Photorefractive Materials: Fundamental Concepts, Holographic Recording and Materials Characterization

PHOTOREFRACTIVE MATERIALS PHOTOREFRACTIVE MATERIALS Fundamental Concepts, Holographic Recording and Materials Charact...

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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


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


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


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


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


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


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).


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


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


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


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.


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

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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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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


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


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


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


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.


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


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


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


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


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


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


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


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


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


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


light

light

49

light

BAND GAP

CONDUCTION BAND

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

+

overall overall negative positive charge 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

+

+

+


E

E

----+

+

+

+

E

E

-----

+


+

+

+

+

---

+


+


VALENCE BAND

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

50


light

light

light

E

E


E

E




E

E



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


E


E


E


E


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.


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


∆

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.


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


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


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


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


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


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


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


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


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


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

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