Acousto-optics (Optical Science and Engineering)

IICOUSTO-OPTICS

OPTICAL ENGINEEFUNG Series Editor

Brian J. Thompson Distinguished UniversityProfessor Professor of Optics Provost Emeritus University of Rochester Rochester, New York

1. Electron and Ion Microscopy and Microanalysis: Principles and Applications, Lawrence E. Murr 2. Acousto-OpticSignalProcessing:TheoryandImplementation, edited by Norman J. Berg and John N. Lee 3. Electro-optic and Acousto-Optic Scanning and Deflection, Milton Gottlieb, Clive L. M. Ireland, and John Martin Ley 4. Single-ModeFiberOptics:PrinciplesandApplications, Luc B. Jeunhomrne 5. PulseCodeFormats for FiberOptical Data Communication:Basic Principles and Applications, David J, Morris 6. Optical Materials: An Introduction t o Selection and Application, Solomon Musikant 7. Infrared Methods for GaseousMeasurements:TheoryandPractice, edited by Joda Wormhoudt 8. LaserBeamScanning:Opto-MechanicalDevices,Systems,and Data Storage Optics, edited by Gerald F, Marshall 9. Opto-Mechanical Systems Design, Paul R. Yoder, Jr. 10. Optical Fiber Splices and Connectors: Theory and Methods, Calvin M. Miller with Stephen C. Mettler and Ian A. White 1 1. Laser Spectroscopy and Its Applications, edited by Leon J. Radziem-, ski, Richard W. Solarz and Jeffrey A. Paisner 12. InfraredOptoelectronics:DevicesandApplications, WilliamNunley and J. Scott Bechtel 13. Integrated Optical Circuits and Components: Design and Applications, edited by Lynn D. Hutcheson 14. Handbook of Molecular Lasers, edited by Peter K. Cheo

15. Handbook of Optical Fibers and Cables, Hiroshi Murata 16.Acousto-Optics, AdrianKorpel 17. Procedures in Applied Optics, John Strong 18. Handbook of Solid-state Lasers, edited by Peter K. Cheo 19.OpticalComputing: Digital andSymbolic, edited by RaymondArrathoon 20. Laser Applications in Physical Chemistry, edited by D. K. Evans 21. .Laser-Induced Plasmas and Applications, edited by Leon J. Radziernski and DavidA. Crerners 22. Infrared Technology Fundamentals, Irving J. Spiro and Monroe Schlessinger 23. Single-Mode Fiber Optics: Principles and Applications, Second Edition, Revised and Expanded, Luc B. Jeunhornrne 24. Image Analysis Applications, edited by Rangachar Kasturi and Mohan M. Trivedi 25. Photoconductivity: Art, Science,andTechnology, N. V. Joshi 26. Principles of Optical Circuit Engineering, Mark A. Mentzer 27. Lens Design, Milton Laikin 28. Optical Components,Systems,andMeasurementTechniques, Rajpal S. Sirohi and M. P. Kothiyal 29. Electronand IonMicroscopy andMicroanalysis:Principlesand Applications, Second Edition, Revised and Expanded, Lawrence E. Murr 30. Handbook of Infrared Optical Materials, edited by Paul Klocek 31 Optical Scanning, edited by Gerald F.Marshall 32. Polymers for Lightwave and Integrated Optics: Technology and Applications, edited by Lawrence A. Hornak 33. Electro-Optical Displays, edited by Moharnrnad A. Karirn 34. Mathematical Morphology in Image Processing, edited by Edward R. Dougherty 35.Opto-Mechanical SystemsDesign:SecondEdition,RevisedandExpanded, Paul R. Yoder, Jr. 36. Polarized Light: Fundamentals and Applications, Edward Collett 37. Rare Earth Doped Fiber Lasers and Amplifiers, edited by Michel J. F. Digonnet 38. Speckle Metrology, edited by Rajpal S. Sirohi 39. OrganicPhotoreceptors for Imaging Systems, Paul M. Borsenberger and David S. Weiss 40. Photonic Switchingand Interconnects, edited by Abdellatif Marrakchi 41. Designand Fabrication of Acousto-Optic Devices, edited by Akis P. Goutzoulis and Dennis R. Pape 42. Digital Image Processing Methods, edited by Edward R. Dougherty 43. VisualScienceandEngineering: Models andApplications, edited by D. H. Kelly 44. Handbook of Lens Design, Daniel Malacara and Zacarias Malacara 45. Photonic Devices and Systems, edited by Robert G. Hunsperger

.

46.InfraredTechnology Fundamentals:SecondEdition,RevisedandExpanded, edited by Monroe Schlessinger 47. Spatial Light Modulator Technology:Materials,Devices,andApplications, edited by Uzi Efron 48. Lens Design: Second Edition, Revised and Expanded, Milton Laikin 49. Thin Films for Optical Systems, edited by Francois R. Flory 50. Tunable Laser Applications, edited by F. J. Duarte 5 1. Acousto-Optic Signal Processing: Theory and Implementation, Second Edition, edited by NormanJ. Berg and John M. Pellegrino 52. Handbook of Nonlinear Optics, Richard L. Sutherland 53. Handbook of Optical Fibers and Cables: Second Edition, Hiroshi Murata 54. Optical Storage and Retrieval: Memory, Neural Networks, and Fractals, edited by Francis T. S. Yu and Suganda Jutamulia 55. Devices for Optoelectronics, Wallace B. Leigh 56. Practical Design and Production of Optical Thin Films, Ronald R. Willey 57. Acousto-Optics: Second Edition, Adrian Korpel

Additional Volumes in Preparation

SECOND EDITION

ADRIAN KORPEL

Korpel Arts and Sciences Iowa City, Iowa

.Marcel Dekker, Inc.

New York. Basel Hong Kong

Library of Congress Cataloging-in-Publication Data

Korpel, Adrian Acousto-optics l Adrian Korpel. -2nd ed. p. cm. -(Optical engineering ;57) Includes index. ISBN 0-8247-9771-X (alk. paper) 1.Acoustooptics. I. Title. II. Series:Opticalengineering (Marcel Dekker, Inc.) ;v. 57 QC220.5.K67 1996 621.382'84~20

96-41103 CIP

The publisher offersdiscounts on this book when ordered in bulk quautities. For more information, write to Special SalesD'rofessional Marketing at the address below.

This book is printed on acid-free paper.

Copyright 0 1W by MARCEL DEKKER, INC. AU Rights Reserved. Neither this book nor any part may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, microfilming, andrecording,orbyanyinformationstorageandretrievalsystem,without permission in writing from the publisher. MARCEL DEKKER,INC. 270 Madison Avenue, New York, New York 10016 Current printing (last digit): 1 0 9 8 7 6 5 4 3 2 1

PRINTED IN TRE UNITED STATES OF AMERICA

To

Loni and Pat in loving memory

This Page Intentionally Left Blank

From the Series Editor

Acousto-optics is an important subfield of optical science and engineering, and as such is quite properly well represented in our series on Optical Engineering. This representation includes the fundamentals and important applications including signal processing, scanning, spectrum analyzers, tuned filters, and imaging systems. The fundamentals and the underpinning for these and other applications were thoroughly covered in thefirst.edition of Acousto-Optics, which was published in our series in 1988 (volume 16). Now we are pleasedto bring out a new edition of this important treatise that covers both the basic concepts and how these basic concepts are used in practical devices, subsystems,, and systems. Many new sections have been added to both the theoryand practice parts of the book. As the editor of this series I wrote in the foreword of another book that acousto-optics is a confusing termto the uninitiated, but it, in fact, refers to a well-documented subfield ofboth optics and acoustics. It refers, of course, to the interactions of light and sound, between lightwaves and sound waves, and between photons and phonons. More specifically, it refersto the control and modification of a light beam by an acoustic beam. Thus, wehave devices that use sound to modulate, deflect, refractand diffract light. Acousto-Optics takes the mystery and confusion out of the subject and provides a firm foundation forfurther study and applications. Brian J. Thompson V

This Page Intentionally Left Blank

Preface to the Second Edition

Since 1988, when the first edition of this book appeared, acousto-optics - at one point declared moribund by many- has gone through a minor renaissance, both in theoryand in practice. Part of this renaissance is dueto much increased contact with scientists from the former Soviet Union, whose original and sophisticated research gave a new impetus to the field. Another important factor is the increased interest in signal processing and acoustooptic tunable filters. Finally, the development of more powerful formalisms and algorithms has also contributed substantially. In this second edition I have tried to take these developmentsinto account. At the same time I have put more emphasis on applications and numerical methods. The aim of the book is, however, unchanged: to generate more insight rather than supply convenient recipes. The heuristic approach (Chapter 3) has remained largely unchanged, except for a new, separate sub-section on the near-Bragg regime. In the formal .approach (Chapter 4), I have added a section on interaction with curved sound wave fronts, because far more is known now than in 1988. In Chapter 5, four new sections on the numerical approach have been addedto the original six. These include the carrier-less split step method, which is the most powerful and convenient simulation tool available. The .number of selected applications discussed (Chapter 6) has also been increased by four, vii

viii

Preface to the Second Edition

and a substantial treatment of fundamental signalprocessing isnow included. Similarly, a fairly complete discussion of beam steering has been added to the section on light deflectors. The Quasi-Theorem, a particular favorite ofmine, has beengiven its own section. The remaining new application sections deal with spectrum analysis, optical phase and amplitude measurements, and schlierenimaging. Anotherimportant application, acousto-optic tunable filters, is treated in a new section following anisotropic Bragg diffraction in Chapter 8. Anoverviewof spectral formalisms commonly used in the analysis of interaction has also been added to this chapter. The appendix now contains, in addition to the summary ofresearch and designformulas, a brief description of the stationary phase method. I would like to thank all my colleagues, especially in Russia, for interesting and sometimes heated discussions. Together with my students' persistent questioning, these have greatly helped me in writing this second edition. Adrian Korpel

Preface to the First ,Edition

It is only fair that a writer show his prejudices to the reader so as to give timely warning. My prejudices are those of an engineerlapplied physicist- I preferBabylonianoverEuclideanmathematics, to borrow Richard Feynman's classification. Simple pictures and analogies fascinate me, while I abhor existence theorems, uniqueness proofs, and opaque equations. On the other hand, I do realize that one person's mathematics is another person's physics, and some of my best friends are Euclideans. Many a time they have shown my intuition to be wrong and humbled me, but each timeI managed to turn their symbolism into my reality by way of revenge. In that same context, I owe a debt of gratitude to my students who, not yet having acquired glib and dangerous intuition, ask me many painful but pertinent questions. During my apprenticeship, Iwas fortunate inhaving teachers and associates who insisted on heuristic explanations, so we could get on with the exciting businessof inventing. It was only later, after the first excitement had worn off,that we began the serious businessof figuring out with formal mathematics why we were right or wrong, and what could be doneto further improve or save the situation. It is in this spirit that I decided to write this book: firstto show the reader heuristically how simple it really is, next to present the essentials of the ix

X

to

Preface

the First Edition

formal theory, and finally, in a kind of dialectical synthesis, to develop new ideas, concepts, theories, inventions, and devices. Following this scheme, I have left out many details that, although important in their own right, obscure the essence of the matter as Isee it. However, I have tried to give the appropriate references, so that the reader can readily pursue certain lines of inquiry that are of particular relevance. In my professional career, Ihave found that having two or more points of view at one’s disposal is a necessary condition for both the release and domestication ofwild ideas. With a bit of luck added, a sufficient condition will then exist for making inventions and discoveries. I never feel this so keenly as when, reading about someone else’s invention, the thought occurs: “How beautifully simple: why didn’t I think of it?” If that turns out to be also the reader’s reaction,Iwill have succeeded in what Iset out to do. The question naturally comes up as to what audience this book is aimed at. It is perhaps easierto answer this first in a negative sense: the book may not be suitable - in the sense of providing quick answers - for readers, such as project managers, system engineers, etc., whose interest in acousto-optics is somewhat peripheral to their main activities and responsibilities. The book should, however, be of value to the seriously involved engineerat the device or research level and to the graduate student. More generally it should interest anyone who wishesto really understand acousto-optics (as opposed to knowing about it), because of either scientificcuriosity or creative necessity. I never intended to write a book on the subject at all, being on the whole inclined to avoid honorary, time-consuming commitments. Looking back, however, writing this book was actually a great deal of fun,so I would like to thank my wife, Loni, who made me do it. I also wantto thank my Belgian Euclidean friends, Bob Mertens and Willy Hereman, for their critical support, and my students DavidMehrl and Hong Huey Lin for their hesitantly incisive comments. In addition, I would like to express my appreciation to the Von Humboldt Foundation and Adolph Lohmann for providing me with some quiet time and space for reflection, and to the National Science Foundation for funding a substantialpart of my own research described in this book. As regards the preparation of the manuscript, I thank the computer industry forinventingwordprocessors,Kay Chambers and Margaret Korpel for doing the drawings, Joost Korpel for helping me with computer programming, and Rosemarie Krist, the production editor, for making me stick to my Calvinist work ethic. Most of all, perhaps, Iought to thank my dog, Bear, whose fierce loyalty and boundless admiration kept me going during timesof doubt and frustration. Adrian Korpel

Contents

From the SeriesEditor

Brian J. Thompson

V

Preface to the Second Edition

vii

Preface to the First Edition

ix

1.

Introduction

1

2.

Historical Background 2.1. The Pre-Laser Era 2.2 The Post-Laser Era References

5

3.

The Heuristic Approach 3.1 The Sound Field as a Thin Phase Grating 3.2 The Sound Field as a Thick Phase Grating 3.3 The Sound Field as a Plane-Wave Composition References

5 16 29 35

35 52 66 83

xi

Contents

xii 4.

5.

6.

The Formal Approach 4.1 Introduction 4.2 CoupledModeAnalysis 4.3 NormalModeAnalysis 4.4 TheGeneralizedRaman-NathEquations 4.5Weak ScatteringAnalysis 4.6WeakPlane-Wave Interaction Analysis 4.7 StrongPlane-WaveInteractionAnalysis 4.8 Feynman Diagram Path Integral Method 4.9 Eikonal Theory of Bragg Diffraction 4.10 Strong Interaction with Curved Sound Wavefronts 4.1 1 Vector Analysis References

85

85 86 88 91 93 96 98 106 109 118 125 132

The Numerical Approach 5.1 Truncation of the Raman-Nath Equations 5.2 NumericalIntegration 5.3 ExactSolutions 5.4 MultipleBraggIncidence 5.5 The NOA Method 5.6SuccessiveDiffraction 5.7CascadedBraggDiffraction 5.8 TheCarrierlessSplit-stepMethod 5.9 The Fourier Transform Approach 5.10 Monte Carlo Simulation References

135

Selected Applications 6.1Weak Interaction of Gaussian Beams 6.2 Strong Bragg Diffraction of a Gaussian Light Beam by a Sound Column 6.3 Bandwidth and Resolution of Light Deflector 6.4 Resolution of Spectrum Analyzer 6.5 BandwidthofModulator 6.6 TheQuasiTheorem 6.7 Optical Phase and Amplitude Measurement 6.8Bragg DiffractionIntermodulationProducts 6.9BraggDiffractionImaging 6.10 Bragg Diffraction Sampling 6. l 1 Schlieren Imaging 6.12 Probing of Surface AcousticWaves 6.13SignalProcessing References

169

136 137 139 140 143 144 146 147 154 156 167 169 172 175 182 184 189 193 200 206 219 223 233 236 253

Contents

7.

8.

...

x111

RelatedFields and Materials 7.1 Acoustics 7.2OpticalAnisotropy 7.3 Elasto-Optics References

257

Special Topics 8.1 Anisotropic BraggDiffraction 8.2Acousto-OpticTunableFilters 8.3 Large Bragg Angle Interaction 8.4 Acousto-Optic Sound Amplification and Generation 8.5 Three-DimensionalInteraction 8.6 SpectralFormalisms References

277

Appendix A: Summary of Research and Design Formulas

257 267 270 276 277 285 288 292 295 303 308

377

General Parameters Raman-Nath Diffraction Bragg Diffraction Bragg Modulator Bragg Deflector Weak Interaction of Arbitrary Fields in Terms of Plane Waves Strong Interactionof Arbitrary Fields Eikonal Theory Vector Equations References

311 312 313 314 315 316 317 318 319 320

B The Stationary Phase Method References

327

Appendix

322

Appendix C: Symbols and Definitions

323

Index

327

This Page Intentionally Left Blank

l Introduction

Hyperbolically speaking, the developmentof acousto-optics has been characterized by misunderstanding, confusion, and re-invention. Even the very name “acousto-optics” is misleading: It evokes images of audible sounds focused by lenses, or light emitted by loudspeakers. As to reinvention, perhaps in no other fieldhave so few principlesbeenrediscovered, re-applied, or re-named by so many people. The principal reason for all of this is that acousto-optics, born from the desire to measure thermal sound fluctuations, developed very quicklyinto a purely mathematical, academic subject and then, 40 years later, chan.ged rather suddenly into a technological area of major importance. As an academic subject, acousto-optics has been, and continues to be, studied by countless mathematicians using a dazzling variety of beautiful analytic tools. It has been forced into a framework of rigid boundary conditions and stately canonical equations. Existence and uniqueness proofs abound, and solutions are obtainedby baroque transformations, sometimes for parameter regionsthat have little connections with physical reality. It is true that during the academic phase afew physical experiments were performed, but onecannot escape the feelingthat these served only to check the mathematics-that they were carried out with physical quantities (i.e., real sound, real light) only because computers were not yet available. 1

2

Chapter I

Then, in the1960s when laser light was ready to be used, it became evident that photons, having no charge, were difficult to control and that acoustooptics provided a wayout of the difficulty. After40 years, the academic bride was ready for her worldly groom, but, alas, her canonical garb was to intimidating forhim. He reneged on the weddingand started looking around for girls ofhisown kind. In less metaphoric language:Scientists and engineers began to reformulate the theory of acousto-optics in terms that were more meaningful to them and more relevant to their purpose. The ultimate irony is that acousto-optics started out as a very earthy discipline. The original prediction of light scattering by sound was made in 1922 by Brillouin, a well-known physicist who followed up some work that had been begun earlier by Einstein. Brillouin’s analysis of the predicted effect was stated entirely in terms of physics, unobscured by any esoteric mathematical notions. However, as very few contemporary scientists have the leisure to read papers that are more than 40 years old, and even fewer are able to read them in French, Brillouin’s theories also were re-invented. It is no exaggeration to say that most concepts used in modern acousto-optics (synchronous plane wave interaction, weak scattering, re-scattering, frequency shifting, etc.) were originated with Brillouin. Santayana, the American philosopher, once said, “Those who cannot remember the past are condemned to repeat it.” The context in which he made that remark is definitely broader than acousto-optics, but I take the liberty of nominating the latter subject as a particularly illustrative example of Santayana’s dictum. To follow up on our earlier metaphor, the present situation in the1990s is that both bride and groom have gone their separateways. Occasionally they meet, but their very idioms have diverged to the pointwhere communication is almost impossible. This is a real pity, because both have much to learn from each other, if they would only spend the effort and the time. Thelatter quantity, however, is in short supply generally nowadays, and the former is, as usual, highly directed by temperament: To a mathematician, the real world is not necessarily fascinating;to anengineer, the conceptual worldnot necessarily relevant. This bookaimsat reconciliation or, at least, improved mutual understanding. It provides neither cookbook-type recipes for doing acoustooptics nor esoteric theories irrelevant to engineering. Rather, it attempts to show to the practicing engineer the physics behind the mathematics and so enlarge his mental store of heuristic concepts. In my own experience, it is exactly this process that makesinvention, the essenceofengineering, possible by free associationof the heuristic concepts so acquired. Of necessity, the above philosophy implies that this book will be most useful to those people who have a deep interest or a compelling need to

Introduction

3

really understand acousto-optics at some intuitive level, either to satisfy their curiosity or stimulate their creativity in research or design. As remarked before, a lot of what gavebirth to present-day acousto-optics lies thoroughly forgotten in the dusty volumes of the not too distant past, together with the names of the people who created it. This book therefore opens witha kind of “Roots” chapter, a historical background that attempts to rectify this situation. This is not just to re-establish priority (although I do mention the names of individual contributors rather than hide them between the brackets of locally anonymous references), but also because what happened in the pastis worth knowing, can save much timeand effort, and may lead to genuinely new extrapolation ratherthan stale duplication. It is, of course, impossible to docomplete justiceto everybody involved in the evolution of acousto-optics. I have therefore concentrated on what, in my opinion, are seminalideas,genuine “firsts” but this procedure, of necessity, reflects my own preferences and biases. Yet any other method is out of the question fora book of finite size. Thus, I apologize to those who feel that they or others they know of have unjustly been left out. Please, do .. write to me about it; perhaps such omissions can be corrected in a future edition. In regard to the more technical contents of the book, I decided from the beginning that I did not wish to duplicate either the many already existing practical guides to acousto-optics or the thorough theoretical treatises on the subject. Rather,I have selected and presented the material insuch.a way as to maximize the acquisition of practical insight, that if is not too pedantic a phrase. In keeping with what was said in the preface, I have therefore relied heavily on analogies, case histories, and multiple points of view whenever possible. By the same token, I have tried to present a heuristic explanation prior to giving a rigorous analysis. Chapter 3 provides a heuristic discussion of acousto-optics based on the mixture of ray and wave optics so familiar to most investigators ina hurry to tentatively explain experimental results. Surprisingly, this method turns out to be versatile enough to explain, even quantitatively, most of acoustooptics, including the weak interaction of light and sound fields of arbitrary shape. It also givesinsight into the physicsofmany acousto-optics cookbook rules and models, such as the diffraction criteria, the validity of the grating model,the applicability of x-ray diffraction analogs, etc. Chapter 4 presents the formal treatment of acousto-optics. Many of the heuristic‘ results of the previous chapter will be confirmed here, and the normal mode and coupled mode approach will be contrasted to each other and to more general methods applicable to arbitrary sound and light fields. Among the latter are the eikonal, or ray description of interaction, and the Feynman diagradpath integral formulation.

Chapter l

4

Chapter 5 is devoted to numerical methods. Although this is somewhat removed from the main emphasis of the book, I feel that computers do present unique and convenient opportunities for simulation-and therefore acquisition ofinsight-so that a discussion of numerical methods is warranted and even desirable. Chapter 6 presents selected applications. These are not necessarily chosen for their technological relevance, but rather must be seen as case histories that illustrate the concepts treated in the preceding chapters.An effort has been made to give at least two explanations for each device or phenomenon described in order to bring out the complementary nature of the physical concepts. Chapter 7 is entitled “Materials” and gives a brief overview of the essential elements of acoustics, optical anisotropy, and elasto-optics. The treatment is not intended to be exhaustive, but should give the reader enough of a background to critically read literature of a more specialized nature and address pertinent questionsto the experts in these areas. Chapter 8 discusses some miscellaneous items that fall somewhat outside the scope of the book, but are relevant to device applications or are otherwise of intrinsic interest. In Appendix A, I have given a summary of useful research and design formulas with referencesto relevant sections and equations in the main text. In order to avoid symbol proliferation, I have, where necessary, sacrificed mathematical rigor to notational simplicity in a manner familiar to most physicists and engineers. In addition, a list of symbols with definitions has been provided. Finally, some technical notes about conventions used in this book are needed. As most acousto-optic applications are connected with electrical engineering, I have followed the electrical engineering convention throughout, i.e., the harmonic real quantity e(t, r) is represented by the phasor E(r) such that

e(t,r)= Re[E(r)exp(jwt)] wherej = & l

(1.1)

This is somewhat awkward for physicists who have to get used to the fact that a diverging spherical wave is represented by exp(-jkr) rather than exp(jkr). I feel, however, that this is still more practical than electrical engineers havingto refer to the admittanceof a capacitance as -joC.

2 Historical Background

2.1 THEPRE-LASER€RA In 1922 Ikon Brillouin, the French physicist, was dealing with the question of whether the spectrum of thermal sound fluctuations in liquids or solids could perhaps be determined by analyzing the light or x-rays they scattered [l]. The modelheusedwas one inwhich the sound induces density variations, and these, in turn, cause fluctuations in the dielectric constant. Using a small perturbation approximation to the wave equation (physically, weak interaction), he formulated the problem in terms of a distribution of scattering sources, polarized by the incident light and modulated in space and time by the thermal soundwaves. (This simple model is feasible because the sound frequencyisassumed to bemuchsmaller than the light frequency.) Each scatterer emits a spherical wave and the total scattered field is obtained by summing the contributions of all the sources, emitted at the right instants to reach the points of observation at the time of observation. In short, he used what we would now call a retarded potential Green’s function method, in the first Born approximation. Before applying this method to the entire spectrum of thermal sound fluctuations, Brillouin tried it out on the simplified case of a single sound wave interacting with a single lightwave. Indeed, even before doing that, he used a still simpler geometric picture of this interaction because, in his own 5

Chapter 2

6

words, "One may already anticipate results which will be confirmed later by a more precise calculation." Figure 2.1showsa modern version [2]of Brillouin's geometric picture. It shows the sound waves propagating upward and the light incident from the left. The angles of incidence must be so chosen as to ensure constructive interferenceof the light beams reflected off the crests of the sound wave. This is, of course, also the condition for x-ray diffraction, the derivationof which may be found in any elementary physics textbook. It leads to critical anglesof incidence

h sin(&")= p 2a where p is an integer, A the sound wavelength,and A. the light wavelength in the medium. The angle h",p = 1 is called the Bragg angle. The angle of

l

,,,,,~,, I

Figure 2.1 AcousticBraggdiffractionshowingcritical angles fordown-shifted interaction (top)and upshifted interaction (bottom). (FromRef. 2.) 0 1981 IEEE.

Historical Background

7

reflectionistwice the Braggangle. (For simplicity, Fig. 2.1shows no refraction at the boundary of the medium.) Brillouin himself refers to the analogy of optimal diffraction by a grating in discussing eq. (2.1). He makes the crucial observation, however, that a sound wave is a sinusoidal grating and that therefore we should only expect two critical angles, i.e., for p = + 1 and p = - 1. As for the sound, its velocity is so small comparedto that of the light that, for purposes of analysis,we may suppose itto stand still. Its only effect, according to Brillouin, isto impart a Doppler shift that he calculates to be equal to the sound frequency, positive for p = 1 (lower part of Fig. 2.1, sound moving toward the observerof the scattered beam) and negative for p = - 1 (upper part of Fig. 2.1, sound moving away from the observer). In modern terminology, we speak of the upshifted, or plus one order; the downshifted, or minus one order; and the undiffracted, or zero order. The phenomenon as such as called Bragg

+

dvfraction.

Following the analysis of the geometrical picture, Brillouin carries out his perturbation calculation and finds essentially the same results, provided the volume of interaction is taken to be sufficiently large. The condition he derivesis one of whatwouldnowbecalled synchronous interaction, meaning that successive contributions to the scattered beam be all in phase. The geometrical interpretation of that condition is given by eq. (2.1) for Ipl=l. Another result foundby Brillouin is that, with the assumptionof a simple isotropic change in refractive index through density variations, the scattered light is of the same polarization as the incident light. In a later monograph [3], he shows that this is true to a very good approximation, even in the case of strong interaction where the perturbation theory fails. The underlying physical reason for this behavior that is only the induced change in density of dipoles is considered, not the change in collective orientation or in direction of the individual dipole moment. If we take the latter phenomena into account, we find that, in the most general case, there will also occur an induced birefringencethat, as we will see later,may sometimes be putto good use. Nevertheless, most of the fundamental principles of acousto-optics may be demonstrated by ignoring polarizationand using scalar formulations. Finally, Brillouin suggestedthat the results he had obtained be verified by using manmade sound, according to the piezo-electric method invented by Langevin. The range of sound wavelengths usable for this purpose, he remarked, stretched from infinityto half the wavelength of light (for smaller sound wavelengths, synchronous interaction is no longer possible); for a typical liquid, one would require electrical signals with an electromagnetic free-space wavelength longer than 9 cm. “These conditions are perfectly realizable,” he concluded.

8

Chapter 2

In spiteofBrillouin’soptimism,it took 10 more yearsbefore the experimentshe had suggestedwereactuallyperformed. In 1932 the Americans Debye and Sears [4] and the French team of Lucas and Biquard [5] found that Brillouin’s predictionswere wrong. [2]. It was Figure 2.2 shows a modern drawing of the essential experiment found that (1) the predicted critical angles did not appearto exist and (2) by increasing the sound strength, numerous orders appeared rather than the two expected on the basis of Brillouin’s calculations.In regard to (2), Debye and Sears correctly surmised and calculated that the nonexistence of critical angles was due to the fact that the interaction lengthwas too small. In their own words “Taking into account, however, that the dimensions of the illuminated volume of the liquid are finite it can easily be shown that in our case Bragg’s reflection angle innot sharply definedand that reflection should occur over an appreciable angular range.” In this context, it is interesting that, because of this fact, modern Bragg diffraction devices have any useful bandwidth at all.

nu

TRANSDUCER

f Figure 2.2 MultipleordersgeneratedintypicalDebye-Sears,Lucas-Biquard experiment. (From Ref. 2.) 01981 IEEE.

Historical Background

9

Debye and Sears also worked out a criteria for what is now called the Bragg regime: The ratio LUA2 should be large compared to unity. In their case,whenworkingwith toluene at afrequencyof 10 MHz, alight wavelength of0.5 p, and an interaction lengthof 1 cm, the ratio was about 0.5; this they said with considerable understatement “cannot be considered as large.” In 1967Klein andCook did somenumerical computer simulations and found a somewhat more quantitative criterion [6].Defining a quality factorQ Q=K2LIk=2wLAJA2=2wxDebye-Sears ratio

(2.2)

they concluded that more than 90% of the incident light could be Braggdiffracted, i.e., diffracted into one order, if Q were larger than about 2w. We shall see later that the actual fractional tolerance around the Bragg angle equals about 4 d Q . A value of Qa2w or 4w is therefore often used as a criterion for Bragg-angle operation. Returning now to 1932, Debye and Sears could not satisfactorily explain the presence of multiple orders. They did use a model of the sound column as a periodic phase grating, but didnot obtain results that agreed with their measurements. Wenow know that, although their sound column was too thin to be treated as an analog of an x-ray diffraction crystal, it was too thick to be considered a simple phase grating. Noting that the angular spacing between orders was a multiple of the primary deflection angle 2@E(shown in Fig. 2.2 as AJA for small angles), Debye and Sears next surmised that even and odd harmonics of the sound frequencywerepresentin the medium.Lucas and Biquard, however, pointed out that the vibration modes of the quartz transducer favored odd harmonics only. They themselves favored a model based on calculated ray trajectories. A drawing of such ray trajectories is shown in Fig. 2.3 [ 5 , 7 . In this model, the crests of the waves act as lenses for the incident light, thereby creating a nonsinusoidal amplitude distribution in the exit plane. This, in turn, should leadto multiple orders. As Lucas and Biquard did not calculate the intensity of the orders, their theory could not be confirmed. To a certain extent, they were on the right track, though, as was later confirmed by Nomoto [ 8 ] , who,following up the original idea, managed to obtain intensity distributions of orders that wereinroughagreementwith experiment. The missing element in both theories, however, was the phase of the rays. A complete ray theory that included the phase was ultimately developed by Berry [9]. Although Berry’s theory gives implicit expressions for the amplitude of the orders (even taking into account the number of caustics, a particular ray has crossed), his method is by his own admission too involved for numerical calculations, It does, however, offer a beautiful

Chapter 2

10

+K

l

2

3

4

6

6

Figure 2.3 Raytrajectoriesinsoundfield.Thesoundispropagatingupward. (From Ref. 7.)

example of the elegant solution of a problem by an, at first glance, impossible method. Let us now return to the situation in 1932. Neither Debye-Sears nor Lucas-Biquard had succeeded in explaining the appearance of multiple orders. This was leftto Brillouin, who, ina 1933 monograph [3], put forward the hypothesisthat multiple orders werethe result of rescattering. It was not until much later that this suggestion was followed up quantitatively. In 1980 Korpel and Poon formulated an explicit, physics-oriented theory based on the multiple scattering of the plane waves of light composing an arbitrary light field, by the plane waves of sound composing an arbitrary sound field [lo]. Previously, in 1960 Berry had used the same concept in a formal mathematical operator formalism [g], applied to the rectangular sound column shown in Figs. 2.1 and 2.2. Both theories made use of so-called Feynman diagrams-Berry to illustrate his formalism, and Korpel-Poon to visualize their physical picture. In the preceding paragraph, Ihave purposely juxtaposed the physicsoriented and mathematics-oriented approach. The latter became dominant at around 1932 and is characterized by, among other things, using the model of a rectangular sound column with perfectly straight wavefronts, as shown

Historical Background

11

in Figs. 2.1 and 2.2. Although the model appears to give results that agree fairly well with experiment in certain situations, this is somewhat of a surprise, because it certainly doesnot represent the true nature of the sound field. The latter is characterized by diffraction spreading, with the wavefronts gradually acquiring phase curvature. Evenclose to the transducer, where the wavefronts are reasonably flat, there exist appreciable variations of amplitude due to Fresnel diffraction. The effect of this can be clearly seen in a 1941 schlieren picture by Osterhammel [l l] shown in Fig. 2.4. The first weak scattering calculation for a “real” sound field was carried out by Gordon [l21 in 1966. Even Brillouin, who had started out from very general physical configurations, adopted the rectangular sound column in his 1933 monograph [3] referred to above. In such a guise, the problem reduces to one of wave propagation in perfectly periodic media with straight parallel boundaries. Consequently, Brillouin assumed a solution for the entire light field that consisted of an infinite sum ofwaves with corrugated wavefronts (i.e., periodic in the sound propagation direction) traveling in the direction of incident light (normal to the sound column in his case), eachwave traveling with its own phase velocity. He foundthat each corrugated wavefront could be expressed by a Mathieu function. Hence, in principle, the problem was solved exactly in terms of the eigenmodes of the perturbed medium. In an engineering context, however, that statement does not mean very much, because, due to the recalcitrant nature of Mathieu functions, no numerical results are readily available; they certainly were not in Brillouin’s time. With some exaggeration, it could be said that the only positive statement to be

Figure 2.4 Schlieren picture of sound field closeto transducer. (From Ref 11.)

12

Chapter 2

made about Mathieu functions is that they are defined by the differential equation of which they are a solution. In reality, of course, something more is known about their properties [13]. For instance, they are characterizedby two parametersand are only periodic when these two parameters bear a certain relation to each other. In the physics of Brillouin’s model, this meantthat there existed a relation between the phase velocity of a wavefront whose corrugation was characterized by a certain Mathieu function and the strength of the sound field. This relation was different for each wavefront in the eigenmode expansion, and no ready analytic expressions were available. The only remedy left was to decompose each Mathieu function(i.e., corrugated wavefront)into a Fourier series, take the phase shift due to its propagation into account, and of all wavefronts add up the Fourier coefficients pertaining to the same periodicity. As is not difficult to guess, these coefficients represent exactly the amplitudes of the orders of scattered light. The trouble with this procedure is again that no analytic expressions are available for the coefficients (i.e., amplitudes of the orders) in terms of one of the parameters representing the Mathieu function (i.e., strength of the sound). So Brillouin had to fallback on certain asymptomatic expansions that were only valid for weak sound fields and gave results that can be calculated much easier by the weak scattering method he himself had used in his first paper [l]. The only additional information obtained was that the higher-order terms in the asymptotic expansion could be interpreted as representing the higher orders that were indeed seen in the experiments. I have treated the history of this development in some detail because it explains why other investigators kept looking for easier ways to calculate the amplitude of the orders, in spite of the fact that Brillouin had found an “exact” solution. Brillouin himselfwas perfectly well aware of this, because in a foreword to a paper by Rytov [l41 that described such an attempt, he remarks that ‘‘. . . these Mathieu functions are terribly inconvenient.” Rytov himself observes that, because of the very nature of the problem (spatial periodic modulation of the medium), every rigorous methodto calculate the field must somehowor other leadto Mathieu’s equation. The next generationof researchersby and large abandoned the attempt to find an exact expression for the total field and concentrated instead on finding relations between the amplitudes of the various orders. In other words, they investigated the coupling between plane waves traversing the medium (i.e., the normal modes of the unperturbed medium) rather than the orthogonal eigenmodes of Brillouin (i.e., the normal modes of the perturbed medium). It was not until 1968 that Kuliasko, Mertens, and Leroy [15], using these very same coupled plane wave relations, returned to the total field concept that in their mathematical formalism was represented by a

Historical Background

13

“generating function.” A more complete theory along the same lines was given by Plancke-Schuytens and Mertens [161, and the final stepWas taken by Hereman [17], who,starting directly from Maxwell’s equations and Using a more general approach ‘than Brillouin, derived the same generating function and the same exact solution as Mertens and co-workers. ASRytov had predicted, the solution was expressed in terms of Mathieu functions. However, by this time, more tables were available for these functions and their Fourier coefficients, so that they were no longer so intractable as in Brillouin’s time. It should be noticed in passing that the development sketched above still adhered strictly to-the rectangular soundcolumn model and hence represents a continuation of the mat he ma ti^^" school of acousto-optics.In fact, until the inventionof the laser abruptly pushed acousto-opticsinto the real world, no “physics” school could be saidto exist. As remarked before, after 1932 the main emphasis shiftedto finding more direct ways of calculating amplitudes of separate orders, and a variety of ingenious methods was proposed. I shall limit myself here to seminal developments, of which the first one is undoubtedly the epochal work of Ramanand Nath; In a series of papers [18-221 written during 1935-1936, their theory evolved from a simple thin grating approximation to an exact derivation of the recurrence relations between orders: the celebrated Raman-Nath equations. It is of interest to consider their contribution at some length, because it shows aspects of both mathematical and physical ingenuity as well as rather surprising naivetk. In their first paper [18], they treat athin sound column as a phase grating that the rays traverse in straight lines. Because of the phase shift suffered by each ray, the total wavefront is corrugated as it leaves the sound field. A simple Fourier decomposition then leadsto the amplitude of the various orders. In modem terms, they calculate the angular spectrumof a field subjected to a phase filter [23]. Before beginning their analysis, Raman and Nath acknowledge that their theory bears avery close analogyto the theory of the diffraction of a plane wave (optical or acoustical) incident normally on a periodic surface, developed by Lord Rayleigh[24]. (In retrospect this is aremarkable statement, especiallywhen weseehowcavalierly Rayleighis treated nowadays by his re-discoverers.) They also invoke Rayleigh in order to of reflection and argue that object to Brillouin’s picture of the process as one reflection is negligible if the variation of the refractive index is gradual compared with the wavelengthof light. Therefore, they themselves prefer “simple consideration of the regular transmission of light in the medium and the phase changes accompanying it.”

14

Chapter 2

In their second paper [19], Raman and Nath develop the case of oblique incidence and present some clever physical reasoning that explains why the whole effect disappears for certain angles of incidence: The accumulated phase shift vanishes, because the ray traverses equal regions of enhanced and diminished refractive index. The third paper [20] deals with the Doppler shift imparted to the various orders and also treats the case of a standing sound wave. The latter case is dealt with in a rather complicated way in order to show that even and odd orders show even and odd harmonics of the sound frequency in their Doppler shift and to calculate their contribution. It seems to have totally escaped Raman and Nath that a standing sound wave can be considered a fixed grating whose phase-modulation index varies slowly (i.e., compared to the light frequency) in time. Consequently, the results could have been derived directly from the previous case by substituting a time-varying accumulated phase shift in the place of a fixed one. In their fourth paper [21], Raman and Nath took as a starting point Helmholtz’s scalar equation with spatio/temporal variation of the propagation constant; their fifth paper [22] was similar but dealt with oblique incidence. Making use of the periodic nature of the sound field, they decomposed the total field into orders propagating into specific directions and derived the now famous Raman-Nath equations that describe the mutual coupling of these plane waves by sound. Note that their model was still the mathematical one of a rectangular column of sound. This model is very similar to that of a hologram, with the exception that holographic fringes do not move and may be slanted relative to the sides of the column. In this context, it is of interest that many years later Raman-Nath type diffraction calculations were repeated for thick holograms [25]. In regard to more general configurations, in 1972 Korpel derived recursion relations (generalized Raman-Nath relations) for the various frequency components present in arbitrary sound- and light-field interaction [26]. This was later formulated in terms of coupling between individual components of the angular plane wave spectra of both fields [27]. In the last two papers of the series, Raman and Nath also pointed out that their latest results indicated that, in general, the emerging wavefront was not only corrugated in phase, but, if the grating was thick enough, also in amplitude. As we have already seen before, the latter effect was considered previously by Lucas and Biquard on the basis of ray bending [5], the former effect was used by Raman-Nath themselves in their first paper on straight, phase-delayed rays [18], and the two effects were ultimately combined by Berry in a rigorous ray theory [9]. In a follow-up paper [28], Nath, starting from Maxwell’s equations, showed that a scalar formulation such as had been used before was allowed

Historical Background

15

in view of the great difference between light and sound velocity. He also considered the asymmetry of the diffraction phenomena at oblique incidence and developed some approximate expressions based on the Raman-Nath relations. In the same paper, there is a succinct statement about the difference between the Raman-Nath approach and that of Brillouin. Nash admits that the latter’s analysis is perfect but

. . . leads to complicated difficulties for, to find the diffraction effects in any particular direction, one will have to find the effects due to all the analysed waves. On the other hand, we have analysed the emerging corrugated wave into a set of plane waves inclined to one another at the characteristic diffracted angles. To find the diffraction effects in any particular direction, one has only to consider the plane wave travelling in that direction. It is difficult to find a more lucid summary of the two basic approaches to the problem; it is also unfortunate that this kind of verbal explanation has largely fallen into disuse with the terse scientific “specialese” of the present time. To derive the recursion relations between orders of diffracted light, Raman and Nath, using a fairly extensive mathematical analysis, needed about 14 pages. Van Cittert, using a heuristic physical approach, did it in two pages [29]. His method was simplicity itself: divide the sound field into infinitesimally thin slices perpendicular to the direction of light propagation. Each slice will act as a thin phase grating and, because of its infinitesimal thickness, will generate from each incident plane wave only two additional ones. The amplitudes of the two new waves will be proportional to the amplitude of the incident wave, the strength of the sound field, and the (infinitesimal) thickness of the grating along the angle of incidence. Carry out this prescription for each plane wave in the distribution and the result is an infinite set of differential recursion relations, the Raman-Nath equations. Van Cittert is seldom quoted these days; I suspect that his lack of mathematical sophistication would be considered in very poor taste by many ambitious young scientists. His method, however, has been adopted by Hargrove [30] and later by Nomoto and Tarikai [31] in the form of numerical algorithm based on successive diffraction. From 1936 until the invention of the laser, a great many researchers concentrated on various aspects of what was basically the same mathematical problem: the diffraction of light into discrete orders by a rectangular column of sound. Most of the work concerned itself with obtaining approximations to either the Brillouin or the Raman-Nath formulation of the problem. An exception is the work of Bhatia and Noble [32], who used the novel

Chapter 2

16

i

approach of expressingthe total field by the sum ofthe incident field and the scattered field. The latter,of course, can be expressed as the contributionof the scatterers (i.e., the sound field) acting on the total field itself. Thus, this approach leads to an integral (actually integro-differential) equation that, under the assumption that the scatterersact on the incident field only (Born approximation), had already been solvedto a first orderby Brillouin. As for other investigators, lack of space limits our discussion to the few whose contributions contained somereally novel elements. Extermann and Wannier [33], for instance, derived algebraic recursion relations between the Fourier coefficients of the corrugated wavefronts of Brillouin’s eigenmodes. The condition for solution of these equations leads to the so-called Hill’s (infinite) determinant whose eigenvalues are related to the phase velocitiesof the eigenmodes. Mertens used a method of separation of variables [34] leading once more to Mathieu functions, and Phariseau extended this theory to include oblique incidence[35].Finally, Wagner gave a rigorous treatmentstarting from Maxwell’sequations [36]. What about solutions? At the beginning of the 1960s, the following were known: (1) the strong interaction multiple-order solution for a thin phase grating, derived by Raman and Nath [19]; (2) the strong interaction twoorder solution near the Bragg angle for a thick phase grating, derived by Bhatia and Noble [32] and Phariseau [37l; and (3) the weak interaction -1-1 and - 1 order solution for an arbitrary thickness sound column, first given by David [38].In addition, various approximations forother regions existed that are, however, not relevant to our present purpose. Concerning applications and techniques developed during the pre-laser era, Mayer has given a concise review [39].The standard work remains Bergmann’s Der Ultraschall[40]for those who read German and are lucky

enough to obtain one of the rare copies. An abbreviated English versionhas also been published [41].

2.2

THEPOST-LASER ERA

During the 1960s, the character of acousto-optics changed completely. The invention of the laser created a need for electronically manipulating coherent light beams, for instance deflecting them. As photons have no charge, it is obvious that this canonlybeachieved by electronicallyvarying the refractive index of the medium in which the light travels. This can be accomplished directly through the electro-optic effect,or indirectly through the acousto-optic effect. Thelatter method, however, has certain advantages, which are almost immediately obvious. Deflection, for instance, is as if it were built in through the dependence of the diffraction angle on acoustic wavelength and, hence, acoustic frequency. Frequency shifting, extremely

Historical Background

17

important for heterodyning applications, issimilarly inherent in the diffraction process through the Doppler shift. Modulation should be possible by varying the amplitude of the electrical signal that excites the acoustic wave. And, what is perhaps the most important aspect, the sound cell, used witha modulated carrier, carriesan optical replica of an electronic signal that is accessible simultaneouslyfor parallel optical processing. All of these aspects were ultimately incorporated in devices during the period 1960-1980, a period that is characterized by a truly explosive growth of research and development in acousto-optics. It is usually forgotten, however, that most of these aspects were in some less sophisticated form already used in measurements or applications prior to the invention of the laser. Debye-Sears [4] and Lucas-Biquard [5], for instance, had measured sound velocitiesby measuring angles of diffraction. A particularly ingenious and beautiful method of displayingtwodimensional lociof sound velocities in any direction and for any of the three modes of sound propagation inarbitrary crystals was developed by Schaefer and Bergmann in 1934 [40]. It was based on exciting sound waves in as many modes and asmany directions as possible by the useof a crystal of somewhat irregular shape. The resulting diffracted beams (one for each mode and direction) were focused in the back focal plane of a lens. In this plane then, each pointof light correspondsto a particular mode ina specific direction. A series of picturesso obtained is shown inFig. 2.5. As for the Doppler shift, Ali had measured this in 1936 by spectroscopic methods [42], a measurement that was later repeated by Cummins and Knable using laser heterodyning techniques [43]. Concerning modulation, according to Rytov [14], an acousto-optic light modulatorwas conceived in 1934 by Mandelstam and co-workers, and around the same time a similar device was apparently patentedby Carolus in Germany. The first published description of a light modulator thisauthor is aware of was given by Lieben in 1962 [44]. Parallel processing for display purposes was pioneered by Ocolicsanyi [46] and used in the Scophony large-screen TV system of 1939 [46,47]. A modern version of the latter was demonstrated in 1966 by Korpel and co-workers [48], first in red and black using a He-Ne laser and later in color [49] with the help ofa Krypton laser. In orderto increase the bandwidthof the sound cell in these experiments (limited by virtue of the fact that long efficient interaction lengths decrease the tolerance about the Bragg angle, as already pointed out by Debye and Sears [4]), a device now called a beam-steering deflector [48,50] had to be invented. In such a deflector, the acoustic beam is made to track the requiredBragg angle by means of an acoustic transducer phased array. The first device for signal processing as such was developed by Rosenthal

18

Chapter 2

Figure 2.5 Schaefer-Bergmann patterns of sound waves propagating in X, Y, and Z plane of a quartz crystal. (From Ref40.)

Historical Background

19

[51] who proposed, among manyother things, a (laserless) correlator using a sound cell and a fixed mask.Later, signal processingusing optical heterodyning was demonstrated independentlyby King and co-workers[52] and by Whitman, Korpel, and Lotsoff [53,54]. Since then, interest in this particular application has increased exponentially; extensive tutorialheview articles and references may be found in [55-571. The mechanism of beam deflection was analyzed by Korpel and coworkersin1965[58].Theyderived the now well-known result that the number of achievable resolvable angles was equal to the productof frequency swing and transit time of the sound through the light beam. For linear (i.e., nonrandom) scanning, Foster later demonstrated that this number could be increased by about afactor of 10through theuse ofa traveling wave lens [59]. We have already mentioned beam-steering for larger bandwidth in scanners [48,50]. Lean, Quate, and Shaw proposed a different approach that increased frequency tolerance through the use of a birefringent medium[60]. A scanner of this kind was realized by Collins, Lean, and Shaw [61]. A good review of scanning applicationsmay be found in Ref. 62. The idea of deflecting a beam of light by changing the frequency of the sound leads naturallyto the concept of an acousto-optic frequency analyzer. The only difference between a beam deflector and a spectrum analyzer is that in the former the various frequencies are applied sequentially, whereas in the latter they are introduced simultaneously. In the area of optical signal processing, the spectrum analyzer concept was adopted rapidly with the result that this field is now characterized by two methods of approach: image field processing and Fourier plane processing.As was pointed out by Korpel [S], and later demonstrated with Whitman [63], these two methods (at least when heterodyning is used) are completely equivalent. They only differ in the experimental configuration, because a single lens suffices to transform an image plane into a Fourier transform planeand vice versa [23]. In the field of image display, the spectrum analyzer concept has found application. Successive samples of a TV signal, for instance, can first be transformed electronically into simultaneous radio frequency (RF) bursts, whose frequency encodes positionand whose amplitude encodes brightness. If these samples are now fed into a soundcell, an entire TV line (or part of a line) will be displayed simultaneously in the focal plane of a lens focusing the diffracted beams at positions according to their frequency [64]. It is clear that this method is the complement of the Scophony systemthat visualized an image of the sound cell contents [47]. In yet another context, the entire subject of frequency-position representation is the complete analog of that of frequency-time representation pioneeredby Gabor [65]. So far I have said little about the further development of theory during the post-laser era. This is not so much an oversight as a consequence of the

20

Chapter 2

fact that during this period theory and experiment cannot be put into strictly separated categories. Each new device stimulated further theoretical development and viceversa. This is perhaps best illustrated by the development of the coupled plane-wave conceptand its applications. It has already been remarked before that Brillouin, in his original work [l], stated the conditions for phase synchronous interaction that make acousto-optic diffraction possible. Ina graphical form, this condition is best illustrated by the so-called wave vector diagram, already indicated by Debye-Sears [4], but more formally developed by Kroll [66]. Figure 2.6 shows such a diagram for upshifted interaction in two dimensions. It is obvious that it represents the wave vector condition

where ko (sometimes written ki) represents the incident planewave of light in the medium, k+ the upshifted plane wave of light, and K the plane wave responsible for the process.In physical terms, (2.3) means that there exists a one-to-one correspondence between plane waves of sound and plane waves of light. Now, itiswellknown that eachfieldsatisfyingHelmholtz’s equation can be uniquely decomposed into plane waves (if we neglect evanescent waves for the moment): the angular plane-wave spectrum [23]. This is, of course, also true for the sound field and offers a possibility to study (weak) interaction geometriesby means of this concept. A semi-quantitative consideration of deflection and modulation devices was carried out along these lines by Gordm [12]. According to these considerations, the angular sensitivityof diffraction devices is intimately tied up with the angular spectrum of the acoustic transducer. If, for instance, the primary light beam is incident at an angle for which no sound waves of appropriate strength are presentin the angular spectrum, then no

Figure 2.6 Wave vector diagram for upshifted interaction.

Historical appreciable diffraction effects will be observed. Because the required angle between plane waves of sound and plane waves of light is dependent on the sound frequency, these same considerations can be used to give a rough prediction of the frequency bandwidthof acousto-optic devices. Also, many predictions and observations, made many years ago, allof a sudden acquire an obvious physical interpretation. The reader may remember the Debye and Sears calculationthat indicated that the tolerance about the Bragg angle decreased with the interaction length. This makes excellent sense from a wave interaction point of view, because the width of theangular spectrum is inversely proportional to the length of the acoustic transducer (interaction length). Hence, the larger this length, the smaller the possibility of finding a suitable sound wave to interact with, when the direction of incident light is varied. Debye and Sears had also noticed that, on turning their sound cell, the intensity of the diffracted light would go through many maxima and minima and grow gradually weaker. We now realize that what they were seeing was actually theplane-wave angular spectrumof the soundthat they sampled by turning the sound cell. In 1965 this was confirmed by Cohen and Gordon [67] who repeated quantitatively the experiment with modem equipment. There is another implication hidden in eq. (2.3). As already discussed, there exists a one-to-one correspondence of planewaves of sound and light. Also, it turns out that the amplitude of the diffracted plane wave is proportional to the product of the amplitude of the incident light wave and the interacting soundwave. It therefore follows that the angular plane-wave spectrum of diffracted light should be similar to that of the sound if the incident light has a uniform, wide angular plane-wave spectrum from which interacting waves can be selected. However, as remarked before, the angular plane-wave spectrum of the sound is what makes up the soundfield. Hence, if we illuminate a sound field with a converging or diverging wedge of light (i.e., a wide uniform angular spectrum), then the diffracted light should carry in some way an image of the soundfield. By proper optical processing, it should then bepossible to make this imagevisible. This method of visualization of acoustic fields, now called Bragg diffraction imaging, was proposed and demonstrated by Korpel in 1966 [68]. Some of the first images obtained in this way are shownin Fig. 2.7.Almost to illustrate the convergence of ideas, the same method was independently developed by Tsai and co-workers [69,70] and by Wade [71]. The reader should note that Bragg diffraction imaging is not the same as schlieren imaging. With the latter, one visualizes an axial cross sectionof the sound, with the formera transverse cross section. (In fact, Braggdiffraction imaging is really spatial imaging, because phase is preserved in the process.) Schlieren imaging is, of course, veryimportant in its own right and plays

Figure 2.7 Acoustic images obtained by Bragg diffraction. (From Ref. 68.)

23

Historical Background

an important role in modern optical signal processing, for instance when the contents of a sound cell have to be imaged on a maskor another sound cell. It wasfirstusedin an acousto-optic context by Hiedemannand Osterhammel in1937[72]. In their first experiment, they still useda conventional schlieren stop; later, Hiedemann [73] made use of the fact that ray bending already created an amplitude image, as predicted by Bachem and co-workers [74]. But even without ray bending, a diffraction image (Fresnel image) exists in front the of sound cell, as was first demonstratedby Nomoto [75]. The latter’s technique has been used in a modern setting by Maloney [76]and by Korpel, Laub, and Sievering [77l. Returning now to Bragg diffraction imaging, in order to explain the method more satisfactorily, it was necessary to develop a plane-wave weak interaction theory for arbitrary sound and light fields. This was carried out by Korpel [26,78,79] and later by Carter [80]. The former also developed a formal eikonal theory [26,81]that predicted the amplitude of diffracted rays and confirmed the initially heuristic, ray-tracing method [68]. A further experimental evolution of Bragg diffraction imaging is Bragg diffraction sound probing, developed by Korpel,Kessler and Ahmed[82]. This technique uses a focused light beam as an essentially three-dimensional, phase-sensitive probe of the sound field. A multitransducer sound field recorded in this way is shown in Fig. 2.8. The perhaps final step in this field was taken by Kessler who invented a pulsed version of Bragg diffraction imaging that provided better depth discrimination [83]. I have chosen to describe at some length the evolution of Bragg diffraction imaging as a prime example of the interplay between theory and

la2

1

fs 4OMHZ A = O.lmm f@ 6.0 A x 0.6mm 1 cm out f

Figure 2.8 Recording of multipletransducersoundfieldobtainedby diffraction sampling. (From Ref.82.)

Bragg

24

Chapter 2

practice. The reason is that I am fairly familiar with it through my own involvement, and also that it typifies the stimulating and hectic research environment of that period. Other, more theoretical subjects evolved at a rapid rate also. As for the evolution of the plane-wave interaction theory, for instance, 1976 in Chu and Tamir analyzed the strong interaction of a Gaussian light beam and a rectangular sound beam using plane-wave concepts [84].This analysis was later extended to optical beams of arbitrary profile by Chu and Kong in 1980 [85].These two theories still usedthe nonphysical model for the sound field. A general theory of plane-wave strong interaction for arbitrary sound and light fields had at about the same time been formulated by Korpel in Ref. 27 and was later ,cast in explicit form by Korpel and Poon [lo].This theory, in turn, was used by Pieper and Korpel to calculate interaction with curved sound wavefronts[86]. The wave vector diagramof Fig. 2.6 illustrates phase synchronism leading to upshifted interaction. If all vectors are multiplied by h/2n (where h is Planck’s constant), then that same diagram illustrates momentum conservation in photon-photon collision processes as first pointed out by Kastler in 1964 [87]. In the samequantum mechanical context, the Doppler shift is a consequence of thequantum energy conservation inherent inthe process

where ji and f+ denote light frequencies and F the sound frequency. It is clear that, in the upshifted process, one phonon of sound is lost for every diffracted photon generated. When the photons are of thermal origin, this phenomenon is called Brillouin scattering[88]. The downshifted diffraction processis characterized by the phase synchronism conditions

illustrated in the diagram of Fig. 2.9. In quantum-mechanical terms, the conservation of momentum is described by [26]

which equation isreadilyseen to be equivalent to (2.5) The physical interpretation of (2.6) is that every incident photon interacting with a phonon stimulates the releaseof second phonon. Consequently, the sound is

25

Historical Background

Figure 2.9 Wave vector diagram for downshifted interaction.

amplified and the diffracted photon has a lower energy consistent with its lower frequency. If the sound isof thermal origin, the phenomenonis called stimulated Brillouin scattering. It requires powerful beams of coherent light and was first observed by Chiao, Townes, and Stoicheff [89].The identical effect with manmade sound was observed at the same timeby Korpel, Adler, and Alpiner [90].The latter, in the same experiment, also generated sound by crossing different frequency laser beams within a sound cell. That this should be possible had been predicted by Kastler [86],who also gave a classical explanation of the effect based on radiation pressure. For the sake of historical interest, Fig. 2.10 shows the attenuation and amplification of the sound inherent in acousto-optic interaction and observedin the experiment described in[90].

(a

1

(b)

Figure 2.10 (a) Light-induced amplification of sound (increase is downward). (b) Induced attenuation. The bottom trace represents the light pulse. The time difference between peaks is dueto sound travel time. (From Ref. 90.)

26

Chapter 2

From the above, it will be clear to the reader that the concept of planewave interaction has been very fruitful in generating new physical insights leading to novel devices, theories, and experiments. It is, however, not the only way to approach acousto-optics and, concurrently with its development, the older classical methods were modified in an effort to account better for the physical reality of nonbounded sound fields and finite incident beamsof light. Thus, in 1969 McMahon calculated weak interaction of both Gaussian and rectangular sound and light beams by using a scattering integral [91], Korpel formulated a generalized coupledorder theory for arbitrary sound and light fields [26], and Leroy and Claes extended the Raman-Nath theory to Gaussian sound fields [92]. Numerous other theories were developed, but most of these used modern formulations of either the normal mode approach (Brillouin) or the couple mode approach (Raman-Nath) in the context of a rectangular periodic column. Theyare,inasense, continuations of the pre-laser era mathematical approach and presently more relevant to diffraction by thick holograms than to modem acousto-optics. We will therefore not discuss them further, other than pointing out that some of these theories contain very useful algorithms for numerical calculations in those situations where the rectangular column model applies. Thus, Blomme and Leroy [93] have developed a numerical method based on the eigenvalues for a system of a finite numbers of orders. The essence of this method was originally proposed by Mertens, Hereman, and Ottoy [94]. With the interest in practical applications of acousto-optics, a demand arose for more sensitive acousto-optic materials. Whereas before acoustooptic parameters had mainly been studied in the context of crystallography [95], the emphasis now shifted to device optimization. Smith and Korpel proposed a figureof merit for diffractionefficiency (now called M2) [97] and demonstrated a simple dynamic measurement technique, later modified by Dixon and Cohen to include shear-wave effects [98]. Other figures of merit addressing different aspects such as bandwidth optimization, etc., were also to proposed [12,99]. At the same time, a beginning was made by Pinnow predict elasto-opticconstant from first principles[100,lOl] in a more general way than the classical derivation'from the Lorentz-Lorenz equation [102]. The latter relates the dielectricconstant to the density of dipoles,and it is a simple matter to calculate the change due to isotropic pressure. It is much more difficult to calculate the variation in intrinsic polarizability, especially where shear forces are involved. In the latter case, the existing phenomenological description [103,104] is actually insufficient to account for anisotropic materials, as pointed out by Nelson and Lax [105,106]. Tabulated values of constants and figures of merit may be found in many places [57,62,97,99,101,107-1121.

Historical Background

27

Many of the materials investigated were anisotropic, and it was soon discovered that anisotropic operation led to special advantages in certain cases such as, for instance, the birefringent scanner mentioned before [61]. A systematic investigation of this entire subject area was undertaken by Dixon [l 131. Harris used birefringent collinear Bragg diffraction in a tunable optical filter [114], and Havlice and co-workers used the same mode of operation for Bragg diffraction imaging [l 151. Birefringence was also used by Carlton for the purpose of acousto-optic signal processing [l161 and by Korpel and co-workers for sound field probing [81]. In the latter technique, one of the variants madeuse of the curious phenomena of dynadic birefringence in liquids, previously studied by Ripley and Klein [1171. Fortuitously coinciding with the upsurge of interest in acousto-optics during the 1960s was an accelerated development of ultrasonics technology [l 181. In particular, the area of surface acousticwaves (SAW) was found to have very important applications in the processingof electrical signals[l 191. This, in turn, stimulated new applications of acousto-optics to measurement and detection of such waves. It also made possible entirely new acoustooptic configurations involving the interaction between acoustic surfacer waves and optical surface waves [l201 in overlay-type wave guides. These configurations were found to have large diffraction efficiencies because of the high-power density of both interacting waves, the energy in both being confined to thin layers about one wavelength deep. Another important consideration isthat such configurations are inherently two-dimensional and should ultimatelylendthemselves to incorporation into integrated electronics and integrated optics [121,1221. As the reader may have guessed, the theoryof these kinds of interactions is fundamentally the same thatasof bulk interaction of sound and light. The actual calculations, though, are usually more complicated not only because detailed depth dependencies have to be taken into account, but also because the substrates are frequently piezo-electric (which necessitates considering indirect acousto-electric-optic effects) and birefringent. The heuristic approach we emphasize in this book applies, however, equally well and hence we will not treat separately the subject of integrated acousto-optics. As for the use of acousto-optics to measure surface acoustic waves, it should be notedthat such waves act as a periodic surface grating,diffracting obliquely incident light in the same way as a thin phase grating [24]. Measuring the diffracted light (either in transmission or reflection) then readily leads to an evaluation of strain in or surface perturbations of the substrate. Sucha direct measurement of diffracted orders wasfirst performed by Ippen [123]. However, from a signal-to-noise point of view, it is preferable to measure weak light signal by heterodyning rather than directly [124]. This may be accomplished in various ways. Korpel, Laub, and

28

Chapter 2

Sievering used a simple Fresnel imaging techniqueand detected the running fringes with a grating of the same period[77]. Whitman and co-workers used frequency-shifted reference beams for direct interferometric visualization [l251 or electro-optic heterodyning[126], as did Massey[127]. Adler, Korpel, and Desmares developed a technique in which a focused light beam was deflected by the acoustic surface perturbations and the deflection subsequently detected by a knifeedge-photodiode combination [128]. Figure 2.1 l shows an image, obtained with this method, of an acoustic surface wave entering a double-groove wave guide. It is natural to ask if a technique in which the light beam is narrower than the wavelengthof sound should still beclassified as acousto-optic diffraction. After all, the light is refracted by the gradientof refractive index rather than diffracted by the periodicity inherent in the sound wave. There is, however, a venerable precedent for classifying this phenomenon as acoustooptic diffraction: Lucas and Biquard studied it in their first paper on the subject [5]. Theynoticed that for such a narrow beam, the separate

Figure 2.11 Image of acoustic surface wave guide obtained by coherent light deflection. (FromRef. 128) 0 1968 IEEE.

Historical Background

29

diffraction orders would disappearto be replaced by a smeared-out version of the’ incident beam. Based on their ray-tracing theory, they correctly ascribed this to a periodic refraction of the light as the sound wave moved through the beam. Much later, Whitman and Korpel proposed a unified theory of acoustic surfacewave detection by optical meansand showed that the narrow-beam case could be treated equally well on the basisof diffraction as refraction [1291. Refraction effects were also used by De Maria andDanielson to construct a laser Q spoiler [130]. The periodic reflection effects on which Fig. 2.1 1 is based were later applied by Korpel and Desmares to build an acoustic holographic camera [l311 and by Korpel, Kessler, and Palermo to build a laser-scanning acoustic microscope[1321. As indicated before, we have, in this historical survey, concentrated on seminal contributions only. The interested reader may find more detailed information about past and current developments in numerous review articles and books [2,12,26,57,62,107-111,133-1471.

REFERENCES 1. 2. 3. 4. 5. 6. 7. 8.

9. 10. 11. 12. 13.

Brillouin, L., Ann. Phys. (Paris), 17, 88 (1992). Korpel, A., Proc ZEEE, 69, 48 (1981). Brillouin, L., Actual. Sci. Znd., 59 (1933). Debye, P., and Sears, F. W., Proc. Nat. Acad Sci US.,18,409 (1932). Lucas, R., and Biquard, P., J Phys. Rad, 3, 464 (1932). Klein, W. R., and Cook, B. D., ZEEE Trans., SU-14. 123 (1967). Lucas, R., and Biquard, P., Comptes Rendus, 195, 1066 (1932). Nomoto, O., Bull. KobayashiZnst. Phys. Res., l, 42 (1951). Berry, M. V., The Diffraction of Light by Ultrasound, Academic Press, New York (1966). Korpel, A., and Poon, T.C., J Opt. Soc. Am., 70, 81 7 (1980). Osterhammel, K., Akust. Zeit., 6, 73 (1941). Gordon, E.I., Proc. ZEEE, 54, 1391 (1966). Blanch, G., “Mathieu Functions,” Handbook of Mathematical Functions (M. Abramowitz and I. A. Stegun, eds.), Dover Publications,New York, p. 721

(1965). 14. Rytov, S., Actual. Sci. Znd., 613 (1938). 15. Kuliasko, F.,Mertens, R., and Leroy, O., Proc. Ind Acad. Sci. A, 67, 295 (1968). 16. Plancke-Schuyten, G., and Mertens, R., Physica, 62, 600 (1972). 17. Hereman, W., Academiae Analecta, 48, 26 (1986). 18. Raman, C. V., and Nath, N. S. N., Proc. Indian Acad. Sci., 2, 406 (1935). 19. Raman, C. V., and Nath, N. S. N., Proc Zndiun Acad Sci., 2, 413 (1935). 20. Raman, C. V., and Nath, N. S. N., Proc. Indian Acad Sci., 3, 75 (1936). 21. Raman, C. V.,and Nath, N. S. N., Proc Indian Acad. Sci., 3, 119 (1936).

30

Chapter 2

22. Raman, C. V., and Nath, N. S. N., Proc. Indian Acad. Sci., 3. 459 (1936). 23. Goodman, J. W., Introduction to Fourier Optics, McGraw-Hill, New York (1968). 24. Rayleigh, J. W. S., Theory ofsound, Vol. 11, Dover, New York(1945). 25. Kogelnik, Bell Syst. Tech. L,48, 2909 (1969). 26. Korpel, A., “Acousto-Optics,” Applied SolidState Science, Vol. 3 (R.Wolfe, ed.), Academic Press, New York, p. 71 (1972). 27. Korpel, A., J: Opt. Soc. Am., 69, 678 (1979). 28. Nath, N. S. N., Proc Indian Acad Sci, 4,222 (1937). 29. Van Cittert, F! H., Physica, 4, 590 (1937). 30. Hargrove, L. E., 1 Ac Soc Am., 34, 1547 (1962). 31. Nomoto, O., and Torikai, Y., Acoustica, 24, 284 (1971). 32. Bhatia, A., and Noble, W. J., Proc Roy. Soc. SeK A, 220, 356 (1953). 33. Extermann, R., and Wannier, C., Helv. Phys. Act., 9, 520 (1936). 34. Mertens, R., Simon Stevin,27, 212 (1949/50). 35. Phariseau, I?, Simon Stevin, 33, 72 (1959). 36. Wagner, E. H., Z. Phys., 141, 604 (1955); also, 142, 249 (1955). 37. Phariseau, P., Proc Indian Acad Sci. A, 44, 165 (1956). 38. David, E.,Phys. Zeit., 38, 587 (1937). 39. Mayer, W. G., Ultrasonic News, 9 (1961). 40. Bergmann, L.,Der Ultraschall, Hinel Verlag, Stuttgart (1954). 41. Bergmann, L., Ultrasonics, Wiley, New York (1938). 42. Ali, L., Helv. Phys. Acta, 8, 503 (1935); also, 9, 63 (1936). 43. Cummins, H. Z.,and Knable, N., Proc. ZEEE, 51, 1246 (1963). 44. Lieben, W., J: Ac. Soc Am., 34, 860 (1962). 45. Jeffree, J. H., Television and Short Wave World, May 1936, p. 260. 46. Okolicsanyi, F., Wireless Engineer, 14,527 (1937). 47. Robinson, D. M.,Proc I.RE, 27, 483 (1939). 48. Korpel, A., Adler, R., Desmares, F!, and Watson, W., Proc ZEEE, 54, 1429 (1966). 49. Watson, W. H., and Korpel, A., Appl. Opt., 9, 1176 (1970). 50. Korpel, A., U.S.Patent, 3,424,906 (1969). 51. Rosenthal, A. H., I R E . Trans., UE-8, 1(1961). 52. King, M., Bennett, W. R., Lambert,L. B., and Arm, M., J: Appl. Opt., 6, 1367 (1967). 53. Whitman, R., Korpel,A., and Lotsoff, S., “Application of Acoustic Bragg

Diffraction to Optical Processing Techniques,”Proc Symp. Mod Opt., Polytechnic Press, Brooklyn, New York, p. 243 (1967). 54. Korpel, A., U.S.Patent 3,544,795 (1970). 55. Korpel, A., Acousto-Optic Signal Processing,”Optical Information Processing (Yu, E. Nesterikhin and G. W. Stroke, eds.), Plenum Press, New York,p. 171 (1976). 56. Special issueof IEEE on Acousto-Optic Signal Processing, Jan. 1981, Vol. 69, No. 1. 57. Acousto-Optic Signal Processing(N. J. Berg and J. N. Lee, eds), Marcel Dekker, New York (1983).

Historical Background

31

58. Korpel, A., Adler, R., Desmares, ,.’l and Smith, T. M., ZEEE. J: Quant. El., QE-I, 60 (1965). 59. Foster, L. C., Crumly, C. B., and Cohoon, R.L., .l Appl. Opt., 9, 2154 (1970). 60. Lean, E. G. H., Quate, C. F., and Shaw, H. J., Appl. Phys. Lett., IO, 48 (1967). 61. Collins, J. H., Lean, E. G. H., and Shaw, H.J. Appl. Phys. Lett., 11, 240 (1967). 62. Gottlieb, M., Ireland, C, L. M., and Ley, J. M. Electro-Optic and AcoustoOptic Scanning and Deflection,Marcel Dekker, New York (1983). 63. Korpel, A., and Whitman, R.L., Appl. Opt., 3, 1577 (1969). 6 4 . Korpel, A., Lotsoff, S. N., and Whitman, R.L., Proc ZEEE, 57, 160 (1969). 65. Korpel, A., Appl. Opt., 21. 3624 (1 982). 66. Kroll, N. M., Phys. Rev., 127, 1207 (1962). 67. Cohen, M. G., and Gordon, E.I., Bell System Tech. .l, 44, 693 (1965). 68. Korpel, A., App. Phys. Lett., 9, 425 (1966). 69. Tsai, C. S., and Hance, H.V., 1 Ac Soc. Am., 42, 1345 (1967). 70. Hance, H. V., Parks, J.K., and Tsai, C. S., J: Appl. Phys., 38, 1981 (1967). 71. Wade, G., Landry, C. J.,and de Souza,A. A., “Acoustical Transparenciesfor Optical Imaging and Ultrasonic Diffraction,” Acoustical Holography,Vol. 1 (A. F. Metherell, H. M. A. El-Sum, and L. Larmore, eds.), Plenum, New York, 159 (1969). 72. Hiedemann, E., and Osterhammel, K., Zeit. Phys., 87, 273 (1937). 73. Hiedemann, E., Asbach, H.R., and Hoesch, K. H., Zeit. Phys., 90,322 (1934). 74. Bachem, C., Hiedemann, E., and Asbach, H. R., Zeit. Phys., 87, 734 (1934); also, Nature (London) 133, 176 (1934). 75. Nomoto, O., Proc. Phys. Math. Soc. Jap., 18, 402 (1936). 76. Maloney, W. T., Meltz, G. and Gravel, R. L., ZEEE Trans., SW-15, 167 (1968). 77. Korpel, A., h u b , L., and Sievering, H. C., Appl. Phys. Lett., IO, 295 (1967). 78. Korpel, A., ZEEE, Trans., SW-15, 153 (1968). 79. Korpel, A., J: Ac. Soc. Am., 49, 1059 (1971). 80. Carter, W. H., J: Opt. Soc. Am., 60, 1366 (1970). 81. Korpel, A., “Eikonal Theory of Bragg Diffraction Imaging,” Acoustical Holography, Vol. 2 (A. F. Metherell and L. Larmore, eds.), Plenum Press, New York (1970). 82. Korpel, A., Kessler, L. W., and Ahmed, M., J: Ac Soc. Am., 51, 1582 (1972). 83. Kessler, L. W., ZEEE Trans., SW-19, 425 (1972). 84. Chu, R. S., and Tamir, T., J: Opt. Soc. Am., 66, 220 (1976). 85. Chu, R. S. and Kong, J. A., J: Opt. Soc. Am., 70, 1 (1980). 86. Pieper, R., and Korpel, A., .l Opt. Soc. Am., 2, 1435 (1985). 87. Kastler, M. A., Comptes Rendus Acad Sc. Paris, 260, 77 (1965). 88. Fleyry, P.A., “Light Scattering as a Probe of Phonons and Other Excitations,” Physical Acoustics,Vol. 6 (W. P. Mason and R. N. Thurston, eds), Academic Press, New York (1970). 89. Chiao, R.Y., Townes, C. H., and Stoicheff, B. l?, Phys. Rev. Lett., 12, 592 (1 964). 90. Korpel, A., Adler, R., and Alpiner, B., Appl. Phys. Lett., 5, 86 (1964).

32 91. 92. 93. 94.

Chapter 2

McMahon, D. H., ZEEE Trans., SW-16, 41 (1969). Leroy, O., and Claeys, J. M., Acustica, 55, 21 (1984). Blomme, E., and Leroy, O., Acustica, 57, 168 (1985). Mertens, R., Hereman, W., and Ottoy, J. P., Proc. Ultrasonics International 85, p. 185 (1985). 95. Bhagavantam, S., and Surayanarayana, D., Proc Indian Acad Sci. A, 26.97 (1 947). 96. Bergmann, L.,and Fuess, E., Nature (London), 24,492 (1936). 97. Smith, T. M., and Korpel, A., ZEEEJ @ant. El., QE-l, 283 (1965). 98. Dixon, R. W., and Cohen, M. G., Appl. Phys. Lett., 8, 205 (1966). 99. Dixon, R. W., J Appl. Phys., 38, 5149 (1967). 100. Pinnow, D. A., ZEEE J Quant. El., QE-6, 223 (1970). 101. Pinnow, D. A., “Electro-OpticalMaterials,” CRC Handbook of Lasers (R.J. Pressley, ed.), Chemical RubberCo., Cleveland, Ohio (1971). 102. Von Hippel, A. R.,Dielectrics and Waves, Wiley, New York (1954). 103. Pockels, F., Lehrbuch der Kristall Optik, B. G. Teubner, Leipzig (1906). 104. Nye, J. F., Physical Propertiesof Crystals, Oxford Univ. Press(Clarendon), New York (1960). 105 Nelson, D. F., and Lax, M., Phys. Rev. Lett., 24, 379 (1970). 106. Nelson, D. F., and Lax, M., Phys. Rev. B, 3, 2778 (1971). 107. Uchida, N., and Niczeki, N., Proc ZEEE, 61, 1073 (1973). 108. Damon, R. W., Maloney, W.T., and McMahon, D. H., “Interaction of Light with Ultrasound; Phenomena and Applications,” Physical Acoustics,Vol. VI1 (W. P. Mason and R.N. Thurston, eds.), Academic Press, New York,p. 273 (1 970). 109. Sapriel, J., L’Acousto-Optique, Masson, Paris (1976). 110. Chang, I. C., ZEEE Duns., SW-23, 2 (1976). 111. Musikant, S., Optical Materials,Marcel Dekker, New York (1985). 112. Narasimhamurty, T. S., Photoelastic and Elasto-Optic Propertiesin Crystals, Plenum, New York (198 1). 113. Dixon, R.W., ZEEE J Quant. El.. QE-3, 85 (1967). 114. Hams, S. E., Nieh, S. T. K., and Feigelson, R. S., Appl. Phys. Lett., 17, 223 (1970). 115. Havlice, J., Quate, C. F., and Richardson, B., “Visualization of Sound Beams in Quartz andSapphire Near 1 GHz, “IEEE Symp. Sonics Ultrason., Vancouver, paper 1-4 (1967). 116. Carleton, H. R., Maloney, W.T., and Meltz, G.,Proc ZEEE, 57, 769 (1969). 117. Riley, W. A., and Klein, W. R., J Ac Soc. Am, 45, 578 (1969). 118. Special Issue on Ultrasonics, Proc ZEEE, 53 (1965). 119. Acoustic Surface Waves (A. A. Oliner, ed.), Springer, New York (1978). 120. Kuhn, L. M. L., Dakss, L., and Heidrich, F. P., App. Phys. Lett., 17,265 (1970). 121. Tsai, C. S., ZEEE Trans., CAS-26, 1072 (1979). 122. Lean, E. G., Progress in Optics, Vol. XI (E. Wolf, ed.), North-Holland, Amsterdam, p. 123 (1973). ‘123. Ippen, E. P., Proc ZEEE, 55, 248 (1967).

Historical Background

33

124. Yariv, A., Optical Electronics, Holt, Rinehart andWinston, New York (1985). 125. Whitman, R. L., J Appl. Opt., 9, 1375 (1970). 126. Whitman, R. L., Laub, L. J.,and Bates, W. J., ZEEE Trans., SU-15, 186 (I 968). 127. Massey, G.A., Proc ZEEE, 56, 2157 (1969). 128. Adler, R., Korpel, A., and Desmares, P., ZEEE Trans., SU-15, 157 (1968). 129. Whitman, R. L., and Korpel, A., J Appl. Opt., 8, 1567 (1969). 130. DeMaria, A. J., and Danielson, G.E., ZEEE J Quant. El., QE-2, 157 (1966). Ac Soc. Am., 45, 881 (1969). 131. Korpel, A., and Desmares, P., .l 132. Korpel, A., Kessler, L.W., and Palerrno, P. R., Nature, 232, 110 (1971). 133. Adler, R., ZEEE Spectrum, 4, 42 (1967). 134. Quate, C. F., Witkinson, C. D. W., and Wilson, D. K., Proc ZEEE, 53, 1604 (1965). 135. Gulyaev, Y V., Proklov, V. V., and Shkerdin, G.N., Sox Phys. Usp., 21, 29 (1978). 136. Sittig, E. K., “Elasto-Optic LightModulation andDeflection,” Progress in Optics (E.Wolf, ed.), North-Holland, Amsterdam (1972). 137. Mahajan, V. N., Wave Electron.,2, 309 (1976). 138. Defebvre, A., Rev. Opt., 46, 557 (1967); also, Rev. Opt., 47, 149,205 (1968). 139. Mertens, R., “Fifty Years of Acousto-Optics,” Proc. 1lth Int. Congr. Acoust., Paris, pp. 101-113 (1983). 140. Hereman, W., Mertens, R., Verheest, F., Leroy, O., Claeys, J. M., and Blomme, E., Physicalia Mag., 6,213 (1984). 141. Korpel, A., “Acousto-Optics,” Applied Optics and Optical Engineering,Vol. 6 (R. Kingslake and B. J. Thompson, eds.), Academic Press, New York, p. 89 (1980). 142. Solymar, L.,and Cooke, D. J., Volume Holography and Volume Gratings, Academic Press, New York (1981). 143. Nelson, D. F., Electric, Optic and Acoustic Interactionsin Dielectrics, Wiley, New York (1979). 144. McSkimmin, H. J., “Ultrasonic Methodsfor Measuringthe Mechanical Properties of Liquids and Solids,” Physical Acoustics,Vol. l-A (W. P. Mason, ed.), Academic Press, New York (1964). 145. Spencer, E. G.,Lenzo, P. V., and Ballman, A. A., Proc ZEEE, 55,2074 (1967). 146. Tsai, C. S., Guided Wave Acousto-Optic Interactions, Devices and Applications, Springer Verlag, Berlin (1980). 147. Alphonse, G. A., RCA Review, 33, 543 (1972).

This Page Intentionally Left Blank

3 The Heuristic Approach

In this chapterwe shall approach the problemof light-sound interaction in a heuristic way based largely on physical intuition, analogies,and a curious but fruitful mixture of ray and wave optics. Some of our assumptions may appear somewhat ad hoc and sometimes not very compelling, but all of them can, in fact, be justified a posteriori by performing the appropriate experiments. In short, as in real experimental practice, we will use many hand-waving arguments, secure inthe knowledge that the outcomeis already known. When later onwe develop the subjectin greater depth, our methods will be formalized and our assumptions refined, but right now we need to develop a feeling for what is likely to happen and for the physical reasons why it happens.

3.1

THE SOUND FIELD AS A THIN PHASE GRATING

Before starting the actual analysis, we shall first discuss the notation, assumptions, and conventions used throughout mostof the text. The interaction configurations wewill deal withmostly are twodimensional, with the sound propagating nominally in the X direction and the light in the Z direction. Wewill therefore use the angular conventions showninFig. 3.1, where directions connectedwith the lightfield are indicated by the angle 4, counted positive counterclockwise from theZ axis. 35

.

Chapter 3

36

For the sound field, we use an angle positive clockwise from the X axis. A typical configuration is seen in Fig. 3.2. This shows an idealized sound beam,extendingfrom x=-Wtox=+w, having straight wavefronts separated by the acousticwavelength A and being contained between planes z=O, z=L inside a medium of nominal refractive index no. A plane wave of light, of angular frequency W , characterized inside the medium by a wave vector ko, is incident from the left normalto the sound column. A configuration as described above does not reflect physical reality,not so much because it is a two-dimensional abstraction, but rather because sound will spread by diffraction, and a true single plane wave of light cannot be generated. Nevertheless, many physical configurations approach this ideal one, for instance one in which a not too wide beam of coherent light traverses the sound field close to a uniform amplitude acoustic transducer. In more general terms, we mean a configuration where in the interaction region the sound and the light spread so little that they may effectively be considered plane waves. Such considerations already make us suspect that for a true physical description a decomposition of both fields into plane waves may be useful. We will discuss this point later in Sec. 3.3. Returning now the idealized configuration of Fig. 3.2, we shall further simplify our picture by assuming that the sound field propagates in a medium that is both acousticallyand optically isotropic.The sound field will be represented by the symbol S that is taken to denote longitudinalstrain in a solid or fractional density change (condensation) in a liquid: s(x, z, t)=Re[S(x, z) exp(int)]

(3.1)

where S(x, z) is a phasor with associated frequencyQ. It should be noted at this point that we sometimes (especially in the case of optical quantities) use X

Figure 3.1 Angle conventionsfor the sound field (@and the light field (4).

37

The Heuristic Approach

n Figure 3.2 Planewavenormallytraversing a thin soundcolumn,becoming corrugated in the process, and giving rise to multiple orders of diffracted light.

the concept of a time-varying phasor,e.g., S(x,z, t). In such cases, the time variation of this phasor is assumed to be slow compared to the associated frequency. Thus, in a later chapter, we refer to a sound field whose basic as S(x,z, t)=S(x, z) exp (i6Qt) soundfrequenty Q hasbeenshifted by where it is understood that the associated phasor frequency equals Q. In general, we shall use lowercase letters for the (obviously real) values of real physical quantities and capital letters for complex quantities. [Notethat the latter are not always phasors as defined by (3.1.)] If time and/or space dependence is not explicitly expressedby the notation, indicatedby the text or obvious from the context, will it be assumed that the quantity is constant. Typical examples of notation are for a plane wave traveling in the + X direction

S(x,z) =Sexp( -Jxx)

(3.2)

where the propagation constant K and the radian frequency Q are related through the sound velocity V

Chapter 3

38 For a standingwave, we may write S(x, z)=S c o s ( f i )

(3.4)

In both eqs. (3.2) and 3.4), S is a complex time- and space-independent amplitude:

S=lsl exP(i4s) Although the notation outlined aboveis not completely unambiguous in the strictest mathematical sense (for instance, quantities S@, z, t), S(x,z), and S are denoted by the same symbol), it is compact and convenient for our purposes. At any rate, the correct interpretation will always be clear fromthe context. At this point, we shall not consider in detail the potentially very complicated way in which the sound affects the optical properties of the medium. Rather, we shall assume the simplest case in which a small isotropic change & in refractive index occurs, proportionalto the sound amplitude: &(X, Z, r)=C's(x,

Z,

t)

(3.6)

where C' is a real materialsconstant. We may then write with (3.1) &(x, z, t)=Re[An(x, z) exp(iQt)]=Re[C'S(x, z) exp(iS2t)l (3.7)

For the planewave (3.2), An(.x, z)=An exp(-ilux>

(3.8)

and for the standingwave eq. (3.4) An(x, z)=An cos(fi)

(3.9)

where from (3.5) An=C'S=IAnl = M exp[i($s+

701

if C ' > O

(3.10a)

if Cc0

(3.10b)

In what follows, we chooseC ' > O . The other case is then readily obtainedby our capitalletter replacing by &i-.rr.Note that in agreementwith convention, An may be complex.

39

The Heuristic

As for the light, we adopt the convention that optical parameters such as wavelength A., velocity c, and light propagation constantsk describe the field inside the medium. If vacuum values are meant, we will use a subscript v as in

(3.1 1) where W denotes the radian frequency of the light and no the refractive index. We further assume that the medium is nonmagnetic, i.e.,po=pv. In order to treat in what follows immediately, the cases of a traveling wave and a standingwave simultaneously, we will for the configuration of Fig. 3.2 write (3.6) as &(X, Z,

t)=b(r) cos [Kx-p(r)]

(3.12)

where for a travelingwave

p(t)=Qt+&

(3.13b)

and for a standingwave b(t)=lAnl cos(Qt+&)

(3.14a)

p(r)= -a

(3.14b)

The importance of writing & in the form (3.12) lies in the fact that the temporal variation of b and p is assumed to be extremely slow comparedto the transit time of the light through the sound field. Consequently, the interaction analysis, for both traveling and standing waves, may proceed on the basis of a “snapshot” of the sound field, during which b and p are constant. Their time-dependence may then be inserted a posteriori in the results of the analysis. The light is representedby its electric field: e(x, z, t)=Re[E(x, z, t ) exp (iot)]

(3.15)

Note that for light we use the concept of generalized time-varying phasors from the beginning. This is particularly useful for the optical fieldswe deal with, because many of these are up- or downshifted by multiples nQ of the

Chapter 3

40

sound frequency. Thus, E,,(x, z, t)=E,,(x, z) exp(jnQt) is an often used expression for sucha phasor field, where it is understood that the associated frequency equals o. For simplicity, we shall at present not take polarization into account; hence, e is a scalar quantity. We shall also usea normalization such that for the plane wave E(x, z)=E exp(-jk,x-jk,z) the intensity is given by

I= EEr

(3.16)

where the asterisk denotes the complex conjugate. A s for the frequencies of sound and light, it will be assumed throughout that nlW 1. With the exception of one application in Chapter 8, we will also adhere to the so-called small Bragg angle assumption, i.e.,Klkel. Together with the assumptionof the paraxial propagationof incident soundand light, this means that also the scattered light can be so treated. Hence, in all so that sin Q-@, sin y-3: cos @-l-q2/2, relevant calculations, ye1 and @el cos y- 1-f l 2 . Following the above preliminaries, we are now ready to analyze the phenomenon itself. For that purpose, we first consider the case where the sound column is thin enoughto be considered a thin phase grating (Exactly what is meant by “thin enough” in this context will be discussed later.) The effect of such a thin phase gratingmay be analyzed by the identical methods used in optical processing, i.e., by calculating the phase shift of each optical “ray”and neglecting ray bending and diffraction [l]. We shall call this technique the straight undiffracted ray approach (SURA). The results so obtained are characteristic of so-called Raman-Nath or Debye-Sears diffraction.

3.1.1

Normal Incidence

Let Ei denote the complex electric field strength of the normally incident plane wavein Fig. 3.2, i.e., E(x, 0, t)=E(x, O)=Ei. By our previous assumptions, the total phase shift of the light “ray” (i.e., the partial wavefront of widthAx) at x is given by L

@(x,L, t ) = -k,l6n(x, z, t)dz- kL

(3.17)

0

With (3.12) we find readily e(x, L, t)=-k,Lb(t) COS[KX-/?(~)]-~L

(3.18)

The Heuristic

41

Thus, the electricfield at z = L is given by E(x, L,t)=Ei exp( -jkL)exp

{ -jkYLb(t)~os[fi-p(t)]}

(3.19)

Equation (3.19) represents a spatially corrugated wavefront at the exit of the sound field. This is schematically indicatedby the wavy line in Fig. 3.2. Equation (3.19) is strongly reminiscent of electronic phase modulation that, as most electrical engineers are well aware of, results in many sidebands, the amplitudes of which are given by Bessel functions [2]. Following this lead, it is readily shown that[3] E(x, L,t)=exp(-jkL)C(-J3”EiJn[kvLb(t)]Xexp[-jnfi+jnp(t)] (3.20)

where the summation is fromn = - W to n = + q and J n denotes the nth order Bessel function. The physical interpretationof (3.20) isthat each term in the sum gives rise to one of the set of plane waves or orders into which the exit field can be decomposed.Morespecifically, the nth term causesaplane wave to k, such that knx=nK,i.e., the angleof propagation propagate in the direction Qi, is given by

nK

sin4, =-=k

nil h

(3.21)

Unless indicated otherwise, paraxial propagation will be assumed for all fields involved so that (3.21) may be written

nK

(pn=-”=-

k

nil h

(3.22)

As schematically indicated in Fig. 3.2, the exiting light field thus gives rise to adiscrete spectrum of plane waves or orders ofdiffractedlight, propagating in the directions specified by (3.21). In an actual experiment, each planewave is of necessity a beam of finite width. In retrospect, (3.21) is quite plausible as it is also the basic equation for light diffraction bya grating of period h [4]. The sound fieldin our experiments acts essentially likethat, with the proviso that its nature isthat of a (sinusoidal) phase, ratherthan amplitude grating. This point is actually nontrivial and merits some discussion. The typical gratings mentioned in elementary physics textbooks are amplitude gratings and only generate a multitude of orders because they consist of sharplydefinedlines(e.g.,aRonchiruling). A sinusoidal

Chapter 3

42

amplitude gratingwould only generate two orders in addition to the incident light [l].Our sound wave generates many orders, in spite of its sinusoidal character, because it is a phase grating. There is a complete analogy here to the case of amplitude modulation and phase modulation in electrical engineering. Returning now to (3.20), let us first consider the caseof a traveling sound wave. With (3.13) we find that the nth order contribution to the exit field is given by

where we have written &(x, L, t ) in the form of an oblique plane wave with the origin as a phase reference point and traveling in a direction @,, defined by knx=k sin &=nK

(3.24a)

k,,=k

(3.24b)

COS

@n=[k2-(nK)2]”2

so that within a small phase error of order KZLlk(4, as we shall show later)

The socalled Raman-Nath parameter v [5] denotes the sound-induced peak phase shift

The relative intensityof the various ordersis thus given by

as shown in Fig. 3.3. With eqs. (3.15) and (3.23), we may write the complete expression for the plane wave in the region z>I to which E&, L, t) gives rise to e,@, z, f)=Re{E,expli(o+nQ)t-jkx sin @,-jkz cos&]} (3.28)

It is seen from (3.28) that the nth order is shifted in frequency bynQ, i.e., by n times the sound frequency. The classical explanation of this is the Doppler shift, which may be seen as follows. An observer lookingat the nth

The Heuristic Approach

43

t

2

3

4

V

5

Figure 3.3 Intensity of orders, diffracted by a thin sound column, as a function of the peak phase delay v (Raman-Nath parameter).

order sees the sound-induced radiating dipoles moving upward with sound velocity V. The velocity component in his direction is given by Vsin $,; hence, with eq.(3.21) we find for the Doppler shift (3.29)

Note also from (3.25) that the nth order is shifted in phaseby -n~-/2+n$~ due to the scattering process. Now consider the sound field to be a standing wave. With (3.14) and (3.20), we find that the nth order contributionto the exit field is given by En(x,L, t )=En(f) exp(-jkx sin &-jkL cos $,)

(3.30)

where, to within the same small phase error as (3.25), in E,,(f)=(-ly cos(Qf+$,)]Ei exp[-jna]J,[v

(3.31)

Note that because J,,(O)=O, n&, each order vanishes twice per cycle. This is, of course, no surprise because the entire sound grating, beinga standing wave, vanishes twice per sound cycle. Finally, it is of interest to investigate the limit of weak interaction, i.e., when v+O. Using the small argument approximation for the Bessel functions and neglecting second-order terms in v, we find readily that only three orders(- 1,0, + 1) remain and that fora traveling wave

44

Chapter 3 (3.32)

(3.33)

x exp(- jkx sin4- jkLcos@,+ jQt)

x exp(- jkx sing-, - jkLcosqLI - jQt)

(3.34)

and for a standing wave

(I) 4 -jkL cos 4 +

El(x, L, t ) = -jEi - exp(-ja) x [exp(-jkx sin

+ [exp(-jkx

jQt

+ j & )]

(3.36)

sin4 - jkL cos 4 - jQt - j g s ) ]

(11

&(x, L, t ) = -jE, - exp(-ja)

- jkL cos

x [exp(-jkx sin

+ [exp(-jkx

&,- jkL cos

sin

+ jQt + j&)]

(3.37)

- jQt - j & ) ]

Note that, in the standing wave case, both orders contain up- and downshifted components. This isinagreementwith the amplitude modulation evident in(3.31).There exists, however,an equivalent and useful physical interpretation as .follows. The standing wave of refractive index variation may be thought of as two counter propagating waves, each giving rise to a peak phase shift of v/2, when acting alone. In the limit of weak interaction, the effects of each ofthesetwo wavesmaybeanalyzed + 1 order the upward traveling separately and the results added. Thus, for the wave will upshift the frequency, the downward traveling wave will cause downshifting. The two components so generated can be calculated from (3.33)and (3.34)by replacing v with v/2). The result will be given by (3.36) and (3.37),if the two travelingwaves have been giventhe right phase shiftto account for a.

45

The Heuristic Approach

3.1.2 Oblique Incidence This configuration is shown in Fig. 3.4 where the light is incident at an angle $Q.The ray bundle entering the sound beam at x suffers a total phase shift upon traversal to x’ (3.38)

where &‘(C, t ) denotes the sound-induced refractive index change along the oblique path C. From Fig. 3.4, we see readily that &‘(c,t)=6n(x+C sin $Q, Ccos Qo, t )

(3.39)

or, using the paraxial approximation,

&’(C,

C,

(3.40)

t)=b(t) cos[K(x+$Q~-P(~)l

(3.41)

t)=&(’(x+$Q<,

With (3.12)

&’(C,

X

Z

Figure 3.4 Obliquely incident light and diffracted orders for a thin sound column.

46

Chapter 3

Substituting (3.41) into (3.38) and using cos @O -1 -&2, order in $0

we find to a second

where sinc X = sin(7GX)InX. The obliquely incident electric field may be written E(x, O)=Er exp( -jk@ox)

(3.43)

The exit field E(x, L, t) may now be calculated in a way analogous to eq. (3.19). Taking into account the ray displacement, i.e., replacing x by x’-L@o, and leaving out the prime on x’ in the final result, we find after some tedious algebra E ( x , L, t ) = E, exp -jkL - jk&,x+ 2 x exp{- jk,,Lb(t) sinc

Kx-p(t)-=] 2

(3.44)

}

Like eq. (3.19), eq. (3.44) also represents a corrugated wavefront that may be written as E ( x , L, t ) = exp(- jkx sin

[

e0- jkL cos eo)X(j)”Er

(

x J , k,Lb(t) sinc -

(3.45)

We note that (3.45) is identical to (3.20) but for a term, exp(-jkx sin eO-jkL cos @I), expressing oblique incidence; a phase shift, nKQoL/2, referring to the center of the sound beam; and a multiplying factor, sinc(KQoL/2z), in the argument of the Bessel functions. The factor sinc(K@oL/2z)= sinc[&/(ML)] describes the effect of phase

The Heuristic Approach

47

cancellations along the oblique path of the incident ray. Note that this factor vanishes whenever &=nNL, ie., whenever the incident ray tranverses an integral number of sound wavefronts. In that case, the accumulated phase shift averages out to zero and the diffraction effects disappear. Because of the similarity of (3.20) and (3.43, all qualitative conclusions drawn from the normal incident case apply to oblique incidence. In particular, the incident light is split up into many orders, located symmetrically around it. This is shown in Fig. 3.4, where (3.46) Defining

(YtL)

v’ = v sinc

(3.47)

we find with (3.12) and (3.13) that for a traveling sound wave, the nth order contribution to the exit field, written as a plane wave, is given by En(x, L, t)=En exp(-jkL cos &-jkx sin ipn+jnQt)

(3.48)

where, to within a negligibly small phase error, (3.49) For a standing wave, we find with (3.12) and (3.14) En(X,

L, t)=En(t) exp(-jkL cos @,-jkx sin @n)

(3.50)

where, to within the same negligible phase error, E,(t) = (- j)“ exp(

-jnKq0L 2

Jn[v’ cos(C2t + &)]E,

(3.51)

It should be noted that in the region s>L, the field is found, as in (3.28), by substituting z for L in (3.48). For later use it is of interest to explicitly write the amplitudes E,, of the orders -1, 0, + I in the case of weak interaction with a traveling sound wave

48

Chapter 3

EF Ei

(3.52) (3.53)

E,, = -jEi

(i’)

sin(

y) +

exp( j@$-

2

(3.54)

Using (3.26) and (3.10), we may write for (3.53) and (3.54)

E-, = -jEi

( ]

E,, = -jEi

( -~ 3 (- [-

1 )

C ’S * sinc[ K@OL exp( jQ0L

C’Ssinc K-;L) exp -j;@oL)

(3.55)

(3.56)

where the sign ofC‘ is accounted for. in (3.56) As we will see later, the factor C‘S sinc(K$ol/2lc)exp(-jK$oL/2) represents theangular plane-wave spectrum (radiation pattern) S(j), evaluated at y= -h, of the sound field emanating froma uniform transducer of width L , displaced by L/2 from the origin [l]. Aphysical interpretation of (3.56) is that, of the entire spectrum, only that particular plane wave of sound propagating upward at an angle y= - h [dashed and labeled S(-&)in Fig. 3.41, and being perpendicular to the incident light, causes diffraction effects that result in + l order generation as indicated by (3.56). In addition, the same plane wave causes a contribution to the -1 order, with an amplitude related to the complex conjugate of the sound plane-wave amplitude, as indicated by (3.55). We will return to these aspects later when we develop our plane-wave interaction theory. There, it will become clear that plane waves of sound and light can only interactif their wave vectors intersect almost perpendicularly. In our present treatment, the “almost” is lost because of the simplifying picture of straight undiffracted rays traversing a thin phase grating.

3.1.3

Criteria for Raman-NathDiffraction

We have seen that the use of the SURA approach describes Raman-Nath diffraction as defined by (3,25), (3,31), (3.49), and (3.50). We willnow investigate under which conditions this approach is valid and Raman-Nath behavior is to be expected.

,

49

n e Heuristic Approach

In our analysis so far, we have made the following assumptions:

1. The sound field is thin enough to ignore optical diffraction effects. 2. The sound is weak enough to ignore optical ray-bendingeffects. Let us now try to quantify these assumptions in orderto find their limitsof validity. In regard to 1, consider the ray bundle of width Ax (Fig. 3.2) with which we have “probed” the soundfield for local phase delays. Inour analysis, this ray bundle is assumedto be parallel; in reality it spreads by diffraction. The angle of spread is on the order1A.x; hence, the width of the bundle at z=L is approximately Ax+LA/Ax. A minimum value is obtainedwhen Ax=(LA1I2 and results in a width of 2(LA)ll2 at the exit. It will be clear that this width must, in fact, be much smaller than, say, 1 rad (N2n) of the sound wave length in order for our phase-sampling to be sufficiently localized. It is readily seen that this leads to the condition A2 L((2na

(3.57)

or, in termsof the so-called Klein-Cook parameterQ = L @ / k [6], Q& 1

(3.58)

A physical interpretation of (3.58) along the lines we have followed here was first given by Adler[l. Condition (3.58) is sometimes called the Raman-Nath or Debye-Sears criterion. It should be realized, however, that it is incomplete because it only states the conditions underwhich diffraction effects may be ignored;to also ignore ray bending, a second criterion is required, which we shall now discuss. Figure 3.5 shows a parallel ray bundle of width Ax’ propagating at an angle a; over a distance h’ Because . the gradient of refractive index causes a difference An (where An is real) across the ray bundle, the lower part of the bundle-where the velocity is greatest-has traversed a greater length than the upper part. The differenceAI is found readily from the differential phase shift kAl= k,AnAz‘

(3.59)

As a result of this effect, the original wavefront “a” has rotated to become the wavefront “by’ after traversing a distance A d . The ray bundle now propagates in the directiona,+Aar,where for small

Chapter 3

50

Figure 3.5 Ray,normal to wavefront “a” bentnormal to wavefront “b” by gradient of refractive index.

AI Aar =-

Ax’

(3.60)

From (3.59) and (3.60) we find, lettingAn,Ax’,Az’, A m , Al+dn, dx’, etc., (3.61)

or, invoking the paraxial approximation assumptions, (3.62)

As ar=dxldz, where the ray trajectory is denoted by x(& and n only depends on x , we find finally d2a

1 dn

7=(G-)z

(3.63)

as the equation forthe ray trajectory. Equation (3.63) may also be derived, in a somewhat less obvious but more rigorous manner,fromgeometrical optics [S]. A non paraxial exact treatment of ray bending can be developed in terms of elliptic functions[9];

The Heuristic Approach

51

for our purposes, it suffices to apply (3.63)to some special regions of the sound field. For simplicity, consider normal incidencewith the refractiveindex variation given by (3.9),with An real and a,=O. The maximum gradient of refractive index is

(2)mm =

Near the straight part of the sinusoidal variation of n, at a suitably chosen instant of time, the ray trajectory is then given, from(3.63)and (3.64),by x=

(-3

A?lz2

(3.65)

where x=O refers to the zero pointof the wave form. it will be clearthat the total displacement of the ray during traversal of the width L of the sound field must be much smaller than, say, 1 rad ( N 2 a of the sound wavelength, in order forour ray bundle to be sufficiently straight. Hence, the criterion for ignoring ray bending can be derived readily: (3.66)

In terms of the Klein-Cook parameter Q and the Raman-Nath parameter v, (3.66)may be written (3.67)

Qv42

The ray displacement interpretation of(3.67)was first given by Rytov[lo]. Another type of ray bending occurs near the maxima of the refractive index variation. Here, the sound wave acts as a lens focusing the incident rays. For our straight ray analysis to be valid, it is necessary that the focal point lies well outside the sound field. The effect may be calculated as follows. Near the maximum of n, at a suitably chosen instant of time, the refractive index is given by Ancos(Kx), and hence dn -=dx

kxn sin(Kx) = -K2Anx

(3.68)

in the immediate neighborhood of the maximum, defined by x=O in the coordinate system used.The ray trajectories follow from(3.63)and (3.68)

52

Chapter 3

(3.69)

Equation (3.69) indicates periodic focusing for rays close to the axis, a phenomenon that also follows from the rigorous ray theory[9] and is clearly visible in Fig. 2.3. The first focus occur& at a distance

3

~ K ( A ~ Z I"~ ) " ~

LP=[

It

(3.70)

where it has been assumed that An4no. Imposing the conditionthat L,bL lead6 to the criterion QV(<, It2

(3.71)

Essentially, (3.67) and (3.71) express the same kind of condition, with (3.71) being somewhat less stringent. The criterion for ray bending to be negligible may thus be written conservatively Qv4

1

(3.72)

In summary then, the criteria that have to be satisfied in order to be able to analyze the sound field with the straight undiffracted ray approach are given by Q 4 1 and Qv4 1. Ipso facto, these are also criteria for Raman-Nath behavior to occur. It seems natural to ask whether the ray analysis can be extended by exact ray trajectories. This calculating the accumulated phase shift along the is indeed possible, and relevant theories may be found in Refs. 1 1 and 12. Because such theories take ray bending into account but not diffraction, they are still subjectto the criterionQ 4 1.

3.2

THE SOUND FIELD AS A THICK PHASE GRATING

If the conditions Q 4 l and Q v a l are no longer satisfied, then the SURA method may no longer be applied. However, the width of the sound column may be subdivided in thin slices parallel to X , each of which satisfies the conditions for applying the SURA approach. This is the basis ofVan Cittert's analysis [13]. The slices are taken to be of infinitesimal thicknessdz and hence act as weak thin phase gratings that generate two additional

The Heuristic Approach

53

orders (3.53-3.56) for each incident one. This scheme, which we shall call the cascaded thin grating approach (CTGA), is symbolically illustrated in Fig. 3.6, where two slicesare shown, separatedby a finite distance for clarity. The beauty of Van Cittert's scheme is that use can be made of the thin phase grating results already known, and that these may now be considered exact because the gratings are infinitesimally thin. For the traveling wave case (the only one treated here), we may thus go directly to (3.55) and 3.56), realizing that inVan Cittert's scheme the nth order at the point z is contributed to by downshifting from the(n+ 1)th orderand upshifting from the (n- 1)th orderat that same point. The contribution d-En(z), downshifted from En+l(z),may be obtained directly from (3.55) by letting E- 1+exp( -jkz cos 4,,)d-En(z)

(3.73a)

E,+exp( -jkz cos qh + l)En+l (z)

(3.73b)

L+dz

(3.73c)

Hence, to within a first order indz, exp( -jkz cos qn)d-En(z) = -0.5jEn+l(z)k,C'S*dz-exp(-jkzcos @,+l)

t X

Figure 3.6 approach.

Successivediffractionmodelusedinthecascadedthingrating

(3.74)

Chapter 3

54

The forward-projecting phase terms in (3.73a) and (3.73b) are needed because the slice dz underconsideration is located at z,not at the origin as in the formulations (3.55) and 3.56). The contribution d+En, upshifted from En-1, is round in the same way from (3.56). exp(-jkz cos &)d+En(z) =-O.SjEn-l(z)k&’S dz.exp(-jkz

COS 6 - 1 )

(3.75)

To conform withthe existing literature [14],we shall adopt the notation kC k,C’ = 2

(3.76a)

so that

(3.76b) Adding the two contributions specified in (3.74) and (3.75), we find with (3.76a) dEn = -0.25jkCSE,, dz

exp[- jkz(cos@,-, -cos@,,)]

(3.77)

- 0.25jkCS* Entlexp[-jkz(cos@ntl -cos@,,)] Note that in (3.77) the En’s denote the amplitude of plane waves with a phase reference point at the origin. Thus, for C=O the solution of (3.77) is given by E,=constant. In our formulation, the propagation through the unperturbed medium is already taken into account. It is useful to define a virtualfield at the origin in terms of such “back projected” waves: E(&, t; z)=XEn(z) exp(-jk sin

&x+jnflt)

(3.78)

Note that in (3.78), z appearsas aparameter that refers to the position (cross section at z) of the actually existing field that is to be back-projected. This actually existing field is, of course, given by forward projecting the waves again:

E(x, z, t)=XEn(z) exp(-jkz cos

&-jkx sin &+jnnt)

We will elaborate on the concept of virtual fields later in Sec. 3.3.3.

(3.79)

55

The Heuristic Appmach In most formulations of the problem (including Van Cittert's),a somewhat artificial fieldY is used instead of a plane wave En(z) exp(-jkz $n)=Y,,(z) cos exp(-jkz cos

$0)

(3.80)

In termsof Yn,(3.77) becomes dyn dz

jk(cos$,,

"

-cos$,,)Y,, = -0.25jkCSY,,-, - 0.25jkCS * 'P,,-]

(3.81)

Written as in(3.81), the equations are similar to the well-known Raman-Nath equations [l 5,161. Klein and Cook [l71 state them in the following form, accurateto a second order inQin: (3.82) where it is assumed that S=-~lq,denoting a sound field of the form IS1 sin(S2t-fi). The unwary reader, trying to read the original source material, should be warned against the notational chaos that awaits him. Not onlyis the definition (3.80) often used, but in addition, Raman and Nath wrote n for -n (a practice still used by most European researchers) and used a rotated coordinate system with the 2 axis aligned with the incident light. To make matters worse, A* often stands for A, p 1 for I A n l , o* for Q, etc. In addition, equations are frequently normalized, with {=vz/2Lor c=vz/L, and various parameters are introduced such as p (or n)=Q/v (sometimes called regime parameters), a=-OSqb/$~,~=v/2L.Also, it is quite customary not to state the phasor convention used,nor which subscripts refer to the medium and which (if any)to free space. To write(3.82)in canonical form, many investigators, starting with Raman-Nath, define {=-vz

(3.83a)

p=Q

(3.83b)

L

V

so that (3.82) becomes

Chapter 3

56

(

2d Y,, +(Y,,-Yn+l) , = j n p n + -' Y ) y n d5

(3.84)

In this book, we shall stick to the formulation (3.77) because ( 1 ) it already takes conventional propagation effects into account; (2) it is physically obvious in the sense that itidentifies the contributions to E, from neighboring orders, with the difference of the cosine terms indicating the degree of phase mismatch; and ( 3 ) it is a special case of a generalized equation valid for any sound field, not just a sound column. The latter point is due to the fact that although the CTGA method is apparently heuristic, it does not really make any simplifying assumptions other than the scalar nature of the field and the paraxial modeof propagation. Neither ray bending nor diffraction are ignored. It should therefore not come as a surprise that the final equations (3.77) may be directly derived from the scalar wave equation (in fact, this was already done by Raman and Nath in their original treatment [15,16]),nor that they may be generalized so as to apply to arbitrary sound fields. We shall return to this point later.

3.2.1 TheRaman-NathRegime As already pointed out, the difference of cosine terms in (3.77) indicates the lack of phase matchingthat characterizes the interaction.To a second order in Klk and qb, we may write with (3.46)

):(

cos &+l -cos $, = - - $0 -

cos (pn-, -cos

(-

(3 +(7)($1

en= - -

$0

-

(3.85)

(3.86)

Let us now assume that the phase mismatch between neighboring orders over the maximum length L ismuch smaller than, say, 1 rad for all significant orders n, i.e., 2

2

k($) Lc(l

and

kn($).L((l

(3.87)

Which is the highest order that we should take into account in (3.87)? We anticipate that, in the absence of phase mismatch, we will obtain much the

The Heuristic Approach

57

same resultas before for small L. Because the order amplitudes will be given by Jn(v), we can guess that the numberofsignificant orders willbe proportional to v. (This, by the way,is the exact analog of frequency modulation in radio engineeringwhere the number of significant sidebands is proportional to the phase modulation index [2].) With that assumption and remembering that Q=K2/Llk,we may write for (3.87)

Qv41

Q41

and

(3.88)

Condition (3.88) is, however, the exact condition for which the SURA method is applicable. Hence, we conclude (1) that assuming perfect phase synchronism in the interaction is equivalent to using straight undiffracted rays and (2) that the results of the present analysis should be identical to those of the previous one. With (3.85-3.87), we may writefor (3.77) dEn - -0.25jkCSE,,-, exp(j&b0z) dz - 0.25jkCS * E,,+,exp(jK&,z)

"

(3.89)

together with the boundary values Eo(O)=Ei (3.90)

and

En(O)=O

Denoting

we may write (3.89) with (3.5) (3.92) If we use the recurrence relations for the Bessel functions [3], it may be shown [l41 after tedious algebrathat the functions sin

Tn=exp(jnbz)J,(a

satisfy the recurrence relation dTn - 0 . 5 exp(2jbz)Tn-, ~ - 0 . 5 ~exp(-2jbz)Tn+, dz

"

(3.94)

Chapter 3

58

Comparison of (3.94) and (3.92) suggests equating

Tn= EL;

a = 0.5kCISI; 2b

= -K$,,

(3.95)

so that the solution of (3.92) may be given directly in terms of EA or, with (3.91), in terms of E n (3.96)

where we have used (3.47) and (3.76). As surmised, our present solution (3.96) is identical with the one (3.49) derived by the SURA method, thus confirming the equivalence of the criteria for phase synchronism with those for straight undiffracted ray analysis.

3.2.2 The BraggRegime If the interaction length is long enough, the phase mismatch expressed by will prevent any cumulative contribution the cosine difference terms (3.77) in from neighboring orders, and the net effect is that no diffraction occurs at all. There exist, however, two conditions in which there is phase matching between the incident light (at angle h) and one neighboring order. This occurs when either cos

$1

(3.97)

in which case, the + l order interacts synchronously with the incident light or when cos $o=cos

$-l

(3.98)

in which case,the interaction takes place with the - 1 order. With (3.46) we find that (3.97) is satisfied when (3.99)

and (3.98) when $o

+K =”= 2k

+$B

(3.100)

59

The Heuristic Approach

Equations (3.99) and (3.100) correspond to the configurations for upand downshifted Bragg diffraction. They are illustrated schematically in Figs. 3.7a and 3.7b. From the symmetry of the interaction, it looksas if the diffracted order is reflected off the moving wavefronts of sound in the same way that x-rays are reflected off atomic planes. Indeed, if one uses this analogy and imposes the condition that rays reflected off successive crests add up constructively, conditions (3.99)and (3.100) are derived readily[lS]. Nevertheless, this simple heuristic model should be viewed with suspicion because, unlike a crystal where the x-rays .are scattered off well-defined discrete planes, a sound column is, in essence, a continuous distribution of “reflectors” Hence, there isno a priori physical reasonwhy it should behave in the same fashion as a crystal. In fact, we have already seen in Ch. 2 that acousto-optic scattering doesnot (at least to a first order) show the multiple Bragg angles typical of x-ray diffraction. Substituting (3.97) into (3.77) and neglecting all order but 0 and +l, we find the equations for upshifted scattering

I

I

a

EO 4

L

E I’i

E -1

b Figure 3.7 (a) Upshifted Bragg diffraction. (b) Downshifted

diffraction.

.110)

Chapter 3

60

d-4 = -0.25jkCSEo -

(3.101)

dEo - 4 . 2 5 jkCS* E,

(3.102)

dz

"

dz

It isnow readily found with (3.76), by imposing the proper boundary conditions, that inside the sound column

El = -j exp(j4,)Ei sin

(3.103)

(3.104) Similarly, with (3.98) the equationsfor downshifted scattering are

- -0.25 jkCS * Eo

"

dz

dE0 = -0.25jkCSE-, dz

(3.105)

(3.106)

with the solutions inside the sound column E-, = -j exp(- jqjs)Eisin ( O . 3

(3.107)

Eo = E, cos(") 0.5vz

(3.108)

Eo to Ei Note that a complete periodic transfer of power takes place from and vice versa. The intensities of the two orders leaving the sound column i.e., at z=L, are in both cases given by ZO'Zi

COS*(0.5V)

1+1,-1=Ii sin2(0.5v) This behavior isshown in Fig. 3.8.

(3.109)

The Heuristic Approach

61

1.2

1.o

0.8 0.6

0.4

0.2 0.0

4

2

0

V

6

8

10

Figure 3.8 Power exchange betweenorders in pure Bragg diffraction.

3.2.3

TheNear-BraggRegime

In many cases, it is of interest to investigate Bragg diffraction behavior near, but not at, perfect Bragg angle incidence. For upshifted interaction, this may be done be setting 11)

&o=-@B+A@

It is readily shownthat in this case &-COS

COS

$1

=AHNk)

(3.112)

Using (3.77), the equations coupling the orders 0 and + 1become dEl = -0.25 jkCSE,, exp(- jK.zA@) dz

(3.1 13)

dE0 = -0.25jkCS * E , exp(jKzA@) -

(3.1 14)

dz

We now introduce new dependent variablesEO’ and El’: E , = E; exp(

T)

(3.115)

Chapter 3

62

(3.1 16) Substituting (3.115) and (3.116) into (3.113) and (3.114), we find (3.117)

dE6 -+ dz

AI$ jK-EE,=-jaS*E; 2

(3.118)

where a= KC14= -kpno2I4. Substituting a time dependenceexp(jr.z) for both El' and Eo' into (3.117) and (3.1 18),

Eh=A exp(irz)+B exp(-jrz)

(3.i 19)

E{ = C exp(irz) +D exp( -jrz)

(3.120)

and setting the determinantof the resulting set of linear equations equalto zero, we find (3.121) The constants A , B, C, and D may be found from the boundary conditions. After some algebraand application of (3.1 15)and (3.1 16), we finally findfor El and EO (3.122)

-jkA4 2r

]

sin(rz) exp( jKF4)

(3.123)

Using the notation alSlL=v/2, we find for the intensities at the output of the sound cell (z=L)

63

The Heuristic Approach

h(;)

(3.124)

- z,

(3.125)

2

I, =

Io = Zj"

Note that the power transfer from IO to I1 and vice versa is now quasiperiodic and initially incomplete. This is shown in Fig. 3.9 for the intensities z=L. at the exit of the sound column It is readily checkedthat the interactionbetween l o and 1-1is expressed by exactly the same relations (3.124) and (3.125) if the angular deviation is defined such that (3.126)

h=q ~ + A q ' 3.2.4

Criteria far Bragg Diffraction Behavior

Our ignoring of orders other than those that interact phase-synchronously is based on the assumption that the term (Klk)2k in (3.85) and (3.86),

1.d

0.8

0.6 0.4

0.2

0.0

0

2

4

V

6

l0

Figure 3.9 Partialpowerexchangeinoff-angleBraggdiffraction.Thedeviation from the Bragg angle equals0.25 NL.

64

Chapter 3

multiplied with an effective interaction lengthLe, must be much larger than 1 rad. If the sound field is weak, then the minimum Le=L, and the condition becomes simply Q 91

(3.127)

In general, however,the effective interaction length shouldbe defined as that length in which complete power transfer has taken place in pure Bragg diffraction. According to (3.108), this occurs when 0.5vLdL=d2, i.e., when L, is of the order of L h . It is readily seen that the condition then becomes Q/vS1

(3.128)

In summary, wemaysay that for Bragg diffraction to occur, both conditions must be satisfied, i.e.,Q%l and Q/v%l. A more mathematically oriented discussion of the various criteria may be found in Refs. 19-21. It should be notedthat in practical devices, v e x so that (3.127) suffices as the sole criterion. Klein and Cook 161 have calculated the power in the first diffraction peak as a function of Q. This is shown in Fig. 3.10, in which it follows that more than 90% of the light is diffracted when Q>6.

1 .o 0.8

0.6 0.4

0.2

0

2 3 4 6 810

Q

20

40

l00

Figure 3.10 Maximumdiffractionefficiencyas a function of Q. (From Ref 6.)

0 1967 IEEE.

I

The Heuristic Approach

3.2.5

65

WeakInteraction

If we make the assumptionthat the interactionis very weakso that only the + l and - 1 orders are generated, and the 0 order is unaffected, then (3.77) becomes with (3.85) and (3.86)

dJ% = -0.25jkCSE, exp -

(3.129)

dz

dE-1

- -0.25jkCS * Ei exp

(3.130)

"

dz

These equations may be integratedto give with (3.76b) E-1( L )= -jEi

('f')* [ C'S

sinc KL(@&-

[

x exp

@B

'3

(3.131)

(3.132)

+gB)]

It is interesting to compare (3.131) and (3.132) and (3.55) and (3.56), and with the subsequent interpretation following the latter equations. It is clear that this interpretation should be revised slightly in the following way. In (3.132) the sinc term represents the angular plane-wave spectrum of the sound field evaluated at Y=-(@+@B), not at y=-& as in (3.56). It is that particular plane wave in the spectrum that could be interpreted as being responsible for generating the E1 order. As shown in Fig. 3.1 1 the wave vector of this wave, labeled ,!?[-(+O+@B)], makes an angle of d 2 - @ B with the incident light, not d 2 as would follow from (3.56). The angle @B is the slight (but very important) correction remarked on in the discussion (3.55). of This time the correctionwas not lost in the assumptions, because (3.77) does take diffraction into account, in contrast to the SURA method that led to (3.56). It will be seen in the next section that the angle d2-@Bis precisely the one required by conservation of momentum conditions. Using the same reasoning as above in connection with (3.131), we find that an angle of

66

Chapter 3

Figure 3.11 Interpretation of diffraction by a thin sound column, using angular plane-wave spectrum concepts. d2+&I is required to generate the - 1 order. This also will be seen to be required by quantum-mechanical considerations of the photon-phonon collision process.

3.3 THE SOUND FIELD AS A PLANE-WAVE COMPOSITION In this section, we shall first define formally what is meant by a plane-wave composition, next discuss the interactionof sound and light on the basis of wave vectordiagrams, and finally formulate a heuristic plane-wave interaction theory, using some results obtained previously to quantify the formalism.

3.3.1 The AngularPlane-WaveSpectrum In the region morethan a few wavelengths away froma real source (suchas a transducer), any field satisfying the wave equation may be decomposed into plane waves [l]. In our treatment, we assume that the sound fields travel nominally in the + X direction and the light fields in the + Z direction, i.e., K,>O and k,>O; in other words, we ignore reflected waves. As for the sound field, that is just an experimental condition; as for the light field, it is equivalent to assuming that the gradient of &I is very small

The Heuristic

67

and that, in any case,. there exists no phase-synchronous interaction between incident light and reflected light. This is, of course, no longer true for large Bragg angles, butwe do not consider such cases as yet. At any rate, with the conditions mentionedabove and assuming paraxial propagation, an arbitrary light field cross section along the line perpendicular to the Z axis at the point zmay be written as a spectrum l?($; z) of plane waves.If we use the origin as a phase reference, z)exp(-jkx sin$- jkz cos$)d(k$/2lt) E(x, z)= pi($;

(3.133)

where E(x,z) isto be thought of as a cross section at z parallel to x.Note that the amplitude spectrum of the plane waves, E($; z), is assumed to be a function of z. This is because generally such a spectrum refers to a'particular scattered order, sayEn($;z) for the nth order, and continuously evolves on its z) is a transit through the interaction region. Inthat regard, &C$; generalization of the z-dependent plane-wave amplitude E&) used before. In the unperturbed medium, l?($; z) is constant and can be written as E($). For the incident light, for instance,Et($; z)=&$). It should be stressed that this independence of z only applies if the angular spectrum is defined with the origin as a phase reference, as we do here. It is quite common in the literature, however,to use a local phase reference forl?($;z)i.e., the point(0, z) itself. This is, for instance, the convention followed in [l]. In that case, (3.133) is simpler in that the exp(-jkz cos $) factor disappears. However, that same factor now reappears as a propagator for the spectrum in the unperturbed medium. We shall in what follows adhere to the convention of using the origin as a phase reference. The plane-wave spectrumso defined is sometimes called the virtual plane-wave spectrum. More details of spectral formalisms will be found in Sec.8.6. Using the paraxial assumptionand rearranging terms,we may write

(::)

E(x, z)= r k ( $ ;z) exp(-jkz cos$) exp(-jkx$) d -

. (3.134)

It is clear that (3.134) represents a Fourier transform between E(x, z) and l?($; z) exp(-jkz cos $), with transform variables x and k$/2lt. The latter quantity is related to the more formally used spatial frequencyf, by (3.135) so that (3.134) may also be written in symbolic form

Chapter 3

68

where 3 denotes the Fourier transform operator. The notation wAfxwe shall often leave out for brevity. With the paraxial assumption,we may also write

[

-

exp(-jkz cos) I exp -jkz

(3.137)

[l-2@*)1

It should be noted that, strictly speaking, the integral limits in (3.133) should be -l/A and +l/& as this is imposed by sin $
l?($;z) = exp(jkz cos$)jmE(x, z)exp(+jkx$)dx

(3.138)

S

or symbolically

E($; =exp(jkz z) cos

$)F~[E(x,z)],

fX+A

(3.139)

where 9 - l denotes the inverse Fourier transform operator. For the sound field we have similar expressions. In this case, however, the plane-wave spectrum Sufi does not change with the distance traveled. With the origin again as the phase reference

-

S(x, z) = r s ( y ) exp(-jKz sin y - jlvx cos y ) d where S(x, z) is to be thought of as a cross section through x parallel to z. Using the paraxial approximation and rearranging terms, we may write

(F):

xexp(-jkiy) d

(3.141)

It is seen that S(x, z) and 11'(fiexp(-jlvx cos fi form a Fourier transform pair with zand K742a as the transform variables.The spatial frequency F, is

69

The Heuristic Approach

defined by -="= 2a A

(3.142)

F,

Equation (3.141) may be written symbolically (3.143)

The inverse relation is &y) = exp(jKx cos y

rS(x,

J-

z) exp(+jfiy) dz

= exp(jKx cos y)s"[S(x, z)], F, 4 y/A

(3.144)

As we have stressed before, it is useful to define virtual (back-projected) optical fields JP(x; z) at the origin along the X axis. The "z" in E(")(x; z) is to be thought of as a parameter that indicates the location of the backprojected cross section E(x, z). Another way of saying this is that such virtual fieldswhen propagated (forward-projected) in the unperturbed medium define actually existing fields E(x, z) at z. Now, E(x, z) is also defined by (3.133) that contains the forward projection [term with exp(-jkz cos 4)] of the angular spectrumE(@;z). It is, therefore, easyto show that the following relationships exist.

E("(x; z)= r k ( 4 ; z ) exp(-jkxq) d a

(3.145)

The concept of virtual fields is useful only for those fields, like &(x,n#i, z), whose plane-wave spectrum is continuously modified by passagethrough the perturbed medium. For fields like Edx, z) that are connected with the unperturbed medium and whose angular spectrum is therefore independent of z, all back-projected fields are identicalby definition E?"'(x;z)=Ei(x, 0).

Chapter 3

70

3.3.2 The Wave Vector Diagram As already mentioned in Chapter 2, the basiclight-sound interaction mechanism can be illustrated by means of wave vector diagrams. These diagrams refer quantum mechanically to the conservation of momentum in photon-phonon collisions and classically to phase-synchronous interaction. Figures 3.12(a) and 3.12(b) show the basic diagrams (in two dimensions) for up- and downshifted interaction. The. relevant k vectors belongto waves that differ in frequency by a very small relative amount Qlo and hence are essentially of equal length. Thus, the wave vector triangle is isosceles, and the Bragg angle condition follows immediately: K

@B

=%

for @B((l

(3.147)

a

Figure 3.12 (a) Wave vector diagram for upshifted Bragginteraction. for downshifted interaction.

(b) Same

71

me Heuristic Approach

Because only one plane wave of sound is shown, the diagrams of Fig. 3.12 refer to the pure Bragg regime where no plane waves of sound in different directions are available to cause rescattering into higher orders. Whether or not this condition in fact pertains experimentally depends on the width of the angular spectrum of sound and the strength of the interaction. Figure 3.13, for instance, shows a very wide spectrum of sound waves (a) that, at high sound levels, causes considerable rescattering (b). Thus, K2 scatters kl into k2, K3 scatters k2 into k3, etc. Note that the scattered orders are separated by 2 @as~was found before. For a sound column centered at L/2 and of length L, the angular planewave spectrum can readily be calculated from (3.144) with x=O. (3.148) and is seen to be of angular width =NL. As said before, rescattered orders are separated by ~ @ B = W AHence . the potential to generate many orders requires that N L S N L . This leads us back again to the Raman-Nath condition Q . 1 , which we have encountered before. In other words, from a

0

k-3 a

b

Figure 3.13 (a) Multiple-order generation by rescattering through (b) wide angular spectrum of sound (From Ref. 14.)

72

Chapter 3

plane-wave interaction point of view, the potential occurrence of many orders in a thin sound column (Raman-Nath regime) is due to the fact that enough different plane wavesof soundare available to cause rescattering. Now, Fig. 3.13 should not be interpreted too literally: rescattering is indeed involved, but the process is not simply one in which, say, first E1 is generated by means ofKI, then E2 by means of K2, etc. Such a picture would L, be equivalentto a cascaded interaction of successive sound cells of width each one acting on a particular order generatedby the preceding one. That is obviously not the real physical situation: there is only one interaction region of length L. In reality, the situation is more complicated and, as we will see later, can only be fully explained in terms of scattering paths (Feynman diagrams) that illustratewhat happens inside that single interaction region. Nevertheless, the diagram of Fig. 3.13 is a useful guide to understanding the physical mechanisms behind the phenomena of multipleorder generation.

3.3.3

Plane-WaveWeakInteractionFormalism

In weak interaction, rescattering doesnot occur and only the three orders0, + 1, and - 1 are involved. This is evident from the single incident plane-wave case described in (3.32-3.34).Is it possibleto say anything of the strengthof the interaction in the incident plane-wave spectrum case, i.e., to derived a quantitative formalism? Here, wehave to proceed very cautiously, The diagrams of Fig. 3.12 describe a global interaction; hence, they tell us nothing about whatgoes on inside the interaction region. We might, however, at this point inour heuristic analysis be able to at least describe the outgoing plane-wave spectrum of the light (i.e., the light field at +m) in terms of the incident spectrum and the angular plane-wave spectrum of the sound. As for the light, we shall therefore use the spectra EO($; +W), ,!?I($; +m), and E-I($; +W) as defined in Sec. 3.3.1. Such spectra could be deduced,by means of (3.139), from measurements outside the interaction region,z=m. at To lighten the notation, we write these spectra as EO($), El($), and I?-I($). Theydefine, through (3.145 , virtual (i.e.,back-projected)lightfields EV'(x; m), E'['(x; W), and E(!'](x; m), along the X axis at the origin. As explained before, E(')(x;z) means "back-projected from z." For brevity, we write Eb"'(x),&'(X), and E?'l(x), for the fields back-projected from infinity. Note that the propagationof Ei only pertainsto the unperturbed mediumso that Ei($; Z ) = E i ( @ ) and E!")(x, Z)=Ei(X,0) for all z. This was explained in connection with (3.145) and (3.146).

The Heuristic Approach

73

The virtual phasor quantities corresponding to the virtual fields are given by E'"0 (x;f)=EP(x)

(3.149)

Ef)(x; (3.150) f)=Ef)(x)exp(jkQt) where k=l, -1. The virtual fields, when propagated forward in the unperturbed medium, will uniquely describe the situation to the right, at z=w, of the interaction region. Thus, these fields describe the final results of the interaction. Our present formalism (i.e., the wave vectordiagram) cannot describe what really goes on inside the interaction region. Outgoing plane-wave spectra or, alternatively, virtual fields are all we can calculate and all we really need at present. In what follows, we shall assume weak interaction only, so that the 0) incident light isnot depleted andhence~o($)=~i($),ES"'(x)=El')(x)=Eo(x, =Ei(X, 0). Now, in accordance with the wave vector diagramof Fig. 3.12(a), it seems reasonableto try a first-order interaction description of the form (3.151) where CI is an interaction constant. Equation (3.151)isvisualized symbolicallyinFig.3.14(a). Inorder to find Cl, letusanalyze the configuration of Fig. 3.11 for which we already know the weak interaction result from (3.132). The obliquely incident field is given by (3.43) as &')(x)=Ei(x, O)=Ei (3.152) exp(-jk@ox)

Using this in (3.146)and remembering that &$), we obtain (3.153) so that

(3.154) From (3.148) we find (3.155)

Chapter 3

74

a

X

b Figure 3.14 (a) Wave vectors used in weak upshifted plane wave interaction formalism. (b) Same for downshifted interaction.

The Heuristic Approach

75

so that (3.151) becomes

or using the relations between$, qb, and $B implied by the delta function:

kl($1= CISLsin{ ( $ o +m&] h

+$&l

-jK(@o 2

(3.157)

($-$;2$d] Substituting (3.157) into (3.145), we find that (3.157) defines the following virtual field at the origin:

x exP[-jk(@Oi-2$B)xl

We recognize (3.158) as a plane wave propagating at an angle (qb+2$~)and amplitude given by the x-independent part of (3.158). This amplitude is to be equated withE@) in (3.132). We then find

c,=--jkC 4

(3.159)

so that (3.151) may be written

For the downshifted case [Fig.3.14(b)], we find similarly 8-,($))=-0.25jkC,!?(-$-$~)&$+2$~)

(3.161)

As will be shown later, the heuristically obtained relations (3.160) and (3.161) are in perfect agreement with those derived formally from the wave equation. They enable us to calculate the weak interaction of arbitrary sound and light fields.

Chapter 3

76

Some interesting properties and applications follow directly from (3.160) and (3.161). We will discuss three of these here, taking (3.160) as our standard equation. Invariance of Oberved DiffHcted Light Intensity Patternto an X Displacement of the Incident Light Beam

Let Ed$) correspond to a finite light beam of some sort, centered at the origin, with its electric field along the X axis given by Ei(x, 0). This will result in a diffracted spectrum ,??I($) whose corresponding virtual field ) the Xaxis can be calculated from (3.145) (beam) E ~ ) ( xalong Now, let us move the center of incident beam up to X = X O . Its angular spectrum is then given by (3.162) If we substitute this into (3.160), it is readily seen that the angular spectrum of the diffracted beam is given by

where we used the relation $B=A/~A.. With (3.145) we find that the virtual diffracted beam itself is now given by E(') I (x-XO) exp(-jfio)

(3.164)

i.e., it is displaced by the same amount as the incident beam without undergoing a change in shape. The only effect is a phase shift by -&o as a result of the incidentbeam intersecting the sound field at a different location. Thus, from an experimental point ofview, the appearance (i.e., the intensity pattern) of the diffracted beam in weak interaction is invariant to a displacement of the incident beam along the z axis. Measurement of the Sound Field Radiation Pattern

Let E,($) represent a plane wave incident at an angle $0.Its field along the X axis is given by (3.165) E,(x, -jk&) 0)=E/ exp(

The Heuristic Approach

77

With (3.139) we find that (3.166)

With (3.160) we find that the diffracted planewave is represented by (3.167)

Hence, it propagates in the direction & + 2 4 ~ ,as is to be expected. By measuring its intensity while varying &, the sound field angular power spectrum can be plotted, as was first demonstrated by Cohen and Gordon v21 A more detailed analysis would, of course, take into account that the probing field is a beam, not a planewave. From (3.160) it follows readilythat this makes little differenceas long as the angular width of the probing beam is much smallerthan thatof the sound field.

Bragg Diffraction Imaging Let &e) have a very large angular width so that, relative to q$),we may consider it to be a constant. It follows from (3.160) that, apart from a fixed angular displacement and an inversion, the diffracted angular spectrum is a replica of the angular spectrum. Hence, by performing an optical Fourier transform, it oughtto be possibleto obtain a pictureof the sound field. This method, called Bragg diffraction imaging, was first demonstrated by Korpel [23]. Detailed descriptions may be found in Refs. 14 and 24-27. We will return to it in Sec. 6.7.

3.3.4 Strong Bragg Diffraction of a Broad Light Beam In this two-dimensional case, a light beam of arbitrary cross section is incident at the Bragg angle on a thick sound column of width L (see Fig. 3.15). For weak interaction this case can, of course,be treated according to Sec, 3.3.3. For strong interaction we assume that the angular spectrum of the incident light is sufficiently narrow so that each incident plane wave in E,(@ is subjectedto near-Bragg angle diffractionand gives rise to a single diffracted wave E1(4+24~).Conversely, each plane wave E I ( ~is) generated from a single corresponding plane wave Ei(4-24B). The latter angleisequivalent to the incidentangle & usedin the treatment of

Chapter 3

78

I L L -

Figure 3.15 Interaction of broad light beam with rectangular sound column.

near-Bragg anglediffractioninSec.3.2.3.According to (3.1 ll), the quantity A$ then equals &+$B=$-$B. Pursuing this reasoning, eqs. (3.122) and (3.123) apply to our present problem if E1 is replaced by El($), EO by &($-2$B), and A$ by $-$E. We find for the plus one order atthe exit of the sound cell

79

The Heuristic

where l ? ~ ( @ = l ? ~ ( &w)=l?~(tj, L), because the perturbed medium ends at z=L. A rigorous treatment will be given in Sec. 4.2.4. A practical example for a Gaussian beam is to be found in Sec. 6.2. Note that for vanishingly small v, (3.168) may be written as

- 0.25jkCSL

sindK(qj-@,)L/2

mGzE

(3.169)

in accordance with (3.160).and (3.155). The method outlined here was proposed independently by Magdich and Molchanov [28] and by Chu and Tamir [29]. A similar method for the near field has been used by Chu and Tamir [30], and was later extended by Chu, Kong, and Tamir to the far field [31]. The selective angular depletion of the incident plane-wave spectrum, as evident from (3.168), has interesting consequences,such as beam shapedistortionand less thanoptimal diffraction. We shall discuss these effects briefly in Sec. 6.2.

3.3.5

Diffraction by a Profiled Sound Column

The case ofa two-dimensional profiled sound column subject to plane-wave incidence in the Raman-Nath region has been solved approximately by Leroy and Claeys [32] through a perturbation expansion, using a modified Raman-Nath equation. For a Gaussian sound beam, they find a negligible difference with a rectangular profile as long as the cumulative phase shifts induced by the acoustic fields are identicaland the width of the Gaussian is comparable to the width of the rectangular profile. This is intuitively plausible in the context of the SURA method, discussed in Sec.3.1. Mathematically, the case of a profiled sound column can be treated readily with Van Cittert’s method, discussed in Sec.3.2. A glance at Fig. 3.6 makesit clear that successive gratings do not have to have the same amplitude or phase. Variations of these parameters with z are simply taken into account by replacing S with S(z) in (3.77). Note, however, that this does not mean that an arbitrary sound field can be treated, because the method cannot deal with the case S=S(x, z). Only interaction with profiled (i.e. nonspreading) sound fields can be analyzed thisway. We will treat the case of an amplitude-profiled soundfield first, because it has a general solution. Starting from (3.77), invoking phase synchronism for all orders, and for

Chapter 3

80

simplicity settingS(z)=IlS(z)l and 0 0 , we find

dEn = 0.25kC1S(z)l(En_,- En+1) dz

(3.170)

Remembering that k~S(z)~/4=kylArz(z)1/2, we introduce an accumulated phase shift

V(z) = 0.5f kC]S(z)ldz

(3.171)

Substituting (3.171) into (3.170), we find

dEn - 0.5(En-,- En+l)

"

dF

(3.172)

which, beingthe recursion formula for the Bessel functions, gives E n =Edn( f)

(3.173)

It is clear that for a rectangular profileV(L)=v. The method outlined above was used by Pieper, Korpel, and Hereman [33] in the investigation of an acousticallyapodizedBraggdiffraction configuration. In that study, it was shown numericallythat the method gave correct results if the Bragg diffraction criteria were satisfied. It was also shown that Bragg behavior, defined as the complete periodic or quasiperiodic exchange of energy between two orders, was still maintained for large v, even when one of the Bragg criteria ( Q / v % l ) was violated. It is thought that this is due to suppression of the acoustic side lobes through apodization. At the time of this book's writing, no satisfactory analytic theory was available for this region, either for apodized or rectangular sound fields. Foraphase-profiled sound fieldthereexists no general solution. Sharangovich C341 has given an explicit solution for interaction of a plane wave of light with a cylindrical wave front of sound in an anisotropic medium. We shall here give the derivation in simplified form [35], ignoring anisotropy. A converging sound fieldis shown in Fig. 3.16. In the interaction region, defined by a relatively narrow beam of light, the radiusR0 of the wave front is assumed to be constant, and the soundfield is defined as

81

The Heuristic Approach

Figure 3.16 Interaction configuration for curved sound wave front.

exp(- j K x ) , 0 Iz IL

(3.174)

where \~rdenotes the relative phase at the edges of the sound field. Assuming Bragg interaction, we now use (3.101) and (3.102) with S=S(z), as explained before:

(3.175)

(3.176)

where a= KCl4. The above two equations can be manipulated to give

Chapter 3

82

(3.177)

A new variable is now introduced: q=-jx(z-T) K L

2

(3.178)

so that (3.177) becomes

(3.179)

Equation (1.179) is thestandardform of theequation defining the degenerate hypergeometric function [36] (also called Kummer's confluent hypergeometric function[37]) given by d2y JY x-+(q-x)"py=o C1.Y dx

(3.180)

with x=h, g= 112, p=a2Sn21(2jK/Ro). Equation (3.180) has the solution

where Q, denotes the degenerate hypergeometric function.In the present caseq= 112 so that x1-q+6q. Now, according to (3.178) 6q has two roots, which, as it turns out, both have to be used [35]:

83

The Heuristic Approach

0

4

a

12

16

Raman Nath parameter v

Figure 3.17 Degenerate hypergeometric solution for interaction withcurved sound wave front, and comparison with eikonal-Feynman diagram method. (From Ref. 35).

A similar treatmentmay be applied to El. A and B may now be found from the boundary conditions Eo(O)= 1 and El(O)=O. The resulting expressions are extremely cumbersome-to the point where it is perhaps easier to numerically solve the differential equation, rather than use tabulated values ofthe degenerate hypergeometric function. Figure 3.17 shows the power in the first order at the output of the cell as a function of the Raman-Nath parameter v [35]. The edge phase is taken to be I,U=~TC. Comparison is made with the approximate solution by the eikonal-Feynman diagram technique, which will be discussed in Sec. 4.9.

REFERENCES 1. Goodman, J. W., Introduction to Fourier Optics, McGraw-Hill, New York (1 968). 2. See, for instance,S. Haykin. Communication Systems, Wiley, New York (1983). 3. Olver, F. W. J., "Bessel Functions of Integer Order," in Handhook of

Mathematical Functions (M. Abramowitz and I. A.Stegun, eds.),Dover, New York, p. 721 (1965). 4. See, for instance,D. Halliday and R. Resnick, Fundarnentuls qfPhysics, Wiley, New York ( 1981 ). 5. Raman, C. V., and Nath, N. S. N., Proc. Indian Acad. Sci., 2,406 (1935). 6. Klein, W. R., and Cook, B. D. IEEE Trans., SW-14, 123 (1967). 7. Adler, R.,IEEE Spectrum. 4.42 (1967). 8. Horn. M., and Wolf, E., Principles Of Optics, Pergamon. New York (1965). 9. Lucas, R., and Riquard, J. Phys. Rud., 3,464 (1932).

P..

Chapter 3

84

10. Rytov, S. Actual. Sci. Ind., 613, (1938). 11. Berry, M. V., The Diffraction of Light by Ultrasound,Academic Press, New York (1966). 12. Defebvre, A., Pouliquen, J., and SCgard, Comptes Rendues Acad. Sc. Paris. B, 265, 1(1967). 13. Van Cittert, P. H., Physica, 4, 590 (1937). 14. Korpel, A. “Acousto-Optics,” in Applied Solid State Science, Vol. 3 (R. Wolfe, ed.), Academic Press, New York, p.71(1972). 15. Raman, C. V., and Nath, N. S. N., Proc. Indian Acad. Sci., 3,459 (1936). 16. Nath, N. S. N., Proc. Indian Acad. Sci., 4,222 (1937). 17. Klein, W. R., and Cook, B. D., IEEE Trans., SU-14, 123 (1967). 18. Yariv, A. Introduction to Optical Electronics, Holt, Rinehart and Winston, New York (1971). 19. Gaylord, T.K. and Moharam, M. G., Appl. Opt., 20,3271 (1981).

20. Solymar, L., and Cooke, D. J., Volume Holography and Volume Gratings, Academic Press, New York,(1981). 21. Hariharan, P., Optical Holography,Cambridge University Press, New York 22. 23. 24. 25. 26. 27.

(1984).

Cohen, M. G. and Gordon, E. I., Bell System Tech. J., 44,693 (1965). Korpel, A., App. Phys. Lett., 9,425 (1966). Korpel, A., IEEE Trans., SU-15, 153 (1968). Tsai, C. S. and Hance, H. V., .lAc. Soc. Am., 42,1345 (1967). Hance, H. V., Parks, J.K., and Tsai, C. S., .lAppl. Phys., 38,1981 (1967). Wade, G., Landry, C. J.and de Souza, A. A., “Acoustical Transparencies for Optical Imaging and Ultrasonic Diffraction,” inAcoustical Holography,Vol. 1 (A. F. Metherell, H. M. A. El-Sum, and L. Larmore, eds.), Plenum, New York, p. 159 (1969). 28. Magdich, L. N. and Molchanov, V. Y., Opt. Spectrosc, 42,299 (1977). 29. Chu, R. S. and Kong, J. A., .lOpt. Soc. Am., 70, 1(1980). 30. Chu, R. S., and Tamir, T., .lOpt. Soc. Am., 66, 220 (1976). 31. Chu, R. S., Kong, J.A., andTamir, T., .lOpt. Soc. Am., 67, 1555 (1977). 32. Leroy, O., and Claeys, J. M., Acustica, 55,22 (1984). Opt. Soc. Am., 3, 1608 (1986). 33. Pieper, R., Korpel, A., and Hereman, W., .l 34. Sharangovich, S. N., Sox Phys. Tech. Phys., 36,61 (1991). 35. Yuh-Ming Chen, Acousto-Optic Interactionin Arbitrary Sound Fields,Ph.D. Thesis, The University of Iowa, 1994.

4 The Formal Approach

4.1

INTRODUCTION

This chapter isorganizedinmore or lesschronological order of development. First we discuss the conventional theories applying to a sound column: the coupled mode analysis and the normal mode analysis. The former uses the eigenmodes of the unperturbed medium, i.e. plane waves that become coupled by the action of the sound. This analysis ultimately leads to the Raman-Nathequations. The second approach uses the eigenmodes of the perturbed medium, which turn out to be described by Mathieu functions. Next we developgeneralizedRaman-Nath equations that apply to arbitrary light and sound fields in two or three dimensions. Formal integral solutions forweak scattering aregiven and then recast using the plane-wave spectra of the participating fields. A strong plane-wave interaction theory of arbitrary fields, restricted to two dimensions, is dealt with next, and leads to the concept of Bragg lines along which order transitions are mediated. Feynman diagram techniques then provide an explicit solution in the form of path integrals. The eikonal theory of acousto-optics leads to ray tracing techniques and ray amplitude calculations. By combing it with Feynman diagram methods, 85

Chapter 4

86

it may be applied to strong interaction. As an example, we treat interaction with curved wavefrontsof sound. Finally, we discuss a vectorial treatment of acousto-optics based on Maxwell's equations.

4.2

COUPLED MODE ANALYSIS

We shall, in what follows, still assume that the interaction can be described by scalar quantities; later on, wewill investigate the validity of this assumption in more detail. We start from the wave equation for the electricfield, e, in two dimensions, X and Z ,

where uv and cVare the permeability and permittivity of free space, and p is the electric polarization. We assume that the polarization is linearly related to the electric field bya sound-induced time-varying electric susceptibility

where XI is real and may be negative. Hence,

The relative dielectric constant is given by El0

=(l+xo)=d

(4.4)

so that &)*=[l +x(t)]=[I +XO+XI cos(Q-Kx+$) (4.5) Assuming that IX,I
where the + sign corresponds to C ' > O and the - sign to C ' < O (see Eq. 3.10).

The FormalApproach

87

Substituting (4.3) into (4.1), we find with (4.4) and (4.7)

d2e d2e +-ax2 d z 2

pv&,nid2e dt

It is now assumedthat @+Q, so that (4.8) may be written d 2 e pv&,no2d2e d2e +-ax? dz2 dt

= + p V & , n ~ [ 2 n ~ ~ ~ A n ~ c o s ( i 2 t -d2e K x + @ ) ] ~ - - ; (4.9) &Writing the electric fieldas a discrete planewave composition e=C0.5En(z)exp[-jkzcos@,,-jkxsin

@n+j(o+nQ)t]+CL

(4.10)

assuming that the variation of En(z) is relatively slow (4.1 l ) and substituting (4.10) into (4.9), we find upon equating equal frequency , invoking the small angle components, using the relation c , , = @ & ~ ) - ~ . ~and assumption @n+1=@n+2@~

(4.12a)

that, with (3.10) and (3.76a). dEn -=

dz

-0.25 jkCSE,,-, exp[-jkz(cos@,,,_I -cos@,,)]

- 0.25jkCS * E,,+Iexp[-jkz(cos@,,+, -cos@,,)]

(4.13)

Equation (4.13) is the plane wave form of the Raman-Math equations, derived earlier (3.77) by the heuristic CTGA method. As already remarked, [l]. it is usually written in the form (3.82) through the transformation (3.80)

88

4.3

Chapter 4

NORMAL MODE ANALYSIS

We shall for clarity limit ourselves to the case of perpendicular incidence and follow the original treatment by Brillouin [2], adapted to a traveling sound wave.We start out by assuming that (4.9) is satisfied by a discrete composition of normal modes. These modes are nonplane waves e, that travel in the 2 direction with propagation constants k,=a,k, where the an are as yet undetermined constants. The wave cross sections or profilesf, must be periodic functions of the total argument, Qt-Kx+& of the sound field, Hence,

The constant a,,should not be confused with the phase a of a standing sound wave, used earlier in(3.14b). Upon denoting Qt-K~+$=27

(4.15)

(4.9) becomes (if we choosethe positive sign for simplicity) (4.16) Substituting (4.14) into (4.16),using(4.15), find

and assumingthat

1,we

(4.17) where the mode numbern has been omitted. Equation (4.17) may bewritten in the canonical form [3] (4.18) where (4.19)

The Formal Approach

89

(4.20) The solutions of(4.18) are the so-called Mathieu functions. Those relevant toour problem are periodic for a countably infinite set of characteristic q-dependent valuesa= a&) that yield even periodic function f,=ce,(q,q), n=O, 1, 2, 3, . . ., and u=b,(q) that yield odd periodic solutions f,=se,(q,q), n=1, 2, 3, . . . The period of the functions equals T or 2n depending on whether n is even or odd. For physical reasons, the functions to be used in the present application must have the same symmetry and periodicity (z)as cos 277; hence, only the solutions cezn(q,q) are suitable. For large q, the ce%(q,q) look very much like distorted harmonicsof cos(277) [4], as shown inFig. 4.1 for ce4 with q= 10. For smallq, it may be shown [3]that ce(q,q)-D2,cos(2q), where D2,, ais normalization constant. The characteristic values U, determine the a,, through (4.19), and these in turn determine the propagationconstants k,=Gk of the correspondingwaves. Because the Mathieu functions are orthogonal and normalized it should be possible to find the coefficients C, appropriate to a particular boundary condition [e.g., e(x, 0, 0)= constant for normal incidence] using standard techniques. In practice, however,this is a cumbersome procedure because the Mathieu functions are rather intractable and appear to have very few “nice” analytic properties. For one thing,an analytic relation between a and q does not exist; one has to use numerical calculations or existing tables to find a (and hence a) from q. A partial plot of a vs. q for some cezr solutions is shown in Fig. 4.2. After the a,,s and cns are found, the normal modes with profilesce%(q,q)

-1.2 I

0

I

77

I

277

rl

Figure 4.1 Typicalnormalmodecrosssection for q=10. Theinterval 0-27r corresponds to one sound wavelength. (Adaptedfrom Ref. 4.)

Chapter 4

90

a 4

Figure 4.2 Relationbetweenthecharacteristicnumber Raman-Nath parameter q. (Adapted from Ref. 4.)

“a” andnormalized

must be propagated, each with its own.phase velocity a , d k to the point z=L. There, they are to be summed in order to find the total field at the sound column boundary. The radiated orders (angular spectrum) for z>L can then be calculatedby an inverse Fourier transformas in (3.129). The equivalence of the coupled mode theory and the normal mode theory was first shown by Berry [5] and by Mertens and Kuliasko [6] using a methoddevelopedin collaboration withLeroy [7]. More extensive investigations, including the case of oblique incidence, were carried out later by the Plancke-Schuyten, Mertens, and Leroy group [8-lo]. Hereman [l l] has given a detailedand up-to-date treatmentof the normal mode approach, its solutions, and its relation to the coupled mode technique. He has also analyzed more rigorously the case of oblique incidence, which is much more complicated and requires the use of Mathieu functions of fractional order. A thorough discussion of Mathieu functions as such may be found in Refs. 12 and 13. From an engineering/physics point ofview, the normal mode ‘approach is difficult to use and offers little opportunity for gaining physical insight. Consequently, we will not further pursue it in this book. Before proceeding with the next topic, it should be pointed out that the normal mode theory (modal theory) has also been investigated from the point of view of microwave concepts and analogies. Thus, Bragg diffraction conditions are typically illustrated by the intersection of modal dispersion

91

The Formal Approach

curves (U-p diagrams), and Bragg interaction devices are often compared with directional couplers. This treatment, although stillrestricted to rectangular sound columns (usually embedded in different mediaon either side)lendsitselfvery well to largeBraggangleconfigurations. It is excellently suited for readers with a microwave background because of the use of already familiar concepts. We, however, will not pursue itin this book and consequently not list all the pertinent investigations and researchers in will nonetheless be found this area.A good description plus many references in the articleby Chu and Tamir [14].

4.4

THEGENERALIZEDRAMAN-NATHEQUATIONS

We start from (4.1), but first extent it to three dimensions for greater generality d2p V 2e - p , . ~d2e , , = p,,dt’ dt -

(4.21)

where

v

2

d 2 d2 d2 =-+-+d.u2

a

2

dZ2

(4.22)

Generalizing the treatment of Sec. 4.1, we write (4.23)

where r is a three-dimensional position vector. Also (4.24)

and (4.25)

so that, assuming x19 1, we find (4.26)

where, according to the generalizationof (3.6),

92

Chapter 4 (4.27)

&(r, t)=C‘s(r, t )

Substituting (4.23), (4.24), (4.26),

and (4.27) into (4.21), we find

d2e d2e V:e - p y & , n ~ = 2pv~,no~’s(r, t)at2

(4.28)

where we have assumed, as before, that the temporal variationof the sound field is slow comparedto that of the light field. In two dimensions, (4.28) becomes d2e

d2e

V:e - b&,n,2 -= 2 p v ~ , n o ~ ’ s t( )p-, dt2

(4.29)

where

and p is a two-dimensional position vector in the X-Z plane. Just as we derived the Raman-Nath equations from(4.9), we now want to derive more general relations from (4.28) and (4.29), referring to more general sound fields. Consequently, wenow write the electric field as a composition of ordersn that, unlike before, are not necessarily planewaves e(r, t)=COSE,(r)expp(o+nQ)t]+c.c.

(4.30)

For the sound field,we write s(r, t)=0.5S(r)exp(@t)+c.c.

(4.3 1)

If we substitute (4.30) and (4.31) into (4.28), it follows readily that (4.32) V2E,(r)+k2E,(r)+O.5k2C~(r)E,-~(r)+O.5k2CS*(r)E,~1(r)=O

The two-dimensional versionof (4.32) is V~E’En(p)+k2E,(p)+O.5k2CS(p)En-~(p)+O.5k2CS*(p)E~+~(p)=O(4.33)

We shall call (4.32) and (4.33) generalizedRaman-Nath equations because if, in (4.33), E , ( p ) is assumed to be a plane-wave propagating nominally in the 2 direction, i.e.,

The Formal Approach Eh)*E,(z)exp(-jkx

93

sin@,-jkz cos&

(4.34)

and (4.35)

S(p)=Sexp(

then (4.12) and (4.13) are retrieved as a special case, provided a slow spatial variation of En(z) is again assumed.

4.5

WEAK SCAlTERlNG ANALYSIS

If we ignore rescattering, i.e., we assume that only the incident light causes scattering, while remaining itself unaffected, i.e., Eo(r)=Ei(r), En(r)=O for Inl> 1, then it follows from(4.32) that the + 1 order satisfies the equation

V2E1(r)+k2EI(r)=-0.5k2CS(r)E,{r)

(4.36)

which, if we assume k r k , has the formal solution[l 5,161

E,(r)= -

S(r’)Ei(r’)dx’dy’dz’

(4.37)

where

The configuration relevantto (4.37) is shown in Fig.4.3. For the - 1 order, we find a similar solution:

In most cases, we are interested in calculating the field far from the interaction region. Then,Irls-r‘l and [l61

(4.39)

where ur is a unit vector in the direction r.

Chapter 4

94

X OBSERVATION POINT

r SCATTER ING POINT

Z

Y Figure 4.3 integral.

Configurationandnotations used inthree-dimensionalscattering

Hence, in both (4.37) and (4.38). (4.40) Using (4.40)in (4.37). we find k‘C exp(-jkr) r

E,(r)= 8K -S

I-

jjexp(~kr’.u,)s(r’)E,(r’)dx’d~’d~’ (4.41)

for ~ r ~ ~For ~ the r ‘ ~- 1. order. we obtain a similar expression, with S being replaced by S*. In two dimensions, X and Z , (4.36) becomes

vfE,(P)+k’E,(P)=-0.5k’CS(P)Ei(p) the standard solution of which is given by [ 151

(4.42)

The Formal Appmach

95

where

and HOis the Hankel function of the zeroth order and thefirst kind [17,18]. The configuration relevant to (4.43)is shown in Fig.4.4. For klp-p’l%l, i.e.. many wavelengths away from the induced cources in the interaction region, we may write

Far from the interaction region p%,p‘, applying (4.39)to two dimensions, we find with (4.43)and (4.44)

For E- ](p),we find a similar expression withS* instead of S . The weak scattering solutions derived above are not as easy t o apply as may appear at first glance. This is because both S and E: are propapting fields and can only be written down explicitly in very few cases. One such case is a Gaussian beam [19]. but even here the resulting expression is too complicated for convenientuse. Gordon [20] has calculated threedimensional scattering between a Gaussian light beam and a rectangular sound beam, and McMahon [21] determined the interactions hetween X OBSERVATION POINT

Figure 4.4 Configuration and notations in two-dimensional scattering intcgraI.

96

Chapter 4

rectangular beamsand Gaussian beams. The difficulties of evaluating (4.41) and (4.45) are to a certain extent artificial. Both S and E satisfy the same kindof wave equation intwo or three dimensions.Hence,given unidirectional propagation, if S and E are known across one plane (line), they are in principle known throughout the entire volume (plane) [22]. Why then should one have to use a volume (surface) integral in the scattering calculation as if S and E were arbitrary distributions? The answer is that one need not do so. As we have already shown heuristically and will now prove formally, for weak interaction in two dimensions the angular plane-wave spectrum of the scattered fieldcan be simply expressed by the product of the angular spectra of incident light and sound. Because each of these spectra may be found from the field distribution across a line, no surface integrals need to be used in this approach and, in two dimensions at least, the problem is simplified considerably. In three dimensions the situation is more complicatedbecause one incident wave vector of light can interact withan entire coneof sound-wave vectors, resulting in a cone of diffracted light wave vectors (this may be easily seen byrotating a typicalwave vector diagram about the incident light wave vector). Although an explicit formulation is available [22], it is mainly conceptual andnot easy to apply. We will return to the full threedimensional problem in Sec.8.5.

4.6 WEAK PLANE-WAVE INTERACTION ANALYSIS We limit ourselvesto two dimensions, Xand 2,and start from (4.45) that is strictly valid forp m . The left-hand sidewe write as an angular plane-wave spectrum of waves propagating nominally in theZ direction, to the right of the interaction region (4.46)

If we use stationary phase methods[[23] and Appendix B], it may be readily shown from(4.46) that, for p=m, (4.47)

where is the angle of the

wave normal to the point p.

97

The Formal Approach Writing S@’)=J$(y)exp(-jKz’siny-

jlYx’cosy)d

($)exp(-jkx’sin$- jkz’cos4)d pr-uP=x’sin 4,,+z‘ cos $,

(4.48)

(4.49) (4.50)

the integral in(4.45) becomes

x exp[jx‘(k sin - K cosy - k sin $)]dx‘ x exp[jz’( k cos q5,, - k cos 4 - K sin y)]dz’

(4.51)

With the paraxial approximation,(4.5 1) may be written:

Making use of the relation between angles implied by the first delta function we can write the second one as8(-@,,/A+O.5A/A2-y/A) so that the total integral becomes (4.52)

Chapter 4

98 Substituting (4.52)and (4.47) into (4.45), we find

(4.53)

Using the same method forE- I gives ~-1(@)=-0.25jkC~(-@-@~)~~@+2@~)

(4.55)

Comparison of(4.54),(4.55)with(3.160),(3.161)shows them to be identical, thus confirming the heuristic reasoning used in Sec. 3.3.3.

4.7

STRONGPLANE-WAVEINTERACTIONANALYSIS

Following Korpel [24], we will start from the two-dimensional generalized Raman-Nath equations (4.33) and write the light and sound fields inside the active medium as plane-wave compositions. For the sound field, we use (4.48); for the incident light field, we use (4.49), but the light fields generated inside the medium, i.e., &(p), we describe by z-dependent angular spectra

(9

- jkz cos 4) d -

(4.56)

Note that, as explained in Sec. 3.1.1, the E,,(@) are a generalization of the plane-wave amplitudes En(:) used in Ch. 3and Sec. 4.1. Substituting (4.56)into (4.33) and assuming that (4.57) we find

The FormalApproach

-[2jkcos@[

99

'En~'z)]exp(-jkxsin@- jkzcos@)d

jscn$-,(4; z)

+ 0.5k2Cj

xexp(-jkxsin@- jkzcos@-jKzsiny-jficosy) ~d[$d(:)+OSk~Cljs'(y),!?~+~(@;~) 4

xexp(-jkxsinq- jkzcos@+ jKzsiny+jficosy)

(4.58)

xd(3d(--)=O @ Y We now multiply both sides of (4.58) by exp(jknxx), whereknx=kSin$, and@n is a specific angle in the spectrum of E n . Subsequently, we integrate from x= -m to x=+m.Setting cos @= 1 in the amplitude factor, we find for the first integral in (4.58). (4.59) The second integral becomes,to a second order inf in the phase term,

where we neglect a term( K / k ) f in the argument of&I

and define

The third integral becomes, similarly,

(4.62)

Chapter 4

100 where

(4.63) and the same approximations have been made as before. Combining (4.59), (4.60), and (4.62), we find

x exp(- j& sin y - jkz cos

- j&qn-,Y2 2

x jkzcos@,,)d(;)

x exp(jfi sin y - jkz cos

X)

+ j&4,,+Iy2 2 -

(4.64)

x jkz cos $,,)d(

With (4.61) and (4.63), it may be readily shownthat k COS 4n-k

COS

Qin-1~-K(Qin-4~)

(4.65)

k COS 4 n - k

COS Qin+lS(Qin+4~)

(4.66)

Substituting (4.65) and (4.66) into (4.641, We find I

dZ

= -0.25 jkCk,,-l ($,, - 24.9; z)

(4.67)

The

Approach

101

where we have written siny foryin order to facilitate the interpretation and have left out phase contributions -jk$Ef/2 in the terms on the right-hand side of (4.64). If we now introduce coordinates

%+l

= z($n

+$B)

(4.68b)

then it is seen readily, with(4.48), that (4.67) may be written

(4.69) where we have replaced qj,, by $. An alternative derivation of (4.69), based on graphical constructions, is given in Ref. 24. Note that the validity of (4.69) is subject to the condition for the relevant that the neglected phase term be small, i.e., Kz,,,,,$~~y~/241 range of 7 Equation (4.69) is the basic equation describing strong, small Bragg angle interaction between an arbitrary light field and an arbitrary sound field. It has a very simple and obvious physical significance that we shall now discuss. First, we note that the angular spectrumof each order is directly coupled to that of the neighboring orders only. Second, each plane wave of one particular orderis coupled to two specific plane waves in neighboring orders, i.e., those that are moving in directions different by &2$E. All this is to be expected from thewave vector diagrams discussed before. The characterof the coupling is best interpretedby means of the diagram shown in Fig. 4.5. In this diagram, each plane wave (i.e., one from the spectrum of each order)is representedby a solid line. The sound fields along the dashed bisectors (called the Bragg lines) are given by S(xL1, z) and S(X;+I, 2). Thus, the interpretationof (4.69) is that in each pointz, there are two contributions to l?”. The first one is an upshifted contribution; it originates from &-I and is caused by the sound amplitude at (&I, z) on the bisector between l?,, and &-I. Similarly, a downshifted contribution arises from &=I, mediated through the conjugate of the sound amplitudeat (x;+,, z) on the Bragg line between E,, and &,+I. This interpretation is intuitively very appealing as the Bragg lines are at exactly the correct angles for interaction of the light, if they are regarded as “pseudo” wavefronts S(x2-I, z)and S(X;+I, z).

I02

Chapter 4

X

Figure 4.5 Diagram illustrating coupling between plane waves of adjacent orders

E,, through sound field along (dashed) Bragg lines.

To make the notation in(4.69)somewhatmore compact, we shall introduce the following notation for coupling factors: A quantity that upshifts from the (n-1)th order to the nth order we shall call S L I ;one that downshifts fromthe (n+ I)th order to the nth order we shall denote by S;+,. Thus, with (4.69),

These quantities are also indicated in Fig. 4.5.The rule to remember is that a minus superscript always refers to the conjugate of a sound amplitude.

The Also, because of the physical interpretation,the quantities S; and S;+, refer to the same Bragg line; hence, S: = &+I)*. This may be checked from (4.70) with (4.63). With (4.70a)and (4.70b), (4.69) may be written in short-hand notation (4.71) where a=0.25kC

(4.72)

The symbol “a” used here should not be confused with that used in the characterization of Mathieu functions (4.19). As sharp boundaries no longer exist inour arbitrary fields configuration, the boundary conditions mustnow be written

E,($;-~)=O

for n z ~

(4.73a)

&(4;-c9=E@)

(4.73b)

For the discussion to follow, it is convenient to combine (4.71) and (4.73) in the following integral form. n#0

for

for n = 0

(4.74)

.

(4.75)

It is of interest to consider the case of a single plane waveof light E,(z)exp( -jkx sin @,,-jkz cos A I ) in each spectrum of order n, proceeding at an angle inside the interaction medium. In that case, it is readily shown, by using (3.138),that (4.76) For the terms in the right-hand side of (4.69). we then find

-

E,+, (4 - 24H; =) =

E,,”,(

m 4- 24H -$,,-l

1

(4.77a)

9)

104

Chapter 4 (4.77b)

Substituting (4.76) and (4.77) into (4.69), we find by equating arguments of delta functions the usual relations (4.12) between &S and also dEn - -jaE,,-,S,+_,- jaE,+,S;+, dz

"

(4.78)

[It will be clear that (4.78), including boundary conditions, may be written in integral form.It then becomes similarto (4.74), without the tildes.] Not surprising, (4.78) is similar in form to (4.74), because it basically expresses the same individual plane wave interaction. Nevertheless, it is important to keep in mind the difference in formalism as expressed by (4.76) that relates a plane-wave spectral density to a corresponding plane-wave amplitude., One particular case of (4.78) we have already considered often is the artificial one in which the sound field is a rectangular column extending from z=O to z=L. In that case, -j&) S(x,z)=S exp( from which it follows readily, with (4.70) and (4.63), that

Substituting (4.80a,b) into (4.78), we find

(4.81) With (4.65), (4.66), and (4.72) this may be written as

In (4.82) we recognize our version (3.77) of the Raman-Nath equations that thus appear to be a special case of the presently discussed more general theory, as they shouldbe.

The Formal Approach

105

Before proceeding with the explicit solution, we shall, as an example of the application of (4.71) and (4.73), treat the case of upshifted Bragg diffraction by a rectangular sound column. The heuristic version of this was discussed in Sec. 3.3.2. In agreement with that discussion, we assume that only G(@) and El(@)are coupled. Accordingto (4.71) and (4.72), we write

a@)

e@,,.).

is short for where The Bragg lines corresponding to the coupling factors S$ and S- are showninFig. 4.6. Theydefine the sound fields S[z($-@),z] and S*[Z(@+@B),Z), as indicated in the figure. With the sound field given by (4.79), we find readilythat S,' = Sexp[-jKx(@- @ B ) ]

(4.85a)

IX

X

-

Z

2

a Figure 4.6 (a) Bragglinepertaining to upshiftedscatteringinto Similarfordownshiftedscatteringinto (4).

81 (4). (b)

106

Chapter 4

Substituting (4.85) into (4.83), (4.84), and in the second resulting equation replacing 4 by 4 - 2 4 ~on both sides, we find

a ~ * ( 4 ) / a z = - 0 . 2 5 j k ~ ~ ~ ( 4 - 2 4 B ~ ) e x p [ - ~ ~ ~ ( 4 - 2 ~ B ) 1 (4.86)

Comparing (4.86) and(4.87) with (3.1 13)and (3.1 14), we find that they are identical if the substitutions are madethat were discussed in Sec. 3.2.2.The final result is then identical to the one found in the heuristic treatment, given by (3.168).

FEYNMAN DIAGRAM PATHINTEGRAL METHOD

4.8

We will now proceed to derive the exact solutionof (4.74) and (4.75) by the diagrammatic methods developedby Korpel and Poon[25]. The basic idea is to reduce the recursive relation (4.74) by repeated applications until one of the terms on the right-hand side is reduced to contain l%. An example will make this procedure clearer; it is symbolically expressed by Fig. 4.7. Suppose we are interested in&. According to (4.74). this quantity is contributed to by both & , and l&. Let us arbitrarily choose theE 4 contribution (solid line) and leave l& (dashed line) until later.Now, E4 is contributed to by & and l&.In the example shown, the l% contribution is chosen and the Es contributions left until later. In the next step, we arbitrarily choose &, etc. This process 7 6

5

0 -1

-2 -3

No. of Steps Figure 4.7 Typical Feynmandiagrams illustrating various routes of scattering from the zeroth order into the fifth order. (From Ref. 25.)

107

The Formal Approach

cowinues (dotted line) until we reach l%. Now, according to (4.73,there are three contributions to choosefrom. We select the knownone, &=kinc (arrow), leaving the other two for later. It will now be clear that the dotted line from point A on level zero to the final destination on level 5 is a valid contribution, one of infinitely many. Some other paths starting at B, C, and D are shown as examples of alternative routes. It will be clear that the shortest possible path has exactly five steps, moving along a diagonal line from0 through 1,2, 3 and 4 to 5. The longest paths consist of infinitely many steps and wander all overthe grid of Fig. 4.7 (perhaps crossing level5 many times) before terminatingon level 5. Such a procedure, which consists of enumerating all different pathways from the initial to the final state, and summing their probabilities (sum over histories) was first developedby Feynman [26]within the context of his own unique development of quantum mechanics. Diagrams, similar to those shown here, that symbolize the various possibilities are called Feynman diagrams. A good introduction to such diagrams and their applications may be found in Refs. 27 and 28. It may be readily shown that the contribution to along the dotted path in Fig. 4.7 is given by

(4.88) JQ

J-

where m is the number of steps(13 in this example). In cases where single plane waves are involved [see (4.7811, it will be clear that (4.83) represents one of the contributions to E,,, provided 6.is replaced by Et. Equation (4.88) represents a typical path integral: the exact solution consists of summingan infinity of such integrals.In symbolic terms.

kn(qkz ) = x ( a l 1 path integrals) (4.89) Written as in (4.89). the solution looks deceptively simple; in practice, of course, the path integrals can only rarely be evaluated analytically. The beauty of Feynman's technique lies rather the in fact that, frequently, certain path integrals may be ignored because physical institution tells us that their contribution is negligible. An acousto-optic example of this may be found in Ref 29, one in a completely different field (the impulse response of acoustic

Chapter 4

108

transducers in Refs. 30 and 31. Syms has applied path integrals to optical problems involving chain matrices [32]. We shallnowapply the path integral method to upshiftedBragg diffraction by a rectangular sound column of a single plane wave of light, incident at the negative Bragg angle. Ifwe use (4.79), it may be shown readily with (4.80)that

Si-l = Sexp{-jKz[$o (4.90a) -l)$,]} + (2n (4.90b)

S;, =S*exp{jKz[$o+(2n+1)$,]}

where we have used the relation (4.90~)

4=&+2n$~ As &= -$B, we may write for (4.90a) and (4.90b)

[

S;-, = Sexp -jQ(n

-1)-

11

(4.91a)

(4.91b) For ideal Bragg diffraction, Q=m, and consequently any path integral containing Q in an exponential term vanishes because of infinitely rapid oscillations along the integration path. From (4.91), it is then clear that the only effective coupling factors are those that connect levels 0 and + l , i.e., S,' = S

(4.92a)

S; = S *

(4.92b)

This situation is illustrated graphically in Figs. 4.8(a) and 4.8(b), which show some of the infinitely many contributions to the levels l and 0, respectively. (Note that all paths only involve these two levels.) For simplicity, takingS to be real, we find readily that E, ( L )=

2

r'

(- j u ) 21f!S dzZ1 S dz2r-l.

I=O

0

0

I,"SEidzl (4.93)

109

n e Formal Approach

'

0

'm a

b

Figure 4.8 (a) Five-step scattering path from zeroth to first order. path from zeroth order back into itself.

(b) Six-step

where we have usedthe relation (4.94)

Similarly, we obtain for the + 1 order

In (4.93) and (4.95), we recognize the well-known expressions for Bragg diffraction interaction, thus confirming the Feynman diagram technique. The caseof Raman-Nath diffractionmay beconfirmed in a similar way 1251. 4.9

EIKONAL THEORY OF BRAGG DIFFRACTION

The goal of the eikonal theory is to justify intuitive ray tracing techniques and to quantify themby calculating the amplitude of the diffracted ray.

l10

Chapter 4

At first glance, it seems rather strange that a ray tracing theory could be developed within the context of acousto-optic c/$fractiun, but we may convinceourselvesof that by the Gedanken experiment illustrated in Fig. 4.9 (borrowed from Ref. 33). The drawing shows a typical upshifted Bragg diffraction experiment, using sound and light beams of finite width. As we have remarked often before, such a picture is essentially unrealistic because of the diffraction of both sound and light that makes both beams diverge. This latter effect, however, disappears whenever k 0 , h-0 (i.e., k=m, K=m). The width of the beams may then be made infinitesimally small, in which case we call them rays. In the limit of infinitesimally small wavelengths, we may thus replace the upshifted beam interaction of Fig. 4.9 by the ray interaction of Fig. 4.10(a), with Fig. 4.10(b) showing the case for downshifted Bragg interaction. The next step consists of applying the ray tracing method to local interaction situations, e.g, the interaction ofan incoming ray with a curved wavefrontof sound. Figure 4.11shows a typical and self-explanatory example. It is taken from an article by Pieper and Korpel [34] dealing with strung local interaction through a combination of eikonal theory and Feynman diagram techniques. Inthe present chapter,we will limit ourselves, however, to weak interaction. X

Figure 4.9 Gedanken experiment illustrating development of ray-tracing method. (From Ref. 33.)

111

The Formal Approach

INCIDENT RAY

I

DIFFRACTED RAY

SOUND RAY

SOUND RAY

a

b

Figure 4.10 Ray-tracingdiagramsfor(a)generation generation of downshifted ray. (From Ref.33.)

of upshifted rayand (b)

Figure 4.11 Localdownshiftedinteractionwithacurvedwavefront (From Ref. 34.)

of sound.

Chapter 4

112

Following Korpel [22,33], we start with the generalized Raman-Nath equations [4.32] for EOand El, and the weak interaction assumption that Eo(r)=Ei(r)

V2El(r)+k2E1(r)=-0.5k2CS(r)Et(r)

(4.96)

Using the eikonal functions Y(r) [35], which have the significance that Y(r)=constant represents a geometrical wavefront,we write

E~(r)=IE~(r)lexp[-jkUl,(r)]

(4.97a)

Substituting (4.97) into (4.96), we find

(4.98)

where A(r)=k~(r)+K~(r)-kYl(r)

(4.99)

It should be noted that the incident sound and light fields S(r) and E@) satisfy an equation identical in form to (4.98) with the driving term on the right-hand side missing. If we follow the usual geometrical optics treatment [35],valid for k, K = w , the coefficients of (l/k)O, (Ilk)', (1/k)2, and ( U r n o , (Urn1,(1/Q2 are separatelyset equal to zero. For the (l/k)O, ( l / K ) O coefficients, this results in the eikonal equations for the homogeneous medium: IVY,{r)l= 1

(4.1OOa)

IVYs(r)l= 1

(4.1OOb)

which signify that the rays of these fields propagate in straight lines [35] in the direction of the unit vector

s=w

(4.1Ooc)

The Formal Approach

113

[Note that in the conventional treatment, the right-hand side of (4.100a) equals “no” rather than unity. In our analysis, this is already taken into account by the definition (4.97b), in which k refers to the propagation constant inside the medium. Also note that the s notation for ray directions and lengths should not be confused with the sound field s ( ~ , t ) nor , with the subscript “S” that refers to the sound.] Setting the coefficient of (Ilk) equal to zero eventually leads to the transport equation for the field strength along the rays. In the case of Ei, it can then be shown readily [35] that

le

IEi(s)l=(Ei(s,)Iexp -0.5

V 2 Y i ( s ds )

1

(4.101)

where S is the distance measured along the ray from an arbitrary point so. Equation (4.101) essentially expresses conservation of energy ainray pencil. Returning now to (4.98) and following the same approach,we first obtain IVYl(r)l=l

(4.102)

expressing the fact that the diffracted rays also propagate in straight lines. To find the transport equation for lE11,we again select the coefficient of (llk), but this time equate to it the driving term ratherthan to zero (as in the case of Ei):

IEl(r)lVzYl(r)+2W~(r).VIE,(r)l = -OSjkqS(r)lE,{r)(exp[-jA(r)]

(4.103)

Using the distances along the diffracted ray ratherthan the position vector r, we write

so that (4.103) becomes

IEl(s)lV2Y(s)+2alEl(s)llas = -0.5jkqs(s)llEl(s)lexp[-jAOl

(4.105)

As pointed out before, without the right-hand driving term the solution would be of the form (4.101).In our case, we shall try a variation

[ e

IEl(s)l = IE,’(s)lexp -0.5

1

V2Yl(s)ds

(4.106)

114

Chapter 4

where the coefficient of the exponential term isno longer a constant, and so is an as yet undetermined pointon the diffracted ray. Substituting (4.106) into (4.105), we find

(4.107)

from .which IE;(s)l= ”0.25j k C r IS(s)llE,(s)lexp[-jA(s)] sb

1

(4.108)

V2yli(s’)ds’ ds+E,(sb) where Sb is another point on theray. A logical choice is (4.109) so that (4.108) becomes

IE;(s)I = -0.25jkCS’a~~(s)lJE,cs>lexP[-iAcs,l

[d

x exp 0.5 V2yli(s’)

I

d r ‘ ds

(4.110)

From (4.99) it will be clear that, when k, Kern, the term exp[-jA(s)] will oscillate infinitely- rapidly. Mathematically, therefore, (4.1 10) may, in the limit, be solved exactly with the method of stationary phase [13, 231. The stationary phase pointA may be found by solving (4.1 11)

which may be written

Vyl1(s)*VA(s)=O

(4.112)

With (4.99) and (4. 1 0 0 ~we ) ~ find for (4.112) s1-(Kss+ksi)=k

(4.1 13)

where ss, Si, and SI are unit vectors in the direction of the sound ray, the incident light ray, and the diffracted lightray.

115

The Formal Approach

It may be shown [22,23] that the only solution to (4.1 13) satisfying the “one ray through one point” conditionis given by

In (4.1 14) we recognize a local wave vector triangle, as is obvious from the construction in Fig. 4.12. Thus, (4.114) defines a locus of ray-crossing interaction points and hence proves the validity of the heuristic method illustrated in Fig.4.10. We will return to the actual use of the method later. It next remains to calculate the amplitudeof the diffracted ray. Let us, for simplicity, equate the pointsa on the diffracted ray with the stationary phase point A shown in Fig. 4.12. According to the method of stationary phase [23], the solutionof (4.1 10) is then given by IE’l(s)l=O

(4.1 15)

for scs,

and (E;(s)(= -0.25 jkC(s(sa)~~Ei(sa)lexp[-j,4(sa)-$] (4,116)

for s>sa and (C32Al~s2)sa>0. For )E’ ](S)] to be a positive real quantity, itis necessary that A&)= -3d4 if C>O and + d 4 if CcO. The phase of E‘] then follows directly from(4.99). Hence, at the interaction point A , the amplitude of the generated diffracted ray is given by

(4.1 17)

after which the amplitude varies along the ray according to (4.106). If, at the stationary phase point, ~3~A(s)lC3s~
A($@)=-

4

3?r 4

i f C > O or - i f C C 0

(4,118)

116

Chapter 4

Figure 4.12 Wave vector triangle diagram in localinteraction. (From Ref 33.)

and

The case of downshifted interaction maybe treated similarly, the only difference being that ys(r)in (4.99) is to be replaced by -ys(r)with a subsequent change of(4.1 14)to read -Kss+ks*=ks-1

(4.120)

as is to be expected. We started out with the theory sketched from above (4.32) that applies to three dimensions. It is clear that by starting from (4.33), entirely analogous results would have been obtained in two dimensions, through replacing r by p. It is of interest to briefly consider an actual application of the acoustooptical eikonal theory. Figure 4.13 shows a two-dimensional configuration in which a cylindrical wave of sound interacts with a ray of light at the stationary phase point A . It is assumed that the Bragg angles 4~ are very small so that to all intents and purposes the S axis (the S here not to be confused with the complex sound amplitude) coincides with the X axis, i.e.,

a/aEa/ax.

117

The Formal Approach

Figure 4.13 Two-dimensional upshifted interaction diagram used to calculate amplitude of ray diffracted by curved wavefront ofsound.

According to (4.110), we must now calculate a2A/ax2.From (4.93) it is clear that, apart from multiplicative factors, A(p) contains two plane wavefronts Yl(p), Yz{p)and one curved wavefront Ys(p).For the first two, the curvature a2Y/ax2is evidently zero. The sound eikonal function is given bY (4.121)

Ys(P)=P

and, in the neighborhood of A , may be written 0 . 5 ~ ~ R

Ys(x) = ( R 2+ x ’ ) ~ . ’ R + -

(4.122)

Hence, (4.123)

Chapter 4

118

With (4.123), (4.99), and (4.117), we find readily the diffracted light amplitude at A

If we compare (4.124) with the weak interaction version( v a 0 ) of (3.1 lo), it is easily seen, with v=kCLIq2, that we may define an effective interaction length LI It will be intuitively clear that the eikonal theory becomes less accurateif there are significant variations in sound amplitude over the length LI, for instance, if the curved sound wavefront is truncated to a length smallerthan LI. More details may be found in Ref. 34. Note that in the pure geometric optics, limit ha0 and the effective interaction length becomes infinitesimally small, so that we may now speak of interaction points.

=m.

4.10

STRONG INTERACTION WITH CURVED SOUND WAVEFRONTS

Having at our disposal a Feynman diagram theory for strong interaction (with path integrals that are difficult to evaluate) and an eikonal theory of weak interaction, it is reasonable to ask whether these two approaches could be combined into an eikonal theory for strong interaction. The first step toward this goal was taken by Pieper and Korpel [34], who assumed single Bragg interaction with a curved sound wavefront similar to Fig. 4.13, but for downshifted interaction. Their general configuration is shown in Fig. 4.14. Downshifted interaction isinvolvedwith the point shown the only interaction point on the wavefront. The different Feynman diagrams leading from the zeroth order back to itself or to the - 1 order are shown in Fig. 4.15. The sound is modeledas S(x, z)= S(z) = So exp

(4.125)

where L is the length of the interaction region (z=O to z=L), and

{=(.-g)

(4.126)

119

The Formal Approach

Figure 4.14 Ray interaction with curved sound wavefront. (From Ref

34.)

0

1

, I - 1

I

I I

I I I I I

2

3

21

22

23

L

Figure 4.15 Feynman diagrams for generation of the -1 order. (From Ref. 34.)

Chapter 4

120

It is assumed that the incident light beam is narrow and the incident angle small, so that within the interaction region the curvature of the sound wavefront is substantially constant. In other words, the sound field is a phase-profiled column. The coupling factors S21 and S6 that mediate the transition from orders -1 to 0 and 0 to -1 (see Secs. 4.6 and 4.7) are given bY S2 1 =Soexp(+&)

(4.127)

Sij=Soexp(-&)

(4.128)

where (4.129) Let us now lookat a typical Feynman diagrampath integral, for example, the one labeled3 on Fig. 4.15. With the notation of (4.88), we may write for the path integral

4

(4.130)

J

x exp[-jy(z)]dz dz, dz2 0

We now apply the stationary phase method (Appendix B) to the first integral (in z) of (4.130). With (4.128) it is easily found that the stationary phase point z2 is given by (4.131) As can be seen from Fig. 4.14, this represents exactly the local interaction point arrived at by ray tracing. The value of the first integral may be readily found by applying the stationary phase method with (4.132)

121

The Formal Approach

where U(ZI-Z,) is a unit step function, indicating that the stationary phase point must be within the integration interval (z1>zs) for the integral not to vanish. Now, if (4.132) is substituted into (4.130) and the integral over ZI is evaluated on a stationary phase basis, we'll find the same stationary phase point as before, and

pexP[+iv(Zl)l~exP[-iv(z)ldz 0

d.1 =

im ~ ( O ) ~ (- zG ,>

0

1 =-A&U(z,-z,)

(4.134)

2

where we have used the property U(O)=1/2. Proceeding in this manner it is easily shown from(4.130) that

l

x exp[-jyl(z)]dz dz, dz2

(4.135)

0

By summing over all odd-numbered paths 1, 3, 5, etc., in Fig. 4.15, we find finally

so that

where (4.137)

and v=2aS0L is the Raman-Nath parameter.

122

Chapter 4

In the same waywe can evaluate the sum of the integrals involving all even numbered paths to obtain (4.138) Note that IE-1I2+Eo2=1. Note also that the phase of the (conjugate) sound fieldispreservedin the diffracted light through thefactor exp[-jw(zs)]. For b e 1 (using order+ 1rather than - 1and setting z,=O as in Fig. 4.13) we again find the expression for lE-11 (4.124) of the previous section. In Ref. 36 the concept of local interaction is heuristically extended to multiple interaction points. A typical configuration is shown in Fig. 4.16. Interaction points A, B, C, and D are found by simpleray tracing techniques, discussed before. The converging sound field is modeled as a phase-profiled beam with constant phase curvature (4.139)

i

Figure 4.16 Multiple interactionpointsoncurvedsound

Ref. 36.)

wavefronts. (From

123

The Formal Approach

where y = K h is the edge phase as shown in Fig. 4.16. As with a flat sound wavefront, a Q factor is defined as Q = -K

~ L

(4.140)

k

It will be seen from Fig. 4.16 that the interaction points are separatedby 2Rqb. The number of potential interaction points is thus given by L / ( ~ R ~ B , which may be shown to equal 8lyl/Q. The number of actual interaction points and their locations depend on the angle of incidence. If, for example, the incident ray is rotated clockwise, then the interaction points are as well, as shown in Fig. 4.17. It is assumed in Ref36 that the interaction points are independent if their separation 2Rqb is substantially greater than the effective local interaction lengthThismaybeexpressedbythe condition that the separation divided by the interaction lengthis much greaterthan 1: (4.141)

It is further assumed that the interaction at independent points is determined by a diffraction efficiency 7 based on (4.136):

Ix

I

I I I

\\\

sound

Figure 4.17 As Fig. 4.16, but with oblique incidence of the light.(From Ref. 36.)

Chapter 4

124

(4.142) The heuristic conjectures of Ref. 35 were confirmed by simulation with an effective numerical algorithm to be discussed in Sec.5.8. Figure 4.1 8 shows a simulation for a configuration with Q=50, ty=50, v=27r, and qhinc=-4@B. There are 8 potential interaction points-the interaction length L=16Ro@~-and the spacing between them equals the in (4.141)]. In spite of this the effective interaction length [i.e., Q/4*=1 interaction points appearto be largely independent, as the measured powers in the different orders are in close agreement with (4.141). Note that the “growth length” of each order roughly equals their separation, and hence equals the effective interaction length. In Fig. 4.19 the same parameters apply, but the incident ray is rotated counterclockwise by 8 @ ~The . shift of the interaction points in Fig.4.19 is found to have the same value (one-half as conjectured. the interaction lengthof 16@~) References 37 and 38 give a rigorous treatment of multiple plane-wave interaction and prove that the geometrically determined points coincide with the stationary phase points in the corresponding path integrals. It is also shown that certain Feynman diagrams are forbidden, namely those that would make rays go backward through interaction points already passed. A

0.20 0.18

L

m

C

0)

.-c

0.16 0.14 0.12 0.10 0.00 0.06 0.04 0.02 0.00 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.0 0.9 1.0 1.1 1.2

Normalized interaction length

Figure 4.18 Simulationresults for configuration of Fig. 4.17 with 4nc=-44B. (From Ref. 36.)

125

The Formal Approach

Normalized interaction length

Figure 4.19 As Fig. 4.18, but with & = + 4 4 ~ . (From Ref. 36.)

typical allowed Feynman diagram representing progressive scattering into the fourth order is shown in Fig. 4.20. Reference 37 also gives some more comparisons between the eikonal-Feynman theory .and the numerical algorithm of Sec. 5.8. Two of these, applying to the configuration of Fig. 4.17, are shown in Figs.4.21 and 4.22. In summary, the central feature of the eikonal-Feynman theory is the multiple interaction at local interaction points. It would appear that extension to the general case should be relatively straightforward in those cases where phase curvature can be defined and the interaction points are separated by at least one effective local interaction length. A necessary condition is, of course, always that the total interaction length L be smaller (This condition isobviously than the local interaction length flagrantly violated for flat wavefronts of sound, as in a sound column; the eikonal theory cannot be used in such cases.)

mo.

4.11 VECTOR ANALYSIS So far we have tacitly assumed that our equations could be writtenin scalar

form. In most treatises, this has conventionally been the case, although some important historical exceptions exist [39,40]. The aim of the present section is to derive the theory of acousto-optic interaction by starting from Maxwell’s equations, and then discuss under which condition the scalar

Chapter 4

126

0 W

k steps

all at z0 j steps Figure 4.20 Typical allowed Feynman diagram for progressive scattering into the fourth order. (From Ref. 38.)

1.o 7

0.8

-

0.6

5

g

0.4

..$a. -

0.2

2

0.0 0

z 0

(1

10

20

30

Raman Nath parameter v

Figure 4.21 Powerin - 1 orderatthe exit of theconfiguration of Fig. 4.18. (From Ref. 38.)

The Formal Approach

127

1.o 0

0.8

-g

0.6

al

L

0

L

W

2Q

-W

m

2

0.4 0.2

0.0 0

10

20

30

Raman Nath parameterv

Figure 4.22 Power in order 0 at the exit of configuration of Fig. 4.19. From Ref. 38.)

formulation used so far is correct. Specific polarization phenomena having to do with the anisotropy of the elasto-optic effect will be discussed later. Here, we will still adhere to the simple assumption that the change in the electrical permittivity is a scalar function of the strain (condensation),i.e., it is supposed to be independent of the polarization of the electric field, and causes no cross-coupling of polarization states. Consequently,we shall write for the time-varying permittivity E(r, t)=a+&’(r, t)

(4.143)

E‘@, t)=EoCs(r, t )

(4.144)

where

so that (4.145) where the subscript r refers to the relative dielectric constant.Hence,

&(r, t)=0.5noCs(r, t )

(4.146)

a relation we have used before in (3.6) and (3.76b). We shall next assume that the incident field (as it would exist in the

Chapter 4

128

absence of any sound field) is source-free in the interaction region and hence satisfies the Maxwell equations Vxhi(r,t)=Eo-

dt

0

Ai(r,

(4.147)

(4.148) Vqei(r, t )=0

(4.149)

V-hdr, t)=O

(4.150)

where we have assumedthat b = p V . When the soundfield is present, we may think of thetotal field as the sum of the incident field plus a scattered field e(r, t)=ei(r, t)+e'(r, t )

(4.151)

h(r, t)=hi(r, t)+h'(r,

(4.152)

t)

The total field must satisfy the relations

d

V x h(r, t ) = -[&(r, t)e(r, t ) ]

(4.153)

Vxe(r, t)= -pdh(r, t)/at

(4.154)

V*[E(r,t)e(r, t)]=O

(4.155)

V-h(r, t)=O

(4.156)

dt

It is relatively simpleto derive the following relations for the scattered field: w r yt ) V x h'(r, t ) = -

(4.157)

w r y2) V x e'(r, t ) = -pLy -

(4.158)

V.d'(r, t)=O

(4.159)

dt

r3r

The Formal Approach

129

V.h’(r, t)=O

(4.160)

W , t)=Eoe‘(r, t)+p’(r, t )

(4.161)

p’@, ?)=E’@, t)[eXr, ?)+e’@, tll

(4.162)

where

It is thus clear that the essential part of the scattered field is caused by an electric polarization p’ that, in turn, is induced by the time-varying permittivity d(r, t ) acted upon by the total field. To simplify matters, we introduce the Hertz vectorn’(r, t ) of the scattered field, in terms of which bothh‘(r, t ) and e‘(r, t ) can be expressed &’(h t ) h’(r, t ) = E ~ x V-

(4.163)

at

e’(r, t ) = V[V d(r, t ) ]- &EO

d2d(r, t )

In order to satisfy (4.157)-(4.160), T‘(r, V2d(r, t)] -

t ) must

d2d(r, t ) =--p’(r, t ) dt2

(4.164)

at then satisfy [22,40]

(4.165)

EO

which has the explicit solution (4.166) where R=lr-r‘l, l/c=d(p~,dV is a volume element of r‘ space, and the integration extends over the entire interaction region. Substituting (4.162) and (4.164) into (4.166), we find readily

130

Chapter 4

Equation (4.167) is an integro-differential equation fore’ and was first used by Bhatia and Noble in their research [35,42]. A weak interaction approximation correspondsto neglecting the term withe‘ in the right-hand side of (4.167). The fact that e’ and ei need not necessarily have the same polarization is due to the VV. operator. Let us investigate this more closely. In the two-dimensional configurationswe have most often used, nothing changes in the Y direction; hence, a/ay=O. If now the polarization of the incident lightis in the same direction, then (4.168)

Thus, the VV. operator drops out of the equation,and, according to (4.167), the scattered light will have the same polarization as the incident light, namely in theY direction, perpendicularto the plane of interaction. In a general three-dimensional configuration, we compare the terms d2(&’ei)

W

O

(4.169)

z = k2E‘ei - pvEOo2&‘ei dt

(where we have neglected the temporal variation of with

E‘

relative to that of e;)

V(V*E’~~)=(V~E’)~;+~(V.~~)(VE’)+E‘V(V*~~)

(4.170)

The first term is,at most, of order K%jed and the second one always smaller than kKEle;l. As it is assumed that K e k , both terms are small relativeto the a2/at2 term in (4.167) and may be neglected. That leaves us with the third term, i.e., E’V(V-ei). We have used the paraxial assumption that the light nominally propagates in the 2 direction. Because ei must be perpendicular to this direction, V*ei is-at most-of the order ksinqlej, where $J is the maximum angle a plane wave in the spectrum ei makes with the 2 axis. Hence, E’V(V*ei)is-at most-of the order k2sin$Jeil.The paraxial approximation assumes that sinqM1; hence this term also may be neglected. The same reasoning applies to e’. In summary then, for our assumptions of small Bragg angleand paraxial propagation, theoperator VV. in (4.167) may be neglected irrespectiveof the incident polarization, and (4.167) reverts to a scalar equation. In the first Born approximation, we may write

The Formal Approach

131

where we have ignored the time variation of E' relative to that of e;. Relating E' to S through (4.145)"using the conventional phasor notation for e', ei, and S as exemplifiedby(4.30) and (4.31),replacing R by lr-r'l, and assuming that !2
(4.173)

(4.174) The total field e(r, t ) follows from n(r, t ) in the usualway: d2n(r,t ) e(r, t ) = V[V x(r, t)]- pveodt

(4.175)

Substituting (4.162) into (4.166), we find with (4.175) (4.176) Using the same arguments as before regarding polarization, we may again neglect the operator VV., both in (4.175) and (4.176), so that, using (4.143) and differentiating (4.176) twice with respectto time, we obtain (4.177)

132

Chapter 4

where e(r, t ) now refers to anarbitrary polarization component.Upon using (4.144), (4.145), and (4.27), it is readily seenthat (4.177) reverts to (4.28). Insummary then, wehaveseen thatour previouslyused scalar formulation is correct in all respects, providedthat either (1) we have a two dimensional X - 2 configuration with Y polarization, or (2) that K&, the sound propagates paraxially in X, and the light in 2.As a historical footnote, it should be mentioned that the relativeinsensitivity to polarization was first pointed out by Brillouin [39]. It should once more be stressed that all of the above, strictly speaking, only apply when there exists a simple scalar isotropic relationship between the change in permittivity and the strain. This is hardly ever the case in practice; even certain liquids exhibitan anisotropy due to localized velocity gradients. Nevertheless, by suitable choice of crystal orientation and light polarization, most of the theory developed so far can be applied. By the same token, contrary configurations may give rise to interesting and useful polarization effects. We will discuss some of these later.

REFERENCES 1. Klein, W. R.,and Cook, B. D., IEEE Trans., SU-14, 123 (1967). 2. Brillouin, L., Actual. Sci. Znd., 59 (1933). 3. Blanch, G., “Mathieu Functions,” in Handbook of Mathematical Functions (M. Abramowitz and I.A. Stegun, eds.), Dover, New York, p. 721 (1965). 4. Jahnke-Emde, Tables of Higher Functions, B.G. Teubner, Leipzig (1952). 5. Berry, M. V., The Diffraction of Light by Ultrasound,Academic Press, New York (1966). 6. Mertens, R.,and Kuliasko, F., Proc. Ind. Acad Sci. A, 67, 303 (1968). 7. Kuliasko, F., Mertens, R., and Leroy, O., Proc Ind. Acad Sci. A, 67, 295 (1968). 8. Plancke-Schuyten, G., Mertens, R., and Leroy, O., Physica, 61,299 (1972). 9. Plancke-Schuyten, G., and Mertens, R.,Physica, 62,600 (1972). 10. Plancke-Schuyten, G., and Mertens, R.,Physica, 66,484 (1973). 11. Hereman, W,, Academiae Analecta (Belgium), 48 (1986). 12. Arscott, F. M., Periodic Diffirential Equations, Pergamon Press, Oxford (1964). 13. McLachlan, N. W., Theory and Applicationsof Matthieu Functions, Dover, New York (1964). 14. Chu, R.S., and Tamir, T.,IEEE Trans., MTT-18,486 (1970). 15. Papoulis, A., Systems and Transforms with Applications in Optics, McGraw-Hill, New York (1968). 16. Kyrala, A., Theoretical Physics,W. B. Saunders, Philadelphia (1967). W. J., “BesselFunctions of Integer-Order,”in Handbook of 17.Olver,F. Mathematical Functions (M. Abramowitz and I. A. Stegun, eds.), Dover, New York, p. 355 (1965).

The Formal Approach

133

18. Spiegel, M. R., Mathematical Handbook.Schaum’s Outline Series, McGraw-Hill, New York (1968). 19. Yariv, A., Optical Electronics, 3d Ed., Holt, Rinehart and Winston, New York (1985). 20. Gordon, E. I., Proc IEEE, 54, 1391 (1966). 21. McMahon, D. H., IEEE Trans., SU-16,41 (1969). 22. Korpel, A., ‘Acousto-Optics,” in Applied Solid State Science, Vol. 3 (R. Wolfe, ed.), Academic Press, New York, p. 7 1 (1972). 23. Erdelyi, A., Asymptomatic Expansions, Dover, New York (1956). 24. Korpel, A., J Opt. SOCAm., 69,678 (1979). 25. Korpel, A., and Poon, T.C., J Opt. Soc. Am., 70, 8 17 (1980). 26. Feynman, R. P., Rev. Mod. Phys., 20,367 (1948). 27. Mattuck, R. D., A Guide to Feynman Diagrams in the Many Body Problem, McGraw-Hill, New York (1967). 28. Schulman, L. S., Techniques and Applications of Path Integration, Wiley, New York (1981). 29. Poon, T. C., and Korpel, A., J: Opt. Soc. Am., 71, 1202 (1981). 30. Banah, A. H., Korpel, A., and Vogel, R. F., Proc. IEEE Ultrason Symp., p. 444 (1981). 31. Banah, A. H., Korpel, A., and Vogel, R. F., .l A. Soc. Am., 73,677 (1983). 32. Syms, R. R. A., Appl. Opt., 25,4402 (1986). 33. Korpel, A., “Eikonal Theory of Bragg Diffraction Imaging,” in Acoustical Holography, Vol. 2 (A. F. Metherell and L. Larmore, eda), Plenum Press, New York .(1970). . 34. Pieper, R. and Korpel, A., J: Opt. SOCAm., 75, 1435 (1985). 35. Born, M., and Wolf, E., Principles of Optics, Pergamon Press, New York (1965). 36. Korpel, A., Opt. Eng., 31,2083 (1992). 37. Chen, Y-M., Acousto-optic interaction in arbitrary soundjields, Ph.D. Thesis, University of Iowa, December 1994. 38. Chen, Y-M., and Korpel, A., .lOpt. SOCAm., 12 (A): 541 (1995). 39. Brillouin, L., Ann. Phys. (Paris), 17,88 (1922). 40. Wagner, E. H., Z. Phys., 141,604,622 (1955); also, 142,249,412 (1955). 41. Stratton, J. A., Electromagnetic Theory, McGraw-Hill, New York (1941). 42. Bhatia, A. B., and Noble, W. J., Proc Roy. SOCA, 222,356 (1953).

This Page Intentionally Left Blank

5 The Numerical Approach

Historically, numerical solutions first emerged for the rectangular sound column in the region where neither Raman-Nath diffraction nor Bragg diffraction applies. Most techniques solve a truncated system of RamanNath equations written in a suitable normalized form. We will show this truncation first and then take a quick look at the subsequent numerical integration. If less than six orders are involved, exact solutions are possible, and this is also the case for two orders with incidence at twice the Bragg angle. Higher, multiple Bragg angle incidence is solved numerically. A more physics-oriented approach uses successive diffraction along the lines of Van Cittert’s analysis. In the cascaded Bragg diffraction method, the interaction region is divided in successive Bragg diffraction modules. This technique is especially advantageous for phased array configurations. The split-step method, arguably the most powerful one, uses a modified beam propagation algorithm and may be applied to arbitrary fields. The Fourier transform technique is based on a systems approach to the problem with suitably defined transfer functions. Finally, the Monte Carlo simulation dependson a quantum-mechanicalinterpretation of the interaction process.

135

Chapter 5

136

5.1 TRUNCATION OF THE RAMAN-NATH EQUATIONS Writing D = -d dz

the general equation in the Raman-Nath formulation is given by [see (3.82)]

It should be remembered that (5.3) specificallyrefers to a sound field Iqsin(Qt-Ki), i.e., S=-~lqor 4=-7d2 [see (3.4)]. If now the truncation is from the Lth order up to the Mth order with M30,

LSO,

L+"

N=(M-L+l)

(5.4)

then the truncated systemstarts with the top equation

This is followed by (M-L-1) equations of the type (5.3), after which the system is truncated with the bottom equation

The total numberofequations, N , equals (M-L+1). conditions for the set(5.3-5.6) are

v401= VncGkO

The boundary (5.7)

where Sk0=0 if k&, and GM= 1 if k=O. The equations one finds in the literature are essentially of the type (5.3-5.6). They may (confusingly) however, vary in detail because of the diverse conventions and normalizations mentioned before [see remarks following (3.82)] and also because of the choice of

The Numerical Approach

137

An important property of the truncated set is that M

Cykty;= constant = wincwi'.c k=L

indicating the somewhat surprising fact ofpower conservation in an incomplete system. Equation (5.8) may be proven readily by multiplying with v k each equation characterizedby (D+jgk)yk, adding to it its complex conjugate, and finally adding all equations. This leads to c D ( v k v k ) = o , k from which (5.8) follows.

5.2

NUMERICALINTEGRATION

The truncated system may be integrated numerically by letting dz+Az, dvk+Ayk, etc. This was first done by Gill[l], who also incorporated higher Taylor corrections inhis program to ensure stability. A more recent numerical integration without higher-order corrections has been presented by Klein and Cook [2].They state that for sufficient accuracy the number of steps P= L/& in thez domain must satisfy

P> lO0v

(5.9)

and also (5.10) where nmaxdenotes the highest order containing any significant amount of light. In practice, accordingto Klein and Cook, In,l equals approximately2v for small Q and decreases with increasing Q.Some of the numerical results from Ref. 2 follow. Figure 5.1 shows nonideal Raman-Nath diffraction for perpendicular incidence ( Q = 1, QvS6); Fig. 5.2 illustrates nonideal Bragg diffraction (Q=5, Qh31.2). Note that in Fig. 5.1 the true zeros, characteristic of the Bessel function solution for the idealcase, disappear for large v. In Fig. 5.2 the periodic power exchange between the two main orders is no longer complete, indicating the presenceof additional orders. Numerical investigation may, of course, also be used for the generalized equations (4.78). They have, infact, been so treated by Pieper and Korpel[3] in the contextof strong interactionwith a converging sound beam.

138

Chapter 5

1.o 0.8 0.6 0.4

0.2 0

0

1

2

3

4

5

6

V

Figure 5.1 NonidealRaman-Nathdiffraction (Q=I, QvC6)forperpendicular incidence. (Adapted from Ref.2.) 0 1967 IEEE

h

139

The Numerical

5.3

EXACT SOLUTIONS

If the truncated set(5.3-5.6) contains fewer than five independent equations, exact solutions may be found. This is readily seen from the characteristic equation of the system obtainedby setting yk= Ckexp(pz) and solving forp . The resulting equation is, at most, of the fourth degree and hence may be solved in terms of radicals[4]. The simplest exampleof such a set is that for which M = l , L=O or M=O, L= -1. This set has two equations, the solutions of which, for h=&$B, correspond to up- or downshifted Bragg interaction, respectively [(3.103), (3.104);(3.107),(3.108); $s=-7d2]. Ifdeviatesslightly from the exact Bragg angle, the solution is given by (3.168). An exact solution for downshifted Bragg interaction,taking into account orders -2, - 1, 0, + 1, has been given by Jaaskelainen in a study of thick transmission gratings [5]. Figure 5.3 (solid line) shows the solution so obtained in the nonideal Bragg case [Q=& Q/vSl]. Note the considerable deviation from ideal Bragg behavior (dashed line). The open circles data are points calculated by the present author using Klein and Cook’s numerical integration scheme [appliedto (4.78)] with 1000 Az slices and 17 orders.

1.0

-

0.8

-

0.6

-

0.4

-

0.2

-

1, /I;

V

Figure 5.3 First-order diffraction in the nonideal Bragg region @=S, Q/v>O.5, solid line) compared with the ideal case (dashed line). The open circles are data 17 orders points calculated by the present author using numerical integration with present. (Adapted from Ref.5.)

140

Chapter 5

A calculation similar to the one above was performed by Blomme and Leroy [6] for incidence near the Bragg angle. In previous investigations, these same authorshad treated oblique incidence for four orders [7], and perpendicular incidence for five [8] and seven orders [g]. In the latter two cases, the number of independent equations turns out to be three and four, respectively, due to symmetry conditions pertainingto normal incidence, i.e., y-n=(- 1)nVn.

5.4

MULTIPLE BRAGG INCIDENCE

It has been found that appreciable power transfer between two orders is possible not only for Braggangleincidence, but also for incidence at multiple Bragg angles. If the light impinges at an angle of'm+B, then a strong interaction appears to exist between orders 0 and -m, or 0 and +m, respectively. Naturally, the intermediate orders must mediate this power transfer; a direct m-phonon interaction would require the mth harmonic of the sound frequencyto be somehow generated, a process that is excluded in the experiments and numerical simulations showing this effect. Consequently, a satisfactory physical interpretationof the phenomenon is as yet lacking, although various exact solutions have been obtained. Alferness [lo] analyzed the diffraction efficiency of thick holograms operating in the second-order Bragg regime and concluded that 100% diffraction efficiency was possible. The analogous case for acousto-optics was treated by Poon and Korpel [ll], whousedFeynman diagram methods and took into account the orders 0, 1, and 2 (i.e., M=2, L=O). For incidence at the angle y 0 = - 2 @ ~their , results were in agreement with the analytic solution of the truncated set obtained by Alferness. The results may be summarized as follows: (5.1 1)

{

K)

I2=0.25 ( c o s ~ - c o s ~ + ) 2 - sinx-sin<

11

(5.12) (5.13)

where (5.14)

h

141

The Numerical

5'7 Q

(5.15)

Complete transfer of power into the second order is only possible for values of Q=4mz, (5.16) m is an integer

]

0.5

v=2z( 4n2- m2

formodd

(5.17a)

(5.17b) where n is a positive integer. It is interesting to note, inpassing, that the requiredvalues of Q correspond to the incident ray's intersecting exactlym wavelengths of sound. The exact expressions (see [ll]) indicate that for other values of Q not exactly equal to 4mz, an appreciable amount ofenergymaystillbe transferred. Figure 5.4shows theoretical (solid line) and experimental (circles) maximum diffraction efficiencies for the acousto-optic case [l l]. Figure 5.5 illustrates diffraction efficiency as a function of v for Q=14. In both cases, there is a reasonable quantitative agreement between theoryand experiment. It is intriguing that, as follows from Alferness' data, the + l order, although nonexistent at the sound cell exit, reaches an intensity of about 25% inside the cell. Benlarbi and Solymar [l21 have givenan interesting analysisof still higher Bragg angle incidence. They use a truncated system with M=O and L= -n. Making somewhat ad hoc assumptions about slow spatial variations in yo and W-,,, they arrive at two first-order equations coupling these quantities. In our terms, their solution is given by zo=cos2(o.5~q~v) I-,=~in~(OS)qlv)

(5.19)

where (5.20)

Chapter 5

142

0.2 0.4 14

12

0

I 1

10

1

1

1

1

1

1

16

'

Q

1

18

1

1

20

1

1

22

1

1

24

Figure 5.4 Maximum diffraction efficiency of second-order interaction as a function of Q. The solid line indicates the theoretical values, the open circles are experimental data points. (Adapted from Ref. 11.) 0 1981 IEEE.

0.6

t

t

l

/'

'L

0

0

V

Figure 5.5 Diffraction efficiency of second-order interaction as a function of v for Q=14. The solid line indicates the theoretical values, the open circles are experimental data points. (Adapted fromRef. 11.) 0 1981 IEEE.

The Numerical Approach

143

For n= 1, we do indeed find back the conventional Bragg diffraction behavior. For n=2, the expressions do not appear to agree with those of Alferness [lo] or Poon and Korpel [l l], although the position of the first maximum is almost the same. It is claimed that the findings do agree with the results of Chu and Tamir [13], the latter results being derived in quite a different manner. Also, the present author’s numerical simulations appear to agree fairlywell with the theory.

5.5

THE NOA METHOD

The process of solving the set of equations (5.3-5.6) has been formalized by Hereman, Mertens, and Ottoy for normal light incidence [l41 and oblique incidence [15]. They gave it the nameNOA method for nth order approximation, indicating the fact that their set is always truncated by orders -N and N, i.e., M= N, L= -N. For perpendicular incidence, the system is simplified because of the symmetry conditions mentioned before, i.e., ty,=(-l)nty-,. The essence of the method consists ofassuming solutions of the kind C,exp(jsz). Substituting this into the set (5.3-5.6) will lead to N+ 1 simultaneous linear algebraic equations forC,. For this systemto have a nontrivial solution,it is necessary that

+

Det(M-sr)=O

(5.21)

where M is a matrix formed from the coefficients of the algebraic equation set, and I is the unit matrix. Expression(5.21) leads to an equation of degree ( N + l ) in S, the roots (eigenvalues) of which are found numerically. The complete solution then followsin the standard wayby finding the eigenvectors corresponding to the eigenvalues [l61 and imposing the usual boundary conditions. A variant of the technique [l41 usesHeaviside’s operational method [l71 and is claimed to be 25% faster in regard to computer time. Figure 5.6 compares NOA (N=7) results for Q=1.26 (solid line) with experimental results (circles) obtained by Klein and Hiedemann [l 81, and with ideal Raman-Nath behavior (dashed line). It will be seen that there is excellent agreement. The NOA technique is relatedto a method developed earlier by Leroyand Claeys [19], who used Laplace transforms in combination with a matrix theory to arrive at an expansion in Bessel functions. Laplace transforms have also been used by Poon and Korpel [20] for the calculation of path

144

Chapter 5

0

2

4

v

(i

8

10

Figure. 5.6 NOAzerothorderpredictions far Q'1.26 andnormalincidence (solid line), compared with experiment (circles) and ideal Raman-Nath behavior (dashed line). (Adapted from Ref14.)

integrals in the Feynman diagram approach, The m t b d ofchoice, however, appears to be the NOA eigenvalue technique or its operational variant.

5.6 SUCCESSIVE DIFFRACTION This method, developed by Hargrove [21], is a filiitc increment version of Van Cittert's cascaded thin grating analysis [22] discwsed in Sec, 3.2. The increments Az are chosen small enough that for each thin grating, the Raman-Nath Bessel function solution applies. The orders generklted by a particular grating are rediffracted by the next one, A disadvantage of the method is that Bessel functions have to be handled by the computer. Nevertheless, the technique appears to give results in good agreement with experiment. Concerning these comparisons, Hargrove makes the interesting observation that usually only agreement in order intensity is looked for, not in relative phase. The phases are, however, veryimportant if one is interebted

145

The Numericdl Approach

in the total light Aeld at the exit of the sound cell and, in fact, may well determine its overall shape. Hargrove gives an example in which, forQ=0.31 and u=4, the calculated intensities agreeto within 1% with those obtained in the pureRaman-Nathcase. There are,however,marked deviations in relative phase. The effect of this becomes clear when all the orders are put together at the exitof the sound cell to give the overall intensity distribution IT, as shown in Fig. 5.7. It is obvious that the fieldis not just phasecorrugated as it would be (by definition) for the ideal Raman-Nath case. In fact, considerable intensity modulation is seen to occur. Notice that the highest intensity peak occurs at x=11/4, i.e., at the point where the sin(&) sound field is a maximum and the rays are focussedby refraction. The successive diffraction methodjust discussedwasdeveloped independently by Hance [23], who gives an interesting proof in which, in the limit of Az+O, Q+O, the classical Raman-Nath solution is retrieved. Pieper and Korpel [24] developed a successive diffraction method using increments thick enough to satisfy Bragg conditions. In the next section we will elaborate on this method.

x /A Figure 5.7 Intensitymodulation of theemergingwavefrontfor Q=0.31, v=4, calculated by the successive diffraction method. (Adapted from Ref. 21 .)

Chapter 5

146

5.7

CASCADEDBRAGGDIFFRACTION

Following Ref. 24 we assume near Bragg diffraction into the -1 order. The angle of incidenceqk, differs from the exact Bragg angle#B by an angle A$: qk,O=$B+A@

(5.22)

We have treated a similar case for the+ l order in Sec. 3.2.3 In the present case we find that (5.23)

Substituting (5.22) into (3.77), and limiting ourselves to orders - 1 and 0, we obtain the following coupled equations: dE-l = - j u s * Eo exp(jKiA4) dz

(5.24)

dE0 = -jaSE-, exp(- jKzA4) -

(5.25)

dz

It shauld be remembered that the phase reference in(5.24) and (5.25) is at the beginning of the first sound cell. Because we are dealing with a cascading of sound cells, it is more convenientto use a sliding phase reference pointz for the solutions, so that at the input of the next cell we automatically have the correct phase reference.In Ref. 24 it is shownthat the coupled equations for the ithcell then become dE-li A4 E-li - -jusi * Eo, - j K dzi 2

"

(5.26)

(5.27)

where the point Zi=O denotes the left (input) edge of the ith cell. Solving (5.26) and (5.27) subject to the appropriate boundary conditions, it may be shown [24] that the ith cell is characterized by a transmission matrixTi: (5.28)

The Numerical Approach

147

where

rill=cos(yiLi)+j-sin(yiLi)

(5.29)

si riI2 = -ja-sin(yiLi)

(5.30)

2Yi

Yi

S; * ri21 = -ja-sin(yiLi)

(5.31)

Yi

riZ2 = cos(yiLi)-j-sin(yiLi) KA@ 2Y i

(5.32)

where (5.33)

The authors apply their matrix method to a beam steering configuration where the sound fieldsSi in each cell form a phased array. Figure 5.8 shows the frequency behaviour of a four-cell beam steering array (solid line) and compares it with a simulation using a multiple-order numerical technique (dashed line). The Q of each of the cells equals 2n at the center frequency, and the Bragg angle condition is exactly satisfied at that frequency (F=l). Figure 5.9 shows the behaviour of the same device withan angular offset of 0.1 Bragg angle at the center frequency. The accuracy of the method is foundto increase with the number of cells used.

5.8

THECARRIERLESSSPLIT-STEP METHOD

This method is a modified form of the beam propagation or split-step type Fourier transform technique [25,26]. In the latter method the medium is divided into many thin slices Az, and the field is propagated twice, in a space-marching fashion, through each slice. In the first step, the mediumisconsidered to be homogeneous and diffracting. Consequently this step involves propagating the local plane-wave spectrum through the distance Az and calculating the propagated field from the propagated plane-wave spectrum.

148

Chapter 5

F

Figure 5.8 Variation of diffracted intensities (orders0 and - 1) with normalized frequency F for a four-cell beam steering phased array, as obtained by matrix multiplication (solid line) and numerical integration(dashed line), (FromRef. 24.)

In the next stepthe slice is cansidered to be inhomogeneous and nondiffracting, i.e., it is treated as a phase filter, The field emerging from this second transit isthus obtained by multiplying the previous propagated field by the phase function that represents the pertubation of refractive index. The final field is taken to the next d c ie and the process repeated until the entire interaction region has been traversed. A compact mathematical treatment of this process may be foundin Ref. 27. In two dimensions the technique may be described by the following operator equation [28]:

where M is the phase filter function describing the inhomogeneity: M=exp[-jk,&(x, z)Az]

(5.35)

149

The Numerical Approach

-

Figure 5.9 Variation of diffracted intensities (orders 0 and 1) with normalized frequency F for a four-cell, beam steering phased anay, the withincidence light offset by 0.1 Bragg angle. M refers to matrix multiplication andD to numerical integration. (From Ref. 24.)

and &I represents the pertubation of the refractive index. The multiplier denotes the so-called plane-wave propagator and may be written as

H = exp(-jk,Az)

H

(5.36)

where (5.37)

for paraxial propagation. The extension to three dimensions is straightforward.

150

Chapter 5

The algorithm just described has been applied to the propagation of a beam in a grating [29]and could, of course, be used in the case where the grating is a sound field. A simplification of the method for sound fields has been described by Venze, Korpel, and Mehrl [28].They apply a priori knowledgeof the properties of the interaction in the individual slices to speed the execution of the algorithm. It is known that, to a first order, in a thin phase grating the light is split into three orders that travel the small distance Az to the next slice. The authors now use the well-known fact that the three orders are separated in direction by twice the Bragg angle. This makes it possible to ignore the spatial carrier of the sound and concentrate on the envelope (profile) ofthe sound only. Because of the greater generality of the method, we shall discuss it in 28. some detail, following the (two-dimensional) treatment outlined in Ref. The pertubation 6n of (5.35)is, in the case of sound-induced refractive index variations, a function of time and space: &(x, z, t)=C’s(x, z, t)

(5.38)

where C ‘ = - O . ~ ~ ~ O ~ = O . ~as~defined O C , before, and s(x, z, t ) is the (real) sound amplitude. S(X,z, t)=O.SSe(x, z)exp( - j f i ) exp(jf2t) +0.5Se*(~,Z) exp(jfi) exp(-jQt)

(5.39)

where S e is the complex envelope ofa sound field propagating nominally in the X direction. A snapshot of the sound field is usedat t=O (other times may be chosen; We’ll return to that issue later)so that, substituting (5.38)into (5.39),we find &(x,z)=OSC’S,(x, z) exp(-jfi)+OSC’S,*(x, z)exp(jfi)

(5.40)

The expression forM may be approximated by (5.41)

M=l -jk,&(x, z)dz

if we make the assumptionthat Ik,Sn(x, z)dzle 1. Substituting (5.40) into (5.41), we find for the first operation ME(x, Z) in (5.34) ME(x, z)=E(x, z)[l -0.SjkvC’AzSe(~, Z) exp(-jfi)

-O.Sjk,C‘&S,*(x,

z) exp(jfi)]

(5.42)

151

The Numerical

The next operation to be performed in the split-step method to is take the Fourier transform of (5.42). Use is made of the following property of the Fourier transform:

.r-'[g(x) exp(-jKi)]=.r"[g(x)] with

k,+k,-K

(5.43)

We shall denote this by . ~ - , r - ~ [ g ( xwhere )] .?" is a "shift operator" that replaces k, by k,-K. Similarly, the shift operator replaces k, by k,+K. Using (5.43) in the evaluationof the next step ,F-'ME(x, z) in (5.34), we find from (5.42) .S/?+

Y - ~ M E ( Xz)=.F-'[E(x, , z)~-o.~~~"~c'.v-.~-'[E(x, z)Se(x, z)]

- 0 . 5 j k , ~ C r . v + . r - ' [ E ( xz)Se*(x, , z)] (5.44) Refemng back to (5.34), the split-step method may nowbe completed by multiplying the propagation factor H and then performing the forward Fourier transformon the result. The method is readily implemented on a computer. A flowchart of the main propagationloop of the program is shown in Fig. 5.10.

next AZ

4

Figure 5.10 Main propagation loop o f the carrierless algorithm. (From Ref'. 28)

152

Chapter 5

Figure 5.1 1 shows the simulated evolutionof a Gaussian beam incidentat the negative Bragg angle ona sound column of width z=L. The maximum sound amplitude corresponds to a Raman-Nath parameter v=a and the Klein-Cook parameter Q equals 13.1. Note the depletion of the center of the zeroth order at the center of the sound cell, asto be expected at v=z. Figure 5.12 shows data points at the exit of the sound cell and compares them with the solid line that is the result of an analytical calculation (see Sec. 3.3.4 and its applicationto a Gaussian beam in Sec.6.2). The authors of Ref. 28 also performed physical experiments in addition to simulation and numerical calculation. The results show very good mutual agreement. It should be pointedout that, although three orders are used in each small step Az of the algorithm of Fig. 5.10, the final result will show all orders generated. This is demonstrated in Ref. 30 which treats interaction with a cylindrical sound wavefront. Figure5.13 is taken fromRef. 30 and shows the

Figure 5.11 Simulation plot of the intensity of the angular spectrum of the total 28.) field at various positions along the interaction length. (From Ref.

The Numerical Approach

153

Figure 5.12 Analytical and simulation plots of the angular spectrum intensityof the total field at the exitof the sound cell, whereV=T, Q=13.1. (From Ref.28.)

0.50 0.45 0.40

0.35 0.30 0.25 0.20 0.15

0.10

0.05 0.00 0.0 0.1 0.2 0.3 0.4 0.50.6 0.7 0.8 0.9 1.0 1.1 1.2

Normalized interaction length

Figure 5.13 Simulatedevolution o f threeorders of diffractedlightinthe interaction with a converging sound wave. (From Ref30.)

Chapter 5

154

evolution of three diffracted orders in the strong interaction of a beam of light with a converging sound field. Note that the orders appear to be generated at specific positions within the interaction regions, in the order -1, + l , and +2. In summary, the carrierless split-step method is, when extended to three dimensions, a very powerful simulation technique for strong acousto-optic interaction of arbitrary fields. If the sound field is known only at the transducer surface, the conventional split-step method may be applied to calculate the sound S(x,z) necessary for the carrierless algorithm. Finally, a word about the time dependence of &(x, z, t ) , which was eliminated in the foregoing by setting t=O. Other values may be chosen to show the (periodic) evolution in time of the diffracted field. In general, with well-separated orders, this should not show anything new-other than that the orders are shifted in frequency. Where orders overlap, the beat notes between the order frequencies should show up, but this has not yet been demonstrated.

5.9 THE FOURIER TRANSFORM APPROACH This technique, developedby Tarn and Banerjee [31-331, uses the notion of a slowly varying transverse profile (envelope) of the light field. Thus the phasor of the rnth order optical field at any point in the sound cell is represented by &(x, z)= ye,m(x, z) exp(-jkz

COS

Qim-jkx sin 6 )

(5.45)

where Qim denotes the nominal direction of the mth order, and itsslowly varying profile (envelope).The incident light atz=O is describedby

and sin

= sin &,c

K +k

(5.47)

Now, rather than working directly with the profile of the light orders, the authors introduce the local spectrum of these profiles: (5.48)

h

155

The Numerical ea

w e ,m

=.

~

~

e

1,=mJye.m(x, z)exp(-jk,x)dx

(5.49)

+

It is now assumed that both and iP change only slowly withz and that the sound field in the interaction regionmay be written as exp(-jlux) S(X,z)=Se(Z)

(5.50)

In other words, the sound profile &(z) does not vary significantly with x inside the interaction region; a profiled sound beam is used as a model. In most cases this isa satisfactory model. A final assumption is that the configuration is such that Bragg operation can be assumed, i.e., only two orders (0 and - 1 in this case) interact. With the above assumptions, the authors find, upon substituting the various expressions in the waveequation

(5.51)

(5.52)

The boundary conditions pertainingto (5.51) and (5.52) are Ye,o(kx, O)=yinc(kx)

(5.53)

ye,-l(kx,o)=O

(5.54)

In (5.51) and (5.52) the first terms on the right-hand side express the propagation of Ye,o and Y~,-Iin the absence of the sound. Thus, these terms describe the effect of diffraction. The second terms describe the interaction with the sound. In Ref. 33 the authors apply their formalism to interaction of a Gaussian light beam with waist W O with a diverging Gaussian sound beam, which in the interaction regionis approximated by

Chapter 5

156

where W denotes the waist of the sound beam, and R its radius of phase curvature. Substituting (5.55) into (5.51) and (5.52), the authors then solve the equations numerically by the Runge-Kutta method. The fixed parameters are w0=2h, W = q Q=8,f,,,,d=24 MHz, h=0.128 mm and A=0.632 p. Figure 5.14 shows (a) the profile of the zeroth orderand (b) that of the - 1 order as a function of a (=v, the Raman-Nath parameter) and Xlwo at the exit of the sound cell, for a radius of curvature R=40 m. Figure 5.15 shows a similar case withR=4 m, and Fig. 5.16 with R= 1 m. Note how with R=40 m, the behavior with v is very close to that of the conventional case. Maxima of the - 1 order appear at v=lt and 3n, minima at 2n and 4n. For the smaller radii of curvature R=4 m and R = l m, the periodic behavior has disappeared, and the first maxima in the - 1 order appears for increasingly larger values of v.

5.10

MONTE CARLO SIMULATION

A s pointed out in Sec. 4.8, two-dimensional strong acousto-optic interaction of arbitrary sound and light fields is, in principle, solvable by the use of Feynman diagrams and path integrals. Figure 5.17, taken from Ref. 34, shows the general configuration with the light propagating nominally in the 2 direction and the sound in the X direction. The arrows labeled n- 1, n, n+l denote the direction of scattered, upshifted orders generated from a particular plane wave in the plane-wave spectrum of the incident light. The dotted lines, bisecting the angles between consecutive orders, are called Bragg lines. As explained in Sec. 4.7, they play an important role in the scattering process in that the sound field along their length mediates the interaction between neighboring orders. The formalism expressing this is given by the recurrence relations (4.74) and (4.75), where, with reference to Fig. 5.17, we have S+,- 1=S(&-

(5.56)

1,z)

S-n+l=S*(Xfl+l,

z)

(5.57)

The constant a is a material constant:

4

1 1 a=-kC=--kn 4

2p

(5.58)

The Numerical Approach

157

Figure 5.14 Profile of the (a) zeroth order and (b) -1order as a function of a (=v, the Raman-Nath parameter) andXIWOat the exitof the sound cell, for a radius

of curvature R=40 m. Other parameters are wo=2A, W = m , Q=8, &und =24 MHz, A=0.128 mm, and il=0.632 p m . (From Ref. 33.)

158

Chapter 5

0.00

<

2.09

Figure 5.15 As Fig. 5.14, but with R=4 m.(From Ref. 33.)

The Numerical Approach

159

2.99

Figure 5.16 As Fig. 5.14, but with R = l m.(From Ref. 33.)

160

Chapter 5

Figure 5.17 General interaction configuration with Bragg lines. (From Ref. 34.)

As discussed in Sec.4.7 by repeated successive application of (4.74),E, can be expressed as an infinite sum of path integrals, the paths themselves being illustrated by Feynman diagrams. One way of depicting these diagrams is shown in Fig. 4.7, but a way more suitedto the present treatment is shown in Fig. 5.18. The scattering is from the incident light to the fourth order through the intermediate orders indicated. The path amplitude for this path isbygiven &(k) = (-jaI6 jS:dz5 ]S:dz, (5.59)

Figure 5.18 TypicalFeynmandiagramforscatteringintotheplusfourthorder. (From Ref. 34.)

h

The Numerical

161

where k is an ordering label for classifying paths leading from B;,, to &. The total scattering amplitude is given by a summation over the infinity of such paths:

k4= 2k4(k)

(5.60)

k=l

In the Monte Carlo method to be discussed, the evaluation of path integrals such as(5.59) has its basis in the interpretation of factors-jaSo+, -ja&+, -jaS2+, etc., as essential parts of quantum-mechanical probability amplitude densities. Thus, in eq. (5.59), the probability amplitude of a transition from order 0 to 1 during dz isgivenby - j a S t d z , and the probability amplitude ofa photon existing in order1 and ZI is given by

-j a

]S:dz

(5.61)

where Einc is normalized to unity for simplicity. The probability amplitude of a photon making a transition from 1 to 2 between 21 and zl+dz is -jaS:dz provided there is a photon available in 1. But the probability of the latter is given by expression (5.61). Hence, the overall probability amplitude of a transition between ZI and a + d z is expressed by (5.62) Continuing the argumentin the same fashion, we arrive at (5.59), i.e., the probability amplitude of a photon existing in level 4 at z. The authors of Ref. 34 add the following noteto their discussion: Naturally this picture must not be taken to seriously. At best it is a metaphor for the quantum mechanicalbehavior of the photon, a behavior that cannot be described by any classical picture. (For one thing, the probability amplitudes from0 to 4 must be added vectorially, as if the photon traversed all paths simultaneously.) However, when used with caution, the model isof considerable valueinguiding the probabilistic simulation. A further discussion of this subject may be found in Ref. 35. In the simulation the authors divide the interaction region in N slices, as shown in Fig. 5.19(a). At any discrete point 1,2, 3, . . . N,. the photon may

162

Chapter 5

LIN

1

3

2

4

N

L

4

W

a

I

:+p m

0

l

b 1 """"""""".._______.___.___..

0 m

1

m

2 C

1

0 m

1

m

m

2

3

d

Figure 5.19 (a) Transition points. (b) Transition from order 0 to 1 at m1 in pure Bragg diffraction. (c) Transition backto 0 at mz.(d) Final transitionto order 1 at mg. (From Ref. 34.)

make a transition from the level it is in . Figures 5.19(b, c, and d) show typical transitions for exact Bragg diffraction in a sound column where all S,,+ and S,- have the same value of SOand there are only two levels. Hence, the probability of transition p = -jSodz= -jSo(WN). The authors prove that for smallp (i.e., N95)the final resultsof such a modelare in agreement with the known laws for Bragg diffraction. In the actual simulation a difficulty arises: The quantum-mechanical probability of a transition equals p , and of no transition equals unity. Obviously 1+IpI2> 1. This does not lead to difficulties in the quantummechanical model, because the total probability doesnot have to equal unity until all parts are summed vectorial&. However, in a simulation with classical

h

163

7'he Numerical

computers, if the probability of a transition is IpI, then the probability of no transition is automatically l-bl. The authors get around this difficulty as shown in Fig. 5.20, illustrating how the transition probabilities are handled in each transition pointof Fig. 5.18. The numbers in brackets indicate probabilities, while the numbers in parentheses indicate weighting factors. Thus, a nontransitioning photon withunavoidable(classical) transition probability of 1-bl isgiven a weighting factor of (l-bl)" to compensate for that. In a transition the photon is "multiplied" by -j to account for the imaginary probability amplitude. Whether a photon makes a transition at m is determined by a uniform random number generator, that generates a number between zero and 1. This number is compared with the value of in (5.58), and if it is found to be smaller, the photon changes its level number (label0 or 1) while being multiplied by -j. In the other case, the photon keeps its level number and is multiplied by (1-bl)-*.The process is repeated at the next slice, and after N steps the photon is stored in the appropriate bin (0 or 1) with its

."""".""""."" n

a

b

Figure 5.20 Simulation of quantum-mechanical transition probability amplitudesbyclassicalcomputersforBragg diffraction. The bracketsindicate classical probabilities, the parentheses indicate weighting factors. (a) up-transition, (b) down-transition. (From Ref.34.)

Chapter 5

0 .o

1 .o

NORMALIZED SOUND AMPLITUDE

Figure 5.21 Bragg diffraction simulation with 10,000 photons and 100 divisions. (From Ref. 34.)

1.0

-

z

g Lu

+

ORDER0

I

ORDER1

-PREDICTED

0

---PREDICTED l

0.8-

c

z

c

5

0.6-

A

2“r

0.4

-

0.2

-

I

U

9

II i

c 0.0. d 0 .o

d

4

l .o

NORMALIZED SOUND AMPLITUDE

Figure 5.22 Near-Braggsimulationwith A @ B = M ~ L10,000 , photons,and 1000 divisions. (From Ref.34.)

The Numerical Approach

165

a

b

. C

Figure 5.23 Raman-Nathsimulationfor M=9, 10,000 photons, and 1000 divisions. Orders0 and 1 are shown. (From Ref. 34.)

proper phase and weighting factor. After this the next photon is processed. In the final step, weightedphotons in each bin are added vectorially, squared (to calculate the power), and their number normalized to the square of the total number of photons used in the simulation. A typical result for 10,000 photons and 100 steps is shown in Fig. 5.21. The dashed and solid lines are the values predicted from conventional Bragg diffraction theory. The agreement is very close,but the results begin to deviate if the Raman-Nath parameters v774 corresponding to a normalized sound amplitude of 1. It is found that the number of photons used in the simulation is more critical than the number of steps. In the example discussed above,the probability densityp does not depend on the transition point m. The authors of Ref 34 also show a simulation for near-Bragg diffraction, where this is no longer the case (see Sec. 3.2.3). Figure 5.22 shows the result foran offset angle ofM4L.

166

Chapter 5

1.0 r+,

0

g -

S

8 z

0.4 0.4-

0.2 0.2U

'

0.0 0.0

.*,

*

9 9

.d

1 .O

NORMALIZEDSOUND AMPLITUDE

Figure 5.24 Raman-Nath simulation for M=9, 10,000 photons, and 1000 divisions. Orders 2 through 4 are shown. (From Ref. 34.)

r

1

+

n

+ a.

Figure 5.25 Raman-Nath simulation with parameters that are the same as Fig. 24, but orders 2,3, and 4 are shown.

The Numerical Approach

167

For the general case, more than two orders must be considered. Figure

5.23 shows the transition probabilities and weighting factors in the case of Raman-Nath diffraction with the orders ranging from -M to +M. When the highest orders are reached only down- (or up-) transitions are possible. Figure 5.24 shows simulation results for orders 0 and 1 , while Figure 5.25 shows orders 2 through 4. In summary, this simulation method appears to be versatile and accurate, although perhaps not as convenient to use as the carrierless algorithm. In contrast to the latter, it is notsuitable for three-dimensional simulation.

REFERENCES 1. Gill, S. P., Office of Naval Res., Contract Nom-1866 (24) NR-384-903,Tech. Memor., 58 (1 964). 2. Klein, W. R., and Cook, B. D., IEEE Trans., SU-14, 123 (1967). 3. Pieper, R., and Korpel, A., .lOpt. Soc. Am., 2, 1435 (1985). 4. Spiegel, M. R., Mathematical Handbook., Schaum’s Outline Series, McGrawHill, New York(1968). 5. Jaskelainen, T., Nuovo Cimento,26,439 (1979). 6. Blomme, E., and Leroy, O., Acustica, 59, 182 (1986). 7. Blomme, E., and Leroy, O., Acustica, 58,4 (1985). Ac Soc India, 11, 1 (1983). 8. Blomme, E.,and Leroy, O.,..l 9. Blomme, E., and Leroy, O., Acustica, 57, 170 (1985). 10. Alferness, R., .lOpt. Soc. Am, 66, 353 (1976). 11. Poon, T. C., and Korpel, A., Proc 1981 Ultrason. Symp., 751 (1981). 12. Benlarbi, B., and Solymar, L., Int. .lElect, 48,361 (1980). 13. Chu, R. S., and Tamir, T., IEEE Trans., MTT-18,486 (1970). 14. Mertens, R., Hereman, W., and Ottoy, J. P., Proc Ultrason Int., 85,422 (1985). 15. Mertens, R., Hereman, W., and Ottoy, J. P., “The Raman-Nath Equations Revisited 11. Oblique Incidence ofthe Light-Bragg Reflection,” in Proc. Symp. Ultrason. Int., London (1987). 16. Franklin, J. N., Matrix Theory, Prentice-Hall, New York (1968). 17. Jeffreys, H., and Jeffreys, B., Methods of Mathematical Physics, 3d ed., Cambridge University Press, Cambridge, Chs. 7 and 8 (1966). 18. Klein, W. R., and Hiedemann, E. A., Physica, 29,981 (1963). 19. Leroy, O., and Claeys, J. M., Wave Motion, 6, 33 (1984). 20. Poon, T. C., and Korpel, A., Optics Lett., 6, 546 (1981). 21. Hargrove, L. E., J Ac Soc. Am., 34,1547 (1962). 22. Van Cittert, P.H., Physica, 4, 590 (1937). 23. Hance, H. V., “Light Diffraction by Ultrasonics Waves as a Multiple Scattering Process,” Tech. Rept. Physics, 6-74-64-35, Lockheed Missile and Space Co., Sunnyvale, Calif, (July 1964). 24. Pieper, R.J., and Korpel, A., Appl. Opt., 22,4073 (1983). 25. Hardin, R., and Tappert, F., SZAM Rev., 15,423 (1973).

168

Chapter 5

26. Korpel, A., Lonngren, K. E., Banerjee, P. P., Sim, H. K., and Chatterjee, M. R., .lOpt. Soc. Am., 3 (B), 885 (1986). 27. Korpel, A., Lin, H. H., and Mehrl, D. J., J: Opt. Soc Am., 6 (A), 630-635. 28. Venzke, C., Korpel, A., and Mehrl, D., App. Opt., 31, 656 (1992). 29. Yevick, D., and Thylen, L., .lOpt., Soc, Am., 72, 1084 (1982). 30. Korpel, A., Opt. Eng., 31,2083-2088 (1992). 31. Banerjee, P. P,and Tam, C. W., Acusticu, 74, 181(1991). 32. Tarn, C. W., and Banerjee, P. P., Opt. Comm., M,481 (1991). 33. Tam, C. W., Banerjee, P. P,and Korpel, A., Opt. Comm., 104,141 (1993). 34. Korpel, A., and Bridge, W., J: Opt. Soc Am., 7 (A), 1505 (1990). 35. Korpel, A., Appl. Opt.,26, 1582 (1987).

6 Selected Applications

In this chapter wewill discuss the application of the theory to many examples, chosen mainly for their suitability in this context, and not in the first place for their relevance to current technology. Examples of the latter kindcanbe found inmanyof the available bookson acousto-optic applications (e.g.,Ref. 1). Aneffort has beenmade to analyzeeach application fromtwo different pointsof view, for instance from the points of view of modulation detection and sideband heterodyning in modulators, plane-wave availability, and phase asynchronism in beam deflectors, etc The last section is devoted to signal processing, a subject which has become very popular recently. The reader will find the basic principles of image plane processing and frequency plane processing reviewed in detail. Some common architectures are discussed briefly.

6.1 WEAK INTERACTION OF GAUSSIAN BEAMS In this section we shall investigate the two-dimensional interaction of an incident Gaussian light beam with a Gaussian sound beam. The relevant configuration is shown in Fig. 6.1. For simplicity, we shall locate the waists of both beams symmetrically about the origin. To within the paraxial approximation, the fieldof the incident beam alongX is given by 169

170

Chapter 6 X I

Z

Figure 6.1 Interaction geometryof Gaussian light and sound beams.

E i ( x , 0) = Eiexp(-jk&x) exp

($-)

where W i is the beam radiusto the lle amplitude point. The sound beam is given by

(3

S(0, z) = S exp

For our calculation, we shall use the plane-wave weak interaction formalism as expressed by (4.54). To that purpose, wevmust first calculate the angular plane-wave spectra g(@)and f(<.n, according to (3.138) and (3.144). After some algebra, it follows readily tHat

171

Selected Applications

With (4.54) we find for incidenceat the appropriateBragg angle h=-$B

k,(4) = -0.25jkCmswiSEiexp

-(K’w,’

+k2w:)(4-eB)* 4

Comparison of (6.5) with (6.3) shows that outside the interaction region, a Gaussian beam propagating at an angle @B and having a waist W Igiven by

El($) represents

It will be clear from (6.5) that if kwi*KWs (i.e., the diffraction spread of the light beam is much less than that of the sound beam), then W l s W i . This is the usual case in beam deflectors, etc.,and approximates a configuration in which the incident beam is a plane wave, as usually assumed. If, on the other hand, kwi4KWs (i.e., the incident light may now better be described as a converging or diverging beam), then W I=Kw,lk. Thus the emerging beam is a demagnified (xlk) replica of the sound beam. This is a typical case of Bragg diffraction imaging [2], already discussed in more general terms in Sec.3.3.3 and tobe analyzed in detail in Sec. 6.9. Let us next calculate the power the in diffracted beamand how it depends on W, and Wi. It may be readily shownthat

First, consider the case where the width of the sound beam isWe fixed. write (6.7) as

It is clear that an asymptotic maximum will be reached when w r ) q i.e., when the incident light beam resembles a plane wave. Apparently, the best strategy is to concentrate all the incident power into one plane wave interacting with the strongest plane waveof sound in the spectrum. However, not much change occurs as long as KWJwi
172

Chapter 6

rapidly, in proportion to the ratio of sound diffraction spread to light diffraction spread. If the width ofthe light beam is fixed, similar considerations can be shown to apply. The best strategy is to concentrate all the sound power into one plane wave interacting with the strongest plane wave of light, but, again, there is not much change until the sound beam is made so small that a substantial number of plane waves of sound do not find any light waves to interact with. The above analysis can be repeated for the more practical case of a rectangular sound beam. The calculation is somewhat more involved, but the results are basically the same. The general effect of beam width on scattered power for this case was analyzed by Gordon [3], who used a threedimensional volume scattering formalism.

6.2

STRONG BRAGG DIFFRACTION OF A GNJSSlAN LIGHT BEAM BY A SOUND COLUMN

The general configuration is as shown in Fig. 3.15, and the general theory has been described in Sec.3.3.4 and 4.6 for the arbitrary 6mss section of the incident light beam. We will here analyze the specific case of a Gaussian incident beam as specified by (6.1), with the angular plane-wave spectrum given by (6.3), with &=-@B. Assuming, for convenience, that S = J ~ Swe ~, find upon substituting (6.3) into (3.168) and normalizing relative to

I&-@B)I

Note that &-@B) represents the maximum plane-wave amplitude (density) in the angular spectrum. For@=@B and v = z , it follows that &@B)=&{-+B), i.e., the center incident plane wave has been completely diffracted. This obviously must leave a hole in the plane-wave spectrum of the zeroth-order diffracted light. The phenomenon is shown in Fig. 6.2 that depicts, for the zeroth order, the results of a numerical calculation performed by Chu, Kong, and Tamir [4]. Figure 6.3 shows the corresponding - 1 order. Beam profiles other than Gaussian also have been analyzed. Results may be found in Ref. 5.

173

Selected Applications

I

-.l2

-.08

-.04

0

.04

.08

.l 2

Figure 6.2 Angular plane-wave spectrum of the zeroth order beam forv=z(solid 4.) line) and v = d 2 (dashed line). (Adapted from Ref.

Apart from beam distortion, the process of selective diffraction also causes decreased diffraction efficiency for the entire beam. This effect has been calculated by Magdich and Molchanov [q.The decrease in diffraction efficiency depends strongly on the parameter

which denotes the ratio of angular widths of incident light and sound. Figure 6.4 shows the relative power in the diffracted + l order beam as a function of electrical power appliedto the acoustic transducer.It is seenthat for all values of a, the maximum diffraction efficiency occurs at the same power level. This level should, according to (6.9), correspond to v=z. For increasing angular spread in the incident light, the maximum diffraction efficieficy decreases, as more and more plane waves of light are diffracted kds efficiently. Note that Fig. 6.4. shows reasonable agreement between €heoryand experinient.

Chapter 6

174

t

4 -.08

-12

-.04

I

0

.04

.OB

4

.l2

Figure 6.3 Angularplane-wavespectrum of the -1 orderbeamfor 4.) line) and v = d 2 (dashed line). (Adapted from Ref.

0

3

6

9 P (Watts)

12

V=K

(solid

15

Figure 6.4 Relativepowerinfirst-orderdiffractedbeamasafunction 6.) electrical power applied to the transducer. (From Ref.

of

Selected Applications

6.3

175

BANDWIDTH AND RESOLUTION OF LIGHT DEFLECTOR

The conventional configuration used for analysis is shown in Fig. 6.5. An infinite plane wave of light with amplitude Ei is incident at -@B,, and a + 1 order beam with amplitude El, is generated at +@B,. The subscript “C” refers to the center frequency F,. By changing the frequency of operation, the diffracted beam, (now called El) is made to change its direction to 2@bb-@Bc. The angle of incidence is fixed, and the Bragg angle condition is satisfied for each frequency by an appropriate planewave at F -(@B-@B,) in the angular spectrumof the sound. For weak interaction, the process may be analyzedby the use of (3.160). For @o=-@B, we find, using (3.128)to calculate Bi(@), (6.11) where q3~=K/2k=Q/(2kV)is the independent variable, proportional to Q. The diffracted light is obviously a plane wave propagating in the expected

Figure 6.5 Beam deflector showing plane-wave Ei incident at the correct Bragg angle @B, for center frequency operation. The diffracted wave El is shown at an arbitrary operating frequencyF, for which the appropriate Bragg angle equals@B.

176

Chapter 6

I

(6.12) Denoting

M= K- Kc

(6.13)

we find that the zerosof E1 occur at (6.14) corresponding to the case where the interacting planewaves of sound fall on the first (positive and negative) zeros of the angular plane-wave spectrum. For largeL, the solutionof (6.14) is approximately given by (6.15) Defining the bandwidth B as one-half of the frequency difference between zeros, we find from (6.15) (6.16)

A different argument, appropriate for the sound column picture, is based on Fig. 6.6. The column is considered to approximate a plane wave whose interaction is represented by the wave vector diagram kt, kl, K,. The end points of all diffractedwave vectors must lieon the circle throughA and B, with 0 as the center. If the sound frequency of the column is raised from f, tof, the lengthof the soundwave vector increases fromKc to K,its direction remaining the same, because it represents the entire sound column in this model. Ideally, the new wave vector trianglek;,kl, K should be closed,but, as shown, there existsa deficiency Ak. This deficiency leads to a cumulative phase mismatch LAk in the interaction,and as a result thetotal contribution from E to E1 vanishes when

LAk= r 2 a

I!

Selected Applications

177

\

/hk

0

Figure 6.6 Wave vector diagram showing phase match at center frequency and phase mismatch Ak at frequency F,corresponding to propagation constantK.

It may be shown readily from Fig. 6.6 that (6.18)

Substituting (6.170) into (6.18), we find forthe first zeros (6.19)

Note that this is identical to (6.15); hence, the phase mismatch reasoning leads to the same bandwidth prediction as the angular plane-wave spectrum argument. From a point of view of efficiency, it is important to make L,and hence Q, large. As shown by (6.16), this will adversely affect the bandwidth. A way around this difficulty has been found by having the sound beam track the required Bragg angle when the frequency changes.

178

Chapter 6

The principle of such a beam-steering phased array is illustrated in Fig. 6.7. The stair-step device has four individual transducers [7,8] alternatively energized in opposite phase, as indicated by the + and - signs. The height of the stair-steps equals one-halfof the wavelength & at the center frequency&. Thus, when the device is operatedat the center frequency, asin Fig. 6.7(a), the effective wavefrontp propagates vertically upward.The light is incident at the center frequency Bragg angle $B= so as to cause optimal generation of the +1 order. If nowthe frequency is reduced, then the required

Figure 6.7 (a) Stair-step'phased array operating at the center frequency fc. (b) Same phased array operating at a slightly lower frequency. (c) Direction of the incident light relative to rotated wavefront p .

Selected Applications

179

Bragg angle is smaller. This is automatically brought about by the effective wavefront rotating clockwiseby an angle @l, as indicated in Figs. 6.7 (b and c). It is readily seen from the figures that the angleof rotation is given by (6.20) As seen from Fig. 6.7 (c), the angle of incidence now equals @ B ~ - @ ~The . required angleof incidence, of course, equals+B, so that the angle of error @e is given by

The spacing between the transducers is now chosenso that dc$eldA=O at A=&. With (6.20)and using the conventional definition for the Bragg angle, we may write (6.22) from which it follows readilythat (6.23) Thus tomake this zero, the design requires that

(6.24) It is interesting to calculate the weak interaction behavior of the phased array and compare it with the conventional sound cell. This is very simple, because we have done most of the work already in connection with (6.11) and (6.12). In fact, a little reflection will showthat we only have to change the argument of the sinc function in (6.12) (representative of the angular spectrum of the sound field) in the following way: @Bc-@B+@Bc-@t-+B=@e. With (6.22), (6.24),and (6.12), we then find readily (6.25)

Chapter 6

180

If again we define the bandwidthas one-half the difference between zeros of the sinc function,we find (6.26)

We should compare (6.26) with (6.16). If, for example, Q=16n, then the fractional bandwidth for the conventional deflector equals 25%, that of the phased array deflector%YO. The first phased array deflector was used in a laser television display system developed at the Zenith Radio Corporation in 1966 [9].Figure 6.8 shows the transducer array used in that system. Figure 6.9 shows plots of diffracted light intensityvs. cell orientation for three distinct frequencies: (a) refers to the phased array of Fig. 6.8, (b) to a comparable but conventional transducer. Finally, Fig. 6.10 shows the frequency response of the phased array at low power input. More details about phased arrays may be found in books or articles devoted to applications (see, e.g., Ref. 10). The resolution of the deflector used as a scanner can be expressed as the number N of resolvable angles within the scan.In air the diffraction angle& of the light beamof width L can be .expressed &J=-&,c L

Figure 6.8 Phasedarraytransducerusedin (From Ref. 9.)

(6.27)

1966 Zenith laser display system.

181

Selected Applications

t

YIUIRAMANS

YILLIRAMUS

Figure 6.9 Plot of diffracted light intensity vs. cell orientation for (a) the phased 9.) array of Fig. 6.8 and(b) a comparable but conventional transducer. (From Ref.

Figure 6.10 Frequencyresponse of thephasedarraylightdeflector (From Ref. 9.)

of Fig. 6.8.

182

’

Chapter 6

and the total scan angle is given by (6.28) Hence; (6.29) where ‘t is the transit time of the sound cell. The expression (6.29) was first demonstrated experimentallyby Korpel et al. [l l]. 6.4

RESOLUTION OF SPECTRUM ANALYZER

In a light deflector operating in the scanning mode, the transducer is sequentially energized by adescending or ascendingseries of sound frequencies. That same device may be used as a spectrum analyzer if the frequencies are applied in parallel. This is shown schematically in Fig. 6.11. A wide beam (width L) of coherent light, represented schematically by the ray marked “a”, is incident upon a Bragg cell. Frequencies f l y f2, . . . f, generate upshifted rays that are focused by the lens with focal length F in points XI,x2 . . .),x of back focal plane X. (Refraction of the rays on the surfaces of the sound cell is ignored in the drawing.) The brightness distribution I(x) in that plane is then proportional to the power spectrum Pv) of the electronic signal according to the following rule:

I

1-1

L I

1-1

l

1-1

A

/

I I=I

I 1-1

’y f ,

f*

...L

II tI

\I

v

x-

II

I

II

Figure 6.11 Operation of spectrum analyzer. Power at sound frequenciesfi, . . .. ,fm is displayed as optical brightnessin points XI, x2, . . .. ,Xm.

ji,

183

Selected

I(x)

a

P( f ) withx = F- f

v,

(6.30)

Now, the brightness distribution around a particular position x, is of course not a point, as indicated in the ray drawing. Instead it consists of the diffraction pattern caused by the entrance pupil of the lens. If this is determined by the length L of the sound cell, thenit may be written as 2

I,(x -x,)

a

sinc

(6.31)

where h, denotes the free space wavelength. Let us denote the width of this pattern in the conventional way as half the distance between the first zeros of the sinc function: (6.32)

This width determines an effective resolution Af of the sound cell. From (6.30) we find that

a v,

h=F-Af

(6.33)

Comparing (6.32) and (6.33), we find finally (6.34)

This is a very plausible result, because it seems reasonable that the frequency resolution should be inversely proportional to the transit time (observation time) of the sound. It should be remarked that the light distribution in theX plane does not just represent the power spectrum of the input signal; if we concentrate on the optical amplitude, then this distribution is, in a profound sense, the frequency spectrum (Fourier transform), because not onlycomplex amplitude, but also frequency itself (through upshifting) is conserved. We will return to that aspect in Section6.13 on optical processing. The resolution of the spectrum analyzer may also be expressed as the number N of resolvable frequencies within the bandwidthB. With (6.34) we find readily that

1S4

Chapter 6

N=Bz

(6.35)

We find that this expression is identicalto (6.29), which gives the number of resolvable points of the deflector. This should not be a surprise, because, as stated before, the only difference between the two devices is the serial vs. parallel frequency input.

6.5

BANDWIDTHOFMODULATOR

The essential configuration of a Bragg modulator is shown in Fig. 6.12.The sinusoidal pattern is a symbolic representationof a carrier with wavelength A, modulated by a sinusoidal signal of wavelength h. The AM modulated carrier may be written as X

Figure 6.12 A typicalBraggmodulatorconfiguration.Thesinosoidalpattern symbolizes the modulated acoustic signal.

185

Selected $(x,t ) = q l +m ~~~(Qrnt-Kmx)] cos(Qt-Kk) (6.36)

where S is real, m is the modulation index, Q, the modulation frequency, and K,,,=2lclh, the propagation constantof the envelope. As the signal moves through the incident beam (amplitude Ei, angle of incidence 4 ~ ~ = x / 2 kthe ) , + l order diffracted beam is modulated in power. When a,,,increases (hence, A m decreases), a larger part of the modulation cycle with appear within the width D of the incident beam. It appears intuitively plausible that the modulation index of E1 will then decrease to reach a zero whenD=&. We will analyze the modulator response in two different ways that are often thought to be incompatible and todescribe different effects.In the first method, we integrate the locally diffracted power in order to calculate (6.37) As before, we assume weak interaction for convenience, although the analysis may equally well be carried out for strong interaction. From (3.1 lo), we find for smallv I,

IiV2 =4

(6.38)

We are here concerned with a modulated beam; hence, we write v=v(x, t)= Vo[l +m cos(Q,t-K,x)] (6.39)

where

v,=-kCLS 2

(6.40)

The total integrated power inthe diffracted beam is given by

S(?)= 0.25r

Dl2

-D/2

Iiv(x,t)’ dx

(6.41)

Assuming m<<1, we find (6.42)

Chapter 6

186

where (6.43)

Pi=IiD

and

(-

m,=2msinc

k D )

As Km=2dh,=QmlV, it is clear that the effective modulation index m1 decreases with increasing modulation frequency. The first zero will be reached when D = & , , as may be verified from ( 6 4 , and is in accordance with physical intuition. The above analysis is essentially one of localEy sampling the diffracted power. It is justified as long as the sampling ray bundlesare not so small that they overlapby diffraction inthe interaction region.The condition for this is similar to the one we have encountered in using theSURA method (see Sec. 3.1.3), but this time with respect to the modulation wavelength h, rather than.the carrier wavelength.Replacing A by h,in (3.57), we find ultimately

(6.45)

As an example, consider a sound cell operating at 50 MHz with Q=lO. The local sampling analysisis then valid for(Q,JQc)2*0.1, say, Fm<5 MHz. For higher modulation frequencies, the analysis must proceed theon basis of angular plane-wave spectra and becomes increasingly complicated, even more so as the nature of the optical detector has to be taken into account. The latter point will become clearer from the alternative analysis to be presented next. The essence of the alternative method is shown in Fig. 6.13. Instead of one diffracted beam, three are shown, this time with amplitudes El+, EIO, and El-. The center beam correspondsto diffraction by the carrier and has a frequency @+Q, where o is the light frequency. The two other beams correspond to diffraction by the sidebands at QkQ, of the modulated signal. They have frequencies ofo+Q+Qm for El+, and O+Q-Qm for El-. Our analysis now proceeds as follows. Imagine a photodetector positioned perpendicularly to the El0 beam so as to intercept all three diffracted beams. Let us ignore, for the time being, the fact that the beams, propagating at slightly different angles, do not overlap exactly. The detector is assumedto deliver a current proportionalto the total integrated power. Leavingout the

'

187

Selected Applications X

Figure 6.13 Interpretation o f Bragg modulation in terms of sideband-carrier mixing. (identical) z dependence of the three beams and the common frequency shift Q, we find i(t) = S’D121E,o +E,+exp(- jkA@x+ jQ,t -Dl2

2

+E,- exp(+jkAqx - jQmt)l dx where A+ is the angle between the beams

A @ = -K

m

k

as followsdirectly from Bragg angle consideration [see (6.3)].

(6.46)

188

Chapter 6

The total sound signal (6.36) may be written as S(X,t)=S COS ( Q t - f i ) + 0 . 5 m S cos[(Q+Qm)t -(K+K,)x]+OSmS co~[(Q-Qm)t-(K-Km)x]

(6.48)

The last two terms in (6.48) represent the sidebands that diffract El+ and E l - . Treating each term (sound field) in (6.48) separately and assuming weak interaction, we find with (3.103) and (6.39) EIO= E1+=E1-=-0.25jmViE~

Substituting (6.49) and (6.50) into (6.46), we find

[

i(t) = 0 . 2 5 q b 2 1 + 2m sin .(kY)]

(6.51)

With (6.47) we find that (6.51) is identical with (6.42). Thus, the second method gives the same results as the first one, as must be implicit in the mathematics. Rather more important than the formal equivalence, though, is the interpretation of the new method. To see that, we write (6.46) in the following way: i(t)=io+ir(t)

(6.52)

io= EoEi dx

I

(6.53)

il(t)=Re[Ip

(6.54)

where the phasor I*, representing the ac current from the photo detector, is given by

I

I

Ip= 2 E,',E,+exp(jkA4x) dx + 2 EloE;- exp(-jkA4x) dx

(6.55)

Each of the RH terms of (6.55) represents the mixing of two light fields at frequencies differing by a m . These fields are, of course, the carrier beam ( E I o )with the upper sideband ( E l + ) beam, and the carrier beam with the lower sideband (E1 -) beam. The mixing beams are inclinedat an angle A+,

Selected

189

and, as a result, the mixing efficiency is reduced until, at A+=AID, the beat frequency disappears. The importance of the above interpretation is that it makes it clear that the detector output may vanish while the diffracted light connected with the modulation process is still present in the total beam. The information may, in fact, berecovered again by narrowing the photodetector aperture, or covering it with a periodic transmission maskof the form(1+cos &x). The stroboscopic effect of such a mask on the running (intensity) interference pattern 1+2m cos(Qmt-Kmx), brought about by the interfering beams, results in an output current at frequency Q,. Thus, in order to discuss modulation efficiencyand bandwidths, the detector should be specified. A detector-independent bandwidth also existsand comes into play when the bandwidth of the device as a defector is exceeded and the diffracted beams El+ and El- can no longer be generated. The criterion for that has been derived in Sec. 6.3. At the beginning of our discussion, we had tacitly ignored beam spreading and separation. That this may be done is a consequence of a remarkably simple theorem that is, however, so important to the application of signal-processing sound cellsthat we devote the next sectionto it. Finally, it will be clear from our analysis that the modulation bandwidth may be increased by decreasing the widthD of the incident beam. However, if this is pushed to the point where the angular spread of the light beam begins to exceed that of the sound beam, diffraction efficiency begins to decrease, as was shown inSec. 6.1.

6.6 THE QUASI THEOREM The quasi theorem gets its name from the fact that it usually, butnot always, applies. It is best discussed by using a concrete example, e.g., the mixing of a carrier at o with upper and lower sidebands at o+Qm(the common frequency shift Q brought about by a sound carrier, as in Sec. 6.5, has been absorbed in 0). The relevant configuration is shown in Fig.6.14. An infinite extent one-dimensional photodetector is located at z1 and collects all three beams. Subsequent heterodyning results in a time-varying output il(t) that contains frequency componentsat Qm and 251,. The first component comes about by the mixingofeach sideband with the carrier. The second component results from the mutual mixing of the two sidebands.It was not No such considered in Sec.6.5 because of the assumption IE1-1, IEI+I~IEIo>. restrictions are necessary here. We are interested in what happens whenwe move the photodetector to a new position on the Z axis, say 22. At first glance, it must appear that il(t)

190

Chapter 6 X

Figure 6.14 Heterodyning of three beams as an illustration of the quasi theorem.

will decrease, because two beams no longer overlap. That would be the wrong conclusion though; the beams do, in fact, overlap morethan shown, having spread by diffraction. The first part of the quasi theorem (and his part is actually not quasi but strictly valid) says But for a time delay,il(t) is independent of the position of the (infinite) photodetector. The physical reason for this is that a photodetector basically records the arrival sequence of photons.This sequence cannot be changed, only delayed, by moving the photon counter, as long asall photons are collected. (Hence, wehave the theoretical requirement for an infinite photodetector.) A mathematical proof plus some remarkable consequences of this may be found in [12]. The secondpart of the quasi theorem says h(t) also does not change if transparent elementsare put in the path of the light beams, provided (1) that each element covers all the relevant fields completelyand (2) that the elements arenot highly dispersive.

It is proviso (2) that accounts for the “quasi” nature of the theorem. A dispersive element may cause temporal bunchingor debunching of photons,

Selected Applications

191

thus changing the character of il(t). For instance, a properly designed and positioned Fabry-Perot interferometer may suppress the two sidebands in transmission and hence make i l ( t ) disappear. A less obvious way to defeat the quasi theorem is given by the following example. Imagine three parallel beams, with El+ and El-, at some reference time, in phase withEOand also assume that IEI+I,IE1-IeIEol. The total beamisobviously amplitudemodulated, and, consequently, a sinusoidal component i l ( t ) will be of the form il cos Qmt, if we neglect 2Qm terms by our previous assumption. If now a highly dispersive element is introduced that phase shifts E1+ and El- by 90"relative to EO,then the amplitude modulation changes to phase modulation and il(t) vanishes. Obviously, this requires some nontrivial and dedicated experimental effort, and hence in almost all other experiments proviso (2) is satisfied,and, subject to proviso (l), the theorem applies. Let us test this on the configuration shown in Fig. 6.14. In the first experiment, we put the photodetector at z1=0 in the plane of the aperture. The beams are then well-defined, and, following the same reasoning as in Sec. 6.5, we may write il(t)=i+(t)+i-(t)

]

i+(t)=Re[I,+

)]

i-(t)=Re[I,-

E:

I,,+= 2

(6.56)

E;,E,+ exp(jkA@x)dx

(6.59)

E,,E,' exp(jkA@x)dx

(6.60)

Dl2

Ip- = 2

ID12

It is readily shownthat I, = 2DE,',EI+ sinc (D?)

(6.61)

Ip = 2DE,,E; sinc (D?)

(6.62)

In the second experiment, we put the photodetector in the back focal plane of a lens of focal length, f, which accepts all three beams. The configuration is shown in Fig. 6.15. For convenience, the aperture is situated

192

Chapter 6

Figure 6.15 As in Fig. 6.14, but with lens inserted as Gedanken experiment.

in the front focal plane of the lens, but that is not necessary for the argument. In the back focal plane of the lens, three Fourier transforms of the aperture D are formed, as shown [13]. The three amplitude distributions are given by

F+= F. =

[3) [3 1 [3) +

sine[(x fA#)$]

(6.63)

sin(%)

sinc[(x fA#)$]

F- =

(6.65)

The heterodyne currents are given by Ip+= 25 F;F+ dx

I

Ip = 2 FiF: dx

(6.66)

(6.67)

193

Selected

Carrying out the integration [14],we retrieve, after some algebra, expressions (6.61)and (6.62), thus provingour point. Note that, in the context of experiment1, the interpretation of (6.61) was that with increasing A#, there is increasing phase cancellation across the surface of the photo conductor. Eventually, this results in il(t) vanishing when A#=A/D, i.e., when the spatial variation in the phase of the locally generated current reaches2n. In the context of experiment 2, the separation between the focused sinc patterns increases, resulting in less overlap and hence a smaller heterodyne current. Eventually, whenA#=A/D, the separationis such that the maximum of one pattern fallson a zero of the neighboring one. The sinc functions are now orthogonal (asin conventional sampling [13]), and again the heterodyne current vanishes. Finally, we should note a trivial but important fact. If at the output of any sound cell all diffracted and undiffracted beams are heterodyned together (if desired after being processed by an optical system satisfying the quasi theorem), then the resulting heterodyne current vanishes. This is so because a sound cell does not absorb photons and hence cannot generate temporal variations in the overall photon flux. Of course, spatial variations in the photon flux do occur, and these maybe transformed into temporal variations by suitable amplitude masks. An interesting example will be found in the Fresnel field signal processor described in Ref. 15.

6.7

OPTICAL PHASE AND AMPLITUDE MEASUREMENT

In optics we can only measure the time-varying flux of photons arrivingat the detector, i.e., we always measure intensity in some formor another. This is in contrast to other fields such as acousticsor the lower frequency region in the electromagnetic spectrum, where it is possible to measure the field quantities directly (strain, pressure, current, voltage, etc.). In optics we must resort to superimposing coherent fields to measure amplitude and phase. For example, we may obtain phase information from the location or shift of interference fringes. If the fields areat slightly different frequencies, then in the total power (which may be measured with a photodetector), there is a time-varying component at the difference frequency that gives information about the field amplitudes. Considerthe local output current caused by the mixing of a reference fieldand a “signal” field on a photodetector placedin some X-Y plane: i(t, X, U)a 1 =1

I*

~ s+

Y ) e x p ( ~ s t+) Er (x,Y ) exp(jmrt)12

~ (X, s

l2

IEr + 2 ~ e {(X, ~ y)Er s (X, Y ) exp[j(ar - u s >t11

(6.68)

194

Chapter 6

where the subscripts ‘S’ and ‘r’ refer to signal and referencefields, respectively. It isclear that the current componentat the difference frequency carries information about the phase and amplitude distributionof the signal field relative to the reference field. The latter could be made uniform to simplify matters. Naturally, the time-varying current must be integrated over the entire photodetector, and in that operation local information vanishes, unless the photodetector is small or the reference field is limited in size, e.g., if it is a focused spot. In that case the signal field may be sampled by moving that spot around. This is exactly what is being done in the “phase and amplitude sampler” developed by Korpel and Whitman [16]. In that device a Bragg diffraction sound cell plays an essential role, because it is used for both scanning and frequency shifting of the reference beam. The experimental set-up is shown in Fig.6.16. The beam from a laser is split into two beams. The upper beam goes through a beam-perturbing object (e.g., a transparency) that creates the signal field incident on a largearea photodiode. The lower part of the beam is fed into a Bragg light deflector, the upshifted order of which forms the scanningreferencebeam. This beamisfocusedbylens L and semitransparent mirror M onto the photodiode. The frequency of the reference !““”l

I.F. FILTER ANDAMP. SWEEP FREQUENCY GENERATOR

VIDEO DETECTOR

GENERATOR

T.V. DISPLAY

Figure 6.16 Experimentalset-upforphaseandamplitudescanning. Ref. 16.)

(From

Selected

195

beam is upshifted by the time-varying sound frequency A(?).It is clear that all the elements we have discussed are present in this configuration. The current at &(t) is mixed with a local oscillator at fs(t)+f;., where f;. is an intermediate frequency. The amplitude of this signal is displayed on a TV monitor that isscannedinsynchronismwith the reference spot. (The relatively slow vertical scanning of the reference spot is accomplished by a mechanical mirror.) For the sake of simplicity, no phase measurements (requiring synchronous detection) were attempted in this experiment. It is clear that with lensL focused on the plane of the photodiode, the field at that plane will be displayed. What is not so obvious is that if L is focused onto some other plane, the field inthat other plane will be displayed in spite of the fact that it does not coincide with the plane of detection. This is a direct consequence of the quasi theorem that we discussed in Sec. 6.6. It does not matter where the photodetector is located as long as all the, light is collected. Figure 6.17 makes this beautifully clear. It shows the field just behind the beam-perturbing object, a Ronchi grating in this case. This was achieved by using forL a negative lens with itsvirtual focus at the plane of the grating. In a second experiment, the lens L was removed altogether, i.e., the reference beam came to a focus at infinity. As a result the far field of the signal beam ought to be displayed. This is clearly the case, as shown in Fig. 6.18.

Figure 6.17 Display of optical field in the plane of a Ronchi grating. (From Ref. 16.)

196

Chapter 6

Figure 6.18 Display of the far field of the light transmitted through grating. (FromRef. 16.)

a Ronchi

The sensitivity of the method is surprising. For shot noise-limited operation the minimum detectable power (for a signal-to-noise ratio of unity) is given by

p& =-eBN a

(6.69)

where e is the electronic charge, B the bandwidth of the system (6 MHz in the experiment), N the number of resolvable spots (lo"),and a the detector sensitivity (0.37 m). For the experiment the calculated sensitivity was 0 . 2 6 ~ W. The measuredsensivitywas 1 . 5 ~ W, asthesystemwas not shot noise limited. It is clear that Bragg diffraction cells are important elements in any amplitude or phase measurement system, not so much because they make scanning possible, but rather because they are convenient frequency shifters. In the example to follow, a Bragg cell is used in a non-scanning phase detector to measure extremely small dynamic oscillations [ l l . Figure 6.19 shows the optical configurationand related electronic components. A laser beam at frequency fo is directed through a Bragg cell where a portion of the light (the zeroth order)is passed undeflectedto lens L, which focuses it on a vibrating quartz wedge, activated by a signal generator at frequency fm. This imposes upon the reflectedlightbeamaphase modulation atfm of modulation index m=2kvu, where U is the amplitude of

197

Selected Applications LENS

SIBGEM.

4

L

WATER FILLED

-Cf0

-

'0 'S

XI

RECORDER

-Y

DRIVE

M2

1

X.

Figure 6.19 Experimental set-up for the measurement of small vibrations. (From Ref. 17.)

the normal surface displacement. The reflected beam passes once more through the sound cell, where a portion of it is deflected and upshifted by the sound wave of frequency& present in the cell. This part of the signal beam, now at frequency fo+&then falls on a photodiode. It is there heterodyned with a reference beam fo-f,. This reference beam is derived from the original downshifted part of the laser beam at frequency fo-f,, whichisfocusedby L onto a stationary mirror M2 and sent back undeflected through the sound cell. The similar treatment of reference and signalbeamensures that theyoverlap at the photodiode in the same direction and with the same wavefront curvature. The photodiode now generates a difference signalat frequency 2&, phase modulatedat frequency fm. Therefore, the diode output current contains sidebandsat 2fs&fm,which, for the small modulation index involved, have amplitudes proportional to m=2kvu. One of the sidebands is selected and amplified by a receiver and detected with a square law detector. In order to achieve a high signal-tonoise ratio, the soundcell signal at& is amplitude modulatedat 1000 Hz to obtain a 1000-Hzmodulated final signal at the output of the square law detector. This is then further processed in a narrow band-tuned amplifier and fed into the Y axis of an X-Y recorder. The X axis is synchronized with the drivethat moves the quartz wedge through the focused beam. Figure 6.20 shows the surface displacement squared vs. position on the wedge atfm=8.5 MHz. The modeof vibration is localized (trapped) near the point on the wedge where the thickness equals one-half the wavelength of the sound wave. It was estimated that peak surface displacement in this

198

Chapter 6

Figure 6.20 Relativesurface displacement squared vs. position on the quartz wedge atfm=8.5MHz. (From Ref. 17.)

experiment amounted to approximately 0.6 A. The laser power in both reference and signal beam was0.15 mW. The predetection bandwidthB1 was 57 kHz and the post-detection bandwidth B2 was 15 Hz. The operationwas limited by thermal noise. For shot noise-limited operation, the minimum detectable surface displacement was calculated to be (6.70)

where B=%, e is the electronic charge,a the photodiode sensitivity,and PI the laser power. For the values used in the experiment this value was calculated to be 2 . 6 ~ A. Yet another technique for measuring phase was invented by L.Laub [18, 191. The method is illustrated in Fig. 6.21, which is taken from the patent description. Referringto the figure, laser beam(12) is directed through a halfsilvered mirror (32) onto a Bragg cell (44).The cell is activated by a swept frequency generated by ramp generator (40) in order to scan the diffracted beams (16) and (18) across the (partially) reflecting topological object (26). Beams 16 and 18 are generated by two closely spaced frequencies fs+f m , where& is a modulation frequency originating in the oscillator (40). The frequencies of the beams arefo+fs+fm, where f o is the optical frequency. The frequency f m is chosen such that the diffraction spots of focused beams 16 and 18, separated by Ax, have a large area of overlap. Beams 16 and 18 are

Selected Applications

199

Figure 6.21 Scanning phase proflometry set-up. (From Ref. 18.)

reflected toward the soundcell with a phase difference A9=2kAh, of where Ah is the difference in height between the centers of the focused spots. Thus A@=2k(dhldx)Ax,as shown in Fig.6.22. At the soundcell each reflected beam is split into two other beams, such that after reflection off the half-silvered mirror (32), the final field at the photodetector (36) contains frequencies f$+2f,, f++f+2fm, and f$+fs-2fm. The phase of the superheterodyne currents at 2fm and 4fm contain the information about Aq5, i.e., about the topological gradient dhldx. This information is extractedby a phasemeter (50) being fed 2fm as a reference signal through a multiplier (52). After integration by a circuit (54), the gradient isdisplayed onan oscilloscope (58) whose horizontal sweep is synchronized with the ramp generator (40). Vertical scanning is accomplished by a translator (64), driven by a motor (62), and synchronized with aDC ramp generator (66). Figure 6.23 shows a drawing of an actual photograph of the oscilloscope display. The object examined consisted of a smear of red blood cells on a glass plate. The maximum height of the phase profile traces is less than 1 pm,the average width of the blood cells is approximately8 pm. The Laub system is basically a common-path differential interferometer and therefore relatively insensitive to vibration compared with the previous system discussed (Fig. 6.19).

200

Chapter 6

Figure 6.22 Detail near foci of sampling beams. (From Ref. 18.)

Figure 6.23 Oscilloscope display of red blood cells obtained with the system of

Fig. 6.21. (From Ref. 18.)

6.8

BRAGG DIFFRACTIONINTERMODULATIONPRODUCTS

In a typical Bragg diffraction signal processor, such as

that shown in Fig.

6.24, and to be further discussed in Section 6.13, numerous frequencies are

present and may have to be processed individually. At weak interaction levels, there is no difficulty with this, but with stronger interaction, multiple scattering may occur, leadingto third-order intermodulation products. Thus, a frequencyf2 may first upshift the light, followed by afi downshift plusa h

20 1

Selected Applications

Mirror Mirror

Loca I scillator Beam

L.O. Optical System " "

Figure 6.24 Prototype coherent acousto-optic signal processor. (From Ref.49.)

/

Figure 6.25 Wave vectordiagramillustratingthird-orderacousto-opticintermodulation in Bragg diffraction.

Chapter 6

202

upshift. As shown in Fig. 6.25, the resulting spurious diffracted light is upshifted by a total of 2fi-fi. (An identical mechanism will also result in the generation of a spurious component at 25 -h.) Exactly how serious this effect is depends, of course, on the proximity of the two frequencies relative to the Bragg diffraction bandwidth. To simplify matters, we shall assumefi to be close t o h so that both frequencies induce on-angle Bragg diffraction. The Feynman diagram discussed in Sec. 4.8 is a convenient tool for analysis of this case. The diagram relevant to the present case is shown in Fig.6.26. Because of our assumption, the coupling factors that connect the levelsasare, in (4.92), S:=$, Si=Si with j = 1,2, depending on the frequencies involved. With (4.95) we find that the plane-wave amplitude at level 1 is given by

With (4.94) we may write this (6.72) VI = v2= v, we have For equal amplitude sound signals,

(6.73)

2f2-f, f2

f2- f,

0 ' Figure 6.26 Feynmandiagramillustratingacousto-opticintermodulationin Bragg diffraction.

Selected Applications

203

where (6.74)

is thepower diffraction efficiency for$ orf2 by themselves. Thus, ifq=lo%, the power in the third-order intermodulation product will be down by 1/3600, or about 35.5 db relative to the main signal. It is of interest to note that in Raman-Nath diffraction two additional paths exist, as shown in Fig. 6.27. Because again the coupling coefficient for each transition equalsS or P, we find readily that the total contribution is three times that given by (6.73). Hence, intermodulation products are likely to be more severe in Raman-Nath diffraction. It will be clear that in the formalism used here, many more complicated intermodulation scenarios-three-tone intermodulation, off-angle Bragg diffraction with limited bandwidth, etc.-may be analyzed in a way that is convenient and physically plausible. For comparison, a more conventional treatment may be found in Ref.20. Now, in accordance with our dual approach, let us analyze acousto-optic +2

-2 f 2

2f2- f,

+l{

O{

-1

\

f,

f2-f:,

\

0

- f,

Figure 6.27 Feynmandiagramwithtreepaths,illustratingacousto-opticintermodulation in Raman-Nath diffraction.

Chapter 6

204

intermodulation from a different point of view. If two frequencies fi and fi are simultaneously present, with equal amplitudes, then the total sound signal in the Bragg cell may be written as S(",

t)=Re[S exp@&t-jKIx)+S exp(j&t-jK2x)]

(6.75)

which may bewritten, if we assume S to be real, t s(x,t)=2Scos (n,-al)Z-(K,-Kl)Z

"I

"I

(6.76)

xcos n,+n1"(K,+K1)5 [ 2 t

Hence, the signalin the sound cellmaybeconsidered a carrier at a frequency (l22+*1)/2, 100% amplitude modulated by a modulation signalat frequency (Q2-R1)/2. If the modulation frequency is small enough [see (6.45)], then from (3.103) with &=O, z = L and using a time- and spacewe fhd dependent v(x, ?)=v cos[(!&-Ql)t/2-(K~-K1)~/2],

[":)'I

El(", L, t) = -jEi sin

(6.77)

where we assume v=kCLS/2e 1 The first term in (6.77) represents contributions to the first order at both f+ and fi. This may be seen readily from the first terms in the diffracted light

"1 "I

-Kl)-

2

-jEi(f)exp[jot+/nlr+(Kz-Kl)-

) + e . .

2

(6.78)

Selected Applications

205

The K terms, which indicate the spatial modulation of the diffracted light, automatically guarantee the correct directions for the orders at o+a1 and w+a2. The second term in (6.77) leads to the other terms

(6.79)

If we concentrate on thepart with 3(!&-!21)t/2, it may be seen readilythat it gives riseto anintermodulation productof the form

with a similar expression in 2Q1-Q2. The K term in (6.80), expressing spurious spatial modulation of the diffracted light, again guarantees that this particular contribution appears in the correct spurious order. It is clear that the amplitudeof (6.80) is identical to that of (6.72), ifin thelatter we set v1 = v2= v. We thus find perfect agreement with the earlier point of view: intermodulation caused by the nonlinear [sin(v/2)] of a sound cell may equally well be explained by multiple scattering. The former model, however, is limited to small frequency differences [see (6.45)]; the latter model has no such limitations, provided we use the proper form of the coupling factors [(4.70a) and (4.70b)l. Finally, in Raman-Nath diffraction, the first model would use a nonlinear response of the kind

YVJ

J1(v)=--- +... 2v 2 2

(6.81)

instead of the Bragg diffraction response (6.82)

206

Chapter 6

It follows that (6.81) indicates an intermodulation contribution (the cubic term) three times higher than (6.82), in accordance with the predictions of the multiple scattering model.

6.9BRAGGDIFFRACTIONIMAGING In Chapter 2we introduced the concept of Bragg diffraction imaging [21]as a natural consequence of the one-to-one correspondence between wave vectors of light and sound, implicit in the wave vector diagram in two dimensions. In the present section, we will investigate this notion in more depth, both on the basis of plane-wave interaction and of ray tracing. We will also perform a scattered amplitude eikonal calculation in detail as an example of general procedure.

6.9.1Plane-WaveInteractionAnalysis In Sec. 3.3.3 it was pointed out, in connection with (3.160), that if the incident light field has large angular spread,the scattered light field‘s planewave composition mirrors that of the sound field. Hence, the scattered light somehow carries an image of the sound field that presumably could be retrieved by the proper optical processing. A typical experimental configuration is shown in Fig. 6.28, where the incident light is convergent and focused to a line perpendicular to the paper at the origin P outside the (dotted) sound cell. The sound emanates from a line source Q, located at , x zq, to interact withthe light in the regionof the sound cell around the point M . The object ofour analysis is to find out where Q is imaged, and how the image quality is affected bythe angular widthof the incident wedge of light. To simplify matters, we assume that the focused lightspot at P is Gaussian: (6.83) where EO is real for simplicity. The angular plane-wave spectrum with reference pointP is, according to (3.139) with z=O, given by

A

or, with (6.83),

(6.84)

207

Selected Applications

Figure 6.28 Braggdiffractionimagingconfigurationshowingsoundsource and incident light focused atP.

Q

(6.85) The sound at Q we model by an idealized Huygens source in the line z=z,:

S(Q) = $$(X

- xq),

z = zq

(6.86)

The sound angular spectrum, with Q as areference point, is then given by

Taking P as a reference point we find (keeping the convention for y in mind; see Fig. 3.1)

i(y, P)= i(y ) = iq e x p ( i h q cosy + j a qsin y )

(6.88)

Chapter 6

208

The + l order scattered light spectrum (Pas reference point) may now be determined from(3.160)

(6.89)

Now let us see if there exists a point where all the plane waves of l?*(+)are in phase again.In other words, we look for a reference pointV such that (6.90) L

Now l?*(@, V ) and l?*(@, P) are related by a simple wavepropagator

E,(Q, V )= E,(Q, P ) exp(- jkx, sin Q - jkzy cosQ)

(6.91)

With (6.89) and (6.91), it follows from(6.90) that f i q

cOS($~-@)+fi~ Sin(@B-Q)-kxv sin Q-kz,

cos Q=O

(6.92)

for all Q. Equating cos Q and sin Q coefficients separately, we find after some trigonometry (6.93)

(6.94)

Equations (6.93) and (6.94) will be recognized as a clockwise rotation by ( d 2 - Q B ) of a vector (zq, xq) into a vector (zv,x") plus a scalingby K/k=NA. Hence, these are the imaging rules that project the sound point Q into its image V. At the same time, the diffracted light going through Vpropagates nominally in the direction 4=2@B, as indicated by the maximum of 81(Q,V) in (6.90). The imaging rules are illustrated in Fig. 6.29. The point V is the

Selected Applications

209

X

I

x‘

Figure 6.29 Illustration of imaging rules for upshifted @+V)and downshifted @+V‘) interaction.

image for the- 1 order. The calculation is entirely similar, again resulting in a scaling by Klk,but this time a counterclockwise rotation by d2-4~. From (6.90) it is clear that the imageof Q is degraded in the sense that it is no longer a delta function like the original sound source object. Rather, it has the shapeof the focus of the incident light beam that constitutes, in fact, the impulse response of the system. This may be seen as follows. The light field along the X‘ axis through the point V expressed as a function of x’=x-xv is found by taking the Fourier transform of (6.90) (with Fourier transform variables x’ and qYA)

E m =m 4 , v

1

S)

= -0.25jkC&E,, exp(

exp(- jkx’24,)

(6.95a)

Note that the profile is indeed identical to that of the incident focus, whereas the amplitudeis proportional to that of the Huygens sound source.The term exp(-jkx’24~) expresses the fact that the scattered light propagates nominally in the $=24B direction.

210

Chapter 6

For downshifted diffraction,one finds similarly (6.95b) where x " = x " x ~ . A more general analysisof the Bragg diffraction imaging process may be found in Ref. 22.

6.9.2 Eikonal Analysis We will now try to find evidence of imaging by applying the ray-tracing of diagram of Fig. 4.10 for upshifted diffraction. Figure 6.30 shows a wedge rays emanating from the Huygen source of sound at Q. The central ray (solid line) intersects the central ray of the optical ray bundle, incident from the left, in the interaction pointA, at the correct angled 2 + @for ~ upshifted

Figure 6.30 Ray-tracinganalysis of Braggdiffractionimaging. Q is the sound source, the incident light is focused atP,the diffracted light atV. The circle C is the locus o f interaction points.

211

Selected Applications

Bragg diffraction. A diffracted ray bundle leavesA with is central rayat the correct diffraction angled2+@Bwith respect to the sound ray. We now look for the locusof other interaction points such asA' and A". It is readily seen that such points mustlie on acircle (C), as this guarantees constant interaction angles. As for the diffracted rays, the angular invariance means that they must all go through one point (V),also located on the circle. Thus, V represents the image ofQ. The imaging rules may be readily derived from Fig. 6.30 as follows. Considering AVAP and AVNP, we see that LV N P = L V A P = ~ @Hence, B, from AVNP we find that VP=2CN sin [email protected] we consider APCQ, it is clear that L P C Q = ~ L P A Q = ~ ( ~ ~ - Thus, @ B ) .LCQP=LCPQ=@B. It follows that QP=2CQ COS@B=~CN COS@B. Hence, VPlQP=2 sin@B=Hk,which confirms our earlier prediction about scaling. It is further seenreadily that L V P Q + L V A Q = z . As L V A Q = d 2 + @ s , it follows that L V P Q = ~ ~ - @inB accordance with results from the wave theory. For downshifted imaging, a diagram similar to Fig.6.30maybe constructed and the appropriate imaging rules derived[23]. We have noted in Sec. 6.9.1 that the resolution in the sound image is limited by the size of the focused spot of light. This is also evident from Fig. 6.30. If, for instance, the incident lightis limited to a uniform wedge of apex angle 2+, then thisis also true of the diffracted light. Thus, the resolution in the demagnified image is of the order of (U2) sin 4, corresponding to an effective resolution of(M2) sin @ in the sound of field. Having demonstrated the usefulness of Bragg diffraction ray tracing, we will next illustrate the other aspect of eikonal theory, i.e., the calculation of the scattered amplitude. The starting point is (4.1 17)or (4.1 19). Let us first calculate

"as,=!E' &(

+ -kd'Y, " as2

Kd'Y, as2

kd'Y, ds2

)l

(6.96) A

in accordance with (4.99), with the eikonal functions defined by (4.97). The corresponding cylindrical wavefronts are shown in Fig. 6.31, from whichit follows readily that %=-pi

(6.97a)

Ys=

(6.97b)

+PS

Y1= -p1

(6.97~)

1 are drawn from the respective where the position vectors pi, ps,and p

212

Chapter 6 $1

X'

Figure 6.31 Diagram for eikonal calculationof Bragg-diffracted light.

sources R Q, and K By transforming eqs. (6.97) to a coordinate system with origin at V and axesin the Z'-X' direction, it maybe found, with d/ds=dl&', that (6.98a)

(6.98b)

(6.98~) , , and pq represent the distance fromA to P and Q, respectively. The where p distance AV is denoted by p".From Fig.6.31 it is further seen by considering AAPD, AAVD, and AAQD that

213

Selected

pp=2pccos a=2pc COS@-$E) sin(a+$)=2pc pq=2pc

sin p

(6.99a) (6.99b)

where P=a+OB, and pc is the radiusof circle C. Subtracting (6.99~)from (6.99a) and using sin $~=K/2k,we find

,

With (6.100), (6.98), and (6.96), it then follows that

$1

=Kpvcos2 $B A PPPl7

(6.101)

and IS(s,)l=lS(A)I. As we use an Our next step is to find ~E,(sa)~=~E8(A)~ eikonal theory, i.e., k, K+-=, the fields may be found directly from the angular plane-wave spectra through(4.47). With (6.85) we have (6.102) where is the angle of the ray through A . (Actually, &=O for the centerray of Fig. 6.30, but we shall leave the expression as is.) With (6.87) it follows that

Substituting (6.101) and (6.103) into (4.1 17), we finally amve at (6.104) Now, the same expression willbe found if westart from different interaction points A', A", etc. We mustthenchange to f l u , etc.As $=$', the last term in (6.104) may then be generalized to e~p(-2k~w~$',,~/4) where "a" stands for any interaction point. Ifwe compare (6.104) with (6.102), it will be clearthat the former expression refers to a Gaussian identicalto the waist of the incident lightbut located at V. From the same comparison, it is clear

214

Chapter 6

that the modulus of the amplitude is given by 0.25kC&Eol, in complete agreement with (6.95). The phase ofEI(A) may be found from the fact that a = - 3 ~ / 4 f o r C>O [see (4.116)]. With (4.99), (6.97), and (6.100), it follows that -3n -kpp + Kp, - kY: (A) = -kpy - kY1 (A) = 4

(6.105)

and, hence, the phase of &(A) is expressed by a factor proportional to exp(ikp,), in agreement with (4.47), if we remember that EI is converging rather than diverging. In summary, we have shown that the eikonal theory not only provides a convenient ray-tracing formalism, but also correctly predicts the amplitude of the scattered ray. More information may be found in Ref. 24. A raytracing analysis of axial cross-section imaging (as frequently used in signal processing) may be found in Ref. 25, whereas Refs. 26and 27 analyze signal processing applications of transverse cross-section imaging, i.e., the type discussedabove. Experiments demonstrating the spatial nature of the imaging, as in a reconstructed hologram, are discussed in Ref. 28.

6.9.3

Imagingin Three Dimensions

So far we have assumed that the imaging is two-dimensional, i.e., all "rays

are parallel to the 1-2 plane. On this assumption, any imaging in the Y direction just consists of a stackingof two-dimensional images. If this were true, then the resolution in the Y direction would be better than in the X direction, because the angular aperture of the incident light would not be relevant. Indeed, inspecting Fig. 6.32 we clearly see the difference in the resolution, as the Fresnel pattern of the illuminating sound field is only visible in the Y direction (horizontal lines). Based on the above, the assumption of stacked imagescannot be too bad, yet we knowthat not all rays can be horizontal: detail of the soundfield in Y must lead to diffracted sound rays in theY-Z plane. In Ref. 29 the three-dimensional situation has been analyzedon the basis of diffracted ray tracing. Figure 6.33 shows the situation for downshifted imaging. The incident light is a cylinder beam focusedinto a line 00'. The sound originates from the point source S. In the X-Z plane we recognize the familiar circle of interaction points for horizontal rays. Diffracted rays in this plane cometo a focus inS-. SA' and SB' are sound rays out of the X-Z plane. They interact with incident rays CO' and D'O', giving riseto diffracted raysA'E' and B'F. To

Selected Applications

215

c Figure 6.32 Transinosonifiedacousticimage of Silver Angel fish. Notethe difference in resolution of the Fresnelfield background. (From Ref.31.)

a first approximation, for small angles 6, it may be shown that all such diffracted rays intersect the lines S-F and E’F’. Hence, the imaging is astigmatic, withS-F and E ’ F forming the verticaland horizontal linefocus, respectively. The vertical focus is located at the same position as for twodimensional imaging; the horizontal focus for downshiftedimagingis located downstream along the central diffracted ray at a distance( N I )times the distance SB from the source S to the incident light. Therefore the “vertical” angular aperturesof the diffracted rays of light and sound are in the same ratio as the wavelengths, i.e., BB’lSB-(NA)BB’lBF’.Hence, but for aberrations not covered by the theory (large e),the vertical aspect (1:l) of the sound field image suffers no degradation due to the limited angular aperture of the incident light. Indeed it has been foundthat resolution of the order of one wavelength of sound may be achieved inthe Y direction [30]. Figure 6.34 shows the ray tracing for upshifted imaging. Note that the horizontal line focusE’F is virtual in this case.

216

Figure 6.33 Ref. 29.)

Chapter 6

Ray tracing of three-dimensionaldownshiftedimaging.(From

F

Figure 6.34 Ray tracing of three-dimensional upshifted imaging. (From Ref.29.)

Selected Applications

217

In order to remove the astigmatism of Brsgg diffraction imaging, a suitable lens system must be employedto merge the horizontal and vertical line focus. At the sametime the severe anamorphism (the horizontal magnification is %A, the vertical one unity) must be compensated for. The simplest way to dothis is by a simple cylinder lensthat images the point Sonto the lineE'F, with the proper magnification of NA. An interesting question is what will happen if the illuminating beam has the more practical shapeof a cone rather than a wedge. It will be seen that the symmetry inherent in this configuration allows the rotation ofthe figure in the diagram of Fig. 6.30 about the axis QI!Thus, rays that are out of the plane form additional images Q on a circle through V with radius pf as indicated [31]. For small angle approximationsit can be shownthat in sucha system again two line foci occur[32]. References 32 and 33 describe and analyze byray methodsan experimental system with almost spherical illumination, i.e., the main illumination lens is spherical and a weak cylinder lens mergesthe two foci. The sound field originates from eight transducers spaced in the horizonal ( X ) direction. One of the inner transducers is defective. A wave-theory analysis of this systemmay be found in Ref.34, Figure 6.35 shows the brightness distribution (without the weak cylinder lens) in some horizontal cross sections (X-?) near the vertical focus, while Fig. 6.36 shows the distribution in some vertical crms sections (Y-2) near

7 mm beyond vert. focus

-

Profiles are Individually Normalized

Figure 6.35 Horizontal cross sections near the vertical focus. (From Ref. 32)

Chapter 6

218

the horizontal focus. Note that in the center horizontal cross section the resolution is sufficient to show up the defective transducer, but not enough to clearly separate theother ones. Figures 6.37 and 6.38 show horizontal and vertical cross sectionswith the foci made to coincide. In this final experiment the spherical lens aperturek

horiz. locus Profiles are Individually Normalized

15(mm) : C Y

9 mm in front of

f

Figure 6.36 Vertical cross.sections near the horizontal focus. (From Ref 32.)

l.O?

I 0

5

10

;?

15 20 25 Position (mm)

30

35

Figure 6.37 Horizontal image distributionwith the foci made to coincide. (From Ref. 32.)

Selected Applications

I .

0.0-1.' 0

.

219

> 5

10

15

20

Position (mm)

Figure 6.38 Verticalimagedistributionwiththefocimade Ref. 32.)

to coincide. (From

increased by a factor of 2, and hence the horizontal resolution doubles.The individual transducersnow show up clearly.

6.10

BRAGG DIFFRACTION SAMPLING

In the area of acoustics in general, there exists a need for small nonmechanical probes with which to sample a sound field locally without disturbing it. The question naturally arises whether acousto-optics could satisfy this need. If the sound field isnot too wide ( e e l ) , one may think of probing it with a parallel light beam and recording the power in a single diffracted order. According to (3.173), what is measured is then the sound field integrated over the interaction path, rather than its valueat one specific point. Moreover, the phase of the sound can never be retrieved from a power measurement. The phaseretrievalproblem might beovercome by heterodyne detection, perhaps along the lines indicated Ref in 15. As for the integrating effect, could that be avoided by not using a parallel beam, but instead focusing the incident light? Would this not create something like a narrow parallel beam of finitelength in the focalregion? It is not immediately clear how to answer this question, but the notion is worth following up. Now, in Bragg diffraction imagingwe have all the elements necessary for a Gedanken experiment. With referenceto Fig. 6.39, we see in the focal plane

220

Chapter 6

figure 6.39 Heterodyne model of sound probing.

of the incident wedge of light the undiffracted light focused to a sinc type pattern at P,while at V1 and V2 are images of sound sources at QIand Q2. The latter are up-and downshifted, respectively, bythe sound frequency Q. If somehow P could be made to coincide with V1 on a photodetector, the resulting RF current wouldpresumably carry information about the amplitude and phase of the sound at Ql.Also, because of the narrow local oscillator spot at VI, the effective resolution would, as in imaging, again be determined by the angular apertureof the incident light, i.e., be equal to N2 sin q5 for the focused light beam.This scheme appears promising but forthe required coincidence of P and V I . Some further reflectionwillreveal, however, that we already have overlapping spots, namely Pinitself wherethe imagesof the sound field at P overlap with each other and with the undiffracted light. Both images are offset in frequency from the zeroth order

Selected

22 I

by the same amount; presumably, all we have to dois just heterodyne P with the now coinciding VI and V2, i.e., find the RF component in the current of out in a photodetector collectingall the light. Unfortunately, as was pointed Sec. 6.6, this will not work. A sound field does not absorb photons,hence, a modulation of all the light passing through it cannot occur. In a more restricted context: the signals at VI and VZ are, according to (6.95), in quadrature to the carrier,thus effectively constituting a phase modulation of the undiffracted light. The result isthat the heterodyne current from VI and P exactly cancelsthat from V2 and P. Is there a way to frustrate this cancellation? It turns out that there exist two ways to do this. The first method makes use of fact the that the nominal directions of light propagation from VI and V2 differ by 4@3. This, in turn, means that in the far field the patterns do not overlap precisely. Although the total current will still be zero,it is now possibleto use a selective mask in front of the photodiode to suppress one of the cancelling heterodyne currents, thereby frustrating the cancellation. A knife edge or a graded neutral density filter will satisfy the requirements. Another way to look at this is illustrated in Fig. 6.40. This shows our initial picture of a narrow (width=w) parallel light beam of finite length l that makes up the focal region of the incident light. The soundwave propagating through this beam will make it deflect periodically,as indicated by the thinlines. A knife edgeK placed in front of a photodetector PD will periodically obstruct the light beam moreor less, thus causing an RF component in theoutput current i(t). Both the heterodyning and the periodic deflection model are equivalent and have been used with much success in the acousto-optic monitoring of surface acousticwaves. We will return to that topic shortly. A completely different way of detection is to rely on polarization effects. Under certain circumstances(to be detailed later), the scattered light has a polarization component perpendicular to that of the incident light. By the use of a quarter-wave plate, the phaseof the diffracted orders is changedby d2, and by means of an analyzer, a common component is selected for heterodyning. The basic principle is identical to that used in the Zernike phase contrast microscope [ 131 or in the electronic detection of phasemodulated signals [35]. A typical configuration is shown in Fig. 6.41. Note that the incident beam forms a cone rather than a wedge of light. In Sec. 6.9.3 we have argued that such a cone would cause the diffracted images to be ring-shaped. However, the diameter of the ring for images of P itself is zero; hence, there isno deleterious effect forour particular application. To analyze the complete device shown in Fig. 6.41, we first assume ideal operation of the quarter-wave plate plus analyzer section, i.e., we assume that the phase of the diffracted light has been shifted by 90" somehow and that all the light is interceptedby a photodetector. In the figure, the latter is

222

Chapter 6

PD

L i (t)

Figure 6.40 Periodic deflection model of sound probing.

Quarter-wave

S(P1

Sound

X‘

Figure 6.41 Actual sound probing configuration. (From Ref. 36.)

placed in the X-Z plane. According to what was discussed before in Secs. 6.4 and 6.5, the relevant RF current is then given by

I,, II{E:(x,z)[jEl(x,z)]+E,(x,z)[-jE-,(x, z)]*}dxdz

(6.106)

223

Selected

where the js refer to the 90" phase shift imparted by the quarter-wave plate. Now, E1 may be calculated in the firstBorn approximation from the volume scattering integral (4.41), and E-I may be obtained in a similar way. Full details of this calculation may be found in Ref. 36. The final result is that I,, may be expressedas follows (6.107) where the integration isover the entire interactionvolume. Expression (6.107) shows that the sound field is indeed being sampled in amplitude and phase by a three-dimensional sampling function, i.e.,IEi(r)l2. With reference to Fig. 6.40, the effective resolution is given by 1'/2, i.e., that distance away from focus where the width W' of the incident cone equals about one wavelength of sound. Phase cancellation in S(r) will then cause the contributionto the integral in (6.107)to vanish at that point. It is easyto see that by this criterion the resolution is of the order of M2 sin@,in agreement with earlier predictions. In the periodically deflected beam model, the same distance 1'12 sets a limit to net effective deflection,and, hence, the same approximate resolution is predicted. However, a precise analysis on the basis of this simple model is impossible. It cannot treat regions far out of focus whereW'> N2. In Fig. 2.8 we have already shown a plot of a sound field cross section obtained with the method of Fig. 6.41. A two-dimensional image is shown in Fig. 6.42.

6.11

SCHLIEREN IMAGING

Schlieren imaging makes possiblethe visualization of inhomogeneities inan otherwise uniform medium. Typical examples areair bubbles and striations (Schlieren in German) in glass and density variations of the air in a wind tunnel. A typical schlieren set-up is shown in Fig. 6.43. Referring to the figure, a parallel bundle of light, exemplified by raysc, d, and e, is incident from the left on the glass block U.By means of lenses L1 and L2 with focallengthf, the block (i.e., the representative cross section a) is imaged onto the plane b. A schlieren stop V in the center of focal plane g stops all directrays like c and d. Thus, in the absence of scattering, the image plane b will be dark. Ray e is scatteredby air bubble A (actually, an infinity of rays is generated in the scattering process), travels past schlierenstop V, and generates an image of A at A'. Hence, inhomogeneities in the medium Uwill be imaged as bright objects on dark a background (dark field imaging).

224

Figure 6.42 Two-dimensionalimageobtainedwiththeapparatus (From Ref. 36.)

of Fig. 6.41.

225

Selected Applications

Figure 6.43 4f configuration for schlieren imaging.

It is obvious that a propagating sound field may be regardedas a special distribution of refractive index variations and hence visualized by schlieren methods. This was first accomplished by Toepler in 1867 [37]. His basic method hasbeen used for acoustic visualizationof transducers ever since, as was mentioned in the introduction. The problem at hand is to analyze whatwe see exactly when we visualize a sound field by schlieren methods. The simplest way to attack the problem is to use the straight undiffracted ray analysis (SURA) that was developed in Chapter 3. Let us assume that in Fig. 6.43 a three-dimensional sound field s(x, y, z, t ) propagates in theXdirection, contained within the boundaries of the glass block. This will give rise to a refractive index variation&(x, y, z, t), which, according to the three-dimensional generalizationof (3.6), is given by &(x, y, z, t)=C’s(x, y,

2 9

t)

(6.108)

According to the three-dimensional generalization of (3.17), the phase8 of the light at the output of the glass block is given by L

8(x, y L, t ) = -kv J&(x, y, z, t ) dz

(6.109)

0

where we have left out constant terms. If Ei represents the phasor of the uniform, normally incident light field, then the field E(x, y, L, t ) is given by

Chapter 6

226

if it is assumedthat lela1 (weak interaction). Let us now model the sound fieldas 1

s(x, y , z, t ) = -&(x, y , z) exp(jQt - jKx) + C.C. 2

(6.11 1)

where S, denotes the complex sound profileand c c stands for the complex conjugate. Upon substituting (6.1 11) into (6.108), the result into (6.109), and the overall resultinto (6.1 lo), we find y , L, t ) = Ei L

--1 jkvC’Ei exp(-j&) exp(jQt)jSe(x,y , z) dz 2 0 L

1 --jkvC’Ei exp(jKx) exp(-jQt)jSi(x,y, z) dz (6.112) 2

0

We recognize theh t term in (6.112) as the background light, the second term as the upshifted scattered light and the third term as the downshifted scattered light. These three components will come to separate foci in the focal plane g of Fig. 6.43 due to the fact that their nominal directions of propagation differ by twice the nominal Bragg angle. We now replace the schlieren stop of Fig. 6.43 by a knife edge that blocks both. the background light and the downshifted light. The phasor field of the upshifted light in plane b is thengiven by L

1 E+(x,y ) = -- jkvC’Ei exp(-jKx) exp(jQt)lS.(x, y, z) dz 2 0

(6.1 13)

where we have ignored the image inversionby the 4f system. Whatwe see in plane b is the image intensity (6.1 14)

It is not immediately obvious what (6.1 14) represents. However, if S, is predominantly independent of z-say, the sound is generated by a long ( 2 ) transducer of limited height (-the schlieren picture shows the evolution

227

Selected

in the Y direction of a one-dimensional sound field along the propagation direction:

This evolution is evident, for example, in Fig. 2.4, where the left side of the picture represents the IFresnel field of the sound and the right side evolves graduallyinto the Fraunhofer field (far field). In the general case, the interpretation of (6.114) is not self-evident. It should be noted though that (6.113) represents a projection in the sense of tomography. Projections at different angles may be obtained by rotating the sound cell about the X and Y axes. By proper tomographic processing and addition of such projections, it should be possible to reconstruct the entire three-dimensional sound field. In Ref. 38 it is argued and demonstrated that, in two dimensions, such a reconstruction is equivalent to Bragg diffraction imaging. The method we have used to analyze schlieren imaging (SURA) is no longer applicable at high frequencies;it runs into the same difficulties as we have encountered with simple sound column configurations where @>l. How then shall we proceed in this potentially useful high-frequency case? A straightforward approach is to use the three-dimensional plane-wave interaction analysisto be discussed laterin Sec. 8.5. A treatment along those lines may be found in Ref. 39. However, it is more instructive to start from first principles based on the wave vector diagram. Such an approach has been followed in Ref. 40. Figure 6.44 is a drawing from that reference illustrating wave vectors of soundand light participating inthe interaction.

X'

X

Figure 6.44 Wave vector diagram for three-dimensional schlieren interaction. (From Ref.40.)

228

Chapter 6

In the drawing the sound propagates nominally in the X direction and the light in the Z direction. The incident light is a plane wave characterized by the vector QO. The sound is three-dimensional, and vectors OA, OB, and OC represent some plane waves in the (continuous) spectrum of the sound. These particular vectors havebeen chosen so as to form wave vector diagrams with the incident plane wave vector QO, giving upshifted diffracted vectors QA, QB, and QC. The wave vector triangle QOA lies in the X-Z plane, and the other two are generated by rotating QOA about the Z axis. It is clear that in general the sound vectors selected in the interaction form a continuum that lies on the hollow cone OABC. By the same token the diffracted light vectors lie on the hollow cone QABC. We shall denote the fictitious sound field that is represented by the selected sound vectors on the hollow cone by S+, and the (real) corresponding diffracted light field by E+. Their angular spectra will be denoted by S+and 8,. Now, two points are worth noting: 1. The selected sound vectors on the hollow cone with apex at 0 all have the same z component K,. The same applies to the upshifted vectorskz on the hollow cone with apex at Q . Therefore, both these fields represent diffraction-jree patterns (generalized diffraction-free beams[41]) that do not spread by propagation. 2. The diffracted light vectorskd and the selected sound vectorsK have the same X and Y components (kd,=K, and kdy=Ky), and hence the two corresponding fields have the same pattern (E+ QT S+). This pattern, of course, is the schlieren imageof S. In Ref. 39 it has been shown experimentally that indeed the schlieren image does not suffer propagational diffraction. Figure 6.45, taken from that reference, shows how the schlieren pattern of part of a sound field is invariant to propagation. Figure 6.46 shows the amplitude of the selected sound spectrum along the circle bounding the hollow cone. Now let us derive the form of the selected sound field S+, which, as we have seen, is identical to the schlieren image. The angular spectrum S+ follows fromthe angular spectrumS by a filter operation

S+=R+$

(6.116)

where the filterR+ expresses the factthat the K, components are constant:

KZ=- K sin @B

(6.117)

229

Selected

.Figure 6.45 Experimental schlieren image of a sound cell formed by a 4f system at various distances z from the image plane: (a) z=O, (b) z=2J (c) z=4J (d) z=6J The light is incident so as to maximizethe complexity of the image. The part of the image near the transducer is not shown. (FromRef. 40.)

as follows from the triangleQOA in Fig. 6.44. Following Ref.42 we now use K, and K, as the spectrum variables rather than YA and ?/A as we did in Sec. 3.3 [for a definitionof yand V, see Fig. 8.12@)].The filterR+may now be defined as l ? +

+

=2n6(K, K sin $B)

(6.118)

where 2n is an appropriate scaling factor.’ The soundfield S+at some arbitrary x is given by

s+=s(PS+)

(6.119)

where s denotes the Fourier transform and P the propagator, i.e., the factor P=exp(-jlY,x) that accounts for the phase shift upon propagation.

230

Chapter 6

Figure 6.46 Amplitude of the selected sound spectrum along the circle bounding the hollow cone of diffracted rays. The width of the pattern is due to the finite illumination aperture. (From Ref.40.)

Selected Applications

23 1

[See (3.13 1) with K cos y=K,]. By the same token

S+=P-' 3-

-' (S+)

(6.120)

We now substitute (6.116) into (6.1 19):

s+=s ( P A+

(6.121)

and then apply (6.120) to S, instead of to S+, and substitute into (6.121):

s+=s[P A+P-'

s -'(S)]

(6.122)

because the operatorsE A+,and P-' commute, (6.122) may be written as

s+=s[R+F" (S)]

(6.123)

Finally, using the convolution propertyof the Fourier transform,we get

where * denotes two-dimensional convolution. It is seen readily that (6.124) results in the following expression for the schlieren image 00

exp(- j k i sin $B )S(x,y, z)dz

S+= exp(j k i sin &)

(6.125)

d

In our case of a limited sound field as in Fig. 6.43 the integration limits may be replaced by0 and L. Now the sound phasorS is related to the sound profileSe, used in(6.1 1 l), as follows:

1 2 1 = -S(x, y,z)exp(jQt) + C.C. 2

~(x,y,z,t)=-S,(x,y,z)exp(jC2t-jlYx)+c.c.

(6.126)

so that

S=Se exp(-jKi)

(6.127)

Chapter 6

232

Substituting (6.127) into (6.125) gives

(6.128)

For low-frequency operation where QGl, it maybeshown that the (6.128) varies but little overz. In exponential term under the integral sign in that case S+ = exp(-j&l

-

Se(x,y, z) dz

(6.129)

Thus, (6.128) predicts a schlieren image identical to (6,113), which was derived‘with the SURAmethod valid for Q e l . This gives us some confidence in the correctness ofour analysis. At high frequencieswe must use (6.128) or (6.125). Let us consider in the latter an integration along the Bragg line 2’ (Fig. 4.5) that is inclined at an angle (PB with respect to the Z axis: zf =z, yf=y,x’=x+z sin +B

(6.130)

As a first approximation it may be assumed that S(xf, yf, z)=exp[-jfi sin

(PB]s(% y, z)

(6.131)

i.e., the amplitude 1 4 does not change appreciably over this small distance. Then (6.125) may be written as (6.132)

Thus, the schlieren image, even at high frequencies, is seento consist of an integration of the sound field,but along Bragg lines rather than along lines parallel to the Z axis. Most important is that S+ still constitutesa projection in the tomographic sense. To a first order in OB, the diffracted light field has the form of (6.132) multiplied by the interaction term- */zjkvC’=-l/&C=-ju [39,42]: on

S(x’, y’, z’)dz’

E+(x,y)= -juE, 4

(6.133)

233

Selected .Applications

6.12

PROBING OF SURFACE ACOUSTIC WAVES

Surface acoustic waves (SAWS) have become of increasing importance in signal processing [43], and a noncontacting probeis needed for checking of diffraction, reflections, etc. Similar to the situation in bulk waves discussed in Sec. 6.10, there exist two ways of acousto-optically probing an acoustic surface wave. In the first method, thewave is considered to form a reflective phase grating on the surface of a mirrorlike substrate. A relatively broad beam of coherent light is incident at an angle. The reflectedbeamis accompanied by two diffracted orders, one of which is measured by a photodetector [44].Because the height of the corrugated surface is very small, of the order of 10-lo m, it acts as an ideal Raman-Nath grating, but diffracts only very little light. Therefore, the dominant noise in the detection system is of thermal origin, and, consequently, the technique has all the disadvantages of direct detection [45]. Heterodyne detection with its inherent shot noise (quantum noise) character is inherently superior, and as discussed in Sec. 6.10.This time, fortunately may be used in the same way however, a polarization method is not feasible, and knife edge detection is commonly used. As before, the explanation may be given either in terms of periodic beam deflection or in terms of heterodyning. Because the SAW interaction is much simplerthan the bulk interaction discussed in Sec.6.10,

' / l /

/

/SUBSTRATE

/ / /

//

Figure 6.47 Probing of surface acoustic waves by focused light beam and knife edge (heterodyne) detection.

234

Chapter 6

we shall give a brief analysis of the former at this,time. A typical setup is shown in Fig. 6.47. A beam of coherent lightof total power P is almost normally incidenton a surface acoustic wave of depth h. When the wave moves through the beam focus, it will periodically, at the sound frequency, deflect the reflected beam by ?A@, where A@=2hK

(6.134)

The spot (far field)on the photodetector isof angular width28, and, hence, the fractional peak variationof the input power is ofthe orderof AN20. If a is the sensitivity ofthe photodetector in amp/W, the peak RF current out of the diode equals (6.135)

From (6.135) it is clear that, because K=QlV, the sensitivity increases linearly with the acoustic frequency. A limit is reached, however, when the wavelength A becomes equal to the sized of the focused spot, because at that point the net effect just averages out. Thus, we may expect maximumcurrent when d ~ N 2i.e., , 8=iv2d=AIA. From (6.135) we then find that

Now let uslook at the heterodyne model. This time, the interpretation of Fig. 6.47 is different. The dotted lines now do not represent a periodically deflected beam, but rather two diffracted beams separated by A@=AIA.The distance from the focus to the photodetector is denoted by E. The fields on the photodetector are sketched in Fig. 6.48. For simplicity, the diffracted spots are shown as squares of size 281x281 rather than as circles. The undiffracted center spot0 with amplitudeEi partially overlapsthe upshifted diffracted spot O+ with amplitude E+ and the downshifted spot 0-with amplitude E-. The relative displacementsof the spots areIA4=1ivA. E+ and E- are given by (6.137)

where the peak phase shiftv (Raman-Nath parameter) is givenby v=2kh

(6.138)

.

235

Selected Applications

Figure. 6.4% Heterodynecurrentcontributions on thesurface of the photodetector.

Wherever there is overlap with current densities are givenby

the undiffracted beam, the heterodyne

Z h = aE,,EI, = j ( f ) l E o r

(6.139a)

(6.139b) It is clear from (6.139) that in the regionI, where all three beams overlap, the plus and minus contributions cancel locally. In regions IV and V, there is no overlap; hence, no current. Finally, the output from I1 cancelsthat from I11 unless the latter region is shielded from the light by a mask. This is, however, precisely what the knife edge K does. With K in place, the total current is then equalto the output from I1 alone:

I, =Zk x ( 2 l O ) x ( ~ ) = - j ( f ) / E 0 ~

h ~2x1'~-

(6.140)

Realizing that the total power P is given by P=IE01*(281)~

(6.141)

236

Chapter 6

we find with (6.138)and (6.140) (6.142) which is equivalent to (6.135), but, in addition, gives some information about the phase of the current. The factor -j refers to the fact that the maximum current is caused for maximum slope (under the focus) of the SAW, a factwhich we did not take into account in the deflector model. it is readily seen from Fig. 6.48 that As for the maximum current IIplmax, this occurs when area I1 reaches the maximum value of le. This happens when the displacement lUL equals le, the same conditionwe derived before. Hence, the two models are equivalent. However, only the heterodyning model reveals what happens when d>N2, namely, a complete cutoff when d=A. After this, i.e., whend>A, no more signal is generateddue tothe lack of overlap of the heterodyning fields. In the near field, however, the beams do overlap, and running amplitude fringe patterns occur periodically due to Talbot imaging [13]. A matching amplitude grating, placed in one of these positions, followed by a photodetector, will again produce a heterodyne current [151. A complete review of various kinds of heterodyne probing may be found in Ref. 46.

6.13 SIGNAL PROCESSING Acousto-optic devices are widely used for signal processing. Most nontrivial applications makeuse of the fact that, with the information signal modulated on the acoustic carrier, the acou_sto-optic cell acts as a temporary storage medium that can be addressed in parallel by a wide optical beam. The diffracted light then carries a moving “image” of the modulation signal. As was pointed out in Chapter 1, parallel processing for display purposes was pioneered in the 1930s and further developed in the 1960s. More general signal processing evolved gradually in the 1960s and 1970s. This, too, has been reviewed in Chapter 1. Acousto-optic signal processing may take placein the image plane or the frequency plane of the associated optical system. Detectionof the resulting signals may be by baseband or by heterodyning techniques. Integration for higher resolution may be over space or overtime.Finally, the second dimension may be used to further increase frequency resolution, or to display special timefrequency distributions. In this section we shall concentrate on the fundamental mechanisms common to all these techniques. An extensive discussion of architectures may be found in Ref. 47.

Selected Applications

237

6.13.1 Image Plane Processing Figure 6.49 shows how a Bragg cell may be thought of as containing a moving image of a modulated sound beam. The electronic signal to the transducer is represented by the analytic signal e‘(t) exp(in,t), where Q, denotes the carrier frequency and e’(t) is the modulation signal. The real physical signal at the transducer is given by Re[e‘(t) exp(ind)l. BY allowing e’(t) to be complex, both phase and amplitude modulation can be modeled. For general phase modulation,we write e’(t)=expp(t)]

(6.143)

where, for an FM signal,
(6.144)

and for a chirp signal (6.145)

Figure 6.49 Bragg cell carrying an amplitude image of the modulation signal

e(-x+ V&). (left) Upshifted interaction; (right) downshifted interaction. (From

Ref. 48.)

238

Chapter 6

For conventional AM modulation, we write e'(t)= 1 +acos(R,t)

(6.146)

The analytic signal in the soundcell may be written as So(t) = d[t -

$1

exp(jRct- jKx) (6.147) = e(x - Ct) exp(jR,t

- jKx)

as indicated in Fig. 6.49. As discussed in Sec. 6.11, we may write for the phasor of the upshifted, scattered light El(t) 0~ jEe(-x+ Vst) exp(jL$t-jkpx)

(6.148)

where E is the incident light and p the nominal Bragg angle. Note that in Fig. 6.49 (left), taken from Ref. 48, the scattered light is indicated by its analytic signal, which includes the factor exp0.m). In (6.148) the exponential part expresses a plane wave travelling upward at an angle p, and having been upshifted in frequency by Rc.The first factor indicates that the light carries an amplitude image of the modulation signal. For downshifted interaction, the situation is as in Fig. 6.49 (right), and the scattered light may be expressed as El 0~

jEe*(-x+ Vst) exp(-jR,t+jkpx)

(6.149)

In the rest of the discussion, we will emphasize upshifted operation. One of the simplest signal processing operations to perform is convolution with a fixed mask pattern. The principle of operation is shown in Fig. 6.50. As shown, an image of the sound amplitude in upshifted light is formed on the amplitude mask g(x), i.e., the phasor amplitude distribution immediately to the left of the mask is given by En]-a jEe(x+ Vst) exp(jact+jkpx)

(6.150)

and immediately to the right of the mask the field is Err,+ 0~ jEe(x+ Vst)g(x) exp(jR,t +jkpx)

(6.15 1)

A photodetector with pinhole aperture is situated in the focal plane of the lens L3 and centered on the diffracted beam. The offset of the pinhole

Selected Applications

239

Sound in cell: e(c"+\t ex pj(Q2 - K x I

transparency

current from photo detector

Pinhole on axis

Signal to transducer

e'( t

in diffracted light

expj(ln, t

je ( x + V , t ) e x p [ j ( o t ~ , ) t + j k p x ]

Figure 6.50 Convolution of signal carried by Bragg cell with a fixed mask. (From Ref. 48.)

corresponds to an angle - p in the Fourier transform plane. The amplitude at the aperture Eapis therefore given by the inverse Fourier transform evaluated at that angle: (6.152) Finally, the current i(t) out of the photodetector is proportional to the square of the amplitude Eap.

Ih(x + I-

i(t) =

l2

V't)g(x) dx

(6.153)

Note from Fig. 6.50 that in practice the function e(x+V,t) is limited by the aperture to a range x= -D/2 to x=D/2. Thus the integration limits in (6.153) should be replaced by these proper limits. It is readily seen that (6.153) represents a finite convolution if we use a reflected coordinate system for the mask plane so that g(x) is replaced by g ( - x ) . Then (6.153) may be written as

Chapter 6

240

(6.154) If no pinhole is used in front of the photodetector, as shown in Fig.6.51 (left), then the currentis given by i(t) = jle(x +V’t)g(-x) dxrdx

(6.155)

d

Equation (6.155) represents a less useful output than (6.154), but even (6.154) is limitedto positive values. If the pinhole is not centered on the diffracted beam, but offset by an angle as shown in Fig. 6.51 (right), then the current is given by

If, using (6.149), we applythe same reasoningto the downshifted light,we find that (6.153) must be replaced by all light collected

I 3

Current from photodetector: 2

j ( e ( x t + x ) g ( x ) l dx

24 1

Selected

i(t)

a / j e * ( x + V s t ) gdx (x)

(6.157)

4

with the pinhole aperture being placed in the center of the downshifted beam, and g(x)referring to the nonreflected X coordinate. The fact that the conjugate signal appears in(6.157)makes this a correlation rather than a convolution. Here, too, we are limited to positive values of the output. Moreover it is difficult to make g ( x ) a complex function; phase-shifting masks arenot easy to construct. To achieve a more general phaseand amplitude mask function, we use the heterodyning technique shown in Fig. 6.52. Here the undiffracted light is not stopped, but modified by an optical filter to appear in plane c (the former mask plane) as reference field &(x). For the sake of simplicity, a simple symbolic light path has been shown in Fig. 6.52.It goes without sayingthat in actuality the path could be quite involved and the optical filter quite complex. If we now observethe currentat Q, delivered by the photodetector (without pinhole) as a result of the heterodyning of the reference field and the Bragg diffracted field,we find W

I ( Q , ) a E: ( x ) e ( x+ Vst)exp(jkj3x) dx

(6.158)

a

Sound in cell eCrc+Vst )expj(fiCt-Kx)

Signal to transducer e'(t)exp(jilct)

yeterodyne current a t L?c:(E,(xle(Vst+x)exp(jk~x)d~ Reference field

for shaping reference field

Image of sound cell in diffracted light j e (~+~,t)exp[j(wtSl,)ty'kpx]

Figure 6.52 Processing by heterodyning with reference field. (From Ref.48.)

242

Chapter 6

where I(aJis a time-varying complex amplitude (time-varying phasor). It will be clearthat we have a wide choice forthe 'effective mask function" geff(x) [compare with (6.153)]: (6.159)

geff(X)=Er*(x)exp(ikDx)

Moreover, theoutput is not squared, but directly proportionalto the finite correlation of e and gd. Thus, if for example, E,(X)=er(x) exp(ikpx) (i.e., propagating in the same direction as the upshifted beam), then (6.160) which represents atrue correlation. Note, by the way, that a correlation can alwaysbechanged into a convolution by the use of a reflected coordinateand a conjugate operation, i.e., the convolution of h(x) and g(x) is equal to the correlation of h(x) and g*(-). To derive (6.158) we have implicitly applied the quasi theorem (Sec. 6.6), because we superimposed the reference fieldand the upshifted field in plane c, although the photodiode (assumed infinite) is located in plane d, which is an inverse Fourier transform plane ofc. According to the quasi theorem, it makes no difference where we calculate the heterodyning current. Had we calculated thiscurrent in plane d,we would have found

-

I(a,)a Is-'*[e(x) exp(jk/3x)]s"[E~(x)]

(6.161) dx'

S

where x' is the coordinate in plane d, proportional to the spatial frequency f x . Using Parseval's theorem and the general properties of the Fourier transform, it is readily shown that (6.161) is indeed equivalent to (6.158). Thus, image plane processing and frequency plane processing are quite equivalent, and in most cases the simple insertion an ofappropriate lens will transform one into the other. Nevertheless, it is instructive to consider frequency plane processing in its own right. We will take up that subject in the next section.

6.13.2 FrequencyPlaneProcessing We will now analyze the operation of the Bragg cell of Fig. 6.49 in terms of frequencies, i.e., we considerthe soundcell to be a frequency analyzer rather than a signal delay line addressed by light.

243

Selected

First we write the modulation signal e'(t) of Fig. 6.49 in terms of its frequency spectrumS(Q,,,): (6.162)

As showninFig. 6.53, alens L1 focuses both the diffracted and undiffracted light in planeb. In particular, the light diffracted by the carrier at Q, is focusedat P,corresponding to a deflection angleof p relative to the horizontal, or 2p relative to the incident light. The lightat P,therefore, has a frequencyw+Q, and anamplitude proportionalto the carrier, i.e.,to S(0). Now, a frequency Qm of e'(t) will correspond to a frequency Qc+Qm of the total signal in the sound cell. This frequency will diffract atlight a somewhat larger angle with respectto the incident light: Q m A( 2p) = -

kV,

Figure 6.53 Braggcellasspectrumanalyzer (From Ref. 48.)

(6.163)

in frequencyPlaneProcessing.

244

Chapter 6

The light diffractedby this component will cometo a focus inQ such that the distance PQ=xm’ is given by (6.164) where f is the focal length of lens L].Thus, in Q there exists a light amplitude proportionto S(Qm) with a frequency ofo+Q,+Qm. Continuing this reasoning, wefind that the entire spectrum of e f ( t ) exp(iR,t) is displayed around point P . Not only are the amplitudes S(Qm) correctly represented by the light amplitudes, but there exists also a one-toone frequency correspondence. What is displayed around P is the actual physical spectrum of e’(t) exPGact), each frequency being superimposed upon the light frequencym.Thus, the phasor light field E(x, t ) along b may now be written as

j:

E(x, t ) = j exp(jQct) S(Q,)S(x -x,,)

exp(flmt)dx,,

(6.165)

’,x and Qm is as in (6.164). where the relation between Expression (6.165) is approximate because no account has been taken of the light spreading dueto the finite aperture sizeD. If this is done, the delta function in(6.165) must be replaced by a sinc function:

6(x -x&)+sinc

nD(x - XL)

f;l

(6.166)

Now assume that a reference field&,(X) at frequency o is also incident at plane b and that all the light is collectedby a photodetector, as indicated in Fig. 6.54, which shows one particular component at Qm being focused in plane b. It is readily seenthat the time-varying amplitude of the total phasor heterodyning current at Q, is given by ea

I(Qc, t ) = JS(Qm)E;(x’)S(x*-x&) exp(ji2,t) dx’dx&

(6.167)

which may be writtenas (6.168)

Selected Applications

Diffracted light field jexP(i(w+S1~+S1,)t)S(S1;Zm)

245

Diffracted light field

je(-x+Vst)exp[j(W+fiJt+jk~x]

Figure 6.54 Relation between frequency plane (left) and image plane (right) processing. (From Ref.48.)

where dxm’ a a,,, [see (6.163)] and (6.169)

Let now S‘(&) represent the frequency spectrumof l(Qc,

t), i.e. (6.170)

Comparing (6.168) and (6.170) we find S’(Qm)aS(&)Eh*(Qm)

(6.171)

Equation (6.171) sums up the essence of frequency plane processing: the reference fieldEb*(Qm) acts asan arbitrary phase and amplitude filter for the signal spectrumS(Qm). Figure 6.55 shows a practical system for frequency plane processing[49]. Although the light paths are somewhat complicated, the reader will have no difficulty recognizing the essential elements of Figs. 6.53 and 6.54. The reference field E h is generated by the box labelled “L.O. Optical System,” which operates on the diffracted light in a separate optical path. In a specific

Chapter 6

246

application, the optical system consists of a simple lens as shown in Fig. 6.56. This causes a curved reference field:

[

Eb = exp - j - ( x 2kR

-x,)

'1

(6.172)

where x , denotes the Xcoordinate ofthe focus formed by the lens, a distance R in front of the detector plane. The coordinate x is related to the temporal

Figure 6.55 A practical system for frequency plane processing. (From Ref.49.) Local Oscil

I

Figure 6.56 Details of the reference field used in Fig. 6.55. (From Ref. 49.)

247

Selected Applications

sound frequenciesfprojected on this plane by lens L1 with focal length 11 in the followingway: (6.173)

Substituting (6.173) into (6.172), we find (6.174)

Thus, the referencefieldisequivalent to asimple parabolic filter or dispersive delay line. The dispersion can be adjusted by changing R, i.e., varying the position of reference lens L2. Some results obtained with this device are shown in Fig. 6.57 in which pulse stretching (a measure of dispersion) is plotted vs. radius of curvature of the reference field.

“Theoretical 2

0

5

I

1.5

2

W

3

3.5

Experimental

4

4.5

5

Figure 6.57 Pulsebroadening vs. referencefieldradius Ref. 49.)

R

(cm)

of curvature.(From

248

Chapter 6

6.13.3 TimeIntegratingProcessing In an interesting variant upon the schemes outlined above, the illuminating light is modulated and the integration is carried out over time rather than space [50, 511. In the simplest example, the configuration of Fig. 6.50 is slightly modified by re-imaging the field immediately to the right of the mask (plane c) upon image plane d by means of lens Ls. This is shown in Fig. 6.58. In image plane d the image of c falls upon an integrating photodetector arrayA . Assume that the illuminating light amplitude is given by Ein(t), then with (6.151) the field E,+ is given by E m + cc jEin(t)e(X+ V.J)g(x)

exp(iln,t+jkpx)

(6.175)

Ignoring imageinversion by the lens, the field at the integrating photodetector is proportional to this. The local accumulated charge at the array q(x) may be written as (6.176)

\A

2f

Em+

Figure 6.58 Modification of the configuration of Fig. 6.50. The plane c is imaged upon plane d.

Selected

249

where T is the integration time of the device, and the mask has been removed [g(x)= l]. For conveniencewe shall write(6.176) as (6.177)

where u =xlV.. It is clear that (6.177) represents a correlation with the correlation shift given by -U. The values are given by the charges on the diodes located at x=uV., and are read out of the diode array in a time-sequential fashion, by appropriate electronics. As follows from (6.177) the correlationis necessarily restricted to positive functions. However, negative functions are allowed by proper biasing. To that purpose we choose

le(t+u)I2= V22[1+s2(t)] where sl(t),s2(t)>

- 1. If it is

(6.179)

assumedthat (6.180)

0

then (6.177) may be written as T

q(u)aIs,(t)s*(t+u)dt+biasterm

(6.181)

0

which indicates atrue correlation of real functionsover a finite time interval. of It is also possibleto perform Fourier transforms with the configuration Fig. 6.58. To achieve that, we choose s2 as a chirp and SI as the same chirp modulated by the signals(t) to be analyzed: s1(t)=s(t)

coss(uhnt+at2)

s2(t)=cos(uhnt+at2) substituting (6.182) and (6.183) into (6.181), we find

(6.182) (6.183)

Chapter 6

250 T

q(u) Is(r)cos(2w,r 0

+ w,u + 2ar2 +au2 +2atu) dt (6.184)

i

+

0

s(r)cos(2aru + w,u

+au2)dt + bias

term

NOWwe choose ~ @ 1 2 ~ ~ t l mand a x ,make 12autlmaxof the order of the bandwidth of s(r). In that case the first term in(6.184) integrates out to zero and q(u) may be written T

exp(-jw,u-

I

juu2)fs(r)exp(-j2atu) dr 0

(6.185)

We recognize in the integral the finite Fourier transform S(0=2au) of s(t). Let us write exppfl2ua)l

S(2ua)=IS(2ua)l

(6.186)

then (6.184) becomes

Equation (6.187) indicates that a chirp-type fringe pattern cos[wmu+au2 -@(2au)] develops across the photodiode array with the fringe amplitude modulated by IS(2au)l and its phase by 4(2au). With appropriate electronic processing both amplitude and phase can be retrieved; hence, in this mode of operation atrue Fourier transform can be obtained. It is noteworthythat the frequency resolution6f= UT, i.e., it depends on T rather than on the transit time z of the sound through the light beam, as is the case with space integrating configurations. The frequency range isbyset Af=2~1ul=2a(D/V,)=2ag where D is the aperture of the sound cell. The chirp rate mustbelimited so thatthe maximum range 2aT of the instantaneous frequency doesnot exceed the bandwidth B of the soundcell: 2aTSB. Putting everything together we iind that N=AflS f S B g where N is the number of resolvable frequencies. This number is still the same as in the space integrating configuration;however, in the time integrating method, a trade-off between Sf and Af is possible by varying a and T. So far, the Y dimension has not been used in any of the signal processors discussed. A simple example of its use is the extension of the Fourier transformer just discussed to parallel operation [52, 531. Figure 6.59 shows such an arrangement taken fromRef. 53.

25 1

Selected Applications 2-0 TIME INTEGRATWG DETECTOR

LED

- 0

PI

LI

L2

L3

L4

p3

h(0

TOP VIEW

SIDE VIEW

Figure 6.59 Multichannel time-integratingspectrum analyzer. (From Ref. 53.)

In the horizontal plane (X)the operationis much as described above, with s(t)=sn(t)modulating the chirp of the incident light. The subscript n refers

to an array of independent, light-emitting diodes (LEDs) arranged in the Y direction. As shown in the top view, lens L2 collimates all light sources horizontally to wide beams for illumination of the sound cell in plane P2. Lenses L2 and L3 image the sound cell horizontally upon plane P3, which contains a two-dimensional arrayof time integrating detectors. The side view shows how in the vertical direction ( Y ) ,lens L1 collimates the individual light source to narrow beams passing through the small height of the sound cell. LensL4 re-images theLEDs in the vertical direction upon the arrayof detectors. It will be clear that this configuration performs a Fourier transform operation on a numberof signals sn(t)in parallel. Instead of using the Y axis for parallel operation, a second sound cell may be aligned along it, as shown in Fig. 6.60, also taken from Ref. 53. Careful

252

Chapter 6

LI

L2

PO

PI

p2

p3

Figure 6.60 Time-integrating triple product processor. (From Ref. 53.)

analysis will show that in this configuration the signal a(t) from the light source LED will be multiplied by b(t-xlV,)=b(t-ut) of the horizontal sound cell and c(t-ylVs)=c(t-u2) of the vertical sound cell. (No bias voltages are shown in the drawing,but it is assumed they have been applied appropriately.) The integrated chargeq(u1,u2) may be written as T

0

A device characterized by (6.188) is called a triple product processor and may be used for many different purposes [54]. In one interesting application [53] the signals a, b, and c are chosen as follows: a(t)=s(t) cos(oht+at2)

(6.189)

Selected

253

b( t) =S( t )

(6.190)

c(t)=cos(aht+at2)

(6.191)

By doing the same kind of analysis as we did on (6. 18l), and making suitable assumptions aboutah, we find q(u) a cos[ahu+au2-O(2au)]lF(2au)(

(6.192)

where IF1 and 8 are the magnitude and phase of the so-called ambiguity function (6.193) Notice that F displays correlations along U I (Xdirection) and frequencies along 242 ( Y direction). It is also possible to display coarse frequency resolution along UI and fine frequency resolution alongut. For this the signalsb and c in (6.188) have to be a fast chirpand a slow chirp, respectively [55]. For more detailed information about acousto-optic processors for time-frequency representations, see Ref 56. Bi-spectral processing is treated in Ref. 57, and a processor for synthetic aperture radar is discussed in Ref. 58.

REFERENCES 1. Gottlieb, M., Ireland, C. L. M., and Ley, J. M. Electro-optic and Acousto-Optic Scanning and Deflection,Marcel Dekker, New York (1983). 2. Korpel, A., ZEEE Spectrum, 5,45 (1968). 3. Gordon, E. I., Proc ZEEE, 54, 1391 (1966). 4. Chu, R.S., Kong, J. A., and Tamir, T., J; Opt. SOCAm., 67, 1555 (1977). 5. Chu, R.S. and Kong, J. A., .lOpt. SOCAm., 70, 1 (1980). 6. Magdich, L.N. and Molchanov, V.Y., .lOpt. Spectrosc., 42,299 (1977). 7 . Korpel, A., Adler, R.,and Desmares, P., Paper 11.5, International Electron Devices Meeting, Washington, D.C.(1965). 8. Korpel, A., U.S. Patent 3,424,906, Dec. 30, 1965. 9. Korpel, A. Adler, R.,Desmares, P., and Watson, W. App. Opt., 5, 1667 (1966). 10. Goutzoulis, A. P. and Pape, D. R.,eds. Desing and Fabrication of Acousto-optic Devices, Marcel Dekker, New York(1994). 11. Korpel, A., Adler, R.,Desmares, P., and Watson, W., ZEEE .l Quantum Electron., QE-I, 60 (1965).

254

Chapter 6

12. Korpel, A., and Whitman, R.L., Appl. Opt., 8, 1577 (1969). 13. Goodman, J. W., Introduction to Fourier Optics, McGraw-Hill, New York (1968). 14. Gradhsteyn, I. S. and Ryzhik, 1. M., Table of Integrals, Series and Products, Academic Press, New York(1965). 15. Korpel, A., Laub, L. J.,and Sievering, H.C., App. Phys. Lett., 10,295 (1967). 16. Korpel, A. and Whitman, R.L., .lOpt. SOCAm., 8,1577 (1969). 17. Whitman, R. L. Laub, L. J.,and Bates, W. J., ZEEETrans. Sonics Ultrasonics, SU-15, 186 (1968). 18. Laub, L. J., “Apparatus and methods for scanning phase profilometry,” U.S. Patent 3,796,495,March 12, 1974. 19. Laub, L. J., Paper ThB16,Meeting Opt. Soc. Am., New York, Spring 1972. 20. Hecht, D. L., IEEE Trans., SU-24, 7 (1977). 21. Korpel, A. App. Phys. Lett., 9,425 (1966). 22. Korpel, A. ZEEE Trans., SU-15, 153 (1968). 23. Korpel, A. “Acousto-Optics,” in Applied Solid State Science, Vol. 3 (R. Wolfe, ed.), Academic Press, New York(1972). 24. Korpel, A. “Eikonal Theory of Bragg Diffraction Imaging,” in Acoustical Holography, Vol. 2 (A. F. Metherell and L. Larmore, eds), Plenum, New York (1970). 25. Korpel, A. and Young, A., Acta Polytechnica Scandinavica, Applied Phys., 150, 221 (1985). 26. Szengessi, 0.I., Proc ZEEE, 60, 1461 (1972). 27. Korpel, A. Proc SPIE, 232,90 (1980). 28. Korpel, A. Znt. 1 Nondest. Test., I , 337 (1970). 29. Korpel, A. 1 Ac. SOC Am., 49, 1059 (1971). 30. Smith, R. A., Wade, G., Powers, J.,and Landrey, C. J., 1 Ac SOCAm., 49,1062 (1971). 31. Korpel, A. “OpticalImaging of UltrasonicFields by Acoustic Bragg Diffraction,” Ph.D Thesis, Universityof Delft, The Netherlands (1969). 32. Korpel, A. and Mehrl, D. J., App. Opt., 28,43534359(1989). 33. Korpel, A. and Mehrl, D.J., Proc 1988 ZEEE UltrasonicsSymposium, pp. 735-737 (1988). 34. Mehrl, D.J. Liu, Z. C., and Korpel, A., App. Opt., 32 5112-5118 (1993). 35. Haykin, S. Communication Systems, Wiley, New York (1978). 36. Korpel, A., Kessler, L. W., and Ahmed, M., .lAc Soc. Am., 51, 1582 (1972). 37. Toepler, A. Poggendofl7s Ann., 131,33,180 (1867). 38. Chen, Y. M., Ph.D Thesis, University of Iowa (1994). 39. Mehrl, D. Korpel, A., and Bridge, W. App. Opt., 29,47664771(1990). 40. Korpel, A., Yu, T.T.,Snyder, H. S. and Chen, Y M., .l Opt. Soc Am., II(A), 2657-2663 (1994). 41. Durnin, J. 1 Opt. SOCAm., 4(A), 651 (1987). 42. Korpel, A., Mehrl, D., and Lin, H.H., Proc 1987IEEE Ultrasonics Symposium, pp. 515-518 (1987). 43. Acoustic Surface Waves (A. A. Oliner, ed.), Springer, New York(1978). 44. Ippen, E. P., Proc ZEEE, 55,248 (1967).

Selected Applications

255

45. Yariv, A, Optical Electronics,Third Ed., Holt, Rinehart andWinston, New York (1985). 46. Whitman, R. L. and Korpel, A. Appl. Opt., 8, 1567 (1969). 47. Berg, N. J. and Pellegrino, J. M., eds, Acousto-Optic Signal Processing, Marcel Dekker, Inc., New York, 1995. 48. Korpel, A. “Acoustic-optic Signal Processing”Optical Information Processing E. Nesterikhin and G. W. Stroke, eds.), Plenum Press, New York, p.171 (1976). 49. Whitman, R., Korpel, A., and Lotsoff, S., “Application of Acoustic Bragg Diffraction to OpticalProcessing Techniques,” Proc.Symp. Mod. Opt., Polytechnic Press, Brooklyn, New York, p. 243 (1967). 50. Sprague, R. and Koliopoulis C., App. Opt., IS, 89 (1976). 51. Sprague, R. A., Opt. Eng., 16,467 (1977). 52. Casasent, D. and Psaltis, D., App. Opt., 19,2034 (1980). 53. Psaltis, D. Proc. ZEEE, 72,962 (1984). 54. Lohmann, A. Proc. ZEEE, 72,989 (1984). 55. Lee, J. N., and Vanderlugt, A., Proc ZEEE, 77,1528 (1989). 56. Nathale, R. A., Lee, J. N., Robinson, E. L., and Szu, H. H., Opt. Lett., 8, 166 (1983). 57. Kauderer, M. H., Backer, M. F., and Powers, E. J., App. Opt., 28,627 (1989). 58. Psaltis, D., and Wagner, K., Opt. Eng, 21 ,822 (1982).

v.

This Page Intentionally Left Blank

7 Related Fields andMaterials

In this chapter we want to ultimately arrive at a more precise description of the intrinsic acousto-optic properties of liquids and solids than the simplified one we have used so far. Before doingthat, however, we shall give a brief review of acoustics and anisotropic optics in order to provide the necessary foundation.

7.1

ACOUSTICS

Although the word “acoustics” originally was synonymous with “audible acoustics,” the technical meaning has shifted graduallyto the point that the term now also connotes ultrasonic propagation in liquidsand solids in the frequency range from l to 4 GHz. In liquids, an acoustic wave consists of density variationsAplpo (called condensation) brought about by hydrostatic pressure [l]. There exists a simple relationship between excess pressure p and condensation S:

Sound waves in liquids are longitudinal,i.e., the particle displacement, say5, is in the direction of propagation, say X. The situation for an infinitesimal 257

258

Chapter 7

cube of liquid is shown in Fig. 7.l(a). Applying Newton's second law yields

where p has been approximated by po. Also, the fractional density change equals the fractional change in displacement:

Combining (7.1),

(7.2) and , (7.3)gives the wave equation

with travelling wavesolutions of the form

where the sound velocity Vis given by

Note from (7.1) that A, the bulk modulus, signifies stiffness, i.e., resistance against compression. A longitudinal wave in an isotropic solid or along certain specific directions in a crystal is characterized by equations similar to (7.1-7.6), with (7.1)written as

where, in comparison with the parameters pertaining to liquid, 0 1 1 = -p, 611=-s=d@Sx, and CII=A.The subscript l1 in 0 denotes a stress in the 1(X) direction acting on a plane perpendicularto the 1(X) axis. Similarly, 611 denotes a fractional extension (tensile strain) in the 1 direction of a line element along the same direction. The parameter c11 is called a stiffness constant. Alongother directions, the subscripts 22 and 33 are used. It will be clear that in an isotropic mediumcll=c22=cg3.

Related Fields and Materials

259

Figure 7.1 (a) Compression of elementary cube in longitudinal wave motion. (b) Distortion and sideways displacement of cube by shear wave.

Chapter 7

260

It is the simple kind of wave considered so far that we have used in our acousto-optic interactions. Beforewe discuss more complicated wave motions, it is of practical interestto derive some simple relations concerning power flow and particle velocity. With (7.3) the particle velocity u=dglSt may be written as U = - - jd SdX

dt so that with (7.5) U = U0 cos(Qt-cfi)

(7.9)

where

U, =sa"=s,v K

(7.10)

In atypical isotropic solid such as glass, the strain is limited to approximately so that, with V d x lo3 ds, particlevelocities are limited to -0.4 d s . The total kinetic and potential energy stored per unit volume in thewave is given by J W = O.5PoUo2 -

m3

(7.1 1)

and is related to the power flowor intensity I,( ~ / m * as)follows: W="1

V

(7.12)

[Equation (7.12) is most easily remembered by thinking of,awaveof intensity, I, filling up a cube of1 X 1X 1 meter with energyW during the time (1/V) it takes thewave to traverse the cube.] Combining(7.10-8.12) yields I , = 0.5povu;(7.13) = 0.5pov3s; The quantityro V is called the mechanical impedance of the medium. If(7.6) is substitutedinto (7.13), we find with (7.1) and writing V= V / V

26 1

Related Fieldsand Materials

OSP; I, =POV

(7.14)

or the equivalent expression for solids 0.5T; I, =POV

(7.15)

where POand TOdenote peak values of pressure and stress. Note that (7.13-7.1 5). are analogousto the electromagnetic case withpoV taking the place of the intrinsic impedanceq= U0 substituting forHo, and TOor POsubstituting forEO.From (7.13-7.15) it follows that

m,

TO(or Po)=po VU0

(7.16)

The analogy is very useful; it enables, for instance, the calculation of (perpendicular incidence) reflection coefficients between different mediaby using the corresponding electromagnetic relations. Equation (7.13) is of great importance in acousto-optic applications as SO is directly proportional to An and hence to the Raman-Nath parameter v=kAnL. We will discuss that in more detail later, but it will be obvious already from (7.13)that media withlow Vwill, in general, require less power to achieve a certain SO,and hence An. Such media are therefore more efficient from an acousto-optic pointof view. Comparing, for instance, glass and water, we find that typically Vglass~3Vwater,and also pglass~3.5&ater, SO that Is,glassZlOOIs,water for equal strain (condensation). Unfortunately, acoustic losses (which generally increase with frequency) are much higher in liquids than in solids, so that the improved efficiency can only be realizedat relatively low frequencies, e.g.,fc20-50 MHz for water. In a longitudinal (dilatational) wave, the elementary cube is made to change its volume only,not its essential shape as is evident from Fig. 7.l(a). In shear (distortional) waves, the cube changes its shape only. This is illustrated in Fig. 7.l(b), which shows a distorted and displaced cube. Note that the displacement q is perpendicular to the direction of propagation X. The active force actingon the element in theY direction is the result of shear forces 0 2 1 acting to the left and the right (7.17) Note that the subscript21 indicates a force acting in the Y) 2( direction on a

Chapter 7

262

plane perpendicular to the 1(X) direction. Note also that, with respect to Fig. 7.1, the usual condition of linear vibration holds, i.e., 012=021 [2]. The total distortion anglea+P=&=& is called the engineering shearstrain (in most nonengineering conventions, half the total angle is called the shear strain and denoted &I), and the following relation applies: 021=c66&1

(7.18)

where C66 (the notation will be clear later) is a rigidity constant.It is readily seen from Fig.7.l(b) that, for small anglesa and p, (7.19) It should be noted that the quantity a-& called the rotation, plays a negligible role in acoustical dynamics[2]. In what follows, we shall often denote XJXI, ~ 3 x 2 ZJX~, , g+, 7 7 3 5 2 , From (7.17-7.19) we find, realizingthat d2t/dydx=0,

c+&.

(7.20) leading to shear or transverse waves of the kind (7.21) where V,, the shear velocity, is given by (7.22) As for particle velocity and powerflow, relations analogous to (7.8-7.12) apply, whereas formal analogies to the electromagnetic field may be drawnin the sameway as before. As far as acousto-optics is concerned, it is important to note that longitudinal wave motion causes changes in the densityof dipoles and may cause changes in the orientation of dipoles, whereas shear waves only affect the orientation. In liquids, where hydrostatic pressure is the active agent, only density changes .are involved, apart from peculiar effects due to streaming of the liquid,to be discussed later.

Related Fields and Materials

263

It will be clear that in the general case of a crystal, the stress-strain relations are more complicated than (7.7) and (7.10). Internal oriented bonding forces will cause complex cross-coupling of motions such that the generalized form of Hooke’s law is given by [3]

(7.23)

An often usedconstant is Young’s modulus l? (7.25) which represents the stiffness of a cube free to contract laterally when extended longitudinally; it determines the velocity of propagation in a rod thin compared to the wavelength. Also frequently used is Poisson’s ratio v that is the ratio of lateral contraction to longitudinal extension in theabove example:

v=-

a + P)

With respect to the rather cumbersome notation of (7.23), it should be remarked that a more efficient abbreviated notation is also in use and is

264

Chapter 7

so that (7.23) becomes

where repeated indices are to be summed over. The cgs are generally called stiffness constants.The inverse relationto (7.28) is

where the sus are called compliance constants. An extensive list of elastic constants may be found in Ref.5. To analyze acoustic wave propagation in crystals, the matrix relations (7.28) have to be used rather than the simple scalar relations we have used before. This makes for a much more complex situation. For an arbitrary propagation direction in the crystal, there exist generally two quasi-shear waves (particle motion not quite transverse) and one quasi-longitudinal wave (particle motion not quite longitudinal) [2]. For specific directions along the crystal axes, the waves may be proper longitudinal and shear waves. In an isotropic solid, there exist in every direction one longitudinal wave and infinitely many shear waves(i.e., the transverse direction of particle motion or polarization is arbitrary). In liquids, only longitudinal waves can propagate. The three types of wave motion possible in an arbitrary direction in a crystal havemutually orthogonal displacements and different phase velocities. The latter dependon the direction of propagation.It is customary to show this graphically by the construction of a so-called slowness surface that essentially marks the endpoints of wave vectors. (As K=WV, a simple scaling I K / f l l = 1/V transforms the locusof wave vectors into such a slowness surface.) An example is given in Fig. 7.2 that shows the cross sections of three slowness surfaces with the cubeface of a GaAs crystal [2] (piezoelectric effects ignored). The three curves shown refer to the three

265

Related Fieldsand Materials PURE SHEAR,

QUASILONGITUDINAL

[1001

Figure 7.2 Slowness curves in cube face of GaAs.(Adapted from Ref. 2.)

possible modes discussed before, in this case two quasi-waves and one pure wave. It should be noted that the amount of diffraction (spreading) of a beam propagating in a particular medium is proportional to the curvatureof the slowness surface. Thus, the shear wave “beam” represented by point B in Fig. 7.2 exhibits the same amount of diffraction as in the isotropic case because it lies on a circular slowness surface. On the other hand, the quasilongitudinal wave at A exhibits less diffractionthan would correspond to an isotropic circle throughthat point. By the same token, the quasi-shearwave at C exhibits more diffraction. It is clear that such effects are of great importance in acousto-optic applications, as they may well determine the interaction efficiency and useful length of a soundcell. Another important effect of anisotropy isthat, in general, the direction of wave propagation isnot the sameas that of energy propagation.The latter is always perpendicular to the slowness surface, whereas the former is in the direction of the radius. Thus, in point D of Fig. 7.2, S, is the propagation direction (wave normal), whereas se designates the energy flow (Poynting

266

Chapter 7

vector). The severe “beam walk-off’ that this effect may cause is strikingly illustrated in Fig. 7.3 [6]that shows a quasi-longitudinal wave in quartz propagating at a slant from the generating transducer at left bottom. Note that the wavefronts arenot parallel to the direction (se) the beam as a whole is travelling in. In addition to the bulk waves discussed so far, there also exists a surface acoustic wave [7]. This is a mixed shear and longitudinal evanescent wave that clings to the surface of a solid.Its amplitude decays rapidly (i.e., within one wavelength) away from the surface, and it has generally the lowest velocities of allacoustic waves. Recently,it has become of importance alsoto acousto-optics [8], where it is used to interact with optical thin film guided waves in planar configurations. As in electromagnetics, it is possible to guide acousticwaves in plates, rods [9], and thin surface strips [8]. The analysis of such structures is, however, much more complicated than inelectromagnetics,because of the phenomenon of mode conversionat boundaries.

Figure 7.3 Quasi-longitudinal wave in quartz. (From Ref. 6.)

Related

7.2

and Materials

267

OPTICAL ANISOTROPY

As in acoustics, the crystal structure imposes severe constraints on the possible modes of propagation. Thus, optical propagation in an arbitrary direction is, in general, only possible for two plane waves with well-defined directions of B, H, E, and D. Although B=pH, because of the absence of magnetic anisotropiceffects, the relation betweenD and E is more involved: Dj=qiEi

i, j = 1,2, 3

(7.30)

where again summation is over repeated indices. The consequence of (7.30) is that, for the two possible plane waves referred to above, E and D are not in the same direction. It may be shown [lo] that the wave normal S, (i.e., the direction of the wave vector k) is perpendicular to the plane of D and H, whereas the direction se of the Poynting vector is perpendicular to the plane of E and H. The general situation is illustrated in the diagram of Fig. 7.4. The two possible wave motions in the direction S, do have orthogonal directions of D but not necessarily of E. Also, their phase velocities are, in general, different.As in acoustics, it is possible to construct “slownesssurfaces’’(calledindex surfaces in optics) consistingof the endpoints of wave vectors whose length depends on direction. At any point, the direction of the Poynting vector is perpendicular to the index surface. It will be obvious that such wave vector

Figure 7.4 Field vector configuration for anisotropic propagation.

268

Chapter 7

surfaces are of great importance in establishing Bragg conditions in acoustooptics. We will return to this later. There exists a convenient construction, due to Fresnel, for finding phase velocities and polarizations of D of the two waves possible in any given direction. It depends on the fact that, by choosing a coordinate system along the principal dielectric axes of the crystal, eq.(7.30) maybe expressed in the simple form D1

DF

IEI

(7.31a)

~22E2

(7.31b)

The construction referred to above now consists of the following. First, construct the ellipsoid (7.32) where n?I=&iII&, n22=~22/~0, and n33=~33/~o. The ellipsoid defined by (7.32) and shown in Fig. 7 4 a ) is called optical indicatrix, index ellipsoid, or ellipsoid of wave normals. Next, consider a wave propagating in the direction S, and construct a plane through 0 perpendicular to S, as shown in Fig. 7.5(b). The intersection of this plane with the index ellipsoid is the ellipse L. The directions of the two possible dielectric displacements D, and Db now coincide with the major and minor axes of L. The appropriate refractive indices for the two allowed plane waves are given by the lengths of the two semi-axis. Thus, wave “a” is characterized by na=a, and wave “6” by nb=b. Both k, and kb are, of course, in the direction of S., In general, there exist two directions spl and sPz for which the intersection L degenerates into a circle. These directions are called the optical axes of the crystal. A wave propagating in the direction of an optical axis is degenerate, i.e., its D polarization is arbitrary in the plane of L, with no effect on propagation velocity. In cubic crystals and amorphous media such as glass and plastic, rill =n22=n33, and the ellipsoid degenerates into a sphere. Each direction forms an optical axis, and the medium is called isotropic.It is media of this kind that we have considered so far. When two of the ellipsoid’s semi-axis are equal, say n11=n22=n0, n33=ne, the medium is called uniaxial. The optical axis is in the 3 direction, no is called the ordinary refractive index, ne the extraordinary refractive index. The corresponding waves are called

269

Related Fieldsand Materials

zI

a

Figure 7.5 (a)Indexellipsoid. characteristics in direction.,S

b

(b) Constructionforfindingpropagation

ordinary and extraordinary waves. If none, it is called negative uniaxial. Because, as we will see later, uniaxial crystals have rather unique acousto-optic properties, we will consider them in some more detail here. Figure 7.6(a) shows a positive uniaxial crystal with a wave propagating in the S, direction in the Y-Z plane, at an angle 8 relative to the optical axis. It is seen readily that of the two allowed waves, the ordinary one (a) is always subject to the same refractive index na=no. The extraordinary wave (b), however, sees a refractive index nb that depends on 8. Let us now plot the loci of na and nb when 8 varies from0 to 2z. These are shown in Fig. 7.6(b). As na is constant, it lies on the circlena(8)=no, while it is readily proved[l l] that nb(8) is defined bythe ellipse (7.33)

It will be clearthat curves no(@ and nb(e) are the cross sections with Y-Z the plane of the index surfaces discussed before. Multiplying n, and nb with ko

Chapter 7

270

IZ

a

b

Figure 7.6 (a) Wave propagation in direction S, in the YZ plane of positive uniaxial crystal. (b) Angular dependence of ordinary and extraordinary refractive index on angle of propagation.

results in two curves giving the loci ofwave the vectors k,and kb. As we will see later, such curves are of great convenience in the analysis of anisotropic acousto-optic diffraction.

7.3

ELASTO-OPTICS

So far we have assumedthat there exists a simple scalar relation between the refractive index variationAn and the sound amplitudeS: A?l=c'S

(7.34)

To the extent that S stands for the condensation in a liquid, thisis, in fact, quite true. In that case, C' may be found from the Lorentz-Lorenz relation v21

n2 -1 ANa n2 +2 =

where A is a constant, N the density of dipoles,

(7.35)

and a the molecular

ields

Related

and Materials

27 1

polarizability. Following Pinnow [13], we differentiate (7.35) with respect to N.Taking into account that N a p,we find (7.36) where (7.38) The condensation S with which we have characterized a longitudinal sound wave is related to the density changeby (7.39) with (7.34), (7.36), and (7.39), letting Ap+dp, h 4 n , we find (7.40)

It is clear that (7.40) represents both the effect of a change in density of dipoles as well as the change in molecular polarizability due to compression as expressed by the factor h.In liquids the latter effect is small,and hence C' is approximately given by (7.40) withh = O . For water (n= 1.33), we find C'=O.31; for mostother liquids, the value is closeto this. In isotropic solids, the factor Ilo is not always negligible; more important, however, is the fact that in such solids, AO ispolarization-sensitive: the mediumbecomes birefringent under stress, This is, of course, plausible as the dipoles tend to line up parallel to the strain. In crystals, the situation is far more complicated. In the first place, the dipoles tend to move along constrained directions whenever a strain is applied. This means that in regard to the factor A, the polarization sensitivity is a complicated (although linear) function of the various strains, and the latter themselves, according to Sec. 7.1, depend on the acoustic propagation direction in a nontrivial way. Also, pure shear waves do not cause any density variation at all; any effect they exert must come about through a direct (anisotropic) change in the polarizability, In view of the above, it is clear that the simple relation (7.36) is no longer sufficient in the general case.It must be replacedby a tensor relationthat expresses the effect

Cha#ter 7

272

of dilatational or distortional strain in any directiori on the propagation of light of any polarization in any other direction. The vehicle for doing so is the strain-induced deformation of the index ellipsoid discussed in Sec. 7.2. In its most general form, this ellipsoid may be writtenas (7.41) where

XI=X,

m=y, x3=z,

n l = h l l , n2=n22, n3=n33, n4=n23,

ns=n31, and

n6=n12.

As we have seen before,for the unperturbed ellipsoid (7.32) when referred to the principal dielectricaxes, the coefficients, n4, n5, and n6 vanish. However, upon the application of strain, they may acquire a finite value. By the same token, the coefficientsnl, n2, and n3 may change in value. We may thus say that strains both distort the ellipsoid and rotate it in space. All this may be succinctly expressed bythe relations (7.42) where the repeated index convention applies,and the S,s are the abbreviated strains as defined in Sec.7.1. The material constantsp are called strain-optic coefficients. The number of independent p s varies from 36 in a triclinic crystal to three or four in a cubic crystal. Inan isotropic solid, only twop s are neededto describe all phenomena:

(7.43) with allother coefficients zero. In actual fact, not even(7.42)issufficientlygeneral to describe all phenomena. Nelsonand Lax [l 5,161 have shownthat not only shear but also ( l / n 2 ) if the mediumis optically local rotation maycausechangesin birefringent. (With respect to Fig. 7.l(b), a shear is defined as a+P, a rotation as a-P, the latter signifying a rotation of the centerpoint of the distorted cube.) The necessary additional parameters are directly related to the coefficientsn1-n6 [15,17]. It is of interest to consider an application of (7.42). Let us assume that in an amorphous medium, a longitudinal sound wave is propagating in the+X direction [Fig.7.7(a)], so that the appropriate strain is denoted by r?~~ldxl=Sl. Let the light propagate nominally in the + Z direction and be

273

Related Fields and Materials

a

Figure 7.7 (a) A longitudinal wavepropagating propagating alongX with particle motionin Y.

along X. (b) Shearwave

polarized along theX axis. It is evident that we have to consider changesin i.e.,pll is the appropriate coefficient. We find with (7.42) nl due to SI,

If the light is polarized in the Y direction, the relevant change is in the appropriate coefficient p21:

n2

and

274

Chapter 7

An2

=- O S ~ : ~ , , S ,

(7.45)

Evidently, the acousto-optic scattering coefficient will depend on the optical polarization. This makespossible an interesting effect.If the light is polarized in the direction at 45" to the X axis, i.e., X in Fig. 7.7(a), it is readily seen that the effective scattering is determinedby

However, if p1lfp21, light will now also be scattered into the orthogonal polarization Y' with an effective strain-optic coefficient [l71

Both (7.46) and (7.47) may be derived easily by considering polarization components along Y and 2. An example of pure orthogonal scattering is shown in Fig. 7.7(b). Here, a shear wave propagates in the +X direction with particle motion along Y (i.e., d&/dXl+a&dX2=S6). As the light propagates in +Z and hence is polarized in theX-Y plane, the affected coefficients of the optical indicatrix are n1, nz, and n6 with corresponding strain optic coefficients p16, p26, and p a . According to (7.43), the first twop s vanish in an amorphoussolid. The coefficient p6 creates a term 2 A ( l / n 6 ) x 1 x 2 = 2 p ~ S a x ~inx the ~ index ellipsoid of (7.41). It may be shown that this changes the indicatrix from a sphere (amorphous solid) to an ellipsoid with axes along X ' , Y', and 2 [l 13. The change in lengthof the initially equalX and Y' semi-axis is given by

If now the light is polarized in either the X or Y direction, the scattered light with polarization in the same direction vanishes because of the opposite signs of A n x * and Any,. Light is, however, scattered with polarization in the orthogonal direction. Its amplitude is determinedby an effective strain-optic coefficientp66=0.5(pll-p12). Note that a shear wave with particle motion in the Z direction (i.e., d&/dx1+d&/dx3=S5) wouldaffect n l , n2, and n6 through p15, p25, and p65), all of which are zero according to (7.43). Such a shear wave would thus not cause any scattering. This is somewhat plausible when it is realized that in that case, the distorted rectanglesof Fig. 7.l(b) lie in the X-2 plane and not in the plane of optical polarization.

Related

275

and Materials

From the examples above, it will be clear that the analysis of the general interaction case can be quite complicated, with the effective strain-optic coefficient usually a linear combination of the ones defined by (7.42) [18]. Tabulated values of the latter coefficients may be found in many review articles and application-oriented books [14,17-201. For actual applications, it is convenient to use a figure of merit that Pd indicates the relative efficiency of a material in terms of diffracted power per unit sound intensityI,. For a weak interaction,we have pda

(&I2

(7.50)

and, from (7.44) and (7.49, ignoring tensor aspects, n6p2SZ

(7.51)

With (7.13) we find (7.52) where M2 is a commonly used figure of merit [21] that lends itself to easy experimental evaluation [21,22]. Other figures of merit, M I and M3, are more specifically relatedto special devices such as light deflectors [18,23]. In this book, wehave throughout used simple constants C' (3.6), (3.10) and C (3.76) to denote the relation between An and S. It will be seen with (7.44) that

C = -0.5n3p

(7.53)

C= -n2p

(7.54)

where p is the one opto-elastic constant characterizing liquids. Some liquids, however, exhibit dynamic birefringence under the action of a sound field [24].Wellabove the relaxation frequency of the liquid, it acts as an amorphous solid in the sense that it exhibitstwo coefficientspi 1 and p21 such that 2&

PI1 - P21 =n37

(7.56)

where 'tis the relaxation timeand Ssthe so-called Sadronconstant [25].

Chapter 7

276

REFERENCES 1. Thurston, R. N., “Wave Propagation in Fluids and Normal Solids,” in Physical Acoustics, Vol. IA (W. F! Mason, ed.), Academic Press, New York (1964). 2. Auld, B. A., Acoustic Fields and Waves in Solids,Wiley, New York (1973). 3. Kolsky, H.,Stress Waves in Solids, Dover, New York (1963). 4. “Standards on Piezoelectric Crystals,” Proc IRE, 3 7 1391 (1949). 5. Randolf-BornsteinTables, NewSeries(Hellwege, K. H. and Hellwege,A. M., eda), Springer Verlag, New York(1979). 6. Staudt, J. H. and Cook, B. D., J: Acoust Soc. Am., 41: 1547 (1967). 7. Acoustic Surface Waves(Oliner, A. A. ed.), Springer Verlag, New York (1978). 8. Tsai, C. S., ZEEE Trans., CAS-26 1072 (1979). 9. Meeker, T. R. and Meitzler, A. H., “Guided Wave Propagation in Elongated 10. 11. 12. 13. 14.

Cylinderand Plates,” in PhysicalAcoustics, Vol. IA(Mason, W. P. ed.), Academic Press, New York(1964). Born, M. and Wolf, E., Principles of Optics, Pergamon, New York (1965). Yariv, A., Optical Electronics, Holt, Rinehartand Winston, New York (1985). Von Hippel, A. R., Dielectrics and Waves, Dover, New York (1956). Pinnow, D. A., ZEEEJ: Quant. Electron., QE-6:223 (1970). Nye, J. F., Physical Properties of Crystals, Oxford University Press, New York

(1 960). 15. Nelson, D. F. and Lax, M., Phys. Rev. Lett., 24: 379 (1970). 16. Nelson, D. F. and Lax, M., Phys. Rev. B, 3: 2778 (1971). 17. Korpel, A., “Acousto-Optics,” in Applied Solid State Science (Wolfe, R. ed.), Academic, New York (1972). 18. Dixon, R. W., J: Appl. Phys., 38 5149 (1967). 19. Musikant, S., Optical Materials, Marcel Dekker, New York (1985). 20. Gottlieb, M.,Ireland, C. L. M. and Ley, J. M., Electro-Optic and Acousto-Optic Scanning and Deflection,Marcel Dekker, New York(1983). 21. Smith, T. M. and Korpel, A., ZEEEJ: Quant. Electron., QE-1: 283 (1965). 22. Dixon, R. W. and Cohen, G. M.,Appl. Phys. Lett., 8: 205 (1966). 23. Gordon, E. I., Proc ZEEE, 5 4 1391 (1966). 24. Riley, W. A. and Klein, W. R., . l Acoust. Soc Am., 4 5 578 (1969). 25. Jerrard, H.C., Ultrasonics, 2: 74 (1964).

8 Special Topics

In this chapter we willdiscussbrieflysome aspects of acousto-optic diffraction that fall somewhat outside the scope of this book, but that are of relevance to device applications or are otherwise of intrinsic interest. The chapter also contains a complete three-dimensionalweak interaction formalism using the plane-wave spectraof arbitrary fields.

8.1 ANISOTROPICBRAGGDIFFRACTION In Sec. 7.3 it was pointed out that a polarization change in scattering is likely to occur upon interaction with a shear wave. If the medium is optically anisotropic, then the incident and scattered k vectors may be of unequal lengths. This changes the condition for Bragg diffraction in some essential aspects, which we will now investigate. Let the interaction mediumbe a positive uniaxial crystal with the optical indicatrix oriented as shown in Fig. 7.6(a). We assume that a pure shear wave with polarization along Y propagates in the direction of the optical (2) axis. The shear wave causes orthogonal scattering of an extraordinary incident wave, propagating in the Y-Z plane, intoan ordinary wave propagating in the same plane.It will be clear that the wave vector triangle must now be constructed subject to the constraints imposed by Fig. 7.6(b). 277

278

Chapter 8

Such a construction is shown in Fig. 8.l(a). Note that for a given k,two kls are possible:kla mediated by Id, and k l b mediated by b.When $i increases, Ilr, will decrease, indicating Bragg angle behavior for frequencies down to zero. In this limit, ki and kla are parallel and opposite to Id. Another interesting aspect of anisotropic diffraction is that multiple forward scattering, and therefore Raman-Nath diffraction, is, in general, not possible. In Fig. 8.l(a), for instance, there does not exist, in general, a second wave vector Kb, equal in length to K, and directed upward, that would rescatter the kl, on the circle into k'l, on the ellipse. It is also puzzling at first glance that kl, represents upshifted light (positive Doppler

a

b

Figure 8.1 Anisotropicinteractioninpositiveuniaxialcrystal.(a)Shearwave along opticalQ axis, light wave inX- Y plane. (b) All waves inX- Y plane.

Special Topics

279

shift), whereas it is clearly directed downward with respect to the sound (negative diffraction angle). The paradox is resolved when we realize that the sound “sees” the incoming lightki as upshifted by a larger amount than the subsequent downshift. In other words, the total diffraction angle is positive. From all of this, it appears that new and unexpected effects are possible in anisotropic diffraction. A popular confimration is one in which, with referenceto Fig. 7.6(a), the sound and light waves all propagate in theX-Y plane perpendicular to the optic axis. It is evident in this case that the two appropriate refractive indices, no and ne, do not depend on the direction of light propagation. A diagram analogous to Fig. 8.1(a) is shown in Fig.8.l(b). It is clear that

and it hasbeen shown by Dixon[l] that

Note that if ne=no, eqs. (8.2) and (8.3) revert to the familiar ones for isotropic Bragg diffraction. An interesting configuration is shown in Fig. 8.2(a). Here, the interaction is collinear (Bragg angle equals 90°), yet occurs at a relatively low frequency

m i c a l values for F’in are of the order of 10-100 MHz [l]. The maximum frequency for which interaction can take place applies to the collinear scattering configuration of Fig. 8..2(b).

Finally, Fig. 8.3(a) shows a scattering configuration that is particularly suited for wide bandwidth beam steering [2,3]. The center frequency,

280

Chapter 8

a

b

Figure 8.2 Collinear interaction in uniaxial crystal. (a) frequency.

Low frequency, (b) high

corresponding to K,,is given by

The wide bandwidth in deflector applications is due to the fact that K, is tangential to the inner circle. Hence, to a first order, its direction does not change when Q deviates from Q,. For a given width of the angular planewave spectrum of the sound, the applied frequency may varyby a relatively

28 1

Special Topics

a

b

Figure 8.3 Tangential interaction in uniaxial crystal. (a) Basic configuration at center frequency,(b) explanation of wide bandwidth, and (c) second-order operation.

282

Chapter 8

C

Figure 8.3 (Continued).

large amount before no more acoustic plane waves are available for interaction. This may be seen fromthe construction in Fig.8.3(b), similar to the one used before in isotropic Bragg diffraction (Fig. 6.6).As in Sec. 6.3 [eq. (6.131, we assume that the bandwidth is determined by the condition L&= k2z.From Fig. 8.3(b) if follows that

giving, with criterion(6.13,

-=(!E) B

0.5

K For the relatively large Q to which (8.8)applies, we see that the bandwidth has indeed improved over that of an isotropic deflector(6.16). So far, we have, in our discussion of anisotropic deflection, dealt with linear polarization only. In certain crystals, the birefringent aspects of propagation and acousto-optic diffraction apply to clockwise and counterclockwise polarization components rather than orthogonal linear polarization components. This is called optical activity. A case in point is Te02 (paratellurite)that is used extensivelyfor low-frequency “birefringent”

283

Special Topics

beam deflection [4]. An interesting application of Te02 is described by Chang and Hecht [5]. To increase resolution in deflection, they use the second-order mode of Bragg diffraction, shown in Fig. 8.3(c). The incident light is scattered into kl byK1, and subsequently rescattered by K2 (=K1) into k2. It is of interest to analyze this with the tools developed in the preceding chapters. It is obvious that we are dealing with a case of two simultaneous pure Bragg interactions. With a combination of (3.101), (3.102) and (3.105), (3.106);we find dE2

- -0.25jkCSE,

"

dz

(8.9)

dE1 = -0.25jkCS * E2 - 0.25jkCSE,

(8.10)

dEo - -0.25jkCS * E, dz

(8.11)

dz

"

Note that eqs. (8.9) and (8.11) are conventional coupling equations and quite analogous to (3.101) and (3.102). Equation (8.10) is a combination of both the latter equations and describes simultaneous contributions to E1 from E2 and Eo. The solutionsto (8.9-8.1 1)may be written, with the proper boundary conditions,

Eo= EiCOS'

z)

--

[2k

(8.12)

(8.13)

(8.14)

where it is assumed that the interaction lengthis limited to z=L. It is of interest to compare these expressions with (3.103), (3.104) for firstorder Bragg diffraction. We notice that 100% diffraction is still possible; the required value of v ( v = d 2) is, however, larger by a factor d 2 than the value needed ( v = z ) for first-order Bragg diffraction. Also note that the

284

Chapter 8

maximum of lE# equals 0.5lEf at v=dg 2). The other half of the powerat that point is equally divided between the zeroth and the first order.A plot of this interesting behavior is shown in Fig. 8.4.As for the bandwidth of the second-order deflector, the same construction may be appliedto Fig. 8.3(c) as used in Fig. 8.3(b).It is seen readilythat the bandwidth is governed by the same considerations as in a first-order deflector if the frequency sensitivity of kl is ignored.We find

a factor of two smaller than the first-order case(6.16). Thus, the benefits of larger deflection angle are nullified by the decrease in bandwidth, and the total number of resolvable points remains the same. If in the configurationof Fig. 8.3(c), ki and kl are interchanged, then two orders, kz (upshifted) and kl (downshifted), are generated simultaneously due to the degeneracyof the Bragg angle condition. A theoretical analysisof this case, along the lines of (8.9-8.11), has been given by Warner and coworkers [4]. In light of the above examples, it will be evident that a comprehensive theory of anisotropic light diffractionwould probably betoo unwieldy to be of much use. Nevertheless,it is of interestto see how sucha theory could be constructed from basic notions of induced cross-polarization.An attempt at such a formulation has been made by Cherykin and Cherkov [q,to whom the reader is referred for further details.

1.o 0.8

0.6 0.4

0.2 0.0 Figure 8.4 Interaction behavior in anisotropic second-order Brag diffraction.

Special

8.2

285

Topics

ACOUSTO-OPTICTUNABLE FILTERS

In acousto-optic tunable filters, use is made the of inherent selectivityof the diffraction process to electronically move the optical passband. Most filters of this kind use anisotropic interaction, whichis the reason they are discussed in this chapter. Before analyzing a typical anisotropic filter, let us consider two simple isotropic configurations. Perhaps the simplest wayof making a tunable optical filter or spectrometer is shown in Fig. 8.5. It depends for its operation on the change in direction of the deflected beam when the light wavelength changes, and functions asfollows. A parallel beam of quasi-monochromatic light, ray a in the figure, is incident at an appropriate angle on a low-Q Bragg cell operating at a frequency$ The width of the beam is limitedto D by the entrance pupilof the sound cell. The diffracted beamb is focused by a lens of focal length F on a pinhole in the focal plane, situated at XO. For the diffracted beamto fall on the pinhole the wavelength of the light must satisfy the relation (8.16) or

r+ L

4

I

Figure 8.5 Isotropic acousto-optic filer using a pinhole for selectivity.

Chapter 8

286

In deriving these equations, we ignore the refraction of the light when leaving the sound cell. By changing the frequency of the sound (and hence A), a desired optical wavelength can be made to pass through the pinhole. In other words, Fig. 8.5 represents an electronically tunable optical filter. The width M of the passband may be estimated as follows. Thespot size formed by the focused beam b is approximately FAJD. A change M moves the center of thespot by F A ( ~ ~ B ) = F A NThus, A. the passband is determined by the condition FAA --FA

"

A

D

or

AA=-AA D

(8.18)

The spectral resolutionR=AJAA is then given by R = -D A

(8.19)

In the device of fig. 8.5, the pinhole aperture is essentialto the operation. The selectivityof the soundcell itself plays no role, as the cell has a low Q. It is, however, possible to reverse this situation by removing the pinhole and relying on the Q of the sound cell instead. The maximum deviation AA is then determined by the condition that the change in the Bragg angle equal the angular widthof the sound spectrum: A A "_ 2A

A

-L

(8.20)

or (8.21)

A similar condition determines the acceptance angle v/ of both the above devices: A L

(8.22)

A more common configmatichi uses collinear anisotropic interaction, as shown in Fig. 8.6. The appropriate condition for interactionis given by eq. (8.4):

I

Special Topics

287

Figure 8.6 Collinear on-axis and off-axis upshifted operation.

h V

f =-

4

(8.23)

where An=ne “ n o The tolerance on the interaction consists off being indeterminate to the extent Af= l/- V/L, where z is the sound transit time and L the collinear interaction length. With(8.23)we find (8.24)

where

R=-L h &

(8.25)

The acceptance angle of the device may be estimated from Fig.8.6. This K) and shows upshifted interactionboth for the on-axis case (sound vector the off-axis case (sound vector K’,angle I,@. It is readily seenthat

288

Chapter 8

(8.26)

Hence, (8.27)

But 2w v 2w K,"Ka=AKa=--27rAf "-=v V L L

(8.28)

From (8.27)and (8.28) it follows that (8.29)

Because of the square root dependence, this device has a relatively wide acceptance angle. This is also intuitively obvious from Fig. 8.6, which shows that &' changes but slowly when vincreases. Figure 8.7 shows a typical anisotropic collinear filter, together with its tuning curve[A.More detailed informationmay be found inRefs. 8-1 1.

8.3

LARGEBRAGGANGLEINTERACTION

In the case of large Bragg angles, the direction of the incident light is closer to parallel than perpendicular to the sound. A typical interaction configuration for downshifted scattering is shown in Fig. 8.8(a), and for upshifted scattering in Fig. 8.8(b). In the analysis to follow we shall limit ourselves to downshifted scatteringat the exact Bragg angle. Following [12], we firstnote that the boundary conditions for this case are given by Eo=Ei

for x=O

(8.30a)

E-I=O

for x=L

(8.30b)

We start from the generalized two-dimensional Raman-Nath (4.33), leaving out all ordersother than 0 and -1:

equations

289

Special Topics REJECTED LIGHT

ACOUSTIC

ON

INCIDENT LIGHT

q-15

il

fSELEtTED L I y T

-c

POLARIZER

140s

2 z

x>W

130

-

120

-

110

-

100

-

90

-

I

I

I

I

I

I

I

I

1

80-

3

S

ANALYZER

IEZOELECTRIC

70

-

w 50-

40

-

3020

I 350

400

I

I

I

I

I

I

450

500

550

600

650

700

J

WAVELENGTH, nm

Figure 8.7 Collinear CaMo04 acousto-optic tunable filter. (From Ref. 7.)

290

Chapter 8

a

c

ound

L

V

Figure 8.8 ConfigurationforlargeBragganglescattering.(a)Downshifted interaction, (b) upshifted interaction. (Adapted from Ref.12.)

V:E~(p)+kZE,(p)+0.5kzCS(p)E_,(p)=0

(8.31)

(8.32)

In analogy to (4.34),we now assume &(p) and E-l(p) to be plane waves propagating the X direction with amplitudes Eo(x)and E-I(x). The sound wave is as given by (4.35).Assuming a rectangular sound column model and

29 1

Special Topics

slow variations of Eo(x) and E-l(x), i.e., ignoring second derivatives,we may readily derive the following coupled equations from (8.31) and (8.32): -jkCS -dEo ="dx 4 sin#B

(8.33)

E-l

dE-l - +jkCS * E O dx 4sin$,

(8.34)

"

With the boundary conditions (8.30a) and (8.30b), we find the following solution: Eo = Ei

cosh[kCISl(L -x) /4sin #B] cosh[kClSIL/4 sin#B]

jm

E-1= -

S*

Ei

sinh[kCISI(L-x)/4sin $B] cosh[kCISIL/4 sin# B ]

We note that the interaction behavior, shown in Fig. 8.9, is of an essentially different character than in the caseof small Bragg angles. There no is longer any periodic behavior, but rather a gradual transfer of energy from EO backward into E- 1. This transfer reaches lOO?! only for L +CO. Upshifted interaction [Fig. 8.8(b)] shows a similar behavior starting from x=O rather than x=L.

0.2 Q 4 1 ,\;

,

0.0

Figure 8.9 Interaction behavior of large Bragg angle scattering.

292

Chapter 8

8.4 ACOUSTO-OPTIC SOUND,AMPLIFICATION AND

GENERATION

In Chapter 3 we discussed the possibility of sound amplification and generation, implied in the quantum-mechanical interpretation of downshifted interaction, Energy conservation on a per quantum basis written as

appears to hint at both possibilities because one photon is released to the sound for every photon generated in the downshifted light. Overall energy conservation may be writtenas (8.38)

P,'+ P/I +P,'= 0

where the primed PSdenote net powers flowing into the interaction region. Let Nj(j=O, - 1,S) be the corresponding net numberof quanta per second flowing into this region at frequency mi. Then, j = O , -1,

P)=NjhWjy

S

(8.39)

where 0s denotes a. Now (8.37) implies that the N,s are equal as there exists a one-to-one quantum exchange. Thus, we may write

@o

@-I

@S

From (8.39) and (8.40), it then follows that (8.41)

(8.42)

Note that multiplying (8.41) by W 0, (8.42) by @-I, and subtracting the latter from the former gives(8.38). Equations (8.41) and (8.42) are derived here from quantum-mechanical considerations. They can be extendedto include more frequencies[131, such

Special Topics

293

as would, for instance, occur in Raman-Nath diffraction. Rather startling perhaps is the fact that (8.41) and (8.42) can be derived classically for a general system of nonlinear interactions through a nondissipative process. Details may be found in the classical papers of Manley and Rowe [14]. The equations (8.41) and (8.42) are aspecialcase ofwhat are called the Manley-Rowe relations. It is of interest to calculate the actual sound amplification for a practical configuration, sayalow-frequency (-40 MHz) Braggcellused for modulation. Such cells are characterized by diffraction efficiencies 7 on the order of 100% per 0.1W of sound power. Thus, (8.43) where CSis of the order of 10 W-1. With regard to (8.42), the amount of power U s delivered to the sound beam equals-PIs, and by the same token P - I equals -P'-l. Hence,

Combining (8.43) and (8.44), we find (8.45) In an experiment at 45 MHz by Korpel and co-workers [15], the following parameter values applied: Pi G lo3W, C,2 10, ws/w-l G APs/Ps2 The results of the experimenthavebeendiscussedin Chapter 2 and presented graphically in Fig. 2.10.An experiment by Chiao and co-workers [ l q has demonstrated amplificationof thermal sound (stimulated Brillouin scattering) in quartz at a frequency of 30 GHz and a Bragg angle of go", at which all beams are collinear. It is clear from (8.45) that for such high frequencies the generated sound power may be much larger; frequently, it is, in fact, the source of internal damage in crystals subject to intense laser pulses. The factthat the Manley-Rowe relations (8.41) and (8.42) may be derived classically makes us suspectthat there may be a classical explanation for the sound amplification discussed above. This is indeed the case. The force driving the sound field is wave a of radiation pressure causedby the moving interference pattern of incident and diffracted light [12, 17-19]. For large angle downshifted Bragg interaction [see Fig. 8.8(a)], it is found that [l21

294

Chapter 8

Is

=Is0

r(L-X) COS’ r L

COS’

(8.46)

(8.47) with 0.5

r

(:I,)

(8.48)

where I,o is the initial sound intensity, I, is the amplified sound intensity,and it has been assumedthat the incident light intensity hasnot been noticeably depleted. A plot of I - ] and I, as a function of x,for TL=0.7, is shown in Fig. 8.10. Note that the sound increasesin the “Xdirection, the scattered light in the -Xdirection. For rL=ld2, the sound and diffracted light intensities become infinite, and the devicenow functions as abackward wave oscillator. However, the assumption of constant pump intensity is violated at that point, and the simplified theory breaks down.It does, however, indicatethat a threshold effect exists in stimulated Brillouin scattering. As a h a 1 point of interest, it should be noticed that the Manley-Rowe relations are satisfied locally in the interaction described by (8.46) and (8.47). It may be shown readily that

2 ,

Figure 8.10 Interactionbehavior of amplifiedsoundandscatteredlightin optically pumped backward wave amplifier.

Special Topics

295

(8.49)

In isotropic substances, collinear sound generation is possible for high frequencies only. The 90" Bragg angle requires that K=2k or ws=(Vlc)w. Typical sound frequencies in solids for visible light pumping are of the order of 10'O Hz. By using anisotropic diffraction in crystals, as in Fig. 8.6, it is possible to generate sound at lower frequencies (-lo8 Hz). Details may be found in Refs. 20and 21.

8.5

THREE-DIMENSIONALINTERACTION

So far, we have limited our discussion by and large to two-dimensional interaction, although mostof the basic equations, e.g., (4.32), (4.41), (4.108), are formulated in three dimensions.It is, however, very difficultto work out the consequences of the full dimensional equations. The simple rectangular sound column, being such a convenient fiction in two dimensions, to fails be even that when a dimension is added. The plane-wave interaction formalism loses its nice one-to-one correspondence aspect; as remarked before, one incident plane wave may nowinteract with a sound of sound and light wave vectors, as shown in Fig. 8.11(a). Consequently, even for weak interaction, simple wave interaction equations such as(4.54) and (4.55) do not exist. As pointed out in Sec. 6.1, if it is really necessary to analyze three-dimensional configurations, the volume scattering integral(4.37) is used [21, 221. Yet this approach seems cumbersome, even more so when it is realized that, in principle, a single integration should suffice. This is so because the soundmodulated polarization over which the spatial integral extends is itself proportional to the product of sound and light fields. These fields satisfy Helmholtz's equation, and, hence, their N3 spatial values are completely determined by their N values in onearbitrary cross section, or, alternatively, their N values in their angular plane-wave spectrum. In short, the (F)* values specifying the combined fields in the volume integral exceed the necessaryminimum of (P)2by N. Thus, by using an appropriate formalism, it should be possible to reduce the triple integration to a single one. It is plausible to try and develop such a formalism by using angular plane-wave spectra. The fact that any particular scattered plane wave is contributed to by a one-dimensional distribution of sound and light waves [i.e., those with wave vectors on the cone of Fig. 8.1 l(a)] would presumably A theory along account for the one integration remaining in this formalism. these lines, but in a somewhat conceptual form, was developed by Korpel

296

Chapter 8

IZ

Y

b

Figure 8.11 (a) Cone of incident light vectors ki, interacting with cone of sound vectors K to generate scattered light vector kl. (b) Interpretation of corresponding mathematical formalism.

Special Topics

297

[l21 using the Hertz vector formulation of Sec. 4.1 1. The basic equations derive from (4.145),(4.163), and (4.166) with the weak scattering assumption that the scattered light e’ is very weak relative to the incident light ei. For time-harmonic quantities, we find in terms of the arbitrary spatial component phasorsrI+(r), rI-(r), S@),and E&).

n+(r)= &j

exp(-jkR) dz,,

(8.50)

(8.51)

+

where dz,,=dx’dy’dz’, and the subscripts and downshifted light. Next, all fields are decomposedinto plane waves:

-

refer to up- and

S(r) = JG,(K) exp(-JK.r) d o K

(8.52)

Ei(r)= JGi(k) exp(-/k-r) d o k

(8.53)

n+(r)= JG+(k)exp(-Jk.r)

do,

(8.54)

n-(r)= JG-(k) exp(-Jk.r)

dok

(8.55)

where doK and dok are infinitesimal areas on the half-spheres that form the loci of the endpoints of K and k. We recognize in the Gs the plane-wave spectra of the interactingfields. Analogously to the development in Sec.4.5, it can now be shown that far away from the interaction region (r-w) (8.56)

298

Chapter 8

Substituting (8.52) and (8.53) into (8.50) and (8.51), evaluating the integral for R+w, and comparing the result with (8.56) and (8.57), we find that

The mutual relationof the vectors k, kB, and Kg is illustrated in Fig.8.1 l(b). As expected, the final results involvean integration over a one-dimensional distribution of interacting wave vectors lying on a cone. Although (8.58) and (8.59) give a compact account of three-dimensional interaction, these equations are not immediately applicable to real-life situations. Recently, a more practical version was developedby Korpel and co-workers [23]. In this formalism, the angular plane-wave spectra E(@,$’) and S(y, 7‘) are defined as in (3.133) and (3.144) (with the assumption of weak interaction), extended to two dimensions. Paraxial propagation is assumed, the sound along X, the light along Z. The angles 4, Cp,’ y, y ‘ are shown in Fig. 8.12(a) and 8,12(b). Following a method similar to the one just discussed and retaining terms downto second order in 4, Cp’, y , y ’, the following interaction equations are obtained.

A geometric interpretation of (8.60) is shown in Fig. 8.13 for a specific direction of incident light. The angles $ and @’ in (8.60) are denoted by $1

Special

IY

299

Topics

X

IY

X

b

Figure 8.12 Propagationanglesandwavevector (From Ref. 23).

for (a) sound and (b) light.

300

Chapter 8

Figure 8.13 Interpretation of paraxial mathematical formalism. (From Ref 23.)

(8.62)

@:=($)Yt

(8.63)

Special Topics

301

From the argumentof S, it follows that y=-m,+”-y9’+(k)y’’/2 K

With (8.62), (8.63), eq. (8.64) may also be written as

F -@B

(8.65)

It will be seen from Fig. 8.13 that (8.63) defines the cone, with apex P,of scattered wave vectors, whereas (8.65) defines the cone, with apex 0, of sound vectors. It may be shown readily that for two-dimensional a situation characterized by (8.66)

Equations (8.60) and (8.61) revert to the two-dimensional interaction equations (3.160) and (3.161) derived earlier. In many cases, (8.60) and (8.61) may be written in a simpler form. If the angular spectra of sound and light are characterized by angular widths A@, A#, A x A y ’ ,then it may be shown that the (K/k)y‘*terms may be ignored if (8.68) A sufficient condition for ignoring the y ’ $ l term is (8.69)

while the term with(Klk)y may be ignored if (8.70)

where the notation(A@, Aj)min means the smaller ofA@iy Ayl

302

Chapter 8

If, as is often the case, (8.68-8.70) are all satisfied, then

k-l(@,@') = - 0 . 2 5 j k C ~ i ( $ + 2 @ B , @$ *'()- @~- m @B, -cD

y')

($1

(8.72)

A typical example of an interaction configuration that may be analyzed conveniently is the interaction of two Gaussian beams shown in Fig. 8.14. The sound beam with axis in the X-Y plane and waist L (2L between lle amplitude points) at ys causes diffraction of the incident light beam with axis X - 2 and waist w at the origin. The following expressions apply: S(0, x, y , z ) = So exp

1

(8.73)

Figure 8.14 Interaction of Gaussian sound and light beam. (From Ref. 23.)

Specia1 Topics

303

(8.74)

(8.75)

(8.76) L

It is readily shown that (8.68-8.70) apply if U’ <<

(8.77)

L

If (8.77) is assumed, the substitution of (8.74) and (8.76) into (8.60) yields

-k2w2($- 2@8- @i)2- k2w2qS2- K2L‘(-@, +@)2 4

1

(8.78)

It is readily seen that a maximum occurs for @i= - @B, @ = @ B as is to be expected. Upon changing ys, the amplitude of this wave, is then seen to trace out the y-dependence of the sound beam.

8.6

SPECTRAL FORMALISMS

In the usual definition of the plane-wave spectrum, the phase reference is located at the observation point on the nominal axis of propagation, say, the 2 axis. We will call such a spectrum with a shifting phase reference point a local angular spectrum. The components of this spectrum change in phase when the field propagates. This change expresses, of course, what is commonly called diffraction. In a nonhomogeneous or nonlinear medium, there will be additional phase and/or amplitude changes imparted to the components by material interaction or self-interaction. In acousto-optics configurations, such changes come about through interaction with a sound field. Two kinds of effects then should be kept in mind: diffraction effects and interaction effects. The local plane-wave spectrum takes both into account in a straightforward manner.

ussed

304

Chapter 8

The virtual plane-wave spectrum is the local plane-wave spectrum at z back-propagated to the origin through the unperturbed medium. By definition its value does not depend on z if no interaction mechanism is involved. If interaction of some kind takes place, then the virtual plane-wave spectrum changes with z, i.e., it depends on which particular local planewave spectrum is back-propagated. The advantage of using this kind of spectrum is that diffraction effects are already included by its very definition. The propagation equations then describe interaction effects only, and are generally easier to handle. Its application to acousto-optic interaction has been 3.3. f It is sometimes more convenientto consider the spatial profile of the field only, rather than including the spatial carrier. Such an approach makes it possible, for example,to devise algorithms for acousto-optic interaction that circumvent the necessity of processing the high resolution spatial carrier [24]. The spectrum of the profile is in all essential respects analogous to the spectrum of a modulating signal in communications [25]. Like the latter, it suffices for the analysis of signal processing, here to be understood as interaction and propagation.A discussion of this method-sometimes called the Fourier transform Approach-in the context of acousto-optics may be found in Refs.26 and 27. The spatial profile spectrum discussed above is a local spectrum in the same sense as the conventional angular plane-wave spectrum. Like the latter, it too may be back-propagated to the origin.It then becomes avirtualprofile spectrum, implicitly accounting for diffraction effects in the unperturbed medium, and explicitly representing interaction in the perturbed medium. A unified treatment of spectral formalisms, with relevance to acoustooptics has beengiveninRef. 28. In this section we shall follow that treatment closely.

8.6.1 The Local and the Virtual Plane-Wave Spectrum In Sec. 3.3.1 we defined the plane-wave spectrum for two dimensions in terms of the angle 4. For the present discussionwe will extend the definition to three dimensions and use the wave vector components kx and ky rather than the angles 4 and 4':

Special Topics

305

where (8.80)

for paraxial propagation. The spectrum so defined is called thevirtual spectrum, because, according to (8.79), the local field atz is obtained by propagating allthe plane waves in the spectrum to z as if there were no interaction. The virtual plane-wave spectrum is obtained by back-propagating the actual local plane-wave spectrum to the origin asif the medium were unperturbed:

(8.81)

where the local planewave spectrum is definedby

(8.82)

with the inverse

(8.83)

According to (8.81) and (8.82), the relation between the virtual spectrum and the local spectrum is givenby (8.84)

If no interaction is involved, the virtual plane-wave spectrum &kx, ky;z) 2. If there is interaction, then it depends on which particular plane-wave spectrum is being back-propagated, i.e.,it depends on z. The advantage of using this kind of spectrum that is diffraction effects are included by its very definition. This isnot the case with the local spectrum

is independent of

306

Chapter 8

A(kx,ky; z). Each of the plane waves in the spectrum propagates as exp( -jk,z) in the nominal propagation directionso that

The quantity H is called the propagator. It specifies the evolution of the spectrum in the absence of interaction. For the virtual spectrum the propagator equals unity in the absence of interaction. For paraxial propagationwhere lkxl,Ikylekwe may write (8.86)

8.6.2 The

Local and Virtual Spatial Profile Spectrum

It is sometimes more convenientto consider only the spatial profile and take the spatial carrier for granted. Such an approach makes it possible, for example, to devise algorithms for acousto-optics interactions that circumvent the necessity process the high frequency carrier [24]. This is implemented by writing

where E e is the spatial profile.In communication theory thisquantity would [25]. be called the complex envelope The local profile spectrumis defined as

with the inverse

Substituting (8.87) into (8.88), we fmd readily, after comparing with(8.84) z)z)=exp(ikz)A(kx, ky;ky; Ae(kx,

(8.90)

The propagationlaw for A , follows readily from that of A by substitution (8.90) into (8.85):

307

Special Topics

(8.91)

where

H,(k,,k,)=exp(jkz-jkz,z)=exp

( y;

j-+j-

7;)

(8.92)

for paraxial propagation. A virtual profile spectrum may now be defined by back-propagation:

-

A,&, k,,; z) = exp

(8.93)

Substituting (8.90) into (8.93) and then (8.84) into the result, we find Ae(kx, ky; z)=E(kx, ky; Z)

8.6.3

(8.94)

SpecialAcousto-OpticProfileSpectra

Here we are concerned with spectra of a particular order n that propagate in a particular direction k,. It then seems reasonable to define spatial profiles yen in the following way:

En(x,y, z)=

v&,

y, z) exp(-ik,xx-jk,yy-jk,zz)

(8.95)

By taking the Fourier transform of (8.93), we find

(8.96)

where YfW=9”’(yot)

(8.97)

We readily find the propagation law for Y e n from (8.95) by invoking the propagation law for A, [(8.91) with A , replaced by A,,]:

308

Chapter 8

where

in the paraxial approximation. The propagatorHenis sometimes called the transfer function for propagation [27].

A virtual acousto-optic profile spectrum can be defined in the usual way by back-propagating: qen

(kx, k y ;Z )

For convenience, the various quantities introduced, their mutual relations, and their propagators are shown Table in 8.1. Table 8.1 List of Various Quantities Describing Propagation, Their Definitions, Mutual Relations, and Propagating Factors [Adapted fromRef 28.1 Quantity

Propagator (transfer function)

Definition real physical field e=Re[E exp(jot)] g=E, exp(-j kz) E n = v m exp(-jk, r) A=9-'+[E] E=A exp(jk, z) Ae=Y-'[Et]=A exp(jkz) y=Y"[ven]

exp[-j(k-kxz/2k-kyz/2k)z] 1 exp~(kxz+kyz)z/2k] e~plj(k~~+2k,,~k, +kyz+2kn,ky)z/2k]

&A,

exp[-j(kxz +ky2)z/2k]=B qm=6(kx knXky kny;z) = Y e n exp[-j(kx2+2knXkx +ky2+2knyky)z/2k]

+

+

1 1

REFERENCES 1. Dixon, R. W., IEEE .lQuunt. Electr., QE-3: 85 (1967). 2. Lean, E. G. H., Quate, C. F. and Shaw, H. J. App. Phys. Let., IO: 48 (1967). 3. Collins, J. H., Lean, E. G. H. and Shaw, H. J. App. Phys. Lett., 11: 240 (1967). 4. Warner, A. W., White, D. L. and Bonner, W. A., .l Appl. Phys., 43: 4490 (1972).

Special Topics

309

5. Chang I. C. and Hecht, D. L.App. Phys. Lett., 27: 517 (1975). 6. Parygin, V. N. and Chirkov, L. E. Sov. 2 Quant. Electr., 5: 181 (1975). 7. Chang, I. C. Acousto-optic tenable feelers, inAcousto-optic Signal Processing (N. J. Berg and Y N. Lee, eds.) Marcel Dekker, W, 1983. 8. Harris, S. E. and Wallace, R. W. .lOpt. Soc Am. 5 9 774 (1969). 9. Xu Jieping and Stroud, R., Acousto-Optic Devices,Wiley, New York (1992). 10. Goutzoulis, A. P. and Pape, D. R. (eds.). Design and Fabrication of AcoustoOptic Devices, Marcel Dekker, New York (1989). 11. Magdich, L. N. and Molchanov, V. Ya. Acoustooptic Devices and Their Applications, Gordon & Breach, New York(1959). 12. Korpel, A. “Acousto-0ptics,”in Applied Solid State Science,Vol. 3 (R. Wolfe, ed.), Academic Press, New York,1972. 13. Weiss, M. T. Proc IRE, 4 5 113 (1957). 14. Manley, J. M. and Rowe, H. E. POCIRE, 44: 904 (1956). 15. Korpel, A., Adler, R. and Alpiner, B. App. Phys. Lett., 5: 86 (1964). 16. Chiao, R. Y, Townes, C. H. and Stoicheff, B. F! Phys. Rev. Lett., 12: 592 (1964). 17. Quate, C. F., Wilkinson, C. D.W. and Winslow, D. K. Proc IEEE, 53: 1604 (1965). 18. Kastler, A. C. R Acad. Sci. B, 259 4233 (1964); 260:77 (1965). 19. Caddbs, D. E. and Hansen, WW . . Lab. Phys. M. L.Rept. 1483 (1966). 20. Hsu, H. and Kavage, W. Phys. Lett., 15: 206 (1965). 21. Piltch, M. and Cassedy, E. S. Appl. Phys. Lett. 17: 87 (1970). 22. Gordon, E. I. Proc. IEEE, 54: 1391 (1966). 23. Korpel, A., Lin, H. H. and Mehrl, D. J. .lOpt. SOC Am.A., 4: 2260 (1987). 24. Venzke, C., Korpel, A. and Mehrl, D. App. Opt., 31: 656 (1992). 25. Haykin, S. Communication Systems, 3rd Ed., Wiley, New York (1994). 26. Tam, C. W. A Spatio-temporal Fourier Transform Approachto Acousto-optic Interactions, Ph.D. Thesis, Syracuse University(1991). 27. Banerjee, P. P. and Tarn, C. W. Acustica, 74 181 (1991). 28. Korpel, A., Banerjee, P. and Tam, C. W. Opt. Comm, 97: 250 (1993).

This Page Intentionally Left Blank

Appendix A Summary of Research and Design Formulas

GENERAL PARAMETERS (Fig. A . l ) K=21r -

A denotes soundwavelength.

k = 21r h

ildenotes light wavelength in medium.

Q =GL h

Klein and Cook parameter.

v=k,AnL

Raman-Nath parameter. k , denotes propagation constant in vacuum. See also after P.

An= -0.5nipS0

Sound-induced peak refractive index also after V.

no

index Refractive

P

Appropriate elasto-optic coefficient 7.3).(Sec.

s0=(2l~~" v-3y.5

Amplitude of condensation Ap/p or strain (Sec. 7.2).

A

change. See

of medium.

31 1

312

Appendix A

L

L

4

Figure A.l

Is

Sound intensity.

Po V

Density of medium. Sound velocity.

Figure of merit (Sec. 7.3).

P

lvl=kv(

Acoustic power radiated by transducer.

)

0.5M2PL

h

2k=X @BzK2A

Bragg angle. For large angles @B*sin

RAMAN-NATH DIFFRACTION (Fig. A.2) Conditions:

Q . 1 ,

Qv41

Diffraction angles:

+n=+o+2n+~ Diffracted orders: In=lEn12=I&In(v')12

@E,

313

Summary of Research and Design Formulas

Figure A.2

where v’ = v

sine(

WOL

A plot of ( A 3 with +0=0, is shown in Fig. 3.3. For a standing sound wave

Refs.: Sec. 3.1, eqs (3.13), (3.14), (3.45-3.47), (3.49). profiled sound column, see Sec. 3.35, eq. (3.173).

For diffraction by a

BRAGG DIFFRACTION (Fig. A.3) Conditions:

Q 1, Q/v ))l and A$ small enough so that only Bragg orders are generated. Diffracted orders:

3 14

Appendix A

-l

.

Figure A.3

Plots of (A.7) and (A.8) are shown in Fig. 3.9. For conventional Bragg diffraction, A@=O and

(A. 10)

Plots of (A.9) and (A.10) are shown in Fig. 3.8. Identical expressions apply for the - 1 order. Refs.: Sec. 3.2.3, eq. (3.124). For diffraction of a profiled light beam, see Sec. 3.3.4, eq. (3.168) and (3.169).

BRAGG MODULATOR (Fig. A.4) Relative (small) modulation index for weak interaction: (A.ll)

where Km=2dAmand A,,, is the modulation wavelength. An identical expression applies for - 1 order operation. Refs.: Sec. 6.5, eqs. (6.44), (6.51).

Summary of Research and Design Formulas

315

Figure A.4

BRAGG DEFLECTOR (Fig. A.5) Deflection angle:

(A. 12) ~ the Bragg angle at the center frequency Fe. where @ B is

Deflected intensity for weak interaction as a function of F:

(A.13) Refs.: Sec. 6.3, eqs. (6.12), (6.14-6.16). Bandwidth (one half-frequency difference between zeros):

(A. 14) where QCis the Q at center frequency. Identical expressions apply for the -1 order. For beam steering deflectors, see Eq.6.25. For anisotropic deflectors, see Sec. 8.1.

316

Appendix A

Figure A.5

WEAK INTERACTION OF ARBITRARY FIELDS IN TERMS OF PLANE WAVES Two-Dimensional Interaction (Fig.A.6)

81(g)=-O.25jkC~(-g+g~)BXg-2g~)

(A. 15)

8-1(g))=-o.25jkcS"*(-g-gB)Bxg+2gB)

(A. 16)

where C=-pn,2 and S" and B are angular plane-wave spectra with phase reference at the origin. An interpretation of (A.15) and (A.16) is shown in Fig. 3.14. Refs.: Sec. 3.3, eqs (3.138), (3.144), (3.160), and (3.161).

317

Summary of Research and Design Formulas

Three-Dimensional Interaction (Fig. A.7) &(@,@’)=-0.25jkCI& -@+$B

“ l

-Y@’+-Y’’,Y’ k

(A. 17)

d(y’lA)

L

For the definition of 4, (8.61).

(A. 18)

@‘, yand y‘, see Fig. 8.12. Refs.: Sec. 8.5, eqs. (8.60),

STRONG INTERACTION OF ARBITRARY FIELDS Two-Dimensional Case in Terms of Plane Waves

di

-

m,

-= -jaE,,-lSi-l- jaE,,+IS;+l

6%

”

Figure A.7

4

(A. 19)

Appendix A

318 where (see Fig. 4.5)

(A.20)

and a=0.25kC=-0.25kpn$ Refs.: Sec. 4.7. For explicit solution by path integrals, see Sec. 4.8.

Two-Dimensional in General Terms

V,2E,(p)+k2E,(p)+0.5k2CS(p)~~-~(p) +0.5k2CSr(p)E,+l(p)=0

(A.22)

Three-Dimensional in General Terms

v~,(r)+k~E,(r)+O.Sk~CS(r)E,-,(r) +O.5k2CP(r)En+l(r)=O

(A.23)

Refs.: Sec. 4.4, eqs. (4.32), (4.33).

EIKONAL THEORY Amplitude of diffracted ray at interaction point:

where the interaction pointS, is defined by the local ray triangle (Fig. 4.12).

kss+ksi=ksl

(A.24)

Summary of Research and Design Formulas

319

Also, (A.25)

(A.26) (A.27) (A.28) and A&)= -?if

C>O, T if C
4

- 1 order.

Refs.: Sec. 4.9;eqs. (4.99), (4.117), and (4.1 19); Sec.6.9.2.

VECTOR EQUATIONS (A.29)

(A.30) (A.31)

e(r, t)=ef, t)+e’(r, t)

d2n(r,t ) e(r,t)=V[V,~(r,t)]-~€~-

(A.32)

at2

ei(r,t)=VIV.ni(r,t)]-p$o

e’(r,t)=VIV.n‘(r,t)]-pU,&o d(r, r)=EoCs(r, t ) Refs.: Sec. 4.1 1.

d2ni(r,t ) dt2

(A.33)

dZd(r,t ) dt2

(A.34) (A.35)

320

Appendix A

REFERENCES 1. Grottlieb, M., Ireland, C. L. M.. and Ley, J. M. Electro-Optic and Acousto-

Optic Scanning and Deflection,Marcel Dekker, New York (1983).

2. Korpel, A. “Acousto-Optics”, inApplied Solid State Science, Vol. 3 (R.Wolfe,

ed.), Academic Press, New York (1972).

Appendix B The Stationary Phase Method

The stationary phase method appliesto the evaluation of the following type of integral [l]:

I = Limk,,

i

h(x) exp[jkg(x)] dx

4

where h(x) and &x) are real functionsof x and k is a real constant. Now, in any part of the integration interval where dg/dx# 0, the phase angle kg(x) will go through many multiples of 2a if k becomes very large. In that case, Re{explikg(x)]}=cos[kg(x)] will oscillate very rapidly and, unless h(x) has discontinuities, this will completely average out the real part of the integral (B.1). The same holds for the imaginary part of the integral involving sin[kg(x)]. The only point on X where there is a contribution to (B.l) is the so-called stationary phase pointx,, where dg/dx=O. Expanding g(x) around the stationary phase point x,, the integral (B.l) may be written

where g"(x) = dg/dx. 32 1

322

AppendixB

Substituting a new variable x‘=x-xp into (B.2), we find (€3.4) using the following integral formulap], [exp(-p’x’

(:j2)

kqx) dx = -exp P

Re(p’) 2 0

The integration rangeof (B.l) need not always be - W to QJ.If k+ QJ,then any integration range is okay, as long as it includes x,. If there are more stationary phase points, their contributions have to be summed.

REFERENCES 1. Erdelyi, A. Asymptotic Expansions, Dover, New York (1954). 2. Gradshteyn, I. S. and Ryzhik, I. M. Table of Integrals, Series and Products, Academic Press,New York (1965).

Appendix C Symbols and Definitions"

Local angular planewave spectrum. Local profile spectrum. Material constant: a=kCl4. Light velocity in medium. Light velocity in vacuum. Material constant such that An= C'S; C'= -0.5Pno3

Refractive index variation. Amplitude of plane wave of refractive index variation; may be complex: An= C 'S. Generalized light phasor with time variation slow relative to the associated frequencyo Spatial profile of light field. Phasor referring to the general nth-order field.The associated frequency equals o : En(x, z, t)=En(X, z) exp(jnR t). Phasor referring to general nth-order field. The associated frequency equals o+nQ. See above. 323

324

Appendix C

As above. En@) Ef"'(x, t; z)

As above three indimensions. Virtual phasor field along the X axis,i.e.,back-projected physical field E(x, r, z).

EP)(x, t; z) En(")(x; z)

Asabove but referring to nth order: En(")(x, t; z)=En(")(x;z) exp(jnL2 t). Seeabove.Back-projectedfieldEn(x, 2).

&(")(x)

Short for En(")(x; m),nn'.

EP)(x)

Short for EP)(x; -m)=E,(x,

En

Amplitude of nth-order plane wave, with the origin as a phase reference point.

En(z)

Virtual plane-wave amplitude. The corresponding real physical fieldis &(X, Z)=En(z)X exp(-jkx sin P-jkz cos $,,). The corresponding virtual (back-projected) field is En(")(x, Z)=En(z)X exp(-jkx sin e n ) .

-5

Amplitude of incident plane wave. Sometimes shorthand for z) when clear from the context. the incident light field Ei(x,

E($; 2)

Virtual angular plane-wave spectrum oflightwithphase reference at the origin. The corresponding physical field is

0).

k sin$ x exp(-jkx sin$ - jkz cos 9) d 2K

The corresponding virtual (back-projected) field is E'"(x;

2) =

p($;2) k sin$ x exp(-jkx sin$) d 2K

&(e)

Short for E n ( @ , W), n# i.

J%e)

Short for E,(@, -m); sometimesused from context.

EO

Dielectric constant medium. of

E '(r,

0

forEl(@, z)whenclear

Time-varying part ofdielectric constant.

325

Symbols and Dejnitions EO+&

'(r, t).

Dielectic constant of vacuum. Fourier transform operator. Inverse Fourier transform operator. Angle used for optical waves; positive counterclockwise with respect to the 2 axis. Propagation angleof nth order: 4 n = $ o+n@ B . Incident angles. Bragg angle:4 B z 0% IA. Phase of the sound. Angle used for sound waves; positive clockwise with respect to the X axis. Light intensity of the nth-order planewave: In=lEnI2. Light propagationconstant in medium. Sound propagationconstant. Light propagationconstant in vacuum. Light wavelength in medium. Light wavelength in vacuum. Sound wavelength. Permeability of vacuum. Refractive index: n(x, z,t)=no+6n(x, z,t). Constant part of the refractive index. Elasto-optic coefficient. Three-dimensional position vector. Two-dimensional position vector. Generalized sound phasor with time variation slow relative to d.The associated frequency equalsR. Time-independentsound equals d.

phasor. Associatedfrequency

Appendix C

326 S

Plane-wave sound amplitude; sometimes shorthand for the field S(x,z) when clear from the context. Angular plane-wave spectrum of sound with phase reference at the origin. The physical fieldat (x,z) is given by

Ksin y 2n

x exp(- jivz sin y - jKx cos y ) d -

,+e z)=S*[z($,-e

S(x2,z)=s[z($ S(&,

B),

21. Sound field along a Bragg line.

E),

z]. Conjugate sound field along a

Bragg line. V

Raman-Nathparameter, peak phase shift: v=kGJC'IISI =kLIqIqI2=2aLIq=KvAn L. Accumulated peak phase shift :v'(z) = 0 . 5 z l klCllS(z)ldz.

V

Sound velocity. Susceptibility of medium. Peak time-varying susceptibility. Point on Bragg line:xn+=z($n+$

B).

"-4

B).

Point on Bragg line:;x =z($ Eikonal function. kY,{r)+KYls(r)-kYI(r). Angular light frequency. Angular sound frequency.

Index

Acoustic holography, 29 Acoustic microscope, 29 Acoustic surface waves (see Surface acoustic waves) Acoustics, 257-266 Acousto-optic tunable filters, 285-288 anisotropic, 287 isotropic, 285 Amplification, of sound by acousto-optics,25,292-295 Amplitude, of diffracted ray, 23, 11 5 Angle convention in three dimensions, 299 in two dimensions,36 Angular planewave spectrum, 20,21,71,304 coupling ofwaves in, 101 of light field, definition, 67 rescattering in, 71 of sound field, definition, 68 Anisotropy in acoustics, 264-265

[Anisotropy] in optics, 267-270 Backward wave amplifier, 294 Beam deflector, 16, 19, 175-182 using birefringence, 27,279 Beam steering, 19,178-181 Born approximation, 15 Bragg angle, 6 incidence at multiple values of, 140-143 large values of, 288,292 tracking of, 19 Bragg diffraction, 6-8, 58-61 in anisotropic crystals,277-285 criteria for, 63,64 in downshiftedlupshifted configuration, 59, 70 eikonal theory of, 109-118 equations, 60 higher order, 140-143 in birefringent medium, 283

327

Index [Bragg diffraction] imaging, 21,22,206-219 in pulsed mode, 23 rules of, 209 by sampling, 23,219-223 in three dimensions, 214-219 using birefringence,27 sound field probing by, 23,219-223 Bragg lines, 102,105,160 Brillouin scattering, 24 Broad light beaminteraction, 77-79 Condensation, 257 Coupled modes, 26,86 Coupled waves, 20,21 Correlator, 19 Corrugated wave front, 37 Curved sound wave fronts strong interaction, 80-83, 118-125 weak interaction, 116-1 18 Debye-Sears criterion (see Raman-Nath criterion) Doppler shift ofscattered light, 14, 17,24,42 Dynamic birefringence, 275 Eikonal theory, 23 of Bragg diffraction, 109-118 of Braggdiffraction imaging, 210-214 Elasto-optic coefficient, 26,272 Elasto-optics,270-275 Electro-opticeffect, 16 Energy, conservation of, in interaction, 24,292 Feynman diagrams, 10 in curved soundwave front interaction, 118 in ideal Bragg diffraction, 108, 109 in intermodulation products, 203 in Monte Carloalgorithm, 160 in scattering calculations, 106, 107 Figure of merit, 26 Fourier plane, processing in, 19, 242-248

Fourier transform, use in angular plane wave spectrum, 67-69 Fourier transform method, 154-156 Fresnel diffraction, 11 Fresnel image,23,28 Frequency shifting, 17 for electro-optic heterodyning, 28,246 for interferometricvisualization, 28 Gaussian beam strong interaction with sound column, 172-175 weak interaction, 169-172, 302, 303 Generating function, 13 Generation, of sound by acousto-optics,25,292-295 Grating cascaded, for analysis, 52-56 induced by sound wave, 7, 13,27,40-44 induced by surface acoustic wave, 27 Helmholtz equation, 14 Hertz vector, 129 Heterodyning, 17, 19,28 as explanation of modulator, 186 in SAW probing, 233-236 in signal processing, 241,248 in sound field probing,219-223 Hologram, 14,26 Hooke’s law, 263 Image plane, signal processing in, 19,237-242 Index ellipsoid, 268 Intermodulation products, 200-206 Klein-Cook parameter, 9,49 Longitudinal wave, 257 Lorentz-Lorenz relation, 26,270 Manly-Rowe relations, 293 Mathieu functions, 11-13, 16, 89

329

Index Measurements of optical fields, 193-196 Modal theory, 90 Modulator, 17 bandwidth of, 184-189 Momentum, conservation of, in interaction, 24 Monte Carlo simulation, 156-1 67 Multiple scattering, 71, 106 NOA method, 143, 144 Normal modes, 11,26,88-90 Numerical approach, 135-168 by carrierless split-step method, 147 by eigenvalues, 143 by Fourier transforms, 154 by integration, 137 by matrix multiplication, 146 by Monte Carlo simulation, 156 for multiple Bragg incidence, 140 by successive diffraction, 144 Oblique incidence,4 5 4 8 Optical axes, 268 Optical indicatrix (see Index ellipsoid) Orthogonal scattering, 273-274,277 Parallel processing, 17,236 Paraxial propagation, 40 Path integrals, 106-109 Phased array deflector(see Beam steering) Phase modulation, 42 Phase shift accumulated, through sound field, 80 through sound beam, 45 Phase synchronism, 20,58 Phasor definition of, 36 generalizationof, 39 Photon-phonon collision, 24,292

Plane wave analysis of Braggdiffraction imaging, 206,210 correspondence, 21 coupling, 70-72 spectrum (see Angular plane wave spectrum) strong interaction in two dimensions, 98-106 weak interaction in three dimensions, 295-303 weak interaction in two dimensions, 23,72-77,96-98 Polarization, of scattered light, 125 Poisson’s ratio, 263 Poynting vector, of sound, 265 Profiled light beam,77-79 Profiled sound column,79-83 Profilometer, 198,200 Q (see Klein-Cook parameter) Q spoiler, 29 Quasi theorem, 189-193 Radiation pattern, of sound, 71 measurement of, 21,76 Raman-Nath, criterion, 49 equations, 13, 15, 55, 56, 87 generalized, 14,92 truncation of, 136 normalized parameter, 90 parameter, 42 regime, 56 Ray amplitude ofdiffracted ray, 23, 115 bending, 14,49, 50, imaging by, 23, focusing, 51, 52, theory (see Eikonal theory) tracing diagrams, 111 in Bragg diffraction imaging, 210 in curved wavefront interaction, 11 1, 122

330

[Ray1 trajectories, 10 Recursion relations, 15, 57 Rescattering (see Multiple scattering) Sadron constant, 275 SAW (see Surface acoustic waves) Scattered field, calculation of, 129 Schaefer-Bergmann patterns, 18 Schlieren imaging ofsound fields, 23,223-233 diffraction free patterns in, 228 and tomography, 227 Scophony system, 17, 19 Shear wave, 262 Signal processing heterodyning in, 241-248 in frequency plane,242-248 in image plane,237-242 with time integration, 248-253 spectrum analyzer, 251 triple product processor, 252 use of Quasitheorem in, 190 Slowness surface, 264 Sound visualization by Bragg diffraction imaging, 21, 22, 206-219 by Bragg diffraction sampling, 23,219-223 by Schlieren imaging, 223-233

Index Spectra, 303-308 local planewave, 305 profile, 306-307 acousto-optic profile, 307-308 virtual plane wave, 304 Spectrum analyzer, 19, 182-184,251 Split-step method, 147-150 carrierless for acoustooptics, 150-1 54 Standing wave, of sound, 38 Stationary phase method, 321-322 Surface acoustic waves, 266 probing of, 233-236 visualization of,28 Transition probability, 161, 162 Uniaxial crystal,268-269 anisotropic interaction in, 278 beam deflector using, 282 collinear interaction in, 280 Vibration measurement, 196-198 Virtual field, 54,69,73 Wave vector diagram, 20,25,70 for beam deflector, 177,280 X-ray diffraction, 6, 9 Young’s modulus, 263