average dispersion loses its accuracy, especially in heavily dispersion-managed systems in which IAp2/fiFI >> 1. The excess power necessary to support soliton transmission in dispersion-compensated systems arises from the interplay between GVD and SPM (Smith, b o x , Doran, Blow and Bennion [1996]). This increase in power can be avoided by compensating GVD with a dispersive medium having a negative value of n2, such as a semiconductor waveguide operating close to, but below, the two-photon resonance (Pare, Villeneuve, BClanger and Doran [ 19961). A general method capable of determining the optimum design of a dispersioncompensated system with N fiber segments is difficult to establish because of the multiple effects which should be taken simultaneously into account. In one approach (Ohhira, Hasegawa and Kodama [1995]), M fiber segments of the same length but having a Gaussian distribution of GVD are grouped in pairs. Each pair is composed of two segments of nearly opposite GVD values such that the average GVD has a small negative value for each pair. In a field experiment on the Tokyo optical-loop network (Nakazawa, Kimura, Suzuki, Kubota, Komukai, Yamada, Sugawa, Yoshida, Yamamoto, Imai, Sahara, Nakazawa, Yamauchi and Umezawa [ 1995]), the concept of dispersion allocation has been applied successfully to demonstrate soliton transmission at 10 Gbis over 2000km, despite the fact that the installed fibers were not originally intended to support solitons. In another investigation (Gabitov and Turitsyn [ 1994]), it was found that because of dispersion compensation, solitons undergo periodic

230

[IY

SOLITON COMMUNICATION SYSTEMS

5

8

oscillations of not only peak power (as in the average-soliton regime) but also in width and shape. Because of rapid oscillations of the pulse superposed on the soliton-like evolution over long distances, such pulses are referred to as a “breathing soliton.” It is worth pointing out that the latter result also applies to systems making use of the NRZ format. Lowering of average GVD 1/3?1 through dispersion compensation not only reduces pulse distortion but also lowers the timing jitter (Kubota and Nakazawa [1992], Tang and Chin [1995]). Pulse distortions can be reduced to such an extent in dispersion-compensated systems that considerable variations in the input peak power of the solitons can be tolerated (Nakazawa, Kubota, Sahara and Tamura [ 19961). Surprisingly, the timing jitter in dispersion-compensated system is reduced to a value even smaller than the value calculated by using the average value of dispersion (Smith, Forysiak and Doran [ 19961). This behavior is a consequence of the fact that the average power launched into the fiber increases relative to a system having a uniform dispersion of same Even though less dramatic in dispersion-compensated systems than in constant-dispersion systems, with a reduction in IfiYI is associated a reduction of the soliton peak power P,, which may degrade the signal-to-noise ratio (SNR) at the receiver (Smith and Doran [1995a]) enough to increase the BER. The optimum dispersioncompensation scheme that minimizes the BER must thus take into account these two different sources of error (timing jitter and signal-to-noise ratio). In an experiment performed at 20 Gbls over transoceanic distances (Suzuki, Morita, Edagawa, Yamamoto, Taga and Akiba [ 19951) using guiding filters and a transmission fiber having pZ = -0.25ps2/km, the maximum distance of error-free transmission (BER < lop9) was achieved by reducing the average dispersion lfiyl over each span to a value as low as -0.025ps2/km. One should note, however, that when dispersion compensation is applied only once in the entire link (at the receiver; i.e., post-transmission compensation), the best performance is obtained by compensating only 50% of the total dispersion of the communication line (Forysiak, Blow and Doran [ 19931, Chandrakumar, Alouini, Thomine, Georges and Pirio [ 19941). The latter limit appears to be due to soliton broadening resulting from the large value of GVD of the DCF necessary for the compensation of the dispersion accumulated over the entire link.

fir.

8.2. DISPERSION PROFILING

Besides its application for dispersion compensation, the concatenation of fiber segments with different GVD values represents a convenient alternative (Ren and Hsu [ 19881, Chi and Lin [1991]) to the development of fibers with continuously

Iv, 9 81

DISPERSION MANAGEMENT

23 I

varying dispersion (Bogatyrjov, Bubnov, Dianov and Sysoliatin [ 19951). It turns out that as few as four or five fiber segments may satisfactorily emulate the exponential GVD profile of a loss-matched DDF (Forysiak, b o x and Doran [ 1994a1, Mamyshev and Mollenauer [ 19961) required for high-bit-rate soliton systems (see $ 7 ) . How should one select the length and GVD of each fiber segment for emulating a DDF? The answer is not obvious, and two methods to optimize the segments length and their GVD values have been proposed. This subsection discusses and compares the two methods. One method is based on the minimization of the quantity (Forysiak, Knox and Doran [ 1994b1)

where M is the number of segments, P,;(z) = P, exp(-aoz) is the actual soliton peak power in the ith fiber segment while F$' is the corresponding peak power of an average soliton. We refer to this method as minimization of power deviations (MPD). An example of the dispersion profile for M = 5, emulating a 50-kmlong DDF of average dispersion b,"" = -I .27 ps2/km and obtained by minimizing eq. (8.2), is shown in fig. 10; lengths of the fiber segments and their GVD values are listed in table 1. The second method divides the total cumulative dispersion ~ Y L(Mamyshev A and Mollenauer [ 19961) into M equal parts. Each fiber segment is assumed to have a cumulative dispersion of @,""L*/M. The length and GVD of each segment is computed by applying the condition for having an average soliton in each section. We refer to this method as the equipartition of cumulative dispersion (ECD) method. One can easily show that the GVD values and the lengths of each section should satisfy the following requirements:

1

{

Li= -- In 1 a

[

I)$-(

1 - exp

} Ly,

(8.4)

where i = 1, ..., M and L r = TOZ/lpTI. The results for this method obtained for the same DDF used to illustrate the MPD method are shown in fig. 10 and table 1. Although the two staircase-like GVD profiles follow quite closely the ideal exponential profile and reduces to it for M + 00, they are not identical. The issue of the optimum profile that

232

[IV, § 8

SOLITON COMMUNICATION SYSTEMS

1

3.5

'

'

I

3.0 2.5 h

E

.-.Y

2.0

v1

a 1.s

W

1.o

0.5

0.0

'

'

0

10

20

30

50

40

Distance (km) Fig. 10. Two different models of approximating the exponential dispersion profile of a DDF with multiple fiber segments having a constant dispersion. ECD: equipartition of cumulative dispersion; MPD: minimization of power deviations.

Table I Comparison of the MPD and ECD methods for emulating a SO-km-long loss-matched DDF

M PD

Segment

ECD

I82 I (ps2/km)

Length (km)

1821 ( P S 2 / W

6.40

2.83

4.3 1

2.95

7.50

2.0s

5.38

2.37

9.07

1.40

1.17

1.78

Length (h)

1 1.46

0.88

10.78

1.18

15.57

0.47

22.36

0.57

emulates a DDF best is far from being completely understood. On physical grounds, such a profile should generate the least amount of dispersive waves and, for WDM systems (see $9.1), minimize asymmetric collisions. In a study of a two-step dispersion profile (Georges and Charbonnier [ 19961) for a singlechannel transmission, the best profile that minimized the emission of dispersive waves was found to be generally different from the two profiles described above.

IV,

P

91

CHANNEL MULTIPLEXING

233

For WDM systems, numerical simulations (Kolltveit, Hamaide and Audouin [ 19961) show that the two-step dispersion profile that minimizes the timing jitter appears to be closer to the profile obtained by using the MPD method than the profile given by the ECD method (eqs. 8.3 and 8.4).

0

9. Channel Multiplexing

It should be clear from the preceding discussion that increasing the bit rate of long-haul soliton communication systems beyond the limiting value (about 20 Gb/s) imposed by the average-soliton regime represents a serious challenge as the solitary nature of the optical soliton becomes increasingly difficult to preserve. The use of a densely-packed train of solitons that multiplexes several channels through time-division multiplexing (TDM) with a total bit rate > 20 Gb/s generally results in pulse distortion and emission of dispersive waves because of strong interaction among solitons. As discussed in previous sections, several system or fiber designs can be used to reduce these troublesome effects. However, the narrow spectral bandwidth of a 20-Gb/s soliton train ( M 0.2 nm) relative to the bandwidth of erbium-doped fiber amplifiers (-30 nm) may suggest the use of wavelength-division multiplexing (WDM). Moreover, since a singlemode fiber supports two polarization modes, polarization-division multiplexing (PDM) can also be used to enhance the transmission capacity of optical fibers. This section is devoted to a discussion of WDM and PDM, with emphasis on the design issues. 9.1. WAVELENGTH-DIVISION MULTIPLEXING

A WDM soliton system transmits several soliton trains superimposed on each other in the time domain, but distinguishable through their different carrier frequencies. A new issue, specific to WDM systems, is related to the occurrence of collisions between solitons of different channels. Such collisions can affect considerably the system performance. Fortunately, one can extract most of the dynamical features associated with soliton collisions in WDM systems by considering multiple painvise collisions occurring between the two solitons of different channels. If the normalized frequency separation between two channels is Wch, related to actual channel spacinghh as Wch = 2nTQfch, the field envelope for one of the solitons can be written as (see eq. 2.7): u,(E,t)= uosech[t-(TB + WchE)/2] e x p [ i ( ~ ~ h d 2 + @ ) ] , (9.1) where ZB = TB/To is the normalized bit-slot duration. The field u2 for the soliton in the other channel is obtained from the above equation by replacing

234

SOLITON COMMUNICATION SYSTEMS

[Iv, § 9

ZB and I&+, by -ZB and -f&h, respectively. Note that the &dependent phase in eq. (2.7) is omitted in eq. (9.1), since it does not result in any phase difference between the two solitons. The actual angular frequencies of the two channels are ~0 f m c h T,/2 (see eq. 2.1). Since dispersion management plays an important role in the design of WDM systems, we use eq. (7.2) (which allows varying dispersion) for soliton propagation, but we neglect the higher-order terms by setting TR = 6 = 0. The resulting equation becomes

.du I

aZU ia , + lu12u = --u, ag + -2p ( E )ar 2

1-

where p(E) is the normalized GVD profile. Similar to the case of a DDF, one can transform eq. (9.2) to a perturbed NSE. However, instead of the transformation of eq. (7.3), it is more convenient to use the transformations (Mollenauer, Evangelides and Gordon [1991]) u’ = u f l and dE’ = p(E)dE, where r(E)= exp(-aE) is the accumulated fiber loss over a distance E. In the transformed variables, eq. (9.2) becomes

From now on, primes on u’ and where r(E) = r(E)@(E). notational convenience.

6’ are omitted for

9. I . 1. Collision-induced frequency shgts

Consider a collision between two solitons separated in frequency by w c h . If we replace the field u by u I + u2 in eq. (9.3), we obtain an equation with many terms oscillating at various frequencies. By separating the terms according to frequency components and neglecting the four-wave mixing (FWM) terms (frequency components appearing at the difference and sum of channel frequencies), we obtain the following two coupled equations for U I and u2:

These equations are identical to the coupled NSEs obtained for the interaction of two copropagating pulses through cross-phase modulation (XPM) (Agrawal

IV, 9: 91

235

CHANNEL MULTIPLEXING

[1995a]). The second nonlinear term in eq. (9.4) is responsible for the XPM and is the origin of perturbations of solitons in a WDM system. The XPM term is important only when two solitons overlap temporally during a collision. By assuming that the two solitons preserve their nature during the collision, one can use the APT theory of $2.3. The results show that the carrier frequency of the two soliton changes during collision by the same amount, but in the opposite directions. The frequency shift for the faster moving soliton is given by (Mollenauer, Evangelides and Gordon [ 19911):

For a constant-dispersion and lossless fiber (7 = l), integrations of eq. (9.6) can be performed analytically, leading to the following expression for the frequency shift:

Since the soliton frequency returns to its original value in the absence of fiber loss, it is useful to define a collision length Lcoll as the distance over which the two solitons overlap before the faster moving soliton overtakes the slower. It is difficult to be precise of the instant at which a collision begins or ends. A commonly used convention uses 2Ts for the duration of the collision by assuming that a collision begins and ends when the solitons overlap at their halfpower points. Since the relative speed of the two solitons is (I/& IwF/To)-', the collision length is given by Lcoll = (2Ts) (IpZlw,mha"/Tg)-',or

where the relations T, = 1.763 TO,B = (2qoTo)-' and w,"h" used.

= 2 nfcnhlaX

have been

9.1.2. Limitations on WDM channels

What is the impact of the frequency shifts occurring during a soliton collision? Recalling from eq. (2.7) that the velocity of a soliton changes with its frequency, it is evident that the collision changes the soliton velocity. Figure 1 l(a) shows changes in the soliton velocity 6f for the faster moving soliton during the

236

"§

SOLITON COMMUNICATION SYSTEMS

9

600 h

2

400

E 200 0

-2

-1

0

1

2

0

I

3

2

h'/qN1

Fig. 1 1 . (a) Change in velocity (frequency) of a soliton during a collision with another soliton in a different WDM channel 75 GHz away in a lossless fiber. (b) Residual frequency shift pertaining after collision because of lumped amplification (after Mollenauer, Evangelides and Gordon [1991]). The dispersion D alternates between 0.5 ps/(km-nm) and 1.5ps/(h-nm) every Lpert = 20 km in the = 50ps and the upper curve. For the lower curve, Lpert = 4 0 h .For both graphs, LA = 20 km, average dispersion is 1 ps/(km-nm).

r,

collision. The maximum velocity change occurs at the point of maximum overlap and is given by:

The maximum velocity change depends on the relative phase of the colliding solitons (Wai, Menyuk and Raghavan [1996]). It is maximum for two in-phase solitons, as assumed for eq. (9.9), and becomes minimum for two initially outof-phase solitons. At the end of the collision, each soliton recovers the original frequency and velocity it had before the collision. Changes in the soliton velocity result in temporal shifts. In fact, collisions in a lossless fiber leave the soliton amplitudes and velocities unaffected, but change their positions and phases (Zakharov and Shabat [ 19711). The temporal shift is easily calculated by integrating eq. (9.7) over 5, and leads to the simple expression:

6z = 4/w&

(9.10)

Since collision-induced temporal shifts are bit-pattern dependent, different solitons of a channel shift by different amounts. This feature implies that soliton collisions can induce a timing jitter even in lossless fibers. The situation is worse in practical soliton systems in which fiber loss is compensated periodically through lumped amplifiers. The reason is that soliton collisions are affected adversely by the loss-amplification cycle. Mathematically,

IV,

5 91

CHANNEL MULTIPLEXING

237

the dependence of Y ( E ) in eq. (9.6) changes the frequency shift. As a result, solitons do not always recover their original frequency and velocity after the collision is over. Similar behavior occurs if the fiber dispersion changes over the collision length. Figure 1 I(b) shows the residual frequency shift remaining after a complete collision of initially well-separated solitons as a function of the ratio Lcoll/Lpert.In fig. ll(b), in addition to the cycle loss-amplification every LA = 20 km, the characteristic dispersion D alternates between 0.5 ps/(km-nm) and 1.5 ps/(km-nm) every 20 and 40 km. Lpertis thus either the amplifier spacing LA or the distance over which the dispersion changes, depending on which one is larger. For the sake of simplicity we replace Lpenby LA, neglecting dispersion variations. The residual frequency shift increases rapidly as LColl approaches LA. When collisions occur over several amplifier spacings, the effects of loss-amplification cycles begin to average out, and the residual frequency shift decreases; it virtually vanishes for Lcoll > 3LA. Since the collision length LC0llis inversely related to the channel spacing wch, the condition LcOll> 3LA sets a limit on the maximum separation between the two outermost channels of a WDM system. As a result, the number N of WDM channels is limited to:

(9.11) One may think that the number of channels can be increased by reducing the channel spacing wch. However, as the channels become closely spaced, the overlap of soliton spectra results in interchannel crosstalk for wch < 4Aw,, where Aw, is the spectral width of solitons (Benner, Sauer and Ablowitz [ 19931, Hasegawa and Kodama [ 19951, Wai, Menyuk and Raghavan [ 19961). Another constraint on the channel spacing is imposed by optical filters which typically require a channel spacing wch 3 5Aw, = 1.763 (10/n) to minimize interchannel crosstalk. By using wch = 5 Aws, eq. (9.1 1) approximately becomes

N < LD/(SLA).

(9.12)

Since the amplifier spacing in the average-soliton regime is typically 10% of the dispersion length, the number of WDM channels is limited by soliton collisions to two or three channels unless amplifier spacing is reduced to impractical values. 9.1.3. Timing jitter

In addition to the conventional sources of timing jitter discussed in 94.2 for a single isolated channel, WDM systems suffer from additional sources of

238

SOLITON COMMUNICATION SYSTEMS

[IV, § 9

jitter specific to WDM. First, each collision of solitons generates a temporal shift (see eq. (9.10) of the same magnitude for both solitons but in opposite directions. Although the temporal shift 6t scales as 0;;and decreases rapidly with increasing wch, the number of collisions increases linearly with wch. As a result, the total time shift after transmission scales as 0.; Second, the number of colIisions that two neighboring solitons in a given channel undergo is slightly different. This difference arises because adjacent solitons in a given channel interact with two different soliton sequences shifted by one bit period T, from each other. Thus, a relative time shift appears among solitons of the same channel that becomes a source of timing jitter in WDM systems because of its dependence on the bit patterns of the copropagating channels (Jenkins, Sauer, Chakravarty and Ablowitz [ 19951). Third, collisions involving more than two solitons can occur and should be considered. In the limit of a large channel spacing (negligible overlap of soliton spectra), multi-soliton interactions are well described by painvise collisions in lossless fibers (Zakharov and Shabat [ 19731, Chakravarty, Ablowitz, Sauer and Jenkins [ 19951). This unique property of solitons allows the calculation of the timing jitter by a summation of pairwise interactions. Two other mechanisms of timing jitter should be considered for realistic WDM systems operating in the average-soliton regime. As discussed earlier, energy variations due to period loss-amplification cycles make collisions asymmetric when Lcoll is shorter than or comparable to the amplifier spacing LA. Asymmetric collisions leave residual frequency shifts which temporally shift the solitons all along the fiber link because of a change in its velocity. This mechanism can be made ineffective by ensuring that LcOll exceeds LA. The second mechanism produces residual frequency shifts through incomplete collisions occurring when solitons from different channels overlap at the input of the transmission link. Since the frequency shift acquired during the first half of a collision is cancelled during the second half in a complete collision, two initially overlapping solitons at the line input suffer an incomplete collision and experience a residual frequency shift. For instance, two solitons of different channels injected synchronously into the fiber link will acquire a net frequency shift of 4/(30,h) since the first half of the collision is absent. Such residual frequency shifts are generated only over the first few amplification stages, but pertain over the whole transmission length and become an important source of timing jitter (Kodama and Hasegawa [1991], Aakjer, Povlsen and Rottwitt [1993]). Their magnitude can be reduced by appropriately delaying each channel to minimize temporal overlaps at the injection point. Similar to the case of single-channel systems, optical filters can be used

IV,

5

239

CHANNEL MULTIPLEXING

91

-0.25 -0.15 u x E

a2

-0.05

a

E

& 0.05 E 0

=

c)

0

0.15

VJ

0.25 t . .

-2

.' ...' 0

2

.

.

.

4

..

. .' '

6

8

'

. . -0.3 10

./Ls Fig. 12. Effect of optical filters on solitons velocities and time shifts during a collision. Dashed lines show the expected behavior in the absence of filters (after Mecozzi and Haus [1992]).

to reduce the timing jitter in WDM systems (Mollenauer, Lichtman, Harvey, Neubelt and Nyman [1992], Mecozzi and Haus [1992]). Typically, FabryPerot-type filters are used since their periodic transmission windows allow simultaneous filtering of all channels. For best operation, the mirrors'reflectivities are kept low (-20%) to reduce the finesse, resulting in a low contrast. Lowcontrast filters remove less energy from solitons of each channel but are nearly as efficient as filters with higher contrast. Their use allows the channel spacing to be close to its minimum possible value (w,h e 5 Am,) (Golovchenko, Pilipetskii and Menyuk [ 19961). Figure 12 shows the effect of filtering on the center frequency and position of a solitons during a collision between two solitons of different channels. The collision length Lcoll is assumed to be much larger than the amplifier spacing LA so that, in the absence of filters (dotted lines), each soliton recovers its initial frequency after the collision. When filters are inserted periodically, both the frequency and temporal shifts are altered by the filter (solid lines) because of the force exerted by the filter that tries to move the soliton frequency toward the filter transmission peak. The net result is to reduce considerably the temporal shift that occurs normally in the absence of filters (Mecozzi and Haus [1992]). Residual frequency shifts due to incomplete collisions are also damped by optical filters, reducing their effect on the timing jitter. Filtering can also relax the condition in eq. (9.1 1) by allowing Lco1l to approach LA (Midrio, Franco, Matera, Romagnoli and Settembre [ 19941) and thus helps to increase the number of channels in a WDM system. However, in the latter case, the transmission distance is generally

240

SOLITON COMMUNICATION SYSTEMS

[IV, § 9

below transoceanic distances due to accumulated residual frequency shifts unless additional timing-jitter controls are used. One way to control the accumulated frequency shift is to apply the technique of synchronous modulation to WDM systems (Desurvire, Leclerc and Audouin [1996], Leclerc, Desurvire and Audouin [1996]). In a WDM experiment involving four channels, each operating at 10 Gb/s, transmission over transoceanic distances has been achieved by using synchronous modulators every 500 km (Nakazawa, Suzuki, Kubota, Kimura, Yamada, Tamura, Komukai and Imai [ 19961). When the synchronous modulators were inserted every 250 km, 3 channels each operating at 20 Gb/s could be transmitted error-free over transoceanic distances (Nakazawa, Suzuki, Kubota and Yamada [ 19961). To implement these schemes, demultiplexing was necessary to isolate each channel. Theses schemes also took advantage of dispersion management described below.

9.1.4. Dispersion management

The discussion so far assumes that the GVD of the fiber link is constant. As discussed in 4 8, the performance of a single high-speed channel making use of TDM can improve significantly if the DDF or some other dispersionmanagement technique is used. One may ask whether WDM systems can also benefit from dispersion management, and the answer is resoundingly yes! In fact, it appears that dispersion management is essential if a WDM soliton system is designed to transmit more than 2 or three channels. It turns out that even the use of just two fiber segments (one relatively short and a longer one) can benefit WDM systems operating with Lcoll M LA (Wabnitz [ 19961). As discussed before, the limitation on the number of channels (see eq. 9.1 1) is imposed by the condition that Lcol, > 3LA arising from the lumped-amplification scheme used in practice. As seen in eq. (9.3), the effect of loss can be cancelled by tailoring the fiber dispersion according to p(E) = r(E)= exp(-aE), the same exponential profile encountered in 9: 7. Such a GVD profile makes soliton As a result, no collisions to be symmetric again, irrespective of the ratio Lcoll/L~. residual frequency shifts occur after soliton collision in systems using DDFs. As a practical alternative to DDFs, the staircase approximation can be used for the exponential profile, making it possible to use multiple constant-dispersion fibers (Hasegawa and Kumar [1996]). An experiment in 1996 took advantage of such a technique and achieved transoceanic transmission of seven 10-Gb/s channels by using only four fiber segments in a recirculating fiber loop (Mollenauer, Mamyshev and Neubelt [1996]). The lower limit on collision length in such a

rv, 5 91

CHANNEL MULTIPLEXING

24 1

system is given approximatively by Lco,l > L*/(3M), where M is the number of segments. The above discussion is based on the use of eqs. (9.4) and (9.5), which neglect FWM terms. However, FWM may affect soliton propagation in WDM systems considerably (Mamyshev and Mollenauer [ 19961) especially if constantdispersion fibers are used. FWM generally results in a transfer of energy among channels and an enhancement of noise at specific frequencies. These effects become important when more than two channels are present. For the case of an equally-spaced three-channel WDM system, FWM occurring during the simultaneous collision of three solitons leads to permanent frequency shifts for the slowest and the fastest moving solitons as well as energy exchange among all three solitons (Evangelides and Gordon [ 19961). If DDFs or their equivalents are used, spectral sidebands generated through FWM in the first half of the collision are totally reabsorbed during the second half of the collision, strongly reducing the effects of FWM on WDM systems. 9.2. POLARIZATION MULTIPLEXING

Since a single-mode optical fiber supports two orthogonal states of polarization for the same fundamental mode, a new kind of multiplexing, known as the polarization-division multiplexing (PDM), can be used to nearly double the capacity of fiber-optic communication systems. In PDM, two channels at the same wavelength can be transmitted through the fiber such that their pulse trains are orthogonally polarized at the fiber input. One may think that such a scheme cannot work unless polarization-maintaining fibers are used since the polarization state changes randomly in conventional fibers because of random birefringence fluctuations. However, even though the polarization state of each channel does change at the end of the fiber link, it changes in the same manner for both channels, preserving their orthogonality. Each channel can than be isolated using simple optical techniques. While implementing PDM for soliton bit streams, a phenomenon known as soliton self-trapping mediated by the XPM tends to destroy the orthogonal nature of the two bit streams. For this reason, one typically implements PDM (Evangelides, Mollenauer, Gordon and Bergano [ 19921) by interleaving the two soliton streams in the time domain (TDM) such that the two neighboring solitons have orthogonal states of polarization. Since soliton interaction is much weaker in that case, the main advantage of using PDM lies in the reduction of solitonsoliton interaction, which can be further reduced (Wabnitz [1995a]) by using sliding-frequency filters. The effective bit rate increases simply because solitons

242

SOLITON COMMUNICATION SYSTEMS

"5 9

can be packed more tightly when the PDM technique is used (De Angelis, Wabnitz and Haelterman [ 19931). The most important factor limiting the performance of PDM in both linear systems ( N U ) and soliton systems is the fiber birefringence. As mentioned previously, even the best optical fibers exhibit residual birefringence that varies along the fiber (typically on a scale smaller than one kilometer) because of stress and core-diameter variations. Associated with fiber birefringence is a relative delay between the two polarization components of a PDM signal (PMD). PMD seriously limits the use of PDM for linear systems making use of the NRZ format. The limitation for linear systems arises because of the frequency dependent of PMD, resulting in pulse depolarization (different parts of the pulse have different polarizations). However, the situation is different for solitons. The natural tendency of a soliton to preserve its integrity under various perturbations also holds for perturbations affecting its state of polarization. Unlike linear pulses, the state of polarization is kept constant across the entire soliton (no depolarization within the pulse), and the effect of polarization perturbations is to induce a small change in the state of polarization of the entire soliton (a manifestation of its particle-like nature). Such resistance of solitons to PMD, however breaks down for large values of the PMD parameter D,. The breakdown limit has been numerically estimated to be (Mollenauer, Smith, Gordon and Menyuk [ 19891): D, 5 0.3 D'I2,

(9.13)

where D, is the PMD parameter [see eq. (4.1 1) and the following text] expressed in ps/& and D is the dispersion parameter in units of ps/(nm-km). Since typically D, < 0.1 ps/& for high-quality optical fibers, D must exceed 0.06ps/(nm-km). In practice, D is larger than 0.1 ps/(nm-km) and PMD is a minor problem for most soliton communication systems. There are two additional mechanisms which can generate timing jitter through fiber birefringence (Mollenauer and Gordon [1994]). The first one is due to accumulation of the delay between two orthogonally polarized solitons in a birefringent fiber. Fortunately, because of random birefringence variations, such a timing jitter tends to average out over long transmission distances. However, orthogonally polarized solitons can experience significant temporal shifts locally during their propagation owing to random birefringence fluctuations. Such local temporal shifts may bring a soliton pair close enough to enhance the solitonsoliton interaction and thus prevent the recovery of the polarization states. For this reason, it is important to ensure that solitons do not deviate too much from

Iv, 5 91

CHANNEL MULTIPLEXING

243

the center of the bit slot at any point along the fiber link. The second source of PMD-induced jitter originates from amplifier noise and has already been discussed in 9 4.2.2. The ASE, being of random polarization, tends to randomize the state of polarization of individual solitons. This randomization gets converted to timing jitter by the random birefringence fluctuations. Equation (4.1 1) shows that the resulting timing jitter is relatively small compared with other sources of timing jitter for high-quality fibers. An important consideration in the development of PDM systems is related to polarization-dependent loss or gain. If a communication system contains multiple elements which amplify or attenuate differently the two polarization components of a soliton, the soliton polarization state is altered, and the information coded by PDM is degraded. In fact, for the worst situation in which the soliton polarization is oriented at 4.5" from the low-loss direction, the soliton state of polarization is aligned with the low-loss direction after 30-40 amplifiers (typically over 2000km for a loss difference of 0.165 dB between the low- and high-loss axes) (Widdowson, Lord and Malyon [1994a]). Even though the axes of polarizationdependent gain and loss are likely to be distributed evenly along the link, such effects may still become an important source of timing jitter. An extension of the PDM technique, called polarization-multilevel coding or polarization-shift keying, has been suggested (Midrio, Franco, Crivellari, Romagnoli and Matera [ 1995, 19961). In this coding technique, the information coded in each bit is contained in the angle that the soliton state o f polarization makes with one of the principal birefringence axes. For instance, by dividing 180" into 16angles, one can store 4 bits of information (z4 = 16) into a single soliton. As for PDM, the polarization-multilevel coding technique is limited by the random variation of fiber birefringence and by randomization of the polarization angle by the ASE noise and has not yet been implemented in practice because of its complexity. To maximize the capacity of a soliton communication system, one can combine PDM with WDM. The new effect that must be considered is the collisions between two orthogonally polarized solitons. Such collisions have been considered (Manakov [I 9741). The results show that such collisions generally lead to considerable changes in polarization state of each soliton except for the cases in which the polarization states are initially parallel or orthogonal. Even though all soliton pairs in a PDM-WDM system initially have either parallel or orthogonal polarizations, random fluctuations of solitons'polarization (because of birefringence fluctuations and amplifier noise) are sufficient to provide the seed from which a rapid scrambling of the polarization state of individual solitons

244

SOLITON COMMUNICATION SYSTEMS

[IV,

5

10

occurs (Mollenauer, Gordon and Heismann [ 19951). The combination of PDM and WDM may thus be difficult to implement in practice.

tj 10. Dark-Soliton Communication Systems

As mentioned earlier, the NSE can be solved by the IST even in the case of normal dispersion (Zakharov and Shabat [ 19731, Hasegawa and Tappert [ 1973b1). However, the intensity profile of the resulting solitons appears as an intensity dip in a uniform background, and it is the dip that remains unchanged during propagation inside the fiber (for a review of dark solitons, see Kivshar [1993]). Although dark solitons were discovered in the 1970s, it is only recently that they have been considered for optical communications. This section describes their properties and the progress realized so far in their use for long-haul transmission. 10.1. DARK-SOLITON CHARACTERISTICS

The general form of the first-order dark soliton obtained by solving eq. (2.5) for > 0) with a = 0 is given by (Kivshar and Yang the case of normal GVD [ 19941, Kivshar, Haelterman, Emplit and Hamaide [1994]):

(a

where <=

~(T-KE),

q=uocosQ,,

K=ugsinQ,,

(10.2)

uo is the amplitude of the CW background, Q, is the so-called internal phase angle, and Q and K are the amplitude and the velocity of the dark soliton, respectively. One important difference between bright and dark solitons is that the velocity K of a dark soliton depends on its amplitude q through the internal phase angle Q,. For Q, = 0, eq. (10.1) reduces to [!id(<, T)[ = uotanh(u0t) and shows that the soliton power drops to zero at the center of the dip. Such a soliton is referred as the black soliton. When @ + 0, the intensity does not drop to zero at the dip center, and such solitons are referred to as gray solitons. Another interesting feature of dark solitons is their phase profile. In contrast with bright solitons which have a constant phase, the phase of dark solitons changes across its width. Figure 13 shows the intensity and phase profiles for several values of @. For a

IY

9 101

245

DARK-SOLITON COMMUNICATION SYSTEMS

1 .o

0.8

L

g 0.4 Q)

a 0.2 0.0

-4

-2

0

2

Time z

4

Time I:

Fig. 13. Intensity and phase profiles of dark solitons for various values of the internal angle 4.

black soliton (@= 0), a phase shift of n occurs exactly at the center of the dip, The CW backgrounds on each side of a black soliton are thus exactly out of phase. A phase shift also occurs for gray solitons, but it is less dramatic as it is gradual and smaller than n. Dark solitons have been observed experimentally by creating a narrow dip within a relatively broad optical pulse that acts as the CW background (Emplit, Hamaide, Reynaud, Froehley and Barthelemy [ 19871, Krokel, Halas, Giuliani and Grischowsky [ 19881, Weiner, Heritage, Hawkins, Thurston, Kirschner, Leaird and Tomlinson [ 19881). Numerical simulations show (Tomlinson, Hawkins, Weiner, Heritage and Thurston [ 19891) that the central dip can propagate as a dark soliton in spite of the nonuniform background as long as the background intensity is uniform in the vicinity of the dip. Higher-order dark solitons do not follow a periodic evolution pattern similar to that shown in fig. 1 for the higher-order bright solitons. The numerical results obtained by solving the NSE with the initial condition Ud(0, z) = uo tanh t [@= 0 in eq. (10.2)] show that for U O > 1, the input pulse forms a fundamental dark soliton by narrowing its width while ejecting several dark-soliton pairs in the process (Zhao and Bourkoff [ 1989a,b]). 10.2. DARK-SOLITON ADVANTAGES

Although the formation of dark solitons in fibers was demonstrated as early as 1987 (Emplit, Hamaide, Reynaud, Froehley and Barthelemy [ 19871, Krokel, Halas, Giuliani and Grischowsky [ 19881, Weiner, Heritage, Hawkins, Thurston, Kirschner, Leaird and Tomlinson [ 1988]), their transmission over long fiber lengths attracted little attention until 1995 (Nakazawa and Suzuki [ 1995a1). This

246

SOLITON COMMUNICATION SYSTEMS

IN

§ 10

delay can be attributed to a variety of factors. First, it is relatively difficult to generate and detect dark solitons in comparison with bright solitons. Second, most fibers are designed to have anomalous GVD at the wavelength of minimum loss while dark solitons require normal dispersion. In addition to these technical challenges, the use of dark solitons may have been delayed by a conceptual bias towards bright solitons since localized optical pulses come closer to our intuitive notion of “bits of information.” However, the properties of dark solitons suggest that there may be intrinsic advantages to using them for long-haul communications. Like bright solitons, dark solitons are very robust under perturbations of their parameters such as the width and the shape of the intensity dip (Zhao and Bourkoff [1989a,b]). However, some differences do exist. In contrast to bright solitons, the interaction between a pair of black solitons is always repulsive. This is expected owing to the phase difference of 76 that should be maintained between two black solitons as a result of the phase jump in the middle of the dip. The interaction force between black solitons also decreases twice as fast with increasing separation than the force between bright solitons. Moreover, the Gordon-Haus jitter of black solitons is also reduced by a factor of two relative to the case of bright soliton (Hamaide, Emplit and Haelterman [1991], Kivshar, Haelterman, Emplit and Hamaide [ 19941). There are also some negative aspects of solitons for long-haul transmission, mainly because of the CW background since the optical signal is “on” except for a short duration within each “1” bit. Since the average power is relatively high, stimulated Brillouin scattering (SBS) can become quite important and must be controlled. Also, periodic variations in the signal power because of the lumpedamplification scheme result in a FWM process that leads to an exponential growth of spectral sidebands in the signal spectrum. One can prevent sideband generation by operating well within the average-soliton regime (i.e., LA << L D )or by inserting optical filters periodically (Kim, Kath and Goedde [ 19961). However, because of the CW background, filtering of dark solitons tends to destroy the soliton along with the sidebands. Apparently, filtering of dark solitons can only be used in combination with other mechanisms of soliton control (Kim, Kath and Goedde [ 19961). As for bright solitons, timing jitter remains an important issue for dark solitons and must be controlled externally. A new mechanism of timing jitter for dark solitons is related to the coupling between the soliton amplitude 7 and the velocity K (see eqs. 10.1 and 10.2). Because of such a coupling, amplitude fluctuations lead directly to timing jitter, in sharp contrast to bright solitons for which amplitude fluctuations need to be converted to frequency fluctuations by

IVI

LIST OF SYMBOLS

247

the Raman effect to lead to any timing jitter. It is thus essential to stabilize the dark-soliton amplitude. Under certain conditions, this stabilization can be achieved by using the saturation properties of the in-line amplifiers (Matsumoto, Ikeda and Hasegawa [1995]), and a numerical investigation has confirmed this feature by using a nonlinear amplifying loop mirror (Caballero and Souza [ 19961). Such saturation also reduces interactions between dark solitons by keeping each dark soliton near the center of the bit slot. Since amplitude and phase profiles are also related for dark solitons, synchronous phase modulation can also be used to control the soliton position, and hence reduce soliton interaction (Maruta and Kodama [1995]). Several techniques can be used to generate dark solitons, including electric modulation of an electro-optic element inserted in one arm of a Mach-Zehnder interferometer (Zhao and Bourkoff [1992]), conversion of a NRZ signal into a RZ signal (by using the clock signal) and then into dark solitons by using a balanced Mach-Zehnder interferometer (Nakazawa and Suzuki [ 1995b]), and nonlinear conversion of a beat signal in a DDF (Richardson, Chamberlin, Dong and Payne [ 19941). In a 1995 experiment (Nakazawa and Suzuki [1995a]) in which a 10Gb/s signal was transmitted over 1200 km by using dark solitons, the CW background was modulated to broaden its spectral width so that SBS did not deplete the signal. The transmission distance was found to be limited by the asymmetry of the dark soliton originating from the time response of the electronic circuit used to generate them. Significant improvements are expected with the development of sources capable of generating a dark-soliton bit stream with little amplitude and width fluctuations.

List of Symbols2 a

Fast varying amplitude of an average soliton (N)

A

Electric field representing a pulse Effective area of the fiber transverse mode

A eff B

B Cf

Soliton amplitude rescaled to local dispersion (N) Bit rate Curvature of the filter central frequency

(N) denotes a normalized quantity.

248

SOLITON COMMUNICATION SYSTEMS

Dispersion parameter Polarization characteristic dispersion parameter Electric field vector Energy of a soliton at the fiber input Fourier Transform of the function f ( t ) Soliton pre-amplification factor Fraction of the bit period Maximum channel spacing in WDM systems (N) Power gain factor (N) Power gain factor in presence of pump absorption (N) Gain necessary to compensate fiber loss over one amplifier spacing Planck constant L

Length of a communication line Amplifier spacing or distance between pump stations Dispersion length (= T:/ Ip2 1) Average dispersion length (= T i /

I/3yI)

Period of perturbation during a solitons collision in a WDM system Soliton period (= nLD/2) Collision length of two solitons Soliton order Nonlinear refractive index Linear refractive index at n(w) Number of photons contained in a soliton Spontaneous emission factor Peak power of a soliton injected into a fiber Peak power of a soliton of order N Peak power of a soliton Soliton separation (N) Reflectivities of the mirrors in a Fabry-Perot interferometer Self-steepening factor (N) Time Time normalization constant

LIST OF SYMBOLS

249

Polarizability relaxation time Bit period Transmission amplitude of a filter in the spectral domain Transmission amplitude of a modulator in the time domain Soliton characteristic width Raman characteristic time Soliton full width at half maximum Maximum soliton width in the adiabatic regime U

Optical field (N) Soliton amplitude (N) Amplitude of the CW background of a dark soliton (N) Dark soliton field (N) Optical field of a soliton in the zth channel of a WDM system (N) Soliton optical field (N) Slowly varying part of an average soliton (N) Renormalized soliton amplitude in dispersion-varying systems (N) Optical field describing the average soliton (N) Group velocity Propagation distance Fiber loss at the signal wavelength (N) Fiber loss at the signal wavelength Fiber loss at the pump wavelength Propagation constant or wave number Inverse of the group velocity Group-velocity dispersion (GVD) parameter Third-order dispersion (TOD) parameter Average dispersion of a dispersion-managed system Nonlinear coefficient Effective nonlinear coefficient (N) Third-order dispersion (N) Free-spectral range of a Fabry-Perot interferometer Perturbation to the NSE

250

SOLITON COMMUNICATION SYSTEMS

Vacuum permittivity Distance normalized to LD (N) Amplifier or pump station spacing (N) Position of the filters (N) Position of the modulators (N) Dark soliton amplitude (N) Renormalized distance in dispersion-varying systems (N) Dark soliton velocity (N) Wave1ength Optical frequency Carrier optical frequency Amplitude of soliton spectrum Soliton phase Dark soliton internal phase Standard deviation of variable i Polarization-induced timing jitter Acoustically-induced timing jitter Retarded time normalized to To (N) Polarization relaxation time (N) Bit period (N) Time delay (N) Raman characteristic time (N) Full width at half maximum of a soliton (N) Deviation from wo (N) Optical carrier angular frequency Channel spacing in WDM systems (N) Maximum channel spacing in WDM systems (N)

\

Soliton . . an 5~ ular fre.f.. uenc.y(N . . mean ~

~

Rate of sliding of filters in the sliding-filter technique Normalized distance for dark soliton (N) Shift of the soliton mean frequency due to a collision Difference in GVD between two fiber segments

LIST OF ACRONYMS

IVI

A%

Soliton spectral width

A Y,, Aw

Relative frequency of the nth sideband

A w,

Normalized soliton spectral width (angular frequency)

Spectral width (angular frequency)

List of Acronyms

APT

Adiabatic perturbation theory

ASE

Amplified spontaneous emissior?

BER

Bit-error rate Continuous wave

cw DCF DDF ECD

Dispersion-compensating fiber Dispersion-decreasing fiber Equipartition of cumulative dispersion

EDFA

Erbium-doped fiber amplifier

FSA

Fast saturable absorber

FWHM FWM

Full width at half maximum

GVD IST

Group-velocity dispersion

MPD NALM NOLM

Minimization of power deviation Nonlinear amplifying loop mirror

NRZ

Non-return to zero format

NSE

Nonlinear Schrodinger equation

OPC

Optical phase conjugation

PDM

Polarization-division multiplexing

PMD

Polarization mode dispersion

RZ

Return to zero format

SBS

Stimulated Brillouin scattering

SNR SPM

Signal to noise ratio

Four-wave mixing Inverse scattering theory

Nonlinear optical loop mirror

Self-phase modulation

25 1

252

SOLITON COMMUNICATION SYSTEMS

SSFS

Soliton self-frequency shift

SVEA

Slowly-varying-envelope approximation

TDM

Time-division multiplexing

WDM

Wavelength-division multiplexing

XPM

Cross-phase modulation

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OLE KELLER Institute of Physics, Aalborg Universip, Pontopiddanstrczde 103. 9220 Aalborg 0st, Denmark

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INTRODUCTION . . . . . . . . . . . . . . . . . . .

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3 . LOCAL FIELDS IN MESOSCOPIC MEDIA WITH STRONGLY LOCALIZED ELECTRON ORBITALS . . . . .

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LOCAL FIELD ELECTRODYNAMICS IN QUANTUM WELLS AND THIN FILMS . . . . . . . . . . . . . . . . . .

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5. 2D SPATIAL CONFINEMENT OF LIGHT BY OPTICAL PHASE CONJUGATION. . . . . . . . . . . . . . . . . . . . 324

ACKNOWLEDGEMENTS . . . . . . . . . . . . . . . . . .

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REFERENCES . . . . . . . . . . . . . . . . . . . . . . .

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Q 1. Introduction To describe the optical properties of condensed matter systems, it is often necessary to go beyond the framework of macroscopic electrodynamics, which is based on the macroscopic Maxwell equations and appropriate phenomenological constitutive equations. A program for determining the range of validity of the macroscopic Maxwell equations was initiated by Lorentz [ 18781, who “separated matter and ether”. The theoretical investigations of Lorentz were based on the hypothesis that the seat of the electromagnetic field is empty space and that the field is created by atomistic electric charges. According to Lorentz, the electromagnetic fields act back on the atomistic charges, and force these to move in the prevailing field in a manner that can be described by Newton’s law of motion. With the birth of quantum mechanics it became clear that the (nonrelativistic) dynamics of electrons and nuclei had to be described by the (manybody) Schrodinger equation instead of by Newtonian mechanics. The framework for describing the microscopic fields nowadays is thus given by the so-called microscopic Maxwell-Lorentz equations, in which only the microscopic charge and current densities appear. To determine the particle response, the (many-body) Schrodinger equation with the local vector and scalar potential must be used. The key to a rigorous calculation of the microscopic (local) electromagnetic field which is rapidly varying in both space and time hence is a self-consistent solution of the coupled Maxwell-Lorentz and (many-body) Schrodinger equations for the medium (model) under consideration. Once the local field has been determined with sufficient accuracy, all optical properties of the system, linear as well as nonlinear, may be obtained. Although most studies of local-field effects in the optics of condensed matter systems are based on the afore-mentioned framework, there are cases in which it is necessary to quantize the electromagnetic field. In these cases, the classical Maxwell-Lorentz equations must be replaced by the corresponding equations among the various field operators. It is not possible to review here all aspects of the theory of local fields in condensed matter systems. An excellent account of the local-field foundation of the macroscopic electromagnetic theory of dielectric media, including also a good description of the Lorentz program and Planck’s theory of dispersion, is given in the review article by Van Kranendonk and Sipe [1977]. A comprehensive 259

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description of the foundation of macroscopic electrodynamics is presented in the books by De Groot [1969] and De Groot and Suttorp [1972]. For readers interested in the close relation between the dielectric function of condensed matter systems and local-field electrodynamics, commendable approaches can be found in the book edited by Keldysh, Kirzhnitz and Maradudin [1989]. In the present review, particular attention is paid to those aspects of the local-field theory which are of particular importance in the optics of mesoscopic media, and it is emphasized that a fruitful interplay can occur in the mesoscopic domain between local-field effects typically observed only in atoms (molecules) or in bulk systems of condensed matter. It appears that such an interplay may lead to a new and better understanding of certain aspects of local-field optics. Although local-field electrodynamics, in one version or another, has been an underlying issue in many theoretical analyses of the optical properties of mesoscopic media over the years this has not always been stressed (or appreciated) explicitly, and therefore it might also be appropriate to review important aspects of the theory of mesoscopic optics from the local-field point of view. Intuitively, it is appealing to use a Green’s function formalism to describe the spatial and temporal properties of the electromagnetic field generated by a given current density distribution. In such a description, the Maxwell-Lorentz equations are turned into an integral equation between the local field and the current density induced by the prescribed external field. The kernel of the integral equation is a dyadic electromagnetic propagator (Green’s function), and by means of a microscopic version of the famous Ewald-Oseen extinction theorem (Ewald [ 19151, Oseen [ 19 151, Wolf [ 1973]), the homogeneous term can be identified as the field originating in the external sources driving the medium under study. The electromagnetic propagator consists of a divergencefree (transverse) part and a rotational-free (longitudinal) part. The longitudinal part contributes to the local-field electrodynamics only in regions of space where the longitudinal part of the induced current density is different from zero. Bearing in mind that, in a statistical sense, (condensed) matter in a quantum mechanical description forms a continuum, the longitudinal part of the localfield response plays an important role, at least in media with strongly delocalized electronic orbitals. The longitudinal response is non-retarded and so is a part of the transverse response, and hence a part of the transverse electromagnetic propagator. By keeping only the non-retarded parts of the electromagnetic propagator, local-field phenomena arise solely from so-called self-field effects. The self-fields are always attached to the matter (particle) field and thus enable one to describe only the non-radiative part of the local field in the medium. The remaining part of the electromagnetic propagator describes retarded local-

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field interactions and leads to radiative effects and fields which may be detached from the matter field. It is well known from the electrodynamics of simple atoms, and in agreement with the basic framework of nonrelativistic quantum electrodynamics (cf., e.g., Cohen-Tannoudji, Dupont-Roc and Grynberg [ 19891) that an atom only “sees” its own transverse field, and if this feature is generalized and used in the framework of linear response theory (Kubo [1957], Pines and Nozieres [ 19661, Martin [1967], Keldysh [ 19891) the only constitutive equation of fundamental importance in microscopic electrodynamics (optics) gives an integral relation between the induced current density and the transoerse part of the prevailing local field, plus a possible longitudinal part of the prescribed external field (Keller [ 1996a1). The kernel appearing in the linear constitutive equation is the many-body conductivity tensor of the medium. By combining the integral relation between the field and current density with the constitution relation, a loop (integral) equation can be established for the transverse part of the local field (Keller [ 1996a1). Within the limited framework of self-energy electrodynamics (Banvick [ 19781, Barut and Kraus [ 19831, Barut and Van Huele [ 19851, Passante and Power [ 19871, Barut and Dowling [ 19871, Barut, Dowling and Van Huele [1988], Blaive, Barut and Boudet [1991], Barut and Blaive [ 1992]), the transverse loop dynamics leads in atomic optics to quite accurate results for the Lamb shift and the rate of spontaneous emission (Crisp and Jaynes [ 19691). Once the self-consistent solution for the transverse part of the local field has been obtained, the induced longitudinal field can be determined (i.e., there is no loop problem). For atomic, molecular or mesoscopic quantum dot systems with effectively only one (or a few) mobile electrons the above-mentioned way of calculating local fields can often be implemented in a successful manner, but for systems containing more than a few mobile electrons the lack of sufficient knowledge of the many-body wavefunctions makes it impossible to carry out this type of rigorous calculation. In the popular random-phase-approximation (RPA) approach (Pines and Bohm [1952], Bohm and Pines [1953], Pines and Nozieres [ 19661, Mahan [ 1990]), the much simpler RPA-conductivity tensor is used and a loop equation is consequently set up for the total local field. It may happen particularly in mesoscopic interaction volumes that the longitudinal part of the induced field even at optical frequencies dominates the internal electrodynamics of the system. In such cases one may use the density-functional approach (Hohenberg and Kohn [1964], Kohn and Sham [1965], Peuckert [1978], Zangwill and Soven [1980], Lundqvist and March [1983], Mahan and Subbaswamy [ 19901, Gross, Dobson and Petersilka [1996]) to calculate the local field. The density-functional theory, which to a certain extent allows one to incorporate exchange and correlation effects in a one-electron scheme, leads to

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an integral equation for the longitudinal part of the local field (or equivalently, the local scalar potential). Being a scalar theory, the density-functional approach at best allows one to calculate the non-retarded part of the local field, only. For a condensed matter system of macroscopic extent it is in general impossible to solve the integral equation for the (transverse part of the) local field without making (too) severe model approximations. The possibilities for performing a rigorous microscopic calculation of the local field (at least within the framework of a one-particle approximation) are much better for a mesoscopic system. The reason for this originates in the fact that the contribution to the conductivity tensor from a given optical transition can be broken into a dyadic product of two vectors belonging to the observation and source coordinates, respectively. In turn, this implies that the integral equation problem for the local field can be converted into a matrix problem (Cho [1991], Keller [1995a], Ohfuti and Cho [ 19951). For a mesoscopic system it is often so that only a rather limited number of optical transitions participate in the electrodynamics and therefore the dimension of the relevant matrix is so small that the problem may be handled numerically. The retarded part of the local-field interaction at distance exhibits an interesting complimentary between so-called quantum nonlocality and electromagnetic nonlocality. For space-like events the interaction is thus intermediated by the fundamental nonlocal character of all couplings in quantum mechanics, and this interaction obeys a characteristic R-3-dependence, R being the distance between the source and observation points. When the light pulse sent out from the source point reaches the point of observation, the nonlocal quantum mechanical interaction is destroyed and replaced by the usual electromagnetic interaction. This interaction exists only on the light cone and exhibits a characteristic R-' (far-field) dependence. In recent years the interest in near-field effects in optics has grown so dramatically that near-field optics now appears as a new and fruitful branch of physical optics (Pohl, Denk and Lanz [1994], Lewis, Isaacson, Harootunian and Murray [1984], Fischer [ 19851, Pohl and Courjon [ 19931, Nieto-Vesperinas and Garcia [1996], Girard and Dereux [1996]). One of the basic goals in near field optics is to achieve a spatial resolution on the atomic length scale. When optics are used to probe features on a length scale much smaller than that of the externally impressed field, it is no longer possible to use macroscopic electrodynamics for the analyses. In the mesoscopic domain local-field effects play a crucial role, and by a careful local-field analysis of the short-range (R-3) and long-range (K' ) interactions it is possible to establish a rigorous microscopic framework (Keller [ 1994, 1996b1) for the point-dipole model (Labani, Girard,

Y § 11

INTRODUCTION

263

Courjon and Van Labeke [1990], Girard and Courjon [1990], Girard and Boujou [ 199 1, 19921, Keller, Xiao and Bozhevolnyi [ 1993a1, Xiao, Bozhevolnyi and Keller [ 19961) and the refractive-index approach (Greffet, Sentenac and Carminati [ 19951, Carminati and Greffet [ 1995a,b], Martin, Girard and Dereux [ 1995a,b], Carminati, Madrazo and Nieto-Vesperinas [1994]) so often used to describe optical near-field phenomena. Among the mesoscopic systems, quantum-well structures have been the most extensively studied up to now, and from a theoretical point of view these structures are particularly simple to deal with because the local-field analysis in both the linear and nonlinear cases effectively reduces to a study of a onecoordinate integral-equation problem. So far, local-field calculations have been used to investigate the optical properties of single quantum wells (Dahl and Sham [19771, Ando, Fowler and Stern [ 19821, Tselis and Quinn [ 1984a1, Das Sarma [1984], King-Smith and Inkson [1987], Jain and Das Sarma [1987], Keller and Liu [ 1992-19951, Wendler and Kandler [ 19931, Luo, Chuang, Schmitt-Rink and Pinczuk [1993], Keller [1993], Liu and Keller [1993], Liu and Keller [1994, 1995a,b], Keller, Lju and Zayats [ 19941, Keller, Zayats, Liu, Pedersen, Pudonin and Vinogradov [ 19951, Keller and Chen [ 19951, Zaluzny [ 19951, Zaluzny and Bondarenko [ 19961, Chen and Keller [ 1997]), two electromagnetically coupled wells (Chen and Keller [ 1996]), multiple quantum wells (Andreani and Bassani [ 19901, Schafer and Henneberger [ 19901, Andreani [ 19951, Liu [ 19951, Knorr, Stroucken, Schulze, Girndt and Koch [ 19951, Stroucken, Anthony, Knorr, Thomas and Koch [1995], Stroucken, Knorr, Anthony, Schulze, Thomas, Koch, Koch, Cundiff, Feldmann and Gobel [ 19951, Stroucken, Knorr, Thomas and Koch [1996]), and quantum lattices (Visscher and Falicov [1971], Fetter [1974], Haupt and Wendler [1987], Bloss [1983], Bloss [1984a,b], Tselis and Quinn [ 1984b], Wasserman and Lee [ 19851, King-Smith and Inkson [ 1987]), and studies have been performed on both semiconducting and metallic systems. For a review of the optics of superlattices the readers is referred to Raj and Tilley [ 19891. The linear and nonlinear opticai properties of metallic overlayers on metal substrates have also been analyzed using the local-field concept. In the absence of external fields the integral equation for the local field enables one to determine the so-called local-field resonances of the mesoscopic system under investigation. These resonances, which can be characterized as the spatial eigenmodes of the many-body system, allow one to obtain a unified view of optical resonances (eigenmodes) in coupled electron-field systems. Although not widely known, one finds among the local-field resonances in quantum-well systems the electromagnetic surface wave dispersion relation for y-polarized light (Boardman [1982], Agranovich and Mills [ 19821, Forstmann

264

LOCAL FIELDS IN LINEAR AND NONLINEAR OPTICS OF MESOSCOPIC SYSTEMS

[y 5

1

and Gerhardts [ 19861, Raether [ 19881) and the s- and p-polarized eigenmode conditions for radiative and non-radiative guided modes in slab structures (Yariv and Yeh [ 19841, Lee [ 19861) when the thickness of the quantum well tends to be macroscopic. To describe the p-polarized local-field electrodynamics in quantum wells one often uses the Feibelman theory (Feibelman [1975a,b, 1976, 1981, 19821, Bagchi [ 19771, Bagchi, Barrera and Rajagopal [ 1979]), originally developed to describe among other things the linear reflectivity of light from free metallic (or semiconducting) surfaces with a smooth electron density profile. The Feibelman approach for p-polarized fields is based on a self-field approximation for the electromagnetic propagator and thus can be classified as a simplified non-retarded (scalar) theory. For s-polarized fields the Feibelman approach necessarily goes beyond the non-retarded approximation, but the local-field effects are generally small. In the full (self-consistent) scalar theory, one starts from the Poisson equation and derives an integral equation between the scalar potential and the induced electron density (Rice, Schneider and Strassler [ 19731, Dasgupta and Fuchs [1981], Wood and Ashcroft [1982], Ekardt [1984], Beck [ 19841, Puska, Nieminen and Manninen [ 19851, Ekardt [ 19851, Zaremba and Persson [1987]). If this relation is turned into an integral equation among the longitudinal part of the local field and the induced current density, it appears that the electromagnetic propagator besides the self-field part used by Feibelman also induces a non-retarded nonlocal part, which technically may be obtained from the retarded transverse propagator letting the speed of light tend towards infinity. In the complete vector theory (Ohfuti and Cho [1993, 19951, Cho, Ishihara and Ohfuti [ 19931, Ishihara and Cho [1993], Keller [1995b,c, 1996a]), which contains the Feibelman and scalar theories as special cases, full account is taken of retardation effects. In the present review the results obtained for the p-polarized reflectivity using the three afore-mentioned approaches are compared for a specific system; namely, a semiconducting quantum well containing only two bound states. Particular attention is devoted to the frequency dependence of the reflectivity in the vicinity of the electronic transition frequency. Quite recently, there has been an interest in the possibilities for phaseconjugating the outgoing field from an optical near-field microscope, and it seems that light foci with a halfwidth substantially smaller than allowed by classical (scalar) diffraction theory can be produced by phase-conjugating also part of the evanescent spectrum of the outgoing field (Bozhevolnyi, Keller and Smolyaninov [ 1994, 19951). To understand the spatial confinement of the phaseconjugated field it appears necessary to make use of a local-field formalism for both the source field and the phase-conjugated field. The first theoretical

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265

paper in this field was published by Agarwal [1982], who studied the optical response of an atom placed in front of a phase-conjugating mirror. In the wake of the important paper by Agarwal, a number of additional theoretical analyses have been carried out (Hendriks and Nienhuis [1989], Milonni, Bochove and Cook [1989], Arnoldus and George [ 1990, 19951, Keller [ 1992, 1996~1,Lozovski and Arnoldus [ 19961, Andersen and Keller [1996], Vohnsen and Bozhevolnyi [1996]), but only quite recently has the phase conjugation of the evanescent modes been taken into account (Agarwal and Gupta [1995], Keller [1996d], Xiao [ 1996a,b]). Up to now, the standard scheme for seeking extreme optical photon localization has been based on multiple scattering of electromagnetic fields from impurities or surface roughness (see, e.g., Sheng [1990], Lagendijk and van Tiggelen [1996]), and the theory has to some extent been patterned from the Anderson localization theory for electrons (Anderson [ 19581). Spatial localization by near-field phase conjugation might offer a new route to extreme two-dimensional confinement of light (Keller [ 19971).

8

2. Local Fields and Nonlocal Optics

2.1, ELECTROMAGNETIC PROPAGATOR APPROACH

To study local fields in condensed matter it is necessary ips0 fact0 to go beyond macroscopic electrodynamics, and therefore take as a starting point the microscopic Maxwell-Lorentz equations, which, upon a convenient transformation to the frequency (u)domain, take the form

where I?(<w) and $ ( t o ) are the microscopic electric and magnetic field vectors, respectively, and j(<w ) and p(< w ) are the microscopic (many-body) current and charge densities. Since the local-field electrodynamics is qualitatively different for the divergence-free (transverse) and rotational-free (longitudinal)

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[y 5

2

parts of the field it is adequate already at this stage to divide the fields and current density into their transverse (T) and longitudinal (L) parts:

and

a

a

a

where . f ? ~ = 9 . JT = 0 and x 2~ = x j~ = 0. The Maxwell-Lorentz equation in (2.4) readily shows that the magnetic field has no L-component, so that $ = &. Although the division at the vector fields into T and L-parts is not relativistically invariant, such a division is certainly adequate in the nonrelativistic domain where the Coulomb gauge can be used to study electrodynamic phenomena in an economic manner (see, e.g., CohenTannoudji, Dupont-Roc and Grynberg [ 19891). Also from a field-quantized point of view this division is fruitful. Although essentially all investigations of local field effects in (mesoscopic) condensed matter systems have been carried out in the framework of semiclassical electrodynamics, the field-quantized aspects are certainly of interest. The T-L division splits the microscopic Maxwell-Lorentz equations into a set

describing the transverse dynamics, and a set (2.9) (2.10) governing the longitudinal dynamics. By combining eqs. (2.9) and (2.10), one can obtain the local charge conservation condition, . j~ = iwp. A combination of eqs. (2.7) and (2.8) leads to the following wave equation for the transverse part of the electric field:

a

where qo = w/cg is the vacuum wave number of the field. Since we shall describe the matter field on the basis of quantum mechanics, it is correct to consider the

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261

transverse current density field in eq. (2.11) as a continuum field, rigorously speaking in the quantum statistical sense, of course. It is adequate at this point to turn eq. (2.1 1) into an integral relation (Keller [1996a]),

between the continuum fields &(< W) and &(< 0).The field EF*(<w ) is identical to the transverse field radiated by external (ext) sources located outside the spatial domain, VT, in which the transverse current density is different from zero. The identification of l?Ft(<w ) as the transverse field originating in external current density sources can be established rigorously by means of a microscopic version (Keller [ 1986, 1996a1) of the original Ewald-Oseen extinction theorem (Ewald [ 19 151, Oseen [ 19151) extended and studied more than half a century later by many others (Sein [1970], Aganval, Pattanayak and Wolf [1971], Pattanayak and Wolf [1972], De Goede and Mazur [1972], Pattanayak [1973], Birman [1982], Bullough and Hassan [ 19831). The integral in eq. (2.12) extends over the volume occupied by the transverse current density, cutting out from VT a small spherical exclusion volume E(J) centered on the space point r' at which we want to calculate the local field, &(< w). After having carried out the spatial integration one lets the exclusion volume shrink to zero while keeping its spherical form. The local field response at Jstemming from the transverse current density at the same point is given by the &(J; w)/(3i~00)and may be called the transverse self-field response. This part of the response is nonretarded, and we shall return to its physical interpretation in fj2.5. The retarded part of the field, originating in &(?'; o),is described by the electromagnetic vacuum propagator &(J- J'; w). It is often convenient to formally re-introduce the current density itself, j ( <8 ), in eq. (2.12) instead of its transverse part. For the retarded part of the response this can be done by replacing Eo(J- J'; w ) by -T its transverse part, Do(?- J'; w), and hence (see, e.g., Keller [1996a]):

&(?- F'; W ) .&(J';W ) d

7'; 0). j(J'; W ) d3#.

(2.13) The integration on the right hand side of eq. (2.13) extends over the spatial domain V of j(J';w), and I have omitted the reference E + 0 since the

268

LOCAL FIELDS IN LINEAR AND NONLINEAR OPTICS OF MESOSCOPIC SYSTEMS

[v 5 2

singularity in Bi(7- 7’; w ) at 7’ = r‘ is so weak that the integral is absolutely w ) in the form convergent. We rewrite the transverse self-field (SF) term, i+F(<

(2.14) where xT(7- 7‘; w ) is the dyadic transverse delta function. By combining eqs. (2.12)42.14), we thus obtain: Z T ( < 0)=

EF‘(< w ) -iaw/‘[B;(?-

7’; w ) + zT(7-F’; w ) ] .j(r“;w)d3r’, (2.15)

where (2.16) is the so-called transverse electromagnetic self-field propagator. The relation in eq. (2.9) between the longitudinal current density and field can also be written in a propagator form where the current density itself enters. Hence,

EL(
1‘

zL(77 ’ ; w ) . j ( 7 ’ ; w ) d3r’,

(2.17)

where gJ-7’;w)

H

1= +L(?-r‘’)

(2.18)

40 tl

is the so-called longitudinal electromagnetic self-field propagator, 6 L ( 7 - 7’) being the dyadic longitudinal delta function. Denoting the Dirac delta function and the unit tensor by 6(7 - 7’) and i?, respectively, one has the relation b ~ ( 7 -7 ’ ) + ‘ B L ( ~ -7’) = i??6(7-7’). By adding eqs. (2.15) and (2.17), it becomes apparent that the integral relation between the total local field, ,?(< w ) and the current density, j ( 7 ; w), can be written in the compact Green’s function form, tt

s”

(2.19)

Go(?- 7 ’ ; W ) = Do(?- 7’; O ) + zT(7-7’; 0) + FL(r‘-7’; W )

(2.20)

.!?(< w ) = gyt(<w) - i b w

Go(?- 7’; w ) . j ( 7 ’ ; w ) d3r’,

where *

ttT

is the full electromagnetic propagator containing retarded as well as non-retarded parts. For brevity, I have omitted writing the V at the top of the integral sign.

v, 9 21

LOCAL FIELDS AND NONLOCAL OPTICS

269

2.2. ELECTRODYNAMICS WITHIN THE FRAMEWORK OF LINEAR RESPONSE THEORY

The integral equation in eq. (2.19) relates two so-far unknown continuum fields, namely the local electric field and the induced current density. In order to determine the prevailing field, or the current density, one needs an additional so-called material relation between I!?(< w ) and J(r‘’;w). Linear optics is based on a linearization of the field-matter interaction. For media possessing physical properties which are (or can be approximately considered to be) translationally invariant in time, the most general linear field-matter coupling takes the form (see, e.g., Keller [ 1996a1)

in the mixed Fourier representation, provided the transverse (longitudinal) current density domains of the source and system do not overlap. The material relation in eq. (2.12) can be derived starting from the many-body (MB) Schrodinger (or Pauli) equation. The reason that only the transverse part of the local field enters the constitutive equation originates in the fact that in the Coulomb gauge the longitudinal electric field is eliminated as a dynamical field variable in favor of the particle position variables (Schiff [1968], Atkins and Woolley [ 19701, Woolley [ 197I , 19751, Babiker, Power and Thirunamachandran [ 19731, Power and Thirunamachandran [ 19781, Cohen-Tannoudji, Dupont-Roc and Grynberg [ 19891). The quantity intermediating in a spatially nonlocal manner the transverse electric field at r” to the induced current density at r’ is a part of the many-body conductivity tensor zMB(r‘,r‘’; w). I have extracted from this tensor a part, z:(r‘,7’; w), giving the transverse current density response to I$, and another part, zY:(r‘,r‘’;w) accounting for the induced longitudinal current density. By combining eqs. (2.15) and (2.2 l), it appears that the transverse part of the local field satisfies the inhomogeneous integral equation

with a kernel

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LOCAL FIELDS IN LINEAR AND NONLINEAR OPTICS OF MESOSCOPIC SYSTEMS

[y 3 2

The formal solution of eqs. (2.22) is given by E T ( < O )=

J“

rTT(J,

J’; W ) .@(J’;

W ) d3r’,

(2.24)

where the nonlocal transverse local-field factor (tensor) FTT(F, 7 ’ ; o),which relates the transverse part of the local field to the transverse part of the source field, is to be obtained from the integral equation u

u

I‘TT(?, J ’ ; O)= 6T(J- J ’ ) +

.I

RTT(J,J ” ; W ) . ? T T ( J ” , J ’ ; W ) d3r’’.

(2.25) Once the solution for the transverse part of the local field has been obtained, the longitudinal field can be calculated by direct integration from

EL(< o)=

.I-

~ L T ( J7 ,’ ; w ) . EXt(?/;w ) d3r’,

where the LT local-field tensor tt

rLT(F,

J’; w) =

J

?LT(?,

7 uMB OLT

lEo0

(2.26)

7’; w ) is given by

-

(r7r + ’ I ; w ) . YTT(J”, 7’; w ) d3rl/,

(2.27)

The result in eqs. (2.26) and (2.27) is obtained readily by combining eqs. (2.9), (2.21) and (2.24). Although the source field in optics is usually transverse, we have realized above that the induced current density in the system probed attains not only a transverse part but also a longitudinal part which in turn gives rise to a longitudinal component in the local field. This mixing of the transverse and longitudinal dynamics is of utmost importance in mesoscopic interaction volumes of condensed matter. In macroscopic optics the longitudinal part of the current density is often so small that it can be neglected. In passing, one should note that FTTand TLTare causal field-field response functions, hence satisfying the Kramers-Kronig relations. It may happen that the source particles are so close to the medium under study that the transverse current density volumes overlap. The driving (external) field in such cases contains a longitudinal component, EYt,which we in the spirit of response theory consider as a prescribed quantity. The external part of the longitudinal field now enters the many-body Schrodinger or (Pauli) equation and in the linear approximation gives rise to an extra current density,

I(<0 ) =

I[?,”,“(?, + zy:(J, .Eyt(?’; J‘; w)

J’; w)]

w ) d3r’,

(2.28)

in addition to the one in eq. (2.21). As above, I have extracted from the relevant conductivity tensor a part 8,”:(J, F’; w ) giving a transverse current density

v, § 21

27 1

LOCAL FIELDS AND NONLOCAL OPTICS

response, and a part zr:(7, 7’; w ) responsible for the longitudinal current density response originating in In compact form, the two material relations can be written as one:

e.

I(<W ) =

s

0

(7,7’; W ) . ( g ~ ( 7W’ ;) + i?pt(7’;0)) d3/,

(2.29)

eMB

where the four ( T T , TL,LT, LL) conductivity tensors can be formally obtained from zMB(y,J’; 0)via

(2.30)

A , B = T or L.

Although the external field has a longitudinal component, the loop equation is still only for the transverse part of the local field, and now takes the form 0)=

I$(<

EyLm(<W ) +

s

ETT(7,

7 ’ ; O). &(7’;

W ) d3r‘.

(2.3 1)

By comparison to eq. (2.22) it is seen that the integral equation problem is the same as before, except that the transverse driving field, i?yt, has been replaced by a new effective (eff) one:

Ey:,(< 0)= k“(< 0) - i b o / [ B : ( J - J”; 0)+ ~ ~ ( J”; 7 0)l -

.zyF(7”,7’; W ) . p ( 7 ’ ;0)d3y” d3r’.

(2.32)

The new driving field is also transverse, but it contains a contribution from the longitudinal part of the external field. The structure of eq. (2.32) shows that the new contribution originates in the transverse current density generated by I??‘, HT and signalled by the transverse propagator Do + gT.The formal solution of the integral equation in eq. (2.3 1) is ZT(c W ) =

J-

r T ~ ( 7 7, ‘ ; W ) .,G$fi(J‘;

cf. eqs. (2.22) and (2.24). With a knowledge of &(< the local field can be determined from EL(<

+J-

w)=

l&OW

(2.33)

W ) d3d;

0)the

longitudinal part of

F(<0)+ -f Er:(F, 7’; 0). f?rt(7’; W ) d3r’

/ Ep(7,

I&oW

7’; W ) . T T T ( J ’ , 7 ” ; W ) . E(&(7”;

0) d3r” d3r’.

(2.34)

In the preceding part of this section, I have underlined the circumstance that to determine the local electric field in linear optics one must solve a loop problem

272

LOCAL FIELDS IN LINEAR A N D NONLINEAR OPTICS OF MESOSCOPIC SYSTEMS

[v, 9

2

(integral equation) for only the transverse part of the prevailing field. Although the aforementioned fact is founded on many-body response theory, it is clear that the situation may change if the many-body approach is abandoned in favor of a single-particle approach. In mesoscopic electrodynamics, in practice one almost always must resort to one or another kind of one-particle approach. In the popular random-phase-approximation (RPA) approach, one assumes that the particle’s (electron’s) response is to the total local field g(7;w ) = &(T; u)+ ZL(< u), instead of to the field g ~ ( < w )+ If we denote the associated trRPA one-particle conductivity tensor by o (?, T’; w) the constitutive relation in eq. (2.29) is replaced by

e(
(2.35)

J

By combining eqs. (2.19) and (2.35) and allowing the external fields to also have a longitudinal component, i.e., pt(< w ) = gxt(<w ) + gyt(<w), the loop equation takes the form

g(<w ) = et(< w )+

s

K

(?, ?’;

0). I?(?’; w )d3r‘,

(2.36)

uRPA

with the RPA kernel given by

The replacement of the many-body conductivity by the much simpler RPA conductivity thus has the consequence that the fundamental loop equation for the transverse field is turned into an integral equation for the total field. In condensed-matter physics, low-frequency electrodynamic interactions are usually dominated by the longitudinal part of the local field. Also at optical frequencies it may happen - particularly in mesoscopic media - that the longitudinal coupling is the most prominent. If this is the case it is ofien possible to rely on a so-called density-functional calculation when studying the dynamics of the mobile particles, in the discussion below electrons with electric charge -e. Examples of application of the density-functional approach to clusters, metallic surfaces and semiconductor heterostructures may be found, respectively, in Ekardt [ 1984, 19851, Puska, Nieminen and Manninen [ 19851, Beck [1987], Ekardt and Penzar [1991], Rubio, Balbas and Alonso [1992], Pacheco and Ekardt [1992a,b, 19931, Alonso, Rubio and Balbas [1994], in Dobson and Harris [ 1986-19881, Liebsch [1987], Gies and Gerhardts [1987,

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LOCAL FIELDS AND NONLOCAL OPTICS

19881, Kempa and Schaich [1988], and in Ando [1977a,b], Yi and Quinn [1983], Wendler and Grigoryan [1989]. To see how the density-functional approach fits into the framework established above, let us start from the relation

N(7; w ) =

.I

x~s(7, 7’; w)[cp(F‘; w ) + qXc(F‘; w)] d3r‘,

(2.38)

which states that in the linear regime the correctly induced electron density N(< w ) can be reproduced in an appropriate single-particle potential that is the sum of the local-field potential (2.39) and the so-called exchange-correlation potential, cpxc(F’;0).In eq. (2.39), cpext(F; w ) is the potential of the external source. In the material relation in eq. (2.38), x~s(7,F’;w ) is the density-density response function of the noninteracting (single-particle) Kohn-Sham (KS) ground state. The exchangecorrelation potential is defined as the potential one must add to the localfield potential to obtain the correct electron density. By introducing the formal nonlocal relation cpxc(7;w ) = / h c ( 7 ,F’;

(2.40)

w)N(?’; w ) d3rr

between the exchange-correlation potential and the induced electron density, one is led to the following integral equation for the longitudinal part of the local field: &(< w ) = ZT&(.‘; w )

+?e

I{pa’&(?,

F’; w ) - xi;(;, F’; w)]} . EL(?’;w ) d3r’,

(2.41)

with the effective driving field given by

I??:*(.‘;

“S {aV’h&(7,

7’; w)-Ac(7,7 ’ ; w)]} .

w)= -

e

p(?’; w ) d3r’. (2.42)

If one compares eq. (2.41) with the corresponding many-body result 1 +-l&OU

HMB

+ +/.

(Y,r ,w) .

(sLL

w ) d3r’,

(2.43)

obtained from eq. (2.34) upon neglect of the transverse dynamics, it appears that the price one has to pay when switching from the longitudinal manybody conductivity to single-particle response functions is the replacement of the

214

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[y 5

2

direct determination of EL(;) from the right-hand side of eq. (2.43) by a selfconsistent loop calculation. The quantity (4n /?- ?‘/)-’ appearing in eq. (2.39) is just the non-retarded version of the electromagnetic scalar Green’s function do(?- 7’; w)= exp(iq0 IJ- 7’1)/(4x - ; 1 ?’I), as one might have anticipated. 2.3. LOCAL-FIELD CALCULATIONS IN MESOSCOPIC MEDIA

The description of the linearized local-field electrodynamics presented in the two preceeding sections has not been directed specifically towards the optics of mesoscopic media, or mesoscopic interaction volumes being parts of a macroscopic system. From an optical point of view, a mesoscopic medium is a medium for which the linear extension in at least one direction is much smaller than the optical vacuum wavelength(s), Ao, used to investigate the medium. For a mesoscopic particle (quantum dot, cluster, molecule, atom, . . .), the mobile electrons are confined (in three dimensions) to a volume of linear extension much less that the optical wavelength. For an ultrathin string, or a quantum wire, the linear dimensions of the cross section are much less than &,, whereas the length of the string is macroscopic. This means that the electrons are subjected to spatial localization in two dimensions in the string case. In an ultrathin film, or a quantum well, the film thickness is much smaller than Ao, and the mobile particles are subjected to strong confinement in one direction only. The localization of the electron motion to a linear scale (in one, two, or three dimensions) much less than the vacuum wavelength may lead to a significant simplification of the general scheme of calculation. To realize this, let us consider the case of a mesoscopic particle subjected to a purely transverse source field. By combining eqs. (2.21) and (2.24), one obtains

j(? 0)=

PMB iJTT (Y,Y

;w )

-I

uMB +oLT (7,F’; o)]. ?TT(?’,

7”; W) . zFt(F’’; w)d3r” d3r’. (2.44)

Since the external field, J?Ft(7; w), in contrast to the local field, &(< w), is slowly varying across the mesoscopic particle it is justified to make a Taylor series expansion of @(?’’; w) in eq. (2.44) around the center of mass of the particle. If this is located at ?” = Jo,one thus has in lowest order:

(2.45)

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In cases where it is sufficient to stop at the lowest order, the mesoscopic particle acts as an electric-dipole (ED) receiver for the external field. I stress here that a Taylor series expansion of the local field, &(?’; w), in eq. (2.21) is usually meaningless since &-(?I; w ) in most cases is varying rapidly in space across the particle domain. If the source field also contains a longitudinal part, one cannot in general employ a Taylor series expansion on the zrt-part since this will often be varying rapidly across the particle. Let us assume that the loop equation for the transverse part of the local field inside the particle has been solved. For a given many-body conductivity, this in turn means that the induced current density j ( <w ) in eq. (2.29) is known. In the domain, v T , outside the transverse current density domain ( V T )the prevailing field may then be calculated from the equation

I?(< 0)= ,!??‘(<W ) - i b w

J Bi(?-

7 ’ ; W ). j ( J ‘ ; 0)d3#,

JE

PT;

(2.46) cf. eqs. (2.19) and (2.20). Since the transverse electromagnetic propagator, ttT Do(?- J ‘ ; w), is slowly varying across the current density domain, it is adequate to make a Taylor series expansion of 8;around J’ = Jo. To lowest order, this gives

Z(C 0)= ,!??‘(<0)- ~ w * B ; ( J -Yo; 0). d ( w ) ,

(2.47)

where

p’(w)=

j ( J ‘ ; w)d3r’

(2.48)

is the electric dipole moment of the induced polarization in the particle. In the approximation of eq. (2.47), the mesoscopic particle emits light as an electric dipole radiator. If the approximation for j(cw ) given in eq. (2.45) is used also, the mesoscopic object behaves both as an electric dipole receiver and radiator. On the optical side, so to speak, the local-field calculation is simplified in relation to the general case by the fact that the linear extension of the mesoscopic medium in at least one of its dimensions is much smaller than the optical wavelength of the source field. On the electronic side, a simplification may occur in a rigorous microscopic description due to the fact that the quasicontinuum of electronic states may be replaced by a distinct level discretization (in at least one quantum number) for a mesoscopic object. To grasp the main idea in a first principle local-field calculation in a mesoscopic medium, let us

276

LOCAL FIELDS IN LINEAR AND NONLINEAR OPTICS OF MESOSCOPIC SYSTEMS

[v, 5 2

start by considering the structure of the linear many-body conductivity tensor, HMB + +, a (Y,Y ; 0 ) .Albeit one has to resort to one or another kind of one-particle approximation in a specific calculation, the main approach described below can essentially be carried over to the one electron case. To keep the analysis as general as possible the many-body framework is preferred below. Neglecting the usually small spin conductivity, one has (2.49) where -MB

Odia (f,7 ;w )=

ie2 -No(f) mu

a(?-

H

7)U,

(2.50)

is the diamagnetic (dia) part of the conductivity, and (2.51) is the paramagnetic (para) part. In eq. (2.50), No(?) denotes the many-body electron density in the field-unperturbed state, and m the electron mass. In eq. (2.51) the double summation runs over the various many-body energy eigenstates, these being denoted by capital letters I , J , etc. For an energy eigenstate A , the many-body wave function is named YA,the energy EA, and the probability that this state is occupied PA. In eq. (2.5 1) also appear the many-body transition current densities jI-,J(F) and JJ-I(~’).In explicit form, the transition current density from state A to state B is given by

(2.52) at space point F. The vector FO(F~) denotes the different electron coordinates, and the summation C , and product run over all electron coordinates. The paramagnetic part of the conductivity given in eq. (2.51) is built up of a double sum of terms, and in each of the terms a set of two levels, say I and J , is involved. Furthermore, the tensor belonging to a given term ( I ,J ) is just the tensor product of the two transition current densities, $+J(F) and ~ J - I ( ? ’ ) , one taken at the point f , the other at F’. The fact that the two vectors J,,J(F) and y~-l(r‘’)entering a given term in the double sum are functions only of r‘ and F’, respectively, is of utmost importance for the calculation of local fields

nB

v, 9 21

LOCAL FIELDS AND NONLOCAL OPTICS

217

in mesoscopic media as we shall see below, where the accompanying physical explanation will also be given. Since the above mentioned separability of the r' and 7'- dependencies also leads to a substantial simplification of the localfield calculation, it is of interest to seek a rewriting of the expression for the diamagnetic part of the conductivity. As it stands in eq. (2.50), this part does not display separability. Not only do we want to bring zzf(r',7'; w )into a separable form, we also want to look for a form resembling as closely as possible that of the paramagnetic part. To achieve the aforementioned goal, let us start from the expression

used by Keller [ 1996al to discuss the gauge invariance of the linear many-body response theory. In this equation N ( 7 ) and N(7') are the many-body number operators taken at 7 and r", respectively. The relation in eq. (2.53) appears adequate from the outset since it contains the characteristic diamagnetic form iVo(?)G(r'- 7') (cf. eq. 2.50), and because the individual terms on the left hand side exhibit the kind of ?and 7' separability we seek. To put the transition current densities on the scene, we use the commutator relation

where GF is the field-independent [free (F)] part of the Hamiltonian operator of the mesoscopic system, and &(?) is the field-independent part of the manybody current density operator. By taking the IJth matrix element ( ( I ).. .I J ) ) of eq. (2.54), one immediately finds

and upon use of this (and the analogous one for (J lN(r')\ I ) ) , eq. (2.53) can be written as

a' . V[No(?)G(F- F')]

278

LOCAL FIELDS IN LINEAR AND NONLINEAR OPTICS OF MESOSCOPIC SYSTEMS

[v, § 2

a'

On the right hand side of this equation the operator can be moved out in front of the double summation sign. Upon doing this, one realizes that

(2.57) One might add a divergence-free (in the ?'-coordinates) vector field to the right hand side of eq. (2.57) and still fulfil eq. (2.56). Here, we disregard such a possibility. Using the tensor identity 9 . (A'& = (ad).$ + (2. one realizes that

a)$,

V[No(F)6(7- F')]

=

a . [N0(F)G(7- 7')D],

(2.58)

a

operator on the right hand side of eq. (2.56) out in front and by moving the of the double summation sign, a comparison of eqs. (2.57) and (2.58) gives

neglecting again a possible rotational-free (in the ?-coordinates) vector field. Interchanging the dummy summation variables I and J , the relation in eq. (2.59) enables us to express the diamagnetic part of the conductivity in the form sought: (2.60)

By adding eqs. (2.5 1) and (2.60), one can write the many-body conductivity for instance as follows [a factor of two stemming from the spin summation may be taken out from the probability factors (PJ,P,) and placed in front of the summation signs, if more adequate]:

(2.6 1) If the field-matter interaction is at resonance at a particular transition, say J 4 I , so that hu = El - EJ (El > E J ) the contribution to BMBfrom this transition is purely paramagnetic.

YP21-

LOCAL FIELDS AND NONLOCAL OPTICS

279

Let us now return to the loop equation for the transverse part of the electromagnetic field. Thus, by combining eqs. (2.22), (2.23), and the TT-part of eq. (2.61), it appears that (2.62)

where

and

(2.64) In eq. (2.63) [and (2.64)], jT-,J denotes the transverse part of .&-J The result in eq. (2.62) is of central importance for our understanding of local-field phenomena in mesoscopic systems, and also allows one to establish an effective scheme for carrying out local-field calculations in many specific cases. The physical interpretation of the result in eq. (2.62) is interesting because of its simplicity. Hence, to obtain the transverse part of the local field at the point 7, one must add to the transverse external field the transverse part of the induced field, This part is given by the terms CIJF;~JI, and thus consists of individual contributions from the various electronic transitions. For a given pair of states, say I and J , one finds two field contributions: one related to the I -+ J transition, and one to the opposite transition J -+ I . Let us now consider the contribution from the I -+ J transition. When the many-body system undergoes this transition, a transverse current density distribution jT-,I is generated in space. From each point (J’) in this distribution an electromagnetic response, signalled by the propagator Ei(F- 7’; W ) + zT(F- 7’; w),is generated at the observation point 7 as described by the vector field F$(<w). The amplitude strength with which the field contribution from the transition I -+ J is emitted is proportional to the work carried out by the yet unknown local field in order to excite the many-body system the opposite way, i.e., from state J to state I . This work is proportional to the quantity PJI(W) given in eq. (2.64). The reason that the strengths of the contributions from the various transitions are unknown originates in the fact that the different transitions are coupled together by the (&-+I).

280

LOCAL FIELDS IN LINEAR AND NONLINEAR OPTICS OF MESOSCOPIC SYSTEMS

[y 8 2

induced local field. To determine the prevailing field in a self-consistent manner, eq. (2.62) is inserted into eq. (2.64). This leads to the following matrix equation problem for the unknown field strengths:

bJI(0)-

N$(w)@OP(m)

=

HJl(w),

(2.65)

w)d3r,

(2.66)

OP

where A&w)

=

F&,(<

/j;-,(F).

@'(ew )d3r:

(2.67)

where the quasi-continuum of electronic levels is replaced by a limited number of discrete levels, the dimension of the matrix Nzp(m) is usually so small that the matrix problem in eq. (2.65) may be handled by numerical methods. In the absence of local-field effects the resonance condition for the system is given by h w + E j -El

=

(2.68)

0,

provided that irreversible damping mechanisms are negligible. The resonances appearing via eq. (2.68) coincide with the zero points of the various denominators of the many-body conductivity (see eq. 2.61). These Bohr transitionfrequency resonances are modified by the local-field coupling between the various transitions. In the presence of local-field effects the resonance condition is given by Det{h,op - N$(w)}

= 0,

(2.69)

where Det{...} means the determinant of {...}, and SJ,,op is the Kronecker delta. The associated resonance (RES) values PiEsof the @J'S are obtained from

p(w ) -

c

N&( w)@gS(w ) = 0.

(2.70)

0,p

The local-field resonances are associated with the self-sustaining solutions I??~'(F; w ) for the local field, so that @ES(<

w)=

cF;(c

w)p;E"(w).

(2.71)

IJ

The resonance field is the solution to eq. (2.62) in the absence of the external source field.

v, 9: 21

LOCAL FIELDS AND NONLOCAL OPTICS

28 1

2.4. RETARDED LOCAL-FIELD INTERACTION AT DISTANCE: SPACE AND TIME-LIKE

COUPLINGS

In 9: 2.1 an electromagnetic propagator formalism was employed to establish in the space-frequency domain a physically intuitive relation between the prevailing current density at space point 7’ and the local electric field at 7. The fundamental integral relation between the aforementioned vector fields was given in eq. (2.19), and the associated propagator was shown to consist of the parts given in eq. (2.20). To gain further insight into the local-field problem it is useful to analyze the structure of the electromagnetic propagator in the time-space domain. I divide this analysis into two parts: one describing so-called retarded interactions, and another one associated with non-retarded effects. The analysis of the retarded dynamics is given below, and the non-retarded interaction is investigated in the subsequent section. First of all, it is necessary via the propagator 80(7-7 ’ ; o)to give a precise definition of what is meant by retarded and non-retarded interactions. Hence, non-retarded interactions are those which are independent of the speed of light. In the Green’s function formalism, these can by picked out letting co 4 00 in u Go(?- 7 ’ ; w). In this limit the dominating terms, which diverge as ci, are the - 7’; o)and FL(7 - 7’; (0). The ci-divergence after two self-field terms zT(7 multiplication with (see eq. 2.19) results in precisely the electrostatic factor b c i = ~ i ’The . physical interpretation of the non-retarded local field phenomena is discussed in 92.5. The retarded part of the interaction is described by the -T transverse electromagnetic propagator Do(?‘- F’; o),which contains terms of order c: and lower. In the space-frequency domain the transverse propagator is known to have the explicit dyadic form (see, e.g., Keller [ 1996a1)

(2.72)

-

where R = F- J ’ , R = 17- 7’1, and .;? = (7- 7’)/ 17- 7‘1. As before, qo = d c o denotes the vacuum wave number of light. The transverse propagator consists in the frequency domain of a far-field part (2.73)

282

LOCAL FIELDS IN LINEAR AND NONLINEAR OPTICS OF MESOSCOPIC SYSTEMS

[y 2

proportional to R-‘, a middle-field part

proportional to R-2, and a near-field part

which is proportional to R-3. In the limit qoR asymptotic form HT

-

+

0, the propagator has the

1

+ ZzZz)+ -23 U3 + O(qoR)

,

(2.76)

a result which confirms that only the self-field terms survive in the electrostatic ++T limit qo -+ 0 (Do K c: in this limit). The result in eq. (2.76) also demonstrates that 0”: diverges as R-’ when R 4 0 , and this confirms that the integral on the right-hand side of eq. (2.13) is absolutely convergent, as already pointed out in the text below this equation. ++T . Although only the full propagator Do IS a physically meaningful quantity in the time-space domain, it is instructive first to analyze separately the time-space structure of the far-, middle- and near-field propagators. Before doing this, let us write down the time-space relation between the retarded (RET) part of the local field, EFET(7, t ) , at the spacetime point (F, t ) and the induced current density at (?’, t’) . As indicated, the retarded part of the local field is transverse, and given by E?ET(7,

t ) = p ( 7 ,t )+

s

Do(?- 7 / ,t - t’) . -T

&7(7’,t/) 3 / d Y dt’, at‘

(2.77)

cf. eqs. (2.19) and (2.20). Using the notation t = t - t’ (and as before 2 = 7- 7’), and the generic form (2.78) for the relation between the Fourier transform pairs Z ( 2 , r ) and Z(2; w ) , it readily appears that the far-field part of the propagator is given by (2.79) in the time-space domain. The result in eq. (2.79) is the familiar one, stating that in the far field the source and observation points are coupled by an outgoing

v, I21

283

LOCAL FIELDS AND NONLOCAL OPTICS

///

........ ... ....... .

.....................

, I

, R -

L

=0

Fig. I . The middle-field and near-field parts of _th_e transverse e1ectrom:gnetic propagator, = [co/(8nR2)](E- 32~Zz)m(R, t) and N(R, t) = [co/(8nR2)](U- 3Z*Zp)n(R, t), respectively, depend on the, time delay T = t - t’ through the functions m(R, t) = sgn(Wc0 - 5 ) and n(R, t) = (cot/R)sgn t + (cgt/R - 1)2gn(,Wco - t),and above are shown m(R, t). n(R, t) and m(R, t)+ n(R,t) as functions of t.The M + N-propagator gives rise to a response which is causal (tz 0 ) , space-like (t < WCO), and retarded (CO finite).

k(Z,t)

spherical light pulse propagating with the speed of light and with the electric field concentrated on the spherical delta function shell S(Wc0 - t),the light cone. Since (2.80) it is realized that the middle-field propagator takes the form

where 0 is the Heaviside unit step function. The time (t)dependence of the the t-dependent middle-field is shown in fig. 1. For time-like interuals (t > WCO) factor [@(R/co- t)- O(t- R/co)] is equal to - 1 , and for space-like interuals (t < Wco)it is + 1. Although space-like events cannot be coupled by a light pulse propagating on the light cone, space-like events can interact via the inherent nonlocal structure of quantum mechanics. I shall return to this point in 92.5. However, if t < 0, the relation between aJ’(7’,t’)/dt’ and f?FET(F,t ) does not obey the principle of causality, and this is also irreconcilable with quantum nonlocality. Altogether, this means that in the time-space domain the middle

284

[y 9 2

LOCAL FIELDS IN LINEAR AND NONLINEAR OPTICS OF MESOSCOPIC SYSTEMS

field alone behaves unphysically, at least for t < 0. The near-field part of the propagator is readily obtained using the result (2.82)

Hence, in terms of Heaviside unit step functions the time-space form of given by

3(i,t) =

3 is

-

{

(:

x t[@(t)-O(-t)]- --t

(:

0 --t

1

(

311

- 0 t--

.

(2.83)

A schematic illustration of the time dependence of the near-field propagator is also shown in fig. 1. In the non-causal region (t < 0), one has tt'

N ( R , Z)

=

-(CO5 8nR2

-

3e;ie;i) = -B(i,t),

t

<0

(2.84)

and this shows that the non-causal behavior of the middle field is cancelled by that of the near field, i.e., k ( i ,t)+ E ( i , t)= 5 for t < 0, a satisfactory result. For time-like intervals (t > WCO), one has R N ( R , t)= co (D - 3Z&j) = t), t > -, (2.85) 8nR2 CO and again the sum of the middle- and near-field effects is zero,

-A@,

t t -

F ( i ,t)+ B(i,t)= 8

for

z > wc0.

We have thus reached the important conclusion that that middle and near-field effects together cannot contribute to retarded electromagnetic interactions for time-like events, nor do these effects together break the principle of causality. In the space-like region (0 < t < WCO) the near field is given by tt-?

N ( R , t)= 8nR3 and since in this time interval

(2.86) CO

t t - . R 0 < z < -, M ( R , t )= co (5-327&), 8xR2 CO the physically relevant propagator becomes

G(2,t)+3(2,t)= 0C2t -(U 4nR3

t t

-

(2.87)

R 3e;ie;?), 0 < t < -.

(2.88)

CO

The time dependence of k + 3 is shown in fig. 1. It appears from eq. (2.88) that the middle and near field together only gives rise to a near-field distance

v, § 21

LOCAL FIELDS AND NONLOCAL OPTICS

285

dependence (- R-3) in the time-space domain. In 5 2.5, it is shown that the nonretarded self-field interaction also has a typical R-3-dependence. Altogether, the conclusion is to be that only far- and near-field effects exist in the time-space description of local-field phenomena. The afore-mentioned considerations have thus demonstrated that the retarded (and I stress the word retarded) part of the local-field interaction is given by the transverse time-space propagator

-

t+T I * Do(R; t) = --(U 4nR

c2t

+-qU 4nR3

-ZjZj)

tt

6

- 3ZEZj) 0(t)0

(2.89)

The result in eq. (2.89) reflects an interesting complimentarity between what one might call quantum nonlocality and electromagnetic nonlocality. Hence, for space-like events, interactions are intermediated by quantum nonlocality which obeys a characteristic R-3-dependence. These interactions for a fixed distance separation grow linearly in time, i.e., they are proportional to t.When the light pulse sent out from the source point ?’(at t’) reaches the point of observation (J), at time t = t’ + WCO, the (retarded) nonlocal quantum mechanical interaction is destroyed and replaced by a nonlocal electromagnetic interaction. This interaction exists only on the light cone, given by S(R/co - t),and exhibits the characteristic far-field dependence, R-’ . A schematic illustration of this complimentarity in local-field interactions is shown in fig. 2. Although the nonlocal quantum interaction is not limited in time by the speed of light, the effect described by the last term of eq. (2.89) does depend on the velocity of light and is a retarded, but causal, phenomenon. For finite time intervals t = t - t’ > 0, there is no singularity in the near-field term since the step function O(R/co - t) removes a spherical region around R = 0 from the propagator (cf. fig. 2). Provided that the time development of the induced current density (at the point J ’ ) is so slow that it can be neglected over the time interval ( t - WCO, t), a situation which may occur in the near-field for a two-level atom performing (a prescribed) dynamical evolution between the two states, only the integrated effect of the near-field propagator will be of importance. Since

-

(2.90) it appears that the retarded nonlocal quantum interaction is proportional to [c$/(4nR3)][R2/(2cg)] = (8nR)-’. This means that the quasistatic transverse (retarded) interaction in the limit qoR 4 00 has a R-’-distance dependence,

286

[y 9 2

LOCAL FIELDS IN LINEAR AND NONLINEAR OPTICS OF MESOSCOPIC SYSTEMS

Fig. 2. Schematic illustration showing the retarded electrodynamic response in the near-field ( R - 3 ) zone of an atom (or a mesoscopic particle with strong electron confinement). The atom (indicated by the black dot e) starts to decay at time t', and at time t the light pulse (indicated by the thin black shell 0) has reached a distance R = co(t- t') away from the atom. In the near-field domain lying in front of the light pulse (shaded region), source and observation points are coupled only via space-like quantum effects. In the domain behind the light pulse (white region), the nonlocal quantum mechanical coupling is destroyed. CrT

a result that is in agreement with the asymptotic (q0R + 0 ) form of Do given in eq. (2.76). Before finishing this section, let us briefly compare the results derived above with those which would appear if one had wrongly used the standard (textbook) propagator given by (see, e.g., Born and Wolf [ 19701)

This propagator contains a longitudinal self-field term gL($; o)that gives rise to a non-retarded response and a cumbersome singularity of the order R-3 at R = 0. This singularity is removed in the retarded transverse interaction where the nearfield factor e'40R/(q$?3)is replaced by (eiqoR- l)/(qiR3), as we have seen above. The singularity at R = 0 still exists in the non-retarded part of the interaction, but here it can be accounted for as we shall realize in the subsequent section where the physical origin of the self-field terms is discussed. The textbook propagator, Bi + gL,in eq. (2.90) also has the problem that the near-field term does not exist in the time domain (the Fourier transform of exp(i: o ) / w 2 ) does not exist). Moreover, the middle-field term exhibits in the time domain a non-causal

v, § 21

287

LOCAL FIELDS AND NONLOCAL OPTICS

behavior (as before), which however cannot be cancelled by that of the nonexisting near field in this case. A heuristic textbook argumentation tells us that ttT retardation is always present in the textbook propagator, Do@;w ) + FL(i;w), since it contains the factor exp[i(qoR - or)]. This appears to be misleading since the argument holds only for strictly monochromatic fields which do not exist. In all cases one must transform the mixed space-frequency domain propagator back to the space-time domain, and this operation changes the conclusion, as we have realized.

-

2.5. NON-RETARDED SELF-FIELD INTERACTION

In the non-retarded limit (co + 00) only the self-field parts gT(F- F'; w ) and gL(F- F'; w ) of the electromagnetic propagator given in eq. (2.20) survive. If one neglects retardation effects, the relation between the local field and the transverse and longitudinal parts of the induced current density takes the form H

where a superscript SF has been put on the field to stress that eq. (2.92) holds only in the self-field (SF) approximation. One should emphasize here that the relation between the self-field, &F(< w), and the two parts of the current density, &(<w) and j ~ ( < w ) ,is a spatially local one, as it must be in the absence of retardation effects, whether these are due to electromagnetic or quantum mechanical nonlocality. This does not mean, however, that the relation between the self-field and the total current density, j ( <w ) = &(< w ) + &(< w), is of local nature. After all, this is not to surprising since the transverse (longitudinal) field originates in the transverse (longitudinal) part of the current density, and since the T-and L-electrodynamics differ by a factor of three. The spatially nonlocal relation between EsF(<w ) and j ( <w ) may be obtained starting from the expressions (see, e.g., Van Kranendonk and Sipe [1977])

&(< w ) = PV J' v x

[a x

(

I),/.-

j ( 7 ;w ) 4n ,F-

d3r' + t j ( <w),

(2.93)

where PV stands for principal value. The conditionally convergent integrals in eqs. (2.93) and (2.94) must be performed with a small sphere of radius E centered

288

LOCAL FIELDS IN LINEAR AND NONLINEAR OPTICS OF MESOSCOPIC SYSTEMS

[V,

5

2

on r' excluded at the outset from the domain of integration and then afterwards for E + 0. Since JT+ j L = 3, the two integrals are of course equal and can conveniently be written in the form ( j =~j++ j ) (2.95) w) The physical interpretation of f(<w ) appears readily if one divides j(< by the factor i h w . Thus, the quantity J(<w)/(i~ow) may be recognized as the quasi-static expression for the electric field stemming from a continuous distribution of electric dipoles having the spatial density ij(7'; w ) / w . By combining eqs. (2.92)-(2.95), it appears that

It is the quasi-static electric-dipole coupling which results in a non-retarded but nonlocal relation between ,!?SF(< w ) and I(<w). The self-field approximation in eqs. (2.92), possibly with the allowance of a longitudinal component w ) in the external field also, has been the starting point for many studies of the local-field electrodynamics of mesoscopic media in recent years. Even though retardation effects are neglected, the localfield problem attached to eqs. (2.92) is difficult to solve in general. Within the framework of the so-called scalar response formalism, only the longitudinal part of the induced current density is kept. This means that the starting point is the relation (Keller [ 19941)

,?rt(<

EfF(<w>= Ert(< w ) + (ieow)-'JL(<

w)

(2.97)

among the longitudinal quantities. Using the identity

eq. (2.97) leads to the following relation between the scalar potentials (cp, wext) and the induced electron density ( N ) :

's

q(< w ) = glext(v'; 0)- EO

g(7- 7 ' ; w)N(J';0)d3r',

(2.98)

v, 9: 21

LOCAL FIELDS AND NONLOCAL OPTICS

where g(7- 7’; w ) = (416 17scalar propagator

?’I)-’

289

is the non-retarded (Coulomb) part of the

(2.99) By combining eq. (2.98) with the two linear constitutive equations N(< w ) =

N ( 7 ; 0)=

J

I

xext(7, 7’; w)cpext(7’;W ) d3r’,

(2.100)

~ ( 77 ,’ ; ~)cp(7’; w ) d3r’,

(2.101)

one can obtain the following integral equation for the dynamic (external) densitydensity correlation function xext(7, 7‘; w )

x y 7 , 7 ’ ; w ) = x(7,7’; 0)

where ~ ( 77‘; , w ) is the independent-particle density-density correlation function. The integral equation above, possible with g replaced by EO

G(7,7 ’ ; 0) = g(7- 7 ’ ; W ) - -fxc(7, F“; W ) (2.103) e to take into account exchange and correlation effects (Garcia-Moliner and Flores [1979], Erdahl and Smith [1987], Mahan and Subbaswamy [1990], Gross, Dobson and Petersilka [1996]), is the starting point for most scalar response formalisms for, e.g., the (isotropic) polarizability of small (mesoscopic) particles (Ekardt [ 19841, Beck [ 19841, Puska, Nieminen and Manninen [ 19851, Ekardt [ 19851, Zaremba and Persson [ 19871). One is usually forced to adopt for the xc-part of the scalar propagator the local form (-Eo/eYxc(7, 7’; w ) = [dcpxc/dn],=~,~~~6(7r“).In the context of the polarizability of mesoscopic particles, the scalar formalism usually works best a low frequencies, but may in some cases also give a fair description at high frequencies, particularly around the plasma frequency. For observation points far from the current density distribution, which we assume is located in the vicinity of r“ = 6, the self-field approximation gives the asymptotic result

(2.104) It is seen that the asymptotic tail of the self-field exhibits a characteristic r-3dependence. This was, of course, to be expected due to the fact that the transverse

290

LOCAL FIELDS IN LINEAR AND NONLINEAR OPTICS OF MESOSCOPIC SYSTEMS

and longitudinal delta functions entering gT(? - F’; u)and given by

[v, 5 2

gL(F- ?’; u)are (2.105)

and

&(a)

u

=

1 4nR3

1 3

-6(?)E+ -(D-3qi&).

(2.106)

-

t+T

The transverse timespace propagator Do (R, t)LJO exhibits a characteristic RP3distance dependence for small R, but before a detailed comparison can be made one must transform the self-field relation in eq. (2.92) to the time domain (Keller u T [1997]). In the frequency domain, Do(R;w) does not show a R-3-behavior at small R; cf. eq. (2.76). To examine the self-field approximation in the time-space domain, we start from eq. (2.96) which in the time domain takes the form

-

aZsF(?,t ) - aEyy?, t ) L~(?, t) -

-

at

at

9% (2.107)

If one assumes that the interaction between the external field and the mobile electrons of our mesoscopic medium starts in the remote part ( t + -m), integration of eq. (2.107) gives 5 9Eo

ZSF(F,t ) = f?Ft(?,t ) - -

/

t

j(?,t‘) dt’

-a2

The induced current density must be obtained from the quantum mechanical mean value

j(?, t)= (Y

IT(?, t)l Y) ,

(2.109)

where

T(?,t ) = --e 2m

c([eu+ eAT(Fu,

t)]

a(?- ?u) + a(?- TU)[FU+ eA&,

t)]

}

a

(2.1 10) is the current-density operator of the electron system, Su and FU denoting the momentum and position operator of electron number a, respectively. In

v, § 21

LOCAL FIELDS AND NONLOCAL OPTICS

291

eq. (2.1 10) we have neglected the small spin part of the current-density operator. The terms in eq. (2.1 10) containing the transverse vector potential 2~are needed in order to assure the gauge invariance of quantum electrodynamics, and they give rise to so-called diamagnetic effects. In many situations, the diamagnetic terms lead only to small local-field corrections; for simplicity, let us neglect these phenomena below. If we also assume that the many-body system is in a pure state (2.111) I

where I!€’[) is the Ith (bound) many-body energy eigenfunction of the mesoscopic medium, the induced current density will be given by (2.1 12)

In eq. (2.1 12) appears the transition current density A+.,(?) already introduced in eq. (2.52). Since the wave functions I!&) can be taken as real, only terms with I # J in eq. (2.1 12) contribute. The current density depends implicitly on the local field through the various C / ( t )coefficients. Since in the present context we are only interested in the spatial structure of the self-field, there is no need to address here the problem of the local-field dependence of the CI(t)’s.By inserting eq. (2.1 12) into eq. (2.108), one finally obtains

(2.113) It appears from this equation that the contribution to the induced self-field, zSF(r‘;f) - zFt(J;t), from a selected transition, say I + J , undergoes the same time development in every point inside the spatial domain of the transverse ((IT+J(.‘)) [or equivalently longitudinal (1k+J(?))] part of the relevant transition current density (I/+,).It is the inherent nonlocal nature of quantum mechanics that causes this identical time development in each space point, and this exotic nonlocality can be traced back to the fact that the contribution C[(t)C;(t)&+,(F) from the individual transitions has “standingwave character”, i.e., a product of a time (C/(t)C;(t))and a space (J+J(?))

292

LOCAL FIELDS IN LINEAR AND NONLINEAR OPTlCS OF MESOSCOPIC SYSTEMS

[V,

5

3

@ +

Fig. 3. Schematic illustration of the electrodynamic response of an atom (or a mesoscopic particle) in the near-field (R-3) zone (shaded region). The total response consists of a self-field part (indicated at left) and a retarded part (at right). The self-field part gives rise to a standing-wave-like growth of the field in the entire near-field zone, and the dynamical development of the retarded response is as described in fig. 2.

factor. The quantum mechanical self-field dynamics thus is not subjected to any speed-of-light limitations or correlations (space-like or time-like). The term 5 $ - - . ~ ( F ) / ( 9 & ) in eq. (2.1 13) is responsible for the so-called contact interactions. Readers familiar with studies of the magnetic interaction between the nuclear spin and the electron spin in an atom may remember that a similar contact interaction between magnetic densities occurs in hyperfine-structure calculations (Cohen-Tannoudji, Diu and Laloe [ 19771). A schematic illustration of the retarded local-field interaction and the self-field dynamics is shown in fig. 3. Q 3. Local Fields in Mesoscopic Media with Strongly Localized Electron Orbitals 3 . I . SHORT- AND LONG-RANGE INTERACTIONS IN ELECTRONICALLY DECOUPLED

MOLECULAR SYSTEMS

Let us now investigate the local-field problem in a mesoscopic medium composed of molecules which to a high degree of accuracy can be considered as electronically decoupled, and let us restrict our analysis to the randomphase-approximation description in which the constitutive relation is given by eq. (2.35). By numbering the individual molecules by the index j , the conductivity tensor in eq. (2.61) takes the form

v, I 31

MESOSCOPIC MEDIA WITH STRONGLY LOCALIZED ELECTRON ORBITALS

293

where

is the single-particle conductivity tensor of molecule number j. The quantities the transition current densities between the oneelectron energy eigenstates m and n. The explicit expression for the transition current density from state a to state p is given by

jnn,(?)and jmn(?’) denote

(3.3)

where qaand q p are the wave functions belonging to the two energy eigenstates, the respective energies being denoted by E , and ED. The probabilities that these states are occupied are given by the Fermi-Dirac distribution factors f a a n d h . In order not to overload the notation, the reference to w in the arguments of the response functions has been omitted, and below I shall continue to do so in the relevant quantities. The electronic decoupling of the molecules implies that the transition current densities~,,(?) andil,n,(F’) are different from zero only if r‘ and ?’ are located inside molecule numberj. The local field in the aforementioned situation is given by

where

and

et(?)

= Ex‘(7). Although the The external field is still transverse, i.e., molecules are electronically decoupled, the field strengths fim,n, with which molecule number i radiates still depend on the presence of the remaining

294

LOCAL FIELDS IN LINEAR AND NONLINEAR OPTICS OF MESOSCOPIC SYSTEMS

[V, $, 3

molecules, identified by the running index j . In fact, the field strengths satisfy the set of algebraic equations J

O~,p~

where the coupling matrix NT;? is given by N;;

. pplol(~) d3r,

= Jym,n.(p)

(3.8)

and Hm,n,= / y m , n c ( F ). p t ( J )d31:

(3.9)

Terms in the coupling matrix for which i = j represent the dynamic self-coupling within the individual molecules, and terms with i # j are associated with the mutual interaction between different molecules. In view of the discussion of the retarded and non-retarded parts of the electromagnetic propagator given in 5 2.4 and 9 2.5, it is appealing to divide the @,,lml's of eq. (3.5) into so-called short-range (SR) and long-range (LR) parts: (3.10) where

and

By inserting eq. (3.10) into eq. (3.4), the result for the local field can be written in the form J

m ~"I,

J

'"I"'I

The coupling matrix NT;: necessary for the determination of the various field strengths also consists of additive contributions from the short- and long-range field: N;;:

= N;;:'

(SR) +

Nl;yr (LR).

(3.14)

The explicit expressions for the SR part of NT;Yi will be given in the following section.

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295

3.2. OPTICALLY DILUTE MOLECULAR SYSTEMS; POINT-PARTICLE MODEL

In order to deepen our understanding of the local-field problem in molecular media it is of interest to study some particularly simple cases. In this section we hence consider a particle system in which the optically polarizable molecules are so far apart that the short-range interaction between different molecules can be neglected. I name such a particle arrangement an optically dilute molecular medium. Inside a given molecule, say number i, the short-range contribution is much larger than the long-range contribution from the same molecule. It is therefore safe in most cases to neglect t h e j = i LR term when calculating the local field inside molecule number i. In the present context it is not possible to neglect the LR terms with j + i , albeit they are also small, because this would destroy the local-field phenomenon we are investigating. In passing I emphasize that the neglect of retardation effects in the electrodynamic interaction of a given molecule with its own field implies that the spontaneous emission from the molecule is suppressed (Crisp and Jaynes [1969], Aganval [1974]). In our semiclassical approach, where the electromagnetic field is unquantized, the spontaneous emission originates in the classical radiation reaction hidden in the wU/(6xico)-part of the Bi-propagator in eq. (2.76). Though the first term + Z&)/(SnR)] in eq. (2.76) is of the order c: compared to the radiativereaction term [w8/(6nico)] which is of the order ci’, it is usually less important because it is in phase with the self-field propagators which are of the order c:. The radiative-reaction term is out of phase with the self-field terms. Based on the considerations above, it appears that the local field at the observation point 6 located inside the ith molecule is given implicitly by

[(U

(3.15) where

is the so-called background (B) field consisting of the external field plus the longrange fields produced by the other ( j + i) molecules. Despite the fact that the

296

LOCAL FIELDS IN LINEAR AND NONLINEAR OPTICS OF MESOSCOPIC SYSTEMS

[v, 5 3

background field at ?Ldepends on the unknown local-field distribution (E(Y;‘)) inside the variousj-molecules, we assume in a first approximation that iB(<) is a prescribed field. Following next the standard scheme for loop calculations of the local field, it appears that

z(<)zB(ri)+ C =

(3.17)

Brn,n,F:i,(C).

m,,n,

The unknown Brn,,,,’sare to be obtained from the matrix problem (3.18)

and

H:,~,

JYmini( d3ri.

=

(3.20)

In the next section we shall analyze the self-coupling scheme a bit further. In the present context it is sufficient to use the fact that the relation between the local field and the background field can be written in the form

1Ti(c,

v‘, ’) .EB(6I ) d3rl,

.I?(< =)

where

Ti(c,6

I )

(3.21)

is the nonlocal field-field response tensor of molecule number

i. By inserting eq. (3.21) into eq. (3.16), one obtains

p @ =) F t ( y ; . )

/ Bi(<

-ipoo

-

<) .

zL(<, JI) .

d’r; d3r;,

(3.22)

i(4

where

$B( ?, , ; I .i) =

1z’(<,?y).T’(?y,

F,:) d3ry

(3.23)

is the so-called background conductivity tensor. From a physical point of view this tensor relates in a linear and nonlocal fashion the current density to the background field in given molecule:

I;(<)=

J zi(5,

y;)

.gB(Fj)d35!

for molecule number j (Keller [ 19941).

(3.24)

v, a

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MESOSCOPIC MEDIA WITH STRONGLY LOCALIZED ELECTRON ORBITALS

297

When studying local-field phenomena in molecular media, one often adopts the so-called point-particle model from the outset (Van Kranendonk and Sipe [ 19771). In this model it is assumed that the individual molecules are point-like quantities from an optical point of view. In turn this means that the induced current density of a given molecule is confined spatially to a single point. It is often stated that the point-particle model is valid because the optical wavelength (read wavelength of the external field) is large compared to the linear dimensions of the individual molecules. Such an argument is not rigorous a priori, since after all it is the prevailing local field that drives the dynamics of the individual molecules and this field may contain components which vary rapidly in space even on the molecular length scale. The heuristic point-particle model also leaves open the intriguing question of how to share the electrodynamics between the polarizability of the molecule and the near-field part of the electromagnetic propagator, as we shall see below and in the subsequent section. For optically dilute molecular media (mesoscopic or macroscopic) the integral equation in (3.22) allows one to (i) demonstrate that a meaningful point-particle model does exist and (ii) establish a rigorous basis for its use. Hence, since the transverse -T vacuum propagator Do(<- 5) is slowly varying in space on the length scale of most molecules it appears from eq. (3.22) that the background field is also slowly varying across the individual molecules. If one therefore neglects the variations of and iB across the molecular domain, eq. (3.22) is reduced to the algebraic form

Bi

i B ( i ; ) = EeX‘(ii)

+

c?(a, i/)P ( i j ) , -

.

(3.25)

.A*,)

with tt-

+

T(R.I - R . )

-T

- -2j).

= -’ ikooDO(R,

s-J-

uB(r,,7;) d’r: d’r,.

(3.26)

In the scheme above the positions of the various point molecules are given by the In terms of the electric-dipole-electric(center of mass) coordinates iiand i,. dipole (ED-ED) polarizability (3.27) the coupling tensor in eq. (3.26) can be written as t+-

t+T

-

T(R,-i,) = - p f J o 2 D o ( R ; - i j ) . E’(w).

(3.28)

Together, eq. (3.25) and (3.28) form a rigorous basis for electric point-dipole descriptions of the local-field electrodynamics in optically dilute molecular

298

LOCAL FIELDS IN LINEAR AND NONLINEAR OPTICS OF MESOSCOPIC SYSTEMS

[y 8 3

media. In the present example all the near-field electrodynamics is described in terms of a local-field dressed ED-ED polarizability given by

where

E i ( w )= i

1

F’(Y;.,Y;.’)d3rjd3ri

(3.30)

is the bare polarizability and a’(w) is the local-field correction to 2;. Using eq. (3.2), one can show that

where (3.32) is the electric-dipole transition matrix element for excitations from state a to state 6. The expression in eq. (3.31) is the standard expression for the bare (local-limit) polarizability. Omitting the diamagnetic factor zIw/(~,, - E ~ , )the expression for Ec is reduced to the standard expression for the bare paramagnetic polarizability (Wood and Ashcroft [ 19821). If the ED-ED point-particle model is insufficient - and this it will be if Fnlm, = 0’ for all relevant (nim;)t*T

transitions - a Taylor series expansion of Do and I ? ~ around ij can be used in eq. (3.22). Such a procedure leads to a multipole [electric dipole (ED), electric quadrupole (EQ), magnetic dipole (MD), . . .] treatment of the pointparticle absorption and radiation. The dressed polarizability hence obtains extra terms (ED-EQ, ED-MD, EQ-ED, MD-ED, EQ-EQ, EQ-MD, MD-EQ, MDMD, etc.). Studies of the local-field correction to the optical polarizability of mesoscopic particles have been carried out by Keller, Xiao and Bozhevolnyi [ 1993b, 19951. The linear and nonlinear optical properties of micro-crystallites and quantum dots have been investigated extensively in the last decade using the bare (electric-dipole) polarizability (A1.L. Efros and A.L. Efros [ 19821, Brus [1984], Schmitt-Rink, Miller and Chemla [1987], Hu, Koch and Tran Thoai [1990], Hu, Lindberg and Koch [1990], Koch, Hu, Fluegel and Peyghambarian [ 19921, Adolph, Glutsch and Bechstedt [ 19931, Willatzen, Tanaka, Arakawa

v, P

31

MESOSCOPIC MEDIA WITH STRONGLY LOCALIZED ELECTRON ORBITALS

299

and Singh [1994]). Ruppin [1975, 19761 included the effect of nonlocality in the electronic response of small metal particles, and without resorting to the electric-dipole approximation for the polarizability Broido, Kempa and Bakshi [1990] and Darnhofer, Rossler and Broido [1995] investigated the effect of longitudinal screening using the RPA-approximation. The microscopic vectorial theory (Cho [ 19911, Keller [ 1996a1) was used by Keller, Xiao and Bozhevolnyi [1993b], Takagahara [1993], Cho, Ishihara and Ohfuti [1993], Keller [1994], Cho, Nishida, Ohfuti and Bellequie [1994], and Keller and Garm [1994, 1995, 1996a,b], in their studies of the optical properties of small particles, quantum dots and microspheres. 3.3. MESOSCOPIC MEDIA DOMINATED BY SHORT-RANGE INTERACTIONS

Let us consider a mesoscopic medium in which the optically polarisable molecules are packed so densely that the induced near-field interaction is appreciable not only between neighbouring molecules but also between more distant molecules. Such a situation typically occurs if the linear extension of the medium is many times less than the vacuum wavelength(s) of the external field. If this is the case it is often righteous to neglect retardation effects. Exceptions to this are the cases of spontanous emission and local-field resonances. We still assume that the electron orbitals of the different molecules do not overlap. If retardation effects are neglected, the local field inside molecule number i (at position 6 ) is given by

where the background field (3.34) is taken as the sum of the external field and the longitudinal and transverse self-fields stemming from the other ( j # i ) molecules of the medium. The transverse and longitudinal parts of the induced current density in molecule number a ( a = i, j ) are denoted by and j:. Before proceeding with the localfield calculation let us seek a simplification of the expression for the background field. Thus, by utilizing eqs. (2.93k(2.95), one gets (3.35)

3 00

LOCAL FIELDS IN LINEAR AND NONLINEAR OPTICS OF MESOSCOPIC SYSTEMS

[v 5 3

Since there is no electronic overlap between molecule i and j , the so-called contact current density j j ( 6 )is zero. We write the remaining current density appearing in eq. (3.35) in the form

where (3.37)

is the asymptotic (16 - $1 00) part of T.'(rf.). The background field is now simplified by assuming that only the asymptotic part of the T and L self-fields of the various molecules overlap each other. This means that ATj(6)is essentially zero. With this assumption the background field is simplified to .--)

(3.38)

In some cases, the approximation

I.(<)+ 3$(r;)

M

(3.39)

-2&(r7)

is extremely good for electronically decoupled molecules. A combination of eqs. (3.33), (3.37) and (3.38) allows one to establish a rigorous basis for the non-retarded point-dipole interaction model used in numerous studies in near-field optics. Thus, from the considerations of 4 3.2 it appears that the solution of eq. (3.33) is given by eq. (3.21), and this in turn implies that the current density induced inside molecule numberj is given by eq. (3.24). Since the background field is essentially the same everywhere inside the integral of the induced current density this molecule, i.e., I?'($ ') M zB(Z,), becomes

.I

j'($)d3rj = -iwzJ(W).

EB(ij),

(3.40)

where the ED-ED polarizability, Z'(W),is given by eq. (3.27). If eq. (3.40) is inserted into eq. (3.37), the asymptotic current density j&(<) is given in terms of the background field f?"(ij). If this relation then is inserted into eq. (3.38),

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MESOSCOPIC MEDIA WITH STRONGLY LOCALIZED ELECTRON ORBITALS

301

and one takes 6 = &, we obtain the following relation between the background fields on the various molecules: (3.41) with (3.42)

By letting the index i in eq. (3.41) traverse all possibilities one generates a set of linear and inhomogeneous algebraic equations among the unknown i = 1,2,. . . ,N , on the position of the N molecules of background fields the mesoscopic system under consideration. Written in supertensor notation, the formal solution of eq. (3.41) takes the form

iB(ai),

(3.43) where

(3.44)

are the relevant supervectors, and

and

g=

(3.46)

3 02

LOCAL FIELDS IN LINEAR AND NONLINEAR OPTICS OF MESOSCOPIC SYSTEMS

[V, 5 3

are supertensors. In this particular case, the resonance condition for the local field is given formally by Det{&! - %} = 0;

(3.47)

cf. eq. (2.69). For a given optical frequency the resonance condition in eq. (3.49) may be fulfilled by moving (some of) the molecules around until the resonance condition is met. In the context of near-field optics, such a resonance is sometimes named a configurational resonance (Keller, Xiao and Bozhevolnyi [ 1993a,b]). Expressed in terms of the longitudinal self-field has the explicit form propagator EL(&- j,), which since Zi #

a,

(3.48) the coupling tensor

@(ai- R,)-

@(& - aj)is

=-

2,) Z'(0). I

(3.49)

In near-field optics one often uses a phenomenological (electric) point-dipole model in the theoretical analyses (Labani, Girard, Courjon and Van Labeke [1990], Girard and Courjon [1990], Girard and Boujou [1991], Keller, Xiao and Bozhevolnyi [ 1993a1, Xiao, Bozhevolnyi and Keller [ 19961, Girard and Dereux [1996]). The model is phenomenological in the sense that one assumes from the outset that the radiating units (atoms, molecules, . . . ) are point entities, and that the electric-dipole polarizabilities are the bare ones [ Z i ( w ) of eq. (3.30)]. The phenomenological approach leads to a coupling tensor - E j ) = - b 0 2 z L ( i i - 2;). Z ~ ( WBY) . comparison to eq. (3.49), it appears that the distance dependence and the tensorial form of the near-field propagator used in the phenomenological approach are in agreement with the rigorously obtained result. The bare polarizability, however, does not appear in the rigorous approach as one might have anticipated. Provided that one makes the replacement ??{(u) =+ in the phenomenological coupling tensor, the electric pointdipole model is based on a firm foundation, and the limits for its application follow from the approximations done in the rigorous derivation.

@(a,

i@(o)

3.4. LINEAR SHORT-RANGE INTERACTIONS IN TWO-LEVEL HYDROGEN-LIKE

(Is ++ 2pz) SYSTEMS

In recent years, a number of papers has appeared in which local-field effects in condensed media have been studied starting from the so-called MaxwellBloch formulation (Allen and Eberly [1975], Ben-Aryeh, Bowden and Englund

Y ii 31

303

MESOSCOPIC MEDIA WITH STRONGLY LOCALIZED ELECTRON ORBITALS

[1986], Bowden and Dowling [1993], Bowden [1995]), and in which it is assumed that the scattering units are two-level atoms. Using a generalized set of Bloch equations, where local-field effects are included in a quasistatic near (i.e., short-range) dipole-dipole (ED-ED) correction parameter, predictions concerning e.g. linear and nonlinear spectral shifts have been put forward (see, e.g., Friedberg, Hartmann and Manasseh [ 1973, 1989, 19901, Rosa-Franc0 [ 1987, 19901, Maki, Malcuit, Sipe and Boyd [1991], Hartmann and Manasseh [1991]). Novel implications for optical bistability, self-induced transparency, self-phase phase modulation, optical switching and adiabatic inversion have been treated. In quantum optics, near dipoledipole effects have been studied in the context of the quantum statistics of superfluorescence, amplified spontaneous emission, and various quantum-induced coherence effects. For an extended list of references to the afore-mentioned phenomena, the reader is conferred to Bowden [ 19951. To elucidate in the context of this article the role of short-range localfield effects in an assembly of two-level atoms (particles), I shall here use the formalism described in the previous section, and thus, for simplicity, limit myself to the linear electrodynamics. To be specific, it is further assumed that the two levels are hydrogen-like Is and 2p2 levels. By denoting the lower (Is) level by index 1 and the upper (2p2) level by index 2, the transition current density from state 1 to state 2 is in spherical (Y,8, y?)-coordinates with the local unit vectors Zr, .Zo and (Zv)given by Z ~ ( Y6), = K [ ( I+ fbr)zr cos 6 - 20 sin

(3.50)

where K = eh/(8ifirna:) and b = 3/(2ao), with a0 the (effective) Bohr radius. Due to the infinitesimal rotational symmetry around the z-axis, the transition current density, as indicated, depends only on the radial (Y) and azimuthal (0) coordinates. In writing down the expression for.T,2(rr0) we have assumed that the center of mass of the selected atom is located at the origin. Since51 = -52, it is a straightforward matter to show that the conductivity tensor of the selected two-level atom is given by

=32

(3.5 1)

leaving out the reference to the particle number. Using eq. (2.29) [the manybody and single-particle conductivity tensors are identical for systems (atoms) with only one mobile electron] and eq. (3.50), it appears that the current density induced in the atom under consideration is given by

J(< w ) = A(w&(?),

(3.52)

304

LOCAL FIELDS IN LINEAR A N D NONLINEAR OPTICS OF MESOSCOPIC SYSTEMS

[v, 8 3

where (3.53)

To determine the transverse and longitudinal parts of j ( <w ) one needs only to calculate the T and L parts of the transition current densityA2. Doing this one finds Z ( r , 8)

= 712(r, 6) =

5 { 3

-7lk 6)

2 -(Zo (br)3

sin 8 + 2Zr cos 8)

with 1 2 2 F(r) = - + __ + br (br)2 (br)3'

(3.55)

Since we have started from a finite-sized atom the T and L parts of the induced current density (as well as the current density itself) have no singularity at r = 0, even though this is not readily seen by taking a glance at eqs. (3.54) and (3.55). The asymptotic part (,fm(< 0)) of the current density W) = &F; w ) - fj(< w ) = +I(?; W) - &(F; w ) is given by

c?<<

&,(r,

8; w)

j L ( r , 8; w ) = - j k ( r , 8; 0) 8KA 1 - (Zo sin 6' + 2Zr cos O), 3 (br)3 = -

(3.56)

and hence exhibits the characteristic dependence. The asymptotic part of the transverse (or longitudinal) current density falls off at a much slower rate with the distance from the nucleus than the current density itself, which apart from an algebraic factor exhibits an exponential decay (- e-"); cf. eq. (3.50). Since 32n KA J(< w)d3r = --2: 3 b3

(3.57)

it readily appears that the asymptotic part of the induced current density in the selected two-level atom can be written in the form (3.58) in agreement with the general result given in eq. (3.37). The linear two-level short-range dynamics hence is brought in contact with the general development presented in 9 3.3.

v, D 41

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LOCAL FIELD ELECTRODYNAMICS IN QUANTUM WELLS AND THIN FILMS

305

4. Local Field Electrodynamics in Quantum Wells and Thin Films

4.1. SINGLE QUANTUM WELLS WITH 2D BLOCH AND FREE-ELECTRON DYNAMICS

Up to this point, I have described the local-field electrodynamics in mesoscopic media where the mobile electrons are subjected to quantum confinement in all three spatial directions. Within the framework of linear many-body response theory one is led to an integral-equation problem for the transverse part of the local field that involves all three spatial coordinates (see, e.g., eq. 2.22). A simpler local-field problem occurs in quantum-well systems where the electron motion is subjected to quantum confinement in only one of the spatial Cartesian coordinates, say z. The basic integral equation for the local field now involves only the z-coordinate, and if the width of the well is sufficiently small only a few bound states exist in the well. In turn, this means that when the integral equation is converted into a matrix-equation problem as described, e.g., in $2.3, the dimension of the coupling matrix is so small that numerical (or sometimes even analytical) inversion and diagonalization are possible. In metallic and semiconducting quantum-well systems the electron dynamics parallel to the well plane is not subjected to essential quantum confinement effects, and this leads in many cases to an interesting interplay between the two-dimensional (2D) Bloch or free-electron electrodynamics in the plane of the well and the atom-like dynamics perpendicular to the well plane. Since the amount of research literature dealing with the various aspects of the linear and nonlinear optical properties of quantum-well structures is quite comprehensive, cf. the references given in the introductory section, only a number of studies related directly to the local-field problem will be described below. To simplify the description we shall use the one-electron approximation to calculate the electrodynamics of the electrons in the quantum well, and we shall assume that the effective one-electron potential, V(r'), of the field-unperturbed Schrodinger equation has the form

where 711= ( x , y , 0) is the position vector in the plane of the well. The ansatz in eq. (4.1), which states that the potential is composed of a sum of a part V I ( ~ I ) which depends on 31 only and a part V_L(Z)which is a function of the depth coordinate z in the well, certainly is an approximation to the real potential which contains a cross-coupling term between the Fll and z-directions. The periodicity

306

LOCAL FIELDS IN LINEAR AND NONLINEAR OPTICS OF MESOSCOPIC SYSTEMS

[v, 5 4

of the screened ionic potential in the coordinates parallel to the quantum-well plane implies that

+a,,>

VII(7II) = q(FlI (4.2) for all in-plane lattice vectors i l l = (RllA,Rllv,O). With the assumption in eq. (4.1) and the criterion of crystalline periodicity in eq. (4.2) the energy eigenstates of the time-independent Schrodinger equation may be chosen in the form

-

11,nmkll - (7) = eikll.‘~~urn(~~l>~,(z),

(4.3)

where urn(71l)= urn(ql+ i l l ) for all i l l . In the plane-wave factor exp(iL1l . y?~), the vector LII= ( k ~ lkllu, ~ , 0) is an electron wave vector lying in the plane of the quantum well. Using the abbreviations a = {n,m, i l l } , and 0= {n’,m’, the transition current densities fulfil the relations

Lll},

JBa(J+

a,,) -

- -

= e’(ki1-k

il),RlljBa(7),

(4.4)

z,~)= e ~ ( ~ ~ ~ - ~ ~ ~ ) . ~ ~ ~ J ~ B ( ~ ) .

J ~ ~ (+J ’

(4.5) It is therefore readily realized that the conductivity tensor is invariant against 2D lattice-vector displacements, i.e., t*

+ -.I

a ( r , r ) = Z(F+i\l,7’+iIl),

(4.6) as one would have anticipated immediately in view of the periodic-potential condition in eq. (4.2). By a Fourier analysis of the conductivity tensor in the xy (x‘y’)-coordinates, viz.,

s

- -

- I

- I

3(7,7’) = ( 2 ~ ) - ~~ ( z , z ’ ; ~ ~ ~ , ~ ~ l ) 11e d2qll ’ q ~d2qll, ~ ’ r ~ ~ e ’ q ~(4.7) ~’r and use of eq. (4.6), it appears that the mixed Fourier amplitude must fulfil the condition

a(z,z’; ill,ill)= z ( z , z ‘ ; ~ l l , ~ ~ l ) e l ( ~ ~ l + ~ ~ l ) . a ~ ~ .

H

(4.8)

This implies that $11 + iil

=

611,

(4.9)

where 611is a 2D reciprocal lattice vector in the plane of the quantum well. Eliminating qil by virtue of eq. (4.9), one may write the mixed conductivity tensor in the form

(2Xl2

~ ( Z , Z ’ ; ~ I I ,=~ \ ~ )

C ~ ( z , z ’ ; i l l6,11- ~ 1 1 ) ~ ( ~ 1 1 + $ 1 1 611). -

(4.10)

61,

The result in eq. (4.10) leads to a substantial simplification of the constitutive equation. The adopted single-particle approach plus the fact that we are usually

v, I 41

LOCAL FIELD ELECTRODYNAMICS IN QUANTUM WELLS AND THIN FILMS

307

dealing with a substantial number of electrons in quantum-well systems make it natural to use the RPA version of the constitutive equation given in eq. (2.35). and coordinates brings this equation into the A Fourier transform in the form

?Il

(4.1 1)

which upon use of eq. (4.10) is reduced to

In the mixed Fourier representation the integral relation between the local field and the induced current density becomes

Together, eqs. (4.12) and (4.13) constitute the starting point for specific calculations of local-field effects in quantum-well systems. By means of the tensorial kernel

Z(z, z’; 4‘11, GI1 - 4‘11) = - i b w

J-

~ o (-z”; z

4‘11) . Z(z”, z’; 4‘11, GI1 - 4‘11) dz”, (4.14)

the integral equation for the local field can be written as follows:

II

(4.15) If this equation is compared to the general FWA version in eq. (2.36), it appears that the 2D bulk character of the response in the x and y-coordinates implies that instead of a complete determination of the x and y-dependencies (or equivalently the 4‘11 and dependencies) of the local field only a self-consistent solution for the prevailing field for a discrete set of wave vectors 4‘11 - 611(V Gll ) is needed. In the qualitatively important case where the external field consists of only one wave-vector component, named q i , and the electronic first-order Bragg scattering from one reciprocal lattice vector, = -gi, dominates, eq. (4.15)

iil

3 08

[v 4 4

LOCAL FIELDS IN LINEAR AND NONLINEAR OPTICS OF MESOSCOPIC SYSTEMS

is reduced to a set of two coupled integral equations among the mixed Fourier amplitudes Z(z; and Z(z; + viz.,

;i (?I),

4‘1)

+JR(z,z’;4‘1,-

Go11 - 4‘1) Z ( d ;4‘; + 61)dz’,

(4.16)

’

and Z(z;

4‘1 + 61)

=

4‘1 + GI, -4‘1) . Z(z’; 4‘;) dz’ + /‘i(z,z’; 4‘; + GI, -6; - 4‘1) . g(z’; 4‘; /‘i(z,z’;

+ 6;) dz’. (4.17)

In the above-mentioned two-band case, it is possible to obtain analytical expressions for the relevant conductivity tensors o(z”,z’;4‘;;,-4‘r;), z(Z”,z’;;p;

-;I),

H

+q),

q,-q

( z ; 4‘11) +

J Z ( z , z’; 4‘1,) .E(z’; 4‘11) dz’,

cJ(z’’,z’;4‘;; + Z(z”,z’;4‘;; + -4‘;;) and afterwards make a separation of the various kernels into z- and z‘-dependent parts. Using the technique described in 4 2.3, the integral equations (4.16) and (4.17) may next be transformed into a set of two coupled but linear matrix equations among the unknown field strengths. The above-mentioned formalism is needed in studies of (i) dynamical light scattering from quantum wells, and (ii) rotational effects related to parametric optical second-harmonic generation from simple metallic or semiconducting quantum wells exhibiting essential Bloch dynamics. A review of the optical second-harmonic generation from interfacial structures can be found in Richmond, Robinson and Shannon [ 19881, where also the ‘fingerprints’ of the Bloch dynamics are discussed. The overwhelming majority of the theoretical analyses of the optical properties of quantum-well systems has been based on the assumption that the in-plane electron dynamics is free-electron-like. With this assumption the loop equation for the local field becomes H

E(z; 4‘11)

=P

(4.18)

with H

K(z, z’; 4‘11)

=-ihw

J-

Go(z- z”; 4‘11) .?3(z”, z’; 4‘11) dz”.

(4.19)

As indicated only one wave vector ($11) is needed in the conductivity tensor due to the fact that the electron response to the field depends only on yil -

?Il.

v, 5 41

LOCAL FIELD ELECTRODYNAMICS IN QUANTUM WELLS AND THIN FILMS

3 09

The information one can obtain from optical studies of quantum wells is in general more rich in details in nonlinear reflection (transmission) experiments than in linear ones. To realize this, let us just consider the formal solution of eq. (4.18): (4.20) To obtain the explicit expression for the field-field response tensor, t?t(z, z'; 4'11) the kernel-separation techniques described in 9 2.3 may be used. In linear optics the external (plane-wave) field is essentially constant across the well. This means that the information one may hope to extract about the local field is the one appearing in the relatively structure-less second-rank tensor

(4.21) As a simple paragon of a nonlinear local-field problem, it is instructive to consider the optical second-harmonic generation from a quantum well in the parametric approximation. In this case, the external field, which drives the dynamics at the second-harmonic frequency, 213, is given by

(4.22)

where j ( z ' ;2q'll) is the forced part of the second-harmonic current density. In contrast to the linear case, the external field in eq. (4.22) is rapidly varying in space across the well in cases where l ( z ' ; 2q'll) has a nonvanishing component in the scattering plane. In these cases it is often sufficient to keep the non-retarded self-field part of the electromagnetic propagator (cf. the discussion in 5 2.5), so that eq. (4.22) is reduced to the local form (4.23)

.Zz being a unit vector in the positive z-direction. The forced current density is only different from zero inside the quantum well, and therefore Eext(z;2?jl) is rapidly varying across the well. Expressed in terms of the nonlinear conductivity

310

LOCAL FIELDS IN LINEAR AND NONLINEAR OPTICS OF MESOSCOPIC SYSTEMS

[\i 9 4

response tensor E(z,z',z";2q'll) and the external field iext(z; 4'11) acting on the quantum well at the fundamental frequency, J(z;2q'll) is

j(z;241) = X(z;24'11) : P

( z ;4'11)Pf(z;4'11)

where the special symbol 8 is meant to indicate that to obtain the tensor element (28TT),jkone must perform the double summation Lm rrnkro ). In the second-harmonic case, the local-field information appears in the structure-rich third-rank tensor

(xi,,

J"

p2(z;2ill) = ( 2 i ~ w ) - '

~ ( z , z ' ;2;ll).

zZzz. &z';

(4.25)

24'11) dz'.

It often happens that the quantum well is placed on the top of a substrate. In such cases, the loop equation for the local field takes the form

Z(z; 4'11) = ZB(z;i l l ) - iku

I

E(z,z"; ~~~)i?(zf',z~;~~~) . i ( z ' ;4'11) d ~ "d~'

(4.26) in the linear regime. The external field, here named the background (B) field, @(z;q11), now consists of the sum of the incident field and the field reflected from the substrate, and the related Green's function u

G(z,z';4'11)

tt

=

Go(z - 2'; 4'11) + 'i(z+ z';4'11) c)

(4.27) tt

is composed of a free-space part, Go, and an indirect part, I , accounting for field propagation including reflection from the vacuumhubstrate interface. A particular example showing recent experimental data for the p-polarized reflectivity ( R p ) of aluminum quantum wells of various thicknesses deposited on glass substrates is shown in fig. 4. 4.2. LOCAL-FIELD RESONANCES AND EIGENMODES

In 9 2.3, I discussed the resonance condition for the local field in the framework of many-body electrodynamics and I emphasized that in the absence of local-field effects the resonance condition is reduced to the Bohr condition for electronic transition between the possible many-body energy eigenstates

v, I 41

311

LOCAL FIELD ELECTRODYNAMICS IN QUANTUM WELLS AND THIN FILMS

1.0 I

I

I

I

I

I

I I

I

I

I

I

I I

0.8

I

I

_I _ _ _

I I

1

cr" I

.k -.-

I

0.0 0

Y

j'

I

I

I

I

I

I

I

I

I

I

1

1

I

1

20

40

60

80

100

120

d [A1 Fig. 4. Experimental results (dots) showing the p-polarized linear reflectivity (R,,) of an Al quantum well deposited on a glass substrate as a function of the well thickness ( d ) . The incident electromagnetic field has a wavelength 4 = 9.2pm and the angle of incidence is 0 = 7'. The lines connecting the various dots are drawn just to guide the eye, and the experimental uncertainty is less than the dot size in each of the measurements. For small thicknesses (d 5 50A) the flat Rpresponse is dominated by diamagnetic effects. The pronounced thickness variations in Rp appearing in the region d N 90 A is the fingerprint of electronic quantum confinement effects. A minimum in the reflectivity is observed in the range -7&90A, and for larger thicknesses a gradual increase of Rp towards its bulk value(with superimposed confinement determined fluctuations) is found. (After Tamez and Keller [ 19971.)

of the (mesoscopic) medium, provided irreversible damping mechanisms are neglected. Because of the inherent simplicity of the integral equation for the local field in quantum-well systems it is of interest to discuss the local-field resonance condition in some detail. This discussion also allows us to establish a bridge between the local-field resonance concept and the collective eigenmodes (polaritons, plasmons, radiative and non-radiative wave guide modes, etc.) familiar to many researchers within the field of physical optics of condensed matter systems. For the afore-mentioned purpose it is sufficient to consider the specific case where a single quantum well exhibiting confined free-electron dynamics is deposited on top of a semi-infinitely extended dielectric substrate. Starting thus

312

LOCAL FIELDS IN LINEAR AND NONLINEAR OPTICS OF MESOSCOPIC SYSTEMS

[y 5 4

from eq. (4.26), it appears that in resonance the local field, iRES(z;ill),must satisfy the homogeneous integral equation

(4.28) If the substrate is optically isotropic, or anisotropic but with one of the principal axes of the dielectric tensor oriented perpendicular to the scattering plane, the local-field resonance condition separates into two: one for s- and one for p-polarized fields. For the p-polarized case the resonance condition in eq. (4.28) leads to a set of coupled homogeneous integral equations among the x- and z-components of the resonant field, assuming the scattering plane to coincide with the xz-plane. The coupling originates in (i) the off-diagonal elements of the conductivity tensor and (ii) the inevitably present cross coupling in the electromagnetic propagator. For metallic and semiconducting quantum wells the xz cross coupling in the conductivity is usually a weak effect in the optical regime where the magnitude of the electromagnetic wave vector along the plane of the quantum well is much less than the Fermi wave number ( k ~of) the electrons, i.e., 411 << kF. It may happen that the radiation from the z-component of the induced current density does not significantly affect the electron dynamics along the well. In such cases the cross-coupling term involving G,(z,z”; 4‘11) in eq. (4.28) may be neglected. A neglect of the x-component of the local field generated by the current-density oscillations in the z-direction leads to a resonance condition involving only the electron dynamics in the plane of the well. Since the dynamics in the z-direction in the present situation is enslaved by the dynamics in the xdirection, one can obtain a separate resonance condition for the z-component of the field. This is done by considering the local field generated (via the radiative Gzxcoupling) in the z-direction by the induced current density flowing along the well as a part of the prescribed external field. A separate resonance condition for the z-dynamics identical to the one mentioned above could, of course, also have been obtained starting from the assumption that the current density along the well does not affect the dynamics in the direction perpendicular to the well plane. Altogether, it appears that the resonance condition for the p-polarized part of the local field is split into two: one for the electron dynamics in the plane of the well and one for the out-of-plane dynamics, provided that electronic (axz,azx) and electromagnetic (G,,, Gzx)cross-coupling effects are of no importance. If the optical diamagnetic response is the dominating one, as will often be the case for metallic and semiconducting quantum wells subjected to radiation

v, ii

41

LOCAL FIELD ELECTRODYNAMICS IN QUANTUM WELLS AND THIN FILMS

313

having frequencies in the mid- and far-infrared parts of the electromagnetic spectrum, the resonance condition for the x-component of a p-polarized field is given by (Keller and Liu [1994]) 401El + Kl. = Nxx(w)q;KI,

(4.29)

where (4.30) with a = b e 2 w z / [ m * ( i+ wz)]. In eq. (4.29), q; = [(w/co)' - qi]'l2 and KI = [ ( o / c ~ )-~qi]'/2 ~ l denote the wave vector components perpendicular to the well plane in the vacuum and substrate, respectively, the quantity EL being the relevant dielectric constant of the substrate. In eq. (4.30), d is the thickness of the quantum well, and N , is the concentration of ions in the well. A possible local and isotropic high-frequency contribution to the dielectric function of the equantum well, originating in ionic excitations, vertical interband transitions, etc., is contained in E,(o). The intraband relaxation time and effective mass of the electrons are denoted by z and m*, respectively. In the limit d 4 0, eq. (4.29) is reduced to the form + K L = 0, or equivalently, 411 =

;(*)

I /2

(4.3 1)

The relation in eq. (4.31) may be recognized as the dispersion relation for electromagnetic surface waves on the bare vacuumhubstrate interface (Boardman [1982], Agranovich and Mills [1982]). In the presence of a quantum well on this interface, the dispersion relation is modified to the form given in eq. (4.29). The resonance condition for the z-component of the local field takes the form (Keller and Chen [1995])

401 El. + KI

= NZZ(W)Y;I,

(4.32)

where (4.33) The integral in eq. (4.33) extends over the quantum well, and it appears that the integrand for small electron scattering rates exhibits a resonance in the vicinity of

314

LOCAL FIELDS IN LINEAR AND NONLINEAR OPTICS OF MESOSCOPIC SYSTEMS

[V,

54

the screened local plasma frequency GP(z) = [n(z)e2/(rn*EOE,)]"~; cf. Forstmann and Gerhardts [ 19861. The local-field effects accompanying the induced density variations of the electron perpendicular to the well plane leads to a resonance in the dispersion relation above the bulk plasma frequency. In semiconducting quantum wells of GaAs the resonance is located in the infrared and it moves upwards in frequency with increasing well thickness (Keller and Chen [ 19951). The resonance condition for the local field given in eq. (4.28) also contains the eigenmode condition (dispersion relation) for radiative and non-radiative modes in symmetric and asymmetric slab waveguides in the macroscopic limit. For a homogeneous and isotropic (thin) film, one inserts in eq. (4.28) the following local expression for the conductivity tensor: *--t

a(z, z'; 4'1,,W) = 8a(w)S(z - z').

(4.34)

The resonant field iRES(z;4'11, W) which solves the homogeneous integral equation for such a model conductivity will be denoted by gM(z;4'11, w), where the superscript M reminds us of the fact that ZM is what is called the macroscopic (M) field. It is certainly not an easy task to obtain a rigorous framework for eq. (4.34) starting from the microscopic local-field formalism. Before proceeding let me emphasize that the macroscopic approach is not necessarily based on the assumption of free-electron-like dynamics parallel to the slab as one might be tempted to believe since eq. (4.18) [and therefore eq. (4.28)] emerged from the 2D Bloch formalism in the limit where the electron dynamics became free-electron-like. In the general case, the conductivity tensor a ( r , r ; w ) is replaced by 8 .'- 7 ' ; ~ )if the medium is assumed to be homogeneous. In the spatial Fourier domain, the conductivity response function of the homogeneous medium is z(q,w ) (cf. Agranovich and Ginzburg [1984], Keldysh, Kirzhnitz and Maradudin [ 1989]), and in the local (4' -+ 0) limit this becomes Z(q + 0 , ~ = ) z ( u ) , or, if isotropy is also assumed, a(w)U. In the mixed space-frequency domain, the last expression reads a(w)"u(? - .'I), and from this eq. (4.34) appears. In dielectric or weakly (semi)conducting media where the point-particle model can be used, so as to obtain an algebraic set of equations for the background field at the various molecular (ion) positions, a rigorous framework for a macroscopic theory can be established if the medium under consideration contains a huge number of molecules in a volume of linear extension comparable to the length scale over which the retarded part of the transverse vacuum propagator varies. In such cases, the background field in eq. (3.25) ends up playing the role of the macroscopic field. The relevant t+T electromagnetic propagator is now Do( J - J ' ) , the self-field parts being included H

4

4'

H

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LOCAL FIELD ELECTRODYNAMICS IN QUANTUM WELLS AND THIN FILMS

315

in the dressed polarizability; cf. eqs. (3.27) and (3.28). In the metallic case, the conduction electron density, n(z), is given by (see, e.g., Feibelman [1982]) (4.35) in the low-temperature ( T + 0 K) approximation. Above, I/&(z) and E, denote the wave function and energy of the nth stationary state, and EF is the Femi energy. When the width of the quantum well becomes larger, the number of bound states also increases, and if the wave functions of the conduction electrons are assumed to be free-electron-like, the conduction electron density tends to be independent of z except near the potential barriers. For macroscopic film thicknesses the optical diamagnetic response hence appears as the response of a homogeneous (and isotropic) medium. If the expression in eq. (4.34) is inserted into eq. (4.28), the resonance condition for the macroscopic field becomes (4.36) To illustrate that the field resonance condition in eq. (4.36) is linked to the eigenmode condition for radiative and non-radiative modes in a slab waveguide, let us consider for heuristic purposes the case where there is vacuum on both sides of the film. In this case, the indirect part, ‘i(z + z’; all), of the propagator in eq. (4.27) is zero. The remaining free-space part is given by

where sgn(z’ - z) = f l for z’ - z P 0. The last part of eq. (4.37) is the nonretarded self-field part. For s-polarized light the resonance condition hence takes the form

316

[v 3 4

LOCAL FIELDS IN LINEAR AND NONLINEAR OPTICS OF MESOSCOPIC SYSTEMS

To solve the integral equation in (4.38) we use the ansatz EF(z; 4'11)

+ Bye-iQL".

= AyeiQLr

(4.39)

By inserting eq. (4.39) into eq. (4.38), the left hand side of this equation becomes a linear combination of four terms with the following z-dependences: eiQ12, e-iQLz, eiqyz, and e&'. Since this combination must vanish for all values of z, the coefficients to the four terms must all be zero. This gives for both the eiQlz and e-iQlz coefficients (Ay and By different from zero) the condition Q i = (qy)2+ ibuooa(w), or equivalently, (4.40) ) relative dielectric constant of the film. The where E ( W ) = 1 + i a ( w ) / ( ~ ~isw the relation in eq. (4.40) may be recognized as the squared dispersion relation for electromagnetic bulk waves in the film. Setting the two remaining coefficients to zero, one obtains the following set of equations:

The condition for having a nonvanishing solution for (Ay, By) is therefore (4.42) The result in eq. (4.42) gives a relation between 411 and o which in implicit form is precisely the dispersion relation, 411 = q11(o),for guided eigenmodes (radiative and non-radiative) in a symmetric slab structure; see, e.g., Yariv and Yeh [1984] and Wallis and Stegeman [1986]. The plus and minus sign represent the antisymmetric and symmetric modes, respectively. If the medium surrounding the film is not vacuum but is an optically homogeneous and isotropic medium with a local response, one needs only to replace co by the actual phase velocity of light in this medium, and divide E ( W ) by the dielectric constant of the surrounding medium. For p-polarized fields a calculation similar to the one outlined above can be carried out, starting with the ansatz

E y ( z ;4'11)= AieiQL'+ Bie-'QLz, i = x,z,

(4.43)

and using the fact that . ZM(x,z) = 0 implies that A,/A, = -q11/Ql and BJB, = qll/Ql. As the outcome, one obtains (i) again the bulk wave dispersion

Y 9 41

LOCAL FlELD ELECTRODYNAMICS IN QUANTUM WELLS AND T N l N FILMS

317

relation in eq. (4.40), and (ii) from the condition that the appropriate determinant must be zero, the following eigenmode condition for the radiative and nonradiative guided waves in a symmetric structure (Yariv and Yeh [1984], Wallis and Stegeman [ 19861) (4.44) If the indirect term of the Green’s function in eq. (4.27) is kept, a straightforward but somewhat tedious calculation shows that the resonance condition for the macroscopic field in eq. (4.36) leads to the dispersion relation for guided waves in an asymmetric slab structure. 4.3. NON-RETARDED DYNAMICS: SELF-FIELD AND SCALAR THEORIES

In the mesoscopic region where the film (quantum well) thickness is considerably smaller than the wavelength(s) of the background field, it appears natural to neglect electromagnetic retardation effects in order to simplify the solution of the fundamental integral equation for the local field (eq. 4.26). For s-polarized dynamics, such a step cannot be taken because this would imply that all local-field effects disappear. That this is so follows readily from the facts that (i) the s-polarized amplitude reflection coefficient for the vacuum-substrate interface, r, = (4; - K l ) / ( q ? + KL), tends to zero for co + 00 so that the indirect part of the propagator Iyy(z + z’, gll) + 0, and (ii) -i~ccooG~~y(z - z’; 4‘11) -+ 0. Together, (i) and (ii) imply -ihwC,,(z,z’; 4‘11) + 0, and therefore Ey(z; 4‘11)+ EF(z;4‘11) in eq. (4.26). This may not be as bad as it seems because local-field corrections are often small (negligible) for s-polarized fields. For p-polarized fields, the adoption of a non-retarded approach would be particularly problematic close to localfield resonances in systems with small electronic dampings. In order to discuss the quantum-well dynamics in the non-retarded limit, it is adequate to slightly convert the formulation of the integral equation problem. Thus, if the formal solution of eq. (4.26) is written in the form (4.45) it is realized that the field-field response tensor T(z, z’; 4‘11) the tensorial integral equation t*

T(z, 2 ’ )

=

“u(z - z’) +

s

z ( z , z”) . F(z”, z’) dz”,

= F(z, z’) must fulfil (4.46)

318

LOCAL FIELDS IN LINEAR AND NONLINEAR OPTICS OF MESOSCOPIC SYSTEMS

with a kernel E(z, z';

a,,)

[y 5

4

t--1

E

K(z, z') given by

E(z, z") . z(z", z') dz"

(4.47)

in a notation where the reference to 4'11 has been left out from the arguments of ?? and 3, also. At the core of the problem now lies a study of the solution of eq. (4.46). In order to investigate the s- and p-polarized linear reflectivities from free metallic (or semiconducting) surfaces with a smooth electron density profile, Feibelman established in the 1970s and 1980s an important nonretarded approach which allows one to express the corrections to the sharpsurface s-polarized (11) and p-polarized (I)reflection coefficients through two complex parameters dll and d l , now usually referred to as the Feibelman d parameters (see, e.g., Feibelman [ 19821, Forstmann and Gerhardts [19861). The Feibelman approach is based essentially on three assumptions: (i) a neglect of off-diagonal elements in the microscopic conductivity tensor, (ii) a self-field approach for the electromagnetic Green's function, and (iii) a neglect of the spatial variation of the background field across the profile region. Although an electromagnetic propagator formalism was not used in the original works by Feibelman, assumption (ii) is hidden implicitly in his assumption that the local field along the surface is identical to the parallel component of the background field in the profile region. Since 411 << k ~ k~ , being the Fermi wave number of the electrons, it is often safe to neglect the off-diagonal elements (axz,an) of ++ u . Doing this, and using the self-field approximation for the propagator, i.e., u

G(z,z')

N

Z(Z -z')

=

(g

S(Z - Z')Z;e',,

(4.48)

eqs. (4.45)44.47) give Ei(z) = EF(z), E,(z)

=

i = x,y,

] Tzz(z,z')E:(z')

dz'.

(4.49) (4.50)

Upon introduction of the zz-component &(z, z') of the relative dielectric tensor E (z, z'), t*

&(Z, z')

= 6(z - z')

i

+ -&(Z, z'), EO 0

(4.5 1)

v, ii

41

LOCAL FIELD ELECTRODYNAMICS IN QUANTUM WELLS AND THIN FILMS

319

it follows that the field-field response function T,(z, z‘) must be determined from the integral equation

I

E,,(z, z”)rzz(z”,z’) dz” = 6(z - z’).

(4.52)

By combining eqs. (4.50) and (4.52), it appears that the inverse relation of eq. (4.50) is

E,B(z) =

J E ( Z , Z’)E,(Z’) dz’.

(4.53)

The integral equation in eq. (4.53) is the essential one in the Feibelman theory. In contrast to what is written in many textbooks, it is Tzz(z,z’)= E;’(Z,Z’) which plays the role of a causal response function, and not E,,(z,z’). Although the Feibelman theory was developed to deal with surface corrections to the Fresnel reflectivities, over the years it has been used to study the reflection from thin film (quantum-well) overlayers on various kinds of substrates. The Feibelman theory is certainly a non-retarded theory, but not a complete -T one since the remaining part of Go, namely Do = EO- g, and the indirect propagator ‘ialso contribute to the non-retarded (NR) dynamics. In a complete non-retarded formalism, one thus must use a propagator t*

with

x [exex- ZzZz + i sgn(z - zt)(ZxZz+ 2zZx)],

(4.55)

and

in dyadic notation, and with unit vectors along the Cartesian axes denoted by Zi (i = x , y , z ) . In the non-retarded limit, the p-polarized vacuumhbstrate l 1). amplitude reflection coefficient is given by r:” = ( E L - l ) / ( ~ + In the afore-mentioned description of the non-retarded electrodynamics, the tensorial Green’s function of the full set of microscopic Maxwell-Lorentz equations is obtained and afterwards the non-retarded kernel of the integral equation

320

LOCAL FIELDS IN LINEAR AND NONLINEAR OPTICS OF MESOSCOPIC SYSTEMS

[y 9: 4

for the field-field response tensor is calculated, letting co + 00 in the propagator. In the literature, an alternative route is usually followed in studies of the nonretarded electrodynamics. In the alternative approach one thus starts by taking the non-retarded (CO + 00) limit in the microscopic Maxwell-Lorentz equations, and thereafter one determines the associated propagator. The starting point for this, maybe more familiar, approach is the quasi-static set of microscopic Maxwell. &<w) = p(< W ) / E O and 9 x E(< w ) = 6. The Lorentz equations, i.e., last equation shows that the quasi-static (longitudinal) field may be obtained w). The scalar from a scalar potential q(< o)via i?(< w ) (= &(< o))= potential in turn satisfies the Poisson equation V2#(7;w ) = -p(< w ) / ~In. the mixed Fourier domain the Poisson equation becomes

v

-a#(<

(4.57)

where N(z; 4‘11) is the corresponding induced (many-body) electron density. With the boundary conditions q(-00;411) = q(oo;?11) = 0, the solution of eq. (4.57) can be written in the compact form (4.58)

To determine the associated non-retarded electromagnetic propagator, one must introduce E(z; 4‘11) and j(z’; 4‘11) in eq. (4.58) instead of q(z; 4‘11) and N(z’; 4‘11). By inserting eq. (4.58) into the mixed Fourier-domain relation J%;

d

4‘11 1 = - ( q + e;z)$(z;

(4.59)

S‘ll),

one obtains the temporary result ~ ( z4‘1;1) =

2EO

/

03

[iz;,,

+ zzsgn(z/- z>le-qll~z--~’ IN(^'; 4‘11) dz’,

(4.60)

-03

where ZiIl = i11/q11. Using the equation of continuity in the mixed Fourier domain, i.e.,

(iq‘ll +

d z2-1 dz

.j ( z ; 4‘11)

= -ieoN(z;

4‘111,

(4.6 1)

to eliminate the electron density from eq. (4.60), and afterwards the relations

(4.62)

v, ii

41

LOCAL FIELD ELECTRODYNAMICS IN QUANTUM WELLS AND THIN FILMS

32 1

and

= 411

/

oc)

eC11 ~z~z’~.Zz . j(z’; 4‘11) dz’ - 2.Z; . j(z; ?11),

(4.63)

--w

one obtains finally:

(4.64) By writing this equation in the standard form Z(z; 4‘11)

= -ibw

/

00

--w

[ B r ( z - z’; 4‘11) + T(z - z’)] . j(z’; 4‘11) dz’,

(4.65)

it appears that the propagator emerging starting from the Poisson equation is g R ( Z- z’; 4‘11)

+ E(z - z’)

+6(z

-

z’)e;z;

1

(4.66)

.

If 4‘11 = qll.Zx,so that Zqll =ex, the propagator in eq. (4.66) coincides with the one in eq. (4.54) neglecting the indirect part. Starting from the Poisson equation one thus ends up with the same propagator as if one has begun with the complete Maxwell-Lorentz equations in the non-retarded limit. The interconnections between the various local-field theories are shown in diagrammatic form in fig. 5. If one transforms eq. (4.65) (with eq. 4.66) back to the space domain, it takes a form identical to that of eq. (2.17):

EL(F)= - i b w

s

EL(?- 7’) . j ( 7 ’ ) d3y‘.

(4.67)

The longitudinal propagator (4.68) ++a

deviates from the longitudinal delta-function dyade dL(7 - 7’) by merely a constant, ( c ~ / w ) The ~ . longitudinal propagator and the longitudinal delta

322

LOCAL FIELDS IN LINEAR AND NONLINEAR OPTICS OF MESOSCOPIC SYSTEMS

MaxwellLorentz Eqs.

”

Curmnt-current

mrdation fund.

Co-m

,*

Scalar theoty

ii-6“

[y 5 4

Non-retarded vectortheory

.

I4,,+6

Retarded vectortheory

&-6

-

Theory of spatial confinement

Self-fiild theory

Fig. 5 . Diagrammatic survey showing the connection between the retarded vector theory, the scalar theory and the self-field theory. The equivalence of the non-retarded vector theory and the scalar theory is indicated by the bar connection between the respective boxes.

function have been marked by a dot ( 0 ) here to emphasize that in the present context these are given in so-called disk contraction form:

For quantum-well systems excited by a single plane-wave field component, all relevant p-polarized vector fields, ?(x, z) = (V,, 0, Vz),have the generic form ?(x,z) = 411) exp(iql1x). The longitudinal, ~ L ( x , z=) ~ L ( Z 4; 11) exp(iqllx), and + PT)can be obtained transverse, ~ T ( xZ), = ~ T ( z4;11) exp(iqllx), parts of formally via

a(,;

v(= v~

(4.70) and (4.7 1) where ‘-8,(z - z’; 411) and

++

~ T ( z- z‘; 411)

may be considered as the relevant longi-

YP

41

LOCAL FIELD ELECTRODYNAMICS IN QUANTUM WELLS AND THIN FILMS

323

tudinal and transverse delta-function dyades in the mixed Fourier representation. It appears from eqs. (4.55) and (4.65H4.68) that u

6 L(Z - z’; 411) = e’Z’b(z - z’)

’ -tI

‘I1 +-e 2

I [ZxZx- ZzZz + isgn(z - z’)(ZxZ’ + Z2Zx)],(4.72)

and therefore, tt

6T(Z-Z’;qll)

=

Z;Z;b(Z-z’) -3e-411 2

P I [zxzx- Z ~ +Z isgn(z ~ - Z‘)(Z~Z’ + Z’Z~)],(4.73)

utilizing ZT(Z- z’;q11) + ~ L ( -Zz’;qll) = (ex& + ZzZz)b(z - z’) in the 2D subspace. With eqs. (4.72) and (4.73) inserted into eqs. (4.70) and (4.71), respectively, an explicit calculation readily shows that the L and T parts of the vector field V(x,z) satisfy the conditions d (iqllez- ex-) . C(z; 411) = 0,

(4.74)

d +eZz). VT(z;qll)=o,

(4.75)

dz

(iqllex

for being rotational- (9 x VL(X,z) = 6) and divergence-free (9. ~ T ( xz), = 0). is found self-consistently as the linear The density perturbation, N(z; response to an appropriate single-particle potential @ = vex‘+ qH+ qXc, i.e.,

all),

N(z;a11) = S ~ K S ( Z , Z ’ : h l ) [ ( p e x t ( Z ’ ; a I / ) +

qH(z’;‘?ll)+ %c(z’;q11)1h‘,

(4.76) where xKs(z,z’; 4‘11) is the so-called single-particle Kohn-Sham (KS) densitydensity response function (Kohn and Sham [ 19651) in the mixed representation. The quantity q H ,which is named the Hartree (H) potential (Hartree [1928]), is the one given in eq. (4.58), and vxc(z; 4‘11)

=

1

fXC(ZY z’;

a11”’;

4‘11) h’

(4.77)

is the exchange-correlation (xc) potential, which as indicated is taken to be linear (but nonlocal) in the density. By combining eqs. (4.58) [(p = qH],(4.76),

324

LOCAL FIELDS IN LINEAR AND NONLINEAR OPTICS O F MESOSCOPIC SYSTEMS

[y 9 5

and (4.77) one obtains the following inhomogeneous integral equation for the density: J

(4.78) with a scalar kernel

In many treatments the exchange-correlation term is omitted. If a substrate is present, the external potential must be taken as the sum of the potential of the incident and reflected fields. In a consistent non-retarded description the plane-wave background (external) field, for simplicity here taken identical to the incident field , o

Zext(z;q11ZX)= EO[Z;

-

(q~/ql~)~~]e’q~’,

(4.80)

which is transverse, assuming the sources to be located outside the quantum well, should be approximated by its non-retarded form j j eNR( xt

z;qllZx) = EO(& - iZx)e-qllz,

(4.81)

which is both divergence-free and rotational-free. It follows from eq. (4.59) that the associated external potential may be chosen as (4.82) For quantum wells and thin films (q11d << l), the external potential acting on the electrons of the well (film) is vex‘‘v (Eo/q11) - Eoz, or just -EOZ since the (unphysical) constant does not give rise to any induced electron density. A numerical comparison of the frequency dependence of the p-polarized linear reflectivities (Rp) of a two-level GaAs quantum-well system obtained using the self-field theory, the non-retarded scalar theory with and without the self-field correction, and the retarded vector theory is presented in fig. 6 .

0 5.

2D Spatial Confinement of Light by Optical Phase Conjugation

5.1. SOURCE FIELD OF A MESOSCOPIC PARTICLE: ATTACHED AND DE-ATTACHED

PARTS

When the many-body system of a mesoscopic particle (or molecule or atom) undergoes a radiative transition from a higher to a lower lying energy eigenstate

v, P

51

2D SPATIAL CONFINEMENT OF LIGHT BY OPTICAL PHASE CONIIJGATION

do9

0.10

0.11

0.12

0.13

0.12

0 13

325

?iw [ev] 0.15

I

0.09

0.10

0 11

fiw [ ev]

Fig. 6 . Linearp-polarized reflectivity ( R p ) of a two-level GaAs quantum well calculated as a function of the photon energy (Am) for two different angles of incidence: 8 = 70' (top) and 8 = 85" (bottom). The solid curves represent the results obtained on the basis of the retarded vector theory. Using a scalar theory without inclusion of self-field effects one obtains the dotted peaks to the left. In such an approach the peak position is located at the electronic resonance energy, &2 - = 100meV. If self-fields are included in the scalar theory, the resonance peak is blue-shifted, and the reflectivities are given by the dotted curves to the right. In this particular example the self-field theory and the full scalar theory give essentially the same result for the frequency dependence of Rp. Although the thickness of the quantum well (d = 130A) is three orders of magnitude less than the optical wavelength, it appears that electromagnetic retardation gives rise to an appreciable reduction of the reflectivity near the local-field resonance, in particular for the largest angles of incidence. The blueshift of the resonance is essentially due to the self-field electrodynamics. The present calculations were carried out using infinite- barrier wave functions; the distance of the lower (occupied) level from the Fermi energy was given by EF - &I = 50meV, a conduction electron density per unit area (surface density) of N, = 1 . 3 9 10" ~ c N 2 was taken, and a relaxation energy h/t = 3 meV was assumed.

an electromagnetic field is emitted. We say that the field is emitted from the particle, and usually we do not worry about what in the geographicaI sense more precisely is meant by the word from. In most cases we are satisfied with a description in which only two important dynamical aspects are followed, namely,

326

LOCAL FIELDS IN LINEAR AND NONLINEAR OPTICS OF MESOSCOPIC SYSTEMS

[y 5

5

(i) the time development of the quantum state of the relevant (many-body) wavefunction, and (ii) the time and space pattern of the far-field radiation from the particle. In 9 2.4 it was realized (cf., e.g., eq. 2.89) that the retarded field from a given source point in the time domain consists of a far-field part and a nearfield part. In the standard scalar diffraction theory only the far-field part of the field is present, and from this theory one is led via the Fraunhofer approach to the Rayleigh criterion for the spatial resolution. Not only in Fraunhofer diffraction but also in Fresnel diffraction, one starts from the far-field approximation. This means that in so-called classical optics, statements about the spatial resolution (and the field confinement) certainly break down when R / & 5 (2n)-',i.e., when the distances ( R ) involved are one to ten times smaller than the vacuum wavelength (&) of the field. It would not be correct to claim that the importance of middle- and near-field effects was not already appreciated already long time ago. Thus, Hertz noticed that the field of a radiating dipole decays with R-3 rather than R-' near the dipole, and Sommerfeld studied the influence of near-field effects on the radiation properties of a dipole antenna near ground (Sommerfeld [1909]). In the context of scattering from small particles (van de Hulst [1957], Kerker [ 19691, Bohren and Huffman [ 1983]), a subject initiated by the works of Lorenz, Rayleigh and Mie, it is also necessary to go beyond the far-field approximation (Rayleigh [ 1871, 18991, Lorenz [ 18801, Mie [ 19081). Of direct importance in the present context, however, is the work of Synge who in 1928 suggested that the use of narrow apertures in optical microscopy would allow one to overcome the classical diffraction limit (Synge [1928]). Also, to understand the transmission of narrow apertures, Bethe invoked nearfield considerations (Bethe [ 19441). By means of microwaves, Ash and Nichols achieved a spatial resolution of -ho/60 (Ash and Nichols [1972]). The works of Synge, Bethe, Ash and Nicols and others may be considered as forerunners of near-field optics, a discipline in which the spatial confinement of light plays a decisive role. Experimental studies in near-field optics have demonstrated in recent years that it is indeed possible to overcome the Rayleigh criterion (see, e.g., Pohl and Courjon [1993], Nieto-Vesperinas and Garcia [1996]). In most of the theoretical investigations in near-field optics, among others those cited in the introduction, the point-dipole model was used together with the textbook c*T propagator, D,(R; o)+ gL(g; 0). One of the main goals ahead of us in nearfield optics seems to be an investigation of the possibilities for reaching a spatial resolution on the atomic length scale. It appears to me that in order to obtain a rigorous answer to the above-mentioned issue it is necessary to address the field-confinement problem itself from the point of view of microscopic

-

v, P

51

2D SPATIAL CONFINEMENT OF LIGHT BY OPTICAL PHASE CONJUGATION

321

(quantum) electrodynamics. Although the point-dipole model is an important one, it is obvious that it cannot provide us with any “deep” answer to the problem because of its inherent (and unphysical) delta-function singularity. The analyses in 9 2 and 9 3 have demonstrated that the self-field electrodynamics plays an indispensable role for the confinement problem, and in fact sets the limit for the degree of spatial confinement one can hope to achieve. The dynamics originating in the self-fields does not involve any retardation effects and is closely related to the fundamental nonlocal nature of quantum mechanics. Therefore, one may argue that the confinement problem - and thus the spatial localization of photons in a coupled field-matter system - is linked intimately to the nonlocality of quantum physics. In passing, I would like to mention that the concept of a localized photon in free space has been of interest for many years, and in the wake of the works of Newton and Wigner [ 19491 and Wightman [ 19621, which indicate the nonexistence of a local wavefunction, Jauch and Piron [ 19671 and Amrein [ 19691 took up the localization problem and introduced concepts more general than that of a wavefunction. Mandel [ 19661, Tatarski [ 19721, Cook [ 1982a, 1982b], and Pike and Sarkar [I9861 have also carried out theoretical studies of the localization problem of few photons. Until recently, the standard scheme for seeking extreme photon localization has been based on multiple scattering of electromagnetic fields from impurities or surface roughness, and is to some extent patterned from the Anderson localization theory for electrons, as mentioned in the introduction of this article. In 1982, Aganval published an important paper in which he studied the optical response of an atom placed in front of a phase-conjugating mirror (Aganval [ 19821). Following Aganval’s suggestion, Hendriks and Nienhuis [1989], Milonni, Bochove and Cook [1989], Arnoldus and George [1990] and the present author (Keller [ 19921) made further analyses. The work of Agarwal inspired me in 1990 to suggest an alternative route to spatial field confinement. Thus, instead of using a simple atom as the radiation source, which would be very ambitious from an experimental point of view, the tip of an optical near-field microscope might be used. The tip should be placed so near the phase-conjugating mirror that a broad part of the angular spectrum of the tip can penetrate the phase conjugator. The phase-conjugated field forms a narrow focus on the position of the tip, and if the memory time of the phase conjugator is sufficiently long it is possible to remove the tip of the microscope and thus to observe a small light spot above the surface for some time. The afore-mentioned idea for obtaining subwavelength 2D-confinement of light was mentioned briefly by Keller [1992], and has been discussed in some detail

328

LOCAL FIELDS IN LINEAR AND NONLINEAR OPTICS OF MESOSCOPIC SYSTEMS

[v, 0 5

recently (Keller [ 1996c,d, 19971). The first experiment demonstrating phase conjugation of the outgoing field from the fiber tip of a near-field microscope was performed by Bozhevolnyi, Keller and Smolyaninov [ 19941. The first experiment already led to the production of light foci substantially smaller in extension than allowed by classical diffraction theory. Further experiments have subsequently been performed by Bozhevolnyi, Keller and Smolyaninov [ 19951, and Bozhevolnyi and Smolyaninov [1995], and a macroscopic theory for the phenomenon has been put forward recently (S.I. Bozhevolnyi, E.A. Bozhevolnyi and Berntsen [1995]). In order to form a focus of subwavelength extension it is necessary to phase-conjugate not only the so-called propagating part of the angular spectrum but also at least a part of the evanescent spectrum. A simple theoretical framework for phase conjugation of evanescent waves has been established recently by Agarwal and Gupta [I9951 and by Keller [1996d]. Let us now consider a mesoscopic particle located in front of a phaseconjugating mirror, and let us put in a Cartesian coordinate system in such a manner that the center of mass of the mesoscopic object is located at (O,O,-d), with d > 0, and so that the surface of the mirror lies in the z = 0 plane. When the mesoscopic particle is excited by an external field, it starts to radiate, and a part of the angular spectrum emitted hits the phase conjugator and becomes available for phase conjugation. Before considering the phase conjugation process itself, it is necessary to analyze the angular spectra of the fields attached to and radiated from the mesoscopic particle. The attached field, or as we have named it previously the induced self-field, is given by

EsF(<W) = -&ow

s

[ffT(7-7’; W)+ ffL(7-7’; w)] .J(7’;W)d3r’,

which in the mixed Fourier domain can be written in the form

(5.1)

Y I 51

329

2D SPATIAL CONFINEMENT OF LIGHT BY OPTICAL PHASE CONJUGATION

*

t)

If the Weyl expansions of aT(z- z’; 4‘11) and SL(Z- z’; 4‘11) given in eq. (5.3) are inserted into eq. (5.2), the angular spectrum of the attached field is seen to be iSF(z;4‘11, w )

+% 3

e-411 +’I

[ziII

-

.J(z’;4‘11, U )dz’

)I

z2z2+ isgn(z - z’)(zill e2+ z ~ z ~ ~ ~

1

(5.4)

To determine the angular spectrum of the retarded (radiated) source field, it is adequate first to consider the Weyl expansion of the transverse propagator *T Do(?- 7’; w). Thus, by means of the plane-wave expansion

it readily appears that the transverse propagator may be represented in the integral form

sy$H

Do(z H T - 2 ’ ; 4‘11, w ) = (2n)-’

4 4 ei q l ( z - z ’ ) d 4 1 ,

(5.6)

in the mixed Fourier domain. In the complex ql-plane, the integrand in eq. (5.6) has first-order poles at 4 1 = fq: = ic(qi - q i ) ” 2 . For 411 < 40, these are located on the real axis, and for 411 > 4 0 on the imaginary axis. Contour integration t)T

allows one to obtain an explicit expression for Do(z - z’; 4‘11, w ) :

(g) e; 2

x

[(q~)24$ll + q p z e ;-

zZ

x ZfIl

z;,,x

22

11.

- q ; q sgn(z - z ’ > ( ~ q ~+~zz~ill

(5.7) The retarded propagator given in eq. (5.5) with eq. (5.6) and (5.7) inserted may be divided conveniently into two parts, i.e.,

Bi(7- 7’; 0)= (27c-2

s

t*T, prop(Z - z’;

411 540

Do HT ev

D,‘ ’40

(z -2’;

qll,o)eiill

)

d2411

qI1,~ ) e i ~ l l ~ ( Fd2qll. l l ~ ~ ~ (5.8) l)

330

LOCAL FIELDS IN LINEAR AND NONLINEAR OPTICS OF MESOSCOPIC SYSTEMS

[y 9

5

The first part consists of the spatial spectrum of so-called propagating (prop) waves, i.e waves for which qy = [ ( U / C O ) ~ - qi]1’2 is real. The related 4‘11integration is performed inside the circle of radius 411 = d c o . The associated prop

-T,

mixed Fourier amplitude Do (z - z’; 4‘11, w ) is given by eq. (5.7), remembering that q; is real and positive. The second part is constituted by the spectrum of so-called evanescent (ev) waves for which q y is purely imaginary. The corresponding integration in eq. (5.8) is carried out over the part of the 4‘11domain that lies outside the circle 411 = d c o . The mixed Fourier amplitude of the evanescent part of the transverse propagator is obtained readily from eq. (5.7) by setting q? = ia; where a: = [qi - ( O / C O ) ~ ] ” ~ > 0. Thus,

The retarded part of the source field, which is identical to the radiated field from the mesoscopic particle, is thus given by

in the mixed Fourier domain, inserting the expressions in eqs. (5.7) and (5.9) for the propagating and evanescent parts of the retarded propagator, respectively. For field points outside the induced current density domain of the mesoscopic particle, it appears from eqs. (5.4), (5.9) and (5.10) that

E:TRET(z;qll,w ) Iyl -’oo

=

;isF(z;4‘ll, 0).

1)

(5.1 1)

This shows (neglecting the factor that the z-dependence of the retarded components of the field for large 411 tends to be just the same as those of the selffield. Since the radiation from the source is described by the retarded transverse ++T propagator Do which contains both propagating and evanescent modes, one would basically hold the view that not only propagating but also evanescent

v, I 51

2D SPATIAL CONFINEMENT OF LIGHT BY OPTICAL PHASE CONJUGATION

33 1

waves can be phase-conjugated, and recent experiments (Bozhevolnyi, Keller and Smolyaninov [ 19941) have confirmed this assertion. However, one should remember that in order to phase-conjugate larger and larger 411-components, one must bring the source closer and closer to the mirror. When the source is sufficiently close to the phase conjugator the extension of the tail of the attached source field, which is proportional to exp(-qll Iz - z’l) for a given 411-value tends to overlap the mirror. When this happens the entire picture of a possible phaseconjugation process must be completely re-examined. 5.2. CONFINEMENT BY MEANS OF AN IDEAL PHASE CONJUGATOR

A qualitative analysis of the possibilities of 2D spatial confinement of light via optical phase conjugation can be made assuming (i) the electron confinement in the source to be perfect, and (ii) the phase conjugator to be ideal. Although both of these assumptions can be relaxed, we shall use them below to keep the mathematical framework simple. Complete electron confinement means that I(?’;w ) = jo(w)6(?’ + d&), and therefore the retarded part of the source field is given by HT

EF(7;w ) = - i ~ w D o ( F + d & ;w ) .-?o(w).

(5.12)

In order to apply the concept of “an ideal phase conjugator” to the retarded field it is adequate at this stage to use the Weyl expansion for the propagator -T Do (J + d&; w), and distinguish explicitly between homogeneous (propagating) and inhomogeneous (evanescent) modes, i.e.,

E,““(<

+

0 )= P o r n ( < 0 ) P h ( <

w).

(5.13)

The phase-conjugating mirror is assumed to occupy the halfspace z 2 0, and for points in the strip -d < z < 0, it is realized by means of eqs. (5.7) and (5.8) that the homogeneous part of the field incident on the phase conjugator is given by

where q; = (qi - qi)”2 _> 0, and i = + q l & . The inhomogeneous part of the incoming field may be obtained directly from eq. (5.14), by making the

332

LOCAL FIELDS IN LINEAR AND NONLINEAR OPTICS OF MESOSCOPIC SYSTEMS

substitution q l 4 ia!, where a; of integration to 411 > 40. Hence,

=

[v, 5 5

(qi - qi)"2 > 0, and changing the domain

-d < z < 0.

(5.15)

If one denotes the phase-conjugated (PC) replica of the incident field by i?p~(?;w), a so-called ideal phase conjugator is characterized by the following condition on the mirror surface

The (complex) quantity rpc( w ) is the amplitude reflection coefficient of the PC-mirror. Although rpc(w) may depend on the frequency of light, for an ideal phase-conjugating mirror it is assumed that the response is independent of the state of polarization of the incident field and that all wave-vector components constituting the incoming field are phase-conjugated in the same manner. The condition in eq. (5.16) is certainly an idealization, in particular when it comes to the study of the phase conjugation, here degenerate four wave mixing, of the evanescent components. A part of the angular spectrum of the evanescent waves gives rise to modes which inside the phase conjugator decay rapidly with the distance from the mirror surface, and to calculate the phase-conjugated response of these modes a microscopic nonlocal theory is required. Elements of such a theory which allows one to treat the (surface) phase conjugation on a length scale much less than the optical wavelength and also degenerate four wave mixing in quantum wells (and thin films) have been established recently (Andersen and Keller [ 19961). In passing, one should notice that the phase-conjugated field at the surface of the mirror is thus obtained by taking the complex conjugate of the spatial part of the incoming field only (and multiplying it by ~ p c ( o ) ) By . combining eqs. (5.14) and (5.16) it appears that the phase-conjugated replica of the homogeneous field is given by i;;m(?ll,z

=

0; w ) = -(4n&gw)-'rpc(w) (5.17)

Y 9: 51

2D SPATIAL CONFINEMENT OF LIGHT BY OPTICAL PHASE CONJUGATION

at the surface, or equivalently upon making the substitution

333

4'11 + -4'11

Eigm(Fll,Z= 0; 0)= -(~X€OW)-~YPC(W) (5.18)

(5.19) is the mirror (M) image of the (real) incident wave vector 4' in the surface plane. The phase-conjugated field moves backwards into the halfspace z < 0, and to find the field at an arbitrary observation point F, each plane-wave component in eq. (5.18) must be multiplied by the propagation factor exp(-iqlz). Altogether, it thus appears that the phase-conjugated replica of the homogeneous part of the incident field takes the form

(5.20) To obtain the phase-conjugated replica of the inhomogeneous field at the mirror surface, eq. (5.15) with z = 0 is inserted into eq. (5.16). This gives

@;(ql,

z = 0; W ) = -i(4~.5,o)~'rpC(o)

(5.21)

(5.22) where

2 = 4'11 + iaiZz

(5.23)

is the complex wave vector of an inhomogeneous plane wave decaying in the positive z-direction and having a (real) wave vector component 4'11 along the

334

LOCAL FIELDS IN LINEAR AND NONLINEAR OPTICS OF MESOSCOPIC SYSTEMS

[v, 9: 5

surface. The inhomogeneous part of the phase-conjugated field also moves backward into the halfspace z < 0, and to find the field at an arbitrary point ?, each plane-wave component in eq. (5.22) is multiplied by the factor exp(@z) 40. in order to fulfil the vacuum dispersion relation for a field having 411 Introducing the mirror image (5.24)

<M = i l l - i&ZZ

of the complex wave vector in eq. (5.23), the phase-conjugated replica of the inhomogeneous part of the source field is given by

in the vacuum halfspace. A schematic illustration of the principle of the field confinement by means of phase conjugation of an outgoing spherical wave is presented in fig. 7. It is illuminating to analyze the phase-conjugated field along the negative z-axis, since particularly simple analytical expressions can be obtained in this case. Thus, by adding eqs. (5.20) and (5.25), and setting 711= 6, it appears that the phase-conjugated field for observation points on the z-axis is given by

EPC(F],= 6, z; 0)= -(4n&()w)-'rpc(w) Hinh

x j ; ( w ) . (rhom(z;0)+ I

( z ; ~ ) ,) z < 0,

(5.26)

where (5.27) and (5.28) If one introduces polar coordinates in the q'li-plane, it is a straightforward matter to perform the double integrations in eqs. (5.27) and (5.28). Using the abbreviation

a+= iqo(z + d ) ,

(5.29)

Y 5 51

335

2D SPATIAL CONFINEMENT OF LIGHT BY OPTICAL PHASE CONJUGATION A

. Fig. 7. Schematic illustration showing the phase conjugation of the outgoing (spherical) field from a mesoscopic particle. The domain occupied by the electrons (or, rigorously speaking, the induced electronic transition current densities) of the mesoscopic particle (or atom, molecule, . . . ) is indicated in black, and the thick white arrows represent the wave vectors of the counterpropagating waves in the phase conjugator. The shaded regions show the extensions of the transverse (or equivalently longitudinal) parts of the transition current densities of the mesoscopic particle and phase conjugator. In the phase conjugation process not only propagating (homogeneous) waves but also those evanescent (inhomogeneous) modes which have wave vectors smaller in magnitude than the reciprocal linear extension of the transverse current density domain of the mesoscopic particle are phase conjugated. From a qualitative point of view the smallest light-field focus attainable is thus comparable in magnitude to the extension of the transverse current density domain of the source particle. The outgoing @robe) field of the mesoscopic particle is indicated by the thin dotted arrows, and the solid arrows show the flow pattern of the phase conjugated replica.

the homogeneous contribution in eq. (5.27) may thus be written in the form horn

H

I

(z;w)=qi

{ ;-

-(U-ZzZz)

1 - 2eC+

2e-"+ 0:

+ 2(1- e-"+) .I (5.30)

a? ++horn

The diagonal form of the tensor I ( z ; o ) shows that current oscillations of the source parallel to the mirror surface on the z-axis give rise only to a phase-conjugated field parallel to the surface. If the source dipole oscillates perpendicular to the plane of the mirror, f?pc is also along the z-direction. To obtain the homogeneous part of the phase-conjugated field on the position where

336

LOCAL FIELDS IN LINEAR AND NONLINEAR OPTICS OF MESOSCOPIC SYSTEMS

[V,

5

5

the electrons are confined, i.e., for a+ = 0 ( z = -d), one takes the limit a+ 4 0 in eq. (5.30). This gives -horn

(z = -d.w)

I

=

23qoa. 3

(5.3 I )

As one would expect, the homogeneous part of the phase-conjugated response on the particle position is isotropic and independent of the distance of the dipole from the surface. The result in eq. (5.31) has a simple interpretation in terms of the socalled radiation-reaction (RR) field, ~ R R ( u ) . It is well known (see, e.g., Van Kranendonk and Sipe [1977]) that this field is associated with the term q o 8 / ( 6 n i ) of the electromagnetic propagator in eq. (2.76). This means that the radiation-reaction field on the source particle is given by

(5.32) in the frequency domain. By combining the homogeneous part of eq. (5.26) with eqs. (5.31) and (5.32), it then follows that

ZFzrn(Fil= 0 , =~-d;

U)= QC(O>~&(W).

(5.33)

The homogeneous part of the phase-conjugated field thus gives rise to a phaseconjugated radiation-reaction on the particle. By carrying out the integrations in eq. (5.28), it is realized that the inhomogeneous contribution is given by -inh

I

( z ; w ) = q;

{:-

-(U-e"e'z)

(5.34)

where

a-

=

iqo(z - d ) .

(5.35)

On the position of the dipole the inhomogeneous part of the phase-conjugated field is thus given by

Elnh PC(

I/

-

0,z = -d; w ) =

(5.36) In contrast to the homogeneous response, the phase-conjugated inhomogeneous one is anisotropic, i.e., the components parallel and perpendicular to the surface

VI

REFERENCES

337

behave differently. The results presented in eqs. (5.33) and (5.36) are identical to those given by Aganval and Gupta [1995]. It appears from eq. (5.34) that the inhomogeneous part of the phase-conjugated response has only near-field (and far-field (- a?) components on the z-axis, whereas the homogeneous part also exhibits a middle-field (- a;’) component (see eq. 5.30). It is obvious from eq. (5.34) that destructive interference between the near- and far-field components leads to a cancellation of the inhomogeneous response parallel (11) to the surface at the position (5.37) provided that wd < c o d % Such a cancellation cannot occur for the response perpendicular to the plane of the mirror. If the cancellation is to occur at the position of the dipole (z//= -d), the vacuum wavelength of the field (&) and the dipole distance from the surface must fulfil the condition A0 = 2?tfid, which means that the dipole must be placed at a distance from the surface that is an order of magnitude less than the wavelength of the radiated field.

Acknowledgements

I am indebted to Prof. Emil Wolf for inviting me to write on a subject so close to my heart, and to all the scientists who over the years had the kindness to make their findings and articles available to me, thus helping me to write this review.

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TUNNELING TIMES AND SUPERLUMINALITY BY

RAYMOND

Y. CHIAO

Department of Physics, Uniuersity of California at Berkeley, Berkeley, CA 94 720, USA

AND

AEPHRAIM M. STEmBERG Department of Physics, Uniuersity of Toronto. Toronto, ON MSS IA 7, Canada

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§ 1. INTRODUCTION . . . . . . . . . . . . . . . . . . .

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0 5 . NEW THEORETICAL PROGRESS . . . . . . . . . . . . § 6 . TUNNELING JN DE BROGLIE OPTICS . . . . . . . . . .

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§ 8. WHY IS EINSTEIN CAUSALITY NOT VIOLATED? . . . .

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§ 9 . CONCLUSION

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ACKNOWLEDGMENTS . . . . . . . . . . . . . . . . . . .

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“Time, time, time, see what’s become of me.” -

0

I? Simon -and A. Garf‘unkel

1. Introduction

How long does it take for a particle to tunnel through a barrier? This simplesounding question has provoked much controversy over the past six decades, ever since the phenomenon of tunneling (i.e., barrier penetration) was first predicted to occur in quantum mechanics. Although tunneling has by now been observed in many physical settings, and has even been applied in many useful devices - such as the Esaki tunnel diode, the Josephson junction, and the scanning tunneling microscope - the speed of the tunneling process remains controversial. One reason for this is that some theories for the tunneling time predict - and some experiments confirm - that the time is so short that (in a sense to be defined more precisely below) the tunneling process is superluminal. The tunneling time question is not only of scientific, but also of technological interest. It is important to know if there is any limitation on the speed of electronic and photonic devices arising from the speed of the tunneling process. Although the tunneling of electrons seems to be more important at the present time for practical devices, the tunneling of photons is central to such devices as fiber couplers, laser output couplers, and scanning photon tunneling microscopes, to name a few examples in optics I . Many conflicting theoretical predictions have been made concerning the tunneling time, and as yet no consensus has emerged as to the theoretical answer. However, the situation is changing rapidly because many experiments, mainly in optics, have now been performed to measure the tunneling time, and the purely theoretical debate has been transformed into one in which actual data can be brought to bear on the question. In the process, it has become clear that one must make a careful operational definition of exactly how the measurement of the tunneling time is actually performed. To show that a clear operational definition is in fact possible at all, we give here one example (others may also be possible): Suppose that two particles were produced simultaneously from a radioactive decay. One particle tunnels through a barrier towards a detector, and

I We use the term “optics” throughout this chapter to refer to all electromagnetic propagation phenomena, including not only those in the visible but also in the microwave region of the spectrum.

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[VL

B

1

the second particle propagates through the vacuum towards a second detector. If the two particles have the same speed, and the two detectors were placed at an equal distance from the radioactive source, the time delay between the two “clicks” registered in coincidence by these detectors would then constitute a clear operational measure of a tunneling time (Steinberg, Kwiat and Chiao [1993]). However, different experimental setups may measure different tunneling times, and the answer to the tunneling time question may differ from experiment to experiment. In particular, one must distinguish carefidly between a tunneling arrival time, which measures how long it takes a particle to cross the barrier and reach the detector, and a tunneling interaction time, which measures how long the tunneling particle interacts with the barrier while crossing it. While classical intuitions lead us to take for granted that these two times ought to be identical, there is no physical law which guarantees this. In fact, as we shall see, Gedankenexperiments designed to measure one or the other of these quantities will in general not agree in quantum mechanics. T h s is a subtle but important distinction. Measurements of tunneling times by photons possess certain advantages over those by electrons or other particles, stemming mainly from the fact that the wavelength of visible light is much larger than the de Broglie wavelength of massive particles. (Only at temperatures on the order of microkelvins does the thermal de Broglie wavelength of even a light atom approach microns; see 5 6.) This makes the relevant physical dimensions of the tunneling barrier much larger, and hence makes the barrier much easier to fabricate for photons than for electrons. The photon also possesses an internal degree of freedom, namely its spin, which could be used as an internal clock during the tunneling process. However, electrons possess certain advantages over photons, the main one being that they possess an electric charge, and therefore that they interact with other charged particles strongly, thus allowing an easier measurement of the tunneling interaction time. There are three main types of tunnel barriers for photons which have been used in tunneling-time experiments: (i) periodic dielectric structures excited inside their band gap or stop-band, (ii) frustrated total internal reflection (FTIR) in glass or dielectric prisms, and (iii) waveguides beyond cutoff, which have been studied so far using microwaves only. The first type of barrier arises from Bragg reflections from the periodic structure, which leads to an evanescent (i.e., exponential) decay of the wave amplitude when the frequency is within the forbidden band gap (or “photonic band gap”, Yablonovitch [1993], John [1991]) at the first Brillouin zone edge, analogous to the evanescent propagation of electron waves with energies inside the band gap of a Kronig-Penney model. It should be noted that within a large bandwidth near the midgap region, the periodic structure is

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nondispersive (i.e., the group velocity approaches a constant), so that tunneling wave packets which are tuned to midgap, although much attenuated in amplitude, can remain essentially undistorted upon transmission through the barrier. The second type of barrier (FTIR) arises from the coupling of an evanescent wave in the spatial gap between a pair of glass prisms when a beam of light is incident on the interface between the prisms beyond the critical angle (Zhu, Yu, Hawley and Roy [ 19861). The third type arises from the evanescent wave inside a waveguide whose dimensions are too small to allow the propagation at the incident frequency. There is negligible dispersion of the tunneling wave packet in FTIR, but waveguides beyond cutoff are highly dispersive. There are other situations besides the three mentioned above, in which tunneling-like phenomena occur in optics; for example, wave propagation below the plasma cutoff frequency, or into the shadow region of a sharp edge by diffraction, or outside the allowed orders of diffraction gratings or Fabry-Perots, or inside absorption lines. These are all “classically” forbidden phenomena, i.e., all are forbidden at the level of geometrical or ray optics, but all can actually occur at the level of physical or wave optics. Some of these phenomena have also been shown experimentally to be superluminal. There has recently been a second controversy which has arisen as a result of those experiments in which superluminal group delays through tunnel barriers have been observed. This controversy is centered around a different, but related, question: Can one send signals, that is, information, through a tunnel barrier faster than the vacuum speed of light? This controversy has been sharpened by the claim by Nimtz and his co-workers that they have actually transmitted Mozart’s 40th Symphony as a radio signal through a microwave tunnel barrier at a speed much faster than c (Heitmann and Nimtz [ 19941). We shall show that there has in fact been no violation of Einstein causality in these and closely related experiments. Therefore the implication that their experiments somehow call causality into question is in our opinion unfounded. In light of the second of these controversies, we have decided to include in this review a discussion of the problem of superluminal group velocities which have been predicted for the propagation of wave packets tuned to transparent spectral regions of media with inverted atomic populations. We shall discuss two cases: superluminal wave packets tuned close to zero frequency, and those tuned close to an atomic resonance with gain in it. In the latter case, optical tachyon-like propagation of collective atom-photon excitations is predicted to occur. These new kinds of superluminal propagation effects can occur over much longer distances than for tunneling. Hence they will force us to understand the meaning of causality, the definition of a signal, and the nature of information.

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[VL 5 2

2. A Brief History of Tunneling Times

Despite its absence from the overwhelming majority of textbook descriptions of tunneling (one early exception being Bohm [ 195I]), the tunneling-time problem has a long and illustrious history. At the heart of the problem is the fact that the kinetic energy of a particle inside a tunnel barrier is negative, so that a semiclassical estimate of its velocity becomes imaginary. This makes it impossible to make the naYve first approximation that the duration of a tunneling event is the barrier width divided by the velocity Within a few years of the first predictions of tunneling, discussions appeared of the time spent by a particle in a “forbidden” region, and of the use of the stationary phase approximation to calculate properties of tunneling wave packets (Condon [1931], MacColl [1932]). By 1955, Wigner published a paper discussing the relationship between scattering phase shifts and the delay time, making explicit the connection between these quantities and the principle of causality (Wigner [ 19551). As is well-known in electromagnetism, the frequencyderivative of the transmission phase shift yields the time-of-arrival of a wellbehaved wave packet peak; we term this quantity the “group delay”, by analogy with the group velocity calculated by the same method of stationary phase2:

w.

where @t is the phase of the tunneling transmission amplitude. (For a free particle with a real momentum hk, we have &(x) = kx, and the above relation yields t,(x) = x/(dw/dk), where the denominator is the familiar expression for the group velocity.) Wigner and his student Eisenbud [1948] applied the interpretation of the derivative of the scattering phase shift as a time delay to the problem of scattering (which of course includes tunneling as a special case), and Wigner observed that “the ‘retardation’ cannot assume arbitrarily large negative values, in classical theory it could not be less than -2a”, where a is the radius of the scattering potential; in other words, a classical particle cannot leave the scattering center before it arrives. Wigner noted that “It will be seen that the wave nature of the particles does permit some infringement of [this

It is important to note, however, that many workers use the terminology “phase time”. We avoid this, as the confusion between phase and group velocities has occasionally clouded the causality issues which plague the tunneling-time controversy. By ‘retardation’, Wigner refers to the group delay relative to the time for free propagation, expressed in units of distance.

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inequality]”. It is primarily with this very infringement that we are concerned here. Does wave mechanics truly allow particles to exit a barrier before they enter it, and in particular, do such effects violate relativistic causality? One could reasonably suspect that the non-relativistic nature of the Schrodinger equation is at fault here, but more careful analyses using the Dirac equation show that such superluminal transmission (which occurs in cases where all relevant energy scales are far less than the electron rest mass in the first place) persists (Leavens and Aers [1989]). The conflict is made even more clear by turning to optical analogs of tunneling, as the same problems arise with Maxwell’s (fully relativistic) equations, and since one begins in the relativistic regime, it is relatively easy to achieve conditions under which the group delay is predicted to be superluminal. Of course, superluminal and even negative group velocities were already known to occur in electromagnetism, and had been reconciled with causality by Sommerfeld and Brillouin (Brillouin [1960]). Their work showed that no real signal could propagate faster than the vacuum velocity of light c in any medium obeying the Kramers-Kronig relations, even in regions of anomalous dispersion. In these regions, the absorption and the strong frequency-dispersion cause the stationary-phase approximation to break down, as an incident pulse is distorted beyond recognition and no single transmitted peak may be observed. Conventional wisdom has it that such a breakdown occurs in every limit where the group velocity exceeds c. Nevertheless, as early as 1970, Garrett and McCumber showed theoretically that for short enough interaction lengths, absorbing media could indeed transmit undistorted (but attenuated) Gaussian pulses at superluminal, infinite, or even negative group velocities (Garrett and McCumber [ 19701). The experimental verifications of these predictions will be discussed in 9 4.2. As we shall see, these effects are consistent with relativistic causality, and no signal is in fact conveyed faster than light. Even before this work on anomalous dispersion (which is still not as widely known as it deserves to be), the question of superluminal wave-packet transmission in tunneling was put on a firmer footing by Hartman [1962]. Hartman was not satisfied by MacColl’s 1932 observation that there is “no appreciable” delay in tunneling4, and he was concerned about the effects of preferential transmission of higher energy components in a wave packet. In a

As alluded to earlier, it is the imaginary momentum which leads to difficulty. If k is imaginary, i.e., if the wave function decays exponentially according to Y(x)= Y(0)e-KX, the phase Q becomes a constant, and the group delay of eq. ( I ) vanishes, apart from effects due to boundary conditions.

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[W P 2

rigorous treatment of the tunneling of wave packets through a rectangular barrier, he indeed found that for very thick barriers, such distortion occurred that no peak could be identified which might appear at the group delay time. For thin barriers, his results were in agreement with the stationary phase prediction, but there was no conflict with causality. Roughly speaking, the prediction is that for thicknesses smaller than one decay length of the evanescent wave (d < U K ) ,a transmitted particle of energy much less than the barrier height ( E << Vo; k << ko) will appear to have travelled at its initial velocity of hWm. This delay is related to the and anti-evanescent (e+Kx) fact that phase is accumulated as the evanescent (0’) waves change in relative size, as the two have different (but constant) phases. For thicker (i.e., “opaque”) barriers ( ~ >> d l), there is no phase change across most of the barrier, since the wave function is dominated by real exponential decay. , time it would take The group delay thus saturates at the finite value 2 m / h k ~the . the free incident particle to traverse two exponential decay lengths 1 / ~ Hartman ~ small confirmed that for intermediate barrier thicknesses, larger than 1 / but enough that the pulse was not distorted significantly, this saturation effect did indeed occur. As the distance traversed continues to grow, but the time required to traverse it remains roughly constant, it is clear that one eventually reaches a regime where the apparent propagation speed exceeds c. (Recently, Low and Mende [ 1991 ] argued that an actual measurement of such an anomalously short (tunneling) traversal time could not be made. However, Deutch and Low [1993] modified this conclusion in the case of relativistic particles.) We now know that there are a number of cases in both electromagnetism and quantum mechanics where the naive application of a causality limit to the description of a wave packet’s propagation may fail. The question is no longer whether the method of stationary phase is valid, but rather whether it is unique. Are there perhaps a multiplicity of timescales which describe tunneling? Does the superluminal appearance of a wave packet peak imply an anomalously short “dwell time”, or some other “interaction time”, for the particle inside a forbidden region? What does it say about the transport of energy, or of information? What of the fundamental limit on the speed of a device whose operation depends on tunneling ? Over the years, and particularly since the 1980s, numerous proposals have been made for other “tunneling times” which might best describe the duration of the tunneling process, rather than just the time of appearance of a wave packet peak. Biittiker and Landauer, in particular, stressed that no physical law guarantees that an incoming peak turns into an outgoing peak (Biittiker and Landauer [ 19821). They and other workers have argued strenuously that the group delay is not a physically significant timescale. This dispute becomes subjective,

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of course, as recent experiments have shown unequivocally that in measurements of arrival time, the group delay is indeed significant. Other experiments which involve tunneling in solid state physics seem to be best described by the BiittikerLandauer or Larmor timescales (Gueret, Baratoff and Marclay [ 19871, Gutret, Marclay and Meier [ 1988a,b], Esteve, Martinis, Urbina, Turlot and Devoret [1989], Landauer [1989]). We are thus left in the uncomfortable situation of being unable to identify a unique timescale for tunneling, which forces us to analyze each conceivable experimental situation separately. The continued work on tunneling times is driven largely by the hope that this potentially infinite number of timescales can be reduced to a manageably finite handful of definitions, whose relationships and physical significances can be pinned down precisely. Although this project is by no means complete, recent work leaves us hopeful that this goal is not an unreasonable one, and that we will soon arrive at a fuller understanding of tunneling and related phenomena. Most optical experiments on tunneling times have studied the group delay; in general, it is more straightforward to measure the arrival time of a photon or an electromagnetic wave than to measure the duration of its interaction with some barrier. In a complementary fashion, studies of tunneling in solid-state physics have so far been unable to observe the group delay, but have lent support to certain other proposed times. Here we focus primarily on the former, but we will also discuss to some extent other candidate times and the outlook for future experiments on them. While it is impossible in this context to provide a full description of every theory that has appeared on the question of tunneling times, there are certain leading contenders with which it is useful to be familiar. The “dwell time” zd seems the most straightforward answer to the question “How much time does a particle spend in the barrier region?” It can be defined alternately for the timedependent or the time-independent case. In the former, its natural statement is as the time-integral of the instantaneous probability that the particle is inside the barrier (assumed to extend from -d/2 to d/2):

In the latter case, it is simply the probability density within the barrier, divided by the incident flux Jin:

Td

(time-independent)

=Jin

-d2

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In the limit of a monochromatic wave packet, these two formulas yield the same result, although for packets of finite extent, the corrections may be important (Hauge, Falck and Fjeldly [ 19871). The importance of definitions in the quantum regime cannot be overexaggerated. In the classical limit, td (the time spent within the potential step) and rg (the time between arrival at the leading edge of the step and departure from the trailing edge) are of course identical, and equal to d/u = md/hk. There is only one sensible quantity to term the “traversal time” in this case. It is because in the quantum limit these different definitions, equivalent in all familiar, classical regimes, yield different answers that there is no unambiguous recipe for providing an experimental prescription for determining “the tunneling time” quantum mechanically. This difficulty has been traced most frequently to two characteristics of quantum mechanics. One is the fact that time is not an observable: there is no Hermitian operator corresponding to the time of arrival, or to the duration of an interaction. The other crucial characteristic is that unlike classical mechanics, quantum mechanics (or wave mechanics, more broadly) does not contain welldefined trajectories with determined durations. A particle’s traversal of a barrier may be described as a Feynman path-integral (or Huygens’-Principle sum) over every possible trajectory linking its emission and its subsequent detection (Fertig [1990, 19931, Sokolovski and Baskin [1987], Sokolovski and Connor [1990, 19931, Hanggi [ 19931). Since the different trajectories in general have different durations, we see that we should not necessarily hope to find a precisely defined interaction time for a quantum particle. Nonetheless, we are free to consider specific experiments and ask by what timescales they are governed. In simple cases, we can perform the full quantummechanical analysis in order to arrive at a result. If we are fortunate, we may discern certain patterns in these results which will allow us to make inferences about problems too complicated for exact solution. At the least, by understanding some of these timescales, we hope to pin down the limits of validity of various approximations, such as the assumption that external degrees of freedom follow adiabatically the evolution of the tunneling particle, or in the opposite limit, remain unaffected by the motion of the particle. The dwell time may appear unsatisfactory as a candidate for several reasons. Foremost, it is a characteristic of an entire wave function, comprising both transmitted and reflected portions. One might well expect that transmitted and reflected particles could spend differing amounts of time in the barrier. (Without a doubt, one would expect them to spend different amounts of time on the far side of the barrier - a finite amount for the transmitted particles and none for the

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reflected ones - whereas the formulations of eqs. (2.2) and (2.3) leave no room to introduce this distinction.) Its definition is so natural that many researchers have argued that it must at least reflect the weighted average of transmission and reflection times, t d = ltI2tt + lrI2rr (with t and r the transmission and reflection amplitudes, respectively), but even this assertion has been disputed hotly (Hauge and Stervneng [ 19891, Biittiker [ 19901, Sokolovski and Baskin [ 19871, Sokolovski and Connor [1990, 19931, Olkhovsky and Recami [1992], Landauer and Martin [ 19941). The second seeming problem with the dwell time is one it shares with the group delay. It is not guaranteed to be greater than the barrier thickness upon the speed of light, d/c. In fact, in the low-energy limit k + 0, the wave is almost entirely reflected by the first interface, and I YI2 is negligible in the barrier, leading t d to vanish as 2 m k h ~ k ; . Biittiker and Landauer [1982] have been the great champions of looking beyond the group delay and the dwell time to definitions related more closely to the kinds of experimental questions which might concern us. In their 1982 paper, which is widely viewed as having rekindled the tunneling-time fire, they proposed a Gedankenexperiment which would allow one to infer the duration of the tunneling process. Consider a particle tunneling through a rectangular barrier. Now modulate the height of the barrier by a small amount, at some relatively low frequency SZ. Clearly, the transmission is lowest when the barrier is highest, and vice versa. But now imagine that SZ becomes greater and greater, until SZ >> l/tt, that is, until the barrier goes through more than one oscillation during the “duration” tt of the tunneling event. Naturally, the modulation of the transmitted wave will be washed out. Biittiker and Landauer therefore solved the problem of the oscillating barrier, and looked for this critical frequency SZ,. They then postulated that the traversal time was t g = ~ I/&&. When the calculation was performed in the opaque limit (Kd >> l), they found the following result:

This is a striking result. Recalling that the local wavevector inside the barrier is i K , we see that this is exactly the time we would expect from a semiclassical or WKB approach ( m d / m ) - aside from the fact that we find a real number here, despite the imaginary value of the wavevector. Due to the similarity of the formulas, the Biittiker-Landauer time is also frequently referred to as the “semiclassical time”. (Far above the barrier, both tg and t d in fact approach the semiclassical time ts = md/filkl.) Since this time is proportional to d , it rarely becomes smaller than d/c; in fact, it would only do so for m/hK > c, which is the

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relativistic limit, where the Schrodinger equation should not be expected to be valid. (In reality, for geometries more complicated than the rectangular barrier, it has been noted that this time may vanish identically, leading once more to causality problems (Biittiker and Landauer [ 19851, Stovneng and Hauge [ 19891, Martin and Landauer [1992], Steinberg, Kwiat and Chiao [1993]).) While above the barrier, the semiclassical time closely resembles the group delay (missing only the oscillations due to multiple reflections at the barrier edges, which become insignificant in the WKB limit), it looks nothing at all like zg below the barrier, diverging when E = VO(where VO= h2ki/2m is the height of the barrier) and falling in the opaque limit (Kd >> 1) to md/hko as opposed to diverging like zg 2m/hkko. The group delay diverges for k -+ 0, but is independent of d; the Buttiker-Landauer time is well-behaved as k + 0, and is proportional to d. Biittiker went on to consider another “clock”, to see if different types of perturbations would bring to light the same timescale. Expanding on work due to Baz’ [ 19671 and Rybachenko [1967], he considered an electron tunneling through a barrier to which a small magnetic field B = Boi is confined. Suppose the electron’s spin is initially pointing along P.The magnetic field causes it to = 2p~Bo/h,where ,UB is precess in the x-y plane at the Larmor frequency the Bohr magneton. If one measures the polarization of the transmitted electron, one will find it to have precessed through some angle O,, and nothing could be more natural than to ascribe this to precession at @ for the duration ,z of the tunneling event, leading to the “Larmor time” zy = O,/a+,. This time turned out to be equal to the dwell time Td, including the latter’s superluminal behavior at low energies. (For cases other than the simple rectangular barrier, these two times do not remain equal. Hence some workers (Hauge and Stovneng [1989]) have argued that they are conceptually quite distinct quantities.) Biittiker’s insight was that this early expression for the Larmor time made the implicit assumption that by taking the Bo 0 limit, one could neglect the tendency of the electron to align itself with respect to the magnetic field. In reality, due to the interaction Hamiltonian ‘Hint = +2pBB&, a spin-up electron sees an effective potential with a higher barrier than that seen by a spin-down electron, and therefore has a lower transmission probability. As the 2-polarized electrons are equal superpositions of S, = & 1/2, this preferential transmission will tend to rotate the polarization out of the x-y plane towards the negative z-axis, so that the transmitted electron beam is slightly spin-polarized antiparallel to the applied B field. Biittiker showed that both this out-of-plane rotation and the in-plane precession were first-order in Bo, and furthermore, that the former dominated the latter in the opaque limit. Defining a second Larmor time related to the polar rotation according to tz= &/@, he found this timescale --f

--f

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to reproduce the m d h K behavior he and Landauer had already calculated by considering the modulated barrier. Suggesting that the true interaction time should take into account the full three-dimensional rotation of the electron’s spin, he proposed that the interaction time was txG t + tz.We refer to this time as “Buttiker’s Larmor time” TB. It agrees with the oscillating-barrier result t B L in both the low- and high-energy limits. A fair number of other approaches had been tried by 1990, mostly yielding combinations of the timescales already described: the group delay, the dwell time, the in-plane Larmor time, the Buttiker-Landauer (or semiclassical) time, or Buttiker’s Larmor time. For example, a Feynman-path approach in which the duration of all relevant paths was averaged with the weighting factor exp{iS[x(t)]/h} yielded the “complex time” Z, = ty- it, (Sokolovski and Baskin [ 19871, Sokolovski and Connor [ 1990, 19931, Fertig [ 1990, 19931, Hanggi [ 19931, Sokolovski [ 19951). It is easy to observe that the magnitude of this time is Buttiker’s Larmor time, while its real and imaginary parts are (for rectangular barriers) the dwell time and minus the semiclassical time, respectively. (An earlier approach (Pollak and Miller [1984], Falck and Hauge [1988]) yielded a similar complex time, whose real part was the group delay, rather than the dwell time.) Despite this telling relationship, many found the concept of a complex time to be unphysical and rejected it out of hand. The similarity of such different approaches can be traced to a particularly convenient functional form (Buttiker [ 19831, Landauer and Martin [ 19941) in which they can be written:

r

a

zg = h-arg(t),

t, =

dE d -A-arg(t) aVO d -h---Injtj

z,

ih-lnt,

T,

=

=

avo

= td

--f

tBL

i

tg =

cv in WKB limit,

in opaque limit,

d

avo

The group delay is the derivative of the transmission phase with respect to the particle’s energy, while the in-plane Larmor time is the derivative with respect to the barrier height. Since the out-of-plane Larmor precession arises from preferential transmission of anti-aligned rather than aligned spin components, it can be expressed similarly as a frequency-derivative of the transmission probability.

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2

This reflects the theoretical situation when optical experiments on tunneling began returning results around 1990. More recent work has begun to shed some light on why the different times are related in the way they are, and how one might physically interpret the real and imaginary parts of a complex time. This approach, and possible experiments, will be discussed in 9; 5 and 9; 6. At least one other principal theoretical approach deserves mention, and this one is sufficiently distinct that we have saved it for the end. It is clear that in classical mechanics, a particle follows a well-defined trajectory, and that such a trajectory can be defined as a certain approximation to the motion of a quantum-mechanical wave packet in the classical limit. The breakdown of such a notion leads to the difficulties regarding the quantum-mechanical tunneling time, in particular to the fact that a time defined in terms of wave packet arrival no longer need coincide with a time defined in terms of a clock which evolves while the particle is within the barrier. The most familiar treatment of trajectories in quantum mechanics is the Feynman path integral discussed above, according to which a particle follows every possible trajectory with a given weighting There is nevertheless a very different proposal for incorporating trajectories into quantum mechanics. This is the pilot wave model of Bohm and de Broglie (Bohm [1952], Holland [ 19931). This deterministic interpretation of quantum mechanics invokes a dual reality, consisting both of the wave function Y (determined in the usual manner) and of a particle with a perfectly welldetermined position. An ensemble of particles with initial positions described by the probability distribution P ( x , 0) = 1 Y(x, 0)12 evolves deterministically according to the hydrodynamic equation of motion

’.

x(t)

h

= -i--VY(x,

m

t),

(2.10)

which is sufficient to ensure that at all later times, the Born interpretation of I Y l2 will remain valid. Although the ensemble as a whole is described by a wave function, and does not possess a unique traversal time, each individual particle follows a classical trajectory whose duration may be calculated. This approach has been followed by various workers (Dewdney and Hiley [1982], Leavens [1990], Leavens and Aers [1993]), and has been shown to have interesting

Note that this weighting affords a rigorous prescription for calculating transition amplitudes, but no established recipe existed for defining a “duration”, in the absence of a clear operational definition of the latter. It is a pleasant surprise then that the natural extension proposed by Sokolovski and Baskin [1987], Sokolovski and Connor [1990, 19931 and Hanggi [I9931 agrees at any rate with other more or less justifiable definitions.

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relationships to the times already discussed. In general, however, the physical significance of these trajectories remains an issue of some contention (see, for example, Englert, Scully, Sussmann and Walther [ 19921, Durr, Fusseder, Goldstein and Zanghi [ 19931, Dewdney, Hardy and Squires [ 19931, Steinberg, Kwiat and Chiao [ 19941). The one feature of the Bohm approach which makes it somewhat haunting is that different Bohmian particles from the same ensemble may not cross each other’s trajectories, thanks to the single-valued velocity function given above. This implies that all transmitted particles originate earlier in the wave packet than all reflected particles. Given the superluminal behavior of tunneling peaks, it is striking that the particles which form the transmitted peak do not, under this interpretation, originate in the incident peak, but rather earlier in time. Later on, we shall see a similar feature in the classical-wave (pulse-reshaping) description of tunneling.

Q 3. Tunneling and Its Optical Analogs We establish here on a more formal basis the analogy between electron and photon tunneling (Chiao, Kwiat and Steinberg [ 19911). From Maxwell’s equations for classical electromagnetic fields, one can derive the wave equation in an inhomogeneous but isotropic medium, which for a monochromatic wave in the scalar approximation reduces to the Helmholtz equation,

V2&+ {n(x,y,z)2u2/C*}& = 0 ,

(3.1)

where & is the scalar amplitude of the electric field, n(x,y,z) is the index of refraction of the medium at w, the angular frequency of the wave, and c is the speed of light in the vacuum. The coefficient of & in the second term (the curly brackets) represents the square of the local wavevector. This equation is formally identical to the time-independent Schrodinger equation for the electron,

v2Y + ( 2 m / f i 2 ) { ~~ ( x , yz)}, Y = o ,

(3.2) where Y is the wavefunction of the electron, m is its mass, V(x,y,z) is the potential energy, and E is the total energy. This identification is exact if we make the following identification6: -

n(x,y, z) H {2m[E- V ( x , y ,z)]}”2c/ho .

(3.3)

Note, however, that the correspondence depends explicitly on w , and thus is only exact over restricted bandwidths. A dielectric interface may have a reflectivity which tends to a constant less than one as the photon energy vanishes, while a step potential will always have reflectivity tending to unity as the electron energy vanishes. It can therefore be subtle to connect Kramers-Kronig-style arguments for photons to those for electrons.

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' 0 .

+output

u unit cell Fig. I . Periodic stack of quarter-wave dielectric layers composed of alternating high- and low-index media, i.e., the ID photonic band-gap material.

Tunneling barriers can arise in regions of space where E < V(x,y,z), which correspond to evanescent wave regions, where the effective index of refraction n(x,y,z) is imaginary. Several situations in optics give rise to such evanescent waves, and hence to photon tunneling. All involve propagation of waves beyond some sort of cutoff, such as the cutoff at a photonic band gap edge, the cutoff at the critical angle for total internal reflection, or the cutoff of a constricted waveguide. As our first example of an optical tunneling barrier, we consider the evanescent wave propagation of electromagnetic waves inside a ID photonic band gap, since there is an obvious analogy to the evanescent propagation of electrons inside the band gap of the Kronig-Penney model for periodic electronic structures. Let the photonic band-gap material be composed of two media with nl > n2, described by n(x,y,z) = nl for all ma

Al

5 x < ma + -,

4 nI

where m = 0, 1,2,. . ., where A. is the vacuum wavelength, and where the lattice constant a of the unit cell is given by a = [n;'

+~,'IA~M.

(3.5)

This periodic dielectric stack is illustrated by fig. 1, and is equivalent to a dielectric mirror consisting of a periodic stack of alternating high- and low-index

VL

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36 I

w

+k 1st Brillouin ninc cdgc

Fig. 2. Dispersion relation for the 1 D photonic band-gap material, where the midgap frequency is q = 2nc/4.

quarter-wave layers. By eq. (3.3), we see that this is equivalent to the problem of an electron in a periodic potential, which can be approximated by the KronigPenney model (Ashcroft and Mermin [1976]). There results a band gap at the first Brillouin zone edge (see fig. 2) which arises from Bragg reflections off the periodic planes between the index strata. Hence the propagation of light inside the band gap becomes evanescent. As a second example, we consider the case of frustrated total internal reflection (FTIR). Consider two right-angle glass prisms, which are placed with their hypotenuses in close proximity, so that coupling through the exponential tail of the light wave (for incidence angles beyond the critical angle) allows the leakage of light from one glass prism into the other through an air gap (see fig. 3). This case is easier to connect with textbook descriptions of tunneling, and has also been used in a number of recent experiments on tunneling times. In the case of TE- or s-polarized light incident in the x-y plane on a glass-air interface at an angle H (see fig. 3), we can take out the dependence of the electric field &B on time and on y (the direction parallel to the interface) as follows:

in all three regions, where k = nw/c is the wavevector in the glass. For s-polarization, where the electric field vector is perpendicular to the plane of incidence, & and thus Y are continuous across the boundaries. If we assume a magnetic permeability of p = 1 in all three regions, then the magnetic field B is continuous as well, and this leads to the continuity of Y ' ( x ) = dY/dx.

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TUNNELING TIMES AND SUPERLUMINALITY

(n

sin

e = sin e’)

d

Fig. 3. Glass-air-glass interface with light rays drawn for the case of (a) tunneling through the air gap in frustrated total internal reflection (FTIR) when 0 > H,, and (b) “classically allowed” transmission when 0 < 0,.

These boundary conditions are the same as those for the one-dimensional Schrodinger wave function Y(x) at a step discontinuity in the potential V ( x ) . The electromagnetic wave equation reduces to

+ ( c ~ / c ) ~cos2 ( n ~H } Y Y” + (W / C ) ~{ 1 - n2 sin2 H } Y Y”

=

o

=

0 in the air gap ,

in glass regions

(3.7)

where the coefficients of Y in the second terms represent the squares of the x-components of the wavevectors in the glass and in the air gap, respectively. Equation (3.7) has exactly the form of the one-dimensional Schrodinger equation for an electron in a rectangular barrier of height Vo and a width equal to the width of the air gap (see fig. 3), when we draw the equivalences

2mE/h2

@ (u/c)2 { n2 cos2 0)

2m(E - V,)/h2 w ( w / c ) ~1(- n2 sin2 0)

(3.8)

It is clear from this correspondence that the critical behavior at E = Vo is analogous to that at the critical angle H = 0, = sin-’(l/n), and that for given

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TUNNELING AND ITS OPTICAL ANALOGS

dielecuic-

363

dielecuic-

air

tunnel barriei

Fig. 4. Microwave tunnel barrier consisting of an air gap section between between two dielectricfilled sections of a rectangular wave guide.

electron mass and photon frequency, a precise one-to-one mapping can be made between the parameters E and VOof the electron experiment and the parameters 0 and n of the photon experiment. In addition, in the classically allowed regime E > VO, the velocity of the electron inside the barrier is proportional to ( E - Vo)1’2in the classical (i.e., WKB) limit. When eq. (3.8) is used to transform this into the analogous photon variables, this electron velocity is seen to be proportional to cos #‘, where #’ is the angle of the refracted beam of light inside the air gap in the “classical” (ie., geometrical optics) limit for the photon (see fig. 3b). Thus the electron traversal time mimics exactly this “ray optics” behavior of the corresponding photon traversal time (Steinberg and Chiao [I 994a1). This is true in spite of the fact that their dispersion relations E ( p ) are quite different. As a third example, we consider a wave guide beyond cutoff. In order to avoid the complications of the fringing fields associated with a sudden decrease and increase in wave guide width, which is usually utilized in microwave experiments on the tunneling time, we analyze here instead the simpler case introduced by Martin and Landauer [1992], who considered a dielectric-filled wave guide interrupted by a rectangular air gap which serves as the barrier (see fig. 4). For simplicity, consider the TElo mode of this wave guide. The dispersion relations come from the relationships

+ k,’ k,’ + k,“ k,’

=

n2w2/c2 for the dielectric-filled sections,

=

w2/c2 for the air gap,

(3.9)

where n is the index of refraction associated with the dielectric, and where the conducting boundary conditions impose the condition k, = n/u (a being the

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W

Fig. 5. Dispersion relation for the TElo mode of the rectangular wave guide.

width of the wave guide) for the TEio mode. Therefore the dispersion relation of the wave guide in the air gap is of the form (see fig. 5 ) (3.10)

where o,= n d a . If the frequency of the wave is chosen to be below this cutoff, but above the cutoff frequency of the dielectric-filled section, then ki will be imaginary, while k, is real, and this wave guide configuration becomes a good analog for the tunneling of an electron through a one-dimensional rectangular barrier. The group delay for this wave guide geometry has been calculated by Martin and Landauer [ 19921.

5

4. Optical Experiments on Tunneling Times

4. I . CARNIGLIA AND MANDEL'S FTIR EXPERIMENT

An early optical experiment measuring the phase shifts which occur in frustrated total internal reflection (FTIR) was performed for both the TM and TE polarizations of the incident light (Carniglia and Mandel [1971a,b]). In a theoretical analysis of their experiment, Carniglia and Mandel calculated the time of arrival of the phase front of the evanescent wave at a point straight across the gap at a minimum distance from the point of incidence. Although their work did not directly address the problem of tunneling times, their results did bear indirectly on the question of whether or not the group delay saturates with increasing barrier thickness.

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Using a modified Rayleigh interferometer, they measured the phase shift accumulated by an evanescent electromagnetic wave after it crosses the air gap between the two glass prisms. Because the evanescent wave propagates parallel to the glass-to-air interface, this wave can penetrate into a direction normal to the interface without much change of phase, since the dominant exponential decay of the evanescent wave amplitude is a real function. This was confirmed in their first experiment, in which they showed that for TM polarization the phase shift saturated at the theoretically predicted (asymptotic) value of =

tan-’

cos2 B - n2(n2sin2 B - I )

2n cos H(n2 sin2 o - 1)1’2

(4.1)

which is independent of the width of the gap (i.e., the barrier thickness), in the opaque or thick-barrier limit. Since the derivative of the phase with respect to the frequency is the group delay, their observation implied that the group delay should also saturate, and thus become independent of the barrier width. Thus their experimental result was consistent with the theoretical conclusion reached earlier by Hartman [ 19621. Since there should be a crossover point beyond which the saturated group delay becomes less than the light-transit time across the barrier, these early experimental and theoretical papers already implied that the tunneling group delay should become superluminal for sufficiently thick barriers. In fact, since eq. (4.1) is independent of frequency, the saturated group delay is approximately zero. This implied that superluminal group delays should be easily achievable. There is an additional contribution to the group delay arising from a lateral shift of the beam due to the Goos-Hanchen shift (Steinberg and Chiao [1994a]). This shift has been observed recently in the transmitted beam in FTIR by Balcou and Dutriaux (see 9 4.1 l), and used by them to measure one of their two tunneling times. However, in Carniglia and Mandel’s original experiment, the beam width was 6cm, which was so large that they could not observe this shift. 4.2. ABSORPTIVE MEDIA WITH ANOMALOUS DISPERSION

In another optical context, superluminal group delays were also predicted theoretically and observed experimentally, namely in the region of anomalous dispersion near the center of an absorption line. Although this is not related directly to the question of tunneling times, many aspects of this earlier controversy concerning superluminal group velocities reappear in the tunnelingtime controversies. In 1970, Garrett and McCumber returned to an old problem

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[VL

s; 4

first considered by Sommerfeld [ 19071. They showed theoretically that for short lengths, absorbing media could transmit undistorted (but attenuated) Gaussian pulses at superluminal, infinite, or even negative group velocities (Garrett and McCumber [1970]). This arose from the fact that the group velocity, which is given by the expression

can have a vanishing denominator in regions of anomalous dispersion, where d Re n/dw is large and negative, i.e., near the center of a strong absorption line. The stationary phase approximation does not automatically break down for smooth Gaussian pulses, in contrast to the case of signals with a discontinuous front considered by Sommerfeld and Brillouin (Brillouin [ 19601). Garrett and McCumber showed that an incident Gaussian wave packet can be reshaped by the absorption process (in which the later parts of the wave packet would be absorbed to a greater extent than the earlier parts) in just such a way as to produce a smaller, but undistorted Gaussian wave packet at the exit face of the medium. (In tunneling, a similar pulse-reshaping occurs, except that the process of absorption is replaced by the process of attenuation due to reflection from the barrier.) The peak of the pulse thus appears to have moved at a superluminal group velocity inside the medium (or a barrier). Tanaka [1989] later extended their work using the saddle point method. He showed that the propagation of a wave packet into an anomalous dispersion medium is characterized by three successive spatial regions with negative, superluminal, and subluminal group velocities, respectively. Chu and Wong [ 19821 verified experimentally that the superluminal behavior of the group velocity as predicted by these theories actually occurred for weak picosecond laser pulses propagating near the center of the bound A-exciton line of a GaP:N sample. Segard and Macke [ 19851 also confirmed these predictions in the propagation of millimeter wave pulses through a gas cell of OCS near the 97GHz J = 7 + 8 transition. Furthermore, both groups observed negative group velocities. The meaning of a negative group velocity is that the peak of the transmitted wave packet leaves the exit face of the gas cell before the peak of the incident wave packet enters the entrance face of this cell, in seeming defiance of our usual notions of causality. However, this effect can again be understood in terms of a pulse reshaping of the Gaussian wave packets due to absorption, and is perfectly causal (see Q 8). These experiments demonstrated that the group velocity, even when it exceeds c, approaches infinity, or becomes negative,

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possesses a definite physical meaning, since there exist definite operational procedures, which have in fact been carried out in practice, to measure these counterintuitive group velocities. These facts fly in the face of conventional wisdom7, which tells us that when the group velocity becomes superluminal, it has no longer any appreciable physical significance, or that somehow it is just not a useful concept. 4.3. THE MILWAUKEE GROUP

Starting in 1989, a group at Marquette University in Milwaukee began to generate a fair amount of controversy by publishing papers with titles as provocative as “Transmit radio messages faster than light”. Needless to say, these articles were greeted with a great deal of skepticism, not mitigated by the fact that they seemed to harbor a confusion between phase and group velocities (Giakos and Ishii [1991a-c], Ishii and Giakos [1991], Stephan [1993]). Most physicists remained blissfully unaware of the argument, which nevertheless raged for a time in Microwave and Guided Wave Letters. The claims were twofold. The authors pointed out that for an electromagnetic wave propagating in free space, the phase velocity measured at an angle 0 to the propagation direction is C / C O S ~ > c. They then claimed to have measured the arrival time for a microwave pulse in this geometry, and found it to be described by this superluminal phase velocity. They also did an experiment in a waveguide, presenting similar conclusions. Although they made no attempt to connect these findings to the phenomenon of tunneling, and their claims were not widely accepted, it is interesting to note that under certain conditions, such setups can indeed be shown to be analogous to tunneling, and to be described by time delays which in the appropriate limits become superluminal. 4.4. THE FLORENCE GROUP, PART I

Similar experiments were being carried out in a different spirit at the Istituto di Ricerca sulle Onde Elettromagnetiche del Consiglio Nazionale delle Ricerche in Florence at about the same time (Ranfagni, Mugnai, Fabeni and Pazzi [ 19911). Ranfagni and co-workers were looking specifically at microwave transmission in waveguides beyond cutoff, whose mathematical equivalence to quantummechanical tunneling has already been noted. Aware of the controversy over

’ See for example p. 23 of Born and Wolf [1975], or p. 302 of Jackson [I9751

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TUNNELING TIMES AND SUPERLUMINALITY

PI, 4 4

tunneling, they hoped to resolve the issue by measuring the transmission delay time and comparing it to the group delay, the semiclassical or Biittiker-Landauer time, and Buttiker’s Larmor time. Their initial results were for an abrupt step being transmitted through a 10-cm-long waveguide with a cutoff of 9.494GHz, as much as 43MHz above the incident frequency. Complicated by the abrupt (roughly 5 ns) turn-on of their step and by the dissipation in the waveguide, their results were inconclusive, but showed rough agreement with the semiclassical time. Theoretical work taking dissipation into account (Ranfagni, Mugnai, Fabeni and Pazzi [ 19901, Mugnai, Ranfagni, Ruggeri and Agresti [ 19921) yielded reasonable agreement with the experimental data. Refinements of this experiment (Ranfagni, Mugnai, Fabeni, Pazzi, Naletto and Sozzi [ 19911) improved the signal-to-noise ratio, allowing good data to be obtained as far as l00MHz below the cutoff of a 15-cm narrowed waveguide segment. These data clearly contradicted the divergent behavior of the semiclassical time at cutoff, and seemed to agree better with the group delay than with the other candidate times. The barrier was not thick enough, however, for the contradiction between the group delay theory and the naive application of the causality principle to be checked directly. The Florence group also indirectly studied z, the out-of-plane portion of the Larmor time (equivalent to the imaginary part of the complex times discussed earlier), and were able to confirm that it behaved as predicted as well. Their conclusions were therefore appropriately cautious: “. . . there is agreement between the experiments and the appropriate theoretical models. This fact . . . leaves the identification of the tunneling time ambiguous”. Furthermore, in this series of experiments, it was impossible to directly test the question of superluminality. 4.5. THE COLOGNE GROUP, PART I

While Ranfagni’s group was working to extend their step-function transmissiontime measurements further below cutoff to adjudicate between the semiclassical and group-delay theories, a group in Cologne was also using microwaves to study tunneling, aiming in particular to test the prediction of superluminal traversal. In their initial experiments (Enders and Nimtz [1992]), they used a network analyzer to measure the transmission phase shift through a narrowed waveguide at different frequencies. They inferred a group delay by fitting their phase data to a smooth curve, and subsequently performing a Fourier transform to predict the delay for a hypothetical pulse. For the longest barrier they used, lOcm, they calculated a group delay of 130 ps, which would correspond to transmission at about 2.5 times the speed of light. They also observed, in agreement

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with the saturation effect predicted by Hartman, that barriers of different lengths yielded essentially the same phase shifts. In their early work, technical considerations made direct time-measurements less reliable than the phase measurements. In 1993, however, they reported time-domain measurements confirming the frequency-domain results, under the slightly misleading title “Zero-time tunneling of evanescent mode packets” (Enders and Nimtz [ 19931). In this experiment, they used a Hewlett-Packard synthesizer to produce sharponset pulses (rise times of a few nanoseconds) with carrier frequencies near 8.65GHz, allowed the waves to tunnel through a 6-cm barrier formed by a waveguide section with a 9.49GHz cutoff frequency (with an attenuation of 40dB), and then used a Hewlett-Packard transition analyzer to detect the transmitted envelope and compare it with that of a wave which traversed a 40 dB attenuator (whose effect on the group delay was verified independently to be negligible), but no barrier region. Due to the large bandwidth of their pulses, they saw a fair amount of distortion, and complicated features, but over much of the step, they found a propagation delay which appeared to be small relative to the 0.2-11s free-space propagation delay. They took this as final confirmation that the microwaves traversed the narrowed waveguide superluminally (indeed, with zero delay, since in a sense all the residual group delay may be attributed to edge effects, i.e., impedance mismatch between the waveguide segments). 4.6.THE BERKELEY GROUP

While most work on optical tunneling was going on with classical electromagnetic waves, typically in the 10GHz range, at Berkeley we had proposed to perform a test of optical tunneling that would stress the single-particle aspects of the effect. Quantum electrodynamics predicts that for purely linear optical effects, such as those considered in this chapter, single photons exhibit the same behavior as classical pulses (Glauber [1965]). In fact, one may consider the (properly normalized) pulse profile as the single-photon wave packet8. It is possible to construct creation and annihilation operators for any pulse mode which is a solution of Maxwell’s equations, simply by superposing operators

Although the existence of “wave packets” for photons is controversial, it is possible by limiting oneself to cases where photon number is conserved and to the paraxial limit to consider the positivefrequency part of the electric field E + ( r ,f ) analogous to a quantum wave function, bearing in mind that the detection probability is proportional to E ( r , t ) E+(r,t ) = ( E + ( r ,t ) l 2 , in analogy with the standard Born interpretation of the electron wave function (Deutsch [1991], Deutsch and Garrison [ 199 I]).

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PI,

5

4

for the plane-wave modes (Deutsch [ 19911, Deutsch and Garrison [1991]). Propagation effects are then governed by the classical wave equations, and quantization merely affects detection statistics and higher-order effects. Having already shown (Steinberg, Kwiat and Chiao [ 1992a1) that single-photon wave packets travelled at the group velocity in media with normal dispersion, we decided to extend this work to the case of tunneling. Our original proposal (Chiao, Kwiat and Steinberg [1991]) discussed the analogy between frustrated total internal reflection and one-dimensional electron tunneling, but we eventually settled on a 1D photonic band gap as a more appropriate medium for tunneling. A dielectric mirror consists of alternating quarterwave layers of high and low-index glasses, leading to constructive interference for reflection and destructive interference for transmission. Such a structure can be thought of as an analog of the Kronig-Penney model for a band gap in condensed-matter physics, and in fact there has been much work, both theoretical and experimental, on photonic band gaps (Yablonovitch [1993], John [1991]). The effective wave vector, or “quasimomentum”, of light inside the band gap is imaginary, and we confirmed by direct numerical calculation that this qualitative similarity was sufficient to create the same saturating effect and superluminal transmission as tunneling through a rectangular barrier. It is important to note that there is no direct analog to the tunnel regime ( E < VO)for light; as shown in tj 3, the analogy between the Schrodinger and Helmholtz equations leads to an effective index n(x,y,z) = {2m[E - V ( ~ , y , z ) ] ) ” ~ c / hwhich o, would be imaginary in any regions where E < V . Each microscopic (quarter-wave) region of the dielectric mirror is a region of allowed propagation, and it is only the Bragg reflection arising from the periodic spacing which makes the mirror as a whole a “forbidden region”. The wave function can be written according to Bloch’s theorem as a periodic Bloch function uk(r) times a plane wave exp{ik . Y}; inside the band gap, k becomes imaginary, leading to an exponentially decaying field envelope, but uk(r) is still a sinusoidally oscillating function. For our barrier, we chose an 1 1-layer mirror, with alternating indices of refraction of 1.41 and 2.22. At the design wavelength of 702nm, this mirror had a transmission that dropped to about 1%; the band gap extended from 600 nm to 800 nm, over most of which range the group delay was smaller than d/c = 3.6 fs. The stationary phase approximation predicted that the group delay near midgap would saturate at approximately 1.7 fs. This structure had several other advantageous features. Unlike the microwave experiments, it involved negligible dissipation, and no dispersion outside the tunnel barrier. Furthermore, both the transmission probability and phase are very flat functions of frequency near midgap, so there is essentially no wave-packet distortion. Finally, the

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symmetry of the problem makes the semiclassical time vanish identically at midgap, which emphasizes that even that time cannot solve all causality problems, and allowed us to distinguish it quite easily from the group delay time. Of course, direct electronic measurement of femtosecond-scale delays is not possible. We therefore used a nonlinear optical effect discovered by Hong, Ou and Mandel [1987], which can be thought of roughly as a time-reversed variant of the nonlinear autocorrelation technique for femtosecond laser pulses (which has also been applied to the tunneling problem by the Vienna group; see below). This effect relies on spontaneous parametric down-conversion, a ~ 1 absorbs a pump photon at process in which a crystal with a ~ ( nonlinearity ~0 and emits in its place a pair of photons (conventionally termed “signal” and “idler” despite the fact that in these experiments they are indistinguishable) at frequencies spread symmetrically about a / 2 , energy conservation being assured by the anticorrelation of the two photons’ frequencies. The photons are emitted simultaneously to within their coherence lengths, and as the latter are only constrained by the phase-matching bandwidth and subsequent filters, one finds correlation times as short as 15 fs. If the two photon wave packets meet simultaneously at opposite sides of a 50/50 beam splitter, a quantum interference effect related to Bose statistics causes them to exit the beam splitter along the same (randomly chosen) direction; detectors placed at the two exit ports of the beam splitter will never register photons simultaneously. On the other hand, if the two photons arrive at different times, each will make an independent choice at the beam splitter, leading to coincidence counts in half of the cases. Thus by changing the path length of one photon’s trip until the coincidence rate is minimized, one can ensure that the photons are meeting simultaneously at the beam splitter (Hong, Ou and Mandel [ 19871, Steinberg, Kwiat and Chiao [1992b], Jeffers and Barnett [1993], Shapiro and Sun [ 19941). If an obstruction such as a tunnel barrier is placed in one arm of the two-photon interferometer, the coincidence dip recorded as a function of external path length will shift, and this shift is a measure of the delay time for traversing the barrier. It is interesting to note that these experiments are typically performed with a continuous-wave argon laser as the pump, so the state of the light is in fact stationary in time. It is only the correlations between the photons which have the very fast ( I 5 fs) time-dependence. Once a photon is detected, it is possible to say that its twin has “collapsed” into a 15-fs wave packet, but prior to that time, the system is better seen as a superposition of 15-fs wavepackets with centers at every possible position. Other than the single-photon aspects, which were predicted theoretically not to modify the propagation times, this technique has some interesting advantages

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Fig. 6. Experimental setup for determining single-photon propagation times through a multilayer dielectric mirror.

relative to such approaches as classical white-light interferometry or nonlinear cross-correlation. Since the nonlinear effect is used only before the tunnel barrier, extremely low intensities may be used at the level of the sample; we typically counted on the order of lo5 photons per second, by using tens of milliwatts of 351 nm light from an argon laser as a pump. As discussed by Steinberg, Kwiat and Chiao [1992a,b], Jeffers and Barnett [I9931 and Shapiro and Sun [ 19941, first-order effects of group-velocity dispersion cancel out, allowing high resolution to be retained even in the presence of material dispersion. Finally, in contrast to standard interference techniques, this method relies only on detection of photon pairs, so the fringe visibility is not reduced by the low transmission through the tunnel barrier; interference occurs between two balanced Feynman processes, each of which involves only one tunneling event. Only the total count rate drops, leading to a f i dependence for the uncertainty, which we countered by averaging a large number of I-hour data runs. By scanning across the coincidence dip while periodically inserting and removing the band gap coating (see fig. 6), we were able to measure the shift due to the propagation delay to better than 1 fs (Steinberg, Kwiat and Chiao [ 19931). We also noted that as predicted, the shape of the coincidence dip (a direct measure of the overlap of the two wave packets) did not change significantly due to the presence of the barrier. In the first iteration of our experiment, we found the arrival time for propagation of a single photon through the 1.1 pm coating to

373

OPTICAL EXPERIMENTS ON TUNNELING TIMES 100%

8

80%

60%

2 c 3.

_.

YI

40%

g

20 %

0% 0’

20‘

10’

30’

40’

50’

60’

70’

80‘

YO’

Angle

Fig. 7 . Left axis: measured delay for mirrors I (squares) and 2 (circles) as a function of angle of incidence, to be compared with the group delay and with Biittiker’s Larmor time. Right axis: transmission versus angle of incidence. All curves for p-polarization.

*

be earlier by 1.47 0.21 fs than the arrival time for propagation through 1 .I pm of air. This 7-standard-deviation result confirmed the superluminality of singlephoton tunneling. It would correspond to an effective tunneling velocity of 1 . 7 ~ . It differs from the stationary-phase prediction of 1.9fs by about two standard deviations, and demonstrated immediately that the semiclassical time (which vanishes at midgap) was inadequate for describing wave packet propagation. In a later extension of this experiment, we studied the frequency-dependence of the tunneling time (Steinberg and Chiao [1995]). Since it was not feasible to change the frequency of the photons in our interferometer, we changed the angle of incidence on the multilayer dielectric, thus altering the Bragg condition. In this way, we were able to scan from midgap nearly to the band-edge. We confirmed the qualitative behavior of the group delay, with absolute agreement generally better than 0.5 fs (see fig. 7). We were able to show that not only the semiclassical time but also Buttiker’s Larmor time failed to describe the propagation effects9.

It is important to realize that these theories are not intended to describe propagation, but rather other aspects of tunneling. However, many researchers, made uncomfortable by the superluminal predictions of stationary phase, have expressed the expectation that these “interaction” times would in fact give the correct, subluminal time of arrival of a wave packet peak. Thus we did not disprove Buttiker’s and Landauer’s theories, but only demonstrated that their validity could not be extended to describe pulse propagation.

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[VL 9 4

This refined data set showed a clearly visible change of sign of the pulse shift as the barrier was tuned from a regime of superluminal transmission to a subluminal one. 4.7. THE FLORENCE GROUP, PART I1

In 1993, Ranfagni and co-workers, having become aware of the work of Ishii and Giakos, performed a new set of intriguing experiments (Ranfagni, Fabeni, Pazzi and Mugnai [1993]). They first repeated the latter’s experiments on signal propagation in waveguides above cutoff, and found no evidence for any causality violation; while the phase velocity was indeed superluminal, the “signal” (their relatively abrupt step-modulated wave) travelled at the group velocity. They subsequently studied the claim of superluminal propagation in free-space. They measured a propagation speed of c for microwaves travelling between two horns which faced one another. When the receiver was translated perpendicularly to the propagation direction, however, they confirmed the surprising result that although the distance between the horns was increasing, the delay time displayed an initial decrease. In a mathematical analysis, they argued that this effect could be understood by analyzing the diffraction of the microwave out of the square aperture of the transmitter. The receiver was observing ‘‘leaky’’evanescent waves in the shadow region of the near-field diffraction pattern. It is fascinating to note that the exponential decay of the field amplitude into this shadow region provides a qualitative analogy to tunneling. It begins to seem that exponential decay whether due to absorption, tunneling, band gaps, or diffraction - leads in general to anomalous delay times. In the simplest cases, the imaginary wave vector is understood to lead to superluminal delays because no phase is accumulated along the propagation direction; in the newer examples where it is only an envelope which decays exponentially, the superluminality was not anticipated originally. More recently, the Florentines have continued studying diffraction effects, this time using evanescent waves produced by a grating formed of metal strips (Mugnai, Ranfagni and Schulman [1997]). One of the evanescent modes was coupled through a paraffin prism onto a receiver (in analogy with the use of a second prism in frustrated total internal reflection). They have predicted that the group velocity will be superluminal in this case, as in the other examples of evanescent waves we have discussed. Experimentally, however, they were limited to measuring the phase shift at various frequencies, rather than performing a direct time measurement. They inferred the group velocity by numerically differentiating the resulting shift with respect to frequency (thus assuming the validity of the stationary-phase approximation), and the result

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they obtained suggested a time advance on the order of 50ps over a distance of 3 cm, i.e., an effective velocity of about 2c. These results, aside from being indirect, suffer from an amplification of the technical noise in the phase measurement. Ranfagni and co-workers are currently working on performing true time-dependent versions of this study. 4.8.THE COLOGNE GROUP, PART I1

In 1994, the Cologne group extended their experiments to several new and interesting cases. Unfortunately, at the same time they extended their interpretational comments (which had been somewhat vague up to that point) to what could be interpreted as a nearly direct contradiction of Einstein causality, stating for example that “the superluminal propagation of frequency-limited signals by tunneling modes is possible”. In order to sharpen up the debate over the meaning of signal propagation (somewhat clouded in much of the literature by the consideration of admittedly idealized situations involving infinitely high-frequency components and analytic wave forms), they encoded Mozart’s 40th Symphony on a microwave signal which they claimed subsequently to have transmitted at 4 . 7 ~ . Since many of these disputes frequently boil down to semantics, and since the workers involved have nonetheless found it impossible to find working definitions which removed all disagreement, it is perhaps best to quote the Cologne group directly (Heitmann and Nimtz [1994]): “The signals considered in the microwave experiments were unlimited in time and not Gaussian. Therefore Enders and Nimtz have never claimed that the front of a signal has travelled at superluminal speed. However, they have stated that the peak and the rising edge of a frequency band limited wave packet propagate faster than c through a barrier. This result corresponds to a superluminal group and signal velocity and it was recently used to transmit Mozart’s Symphony No. 40 through a tunnel of 114 mm length at a speed of 4 . 7 ~ ” . In fact, as will be seen below in our discussion of causality and superluminality (see $8), this appearance of a wave form faster than c is in itself nothing surprising. This becomes particularly clear when one considers the timescales involved. The time advance being discussed is well under 1 ns in Nimtz’s experiments. An acoustic wave form, on the other hand, has a useful bandwidth on the order of 20 kHz, which is to say that no significant deviation from a loworder Taylor expansion occurs in less than about 50ps. To predict where the wave form would be 50ps in advance requires little more than a good eye; to predict it 1 ns in advance hardly even requires a steady hand. As was already

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suggested by Chiao, Kwiat and Steinberg [I9931 and Steinberg [1994, 1995~1, and recently made more explicit by Kurizki and Japha [ 19931 and Diener [ 19961, the interference at work in tunneling has the effect of advancing the incident wave form due to the first derivative term of Taylor’s theorem l o . Hence even though the transmitted wave mimics the future behavior of the incident wave impressively well, it does so without any need for information about the later behavior of the incident field. The already existing information at any given time is more than sufficient to make an educated guess about what is to come a short time later, and a tunnel barrier does no more than act as an analog computer for this purpose. All the same, this ability (particularly when coupled with amplification, as will be discussed below) does provide an interesting way to advance the triggering of a fixed-discriminator-level detection system, and may not be without technological application. Of course, it becomes even more surprising when we are not merely arguing about the shape of a classical wave form, but the unique time of arrival (i.e., the “click”) of an individual quantum particle. Since this latter quantity is tied inextricably to interpretational issues (such as the frequently invoked “instantaneous” collapse), no solution is likely to be forthcoming soon. Leaving aside these interpretational issues for the moment, the recent series of experiments in Cologne extend the microwave work to new barriers, including an analog of the periodic-dielectric structure first studied at Berkeley. Although some of them rely again on phase measurements, and the signal-to-noise ratio remains dubious, they provide an elegant confirmation, and reach effective speeds of several times that of light. Furthermore, Nimtz and co-workers have been able to verify again the thickness-independence of the tunneling time in the opaque limit. Finally, since microwave experiments are plagued by effects of dissipation in the waveguides, they have performed interesting studies on tunneling in the presence of dissipation, which has also been analyzed in various other frameworks (Nimtz, Spieker and Brodowsky [ 19941, Mugnai, Ranfagni, Ruggeri and Agresti [ 19941, Raciti and Salesi [1994], Steinberg [ 1995b], Brodowsky, Heitmann and Nimtz [ 19961).

’”

I F destructive interference is set up between part of the wave travelling unimpeded and part which has suffered a small delay Af due to multiple reflections, one has Yout(t) = Y,”(t) - EYln(f- A t ) RZ ( I - E ) Y , , , ( t ) + EAt d Yln(t)/dt N ( I - E ) lyn(t+ EAf/(I - E ) ) , which is already a linear extrapolation into the future. In cases where the dispersion is sufficiently flat, as in a bandgap medium, the extrapolation is in fact surprisingly better than this first-order approximation. As was suggested in Steinberg [ 1995~1and recently discussed more rigorously by Lee and Lee [ 19951 and Lee [ 19961, this implies that even a simple Fabry-Perot interferometer exhibits superluminality when excited off resonance

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4.9. THE VIENNA GROUP

The Berkeley work, in which multilayer dielectric mirrors functioned as photonic band gap media and hence as effective tunnel barriers, was extended by the ultrafast laser group at the Vienna Technical University in 1994. By using 12-fs laser pulses and standard nonlinear-optical autocorrelation techniques, they benefitted from a better signal-to-noise ratio than the single-photon counting experiments, and were therefore able to study barriers of lower transmission. Of course, in so doing, they were only able to study classical electromagnetic pulses, disregarding the single-particle features, but as we have discussed, the single-photon arrival times had been seen to be quite well described by Maxwell’s equations. Since such group-delay measurements are incapable of addressing deeper issues of particle-wave duality (for these, “clocks” such as the Larmor clock to be discussed further below are essential), the sacrifice is not a great one. Spielmann, Szipocs, Stingl and Krausz [I9941 used 12-fs FWHM sechsquared optical pulses with energies of about 1 nJ at a repetition rate near 100 MHz to measure transmission times through quarter-wave stacks of 6, 10, 14, 18, and 22 layers, with transmissions ranging from 30% to 2 . (compare the 1 1-layer Berkeley structure with its 1% transmission, near the noise limit for that experiment). A freely-propagating pulse was compared with one which had to traverse the coating being studied, and the two pulses were subsequently superposed in a non-collinear geometry in a BBO crystal to generate secondharmonic light and thus a background-free cross-correlation signal. Since the required time resolution was of the order of 1 fs, while the bandwidth-limited pulses were 10 to 15 times longer, a multishot averaging technique was used. This requires extremely high stability of the pulse parameters, which the Vienna group achieved thanks to a mirror-dispersioncontrolled Tixapphire laser (Stingl, Spielmann, Krausz and Szipocs [ 19941). This laser generated bandwidth-limited pulses at 800nm, with close to 1% stability in the frequency-doubled output. They split each pulse in two parts, which were superposed in the nonlinear crystal after one part traversed the dielectric coating while the other propagated in air. The cross-correlation signal varied as a function of the degree of overlap of the two pulses in the crystal. By adjusting the path-length difference to put themselves on the edge of the output signal, and then switching the coating between the two arms of the correlator, the researchers were able to measure small shifts in the pulse position caused by the coating (Spielmann, Szipocs, Stingl and Krausz [1994]). Great care was taken to eliminate systematic errors

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due to the change in the shape of the cross-correlation signal occasioned by the insertion and displacement of the sample, due to drifts and fluctuations of pulse parameters, etc. The experimenters thus obtained results with statistical uncertainties of f 0 . 3 fs, and by studying progressively thicker samples, they were able to confirm the prediction that the time delay should saturate at a finite value even as the thickness of the sample continued to grow. For the thickest sample studied, they found an advance of about 6 fs over free propagation in air. However, their results showed a systematic deviation from the stationary-phase prediction of about one and a half femtoseconds; this discrepancy is not yet understood. They did observe that the 28-THz bandwidth pulses from their laser were somewhat distorted, at least by the 22-layer barrier, based on interferometric autocorrelation traces. The pulse width decreased from 12 fs to 6.5 fs, consistent with the effectively increased bandwidth due to the lower transmission at the center of the pulse spectrum than in the wings. Since the wings also have a longer group delay than the center frequency, it is possible that the observation of slower-than-predicted traversal is in part due to the preferential transmission of these slower components, but a full explanation has not yet been given. 4.10. DEUTSCH AND GOLUB’S LARMOR-CLOCK EXPERIMENT

Deutsch and Golub [I9961 performed an experiment to measure the Larmor tunneling time for photons. Their experiment utilizes an analogy between the spin of an electron and the spin of a photon, whose polarization state can be described by a point on the Poincari sphere given by the Stokes parameters S. The equation of motion for the Stokes parameters for a beam of light propagating along the x-axis through a medium with an anisotropic refractive index is given by dS/dx

=

QxS,

(4.3)

where D is the precession rate of the tip of the S vector on the Poincare sphere arising from the anisotropic index of refraction. This equation is formally identical to the one describing the precession of the tip of the electron spin vector u on the Bloch sphere arising from an applied magnetic field

when the optical precession rate D is identified (apart from a proportionality constant) with the rate of Larmor precession D,.

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This analogy between electron and photon spin precession led Deutsch and Golub to suggest an optical implemention of the Larmor-clock measurement of the tunneling time of Baz’ and Rybachenko (latter corrected and generalized by Biittiker). The basic idea is to replace electrons with photons, and to replace a uniform magnetic field confined to the electron tunnel barrier region with a uniform birefringent medium confined to the corresponding optical tunnel barrier. Thus, instead of utilizing the precession of the electron spin as an internal clock to measure the Larmor tunneling time, they utilized the precession of the S vector of the photon as an internal clock. In their experiment, they used frustrated total internal reflection between two glass prisms as the tunnel barrier. The gap between the prisms, which served as the tunnel barrier, was filled with a birefringent fluid (a liquid crystal). There are a number of advantages in performing an experiment using photons to measure the Larmor tunneling time. In contrast to the case of electrons, it is easy to confine the region for photon spin precession to the region of the barrier, by simply restricting the birefringent fluid to the region of the gap, whereas it is hard to confine the magnetic field to the region of the tunneling barrier for electrons. Also, since the photon is neutral, complications inherent in electron tunneling-time measurements associated with image charges induced in the faces of the tunnel barrier could be avoided. Moreover, the interaction between the photons is negligibly weak, in contrast to the strong Coulomb repulsion between the electrons inside the barrier. Exploiting these advantages, Deutsch and Golub successfully completed their experiment to measure the Larmor tunneling time, with the result that the theoretical predictions of Buttiker for the Larmor time were qualitatively confirmed. However, in a critical examination of their own experiment, Deutsch and Golub pointed out a weakness: the Larmor tunneling time is based ultimately on an arbitrary definition that is, in their words, “not a physical scale that emerges naturally, or that is needed to calculate the results of measurements”. They pointed out another possible weakness: the process as measured by the Larmor clock is a stationary one involving only a single energy or frequency of the photon. It has been argued that the tunneling time cannot have any meaning for stationary processes, which have no beginning or ending (Falck and Hauge [ 19881, Gasparian and Pollak [1993], Gasparian, Ortuiio, Ruiz, Cuevas and Pollak [ 19951, Krenzlin, Budezies and Kehr [ 19961). However, we shall see that stationary processes can in fact give indirect information on tunneling times in 2D situations, as has been demonstrated by the continuous-wave experiments of Balcou and Dutriaux (see the next section). In her PhD thesis, Deutsch gave a theoretical treatment of the nonstationary

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problem of an interacting photon system inside the tunneling barrier, interacting via a third-order nonlinear optical susceptibility confined to the region of the barrier. The basic idea is that when one photon is inside the barrier region, it causes a refractive-index change through the nonlinear susceptibility, which tends to exclude (for the repulsive sign of the nonlinearity) the presence of a second incident photon which is about to enter the barrier. The tunneling time was defined as the duration over which the second photon tends to be excluded by the first photon. Thus one could determine the tunneling time through Glauber’s two-photon correlation function, as applied to a nonlinear beam splitter used as a model for the tunneling barrier. The result of the calculation was a certain correction term in the two-photon correlation function which arose from the nonlinearity. She made an identification of the resulting tunneling time with the dwell time. However, as many workers have pointed out (Hauge and Stervneng [ 1989]), the dwell time cannot distinguish between reflected and transmitted particles, and hence cannot be regarded as a genuine tunneling time; we will see in $ 5 how one might hope to get around such objections. 4. I 1 . BALCOU AND DUTRIAUX’S FTIR EXPERIMENT

Tunneling times have been measured recently in frustrated total internal reflection (FTIR) by Balcou and Dutriaux [1997]. The idea of this beautifully simple experiment is to utilize both the lateral displacement and the angular deflection of the transmitted light beam (which is composed of the tunneling photons), as a simultaneous measurement of two different kinds of tunneling times, which turned out to be the group delay and the semiclassical time. These two tunneling times correspond to the real and imaginary parts of a complex time related closely to that of the Larmor times of eq. (2.9). In $ 5 , we shall see that it is possible to delineate clearly the physical significances for these two different times. Let us define the x-axis as the direction normal to the interface between the prisms and y-axis as the direction parallel to the interface in the plane of incidence (see fig. 8). This 2D FTIR tunneling geometry has been analyzed previously by Steinberg and Chiao [1994a] and by Lee and Lee [1997]. During the tunneling process which occurs in the x-direction, the wave packet continues to propagate in the y-direction, since its y-component of momentum is conserved. Balcou and Dutriaux argue heuristically that one expects the propagation velocity along the y-axis to be uniform during tunneling, and that, therefore, this would result in a lateral displacement D along the y-direction which would be proportional to some unknown temporal delay due to tunneling.

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Fig. 8. Schematic of Balcou and Dutriaux’s frustrated total internal reflection experiment to measure two tunneling times. These two times are inferred from the Goos-Hanchen shift D , and from the angular deflection of the transmitted beam 60, respectively.

Calculations (Ghatak, Shenoy, Goyal and Thyagarajan [ 19861, Ghatak and Banerjee [ 19891) show that after the wave packet has finished tunneling through the interface in the opaque limit, it is the group delay zg which causes the lateral displacement of the transmitted wave packet along the y-direction by an amount D = u y t g , where uv = c/n sin 0 is the y-component of the velocity of the wave packet (0 being the angle of incidence). This lateral shift of the transmitted light beam turns out to be identical to the well-studied Goos-Hanchen shift. Therefore, Balcou and Dutriaux infer that a measurement of the displacement D will lead to the tunneling time T~ = D[c/n sin

el-’.

(4.5)

In addition to this lateral displacement, there is also an angular deflection of the transmitted beam, which arises from its finite beam size. Due to diffraction, the finite width of the incident beam of light leads to some finite spread in the angles of its wave vectors. Larger angles are transmitted less than smaller angles, since they are farther away from the critical angle. This causes a preferential transmission of the smaller angle components of the incident beam, which leads to a deflection of the transmitted beam slightly towards the normal. This is analogous to the effect associated with Biittiker’s Larmor time in which there is a preferential transmission of electron spins aligned antiparallel to the magnetic field, which leads to a slight spin polarization of the transmitted beam.

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Calculations similar to those above show that this preferential transmission leads to an effective angular frequency of rotation of the beam at the rate

where WR is the beam Rayleigh length. Balcou and Dutriaux therefore infer that a measurement of the angular deflection 66’ will yield the tunneling time

where TL, the so-called “loss time”, approaches the semiclassical time of Buttiker and Landauer for opaque barriers. The two tunneling times zg and zL turn out to be identical to the real and imaginary parts of the complex tunneling time introduced by Pollak and Miller [ 19841, z,

=

zs + 1ZL = -1-

.aInt

aw

(4.8)

where t is the complex transmission coefficient of the tunnel barrier. Balcou and Dutriaux obtained experimental data which agree well with the above theory for the two tunneling times. In particular, they have demonstrated not only that the group delay saturates with increasing barrier thickness (the Hartman effect), but also that the semiclassical time increases linearly with this thickness. However, they interpreted the semiclassical time as the one “most relevant to describe the physics of tunneling”, in contrast to the group delay. They do so for two reasons. First, the semiclassical time “yields only subluminal velocities so that the causality principle is explicitly obeyed”, in contrast to the group delay, which yields superluminal velocities. Second, the group delay is dependent on the boundary conditions, and differs considerably for TM and TE polarized light, whereas the semiclassical time is independent of these boundary conditions. They argue that since a tunneling time should be independent of boundary conditions (it should depend only on what happens in the interior of the barrier), this singles out the semiclassical time as the true tunneling time. In answer to their first point, in point of fact the semiclassical time under certain circumstances can also be superluminal, a point which they failed to recognize. In the case of the 1D photonic band gap discussed earlier, the semiclassical time is zero at midgap (Martin and Landauer [ 19921, Steinberg, Kwiat and Chiao [ 1993]), which is a behavior even more superluminal than that predicted by the group delay for this kind of barrier. In answer to their second point, boundary conditions are in fact very important for tunneling. Again, in the example of evanescent waves in the 1D photonic

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band gap, it is the Bragg reflections from the periodic dielectric boundaries which give rise to the band gap, and hence tunneling. These reflections would of course vanish if there were no boundary conditions necessary for the partial reflections at the interfaces between the successive dielectrics, and tunneling would disappear. More generally, tunneling is a wave-interference phenomenon. Since boundary conditions are important for determining this interference, it is unreasonable to demand that the tunneling time be independent of boundary conditions. Hence, as should become clear in the following section, we disagree with their conclusion that it is the semiclassical time, not the group delay, that is the one “related solely to tunneling”. Rather, we believe that their results constitute experimental evidence for the simultaneous existence of these two tunneling times in the same barrier.

5

5. New Theoretical Progress

One commonly cited reason for the difficulty of defining a tunneling time unambiguously is the fact that time in quantum mechanics does not have the status of a Hermitian operator, and can thus not be measured directly. This is not an airtight objection, since most physical measurements are in fact indirect: we say we have measured the position of a particle when what we may in fact have observed is which element of a CCD array absorbed photons scattered by the particle and then focused. Even in classical mechanics, one never measures “the time of a particle”, or even “the time of an event”, but a quantity such as the angle through which a stopwatch hand rotates if it is started by the particle’s entry into a region and stopped by its exit from that region. When many different operational definitions of this sort yield the same result, we feel justified in calling the quantity we have found “the time”; if, as in the tunneling case, different measurements yield different results, we must be more cautious. In quantum mechanics, it is straightforward to define an operator 0, which is 1 if the particle is in the barrier region and 0 otherwise. Such a projection operator is Hermitian, and may correspond to a physical observable. Its expectation value simply measures the integrated probability density over the region of interest- it is this expectation value divided by the incident flux which is referred to as the dwell time. Thus the central problem is not the absence of an appropriate Hermitian operator, but rather the absence of well-defined histories (or trajectories) in standard quantum theory. For example, the dwell time measures a property of a wave function with both transmitted and reflected portions, and does not display a unique decomposition into portions

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corresponding to these individual scattering channels. Some workers calculate the expectation value not for the initial state but rather for the final state (van Tiggelen, Tip and Lagendijk [1993]). This answers the question no better than does the usual dwell time; instead of discarding information about late times, it discards information about early times. Approaches relying on projector algebra in general have been analyzed by Muga, Brouard and Sala [ 1992bI and Leavens [ 19951. Other related approaches follow phase space trajectories (Muga, Brouard and Sala [ 1992a]), Bohm trajectories (Dewdney and Hiley [ 19821, Leavens [ 1990, 19931, Leavens and Aers [ 199 I , 19931, Leavens, lannaccone and McKinnon [1995], Leavens and McKinnon [1995]), or Feynman paths (Sokolovski and Baskin [ 19871, Sokolovski and Connor [ 1990, 1993, 19941, Hanggi [ 19931, Fertig [ 1990, 19931). No consensus has been reached as to the validity and the relationship of these various approaches. ideally, transmission and reflection times t T and T R would, when weighted by the transmission and reflection probabilities Ill2 and l y I 2 , yield the dwell time Td:

this relation has served as one of the main criteria in a broad review of tunneling times (Hauge and Stsvneng [1989]), but has also been criticized (see, for example, Landauer and Martin [ 19941). However, a formalism due to Aharonov, Albert and Vaidman [1988] and Aharonov and Vaidman [ 19901 shows how to analyze “conditional measurements” in quantum mechanics; that is, how to predict outcomes of measurements not for entire ensembles, but for subensembles determined both by state preparation and by a subsequent postselection. In the case which concerns us, the state is prepared with a particle incident from the left, and selected to have a particle emerging on the right at late times. Due to the time-reversibility of the wave equation, results of intervening measurements depend both on the initial and the final state. This formalism relies only on standard quantum theory, and yields a result that is completely general for any measurement arising from a von Neumann-style measurement interaction, in the limit where the interaction strength is kept low enough to avoid irreversibly disturbing the quantum evolution. This low strength implies great measurement uncertainty on any individual shot, but an average may be calculated for a large number of data runs. We have recently shown (Steinberg [1995a,b]) how to apply this formalism to tunneling, and the time we find is identical to the complex time of Sokolovski, Baskin, and Connor, rc. But thanks to the “weak measurement” formalism, it becomes clear what the physical significance of the real and imaginary

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parts is: the real part (the in-plane Larmor time) quantifies how strongly the tunneling particle will affect a clock with which it interacts; this is the portion which corresponds to a classical measurement outcome. The imaginary part, on the other hand, describes the amount of back-action the measuring apparatus will exert on the particle (the sensitivity of the tunneling probability to small perturbations, in other words, as in Biittiker’s out-of-plane Larmor rotation). While the former effect remains constant as the measurement is made weaker and weaker, the back-action may be made arbitrarily small by resorting to extremely “gentle” (and consequently uncertain) measurements. Among other attractive properties, these conditional times automatically satisfy eq. (5.1). The generality of the times obtained in this way suggests that it may be possible to apply them to a broad variety of problems, at least approximately, even in cases where exact solution would be intractable. It has already been shown that not only are the Larmor times a clear subset of these “conditional times”, but that the counter-intuitive effects of absorption on light propagating through layered media can be understood qualitatively by application of these complex times (Steinberg [1995b]). The equivalence of ZBL and -1m tcmakes sense given that the oscillating-barrier approach in fact studies the sensitivity to perturbations in the barrier height. The direct connection to measurement outcomes lifts the ambiguity present in other “projector approaches” and the Feynman-path formalism. Finally, it is possible using these methods to calculate conditional probability distributions for transmitted or reflected particle positions as a function of time, and to directly investigate questions about whether tunneling particles spend significant lengths of time in the center of the barrier, whether only the leading edge of the wave is transmitted, etc. Since these probability distributions may have large values on both sides of the barrier simultaneously, and independent “weak measurements” can be shown to add linearly (unlike “strong” measurements of non-commuting observables), it is interesting to speculate about whether a statistical demonstration that during tunneling, a particle is “in two places at once” might be possible. Work continues on all of these issues. Extensions are also underway to analyze whether one can go a step beyond these expectation-value-like tunneling times and calculate higher moments, or entire distributions (Iannaccone [ 19961).

9

6. Tunneling in de Broglie Optics

Tunneling was, of course, discussed per se for electrons before the analogy to optical effects was drawn. However, it is an effect that is quite general to wave

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propagation. Future promising directions for studying tunneling rely on a variety of particles and barriers with their own particular advantages and difficulties. Recently, workers at Kyushu University and the Research Reactor Institute in Osaka have used a neutron spin echo instrument to measure Larmor precession (and thus Larmor times) for neutrons traversing a magnetic layer (Hino, Achiwa, Tasaki, Ebisawa, Akiyoshi and Kawai [ 19961). Preliminary results appear to agree well with theory, even near the critical angle for total reflection of the neutrons, and there is every reason to expect more interesting data to come from studies of neutron tunneling. Ballistic transport and even refraction of electrons in heterostructures has been described theoretically (Gaylord, Henderson and Glytsis [ 19931) and observed experimentally (Spector, Stormer, Baldwin, Pfeiffer and West [ 19901). It is clearly feasible to extend these geometries and observe frustrated total internal reflection of electrons. As discussed by Steinberg and Chiao [ 1994a1, there is a number of interesting similarities and differences between tunneling of massive and massless particles and between one- and two-dimensional tunneling. Future studies with ballistic electrons ought to be able to shed new light on aspects of the tunneling problem (Lee and Lee [ 19951). They will also be closer to areas which are likely to be of technological impact (Spector, Stormer, Baldwin, Pfeiffer and West [ 19901, for example, have demonstrated a new kind of electronic switch relying on electron refraction). Atoms also display wave properties. For a number of years now, atom interferometers have been in operation, and recently both Bose-Einstein condensation and a coherent pulsed output coupler for such matter waves have been observed (Anderson, Ensher, Matthews, Wieman and Cornell [ 19951, Mewes, Andrews, Kurn, Durfee, Townsend and Ketterle [ 19971, Andrews, Townsend, Miesner, Durfee, Kurn and Ketterle [1997]). The tunneling of such composite particles is in a sense even more striking than that of photons, neutrons, or electrons. The wealth of internal degrees of freedom of an atom also makes it an attractive candidate for studying a variety of “interaction times”. With the latest lasercooling and -trapping techniques, atoms may now be produced with de Broglie wavelengths significantly larger than an optical wavelength, meaning that tunnel barriers can be constructed from tightly focussed light beams, making use of the repulsive dipole force (Steinberg, Thompson, Bagnoud, Helmerson and Phillips [1996]). Auxiliary probe beams interacting with the atoms while in or near the tunnel region could be used to make the atoms fluoresce (Japha and Kurizki [1996a]), or to optically pump them, or (in order to avoid any dissipation) to induce Raman transitions. By looking at atoms transmitted through such beams, at Toronto we plan to study a number of interaction times, as well

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as their position- and spatial-dependence, as discussed by Steinberg [ 1995a,b]. Multiple simultaneous probe beams would allow one to investigate further issues of locality and the “reality of the wave function”. We are also studying the conjecture that position-dependent magnetic fields, which can rapidly tune atoms through either Raman or RF resonances (which can be extremely narrow on the scale of feasible Zeeman shifts even over length scales much smaller than an optical wavelength (Thomas [1994])), can be used to create extremely thin interaction regions which will lead to quantum reflection and tunneling once the de Broglie wavelength is longer than the interaction length (cf. Kurizki [ 19971). Such mechanisms would allow even more sensitive studies, as well as extensions to more complicated geometries, such as thin Fabry-Perot cavities for atoms. Tunneling of atoms has already been observed in a very different context. Investigating the behavior of ultracold atoms in a standing wave, Raizen’s group at the University of Texas has observed a number of fascinating effects related to the band structure of the atoms’ center-of-mass motion in a periodical optical potential, including the analog of Landau-Zener tunneling when the optical potential is accelerated fast enough that the atoms begin to tunnel to a higher band (Niu, Zhao, Georgakis and Raizen [ 19961).

8

7. Superluminality and Inverted Atoms

The fact that superluminal wave packet propagation through tunneling barriers has been observed experimentally leads naturally to the following question: Are there any other situations in physics where such superluminal behavior can arise? Of course it would be nonsensical to ask: Can light go faster than light? But it does make sense to ask the question: Can light in a medium go faster than light in the vacuum? Surprisingly, the answer to this question is “yes” in at least one instance other than in tunneling, namely, when off-resonance pulses propagate through a medium with inverted atomic populations; that is, when wave packets are tuned to a transparent spectral region outside of the gain line (Chiao [ 19961). There are two situations in which closely related superluminal propagation effects appear in media with atomic population inversion. In the first situation, a steady-state one, an index of refraction model of the medium leads to an accurate description of the behavior of the system. When a two-level system is pumped steadily so that it becomes inverted, the real part of the linear susceptibility of the inverted two-level medium suffers a sign change relative to that of an uninverted medium, leading to superluminal group velocities in transparent spectral windows far away from resonance (Chiao [ 19931). In the second situation, a transient

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one, the sudden inversion of the two-level system by a pulsed pump gives insight into the collective behavior of the system. Undamped atomic polarization waves are coupled strongly to electromagnetic waves, and this coupling leads to tachyon-like collective excitations, i.e., normal modes of the coupled atomradiation system which exhibit a tachyon-like dispersion relation near resonance (Chiao, Kozhekin and Kurizki [1996]). It should be noted at the outset that these situations will lead to superluminal propagation phenomena which are much more dramatic than those which occur in tunneling, since no appreciable attenuation or reflection of the wave packets will occur in these dilute, transparent media, and consequently the distances over which superluminal propagation occurs can be much larger than those that occur in tunneling barriers. As an example of the first, steady-state situation, we shall focus on the special case of superluminal propagation of finite-bandwidth pulses through a population-inverted medium, whose carrier frequencies are much lower than resonance. Although superluminal propagation also occurs near the resonance line”, it is much simpler to understand the very-low-frequency case first. The refractive index of a two-level medium can be obtained from the usual Lorentz model, which yields (Jackson [1975], Kittel [1986])

where y is a (small) phenomenological linewidth, ~0 is the resonance frequency of the medium, and upis “the effective plasma frequency”, a measure of the strength of the coupling between the atoms and the radiation field, which is given bY

q,= (-4nwf

N e2/ml‘/2.

(7.2)

The Lorentz model has been generalized to include the possibility of population inversion, based on the density-matrix equations of motion for the two-level atom (Boyd [ 1992]), by introducing into eq. (7.2) the fractional atomic population inversion w, which is given by

Nu being the number density of atoms in the upper level, N, being the number density of atoms in the lower level, and N = Nu + N, being the total number ” An experiment is presently being performed at Berkeley using the stimulated Raman effect in rubidium vapor to demonstrate these resonantly enhanced superluminal group velocities (Chiao [1994], Chiao, Bolda, Bowie, Boyce, Garrison and Mitchell [ 19951).

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Fig. 9. Real part of the refractive index versus frequency for a completely inverted two-level atomic medium (solid line for w = + I ) , compared with that for the same medium with completely uninverted populations (dashed line for w = - I ) .

density of atoms in the two-level system. As usual, e is the electron charge, and m is the electron mass. The single-atom oscillator strength of the transition between these two levels is given by

where E, and El are the energies of the upper and lower states of the atom, respectively, and (u(xjl) is the transition matrix element between these two states. In the special case when all the atoms are in the lower level (w = -I), the effective plasma frequency is real, but when there is complete population inversion and all the atoms are in the upper level (w= +I), the effective plasma frequency becomes imaginary. When one completely inverts the system, the inversion process can be thought of as an interchange of the two energy levels of the atom E, and El, thus leading effectively to a sign change in the oscillator -f upon a complete strength given by eq. (7.4). Thus for each atom, f inversion of the system. Now let us consider the typical situation in which the inequalities y << up<< wo are obeyed. A plot of the real part of eq. (7.1) is shown in fig. 9. The extreme case of w = -1, with all the atoms in the lower level, where there is maximum absorption, is represented by the dashed line, and the opposite extreme case of w = + I , with all the atoms in the upper level, where there is maximum gain, is represented by the solid line. Note that the nature of the dispersion has been --f

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TUNNELING TIMES A N D SUPERLUMINALITY

PI, 4 7

reversed for these two cases - regions of normal dispersion are interchanged with regions of anomalous dispersion upon the inversion of atomic populations, and vice versa. Also note that the sign in front of the second term under the square root in eq. (7.1) reverses upon population inversion - it is positive for the uninverted medium, but it becomes negative for the inverted medium. The physical meaning of this second term is that it represents the complex, frequencydependent susceptibility of the medium (apart from a constant of 4 ~ ) .This complex susceptibility reverses sign upon an inversion of population. Hence the imaginary part of the susceptibility reverses sign, which indicates the passage of the system from absorption into amplification. The real part of the susceptibility also reverses sign (Chiao and Boyce [ 1994]), which indicates the passage of the system from subluminality into superluminality in transparent spectral regions far away from resonance. In particular, as a result of this sign change, the index of refraction near zero frequency passes from a value greater than unity, over to a value less than unity given by

n(0) = ( 1

-

lup12/u~2)”* <1

(7.5)

This result is valid whenever a strong, low-frequency resonance dominates the zero-frequency sum rule, e.g., when there exists an inverted population in the 24 GHz ammonia resonance used in the first maser (Chiao [ 19961). From eq. (7.1) it also follows that the slope d[Ren(w)]/dw approaches zero as (LI + 0. Since the resulting group velocity dispersion vanishes near zero frequency, the medium is essentially dispersionless near DC (see fig. 9), a fact which is true for both the inverted and the uninverted media. Now consider the propagation of a classical, finite-bandwidth pulse, for example, a Gaussian wave packet, whose carrier frequency and spectrum lie far below the resonance frequency of the two-level atom. Let this wave packet be incident upon a population-inverted medium. The amplitude of this wave packet will be chosen sufficiently small so that only the linear response of the medium to this weak perturbation need be considered. The fact that the index n(0) < 1 is less than unity means that the phase velocity

up(0) = c/n(0) > c

(7.6)

is greater than the vacuum speed of light c. It is well known that the phase velocity can exceed c without any violation of special relativity. (The phase velocity, which is the velocity of the zero-crossings of the carrier wave, characterizes the motion of a pattern which carries no information with it.)

VI,

P 71

SUPERLUMINALITY AND INVERTED ATOMS

39 I

More surprisingly, here as zero frequency is approached, the group velocity 'g(O)

=

(

dRek(o) do

)-'

0-0

-I

is equal to the phase velocity, and is therefore also superluminal: The group oelocity also exceeds the oacuum speed of light. Furthermore, there is negligible distortion of the pulse during its propagation, as the group velocity dispersion vanishes at low frequencies. Conventional wisdom tells us that the group velocity, which is the velocity of the peak of the pulse, is the true signal velocity, in contradistinction to the phase velocity, since normally energy transport is characterized by the group and not the phase velocity. If we were to cling to this definition of signal velocity, then we would be forced to accept signal velocities faster than light. However, special relativity is in fact not violated by these superluminal group velocities, as we shall see in the next section. Unlike a medium in its ground state, the inverted medium can temporarily loan part of its stored energy to the forward tail of the wave packet, in a pulsereshaping process which moves the peak of the wave packet forward in time. One can think of this pulse-reshaping process as the virtual amplification of the forward tail of the wave packet, followed by the virtual absorption of the peak, resulting in an adoancement of the wave packet. This is a reversal of the pulse-reshaping process produced by the uninverted medium, in which the peak of a wave packet first undergoes virtual absorption, followed by the virtual amplification of its trailing tail, resulting in a retardation of the wave packet. Energy is loaned by the medium to the wave, or vice versa, in the inverted and the uninverted cases, respectively, so that the energy in the pulse remains unchanged in both kinds of pulse-reshaping processes in these transparent media. Thus the energy velocity, as defined by Sommerfeld and Brillouin (Brillouin [1960]), is also superluminal for the inverted medium near zero frequency

where ( S ) is the time-averaged Poynting vector, ( u ) is the time-averaged energy ) the zero-frequency dielectric constant. This is a reversal density, and ~ ( 0 is

392

TUNNELING TIMES AND SUPEKLUMINALITY

[VL

P7

of the case of the uninverted medium, where the energy velocity is of course subluminal. The Sommerfeld and Brillouin energy velocity is usually interpreted as the velocity of energy transport by the propagating wave packet. However, there is controversy concerning the proper definition of the energy velocity (Schulz-DuBois [1969], Loudon [1970], Oughstun and Shen [ 19881, Diener [ 19971); after all, in addition to the purely electromagnetic energy density, there is energy stored in the inverted medium itself. Still more surprisingly, the “signal” velocity of Sommerfeld and Brillouin, which they defined arbitrarily as the propagation velocity of the first point of half-maximum wave amplitude, is the same here as the group velocity, since there is little distortion of the shape of the wave packet during its propagation. However, we shall see that it is highly misleading to call this the “signal” velocity. Since dispersion is negligible in this large, transparent spectral window stretching from DC to the low-frequency side of resonance, all of the above wave velocities, including the so-called “signal” velocity, are faster than c. It should be emphasized that any arbitrary, low-frequency finite-bandwidth wave form, e.g., Rachmaninov’s 3rd Piano Concerto, and not merely Gaussian wave packets, will propagate faster than c with negligible distortion, so that a complicated wave form can also be advanced to earlier times at the output face of the inverted medium. Recently, some of these counterintuitive effects have been observed in an experiment with very low frequency bandpass electronic amplifiers (Mitchell and Chiao [ 19971).Negative group delays were observed, in which pulses transmitted through a chain of amplifiers were aduunced with little distortion by several milliseconds, i.e., the transmitted peak left the output port of the amplifier chain before the incident peak arrived at the input port. Howcver, the behavior of abrupt “fronts” and “backs” showed that causality was in fact not violated. The nervous reader may ask at this point how it is possible to avoid a violation of special relativity. A brief answer is that the front velocity of Sommerfeld and Brillouin in the case of a medium with inverted populations is still exactly c, as it is also in the case of tunneling. This will be shown in detail in the next section. We shall further see that the front velocity, and not the so-called “signal” velocity of Sommerfeld and Brillouin, should be identified as the true signal velocity, and this fact will prohibit any genuine information from being communicated faster than c. The reader may also object to our use of the Lorentz model, which after all is merely a model. However, the above results can also be shown to follow very generally from the Kramers-Kronig relations, which are themselves consequences of causality and linearity. These results must therefore transcend all models (Chiao [1993]. In general, the Kramers-Kronig relations (i.e., the very

SUPERLUMINALITY A N D INVERTED ATOMS

+

INPUT

393

OUTPUT+

........

Fig. 10. A linear array of undamped Lorentz oscillators l o r calculating the polanton-like (for uninverted atoms) and the tachyon-like dispersion relations (for invertcd atoms), for a strongly coupled atom-radiation system.

requirement of causality itself) demand that superluminal group velocities arise in any dispersive medium (Bolda, Chiao and Garrison [1993]); in particular, they must arise in any medium with gain. As an example of thc second, transient situation, we shall focus on the special case of tachyon-like propagation of wave packets through a population-inverted medium at frequencies close to resonance. Although the theory for the tachyonlike excitations of this medium was originally worked out starting from the sineGordon equation for the fully nonlinear problem of the coupling between the two-level atoms and the radiation field (Chiao, Kozhekin and Kurizki [ 1996]), we present here a simplified, linearized version of this theory, which brings out more directly the essential features. Our goal is to calculate the dispersion relations for small-amplitude excitations of the strongly coupled atom-field medium, and show that tachyon-like excitations emerge naturally as the normal modes of an undamped medium composed of atoms with suddenly inverted populations. Consider a long collection of Lorentz oscillators with a uniform density along the z-axis (see fig. 10). (There are no mirrors at the ends of this medium.) We shall focus on the special case of undamped motions of these oscillators. Such a system is a good model for two-level atoms in their ground states (Burnham and Chiao [1969]), but can be generalized easily to the case of atoms with inverted populations (see eq. 7.2). The two equations which describe the coupled atomradiation system are (i) Maxwell’s equations in the form of the wave equation

and (ii) the undamped simple harmonic equation of motion for the Lorentz oscillators, d2X ~

d t2

+ w;x

=

eE m

-.

(7.10)

394

TUNNELING TIMES A N D SUPERLUMINALITY

[VL

a7

Here P = Nex is the polarization of the medium ( N being the number density of Lorentz oscillators and x being the displacement from equilibrium of a given oscillator), and m is the natural resonance frequency of the oscillators. In order to calculate how a wave packet will propagate through this system, we shall use the slowly-varying envelope ansatz (SVEA)

E

=

€(z, t ) exp[i(koz - m t ) ] ,

P

=

P(z, t ) exp[i(koz - coot)],

x = x(z, t ) exp[i(koz - coot)],

(7.1 1 )

where &(z, t), P(z, t ) , x(z, t ) are all slowly varying envelopes which modulate the common, fast plane-wave factor, exp[i(koz - (%t)],and where by definition, ko = m/c is the vacuum wave number of the uncoupled waves. Neglecting the second derivatives of the slowly-varying amplitudes, we obtain two first-order partial differential equations (PDE’s): oo 4nm2Ne (7.12) 2iko- +2i-= X d Z c2 a t C2 e& (7.13) m which are the linearized Maxwell-Bloch equations. Taking the partial derivative with respect to time of the first of these equations, and eliminating d x / d t by means of the second equation, we obtain a PDE for the electric field envelope:

a&

a&

d2& 1 d2& 1 w2 p& = 0. (7.14) azat cat2 4 c To include the possibility of population inversion, we use the effective plasma frequency up given by eq. (7.2) 1 2 . In order to find the dispersion relations, we substitute into this PDE the plane-wave ansatz -+--++-

€

=

A exp[i(Kz

-

a t ) ],

where K = k - and 52 (quadratic) equation Q2

-KcQ-

fa+,’

=

(7.15)

= o w; this converts eq. (7.14) into the algebraic -

(7.16)

0.

The solution of this quadratic equation yields the dispersion relations

52

I /2

=

~ Kf C ( K 2 c 2+ wp2) ,

(7.17)

which are plotted in fig. 1 1 . In the case of uninverted atoms (w = - I ) , a+, is real, and we recover polariton-like dispersion relations, whereas in the case of inverted

’*

The definition of the effective plasma frequency used here differs from that used in Chiao, Kozhekin and Kurizki [I9961 in that the factor of (-w)there has been absorbed into fop2 here.

VI,

71

SUPERL.UMINAL1TY AND INVERTED ATOMS

395

Fig. I I . Dispersion relations for the coupled atom-radiation system as calculated from the undamped Lorentz model for uninverted atoms (dashed curve), corresponding to polaritonic branches with w = -1, and for inverted atoms (solid curve), corresponding to tachyonic branches with w = +I.

atoms (w = +I), cup is imaginary, and we find tachyon-like dispersion relations. The tachyonic branches have group velocities which are always faster than c (but which approach c far from resonance), infinite at the turning points A+ and A _ , or negative as resonance is approached. Computer simulations indicate that negative group velocities in gain media also have a well-defined physical meaning (Bolda, Garrison and Chiao [ 19941). However, under no circumstances can these tachyonic excitations outrace the front (Aharonov, Komar and Susskind [ 19691, Chiao, Kozhekin and Kurizki [1996]). The wave-number gap between A , and A - is a gap of instability arising from population inversion. Vacuum fluctuations with frequency components inside this gap can trigger spontaneous emission, and hence superfluorescence. However, spontaneous emission does not prevent superluminality. It has been shown that the typical delay time for the onset of superfluorescence in realistic media (Bolda [ 19961) is much longer than the passage time for a typical tachyonic excitation, so that the population inversion does not disappear due to the emission of a superfluorescent pulse before the tachyonic excitation has had a chance to finish propagating through the medium. This should make experiments to observe tachyon-like excitations possible, and an experiment has been commenced at Berkeley to demonstrate the existence of these excitations in ammonia gas pumped by a carbon dioxide laser, on the same transition used in the first maser by Gordon, Zeiger and Townes [ 19541. It has also been shown that the effective plasma frequency is directly proportional to the effective mass of the corresponding collective excitation; hence a polariton-like excitation possesses a real effective mass, but a tachyon-

396

TUNNELING TIMES AND SUPERLUMINALITY

W, Z

8

like excitation possesses an effective mass which is imaginary, which is the basis for calling them “tachyonic” (Chiao, Kozhekin and Kurizki [ 19961). However, it should be emphasized that these tachyonic excitations should be viewed as quasiparticles in a medium, like phonons, and not as true particles in the vacuum, like photons.

8

8. Why Is Einstein Causality Not Violated?

The question naturally arises whether Einstein causality is or is not violated by the superluminal behavior exhibited in tunneling or in population-inverted media. In the case of tunneling, numerous theoretical analyses have shown that there is in fact no contradiction with causality (see, for example, Deutch and Low [ 19931, Hass and Busch [1994], Azbel [1994], Wang and Zhang [I9951 and Japha and Kurizki [1996b]). Let us first make some qualitative remarks concerning this question, and then return to some more rigorous, quantitative considerations. We shall restrict our attention here to classical electromagnetic signals, for example, voltage wave forms displayed on an oscilloscope. Also, we shall assume the total absence of noise in the following section. However, the fundamental considerations of causality given below for classical electromagnetism should be generalizable to quantum field theories (Eberhard and Ross [ 19881). The qualitative discussion starts with the observation that there is no information contained in the peak of an analytic wave packet which is not already present in its forward tail. For example, the behavior near a peak of an analytic wave form, e.g., of a Gaussian wave packet, could have been predicted by Taylor’s theorem from the earlier behavior of its forward exponential tail (i.e., using the knowledge of all the derivatives of the earlicr portions of the wave form, we could extrapolate to all later portions; in particular, we could in principle predict the exact moment of arrival for the peak of the wave form 1 3 ) . Therefore there is no real surprise when the peak eventually arrives. New information is communicated only when there is an unexpected change, such as a discontinuity, whose arrival time cannot be inferred from the past behavior of the wave.

l 3 Pulse reshaping mechanisms, such as the virtual amplification of the forward tail followed by the virtual absorption of the peak of the Gaussian wave packet which reproduces the shape of this wave packet, can therefore advance the peak forward in time in a completely predictable and causal manner.

VL 5 81

WHY IS EINSTEIN CAUSALITY NOT VIOLATED!

397

A simple example of such a discontinuity is that of a step-modulated sine wave, i.e., a jump discontinuity or “front”, which Sommerfeld and Brillouin used in their study of precursors. Their wave form thus has a sharp jump from zero to finite intensity at the front. They found that no features of their solution, including their precursors, could ever overtake this front. In contrast to the peak of the Gaussian wave packet, the arrival of the front could never have been predicted from any prior information, and hence the front in this example constitutes a genuine signal, i.e., new information. However, any point of nonanalyticity in a wave form, such as a jump discontinuity in some higher derivative, and not just a jump discontinuity in the wave amplitude such as the front of Sommerfeld and Brillouin, can serve as a carrier of genuinely new information. Any such point of nonanalyticity is always preserved upon transmission by any linear, causal system, as we shall demonstrate below. Nonunabtic wave forms, for example, piecewise analytic functions joined smoothly at given points of nonanalyticity, have Fourier components which fall off algebraically in the high-frequency limit (the higher the order of the derivative jump, the larger the negative exponent of the frequency in this fall-off). It is the response in the infnite-frequency limit of the system that ultimately determines the propagation speed of the points of nonanalyticity, and hence of truly new information. Since the propagation of infinite-frequency components of a disturbance occurs at the vacuum speed of light, i.e., at Sommerfeld and Brillouin’s front velocity, this is also the velocity of propagation of the points of nonanalyticity, and hence of genuine information. It is fundamentally for this reason that Einstein causality cannot be violated under any circumstances, either in the tunneling barrier or in population-inverted media. The rigorous, quantitative considerations start with a “black box” which locally relates an input to an output wave form by means of a linear transfer function T(t),via the equation

where t is a delay time, & ( t ) is an arbitrary input function, and fout(t) is the resulting output function. For example, the inputf;,(t) could represent an electric field applied to an atom, whose polarizability would be represented by T ( z ) , and the outputh,,(t) would represent the dipole moment response of the atom produced by the electric field. It should be stressed that & ( t ) and fout(t) can represent any of the higher derivatives of the wave form, as well as the wave form itself. This follows directly from the linearity of eq. (8.1).

398

TUNNELING TIMES AND SUPERLUMINALITY

PI,S; 8

The principle of causality demands that the integrand must vanish for z < 0 in eq. (8. I), since any effect (e.g., the atomic dipole moment) must not precede its cause (e.g., the applied electric field). This necessitates that T ( t ) = 0 for all t < 0 .

(8.2)

When eq. (8.1) is Fourier tranformed into the frequency domain, it becomes

fo”t(W) =

mJ%4 7

(8.3)

where the tildes denote Fourier transforms. The complex frequency transfer function T(w), as a consequence of eq. (8.2), must satisfy the condition that

-

T ( w )is analytic for all Im w > 0 ,

(8.4)

i .e., the complex frequency transfer function must be analytic in the upper half frequency plane (UHP), which is an expression of causality equivalent to eq. (8.2). This leads to the Kramers-Kronig relations for T ( w ) (Landau and Lifshitz [1960]). Now suppose that the functionf;,(t) has a front in it at the time to, so that

Jn(t)

=

0 for all t < to .

(8.5)

Then the Fourier transform of this function must satisfy the condition that f;n(w) is analytic for all Im o > o ,

(8.6)

i.e., the Fourier transform of the input function must be analytic in the UHP. Since each of its factors are analytic in the UHP, it follows that the product

-

h u t ( w ) = ;ir(w)f;n(w)is analytic for all Im w > o ,

(8.7)

i.e., the Fourier transform of the output function must also be analytic in the UHP. Therefore using the inverse Fourier transform, we obtain the result

where it can be shown that tE, = to for any “black box” that has a negligible spatial extent. This proves that fronts in the input survive the transfer through any “black box” which is linear and causal: Fronts are preserued in the output. Therefore, although there is no physical law which guarantees that an incoming

VL 9 81

WHY IS EINSTEIN CAUSALITY NOT VIOLATED’?

399

peak turns into an outgoing peak, there is a physical law namely causality, that guarantees that an incoming front turns into an outgoing front, even when the front carries little energy or probability. Using linearity, we can generalize this result to any point of nonanalyticity, for example, a jump discontinuity in some higher derivative of the wave form. Using the superposition principle, which also follows from the linearity of the system, we can further generalize this to all the points of nonanalyticity to, t l , t 2 , . . . in the wave form. Motivated by these considerations, we shall define a signal as the complete set of all the points of nonanalyticity {to, t l , t 2 , . . .}, together with the values of the input functionJn(t) in a small but finite interval of time inside the domain of analyticity immediately following these points. It should be emphasized that this definition leads to a signal velocity that differs from the conventional one given by the group velocity. The principle of causality makes this new definition necessary. However, we are making idealizations, in particular, in assuming the highest possible detector sensitivity and the perfect noiselessness of the system, in formulating this fundamental definition, but this may not be a practical definition under all circumstances. The generalization of this argument to propagation through any spatially extended “black box” that is linear and causal, is straightforward (Jackson [1975]). For an input with a single point of nonanalyticity at to given by

&(t)

=

0 for all t < to ,

(8.9)

the output must satisfy the condition that fout(t) = 0 for all t - d/c < to ,

(8.10)

where d is the distance from the input face to the output face of the “black box”. Using the definition given above, we conclude that genuine signals cannot propagate faster than c. In fact they propagate exactly at c, i.e., at the front velocity. Thus Einstein causality, i.e., special relativity, is not violated. Although at a fundamental level no genuine signal can be transmitted faster than light, at a practical level there are situations in which useful temporal advances of a wave form are possible. For example, unwanted positive group delays arising from normal dielectric media in the system may be compensated by negative group delays, but only up to the limit permitted by Einstein causality (Chiao, Boyce and Garrison [ 19951, Steinberg and Chiao [ 1994b1). In another example, a detector followed by a discriminator with a fixed trigger level can register the arrival of a pulse earlier with the aid of an amplifier than without

400

‘TUNNELING TIMES AND SUPERLUMINALITY

PI,5 9

it, but again only up to the Einsteinian limit (Chiao [1996], Mitchell and Chiao [ 19971). The meaning of superluminal group velocities was also considered recently by Diener [ 19961. He also concluded that superluminal group velocities cannot be interpreted as a velocity of information transfer. The method he used to reach this conclusion was different, being based on the Green’s function and its application to the analytic continuation of the pulse shape using information only within the past light cone. However, Diener continued to interpret subluminal group velocities as signal velocities, whereas we believe that the same definition for “signal” should in principle be consistently applied to both superluminal and subluminal cases.

9

9. Conclusion

We thus see that a relatively old debate over how long the tunneling process takes has begun to shed new light on a variety of issues, in no small part thanks to the realization that the analogy between electromagnetic and Schrodinger wave equations permits the same phenomenon to be studied in optics rather than in the solid state. We are developing a new understanding of the limits imposed by causality on various propagation speeds, and have relearned that a group velocity, and even the motion of a real, well-behaved wave packet peak, can in fact be greater than c. We see also that time in quantum mechanics is not a simple issue: a given process may have not a single duration, but a set of different timescales describing its various aspects. When the problem is studied in the light of particle-wave duality, where the actual time of arrival of individual quanta is on average earlier than what would be expected from a nai’ve application of causality principles, we come up against one of the central problems of quantum mechanics -the extent to which one can discuss quantities which have not been measured directly, such as the past history of a particle we observe at the present time. This applies to single-photon wave packet propagation both in tunneling and in gain media. In the case of tunneling, there is no clear way to separate “to-be-transmitted” and “to-be-reflected” portions, nor to answer the question of where a particle is save in a probabilistic manner. Yet a quantum particle may be forced eventually into a purely transmitted or reflected state, and the question of how much effect it has had on devices placed in its path (or how much effect thcy have had on it) is certainly a reasonable, and an important, one to ask. The superluminality of the tunneling process should also be a relevant consideration in fundamental questions concerning the nature of Hawking

VI1

REFERENCES

40 1

radiation from an evaporated black hole, and of similar radiative processes which involve the tunneling of particles through an event horizon (Massar and Parentani [ 19971). Closely related are the questions raised here: what constitutes a signal, i.e., what is information at the quantum level? Aside from their fundamental interest, the answers to such questions are crucial for responding to questions such as what the maximum speed of a tunneling device might be. Work continues on these issues at both the experimental and the theoretical level, and in both arenas, optical versions of tunneling and other superluminal phenomena have been and will continue to be of great value to the debate. Not only should we expect this work to teach us more about the fundamental nature of the tunneling process, but about some of the deepest mysteries of quantum mechanics.

Acknowledgments

We thank Jack Boyce, John Garrison, Eivind Hauge, Rolf Landauer, John McGuire, and Morgan Mitchell for helpful discussions, and especially Jack Boyce for his help in producing the manuscript. This work was supported by the O N R under Grant No. N000149610034.

Note added in proof

After this review was written, another review on a similar subject was published by Nimtz and Heitmann [1997](Prog. Quantum Electron. 21, 81). These authors deny the central significance of the front velocity for signals. For the reasons given in 5 8, we believe that their point of view is fundamentally incorrect.

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AUTHOR INDEX FOR VOLUME XXXVII

A Aakjer, T. 238 Abe, S. 33 Ablowitz, M.J. 237, 238 Aceves, A.B. 207, 209 Achiwa, N. 386 Adolph, B. 298 Aers, G.C. 351, 358, 384 Afanasjev, VV 209, 217, 221 Agarwal, G.S. 49, 50, 265, 267, 295, 327, 328, 337 Agranovich, VM. 263, 313, 314 Agrawal, G.P. 187, 189, 196, 211, 212, 214, 216-219, 222-224, 226, 234 Agresti, A. 368, 376 Agudin, J.L. 67 Aharonov, Y 384, 395 Ahrenkiel, R.K. 86 Ainslie, B.J. 220 Akhmanov, S.A. 45 Akiba, S. 209, 211, 229, 230 Akiyoshi, T. 386 Albert, D.Z. 384 Alda, J. 15, 17, 40, 41 Aleksandrov, VV 123-1 25, 131, 132 Alexander, R. 68 Allan, M.P. 169 Allen, L. 302 Alms, G.R. 161, 162, 172 Alonso, J.A. 272 Alouini, M . 3 . 230 Alpert, S.S. 136 Altarelli, M. 61, 87 Al’tshuler, S.A. 129, 130 Amrein, W.O. 327 Anan’ev, Yu.A. 18 Andersen, H.C. I69 Andersen, T. 265, 332

Anderson, M.H. 386 Anderson, P.W. 265 Ando, T. 263, 273 Andreani, L.C. 263 Andrew, A.F. 178 Andreeva, T.L. 118, 119, 124 Andrekson, P.A. 214, 220 Andrews, M.R. 386 Andrieux, J.C. 53 Andronov, A.A. 122 Anisimov, M.A. 140 Anthony, C. 263 Antos, A.J. 220 Arai, M. 128 Arakawa, Y. 298 Amoldus, H.F. 265, 327 Artigas, D. 42 Asakura, T. 12, 60, 62, 73, 77, 78, 80-82, 91 Ash, E.A. 326 Ashcroft, N.W. 264, 298, 361 Aspnes, D.E. 60 Atkins, PW. 269 Atkinson, D. 2 17 Aubin, G. 208, 210 Audouin, 0. 191, 2 10, 233, 240 Auld, B.A. 46, 47 Azbel. M.Ya. 396 B Babiker, M. 269 Bagchi, A. 264 Bagini, V 34 Bagnoud, V 386 Bakshi, F! 299 Balbis, L.C. 272 Balcou, Pb. 380 Baldwin, K.W. 386 Bamler, R. 12

407

408

AUTHOR INDEX FOR VOLUME XXXVII

Banerjee, S. 381 Banyai, W.C. 47 Baratoff, A. 353 Barlow, A.J. 174 Barnett, S. 1 1 Barnett, S.M. 371, 372 Barrera, R.G. 264 Barrett, H.H. 13 Barshan, B. 33 Bartelt, H.O. 12 Barthelemy, A. 245 Barut, A.O. 261 Banvick, J. 261 Baskin, L.M. 354, 355, 357, 358, 384 Bassani, E 61, 62, 70, 91, 263 Bastiaans, M.J. 7, 9-12, 17, 18 Basu, R.S. 140, 141 Bauer, D.R. 161, 162, 172 Baz', A.I. 356 Bechstedt, F. 298 Beck, D.E. 264, 272, 289 Beck, M. 45, 47, 48, 51 Bekki, N. 217 Bekshaev, A.Ya. 18 Belanger, N. 215 Belanger, P.-A. 40, 41, 215, 229 Bell, R.J. 68 Bellequie, L. 299 Ben-Aryeh, Y. 302 Ben-Reuven, A. 169 Benedek, G.B. 118 Benner, A.F. 237 Bennion, I. 217, 229 Bergano, N.S. 241 Bernabeu, E. 15, 17, 40, 41 Berne, B.J. 97, 165, 172 Berntsen, S. 328 Berriel-Valdos, L.R. I3 Bertilsson, K. 220 Bertolotti, M. 97 Bertrand, P. 53 Bescos, J. 13 Bethe, H.A. 326 Bezot, P. 162, 169, 175 Bhattacharjee, J.K. 140, 141 Bialynicki-Birula, I. 50 Bialynicki-Birula, Z. 50 Bigo, S. 210 Biotteau, B. 191 Birman, J.L. 267

Bjarklev, A. 220 Blaive, B. 261 Bloembergen, N. 91 Bloom, D.M. 46, 47 Bloss, W.L. 263 Blow, K.J. 210, 211, 229, 230 Boardman, A.D. 263, 313 Bochove, E.J. 265, 327 Bogatyjov, V.A. 222, 23 1 Bohm, D. 261, 350, 358 Bohmer, B. 53 Bohren, C.F. 60, 326 Bolda, E.L. 388, 393, 395 Boley, C.D. 118 Bondarenko, V; 263 Boon, J.P. 97, 140 Born, M. 11, 24, 59, 286, 367 Borovik-Romanov, A S . 97 Boudet, R. 261 Boujou, X. 263, 302 Bourkoff, E. 245-247 Bowden, C.M. 302, 303 Bowie, J. 388 Boyce, J. 388, 390, 399 Boyd, R. 222 Boyd, R.W. 203, 303, 388 Bozhevolnyi, E.A. 328 Bozhevolnyi, S.I. 263-265, 298, 299, 302, 328, 331 Brauman, J.I. 161, 162, 172 Bremmer, H. 7 Brenner, K.H. 12 Brillouin, L. 104, 351, 366, 391 Brodowsky, H.M. 376 Broido, D.A. 299 Brouard, S. 384 Brun, E. 191 Brus, L.E. 298 Brya, W.J. 130 Bryan, R.K. 73 Bubnov, M.M. 222, 231 Buckland, E.L. 203 Budezies, J. 379 Bugnolo, D.S. 7 Bullough, R.K. 267 Burg, J.P. 74 Burnham, D.C. 393 Burstyn, H.C. 140 Busch, P. 396 Biittiker, M. 352, 355-357

AUTHOR INDEX FOR VOLUME XXXVll

C Caballero, P. 247 Cabannes, J. 102 Cahil, K.E. 49 Cannell, D.S. 141 Cardona, M. 97 Carminati, R. 263 Carniglia, C.K. 364 Carter, S.J. 38 Cartwright, N.D. 53 Caspers, W.L. 61, 69 Cazabat-Longequeue, A.M. 118 Cerullo, G. 41 Chaban, I.A. 121, 174, 177, 178 Chaikov, L.L. 136-138, 141-149 Chakravarty, S. 238 Chalyii, A.V. 135 Chamberlin, R.P. 222, 247 Chan, A.K. 42 Chan, D.A.S. 71 Chandrakumar, V. 230 Chandrasekharan, V: 107 Chapell, P.J. 169 Charbonnier, B. 232 Chechetkina, E.A. 122 Chemla, D.S. 298 Chen, S.H. 140 Chen, X. 263, 313, 314 Chen, Y. 42 Chernyshova, E.O. 161 Chi, S. 209, 222, 230 Chiao, R.Y. 348, 356, 359, 363, 365, 370373, 376, 380, 382, 386-388, 390, 392-396, 399, 400 Chin, M.-K. 230 Chirkin, A.S. 45 Cho, K. 262, 264, 299 Chu, B. 97 Chu, P.L. 206 Chu, S. 366 Chuang, S.L. 263 Chui, C.K. 42 Citrovsky, A. 141, 142 Claasen, T.A.C.M. 6, 7, 9, 15 Clark, N.A. 141 Clouter, M.J. 128, 129 Coblentz, D. 214 Cohen, E.G.D. 128 Cohen, L. 3, 6, 49, 50 Cohen-Tannoudji, C. 261, 266, 269, 292

Collins, D.M. 73 Condon, E.U. 350 Conner, M. 12 Connor, J.N.L. 354, 355, 357, 358, 384 Constantine, PD. 220 Cook, R.J. 265, 327 Cooper, J. 51 Cornell, E.A. 386 Cottam, M.G. 97 Coutjon, D. 262, 263, 302, 326 Crisp, M.D. 261, 295 Crivellari, M. 243 Crosignani, B. 97 Cuevas, E. 379 Cummins, H.Z. 113, 136, 140, 175 Cundiff, S.T. 263 D Dahl, D.A. 263 Daniel, G.J. 73 Darnhofer, T. 299 Das Sarma, S. 263 Dasgupta, B.B. 264 Davey, S.T. 220 Davidovich, L.A. 141 De Alfaro, V. 86 De Angelis, C. 209, 242 De Goede, J. 267 De Groot, S.R. 8, 260 De Silvestri, S. 41 Debye, P. 111 Dembovskii, S.A. 122 Denk, W. 262 Dereux, A. 262, 263, 302 Desem, C. 206 Desurvire, E. 210, 240 Deutch, J.M. 352, 396 Deutsch, I.H. 369, 370 Deutsch, M. 378 Devaux, F. 2 10 Devlin, G.E. 130 Devoret, M.H. 353 Dewdney, C. 358, 359, 384 Dexter, D.L. 61, 86, 87 Dianov, E.M. 203, 204, 222, 231 Diener, G. 376, 392, 400 Dil, J.G. 97 Dios, F. 42 DiPorto, P. 97 Diu, B. 292

409

410

AUTHOR INDEX FOR VOLUME XXXVII

Dobson, J.F. 261, 272, 289 Dong, B.Z. 12 Dong, L. 222, 247 Doran, N.J. 207, 211, 217, 229-231 Dorfman, J.R. 128 Dorfmiiller, T. 175 Douglas, A.E. 1 I 9 Dowling, J.P. 261, 303 Drabovich, K.N. 45 Dragoman, D. I I , 12, 16-19, 23, 24, 26, 30, 33, 34, 36, 39, 41, 44, 4 6 4 8 Dragoman, M. 4 6 4 8 Du, M. 42 Dufour, C. 122 Duguay, M.A. 155 Dulong, I? 98 Dung, J.-C. 209 Dupont-Roc, J. 261, 266, 269 Durfee, D.S. 386 Diirr, D. 359 Dutriaux, L. 380 Dyakonov, A.M. 131, 132

E Eastman, D.P. 118 Easton, R.L. 13 Eberhard, I?H. 396 Eberly, J.H. 302 Ebisawa, T. 386 Edagawa, N. 209, 21 1, 229, 230 Efros, A.L. 298 Efros, ALL. 298 Ehrenreich, H. 60 Eichmann, G. 12 Einstein, A. 98, 100, 101 Eisenbud, L. 350 Ekardt, W. 264, 272, 289 Ellis, A.D. 21 1, 222, 227 Emplit, Ph. 244-246 Enders, A. 368, 369 Englert, B.-G. 359 Englund, J.C. 302 Enright, G . 163, 164 Ensher, J.R. 386 Erdahl, R. 289 Erginsav, A. 174 Essiambre, R.-J. 217, 218, 222-224, 226 Esteve, D. 353 Evangelides, S.G. 207, 208, 212, 213, 234236, 241

Evans, A.F. 222 Ewald, PI? 260, 267

F Fabelinskii, I.L. 97, 99-103, 110, 112-1 15, 117, 120, 121, 128, 141-155, 157-160, 163-167, 169, 171, 173-178 Fabeni, P. 367, 368, 374 Falck, J.I? 354, 357, 379 Falicov, L.M. 263 Faridani, A. 51 Favre, F. 198, 207 Feibelman, P.J. 264, 3 15, 3 18 Feix, M.R. 53 Feldmann, J. 263 Fermann, M.E. 21 I Ferrell, R.A. 140, 141 Fertig, H.A. 354, 357, 384 Feshbach, H. 63, 90 Fetter, A.L. 263 Filatova, L.S. 169 Firth, W.J. 210, 211 Fischer, U. 262 Fisher, I.Z. 136, 154 Fisher, M. 136, 140 Fjeldly, T.A. 354 Flannery, B.P. 79 Fleury, PA. 97, 140 Flores, F. 289 Flubacher, P. 120, 158 Fluegel, B. 298 Forstmann, F. 263, 314, 318 Forysiak, W. 229-23 I Fowler, A.B. 263 Franco, P. 193, 239, 243 Frenkel, J. 159 Friedberg, R. 303 Froehley, C. 245 Fubini, G. 86 Fuchs, R. 264 Fujimoto, J.G. 45 Furlan, G. 86 Furya, K. 62, 88 Fusseder, W. 359 Fytas, G. 175

G Gabitov, I.R. 229 Gammon, R.W. 129 Garcia, N. 262, 326

AUTHOR INDEX FOR VOLUME XXXVII

Garcia-Moliner, F, 289 Garm, T. 299 Garrett, C.G.B. 351, 366 Garrison, J.C. 369, 370, 388, 393, 395, 399 Case, R. 3, 15 Gasparian, V. 379 Gaylord, T.K. 386 Gelbart, M.W. 135 Georgakis, G.A. 387 George, T.F. 265, 327 Georges, T. 193, 198, 207, 208, 230, 232 Gerhardts, R.R. 263, 272, 314, 318 Gershon, N.D. 169, 174 Geschwind, S. 130 Ghatak, A.K. 381 Giakos, G.C. 367 Gies, P. 272 Ginzburg, VL. 99, 110, 114, 115, 140, 314 Girard, C. 262, 263, 302 Gires, G. 155 Girndt, A. 263 Giuliani, G. 245 Glauber, R.J. 49, 50, 369 Glunder, H. 12 Glutsch, S. 298 Glytsis, E.N. 386 Gobel, E.O. 263 Godil, A.A. 46, 47 Goedde, C.G. 246 Goedecke, G.H. 67 Goethals, A. 5, 21, 24 Goldstein, S. 359 Golovchenko, E.A. 208, 239 Golub, J.E. 378 Gonzalo, C. 13 Good Jr, R.H. 61, 69 Gordon, J.P. 187, 191, 196, 201, 202, 205, 207,208, 213, 216, 219,220, 234-236, 241, 242, 244, 395 Gori, F. 34 Gotze, W. 122 Goyal, I.C. 381 Granieri, S. 33, 37 Greffet, J.-J. 263 Greytak, T.J. 1 18 Griffin, A. 97, 128 Grigoryan, VG. 273 Grigoryan, VS. 212 Grischowsky, D. 245 Gross, E.F. 104, 113, 119, 161

41 1

Gross, E.K.U. 261, 289 Grudinin, A.B. 2 17 Griindler, R. 88 Grynberg, G. 261, 266, 269 GuBret, P. 353 Guillemin, V 4 Gull, S.F. 73 Gundersen, S. 220 Giintherodt, G. 97 Gupta, A.K. 12 Gupta, S.D. 265, 328, 337 Gurevich. V.L. 132 H Haase, T. 16 Haberl, F. 21 1 Hadjichnstov, G.B. 61, 71 Haelterman, M. 242, 244, 246 Hagan, D.J. 61, 64, 71 Halas, N.J. 245 Hallem, R.I. 169 Halperin, 8.1. 140 Hamada, S. 131 Hamaide, J.-P. 191, 233, 244-246 Haner, M . 207, 213, 214 Hanggi, P. 354, 357, 358, 384 Hansen, J.W. 155 Hanson, S.G. 5 Hara, E.H. 118 Hardy, L. 359 Hariharan, P. 122 Harootunian, A. 262 Harper, P. 217 Harris, G.H. 272 Hartman, T.E. 35 I, 365 Hartmann, S.R. 303 Hartree, D.R. 323 Harvey, G.T. 207, 213, 239 Hasegawa, A. 187, 191, 194, 199, 207, 208, 21 I , 212, 217, 229, 237, 238, 240, 244, 247 Hasegawa, T. 61, 71, 83 H a s , K. 396 Hassan, S.S. 267 Hauge, E.H. 354-357, 379, 380, 384 Haupt, R. 263 Haus, H.A. 187, 198, 201, 207, 239 Hawkins, R.J. 245 Hawley, D. 349 Haykin, S. 75, 76 Heffner, H. 91

412

AUTHOR INDEX FOR VOLUME XXXVlI

Heismann, F. 244 Heitmann, W. 349, 375, 376, 401 Helmerson, K. 386 Henderson, G.N. 386 Hendnks, B.H.W. 265, 327 Henneberger, K. 263 Heritage, J.P. 245 Hermann, G.T. 47 Herzfeld, K.F. 121 Hesse-Bezot, C. I69 Hiley, B.J. 358, 384 Hino, M. 386 Hirlimann, C. 38 Hochreiter, H. 21 1 Hodgson, N. 16 Hofer, M. 211 Hohenberg, P. 261 Hohenberg, P.C. 140 Holland, P.R. 358 Hong, C.K. 371 Hopf, EA. 71 Houston, W.V. 122 Hsu, H. 222, 230 Hu, Y.Z. 298 Huffman, D.R. 60, 326 Hulthkn, R. 67 Hurd, D.L. I 15, 116 Hutchings, D.C. 61, 64, 71 Hynne, F. 175 I Iannaccone, G. 384, 385 Ikeda, H. 211, 217, 247 Imai, T. 229, 240 Imajuku, W. 21 1 Ingrad, K.Y. 122 Inkson, J.C. 263 Ippen, E.P. 45 lsaacson, M. 262 lsakovich, M.A. 1 1 1 , 121, 174, 177 Ishihara, H. 264, 299 Ishii, T.K. 367 Islam, M.N. 196, 219, 220 Iwai, T. 12 lwasa, Y. 61, 71, 83 Iwatsuki, K. 214

J Jackson, J.D. 367, 388, 399 Jahoda, FC. 67

Jain, J.K. 263 Jany, P. 141, 142 Japha, Y. 376, 386, 396 Jauch, J.M. 327 Jaynes, E.T. 73, 261, 295 Jeanny, E. 2 10 Jeffers, J. 371, 372 Jenkins, R.B. 238 Jezierski, K. 69 John, S. 348, 370 Johnston, R.G. 141 K Kadanoff, L.P. 140 Kalpouzoz, G.A. 155, 156 Kamp, L.P.J. 40 Kandler, E. 263 Karpman, VI. 193, 205, 206 Karttunen, K. 60 Kath, W.L. 246 Kauffman, M.T. 47 Kawachi, M. 214, 218 Kawai, T. 386 Kawasaki, K. 140 Kayte, R.H. 122 Kean, P.N. 217 Kehr, K.W. 379 Keldysh, L.V. 260, 261, 314 Keller, 0. 261-265, 267, 269, 277, 281, 288, 290, 296, 298, 299, 302, 3 11, 3 13, 3 14, 327, 328, 331, 332 Kemp, M.C. 73 Kernpa, K. 273, 299 Kenney-Wallace, G.A. 155, 156 Keren, E. 15 Kerker, M. 97, 326 Kerr, F.H. 33 Kesler, S. 75 Ketolainen, P. 60 Ketterle, W. 386 Keyes, T. 169 Khasanov, A.Kh. 130 Khmelev, A.K. 123, 124 Khokhlov, K.V. 45 Khvostikov, I. I13 Kiefte, H. 128, 129 Kim, A.D. 246 Kim, Y.S. 3, 50 Kimura, Y. 210, 213, 229, 240 King, EW. 60, 62, 70, 88, 89, 91

AUTHOR INDEX FOR VOLUME XXXVlI

King-Smith, R.D. 263 Kircheva, P.P. 61, 71 Kirkpatrick, T.R. 128 Kirschner, E.M. 245 Kirzhnitz, D.A. 260, 314 Kishida, H. 61, 71, 83, 84 Kitoh, T. 214, 218 Kittel, C. 388 Kivelson, D. 161, 169 Kivshar, Y.S. 244, 246 Knaap, H.F.P. 118 Knorr, A. 263 Knox, EM. 217, 229, 231 Kobayashi, T. 72 Koch, M. 263 Koch, S.W. 263, 298 Kochelaev, B.I. 129 Koda, T. 61, 71, 83, 84 Kodama, Y. 187, 191, 199, 207, 209, 229, 237, 238, 247 Kogan, S.M. 61, 69 Kohn, W. 261, 323 Kolesnikov, (3.1. 159, 160 Kolltveit, E. 233 Kolner, B.H. 46, 47 Komar, A. 395 Komarov, L.I. 136 Komukai, T. 229, 240 Konno, H. 43, 44 Kostenbauder, A.G. 5 Kostka, R. 16 Kovalenko, K.V. 142-146, 158, 159, 163, 166, 177 Kovrigin, A.I. 45 Kozhekin, A.E. 388, 393-396 Kozyrev, B.M. 129 Krarners, H.A. 60 Kraus, J. 261 Krausz, E 377 Kreines, N.M. 97 Krenzlin, H.M. 379 Krishnan, R.S. 120 Krivokhizha, S.V 121, 141-149, 158, 159, 163, 166, 177 Krokel, D. 245 Kronig, R. 60 Kruskal, M.D. 187 Kubo, R. 261 Kubota, H. 209, 210, 213, 221, 227, 229, 230, 240

413

Kumar, S. 212, 240 Kurizki, G. 376, 386-388, 393-396 Kurn, D.M. 386 Kurokawa, K. 221 Kwiat, P.G. 348, 356, 359, 370-372, 376, 382

L Labani, B. 262, 302 Lagendijk, A. 265, 384 Lai, C.C. 140 Lai, Y. 198, 207 Lakoza, E.L. 135 Lallemand, P. 97, 118 Laloe, F. 292 Lamb, H . 107 Lancis, J. 23 Landau, L.D. 64, 100, 1 1 1 , 149, 152, 159, 398 Landauer, R. 352, 353, 355-357, 363, 364, 382, 384 Landsberg, G.S. 127 Lanz, M. 262 Larsen, G.A. 141 Lavi, S. 15 Law, B.M. 129 Lax, M. 50 Leadbetter, A.J. 120, 158 Leaird, D.E. 245 Leavens, C.R. 351, 358, 384 Leclerc, 0. 240 Lederer, F. 229 Lee, B. 376, 380, 386 Lee, D.L. 264 Lee, M.H. 67 Lee, S.-Y. 68 Lee, W. 376, 380, 386 Lee, Y.I. 263 Leonhardt, U. 50, 52, 53 Leontovich, M.A. 99, 103, 107, 113, 119, 121, 122, 126-129, 152, 157, 164, 165, 169 Lester, C. 220 Levanyuk, A.P. 140 Lewis, A. 262 Li, G. 175 Li, W.B. 129 Li, Y. 12 Lichtenberg, A.J. 5 , 24 Lichtman, E. 207, 213, 239 Liebsch, A. 272

414

AUTHOR INDEX FOR VOLUME XXXVll

Lifshitz, E.M. 64, 100, 1 1 1, 159, 398 Lin, M.-C. 222, 230 Lin, Q. 15, 17 Lindberg, M. 298 Lipsworth, E. 136 Litovitz, T.A. 121 Liu, A. 263, 3 13 Lockwood, D.J. 97 Logan, R.A. 214 Loh, W.H. 217 Lohmann, A.W. 12, 23, 33, 34, 37, 47 Lomdahl, P.S. 43, 44 Lord, A. 243 Lorentz, H.A. 259 Lorenz, L. 326 Lotshaw, W.T. 155, 156 Loudon, R. 392 Louisell, W.H. 50 Low, F.E. 352, 396 Lozovski, V.Z. 265 Luchnikov, A.V. 203, 204 Lugovaya, O.A. 141, 142 Lundqvist, S. 261 Luneburg, R.K. 4, 10 Luo, MS-C. 263 M Ma, R.J. 175 Ma, Sh.K. 140 MacColl, L.A. 350 Machta, J. 128 Macke, B. 366 Madrazo, A. 263 Magni, V. 41 Mahan, G.D. 261, 289 Maker, P.D. 155 Maki, J.J. 303 Malcuit, M.S. 303 Malomed, B.A. 2 17 Malyon, D.J. 21 I , 243 Malyugin, A.V. I 18, 124 Mamyshev, P.V. 204, 21 3, 23 I , 240, 241 Manakov, S.V 243 Manasseh, J.T. 303 Mandel, J. 327 Mandel, L. 64, 364, 371 Mandelstam, L.I. 103, 104, 113, 119, 121, 122, 124, 125, 128, 129, 134 Manninen, M. 264, 272, 289 Manogue, C.A. 89

Maradudin, A.A. 260, 314 March, N.H. 261 Marclay, E. 353 Marcuse, D. 201 Marcuvitz, N. 39 Marshall, I.W. 220 Martin, O.J.F. 263 Martin, P.C. 61, 261 Martin, Th. 355-357, 363, 364, 382, 384 Martinez-Herrero, R. 14, 15, 17, 18, 25, 38, 40 Martinis, J.M. 353 Martynenko, L.Ph. 123 Maruta, A. 187, 212, 247 Mash, D.I. 113, 117, 158 Maslov, E.M. 193 Massar, S. 401 Matera, E 239, 243 Matsuda, T. 229 Matsumoto, M. 211, 217, 247 Matsumoto, T. 73 Matthews, M.R. 386 May, A.D. 118 Mayer, G. 155 Mazur, P. 128, 129, 267 McBride, A.C. 33 McCumber, D.E. 351, 366 McFee, J.H. 131 Mclntyre, D. 140 McKinnon, W.R. 384 McMorrow, D. 155, 156 Mecozzi, A. 207, 209, 214, 239 Mehta, C.L. 50 Meier, H. 353 Mejias, P.M. 14, 15, 17, 18, 25, 38, 40 Meklenbrauker, W.F.G. 6, 7, 9, 15 Mende, P.F. 352 Mendlovic, D. 33, 34 Menyuk, C.R. 208, 209, 236, 237, 239, 242 Mermin, N.D. 361 Mewes, M.-0. 386 Mezincescu, G.A. 61 Midrio, M. 193, 207, 239, 243 Mie, G. 326 Miesner, H.-J. 386 Migus, A. 1 1 Miki, Sh. 131 Miller, D.A.B. 298 Miller, W.H. 357, 382 Mills, D.L. 263, 3 13

AUTHOR INDEX FOR VOLUME XXXVIl

Milonni, P.W. 265, 327 Mitchell, D.J. 42 Mitchell, M.W. 388, 392, 400 Molchanov, VA. I 10, 112 Mollenauer, L.F. 187, 196, 202-204, 207, 208, 212, 213, 219, 220, 231, 234-236, 239-242, 244 Mollow, B.R. 50 Montalant, T. 208, 210 Moores, J.D. 207 Morhange, J.F. 38 Morita, I. 229, 230 Morozov, VV 117 Morrison, J.A. 120, 158 Morse, P.M. 63, 90 Moulu, J. 208, 210 Mountain, R.D. 150, 169 Mourgues, G. 53 Mozhaev, VG. 123, 124 Muga, J.G. 384 Mugnai, D. 367, 368, 374, 376 Mukunda, N. 12, 24 Murray, A. 262

N Naka, A. 229 Nakazawa, H. 229 Nakazawa, M. 209-211,213,214,218, 221, 221, 229, 230, 240, 245, 247 Nalesso, G. 209 Naletto, G. 368 Namias, V 33 Narcowich, F.J. 50 Nasalski, W. 41 Nash, P.L. 68 Nazarathy, M. 46 Neira, J.L.H. 15, 18 Nemes, G. 16 Nemoto, S. 41 Neubelt, M.J. 204, 207, 212, 213, 239, 240 Newhouse, M.A. 220 Newton, T.D. 327 Nichols, G. 326 Nieminen, R.M. 264, 272, 289 Nienhuis, G. 265, 327 Nieto-Vesperinas, M. 262, 263, 326 Nieuwoudta, J.C. 129 Nimtz, G. 349, 368, 369, 375, 376, 401 Nishi, S. 214 Nishida, M. 299

415

Niu, Q. 387 Nortier, B. 208 Noz, M.E. 3, 50 Nozieres, P. 261 Nussenzveig, H.M. 61, 62, 87 Nyman, B.M. 207, 212, 213, 239

0 O’Connell, R.E. 50 Ohfuti, Y. 262, 264, 299 Ohhira, R. 229 Ojeda-Castaneda, J. 23 Olkhovsky, VS. 355 Olsson, N.A. 214 Onciul, D. 17, 18, 26, 3 1 Onural, L. 33 Oppenheim, I . 128, 174 Omstein, L.S. 136 Ortuiio, M. 379 Oseen, C.W. 260, 267 Ostrowsky, N. 169 Ou, Z.Y. 371 Oughstun, K.E. 392 Oxtoby, D.W. 135 Ozaktas, H.M. 33, 34

P Pacheco, J.M. 272 Pagonabarraga, I. I29 Palik, E.D. 80, 82 Palumbo, L.J. 67 Papoulis, A. 15, 34 Park, C. 40, 41, 229 Parentani, R. 401 Parks, T.W. 53 Passante, R . 261 Pattanayak, D.N. 267 Paul, H. 50, 52, 53 Paye, J. 11, 39, 45 Payne, D.N. 217, 222, 247 Pazzi, G.P. 367, 368, 374 Pecora, R. 97, 165, 169, 172 Pecora, R.J. 161, 162, 172 Pedersen, K. 263 Peiponen, K.-E. 60-62, 66, 69-73, 77, 78, 80-82, 84, 85, 88, 90, 91 Pender, W.A. 222 Penney, R. 128, 129 Penzar, Z. 272 Pershan, P.S. 91

416

AUTHOR INDEX FOR VOLUME XXXVII

Persson, B.N.J. 264, 289 Pesin, M.S. 120, 155 Petersilka, M. 261, 289 Petit, A. 98 Peuckert, V 261 Peyghambanan, N. 298 Pfeiffer, L.N. 386 Philipp, H.R. 60 Phillips, W. 386 Pike, E.R. 327 Pilipetskii, A.N. 203, 204, 208, 239 Pinczuk, A . 263 Pines, D. 261 Piquero, G. 15, 17, 25 Pino, F. 208, 210, 230 Piron, C. 327 Placzek, G. 102, 149, 152 Platzeck, A.M. 67 Pohl, D.W. 262, 326 Pokrovskii, VL. 140 Poladian, L. 42 Pollak, E. 357, 382 Pollak, M. 379 Pons, A. 23 Porras, M.A. 40, 41 Potapova, Yu.B. 123, 124 Potashinskii, A.Z. 140 Povlsen, J.H. 220, 238 Power, E.A. 261, 269 Presby, H.M. 214 Press, W.H. 79 Price, P.J. 61, 69 Procaccia, I. 128 Prochaska, R. 15 Prokhorov, A.M. 203, 204 Pudonin, EA. 263 Puska, M.J. 264, 272, 289

Q Quentrec, 9. 169, 175 Quinn, J.J. 263, 273

R Raciti, F. 376 Raether, H. 264 Raghavan, B. 236, 237 Raizen, M.G. 387 Raj, N. 263 Rajagopal, A.K. 264 Raman, C V 119

Ramaswamy, M. 45 Ranfagni, A. 367, 368, 374, 376 Rank, D.H. 118, 119 Rao, B.VR. 113, 119 Rasigni, G. 69 Rasigni, M. 69 Raybon, G. 212 Rayleigh, Lord 107, 1 I 1, 326 Raymer, M.G. 45, 47, 48, 51 Recami, E. 355 Ren, Q. 222, 230 Reynaud, F. 245 Rice, M.J. 264 Richardson, D.J. 222, 247 Richmond, G.L. 308 Ridener, EL. 61, 69 Risken, H. 51 Robinson, J.M. 308 Roesler, D.M. 83 Romagnoli, M. 193, 207, 239, 243 Romanov, VP 135, 161, 169 Ronis, D. 128 Rosa-Franco, L. 303 Rosetti, G. 86 Ross, R.R. 396 Rossler, U. 299 Rottwitt, K. 220, 238 Rouch, J. 140 Rowland, D.R. 42 Roy, R. 349 Rozhdestvenskaya, N.B. 165 Rubi, J.M. 129 Rubin, R.L. 119 Rubio, A. 272 Ruggen, G.J. 49 Ruggeri, R. 368, 376 Ruiz, J. 379 Ruppin, R. 299 Rybachenko, VF. 356 Rytov, S.M. 103, 150, 151, 169, 170, 172, 173 S Sabirov, L.M. 164-167, 173 Sahara, A. 229, 230 Saito, S. 229 Sala, R. 384 Salesi, G. 376 Sanchez, M. 15, 18 Sandercock, J.R. 122

AUTHOR INDEX FOR VOLUME XXXVII

Santagiustina, M. 209 Santarsiero, M. 34 Saphonov, M.V. 123 Sarkar, S . 327 Saruwatari, M. 214 Sasnett, M.W. 19 Sauer, J.R. 237, 238 Savage, C.M. 155 Scandolo, S. 61, 62, 70, 91 Schafer, W. 263 Schaich, W.L. 273 Schiff, L.I. 269 Schmitt-Rink, S. 263, 298 Schneider, W.R. 264 Schulman, L.S. 374 Schulz-DuBois, E.O. 392 Schulze, A. 263 Scott, J.F. 140 Scully, M.O. 359 Searby, G.M. 162 Seeds, A.J. 217 Segall, B. 60 Segard, B. 366 Segre, P.N. 129 Sein, J.J. 267 Sen, D. 122 Sengers, J.V. 129, 140, 141 Sentenac, A. 263 Serkin, VN. 221 Serna, J. 14 Settembre, M. 239 Shabat, A.B. 190, 236, 238, 244 Sham, L.J. 261, 263, 323 Shan, X. 227 Shannon, C.E. 73 Shannon, VL. 308 Shapiro, J.H. 371, 372 Sheik-Bahae, M. 61, 64, 71 Shen, S. 392 Shen, Y.R. 70 Sheng, P. 265 Shenoy, M.R. 381 Shenoy, R.G. 53 Sheridan, J.T. 33 Shifrin, K.S. 97 Shinder, 1.1. 141 Shore, K.A. 71 Shubin, A.A. 127, 147-149 Shustin, O.A. 110, I12 Sicre, E.E. 23, 33, 37

417

Siegman, A.E. 5, 16, 19, 25 Siggia, E.D. 128 Simon, R. 12, 24 Simpson, J.R. 214 Sindoni, 0.1. 67 Singh, J. 298 Sipe, J.E. 259, 287, 297, 303, 336 Sixon, P. 162 Skilling, J. 73 Smet, E 61, 69 Smet, I? 61, 69 Smith, D.Y. 61, 62, 64, 65, 67, 68, 86-89 Smith, K. 187, 203, 210-212, 219, 242 Smith, N.J. 207, 210, 211, 217, 229, 230 Smith, R.W. 131 Smith Jr, V.H. 289 Smithey, D.T. 51 Smoluchovski, M. 122 Smolyaninov, 1.1. 264, 328, 33 I Snyder, A.W. 42 Sobyanin, A.A. 140 Soffer, B.H. 47 Sokolovski, D. 354, 355, 357, 358, 384 Solov’ev, V.A. 161, 169 Solovev, V.V 205, 206 Sommerfeld, A. 326, 366 Sonnenberg, H. 91 Sorensen, C.M. 141 Souza, R.F. 247 Soven, P. 261 Sozzi, C. 368 Spector, J. 386 Spieker, H. 376 Spielmann, Ch. 377 Spirit, D.M. 220 Squires, E.J. 359 Snnivas, M.D. 49 Stanley, H.E. 140 Starodumov, A.N. 203 Starunov, VS. 113, 117, 154-156, 158-160, 164-167, 169, 171-176 Stegeman, G.I. 71, 316, 317 Stegeman, G.I.A. 159-161, 165, 166, 168 Steinberg, A.M. 348, 356, 359, 363, 365, 370-373, 376, 380, 382, 384-387, 399 Stentz, A.J. 222 Stephan, K.D. 367 Stern, F. 263 Sternberg, S. 4 Stingl, A. 377

418

AUTHOR INDEX FOR VOLUME XXXVII

Stoicheff, B.P. 120, 158-161, 163, 164, 168 Stolen, R.H. 187, 196, 219 Stormer, H.L. 386 Stavneng, J.A. 355, 356, 380, 384 Strassler, S. 264 Streibl, N. 23 Stroucken, T. 263 Subbaswamy, K.R. 261, 289 Sudarshan, E.C.G. 12, 24, 50 Sugawa, T. 229 Sukhorukov, A.P. 45 Sun, K.X. 371, 372 Sunanda Bai I13 Susskind, L. 395 Sussmann, G . 359 Suttorp, L.G. 8, 260 Suydam, B.R. 40 Suzuki, K. 209,210,213,214,2l8,229,240, 245, 247 Suzuki, M. 209, 21 1, 229, 230 Swift, J. 140 Swinney, H.L. 136, 140 Synge, E.H. 326 Sysoliatin, A.A. 222, 23 1 Szipocs, R. 377

T Taga, H. 209, 21 I , 229, 230 Tai, K. 217 Tajima, K. 221 Takada, A . 21 I Takagahara, T. 299 Takaya, M. 210, 213 Tamez, R.-V. 31 1 T a m , I.E. 114 Tamura, K. 230, 240 Tanaka, H . 209, 21 1 Tanaka, M . 366 Tanaka, T. 298 Tanbun-Ek, T. 214 Tang, X.-Y. 230 Tanji, T. 73 Tao, N.J. 175 Tappert, F. 187, 244 Tartaglia, P. 140 Tasaki, S. 386 Tatarski, V.I. 327 Teague, M.R. 14, 16 Terasaki, A. 72 Terhune, R.W. 155

Teukolsky, S.A. 79 Thirunamachandran, T. 269 Thomas, J.E. 387 Thomas, P. 263 Thomine, J.-B. 208, 210, 230 Thompson, R. 386 Thurston, R.N. 245 Thyagarajan, K. 381 Ticknor, A.J. 13 Tiganov, E.V. 113, 158, 160, 166, 169 Tilley, D.R. 263 Tip, A. 384 Toda, H. 208 Tokunaga, E. 72 Tokura, Y. 6 I , 7 1, 83 Toll, J.S. 62 Tomlinson, W.J. 245 Tonomura, A. 73 Torner, L. 129 Townes, C.H. 395 Townsend, C.G. 386 Trabocchi, 0. 33, 37 Tran Thoai, D.B. 298 Treacy, E.B. 45 Tremblay, A.-M.S. 128 Tremblay, C. 128 Tsay, S.J. 161 Tselis, A.C. 263 Tsuboi, T. 62, 91 Turitsyn, S.K. 229 Turlot, E. 353

U Uda, T. 217 Umeno, M. 131 Umezawa, M. 229 Urbina, C. 353 Uzunov, I. 229 V Vaccaro, J.A. 53 Vaidman, L. 384 Vaittinen, A. 66 Valishey R.M. 130 Vallee, R. 217 van de Hulst, H.C. 97, 326 Van den Bos, A. 76 van der Zwan, G. 128, 129 van Groenendael, A. 61, 69 Van Huele, J.F. 261

AUTHOR INDEX FOR VOLUME XXXVll

Van Kranendonk, J. 259, 287, 297, 336 Van Labeke, D. 262, 302 van Stryland, E.W. 61, 64, 71 van Tartwijk, G.H.M. 222 van Tiggelen, B.A. 265, 384 Vartiainen, E.M. 60-62, 66, 71-73, 77, 78, 80-82, 84, 85, 91 Vasileva, 0.1. 13 1 Vause, C.A. 138, 142, 147 Velasco, V.R. 123 Velichkina, T.S. 119, 123-125, 131, 132, 147 Velicky, B. 67 Venkatesvaran, C.S. 113, 117, 119 Vetterling, W.T. 79 Villani, A. 62, 88 Villeneuve, A. 229 Vinen, W.F. 115, 1 I6 Vinogradov, E.A. 263 Visscher, P.B. 263 Vladimirskii, VV 128, 129 Vodolazskaya, I.V. 123, 125 Vogel, K. 51 Vohnsen, B. 265 Volterra, V. 169 Voronkova, V.1. 123, 125 Vuks, M.F. 158 Vysloukh, V.A. 221 W Wabnitz, S. 193, 207, 209, 210, 214, 229, 240-242 Wai, P.K.A. 236, 237 Wakita, K. 13 1 Walker, J.S. 138, 142, 147 Wallis, R.F. 316, 317 Walmsley, I.A. 45, 47, 48 Walther, A. 3 Walther, H. 359 Wang, C.H. 175 Wang, S. 15, 17 Wang, Y.P. 396 Wasserman, A.L. 263 Weber, H. 13, 16 Wecht, K.W. 214 Weiner, A.M. 245 Wen, S. 209 Wendler, L. 263, 273 West, K.W. 386 Westin, E. 209 Wetling, W. 13 I

419

White, D.L. 131 Widdowson, T. 2 1 I , 222, 227, 243 Wieman, C.E. 386 Wiggins, T.A. 118 Wightman, A.S. 327 Wigner, E. 3, 49, 50 Wigner, E.P. 50, 327, 350 Willatzen, M. 298 Williams, D.L. 220 Wilson, R. 178 Wiltzins, P. 141 Wolf, E. 11, 24, 32, 49, 50, 59, 64, 260, 267, 286, 367 Wong, S. 366 Wong, V. 45, 47, 48 Wong, W.S. 187 Wood, D. 21 1 Wood, D.M. 264, 298 Woolley, R.G. 269 Wooten, F. 59, 86

X Xiao, M. 263, 265, 298, 299, 302

Y Yablonovitch, E. 348, 370 Yakovlev, LA. 123-125, 131, 132 Yakovlev, N.L. 123 Yamada, E. 209-211, 213, 214, 218, 221, 229, 240 Yamagishi, H. 208 Yamamoto, S. 209, 21 1, 229, 230 Yamamoto, T. 229 Yamauchi, 0. 229 Yang, X. 47, 244 Yanovskii, V.K. 123, 125 Yanv, A. 264, 316, 317 Yeh, P. 264, 3 16, 3 17 Yeh, Y. 136 Yi, K.S. 273 Yip, S. 118 Yoshida, E. 214, 218, 229 Young, R.H. 69 Yu, A.W. 349 Yukov, E.A. 119 Yura, H.T. 5 Z Zabusky, N. 187 Zakharov, V.E. 190, 236, 238, 244

420 Zaluzny, M. 263 Zamb, J. 174 Zarnir, E. 169 Zanghi, N. 359 Zangwill, A. 261 Zaremba, E. 264, 289 Zayats, A . 263 Zeiger, H. 395

AUTHOR INDEX FOR VOLUME XXXVll

Zemike, F. 136 Zhang, D.L. 396 Zhang, J. 175 Zhao, W. 245-247 Zhao, X.-G. 387 Zhu, S. 349 Zimerman, A.H. 62, 88 Zubkov, L.A. 135, 165

SUBJECT INDEX FOR VOLUME XXXVII

A

acoustic wave 110-1 12 amplified spontaneous emission (ASE) 208 Anderson localization 265 autocorrelation function 74

elastic thermal waves I 10-1 12 erbium-doped fiber amplifier 197 Esaki tunnel diode 347 Ewald-Oseen extinction theorem 260, 267

198,

F f-sum rule 61 Fabry-Perot filter 208, 209, 239 - - interferometer 122, 376 far-field approximation 326 Feibelman d parameters 3 18 theory 3 19 Fermi-Dirac distribution 293 Feynman path-integral 354 fiber-optic communication system 194 four-wave mixing 227 Fourier transform, fractional 33 Fraunhofer diffraction 326 Fresnel approximation 33 - formula 67 - transform 34 front velocity 397 frustrated total internal reflection (FTIR) 348, 364,380

B Bloch function 370 Bloch’s theorem 370 Bragg’s condition 106, 373 Brillouin zone 348 Burg’s maximum entropy method 74, 75, 77 Biittiker-Landauer time 355-357, 368 Buttiker’s Larmor time 357, 368, 373

-

C Cabannes’ correction factor 102 Cauchy-Riemann condition 62 Choi-Williams distribution 6 coherent anti-Stokes Raman scattering 70 - - - - spectrum (CARS) 61, 73, 91 correlation function 7 critical phenomena 136 D de Broglie wavelength 348 Debye spectrum 125 -wave 125 Dirac equation 351 dispersion-decreasing fibers 22 1-227 - relations in nonlinear optics 69, 72 distributed amplification 219

G Gauss-Schell beam 9, 16 --field 12 Gaussian beam 16, 24, 41 - field 20, 23 - light source 27 geometrical optics 3 Goos-Hanchen shift 365, 381 Gordon-Haus jitter 201, 202, 224 graded-index optical fiber 33 Green’s function 268, 319 group velocity 350, 366, 367, 374, 375 dispersion (GVD) 189

E eight-port homodyne detector 53 Einstein causality 375, 396, 399 - formula I01

- -

42 I

422 guaiacol-glycerol solution

SUBJECT INDEX FOR VOLUME XXXVll

137, 142-148

H Hamiltonian equations 4 Hartley transform 13 Hartman effect 382 Hartree potential 323 Helmholtz equation 5, 359, 370 - -, in a transverse inhomogeneous medium 11 Hilbert transforms 62-64

J Jordan’s lemma 68 Josephson junction 347 K Kerr effect 151, 155, 156 -medium 39 - nonlinearity 203 Kohn-Sham density-density response function 323 - - ground state 273 Kramers-Heisenberg model 63 Kramers-Kronig relations 60-72, 270, 351, 392, 398 - - _ in absorption spectroscopy 64-67 - - _ in reflection spectroscopy 67-69 Kronig-Penney model 348, 360, 361, 370

L Lamb shift 261 Landau-Placzek relation 148-1 5 1 Landau-Zener tunneling 387 Larmor clock 377-379 - frequency 356 -time 356, 357, 381 tunneling time 378, 379 light cone 283 linear response theory 269 Liouville theorem 5 local field 257-292 - electrodynamics in quantum well 305324 - - in mesoscopic media 292-304 - - resonances 263

Maxwell equations 188, 259 Maxwell-Lorentz equations 259, 260, 265, 266, 319-321 Maxwellian velocity distribution 1 18

N nonlinear fiber 38 Schrodinger equation (NSE) 44, 188-190, 216 _ - _ - , soliton solution of 42 -

0 optical bandpass filter 207-209 homodyne tomography 51 phase conjugation 225-227, 324

-

P Page distribution 6 parametric amplifier 50 paraxial approximation 5, 10 partial coherence 3 partially coherent field 9 source 31 particle-wave duality 377 Pauli equation 269, 270 phase-conjugating mirror 265, 327 - retrieval in optical spectroscopy 73-86 - space, light propagation 4 - velocity 367 photon localization 327 photonic band gap 348, 370, 382 pilot wave model 358 Planck’s law 98 - theory of dispersion 259 polarization-division multiplexing (PDM) 241 -244 power spectrum 74 - -

-

-

M Mach-Zehnder interferometer 247 Margenau-Hill distribution 6

Q Q function 50, 52, 53 quantum dot 274 - electrodynamics 369 - nonlocality 283 -well 263, 264, 305-324

R radiometry 3 Radon transform 13 - -, inverse 47, 48 Raman effect 216, 224, 225

SUBJECT INDEX FOR VOLUME XXXVll

random-phase approximation (RPA) 261, 272, 292 Rayleigh length 382 - scattering 97 -wave 123 Rihaczek distribution 6 Roesler method 83 S saddle point method 366 scanning tunneling microscope 347 secant hyperbolic pulse 189 self-phase modulation (SPM) 189 signal velocity 375 slowly-varying-envelope approximation (SVEA) 188 soliton 42, 43, 187, 196, 218 -, average I99-2 15 -, dark 190, 244-247 -, in fibers 188-194 -, information transmission with 194-1 96 - based communication system 194-198, 201, 209, 215, 220 - interaction 204 stationary phase approximation 366 Stokes equation I07 - parameters 378 sum rules 86-91 - - in linear optics 86-90 - - in nonlinear optics 90, 91

423

superluminal propagation 374, 375 351 synchronous modulators 209-2 1 1

- wave-packet

T tachyon-like excitations 388 thin film 305-324 Thomas-Reiche-Kuhn sum rule 61, 86 time-division multiplexing (TDM) 233 tunneling in de Broglie optics 385-387 -time 347, 348, 379-383 - -, brief history of 350-359 - -, optical experiments on 364383

W wavelength-division multiplexing (WDM) 233-241 Weyl expansion 329 Wigner distribution function (WDF) 1-53 _ - - _ , complex field reconstruction from 4449 - - _ - , fractional 32-38 _ - - - , in quantum optics 49-53 - - _ _ , optical system characterization with 20-26 - - _ _ , propagation law for 10, 18, 39 - - _ _ , properties of 7-9 - - _ _ , representation of the coupling efficiency 26-32 - - _ _ , transformation law for 9, 16, 18, 36 - - _ - , transport equation for 10, 1 I , 40, 41

This Page Intentionally Left Blank

CONTENTS OF PREVIOUS VOLUMES

VOLUME I(1961)

I I1 111

IV V V1 VII VIII

The Modem Development of Hamiltonian Optics, R.J. PEGIS 1- 29 Wave Optics and Geometrical Optics in Optical Design, K. MIYAMOTO 31- 66 The Intensity Distribution and Total Illumination of Aberration-Free Diffraction Images, R. BARAKAT 67-1 08 Light and Information, D. GABOR 109-1 53 On Basic Analogies and Principal Differences between Optical and Electronic Information, H. WOLTER 155-21 0 21 1-251 Interference Color, H. KUBOTA 253-288 Dynamic Characteristics of Visual Processes, A. FIORENTINI Modem Alignment Devices, A.C.S. VANHEEL 289-329 VOLUME I1 (1963)

Ruling, Testing and Use of Optical Gratings for High-Resolution Spectroscopy, G.W. STROKE I1 The Metrological Applications of Diffraction Gratings, J.M. BURCH Ill Diffusion Through Non-Uniform Media, R.G. GIOVANELLI IV Correction of Optical Images by Compensation of Aberrations and by Spatial Frequency Filtering, J. TSUJIUCHI V Fluctuations of Light Beams, L. MANDEL VI Methods for Determining Optical Parameters of Thin Films, F. ABELBS I

1- 72 73-1 08 109-129

131-180 181-248 249-288

VOLUME 111 (1964) I The Elements of Radiative Transfer, F. KOTTLER I1 Apodisation, P. JACQUINOT, B. ROIZEN-DOSSIER I11 Matrix Treatment of Partial Coherence, H. GAMO

1- 28 29-186 187-332

VOLUME IV (1965)

I Higher Order Aberration Theory, J. FOCKE I1 Applications of Shearing Interferometry, 0. BRYNGDAHL Ill Surface Deterioration of Optical Glasses, K. KINOSITA IV Optical Constants of Thin Films, I? ROUARD, P. BOUSQUET V The Miyamoto-Wolf Diffraction Wave, A. RUBINOWICZ VI Aberration Theory of Gratings and Grating Mountings, W.T. WELFORD VII Diffraction at a Black Screen, Part I: Kirchhoff’s Theory, F. KOTTLER 42 5

1- 36 37- 83 85-143 145-197 199-240 24!-280 28 1-3 14

426

CONTENTS OF PREVIOUS VOLUMES

VOLUME V (1966)

I I1 III IV V VI

Optical Pumping, C. COHEN-TANNOUDJI, A. KASTLER I- 81 Non-Linear Optics, PS. PERSHAN 83-144 Two-Beam Interferometry, W.H. STEEL 145-197 Instruments for the Measuring of Optical Transfer Functions, K. MURATA 199-245 Light Reflection from Films of Continuously Varying Refractive Index, R. JACOBSSON247-286 X-Ray Crystal-Structure Determination as a Branch of Physical Optics, H. LIPSON, C.A. TAYLOR 287-350 VII The Wave of a Moving Classical Electron, J. PICHT 351-370 VOLUME VI (1967) Recent Advances in Holography, E.N. LEITH,J. UPATNIEKS 1- 52 Scattering of Light by Rough Surfaces, P. BECKMANN 53- 69 Measurement of the Second Order Degree of Coherence, M. FRANCON, S. MALLICK 71-104 Design of Zoom Lenses, K. YAMAJI 105-1 70 Some Applications of Lasers to Interferometry, D.R. HERRIOT 1 7 1-209 Experimental Studies of Intensity Fluctuations in Lasers, J.A. ARMSTRONG, A.W. SMITH 21 1-257 H. SAKAI VII Fourier Spectroscopy, G.A. VANASSE, 259-330 Vlll Diffraction at a Black Screen, Part 11: Electromagnetic Theory, F. KOTTLER 331-377 I I1 I11 IV V VI

VOLUME VII (1969) Multiple-Beam Interference and Natural Modes in Open Resonators, G. KOPPELMAN 1- 66 Methods of Synthesis for Dielectric Multilayer Filters, E. DELANO, R.J. PEGIS 67-137 Ill Echoes at Optical Frequencies, I.D. ABELLA 139-1 68 IV Image Formation with Partially Coherent Light, B.J. THOMPSON 169-230 V Quasi-Classical Theory of Laser Radiation, A.L. MIKAELIAN, M.L. TER-MIKAELIAN 231-297 VI The Photographic Image, S. OOUE 299-358 VII Interaction of Very Intense Light with Free Electrons, J.H. EBERLY 359415 I

II

VOLUME VIII (1970) Synthetic-Aperture Optics, J.W. GOODMAN 1- 50 The Optical Performance of the Human Eye, G.A. FRY 51-131 133-200 Light Beating Spectroscopy, H.Z. CUMMINS, H.L. SWINNEY Multilayer Antireflection Coatings, A. MUSSET,A. THELEN 20 1-23 7 2 39-294 Statistical Properties of Laser Light, H. RISKEN Coherence Theory of Source-Size Compensation in Interference Microscopy, T. YAMAMOTO 295-34 1 VII Vision in Communication, L. LEV1 343-372 VIII Theory of Photoelectron Counting, C.L. MEHTA 373440 I I1 111 IV V VI

VOLUME IX (1971) I

Gas Lasers and their Application to Precise Length Measurements, A.L. BLOOM Picosecond Laser Pulses, A.J. DEMARIA 111 Optical Propagation Through the Turbulent Atmosphere, J.W. STROHBEHN IV Synthesis of Optical Birefringent Networks, E.O. AMMANN 11

I- 30 31- 71 73-1 22 123-1 77

CONTENTS OF PREVIOUS VOLUMES

427

V Mode Locking in Gas Lasers, L. ALLEN,D.G.C. JONES 179-234 VI Crystal Optics with Spatial Dispersion, V.M. AGRANOVICH, V.L. GlNZBURG 235-280 VII Applications of Optical Methods in the Diffraction Theory of Elastic Waves, K. GNIADEK, J. FETYKIEWICZ 281-310 VIII Evaluation, Design and Extrapolation Methods for Optical Signals, Based on Use of the Prolate Functions, B.R. FRIEDEN 31 1 4 0 7

VOLUME X (1972)

I II 111 IV V V1 VII

Bandwidth Compression of Optical Images, T.S. HUANG The Use of Image Tubes as Shutters, R.W. SMITH Tools of Theoretical Quantum Optics, M.O. SCULLY, K.G. WHITNEY Field Correctors for Astronomical Telescopes, C.G. WYNNE Optical Absorption Strength of Defects in Insulators, D.Y. SMITH,D.L. DEXTER Elastooptic Light Modulation and Deflection, E.K. SlTTlG Quantum Detection Theory, C.W. HELSTROM

1- 44 45- 87 89-135 137-1 64 165-228 229-28 8 289-369

VOLUME XI (1973) 1

I1 111 IV V VI VII

Master Equation Methods in Quantum Optics, G.S. ACARWAL Recent Developments in Far Infrared Spectroscopic Techniques, H. YOSHINAGA Interaction of Light and Acoustic Surface Waves, E.G. LEAN Evanescent Waves in Optical Imaging, 0. BRYNGDAHL Production of Electron Probes Using a Field Emission Source, A.V. CREW Hamiltonian Theory of Beam Mode Propagation, J.A. ARNAUD Gradient Index Lenses, E.W. MARCHAND

1- 76 71-122 123-166 1 67-22 1 223-246 247-304 305-337

VOLUME XI1 (1974)

I II

Self-Focusing, Self-Trapping, and Self-phase Modulation of Laser Beams, 0. SVELTO 1- 5 1 Self-Induced Transparency, R.E. SLUSHER 53-100 Ill Modulation Techniques in Spectrometry, M. HARWIT, J.A. DECKER JR 101-1 62 1v Interaction of Light with Monomolecular Dye Layers, K.H. DREXHAGE 163-232 V The Phase Transition Concept and Coherence in Atomic Emission, R. GRAHAM 233-286 287-344 VI Beam-Foil Spectroscopy, S. BASHKIN

VOLUME XI11 (1976) On the Validity of Kirchhoff’s Law of Heat Radiation for a Body in a Nonequilibrium I - 25 Environment, H.P. BALTES I1 The Case For and Against Semiclassical Radiation Theory, L. MANDEL 27- 68 111 Objective and Subjective Spherical Aberration Measurements of the Human Eye, W.M. ROSENBLUM, J.L. CHRISTENSEN 69- 91 IV lnterferometric Testing of Smooth Surfaces, G. SCHULZ, 93-167 J. SCHWIDER V Self-Focusing of Laser Beams in Plasmas and Semiconductors, M.S. SODHA, A.K. GHATAK, V.K. TRIPATHI 169-265 VI Aplanatism and Isoplanatism, W.T. WELFORD 267-292 1

42 8

CONTENTS OF PREVIOUS VOLUMES

VOLUME XIV (1 976)

I I1 Ill IV V VI VII

The Statistics of Speckle Patterns, J.C. DAINTY High-Resolution Techniques in Optical Astronomy, A. LABEYRIE Relaxation Phenomena in Rare-Earth Luminescence, L.A. RISEBERG, M.J. WEBER The Ultrafast Optical Kerr Shutter, M.A. DUGUAY Holographic Diffraction Gratings, G. SCHMAHL, D. RUDOLPH Photoemission, P.J.VERNIER Optical Fibre Waveguides - A Review, P.J.B. CLARRICOATS

I - 46 47- 87 89-159 161-1 93 195-244 245-325 327402

VOLUME XV (1977)

r

Theory of Optical Parametric Amplification and Oscillation, W. BRUNNER, H. PAUL I1 Optical Properties of Thin Metal Films, P. ROUARD, A. MEESSEN III Projection-Type Holography, T. OKOSHI IV Quasi-Optical Techniques of Radio Astronomy, T.W. COLE V Foundations of the Macroscopic Electromagnetic Theory of Dielectric Media, J. VAN KRANENDONK, J.E. SIPE

1- 75 77-1 37 139-185 187-244

245-350

VOLUME XVI (1978) I II

I11 IV V VI VII

Laser Selective Photophysics and Photochemistry, VS. LETOKHOV 1- 69 Recent Advances in Phase Profiles Generation, J.J. CLAIR,C.1. ABITBOL 71-1 I7 Computer-Generated Holograms: Techniques and Applications, W.-H. LEE 1 19-232 Speckle Interferometry, A.E. ENNOS 233-288 Deformation Invariant, Space-Variant Optical Pattern Recognition, D. CASASENT, D. PSALTIS 289-356 Light Emission From High-Current Surface-Spark Discharges, R.E. BEVERLY 111 35741 I Semiclassical Radiation Theory Within a Quantum-Mechanical Framework, I.R. SENITZKY 4 I3448

VOLUME XVII (1980) Heterodyne Holographic Interferometry, R. DANDLIKER 1- 84 Doppler-Free Multiphoton Spectroscopy, E. GIACOBINO, B. CAGNAC 85-161 Ill The Mutual Dependence Between Coherence Properties of Light and Nonlinear Optical Processes, M. SCHUBERT, B. WrLHELMi 163-238 239-277 IV Michelson Stellar Interferometry, W.J. TANGO, R.Q. ' h s s V Self-Focusing Media with Variable Index of Refraction, A.L. MIKAELIAN 279-345 I II

VOLUME XVIII (1 980)

I II

Graded Index Optical Waveguides: A Review, A. GHATAK, K. THYAGARAJAN 1-126 Photocount Statistics of Radiation Propagating Through Random and Nonlinear Media, J. PERINA 127-203 Ill Strong Fluctuations in Light Propagation in a Randomly Inhomogeneous Medium, vr. TATARSKII, VU. ZAVOROTNYI 2 04-2 56 IV Catastrophe Optics: Morphologies of Caustics and their Diffraction Patterns, 257-346 M.V. BERRY,C. UPSTILL

CONTENTS OF PREVIOUS VOLUMES

429

VOLUME XIX (1981)

I

Theory of Intensity Dependent Resonance Light Scattering and Resonance I- 43 Fluorescence, B.R. MOLLOW I1 Surface and Size Effects on the Light Scattenng Spectra of Solids, D.L. MILLS, K.R. SUBBASWAMY 45-137 I11 Light Scattering Spectroscopy of Surface Electromagnetic Waves in Solids, S. USHIODA 139-21 0 21 1-280 IV Principles of Optical Data-Processing, H.J. BUTTEKWECK V The Effects of Atmospheric Turbulence in Optical Astronomy, E RODDIER 281-376

VOLUME XX (1983)

I

Some New Optical Designs for Ultra-Violet Bidimensional Detection of Astronomical Objects, G. COURTZS, P. CRuvELLIE& M. DETAILLE, M. SAYSSE I- 61 B. COLOMBEAU, I1 Shaping and Analysis of Picosecond Light Pulses, C. FROEHLY, M. VAMPOUILLE 63-153 111 Multi-Photon Scattering Molecular Spectroscopy, S. KIELICH 155-26 I 263-324 IV Colour Holography, P. HARIHARAN V Generation of Tunable Coherent Vacuum-Ultraviolet Radiation, W. JAMKOZ, B.P. STOICHEFF 325-380

VOLUME XXI (1984) Rigorous Vector Theones of Diffraction Gratings, D. MAYSTRE 1 I- 67 69-2 16 I1 Theory of Optical Bistability, L.A. LUCIATO 217-286 111 The Radon Transform and its Applications, H.H. BARRETT D.W. SWEENEY287-354 IV Zone Plate Coded Imaging: Theory and Applications, N.M. CEGLIO, V Fluctuations, Instabilities and Chaos in the Laser-Driven Nonlinear Ring Cavity, J.C. ENGLUND, R.R. SNAPP,W.C. SCHIEVE 3.55428

VOLUME XXll (1985) Optical and Electronic Processing of Medical Images, D. MALACARA I- 76 77-144 W.A. VAN DE GRIND,P. ZUIDEMA Quantum Fluctuations in Vision, M.A. BOUMAN, Spectral and Temporal Fluctuations of Broad-Band Laser Radiation, A.V MASALOV145-196 Holographic Methods of Plasma Diagnostics, G.V. OSTROVSKAYA, Yu.1. OSTROVSKY197-270 Fringe Formations in Deformation and Vibration Measurements using Laser Light, I. YAMACUCHI 271-340 341-398 VI Wave Propagation in Random Media: A Systems Approach, R.L. FANTE

I I1 III IV V

VOLUME XXIII (1986) Analytical Techniques for Multiple Scattering from Rough Surfaces, J.A. DESANTO, G.S. BROWN 1- 62 63-1 1 1 I1 Paraxial Theory in Optical Design in Terms of Gaussian Brackets, K. TANAKA R.P. NETTERFIELD 113-182 111 Optical Films Produced by Ion-Based Techniques, P.J. MARTIN, 183-220 IV Electron Holography, A. TONOMURA V Principles of Optical Processing with Partially Coherent Light, F.T.S. YU 221-275 I

430

CONTENTS OF PREVIOUS VOLUMES

VOLUME XXIV (1987) I II III IV V

Micro Fresnel Lenses, H. NISHIHARA, T. SUHARA Dephasing-Induced Coherent Phenomena, L. ROTHBERC Interferometry with Lasers, I? HARIHARAN Unstable Resonator Modes, K.E. OUCHSTUN Information Processing with Spatially Incoherent Light, I. GLASER

I- 37 39-101 103-1 64 165-387 389-509

VOLUME XXV (1988) Dynamical Instabilities and Pulsations in Lasers, N.B. ABRAHAM, P. MANDEL, L.M. NARDUCCI 1-190 I1 Coherence in Semiconductor Lasers, M. OHTSU,T. TAKO 191-278 Ill Principles and Design of Optical Arrays, WANCSHAOMIN, L. RONCHI 279-348 IV Aspheric Surfaces, G. SCHULZ 349415 I

VOLUME XXVl (1988) Photon Bunching and Antibunching, M.C. TEICH,B.E.A. SALEH Nonlinear Optics of Liquid Crystals, I.C. KHOO Ill Single-Longitudinal-Mode Semiconductor Lasers, G.P. ACRAWAL IV Rays and Caustics as Physical Objects, Yu.A. KRAVTSOV V Phase-Measurement Interferometry Techniques, K. CREATH I

II

1-104 105-1 61

163-225 227-348 349-3 9 3

VOLUME XXVll (1989)

I

The Self-Imaging Phenomenon and Its Applications, K. PATORSKI 1-108 Axicons and Meso-Optical Imaging Devices, L.M. SOROKO 109-1 60 111 Nonimaging Optics for Flux Concentration, I.M. BASSETT, W.T. WELFORD, 161-226 R. WINSTON IV Nonlinear Wave Propagation in Planar Structures, D. MIHALACHE, M. BERTOLOTTI, 221-3 13 C. SIBILIA V Generalized Holography with Application to Inverse Scattering and Inverse Source 3 15-397 Problems, R.P. PORTER II

VOLUME XXVIll (1990) I II

Digital Holography - Computer-Generated Holograms, 0. BRYNCDAHL, F. WYROWSKI 1- 86 Quantum Mechanical Limit in Optical Precision Measurement and Communication, Y. YAMAMOTO, S. MACHIDA,S. SAITO,N. IMOTO,T. YANACAWA, M. KITACAWA, G. BJORK 87-1 79 Ill The Quantum Coherence Properties of Stimulated Raman Scattering, M.G. RAWER, 181-270 LA. WALMSLEY IV Advanced Evaluation Techniques in Interferometry, J. SCHWIDER 271-359 V Quantum Jumps, R.J. COOK 361416

CONTENTS OF PREVIOUS VOLUMES

43 I

VOLUME XXIX (1991)

I

Optical Waveguide Diffraction Gratings: Coupling between Guided Modes, I- 63 D.G. HALL I1 Enhanced Backscattering in Optics, Yu.N. BARABANENKOV, Yu.A. KRAVTSOV, V.D. OZRIN,A.I. SAICHEV 65-197 199-29 1 I11 Generation and Propagation of Ultrashort Optical Pulses, I.P. CHRISTOV IV Triple-Correlation Imaging in Optical Astronomy, G. WEIGELT 293-3 19 V Nonlinear Optics in Composite Materials. 1. Semiconductor and Metal Crystallites in Dielectrics, C. FLYTZANIS, F. HACHE,M.C. KLEIN,D. RICARD,PH. ROUSSIGNOL 3 2 1 4 1 1

VOLUME XXX (1992) Quantum Fluctuations in Optical Systems, S. REYNAUD, A. HEIDMANN, E. GIACOBINO, I- 85 C. FABRE I1 Correlation Holographic and Speckle Interferometry, Yu.1. OSTROVSKY, V.P SHCHEPINOV 87-135 I11 Localization of Waves in Media with One-Dimensional Disorder, V.D. FREILIKHER, S.A. GREDESKUL 137-203 IV Theoretical Foundation of Optical-Soliton Concept in Fibers, Y. KODAMA, A. HASEGAWA 205-259 V Cavity Quantum Optics and the Quantum Measurement Process., P. MEYSTRE 261-355 I

VOLUME XXXI (1993) Atoms in Strong Fields: Photoionization and Chaos, P.W. MILONNI, B. SUNDARAM 1-137 Light Diffraction by Relief Gratings: A Macroscopic and Microscopic View, E. POPOV 139-1 87 111 Optical Amplifiers, N.K. DUTTA,J.R. SIMPSON 189-226 227-26 1 IV Adaptive Multilayer Optical Networks, D. PSALTIS,Y. QIAO V Optical Atoms, R.J.C. SPREEUW, J.P. WOERDMAN 263-3 19 VI Theory of Compton Free Electron Lasers, G. DATTOLI, L. GIANNESSI, A. RENIERI, A. TORRE 321412 I I1

VOLUME XXXII (1993)

I- 59 Guided-Wave Optics on Silicon: Physics, Technology and Status, B.P. PAL Optical Neural Networks: Architecture, Design and Models, F.T.S. Yu 61-144 111 The Theory of Optimal Methods for Localization of Objects in Pictures, L.P. YAROSLAVSKY 145-20 I IV Wave Propagation Theories in Random Media Based on the Path-Integral Approach, J. GOZANI, V.1. TATARSKII, VU. ZAVOROTNY 203-266 M.I. CHARNOTSKII, V Radiation by Uniformly Moving Sources. Vavilov-Cherenkov effect, Doppler effect in a medium, transition radiation and associated phenomena, V.L. GINZBURG 267-3 12 VI Nonlinear Processes in Atoms and in Weakly Relativistic Plasmas, G. MAINFRAY, C. MANUS 313-361 I

11

432

CONTENTS OF PREVIOUS VOLUMES

VOLUME XXXIII (1994) I

The Imbedding Method in Statistical Boundary-Value Wave Problems, V.1. KLY-

1-127 A. LUKS 129-202 Quantum Statistics of Dissipative Nonlinear Oscillators, V PERINOVA, 111 Gap Solitons, C.M. DE STEM, J.E. SlPE 203-260 IV Direct Spatial Reconstruction of Optical Phase from Phase-Modulated Images, VI, VLAD,D. MALACARA 261 -3 17 V Imaging through Turbulence in the Atmosphere, M.J. BERAN,J. OZ-VOGT 3 19-388 VI Digital Halftoning: Synthesis of Binary Images, 0. BRYNGDAHL, T. SCHEERMESSER, F. WYROWSKI 389463 ATSKIN

II

VOLUME XXXIV (1995) I 11

Quantum Interference, Superposition States of Light, and Nonclassical Effects, V BUZEK,P.L. KNIGHT Wave Propagation in Inhomogeneous Media: Phase-Shift Approach, L.P. PRES-

1-158

NYAKOV 159-181 I11 The Statistics of Dynamic Speckles, T. OKAMOTO, 183-248 T. ASAKURA IV Scattering of Light from Multilayer Systems with Rough Boundaries, 1. OHLIDAL, 249-33 I K. NAVRATIL, M. OHL~DAL V Random Walk and Diffusion-Like Models of Photon Migration in Turbid Media, G.H. WEISS 333402 A.H. GANDJBAKHCHE,

VOLUME XXXV (1 996) Transverse Patterns in Wide-Aperture Nonlinear Optical Systems, N.N. ROSANOV I 60 I I Optical Spectroscopy of Single Molecules in Solids, M. ORRIT,J. BERNARD, R. BROWN,B. LOUNIS 61-144 111 Interferometric Multispectral Imaging, K. ITOH 145-1 96 IV lnterferometric Methods for Artwork Diagnostics, D. PAOLETTI,G. SCHIRRIPA SPAGNOLO 197-255 V Coherent Population Trapping in Laser Spectroscopy, E. ARIMONDO 257-354 VI Quantum Phase Properties of Nonlinear Optical Phenomena, R. TANAS,A. MIRATs. GANTS~G 355446 NOWICZ, 1

-

VOLUME XXXVI (1996)

I

Nonlinear Propagation of Strong Laser Pulses in Chalcogenide Glass Films, 1- 47 V. CHLJMASH, I. COJOCARU, E. FAZIO,F. MICHELOTTI, M. BERTOLOTTI I1 Quantum Phenomena in Optical Interferometry, P. HARIHARAN, 49-128 B.C. SANDERS 111 Super-resolution by Data Inversion, M. BERTERO, 129-1 78 C. DE MOL IV Radiative Transfer: New Aspects of the Old Theory, YuA. KRAVTSOV,L.A. APRESYAN 179-244 V Photon Wave Function. I. BIALYNICKI-BIRULA 245-294

CUMULATIVE INDEX - VOLUMES I-XXXVII

ABELZS,F., Methods for Determining Optical Parameters of Thin Films II, ABELLA,I.D., Echoes at Optical Frequencies VII, ABITBOL, C.I., see Clair, J.J. XVI, Dynamical Instabilities and Pulsations ABRAHAM, N.B., J? MANDEL,L.M. NARDUCCI, in Lasers xxv, AGARWAL, G.S., Master Equation Methods in Quantum Optics XI, AGRANOVICH, VM., VL. GINZBURG, Crystal Optics with Spatial Dispersion IX, G.P., Single-Longitudinal-Mode Semiconductor Lasers XXVI, AGRAWAL, G.P., see Essiambre, R.-J. XXXVII, AGRAWAL, ALLEN,L., D.G.C. JONES,Mode Locking in Gas Lasers IX, AMMANN, E.O., Synthesis of Optical Birefringent Networks IX, APRESYAN, L.A., see Kravtsov, Yu.A. XXXVI, ARIMONDO, E., Coherent Population Trapping in Laser Spectroscopy XXXV, ARMSTRONG, J.A., A.W. SMITH,Experimental Studies of Intensity Fluctuations in Lasers VI, ARNAUD, J.A., Hamiltonian Theory of Beam Mode Propagation XI, T., see Okamoto, T. XXXIV, ASAKURA, T., see Peiponen, K.-E. XXXVII, ASAKURA, BALTES,H.P., On the Validity of Kirchhoff's Law of Heat Radiation for a Body in a Nonequilibrium Environment BARABANENKOV, Yu.N., Yu.A. KRAVTSOV,VD. OZRIN,A.I. SAICHEV,Enhanced Backscattering in Optics BARAKAT, R., The Intensity Distribution and Total Illumination of Aberration-Free Diffraction Images BARRETT,H.H., The Radon Transform and its Applications BASHKIN, S., Beam-Foil Spectroscopy BASSETT,I.M., W.T. WELFORD, R. WINSTON, Nonimaging Optics for Flux Concentration BECKMANN, P., Scattering of Light by Rough Surfaces BERAN,M.J., J. OZ-VOGT,Imaging through Turbulence in the Atmosphere J., see Orrit, M. BERNARD, BERRY,M.V, C. UPSTILL,Catastrophe Optics: Morphologies of Caustics and their Diffraction Patterns BERTERO,M., C. DE MOL, Super-resolution by Data Inversion BERTOLOTTI, M., see Mihalache, D. BERTOLOTTI, M., see Chumash, V BEVERLY 111, R.E., Light Emission From High-Current Surface-Spark Discharges BIALYNICKI-BIRULA, I., Photon Wave Function BJORK,G., see Yamamoto, Y. 43 3

249 139 71 1 1 235 163 185 179 123 179 257

211 247 183 57

XIII,

I

XXIX,

65

I, XXI, XII, XXVII, VI, XXXIII, XXXV,

67 217 287 161 53 319 61

XVIII, XXXVI, XXVII, XXXVI, XVI, XXXVI, XXVIII,

257 129 227 1

357 245 87

434

CUMULATIVE INDEX

~

VOLUMES I-XXXVII

BLOOM,A.L., Gas Lasers and their Application to Precise Length Measurements BOUMAN, M.A., W.A. VAN DE GRIND,F! ZUIDEMA, Quantum Fluctuations in Vision BOUSQUET, P., see Rouard, F! BROWN,G.S., see DeSanto, J.A. BROWN,R., see Orrit, M. BRUNNER, W., H. PAUL,Theory of Optical Parametric Amplification and Oscillation BRYNGDAHL, O., Applications of Shearing Interferometry BRYNGDAHL, O., Evanescent Waves in Optical Imaging BRYNGDAHL, O., F. WYROWSKI, Digital Holography - Computer-Generated Holograms BRYNGDAHL, O., T. SCHEERMESSER, F. WYROWSKI, Digital Halftoning: Synthesis of Binary Images BURCH,J.M., The Metrological Applications of Diffraction Gratings BUTTERWECK, H.J., Principles of Optical Data-Processing BUZEK, V., P.L. KNIGHT,Quantum Interference, Superposition States of Light, and Nonclassical Effects

IX, XXII, IV, XXIII, XXXV, XV, IV, XI, XXVIII,

1

77 145 1

61 1 37 167 1

XXXIII, 389 II, 73 XIX, 211 XXXIV.

I

XVII, CAGNAC, B., see Giacobino, E. CASASENT,D., D. PSALTIS,Deformation Invariant, Space-Variant Optical Pattern XVI, Recognition CEGLIO, N.M., D.W. SWEENEY, Zone Plate Coded Imaging: Theory and Applications XXI, V.U. ZAVOROTNY, Wave Propagation CHARNOTSKII, M.I., J. GOZANI,V.I. TATARSKII, Theories in Random Media Based on the Path-Integral Approach XXXII, CHIAO,R.Y., A.M. STEINBERG, Tunneling Times and Superluminality XXXVII, XIII, CHRISTENSEN, J.L., see Rosenblum, W.M. CHRISTOV, I.P., Generation and Propagation of Ultrashort Optical Pulses XXIX, V., I. COJOCARU,E. FAZIO,F, MICHELOTTI, M. BERTOLOTTI, Nonlinear CHUMASH, XXXVI, Propagation of Strong Laser Pulses in Chalcogenide Glass Films CLAIR,J.J., C.I. ABITBOL,Recent Advances in Phase Profiles Generation XVI, CLARRICOATS, P.J.B., Optical Fibre Waveguides - A Review XIV, COHEN-TANNOUDJI, C., A. KASTLER,Optical Pumping v, COJOCARU, I., see Chumash, V: XXXVI, COLE,T. W., Quasi-Optical Techniques of Radio Astronomy XV, XX, COLOMBEAU, B., see Froehly, C. COOK,R.J., Quantum Jumps XXVIII, COURT~S, G., P. CRUVELLIER, M. DETAILLE, M. SAi'SSE, Some New Optical Designs xx, for Ultra-Violet Bidimensional Detection of Astronomical Objects CREATH,K., Phase-Measurement Interferometry Techniques XXVI, CREW,A.V., Production of Electron Probes Using a Field Emission Source XI, F!, see Courtes, G. xx, CRUVELLIER, CUMMINS, H.Z., H.L. SWINNEY, Light Beating Spectroscopy VIII,

85 289 287 203 345 69 199 1 71 327 1

1 187 63 361 1 349 223 1 133

DAINTY, J.C., The Statistics of Speckle Patterns XIV, I DANDLIKER, R., Heterodyne Holographic Interferometry XVII, 1 DATTOLI,G., L. GIANNESSI, A. RENIERI, A. TORRE,Theory of Compton Free Electron XXXI, 321 Lasers DE MOL, C., see Bertero, M. XXXVI, 129 DE STERKE,C.M., J.E. SIPE,Gap Solitons XXXIII, 203 DECKER JR, J.A., see Hanvit, M. XII, 101 E., R.J. PEGIS,Methods of Synthesis for Dielectric Multilayer Filters VII, 67 DELANO, DEMARIA, A.J., Picosecond Laser Pulses IX, 31

43 5

CUMULATIVE INDEX - VOLUMES I-XXXVII

DESANTO,J.A., G.S. BROWN,Analytical Techniques for Multiple Scattering from Rough Surfaces XXIII, DETAILLE, M., see Courtk, G . xx, DEXTER,D.L., see Smith, D.Y. X, DRAGOMAN, D., The Wigner Distribution Function in Optics and Optoelectronics XXXVII, DREXHAGE, K.H., Interaction of Light with Monomolecular Dye Layers XII, DUGUAY, M.A., The Ultrafast Optical Kerr Shutter XIV, XXXI, DLI~TA, N.K., J.R. SIMPSON, Optical Amplifiers EBERLY,J.H., Interaction of Very Intense Light with Free Electrons VII, ENGLUND, J.C., R.R. SNAPP,W.C. SCHIEVE, Fluctuations, Instabilities and Chaos in the Laser-Driven Nonlinear Ring Cavity XXI, ENNOS,A.E., Speckle Interferometry XVI, Soliton Communication Systems ESSIAMBRE, R.-J., G.P. AGRAWAL, XXXVII, FABELINSKII, I.L., Spectra of Molecular Scattering of Light XXXVII, FABRE,C., see Reynaud, S. xxx, FANTE,R.L., Wave Propagation In Random Media: A Systems Approach XXII, FAZIO,E., see Chumash, V XXXVI, I, FIORENTINI, A,, Dynamic Characteristics of Visual Processes Nonlinear Optics FLYTZANIS,C., F. HACHE,M.C. KLEIN,D. RICARD,PH. ROUSSIGNOL, XXIX, in Composite Materials. 1. Semiconductor and Metal Crystallites in Dielectrics FOCKE,J., Higher Order Aberration Theory IV, VI, FRANCON, M., S. MALLICK, Measurement of the Second Order Degree of Coherence FREILIKHER, VD., S.A. GREDESKUL, Localization of Waves in Media with OneDimensional Disorder xxx, FRIEDEN, B.R., Evaluation, Design and Extrapolation Methods for Optical Signals, Based on Use of the Prolate Functions IX, FROEHLY, C., B. COLOMBEAU, M. VAMPOUILLE, Shaping and Analysis of Picosecond Light Pulses XX, VIII, FRY,G.A., The Optical Performance of the Human Eye GABOR,D., Light and Information GAMO,H., Matrix Treatment of Partial Coherence GANDJBAKHCHE, A.H., G.H. WEISS,Random Walk and Diffusion-Like Models of Photon Migration in Turbid Media GANTSOG, Ts., see TanaS, R. Graded Index Optical Waveguides: A Review GHATAK,A., K. THYAGARAJAN, GHATAK,A.K., see Sodha, M.S. GIACOBINO, E., B. CAGNAC,Doppler-Free Multiphoton Spectroscopy GIACOBINO, E., see Reynaud, S. GIANNESSI, L., see Dattoli, G. VL., see Agranovich, VM. GINZBURG, GINZBURG, VL., Radiation by Uniformly Moving Sources. Vavilov-Cherenkov effect, Doppler effect in a medium, transition radiation and associated phenomena GIOVANELLI, R.G., Diffusion Through Non-Uniform Media GLASER,I., Information Processing with Spatially Incoherent Light Applications of Optical Methods in the Diffraction GNIADEK,K., J. PETYKIEWICZ, Theory of Elastic Waves GOODMAN, J.W., Synthetic-Aperture Optics GOZANI, J., see Chamotskii, M.I.

1 1

165 1

163 161 189 359 355 233 185 95 1 341 1 253 321 I 71 137 311 63 51

I, 109 Ill, 187

XXXIV, xxxv, XVIII, XIII, XVII, xxx, XXXI, IX,

333 355 1 169

85 1

321 235

XXXII, 267 11, 109

XXIV, 389 IX, 281 I VIII, XXXII, 203

43 6

CUMULATIVE INDEX - VOLUMES I-XXXVII

GRAHAM,R., The Phase Transition Concept and Coherence in Atomic Emission GREDESKUL, S.A., see Freilikher, V.D.

XII, 233 XXX, 137

XXIX, 321 HACRE,F., see Flytzanis, C. HALL, D.G., Optical Waveguide Diffraction Gratings: Coupling between Guided XXIX, I Modes XX, 263 HARIHARAN, P., Colour Holography XXIV, 103 HARIHARAN, P., Interferometry with Lasers XXXVI, 49 HARIHARAN, P., B.C. SANDERS, Quantum Phenomena in Optical Interferometry XII, 101 HARWIT,M., J.A. DECKERJR, Modulation Techniques in Spectrometry XXX, 205 HASEGAWA, A,, see Kodama, Y. xxx, 1 HEIDMANN, A,, see Reynaud, S. X, 289 HELSTROM, C.W., Quantum Detection Theory VI, 171 HERRIOT, D.R., Some Applications of Lasers to Interferometry HUANG,T.S., Bandwidth Compression of Optical Images I x, IMOTO,N., see Yamamoto, Y. ITOH,K., Interferometric Multispectral Imaging JACOBSSON, R., Light Reflection from Films of Continuously Varying Refractive Index P., B. ROIZEN-DOSSIER, Apodisation JACQUINOT, JAMROZ,W., B.P. STOICHEFF, Generation of Tunable Coherent Vacuum-Ultraviolet Radiation JONES,D.G.C., see Allen, L.

XXVIII, 87 XXXV, 145 V, 247 29

Ill,

XX, 325 IX, 179

KASTLER, A., see Cohen-Tannoudji, C. v, KELLER,O., Local Fields in Linear and Nonlinear Optics of Mesoscopic Systems XXXVII, KHOO,I.C., Nonlinear Optics of Liquid Crystals XXVI, KIELICH, S., Multi-Photon Scattering Molecular Spectroscopy XX, KINOSITA, K., Surface Deterioration of Optical Glasses IV, KITAGAWA, M., see Yamamoto, Y. XXVIII, XXIX, KLEIN,M.C., see Flytzanis, C. V.I., The Imbedding Method in Statistical Boundary-Value Wave Problems XXXIII, KLYATSKIN, KNIGHT, P.L., see BuZek, V XXXIV, KODAMA,Y., A. HASEGAWA, Theoretical Foundation of Optical-Soliton Concept in Fibers XXX, G., Multiple-Beam Interference and Natural Modes in Open Resonators VII, KOPPELMAN, KOTTLER, F,, The Elements of Radiative Transfer Ill, KOITLER,F., Diffraction at a Black Screen, Part I: Kirchhoff’s Theory IV, KOTTLER, F., Diffraction at a Black Screen, Part II: Electromagnetic Theory VI, KRAVTSOV, Yu.A., Rays and Caustics as Physical Objects XXVI, KRAVTSOV, Yu.A., see Barabanenkov, Yu.N. XXIX, KRAVTSOV, Yu.A., L.A. APRESYAN, Radiative Transfer: New Aspects ofthe Old Theory XXXVI, KUBOTA,H., Interference Color I,

LABEYRIE, A,, High-Resolution Techniques in Optical Astronomy LEAN,E.G., Interaction of Light and Acoustic Surface Waves LEE, W.-H., Computer-Generated Holograms: Techniques and Applications LEITH,E.N., J. UPATNIEKS, Recent Advances in Holography LETOKHOV, V.S., Laser Selective Photophysics and Photochemistry LEVI,L., Vision in Communication

XIV, XI, XVI, VI, XVI, VIII,

I 257 105 155 85 87 321 1 1

205 1

I 281 331 227 65 179 211

47 123 119 1 1

343

437

CUMULATIVE INDEX - VOLUMES I-XXXVII

LIPSON,H., C.A. TAYLOR, X-Ray Crystal-Structure Determination as a Branch of Physical Optics V, 287 Lou~rs,B., see Orrit, M. XXXV, 61 LUGIATO, L.A., Theory of Optical Bistability XXI, 69 XXXIII, 129 LuKS,A,, see Peiinovi, V MACHIDA, S., see Yamamoto, Y. MAINFRAY, G., C. MANUS,Nonlinear Processes in Atoms and in Weakly Relativistic Plasmas MALACARA, D., Optical and Electronic Processing of Medical Images MALACARA, D., see Vlad, W. MALLICK, S., see FranGon, M. MANDEL, L., Fluctuations of Light Beams MANDEL, L., The Case For and Against Semiclassical Radiation Theory MANDEL, P., see Abraham, N.B. MANUS,C., see Mainfray, G. MARCHAND, E.W., Gradient Index Lenses MARTIN, P.J., R.P. NETTERFIELD, Optical Films Produced by Ion-Based Techniques MASALOV, A.V, Spectral and Temporal Fluctuations of Broad-Band Laser Radiation MAYSTRE, D., Rigorous Vector Theories of Diffraction Gratings MEESSEN, A,, see Rouard, P. MEHTA,C.L., Theory of Photoelectron Counting MEYSTRE, P., Cavity Quantum Optics and the Quantum Measurement Process. MICHELOTTI, F., see Chumash, V MIHALACHE, D., M. BERTOLOTTI, C. SIBILIA, Nonlinear Wave Propagation in Planar Structures MIKAELIAN, A.L., M.L. ER-MIKAELIAN, Quasi-Classical Theory of Laser Radiation MIKAELIAN, A.L., Self-Focusing Media with Variable Index of Refraction Surface and Size Effects on the Light Scattering MILLS,D.L., K.R. SUBBASWAMY, Spectra of Solids MILONNI, P.W., B. SUNDARAM, Atoms in Strong Fields: Photoionization and Chaos MIRANOWICZ, A,, see TanaS, R. K., Wave Optics and Geometrical Optics in Optical Design MIYAMOTO, MOLLOW,B.R., Theory of Intensity Dependent Resonance Light Scattering and Resonance Fluorescence K., Instruments for the Measuring of Optical Transfer Functions MURATA, MUSSET,A,, A. THELEN, Multilayer Antireflection Coatings NARDUCCI, L.M., see Abraham, N.B. NAVRATIL, K., see Ohlidal, 1. NETTERFIELD, R.P., see Martin, P.J. NISHIHARA, H., T. SUHARA, Micro Fresnel Lenses

XXVIII,

87

XXXII, 313 XXII, 1 XXXIII, 261 VI, 71 11, 181

XIII, xxv, XXXII, XI, XXIII, XXII, XXI, xv, VIII, XXX, XXXVI,

27 1

313 305 113 145 1 77 373 261 1

XXVII, 227 VII, 23 1 XVII, 279 XIX, 45 XXXI, 1 x x x v , 355 I, 31 XIX, I V, 199 VIII, 201 xxv, 1 XXXIV, 249 XXIII, 113 XXIV, 1

OHLIDAL, I., K. N A V ~ T IM. L , OHLIDAL, Scattering of Light from Multilayer Systems XXXIV, 249 with Rough Boundaries XXXIV, 249 OHLIDAL, M., see Ohlidal, 1. XXV, 191 OHTSU,M., T. TAKO,Coherence in Semiconductor Lasers XXXIV, 183 OKAMOTO, T., T. ASAKURA, The Statistics of Dynamic Speckles XV, 139 OKOSHI, T., Projection-Type Holography VII, 299 OOUE,S., The Photographic Image

43 8

CUMULATIVE INDEX

-

VOLUMES I-XXXVll

OKRIT,M., J. BERNARD,R. BROWN,B. LOLJNIS,Optical Spectroscopy of Single Molecules in Solids XXXV, 61 G.V., Yu.1. OSTKOVSKY, Holographic Methods of Plasma Diagnostics XXII, 197 OSTKOVSKAYA, OSTKOVSKY, Yu.I., see Ostrovskaya, G.V. XXII, 197 OSTKOVSKY, Yu.I., VP. SHCHEPINOV, Correlation Holographic and Speckle Interferometry x x x , 87 OUGHSTUN, K.E., Unstable Resonator Modes XXIV, 165 XXXIII, 319 OZ-VCCT, J., see Beran, M.J. XXIX, 65 OZKIN,V.D., see Barabanenkov, Yu.N. PAL,B.P., Guided-Wave Optics on Silicon: Physics, Technology and Status XXXII, PAOLETTI,D., G. SCHIKKIPA SPACNOLO,Interferometric Methods for Artwork Diagnostics XXXV, K., The Self-Imaging Phenomenon and Its Applications XXVII, PATOKSKI, PAUL,H., see Brunner, W. xv, PEGIS,R.J., The Modern Development of Hamiltonian Optics 1, PEGIS,R.J., see Delano, E. VII, PEIPONEN, K.-E., E.M. VARTIAINEN, T. ASAKURA,Dispersion Relations and Phase Retrieval in Optical Spectroscopy XXXVII, PERINA,J., Photocount Statistics of Radiation Propagating Through Random and Nonlinear Media XVIII, PERINOVA, V., A. LuKS, Quantum Statistics of Dissipative Nonlinear Oscillators XXXIII, PEKSHAN, P.S., Non-Linear Optics V, IX, PETYKIEWICZ, J., see Gniadek, K. PICHT,J., The Wave of a Moving Classical Electron V, POPOV,E., Light Diffraction by Relief Gratings: A Macroscopic and Microscopic View XXXI, PORTER,R.P., Generalized Holography with Application to Inverse Scattering and Inverse Source Problems XXVII, PRESNYAKOV, L.P., Wave Propagation in Inhomogeneous Media: Phase-Shift Approach XXXIV, XVI, PSALTIS, D., see Casasent, D. PSALTIS, D., Y. QIAO,Adaptive Multilayer Optical Networks XXXI, QIAO,Y., see Psaltis, D.

1

197 1 1 1

67 57 127 I29 83 281 351 139 315 I59 289 227

XXXI, 227

M.G., I . A . WALMSLEY, The Quantum Coherence Properties of Stimulated RAYMER, XXVIII, 181 Raman Scattering A,, see Dattoli, G. RENIERI, XXXI, 321 REYNAUD, S., A . HEIDMANN, E. GIACOBINO, C. FABRE,Quantum Fluctuations in Optical 1 Systems xxx, XXIX, 321 RICARD, D., see Flytzanis, C. RISEBEKG, L.A., M.J. WEBEK,Relaxation Phenomena in Rare-Earth Luminescence XIV, 89 RISKEN,H., Statistical Properties of Laser Light VIII, 239 RODDIER, XIX, 281 F., The Effects of Atmospheric Turbulence in Optical Astronomy 111, 29 ROIZEN-DOSSIER, B., see Jacquinot, P. RONCHI, L., see Wang Shaomin XXV, 279 ROSANOV,N.N., Transverse Patterns in Wide-Aperture Nonlinear Optical Systems xxxv. 1 W.M., J.L. CHKISTENSEN, Objective and Subjective Spherical Aberration ROSENBLUM, Measurements of the Human Eye XIJI, 69 ROTHBEKG, L., Dephasing-Induced Coherent Phenomena XXIV, 39 Optical Constants of Thin Films IV, 145 ROUAKD, P, P. BOUSQUET,

CUMULATIVE INDEX

~

439

VOLUMES I-XXXVII

ROUARD,P., A. MEESSEN,Optical Properties of Thin Metal Films ROUSSIGNOL, PH., see Flytzanis, c. RUBINOWICZ, A,, The Miyamoto-Wolf Diffraction Wave RUDOLPH, D., see Schmahl, G.

xv, XXIX, IV, XIV,

77 321 199 195

XXIX, SAICHEV, A.I., see Barabanenkov, Yu.N. SAISSE,M., see Courtts, G. xx, SAITO,S., see Yamamoto, Y. XXVIII, VI, SAKAI,H., see Vanasse, G.A. SALEH,B.E.A., see Teich, M.C. XXVI, SANDERS, B.C., see Hariharan, P. XXXVI, XXXIII, SCHEERMESSER, T., see Bryngdahl, 0. XXI, SCHIEVE, W.C., see Englund, J.C. G., see Paoletti, D. XXXV, SCHIRRIPA SPAGNOLO, SCHMAHL, G., D. RUDOLPH, Holographic Diffraction Gratings XIV, SCHUBERT, M., B. WILHELMI, The Mutual Dependence Between Coherence Properties XVII, of Light and Nonlinear Optical Processes SCHULZ, G., J. SCHWIDER, Interferometric Testing of Smooth Surfaces XIII, xxv, SCHULZ, G., Aspheric Surfaces XIII, SCHWIDER, J., see Schulz, G. XXVIII, SCHWIDER, J., Advanced Evaluation Techniques in Interferometry X, SCULLY, M.O., K.G. WHITNEY, Tools of Theoretical Quantum Optics I.R., Semiclassical Radiation Theory Within a Quantum-Mechanical SENITZKY, Framework XVI, SHCHEPINOV, V.P., see Ostrovsky, Yu.1. XXX, XXVII, SIBILIA, C., see Mihalache, D. SIMPSON, J.R., see Dutta, N.K. XXXI, XV, SIPE,J.E., see Van Kranendonk, J. XXXIII, SIPE,J.E., see De Sterke, C.M. X, SITTIG,E.K., Elastooptic Light Modulation and Deflection XII, SLUSHER, R.E., Self-Induced Transparency VI, SMITH,A.W., see Armstrong, J.A. X, SMITH,D.Y., D.L. DEXTER,Optical Absorption Strength of Defects in Insulators SMITH,R.W., The Use of Image Tubes as Shutters XXI, SNAPP,R.R., see Englund, J.C. V K , TRIPATHI, Self-Focusing of Laser Beams in Plasmas SODHA,M.S., A.K. GHATAK, XIII, and Semiconductors XXVII, SOROKO,L.M., Axicons and Meso-Optical Imaging Devices XXXI, SPREEW,R.J.C., J.P. WOERDMAN, Optical Atoms V, STEEL,W.H., Two-Beam Interferometry XXXVII, STEINBERG, A.M., see Chiao, R.Y. XX, STOICHEFF, B.P., see Jamroz, W. IX, STROHBEHN, J.W., Optical Propagation Through the Turbulent Atmosphere STROKE,G.W., Ruling, Testing and Use of Optical Gratings for High-Resolution II, Spectroscopy SUBBASWAMY, K.R., see Mills, D.L. XIX, XXIV, SUHARA,T., see Nishihara, H. SLNDARAM,B., see Milonni, P.W. XXXI, SVELTO, O., Self-Focusing, Self-Trapping, and Self-phase Modulation of Laser Beams XII, XXI, SWEENEY,D.W., see Ceglio, N.M. V111, SWINNEY, H.L., see Cummins, H.Z.

65 1 87 259 1

49 389 355 197 195 163 93 349 93 271 89 413 87 227 189 245 203 229 53 211 165 45 355

x,

169 109 263 145 345 325 73 1

45 1

I

I 287 133

440

CUMULATIVE INDEX

-

VOLUMES 1-XXXVll

XXV, 191 TAKO,T., see Ohtsu, M. TANAKA, K., Paraxial Theory in Optical Design in Terms of Gaussian Brackets XXIII, 63 TANAS,R., A. MIRANOWICZ, Ts. GANTSOC,Quantum Phase Properties of Nonlinear Optical Phenomena x x x v , 355 XVII. 239 TANGO,W.J., R.Q. 7\urss, Michelson Stellar Interferometry TATARSKII, VI., VU. ZAVOROTNYI, Strong Fluctuations in Light Propagation in a Randomly Inhomogeneous Medium XVIII, 204 XXXII, 203 TATARSKII, VI., see Chamotskii, M.I. V, 287 TAYLOR, C.A., see Lipson, H. XXVI, 1 TEICH,M.C., B.E.A. SALEH,Photon Bunching and Antibunching VII, 231 TER-MIKAELIAN, M.L., see Mikaelian, A.L. VIII, 201 THELEN, A., see Musset, A. VII, 169 THOMPSON, B.J., Image Formation with Partially Coherent Light 1 XVIII, THYAGARAJAN, K., see Ghatak, A. XXIII, 183 TONOMURA, A,, Electron Holography XXXI, 321 TORRE,A,, see Dattoli, G. XIII. 169 TRIPATHI, VK., see Sodha, M.S. TSUJIUCHI, J., Correction of Optical Images by Compensation of Aberrations and by 11, 131 Spatial Frequency Filtering XVII, 239 Twlss, R.Q., see Tango, W.J. UPATNIEKS, J., see Leith, E.N. UPSTILL, C., see Berry, M.V USHIODA,S., Light Scattering Spectroscopy of Surface Electromagnetic Waves in Solids

VI, 1 XVIII, 257 XIX, 139

XX, VAMPOUILLE, M., see Froehly, C. XXII, VANDE GRIND,W.A., see Bouman, M.A. I, VANHEEL,A.C.S., Modem Alignment Devices VAN KIJANENDONK, J., J.E. SIPE, Foundations of the Macroscopic Electromagnetic XV, Theory of Dielectric Media VANASSE,G.A., H. SAKAI,Fourier Spectroscopy VI, XXXVII, VARTIAINEN, E.M., see Peiponen, K.-E. VERNIER, P.J., Photoemission XIV, VLAD,Vl., D. MALACARA, Direct Spatial Reconstruction of Optical Phase from PhaseModulated Images XXXIII, WALMSLEY, I.A., see Raymer, M.G. WANGSHAOMIN, L. RONCHI, Principles and Design of Optical Arrays WEBER,M.J., see Riseberg, L.A. WEIGELT, G., Triple-Correlation Imaging in Optical Astronomy WEISS,G.H., see Gandjbakhche, A.H. WELFORD, W.T., Aberration Theory of Gratings and Grating Mountings WELFORD,W.T., Aplanatism and Isoplanatism WELFORD, W.T., see Bassett, I.M. WHITNEY, K.G., see Scully, M.O. WILHELMI, B., see Schubert, M. R., see Bassett, I.M. WINSTON, WOERDMAN, J.P., see Spreeuw, R.J.C. WOLTER,H., On Basic Analogies and Principal Differences between Optical and Electronic Information

XXVIII, XXV, XIV, XXIX, XXXIV, IV, XIII, XXVII, X, XVII, XXVII, XXXI,

63 77 289 245 259 57 245 261 181 279 89 293 333 241 267 161

89 163 161

263

I, 155

CUMULATIVE INDEX

~

44 1

VOLUMES I-XXXVII

WYNNE, C.G., Field Correctors for Astronomical Telescopes F., see Bryngdahl, 0. WYROWSKI, WYROWSKI, F,, see Bryngdahl, 0.

X, 137 XXVIII, I XXXIII, 389

YAMAGUCHI, I., Fringe Formations in Deformation and Vibration Measurements using Laser Light XXII, K., Design of Zoom Lenses YAMAJI, VI, YAMAMOTO, T., Coherence Theory of Source-Size Compensation in Interference Microscopy VI11, M. KITAGAWA, YAMAMOTO, Y., S. MACHIDA, S . SAITO, N. IMOTO, T. YANAGAWA, G. BJORK, Quantum Mechanical Limit in Optical Precision Measurement and XXVIII, Communication YANAGAWA, T., see Yamamoto, Y. XXVIII, YAROSLAVSKY, L.P., The Theory of Optimal Methods for Localization of Objects in XXXII, Pictures YOSHINAGA,H., Recent Developments in Far Infrared Spectroscopic Techniques XI, XXIII, Yu, F.T.S., Principles of Optical Processing with Partially Coherent Light XXXII, Yu, F.T.S., Optical Neural Networks: Architecture, Design and Models ZAVOROTNY, V.U., see Charnotskii, M.I. V.U., see Tatarskii, V.I. ZAVOROTNYI, ZUIDEMA, P., see Bouman, M.A.

271 105 295

87 87 145

77 221 61

XXXII, 203 XVIII, 204 XXII, 77

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