Photorefractive Materials: Fundamental Concepts, Holographic Recording and Materials Characterization

PHOTOREFRACTIVE MATERIALS

PHOTOREFRACTIVE MATERIALS Fundamental Concepts, Holographic Recording and Materials Characterization

JAIME FREJLICH Universidade Estadual de Campinas Instituto de Fı´sica-Laborato´rio ´ ptica Campinas-SP BRAZIL de O

WILEY-INTERSCIENCE A John Wiley & Sons, Inc., Publication

Copyright ß 2007 by John Wiley & Sons, Inc. All rights reserved Published by John Wiley & Sons, Inc., Hoboken, New Jersey Published simultaneously in Canada No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, scanning, or otherwise, except as permitted under Section 107 or 108 of the 1976 United States Copyright Act, without either the prior written permission of the Publisher, or authorization through payment of the appropriate per-copy fee to the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, (978) 750-8400, fax (978) 750-4470, or on the web at www.copyright.com. Requests to the Publisher for permission should be addressed to the Permissions Department, John Wiley & Sons, Inc., 111 River Street, Hoboken, NJ 07030, (201) 748-6011, fax (201) 748-6008, or online at http://www.wiley.com/go/permission. Limit of Liability/Disclaimer of Warranty: While the publisher and author have used their best efforts in preparing this book, they make no representations or warranties with respect to the accuracy or completeness of the contents of this book and specifically disclaim any implied warranties of merchantability or fitness for a particular purpose. No warranty may be created or extended by sales representatives or written sales materials. The advice and strategies contained herein may not be suitable for your situation. You should consult with a professional where appropriate. Neither the publisher nor author shall be liable for any loss of profit or any other commercial damages, including but not limited to special, incidental, consequential, or other damages. For general information on our other products and services or for technical support, please contact our Customer Care Department within the United States at (800) 762-2974, outside the United States at (317) 572-3993 or fax (317) 572-4002. Wiley also publishes its books in a variety of electronic formats. Some content that appears in print may not be available in electronic formats. For more information about Wiley products, visit our web site at www.wiley.com. Library of Congress Cataloging-in-Publication Data Frejlich, Jaime, 1946Photorefractive materials : fundamental concepts, holographic recording, and materials characterization / by Jaime Frejlich. p. cm. ‘‘A Wiley-Interscience publication.’’ Includes bibliographical references and index. ISBN-13: 978-0-471-74866-3 ISBN-10: 0-471-74866-8 1. Crystal optics. 2. Photorefractive materials. I. Title. TA418.9.C7F74 2007 620.10 1295–dc22

Printed in the United States of America 10 9 8

7 6 5

4 3 2 1

2006046394

To the memory of my son Gabriel

CONTENTS

LIST OF FIGURES LIST OF TABLES PREFACE ACKNOWLEDGMENTS

xiii xix xxi xxiii

I FUNDAMENTALS

1

1 ELECTRO-OPTIC EFFECT

5

1.1 Light propagation in crystals 1.1.1 Wave propagation in anisotropic media 1.1.2 General wave equation 1.1.3 Index ellipsoid 1.2 Tensorial Analysis 1.3 Electro-optic effect 1.3.1 Sillenite-type crystal 1.3.2 Lithium niobate 1.3.3 KDP-(KH2 PO4 ) 1.4 Concluding Remarks

5 6 6 7 9 10 11 16 17 18

vii

viii

CONTENTS

2 PHOTOACTIVE CENTERS AND PHOTOCONDUCTIVITY 2.1 Photoactive centers: Deep and shallow traps 2.1.1 Cadmium telluride 2.1.2 Sillenite-type crystals 2.1.3 Lithium niobate 2.2 Photoconductivity 2.2.1 Localized states: traps and recombination centers 2.2.2 Theoretical models 2.2.2.1 One-center model 2.2.2.1.1 Steady state under uniform illumination 2.2.2.2 Two-center/one-charge carrier model 2.2.2.2.1 Steady state under uniform illumination 2.2.2.2.2 Light-induced absorption 2.2.2.3 Dark conductivity and dopants 2.2.3 Photoconductivity in bulk material 2.3 Photochromic effect 2.3.1 Transmittance with light-induced absorption

19 21 21 23 26 26 27 29 33 34 35 36 38 38 39 39 40

II HOLOGRAPHIC RECORDING

45

3 RECORDING A SPACE-CHARGE ELECTRIC FIELD

47

3.1 Index of refraction modulation 3.2 General formulation 3.2.1 Rate equations 3.2.2 Solution for steady state 3.3 First spatial harmonic approximation 3.3.1 Steady-state stationary process 3.3.1.1 Diffraction efficiency 3.3.1.2 Hologram phase shift 3.3.2 Time-evolution process: Constant modulation 3.4 Steady-state nonstationary process 3.4.1 Running holograms with hole–electron competition 3.4.1.1 Mathematical model 3.5 Photovoltaic Materials 3.5.1 Uniform illumination: @N =@x ¼ 0 3.5.2 Interference pattern of light 3.5.2.1 Influence of donor density

50 54 56 56 59 62 63 64 65 67 71 75 79 80 81 82

CONTENTS

4 VOLUME HOLOGRAM WITH WAVE MIXING 4.1 Coupled wave theory: Fixed grating 4.1.1 Out of Bragg condition 4.2 Dynamic coupled wave theory 4.2.1 Combined phase-amplitude stationary gratings 4.2.1.1 Fundamental properties 4.2.1.2 Irradiance 4.2.2 Pure phase grating 4.2.2.1 Time evolution 4.2.2.1.1 Undepleted pump approximation 4.2.2.1.2 Response time with feedback 4.2.2.2 Stationary hologram 4.2.2.2.1 Diffraction 4.2.2.3 Steady-state nonstationary hologram with bulk absorption 4.2.2.3.1 Diffraction efficiency 4.2.2.3.2 Output beams phase shift 4.3 Phase modulation 4.3.1 Phase Modulation in dynamically recorded gratings 4.3.1.1 Phase modulation in the signal beam 4.3.1.1.1 Unshifted hologram 4.3.1.1.2 Shifted hologram 4.3.1.2 Output phase shift 4.4 Four-wave mixing 4.5 Final remarks

5 ANISOTROPIC DIFFRACTION 5.1 Coupled wave with anisotropic diffraction 5.2 Anisotropic diffraction and optical activity 5.2.1 Diffraction efficiency with optical activity r 5.2.2 Output polarization direction

ix

85 85 88 89 90 92 93 95 95 96 98 101 104 108 110 112 114 118 119 119 120 120 122 123

125 125 127 128 130

6 STABILIZED HOLOGRAPHIC RECORDING

131

6.1 Introduction 6.2 Mathematical formulation 6.2.1 Stabilized stationary recording 6.2.1.1 Stable equilibrium condition

131 133 136 137

x

CONTENTS

6.2.2 Stabilized recording of running (nonstationary) holograms 6.2.2.1 Stable equilibrium condition 6.2.2.2 Speed of the fringe-locked running hologram 6.2.3 Self-stabilized recording with arbitrarily selected phase shift 6.3 Self-stabilized recording in actual materials 6.3.1 Self-stabilized recording in Sillenites 6.3.2 Self-stabilized recording in LiNbO3 6.3.2.1 Holographic recording without constraints 6.3.2.1.1 Space-charge electric field build-up 6.3.2.1.2 Hologram phase shift 6.3.2.1.3 Diffraction efficiency with wave mixing 6.3.2.2 Self-stabilized recording 6.3.2.2.1 Effect of light polarization 6.3.2.2.2 Glass plate-stabilized recording

138 140 140 141 144 144 145 145 147 148 149 152 157 159

III

MATERIALS CHARACTERIZATION

163

7

NONHOLOGRAPHIC OPTICAL METHODS

165

7.1 Light-induced absorption 7.2 Photoconductivity 7.2.1 Alternating current technique 7.2.1.1 Wavelength-resolved photoconductivity 7.2.2 Modulated photoconductivity 7.2.2.1 Quantum efficiency and mobility-lifetime product 7.3 Electro-optic coefficient

165 170 171 174 176

8 HOLOGRAPHIC TECHNIQUES 8.1 Direct 8.1.1 8.1.2 8.1.3 8.1.4

holographic techniques Energy coupling Diffraction efficiency Holographic sensitivity Hologram recording and erasure 8.1.4.1 Dark conductivity 8.1.5 Hole–electron competition

178 179 181 181 182 185 186 188 190 191

CONTENTS

8.2 Phase modulation techniques 8.2.1 Holographic sensitivity 8.2.2 Holographic phase-shift measurement 8.2.2.1 Wave-mixing effects 8.2.3 Photorefractive response time 8.2.4 Selective two-wave mixing 8.2.4.1 Amplitude and phase effects in GaAs 8.2.5 Running holograms 8.2.6 Photo-electromotive-force techniques 8.2.6.1 Holographic photo-emf

xi

194 194 196 197 197 201 203 206 212 212

9 SELF-STABILIZED HOLOGRAPHIC TECHNIQUES

225

9.1 Holographic phase shift 9.2 Fringe-locked running holograms 9.2.1 Absorbing materials 9.2.1.1 Low-absorption approximation 9.2.2 Characterization of materials 9.2.2.1 Measurements 9.2.2.1.1 Hologram speed Kv 9.2.2.1.2 Diffraction efficiency 9.2.2.2 Theoretical fitting 9.3 Characterization of LiNbO3 :Fe

225 229 230 232 232 232 232 233 234 238

IV APPLICATIONS 10

243

VIBRATIONS AND DEFORMATIONS

245

10.1 10.2

246 246 246 248 249

Measurement of Vibration and Deformation Experimental Setup 10.2.1 Reading of Dynamic Holograms 10.2.2 Optimization of illumination 10.2.2.1 Target illumination 10.2.2.2 Distribution of light among reference and object beams 10.2.3 Self-stabilization Feedback Loop 10.2.4 Vibrations 10.2.5 Deformation and tilting

249 251 252 255

xii

11

CONTENTS

FIXED HOLOGRAMS

259

11.1 11.2 11.3 11.4

259 260 260 262

Introduction Fixed holograms in LiNbO3 Theory Experiment

V APPENDICES A

267

DETECTING A REVERSIBLE REAL-TIME HOLOGRAM

269

A.1

270 270 270 271

A.2

Naked-eye detection A.1.1 Diffraction A.1.2 Interference Instrumental detection

B DIFFRACTION EFFICIENCY MEASUREMENT: REVERSIBLE VOLUME HOLOGRAMS B.1

Angular Bragg selectivity B.1.1 In-Bragg recording beams B.1.2 Probe beam B.2 Reversible holograms B.3 High index of refraction material

273 273 274 274 278 278

C EFFECTIVELY APPLIED ELECTRIC FIELD

281

D PHYSICAL MEANING OF SOME FUNDAMENTAL PARAMETERS

283

D.1

D.2

Debye screening length D.1.1 Temperature D.1.2 Debye screening length Diffusion and mobility

E PHOTODIODES E.1 E.2 E.3

Photovoltaic regime Photoconductive regime Operational amplifier operated

283 283 284 285 287 289 290 291

BIBLIOGRAPHY

293

INDEX

305

LIST OF FIGURES

I.1 Lithium Niobate Crystal 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8

2 8 8 11 12 12 13 13

1.9 1.10 1.11

Refractive index ellipsoid Plane wave propagation Crystallographic axes of a sillenite Raw Bi12 TiO20 boule From raw BTO to ready-to-use crystal sample Sillenite crystals Index of refraction of BTO Crystallographic axes for Bi12 SiO20 and unidirectional applied electric field Sillenite crystal under applied field Lithium niobate crystal Modified birefringence of lithium niobate crystal

2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 2.10 2.11

Energy diagram for a typical CdTe crystal doped with Vanadium Dark conductivity for CdTe States in the BTO band gap Intrinsic semiconductor Doped semiconductor Quasi-stationary Fermi levels in doped semiconductor Recombination centers Traps One-center, one single species model One-center, one single species model, under the action of light Electrons and holes photoexcited

22 23 25 27 28 28 29 29 30 31 31

14 15 16 17

xiii

xiv

LIST OF FIGURES

2.12 2.13 2.14 2.15 2.16 2.17

Only electrons photoexcited Photochromic effect Crystal with electrodes Light-induced absorption Arrhenius curve for photochromism in BTO Photochromic effect in a Bi12 TiO20 crystal

32 36 38 41 41 42

3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 3.10 3.11

Photoactive centers schema Excitation of charges Space-charge modulation Crystal lattice deformation Index of refraction modulation Holographic setup Generation of an interference pattern of fringes Light excitation of electrons to the CB Generation of an electric charge spatial modulation Generation of a space-charge electric field modulation The electric field modulation produces an index of refraction modulation Reading the recorded grating The grating is erased during reading . . .Until all recording is erased Space-charge field grating Space-charge electric field without externally applied field for a pattern of fringes with different modulation m Simulated recording and erasure of a space-charge field Index of refraction modulation Simulation of holographic phase shift Running hologram Real part of the photorefractive space-charge field Imaginary part of the photorefractive space-charge field Square modulus of the photorefractive space-charge field Quality factor of a running hologram Quality factor versus K Effective field in a running hologram One-species/two-valence/two-charge carrier model Two-species/two-valence/two-charge carrier model Hole-electron competition on different photoactive centers under the action of low energetic photon recording light Short- and open-circuit operating schema for LiNbO3

48 49 49 50 50 51 51 52 52 52

3.12 3.13 3.14 3.15 3.16 3.17 3.18 3.19 3.20 3.21 3.22 3.23 3.24 3.25 3.26 3.27 3.28 3.29

3.30 4.1 4.2 4.3 4.4 4.5

Reading a volume hologram Recording a volume hologram Bragg condition Amplitude coupling in two-wave mixing Phase coupling in two-wave mixing

53 53 53 54 55 58 63 63 65 67 70 71 71 72 72 73 73 74

74 81 86 86 87 92 93

LIST OF FIGURES

4.6 4.7 4.8 4.9 4.10 4.11

xv

Hologram erasure with positive gain feedback Hologram erasure with negative gain feedback Hologram erasure with positive phase feedback Hologram erasure with negative phase feedback Transient effect on a running hologram Computed running hologram Z as a function of Kv for K ¼ 0:5 mm1 Computed running hologram Z for K ¼ 2 mm1 Computed running hologram Z as a function of Kv for K ¼ 10 mm1 Computed running hologram Z as a function of Kv for K ¼ 20 mm1 Running holograms in thick BTO: tan j versus Kv (rad/s), for K ¼ 2 mm1 Running holograms in thick BTO: tan j versus Kv for K ¼ 11 mm1 Running holograms in thin BTO: tan j versus Kv for K ¼ 11 mm1 Running holograms in thin nonabsorbing BTO: tan j versus Kv for K ¼ 1 mm1 Phase modulation setup Wave mixing schema showing the hologram phase shift f and the phase shift j between the transmitted and diffracted beams at the crystal output Degenerate four-wave mixing

121 123

5.1 5.2 5.3 5.4 5.5

Input and output light polarization Actual input and output polarization Polarization of the diffracted and transmitted beams Orthogonally polarized beams at the output Parallel-polarized beams at the output

127 127 129 129 130

6.1

Scanning electronic microscopy image of a 1D hollow sleeves structure Scanning electronic microscopy image of a 2D-array Scanning electronic microscopy image of a blazed grating Self-stabilized recording: Block-diagram Self-stabilized recording: Actual setup Noise propagation Fringe-locked running hologram: block-diagram Fringe-locked running hologram: actual setup Fringe-locked running hologram: experimental data Schema of the self-stabilized setup in Fig. 6.8 modified to operate with arbitrarily selected j Transverse optical configuration for BTO Self-stabilized recording in BTO: experimental data Self-stabilized recording on BTO:Comparison

4.12 4.13 4.14 4.15 4.16 4.17 4.18 4.19 4.20

4.21

6.2 6.3 6.4 6.5 6.6 6.7 6.8 6.9 6.10 6.11 6.12 6.13

99 99 99 100 100 110 111 112 113 114 115 116 117 118

132 132 133 135 135 136 138 138 142 143 145 146 146

xvi

LIST OF FIGURES

6.14 6.15

Experimental setup Computed Z as a function of 2kd for non stabilized recording in LiNbO3 :Fe with different degree of oxidation Computed Z as a function of 2kd and f, for b2 ¼ 1 Computed Z as a function of 2kd and f, for b2 ¼ 10 Computed IS2 , with  ¼ 0 as a function of 2kd Computed evolution of f, IS in arbitrary units, and Z as functions of 2kd for self-stabilized conditions (IS2 ¼ 0) and b2 ¼ 1:1 Computed evolution of f, IS in arbitrary units, and Z as functions of 2kd for self-stabilized conditions (IS2 ¼ 0) and b2 ¼ 10 Self-stabilized recording in a less oxidized crystal with b2  1 Self-stabilized recording in an oxidized crystal with b2 ¼ 1 Self-stabilized recording in an oxidized crystal (sample LNB1) with b2 ¼ 12 Overall beam IG produced by the interference of the recording beams transmitted and reflected by a thin glassplate Measurement of the running hologram speed Two self-stabilized recording experiments on the same LiNbO3 : Fe sample with ordinarily and extraordinarily polarized light Recording setup stabilized on a nearby placed glassplate Glassplate-stabilized experimental data for the recording on an oxidized sample (LNB1) with b2  1 Mathematical simulation of non self-stabilized recording with b2 ¼ 1 Evolution of Z and scattering during stabilized holographic recording with and without self-stabilization in LiNbO3 :Fe

6.16 6.17 6.18 6.19

6.20 6.21 6.22 6.23 6.24 6.25 6.26 6.27 6.28 6.29 6.30 7.1 7.2 7.3 7.4 7.5 7.6 7.7 7.8 7.9 7.10 7.11 7.12 8.1 8.2 8.3 8.4

147 150 151 151 153

154 154 156 156 157 157 158 158 159 160 161 161

Evolution of absorption coefficient in undoped B12 TiO20 Light-induced absorption in BTO Light-induced absorption of Bi12 TiO20 at 514.5 nm Photoluminescence in BTO Absorption coefficient-thickness ad measured for three different BTO samples Experimental setup: Photoconductivity Photocurrent in BTO Wavelength-resolved photoconductivity setup Wavelength-resolved photoconductivity for BTO Wavelength-resolved photoconductivity for BTO Wavelength-resolved photoconductivity for BTO Modulated photocurrent data

166 167 168 169

Holographic setup Energy transfer Exponential gain Hologram erasure in BTO:Pb

182 183 184 189

169 172 173 174 175 175 176 177

LIST OF FIGURES

8.5 8.6 8.7 8.8 8.9 8.10 8.11 8.12 8.13 8.14 8.15 8.16 8.17 8.18 8.19 8.20 8.21 8.22 8.23 8.24 8.25 8.26 8.27 8.28 9.1 9.2 9.3 9.4 9.5 9.6 9.7 9.8 9.9 9.10 9.11

Hologram erasure in LiNbO3 :Fe Hologram relaxation in the dark Hologram erasure with hole-electron competition in BTO:Pb Hologram erasure with hole-electron competition in BTO:Pb Hologram erasure with hole-electron competition in undoped BTO Holographic sensitivity Second harmonic evolution for KNSBN:Ti Phase shift data for BTO at 514.5 nm Two-wave mixing experiment in GaAs Response time in GaAs Selective two-wave mixing experiment in GaAs Two-wave mixing experiment in GaAs with polarization selection at the output First I  and second I 2 harmonic terms in GaAs with gratings of multiple nature Running hologram setup Running hologram simulation for K ¼ 2:55 mm1 Running hologram simulation for K ¼ 11:3 mm1 Running hologram data for Z Running hologram data for the phase-shift First temporal harmonic of the holographic current Photocurrent as a function of the frequency Holographic current setup schema First harmonic component of the holographic photocurrent for low frequencies First harmonic component of the holographic photocurrent for high frequencies ji j data ploted as a function of =2p, for K ¼ 1:1 rad Typical time evolution of the VX and VY signals at the starting of the recording process in Bi12 TiO20 Hologram phase shift vs. electric fiel at the starting of the recording process in Bi12 TiO20 Phase shift data for Bi12 TiO20 at 532 nm Fringe-locked running hologram data for BSO Fringe-locked running hologram experiment: Frequency detuning Fringe-locked running hologram: Hologram speed Kv and Z experimentally measured on an undoped Bi12 TiO20 sample 3D representation of Diffraction efficiency and Kv experimentally measured data 3D representation of Diffraction efficiency with theoretical fit Characterization of reduced LiNbO3 Characterization of reduced LiNbO3

xvii

190 191 192 193 193 195 196 197 199 200 204 204 205 206 208 209 211 211 218 219 220 221 221 222 227 227 229 230 233 234 235 236 237 239 240

xviii

LIST OF FIGURES

9.12

Characterization of oxydized LiNbO3

240

10.1 10.2 10.3

Schematical diagram of the experimental setup Lateral view of the setup Simplified schema showing the distribution of light between reference and object beams Optimization of the target illumination Loud-speaker membrane (left) excited with a 3.0 kHz Amplitude of vibration at a point of local maximum in the membrane of a loud-speaker as a function of the applied voltage Amplitude of vibration at two different points of local maximum in the membrane of a loud-speaker as a function of the applied voltage Time-average holographic interferometry pattern of a thin phosphorous-bronze metallic plate Vibration pattern of a thin metallic plate at 600Hz Vibration pattern of a thin metallic plate at 800Hz Double exposure holographic interferometry of a tilted rigid plate Double exposure holographic interferometry of a tilted rigid plate Double exposure holographic interferometry of a tilted rigid plate

247 248

Experimental setup Evolution of I  and I 2 during high temperature self-stabilized holographic recording Diffraction efficiency of the overall grating during white-light development

263

277

B.3

Diffraction efficiency as a function of out-of-Bragg angle v, computed from Eq.(B.15), as a function of v for in-Bragg condition Measurement of diffraction efficiency of volume holograms

C.1

Effective field coefficient

282

D.1

Diffusion and mobility

285

E.1 E.2

np-junction showing the depletion layer np-junction showing the depletion layer including the intrinsic layer and a diagram of the Schottky potential barrier pn-junction showing the depletion layer and a diagram of the Schottky potential barrier Photovoltaic mode operation for photodiodes Photoconductive mode operation for photodiodes Operational amplifier operated photodiode

288

10.4 10.5 10.6

10.7

10.8 10.9 10.10 10.11 10.12 10.13 11.1 11.2 11.3 B.1 B.2

E.3 E.4 E.5 E.6

249 250 253

254

254 255 255 256 256 257 257

264 265

277 279

288 289 290 290 291

LIST OF TABLES

1.1

Index of refraction of KDP

17

3.1

Photovoltaic transport coefficient

80

6.1

LiNbO3 :Fe samples

155

7.1 7.2 7.3

Absortion parameters for pure and doped BTO Photoconductivity and derived parameters for BTO Typical parameters of some pure and doped BTO samples

167 174 179

8.1 8.2 8.3 8.4 8.5

Properties of KNSBN:Ti sample Holographic Sensitivity and Gain Hole-electron competition in Pb-doped BTO Running hologram Holographic photocurrent

185 188 192 212 222

9.1 9.2

Initial phase shift for Bi12 TiO20 Fitting of experimental data of undoped Bi12 TiO20 with K ¼ 7:35 mm1 Parameters for LiNbO3 samples LiNbO3 : Material parameters

228 237 241 241

Fixed diffraction efficiency

265

9.3 9.4 11.1

xix

PREFACE

This book is meant to present an overview of the basic features and properties of photorefractive materials in a wide array of topics, while striving to maintain a coherent setting in the hope that it will be of interest for graduate students who are taking up the subject as well as for advanced students who are familiar (at least in part) with this field. We hope this book may also be of interest for senior researchers willing to review the fundamentals of photorefractives and gain a deeper insight into more specialized topics like material characterization and self-stabilized holographic recording techniques. We welcome also the curious student interested in having a glance at this complex and fascinating research field. The book is divided into four parts and an appendix. The Part I, called ‘‘Fundamentals,’’ is a review of the basic properties of the electro-optical effect and of photoconductivity, with some attention to photochromism. These properties are important for the understanding of materials’ response to the action of light. Part II is entitled ‘‘Holographic Recording’’ and deals with the buildup of a space-charge electric field, the associated volume hologram, the diffraction of light by this hologram, and the mutual interaction between the recording beams and the hologram being recorded, a phenomenon called wave mixing. Special attention is here devoted to the analysis of a feedback-controlled (or self-stabilized) holographic recording allowing a high degree of control of the recording process. Part III, ‘‘Materials Characterization,’’ then follows, in which we describe the use of optical (mainly holographic) techniques for the characterization of photorefractives and for the measurement of some of their fundamental parameters. Mixed techniques like photoconductivity are also described. We have here emphasized the use of phase modulation in two-wave mixing and self-stabilized holographic recording techniques. xxi

xxii

PREFACE

Part IV is ‘‘Application,’’ where we describe in detail two well-known paradigmatic applications: measurement of vibrations and deformations and fabrication of diffractive fixed holographic optical components. A large number of applications of photorefractives have already been reported in the scientific literature, but we believe that those referred to above will suffice to hint at the practical possibilities of these materials because they involve two widely different fields (image processing on the one hand and holographic optical component fabrication on the other) and two very different materials, one of which is a slow, highly diffractive material and the other a fast, poorly diffractive one. The Appendix includes some topics of practical interest for the beginner who is willing to start with laboratory experiments involving photorefractives. We therefore discuss here the particular features of holograms in reversible real-time recording materials and describe in detail how to detect and reliably measure them. A section devoted to photodiodes for the measurement of light is also included here because of their practical relevance for laboratory experiments. The reader will find here a section in which we discuss the physical meaning of a couple of parameters widely used in the scientific literature and frequently referred to in this book. We have included, whenever possible, a large amount of illustrative first-hand data from experimental results, mainly obtained by ourselves and by our co-workers as well as by the many generations of graduate students who have prepared their theses in our laboratory. The close relationship between theory and experiment throughout this book has also forced us to deal solely with the materials and experiments with which we have some direct practical experience. We believe such laboratory results, despite the incertitude and limitations inherent in experimental work itself, will be of interest to realistically illustrate the distinct theoretical topics developed in this book and, at the same time, stimulate those who are rather fond of experimental research. A large number of important subjects—photorefractive polymers, quantum wells, photorefractive photonics, resonators, phase conjugation, image processing, data storage, and solitons, among many others—are not mentioned in this book. There are specialized publications devoted to these subjects to which the interested reader should refer. We believe that, despite the rather limited scope of this book (compared to the whole research field of photorefractives and their applications), the basic ideas discussed here should provide with minimum necessary introductory background for further continuing the adventure in the wonderful world of photorefractive materials research. We are aware of the huge amount of scientific literature already available on the subject of photorefractive materials and, despite the inclusion of the many references cited in this book, we apologize in advance for the important references that will certainly be missing. JAIME FREJLICH CAMPINAS, BRAZIL APRIL 2006

ACKNOWLEDGMENTS

This book is the result of direct and indirect cooperation of collegues from Brazil and all over the world who have contributed with their experience, work, and advice, as well as the graduate students working on their thesis and undergraduate students starting their experimental work in our laboratory in the Instituto de Fı´sica of the Universidade Estadual de Campinas, Campinas-SP, Brazil. My warm acknowledgments to: Klaus H. Ringhofer { Karsten Buse Detlef Kip Jean Claude Launay Luis Arizmendi Agnaldo A. Freschi Paulo Magno Garcia Marcelo C. Barbosa Bertrand Sugg Renata Montenegro

Eckhard Kra¨tzig Alexei A. Kamshilin Shaopin Bian Christophe Longeaud Jesiel F. Carvalho Marvin Klein Luis Mosquera Eduardo A. Barbosa Pedro Valentim dos Santos Nilson R. Inocente Junior

Lucila Cescato Ekaterina Schamonina Romano A. Rupp Mercedes Carrascosa Antonio C. Hernandes Victor V. Prokofiev Ivan de Oliveira Paulo Acioly Marques dos Santos

without whose direct or indirect cooperation this book could not have been written.

xxiii

PART I

FUNDAMENTALS

2

FUNDAMENTALS

Figure I.1. Naturally birefringent uniaxial lithium niobate crystal view under converging white light between crossed polarizers with its c-axis (optical axis) laying perpendicular to the plane (upper) and on the plane (lower).

FUNDAMENTALS

3

INTRODUCTION Photorefractive crystals are electro-optic and photoconductive materials. An electric field applied to an electro-optic material produces changes in its refractive index, a phenomenon also called Pockel’s effect. On the other hand, photoconductivity means that light of adequate wavelength is able to produce electric charge carriers that are free to move by diffusion and also by drift under the action of an electric field. In the case of photorefractive materials the light excites charge carriers from localized states (photoactive centers) in the forbidden band gap to extended states (conduction or valence bands) where they move, are retrapped and excited again, and so on. During this process the charge carriers progressively accumulate in the darker regions of the sample. In this way, charges of one sign accumulate in the darker regions while leaving charges of the opposite sign in the brighter regions. This spatial modulation of charges produces an associated space-charge electric field. The combination of both effects gives rise to the so-called photorefractive effect: The light produces a photoconduction-based electric field spatial modulation that in turn produces an index of refraction modulation via the electro-optic effect. This change can be reversed by the action of light or by relaxation even in the dark. The action of light on a photosensitive material may produce changes in the electrical polarizability of the molecules, and by this means a change in the complex index of refraction will result. This change may be sensible or not depending on the wavelength spectral range analyzed. The imaginary part of the index (the extinction coefficient, related to absorption) or the real part (the so-called ‘‘index of refraction’’ itself) may be more affected when observed in a certain wavelength spectral range. This is the case of dyes, some silver salts, chalcogenic glasses, photoresists, and other materials. When sensible changes occur in the real part of the complex index of refraction, these materials are also called ‘‘photorefractives’’ because they actually show changes in the real refractive index under the action of light. These changes can be reversible or not. What is the essential difference between these processes and those we have mentioned before and we are dealing with in this book? The difference is that the latter always involve the establishment of a space-charge electric field and the production of index of refraction changes via the electro-optic (or Pockels) effect. We should therefore rather call them ‘‘photo-electro-refractive’’ materials instead of just using the ‘‘photorefractive’’ label. However, the latter generic name is so widespread nowadays in the scientific literature that it would be hard to change it now. In this book we shall therefore use the term ‘‘photorefractive’’ only, but the reader should be aware that materials of different nature are usually referred to under this same label. Chapter 1 contains a review of the electro-optic effect including a little bit of tensorial analysis. The effect of an applied electric field over the index ellipsoid of some usual electro-optic crystals is analyzed so that the reader may become familiar with these procedures. We hope these examples will enable the reader to properly handle different materials and optical configurations. Chapter 2 deals with photoconductivity and light-induced absorption and their relation with the localized states (photoactive centers) in the forbidden band.

CHAPTER 1

ELECTRO-OPTIC EFFECT

The electro-optic effect and photoconductivity are the fundamental phenomena underlying the photorefractive effect. Most photorefractive crystals are anisotropic (their properties are different along different directions), and even those that are not become anisotropic under the action of an externally applied electric field. Therefore, we shall start with a review of light propagation in anisotropic media. These materials usually exhibit a piezoelectric effect, too [Yariv, 1985, Shepelevich et al., 1990, Stepanov et al., 1998] but, for the sake of simplicity, we shall not consider it here. The electro-optic effect in photorefractive materials is of the highest importance because it is at the origin of the ‘‘imaging’’ of a space-charge field modulation into an index of refraction modulation. In fact, the buildup of a holographic grating in photorefractive materials consists of the spatial modulation of the index of refraction in the volume of the sample. In these materials such a modulation arises from the buildup of a modulated space-charge field that in turn modulates the index of refraction via the electro-optic effect.

1.1

LIGHT PROPAGATION IN CRYSTALS

Crystals are in general anisotropic, that is to say, they have different properties for the light propagating along different directions.

Photorefractive Materials: Fundamental Concepts, Holographic Recording and Materials Characterization, By Jaime Frejlich Copyright # 2007 John Wiley & Sons, Inc.

5

6

ELECTRO-OPTIC EFFECT

1.1.1

Wave Propagation in Anisotropic Media

Let us start with the general vectorial relations ~ ¼ e0 ~ D E þ~ P ~ ~ wE P ¼ e0 ^

ð1:1Þ

~ P ¼ e0 w~ E

ð1:3Þ

ð1:2Þ

P, where e0 ¼ 8:82  1012 coul/(mV) is the permittivity of vacuum. The quantities ~ ~ ~ are the polarization, electric field, and displacement fields, respectively, E, and D w (polarizability) being a tensor that, for isotropic media only, can be written as with ^ a scalar, thus simplifying the relation in Equation (1.2)

The relation in Equation (1.2) can also be written as 2 3 2 32 3 w11 w12 w13 P1 E1 4 P2 5 ¼ e0 4 w21 w22 w23 54 E2 5 P3 w31 w32 w33 E3

ð1:4Þ

and also

~ ¼ e0 ð^ D 1þ^ wÞ~ E where ^ 1 and ^ w are tensors that are written as: 2 2 3 w11 1 0 0 ^ ^ w ¼ 4 w21 1 ¼ 40 1 05 w31 0 0 1

ð1:5Þ

w12 w22 w32

3 w13 w23 5 w33

ð1:6Þ

Let us recall that there is always a set of coordinate axes, called ‘‘principal axes,’’ where ^ w assumes a diagonal form 2 3 0 w11 0 ^ ð1:7Þ w ¼ 4 0 w22 0 5 0 0 w33

1.1.2

General Wave Equation

The equation describing the electromagnetic wave, in nonmagnetic and noncharged media, can be deduced from the Maxwell’s equations ~ @H @t ~ @ E @~ P ~ ~ ¼ e0 þ þ J with ~ J ¼ s~ E rH @t @t 1 P r~ E ¼  r~ e0 ~¼0 rH r~ E ¼ m0

ð1:8Þ ð1:9Þ ð1:10Þ ð1:11Þ

LIGHT PROPAGATION IN CRYSTALS

7

In a system of principal coordinate axes it is P1 ¼ e0 w11 E1 P2 ¼ e0 w22 E2 P3 ¼ e0 w33 E3 1.1.3

D1 ¼ e11 E1 D2 ¼ e22 E2 D3 ¼ e33 E3

e11 ¼ e0 ð1 þ w11 Þ e22 ¼ e0 ð1 þ w22 Þ e33 ¼ e0 ð1 þ w33 Þ

ð1:12Þ

Index Ellipsoid

We shall write the expressions for the electric we and magnetic wm energy densities in electromagnetic waves as [Born and Wolf, 1975] X 1 ~¼1 Ek Ekl El ED we ¼ ~ 2 2 kl

1 ~ ¼ 1 mH 2 wm ¼ ~ BH 2 2

ð1:13Þ

and write the Poynting formulation for the energy flux as ~ ~ S¼~ EH

ð1:14Þ

After adequate substitutions and transformations taking into account Maxwell’s equations we get, for the principal coordinate axes, D2x D2y D2z þ þ ¼ 8e0 pwe ¼ constant Ey Ex Ez

Ex  E11 ¼ 1 þ w11 Ey  E22 ¼ 1 þ w22 Ez  E33 ¼ 1 þ w33

ð1:15Þ

Following the definitions Dx x ¼ pffiffiffiffiffiffiffiffiffi we e0 Dy y ¼ pffiffiffiffiffiffiffiffiffi we e0 Dz z ¼ pffiffiffiffiffiffiffiffiffi we e0

with

n2x ¼ Ex ¼ ex =e0

n2y ¼ Ey ¼ ey =e0 n2z ¼ Ez ¼ ez =e0

we get the indicatrix formulation x2 y2 z2 þ þ ¼1 n2x n2y n2z

ð1:16Þ

where nx , ny and nz are the index of refraction along coordinates x, y, and z, respectively, as represented in Figure 1.1. To use this ellipsoid to analyze the

8

ELECTRO-OPTIC EFFECT

z nz

y ny

nx x

Figure 1.1. Refractive index ellipsoid.

propagation of a plane wave with propagation vector ~ k we just intersect the indicatrix with a plane orthogonal to the vector ~ k. An elliptic figure results where the extraordinary ne and ordinary n0 indexes, for this wave, are found from the intersection with the corresponding direction of vibration of the electric field as shown in Figure 1.2. In Section 1.2 we shall analyze Equation (1.16) in a more general form.

z nz k

ne

y nx

no

ny

x

Figure 1.2. Refractive indices for a plane wave propagating in an anisotropic medium.

TENSORIAL ANALYSIS

1.2

9

TENSORIAL ANALYSIS

Let us write the general equation [Nye, 1979] i¼N;j¼N X i¼1;j¼1

Sij xi xj ¼ 1

or Si;j xi xj ¼ 1

ð1:17Þ

where xi and xj are variables and Sij are coefficients. If we assume that Sij ¼ Sji , then Equation (1.17) turns into the general ellipsoid representation: S11 x21 þ S22 x22 þ S33 x23 þ 2S12 x1 x2 þ 2S13 x1 x3 þ 2S23 x2 x3 ¼ 1

ð1:18Þ

Equation (1.18) can be transformed into new coordinate axes x0i , by using the axes rotation transformation matrix, as follows x01 ¼ a11 x1 þ a12 x2 þ a13 x3

x02 ¼ a21 x1 þ a22 x2 þ a23 x3 x03 ¼ a31 x1 þ a32 x2 þ a33 x3

ð1:19Þ

which can be written in a matricial form 3 2 x01 a11 4 x02 5 ¼ 4 a21 x03 a31 2

a12 a22 a32

32 3 x1 a13 a23 54 x2 5 x3 a33

ð1:20Þ

From the matricial relation above we should deduce that it is also 3 2 a11 x1 4 x2 5 ¼ 4 a12 a13 x3 2

a21 a22 a23

32 0 3 x1 a31 4 5 x02 5 a32 x03 a33

ð1:21Þ

The relation above can be written in the form xi ¼ aki x0k

xj ¼ alj x0l

ð1:22Þ

which substituted into Equation (1.18) leads to Sij xi xj ¼ Sij aki alj x0k x0l ¼ S0kl x0k x0l

ð1:23Þ

where S0kl are the coefficients in the new coordinate system. An ellipsoid can be used to describe any symmetric tensor (Sij ¼ Sji ) of second order and is specially useful to decribe any property in a crystal that should be represented by a tensor. An important

10

ELECTRO-OPTIC EFFECT

property of an ellipsoid is the presence of ‘‘principal axes’’ in which case Equation (1.18) can be simplified to

S11 x21 þ S22 x22 þ S33 x23 ¼ 1

1.3

)

S11 4 Sij ¼ 0 0 2

0 S22 0

3 0 0 5 S33

ð1:24Þ

ELECTRO-OPTIC EFFECT

The indicatrix in Equation (1.16) is an ellipsoid in a principal coordinate axes system. Its general formulation is [Nye, 1979] Bij xi xj ¼ 1

with

Bij ¼

1 Eij

ð1:25Þ

The slight variation in the refractive index produced by an electric field can be described by the third-order electro-optic tensor rijk (in the range of 1012 m=V for most materials) through the relation Bij ¼ rijk Ek from Bij ¼ Bji ) rijk ¼ rjik

ð1:26Þ ð1:27Þ

The B tensor can be written as 2

3

2

3

B11

B12

B13

B1

B6

B5

6 4 B21 B31

B22 B32

7 6 B23 5 ¼ 4 B6 B33 B5

B2 B4

7 B4 5 B3

ð1:28Þ ð1:29Þ

The electro-optic relation is therefore simplified to Bi ¼ rij Ej

ði ¼ 1; 2; 3; 4; 5; 6; j ¼ 1; 2; 3Þ

ð1:30Þ

or explicitly written as 3 2 B1 r11 6 B2 7 6 r21 6 7 6 6 7 6 6 B3 7 ¼ 6 r31 6 7 6 4 ... 5 4... B6 r61 2

3 r12 r13 2 3 r22 r23 7 7 E1 7 r32 r33 74 E2 5 7 . . . . . . 5 E3 r62 r63

ð1:31Þ

11

ELECTRO-OPTIC EFFECT

y z

(001)

(110)

E3 E2

X3

x

X2

E1 X1

Figure 1.3. Crystallographic axes of a sillenite and an applied 3D electric field.

Let us assume that an electric field is applied, with components E1 ; E2 ; E3 as shown in Figure 1.3 so that Equation (1.25) turns into: ðB1 þ r11 E1 þ r12 E2 þ r13 E3 Þx21 þ ðB2 þ r21 E1 þ r22 E2 þ r23 E3 Þx22

þ ðB3 þ r31 E1 þ r32 E2 þ r33 E3 Þx23 þ ðB4 þ 2r41 E1 þ 2r42 E2 þ 2r43 E3 Þx2 x3 þ ðB5 þ 2r51 E1 þ 2r52 E2 þ 2r53 E3 Þx1 x3 þ ðB6 þ 2r61 E1 þ 2r62 E2 þ 2r63 E3 Þx1 x2 ¼ 1

ð1:32Þ

We are interested in the slow index of refraction buildup produced by the slow accumulation of electric charges. Therefore all the electro-optic coefficients referred to in this chapter are the low-frequency ones only. In the following sections we shall see what Equation (1.32) looks like for some particular materials. 1.3.1

Sillenite-Type Crystal

The well-known crystals of this family are: Bi12 GeO20 (BGO), Bi12 SiO20 (BSO), and Bi12 TiO20 (BTO). They belong to the cubic noncentrosymmetric crystal point class 23 and are piezo-electric, electro-, and elasto-optic and optically active. BTO is the crystal having the lowest optical activity (optical activity is undesirable for most applications) but is also the most difficult to grow because the chemical composition of the melt and the crystal are different—noncongruent. These crystals are usually grown using the so-called ‘‘top seed solution growth’’ (TSSG) that can be considered a modification of the Czochralski technique. Growing is more easily carried out along the [001]-crystal axis, and during growing there are frequently variations in the growing rate that produce the characteristic striations along the growing direction as shown in the picture in Figure 1.4. The latter result in small variations in the crystal composition and associated index of refraction changes as well. To avoid this index of refraction modulation being too visible through the polished (110)-face (the usual configuration employed for holographic recording) the latter should be cut slanted to these striations as illustrated in Figure 1.5. The axes in the sample are

12

ELECTRO-OPTIC EFFECT

Figure 1.4. Raw Bi12TiO20 boule grown by TSSG technique. The crystal was grown along the [001]-axis and the striations are clearly perpendicular to this axis.

[001] [100]

(0

11

)

[001] (110)

Figure 1.5. From raw Bi12TiO20 boule to ready-to-use crystal sample. Schematic representation of a raw crystal boule with its striations, indicating the way it will be sliced (top left); already sliced crystal with striations not perpendicular to the (011)-face (top right); ready-to-use crystal with renamed axes (bottom).

ELECTRO-OPTIC EFFECT

13

Figure 1.6. Undoped sillenite crystals. Bi12SiO20 crystal with (110)-surface cut and polished (center), raw Bi12TiO20 crystal boule grown along [001]-axis and showing striations on the lateral surfaces with both opposite (001)-faces cut and polished (left) and Bi12TiO20 crystal with (110)-face cut and polished, longer direction along [001]-axis (right).

conveniently renamed, accounting on its cubic and isotropic nature in which case the axes [001], [010], and [100], for example, can be interchanged. In the slantedsliced sample in Figure 1.5, the striations are not visible through the polished (110)-face. Figure 1.6 shows actual crystal samples. 2.70

n

2.65

2.60

2.55

2.50 450

500

550

600

650

700

λ (nm)

Figure 1.7. Index of refraction of BTO which is formulated by n ¼ 0:00863=l4 þ 0:0199=l2 þ 2:46 [Riehemann et al., 1997].

14

ELECTRO-OPTIC EFFECT

y z

(001)

(110) X3

X2 E

X

X1

Figure 1.8. Bi12SiO20-type cubic crystal orientation and its crystallographic axes X1 ; X2 and X3 . The electric field E applied along the ‘‘x’’-direction is also shown.

The electro-optic tensor of this crystal family in the principal axes coordinates [X1 ; X2 ; X3 ] has the following elements [Grousson et al., 1984]: r41 ¼ r52 ¼ r63  5  1012 m=V

ð1:33Þ

all other elements being zero. In the absence of electric field (E ¼ 0) the ellipsoid is x21 þ x22 þ x23 ¼1 n20

ð1:34Þ

showing that we are dealing with an isotropic crystal. Applying an electric field along direction ‘‘x’’ as indicated in Figure 1.8, we have the field components: pffiffiffi 2 E3 ¼ 0 ð1:35Þ E1 ¼ E2 ¼ E 2 so that the index ellipsoid is modified to: x21 x22 x23 þ þ þ 2r41 E1 x2 x3 þ 2r52 E2 x1 x3 ¼ 1 n20 n20 n20

ð1:36Þ

pffiffiffi 2 x21 x22 x23 ðx2 x3 þ x1 x3 Þ ¼ 1 þ 2 þ 2 þ 2r41 E 2 2 n0 n0 n0

ð1:37Þ

or

Let us now rotate the system from coordinates X1 ; X2 ; X3 to coordinates X; Y; Z pffiffiffi 2 ð1:38Þ x ¼ ðx1 þ x2 Þ 2 pffiffiffi 2 ð1:39Þ y ¼ ðx2  x1 Þ 2 z ¼ x3 ð1:40Þ

ELECTRO-OPTIC EFFECT

h z

h

h

z

z

nz

nh

nz

45°

x x

45°

15

x

nh no E

no z

E

z

z

Figure 1.9. Principal coordinate axes system Z   arising by the effect of an electric field E applied along the ‘‘x’’-axis, as shown in Fig. 1.8.

which substituted into Equation (1.37) with rearranging gives x2 y2 z2 þ þ þ 2r41 Exz ¼ 1 n20 n20 n20

ð1:41Þ

To eliminate the above term in ‘‘xz’’ it is necessary to carry out another rotation, now in the ‘‘x-z’’ plane as shown in Figure 1.9 pffiffiffi 2 x ¼ ðZ þ Þ 2 pffiffiffi 2 z ¼ ðZ  Þ 2

ð1:42Þ ð1:43Þ

which substituted into Equation (1.41) gives the relation



2

   1 y2 2 1 ¼1  r E þ Z þ r E þ 41 41 n20 n20 n20



ð1:44Þ

which means that the refractive indexes along the new axes , Z, and y are: 1 n ¼ n0 þ n30 r41 E 2 1 nZ ¼ n0  n30 r41 E 2 ny ¼ n 0

ð1:45Þ ð1:46Þ ð1:47Þ

for n0  n30 r41 E=2. The wavelength dependence of n0 for BTO is reported in Figure 1.7.

16

ELECTRO-OPTIC EFFECT

Exercise: Following the mathematical development above, show that for an electric field E along the axis [001] the principal axes of the index ellipsoid are directed along x, y, and z with the index ellipsoid having the form     1 1 z2 x2 2 þ r63 E þ y2 2  r63 E þ 2 ¼ 1 ð1:48Þ n0 n0 n0 thus meaning that, in the input crystal plane (110) that is also the x-z plane, the index of refraction changes only along x and is constant along z. GaAs, InP, and CdTe are also cubic noncentrosymmetric crystals though belong to the point class 43m but have the same electro-optic tensor structure as sillenites, that is to say, all elements are zero except r41 ¼ r52 ¼ r63 ¼ 1:72 pm=V for GaAs r41 ¼ r52 ¼ r63 ¼ 1:34 pm=V for InP r41 ¼ r52 ¼ r63 ¼ 5:5 pm=V for CdTe

ð1:49Þ

ð1:50Þ ð1:51Þ

The 43m symmetry, however, guarantees that there is no optical activity. The index of refraction of CdTe varies from 2.86 at l ¼ 1:06 mm to 2.73 at l ¼ 1:55 mm and follows the relation [Verstraeten, 2002]: n2 ¼ 4:744 þ 1.3.2

2:424l2 l2  282181:61

ð1:52Þ

Lithium Niobate

The electro-optic tensor in the principal axes system [X1 ; X2 ; X3 ] for this material has zero elements everywhere except the following [Weis and Gaylord, 1985]: r12 ¼ r22 ¼ r61  6:8 pm=V r13 ¼ r23 ¼ 10:0 pm=V r33 ¼ 32:2 pm=V

r42 ¼ r51 ¼ 32 pm=V

ð1:53Þ

For an electric field E3 applied along axis x3 as shown in Figure 1.10, tensorial Equation (1.32) becomes:       1 1 1 2 2 ð1:54Þ þ r13 E3 x1 þ 2 þ r13 E3 x2 þ 2 þ r33 E3 x23 ¼ 1 ne n20 n0 x1

c x3 x2

E3

Figure 1.10. Lithium niobate crystal with an applied electric field along the photovoltaic c-axis.

ELECTRO-OPTIC EFFECT

n0 + Dn2

17

n0 – Dn2

ne – Dn3

ne + Dn3 n0 – Dn1

n0 + Dn1

E3

E3

Figure 1.11. Lithium niobate crystal ellipsoid (black) and its modified (gray) size by the action of an applied field in opposite directions (left and right pictures) along the c-axis.

with n0 ¼ 2:286 and ne ¼ 2:200 at l ¼ 633 nm [Yariv, 1985] and the following relations   1 1 n3 ¼ 2 3 ðn1 Þ ¼ r13 E3 ) ðn1 Þ ¼  0 r13 E3 2 2 n1 n0   1 1 n3  2 ¼ 2 3 ðn2 Þ ¼ r13 E3 ) ðn2 Þ ¼  0 r13 E3 2 n2 n0   1 1 n3  2 ¼ 2 3 ðn3 Þ ¼ r33 E3 ) ðn3 Þ ¼  e r33 E3 2 ne n3



ð1:55Þ ð1:56Þ ð1:57Þ

and the index ellipsoid is modified as shown in Figure 1.11. 1.3.3

KDP-(KH2 PO4 )

This crystal is actually not a photorefractive one but is included here as an example of electro-optic tensor somewhat similar to that of sillenites. It has the following electrooptic tensor: 2

0 6 0 6 6 0 rij ¼ 6 6 r41 6 4 0 0

0 0 0 0 r52 0

3 0 0 7 7 0 7 7 0 7 7 0 5 r63

r41 ¼ r52 ¼ 8:6 pm=V

r63 ¼ 10:6 pm=V

The index of refraction for this material is reported Table 1.1.

TABLE 1.1. Index of Refraction of KDP l ðnmÞ 546 633

n0

ne

1.5115 1.5074

1.4698 1.4669

ð1:58Þ

18

ELECTRO-OPTIC EFFECT

The indicatrix equation formulated in the principal coordinate (crystallographic) axes X1 , X2 , and X3 , as represented in Figure 1.8, is x21 x22 x23 þ þ þ 2r41 E1 x2 x3 þ 2r52 E2 x1 x3 þ 2r63 E3 x1 x2 ¼ 1 n20 n20 n2e

ð1:59Þ

Let us assume an externally applied field E3 along axis x3 only. In this case we should proceed as for the case of Bi12 SiO20 in Figure 1.9 to get the following ellipsoid 

2

   1 y2 2 1 þ Z þ  r E þ r E ¼1 63 3 63 3 n2e n20 n20



ð1:60Þ

with 1 n ¼ n0 þ n30 r63 E3 2 1 nZ ¼ n0  n30 r63 E 2 ny ¼ ne

1.4

ð1:61Þ ð1:62Þ ð1:63Þ

CONCLUDING REMARKS

The aim of this chapter was just to recall some fundamental properties of optically anisotropic materials and the way an electric field is able to modify the index ellipsoid via the electro-optic effect. We have briefly shown how to calculate these effects in a few kinds of crystals having different electro-optic tensors. We hope these examples will enable the reader to understand how to operate on different materials, different crystals, and different optical configurations.

CHAPTER 2

PHOTOACTIVE CENTERS AND PHOTOCONDUCTIVITY

Photorefractives are electro-optic and photoconductive [Rose, 1963] materials, which means that electrons and/or holes are excited, by the action of light, from photoactive centers (donors or acceptors) somewhere inside the forbidden energy band gap to the conduction band (CB) (electrons) or valence band (VB) (holes), where they accumulate and diffuse away under the action of the diffusion gradient or are drifted in the presence of an externally applied electric field. After moving along an average diffusion length LD (or drift length LE in the case of an applied field) they are retrapped somewhere else, excited again, and retrapped again, and so on. Such a process leads, in the presence of a spatially modulated intensity of light onto the material, to charge carriers being progressively accumulated in the less illuminated regions whereas the more illuminated regions become oppositely charged. Such a spatial modulation of charged traps produces separation of electric charges and an associated electric (space-charge) field that is able to modify the index of refraction via electro-optic effect as explained in Section 1.3. The movement of charges under the action of the diffusion gradient is opposed by the growing space-charge field until an equilibrium is achieved. The presence of defects forming localized states in the band gap is therefore absolutely necessary to enable building up the space-charge field that is at the basis of the photorefractive effect. These defects may arise from doping (Fe in LiNbO3, for example) and are called ‘‘extrinsic.’’ Or they may be the so-called ‘‘intrinsic’’ defects, produced during the growing process, that result from missing atoms or atoms occupying the position of other different atoms in the crystalline structure. To Photorefractive Materials: Fundamental Concepts, Holographic Recording and Materials Characterization, By Jaime Frejlich Copyright # 2007 John Wiley & Sons, Inc.

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PHOTOACTIVE CENTERS AND PHOTOCONDUCTIVITY

confirm the role of defective growing in the final crystal properties, some researchers have already reported [Leigh et al., 1994; Wiegel and Becla, 2004] that Bi12SiO2 grown by hydrothermal methods produces almost perfect intrinsic crystals without photochromic and photorefractive properties whereas Czochralski and Bridgman– Stockbarger techniques, using the same raw chemicals, produce (defective enough) crystals with photorefractive properties. The interference of coherent laser beams is able to produce sinusoidally modulated patterns of light with small spatial periods of the order of the wavelength dimension of the recording light. Such small periods produce rather large diffusion gradients and consequently large opposing space-charge fields can be obtained in this way. Space-charge fields of a few kV/cm are easily produced in this way, and consequently rather large overall index of refraction changes can be observed. These effects may be produced by light with photonic energy hn high enough to excite charge carriers but lower than that of the band gap (Eg ) so that the material is rather transparent to this radiation. Therefore the recording is carried out in the whole material volume, and the recording beams are also able to detect the effect of their own action: They are refracted or diffracted by the index of refraction variation they are producing themselves in (almost) real time on the material volume. Of course, the whole process in the material volume depends on the distribution of light inside it, so that the bulk absorption and the light-induced absorption (if ever existing) effects must be accounted for. The spatial modulation of charge is in fact represented by a spatial distribution of acceptors that have received an electron and donors that have lost one. The dielectric polarizability of such filled and emptied photoactive centers (traps1) is not necessarily the same as that of their initial state. This means that the real (index of refraction itself) and the imaginary (extinction coefficient or absorption) part of the complex index of refraction may be also modulated via trap modulation in the material’s volume. Such so-called trap-arising index of refraction and absorption modulation are related to the electro-optical-based effect, but they are nevertheless additional effects of a different nature. We shall see further on that the index of refraction modulation arising from trap polarizability modulation and that arising from space-charge modulation are mutually p=2-phase shifted and both are, in general, also shifted from the recording pattern of light. There is still the possibility of finding an additional local index of refraction and absorption modulation effect arising from the direct action of light on the material without any relation to charge carrier excitation and trapping modulation. Both the trap-arising and the electric field-arising index of refraction modulations are essentially originating from a spatial modulation of electric charges but have different sizes, properties, and characteristics. The buildup of such a space-charge modulation is determined by the dynamics of electric charge transport in the material and is characterized by a time constant that depends, among other parameters, on the Maxwell (or dielectric) relaxation time tM that in turn is inversely proportional to the conductivity s. The relation between the holographic buildup time and the 1

Unless otherwise stated we shall use the term ‘‘trap’’ in its more general sense, meaning localized states in the band gap that are able to receive charge carriers.

PHOTOACTIVE CENTERS: DEEP AND SHALLOW TRAPS

21

conductivity makes holography a particularly interesting technique for the measurement of conductivity. In practice, however, these relations are somewhat more complicated because the recording and erasure of holograms are influenced by self-diffraction effects. The conductivity may also vary along the interelectrode distance because of the difficulty of avoiding nonuniform light distribution on the sample and may certainly also vary along the crystal thickness because of the nonuniform distribution of light produced by the bulk optical absorption effect [de Oliveira and Frejlich, 2000] the size of which will depend on the kind of material, the particular sample, and the light wavelength. The reader may foresee the difficult task that may be involved in the analysis of the experimental data depending on the characteristics of the sample under analysis. A sample exhibiting a behavior that can be understood using the so-called ‘‘one-center/two-valence/one-charge carrier’’ model is simple to analyze. However, some materials may require a ‘‘two-center’’ or a ‘‘one-center/three-valence’’ model, etc. [Buse, 1997]. Shallow and deep traps may coexist, and even hole-electron competition may appear. The mathematical model may become so complicated as to prevent a quantitative analysis unless considerable simplifications are accepted. This chapter starts with a brief description of photoactive centers in the band gap for some paradigmatic materials—CdTe, Bi12TiO20, and LiNbO3:Fe—in order to point out their complex nature and provide a more realistic background for better understanding the description of the theoretical models in the following sections.

2.1

PHOTOACTIVE CENTERS: DEEP AND SHALLOW TRAPS

In the following sections we shall describe some well-known photorefractive materials to illustrate the physical model involved as well as to provide some information about these materials, which will be studied in Part III of this book. 2.1.1

Cadmium Telluride

CdTe is a large band gap (1.6 eV at 4K to 1.5 eV at 300K [Verstraeten, 2002]) semiconductor of the II–VI family with a face-centered cubic structure, binary analog to diamond. The Cd-Te bonds are sp3-type atomic hybrid orbitals. Each atom is surrounded by a tetrahedron of the other atom species [Verstraeten, 2002]. CdTe is a well-studied material and will be analyzed here as an example in order to understand the effect of dopants (deep and shallow traps) in the properties of materials. Pure intrinsic CdTe is theoretically very resistive with very low dark conductivity. It exhibits intrinsic defects that are believed to be a Cd vacancy (VCd ) at about 0.4 eV above the VB, acting as an acceptor, and a Te occupying a Cd vacancy (Te in Cd antisite represented by the symbol TeCd ) at about 0.23 eV below the CB, acting as a donor [Verstraeten et al., 2003]. There are also extrinsic defects like Fe, Mn, etc. Cd vacancies give the p-type character to the dark conductivity. It is possible to increase the number of such vacancies by annealing under vacuum. It is also possible to reduce these Cd vacancies by annealing under Cd vapor atmosphere, but it is not

22

PHOTOACTIVE CENTERS AND PHOTOCONDUCTIVITY

CB 0.23 eV

1.6eV

Te+/ Te2+

V3+

EF

V2+

0.68eV 0.62eV VCd 0.4eV

VB

Figure 2.1. Energy diagram for a typical CdTe crystal doped with Vanadium, with the Te in the Cd anti-sites at 0.23eV below the CB and the Cd vacancies 0.4eV above the VB. [Verstraeten et al., 2003].

possible to completely eliminate them. In principle it is possible to perfectly compensate the Cd vacancies near the VB with TeCd donors close to the CB: The electrons from the latter will fill in the Cd vacancy acceptors, and dark conductivity will be strongly reduced. Once more this is not practical because one or the other defect will be always in excess. The excess of Te donors or Cd vacancy acceptors, however, can be compensated by doping with vanadium. In the absence of other dopants the V2þ =V3þ level is near the middle of the band gap and is considered to be a deep trap. For a sufficiently large concentration of V2þ and V3þ the Fermi level is pined to this V2þ =V3þ energy level. In the realistic example of Figure 2.1 the Fermi level is thus located at EF ¼ 0:68 eV above the VB with the V2þ slightly below (0.62 eV) and the V3þ slightly above EF . The Fermi level is here shown crossing the the V2þ distribution close to its upper end as well as the lower end of the V3þ distribution, thus indicating that the latter is a little bit filled with electrons whereas the former is a little bit emptied of electrons. The Fermi level is closer to the VB than to the CB so that dark conductivity is still predominantly by holes. In the presence of a small excess of either TeCd or Cd vacancies, the electrons from donors or the holes from acceptors are fixed in the deep vanadium level and by this means the free charge carriers can be strongly reduced, that is to say, dark conductivity can be reduced by doping CdTe with vanadium. Figure 2.2 shows the Arrhenius [Pillonnet et al., 1995] curve for a particular sample having a Fermi level almost exactly in the middle of the band gap. In this case the dark conductivity is probably the lowest possible one for this material. Under illumination we should expect the density of holes and electrons photoexcited to be similar. This, however, is not the case because the mobility of electrons is roughly 10-fold higher than for holes (me  10mh ). Furthermore, the density of V2þ

PHOTOACTIVE CENTERS: DEEP AND SHALLOW TRAPS

23

–3,00E+00 1,80E+00 2,00E+00 2,20E+00 2,40E+00 2,60E+00 2,80E+00 3,00E+00 3,20E+00 3,40E+00

–4,00E+00

logsigma

–5,00E+00 –6,00E+00 –7,00E+00 –8,00E+00 –9,00E+00 –1,00E+01

1000/T

Figure 2.2. Dark conductivity (arb. unit) measured at various temperatures for a CdTe:V crystal (labelled CdTeBR16B3) produced and measured by Dr. J.C. Launay, ICMCB-Bordeaux, France. From the Arrhenius plot the energy of the Fermi level EA ¼ 0:83eV is computed.

is usually larger than that of V3þ centers, thus increasing the influence of electrons in the process. 2.1.2

Sillenite-Type Crystals

For the case of photorefractive sillenites we shall focus on Bi12 TiO20 crystals. The band gap energy for all Bi12 GeO20 (BGO), Bi12 SiO20 (BSO), and Bi12 TiO20 (BTO) was determined to be Eg ¼ 3:2 eV (l  400 nm) at room temperature [Oberschmid,1985]. The same value was found for Bi12 GaO20. The fact that the absorption edge is the same for all four materials can be explained by assuming that an identical Bi-O lattice in all these crystals is responsible for the band gap. Their yellow color is due to a broad absorption shoulder between 2.3 and 3.2 eV. Such an absorption center may be due to an incorrect occupation of an M (M  Ge,Si,Ti) site in the oxygen tetrahedron by a Bi atom (BiM ): an antisite defect. The density of these centers in Bi12 GeO20 is lower (2- to 3-fold) than in Bi12 SiO20 , which in turn is lower than for Bi12 TiO20. This is also assumed [Oberschmid, 1985] to be a consequence of the Ge atoms being bound 0.1 eV stronger than for Si and the latter in turn being bound 0.16 eV stronger than Ti does. To meet electrical neutrality, the BiM center may be a Bi3þ ion with a bound electron defect hþ that is assumed to be resonantly distributed among the four oxygens in the tetrahedron around the BiM in the M vacancy: Bi3þ þ hþ . The latter defect is, at the same time, acting as electron donor þ 5þ  ðBi3þ M þ h Þ ! BiM þ e

ð2:1Þ

and as electron acceptor (hole donor) þ  3þ ðBi3þ M þ h Þ þ e ! BiM

ð2:2Þ

24

PHOTOACTIVE CENTERS AND PHOTOCONDUCTIVITY

3þ 5þ þ Bi3þ M and ðBiM þ h Þ are absorbing centers, whereas BiM is not. Sillenites exhibit dark p-type conductivity that is assumed to arise from the fact that the (Bi3þ þ hþ ) centers are closer to the VB than to the CB. The activation energy of these electron acceptor or hole donor centers was measured with impedance spectroscopy at high temperature that led to 0.99 eV for BTO [Lanfredi et al., 2000] and 0:48  0:02 eV for Bi12 GaO20 [Lobato et al., 2000]. It was measured to be 1.1 eV for BGO and BSO [Grabmaier and Oberschmid, 1986]. Direct measurement of dc dark conductivity in the range 50–130 C gave 1.06 eV for BTO [Marinova et al., 2003]. A hologram was recorded with 514.5-nm wavelength (probably in these deep centers), and its relaxation in the dark was measured [dos Santos et al., 2006] at different temperatures (from about 40 C to 90 C) to construct an Arrhenius curve (such as the one shown in Fig. 2.2 for CdTe) from which data an activation energy of 1.05 eV was obtained. This energy is close to that measured by several researchers but for plain dark p-type conductivity as reported above. This means that holographic relaxation in the dark is probably due to p-type conductivity. However, the photoconductivity þ of these materials is n type, probably arising from the same (Bi3þ M þ h ) centers at about 2.2 eV below the CB, which are now acting as electron donors. The n-type nature of this material under the action of light may arise from the larger cross section of donors in the 2.2-eV level for photons, the larger mobility of electrons in the CB, or a combination of both effects. Holograms can be recorded with light in the wavelength range of 514.5 nm(2.4 eV) to 780 nm(1.6 eV) but not with 1064 nm(1.16 eV), at least in undoped BTO. The holographic recording with light in this range is of n type, although there is no evidence of a populated electronic donor center at (or closer than) 1.6 eV below the CB so that we believe that direct (without preexposure) holographic recording with l ¼ 780 nm light occurs either a) directly at the Fermi level at 2.2 eV below the CB with a two-step mechanism to enable the electrons to be first pumped to an intermediate level between the Fermi level and the CB and from there on to the CB or b) at an intermediate level between the Fermi level and the CB that is populated from the Fermi level simultaneously during recording. In preexposure conditions instead, an intermediate level is probably filled from the Fermi level by the preexposure light and then the 780-nm-light holographic recording is carried out by electron excitation between this previously filled level and the CB. These materials also exhibit a strong photochromic darkening effect on illumination with light of wavelength at least in the 514.5- to 780-nm range, although the effect is decreasing with increasing wavelength. Photochromic darkening is a rather strong effect but a slow process that saturates at comparatively low light intensities, at least for the 532-nm and 514.5-nm wavelengths. This photochromic effect cannot be explained by the simple one-center model. In fact, the one-center model assumes that moderately low-intensity light onto the sample will not significantly change the total-to-acceptor trap density ratio but will just produce a spatial modulation in its value the spatial average of which will remain constant so that no photochromic effect can be detected under a uniform illumination. The twocenter model instead may allow for a kind of light-induced absorption coefficient or photochromic effect as will be seen below.

PHOTOACTIVE CENTERS: DEEP AND SHALLOW TRAPS

25

Modulated photoconductivity, photochromic measurement, and holographic recording, among other experiments, have indicated the presence of several localized states in the band gap of undoped BTO, among which is a shallow empty level at 0.42 eV (probably below the CB) that is responsible for photochromism and an electron donor center at 2.2 eV below the CB. Dark p-type conductivity was associated to an activation energy of about 1 eV; the latter is probably referred to the VB and, according to the 3.2-eV band gap, is probably the same electron donor level at 2.2 eV below the CB. This is probably the Fermi level associated with the position þ of the electron donor/acceptor (Bi3þ Ti þ h ) center referred to in Equations (2.1) and (2.2). At least a couple (or more) of empty levels (one certainly at 2.0 eV) should also be present between the 2.2 eV Fermi level and the CB to explain holographic recording using light with photonic energy as low as 1.6 eV. Other levels at 0.10, 0.14, and 0.29 eV, either located below the CB or above the VB, were detected by modulated photocurrent (MPC) techniques. Electron donor levels farther than 2.2 eV from the CB were also detected by wavelength-resolved photoconductivity (WRP). A possible representation [Frejlich et al., 2006a] of some of the relevant states in the band gap of undoped BTO is shown in Figure 2.3, where the 0.42-eV level conduction band 0.10 eV 0.14 eV 0.29 eV 0.42 eV

1.3 eV 1.4 eV 3.2 eV 1.7 eV 2.2 eV 2.0 eV

2.4 eV 2.5 eV 1 eV 2.7 eV 2.8 eV valence band

Figure 2.3. Possible position of some relevant localized states in the band gap of nominally undoped B12 TiO20 crystal.

26

PHOTOACTIVE CENTERS AND PHOTOCONDUCTIVITY

responsible for photochromism is shown as well as the 0.10, 0.14, and 0.29 eV that were arbitrarily placed close to the CB. Despite the practical interest in sillenites and the large number of publications on these materials, their actual nature is still poorly known and is a subject of active research, so the model of localized states in the band gap represented in Figure 2.3, as well as the nature of most of the photoactive centers involved, should be considered as tentative representations subject to revision. 2.1.3

Lithium Niobate

The deep trap centers in this material are known to be Fe2þ =Fe3þ at approximately 1 eV below the CB. There are also shallow traps due to defect Nb4þ Li centers [Malovichko et al., 1999] producing polaronic2 electron conduction with an activation energy 0.1–0.4 eV. There is still ionic conductivity (in as grown and in hydrogen doped) predominantly due to Hþ ions with characteristic activation energy of 1.2 eV. Hydrogen is located in the oxygen planes along the O–O bond, and its relative contents can therefore be avaluated as the strength of the OH stretching vibration absorption line near 2.87 mm [Vormann et al., 1981]. Above 70–80 C the ionic conductivity largely prevails over the Fe2þ electron detrapping-based dark conductivity. At temperatures below 60 C, dark conductivity is predominantly due to polaronic electrons from Nb4þ Li centers. For iron concentration higher than 0.05%wt Fe2 O3 , dark conductivity is predominantly due to tunneling of electrons between localized iron sites without significant influence of band transport [Nee et al., 2000].

2.2

PHOTOCONDUCTIVITY

The conductivity basically depends on the concentration of free charge carriers (electrons or holes) in the extended states (conduction or valence bands). In the presence of a relatively large band gap, as is the case with most photorefractive materials, the density of free carriers in the extended states largely depends on the number and quality of localized (photoactive) states in the band gap. We shall first analyze the effect of these localized states and then discuss two simple models referred to in the literature: the one- and two-center models. We shall then focus on the way the photoconductivity should be measured and the relation between the photocurrent and the photoconductivity. We shall also analyze the photochromic effects arising from the two-center model and relate the measured quantities (conductivity and absorption coefficients) with the theoretical parameters derived from the theory. 2

The electron placed in a elastic or deformable lattice produces a strain in the lattice. The electron plus the associated strain field is called a polaron. The displacement of this associated field increases the effective mass of the electron: For the case of KCl, for example, the electron mass is increased by a factor of 2.5 with respect to the band theory mass in a rigid lattice [Kittel, 1996].

PHOTOCONDUCTIVITY

CB

27

EC

EF = Eg/2 EF

Fermi Eg

EV VB

0

1

Figure 2.4. Intrinsic semiconductor: Fermi level for an intrinsic semiconductor and its ‘‘energy vs. occupation-of-states diagram’’ (right side). The electrons are represented by . and the holes by .

2.2.1

Localized States: Traps and Recombination Centers

It is worth recalling that, in an intrinsic semiconductor, the Fermi lever is exactly in the middle of the band gap as illustrated in Figure 2.4, with roughly 100% electronoccupied states below the Fermi level and zero above. The density of free electrons n in the conduction band (CB) and free holes h in the valence band (VB) are determined by the relations n ¼ NC eðEC EF Þ=kB T

h ¼ NV eðEF EV Þ=kB T

ð2:3Þ

where kB is the Boltzmann constant, T is the absolute temperature, NC and NV are the density of states (DOS) at the bottom and at the top of the CB and VB, respectively, EC and EV are the corresponding energies, and EF is the energy of the Fermi level. In the presence of a sufficiently large density of impurities, the Fermi level may be pinned to the position of these impurities, as illustrated in Figure 2.5, where all donor levels above the Fermi level are empty in equilibrium (as expected) in the dark, as depicted by the occupation of states (from zero to one) diagram shown on the right-hand side. In the example of Figure 2.5 the density of free holes is larger than that of free electrons because EF is closer to the VB than to the CB and we have assumed that NC  NV . This situation can be changed by the action of light. In fact, under the action of sufficiently energetic photonic light, charge carriers are excited so that initially empty localized levels become populated and the density of free carriers in the CB and/or VB also increases. To be able to account of these changes and still allow Equation (2.3) to be verified, steady-state Fermi (or quasi-Fermi) levels for electrons Efn and for holes Efp are defined [Rose, 1963] as depicted in Figure 2.6 with the occupation of states accordingly modified as represented by the righthand diagram [Simmons and Taylor, 1971]. The density of free carriers is now

28

PHOTOACTIVE CENTERS AND PHOTOCONDUCTIVITY

CB

EF acceptors

Eg

Fermi donors 0

VB

1

Figure 2.5. Doped semiconductor: Fermi level pined at the position of the dopant in the band gap. On the right-hand side is the ‘‘energy vs occupation-of-states’’ diagram.

as follows n ¼ NC eðEC Efn Þ=kB T

h ¼ NV eðEfp EV Þ=kB T

ð2:4Þ

In this condition, charge carriers in localized states between Efn and Efp are stable and remain in these states for a long time until recombination with an oppositely charged carrier. These levels are therefore called ‘‘recombination centers’’ and are illustrated in Figure 2.7. Localized states outside the Efn –Efp energy band easily relax their charge carriers to the nearest extended states and are called ‘‘traps,’’ as illustrated in

ILLUMINATION CB Efn EF

Fermi Eg

Efp

0

1

VB

Figure 2.6. Doped semiconductor: Fermi Ef and quasi-stationary Fermi levels upon illumination. The ‘‘energy vs occupation-of-states’’ graphics is shown on the right-hand side.

PHOTOCONDUCTIVITY

29

CB band-to-band recombination

recombination Efn

EF Fermi Efp

recombination

VB

Figure 2.7. Recombination centers.

Figure 2.8. In the case of sillenites, recombination centers produced by the action of light remain (at least partially) like that for hours, days, or weeks in the dark. 2.2.2

Theoretical Models

The behavior of all three examples (sillenites, CdTe, and LiNbO3:Fe) described in Section 2.1 can be adequately generalized by assuming a single species (one ion) with two different valence states like Fe2þ =Fe3þ for the case of lithium niobate in Section 2.1.3 or Bi3þ =Bi5þ for the case of sillenites in Section 2.1.2 and V2þ =V3þ for the case of CdTe in Section 2.1.1. Donors and acceptors are incorporated and/or formed in the material during the growing of the crystal in an electrically neutral local environment.

CB

trapping EF Efn Fermi Efp trapping VB

Figure 2.8. Traps.

30

PHOTOACTIVE CENTERS AND PHOTOCONDUCTIVITY

BAND GAP

CB

+ + + + + - + - + - + - +- + + +- + + +- - -

VB

Figure 2.9. Schematic representation of a material with one center (one single species) with two valence states (electron donors and electron acceptors) on two correspondingly slightly different localized states in the band gap. Electron acceptors are here represented as positively charged so that a nonphotoactive negative ion should be close to it in order to produce electrical neutrality at equilibrium for the as-grown crystal.

That is, in thermal equilibrium (in as-grown crystals) ions in their different valence states are therefore stabilized by an adequate environment to produce local electric neutrality so that they are certainly located at different energy positions in the band gap, as schematically illustrated in Figure 2.9, with acceptors above and donors below the Fermi energy level. Depending on their density these two levels may relevantly contribute to define the position of the material’s Fermi level. These donors and acceptors are intentionally shown in Figure 2.9 to be distributed along a finite energy bandwidth in the band gap, thus emphasizing that they do not occupy one energy position but a narrow energy band. Under the action of light of adequate wavelength, electrons are shown in Figure 2.10 to be excited from donors to the conduction band (CB), diffuse, or drifted (if there is an external electric field), and, after some time (photoelectron lifetime), they are likely to be retrapped somewhere else in available acceptors, be excited again, and so on. On average, the density of electrons in the CB increases by the action of light so that the n-type photoconductivity increases, too. A similar situation is described in Figure 2.11 where, besides electrons, holes are also excited [but to the valence band (VB)] by the light. The photoconductivity is here produced by electrons and holes, although electrons appear here to predominate. In other cases holes could predominate or the photoconductivity could even be due only to holes without electrons participating in the process.

PHOTOCONDUCTIVITY

31

CB -

-

BAND GAP

-

+ + + + + + - + - + - - + + + + + +

-

+ - + +

+ + -

VB

Figure 2.10. Under the action of light (of adequate wavelength) electrons are excited to the CB thus increasing the electron density in the CB and therefore increasing the n-type (photo)conductivity. In the CB they diffuse (or are drifted if there is an externally applied electric field) and are retraped (on the available acceptors) again and re-excited and so on.

CB

-

BAND GAP

-

+

-

+

+ + + + + + - - + + - - +- + - + - ++ + +

+

+

+ VB

Figure 2.11. In this example, under the action of light, electrons and holes are excited to the CB and VB respectively so that the photoconductivity is due to electrons and holes. In this case electrons do predominate but it could be the opposite as well, or even be only holes to be excited and the photoconductivity be of type-p.

32

PHOTOACTIVE CENTERS AND PHOTOCONDUCTIVITY

BAND GAP

CB

+

+

+ + + + - - +- - +- - - - - +- ++- - - +- + +

negative positive

+

negative

positive

negative

VB

Figure 2.12. Under nonuniform light, negative charges (in this case we assume to be electrons only) accumulate in the darker (less illuminated) regions leaving behind, in the more illuminated regions, opposite (positive here) charges.

Under nonuniform illumination, as shown in Figure 2.12, the charge carriers excited to the extended states (CB or VB) diffuse and/or are drifted and progressively accumulate in the darker (less illuminated) regions where the excitation rate by light is lower than in the brighter regions. It is important to realize that to produce an overall accumulation of electric charge in the illuminated volume it is necessary to have both donors and acceptors already available in adequately large concentrations. Otherwise, the excited charge carriers (electrons or holes) would have nowhere to be retrapped but to return back to emptied donors (for electrons) or filled acceptors (for holes) in the illuminated volume of the material where they were excited from. After the light is switched off, the (deep) trapped electrons remain where they are because thermal excitation is very low to excite them back to the CB at a sensible rate. The result is that the regions that were illuminated become positively charged whereas those that were less illuminated become negatively charged. If the charge carriers were holes, instead of electrons, the spatial distribution of charges would obviously be the opposite one. If charge carriers were both electrons and holes instead, without an externally applied field, there should be a mutual partial compensation of the spatial charge distribution in the material and even no charge accumulation at all could ocur in the hypotetical case of both electrons and holes being equally effective in the process. The participation of electrons and/or holes in this process is dependent on the presence of an externally applied electric field, the respective density of donors and acceptors, their respective cross section coefficients for the

PHOTOCONDUCTIVITY

33

illumination wavelength, and the mobilities of holes and electrons in their respective extended states (CB or VB). It is important to point out that the overall electric charge density variation, resulting from a local illumination, is due to the local (spatial) variation of donor/ acceptor densities independent of this variation being produced by eletrons, holes, or both in different proportions. That is, for one single system of donor/acceptor level, one single structure results even if donors and acceptors are placed on different energy levels. For the sake of simplicity we may sometimes show such donors and acceptors on the same energy level in the band gap just to emphasize the fact that we are handling one single species leading to one single structure of spatial trap modulation. For the case of sillenites and at least for undoped Bi12 TiO20 it is known that at least two distinct gratings are recorded under usual conditions. A single species (or single center) with two valence states cannot explain such a behavior, and more than one species should therefore be involved. In this case two independent modulated photoactive (two centers) systems can be produced by the action of light, and two gratings can be recorded each one of them involving electrons and/or holes as for the case of one single species discussed above. The particular expressions for the density of free electrons in the conduction band and for the photo- and dark conductivity depend on the theoretical model used to describe the material behavior. Here we shall analyse the two simplest models: one center and two center, always with one single charge carrier. Section 3.4.1 in Chapter 3 discusses the case of two different species (photoactive centers), one based on electron transport and the other based on hole transport. Much more complicated structures can be analized following the procedures used to handle these few rather simple examples. 2.2.2.1 One-Center Model. For the simplest one-center/one-charge carrier þ ) is assumed, in which case model, one single type photoactive center (ND1 , ND1 the charges and space-charge electric field are determined by the rate, continuity, and Poisson equations below @N ðx; tÞ ¼ G  R  ðr ~j Þ=q @t

ð2:5Þ

@NDþ ðx; tÞ ¼GR @t G ¼ ðND 

ð2:6Þ NDþ ðx; tÞÞ



sI þb hn



ð2:7Þ

R ¼ rNDþ ðx; tÞN ðx; tÞ

ð2:8Þ

~j ¼ eN ðx; tÞm~ Eðx; tÞ  qDrN ðx; tÞ

ð2:9Þ

r  ðEeo~ Eðx; tÞÞ ¼ qðNDþ ðx; tÞ  N ðx; tÞ  NA Þ

ð2:10Þ

34

PHOTOACTIVE CENTERS AND PHOTOCONDUCTIVITY

where e is the absolute value of the electron charge. For holes it is q ¼ e (and accordingly we should substitute NDþ by ND and so on), and for electrons it is q ¼ e. N is the free electron density in the conduction band, NDþ ðx; tÞ, ND are the density of ionized empty (electron acceptors) and total photoactive centers, respectively, and NA is the concentration of nonphotoactive negative ions that compensate for the initial positively NDþ -charged traps. Equations (2.5) and (2.6) are charge conservation equations, Equation (2.9) describes the charge current in terms of drift (the first term) and diffusion (the last term), Equations (2.7) and (2.8) describe photoelectron generation and recombination, respectively, and Equation (2.10) is the Poisson’s relation between charge density and electric field. The mobility and diffusion constant for electrons are m and D; respectively, s is the effective cross section for photoelectron generation, b is the thermal photoelectron generation coefficient, and E is the dielectric constant of the crystal. 2.2.2.1.1 Steady State Under Uniform Illumination. For steady-state equilibrium under uniform illumination, all time and spatial derivatives are zero, so that it is G ¼ R, r ~j ¼ 0 and r  ~ Eðx; tÞ ¼ 0. From Equations (2.7) and (2.8) we get ðND  NDþ Þ



sI þb hn



¼ rNDþ N

ð2:11Þ

with t  ðrNDþ Þ1

ð2:12Þ

being the photoelectron lifetime that is substituted into Equation (2.11) and leads to N ¼ ðND  NDþ Þt



sI þb hn



ð2:13Þ

We may describe the photoelectron generation, in terms of the absorbed light, as follows ðND  NDþ Þs ¼ a

ð2:14Þ

which substituted into Equation (2.13) leads to N ¼t

aI þ ðND  NDþ Þbt hn

dIabs dI ¼  ¼ aI dz dz

ð2:15Þ

with I ¼ I0 eaz where I0 is the incident irradiance and z is the coordinate along the crystal thickness. In this case aI is the effectively absorved irradiance per unit volume at z and  is the quantum efficiency for photoelectron generation. The

PHOTOCONDUCTIVITY

35

parameter t in Equation (2.12) is a constant if we assume that the effect of the light on NDþ and ND  NDþ is weak enough not to significantly affect their values. The concentration of free electrons in the conduction band in the dark and under the action of light are, respectively, N d ¼ ðND  NDþ Þbt N ph ¼ ðND  NDþ Þ

sI t hn

ð2:16Þ ð2:17Þ

Note that for a spatially uniform and constant illumination I0 it is 0

N ph ¼ ðND  NDþ Þt

sI0 hn

ð2:18Þ

The general expression for the conductivity is s ¼ qmN

ð2:19Þ

and the corresponding expressions for the photo- and dark conductivity are sph ¼ eðND  NDþ Þ

sI mt hn

sd ¼ eðND  NDþ Þbmt

ð2:20Þ ð2:21Þ

So far we have been dealing with the ‘‘one-center/one-charge carrier’’ model only. 2.2.2.2 Two-center/one-charge carrier model. This model is essentially related to the presence of shallow traps ND2 , as represented in Figure 2.13, which are certainly influencing the electrical conductivity in these materials. We assume þ represent the total density of the deep centers and [Buse, 1997] that ND1 and ND1 þ represent the density of the empty deep centers, respectively, whereas ND2 and ND2 the same but for the shallow centers, with ND2 being small enough so that the þ þ =ðND2  ND2 Þ may be strongly affected by the action of light, which is ratio ND2 not the case for the deep traps. We shall also assume that only one single type (electrons) of charge carrier is involved here. In equilibrium conditions shallow traps are empty, and under the action of light they start to be filled with electrons þ pumped from the deeper donor centers ND1 . The filled shallow traps ND2 –ND2 þ have a much larger absorption cross section than the filled deep traps ND1 –ND1 so that their filling produces a considerable increase in the overall absorption coefficient. This is the origin of the light-induced photochromic darkening. As the irradiance of the light increases, the density of filled shallow traps increases too, until nearly all of them are filled and the light-dependent absorption coefficient saturates.

36

PHOTOACTIVE CENTERS AND PHOTOCONDUCTIVITY ILLUMINATION

conduction band CB – – +

–

– + – + + – +

+

–

+

–

+

–

–

+ – + + – +

conduction band CB

–

–

+

–

+ + – + + – + + –

+

–

–

+

–

+

–

+

+

+

+

–

+

–

–

+

–

+

+

–

–

+

–

– +

–

–

+

–

+

–

E valance band VB –

– nonphotoactive

valance band VB + + ND1 acceptor + N+D2 acceptor

+

ND1–ND1 donor

– nonphotoactive

–

+

+ ND1 acceptor + N+D2 acceptor

+ ND1–ND1 donor

Figure 2.13. Photochromic effect and the Band-transport model. On the left side we are representing deep photoactive centers (acceptors and donors) and shallower centers close to þ . In this figure electron acceptors, the CB, with empty donors (acceptors) only, labelled ND2 both for deep and for shallow centers, are represented as positively charged so that a non photoactive negative ion should be close to these charged acceptors to ensure local electric neutrality. On the right side we see that under the action of light (represented by the arrows), the electrons are excited into the conduction band. Some of the electrons are retrapped to the þ þ þ ND1 and some others to the ND2 centers. The latter ones, that slowly relax to the deeper ND1 centers in the dark, have a higher light absorption coefficient and are therefore responsible for the photochromic darkening effect.

2.2.2.2.1 Steady State Under Uniform Illumination. In this case the rate equations in Section 2.2.2.1 should also include those for the shallow traps, always excluding spatial derivatives, as follows þ @ND1 ¼ G1  R1 @t

ð2:22Þ

þ @ND2 ¼ G2  R2 @t

ð2:23Þ

@N ¼ G1 þ G2  R1  R2 @t

ð2:24Þ

with Gi ¼ ðNDi 

þ NDi Þ



siI þ bi hn



and

þ Ri ¼ ri NDi N

i ¼ 1; 2

ð2:25Þ

þ þ For the limit conditions where either ND2 or ND2  ND2 are zero or close to zero, it is probably not possible to find the equilibrium by zeroing Equation (2.23). Instead, the equilibrium value for N may be found out by stating the rate equation for N þ @N @ND1 @N þ þ D2 ¼ @t @t @t

ð2:26Þ

PHOTOCONDUCTIVITY

and assuming the quasi-equilibrium condition

N ¼

þ ðND1  ND1 Þ

s I 1

hn

@N @t

37

¼ 0, so that we get

   þ Þ shn2 I þ b2 þ b1 þ ðND2  ND2 þ 1=t1 þ r2 ND2

ð2:27Þ

The corresponding expression in Equation (2.17) for this model should be written as N ph ¼

þ þ ðND1  ND1 Þs1 þ ðND2  ND2 Þs2 I þ hn 1=t1 þ r2 ND2

ð2:28Þ

We have two limit situations for Equation (2.27), always assuming that we are far from saturation for the deep traps (ND1 ):

The case where the irradiance is large enough to reach shallow trap saturation

þ ND2 )0

so that

N )

þ ðND1  ND1 Þ



   s1 I s2 I þ b1 þ ND2 þ b2 hn hn 1=t1

ð2:29Þ

The case where the irradiance is weak enough for the shallow traps to be empty   s1 I þ ðND1  ND1 Þ þ b1 hn þ ND2 ð2:30Þ ) ND2 so that N ) 1=t1 þ r2 ND2 The discussion above shows that the conductivity varies between two levels, with a lower value for low irradiances in Equation (2.30) and a higher value for a larger irradiance in Equation (2.29). In the general case, however, the photo- and dark conductivity can be calculated from Equation (2.27) as

sph ¼ em sd ¼ em

þ þ ½ðND1  ND1 Þs1 þ ðND2  ND2 Þs2  I þ hn 1=t1 þ r2 ND2

ð2:31Þ

þ þ ½ðND1  ND1 Þb1 þ ðND2  ND2 Þb2  þ 1=t1 þ r2 ND2

ð2:32Þ

It is still possible to be in the presence of hole–electron competition, in which case the formulation above should be modified. It is interesting to analyze the meaning of Equation (2.32): It states that after having been strongly illuminated, the dark conductivity is higher than its steady state in the dark. That is, illumination affects the dark conductivity too, which evolves until its lower steady-state value is reached.

38

PHOTOACTIVE CENTERS AND PHOTOCONDUCTIVITY

2.2.2.2.2 Light-Induced Absorption. Comparing Equation (2.28) with Equation (2.15) it becomes clear that a in the latter equation should, for the present model, be substituted by a ) a0 þ ali ðzÞ

ð2:33Þ

where a0 is a constant and ali (the so-called ‘‘light-induced’’ absorption) is an irradiance-dependent quantity that are defined as þ Þs1 a0 ¼ ðND1  ND1

ali ¼ ðND2 

þ ND2 Þs2

ð2:34Þ ð2:35Þ

2.2.2.3 Dark Conductivity and Dopants. As analyzed in Section 2.2.2.2, shallow photoactive centers are responsible for a higher dark conductivity immediately after the recording illumination is switched off, and in this way dark stability of recorded information is rapidly degradated. Such an effect may be compensated by the action of dopants in a deep level in the energy band gap as illustrated for the case of CdTe crystals [Verstraeten et al., 2003]. Such crystals exhibit shallow centers at approximately 0.2 eV below the CB and also at approximately 0.4 eV above the VB that are responsible for an enhancement of dark conductivity. The introduction of V3þ -V2þ impurities at a deep level, roughly in the middle of in the energy band gap, considerably reduces the influence of the shallow centers’ effect by acting as a sink for the electron donors and filling up the hole donors and, by this means, considerably reducing shallow trap-arising free charge carriers (electrons in the CB and holes in the VB) in the dark. This is also probably the case of BTO doped with Ru [Marinova et al., 2003], where sd decreases more than threefold from undoped to ½Ru ¼ 1019 cm3 Ru-doped samples.

Figure 2.14 Typical crystal view, in the so-called transverse configuration, with the electrodes (dark surfaces) separated by a distance l, the thickness (along the light propagation) is d, and the height is h.

PHOTOCHROMIC EFFECT

2.2.3

39

Photoconductivity in Bulk Material

In bulk samples as the one represented in Figure 2.14 the irradiance along the sample thickness (z-axis) varies considerably, and therefore the photoconductivity also varies. The measured overall photocurrent is therefore a kind of weighted average along the sample thickness that is related to the photoconductivity that we want to calculate. The z-dependence photoconductivity sph can be written as: sph ðzÞ ¼ qN ph ðzÞm

N ph ðzÞ ¼ ta

Ið0Þ az e hn

ð2:36Þ

where N ph is the density of electrons due to the action of light and is derived from Equation (2.17). For materials exhibiting light-induced absorption, the a in Equation (2.36) should be substituted by the expression in Equation (2.33). In any case the value of interest is the so-called photoconductivity coefficient hn

sph ð0Þ ¼ qmtða0 þ ali ð0ÞÞ Ið0Þ

ð2:37Þ

that is related to fundamental parameters of the crystal where all the quantities (s, I, and a) are computed at the input plane z ¼ 0 inside the crystal. 2.3

PHOTOCHROMIC EFFECT

As already pointed out above, photochromic effects are not expected for the onecenter model. Let us therefore refer to the two-center model including shallow traps. In this case the light-induced absorption ali has already been formulated in Equation (2.35) and is a function of the irradiance I. Photochromic effects are easy to measure and may give valuable information to be compared with that obtained from photoconductivity. To compute ali , we substitute Equation (2.27) into Equation þ , (2.23) and find the stationary equilibrium (that is the null time derivative) for ND2 þ þ which is only possible if we are far from the extremes ND2 ¼ ND2 or ND2 ¼ 0. In this case we get an expression   s1 I þ ND2 r2 ðND1  ND1 Þ þ b1 hn þ   ND2  ND2 ¼ 1 þ þ b2 =t1 þ r2 ðND1  ND1 Þb1 þ r2 ðND1  ND1 Þs1 =ðhnÞ þ s2 =ðhnÞ I t1

ð2:38Þ

that substituted into Equation (2.35) results in an expression for the light-induced absorption ali ¼

aI þ d bI þ c

ð2:39Þ

40

PHOTOACTIVE CENTERS AND PHOTOCONDUCTIVITY

s1 hn 1 s2 þ s1 Þ þ b  r2 ðND1  ND1 hn t1 hn b2 b þ c  þ r2 b1 ðND1  ND1 Þ 2 t1 t1 þ d  r2 ND2 ðND1  ND1 Þs2 b1  0 þ a  r2 ND2 ðND1  ND1 Þs2

ð2:40Þ ð2:41Þ ð2:42Þ ð2:43Þ

the limit values of which are lim ali ¼ I!0

lim ali ¼

I!1

þ d r2 ND2 s2 ðND1  ND1 Þt1 b1 b þ ¼ Þt1 1  0  r2 ND2 s2 ðND1  ND1 þ b2 c b2 þ r2 b1 t1 ðND1  ND1 Þ þ a ðND1  ND1 Þt1 r2 s1 ¼ ND2 s2 þ b ðND1  ND1 Þt1 r2 s1 þ s2

ð2:44Þ ð2:45Þ

where the approximated values above indicate that we assume that photoelectrons are mainly generated by the action of light on the deep traps and that thermally excited electrons are only produced from the shallow centers. In this case Equation (2.39) is simplified to ali ¼

aI bI þ c

ð2:46Þ

The typical darkening light-induced absorption in undoped Bi12 TiO2 (BTO) is observed in Figure 2.15, and the activation energy of these photochromic centers was measured, for the case of BTO (sample labeled BTO-8) by saturating the sample at 514.5 nm and then measuring the photochromic effect relaxation in the dark, using the Arrhenius [Pillonnet et al., 1995] law as shown in Figure 2.16, from which data it was found to be [dos Santos et al., 2002] 0.42 eV. 2.3.1

Transmittance with Light-Induced Absorption

The formulation of the transmitted light in the presence of light-induced absorption follow the usual pattern

b

ZI t I0

dI ða0 b þ aÞI þ a0 c a0 þ ali ¼ ¼ ða0 þ ali ÞI dz bI þ c ðbI þ cÞ dI ¼ dz I½ða0 b þ aÞI þ a0 c dI þc ða0 b þ aÞI þ a0 c

ZI t I0

dI ¼ d I½ða0 b þ aÞI þ a0 c

a=b ða0 þ a=bÞI0 þ a0 c=b It ln þ ln ¼ a0 d t I0 a0 þ a=b ða0 þ a=bÞI þ a0 c=b

ð2:47Þ

PHOTOCHROMIC EFFECT

41

Figure 2.15. Light-induced absorption spots produced in the center of an undoped Bi12 TiO20 crystal by the action of a thin l ¼ 532 nm laser line beam; the second spot is due to the beam reflected from the rear crystal face.

Figure 2.16. Photochromic relaxation time for BTO as a function of inverse absolute temperature. Arrhenius data fitting leads to an activation energy of 0:42  0:02 eV.

42

PHOTOACTIVE CENTERS AND PHOTOCONDUCTIVITY

0.6 y = +0.00142x1 +0.00345, max dev:2.69E–4, r2 = 1.00

Pt (µW)

0.4

0.2 y = +5.70E–4x1 + 0.0899, max dev:0.00, r2 = 1.00

0 0

200

400

600

800

P0 (µW)

Figure 2.17. Transmitted versus incident power for a thin (1.3 mm radius: P ¼ 800 mW corresponding to I ¼ 150 mW=m2 ) gaussian cross section uniform beam of 532 nm wavelength, shining a photorefractive Bi12 TiO20 crystal labelled BTO-010, of thickness 8.1 mm. Both beams are measured in the air outside tha sample. The data in the graphics are fit by a linear equation for the limits P0 ! 0 (black line) and P0 ! 1 (gray line) as shown in the graphics.

For the limit conditions we found some simple expressions: I t ¼ I0 ea0 d t

I ¼ I0 e

I0 ) 0

for

ða0 þabÞd

for

ð2:48Þ

I0 ) 1

ð2:49Þ

Figure 2.17 shows the transmittance curve for the BTO-010 crystal (circles) using the 532-nm wavelength. The angular coefficients are 0.00142 for the low irradiance limit Pt =Po ¼ ð1  RÞ2 ea0 d ¼ 0:00142 ) a0 ¼ 754:4 m1 and 5:70  104 for the high irradiance limit a

Pt =Po ¼ ð1  RÞ2 eða0 þbÞd ¼ 5:7  104 ) a0 þ

a ¼ 867:7 m1 b

and the reflectance R being R

ðn  1Þ2

ðn þ 1Þ2

ð2:50Þ

PHOTOCHROMIC EFFECT

43

In most photorefractive materials, as for the case of sillenites, the index of refraction is rather high, as reported in the graphics of Figure 1.7, so that losses by reflection should be carefully accounted for. From these data the parameters a0 and a=b in Equations (2.48) and (2.49) can be computed. Thus we get a0 ¼ 754:4 m1 and a=b ¼ 112:8 m1 . A more sophisticated mathematical fit of Equation (2.47) should allow us to adjust the theory over the entire range and get the parameter c=b in Equation (2.47) as well.

PART II

HOLOGRAPHIC RECORDING

46

HOLOGRAPHIC RECORDING

INTRODUCTION This second part of the book is devoted to holographic recording in photorefractive materials. These materials are particularly interesting for holographic recording, and many applications in this field and related fields have been and are currently being developped. Some of their advantages over other photosensitive recording materials are: almost real-time optical recording, reversibility, indefinite number of recordingerasure cycles, and very high spatial resolution. Also, the final recording state does not depend on the irradiance and on the total energy (time-integrated irradiance) but on the pattern of light modulation, and this is particularly interesting for recording with low levels of irradiance as is usually the case for image processing applications. Chapter 3 describes the recording of a space-charge electric field without caring about the associated index of refraction modulation, whereas Chapter 4 is devoted to the buildup of an index of refraction modulation in the material’s volume, that is to say, a phase volume hologram. Because of the real-time nature of the recording process, the hologram does diffract the recording beams during recording, thus modifying their relative amplitudes and their mutual phase-shift, which also modify the hologram being recorded and in turn further modifies the recording beams and so on. This kind of feedback process, called wave mixing or self-diffraction, is characteristic of real-time reversible recording materials and is also dealt with in Chapter 4. Holograms recorded in some materials show diffracted light having a polarization direction different from that of the transmitted light, and this subject is treated in Chapter 5. Chapter 6 is the last chapter in this part and describes a practical feedbackcontrolled stabilized holographic recording procedure that requires no external reference for stabilization and reduces environmental perturbations during recording, thus strongly improving the recording process. The process and its application to a couple of very representative materials are described in detail.

CHAPTER 3

RECORDING A SPACE-CHARGE ELECTRIC FIELD

This chapter is focused on the mechanisms responsible for the buildup of a modulated space-charge electric field under the action of a modulated pattern of light projected onto the sample, without considering wave-mixing effects, that is, without caring about the diffraction of the recording beams by the index of refraction modulation associated with the space-charge field hologram being built up. The theoretical model is based on the Band Transport theory and rate equations proposed by Kukhtarev and co-workers [Kukhtarev et al., 1979a, 1979b]. Before starting with the mathematical development let us qualitatively describe the processes involved. The material is usually characterized by a relatively large energy band gap compared to the recording light so the latter can go through the whole sample volume. Inside the band gap there are one or more localized states (photoactive centers) from where electrons and/or holes can be excited to the conduction band (CB) or to the valence band (VB) as illustrated by Figure 3.1. To ensure electrical neutrality in equilibrium, charged donors or acceptors should be in the close vicinity of an oppositely charged nonphotoactive ion. Under the action of a modulated pattern of light onto the crystal, electrons (for the sake of simplicity we shall assume that only electrons are involved) are excited to the CB, where they diffuse along the direction of their concentration gradient with a characteristic diffusion length distance, are retrapped again, are reexcited, and so on. After some time electrons are accumulated preferentially in the less illuminated regions because there they are less efficiently excited than anywhere else. A spatial distribution of electric charge is therefore built up, with exceeding positive charges being left in the Photorefractive Materials: Fundamental Concepts, Holographic Recording and Materials Characterization, By Jaime Frejlich Copyright # 2007 John Wiley & Sons, Inc.

47

48

RECORDING A SPACE-CHARGE ELECTRIC FIELD

light

light

light

BAND GAP

CONDUCTION BAND

- +- +- +- +- +- +- +- +- +- +- +- +- +- +- +-

+

VALENCE BAND

empty trap ACCEPTOR

filled trap DONOR

+ nonphotoactive ion

Figure 3.1. Photoactive centers inside the band gap. There are filled traps ND  NDþ (electron-donors), empty traps NDþ (electron-acceptors) and nonphotoactive ions (þ) to provide with local charge neutrality.

illuminated regions and negative charges in the less illuminated regions as illustrated in Figure 3.2. The spatial modulation of charge produces an associated space-charge electric field modulation, as illustrated in Figure 3.3, which is p=2 phase-shifted to the spatial modulation of charge because of the well-known Poisson’s equation relating charge and electric field. If the material is electro-optic besides being photoconductive, then the space-charge field modulation produces a corresponding modulation in the index of refraction, in phase with the field, as already described in Section 1.3 and illustrated in Figures 3.4 and 3.5. Under the action of an externally applied field the electrons move because of the electric drift besides the action of diffusion concentration gradient. Because of the nonsymmetric action of the drift, the resulting spatial modulation of charge is no longer in phase with the pattern of light modulation. A sinusoidal pattern of light that can be used for holographic recording is produced by the simple interferometric (or holographic) setup schematically illustrated in Figure 3.6. It is worth pointing out a general property of any holographic setup: The angular deviation a of the input laser beam produces a linear deviation of the pattern of fringes at the recording plane that is proportional to a2  L [Mollenauer and Tomlinson, 1977], where L is the optical path difference between the two interfering beams. To reduce such an inestability it is therefore highly recommended to reduce L as much as possible. The choice of L also depends on the coherence length of the laser in the setup: The latter should be much

RECORDING A SPACE-CHARGE ELECTRIC FIELD

light

light

49

light

BAND GAP

CONDUCTION BAND

- - +- + + +- - +- - +- + + +- - +- - +- + + +- - +-

+

overall overall negative positive charge charge

overall negative charge

overall positive charge

overall negative charge

overall positive charge

VALENCE BAND

-

empty trap ACCEPTOR

+

filled trap DONOR

nonphotoactive ion

Figure 3.2. Under the action of light the electrons are excited from the traps into the conduction band where they diffuse and are retrapped in the darker regions. A space modulation of electric charge results with overall positive charge in the illuminated and negative charge in the less illuminated regions.

longer than L, otherwise a poor pattern of fringes contrast or no fringes at all may be produced. The whole recording and reading process can be schematically described by Figures 3.7 to 3.14. The above qualitatively described processes will be developed in the following of this chapter on a quantitative mathematical basis. light

light

light

CONDUCTION BAND

BAND GAP

E

---

+

+

E

+

+

+

overall overall negative positive charge charge

E

E

----+

+

+

+

E

E

-----

+

overall overall negative positive charge charge

+

+

+

+

---

+

overall overall negative positive charge charge

+

overall negative charge

VALENCE BAND

Figure 3.3. The charge distribution produces a space-charge electric field modulation.

50

RECORDING A SPACE-CHARGE ELECTRIC FIELD

light

light

light

E

E

overall overall negative positive charge charge

E

E

overall negative charge

overall negative charge

overall positive charge

E

E

overall positive charge

overall negative charge

crystal lattice deformation

Figure 3.4. The electric field modulation may produce deformations in the crystal lattice.

3.1

INDEX OF REFRACTION MODULATION

Let us think about the way in which the space-charge field modulation may act on the index ellipsoid in order to produce the index of refraction modulation that is necessary to produce a volume grating in a photorefractive material. Let us take the example of sillenites as represented in Figure 1.9 and described in Equations (1.44)– (1.47). It is easy to understand that for any polarization direction of the reading beam (the incident beam that is diffracted by the grating in the crystal’s volume) the index

light

E

light

light

E

overall overall negative positive charge charge

E

overall negative charge

E

overall positive charge

E

overall negative charge

E

overall positive charge

index of refraction modulation

Figure 3.5. If the photoconductive material is also electro-optic, that is to say that it is photorefractive, the space-charge field may produce an index of refraction modulation in the crystal volume that is in-phase (or counterphase) with the space-charge field modulation and is p=2-shifted to the recording pattern of light.

INDEX OF REFRACTION MODULATION

Sh1

51

Sh3 BS

M1

laser

Sh2

M2

C D1

D2

Figure 3.6. Holographic setup: A laser beam is divided by the beamsplitter BS, reflected by mirrors M1 and M2 and interfering with an angle 2y. A sinusoidal pattern of light, as described in the text, is produced in the volume where these two beams interfere. A photorefractive crystal C is placed in the place where this pattern of light is produced. The irradiance of the two interfering beams are measured behind the crystal using photodetectors D1 and D2. Shutters Sh1, Sh2 and Sh3 are used to cut-off the main and each one of the interfering beams, if necessary.

of refraction will be changing as the space-charge field (~ E in Fig. 1.9) is changing also in value and sense. However, it is not so obvious to understand why the index of refraction modulation is invariant for any polarization direction of the reading beam, as far as the electro-optic configuration represented in Figures 1.3, 1.8, and 1.9 is

LASER BEAM

LASER BEAM ∆

INTERFERENCE PATTERN OF FRINGES OF PERIOD ∆

Figure 3.7. Generation of an interference pattern of fringes.

52

RECORDING A SPACE-CHARGE ELECTRIC FIELD

∆

Figure 3.8. Light excitation of electrons to the CB in the crystal.

+ + + + +

+ + + + +

+ + + + +

+ + + + +

+ + + + +

+ + + + +

Figure 3.9. Generation of an electric charge spatial modulation in the material.

+ + + + +

+ + + + +

+ + + + +

+ + + + +

+ + + + +

+ + + + +

Figure 3.10. Generation of a space-charge electric field modulation.

INDEX OF REFRACTION MODULATION

53

Pattern of fringes

+ + + + +

+ + + + +

+ + + + +

+ + + + +

+ + + + +

+ + + + +

Index of refraction modulation

Figure 3.11. The electric field modulation produces a index of refraction modulation (volume grating) via electro-optic effect.

incident beam

+ + + + +

+ + + + +

+ + + + +

+ + + + +

+ + + + +

+ + + + +

transmitted beam

diffracted beam

Figure 3.12. The recorded grating can be read using one of the recording beams which is transmitted and diffracted...

incident beam

+ + + +

+ + + +

diffracted beam

+ + + +

+ + + + +

+ + + +

+ + + +

transmitted beam

Figure 3.13. ...And the grating is erased during reading...

54

RECORDING A SPACE-CHARGE ELECTRIC FIELD

incident beam

transmitted beam

Figure 3.14. ...Until all recording is erased.

concerned. In fact, any lineraly polarized reading beam may be decomposed in two eigenwaves propagating along each one of the two principal axes Z and  in Figure 1.9. In one grating period the space-charge field will change from the maximum value along x (Fig. 1.9) to the maximum along the other direction. Therefore, the index of refraction variation along axis Z and along axis  in one grating spatial period will be, in absolute values, the same: 1 1 jn j ¼ jnZ j ¼ n30 r41 E  ðÞ n30 r41 E ¼ n30 r41 E 2 2

ð3:1Þ

In conclusion, as the index of refraction modulation along any of the two principal axes is the same, the proportion of the incident reading wave that is decomposed and propagated along each one of the principal axes does not affect at all the overall phase modulation of the reading wave. Therefore, the diffraction efficiency measured with the reading beam, as far as the index of refraction modulation is concerned, will be invariant with the direction of polarization of the reading wave. It should be noted that this conclusion is independent of optical activity or effect of any other nature on the diffraction efficiency. This result has been experimentally reported and also theoretically demonstrated on a more quantitative basis by several authors [Apostolidis et al., 1985; Mallick et al., 1987; Marrakchi et al., 1986]. Exercise : Following the development above, show that, differently than reported for the configuration in Figure 1.9, the diffraction efficiency for the case of lithium niobate, in the electro-optical configuration represented in Figures 1.10 and 1.11, depends on the polarization direction of the reading beam. 3.2

GENERAL FORMULATION

We shall here analyze the charge transport and associated equations for the particular case of an interference pattern of light being projected onto the sample as schematically illustrated in Figure 3.15.

55

GENERAL FORMULATION

space-charge field grating and associated hologram x R

S(0)

2θ

z

φ

S

∆

pattern of light

R(0)

K = 2Π/∆

d

Figure 3.15. Sinusoidal pattern of fringes and resulting space-charge field grating.

The interference of two plane waves of complex amplitudes of the form ~ ~ Sð0Þ ¼ ~ S0 eiðkS ~x þ fotÞ

ð3:2Þ

~ ~ Rð0Þ ¼ ~ R0 eiðkR ~x  otÞ

ð3:3Þ

produces a pattern of light onto the sample that is represented in Figure 3.15 and is described by I ¼ j~ Sð0Þ þ ~ Rð0Þj2 "

I ¼ ðj~ S0 j þ j~ R0 j Þ 1 þ 2 2

2

~ S0  ~ R0

jS0 j2 þ jR0 j2

cosðKx þ fÞ

#

ð3:4Þ

I ¼ I0 ½1 þ jmj cosðKx þ fÞ

ð3:5Þ

I ¼ I0 þ I0 =2½meiKx þ m eiKx 

ð3:6Þ

with the following definitions: IS0  jS0 j2

IS  jSj2

IR0  jR0 j2

m  jmjeif

jmj  2

IR  jRj2

I0 ¼ IR0 þ IS0

ð3:7Þ

with S0 R0 2

jS0 j þ jR0 j2

cosð~ S0 , ~ R0 Þ

ð3:8Þ

and K  2p= ¼ 2k sin y with k ¼ jk~S j ¼ jk~R j

ð3:9Þ

where o is the angular frequency of the light, k~S and k~R are the corresponding propagating vectors, 2y is the angle between the interfering beams, and  is the spatial period of the sinusoidal pattern of fringes. This pattern of light is projected on

56

RECORDING A SPACE-CHARGE ELECTRIC FIELD

the material in order to record an elementary hologram (grating), where S, IS and R, IR are the complex amplitudes and corresponding irradiances of each one of the two interfering beams. The index ‘‘0’’ indicates their values at the input plane. The quantity m is the so-called complex pattern of light fringes modulation. 3.2.1

Rate Equations

Unless otherwise stated the simplest ‘‘one-center two-valence one-charge carrier model’’ will be assumed as depicted in Figs. 3.1 to 3.5. The equations for this model were already formulated in Equations (2.5)–(2.10) as follows @N ðx; tÞ ¼ G  R  ðr ~j Þ=q @t @NDþ ðx; tÞ ¼GR @t   sI G ¼ ðND  NDþ ðx; tÞÞ þb hn

R ¼ gNDþ ðx; tÞN ðx; tÞ ~j ¼ eN ðx; tÞm~ Eðx; tÞ  qDrN ðx; tÞ

Eðx; tÞÞ ¼ eðNDþ ðx; tÞ  N ðx; tÞ  NA Þ r  ðEe0~

where e ¼ 1:6  1019 coul. For the case in which the charge carriers are electrons it is q ¼ e. For the case of holes it is q ¼ e with ND and ND  ND instead of NDþ and ND  NDþ , and NAþ instead of NA . 3.2.2

Solution for Steady State

We should now find a solution of the rate equations in Section 3.2.1 for the steady state. In this case all the time derivatives are zero, so that we can rewrite the rate equations as GR¼0 r ~j ¼ 0 ) j ¼ j0 ¼ constant

ð3:10Þ

N  N 0 ð1 þ m cos KxÞ

ð3:12Þ

ð3:11Þ

in which case we get

N0 

ND  NDþ gNDþ

s I0 hn

ð3:13Þ

where the dark excitation (b) was neglected and the spatial modulation of the trap ratio ðND  NDþ Þ=NDþ was also neglected compared to the pattern of fringes

GENERAL FORMULATION

57

modulation. In this way the density of free electrons N in the CB follows exactly the spatial pattern of the light as derived in Equation (3.12). From Equation (2.9) with j ¼ j0 we get j0 ¼ eN ðx; tÞmEðx; tÞ  qD

@N ðx; tÞ @x

ð3:14Þ

and substituting N and its spatial derivative by their expressions from Equation (3.12) we get an expression for the space-charge electric field E¼

j0 KD sin Kx þm emN 0 ð1 þ m cos KxÞ m 1 þ m cos Kx

ð3:15Þ

where E and j0 are the values of the respective x-component vectors that are the only ones in our unidirectional geometry. The integration of Equation (3.15) may help simplifying the above relations. In fact, the applied external voltage V0 is V0 ¼

ZL

ð3:16Þ

Edx

0

and the terms in Equation (3.15) are ZL 0

ZL 0

1 L dx ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ m cos Kx 1  m2 sin Kx dx ¼ 0 for 1 þ m cos Kx

ð3:17Þ

L  2p=K

ð3:18Þ

Accordingly we should write E0  V0 =L ¼

j0 1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffi emN 0 1  m2

ð3:19Þ

which substituted into Equation (3.15), together with the expression above, gives the electric field expression pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1  m2 sin Kx þ mED E ¼ E0 1 þ m cos Kx 1 þ m cos Kx

ð3:20Þ

58

RECORDING A SPACE-CHARGE ELECTRIC FIELD

with ED 

KD kB T ¼K m e

ð3:21Þ

where ED is the diffusion-arising space-charge field and E0 is the externally applied electric field . The result above shows that the sinusoidal pattern of fringes does not lead, in general, to a sinusoidal space-charge field. For the particular case of small pattern of fringes modulation (jmj 1), however, Equation (3.20) can be approximated to pffiffiffiffiffiffiffiffiffiffiffiffiffiffi ED sin 2Kx E  E0 1  m2 ð1  m cos KxÞ þ mED sin Kx  m2 2

ð3:22Þ

10

10

1.0

1.0

5

5

0.5

0.5

0

0

–5

–5 0.5

1.0 x (au)

E (au)

–10 0

–10 2.0

1.5

E (au)

E (au)

which contains the first and second harmonic terms in Kx. For a sufficiently small m, however, the second harmonic in m2 can be neglected. Figure 3.16 shows the theoretically computed shape of the space-charge field for different pattern of fringes visibility m: It is obvious that the field is completely asymmetric for m ¼ 0:99 and is rather sinusoidal for m ¼ 0:30. That is, we should rather consider a ‘‘first spatial approximation’’ only for m  0:3.

0

0

–0.5

–0.5 –1.0 0

0.5

0.4

0.4

0.2

0.2

0

1.5

–1.0 2.0

0

–0.2 –0.4 0

1.0 x (au)

–0.2 0.5

1.0 x (au)

1.5

–0.4 2.0

Figure 3.16. Space-charge electric field without externally applied field for a pattern of fringes with modulation m ¼ 0:99 (left), 0.60 (right) and 0.30 (center).

59

FIRST SPATIAL HARMONIC APPROXIMATION

3.3

FIRST SPATIAL HARMONIC APPROXIMATION

The procedure in Section 3.2.2 allows one to compute the space-charge field for an arbitrarily large pattern of fringes contrast m, but the calculation is limited to finding out the final stationary state only. In this section we will limit ourselves to m 1 but will be able to develop an expression for the temporal evolution, too. If the light modulation onto the crystal, as described by Equation (3.8), is sufficiently small (jmj 1), we may assume that N ðx; tÞ, NDþ ðx; tÞ, and the space-charge electric field Eðx; tÞ are all periodic real functions of coordinate x and may be described by their first Fourier series development term, the so-called ‘‘first spatial harmonic approximation,’’ as follows: N ðx; tÞ ¼ N 0 þ N 0 =2½aðtÞeiKx þ a ðtÞeiKx 

ð3:23Þ

NDþ ðx; tÞ ¼ NDþ þ NDþ =2½AðtÞeiKx þ A ðtÞeiKx 

ð3:24Þ

Eðx; tÞ ¼ E0 þ ð1=2Þ½Esc ðtÞeiKx þ Esc ðtÞeiKx 

NDþ

¼

NA

þ N0 

NA

ð3:25Þ ð3:26Þ

Substituting Equations (3.6) and (3.24) into Equation (2.7) one can write the generation term as: Gðx; tÞ ¼ G0 þ

G00 ½gðtÞeiKx þ g ðtÞeiKx  2

ð3:27Þ

where G0 ¼ ðND  NDþ Þ



G00 ¼ ðND  NDþ Þ



gðtÞ ¼

sI0 þb hn



sI0 þb hn



1

¼

1 NDþ sI0 =ðhnÞ ðAðtÞm þ AðtÞ mÞ þ 4 ND  ND sI0 =ðhnÞ þ b



ð3:28Þ

N0 t

ð3:29Þ

msI0 =ðhnÞ NDþ  AðtÞ sI0 =ðhnÞ þ b ND  NDþ

ð3:30Þ

For the case of small fringes visibility jAðtÞ mj 1 the expressions above are simplified to G0  G00 ¼ N 0 =t

ð3:31Þ

By substituting Equations (3.23) and (3.24) into Equation (2.8) an expression for the retrapping is also obtained: Rðx; tÞ ¼ R0 þ

R0 ½rðtÞeiKx þ r ðtÞeiKx  2

ð3:32Þ

60

RECORDING A SPACE-CHARGE ELECTRIC FIELD

where R0 ¼ gNDþ N 0 ¼

N0 t

ð3:33Þ

and rðtÞ ¼ aðtÞ þ AðtÞ

ð3:34Þ

For quasistationary conditions defined as @N ðx; tÞ ¼0 @t and substituted into Equation (2.5) we deduce G  R ¼ ðr ~j Þ=q

~j ¼ eN ðx; tÞm~ Eðx; tÞ  qDrN ðx; tÞ

ð3:35Þ

Substituting the corresponding terms in eiKx from Equations (3.27) and (3.32) into the expression in Equation (3.35) we get G0 R0 e Esc ðtÞ e N0 N0 gðtÞ  rðtÞ ¼ mN0 iK aðtÞiK  D ðiKÞ2 aðtÞ þ mE0 2 2 2 2 q 2 q which is rearranged to get aðtÞ explicitly as aðtÞ ¼

e=q Esc ðtÞiKmt þ msI0 =ðsI0 þ bÞ  AðtÞND =ðND  NDþ Þ 1 þ ie=q KtmE0 þ K 2 Dt

ð3:36Þ

Following the same procedure for Equation (2.6) we get NDþ

@AðtÞ N 0 sI0 m=ðhnÞ N 0 ND N0 aðtÞ  AðtÞ  ¼ t sI0 =ðhnÞ þ b t ND  NDþ t @t

ð3:37Þ

Also substituting the expressions in Equations (3.24) and (3.25) into Equation (2.10) with the assumption N 0 NDþ  NA , and solving for the terms in eiKx only, we get iKEe0 Esc ðtÞ  qNDþ AðtÞ

ð3:38Þ

Combining Equations (3.37) and (3.36) we get an equation only in AðtÞ: NDþ

@AðtÞ N 0 sI0 m=ðhnÞ N 0 ND ¼  AðtÞ @t t sI0 =ðhnÞ þ b t ND  NDþ

msI0 þ þ N 0 e=q Esc ðtÞiKmt  hn =ðsI0 =ðhnÞ þ bÞ þ AðtÞND =ðND  ND Þ þ t 1 þ ie=q KLE þ K 2 L2D

61

FIRST SPATIAL HARMONIC APPROXIMATION

Substituting AðtÞ above by its expression in Equation (3.38) we get



iKEe0 @Esc N 0 ¼ q @t t



 sI0 m=ðhnÞ iKEe0 Esc ND þ ð1 þ ie=q KLE þ K 2 L2D Þ sI0 =ðhnÞ þ b qNDþ ND  NDþ 1 þ ie=q KLE þ K 2 L2D

ND iKEe0 Esc sI0 m=ðhnÞ þ ie=q KmtEsc þ sI0 =ðhnÞ þ b ND  NDþ qNDþ þ 1 þ ie=q KLE þ K 2 L2D After rearranging terms the resulting expression for the space-charge electric field becomes:  @Esc ðtÞ q 1 msI0 =ðhnÞ ðE0  ie=q ED Þ þ Esc ðtÞ ¼ @t e tM ð1 þ K 2 L2D þ e=q iKLE Þ sI0 =ðhnÞ þ b 

e ND KEe0 KD e ND KEe0 E þi  1þ q ðND  NDþ ÞNDþ e m q ND  NDþ eNDþ



ð3:39Þ

where LD 

pffiffiffiffiffiffi Dt

LE  mtE0

ð3:40Þ ð3:41Þ

are the diffusion and drift length, respectively, and t  ðgNDþ Þ1

ð3:42Þ

tM  Ee0 =ðemN 0 Þ

ð3:43Þ

are the free electron lifetime and Maxwell (or dielectric) relaxation time, respectively, with kB being the Boltzmann constant. If we define an effective trap concentration as ðND Þeff  NDþ ðND  NDþ Þ=ND

ð3:44Þ

and substitute the above definitions into Equation (3.39), a simple formulation results: meff ðE0  ie=q ED Þ þ Esc ðtÞð1 þ K 2 l2s þ i qe KlE Þ @Esc ðtÞ ¼ @t tM ð1 þ K 2 L2D þ ie=q KLE Þ

ð3:45Þ

62

RECORDING A SPACE-CHARGE ELECTRIC FIELD

where meff  msI0 =ðhnÞ= K 2 l2s  ED =Eq ¼ KlE  E0 =Eq ¼ Eq 

eðND Þeff KEe0

  sI0 þb hn

K 2 Ee0 kB T e2 ðND Þeff

KEe0 E0 eðND Þeff

ð3:46Þ ð3:47Þ ð3:48Þ ð3:49Þ

where Eq represents the saturation space-charge field and ls is the Debye screening length. For the particular case where the charge carriers are electrons, it is q ¼ e and Equation (3.45) simplifies to @Esc ðtÞ meff ðE0 þ iED Þ þ Esc ðtÞð1 þ K 2 l2s  iKlE Þ ¼ @t tM ð1 þ K 2 L2D  iKLE Þ

ð3:50Þ

or tsc

@Esc ðtÞ þ Esc ðtÞ ¼ mEeff @t

ð3:51Þ

with Eeff 

E0 þ iED 1 þ K 2 l2s  iKlE

ð3:52Þ

1 þ K 2 L2D  iKLE 1 þ K 2 l2s  iKlE

ð3:53Þ

and: tsc  tM

For the following, unless otherwise stated, we shall always assume that the charge carriers are only electrons. Figure 3.17 shows the evolution of Esc during recording and erasure, in arbitrary units (au) with E0 ¼ 0 and tsc ¼ 10 au, as computed from Equation (3.51). 3.3.1

Steady-State Stationary Process

For stationary steady-state conditions, it is @Esc ðtÞ=@t ¼ 0, which substituted into Equation (3.51) gives the stationary space-charge field: Esc ðt ! 1Þ ¼ Esc ¼ meff Eeff

ð3:54Þ

63

FIRST SPATIAL HARMONIC APPROXIMATION

1.0 recording

ESC (t) (au)

0.8 0.6

erasure

0.4 SC

0.2 0

0

10

= 10 (au)

20 30 Time (au)

40

50

Figure 3.17. Simulated recording (from 0 to 20 au) and erasure (from 20 to 50 au) of a space-charge field with E0 ¼ 0 and tsc ¼ 10 au.

Unless otherwise stated we shall from here on always assume meff ¼ m. We shall also understand that ‘‘stationary’’ means that it is fixed in space, whereas ‘‘steadystate’’ means that it has reached an equilibrium with a time-invariant formulation, even if it includes a function of time, like a wave function. 3.3.1.1 Diffraction Efficiency. Figure 3.18 represents a pattern of fringes producing a spatial modulation of charges and an associated space-charge field of amplitude Esc that produces, via linear electro-optic effect, an index of refraction modulation of amplitude n1 , as defined, for example, in Equations (1.45) and (1.46). The index of refraction modulation is always in phase or counterphase

1.0 0.5

IRRADIANCE

0 –0.5 –1.0

0.5

0

1.0

1.0

1.5

ρ

DENSITY OF CHARGES

+

+

0

– 1.0

0

0.5

–0.5 0.5

⇒

90°

⇐

1.0

–1.0

1.5

SPACE-CHARGE FIELD ∆n

0.5 0 –0.5 –1.0

0

0.5

1.0

1.5

Figure 3.18. Index of refraction modulation arising in the crystal volume. The upper figure shows the pattern of light fringes projected onto the crystal, the mid figure shows the resulting charge density and the lower figure shows the spacial-charge field and index of refraction modulation. All vertical coordinates are in ‘‘arbitrary units’’.

64

RECORDING A SPACE-CHARGE ELECTRIC FIELD

with the space-charge field and represents a volume phase grating or hologram. The latter diffraction efficiency (Z) is computed, as described in detail in Chapter 4, from the well-known Kogelnik [Kogelnik, 1969] formula: 

ð3:55Þ

n1 ¼ ðn3 =2Þreff Esc

ð3:56Þ

Z ¼ sin

2



pn1 d l cos y

with

where reff is the effective electro-optic coefficient for the given crystal configuration, Esc is the amplitude of the space-charge electric field modulation, n is the average refractive index, l is the illumination wavelength, 2y is the angle between the incident beams inside the crystal, and d is the crystal thickness. Equation (3.55) assumes the simplifying approximation of a uniform index of refraction modulation along the sample’s thickness. 3.3.1.2 Hologram Phase Shift. The phase position (f) of the recording pattern of fringes is given by the phase of the complex modulation m in Equation (3.8), whereas that of the resulting hologram is given by the phase of the complex Esc . The so-called photorefractive hologram phase shift fP , which is the phase difference between the recording pattern of light and the resulting photorefractive hologram, is the phase of the complex quantity Esc =m. For steady-state conditions, as in Equation (3.54), where Esc ¼ mEeff , the phase shift fP is computed as tan fP ¼

=fEeff g
ð3:57Þ

From Equation (3.52) we get 1 E0 =ED ¼ tan fP 1 þ K 2 l2s þ K 2 l2s ðE0 =ED Þ2 Þ

ð3:58Þ

It is straightforward to show that fP ¼ p=2 for E0 ¼ 0 and fP ! 0 or p for E0 =ED  1. The maximum value of Equation (3.58) and the corresponding abscissa are respectively 

1 tan fP



1 ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 ð1 þ K 2 l2s ÞK 2 l2s M

and

½ðE0 =ED Þ2 M ¼ 1 þ

1 K 2 l2s

ð3:59Þ

that both depend on K 2 l2s / 1=ðND Þeff : The lower K 2 l2s the higher ½E0 =ED M and the closer fP to 0 (or 180 ), as illustrated in the simulation of Figure 3.19.

FIRST SPATIAL HARMONIC APPROXIMATION

65

0.9 Kls=0.5

1/tan φP

0.6

0.3

0

Kls=1

0

2

4

6

8

10

E0/ED

Figure 3.19. Mathematical simulation of 1/tan fp plotted as a function of E0 =ED for Kls ¼ 0:5 and for Kls ¼ 1.

3.3.2

Time-Evolution Process: Constant Modulation

For the general case, the hologram being recorded (because of the growing of a space-charge electric field modulation Esc ) modifies the pattern of light throughout the crystal volume so that the light modulation m is not at all constant but varies along the crystal thickness. In this case Equation (3.51) can be solved with the help of the coupled wave theory, as we shall show in Chapter 4. We shall here assume, however, that m in Equation (3.51) is constant. This may be approximately true for low diffraction efficiency and for no sensible energy exchange between the interfering beams as they propagate through the sample thickness. In this case Equation (3.51) is easily solved to give: Esc ðtÞ ¼ mEeff ð1  et=tsc Þ

ð3:60Þ

We shall here take into consideration the fact that tsc , Eeff , m, and consequently Esc, are all complex quantities, so that we should explicitly write



Esc ðtÞ Esc ðtÞ Esc ðtÞ þ i= ¼< m m m

2 Esc ðtÞ < ¼ 
=



Esc ðtÞ m



 =fEeff get
ð3:61Þ

ð3:62Þ

2

¼ =fEeff g½1  et
þ
ð3:63Þ

66

RECORDING A SPACE-CHARGE ELECTRIC FIELD

where


ð1 þ K 2 L2D Þð1 þ K 2 l2s Þ þ K 2 L2D ð1 þ

K 2 l2s Þ2

E02 ED E q

E2 þ 02 Eq

E0 E0  K 2 L2D Eq ED =ftsc g ¼ tM E2 ð1 þ K 2 l2s Þ2 þ 02 Eq

ð3:64Þ

ð3:65Þ

and
E0

ð3:66Þ

ð1 þ K 2 l2s Þ2 þ E02 =Eq2 ED ð1 þ K 2 l2s Þ þ E02 =Eq ð1 þ K 2 l2s Þ2 þ E02 =Eq2

ð3:67Þ

The evolution of Esc in modulus (jEsc j2 / Z) and phase is described by: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 jEsc ðtÞj ¼ jmjjEeff j 1 þ e2t
ð3:68Þ

=fEsc ðtÞ=mg
=fEeff g  ½
2 2

ð3:69Þ

We can compute the expressions for the initial and the stationary conditions: 2

tan fP ð0Þ ¼

tan fP ð1Þ ¼


Esc ð1Þ ¼ mEeff

2

Esc ð0Þ ¼ 0

for t ¼ 1

for t ¼ 0 ð3:70Þ ð3:71Þ

where the latter expression is, of course, the same as the one in Section 3.3.1.2. The expression in Equation (3.70) can be more easily deduced from Equation (3.50) for the limit t ! 0 (and consequently Esc ! 0) as Esc ðtÞ ¼ m

t E0 þ iED 2 tM 1 þ K L2D  iK 2 L2D E0 =ED

ð3:72Þ

STEADY-STATE NONSTATIONARY PROCESS

67

RUNNING HOLOGRAM V I n X

V

RESONANCE SPEED VO I

n

X VO

Figure 3.20. Schematic description of running hologram generation in photorefractives. A moving pattern of fringes onto the sample produces a synchronously moving volume hologram that reaches a maximum amplitude at a resonance speed.

so that 1
ð3:73Þ

The expressions in Equations (3.58) and (3.73), for the steady state and the inital hologram phase-shift phases respectively, can be used to determine some of the materials parameters, as discussed in Sections 8.2.2 and 9.1 and reported elsewhere [Garcia et al., 1989; Freschi et al., 1997]. Note that the steady-state phase shift in Equation (3.58) and the initial phase shift in Equation (3.73) have the same formulation except that K 2 L2D and K 2 l2s are interchanged. Also, it is shown below in Section 3.4 that the hologram phase shift of running holograms at resonance has exactly the same formulation as that of the initial phase shift in Equation (3.73).

3.4

STEADY-STATE NONSTATIONARY PROCESS

Running holograms in photorefractives were first reported in Bi12SiO20 crystals by Huignard et al. [Huignard and Marrakchi, 1981a], who pointed out the resonant behavior of the two-wave mixing amplitude gain and demonstrated its interest for coherent beam amplification and vibration measurement. Stepanov et al. [Stepanov et al., 1982] further developed the subject by reporting illustrative experimental

68

RECORDING A SPACE-CHARGE ELECTRIC FIELD

results and by establishing a sound theoretical basis to explain the main features. Refregier et al. [Refregier et al., 1985] analyzed these holograms with special attention to amplitude gain for sillenites and showed that the resonance velocity condition was particularly suitable for amplitude gain not only because of its characteristic large diffraction value but also because it exhibits a nearly 90 hologram phase shift that optimizes amplitude coupling. The research extended to semi-insulating semiconductor photorefractives like GaAs and InP [Valley, 1984; Brost et al., 1998; Imbert et al., 1988; Kumar et al., 1987]. The whole subject of moving holograms in photorefractives has been analyzed under the general approach of the so-called space-charge waves formalism [Sturman et al., 1993, 1995; Aubrecht et al., 1995; Pedersen and Johansen, 1999]. Running holograms were experimentally detected and measured in photorefractive materials under the action of a moving pattern of fringes in the presence of an externally applied electric field [Stepanov and Petrov, 1988]. Just to start understanding the matter before going to equations, let us think of an already recorded space-charge field with charge carriers being continuously excited to the CB and preferentially drifted (because of the external field) and retrapped along one sense of the applied field direction. This is just like inducing the grating to move along that sense. But the pattern of fringes is stationary so that the hologram does not move. If we allow the pattern of fringes to move along with the charge carriers (as illustrated in Figure 3.20) we will allow the grating to move. An optimum speed exists that depends on the drifting force of the external field and the response time of the material. By moving the pattern of fringes with such a speed a resonance is achieved where the moving grating amplitude is maximum. If the pattern of fringes moves faster or slower than this resonance speed, the recorded hologram will certainly follow the pattern of fringes speed, but a weaker grating will result. Let us now put these ideas into equations. We can theoretically find a solution for the moving holograms by just solving the same fundamental set of Equations (2.5)–(2.10), where the stationary pattern of fringes in Equation (3.5) is substituted by a moving pattern of fringes of the form I ¼ I0 ð1 þ jmj cosðKx  Kvt þ fÞÞ ¼ I0 þ ðI0 =2Þ½m expðiKx  iKvtÞ þ m expðiKx þ iKvtÞ

ð3:74Þ

where v is the speed of the fringes moving along the x-axis. In this case the development in Section 3.3 leads to a general equation of the form @Esc ðtÞ þ Esc ðtÞ ¼ mEeff eiKvt @t E0 þ iED 1 þ K 2 L2D  iKLE ¼ t ¼ t sc M 1 þ K 2 l2s  iKlE 1 þ K 2 l2s  iKlE tsc

with

Eeff

ð3:75Þ

Note that the expressions for Eeff and tsc above are the same as those for the stationary hologram in Equations (3.52) and (3.53), respectively. The difference is

STEADY-STATE NONSTATIONARY PROCESS

69

that here an exponential term eiKvt is factoring the expression for Eeff . The expression for tsc can be written as 1 1 1 þ K 2 l2s  iKlE ¼ oR þ ioI ¼ tsc tM 1 þ K 2 L2D  iKLE

ð3:76Þ

where the real and imaginary parts are explicitely formulated as oR ¼ oI ¼

1 ð1 þ K 2 l2s Þð1 þ K 2 L2D Þ þ KlE KLE tM ð1 þ K 2 L2D Þ2 þ K 2 L2E 1 KLE  KlE tM ð1 þ K 2 L2D Þ2 þ K 2 L2E

ð3:77Þ ð3:78Þ

The general solution of the differential equation in Equation (3.75) is  trans o t io t st iKvt Esc ðtÞ ¼ mEsc e þ Esc e R e I

with

st Esc ¼

Eeff ðoR þ ioI Þ oR þ ioI  iKv

ð3:79Þ

ð3:80Þ

The first term in Equation (3.79) is the steady-state solution. The second term is the transient solution that amplitude decays with a time constant oR . Note that oI is the natural oscillation frequency of the system, that is, the resonance frequency. The maximum value for the nonstationary (running) hologram is therefore achieved for Kv ¼ oI

ð3:81Þ

The quality factor for this resonant system is defined in the usual way as Q¼

oI K 2 L2D  K 2 l2s E0 ¼ 2 2 2 2 2 2 2 2 2 2 oR ð1 þ K ls Þð1 þ K LD Þ þ K LD K ls E0 =ED ED

ð3:82Þ

st The unit-modulation steady-state amplitude Esc in Equation (3.80) can also be written as st ¼ Esc

ð1 þ

K 2 L2D

E0 þ iED  iKLE ÞtM ðoR þ ioI  iKvÞ

ð3:83Þ

and as [Stepanov and Petrov, 1988] st Esc ¼

1þ

K 2 l2S

E þ iED ;  iKlE  itM Kvð1 þ K 2 L2D  iKLE Þ

ð3:84Þ

70

RECORDING A SPACE-CHARGE ELECTRIC FIELD

where its real and imaginary parts are st
E0 ð1  K 2 LE vtM Þ  ED ½KvtM ð1 þ K 2 L2D Þ

ð1 þ K 2 l2s  K 2 LE vtM Þ2 þ ½KlE þ KvtM ð1 þ K 2 L2D Þ2 ED ð1 þ K 2 l2s  K 2 LE vtM Þ þ E0 ½KlE þ KvtM ð1 þ K 2 L2D Þ ð1 þ K 2 l2s  K 2 LE vtM Þ2 þ ½KlE þ KvtM ð1 þ K 2 L2D Þ2

ð3:85Þ ð3:86Þ

which are particularly interesting because they determine, from Equation (3.57), the hologram phase shift fP 1 ½1 þ K 2 L2D þ K 2 L2D ðE0 =ED Þ2 ðoI  KvÞ þ oR ðE0 =ED Þ ¼ tan fp ½1 þ K 2 L2D þ K 2 L2D ðE0 =ED Þ2 oR  ðE0 =ED ÞðoI  KvÞ

ð3:87Þ

with its value at resonance (Kv ¼ oI ) being 

1 tan fP



Kv ¼ oI

¼

E0 =ED 1þ

K 2 L2D

þ K 2 L2D ðE0 =ED Þ2

ð3:88Þ

which is exactly the same as the expression for the initial phase in Equation (3.73). Figures 3.21–3.23 represent the theoretical values for the real and imaginary st , respectively, computed from Equation (3.80). parts and the square modulus of Esc It should be noted that Figure 3.23 actually shows a resonant behavior with a characteristic resonance frequency oI and a dissipative term oR with a quality factor Q ¼ oI =oR. Figure 3.24 shows the variation of Q as a function of LD and ls , whereas Figure 3.25 shows its dependence on K, for some typical parameter values.

1×106 500000

–10

–5

5

10

–500000 –1×106 st Figure 3.21. Computed real part of Esc (V/m) from Eq. (3.85), with Kv in rad/s in the horizontal axis, and the hypothetical parameters: LD ¼ 0:20 mm, ls ¼ 0:02 mm, K ¼ 10 mm1 , E0 ¼ 106 V/m, for the l ¼ 514:5 nm with a ¼ 11:5 cm1 ,  ¼ 0:5 and an intensity inside the front crystal plane I ¼ 100 W=m2 . Note that
STEADY-STATE NONSTATIONARY PROCESS

71

2×106 1.5×106 1×106 500000

–10

–5

5

10

st Esc

Figure 3.22. Computed imaginary part of (V/m) from Eq. (3.86) with Kv in rad/s in the horizontal axis and same parameters as in Figure 3.21. Note that resonance occurs at Kv  5 rad/s. st is zero is very close (although Note that the value of Kv where the real part of Esc not exactly) to the value where its imaginary part and its modulus are maximum as illustrated in Figure 3.26. These features are quite interesting and have practical consequences because the modulus is related to the diffraction efficiency and the imaginary part is connected to amplitude coupling in two-wave mixing (TWM) experiments, as analyzed in the following chapters.

3.4.1

Running Holograms with Hole–Electron Competition

It is possible to have electrons and holes excited simultaneously, by the action of light, in order to produce their correspondingly associated space-charge field gratings. If both electrons and holes are excited from the same photoactive species in the band gap, as illustrated in Figure 3.27, there is one single spatial trap (charges) modulation and the recording process follows a one-exponential law [Valley, 1986]

5×1012 4×1012 3×1012 2×1012 1×1012

–10

–5

5

10

Kv (rad/s)

st 2 Figure 3.23. Plot of jEsc j / Z for the hypothetical parameters: LD ¼ 0:20 mm, ls ¼ 0:02 mm, 1 K ¼ 10 mm , E0 ¼ 106 V/m, l ¼ 514:5 nm with a ¼ 11:5 cm1 ,  ¼ 0:5 and an intensity inside the front crystal plane I ¼ 100 W=m2 . In this case oI ¼ 5:1 rad/s and Q  2.

72

RECORDING A SPACE-CHARGE ELECTRIC FIELD

4 3 Q 2 1 0

0.05 0.04 0.03

0 0.02

0.2 0.4 LD (µm)

LS (µm)

0.01

0.6 0.8 1

0

Figure 3.24. Plotting of Q as a function of LD (LD-axis) and ls (LS-axis) for E0 ¼ 106 V/m, K ¼ 10 mm1 , l ¼ 514:5 nm with a ¼ 11:5 cm1 ,  ¼ 0:5 and an intensity inside the front crystal plane I ¼ 100 W=m2 .

where the characteristic exponential time depends on properties of both traps and carriers [Valley, 1986]. In the absence of external field the movement of these charge carriers is controlled by diffusion. In this case there should be one charge carrier predominating over the other in order for an effective space-charge modulation to be built up. Otherwise, both carriers will compensate each other and no actual charge separation will occur. Instead, if electrons and holes are excited from different species (photoactive centers), as illustrated in Figure 3.28, there may be an effective buildup of two physically distinct (each one on different centers) opposite-sign

Quality factor Q 2 1.5 1 0.5 K (µm–1) 2

4

6

8

10

Figure 3.25. Plotting of Q as a function of K, from Eq. (3.82), for typical values LD ¼ 0:15 mm, ls ¼ 0:03 mm, and different applied electric fields from 5  105 V/m, 7  105 V/m, 10  105 V/m to 15  105 V/m, represented by the progressively increasing size of the dashed lines, respectively.

STEADY-STATE NONSTATIONARY PROCESS

73

2 1.5 1 0.5 0 –0.5 –1 –5

0

5 Kv (rad/s)

10

15

Figure 3.26. Mathematical simulation of
gratings and the overall recording dynamics may follow a two-exponential law [Valley, 1986], one for each one of the species, which in this case also correspond to two different charge carriers. There is still the very realistic possibility that both electrons and holes are excited from both photoactive species with holes dominating one and electrons dominating the other. It is still possible that the two species are able to allow holes and electrons to be excited but these species are placed somewhat differently in the band gap with the photonic energy being small enough to allow exciting only electrons from one species and only holes from the other as schematically depicted in Figure 3.29. ILLUMINATION CB

–

+

–

VB – – NA

+

+

–

nonphotoactive

–

+

–

–

+

+ –

–

–

–

+ + +

–

+

+ N+D acceptor

+

–

+

–

+

+

ND–ND donor

Figure 3.27. One-species/two-valence/two-charge carrier model contributing to charge trasport: One single spatial trap modulation structure is produced.

74

RECORDING A SPACE-CHARGE ELECTRIC FIELD

ILLUMINATION CB

–

–+

+

–

–+–

–

–+

–+ – –+

– + +

–

–

N+A2 nonphotoactive

+ +

–

+

+ N+D acceptor + ND2–ND2 acceptor

– NA– nonphotoactive

–+

+

–

+

+

VB

+

+

+

+

+–

–

–

– –+

+

+

+ +

–

–

ND–N+D donor + donor – ND2

Figure 3.28. Two-species/two-valence/two-charge carrier model contributing to charge transport: Two distinct spatial trap modulation structures are produced.

CB

–

– – –

+

+

+

++ + –

–

+

+ –

+

–

+

–

+

–

– –

–

+

+ –

–

+

+ –

+

+ –

+ –

VB

Figure 3.29. Hole–electron competition on different photoactive centers under the action of low energetic photon recording light: Only charge carriers close to the CB (electrons) and to the VB (holes) can be excited but neither electrons can be excited from the hole-donor level nor holes from the electron-donor level, because of energy considerations. In this case and electron-base hologram is recorded in the level closer to CB and same for holes in the level close to VB. However, electrons progressively accumulate in the (deeper) level closer to the VB and holes accumulate in the level close to the CB, where they cannot be re-excited again because the recording light is not energetic enough. The recording is progressively decreasing, because of the decrease in the corresponding charge-carriers, until a steady state is achieved because of the exhaustion of any one of the two levels.

STEADY-STATE NONSTATIONARY PROCESS

75

3.4.1.1 Mathematical Model. We shall here handle the case where a holebased and an electron-based independent grating are recorded (each one on different species or centers, with holes and electrons originating from different centers) and for the case where there is a mechanism for providing with reversibility. The simultaneous presence of electron and hole photoactive centers produces an electrically coupled system of equations that, under certain conditions, can be analytically solved. Considering the formulation in Equation (3.74) for the pattern of fringes being I ¼ I0 ð1 þ jmj cosðKx  Kvt þ fÞÞ

¼ I0 þ ðI0 =2Þ½m expðiKx  iKvtÞ þ m expðiKx þ iKvtÞ

and from the above assumption that electrons and holes are originated from different and independent photoactive centers, we should write the same system of fundamental Equations (2.5)–(2.8) for the electrons @N ðx; tÞ ¼ G  R þ ðr  ~ JÞ=e @t @NDþ ðx; tÞ ¼GR @t ~ J ¼ eN ðx; tÞm~ Eðx; tÞ þ eDrN ðx; tÞ   sI þ G ¼ ðND  ND ðx; tÞÞ þb hn R ¼ gNDþ ðx; tÞN ðx; tÞ

and an independent similar system of equation for holes @Pðx; tÞ ¼ G2  R2  ðr  J~2 Þ=e @t  @ND2 ðx; tÞ ¼ G2  R2 @t Eðx; tÞ  eD2 rPðx; tÞ J~2 ¼ ePðx; tÞm2~   s2 I  ðx; tÞÞ þ b2 G2 ¼ ðND2  ND2 hn  R2 ¼ g2 ND2 ðx; tÞPðx; tÞ

ð3:89Þ ð3:90Þ ð3:91Þ ð3:92Þ ð3:93Þ

The coupling between holes and electrons arises from the Poisson equation  e0 Er  ~ E ¼ eðNDþ þ P  NA  ND2  N þ NBþ Þ

ð3:94Þ

76

RECORDING A SPACE-CHARGE ELECTRIC FIELD

We neglect thermal excitation and assume a linearized set of equations for all parameters involved, of the same form as those in Equations (3.23)–(3.25), now also including similar expressions for the holes and their photoactive centers, which are represented by the subindex ‘‘2’’ whereas ‘‘1’’ is for the electrons. The solution of the above system of coupled equations starts with the usual assumption of quasistationary condition: @N =@t  @P=@t  0. In this case we are able to write þ Þ G1  R1 ¼ ðND1  ND1

s1 I0 iKvt s1 I0 þ þ me N A1 ðtÞ  N 0 ND1  A1 ðtÞg1 =2 2hn 2hn D1

þ  N 0 a1 ðtÞND1 g1 =2  G2  R2 ¼ ðND2  ND2 Þ

ð3:95Þ

s2 I0 iKvt s2 I0 þ  me N A2 ðtÞ  P 0 ND2  A2 ðtÞg2 =2 2hn 2hn D2

  P 0 a2 ðtÞND2 g2 =2

ð3:96Þ

r ~j1 ¼ r  ðN 0 m1 Esc ðtÞ=2 þ N 0 a1 ðtÞm1 E0 =2Þ  D1 K 2 N 0 a1 ðtÞ=2 e r ~j2 ¼ r  ðP 0 m2 Esc ðtÞ=2 þ P 0 a2 ðtÞm2 E0 =2Þ þ D2 K 2 P 0 a2 ðtÞ=2 e

ð3:97Þ ð3:98Þ

Substituting Equations (3.96) and (3.98) into Equation (3.89) and proceeding in a similar way with the electrons we get s1 I0 m iKvt s1 I0 þ þ þ N A1 ðtÞ  N 0 ND1 e A1 ðtÞg1  N 0 a1 ðtÞND1 g1  hn D1 hn em N 0 þ em N 0  ND1 A1 ðtÞ þ 1 ND2 A2 ðtÞ  iKN 0 a1 ðtÞm1 E0 ¼ D1 K 2 N 0 a1 ðtÞ  1 Ee0 Ee0

þ Þ ðND1  ND1

s2 I0 m iKvt s2 I0    e N A2 ðtÞ  P 0 ND2 A2 ðtÞg2  P 0 a2 ðtÞND2 g2  hn hn D2 em P 0  em P 0 þ A2 ðtÞ þ 2 ND1 A1 ðtÞ þ iKP 0 a2 ðtÞm2 E0 ¼ D2 K 2 P 0 a2 ðtÞ  2 ND2 Ee0 Ee0

þ ðND2  ND2 Þ

From the above two equations we get the relations between ai ðtÞ and Ai ðtÞ: 

þ a1 ðtÞ½N 0 ND1 g1 D1 K 2 N 0 þ iKN 0 m1 E0 ¼ A1 ðtÞ

þ

em1 N 0  s1 I 0 þ mðND1  ND1 ND2 A2 ðtÞ  ÞeiKvt Ee0 hn

 g2  D2 K 2 P 0  iKP 0 m2 E0  ¼ A2 ðtÞ a2 ðtÞ½N 0 ND2

þ

em2 P 0 þ s2 I 0  mðND2  ND2 N A1 ðtÞ  ÞeiKvt Ee0 D1 hn

s1 I 0 þ em N 0 þ þ ND1 þ N 0 ND1 g1  1 ND1 hn Ee0



ð3:99Þ   s2 I0  em P 0   ND2 þ P 0 ND2 g2  2 ND2 hn Ee0 ð3:100Þ

STEADY-STATE NONSTATIONARY PROCESS

77

Substituting Equation (3.95) into Equation (2.6) and proceeding similarly with the equations for holes, we get the two equations þ ND1

@A1 ðtÞ @t

¼ A1 ðtÞ  ND2

@A2 ðtÞ @t

¼ A2 ðtÞ



 s1 I 0 þ þ s1 I 0 þ þ ND1 þ N 0 ND1 meiKvt  N 0 ND1 Þ g1 þ ðND1  ND1 g1 a1 ðtÞ hn hn



 s2 I 0   s2 I 0   ND2 þ P 0 ND2 meiKvt  P 0 ND2 Þ g2 þ ðND2  ND2 g2 a2 ðtÞ hn hn

Substituting the values of a1 ðtÞ and a2 ðtÞ in the equations above by their expressions computed from Equations (3.99) and (3.100) and also using the linearized relation þ  A1 ðtÞ  ND2 A2 ðtÞÞ iKEe0 Esc ðtÞ  eðND1

from Equation (3.94), we get the two coupled differential equations @Esc1 ðtÞ þ Esc1 ðtÞ ¼ mEeff1 eiKvt  k12 Esc2 ðtÞ @t @Esc2 ðtÞ tsc2 þ Esc2 ðtÞ ¼ mEeff2 eiKvt  k21 Esc1 ðtÞ @t

tsc1

ð3:101Þ ð3:102Þ

with the usual parameter definitions ¼

1 þ K 2 l2s1  iKlE1 ð1Þ ð1Þ ¼ oR þ ioI tM1 1 þ K 2 L2D1  iKLE1

ð3:103Þ

¼

1 þ K 2 l2s2  iKlE2 ð2Þ ð2Þ ¼ oR þ ioI tM2 1 þ K 2 L2D2  iKLE2

ð3:104Þ

Eeff1 ¼

E0 þ iED1 1 þ K 2 l2s1  iKlE1

ED1 ¼ K

ð3:105Þ

Eeff2 ¼

E0 þ iED2 1 þ K 2 l2s2  iKlE2

ED2 ¼ K

1 tsc1 1 tsc2

1 1

kB T e kB T e

ð3:106Þ

KLE1 ¼ K 2 L2D1 E0 =ED1

ð3:107Þ

KLE2 ¼ K 2 L2D2 E0 =ED2

ð3:108Þ

KlE1 ¼ K 2 l2s1 E0 =ED1

ð3:109Þ

KlE2 ¼ K 2 l2s2 E0 =ED2

ð3:110Þ

78

RECORDING A SPACE-CHARGE ELECTRIC FIELD

and the new definitions 1=k12 ¼ 1 þ K 2 l2s1  iKlE1

ð3:111Þ

Esc ðtÞ ¼ Esc1 ðtÞ þ Esc2 ðtÞ

ð3:113Þ

1=k21 ¼ 1 þ

K 2 l2s2

 iKlE2

ð3:112Þ

with

Let us search for a steady-state solution of Equations (3.101) and (3.102) of the same form as in Equation (3.79): st iKvt Esc1 ðtÞ ¼ mEsc1 e

and

st iKvt Esc2 ðtÞ ¼ mEsc2 e

that substituted into the coupled equations give st st st ¼ Esc1 þ Esc2 Esc

i 

h ð2Þ ð2Þ ð1Þ ð1Þ oR þ ioI oR þ ioI ð1  k21 Þ  iKv     ¼ Eeff1 ð2Þ ð2Þ ð1Þ ð1Þ ð1Þ ð2Þ ð1Þ oR þ ioI  iKv oR þ ioI  iKv  oR þ ioð2ÞI oR þ ioI k12 k21 i 

h ð1Þ ð2Þ ð1Þ ð2Þ oR þ ioI oR þ ioI ð1 k12 Þ iKv     þ Eeff2 ð2Þ ð2Þ ð1Þ ð1Þ ð2Þ ð2Þ ð1Þ ð1Þ oR þ ioI  iKv oR þ ioI  iKv  oR þ ioI oR þ ioI k12 k21 ð3:114Þ

where the effect of holes and electrons on the space-charge field grating are coupled. If we assume that the coupling constants are sufficiently small so that k12  k21  0; Equation (3.114) can be written as a pair of independent terms, each one depending on one type of charge carrier only

st Esc ¼ Eeff1

ð1Þ

ð1Þ

oR þ ioI

ð1Þ

ð1Þ



oR þ ioI  iKv

þ Eeff2

ð2Þ

ð2Þ

oR þ ioI

ð2Þ

ð2Þ



oR þ ioI  iKv

ð3:115Þ

It is interesting to find out what the expression in Equation (3.114) would look like for the steady-state stationary limit condition where Kv ¼ 0. In this case we have Equation (3.114) simplified to st Esc ¼ Eeff1

I  K21 I  K12 þ Eeff2 I  K12 K21 I  K12 K21

ð3:116Þ

PHOTOVOLTAIC MATERIALS

79

In the absence of externally applied field Equation (3.116) becomes st ¼K Esc

kB T K 2 l2s1  K 2 l2s1 2 e K 2 ls1 þ K 2 l2s2 þ K 2 l2s1 K 2 l2s2

ð3:117Þ

and for nonsaturated conditions (K 2 l2s1 1 and K 2 l2s2 1) we get st Esc ¼K

kB T K 2 l2s2  K 2 l2s1 e K 2 l2s1 þ K 2 l2s2

ð3:118Þ

which leads to zero effective space charge field if the Debye lengths are similar for electrons and holes. If the concentration of one of the centers (for ex. holes) is much lower than the other (K 2 l2s2  K 2 l2s1 ) instead, the overall space-charge field is dominated by the nondepleted centers, as expected st Esc ¼K

kB T 1 e 1 þ K 2 l2s1

ð3:119Þ

which in this case are the electron-based centers. More complicated situations including wave mixing and bulk light absorption effects, which usually occur in photorefractive materials, do not usually lead to analytical solutions. The solution for a particular simple case is treated in Section 8.2.5.

3.5

PHOTOVOLTAIC MATERIALS

Space-charge electric field buildup in photovoltaic crystals exhibits specific features that need special attention. Let us write the continuity and the Gauss equation @r ¼0 r ~j þ @t ~¼r rD

ð3:121Þ

~j ¼ qD @N þ ^xkph Iabs þ ½sd þ sph ^xEðxÞ d @~ x

ð3:122Þ

ð3:120Þ

with

where the first, second, and third terms in the right-hand side of Equation (3.122) are the diffusion, photovoltaic, and ohmic components, respectively, sd is the dark conductivity, sph is the photoconductivity, and Iabs is the light intensity absorved by

80

RECORDING A SPACE-CHARGE ELECTRIC FIELD

TABLE 3.1. Photovoltaic Transport Coefficient jph Dopant

Fe

wavelength (nm) kph  109 A cm/W

514.5 3.0

Cu 472.7 4.8

514.5 0.55

472.7 0.96

from [Glass et al., 1974]

the sample. The ^x is the unit vector along coordinate axis x. The kph is the photovoltaic transport coefficient that was found [Glass et al., 1974] to depend on the nature of the absorbing center and the wavelength as reported in Table 3.1. From Equations (3.120), (3.121), and (3.122) above we get ~ @D r  ~j þ @t

!

¼0

ð3:123Þ

and ~ ~jðx; tÞ þ @ Dðx; tÞ ¼ ~ j0 @t

ð3:124Þ

For the following we shall assume all vectors along x-coordinate only.

3.5.1

Uniform Illumination: @N =@x ¼ 0

We shall analyze the electric field buildup with uniform illumination for a shortcircuited and for an open-circuited sample:

Short Circuit: E ¼ D ¼ 0 j0 ¼ kph Iabs =d

ð3:125Þ

Open Circuit: j0 ¼ 0 Ee0

kph Iabs @E ¼0 þ sE þ d @t

ð3:126Þ

so that: E ¼ Eph ð1  expðt=tsc ÞÞ

ð3:127Þ

where: Eph ¼

kph Iabs sd

and

tsc ¼ Ee0 =s;

s ¼ sd þ sph

ð3:128Þ

PHOTOVOLTAIC MATERIALS

81

From the expressions for Iabs in Equation (2.15) and for s  sph in Equations (2.20) and (2.12) we may rewrite the expression for Eph as Eph ¼

kph hnr þ ND me

ð3:129Þ

Note that the open circuit leads to a progressive electric polarization under uniform light illumination that is opposite to the photovoltaic field Eph and may therefore even compensate the latter, preventing any further optical recording [Huignard et al., 1975]. 3.5.2

Interference Pattern of Light

The presence of a modulated pattern of light IðxÞ ¼ I0 ð1 þ m cosðKxÞÞ and the resulting self-diffraction effects do not allow for an analytical solution in this case. To get an analytical solution, self-diffraction as well as diffusion (qD@N =@x ¼ 0) are neglected

Open circuit: j0 ¼ 0 Ee0

kph Iabs @E ð1 þ m cosðKxÞÞ ¼ 0 þ sE þ d @t

ð3:130Þ

so that E ¼ Eph ð1 þ m cosðKxÞÞð1  expðt=tsc ÞÞ

Short circuit:

R‘ 0

ð3:131Þ

E  dx ¼ 0

In this case the crystal is usually short-circuited with conductive silver glue as represented in Figure 3.30. Integrating the expression j0 ¼ Ee0

kph Iabs @E ð1 þ m cosðKxÞÞ þ sE þ d @t

C

ð3:132Þ

C

Figure 3.30. Short-circuit schema using conductive silver glue to electrically connect the opposite faces along the photovoltaic axis ~ c (left) and open-circuit schema, without any electrical connection (right).

82

RECORDING A SPACE-CHARGE ELECTRIC FIELD

from x ¼ 0 to x ¼ ‘, where ‘ is the interelectrodes distance, we get the expression in Equation (3.125) j0 ¼

kph Iabs d

Equating Equations (3.132) and (3.125) we get an expression for the electric field: E ¼ mEph ð1  expðst=ðEe0 ÞÞÞ cosðKxÞ

ð3:133Þ

3.5.2.1 Influence of Donor Density. The formulation of the photovoltaic effect as stated in Equation (3.122) does not take into account the influence of the electron donor density that is also involved. The correct formulation of the photovoltaic effect in the current density instead is: ~j ¼ qD @N þ ^x qL~ph sðND  NDþ Þ I þ ^x qN mE @~ x hn Lph ¼ mtEph

ð3:134Þ ð3:135Þ

where Lph is formulated in a way similar to LE ¼ mtE0 in Equation (3.41). All calculations can be repeated now with Equation (3.134) instead of Equation (3.122), all other hypothesis being maintained. The space-charge field time-derivative expression in Equation (3.50) now becomes:   N þ sI0 meff ðE0 þ Eph þ iED Þ þ Esc ðtÞ 1 þ K 2 l2s  iKlE  iKlph D @Esc ðtÞ ND sI0 þ b ¼ @t tM ð1 þ K 2 L2D  iKLE Þ

ð3:136Þ

with Klph ¼ Eph =Eq

meff ¼ msI0 =ðsI0 þ bÞ

ð3:137Þ

or written in a more compact form tsc

@Esc ðtÞ þ Esc ¼ meff Eeff @t

ð3:138Þ

with Eeff 

E0 þ Eph þ iED N þ sI0 1 þ K 2 l2s  iKlE  iKlph D ND sI0 þ b

ð3:139Þ

PHOTOVOLTAIC MATERIALS

83

with Klph ¼ Eph =Eq

meff ¼ msI0 =ðsI0 þ bÞ

ð3:140Þ

and 1 ¼ oR þ ioI tsc

oR 

1 1 þ K 2 l2s tM 1 þ K 2 L2D

oI 

ð3:141Þ

Klph 1 NDþ sI0 2 tM ð1 þ K 2 L2D Þ ND sI0 þ b

ð3:142Þ

Note that oI represents the phase shift speed of the grating under uniform illumination, so that the total phase shift during the characteristic time tM is, from Equation (3.141): tM oI ¼

Klph ð1 þ

K 2 L2D Þ2

NDþ ND

ð3:143Þ

CHAPTER 4

VOLUME HOLOGRAM WITH WAVE MIXING

The space-charge electric field modulation that is produced by the action of a pattern of light as described in Chapter 3 also produces an associated real-time index of refraction modulation due to the electro-optic effect analyzed in Chapter 1. As a consequence, a phase grating results that, because of its real-time nature, diffracts the light during the recording process itself, thus modifying the recording pattern of light that in turn further affects the recorded grating and so on in a feedback process. This feedback phenomenon is known as ‘‘self-diffraction’’ or ‘‘wave mixing’’ and is the subject of this chapter.

4.1

COUPLED WAVE THEORY: FIXED GRATING

We shall first make a short review of the coupled wave theory that deals with light diffraction by a fixed-volume hologram and then extend this theory for the case of a dynamic reversible recording material where diffraction and recording occur simultaneously and therefore a feedback is established between both processes. Following Kogelnik theory [Kogelnik, 1969], let us represent a fixed volume grating ~ thickness d, and a reading beam of amplitude R incident of period , wavevector K, at the angle y (always measured inside the grating volume), as described in Figure 4.1. The reading beam is one of the two beams previously used to record this same hologram as shown in Figure 4.2, where the index of refraction modulation

Photorefractive Materials: Fundamental Concepts, Holographic Recording and Materials Characterization, By Jaime Frejlich Copyright # 2007 John Wiley & Sons, Inc.

85

86

VOLUME HOLOGRAM WITH WAVE MIXING

HOLOGRAM x

R θ

R(0)

z

θ

θ S

∆ d

K = 2Π/∆

Figure 4.1. Reading the recorded hologram with one of the recording beams with y measured inside the recording material volume.

pattern is shifted by fP referred to the recording pattern of light as represented in Figure 4.2. We also assume that the average index of refraction of the grating is the same as that of the surrounding medium. Let us also assume the presence of a fAshifted fixed-amplitude grating (not represented in the figure). Let us write the wave equation inside the material [Fowles, 1975] r2  þ k 2  ¼ 0

ð4:1Þ

with   o2 s k ¼ 2 1þwþi c oe0 2

n2 ¼

c2 s ¼1þwþi v2 oR e0

E1þw

ð4:2Þ

Kogelnik writes the above relations in the form k2 ¼

o2 E þ ioms c2

b2 ¼

o2 E0 c2

mcs0 a ¼ pffiffiffiffi 2 E0

ð4:3Þ

hologram S(0)

x

2θ

φ

R(0)

pattern of light

R

z ∆ d

S K = 2Π/∆

Figure 4.2. Recording a fixed volume index-of-refraction hologram that is phase shifted by f ¼ fP referred to the recording pattern of fringes.

COUPLED WAVE THEORY: FIXED GRATING

87

The index of refraction and amplitude gratings are represented by the modulations E1 and s1 in the dielectric constant E and in the conductivity s, respectively, as follows ~

~ ~ r þ fP Þ ¼ E0 þ E1 E ¼ E0 þ E1 cosðK

~

eiðK~rþfP Þ þ eiðK~rþfP Þ 2 ~

~ ~ s ¼ s0 þ s1 cosðK r þ fA Þ ¼ s0 þ s1

ð4:4Þ

~

eiðK~rþfA Þ þ eiðK~rþfA Þ 2

ð4:5Þ

Substituting Equations (4.4) and (4.5) into the formulation for k2 in Equation (4.3) we get the expression ~

~

k2 ¼ b2 þ i2ba þ 2bðkþ eiK~r þ k eiK~r Þ

ð4:6Þ

where kþ and k are defined below. Searching a solution for Equation (4.1), having the form ~

 ¼ RðzÞei~r~r þ SðzÞeid:~r

ð4:7Þ

with the Bragg condition (see, e.g., p. 388 in [Yariv, 1985]), ~ þ~ K d ¼~ r

ð4:8Þ

with j~ rj ¼ j~ dj ¼ 2p=l

and

K ¼ 2p=

represented in Figure 4.3 with the assumption of a weak coupling @2R @2S  2 0 @z2 @z

ð4:9Þ

and following the development of Kogelnik, for TE polarization, we get the

ρ K

δ

Figure 4.3. Bragg condition where ~ r and ~ d are the incident beam and the diffracted beam ~ is the grating wave vector. wave vectors respectively (or vice versa) and K

88

VOLUME HOLOGRAM WITH WAVE MIXING

coupled equations cos y

@R þ aR ¼ ikþ S @z

ð4:10Þ

@S þ aS ¼ ik R @z

ð4:11Þ

cos y with

    1 oE1 ifP ms1 c 1 on ifP e þ iaeifA pffiffiffiffi e þ i pffiffiffiffi eifA ¼ E0 4 c E0 2 c     1 oE1 ifP ms1 c 1 on ifP k ¼ þ i pffiffiffiffi eifA ¼ þ iaeifA e pffiffiffiffi e E0 4 c E0 2 c kþ ¼

ð4:12Þ ð4:13Þ

where

E1 n ¼ pffiffiffiffi 2 E0

mcs1 a ¼ pffiffiffiffi 2 E0

ð4:14Þ

From the equations above Kogelnik showed that the diffraction efficiency of an unslanted grating of purely index-of-refraction nature is Z ¼ sin2



pn d l cos y



ð4:15Þ



ð4:16Þ

whereas for a purely absorption grating it is Z ¼ sinh

2



a d 2 cos y

Both Equations (4.15) and (4.16) do not consider the effect of average bulk absorption that is consistent if we define Z ¼ I d =ðI d þ I t Þ in terms of the beams (transmitted I t and diffracted I d ) behind the crystal.

4.1.1

Out of Bragg Condition

The calculations above assume that the incident reading beam exactly matches the Bragg condition represented by Equation (4.8), which can be also written as 2k sin y ¼ K

ð4:17Þ

However, the incident beam can be shifted away from this condition because of a mismatch y of the incidence angle or because of a mismatch in the wavelength l

DYNAMIC COUPLED WAVE THEORY

89

or both. Both parameters are related by Equation (4.17), and such a relation can be explicitely formulated by derivation of this equation dy K ¼ dl 4p cos y

ð4:18Þ

It is possible to show [Kogelnik, 1969] that the diffraction efficiency for slightly out-of-Bragg conditions can be accounted for by introducing the mismatch parameter x ¼ yKd=2 ¼

lK 2 d 8p cos y

ð4:19Þ ð4:20Þ

in the modified formulation for Z pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sin2 n2 þ x2 Z¼ 1 þ x2 =n2

ð4:21Þ

n

ð4:22Þ

pn d l cos y

Photorefractive crystals allow recording gratings with rather large Kd that result in a sensibly high angular and wavelength selectivity as deduced from Equations (4.19) and (4.20). Their high Bragg selectivity together with their adaptability (because of their real-time and reversible recording properties) make these materials particularly suitable, for exemple, as efficient filters in extended-cavity semiconductor lasers for improving single-mode laser operation [Godard et al., 2004] Exercise: What is the Bragg angular mismatch reducing the diffraction efficiency to half its original 100% diffraction efficiency value for a 2-mm-thick purely index of refraction grating with 0.5-mm spatial period grating?

4.2

DYNAMIC COUPLED WAVE THEORY

For the case of a dynamic recording medium such as photorefractive crystals, the coupling between the interfering beams is not characterized by constant parameters like kþ and k . Instead, a feedback mechanism is present in this case, relating the holograms being recorded and the diffraction of beams being used for recording, a phenomenon called ‘‘self-diffraction’’ (or wave mixing) that does not exist for fixed gratings and is therefore not accounted for in the original Kogelnik formulation. To understand these differences, let us recall the expressions of the amplitude of the index of refraction modulation as expressed in Equations (1.45) and (1.46) and in

90

VOLUME HOLOGRAM WITH WAVE MIXING

Equations (1.55)–(1.57) which can be generalized as n ¼ 

n3 reff jEeff j 2

ð4:23Þ

where reff and Eeff are the effective values for these parameters. Let us also recall that the steady-state amplitude of the electric field generated by holographic recording in a photorefractive material is given by mEeff as reported in Equation (3.54), where Eeff is constant but m is the visibility of the recording pattern of light fringes, which may vary along the sample thickness. It is therefore necessary to write Equation (4.23) as n ¼ mn1

n1  

n3 reff jEeff j 2

ð4:24Þ

to explicitly show the dependence of the index of refraction modulation n on the fringe visibility m. 4.2.1

Combined Phase-Amplitude Stationary Gratings

Therefore, for the general case of a dynamic grating exhibiting at the same time a phase modulation (with index of refraction amplitude modulation n1 ) fP -shifted and an amplitude modulation (with amplitude a1 ) fA -shifted to the interference pattern of light onto the crystal, the Kogelnik’s formulations in Equations (4.10) and (4.11) are straightforwardly modified into @R þ aR ¼ ikþ mS @z @S cos y þ aS ¼ ik m R @z

cos y

ð4:25Þ ð4:26Þ

with: pn1 ifP a1 e þ i eifA l 2

ð4:27Þ

pn1 ifP a1 e þ i eifA l 2

ð4:28Þ

kþ ¼ and: k ¼ and:

n1 ¼ n3 reff jEeff j=2

ð4:29Þ

where fP is the phase of the complex parameter Eeff and is therefore the phase shift between the recording pattern of fringes and the recorded grating. We have here

DYNAMIC COUPLED WAVE THEORY

91

assumed that a is the average bulk light absorption, and a is the absorption modulation, which we assume to be proportional to m as for the case of n. Such an assumption is rather reasonable if we consider that the modulation a also arises from the spatial modulation of traps in the sample. Both n1 and a1 represent the maximum possible respective modulation amplitudes that are achieved for m ¼ 1. For the case of a uniform light background (Ib ) the pattern of light onto the crystal may be written as: I ¼ Ib þ I0 ½1 þ jmj cosðKx þ fÞ   I0 I ¼ ðIb þ I0 Þ 1 þ jmj cosðKx þ fÞ Ib þ I0

ð4:30Þ ð4:31Þ

0 everywhere. With the simple so that in this case jmj should be converted to jmj Ib IþI 0 az= cos y , the bulk absorption term transformation S ! Se pffiffiffiffiaS may be eliminated pffiffiffiffi in Equation (4.26), and similarly for R. Substituting S ¼ IS eicS and R ¼ IR eicR into the above simplified equations and comparing the imaginary and real terms, the following set results:

kI 4IR IS @IR ¼ þ cos y IR þ IS @z

ð4:32Þ

@IS kI 4IR IS ¼  @z cos y IR þ IS

ð4:33Þ

kR @cR 2IS ¼ þ @z cos y IS þ IR

ð4:34Þ

@cS kR 2IR ¼  cos y IS þ IR @z

ð4:35Þ

where kIþ , kI , and kRþ and kR are the imaginary and real terms, respectively, of kþ and k . Equations (4.32)–(4.35) may be rearranged to: @ðIR þ IS Þ 4a1 cos fA IR IS ¼ cos y IR þ IS @z

ð4:36Þ

@ðIS  IR Þ 8pn1 sin fP IR IS ¼ l cos y IR þ IS @z

ð4:37Þ

@ðcS  cR Þ pn1 cos fP IS  IR a1 sin fA ¼2  @z l cos y IR þ IS cos y

ð4:38Þ

@ðcS þ cR Þ pn1 cos fP a1 sin fA IS  IR þ ¼ 2 l cos y cos y IS þ IR @z

ð4:39Þ

92

VOLUME HOLOGRAM WITH WAVE MIXING

Figure 4.4. Amplitude coupling in two-wave mixing: In this example the weaker beam receives energy from the stronger, but could also be the other way round.

4.2.1.1 Fundamental Properties. The analysis of Equations (4.36)–(4.39) allows one to formulate the fundamental properties of combined amplitude-phase gratings. Let us discuss them in detail.

Energy conservation: Energy conservation means that, not considering bulk absorption, IR þ IS is constant along z in the sample thickness. Energy conservation holds for any condition making zero the right side in Equation (4.36): Energy is conserved for a 90 -shifted amplitude grating (cos fA ¼ 0) or in the absence of any amplitude grating (a1 ¼ 0).

Energy exchange or amplitude coupling: Energy exchange as described by Equation (4.37) is dependent on the imaginary part of the phase grating only and is illustrated in Figure 4.4. The amplitude grating has no effect at all. If there is no phase grating, there is no possibility for energy to be exchanged from one beam to the other.

Phase shifting or phase coupling: The shifting of the interference pattern phase planes is described by the evolution of cS  cR and is illustrated in Figure 4.5. The phase difference (hologram phase shift) between the recording pattern of light and the hologram being recorded is determined by the material and experimental parameters as described in Equation (3.57). This means that the hologram will follow the shifting of the recording pattern of light in order to keep constant the holographic phase shift. Accordingly, both the pattern of light and the recorded hologram will be synchronously shifted. Equation (4.38) shows that there is no hologram phase shifting, in the following cases: a) in-phase (or counterphase) amplitude and 90 -shifted phase grating or b)

DYNAMIC COUPLED WAVE THEORY

93

Figure 4.5. Phase coupling in two-wave mixing: The pattern of fringes and associated grating are progressively shifted. The picture shows some degree of amplitude coupling, too.

in-phase (or counterphase) amplitude grating and in-phase phase grating (meaning no beam coupling) with equal input irradiance beams (IS0 ¼ IR0 ). Either one of these conditions will make the right-hand side in Equation (4.38) zero, thus meaning that the phase difference remains constant through the crystal thickness so that there is no phase coupling and therefore no hologram ‘‘bending’’ due to self-diffraction. 4.2.1.2 Irradiance. The general case of mixed phase-amplitude gratings as described by Equations (4.32)–(4.35) do not verify either energy conservation or phase uncoupling. Multiplying Equation (4.32) by kI and Equation (4.33) by kIþ and adding both equations we get: kI

@IR @IS @ðkIþ IS  kI IR Þ þ kIþ ¼ ¼0 @z @z @z

ð4:40Þ

Equation (4.40) above shows that the quantity J ¼ kIþ IS  kI IR is a constant and may play the same role as energy conservation in the solution of coupled Equations (4.32)–(4.35). Unfortunately, the general solution in this case does not provide with an explicit analytic formulation for IS and for IR. Therefore, substituting J into Equation (4.33) and rearranging terms we get cos y

kIþ IS  J @IS ¼ 4kI IS @z IS ðkIþ þ kI Þ  J

ð4:41Þ

94

VOLUME HOLOGRAM WITH WAVE MIXING

Rearranging terms and integrating from the input (d ¼ 0 and IS0 ) to the output (d and IS ) we get the following expression: cos y kI

ZIS

IS0

IS ðkIþ þ kI Þ  J dIS ¼ 4 IS2 kIþ  J IS

Zd

dz

ð4:42Þ

dIS ¼ 4d IS2 kIþ  J IS

ð4:43Þ

0

where cos yðkIþ

þ

ZIS

ZIS

IS0

dIS J cos y  IS kIþ  J kIþ

Z

dIS 1 ¼ I lnðIS kIþ  J Þ IS kIþ  J kþ

ð4:44Þ

kIþ IS dIS 1 ln ¼ kIþ IS  J J J IS þ IS2 kIþ

ð4:45Þ

kI Þ=kI

IS0

Knowing that

and Z

and substituting Equations (4.44) and (4.45) into Equation (4.43) and rearranging terms we get the final solution:   r IS IS r  1 þ 1 ¼ expð4dkIþ = cos yÞ ð4:46Þ IS0 IS0 b2 A similar solution is found for the beam in the other direction: IR IR0

 r  2 IR b þ 1 ¼ expð4dkI = cos yÞ  1 r IR0

ð4:47Þ

where r  kIþ =kI

and

b2  IR0 =IS0

For the particular case of a1 ¼ 0, Equations (4.46) and (4.47) become: IS ¼ IS0

1 þ b2 1 þ b2 expðdÞ

ð4:48Þ

1 þ b2 b2 þ expðdÞ

ð4:49Þ

and IR ¼ IR0

95

DYNAMIC COUPLED WAVE THEORY

which are the same expressions found below for the case of pure photorefractive holograms in Equations (4.79) and (4.89), with  as defined in Equation (4.74).

4.2.2

Pure Phase Grating

We shall discuss the particular case where a1 ¼ 0, in which case we deduce, from Equation (4.36), that energy conservation holds for a pure phase grating, so that we may write IR þ IS ¼ IR0 þ IS0 ¼ I0

ð4:50Þ

4.2.2.1 Time Evolution. Let us consider the buildup of a space-charge electric field and the associated index of refraction modulation and corresponding phase grating. For a purely phase-modulated (a1 ¼ 0) grating and subtituting the expression of n1 in Equation (4.24) into Equations (4.25)–(4.29), we should write m kðtÞ ¼ m kþ ðtÞ ¼ m k  ðtÞ ¼ 

pn3 reff Esc ðtÞ 2l

ð4:51Þ

Now kþ ðtÞ and k ðtÞ are associated with the index of refraction modulation amplitude for jmj ¼ 1. Assuming no absorption at all, neither bulk nor modulation effects, the Kogelnik coupled wave equations are now written as: @R ¼ imkðtÞS @z @S cos y ¼ im k ðtÞR @z

cos y

kðtÞ  ½kþ ðtÞa1 ¼0 k ðtÞ  ½k ðtÞa1 ¼0

ð4:52Þ ð4:53Þ

Time-derivating Equation (4.53), substituting Equation (4.51) into the latter, and considering Equation (3.51) with Equation (3.8), we get the following expression for S: cos y

@ 2 S cos y @S 2jRj2 S pn3 reff Eeff pn3 reff Esc ðtÞ @R ¼ i þ þi 2 2 2l @z@t tsc @z 2l @t tsc ðjRj þ jSj Þ

ð4:54Þ

Following a similar procedure we get an expression for R, too:

cos y

@ 2 R cos y @R 2jSj2 R pn3 reff Eeff pn3 reff Esc ðtÞ @S ¼ i þ þi 2l @z@t tsc @z 2l @t tsc ðjRj2 þ jSj2 Þ

ð4:55Þ

96

VOLUME HOLOGRAM WITH WAVE MIXING

4.2.2.1.1 Undepleted Pump Approximation. If we assume energy conservation I0 ¼ I ¼ jRj2 þ jSj2 that is reasonable for a pure index of refraction grating and state the so-called ‘‘undepleted pump approximation’’ (jRj2  I, I constant) condition, we deduce the following relations: m¼

2RS  2S =R I

@S I @m  @z 2R @z      1 @R  1 @R    1  R @t  R @z  1

jEsc ðtÞj  jmEeff j

ð4:56Þ ð4:57Þ ð4:58Þ ð4:59Þ

which substituted into Equation (4.54), dividing the whole by R, and computing the conjugate results in @2m 1 @m m pn3 reff Eeff ¼0 þ þi @z@t tsc @z tsc l cos y

ð4:60Þ

because     Esc ðtÞ @R  mEeff @R      R @t   R @t  jmEeff j

ð4:61Þ

A similar expression is found for S:

@ 2 S 1 @S 1 pn3 reff Eeff S ¼0 þ þi @t@z tsc @z tsc l cos y

ð4:62Þ

There is not a similar formulation for the pump beam R because the right-side term cannot be neglected in Equation (4.55). However, in the case that R is the weak beam and S is the pump, another couple of differential equations can be found: @ 2 mR 1 @mR m pn3 reff Eeff þ i ¼0 @z@t tsc @z tsc l cos y

ð4:63Þ

@2R 1 @R 1 pn3 reff Eeff R¼0 þ i @t@z tsc @z tsc l cos y

ð4:64Þ

where mR means the modulation for the case that R is the weak beam.

DYNAMIC COUPLED WAVE THEORY

97

The differential Equations (4.60)–(4.62) may be written in the general form [Cronin-Golomb et al., 1987; Cronin-Golomb, 1987; Horowitz et al., 1991]: @ 2 Aðz; tÞ 1 @Aðz; tÞ þ  bAðz; tÞ ¼ 0 @z@t tsc @z

b ¼ i

pn3 reff Eeff ltsc cos y

ð4:65Þ

with a solution of the form: Aðz; tÞ ¼ A0 ðt0 ÞA1 ðz; tÞ expðt=tsc Þ

ð4:66Þ

which substituted into Equation (4.65) gives @ 2 A1 ðz; tÞ  bA1 ðz; tÞ ¼ 0 @z@t

ð4:67Þ

With the change of variable a ¼ bzt, we get the second-order differential equation in one single variable: a

@ 2 A1 @A1  A1 ¼ 0 þ @a2 @a

ð4:68Þ

Equation (4.68) may be transformed a Bessel equation by just making the pffiffiffi intopffiffiffiffiffiffi following variable change  ¼ i2 a ¼ i2 bzt: @ 2 A1 1 @A1 þ A1 ¼ 0 þ @ 2  @

ð4:69Þ

whose solution is the zero-order Bessel function: pffiffiffiffiffiffi A1 ¼ J0 ði2 bztÞ

ð4:70Þ

From Equations (4.69) and (4.70) we get the solution for Equations (4.60) and (4.62): rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  8kzt expðt=tsc Þ i tsc cos y rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  kzt 8i expðt=tsc Þ mR ðor RÞ ¼ A0 ðt0 ÞJ0 tsc cos y m ðor S Þ ¼ A0 ðt0 ÞJ0

ð4:71Þ ð4:72Þ

where we have defined

k ¼ lim kðtÞ ¼ t!1

pn3 reff Eeff 2l

ð4:73Þ

98

VOLUME HOLOGRAM WITH WAVE MIXING

from Equations (4.51) and (3.54). Note that the holographic phase shift in Equation (3.57) can now be written as tan fP ¼

 4I fkg 4
ð4:74Þ

Note that energy and phase coupling are associated to  and g respectively as deduced from Equations (4.37) and (4.38) respectively. 4.2.2.1.2 Response Time with Feedback. The coupled wave theory shows that diffraction of beams along the directions S and R, due to the grating being recorded, results in energy transfer from one of these beams into the other, a process called ‘‘two-wave mixing.’’ If a grating is erased, using one of the beams (let us say R), which transfers energy to the other beam (S), the erasure can be considered a positive feedback process. In fact, the signal beam S is increased (‘‘amplified’’) because of this transfer of energy and by this means the erasure is slowed down. In the opposite case, if R is used to erase but energy transfer occurs from S to R, there is a negative feedback because the signal beam S is not amplified but reduced and the hologram erasure is speeded up. The transfer of energy from one beam to the other is determined by the direction of propagation of the beam and the crystal parameters, that is, determined by the sign of k or k in the corresponding coupled wave equation. As for the case of amplifiers in electronic circuits, negative feedback results in the increase of the frequency bandwidth, that is, results in a faster response. And the opposite occurs for positive feedback. This similarity between photorefractive two-wave mixing and electronic amplification has been already pointed out [Horowitz et al., 1991; de Oliveira and Frejlich, 2001a]. The matter can be mathematically analyzed by assuming an erasure processe where the weak beam is S, which evolution is described by Equation (4.71) with adequately selected constants (A0 ðt0 Þ ¼ jS ð0Þj) given by the corresponding boundary conditions. The corresponding intensity evolution is therefore given by  2 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi   kzt t=tsc   8i jSj ¼ A0 ðt0 ÞJ0 e  t cos y 2

sc

 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2   t2Rftsc2g kzt 2  e jtsc j 8i ¼ jA0 ðt0 Þj J0 t cos y  sc

ð4:75Þ

Figures 4.6–4.9 show the numerical plotting of Equation (4.75) for jA0 ð0Þj2 ¼ 1 and for some usual values for a BTO crystal with E0 =ED ¼ 2 and K ¼ 12 mm1 :
DYNAMIC COUPLED WAVE THEORY

99

1 0.8 0.6 0.4 0.2

2

4

6

8

10

2

Figure 4.6. Numerical plotting of jSj vs the normalized time t=jtsc j, from Eq. (4.75) for d ¼ 1,
2

4

6

8

10

2

Figure 4.7. Numerical plotting of jSj vs the normalized time t=jtsc j, from Eq. (4.75) for d ¼ 1,
2

4 2

6

8

10

Figure 4.8. Numerical plotting of jSj vs the normalized time t=jtsc j, from Eq. (4.75) for gd ¼ 1,
100

VOLUME HOLOGRAM WITH WAVE MIXING

1 0.8 0.6 0.4 0.2

1

2

3

4

5

2

Figure 4.9. Numerical plotting of jSj vs the normalized time t=jtsc j, from Eq. (4.75) for gd ¼ 1,
400

1.1 200 1.0 0

0.9 0

2

4

6

8

10 0.55

600 Ramp Voltage (V)

η (%)

1.2

0.50 400

0.45

η (%)

Ramp Voltage (V)

this case, a perturbation is automatically established by the starting of a ramp-shaped voltage (thick curve) applied to a PZT-supported mirror (in order to produce the necessary detuning Kv for the running hologram generation) in the setup. We see how fast the running hologram diffraction efficiency (thin oscillating curve) evolves

0.40 200

0.35 0

2

4

6

8

10

Time (s)

Figure 4.10. Transient effect of a perturbation, in the form of a ramp voltage (thick curve) applied to the PZT-supported mirror in the holographic setup, on the diffraction efficiency (thin curve) of a running hologram recorded in a photorefractive BTO-crystal using the 514.4 nm wavelength. The diffraction efficiency evolution to equilibrium is faster for the negative-gain (lower graphics, with K ¼ 2:55 mm1 ) than for the positivegain (upper graphics with K ¼ 4:87 mm1 ) experiment. In both cases the applied external field is E0  7:5 kV=cm, the total incident irradiance is I0  22:5 mW=cm2 and the beams ratio is b2  40. Reproduced from [de Oliveira and Frejlich, 2001a].

DYNAMIC COUPLED WAVE THEORY

101

to equilibrium after the setup is perturbated by the starting of the mirror movement: The negative-gain experiment (lower graphics) shows a much faster and less oscillatory evolution to equilibrium than the positive-gain experiment in the upper graphics, in agreement with the theoretical predictions. 4.2.2.2 Stationary Hologram. Substituting Equation (4.50) into Equation (4.33), where a pure phase grating is considered, we get: @IS 4kI IS ðI0  IS Þ ¼  @z cos y I0

ð4:76Þ

with the variable substitution t ¼ 1=IS we get @t 4kI 4kI tþ ¼0  I0 cos y @z cos y

ð4:77Þ

where the solution is I

t ¼ t0 e4k z= cos y þ 1=I0

ð4:78Þ

From the boundary conditions we get IS ðzÞ ¼ IS0

1 þ b2 1 þ b2 ez

ð4:79Þ

where

4k  ¼ 4k0 sin fP = cos y ¼ = cos y

ð4:80Þ

with k ¼ k0 eifP ¼ kþ ¼ ðk Þ ¼

pn3 reff Eeff 2l

ð4:81Þ

and k0 ¼ jkj ¼

pn1 ; l

b2 ¼ IR0 =IS0

Eeff above is the complex parameter described in Equation (3.52), and fP is the phase shift of the pure phase hologram. We shall only consider an index of refraction modulation that, in photorefractive crystals, arises from the electro-optic properties of these materials. In fact, a modulated pattern of space-charge electric field is produced under the action of a corresponding pattern of fringes of light as already described in Chapter 3. As a consequence of the electro-optic effect a corresponding modulated pattern of index of refraction is produced the amplitude n of which is,

102

VOLUME HOLOGRAM WITH WAVE MIXING

according to Equation (4.23) for the stationary case (t ! 1): n eifP ¼ 

n3 reff Esc 2

ð4:82Þ

which substituted into Equation (4.27) and (4.28), for a purely phase-modulated (a ¼ 0) grating may be written as mk ¼ mkþ ¼ m k  ¼ 

pn3 reff Esc 2l

ð4:83Þ

Now kþ and k are associated with the index of refraction modulation amplitude for jmj ¼ 1. Substituting Equation (4.50) into Equation (4.35) we find an equation for cS : @cS g I0  IS ¼ @z 2 I0

ð4:84Þ



4k g¼< cos y

ð4:85Þ

where

Note that the hologram phase shift defined in Equation (3.57) and written as Equation (4.74) can now be written, from Equations (4.80) and (4.85), also as tan fP ¼ =g

ð4:86Þ

Substituting Equation (4.79) into Equation (4.84) and rearranging terms we get the differential equation:   @cS g 1 ¼  1 @z 2 1 þ b2 ez

ð4:87Þ

whose solution is: cS ðzÞ ¼ cS ð0Þ 

gz þc 4

with c¼

1 ez=2 þ b2 ez=2 ln 2 tan fp 1 þ b2

ð4:88Þ

DYNAMIC COUPLED WAVE THEORY

103

We find a similar set of equations for IR and cR : 1 þ b2 b2 þ ez gz cR ðzÞ ¼ cR ð0Þ   c 4 IR ðzÞ ¼ IR0

ð4:89Þ ð4:90Þ

From Equations (4.88) and (4.90) an expression is found cS ðzÞ  cR ðzÞ ¼ cS ð0Þ  cR ð0Þ þ

1 ðb2 þ ez Þ2 ln 2 tan fp ð1 þ b2 Þ2 ez

ð4:91Þ

that describes the pattern of light phase planes and consequently the hologram phase planes also. Equation (4.91) represents a pattern of fringes that is being continuously shifted from the input cS ð0Þ  cR ð0Þ to the output cS ðdÞ  cR ðdÞ and is therefore leading to a ‘‘bent’’ hologram. This is a direct consequence of phase coupling represented by Equation (4.38). For the case of materials exhibiting optical activity we can show that the coefficients kIþ and kI in the right-hand side of Equations (4.32)– (4.35) are not constants but are functions of z [Foote and Hall, 1986]. In this case the differential equations in Equations (4.76) and (4.84) should be represented as: @IS ðzÞ IR ðzÞIS ðzÞ ¼ ðzÞ @z IR ðzÞ þ IS ðzÞ

ð4:92Þ

@cS ðzÞ gðzÞ IR ðzÞ ¼ @z 2 IR ðzÞ þ IS ðzÞ

ð4:93Þ

where the explicit dependence of the real gðzÞ and imaginary ðzÞ parts of the coupling constant on the crystal thickness ðzÞ are indicated. Equation (4.92) can be written as: @t ðzÞ þ ðzÞt ¼ @z I0

t

1 IS ðzÞ

I0 ¼ IS ð0Þ þ IR ð0Þ

ð4:94Þ

whose general solution is [Boas, 1983] t¼



t0 þ

Z

ðzÞ e I0

R

ðzÞdz

 R dz e ðzÞdz

ð4:95Þ

Rearranging terms we get the general formulation IS ðzÞ ¼ IS0

1 þ b2 R 1 þ b2 expð ðzÞdzÞ

ð4:96Þ

104

VOLUME HOLOGRAM WITH WAVE MIXING

where the amplitude coupling is shown to depend on the integral of ðzÞ. For the case of the phase and for the undepleted pump approximation (IR ðzÞ  IS ðzÞ) we can write Equation (4.93) as @cS ðzÞ gðzÞ  @z 2

ð4:97Þ

where the dependence of phase coupling on the integral of gðzÞ is obvious. The results in Equations (4.96) and (4.97) show that for the case of  and g varying along the crystal thickness, their influence on amplitude and phase coupling are represented by their corresponding integrals. That is, the simple z and gz products should be substituted by their integrals. This conclusion is in fact a general one that may be applied whenever  and g are dependent on the sample thickness. 4.2.2.2.1 Diffraction. Let us recall the formulation of the coupled wave equations in Equations (4.25) and (4.26) @RðzÞ k ¼i mðzÞSðzÞ @z cos y mðzÞ ¼

@SðzÞ k ¼i mðzÞ RðzÞ @z cos y

ð4:98Þ

2S ðzÞRðzÞ ; I

where absorption has been neglected. The solution for the corresponding intensities and phases were already computed in Section 4.2.2.2. We shall now investigate the situation when the already written (that is, mðzÞ is fixed) grating is read by another couple of waves RðzÞ and SðzÞ that are identical to the corresponding writing ones. Such a formulation is necessary to allow one to compute the diffracted beam (and therefore the diffraction efficiency) that is to be measured when a probe beam, different but in principle identical to the recording beam, is diffracted by the grating without erasing it, so that Equations (4.98) should be written as @RðzÞ k ¼i mðzÞSðzÞ @z cos y @SðzÞ k mðzÞ RðzÞ ¼i cos y @z

ð4:99Þ ð4:100Þ

When reading with the reference beam RðzÞ the boundary conditions Rð0Þ ¼ 1 and Sð0Þ ¼ 0 should be fulfilled. Likewise, when reading out the same grating with the signal beam SðzÞ, we need the solutions of the same equations fulfilling the boundary conditions Rð0Þ ¼ 0 and Sð0Þ ¼ 1. How to find this fundamental system of solutions will be shown later. But as soon as these solutions are available, the

DYNAMIC COUPLED WAVE THEORY

105

reference R and the signal S beams can be written in the form R ¼ Rð0Þ RR þ Sð0Þ RS

ð4:101Þ

S ¼ Rð0Þ S R þ Sð0ÞS S

ð4:102Þ

where RR is the transmittance and RS is the diffraction coefficient, respectively, for beam R and similarly for S. 4.2.2.2.1.1 UNSHIFTED HOLOGRAMS. We shall here show how to evaluate the expressions in Equations (4.101) and (4.102) for the simple case of an unshifted (local) grating. Let us assume an unshifted grating, f ¼ 0 (or  ¼ 0), which is just a homogeneous tilted grating. From Equation (4.91) we compute, for  ¼ 0, the phases 1 c ¼ kz 2

k ¼

g IS ð0Þ  IR ð0Þ 2 I

cR ¼ cR ð0Þ 

g IS ð0Þ z 2 I

g IR ð0Þ z cS ¼ cS ð0Þ  2 I

ð4:103Þ

and therefore g IS ð0Þ I z

R ¼ Rð0Þei2

g IR ð0Þ I z

S ¼ Sð0Þei2

ð4:104Þ

so that the intensities are fixed: IR ¼ IR ð0Þ and IS ¼ IS ð0Þ. The modulation, however, depends on the crystal depth: mðzÞ ¼ mð0Þeikz

ð4:105Þ

We insert the expression of the modulation in Equation (4.105) into Equations (4.99) and (4.100) and, taking into account that f ¼ 0 so that k= cos y ¼ g=4, we obtain @R g ¼ i mð0Þeikz S @z 4

ð4:106Þ

@S g ¼ i mð0Þ eikz R @z 4

ð4:107Þ

Differential equations with constant coefficients are obtained by the transformations ^ ikz=2 R ¼ Re

^ ikz=2 S ¼ Se

106

VOLUME HOLOGRAM WITH WAVE MIXING

that result in the coupled equations @ R^ g ¼ i mð0ÞS^ @z 4

ð4:108Þ

@ S^ g ¼ i mð0Þ R^ @z 4

ð4:109Þ

which are easily solved. The result for diffraction of the reference wave from the dynamic grating is the solution of Equations (4.106) and (4.107) with the boundary conditions RR ð0Þ ¼ 1 and S R ð0Þ ¼ 0: RR ¼

IR ð0Þ ig IS ð0Þz IS ð0Þ ig IR ð0Þz e 2 I þ e2 I I I

g IR ð0Þ g IS ð0Þ 1 S R ¼ mð0Þ ½ei2 I z  ei2 I z  2

RR ð0Þ ¼ 1 S R ð0Þ ¼ 0:

ð4:110Þ

From this we find the diffraction efficiency Z ¼ jS R j2 ¼ jmð0Þj2 sin2

gz 4

ð4:111Þ

This is Kogelnik’s formula [Kogelnik, 1969] with n ¼ jmð0Þj

gz 4

1 x ¼ c ¼ kz 2

ð4:112Þ

For modulation m ¼ 1 we have x ¼ 0. There is no tilting, and we obtain a simple sin2 function. For small modulation jxj  jnj. To obtain the corresponding formulae for diffraction of the signal wave it is enough to observe that Equations (4.106) and (4.107) are invariant under R $ S, R $ S, and k $ k; g therefore remains the same. Therefore the diffraction efficiency is not changed and g IS ð0Þ g IR ð0Þ 1 RS ð0Þ ¼ 0 RS ¼ mð0Þ½ei2 I z  ei2 I z  2 IS ð0Þ ig IR ð0Þz IR ð0Þ ig IS ð0Þz SS ¼ S S ð0Þ ¼ 1 e 2 I þ e2 I I I

ð4:113Þ

4.2.2.2.1.2 PHASE-SHIFTED HOLOGRAMS. We shall now deal with an arbitrarily phaseshifted grating. We shall start again from Equation (4.98) and consider again a fixed (mðzÞ) grating. Note that R and S still solve this system. The method to obtain a second, linearly independent solution of Equation (4.98) can be found in any textbook on ordinary differential equations. Here, however, it is enough to take the

DYNAMIC COUPLED WAVE THEORY

107

complex conjugate of Equation (4.98) to find that R ¼ S

S ¼ R

ð4:114Þ

is a second solution of Equations (4.99) and (4.100). Because the determinant of these two solutions is equal to the intensity I, they are indeed linearly independent. Now Rð0Þ =I times the old solution vector ðR; SÞ minus Sð0Þ=I times the new solution vector ðR; SÞ gives the solution corresponding to the diffraction of the reference wave: RR ¼ ½Rð0Þ R þ Sð0ÞS =I

S R ¼ ½Sð0ÞR þ Rð0Þ S=I

RR ð0Þ ¼ 1

ð4:115Þ

S R ð0Þ ¼ 0

The solution corresponding to the diffraction of the signal wave is obtained by the above symmetry: RS ¼ S R ¼ ½Sð0Þ R  Rð0ÞS =I

RS ð0Þ ¼ 0

S S ¼ R R ¼ ½Rð0ÞR þ Sð0Þ S=I

ð4:116Þ

S S ð0Þ ¼ 1

The diffraction efficiency is hereby given by  g b2 cosh 2 z  cos 2 z Z ¼ jS R j ¼ jRS j ¼ 2 :   1 þ b2 b2 e 2z þ e 2z 2

2

ð4:117Þ

It is interesting to compute lim Z ¼

d!0



2b 1 þ b2

 3 2 2 " 2  2 # d gd pn reff 2 2 2 þ ¼ jmj jEeff j d 4 4 2l cos y0 with

jmj2 ¼



2b 1 þ b2

2

ð4:118Þ

which is the well-known Kogelnik formula [Kogelnik, 1969] for Z 1. It is possible to verify that R ¼ Rð0ÞRR þ Sð0ÞRS

ð4:119Þ

S ¼ Rð0ÞS R þ Sð0ÞS S

ð4:120Þ

and also that jRR j2 þ jS R j2 ¼ jRS j2 þ jS S j2 ¼ 1

108

VOLUME HOLOGRAM WITH WAVE MIXING

The phase shift j between the transmitted and the diffracted beams along the same direction at the hologram output can be computed from Equation (4.119) or Equation (4.120). From the latter, for exemple, we get tan j ¼

=fRð0ÞS R Sð0Þ S S g
ð4:121Þ

Verify that, substituting the parameters above by their expressions in Equation (4.116) and using the expressions in Equations (4.89) and (4.79) we obtain

tan j ¼ 

g sin z 2   1  b2  g  cosh z  cos z þ sinh z 2 2 2 2 1þb

ð4:122Þ

which relates the output phase shift j with the material parameters  and g and the experimental parameter b2. 4.2.2.3 Steady-State Nonstationary Hologram with Bulk Absorption. In the presence of wave mixing, the mathematical model describing the generation of running (nonstationary) holograms is the same as in Section 3.4 except that wave mixing effects should be now accounted for. The diffraction efficiency, for example, is now formulated in terms d and gd as described in Equation (4.117), where  and st in Equation (3.80). g are computed from the expression of Esc Bulk light absorption does not affect the way a stationary (not moving) hologram is recorded except for the fact that its buildup is slower. In fact the average light intensity decreases along the crystal thickness z, and the Maxwell (or dielectric) relaxation time tM (that is, determining the recording response time) in Equation (3.43) is not any more a constant but varies along z as: tM ðzÞ ¼ tM ð0Þeaz

tM ð0Þ ¼

Ee0 hn qmtI0 a

ð4:123Þ

Consequently, the deeper layers in the sample are slower, because progressively less light is left for recording, because of absorption. st st g and g /
ar eaz Kv þ cr

ae2az ðKvÞ2 þ beaz Kv þ c

ð4:124Þ

109

DYNAMIC COUPLED WAVE THEORY

st =fEsc g¼

ai eaz Kv þ ci

ae2az ðKvÞ2 þ beaz Kv þ c

ð4:125Þ

a ¼ ½K 2 L2E þ ð1 þ K 2 L2D Þ2 tM ð0Þ2

ð4:126Þ

b ¼ 2tM ð0Þ½K 2 l2s  K 2 L2D 

ð4:127Þ

E0 ED

c ¼ ð1 þ K 2 l2s Þ2 þ K 2 l2E ar ¼ ½ð1 þ c r ¼ E0

K 2 L2D ÞED

ai ¼ E0 tM ð0Þ ci ¼ E0 KlE þ ED ð1 þ

tM ð0Þ ¼

Ee0 ðkB T=qÞhn qL2D aI0 ð0Þ

þ KLE E0 tM ð0Þ

K 2 l2s Þ

ð4:128Þ ð4:129Þ

ð4:130Þ

ð4:131Þ

ð4:132Þ ð4:133Þ

The parameters  and g in Equations (4.80) and (4.85) are accordingly written as st g ðzÞ ¼ 4w=fEsc

ð4:134Þ

gðzÞ ¼

st 4w
ð4:135Þ

pn3 reff 2l

ð4:136Þ

with w¼

and are varying along the crystal thickness z. It is possible to argue that if ðzÞ and gðzÞ are not constants, their products d and gd should be substituted everywhere (particularly in the expression for Z in Equation (4.117) by their integrals as already explained in Section 4.2.2.2 for the case of optical activity. In the present case the integrals are Zz¼d

z¼0

    bci 2 2aKveaz þ b z¼d  pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi arctan pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðKv; zÞdz ¼ d¼ 4w ai  2c a 4ac  b2 4ac  b2 z¼0 " #z¼d ci e2az ln þ 4w 2ac aðKvÞ2 e2az þ bKveaz þ c z¼0

Zz¼d

z¼0

ð4:137Þ

    bcr 2 2aKveaz þ b z¼d pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi arctan pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi gðKv; zÞdz ¼ gd¼ 4w ar  2c a 4ac  b2 4ac  b2 z¼0 " #z¼d cr e2az ln þ 4w 2ac aðKvÞ2 e2az þ bKveaz þ c z¼0

ð4:138Þ

110

VOLUME HOLOGRAM WITH WAVE MIXING

with the condition 4ac  b2 . The results above indicate that the formulations of Z and f should be correspondingly revised, which is the subject of the following sections. 4.2.2.3.1 Diffraction Efficiency. If bulk absorption is considered, the usual expression for diffraction efficiency, reported in Equation (4.117), under selfdiffraction effects should be further modified to Z¼

 2b2 coshðd=2Þ  cosðgd=2Þ  2 2 d=2 1þb b e  þ ed=2

ð4:139Þ

 and gd as already defined. It is interesting to see how the different material with d parameters affect Z. For a hypothetical experimental condition using typical experimental parameters (E0 ¼ 500 kV=m, I0 ¼ 200 W=m2 , d ¼ 2:05 mm, b2 ¼ 50; and l ¼ 514:5 nm) and typical parameters for an undoped BTO crystal (a ¼ 1165=m and reff ¼ 5:6 pm=V) Z was computed (ordinates) for different K (ranging from K ¼ 20 to 0:5 mm1 ) as a function of the detuning Kv (horizontal axis in rad/s) and plotted on Figures 4.11–4.14. From these figures we draw the following conclusions:

Quantum efficiency  has no influence on the Z peak value. It just acts on the position of the peak and the shape of the curve; The lower K, the larger its influence.

Figure 4.11. Computed running hologram Z as a function of Kv (rad/s) for K ¼ 0:5 mm1 and different material parameters.

DYNAMIC COUPLED WAVE THEORY

111

Figure 4.12. Computed running hologram Z as a function of Kv (rad/s) for K ¼ 2 mm1 and different material parameters.

ls , which always appears as K 2 l2s , has no influence either on the peak position or on the shape of the curve. It just acts on the peak size, but has no influence for K 2 l2s 1, that is, for the far from saturation condition, as expected.

LD , which always appears as K 2 L2D , acts on the peak position and the shape of the curve, increasing the Kv value for the peak and widening the curve for increasing LD . For K 2 L2D  1 LD has no influence at all. The physical meaning of the above-referred features is usually not easy to seize. The increase of the abscissae of the Z peak with the increasing LD , for instance, is easy to understand beacause the latter is related to the carrier mobility L2D ¼ Dt ¼ ðkB T=qÞmt. That LD is having no effect on the hologram movement for K 2 L2D  1 is probably due to the fact that, for LD large compared to the grating period, the distribution of photoelectrons is somewhat randomized and the actual value of LD no longer has importance on the dynamics of the process. The increase in the resonance speed (the position of the peak) as  increases is also reasonable because the latter increases the light effectively involved in the process, and by this way tM ð0Þ is decreased and the material becomes faster. The large influence of ls is something easy to understand, too: Increase in ls means a

112

VOLUME HOLOGRAM WITH WAVE MIXING

Figure 4.13. Computed running hologram Z as a function of Kv (rad/s) for K ¼ 10 mm1 and different material parameters.

reduction in the effective photoactive center concentration, and the peak of Z therefore becomes also limited by the lack of charge carriers to build up the charge modulation in the crystal. 4.2.2.3.2 Output Beam Phase Shift. In the presence of bulk light absorption, there is no one single value for f, but it varies along the sample’s thickness because of the variation of  and g as described in Equations (4.134) and (4.135). The phase shift j between the transmitted and diffracted beams behind the sample is affected, too. Its formulation in the presence of self-diffraction is discussed in Section 4.3.1.2 and by Frejlich et al. [Frejlich et al., 1997]. As for the case of Z, the formulation of j in Equation (4.122) should be also modified accordingly in the presence of bulk  and g, respectively, leading to absorption by susbtituting  and g by  tan j ¼ 

sinðgd=2Þ 1  b2   ðcoshðd=2Þ  cosðgd=2ÞÞ þ sinhðd=2Þ 1 þ b2

ð4:140Þ

DYNAMIC COUPLED WAVE THEORY

113

Figure 4.14. Computed running hologram Z as a function of Kv (rad/s) for K ¼ 20 mm1 and different material parameters.

To understand the influence of the different material parameters and experimental conditions on j, Equation (4.140) was numerically computed for a typical BTO crystal, for the 514.5-nm wavelength, and for the same parameters as for Z (E0 ¼ 500 kV=m, Ið0Þ ¼ 200 W=m2 , and b2 ¼ 50), always for a negative two-wave mixing amplitude gain. Comparing Figure 4.15 with Figure 4.16, we see that the effect of LD is lower for K 2 L2D  1, as already pointed out in Section 4.2.2.3.1 for Z. There is apparently some optimum value of K that enhances the effect of LD for some value of Kv, as seen in Figure 4.15. No definite conclusions can be drawn instead, about the effects of ls and  on j. Figure 4.17 shows a simulated result for a hypothetical very thin crystal, where self-diffraction effects can be neglected: In this case positive (not shown) and negative gain give the same result. Figure 4.18 is also for a hypothetical thin but low-absorption a ¼ 1 m1 sample so as to be able to neglect the effect of absorption on the movement of the hologram. These figures show that in these conditions tan j becomes constant, for Kv sufficiently different from zero, with its value only depending on LD , whatever the value of K. The influence of ls and

114

VOLUME HOLOGRAM WITH WAVE MIXING

Figure 4.15. tan j versus Kv (rad/s), computed for K ¼ 2 mm1 and different material parameters, for a typical BTO crystal 2.05 mm thick and a ¼ 1165 m1 .

 is clearly negligible here. This result is in agreement with Equation (3.87). It appears that, in the absence of self-diffraction and absorption, the movement of the hologram (and consequently the value of the hologram phase shift) is only dependent on the charge carrier diffusion length LD : In this case there is no such ‘‘randomization’’ of the charge carrier distribution in the CB (see comments in Section 4.2.2.3.1) because the whole grating is moving along with the charge carriers. The whole set of Figures 4.15 to 4.18 shows that for Kv ¼ 0 (that is, stationary holograms) the only parameter of relevance is ls , in agreement with Equation (3.58).

4.3

PHASE MODULATION

Phase modulation in two-wave mixing is produced by phase modulating (with amplitude cd ) one of the interfering beams with (angular) frequency  ~ ~ Sð0Þ ¼ ~ S0 eiðkS ~xþfotþcd sin tÞ

ð4:141Þ

PHASE MODULATION

115

Figure 4.16. tan j versus Kv (rad/s), computed for K ¼ 11 mm1 and different material parameters, for a typical BTO crystal 2.05 mm thick and a ¼ 1165 m1 .

whereas the other beam remains unchanged ~ ~ Rð0Þ ¼ ~ R0 eiðkR ~xotÞ

ð4:142Þ

In this case the interference pattern of light in Equation (3.5) becomes (assuming S~0  R~0 ¼ S0 R0 ) Iðx; tÞ ¼ I0 þ I0 jmj cosðKx þ f þ cd sin tÞ

ð4:143Þ

which represents a sinusoidal pattern of light vibrating with frequency  and phase ~ that is also parallel to the coordinate axis ~ amplitude cd along K x. If tsc 1, then the hologram is faster than the movement of the pattern of light so that it follows the pattern and is recorded (and erased) continuously. Both hologram and pattern are moving simultaneously and, except because it is moving, the recorded grating is the same as if it were recorded from a standing pattern of light. If tsc  1 instead, the recording is much slower and cannot follow the movement of the pattern. The result is a hologram produced by a pattern of light that is the time average of the actual moving one. For intermediate cases, the corresponding differential equation must be taken into account and the strength of the resulting hologram will depend on

116

VOLUME HOLOGRAM WITH WAVE MIXING

Figure 4.17. tan j versus Kv (rad/s), computed for K ¼ 11 mm1 and different material parameters, for a typical BTO crystal 0.0205 mm thick and a ¼ 1165 m1 .

the relation between  and 1=tsc . This case is analyzed in Section 8.2.3 and is used to measure the response time tsc of the recording material. We shall here focus on the second case, when tsc  1. In this case it is necessary to compute the time average of the pattern of light 1 hIðx; tÞi  lim t0 !1 t0

Zt0

Iðx; tÞdt

0

hIðx; tÞi ¼ I0 þ I0 jmj cosðKx þ fÞhcosðcd sin tÞi  I0 jmj sinðKx þ fÞhsinðcd sin tÞi

where t0  1=. Developing the time-dependent terms in Bessel series and making the time average hcosðcd sin tÞi ¼ J0 ðcd Þ þ 2J2 ðcd Þhcos 2ti þ 2J4 ðcd Þhcos 4ti þ . . . ¼ J0 ðcd Þ

ð4:144Þ ð4:145Þ

and hsinðcd sin tÞi ¼ 2J1 ðcd Þhsin ti þ 2J3 ðcd Þhsin 3ti þ . . . ¼0

ð4:146Þ

ð4:147Þ

PHASE MODULATION

117

Figure 4.18. tan j versus Kv (rad/s), computed for K ¼ 1 mm1 and different material parameters, for a typical BTO crystal 0.0205 mm thick and an hypothetically low a ¼ 1 m1.

and substituting the results above into the expression for hIðx; tÞi results in hIðx; tÞi ¼ I0 þ I0 jmjJ0 ðcd Þ cosðKx þ fÞ

ð4:148Þ

where Jn ðxÞ represents the ordinary Bessel function of order n (n integer). Equation (4.148) means that the pattern of light behaves as if it were standing with an effective fringe modulation of jmjJ0 ðcd Þ instead of just jmj. The phase modulation experimental setup is schematically depicted in Figure 4.19. In these conditions the phase-modulated beam ‘‘S’’ is not modified by the recorded hologram because the latter is too much slow for that. Accordingly, the transmitted and diffracted ‘‘S’’ beams behind the crystal exhibit the same cd sin t phase modulation imposed at the crystal input. The overall irradiances behind the sample are formed by the coherent addition of the phase-modulated transmitted ‘‘S’’ plus the diffracted (nonmodulated) ‘‘R’’ beam, for IS, and the phasemodulated diffracted ‘‘S’’ plus the nonmodulated transmitted ‘‘R’’ beams for IR. Because of the nonlinear relation between irradiance and phase, such a dither signal of frequency  gives rise to multiple harmonics in  that may be detected in IS and IR , at the sample’s output, with phase-sensitive frequency-tuned lock-in amplifiers. Phase modulation can be used for operating a stabilized holographic recording setup

118

VOLUME HOLOGRAM WITH WAVE MIXING

+V ° M BS

IS

IR° IS°

PZT

OSC Ω

IR BTO

+

D

HV

LA Ω

LA 2Ω

Ω

V

2Ω

V

Figure 4.19. Phase modulation setup: BS: beamsplitter, PZT piezoelectric-supported mirror, D: photodetector, LA- and LA-2: lock-in amplifiers tuned to  and 2 respectively, HV high voltage source for the PZT, OSC oscillator to produce the dithering signal.

as developed in Chapter 6. It can be also used in different ways to characterize material parameters, which is the subject of Part III of this book. 4.3.1

Phase Modulation in Dynamically Recorded Gratings

Diffraction from a dynamically recorded space-charge grating cannot be modeled easily, because in general it will not be homogeneous and it will be tilted and bent. Kogelnik’s formula [Kogelnik, 1969] in Equation (4.15) is only valid for the diffraction from a homogeneous nontilted grating, and the question arises as to how to generalize his results to the case of a phase-modulated dynamic grating. We shall here decribe in detail the accurate formulation of temporal harmonic components in phase-modulated photorefractive two-wave mixing (TWM). We shall show how to obtain the general solution of the problem of diffraction from a fixed dynamic grating (described by a system of linear ordinary differential equations) by exploiting the solution obtained from solving the nonlinear two-wave mixing equations. From these results we shall derive analytical expressions for the first and second temporal harmonics of the signal output beam. According to the assumptions above we shall henceforth assume that the pattern of light and the corresponding hologram are not affected by the oscillating pattern of light (with tsc  1), except for the fact that the fringe visibility is now J0 ðcd Þjmj instead of jmj. We shall focus first on the simple particular case of f ¼ 0, and then on the more complex general case of arbitrarily f-shifted holograms. Approximate expressions relating the first and second temporal harmonic terms to the hologram phase shift f [Garcia et al., 1989] and accurate formulations for particular conditions such as equal-incident beams [Hofmeister et al., 1993] or

PHASE MODULATION

119

undepleted pump approximation [Cudney et al., 1992] have already been published. We describe below an accurate general formulation relating the fundamental photorefractive material parameters to the temporal harmonics in a two-wave mixing phase modulation experiment. 4.3.1.1 Phase Modulation in the Signal Beam. We shall now investigate the development of the expressions for the dynamic coupled wave in Section 4.2.2.2 when the amplitude of the signal beam oscillates in the form eicd sin t with an angular frequency large relative to the reciprocal holographic relaxation time of the crystal tsc  1. In this case Equations (4.101) and (4.102) should be substituted by the corresponding R ¼ Rð0ÞRR þ Sð0Þeicd sin t RS

ð4:149Þ

S ¼ Rð0ÞS R þ Sð0Þeicd sin t S S

ð4:150Þ

¼ S þ Sð0ÞS S ðeicd sin t  1Þ

ð4:151Þ

Note that we have neglected the twice-diffracted modulated beam at the output along the direction of the directly transmited modulated beam. Such an approximation may not be posible for sufficiently highly diffractive gratings, and the exact handling of this case has been reported by Ringhofer and co-workers [Ringhofer et al., 2000]. Expanding the expression in Equation (4.150), to compute the intensity IS ¼ jSj2 of the signal beam in terms of c2d , allows one to find harmonic terms in t IS ¼ IS þ IS sin t þ IS2 cos 2t þ . . .

ð4:152Þ

with IS ¼ 4J1 ðcd Þ=fS Sð0ÞS S g

2

IS2 ¼ 4J2 ðcd ÞðRfS Sð0ÞS S g  IS ð0ÞjS S j Þ

ð4:153Þ ð4:154Þ

4.3.1.1.1 Unshifted Hologram. To evaluate Equations (4.153) and (4.154) for the special case of f ¼ 0 (unshifted) and according to the results from Section 4.2.2.2.1.1 we must calculate S Sð0ÞS S ¼

g IS ð0Þ ½IS ð0Þ þ IR ð0Þei2z  I

and jS S j2 ¼ 1  Z ¼ 1  jmð0Þj2 sin2

gz 4

120

VOLUME HOLOGRAM WITH WAVE MIXING

From this we obtain IR ð0ÞIS ð0Þ g sin z I2 2 IR ð0ÞIS ð0Þ IR ð0Þ  IS ð0Þ 2 g sin z IS2 =I ¼ 2J2 ðcd Þ I2 I 4 IS =I ¼ 4J1 ðcd Þ

ð4:155Þ ð4:156Þ

For the ratio of the intensities we obtain IS2 J2 ðcd Þ IR ð0Þ  IS ð0Þ g ¼ tan z  J1 ðcd Þ I 4 IS

ð4:157Þ

4.3.1.1.2 Shifted Hologram. Again assuming that the signal beam amplitude oscillates in the form eicd sin t , with tsc  1, we first need the results in Section 4.2.2.2.1.2 to compute the expressions

S Sð0ÞS S ¼ IS ð0Þ and

g







IR ð0Þei2z þ IS ð0Þe 2 z IR ð0Þe 2z þ IS ð0Þe 2z

g   IR ð0Þ2 e 2z þ IS ð0Þ2 e 2 z þ 2IR ð0ÞIS ð0Þ cos z 2 jS S j2 ¼ 1  Z ¼   ½IR ð0Þe 2 z þ IS ð0Þe 2 z I From this we find g b2 sin z  2 IS ðzÞ ¼ 4J1 ðcd ÞIS ð0Þ 2 z  b e 2 þ e2z and   g b2 b2 e 2 z  e 2 z þ ð1  b2 Þ cos 2 z 2 IS ¼ 4J2 ðcd ÞIS ð0Þ   1 þ b2 b 2 e 2 z þ e 2 z The ratio of these intensities is g  2 2 2 z 2z IS2 J2 ðcd Þ b e  e þ ð1  b Þ cos 2 z ¼ g J1 ðcd Þ IS ð1 þ b2 Þ sin z 2

ð4:158Þ

ð4:159Þ

ð4:160Þ

Note that Equation (7b) in [Hofmeister et al., 1993] is just a particular case of Equation (4.160) for b2 ¼ 1. 4.3.1.2 Output Phase Shift. We may also directly operate on Equation (4.150) to describe the total irradiance at the output along the S-beam direction IS ¼ jSj2 ¼ jRð0ÞS R þ Sð0Þei cd sin t S S j2 pffiffiffiffiffiffiffiffiffiffiffi pffiffiffi ¼ jRð0Þ Z þ Sð0Þ 1  Zeijþicd sin t j2

ð4:161Þ

PHASE MODULATION

IR

0

IS

121

φ

tran

sm

0 IR

diff

itte

d

rac t

pattern of fringes

IS

ed

shifted by ϕ hologram

Figure 4.20. Wave mixing schema showing the hologram phase shift f and the phase shift j between the transmitted and diffracted beams at the crystal output.

where IS is the irradiance along the incident beam S, measured behind the crystal, with Sð0ÞS S Rð0ÞS R ¼ jSð0ÞS S Rð0ÞS R j eij

ð4:162Þ

IR0 ¼ jRð0Þj2

ð4:163Þ

IS0 ¼ jSð0Þj2

and pffiffiffi Z ¼ jS R j

pffiffiffiffiffiffiffiffiffiffiffi 1  Z ¼ jS S j

ð4:164Þ

as already defined in Section 4.2.2.2 and j representing the phase shift between the transmitted and diffracted beams behind the crystal as shown in Figure 4.20, where the hologram phase shift f is also shown. Equation (4.161) is formulated in terms of paramenters directly measured at the input and output of the sample. Developing Equation (4.161), one gets a phenomenological formulation ISdc ¼ jS0 j2 ð1  ZÞ þ jR0 j2 Z þ 2ðIS0 þ IR0 Þ qffiffiffiffiffiffiffiffipffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi IS ¼ 4J1 ðcd Þ IS0 IR0 Zð1  ZÞ sin j qffiffiffiffiffiffiffiffipffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi IS2 ¼ 4J2 ðcd Þ IS0 IR0 Zð1  ZÞ cos j

tan j ¼ 

IS J2 ðcd Þ IS2 J1 ðcd Þ

pffiffiffiffiffiffiffiffiffiffiffipffiffiffi b J0 ðcd Þ 1  Z Z cos j b þ1 2

ð4:165Þ ð4:166Þ ð4:167Þ ð4:168Þ

122

VOLUME HOLOGRAM WITH WAVE MIXING

Substituting Equation (4.160) into Equation (4.168) we get the following expression for the output beam phase shift

tan j ¼ 

g sin z 2   1  b2  g  cosh z  cos z þ sinh z 2 2 2 2 1þb

ð4:169Þ

which is, of course, the same expression already reported in Equation (4.122), although computed following a different procedure. The above formula specializes to 1  b2 g tan j tan z ¼ 1 2 4 1þb

ð4:170Þ

for fP ¼ 0 (so-called local materials) and to tan j tan fP ¼ 1

ð4:171Þ

for b2 ¼ 1 with z 1 and gz 1, that is, in the absence of phase coupling, but for any fP. Another interesting special case is the one of thin crystals, which corresponds to the exclusion of self-diffraction effects, in which case it also simplifies to tan j tan fP ¼ 1

for

z!0

ð4:172Þ

Exercise: Verify that the negative sign in Equation (4.166) is in agreement with the negative sign in Equation (4.153). Note that the sign depends on the way j is defined, so be sure that it is defined in the same way in both formulations.

4.4

FOUR-WAVE MIXING

So far we have been dealing with the interference of two waves only, either with the same (degenerate) or with slightly different (nearly degenerate) frequencies. These two waves produce a hologram in a nonlinear (in our case a photorefractive) material, and the resultant hologram diffracts the recording waves. There is a feedback between the recording waves and the nonlinear media, and by means of this interaction the amplitudes and/or phases of the recording waves are changed: This is ‘‘two-wave mixing,’’ or TWM. We can similarly mix four waves instead of two and this is called ‘‘four-wave mixing,’’ or FWM. If all the waves involved have the same frequency it is known as ‘‘degenerate’’ FWM, or DFWM. The mathematics is somewhat more involved than for TWM, but the phenomena are essentially the

FINAL REMARKS

R

R

123

P

S

S∗

S

Figure 4.21. Degenerate four-wave mixing showing the signal S and reference R beams interfering to produce a real-time hologram in the nonlinear material (left); Then a pump beam P identical to R but much stronger and counter propagating is diffracted by the already recorded hologram and the diffracted beam is the conjugate S of the signal S beam, reflecting back along the same incidence direction.

same. In this case waves S and R, mutually coherent and having the same temporal frequency, interfere in the material, and a real-time (or almost) hologram arises as represented in Figure 4.21. A pump beam P with same wavefront shape as R and the same temporal frequency (not necessarily coherent with R) but usually much stronger and propagating in the opposite direction is diffracted by the real-time hologram as shown in the right-hand side of Figure 4.21. The diffracted P beam is S , which represents the conjugate of S. Our simplified picture shows a hologram arising only from the interference of S and R, which is true if the pump P is not coherent with the former two beams. The whole is behaving as a so-called ‘‘phaseconjugate’’ mirror, and Figure 4.21 schematically describes its behavior: The incident wave S is phase conjugated and reflected back exactly along its incidence direction. FWM and DFWM is extremely interesting for number of applications, but they are rarely used for material characterization, so we shall not expand further on this subject. More details about this interesting subject can be found, for example, in [Yariv, 1988] among many other books.

4.5

FINAL REMARKS

Before closing this chapter we would like to draw the reader’s attention to the fact that photorefractives exhibit unique features—the adaptive and multiplicative characteristic of real-time recording, the low-pass filtering arising from the finite material response time, the amplitude coupling or energy transfer derived from the phase-shifted nature of photorefractive recording—that have been at the root of many applications focused on image and signal processing (see, e.g., [Khoury et al., 1991; de Montchenault et al., 1987; Anderson et al., 2001] among many other references) and make photorefractive matarials a fertile field for applied research.

CHAPTER 5

ANISOTROPIC DIFFRACTION

Some crystals exhibit anisotropic diffraction, that is, the polarization of the incident and the diffracted light have different directions. This is the case, among others, of sillenite-type crystals, and this feature derives from the structure of their electrooptic tensor. In fact, let us recall the expressions for the index of refraction along axes , Z; and y in Section 1.3.1, which we now designate, in a more conventional way, as x; y, and z, respectively. 1 nx ¼ n0 þ n30 r41 E 2 1 ny ¼ n0  n30 r41 E 2 n z ¼ n0 5.1

COUPLED WAVE WITH ANISOTROPIC DIFFRACTION

Let us now analyze the expression of the coupled wave equations for a pure phase grating in Equations (4.52) and (4.53), where the coupling constant k, as formulated in Equation (4.81), should be now written in tensorial form as 2 3 1 0 0 3 n r41 pE 4 ^¼ 0 ð5:1Þ 0 1 0 5 k 2l 0 0 0 Photorefractive Materials: Fundamental Concepts, Holographic Recording and Materials Characterization, By Jaime Frejlich Copyright # 2007 John Wiley & Sons, Inc.

125

126

ANISOTROPIC DIFFRACTION

In this case coupled Equations (4.52) and (4.53), neglecting absorption (a ¼ 0) and simplifying to jmj ¼ 1, look like: @~ RðzÞ ¼ i^ k~ S @z @ S~ ðzÞ ¼ i^ k~ R cos y @z

ð5:2Þ

cos y

ð5:3Þ

If we define ~ r and~ s as the unit vectors indicating the polarization direction of ~ R and ~ S, respectively, we should write Equations (5.2) and (5.3) as: @R ^~ ¼ ið~ rk sÞS @z @S ^~ ¼ ið~ sk rÞR cos y @z

ð5:4Þ

cos y

ð5:5Þ

where 2 32 cos g 1 n30 r41 pE 4 5 4 ^~ ~ rk s¼ sin g : 0 2l 0 0

0 1 0

32 3 0 cos a 0 54 sin a 5 0 0

ð5:6Þ

with~ sðcos a; sin a; 0Þ and~ rðcos g; sin g; 0Þ This means that the value (modulus) of the effective coupling constant ^~ ~ rk s¼

n30 r41 pE n3 r41 pE ðcos g cos a  sin g sin aÞ ¼ 0 cosðg þ aÞ 2l 2l

ð5:7Þ

is maximum for: cosðg þ aÞ ¼ 1 ^~ ½~ rk smax ¼ k ¼

n30 r41 pE 2l

for

a ¼ g

ð5:8Þ

The diffracted light, with a polarization direction verifying the conditions above, is optimized and will develop over all other possibilities. Equation (5.8) means that the polarization directions of the incident and the diffracted beams are symmetric relative to the coordinate axis (x or y) in the crystal incidence plane as illustrated in Figure 5.1. Note that in the case of a sillenite-type crystal with an electric field applied as illustrated in Figure 1.8, the principal coordinate axes of the index ellipsoid are 45 -rotated as illustrated in Figure 1.9, so that in this case the actual picture stands as represented in Figure 5.2, which is 45 -rotated to that of Figure 5.1,

ANISOTROPIC DIFFRACTION AND OPTICAL ACTIVITY

R(z)

S(z)

y

^

127

^

s

r

α γ

x α =–γ

Figure 5.1. Input and output light polarization.

S(z) ^

s y

x α γ α = –γ

R(z) ^

r

Figure 5.2. Input and output polarization referred to actual principal axes coordinates.

but does not change the fact that the output polarization directions of the incident and diffracted beams are simetric along (the new) axes x and y. Exercise: Repeat the above procedure for lithium niobate (see Section 1.3.2) and verify that there is no anisotropic diffraction for this material.

5.2

ANISOTROPIC DIFFRACTION AND OPTICAL ACTIVITY

A possible solution for coupled Equations (5.4) and (5.5) is RðzÞ ¼ R0 cosðkzÞ þ iS0 eif sinðkzÞ if

SðzÞ ¼ iR0 sinðkzÞ þ S0 e cosðkzÞ

ð5:9Þ

ð5:10Þ

Assuming kz 1, the components polarized along g and a ¼ g are, respectively, Rg ðzÞ ¼ R0 cosðkzÞ

Rg ðzÞ ¼ iS0 eif sinðkzÞ

Sg ðzÞ ¼ S0 eif cosðkzÞ Sg ¼ iR0 sinðkzÞ

ð5:11Þ

ð5:12Þ

128

ANISOTROPIC DIFFRACTION

5.2.1

Diffraction Efficiency with Optical Activity q

From Equations (5.11) and (5.12) we can write dSg ¼ iRg kdz Rg ¼ R0 cosðkzÞ

ð5:13Þ ð5:14Þ

with the x and y components at the crystal output z ¼ d being dSg x ¼ iR0 k cosðkzÞ cos½g þ rðd  zÞdz

ð5:15Þ

dSg y ¼ iR0 k cosðkzÞ sin½g þ rðd  zÞdz

ð5:16Þ ð5:17Þ

After factoring the trigonometric functions above and integrating we get   iR0 k sinðkz þ 2rz þ g0  rdÞ sinðkz  2rz  g0 þ rdÞ z¼d SðzÞg x ¼ þ 2 k þ 2r k  2r z¼0

ð5:18Þ

  iR0 k cosðrd  g0  kz  2rzÞ cosðrd  g0 þ kz  2rzÞ z¼d SðzÞg y ¼ ð5:19Þ þ 2 k þ 2r k  2r z¼0 g ¼ g0 þ rz

g0 ¼ gð0Þ

ð5:20Þ

Assuming k 2r we have SðdÞg x ¼

iR0 k sinðkd þ rd þ g0 Þ  sinðg0  rdÞ  sinðkd  2rd  g0 Þ 2 2r

ð5:21Þ

SðdÞg y ¼

iR0 k cosðkd þ rd þ g0 Þ  cosðrd  g0 Þ þ cosðg0 þ rd  kdÞ 2 2r

ð5:22Þ

that can be written as iR0 k 2 cosðkdÞ sinðrd þ g0 Þ þ 2 sinðrd  g0 Þ 2 2r iR0 k 2 cosðkdÞ cosðrd þ g0 Þ  2 cosðrd  g0 Þ SðdÞg y ¼ 2 2r SðdÞg x ¼

ð5:23Þ ð5:24Þ

ANISOTROPIC DIFFRACTION AND OPTICAL ACTIVITY

[001]

129

d 70°

d 70°

t t

ρd = 20° [010] t

[110]

[100]

t

Figure 5.3. General illustration of the polarization direction of the transmitted and diffracted beams through a crystal with optical activity and anisotropic diffraction. At mid-crystal thickness the polarization directions of the transmitted and diffracted beams are 10 shifted from the [110] and [001] axes respectively.

where R2 k 2 ISdiff ¼ jSðdÞg x j2 þ jSðdÞg y j2 ¼ 0 2 ½sinðrdÞ2 r   2 I diff sinðrdÞ Z ¼ S 2  ðkdÞ2 rd jR0 j [001]

d d

ρd = 20°

ð5:25Þ ð5:26Þ

90°

90°

t t

[010] t

[110]

[100] t

Figure 5.4. Transmitted and diffracted beams orthogonally polarized at the output through a crystal with optical activity and anisotropic diffraction. Assuming rd ¼ 20 , the incident beam’s polarization direction at the input plane should be 10 referred to the [110]-axis.

130

ANISOTROPIC DIFFRACTION

[001] d d

t t

ρd=20°

[010] t [110]

[100] t

Figure 5.5. Transmitted and diffracted beams parallel-polarized at the output through a crystal with optical activity and anisotropic diffraction. Assuming rd ¼ 20 , the incident beam’s polarization direction at the input plane should be 35 referred to the [110]-axis.

5.2.2

Output Polarization Direction

In Section 5.2.1 it was stated that g0 is the angle of the polarization direction of wave ~ RðzÞ at the input. Let as then assume as to be the corresponding angle (see Fig. 5.2) for ~ SðzÞ at the output. From Equation (5.23) and Equation (5.24) we may compute as as follows tan as ¼

SðdÞg y

SðdÞg x

¼  tan g0

ð5:27Þ

always for kd 1. Figures 5.3 to 5.5 illustrate some typical results for a crystal with rd ¼ 20 and interfering beams with same input polarization direction.

CHAPTER 6

STABILIZED HOLOGRAPHIC RECORDING

6.1

INTRODUCTION

Holographic setups are extremely sensitive to environmental perturbations (thermal drifts, air currents, mechanical vibrations, etc), and this fact makes it difficult to obtain reproductible holographic recordings, unless the recording time is much smaller than the period of the perturbations. The characteristic recording time, in photorefractive materials at least, is roughly inversely proportional to the average irradiance onto the crystal, so that stable holograms may only be produced with high-intensity laser beams. High intensity, however, is rarely achieved in many applications like image processing experiments, for example, where the light scattered back from the target is usually weak. For a Bi12 TiO20 sample illuminated by 200–300 mW=cm2 in the 514-nm wavelength the recording time is of the order of a few seconds, and it is still larger for the 633-nm wavelength. The simplest way to overcome holographic instability is to perform a stabilized holographic recording. It is still better to carry out ‘‘self-stabilized’’ recording, where the hologram being recorded is used itself as a reference to stabilize the relative position of the recording pattern of fringes. Although not related to photorefractive materials, Figures 6.1 and 6.2 clearly show the possibilities of self-stabilized recording in microelectronics-grade photoresists: The large height-to-width ratio of the structures shown in these figures could never have been obtained without using self-stabilized holographic recording. In fact, this technique allows one to firmly fix the recording pattern of light in order Photorefractive Materials: Fundamental Concepts, Holographic Recording and Materials Characterization, By Jaime Frejlich Copyright # 2007 John Wiley & Sons, Inc.

131

132

STABILIZED HOLOGRAPHIC RECORDING

Figure 6.1. Scanning electronic microscopy image of a 1D hollow sleevs structure first recorded on photoresist film, then metallic vacuum deposited and finally washed away form all remaining photoresist to produce the hollow metallic structures. Produced and ´ ptica, Instituto de Fı´sica, Universidade photographed by Lucila Cescato, Laborato´rio de O Estadual de Campinas, Brazil.

to obtain the very sharp spatial light contrast that is necessary to produce these structures. Another feature of stabilized recording is shown in Figure 6.3, where the two first spatial harmonics (with adequate spatial frequency, relative amplitude and mutual phase shift) of a ‘‘sawtooth’’ profile were recorded on a photoresist film. The profile does not look much like a sawtooth, but its behavior under diffraction definitely does, because the diffraction properties rely, above all, on the first few harmonics

Figure 6.2. Scanning electronic microscopy image of a 2D-array holographically recorded and chemically developed on photoresist film. Produced and photographed by Lucila ´ ptica, Instituto de Fı´sica, Universidade Estadual de Campinas, Cescato, Laborato´rio de O Brazil.

MATHEMATICAL FORMULATION

133

Figure 6.3. Scanning electronic microscopy image of a blazed grating made by the holographic recording of the first and the second spatial harmonic components of a sawtoot´ ptica, shape profile on a photoresist film. Produced and photographed at Laborato´rio de O Instituto de Fı´sica, Universidade Estadual de Campinas, Brazil and published in ref. [Cescato et al., 1988].

[Cescato et al., 1988]. Stabilization here has a double purpose: avoiding perturbations on the recording setup and also adequately fixing the mutual phase shift between the two spatial harmonics during recording. Pioneer research leading to the present self-stabilized holographic recording was proposed by Neumann and Rose [Neumann and Rose, 1967] by 1967. They first proposed to amplify the recording interference pattern of fringes with a microscope objective and project this amplified pattern onto a photodetector in order to operate an electronic feedback loop to stabilize this recording pattern of fringes. MacQuigg [MacQuigg, 1977] further improved this technique by using an auxiliary (previously recorded) hologram, instead of a microscope objective, for amplifying the pattern of fringes on the photodetector. In this way a much brighter amplified pattern of fringes was obtained. Additionally he phase modulated one of the interfering beams to produce a temporal harmonic term, in one of the output beams, to be detected with a phase-selective frequency-tuned amplifier known as a‘‘lock-in’’ amplifier and to be used as error signal in the feedback stabilization loop.

6.2

MATHEMATICAL FORMULATION

The plain stabilized recording is a particular simple case where one uses an external reference (a fixed hologram or a glass plate adequately placed by the side of the sample) that is able to produce an interference pattern of light that can be used to operate the feedback loop. In this chapter we shall focus on the self-stabilized recording because it is the more complex situation and is also the most interesting procedure for holographic

134

STABILIZED HOLOGRAPHIC RECORDING

recording. This technique can also be used with nonreversible materials like photoresists [Cescato et al., 1987; Cescato and Frejlich, 1988], for example. The self-stabilization procedure does not use an external reference. This technique is based on phase modulation, as described in Section 4.3, where a modulation of amplitude cd and angular frequency  ( much larger than the frequency response of the hologram) is produced in the phase of one of the two interfering beams (of irradiances IR and IS ) in the holographic setup. By this means, the phase shift j between the transmitted and diffracted beams behind the sample is correspondingly modulated, so that the expression of the overall irradiance along the direction IS behind the sample can be written as IS ¼ IS0 ð1  ZÞ þ IR0 Z þ 2

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiqffiffiffiffiffiffiffiffi Zð1  ZÞ IS0 IR0 cosðj þ cd sin tÞ

ð6:1Þ

where IR0 and IS0 are the values at the input. Because of the nonlinear relation between j and IS , harmonic terms in  appear where the amplitude of the first and second ones were already formulated in Equations (4.166) and (4.167): pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiqffiffiffiffiffiffiffiffi IS ¼ 4J1 ðcd Þ Zð1  ZÞ IS0 IR0 sin j pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiqffiffiffiffiffiffiffiffi IS2 ¼ 4J2 ðcd Þ Zð1  ZÞ IS0 IR0 cos j

with j described in Equation (4.122) as

tan j ¼ 

g sin z 2   1  b2  g  z  cos z þ sinh z cosh 2 2 2 2 1þb

For nonphotovoltaic materials in the absence of external electric field the hologram phase shift (phase difference between the recording pattern of fringes and the resulting hologram) is fP ¼ p=2 (with tan fP  =g), which substituted in the expression above for tan j leads to j ¼ 0 (or p). This means that, in equilibrium, it should be IS ¼ 0 and we can therefore use IS as error signal to operate a stabilization system in order to keep the holographic setup actively fixed to this j ¼ 0 condition. Unless otherwise stated, we shall henceforth assume that IS is always used as error signal in self-stabilization experiments. The setup is schematically represented in the block diagram of Figure 6.4 and the schema of the actual setup in Figure 6.5. The effect of a phase perturbation (noise) jN on the two-wave mixed output is illustrated in Figure 6.6. The photodetector D transforms the overall irradiance IS at the crystal output into an electric signal the harmonic term amplitudes of which in  and 2 are, respectively, VS ¼ Kd IS

VS2

¼

Kd2 IS2 ;

ð6:2Þ

ð6:3Þ

MATHEMATICAL FORMULATION

135

ϕN ϕf

PM v(t) OSC

ϕ

HOLOGRAPHIC SETUP

Is

+ Vf LA–Ω

HV

D

Figure 6.4. Block-diagram of a self-stabilized setup: D photodetector, LA- phase sensitive lock-in amplifiers tuned to , HV voltage source for the phase modulation device PM, OSC oscillator at frequency . The output phase shift, feedback and noise phases are j, jf and jN respectively. Reprinted from [Barbosa et al., 2002]

with Kd and Kd2 being the photodetector responses to signals with frequencies  and 2, respectively. A -tuned lock-in amplifier LA- selects the first harmonic term and produces a demodulated and amplified signal VC ¼ A VS

ð6:4Þ

which is the correction signal in the feedback loop, where A is the amplification. This signal is fed to the voltage source HV, which produces an electric feedback signal Vf ¼ K0 VC

ð6:5Þ

where K0 is the HV amplification. The signal Vf acts on the phase modulator device PM (in this case a piezoelectric supported mirror PZT), which produces a correction

M er

IS

0

las

D

IR C

BS 0

IS

IR PZT Ω OSC

Vd

+

Vf

HV

Vc

LA Ω

LA 2Ω 2Ω

VS

Figure 6.5. Schematic description of the actual self-stabilized holographic recording setup: C photorefractive crystal, D photodetector, LA and LA2 phase sensitive lock-in amplifiers tuned to  and 2 respectively, HV high voltage source for the piezoelectric supported mirror PZT acting as phase modulator, OSC oscillator at frequency .

136

STABILIZED HOLOGRAPHIC RECORDING

ϕΝ

NOISE 0

φ + ϕN

R

S

tran

sm

0 R

diff

itte

d

rac t

pattern of fringes

ed

S mutually shifted by ϕ + ϕN

hologram

Figure 6.6. Schematic description of the effect of noise on the two-wave mixing in the holographic setup.

feedback phase jf on the holographic setup 0 Vf ¼ A sin j jf ¼ KPM

qffiffiffiffiffiffiffiffipffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0 K0 A Kd 4J1 ðcd Þ IS0 IR0 Zð1  ZÞ A ¼ KPM

ð6:6Þ ð6:7Þ

0 where KPM is the voltage-to-phase response at the PM for   0. At the same time an oscillator OSC produces a small ac voltage of frequency  of the form

vðtÞ ¼ Vd sin t

ð6:8Þ

that is added to Vf and is fed to the PM to produce the phase modulation of frequency  and amplitude  Vd cd ¼ KPM

ð6:9Þ

that is necessary to produce the phase modulation that is represented in Equation  is the voltage-to-phase response of the PM at . (6.1). The quantity KPM 6.2.1

Stabilized Stationary Recording

In the absence of feedback, the output phase in the setup can be written as 0 V0 j ¼ c0 þ cH  KPM

0 V0 fP ¼ cH  KPM

ð6:10Þ

ð6:11Þ

MATHEMATICAL FORMULATION

137

where cH is the (phase) position of the recorded hologram, V0 is the dc bias voltage 0 applied to the PM, KPM V0 is the pattern of fringes position, and their difference is the so-called hologram phase shift fP . The c0 is a correction term that depends on the nature of the hologram, the value of fP , and the degree of phase coupling [Kukhtarev et al., 1979a]. In steady-state conditions fP and c0 are constants, and the corresponding steady-state value of j is also a constant j0 j0 ¼ c0 þ fP

ð6:12Þ

The term c0 is implicitely contained in Equation (4.122) and the expression for fP in Equation (4.86) g sin z  2 tan fP ¼ ð6:13Þ tan j0 ¼   2  g 1b  g  cosh z  cos z þ sinh z 2 2 2 1 þ b2 I where j was substituted by its non-feedback-constrained value j0 . Note that g in Equation (4.85) and  in Equation (4.80) are proportional to the components of the space-charge electric field amplitude, which are in-phase and p=2-shifted, respectively, to the pattern of fringes. For the particular case when fP ¼ p=2, it is g ¼ 0 and consequently j0 ¼ 0 (or p) and implicitly c0 ¼ p=2. For other values of fP the corresponding values for j0 can be computed from Equation (6.13). In steady state under feedback conditions, the expression of jf in Equation (6.6) is substracted (negative feedback) from Equation (6.10) to give the steady-state feedback equilibrium value jeq 0 jeq ¼ c0 þ cH  KPM V0  j f

with

jf ¼ A sin jeq

ð6:14Þ

where jeq is, in general, different from the non-feedback-constrained j0 one (jeq 6¼ j0 ) except for the case j0 ¼ 0 (that is, fP ¼ p=2), in which case it is also jeq ¼ 0 in Equation (6.14). This means that under feedback constraint the system will be in stationary equilibrium only for fP ¼ p=2. Otherwise the feedback will force the pattern of fringes and associated hologram to move because of the mismatch between jeq and j0 . 6.2.1.1 Stable Equilibrium Condition. For the stationary (nonmoving hologram) case, where j0 ¼ 0, it is still necessary to analyze the stability of the equilibrium condition jeq ¼ 0. In the presence of a phase perturbation jN in the setup, Equation (6.14) becomes 0 V0  jf j ¼ jN þ c0 þ cH  KPM

ð6:15Þ

A stable equilibrium condition requires that dj=djN  0, which substituted into Equation (6.15) results in   dj 1 1 0 ð6:16Þ ¼ ¼ djN jeq 1 þ A cos j jeq 1 þ A

138

STABILIZED HOLOGRAPHIC RECORDING

ϕN PM

ϕf

ϕ

HOLOGRAPHIC SETUP

v(t)

Is

+

OSC

Vf INT

HV

LA–Ω

D

Figure 6.7. Block diagram of fringe-locked running hologram setup. Same as for Fig. 6.4 with the addition of an integrator INT at the output of the lock-in amplifier. Reproduced from [Barbosa et al., 2001]

where j ¼ jeq  0 was assumed in order to enable the use of IS as error signal. The condition in Equation (6.16) is actually verified for a large negative (A  1) feedback. 6.2.2

Stabilized Recording of Running (Nonstationary) Holograms

Moving holograms or space-charge waves are described in Section 3.4. If the pattern of light is moving with the resonance speed characteristic for the hologram in the crystal (that is, Kv ¼ oI ) a maximum in diffraction efficiency is reached [Stepanov and Petrov, 1988] as shown in Figure 3.23. The operation of the feedback in order to produce running holograms in a self-stabilized way is the central point here. Such holograms are also known as ‘‘fringe-locked’’ running holograms. As discussed above, a nonstationary (running) hologram is automatically established when j0 6¼ jeq . The latter relation is verified when j0 6¼ 0. In this condition the hologram is forced to be erased and rewritten continuously somewhere ahead, and by this means a continuous movement occurs. The block diagram of this new experimental setup is depicted in Figure 6.7, whereas the actual setup schema is depicted in Figure 6.8. An integrator was included here between the lock-in amplifier output and M er

IS

0

las

D

IR C

BS 0

IS

IR

PZT Ω OSC

Vd

+

Vf

HV

VC

INT

LA Ω

LA 2Ω 2Ω

VS

Figure 6.8. Schematic actual setup for self-stabilized running hologram recording: same as for Fig. 6.4 with the addition of an integrator INT at the output of the lock-in amplifier.

139

MATHEMATICAL FORMULATION

the voltage source HV so that the correction feedback phase jf is not described any more by Equation (6.6) but by the integral A jf ¼ ti

Zt

ð6:17Þ

sin j dt

0

where ti is a factor arising from the integrating circuit. This jf is required to produce the mismatch between jeq and j0 that is necessary for running hologram generation and at the same time to fulfill the feedback loop condition jeq  0 that is determined by the use of IS as error signal in the feedback. The expression for jeq under these new feedback constraints can be formulated as jeq ¼ c0 þ cH 

0 KPM V0

A  ti

Zt

sin jeq dt

ð6:18Þ

0

The steady state condition is represented by djeq dc0 dcH þ  kf sin jeq ¼ 0 ¼ dt dt dt

ð6:19Þ

where kf  A=ti . The equilibrium condition dc0 =dt ¼ 0 should be considered to get the expression for the hologram speed oH ¼

dcH ¼ kf sin jeq dt

ð6:20Þ

But the hologram speed oH is a function of the mismatch between jeq and j0 , so a relation can be stated in the form oH ¼ f ðjeq  j0 Þ

with

f ð0Þ ¼ 0

ð6:21Þ

where f ðjeq  j0 Þ is a function depending on material and experimental parameters. From Equations (6.20) and (6.21) we should write oH ¼ f ðjeq  j0 Þ ¼ kf sin jeq

ð6:22Þ

showing that kf ! 1 leads to jeq ! 0, in which case Equation 6.22 becomes lim oH ¼ f ðj0 Þ ¼ kf sin jeq

kf !1

ð6:23Þ

which means that for a sufficiently large amplification kf the hologram speed oH is independent of kf and dependent on the unconstrained equilibrium value j0 through the functional relation f ðj0 Þ. The latter function will be discussed further on.

140

STABILIZED HOLOGRAPHIC RECORDING

6.2.2.1 Stable Equilibrium Condition. As for the case of stationary holograms, we need to analyze the stability of the equilibrium, that is, the way the feedback loop reduces the effects of both a noise (see Fig. 6.6) jN on the phase and a noise oN on the speed, near the equilibrium position. For this purpose let us write the differential Equation (6.19), describing the output phase shift jeq under feedback at equilibrium, as djeq þ kf jeq ¼ oH dt

ð6:24Þ

where we have assumed sin jeq  jeq 1, in which case the general solution is [Boas, 1983]: jeq ¼ e

R

kf dt



jN þ

Z

If we assume kf to be independent of time it is of integration by parts we can write: Z

oH ekf t dt ¼

oH e R

R

kf dt

dt



ð6:25Þ

kf dt ¼ kf t. Also, using the theorem

€H oH kf t o_ H kf t o e  2 e  3 ekf t  . . . þ C kf kf kf

which simplifies to Z

oH ekf t dt 

oH kf t e þC kf

for

o_ H k2f

and the expression for Equation (6.25) becomes:   oH oN kf t jeq  e þ ji þ kf kf

ð6:26Þ

ð6:27Þ

where we have written C  oN with jN and oN being constants arising from the solution of the homogeneous differential equation and consequently representing the transient solutions. The quantities jN and oN can be thought to be perturbations or noises on the phase and on the speed, respectively. For a large negative amplification (kf  1) the term in square brackets (where noises jN and oN are included) is rapidly vanishing. The remaining stationary term oH =kf represents the steady-state (inhomogeneous differential equation) solution of the nonperturbed system. For a sufficiently large kf it is oH =kf  0 and consequently jeq  0 in Equation (6.27), as required for the feedback operation. 6.2.2.2 Speed of the Fringe-Locked Running Hologram. Equation (6.23) states that the speed of the hologram is a function of f ðj0 Þ, where j0 is the phase shift between the transmitted and diffracted beams along the same direction behind the crystal, in equilibrium, without feedback. On the other hand, the actual

MATHEMATICAL FORMULATION

141

expression for the self-stabilized (also known as fringe locked) running hologram speed can be computed from IS ¼ 0

ð6:28Þ

a condition that is inherent to the use of IS as error signal in the feedback stabilization loop. The condition above means, from Equations (4.158), (4.85), and (4.81), that g /
ð6:29Þ

For the case of a running hologram, however, the expression Eeff above should be st subtituted by Esc as formulated in Equation (3.80), where the bulk light absorption effect on the hologram speed has been neglected. Then from st g¼0 g /
ð6:30Þ

and its explicit expression in Equation (3.85) we get Kv ¼

1 E0 =ED 2 tM ð1 þ E0 =ED2 ÞK 2 L2D þ 1

ð6:31Þ

From Equation (6.31) we see that Kv depends on both ED /  and E0 / g which in turn determine j0 in Equation (6.13), so that Kv itself is implicitly determined by j0 and the theoretical statement in Equation (6.23) is therefore justified. The effect of bulk absorption on the hologram speed is more difficult to analyze and is studied in Section 9.2.1. To experimentally verify the independence of Kv on the settings of the feedback loop, the Kv was measured for different values of kf on a BTO crystal using the 514.5-nm wavelength and a nominally applied field of 4.7 kV/cm, as shown in Figure 6.9. This figure shows that for a 50-fold variation in kf (in arbitrary units) the Kv does vary by only 1:5%, a variation that is roughly of the order of magnitude of data dispersion in the experiment. The increase of kf in Equation (6.23) just makes jeq approach zero without sensibly affecting oH as predicted by the theory and here experimentally confirmed. Nevertheless, the adequate choice of kf is of the highest practical relevance in the sense that the operation of the feedback is very sensitive to the amplification in the loop: A low amplification may be not enough, whereas a too large amplification may produce instabilities and even drive the setup to oscillatory behavior. 6.2.3 Self-Stabilized Recording with Arbitrarily Selected Phase Shift The self-stabilized holographic recording setup described above suffers from a limitation: The phase shift j between the transmitted and diffracted beam behind the sample is fixed either to j ¼ 0, p if the first harmonic term V  is used as error signal, or to j ¼ p=2 if the second harmonic term V 2 is used instead.

142

STABILIZED HOLOGRAPHIC RECORDING

0.100

Kv (rad/s)

0.095

0.090

0.085

0.080

0

20

40

60

κf (au)

Figure 6.9. Fringe-locked running hologram speed: Kv (rad/s) vs feedback amplification kf (arbitrary units) in a fringe-locked running hologram experiment carried out on an undoped Bi12 TiO20 crystal using the 514.5 nm wavelength, with E0 ¼ 4:7 kV/cm, IR0 ¼ 533 mW/cm2, IS0 ¼ 20 mW/cm2, =ð2pÞ ¼ 2:1 kHz, K ¼ 7:55 mm1 and cd  0:011 rad. Reproduced from [Barbosa et al., 2002]

We shall now describe a modification in the electronics of the feedback stabilization loop that allows one to actively fix j to any value at will. The modified setup is schematically represented in Figure 6.10. Let us recall the expressions of the first and second harmonic terms, in the output voltage v D ðtÞ from the photodetector D, as derived from Equations (4.166) and (4.167) and Equations (6.2) and (6.3): v  ðtÞ ¼ V0 sin j sinðt þ E1 Þ

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiqffiffiffiffiffiffiffiffi Zð1  ZÞ IS0 IR0 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiqffiffiffiffiffiffiffiffi  Kd2 4J2 ðcd Þ Zð1  ZÞ IS0 IR0

V0  Kd 4J1 ðcd Þ

v 2 ðtÞ ¼ V02 cos j cosð2t þ E2 Þ V02

ð6:32Þ ð6:33Þ

where Kd and Kd2 are the irradiance-to-voltage conversion at the photodetector D for frequencies  and 2, respectively, and E1 and E2 are the phase shifts due to the electronics. The signal v D ðtÞ is, on one side, filtered to extract the 2 component (bandpass filter BP 2), which is amplified (by the amplifier A), and the result is the signal v 2 ðtÞ ¼ V 2 a cosð2t þ EÞ cos j

ð6:34Þ

The same signal v D ðtÞ is also filtered to extract the  component (bandpass filter BP ), which is phase shifted with PS and then multiplied with the signal from the oscillator v  ðtÞ ¼ v d sin t

ð6:35Þ

MATHEMATICAL FORMULATION

143

M r

IR C

BS PM HV

Vf

vd(t)

0

IS

+

M VC

D

IS

0

e las

vdsin Ωt OSC Ω

IR

BP Ω vdsin Ωt

BP 2Ω

X A

INT PS θ

LA 2Ω

VX

VY

v1(t)

+

v2(t)

v+(t)

Figure 6.10. Schema of the self-stabilized setup in Fig. 6.8 modified to operate with arbitrarily selected j: PM is a generic phase modulation that could also be the PZT, BP and BP 2 are bandpass filters tuned to  and 2 respectively,  is a function multiplier, PS is a phase shifter, LA 2 is a dual-phase lock-in amplifier tuned to 2 with orthogonally shifted outputs X and Y, with all other components as already described in Fig. 6.8.

A second harmonic term v 1 ðtÞ arises from this operation v 1 ðtÞ ¼ V 2 sinð2t þ EÞ sin j þ dc

ð6:36Þ

where the low-frequency dc terms can be neglected because they will be filtered out in the next step by the lock-in amplifier LA-2 tuned to 2. The phase shift E in v 1 ðtÞ and v 2 ðtÞ is the same because it is possible to adjust it by means of the PS on the v 1 ðtÞ signal above. The signals v 1 ðtÞ and v 2 ðtÞ are also adjusted to have the same amplitude (by means of the amplifier A) and then added to get the signal v þ ðtÞ ¼ V0 sinð2t þ EÞ sin j þ V0 cosð2t þ EÞ cos j ¼ V0 cosð2t þ E  jÞ

ð6:37Þ

The signal above is fed to the double phase lock-in amplifier LA-2, which allows one to measure the two components VX ¼ V0 sinðj  jS Þ

ð6:38Þ

jS  E þ y

ð6:40Þ

VY ¼ V0 cosðj  jS Þ with

ð6:39Þ

where y is the phase shift selected for the reference signal in the lock-in amplifier.

144

STABILIZED HOLOGRAPHIC RECORDING

The advantage of this new signal processing is the fact that we are able to include the phase shift j between the transmitted and diffracted beams behind the sample into the temporal argument of the second harmonic term so that we are now able to operate with the phase j  jS instead of simply j, where jS is adjusted at will by acting on the reference phase shift y in the lock-in amplifier. Now it is possible to select the signal VX as error signal, for example, in which case the system will automatically set the argument of the sin to zero so that the phase shift will be set to j ¼ jS ¼ E þ y

ð6:41Þ

This abitrary phase shift stabilized setup is interesting not only for photorefractives but for classic optical interferometry in general. For the particular case of photorefractives, as we shall see in the following sections, the output phase shift j is not constant throughout the recording process except for the case of p=2shifted holographic phase shift (fP ¼ p=2). It is therefore not possible, in general, to adjust the operating jeq in the setup to the unconstrained j in order to avoid phase mismatching and keep the hologram self-stabilized without moving. It is, however, possible to measure j during the recording process, as already proposed elsewhere [Nakamura et al., 1997], and continuously feed this information to the feedback stabilization system to keep the system stationary self-stabilized.

6.3

SELF-STABILIZED RECORDING IN ACTUAL MATERIALS

We shall apply self-stabilization for holographic recording on two widely different materials: Bi12 TiO20 and LiNbO3:Fe. In both cases we shall see that self-stabilized recording not only reduces external perturbations on the setup but sometimes also modifies the recording process itself. For the case of LiNbO3:Fe, for example, this procedure allows one to produce a 100% diffraction efficiency volume grating for widely different conditions for the sample and for any recording beams ratio b2 , which is not always possible in a non-self-stabilized regime. To understand this feature one should figure out that the feedback-driven pattern of fringes movement is automatically adjusted to produce the required hologram phase shift to achieve Z ¼ 1, whatever the material and experimental conditions. The present conclusions may certainly be extended to other photorefractive materials besides lithium niobate provided their holographic phase shift and coupling effects are adequately considered. Section 6.3.1 on self-stabilized recording in sillenites will focus on the ability of the setup to cope with environmental perturbations, whereas Section 6.3.2 on LiNbO3 is directed toward illustrating the way in which self-stabilization may interfere in the recording process itself regardless of the always underlying ability to reduce external perturbations, which is extremely important for long-term duration recording materials, as this one certainly is. 6.3.1

Self-Stabilized Recording in Sillenites

The presently analyzed self-stabilization holographic recording may be used to record holograms in undoped BTO crystals with the setup schematically shown in

SELF-STABILIZED RECORDING IN ACTUAL MATERIALS

145

[001] K

[110]

Figure 6.11. Transverse optical configuration for holographic recording on BTO: The incident beams, incidence plane and pattern-of-fringes onto the input crystal face are shown, ~ being perpendicular to the [001]-axis and parallel to the [110]with the holographic vector K axis.

Figure 6.5, with the crystal in the transverse optical configuration as shown in Figure 6.11. For a 90 -shifted photorefractive hologram (which is the present case for a nonphotovoltaic crystal without externally applied field) we deduce from Equation (4.86) that it should be g ¼ 0, and from Equation (4.158) we get I  ¼ 0 pffiffiffi whereas from Equation (4.159) we know that I 2 / Z. This means that the I  signal may be used as ‘‘error signal’’ in our negative-feedback stabilization loop. In this case the feedback loop actively keeps the hologram and the pattern of light in the stable 90 -shifted position, and the recording proceeds in a self-stabilized mode. Because the free and feedback-constrained conditions are the same (j0 ¼ jeq ¼ 0 as deduced from Equation (4.122)), a stationary (nonmoving) hologram is recorded. We have already demonstrated (see Section 6.2.1.1) that such a system is in stable equilibrium so that each time a perturbation shifts the system away from 90 , a correction signal acting on the piezo-mirror drives back the system to the I  ¼ 0 stable position. A lock-in amplifier tuned to 2 may be used to follow pffiffiffi the evolution of the diffraction efficiency profiting from the relation I 2 / Z. The good performance of self-stabilized holographic recording on BTO and the usefulness of using the VS2 / IS2 harmonic to follow the recording are evident from the results reported in Figures 6.12 and 6.13. 6.3.2

Self-Stabilized Recording in LiNbO3

Self-stabilized recording in a strongly photovoltaic material such as LiNbO3 is also possible. In this case, however, it is fP  180 (that is,   0) instead of p=2. In the absence of self-diffraction Equation (4.172) shows that fP  p leads to j  p=2 and to IS2  0 in Equation (4.167). In this case IS2 should be used as error signal in the feedback stabilization loop, as shown in Figure 6.14, instead of IS ; as was the case for sillenites. 6.3.2.1 Holographic Recording Without Constraints. Let us first analyze the holographic recording without feedback constraints, that is, without selfstabilization. The effect of self-diffraction on the IS2 -term as represented by Equation (4.159) is more complicated than for IS because IS2 is not zero, even for   0, unless IS ¼ IR . Additionally, holographic recording (without feedback)

146

STABILIZED HOLOGRAPHIC RECORDING

Signal (Volts)

1.2 0.8 0.4 0 –0.4 0

1

2

3

4

5

0

1

2

3

4

5

Signal (Volts)

1.2 0.8 0.4 0 –0.4 Recording Time (min)

Figure 6.12. Self-stabilized recording in a Bi12 TiO20 crystal: The upper figure shows the evolution of the VS (thin black line) and the VS2 (thick red line) when the stabilization is off. The lower figure shows the evolution of both signals when VS is used as error signa in which pffiffiffi case VS2 / Z. The recording was with l ¼ 633 nm with IR0 ¼ 0:52 mW/cm2 and 0 IS ¼ 11 mW/cm2, interfering with an angle 2y ¼ 60 on a 10 mm-thick crystal with the ~ perpendicular to the [001] pattern-of-fringes on the (110) plane and the hologram vector K axis and parallel to [110].

in photorefractive crystals under externally applied electric field exhibits amplitude and phase coupling, as already reported in Section 4.2.1, that lead, in general, to a tilted hologram [Buse et al., 1995]. This is also true, in general, for photovoltaic lithium niobate crystals [Tao et al., 1994]. Tilted holograms are automatically out 0.15

V2Ω (au)

Nonstabilized

Stabilized

0.10

0.05

0

5

10

15

20

Time (s)

Figure 6.13. Second harmonic evolution during holographic recording in a nominally undoped photorefractive BTO crystal with the self-stabilization off (left side) and on (right side), for IR0 þ IS0 ¼ 12 mW/cm2, using the l ¼ 514:5 nm laser line and K  4:5 mm1 . Reproduced from [Barbosa et al., 2002]

SELF-STABILIZED RECORDING IN ACTUAL MATERIALS

147

M r

IS

0

e las

D

IR C

BS 0

IS

IR

PZT Ω

Vd

OSC

+

Vf

HV

VC

INT

LA 2Ω

LA Ω Ω

VS

Figure 6.14. Experimental setup: BS beamsplitter, C: LiNbO3:Fe crystal, M mirror, PZT pzt-driven mirror, OSC signal generator, HV high voltage source, INT integrator, D1;2 detectors, LA- and LA-2 lock-in amplifiers tuned to  and 2, respectively.

of Bragg (always referred to the direction of any of the recording beams). Tilting arising from phase coupling prevents the achievement of 100% diffraction efficiency [Frejlich et al., 1997] unless the crystal is rotated, after recording, in order to adjust to the Bragg condition. In reduced LiNbO3:Fe crystals, phase coupling can be avoided by using recording beams of equal irradiance [Tao et al., 1994] as explained in Section 4.2.1, and in this condition 100% diffraction efficiency may be obtained [Freschi and Frejlich, 1994]. For oxidized samples, however, it is not possible to avoid phase coupling so that, in general, 100% diffraction efficiency cannot be obtained except for very particular conditions. 6.3.2.1.1 Space-Charge Electric Field Build-Up. Because the self-stabilized setup produces a running hologram, in general, we shall here refer to a moving (with ~ space-charge electric field grating, that is, speed v along the grating vector K) neglecting self-diffraction effects, ruled by a differential equation [Miguel et al., 2000; Aguilar et al., 1997; Jariego and Agullo´-Lo´pez, 1990] tsc

@Esc ðtÞ þ Esc ðtÞ ¼ mEeff eiKvt @t

ð6:42Þ

with Eeff ¼

1þ

Eph þ iED

K 2 l2s



Nþ iKlph NDD



Eph Nþ

1  iKlph NDD

ð6:43Þ

and: 1=tsc ¼ oR þ ioI oR ¼

1 1 þ K 2 l2s tM 1 þ K 2 L2D



1 / tM

ND  NDþ NDþ

ð6:44Þ ð6:45Þ

148

STABILIZED HOLOGRAPHIC RECORDING

Klph Klph NDþ 1 NDþ   tM ND tM ð1 þ K 2 L2D Þ2 ND Eph ND /  Eq ND  NDþ

oI ¼  Klph

ð6:46Þ ð6:47Þ

which includes the effect of a moving grating [Refregier et al., 1985; Stepanov and Petrov, 1988; de Oliveira and Frejlich, 2001a]. The differential equation above is essentially the same as in Equation (3.138) with an additional moving pattern term and excluding the external field E0 . NDþ and ND are the concentration of the empty (electron acceptors Fe3þ ) and the total (acceptors Fe3þ plus donors Fe2þ ) photoactive centers in the sample, respectively. The approximate relations in Equations (6.43), (6.45), and (6.46) derive from the usual assumptions for LiNbO3:Fe: a) far from photoactive center saturation, that is, K 2 l2s 1, b) diffusion length short compared to the grating period as stated by K 2 L2D 1, and c) photovoltaic field much larger than the diffusion field, say, Eph  ED . The proportionality relation in Equation (6.45) is derived from Equations (3.43) and (2.13), whereas that in Equation (6.47) is derived from Equations (3.129) and (3.49). The solution of Equation (6.42) for recording is st iKvt st ðoR þioI Þt e þ mEsc e Esc ðtÞ ¼ mEsc st Esc 

Eeff ðoR þ ioI Þ oR þ iðoI  KvÞ

ð6:48Þ ð6:49Þ

where the first right-hand term represents the stationary space-charge wave moving synchronously along with the pattern of fringes, whereas the second term represents the transient effect fading away with a time constant 1=oR . 6.3.2.1.2 from

Hologram Phase Shift. The hologram phase shift is always computed tan f ¼ =g

with 2pn3 reff =fEsc ðtÞ=mg ¼ =f4k= cos yg l cos y 2pn3 reff
¼

ð6:50Þ ð6:51Þ

as described in Equations (4.80) and (4.85). In the present case, for a quasist was stationary (slowly time-dependent) steady-state recorded grating, Esc substituted by the slowly varying time function Esc ðtÞ=m for the calculation of  and g above.
SELF-STABILIZED RECORDING IN ACTUAL MATERIALS

149

In the absence of self-stabilization the pattern of fringes onto the sample is a stationary one, and therefore only a stationary (v ¼ 0) hologram arises. In this condition and for the case of reduced LiNbO3:Fe crystals it is possible to assume that Klph NDþ =ND 1, in which case oI oR , which substituted into Equation (6.48) leads to Esc ðtÞ=m  Eph ð1  et=tM Þ

ð6:52Þ

In this case (see Eqs. (6.50) and (6.51))   0, and therefore it is f  0; p that characterizes a local hologram. For the case of oxidized samples instead, and still for stationary (v ¼ 0) holograms, it is Esc ðtÞ=m  Eph

1 þ iKlph ðNDþ =ND Þ

1þ

K 2 l2ph ðNDþ =ND Þ2

þ

ð1  et=tM eiKlph ðND =ND Þt=tM Þ

ð6:53Þ

which substituted into the expression for the hologram phase shift in Equation (4.86) with Equations (6.50) and (6.51) leads, in general, to f 6¼ 0; p. In fact, the holographic phase f here depends (among other parameters) on the degree of oxidation NDþ =ND and may therefore considerably differ from 0 and p. 6.3.2.1.3 Diffraction Efficiency with Wave Mixing. In the presence of selfdiffraction (wave mixing) there is, in general, amplitude and phase coupling between the interfering beams, in which case the expression for the diffraction efficiency (measured along the direction of any one of the recording beams) of a quasi-steady-state (that is, slowly time-dependent) recorded grating may be assumed to be formulated as for the steady-state case in Equation (4.117) ZðdÞ ¼ 2

b2 cosh d=2  cos gd=2 1 þ b2 b2 ed=2 þ ed=2

with  and g as defined in Equations (6.50) and (6.51), with b2 ¼ IR0 =IS0 , IR0 and IS0 being the irradiances of the incident beams. The computed value of Z above is shown in Figure 6.15 as a function of 2kd for samples with different degree of oxidation: a reduced sample (leading to a grating with f ¼ p) and two others with increasing degree of oxidation, each one for different values of b2 . Here the coupling constant k in Equation (4.81) is pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi k  2 þ g2 =4 ð6:54Þ

as deduced from Equations (6.50) and (6.51) and is here used to represent the size of the index of refraction modulation in a sample of thickness d. Figure 6.15 shows that, for f ¼ p (which is the case of a reduced sample), Z is oscillating and reaches Z ¼ 1 only for b2 ¼ 1. This result can be straightforwardly obtained by substituting  ¼ 0 into the expression for Z above, which leads to ZðdÞ ¼

4b2 ð1 þ b2 Þ2

sin2 ðkdÞ

ð6:55Þ

150

STABILIZED HOLOGRAPHIC RECORDING

1 0.8

η

0.6 0.4 0.2 0 0

2

4

6

8

10

12

0

2

4

6

8

10

12

0

2

4

6 2κd

8

10

12

1 0.8

η

0.6 0.4 0.2 0

1 0.8

η

0.6 0.4 0.2 0

Figure 6.15. Computed Z as a function of 2kd for non stabilized recording in LiNbO3:Fe with different degree of oxidation: a reduced sample with f ¼ p (top), an oxidized sample with f ¼ 2:8 rad (middle) and a still more oxidized sample with f ¼ 2:5 rad (bottom). The figures were computed for b2 ¼ 1 (thick curve), 2 (thin curve) and 10 (dashed curve). Reproduced from [de Oliveira et al., 2004]

which is the general result [Kukhtarev et al., 1979a; Kogelnik, 1969] for a tilted (out of Bragg) uniform stationary grating where, for b2 6¼ 1, it is always Z < 1. For the special case of b2 ¼ 1 an in-Bragg grating results instead, and Z ¼ 1 may be achieved. A more general view is shown in Figures 6.16 and 6.17, where Z is plotted

SELF-STABILIZED RECORDING IN ACTUAL MATERIALS

30

151

2κ d

20

10 0 1 0.6 0.3 h 3

0

2 1 0

f

Figure 6.16. Computed Z as a function of 2kd and f, for b2 ¼ 1.

as a function of 2kd and f for b2 ¼ 1 and for b2 ¼ 10, respectively. These figures confirm that it is possible to reach Z ¼ 1 at f ¼ 0 or p, only for b2 ¼ 1, and also at some discrete values of f for b2 ¼ 10. From these figures one may induce that, for any b2, it is always possible to achieve Z ¼ 1 but for discrete values of f only. Reproduced from [de Oliveira et al., 2004] 2κd

30

20 10 0 1 0.6 0.3 3

h

0

2 1 0

f 2κd

30

20 10 0 1 0.6 0.3 3

h

0

2 1 0

f

Figure 6.17. Computed Z as a function of 2kd and f, for b2 ¼ 10. The plane Z ¼ 0:98 superimposed in the lower picture is a guide for the eyes only. Reproduced from [de Oliveira et al., 2004]

152

STABILIZED HOLOGRAPHIC RECORDING

6.3.2.2 Self-Stabilized Recording. In a self-stabilized regime, Z ¼ 1 may be achieved for any b2 and for any value of f. This is possible because in this case f is automatically adjusted to the required value, by the feedback operation, as will be shown below. Self-stabilized holographic recording arises from the action of a negativefeedback optoelectronic loop that forces the phase between the transmitted and the diffracted beams behind the sample to be fixed at a particular value. The latter value is, in general, different from the open loop (non-self-stabilized regime) value so that the hologram is erased to be written at a different position to comply with the feedback constraint. The result is the establishment of a continuously moving hologram the speed of which depends on the material and the feedback conditions. The phase modulating setup produces the harmonic terms already reported in Equations (4.166) and (4.167) pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiqffiffiffiffiffiffiffiffi IS ¼ 4J1 ðcd Þ Zð1  ZÞ IS0 IR0 sin j pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiqffiffiffiffiffiffiffiffi IS2 ¼ 4J2 ðcd Þ Zð1  ZÞ IS0 IR0 cos j

with the formulation for j in Equation (4.122) tan j ¼ 

g sin z 2   1  b2  g  cosh z  cos z þ sinh z 2 2 2 1 þ b2

Substituting the expression for Z in Equation (4.117) and for j in Equation (4.122) into the expressions above for the first and second harmonics in , and rearranging terms, we get the formulations already reported in Equations (4.158) and (4.159) IS ¼ 4J1 ðcd ÞIS ð0Þ

b2 sin gd=2 b2 ed=2 þ ed=2

IS2 ¼ 4J2 ðcd ÞIS ð0Þ

b2 b2 ed=2  ed=2 þ ð1  b2 Þ cos gd=2 1 þ b2 b2 ed=2 þ ed=2

The signal IS2 (instead of IS as was the case for sillenites) is selected out from the overall irradiance IS behind the crystal, amplified with a phase-selective 2-tuned lock-in amplifier, and used as error signal in the feedback as represented in Figure 6.14. For the particular case of a stationary grating (that is, the case of v ¼ 0, without feedback) in a reduced sample, it is oI  Klph NDþ =ND  0. In this case Equation (6.53) shows that Esc ðtÞ=m is a real quantity so that it is f ¼ 0; p and therefore   0. In this case IS2 is plotted in Figure 6.18 (with  ¼ 0) as a function of 2kd for b2 ¼1, 2, and 10 where we see that, only for b2 ¼ 1, it is always IS2 ¼ 0 and consequently, from the expression for IS2 above, we see that it is j ¼ p=2. This means that for a reduced crystal and b2 ¼ 1 one can use IS2 as error signal to operate

SELF-STABILIZED RECORDING IN ACTUAL MATERIALS

153

Is2Ω (au)

1.5

1

0.5

0 0

1

2 2κd (rad)

3

4

5

Figure 6.18. Computed IS2 (in arbitrary units), with  ¼ 0 (that is to say f ¼ 0, p) as a function of 2kd for b2 ¼ 1 (dashed curve), 2 (thin curve) and 10 (thick curve). Reproduced from [de Oliveira et al., 2004]

the active stabilization setup. Additionally, because in this case the open-loop and closed-loop values of j are the same (j ¼ p=2), a stationary (nonmoving) selfstabilized hologram results, but only for b2 ¼ 1. For oxidized crystals and/or for any crystal with b2 6¼ 1, it is j 6¼ p=2, in general, and in these cases it is I 2 6¼ 0. However, even in this case it is also possible to use IS2 as error signal to operate the self-stabilized setup. To understand this possibility we shall realize that the condition IS2 ¼ 0 in its expression above means that j ¼ p=2. Substituting the latter value into the expression for tan j above, we get the feedback-constrainted relation 1  b2 ðcosh d=2  cos gd=2Þ þ sinh d=2 ¼ 0 1 þ b2

ð6:56Þ

which substituted in turn into Equation (4.117) leads to ZðdÞ ¼

b2 ed  1 b2  1 b2 þ ed

for

b2 6¼ 1

ð6:57Þ

For b2 ¼ 1 instead, the j ¼ p=2 condition in the expression for tan j leads to d=2 ¼ 0 and the corresponding Z in Equation (6.55) becomes ZðdÞ ¼ sin2 gd=4

for

b2 ¼ 1

ð6:58Þ

From Equations (6.57) and (6.58) it is clear that under self-stabilized conditions it is always possible to achieve Z ¼ 1 when ed=2 ¼ b2 for b2 6¼ 1 and when gd=4 ¼ p=2 for b2 ¼ 1. To illustrate this important result we simulate the evolution of Z, f, and IS during self-stabilized recording for b2 ¼ 1:1  1 in Figure 6.19 and for b2 ¼ 10 in Figure 6.20. Both figures show a result that can be straightforwardly deduced from

154

STABILIZED HOLOGRAPHIC RECORDING

φ (rad) 3.2

1.0 0

φ

0.8

3.0 η

–0.2

0.6 η

2.8

Ω

0.4

S

2.6

2.4

–0.4 0.2 0

–0.6 0

1

2

3

2 κd

Figure 6.19. Computed evolution of f (o), IS (&) in arbitrary units and Z (r) as functions of 2kd for self-stabilized conditions (IS2 ¼ 0) and b2 ¼ 1:1. Note that f  p throughout. Reproduced from [de Oliveira et al., 2004]

Equation (4.166): As far as j is actively fixed to p=2, IS ¼ 0 means that Z ¼ 0 or 1 and its maximum (in absolute value) occurs for Z ¼ 0:5. Another important feature is shown in Figure 6.19: For b2  1 it is f  p (in self-stabilized regime), which is also the (open loop, that is, without feedback) equilibrium value for a stationary grating in a reduced sample. Because of this fact the recording pattern of fringes remains stationary (at a fixed position in space) during recording on a reduced φ (rad)

3.0

0.5

1.0 φ

0 –0.5

2.4

0.8 0.6

Ω

2.8

2.6

η

η

3.2

S

–1.0

0.4

–1.5

0.2

–2.0

0 0

1

2

3

4

2 κd

IS

Figure 6.20. Computed evolution of f ( ), (&) in arbitrary units, and Z (r) as functions of 2kd for self-stabilized conditions (IS2 ¼ 0) and b2 ¼ 10. Note that f rapidly shifts away from p during recording. Reproduced from [de Oliveira et al., 2004]

SELF-STABILIZED RECORDING IN ACTUAL MATERIALS

155

TABLE 6.1. LiNbO3:Fe Samples Sample

Thickness (mm)

[Fe ] / [Fe ]

[Fe] 10 =cm3

LNB5 LNB4 LNB3 LNB2 LNB1

0.85 0.35 1.39 0.96 1.5

0.03 0.013 0.013 0.0037 0.0021

2 20 2 2 2

2þ

3þ

19

[Hþ] 1018 =cm3 — 6.4 0.32 22 0.34

crystal for b2 ¼ 1. For b2 6¼ 1 instead, f is neither p nor even constant throughout the recording process, as shown in Figure 6.20. In this case, for b2 6¼ 1 and/or oxidized crystals, f is certainly not equal to its open-loop equilibrium value and because of such a phase mismatch, a running hologram is established as already reported. To illustrate the theoretical development above, some holograms were recorded in LiNbO3:Fe crystals with different degrees of oxidation as described in Table 6.1. All crystals were short-circuited with conductive silver glue and illuminated on all their volume. The recording was always carried out with K  10=mm, using the extraordinarily polarized 514.5-nm wavelength line of an argon laser, and the selfstabilization was operated as described in previous sections but now using the IS2 as error signal. The diffraction efficiency here is defined as Z ¼ I d =ðI d þ I t Þ where I d and I t are the diffracted and the transmitted irradiances, respectively, and are always measured using the in-Bragg recording beam as described in Appendix B. Such a definition allows one to get rid of bulk absorption and interface losses. It is also possible to compute Z from the IS signal during self-stabilized recording because, as mentioned above, in this condition IS (corrected from scattering) is maximum at Z ¼ 0:5 and is IS ¼ 0 at Z ¼ 1. Any (or both) of these conditions are used to adjust Equation (4.166) and from the latter any intermediate value for Z may be computed. Figure 6.21 reports the results for the less oxidized sample LNB5. In this case, Z  1 was directly measured, from the diffraction of the recording beam and from the evolution of IS , at the end of the recording cycle. The difference in behavior while using b2 > 1 or b2 < 1 arises from the nonsymmetric dependence of Equation (6.57) on the sense of energy transfer: In fact, Z ¼ 1 is only possible for b2 > 1 if d > 0 and for b2 < 1 if d < 0, where the former was the actual case in the experiment described here. Self-stabilized recordings on the most oxidized sample (LNB1), for b2  1 and for b2  12, are reported in Figures 6.22 and 6.23, respectively. The small secondary peak in Figure 6.23 is probably due to some oscillatory kinetics and was only observed in these conditions. In both cases Z ¼ 1 is achieved, in agreement with theory. The recording time needed to achieve Z ¼ 1 increases as b2 shifts away from 1 and also increases with the degree of oxidation as observed from Figures 6.21– 6.23. The movement of the recorded hologram (and corresponding recording pattern

156

STABILIZED HOLOGRAPHIC RECORDING

8 Ω

IS (au)

6

4 2 2Ω

IS

0 0

50

100

150

200

Time (s)

Figure 6.21. Self-stabilized recording in the less oxidized crystal (sample LNB5) with b2  1ðIR0 ¼ 141:1 W=m2 and IS0 ¼ 116 W=m2 ). The evolution of IS during the selfstabilized holographic recording experiment and the error signal I 2 are shown both in arbitrary units. At the end of the cycle Z ¼ 1 was measured. Reproduced from [de Oliveira et al., 2004]

of fringes), by the effect of self-stabilization, can be measured with the interference pattern IG produced by the reflection and transmission of some of the recording beams on a small glass plate fixed by the side of the crystal as schematically represented in Figure 6.24. The output beams through the crystal are used in the usual way for operating the self-stabilization loop. The time evolution of IG and the Kv computed from these data for a typical self-stabilized experiment are shown in Figure 6.25. The same experiment was carried out on a reduced sample 10

I

Ω

S

5

I 0

0

2Ω S

2000

4000

6000

Time (s)

Figure 6.22. Self-stabilized recording in an oxidized crystal (sample LNB1) with b2  1ðIR0 ¼ 113:5 W=m2 and IS0 ¼ 108:1 W=m2 ) showing the evolution of the IS (in arbitrary units). The Z ¼ 1 value by the time IS reaches zero was qualitatively verified. Reproduced from [de Oliveira et al., 2004]

SELF-STABILIZED RECORDING IN ACTUAL MATERIALS

157

4

Ω

IS

2

0 0

5000

10000

15000

20000

Time (s)

Figure 6.23. Self-stabilized recording in an oxidized crystal (sample LNB1) with b2 ¼ 12 ðIR0 ¼ 243:2 W=m2 and IS0 ¼ 20:3 W=m2 ) showing the evolution of the IS (in arbitrary units). The Z ¼ 1 value by the time IS reaches zero was qualitatively verified. Reproduced from [de Oliveira et al., 2004]

(labeled 758-1) in similar conditions with the speed being Kv < 0:03 rad/min, thus experimentally confirming our conclusions about the different behavior of oxidized and reduced samples. 6.3.2.2.1 Effect of Light Polarization. LiNbO3 is a naturally birefringent crystal with widely different ordinary and extraordinary index of refractions and electrooptic coefficients as already discussed in Section 1.3.2. These parameters are involved in the expression of Z as, for example, in Equation (6.58) Z ¼ sin2 gd=4

C

G

IG

Figure 6.24. Overall beam IG produced by the interference of the recording beams transmitted and reflected by a thin glassplate G adequately placed close to the photorefractive crystal C being studied.

0.6

0.6

0.5

0.5

0.4

0.4

0.3

0.3 0

1000

2000

3000

Kv (rad/min)

STABILIZED HOLOGRAPHIC RECORDING

IG (au)

158

4000

Time (s)

Figure 6.25. Measurement of the running hologram speed for the sample LNB1, b2  1, IS0 þ IR0  17 mW=cm2 and K ¼ 10=mm. The oscillating shape curve is the interference of the transmitted plus reflected beams in a glassplate fixed close to the sample. Its decreasing amplitude is due to scattering of light in the sample. The filled circles represent the computed pattern-of-fringes speed, corrected from scattering and the dashed curve is only a guide for the eyes. Reproduced from [de Oliveira et al., 2005]

where 1 gd=4 ¼ pn30 r13 jEsc j for ordinarily polarized light 2 1 gd=4 ¼ pn3e r33 jEsc j for extraordinarily polarized light 2

ð6:59Þ ð6:60Þ

2.5 EXTRAORDINARY 2.0

IΩ (au)

1.5

ORDINARY

1.0

0.5

0 0

500

1000

1500

Time (s)

Figure 6.26. Two self-stabilized recording experiments on the same LiNbO3:Fe sample (LNB3) with ordinarily and extraordinarily polarized l ¼ 514:5 nm light and b2  1, all other experimental conditions being similar. Reproduced from [de Oliveira et al., 2005]

SELF-STABILIZED RECORDING IN ACTUAL MATERIALS

159

We already know that self-stabilized recording is limited to Z ¼ 1, that is, to gd=4 ¼ p=2. On the other hand, it is ðn3e r33 Þ=ðn30 r13 Þ  3 in the visible spectral range. This means that self-stabilized recording with ordinarily polarized light allows achieving roughly a threefold higher space-charge modulation jEsc j than operating with extraordinarily polarized light. Figure 6.26 reports two experiments carried out with the same LiNbO3:Fe crystal (sample LNB3) in similar conditions except that one was with ordinary and the other with extraordinary recording beams where the latter is roughly fourfold faster than the former. More details about the use of light polarization to improve the index-of-refraction modulation recorded in LiNbO3 was published elsewhere [De Oliveira et al., 2005]. The larger time (4-fold) compared to the (3-fold) space-charge ratio is probably due to the exponential relation between both parameters during recording. It is interesting to point out that, while it was very easy to record holograms in this sample, it was almost impossible to record a hologram in sample LNB4 that has the same oxidation degree but 10-fold larger Fe concentration. Only a weak hologram could be recorded in LNB4 which was erased in a few minutes even in the dark. Some researchers have alredy reported [Nee et al., 2000] this particular behavior of highly Fe-doped crystals: they believe that the short distance between highly concentrated Fe photoactive center traps allows electrons to tunnel among these centers and in this way the electric charge distribution could not be produced or at least could not be kept in place for a sufficiently long time. 6.3.2.2.2 Glass Plate-Stabilized Recording. The stabilization can be also operated in non-self-stabilized mode by using the above-referenced small glass plate of Figure 6.24 in the setup schematically depicted in Figure 6.27: The transmitted R beam and the phase-modulated S beam reflected from this glass plate, both propagating along R, mutually interfere, producing harmonic terms in . IΩ S M

R

SE

4 51

LA-Ω

D1

IS

nm

I 0R

LA

LA-2Ω C

BS

G

I 0S

IG D2

Ω OSC

I 2Ω S

LA-Ω

PZT

+

IΩ S HV

INT

Figure 6.27. Recording setup stabilized on a nearby placed glassplate G, all other elements being the same as described in Fig. 6.14. Reproduced from [de Oliveira et al., 2004]

160

STABILIZED HOLOGRAPHIC RECORDING

1.2

Ω

IS

(a.u)

0.9

0.6

0.3 Ω

IG

η=85%

0 0

20

40 60 Time (min)

80

100

Figure 6.28. Glass plate-stabilized experimental data for the recording on an oxidized sample (LNB1) with b2  1 and IS in arbitrary units. The error signal IG through the glassplate is also shown. At the point where IS ¼ 0 it was measured Z ¼ 0:85. Reproduced from [de Oliveira et al., 2004]

In this case either the first IG or the second harmonic IG2 terms in IG can be used as error signal in a feedback opto-electronic loop to keep the pattern of fringes stabilized in relation to the glass plate. At the same time we may use the harmonic terms independently measured through the sample (e.g., IS ) to obtain information about the evolution of the holographic recording. We shall call this one a freestabilized (or stabilized-non-self-stabilized) recording because the stabilization is not referred to the hologram itself: The recording pattern of fringes is fixed in space because of the glass plate-operated feedback, but the recording occurs without constraints because of the absence of self-stabilization. Therefore the recording pattern of fringes is stabilized in space, but the recording process itself is not affected in any way. Such a recording was carried out on the oxidized sample LNB1 for b2  1, and the result is reported in Figure 6.28. Differently from self-stabilization, the IS ¼ 0 condition here does not necessarily mean that Z ¼ 0 or Z ¼ 1, because in this case j is not actively fixed so that it is free to vary: It might be IS ¼ 0 just because of sin j ¼ 0. In fact, it was measured as Z ¼ 0:85 at I  ¼ 0 in Figure 6.28 and, although not shown in this figure, Z ¼ 1 was never achieved in this experiment. Such a result for oxidized crystals is in agreement with information from Figures 6.16 and 6.17, where it is obvious that it is not possible to get Z ¼ 1 for b2 ¼ 1 unless f ¼ 0 or p, which is not the case for oxidized samples. To further explain these facts, a mathematical simulation is shown in Figure 6.29, where we plot the expressions for Z in Equation (4.117), for j from Equation (4.122), and for IS in Equation (4.166), using tan f ¼ 2:8, which corresponds to one of the examples in Figure 6.15. This simulation apparently qualitatively explains the main features in Figure 6.28. For the same sample LNB1 and the same experimental conditions but for b2 ¼ 12 the glass plate-stabilized

161

SELF-STABILIZED RECORDING IN ACTUAL MATERIALS

1.25 1 0.75 0.5 0.25 0 –0.25 0

2

4

2κd

6

8

10

Figure 6.29. Mathematical simulation of non self-stabilized recording with b2 ¼ 1. The thick curve is Z, the thin curve is IS and the dashed is j, for tan f ¼ 2:8 that seems to qualitatively fit data for LNB1 in Fig. 6.28. Reproduced from [de Oliveira et al., 2004]

experiment didn’t work: The recording was so noisy that stabilization (and recording) in these conditions was impossible. Self-stabilized recording in highly diffractive materials exhibiting phase coupling (hologram bending) as is the case of photovoltaic LiNbO3 crystals has additional advantages besides the main one (reducing environmental perturbations) as already reported elsewhere [Garcia et al., 1996]:

1.0

0.20

1.0

0.20

A

B 0.8

0.8 0.15

0.15 η 0.6

0.6 η

0.10

0.10 0.4

0.4 0.05

0.05 0.2

PSL

0

0 0

10

20

Time (min)

30

0.2

PSL

0

0 0

10

20

30

Time (min)

Figure 6.30. Evolution of Z and scattering during stabilized holographic recording with (figure B) and without (figure A) self-stabilization in LiNbO3:Fe using l ¼ 514:5 nm with IR0 =IS0  16 and IR0 þ IS0  4 mW/cm2. The diffraction efficiency Z is not considering bulk light absorption and PSL is the scattered light (in %). Reproduced from [Garcia et al., 1996]

162

STABILIZED HOLOGRAPHIC RECORDING

Reduces hologram bending: In fact, the use of I 2 as error signal leads to j ¼ p=2 as reported in Equation (4.167), and in this case it is always possible to achieve Z ¼ 1 as reported in Section 6.3.2.2. However, if the hologram is bended (out of Bragg) it is not possible to achieve Z ¼ 1, as is the case in Figure 6.28 and also in the simulation of graphics A in Figure 6.29. The latter graphics shows the evolution of Z during holographic recording where the setup is stabilized on a glass plate by the side of the sample: This recording is therefore stabilized but non-self-stabilized, and Z starts growing but after some time it starts decreasing because it progressively gets out of Bragg condition. The graphics B instead shows a self-stabilized experiment where Z steadily grows up to the limit (because of self-stabilization constraints) of Z ¼ 1. We should conclude that, by keeping j ¼ p=2, self-stabilization somehow keeps the hologram in Bragg during recording.

Reduces scattering: Light scattering arises from the diffraction of randomic gratings produced by the interference of light scattered in defects inside the crystal. The movement of the recording pattern of light produced by selfstabilization maximizes the recording of this pattern of light and sensibly reduces the recording of other holograms that are not related to this one, as is the case of those producing scattering. This effect is clearly verified comparing graphics A and B in Figure 6.30.

PART III

MATERIALS CHARACTERIZATION

164

MATERIALS CHARACTERIZATION

INTRODUCTION Properties of practical interest like sensitivity, diffraction efficiency, energy transfer (amplitude coupling), phase coupling, etc. depend on material parameters like the diffusion length LD , the Debye screening length ls , the quantum efficiency  for photoelectron excitation for the recording wavelength, and other parameters like light absorption coefficient, optical activity (if any), etc. It therefore becomes a matter of paramount importance to get information about these parameters. The research on the presence and the nature of the localized (photoactive) levels in the band gap of photorefractive materials is an important subject, too, because these photoactive levels are the ones where the spatial modulation of charges occurs under the action of light. Such a modulation is the starting point of optical recording in photorefractives. The adequate characterization of these localized states in the band gap will certainly allow one to better understand the optical recording process in a particular material. The characterization of materials, including research on their photoactive centers, is the objective of this third part of the book. We have chosen optical methods, mainly holographic ones, for materials characterization because photorefractives are photosensitive materials and, on the one hand, their interaction with the light is at the basis of the processes we want to study and, on the other hand, holography underlies most of their practical applications. Nonholographic optical methods, with emphasis on photoconductivity, are dealt with in Chapter 7. Holographic techniques are the subject of Chapters 8 and 9, where particular attention is devoted to phase modulation and self-stabilized recording techniques, which are rarely described in a comprehensive way in the scientific literature. Although these techniques are of general interest, their application will be limited here to a few paradigmatic materials—those about which we can provide reliable first-hand experimental results. Self-stabilization is separately described in Chapter 9 because of the extent and the complex nature of this subject. The objective of this part of the book is to give an overview of the possibilities of optics in general, and holography in particular, for materials characterization. We should point out that most of these techniques are not restricted to photorefractives but may be applied to other photosensitive materials as well. The reader should bear in mind that the actual values here reported for some material parameters should be handled with caution because they may vary from one sample to the other as they depend on the fabrication technique and raw chemicals used to produce a particular sample.

CHAPTER 7

NONHOLOGRAPHIC OPTICAL METHODS

We shall discuss a few methods involving optics, to some extent, with the exclusion of holography, which will be dealt with in the next chapters.

7.1

LIGHT-INDUCED ABSORPTION

This subject was alredy theoretically developed in Chapter 2, where Equation (2.46) describes the light-induced absorption coefficient ali , the limits of which are lim ali ¼ 0 I!0

lim ali ¼ ND2 s2

I!1

þ ðND1  ND1 Þt1 r2 s1 þ ðND1  ND1 Þt1 r2 s1 þ s2

as described in Equations (2.44) and (2.45), respectively. In this case the transmitted irradiance I t is computed from the incident I0 as follows dI ¼ ða0 þ ali ÞI dz

a0 þ ali ¼

ða0 b þ aÞI þ a0 c bI þ c

which was solved in Section 2.3.1, leading to Equation (2.47), which can be

Photorefractive Materials: Fundamental Concepts, Holographic Recording and Materials Characterization, By Jaime Frejlich Copyright # 2007 John Wiley & Sons, Inc.

165

166

NONHOLOGRAPHIC OPTICAL METHODS

rearranged as a=b ða0 þ a=bÞI0 þ a0 c=b It ln þ ln ¼ a0 d I0 a0 þ a=b ða0 þ a=bÞI t þ a0 c=b

ð7:1Þ

and can be simplified for the limit conditions I t ¼ I0 ea0 d I t ¼ I0 e

for

ða0 þabÞd

I0 ) 0 for

ð7:2Þ

I0 ) 1

ð7:3Þ

The measurement of bulk and light-induced absorption is carried out in a simple way: A light beam with uniform cross section irradiance is projected on the sample, and the transmitted irradiance I t is measured as a function of the incident I0 . Both I0 and I t values must be corrected for their respective values inside the sample as follows Ið0Þ ¼ I0 ð1  RÞ

cos y cos y0

and

IðdÞ ¼

I t cos y 1  R cos y0

ð7:4Þ

where R is the interface reflectance, y and y0 are the incidence angles outside and inside the sample (which are zero in our case), respectively. The whole light at the crystal output is collected with a lens and focused on a photodetector behind the crystal in order to be free from the possible lenslike effect produced by thick, high index of refraction (n  2:6) samples. Figure 7.1 shows the evolution of light-induced absorption (photochromic darkening) on a typical undoped Bi12 TiO20 crystal at a fixed incident irradiance. Steady-state results are shown in Figure 7.2 on the same sample, where the nonlinear 760

α (m–1)

720 680 640 600 560 0

25

50 75 Time (min)

100

125

Figure 7.1. Evolution of the absorption coefficient in an undoped B12 TiO20 crystal (labelled BTO-010) under uniform illumination of I0  2 mW=cm2 at l ¼ 532 nm.

LIGHT-INDUCED ABSORPTION

167

It (W/m2)

0.12

0.08

0.04

0 0

30

60 I0 (W/m2)

90

120

Figure 7.2. Light-Induced absorption: Transmitted I t versus incident I 0 irradiances measured using an uniform beam of 532 nm wavelength on the same sample BTO-010 as in Fig. 7.1. The dashed lines are the best fit at the limit I ! 0 (with an angular coefficient of 0.00299) and for saturation with an angular coefficient of 8:78  104 . Reproduced from [Mosquera et al., 2001]

relation between the incident and transmitted irradiances is evident. Data fit to Equation (7.1), with R ¼ 0:2 and y  0, give a0 ¼ 662 m1

a ¼ 198 m1 b

c ¼ 0:75 W=m2 b

ð7:5Þ

The above results can be formulated in terms of the two-center model parameters in Equations (2.39)–(2.43) as: þ Þs1 a0  ðND1  ND1

a ¼ b c=b 

ð7:6Þ

þ ðND1  ND1 Þt1 r2 s1 ND2 s2 þ ðND1  ND1 Þt1 r2 s1 þ s2

ð7:7Þ

b2  ho þ t1 r2 ðND1  ND1 Þs1

ð7:8Þ

Other samples were measured, and the results are displayed in Table 7.1. Data reported in this table show that all doped and undoped BTO crystals exhibit a large light-induced absorption effect that is characterized (at saturation) by the a to b ratio. This effect is a rather slow one compared to the recording of a photorefractive grating in these materials. For an irradiance much lower than c=b there is a TABLE 7.1. Absorption Parameters for Pure and Doped BTO for k ¼ 532 nm Sample 1

a0 (m Þ a=bðm1 Þ c=b ðW=m2 Þ

BTO-010

BTO-011

BTO-013

BTO:Ce

BTO:Pb

662 198 0.75

658 242 0.7

583 226 0.34

430 138 0.7

473 250 0.29

168

Absorption coefficient α (m–1)

NONHOLOGRAPHIC OPTICAL METHODS

1100

1100

1000

1000

900

900

800

800

700

0.1

1 I0 (W/m2)

10

100

700

0

25

50

75

100

I0 (W/m2)

Figure 7.3. Light-induced absorption of undoped Bi12 TiO20 (sample labelled BTO-013) at l ¼ 514:5 nm as a function of the incident irradiance measured in the air. The left-hand side graphics is in semi-log scale for detailed view at low irradiances The continuous curve on the right-hand side graphics is the best fitting of Eq. (2.46) with the following parameters: a0 ¼ 789 m1 , a ¼ 1:4  106 m/(s2 W), b ¼ 4:91  109 m2 /(W s2 ) and c ¼ 7:48  109 s 2 .

negligible light-induced absorption effect, whereas for a much larger irradiance the absorption becomes nearly saturated as shown in Figure 7.3. This limit irradiance is comparatively weak (a few hundred mW/cm2) for all doped and undoped BTO, a fact that allows one to assume a nearly saturated light-induced absorption condition, even for the moderately large irradiances usually employed, provided that the experiment is allowed to last for a sufficiently long time to reach equilibrium. The measurement of absorption may be complicated by the presence of luminescence effects. Luminescence occurs when electrons are excited to the CB and undergo a radiative decay to intermediate states in the band gap, emitting correspondingly associated photons. The latter are less absorbed than the ones used to excite the electrons, and they are detected at the sample output. This luminescence-arising radiation may be very relevant, especially when weak direct transmitted light is observed, as is the case near the band edge where the direct absorption is strong. This fact is illustrated in Figures 7.4 and 7.5. Figure 7.4 shows the spectrum of a quasi-monochromatic LED, centered at l ¼ 408 nm (left-side peak), illuminating a 2.8-mm-thick undoped BTO crystal; the spectrum of the light behind the sample is represented by the wider peak at the right-side and centered at about 570 nm. The input light centered at 408 nm is not detected at all at the output because this radiation is very close to the band gap limit so that it is completly absorbed. The 570-nm-centered output light probably arises from luminescence. Figure 7.5 shows the absorption-thickness product ad measured for two samples (labeled BTO-Q and BTO-8) in a spectrophotometer where the detector is placed

LIGHT-INDUCED ABSORPTION

169

8000

Irradiance (au)

6000

4000

2000

0

–2000 300

500

700 λ (nm)

900

1100

Figure 7.4. Photoluminescence in BTO-008. The dashed is the spectrum of the light of a LED illuminating the BTO crystal sample. The continuous curve is the spectrum of the light measured at the crystal output, very close to it. A luminescent peak at 570 nm (2.2 eV) appears.

5 BTO-008 BTO Q BTO-8

4

αd

3

2

1

0

400

600

800

λ (nm)

Figure 7.5. Absorption coefficient-thickness ad measured for three different BTO samples (BTO-8, BTO-Q and BTO-008) as a function of wavelength. BTO-8 and BTO-Q were measured in a standard spectrophotometer whereas BTO-008 was measured with a photodetector placed about 1 cm behind the crystal.

170

NONHOLOGRAPHIC OPTICAL METHODS

comparatively far away from the sample at the output. The curve for BTO-008 instead was measured with a detector at about 10 mm behind the sample. Note that the latter curve shows an apparent paradoxical decrease in ad for wavelengths with photonic energies larger than that of the band gap (at about 400 nm), which is due to the fact that there is an increasing contribution from luminescence effects. In fact, luminescence is a scattering process so its irradiance decays as the inverse distance square, so that it is much easier to be detected the closer the detector is to the sample.

7.2

PHOTOCONDUCTIVITY

For a sample with electrodes in the so-called transverse configuration as shown in Figure 2.14, the actual photocurrent at z can be written in differential form as diph ðzÞ ¼ sph ðzÞEhdz

ð7:9Þ

where E is the applied electric field and h is shown in Figure 2.14. The expression of sph ðzÞ is reported in Equation (2.36) to be sðzÞ ¼ qmtða0 þ ali ðzÞÞ

Ið0Þ ða0 þali ðzÞÞz e ho 

ð7:10Þ

where a was substituted by a0 þ ali ðzÞ and a by a0 þ ali ðzÞ. Ið0Þ is the incident irradiance at the input surface inside the crystal. The overall photocurrent is computed by substituting Equation (7.10) into Equation (7.9) and integrating throughout the crystal’s thickness d: iph

Ið0Þ ¼ Ehqmt ho 

Zz¼d

½a0 þ ali ðzÞeða0 þali ðzÞÞz dz

ð7:11Þ

z¼0

Substituting the actual expression for ali in Equation (2.46) into Equation (7.11), the integration can be carried out either analytically or numerically. Otherwise, assuming the light-induced absorption is constant throughout the crystal’s thickness, the integral is easily computed as iph ¼ Ehqmt

 a0 þ ali Ið0Þ 1  eða0 þali Þd a0 þ ali  ho

ð7:12Þ

To get a specific intrinsic parameter for the material we shall define a photoconductivity coefficient using the expression in Equation (2.37) but considering the irradiance in terms of flux of photons as follows: s ¼ sph ð0Þ

o h ¼ qmtða0 þ ali Þ Ið0Þ

ð7:13Þ

PHOTOCONDUCTIVITY

171

Comparing Equations (7.13) and (7.12), we conclude that it is also possible to write s ¼ sph ð0Þ

iph  ho ho a0 þ ali ¼ Ið0Þ hV=‘ Ið0Þ 1  eða0 þali Þd

ð7:14Þ

in terms of experimentally measured quantities where V is the applied voltage and l the interelectrode distance. It is possible to verify that for ða0 þ ali Þd 1 the expression above is independent of optical absorption, which is an interesting result because in these conditions the absorption coefficient is difficult to measure and is associated with a relatively large uncertainty.  represents the efficiency for a photon to excite an electron to the CB, whereas a is the fraction in the absorption that is effectively related to photoelectron excitation. Both terms are adequate for describing the distribution of photoactive centers in the band gap. 7.2.1

Alternating Current Technique

We shall focus here on an ac method that facilitates the detection of small photocurrent signals in a much larger nonphotoconductive current. The method is based on the use of a time-modulated spatially uniform illumination and the detection of the associated current with a phase-sensitive frequency-tuned lock-in amplifier. If we illuminate a photoconductive (not necessarily a photorefractive) material with a spatially uniform sinusoidally (amplitude) modulated light of angular frequency  and contrast jmj I ¼ I0 ð1 þ jmj cosðtÞÞ

ð7:15Þ

Equation (2.5) turns into the differential equation @N ðx; tÞ a þ N =t ¼ I0 ð1 þ jmj cos tÞ @t hn

ð7:16Þ

where the dark conductivity is neglected. Its solution is N ¼ N 1 cosðt þ fs Þ þ N 0 N0 

a I0 hn

ð7:17Þ ð7:18Þ

1 N 1  N 0 jmj pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ  2 t2

ð7:19Þ

tan fs  t

ð7:20Þ

Therefore a time-modulated photoconductivity of the form jmj s ¼ sph ð1 þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi cosðt þ fs ÞÞ 1 þ 2 t2

tan fs ¼ t

ð7:21Þ

172

NONHOLOGRAPHIC OPTICAL METHODS

+ HV L SF LASER BTO CH C R OA

LA

Figure 7.6. Experimental setup: Schematic setup for the electric measurement of dark and photoconductivity. The laser beam is chopped CH at frequency  and the beam is filtered and expanded using a spatial filter SF and collimated using a lens L. The chopped expanded and uniform beam shines the sample that produces a photocurrent under the action of a voltage HV. An operational amplifier OA with a feedback resistance R and capacitor C transforms the current into a voltage that is read using a -tuned lock-in amplifier LA (for the case of photoconductivity) or a simple dc voltmeter (for the case of dark conductivity). Reproduced from [Mosquera et al., 2001]

results [Gerwens et al., 1997], with t being the photoelectron effective lifetime as defined in Equation (2.12) provided that we can neglect the response time of the measuring circuit itself. In fact, the response time of the measurement circuit can be neglected because the sample’s resistance (usually very high in most photorefractive materials even under illumination) is not related with the photocurrent generation as deduced from Equation (7.16). Also, the input resistance of the operational amplifier OP-AMP used to convert the photocurrent into a voltage in Figure 7.6 is always very low, and the associated response time is accordingly very low, too. The output OPAMP resistance instead is usually very large, but there are instrumental features able to strongly reduce this output impedance and thus reduce the associated response time. From the development above it seems that ac-photocurrent measurements are likely to be limited by charge carrier lifetime in the extended states rather than by the response time of the instrument itself. It is easy to show that, for a rectangular time-modulated (with fundamental angular frequency ) spatially uniform illumination, the dc plus fundamental term of the irradiance has the form 2 I ¼ I0 þ I0 cosðtÞ p

ð7:22Þ

PHOTOCONDUCTIVITY

173

3000 43

pA

30

2000

20

iph (pA)

10 0

0

0.2

W/m2

0.4

0.6

1000

0

0

5

10 I(0) (W/m2)

15

20

Figure 7.7. Photoconductivity: Photocurrent (in picoamperes) as a function of the incident irradiance on the input plane inside the crystal Ið0Þ, measured using a time-modulated uniform 532 nm wavelength laser beam onto the BTO-010 crystal with 2000 V applied to the sample. The corresponding time modulated photocurrent is measured using a synchronously frequency-tuned lock-in amplifier where are data for a sample that has been kept in the dark for a long time and are data for the previously light-saturated crystal. The dashed line for the

is the best fit for the final linear range that gives and angular coefficient of 138 pA.m2 /W. The dashed line for the in the inset represents the best fit for the nonexposed sample for the limit Ið0Þ ! 0 condition giving an angular coefficient of 69.2 pA.m2 /W. From these data the values in Table 7.2 were computed for BTO-010. Reproduced from [Mosquera et al., 2001]

If this chopped light is shining on a photorefractive (or simply photoconductive) crystal, a time-modulated photocurrent also results where its fundamental harmonic term has the form 2=p iph ðtÞ ¼ iph pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi cosðt þ fs Þ 1 þ  2 t2

tan fs ¼ t

ð7:23Þ

that can be measured pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiwith a lock-in amplifier tuned to  as depicted in Figure 7.6. The term 1= 1 þ 2 t2 is experimentally determined for the frequency  used in the experiment. Figure 7.7 shows a typically measured photocurrent iph vs. Ið0Þ for the BTO sample labeled BTO-010. The open circles represent the photocurrent measured for the nonexposed sample, whereas the filled circles show the data for the sample having just been previously exposed to saturation. The overall measured photocurrent iph corresponds to I0 (or more precisely to Ið0Þ, which is its value inside the sample) and is related to the material’s parameters by Equation (7.12). From these experimental data the photoconductivity coefficient in Equations (7.13) or (2.37) can be computed. Results are displayed in Table 7.2. For comparison the photoconductivity was also measured at 514.5 nm (BTO-011) and at 633 nm (BTO-010), showing a large dependence on the wavelength, probably due to

174

NONHOLOGRAPHIC OPTICAL METHODS

TABLE 7.2. Photoconductivity and Derived Parameters for BTO at 532 nm Sample

BTO-010

sph ð0Þ lim Ið0Þ ð1012 m=ðWÞÞ I!0 s ð0Þ lim ph ð1012 m=ðWÞÞ I!1 Ið0Þ 2

mt (m =V  10  a b

12

)

b

47.5 (1:8 )

BTO-011

BTO-013

52.2

65.5

0.56

53.5

317.7

8.72

230.7

3a

b

122.6 (7:2 )

127.4 (10 ) b

0.88 (0:47 ) 0.19 (0:16b )

0.72 0.25

2.58 0.10

BTO:Ce BTO:Pb

0.14 0.02

1.66 0.16

l ¼ 514:5 nm [de Oliveira and Frejilich, 2000] l ¼ 633 nm

the wavelength dependence of the characteristic effective cross sections s1 and s2 . Some results for doped crystals are also shown in Table 7.2. 7.2.1.1 Wavelength-Resolved Photoconductivity. Figure 7.8 shows a typical setup for wavelength-resolved photoconductivity measurement, although the data shown in this section were obtained in a different setup using discrete quasi-monochromatic sources of light [Montenegro et al., 2006]. The dependence of the photoconductivity coefficient, defined in Equation (7.13), on the light wavelength may provide important information about the photoactive center levels in the band gap. Figures 7.9–7.11 show such data for an undoped BTO crystal, as a function of  ho ¼ hn for the irradiance light, for two different conditions: for the sample (partially) relaxed in the dark for 2 h at 80–90 C and for the same sample after being preexposed to 2.3-eV irradiance. The preexposure was carried out

Figure 7.8. Typical wavelength-resolved photoconductivity experimental setup: White light source and housing (1), commercial monochromator adapted for stepping-motor driving (2), mechanical chopper (3) and home-made housing containing the sample with shielded electrodes with connection cables and photodetectors (4).

PHOTOCONDUCTIVITY

175

10

σ (10–28 sm/Ω )

0.1

0.001

0.00001 1.0

1.5

2.0

2.5

3.0

3.5

hν (eV)

Figure 7.9. Photoconductive coefficient sð0Þhn=Ið0Þ (in sm/ logarithmic scale) as a function of hn (in eV) of the light onto an undoped Bi12 TiO20 sample (labelled BTO-008) for: pre-exposed for 2 min to 2.3 eV ( ) and thermally relaxed for 2 hours at 80–90 C ( ).

immediately before each one of the individual measurement for each one of the different wavelengths. The measurement was always carried out from the highest to the lowest wavelength. Electrons can be excited, by the action of a sufficiently energetic photonic light, from the VB to a localized acceptor state in the band gap, thus producing an equivalent number of holes free to move in the VB. Also, electrons can be excited from a localized donor state in the band gap to the CB, where they are free to move. In both cases the (photo)conductivity increases because of the increase of free carriers in the extended (VB and/or CB) states. Each time the photonic energy is

σ (10–28 sm/Ω )

0.3

0.2

0.1

0 1.2

1.6

2.0

2.4

hν (eV)

Figure 7.10. Photoconductive coefficient sð0Þhn=Ið0Þ (in sm/) as a function of hn (in eV): Detailed view of Fig. 7.9.

176

NONHOLOGRAPHIC OPTICAL METHODS

Figure 7.11. Photoconductive coefficient sð0Þhn=Ið0Þ as a function of hn (in eV): Detailed view of Fig. 7.10.

large enough to produce such an electronic transition, a corresponding increase in the measured photocurrent should be detected. The wavelength-resolved photocurrent should therefore have a steplike shape, as far as it is adequtely resolved. This technique is therefore a potential powerful tool for the study of localized states in the band gap. It is interesting to realize that the preexposed sample curve in Figures 7.9–7.11 exhibits clear slope variations in the range 1.2 to 1.9 eV and then a large one by 1.9–2.0 eV and a much larger at about 2.2 eV. The (partially) relaxed sample shows similar features although with much smaller steps, except for the very steep increase at about 2.2 eV. In both cases a strong increase in photoconductivity appears by 2.5 eV, and from there on the photocurrent is steadily increasing without showing resolved steps. We may therefore conclude that this material has a strongly populated electron donor level at 2.2 eV that is probably determining its Fermi level. Another one is also present at 2.5 eV. There are also empty donors, closer to the CB, that are weakly detected unless they are previously (at least partially) filled by preexposure. In the present case some of such centers are probably the ones appearing in the preexposed sample at 1.4–1.5 eV, 1.7–1.8 eV, and 1.9–2.0 eV probably below the CB. 7.2.2

Modulated Photoconductivity

This method [Oheda, 1981; Longeaud and Kleider, 1992; Kleider and Longeaud, 1995] consists of illuminating the sample with a time-modulated (frequency o) spatially uniform strong light I ¼ Idc þ Iac cos ot

ð7:24Þ

with fixed photonic energy slightly higher than that of the band gap in order to produce band-to-band charge carrier excitation. The dc irradiance fixes the

PHOTOCONDUCTIVITY

177

Nc/µ (cm–2VeV–1)

1011

0.29 ev 1010

109 130 K ≤ T ≤ 260 K CNbe = 2.5 × 1011 s–1 108 0.15

0.20

0.25

0.30

0.35

0.40

0.45

0.50

| Ebe – Eω | (eV)

Figure 7.12. Modulated photocurrent data of an undoped Bi12 TiO20 crystal, where different colours correspond to different temperatures. From Dr. Christophe Longeaud, Supelec-Paris, France.

recombination process, whereas the ac irradiance determines the trapping and release process, thus giving the quantity NðEÞC 2 sin f AqxG ¼ m pkB T Iac

ð7:25Þ

with N being the equivalent density of states at energy E, C the capture coefficient of these states, A the the cross-sectional area for photocurrent flowing, x the applied electric field, G the charge carrier generation rate, and f the phase shift between the excitation light and resulting photocurrent. A relation between E and the excitation frequency o exists: jEbe  Ej ¼ kB T logðn=oÞ

ð7:26Þ

n ¼ C Nbe

ð7:27Þ

where Nbe is the equivalent density of states at the limit (bottom of CB or top of VB) of the corresponding extended state where the energy is Ebe and n is called the ‘‘attempt-to-escape’’ frequency. The latter, together with the energy (E) of the state in the band gap, allows computation of the relaxation time trlx of charge carriers in this state by 1=trlx  en ¼ neE=kB T

ð7:28Þ

The peaks should superpose at all temperatures, as illustrated in Figure 7.12, so that the experiment is repeated at different T. To have the peaks superposed at different

178

NONHOLOGRAPHIC OPTICAL METHODS

T one must adjust the value of n [Longeaud et al., 1999]. The value of the difference Ec  E, however, does not mean that the particular state is necessarily below the CB: It may be above the VB as well.

7.2.2.1 Quantum Efficiency and Mobility-Lifetime Product. The lifetime t as defined in Equation (2.12) is independent of the light only for sufficiently low irradiances in order not to sensibly change the density of free acceptors NDþ . This is the case for usual applications to holography or image processing experiments where powerful pulsed lasers are not necessary. In photochromic materials instead, the photoelectron lifetime may be sensibly affected, even by rather weak irradiances, as easily deduced from the expression of the effective lifetime þ 1=t ¼ 1=t1 þ r2 ND2

ð7:29Þ

in Equation (2.28). Photoconductivity and light-induced absorption allow one to compute some relevant material’s parameters. Note that t in Equation (2.37) is light dependent, as deduced from Equation (7.29). If we assume that r2 ND2 t1 1 in Equation (7.29) (that is, we neglect the influence of the shallow traps in the photoelectron lifetime), the t in Equation (2.37) may be assumed to be a constant, in which case we may compute the mobility-lifetime term as follows hn q

sph ð0Þ sph ð0Þ  lim lim I!1 Ið0Þ I!0 Ið0Þ





lim ali ¼ mt

I!1

ð7:30Þ

From Equation (2.37) and from the knowledge of mt obtained in Equation (7.30),  can be also computed as 

sph ð0Þ I!0 Ið0Þ

lim

 qmta0  ¼ hn

ð7:31Þ

The term mt computed from Equation (7.30) and the  computed from Equation 12 2 m ffi=V value for BTO-011 (7.31) are displayed in Table 7.2. The mt ¼ 0:72  p10 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi leads to a charge carrier diffusion length LD ¼ mtkB T=q ¼ 0:14 mm (kB is the Boltzman constant, T is the absolute temperature, and q is the charge of the electron) that is in excellent agreement with the value LD ¼ 0:14  0:01 mm reported elsewhere [Frejlich et al., 2000] for this sample. The definition of  in Equation (7.6) is somewhat different from the definition of 0 0 ða0 þ ali Þ ¼ a0 þ ali

ð7:32Þ

that is implicitely used in most papers about photorefractives. The present value  ¼ 0:25 for BTO-011 in Table 7.2 can be converted into that used in [Frejlich et al.,

ELECTRO-OPTIC COEFFICIENT

179

TABLE 7.3. Typical Parameters of Some Pure and Doped Bi12 TiO20 (BTO) Samples a0 þ a=b (m1 ) at l (nm)

r (deg/mm) at l (nm) Sample BTO Bi12 Ti0:9 Ga0:1 O20 Bi12 Ti0:7 Ga0:3 O20 Bi12 GaO20 BTO:Ce BTO:Pb BTO:V

r41 (pm/v) 5.7–5.2 5.5 5.6 4:8  0:1 4.7 4.1–4.2 5:8  0:2

E low freq. 47

633 6:7  0:3 7:5  0:3 9:7  0:2 18  0:2 5.9 5.5 4.5

532

514.5

633

532

514.5

— — — — — — —

12 — — — — 11.5 —

40–90 — — — — — 135

850 — — — 570 720 —

1160 —

— — —

2000], for large irradiances, by 0 ¼ ða0 þ a=bÞ=ða0 þ a=bÞ ¼ 0:45, which is only 20% higher than the value (0.36) already reported [Frejlich et al., 2000] for the same sample at the same wavelength.

7.3

ELECTRO-OPTIC COEFFICIENT

The electro-optic coefficient is one of the most important parameters of photorefractive materials. Although an effective value of the electro-optic coefficient can be obtained from the measurement of diffraction efficiency of the recorded holograms, direct optical nonholographic methods are very simple to carry out. They are based on the measurement of the change in the ellipticity of a linearly polarized light going through a slab of the material under analysis with an applied transverse electric field on it. Materials exhibiting optical activity (like sillenites) are somewhat more difficult to measure because the effect of birefringence and optical activity both act on the ellipticity of the light and the effect of birefringence should be separately evaluated. Special techniques for such evaluation were published by Henry et al. [Henry et al., 1986] and by Bayvel et al. [Bayvel et al., 1988], which we will not describe in this book. The interested reader should address these or other publications on this subject. Typical results measured for pure and doped BTO crystals are shown in Table 7.3.

CHAPTER 8

HOLOGRAPHIC TECHNIQUES

Most applications of photorefractive crystals involve holographic recording—phase conjugation, associative memories, light amplification, novelty filters, spatial filtering, image processing, holographic interferometry, etc. It therefore becomes obvious that these holographic techniques are themselves most suitable for materials characterization as far as recording is concerned. Some of these techniques are well known and have been used extensively in the past: hologram recording and hologram erasure time constants, diffraction efficiency and amplitude gain in twowave mixing, etc. These and similar techniques might be considered as direct holographic techniques and are analyzed in Section 8.1. Other methods requiring more sophisticated detection techniques are considered in Section 8.2, ‘‘Phase Modulation Techniques.’’ We shall assume throughout that we are dealing with ‘‘thick’’ holograms in order to verify the Bragg selectivity condition and therefore have to deal with one single diffraction order only.

8.1

DIRECT HOLOGRAPHIC TECHNIQUES

The measurement of experimental quantities like diffraction efficiency, amplitude gain, holographic sensitivity and the time constant for recording and for erasure, among many other possibilities, are important because they will determine the applications of a certain material. These quantities will, at the same time, allow one

Photorefractive Materials: Fundamental Concepts, Holographic Recording and Materials Characterization, By Jaime Frejlich Copyright # 2007 John Wiley & Sons, Inc.

181

182

HOLOGRAPHIC TECHNIQUES

Sh1

Sh3 BS

M1

laser

Sh2

M2

C D1

D2

Figure 8.1. Holographic setup. A laser beam is divided by the beamsplitter BS, reflected by mirrors M1 and M2 and intefering with an angle 2y. A sinusoidal pattern of light is produced in the volume where these two beams interfere. A photorefractive crystal C is placed in the volume where this pattern of light is produced. The irradiance of the two interfering beams is measured behind the crystal with photodetectors D1 and D2. Shutters Sh1, Sh2; and Sh3 are used to cut off the main and each one of the interfering beams if necessary.

to compute several fundamental material parameters such as diffusion length, Debye length (and density of donors), photocarriers mobility, quantum efficiency, dark conductivity, photoconductivity coefficient, etc. We shall mention below just a few examples of these direct methods. These measurements are carried out with a simple holographic (or interferometric or twowave mixing) setup such as the one schematically represented in Figure 8.1. 8.1.1

Energy Coupling

Energy coupling appears as light diffracts through a phase grating because, in general, some light is transferred from one to the other beam direction at the output behind the grating according to the equations IS ¼ IS0 ð1  ZÞ þ IR0 Z þ 2 IR ¼ IR0 ð1  ZÞ þ IS0 Z  2

qffiffiffiffiffiffiffiffipffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi IS0 IR0 Zð1  ZÞ cos j

qffiffiffiffiffiffiffiffipffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi IR0 IR0 Zð1  ZÞ cos j

ð8:1Þ ð8:2Þ

For such an energy transfer to occur it is necessary that j ¼ 6 p=2. In the case of photorefractive materials this is usually the case. Also, energy coupling in photorefractives is rather efficient because if energy is transferred in the adequate direction (from the

DIRECT HOLOGRAPHIC TECHNIQUES

183

strongest to the weakest beam), the pattern of fringes visibility steadily increases from the input to the output, thus producing a nonlinear enhancement that is not evident in the simple formulation of Equations (8.1) and (8.2) but is explicitly included in the expression Equation (4.79) IS ðdÞ ¼ IS0

1 þ b2 1 þ b2 ed

For b2  1 the expression above simplifies to IS ðdÞ ¼ IS0 ed where d is the sample’s thickness and  here assumes the role of an exponential gain. Computing  from the equation above is particularly simple and precise because only one single photodetector is used: the one along direction S. Then measuring the IS ðdÞ without and with the other beam shining on the sample one is able to compute IS0 and IS ðdÞ, always measured behind the sample, and from these data compute . Figure 8.2 shows experimental data of energy transfer in a TWM experiment on BTO. Both IS ðdÞ and IR ðdÞ are measured simultaneously and with one and the other beam being switched off as indicated in the figure.

Intensity (a.u.)

0.25

0.15

0.05

0

20

40

60

Time (s)

Figure 8.2. Energy transfer between interfering l ¼ 633 nm beams in the two-wave mixing experiment, represented in Fig. 8.1, on a BTO crystal (2.8 mm thick). The figure shows the overall irradiance at the crystal output with both beams onto the sample (shutters Sh1, Sh2, and Sh3 open) and when one (all open and Sh3 switched off) and the other (all open and Sh2 switched off) beam are alternatively switched off. From these data, and knowing the input recording beams irradiance ratio b2 ¼ 1:5, it is possible to compute the exponential gain coefficient  and also Z.

184

HOLOGRAPHIC TECHNIQUES

Energy coupling has been already analyzed for a pure phase grating in Section 4.2.2.2, where the exponential energy (or amplitude) gain coefficient  was defined in Equation (4.80) in terms of the imaginary part of Eeff ¼

2pn3 reff =fEeff g l cos y0

ð8:3Þ

Substituting Equation (3.67) into Equation (8.3), in the absence of external field (E0 ¼ 0), the energy gain can be written as ¼

2pn3 reff ED l cos y0 1 þ K 2 l2s

ð8:4Þ

From the expression above we deduce that the maximum for  is M ¼

n3 reff pkB T ls lq cos y0

for

Kls ¼ 1

ð8:5Þ

which is achieved for K 2 l2s ¼ 1. By measuring  in a TWM experiment, as a function of K, one may find its maximum value M , and from the relations in Equation (8.5) one can compute ls , too, from the simple relation K 2 l2s ¼ 1. From the value of ls (and some other material parameters) we can compute the effective trap concentration ðND Þeff from Equation (3.47). Additionally, the reff can be computed from Equation (8.5) if the index of refraction n is known. A practical consequence stems from Equation (8.5): Maximum energy transfer can be achieved by adequately chosing the hologram wave vector to be K ¼ 1=ls . Figure 8.3 illustrates the above-referenced procedure as applied to the Ti-doped KNSBN crystal described in Table 8.1 [Bian and Frejlich, 1995] with extraordinary

Γ (cm–1)

6

4 ΓM = 5.1 cm–1

2

θM = 12.4°

0 0

10

20 θ (degrees)

30

40

Figure 8.3. Exponential gain coefficient  as a function of the external incidence angle y ~ measured for a KNSBN:Ti crystal with its optical c-axis parallel to the grating vector K. Holographic recording is carried out with extraordinarily polarized (polarization direction along c-axis) 514.5-nm-wavelength laser line. Reproduced from [Bian and Frejlich, 1995]

185

DIRECT HOLOGRAPHIC TECHNIQUES

TABLE 8.1. Properties of a KNSBN:Ti Sample (Kx Na1x Þ2m (Sry Ba1y )1m Nb2 O6 0.36% wt 5:5 mm  5:1 mm  4:9 mm Along 5.5-mm side 0:15 cm1  2:31  2:28

KNSBN formula Ti-dopant dimensions c-Axis a (514.5 nm) n0 (514.5 nm) ne (514.5 nm) *

from Red Optics/USA datasheet.

~ parallel to the c-axis. In this case polarized 514.5-nm light and the grating vector K 1 M ¼ 5:1 cm , and the corresponding ls ’ 0:20 mm are obtained. Note that, because of the polarization direction of the recording beams lying in the incidence plane and making an angle 2y0 between them as measured inside the sample, all expressions for  above should be factored by cos 2y0. Exercise: 8.1.2

From data in Fig. 8.2 show that   0:68 cm1 .

Diffraction Efficiency

Diffraction efficiency Z is conveniently computed, from the diffracted I d and the transmitted I t irradiances measured in the output beams behind the sample, as Z¼

Id Id þ It

ð8:6Þ

where bulk absorption and interface losses are conveniently not included so that the diffraction phenomenon itself is better analyzed. This measurement procedure, however, requires one of the recording beams to be switched off during the time the transmission and diffraction of the other beam are measured. This time should be short compared to the response time of the material under analysis for the hologram not to be sensibly erased in the meantime. It is also interesting to use energy coupling experiments to measure Z. To do this it is necessary to measure the two-wave mixed irradiance along one of the directions behind the sample, let us say IS as formulated by Equation (8.1) IS ¼ IS0 ð1  ZÞ þ IR0 Z þ 2

qffiffiffiffiffiffiffiffipffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi IS0 IR0 Zð1  ZÞ cos j

The irradiance is also measured at the moment the other beam is switched off ðIS ÞI 0 ¼0 ¼ IS0 ð1  ZÞ R

ð8:7Þ

186

HOLOGRAPHIC TECHNIQUES

From the two equations above we write IS  ðIS ÞI 0 ¼0 R

ðIS ÞI 0 ¼0 R

sffiffiffiffirffiffiffiffiffiffiffiffiffiffiffi IR0 Z I0 Z þ 2 R0 ¼ 0 cos j IS 1  Z IS 1  Z

ð8:8Þ

From Equation (8.8) Z can be computed if one knows the input irradiance ratio b2 ¼ IR0 =IS0 and the j, which is usually 0 or p for nonphotovoltaic materials in the absence of applied field. We assume parallel-polarized diffracted and transmitted output beams. This method is very interesting for relatively thick samples because in these cases the lack of perfect planicity of the sample’s surfaces may lead to a lenslike effect and the transmitted and diffracted beams may be focused/defocused while going through the sample. In this case it is difficult to compare the irradiances of the diffracted and transmitted beams, along the two different directions behind the sample, in order to carry out the classic measurement of Z from Equation (8.6). The presently reported energy transfer method instead only requires the measurement along one single beam, always using one single photodetector measuring at one single position behind the crystal. The measurement of Z for thick samples and reversible materials is discussed in detail in the Appendix. Exercise: From Equation (8.8) and data in Figure 8.2, compute Z. Verify the compatibility between this result and the value of  computed in the exercise of Section 8.1.1. 8.1.3

Holographic Sensitivity

The speed at which holograms are recorded can be characterized by the so called ‘‘sensitivity’’ S that is defined as the refractive index variation (jnj) per unit absorbed energy in the unit crystal volume (Iabs =d) per unit light pattern modulation (jmj) at the initial recording stage (t ¼ 0) [Gu¨nter and Huignard, 1988]   d @jnj S¼ jmjIabs @t t¼0

ð8:9Þ

where jnj is computed from Equation (4.82). Assuming a quasi-stationary equilibrium we should directly include the time dependence as jnðtÞj ¼ n3 reff jEsc ðtÞj=2

ð8:10Þ

and @jnðtÞj @t



n3 reff @jEsc ðtÞj ¼ 2 @t t¼0



t¼0

ð8:11Þ

DIRECT HOLOGRAPHIC TECHNIQUES

187

From Equation (3.51), and assuming Esc ð0Þ ¼ 0, we compute @Esc ðtÞ @t



t¼0

¼ mEeff =tsc

ð8:12Þ

Substituting Equations (8.11) and (8.12) into Equation (8.9) we get   n3 reff kB T K sd qmt þ S¼ 2Ee0 q 1 þ K 2 L2D aI hn

ð8:13Þ

The plot of S as a function of K or the incidence angle y [Moharan et al., 1979] is similar in shape (see Fig. 8.10) to that of  in Figure 8.3 except for the fact that S is maximum for KLD ¼ 1: SM ¼

  n3 reff kB T sd qmt þ ELD 4qe0 aI hn

for

KLD ¼ 1

ð8:14Þ

In Equation (8.13) we have substituted the corresponding expressions for tsc in Equation (3.53), for tM in Equation (3.43), and for Eeff in Equation (3.52) with E0 ¼ 0 as well as the definition of ED in Equation (3.21). If we assume that Ia  I0 a throughout the sample’s thickness and substitute it in Equation (8.13), the linear plot of S as a function of 1=ðaI0 Þ allows computation of some material parameters. In fact, the corresponding angular coefficient and independent term are, respectively, n3 reff kB T K sd 2Ee0 q 1 þ K 2 L2D

n3 reff kB T K qmt 2 2 2Ee0 q 1 þ K LD hn

ð8:15Þ ð8:16Þ

and from these parameters sd and mt can be easily computed. Table 8.2 shows some values for S M (and also for M ) for different materials, measured by ourselves or from the literature, for comparison. A question of practical interest arises: How do we measure @n @t in order to pffiffiffi calculate S? We may do it from the evolution of Z. In fact, for the initial stage of recording, Z can be written, from Equation (4.118), as   mpn3 reff Eeff d2   Z¼ 2l cos y0 

The equation above derives from stationary conditions where, from Equation (3.54), we get Esc ¼ mEeff

188

HOLOGRAPHIC TECHNIQUES

TABLE 8.2. Holographic Sensitivity and Gain for Some Materials



S M (107 cm3 =mJ)

KNSBN KNSBN:Ti þþ KNSBN:Cu KNSBN:Cr SBN:Rhy BSO BTO BaTiO 3 LiNbO3 :Feyyy GaAszz CdTe:V

1.7 4.1 0.6 2.3 0.02 20yy 7.7 5.4 0.1 500 4000?

M (cm1 ) 2.1 5.1 7.2 11 70 3yy 8 50 0.4zzz 1??

r eff (pm/V) 23.4 55.1 27.4 92.8  5.0 5.6 97þ 30.8z 1.7 5.5

Unless stated otherwise, data refer to extraordinary polarization at l ¼ 514:5 nm, except for GaAs and CdTe, which is l ¼ 1064 nm * described in Table 8.1; þ Red Optronics Co. datasheet; [Tomita et al., 1993]; *** [Gu¨nter and Huignard, 1988]; þþ [Bian and Frejlich, 1995]; y [Vazquez et al., 1992]; yy [Frejlich et al., 1990]; yyy [Garcia et al., 1995]; z [Yariv, 1985]; zz [Bian et al., 1995]; zzz [Klein, 1984]; ? [Marfaing, 1999]; ??[Delaye et al., 1997]

If we assume a quasi-stationary time evolution process where we could substitute Esc pffiffiffi in the above expression by Esc ðtÞ, then the expression for Z becomes time dependent as follows     pffiffiffiffiffiffiffiffi pn3 reff Esc ðtÞd pdnðtÞ ¼  ZðtÞ ¼  2l cos y0   2l cos y0 

ð8:17Þ

We can therefore compute

 pffiffiffi  pffiffiffi    @ Z @ Z @jnj ¼ @t t¼0 @jnj t¼0 @t t¼0   pd @jnj ¼ 2l @t t¼0

ð8:18Þ

where cos y0  1. Equation (8.18) above shows that the time derivative of the index of refraction modulation can be computed from the measured square root diffraction efficiency evolution at the beginning of the recording.

8.1.4

Hologram Recording and Erasure

Holography is particularly useful for the measurement of conductivity and photoconductivity provided it is possible to record a hologram on the material

DIRECT HOLOGRAPHIC TECHNIQUES

189

under analysis. In fact, the hologram characteristic time tsc in Equation (3.53) is proportional to tM , which according to Equation (3.43) is inversely proportional to the conductivity s  N em and directly proportional to the dielectric pemittivity. Therefore, aside from parameters K 2 L2D , K 2 l2s , and similar ones, the relaxation of a hologram, under the action of light or in the dark, allows computation of the photoconductivity or the dark conductivity, respectively, in the sample’s volume without need of electrodes at all, as far as the dielectric constant of the material is known, which is usually the case. Holographic erasure has an obvious advantage over recording: It is free from environmental phase perturbations on the setup. Figure 8.4 shows the erasure with l ¼ 633 nm of a previously recorded hologram (using the same wavelength) in a Pb-doped BTO crystal. In this case it is Z 1, so that Z here can be approximated by Equation (4.118). The erasure, however, fits better a double exponential of the form Z ¼ jA1 et=tsc1 þ A2 et=tsc2 j2 þ d

ð8:19Þ

than a single exponential, thus indicating that more than one photoactive center is participating in this process. The parameter d here accounts for background illumination. Figure 8.5 instead shows the white light erasure of a previously recorded (using 514.5-nm light) hologram in Fe-doped LiNbO3 crystal. The measurement is carried out with short pulses of one of the 514.5-nm wavelength recording beams. In this

0.4

Id (au)

0.3

0.2

0.1

–50

0 Time (s)

50

Figure 8.4. Hologram erasure in BTO:Pb. The graph shows the erasure of a hologram in a Pb-doped BTO crystal. The hologram was recorded and erased using l ¼ 633 nm light. The erasure is monotonically decreasing but is not a single exponential. It fits better two (probably electron-based exponentials) as represented in Eq. (8.19) with A1 ¼ 0:37, A2 ¼ 0:28, tsc1 ¼ 34:0 s, tsc2 ¼ 5:47 s, d ¼ 0:0078.

190

HOLOGRAPHIC TECHNIQUES

80

η (%)

60

40

20

0 0

200

100

300

Time (min)

Figure 8.5. White light hologram erasure in LiNbO3 :Fe. The erasure data ( ), measured using one of the 514.5-nm recording beams, adequately fit a single exponential (dashed curve) law as described by Eq.(8.20) with a ¼ 1:06 rad and b ¼ 180 min.

case Z is rather large and its expression is that reported in Equation (4.111), with the time evolution being described by Z / sin2 ½að1  et=b Þ

ð8:20Þ

The rather good data fit to a monoexponential expression in Figure 8.5 indicates that, differently than for sillenites, a single photoactive level is actually involved in holographic recording on LiNbO3 . 8.1.4.1 Dark Conductivity. For the particular case of dark relaxation, which is an energy-barrier controlled phenomenon, the relaxation time follows an Arrheniustype exponential law kEaT

tsc ¼ t0sc e

B

ð8:21Þ

that allows computing the energy of the barrier-controlling energy Ea as illustrated for an undoped BTO (sample BTO-8) in Figure 8.6. The resulting Ea  1 eV is close to the p-type dark conductivity already measured on this material with purely electric techniques. This fact indicates that whatever the localized states where holograms are recorded, they are erased in the dark via hole conductivity. The practical measurement of the evolution of Z in the dark, which is required here, is somewhat complicated by the fact that light is always required to measure Z. It is always possible to use very weak and very short light pulses at sufficiently large intervals and always correcting the effect of these short pulses of light. It is not recomended to use an auxiliary beam of different l (usually a much larger one) to measure Z because it is difficult to match Bragg conditions between the beam and the grating being measured on one side, and because no one knows what might be the effect of such wavelength radiation on the complex nature of some materials.

DIRECT HOLOGRAPHIC TECHNIQUES

191

τsc (min)

100

10

1 2.8

3.0

3.2 1000/T (K–1)

Figure 8.6. Hologram relaxation in the dark. Exponential time as a function of inverse temperature for hologram relaxation in the dark. The hologram was recorded using l ¼ 514:5 nm light onto an undoped BTO sample (BTO-8) approximately 1 mm thick. Diffraction efficiency was measured from time to time using one of the in-Bragg recording beams during a very short time and correcting data for the effect of exposure to light. From the Arrhenius-type curve an activation energy of 1.04 eV was computed.

8.1.5

Hole–Electron Competition

The erasure of a hologram under the action of light is a powerful means to measure photoconductivity and to put into evidence the presence of hole–electron competition. In fact, as described in Section 3.4.1, the erasure of a combined hologram, made of electrons on one photoactive level and of holes on another level, without applied external field leads to the characteristic shape shown in Figure 8.7 because there are present two distinct mutually p-shifted holograms with widely different characteristic exponential decay times. The first rapid decay corresponds to the erasure of the faster electron-based hologram. The relative maximum roughly characterizes the size of the slower hole-based hologram. The slow decay at the end describes the hole-based hologram decay. Note that a combined photorefractive (refractive index modulation) plus photochromic (absorption modulated) hologram may produce a pattern of erasure similar to that shown in Figure 8.7 on one of the directions behind the crystal but not in both (see Eqs.(4.27 and 4.28)), which is the case of hole–electron competition. Note also that the erasure of a hologram recorded by electrons and holes on the same photoactive centers level leads to a monotonic decay because there is not two but one single grating being recorded and erased. The curves in Figure 8.7 show the erasure of previously recorded holograms using 633-nm-wavelength beams, previously exposed to uniform 532-nm light. These curves were fitted with e

h

Z ¼ jAet=tsc eij  Bet=tsc j2

ð8:22Þ

192

HOLOGRAPHIC TECHNIQUES

0.45

1

2

η (au)

0.30

0.15

0 0

50

100

150

300

Time (s)

Figure 8.7. The graph shows the erasure of a hologram in a Pb-doped BTO (same sample as in Fig. 8.4) measured along both directions at the crystal output. Both erasure curves (squares and circles) are artificially shifted in time for better observation. The crystal was preexposed for a few minutes to a uniform light at l ¼ 532 nm. Preexposure was switched off immediately before holographic recording started using an He-Ne laser line of l ¼ 633 nm. The hologram was erased with one of the in-Bragg recording beams. No external electric field was applied. Eq. (8.22) was fit (continuous curves) to data, and the resulting parameters are reported in Table 8.3.

where A and B are the amplitudes of the electron- and hole-based holograms, tesc and thsc are their respective holographic response times, and j is an arbitrary phase shift between both gratings (aside from the p phase shift naturally arising from their opposite electric charge nature) produced by environmental perturbations. The results from fitting on curves 1 and 2 is shown in Table 8.3. The same experiment (recording and erasure with l ¼ 633 nm and preexposure with l ¼ 532 nm) on undoped BTO always led to a monotonically decreasing Z only. Figure 8.8 shows the effect of short-wavelength preexposure on holographic recording (and erasure) with l ¼ 780 nm. The experiment is similar to the one carried out with l ¼ 633 nm and shown in Figure 8.7. Figure 8.9 illustrates the effect of preexposure (with light of different wavelength) on the holographic recording with 780-nm radiation on undoped BTO: A hologram is recorded on undoped BTO using a laser at l ¼ 780 nm, and TABLE 8.3. Hole–Electron Competition in BTO:Pb: Fitting of Hologram Erasure Data Curve 1 2

A (au)

B (au)

B/A

tesc (s)

thsc (s)

thsc =tesc

j (rad)

0.72 0.71

0.35 0.36

0.49 0.51

3.2 3.6

172 184

54 51

0.89 0.87

DIRECT HOLOGRAPHIC TECHNIQUES

193

Figure 8.8. This figure shows the erasure of holograms in Pb-doped BTO (same sample as in Fig. 8.4) recorded during 2 min with a diode laser of 780-nm wavelength, observed along the reference beam direction (left-hand graph) and along the signal beam (right-hand graph) using one of the recording beams. The curves showing a local maximum result from 3-min preexposure at 524-nm light from a LED, whereas the monotonically decreasing ones are without preexposure and actually verify a monoexponential law with a tsc  100 s. The fit of the curves exhibiting hole–electron competition showed an electronic grating characteristic time of tesc  13–16s and a corresponding value thsc  35tesc for the hole grating. 0.3 476 nm 634 nm 670 nm 593 nm 524 nm

η (au)

0.2

0.1

0

0

50

100

150

200

250

Time (s)

Figure 8.9. The graph shows the erasure of holograms in undoped BTO under 10–15 min  1 mW=cm2 preillumination with light of different wavelengths as indicated in the graph. The recording and erasure were always carried out with l ¼ 780 nm. Measurement along the other direction behind the crystal showed similar shapes. Erasure curves are artificially shifted in time for better observation.

194

HOLOGRAPHIC TECHNIQUES

then the hologram is erased, using one of the recording beams. The erasure curves are shown in Figure 8.9 for 10- to 15-min preexposure with spatially uniform light of different wavelengths. For preexposure with low-energy photons (l  670 nm) the erasure is monotonically decreasing, showing that either there is no hole–electron competition or it occurs at the same photoactive level. For shorter-wavelength preexposure the erasure shows the typical pattern of hole–electron competition on different photoactive levels such as that in Figure 8.7. Note that direct recording and erasure with the more energetic light (e.g., at 514.5 nm or 633 nm) always shows a monotonically decaying erasure curve, for undoped BTO, no matter whether there is preexposure or not, independently of the photonic energy of preexposure. We know that this more energetic radiation allows the simultaneous recording of hole- and electron-based holograms so that we may conclude that both type of gratings do exist but are probably on the same level in the band gap. Similar results were already reported [Odoulov et al., 1994] also for Bi12 TiO20 but for l ¼ 1064 nm recording and white light preexposure.

8.2

PHASE MODULATION TECHNIQUES

In this section we describe some examples illustrating the possibilities of phase modulation (described in Section 4.3) for materials characterization. The use of phase modulation has two main advantages:

It is a real-time nonperturbating technique that allows measurements to be carried out without disturbing the recording itself.

It allows, in general, operation in self-stabilized recording mode where either the first (I  ) or the second (I 2 ) harmonic term is used as error signal for operating the stabilization, in which case the other harmonic is available for measurement. In this condition the phase shift j between the transmitted and the diffracted beams along the same direction at the sample output is known a priori because it is fixed by the stabilization loop. 8.2.1

Holographic Sensitivity

The relevance of holographic sensitivity both for material research and for applications was discussed in Section 8.1.3. We will just point out here the utility of phase modulation techniques for the practical measurement of this quantity [Bian and Frejlich, 1995]. In Section 8.1.3 we showed how it is possible to compute S from the evolution of Z. We shall here show how to compute S from the time evolution of the I 2 signal, when the I  is used as error signal in a self-stabilization setup, in which case we should write  2   pffiffiffi  2  @ Z @I @I ¼ pffiffiffi @t t¼0 @ Z t¼0 @t t¼0

ð8:23Þ

PHASE MODULATION TECHNIQUES

pffiffiffi where @I 2 =@ Z is computed from Equation (4.167) as  2  qffiffiffiffiffiffiffiffi @I ¼ 4J2 ðcd ÞJ0 ðcd Þ IS0 IR0 pffiffiffi @ Z t¼0

195

ð8:24Þ

where Z in Equation (4.167) was substituted by ZJ02 ðcd Þ to take into account the effect of pattern of fringes vibration onto Z, so that Z represents here its nonperturbated value. Also, we substitute j ¼ 0 into the expression for I 2 because the setup is self-stabilized. Self-stabilization strongly contributes to reduce phase perturbations and improve measurement dispersion, but one should keep in mind that the j ¼ 0 value fixed by self-stabilization should correspond to the unconstrained recording condition because otherwise the recording would be modified by selfstabilization itself, as discussed in Chapter 6. Substituting Equations (8.24) and (8.18) into Equation (8.23) we get  2    qffiffiffiffiffiffiffiffi @I pd @jnj 0 0 ð8:25Þ ¼ 2J2 ðcd ÞJ0 ðcd Þ IS IR @t t¼0 @t t¼0 l cos y0 From Equations (8.25) and (8.9) and the definition of jmj we get  2  l cos y0 1 @I pffiffiffiffiffiffiffiffi S¼ paI0 d 4J2 ðcd ÞJ0 ðcd Þ IS0 IR0 @t t¼0

ð8:26Þ

The use of I 2 therefore allows measurement of the sensitivity during holographic recording without perturbating it. Figure 8.10 shows the experimental sensitivity measured at l ¼ 514:5 nm for the same KNSBN:Ti sample reported in Table 8.1 and the same optical configuration and experimental setup as for Figure 8.3. S was computed from Equation (8.26) and the evolution of I 2 for the limit of t ! 0. A typical experimental plot of V 2 / I 2 –7

5 × 10

3 × 10–7

3

S (cm /mJ)

4 × 10–7

2 × 10–7

SM = 4.1×10

–7

1 × 10

–7

3

cm /mJ

θM = 13.0°

0 0

10

20

30

40

θ (degrees)

Figure 8.10. Photorefractive sensitivity S data ( ) as a function of the external incidence angle y for the KNSBN:Ti sample of Table 8.1 in the same optical and recording configuration as in Fig. 8.3. From these data we compute LD ¼ 0:18 mm. Reproduced from [Bian and Frejlich, 1995]

196

HOLOGRAPHIC TECHNIQUES

8

4

V

2Ω

(Volts)

6

2

0

0

100

200

300 t (sec)

400

500

Figure 8.11. Second harmonic evolution for KNSBN:Ti for the same sample and experimental conditions as for Fig. 8.10 with y ¼ 15 and IS0 þ IR0  3 mW=cm2 . Reproduced from [Bian and Frejlich, 1995]

for this material is shown in Figure 8.11, where the low data dispersion is due to the fact that the setup was self-stabilized with the first harmonic term I  as error signal, as discussed in Chapter 9. From data in Figure 8.10 the maximum sensitivity S M ¼ 4:1  107 cm3 =mJ and corresponding LD ¼ 0:18 mm are computed. In this case, as for the case of  in Section 8.1.1, all expressions above for S should be also factored by cos 2y0. 8.2.2

Holographic Phase-Shift Measurement

The explicit formulation of the phase shift fP of the stationary space-charge electric field grating in Equation (3.58) allows one to compute ls , from the experimental plot of tan fP vs. E0 . The parameter ls is related to the effective density of traps as indicated by Equation (3.47). The measurement of fP itself is also of relevance because it allows one to know whether we are dealing with a purely diffusion-arising recording mechanism (fP ¼ p=2), a photovoltaic material (fP  0 or p), or some mixed effects. In fact, this technique has already been used to show that Fe-doped BaTiO3 crystals exhibit photovoltaic effects [Holtmann et al., 1992]. The problem is that fP is hardly directly available [Rupp, 1986, 1987] from the experiments. One possibility is to compute fp from the output beam phase shift j. In the absence of wave mixing, the simple relation in Equation (4.172) 1 ¼  tan j tan fp holds, where tan j can be computed from Equation (4.168) so that we should write 1 IS J2 ðcd Þ ¼ 2 tan fp IS J1 ðcd Þ

ð8:27Þ

where the quantities in the right-hand side in Equation (8.27) are measured in a phase modulated two-wave mixing experiment.

PHASE MODULATION TECHNIQUES

197

Note that self-stabilization is not recommended here because it will, in general, affect the recording process and consequently the phase shift would be also affected. It is possible, however, to stabilize the recording pattern of fringes using the interference of the transmitted and reflected beams on a small glass plate tightly fixed by the side of the sample under analysis as discussed in Chapter 9. 8.2.2.1 Wave-Mixing Effects. In the presence of self-diffraction effects, however, the simple relation in Equation (4.172) does not hold. The much more complicated relation in Equation (4.122) should be used instead, where fP is implicitly indicated by the parameters  and g and their relation tan fP ¼ =g. Figure 8.12 clearly illustrates this point: The directly experimentally obtained tan j vs. E0 data (squares) are plotted; the 1= tan fp computed from tan j with Equations (4.122), (4.80), (4.85), (4.81), (3.52), and (4.86) is also plotted in the same figure. We see that, except for low E0, tan j 6¼ 1= tan f, as discussed above. The continuous curve is the fitting of Equation (3.58) that is clearly in good agreement with the 1= tan fp data. From this fit we get ls . 8.2.3

Photorefractive Response Time

It is possible to use phase modulation techniques in two-wave mixing (TWM) experiments for hologram response time measurement, as an alternative to conventional hologram erasure techniques [Bylsma, 1988]. The basic idea in this method

10 8

tan ϕ

6 4 –1/ tan φ 2 0

0

2

4

6

8

10

E (kV/cm)

Figure 8.12. Evolution of the 1= tan f ðwith f ¼ fp Þ accounting on self-diffraction effects as described in the text ( ), as a function of the applied field E for a 2-mm-thick nominally undoped Bi12 TiO20 sample with K ¼ 7:08 mm1 , b2 ¼ 9, and I0  4 mW=cm2 using the 514.5-nm wavelength laser line. The crystal is in a transverse electro-optical configuration with the (110)-plane perpendicular to the incidence plane and the [001]-axis ~ Data fitting leads to ls ¼ 0:027 mm. In the same figure perpendicular to the grating vector K. is plotted (&) the directly measured tan j. Reproduced from [Frejlich et al., 1997]

198

HOLOGRAPHIC TECHNIQUES

is simple: A small phase modulation in one of the interfering beams makes the interference pattern onto the crystal vibrate with the modulation frequency as described in Section 4.3. For a frequency much smaller than the frequency response of the crystal, the recorded hologram vibrates synchronously with the pattern of light and no modulation signal is detected along any of the two-wave mixed beams behind the crystal. For a much higher modulation frequency instead, the hologram is comparatively too much slow to move while the pattern does oscillate and therefore the resultant modulation signal is independent from the crystal response and dithering frequency. For intermediate frequency values, the modulation signal behind the crystal depends on the crystal response: from these data its response time may be obtained. In a TWM setup like the one depicted in Figure 4.19, one of the interfering beams (S in this case) is phase modulated with angular frequency  and phase amplitude cd . The interference pattern of light onto the crystal is therefore vibrating with frequency  as described before in Equation (4.143). Iðx; tÞ ¼ I0 þ I0 jmj cosðKx þ f þ cd sin tÞ The light pattern modulation jmj is assumed to be constant throughout the sample thickness. The time evolution of the space-charge electric field modulation amplitude Esc , for a purely diffusion-arising (that is, without external field E0 ¼ 0) photorefractive hologram produced by this vibrating pattern of light, is ruled by Equations (3.51)–(3.53) @Esc Esc imðtÞED þ ¼ @t tsc tsc ð1 þ K 2 l2s Þ

ð8:28Þ

with E0 ¼ 0. From the expression of the modulated pattern of fringes above, the time dependence term for the light modulation is written in a complex notation as mðtÞ ¼ jmj expðicd sin tÞ ¼ jmj

1 X

L¼1

JL ðcd Þ exp iðLt þ fÞ

ð8:29Þ

Substituting the expression for mðtÞ above into Equation (8.28) we find the solution for Esc ðtÞ: Esc ðtÞ ¼ ijmj

1 X ED JL ðcd Þ exp iðLtÞ 1 þ K 2 l2s L¼1 1 þ iLtsc

ð8:30Þ

For a GaAs crystal with its [001]-axis perpendicular to the incidence plane, the ~ and the interfering beams being linearly polarized along [110]-axis parallel to K, the [001]-axis direction, as depicted in Figure 8.13, the diffracted beams behind the ~ direction, which is orthogonally polarized crystal are linearly polarized along the K to the transmitted beams [Marrakchi et al., 1986; Liu and Cheng, 1989]. The overall irradiance along the direction S behind the crystal may be therefore written as pffiffiffi IS  j^sS0 expðicd sin tÞ þ i^r R0 Zeiðjp=2Þ cos gj2

ð8:31Þ

PHASE MODULATION TECHNIQUES

199

[001] K

[110]

Figure 8.13. Two-wave mixing experiment in a photorefractive GaAs intrinsic crystal with mutually orthogonally polarized diffracted and transmitted beams. The polarization direction is represented by the black arrows. The input and transmitted beam polarization is along [001]-axis, whereas the diffracted beam is perpendicular to [001]-axis.

^s and ^r are unit vectors parallel to the polarization of the transmitted and diffracted beams, respectively, with ^s  ^r ¼ 0 in this particular experiment. For low diffraction efficiency jZj 1 and from Equation (4.118) it is pn1 d pffiffiffi iðjp=2Þ m Ze l cos y

mn1 ¼ n30 reff Esc =2

ð8:32Þ

1 X i ED JL ðcd Þ exp iðLtÞ n1 ¼ n30 reff jmj 2 2 1 þ K ls L¼1 1 þ iLtsc 2

ð8:33Þ

and

From Equation (8.30) n1 can be written as

where n0 is the bulk index of refraction and reff is the electro-optic coefficient. Substituting the modulated beam in Equation (8.31) by its Bessel development S0 expðicd sin tÞ ¼ S0

1 X

N¼1

JN ðcd Þ expðiNtÞ

ð8:34Þ

and inserting Equations (8.33) and (8.34) into Equation (8.31), the resultant expression is developed into pdn30 reff ED jmj S0 R0^s  ^r 1 þ K 2 l2s l cos y 1 X Ltsc sin½ðN  LÞt  cos½ðN  LÞt JN ðcd ÞJL ðcd Þ  1 þ ðLtsc Þ2 N;L¼1

IS ¼ C þ

ð8:35Þ

where C is a constant. For cd 1, Equation (8.35) may be limited to the secondorder Bessel function, in which case we should also substitute the Bessel functions by their approximate expressions: J0 ðcd Þ ’ 1 J1 ðcd Þ ’ cd =2

J2 ðcd Þ ’ c2d =8

ð8:36Þ

200

HOLOGRAPHIC TECHNIQUES

40

20

I

2Ω

(a.u.)

30

10

0 0

1000

2000

3000

4000

5000

Modulation frequency Ω/2π (Hz)

Figure 8.14. Second harmonic response curves for an undoped semi-insulating GaAs crystal illuminated with a 1.06-mm laser wavelength line, with jmj ¼ 1 and an angle g ¼ 10 between ~ Theoretical fit to data the transmitted beam polarization direction and the grating vector K. 1 2 (&) for K ¼ 3:5 mm and I0  118 mW=cm lead to tsc ¼ 0:22 ms; fit to data ( ) for K ¼ 2:1 mm1 and I0  168 mW=cm2 lead to tsc ¼ 0:1 ms.

In this case the final expression for the second-harmonic in t is obtained ½IS2 n1 ¼ j½IS2 n1 j cosð2t þ fn1 Þ j½IS2 n1 j ¼ jmjS0 R0^s  ^r c2d tan fn1 ¼

2ðtsc Þ2  1 3tsc

pdn30 reff ED

2l cos yð1 þ

2 t2sc ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi p K 2 l2s Þ ð1 þ 2 t2sc Þð1 þ 42 t2sc Þ

ð8:37Þ ð8:38Þ ð8:39Þ

From Equation (8.38) the response time tsc may be computed. Experimental results for a GaAs semi-insulating crystal are shown in Figure 8.14. A similar approach may be used for computing the amplitude of the first harmonic for an index of refraction (of amplitude n01 ) grating resulting from the photoactive center modulation during photorefractive recording (that is different from the strictly speaking photorefractive index of refraction grating) and is in phase with the pattern of light onto the sample. In this case we should assume that

where

NDþ AðtÞ

n01 / NDþ AðtÞ

ð8:40Þ

is related to Esc ðtÞ in Equation (3.38)

iKEe0 Esc ðtÞ  qNDþ AðtÞ

It is therefore enough to substitute Esc ðtÞ in Equation (8.30) into the expression above to get n01 / NDþ AðtÞ ¼ jmj

1 X Ee0 kB T K 2 JL ðcd Þ exp iðLtÞ q q 1 þ K 2 l2s L¼1 1 þ iLtsc

ð8:41Þ

PHASE MODULATION TECHNIQUES

201

with 1 X pd Ee0 kB T K 2 JL ðcd Þ pffiffiffiffiffiffi ijn0 exp iðLtÞ Zn01 e 1 / jmj 2 2 l cos y q q 1 þ K ls L¼1 1 þ iLtsc

ð8:42Þ

Processing similarly as above but for the first harmonic in t, we get an expression for an in-phase index of refraction grating j½IS n0 j / 2cd jmjS0 R0 1

Ee0 kB T K 2 pd tsc pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi q q 1 þ K 2 l2s l cos y 1 þ 2 t2sc

ð8:43Þ

The same procedure above is employed for computing jIS2 j for a pure amplitude (photochromic) grating in phase, associated to the photorefractive effect (that is, arising from photoactive trap absorption modulation). In this case a0 d pffiffiffiffiffi ijA ZA e ¼ 1 2 cos y with

for

jZA j 1

a01 / NDþ AðtÞ

ð8:44Þ ð8:45Þ

and the final result is j½IS2 a0 j / 1

c2d Ee0 kB T K 2 d 2 t2sc ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi p jmjS0 R0 q q 1 þ K 2 l2s cos y ð1 þ 42 t2sc Þð1 þ 2 t2sc Þ 2

ð8:46Þ

Note that for the case of a nonphotorefractive index of refraction and an absorption grating, there is no anisotropic diffraction so that the transmitted and diffracted beams are always parallel-polarized and ^s  ^r ¼ 1, which is not the case for the photorefractive index of refraction grating. Exercise: Prove that it is ½IS n1 ;a0 ¼ 0 for the photorefractive and associated 1 photochromic gratings, and that it is ½IS2 n0 ¼ 0 for the associated nonphotore1 fractive index grating. 8.2.4

Selective Two-Wave Mixing

Selective two-wave-mixing (S2WM) [Frejlich et al., 1992] is a simple two-wave mixing (TWM) experiment where the irradiance along both directions behind the crystal are simultaneously measured with the same phase-modulation techniques described above. This method takes advantage of differences in symmetry properties of amplitude and phase gratings concerning energy exchange as described in Section 4.2.1. In fact, any phase shift of the pattern of light referred to a phase grating produces an asymmetric change in the energy of the beams along both directions behind the sample: If the intensity increases in one direction, it necessarily decreases

202

HOLOGRAPHIC TECHNIQUES

in the other, because of energy conservation. For an amplitude grating instead, energy conservation does not verify, and the intensity change in both directions is the same: A phase shift produces an increase or a decrease in the intensity along both directions at the same time. For low-diffraction efficiency gratings we may neglect phase coupling and assume that the effects of simultaneous amplitude and index of refraction gratings (both in and out of phase from the recording pattern of fringes) are not coupled [Frejlich et al., 1992], so that the expression for the output irradiance in Equation (8.31) can be approximately written as pffiffiffiffiffi pffiffiffiffiffi pffiffiffiffiffi IS  j^sS0 eicd sin t þ i^r R0 ZP eifP þ i^sR0 Zn eifn þ ^sR0 Za eifa j2

ð8:47Þ

pffiffiffiffiffi pffiffiffiffiffi pffiffiffiffiffi IR ¼ j^r S0 eicd sin t þ i^sS0 ZP eifP þ i^r S0 Zn eifn þ ^r S0 Za eifa j2

ð8:48Þ

where the subindices P, n, and a indicate the diffraction efficiency and the corresponding holographic phase shift of a photorefractive index of refraction grating, a nonphotorefractive index grating, and an absorption grating, respectively. Similarly we find for the other direction

Developing Equations (8.47) and (8.48) in terms of t and limiting ourselves to the first and second harmonic terms, we get the final expressions for these terms measured along both directions behind the sample: pffiffiffiffiffi pffiffiffiffiffi pffiffiffiffiffi IS ¼ 2cd jS0 jjR0 jð Za sin fa þ ^s  ^r ZP cos fP þ Zn cos fn Þ pffiffiffiffiffi pffiffiffiffiffi pffiffiffiffiffi IR ¼ 2cd jS0 jjR0 jð Za sin fa  ^s  ^r ZP cos fP  Zn cos fn Þ

IS2 ¼ IR2 ¼

c2d pffiffiffiffiffi pffiffiffiffiffi pffiffiffiffiffi jS0 jjR0 jð Za cos fa  ^s  ^r ZP sin fP  Zn sin fn Þ 2 c2d pffiffiffiffiffi pffiffiffiffiffi pffiffiffiffiffi jS0 jjR0 jð Za cos fa þ ^s  ^r ZP sin fP þ Zn sin fn Þ 2

ð8:49Þ ð8:50Þ ð8:51Þ ð8:52Þ

All cross terms (corresponding to twice-diffracted beams) above were neglected because diffraction efficiencies involved are assumed to be very small. The difference  and the sum  between corresponding terms above leads to pffiffiffiffiffi pffiffiffiffiffi   IS  IR ¼ 4cd jS0 jjR0 jð^s  ^r ZP cos fP þ Zn cos fn Þ I pffiffiffiffiffi pffiffiffiffiffi 2 I  IS2  IR2 ¼ c2d jS0 jjR0 jð^s  ^r ZP sin fP þ Zn sin fn Þ pffiffiffiffiffi I  IS þ IR ¼ 4cd jS0 jjR0 j Za sin fa pffiffiffiffiffi I2  IS2 þ IR2 ¼ c2d jS0 jjR0 j Za cos fa

ð8:53Þ ð8:54Þ ð8:55Þ ð8:56Þ

From the set of equations above we see that the difference terms are a function of index of refraction gratings only (photorefractive and nonphotorefractive), whereas the addition terms depend only on amplitude gratings. This is a consequence of the

PHASE MODULATION TECHNIQUES

203

difference in symmetry properties of these kind of gratings as already stated in Section 4.2.1. Such properties are very important for separately measuring amplitude and phase effects in a continuous nondestructive way. S2WM also allows operation of a self-stabilized holographic setup just using amplitude or just phase effects, or stabilizing the setup on the amplitude grating while following the evolution of the phase grating recording, etc. It is possible to still further specialize Equations (8.53)–(8.56) for the case of low-diffraction efficiency coefficients when phase-coupling effects can be neglected: In this case, photorefractive gratings (in the absence of externally applied field) are nonlocalized with fP  p=2. Instead, the other two gratings, arising from either the photoactive trap modulation or direct modulation of the material by the action of light, are usually localized gratings with fn ¼ fa  0 (or p). Substituting these values into Equations (8.53)–(8.56) we get pffiffiffiffiffi   4cd jS0 jjR0 j Zn I

2 I



pffiffiffiffiffi c2d jS0 jjR0 j^s  ^r ZP

pffiffiffiffiffi I ¼ 4cd jS0 jjR0 j Za sin fa  0 pffiffiffiffiffi I2  c2d jS0 jjR0 j Za

ð8:57Þ ð8:58Þ ð8:59Þ ð8:60Þ

A similar S2WM technique was first reported by Boothroyd and co-workers [Boothroyd et al., 1989], which is based on the symmetric-antisymmetric effects of phase and absorption gratings for a running hologram moving along one direction and along the opposite one. 8.2.4.1 Amplitude and Phase Effects in GaAs. GaAs crystals (see Section 1.3.1) have an electro-optic tensor identical to that of sillenites, but they do not have optical activity [Walsh et al., 1987]. This fact largely simplifies experiments with these materials. In sillenites, GaAs, and similar materials, photorefractive index of refraction effects can be distinguished from amplitude and nonphotorefractive effects of any other nature by the fact that the former exhibits anisotropic diffraction whereas the latter do not. For the (110)-crystal cut with the [001]-axis orthogonal to ~ and the incident beam polarized the incident plane containing the grating vector K, along the [001]-axis, as shown in Figure 8.13, the photorefractive diffracted output beam is orthogonally polarized to the transmitted beam [Marrakchi et al., 1986]. The amplitude and nonphotorefractive index of refraction gratings, if any, have parallelpolarized diffracted and transmitted beams. The measurement of polarization properties at the output, therefore, may allow one to distinguish between both kinds of gratings. This technique, however, will not enable one to distinguish amplitude effects from nonphotorefractive index of refraction modulation, such as that due to photorefractive trap center modulation, or any other effect leading to a modulation in electrical polarizability not related to electro-optic properties. S2WM instead is specifically concerned with symmetry in energy exchange and may therefore allow one to separate amplitude from phase effects, no matter the origin of the latter. The two-wave mixing experiment indicated in Figure 8.15 to be carried out on a GaAs

204

HOLOGRAPHIC TECHNIQUES

Figure 8.15. Two-wave mixing experiment in a photorefractive GaAs intrinsic crystal with incident and transmitted beams polarized along the [001]-axis of the GaAs crystal. The polarization of the diffracted beams (the shorter arrows) at the crystal output depends on the nature of the diffraction grating in the GaAs. A polarizer (P) and two photodetectors with a summation/substraction device produce the adequate electric signal for TWM processing.

sample could allow the use of S2WM and polarization properties to detect the simultaneous presence of amplitude and photorefractive phase effects as already reported elsewhere [Bian et al., 1995]. We shall here only report the data measured along one single direction behind a polarizer placed at the sample output as depicted in the schema of Figure 8.16, where the first harmonic in Equation (8.49) becomes pffiffiffiffiffi pffiffiffiffiffi pffiffiffiffiffi IS ¼ 2cd jS0 jjR0 jðsin2 g Za sin 0 þ sin g cos g ZP cos p=2 þ sin2 g Zn cos 0Þ ð8:61Þ pffiffiffiffiffi IS ¼ 2cd jS0 jjR0 j sin2 g Zn ð8:62Þ

and the second harmonic in Equation (8.51) becomes IS2 ¼ IS2 ¼

c2d pffiffiffiffiffi pffiffiffiffiffi pffiffiffiffiffi jS0 jjR0 jðsin2 g Za cos 0  sin g cos g ZP sin p=2  sin2 g Zn sin 0Þ ð8:63Þ 2 c2d pffiffiffiffiffi pffiffiffiffiffi jS0 jjR0 jðsin2 g Za  sin g cos g ZP Þ 2

ð8:64Þ

where g is the angle between the polarization direction transmitted by the polarizer behind the sample and the input beams polarization along the [110]-axis of the [001]

[001]

g

K

[110] [110]

Figure 8.16. Two-wave mixing experiment in a photorefractive GaAs intrinsic crystal as for Fig. 8.13 but with a polarizer at the crystal output where its transmitted polarization direction makes an angle g with the crystal axis [110].

PHASE MODULATION TECHNIQUES

0.20

0.05 0.04

0.15

2Ω

0.03 0.10 IΩ

0.02

IΩ (au)

2Ω

(au)

I

I

205

0.05

0.01

0

0 0

30

60

90

γ (degrees) 

Figure 8.17. Plot of the first I (Eq.(8.62)) and second I 2 (Eq.(8.64)) harmonic terms after fitting the corresponding actual data in GaAs as a function of the polarization angle g behind the crystal (see Fig. 8.16) during steady-state multiple nature holograms recorded with l ¼ 1064 nm and K ¼ 2:1 mm1 .

GaAs. The experimental first and second harmonics were measured on an intrinsic photorefractive GaAs crystal, in the setup schematically depicted in Figure 8.16, as a function of the polarization angle g. These data are accurately fit by Equations (8.62) and (8.64), respectively, and the latter fittings are reproduced in Figure 8.17. From these fittings were obtained the values of all three different kinds of gratings simultaneously present in this material after holographic recording using pffiffiffiffiffi l ¼ 1064 nm light: a photorefracive grating with ZP ¼ 1%, a nonphotorefractive pffiffiffiffiffi index of refraction grating with Zn ¼ 0:20%, and an absorption grating with pffiffiffiffiffi Za ¼ 0:05%. More details can be found in the literature [Bian et al., 1995]. In this case it was not necessary to use the S2WM technique to separately measure the three different gratings because the conditions were particularly simple, but in more complex cases S2WM should be required. The different effect of index of refraction and absorption gratings on the first and second harmonic terms in phase-modulated TWM has been already reported for separate detection of these two effects in photorefractive quantum wells [Shimura et al., 2004; Ramos-Garcia et al., 2003]. The symmetric/asymmetric diffraction effects in TWM underlying the S2WM technique have already been used to assess a purely (or largely predominant) photorefractive nature to the dark buildup of a relatively large grating in BaTiO3 after holographic recording with l ¼ 488 nm and switching off the recording beams [Buse et al., 1993]. This technique was also used [Freschi and Frejlich, 1994] to separately detect the relatively weak absorption grating (compared to the simultaneosuly recorded photorefractive grating) in Fe-doped LiNbO3 and use this grating as a reference for self-stabilized recording of a photorefractive hologram with an index of refraction

206

HOLOGRAPHIC TECHNIQUES

modulation exceeding the value for Z ¼ 1, which would have been impossible to achieve for self-stabilized recording on the photorefractive hologram itself, because of the upper Z ¼ 1 limitation of this recording technique, as already discussed in Section 6.3.2. The use of S2WM has also enabled the detection of the simultaneous presence of amplitude effects and photorefractive hole–electron competition in an undoped Bi12 TiO20 crystal at rather low irradiance level (200 to 300 mW=cm2 ) [Frejlich and Garcia, 1992]. 8.2.5

Running Holograms

Running holograms were discussed theoretically in Section 3.4. These kinds of holograms have been extensively applied to image processing and other applications but hardly used for material characterization, probably because of the inherent complexity of their nature and the number of independent parameters characterizing the process. Here we shall focus on the possibilities of running holograms as a tool for materials characterization. To study these moving holograms experimentally it is very convenient to use phase modulation techniques that allow measurement of several properties of the hologram without perturbating the process. The adequate setup is schematically depicted in Figure 8.18, where the first and second temporal harmonic signals are used to compute at the same time the diffraction efficiency Z and the phase shift j between the beams behind the sample. The oscillator produces the dither signal of frequency  that is used for detection, and the high-voltage source is used to feed a sawtooth electric signal to the piezo-mirror so as to produce a moving pattern of fringes onto the sample. It has been already reported [Brost et al., 1998; Webb and Solymar, 1991] that bulk optical absorption and optical activity strongly influence the amplitude gain in running holograms. It was also shown [Shamonina et al., 1997] that the typical asymmetric

Figure 8.18. Experimental setup for the generation and measurement of running holograms.

PHASE MODULATION TECHNIQUES

207

shape of the diffraction efficiency vs: velocity curve is mainly due to the bulk absorption and the higher spatial harmonic components whenever present. For the purpose of materials characterization, however, it is interesting to use a low value for the pattern of fringes modulation coefficient in order to fulfill the first spatial harmonic approximation [Hall et al., 1985] that leads to a comparatively simple set of equations, as reported in Section 3.3, describing the wave coupling in the crystal volume that facilitates the moving grating analysis [Brost et al., 1994]. It is also convenient to use a relatively thin sample in order to be able to neglect the combined effect of optical activity (for the case of sillenites) and birefringence [Shamonina et al., 1997]. We shall show some results from experiments that were carried out on a thin photorefractive Bi12 TiO20 (BTO) crystal and measure the diffraction efficiency and the output beam phase shift as a function of the pattern of fringes movement. We show that the analysis of the experimental results might lead to erroneous conclusions because of the presence of even low concentrations of minoritary hole photoactive centers in this sample. Following the considerations in the paper by Shamonina et al. [Shamonina et al., 1997] we shall neglect birefringence and optical activity (as we are concerned with a thin BTO sample), take into consideration bulk absorption (which is particularly large in BTO for the 514.5-nm wavelength), and consider self-diffraction effects. Charge carrier excitation and transport in photorefractives are complex phenomena sometimes involving more than one type of photoactive center [Valley, 1983], centers with more than one valence state [Buse, 1997], and hole–electron competition [Strohkendl et al., 1986; Pauliat et al., 1987]. We shall here assume the simplest ‘‘one center and one charge carrier’’ model where the possible presence of holes will be treated as a perturbation. Experimental conditions will be looked for to minimize such a perturbation. In these conditions the steady-state space-charge electric field is described in Section 3.4. In the presence of self-diffraction the diffraction efficiency is described by Equation (4.117), which for a sufficiently small crystal thickness d simplifies to Equation (4.118)  2  2 ! d gd 2b 2 þ jmj ¼ Zm 4 4 1 þ b2 The phase shift j between the transmitted and diffracted beams behind the sample is reported in Equation (4.122). In the presence of bulk absorption, however, the expressions for Z and tan j should be accordingly modified as described in Section 4.2.2.3. We shall here focus on sillenites that are known to exhibit hole conductivity in the dark and electron conductivity under the action of light [Gu¨nter and Huignard, 1988]. It is not possible to exclude some degree of hole–electron competition in the deep trap level, too, at least for some samples [Pauliat et al., 1987; Frejlich and Garcia, 1992]. As shown below even a comparatively small density of hole photoactive centers may considerably affect the shape of the diffraction efficiency curve. To mathematically simulate this effect we assume the simple model in Section 3.4.1.1 where independent photoactive deep centers for electrons and for holes are present without any interaction among them, except for the fact that at each

208

HOLOGRAPHIC TECHNIQUES

point in the crystal volume the charge density is built up by the superposition of both holes and electrons. For the sake of simplicity we shall here assume that the effective st can be separately computed for electrons and for holes, as formulated in field Esc Equation (3.115), using their corresponding assumed material parameters (LD , ls etc.). The integrals in Equations (4.138) and (4.137) are computed for electrons (ge e ) and for holes ( h ), and the expressions for the diffraction efficiency in and  gh and  Equation (4.139) and phase shift in Equation (4.140) can be thus computed separately for the electrons, for the holes and for the superimposed hole-electron e þ  h ). condition (using the quantities  ge þ gh and  Such a simulation for Z and tan j was computed assuming some typical parameters for the electron photoactive centers in BTO and assuming a much lower quantum efficiency and concentration for holes: 100-fold lower quantum efficiency and approximately 25-fold lower concentration representing a roughly 5-fold higher Debye length, with the condition LD > ls . The result is depicted in Figure 8.19 for

0.15

h

0.01

0.005

0 –5

0

5 Kv (rad/s)

15

5 Kv (rad/s)

15

tan j

4

2

–5

0

Figure 8.19. Diffraction efficiency (top) and tan j (bottom) as a function of Kv computed with the experimental parameters K ¼ 2:55 mm1 , a ¼ 11:65 cm1 , xE0 ¼ 4:55 kV=cm, and I0 ¼ 17:5 mW=cm2 . The material parameters are LD ¼ 0:22 mm, ls ¼ 0:03 mm, b2 ¼ 40, and  ¼ 0:4 for electrons (continuous curve) and LDh ¼ 0:16 mm, lsh ¼ 0:15 mm and h ¼ 0:004 for holes (dashed curve). The resulting electron-to-hole diffraction efficiency ratio at Kv ¼ 0 is Ze =Zh  2:4. The thick continuous curve is the overall result. Reproduced from [de Oliveira and Frejlich, 2001b]

PHASE MODULATION TECHNIQUES

209

Figure 8.20. Diffraction efficiency (top) and tan j (bottom) as a function of Kv computed with K ¼ 11:3 mm1 . All other experimental and material parameters and the meaning of thick, thin, and dashed curves are the same as for Fig. 8.19, with Ze =Zh  17 for Kv ¼ 0. Reproduced from [de Oliveira and Frejlich, 2001b]

K ¼ 2:55 mm1 and in Figure 8.20 for K ¼ 11:3 mm1, which were the values for K in our experiment. In the latter figure we see that even such a small hole photoactive center concentration (with an electron-to-hole hologram diffraction efficiency ratio of Ze =Zh  17 at Kv ¼ 0) is able to produce a pronounced effect on the shape of the overall Z. Something similar is seen in Figure 8.19 for K ¼ 2:55 mm1, except that here the effect of holes is not so weak: Ze =Zh  2:4. In both cases the effect of holes becomes weaker as we go farther away from Kv ¼ 0 along increasing Kv. The different effect of hole–electron competition for different values of K is due to the fact that for small K values it is K 2 l2s 1 both for holes and for electrons. This means that the hologram buildup is not limited by its respective photoactive center density (see Eq.(3.47)) so that similarly strong holograms result both for holes and for electrons as seen in Figure 8.19. For a larger value of K instead, it should still be K 2 l2s 1 for electrons but not for holes, which may approach saturation and develop a weaker hologram as seen in Figure 8.20. The effect of holes on the tan j as seen in Figures 8.19 and 8.20 is much less relevant (except near Kv ¼ 0) than for Z. As for the latter, the effect of hole–electron competition becomes negligible for

210

HOLOGRAPHIC TECHNIQUES

Kv > 0 sufficiently far from the origin. It is also interesting to note that it is not reliable to compute LD for the case where K 2 L2D  1 because in these conditions (the movement of charge carriers being much larger than the fringes period) there is a kind of randomization of the charge carriers in the volume of the sample, and the movement of the hologram is no longer dependent on the value of this parameter. This conclusion is supported by numerical simulations, too: Some simulations have shown (although the physical reason is not yet evident) that tan j is also not much dependent on LD whatever the value of K. It is clear that, with the restictions discussed above, both Z and tan j data should be used to compute the material parameters, mainly for Kv > 0 to minimize eventual hole perturbations. A two-wave mixing running hologram experiment was carried out using the 514.5-nm-wavelength laser line and a 2.05-mm-thick nominally undoped photorefractive Bi12 TiO20 crystal growth by the TSSG technique [Prokofiev et al., 1995] (a ¼ 11:65 cm1 ) with the [001] crystal axis perpendicular to the incidence plane ~ parallel to the [110]-axis in a configuration similar and the hologram wave vector K to the one depicted in Figure 8.13 for GaAs. An external electric field is applied to ~ by means of silver-painted electrodes. The interfering incident the crystal along K beams are expanded and collimated so that uniform irradiances over the sample (less than 10% variation) result. The beams are linearly polarized with their polarizations selected to be at 45 to the [001]-axis at the midcrystal plane, in which case the transmitted and diffracted beams behind the crystal are parallel polarized [Mallick and Roue`de, 1987] as illustrated in Figure 5.5. The setup is adjusted so that energy is transferred from the signal (IS ) to the pump (IR ) beam in order to characterize a negative gain process. Negative gain produces a lower Z but was shown [de Oliveira and Frejlich, 2001a] to lead to more stable experiments that facilitate the measurement. A piezoelectric-supported mirror (PZT) fed with an electric ramp signal of adjustable slope produces a detunning Kv on one of the interfering beams. To produce negative Kv values the sense of the applied field E0 is just reversed. Data measured at Kv ¼ 0 were verified not to depend on the direction (direct or reversed) of the applied field E0 . The diffraction efficiency Z and output phase shift j can be written in terms of the first and second harmonic terms as

Z¼ tan j ¼

1 IS IR ðKd Þ2

2  4

V  A 2v d KPZT

 V  A2 KPZT vd 2  4 V A





2

þ

2V 2  Þ2 A2 ðv d KPZT

!2 3 5

ð8:65Þ ð8:66Þ

where the parameters are defined in Section 6.2. Figures 8.21 and 8.22 show typical experimental results (spots) for Z and tan j, respectively, as functions of Kv for K ¼ 2:55 mm1 , b2 ¼ 41:2, and I0 ¼ 22:5 mW=cm2 . From Figure 8.21 we got LD ¼ 0:14 mm and  ¼ 0:45, whereas from Figure 8.22 we got  ¼ 0:41. Note that the large spots for Kv  0 were actually used for fitting whereas the small spots (Kv < 0 side) are just included to appreciate their differences with the theoretical fit.

PHASE MODULATION TECHNIQUES

211

h

0.01

0.005 –5

0

5 Kv (rad/s)

10

15

Figure 8.21. Diffraction efficiency Z experimental data (spots) as a function of detunning Kv and best theoretical fit (continuous curve) to Eq.(4.139) for x ¼ 0:96, K ¼ 2:55 mm1 , E0 ¼ 7:3 kV=cm, b2 ¼ 41:2, and I0 ¼ 22:5 mW=cm2 . The resulting best-fitting parameters are LD ¼ 0:14 mm and  ¼ 0:45. Data for Kv < 0 (small spots) were not used for the fit. Reproduced from [de Oliveira and Frejlich, 2001b]

It was not possible to fit the whole experimental data (small and large spots) set to theory in any of our experiments. It is well known that the effective field inside the sample may be different from its nominal value E0 [Grunnet-Jepsen et al., 1995; Freschi et al., 1997] so that the latter should be substituted by xE0 everywhere, with x being an experimentally evaluated effective field coefficient as discussed in Appendix C. In this work x was computed from an auxiliary experiment where Z was measured as a function of E0 at v ¼ 0 and the theory was fit to these data to get the corresponding x values in Table 8.4. Other experiments were carried out to measure Z and tan j for the same K ¼ 2:55 mm1 and also for K ¼ 11:3 mm1. The theoretical fitting to the data from each one of these experiments allows one to obtain some material parameters that are

4

tan j

3

2 1

0

5

10

Kv (rad/s)

Figure 8.22. tan j experimental data (spots) as a function of Kv for the same conditions as in Fig. 8.21, with Eq. (4.140) (continuous curve) being fit to data (large spots) and the resulting parameter being  ¼ 0:41. Data for Kv < 0 (small spots) are also not considered for the fit here. Reproduced from [de Oliveira and Frejlich, 2001b]

212

HOLOGRAPHIC TECHNIQUES

TABLE 8.4. Running Hologram: Experimental Results for Undoped BTO and l ¼ 514:5 nm Diffraction Efficiency K mm1 2.55 2.55 11.3 11.3

I0 mW=cm2 22.5 22.5 24.3 19.4

b2 41.2 39 30.8 26.7

LD mm 0.14 0.14 — —

ls mm — — 0.015 0.028

Phase shift

 mm

LD

ls mm

0.45 0.63 — —

— — — —

— — 0.032 0.036

 0.41 0.48 0.38 0.30

Zðv ¼ 0Þ x 0.96 0.78 0.90 0.90

displayed in Table 8.4 together with their corresponding x values. The average results for this sample were LD  0:14 mm, ls ¼ 0:03  0:01 mm, and  ¼ 0:4  0:1. 8.2.6

Photo-Electromotive-Force Techniques

The photo-electromotive-force (photo-emf) is produced in photoconductors where, under the action of light, a distribution of free charge carriers in the extended states (conduction and/or valence band) is produced and an electric charge distribution and associated space-charge electric field are built up as described in Chapter 3. If the pattern of light is moved faster than the response of the space-charge field but slower than the lifetime of free charges in the extended states, the free charges will follow the movement but the space-charge field will not. In this way the free charges will not be in equilibrium any more and a current will appear. The photo-emf is not necessarily concerned with holography in the sense that it does not require the recording of an index of refraction modulation hologram at all but it does rely on the projection of an interference (or holographic) pattern of fringes in the sample volume to establish a space-charge electric field modulation instead. It is described in this section just because it is produced by the interference of two beams. The so-called holographic photo-emf is directly arising from phase modulation in an interferometric setup. 8.2.6.1 Holographic Photo-emf. In the absence of an externally applied electric field on a nonphotovoltaic photorefractive material, the recorded space-charge field modulation and the free charge carrier distribution in the conduction band are mutually p=2-phase shifted so that the electric current averaged along the interelectrode distance is zero. However, if the pattern of fringes is moved along the grating wave vector, the above-mentioned phase shift is modified and a current may appear in the form of a pulse or an ac or dc signal, according to the way the pattern of fringes ~ along the interelecis moved. Let us assume a sinusoidal pattern of fringes with K trode coordinate x, as described by Equation (3.6) I ¼ I0 ð1 þ jmj cosðKx þ fÞÞ ¼ I0 þ ðI0 =2Þ½meiKx þ m eiKx 

PHASE MODULATION TECHNIQUES

213

~ with amplitude , so that m above with the fringes sinusoidally vibrating along K, should be substituted as follows m ) mðtÞ ¼ jmjeif eiK sin t

ð8:67Þ

Similarly to the development in Section 8.2.3, the following relation holds eiK sin t ¼

þ1 X

l¼1

Jl ðKÞeilt

ð8:68Þ

where Jl ðÞ is the ordinary Bessel function of order l. In this case we should write the modulation in Equation (8.67) as mðtÞ ¼ jmjeif

þ1 X

l¼1

Jl ðKÞeilt

ð8:69Þ

Let us assume the first spatial harmonic approximation (see Section 3.3) that allows one to consider the linearized expressions in Equations (3.23)–(3.25) N ðx; tÞ ¼ N 0 þ N 0 =2½aðtÞeiKx þ a ðtÞeiKx 

NDþ ðx; tÞ ¼ NDþ þ NDþ =2½AðtÞeiKx þ A ðtÞeiKx 

Eðx; tÞ ¼ E0 þ ð1=2Þ½Esc ðtÞeiKx þ Esc ðtÞeiKx 

The equations above substituted into Equations (2.5)–(2.10) were shown to lead to the following relations for Esc ðtÞ in Equation (3.51), for aðtÞ in Equation (3.36), and for AðtÞ in Equation (3.38) as follows tsc

@Esc ðtÞ þ Esc ðtÞ ¼ mðtÞ Eeff @t e=q Esc ðtÞiKmt þ mðtÞ sI0 =ðsI0 þ bÞ  AðtÞND =ðND  NDþ Þ aðtÞ ¼ 1 þ ie=q KtmE0 þ K 2 Dt þ iKEe0 Esc ðtÞ  qND AðtÞ

From the last two equations above and Equation (8.69) we get aðtÞ ¼ i

þ1 K 2 L2D  K 2 l2s Esc ðtÞ jmjeif X þ Jl ðKÞeilt 1 þ K 2 L2D ED 1 þ K 2 L2D l¼1

ð8:70Þ

where we have assumed that sI0  b and E0 ¼ 0. On the other hand, the solution of the differential equation above for Esc ðtÞ is Esc ðtÞ ¼ jmjEeff

þ1 X Jl ðKÞeilt 1 þ iltsc l¼1

ð8:71Þ

214

HOLOGRAPHIC TECHNIQUES

where we have assumed that the electric grating has been recorded for the pattern of fringes fixed at f ¼ 0. Substituting Equation (8.71) into Equation (8.70) we get the expression aðtÞ ¼

þ1 þ1 jmjeif X K 2 L2D  K 2 l2s Eeff X Jl ðKÞeilt Jl ðKÞeilt  i jmj 2 2 2 2 ED l¼1 1 þ iltsc 1 þ K LD l¼1 1 þ K LD

ð8:72Þ

Its is possible to show [Stepanov, 2001] that the total current density flowing through ~ can be written as the electrodes at the ends of the sample, along K, 1 jðtÞ ¼ L

ZL

emN ðx; tÞEsc ðx; tÞdx

0

where L is the interelectrode distance. In our special case the formula above simplifies to   N0 Esc ðtÞ Esc ðtÞ aðtÞ þ aðtÞ jðtÞ ¼ em 2 2 2

ð8:73Þ

We should also write aðtÞ and Esc ðtÞ in Equations (8.72) and (8.71) in terms of their harmonic components in  as follows aðtÞ ¼ Esc ðtÞ ¼

þ1 X

al eilt

ð8:74Þ

l ilt Esc e

ð8:75Þ

l¼1 þ1 X

l¼1

with the following first few parameters for aðtÞ a0 ¼

jmjeif K 2 L2D  K 2 l2s Eeff J0 ðKÞ jmj J ðKÞ  i 0 2 2 2 2 ED 1 þ K LD 1 þ K LD

ð8:76Þ

a ¼

jmjeif K 2 L2D  K 2 l2s Eeff J1 ðKÞ J ðKÞ  i jmj 1 2 ED 1 þ itsc 1 þ K 2 LD 1 þ K 2 L2D

ð8:77Þ

a2 ¼

jmjeif K 2 L2D  K 2 l2s Eeff J2 ðKÞ jmj J2 ðKÞ  i 2 2 ED 1 þ i2tsc 1 þ K 2 L2D 1 þ K LD

ð8:78Þ

al ¼

jmjeif K 2 L2D  K 2 l2s Eeff Jl ðKÞ J ðKÞ  i jmj l 2 2 2 2 ED 1 þ iltsc 1 þ K LD 1 þ K LD

ð8:79Þ

PHASE MODULATION TECHNIQUES

215

and for Esc ðtÞ 0 Esc ¼ jmjEeff J0 ðKÞ  Esc ¼ jmjEeff 2 ¼ jmjEeff Esc l ¼ jmjEeff Esc

J1 ðKÞ 1 þ itsc

J2 ðKÞ 1 þ i2tsc

Jl ðKÞ 1 þ iltsc

ð8:80Þ ð8:81Þ ð8:82Þ ð8:83Þ

The expression in Equation (8.73) can be also written in terms of its harmonics, in the same way as already done for E, N , and other parameters, as jðtÞ ¼

j0 j it j2 i2t þ e þ e þ . . . þ cc 2 2 2

ð8:84Þ

where the few first coefficients are s0 0 0 0   2 2 ða ðEsc Þ þ ða0 Þ Esc þ a ðEsc Þ þ ða Þ Esc þ a2 ðEsc Þ þ ða2 Þ Esc þ . . .Þ 2 ð8:85Þ s 0   0 0  Þ þ ða0 Þ Esc þ a ðEsc Þ þ ða Þ Esc þ a2 ðEsc Þ j ¼ ða0 ðEsc 2 j0 ¼

2 2  þ a ðEsc Þ þ ða Þ Esc þ ða2 Þ Esc . . .Þ

j2 ¼

s0 2 2   ða0 ðEsc Þ þ ða0 Þ Esc þ a ðEsc Þ þ ða Þ Esc 2 0 0 þ ða2 Þ ðEsc Þ þ a2 ðEsc Þ . . .Þ

ð8:86Þ

ð8:87Þ

where s0 ¼ emN 0 . After substituting and rearranging terms we get the following expression for the dc component j0 ¼ 

s0 jmj J0 ðKÞ2 J1 ðKÞ2 Eeff m þ Eeff m ðE m þ E mÞ  s jmj eff 0 eff 2 1 þ K 2 L2D 1 þ K 2 L2D 1 þ 2 t2sc

 s0 jmj

J2 ðKÞ2 Eeff m þ Eeff m 2 2 2 2 1 þ K LD 1 þ 4 tsc

ð8:88Þ

as well as for the first harmonic j ¼

s0 jmj J0 ðKÞJ1 ðKÞ itsc ðEeff m  Eeff mÞ 2 1 þ K 2 L2D 1 þ itsc þ

s0 jmj J1 ðKÞJ2 ðKÞ i3tsc ðEeff m  Eeff mÞ 2 2 2 1 þ K LD ð1  itsc Þð1 þ i2tsc Þ

ð8:89Þ

216

HOLOGRAPHIC TECHNIQUES

and for the second harmonic j2 ¼  s0

jmj J0 ðKÞJ2 ðKÞ ðEeff m þ Eeff mÞ 2 1 þ K 2 L2D



s0 jmj J0 ðKÞJ2 ðKÞ Eeff m þ Eeff m 2 2 2 1 þ i2tsc 1 þ K LD

þ

s0 jmj J1 ðKÞ2 Eeff m þ Eeff m 2 2 2 1 þ K LD 1 þ itsc

þ is0 jmjJ0 ðKÞJ2 ðKÞ i

K 2 L2D  K 2 l2s jEeff j2 m  m ED 1 þ i2tsc 1 þ K 2 L2D

s0 jmj K 2 L2D  K 2 l2s jEeff j2 m  m J1 ðKÞ2 ED ð1 þ itsc Þ2 2 1 þ K 2 L2D

ð8:90Þ

In the absence of external field (E0 ¼ 0) it is Eeff ¼

iED 1 þ K 2 l2s

which substituted in the equations above lead to the following simplified expression for the dc term: j0 ¼  s0 jmj2

J0 ðKÞ2 ED sin f ð1 þ K 2 L2D Þð1 þ K 2 l2s Þ

 2s0 jmj2

J1 ðKÞ2 ED sin f 2 2 2 2 ð1 þ K LD Þð1 þ K ls Þ 1 þ 2 t2sc

 2s0 jmj2

J2 ðKÞ2 ED sin f ð1 þ K 2 L2D Þð1 þ K 2 l2s Þ 1 þ 42 t2sc

ð8:91Þ

as well as for the first harmonic j ¼  s0 jmj2  s0 jmj2

J0 ðKÞJ1 ðKÞ tsc ED cos f 2 2 1 þ K LD 1 þ itsc 1 þ K 2 l2s J1 ðKÞJ2 ðKÞ 3tsc ED cos f 2 2 1 þ K LD ð1  itsc Þð1 þ i2tsc Þ 1 þ K 2 l2s

ð8:92Þ

PHASE MODULATION TECHNIQUES

217

and for the second harmonic term

j2 ¼  s0 jmj2

J0 ðKÞJ2 ðKÞ ED sin f 1 þ K 2 L2D 1 þ K 2 l2s

 s0 jmj2

J0 ðKÞJ2 ðKÞ ED 1 sin f 1 þ K 2 L2D 1 þ K 2 l2s 1 þ i2tsc

 s0 jmj2

J1 ðKÞ2 ED 1 sin f 1 þ K 2 L2D 1 þ K 2 l2s 1 þ itsc

 2s0 jmj2 J0 ðKÞJ2 ðKÞ þ s0 jmj2 J1 ðKÞ2

K 2 L2D  K 2 l2s ED 1 sin f 1 þ K 2 L2D 1 þ K 2 l2s 1 þ i2tsc

K 2 L2D  K 2 l2s ED 1 sin f 1 þ K 2 L2D 1 þ K 2 l2s ð1 þ itsc Þ2

ð8:93Þ

To understand the meaning of Equations (8.91)–(8.93) it is necessary to keep in mind that in the absence of perturbations it should be f ¼ 0, in which case the dc and the second harmonic are null. Note also that s0 depends on the irradiance and may therefore vary along the sample thickness in absorving materials. The first harmonic can be also written in its binomial form as follows:  j ¼ j R þ ijI

j R ¼ tsc J1 ðKÞ 

ð8:94Þ ½2J0 ðKÞ þ 3J2 ðKÞð1 þ 22 t2sc Þ  J0 ðKÞ ð1 þ 2 t2sc Þð1 þ 42 t2sc Þ

s0 jmj2 ED cos f ð1 þ K 2 L2D Þð1 þ K 2 l2s Þ

2 2 j I ¼  tsc J1 ðKÞ

ð8:95Þ

3J2 ðKÞ þ J0 ðKÞð1 þ 42 t2sc Þ s0 jmj2 ED cos f ð1 þ 2 t2sc Þð1 þ 42 t2sc Þ ð1 þ K 2 L2D Þð1 þ K 2 l2s Þ ð8:96Þ

The real part of the first harmonic is j it ðj Þ it j þ ðj Þ j  ðj Þ e cos t þ i sin t e þ ¼ 2 2 2 2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2   2 ¼ j cos t  j sin t ¼ ðj R Þ þ ðjI Þ cosðt þ j Þ R I


ð8:97Þ

218

HOLOGRAPHIC TECHNIQUES

(au) 0.3 0.25 0.2 0.15 0.1 0.05 1

2

3

4

K∆ (rad)

Figure 8.23. jj j (in arbitrary units) as a function of the vibration amplitude K (in radians) for tsc ¼ 1000 rad, 5 rad, 1 rad, and 0.1 rad, from the finest to the coarsest dashed curves, respectively, always without externally applied field.

The actually measured value (using a lock-in amplifier tuned to  frequency, for example) is the amplitude of the above real signal, that is, qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2  2 ð8:98Þ j j  j ¼ ð j R Þ þ ð jI Þ

The latter theoretical equation is plotted in Figure 8.23, as a function of K, for some particular conditions. The expression of jj j is particularly interesting for the case of low tsc (tsc 1), where it assumes the form 2 j j 0 j  Ajmj ED cos f J1 ðKÞðJ0 ðKÞ þ 3J2 ðKÞÞ

ð8:99Þ

with A

ð1 þ

tsc s0 ¼ Ee0

s0 tsc 2 K L2D Þð1

þ

K 2 l2s Þ



Ee0 ð1 þ K 2 l2s Þ2

1 þ K 2 L2D  iKLE 1 þ K 2 L2D  Ee 0 1 þ K 2 l2s  iKlE 1 þ K 2 l2s

ð8:100Þ ð8:101Þ

where the approximate relations in the right side above are for no externally applied field, E0 ¼ 0. Note that A in Equation (8.100) is independent of the irradiance and the light absorption, and consequently the expression in Equation (8.99) is also constant along the sample’s thickness in these conditions. For the opposite limit condition tsc  1 we also get a simplified formulation of j j j A jmj2 ED cos f J0 ðKÞJ1 ðKÞ ð8:102Þ j j 1j  tsc that does not depend any more on  as seen in Figure 8.24.

PHASE MODULATION TECHNIQUES

219

(au) 0.34 0.32 2

4

6

8

10

Ω

SC

(rad)

0.28 0.26 0.24 0.22

Figure 8.24. Computed jj j (in arbitrary units) as a function of tsc in rad for a fixed amplitude K ¼ 1:1 rad.

Holographic photo-emf, in a two-wave interferometric setup [Trofimov and Stepanov, 1986; Petrov et al., 1986, 1990; Mosquera and Frejlich, 2002], allows transducing the phase modulation in one of the interfering beams into an oscillating pattern of fringes projected onto a suitable photorefractive or just a plain photoconductive material. This oscillation produces an alternating current in the photoconductive sample that depends on the amplitude of the pattern of fringes oscillation, among other parameters, and may be therefore used, with adequate scaling, to measure the longitudinal oscillation amplitude of the device producing the phase modulation. This technique is a self-calibrating one because the size of the signal is easily related to the spatial period of the pattern of fringes that is straightforwardly computed from the geometry of the setup. Self-calibration has facilitated the application of this technique to widely different fields [Wang et al., 2002; Murfin and Dewhurst, 2002], including the measurement of mechanical vibration amplitude [Stepanov et al., 1990] and materials characterization [Sochava et al., 1993, 1994; Korneev et al., 1999]. We shall here focus on this latter point because it is the subject of this part of the book. The experimental setup is schematically shown in Figure 8.25, where a sinusoidal pattern of fringes is projected onto the (110) crystallographic plane of an undoped BTO crystal, with its [001]-axis perpendicular ~ The angle between the interfering beams (in air) to the plane of incidence and to K. is 51 in this case, and the wavelength is l ¼ 0:5145 mm. The visibility coefficient jmj of the pattern of fringes is computed from the ratio of amplitudes of the interfering waves, taking also into account their polarization directions that are in the plane of incidence. A piezoelectric-supported mirror (PZT), placed in one of the interfering beams, is driven by a sinusoidal voltage vðtÞ ¼ v d sin t

ð8:103Þ

which produces a corresponding phase modulation of amplitude (in radians) K K ¼ KPZT v d where KPZT is the response of the piezoelectric.

ð8:104Þ

220

HOLOGRAPHIC TECHNIQUES

PZT 0

R-feedback (100)

coherent laser beams

|001|

IR

BTO

0

IS

+ – output OA

Figure 8.25. Holographic current setup schema. A laser beam of 514.5-nm wavelength is divided in two, filtered, expanded, collimated, and made to interfere over the BTO sample. A piezoelectric-supported mirror PZT in one of the beams is vibrating with angular frequency . A lock-in amplifier measuring current, and schematically represented by the operational amplifier with feedback, is tuned to  to measure the first harmonic component i of the ~ direction in the sample’s volume. Reproduced from [Mosquera and photocurrent along the K Frejlich, 2002]

Light absorption effects cannot be neglected in most photorefractive materials. In fact, we need to consider also the effect of absorption on the holographic response time, besides its obvious effect on the photoconductivity. Because of bulk absorption the irradiance decreases exponentially along the sample’s thickness coordinate z so that irradiance-dependent quantities like photoconductivity and holographic response time also vary along z as follows: s0 ðzÞ ¼ s0 ð0Þeaz

ð8:105Þ

tsc ðzÞ ¼ tsc ð0Þeaz

ð8:106Þ

where s0 ð0Þ and tsc ð0Þ are the values at the input plane inside the sample and a is the effective absorption coefficient. The expression for j in Equations (8.94)–(8.96) is therefore also dependent on z, because of its dependendence on s0 ðzÞ and tsc ðzÞ, and should be rather written as j ðzÞ to explicitly indicate this dependence. The experimentally measured photocurrent value, accounting for irradiance decrease along the sample’s thickness, is



ji j ¼ H

Zd

jj ðzÞjdz

ð8:107Þ

0

where H is the height and d is the thickness of the sample. The first harmonic amplitude value ji j was measured, using the direct current measurement facilities of an EG& G model 5210 lock-in amplifier, as a function of K for different fixed values of . Typical results for the same sample are shown in Figures 8.26 and 8.27. The curves in Figure 8.26 represent the best fit of the theoretical equations to data (spots). The same procedure was followed for data in Figure 8.27, although the

PHASE MODULATION TECHNIQUES

221

Figure 8.26. First harmonic component of the holographic current ji j data (spots) as a function of the K for I0 ¼ IR0 þ IS0 ¼ 455 W=m2. The continuous curves are the best fit to theory, from =2p ¼ 980 Hz (thickest continuous) to 3.5 Hz (thinest dashed). Data for 980, 546, and 349 Hz are omitted because are close to data for 152 Hz.

fitting curve is not shown here. In both cases there is an excellent agreement between theory and experimental data. Note that the maxima of all high-frequency curves in Figure 8.27 occur at K ¼ 1:1, in agreement with the position of the maximum for the product J0 ðKÞJ1 ðKÞ in Equation (8.102). The maxima for the curves in Figure 8.26 instead occur at K ¼ 1:1 for the higher frequencies and progressively shift to higher values of K for the decreasing frequencies. This is

2.0

i Ω (nA)

1.5

1.0

Ω = 129 hz Ω = 268 hz Ω = 313 hz Ω = 496 hz Ω = 695 hz

0.5

0

0

1

2

3

K∆ (rad)

Figure 8.27. First harmonic component of the holographic current jj j data (spots) as a function of K for I0 ¼ IR0 þ IS0 ¼ 177 W=m2. All data fit the same curve, not shown. Reproduced from [Mosquera and Frejlich, 2002]

222

HOLOGRAPHIC TECHNIQUES

Figure 8.28. ji j data (spots) ploted as a function of =2p, for K ¼ 1:1 rad: Ce-doped BTO (thickest curve), Pb-doped BTO (thinnest curve), and undoped BTO (midthickness curve).

probably due to the fact that the maximum ðfor the term J1 ðKÞðJ0 ðKÞþ 3J2 ðKÞÞÞ for the low-frequency limit expression represented by Equation (8.99) occurs at K  2, so that the position of the maximum will depend on the frequency . Experimental ji j data obtained for different frequencies at a fixed K ¼ 1:1 and fixed irradiance from Figure 8.26 are plotted in Figure 8.28. Curves for Ce-doped and Pb-doped BTO obtained in the same way are also shown in this figure. It is interesting to note that the shape of these curves is quite similar to the theoretical curve in Figure 8.24. Note that, for the high-frequency range, the effect of bulk light

TABLE 8.5. Holographic Photocurrent BTO I0 I(0) m d L H a514:5 nm A ls tsc ð0Þ A=tsc ð0Þ s0 ð0Þð1 þ K 2 L2D Þ=Ið0Þ

2

(W/m ) (W/m2 ) (mm) (mm) (mm) (m1 ) (1010 F=m) (mm) (ms) (1010 F=sm) ð1010 m=ð WÞ

460 360 0.6 2.05 6.2 7.0 1290 2.51 0.05 8.7 260 1

BTO:Ce 443 346 0.55 6.05 6.05 7.55 932 0.15 0.2 1.11 130 2

BTO:Pb 443 346 0.55 6.0 6.05 7.5 1013 1.86 0.07 7.6 250 1

PHASE MODULATION TECHNIQUES

223

absorption in Equation (8.107) is just a term Zd

eaz dz ¼

1  ead a

ð8:108Þ

0

From the results in this section it is clear that our theoretical model adequately describes the experimental phenomena involved. To illustrate the use of this technique for materials characterization, the theoretical Equation (8.107) (continuous curve) is fitted to the experimental ji j data (spots) in Figure 8.28, and from this fitting we are able to find out the parameters A and tsc , or just A=tsc in the high-frequency limit. The experimental and material parameters from fitting are reported in Table 8.5. Note that ls is mainly determined from A, which are reported in following lines in Table 8.5. The s0 ð0Þ or s0 ð0Þ=Ið0Þ instead (although associated with K 2 L2D , which means that an auxiliary experiment is still necessary to compute LD ) is derived either from tsc ð0Þ or from A=tsc . Note that the latter can be directly obtained from the high-frequency range without caring about the frequency-dependent part of the curve.

CHAPTER 9

SELF-STABILIZED HOLOGRAPHIC TECHNIQUES

This chapter will show some of the many interesting possibilities of self-stabilized holographic recording for the measurement of photorefractive materials parameters, although this technique is not limited to photorefractives. We shall give a few examples only, focusing on the few materials for which we are able to give direct first-hand actual experimental data.

9.1

HOLOGRAPHIC PHASE SHIFT

The phase shift between the pattern of light onto a photorefractive crystal and the resulting hologram at the very beginning of the recording process (fI ) as described in Equation (3.73) is different from that at steady state in Equation (3.58). The former expression has been also shown to describe the phase shift in running holograms at resonance as reported in Equation (3.88) and allows one to compute the diffusion length of photoexcited charge carriers. Differently than for the steady-state case, the value fI can be easily obtained from j in Equation (4.172) because at the very beginning of the recording process the hologram is very weak so that selfdiffraction effects can be neglected, as described in Equation (4.118), and in this case the simple relation j ¼ fI þ p=2 holds [Staebler and Amodei, 1972] for jj 1 and jgj 1.

Photorefractive Materials: Fundamental Concepts, Holographic Recording and Materials Characterization, By Jaime Frejlich Copyright # 2007 John Wiley & Sons, Inc.

225

226

SELF-STABILIZED HOLOGRAPHIC TECHNIQUES

Measurement of the holographic phase shift under self-stabilized conditions is, unfortunately, not possible because self-stabilization acts directly on the output phase j and therefore also on the hologram phase shift itself. It is, however, possible to stabilize the pattern of fringes by using a reference placed close to the sample to be measured and in this way minimize perturbations and get much more reliable results. We shall here illustrate the use of such an external reference-based stabilization for the measurement of the so-called initial phase shift described in Section 3.3.2. The external reference is produced by a small, thin glass plate, firmly fixed by the side of the crystal as illustrated in Figure 6.27. The interference of the transmitted and reflected beams generated by the glass plate exhibit harmonic terms in , as for the beams through the crystal. The first term is selected with a lock-in amplifier tuned to , electronically integrated (although integration is not fundamental in this case), amplified, and used to operate a feedback stabilization loop. The gain–bandwidth product of the feedback loop is set to fix the light fringe pattern in a time interval much shorter than that required to measure with the lock-in amplifier. To illustrate the method and the advantages of stabilized recording techniques, even for the measurement of the phase shift, we shall describe the measurement of the initial phase shift for a 2.05-mm-thick nominally undoped Bi12 TiO20 crystal [Freschi et al., 1997]. In this sample the charge carriers are electrons without any noticeable hole competition. The experiment was carried out with the [001]-crystal ~ that is parallel to the applied electric field axis perpendicular to the grating vector K direction, with the pattern of fringes projected onto the (110)-crystal face. The recording was carried out with the 532-nm-wavelength laser line, with the input beam polarization chosen to have the transmitted and diffracted beams approximately parallel–polarized behind the crystal [see Chapter 5]. The room temperature was kept fixed to 22  1 C. The first I  and second I 2 harmonic terms of the irradiance behind the sample were separately detected with two lock-in amplifiers (tuned to  and 2, respectively) for computing j from Equation (4.168) in Section 4.3.1.2. In the present experiment we used an electronic processing circuit and a two-phase lock-in amplifier that essentially produces two signals in quadrature [Freschi and Frejlich, 1995] VX ¼ V0 sin j and pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi V0 ¼ k Zð1  ZÞ

VY ¼ V0 cos j

ð9:1Þ ð9:2Þ

but the usual V  / I  and V 2 / I 2 signals, reported in Section 4.3.1.2, could be used as well. Here k is a known setup constant and Z is the diffraction efficiency of the hologram. In the present experimental conditions self-diffraction can be neglected and the (initial) holographic phase shift is straightforwardly computed from Equation 4.172 as 1= tan fI ¼  tan j ¼ VX =VY

ð9:3Þ

Figure 9.1 illustrates typical VX and VY data recorded in a time interval of approximately one-tenth of the hologram response time tsc . Figure 9.2 shows the

HOLOGRAPHIC PHASE SHIFT

227

Figure 9.1. Typical time evolution of the VX and VY signals (dots) at the initial stage of the recording process in Bi12 TiO20 for E ¼ 0 ðaÞ and E ¼ 3:15 kV=cm ðbÞ. The ratio between the angular coefficients of the linear fittings (continuous curves) are used to compute j. The diffraction efficiencies at t ¼ 1:2 s are Z  3  105 ðaÞ and Z  5  105 ðbÞ, whereas the minimum detectable signal was estimated to correspond to Z  107 . Reproduced from [Freschi et al., 1997]

0.50

0.50

0.40

0.40 0.30

tan ϕ

0.30 A

B

0.20

0.20

0.10

0.10

0

0

0

1

2

3 4 5 E (kV/cm)

6

7

C

0

1

2

3 4 5 E (kV/cm)

6

7

Figure 9.2. Computed initial tan j vs. applied electric field data (spots) in Bi12 TiO20 . The best fitting theory is represented by the continuous curves. Curve A represents nonstabilized, whereas curves B and C represent stabilized experiments. Experimental parameters and the values for LD and x computed from data fitting are reported in Table 9.1. Dashed lines were plotted in curve B for LD ¼ 0:13 mm (upper) and for LD ¼ 0:14 mm (lower) and similarly in curve C for LD ¼ 0:13 mm (upper) and for LD ¼ 0:15 mm (lower), to approximately indicate the precision of the measurement. Reproduced from [Freschi et al., 1997]

228

SELF-STABILIZED HOLOGRAPHIC TECHNIQUES

TABLE 9.1. Initial Phase Shift: Experimental Parameters from Data Fitting on Undoped Bi12 TiO20 Curve A B C

I0R (mW=cm2 ) I0S 2.8 0.42 0.32

0.12 0.025 0.020

K (mm1 ) 7.07 7.07 11.27

tsc (s)

x

LD (mm)

0.6 3.5 12

0.73 0.86 0.87

0.15 0.135 0.14

Curves A, B and C refer to Fig. 9.2. tsc is the hologram response time measured for E ¼ 0; x is the electric field correction coefficient.

tan j data, computed from VX and VY signals as the ones shown in Figure 9.1 and plotted against the applied electric field for three independent experiments. Unlike curves B and C, data plotted in curve A were measured with the stabilization loop switched off. The theoretical Equation (3.73) is fit to all experimental data, where the electric field E is substituted by its effective value xE, as discussed in Appendix C, for the region in the crystal where the measurement is carried out. The best theoretical fittings in Figure 9.2 are represented by the continuous curves that lead to a diffusion length value of LD and a parameter x for each experiment, which are listed in Table 9.1. Computing j in the nonstabilized experiment (curve A, measurement time  60 ms) produces much larger data dispersion than for the stabilized experiments (curves B and C). Although all three experiments show a good agreement for the computed LD parameter, the scattered data in curve A do not lead to a good estimation for the effective field coefficient x, thus showing the relevance of stabilization techniques. Dashed lines in curves B and C were plotted to give an idea of the data dispersion effect on LD measurement precision. These lines were plotted just introducing small variations in the best-fitted LD value in order to wrap up most of the corresponding experimental data. We compared our results, for the same sample, wavelength, and room temperature, with those obtained from two well-known techniques: measurement of the hologram time constant vs. spatial frequency [Feinberg et al., 1980], giving LD ¼ 0:12  0:04 mm [dos Santos et al., 1989], and the measurement of the holographic sensitivity vs. spatial frequency, as in Section 8.1.3 and published elsewhere [Bian and Frejlich, 1995], leading to LD ¼ 0:15  0:04 mm. Results from both these techniques are clearly consistent with those reported in Table 9.1. The same stabilization technique reported above was also used for the measurement of the stationary phase shift, results of which are reported in Figure 9.3. This figure shows experimental phase-shift data for the same Bi12 TiO20 sample and the same configuration as for data in Figure 8.12 but for l ¼ 532 nm. For stationary phase shift, however, the calculations are more complex (as already discussed in Section 8.2.2) than for the initial phase because of self-diffraction effects that are almost absent for the latter.

FRINGE-LOCKED RUNNING HOLOGRAMS

229

1.5

tan ϕF

0.5

–0.5

–1.5 –1.0×106

0

–0.5×106

0.5×106

1.0×106

E (V/m)

Figure 9.3. Phase shift j ¼ jF vs. applied electric field (E0 ) data (circles) for a 2.05-mmthick Bi12 TiO20 crystal and grating vector K ¼ 5:5 mm1 for b2 ¼ 30, and 532-nm wavelength, with a ¼ 8:5 cm1 . The continuous curve is the best fit of theoretical equation in Eq. (4.122) that leads to ls ¼ 0:03 with a field factor x  0:74.

9.2

FRINGE-LOCKED RUNNING HOLOGRAMS

Self-stabilized or fringe-locked running holograms, as described in Section 6.2.2, were recorded on undoped Bi12 TiO20 and Bi12 SiO20 crystals. The expression for the hologram speed in Equation (6.31) can be also written as v¼ with and

2v M EM E0 2 E02 þ EM

2 EM ¼ ED2 ½1 þ ðKLD Þ2 

2EM v M ¼

q  Iabs hne0 E K 2 d

ð9:4Þ ð9:5Þ ð9:6Þ

to better put into evidence the maximum speed v M and associated field EM . The equation above is valid in the absence of bulk light absorption in the crystal (see Section 4.2.2.3) only. Note that the adaptive running hologram speed in Equation (9.4) is different from the ‘‘free’’ resonance running hologram speed in Equation (3.81). It is easy to realize that v in Equation (9.4) becomes independent of LD for both the cases KLD 1 and KLD  1, for the same reasons as developed in Section 8.2.5. Figure 9.4 shows some experimental results for a Bi12 SiO20 crystal, showing the theoretical fitting of Equation (9.4) to experimental data, which allows computing important transport parameters like the diffusion length LD and the quantum efficiency . Experiments carried out on this Bi12 SiO20 sample in different conditions give very reproductible LD ¼ 0:19 mm results with a precision better than

230

SELF-STABILIZED HOLOGRAPHIC TECHNIQUES

50

v (10–3 µm/s)

40 30 20 10 0 0

2

4

6

8

10

E0 (KV/cm)

Figure 9.4. Fringe-locked running hologram speed vs. applied electric field for a 1.71-mmthick Bi12 SiO20 crystal with a ¼ 3 cm1 for the 514-nm wavelength with m  0:3; IS ¼ 12 mW=cm2 , IR ¼ 440 mW=cm2 , and K ¼ 4:24 mm1 . Theoretical fit (continuous curve) to experimental data (o) leads to LD ¼ 0:19 mm and 0:46    0:6 with an estimated field factor of 0.87. Reproduced from [Frejlich et al., 1989, Frejlich et al., 1990].

5% and a value for  ranging from 0.6 to 0.46 [Frejlich et al., 1989, 1990]. It is unnecessary to point out that the use of adaptive techniques provides quite reliable and reproductible results. 9.2.1

Absorbing Materials

Moving holograms in strongly absorbing materials was studied in Section 4.2.2.3, where we showed that in this case the parameters d and gd should be replaced everywhere by their corresponding values integrated along the sample thickness, as described by Equations (4.137) and (4.138) Zz¼d

z¼0

    bci 2 2aKveaz þ b z¼d  pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi arctan pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðKv; zÞdz ¼ d¼ 4w ai  2c a 4ac  b2 4ac  b2 z¼0 " #z¼d ci e2az ln þ 4w 2ac aðKvÞ2 e2az þ bKveaz þ c z¼0

Zz¼d

z¼0

    bcr 2 2aKveaz þ b z¼d pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi arctan pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi gðKv; zÞdz ¼ gd¼ 4w ar  2c a 4ac  b2 4ac  b2 z¼0 " #z¼d cr e2az ln þ 4w 2ac aðKvÞ2 e2az þ bKveaz þ c z¼0

FRINGE-LOCKED RUNNING HOLOGRAMS

231

In this case the adaptive running hologram feedback condition is not any more g /
Zz¼d

gdz /

0

Zz¼d

st
ð9:7Þ

0

Substituting the integral above by its expression in Equation (4.138) we get Zz¼d

z¼0

    bcr 2 2aKveaz þ b z¼d pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi arctan pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi gðKv; zÞdz ¼ gd¼ 4w ar  2c a 4ac  b2 4ac  b2 z¼0 " #z¼d cr e2az ln þ 4w ¼0 2ac aðKvÞ2 e2az þ bKveaz þ c z¼0

ð9:8Þ

After integrating, for 4ac  b2 and rearranging terms we get the final formulation pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4a1 c1  b21 tM ð0Þ Kv ðead  1Þ

2c1 þ 2a1 tM ð0Þ2 K 2 v 2 ead þ b1 tM ð0Þ Kvðead þ 1Þ " pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi !# f 4a1 c1  b21 1 a1 tM ð0Þ2 K 2 v 2 e2ad þ b1 tM ð0Þ Kvead þ c1 ¼ tan ad  ln 2gc1 þ fb1 2 a1 tM ð0Þ2 K 2 v 2 þ b1 tM ð0Þ Kv þ c1

ð9:9Þ

where the parameters above are conveniently slightly differently defined than in Section 4.2.2.3 as a1 ¼ ðK 2 L2D f Þ2 þ ð1 þ K 2 L2D Þ2 b1 ¼ c1 ¼

g¼

2f ðK 2 l2S

 K 2 L2D Þ ð1 þ K 2 l2S Þ2 þ ðfK 2 l2S Þ2 K 2 L2D f 2 þ K 2 L2D þ 1

f ¼x

E0 ED

ð9:10Þ

ð9:11Þ ð9:12Þ

ð9:13Þ

ð9:14Þ

It is worth pointing out that Equation (9.9) brings about an implicit relation GðKv; E0 Þ ¼ 0

ð9:15Þ

between the speed (v or Kv) and the applied electric field E0 (included in parameters f , a1 , b1 , c1 , and g above) in substitution of the simplified explicit relation in Equation (6.31) for absorptionless materials.

232

SELF-STABILIZED HOLOGRAPHIC TECHNIQUES

9.2.1.1 Low-absorption approximation. For the case ad 1 we can write ead  1 þ ad and also substitute tan x  x for jxj 1 in Equation (9.9) just to verify that the expression obtained KvtM ð0Þ¼

1þ

K 2 L2D

f þ K 2 L2D f 2

for

ad 1

ð9:16Þ

is the same already reported in Equation 6.31 except for the additional factor f , which now includes the coefficient x taking into account the effectively applied electric field value. 9.2.2

Characterization of Materials

The self-stabilized (fringe locked) running hologram is a powerful tool for material characterization, and, under adequate conditions, one single experiment may provide most of the relevant material parameters and also the value of the effectively applied electric field coefficient x. From a fringe-locked experiment it is possible to directly measure the detuning Kv as a function of the applied field E0 . It is also possible to compute the diffraction efficiency Z from the same experimental run. From these two data sets (Kv vs. E0 and Z vs. E0 ) we are able to determine the whole set of parameters LD ; ls , and , plus the experimental coefficient x [Barbosa and Frejlich, 2003]. Although the theoretical analysis of the effect of bulk light absorption on the material response time is not too important to understanding running holograms or to understanding the way self-stabilized running holograms are produced, such an analysis is essential to be able to accurately fit the theoretical equation to the experimental data in absorbing materials so as to enable their characterization. In fact, in view of the large number of parameters involved here, it is necessary to have an accurate theoretical function as well as accurate experimental data to enable fitting with high possibilities of convergence without multiple solutions and minimum uncertainties. In other words: There are too many parameters to be fitted, and they are better fitted as the theoretical model is better adjusted to the experiment and the experimental data are as least dispersive as possible. Let us recall that self-stabilized recording, involving either stationary or nonstationary holograms, inherently produces less dispersive data than nonstabilized recordings. They are even less dispersive than stabilized non-self-stabilized (stabilized on external references different from the hologram itself being recorded) holograms. 9.2.2.1 Measurements. We have already seen the importance of dealing with data with reduced dispersion, so it is worth spending some time to briefly explain how to measure in order to get adequate data for processing. 9.2.2.1.1 Hologram Speed Kv. The detuning Kv can be computed, in a continuous and nonperturbative way, from the movement of the PZT-supported mirror, that is, from the voltage applied to this device, after calibration. This is the easiest and most direct way, although it is not the best way, to measure the hologram

FRINGE-LOCKED RUNNING HOLOGRAMS

233

0.8

Kv (rad/s)

0.6

0.4 0.2 0 –0.2

0

2

4

6

E / ED

Figure 9.5. Fringe-locked running hologram experiment. Frequency detuning Kv (measured from the movement of the PZT-supported mirror) vs. normalized applied field E0 =ED data from a typical fringe-locked running hologram experiment carried out on an undoped Bi12 TiO20 crystal using the 514.5-nm wavelength with K ¼ 7:55 mm1 , IR0 ¼ 21:5 mW=cm2 , and IS0 ¼ 0:45 mW=cm2 .

speed because the movement of the PZT also accounts for the feedback correction of environmental perturbations and steady-state drifts (produced by temperature, for example) on the setup, so that data from PZT are usually rather noisy as seen in the typical results [Barbosa et al., 2001] in Figure 9.5. A better way to carry out such measurements is by using the pattern of interference between the transmitted and reflected beams in a small, thin glass plate placed by the side of the sample as already discussed and shown in Figure 6.24. In this way it is possible to follow the evolution of such fringes as reported in Figure 6.25, and in this way much less dispersive Kv data are obtained, as becomes obvious from the results in Figure 9.6. 9.2.2.1.2 Diffraction Efficiency. The first and second harmonic terms in , respectively, reported in Equations (4.166) and (4.167) are qffiffiffiffiffiffiffiffipffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi and IS ¼ 4J1 ðcd Þ IR0 IS0 Zð1  ZÞ sin j qffiffiffiffiffiffiffiffipffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi IS2 ¼ 4J2 ðcd Þ IR0 IS0 Zð1  ZÞ cos j

that are detected along the IS direction, behind the sample under analysis, using a photodetector and lock-in amplifiers tuned to  and 2, respectively, so that the corresponding output signals VS ¼ AJ1 ðcd Þ

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Zð1  ZÞ sin j

and

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi VS2 ¼ AJ2 ðcd Þ Zð1  ZÞ cos j

ð9:17Þ

234

SELF-STABILIZED HOLOGRAPHIC TECHNIQUES

Kv (rad/s)

1.05

0.07

0.35

0

0

1

2

3

4

5

E0 / ED

Figure 9.6. Fringe-locked running hologram experiment. Frequency detuning Kv (measured from the interference pattern from an auxiliary glass plate) vs. normalized applied field E0 =ED data from a typical fringe-locked running hologram experiment carried out on an undoped Bi12 TiO20 crystal using the 514.5-nm wavelength with K ¼ 8:5 mm1 , IR0 þ IS0 ¼ 52 mW=cm2 , and b2 ¼ 183.

are obtained, where A is the overall amplification that depends on the photodetectors, beam irradiances, amplifiers, and other experimental settings. The VS signal is used as an error signal in the feedback loop so that it is automatically set to 0, by imposing sin j ¼ 0 as a consequence of the feedback condition in Equation (9.7) and the expression of j in Equation (4.140). For nonphotovoltaic crystals, in the absence of externally applied electric field, the equilibrium value is j ¼ 0. However, in the presence of an external field it is in general j 6¼ 0. By imposing the j ¼ 0 constraint, the pattern of fringes is put in movement with a speed v that depends on the mismatch between the actual equilibrium j value and the imposed j ¼ 0 as already discussed in Section 6.2.2. Under steady-state conditions the photorefractive hologram moves synchronously with the pattern of fringes. For j ¼ 0 we have then VS ¼ 0

and

VS2 ¼ AJ2 ðcd Þ

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Zð1  ZÞ:

ð9:18Þ

Therefore it is possible to measure Z from VS2 in a continuous nonperturbative way during recording. Typical experimental results obtained for Z are plotted on the right-hand side in Figure 9.7. The left-hand side shows Kv data, computed as described above, for the same sample and experiment. 9.2.2.2 Theoretical Fitting. The theoretical expression of Z in Equation (4.139), for the imposed feedback condition in Equation (9.7), becomes

Z¼

 2b2 coshðd=2Þ 1 2 2   1 þ b b expðd=2Þ þ expðd=2Þ

ð9:19Þ

235

FRINGE-LOCKED RUNNING HOLOGRAMS

0.05 0.04

0.3 h

Kv (rad/s)

0.4

0.2

0.03 0.02

0.1

0.01 0

1

2

3 E0/ED

4

5

6

0

1

2

3 E0/ED

4

5

6

Figure 9.7. Kv and Z experimentally measured on an undoped Bi12 TiO20 sample (labeled BTO-013) for different values of the normalized applied field, with IR0 þ IS0 ¼ 14 W=m2 ; b2  48; K ¼ 7:55 mm1 , and a ¼ 1041 m1 at 514.5-nm wavelength.

 (see Equation (4.137)) is dependent on E0 and on Kv so that Z in Equation but d (9.19) is implicitly dependent also on E0 and on Kv. The other consequence of the feedback condition, besides leading to Equation (9.19), is to bring about an implicit relation between E0 and Kv as formulated in Equation (9.15) GðKv; E0 Þ ¼ 0 which makes Kv no longer an independent variable but one determined by E0. Such an implicit relation turns the 3D surface represented by Equation (9.19) into a 3D curve actually representing the theoretical formulation of fringe-locked experiments. Because of the 3D nature of the theoretical formulation, and to facilitate data fitting, it is interesting to operate with 3D experimental data, too. Therefore, instead of displaying data in 2D as in Figure 9.7, we display the same data in 3D as shown in Figure 9.8, where the continuous curve is the result of previous data fit in Figure 9.7, but direct experimental data without previous fitting could be plotted instead as well. We may choose not to fit the 3D experimental data with the 3D theoretical curve arising from Equations (9.19) and (9.15) but instead just to use the 3D surface represented by Equation (9.19). The consequence of this choice is that experimental data are fitted with a larger class of functions (a surface instead of a curve), but the handling of this 3D surface is easier than the 3D curve containing the implicit relation GðKv; E0 Þ ¼ 0. The result of such a fitting is shown in Figure 9.9, and the parameters obtained from this fit are reported in Table 9.2. Fitting requires some initial hint (‘‘input’’) as to the parameters we are looking for (LD ; ls ; , and x) in order to obtain associated results (‘‘output’’). A few such inputs and resulting outputs are reported in Table 9.2. The last two inputs lead to unacceptable outputs, either because they lead to impossible (e.g.,  > 1) results or because they lead to unrealistical values for one or more of the parameters we are

236

SELF-STABILIZED HOLOGRAPHIC TECHNIQUES

0

E0/ED 2 4

0.02 η 0.01

0 0.4

0.3 0.2 Kv (rad/s)

0.1 0

Figure 9.8. 3D representation of diffraction efficiency and Kv experimentally measured data (from Fig. 9.7) in a nominally pure photorefractive Bi12 TiO20 crystal 2.35 mm thick and labeled BTO-013, for different values of the applied field E0 , for IS0 þ IR0 ¼ 14 W=m2 ; b2 ¼ IR0 =IS0  48, absorption coefficient a ¼ 1041 m1 ; l ¼ 514:5 nm, and K ¼ 7:55 mm1 .

looking for. The first two inputs (first two rows) are acceptable and actually lead to similar results for all parameters except for , which is found to be either 0.41 or 0.74. Note also that the acceptable outputs are also characterized by a much lower variance coefficient for the fit, thus indicating a statistically better fit compared to the two last ones. Among other means to decide what are good and what are unacceptable results one should bear in mind that 0    1 and 0  x  1, for example. In the present case we believe that the output in the first row is the right one just because both LD and  are closer to the available data in the literature for similar samples (LD ¼ 0:14 mm [de Oliveira and Frejlich, 2001b; Freschi et al., 1997]) and for the same wavelength ( ¼ 0:45 [de Oliveira and Frejlich, 2001b]), although the differences between both acceptable outputs are not very significant for this kind of experiment. Because of the comparatively large number (four) of parameters involved, the fitting is particularly sensitive to the dispersion of the experimental data. Reducing dispersion, that is, obtaining data with higher accuracy, may considerably reduce the number of possible acceptable results. It is also possible that using the actual 3D theoretical curve instead of the larger class of 3D theoretical surfaces may also reduce the number of multiple solutions from fitting.

FRINGE-LOCKED RUNNING HOLOGRAMS

237

0 E0/ED 2 4

0.04

0.03 η 0.02

0.01

0 0.6 0.4 0.2 Kv (rad/s)

0

Figure 9.9. Same experimental data (continuous thick curve) of Fig. 9.8 with the best-fit theoretical 3D-surface from Eq. (9.19). The resulting best-fitting parameters are reported in Table 9.2.

To better understand what the actual possibilities are to obtain reliable values for such a relatively large number of parameters to be fitted it is worth recalling here some facts [de Oliveira and Frejlich, 2001b] about the influence of LD and ls in running hologram phenomena: If K 2 L2D  1, the value of LD does not affect the dynamics of the recording process because in this case the large diffusing length compared to the hologram spatial period somewhat randomizes the position of the TABLE 9.2. Fitting of Experimental Data of Undoped Bi12 TiO20 with K ¼ 7:35 lm1 LD ðmmÞ

Data

ls ðmmÞ



x

Input Output

0.16 0.13

0.04 0.042

0.35 0.41

0.75 0.73

Input Output

0.1 0.094

0.01 0.044

0.4 0.74

0.6 0.75

1 1:26  106

0.1 0.27

0.1 1019

0.001 0.048

0.1 9.33

Input Output(*) Input Output(*)

*Unacceptable output

0.01 9  103

0.1 637 1 1.08

Variance — 1:38  108 — 1:16  108

— 1:6  104

— 1:24  107

238

SELF-STABILIZED HOLOGRAPHIC TECHNIQUES

excited electron in the conduction band. On the other hand, K 2 l2s 1 means that the material is very far from saturation, and therefore the process does not depend on the density of photoactive centers that is related to 1=l2s . If any of the above conditions are fullfilled the fitting will not lead to the parameter involved simply because it is not relevant for the recording process itself.

9.3

CHARACTERIZATION OF LiNbO3:Fe

The first part of this chapter was devoted to illustrating the use of stabilized and selfstabilized techniques for the characterization of fast and low diffractive materials like sillenites. We shall apply now self-stabilized holographic recording for the characterization of a widely different type of materials: a slow and highly diffractive photovoltaic like LiNbO3:Fe. As is usual with photovoltaic crystals, we shall operate in short-circuit mode (as illustrated in Figure 3.30) and record a hologram, without ~ parallel to the c-axis. Let us recall that selfapplied field, with the hologram vector K stabilized recording on LiNbO3:Fe, with any degree of oxidation, is carried out using I 2 as error signal as described in Section 6.3.2. In this case, if we operate with equal irradiance recording beams ðb2  1Þ; Z is described by Equation (6.58) b2 ¼ 1

Z ¼ sin2 gd=4 for

with

g=4 ¼

pn3eff reff jEsc j 2l

and

¼0

ð9:20Þ

Under self-stabilized recording a steady-state nonstationary space-charge field arises of the form st iKvt e Esc ðtÞ ¼ mEsc st Esc 

Eeff ðoR þ ioI Þ oR þ iðoI  KvÞ

as already described in Section 6.3.2.1.1, with Eeff 

E0 þ Eph þ iED

1 þ K 2 l2s  iKlE 

Nþ iKlph NDD



Eph Nþ

1  iKlE  iKlph NDD

as reported in Equation (3.139), where the photovoltaic field Eph is described in Equation (3.129) as: kph Iabs kph hnr 3þ ½Fe   me sd tsc  tM ¼ Est33 e0 =s Eph ¼

2þ

s ¼/I

½Fe  ½Fe3þ 

ð9:21Þ ð9:22Þ

CHARACTERIZATION OF LiNbO3:Fe

239

where Est33 is the static dielectric constant along the c-axis and kph is a photovoltaic transport coefficient [Glass et al., 1974]. The feedback stabilization (I 2 ¼ 0) with b2 ¼ 1 imposes the additional condition (see Section 6.3.2.2) st =fEsc g¼0

ð9:23Þ

Accordingly, the time evolution of Z during self-stabilized recording with b2  1 can be formulated as   3  pffiffiffiffiffiffiffiffi   pneff reff d st t=tsc   m
The I 2 is used as error signal in the feedback stabilization loop and I /

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Zð1  ZÞ ¼ sin½Bð1  et=tM Þ

ð9:25Þ

pn3 reff d st B  eff g m
ð9:26Þ

is used to follow the recording evolution. Figures 9.10 to 9.12 show some experimental results for different samples. It is important to emphasize the interest of self-stabilized recording here: It ensures holographic recording, with minimum environmental perturbations, for the very long recording time required for these very slow materials and forces the recording to occur in such a way as to verify the simple relation in Equation (9.24). 2.5 EXTRAORDINARY

Ω

I (u.a)

2.0 ORDINARY

1.5 1.0 0.5 0 0

500

1000

1500

Time (s)

Figure 9.10. Characterization of reduced LiNbO3. Self-stabilized holographic recording on a d ¼ 1:39 mm-thick crystal (labeled LNB3) using ordinarily and extraordinarily polarized l ¼ 514:5 nm light (b2  1 and IR0 þ IS0  16 mW=cm2 ) with an irradiance absorption a ¼ 7:5 cm1 at this wavelength. The fitting of Eq. (9.25) to experimental I  data gives B and tM as reported in Table 9.3. Reproduced from [de Oliveira et al., 2005]

240

SELF-STABILIZED HOLOGRAPHIC TECHNIQUES

8

6 Ω

IS 4

2 2Ω

IS 0 0

50

100

150

200

250

Time (s)

Figure 9.11. Characterization of reduced LiNbO3. Self-stabilized holographic recording on a d ¼ 0:85 mm-thick crystal using extraordinarily polarized l ¼ 514:5 nm light with IR0 ¼ 141:1 W=m2 and IS0 ¼ 116 W=m2 . Eq. (9.25) was fitted to data, and the resulting parameters are reported in Table 9.3.

From the evolution of I  in Figures 9.10 to 9.12 and the theoretical relations in Equations (9.24) to (9.26) above, it is possible to characterize some important material parameters. To do this it is necessary to keep in mind that the index of refraction is dependent on the light wavelength and that it is, as well as the 8 Ω

I 6

4

2 I2Ω 0 0

2000

4000

6000

Time (s)

Figure 9.12. Characterization of oxidized LiNbO3. Self-stabilized holographic recording on a d ¼ 1:5 mm-thick crystal using extraordinarily polarized l ¼ 514:5 nm light (IR0 ¼ 113:5 W=m2 and IS0 ¼ 108:1 W=m2 ) and fitted with Eq. (9.25). The resulting parameters are reported in Table 9.3.

CHARACTERIZATION OF LiNbO3:Fe

241

TABLE 9.3. Parameters for LiNbO3 Samples S 1012 m3 =J Sample

Pol.

LNB5 LNB1

ext ext ext

LNB3

d (mm)

ad at 514.5 nm

[Fe2þ  [Fe3þ ] 1019 =cm3 =½Fe3þ ]

0.85 1.5 1.39

1.04

2 2

0.03 0.002

2

0.013

ord

B

tM (s)

Exp. Theor.

10.7 4.65 10.2

540 3708 827

6.4

10.6 10.6 10.6

6.46

1872

1.8

3.3

electro-optic coefficient, quite different for ordinary (ord) and extraordinary (ext) light polarization. It is straightforward to deduce from Equation (9.25) that

The maximum of I  is achieved for Bð1  et=tM Þ ¼ p=2.

I  goes down to zero for Bð1  et=tM Þ ¼ p. The simple relations above allow direct computation of B and tM from the experimental data, although it is always possible to compute these parameters from the theoretical equation fit to the experimental data as well. From the B and tM parameters it is possible to compute the so-called sensitivity as defined in Section 8.1.3, which, for the particular case of a photovoltaic crystal, should be written as: S ¼ n3eff reff kph =ð2Est33 e0 Þ

ð9:27Þ

and can be computed from S¼

B l tM 2pmIabs

ð9:28Þ

The parameters B, tM and S, computed for the different samples in these experiments are reported in Table 9.3 together with some information about these samples. General data about LiNbO3 crystals from the literature are reported in Table 9.4.

TABLE 9.4. LiNbO3: Material Parameters n0 ¼ 2.33 ne ¼ 2.25 r13 ¼ 8:6  1012 m=V r33 ¼ 30:8  1012 m=V Est33 ¼ 32 kph  1:7  1011 m=V For l ¼ 514:5 nm

Arizmendi, 1988 Arizmendi, 1988 Yariv, 1985 Yariv, 1985 Yariv, 1985 Sommerfeldt et al., 1988

242

SELF-STABILIZED HOLOGRAPHIC TECHNIQUES

The S values computed from our experimental data in Table 9.3 are shown to be in good agreement with the theoretical values computed from the available data in the literature listed in Table 9.4. Note that the theoretical development here is concerned with the so-called first harmonic approximation, which is verified for m 1 but not necessarily valid for the m  1 value in most of the experiments described here, a fact that may explain the lack of a better agreement between experimental data and theory in Table 9.3. It must be also pointed out that the presence of a strong light scattering effect may have a sensible effect on the measured response time tsc [Maniloff and Johnson, 1993] and may also interfere with a better agreement with theory.

PART IV

APPLICATIONS

244

APPLICATIONS

INTRODUCTION A large number of interesting applications of photorefractive material have been already reported in the scientific literature, and plenty of others are being described at the time this book is written. We do not intend even to mention them here because the highly dynamic nature of the research in this area is likely to render them outdated in the short term. Instead we will focus on two specific applications that we consider particularly illustrative: a) measurement of mechanical vibrations and deformations, concerning fast and low diffractive photorefractive materials, and b) fabrication of diffractive holographic optical components involving slow materials exhibiting highly diffractive possibilities.

CHAPTER 10

VIBRATIONS AND DEFORMATIONS

Holographic interferometry enables the real-time measurement of vibrational modes and static deformations in surfaces using low-power laser illumination and a photorefractive crystal as the recording medium. Since Huignard and co-workers first demonstrated [Huignard et al., 1977] the possibilities of holographic interferometry using photorefractives to measure mechanical vibrations, plenty of publications have appeared [Frejlich and Garcia, 2000] in this field, although it took a long time until an efficient device was made available for this purpose [Georges et al., 2005; Thizy et al., 2005]. In this chapter we describe a conventional setup using a nominally undoped photorefractive Bi12 TiO20 (BTO) crystal where most of the critical elements have been optimized: target illumination and backscattered light collection, distribution of light between the object and the reference beams. The novelty here is the use of self-stabilized holographic recording to improve the setup performance. The use of photorefractive materials as real-time, reversible holographic recording media has been shown to eliminate most of the handicaps of holography, thus providing a practical tool for vibration and deformation measurement. Lowfrequency perturbations and changes in the setup are adaptively coped with because of the relatively fast response of these materials. Higher-frequency perturbations can be compensated by the use of an active stabilization feedback opto-electronic loop as we reported in Chapter 6 and describe in detail in what follows. The efficient illumination of the target surface and the collection of the backscattered light from

Photorefractive Materials: Fundamental Concepts, Holographic Recording and Materials Characterization, By Jaime Frejlich Copyright # 2007 John Wiley & Sons, Inc.

245

246

VIBRATIONS AND DEFORMATIONS

the surface are very important for maximizing the intensity of the holographically reconstructed object wave containing the required information about vibration and deformation. Such optimization needs the retroreflectivity of the target surface to be taken into account. The negative feedback opto-electronic loop used for stabilizing the setup has been rearranged to decrease the level of parasitic signals in it. 10.1

MEASUREMENT OF VIBRATION AND DEFORMATION

Several techniques allow measurement of vibrations and deformations with holography. We report here the time-average holographic interferometry for vibrations [Huignard et al., 1977] and double holographic exposure for deformations and tilting [Collier et al., 1971]. These are general methods that have been adapted to the special features of photorefractive recording media. The good setup performance is due to the particular features of the setup, including efficient target illumination and light distribution, as well as the use of a negative opto-electronic feedback loop for stabilizing the setup. The use of retroreflective painting on the target surface largely contributes to the performance of the setup. 10.2

EXPERIMENTAL SETUP

The experimental setup used in these experiments is described in Figure 10.1 and actually shown in Figure 10.2. The input laser beam (with irradiance I0 ) is divided into a reference and an object beam, using a polarization beam-splitter cube (PBS). The amplitude ratio R between both output beams is controlled with a half-wave retardation plate HWP at the PBS input. The polarization directions of the PBS-exiting beams are made parallel by the use of another HWP at one of the outputs. A low-power microscope objective lens L1 is used to expand the object beam in order to illuminate the whole target surface, which is a loudspeaker in this case. A device formed by a PBS, two HWP, and one quarter-wave retardation plate QWP is used to direct all the light onto the target surface and then allow the whole backscattered collected light through the PBS directly onto the recording photorefractive crystal (BTO) with minimum losses. Two (L2 and L3) photographic good-quality objectives are used to produce a reduced target image onto the BTO and then an adequately sized image onto the CCD camera for observation and/or image acquisition and processing. The reference beam is also directed onto the BTO to interfere with the object beam to produce the required hologram for recording. The hologram is produced in real time in the BTO crystal and at the same time is reconstructed by the same reference beam used for recording it: The diffracted reference beam is actually the reconstructed object beam carrying all the information needed about the target vibration or deformation. 10.2.1

Reading of Dynamic Holograms

The reading of holograms written in real-time reversible recording media, as photorefractive crystals are, requires special techniques because the uniform reference

EXPERIMENTAL SETUP

247

LOUDSPEAKER LASER PZT

PBS

HWP PBS HWP HWP

L1

M

EOM

QWP PBS

PLC L2 PBS M SF

BTO L5 D

P2 L4

HWP PBS HWP

L3 LA

P1

CCD INT

HV

Figure 10.1. Schematic diagram of the experimental setup: PBS: polarizing beam-splitter cube; HWP and QWP: half-wave and quarter-wave retardation plates, respectively; M: first surface mirrors; PZT: piezoelectric supported mirror; PLC: path length compensator; EOM: eletro-optical modulator; SF: spatial filter; BTO: photorefractive Bi12 TiO20 crystal; D: photodetector; P1 and P2: polarizers; CCD: image detector; LA: lock-in amplifier; INT: integrator; HV: high-voltage source for the PZT. Reproduced from [Frejlich and Garcia, 2000]

beam erases the hologram during reading. Several possibilities exist for reading these dynamic holograms, as they are called. We have chosen an efficient technique based on the anisotropic diffraction properties of some crystals among which are the sillenites and in particular the Bi12 TiO20 (BTO) used in these experiments. In fact, under certain experimental contitions, the transmitted and diffracted (holographically reconstructed) beams are orthogonally polarized, following the procedures discused in Chapter 5 and in the literature [Kamshilin and Petrov, 1985]. In this case the diffracted beam carrying the necessary information about vibration and deformation can be separated from the transmitted beam that carries no information, just by using a simple polarizer (P1 in Fig. 10.1).

248

VIBRATIONS AND DEFORMATIONS

Figure 10.2. (Above) Lateral view of the setup: CCD camera (1), output polarizer (2), photographic objective lens for imaging the hologram onto the CCD (3), photorefractive crystal in its nylon holder (4), photographic objective lens for imaging the target onto the crystal (5), target painted with retroreflective ink (6) and 633-nm He-Ne laser (7). (Below) Detailed view of the photorefractive crystal in its nylon holder (center), between the two photographic objective lenses and the ouput polarizer.

10.2.2

Optimization of Illumination

The amount of light available to illuminate the target, record the hologram in the crystal, and read it is limited by the power of the laser source being used. A powerful source is interesting because:

It speeds up the holographic recording because the recording time is roughly inversely proportional to the average light onto the crystal.

The speedup of recording allows one to adaptively cope with perturbations of higher frequency.

It allows illumination of a larger target surface.

EXPERIMENTAL SETUP

249

To optimize the available amount of light we must efficiently illuminate and collect the light from the target and adequately divide the input beam between the object and reference beams in the setup. 10.2.2.1 Target Illumination. The illumination and light collection from the target surface are described in Figure 10.1: A polarization beam-splitter cube PBS, a half-wave retardation plate HWP, and a quarter-wave plate QWP are used. The incident light (TE polarized) is completely reflected toward the target by the PBS, and on its way forth and back from it crosses the QWP twice, thus rotating its polarization direction by 90 and therefore being transmitted through the PBS to the crystal. In this way the limited available light from the laser source is efficiently used. To further improve light collection from the target, the latter is painted with a special thin retroreflective ink film. 10.2.2.2 Distribution of Light Among Reference and Object Beams. Figure 10.3 shows a simplified schema of the light distribution between the reference and the object beams in the setup that allows calculation of the diffracted reference beam intensity IRD that is to be maximized. The latter is computed from the relations below

IRD ¼ ZIR0

Z ¼ Z0 m 2

l0

m¼

2b ð1 þ b2 Þ

b2  IR0 =IS0

R  IS1 =IR1

ð10:1Þ

1

lR

1

M

BS

lS

0

lR

D

target

lR

0 lS

BTO

Figure 10.3. Simplified schema showing the distribution of incident light (I0 ) between reference and object beams: BS, beam-splitter; M, mirror; IR1 and IS1 reference and object beams at the BS output; IR0 and IS0 , reference and object beams effectively incident on the crystal. Reproduced from [Frejlich and Garcia, 2000]

250

VIBRATIONS AND DEFORMATIONS

where Z0 is the maximum diffraction efficiency that can be obtained for a hologram in the crystal. I0 ¼ IS1 þ IR1

IR0 ¼ fIR1

IS0 ¼ gIS1

IRD ¼ 4Z0 I0 f

ð10:2Þ

R f =g ðR þ f =gÞ2 ð1 þ RÞ

ð10:3Þ

In the expression above for IRD, Z0 depends on the BTO, I0 is the available laser irradiance, and f and g depend on mirror M and on the target. The only parameter that it is possible to adjust over a large range in Equation (10.3) is the distribution of light R at the beam-splitter BS. From the relations above we see that IRD is maximum for f =g ¼ R2 ð1 þ

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ 2=R þ 1=R2 Þ

ð10:4Þ

Figure 10.4 shows the values measured for IRD as a function of R for a loudspeaker membrane painted with retroreflected ink. The continuous curve is the best theoretical fitting with Equation (10.3), and from this fitting the value f =g ¼ 1:15 is obtained. From the independently measured f ¼ 0:14 value the effectively collected retroreflected light can be estimated to be g ¼ 0:12 for that target in our setup. As seen from data in Figure 10.4 the maximum value for IRD is obtained for R ¼ 0:61, in good agreement with the value that can be deduced from Equation (10.4).

D IR A.U. (mV)

40

30

20

10

0 2

6

10

R

Figure 10.4. Optimization of the target illumination: IRD , diffracted reference beam measured (arbitrary units) as a function of R ¼ IS1 =IR1 (o), and the best fitting to theory (continuous line). From fitting we get f =g ¼ 1:15 for our presently retroreflective painted loudspeaker membrane. Reproduced from [Frejlich and Garcia, 2000]

EXPERIMENTAL SETUP

10.2.3

251

Self-Stabilization Feedback Loop

The light intensity propagating along the object beam direction, behind the crystal, can be written as: pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiqffiffiffiffiffiffiffiffiffiffi ð10:5Þ IS ¼ IS0 Z þ IR0 ð1  ZÞ  2 g Zð1  ZÞ IS0 IR0 cos j

where g is a parameter depending on mutual polarization and coherence relations between reference and object beams, and all other parameters have the usual meaning in this book. All measurements are carried out behind the crystal, so that bulk absorption need not be considered throughout. If no external electric field is applied to the crystal (the present case) we can show that j ¼ 0. To actively stabilize the setup it is necessary to modulate the phase of one of the interfering beams (the reference one in our case) in order to produce the harmonic term that is selectively detected and amplified for use as error signal in the feedback loop. In fact, a phase modulation of amplitude cd and frequency  (2p10 kHz here) in the phase j of Equation 10.5 will result in harmonic components in  where the amplitude of the first two ones are pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiqffiffiffiffiffiffiffiffiffiffi ð10:6Þ IS ¼ 4 g J1 ðcd Þ Zð1  ZÞ IS0 IR0 sin j qffiffiffiffiffiffiffiffiffiffi p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð10:7Þ IS2 ¼ 4 g J2 ðcd Þ Zð1  ZÞ IS0 IR0 cos j

with J1;2 being the Bessel function of order 1 or 2. The phase modulation in j is produced with the help of an electro-optical modulator (EOM in Figure 10.1) placed in the reference beam. In nonperturbated conditions and in the absence of an externally applied field on the crystal, j ¼ 0 and consequently IS ¼ 0. As soon as a perturbation on the setup appears that is faster than the crystal’s response time, the phase shift becomes j 6¼ 0 and also IS 6¼ 0. Therefore the lattter signal can be used as error signal in the negative-feedback stabilization loop. The operation of the selfstabilization feedback loop is discussed in Section 6.2 and is represented by the same block diagram in Figure 6.7. In this case the integrated error signal is a fundamental feature that largely improves the stabilization performance because it allows keeping the phase-shift condition j ¼ 0 and still coping with steadily growing perturbations. An element contributing to the good performance of stabilization is the choice of an error signal that is obtained from the object beam behind the crystal with the help of the polarizing components HMP, PBS, and HWP at the crystal output in Figure 10.1. Although this sampling of the output object beam will somewhat decrease the final IRD , it will avoid detecting the direct transmitted reference beam that is phase modulated and residually amplitude modulated, to some extent, because of unavoidable misalignment of the EOM. Such an amplitude-modulated (at the same frequency ) signal in the feedback loop would seriously interfere in the stabilization process. The experimental setup used in this experiment is relatively complicated, but its operation is very simple and can be carried out by nonspecialized technicians, once

252

VIBRATIONS AND DEFORMATIONS

the optical components are adjusted and fixed. The only adjustment left for the operator is to place the target in the correct position to have it adequately focused on the TV screen and somtimes to correct the illumination of the target by gently acting on the screw of a mirror in the setup. The analysis of the pattern of fringes can be carried out on its photographic image, or, alternatively, the pattern can be transferred to a personal computer for analysis with adequate standard commercial software. The use of a photorefractive crystal acting as a (nearly) real-time holographic recording reversible medium is essential in this experiment and allows overcoming most of the handicaps of classic holography. Thus the operator can forget that a hologram is being recorded somewhere, and all changes in the target can be observed almost in real time. The setup allows measurement alternatively of vibrations or deformations, with a very simple modification in the operation procedure, without any change in the setup. The photorefractive crystal (undoped Bi12 TiO20 ) used in this instrument has been chosen because of its advantageous properties compared to other possible materials: suitable spectral sensitivity, recording speed, diffraction efficiency, optical quality, availability on the international market, etc, plus other specific properties (anisotropic diffraction) that make it particularly interesting for our purposes. The actively stabilized optoelectronic circuit is also essential to enable the operation of this instrument in a moderately perturbated environment. The use of self-stabilization (and not just external reference-based stabilization) is essential to allow observation of welldefined interference patterns that would be otherwise hard to observe except for ocasional moments during the experiment. 10.2.4

Vibrations

The measurement of vibrations by time-average holographic interferometry is based on the fact that the diffraction efficiency of the hologram recorded (during a time much larger than the period of the vibration under analysis) by the light backscattered from a surface vibrating with amplitude d and frequency  can be written as: ZðdÞ ¼ Z0 jmj2 J02 ð4pd=lÞ

m¼

2b ð1 þ b2 Þ

b2  IR0 =IS0

ð10:8Þ

where Z0 is the diffraction efficiency (using equal intensity recording beams) of the surface at rest, jmj is the value of the visibility of the interference fringes, IR0 , IS0 are the intensities of the reference and object beams incident on the crystal, and J0 is the Bessel function of order zero. The holographically reconstructed target surface image is therefore superimposed on a pattern of dark and bright fringes corresponding to the different maxima and minima of the Bessel function as shown in Figure 10.5. The position of these fringes allows computation of the map of the local values of the amplitude vibration d over the target surface with the help of a table of Bessel functions as shown in the table of Figure 10.5. Note that each point of local

EXPERIMENTAL SETUP

253

Figure 10.5. Loudspeaker membrane (left) drived at 3.0 kHz and analyzed by the timeaverage holographic interferometry technique. The brighter areas are those at rest, the first dark fringe indicates a vibration amplitude of 0.12 mm, the second one 0.28 mm, the third one 0.44 mm, and so on according with data in the table (right) showing the amplitude d of the vibration associated with the minima (for J0 ðxÞ ¼ 0) and maxima in the pattern of fringes. Reproduced from [Frejlich and Garcia, 2000]

maximum amplitude of vibration in the membrane is at the center of a pattern of approximately concentric fringes. To evaluate the performance of this technique, the response of some points of local maximum amplitude of vibration in a loudspeaker membrane as a function of the applied voltage is shown in Figures 10.6 and 10.7 for two different frequencies. The vibration of a thin (0.2 mm thick) phosphorus-bronze metallic plate was also visualized with the above-described real-time holographic interferometry technique. The external plate border was tightly fixed to the external metallic ring of a commercial loudspeaker, using a plastic (PVC) double ring with a clear 79.5-mm internal diameter. The density of the plate was 6.24 g/cm3. The plate was painted with a thin retroreflective ink film to increase the amount of backscattered light collected by the optical setup and focused onto the photorefractive crystal. The

254

VIBRATIONS AND DEFORMATIONS

0.8

Amplitude (µm)

0.6

0.4

0.2

0 100

200

300

Applied voltage (mV)

Figure 10.6. Amplitude of vibration at a point of local maximum in the membrane of a loudspeaker as a function of the applied voltage for a signal of 4.2 kHz. Reproduced from [Frejlich and Garcia, 2000]

loudspeaker was used to excite the plate. Figures 10.8–10.10 show the interference patterns obtained for the frequencies of the electric signals feeding the loudspeaker that lead to the first, third, and fourth normal vibration modes, respectively. The amplitudes of the local maxima can be approximately estimated from the number of fringes and the table in Figure 10.5.

0.8

C Amplitude (µm)

0.6 A 0.4

0.2

0 0

0.2

0.4

0.6

0.8

0.10

Voltage (V)

Figure 10.7. Amplitude of vibration at two different points of local maximum in the membrane of a loudspeaker as a function of the applied voltage for a signal of 1.4 kHz. Reproduced from [Frejlich and Garcia, 2000]

EXPERIMENTAL SETUP

255

Figure 10.8. Time-average holographic interferometry pattern of a thin phosphorus-bronze metallic plate tightly fixed by its external border to a loudspeaker vibrating at 255 Hz. Reproduced from [Frejlich and Garcia, 2000]

10.2.5

Deformation and Tilting

Photorefractive crystals can be used as double-exposure recording media because the recording takes a finite time (inversely proportional to the total amount of light onto the crystal) so that it is possible to record the first image of the target under

Figure 10.9. Time-average holographic interferometry pattern of a thin phosphorusbronze metallic plate tightly fixed by its external border to a loudspeaker vibrating at 600 Hz. Reproduced from [Frejlich and Garcia, 2000]

256

VIBRATIONS AND DEFORMATIONS

Figure 10.10. Time-average holographic interferometry pattern of a thin phosphorus-bronze metallic plate tightly fixed by its external border to a loudspeaker vibrating at 800 Hz. Reproduced from [Frejlich and Garcia, 2000]

study and then the second image of the deformated target, as is usually done in classic interferometry. In the case of photorefractive crystals, however, the latter image has been recorded while the former image begins to fade: At a certain moment in this process both images reach similar intensities and a maximum contrast of the fringes arising from the interference of both wavefronts (first and second object images) is obtained as shown in Figures 10.11–10.13.

Figure 10.11. Double-exposure holographic interferometry of a tilted rigid plate. The smallest diagonal dimension of the small cells printed in the plate is approximately 4.8 mm.

EXPERIMENTAL SETUP

257

Figure 10.12. Double-exposure holographic interferometry of a rigid plate that was less tilted than in Fig. 10.11. The smallest diagonal dimension of the small cells printed in the plate is approximately 4.8 mm.

This pattern of fringes image can be recorded with a CCD TV camera (as is the case here) or any other adequate device, for further processing. The BTO crystal used in these experiments is particularly well suited because it exhibits a rather low dark conductivity that grants no sensible changes in the already recorded image once the light is switched off between both exposures.

Figure 10.13. Double-exposure holographic interferometry of a rigid plate that was more tilted than in Fig. 10.11. The smallest diagonal dimension of the small cells printed in the plate is approximately 4.8 mm.

258

VIBRATIONS AND DEFORMATIONS

Exercise: The fringes in the pattern of Figure 10.11 are not all parallel, thus indicating that there is also a deformation besides the tilting of the plate, but those in Figures 10.12 and 10.13 are rather regular, thus indicating that there was only tilting in these cases. Knowing that the light in the setup was the 633-nm He-Ne laser line and that the distance along the diagonal of the squares printed on the plate is 4.8 mm, calculate the angle of tilting for the three figures referred to above and verify that they are respectively 33 mrad, 15 mrad, and 66 mrad.

CHAPTER 11

FIXED HOLOGRAMS

11.1

INTRODUCTION

Photorefractive materials are essentially reversible real-time recording materials and consequently are not at all suitable for storing information unless the crystal is kept in the dark. It is nevertheless possible to fix holograms in some materials by using special techniques, thus allowing production of volume diffractive optical components to be used in practical applications under illumination. One such technique for hologram fixing is the use of double-doped LiNbO3 [Buse et al., 1998; Ren et al., 2004; Fujimura et al., 2005], where recording occurs at the shallower traps and the hologram is transferred, during recording, to the deeper traps where the hologram cannot be erased during readout. Fixing by inducing ferroelectric domain inversion by the combined action of light and an applied electric field was also demonstrated in Srx Ba1x Nb2 O6 crystals [Horowitz et al., 1993]. Another fixing technique uses high temperature compensation to substitute the initially recorded photosensitive hologram by an opposite-sign complementary ionic (assumed to be Hþ ) nonphotosensitive one in LiNbO3:Fe [Amodei and Staebler, 1971; Carrascosa and Agullo-Lopez, 1990; Yariv and Orlov, 1996; Me´ndez and Arizmendi, 1998; Miguel et al., 2000]. A similar procedure was successfully applied on undoped Bi12 SiO20 [Arizmendi, 1989] and Bi12 TiO20 [McCahon et al., 1989]. The development of a complementary fixed grating has been reported to occur even at room temperature in Bi12 SiO20 [Herriau and Huignard, 1986; Delboulbe et al., 1989]. Photorefractive Materials: Fundamental Concepts, Holographic Recording and Materials Characterization, By Jaime Frejlich Copyright # 2007 John Wiley & Sons, Inc.

259

260

FIXED HOLOGRAMS

The recording and compensation processes may be carried out simultaneously at high temperature, and this way very good results (up to fixed Z ¼ 0:16) were obtained in KNbO3:Fe [Tong et al., 1996]. We shall focus on a modification of the latter technique [Breer et al., 1998], where holographic recording and compensation are simultaneously carried out at moderately high temperature but recording is carried out in self-stabilized mode.

11.2

FIXED HOLOGRAMS IN LiNbO3

Fixed holograms in iron-doped LiNbO3 are made possible by first recording a large grating arising from electron photoexcitation at room temperature and then heating the sample to 120–135 to increase the mobility of positive Hþ ions in the crystal. In this way the Hþ ions completely neutralize the holographically recording space-charge grating. The sample is then allowed to cool down to room temperature, and a strong, spatially uniform white light is projected onto the sample to erase and phase shift the electronic grating to some extent. The result is an overall positive ion grating that is stable to illumination because Hþ is not photosensitive. The result is better if oxidized samples are used [Miguel et al., 2000; de Oliveira et al., 2003, 2005], although the holographic recording is much more time-consuming [de Oliveira et al., 2004]. The use of self-stabilized recording is important for reducing the environmental perturbations during long-time recording, but it does not allow recording of an electronic grating with an index of refraction modulation larger than that corresponding to Z ¼ 1 [de Oliveira et al., 2004]. This means that after compensation and development a lower (sometimes a much lower indeed) efficiency fixed grating results [de Oliveira et al., 2005]. To overcome such a limitation Breer and co-workers [Breer et al., 1998] proposed to carry out the simultaneous recording and compensation at moderately high temperature during self-stabilized holographic recording. In this way, if the operating temperature is adequately chosen, Z ¼ 1 is never reached because of the simultaneous compensation process. In principle the compensated recording can proceed until exhaustion of either Hþ or iron dopant. We describe here use of the above-described simultaneous recording and fixing process in crystals [Frejlich et al., 2006b] that would otherwise result in a rather low fixed hologram with the conventional three-step process (recording at room temperature—compensation at high temperature—development again at room temperature). It is essential here to select the adequate operating temperature: neither too high to compensate almost all the electronic grating so that there would be no diffracted light left to operate the feedback stabilization loop nor too low to slow down the process and make it impractical.

11.3

THEORY

The simultaneous recording and compensation process should be carried out at a temperature high enough to increase the mobility of Hþ (with volume concentration

THEORY

261

Hþ ) and allow compensation of the electron traps ðdonors ND  NDþ ¼ ½Fe2þ  and acceptors NDþ ¼ ½Fe3þ Þ spatial modulation arising from photoelectron excitation þ and retraping. The evolution of the first spatial harmonic components for traps ND1 þ and ions H1 , respectively, are ruled by the following coupled differential equations þ @ND1 ðtÞ þ iKvt ðtÞ þ xH Hþ þ ge ð1 þ xe ÞND1 1 ðtÞ ¼ k m e @t @Hþ þ 1 ðtÞ þ gH ð1 þ xH ÞHþ 1 ðtÞ þ xe ND1 ðtÞ ¼ 0 @t

ð11:1Þ ð11:2Þ

as already reported by Sturman and co-workers [Sturman et al., 1998] and slightly modified here for the case of a recording pattern of light, with visibility m and ~ In fact, the use of a self-stabilized moving with speed v along the grating vector K. holographic recording setup produces, in general, a running pattern of fringes and corresponding running hologram because of the phase mismatch between the unconstrained nonstabilized recorded hologram and the stabilized hologram as already reported in Section 6.2.1 and particularly in Section 6.3.2. The parameters ge ð1 þ xe Þ and gH ð1 þ xH Þ in Equations (11.1) and (11.2) represent the response constants associated to grating buildup by electrons and Hþ ions, respectively, whereas xe and xH represent the corresponding electric coupling, values of which depend on crystal parameters, and k is a constant. The parameters above are defined elsewhere [Sturman et al., 1998] qme n0 Ee0 qmH H0 gH ¼ Ee0 Eph ½Fe3þ  xe  i Eq ½Fe ED ðND Þeff xH ¼ Eq H 0 ge 

ð11:3Þ ð11:4Þ ð11:5Þ ð11:6Þ

where ½Fe  ½Fe2þ  þ ½Fe3þ  and Eq ¼

qðND Þeff KEe0

ðND Þeff ¼ ½Fe2þ ½Fe3þ =½Fe

ð11:7Þ

where mH and H0 are the mobility and average concentration of ions, respectively, q is the value of the electronic charge, E is the dielectric constant, and e0 is the permittivity of vacuum. The solution of the equations above leads to transient and stationary terms þ ¼ Nst eiKvt þ transients ND1

Hþ 1

¼ Hst e

iKvt

þ transients

ð11:8Þ

ð11:9Þ

262

FIXED HOLOGRAMS

We are interested just in the stationary terms, the amplitudes of which are k m ðgH ð1 þ xH Þ  i KvÞ ge gH ðxe þ xH Þ  K 2 v 2  i Kvðge þ gH Þ k m gH Hst ¼  ge gH ðxe þ xH Þ  K 2 v 2  i Kvðge þ gH Þ Nst ¼

ð11:10Þ ð11:11Þ

From Equations (11.10) and (11.11) we compute Hst gH qmH H0 =ðEe0 Þ  ¼ Nst gH ð1 þ xH Þ  iKv qmH H0 =ðEe0 Þ  iKv

ð11:12Þ

where the approximate sign in the right-hand side is for the xH 1 condition. Note that the gratings from electrons and from ions are phase shifted because of the term iKv. Otherwise, for standing holograms (v ¼ 0), the phase shift would be exactly p, that is, counterphase. In any case and for a sufficiently large ion concentration-ion mobility product such as qmH H0  jKvj Ee0

ð11:13Þ

the relation in Equation (11.12) simplifies to Hst  1 Nst

ð11:14Þ

which means that, in these conditions, the electron-based grating can be completely compensated during self-stabilized recording, even for the case of moving holograms, provided the ion concentration and their mobility are high enough. This means that an electron donor trap spatial modulation and a nonphotosensitive ionic spatial modulation that move synchronously with the pattern of fringes are produced. This is the fundamental feature enabling simultaneous recording and compensation at high temperature.

11.4

EXPERIMENT

Simultaneous recording and compensation were carried out on a 1.4-mm-thick Fe-doped LiNbO3 crystal with ½Fe2þ =½Fe3þ  ¼ 0:013, total iron concentration ½Fe  2  1019 cm3 , and total hydrogen ion concentration ½Hþ  ¼ 3:2  1017 cm3 . The sample was short circuited (as usual) with silver conductive glue and was placed in a copper holder surrounding the sample and in good thermal contact with a temperature-controlled heated massive copper cylinder as schematically represented in Figure 11.1.

EXPERIMENT

W

263

C L L

H

S

Figure 11.1. Experimental setup: S: massive copper cylinder with temperature-controlled heating element in direct thermal contact with the copper holder H supporting and surrounding the sample C. A thin Pyrex glass cylinder W to minimize heat losses and thermal convection, around the sample, allows laser beams L to go through. A flat heatisolating plate (not seen) covers the upper cylinder side. Reproduced from [Frejlich et al., 2006b]

A thin, approximately 8-cm-diameter, hollow Pyrex glass cylinder around the sample (with a flat heat-isolating cover that is not shown in the figure) minimizes heat losses and convection but allows the recording laser beams to go through. The recording light was the 514.5-nm expanded and collimated beam from an Arþ laser with fringe modulation jmj  1 and K ¼ 10 mm1 . The recording was carried out in the usual self-stabilized mode already described in Section 6.3.2, which allows one to carry out holographic recording even for hours without being affected by environemental perturbations. The error signal necessary to operate the feedback loop in the setup arises from the diffracted light (interfering with the other transmitted beam along the same direction behind the sample) from the remaining electronic grating that is not completely compensated by the nonphotosensitive ionic grating. It is therefore essential to keep the operating temperature high enough for the compensation to occur efficiently but not too high in order to avoid the recorded grating being completely compensated. A preliminary non-self-stabilized recording experiment was carried out on a more oxidized sample ð½Fe2þ =½Fe3þ   0:006Þ with similar total iron concentration to find out the adequate operating temperature. This experiment showed that 150 C is too high because the stabilization setup was not adequately operating, probably because the ionic compensation of the photosensitive electronic grating was too complete and there was not enough overall remaning grating to diffract light in order to operate the feedback loop in the stabilization setup. The 130 C–135 C temperature apparently allows a much higher remaining hologram that was actually perfectly suitable for operating the feedback. During self-stabilized holographic recording at high temperature the second harmonic (I 2 ) term in the irradiance behind the crystal is used as error signal (see Section 6.3.2) so that it isp kept at approximately zero by the feedback loop, whereas ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi the first harmonic ðI  / Zð1  ZÞÞ shows the evolution of the overall (electronic pffiffiffi plus ionic) hologram Z as reported in Figure 11.2 for a typical experiment. Note that I  begins growing and after a while reaches a roughly constant value, indicating an equilibrium between the electronic grating recording and ionic compensation rates.

264

FIXED HOLOGRAMS

Figure 11.2. Evolution of I  and I 2 during high-temperature self-stabilized holographic recording (and compensation) for a typical experiment. Reproduced from [Frejlich et al., 2006b]

After some time of recording at 130 C–135 C on our 1.4-mm-thick crystal, the whole chamber was allowed to cool down to room temperature. Then the sample was developed using a powerful white light spatially uniform source illuminating both sample sides, and the diffraction efficiency was measured from time to time, during development, using one of the recording beams, which are authomatically in Bragg condition. The variation from 130–135 C during recording to room temperature for development, however, usually produces some mechanical displacements requiring the angular adjustment of the sample to match Bragg conditions before diffraction efficiency measurement. The diffraction efficiency was measured as Z¼

Id Id þ It

ð11:15Þ

where I d and I t are the diffracted and transmitted beams. In this way interface losses and bulk absorption are not affecting Z. Figure 11.3 shows one such measurement leading to a stationary final value of Z  0:66 for the fixed grating. The coupled constant-thickness parameter is computed from the final Z pffiffiffiffiffiffiffi ½kdfix ¼ sin1 ð Zfix Þ

ð11:16Þ

for the fixed grating. The experimental ½kdfix for different recording/compensation times and same sample are reported in Table 11.1. It is interesting to note that the sample used in this experiment reached only Z ¼ 3% after fixing in the usual threestep (recording at room temperature—compensation at high temperature— development at room temperature) procedure, whereas it achieved Z  0:66 ðor ½kdfix ¼ 0:95 radÞ after 1 h of simultaneous self-stabilized recording/compensation

EXPERIMENT

265

70 60

η (%)

50 40 30 20 10 0 0

30

60

90 120 Time (min)

150

180

Figure 11.3. Diffraction efficiency of the overall grating during white light development as a function of development time. Note that the time scale depends on the overall development light intensity on the sample. Reproduced from [Frejlich et al., 2006b]

process. From data in Table 11.1 we may deduce that the sample becomes exhausted after 60 min of recording and that the saturation value for the fixed grating is actually Z  64–66%. A question arises about the possible interference of an absorption grating in the self-stabilization holographic recording process: It is easy to show that there is none. In fact, the absorption grating arises from trap modulation, that is, from the modulation of Fe2þ and Fe3þ concentration in the crystal. It is known however that, in the framework of first harmonic approximation [Gu¨enter and Huignard, 1988] the space-charge field Esc (in phase with the photorefractive grating) and the trap density NDþ (in phase or counterphase with the absorption grating) are related by iKe0 EEsc ¼ qNDþ

ð11:17Þ

TABLE 11.1. Fixed Diffraction Efficiency Recording Time (min) 8.5 17 34 34 60 120 3000

Fixed kd 0.22 0.36 0.52 0.55 0.95 0.95 0.81

Fixed Z (%) 5 11.8 25 26.3 66 66 64

266

FIXED HOLOGRAMS

which shows they are p=2-shifted, that is, the photovoltaic holographic grating is fph  p whereas the absorption grating holographic phase shift is fa  p=2. The corresponding phase shifts between the transmitted and diffracted beams at the crystal output are therefore jph ¼ fph  p=2  p=2 ja ¼ fa ¼ p=2

ð11:18Þ

ð11:19Þ

Because the second harmonic term I 2 / cos f is the error signal in the feedback loop, it is straightforward to realize that the contribution of the absorption grating to I 2 is null, so that we should not expect any interference of an absorption grating to the self-stabilization holographic recording here. The utility of volume hologram fixing for nonvolatile optical memories [Pauliat and Roosen, 1991] and optical component fabrication [Muller et al., 1994] is obvious, and photorefractives (specially LiNbO3) look particularly suitable for these purposes. The high angular and wavelength Bragg selectivity is at the basis of the main interest in these components, but they can also found rather unconventional and interesting applications, for example, as sources of light masks for atomic nanolithograhy [Mu¨tzel et al., 2003].

PART V

APPENDICES

268

APPENDICES

INTRODUCTION These appendices are intended to provide some general and practical tools for those who are willing to start with experimental work and are still not familiar with the problems involved in handling photorefractive materials and measuring some of their basic properties, like diffraction efficiency. There are two final sections, one dealing with a rather theoretical subject (physical meaning of some material parameters) and the other one providing general information about the operation of diode photodetectors that are the widespread and unexpensive tool for light measurement nowadays.

APPENDIX A

DETECTING A REVERSIBLE REAL-TIME HOLOGRAM

While recording a photorefractive hologram one should keep in mind that we are dealing with an almost real-time and reversible process. It is essentially different from recording on a photographic plate or any other nonreversible material. The speed of recording or erasing in photorefractives is roughly proportional to the total recording irradiance, so that, as we try to illuminate the hologram to observe diffraction, we may be erasing it. A low irradiance may slow down the process and by this means facilitate the observation of the phenomenon but, besides weakening the displayed hologram itself, it makes the recording proportionally more exposed to environmental perturbations, thus leading to a more unstable recording and consequently a poorer recording and a weaker hologram. Holographic recording is very sensitive to environmental perturbations, and we can take advantage of this feature to detect the presence of a hologram. Detection of diffraction is particularly difficult in relatively fast and poorly difractive materials like GaAs and Bi12 TiO20 or any other sillenite-type crystal, where diffraction efficiency may be Z  0:01 or even lower. Detecting a hologram is performed differently depending whether one is doing it by direct naked-eye observation or by an instrument-assisted technique. We shall, in the following, briefly describe both cases.

Photorefractive Materials: Fundamental Concepts, Holographic Recording and Materials Characterization, By Jaime Frejlich Copyright # 2007 John Wiley & Sons, Inc.

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270

A.1

APPENDIX A: DETECTING A REVERSIBLE REAL-TIME HOLOGRAM

NAKED-EYE DETECTION

The direct qualitative detection of the actual presence of a hologram can be carried out by naked eye, which is extremely useful because it is the first means we have for guiding our handling and adjusting of the setup for recording. Once a hologram, even a very weak one, is qualitatively detected, instrument-assisted quantitative means can be used to optimize the setup.

A.1.1

Diffraction

For the case of slow and highly diffracting materials like LiNbO3 the detection of a hologram being recorded is very simple because it is enough to switch off one of the recording beams and watch the diffraction of the other one. For faster materials, however, such a simple technique is not possible because the hologram is usually rapidly erased while exposed to one single beam and also because diffraction efficiency in faster materials is usually rather weak so that its visual detection may be jeopardized by the scattering of light from the sample itself or from other parts in the setup. It is also very difficult to detect the diffracted beam along with the transmitted beam propagating along the same direction without switching the latter off: For the example of Z  0:01, the transmitted beam is more than 100-fold larger than the diffracted one! To perform detection during recording it is therefore necessary to somehow reduce the transmitted beam without affecting the input recording beams. Sillenitetype crystals are particularly suited for such a task because of their anisotropic diffraction properties (see Chapter 5) that allow one to adjust the input polarization condition so as to produce diffracted and transmitted beams with different (even mutually orthogonal) polarization directions at the crystal output. In this case it is enough to put a simple polarizer sheet behind the crystal and adjust it so as to minimize the transmitted direct beam through the sample while the other beam is switched off. After the polarizer is adjusted, the other beam is switched on to allow the recording to proceed and, if a hologram is recorded, you should be able to see an unstable fluctuating light behind the polarizer that only appears when both beams are shining on the crystal. If the response time is too fast and/or the environment is not noisy enough, you should swiftly knock down on the setup table to artificially perturb the recording and see some fluctuation in the diffracted light. In practice it is not easy to have exactly ortogonally polarized beams, and it is even not necessary to: If we assume 80 between the transmitted and diffracted beam polarization directions, for instance, instead of 90 , the transmitted beam can be cut off while the diffracted beam will be reduced in 1  cos 10  3% only. A.1.2

Interference

It may sometimes be easier, however, to detect phase perturbations rather than diffraction efficiency variations. Let us recall the expression of the overall irradiance

INSTRUMENTAL DETECTION

271

along the signal beam behind the sample as formulated in Equation (6.1) IS ¼ IS0 ð1  ZÞ þ IR0 Z þ 2

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiqffiffiffiffiffiffiffiffiffi Zð1  ZÞ IS0 IR0 cosðj þ jN Þ

where we have substituted the phase modulation by a phase noise jN and assumed that the transmitted and diffracted beams are parallel polarized and of similar irradiances. In this case, and always for our Z  0:01 example, the diffracted beam (the second term in the right-hand side) is 100-fold lower than the transmitted one (the first term), whereas the interference term (the third term), which is the only term where the phase parameter j shows up, is roughly fivefold weaker than the transmitted beam. It is still hard, however, to see phase fluctuations in the interference term in such conditions without further reducing the transmitted beam. We should do it by just operating with orthogonally (or nearly orthogonally) polarized diffracted and transmitted beam condition. In this way it is possible to use a polarizer almost aligned with the diffracted beam polarization direction (and therefore almost perpendicular to that of the transmitted beam) at the crystal output. The weak diffracted beam is almost unaffected, but the transmitted beam is strongly reduced, the relative size of the interference term is therefore increased, and phase fluctuations in jN and/or in Z in the interference term are likely to be observed. A.2

INSTRUMENTAL DETECTION

By this we mean using a photodetector connected to an oscilloscope to detect fluctuations in the overall beam behind the crystal. Such fluctuations arise from pffiffiffi variations in Z in the diffracted beam (usually rather small) and/or in Z in the interference term. Variations in the phase shift j are also detected in the interference term and may be much faster than those in Z. Instrumental detection does not require a large visibility (that is, a comparatively large interference term) as in the case of direct visual detection because the oscilloscope is able to operate in ac mode so as to reject the dc signal from the stronger transmitted beam, provided the photodetector feeding the oscilloscope does not become saturated by the overall irradiance shining on it. It is always possible to use the more sophisticated phase modulation techniques described in Section 4.3 that are particularly suitable for the detection of the interference term. This technique is very sensitive and allows detection of extremely weak signals from very large background nearly dc signals. This technique is particularly convenient for crystals not exhibiting anisotropic diffraction effects so that the transmitted and diffracted beams are always parallel polarized and there are therefore no possibilities to play with the difference in polarization of the output transmitted and diffracted beams.

APPENDIX B

DIFFRACTION EFFICIENCY MEASUREMENT: REVERSIBLE VOLUME HOLOGRAMS

Diffraction efficiency is an important quantity in holography and is therefore something to be measured to start characterizing the hologram under analysis. Unfortunately, its measurement is usually much harder to carry out than most people believe it to be. The difficulties arise from:

The very high angular Bragg selectivity of the hologram

The rather high average index of refraction of the material

The reversible nature of the recording process B.1

ANGULAR BRAGG SELECTIVITY

Diffraction efficiency of volume gratings has been an active subject of research [Kogelnik, 1969; Marotz and Ringhofer, 1987; Solymar and Cooke, 1981; Yeh, 1993] in the last decades and, besides its academic interest and practical applications, its measurement is of the highest importance for the characterization of photosensitive materials in general and photorefractives in particular. Diffraction efficiency (Z) measurement in volume holograms, however, is usually neither straightforward nor free of errors. A usual source of error arises from the high Bragg angular selectivity instrinsic to thick volume holograms that leads to lower apparent efficiency values.

Photorefractive Materials: Fundamental Concepts, Holographic Recording and Materials Characterization, By Jaime Frejlich Copyright # 2007 John Wiley & Sons, Inc.

273

274

APPENDIX B: DIFFRACTION EFFICIENCY MEASUREMENT

It is usual to measure Z using a so-called ‘‘probe’’ beam either with the same or a different wavelength than the one used for holographic recording. In any case the incidence of this beam with wavelength l should be adjusted to fulfill Bragg condition as discussed in Section 4.1.1 K ¼ 2k sin y

K ¼ 2p=

k ¼ 2p=l

ðB:1Þ

where  is the spatial period of the hologram and y the incidence angle. The advantage of using a probe beam is that the irradiance can be chosen to be weak enough not to sensibly affect the recording/erasure process and a wavelength can even be chosen having a minimum effect on the material so that measurement can be performed even in reversible recording materials without perturbing the already recorded hologram or the hologram being recorded. However, a simple calculation shows that for a 2-mm-thick grating with K ¼ 10 mm1 ð  0:63 mm), for example, the angular Bragg selectivity (reducing Z from 1 to 0.5 in Eq. (4.21)) is approximately 0.1 mrad, which is too restrictive for the usual angular divergence (1 to 2 mrad) of commercial He-Ne lasers. It is possible to show that, except for lowefficiency gratings, the diffracted probe beam is not even proportional to Z, in which case the probe beam technique is not even useful for qualitative purposes. B.1.1

In-Bragg Recording Beams

It follows that for rather thick volume holograms, diffraction efficiency can be measured accurately, using only the recording beams themselves, because they are automatically ‘‘in Bragg’’ whatever their wavefront is. The handicap here arises from the fact that a sensible erasure may occur during measurement in reversible materials such as photorefractives. It is usually not even possible to decrease the irradiance with a filter because by doing so we may slightly change the beam wavefront and such a change may be enough to produce a sensible mismatch with the recorded hologram and therefore show a reduced Z. One possibility to reduce erasure during measurement without using a probe beam is to use a shutter to produce short pulses to minimize light exposure on the photosensitive material. B.1.2

Probe Beam

The diffraction efficiency of purely index of refraction thick volume gratings was formulated by Kogelnik [Kogelnik, 1969] and is ruled by pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sin2 x2 þ n2 Z¼ 1 þ x2 =n2 x  yKd=2 n¼

pn3eff reff Esc 2l cos y

d

ðB:2Þ ðB:3Þ ðB:4Þ

ANGULAR BRAGG SELECTIVITY

275

as reported in Section 4.1.1, where y is the angular departure from the Bragg condition, d is the hologram thickness, and l is the wavelength of the light used to measure diffraction. The value of the grating modulation is n, which is equivalent to jkj as derived from Equation (4.81) because it is here formulated for the particular case of photorefractive materials with reff and neff being the effective electro-optic coefficient and index of refraction, respectively and Esc being the space-charge electric field modulation amplitude. The actually measured diffraction efficiency must take into account the angular spectrum of plane waves of the laser beam used for measurement that, because of the gaussian shape of the beam intensity dsitribution, should be better formulated in a gaussian form, too [Goodman, 1968] 2

A ¼ A0 eðyyÞ =a

2

ðB:5Þ

where a is the angular spectrum bandwidth. For a commercial 10-mW He-Ne laser of 0.7-mm beam diameter from Uniphase, for example, the technical datasheet informs a  1:2 mrad. The practical diffraction efficiency measurement of a grating is computed as a coherent summation along the full spatial angular spectrum. For the case of a low-divergence beam with the rapidly varying phase difference term being averaged out with the stationary phase theorem [Born and Wolf, 1975] the diffraction efficiency expression is simplified to Z ¼ I d =I t0 d

with I ¼ A0

ðB:6Þ

yþp=2 Z

2

2

Z eðyyÞ =a dy

ðB:7Þ

yp=2

t0

and I ¼ A0

Zp=2

2

2

ey =a dy

ðB:8Þ

p=2

where I d is the diffracted irradiance and I t0 represents the whole beam irradiance behind the sample. Of course, it is always possible to expand and carefully collimate the laser beam to closely fulfill the Bragg condition but this procedure is rarely employed because it requires high-quality components, is time-consuming, and is rather cumbersome. To illustrate the calculations above we carried out an experiment on a thickvolume (1.5 mm) holographic grating of K  10=mm recorded in a Fe-doped lithium niobate photorefractive crystal (labeled LNB1 in Table 6.1) using the ordinary polarization of a 541.5 nm wavelength line of an Arþ laser. After being recorded the grating was fixed with the process reported in Chapter 11 that results in the subtitution of the original electronic grating by a nonphotosensitive positive ionic grating. The fixed grating is carefully replaced at the same position in the recording setup with the help of a specially prepared support. The measurement of diffraction efficiency was then carried out on the fixed grating without any risk of partially erasing the grating during measurement, using the same ordinary polarized

276

APPENDIX B: DIFFRACTION EFFICIENCY MEASUREMENT

514.5-nm-wavelength beams previously employed for recording, which ensures full in-Bragg condition, following the method described in Section B.3. The result was Z0514 ¼ 0:352

ðB:9Þ

which, from Equation (B.2), resulted in a grating modulation of n0514 ¼ 0:635

ðB:10Þ

The latter was converted for the 633-nm wavelength n0633 ¼ n0514 514:5=633 ¼ 0:516

ðB:11Þ

using the relation in Equation (B.4), where the relatively small effect of the wavelength on the refractive index was neglected for the sake of simplicity. Substituting the value in Equation (B.11) into Equation (B.2), with x ¼ 0, we got the theoretically fully Bragg-matched value Z0633 ¼ 0:244

ðB:12Þ

for the l ¼ 633 nm ordinarily polarized light. The sample was taken out from the holographic recording setup and placed in an auxiliary setup to measure the diffraction efficiency with an ordinarily polarized direct 633 nm He-Ne laser beam. The experimentally measured value was ½Z0633 exp ¼ 0:16. Substituting the latter value into Equation (B.6) together with the already computed value in Equation (B.11) and solving the corresponding equation with an appropriate algorithm, for y ¼ 0, we found out the beam divergence a ¼ 0:35 mrad for this laser beam. We are now able to check our results for the extraordinary polarization of the 633-nm laser beam. To do this, we rotated the laser for the extraordinary polarization and carried out diffraction efficiency measurements for different values of y as represented by the spots in Figure B.1. Then we converted the previously computed ordinary grating modulation in Equation (B.11) into the extraordinary polarization as follows: ne633 ¼ n0633 br

3

0

br ¼ ðne =n0 Þ ðr33 =r13 Þ cosð2b Þ ¼ 2:57

ðB:13Þ ðB:14Þ

with [Yariv, 1985]: ne ¼ 2:2, n0 ¼ 2:286, r33 ¼ 30:9 pm/V, and r13 ¼ 9:6 pm/V. We replace the resulting ne633 value into Equation (B.6) and plot (continuous curve) the diffraction efficiency in Figure B.1 as a function of y using the previously computed value a ¼ 0:35 mrad that characterizes the angular divergence of the laser beam in this setup. Figure B.1 shows a good agreement between experimental data and theory. Note that the theoretical curve is not mathematically fitted to, but just plotted together with, the experimental data in Figure B.1. The dashed curve in the same figure represents what would have been the theoretically measured value using an hypothetically zero divergence (a ! 0) probe laser beam.

ANGULAR BRAGG SELECTIVITY

277

1 0.8

h

0.6 0.4 0.2

0

0.5

1 q (mrad)

1.5

2

Figure B.1. Diffraction efficiency as a function of out-of-Bragg angle y in mrad for the measured data ( ), theoretically computed for a ¼ 0:35 mrad (continuous curve) and for a ! 0 (dashed curve). Reproduced from [de Oliveira and Frejlich, 2003]

It may be somewhat surprising that such a low angular divergence as a ¼ 0:35 mrad may lead to considerable errors if not adequately considered, as illustrated in Figure B.1. This fact is also clearly illustrated in Figure B.2, where the apparent n modulation computed from the measured (average) Z pffiffiffi n ¼ arcsin Z ðB:15Þ without taking into account the finite angular divergence of the measurement probe beam, is plotted as a function of n. Figure B.2 clearly shows that n is different from n and, still worse, that they are not even proportional except for low values of n.

2

u (rad)

1.5

1

0.5

0 0

0.5

1 u (rad)

1.5

2

Figure B.2. n, computed from Eq. (B.15), as a function of n for in-Bragg condition and same parameters as in Fig. B.1. Reproduced from [de Oliveira and Frejlich, 2003]

278

B.2

APPENDIX B: DIFFRACTION EFFICIENCY MEASUREMENT

REVERSIBLE HOLOGRAMS

The reversibility of photorefractive materials is an interesting and useful property but represents a serious drawback for Z measurement because of erasure during measurement. The use of continuous nonperturbative methods based on the in-Bragg recording beams is the best alternative. In this case phase modulation, with the frequency  of the modulation being very fast tsc  1 compared to the material response time tsc , as described in Section 4.3 is one of the best-suited techniques. This technique allows measurement of the first and second harmonic terms, along any one of the beams behind the crystal, which are formulated in Equations (4.166) and (4.167), respectively qffiffiffiffiffiffiffiffipffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi IS ¼ 4J1 ðcd Þ IS0 IR0 Zð1  ZÞ sin j qffiffiffiffiffiffiffiffipffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi IS2 ¼ 4J2 ðcd Þ IS0 IR0 Zð1  ZÞ cos j tan j ¼ 

IS J2 ðcd Þ IS2 J1 ðcd Þ

If the phase modulation amplitude is sufficently small cd 1, the recording will not be sensibly affected. From these harmonic terms it is possible to compute Z straightforwardly: 

2  2 2 IS IS þ ¼ IS0 IR0 Zð1  ZÞ 4J1 ðcd Þ 4J2 ðcd Þ

ðB:16Þ

The advantage of this technique is its real-time online capabilities. Its main drawback arises from its dependence on some parameters like the frequency response of the phase modulator, which is particularly delicate for the case of piezoelectric modulators because they are not very stable.

B.3

HIGH INDEX OF REFRACTION MATERIAL

Diffraction efficiency measurement, even using the in-Bragg recording beams, may be jeopardized by the relatively large thickness of the sample under analysis, with an enhanced effect due to the relatively high index of refraction exhibited by most photorefractive materials. In fact, in these conditions and for faces slightly deviating from the perfect parallel-plane condition, a lenslike effect is to be expected and the beam through the sample may be focused or defocused in different proportions for each one of both recording beams because they go through slightly different paths along the crystal. It is also possible that the recording beams may not be perfectly collimated ones, as illustrated in Figure B.3. In this case it is flawed to compute Z from the values of diffracted and transmitted irradiances because their values may be strongly dependent on their way through the

HIGH INDEX OF REFRACTION MATERIAL

279

IS DS

crystal

DR

0

IR

0

IS

IR

Figure B.3. Measurement of diffraction efficiency. The recording beams are not collimated, and the sample adds focusing/defocusing effects. The output irradiance along each one of the incident directions is the coherent addition of the transmitted and the diffracted beams. The two different detectors, with different responses, are centered on the same spot of the crystal. Reproduced from [de Oliveira and Frejlich, 2003]

sample and on the position along their propagation direction behind the sample where the detector is placed. It is always possible to carry out these measurements by using the total power of the beams, but for this purpose some focusing lenses should be used at the output of the sample or, alternatively, the recording beams should be reduced in size in order to be entirely collected into the detectors behind the sample. Neither of these possibilities is always practical, mainly if the recording process is in course and the same setup is used for recording and measurement. In fact, a reduced illuminated area in a photovoltaic or a photorefractive crystals under applied field is highly undesirable because of the buildup of screening charges. On the other hand, the use of a focusing lens may be interesting but could interfere with some online processing at the sample output. We shall here show that it is always possible to use the recording beams for directly computing Z even on lenslike samples. Let us assume the overall transmitted plus coherently added diffracted beams along any of the directions behind the sample are written as pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiqffiffiffiffiffiffiffiffiffiffi ðB:17Þ IS ¼ ISt0 ð1  ZÞ þ IRt0 Z  2 Zð1  ZÞ ISt0 IRt0 cos j ffiffiffiffiffiffiffiffiffi ffi q pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi IR ¼ IRt0 ð1  ZÞ þ ISt0 Z þ 2 Zð1  ZÞ ISt0 IRt0 cos j ðB:18Þ

where j is the phase shift between the transmitted and diffracted beams and Z is defined as Z¼

I t0 Z I t0 Z þ I t0 ð1  ZÞ

ðB:19Þ

280

APPENDIX B: DIFFRACTION EFFICIENCY MEASUREMENT

where I t and I d are the transmitted and diffracted irradiances, respectively. We assume high Bragg angular selectivity gratings exhibiting only one diffracted order. The formulation in Equation (B.19) is usually employed [de Oliveira and Frejlich, 2001a; Rupp, 1987; Garcia et al., 1996; Miguel et al., 2000] in order to get rid of bulk absorption, scattering in the sample volume, and interface losses that are not essential to diffraction itself. The responsivity of each one of the detectors at their corresponding positions is KR and KS , respectively, here including their (possible) different nature, electronics, and aging. In this case the voltages measured at each one of the detectors DS and DR are respectively VS ¼ KS IS

VR ¼ KR IR

ðB:20Þ

ðB:21Þ

It is important to realize that whatever the shape of the beams, the transmitted and the diffracted beams (along the same direction) have the same shape because the latter is just the holographic reconstruction of the transmitted wave. That is, each detector is always measuring a wavefront having a constant shape as shown in Figure B.3. To measure Z for the case of uncalibrated photodetectors (only linearity of the response is assumed) and in the presence of lenslike effect, we should proceed as follows: 1. First shut off the incident beam IR0 and let both detectors measure the respective transmitted and diffracted signals. VSS ¼ KS ISt0 ð1  ZÞ VRS

¼

KR ISt0 Z

ðB:22Þ

ðB:23Þ

2. Then shut off the other beam and repeat the measurement on the other detector. VSR ¼ KS IRt0 Z

VRR

¼

KR IRt0 ð1

 ZÞ

ðB:24Þ

ðB:25Þ

In the case that the measurement is carried out during recording or on a reversible recording material the operations above should obviously be carried out fast enough not to allow the hologram to be sensibly erased. Then from Equations (B.22)–(B.25) we compute    VSS VSR 1Z 2 ðB:26Þ ¼ Z VRS VRR which is a second-order equation in Z that is straightforwardly solved without any concern about the detectors’ responsivities or the beams’ shape. The only concern here is about the linear dc response of each detector (including its electronic), which is usually quite accurate for adequately designed photodetectors.

APPENDIX C

EFFECTIVELY APPLIED ELECTRIC FIELD

The use of material characterization techniques requiring the application of an external electric field on the sample has a serious drawback: A considerable discontinuity in the electric field inside the sample may be produced by electrodecontact problems. Some nonuniformity in the electric field may also arise from nonuniform photoconductivity in the sample. In fact, under the action of a gaussianshape spatial cross section light irradiance IðxÞ, as illustrated in Figure C.1, a similar gaussian cross section for the photoconductivity sðxÞ / IðxÞ is produced. A lower value for the associated electric field EðxÞ results at the region where the sample is more illuminated, according to the relation j ¼ sðxÞ EðxÞ ¼ constant as illustrated in Figure C.1. In this case, the field is not constant across the interelectrode distance and cannot be calculated from the applied voltage simply as voltage-to-interelectrode distance ratio. It is therefore necessary to define a parameter x to account for the actual value of the effective field at any point in the sample. It is hardly possible to theoretically predict the value of x: In general, it is experimentally computed from the measurement of some well-known parameter or from the fitting of some quantity the theoretical equation for which is reasonably well known. One such possibility is to measure the electro-optic coefficient at

Photorefractive Materials: Fundamental Concepts, Holographic Recording and Materials Characterization, By Jaime Frejlich Copyright # 2007 John Wiley & Sons, Inc.

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APPENDIX C: EFFECTIVELY APPLIED ELECTRIC FIELD

1.0 0.8

I

0.6 0.4

σ

0.2 0

E 0

0.5

1.0

1.5

2.0

Figure C.1. Effective field coefficient. The figure shows a gaussian cross section irradiance IðxÞ illuminating a photoconductive material in steady-state regime with constant photocurrent jðxÞ ¼ j, showing the resulting photoconductivity distribution sðxÞ and associated electric field EðxÞ. The coordinate x (in arbitrary units) is along the two electrodes on the sample, and all quantities represented (in ordinates) are also in arbitrary units.

different points along the interelectrode distance [Lemaire and Georges, 1992], and its variation allows one to directly deduce the variation of the effectively applied field because the electro-optic coefficient should actually be constant. Let us analyze an example like the one of Equation (3.58) describing the holographic phase shift fP as a function of the applied field E0 ¼ V0 =‘ and other parameters, where V0 is the applied voltage and ‘ is the interelectrode distance. According to the development in this section, the effective electric field at the point we are measuring is probably different from the theoretical value E0 , and a coefficient x should be included to take account of the nonuniform photoconductivity in the material. Equation (3.58) should therefore be written as 1 xE0 =ED ¼ tan fP 1 þ K 2 l2s þ K 2 l2s ðxE0 =ED Þ2 Þ

ðC:1Þ

where the coefficient x should be evaluated from the fit of the equation above to actual experimental data, as reported in Sections 9.1 and 9.2, or with the help of an auxiliary experiment as in Section 3.4.

APPENDIX D

PHYSICAL MEANING OF SOME FUNDAMENTAL PARAMETERS

It is interesting to get some insight on the meaning of some of the parameters that were defined during the development of the fundamental mathematical relations in the first part of this book. We shall focus on the Debye screening length, the diffusion coefficient, and the diffusion length. We shall not provide careful mathematical demonstrations but just discuss their meaning and where they originate from.

D.1 D.1.1

DEBYE SCREENING LENGTH Temperature

A gas in thermal equilibrium has molecules of mass m moving randomly in all directions with velocities u having different values. These velocities are likely to follow a so-called maxwellian distribution that, for one-single dimension model, takes the form 2

1mu

kB ¼ 1:38  1023 J= K f ðuÞ ¼ Ae 2kB T rffiffiffiffiffiffiffiffiffiffiffiffiffi Zþ1 m f ðuÞdu A¼n n¼ 2pkB T 1

ðD:1Þ ðD:2Þ

Photorefractive Materials: Fundamental Concepts, Holographic Recording and Materials Characterization, By Jaime Frejlich Copyright # 2007 John Wiley & Sons, Inc.

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APPENDIX D: PHYSICAL MEANING OF SOME FUNDAMENTAL PARAMETERS

where n is the number of molecules per unit volume and f du is the number of molecules per unit volume with velocities between u and u þ du. The width of the distribution is characterized by the constant T that we call ‘‘absolute temperature.’’ Let us compute the average kinetic energy in this distribution

Eav ¼

Zþ1

1

ðmu2 =2Þf ðuÞdu Zþ1

1

1 ¼ kB T 2

ðD:3Þ

f ðuÞdu

Defining an average velocity uth as 1 Eav ¼ mu2th 2

ðD:4Þ

rffiffiffiffiffiffiffiffi kB T uth ¼ m

ðD:5Þ

we deduce that

We should extend this result to 3D, in which case f ðuÞ and Eav become 12mðu2 þv 2 þw2 Þ=kB T

f ðu; v; wÞ ¼ A3 e



m A3 ¼ n 2pkB T

3 Eav ¼ kB T 2 D.1.2

3=2

ðD:6Þ ðD:7Þ

Debye Screening Length

It is possible to show [Chen, 1984] that a positive charge placed inside (but electrically isolated) an electron plasma (formed by equal density of positive— practically not moving—massive ions forming a stationary positive charge background and lighter negative moving electrons) is surrounded by a cloud of electrons so as to shield it. The size of this cloud is determined by the kinetic energy distribution of the electrons at a given temperature T: the most energetic electrons can stand farther away from the positive charge. This cloud of electrons decreases the electric potential produced by the positive charge following the relation ðrÞ ¼ ð0Þer=ls

l2s ¼

e 0 k B Te n1 q2

ðD:8Þ

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where r ¼ x2 þ y2 þ z2 is the 3D coordinate, ð0Þ is the electric potential at the position of the positive charge, n1 is the volume density of electrons far away from the charge perturbation, and Te is the electron cloud temperature computed from Section (D.1.1). Equation (D.8) shows that ls is the distance where the electric

DIFFUSION AND MOBILITY

285

potential decreases to 1=e of its maximum. In the case of photorefractives, the shielding depends on the effective concentration of photoactive centers so that n1 in Equation (D.8) is substituted by ðND Þeff in Equation (3.44), as deduced from the expression in Equation (3.47), and e0 for vacuum is substituted by Ee0 for the actual material medium.

D.2

DIFFUSION AND MOBILITY

Figure D.1 schematically represents the flux  of electrons going through a volume of stationary neutral atoms with volume density ni and a cross section s for fully absorving the electron momentum. Because of absorption, the flux is decreasing along coordinate x as @ ¼ sni  @x

ðD:9Þ

ðxÞ ¼ ð0Þex=lm

lm ¼

1 ni s

ðD:10Þ

where lm is called the ‘‘mean free path.’’ For electrons with velocity v the mean time between collisions is ðD:11Þ

t ¼ lm =v

Averaging over electrons with all possible velocities we compute the collision frequency average ðD:12Þ

f ¼ ni sv V Γ (x + dx)

A

Γ (x)

dx

Figure D.1. Volume A  dx with fixed ions of volume density ni of characteristic collision cross section s, receiving a flux  of electrons of mass me and velocity v.

286

APPENDIX D: PHYSICAL MEANING OF SOME FUNDAMENTAL PARAMETERS

From the equation of motion including these collisions we compute [Chen, 1984] the mobility and the associated diffusion coefficient for the electrons, which turn out to be jqj me f kB Te D¼ me f

ðD:13Þ

m¼

ðD:14Þ

To estimate the collision cross section (for fully absoved electron momentum) s let us assume a coulombian force F ¼ q2 =ð4pe0 r 2 Þ acting during an average time r0 =v and producing a 90 deviation on the electron, so that ðme vÞ ¼ me v ¼ jFr0 =vj  r0 

q2 4pe0 r0 v

q2 4pe0 me v 2

and s ¼ pr02 

ðD:15Þ ðD:16Þ

q4 16pe20 m2e v 4

ðD:17Þ

and the average collision frequency is f ¼ ni sv 

ni q4 16pe20 m2e v 3

ðD:18Þ

from the definition of conductivity [Feynman et al., 1964] s ¼ mqne ¼ 16pe0 ne From the equations above we conclude

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 kB Te =me ls

rffiffiffiffiffiffiffiffiffiffi kB Te 2 l me s   kB Te 162 p2 e0 kB Te 2 ¼ ni l6s LD ¼ Dt ¼ mt jqj q q 16pe0 m¼ q

ðD:19Þ

ðD:20Þ ðD:21Þ

APPENDIX E

PHOTODIODES

Photodiodes are essentially semiconductor interfaces of n- and p-type (p/n or n/p junction diodes) where electrons diffuse from the n-type to the p-type semiconductor and holes diffuse the other way, so that a depletion layer is formed on both sides of the interface. Because of the depletion layer on both sides of the interface being oppositely charged, a space-charge electric field and associated electric potential barrier V appear, as indicated in Figure E.1. Under the action of light, with photonic energy high enough to produce intrinsic (band-to-band) excitation in the semiconductor, an hole–electron pair is formed. If they are formed outside the depletion layer, they are likely to recombine in a rather short time. However, if they are formed inside the depletion layer, the electron and the hole are drifted along opposite directions because of the space-charge field and a current i0 appears that is proportional to the irradiance I i0 ¼ KI

ðE:1Þ

Sometimes an intermediate intrinsic layer is used (p-i-n diodes), which is intended to enlargen the depletion layer, as seen in Figure E.2, to allow most of the photogenerated electron–hole pairs to be actually produced inside this layer so that they directly contribute to i0 . In the absence of light, a drift current id is also produced (which depends on the temperature and the semiconductors’ nature and doping) by the thermal generation of hole–electron pairs in the depletion layer and is considered Photorefractive Materials: Fundamental Concepts, Holographic Recording and Materials Characterization, By Jaime Frejlich Copyright # 2007 John Wiley & Sons, Inc.

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288

APPENDIX E: PHOTODIODES ––– –– – –– – – – – – – – –– – – –– –

P

V

– – – – – – – – – –

++++ ++ ++++ +++ ++ + +++ ++++ ++ ++++ +++

N

E

d

Figure E.1. np-Junction showing the depletion layer and a diagram of the Schottky potential barrier.

a ‘‘dark noise.’’ The equilibrium is reached when the diffusion current idiff becomes high enough to counterbalance the drift current id , idiff ¼ id

ðE:2Þ

Under the action of a direct electric potential V, as shown in red in Figure E.3, the potential barrier is decreased from V to V  V (from continuous to dashed curve) and the diffusion current increases accordingly as shown in the figure. Under the action of a reverse potential, the potential barrier increases and the diffusion current decreases, following the Shockley equation idiff ¼ id eeV=kB T

ðE:3Þ

and the overall current i under light and reverse polarization is therefore i ¼ id eeV=kB T  id  KI ––– –– – –– – – – – – – – –– – – –– –

P

V

– – – – – – – – – –

i

++++ ++ ++++ +++ ++ + +++ ++++ ++ ++++ +++

ðE:4Þ

N

E d

Figure E.2. np-Junction showing the depletion layer including the intrinsic layer and a diagram of the Schottky potential barrier.

PHOTOVOLTAIC REGIME

289

Figure E.3. pn-Junction showing the depletion layer including the intrinsic layer and a diagram of the Schottky potential barrier. The dashed curve shows the potential barrier under a direct bias potential V indicated by the dashed arrow.

E.1

PHOTOVOLTAIC REGIME

The so-called photovoltaic operation regime is illustrated in Figure E.4. For the socalled loaded setup in A, Equation (E.4) turns into V=RL ¼ id ðeeV=kB T  1Þ  KI

ðE:5Þ

showing the nonlinear relation between the voltage V measured in the load resistance RL and the irradiance I. Linearity is only approximately achieved for V kB T=e. For the so-called open-circuit setup in Figure E.4, B, there is a logarithmic relation between I and the output voltage V: 0 ¼ id ðeeV=kB T  1Þ  KI kB T KI V¼ lnð þ 1Þ e id

ðE:6Þ

C in Figure E.4 shows the so-called ‘‘short-circuit’’ operation, where the current is actually exactly proportional to the irradiance i ¼ id ðe0  1Þ  KI

i ¼ KI

ðE:7Þ

290

APPENDIX E: PHOTODIODES

i

i

V

i0

i

V

i0

i0

RL

A

B

C

Figure E.4. Photovoltaic mode operation for photodiodes. Figure A shows its operation with a load RL , figure B shows the open-circuit operation, and figure C shows the short-circuit operation.

E.2

PHOTOCONDUCTIVE REGIME

The photoconductive operation mode is shown in Figure E.5, where a reverse bias voltage VB is allowed for and the corresponding equation is eðVVB Þ V ¼ id ðe kB T  bigÞ  KI RL

ðE:8Þ

The relation becomes linear only for the approximation V  id  KI RL

for

VB  V

ðE:9Þ

The term id in the right-hand side is the noise. There is not such a noise in the photovoltaic short-circuit setup, so that photoconductive diodes are considered to be noisier than photovoltaic but they are also faster because the reverse bias field also considerably reduces the capacitance of the depletion layer and the time constant RC is also proportionally reduced.

Figure E.5. Photoconductive mode operation for photodiodes. A reverse bias voltage VB (usually VB  V) is applied as shown, in order to increase speed and improve linearity of the response.

OPERATIONAL AMPLIFIER OPERATED

i

291

i

V

i0

Rf + AMP-OP –

V0

–

Figure E.6. Operational amplifier operated photodiode in the short-circuit photovoltaic regime.

E.3

OPERATIONAL AMPLIFIER OPERATED

The use of an operational amplifier (OA) in the photovoltaic short-circuit regime allows transformation of the current output into a voltage with controlable gain, always keeping the short-circuit operating regime, as illustrated in Figure E.6. In fact, the virtual ground at the OA input grants the short-circuit operation of the photodiode (V  0) with the output voltage (V0 ) being proportional to the light irradiance (I): V0 ¼ i Rf / I. Also, the very low OA output resistance is an interesting feature of this photodiode-amplifier device.

BIBLIOGRAPHY

Aguilar, M., Carrascosa, M., and Agullo´-Lo´pez, F. (1997). Optimization of selective erasure in photorefractive memories. J. Opt. Soc. Am. B 14: 110–115. Amodei, J. and Staebler, D. (1971). Holographic pattern fixing in electro-optic crystals. Appl. Phys. Lett. 18: 540–542. Anderson, D., Ao, V. D., Popovic, D., Popovic, Z., Romisch, S., and Sullivan, A. (2001). 70dB optical carrier suppression by two-wave mixing coupling in photorefractive media. Appl. Phys. B 72: 743–748. Apostolidis, A., Mallick, S., RouSde, D., Herriau, J., and Huignard, J. (1985). Polarization properties of phase gratings recorded in a Bi12 SiO20 crystal. Opt. Commun. 56: 73–78. Arizmendi, L. (1988). Simple holographic method for determination of Li/Nb ratio and homogeneity of LiNbO3 crystals. J. Appl. Phys. 64: 4654–4656. Arizmendi, L. (1989). Thermal fixing of holographic gratings in Bi12 SiO20 . J. Appl. Phys. 65: 423–427. Aubrecht, I., Ellin, H., Grunnet-Jepsen, A., and Solymar, L. (1995). Space-charge field in photorefractive materials enhanced by moving fringes: comparison of hole-electron transport models. J. Opt. Soc. Am. B. 12: 1918–1923. Barbosa, M.C. and Frejlich, J. (2003). Photorefractive fringe-locked running hologram analysis in three-dimensional space. J. Opt. A: Pure Appl. Opt. 5: S416–S419. Barbosa, M.C., Mosquera, L., and Frejlich, J. (2001). Speed and diffraction efficiency in feedback-controlled running holograms for photorefractive crystal characterization. Appl. Phys. B 72: 717–721.

Photorefractive Materials: Fundamental Concepts, Holographic Recording and Materials Characterization, By Jaime Frejlich Copyright # 2007 John Wiley & Sons, Inc.

293

294

BIBLIOGRAPHY

Barbosa, M.C., de Oliveira, I., Frejlich, J. (2002). Feedback operation for fringe-locked photorefractive running hologram. Opt. Commun. 201: 293–299. Bayvel, P., McCall, M., and Wright, R. (1988). Continuus method for measuring the electro– optic coefficient in Bi12 SiO20 and Bi12 GeO20 . Opt. Lett. 13: 27–29. Bian, S. and Frejlich, J. (1995). Actively stabilized holographic recording for the measurement of photorefractive properties of a Ti-doped KNSBN crystal. J. Opt. Soc. Am. B 12: 2060– 2065. Bian, S., Frejlich, J., Sugg, B., and Rupp, R. A. (1995). Multiple gratings in semi-insulating GaAs crystal. Opt. Commun. 115: 151–157. Boas, M. L. (1983). Mathematical Methods in the Physical Sciences, 2nd ed. John Wiley & Sons, New York. Boothroyd, S., Chrostowski, J., and O’Sullivan, M. (1989). Two-wave mixing by phase and absorption gratings in saturable absorbers. J. Opt. Soc. Am. B 6: 766–771. Born, M. and Wolf, E. (1975). Principles of Optics, 5th ed. Pergamon Press, Oxford. Breer, S., Buse, K., Peithmann, K., Vogt, H., and Kra¨tzig, E. (1998). Stabilized recording and thermal fixing of holograms in photorefractive lithium niobate crystals. Rev. Sci. Inst. 69: 1591–1594. Brost, G. A., Magde, K. M., Larkin, J. J., and Harris, M. T. (1994). Modulation dependence of the photorefractive response with moving gratings: numerical analysis and experiment. J. Opt. Soc. Am. B 11: 1764–1772. Brost, G., Norman, J., Odoulov, S., Shcherbin, K., Shumelyuk, A., and Taranov, V. (1998). Gain spectra of beam coupling in photorefractive semiconductors. J. Opt. Soc. Am. B 15: 2083–2090. Buse, K. (1997). Light-induced charge transport processes in photorefractive crystals I: Models and experimental methods. Appl. Phys. B 64: 273–291. Buse, K., Adibi, A., and Psaltis, D. (1998). Non-volatile holographic storage in doubly doped lithium niobate crystals. Nature 393: 665–668. Buse, K., Frejlich, J., Kuper, G., and Kra¨tzig, E. (1993). Dark build-up of holograms in BaTiO3 after recording. Appl. Phys. A 57: 437–440. Buse, K., Ka¨mper, S., Frejlich, J., Pankrath, R., and Ringhofer, K. H. (1995). Tilting of holograms in photorefractive Sr0:61 Ba0:39 Nb2 O6 crystals by self-diffraction. Opt. Lett. 21: 2249–2251. Bylsma, R. B. (1988). Photochromic gratings in photorefractive materials. Opt. Lett. 13: 853–855. Carrascosa, M. and Agullo-Lopez, F. (1990). Theoretical modelling of the fixing and developing of holographic gratings in LiNbO3. J. Opt. Soc. Am. B 7: 2317–2322. Cescato, L. and Frejlich, J. (1988). Self-diffraction for intrinsic optical modulation evolution measurement in photoresist. Appl. Opt. 27: 1984–1987. Cescato, L., Mendes, G. F., and Frejlich, J. (1987). Stabilized holographic recording using the residual real-time effect in a positive photoresist. Opt. Lett. 12: 982–983. Cescato, L., Mendes, G. F., and Frejlich, J. (1988). Fourier synthesis for fabricating blazed gratings using real-time recording effects in a positive photoresist. Appl. Opt. 27: 1988–1991. Chen, F. F. (1984). Plasma Physics and Controlled Fusion, 2nd ed. Plenum Press, New York.

BIBLIOGRAPHY

295

Collier, R. J., Burckhardt, C. B., and Lin, L. H. (1971). Optical Holography. Academic Press, New York. Cronin-Golomb, M. (1987). Analytic solution for photorefractive two beam coupling with time varying signal. In First Topical Meeting on Photorefractive Materials, Effects and Devices Los Angeles, pp. 142–145. Cronin-Golomb, M., Biernacki, A. M., Lin, C., and Kong, H. (1987). Photorefractive time differentiation of coherent optical images. Opt. Lett. 12: 1029–1031. Cudney, R. S., Bacher, G. D., Pierce, R. M., and Feinberg, J. (1992). Measurement of the photorefractive phase-shift. Opt. Lett. 17: 67–69. de Montchenault, G. H., Loiseaux, B., and Huignard, J. (1987). Amplification of high bandwidth signals through two-wave mixing in photorefractive Bi12 SiO20 crystals. Appl. Phys. Lett. 50: 1794–1796. de Oliveira, I. and Frejlich, J. (2000). Dielectric relaxation time measurement in absorbing photorefractive materials. Opt. Commun. 178: 251–255. de Oliveira, I. and Frejlich, J. (2001a). Gain and stability in photorefractive two-wave mixing. Phys. Rev. A 64: 033806. de Oliveira, I. and Frejlich, J. (2001b). Photorefractive running hologram for materials characterization. J. Opt. Soc. Am. B 18: 291–297. de Oliveira, I. and Frejlich, J. (2003). Diffraction efficiency measurement in photorefractive thick volume holograms. J. Opt. A: Pure Appl. Opt. 5: S428–S431. de Oliveira, I., Frejlich, J., Arizmendi, L., and Carrascosa, M. (2003). Holographic phase shift measurement during development of a fixed grating in lithium niobate crystals. Opt. Lett. 28: 1040–1042. de Oliveira, I., Frejlich, J., Arizmendi, L., and Carrascosa, M. (2004). Self–stabilized holographic recording in reduced and oxidized lithium niobate crystals. Opt. Commun. 229: 371–380. de Oliveira, I., Frejlich, J., Arizmendi, L., and Carrascosa, M. (2005). Nearly 100% diffraction efficiency fixed holograms in oxidized iron-doped LiNbO3 crystals using self-stabilized recording technique. Opt. Commun. 247: 39–48. Delaye, P., Montmorillon, L., Biaggio, I., Launay, J., and Roosen, G. (1997). Wavelength dependent effective trap density in CdTe: evidence for the presence of two photorefractive species. Opt. Commun. 134: 580–590. Delboulbe, A., Fromont, C., Herriau, J., Mallick, S., and Huignard, J. (1989). Quasinondestructive readout of holographically stores information in photorefractive Bi12 SiO20 crystals. Appl. Phys. Lett. 55: 713–715. dos Santos, P., Carvalho, J., and Frejlich, J. (2006). Photochromism, bleaching and photorefractive recording in undoped in undoped Bi12 TiO20 crystals in the visible and near infrared wavelength range. Opt. Mat. accepted for publication. dos Santos, P. A. M., Garcia, P. M., and Frejlich, J. (1989). Transport length, quantum efficiency and trap density measurement in Bi12 SiO20 . J. Appl. Phys. 66: 247–251. Feinberg, J., Heiman, D., Tanguay, A. R., and Hellwarth, R. W. (1980). Photorefractive effects and light–induced charge migration in barium titanate. J. Appl. Phys. 51: 1297–1305. Feynman, R., Leighton, R., and Sands, M. (1964). The Feynman lectures on Physics, 5th ed. Addison-Wesley Publishing Co., California Institute of Technology.

296

BIBLIOGRAPHY

Foote, P. D. and Hall, T. J. (1986). Influence of optical activity on two beam coupling constants in photorefractive Bi12 SiO20 . Opt. Commun. 57: 201–206. Fowles, G. R. (1975). Introduction to Modern Optics, 2nd ed. Holt, Rinehart and Winston, New York. Frejlich, J. and Garcia, P. M. (2000). Advances in real-time holographic interferometry for the measurement of vibrations and deformations. Opt. Lasers. Engineering 32: 515–527. Frejlich, J., Freschi, A., Garcia, P., Shamonina, E., Gayvoronsky, V., and Ringhofer, K. (2000). Feedback-controlled running holograms in strongly absorbing photorefractive materials. J. Opt. Soc. Am. B 17: 1517–1521. Frejlich, J. and Garcia, P. (1992). Quasipermanent hole-photorefractive and photochromic effects in Bi12 TiO20 crystals. Appl. Phys. A 55: 49–54. Frejlich, J., Garcia, P. M., and Cescato, L. (1989). Adaptive fringe-locked running hologram in photorefractive crystals. Opt. Lett. 14: 1210–1212. Frejlich, J., Garcia, P. M., and Cescato, L. (1990). Adaptive fringe-locked running hologram in photorefractive crystals: Errata. Opt. Lett. 15: 1247. Frejlich, J., Garcia, P. M., Ringhofer, K. H., and Shamonina, E. (1997). Phase modulation in two–wave mixing for dynamically recorded gratings in photorefractive materials. J. Opt. Soc. Am. B 14: 1741–1749. Frejlich, J., Kamshilin, A., and Garcia, P. (1992). Selective two-wave mixing in photorefractive crystals. Opt. Lett. 17: 249–251. Frejlich, J., Montenegro, R., Inocente Junior, N. R., dos Santos, P. V., Launay, J. C., Longeaud, C., and Carvalho, J. F. (2006a). Phenomenological characterization of photoactive centers in Bi12 TiO20 crystals. To be published. Frejlich, J., de Oliveira, I., Arizmendi, L., Carrascosa, M. (2006b). Fixed holograms in irondoped lithium niobate: simultaneous self-stabilized recording and compensation. To be published. Freschi, A. A. and Frejlich, J. (1994). Stabilized photorefractive modulation recording beyond 100% diffraction efficiency in LiNbO3: Fe crystals. J. Opt. Soc. Am. B 11: 1837–1841. Freschi, A. A. and Frejlich, J. (1995). Adjustable phase control in stabilized interferometry. Opt. Lett. 20: 635–637. Freschi, A. A., Garcia, P. M., and Frejlich, J. (1997). Charge-carriers diffusion length in photorefractive crystals computed from the initial hologram phase shift. Appl. Phys. Lett. 71: 2427–2429. Fujimura, R., Shimura, T., and Kuroda, K. (2005). Recording properties of the nonvolatile hologram in Ru and Fe doped LiNbO3 crystal. Trends Opt. Photonic Ser. 99: 582–586. Garcia, P. M., Buse, K., Kip, D., and Frejlich, J. (1995). Self-stabilized holographic recording in LiNbO3: Fe crystals. Opt. Commun. 117: 235–240. Garcia, P. M., Cescato, L., and Frejlich, J. (1989). Phase-shift measurement in photorefractive holographic recording. J. Appl. Phys. 66: 47–49. Garcia, P. M., Freschi, A. A., Frejlich, J., and Kra¨tzig, E. (1996). Scattering reduction for highly diffractive holograms in LiNbO3 crystals. Appl. Phys. B 63: 207–208. Georges, M., Thizy, C., Lemaire, P., Pauliat, G., Roosen, G., Alexeenko, I., Pedrini, G., Ryhon, S., Coates, N., Kelnberger, R., Schubach, H., and Ru¨mmler, R. (2005). Double-pulsed holographic interferometry with photorefractive crystals for vibration and shock analysis. Trends Opt. Photonic Ser. 99: 661–668.

BIBLIOGRAPHY

297

Gerwens, A., Buse, K., and Kra¨tzig, E. (1997). Alternating photocurrents and modulated photoconductivities in photorefractive sillenite crystals. Ferrolectrics 202: 203–210. Glass, A. M., von der Linde, D., and Negran, T. J. (1974). High-voltage bulk photovoltaic effect and the photorefractive process in LiNbO3. Appl. Phys. Lett. 25: 233. Godard, A., Pauliat, G., Roosen, G., and Ducloux, E. (2004). Relaxation of the single-mode emission conditions in extended-cavity semiconductor lasers with a self-organizing photorefractive filter. Appl. Opt. 43: 3543–3547. Goodman, J. W. (1968). Introduction to Fourier Optics. McGraw-Hill Book Company, San Francisco. Grabmaier, B. and Oberschmid, R. (1986). Properties of pure and doped Bi12 GeO20 and Bi12 SiO20 crystals. Phys. Stat. Sol. 96: 199–211. Grousson, R., Henry, M., and Mallick, S. (1984). Transport properties of photoelectrons in Bi12 SiO20 . J. Appl. Phys. 56: 224. Grunnet-Jepsen, A., Aubrecht, I., and Solymar, L. (1995). Investigation of the internal field in photorefractive materials and measurement of the effective electro–optic coefficient. J. Opt. Soc. Am. B 12: 921–929. Gu¨nter, P. and Huignard, J. P. (1988). Photorefractive Materials and Their Applications I, vol. 61 of Topics in Applied Physics. P. Gu¨nter and J.-P. Huignard, Springer-Verlag, Berlin. Hall, T. J., Jaura, R., Connors, L. M., and Foote, P. D. (1985). The photorefractive effect–a review. Prog. Quantum Electronics 10: 77–146. Henry, M., Mallick, S., and Roue`de, D. (1986). Propagation of light in an optically active electro-optic crystal of Bi12 SiO20 : measurement of the electro-optic coefficient. J. Appl. Phys. 59: 2650–2654. Herriau, J. and Huignard, J. P. (1986). Hologram fixing process at room temperature in photorefractive Bi12 SiO20 crystals. Appl. Phys. Lett. 49: 1140–1142. Hofmeister, R., Yariv, A., Kewitsch, A., and Yagi, S. (1993). Simple method of measuring the net photorefractive phase-shift and coupling constant. Opt. Lett. 18: 488. Holtmann, L., Kuper, G., Kra¨tzig, E., and Godefroy, G. (1992). Photovoltaic Holographic Recording in BaTiO3: Fe. Ferroelectrics 126: 51–56. Horowitz, M., Bekker, A., and Fisher, B. (1993). Image and hologram fixing methods with Srx Ba1x Nb2 O6 crystals. Opt. Lett. 18: 1964–1966. Horowitz, M., Kligler, D., and Fisher, B. (1991). Time–dependent behavior of photorefractive two– and four–wave mixing. J. Opt. Soc. B 8: 2204–2217. Huignard, J. and Marrakchi, A. (1981a). Two-wave mixing and energy transfer in Bi12 SiO20 crystals: application to image amplification and vibrational analysis. Opt. Lett. 6: 622–624. Huignard, J. P., Herriau, J.-P., and Micheron, F. (1975). Optical storage in LiNbO3: Fe with selective erasure capabilities. Rev. Phys. Applique´e 10: 417–423. Huignard, J. P., Herriau, J. P., and Valentin, T. (1977). Time average holographic interferometry with photoconductive electrooptic Bi12 SiO20 crystals. Appl. Opt. 16: 2796–2798. Huignard, J. P. and Marrakchi, A. (1981b). Coherent signal beam amplification in two-wave mixing experiments with photorefractive Bi12 SiO20 crystals. Opt. Communo 38: 249–254. Imbert, B., Rajbenbach, H., Mallick, S., Herriau, J., and Huignard, J. (1988). High photorefractive gain in two-beam coupling with moving fringes in GaAs: Cr crystals. Opt. Lett. 13: 327–329.

298

BIBLIOGRAPHY

Jariego, F. and Agullo´-Lo´pez, F. (1990). Monotonic versus oscillatory behaviour during holographic writing in photorefractive photovoltaic materials. Opt.Commun. 76: 169–172. Kamshilin, A. and Petrov, M. (1985). Continuous reconstruction of holographic interferograms through anisotropic diffraction in photorefractive crystals. Opt. Commun. 53: 23–26. Khoury, J., Ryan, V., Woods, C., and Cronin-Golomb, M. (1991). Photorefractive optical lockin detector. Opt. Lett. 16: 1442–1444. Kittel, C. (1996). Introduction to Solid State Physics, 7th ed. John Wiley & Sons, Inc., New York. Kleider, J. P. and Longeaud, C. (1995). Solid State Phenomena. Scitec Publications Ltd, edited by H. Neber-Aeschbacher. Klein, M. B. (1984). Beam coupling in undoped GaAs at 1.06 mm using the photorefractive effect. Opt. Lett. 9: 350–352. Kogelnik, H. (1969). Coupled wave theory for thick hologram gratings. Bell Syst. Tech. J. 48: 2909–2947. Korneev, N., Mayorga, D., Stepanov, S., Veenhuis, H., Buse, K., Kuper, C., Hesse, H., and Kra¨tzig, E. (1999). Holographic and non-steady-state photocurrent characterization of photorefractive barium-calcium titanate. Opt. Commun. 160: 98–102. Kukhtarev, N. V., Markov, V. B., Odulov, S. G., Soskin, M. S., and Vinetskii, V. L. (1979a). Holographic storage in electrooptic crystals. I. Steady state. Ferroelectrics 22: 949–960. Kukhtarev, N. V., Markov, V. B., Odulov, S. G., Soskin, M. S., and Vinetskii, V. L. (1979b). Holographic storage in electrooptic crystals. II. Beam-coupling-light amplification. Ferroelectrics 22: 961–964. Kumar, J., Albanese, G., and Steier, W. (1987). Measurement of two-wave mixing gain in GaAs with a moving grating. Opt. Commun. 63: 191–193. Lanfredi, S., Carvalho, J.F., and Hernandes, A. (2000). Electric and dielectric properties of Bi12 TiO20 single crystals. J. Appl. Phys. 88: 283–287. Leigh, W., Larkin, J., and Brown, R. (1994). Characterization of Czochralski- and hydrothermal-grown Bi12 SiO20 . J. Appl. Phys. 76: 660–666. Lemaire, P. and Georges, M. (1992). Electrooptic coefficient measurements: correction of electric-field inhomogeneities in the transverse configuration. Opt. Lett. 17: 1411–1413. Liu, D. T. H. and Cheng, L. J. (1989). Polarization properties of degenerate four-wave mixing in gaas. J. Opt. Soc. Am. B 6: 1554. Lobato, A., Lanfredi, S., Carvalho, J.F., and Hernandes, A. (2000). Synthesis, crystal growth and characterization of g-phase bismuth titanium oxide with gallium. Mat. Res. 3: 92–96. Longeaud, C. and Kleider, J. P. (1992). General analysis of the modulated photocurrent experiment including the contribution of holes and electrons. Phys. Rev. B 45: 11672– 11684. Longeaud, C., Kleider, J. P., Kaminski, P., Koslowski, R., Pawlowski, M., and Cwirko, R. (1999). Investigation of defect levels in semi-insulating materials by modulated and transient photocurrent: comparison of methods. Semiconductor Science Technology 14: 747. MacQuigg, D. R. (1977). Hologram fringe stabilization method. Appl. Opt. 16: 291–292. Mallick, S. and Roue`de, D. (1987). Influence of the polarization direction on the two–Beam coupling in photorefractive Bi12 SiO20 : diffusion regime. Appl. Phys. B 43: 239–245.

BIBLIOGRAPHY

299

Mallick, S., Roue`de, D., and Apostolidis (1987). Efficiency and polarization characteristics of photorefractive diffraction in a Bi12 SiO20 crystal. J. Opt. Soc. Am. B 4: 1247–1259. Malovickko, G., Grachev, V., and Schirmer, O. (1999). Interrelation of intrinsic and extrinsic defects—congruent, stoichiometric, and regularly ordered lithium niobate. Appl. Phys. B 68: 785–793. Maniloff, E. S. and Johnson, K. M. (1993). Effects of scattering on the dynamics of holographic recording and erasure in photorefractive lithium niobate. J. Appl. Phys. 73: 541–547. Marfaing, Y. (1999). State of the art and prospects of photorefractive CdTe. J. Crys. Grow 197: 707–717. Marinova, V., Lin, S., Sainov, V., Gospodinov, M., and Hsu, K. (2003). Light-induced properties of Ru-doped Bi12 TiO20 crystals. J. Opt. A Pure Appl. Opt. 5: S500–S506. Marotz, J. and Ringhofer, K. H. (1987). Reading volume gratings under the Bragg angle. Opt. Commun. 64: 245–247. Marrakchi, A., Johnson, R. V., and Tanguay, A. R. (1986). Polarization properties of photorefractive diffraction in electrooptic and optically active sillenite crystals (Bragg regime). J. Opt. Soc. Am. B 3: 321–336. McCahon, S., Rytz, D., Valley, G. C., Klein, M. B., and Wechsler, B. A. (1989). Hologram fixing in Bi12 TiO20 using heating and an ac electric field. Appl. Opt. 28: 1967–1969. Me´ndez, A. and Arizmendi, L. (1998). Maximum diffraction efficiency of fixed holograms in lithium niobate. Opt. Mat. 10: 55–59. Miguel, E., Limeres, J., Carrascosa, M., and Arizmendi, L. (2000). Study of developing thermal fixed holograms in lithium niobate. J. Opt. Soc. Am. B 17: 1140–1146. Moharan, M. G., Gaylord, T. K., Magnusson, R., and Young, L. (1979). Holographic grating formation in photorefractive crystals with arbitrary electron transport lengths. J. Appl. Phys. 50: 5642–5651. Mollenauer, L. F. and Tomlinson, W. J. (1977). Piecewise interferometric generation of precision gratings. Appl. Opt. 16: 555–557. Montenegro, R., Inocente Junior, N. R., and Frejlich, J. (2006). New possibilities for the measurement of wavelength-resolved photocondutivity. Rev. Sci. Inst. 77: 043905-1– 043905-6. Mosquera, L. and Frejlich, J. (2002). Enhanced photocurrent in photo-emf experiments in pure and doped absorbing photorefractive crystals. J. Opt. Soc. Am. B 19: 2904–2910. Mosquera, L., de Oliveira, I., Frejlich, J. (2001). Dark conductivity, photoconductivity, and light-induced absorption in photorefractive sillenite crystals. J. Appl. Phys. 90: 2635–2641. Muller, R., Santos, M. T., Arizmendi, L., and Cabrera, J. M. (1994). A narrow–band interference filter with photorefractive LiNbO3. J. Phys. D Appl. Phys. 27: 241–246. Murfin, A. and Dewhurst, R. (2002). Estimation of wall thinning in mild steel using laser ultrasound Lamb waves and a non-steady-state photo-emf detector. Ultrasonics 40: 777–781. Mu¨tzel, M., Rasbach, U., Merschede, D., Burstedde, C., Braun, J., Kunoth, A., Peithmann, K., and Buse, K. (2003). Atomic nanofabrication with complex light fields. Appl. Phys. B 77: 1–9. Nakamura, A., Shimura, T., and Kuroda, K. (1997). Stabilization of an externally-pumped phase conjugator by the control of mean phases of incident beams. Opt. Commun. 135: 337–341. Nee, I., Mu¨ller, M., Buse, K., and Kra¨tzig, E. (2000). Role of iron in lithium-niobate crystals for the dark-storage time of holograms. J. Appl. Phys. 88: 4282–4286.

300

BIBLIOGRAPHY

Neumann, D. B. and Rose, H. W. (1967). Improvement of recorded holographic fringes by feedback control. Appl. Opt. 6: 1097–1104. Nye, J. F. (1979). Physical Properties of Crystals. Oxford University Press, Oxford. Oberschmid, R. (1985). Absorption Centers of Bi12 GeO20 and Bi12 SiO20 Crystals. Phys. Stat. Sol. (a) 89: 263–270. Odoulov, S. G., Shcherbin, K. V., and Shumeljuk, A. N. (1994). Photorefractive recording in BTO in the near infrared. J. Opt. Soc. Am. B 11: 1780–1785. Oheda, H. (1981). Phase-shift analysis of modulated photocurrent: Its application to the determination of the energetic distribution of gap states. J. Appl. Phys. 52: 6693– 6700. Pauliat, G., Allain, M., Launay, J. C., and Roosen, G. (1987). Optical evidence of a photorefractive effect due to holes in Bi12 GeO20 crystals. Opt. Commun. 61: 321–324. Pauliat, G. and Roosen, G. (1991). New advances in photorefractive holographic memories. Int. J. Opt. Comput. 2: 271–291. Pedersen, H. and Johansen, P. (1999). Space-charge wave theory of photorefractive parametric amplification. J. Opt. Soc. Am. B 16: 1185–1188. Petrov, M. P., Sokolov, I. A., Stepanov, S. I., and Trofimov, G. S. (1990). Non-steady-state photo-electromotive-force induced by dynamic gratings in partially compensated photoconductors. J. Appl. Phys. 68: 2216–2225. Petrov, M. P., Stepanov, S. I., and Trofimov, G. S. (1986). Time-varying emf in a nonuniform illuminated photoconductor. Sov. Tech. Phys. Lett. 12: 379. Pillonnet, A., Garoum, M., and Ongano, R. (1995). Arrhenius plots of conductivity for semiconductors provided with discrete donors and acceptors distributed arbitrarily in the gap. J. Phys. D Appl. Phys. 28: 149–159. Prokofiev, V. V., Carvalho, J. F., Andreeta, J. P., Gallo, N. J. H., Hernandes, A. C., Frejlich, J., Freschi, A. A., Garcia, P. M., Maracaiba, J., Kamshilin, A. A., and Jaaskelainen, T. (1995). Growth and characterization of photorefractive Bi12 TiO20 single crystals. Crystal Res. Tech. 30: 171–176. Ramos-Garcia, R., Chiu-Zarate, R., Nolte, D., and Melloch, M. (2003). Measurement of absorption coefficient and refractive index changes without the Kramers–Kronig relation in photorefractive quantum wells of GaAs. J. Opt. A Pure Appl. Opt. 5: S514–S517. Refregier, P., Solymar, L., Rajbenbach, H., and Huignard, J. (1985). Two-beam coupling in photorefractive Bi12 SiO20 crystals with moving grating: theory and experiments. J. Appl. Phys. 58: 45–57. Ren, L., Liu, L., Liu, D., and Yao, B. (2004). Dynamic characteristics of holographic recording and fixing in LiNbO3: Ce: Cu crystal. Opt. Commun. 238: 363–369. Riehemann, S., Rickermann, F., Volkov, V., Egorysheva, A., and von Bally, G. (1997). Optical and photorefractive characterization of BTO crystals doped with Cd, Ca, Ga and V. J. Nonlinear Opt. Phys. Mat. 6: 235–249. Ringhofer, K., Kamenov, V., Sturman, B., and Chernykh, A. (2000). Shaping of photorefractive two-wave coupling by fast phase modulation. Phys. Rev. E 61: 2029–2037. Rose, A. (1963). Concepts in Photoconductivity and Allied Problems. Interscience Publishers, New York. Rupp, R. (1986). Microphotometric investigations of thick refractive index gratings in photorefractive crystals. Appl. Phys. B 41: 153–168.

BIBLIOGRAPHY

301

Rupp, R. A. (1987). Investigation of the dynamic recording process of thick refractive index gratings in photorefractive crystals by the microphotometric method. Appl. Phys. B 42: 21–29. Shamonina, E., Ringhofer, K. H., Garcia, P. M., Freschi, A. A., and Frejlich, J. (1997). Shapeasymmetry of the diffraction efficiency in Bi12 TiO20 crystals: the simultaneous influence of absorption and higher harmonics. Opt. Commun. 141: 132–136. Shepelevich, V., Shandarov, S., and Mandel, A. (1990). Light diffraction by holographic gratings in optically active photorefractive piezocrystals. Ferroelectrics 110: 235– 249. Shimura, T., Grappin, F., Delaye, P., Iwamoto, S., Arakawa, Y., Kuroda, K., and Roosen, G. (2004). Simultaneous determination of the index and absorption gratings in multiple quantum well photorefractive devices designed for lase ultrasonic sensor. Opt. Commun. 242: 7–12. Simmons, J. and Taylor, G. (1971). Nonequilibrium steady-state statistics and associated effects for insulators and semiconductors containing an arbitrary distribution of traps. Phys. Rev. B 4: 502–511. Sochava, S., Buse, K., and Kra¨tzig, E. (1993). Non-steady-state photocurrent technique for the characterization of photorefractive BaTiO3. Opt. Commun. 98: 265–268. Sochava, S., Buse, K., and Kra¨tzig, E. (1994). Characterization of photorefractive KNbO3: Fe by non-steady-state photocurrent techniques. Opt. Commun. 105: 315–319. Solymar, L. and Cooke, D. (1981). Volume Holography and Volume Gratings. Academic Press, London. Sommerfeldt, R., Holtman, L., Kra¨tzig, E., and Grabmaier, B. C. (1988). Influence of Mg doping and composition on the light-induced charge transport in LiNbO3. Phys. Stat. Sol. 106: 89–98. Staebler, D. L. and Amodei, J. J. (1972). Coupled-wave analysis of holographic storage in LiNbO3. J. Appl. Phys. 43: 1042–1049. Stepanov, S. (2001). Photo-electromotive-force effect in semiconductors. In H.S.Nalwa, editor, Semiconductor Devices vol. 2 of Handbook of Advanced Electronic and Photonic Materials and Devices chapter 6, pp. 205–272. Academic Press, San Diego, CA. Stepanov, S., Korneev, N., Gerwens, A., and Buse, K. (1998). Self-diffraction from the free surface relief gratings in a photorefractive Bi12 TiO20 crystal. Appl. Phys. Lett. 72: 879– 881. Stepanov, S., Kulikov, V., and Petrov, M. (1982). Running holograms in photorefractive Bi12 TiO20 crystals. Opt. Commun. 44: 19–23. Stepanov, S. and Petrov, P. (1988). Photorefractive Materials and Their Applications I, vol. 61 of Topics in Applied Physics, chapter 9, pp. 263–289. P. Gu¨nter and J.-P. Huignard, Springer-Verlag Berlin. Stepanov, S., Sokolov, I., Trofimov, G., Vlad, V., Popa, D., and Apostol, I. (1990). Measuring vibration amplitudes in the picometer range using moving light gratings in photoconductive GaAs: Cr. Opt. Lett. 15: 1239–1241. Strohkendl, F. P., Jonathan, J. M. C., and Hellwarth, R. W. (1986). Hole-electron competition in photorefractive gratings. Opt. Lett. 11: 312–314. Sturman, B., Carrascosa, M., Agullo-Lopez, F., and Limeres, J. (1998). Theory of hightemperature photorefractive phenomena in LiNbO3 crystals and applications to experiment. Phys. Rev. B 57: 12792–12805.

302

BIBLIOGRAPHY

Sturman, B., Mann, M., Otten, J., and Ringhofer, K. (1993). Space–charge waves and their parametric excitation. J. Opt. Soc. Am. B 10: 1919–1932. Sturman, B., Shamonina, E., Mann, M., and Ringhofer, K. (1995). Space–charge waves in photorefractive ferroelectrics. J. Opt. Soc. Am. B 12: 1642–1650. Tao, S., Song, Z. H., and Selviah, D. R. (1994). Bragg-shift of holographic gratings in photorefractive Fe: LiNbO3 crystals. Opt. Commun. 108: 144–152. Thizy, C., Stockman, Y., Lemaire, P., Houbrechts, Y., Mazzoli, A., Georges, M., Mazy, E., Tychon, I., Doyle, D., and Ulbrich, G. (2005). Qualification of large reflectors in space environment with a holographic camera based on a BSO crystal. Trends Opt. Photonic Ser. 99: 707–713. Tomita, Y., Bergquist, J., and Shibata, M. (1993). Photorefractive properties of undoped, Cr-doped, and Cu-doped potassium sodium strontium barium niobate crystals. J. Opt. Soc. Am. B 10: 94–99. Tong, X., Zhang, M., Yariv, A., and Agranat, A. (1996). Thermal fixing of volume holograms in potassium niobate. Appl. Phys. Lett. 69: 3966–3968. Trofimov, G. S. and Stepanov, S. I. (1986). Time-dependent holographic currents in photorefractive crystals. Sov. Phys. Solid State 28: 1559–1562. Valley, G. (1983). Erase rates in photorefractive materials with two photoactive species. Appl. Opt. 22: 3160–3164. Valley, G. (1984). Two-wave mixing with an applied field and a moving grating. J. Opt. Soc. Am. B. 1: 868–873. Valley, G. (1986). Simultaneous electron/hole transport in photorefractive materials. J. Appl. Phys. 59: 3363–3366. Vazquez, R., Neurgaonkar, R., and Ewbank, M. (1992). Photorefractive properties of SBN: 60 systematically doped with rhodium. J. Opt. Soc. Am. B 9: 1416–1427. Verstraeten, D. (2002). Les II–VI Photore´fractifs dans la Bande 0.6–1.5 mm pour l’Enregistrement Holographique Dynamique. PhD thesis, Universites de Liege et de Bordeaux— Facultes des Sciences. Verstraeten, D., Longeaud, C., Mahmoud, A. B., von Bardeleben, H., Launay, J. C., Viraphong, O., and Lemaire, P. (2003). A combined EPR and modulated photocurrent study of native defects in Bridgman grown vanadium doped cadmium telluride: the case of the telluride antisite. Semicond. Sci. Technol. 18: 919–926. Vormann, H., Weber, G., Kapphan, S., and Kra¨tzig, E. (1981). Hydrogen as origin of thermal fixing in LiNbO3: Fe. Solid State Commun. 40: 543–545. Walsh, K., Hall, T. J., and Burge, R. E. (1987). Influence of polarization state and absorption gratings on photorefractive two-wave mixing in GaAs. Opt. Lett 12: 1026– 1028. Wang, C.-C., Trivedi, S., Jin, F., Jia, K., He, H., Elliott, G., and Khurgin, J. (2002). Highprecision measurement of optical frequency differences between Q-switched laser pulses using photo-electromotive-force sensors. J. Opt. Soc. Am. B 19: 177–182. Webb, D. and Solymar, L. (1991). The effects of optical activity and absorption on two-wave mixing in Bi12 SiO20 . Opt. Commun. 83: 287–294. Weis, R. S. and Gaylord, T. K. (1985). Lithium niobate: Summary of physical properties and crystal structure. Appl. Phys. A 37: 191–203.

BIBLIOGRAPHY

303

Wiegel, M. E. and Becla, P. (2004). Comparative analysis of electro-optic properties in bismuth silicate grown by the Czochralski, Bridgman–Stockbarger, and hydrothermal techniques. Opt. Mat. 26: 471–478. Yariv, A. (1985). Optical Electronics, 3rd int. ed. Holt, Rinehart and Winston, New York. Yariv, A. (1988). Optical Electronics, 3rd ed. John Wiley & Sons, New York. Yariv, A. and Orlov, S. S. (1996). Holographic storage dynamics in lithium niobate: theory and experiment. J. Opt. Soc. Am. B 13: 2513–2523. Yeh, P. (1993). Introduction to Photorefractive Nonlinear Optics. Wiley Series in Pure and Applied Optics. John Wiley & and Sons, New York.

INDEX

Absorption bulk, in nonstationary holograms, 108 coefficient, 169 lighgt-induced, 38, 40, 165 BTO, 166, 167, 168 Arrhenius BTO, 41, 190 CdTe, 23 Bandgap schema 22, 25 BaTiO3 sensitivity, holographic, 188 Bragg selectivity, 88, 273 BSO sensitivity, holographic, 188 BTO bandgap schema, 25 conductivity, dark, 190 electro-optic coefficients, 14 holograms erasure, 193 hole-electron competition, 191 phase shift, initial, 227

phase shift, stationary, 228 sensitivity, 188 light-induced absorption, 167 mobility-lifetime, 174 parameters, table, 179 photoactive centers, 23 photochromism, 40, 42 energy, activation, 41 photoconductivity, 173, 174 wavelength-resolved, 175 photo-electromotive-force, 222 quantum efficiency, 174 refractive index, 13 running hologram, 210 hole-electron competition, 210 self-stabilized, 229, 237 BTO:Ce light-induced absorption, 167 mobility-lifetime, 174 photoconductivity, 174 photo-electromotive-force, 222 quantum efficiency, 174

Photorefractive Materials: Fundamental Concepts, Holographic Recording and Materials Characterization, By Jaime Frejlich Copyright # 2007 John Wiley & Sons, Inc.

305

306

INDEX

BTO:Pb erasure, hologram, 189 hole-electron competition, 192 light-induced absorption, 167 mobility-lifetime, 174 photoconductivity, 174 photo-electromotive-force, 222 quantum efficiency, 174 Characterization, materials, 163 BTO, 237 energy coupling, 182 holographic erasure, 188, 191 light-induced absorption, 165 lithium niobate, 238 photoconductivity, 170 photo-electromotive-force, 222 running holograms, 206 self-stabilized, 232 sensitivity, holographic, 186 materials, table for some typical, 188 CdTe bandgap schema, 22 conductivity, dark, 23 electro-optic coefficients, 16 photoactive centers, 21 refractive index, 16 sensitivity, holographic, 188 Concentration, effective, 61 Conductivity CdTe, 23 dark, 38, 190 Coupled wave theory dynamic grating, 90 anisotropic diffraction, 125 coupling coefficient, 90 diffraction efficiency, 106, 107, 149 field, effective, 62 nonstationary with bulk absorption, 108 phase, pure, 95 phase shift, 108 properties, 91 fixed grating, 85 Bragg, out of, 88 coupling coefficient, 88 diffraction efficiency, 88 Coupling coefficient, 88, 90 energy, 92, 98, 101, 182 absorption, bulk, 109

phase, 92, 98, 102 absorption, bulk, 109 Czochralski, 11 Debye screening length, 62, 64, 184, 283 Deformation, 255 Diffraction efficiency, 66, 88, 104, 107, 149, 154 absorption, 110 anisotropic, 125 optical activity, 129 polarization, output, 130 measurement, 185, 210, 233, 273 Diffusion length, 61, 187, 285 Electric field effective, 281 photovoltaic, 80 Electro-optic effect, 5, 10 coefficient, 14, 16, 17 measurement, 179 table, sillenites, 179 Energy coupling, 92, 98, 101, 109 measurement, 182 BTO, 183 Fermi level, 22, 24 steady-state, 27 Four wave mixing, 122 GaAs amplitude and phase effects, 203 electro-optic coefficients, 16 response time, 198 selective two-wave mixing, 201 sensitivity, holographic, 188 Gain exponential, 92, 94, 98, 101, 148 absorption, bulk, 109 measurement, 182 KNSBN, 184 Hole-electron competition running hologram, 207, 210 erasure, hologram, 189, 191 Hologram bending, 103, 162 diffraction efficiency, 63, 66, 88, 104, 107, 110, 125, 129, 149, 185

INDEX

erasure, 188 hole-electron competition, 191, 207 fixed, 85, 259 phase coupling, 91, 98, 102 phase modulation, 114, 118 phase shift, 64, 108, 145, 196 measurement, 196 photo-electromotive-force, 212 harmonic terms, temporal, 215 response time, 62, 77 feedback, 98 GaAs, 198 measurement, 197 running, 67, 138, 206 quality factor, 69 absorption, 108 hole-electron competition, 207 recording, 95, 101, 131 self-stabilized, 144 undepleted pump approximation, 96 sensitivity KNSBN, 195 measurement, 186, 194 table, for some materials, 188 time constant, 62 measurement, 197 InP electro-optic coefficients, 16 KDP electro-optic coefficients, 17 refractive index, 17 KNSBN, 184, 195 sensitivity, holographic, 188 table, 185 Kogelnik 85 Length, Debye screening, 62, 64, 184, 283 diffusion, 61, 187, 285 drift, 61 Lifetime, 61 Light-induced absorption BTO, table, 167 equations, 165

Light propagation crystals, 5 anisotropic media, 6 index ellipsoid, 7 wave equation, general, 6 Lithium niobate, 190 donor density, 82 electro-optic coefficients, 16 erasure, holographic, 190 fixed hologrm, 259 phase shift, 148 photoactive centers, 26 refractive index, 17 samples, 155 sensitivity, holographic, 188 transport coefficients, table, 80 Maxwell relaxation time, 61 absorption, bulk, 108 Mobility, 285 mobility-lifetime, 174, 178 Modulation, pattern of fringes, 55 Optical activity, 127 Phase coupling, 92, 98, 102, 148 absorption, bulk, 109 modulation, 114, 118 characterization techniques, 194 harmonic terms, 119, 134 shift, 120, 148 initial, 66 measurement, 196, 210, 226, 228 stationary, 64, 66, 108, 112, 148, 229 Photoactive centers cadmium telluride, 21 lithium niobate, 26 sillenite, 23 Photochromism, 36, 39 activation energy for BTO, 41 light-induced absorption, 38, 40, 165 Photodiodes, 287 operational amplifier, 291 photoconductive, 290 photovoltaic, 289 Photo-electromotive-force, 212 BTO, 222 harmonic terms, temporal, 215

307

308

INDEX

Photoconductivity, 37, 39 BTO, 174, 175, 177 coefficient, 170 localized states, 27 measurement, 170 ac technique, 171 modulated photocurrent, 176 setup, 172 wavelength-resolved, 174 model, theoretical one-center, 33 two-center, 35 Photoluminescence, 169 Polarization output, 131 Probe beam, 275 Quantum efficiency, 34, 110, 174, 178 Recombination centers, 29 Recording hologram refractive index, 95 running, 108, 206 self-stabilized, stationary, 136 refractive index modulation, 50 space-charge field modulation, 47 nonstationary, 67 photovoltaic materials, 79 Refractive index BTO, 13 CdTe, 16 ellipsoid, 7 KDP, 17 lithium niobate, 17 modulation, 50, 90 Running hologram, see: hologram, running SBN sensitivity, hologrphic, 188 Scattering, 161 S2WM, see: Selective two-wave mixing Selective two-wave mixing, 201 GaAs, 203 Self-stabilized recording, 131 arbitrary phase shift, 141 equilibrium condition, 137, 140

formulation, general, 133 lithium niobate, 145, 152, 238 diffraction efficiency, 149 feedback-constraint, 153 harmonic terms, 152 light polarization, effect, 157 phase, 152 scattering, 161 running hologram, 138 absorption, 230 BTO, 237 characterization, for material, 232 sillenites, 229 speed, 140, 232 setup, 135 sillenites, 144 stationary, 136 Semiconductor doped, 28 intrinsic, 27 Sensitivity, holographic measurement, 186, 194 table, for some materials, 188 Setup schema holographic interferometry, 247 holographic recording, 51, 182 phase modulation, 118 photo-electromotive-force, 220 running, 206 self-stabilized, 135, 138, 143, 147 stabilized, 159 photoconductivity, 172, 174 selective two-wave mixing, 204 temperature control, 263 Sillenite crystals, 11 photoactive centers, 23 Stabilized recording, 131 glassplate, 159 Space-charge field effective, 62, 69, 77, 82 first spatial harmonic, 59 recording, holographic, 47 nonstationary, 67, 108 nonstationary, self-stabilized, 147 photovoltaic materials, 79 rate equations, 56 steady-state, approximation, 62 steady-state, general, 57 time-evolution, 65

INDEX

phase shift, 64, 148 initial, 66 steady-state, 66 saturation, 62 Tensorial anlysis, 9 Top seed solution growth, 11

Trap centers, 29 TSSG, see: Top seed solution g rowth Vibrations measurement, 219, 252

309