Landmark Papers on Photorefractive Nonlinear Optics 981021443X, 9789810214432

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LANDMARK PAPERS ON

PH OTO REFRACTIVE NO LINE R OPTIC

Editors

Pochi Yeh University of California, Santa Barbara

Claire Gu The Pennsylvania State Univ.

World Scientific Singapore• New Jersey• London• Hong Kong

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Library of Congress Cataloging-in-Publication Data Landmark papers on photorefractive nonlinear optics I editors, Pochi Yeh, Claire Gu. p. cm. Includes bibliographical references. ISBN 981021443X 1. Nonlinear optics. 2. Nonlinear optics -- Industrial applications. 3. Photorefractive materials. I. Yeh, Pochi, 1948II. Gu, Claire. QC446.2.L36 1995 621.36'9--dc20 95-16548 CIP

While every effort has been made to contact the publishers of reprinted papers prior to publication, we have not been successful in a few cases. Where we could not contact the publishers, we have acknowledged the source of the material. Proper credit will be given to these publishers in future editions of this work after permission is granted. The author and publisher would like to thank the following publishers of the various journals and books for their assistance and permission to include the selected reprints found in this volume: American Institute of Physics American Physical Society Elsevier Science Publishers Gordon and Breach Science Publishers IEEE Nature Optical Society of America SPIE Springer-Verlag Taylor and Francis

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CONTENTS 1

Introduction Part I. Fundamental Photorefractive Phenomena A. M. Glass, "The Photorefractive Effect," Opt. Eng. 17, 470 (1978)

11

G. C. Valley and M. B. Klein, "Optimal Properties of Photorefractive Materials for Optical Data Processing," Opt. Eng. 22, 704 (1983)

21

A. Ashkin, G.D. Boyd, J.M. Dziedzic, R. G. Smith, A. A. Ballman, J. J. Levinstein, and K. Nassau, "Optically-Induced Refractive Index Imhomogeneities in LiNb03 and LiTa03," Appl. Phys. Lett. 9, 72 (1966)

29

F. S. Chen, J. T. LaMacchia, and D. B. Fraser, "Holographic Storage in Lithium Niobate," Appl. Phys. Lett. 13, 223 (1968)

33

N. V. Kukhtarev, V. B. Markov, S. G. Odulov, M. S. Soskin, and V. L. Vinetskii, "Holographic Storage in Electro-Optic Crystals. I. Steady State," Ferroelectrics 22, 949 (1979)

37

N. V. Kukhtarev, V. B. Markov, S. G. Odulov, M. S. Soskin, and V. L. Vinetskii, "Holographic Storage in Electro-Optic Crystals. IL Beam Coupling and Light Amplification," Ferroelectrics 22, 961 (1979)

49

V. L. Vinetskii, N. V. Kukhtarev, S. G. Odulov, and M. S. Soskin, "Dynamic Self-Diffraction of Coherent 742 (1979) Light Beams,'' Sov. Phys. Usp.

53

J. Feinberg, D. Heiman, A. R. Tanguay, Jr., and R. W. Hellwarth, "Photorefractive Effects and Light-Induced Charge Migration in Barium Titanate," J. Appl. Phys. 51, 1297 (1980)

69

V. V. Voronov, I. R. Dorosh, Yu. S. Kuz'minov, and N. V. Tkachenko, "Photoinduced Light Scattering in Cerium-Doped Barium Strontium Niobate Crystals," Sov. J. Quantum Electron. 10, 1346 (1980)

79

v

L. K. Lam, T. Y. Chang, J. Feinberg, and R. W. Hellwarth, "Photorefractive-Index Gratings Formed by Nanosecond Optical Pulses in BaTi03," Opt. Lett. 6, 475 (1981)

83

P. Yeh, "Fundamental Limit of the Speed of Photorefractive Effect and its Impact on Device Applications and Material Research,'' Appl. Opt. 26, 602 (1987)

87

C. Gu and P. Yeh, "Scattering Due to Randomly Distributed Charge Particles in Photorefractive Crystals,'' Opt. Lett. 16, 1572 (1991)

91

Part H. Two-Wave Mixing P. Yeh, "Two-Wave Mixing in Nonlinear Media,'' IEEE J. Quantum Electron. 25, 484" (1989)

97

H. Kogelnik, "Coupled Wave Theory for Thick Hologram Gratings,'' Bell Syst. Tech. J. 48, 2909 (1969)

133

D. L. Staebler and J. J. Amodei, "Coupled-Wave Analysis of Holographic Storage in LiNb03,'' J. Appl. Phys. 34, 1042 (1972)

173

D. L. Staebler and J. J. Amodei, "Thermally Fixed Holograms in LiNb03,'' Ferroelectrics 3, 107 (1972)

181

D. W. Vahey, "A Nonlinear Coupled-Wave Th~ory of Holographic Storage in Ferroelectric Materials,'' J. Appl. Phys. 46, 3510 (1975)

189

J.P. Huignard and A. Marrakchi, "Two-Wave Mixing and Energy Transfer in Bi12Si020 Crystals: Application to Image Amplification and Vibration Analysis," Opt. Lett. 622 (1981)

195

P. Yeh, "Contra-Directional Two-Wave Mixing in Photorefractive Media,'' Opt. Comm. 45, 323 (1983)

199

G. C. Valley, "Two-Wave Mixing with an Applied Field and a Moving Grating,'' J. Opt. Soc. Am. Bl, 868 (1984)

203

Ph. Refregier, L. Solymer, H. Rajbenbach, and J. P. Huignard, "Two-Beam Coupling in Photorefractive Bi12Si020 Crystals with Moving Gratings: Theory and Experiments,'' J. Appl. Phys. 58, 45 (1985)

209

vi

J.M. Heaton and L. Solymar, "Transient Energy Transfer During Hologram Formation in Photorefractive Crystals," Optica Acta 32, 397 (1985)

223

Part HI. Four-Wave Mixing M. Cronin-Golomb, J. 0. White, B. Fischer, and A. Yariv, "Exact Solution of a Nonlinear Model of Four-Wave Mixing and Phase Conjugation," Opt. Lett. 7, 313 (1982)

237

M. Cronin-Golomb, B. Fischer, J. 0. White, and A. Yariv, "Theory and Applications of Four-Wave Mixing in Photorefractive Media," IEEE J. Quantum Electron. QE-20, 12 (1984)

241

N. V. Kukhtarev, T. I. Semenets, K. H. Ringhofer, and G. Tomberger, "Phase Conjugation by Reflection Gratings in Electro-Optic Crystals," Appl. Phys. B41, 259 (1986)

261

A. Bledowski and W. Krolikowski, "Exact Solution of Degenerate Four-Wave Mixing in Photorefractive Media," Opt. Lett. 13, 146 (1988)

267

R. Saxena, C. Gu, and P. Yeh, "Properties of Photorefractive Gratings with Complex Coupling Constants," J. Opt. Soc. Am. B8, 1047 (1991)

271

C. Gu and P. Yeh, "Reciprocity in Photorefractive Wave Mixing," Opt. Lett. 16, 455 (1991)

277

P. Stojkov and M. Belie, "Symmetries of Photorefractive Four-Wave Mixing," Phys. Rev. A45, 5061 (1992)

281

Part

Phase Conjugato:rs and Resonators

P. Yeh, "Photorefractive Phase Conjugators," Proc. IEEE 80, 436 (1992)

287

B. Ya. Zel'dovich, V. I. Popovichev, V. V. Ragul'skii, and F. S. Faizullov, "Connection Between the Wave Fronts of the Reflected and Exciting Light in Stimulated Mandel'shtam-Brillouin Scattering," ZhETF Pis. Red. 15, 160 (1972)

303

vii

J. Feinberg and R. W. Hellwarth, "Phase-Conjugating Mirror with Continuous-Wave Gain,'' Opt. Lett. 5, 519 (1980)

307

J. 0. White, M. Cronin-Golomb, B. Fischer, and A. Yariv, "Coherent Oscillation by Self-Induced Gratings in the Photorefractive Crystal BaTi03," Appl. Phys. Lett. 40, 450 (1982)

311

J. Feinberg, "Self-Pumped, Continuous-Wave Phase Conjugator Using Internal Reflection,'' Opt. Lett. 7, 486 (1982)

315

P. Yeh, "Theory of Unidirectional Photorefractive Ring Oscillators,'' J. Opt. Soc. Am. B2, 1924 (1985)

319

T. Y. Chang and R. W. Hellwarth, "Optical Phase Conjugation by Backscattering in Barium Titanate,'' Opt. Lett. 10, 408 (1985)

325

S. Weiss, S. Sternklar, and B. Fischer, "Double Phase-Conjugate Mirror: Analysis, Demonstration, and Applications," Opt. Lett. 12, 114 (1987)

329

R. W. Eason and A. M. C. Smout, "Bistability and Noncommutative Behavior of Multiple-Beam Self-Pulsing and Self-Pumping in BaTi03," Opt. Lett. 11, 51 (1987)

333

A. M. C. Smout and R. W. Eason, "Analysis of Mutually Incoherent Beam Coupling in BaTi03" Opt. Lett. 12, 498 (1987)

337

M. D. Ewbank, "Mechanism for Photorefractive Phase Conjugation Using Mutually Incoherent Beams," Opt. Lett. 13, 47 (1988)

341

P. Yeh, T. Y. Chang, and M. D. Ewbank, "Model for Mutually Pumped Phased Conjugation," J. Opt. Soc. Am. B5, 1743 {1988)

345

Q.-C. He, "Theory of Photorefractive Phase Conjugators with Mutually Incoherent Beams,'' IEEE J. Quantum Electron. 24, 2507 (1988)

353

V. V. Eliseev, V. T. Tikhonchuk, and A. A. Zozulya, "Double Phase-Conjugate Mirror: Two-Dimensional Analysis,'' J. Opt. Soc. Am. B8, 2497 (1991)

361

viii

Part V. Materials (Growth and Physics) A. M. Glass, D. von der Linde, and T. J. Negran, "High-Voltage Bulk Photovoltaic Effect and the Photorefractive Process in LiNb03," Appl. Phys. Lett. 25, 233 (1974)

371

P. Gunter, U. Fliickiger, J. P. Huignard, and F. Micheron, "Optically Induced Refractive Index Changes in KNb03:Fe,'' Ferroelectrics 13, 297 (1976)

375

G. C. Valley, "Erase Rates in Photorefractive Materials with Two Photoactive Species,'' Appl. Opt. 22, 3160 (1983)

379

M. B. Klein, "Beam Coupling in Undoped GaAs at 1.06 µm Using the Photorefractive Effect,'' Opt. Lett. 9, 350 (1984)

385

M. B. Klein and R. N. Schwartz, "Photorefractive Effect in BaTi03: Microscopic Origins,'' J. Opt. Soc. Am. B3, 293 (1986)

389

F. P. Strohkendl, J.M. C. Jonathan, and R. W. Hellwarth, "Hole-Electron Competition in Photorefractive Gratings,'' Opt. Lett. 11, 312 (1986)

403

M. D. Ewbank, R.R. Neurgaonkar, W. K. Cory, and J. Feinberg, "Photorefractive Properties of Strontium-Barium Niobate,'' J. Appl. Phys. 62, 374 (1987)

407

H. Rajbenbach, A. Delboulbe, and J. P. Huignard, "Noise Suppression in Photorefractive Image Amplifiers,'' Opt. Lett. 14, 1275 (1989)

415

C. Gu and P. Yeh, "Diffraction Properties of Fixed Gratings in Photorefractive Media,'' J. Opt. Soc. Am. B7, 2339 (1990)

419

S. Ducharme, J.C. Scott, R. J. Twieg, and W. E. Moerner, "Observation of the Photorefractive Effect in a Polymer," Phys. Rev. Lett. 66, 1846 (1991)

427

M. H. Garrett, J. Y. Chang, H.P. Jenssen, and C. Warde, "High Photorefractive Sensitivity in an n-Type 45°-cut BaTi03 Crystal,'' Opt. Lett. 17, 103 (1992)

431

M. C. J. M. Donckers, S. M. Silence, C. A. Walsh, F. Hache, D. M. Burland, W. E. Moerner, and R. J. Twieg, "Net Two-Beam-Coupling Gain in a Polymeric Photorefractive Material," Opt. Lett. 18, 1044 (1993)

435

ix

Part VI. Applications a. Image Processing Y. Fainman and S. H. Lee, "Applications of Photorefractive Crystals to Optical Signal Processing," in Optical and Hybrid Computing, H. H. Szu, ed., Proc. SPIE 634, 380 (1986)

441

J. 0. White and A. Yariv, "Real-Time Image Processing via Four-Wave Mixing in a Photorefractive Medium," Appl. Phys. Lett. 37, 5 (1980)

455

J. P. Huignard and A. Marrakchi, "Coherent Signal Beam Amplification in Two-Wave Mixing Experiments with Photorefractive Bi12Si020 Crystals," Opt. Comm. 38, 249 (1981)

459

Y. Shi, D. Psaltis, A. Marrakchi, and A. R. Tanguay, Jr., "Photorefractive Incoherent-to-Coherent Optical Converter," Appl. Opt. 22, 3665 (1983)

465

A. E. Chiou and P. Yeh, "Parallel Image Subtraction Using Phase-Conjugate Michelson Interferometer," Opt. Lett. 11, 306 (1986)

469

P. Yeh, T. Y. Chang, and P.H. Beckwith, "Real-Time Optical Image Subtraction Using Dynamic Holographic Interference in Photorefractive Media," Opt. Lett. 13, 586 (1988)

473

I. McMichael, P. Yeh, and P. Beckwith, "Correction of Polarization and Modal Scrambling in Multimode Fibers by Phase Conjugation," 507 (1987)

477

P. H. Beckwith, I. McMichael, and P. Yeh, "Image Distortion in Multimode Fibers and Restoration by Polarization-Preserving Phase Conjugation," Opt. Lett. 12, 510 (1987)

481

b. Optical Storage P. J. van Heerden, "Theory of Optical Information Storage in Solid," Appl. Opt. 2, 393 (1963)

x

487

J. J. Amodei and D. L. Staebler, "Holographic Pattern Fixing in Electro-Optic Crystals,'' Appl. Phys. Lett. 18, 540 (1971)

495

F. Micheron, C. Mayeux, and J. C. Trotier, "Electrical Control in Photorefractive Materials for Optical Storage," Appl. Opt. 13, 784 (1974)

499

D. L. Staebler and W. Phllips, "Fe-Doped LiNb03 for Read-Write Applications," Appl. Opt. 13, 788 (1974)

503

D. L. Staebler, W. J. Burke, W. Phillips, and J. J. Amodei, "Multiple Storage and Erasure of Fixed Holograms in Fe-Doped LiNb03,'' Appl. Phys. Lett. 26, 182 (1975)

511

F. H. Mok, M. C. Tackitt, and H. M. Stoll, "Storage of 500 High-Resolution Holograms in a LiNb03 Crystal," Opt. Lett. 16, 650 (1991).

515

G. A. Rakuljic, V. Leyva, and A. Yariv, "Optical Data Storage by Using Orthogonal Wavelength-Multiplexed Volume Holograms," Opt. Lett. 17, 1471 (1992)

519

c. Optical Computing P. Yeh, A. E. Chiou, J. Hong, P. Beckwith, T. Chang, and M. Khoshnevisan, "Photorefractive Nonlinear Optics and Optical Computing,'' Opt. Eng. 28, 328 (1989)

525

D. Psaltis, D. Brady, X.-G. Gu, and S. Lin, "Holography in Artificial Neural Networks," Nature 343, 325 (1990)

541

P. Yeh, A. E. T. Chiou, and J. Hong, "Optical Interconnection Using Photorefractive Dynamic Holograms,'' Appl. Opt. 27, 2093 (1988)

547

J. H. Hong, S. Campbell, and P. Yeh, "Optical Pattern Classifier with Perceptron Learning,'' Appl. Opt. 29, 3019 (1990)

551

M. Saffman, C. Benkert, and D. Z. Anderson, "Self-Organizing Photorefractive Frequency Demultiplexer," Opt. Lett. 15, 1993 (1991)

559

D. Z. Anderson and M. C. Erie, "Resonator Memories and Optical Novelty Filters,'' Opt. Eng. 26, 434 (1987)

563

xi

C. Gu, S. Campbell, J. Hong, Q. B. He, D. Zhang, and P. Yeh, "Optical Thresholding and Maximum Operations," Appl. Opt. 31, 5661 (1992)

575

C. Gu, S. Campbell, and P. Yeh, "Matrix-Matrix Multiplication by Using Grating Degeneracy in Photorefractive Media,'' Opt. Lett. 18, 146 (1993)

581

d. Other Applications D. Psaltis, J. Yu, and J. Hong, "Bias-Free Time-Integrating Optical Correlator Using a Photorefractive Crystal,'' Appl. Opt. 24, 3860 (1985)

587

A. E. Chiou and P. Yeh, "Laser-Beam Cleanup Using Photorefractive Two-Wave Mixing and Optical Phase Conjugation,'' Opt. Lett. 11, 461 (1986)

593

P. Yeh, "Photorefractive Coupling in Ring Resonators,'' Appl. Opt. 23, 2974 (1984)

597

I. McMichael and P. Yeh, "Self-Pumped Phase-Conjugate Fiber-Optic Gyro," Opt. Lett. 11, 686 (1986)

603

xii

INTRODUCTION

The photorefractive effect is an optical phenomenon in some electro-optic crystals where the local index of refraction is changed by the spatial variation of the light intensity. Such an effect was first discovered in LiNb03 crystals in the 1960s. The spatial index variation leads to the distortion of the wave front, and such an effect was referred to as "optical damage". The photorefractive effect has since been observed in many other electro-optic crystals, including BaTi03, KNb03, SBN, BSO, BGO, GaAs, CdTe, InP, etc. The photorefractive effect is generally believed to arise from optically generated charge carriers which migrate when the crystal is exposed to a spatially varying pattern of illumination with photons having sufficient energy. Migration of the charge carri~rs due to drift, diffusion, and the photovoltaic effect produces a space-charge separation, which then gives rise to a strong space-charge field. Such a field induces a refractive index change via the electro-optic (Pockels) effect. This simple picture of the photorefractive effect can be employed to explain several interesting optical phenomena in these media. Photorefractive materials are, by far, the most efficient media for the recording of volume dynamic holograms. In these media, information can be stored, retrieved, and erased by the illumination of light in real time. The holographic recording can be employed for 3-D optical data storage with an ultrahigh density. Such a scheme of volume holographic storage offers the unique property of parallel readout with an extremely short access time. In addition to the efficient holographic response, beam coupling known as two-wave mixing (TWM) occurs naturally in photorefractive media. When two beams of coherent radiation intersect inside a photorefractive medium, a stationary index grating is formed. This index grating is spatially shifted by 1f' /2 relative to the intensity pattern. Such a spatial phase shift leads to nonreciprocal energy transfer when these two beams propagate through the medium. The unique property of nonreciprocal energy transfer can be employed for many applications, including laser beam cleanup, photorefractive resonators, nonreciprocal transmission window, biased elements for laser gyros, self-pumped phase conjugators, mutually pumped phase conjugators, otpical interconnection, neural networks, phase conjugate interferometry, etc. It is important to note that the direction of energy flow in TWM is determined by the orientation of the cyrstal. In addition to holographic storage and TWM, photorefractive crystals are also efficient media for four-wave mixing (FWM) which is a generic process for the generation of phase conjugate waves. Optical FWM with various boundary conditions can be employed to construct several different types of phase conjugators including, externally pumped phase conjugators, ring conjugators, self-pumped phase conjugators (SPPC), mutually pumped phase conjugators (MPPC), etc. This book collects more than 80 classic papers in photorefractive nonlinear optics. Almost all the papers are well known to and frequently referenced by many researchers working in this field. These papers stand as landmarks in the history of photorefractive optics. The first part of the book starts from a collection of several papers on fundamental photorefractive phenomena. In this part, several pioneering papers reporting the discovery and the explanation of the photorefractive effect are included. Papers in Part 2 and Part 3 of the book provide the analysis of two-wave mixing (TWM) and four-wave mixing (FWM) processes, respectively. The unique properties of nonreciprocal energy transfer are employed in numerous interesting and important applications. Phase conjugators and resonators are discussed in Part 4. Part 5 covers material growth and physics which are essential for practical applications. The last part of the book is devoted to novel applications including image processing, optical data storage, optical computing, and other applications. Both review papers that are especially helpful to students and beginning researchers, and technical reports that marked breakthroughs in the understanding and applications of photorefractive

effects are collected. "The Photorefractive Effect,'' by Glass, reviewed the various studies of the photorefractive effect which led to improved understanding of the phenomenon in the 1960s and 1970s with special emphasis on measurement techniques. In the early years, most effort was devoted to the study of LiNb03. Later, new materials and effects were discovered in other materials. Valley and Klein in their paper "Optical Properties of Photorefractive Materials for Optical Data Processing,'' reviewed the charge transport model and several important figures of merit. These physical parameters were evaluated as functions of grating period and applied external electric field for Bi12Si020, a fast material with a relatively small electro-optic coefficient; and BaTi03, a slower material with a much larger electro-optic coefficient. Followed by these two reviews, several historically famous papers are included. Ashkin et al. reported the first discovery of the photorefractive effect in their paper "Optically Induced Refractive Index Inhomogeneities in LiNb03 and LiTa03,'' in 1966. In their Abstract, they wrote "The effect although interesting in its own right, is highly detrimental to the optics of nonlinear devices based on these crystals". Shortly after, in 1968, Chen et al. realized that such an effect could be used for optical data storage. In the paper "Holographic Storage in Lithium Niobate,'' they observed that the holograms in these media exhibited high diffraction efficiencies and high resolution, and were thermally erasable. They were the first to suggest that such materials might be useful in highcapacity, erasable optical information storage, processing and display devices. In 1979, Kukhtarev et al. in their paper "Holographic Storage in Electro-Optic Crystals I. Steady State," successfully explained the photorefractive effect in terms of the band transport model and formulated the socalled Kukhtarev equations. At the same time, they also formulated a theory of energy transfer between two coupled beams in their paper "Holographic Storage in Electro-Optic Crystals II. Beam Coupling and Light Amplification". These two papers laid the foundation for later investigations of grating formation and two-wave mixing. The theoretical and experimental works of Soviet scientists in the 1970s were reviewed in the paper "Dynamic Self-Diffraction of Coherent Light Beams,'' by Vinetskii et al. This paper showed that dynamic self-diffraction was substantially different for media with different types of response. Stationary energy transfer between interacting beams was possible in the case of nonlocal asymmetric response, whereas this type of transfer is forbidden in the case of local response. A different theoretical mod~l for the photorefractive effect was suggested by Feinberg et al. in "Photorefractive Effects and Light-Induced Charge Migration in Barium Titanate,'' in 1980. Their model, known as the hopping model, assumed that identical charges migrate by hopping between adjacent sites, with a hopping rate proportional to the total-light intensity at the starting site. The net hopping rate varies with the local electric potential that is calculated selfconsistently from the charge migration pattern. Another important photorefractive phenomenon called "fanning" was explained by Voronov et al. in "Photoinduced Light Scattering in Ce-doped Barium Strontium Niobate Crystals,'' in terms of holographic amplification of light scattered by crystal defects. Further studies revealed that photorefractive gratings could also be formed by optical pulses. Lam et al. in "Photorefractive Index Gratings Formed by Nanosecond Optical Pulses in BaTi03," compared photorefractive gratings written by optical pulses with various pulse length. In 1987, Yeh in his paper "Fundamental Limit of the Speed of Photorefractive Effect and its Impact on Device Applications and Material Research,'' examined the time constant for the formation of an index grating in photorefractive crystals from the point of view of energy deposition needed for the photoexcitation. This approach yielded a quantum limit for the speed of the photorefractive effect. In 1991, Gu and Yeh in "Scattering Due to Randomly Distributed Charge Particles in Photorefractive Crystals," pointed out and analyzed another fundamental problem. Scattering produced by randomly distributed charge particles in these crystals imposes a fundamental limitation on the dynamic range of photorefractive devices.

2

The most interesting difference between photorefractive materials and other holographic recording media is its nonlocal response, i.e., there exist a spatial phase shift between the index grating and the corresponding intensity interference fringes, which leads to an asymmetrical energy transfer between the two writing beams. In the review paper "Two-Wave Mixing in Nonlinear Media," Yeh discussed the coupling of two coherent electromagnetic waves inside a photorefractive medium and a Kerr medium. Both codirectional and contradirectional coupling were considered. The- coupling of two polarized beams inside photorefractive cubic crystals was also included. The similarity among various kinds of two-wave mixing, including stimulated Brillouin scattering and stimulated Raman scattering was pointed out. Historically, the diffraction properties of volume holographic gratings were treated by Kogelnik, in 1969, in "Coupled-Wave Theory for Thick Hologram Gratings". In this paper, formulations were given for the diffraction efficiencies and the angular and wavelength sensitivities of transmission and reflection holograms. The coupled wave theory is valid not only for photorefractive gratings but also for gratings in other thick holographic media. In 1972, Staebler and Amodei published "Coupled-Wave Analysis of Holographic Storage in LiNb0 3," and "Thermally Fixed Holograms in LiNb03". These two papers systematically described holographic recording, diffraction, erasure, and fixing in LiNb03. They suggested a model in which the fixed holograms were associated with ionic charge patterns formed by drift of thermally activated ions in the electric field pattern of the original hologram. In 1975, Vahey in "A Nonlinear CoupledWave Theory of Holographic Storage in Ferroelectric Materials," described the time evolution of the diffraction properties of thick phase gratings in ferroelectrics, particularly in iron-doped lithium niobate. In 1981, Huignard and Marrakchi in "Two-Wave Mixing and Energy Transfer in Bi12Si020 Crystals: Application to Image Amplification and Vibration Analysis," demonstrated the image amplification of a diffuse object. The image intensity transmitted by the crystal was amplified !Ox in the presence of the pump reference beam. In 1983, Yeh in "Contradirectional Two-Wave Mixing in Photorefractive Media," presented an analytical solution to the two-wave mixing with reflection gratings and discussed the nonreciprocal energy transfer resulting from nonlocal response. In 1984, Valley in "Two-Wave Mixing with an Applied Field and a Moving Grating," developed a theory for two-wave mixing in a photorefractive material with an external electric field and a moving grating. In 1985, Refregier et al. in "Two-Beam Coupling in Photorefractive Bh2Si020 Crystals with Moving Gratings: Theory and Experiments," reported large photorefractive coupling constant r:::::: 8-12 cm- 1 in BSO with a moving grating. They showed that resonance effect at the optimum velocity, optimum grating spacing could lead to a 7T /2 phase shift between the space charge field and the intensity fringe pattern. In the same year, Heaton and Solymar in "Transient Energy Transfer during Hologram Formation in Photorefractive Crystals," provided a transient analysis of energy transfer in two-wave mixing. Based on the same mechanisms as two-wave mixing, more optical waves can exchange energy and experience amplification or depletion in photorefractive media. Among the multiwave mixing processes, the most important one is the four-wave mixing (FWM) that plays a key role in optical phase conjugation. The nonlinear coupled wave equations that describe the four-wave mixing process is considerably more complicated than those of two-wave mixing. Analytical solutions were first derived in 1982 by Cronin-Golomb et al. in "Exact Solution of a Nonlinear Model of Four-Wave Mixing and Phase Conjugation," by using several constants of integration of the system. They found that a four-wave mixing phase conjugate mirror might exhibit bistability. In 1984, Cronin-Golomb et al. in a review paper, "Theory and Applications of Four-Wave Mixing in Photorefractive Media," described the development of a theory of four-wave mixing in photorefractive crystals. This theory was solved in the undepleted pumps approximation with linear absorption and without using the undepleted pumps approximation for negligible absorption. Both the transmission and reflec-

3

tion gratings were treated individually. The results were used to analyze several photorefractive phase conjugator mirrors, yielding reflectivities and thresholds. In 1986, Kukhtarev and Semenets in "Phase Conjugation by Reflection Gratings in Electro-Optic Crystals," analytically solved the four-wave mixing problem for the case of reflection gratings. Phase mismatch and self-oscillation effects were also dealt with in the undepleted pump approximation. In 1988, Bledowski and Krolikowski in "Exact Solution of Degenerate Four-Wave Mixing in Photorefractive Media,'' used the grating integral method to obtain closed form solutions for the diffusion dominate case with a phase shift of 7r /2. This method is applicable to asymmetrical angles of incidence, different refractiveindex modulation of interacting waves, and external incoherent illumination of the photorefractive crystal. In 1991, Saxena et al. in "Properties of Photorefractive Gratings with Complex Coupling Constants," further examined simultaneous write-read of dynamic photorefractive gratings for arbitrary phase shifts between the index grating and the intensity pattern. Also in 1991, Gu and Yeh in "Reciprocity in Photorefractive Wave Mixing," considered the validity of the reciprocity theorem as applied to steady-state photorefractive wave mixing. Some constants of integration in four-wave mixing were physically interpreted as results of reciprocity for the first time. In 1992, Stojkov and Belie in "Symmetries of Photorefractive Four-Wave Mixing,'' carried out a symmetry analysis of degenerate four-wave mixing equations in photorefractive crystals. Using underlying SU symmetries, five conserved quantities were found. Optical phase conjugation is an important method for wavefront recovery. Phase conjugation occurs in photorefractive media at a much less operating intensity as compared with other nonlinear optical processes such as SBS. In a review paper by Yeh, "Photorefractive Phase Conjugators,'' published in 1992, various kinds df optical phase conjugators using photorefractive materials were described. These included externally pumped phase conjugators (EPPCs), self-pumped phase conjugators (SPPCs), and mutually pumped phase conjugators (MPPCs). Historically, phase conjugate waves were first generated in the SBS process. In 1972, Zel'dovich et al. in "Connection Between the Wavefronts of the Reflected and Exciting Light in SBS,'' provided a proof that the phase conjugate component was dominant in all back-scattered waves in SBS. Phase conjugate mirrors have unique properties such as retro-reflection and wavefront reconstruction as compared with ordinary mirrors. In 1980, Feinberg and Hellwarth in "Phase Conjugating Mirror with Continuous-Wave Gain," observed cw self-oscillation in an optical resonator formed by phase conjugate mirror and a normal mirror and demonstrated the difference between a phase conjugate resonator and conventional resonators. In 1982, White et al. in "Coherent Oscillation by Self-Induced Gratings in Photorefractive Crystals," demonstrated several optical oscillator configurations including a unidirectional ring oscillator and a self-pumped phase conjugate mirror. Also in 1982, Feinberg in "Self-Pumped Continuous-Wave Phase Conjugator Using Internal Reflection," demonstrated the self-pumped phase conjugate mirror, known as the cat mirror, which was self-starting and did not need any external pump beams or cavities. Self-pumped phase conjugator mirror can also be constructed with a photorefractive crystal and an external cavity. In 1985, Yeh in "Theory of Unidirectional Photorefractive Ring Oscillators," analyzed the gain bandwidth and cavity detuning of a unidirectional photorefractive ring resonator. Besides phase conjugation based on four-wave mixing, backscattering in photorefractive media also generates phase conjugate waves in a similar fashion as SBS. In 1985, Chang and Hellwarth in "Optical Phase Conjugation by Backscattering in the Photorefractive Crystal Barium Titanate," experimentally demonstrated this SBS-type phase conjugate mirror in photorefractive BaTi03. In addition to self-pumped phase conjugators, there is another group of phase conjugate mirrors called mutually pumped phase conjugate mirror. In 1987, Weiss et al. in "Double Phase Conjugate Mirror: Analysis, Demonstration and Applications,'' showed that two inputs to opposite sides of a photorefractive BaTi03 crystal, which might carry different spatial

4

images, pumped the same four-wave mixing process mutually and were self-refracted without any external or internal crystal surface. This resulted in the phase conjugate reproduction of the two images simultaneously. In the same year, Eason and Smout in "Bistability and Noncommutative Behavior of Multiple-Beam Self-Pulsing and Self-Pumping in BaTi03,'' investigated two mutually incoherent beams simultaneously incident from the same side of a BaTi03 crystal. They observed both pulsing and mutually pumped phase conjugation (MPPC). In a separate paper by Smout and Eason, "Analysis of Mutually Incoherent Beam Coupling in BaTi03,'' they further described the mutually pumped phase conjugation with two inputs incident from the same face of the crystal. In 1988, Ewbank in "Mechanism for Photorefractive Phase Conjugation Using Mutually Incoherent Beams,'' reported and explained the bird-wing phase conjugator with BaTi03 which involved more than one interactive regions and reflection at the surface of the crystal. As various configurations of MPPC were demonstrated, theoretical models were also proposed and analyzed. Yeh et al. in "Model for Mutually Pumped Phase Conjugation,'' considered the buildup of two independent counterpropagating oscillations in a ring resonator containing two-wave mixing gain media and explained various kinds of MPPC. He in "Theory of Photorefractive Phase Conjugators with Mutually Incoherent Beams,'' solved the coupled wave equations of MPPC for the cases of both one and two interactions regions. The models of Yeh and He were based on plane wave analysis. In 1991, Eliseev et al. in "Double Phase-Conjugate Mirror - 2-Dimensional Analysis,'' analyzed the spatial information carried by the two pump beams in an MPPC. They argued that the double phase-conjugate mirror (DPCM) was an oscillator in the framework of a one-dimensional theoretical model, but a convective amplifier as a result a more consistent two-dimensional approach. Since the discovery of the photorefractive effect in LiNb03, many inorganic crystals and dopants have been investigated to search for better optical quality and photorefractive response. Recently, organic polymers have also been found to exhibit the photorefractive effect. Besides the general photoinduced index variation, each material has its own properties that affect the dynamics of photorefractive recording and erasure. The first discovered photorefractive crystal, LiNb03, has a large photovoltaic effect. In 1974, Glass et al. in "High-Voltage Bulk Photovoltaic Effect and the Photorefractive Process in LiNb03," demonstrated and analyzed the photocurrents in doped LiNb03 crystals due to a bulk photovoltaic effect with saturation voltages in excess of 100 V. They explained the photovoltaic effect based on the asymmetry of the lattice. In 1976, Gunter et al. in "Optically Induced Refractive Index Changes in KNb03 : Fe,'' investigated the crystal KNb03 which exhibited a large electro-optic coefficient. They found that the index change could be drastically reduced or enhanced by applying a small de field along the ±c axis. In crystals such as B12Si020, there exist two types of photoactive species. In 1983, Valley in "Erase Rates in Photorefractive Materials with Two Photoactive Species,'' derived a model of photorefractivity in which two photoactive ions participated as donors and acceptors. The model showed two decay rates during the optical erasure. The popular semiconductor GaAs is also a photorefractive material which exhibits fast response at infrared region. In 1984, Klein in "Beam Coupling in Undoped GaAs at 1.06 µm Using the Photorefractive Effect,'' observed beam coupling and degenerate four-wave mixing in GaAs with a response time 20 µsat an intensity level of I= 4 W/cm 2 . For wave mixing and phase conjugation, BaTi03 crystal is particularly interesting because of the very large electro-optic coefficient which leads to a large photorefractive gain. In 1986, Klein and Schwartz in "Photorefractive Effect in BaTi03: Microscopic Origins,'' used a number of experimental techniques to identify the photorefractive species in commercial samples of BaTi03. They found that Fe impurities (in the Fe2 + and Fe3+ states) were the predominant photorefractive species. Later, other dopants were added to BaTi03 to enhance the photorefractive effect. Also in 1986, Strohkendl et al. in "Electron-Hole Competition in Photorefractive Gratings,'' derived a band-conduction model for the photorefractive effect in which

5

simultaneous hole and electron conduction was taken into account and explained the anomalous behavior of nearly compensated BaTi03 crystals in beam-coupling experiments. Strontium-barium niobate (SBN) is another high-gain material. In 1987, Ewbank et al. in "Photorefractive Properties of Strontium-Barium Niobate,'' first systematically characterized SBN crystals. While crystals such as BaTi0 3 and SBN can provide high gain, they also amplify noises and cause the fanning problem. In 1989, Rajenbach et al. in "Noise Suppression in Photorefractive Image Amplifers,'' proposed and demonstrated a simple technique capable of significantly enhancing the signal-to-noise ratio of photorefractive amplifiers. Fanning noise was reduced by performing two-wave mixing in off-axisrotating BaTi03 and Bi12Si020 crystals. Since nonreciprocal energy transfer can occur between optical waves inside photorefractive crystals, the diffraction properties of a volume index grating in photorefractive media differ from ordinary Bragg diffraction. In 1990, Gu and Yeh in "Diffraction Properties of Fixed Gratings in Photorefractive Media,'' analyzed the diffraction properties of fixed volume gratings, including the effect of additional photorefractive energy coupling between the incident wave and the diffracted wave. Recently, some organic polymers were found to exhibit photorefractive effect. In 1991, Ducharme et al. in "Observation of the Photorefractive Effect in a Polymer,'' reported the first observation of the photorefractive effect in a polymeric material, the electro-optic polymer bisphenol-A-diglycidylether 4-nitro-1,2-phenylenediamine made photoconductive by doping with the hole-transport agent diethylamino-benzaldehyde diphenylhydrazone. The gratings formed exhibited dynamic writing and erasure, strong electric-field dependence, polarization anisotropy, and estimated space-charge fields up to 26 kV /cm at an applied field of 126 kV /cm. And the improvement of inorganic crystal responses continues. For example, in 1992, Garrett et al. in "High Photorefractive Sensitivity in an n-type 45°-cut BaTi03 Crystal,'' reported the beam-coupling properties of a cobalt-doped oxygen-reduced n-type barium titanate crystal in the 0 degrees and 45 degrees crystallographic orientations. They also found Co-doped BaTi03 was sensitive to infrared light. At the same time, the photorefractive performance of organic polymers has been significantly improved with a rapid pace. Recently, in 1993, Donckers et al. in "Net Two-Beam-Coupling Gain in a Polymeric Photorefractive Materials,'' demonstrated for the first time that the two-beam-coupling gain coefficient exceeds the absorption coefficient in an organic photorefractive system. Photorefractive materials with their unique properties of real-time response and low-intensity operation provide promising candidates for many applications especially in optical data storage and information processing. The review article "Applications of Photorefractive Crystals to Optical Signal Processing," by Fainman and Lee described image amplification, 2-D optical logic, phase conjugation, spatial correlation and convolutions. The potential applications of these elementary optical computing operations to analog optical computing (e.g., matrix inversion, solving matrix eigenvector/ eigenvalue problems), digital optical processing, pattern recognition and image processing were discussed. The earliest application of photorefractive material in image processing was reported by White and Yariv in "Real-Time Image Processing via Four-Wave mixing in a Photorefractive Medium,'' in 1980. They used optical four-wave mixing in a BSO crystal to perform image convolution and correlation. A direct application of energy transfer in two-wave mixing is image amplification. In 1981, Huignard and Marrakchi in "Coherent Signal Beam Amplification in Two-Wave Mixing Experiments with Photorefractive Bi12Si020 Crystals,'' demonstrated image amplification with moving gratings and applied field. In 1983, photorefractive materials were used to convert incoherent image into a coherent one by Shi et al. in "Photorefractive Incoherent-to-Coherent Optical Converter". In 1986, Chiou and Yeh in "Parallel Image Subtraction Using Phase-Conjugate Michelson Interferometer,'' employed the Stokes' relation and phase conjugate mirror to implement optical image subtraction. In 1988, Yeh et al. in "Real-Time Optical Image Subtraction Using Dynamic Holographic Interference in Photorefractive Media," implemented Gabor's image subtrac-

6

tion technique in real-time by using a double Mach-Zehnder interferometer. In 1987, McMichael et al. investigated image distortion in multimode fibers including modal and polarization scrambling and demonstrated image restoration by phase conjugation. Their works were published in "Correction of Polarization and Modal Scrambling in Multimode Fibers by Phase Conjugation," and "Image Distortion in Multimode Fibers and Restoration by Polarization-Preserv ing Phase Conjugation". Among the applications of photorefractive materials, the first proposed and perhaps the most promising one is for optical storage. Early in 1963, Heerden in "Theory of Optical Information 3 Storage in Solid," pointed out the ultimate limit V / >. for the storage capacity, where V was the volume of the storage medium and >. was the recording wavelength. In photorefractive materials, long-term storage requires fixing of the recorded holograms. In 1971, Amodei and Staebler in "Holographic Pattern Fixing in Electro-Optic Crystals,'' demonstrated thermal fixing in LiNb03 and doped Ba2NaNb5015. Another fixing technique, electrical fixing, was reported by Micheron et al. in "Electrical Control in Photorefractive Materials for Optical Storage,'' in 1974. In the same year, Staebler and Phillips in "Fe-Doped LiNb03 for Read-Write Applications,'' demonstrated high-erasure sensitivity in heavily reduced Fe:LiNb03, with a reduced diffraction efficiency. In 1975, Staebler et al. in "Multiple Storage and Erasure of Fixed Holograms in Fe-doped LiNb03,'' reported the recording of 500 fixed holograms, each with more than 2.53 diffraction efficiency. Sixteen years later, in 1991, Mok et al. in "Storage of 500 High-Resolution Holograms in a LiNb03 Crystal," also recorded (at room temperature) 500 high-resolution, uniformly diffracting volume holograms in a single Fe-doped LiNb03 crystal. These holograms were recorded with angle multiplexing. Within the 16 years, crystal quality and spatial light modulators have been significantly improved. Therefore, researchers are now investigating various possibilities to enable practical use of the photorefractive holographic storage. With the improvement of lasers, wavelength multiplexing becomes a reasonable alternative. In 1992, Rakuljic et al. in "Optical Data Storage by Using Orthogonal WavelengthMultiplexed Volume Holograms," proposed and demonstrated a volume holographic data storage scheme that employed counterpropagating reference and image beams and wavelength multiplexing for page differentiation and predicted a reduction in holographic cross talk. Another exciting application of photorefractive materials is for optical computing including neural networks. Yeh et al. in "Photorefractive Nonlinear Optics and Optical Computing,'' reviewed various nonlinear optical phenomena in photorefractive media and selected applications in optical computing. These phenomena included optical phase conjugation, two- and four-wave mixing, and real-time holography. The applications included image amplification and subtraction, logic and matrix operations, and optical interconnection. Paaltis et al. in the Nature article "Holography in Artificial Neural Networks," reviewed optical neural networks where photorefractive gratings served as interconnections while optoelectronic elements served as 'neurons'. Such an optical neural network could realize processes such as learning. Most of the research on optical computing and neural networks is conducted within the last ten years. In 1988, Yeh et al. in "Optical Interconnection Using Photorefractive Dynamic Holograms," proposed and demonstrated a new method of reconfigurable optical interconnection using dynamic holograms in photorefractive crystals. Such a method provided a very high energy efficiency by eliminating the fanout energy loss. In 1990, Hong et al. in "Optical Pattern Classifier with Perceptron Learning," described an optical realization of a singlelayer pattern classifier in which Perceptron learning was implemented to modify the interconnection weights. Novel use of the Stoke's principle of reversibility for light was made to realize both additive and subtractive weight modifications necessary for true Perceptron learning. In 1991, Saffman et al. in "Self-Organizing Photorefractive Frequency Demultiplexer," demonstrated a photorefractive circuit that demultiplexed a beam that had two signals imposed on separate optical carrier frequencies into two beams. The application of the unique photorefractive dynamics was proposed and

7

analyzed by Anderson and Eric in "Resonator Memories and Optical Novelty Filters," in 1987. In 1992, Gu et al. in "Optical Thresholding and Maximum Operations," considered self-oscillations in nonlinear optical four-wave mixing and resonators and suggested the implementation of parallel optical thresholding, comparing, and maximum operations. In .1993, Gu et al. in "Matrix-Matrix Multiplication Using Grating Degeneracy in Photorefractive Media,'' proposed and demonstrated a novel method that utilized grating degeneracy in photorefractive media in conjunction with an incoherent laser array to implement parallel optical matrix-matrix multiplication. Besides applications in image processing, optical storage, and optical computing, there are many other applications of the photorefractive effect. A few representative examples are selected in this book. An acoustic time-integrating correlator was demonstrated using a photorefractive crystal as the time-integrating detector in "Bias Free Time-Integrating Optical Correlator Using a Photorefractive Crystal,'' by Paaltis et al. in 1985. In 1986, Chiou and Yeh in "Laser-Beam Cleanup Using Photorefractive Two-Wave Mixing and Optical Phase Conjugation,'' demonstrated the ability of wavefront conversion in the two-wave mixing process. The insertion of a photorefractive medium in a bidirectional ring laser resonator was proposed and analyzed by Yeh in "Photorefractive Coupling in Ring Resonators,'' in 1984. This analysis provided the theoretical background for the implemen- . tation of a photorefractive optic gyro. In 1986, McMichael and Yeh demonstrated an application of photorefractive phase conjugator for inertial navigation in "Self-Pumped Phase-Conjugate FiberOptic Gyro". Since the discovery of the photorefractive effect, extensive research has been conducted to understand the physical mechanisms, io improve the quality of materials, to study the nonlinear interaction between optical waves, to develop theoretical models for photorefractive systems, and to utilize the materials for practical applications. During the years, thousands of papers have been published on photorefractive nonlinear optics. It is our intention to select some of the papers in this book and to provide an advanced reference book that includes both basic principles and important applications. Although a wide range of previous and current research effort has been included in this book, there are still many important and interesting works that are not covered here. The combination of a textbook, such as "Introduction to Photorefractive Nonlinear Optics" by Pochi Yeh which is mainly for educational purposes, and this book which serves as references for further research and indepth study will provide a comprehensive understanding of various aspects of the field of photorefractive nonlinear optics. We hope that this book will stimulate further interest in this field and lead to continuing development in materials and configurations which will make photorefractive nonlinear optics even more important to practical applications.

Pochi Yeh University of California at Santa Barbara Claire Gu The Pennsylvania State University March 1994

8

Part I Fundamental Photorefractive Phenomena

Optical Properties of Optical Materials

Reprinted with kind permission from the SPIE

The Photorefractive Effect A. M. Glass Bell Laboratories, Holmdel, New Jersey 07733

Abstract Without knowing the nature of the absorbing and trapping defects, nor the origin of the internal field, Chen demonstrated how the photorefractive effect can be used as the basis of a volume holographic memory of extremely high bit density. In subsequent years a great deal of effort has been devoted to identifying the microscopic details of the mechanism in order to optimize the materials for either memory or nonlinear optical applications. The studies have encompassed a wide range of physical studies, including optical spectroscopy of color centers, traps and transition metal ions, controlled crystal preparation, transport properties, electro-optics, holography, integrated optics, nonlinear absorption and transient spectroscopy and have led to the discovery of additional new effects in electro-optic materials -namely the bulk photovoltaic effect and excited state polarization. At the present time we have a reasonably good (though still incomplete) understanding of the photorefractive effect enabling the preparation of materials having a high photorefractive sensitivity with holographic recording sensitivities comparable with silver halide emulsions, as well as materials in which the photorefractive sensitivity is very low or even too small to measure. Photorefractive effects have now been observed in a variety of electro-optic crystals including CdS,4 Ba 2 NaNb 5 O 15 ,s 9 6 Ba1-xSrxNb20y, BaTi03, 7 PLZT ceramics," Bi4Ti30 12 , and Bi 12 (Si,Ge)020 in addition to those mentioned above, and may be considered a general property of electro-optic materials under the appropriate conditions of illumination. In this article the various studies of the photorefractive effect which have led to improved understanding of the phenomenon will be summarized with special emphasis on measurement techniques. Since by far the most effort has been devoted to the study of LiNb03, this review will reflect that bias. Although this material is by no means optimum for most applications it is the detailed study of LiNb0 3 which has been important in arriving at our present understanding of the effect. The first section of this review will survey the techniques used for studying the photorefractive effect. The following sections will describe how our understanding of the basic mechanisms evolved-both the nature of the absorption processes and the electron transport processes. The last part of the review is devoted to a discussion of the present status of various applications of the photorefractive effect and the possibility of future application.

The optically induced change of the refractive index of electrooptic crystals, which was discovered over ten years ago in LiNb03, is now referred to as the photorefractive effect (by analogy with photochromism). Progress in our understanding of the microscopic mechanisms which has led to the development of optical recording sensitivities comparable to that of silver halide emulsions is reviewed. Possibilities for application of the effect to optical memories, holographic interferometry, and integrated optics are considered.

I. Introduction A little over a decade ago it was discovered that an intense blue or green laser beam focused into LiNb0 3 or LiTa0 3 caused a 1 change of the refractive index of these crystals at the focus. This index inhomogeneity distorted the wavefront of the transmitted beam and was first referred to as "laser damage" since it prevented the use of. these important materials for nonlinear applications in this spectral region. Since the crystals can be returned to their initial, homogeneous state by briefly heating above 200 C and since, as we shall see, the origin of the effect is quite different from the reversible catastrophic damage which occurs at much higher power densities, the term "photorefractive effect" has now been adopted to describe the phenomenon. This is now analogous to the well-known photochromic effect which describes light-induced absorption changes. Optical degradation of electro-optic crystals due to lightinduced refractive index changes was also observed in electrooptic K(Ta,Nb)0 3 (KTN) modulators in the presence of an applied electric field. 2 This was attributed to the creation of space charge fields by mobile carriers in the crystal. In LiNb0 3 and LiTa03, however, the effect was observed without an applied field. Nevertheless, Chen 3 found that the index changes could be accounted for in a similar manner, by a model in which photoexcited carriers are displaced along the polar axis to trapping sites under the influence of some "internal field." The resulting space charge fields Ei give rise to an index change Lill· via the J electro-optic effect, i.e.

'----1 - 2 nj 3 rji Ei Llllj

°

(1)

where rji is the linear electro-optic coefficient. After the light is turned off the space charge fields and hence the index change persist for a time determined by the dark resistivity of the crystals (several months in LiNb0 3 ). To explain the observed values of L'lnj ~ 10-3 in LiNb0 3 the magnitude of the postulated in5 ternal field must be greater than or equal to E· ~ 10 V /cm. 3 15 This in turn requires a trapped charge density of ab6ut 10 /cm .

II. Measurement Techniques The photorefractive sensitivity S of a material is best defined as the refractive index change per unit absorbed energy density i.e. L'ln· S= _ J 0!.W

MT-105 received June 2, !97S. 470 I OPTICAL ENGINEERING I Vol. 17 No. 5 I September-October 1978

11

(2)

THE PHOTOREFRACTIVE EFFECT

where W is the incident energy density and a is the absorption coefficient including both linear and nonlinear (e.g. two-photon) absorption. Effects of reflection at the crystal surfaces will be neglected in this paper even though it has been shown 11 that reflections can contribute significantly to the behavior of the holograms near saturation. The usefulness of a sensitivity defined by Eq. (2) lies in the fact that for extrinsic absorption by an impurity color center S depends only on the nature of the absorbing center (the initial and final states) and not on its concentration. For several overlapping absorption bands in the crystal spectrum, the index changes assocjated with each absorption band are additive (i.e. lilltotal = ~~a1 S1 + 0!2 S2 )). Quantitative measurement of S/is normally made either by direct measurement of the refractive index change using holographic techniques or by measurement of the change of birefringence L'l(ni-nj) using conventional compensation techniques. In addition it has been shown recently how optically generated space charge fields may be measured directly using an electrostatic probe, and hence related to Llnj via Eq. ( 1).

lo) B

t>,+ LASER

MIRROR 30%R

Birefringence Measurement

SM~~J[_~DGE

The usual experimental arrangement 3 for measurement of birefringence changes is shown in Figure 1. The crystal polar axis is

Figure 2. Basic experimental arrangements for recording and reconstructing elementary holograms.

POLAR AXIS

is split with beamsplitter B and recombined in the crystal at an angle 28. Interference of the two beams of intensity I 1 and I2 creates a sinusoidal spatial intensity variation which is described by

He-Ne LASER

Ar•

I = 2I 0 ( 1 + mcosKx)

LASER

Figure 1.

r

MIRROR 100%R

(4)

where 2I 0 = I 1 + I 2 and the modulation index m = 2~/ (11 + I 2 ). If the index change is directly proportional to the incident intensity I then the optical interference pattern records a phase hologram of grating vector K (= 4irsin8 /A.), and the magnitude of the sinusoidal component of the index change can be measured from the diffraction efficiency of the grating. For thick phase transmission holograms with a peak-to-peak index change Lln, the diffraction efficiency 71 is 13

Experimental arrangement to measure laser-induced bire-

fringence changes.

normal to the incident beam and at 45° to the axes of the crossed polarizer and analyzer. If the transmission of the probe beam (usually a low power He-Ne laser) is initially set to zero by means of the compensator, then, following an exposure to the intense recording beam, the transmission, neglecting reflections, is given by

71 = e -0d/cos8 sin2 (3)

irllnd 2A.cos8

(5)

The uniform index change corresponding to the de component of the incident intensity is not measured by diffraction, but it will be shown later that this component can have a significant effect on the recording sensitivity. Reconstruction of a thick elementary hologram of just one spatial frequency K is possible with light of wavelength A.', different from that used for recording, A., provided the Bragg condition in the material is satisfied, i.e.

where a is the absorption coefficient and d is the crystal thickness. If the electro-optic coefficients are known then Llni and till· can be independently evaluated. In many cases, such as Li~b0 3 , the change of the ordinary index is small compared to the extraordinary index so that the experiment approximately measures b.n 3 . For accurate measurements the crystal thickness must be less than the near field distance of the focused beam and the diameter of the probe beam must be small compared with the recording beam in the crystal. This technique is limited by the extinction ratio of the crystal and can measure birefringence changes down to about 10-s in good quality electro-optic crystals. This technique has the advantage over the holographic technique in that high intensities are readily available by using focused laser beams. However, thermally induced index changes may be larger than the photorefractive effect during recording so that this method is not so useful for studies of recording kinetics.

K = 4irsin8' /A'

(6)

However, for image reconstruction (many spatial frequencies), readout with the same angle used for recording is generally necessary (although in birefringent materials this can be relaxed as discussed later). The angle and wavelength selectivities are given by

t-.e ~ rl/d and b.A. ~fl, cot 8/d

(7)

For a crystal thickness of 1 cm, and rl = 1 µm, we find t-.e ~ 0.1 mR and b.A. ~ 2 A at A.= 0.5 µm. Thus holograms can be superimposed in the same crystal volume provided the crystal is rotated by t-.e or the recording wavelength is changed by b.A.. If either the crystal thickness or the spatial frequency is reduced such that d < [/, then the Bragg conditions are completely relaxed and holograms cannot be superimposed. The diffraction

Holographic Technique The holographic technique has proved to be the most generally useful method of studying the photorefractive effect although it is perhaps experimentally the most complicated. The basic arrangements 12 are shown in Figure 2. The recording laser beam

September-October 1978

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I Vol. 17 No. 5 I OPTICAL ENGINEERING I 471

A. M. GLASS

efficiency of these thin holograms is 17 = e -oid/cosO J 1 2 ( irL'.nd ) AcosO

the insulating crystals has been measured directly by means of an electrostatic voltmeter. 18 A feedback circuit in the voltmeter nulls the fringing fields in a small hole in a metal probe by making the probe potential equal to the local potential at that point in space. With small holes a resolution down to 0.1 mm has been obtained. This technique allows the sign of the fields giving rise to the index change to be determined directly and is useful for studies of the mechanism leading to the photorefractive effect. However, for studies of the photorefractive sensitivity the optical techniques which have higher resolution are more generally used. If a large area of the crystal is illuminated the electric fields and photocurrents can be directly measured with electrodes attached to the polar faces of the crystal. 19

(8 )

where J 1 is a Bessel function of the first order. As the diffracted intensity becomes large the diffraction efficiency deviates from that expected from Eqs. (5) and (8), in photorefractive materials. This is because the diffracted and undiffracted beams interfere with each other 14' 15 ' 16 during recording (and also during readout and erasure) thereby recording new gratings which may add or subtract from the initial grating. These beam coupling effects are particularly noticeable when the induced index change is such that L'.nd/cos 0 ; I. In thin holograms the problem is further complicated by the higher order diffracted beams. A typical measurement of the diffraction _efficiencl of. a thick cryst.al of iron-doped LiNb0 3. is shown in Figure 3. 1 It 1s clear that mstead of a pure smuso1dal func-

III. Free Carrier Excitation Early Studies For some time after Chen proposed his model, the nature of the excitation and trapping mechanisms remained a mystery. LiNb0 3 and most of the other crystals studied typically had bandgaps of about 4 eV which pointed to extrinsic defects as the origin of free carrier excitation in the green and blue regions of the spectrum. Early suggestions included oxygen vacancies or defects associated with the nonstoichiometric composition of LiNb0 3 single crystals. However crystals obtained from different sources grown from melts of the same stoichiometry differed widely in their damage susceptibility, while variations of crystal stoichiometry from a Li/Nb ratio in the melt from 44% to 54% had no noticeable effect on the damage beKond that found between crystals of the same stoichiometry. 2 Reducing crystals increased the damage susceptibility to a certain extent but no direct relationship with oxygen partial pressure could be established. In any case the absence of F-center absorption in the optical and epr spectra seemed to rule out oxygen vacancies as the origin of the mechanism.

0.6 WRITING WAVELENGTH= 514.5nm WRITING POWER DENSITY=3.2x10-2 w/mm 2

£- 0.5 ~

UJ

0.4

~z

0.3

c::;

0

i=

~ 0.2

er

tt

0 0.1

100 150 200 250 WRITING TIME (SECONDS)

50

Iron Impurities With careful epr measurements it was discovered that Fe impurities are normally present in undoped LiNb0 3 crystals with concentrations typically in the 1-10 ppm range or greater, and that by oxidation and reduction of the crystals the valence state can be varied from Fe 2 + to Fe 3 +. 21 It was subsequently verified that the damage susceptibility correlated with the Fe 2 + content, and that the susceptibility could be controlled over a wide range by suitably doping crystals with Fe impurities to increase the damage, or by heat treatments which controlled the Fe 2 +/Fe 3 + ratio. Thus a microscopic mechanism of the photorefractive effect is the reaction Fe 2 + + hv = Fe 3 + +free carrier

Figure 3. Oscillatory hologram recording characteristic of a 1.66 mm thick iron doped LiNb0 3 • Writing beam polarizations are in the plane of

incidence.

tion as expected from Eq. (5) successive cycles are greatly attenuated. This behavior can be accounted for by means of a dynamic theory 15 , 16 which includes the effect of beam coupling. Studies of the energy transfer between the subject and reference beam during recording, and between the incident and diffracted beams during erasure can provide information on the relative phases of the incident intensity pattern and the recorded grating, which can be helpful 14 in the identification of the carrier transport mechanism during the formation of the gratings. For quantitative studies of the photorefractive effect, the holographic technique is most versatile. The minimum index change that can be measured is typically L'.n ~ 10-6 to 10-7 . Each component of L'.nj corresponding to different experimental geometries and polarization of the read beam can be measured independently. In LiNb0 3 the maximum sensitivity is obtained with K parallel to the polar c-axis and the read beam ir polarized utilizing the r 33 electro-optic coefficient. The photorefractive sensitivity can be measured as a function of spatial frequency by varying the angle to a resolution of A/2 corresponding to 0 = 0. At high spatial frequencies vibration effects become important, but these can be minimized with the experimental arrangement shown in Figure 2(b). The zoom telescopic lens system allows easy variation of(). The kinetics of recording can be studied without interference from thermal effects since for spatial frequencies ~ 10 4 cm- 1 , the thermal relaxation time is~ 10-s seconds.

There is also some evidence that at least in iron-doped crystals that the reverse reaction serves as the trapping mechanism so that the net result of illumination is a spatial variation of the Fe 2 + /Fe 3 + ratio. The absorption spectrum of iron-doped LiNb0 3 !7, 22 is shown in Figure 4. The broad band centered near I µm is a localized d-d shell transition which does not give rise to free carrier excitation. A transition of this kind can only give rise to a transient refractive index change for the duration of the excited state 23 and not a long-lived photorefractive effect. Illumination in the broad band near 20,000 cm- 1 does give rise to a photorefractive effect, and free carrier excitation can be additionally verified by the Rresence of photoconductivity during steady state illumination 9, 24 (for transient illumination pyroelectric effects usually mask photoconductivity). This transition is thus attributed to intervalence transfer from the Fe 2 + ion to Nb 5 + host ions 22 and since the conduction band of LiNb0 3 is based on Nb 5+ orbitals the excited carrier can move through the crystal until it is trapped.

e

Electrostatic Technique The electrostatic potential resulting from charge separation in

472 I OPTICAL ENGINEERING I Vol. 17 No. 5 I September-October 1978

13

THE PHOTOREFRACTIVE EFFECT

to an increased damage susceptibility. In different spectral regions or different materials other absorption mechanisms may prove to be more effective. Even in a material as similar as LiTa0 3 the photorefractive sensitivity is considerably below that of LiNb0 3 for the same impurity concentration (hence is a preferred material for many applications). Possible absorption mechanisms are shown in Figure 5. The

so~--------------------,4x 10-9

a 3

60

7

~

E40

2

I

I K/

!j

\

I

I

20

0

\

\

" \

I

I

I

30

5

10- 11 sec. This was demonstrated 31 with Cr3 + doped LiNb0 3

(14)

474 I OPTICAL ENGINEERING / Vol. 17 No. 5 I September-October 1978

15

THE PHOTOREFRACTIVE EFFECT

Thus the index change given by Eq. (I) is

ti on gradients: dn

Jdiff:eD~

(15)

where n is the instantaneous free carrier concentration and Dis the diffusion coefficient (D : µkT/e). The saturation space charge field in the steady state is then determined by the condition

which gives a photorefractive sensitivity

S:

1

e

2 hw

(njrji )

~

q

2 (~L I + K L2 2

-1/2 )

(16)

Jdiff + Jdrift: 0

This reaches a maximum value when the excited electron travels a distance equal ~o the grath1g spaci?g, KL .1. The ratio rjj/EE 0 is just the polanzat10n-optlc coefficient which has been found to obey the relation rij/EE 0 - P/4 m 2 /C (for rr in m/V and Pin C/m 2 ) for a wide variety of materials P being \he total polarization (spontaneous + field-induced). 33 This estimate is also true for quadratic electro-optic materials such as K(Ta/Nb)0 3 above the Curie temperature provided that the linear electro-optic coefficient rij is replaced by 2ee 0 gi.il

Smax - 0.1 cm 3 /J

(19)

This saturation field is independent of any material parameters. Diffusion transport cannot account for the large anisotropy of the photorefractive sensitivity with individual crystals, nor why different crystals have widely different photorefractive sensitivities. Even with the highest spatial frequencies accessible with visible light of K - I 8x 104 cm -I saturation fields of only a few kilovolts can be achieved-far less than the experimentally observed values in LiNb0 3 . Thus, althoug]J. it has been verified that both drift and diffusion are important contributors to the photorefractive effect in many materials, neither can account for all the observations in LiNb0 3 . In view of these difficulties Johnston 35 proposed that the index change did not result from space charge field variations but from changes of the crystal polarizability due to redistribution of the free carriers. This again proved unsatisfactory since this model also predicted an effect which was too small. In any case, there is now ample evidence that the index change is indeed due to space charge fields.

(17)

As we shall see later, sensitivities approaching this upper limit have already been realized with KTN 29 , Bi 12 Ge0 20 and Bi 12 Si02orn Drift The magnitude of the photocarrier displacement is directly related by Eqs. (12) and (14) to the magnitude of the photocurrent density J. The origin of this photocurrent in LiNb0 3 has been the subject of considerable controversy since the observations cannot be described in terms of any conventional transport mechanism. In materials such as KTN which show large photorefractive effects only when biased with an external electric field Eext• the transport is clearly by photoconductivity

Bulk Photovoltaic Effect (Extrinsic) Conclusive evidence that the photorefractive effect was due to undirectional transport of free carriers along the polar axis of the LiNb0 3 crystals was provided by studies of the spatial frequency dependence of the photorefractive sensitivity of irondoped LiNb0 3 together with measurements of the de photocurrent measured in uniformly illuminated crystals. 19 If the polar faces of the crystals are connected to an electrometer the typical current response when the light is turned on is shown in Figure 7. The initial pyroelectric transient decays with the ther-

(18) Clearly the maximum index change is determined by the magnitude of the applied field while the. photorefractive sensitivity is determined by the free carrier mobilityµ (a : Neµ). When the 11 external field is removed the space charge field relaxes with a time constant T

T:ee 0/a LIGHT

where a here is the total conductivity including both photoconductivity and dark conductivity. During recording only photoconductivity is relevant since the dark conductivity being uniform throughout the crystal does not give rise to space charge fields. Attempts to explain the photorefractive effect in terms of photoconductivity in an internal field have not been satisfactory. The observed index changes as high as 10-3 correspond to space charge fields Esc - 10 5 V /cm which in turn implies the existence of a permanent internal field in the entire crystal volume of this magnitude or greater. This can be immediately ruled out since any field of this magnitude in homogeneous single crystals, regardless of its origin, would relax in the presence of photoconductivity or dark conductivity except in small regions below the crystal surfaces, thereby permanently reducing the photorefractive sensitivity to zero.

OFF

--------i+--- STEADY

~

0

STATE

Cl.

~\---'--~~~~~~~~~+-~~-,:;;..-~~~~~~-

,_

~

~

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LIGHT

ON

::> (.)

Figure 7.

Current response along the polar cMaxis of an iron doped LiNbO, crystal when uniformly illuminated with 0.5145 µm radiation.

ma! relaxation time to a steady state value which persists as long as the crystal is illuminated. The magnitude of this current can be characterized by the equation

Diffusion

(21)

Amodei 34 considered the possibility of generating the high electric fields by diffusion transport in the photocarrier concentra-

where

K

depends on the wavelength of light and the nature of

September-October 1978 I Vol. 17 No. 5 I OPTICAL ENGINEERING I 475

16

A. M. GLASS

the absorbing center. In Fe2+ doped LiNb0 3 K ~ 3x 10-9 Acm/W at 5145 A for a wide range of Fe 2 + concentrations, even though a varies by over two orders of magnitude. At very high dopant concentrations there is some evidence for a decrease of K possibly due to the formation of more complex defects. 36 Steady-state photocurrents along the polar axis of pyroelectric crystals have been known since 193 9 when measurements were made on Rochelle salt. 37 In BaTi0 3 38 the photocurrents were attributed to surface effects while in LiNb0 3 Chen 3 took them to be evidence of internal fields. However, measurements of open circuit photovoltages in excess of l 0,000 volts (l 0 5 V/cm) in homogeneous crystals rule out any explanation in terms of conventional transport processes. Previous observations of larger-than-band gap (but still quite small) photovoltages in striated crystals, and polycrystalline films and ceramics (see Ref. 17) could be explained in terms of cumulative junction effects in these materials, but in LiNb0 3 there is no evidence for the high density of junctions between optically dissimilar regions which would be necessary to account for the high voltages. Crystals grown in different laboratories and with different growth axes all show the same effect. The open circuit photovoltage in LiNb0 3 is consistent with the transport equation in the steady state of J =Ket!+ aE (22) in which E

-

sat-

K

v

Eo ------

v

is independent of the field and intensity: i.e.

KC> NA). In case (2), the terms proportional to (TR./ TE)2 in the numerator and denominator of Eq. (9) dominate and give

depends directly on the steady-state index, exceed a threshold.34 For these purposes BaTi03 may be above threshold, while BSO is below threshold.

7. ACKNOWLEDGMENTS The authors wish to thank R. A. Mullen, R. W. Hellwarth, J. F. Lam, and M. Sparks for helpful conversations. We have benefited from constructive suggestions of the referee and from the Guest Editor, A. R. Tanguay, Jr.

Since 11"" I/ 'YR no and n0 =al/ 'YRNA, 'YR has no effect in this limit, and decreasing NA decreases the response time. In case (3), TR.Tdi/(ro11)>> I and TRiro. ""0.5 µm2• is sufficient. The trap number density is measured from grating decay times. 15 The recombination coefficient and mobility in BaTi03 are difficult to obtain. Mobilities of 2x10- 4 30 to 0.5 cm2/vs2• have been measured in BaTi03 • The dielectric relaxation time measured by Feinberg et al., 15 gives the ratio 'YR/µ, if .and a are known. Also, the short pulse observations in BaTiQ 331 suggest that the recombination time is of the order of a nanosecond. The recombination coefficient of 5X10-8 is consistent with a mobility of 0.5 cm2 /Vs and the other observations. Mobilities as low as 2X 10-4 cm2 /Vs seem completely inconsistent with the photorefractive observations. 15.31 There is, however, still one problem with this set of values. The measured value ofµ= 0.5 cm 2 /Vs is an electron mobility, while Feinberg et aJ.15 found that the sign of the moving charges in their BaTi03 sample was positive, and in general one expects the hole mobility to be considerably smaller than the electron mobility.

Substitution for 11 and TR shows that Te is proportional to 'YRNA 2• In this regime, decreasing NA will decrease the response time and improve the 1% energy requirement and photorefractive sensitivity. In case (4), the dominant term in the denominator of Eq. (9) is (TRI TE)2 ( Tdil 11) while TRI TE < < I so that Te ""

TJ r2E/ r2R ·

Here Te is proportional to y 2 RNA 3,and decreasing NA and 'YR is even more effective in improving performance. It should be noted that all of the conclusions in this paragraph are dependent on grating period and the other parameters of BSO and BaTi03 .

6. CONCLUSIONS The charge transport model and solutions developed by Kukhtarev20 have been used to evaluate four common ODP figures of merit for two sets of parameters-one set thought to be roughly applicable to BSO, a photorefractive material with a small electro-optic coefficient and a fast response time, the other for BaTi0 3 , which has a large electro-optic coefficient and a slow response time. For BSO and BaTi03 with the parameters used here, BaTi03 has by far the largest steady-state index, while BSO has a much shorter response time and somewhat smaller energy requirements for a 1% efficiency; the photorefractive sensitivities are comparable. Other factors should also be considered in com paring these materials. Certain phase conjugation processes and image amplifiers require that the refractive index grating be shifted from the optical interference pattern by 90°. This occurs when E0 = 0 or when Eq < < E0 . Thus, BSO in the applied field limit cannot be used as an image amplifier for grating periods larger than a few micrometers unless an external technique is used to shift the grating. 38 Some of the phase conjugation processes require that the steady-state gain, which TABLE I. Materials

9. REFERENCES I. D. L. Staebler and J. J. Amodei, J. Appl. Phys. 43, !042(1972). 2. D. L. Staebler and W. Phillips, Appl. Opt. 13, 788(1974). 3. F. S. Chen, J. T. LaMacchia, and D. B. Frazer, Appl. Phys. Lett. 13, 223(1968).

forBSO

Parameter

BaTiOa

Background index Dielectric constant

2.5' 568

Electro-optic coefficient

pm/V

Donor number density Trap number density

cm-3 cm-3

1019b,f 101sb.t

cm 2

1.s-10-1sb 2. 10-11b

Photoionization cross section

Recombination coefficient Mobility

cm3/s cm 2 /Vs

a. Gunter.2e b. Peltier and Micheron.23 c. Feinberg et a1.1s

d. Bursian et a1.2 9

e. Landolt·Bornstein32 f. Hou et al.• 1 • Undamped value. Clamped values 820, 28. t s(No - NA)= o::::: o.32a at>..::::: 0.5 ,um.

710 I OPTICAL ENGINEERING I November/December 1983 I Vol. 22No. 6

27

2.48 4300"(•11) 168'(•33) 1640"(rd* 80"(r331•

o.03b

-t 2. 101ac

-t 5 · 10-s Q.5d__

OPTIMAL PROPERTIES OF PHOTOREFRACTIVE MATERIALS FOR OPTICAL DATA PROCESSING 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22.

J.P. Huignard and J.P. Herriau, Appl. Opt. 17, 2671(1978). J. Feinberg, Opt. Lett. 5, 331(1980). L. Pichon and J.P. Huignard, Opt. Comm. 36, 277(1981). J. 0. White and A. Yariv, Appl. Phys. Lett. 37, 5(1980). J.P. Huignard, J. P. Herriau, P. Aubourg, and E. Spitz, Opt. Lett. 4, 21(1979). J. 0. White, M. Cronin-Golomb, B. Fischer, and A. Yariv, Appl. Phys. Lett. 40, 450( 1982). J. Feinberg, Opt. Lett. 7, 486(1982). P. N. Gunter, Opt. Lett. 7, 10(1982). M. D. Levenson, K. M. Johnson, V. C. Hanchett, and K. Chiang,J. Opt. Soc. Am. 71, 737(1981). N. V. Kukhtarev, V. B. Markov, and S. G. Odulov, Opt. Comm. 23, 338(1977). N. V. Kukhtarev, V. B. Markov, S. G. Odulov, M. S. Soskin, and V. L. Vinetskii, Ferroelectrics 22, 949 and 961(1979). J. Feinberg, D. Heiman, A. R. Tanguay, Jr., and R. W. Hellwarth, J. Appl. Phys. 51, 1297(1980). A. Marrakchi and J.P. Huignard, Appl. Phys. 24, 131(1981). C.-T. Chen, D. M. Kim, and D. von der Linde, IEEE J. Quantum Electron. QE-16, 126(1980). J. P. Hermann, J. P. Herriau, and J. P. Huignard, Appl. Opt. 20, 2173(1981). L. K. Lam, T. Y. Chang, J. Feinberg, and R. W. Hellwarth, Opt. Lett. 6, 475(1981). N. V. Kukhtarev, Sov. Tech. Phys. Lett. 2, 438(1976). J. J. Amodei, RCA Review 32, 185(1971). L. Young, W. K. Y. Wong, M. L. W. Thewalt,and W. D. Cornish, Appl.

Phys. Lett. 24, 264( 1974). 23. M. Peltier and F. Micheron, J. Appl. Phys. 48, 3683(1977). 24. G. C. Valley, Erase rates in photorefractive materials with two photoactive species, submitted to Appl. Opt. (1983). 25. G. C. Valley, Short pulse grating formation in photorefractive materials submitted to IEEE J. Quantum Electron (1983). 26. P. Gunter, Phys. Reports 93, 200(1983). 27. H. Kogelnik, Bell Syst. Tech. J. 48, 2909(1969). . . . 28. M. Minden, Hughes Research Laboratories, private communication (1982). . 29. E. V. Bursian, Ya.G. Girshberg, and A. V. Ruzhmkov, Phys. Stat. Sol. (b), 74, 689(1976). 30. G. A. Cox and R. M. Tredgold, Phys. Lett. 11, 22(1964). 31. L. K. Lam, T. Y. Chang, J. Feinberg, and R. W. Hellwarth, Opt. Lett. 6, 475(1981). 32. Landolt-Bornstein, Numerical Data and Functional Relationships in Science and Technology, New Series, K. H. Hellwege,ed., Vol. 11, Chap. 5, p. 578, Springer-Verlag, Berlin ( 1979). 33. J. Feinberg and R. W. Hellwarth, Opt. Lett. 5, 519(1980). 34. J. Feinberg, Opt. Lett. 7, 486(1982). 35. D. von der Linde and A. M. Glass, Appl. Phys. 8, 85(1975). 36. A. M. Glass, Opt. Eng. 17(5), 470( 1978). 37. E. Kratzig and R. Orlowski, Appl. Phys. 15, 133(1978). 38. J.P. Huignard and A. Marrakchi, Opt. Comm. 38, 249(1981). 39. J. P. Huignard, J. P. Herriau, G. Rivet, and P. Gunter, Opt. Lett. 5, 102(1980). 40. A. Krumins and P. Gunter, Appl. Phys. 19, 153(1979). 41. S. L. Hou, R. B. Lauer, and R. E. Aldrich, J. Appl. Phys. 44, 2652(1973).

e

OPTICAL ENGINEERING I November /December 1983 I Vol. 22 No. 6 I

28

711

Reprinted from

Volume 9, Number 1

APPLIED PHYSICS LETTERS

1July1966

OPTICALLY-INDUCED REFRACTIVE INDEX INHOMOGENEITIES IN LiNb03 AND LiTa03 A. Ashkin, G.D. Boyd,]. M. Dziedzic, R. G. Smith A. A. Ballman,].]. Levinstein, K. Nassau (ferroelectric materials; nonlinear optics; E)

Bell Telephone Laboratories, Incorporated Murray Hill, New Jersey (Received 20 May 1966)

We have observed an optically-induced inhomogeneity in the refractive index of crystals of LiNb03 (refs. l and 2), LiTa03 (ref. I) and other ferroelectrics. The effect, although interesting in its own right, is highly detrimental to the optics of nonlinear devices3 - 9 based on these crystals. In LiNb03 and LiTa0 3 the inhomogeneity is easily produced with a focused gas laser beam of several milliwatts in the visible with either ordinary or extraordinary polarization, usually within minutes. An unfocused gas laser also produces an inhomogeneity. A track of inhomogeneity is produced along the path of the beam principally in the extraordinary refractive index. If the inhomogeneity is produced by an extraordinary wave propagating along, the X (hexagonal a axis) or Y crystal axes, the beam is observed to distort predominantly along the Z axis (optic c axis) as shown in Fig. l(a). The inhomogeneity becomes quite evident when the sample is probed or illuminated with a light beam. The inhomogeneity has been produced with the .5J47-µ and .6328-µ. laser and with incoherent visible or ultraviolet light but not with the 1.1526-µ or the CW l.06-µ. laser. The effect is observed to reach an equilibrium state with time (with a spot radius of Wo = 0.03 mm and -IO mW of .5147-µ light it takes about l min) and apart from some initial relaxation it is rather permanent (days or months). On some samples reducing the power by a given factor and increasing the exposure time by the same factor, that is, keeping the total energy constant, yields similar inhomogeneities over a range of 1000 in power. Thus on this time scale the effect integrates. Samples do exist however which do not integrate at all over a similar range. If the effect is put into equilibrium with a small focused spot and then subsequently illuminated with a larger spot (-10 times the radius) of the same wavelength the initial spot is observed to erase. This erasure occurs even though the larger beam produced a smaller net change in index than the original smaller beam. In fact after erasure an additional

smaller beam inhomogeneity could be introduced again inside the large spot. This erasure cannot be accomplished with l.15-µ light. Heating of the crystals to a temperature of =I 70°C, however, causes the effect to relax at a rate faster than its creation. The relaxation does not appear to be caused by pyroelectric fields. An electric field is observed to play a role. Applying a field along the Z axis in a direction which would

A

B

c

D

Fig. l The optic axis is vertical throughout. A, C, D are LiNbO,. with the polarization vector vertical (i.e., +c axis at the top). Bis LlTa03• l(a). Composite of three photos. The original extraordinary wave beam (on the right), distorts due to the onset of the elfect (middle), and then shrinks as the inhomogeneity spreads (left). Note preferential scattering in -c direction. This can be enhanced with an antiparallel electric field applied subsequent to the introduction of the inhomogeneity. l(b). Shows oval region which produces no index inhomogeneity. l(c). Shadows cast by single inhomogeneity tracks (-5-mm long) viewed with extraordinary light in the far field. 1(d). Shadow cast by .25-mm-radius index inhomogeneity as viewed along its axis in the near field.

72

29

Volume 9, Number l

APPLIED PHYSICS LETTERS

1July1966

Using a larger beam of 0.25-mm radius one obtains an inhomogeneity whose shape is more directly distinguishable upon illumination with extraordinary light. In Fig. l(d) an approximately near-field photograph shows that the index varies rapidly along the Z axis and slowly at right angles to it. By probing this inhomogeneity with a smaller probe beam and observing the deflection one can determine the gradient of the index of refraction at each point. The inhomogeneity consists primarily of a region of increasing index of refraction emanating from the beam axis, surrounded by a few ripples in the index of refraction. By observing the interference between ordinary and extraordinary light a change in the extraordinary index of refraction as large as 2 x 10-4 was observed. If caused by the electro-optic effect this implies an internal electric field -10 kV/cm. Bursts of damped acoustic wave trains in the .5 Mc/sec range as well as optical bursts have been observed in LiNb03 while samples are heated or cooled between room temperature and - l 70°C but these effects have not been directly associated with the inhomogeneity. An index inhomogeneity was also observed due to .514 7-µ light incident along the a axis of singledomain BaTi03 • This apparently relaxed within a minute as viewed with extraordinary light. When viewed with a polarizing microscope, however, the region at the center of the beam appeared as a single domain (unaltered by the beam) surrounded by a multiplicity of small antiparallel domains. The crystal could then be repoled 10 with some difficulty. No effect was observed in tourmaline (point group 3m), ZnO, CdS, or GaP. A similar and possibly related refractive index inhomogeneity was also produced in paraelectric KTN 11 in the presence of an applied electric field of several kV/cm. The effect remained visible when the field was removed, though in the absence of electric field the effect could not be produced. The visual pattern differed from that of LiNb03 presumably due to the different symmetry of the crystals. Optically-induced changes in the refractive index of KTN have been previously observed. 12 No self-consistent explanation of the phenomena has been found. The thermal relaxation is suggestive of electro-optic properties previously observed. 3 Pyroelectricity and microdomain reversals must be considered as well as photoionization of color centers, charge transport and trapping phenomena. The coefficients of both the electro-optic6 •7 and photoelastic 13 tensors favor a change in the extra-

flip ferroelectric domains enhances the inhomogeneity produced in a given time. Also the index inhomogeneity is more pronounced on poled singledomain samples 2 than unpoled samples from the same boule. The effect has thus far occurred in all LiNb03 samples tried but in varying degrees. Samples which have been annealed in nitrogen to produce oxygen deficiencies show the effect more strongly and relax at lower temperatures (-105°C). In such a crystal (brown in color), in which the relative absorption at .5147 µand .6328 µdiffered by a factor of 8, the inhomogeneity produced was proportional to the total absorbed power at either wavelength. An increase in the optical absorption of the extraordinary wave was observed to occur after an exposure of about a minute on a brown sample. This bleached out on further exposure leaving only the index inhomogeneity. Another brown sample of lesser absorption was found which never reached a static situation. Extraordinary light deflected by the sample [as in Fig. l(a)] was constantly changing and appeared as if it was deflected outward from the beam center in ripples. In lithium tantalate the effect is very similar to that observed in LiNb03 with one important exception: Samples do exist which show no effect. Some samples show the effect only over part of their volume. A poled LiTa03 (ref. 2) sample showing no effect in an oval shaped region (-5 mm in diam) in the center of the sample is shown in Fig. l(b). The broad horizontal lines (regions of shadow) were produced by traversing a focused .5147-µ laser beam across the sample. The inhomogeneity was photographed by illuminating the sample with a parallel beam of extraordinary laser light and projecting on a screen 50 cm away. The sample, 0.5-mm thick, was tilted to show the depth. The area showing no effect is slightly colored suggesting the presence of some local impurity. The boundary between the two areas could also be observed as a refractive index variation under a polarizing microscope eve,n when the inhomogeneity was absent. The individual inhomogeneity tracks are conveniently photographed as shown in Fig. l(c) by illuminating a LiNb03 crystal having two tracks, along the X and Y directions, with extraordinary light shining along the X direction. The doublelobed pattern is the characteristic far-field pattern that appears for light passing along the axis of the track. The side view of the track passing along the Y direction is visible as a line. Tracks along the Z direction are not as readily visible. 73

30

Volume 9, Number 1

APPLIED PHYSICS LETTERS

ordinary refractive index. A change in the structure14 of the unit cell such as a movement of the Nb atom (or possibly Li but not both) resulting in a change in the polarizability of the unit cell and thus the refractive index may be possible. Work on the mechanism and its possible elimination is continuing. The stimulation, ideas and continued interest of our associates S. C. Abrahams, R. T. Denton, R. Kompfner, S. K. Kurtz and R. C. Miller in this work is gratefully acknowledged. 1

A. A. Ballman,]. Am. Ceram. Soc. 48, 112 (1965). K. Nassau, H.J. Levinstein, G. M. Loiacono, Parts 1 and II, ]. Phys. Chem. Solids 27, (1966), in press. 3 G. D. Peterson, A. A. Ballman, P. V. Lenzo, P. M. Bridenbaugh, Appl. Phys. Letters 5, 62 ( 1964). 2

l July 1966

4 G. D. Boyd, R. C. Miller, K. Nassau, W. L. Bond, A. Savage, Appl. Phys. Letters 5, 234 ( 1964). 5 R. G. Smith, K. Nassau, M. F. Galven, Appl. Phys. Letters 7, 256 (1965). 6 E. H. Turner,Appl. Phys. Letters 8, 303 (1966). 7 P. V. Lenzo, E. G. Spencer, K. Nassau,]. Opt. Soc. Am. 56, Uune, 1966), in press. 8 1. P. Kaminow, Appl. Phys. Letters 8, 305 (1966), and 7, 123 (1965) and erratum 8, 54 (1966). 9 G. D. Boyd, A. Ashkin, Phys. Rev. 146, 187 (1966). 10 R. C. Miller, Phys. Rev. 134, Al313 (1964). "F. S. Chen, J. E. Geusic, S. K. Kurtz, J. D. Skinner, S. H. Wemple,]. Appl. Phys. 37, 388 (1966). 12 S. K. Kurtz, to be published in Bell System Tech.]., and F. S. Chen (private communication). 13 R. W. Dixon, M. G. Cohen, Appl. Phys. Letters 8, 205 (1966). "S. C. Abrahams, J. M. Reddy, J. L. Bernstein,]. Phys. Chem. Solids 27, (1966), in press.

ERRATUM In "Laser Oscillation in Chemically Formed CO," [Appl. Phys. Letters 8, 237 (1966)], M.A. Pollack, Bell Telephone Laboratories, Inc., Murray Hill, New Jersey, the index should correctly read: "CS2 -0 2 system; fl.ash photolysis; new infrared laser lines; molecular spectroscopy; E".

31

Volume 13, Number 7

APPLIED PHYSICS LETTERS

1 October 1968

HOLOGRAPHIC STORA GE IN LITHIUM NIOBATE* F. S. Chen, J. T. LaMacck ia, and D. B. Fraser Bell Telephone Laborato ries, Inc. Murray Hill, New Jersey 07974 (Received 15 July 1968)

Single-c rystal lithium niobate has been used as a holograp hic storage medium. The material undergoes a change in refractiv e indices upon exposure to suitably intense light thus allowing it to act as a pure-pha se, volume-h olograph ic medium requiring no processi ng. The hologram s formed have high diffractio n efficienc ies and are thermall y erasable . The high resolutio n obtained suggests that such material may be useful in high-cap acity, changeab le optical informat ion storage, processi ng and display devices. Previous workers ' have describe d optically -induced refractiv e index inhomog eneities ("optical damage" ) in lithium niobate (LiNbOa). These inhomog eneities pose serious limitatio ns in the use of LiNb0 3 in nonlinea r experime nts requiring high light intensities. However, this same effect can be used to advantage to form a holograp hic recordin g in applications where a material that gives refractiv e index change directly upon exposure , througho ut a relatively thick sample, would be desirable . However, the index change produced must be highly localized , i.e., the resolutio n must be high for the material to be useful. Indeed, we have found that poled single crystals of LiNbOa have a resolutio n in excess of 1600 lines/mm and thus are extremel y interesti ng holograp hic media. Poled single crystals of LiNbOa were placed in the hologram reading and writing system shown in Fig. 1. The crystallo graphic c axis was located in the plane formed by the referenc e and object beams and could take any direction other than approxi mately parallel to the bisector of the beams. The polarizat ion of the writing beams was not critical and could be varied by rotating the .\./2 plate. The results tabulated below were obtained for a transppe nt LiNbOa crystal 1 cm thick, exposed at 4880 A and read out with extraord inarily polarized light at either 4880 A or 6328 A. Similar results were also obtained with crystals of lithium tantalate . The exposure time for maximum diffracte d power at a writing intensity I= 1 W/cm 2 was 100 sec. We have observed that there is a maximum index change, (.o.n)max, correspo nding to a given writing intensity and that (.o.n)max is proportio nal to 1 112 for I ;S 200 W/cm 2 .2 Furtherm ore, there is also a minimum threshold intensity for an observab le effect. A 6328-A laser as shown in Fig. 1 was used to monitor the diffractio n efficienc y, TJ, of the grating during exposure . A plot of the amplitud e of the diffracted wave, which is proportio nal to n 112 , is shown in Fig. 2. The crystals respond in a nearly linear

Li Nb 03 CRYSTAL

Fig. 1.

Experim ental configura tion.

fashion over a large exposure range. Diffracti on efficienc ies up to 40% were observed at recorded spatial frequenc ies of as high as 1600 lines/mm . We have also made reflectio n hologram s with an angle between the beams of 160°. Although the diffractio n efficienc y in this case was only .01%, this indicates a limiting resolutio n in excess of 4000 lines/mm . The diffractio n efficienc y is given 3 by _ • 2 rr.o.nt (1) TJ - sm 2.\. cos e where t = thickness and 28 = angle between the object and referenc e beams. For the condition s of our experime nt the observed value of TJ (40%) implies a .o.n of 2 x 10-•. Relative index changes as high as 10- 3 have been observed for higher power densities in nonholographic experime nts. Thus, these experime nts have barely begun to use the allowable dynamic range of the medium. 0.6

0.5

0.4

~0.3 0.2

0.1

20

40

60

80

100

120

140

160

EXPOSURE TIME (SECONDS)

*Present ed at the October 1968 meeting of the Optical Society of America in Pittsburg h, Pennsylvania.

Fig. 2. Amplitude of diffracte d wavefron t during exposure (.\.write:;" 4880 A, .\.read

= 6328

A).

223

33

Volume 13, Number 7

APPLIED PHYSICS LETTERS

1October1968

The angular selectivity in diffraction from a grating of thickness t and spacing A is approximately t:..e ""' ±

A/t

(2)

We have measured t:..e = ±0.2 mr, compared with i::..e ""'±0.1 mr given by Eq. (2). The difference is due to the angula.r divergence in the reference beam. 2 Under steady illumination of 0.12 W/cm at eforiginal its of e 1/ 4880 A, the grating decays to ficienpy in 10 min. For constant interrogation at 6328 A, the lifetime is much longer. For an illu2 minating beam intensity of 0. 8 W/cm , the efficiency falls off exponentially to 1/e of its original value in approximately 9 h. Under ordinary room illumination, the grating showed only a 10% degradation after 100 h. When the crystal is heated to 170° C all holograms are completely erased. No fatigue has been observed for at least ten write-erase cycles. The material has also been used to write pictorial holograms. Referring to Fig. 1, this was done by placing in one of the beams a lense and subject transparency one focal distance ahead of the crystal. One result is shown in Fig. 3 which includes a photograph of a virtual image reconstruction. The horizontal striae across the reconstruction are due to strain-induced growth striae in the crystal. These striae are also visible in the photograph of the subject as viewed through the unexposed crystal. We have observed that recently available crystals do not show this type of optical inhomogeneity. The observed index changes are believed to be due to the photoexcitation of trapped charges,2 which then drift a short distance before becoming retrapped. Based on the observed resolution, the drift distance must be a small fraction of the micron. The drift field is antiparallel to the spontaneous polarization of the sample, and although the origin of the drift field is not well understood, it is likely due to incomplete compensation of surface dipole charges in these strongly polarized crystals. The space-charge field created by the displaced electrons causes the change in the refractive in-

Fig. 4. Reconstruction showing diffractionlimited resolution which can be obtained with striae-free lithium niobate crystals ("Linobate," Crystal Technology, Inc., MountainView, California). dices via the electro-optic effect. One of the principal pieces of evidence supporting an electro-optic effect is the observation that reconstructions produced by o-ray illumination are only 1/10 as efficient as those produced by e-ray illumination. Based on Eq. (1), this implies (t:..n) 0 (t:..n)e ""' .3. The 5 effective electro-optic coefficient•- for the o ray, n~ r 13 is 1/3 to 1/ 4 as strong as the coefficient for thee ray, 33 , consistent with the above observation. Figure 4 shows the kind of excellent holograph~c reconstruction which can be obtained by using striae-free lithium niobate. It has been observed that these ferroelectrics can be made more susceptible to optically induced index changes by increasing the number of available deep traps. Standard samples of transparent LiNb0 3 have been annealed at 900° C in an oxygendeficient environment and then partially reoxidized in an 02 atmosphere at J.100° C. Such treated crystals are left slightly colored, indicating a reduced condition of the niobium. As a result of such treatment, we were able to obtain still higher diffraction efficiency (42%) in a sample only 5 mm thick with 2 an exposure time of 3 5 sec at 1 W/cm • This amounts to a threefold speed increase and a twofold increase in the saturation value of t:..n. The authors thank K. Nassau and A. A. Ballman for supplying some of the crystals used in these experiments.

n;r

RECONSTRUCTION ORIGINAL Fig. 3. Photo on left is of the subject (8mm transparency) viewed through the LiNb0 3 sample. Photo on right is of the virtual image reconstructed from a hologram in in the LiNbOs.

1 A. Ashkin, G. D. Boyd, J. M. Dziedzic, R. G. Smith, A. A. Ballman, H. J. Levinstein, and K. Massau, Appl. Phys. Letters 9, 72 (1966). 2 F. S. Chen (unpublished).

224

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Volume 13. Number 7

APPLIED PHYSICS LETTERS

3

H. W. Kogelnik, Proceeding::: of the Symposium on Modern optics (Polytechnic ~ss, New York,

l October 1968

4 P. V. Lenzo, E. H. Turner, E.G. Spencer, and A. A. Ballman, Appl. Phys. Letters 8, 81 (1966). 5 E. H. Turner, Appl. Phys. Letters 8, 303 (1966).

1967), pp. 605-617.

35

Ferroelectrics. 1979. Vol. 22. pp. 949-960 0015-0193/79/2203-0949$04.50/0

©Gordon and Breach Science Publishers. Inc., 1979 Printed in Great Britain

HOLOGRAPHIC STORAGE IN ELECTROOPTIC CRYSTALS. I. STEADY STATE N. V. KUKHTAREV. V. B. MARKOV, S. G. ODULOV, M. S. SOSKIN and V. L VINETSKII

Institute of Physics, Academy of Sciences of the Ukranian SSR, 252650, Kiev, USSR (Received August, 15, 1977; in final form January JO, 1979) The non-linear theory of the self-diffraction on the light induced grating of refractive index in electrooptic crystals is developed. The intensities of the diffracted beams. the diffraction efficiency. and the shape of the surfaces of equal index change are calculated analytically for saturation holograms. Holographic storage in nominally pure reduced crystals of LiNb0 3 is studied experimentally. It is shown that the developed theory in diffusion approximation satisfactorily describes the experimental data.

INTRODUCTION

(µis the mobility, £ 0 the external electric field, r the carrier lifetime) becomes comparable or larger than fringe spacing one obtains also the rc/2 shifted HG at the initial stage of recording in the applied field. Phase mismatch between HG and FP in this case is a consequence of large separation of negative and positive charges in the crystal, where the difference between concentrations of the positive and negative charges is comparable with the concentration of the charges of both signs (strict violation of quasineutrality). In the paper of Ninomiya 7 the "dynamic" approach to the problem of holographic recording was formulated. The theory is based on the solution of non-linear wave equation with the dielectric constant depending on the light intensity. But as in the first paper of Nimomiya 7 so in more recent works of Sidorovitch and Stasielko, 8 Magnussen and Gaylord, 9 and Vahey 10 the dependence of the refractive index K on the light intensity I was postulated and not derived from the material equations. In spite of the fact that the "guessed" dependence K = f (I) was not correct in all these papers there are many important qualitative results and conclusions. On the other hand more detailed calculations of the space charge field E sc from the solution of the full set of material equations were presented in Refs. 1115. The results of these papers are applicable only to the thin layers of electrooptic crystals or to the initial stage of recording when energy and phase redistribution between two writing beams may be neglected. In our paper (see Ref. 16) the self-consistent theory of saturated holograms in electrooptic crystals, taking into account both the effects or optical feedback (dynamic theory of Ninomiya) and

It is possible to record the volume phase grating in electrooptic crystal illuminating it by fringe pattern of two coherent light beams. Phase recording in these media is connected with the redistribution of photoexcited electrons in the volume of the crystal and succeeding retrapping on deep centres. Charge transfer may be caused by diffusion due to nonuniform free carrier distribution, 1 and may be connected with the drift in external field, 2 or with the specific to polar crystals photovoltaic effect. 3 The first theoretical treatments of the problem of holographic storage in electrooptic crystals were based on the solution of charge transport equation. 4 • 5 In this approach neither the influence of arising space charge field, nor the effect of selfdiffraction of the writing beams on the holographic grating (HG) were taken into account. It has been shown in this approximation that for diffusion process of recording the diffraction efficiency 17 is independent on the light intensity and increases as an inverse square of fringe spacing 17 oc L - 2• The holographic grating was shown to be shifted by a quarter of a period with respect to the fringe pattern (FP) leading to a considerable energy redistribution between two writing beams. 5 For drift process of recording it was found that the extrema of FP and HG coincide in the volume of the crystal and energy transfer is forbidden. 5 Lately it was shown in the same approximation that the conclusions mentioned above are valid for the case when the specific transport length of free carriers is much less than fringe spacing. 6 If this condition is violated and the drift length ld = µE 0r 949

37

950

N. V. KUKHTAREV et al.

of large-scale fields 22 was not taken into account here. We have also not considered the interference effects due to Fresnel reflections on input and output surfaces of the crystal. 23 All calculations were performed for the case of Hpolarization of writing beams (typical experimental situation for LiNb0 3 crystals). For the usual experimental arrangement when two beams impinge the crystal from the same side the difference between the results for E and H polarizations is not very high because the angle of beam intersection in the crystal is small enough. Some authors believe that correct comparison of experimental results with the theoretical is practically impossible and useless, due to the fact that there are many free parameters to fit in theory and experimental conditions which are not always strictly controlled. 11 • 19 We have chosen for experiments nominally pure (probably lightly Fe doped) crystal of LiNb0 3, reduced in hydrogen at 500°C. The principle mechanism of recording in this crystal must be Amodei's electron diffusion 14 (saturation photovoltaic field 24 in our crystals was no more than 100 V /cm). That is a very interesting object to study and to compare with the theory inasmuch as the theory in diffusion approximation has no free parameters. The dependencies of the diffraction efficiency on various experimental conditions, light intensity, intensity ratio. angle of incidence, were studied and compared with the theory in the first part of this work. The second part of this article 25 is devoted to investigation of energy transfer between two writing beams due to the beam coupling effect. The comparison with the theory shows that diffusion is really the main process of recording in these crystals and also that the developed dynamic theory describes experimental data much better than the theories which neglected optical feedback effect (e.g. see Ref. 14 ).

electric feedback, 11 - 15 was developed for the case of diffusion and drift in external field. In Refs. 17-18 the calculations of the recording kinetics in the same approximation was done. The same approach was also used by Young and Moharam in Ref. 19, published after we had first sent this work for publication. But in Ref. 19 only the numerical calculations are given for some typical experimental situations. In the present paper the complete set of material equations for electrooptic crystal (charge transport, continuity and Poisson equations) and non-linear wave equation were solved to calculate the intensities of two interactions in the crystal light beams and the distribution of electric field modulating the refractive index. Three main processes of charge transport are considered: diffusion, drift in external field and photovoltaic effect. As distinguished in Ref. 19 we have obtained the analytical expressions for steady state, where the time of recording is much greater than all relaxation times of the system (saturation holograms). Inasmuch as we deal with the complete set of equations we can calculate not only the field E 5c(x,z) and intensities of the light beams I+ 1 (x, z) but also electric current and spatial distribution of electrons n(x,z) and charged traps Nfj(x,z) in the crystal without restricting ourselves by the case n(x,z) oc I(x, z) as was done in most previous papers (see Refs. 4, 14, 15 and 20). Considering real electron distribution in the crystal it is possible to overcome the usual difficulty in determining the space charge field for equal writing beam intensities. As distinguished from the result of Refs. 4 and 20 the amplitude of the first term in Fourier series of refractive index distribution for drift and photovoltaic recording is limited by the value of external field or "effective" photovoltaic field. The distributions of electric field E (x, z) and I+ 1 (z) obtained are used for calculation of the holographic grating structure, its position with respect to the fringe pattern and also for diffraction efficiency calculations. It is shown, in particular, that phase mismatch between FP and HG is uniform in the crystal for saturation holograms. In our previous works 18 • 21 it has been shown that in transient case the isophase surfaces of FP and HG are not parallel, which give rise to additional energy transfer between light beams. Three typical experimental situations are considered: short-circuit, open-circuit and applied voltage conditions. provided that all crystal is uniformly illuminated with fringe pattern. Possible appearance

2

2.1

THEORY

Formulation Equations

of the Problem

and Basic

Let us consider the simplest holographic scheme, presented in Figure 1. Two light beams intersect in the volume of the crystal. Electrooptic axis C is oriented along X direction, Cl IX; the bisectrix of the angle between two beams is parallel to the z

38

HOLOGRAPHIC STORAGE-PART I, STEADY STATE

951

polarization) is given by

)(

I dNJ dn -=----Vj;

dt

dt

e

dNf;

- - = (sl + /J) (ND dt

NJ) - yRnNJ;

j=eµn(E-: Vlnn)+plec;

(I)

FIGURE I Schematic of plane wave hologram recording in electrooptic crystal. Solid lines-light intensity maxima; dashed lines-maximum change of refractive index.

V(e0 E)

direction. We introduce the intensity ratio m0 of the writing beams as the intensity of the beam with negative x-component of the wave vector divided on the intensity of the beam with positive x-component (so this quantity may be less and more than unity): I+ 10 is the value of I+ 1 at z = O; m 0 =I_ 10 /IIO' - Fringe pattern into the crystal initiates the phototransitions of electrons from the impurity levels to the conductivity band. Electrons photoexcited into the conductivity band may be shifted by the diffusion, by the external field, or by photovoltaic effect. Under the steady state conditions the flow of photoexcited electrons is compensated by the difference between their generation and recombination. The position and the contrast of the FP inducing photoexcitations in the crystal varies with the crystal depth due to the possible intensity and phase redistribution between two writing beams. In turn, this redistribution forms the HG correspondingly to the FP. To describe this process the non-linear wave equation is introduced with the dielectric constant depending on the light intensity. The crystal is supposed to contain donor and trap centres, the electron transitions taking place from the donors to the conductivity band and ionized donors capturing the carriers with the probabilities (sl + /3) (ND - Nf;) and yRnN!i respectively. Here s is the cross section of photoionization, /3 is the rate of thermal generation, YR is the recombination constant, n and Nf; stand for the concentration of the carriers and ionized donors. Compensative acceptor levels are filled completely by electrons and are not involved in phototransitions (but lead part of the donors to be free from electrons even in the dark). The set of equations for determining the spatial distribution of electric field, modulating the refractive index, E(x,z) and that of the light waves E(x,z) with the light polarized in the plane of incidence (E

where ec is the unit vector along c-axis, I = cl El 2 Ve/4n is the light intensity, Tis the temperature in energy units, e, e0 are the high frequency and static dielectric constants, pl is the photovoltaic current, introduced by Glass et al., 3 p is the photovoltaic constant. The values of e(I) depends on the light intensity I through the linear electrooptic effect:

= 4ne(n +NA - Nf;);

I d 2(eE) _ =0, V2E+c2 dt 2

e;{l) = e;(I + !'.le;); 2/),.K.

Lle; = r;keiEk(I) = - - ' (i,k = x,y, z), K;

here K; = ~are the components of refractive index and r;k the components of electrooptic tensor, and Ek the components of the space-charge field. E is the microscopic field which includes both the space charge and applied fields. Small light absorption in Maxwell equations will be considered in Section 2.4. The electric field of two interacting beams may be written in the form

_ E

r;-

4n

= . ,r;: [e+ 1 y eye

l+ 1 exp i(!fJ+ 1 + kxx - kAz)

+ e_ 1 ~exp i((jJ_ 1 -kxx- k,z)l, where e± 1 are wave vector intensities and Generation transitions:

the polarization vectors, kx 1 k, are the components, and l+P O. The critical size of the instabilities and, correspondingly, the critical spatial frequency, determined by the nonlinear properties of the medium and the power carried by the field, were found. All the leading features of Bragg self-diffraction, including the appearance of the dynamic grating, and the conditions for and size of energy transfer in the scheme shown in Fig. le were analyzed in Ref. 5 which appeared in the same year. The experimental realization of this scheme on the basis of ruby-laser beams interacting in nitrobenzene followed soon after. 21 This was followed by work on stimulated temperature scattering of the Rayleigh-line wing, and stimulated scattering due to absorption (detailed references are cited in Refs. 11 and 12). From the standpoint of self-diffraction, the initially theoretical and subsequently experimental verification of the possibility of self-diffraction of beams of 5>This is not a fortuitous coincidence because the two effects, i.e. , self-focusing and self-diffraction, are due to cubic nonlinearity.

Sov. Phys. Usp. 22(9). Sept. 1979

Vinetskii et al.

54

743

strictly equal frequency in media with local response under nonstationary conditions I 7'2 2 is the most important result, although the reason for the effect was not established at the time (see Sec. 4). BY now, dynamic self-diffraction has become an independent branch of holography. The emergence of dynamic holography as part of the development of holography generally was recently reviewed in Ref. 9. It was noted that the starting point for the development of dynamic holography was the discovery of stationary energy transfer and its interpretation as a consequence of the writing of the shifted grating in lithium niobate crystals. 39 The first systematic theory of this effect was given in Ref. 8, and interest in studies in this field arose in connection with the possibility of holographic transformation of intensity as a means of correcting the laser wave front.11 8.!lo

depends, in addition, on the temporal and spatial characteristics of the writing fields. For example, in the case of pulsed excitation in a time interval that is small in comparison with the excited-state lifetime r, the nonlinearity determined by is independent of r but, in the case of continuous illumination, _does depend on T. This difference between the effects of self-interaction of beams in a medium with delayed or nonlocal response means that the holographic description of self-diffraction becomes more convenient, since it explicitly involves the phase mismatch between the field and grating, which plays a dominant role in the self-diffraction process. 7,s.1s

x

It is important to note that the terminology used to describe self-diffraction is not as yet finally established. The phrase temporary holograms has been widely used in foreign literature since the publication of Ref. 25 and is meant to indicate that the holograms are produced and read in the presence of radiation in the course of which the writing beams become modified and that the holograms decay after the end of the writing process. "Real time" holography has a very similar meaning. "self-diffraction'' 4 and "dynamic holography''9° 2s.s1 are most frequently used in soviet literature and have the same meaning. If the length of the writing pulse is less than or comparable with the response relaxation time of the recording medium, one speaks of "transient holograms (nonstationary holograms). 27

Renewed interest in degenerate four-wave interaction arose in 1977-8 when it was shown that weak beams could be efficiently enhanced and complex conjugate waves could be generated in the stationary state under the conditions of spatial synchronism in a scheme involving crossed beams collinear in pairs. 23•24 The description of self-diffraction from the standpoint of nonlinear optics is the most convenient in the case of the transparency region when the response can be regarded as being instantaneous and the increment ~i; in the Maxwell equation is determined by the nonlinear polarizability p of the form 3

To so-called superposition-state holograms11 4.11s which appear during the interference between excited atomic states 116 are a new and interesting form of dynamic holograms. Superposition holography has been studied mainly theoretically, but the first experimental results on superposition holograms in ruby have already been reported.117

(1.4)

where the variable subscripts i,k,l, m correspond to the Cartesian components x, y, z, and Xiklm is the cubic nonlinearity tensor whose values are standard characteristics of the material. s. 37 Almost simultaneously with the emergence of the nonlinear optics approach (in 1967 25 ), the other, holographic, approach was developed. The holographic approach provided a more graphic interpretation of the effect of self-diffraction and led to qualitatively new results in a number of cases. The example of self-diffraction of two beams of monopulse radiation from a ruby laser in thin films of a solution of a transmitting dye (cryptocyanine) was used to demonstrate that it was possible to form, read, and transform images in real time with the aid of dynamic holograms. In recent years, self-diffraction has been produced and investigated in a large number of reversible detecting media. 26-35

In the general case of a medium with nonzero inertia, the nonlinear polarizability is given by ['(NLJ

(w, r, t)

= i(

[£ (w,

r, r', t, t') E (w, r, r', t, t')IX E (w, r', t'),

(1.5)

where x[E(w,r,r',t,t')E(w,r,r',t,t')] is an integrodifferential operator whose form is determined by all the processes involving the migration of excitation in coordinate and energy spaces, i.e., by the change in level population in the medium. Accordingly, is no longer a standard characteristic of the medium and

x

744

x

2. ENERGY TRANSFER IN DYNAMIC SELF-DIFFRACTION. GENERAL PRINCIPLES

The theory of the phenomenon is based on the solution of Maxwell's equations that include the nonlinear dependence of the permittivity on the light-field amplitude [see (1.3)]. The specific form of this dependence in each particular case is deduced from the set of constitutive equations for the medium. Since the boundary conditions are periodic, the solution is usually sought in the form of a series in terms of the spatial harmonics of the original interference field acting on the medium. The result of this is the appearance of a set of discrete light beams-higher diffraction orders-and there is also a change in the amplitude and phase of the interacting beams. Depending on experimental conditions (thickness of nonlinear layer, angle between the beams, and size of nonlinearity), the higher diffraction orders may appear (thin dynamic hologram) or may be interference-quenched (volume holography and Bragg diffraction). Let us begin by considering the solution for the general case when l-th order diffraction has nonzero intensity. If we write the Z-th order light wave amplitude in the form E 1 = JT, exp(icp1), and the P-th component of the permittivity grating in the form Ep =I Ep Iexp(i ,), we

Sov. Phys. Usp. 22(9), Sept. 1979

Vinetskii et al.

55

744

obtain the following expressions for the intensities / 1 and phases ..= 5890 A, pumped by a continuous dye laser of only 15 ~. 70 The contribution of the phase grating could be increased by departing from the line center at which the grating was of pure amplitude

This is analogous to Raman- Nath diffraction by given periodic structures 73 and occurs when the higher diffraction orders are not interference-quenched within the nonlinear layer for one reason or another. Many researchers have used the appearance of higher diffraction orders to investigate the nonlinearity mechanism. 74,75

5,---,---,---.,--,----,,...-.,----,- --.--,---,--,

it.I'

Self-diffraction, including the appearance of the higher orders, is usually discussed on the basis of a simplified scheme without taking into account the reaction of the change in the refractive index on the writing light beams (this is the nonstationary phase transparency approximation 76 - 78 ). The calculated characteristics of diffraction by a thin thermal grating 76. 17 show that the intensity of the higher diffraction orders is described by a sum of the form (6.2)

o"---'--~-"'~~~89~5-'---'---'.odl'-'58~~~~-""'.""' l,A

FIG. 12. Self-diffraction of four collinear beams propagating in opposite directions in sodium vapor. 69 The intensity of the fourth wave /c 4 J2 is shown as a ftmction of the wavelength of the interacting beam. 752

= k, ±

where J .. and J,,, • 1 are Bessel functions representing Vinetskii et al.

Sov. Phys. Usp. 22(9). Sept. 1979

63

752

w.......

:::-r-~.----.~.----,

I,

0,5

0

FIG. 14. Dynamic self-diffraction with the participation of higher diffraction orders.87 The figure shows the intensity of the first non-Bragg diffraction order as a function of the spatial frequency of the interference field for the cadmium telluride crystal. Solid line-calculated, 86 open circles-experimental.

diffraction in a given direction from each of the writing beams, and Tis the layer transmission. Experimental studies of self-diffraction under the Raman-Nath conditions have been carried out for solutions of organic dyes 2a. 3o.ss and semiconductors. 2"29° 32 •33, 54, 74, st-s4 It was shown that, when the diffraction efficiency was low, reasonable agreement could be a achieved with different variants of the nondynamic theory. 1s- 7s.s 4.s5.ss This can be understood by recalling that the absolute change in the phase difference between the interacting beams becomes negligible when the overall phase change across the nonlinear layer is small. Improved calculations referring to particular experimental situations appeared after this work. For example, nonstationary self-diffraction by thermal gratings was examined in Ref. 85 and by free-carrier gratings in Ref. 74. Raman-Nath self-diffraction was analyzed in Ref. 78 with allowance for nonlinear absorption. The correct description of self-diffraction with allowance for dynamic feedback is difficult because of the increased number of equations. The first non- Bragg intensities [0 3 have been calculated 49 •S6 for self-diffraction in semiconducting crystals produced as a result of the production of pairs of free carriers. Comparison with experiments performed with CdTe crystals in which stationary dynamic holograms were written by monopulse Nd 3' : YAG radiation showed that there was good agreement with theory87 (Fig. 14). 7. APPLICATIONS OF THE SELF-DIFFRACTION EFFECT

The phenomenon of self-diffraction has found extensive applications in both physics and technology. Image writing, reading, and transformation in real time were demonstrated in the very first paper on dynamic holography. 25 Proposals for logic elements based on dynamic holograms were subsequently investigated. s9 The same idea formed the foundation for a method of measuring the duration of ultrashort light pulses writing the grating. This was suggested and carried out in Ref. 91. The characteristics of selfdiffraction or test-beam diffraction by a dynamic hologram can be used to determine the probabilities of different relaxation processes leading to the erasure 753

of the grating (see the review paper given in Ref. 74). Here, we have the possibility of measuring the temperature diffusivity in liquids and solids, 92, 93 the mobilities of free carriers, and the probabilities of recombination processes in semiconducting compounds, the depth of impurity centers participating in the writing and erasure processes, and the investigation of new mechanisms of nonlinearity. For example, the writing of dynamic gratings in record times with the aid of intraband absorption by the carriers was reported in st. s2• Modern technology can be used to measure diffracted radiation of 10- 5 of the intensity of the incident light, which corresponds to the modulation of the optical path difference by amounts of the order of io- 3A., i.e., the situation has been pushed practically to the limit of the optical band. This high sensitivity of the method means that exceedingly weak effects, such as second sound propagation in crystals, can be investigated. 95 •96 The coefficients of diffusion of excited molecules of a dye in liquid crystals 97 and the anisotropy of thermal conductivity in liquid crystals 9S have also been investigated. Self-diffraction can also be used to determine the components of the nonlinear polarizability tensor, which is responsible for self-diffraction, 5S and to investigate internal inhomogeneities in a medium that give rise to the inhomogeneity of the nonlinear polarizability tensor. 6s Four-wave paired collinear interaction can be used to achieve one of the variants of two-photon laser spectroscopy within the Doppler- broadened line with the elimination of background. sa.10 On the other hand, dynamic holography has opened up new possibilities for real-time image processing. Selfdiffraction in self-translucid liquids and gases has been suggested 99.tot as a means of correlational comparison between two continuously-varying specimens. Two beams of light passing through variable transparencies are used to write the Fourier hologram in the focus of a lens and the third, specially shaped beam of the same frequency is used to read and reconstruct the mutual correlation function for the two specimens in one of the diffraction orders. Amplification of coherent light beams, including amplification of beams carrying optical information, is a possible application. The dynamic nature of the process is such that it can be used to amplify time-dependent signals. Image enhancement during the writing of stationary shifted holograms in lithium niobate crystals 41 has been reported as well as nonstationary energy transfer in lithium niobate crystals in an external field 50 and in different four-wave arrangements for paired collinear interaction. 69•7° Figure 15 shows the image of a television testcard enhanced by a factor of ten during the writing of a hologram in a nominally pure lithium niobate crystal (stationary amplification), Another possible application of dynamic self-diffraction is real-time holographic interferometry.1o 2.io 3 Several theoretical treatments 40 •1°4-tos have been reVinetskii et al.

Sov. Phys. Usp. 22(9), Sept. 1979

64

753

and Ai; is the light-induced nonlinear increment in the permittivity. we shall confine our attention to two waves with polarization perpendicular to the plane of incidence xz [i.e., E= (0, E, O)], incident symmetrically on the crystal at an angle of 28 to each other (see Fig. 1). We suppose that the medium is infinite in the x and y directions, so that we can put o/oy =0 in (A.1) and seek the solution in the form of the Fourier series Eu(.r, z, t}=~Cn.(z, t}ei{{l)t-1'zz+nA::i:>,

11.e (.:z:, z, t)-

FIG. 15. Enhancement of the testcard image on a television screen as a result of dynamic self-diffraction by a stationary shifted grating written in a nominally pure lithium niobate crystal.

ported of possible applications of the energy transfer effect in determinations of the parameters of electrooptic crystals. The use of the energy transfer effect in the visualization of phase inhomogeneities in a nonlinear medium and in the correction of amplitude-inhomogeneous light beams has been discussed. 10 8 Jn the case of stationary self-diffraction, 39 the direction of energy transfer depends on the sign of the mobile charge carrier. 39 It has recently been shown109 that, when lithium niobate crystals are excited in the ultraviolet band, the main mobile carriers are holes and not electrons as in the case of excitation in the visible band. Writing in dynamically nonlinear media can also be used to solve the problem of the transformation of complicated wavefronts to a given form and, in particular, to ·correct the angular divergence of real lasers. s,45,so.110 Calculations have shown that, when the writing conditions are correctly chosen, considerable enhancement can be achieved for the acceptor beam with nearly plane wavefront without appreciable distortion.111, 11 2 On the other hand, the generation of complex conjugate wavefronts in Raman-Nath self-diffraction and in the four-wave interaction68 can be used as a means of compensating dynamic phase inhomogeneities of powerful laser amplifying sections 45 and fiberoptic schemes (this is a variant of adaptive optical systems with amplification). Like many other nonlinear effects, self-diffraction can be used to control the duration of the diffracted beam. Compression resulting from the four-wave interaction, which depends on the pump-wave intensity, was reported· in Ref. 67. However, the compression of ultrashort light pulses by "time inversion" of the beam with the complex conjugate wavefront is· much more interesting. 64

APPENDIX The basic set of equations used to describe self-diffraction of light waves in a nonlinear medium includes the Maxwell equations VXE=-+

~~,

VXH=+

D=(e+6e)E,

where 754

E

aa~ +

4

cn j,

(A.1)

J=crE,

is the permittivity averaged over the volume,

::E Ep (z,

(A.2)

t) eiPk:cx.

substitution of (A.2) and (A.1) yields (fork, =k 0 £ cose, k,=k 0 /t sine):

(A.3) Henceforth, we shall consider smooth variations of C 1 with z and t, which is valid provided a;~L

(x) is the quasistatic potential existing in the crystal due to internal charge migration, externally applied fields and intrinsic chemical potentials. Whenever light is present, the gradient of¢ will cause a net drift by hopping of charges in time, away from a stationary background of neutralizing charge (which we assume does not hop). In our model this drift is governed by Eq. (l ), for hopping in all directions. In barium titanate, the omission from the hopping rate of any dependence on the probability of occupation of the final site is consistent with the experimental results. That is, in our experiments the site occupation probabilities appear to be much less than unity, implying that most sites are unoccupied. It is straightforward to account for final-site occupation by appending factors 1-Wm to the rate to hop to site min Eq. (1), should it prove appropriate. To analyze our present experimental results we will need to consider only the case where I" varies from the interference of two "writing" optical beams of the same temporal frequency whose complex electric field amplitudes at position x are exp(1k 1·x) ande2E 2 exp(ik2 ·x). See Fig. I. The complex polarization vectors and e2 are normalized by 1 = l, etc. Therefore, we may write

ii. THEORY We describe here a model for charge migration in photorefractive crystals which, with two parameters, predicts the energy exchanges among two or four optical beams. This exchange, in both transient and steady-state regimes, is shown to depend on the intensities, polarizations, and angles of the beams, and on an applied static electic field for a given orientation of the crystal. Given the charge-migration patterns from our model, one can derive the quasistatic electric fields inside the crystal, and from this the refractive index O], this implies that.fo and hence q is positive. In similar experiments on LiNb0 3 , the direction of energy coupling was opposite to that observed here. 11 Assuming that the pyroelectric determination of the positive direction of the c axis used in that work 12 agrees with the definition used here, the signs of the charge carriers must be opposite in the two materials. However this point should be checked.

There are many ways to determine the sign of the 10 charge carriers in photorefractive materials. • l 1 The sign of the charge carriers in barium titanate was experimentally determined to be positive by energy-coupling experiments, as follows. From Eqs. (18), (13), and (5), the direction of energy coupling will depend only on the sign of the electrooptic coefficient, the orientation of the crystal relative to the writing beams, and on the sign of the charge carriers. In order to avoid any confusion in determining the sign of the charge carriers, a method was used which is independent of the definition of the positive direction of the c axis. The crystal was first poled by painting silver electrodes . on the c-axis faces, slowly heating it to about 5 ·c below the Curie temperature (Tc = 133 'C), and then applying a de poling field of 3.5 kV /cm along the c axis. The crystal was then slowly cooled to room temperature with the poling field still applied to produce a single-domain crystal. For convenience, we define the positive c-axis direction as pointing toward the electrode that had been connected to the negative terminal of the applied de voltage. (The final determination of the sign of the charge carriers is independent of this choice.) Next the sign of the electro optic coefficient was determined to be positive relative to the positive c axis by placing the crystal between crossed polarizers with the c axis oriented at 45' to the direction of polarization. A Babinet-Soleil compensator, also placed between the polarizers, was adjusted to null out the birefringence of the crystal at 515 nm. A de field was applied in the same direction as the previous poling field, and the compensator was adjusted to give a new null that corresponded to a decrease in the difference of the extraordinary and ordinary indices when the applied field was increased. From (13), n.{E) =

n 0 (E) =

D. Phase and absolute magnitude of index grating The magnitude of the steady-state energy coupling between two beams depends on the component of the index grating that is 90° out of phase with the intensity distribution of the two beams (i.e., on the imaginary part of 8k 1). However, the diffraction efficiency R with which the grating scatters a third reading beam depends on the magnitude of the grating and not on its phase with respect to the writing beams. Therefore, by separately measuring the fraction F of two-beam energy coupling and the diffraction efficiency described in Sec. II B, real and imaginary parts of ok in Eq. (19) can be determined. From Eq. (19) with no applied electric field (/ = 0), the real part of ok should be zero~ and

F;::::,2R

when all beams have the same polarizations, m = l, and R 33 k[ cxFsin> 1, the intensity coupling constant 'Y decreases significantly. The time constant r depends on materials as well as on the intensity of the laser beams. The fundamental limit of such a time constant r is discussed next.

D. Speed of Photorefractive Effect-Grating Formation Time As mentioned earlier, photorefractive crystals such as BaTi03, SrxBaI -xNb206 (SBN), Bi12Si020 (BSO), etc., are by far the most efficient media for the generation of phase-conjugated optical waves using relatively low light intensities ( 1-10 W /cm2 ). In addition, these materials also exhibit several interesting and important phenomena such as self-pumped phase conjugation, two-beam energy coupling, and real-time holography. All of these phenomena depend on the formation of volume index gratings inside the crystals [8], [9]. One of the most important issues involved in device applications is the speed of the grating formation (or the time constant r). Such a speed of the light-induced index gratings has been investigated theoretically using Kukhtarev 's model and others, as well as experimentally in various crystals [8]-[10], [21]. The issue of fundamental limit of the speed of photorefractive effect has been a subject of great interest recently. Using Kukhtarev's model, let us examine the four fundamental processes involved in the photorefractive effect in sequence: l) photoexcitation of carriers, 2) transport, 3) trap, and 4) Pockels effect. The photorefractive effect is a macroscopic phenomenon and requires the generation and transport of a large number of charge carriers. We note that without the presence of charge carriers, photorefractive gratings can never be formed, and no matter how fast the carriers can move (even at the speed of light, 3 x 108 m/s ), the formation of index grating is still limited by the rate of carrier generation. Therefore, although each of the four processes involved imposes a theoretical limit on the response time of photorefractive effect, the fundamental limit of the speed of photorefractive effect is determined by the pho-

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stant 'Y and is inversely proportional to the light intensity. Equation (51) is the expression for the minimum time required for the formation of an index grating which provides a coupling constant 'Y. A figure-of-merit for photorefractive material is often defined as

Yol = 10 al = 1

(52) 10-2

Table I lists such parameters for some photorefractive materials. Using such a parameter, the photon-limited time for the index grating formation becomes t

Fig. 4. Signal gain gas a fcnction of {)T for various values of m.

toexcitation of carriers and not by the carrier transport. In other words, the charge carriers must be generated before they can be transported. Any finite time involved in the transport process can only lengthen the formation time of the grating. From this point of view,_ the fundamental limit may also be called the photon-flux limit, or simply the photoexcitation limit. Although the fundamental limit can be derived from (1)-(4) of Kukhtarev's model, it has been recently derived using a relatively simple method [22]. Such a fundamental limit has been confirmed experimentally [23]. In ~he limit when the crystal is illuminated with infinite intensity, the speed of the photorefractive effect will be limited by the charge transport process [24]. Assuming that the separation of a pair of charge particles requires the absorption of at least one photon, we can calculate the energy required to form a given volume index grating. To illustrate this, let us consider the photorefractive effect in BaTi03 . Generally speaking [10], an efficient beam coupling would require a charge carrier density of approximately 10 16 cm- 3 • Such a charge separation would require the absorption of at least 10 16 photons in a volume of l cm 3 . Using a light intensity of l W in the visible spectrum, the photon flux would be approximately 10 19 /s. Thus, assuming a quantum efficiency of 100 percent, it takes at least l ms just to deposit enough photons to create the charge separation. The actual grating formation time can be much longer because not all of the charge carriers are trapped at the appropriate sites. According to the model described in [22], the minimum time needed for the formation of an index grating, which provides a coupling constant of 'Y, is given by t =

(h;)(~)(;J :~ ·1;3r

(51)

where hv is the photon energy, e is electronic charge, 'A is wavelength of light, A is the grating period, aP is photorefractive absorption coefficient, ~ is the quantum efficiency, e is dielectric constant, r is the relevent electrooptic coefficient, and /is the intensity of light. Note that the time constant is directly proportional to the coupling con-

103

=(h;)(~)(;J 7r:!Q'

(53)

Here, we note that this photon-limited time is inversely proportional to the material's figure-of-merit and is proportional to the coupling constant 'Y. We now discuss this photon-limited time for the formation of an index grating which yields a coupling con1 stant of 1 cm- • For materials such as BaTi03 , SBN, BSO, and GaAs, the figure-of-merit Q is of the order of 1 (see Table I) in MKS units (M 2 /FV). In many of the experiments reported recently, hv is approximately 2 eV, ( A/ A) is of the order of 0. 1. We further assume that the photoexcitation absorption coefficient is 0.1 cm -t and that the quantum efficiency ~ is 100 percent. Using these parameters and a light intensity of 1 W /cm 2 , (53) yields a photon-limited time constant of 0.15 ms. This is the minimum time required for the formation of an index grating which can provide a two-wave mixing coupling constant of 1 cm_,. By virtue of its photoexcitation nature, the photorefractive effect is relatively slow at low intensities because of the finite time required to absorb the photons. Table II shows the comparison between the measured time constants with the calculated minimum time from (53). The only way to speed up the photorefractive process is by using higher intensities. Fig. 5 plots this minimum time constant for BaTi03 (or GaAs) as a function of intensity. The photoexcitation process imposes a fundamental limit on the speed of photorefractive effect at a given power level. The time constant given by (51) here is the absolute minimum time required to generate a volume grating of given index modulation. We assume that the transport is instantaneous and the quantum efficiency is unity. Thus the derived time constant is the absolute minimum time. Any finite time involved in the transport process can only slow down the photorefractive process. The fundamental limit discussed here can also provide important guidelines for many workers in the area of material research. For example, if we compare it with the experimental results, we find that the time constant of some materials (e.g., BaTi03 , SBN) is two orders of magnitude larger than the fundamental limit. Thus, the calculation of such a fundamental limit and a simple comparison point out the room for improvement by either in- , creasing the photorefractive absorption or the quantum ef-

491

YEH: TWO-WAVE MIXING IN NONLINEAR MEDIA

TABLE I FIGURE-OF-MERIT FOR SOME PHOTOREFRACTIVE MATERIALS

1'

r

Materials

µm

pm/V

BaTi0 3 SBN Ga As BSO LiNb0 3 LiTa0 3 KNbO,

0.5 0.5 I.I 0.6 0.6 0.6 0.6

'The figure-of-merit

T42

r 33 '12 T41

r 33 r 33 T42

Q

[22] Q'

n

= 1640 = 1340 = 1.43

n, n, n, n n, n, n

= 5

= 31 = 31 = 380

= 2.4 = 2.3 = 3.4 = 2.54 = 2.2 = 2.2 = 2.3

•/•,,

pm/V•,,

Q(MKS)

•1 = 3600 ,, = 3400 '= 12.3 '= 56 ,, = 32 ,, = 45 ,, = 240

6.3 4.8 4.7 1.5 10.3 7.3 19.3

0.71 0.54 0.53 0.17 1.16

0.83 2.2

depends on the configuration of interaction. TABLE II

COMPARISON OF MEASURED TIME CONSTANTS 7 AND THE FUNDAMENTAL LIMIT t

1'

A

Olp

µm

µm

cm- 1

-y cm- 1

7'

Materials GaAs GaAs:Cr BaTi0 3 BSO SBN SBN:Ce

l.06 1.06 0.515 0.568 0.515 0.515

1.0 l.l 1.3 23.0

1.2 4.0 1.0 0.13 0.1 0.7

0.4 0.6 20.0 10.0 0.6 14.0

80 x 10-• 53 x 10-• 1.3 15 x 10- 3 2.5 0.8

aT

•1

1.5 1.5

,• Remarks 45 31 2 2 6 2

x x x x x x

10-• 10-• 10- 3 10- 3 10- 3 10- 3

(25) (26) (27) (28) (29) (29)

and t are time constants at incident intensity of 1 W / cm 2 .

is the calculated time constant by using (53) and assuming a quantum efficiency of 1. number of photons at a given power level. By counting the total number of photons needed for the formation of an index grating, the photon-limited time constant is derived. This time constant is inversely proportional to the light intensity. We further estimated this minimum time constant for some typical photorefractive crystals. Such a fundamental limit provides important guidelines for researchers in the areas of device application and material research. III. PHOTOREFRACTJVE Two-WA VE MIXING IN CUBIC CRYSTALS INTENSITY (W/cm2l

Fig. 5. Fundamental limit of the speed of photorefractive effect of BaTiO., (or GaAs) with coupling constant of I cm- 1

ficiency. There are some materials (e.g., GaAs) whose photorefractive response is close to the fundamental limit, leaving no more room for further improvement by any means (e.g., heat treatment, doping, or reduction.) Recently, highly-reduced crystals of KNb0 3 were prepared which exhibit a photorefractive response time very close to the fundamental limit [23]. In summary, the photorefractive effect is a macroscopic phenomenon. It involves the transport of a large number of charge carriers for the formation of any finite grating. The fundamental limit is the minimum time needed· for the generation of these carriers. The speed is fundamentally limited by the finite time needed to absorb a large

Photorefractive two-beam coupling in electrooptic crystals has been studied extensively for its potential in many applications. Much attention has been focused on materials such as BaTi03 , BSO, SBN, etc., because of their large coupling constants (see, for example, Table II). Although these oxide materials are very efficient for two-beam coupling, they are very slow in response at low operating powers [22]. Recently, several experimental investigations have been carried out to study two-wave mixing in cubic crystals such as GaAs, which responds much faster than any of the previously mentioned oxides at the same operating power [25], [26]. In addition to the faster temporal response, the optical isotropy and the tensor nature of the electrooptic coefficients of cubic crystals allow for the possibility of crosspolarization coupling. Such cross-polarization two-wave mixing is not possible in BaTi03 and SBN because of the optical anisotropy, which leads to velocity mismatch. The velocity mismatch also exists in BSO crystals because of

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the circular birefringence. A number of special cases of two-wave mixing have been analyzed. Recently, a general theory of photorefractive two-wave mixing in cubic crystals was developed [30]. Such a theory predicts the existence of cross-polarization signal beam amplification. These cross-polarization couplings have been observed in GaAs crystals [31]-[33]. In what follows, we describe the coupled-mode theory of photorefractive two-wave mixing in cubic crystals, especially those with point group symmetry of 43m. The theory shows that cross-polarization two-wave mixing is possible in cubic crystals such as GaAs. Exact solutions of coupled mode equations are obtained for the case of codirectional coupling. A. Coupled-Mode Theory Referring to Fig. 6, we consider the intersection of two polarized beams inside a cubic photorefractive crystal. Since the crystal is optically isotropic, the electric field of the two beams can be written as

E=

( sA,

Fig. 6. Schematic drawing of photorefractive two-beam coupling in cubic crystals,

where e1 is a 3 x 3 tensor, cf> is the spatial phase shift between the index grating and the intensity pattern, (} is the angle between the beams inside the crystal, and 10 is given by

10

=

A;' A, +A: AP + B;' B, + B: BP.

(58)

For cubic crystals with point group symmetry of 43m, e 1 is given by

e1

+ j},Ap) exp (-ik, . r)

n4r41 (~z

=

: ~) 2

(59)

Ey Ex 0

+(sB,+p2 Bp)exp(-ik 2 ·7)

(54)

where k1,and k2 are wave vectors of the beams, s 1s a unit vector perpendicular to the plane of incidence, and p1, p2 are unit vectors parallel to the plane of incidence and perpendicular to the beam wave vectors, respectively. Since each beam has two polarization components, there are four waves involved and A,, AP, B,, and BP are amplitudes of the waves. All of the waves are assumed to have the same frequency. In addition, we assume that the crystal does not exhibit optical rotation. In the photorefractive crystal (from z = 0 to z = L ), these two beams generate an interference pattern,

- ... E- =A,* A, E

where r41 = r131 = r312 = r123, and Ex, Ey, and £ 2 are the three components of the amplitude of the space-charge field. Substitution of the index grating equation (56) into Maxwell's wave equation leads to the following set of coupled equations: i

d

2,..,

z

· (A,B: +APB: cos 0)/10

dzd B,

=

p p + AP* AP + B,*Bs + B*B

2

i e-' "' [r,,A, (3

+ r,p,Ap]

2

· (A;' B, +A; BP cos 0)/10

+ [ (A,B;' + APB: j}, . P2) ·exp (iK · 7) + c.c.]

dzd AP

(55)

where K = k2 - k1 is the grating wave vector, and c.c. represents the complex conjugate. We note that there are two contributions to the sinusoidal variation of the intensity pattern. As a result of the photorefractive effect, a space-charge field E'c is formed which induces a volume index grating via the Pockels effect, (56) where e0 is the dielectric permittivity of vacuum, n is the index of refraction of the crystal, r;jk is the electrooptic coefficient, and Ek is the k component ( k = x, y, z) of the space-charge field. The fundamental component of the induced grating can be written ~e = -eoe 1[(A,B;' +APB; cos 0)

·exp (iK ·

'

-d A, = ---;:;- e' [r,,B, + r,p,Bp]

r +cf>)+ c.c.])/1

0

(57)

105

i

=

i> I), one-half of the incident pump energy A, ( 0)

Fig. 8. Intensity of the_ four waves are plotted as functions of distance for various interaction situations. (a) Both incident beams are s-polarized (i.e., (Ap(O) = Bp(O) = 0, c2 /c 1 = 0.1). (b) The pump beam is linearly polarized at an azimuth angle of 30° relative to the s direction, and the signal beam is s-polarized, c2 / c 1 = 0.01.

is coupled to the p component of the signal beam BP, and one-half of the incident signal energy B, ( 0) is coupled to the p component of the pump beam. As a result of the opposite signs in the wave amplitudes ( f < 0), the two contributions to the index grating tend to cancel each other. Thus, when the energy of the p components reaches one half of the incident energies, the coupling ceases. It is possible that the p component of the signal beam BP receives most of the incident pump energy A,. Fig. 8(b) illustrates a case in which the pump beam has both s and p components, whereas the signal beam is s-polarized. We note that for strong coupling ( 'YL >> 1 ), most of the energy of the s component of the pump beam is coupled to the p component of the signal beam. The nonreciprocal transfer of energy is very similar to that of conventional two-wave mixing.

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495

The exact solutions of (77) and (79) are useful when the coupling is strong ( -yL >> 1) and the energy exchange is significant. For the case of weak coupling ( -yL = 11' /2. The case of cl> = 0 corresponds to a pure local response of the material. Although the energy is exchanged back and forth between As and BP as well as between AP and B,, there is no nonreciprocal energy transfer. In other words, there is no unique direction of energy flow as compared with the case when cl> 0. For cases with 0 < Icl> I < 11', nonreciprocal energy transfer is possible according to our so2

*

108

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496

lotions (88) and (89) with maximum energy transfer at ¢ == ±7r/2.

A,A: - B,B; = ASBS - APBP =

C. Contradirectional Cross Coupling Referring to Fig. 7(b), we consider a case of contradirectional coupling which does not permit the parallel coupling to occur. The two beams enter the crystal in such a way that the grating wave vector is along the [001] direction of the crystal. In this configuration, the unit vector sis parallel to [010] and the unit vectors Pi. p2 are perpendicular to [010]. The amplitude of the induced index grating e1 can thus be written, according to (59),

(94)

where E'c is the amplitude of the space-charge field. According to (62) and (94), and after few steps of algebra, the coupling constants can be written

r,,

=

rP1P2

fsp 1

=

fp 1s

=

o

= f,p 2 = f P2, = n r41E'c COS {8/2) 4

(95)

where we assume that the beams enter the crystal symmetrically [see Fig. 7(b)] such that {3 1

=

-{32

=

-{27r/A.)n cos (8/2).

(96)

We now substitute (95) and (96) into the coupled equation (60). This leads to

1z A,= -yBp(A,B: +APB: cos 8)/1

0

~ B,

== -yAp(A: B, +A: BP cos 8)/10

1z AP = -yB,(A,B: +APB:

COS

8)/10

1z BP= -yA (A: Bs +A: BP cos 8)/1 5

where we have taken ¢

0

(97)

= 11" /2, and 'Y is real and is given (98)

Notice that the coupled mode equation (97) is similar to that of (67), except for the signs. The difference in signs is due to the direction of propagation of the pump beam (A,, Ap). As a .result of this difference, the total intensity lo is no longer a constant. According to (97), (A: A, + AP - B: B, - B: BP), which is proportional to the net Poynting power flow along the + z direction, is a constant [37]. There are other constants of integration. These include

A:

A: A, - B; BP =

C1

A: AP - B: B, -

C2

C3

C4.

(99)

Because 10 is not a constant, the integration of the coupled equation (97) is not as simple as that of (67). As of now, there is no closed form solution available. However, numerical techniques can be used to integrate the coupled equations. For the case of no pump depletion, we may treat A, and AP as constants. In this case, the coupling equations for Bs and BP are identical to those of the codirectional coupling, and the solutions are given by (85) and (86). In summary, we have derived a general theory of the coupling of polarized beams in cubic photorefractive crystals. As a result of the optical isotropy of the crystal and the tensor nature of the holographic photorefractive grating, cross-polarization energy coupling occurs. Exact solutions for the case of codirectional coupling are obtained. Such cross-polarization coupling may be useful for the suppression of background noises. D. Cross-Polarization Two-Beam Coupling in GaAs Crystals

Cross-polarization two-beam coupling has been observed in GaAs crystals recently. The experimental results are in good agreement with the coupled-mode theory presented earlier [31]-[33]. In a contradirectional two-beam coupling experiments as described in [33], a 1. 15 µm beam from a He-Ne laser is split into two by a beam splitter. The two beams intersect inside a liquid-encapsulated Czochralski (LEC) grown, undoped, semiinsulating GaAs crystal from opposite sides of the (001) faces [see Fig. 7(b)]. The intersecting angle of two beams is approximately 168 °. The wave vector of the induced index grating is along the [001] crystalline direction. One beam, B, is polarized along the [010] direction (spolarization) using a polarizer, which fits the condition of Bp(O) = 0. The other beam, A, is transmitted through another polarizer (along the [ 100] direction), followed by a half-wave plate, which is used to vary the polarization of the pump beam. The power of beams A and B is 80 mW/cm2 and 1 mW/cm2 , respectively. The GaAs crystal is 5 mm thick. The gain coefficient of the crystal measured with the regular beam-coupling configuration is about 0. 1 /cm. These values fit well with the conditions of no beam A depletion and -yL a, according to ( 118). Also notice that the coupled equations (118) are exactly identical to those of the stimulated Brillouin scattering and stimulated Raman scattering. Solutions for the lossless case had been derived by previous workers [2]. We now derive the solution for the case of lossy nonlinear medium. Using the classical model mentioned above, beam 2 will gain energy from beam l, provided that the phase shift is positive. Thus, according to (l 15), the low-frequency beam will always see gain. The coupled equations ( 118) can be integrated exactly, and the solution is (see Appendix B)

( 115)

;q, A

2kz c2 e

0 :S 0

7r (3 =A cos (0/2) n2 cos.

2

l

.

27r

= -i~;,~~2e-iIA2l2A1

!!._ A1 = - . w2non2

( 119)

= f3!1

g =A cos (0/ ) n2 sm ,

where we recall that r is the response time of the medium. 1) Codirectional Two-Wave Mixing: Now, by using (112) for n and the scalar wave equation and by using the parabolic approximation (i.e., slowly-varying amplitudes), we can derive the following coupled equations:

dzA,

"12

where

( 113)

n10

.

exp (1¢) = 1

(118)

and

-oo

where n20 is the value of index change for the degenerate case. Integration of (113) yields the following expression for n2 exp (i ): n1

d dz 12 = g/112 - a.12

(116)

2

where we assume that w2 = w 1 = w, and k, is the z comk2 cos ponent of the wave vectors (i.e., k, = k 1 cos ~ 0 ~ 0). The parameter 0 is the angle between the two beams. In ( 116), we have neglected the term t:.n 0 •

=

112

where m is the input beam ratio (124)

and 'Y is given by "( = g[/1(0)

+ /2(0)].

(125)

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IEEE JOURNAL OF QUANTUM ELECTRONICS, VOL. 25. NO. 3, MARCH 1989

Substituting (122) and (123) for / 1 and / 2 , respectively, into (119) and carrying out the integrations, we obtain

{3 1

=

l

l

+ m-1

g og ( 1 + m- exp[~ (1 1

1oz,,.....,.-,-......-......,.-.--,-..--.-.--.-..,....-,,.......-,-,--,.....,.-,-,

]

- e-"')] (126)

and

1fi(z) - 1/12(0)

~ +•g [+ l

z(cm)

li ~ -,-~) lJ

1 m "P

Fig. 10. Intensity variation with respect to

(127) Note that according to (126) and (127), the phases of the two waves are not coupled. In other words, these two waves can exchange energy without any phase crosstalk. Such a phenomenon has been known in stimulated Raman scattering for some time, and can be employed to pump a clean signal beam with an aberrated beam. Here, the result can be applied to more cases, including forward stimulated Brillouin scattering. Ifwe neglect absorption (i.e., ex= 0), then / 2 (z) is an increasing function of z and / 1 ( z) is a decreasing function of z, according to (122) and (123), provided 'Y is positive. Transmittance for both waves for the lossless case, according to (122) and (123), is

(L) l + m- 1 / 1 (0) = l + m- 1 exp (-yL)

/1

Ti

5

T-[z(L)_ 2

-

/ 2 (0)

l+m

- 1 + m exp ('YL)

z in the Kerr medium.

written 21rn 20 g =A cos (8/2). 1

-nr + (nr/

( 129 )

where We recall that{} = W2 - W1 is the frequency detuning and r is the grating decay time constant. We notice that the gain coefficient is positive for the beam with lower frequency provided that n20 is positive. Such a gain coefficient is maximized at Or = ± 1. Such a dependence on frequency detuning can be used to measure the time constant r. Some experimental works will be discussed in Section IV-E. 2) Contradirectional Two-Wave Mixing: We now consider the case of contradirectional two-wave mixing in which beam 1 enters the medium at z = 0, and beam 2 enters the medium at z = L. The coupled-mode equations for the beam intensities can be obtained in a similar manner and are written

( 128)

d

'd/1

where m is the incident intensity ratio m = / 1 ( 0) / [z ( 0). Note that T2 > l and T1 < l for positive 'Y. The sign of 'Y is determined by the sign of n 2 and the phase shift . Interestingly, these expressions are formally identical to those of the photorefractive coupling. The major difference is that the 'Y for Kerr media is proportional to the total power density of the waves, according to (125). Fig. IO illustrates the intensity variation with respect to z for the case when g = IO cm/MW, ex = 0.1 cm- 1, 2 / 1 (0) = IOO kW /cm , and 12 (0) = 1 kW /cm 2 • Note that even with the presence of absorption, the intensity of beam 2 increases as a function of z until z = I where the gain equals the loss. Beyond z = le, the intensities of both beams are decreasing functions of z. Similar results were obtained earlier by other workers in a study of stimulated scattering of light from free carriers in semiconductors [50]. According to ( 114), ( 115), and ( 120), the gain coefficient g is a function of the frequency detuning and can be

d -12 = -g/112 dz

where the

+ od2

( 130)

coefficient g is given by

211' ) n 2 sin , 812

g = A sin (

7r /2

< 8

:S 7r.

(

131 )

We notice a slight difference between the two cases as compared with (120) for the codirectional coupling. Here, we recall that 8 is the angle between the positive direction of the two wave vectors. Thus, for codirectional coupling, the angle 8 is always less than 90°, whereas 1r /2 < 8 < 1r for the contradirectional coupling. This is a result of the boundary condition in which we assume that the waves and the medium are all of infinite extent in a plane perpendicular to the z axis. Solutions of (130) for the case of lossless medium are given by

113

501

YEH: TWO-WAVE MIXING IN NONLINEAR MEDIA

/1 (z) /1(0)

-

12 (z) /z(O)

egaz -

section, we investigate the photoinduced index grating in nondegenerate two-wave mixing and focus our attention on the complex Kerr coefficient and the spatial phase shift. Basic equations for the electrostrictive coupling between photons and phonons have been formulated and several theoretical papers on SBS have been published [51] . Most of the earlier work was concentrated on backward-wave coupling. Very little attention was paid to codirectional nondegenerate two-wave mixing. The mathematical formulation of such a coupling in Kerr media including material absorption has been recently solved and is described in Section IV-A. In this section, we focus our attention on the derivation of the photoinduced index grating as well as the relation between the photoelastic coefficient and the electrostrictive constant. The electrostrictive pressure in liquids is given by

1 - p pe-gaz

1 - p

(132)

P

where a and p are constants and are related to the intensities at z = 0,

12(0) p

= /1(0)

(J

= /1(0) - 12(0).

( 133)

The constant a may be regarded as the net power flux through the medium. The solutions of (132) are expressed in terms of / 1 ( o) and /z (0), which are not input intensities. In the contradirectional coupling, we note that the incident intensities are / 1 (0) and lz (L). For interaction L, such that gaL >> 1, the intensity growth for beam 2 is exponential and is given by, according to (124),

l2(L) gaL 12(0) ""---e . ( 1 - p)

(135) 2

where ( E ) is the time average of the varying electric field and is given by (109), and 'Y is the electrostrictive coefficient which is defined as

(134)

'Y

In SBS, / 2 (L) is virtually zero and represents intensity of noises or scattered light. The parameter p = 12 ( 0) / / 1 ( 0) is the phase-conjugate reflectivity of the SBS process. It is always less than unity for two reasons. First, in SBS there is no beam 2 incident at z = L; therefore, p ::;; 1 is required by the conservation of energy. Second, the exponential gain per unit length ga is proportional to the power throughput. A reflectivity of p = 100 indicates a zero power flux and consequently zero gain.

B. Electrostrictive Kerr Effects The Kerr effect arises from several physical phenomena. These include molecular orientation, molecular redistribution, third-order nonlinear polarizability [3], electrostriction, and thermal changes. In liquids such as CS 2 , contribution to the Kerr effect is dominated by the electrostriction. The coupling of two electromagnetic waves via electrostriction has been known for some time and is responsible for SBS. Although this subject has been studied extensively [51], little attention has been paid to the "photorefractive" nature of such a process, which, we believe, can provide a great deal of insight into generalizing the SBS process. For example, there exists a similar spatial phase shift of 90° between the induced index grating and the light interference pattern in conventional SBS [52]. Such a spatial phase shift of 90° is responsible for the energy exchange between the incident wave and the phase-conjugated wave in SBS. In addition, self-pumped phase conjugation in BaTi03 crystals [53], [54] is very similar to the phase conjugation in SBS [55], [56]. The spatial phase shift of 90° can be utilized in other SBS 180° ). In this configurations (e.g., injected SBS at 8

=PG;)

(136)

where p is the density and e is the dielectric constant. As a result of the electrostrictive pressure according to (135) and (109), a density wave in the medium is generated. By solving the isothermal Navier-Stokes equation [51] we obtain the complex amplitude of the induced density wave as .:l

p -

-

~

2 112

-

( 137)

A* A i(Or-K· rJ K1'Y Q~ - inf B I 2 e

where 118 may be regarded as the resonance phonon frequency and is given by (138) 0 8 = vK with v as the velocity of the acoustic wave, and inverse of the phonon lifetime and is given by

I's=riK 2 /p

r8

is the (139)

with 'Y/ as the viscosity coefficient. Using ,iE = 2nE 0 dn and the definition of the electrostrictive coefficient equation (135), we obtain the linear relation between the index grating and the density wave !in = _'Y_ Ap. 2npfo

(140)

Using the complex number representation, the induced index change can be written, according to (112), (141) Substituting (141) for An and (137) for Ap in (140), we obtain the following expression for the complex Kerr coefficient:

*

114

(142)

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IEEE JOURNAL OF QUANTUM ELECTRONICS, VOL. 25, NO. 3, MARCH 1989

Note that this complex Kerr coefficient is a function of the frequency difference between the two waves. At resonance fl = ±08 , the Kerr coefficient is purely imaginary, indicating a 90° phase shift between the index grating and the intensity pattern. The complex Kerr coefficient derived above is different from the traditional one used in self-focusing and selfphase modulation as described in [61] . The Kerr coefficient measured in those experiments may be regarded as the de Kerr coefficient and is related to that of ( 142) by putting fl = 0. Such a de Kerr coefficient is written

n1(fl

'¥2

= O) = - -2-

4npv Eo

> 1 (or g/L >> 1 ), which corresponds to the Kerr regime. However, the diffraction efficiency is zero when KL= 0, according to (171) and (173). At b = 0, (173) reduces to T/ = sin2 11.L, which is the familiar expression of the Bragg cell diffraction efficiency. For such nonlinear Bragg scattering to be seen, the Kerr coupling constant must be comparable with the Bragg coupling constant. Thus the parameter b must be of the order of 1. If b = 1 is used as an example, the Kerr intensity-coupling constant must be g/ = 4K.

We now take a Bragg coupling constant of " = l cm - I as an example and use a nonlinear medium such as CS 2 • From the data available in [51], the Kerr coupling constant g for a Bragg angle of 5° (8 = 10°) is g = 1.5 cm/MW, and the radio frequency required is 640 MHz. Thus, the optical intensity needed for observation of a significant nonlinearity in Bragg scattering, according to the above condition, is approximately 2.7 MW /cm2 • The results show that diffraction efficiency is a nonlinear function of the optical intehsity and can be greatly enhanced by increasing the intensity of the optical wave.

505

YEH: TWO-WAVE MIXING IN NONLINEAR MEDIA

It can be used as a nonlinear device in which high-efficiency diffraction only occurs when the optical intensity is above a threshold.

tensity, whereas the photorefractive gain coefficient is independent of the intensity. Thus, for high-power applications, SBS and SRS can be efficient means for beam coupling. In addition, the frequencies of the idler wave are very different. In SRS, the idler wave is optical phonon. In SBS, the idler wave is acoustic phonon. Also, in photorefractive two-wave mixing, the idler wave is a holographic grating. As a result of the finite frequency of the idler wave, the coupled waves in these three processes are different in frequencies. For SBS and nondegenerate two-wave mixing in photorefractive media, the frequency difference is small so that the two waves propagate at virtually the same speed. In SRS, the large Stokes shift may lead to a significant difference in the phase velocity of the two waves due to dispersion. This may result in a phase mismatch in the wave couplipg. The coupled equations for stimulated Raman scattering are identical to those of the stimulated Brillouin scattering, except for the possibility of dispersion. In fact, it is known that, like SBS, SRS also exhibits phase conjugation [67]. The energy coupling in both SBS and SRS is due to the imaginary part of the third-order dielectric susceptibility [2]. If we examine (10) and (13), we notice that the energy coupling in photorefractive crystal is due to the out-of-phase term of the index grating. This spatial phase shift is 90° in crystals such as BaTi03 , which operates by diffusion only. If we interpret the idler wave in SBS and SRS as a traveling index grating, then the spatial phase shift is also exactly 90° in resonant scattering [see (142)]. In view of the above discussion, we may generalize the meaning of photorefractive effect to include other phenomena such as the Kerr effect. In other words, the generalized photorefractive effect is a phenomenon in which a change of the index of refraction is induced by the presence of optical beams. Thus, we may view SBS and SRS as nondegenerate photorefractive two-wave mixing in nonlinear media.

D. SBS, SRS, and Photorefractive Two-Wave Mixing Thus far we have discussed two-wave mixing in photorefractive crystals and Kerr media. In photorefractive two-wave mixing the frequency difference between the two beams is zero or small (a few Hertz). For two-wave mixing in Kerr media or stimulated Brillouin scattering (SBS), the frequency difference can be as large as a few gigahertz. Energy exchange between two beams also occurs in stimulated Raman scattering (SRS) [2]. The frequency difference between the beams in Raman scattering is in the range of terahertz. There are several common features among the three types of two-wave mixing. All three types of wave mixing show nonreciprocal energy exchange without phase crosstalk. In fact, if we examine their coupled-mode equations (15) and (116), we note that the mathematical formulations are very similar. A fundamental difference exists between these types of two-wave mixing. In SBS and SRS, the gain coefficient (125) is proportional to the total in-

E. Experimental Work It was shown earlier that energy transfer in two-wave mixing requires a finite spatial phase shift between the intensity pattern and the induced index grating. In Kerr media where the response is local, such a spatial phase shift can be induced by the use of moving gratings in the medium. Thus, energy transfer is possible in nondegenerate two-wave mixing in Kerr media. Although the concept of using moving gratings in local media for the energy coupling between two beams had been suggested in the 1970's [41]-[49], no experimental results were reported until recently. In 1986, a steady transfer of energy was observed in a two-wave mixing experiment in atomic sodium vapor [68]. In that experiment, a flash-pumped dye laser was used to pump a cell of sodium vapor that was inserted into a ring resonator. The laser frequency was detuned slightly from the sodium D line. The parametric gain due to the two-wave mixing leads to a unidirectional oscillation in a ring resonator.

b

= -2 10

Fig. 12. Intensity variation of the scattered beam / 2 (z) as a function of z for various values of b.

40

Fig. 13. Diffraction efficiency ~ as a function of the parameter b for various values of Kl.

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IEEE JOURNAL OF QUANTUM ELECTRONICS, VOL. 25, NO. 3, MARCH 1989

506

The frequency of the oscillating beam in the ring resonator was measured and was found to be lower than that of the pump beam. In a later experiment using a CW dye laser, a frequency shift of several MHz's was measured [69]. The frequency shift agrees with our theoretical result [see (120) and (115)], which indicates that the lowfrequency beam gets amplified when the Kerr coefficient is positive. By tuning the frequency of the dye laser to the other side of the D line, an opposite sign of the frequency shift was observed. This indicates the reverse of sign of the Kerr coefficient at this new frequency. In addition, the frequency of oscillation and the intensity of oscillation are functions of the cavity length. Oscillation ceases at cavity lengths when the frequency shifts are less than 8 or more than 50 MHz. Similar observations on the dependence on cavity length were found in photorefractive unidirectional ring resonators [70], [71] . In a two-wave mixing experiment, a fluorescent-doped boric acid glass is used as the nonlinear material [72]. In this experiment, a frequency shift of 0.1 Hz was induced by reflecting one of the beams off a mirror that was translated at a constant velocity by a piezoelectric transducer (PZT). By varying the frequency difference between the beams and monitoring the change in intensity of the probe beam, a time constant of 100 ms was measured. In a similar experiment, a ruby crystal is used as the nonlinear medium [73]. Energy coupling at a frequency shift of up to 500 Hz was observed. A time constant of 3.4 ms was determined by measuring the probe intensity at various frequ~ncy shifts. In addition, a net gain (exceeding the absorption and reflection loss) of more than 50 percent was observed. Recently, energy transfer between two coherent beams in liquid ~rystals has been observed by several workers [74]. The energy exchange is due to the thin holograms in the medium. In these configurations, the scattering of light by the induced grating is in the Raman-Nath regime due to the small interaction length. The presence of higher order scattering terms results in a multiwave mixing that leads to the energy transfer from the strong beam to the weak beam. If the interaction length is increased, the en.ergy transfer will decrease because the interaction will be in the Bragg regime. V.

has been observed by using a BaTi03 crystal pumped with an argon ion or a HeNe laser [54]. Unlike the conventional gain medium (e.g., He-Ne), the gain bandwidth of photorefractive two-wave mixing is very narrow (a few hertz's for BaTi03 ; see also Fig. 4). Despite this fact, the ring resonator can still oscillate over a large range of cavity detuning. This phenomenon was not well understood until a theory of photorefractive phase shift was developed [70]. The theory shows that oscillation can occur at almost any cavity length despite the narrow-band nature of two-wave mixing gain, provided the coupling is strong enough. Such a theory is later verified experimentally by studying the frequency of unidirectional ring oscillation at various cavity detunings [71]. Referring to Fig. 14, we now investigate the oscillation of a ring resonator in which a photorefractive crystal is inserted. Let us focus our attention on the region occupied by the photorefractive crystal and examine the gain due to two-wave mixing. The results of nondegenerate twowave mixing derived in Section II-C can be used to explain the ring oscillation. In a conventional ring resonator, the oscillation occurs at those frequencies (176) which lie within the gain curve of the laser medium (e.g., He-Ne). Here, Sis the effective length of a complete loop, f 0 is a constant, and N is an integer. For S :s; 30 cm, these frequencies ( 17 6) are separated by the mode spacing c / S ~ 1 GHz. Since the width of the gain curve for the conventional gain medium is typically several GHz due principally to Doppler broadening, oscillation can occur at almost any cavity length S. On the contrary, if the bandwidth of the gain curve is narrower than the mode spacing c/S, then oscillation can sustain, provided the cavity loop is kept at the appropriate length. Unlike the conventional gain medium, the bandwidth of photorefractive two-wave mixing is very narrow. Using photorefractive crystals that operate by diffusion only, e.g., BaTi03 , the coupling constant can be written, according to (48)

APPLICATIONS

(177)

The photorefractive coupling of two waves in electrooptic crystals has a wide range of applications. These include real-time holography, self-pumped phase conjugation [53], ring resonators [54], [70], [71], [75], laser gyros [72], nonreciprocal transmission [76], image amplification [20], vibrational analysis [77], and image processing [78], [79], etc. Some of these applications will be discussed in this section.

A. Photorefractive Resonators The coherent signal beam amplification in two-wave mixing can be used to provide parametric gain for unidirectional oscillation in ring resonators. Such oscillation

where 'Y 0 is the coupling constant for the case of degenerate two-wave mixing (i.e., {} = W2 - Wt = 0) and is given by 'Yo=)\ COS (6/2)"

(178)

The parametric two-wave mixing gain is given by, according to (50)

119

/ 2 (L) 1 +m -etL g "" / 2 (0) = l + me -yL e

(179)

YEH: TWO-WAVE MIXING IN NONLINEAR MEDIA

507 1.0~---------------

-1.0 '---'-----'--'---'---'--L--'----'--'-- -'---'---' -4 -8 -2 0 4 6 lh

Fig. 15. Photorefractive phase shift AY, as a function of !lT for various values of m. Fig. 14. Schematic drawing of a unidirectional photorefractive ring resonator.

where we recall that m is the input beam ratio m = 11 ( 0) / 12 ( 0) and L is the length of interaction. Note that amplification ( g > l ) is possible only when 'Y > ex and m > (1 - e-"'L)/(e-"'L - e-~L). Also note that g is an increasing function of m o> and g is an increasing function of L, provided 'Y > ex and

(180) The gain as a function of frequency w2 (or equivalently as a function of 0 = w2 - w1 ), has been plotted in Fig. 4 for various values of m. Note that gain is significant only when Iw2 - w1 IT < l. For materials such as BaTi03 and SBN, T is between 1 and 0.1 s. Thus, the gain bandwidth is only a few hertz. In spite of such an extremely narrow bandwidth, unidirectional oscillation can still be observed easily at "any" cavity length in ring resonators using BaTi03 crystals as the photorefractive medium. Such a phenomenon can be explained in terms of the additional phase shift [(24) and (25)] introduced by the photorefractive coupling. This phase shift is a function of the oscillation frequency and is plotted in Fig. 15 as a function oHlT. For BaTi03 crystals with 'YaL > 47r, this phase shift can vary from - 7r to + 7r for a frequency drift of -6.0T = ± 1. Such a phase shift is responsible for the oscillation of the ring resonator which requires a round-trip phase shift of an integer times 27r. 1) Oscillation Conditions: We now examine the boundary conditions appropriate to a unidirectional ring oscillator. At steady-state oscillation, the electric field must reproduce itself, both in phase and intensity, after each round-trip. In other words, the oscillation conditions can be written -6.if; +

1

k ds = 2N7r

=1

Ar = 2N'7r -

Jk ds

(183)

where N' is an integer chosen in such a way that Ar lies between -7r and +7r, then the oscillation condition (181) can be written

Ai/;= Ar+ 2M7r

( 184)

where Mis an integer. In other words, oscillation can be achieved only when the cavity detuning can be compensated by the photorefractive phase shift. Equations (181) and (182) may be used to solve for the two unknown quantities m = 11 ( 0) / 12 ( 0) and 0 = w2 w 1• If we fix the pump intensity 11 ( 0) and the pump frequency w 1, then (181) and (182) can be solved for the oscillation frequency w2 and the oscillation intensity / 2 (0). Substituting (179) for gin (182) and using (25), we obtain

(185) This equation can now be used to solve for the oscillation frequency OT. For the case of pure diffusion, using (46) for 0 = 7r /2 and (19) and (20), we obtain from (185)

OT=

2Aif; exL-lnR

2(Ar + 2M7r) exL-1nR

(186)

where Ar is the cavity detuning and is given by (183). Substituting (179) for gin (182), we can solve form and obtain

(181)

( 187)

( 182)

Since m must be positive, we obtain from (187) the threshold condition for oscillation

and gR

where Ai/; is the additional phase shift due to photorefractive coupling, the integration is over a round-trip beam path, the parameter R is the product of the mirror reftectivities, and g is the parametric gain of (179). If we define a cavity detuning parameter Ar as

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IEEE JOURNAL OF QUANTUM ELECTRONICS, VOL. 25, NO. 3, MARCH 1989

"fl

= al -

> "f,l

(188)

In R

al• 1

where 'Yt is the threshold parametric gain constant. Since is a function of frequency n' ( 188) dictates that the par:metric gain is above threshold only in a finite spectral regime. Using (177) for "f, (188) becomes

IOr I
"frl + -

1

'Yrl

(2A\b)

2

= Grl

-2.0 -4.0 -1.0

optical phase reproduces itself (to within an integer multiple of 2 7r). The condition on phase is unique because of a significant contribution to the optical phase shift due to nondegenerate photorefractive two-wave mixing. This condition is satisfied at any cavity length if the oscillation frequency is slightly detuned from the pump frequency, since the photorefractive phase shift [(185)) depends on the detuning. The frequency difference n ( = Wz - W1) between the pumping and oscillating beams can be written

( 191)

*

1

2

+ - l [2(A\b + 2M7r)]. 'Yr

1.0

0.6

Fig. 16. Oscillation intensity as a function of cavity detuning Or for various values of -y,, L.

where 'Yr is the threshold parametric gain of (188) for the case when A\b = 0, and Gr may be considered as the 0. According to threshold gain for the case when A\b (191), the threshold gain increases as a function of the cavity detuning Ar. The cavity detuning Ar not only detennines the oscillation frequency [(186)], but also the threshold gain Gr. The Af in (183) is the cavity detuning and is defined between -'Ir and 'Ir. However, the photorefractive phase shift (25) can be greater than 'Ir. When this happens, the unidirectional ring resonator may oscillate at more than one frequency. These frequencies are given by (186), with M = 0, ± 1, ± 2, · · · , etc., and with their corresponding threshold gain given by G,l = "f1l

-0.6

(192)

In other words, for each cavity detuning Ar, the ring resonator can support multimode oscillation, provided the coupling constant 'Yo is large enough. Fig. 16 shows the oscillation intensity, as well as the oscillation frequency as functions of cavity detuning Ar. Note that for larger "{ 0 l, the resonator can oscillate at almost any cavity detuning Ar, whereas for small "f 0 l, oscillation occurs only when the cavity detuning is limited to some small region around Ar = 0. In summary, ring oscillation occurs when the two-wave mixing gain dominates cavity losses and the round-trip

121

n=

(2(Ar

+ 2M7r)/rA]

(193)

where Ar is the cavity-length detuning with respect to an integer multiple of optical pump waves in the cavity, M is an integer, r is the photorefractive time response, and A represents the total cavity loss. There are threshold conditions for oscillation involving cavity loss and gain (taking M to be zero):

IOI : :; IAr\

(l/r)('Yl/A - 1)

:s (A/2)('Yl/A - 1)

112 112

(194) (195)

where 'Y is the degenerate two-wave mixing coupling coefficient, l is the interaction length, and A = -In (RT,Tp) (with R being the product of the reflectivities of the cavity mirrors and output coupler; T, is the transmission through the photorefractive crystal accounting for the absorption, Fresnel reflections, and scattering (or beam fanning); and Tp is the effective transmission through the pinhole aperture). This theory predicts that the unidirectional ring resonator will oscillate at a frequency different from the pump frequency by an amount directly proportional to the cavity-length detuning. Furthermore, in a photorefractive material with moderately low r, the theory postulates a threshold where oscillation will cease if the cavity detuning (frequency difference) becomes too large. Such a theory has been validated experimentally in a BaTi03 photorefractive ring resonator [71). The experiments perfonned to examine the above theory will now be discussed in detail. Fig. 17 shows the experimental setup. A single-mode argon-ion laser (514.5

509

YEH: TWO-WAVE MIXING IN NONLINEAR MEDIA

BS M 0 L

CAVITY DETUNING[%)

= BEAMSPLITTER =MIRROR = OETECTOR =LENS

-8

-12

-4

0

4

8

12

o.s-------------------(b) 0.4

-0.4

10

w;e

~.!. 3 -> 0:>-

""iii wz CJw Z>wz a:wz

N.--=-~~~+-~~~~i.--..~-tt~~--..JD2 L2

"-a: a:w ~I: 1

.. z-o.

\'1)

-COS ()/COS (()o -

which we have expressed here, for Bragg incidence, in terms of the angle of incidence () 0 and the slant angle cf>. Figure 5 indicates lines of constant c as a function of () 0 and cf>. For transmission holograms c is positive (c > O), and for reflection holograms c is negative (c < 0). In the diagram transmission and reflection holograms are separated by the line for c = co • III. TRANSMISSION HOLOGRAMS

In this section we discuss transmission holograms in greater detail. We give algebraic formulas and their numerical evaluations for the diffraction efficiencies and the angular and wavelength sensitivities of dielectric and of absorption gratings. This includes results on the influence of loss and slant.

c =1

C=

1

Fig. 5-The slant factor c as a function of the angle of incidence Oo and the slant angle. c is positive for transmission holograms and negative for reflection holograms.

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THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1969

It is convenient to write the various diffraction formulas in terms of parameter s v and ~' which are redefined for each grating type. In these parameter s are lumped together the constants of the medium (n, a, n 1 , a1 , K), the obliquity factors (cR , cs), the wavelength, the grating thickness d, and the dephasing measure if. By using v and ~' various trade-offs become immediately apparent. We recall that, for transmission holograms, cs is positive and the output signal appears at z = d. Combining equations (30) and (34) we obtain a general formula for the signal amplitude S of a transmission grating

s = -j(~:r •exp (-ad/CR) ·e~ ·sin [v

2 -

~2 ]! /[1

-

~2 /v 2]! 1 (41)

where K is the coupling constant given in equation (9), if the dephasing measure of equation (18), cR and cs are the obliquity factors of equation (23), a is the absorption constant and d the grating thickness. In the above form the parameter s v and ~ are, in general, of complex value. Lossless Dielectric Gratings For completeness we give the formulas for the lossless dielectric grating. For the unslanted case of this grating these formulas have whose prime interest been previously obtained by several workers 20 21 28 this grating type it For • • waves. acoustic by diffraction was light lossless dielectric the For fringes.* slanted of effect the include to is easy a 1 = 0. Equa= a and 7rnJ>. = K constant coupling a have grating we form the in rewritten be can tion (41)

3.1

S

=

-j(~:re-H sin (v + f)lj(l + ~2 /1?)!, 2

(42) ~ =

ifd/2cs

where v and ~ have been redefined and are real-valued. The associated formula for the diffraction efficiency is *Slant was also included in the treatment of dielectric gratings in Ref. 29.

146

WAVES IN THICK HOLOGRAMS

2923 (43)

For significant deviations from the Bragg condition the parameters v and~ are of equal order of magnitude, and we can take v as independent of fl(} or fl}.. without causing an appreciable change in the predictions of equation (43). In this equation the angular and wavelength deviations are represented by the parameter ~ which can be written in the form ~

= AO·Kd sin( - 00 )/2cs (44)

by using equation (18). The angular and wavelength sensitivities of lossless dielectric gratings are shown in Fig. 6, where the efficiencies as given by equation (43) are plotted (normalized) as a function of ~ for three values of v. The figure shows the sensitivity of gratings with v = 7r/4 and a peak diffraction efficiency of 'l/o = 0.5, with v = 7r/2 and a peak efficiency of 'l/o = 1, and with v = 3ir/4 and 'l/o = 0.5. We notice that the half-power points are reached for values near ~ = 1.5: There is some narrowing in the sensitivity curves for increasing values of v, and a marked increase in the side lobe intensity.

Fig. 6-Transmission holograms-the angular and wavelength sensitivity of lossless dielectric gratings with the normalized efficiencies 71 frio as a function of ~.

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THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1969

1.0 . . - - - - - - - - - - - / / - : ; : : , . . . , . . ......." " " " " - - - - - - - - - - - . ,

0.8

/; /;

0.6 0

'"'"'"" 0.4 ~ 0.2

~

~

//

~

D0 = o

~ v

o.___ __,__ _.___ __,__ _,___

-s

=7T /2

,/)

-4

-3

-2

__,__~~-~-~~-~-~

0

-1 /j,

0

2

1

4

3

5

IN DEGREES

Fig. 7 -Transmission holograms-the angular sensitivity of a lossy dielectric grating with v = .,./2 and Do = 2 compared with that of a lossless dielectric grating (Do = 0), for Oo = 30° and (3d = 50.

The above formulas include the influence of slant through the obliquity factors cR and cs . If there is no slant (q, = 7r/2) and if the Bragg condition is obeyed then cR = cs = cos 00 and equation (43) becomes the well known 20 • 21 • 30 (45)

By inserting the above half-power values for~ into equation (44) we obtain simple rules of thumb for the angular and spectral half-power bandwidths of unslanted gratings: 2M1 ~ A/d, 211A.1/A. ~ cot (). A/d.

Lossy Dielectric Gratings Let us first study the influence of loss on the angular sensitivity of a dielectric grating. We assume that there is no slant (ef> = 7r/2) and therefore cR = cs = cos 0. With this and a coupling constant of " = 7rn1 /A. we obtain from equation (41) for the signal amplitude

3.2

S = - j exp (-ad/cos O)·e-H·sin (v2

+ ~ )!/(1 + ~ /1,2)! 2

2

(46) ~ =

iJd/2 cos () = 110·{3d sin 00

where v and~ have been redefined, and~ has been expressed in the needed form with the use of equations (14) and (18). Equation (46) has a form similar to that of equation (42) except

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WAVES IN THICK HOLOGRAMS

for an additional exponential term containing the absorption constant a. This term decreases the peak efficiency and it changes the angular sensitivity of the grating. But this change is very small, even for high loss values, as illustrated in Fig. 7. This figure compares the angular sensitivitities of a lossless grating (D 0 = 0) with that of a grating of high loss (D 0 = 2) for a parameter value of v = 7r/2, a Bragg angle of B0 = 30°, and an optical grating thickness of (:Jd = 27rnd/A. = 50. The loss parameter D 0 was defined as D 0 = ad/cos B0

(47)

which is closely related to the conventional photographic density D (except that D 0 is measured in the direction of the reference wave given by B0 ). A value of D 0 = 2, which is the parameter used for the dashed curve, represents very high loss, with a decrease of the peak efficiency by a factor of about 50. Still, the differences of the two sensitivity curves are very small and consist mostly of an angular shift. The differences are even smaller for larger values of (:Jd (we checked up to {3d = 200), and, of course, for smaller values of D 0 • The main conclusion is that the presence of loss has very little influence on the angular sensitivity of a dielectric transmission grating. This is probably because absorption influences the phase relations between the waves Rand S very little. It agrees with observations by Belvaux. 31 Next let us consider the influence of loss on the efficiency of a slanted dielectric grating. For simplicity we assume Bragg incidence, that is, tJ = 0. The obliquity factors are positive and given by ca = cos B0 and Cs = -cos (Bo - 2cf>). For this case we can write equation (41) for the signal amplitude S in the form

S

=

-j(~:Y ·exp [-!D (l + c)] sin (v2 0

- f)i/(1 -

(48)

v = '1rn1d/X(CaCs)! ~ =

~2 jv2)11

!D0 (l - c)

where we have used the loss parameter D 0 as above in equation (47), and the slant factor c

Do = ad/ca c =Ca/cs

= ad/ cos B0

= -cos B0 /cos (B0

-

2cf>).

Figure 8 shows the diffraction efficiency of slanted grantings as calculated from equation (48). The efficiencies are plotted as a function

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THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1969

1.of-------- --D-o-=-0- ------------ ---i

~========~~=-----::---------_.:.V~=~'IC'~/~2---J 0.1

0.8

0.6

1J

0.2

:=::J~:;:±~~::;;;j ol___l_~_J_~~=====r===::c= 2.0 1.8 1.6 1.4 1.2 1.0 0

0.2

0.4

0.6

0.8

CR/Cs

Fig. 8 -Transmission holograms-the efficiency of lossy dielectric gratings as a function of slant for v = 7r/2. c = cR/cs is the slant factor.

of the slant factor c for various values of D 0 , and for a value of v = 7r /2 which corresponds to the maximum attainable efficiencies. Similar curves for v = 7r/ 4 and v = 371'/ 4 and the same D 0 values are almost identical to the curves of Fig. 8, except that the efficiency scale is reduced to a maximum efficiency of 0.5. This implies that for the range of chosen parameter values the exponential factor in equation (48) dominates in predicting the slant-dependence of the diffraction efficiency. The results show that, for higher efficiencies, the grating prefers small c-values, assuming constant 80 and D 0 • This is a preference of small exist angles for S which means that we get the best efficiency if the signal wave leaves the grating on the shortest possible path after it has been generated. 3.3

Unslanted Absorption Gratings

When one records holograms in conventional photographic emulsions one produces absorption gratings (bleaching can convert this into a dielectric grating). In an absorption grating there is no spatial modulation of the refractive index (n 1 = O) and the coupling is provided by a modulation (a 1) of the absorption constant. We have, then, an imaginary coupling constant K = - jaif 2. In this section we study the efficiencies and the angular and wavelength sensitivities of unslanted absorption gratings where cf> = 7r/2 and cR = Cs = cos 8. From equation (41) we obtain for the signal amplitude

150

WAVES IN THICK HOLOGRAMS

v

= 0i1d/2 cos ()

~

= !Jd/2

cos()~

2927

(49) t:..0·{3d·sin 00 = -!(t:..A./A.)Kd tan () 0

where v and ~ are real-valued, and equation (18) was used to express the parameter ~ again in various forms, showing explicitly the angular deviations t:..O and the wavelength deviation t:..A. from the Bragg condition. For Bragg incidence we have ~ = 0, and obtain from the above a formula for the diffraction efficiency ri of absorption gratings ri

= exp (-2ad/cos 00 ) ·sh2 (a1d/2 cos 00 ).

(50)

As we exclude the presence of negative absorption (gain) in the medium, there is an upper limit for the amplitude a 1 of the assumed sinusoidal modulation, which is a 1 ~ a. The highest diffraction efficiency possible for an absorption grating is reached in the limiting case a 1 = a for a value of ad/cos 00 = ln 3. According to equation (50) this maximum efficiency has a value of 1/max = 1/27, or 3.7 percent. Figure 9 shows values for the diffracted amplitude S of absorption gratings as computed from equation (50) as a function of the modulation amplitude a 1 and for various values of the depth of modulation. For convenience we have again used loss parameters, which are D 0 = ad/cos 00 and D 1 = a 1d/cos 00 • D 1 is a measure for the amplitude of the spatial modulation and D 0 /D 1 = a/a 1 indicates the modulation depth. The dashed curves for constant D 0 show the grating behavior for constant background absorption. We have plotted S on a linear scale in order to identify the regions of linear grating response. Note that a good linear response and relatively good efficiency is obtained if the absorption background is held constant to a value of about Do = 1. Equation (49) predicts also the angular sensitivity and the frequency sensitivity of absorption gratings. Such sensitivity curves are plotted in Fig. 10 for the special case of a 1 = a 0 and values of v = D 1/2 = 1 (dashed) and v = ! In 3 = 0.55. For the latter parameter value the peak efficiency of 3.7 percent is reached, and for v = 1 we have a peak efficiency of 2.5 percent. In the figure the relative efficiencies are plotted as functions of the parameter~. We note that there is very little difference between the sensitivity curves for the two v-values chosen. We have also computed the sensitivity for smaller values of v (0.2, 0.4), but the resulting curves differ so little from the ones shown that we have omitted them from the figure. The sensitivity curves are very

151

THE BELL SYSTEM TECHNICA L JOURNAL , NOVEMBER 1969

2928

...

--------------.

0.20~----------

0.18

0.16

0.14

0.12

Isl

0.10

0.08

0.06

0.04

0.02

1.6

1.2

2.0

D1=a 1d/cos8 0 =2v

Fig. 9. -Transmi ssion hologram s-the diffracted amplitude of an absorptionn grating as a function of the modulation D1 = aid/cos e = 2v for variol\B modulatio depths Di/Do (solid curve) and various bias levels Do = ad/cos 8 (dashed curve).

similar to those of the dielectric gratings with smaller 11-values which are shown in Fig. 6. Again, the half-power points are reached for about in~ = 1.5. But for absorptio n gratings there is no narrowing with low. remains intensity lobe side the and 11, of values creasing Slanted Absorption Gratings In this section we consider the influence of slant on the efficiency of an absorption grating. For simplicity we assume Bragg incidence (tJ. = O), and describe the slant by the obliquity factors CR = cos eo and Cs = cos (e0 - 2.

~

=

~o -

~o

cos Oo

jDo,

(61)

= - M · /3d sin Oo

Do = ad/ COS 00 where ~ is now a complex parameter with ~o describing the angular

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THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1969

deviations and D 0 representing the loss. An evaluation of this formula is shown in Fig. 13, which shows the angular sensitivity of dielectric gratings for various values of the loss parameter D 0 and a grating parameter of v = 7r/2. In constrast to what we have observed in the case of dielectric transmission holograms (Fig. 7), we see here a quite noticeable effect of the grating loss on the sensitivity curves. With increasing loss values the curves broaden in the wings, sharpen somewhat in the center and the side-lobe level decreases. To study the influence of loss on the diffraction efficiencies of dielectric gratings we rewrite equation (55) in the form

S =

(~;Y/{~/v +(I+ ~2/v2) 1 ·coth (v

v = j7rn 1d/A(cRcs) 1 ~

2

+

~2)!l (62)

= tDo(l - c)

where we have written v and ~ as real-valued parameters in a form which is valid for Bragg incidence and for slanted or unslanted gratings. Just as in the case of transmission holograms we have used the loss

Fig. 13 - Reflection holograms-the influence of loss on the angular and wavelength sensitivity of a dielectric grating for v = 71" /2. The normalized efficiencies 'YJ!'YJo are shown. The peak efficiencies are 'Y/o = 0.84 for Do = 0, 'Y}o = 0.64 for Do = 0.5, 'Y/O = 0,28 for Do = 11 and 'Y}o = 0.12 for Do = 2.

158

WAVES IN THICK HOLOGRAMS

2935

parameter D0 and the slant factor c (which is now negative) D0 = ad/ COS 80

c

=

(63)

cR/cs .

In the case of unslanted gratings the parameters v and v

~

= 7r'n1d/A. cos 80

~ =

Do = ad/ cos 80

simplify to (64)

•

The results of a numerical evaluation for unslated gratings are shown in Fig. 14, where the signal amplitude is plotted as a function of v for various values of the loss parameter Do . The curve D 0 = 0 gives the values for lossless gratings, while the others indicate the influence of loss. The behavior of slanted dielectric gratings in the presence of loss is shown in Fig. 15. The curves of this figure are also computed from equation (62) and show the diffraction efficiency as a function of the slant factor for v = 7r/2 and various values of the loss parameter D 0 • For constant D 0 we notice an increase of the efficiency for decreasing values of the slant factor, as in the case of transmission holograms.

0.8

0.6

\SI 0.4

0.2

O"'-~~-'--~~-L~~--'.,--~---,.~~~-'

0

0.2

0.4

0.6

0.8

1.0

v/w = n 1d/ilcose 0

Fig. 14-Reflection holograms-the influence of loss on the diffracted amplitude S of an unslanted dielectric grating. I S \ is shown as a function of 11/1r = n1d/i1 cos 00 for various loss parameters D 0 •

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THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1969

1.0--------------------------.

v = 'lr/2

D0

=o

j 0.1 ==----~~_ r0.8r-:=== 0.2

7)

0.5

1.0 1.5 2.0 o~-~--..L--~--~-~--~--~--':---:"':---::-

o

0.6

0.4

0.2

1.2

1.0

0.8

1.8

1.6

1.4

2.0

lcR/csl Fig.. 15- Reflection holograms-the efficiency of a lossy dielectric grating as a function of slant for.,, = w/2. c = cR/c8 is the slant factor.

Again, for given loss and a given angle of incidence short signal paths through the grating (that is, small exit angles) are preferred for higher efficiencies. 4.3

Unslanted Absorption Gratings

Following the pattern set in the discussion of transmission holograms (Section III), we again describe an absorption grating by an imaginary coupling constant K = -jai/2, and proceed to study the diffraction efficiencies and the angular and wavelength sensitivities of unslanted ( = O) gratings. In this case equation (55) simplifies to S =

-j(~;/; IUv + [(Uv)

2 -

l]~coth (~ 2

v = ja 1 d/2(cRcs)! ~

= Do -

2

-

v )!\ (65)

i~o

where the real-valued parameters D 0 and ~o can be expressed to first order in the angular deviations b.B and the wave-length deviations b.t. by D 0 = ad/ COS Bo ~o

= M·(3d sin B0 = !(b.t./t..)Kd.

160

(66)

WAVES IN THICK HOLOGRAMS

2937

D0 is a loss parameter as before, and ~o is a normalized measure for the angular or the wavelength deviations from the Bragg condition. If the Bragg condition is obeyed equation (65) can be written in the form S = - Di/2[D 0

+ (D~

- DU4)t · coth (D~ - DU4)tj

(67)

where D1 =

2111

= a1d/cos Oo

measures the spatial modulation of the absorption constant (a 1). For the deepest allowable modulation where we have D 1 = D 0 (a 1 = a 0 ), this equation predicts the maximum diffraction efficiency 17max which is possible for reflection holograms with a (sinusoidal) absorption modulation. We obtain 17max = 1/(2 + v'3) 2 , or a maximum efficiency of 7.2 percent for D 0 = D 1 ~ oo. The formula reflects the experimental fact that, for reflection holograms of the absorptive kind, one obtains the largest efficiencies for high photographic densities. Figure 16 shows a numerical evaluation of the above formula. Here the signal amplitude S is plotted as a function of the modulation amplitude D 1 for various levels of loss "bias" D 0 (dashed curves) and for various modulation depths D 0 /D1. An evaluation of the grating sensitivity as predicted by equation (65) is shown in Fig. 17 for the special case of a maximum depth of modulation where D 1 = D 0 • In this figure the (normalized) efficiency is plotted as a function of the parameter ~o for various values of D 1 = D 0 • As in the corresponding grating for the case of transmission holograms (Fig. 10) the sensitivity curves are seen to reach their half-power points for values of about ~o = 1.5. But in the present case of reflection holograms there is a noticeable broadening of the curves with increasing loss values D 1 = Do . 4.4

Slanted Absorption Gratings

In this section we consider the influence of slant on the diffraction efficiency of an absorption grating for reflection holograms. We assume Bragg incidence (?'J = 0) and again use the obliquity factors cR = cos 00 and Cs = -cos (0 0 - 2ct>) to describe the slant (for reflection holograms we have Cs < 0 !) . We find that equation (65) can be used as a formula for the signal amplitude for the present case if we modify the parameters to

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THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1969

o.2s~-------------------------.

0.24

0.20

I

I

I

I

/D 0 =2

I

0.16

I

D0 /D 1 =2

0.12

0.08

Do/D 1 =4 D0 /D 1 =s

0.4

0.8

1.2

1.6

D1=a 1d/cos/10

2.0

2.4

2.8

3.2

=2 v

Fig. 16-Refl.ection holograms-the diffracted amplitude of an absorption grating as a function of the modulation D1 = aid/cos II = 2v for modulation depths Di/Do (solid curve) and bias levels Do = ad/cos () (dashed curve). ~

= !D0 (l - c)

(68)

Do = ad/ cos 80 , C =CR/Cs

where the slant factor c is negative. All these parameters are real-valued in the present case. For a maximum depth of modulation, that is, a 1 = a, there are further simplifications, and we obtain a simple expression for the slant-dependence of the diffraction efficiency "11

=

-c/ll - c

+ (1

- c

+ c )t·coth !D (l 2

0

- c

+c

2

)

1} 2 •

(69)

Figure 18 shows a numerical evaluation of this formula for various values of D 0 = Di . The slant factor value of I c I = l refers to unslanted gratings. In this case the maximum efficiency value '17max = 0.072 is approached for large Do . We note that for values of D 0 below unity

162

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WAVES IN THICK HOLOGRAMS

the efficiencies increase for J c I-values larger than 1 and up to about 3, that is, for relatively large exit angles of the signal wave. 4.5

Mixed Gratings

Mixed gratings are described by a complex coupling constant = 7rn1 />.. - ja1/2 (see Section 3.5). For Bragg incidence (iJ = O) and unslanted fringe-planes ( = O) we can obtain from equation (55) a formula for the signal amplitude of mixed gratings, which is K

S = -jK /

{~ +

2 (K

+cl)!· coth

co: 00 (i + a )!} 2

(70)

where K is of complex value, a is the average absorption constant, d the grating thickness and 00 the angle of incidence. V. AMPLITUDES OF THE DIRECT WAVES

For diagnostic purposes it is often of interest to monitor the change in amplitude of the direct reference wave R, which is depleted because of diffraction into S and absorption. The quantities of interest are the amplitudes R(d) which can be obtained from the analysis of Section 2.2.

~o

Fig. 17 - Reflection holograms-the angular and wavelength sensitivity of an absorption grating for "'' = "' (D1 = Do) and values of Di = 2v = Do = 0.2 (110 = 0.007), D, = 1 (110 = 0.05), and D1 = 2 (110 = 0.068).

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THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1969

2940

0.08 . - - - - - - - - - - - - - - - - - - - - - - - ,

D0

=2.0

0.06

'II

0.04

0.02

2.5

3.0

3.5

Fig. 18 - Refiectjon holograms-the efficiency of an absorption grating as a function of slant for"" = a (Di = Do). c = cR/c8 is the slant factor.

We will give here the general results for transmission and reflection holograms. The notation is that of Section 2.

Transmission Holograms From equations (27) and (33) we get for the constants r, of equation (25) the expressions

5.1

r1 = -i/cs('Y1 - 'Y2)(cR'Y1

+ a)

(71)

r2 = i/cs('Y1 - 'Y2)(cR'Y2 +a).

Using this we can write the output amplitude R(d) of the reference wave in the form (72)

Reflection Holograms For reflection holograms we use equations (27), (37), and (39) to express the constants r; in the form

5.2

r,

=

+ a + jiJ) exp h2d)/ {exp h2d)(a + jiJ + Cs'Y1) - exp ('Y1d)(a + jiJ + Cs'Y2)) (cs'Y2 + a + jiJ) exp hid)/ {exp ('Y2d)(a + jiJ + Cs'Y1) - exp ('Y 1d)(a + jiJ + Cs'Y2)}.

(cs'Y1

r2 = -

164

(73)

WAVES IN THICK HOLOGRAMS

2941

The output amplitude R(d) of the reference wave becomes R(d) = Cs('Y1 - 'Y2)/ I (a

+ jiJ + Cs'Y1) exp (-'Yid) - (a + jiJ + Cs'Y2) exp (-'Y 2d)}.

(74)

More detailed evaluations of the above formulas should follow the pattern prescribed in Sections III and IV. They can be undertaken for the specific case when the need arises. VI. VALIDITY OF THE THEORY

We have tried to make our results as generally applicable as possible. We have allowed for the presence of absorption in the various hologram gratings and for a slant of the fringe planes. But a whole range of assumptions had to be made to make the simple coupled wave analysis possible. It seems appropriate to recount these assumptions to make clear the region of validity of the coupled wave theory. We have assumed that: (i) The electric field of the light is polarized perpendicular to the plane of incidence. However, the appendix gives an extension of the theory to allow also for light of parallel polarization. (ii) A slant of the fringe planes with respect to the z-axis is allowed, except that these planes are perpendicular to the plane of incidence. (This is reflected in the assumption E(x, z), u(x, z).) But this assumption is not made in the generalization which we have given in the appendix. (iii) The spatial modulation of the refractive index and the absorption constant is sinusoidal. (iv) There is a small absorption loss per wavelength and a slow energy interchange (per wavelength) between the two coupled waves. This condition is stated mathematically in equation (7) and justifies neglecting the second derivatives R" and S" in the analysis. (v) There is the same average refractive index n for the regions inside and outside the grating boundaries. If the grating has interfaces with air, then Snell's law has to be used to correct for the angular changes resulting from refraction. (vi) Light incidence is at or near the Bragg angle and only the diffraction orders which obey the Bragg condition at least approximately are retained in the analysis. The other diffraction orders are neglected.

A detailed mathematical justification of assumption vi is outside the scope of our simple analysis. One can advance physical arguments to show that this step limits the validity of the theory to "thick" gratings,

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THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1969

where the phase synchronism between the two coupled waves has enough time to develop a strong and dominating effect. Better definitions of a "thick grating" must come from more accurate theories which are available for special cases. A large amount of work has been done on acoustic diffraction gratings which correspond to the case of30 our In unslanted, lossless, dielectric transmission-hologram gratings. acoustic diffraction one defines the parameter

Q = 2rrt..d/nA

2

(75)

as an appropriate measure of grating thickness. We can regard a grating 21 6 as thick when the condition Q » 1 holds. '1 It appears that the coupled wave theory begins to give good results for values of Q = 10. This is particularly well demonstrated by Klein and his associates in theoretical and experimental work on acoustic gratings for the predictions of both 16 17 34 the peak efficiencies and the angular sensitivities. ' ' We hasten to add that for the majority of practical holograms the parameter Q is larger, and sometimes much larger, than 10. Further checks of the validity of the coupled wave theory are provided by comparisons with accurate computer calculations and with experiments on special examples of gratings. Burckhardt has made computer calculations on unslanted, lossless, dielectric transmission holograms for selected values of grating parameters which are commonly encoun18 19 tered in holography. ' Comparison with the results of the coupled 35 wave theory shows very satisfactory agreement. Measurements by Shankoff and Lin on dielectric transmission holograms prepared with dichromated gelatin yielded diffraction efficiencies approaching 100 percent,- which agrees with the theory (even though there may be some 10 11 uncertainty as to the exact nature of the refractive index variations). • Efficiency measurements on thick absorption gratings for the case36 of 37 transmission holograms were made by George, Mathews, and Latta. ' Efficiencies approaching our predicted maximum value of 3.7 percent were observed. Kiemle has studied unslanted (cf> = O) reflection holograms for the special case of normal incidence (80 = 0) by analyzing equivalent 38 four-terminal networks. His treatment of absorption gratings corresponds to the material we discussed in Section 4.3 specialized to the case of 80 = 0. But Kiemle's value of 2.8 percent for the maximum diffraction efficiency of absorptive reflection holograms does not agree with our prediction of 7.2 percent. This disagreement appears to derive from a set of restrictive assumptions made in Kiemle's work. Experimental observations on absorptive reflection holograms were made by

166

WAVES IN THICK HOLOGRAMS

2943

Lin and Lo Bianco. 9 Efficiency values as high as 3.8 percent were measured, which seems to support the predictions of the coupled wave theory. But further experiments are needed for a good confirmation. VII. CONCLUSIONS

We have discussed a coupled-wave analysis of the Bragg diffraction of light by thick hologram gratings. This approach made it possible to derive simple algebraic formulas for the behavior of various types of holograms, even for the case of high diffraction efficiencies where the incident wave is strongly depleted. The treatment covers transmission holograms and reflection holograms, and it includes the spatial modulations of both the refractive index and the absorption constant. The influence of loss in the grating and of slanted fringes is also discussed. Formulas and their numerical evaluations are given for the diffraction efficiencies and the angular and wavelength sensitivities of various grating types. For special cases we can compare the results of this theory with more accurate computations and with experimental observations. These comparisons give us the confidence to assume that the coupled wave predictions are good for a broad range of practical hologram types. VIII. ACKNOWLEDGMENT

The author would like to acknowledge with thanks useful discussions with C. B. Burckhardt, and the patience of those who have waited so long for the promised completion of this article. APPENDIX

Reduced Coupling for Light Polarized in the Plane of Incidence

In the body of this paper it was assumed that the incident light is polarized perpendicular to the plane of incidence. The purpose of this appendix is to show that we can use the results of the main paper also when the light is polarized in the plane of incidence, provided that we modify the coupling constant "· Such a modification is suggested already by the dynamical theory of X-ray diffraction. As in Section II we start with the wave equation (76)

for the electric field in the grating. Here, in contrast to equation (1), we have described the field by the vector quantity E and have included

167

2944

THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1969

the term \l(\l ·E), which is not necessarily zero. The constant k2 is defined in equation (4). As in the main paper, we assume that only two waves are present in the grating, and put (77)

using the vectors R and S to describe the amplitudes of the reference and signal waves. g and dare the propagation vectors (as in Section II) which point in the direction of the wavenormals. They are related by equation (11). In addition we assume that, both, Rand Sare transverse waves, that is, that the following conditions hold

= 0, (d· S) = 0. (g·R)

(78)

Combining equations (76), (77), and (78) we get, after separating terms with equal exponentials and neglecting second derivatives a2 / az 2 -2jp.R' -2ju.S'

+ joS~ +

((3

+ jgR~ 2 -

u

- 2ja(3R

+ 2K(3S

2ja{3)S

+ 2K(3R

2 -

=0 =0

(79) (80)

where R. and S. are the z-components of Rand S, and the notation of Section II is used. We now make the additional assumption that the polarizations of R and S do not change in the grating and write R(z) = R(z)r,

(81)

S(z) = S(z)s,

where R(z) and S(z) are the scalar amplitudes of the two waves, and r and s are polarization vectors independent of z. These vectors are normalized so that (r·r) = 1,

(s · s)

=

1.

(82)

(s·d)

= 0.

(83)

Because of (78) we have (r·g) = 0,

After forming the dot products of r with eq. (79) and of s with (80) we use equations (81), (82), and (83) to arrive at

+ 2K(3S(r· s) 2jaf3)S + 2K(3R(r· s)

-2jp,R' - 2ja{3R -2ju.S'

+ ((3

2 -

u

2 -

168

= 0

(84)

= 0.

(85)

WAVES IN THICK HOLOGRAMS

2945

As in Section II, we introduce the abbreviations Cn = p,/{3,

Cs

=

(86)

u,/{3,

and (87)

which allow us to write the above equations in the form

+ aR = +(a+ jl})S = cRR'

c8 S'

-jK(r·s)S

(88)

-jK(r·s)R.

(89)

These are coupled wave equations which govern the Bragg diffraction of light polarized parallel to the plane of incidence, and indeed, of light of arbitrary polarization. They are similar in form to the coupled wave equations (21) and (22) which were derived for perpendicular polarization. The only difference is a reduction of the effective coupling constant by the dot product (r·s) of the two polarization vectors. Referring to the gmting_geometry of Fig. 1 we have (r·s) = 1 for light polarized perpendicular to the plane of incidence. For parallel polarization the value of this dot-product depends on the inclination angles, and we have a reduced effective coupling constant Kn given by Ku

= K(r·s) =

-K

cos 2(00

-

cf>).

(90)

We can apply the results of the main paper for parallel polarization if we replace K by Ku • For this polarization there is the trivial case of a Bragg angle of 45° (that is, diffraction angles of 90°) where (r·s) = 0 and the intensity of the diffracted light goes to zero. REFERENCES

l. van Heerden, P. J., "Theory of Optical Information Storage in Solids," Appl. Opt., 2, No. 4 (April 1963), pp. 393-400.

2. Smits, F. M., and Gallaher, L. E., "Design Considerations for a Semipermanent Optical Memory," B.S.T.J., 46, No. 6 (July-August 1967), P,P· 1267-1278. 3. Vitols, V. A., "Hologram Memory for Storing Digital Data, ' IBM Technical Disclosure Bull., 8, No. 11 (April 1966), p. 1581. 4. Pennington, K. S., and Lin, L. H., "Multicolor Wavefront Reconstruction," Appl. Phys. Letters, 7, (August 1965), pp. 56-57. 5. Denisyuk, Y. N., "On the Reproduction of the Optical Properties of an Object by the Wave Field of Its Scattered Radiation," Opt. Spectroscopy, 15, No. 4 (October 1963), pp. 279-284. 6. Stroke, G. W., and Labeyrie, A. E., "White-light Reconstruction of Holographic Images Using the Lippman-Bragg Diffraction Effect," Phys. Letters, 20, (March 1966 ), pp. 368-370. 7. Lin, L. H., Pennington, K. S., Stroke, G. W., and Labeyrie, A. E., "Multicolor Holographic Image Reconstruction with White-light Illumination," B.S.T.J., 45, No. 4 (April 1966), pp. 659-660.

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THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1969

8. Upatnieks, J., Marks, J., and Fedorwicz, R., "Color Holograms for White Light Reconstruction," Appl. PI!ys. Letters, 8, No. 11 (June 1966), pp. 286-287. 9. Lin, L. H., and Lo Bianco, C. V., "Experimental Techniques in Making Multicolor White Light Reconstructed Holograms," Appl. Opt., 6, No. 7 (July 1967), pp. 1255-1258. 10. Shankoff, T., "Phase Holograms in Dichromated Gelatin," Appl. Opt., 7, No. 10 (October 1968), pp. 2101-2105. 11. Lin L. H., "Hologram Formation in Hardened Dichromated Gelatin Films," Appl. Opt., 8, No. 5 (May 1969), pp. 963-966. 12. Chen, F. S., LaMacchia, J. T., and Fraser, D. B., "Holographic Storage in Lithium Niobate," Appl. Phys. Letters, 12, No. 7 (October 1968), pp. 223-224. 13. Close, D. H., Jacobson, A. D., Margerum, J. D., Brault; R. G., and McClumg, F. J., "Hologram Recording in Photopolymer MaterialS,'' Appl. Phys. Letters 14, No. 5, (March 1969), pp. 159-160. 14. Leith, E. N., Kozma, A., Upatnieks, J., Marks, J., and Massey, N., "Holographic Data Storage in Three-Dimensional Media," Appl. Opt., 5, No. 8 (August 1966), pp. 1303-1311. 15. Gabor, D., and Stroke, G. W., "The Theory of Deep Holograms," Proc. Royal Soc. of London, A. 304, (February 1968), pp. 275-289. 16. Klein, W. R., Tipnis, C. B., and Hiedemann, E. A., "Experimental Study of Fraunhofer Light Diffraction by Ultrasonic Beams of Moderately High Frequency at Oblique Incidence,'' J. Acouat. Soc. Amer., 38, No. 2 (August 1965), pp. 229-233. 17. Klein, W.R., "Theoretical Efficiency of Bragg Devices," Proc. IEEE, 54, No. 5 (May 1966), pp. 803-804. 18. Burckhardt, C. B., "Diffraction of a Plane Wave at a Sinusoidally Stratified Dielectric Grating," J. Opt. Soc. Amer., 56, No. 11 (November 1966), pp. 15021509. 19. Burckhardt, C. B., "Efficiency of a Dielectric Grating," J. Opt. Soc. Amer., 57, No. 5 (May 1967), pp. 601-603. 20. Bathia, A. B., and Noble, W. J., "Diffraction of Light by Ultrasonic Waves II," Proc. Royal Soc. of London, 220A, (1953), pp. 369-385. 21. Phariseau, P., "On the Diffraction of Light by Progressive Supersonic Waves," Proc. Ind. Acad. Sci., 44A, (1956), pp. 165-170. 22. Quate, C. F., Wilkinson, C. D., and Winslow, D. K., "Interaction of Light and Microwave Sound," Proc. IEEE, 53, No. 10 (October 1965), pp. 1604-1623. This paper includes a comprehensive bibliography of work on acoustic scattering of light. 23. Gordon, E. I., and Cohen, M. G., "Electro-Optic Diffraction Grating for Light Beam Modulation and Diffraction," IEEE J. Quantum Elec., QE-1, No. 5 (August 1965), pp. 191-198. 24. Batterman, B., and Cole, H., "Dynamical Diffraction of X-Rays by Perfect Crystals,'' Rev. Mod. Phys. 36, No. 3 (July 1964), pp. 681-717. This paper contains a review of the work on the diffraction of X-rays. 25. Saccocio, E. J., "Application of the Dynamical Theory of X-Ray Diffraction to Holography," J. Appl. Phys., 38, No. 10 (September 1967), pp. 3994-3998. 26. Kogelnik, H., "Reconstructing Response and Efficiency of Hologram Gratings," Proc. Symp. Modern Optics, Polytechnic Inst., Brooklyn, March 1967, pp. 605-617. 27. Kogelnik, H., "Hologram Efficiency and Response," Microwaves, 6, No. 11 (November 1967), pp. 68-73. 28. Born, M., and Wolf, E., Principles of Optics, New York: Pergamon Press, 1959, Chapter 12. 29. Bergstein, L., and Kermisch, D., "Image Storage and Reconstruction in Volume Holography,'' Proc. Symp. Mod. Opt., Polytechnic Inst. Brooklyn, March 1967, pp. 655-680. 30. Gordon, E. I., "A Review of Acoustooptical Deflection and Modulation Devices,'' Proc. IEEE, 54, No. 10 (October 1966), pp. 1391-1401. 31. Belvaux, Y., "Influence of Emulsion Thickness and Absorption in Hologram Reconstruction," Physical Letters, 26A, No. 5 (January 1968), pp. 190-191. 32. Nassenstein, H., "A New Hologram with Wavelength-Selective Reconstruction" Physical Letters, 28A, No. 2 (November 1968), pp. 141-142.

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WAVES IN THICK HOLOGRAMS

2947

33. Tien, P. K., un?rublished work. 34. Klein, W.R., 'Light Diffraction by Ultrasonic Beams of High Frequency Near Bragg Incidence," Proc. 5th Congress Int. D' Acoustique, Liege, D24 (September 1965 ). 35. Kogelnik, H., "Bragg Diffraction in Hologram Gratings with Multiple Internal Reflections," J. Opt. Soc. Amer., 57, No. 3, (March 1967), pp. 431-433. 36. George, N., and Mathews, J. W., "Holographic Diffraction Gratings," Appl. Phys. Lett., 9, No. 5 (September 1966), pp. 212-125. 37. Latta, J. N., "The Bleaching of Holographrc Diffraction Gratings for Maximum Efficiency,'' Appl. Opt., 7, No. 12 (December 1968), pp. 2409-2416. 38. Kiemle, H., "Lippman-Bragg Holograms as Periodic Ladder Networks," Frequenz, 22, No. 7 (July 1968), pp. 206-211.

171

Coupled-Wave Analysis of Holographic Storage in LiNbO/ D. L. staebler and J. J. Amodei RCA Laboratories, Princeton, New Jersey 08540 (Received 16 August 1971; in final form 26 October 1971) This paper considers two effects for directly studying thick-phase holograms: (a) coupling between the two laser beams used to record a hologram and (b) interference between a readout beam and the diffracted beam within the hologram. Both effects were observed in experiments using single crystals of undoped LiNb03 • The results demonstrate that the holograms arise from electric field patterns caused by either diffusion or drift of photogenerated free electrons. I. INTRODUCTION

Single-crystal electro-optic materials have been used for the storage of thick-phase holograms. t-4 It is believed that these holograms are recorded through migration of photogenerated free carriers. The migration mechanism, and the cause of the resulting refractiveindex modulation, are subjects of some controversy. Chen et al. 1 initially proposed that in LiNb03 , the free carriers drift under the influence of an internal electric field and are then retrapped at regions of low light intensity; the refractive. index is modulated by an electric field pattern set up by the resulting charge displacement. Thaxter 2 then demonstrated the influence of an applied electric field on hologram storage in Sr 0 • 75 Ba 0 • 25 Nb20 6 , lending support to the "drift" model. Later, Johnson 5 proposed that in LiNb0 3 the internal electric field arises from optically induced polarization changes. He also suggested from Chen's study of "optical damage" effects 6 that the resulting charge displacements modulated the index through changes in the internal polarization, i.e., modulation through variations in the density of occupied traps. Townsend and LaMacchia3 considered this mechanism in their study of hologram storage in BaTi0 3 • Amodei 7 ' 8 has pointed out that charge migration by diffusion may play an important role in holographic recording for sufficiently small grating spacings, and has derived expressions for the electric field patterns generated through drift and diffusion for plane-wave holograms. In the study of these optically induced.phase modulations, knowledge of their shape and phase relationship are quite important, and the above work 8 has shown that there are substantial differences in the patterns generated through drift and diffusion mechanisms. So far, the only experimental determination of the relationship between the shape of the index change and the light-intensity pattern which causes it has been for optical damage induced by a laser beam focused to 0. 3 mm. These results are not necessarily pertinent to hologram storage where the migration process occurs over distances of 10-4 cm or less.

In Sec. III we demonstrate both effects-coupling during writing and interference during readout-using a LiNb0 3 single crystal. The latter effect influences the readout erase time, making it dependent on the sample's orientation in a way consistent with previously observed enhancement of "fixed" holograms in these same crystals. 10 We use these results to show that in our samples the phase gratings result from internal electric field patterns caused by either diffusion or drift of photogenerated free carriers. This coupled-wave technique can also be used to determine the sign of the free carriers. For the undoped LiNb03 crystals used in this work, the carriers are electrons.

IL COUPLED·WAVE THEORY A. Basic Equations

Consider two coherent beams R and S symmetrically incident at an angle e relative to the z axis on a region with periodic variations of refractive index, (1)

that extends from z =0 to z =d as shown in Fig. 1. Although the following analysis is valid for the E vectors normal to the plane of incidence, it can be straightforwardly applied 9 to the more practical case where the polarizations are in the plane of incidence. For convenience, we assume that the average index of refraction is the same inside and outside the grating and that perfect Bragg conditions prevail. The relationship between x, the opticaJ wavelength, and R., the wavelength of the phase grating, is given by A= 2£

sine .

Thus, for this case we can write

This paper develops and demonstrates a technique for directly studying simple phase holograms. Two intriguing effects are considered for this purpose: (a) coupling between the two laser beams used to record the hologram, a coupling induced by the hologram itself, and (b) interference between the readout beam and the diffracted beam within the crystal, an effect that can write a "new" hologram during readout. To the best of our knowledge these have not been previously discussed. J. Appl. Phys., Vol. 43, No. 3, March 1972

In Sec. II we show, using the coupled-wave theory of Kogelnik, 9 that these effects can be used as powerful tools for assessing the relative merits of proposed mechanisms for storing phase holograms in these materials.

R=R(z)exp{-i[(211cos8/X )z+ (11/£.)x]} , S= S(z) exp{-i[(211cos8/X)z - (11/£.)x]} .

(2)

From the coupled-wave-theory analysis by Kogelnik,9 we find that in the region of the simple-phase grating and for perfect Bragg conditions, ddRz(z)=-iKS(z),

1042

173

dS(z)=-iKR(z) dz '

(3)

COUPLED-W AVE ANALYSIS OF HOLOGRAP HIC STORAGE +X

1043

C AXIS

+x

~ •

FIG. 1. Two coherent light beams incident to a simple phase grating with the indicated refractive-index modulation t!.n. The caxis direction refers to the orientation of LiNb0 3 electro-optic crystals as discussed in the text.

y R

--d-

-x where K = 11ni/Xcos8

This has the general solution R(z)=ae'"'+be "'"' ,

(4)

S(z)= -ae'"'+be·i= ± 11/2 the two beams retain their relative phases during recording. Thus the light-intensity pattern is not shifted along the x axis by the grating. The same does not necessarily hold true where writing is proportional to the light intensity. Here, = 0 or n, and from Eq. (5 ), the relative phase changes if the two writing beams have different incident amplitudes. This could "bend" the hologram as it is being written.

J. Appl. Phys., Vol. 43, No. 3, March 1972

175

1045

t:UUPLED-WA VE ANALYSIS OF HOLOGRAPHIC STORAGE 2

2. Drift in an Electric Field

2 IR=IRl =cos (Kz),

In this case, the migration current goes as

ls= lsl 2 =sin2 {Kz) .

J(x)o:E{l+ cos[(211/£)x]} . If E is independent of x, as for an applied electric field,

then the migration current is proportional to the light intensity, and the results of the last section are interchanged. That is, if the index change arises from a modulated electric field via the electro-optic effect, then no interaction occurs. For this case the index modulation is proportional to the migration current which in turn is proportional to the light intensity. Thus, the phase grating is symmetrical with respect to the interference pattern and no interaction occurs. However, when a significant fraction of the trapped charges are removed from regions of high light intensity, this analysis is no longer valid and interactions could occur. Keep in mind that this would occur only during the final stage of writing, i.e. , as the system approaches equilibrium. 8 Storage through variations in occupied-trap densities, however, gives an interaction even during the initial stage of writing. For this case the index modulation is prcportional to the gradient of the light intensity, and maximum coupling occurs. The direction of the resulting energy transfer is determined by the sign of the charge carrier and by the polarity of the field. If the E field is due to photoinduced polarization changes, as suggested by Johnston, qualitatively the same result occurs. However, the amplitude of this field should depend on the light intensity which in this case is sinusoidally modulated. The resulting field modulations would introduce higher orders in the diffraction grating so that the amount of coupling would not necessarily be that given by Eq. (7).

In summary, if the index change arises from an electric field pattern, we find that for recording through freecarrier drift, there is no transfer of energy between the two writing beams, at least at the initial stage of writing. For diffusion effects, however, energy transfer does occur and it increases linearly with time. In this case the direction of this transfer is determined only by the sign of the migrating carriers. On the other hand, if the index modulation arises from variations of occupied-trap densities, then the above situation is reversed, i.e., diffusion leads to no coupling, but drift does.

The diffraction efficiency is given by the value of Is at

z = d, the grating thickness. The interference of these two beams, the incident beam R and the diffracted beam S, forms a light-intensity pattern. Substituting the above amplitudes of R and S into

Eq. (2) we have / 101 , 1

= IR+Sl 2 =1+sin{2Kz)sin[(211/£)x] .

(13)

Assuming that the original phase-grating amplitude is weak enough (n 1 small enough) so that 2Kz < 11/2, then the light-intensity pattern is a positive sine function with an amplitude that monotonically increases with z. This interaction of the incident and diffracted beams can write a "new" phase grating in the normal manner through migration of photogenerated free carriers, a process t)1at can change the net diffraction efficiency. We now determine this change for the two different types of grating discussed in the last section: those which cause an interaction between the two writing beams (¢= ± 11/2), and those which do not(¢= 0 or 11). I. "No-Interaction" Grating

In this case the new index modulation lln' is proportional to the light intensity, so that lln'=n 2 sin[(211/£)x] , where n 2 is proportional to sin{2Kz). The net phase modulation lln, is then lln,=n 1 cos[(211/£)x]+n 2 sin[(211/£)x] .

Since n 2 increases with z, the resultant phase grating becomes "bent" one way or the other as the original hologram is read out. The effect of this bending on the over-all diffraction efficiency will not be considered further other than to point out that it should be the same for readout with the R beam or the S beam. 2. ''Interaction" Grating

In this case the grating is displaced by a quarter wavelength from the light pattern, so that lln '= n 3 cos[(211/£)x] ,

where n 3 is proportional to sin{2Kz ). phase modulation lln, is given by

Thus, the net

lln 1 = {n 1 +n 3 )cos[(211/i.)x] . C. Interactions During Readout

Here we consider the possibility of writing a "new" diffraction pattern during readout of a previously recorded one, and the change of the over-all diffraction efficiency during this process. For readout, the S beam is blocked so that the boundary conditions are S(O) = 0 and R(O) = 1. From Eqs. (5) and (6) the amplitudes of the two beams within a diffraction grating, n =n 1 cos[(211/£)x ], are given by R(z)= cos(Kz),

With amplitudes

S(z)= -i sin{Kz),

In this case, the grating is not "bent" during readout, but its amplitude is changed. Of course this change is

larger at the back of the grating than at the front, but its effect on the over-all diffraction efficiency is easily determined. It can be shown by integration of Eq. (3) that for a phase grating with an amplitude that varies with z the diffracted efficiency depends only on the avera~e amplitude. 13 Thus, it is clear that this "new" hologram can either enhance or diminish the over-all diffraction efficiency depending on the sign of n3. For example, consider the "new" hologram due to a diffusion-induced electric field pattern. From Eqs. (8), (10), and (13), the new grating is J. Appl. Phys., Vot. 43, No. 3, March 1972

176

1046

D.

L. STAEBL ER AND J. J. AMODEI

t>.n'cr (-)qcos[(21 1/£)x]

(S)

OBJECT BEAM

If the free carriers are electrons, then the diffraction efficiency is enhanced during readout.

This enhanceme nt is shown schematica lly in Fig. 3. The first curve shows the fixed charge distributio n of the original phase grating. The resulting electric field is shown by the second curve. The third curve shows the modulation of the index of refraction, where it is assumed that a positive field decreases the index, the case for LiNb03 with the c axis oriented as shown in Fig. 1, i.e. , pointed in the + x direction. When the reference beam R is used for readout, it interferes with the diffracted beam to form the light-inten sity pattern shown by the fourth curve. Note that for the case considered here, i. e. , the readout beam propagatin g in the +x direction of the crystal (see Fig. 1), the maximum of the resulting light-inten sity grating is on the + x side of the index-chan ge maximum. The free electrons generated by the light then diffuse, to accumulate at regions of low light intensity, creating the charge pattern of the last curve. This newly created charge pattern adds to the original charge, and thus increases the over-all diffraction efficiency.

REFERENCE BEAM (R)

FIG. 4. Experimenta l apparatus for recording simple phase gratings in LiNb0 3 •

lowing three conditions: (a) If holes instead of electrons are the mobile carriers, the last curve is reversed, and the net charge decreases. (b) If the experimen t is reversed by using the S beam for readout, i. e. , the beam propagatin g toward the - x side of the crystal, the light-inten sity maximum will be on the -x side of the index maximum. This reverses the phase of the lightmodulation pattern, and the resulting diffusion of electrons decreases the net charge. (c) If the only change is to reverse the crystal so that the c axis points toward the - x direction, then the index change of curve 3 is reversed in phase and the same result occurs.

Note that the opposite effect will occur, i.e. , the efficiency will be diminished during readout, for the fol-

In this section we have discussed two phenomena suitable for the study of optically recorded phase gratings-beam coupling during writing and interferenc e during readout. We now present experimen tal results that demonstra te both effects.

Ill. EXPERIMENTAL RESULTS

Simple phase gratings were recorded and read out in undoped single crystals of poled LiNb03 with the apparatus shown in Fig. 4, using an argon laser operating at 4880 A. The two beams, the object beam S and the reference beam R, intersected at 30° and provided a net power density at the sample of roughly 0. 6 W/cm 2 • The crystal was cut and polished with two parallel faces containing the c axis and was aligned normal to the bisector of the two beams. The E vector of the light was horizontal, i.e. , in the plane of incidence, as was the c axis of the crystal. The detector measured (a) the intensity of the transmitted S beam during writing and (b) the amount of light diffracted from the R beam during readout. These experimen ts were done with the c axis pointed away from the detector as shown in Fig. 4, hereafter referred to as the + c direction, or with the c axis pointing in the opposite or - c direction. The caxis polarity of our crystals was determined from pyroelectric measureme nts. 14

2

3

4

5

FIG. 3. Schematic description of enhancemen t during readout of a phase grating, curve 3, associated with a charge-dens ity pattern, curve 1. The readout and diffracted beams interfere to give the modulated light intensity of curve 4 which then, through diffusion of electrons, makes a "new" charge-dens ity pattern in phase with the original.

Figure 5 shows the intensity of the transmitte d S beam during exposure to both the R and S beams. For the + c direction the intensity increases linearly at first, saturating at nearly twice its original value. If the crystal is reversed to the - c direction, the opposite occurs, i. e. , the S beam goes down. The diffraction efficiency of the phase grating produced during the above exposures is shown in the first set of curves of Fig. 6. These values were obtained by momentari ly blocking the S beam at regular intervals. Note that, at least

J. Appl. Phys., Vol. 43, No. 3, March 1972

177

L. STAEBLER AND J. J. AMODEI

D.

1048

nent KD-so that sincj>= KD/(K~-r K~) 112 , where (K~+ K~) 1 i2 is the net amplitude of the resulting grating. If this amplitude is small enough to make a linear approximation for sin(2Kd), then the interaction is proportional to only K D the diffusion component, and thus is not changed upon application of an electric field .

04 H

- 0.3

0.3

---+c -c

{!. - - - -

u :=

~

200

400

:~ 0

200

::L 0

200

600

75°C I

400

600

100°c I

400

--------:r 600

40

"'~ 10

15

20 400

600

EXPOSURE TIME

800

1000

(sec)

0'--""""''--~-J.~~~~-'-~~~~.....1-~~

0

FIGURE 3 Demonstration of enhancement during readout, due to the writing of a new hologram by the intersection of the readout beam and the diffracted beam within the crystal, and the erasure of this effect as discussed in the text.

200

400

600

READ OUT TIME (sec.) FIGURE 4 Room temperature readout after heating a hologram for 5 minutes at the indicated temperature. using the 0.7 cm sample of LiNb0 3.

this off-angle erasure. The diffraction efficiency decreases to a lower level corresponding to that of the fixed hologram. Comparison with Figure 2 shows that the fixed hologram efficiency is roughly one half of the one in the crystal before heating. The temperature dependence of the fixing process was determined in two ways: (1) by reading out holograms fixed for a given time at different temperatures and (2) by monitoring the thermal decay of the hologram during fixing. In the first method. a sample was heated for 5 minutes at a

the stored hologram (i.e. it decayed normally during readout at room temperature after such heatings). At the highest temperature. 125°C. the sample was completely fixed so that the efficiency was initially zero. and then increased during readout. The 100°C heating, on the other hand. pro- ' duced an intermediate effect; a smaller initial efficiency than for the two lower temperatures. decay of the initial hologram to zero diffraction

183

D. L. STAEBLER AND J. J. AMODEI

110

results of the previous 'pulse annealing' experiment. After complete decay at any of these tempera. tures. the hologram could be brought back by cooling the sample to room temperature and then exposing it to the readout beam. That is. a holograrn can be fixed at any of these temperatures. with a required fixing time that decreases rapidly with increasing temperature. We have found however. that if a sample is heated to 300°C for an hour. no hologram is observed. fixed or otherwise. upon cooling to room temperature. A complete erase of a fixed hologram is also observed if we illuminate the sample with the readout beam while the sample is at a fixing temperature. Similar erasure is accomplished by y irradiation at room temperature. To see if the thermal decay of the hologram was related to dielectric relaxation, we measurecl the electrical conductivity of a sample cut from the same boule as the one used for the hologram measurements. Figure 6 shows the measured c-axis conductivity from l00°C to 300°C. The con-

efficiency. followed by an increase to a saturation value much lower than that of the 125°C test. In each test. the heating and cooling processes generated pyroelectric fields large enough to cause breakdown across the surface of the sample between the c faces. To monitor the thermal decay of a hologram during fixing. recording was done at the fixing temperature. The resulting hologram was then periodically read out using the reference beam taking care to prevent optical erasure. At each temperature used. the hologram decayed exponentially with time, with the time constants shown in Figure 5. They follow an exponential function. TEMPERATURE ( ° C ) 100 90 130 120 110

80

TEMPERATURE ( °C) 300

250

200

150

100

-9

10

-10

c

FROM DECAY TIMES

o

FROM CT MEASUREMENTS

10

I

-II ~10

E u I

c:; -12 -10

,_>> ~ :::i 0

-13

10

z

2.4

0

u

2.9 2B 2.7 1 [TEMPERATUREr ( 1000/° K) 2.5

2.6

-14

10

FIGURE 5 Time constant for thermal decay of a hologram during a fixing at different temperatures.

-I

10

r = r 0 e+E./kT (r is the decay time. k is Boltzmann's constant, Tis the temperature. and E. is an activation energy). with an activation energy of "'1.10 eV. These decay times are consistent with the

1.8

2.8 2.6 24 22 2.0 1 [TEMPERATURET (10001°K)

FIGURE 6 Measured conductivity compared to that expected from the decay times of Figure 5 assuming dielectric relaxation.

184

THERMALLY FIXED HOLOGRAMS IN LiNb0 3

ductivities above 200°C agree with those of Roitberg et al. 9 and. when extrapolated to 1000°C using our measured activation energy. they are in approximate agreement with those of Bergmann 10 and of Jorgensen and Bartlett. 11 Figure 6 also shows the conductivities calculated from the thermal decay measurements of Figure 5. assuming that the decay time of the hologram is given by one half of the dielectric relaxation time. e/ 1 and t 2 < 1 for positive 'Y· The slgn of 'Y depends on the direction of the c-axis. It is interesting to note that these expressions for transmissivity are formally identical to those of the codirectional coupling even though the spatial variations of 11(z) and I 2 (z) with respect to z are very different. The relative phase of the two-waves is obtained by solving eq. (5) in cooperation with eq. (8) and is given by ../1 2 - .J; 1 =t.Bz +constant,

0

z=O

z= L

I Fig. 2. Intensity variation with respect to z on J?hotorefractive crystals. The coupling constant is taken as 'Y = 10- 1 cm and interaction length L is 2.5 mm. The dashed curves are for the lossless case (i.e.,"' =0). The solid curves are obtained by numerical integration incuding the loss(= 1.6 cm-1 ). The dotted curves are the approximate solution (13) .

(12)

1

where .B = kn 1 cos ¢/cos 8 , the phases are defined as A 1(z) = y/ 1(z) exp(i.J; 1) and A 2(z) = yI2(z) exp(i.J; 2). The relative phase varies linearly with z, and thus leads to a change in the grating wave vector by ,B/2 along the z-direction (i.e., the grating wave vector becomes K + ! ,Bz). The nonreciprocal transmissivity of photorefractive media may have important applications in many optical systems. It is known that in linear optical media, the transmissivity of a layered structure (including absorbing material) is independent of the side of incidence (the so-called left-and-right-incidence theorem [12]). Right now, with the photorefractive material available, it is possible to make a "one-way" glass which favors transmission from one side only. Using such a nonreciprocal medium in a ring laser we would certainly be able to suppress the oscillation of one of the oppositely directed traveling waves. Unidirectional oscillation in a ring resonator using co-directional coupling/pumping in BaTi0 3 crystal has been demonstrated [13). The coupled equations ( 4) did not take into account the effect of the bulk absorption of light. The attenuation due to finite absorption coefficient would add a -al1 term on the right-hand side of the first equation in (5), and a +Oil2 term on the right-hand side of the other equation. With these two additional terms accounting for bulk absorption, closed form

solutions are not available [14]. However, eq. (5) can still be integrated numerically. It is found that a very good approximate solution is lj'(z) =

ri=0(z) exp(-a:z),

fi(z) =ri= 0 (z)exp[a:(z - L)].

(13)

Fig. 2 illustrates the intensity variation with respect to z for the case when 'Y = 10- 1 cm, a:= 1.6 cm- 1 , and L = 2 .5 mm. If the loss were neglected (i.e., a: = 0), the transmissivity would be t 1 = 1.81 and t 2 = 0 .15. With a: = 1.6 cm- 1, the transmissivities become t 1 = 1.27 and t 2 =0 .11 according to a numerical integration. The approximate solution (13) would lead tot 1 = 1.21 and t 2 = 0.10. Note that even with the presence of absorption, the transmissivity can still be greater than unity. The approximation is legitimate provided a:

w E

10

1.0

100

A9, µm

Fig. 3. Imaginary part of the space-charge field in BSO as a function of grating period Ag with optimal velocity moving grating, T di/72 = 6 X 10-4, applied field Eo, and trap density NA·

Tdi T2

= E((3 + slo + l'Rno)

30

'7

.-§

(6)

10

41!"eµno

By substituting no = s(Nv - NA)lo/(")'RNA) into Eq. (6), neglecting dark generation of carriers (3 compared with photogeneration slo, and assuming that NAINv « 1, one obtains Tdi

f")'R

Tz

41Teµ;

20

Ag•/Lm

Fig. 4. Gain coefficient as a function of grating period for parameters shown for comparison with.the data of Rajbenbach et al. 16

-=--· Since f is well known in BS0, 28 uncertainty in the ratio /'RIµ; controls uncertainty in Ttl;/T2. Two values of l'Rfµ are given in Table 1. The value 4 X io-10 V cm is based on photoconductivity measurements of the product µTR [TR is the electron recombination time TR = l/(")'RNA)] combined with a photorefractive estimate of NA. 29,34 The value of 2 X 10- 11 V cm was calculated from photorefractive measurements of the time decay of gratings. 30 The correspondence between the parameters measured in Ref. 30 and the parameter T di/T2 can be made by noting that, in the charge-hopping model8 used in Ref. 30, T di/T d equals the square of the ratio of the diffusion length to the Debye screening length. Several interesting features of the space-charge fields enhanced by a moving grating are apparent in Figs. 2 and 3. First, the internal space-charge fields are predicted to be huge compared with those with v = 0 in Fig. 1 (note that the scale changes). Second, the maxima in the curves are shifted to longer periods. In order to make a unique comparison with the results of Rajbenbach et al., 16 one needs the effective electro-optic coefficient reff, the trap density NA, the ratio Tdi/T2 (or l'Rfµ;),

l::,,, DATA E0

=

10 kV/cm

DATA E0

=

6kV/cm

0

Q DATA E0 LINES:

2kV/crn

THEORY w"' 0.01

NA"' 2 x 10

10

15 cm- 1

30

100

Fig. 5. Gain coefficient as a function of grating period based on theory given here (lines) for comparison with values measured by Rajbenbach et al. 16 (data points).

205

Vol. 1, No. 6/December 1984/J. Opt. Soc. Am. B

George C. Valley

effects. The analysis given in Appendix A suggests that this is not the case for steady-state two-wave mixing assisted by a moving grating. Recently, Refregier et al. 35 explained this dependence as a contribution from higher-order Fourier terms (i.e., terms of order ±4 in Eqs. (All) and (Al2) below]. Since these terms may well be large even for small beam ratios, it would appear to be useful to measure only the first-order Fourier contribution to two-wave mixing for comparison with the first-order gain coefficient calculated here.

GaAs NA= 1.5 x 10 15 cm- 3

NO VELOCITY

10

100

APPENDIX A: DERIVATION OF EQUATIONS FOR SPACE-CHARGE FIELD AND OPTICAL FIELD

Fig. 6. Imaginary part of the space-charge field in GaAs as a function of grating period with applied field Eo and no moving grating.

In this appendix we give a derivation of the basic equations for the internal space-charge field and for the optical field for two-wave mixing in a photorefractive material with an applied external field Eo and with a frequency shift between the two beams. The starting point is the set of equations used by Kukhtarev4 (see also Refs. 1, 2, 24, 25):

GaAs NA= 1.5 x 10 15 cm-3

E

Td/T 2

=

3 x 10-5

""::;;

an - aN'}; = - V' • j at at e

w~ 50 E

aN+ __!!_

at

10 A .µm 9

= (/3 + sl)(Nn = eµnE

APPLICATION TO GaAs

Recent results2l,22 show that GaAs has the potential to become an important photorefractive material in the near infrared. Since its tabulated r 41 is even smaller than that of BSO, applied-field and moving-grating techniques are likely to be useful in obtaining large gains. Figure 6 shows the predicted space-charge field with no velocity, whereas Fig. 7 gives the predicted field with optimum velocity. Since 2o = 0.41 kV-1, one predicts, for example, a gain coefficient of 7 cm- 1 at Ag = 10 µm and Eo = 2 kV /cm.

=-

(A2)

- kBTµV'n + pl

V' ·(EE)= 47re(n +NA - Nii) (Poisson's equation), V' 2Eopt

(Al)

Nii) - '(RnNii

(current equation),

100

Fig. 7. Imaginary part of the space-charge field in GaAs as a function of grating period with applied field.

5.

(continuity),

(ion rate equation), j

4.

871

(A3)

(A4)

1 a2 ~ at 2 n~ptEopt

(wave equation),

(A5)

+ r effnf,Escl.

(A6)

n~pt = nt(l

where n is the electron number density and Nn is the total number density of photorefractive donors (both ionized and neutral). N n may be an intentional or an unintentional dopant, such as iron, or may be a defect. Nii is the number density of ionized donors, which also act as traps. NA is the number density of negatively charged ions that compensate for the charge of Nii in the dark. NA is assumed to be a constant and not to take part in the photorefractive process. This restriction is removed in Ref. 36. Nn - Nii is the number density of neutral (filled) donors; j is the current density; e is the charge on the electron; f3 is the dark number-density generation rate; s is the photo cross section; I is total optical irradiance in the crystal; 'tR is the electron-trap recombination rate; µ is the mobility; kB is the Boltzmann constant; T is the temperature; p is the photovoltaic constant; f is the static dielectric constant; E is the sum of the applied and the space-charge fields; nopt is the refractive index; nb is the background refractive index of the crystal; Esc is the magnitude of the space-charge field; and Eopt is the amplitude of the optical field. Solution of Eqs. (Al)-(A6) is most easily performed by solving Eqs. (Al)-(A4) for Esc• assuming a given optical field, and then by substituting the space-charge field into the wave equation by use of Eq. (A6). First, eliminate Nii and n from Eqs. (Al) and (A4) to obtain

DISCUSSION

The charge-transport model of photorefractivity has been used to predict the imaginary part of the space-charge field and hence the gain coefficient for steady-state two-wave mixing with a frequency offset between the beams. The predicted gain coefficients can be adjusted to fit the basic observations of Ref. 16, but some problems remain. In addition to the problems fitting the basic observations of r discussed above, Rajbenbach et al. 16 observed that the gain coefficient depends on the total intensity of the two beams (Fig. 5 of Ref. 16). The theory derived here provides no explanation for this dependence. The dependence of r on the initial ratio of irradiances IA+ 1 (0)1 2 ~A- 1 (0)l 2 found in Fig. 4 of Ref. 16 is also difficult to explain. Rajbenbach et a/. 16 tentatively attribute this to transient energy-transfer effects5 mixed with moving-grating

206

872

J. Opt. Soc. Am. BNol. 1, No. 6/December 1984

George C. Valley

a

aE 47rj 47rJ at=--+-,

,,- -

x

'V ·-(EE)= -47r'V · j. (A7) at Since f is independent of time, Eq. (A 7) may be integrated:

-

-

-

-

aE

47r

471"

M

£

£

(-f'V. E 47re

/

I

I

) / /

/ /

(A9) Fig. 8.

n =no+ n2(z)exp(2ikxx - ikvt) + n-2(z)exp(-2ikxx + ikvt),

Also, the parameter a relates the optical field to the intensity: a= cnb/411", lo= a(iA11 2 + IA-11 2).

Note that additional terms in a that are proportional to reffnT,Esc are required from Eq. (A6) if reffntEsc ~ IA 1A-il/ (IA1l 2 + IA-11 2). Elimination of nz in favor of E 2 yields

(All)

E = [Eo + E2(z )exp(2ikxx - ikvt) + E-2(z)exp(-2ik,x + kvt)]x,

(Al2)

1[-ikv + (l/71) + (lfrR) + (ifrE) + (lfrn)] X [-ikv + (lfrdill - [(i/r2) - (ifrdi)][(-1/TE) + (ifrn)JlE2

(i - ~~)]

+ A-1(z)exp[i(-kxx - k,z +wt) (1 +

~~)]} y,

.

f

+ PxaA1A'..i), -ikun2 = sNnaA1A'.. 1 + (/3 + slo + 'YRno) X

= -(Eo E-2 = E?_,

(A+ Bu)E2 (Al4) A B u

+ (saA1A'.. 1 + "fRn2)(-no-NA) e

(/3

+ sloHNn -

(A18)

iEn)A1A'..i/(IA11 2 + IA- 112), (Al9)

where

(-E2ik,E2 - n 2 ) 47re

2ik, - - - (eµnoE2

A A'..

where T di= f/(47reµno) is the dielectric relaxation time. TE = l/(kgµEo) is the drift time, Tn = e/(µknTki) is the diffusion time, TR = lf"fRNA is the electron recombination time, T1 = 1/(/3 + slo + 2)"Rno), T2 = 1/(/3 + slo + "fRno), and En = knTkg/e is the diffusion field. Again, for the typical irradiances, Tl is much smaller than TR; also, typical frequency shifts kv are of the order of a few hertz, 12 whereas TE is typically less than 10 µsec so that kv « TE - 1. Using these approximations in Eq. (A18) yields

+ eµn2Eo

- 2ikxknTµn2

1

1 1 = - (E o - iEn) - , TdjTR IA11 2 + IA-11 2

(A13)

where kx and kz are the wave numbers in the crystal, 2kx = 27r/ Ag (Ag is the refractive-index grating period), v is the velocity of the moving mirror used to give beams A1 and A_ 1 a frequency shift, w is the optical angular frequency, and k = nbw/c. The coordinate system is shown in Fig. 8. Equations (All)-(Al3) neglect higher-order Bragg components, discussed, for example, in Refs. 6 and 35. Substitution ofEqs. (All)-(Al3) into Eqs. (A9) and (AlO) and retention of the lowest-order Bragg components yields

"k vE2 = - 471" - (eµnoE2

Two-wave-mixing geometry.

n - NA)

1

-1

I

//

I"-------(

- - 'V · (eµnE - knTµ'Vn + pl). (AlO) e Equations (A9) and (AlO) are solved by assuming that n, E, and Eopt are of the form

Eopt = {A1(z)exp[i(kxx - kzz +wt)

-.11

;

(A8)

- = --(eµnE -knTµ'Vn +pl) + - J ,

M

/

where 47rJ/£ is the integration constant. JA is the current in the external circuit that is used to apply the external field E 0 to the crystal over the area A of electrodes. Next, both Nii and j may be eliminated from Eqs. (Al)-(A4) and (A8) to obtain 13

an= (/3 + sl)Nn + (/3 + sl +)°Rn)

-

/ /

Eq

= 1 + (En/Eq) + (iE 0 /Eq), =

(Eo!Eq) - (iT

u.J

E0 =6kV.cm- 1

I

40

.§

(33)

20

where 500

LE =µE0 r, =µEofyRNA (34) is the electron drift length defined in the literature, 23 and

1000

v f.µm s- 1) FIG. 3. Theoretical evolution of the imaginary part of the space charge field amplitude vs the velocity of the fringes for different values of the applied electric field E 0 • A= 20 µm, A.= 568nm,10 = 140 mW cm- 2 •

[ImEs,]b~b"P' [Im Es,

]b~o

=_!_(E•) =20, 2

2

(32)

E0

a very large factor indeed. This is because by moving the fringes we gain in two respects, we increase the modulus of Es, and also we make the phase of Es, correct for amplification as may be seen in Fig. 4. Both Es, and Im Es, reach their maximum at the same velocity and the corresponding phase angle (between the fringe pattern and the refractive index distribution) is 90". Note that for small and large velocities, the phase angle is quite close to 0° and 180", respectively, both being unfavorable for beam coupling. 100

,.)..- -

~

.5

150

showing that Im Es, is proportional both to N ,t, and to A. Since N ,t, decreases with increasing A an optimum fringe spacing may be expected to exist. The optimum, as may be easily proven, occurs, when

7. Validity of the approximations

I

I I

120

60 90

Qj ~

.""

~

40

-&

60

,;; u.J

20

v (µm.s- 11

FIG. 4. Theoretical evolution of the modulus and of the imaginary part of the space charge field vs the fringe velocity. The dashed lines represent the phase angle

'

---

/

80

= -..

eN+A ImEs =--D-'-,

180

ro> 1 (equivalent to saying that the reference beam is undepleted) Eq. (56) simplifies to I

10

~ I

N

6

3

30

'° 20

The formula for the optimum velocity has been previously derived both heuristically [Eq. (16)) and analytically [Eq. (23)), the latter formula reducing to the former one under certain approximations. It may be seen that v0 P, is proportional to I 0 , 1/E 0 and A 2 . The linear relationship with I 0 , may serve to determine the product sND. Following Peltier and Micheron22 in choosing ND= 1025 /m 3 we obtain s = l.06X 10- 5 m 2/Jbymatching theory and experiment in Fig. 6 where Vopt is plotted against I 0 for the thick crystal. At a given value of/0 the dependence ofv0 P, on A and E 0 can be best summarized by plotting the measured frequency shift 8/ = v0 P, I A against 1/E 0 • The theoretical curves are plotted from Eq. (23), showing an almost linear behavior (Fig. 7). With our previous choice of s, the theory is now matched to the experimental results (obtained with the thick crystal for A= 22.7 and 3 µm) also vary linearly but the slope is somewhat smaller and they do not tend towards the origin. The main cause of the discrepancy is very likely to be the attenuation of the crystal not taken into account by the theory.

A=3_µm

0

600 E

a 0

200

150

0.3

B. Spatial frequency dependence of the gain coefficient

V>

100

0.2

FIG. 7. Dependence of the Doppler frequency shift [8UJ = (21T/A lv0P,] on the inverse of the applied electric field l/E0 . 10 = 140 mW cm-'.P= 103, ..t = 568 nm. -, Theoretical curve; \1 f:;, experimental points.

a

::i_

0.1

1/ E0 (cm kv- 11

800

200

!0 (mW cm- 21 FIG. 6. Evolution of the optimum fringe velocity vs the total incident beam intensity on the crystal. The crystal interaction length is I = 10 mm. {!.._= 103, E 0 = 10 kV cm - 1• A = 23 µm, ..t = 568 nm. -Theoretical curve; U experimental points. 52

l

J

A

I

A. Optimum velocity

50

""'

40

In this section we shall use the theory developed to explain a set of experimental results part of which have already been published. 13 •17 The subjects for comparison are the optimum velocity and the gain coefficient.

400

A=22.7_µm

50

IV. COMPARISON BETWEEN EXPERIMENTAL AND THEORETICAL RESULTS

>o

4 A

(57)

I'=-lnro· l

8

The gain coefficient at the optimum velocity is a function of the spatial frequency as it was found before for the thick crystal, 13 and we have now measured similar curves with the thin crystal at the same beam ratio, .B = 103 • All these experimental results are summarized in Figs. 8-10 where r is plotted against A - l for Eo = 106 v/m. Let us first look at the results obtained with the thin crystal, in which case .B = 103 is fairly close to the value yielding saturation, so the second perturbation calculations should be valid and may be expected to give good agreement with the experimental results. Note that the values of the various parameters were chosen in Eq. (31) with hindsight. Those were the values we finally chose in order to have the "best" agreement with the experimental results (we shall return to the choice of parameters in Sec. IV D). There is only one more parameter then to be determined and that is the effective value of the electro12 optic coefficient '•ff· With the choice of '•ff = 0.95 X 101 at agree values experimental and theoretical the , m v.B = 103 , E 0 = 106 V/m and A = 23 µm as may be seen in Fig. 8. There are two theoretical curves, one for small signals (/3-> oo) and one for large signals (/3 = 103 ). The large signal experimental points were directly measured. The small signal point at A = 23, 10, and 3 µm are extrapolations deduced from the curves of I' against .B presented in the next section, as .B-> oo. The agreement between the theoretical curves and the two sets of points is quite good. We have some further large signal experimental results for E 0 =8X105 VIm which are shown in Fig. 9. There are now no free parameters to adjust, so this is now a stricter test of the theory. As may be seen, the agreement is still quite good. Refregier et al.

J. Appl. Phys., Vol. 58, No. 1, 1 July 1985

216

52

12

2000 4

7

1000 500 400 300 200

10 6 3

"

\l

.,

5

5

.,.:

D

6

'-

~

I '7E

.!:! '-

2 D

4

''

100 50 40 30 20

3

4

)..!'

10

2

1

o!Lu.-s~o_._~10-o-'--::15~0-'-:2~0~0'--::2s~o- oo. f3 = IO': experimental points; ---, theoretical curve obtained with the second perturbation.

0,

Let us turn to the experimental results for the thick crystal. They are shown in Fig. 10 for an applied electric field ofE0 = 106 V/m for the same beam ratio as before (/J = 103 ), using the same technique. The small signal theory is given by the first perturbation but we are unable to use the second perturbation for the large signal theory because /3 = 103 is too far from the saturation value. We can, however, use the phenomenological theory as represented by Eq. (55). As will be shown in the next section, the parameter a takes the values of a= 2.8, 1, and 0.5 for A = 23, 10, and 3 µm, respectively. 12 4 JO

/

I

I

5

~

.....

3

'

c. '

"

/§' ,.,_

6

.,....

'--

4

c.

For other values of fringe spacing, we use some judicious interpolations and extrapolations. The small signal points at A = 23, 10, and 3 µm are deduced from the curves of I' against f3, by adding from the corresponding phenomenological points the difference I'fJ-.~ I'13 ~ rn'. Note that, for A = 10 µm, I' is higher than the experimental saturation points of Fig. 12. This difference (0.8 cm- 1 ) may be seen as an example of possible shifts in the value of I' if extracted from to different sets of measurements. Once again we can choose the value of the effective electro-optic coefficient so as to match the large signal theoretical and experimental results at A= 23 µm. We obtain r.tr = 0.6X 10- 12 m/V, which is smaller than the value obtained for the thin crystal. The agreement between theoretical results may again be regarded as good. For E 0 = 8 X 105 V/m, we give the experimental results in Fig. 11. There.are now no experimental data (I' against /3 has not been measured for this applied field), which could be used to obtain the parameters of the phenomenological theory. So, we cannot give, for the present case, a large signal theoretical curve. We give, nevertheless, the small signal theoretical curve which is related to the large signal experimental points in roughly the same manner as in Fig. 10 for E 0 = 106 V/m. C. Beam ratio dependence of the gain coefficient

c.

olL....-'---'50~-'---1~00~.._-1s~o__.~2~0-o~-2='so1 K 1 (mm- 1) FIG. 9. Evolution of the exponential gain r VS the spatial frequency. I= l.27mm, E0 = 8 kV cm- 1,10 = 140mW cm-2,.-i = 568 nm./3= 10': -, theoretical curve rp-. oo ); f3 = IO': f';, experimental points.

53

FIG. IO. Evolution of the exponential gain I' vs the spatial frequency. I = IO mm, E 0 =IO kV cm- 1, I 0 = 140mW cm- 2 ,,t = 568 nm,_/3= IO': 8, experimental points; - , theoretfoal curve f/3---> oo ). /3 = I03 : D Experimental points; --theoretical curve, second perturbation.

J. Appl. Phys., Vol. 58, No. 1, 1 July 1985

The dependence of the gain coefficient I' on the input beam ratio f3 is shown in Figs. 12 and 13 for the thin and thick crystal, respectively. The experimental measured quantity is Yo but it is more conventional to make the comparison for r. The curves for A = 23 µm have been published before, 17 those for A = 10 and 3 µm are new results. The following conclusions may be drawn: (i) For sufficiently large value of/3 all the curves tend to a constant value. (ii) The saturation value I', is always higher for the thin Refregier et al.

217

53

106

14 r---- r--T ---i- --r- -r

2000

8

5 L-

I

I

I

1000

1J

~ 10 5

6

500 400 300 200

10

~ 10 4

5

100 50 40 30 20

4 A

3

A

Ii

i\c73JJm

-,...:'

A"'

5

/

4

2

/_ _ //A~10~~T-

o"~~ J\=3µm

~~----/"''-:_....____

..-.1. .-J..- J..-"" -.l.--' l-"

I

L-

10

I

3 log I>

4

2

10

10 1 10°

6 .

ratio p icient I'as a function of input beam 2 FIG. 13. Exponential gain coeff , A= 568 cmmW 140 = I mm, 10 0 ng A. I= for different grating spaci tical 1 experimental points. - · - ·-,th eore nm, E 0 = 10 kV cm- • 0, [;,, l curve obetica theor -, . terms er d-ord curve obtained by including secon e and space onship /(m) between the fring tained by assuming the relati charge modulations.

can always be regarded as crystal. (iii) The reference beam µm, there is strong depen23 = A being undepleted. (iv} For dence on/3. 13 are the best indicaThe curves shown in Figs. 12 and signal theory breaks ll sma ed ariz line the tions of the fact that ent that the linearevid is it down. With our definition of I' reas the experiwhe t, stan con = I', I'= ized theory gives

14 12 4

10

5 1-

"

4 oo

D D

21 I

0

, particularly for A = 23 µm, mental results clearly indicate f3. on that I' is strongly dependent tioned already in Sec. IV B, men as tal, crys thin the For cted to give quite a good expe be may the second perturbation nt between the experimental approximation. The agreeme n in Fig. 12 is not so good show lines points and the dotted ncy between theory and exmainly because of the discrepa r hand, the continuous othe the On periments as /3--+ro. gical approach give very nolo curves as given by the phenome by choosing the valible poss e mad is This good agreement. ntally measured rime expe the ues of I', so as to agree with nt in itself is no eme agre that s, Thu n. ratio values at satu ry. There is no reason, howcheck on the validity of the theo meter a should give good ever, why any choice of the para choices of a already mene thre the agreement. The fact that nt for the thin crystal is entioned give such good agreeme tion of the phenomenologifica justi couraging. But the best es of a give good agreement cal theory is that the same valu cating that the relationship indi , well as tal for the thick crys good physical basis. a assumed by Eq. (46) must have dotted curves in Fig. 13. the at look may Finally, we be expected to give good only The second perturbation can is indeed born out by the this and f3 of es valu e results for larg results.

I logp

ratio P for as a function of the input beam 2 FIG. 12. Exponential gain I' W cm- ,..t = 568 nm, 140m = I m, l.27m 0 I= A. ng different grating spaci curve 1 ntal points. - · - · -, theoretical Eo = 10 kV cm- • 0, [;,, D, experime theoretical curve obtained by s.-, term r orde d secon ding obtained by inclu charge field between the fringe and space assuming the relationship /(m) modulations. 54

ers D. Choice of the free paramet y as 6 free parameters, man as ry theo We have in the must choose them so We '•tr· namely s, Nv, NA, µ, YR, and theory and experieen betw ncy repa disc the as to minimize divided into two be can lem ment. Fortunately, the prob ND are separate and s ing rmin dete nts rime parts. The expe Refregier et al.

1, 1 July 1985 J. Appl. Phys., Vol. 58, No.

218

54

parameters. In fact, it is from those determining the othe r rmined by matc hing the the prod uct of sNv that can be dete the experimental points to 16) ( Eq. by d sente theory [as repre 3 25 value, which is probcm I 10 in Fig. 6]. If we chose ND = r density, we obtain dono the of ation oxim appr 23 ably a good 2 5 es"= l.6X 10valu the that s == 1.06X 10- m /J. Note ition 26 defin rent diffe a m1 obtained by Valley results from (sv

== shv).

V. DISCUSSION

s A. The poss ibilit y of gene raliz ation e of experimental rang We have prov ided here a wide moving fringes with nts rime expe ling coup results for beam h goes beyo nd whic ry theo in BSO. We have also provided a active materiorefr phot in ling coup beam of existing theories an approxing findi i.e, ts, als by including large signal effec tions. equa rials mate inear nonl the of mate solution of the problems for We shall now briefly review some the moment. at able avail is er answ which no complete

rs is concerned, the As far as the next set of para mete h gives the value whic (28), Eq. is ip rnost impo rtant relationsh s of NA and µ!yR , term in ing spac e fring um optim of the 9 the second pertu rbati on of Kukh tarev et al. secondly a,, the key para mete r in 1. Are the differential equations on nd depe to (45)] and calculations, may be seen [Eqs. (44) sufficiently gene ral? answer is yes. If the Aopt and NA. follows. Take first We believe, for most purposes, the The matc hing proc edur e is then as 21 3 equation for the rate her anot low, is er the pow t is h inpu whic al total optic 41T 10 /m in the form the thin crystal and start with NA21 = aps (perh ssary nece be acceptor density would matc hing theoretically 26 value given by Huig nard et al. by it is difficult to see any e rwis othe but ) ity y dens Valle trap by the ested sugg (in fact, the four-wave mixing results in 21BSO3 y equation and Poisneed for generalization. The continuit but that was due to an suggested in that pape r was l0 /m ibly, at very high Poss tion). Next, assume a son's equation are corre ct as they are. error of 41T in defining the field equa nden t on inpu t depe me beco YR sand powers, the constants t grea ter than the measmall signal value for A 0 P, (somewha . limit 6 power but we are very far from that E = 10 V/m and plot sured large signal value of A 0 p,) at 0 se be mad e general of I' as a e curv l The wave equation may of cour signa large the and l both the small signa al anisotropy and cryst ts, 1 enough by including vectorial effec r choices of Aopt until the function of A - • Repe at it for othe ulated for BSO. form been yet not has 23 = this A to but birefringence e of I' is close maximum of the large signal curv for A = 23, 10, and 3 µm. Plot finally I' as a function of f3 equa tions (small and large vales curv al retic theo 5 have now z. Is the one-dimensional solution of the materials µm. We 1 three for fJ st again I' and , A sufficiently gene ral? ues of I' as a function of matc h the corresponding factory provided we different values of A ) whic h should A one-dimensional solution is satis the amplification is when but n regio experimental results. l signa l smal are in the s of f3 does not value large for sian beam may be ature gaus a , curv the ratio that We find dependent on the inpu t beam e of NA, on the othe r ably, there is no Prob tion. ifica ampl after rted depend strongly on the assu med valu disto severely 23 µmd oes, so it is = atA action theory. w ~ diffr ) al /(I' nsion -. dime (I')f3tworatio 13 hand, the need for a fully fledged om in freed Our t. resul into severr latte beam the the h e up more impo rtant to matc It woulld very likely suffice to divid the thin crystal is for rately for sepa If ed. tion limit ifica r ampl rathe the is NA late choosing al filaments and calcu NA se choo may we then 5% ± of known with an accu racy each filament. within a range of abou t ± 10%. as are choosing NA Some addi tiona l considerations in erica l solu tion of the good agreement for the I' against 3. Is it nece ssar y to have a num follows: (i) We shou ld have being s 5 thing (ii) Othe r materials equa tions ? A - l curve at E 0 = 8.10 V /ma s well. solution valid in the rable because it leads to It seems unlikely that an analytical equal, a low value of NA is prefe r means. (iii) othe by efore, particularly ured Ther d. meas e foun valu be the will to est er higher retr• near whole range of inter well. as al cryst 1, the only soluthick = /J the to for n The agreement shou ld be good if we wish to use beam ratios dow not be far from ve a com pute r ld belie shou We one. µ!yR l of e erica valu num a ting be to (iv) The resul tion appears 28 of kind rent diffe consuming (a from time 6 al. but et d Hou rwar 8.4 X io- NA given by solution would be strai ghtfo ensure that all to ssary nece be will tions itera ents. of urem meas large num ber - 1 or {3, it depends tions are periodic). Note that when we plot I' against A the solutions of the differential equa s value l idua indiv the on tly, sligh also, but /yR onµ only not is only an appr oxim ate ofµ and y. This is because Eq. (28) er harm onic s into the wave take extreme values not 4. Is it nece ssar y to inclu de high formula. However, as long as we do ld mak eµE 0 wou that ility mob the of e tion? valu l equa (say such a smal y said that only the ratio ded harm onic s in the comparable with Vapt ), it can be safel In the pres ent study, we have inclu valThe ts. resul ntal rime expe the rials equations but hing mate the matc of in ts ion µ/yR coun second pertu rbati on solut en so as to be near to the value assu mpti on that the e mad We tion. equa ues ofµ and r R are then21chos wave not into the 26 28 amp litud e of • the • ence ture. influ litera only the will in quoted the large signal effects d, tione men ts poin the all ver, that in ng howe ideri ve, belie At the end, after cons the fundamental component. We by Eq. (31) and appreciagiven carry set may s the e onic chos harm er ly high final we the /J = 1 region the 1 tal and crys 12 thin tion. the equa for wave the mvinto retf =0.9 5x1 0ble power so they should be included 1 12 crystal. r eff = 0.6 x 10- m v- for thick 00

r

Refregier et al. 55

1985 J. Appl. Phys., Vol. 58, No. 1, 1July

219

55

B. Other comments

1. Problem of the electro-optic coefficient There are two unfortunate facts concerning our value of the electro-optic coefficient: (i) it is below that gi~en in the literature derived from different kind of measurements, 29•30 (ii) it is different for the thick and the thin crystal. We can say very little about these discrepancies. One possibility is that our crystals were not perfect and the available amplification was reduced by some inhomogeneities (more inhomogeneities for the thick crystal). A second possibility is that some crystal properties (particularly birefringence) disregarded in the theory make the coupling between the signal and reference beams less effective and will thereby reduce the amplification (more rotation of polarization for the thick crystal hence less amplification). This second possibility is certainly susceptible to analysis but would of course involve adding further complexities to a theory already complicated. A third possibility is the effect of the unwanted parasitic gratings necessarily present. Their effect is no doubt deleterious but it is difficult to estimate the amount of reduction in beam coupling they might cause. Furthermore, a voltage drop across the electrodes is also possible and the constant field inside the crystal may be slightly smaller than the field externally applied. Finally we mention attenuation, which will, no doubt, be more noticeable for the thick crystal. When / 0 decreases along the crystal it means [see Eq. (16)] that the fringe velocity in some part of the crystal may be far away from the optimum, hence total amplification is bound to decrease. 2. Acceptor density, mobility, recombination constant When starting the matching procedure, we were under the impression that we would end up with a fairly wide band for the values of all the parameters. It turns out, and this result depends crucially on the assumption that the second perturbation calculations are valid at f3 = 103 for the thin crystal, that the measured difference in the values of r.PP for small and large signals will determine NA the density of acceptor atoms, with an accuracy of about ± l 0%. We believe that the ratioµ! R is determined from our measurements with about the same accuracy as that of NA, on the other hand, we can say very little about their individual values.

relatively low beam ratio used (/3 = 50). As shown in the present study, such a value considerably reduces the van. ation with A.

VI. CONCLUSION We have shown in this paper that large values of the exponential gain are obtainable when recording with a moving interference pattern in photorefractive BSO crystals (drift recording mode, diffusion field negligible). This technique has resulted in the efficient amplification ( X 103 ) of a low intensity signal beam. The solution ofKukhtarev's equations with a moving grating has shown that there is a resonance effect which at the optimum velocity makes the modulation of n, N t , and E, 0 higher. An optimum of the grating spacing A also exists from the condition EMEq = E"f,. [A 0 P 1 = (2TrEofNA )~µE/eyR]. Under these conditions, the photoinduced space charge field is TT/2 phase shifted with respect to the incident moving fringes, allowing an efficient beam coupling in two-wave mixing configuration. Measured values of the exponential gain F of the order of 8-12 cm- 1 are obtained with the following conditions: E0 = 10 kV cm- 1; A 0 P1 = 23 µm; incident beam ratio /3>: 103 ; A, = 568 nm. The dependence of the gain coefficient Fon the beam ratio f3 is interpreted by adding a second-order term in the Fourier expansion of the photoinduced space charge field. In accordance with the experiments, this theoretical approach shows a decrease of the gain of the photorefractive amplifier when the signal beam intensity increases f/3-" 1). In conclusion, a good agreement between the experimental measurements and the theory is obtained after a correct choice of crystal parameters. The adopted values for these parameters are discussed as well as the limitations of the theory for large signal effects. Finally, the agreement between the theory and experiment could certainly be further improved for any signal beam intensity and grating spacing with a numerical solution of the Kukhtarev equations with moving fringes.

r

3. Previous experiments with moving fringes When the experimental results with moving fringes in BSO were obtained by Huignard and Marrakchi 11 at A,= 514.5 nm, the solution for the refractive index modulation was obtained by taking the convolution of the space charge field with the temporal response of the crystal. This solution, in contrast to the present one, showed no strong dependence on A, nor did the experimental results measured in the rangeA = 3.5-9.5 µm. It is interesting to note that the present theory reduces to that ofHuignard and Marrakchi 11 under certain approximations as it was shown in Sec. III B 4. It is possible that the parameters of the crystal used in their experiments belonged to that range. More probably, the lack of variation with A found experimentally is caused by the 56

J. Appl. Phys .• Vol. 58, No. 1, 1 July 1985

1

P. Gunter, Phys. Rep. 92, 199 (1982). 0. L. Staebler and J. J. Amodei, J. Appl. Phys. 43, 1042 (1972). 3 J. Feinberg, D. Heimann, A. R. Tanguay, and R. L. Hellwarth, J. Appl. Phys. 51, 1297 (1980). 4 J. 0. White, M. Cronin-Golomb, B. Fischer, and A. Yariv, Appl. Phys. Lett. 40, 450 (1980). 'A. Krumins and P. Gunter, Appl. Phys.19, 153 (1970). 6 A. Marrakchi, J.P. Huignard, and P. Gunter, Appl. Phys. 24, 131 (1981). 7 A. M. Glass, A. M. Johnson, 0. H. Olson, W. Simpson, and A. A. Ballman, Appl. Phys. Lett. 44, 948 (1984). 8 M. B. Klein, Opt. Lett. 9, 350 (1984). 9 N. V. Kukhtarev, Sov. Techn. Phys. Lett. 2, 438 (1976); N. V. Kukhtarev, V. B. Markov, S. G. Odulov, M. S. Soskin, and V. L. Vinetskii, Ferroelectrics 22, 949, 961 (1979). 10 0. Rak, I. Ledoux, and J. P. Huignard, Optics Commun. 49, 302 (1984). 11 J. P. Huignard and A. Marrakchi, Optics Commun. 38, 249 (1981). 12 S. I. Stepanov, K. Kolikov, and M. Petrov, Optics Commun. 44, 19 (1982). 13 H. Rajbenbach, J. P. Huignard, and B. Loiseaux, Optics Commun. 48, 247 (1983). 14 Y. Ninomiya, J. Opt. Soc. Am. 63, 1124 (1973). 15 0. W. Vahey, J. Appl. Phys. 46, 3510 (1975). 16 G. C. Valley, J. Opt. Soc. Am. B 1, 868 (1984). 17 Ph. Refregier, L. Solymar, H. Rajbenbach, and J.P. Huignard, Electron. Lett. 20, 656 (1984). 2

Refregier et al.

220

56

"N. Kukhtarev, V. Markov, and S. Odulov, Optics Commun. 23, 338 (1977). '"Note that if instead of the quadratic recombination term R nN ,j in Eq. (6) we had taken a linear one (i.e., N ,j is regarded a constant as has often been done in the literature20 an optimum velocity would still exist but we would not find the optimum fringe spacing as obtained here. In that case (Im Es, )6 ~ •"" would monotonically decrease with fringe spacing in the drift dominated region. "'M. G. Moharam, T. K. Gaylord, R. Magnusson, and L. Young, J. Appl. Phys. 50, 5642 (1979). "G. C. Valley and M. B. Klein, Opt. Eng. 22, 704 (1983). "M. Peltier and F. Micheron, J. Appl. Phys. 48, 3683 (1977). "L. Young, W. K. Y. Wong, M. L. W. Thewall, and W. D. Cornish, Appl.

r

57

Phys. Lett. 24, 264 (1974). The assumption of no higher diffraction orders seems less justified in the present case because we permit (as demonstrated by the second perturbation) variation of the space charge field, at twice the fundamental period. Our main justification for neglecting the higher orders is that they were not observed in the experiments. "H. Kogelnik, Bell. Syst. Tech. J. 48, 2909 (1969). 26 0. C. Valley, Appl. Opt. 22, 3160 (1983). 27 J. P. Huignard, J.P. Herriau, G. Rivet, and P. Gunter, Opt. Lett. 5, 102 (1980). "S. L. Hou, R. B. Lauer, and R. E. Aldrich, J. Appl. Phys. 48, 3683 (1977). 29 R. E. Aldrich, S. L. Hou, andM. L. Harvill, J. Appl. Phys. 42, 493 (1971). 30 P. Pellat-Finet, Optics Commun. 50, 275 (1984). 24

Refregier et al.

J. Appl. Phys., Vol. 58, No. 1, 1 July 1985

221

57

OPTICA ACTA,

1985,

VOL.

32,

NO.

4, 397-408

Transient energy transfer during hologram formation in photorefractive crystals J.M. HEATON and L. SOLYMAR Holography Group, Department of Engineering Science, University of Oxford, Parks Road, Oxford OX1 3PJ, England (Received 3 October 1984) Abstract. The temporal and spatial variation of the beam intensities and space charge electric field during hologram formation in a photorefractive crystal are determined by solving the relevant partial differential equations. Starting with the equations of Kukhtarev, Markov and Odulov [1], analytic and numerical solutions are presented for a range of parameters for the time evolution of the energy transfer between the optical beams, the appearance of the space charge electric field, the phase angle between the interference pattern and the space charge field, and the bending of the grating.

l.

Introduction Photorefractive crystals have been widely studied during the past fifteen years for ' their applications in volume holography and two wave mixing [2]. Light induced changes of refractive index can be used to record highly selective and efficient realtime holograms in crystals including lithium niobate [1, 3], barium titanate, bismuth silicon oxide [ 4, 5, 6], bismuth germanium oxide [7], gallium arsenide (8, 9) and indium phosphide [9]. The formation of a volume hologram in a photorefractive crystal often shows strong transient effects during the initial stages of the recording process: the interaction between the light intensity fringes and the emerging refractive index grating leads to transient energy transfer between the recording beams which eventually tends to a steady-state equilibrium limit [1, 6, 7]. The recording mechanism involves the photoexcitation of electrons which migrate by diffusion or drift to regions of low light intensity where they are trapped. This process gradually sets up an intensity dependent space charge electric field which changes the refractive index via the electro-optic effect [3, 4]. It follows that the space charge field will not usually be in phase with the intensity fringes, and diffraction theory predicts that the energy transfer process depends strongly on the phase difference ( -

2

>-

1.5

u w

tb

"'

1.0

20

30

40

COUPLING STRENGTH Yi

Fig. 5. Reflectivity of a PR phase conjugate mirror versus coupling strength magnitude 1-y/I. The incident pump beams are of equal intensity (r = 1), the intensity of the incident probe beam is 20 percent of the total incident pumping intensity and the phase shift between index grating and the interference fringes is 5°. The fourwave mixing is via the transmission grating.

class of nonlinear parametric processes including four-wave mixing [57]. This solution does not require the finding of conservation laws for the decoupling of the equations. However, it is still only valid for local nonlinear susceptibilities. In the paragraphs below, we find solutions for a system with nonlocal susceptibilities, the photorefractive crystal. It should be pointed out that all of these analyses derive the intensities and not the phases of the various beams. The effects of

245

CRONIN-GOLOMB eta/.: FOUR-WAVE MIXING IN PHOTOREFRACTIVE MEDIA

strong nonlinearities on the phases of the output beams are not yet understood, and are the subject of current theoretical efforts. These effects will be important in considerations of the faithfulness of phase conjugation and the performance of resonators employing phase conjugate mirrors.

The Transmission Grating We develop here the exact solution of (2.9) for four-wave mixing in PR media by the transmission grating with negligible linear absorption [58]. The first step is to write down a set of conservation laws.

A1A2 +A3A4 =c1 =c

(3.la)

A 1A;'-AiA4 =c2

(3.1 b)

I1 + I4 =di

(3.lc)

I2+h=d2

(3.ld)

and D and E are constants of integration. At this point, the problem has been transformed from a set of nonlinear differential equations (2.9) to another set of equations [(3.1) and (3.4)] which may be solved by fitting boundary conditions appropriate to the particular device under consideration. We will first describe the application of this theory to the derivation of the reflectivity of a phase conjugate mirror with externally provided pumps. In this case, the amplitudes of all beams at their respective entrance faces are known. We observe first of all that the power flux /:i. = I 2(/) - Ii (O)I4 (0) is known so we need only solve for D, E, and c to finally obtain the phase conjugate reflectivity. The starting equations are the values of (3.4) at the boundaries z = 0 and z =l. The conservation Jaw (3.la) is also used to express the unknown field quantities A 1(1) and Aj(O) in terms of c, A 3(0), and known fields

where Ci = c, c 2 , di, and d 2 are constants of integration. These relations may be checked easily using the coupled wave equations. With the help of these conservation Jaws, (2.9a) and (2.9b) with zero absorption can be decoupled from (2.9c) and (2.9d). dA1 r * - - =- -[A1di -Ai(Ii +I2)+A2c] dz Io

(3.9a)

(3.9b)

(3.2a)

(3.9c)

(3.2b)

(3.9d)

dA3 r - - = - [A3d2 - A3(I3 + I4) + A!c] dz Io

(3.2c)

dA! r - - = - [A3c* - A!(I3 + I4) + A!d 1 ]. dz Io

(3.2d)

The procedure used to solve the equations is 1) Solve for E in terms of lcJ 2 using (3.9c).

s.. )i/2 ( ( s_ e,, 1 =

E= -

dz

=- -r

[c +(d 1

Io

(3.3a)

-2cT

Noting that I 0 is constant because of the conservation laws, we see that these equations are immediately integrable

S_De-µz - S.,.D- 1 eµz]

=~ [ 2c*(De-µz - D- 1 eµz

D= (

S_Ee-µz - S.,.E- eµz]

=[ 2c*(Ee-µz - E-1 eµz

(3.5)

Q=(f:i.2 +4lcl2)i/2

(3.6)

s± =::. ± Q µ

= rQ/(2Io)

)1/2

+4lcl 2 ITl 2h(O)I2(l) + 2lcJ 2I 4 (0)(1:i. 2 + 4Jcl 2) 1l 2 (T+ T*)

(3.4b)

!:i.=d2-d1

l:i. + (f:i.2 + 41cl2)1/2 + 2lcl2 /I2(/) l:i. - (l:i. 2 + 4lcl 2 ) 1' 2 + 21c1 2/I2(1)

eµ. 1

(3.12)

(lcl 2 - I1 (0) Iz(l)) l!:i.T + (l:i. 2 + 4JcJ 2 ) 112 J2

(3.4a)

where we define the following quantities

(3.11)

4) Solve for lcJ 2 using (3.9b), (3.11), and (3.12). We find that lcl 2 is given by the roots of the following equation:

1

A34(z)

(3.10)

where T =tanh µ/. 3) Solve for Din terms of lcl 2 using (3.9a).

(3.3b)

Aiz(z)

e,, 1•

2) Solve for pin terms of lcl 2 using (3.9d) and (3JO).

By eliminating the term in I 1 + I 2 between (3.2a) and (3.2b), and the term in I 3 + I 4 between (3.2c) and (3.2d), we find the following expressions for A 12 =A 1 /Ai and A 34 =A 3 /A!.

dA12

l:i. + (l:i.2 + 4Jcl2)1/2)1/2 l:i.- (f:i.2 + 4lcl2)1/2

=0.

(3.13)

5) The phase conjugate reflectivity is then given by the squared modulus of (3.11). When considering reflectivity as a function of the various input beam intensities, it is convenient to define the probe

ratio q:

(3.7) (3.8)

246

(3.14)

IEEE JOURNAL OF QUANTUM ELECTRONICS, VOL. QE-20, NO. I, JANUARY 1984

is

so that only two parameters are required to describe the input beams: the probe ratio q and the pump ratio r = 12 (!)/Ii (0). In terms of these parameters we have

Io Ii(O)= (r+l)(q+l) r/0 /z(I)= (r+l)(q+l) qlo

/4(0)= (q+ 1).

(3.15a)

(3.lSb)

,,------------------

-3

(3.15c)

1,/' ,,,,

-4

In Fig. 5, for example, we plot the reflectivity of a phase conjugate mirror as a function of the coupling strength Iril for the case where the phase shift between the grating and the interference fringes is 5 °. The intensities of the two pumping beams are equal (r = 1) and the probe intensity is 20 percent of the total pumping intensity (q = 0.2). The top of the graph corresponds to the reflectivity that would result if all the power of beam 2 were transferred to beam 3. This is the maximum reflectivity consistent with the conservation laws (3.1). The peaks in the curve correspond to the poles in the reflectivity of a phase conjugate mirror with no pump depletion (R = ltanh ( rl/2)1 2 for no phase shift between the grating and interference fringes). We have set the phase shift slightly nonzero to demonstrate the resultant damping of the oscillatory behavior. The peaks bend toward the right, probably since pump depletion causes high reflectivities to demand higher coupling strengths than are required for the same behavior in the undepleted phase conjugate mirror. The bending of the peaks can even be sufficient for bistability, as can be seen in the first two peaks of Fig. 5. In.Fig. 6, we show a contour plot of phase conjugate reflectivity for rl =-3, as a function of both pump and probe ratios. The first point to notice is the region of multistability, a direct result of the nonuniqueness of the solution of (3.13) for a certain range of parameters. Secondly, we observe that the reflectivity can remain finite as the pump ratio tends to infinity. This possibility for phase conjugation in the absence of pumping beam 2 has important consequences for the passive phase conjugate mirrors described below. The basic physical difference between the two solution surfaces of Fig. 6 lies in the relative phases of the two terms A 1 A! and AiA 3 in the interference factor (AiA! +AiA 3 ) which appears in the coupled wave equations (2.9). When the phase conjugate mirror is operating on the main surface, the one which extends over the entire q-r plane, the phase conjugate beam is generated so that the interference pattern formed between itself and beam 2 is in phase with the interference pattern formed between the forward going beams 1 and 4. On the secondary surface, these two terms are 11 out of phase with each other, so that both the grating strength and the reflectivity are diminished. It is often convenient to be able to examine the intensities of the various beams as a function of location z in the crystal. For example, in the next section of this paper, we will consider several devices whose boundary conditions are given by the ratios of intensities of pumping beams. These ratios appear in functions whose zeros must be found to reach a solution.

•

o/. /

,,..,.. -1/

/

/

-~~,~-'--+-'--~-'---!,~~~.,___'--~~--'~~

LOG PUMP RATIO

Fig. 6. Contour plot of phase conjugate reflectivity for -yl = -3 as a function of the pump ratio (/2 (1)//i (0)) and the probe ratio (14 (1)/ [/1 (0) + / 2 (1)] ). The transmission grating is operative.

Occasionally, since the ratio of two negative numbers is positive, these functions will indicate solutions with negative intensities which nevertheless satisfy the boundary conditions. It is important in checking for these spurious solutions to have expressions for beam intensities as a function of z. These may be derived by using the amplitude ratio functions A iz and A3 4 in the conservation laws (3.1). / 1 (z)=/12 (z)

dz - /34(Z) di I - Ii2(z)/34(z)

d 2 - fJ4(z) di 1 - f12(z) fJ4(Z) di - Ii2(z) dz 1 - /n(z) /34(z) di - liz(z)d2 1- 11z(z)/34(z) where I;j

(3.16a) (3.16b)

(3.16c) (3.16d)

=I;/11.

The Reflection Grating Our procedure for the solution of the coupled wave equation (2.18) for the reflection grating differs significantly from that for the transmission grating, mainly because the spatially averaged intensity Io is no longer conserved. We have so far only been able to demonstrate complete solutions for 11/2 and zero phase shift L)] sinh(ln q-

P

q)

+

ao 1 [ 1 + cosh(lnq) (a 12 - a 22) 112 n 1 + cosh(ln q - 2L) - 2.Yz

where

ao =

t

= 0,

J (14)

The authors thank Z. Bialynicka-Birula and M. R. Belie for critical reading of the manuscript. (1 - p2)[l10 - l4of p2

+ l2L -

2(12Ll40)112 References

X sinhL/p],

a 1=

t

[(!10 + l 4o/p 2)cosh 2L

+ l 2L -

1. See, for example, P. Gunter, Phys. Rep. 93, 199 (1982). 2. M. Cronin-Golomb, J. 0. White, B. Fischer, and A. Yariv, Opt. Lett. 7, 313 (1982). 3. M. Cronin-Golomb, J. 0. White, B. Fischer, and A. Yariv, IEEE J. Quantum Electron. QE-20, 12 (1984). 4. M. R. Belie, Phys. Rev. A 31, 3169 (1985). 5. M. R. Belie and M. Lax, Opt. Commun. 56, 197 (1985). 6. M. R. Belie, Opt. Lett. 12, 105 (1987). 7. N. V. Kukhtarev, T. I. Semenets, K. H. Ringhofer, and G. Tomberger, Appl. Phys. B 41, 259 (1986). 8. A. Bledowski and W. Krolikowski, "Anisotropic fourwave mixing in cubic photorefractive crystals," IEEE J. Quantum Electron. (to be published). 9. N. V. Kukhtarev, V. B. Markov, S. G. Odulov, M. S. Soskin, and V. L. Vinetski, Ferroelectrics 22, 949 (1979). 10. S. I. Stepanov and M. P. Petrov, Opt. Commun. 53, 292 (1985). 11. S. I. Stepanov, V. V. Kulikov, and M. P. Petrov, Opt Commun. 44, 19 (1982).

2(12Ll40) 112

X sinh ctNp],

a2 =

t

[-(!10

+ l40/.02)sinh 2L + 2(l2Ll40)112

X cosh L/p], p

+ l 2LJ2

= cos t? 1/cos t? 3,

- (al - a2)1/2 ' al+

q- - - a2

and "( = ')'/(cos t? 1 cos t?2) 112 is the effective coupling constant. For the symmetrical case, i.e., when p = 1, Eqs. (13) and (14) give

269

1047

a reprint from Journal of the Optical Society of America B

Properties of photoref:ractive gratings with complex coupling constants Ragini Saxena, Claire Gu, and Pochi Yeh Rockwell International Science Center, Thousand Oaks, California 91360

Received July 17, 1990; accepted November 29, 1990 We examine simultaneous write-read of dynamic photorefractive gratings for arbitrary phase shifts between index grating and intensity pattern. The nonlinear behavior of diffraction efficiency with read beam intensity, as well as the nonreciprocity of diffraction efficiency with respect to direction of readout, is examined for various values of the phase shift.

1. INTRODUCTION

the intensity pattern. Analytic expressions for the diffraction efficiency are obtained for special values of the phase shift between index change and fringe pattern. Note that analytic solutions exist for standard four-wave mixing with transmission gratings and complex photorefractive coupling constants. 12

Dynamic volume holograms can be efficiently recorded in photorefractive media at low levels of light without the need for any development techniques. The amplitude of such holograms is nonuniform 1 within the crystal because of two-beam coupling effects 2 between the pair of coherent writing beams. Further, the read beam and its diffracted beam induce a new photorefractive grating that may be in or out of phase with the original grating being read. 3 The diffraction is therefore quite different from that of a simple Bragg scattering, and the coupled-wave theory of Kogelnik4 for uniform gratings must be modified to describe the diffraction efficiency 11 of such gratings. A recent study 5 has examined 1) when there is polarization asymmetry during writing and readout. The new photorefractive grating formed by the read beam and its diffracted component is ignored for a reading beam that is weak compared with the writing beams. For this case, 11 is independent of the read beam intensity, and the argument of the sine function in 1) is shown to be a nonlinear function of grating thickness, coupling constants during writing and readout, and input intensity ratio of writing beams. Inclusion of the effects arising from a read beam's having an intensity comparable with that of the writing beams has shown 11 to be a nonlinear function of the read beam intensity and nonreciprocal with respect to readout from the two input ports. 6 Similar nonreciprocity with respect to direction of readout has been observed for fixed volume gratings in photorefractive media. 7 A theoretical explanation8 of this effect was obtained by taking into account the new photoinduced grating formed by the read beam and its diffracted component, besides the fixed grating present initially. The formulation in Ref. 6 was developed for a real photorefractive coupling constant in the coupled-wave equations. In this paper we solve the equations by a method that was previously used for studying forward four-wave mixing and vectorial two-beam coupling in photorefractive media in transmission geometry. 9-11 This method is valid for a complex coupling constant, and we establish the equivalence of the results from the two methods for real coupling constants. The nonlinear and nonreciprocal behavior of 1) is also examined for various phase shifts of the photorefractive grating from 07 40-3224/91/05104 7-06$05.00

2. THEORETICAL FORMULATION The basic interaction geometry is illustrated in Fig. 1. A 2 and B2 are two coherent laser beams of frequency w 2 that intersect within a photorefractive crystal of thickness L. The angle between the two beams is 28 2 in the medium. The photorefractive grating induced by this pair of beams is read by an incoherent third beam A 1 of frequency w1 that is incident at a different angle (8 1 in the medium) for Bragg matched readout. A fourth beam, Bi. is generated by the diffraction of the read beam off this grating. In contrast to phase conjugation by four-wave mixing in transmission geometry, 12 in which the third (read) beam is incident at the Bragg angle from the other side of the nonlinear medium and the phase-conjugate wave propagates opposite the probe beam, here all four beams propagate in the same forward direction in the nonlinear medium. We assume for simplicity that all four waves are plane waves of the same polarization and that the electric field amplitudes of the beams in one direction may be written as EAn(r, t) = %A.(x)exp[i(w.t - kAn • r)]

+ c.c.,

(1)

where An is the complex amplitude of the nth field at steady state, kAn is the corresponding wave vector in the medium, and subscript n = 1, 2. To describe the fields from the other input port, A is replaced by B. The x axis is taken normal to the surface of the medium, and the complex amplitudes are assumed to be functions of x that are due to energy transfer by beam coupling. The total intensity is the sum of the intensity of all four beams and the interference terms arising from the two pairs of coherent beams. The interference pattern induces a spatial modulation of the refractive index of the medium by means of the photorefractive effect 13 that has the form n = nb

+

~2[e-i = 0.

>-

u z

~=

0.8

UJ

u ""-

0

'. ''

...

0.6

•' ''

/

UJ

'

z

2

0.4

,.'

I-

u

'


-

'.'•"

/\

cc

'' '' ''

." '

~= 0

'' ''

'

' '

0.8

(.)

z

LU

u u::

u.

' 0 .6

LU

z 0

e:

>-

0.4

(.)

3.

_,

'

< a: u. u.

NUMERICAL RESULTS

0.2

'

'

Ci

For the general case of a nonzero spatial phase shift between photorefractive grating and fringe pattern, and a read beam intensity comparable with that of the writing beams, the parameter dependence of diffraction efficiency is studied numerically. Figure 2 shows the nonlinear behavior of 77 as a function of the read beam intensity h, (0)

;

'

"2 0

"1 0

10

' ''

'' ''

1,::

~

\!

"3 0

'

'

''

''

'' ·:

20

30

COUPLING STRENGTH y0 L

Fig. 4. Same as in Fig. 3 but for h,(0) = 0.01.

274

"'

40

Saxena et al.

Vol. 8, No. 5/May 1991/J. Opt. Soc. Am. B

yR

=

of parameters in this figure, the slant of the gratings is zero, leading to a Bragg-matched readout with 100% maximum diffraction efficiency. In order to have a better understanding of the nonreciprocal behavior shown in Fig. 3 when q, = Tr/2, we also plot the beam intensities and grating amplitudes as a function of distance in the nonlinear medium. Figure 5(a) shows the intensities of the four beams I.(x) versus x for 'YR = 'Yo= 20 cm- 1, and Fig. 5(b) shows the two grating amplk tudes IG1I = -vt;:;f;,/Io and IG2I = VIA,IB,/10 • Close to the input plane, the new grating G 1 generated by read beam A 1 and its diffracted component B1 is in phase with grating G 2 being read (see the discussion in Subsection 2.A), so that IB,(x) and IG1(x)I increase with x. After some distance, two-beam coupling between the writing beams transfers all the energy from beam A2 to beam B2 , so that the initial grating G2 that was being read is now zero. At this point in space, there is only one grating (Gi) left. The amplified beam B2 starts reading grating Gi, and the direction of energy transfer between the writing beams is now reversed, i.e., beam B2 starts losing energy and beam A2 gains energy. Since scattering of B2 by the grating G 1 is equivalent to readout from the other input port, grating G 1 and the restored grating G2 will be out of

20 cm - ', $ ~" I 2

1.6

-; ~

1.2

>-

':::

i3"'

IB l(X)

0.8

----------->-------------

r-

z

0.4

I

A I

IX)

------------------0.2

0.4

0.6

0.8

DISTANCE x (cm)

(a)

0.5 x

yR = 20 cm_,, $ ~" / 2

~ 0.4

1051

w Cl

::>

r-

_,

0.3

a.

L:

0.2

"':;;:

-;;

r-

-

1--

"'wz 0.6 DISTANCE

--- -----------·---------------IA l(X)

0.8

1--

0.8

'." 0.4

x (cm)

(b) Fig. 5. (a) Intensities J(x) of the four beams as a function of the depth x within the medium when cf> = 7r/2. The coupling constant 'YR is equal to 20 cm- 1, and the interaction length Lis equal to 1 cm. The input intensities of the four beams are IA, (0) = /A 2 (0) = IB 2 (0) = 1, JB (0) = 0. (b) Grating amplitudes IG1(x)I 2 and IG2(x)j as a function of the depth x within the medium. Grating G 1 is induced by the read beam A1 and its diffracted component Bi, while grating G 2 is written by the coherent pair of writing beams A 2 and B 2 • All the parameters are the same as in (a).

IB 1,_ (X) ______________ _ _____________

0 0

0.2

0.4

0.6

0.8

DISTANCE x (cm)

(a) 0.5

""c;x

and a read beam intensity equal to that of the writing beams. For equal-intensity writing beams, a change in the sign of the coupling constant is equivalent to readout from the other input port. Note that q, = 0 (t/J = Tr/2) gives oscillatory (exponential) behavior of T/ as a function of l'YIL, while q, = Tr/6 exhibits a combination of both behaviors, with decaying oscillations. The nonreciprocity is enhanced for q, = Tr/6, while the case of q, = 0 is symmetric with respect to readout from either input port, even when the read beam is as intense as the write beams. Figure 4 is a plot of T/ versus l'YIL for a weak reading beam [JA1(0) = 0.01]. The diffraction efficiency is now a symmetric function of the coupling constant for all values of t/J, and the oscillatory behavior for q, = 0 reaches a peak value of unity, as expected from Eq. (12). For the choice

0.4

yR = - 20 cm - ', $ = "1 2

w Cl

::>

0.3

1--

_,

a.

L: a and m > (1 - e-al)/(e-a l e-1' 1). Also note that g is an increasing function of m (i.e., og/am > 0) and g is an increasing function of l, provided that 'Y >a and

(27)

where M is an integer. In other words, oscillation can be achieved only when the cavity detuning can be compensat ed for by the photorefrac tive phase shift.

321

Vol. 2, No. 12/December 1985/J. Opt. Soc. Am. B

Pochi Yeh

The same oscillation frequency must also satisfy expres~ sion (32). Thus we obtain the following oscillation condi. ···· tion:

,,j=1

2\Aif;\

al- logR

fth.o scattered waves 1 and 3 will exponentially grow in time. This growth is referred to as oscillation. One should note that the absence or the presence of the oscillation in the system is determined only by the level-of nonlinearity f and by the pump beam ratio q but is not dependent on the level of seeds, e. Exponential growth of the scattered radiation takes place for arbitrarily small values of seeds, e, provided that the condition f > fth,o is fulfilled. Moreover, the seeds are needed only as the initial push. Indeed, consider the case in which the seeds for waves 1 and 3 are turned on during some time and then are turned off: e(t) = e for 0 < t < t0 and e(t) = 0 fort > to. The corresponding expressions for the Laplace transforms of waves 1 and 3 are obtained from Eqs. (6) by multiplication of the right-hand sides of these expressions by 1 - exp(-sto). This multiplicand has no poles and does not influence the oscillation conditions. To summarize, in oscillation-type systems the scattered radiation grows exponentially with time from an arbitrarily small level of seeds acting for an arbitrarily small period of time provided that the threshold of oscillation is exceeded. Exponential growth of these waves is saturated owing to the depletion of pumping waves 2 and 4. The characteristics of the stationary nonlinear state practically do not depend on the level of seeds except for the narrow nearel/3 . This is why the analytithreshold region If - fthl cal expressions describing this nonlinear state are usually obtained in the limit of zero seeds (€ = 0).

3. ONE-DIMENSIONAL MODEL

= (l

lnq

A,(l,r, t __. oo) = (3)

Physically, these seeds are due to the fanning of pumping beam 4 into scattered beam 1 and of beam 2 into beam 3. The parameter e +-

where N = 0, ±1, ±2, ... is the integer number. The contribution of each pole corresponds to the time dependence of the scattered waves proportional to exp(sNt). For f < fth,o the real parts of all the poles are negative (s' < 0), and the only nonvanishing contribution will be determined by the pole s = 0 in Eqs. (6):

Aa,; 0 (x, y, t) = VcA2,in,

v(x,y,t = 0) = 0.

'

q + 1[ fth,N = - - lnq q - 1

A 4 ,; 0 (x, y, t) = A 41°l,

A1,; 0 (X, y, t) = VcA4,in,

1

,. _ 27TN(f/fth,N) , In q TSN -

The system of equations (1) and (2) is to be supplemented by the boundary conditions, corresponding to the specification of input amplitudes of the pumping beams and of small seeding values for the amplitudes of the scattered beams at the boundaries of the interaction region: Az.in(X, y, t) = A 210l,

r

TSN = - - -

4.

2 where 12.4 = IA2.•l and l,r is the effective length of the beam overlap region in the x direction. Solving Eqs. (4) and (5), one obtains the following for the Laplace transforms of output amplitudes 1 and 3:

LINEAR TWO-DIMENSIONAL ANALYSIS

We turn now to the analysis of the linearized twodimensional set of equations (1)-(3), which takes into account finite diameters and noncollinearity of the inter-

362

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Vol. 8, No. 12/December 1991/J. Opt. Soc. Am. B

acting beams. Part of the beam overlap region, lying inside a nonlinear medium, in general has a hexagonal shape. We will analyze Eqs. (1)-(3) in the two most representative cases: (a) the beam overlap region lies fully inside a nonlinear medium (Fig. la), and (b) the beam overlap region is mostly outside the nonlinear medium (Fig. lb). For simplicity, we assume below that pumping beams 2 and 4 have the same diameters d and that the transverse distribution of their intensity does not depend on coordinates (i.e., IA2J 2, JA41 2= const). When the beam overlap region is fully inside the nonlinear medium (d/sin 8 < L), it has the shape of a rhombus. We introduce a new system of coordinates (x',y'):

x'

= -x

sin 8 - y cos 8,

y' = x sin 8 - y cos 8,

(10)

Equation (15) is easily solved by the Riemann method, 6 since the corresponding Riemann function for this equation, V(x',y',x",y"), is known: V(x' ' x" ") - I { r +TS) ,y, ,y - O d(l + q)(l

x [q(x" - x')(y" -y')]112 },

aAa/ay'

= -11A2fsin 28,

2

+

VeA

Aa(x', d/2, t) =

4,

(11)

v(x',y',t

2

y'

dy"V

(

d ) a (d ) x:J1',2' y" ay"f 2' y",s . (18)

= 0) = 0.

Using boundary conditions (16) in Eq. (18), one obtains with the help of relations (13) and (14) the following expressions for the Laplace transforms of the scatteredbeam amplitudes at the exit from the beam overlap region:

(12)

A 1(-d/2,y',s)

= VeA 4 F(y',q,s)/s,

A 3(x',-d/2,s) = VeA 2 F(x',q- 1,s)/s,

A 3(x',y',s) = f(x', y', s)A 4 * exp[- d(l +

q~(l + TS) (x' + qy')],

F(y,q,s) = exptl +

q~(l + TS) (x' + qy')].

(1 + q) (1 + TS)

x (14)

ax'ay'

rq - d 2(1 + q) 2(1 + Ts) 2 f =

-

(l 5 ) x

4

8

x exp[d(l +

TS)

z..)112]+ rq d (1 + q) (1 + TS)

(1

y )11

2

2- d

o

]

dz

q)~l +rs) (1 + qz)] J

2rv'QZ 10 [ (l+q)(l+rs).

(20)

The time dependence of the scattered radiation is determined by the inverse Laplace transformation of expressions (19). Formula (20) shows that these expressions do not contain poles with Re(s) > 0, which would correspond to the oscillational development of scattering. The only pole, s = 0, determines the evolution of scattered fields toward a stationary state. In other words, the DPCM is not an oscillator. This was first noted in Ref. 3. Making inverse Laplace transformations of Eqs. (19) and (20), one obtains

(q: + x') ],

(d2' y,, ) = - (Ve) fq (A --;- d(l + q) (1 + Ts) A.*

2

112-y1d

+ q)(l +

+

(].. -

q)~l +TS) [z + q(i - ~)]} J

rq (1

x exp[(l

o,

(~) (:.~)exp[d(l + q~(l +TS) x

dz expL +

rvqz

with the boundary conditions r(x»{•s) =

f

2 x Io [ (1 + q) (1 + rs)

Substituting expressions (13) and (14) for A 3 and A 1 , respectively, in the Laplace-transformed set of equations (2) and (11), one may reduce it to one partial derivative equation for the function f: 8 2f(x',y',s)

q)~l +TS)[ 1 + q(i-i)]}

2rvq

x Io[

, , _ d(l + q) (1 + TS) * 8 , , A1(x,y,s) - r A. --;f(x,y,s) q ay x exp[- d(l +

(19)

where the transfer function F is determined by the relation

(13)

where r = yd/sin 28 and q = IA2/A4 j 2 • The amplitude of the second beam, A 1 , is expressed through the function f by the following relation, which is a consequence of Eqs. (2) and (11):

a

a (x'y' " -d )f ( x" -,s d ) dx"-V ax" ' 'x ' 2 '2

2,

The first step in solving Eqs. (2) and (11) consists in Laplace-transforming these equations and boundary conditions (12) in time (t ..... s). Then, it is convenient to introduce a function f(x', y', s) proportional to the amplitude of beam A 3 :

ay,f

d1

d1

-

VeA

J J x'

and the boundary conditions are written as Ai(d/2, y', t) =

(17)

where Io is the modified Bessel function. The solution of Eq. (15) by the Riemann method makes it possible to express the function fat any point inside the beam overlap region through its values at the boundaries of this region:

where the axes are along the directions of propagation of pumping beams 2 and 4. In the (x',y') frame the beam overlap region is transformed into ad x d square, Eqs. (1) take the form aA 1/ax' = -vA 4 /sin 28,

2499

)

~~l + TS)(qy' + %)](16)

363

2500

J. Opt. Soc. Am. B/Vol. 8, No. 12/December 1991

the total illumination of the crystal, thereby diminishing the effective coupling constant. The coherent nature of amplification in the geometry of the DPCM, combined with the relatively high level of seeds in photorefractive crystals, results in low values of r; corresponding to the observable level of scattering. Thus, for equal-power pumps (q = 1) and for E = 10- 1, 10- 5, and 10-3, the total stationary reflectivity of 1% eorresponds to the value of the coupling coefficient r "" 4.2, 2.9, and 1.4, respectively. The value of seeding, E, that is used in our formulas is a phenomenological parameter and, in general, should depend not only on the crystal but also on the pumping beam distribution, since it determines the relative value of scat. tering that contributes to the growth of a given scattering mode. Besides, the value of scattering itself varies from crystal to crystal. In numerical examples in Section 5 we will use the values of seeding of the order of 10- 5-10- 3, but the effects to be discussed are not sensitive to this particular level of seeding and will occur for any value of E. The dynamics of nonlinear reflectivity R(t) for equal. power pumps (q = 1), calculated with the help of Eqs. (21) and (22), is presented in Fig. 2 as the solid curve. The parameters for this curve are r = 1.76 and E = 10- 3 (total stationary reflectivity of 5%). The dashed curve is the reflectivity given by the one-dimensional nonlinear model. Here r = 2.033, which corresponds to the 5% stationary reflectivity in the framework of this model, and the initial and boundary conditions are Aun = A3.in = 0 and v(x, t = O) = 0.033 [this starting value of v ensures the same initial level of scattering R(O) = 10- 3 as in the twodimensional model]. The characteristic time for reaching a stationary state, as predicted by the one-dimensional model, considerably exceeds that from the two· dimensional one for low-reflectivity regimes. The transverse distribution of the scattered output amplitude A, in a stationary state for r = 1.76 and q = 1, as given by formulas (21) and (22), is presented in Fig. 3. It is inhomogeneous and grows across the diameters of the beams. The physical reason for such behavior is transport of energy out of the interaction region by scattered beams 1 and 3 in the direction of the common component of their group velocities (from bottom to top of Fig. 1). It is this flow of energy out of the interaction region that inhibits oscillation and turns the scattering into a convective instability. For more realistic Gaussian-type pump-

A,(-d/2, y; t) = VEA. cf>(y', q, t), 1

A 3(x; -d/2, t) = VEA2 cf>(x; q- , t),

(21)

where cf>(y,q,t) = 1

+ (1 +

T(t)

q)

J

[

dz exp

0

(1

xIo Hq(+-~ )]1'}1(z)

+ q)z 2 ] 4f (22)

and T(t) = [4rt/(1 + q)r]1 12 • The stationary transverse distributions of the output scattered beams follow from Eqs. (21) and (22) for t = oo. Formulas (21) and (22) clearly demonstrate a qualitative difference between the predictions of one- and twodimensional models. Indeed, integral (22) is convergent for any values of nonlinearity r. This means that after some transient period the scattered radiation will reach its stationary level; that is, it behaves like ordinary fanning or convective stimulated Brillouin scattering (SBS). Had we used the one-dimensional model, we would have obtained exponential growth of the scattered waves provided that the value of the nonlinear coupling coefficient exceeds its threshold value. Formulas (21) and (22) show that this threshold of oscillation does not exist in the two-dimensional description. One may introduce only the notion of observable threshold of scattering, as is done in SBS. It is defined as the value of nonlinearity r corresponding to some small observable level of scattering (say 1%). This threshold is dependent on the value of the seeds, E. In SBS the level of the seeds is small, of the order of 10- 9 , so the threshold value of nonlinearity is Gll "" 20. 7 In photorefractive crystals the relative level of the seeds is higher and the threshold value of nonlinearity r is of the order of several units. Besides providing information about the time and space dependences of the scattered amplitudes, formulas (21) and (22) also permit one to calculate the integral characteristics of scattering, such as the total nonlinear ref!ectivities of pumping beam A.: Rz(t)

= ([A~ 2d)

r:: dylA1(-%•y',t)j2 2

(23)

and analogously R 4 • Formulas (21) and (22) can also describe the dynamics of fanning. Indeed, fanriing is the amplification of weak beam A 1 at the expense of pump A. in the absence of second pump A2 and is obtained from Eqs. (21) and (22) for q = 0. The difference between the coherent scattering in the FWM geometry of DPCM and the fanning is manifested in different levels of amplification of the seeds. Thus, for equal-power pumps (q = 1), the evaluation of integrals (21) and (22) gives the following expression for the stationary nonlinear reflectivity (valid for r » 1): R 2 = R 4 = R "" (e/47Tf 2 )exp(4f);

R,% 5

(24)

whereas, for the fanning one obtains Rr.n = E exp(2f}. The level of fanning of pumping wave A. in the presence of waveA 2 is still lower. Thus, for q = 1, Rran = E exp([}. The reason is that the second pump wave does not act coherently here with the first one and contributes only to

0

--50

100 t/r

Fig. 2. Dynamics of nonlinear reflectivity as given by the twodimensional (solid curve) and one-dimensional (dashed curve) models.

364

EJiseev et al.

Vol. 8, No. 12/December 1991/J. Opt. Soc. Am. B

exp(k 1

-

2501

k 2 ) = [k 2 + pL tan O - f/(1 + q)]

x [k1 + pL tan 0 - f/(1 + q)]" 1.

(27)

For p = 0 Eq. (27) reduces to the relation r

- 0.5

0.5

0

y'/d

ing beams this flow of energy results in the shift of the centers of the output scattered beams from those of the pumping beams (see Figs. 4-7 below). The statement that the DPCM is not an oscillator, illustrated above for the case in which the beam overlap region lies fully inside the nonlinear medium (d/sin 8 < L) is general and can be proved for any relation among the values of d, 8, and L, but the corresponding formulas are cumbersome in the general case. Simple analytical formulas giving an additional insight into the behavior of the DPCM can be obtained in the second limiting case, shown in Fig. lb, in which the beam overlap region is mostly outside the nonlinear medium (d/sin 8 » L). Turning to the analysis of this limiting case, we will not analyze general time-dependent solutions of Eqs. (1)-(3). Since we already know that the DPCM is not an oscillator but a convective amplifier of input seed signals, we will look only for stationary solutions of these equations (a/iit = 0). A convenient way to obtain these solutions in the limit d/sin 8 >> L is to use a Laplace transformation in the coordinate y in Eqs. (1) and (2). Solving for the Laplace transforms of the scattered beams at the exit from the interaction region, one obtains A 1(L/2,p)

p* = (f - r,hlfth(f,h - 2)" 1[(1 - q)/(l

For equal pumps (q = 1) this reduces to p* = (3/2)(f - 2)/L tan 0.

A1(L/2,y)/A2 = Vc(l + q) 2q- 1(f - r,h) x {exp[p*(y

(25)

R., R 4 = exp(2p*d). For large enough diameters of the beams (d >> d°') exponential growth of scattered radiation along the coordinate across beams is soon saturated owing to the depletion of pumping beams 2 and 4, and the transverse intensity distributions of the scattered beams for the rest of the transverse coordinate follow those of the pumps. For d >> d., the DPCM may be described within the framework of a one-dimensional model. A rough estimate for the critical diameter of the interacting beams, d"' can be derived from the relation

(26)

with r = yL/cos 8 and 2

1-q

4

2

1-q

1/2

- fp tan 0

+

(pL tan 0) 2 ]

(30)

For r > r,h scattered radiation exponentially grows along the nonlinear medium boundary. This exponential growth was first discussed in Ref. 4 amid a consideration of SBS in a laser-produced plasma in a geometry similar to that of DPCM. The physical reason for such behavior was discussed above. It is again transport of energy out of the interaction region by scattered beams 1and3 in the direction of the common component of their group velocities. As in the previous case this flow of energy results in the shift of the centers of the output scattered beams from those of the pumping beams (see Figs. 4-7). For small enough diameters of incident beams 2 and 4 their reflectivities are exponentially dependent on their diameters:

- kz) + [fq/(1 + q)][exp(-k2) - exp(-k 1)]}{[k 1 + pL tan () - f/(l + q)] x exp(-k 2 ) - [k 2 + pL tan()

.!.r 1 + q:t [.!.r 2 ( 1 + q)

+ d/2)] - l},

A 3 (-L/2,y)/A 4 = A 1(L/2,y)/A ..

C(p,q) = {(k 1

=

(29)

The inverse Laplace transformation of Jr - r,hJ .. cos 8,

(4)

where kB is Boltzmann's constant. For values of /.LTR > e Ag 2/47r 2kBT, expressions (2)-(4) can be combined to yield

where 8 is the Bragg angle inside the crystal. Thus, from a measurement of r, we can directly determine the space-charge electric field. In the case of GaAs, our knowledge of all required material parameters (determined by independent measurement) is complete enough for us to compare calculated values of r (as a function of grating period) with those determined by direct measurement. In our two-wave mixing experiments, we used a sample of undoped GaAs (grown by the liquid-encapsulated Gochralski technique) with a dark resistivity p = 6.3 X 107 Q cm and an electron Hall mobility µ, = 5800 cm2/V sec. The concentration of ionized EL2 donors was obtained from measurements of conductivity as a function of temperature; we find that NtLz = 1.4 X 1015 /cc at room temperature. 14 The concentration of neutral (filled) EL2, as determined from the measured

(5)

In this limit, the response time becomes independent of J.LTR and strongly dependent on Ag. Using expression (5) with Ag= 1µm,f=12.9, a= 1.2 cm- 1 , and Io = 4 W/cm2 , we calculate T = 28 µsec. This response time is much larger than the dielectric relaxation time (7 di= 1.9 nsec), thus illustrating the importance of the correction factor f(Ag, 0). Note, however, that shorter response times are predicted for larger values of irradiance or grating period. The technique that we have used to study the photorefractive effect in GaAs is two-wave mixing, or beam coupling. 12 In this technique, two beams are incident upon a sample, thereby producing a spatially periodic irradiance pattern. If the resultant refractive-index

386

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OPTICS LETTERS I Vol. 9, No. 8 I August 1984

efficiency were of the order of 0.1 % and were consistent with our measured values of signal-beam gain. Finally, we have studied the transient response of the signal-beam gain by chopping the reference beam with an electro-optic switch and monitoring the time response of the transmitted signal beam by using a fast photodiode. When the switch is opened, the signalbeam intensity increases to a new steady state with a response time (for small values of I'L) that is comparable in magnitude with the grating response time given in Eq. (2). When the switch is closed, the signal decays to its original level with a response time limited only by the detection circuitry. In our experiment (lo = 4 W/cm 2, Ag = 1 µm), we measure a rise time of 20 µsec, in reasonable agreement with the value (28 µsec) calculated earlier by using Eq. (5). In summary, GaAs (and other III-V materials) shows great promise for a number of DFWM applications at wavelengths in the near infrared, especially those requiring high speed or sensitivity. Large steady-state values of weak-beam gain and DFWM reflectivity should also be obtainable by using an applied electric field and techniques to induce a spatial phase shift in the space-charge field pattern.

0.4

0.5

1,0

1.5

2.0

GRATING PERIOD Ag•

3.0

;im

Fig. 3. Two-wave gain coefficient versus grating spacing in undoped GaAs.

absorption coefficient and the photoiomzat10n cross section, 13 was found to be NEL 2 = 1.0 X 1016/cc. The crystal was oriented as shown in Fig. 2. The cross section of the crystal was 6 mm X 5 mm, and the thickness Lwas4mm. The laser source for our experiments was a cw Nd: YAG laser operating in the fundamental transverse mode and having a coherence length of ~ 1 cm. The output beam from the laser was divided at a beam splitter, and the two resulting beams were recombined at the sample in such a way that the input angle could be varied while the path lengths of the two beams were kept equal. Both input beams were s polarized [i.e., along (110)) to exploit the refractive-inde x change induced through the electro-optic tensor component r41· 9 The signal beam had an intensity of 10 mW and a diameter of ~2 mm. The reference beam was expanded with a telescope to a diameter of ~5 mm in order to maintain a uniform interaction region over the length of the crystal as the angle of incidence is varied. The intensity of this beam was made sufficiently large [IR(O) ""10/8 (0)) to ensure that it would remain undepleted in our experiments. A typical plot of signal-beam gain versus grating period is shown in Fig. 3. The curve peaks at a value of Ag equal to the Debye screening length Lr = 11'fkBT/e 2N"/i,L2•12 Thus, from the measured value of Lr, we can determine the value of N"ftL2· We find that N"ii,L2 = 1.3 X 1015, in good agreement with the value determined from conductivity measurements. The measured values of gain are close to those measured at 5145 nm in BSO (Ref. 9) (with no applied field). When we account for the difference in the measurement wavelength, we find that the figure of merit n 3r 4 1 for GaAs is a factor of 2 larger than the effective value for BSO. We have also made a direct measurement of grating efficiency using DFWM. A third input beam was generated with a beam splitter and was incident upon the crystal in a direction counterpropag ating the strong forward wave. The intensity of this backward beam was reduced to avoid depletion of either forward wave. In this situation, the ratio of diffracted to undiffracted backward wave is a direct measure of the grating diffraction efficiency. Our measured values of diffraction

I would like to acknowledge helpful discussions with G.C. Valley, A. T. Hunter,A. M. Glass,andR.A . Mullen and technical assistance by R. H. Sipman. The GaAs samples were provided by H. Kimura, C. Afable, and H. M. Olsen. J.P. Baukus provided the EL2 absorption curve. I am grateful to G. C. Valley for his critical reading of this manuscript.

References 1. A. M. Glass, in Technical Digest of Conference on Lasers and Electro-Optics (Optical Society of America, Washington, D.C., 1983), paper WhI. 2. A. M. Glass, A. M. Johnson, D. H. Olson, W. Simpson, and A. A. Ballman, Appl. Phys. Lett. (to be published). 3. M. Sugie and K. Tada, Jpn. J. Appl. Phys. Hi, 421 (1976). 4. M. Kaminska, M. Skowronski, J. Lagowski, J. M. Parsey, and H. C. Gatos, Appl. Phys. Lett. 43, 302 (1983). 5. A. M. Glass, Opt. Eng. 17, 470 (1978). 6. G. C. Valley and M. B. Klein, Opt. Eng. 22, 704 (1983). 7. J. Feinberg and R. W. Hellwarth, Opt. Lett. 5, 519 (1980). 8. B. Fischer, M. Cronin-Golomb, J. 0. White, A. Yariv, and R. Neurgaonkar, Appl. Phys. Lett. 40, 863 (1982). 9. A. Marrakchi, J.P. Huignard, and P. Gunter, Appl. Phys. 24, 131 (1981). 10. H. Rajbenbach, J.P. Huignard, and B. Loiseaux, Opt. Commun. 48, 247 (1983). 11. N. V. Kukhtarev, Sov. Tech. Phys. Lett. 2, 438 (1976). 12. N. V. Kukhtarev, V. B. Markov, S. G. Odulov, M. S. Soskin, and V. L. Vinetskii, Ferroelectrics 22, 949, 961 (1979). 13. G. M. Martin, Appl. Phys. Lett. 39, 747 (1981). 14. A. T. Hunter and R. Baron, Hughes Research Laboratories, Malibu, California 90265 (personal communication).

387

Reprinted from Journal of the Optical Society of Amcr!ca B, Vol. 3, ~a~e 29:l, February_ 1986 . Copyright© 1986 by the Optical Society of America and reprmted by perm1ss10n of the copyright o"ner.

Photorefractive effect in BaTi03: microscopic origins M. B. Klein and R. N. Schwartz Hughes Research Laboratories. 3011 Malibu Canyon Road. Malibu, California 90265 Received September 16, 1985; accepted October 11, 1985 We have used a number of experimental techniques to identify the photorefractive species in commercial. sample~ of BaTi0 3. We find that Fe impurities (in the Fe'+ and Fe3+ stat~s) are t~e predommant photorefract1ve species. Techniques for optimizing the photorefractive properties of BaT10s are discussed.

1.

INTRODUCTION

had been mechanically poled to eliminate 90° domains. We then electrically poled these samples to control 180° domains. The technique consisted of heating the crystals in oil to 120-130°C, applying a field of a few kilovolts per centimeter and cooling to room temperature. Without further characterization, it is difficult to determine the effectiveness of our electrical-poling procedure. If 180° domains remain in a sample (and they are uniformly distributed), their effect is to reduce all second-order parameters of the crystal in proportion to the number of remaining rno 0 domains. The parameters affected by incomplete poling include the pyroelectric, piezoelectric, electro-optic, and nonlinear optic coefficients. One objective of our measurements is to determine the degree of poling in our samples. The identity and concentration of impurities were determined by both secondary ion mass spectroscopy (SIMS) and spark-source emission spectroscopy (Burgess Analytical Labs, North Adams, Mass.).

Photorefractive materials offer great promise for applications in optical data processing and phase conjugation using degenerate four-wave mixing (DFWM). BaTiOs is a particularly promising material, primarily because th~ very large value of the electro-optic tensor component r42 yields correspondingly large values of grating efficiency, beam-co~pling gain, and four-wave mixing reflec.tivi~y. This permits _the realization of a variety of new apphcat10ns, such as real-time convolution and correlation, edge enhancement, image transmission through fibers, self-pumped phase-conjugate mirrors, and phase-conjugate resonators. In spite of the intense interest in the applications of BaTi03, little is known regarding the species responsible ~or the photorefractive effect in this material. If this species can be identified, we can then hope to optimize the performance of this material through doping and heat treatment. The work described in this paper follows the spirit of the early researchl-3 on the origins of the photorefractive effect in LiNb0 3 as well as a subsequent study of photorefractive centers in Fe-doped LiNb0 3.4 Common elements between this work and ours are the judicious use of evidence from the literature as well as detailed electron paramagnetic resonance (EPR) and optical absorption measurements. In addition, our approach adds another diagnostic technique: photorefractive beam coupling. The preliminary results of our study are given in Ref. 5. The general arrangement of this paper is as follows. In Section 2 the experimental methods are outlined. Section 3 is devoted to developing the general background and models with special emphasis on deep-level defects and electrical transport. The experimental results obtained from beamcoupling measurements, absorption spectra, impurity analysis and EPR measurements are discussed in Section 4. In Se~tion 5 we discuss the interpretation of the data, with attention given to the study of correlations among the measured parameters, and the relationship to existing energylevel calculations. A general discussion of the results and a recommendation for material optimization are given in Section 6.

B. Beam-Coupling Measurements Steady-state beam coupling is a convenient technique for measuring several important parameters in a photorefractive crystal. Our beam-coupling measurements in BaTi03 and their interpretation are described in detail in Ref. 6. We have measured the amplitude and the sign of the beam-coupling gain as a function of grating period in nine BaTi03 samples. The laser source was a cw He-Cd laser operating in the fundamental transverse mode and having a coherence length of ~5 cm. The output beam from the laser was divided at a beam splitter, and the two resulting beams were recombined at the sample in such a way that the input angle and thus the grating period could be varied while the path lengths of the two beams were kept equal. In all our experiments, the grating normal was aligned parallel with the c axis. Both input beams were s polarized (along a crystalline a axis) to exploit the refractive-index change induced through the electro-optic tensor component r13. The signal beam had a power of ~ 10 µ W and a diameter of ~2 mm. The reference beam was expanded with a telescope to a diameter of ~5 mm in order to maintain a uniform interaction region over the length of the crystal as the input angle was varied. The power in this beam was made sufficiently large (~5 mW) to ensure that it would remain undepleted in our experiments. At the resulting level of irradiance (~20 mW /cm 2 ), the photoconductivity is approximately 3 orders of magnitude larger than the dark conductivity, as measured in two of our crystals.

2. EXPERIMENTAL METHODS A. Samples The samples for all our experiments were purchased from Sanders Associates, Nashua, N.H. The crystals as supplied 07 40-3224/86/020293-13$02.00

© 1986 Optical Society of America

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J. Opt. Soc. Am. B/Vol. 3, No. 2/February 1986

C. Electron Paramagnetic Resonance Measurements EPR spectra were recorded at X band with a Varian E-9 spectrometer equipped with a Hewlett-Packard 5342A automatic microwave frequency counter and a Varian nuclear magnetic resonance gaussmeter for magnetic-field measurements. Single-crystal spectra were made by aligning the sample (with faces normal to principal directions) on a flat surface on the shaft of a single circle goniometer. The error in the alignment of the crystal axes relative to the axes of rotation is estimated to be ±3°, whereas the estimated error in the crystal orientation relative to the magnetic field is ± 1° and is determined by the resolution of the goniometer dial. The concentration of spins (e.g., Fe3 +) was obtained by comparison with a calibrated reference sample of powdered Cr 20 3 (Cr3+; g = 1.9796), which was contained in a capillary tube mounted next to the BaTi03 sample. The Cr sample was calibrated against a Varian 3.3 X 10-4 % pitch on KC! standard. The accuracy in the determination of the absolute number of spins (e.g., Fe3+) in the BaTi03 sample is ±50%; the relative accuracy is ±10%. D. Optical Absorption Measurements The absolute absorption coefficient a at specific wavelengths was obtained from laser transmission measurements, with account taken of losses due to Fresnel reflections from the entrance and the exit faces. For each measurement, care was taken to avoid interference effects from multiple reflections within the sample. The crystals were measured with a beam propagating normal to the c axis and with the optical electric-field vector E either parallel or perpendicular to the caxis. For this geometry, photorefractive coupling between the incident and the reflected beams in the crystal can be neglected, since the effective electrooptic coefficient for this interaction is small. The measured values of the absorption coefficient include a contribution from scattering that is thought to be at most 0.1 cm-1 in our crystals. Broadband transmission measurements were made using a Cary 14 spectrophotometer.

3. GENERAL BACKGROUND AND MODELS A. Energy-Level Model The energy levels and notation that we use for our ensuing discussion are shown in Fig. 1. We assume that a single species X, which can exist in two valence states (X and X+), is responsible for the energy states in the BaTi03 band gap. Jn the case of Fe-doped LiNb03 (LiNb03:Fe), X corre-

1

' J_

PHOTOCARRl ERS

CHARGE TRANSPORT (DIFFUSION AND DRIFT)

z SPACE CHARGE

z SPACE CHARGE FIELD, REFRACTIVE • INDEX CHANGE Z WITH PHASE SHIFT

Fig. 2. Gratings in a photorefractive material. The periodic irradiance pattern results from the interference of two waves in the material.

sponds to Fe2+ and x+ corresponds to Fe3+. We designate the concentration of X as N and the concentration of x+ as N+. Other states, which are optically inactive, provide overall charge compensation within the crystal (see Subsection 3.D). It is important to note that electrons or holes (or both) can contribute to the charge transport in our BaTi03 crystals. For electron transport, state X is a donor, or filled, state, and state x+ is an ionized donor, or empty, state. For hole transport, state X + is an acceptor or f"illed state, and state Xis an ionized acceptor or empty state. We further note that the sign of the space-charge field is opposite for the two charge carriers. This changes the direction of the beam coupling and allows us to measure the dominant photocarrier. In association with the energy-level notation given above, we must also define rate coefficients for levels X and x+ and transport coefficients for free electrons and holes. The cross section for photoionization oflevel X (thus creating an electron) is s., whereas the corresponding cross section for level x+ (thus creating a hole) is sh. The mobility for electrons (or holes) isµ, (or Ph). The recombination rate coefficient for electrons at centers X + is 1'., and the corresponding coefficient for holes at centers X is 1'h·

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) 21.5'(1111), 34.7'(_-6.),425' C0L and 60.0' (9). For low-intensity levels, th; two-beam couphng efficiency 1s decreased from its high-intensity value due to the nonzero dark erasure rate of the crystal.

number of different beam crossing angles in nominally undoped SBN sample C. The gain coefficient r saturates at large optical power, and is reduced at small optical power by 10 the fi.nite dark cond.uctivity [ ud = 0.7X 10- (!l cm)-'] ofth1s sample. (This dark conductivity was obtained from transient two-wave mixing measurements, described below.) B. Optimal grating spacing for two-wave mixing in SBN Combining Eqs. (1 )-(6), the gain coefficientr can be written in the form r = [A sin 8 /(1

(cos W;lcos 81 ),

(8)

A. Intensity dependent two-wave mixing Figure 5 shows the dependence of the steady-state twowave mixing gain coefficient r on total optical intensity for a 377

+ B - 2 sin2 8)]

where 8 is the external half angle and 8; the internal half angle between th_e two incident laser beams. [Over the range of external crossing angles 2() used here (0 < 2() < 60°), the internal crossing angle 28; was always less than 25°, and the factor (cos W;/cos 8;) in Eq. (8) varied by less than 7% from unity.] In Eq. ( 8) we have assumed that the hole-electron competition factor47 •48 t(K) is constant with Kin order to simplify the data analysis. Figure 6 shows the mea,sured two-wave mixing gain coefficient r as a function of crossing angle of the optical beams [or grating spacing, i.e., see Eq. ( 1)] in various Cedoped and undoped SBN samples at a A.= 514.5 nm. The gain increases linearly with 8 for small crossing angles, reaching a maximum at 8 = 8peak• and then decreases for larger crossing angles, as predicted by Eq. (8). All data points were taken at sufficiently large optical intensity such that the gain was independent of intensity. The solid curves are a best fit to Eq. ( 8), and yield values for the parameters A and B, which relate to the photorefractive properties as follows. The parameter A is proportional to the effective Pockels coefficient r.tr: Ewbank et al.

J. Appl. Phys., Vol. 62, No. 2, 15 July 1987

410

377

Ag(µm )

1.5

1.0

0.8

0.7

larger, as expected, due to its larg er electro-optic coefficient. Not e also that electron and hole competition47•48 appears to be important, as seen by the vary ing prod uct reff t(K) in the various samples. The fitted valu es for reffs (K) and Neff• corresponding to each curve in Fig. 6, are listed in Table II. The fact that cerium doping does not cause a systematic change in r etrs (K) implies that the pres ence of cerium primarily alters the photorefractive char ge density Neff and not the competition of electrons and hole s.

0.6

10

c. Wavelength dependence of the two-wa ve mixing gain in undoped SBN 10

20

30

40

50

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60

FIG. 6. The two-wave mixing gain coefficient r as a function offull external crossing angle 28 in nominally undoped SBN:61 samples A-C, ceriumdoped SBN:61 samples D-F, and cerium-doped SBN:75 sample G. The solid curves are best fits to the expre ssion in Eq. ( 8). Note that ceriµ m doping enhances the photorefractive coup ling efficiency.

(9) and is determined by the slope of the plot of r vs 28 near (} = 0. The para met er B is related to the effective photorefractive charge density Neff:

B =A.Ko = 41T

~ ~-;:;;sin(} 41T \j EE,jcsT

eak

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10)

P

and is determined by 28peak, the crossing angle at which the gain r(W ) reaches its max imu m value. Comparing the curves for the undoped and doped SBN:61 samples in Fig. 6, it is appa rent that cerium doping the SBN:61 samples causes an increase in the peak-gain crossing angle (}pea k. Acc ordi ng to Eq. ( 10), this indicates that cerium doping increases the effective density of photorefractive charges Neff• In addition , the slope of each curve near the origin is rela ted to the prod uct reff t(K) [see Eq. {9)] . The initial slope of the SBN : 75 sample is noticeably

Figure 7 shows the wavelength dependence of r vs Ag- 1 for nominally undoped SBN:61 sample C. Here, the solid curves are the best fit of the data to Eq. ( 8) with the (cos 28;/ cos 8;) factor taken to be unity and the beam crossing angle 28 converted to inverse grating spacing Ag- 1 via Eq. (I). The gain of the nom inally undoped SBN increases at shor ter wavelengths and approaches that of cerium-doped SBN, indicating an incr ease in the effective number of photorefractive charges Neff in this sample at shor t wavelengths. Figu re 8 compares the wavelength dependence of the effective num ber of photoref ractive charges Neff with the absorption coefficient a for SBN sample Cov er the visible wavelength range. In this SBN sample, Neff is approximately a linearly decreasing func tion of wavelength: Neff = [ - 0.0139 XA (nm ) + 8.4] X 10 16 cm- 3 • In contrast, the wavelength depe ndence of Neff in BaT i0 was -A. - 2 in one sample27 and 3 Neff was the same at 457.9 and 514.5 nm in anot her sample. 28

D- Sign of the effective photore frac tive charge carriers inSB N The sign of the dom inan t photoref ractive charge carrier in SBN was dete nnin ed by com paring the direction of twobeam coupling to the direction of the positive c axis of the crystal. The c-axis direction was verified experimentally, subsequent to poling, by observin g the sign ofa compressionally induced piezoelectric volta ge. Comparing this piezoelectric voltage to the direction of beam coupling in each SBN sample indicated that the sign of the dom inan t photorefrac-

TABLE U. Identification (note that these same ID's are used in Figs. 3, 6, and 9), compositio Sr,B a 1 _xN b20 • The characteriz n x, dopant, and thickness L 6 ation parameters, including the for seven samples of effective photorefractive charge coefficient r,, and the hole/elect density N,,, the product of the ron competition factor i:'CKJ, the effective electro-optic grating formation rate per unit inten ·ation-time prod uctµr R, and the sity, the dark conductivity ad, the diffusion length Ld were obtained mobility/recombinfrom the data in Figs. 6 and 9 at A.= 514.5 nm using Eqs. (9)-( 12). x L N"'" (Xl0 16) r,,t( K) ID Rate /int. (%) Dop. (mm ) adcx 10- 10 J (cm- 3) µTR ( X 10-lO) (pm/ V) (cm 2/Ws ) LJI cm)- 1 (cm 2/V) (A) A 61 5.3 0.7 170 B 0.27 61 7.1 0.92 1.3 4.2 90 330 c 0.17 61 5.5 2.65 I.I 1.2 120 170 D 0.35 61 Ce 1.7 0.69 2:5 5.6 150 380 E 1.85 61 Ce 4.2 0.30 3.4 3.5 120 300 F 1.05 61 Ce 4.9 0.23 1.8 2.9 210 270 G 1.67 75 Ce 6.0 0.20 0.9 2.4 280 250 0.10 0.14 1.7 210

en

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J. Appl. Phys., Vol. 62, No. 2, 15 July 1987

Ewbank eta!.

411

378

limited intensity range, the rate was approximately linear with an intensity of l WI cm2 producing a photorefractive grating in ~ l s. Note that cerium doping in SBN:61 increases the grating formation rate per unit intensity (slope of the lines in Fig. 9) by a factor of ~ 5. The data in Fig. 9 can be used to compute the dark con. ductivity CTd and the product of the mobilityµ and recombin. ation time r R. With no externally applied electric field, and in the limit that the grating spacing is much greater than the diffusion length 50 (i.e., A;>41rµrRk 8 T/e), the photorefractive time response r PR is given by the inverse of the dielectric relaxation rate 1•50 and can be written as

O'--~~~~~~,----~--:-'::-~~-=-':,---~-::-'

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2.0

1.5

1.0

2.5

( 11)

1191 lµm-1)

where (he/A.) is the photon energy. Furthermore, the effective mean-free path or diffusion length Ld of the photorefractive charge carrier can simply be estimated as 51

FIG. 7. The wavelength dependence of the two-wave mixing gain coefficient r VS inverse grating spacing A; 1 in nominally undoped SBN:6 l sample C. More impurity states become accessible at shorter optical wavelengths, contributing to a large photorefractive coupling efficiency.

(12)

tive charge carrier is negative in all of the SBN samples examined thus far, so that the direction of two-wave mixing gain (and the direction of beam fanning) is toward the positive poling electrode c face, in contrast to commercially available26 BaTi0 3 • E. Photorefractive response time of SBN

The photorefractive response time of SBN was determined by measuring the rate of grating formation as a function of the total optical intensity incident on the crystal. Traditionally, one studies grating erasure27 •28 rather than grating formation, since the former has a simple exponential time dependence while the latter does not. 49 However, the grating formation rates are of more practical importance, and here they are arbitrarily defined to be the inverse of the time for the amplified beam in two-wave mixing to reach (1 - e- 1 ) of its steady-state value. In general, the grating formation rate in SBN increased sublinearly with intensity. Figure 9 shows the measured grating formation rate at 514.5 nm in seven different SBN crystals over the intensity range of~ 1-15 W /cm 2 • Over this 2.5

The fitted values for CTd and (µrR) are obtained from the intercept and slope, respectively, of each line in Fig. 9 for every SBN sample. These values, along with the corresponding Ld, are listed in Table II. Note that the values obtained for (µr R ) are self-consistent with the assumption50 Ag >27TLd. Cerium doping ofSBN:61 increases the rate of response and reduces the dark conductivity. The ceriumdoped SBN: 75 sample also shows a reduced dark conductivity, although this crystal's overall response is relatively slow compared to SBN :61 due to the increase of its dielectric constant € 3 • In a direct comparison between SBN and BaTi0 3 , we found that the erasure and formation rates of a BaTi03 crystal obtained from Sanders 26 were a factor of 2 faster than the corresponding rates in the undoped SBN sample C, and were approximately one-third of the rates of the Ce-doped SBN sampleD.

0.4

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'f

5

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x

1i z

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'i

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0.1 0.5

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0 600

FIG. 8. Comparison of the spectral dependence of the effective photorefractive charge density N0 ff (left) and the absorption coefficient a (right) in nominally undoped SBN:61 sample C. The straight line is a linear leastsquare fit to N0 ff vs A. 379

In summary, SrxBa 1 _xNb2 0 6 (SBN) can be grown with sufficient size, optical quality, and extrinsic dopants to be suitable for photorefractive applications. Small signal two-wave mixing gains (erL) exceeding 1000 have been observed in both nominally undoped SBN and cerium-doped SBN. The nominally undoped SBN:61 has an effective photorefractive charge density of~ l X 1016 cm- 3 and a photorefractive grating formation rate per unit intensity of ~0.3 cm2 /W s over the intensity range of~ 1-15 W /cm2 at 514.5 nm. Cerium doping SBN:61 increases the charge density by a factor of - 3 and increases the formation rate by a factor of ~s.

1.0

450

VI. CONCLUSIONS

SBN potentially exhibits a greater flexibility for doping than other photorefractive materials such as BaTi03 , due to its open crystal structure containing vacant lattice sites. Currently, effort is underway to optimize the photorefractive response of cerium-doped SBN in the infrared by changing the crystallographic site occupied by the dopant from twelve- to nine- or sixfold coordination. Although the photorefractive efficiency and formation rate are enhanced by cerium doping, the identification of the valence states of ceriEwbank et al.

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379

INTENSITY IW/cm2)

FIG. 9. The rate of photorefractive grating formation during two-wave mixing in nominally undoped SBN:61 samples A-C, cerium-doped SBN:61 samples D-F and cerium-doped SBN:75 sample Gas a function of incident intensity, for,\ = 514.5 nm and A, = 2 µm. Cerium doping enhances the photorefractive formation rate.

um (e.g., ce+ 3 or ce+ 4 ) and their role in the photorefractive process remains to be determined. ACKNOWLEDGME NTS

This w_ork has, in part, been supported by contracts from DARPA and DARPA/AFWAL /Materials Laboratory. The authors appreciate discussions with A. Chiou, M. Khoshnevisan, J. Oliver, and P. Yeh from Rockwell International Science Center, and thank the referee for helpful suggestions. 1

N. V. Kukhtarev, V. B. Markov, S. G. Odulov, M. S. Soskin, and V. L. Vinetskii, Ferroelectrics 22, 949 (1979); P. Gunter, Phys. Rep. 93, 199 (1982). 'P. Yeh, Proc. Soc. Photo-Opt. Instrum. Eng. 613 ( 1986). 3 J. Feinberg and R. W. Hellwarth, Opt. Lett. 5, 519 (1980); 6, 257 (E) (1981). 4 ]. 0. White, M. Cronin-Golomb, B. Fischer, and A. Yariv, Appl. Phys. Lett. 40, 450 (1982). 'R. Fisher, editor Optical Phase Conjugation (Academic, New York, 1983). 6 J. P. Huignard and A. Marrakchi, Opt. Commun. 38, 249 ( 1981 ). 7 F. Laeri, T. Tschudi, and J. Albers, Opt. Commun. 47, 387 (1983). 8 Y. Fainman, E. Klancnik, and S. H. Lee, Opt. Eng. 25, 228 ( 1986). 9 J. 0. White and A. Yariv, Appl. Phys. Lett. 37, 5 (1980). 0 ' J. Feinberg, Opt. Lett. 5, 330 (1980). 11 E. Ochoa, J. W. Goodman, and L. Hesselink, Opt. Lett. 10, 430 (1985). 12 S. -K. Kwong, G. A. Rakuljic, and A. Yariv, Appl. Phys. Lett. 48, 201 (1986). 13 A. E. Chiou and P. Yeh, Opt. Lett. 11, 306 ( 1986).

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14 Y. Fainman, C.C. Guest, and S. H. Lee, Appl. Opt. 25, 1598 ( 1986). "M. Cronin-Golomb, B. Fischer, J. Nilsen, J. 0. White, and A. Yariv, Appl. Phys. Lett. 41, 219 (1982). 16 R. A. McFar!ane and D. G. Steel, Opt. Lett. 8, 208 (1983). "M. D. Ewbank, P. Yeh, M. Khoshnevisan, and J. Feinberg, Opt. Lett. 10, 282 (1985). 18 P. Pellat-Finet and J. -L. De Bougrenet De La Tocnaye, Opt. Commun. 55, 305 (1985). 19 M. D. Ewbank and P. Yeh, Opt. Lett. 10, 496 ( 1985). '°J.-C. Diels and I. C. McMichael, Opt. Lett. 6, 219 (l 981). 2 'P. Yeh, M. Khoshnevisan, M. D. Ewbank, and J. Tracy, Opt. Commun. 57, 387 (1986). 22 P. Yeh, I. C. McMichael, and M. Khoshnevisan, Appl. Opt. 25, 1029 (1986). 23 D. Anderson, Opt. Lett. 11, 56 (1986). 24 B. H. Soffer, G. J. Dunning, Y. Owechko, and E. Marom, Opt. Lett. 11, 118 (1986). 25 A. Yariv and S.-K. Kwong, Opt. Lett. 11, 186 (1986). 26 Sanders Associates, 95 Canal Street, Nashua, NH 03061. 27 J. Feinberg, D. Heiman, A. R. Tanguay, Jr., and R. W. Hellwarth, J. Appl. Phys. 51, 1297 (1980); 52, 537 (E) (1981). 28 S. Ducharme and J. Feinberg, J. Appl. Phys. 56, 839 ( 1984). 29 D. Rak, I. Ledoux, and J. P. Huignard, Opt. Commun. 49, 302 ( 1984 ). 30 M. B. Klein and G. C. Valley, J. Appl. Phys. 57, 4901 ( 1985). 31 S. Ducharme and J. Feinberg, J. Opt. Soc. Am. B 3, 283 ( 1986). 32 M. B. Klein and R. N. Schwartz,J. Opt. Soc. Am. B3, 293 (1986); M. B. Klein, Proc. Soc. Photo-Opt. Instrum. Eng. 519, 136 (1984). 33 J. B. Thaxter and M. Kestigian, Appl. Opt. 13, 913 (1974). 34 !. R. Dorosh, Y. S. Kuzminov, N. M. Polozkov, A. M. Prokhorov, V. V. Osika, N. V. Tkachenko, V. V. Voronov, and D. K. Nurligareev, Phys. Status Solidi A 65, 513 ( 1981). 35 B. Fischer, M. Cronin-Golomb, J. 0. Whiie, A. Yariv, and R. R. Neurgaonkar, Appl. Phys. Lett. 40, 863 (1982). 36 R. R. Neurgaonkar and W. K. Cory, J. Opt. Soc. Am. B3, 274 ( 1986). 37 G. A. Rakuljic, A. Yariv, and R.R. Neurgaonkar, Proc. Soc. Photo-Opt. Instrum. Eng. 613, HO (1986); G. A. Rakuljic, A. Yariv, andR. R. Neurgaonkar, Opt. Eng. 25, 1212 ( 1986). 38 K. Megumi, H. Kozuka, M. Kobayashi, and Y. Furuhata, Appl. Phys. Lett. 30, 631 (1977). 39 V. V. Voronov, I. R. Dorosh, Yu. S. Kuz'minov, and N. V. Tkachenko, Sov. J. Quantum Electron. 10, 1346 (1980). 40 J. Feinberg, Opt. Lett. 7, 486 ( 1982). 41 G. Salama, M. J. Miller, W.W. Clark, G. L. Wood, and E. J. Sharp, Opt. Commun.59,417 (1986). 42 K. Megumi, N. Nagatsuma, Y. Kashiwada, and Y. Furuhata, J. Mater. Sci. 11, 1583 (1976). 43 The precise transition temperature in BaTi03 from Sanders Associates depends on the impurity level and the oxidation state of the crystal. See Ref. 31. 44 R. J. Pressley, editor, Handbook ofLasers (Chemical Rubber, Cleveland, 1971 ), p. 452. 45 P. Yeh, J. Opt. Soc. Am. B 2, 1924 (1985). 46 I. McMichael and P. Yeh, Opt. Lett. 12, 48 ( 1987). 47 F. P. Strohkendl, J.M. C. Jonathan, and R. W. Hellwarth, Opt. Lett. 11, 312 (1986). 48 G. C. Valley, J. Appl. Phys. 59, 3363 (1986). 49 L. Solymar and J. M. Heaton, Opt. Commun. 51, 76 ( 1984). 50 M. B. Klein, Opt. Lett. 9, 350 (1984). 51 R. A. Mullen and R. W. Hellwarth, J. Appl. Phys. 58, 40 (1985).

Ewbank et al.

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November 15, 1989 I Vol. 14, No. 22 I OPTICS LETTERS

1275

Noise suppression in photorefractive image amplifiers H. Rajbenbach, A. Delboulbli, and J.P. Huignard Laboratoire Central de Recherches, Thomson-CSF, Domaine de Gorbeville, 91404 Orsay Gedex, France Received May 22, 1989; accepted September 8, 1989 We propose and experimentally demonstrate a simple technique capable of significantly enhancing the signal-tonoise ratio of photorefractive amplifiers. The optical noise due to amplified scattered light and multiple interface reflections is removed by performing two-wave mixing in off-axis-rotating BaTiOa and Bii2Si020 crystals. A 20-fold improvement of the signal-to-noise ratio is achieved, and virtually noise-free image amplifiers are demonstrated.

When a weak optical signal is injected into a nonlinear medium illuminated with a strong pump beam, it experiences gain through a two-wave mixing process. A problem common to these amplifiers is the presence of noise sources that corrupt the quality of the emerging amplified optical signal. This problem is particularly disturbing for low-intensity injected signals and contributes to poor signal-to-noise ratios in the detection plane. In photorefractive amplifiers1 the spontaneous optical noise emission and amplification (also called beam fanning) originates from light scattered in or reflected from the crystal imperfections and interfaces. 2•3 A portion of this light propagates in the same direction as the injected signal and is also amplified through the formation of parasite noise gratings with the pump beam. 4 Furthermore, the maxima of noise power obviously occur along the directions of maximum gain, i.e., in the directions that lie within the angular bandwidth of the photorefractive amplifier. Thus, it appears difficult to suppress the noise of photorefractive amplifiers efficiently while maintaining high gain for the signal. In this Letter we propose and experimentally demonstrate the first technique, to our knowledge, capable of significantly reducing the optical noise while maintaining their high gain coefficients. We show that performing two-wave mixing in slowly rotating crystals results in a complete washout of the noise gratings. This phenomenon is explained in terms of a dynamic filtering that profits from the slow buildup time constants of the noise gratings relative to the injected signal grating. First, we estimate the influence of a crystal rotation (angular velocity Q around point 0) on the noise intensity in the output plane [Fig. l(a)]. A noise source located near point A radiates a complex wave front whose component along the direction of the injected signal is represented by a plane wave of intensity IN (A is a crystal imperfection or interface). It is essential to note that the noise sources are bounded to the crystal and consequently will move with it., This is the basis for the process of discrimination between the noise gratings and the injected signal grating. In the Ox coordinate bounded to the rotating crystal, the interference pattern responsible for the formation of a noise grating around point A is given by IN(x, t) = I 0 [1 + mN cos KN(t)x], where KN(t) ~ 211'(0 + Qt)/'A is the 0146-9592/89/221275-03$2.00/0

time-dependent grating wave vector (we have assumed that the angles inside the crystal are small), Io is the total incident intensity (Io~ IPQ), 0 is the angle between the beams inside the crystal, 'A = 'A0/n is the optical wavelength in the crystal with refractive index n, Q is the crystal angular velocity, and mN is the fringe modulation. IN(x, t) can be written in the form IN(x, t) = Io[l + mNcosK(x - VNt)], whereK = 211'/A = 211'0/ 'A is the average wave vector, A is the average fringe spacing, and VN = AQx/'A is the equivalent local fringe velocity around point A in the x direction. Note that VN is dependent on the position of the noise source A in the crystal volume. In dynamic holographic media, the photoinduced index modulation due to the moving

0

l!l

Fig. 1. Schematic of the effect of a crystal angular rotation (0) on (a) a noise grating, where 8 + Qt is the time-varying angle between the fixed pump beam and an internally generated noise beam originating from point A, and (b) the injected signal grating for which the rotation of the crystal yields a tilt of the illumination fringes. (The figures are not to scale.) © 1989 Optical Society of America

415

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OPTICS LETTERS I Vol. 14, No. 22 I November 15, 1989

Fig. 2. Top: The noise distribution in the output plane of photorefractiveamplifiers (Ail= 514nm,J0 = 5mW/cm2, Q = 0). The bright spots are the impact of the transmitted pump beams. Bottom: The time evolution of the noise and amplified signal emerging from photorefractive amplifiers when the pump beam is switched on. A = 2.5 µm and Eo = O for BaTi03, and A= 20 µm and E 0 = 8 kV· cm-I for BSO.

fringes is related directly to the fringe velocity through the relation 5 !lnN(x, t) = mN!ln 0(1 + K2vr?rr?)-lf2 cos[K(x - VNt) + 1/lo + .

(23)

7/ =

Note that y and f3 are related to r by r = y + i2[3. In photorefractive crystals that operate by pure diffusion, f3 = 0. Since wave A 2 is generated by Bragg scattering of wave A 1 from the fixed grating, the phase of A 2 is automatically locked to that of A 1 • Thus the possible solutions to Eqs. (19) and (20) are l:.lfJ = 1T/2

(24)

l:.1/1 = -1T/2.

(25)

I+

Z

I

+

(26)

2

d I1I2 -d I.= y I - I

- ITT

+ 2KVI1I2.

(27)

2

Since I 1 + I 2 is a constant, Eqs. (26) and (27) are mathematically identical to those for nonlinear optical Bragg scattering and can be solved exactly. 26 The solutions, subject to the boundary condition A 2(0) = 0, are written

I1(z) = Io cos2 u,

(28)

I 2(z) = 10 sin2 u,

(29)

where I 0 is the incident intensity and u is given by 26 tan u = (1 - b2)112 - b tan[Kz(l - b•)112],

(30)

b = y/(4K).

(31)

with

The intensity of the diffracted wave a8 a function of z was investigated in a previous study 26 for various values of b. The parameter b is a measure of the strength of the photorefractive grating relative to the fixed grating. It is important to note that b in this paper is independent of the intensity, whereas bis proportional to Io in Kerr media. The diffraction efficiency is a nonlinear function of b. For a crystal with interaction length L, the diffraction efficiency is 7/ =

I.(L) Io =

1+

4~2

-

2 exp(-2KLlbl)] ·

(34)

=

With the boundary condition A 2(0) = 0, we find that ti.I/I = 1T/2 is a proper solution by examining the phase of A 2 in Eq. (11). Substituting l:.1/1 = 1T/2 into Eqs. (17) and (18), we obtain

Z

[

Here we note that the expressions are slightly different from those that appeared in Eqs. (18) and (19) of Ref. 26. Figure 2 depicts the diffraction efficiency as a function of b for different values of KL. For b > 0 the photorefractive grating is in phase with the fixed grating. When KL is small the presence of the photoinduced grating strengthens the grating, and the diffraction efficiency increases as a function of b. When KL = 1T/2 the diffraction efficiency is 1 at b = 0 (no photoinduced grating). The diffraction efficiency decreases as b increases from b = 0, reaches a minimum at b 0.89, and then increases to 1 when bis extremely large. When KL> 7r, the diffraction efficiency will oscillate between 0 and 100% within lbl < 1. This oscillatory behavior can also be seen in Fig. 2 of Ref. 26, since I 1 and I 2 are periodic functions of z for -1 < b < 1. The sign of the parameter b depends on the direction of incidence. Thus if b is positive for an incident angle of 0, it will be negative when the beam is incident at an angle of -0. The different values of 7/ for +b and -b reflect nonreciprocal diffractions caused by TWM. In other words, the diffraction efficiency of fixed gratings in photorefractive media depends on the direction of incidence at Bragg angles. Figure 3 shows the grating amplitude as a function of z. When b > 0, e.g., b = 1, the photoinduced grating is in phase with the fixed grating; therefore the grating is strengthened. When b < 0, e.g., b = -1, the photoinduced grating is out of phase with the fixed grating; therefore the grating is weakened. From the curve for b = 10, we notice that at the beginning the photoinduced grating is in phase with the fixed grating; therefore the grating is strengthened. At the point in space where all the incident power is converted into a diffracted beam, i.e., I 2 = I 0 and I 1 = 0, the photoinduced grating disappears. After this point the photoinduced grating is out of phase with the fixed grating. Physically, this is equivalent to the case in which the beam is incident from the other input port (i.e., at -0). When the photoinduced grating cancels the fixed grating, there is no more diffraction, and thus the grating amplitude stays 0. The curve for b = -10 is the same as the curve for b = 10 after the point 77 = 100%.

and

d I1I2 - ITT -d I,= - y I - I - 2KVI1I2,

4~2

2341

0.9 0.8

C.7 0.6

.

sm2 u(z = L).

(32)

0.5 0.4

When b >> 1 and KLb >> 1, the asymptotic form of 7/ can be written as

0.3 0.2

l 77=1- b2[1- 4b 2 exp(-2KLb)] 2 4 -

=1 -

1 b2[1 - 8b 2 exp(-2KLb)). 4

The asymptotic form for b

~

(33)

4

-

3

-

2

-

1

Fig. 2. Diffraction efficiency as a function of b for transmission phase gratings with different values of KL. Solid curve, KL = 11'/6; dotted-dashed curve, KL = 11'/2; dashed curve, KL = 7r.

- co can be written as

421

2342

J. Opt. Soc. Am. B/Vol. 7, No. 12/December 1990

C. Gu and P. Yeh

Reflection Phase Gratings (n. = 0, /31 /J 2 < O) For reflection gratings, the z components of the two wave vectors are of opposite sign. They are given by /31

10.0

= -{:32 = (2-rr/A)n 0 cos 8.

(36)

With Eqs. (7) and (8), the coupled-mode equations can be written in the form Grating Amplitude

S.O

(37)

d -A2 dz

f* IA 1f 2 2 10

= - - --A2

+ i(K

-

a)

a

i - A1 + -A2. 4 2

(38)

Following similar steps as in transmission phase gratings, the coupled equations for the intensities of the two waves can be found as -5.0

~--~---~-~-~--~--___,

0.0

0.4

0.2

0.6

0.8

\.0

(39)

Fig. 3. Grating amplitudes as functions of z for b equal to 10, 1, 0, -1, and -10 in the case of transmission phase gratings.

Photorefractive media and Kerr media26 can be distinguished by the different expressions for b, i.e., Eq. (31) in this paper and Eq. (16) in Ref. 26. In Kerr media b is a function of intensity; therefore the diffraction efficiency depends on the incident intensity nonlinearly. In photorefractive media there is no such nonlinearity.

(•)

Transmission Absorption Gratings (np = 0, /3 1 /32 > 0) Nonreciprocal diffractions also occur in the case of absorption gratings. When a ;e 0, the coupled-mode equations (10) and (11) cannot be easily solved analytically; however, nonreciprocal behavior can still be seen in the numerical solutions. We have solved the coupled-mode equations (10) and (11) by using the fifth-order RungeKutta method with adaptive step size. In the general case, the parameter b is defined as

b=

(( 4K) 2

'Y

+ a 2 ] 112

(b)

(c)

o.o

(35)

2.0

1.0

3.0

Fig. 4. Intensities of the incident and diffracted beams as functions of (a/4)z for transmission absorption gratings (K = 0) with various values of b. Solid curves, 11 ; dotted curves, I,. (a) b = 1, (b) b = 0, (c) b = -1.

Figure 4 shows the intensities of the incident and diffracted beams as functions of z for various values of b, where the fixed grating is a pure absorption grating (K = 0). Similar to the case of phase gratings, diffraction from one input port (b > 0) .is enhanced and that from the other input port (b < 0) is suppressed, as compared with the case when TWM disappears (b = 0). The diffraction efficiency as a function of b is plotted in Fig. 5 for various values of aL. We note that the nonreciprocity in absorption gratings is quite strong. For fixed gratings with both phase and absorption variations, the diffraction property is a combination of that of the pure phase and pure absorption gratings. As an example, we give the numerical solutions for K = cos(TT/30) and a = 4 sin(-rr/30). Figure 6 shows the intensities of the incident and diffracted beams as functions of z for various values of b, and Fig. 7 shows the diffraction efficiency as a function of b for various values of L. In the numerical calculations, the units of K, a, and z are chosen arbitrarily, since KZ and az are dimensionless. Again, the diffraction is nonreciprocal.

0.6

0,5

0.4

0.3

o.z 0.1

-

5

10

15

Fig. 5. Diffraction efficiency as a function of b for transmission absorption gratings (K = 0) with various values of aL. Solid curve, aL = 0.2?T; dotted curve, aL = 0.4?T; dashed curve, aL = 0.8'1T.

422

Vol. 7, No. 12/December 1990/J. Opt. Soc. Am. B

C. Gu and P. Yeh

.....--........

2343

related to z by ·····- ...

(•)

e"" [cosh(2u) + b sinh(2u)]b = exp[- 2 K(l - b")(z - L)],

····-·-

(43) (b)

0.1.00

where b was given by Eq, (31), Equation (43) must be handled with care when !bl = l, The solution for u as a function of z when b = 1 can be written

Intensity

(o)

1.0

o.o

u = 2K(L - z) - V. + 1/.e-4",

..•·.··

(44)

whereas, when b = -1, (d)

u = 2K(L - z) + V. - 1/.e4 ",

Figure 8 shows the intensities as functions of z for b = 1, b = 0, and b = - L We note that backward coupling of energy is enhanced for the case when b > 0. The diffraction efficiency is given by the expression

(e)

o.o ·········

12(0)

'7 = l,(O) =

Fig. 6. Intensities of the incident and diffracted beams as functions of z for fixed gratings with both phase and absorption variations [K = cos(1T/30) = 0.9945, a = 4 sin(1T/30) = 0.4181] and various values of b. Solid curves, [ 1 ; dotted curves, 12 • (a) b = 1, (b) b = 0.5, (c) b = 0, (d) b = -0.5, (e) b = -l. The units of K, a, and z are chosen arbitrarily, since KZ and az are dimensionless.

(46)

(47)

We note that when KL > Vs u(O) is positive, and e-•• becomes extremely small when 2KL - y, > V.. Therefore, when KL > 'I•, u(O) == 2KL - 114 and '7 == tanh2(2KL - 'I•). The diffraction efficiency approaches 100% exponentially as 2KL - V. >> l; i.e., KL >> o/s, Figure 9(a) shows the diffraction efficiency as a function of KL for b = 0, b = ±1, and also for b = ±2. When b = 0, '7 = tanh 2 KL, When b » i and KLb >> 1, the asymptotic expression for the diffraction efficiency is

0.7 0.6 0.5

I

I

0.3

2

tanh u(z = 0).

Given a KL, u(O) can be solved from Eq, (43), For example, when b = 1, according to Eq. (44), u(O) is given by

0.8

0.4

(45)

,/····--------------------------------------

0.2

(48)

'7 = 1 - 2b exp(-2KLb),

We note that the diffraction efficiency approaches 100%

0.1

-5-4-3-2-1

0

b

Fig. 7. Diffraction efficiency as a function of b for fixed gratings with both phase and absorption variations [K = cos(Tr/30) = 0.9945, a = 4 sin(,,./30) = 0.4181] and various values of L. Solid curve, L = 1T/6; dotted-dashed curve, L = w/2; dashed curve, L = 'IT. The units of K, a, and L are chosen arbitrarily, since KZ and az are dimensionless.

(a)

Intensity

(40)

0.0

In deriving Eqs, (39) and (40), we used q, = 'IT/2 and flt/I = 'IT/2, which is one of the two solutions (flt/I = 'IT/2 or flt/I = -'IT/2) for the equations of phase change. Equations (39) and (40) can be solved analytically. The solutions with the boundary condition 12 = 0 at z = Lare given by 2

l,(z) = C cosh u,

(41)

l2(z) = C sinh 2 u,

(42)

(b)

········-······-···

-

1.or=============J (o)

1.0

0.0 '--~-'--~--'----'---===----' 0.8 0.7 0.3 0.2 0.0

o.•

Fig. 8. Intensities of incident and diffracted beams as functions of z for reflection phase gratings (K = 1, a = 0). (a) b = 1,

where C is the transmitted intensity at z = L, and u is

(b) b

423

= 0, (c) b = -1,

C. Gu and P. Yeh

J. Opt. Soc. Am. B/Vol. 7, No. 12/December 1990

2344

value that corresponds to b = 0. This is because the pho. toinduced grating is proportiona l to IA1A2l/U1 + 12 ) which depends on the sign of b as well as its absolute value' as shown in Fig. 8. This asymmetric behavior also oc~ in the transmission phase gratings, as shown in Fig. 3. Reflection Absorption Gratings (np = 0, /J1 /J 2 < 0) For absorption gratings, closed-form solutions of the coupled equations are not easily available. The diffraction properties of these gratings can be studied by using numerical techniques. We solved the coupled-mode equa. tions (37) and (38) numerically with a nonzero absorption constant. Since the photoinduced grating depends only on the modulation depth, Eqs. (37) and (38) are scaling invariant. In other words, if A1 and A 2 are solutions to Eqs. (37) and (38), cA 1 and cA2 are also solutions to these

Diffroction Efficiency

0.2 b=-2

15.0 ,----.-----.--~-.------.-....,

3.0

2.0

1.0

kl

(a)

0.9

10.0

0.8 Groting Amplitude

0.7 0.6 0.5

5.0

0.4 0.3 0.2

b•I

0.1

b-0 b--1

-

20

-

15

-

10

-

5

to

15

20

0.0 0.0

(b) for reflection phase gratings as efficiency Diffraction (a) 9. Fig. a function of KL' for b equal to 0, ±1, and ±2. (b) Diffraction efficiency as a function of b for reflection .phase gratings with various values of KL. Dotted curve, KL = 1/a; dotted-dashed 1 curve, KL = 'I•; dashed curve, KL = /2.

0.6

0.4

0.2

0.8

1.0

Fig. 10. Grating amplitudes as functions of z for b equal to 10, 1, 0, -1, and -10, in the case of reflection phase gratings.

exponentially as b increases. When b--> -oo, the asymptotic expression for the diffraction efficiency is 7J =

1 b [l - 2 exp(-2KLjb\)]. 4 2

(49)

2 As 2KLlbl » 1, 7J = Y..b for all KL. Figure 9(b) shows the diffraction efficiency as a function of b for various values of KL. Comparison of Figs. 9 and 2 reveals that the diffraction is nonreciproca l for both transmission and reflection gratings. The oscillatory behavior in Fig. 2 does not occur in Fig. 9 because the intensities in Eqs. (41) and (42) are not periodic functions of z. Figure 10 shows the grating amplitude as a function of z. Similar to the case of codirectiona l diffractions, the photoinduced grating is in phase with the fixed grating when b > 0 and out of phase when b < 0, and therefore strengthens or weakens the fixed grating, respectively. We note from Fig. 10 that the grating amplitude for b = 1 and that for b = -1 are not symmetric with respect to the

Intensity

(b)

(o)

0.0

===d b=~......---'-----'--~=== 1.0 0.6

o.o

0.2

0.3

0.5

0.7

Fig. 11. Intensities of incident and diffracted beams as func· tions of z for reflection absorption gratings (K = 0, a = 4). (a) b = 1, (b) b = 0, (c) b = -1.

424

C. Gu and P. Yeh

Vol. 7, No. 12/December 1990/J. Opt. Soc. Am. B

2345

pure diffusion. Bragg mismatch (Ak "' 0) as well as arbitrary phase shift (q, "' TT/2) are important issues that will affect the diffraction properties of fixed gratings in photorefractive media.

o.9 0.8

0.7 0.6

o.s

ACKNOWLEDGMEN TS

0.4

We thank R. Saxena for technical discussions. This work is supported, in part, by U.S. Office of Naval Research contract N00014-88-C-0230.

0.3

0.2 0.1

- w

-

15

-

10

-

0

5

10

15

REFERENCES AND NOTES

20

1. See, for example, G. C. Valley and P. Yeh, eds., Feature on Photorefractive Materials, Effects, and Devices, J. Opt. Soc. Am. B 5, 1681-1821 (1988). Many papers on photorefracts are included. 2. See, for example, P. Yeh, "Two-wave mixing in nonlinear media," IEEE J. Quantum Electron. 25, 484-519 (1989). 3. See, for example, P. Yeh, A. E. Chiou, J. Hong, P. Beckwith, T. Chang, and M. Khoshnevisan, "Photorefractive nonlinear optics and optical computing," Opt. Eng. 28, 328-343 (1989). 4. P. J. van Heerden, "Theory of optical information storage in solids," Appl. Opt. 2, 393-400 (1963). 5. A. E. T. Chiou and P. Yeh, "Parallel image subtraction using a phase-conjugate Michelson interferometer," Opt. Lett. 11, 306-308 (1986). 6. T. Y. Chang, P. H. Beckwith, and P. Yeh, "Real-time optical image subtraction using dynamic holographic interference in photorefractive media," Opt. Lett. 13, 586--588 (1988). 7. P. Yeh, A. E.T. Chiou, and J. Hong, "Optical interconnections using photorefractive dynamic holograms," Appl. Opt. 27, 2093-2096 (1988). 8. A. Chiou and P. Yeh, "Energy efficiency of optical interconnections using photorefractive holograms," Appl. Opt. 29, 1111-1117 (1990). 9. H. Lee, X. G. Gu, and D. Psaltis, "Volume holographic interconnections with maximal capacity and minimal cross talk," J. Appl. Phys. 65, 2191-2194 (1989). 10. D. Psaltis, X. G. Gu, and D. Brady, "Holographic implementations of neural networks," in An Introduction to Neural .and Electronic Networks, S. F. Zornetzer, J. L. Davis, and C. Lau, eds. (Academic, New York, 1990), pp. 339-348. 11. D. Psaltis, D. Brady, X. G. Gu, and S. Lin, "Holography in artificial neural networks," Nature (London) 343, 325-330 (1990). 12. D. L. Staebler and J. J. Amodei, "Thermally fixed holograms in LiNb03 ," Ferroelectrics 3, 107-113 (1972). 13. F. Micheron and G. Bismuth, "Electrical control of fixation and erasure of holographic patterns in ferroelectric materials," Appl. Phys. Lett. 20, 79-81 (1972). 14. F. Micheron, C. Mayeux, and J. C. Trotier, "Electrical control in photoferroelectric materials for optical storage," Appl. Opt. 13, 784-787 (1974). 15. D. von der Linde, A. M. Glass, and K. F. Rodgers, "Highsensitivity optical recording in KTN by two-photon absorption," Appl. Phys. Lett. 26, 22-24 (1975). 16. A. Delboulbe, C. Fromont, J. P. Herriau, S. Mallick, and J. P. Huignard, "Quasi-nondestructive readout of holographically stored information in photorefractive Bi12Si020 crystals," Appl. Phys. Lett. 55, 713-715 (1989). 17. J. Hong, S. Campbell, and P. Yeh, "Optical pattern classifier with perceptron learning," Appl. Opt. 29, 3019-3025 (1990). 18. J. Hong, S. Campbell, and P. Yeh, "Optical learning machine for pattern classification," in Digest of Optical Society of America Annual Meeting (Optical Society of America, Wash· ington, D.C., 1989), paper WJ3. 19. D. Psaltis, D. Brady, and K. Wagner, '~daptive optical networks using photorefractive crystals," Appl. Opt. 27, 17521759 (1988). 20. E. G. Paek, J. Wullert, and J. S. Patel, "Holographic implementation of a learning-machine based on a multicategory perceptron algorithm," Opt. Lett. 14, 1303-1305 (1989).

b

Fig. 12. Diffraction efficiency as a function of b for reflection absorption gratings (K = 0) with various values of aL. Dotted curve, aL = %; dotted-dashed curve, aL = 1; dashed curve, aL = 2.

two equations, where c is a complex constant. Therefore we can reduce the two-point boundary-value problem to the one-point one. Numerical integration starts from z = L, where A 2 = 0 and A 1 is set to be 1. As we go from z = L to z = 0, complex A 1 (z) and A 2 (z) are obtained. Then the intensities/1 (z) and I 2 (z) are normalized by taking the normalized 11 (0) to be 1. For the case of a pure absorption grating (K = 0), Fig. 11 shows the intensities as functions of z. We note that 12 grows from 0 as the diffracted beam propagates from z = L to z = 0. Because of absorption, the growth is slower than that for phase gratings. The incident beam 11 decays as it propagates from z = 0 to z = L because of both energy exchange with 12 and absorption. Figure 12 shows the diffraction efficiency as a function of b for various values of aL. The diffraction property is similar to that in the absence of absorption (Fig. 9), except that because of the loss in the medium the diffraction efficiency cannot reach 100%. The nonreciprocity, in the case of absorption gratings, can be explained by the phase shift between the photoinduced phase grating and the interference pattern. The fixed absorption grating provides a source for generating the second beam when the incident beam satisfies the Bragg condition.

CONCLUSION In conclusion, we analyzed the diffraction properties of fixed gratings in photorefractive media with the consideration of TWM between the incident and the diffracted waves. Both codirectional and contradirectional diffractions were solved, including the presence of both phase and absorption gratings. The results indicate that fixed gratings in photorefractive media exhibit strong optical nonreciprocity, which cannot be explained without considering TWM. Since the photorefractive grating depends on the modulation depth instead of the intensities, there is no optical nonlinearity like that which appeared in the Kerr media. 26 Our analysis can be applied not only to fixed photorefractive gratings but also to those that may be fabricated artificially. In this paper we considered only diffraction properties of fixed gratings with exact Bragg match, and the photorefractive media were assumed to be in the regime of

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21. D. L. Staebler and J. J. Amodei, Coupled-wave analysis of holographic storage in LiNbO,," J. Appl. Phys. 43, 1042-1049 (1972). 22. H. Kogelnik, "Coupled wave theory for thick hologram gratings," Bell Syst. Tech. J. 48, 2909-2945 (1969). 23. R. Saxena, F. Vachss, I. McMichael, and P. Yeh, "Diffraction properties of multiple-be am photorefrac tive gratings," J. Opt. Soc. Am. B 7, 1210-1215 (1990). 24. T. Chang and P. Yeh, "Dark rings from photorefractive conical diffraction in a BaTiOs crystal,'' Proc. Soc. Photo-Opt. Instrum. Eng. 739, 109-116 (1987).

25. N. A. Vainos, S. L. Clapham, and R. W. Eason, "Multiplexed permanent and real-time holographic recording in Photorefractive BSO," Appl. Opt. 28, 4381-4385 (1989). 26. P. Yeh and M. Khoshnevisan, "Nonlinear-optical Bragg scattering in Kerr media,'' J. Opt. Soc. Am. B 4, 1954-1957 (1987). We believe that there may be a slight error in this reference (typesetting error in the asymptotic form of the diffraction efficiency for b » 1). 27. F. Vachss, I. McMichael, M. Khoshnevisan, and P. Yeh, "Enhanced acousto-optic diffraction in electrostrictive media " ' J. Opt. Soc. Am. B 7, 859-867 (1990).

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Reprinted with pennis sion from The American Physical Society PHYS ICAL REVI EW LETT ERS

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Observation of the Photorefractive Effect in a Polymer Stephen Ducharme, J.C. Scott, R. J. Twieg, and W. E. Moerne r

IBM Research Dfoision, IBM Almade n Research Center, San Jose, California 95120-6099 (Received 13 November 1990)

We report the first observation of the photorefractive effect in a polymeric material, the electro-optic polymer bisphenol-A-diglycidylether 4-nitro-1,2-phenylenediam ine made photoconductive by doping with the hole-transport agent diethylamino-benzaldehyde diphenylhydrazone. The gratings formed exhibit dynamic writing and erasure, strong electric-field depende nce, polarization anisotropy, and estimated space-charge fields up to 26 kV /cm at an applied field of 126 kV /cm. Application of similar concepts should provide a broad new class of easily fabricated photoref ractive materials. PACS numbers: 42.65.Hw, 72.20.Jv, 78.20.Jq, 78.65.Hc

Twenty years have elapsed since the discovery of the photorefractive (PR) effect and the early realization of its potential utility; 1 for example, 2 high-density opticaldata storage, many image-processing techniques, phase conjugation, simulations of neural networks and associative memories, and programmable interconnection. The commercial introduction of such technology has been hindered by the difficulty of preparing and processing the inorganic crystals (such as BaTi03 and SrxBa1 - xNb206 to name two examples) which show the effect. The purpose of this Letter is to describe the physical basis for, and properties of, the first of an entirely new class of easily fabricated PR materials, doped nonlinear optical polymers. The PR effect requires the redistribution and trapping of charge in the bulk of a photoconducting insulator or semiconductor under nonuniform illumination. The electric fields associated with the trapped charge alter the index of refraction of the material through the electrooptic (EO) effect, thus producing a phase replica, or hologram, of the optical intensity distribution. The ingredients necessary for producing a PR phase hologram are, therefore, photoionizable charge generator, transporting medium, trapping sites, and dependence of the index of refraction upon space-charge field. Recently, the PR effect was observed in a carefully grown doped organic crystal. 3 In fact, in spite of possibly lower mobilities, organics in general have potential advantages over inorganic crystals since a reasonably large electro-optic response can be attained with far lower dielectric constant. 4 The growth of doped organic crystals, however, is a very difficult process because most dopants are expelled during the crystal preparation. Polymers, on the other hand, can frequently be doped by simply mixing the various functional components; furthermore, they are ideally suited for future thin-film waveguide applications and can have nonlinear coefficients larger than LiNb0 3, for example. 5 We have made a polymeric PR material, the first to our knowledge, in order to illustnrte the generality of this approach for the formulation of new photorefractive ma1846

©

terials. Polymers have been separately made electrooptic by poling guest or attached nonlinear chromophores 6·7 and made photoconducting at virtually any wavelength by doping with charge-generation and -transporting agents. 8•9 In this work, the EO material is the partially crosslinked epoxy polymer bisA-NPDA composed of bisphenol-A-diglycidylether (bisA) reacted with the nonlinear chromophore 4-nitro-1,2-phenylenediamine (NPDA) which has been studied previously in its crosslinked form as a stable second-order material. 10 The polymer had a glass transition temperature T -65 °C 8 and number and weight average molecular weights Mn -2200 and M..,-69 00, respectively. The bisA-NPDA polymer acquires second-order nonlinearity when the NPDA chromophores are oriented in an electric field. It will be shown below that the NPDA also provides optical absorption for the charge-generation process. Monopolar charge transport is facilitated by addition of the hole-transport agent diethylamino-benzaldehyde diphenylhydrazone (DEH) to the polymer to form connected pathways for the hopping motion of holes as is done in the charge-transport layer of electrophotographic photoconductors. 9 In molecularly doped polymers, trapping centers are invariably present due to defects and conformational disorder inherent in the amorphous structure. Therefore, for these initial studies, no additional traps were added. The polymer and DEH were codissolved in the appropriate ratio in PM (propylene glycol monomethyl ether) or PM acetate, stirred thoroughly, and filtered. Samples were made by dripping 1-3 ml of the polymer-DEH-solvent mixture onto two indium-tin-oxide-coated glass plates set on a hot plate, drying at 95 °C for 30 min to remove most of the solvent, and then pressing the two plates together face to face. Fifteen samples 178-533 µm thick prepared in the same manner exhibited similar EO, photocurrent, and PR properties. The separate contributions that the host polymer and the hole-transport agent make to the photoconductivity are illustrated in Fig. I. The absorbance of the sample

199 l The American Physical Society

427

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8 APRIL 1991

PHY SICA L REVI EW LETT ERS

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FIG. I. Absorbance (left ordinate) and photosensitivity with 30 (right ordinate) of the bisA-NPDA polymer mixed nm, 1.4 wt% DEH. Inset: The photosensitivity (at 647.1 of DEH W/cm 2 illumination, 28 kV/cm field) as a function concentration. (The point at 0 wt% is an upper limit.) from in the red (left scale) is not significantly different blue. the in s absorb DEH since r polyme d the undope nic The strong absorption onset is due to the first electro photransition of the NPDA . The inset shows that the ty) tosensitivity upcfI (photoconductivity per unit intensi conincreases rapidly once sufficient DEH is added, k sistent with the establi shmen t of a conductive networ were for charge transpo rt. The measu red photoc urrents ty approximately linear in the bias voltage and in intensi contriant signific from free be to ined and were determ sample butions due to heating. The photosensitivity of a ence depend l spectra a has DEH wt% 30 with made ance of which closely follows the measu red optical absorb optical the sample (Fig. 1, right scale), suggesting that step in first the is phores chromo NPDA excitation of the charge generation. and Usually, epoxy polymers are aligned by poling e then cross-linked at elevated temper atures to produc 10 s prostable nonlinearity. Attemp ts to cure our sample sepaduced a loss of optical quality perhap s due to phase a slow ration. (DEH segregation may also accoun t for 20°C loss of photoconductivity of =50% per week at to not chose we Hence, °C.) 7 at and 50% per 2 months ents cross-link the samples, but to perform most experim field at room temper ature with a de poling electric EO present both to permit externa l control of the response and to simplify sample prepar ation. low We measur e the EO coefficients of the samples at er infrequencies (10-10 000 Hz) using a Mach- Zehnd 11 The observed response was proportional terferometer. of hysto the applied field E o except for a small amoun t or revteresis at fields below 5 kV /cm. Upon changi ng rises in se respon EO ed observ the field, applied ersing the longa matter of minutes to approx imately 90% of its time G.e., many hours) value as the nonlinear chromo obthe of ence depend time slow The . phores are aligned

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e FIG. 2. Right ordinate: Square of the electro-optic respons with n the index of refraction) for p-polarized 632.8-nm 356-µmradiation incident at an external angle of 60° on a nm 647.1 at y efficienc tion Diffrac : ordinate Left thick film. The for the same sample as a function of applied field. Inset: Etc. ratio of the latter to the former, which is proportional to (n 3r,ff,

nonserved EO response confirms that the source of the a Kerr linearity is molecu lar alignm ent rather than steadyeffect. The right ordina te of Fig. 2 shows the e. The state EO response of a 30-wt % DEH mixtur rell effective EO coefficient (neglecting birefringence) 2 lon2 =r33si n a+r 13 cos a is a linear combination of the where gitudin al r3 3 and transverse r 13 - r 23 coefficients field electric applied the of on directi the 3 directi on is the 0 1.63) is (norma l to the film plane) and a =32. l (for n = the angle of propag ation within the film. maThus, three of four necessary ingredients for a PR can be terial (absorption, transpo rt, and EO response) r easily measur ed in our polymer mixtures. The polyme traps, would be expected to supply a large density of , but which may be polymer chain ends of other defects nizthere is no guaran tee that these traps will be photoio le PR able as require d for the produc tion of an erasab 2 and space-charge field. The presence of the PR effect been therefo re a sufficient numbe r of useful traps has Two verified in the polymer films by volume holography. A. at beams g" "writin an Gaussi mutually cohere nt 2 in=647. l nm with equal intensities of 13 W/cm were and tersected in the sample at incidence angles of 30° respectively, thus producing interference 60° Gn the fringes with spacing Ag= l.6 µm oriente d 25° from 12 beam g" "readin , weaker much third, A film plane. Bragg directed opposite to the 60° writing beam was beams. writing the by formed grating the from ted diffrac ratio of The diffraction efficiency 11 was recorded as the writing diffracted to incident reading beam power. The ized p-polar but ation, polariz either be beams could , conreadou t produced ""6 ± 2 times stronge r signals -optic sistent with the expected anisotropy of the electro ·tensor. The influence of E o on 17 is shown in Fig. 2, left scale, where at each point the molecular alignment was allowed 1847

428

VOLUME66, NUMBER 14

PHYSICAL REVIEW LETTERS

to reach steady state in the field for 10 min. The value of 17-(n 3rerr1CLGE,J2'>..) 2 increases rapidly with Eo through increases in both the PR space-charge field E sc and the EO response n 3r err, where L -356 µm and G is a polarization and geometrical factor equal to 0.86. The 2 13 14 PR space-charge field in the standard model • • is

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8 APRIL 1991

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where Eog is the component of E 0 along the grating wave vector, Ed-kgksT/e-1.0 kV/cm, and kg-2;c/Ag =3.9x 10 4 cm- 1• The inset to Fig. 2shows17 divided by (n 3r 0 rr) 2, which should be proportional to Eic. The solid line is a least-squares fit by Eq. (I) with two adjustable 15 parameters: an overall multiplicative factor and A -0.034 which yields the effective density of photorefractive traps Np,-kfeeok 8 T/Ae 2=1.9Xl0 15 cm- 3 (e -2.9). The fit provides a good description of the shape of the experimental data. The maximum Esc reached, at an applied field of 126 kV /cm, from Eq. ( 1) is 26 kV /cm, a value which is larger than is generally attainable in inorganic PR materials. Clearly, the low dielectric constant of the polymer allows a larger field for the small trapped charge density N pr than might be possible in inorganics. The low values of 17, on the other hand, are to be expected due to the short optical path lengths and relatively small EO coefficients of this particular polymer. In preliminary measurements with the more highly nonlinear polymer NNDN-NAN (see Ref. 16 for structure and nonlinear properties) also doped with DEH, 17 values up to 10 - 3 have been observed in 350-µm-thick samples. To further establish that the observed gratings were indeed electro-optic in origin, we recorded gratings at large applied voltage, removed the writing beams, and observed the diffracted signal as the applied voltage was altered, with typical results shown in Fig. 3. From a to b, the grating was recorded for 2.5 min in a field of -84.3 kV /cm which had been applied long in the past. The fluctuations in the signal as it rises to its steady-state value are due to slow changes in the optical paths of the writing beams. At b, both writing beams and the applied field were switched off resulting in a rapid decrease in the signal. We attribute this to initial relaxation of the chromophore alignment, without loss of the space charge (except for the slow erasure by the much weaker reading beam), because the signal returns upon application of a field of either sign (c and din Fig. 3). The dynamics for signal recovery are consistent with that required for reorientation of the nonlinear chromophores and are far slower than the charging time of the circuit, thus ruling out strong Kerr effect contributions. At e, one of the writing beams was turned back on to erase the spacecharge field in a time of order 30 s, which should be compared with the dark lifetime of order 500 s, limited by the dark conductivity of the material. Thus, Fig. 3

0.50

Time (min.)

FIG. 3. Effect of applied field on grating readout by a weak reading beam: a-b, writing the grating in the presence of a (previously applied) negative voltage; b-c, writing beams and applied voltage turned off; c-d, positive voltage applied; d-e, negative voltage applied; and e on, grating erasure by one strong beam.

establishes the relative independence of space-charge formation and erasure from the molecular alignment whose only purpose is to permit readout, and reinforces the interpretation in terms of a photorefractive mechanism. The holographic gratings could be repeatedly written and erased dozens of times without noticeable degradation. The need for large external electric fields during grating recording can be understood because an applied field is necessary to assist in separation of the photogenerated charges. In the absence of an applied field during writing, Eq. (I) shows that E sc is equal to Ed= 1.0 kV /cm which would yield 77= 10-s even with a readout field of 126 kV/cm. As an additional test, we attempted to record gratings with Eo=O and, after blocking the writing beams so that any resulting space charge would remain fixed, applied a large Eo. No signal was observed larger than the approximately (l-S)x!0- 8 background level. This is also consistent with the well-known pro17 nounced dependence of the photogeneration rate and 18 the mobility on the external field in organic photoconductors. The PR effect has a number of unique features which distinguish it from any other known mechanism of grating formation. The clearest single signature of PR grat19 ings, dynamic two-beam energy coupling, has not yet been observed in our polymers because of the low diffraction efficiencies achieved. However, the electricfield dependence of the grating formation and readout, the correlation with photoconductivity and .EO response, and the cyclic erasability eliminate the possibility that

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the observed effect is due to any mechanism other than photorefraction, such as photochromism, photochemical effects, or photophysical changes in molecular structure. Hence, the observations detailed here, taken together, provide unequivocal evidence that the molecularly doped polymer system studied is indeed photorefractive. The ease of formulation of this material ensures that future efforts by synthetic chemists and optical physicsts alike will provide many new examples of photorefractive polymers. From such interdisciplinary studies, the detailed mechanistics, ultimate limitations, and possibilities of this new class of materials can be deduced. We thank B. Reck for polymer synthesis, D. Jungbauer for solvent and poling advice, and G. C. Bjorklund and D. M. Burland for critical support and encouragement. This work was supported in part by the U.S. Office of Naval Research.

1F. S. Chen, J. Appl. Phys. 38, 3418 (1967). Materials and Their Applications. edited by P. Giinter and J.-P. Huignard (Springer-Verlag, Berlin, 1988), Vol. l; Photorefractit•e Materials and Their Applications, edited by P. Giinter and J.-P. Huignard (SpringerVerlag, Berlin, 1989), Vol. 2. 3K. Sutter, J. Hulliger, and P. Giinter, Solid State Commun. 74, 867 (1990). 4S. Ducharme, J.C. Scott, R. J. Twieg, and W. E. Moerner, postdeadline paper to the Optical Society of America Annual Meeting, Boston, Massachusetts, 5-9 November 1990 (unpublished). 5K. D. Singer, M. G. Kuzyk, W. R. Holland, J. E. Sohn, S. J. Lalama, R. B. Comizzoli, H. E. Katz, and M. L. Schilling, Appl. Phys. Lett. 53, 1800 (1988). 6 0. J. Williams, in Nonlinear Optical Properties of Organic Molecules and Crystals I, edited by D. S. Chemla and J. Zyss 2Photorefractfoe

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(Academic, Orlando, 1987), pp. 405-435. 7G. R. Miihlmann, in Organic Materials for Non-linear Optics, edited by R. A. Hann and D. Bloor (Royal Society of Chemistry, London, 1989), p. 275. BM. Lardon, E. Lell-Diiller, and J. W. Weigl, Mol. Cryst. 2, 241 (1967). 9L, B. Schein, A. Rosenberg, and S. L. Rice, J. Appl. Phys. 60, 4287 (1986). IOM. Eich, B. Reck, D. Y. Yoon, C. G. Willson, and G. C. Bjorklund, J. Appl. Phys. 66, 3241 (1989). t ts. Ducharme, J. Feinberg, and R. Neurgaonkar, IEEE J. Quantum Electron. 23, 2116 (1987). 12This geometry was chosen to enhance the sensitivity by providing a reasonable projection of the space-charge field along the film normal, i.e., the poling direction. 13J. Feinberg, D. Heiman, A. R. Tanguay, Jr., and R. W. Hellwarth, J. Appl. Phys. 51, 1297 (1980); 52, 537 0980). ' 4A. Twarowski, J. Appl. Phys. 65, 2833 (1989), derived the E sc expected in the case of organic photoconductors with Onsager geminate recombination. Though the functional form of Esc differs from Eq. (I), the data in Fig. 2 are not sufficiently accurate to distinguish the models. 15Though the data of Fig. 2 were measured on the same sample, considerable scatter in EO coefficients (25%) was observed due to sample inhomogeneity. Also, the EO coefficient of the polymer was approximately 10% higher at 632.8 than at 647.1 nm and separate measurement revealed that the piezoelectric coefficient dii- +0.2 pm/V, which may contain a contribution from electrode attraction, also made a significant 00%-30%) contribution to the observed EO measurements. The data in Fig. 2 are not corrected for these observations, thus an overall fitting factor of =2.5 is not unreasonable. t6o. Jungbauer, B. Reck, R. Twieg, D. Y. Yoon, C. G. Wilson, and J. D. Swalen, Appl. Phys. Lett. 56, 2610 (1990). ' 7T. E. Goliber and J. H. Perlstein, J. Chem. Phys. 80, 4162 0984). 18J. X. Mack, L. B. Schein, and A. Peled, Phys. Rev. B 39, 7500 (1989). 19V. L. Vinetskii, N. V. Kukhtarev, S. G. Odulov, and M. S. Soskin, Usp. Fiz. Nauk 129, 113 (1979) !Sov. Phys. Usp. 22, 742 (1979)].

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103

High photorefractive sensitivity in an n-type 45°-cut BaTiOa crystal M. H. Garrett,* J. Y. Chang, H. P. Jenssen, and C. Warde Center for Materials Science and Engineering, the Department of Materials Science and Engineering, and the Department of Electrical Engineering and Computer Science, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Room 13-3157, Cambridge, Massachusetts 02139 Received September 18, 1991 We report the beam-coupling properties of a cobalt-doped oxygen-reduced n-type barium titanate crystal in the 0° and 45° crystallographic orientations. Oxygen reduction improved the response time of the 0° -cut crystal by a factor of -4.5 without diminishing the beam-coupling gain. The 45° -cut crystal has a peak gain of -38.7 cm- 1, a response time of -21 ms, and a photorefractive sensitivity of 3.44 cm3 /kJ. We infer from response time measurements an equivalent percentage change in the de dielectric constant and the mobility with respect to crystallographic orientation.

Barium titanate (BaTi0 3) has the largest known third-rank electro-optic tensor component of inorganic crystals, r 42 = 1640 pm/v,1 which helps give high beam-coupling gain without applied fields. In the 0°-cut orientation (a 0° cut has crystal faces normal to the a and c axes), it is difficult for beamcoupling experiments to access the r 42 electro-optic coefficient except at steep entrance angles. It is more easily accessed in a 45°-cut crystal when the grating wave vector is at 45° with respect to the c axis. 2 This orientation was previously examined by Ford et al., 2 Pepper, 3 and Ewbank et al., 4 who all found that the beam-coupling gain was substantially higher in the 45° crystallographic cut, up to 65 cm- 1. 4 However, the response time of BaTi03 is usually slow (e.g., 1 sat 1 W/cm 2 ), and thus the photorefractive sensitivity is relatively low. Nevertheless, we note that the figure of merit for BaTi03, n 3r.rr/E, in the 45° cut is larger than that for many photorefractive inorganic crystals at 9.5 pm/V. Using the formulation for the theoretical limit of the response time as described by Yeh, 5 one finds that the response time of BaTi03 could be as fast as 0.1-1.0 ms at l W/cm 2 (depending on the quantum efficiency) in the 45°-cut orientation. We report in this Letter that the 1/e decay time of the diffraction efficiency, or the response time, of cobalt-doped BaTi03 (0° cut) is lowered by a factor of -4.5 when oxygen reduced to -34 ms measured at 1 W/cm 2, while one obtains a beam-coupling gain equivalent to that of the as-grown cobalt-doped crystal. In addition, experimentally the response times are nearly equivalent for the 0° and 45° orientations, which we attribute to the anisotropies of the de dielectric constant and mobility that we imply change in the same manner with crystallographic orientation. The response time of the 45° -cut crystal is slightly faster and also has a much higher gain than that of the 0° -cut crystal. Thus we find that this n-type BaTi03: Co crystal, fabricated into a 45°-cut orientation, has a photorefractive sensitivity comparable with that of other photorefractive oxides. 0146-9592/92/020103-03$5.00/0

The photorefractive sensitivity, defined as the index change per energy absorbed per unit volume, is given in Ref. 6 as S = y}l./47rafr, where y is the beam-coupling gain coefficient, }I. is the wavelength of light, a is the absorption coefficient, I is the incident intensity, and T is the 1/e decay time of a diffracted readout beam that monitors the grating when it is erased from its steady-state value. To determine these parameters for 0°- and 45°-cut BaTi03, we first grew a cobalt-doped 20 parts in 106 (in the melt) BaTi03 boule by using the top-seeded solution growth technique. 7 (Cobalt doping is known to increase the gain. 8 ) An x-ray-oriented 0°-cut sample was extracted from the boule. The crystal was then oxygen reduced to obtain n-type BaTi03 (confirmed by the direction of beam coupling with respect to the polar axis). This sample was fabricated into a single-domain crystal with an etching and electrical poling technique described in Ref. 9. It was polished to an optical quality finish with dimensions 6.90 mm x 5.16 mm x 4.52 mm, where the c axis was parallel to the long dimension. It should be pointed out that oxygen reduction lowered the response time by a factor of -4.5 for this crystal, although we have measured even faster response times in other reduced BaTi03 crystals. Beam-coupling gain and response time measurements were made with each crystal placed in a cuvette filled with silicone oil. To obtain an accurate measurement of the gain in the 0° cut (to avoid beam-fanning effects), we used argon-ion laser illumination at 514.5 nm that was ordinary polarized, and the grating wave vector was parallel to the c axis; thus we accessed the smallest electro-optic coefficient r 13 • The pump beam, with an intensity of 1 W/cm2 , was expanded to a diameter of 2 cm, which completely enveloped the I-mm-diameter signal. The pump-to-signal intensity ratio was set so that pump depletion could be avoided. The Debyescreening wave vector of the 0°-cut crystal was experimentally determined to be k 0 (0°) = 1. 76 x 10 7 m- 1 and the intensity-dependent factor to be © 1992 Optical Society of America

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OPTICS LETTERS I Vol. 17, No. 2 I January 15, 1992

71(/) = 0.51. 10 These values were used to predict the gains for different polarizations and orientations of the grating wave vector with respect to the c axis. The 0°-cut sample was then fabricated into a 45°-cut crystal. 2 The crystal had a thickness of 0.870 mm (which may help to minimize beamfanning effects 4). Beam-coupling measurements were then made with extraordinary and also ordinary polarization, and the bisector of the grating writing beams was normal to the entrance face, which makes the grating wave vector at 45° with respect to the c axis. Measurements of the gain with extraordinary polarization and with the grating wave vector parallel to the c axis for a 0°-cut n-type BaTi03:Co crystal, from the same boule as the other samples and reduced to the same level, were also made to quantify the gain in this configuration as shown in Table 1. A weak He-Ne laser probe beam (,\ = 633 nm) was Bragg matched to the grating, and the temporal evolution of its diffraction was measured to determine response times when the writingbeam ratio was 24:1 and a beam of 1 W/cm 2 erased the steady-state grating. 11 The functional dependence of the electro-optic gain coefficient, including effects of deep and shallow traps, is given by the following equation 10 : 27Tn

3

r,rr

kBT (

'Y = -A-cos(fJ;) _e_

stant in the direction of the grating wave, vector is determined by the transformation E = kg · € . k where € is the relative de dielectric tensor with co~: ponents Eu = E22 = 3600 and E33 = 150, 14 and k is the normalized grating wave vector. Therefore the de dielectric constant for the 45° cut is E(45°) = 1875. The Debye screening wave vector in the 45°-cut crystal is given in terms of the 0°-cut screening wave vector as ko( 45°) = [ E33/E( 45°)] 112ko(0°) = 0.283k 0(0°). Shown in Table 1 are the predicted values of the gain obtained by using the previously described transformations from the 0°-cut data and Eq. (1), and for extraordinary polarization, e 1 · e 2 * = cos(2fJ;). We observe that when the larger electro-optic coefficients r 33 and r.2 are used, regardless of the thickness (3.4 and 0.87 mm), the measured gains are lower than the predicted values. Even lowering the beam ratio by an order in magnitude did not change our experimentally determined gain for the 45°-cut crystal. We believe that this is because of the large amount of fanning that we observed. We expect that even thinner crystals may yield a value for the gain much closer to the predicted value. The Debye screening wave vector for the 45°-cut crystal was determined from gain versus grating period measurements to be k 0(45°) = 5.35 x l06 m- 1, with a peak gain of -38.7 cm- 1 and a response time of -21 ms. The ratio is k 0 (45°)/k 0(0°) = 0.30, which is in good agreement with the predicted value of 0.28. 15 A formulation for the response time, r, shows that Tis directly proportional to the de dielectric constant and is inversely proportional to the mobilitylifetime product µ,T, and the excitation cross sections; i.e., T 10 cm - 1 were observed at 647 nm (writing intensity of 1 W cm-2, applied field of 40 V /Lm- 1 ).

temperature Tg of the films is -40 ·c, considerably below the Tg of pure PVK (212 °C), owing to plasticization by the NLO chromophore and residual solvent. A PR material must both be photoconductive and have a nonzero linear electro-optic coefficient. To demonstrate that our samples are photoconductive, we measured the de photoconductivity for illumination at 676 nm and determined it to be 2.0 (±0.2) x 10- 12 (n- 1 cm- 1)/(W cm- 2 ). To obtain a nonzero electro-optic coefficient r;J, we made the material acentric through poling of the NLO chromophores in an externally applied de electric field. Owing to the low Tg and the relatively small size of the chromophore, poling occurs at room temperature. With a Mach-Zehnder interferometric technique, 9 n 3 r 13 was found to increase linearly with the externally applied field, reaching a value of 2.4 (±0.2) pm v- 1 for E = 40 V µm- 1 measured at A= 830 nm. When the de field is removed, n 3 r 13 goes to zero in 8 kV cm- 1 and a crystal or fringe displacement speed around 5 µm s- 1 at the green line p, =514 nm) of an argon laser.

I. Introduction

Fringe illumination of a photorefractive Bi 12 Si0 20 (BSO) crystal in the blue-green spectral region gives rise to the recording of a real-time phase volume hologram. The low writing and erasure energy ( ls illuminate the 0 BSO crystal. When moving the recording medium at a constant speed v, the spatial and temporal distribution of the incident intensity is given by the following expression:

Rfi +Sfi

r

I

3. Amplitude of the index change shifted component and experimental results

-R3- exp(o:z), 53

- =-(r-o:)S-- - - exp(o:z). dz 2 2 R2 +S2 0

15 August 1981

(8)

0

The waves intensities emerging from the material

(z = /) and solutions of equations (8) are given by the

I(x, t) = 10 [I+ m cos (K(x - vt))];

following relations:

I 0 =IRo +ISo is the total incident intensity; m and A =A./2 sin 80 are respectively the fringe modulation and spacing at the recording wavelepgth A.; 280 is the angle

RB +sfi exp[(r-o:)/], RB +SB exp(rt)

Is=Is 0 -

IR =IR

0

R2 +s2 0 0 ----------exp(-o:/). + sfi exp(rl)

Rfi

K = 211/A,

between the writing beams outside the material. Since the incident fringe pattern is now moving at a constant speed, the steady state value of the photoinduced index change must be derived from a treatment which consider the finite response time of the medium with respect to the displacement speed. Such effects caused by the finite temporal response of the interaction have been also previously encountered in wave mixing experiments with other non-linear materials such as LiNb0 3 , liquid crystal, ruby, Ge [5,14-16]. In the BSO, the photoinduced refractive index under fringe illumination takes the form:

(9)

These relations express an exponential gain on the signal wave when the condition r > o: is satisfied (6]. r is the gain per unit of length induced by the n/2 shifted grating component. This gain arises physically from the self interference in the same direction of the input signal wave S and the diffracted one under phase matching conditions. As proposed in ref. [7], a direct figure of merit of the energy transfer is the effective gain r 0

where n 0 is the crystal refractive index and

ls with reference beam "Yo =ls without reference beam'

t:i.n(x, t)

'Yo=

n (x, t) = n 0

t

JEsc(x, t') o(t - t') dt'.

(11)

Eq. ( 11) is the convolution of the photo induced space charge field Esc at time t' and position x with the crystal temporal response (t). C0 is a constant of proportionality. With the recording conditions used in the experiment (-Ea varies from 1 to 10 kV cm-1; fringe spacing A> 3.5 µm), the space charge field amplitude is proportional to the incident light intensity I(x, t) and to the externally applied electric field E 0 [1, 8] . For the range of spatial frequencies considered, the contribution of the diffusion field ED will be neglected with respect to Ea (E 0 = 450 V cm-1 at A= 3.5 µm; E 0 > 1 kV cm-1 ). Eq. (11) can thus be written as follows:

o

(RB > s6) of the two

'Yo "' exp (rt).

The physical result of the beam coupling and related energy exchange is an increase of the fringe modulation level m (z) when the two waves R and S propagate through the crystal thickness:

m(z)

= C0

0

(RB + sfi) exp (rz) . RB +SB exp (rl)

For larger intensity beam ratio interfering waves r 0 becomes:

+ t:i.n(x, t),

=2 [Is(z)/IR(z)]lf'.2,

using relation (9): m(z) = m 0 exp(~ rz), where m 0 is the incident fringe modulation level due to coherent interference of R 0 and S 0 (m 0 8 kV cm- 1 . Solid line: experi· mental curve. Dashed line: theoretical curve. • A = 3.5 .um; +A =6,um;a A= 9.5,um.

·10

10

20

30

40 U"(µms·I)

Fig. 4. Gain "Yo as a function of the fringes displacement speed.

E0 = 6 kV cm- 1 ; A= 3.5 ,um. Solid line: experimen tal curve. Dashed line: theoretical curve.

the applied electric field. The absorptio n coefficient of BSO, including interfaces reflections, being equal to a= 1.9 cm-1 at A.= 514 nm, the output signal beam is seen amplified for applied electric field E > 8 kV 0 cm-1. (Is= 1.5 Is for £ = 10 kV cm-1). 0 0 The measured exponent ial gain r::::: 2.4 cm-1 for £ 0 = 10 kV cm-1 is inferior to the reported one in LiNb0 3 crystals because of the lower saturation value of the photoind uced index change in BSO crystals (limitatio n of the space charge field amplitud e by the number of trapping centers). Fig. 4 gives the evolution of 'Yo versus the displacement speed for a given value of the applied electric field £ =6 kV cm- 1. As pre0 dicted from the theory, resonance on the energy transfer is obtained for a displacement speed such as Kv r 0 ::::: 1 (r ::::: 120 ms; v = 6 µm s- 1 for a fringe spacing A 0 =3.5 µm). No apparent beam coupling is observed when reversing the crystal (or fringe) displacement. Fig. 5 gives the optimum displacement speed v as a 0 function of the applied electric field E . 0 If we now place a retro-mir ror on the reference beam path (four-wave mixing configura tion), we also measure an appreciable increase in the conjugate beam intensity (factor X 3). Such an influence of the beam coupling process in four-wave mixing experime nts had been previously noted and studied in ref. (17] with LiNb0 3 crystals as a dynamic recording medium or more recently with BaTi0 [18]. 3

electrodes in the 001 direction and the phase shift ¢ is introduced by moving either the fringes or the crystal at a constant speed. The crystal velocity is controlle d by a low speed motor; the fringes displacem ent is obtained with a piezoelectric mirror placed on the pump beam path and driven by a ramp generator . A photodiode measures the intensity of the transmitt ed signal beam with or without the pump reference beam. On fig. 3' the effective gain 'Yo and the exponent ial gain r are plotted versus the applied electric field £ for dif0 ferent fringe spacings. F9r each experime ntal point of the curve, the velocity has been adjusted to the opti· mum value v0 (Kv 0r "'=' 1). For fringe spacings A> 3 µm, 10 remains unchanged, since the saturatio n value of the photoindu ced index modulati on is nearly constant in that spatial frequency range for the "drift" recording mode. For £ 0 = 0, diffusion process gives a low contribution to the beam coupling ('Yo"'=' 1-2 for £ =0). The 0 theoretical curve has been fitted to the experime ntal points with the following data: i/lo =O; Kv r::::: l. Dis0 crepancy with the theoretic al exponent ial variation law for applied field £ > 6 kV cm-1 may be due to 0 several parameters which limit the saturation value of the photoindu ced index change such as: non-unifo rm index modulatio n in the crystal volume, gaussian shape of laser beams, multiple beam reflection s, and saturation of the space charge field for large values of

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JO

non-local response or may be introduced by moving the fringes at a constant speed if the physical effect in. volved in the material gives a local response. This last method can be envisionned for amplification of optical beams in wave mixing experiments using other materi. als such as semiconductors or saturable absorbers where the index changes are initiated with high power densities on the pump beam.

• /\=3.Sµm • /\ = 6 µm

20

"

15 August 1981

-·-·-·-----

References Fig. 5. Optimum fringes displacement speed uo as a function of the applied electric field Eo. Solid line: experimental curve. Dashed line: theoretical curve. • /\ = 3.5 µm; + /\ = 6 µm.

4. Conclusion We have shown in this paper that dynamic holography with highly sensitive BSO crystals makes possible the amplification of CW coherent light beams. Improvement on the energy transfer in this crystal has been obtained by moving the fringes or the crystal at a constant speed. This induces a rr/2 phase shifted component of the index modulation which causes the beam coupling. The exponential gain r derived from the coupled waves equations has been expressed as a function of the saturation value of the photoinduced index change (r = ((4rr~n 5)/(/\ cos 8)] sin 1/1). The measured value of r is 2.4 cm-1 at/\= 514 nm (£0 = 10 kV cm-1 ). This corresponds to an increase of the incident object wave intensity of about 50% after passing through the crystal. The gain does not depend on the incident intensity for usual light levels (10 > l mW cm-2) and is nearly independent on the intensity ratio of the two interfering waves. As analysed in ref. [9] by Vinetskii et al., any interaction material which changes the refractive index under fringe illumination can be seen as a non-linear medium for image or complex wavefronts amplification in wave mixing experiments. The non-linearity is self induced if the material has a

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[ l] J.P. Huignard, J.P. Herriau, P. Aubourg and E. Spitz, Optics Lett. 4 (1979) 21. [2] J.P. Huignard, J.P. Herriau, L. Pichon and A. Marrakchi, Optics Lett. 5 (1980) 436. [3] D.L. Staebler and J.J. Amodei, J. Appl. Phys. 43 (1972) 1042. [4] V. Markov, S. Odulov and M. So skin, Opt. and Laser Tech. April 1979. [5] N. Kukhta.rev, V. '.\1arkov and S. Odulov, Optics Comm. 23 (1977) 338. [6] N. Kukhta.rev, V. '.\1arkov, S. Odulov, M. Soskin and V. Vinetskii, Ferroelectrics 22 (1979) 949, 961. [7] A. Marrakchi, J.P. Huignard and P. Gunter, Appl. Phys. 24(1981) 131. [8] J.P. Huignard, J.P. Herriau, G. Rivet and P. Gunter, Opt. Lett. 5 (1980) 102. [9] V.L. Vinetskii, N.V. Kukhtarev, S.G. Odulov and M.S. Soskin, Soviet Phys. Usp. 22 (1979) 742. [10] Yu.A. Anan'ev, Soviet. J. Quantum Electron 4 (1974) 929. [ll] J. Feinberg, D. Heiman, A.R. Tanguay and R.W. Hellwarth, J. Appl. Phys. 51 (1980) 1297. [12] H. Kogelnik, B.S.T.J. 48 (1969) 2909. [13] V. Kondilenko, V. Markov, S. Odulov and M. Soskin Optic:i Acta 26 (1979) 239. [14] D. Fekete, J. Au Yeung and A. Yariv, Optics Lett. 5 (1980) 51. [15] A.L. Smirl, R.F. Boggess and F.A. Hopf, Optics Comm. 34 (1980) 463. [16] P.F. Liao and D.M. Bloom, Optics Lett. 3-4 (1978). [17] N. Kukht:liev and S. Odulov, Optics Comm. 32 (1980) 183. [18] J. Feinberg and R.W. Hellwarth, Optics Lett. 5 (1980) 519.

Photore fractive incohere nt-to-co herent optical converte r Y. Shi, D. Psaftis, A. Marrakchi, and A. R. Tanguay, Jr.

Y. Shi and D. Psaltis are with California Institute of Technology, Division of Engineering & Applied Science, Pasadena , California 91125; the other authors are with University of Southern California, Departme nts of Electrical Engineering and Materials Science, Optical Materials & Devices Laboratory, Los Angeles, California 90089. Received 17 Septemb er 1983. Sponsored by William T. Rhodes, Georgia Institute of Technology. 0003-6935/83/233665-03$01.00/0. © 1982 Optical Society of America.

Photorefr active materials have been extensively used in recent years as real-time recording media for optical holography.1·2 One prospective application of real-time holography is in the area of optical information processing; for example, the correlation between two mutually incoherent images has recently been demonstrated in real time in a four-wave mixing .geometry. 3 Often, however, the information to be processed exists only in incoherent form. High performance spatial light modulators 4 are thus necessary in many optical information processing systems to convert incoherent images to coherent replicas for subseque nt processing. We report in this Communication the successful demonstration of real-time incoherent-to -coheren t images transduction through the use of holographic recording in photorefractive crystals. Several possible configurations and experimental results are presented. The interference of two coherent beams in the volume of a photorefractive crystal generates nonuniformly distributed free carriers, which are redistributed spatially by diffusion and/or drift in an external applied field. The subseque nt trapping of the free carriers in relatively immobile trapping sites results in a stored space-charge field, which in turn modulates the index of refraction through the linear electrooptic effect. 5 Thus a volume phase hologram is recorded. If the two coherent beams are plane waves, a uniform phase grating is established. An incoherent image focused in the volume of the photorefractive material will spatially modulate the charge distributi on stored in the crystal. This spatial modulation can be transferred onto a coherent beam by reconstructing the holographic grating. The spatial modulation of the coherent reconstructed beam will then be a negative replica of the input incoherent image. tr'he holographic 1 December 1983 /Vol. 22, No. 23 /APPLIED OPTICS

465

3665

Fig. 3.

Photorefractive incoherent-to-coherent conversion of a transparency with grey levels.

Fig. 1. Experimental setup for incoherent-to-coherent conversion with phase conjugation in four-wave mixing. The writing beams I 1 and I 2 and the reading beam I 3 are generated from an argon laser (A = 514 nm). The phase conjugate beam I 4 is diffracted at the same wavelength. The transparency T(x ,y) is illuminated with a xenon arc lamp S and imaged on the BSO crystal with the optical system L 1 through a filter F(;\ = 545 nm). BS is a beam splitter, and Pis a polarizer placed in the output plane.

Fig. 4. Fourier transform of a grid pattern formed after a photorefractive incoherent-to-coherent conversion of the grid pattern pattern image.

(a)

(b) Fig. 2. Incoherent-to-coherent conversion utilizing phase conjugation in four-wave mixing: (a) spoke target; (b) USAF resolution target. The group 3.6, corresponding to a resolution of 14.3 lp/mm, is well resolved. 3666

grating can be recorded before, during, or after the crystal is exposed to the incoherent image. Therefore, a number of operating modes are possible. These include the grating erasure mode (GEM), the preerasure writing mode (PEWM), and the simultaneous erasure writing mode (SEWM), among others. In the grating erasure mode, a uniform grating is recorded by interfering the two writing beams in the photorefractive crystal. This grating is then selectively erased by incoherent illumination of the crystal is then selectively erased by incoherent illumination of the crystal with an image-bearing beam. The incoherent image may be incident either on the same face of the crystal as the writing beams or on the opposite face. When the absorption coefficients of the writing and imagebearing beams give rise to significant depth nonuniformity within the crystal, these two cases will have distinct wavelength-mat ching conditions for response optimization. In the preerasure writing mode, the crystal is preilluminated with the incoherent image-bearing beam prior to grating formation. This serves to selectively decay (enhance) the applied transverse electric field in exposed (unexposed) regions of the crystal. After this preexposure, the writing beams are then allowed to interfere within the crystal, causing grating

APPLIED OPTICS/ Vol. 22, No. 23 / 1 December 1983

466

formation with spatially varying efficiency due to vast differences in the local effective applied field. This technique will also work in the diffusion limit with no external applied field by means of a similar physical mechanism. In the simultaneous erasure writing mode, the conventional degenerate four-wave mixing geometry is modified to include simultaneous exposure by an incoherent image-bearing beam, as shown schematically in Fig. 1. Diffraction by a phase grating in the four-wave mixing configuration has been modeled following two different approaches. 6 •7 Common to both analyses, the diffracted intensity is proportional to both the readout intensity and the square of an effective modulation ratio, in the first-order approximation, and assuming no pump depletion. In addition, a uniform beam incident on the photosensitive medium at an arbitrary angle decreases the modulation ratio and hence the overall diffraction efficiency.s In the SEWM configuration, these effects can be combined with the diffraction of a conjugate beam in a four-wave mixing geometry to perform the incoherent-to-coherent image conversion. In particular, this conversion can be regarded as caused by selective spatial modulation of the grating by spatial encoding of the incoherent erasure beam. It should be noted here that a related image encoding process could be implemented in a nonholographic manner by premultiplication of the image with a grating.9 In our experiments, we have successfully produced incoherent-to-coherent conversions in all three operating mode configurations as well as in several modifications of the basic arrangements described above. We present here experimental results from our implementations of the simultaneous erasure writing mode. The experimental arrangement in one implementation is as shown in Fig. 1. The two plane wave writing beams (labeled I 1 and I 2) are generated from an argon laser ().. "' 514 nm) and interfere inside the photorefractive crystal to create a phase volume hologram. The readout beam I 3 , collinear with I 1 to satisfy the Bragg condition, diffracts the phase conjugate beam I 4 at the same wavelength and with increased diffraction efficiency when a transverse electric field is applied to the electrooptic medium. An incoherently illuminated transparency T(x,y) with intensity J.,(x,y) (either quasimonochromatic or white light) is imaged in the plane of the crystal. The beam splitter BS separates the diffracted signal from the writing beam; the Polaroid filter P in the output plane eliminates the unwanted scattered light to enhance the signal-to-noise ratio. 10 The photorefractive material utilized was a single crystal of bismuth silicon oxide (BSO), cut to expose polished (110) faces, and of dimensions 7.3 X 6.9 X 1.3 mm 3 . A transverse electric field Eo = 4 kV /cm was applied along the [110] axis perpendicular to the polished faces. The carrier frequency of the holographic grating, f = 300 Ip/mm, was within the optimum range for the drift-aided charge transport process. 11 The vertically polarized coherent writing beam and signal intensities werel 1.2 = 0.4 mW/cm2 and ls == 8 mW/cm2, respectively. Figure 2 shows the converted images obtained from two binary transparencies (a spoke target and a USAF resolution target). The illumination was provided by a xenon arc lamp through a broadband filter centered at ).. "' 545 nm (FWHM "' 100 nm). An approximate resolution of 15 lp/mm was achieved without optimizing factors such as the optical properties and quality of the crystal, the depth of focus in the bulk of the medium, the carrier frequency of the grating, and the relative intensities and wavelengths of the various beams. This spatial bandwidth is comparable with that obtained with a PROM 12 or a liquid crystal light valve.13 · The image shown in Fig. 3 demonstrates the capability of

the technique to reproduce many grey levels. To obtain this image, a negative transparency was illuminated with blue light (A"' 488 nm) derived from an argon laser and was focused on the BSO crystal. The holographic grating was recorded with green light ().. = 514 nm) and read out with an auxiliary red beam ().. "' 6328 A). The 2-D Fourier transform formed by a lens after the incoherent-to-coherent conversion of a grid pattern is shown in Fig. 4. The fundamental spatial frequency of the grid was ~1 Ip/mm. The existence of several diffracted orders and the well-focused diffraction pattern are positive indications that the device is suitable for coherent optical processing operations. Although these results are preliminary, they clearly demonstrate the feasibility of real-time incoherent-to-coherent conversion utilizing phase conjugation in photorefractive BSO crystals. This device has potential for incoherent-to-coherent conversion with high resolution, which can be realized by optimizing the optical properties and quality of the crystal, the depth of focus in the bulk of the medium, the carrier frequency of the grating, and the relative intensities and wavelengths of the various beams. In addition, such a device is quite attractive from considerations of low cost, ease of fabrication, and broad availability. With such a device, numerous optical processing functions can be directly implemented that utilize the flexibility afforded by the simultaneous availability of incoherent-to-coherent conversion and volume holographic storage.

The authors would like to thank F. Lum for his technical assistance and Y. Owechko, J. Yu, and E. Paek for helpful discussions. This research was supported in part at USC by RADC under contract F19628-83-C-0031, the Defense Advanced Research Projects Agency, the Joint Services Electronics ProgrlllII, and the Army Research Office and at Caltech by the Air Force Office of Scientific Research and the Army Research Office.

References 1. P. Gunter, "Holography, coherent light amplification, and optical phase conjugation," Phys. Rep. in press (1983). 2. D. M. Pepper, Opt. Eng. 21, 156 (1982). 3. J. 0. White and A. Yariv, Appl. Phys. Lett. 37, 5 (1980). 4. A. R. Tanguay, Jr., in Proceedings, ARO Workshop on Future

Directions for Optical Information Processing, Lubbock, Tex., May, 1980 (1981), pp. 52-77. 5. D. L. Staebler and J. J. Amodei, J. Appl. Phys. 43, 1042 (1972). 6. J. Feinberg, D. Heiman, A. R. Tanguay Jr., and R. W. Hellwarth, J. Appl. Phys. 51, 1297 (1980).

7. N. V. Kakhtarev, V. B. Markov, S. G. Odulov, M. S. Soskin, and V. L. Vinetskii, Ferroelectrics 22, 961 (1979). 8. W. D. Cornish and L. Young, J. Appl. Phys. 46, 1252 (1975). 9. R. Grousson and S. Mallick, Appl. Opt. 19, 1762 (1980). 10. J.P. Herriau, J.P. Huignard, and P. Aubourg, Appl. Opt. 17, 1851 (1978). 11. J.P. Huignard, J.P. Herriau, G. Rivet, and P. Gunter, Opt. Lett. 5, 102 (1980). 12. Y. Owechko and A. R. Tanguay, Jr., Opt. Lett. 7, 587 (1982). 13. P. Aubourg, J.P. Huignard, M. Hareng, and R. A. Mullen, Appl. Opt. 21, 3706 (1982). 1 December 1983 / Vol. 22, No. 23 /APPLIED OPTICS

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Reprinted from Optics Letters, Vol. 11, page 306, May, 1986. . Copyright © 1986 by the Optical Society of America and reprinted by permission of the copyright owner.

Parallel image subtraction using a phase-conjug ate Michelson interferomete r Arthur E. Chiou and Pochi Yeh Rockwell International Science Center, Thousand Oaks, California 91360 Received December 2, 1985; accepted February 20, 1986 A phase-conjugate Michelson interferometer using an internally self-pumped barium titanate crystal as reflectors has been constructed to perform parallel image subtraction, intensity inversion, and exclusive OR logic operation. These operations are independent of the optical path differences and phase aberration.

We report parallel image subtraction, exclusive OR (XOR) logic operation, and intensity inversion using a phase-conjugate Michelson interferometer that consists of a beam splitter and a phase-conjugate reflector in place of the usual interferometer mirrors (see Fig. 1). Such an interferometer is equivalent to the double phase-conjugate interferometer and also exhibits time reversal. 1 Image subtraction has been a subject of considerable interest in signal processing. Electronic digital processing of images is slow because of its serial nature. Optical techniques offer the capability of parallel processing over the entire images. The parallel processing is versatile and inherently faster. The technique of optical image synthesis by the addition and subtraction of the complex amplitude of light was first described by Gabor et al. 2 The basic principle consists of spatially modulating the two images by periodic waves that are mutually shifted by a phase of 180 deg. 3•4 Although there are other techniques of image subtraction, 5-8 interferometers such as the MachZehnder or Michelson offer convenient ways for the addition and subtraction of the complex amplitude of images.9-U In. the interferometric methods the subtraction is obtained by introducing the two images symmetrically in the two arms of the interferometer where a path difference corresponding to 1l' phase shift exists between them. It is known that the interferometers are extremely difficult to adjust and that they cannot easily maintain the fixed path difference. In addition, only the center fringe is useful for image subtraction or addition. In many cases the center fringe is not large enough to cover all the images. The new method of parallel image subtraction by phaseconjugate interferometry described in this Letter eliminates these two problems. In our phase-conjugate Michelson interferometer, a plane wave with amplitude E is divided into two by the beam splitter BS2. Each of these two beams passes through a transparency and is then reflected by a selfpumped BaTi03 phase conjugator. 12- 15 When these two phase-conjugated beams recombine at the beam splitter, image subtraction is obtained at the usual interferometer output port A (see Fig. 1). Let the 0146-9592/86/050306-03$2.00/00

reflection coefficient of the phase conjugator be p. 16 The image intensity at the output port A is given by IA(x,y)

= IEJ 2IPl 2lt*r'T1 (x,y) + r*tT2 (x,y)l 2 ,

(1)

where r and r' are the amplitude reflection coefficients of the beam splitter BS2 for beam incidence from the left and right sides, respectively; t is the amplitudetransmission coefficient; * denotes complex conjugation; and T 1(x, y) and T2(x, y) are the intensity-transmittance functions of transparencies 1 and 2, respectively. Using Stokes's relation17 r't*

+ r*t = 0,

(2)

Eq. (1) for the intensity at the output port becomes IA(x,y) = IEJ~Pl 2RTIT1(x,y)- T2(x,y)l 2 ,

(3) where R and T are intensity reflectance and transmittance, respectively, of the beam splitter BS2. Note that the output intensity is proportional to the square of the difference of the intensity-transmittance functions. The 180-deg phase shift between the two images as dictated by the Stokes's relation plays a crucial role in this image-subtraction technique. The Stokes relation holds for any lossless dielectric mirror and results directly from the time reversal. Such a relation was first predicted by Stokes 18 in the nineteenth century and was not proved experimentally until recently by using phase-conjugate reflectors.1 The phase-conjuAr LASER

514.5 nm

Fig. 1. Schematic diagram illustrating the basic idea of coherent image subtraction and addition by a phase-conjugate Michelson interferometer. © 1986, Optical Society of America

469

May 1986 I Vol. 11, No. 5 I OPTICS LETTERS

307

Fig. 2. Experimental results for the image subtraction and addition by the phase-conjugate Michelson interferometer. The horizontal and the vertical bars are the images of transparencies 1 and 2, respectively, when the illuminating beam for the other arm is blocked. The checkerboard patterns at upper and lower right are the intensity distribution of the coherent subtraction and addition, respectively, of the two images.

performed with weighing factors T and R, respectively. The subtraction, however, is independent of the ratio of R and T. Typical experimental results are shown in Fig. 2. The horizontal and the vertical bars in the upper photos are the images at output port A(see Fig. 1) of transparencies 1 and 2, respectively, when the illuminating beam for the other arm is blocked. The upper right checkerboard pattern represents the coherent subtraction of the two images due to destructive interference when both illuminating beams are present. Note that the intensity distribution where subtraction takes place is fairly uniform and is very close to that of the true dark background (four dark squares where the dark regions of the bars overlap). The lower photos are the corresponding results at output port B where image addition takes place. The image subtracter can also perform logic operation. Consider the case when both transparencies are either 1or0. According to Eq. (3), a complete cancellation would require that these two transparencies be identical. An output intensity of 1 will appear at port A when only one of these two transparencies transmits. Thus such an image subtracter can act as an XOR gate. The truth table of such a logic operation is given in Table 1. In the case when the transparencies are encoded with a matrix of binary data, such an

gation process also eliminates the problems of phase distortion and the critical alignment requirement associated with conventional interferometry. At the image plane B (see Fig. 1) the intensity distribution is given by IB(x, y) = IEJ 2IPl 21TT1(x, y) + RT2(x, y)l 2. (4) In our experiment, an argon-ion laser (514.5 nm) with output power of a few hundred milliwatts is used as the coherent light source. The laser output, after spatial filtering through a 5X microscope objective and a 25-µm-diameter aperture, is expanded and collimated to a I-cm-diameter beam size. The collimated beam is split into two by beam splitters BS2 (intensity transmittance, 64%) to illuminate the two transparencies (the horizontal and the vertical triplet bars from the U.S. Air Force Resolution Chart). They are then redirected and focused ({-number, f/50) onto the a face of a BaTi03 crystal with angles of incidence of approximately 15 and 19 deg. These beams are polarized in the xz plane (i.e., the plane of incidence) and excite only extraordinary wave in the crystal. Both beams self-pump the crystal and are phase conjugated with a reflectance of about 32%. Each of the phaseconjugated beams retraces its incoming path backward through the transparencies, and the two recombine at the beam splitter BS2. Note that the beam splitter BS 1 simply serves physically to separate the output port B from the input. It is not part of the interferometer. At each of the interferometer output ports, a lens ({-number, f /25) is used to image both objects onto the image plane. We have experimentally demonstrated that the two images are subtracted from each other in the image plane A and are added together in the image plane B. These operations are independent of the optical path lengths of the two arms of the interferometer. According to Eq. (4), the addition of the two images is

Table 1. The Logic Operations Represented by the Intensity IA T1

470

T2

IA

0:

IT1 - T21 2

1

1

0

1 0 0

0 1

1 1

0

0

308

OPTICS LETTERS I Vol.11, No. 5 I May 1986

(a)

(b)

(c)

Fig. 3. Experi mental results for the intensi ty inversi arm with the transp arency removed, (b) image of the on: (a) intensi ty distrib ution of the phase-conjugate beam in the first transpa rency in the second arm, and (c) the intensi ty inversion of (b). (a) l1"'IT1 (x,y)l 2 =1, (b)I2 "'IT2(x ,y)l 2, (c)I"' IT1(x,y ) - T2(x,y) l 2 = Jl -T2(x, y)j 2.

terferometer. The experimental results can be explained theoretically by the principle of time reversal. Similar work has been carried out indepe ndentl y by Kwong et al. 19 and was report ed recently. The author s acknowledge technical discussions with M. Khoshnevisan, M. D. Ewbank, and I. McMichael. This research was suppo rted partia lly by the U.S. Office of Naval Research under contra ct no. N00014-85C-0219.

,,, 1 MIN

1---l

l2 = 35µW-

Ji=

19µW-

ZERO-

-l

A

Refer ences 1. M. D. Ewbank, P. Yeh, M. Khoshnevisan, and J. Fein-

-

lbl

Fig. 4. Tempo ral fluctua tion of the intensi ty of each image and their cohere nt subtra ction for (a) a phase-conjug ate Michelson interfe romete r, (b) a Mach- Zehnd er interfe rometer with regular mirrors. !!,, is the signal output at the subtractio n port.

image subtra cter acts as a two-dimensional array of

XOR gates.

A special case of image subtra ction is intens ity inversion, which is obtain ed by removing one of the transparencies so that the transm ittanc e Ti(X, y) becomes unity in one arm. The experimental result is shown in Fig. 3. The tempo ral stabil ity of the subtra cted image intensities is compared in Fig. 4 for the case of an interferometer with a phase conjugator and that of one with regular mirrors. For both cases, the centermost square of the checkerboard patter n (the upper right in Fig. 2) is imaged on the detect or throug h a circula r apertu re with diame ter slightly smaller than the side of the square. The tempo ral evolution of intens ity I1 and I 2 is recorded in turn when the illuminating beam in the other arm is blocked. The intens ity difference .6. is then recorded when both beams are present. The improvement in tempo ral stabili ty of the subtra cted image intens ity is dramatic. In conclusion, we have demon strated coherent image subtraction, intens ity inversion, and exclusive OR logic operation using a phase-conjugate Michelson in-

471

berg, Opt. Lett. 10, 282 (1985). 2. D. Gabor, G. W. Stroke , R. Restrick, A. Funkh ouser, and D. Brumm , Phys. Lett. 18, 116 (1965). 3. See, for examp le,J. F. Ebersole, Opt. Eng.14 , 436 (1975). 4. See, for example, G. Idebetouw, L. Bernar do, and M. Miller, Appl. Opt. 19, 1218 (1980). 5. Y. H. Ja, Opt. Commun. 42, 377 (1982). 6. C. P. Grover and R. Tremb lay, Appl. Opt. 21, 2666 (1982). 7. C. Warde and J. I. Thack ara, Opt. Lett. 7, 344 (1982). 8. G. G. Mu, C. K. Chiang, and H.K. Liu, Opt. Lett. 6, 389 (1981). 9. K. Matsud a, N. Takeya , T. Tsujiuchi, and M. Shima da, Opt. Commun. 2, 425 (1971). 10. K. Patrosk i, S. Yokozeki, and T. Suzuki, Nouv. Rev. Opt. 6, 25 (1975). 11. W. T. Cathey, Jr., and J. G. Doidge, J. Opt. Soc. Am. 56, 1139 (1966). 12. J. 0. White, M. Cronin-Golomb, B. Fischer, and A. Yariv, Appl. Phys. Lett. 40, 450 (1982). 13. M. Gronin-Golomb, B. Fischer, J. 0. White, and A. Yariv, Appl. Phys. Lett. 41, 689 (1982). 14. J. Feinberg, Opt. Lett. 7, 486 (1982). 15. K. R. McDonald and J. Feinberg, J. Opt. Soc. Am. 73, 458 (1983). 16. These two beams enter the BaTiOa phase conjug ator at the same spot with approx imatel y the same angle of incidence and are considered parts of a composite beam. Thus the assump tion of a unique phase-conjugate reflectivity is legitimate. This has also been proved experimentally. 17. Z. Knittl, Optics of Thin Films (Wiley, New York, 1976), p. 242. 18. G. G. Stokes, Camb. Duhl. Math. J. 4, 1 (1849). 19. S. K. Kwong G. A. Rakuljuc, and A. Yariv, Appl. Phys. Lett. 48, 201 (1985).

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OPTICS LET.TERS I Vol. 13. No. 7 I July 1988

Real-time optical image subtraction using dynamic holographic interference in photorefractive media Pochi Yeh, Tallis Y. Chang, and Paul H. Beckwith Rockwell International Science Center. 1049 Camino Dos Rios. Thousand Oaks, California 91360 Received March 1, 1988; accepted April 24. 1988 A method of optical image subtraction is proposed. It provides parallel and real-time amplitude subtraction of two images by using holographic interference in photorefractive media. This new method provides an automatic phase shift of 180 deg between the two images and may be viewed as a real-time implementation of Gabor's imagesubtraction technique. Some experimental results are presented and discussed.

We propose a method of real-time optical image subtraction by using holographic interference in photorefractive media. In addition, we report an experimental demonstration of the subtraction by using such holographic interference in a BaTiOa crystal. Subtraction of two images was first proposed by Gabor et al. 1 by using a successive recording of the two images on a holographic plate with a 180-deg phase shift and the subsequent readout of the composite hologram. This method requires tl1e addition of a 180-deg phase shift, which must remain uniform across the aperture during the second recording. Image subtraction can also be achieved by other techniques.2·3 In most of the optical subtraction techniques, a phase shift of 180 deg must be introduced between the two images. Recently, time reversal in phase conjugation was used to provide such a phase shift. 4 Furthermore, the subtraction of two intensity patterns using phase-conjugate Michelson interferometers was proposed and demonstrated.5- 7 These methods may not be suitable for cascaded operations and reflective objects because of the round-trip propagation requirement. In this Letter we describe a new method that utilizes the simultaneous recording and readout of the two images in a photorefractive medium (or other nonlinear medium) by using a double Mach-Zehnder interferometric arrangement. The 180-deg phase shift is automatically present between the two sets of beams writing the image-bearing hologram by virtue of the Stokes principle. 8 When such a composite hologram is read out by a reference beam, parallel subtraction of the two images is obtained. We also report the experimental demonstration of this method using a BaTi03 crystal. Figure 1 shows the subtraction of two images on transparencies T 1 and Tz. These two transparencies are illuminated with two mutually incoherent light sources. For transparencies the illumination can be from behind the images, as opposed to frontal illumination that is required for reflective objects. There is no limitation on the thickness of the transparencies. The two image-bearing beams are directed toward the two input ports of a Mach-Zehnder interferometer. They are mixed into two beams, which are then recom0146-9592/88/070586-03$2.00/0

bined at a photorefractive crystal (or other nonlinear medium). Let E 1 and E2 be the field amplitudes of the mixed beams. At the photorefractive crystal, these two amplitudes can be written

E 1 = (tT 1A + r'T:zl3)exp(-ik 1 -r), E 2 = (rT 1A + t'T:zl3)exp(-ik 2 • r),

(1)

where A and B are the amplitudes of the two incoherent ilJuminating beams and t, r, t', and r' are the Fresnel transmission and reflection coefficients of beam splitter BS 1 (the prime indicates incidence from the back side of the beam splitter). These two field amplitudes form an interference pattern in the photorefractive BaTi03 crystal. The fundamental component of the intensity pattern can be written

CX::Tt - T2"

r, I I

.., ,,

r. I

r'. t'

e,

PM--

S.Ti03

REFERENCE

Fig. 1. Schematic of a real-time optical image-subtraction system. Transparencies T 1 and T2 imprinted with images are backilluminated by mutually incoherent light sources with amplitudes A and B, and the throughputs are directed to the input ports of a Mach-Zehnder interferometer. Beam splitter BS 1 mixes this light into two beams, E1 and~. which are then interfered in a photorefractive BaTiOa crystal. Spatial information on E2 is removed by pinhole PH. The interference pattern containing image information T1 is 180 deg out of phase with respect to the pattern containing image information T2• When read out by a reference beam that is then sampled through beam splitter 88 2, the intensity pattern will correspond to an algebraic subtraction of the two complex image amplitudes. © 1988, Optical Society of America

473

July 1988 I Vol. 13, No. 7 I OPTICS LETTERS 2 I= (rt•IT 112IAl 2 + t'r'*IT2'21Bl )exp(-iK · r)

+ c.c., (2)

where K = k2 - k1 and we have neglected the cross terms A• B and B* A because A and B are the amplitudes of two mutually incoherent beams. According to the Stokes principle, the Fresnel reflection and transmission coefficients for a lossless beam splitter satisfy the relation (3) rt• + tr'* = 0, into (3) Eq. ing where we have used t' = t. Substitut Eq. (2), we obtain 2 I= rt•(IT112IAl 2 - IT2l2IBl >exp(-iK · r) + c.c. (4) Thus the interference pattern (and recorded hologram) that contains image information T 1 is 180 deg out of phase with respect to that containing image information T 2• This 180-deg phase shift is independent of the path length of the two arms of the MachZehnder interferometer. Thus we have a simultaneous recording of two holograms containing image information with an inherent phase shift of 180 deg. When such a hologram is read out by a reference beam, the diffracted beam that is sampled through beam splitter BS 2 contains information about the difference between the intensities of the two images. To obtain high-contrast image subtraction, it is desirable that we have IAl 2 = IBl 2; note that the amplitude reflection and transmission coefficients (r, r', t) are arbitrary. We now modify the above analysis in order to perform algebraic subtractio n of the two complex amplitudes. If we place a pinhole in one arm of the interferometer, the pinhole will act as a spatial filter and eliminate spatial informati on about the images. Thus, for this case, the field amplitude E2 becomes E 2 = (rA

+ t'B)exp(- ik 2 • r),

587

der to read out the hologram efficiently (i.e., select only the fundamental component of the intensity pattern). The reference beam can be either coherent or incoherent with respect to the two illumination beams, as it simply reads the gratings written in the crystal. This new technique can also be used for the subtraction of two three-dimensional objects. When used for this purpose, the difference must be observed in a volume instead of a plane. In practice, this could be achieved by examining the beam in a scattering medium such as liquid-nitrogen vapor. Since this new method does not require that there be a round-tri p propagation through the transparencies, it can be used for multiple operations, cascaded operations, or even do-loop-type operations. Figure 2 shows the experimental setup used to demonstrate real-time optical image subtractio n using dynamic holographic interference in a photorefractive BaTiOs crystal. The light source is an argon-ion laser operating in a multilongitudinal mode at 515 nm with an output power of a few hundred milliwatts. The Faraday rotator and polarizers isolate the laser from retroreflections. The wedge picks off a small fraction (-4%) of the beam for the reference. The telescope arrangement of lenses L 1 and L2 expands and collimates the beam to a 1.5-cm diameter; the beam is then split into two equal components by beam splitter BS and directed to illuminate transpare ncies T 1 and T 2• A mirror arrangem ent consisting of M2, M3 , and M4 folds one of the beams to ensure that the two beams illuminating the two transparencies are mutually incoherent. Once the image-bearing beams are mixed by

(5)

provided that the de (spatial) components of images T 1 and T 2 are of the same magnitude. The intensity pattern of the interference becomes 2 I= (rt*T1*IAl 2 + t'r'*T2*IBl )exp(-iK · r) + c.c., (6) and when we substitute in Eq. (3) this becomes ·- T2•\B\ 2)exp(-iK · :r) + c.c. I""

(7)

We note that, according to Eq. (7), the holograms now contain the complex field amplitudes of the images instead of the intensities (absolute square of the amplitudes) of the images. Again, because of the Stokes principle, the holographic gratings formed by beams from these two input ports are shifted relative to each other by exactly 180 deg. In other words, the holographic grating that contains complex image T 1 is shifted 180 deg relative to that of complex image T 2 regardless of the path difference. When such a superposition of holograms is read out by the reference beam, the diffracted beam consists of the algebraic difference of the two complex fields. Strictly speaking, the amplitud e of the diffracted beam when observed at the difference plane is proportional to the complex conjugate of (T1 - T2l. The reference beam must be counterpropagating with respect to E 2 in or-

Fig. 2. Experimental setup used to demonstra te real-time optical image subtraction. The laser output is isolated and then split into a reference and a main beam by wedge W. The main beam is split into two components that are made mutually incoherent by having a sufficiently large pathlength difference. These beams illuminate two transparencies T 1 and T 2• The image-bearing beams are mixed on beam splitter BS 1 and made to interfere in the BaTi0 3 crystal. Spatial information is removed from one of the beams by the lens 1 3 and pinhole PH arrangement. The reference beam reading the holograms written in the crystal is sampled at the output of beam splitter BS 2 and imaged onto the screen. FR, Faraday rotator; P's, polarizers; 1 1-Ls. lenses; M 1-M 9 , mirrors. An alternative scheme using selfpumping of the input beams (with no pinhole or reference beam) can also be used for the subtraction.

474

588

OPTICS LEITERS I Vol. 13, No. 7 I july 1988 PROFILE 1

SUBTRACTION PROFILE

PROFILE 2

CASE 1

CASE2

Fig. 3. Photographs of the original images impressed on the input beams to the Mach-Zehnder interferometer and the output subtraction observed at the screen (referring to Fig. 2). In the first case two uniform Gaussian beams are subtracted (exposure time for the subtraction photo is 25 times longer). In the second case a resolution chart image is subtracted from a Gaussian beam.

50% beam splitter BSi. they are tightly focused and interfered in the photorefractive BaTiO:i crystal with a crossing angle of approximately -15°. Spatial information is removed from one of the beams by the lens L., and pinhole arrangement PH. The reference beam reads the holograms written in the crystal and is sampled at the output of beam splitter BS~ and imaged onto a screen. This scheme is difficult to align, especially for the case of images (as opposed to Gaussian beams). An alternative scheme using self-pumping~ of the input beams (with no pinhole or reference beam) can also be used for the subtraction. In this case, the counterpropagating requirement for the reference beam is automatically satisfied. The phaseconjugate reflectivity was -:!0% for both beams. Typical power levels incident on the crystal are of the order of tens of milliwatts for the two writing beams and a few milliwatts for the reference beam. ..\ll beams incident on the crystal are extraordinary. Figure 3 shows photographs of the original beam profiles input into the Mach-Zehnder interferometer and the output subtraction observed on the screen. In the first case two uniform Gaussian beams are subtracted, resulting in a null for the subtraction. One can observe each input profile separately by blocking the other beam. In the second case a U.S. Air Force resolution chart and a Gaussian beam were used as inputs, and the resulting image on the screen clearly shows an image reversal as a result of the optical subtraction. In cone! sion, we have proposed and demonstrated a new me nod of optical image subtraction. It provides par !lei and real-time amplitude subtraction of two com lex images by using holographic interference in phot< ·efractive media. This new method provides

475

an automatic phase shift of 180 deg between the complex images and may be viewed as a real-time implementation of Gabor's image-subtraction technique. The major advantages of this technique over previous image subtraction techniques"- 7 are the amplitude subtraction capability (instead of intensity subtraction) and the ability to process three-dimensional images and perform multiple operations. since there is only a single pass through the transparencies. In addition, the phase information contained in the transparencies can be determined by this technique~ since the output from the interferometer is proportional to the difference in the field amplitudes. For example, if T 1 = l and T 1 e"i'(-1 + iw, for small WI, the output will be proportional to"'·

=

The authors acknowledge helpful disrnssion with their colleagues M. Khoshnevisan and A. Chiou. This work is supported in part by the Office of Naval Research under contract NOOOl-1-85-C-0:219. References I. D. Gabor, G. W. Stroke, R. Restrick, A. Funkhouser, and D. Brumm. Phys. Lett. 18, 116 ( 196.';l. 2. See. for example, ,J. F. Ebersole, Opt. Eng. 14, 4:l6 ( 1975). :l. See, for example. H. K. Liu and T. H. Chao, Proc. Soc. Photo-Opt. Instrum. Eng. 638, .55 ( 1986). 4. M. D. Ewbank, P. Yeh, and .J. Feinberg, Opt. Lett 10, 282 (1985). .'\. A. E.T. Chiou and P. Yeh, Opt. Lett. 11, :J06 ( 1986). 6. A. E.T. Chiou, P. Yeh, and M. Khoshnevisan, Proc. Soc. Photo-Opt. lnstrum. Eng. 613, 201 ( 1986). 7. S. K. Kwong, G. A. Rakuljuc, and A. Yariv, Appl. Phys. Lett. 41!, 201 (1986). 8. G. G. Stokes, Camb. Duhl. Math..J. 4, I 118491. 9..J. Feinberg, Opt. Lett. 7, 486 ( 1982).

Reprinted from Optics Letters, Vol. 12, page 507, July 1987. Copyright © 1987 by the Optical Society of America and reprinted by permission of the copyright owner.

Correction of polarization and modal scrambling in multimode fibers by phase conjugation Ian McMichael, Pochi Yeh, and Paul Beckwith Rockwell International Science Center, 1049 Camino dos Rios, Thousand Oaks, California 91360 Received January 23, 1987; accepted March 30, 1987 When polarized light is incident upon a long multimode fiber, the emerging light is randomly distributed among the spatial and polarization modes. We present experimental and theoretical results demonstrating that recovery of the spatial and polarization modes of the incident light takes place only when a phase conjugator at the fiber output preserves polarization on reflection.

When polarized light is incident upon a long multimode fiber, the emerging light is randomly distributed among the spatial and polarization modes because of modal scrambling. We previously reported preliminary results on the correction of modal scrambling by phase conjugation. 1•2 In this Letter we present recent results of experimental and theoretical investigations into the correction of modal scrambling for the cases when both polarization components are conjugated at the fiber output (polarization-preserving phase conjugator,2·3 PPPC) and when only one polarization component is conjugated (nonpolarization-preserving phase conjugator, NPPPC). We find that recovery of the original spatial and polarization modes takes place only when the reflection from the phase conjugator preserves the polarization of the incident light. However, if the fiber supports a large number of spatial modes and one uses spatial filtering to look only at the light returning in the same spatial mode as the incident light, then one will see nearly complete polarization recovery, even when only one polarization component is conjugated at the output of the fiber. 4•5 This is because light returning down the fiber that is orthogonal in polarization to the input is randomly distributed among all spatial modes. A PPPC produces a time-reversed wave that can correct for modal scrambling. 1•2 We now consider the result of conjugating only one polarization component. The specific case of a linearly polarized incident wave is analyzed graphically in Fig. 1. In what follows we analyze the case of an incident wave of arbitrary polarization. Consider a multimode fiber that supports N modes with wave functions

where An's are the mode amplitudes. The output (z = L) field En can be written as (3)

where Enx and Eny are the x and y components. If modal scrambling is complete, then the energy is distributed equally among the x and y polarizations:

f

2

1Enx1 dxdy =

f

1En)2dxdy.

(4)

Let this field be incident upon a phase conjugator that has unit reflectivity and a polarizer along the x axis in front of it. The reflected field Em is given by Em = (Enx *' O).

(5)

This field can be written as the sum of two components: a component En* that is the phase conjugate of the light emerging from the fiber at z = L and a component En* _1_ that is orthogonal to the conjugate in the sense that f En • En* _1_ dxdy = 0, Em= aE 11* + bEn* _1_,

(6)

where a and b are constants. Scalar multiplying both sides of Eq. (6) by En, integrating over x and y, and using Eqs. (4) and (6), we obtain the results a= l/z, b = l/z, and E1a = (En,, -Euy). With these results, Em becomes

where f3n is the propagation constant and n is the mode number. An incident electric field E1 can be expressed as a sum over the N modes at the input (z = 0):

Em= %En*+ 1/zEn* _1_· (7) This field propagates back through the fiber. If the fiber is linear and lossless, then the conjugate component %En* generates %E1* by time reversal and the orthogonal component 1/2En* _1_ generates %E1* _1_, where E1* _1_ is orthogonal to E1* in the sense that f E1 • E1* _1_ dxdy = 0; note that E1* _1_ ~ (E1*,, -E1* y). Since we assume complete modal scrambling, the energy in E1* _1_ is distributed equally among the x and y polarizations. The returning field at the fiber input becomes

(2)

(8)

n = 1, 2, ... ,N,

(1)

Note that when only one polarization component is" 0146-9592/87 /070507-03$2.00/0

© 1987, Optical Society of America

477

OPTICS LETTERS I Vol. 12, No. 7 I July 1987

508

perform spatial filtering of the returning phase-conjugate signal by closing it to the size of the input beam. The qµarter-wave retarder A/4 can be used in conjunction with the polarizer to eliminate the reflection from the air-fiber interface at the fiber input. Light emerging from the fiber is collimated by a lOX microscope objective L2, focused by lens L3 (f = 0.5 m), and

MMF x

vl-.z

0

E1

l

z=0

*

En

Z

Enx

*p~

=L

l

PCM

~

t ~

E1v

(bl /

I

/

)'.~:2 E1~\

;

I

I

\

I \

'' (cl

112

/

I

I

x

- --' ' ' /

I

'

-

--

',

//Em

Y.

x

;

_.,,. 1/2En"

M3

,,../1 /

Fig. 2. Experimental setup. Light emerging from the multimode fiber MMF is reflected from the PPPC formed by the optical elements PBS, Ml-M4, A/2, and the BaTi0 3 crystal. The effect of conjugating only one polarization component can be observed by blocking one of the outputs from the polarizing beam splitter PBS. After propagating back through the fiber, the light is sampled by beam splitter BS, analyzed by the polarizer P, and measured by detector Dor photographed.

/-

/

/

/

I

i12v'2_

("

Fig. 1. Illustration of what happens when light propagates in a multimode fiber (MMF) and only one polarization component is conjugated at its output. The top section shows a sketch of the optics, and (a)-(c) show the evolution of the polarization of the light as it propagates. The field E1 = x enters the MMF at z = 0. (a) Because of modal scrambling, the output field En is equally distributed among the x and y polarizations and the change in polarization is represented by rotation of E 1 to En. (b) The x component of En passes through the polarizer and reflects from the phase-conjugate mirror PCM to produce Em = Enx *x. This field has a phase-conjugate component, 1/zEn*, and an orthogonal component, 1/zEn* .L· (c) When the conjugate component propagates back to z = 0 it generates 1/zE1* by time reversal (depicted as rotation of 1/zEn* to 1/zE1*), and the orthogonal component generates 1/zE1* .L• where E1* .L is orthogonal to E 1*. Because of modal scrambling, E1* .L is distributed equally among the x and y polarizations. Nate that only one fourth of the energy is recovered in the reconstruction of the input field. The other one fourth is randomly distributed among the spatial and polarization modes. One half of the energy is lost at the polarizer.

POLARIZATION

l.

REFLECTED LIGHT FROM AIR/FIBER INTERFACE AT FIBER OUTPUT

POLARIZATION· PRESERVING PHASE-CONJUGATE MIRROR

PHASE-CONJUGATE MIRROR (NONPOLARIZATION

conjugated at the output of the fiber only one quarter of the energy (i%E1*1 2) is recovered in the reconstruction of the input field. The other one quarter of the energy (i 1hE1*.t 12) is randomly distributed among the spatial and polarization modes. Therefore (1/4 + 1/2 X 1/4) = 3/8 of the original energy returns with the polarization of the input and (1/2 X 1/4) = 1/8 returns orthogonal to it. One half of the original energy is lost at the polarizer. Our experimental setup is shown in Fig. 2. Light from a single-mode argon-ion laser is focused into a step-index multimode fiber (core diameter d = 100 µ,m, numerical aperture 0.3, attenuation 30 dB/km, length 20 m) by lens Ll (focal length f = 5 cm, diameter D = 4 cm). The aperture AP can be used to

PRESERVING)

Fig. 3. Photographs of the returning light as seen at the position of detector D in Fig. 2. 11 corresponds to the polarization of the light incident upon the fiber. The first set of photos was taken with the phase-conjugate mirror blocked, the second set of photos with the PPPC at the fiber end, and the third set of photos with the NPPPC (one of the outputs from the polarizing beam splitter in Fig. 2 is blocked). For the PPPC all the light returns as the phase conjugate (bright spot in the photo of 11 polarization). However, for the NPPPC only one half of the returning light is the phase conjugate; the other half is randomly distributed among the spatial and polarization modes.

478

July 1987 I Vol. 12, No. 7 I OPTICS LETTERS

509

Table 1. Measured and Predicted Power Ratios for Phase-Conjuga tion with Multimode Fibersa Aperture Open Reduced'

PPPC (4.4 ± 1.3) X

10- 3

NPPPC [OJ

0.32 ± 0.02 [1/3J (1.4 ± 0.3) x 10-2 [«lJ

(LO ± 0.4) X 10-3 [OJ

0.42 ± 0.09 [1/2J 0.22 ± 0.03 [1/4]

See Fig. 2; predicted values are in square brackets. is the ratio of the power in the polarization orthogonal to the input to that parallel to the input as measured by the detector (see Fig. 2); PPPC corresponds to the situation shown in Fig. 2; NPPPC corresponds to blocking one of the outputs from the polarizing beam splitter in Fig. 2. 'Aperture size reduced to the input beam size (~4 mm). a

b P J_IP11

reflected from the PPPC 2 formed by the polarizing beam splitter (PBS), mirrors Ml-M4, half-wave retarder 'A/2, and the barium titanate crystal BaTi03• The effect of conjugating only one polarization component can be observed by blocking one of the outputs from the PBS. After propagating back through the fiber, the light is sampled by a beam splitter, analyzed by a polarizer, and measured by a detector or photographed. Figure 3 shows photographs of equal exposure of the returning light as seen at the position of the detector with the aperture open. ~ corresponds to the polarization of the light incident upon the fiber. For the first column, a A/4 retarder is used to avoid the reflection from the air-fiber interface at the fiber input. The first row of photos was taken with the phase conjugator blocked. The grainy circular patterns observed in both polarizations are from the light reflected at the air-fiber interface at the fiber output. The fact that the circular patterns are uniform and equal in intensity indicates complete modal scrambling. The second row of photos was taken with the PPPC at the fiber end. A strong phase-conjugate return can be seen in the center of the photo of the II polarization, and there is little change in the intensity of the grainy circular patterns (in fact, we observed that the backscattering from the output end of the fiber with polarization orthogonal to the input reduces by 5-10% owing to phase conjugation6), indicating that all the light returns as the phase conjugate. The third row of photos was taken with a NPPPC (one component in the PPPC blocked) at the fiber end. The conjugate return is weaker than that for the PPPC, and the intensity of the grainy circular pattern has increased over that produced by the fiber end reflection, indicating that some of the returning light is not the phase conjugate of the input. Table 1 lists measured and predicted power ratios. As before, ~corresponds to the polarization of the light incident upon the fiber. For the PPPC, the measured ratio P 1-IP11 is small, indicating nearly complete correction of polarization scrambling. The ratio decreases when the aperture is closed down to the size of the input beam to provide spatial filtering of the returning phase-conjugate signal. For the NPPPC, with the aperture open, the measured ratio is 0.32 ± 0.02, in agreement with our theory [(1/8)/(3/8) = 1/3]. This indicates that when only one polarization is conjugated at the output of the fiber, one third of the returning light is orthogonal in polarization to the

479

input. When the aperture is closed, the measured ratio decreases to (1.4 ± 0.3) X 10- 2 • This indicates that the returning light with polarization orthogonal to the input is spatially orthogonal to the input. With the aperture open, when we block the central phaseconjugate return with a mask in front of the detector to look only at the grainy circular pattern, the measured power ratio is 1.0 ± 0.1, indicating equal power distribution in the two polarizations. We also compared the total power in the returning light for a PPPC with that for a NPPPC. With the aperture open, the measured ratio PNPPPcfPpppc is 0.42 ± 0.09, reflecting the fact that one half of the light from the fiber output is blocked in the NPPPC. With the aperture closed, the measured ratio is 0.22 ± 0.03, indicating that with the NPPPC only one fourth of the light incident upon the fiber returns as the phase conjugate. In conclusion, we have presented theoretical and experimental results demonstrating that correction of modal scrambling by multimode fibers requires a PPPC. The technique described in Ref. 4 (using a modal filter followed by a multimode fiber, a polarizer, and a non-polarization-preserving phase conjugator) can be used to correct for polarization scrambling when the wave to be conjugated contains much less spatial information than the maximum that can be supported by the fiber and when a reduction of the phase-conjugate signal by a factor of 4 and an increase in noise is tolerable. The authors thank M. Ewbank and A. Chiou for many helpful discussions. This research is supported by the U.S. Office of Naval Research under contract N00014-85-C-0 219.

References 1. I. McMichael and M. Khoshnevisan, in Digest of the

Conference on Lasers and Electro-Optics (Optical Society of America, Washington, D.C., 1985), paper THNl, p. 220.

2. I. McMichael, M. Khoshnevisan, and P. Yeh, Opt. Lett. 11, 525 (1986). 3. P. Yeh, Opt. Commun. 51, 195 (1984). 4. K. Kyuma, A. Yariv, and S. Kwong, Appl. Phys. Lett. 49, 617 (1986). 5. A. Yariv, Y. Tomita, and K. Kyuma, Opt. Lett. 11, 809 (1986). 6. P. D. Drummond and A. T. Friberg, J. Appl. Phys. 54, 5618 (1983).

510

OPTICS LETIERS I Vol. 12. No. 7 I July 1987

Image distortion in multimode fibers and restoration by polarization-preserving phase conjugation Paul H. Beckwith, Ian McMichael, and Poch! Yeh Rockwell International Science Center, 1049 Camino Dos Rios. Thousand Oaks, California 91360

Received February 27, 1987; accepted April 9, 1987 Image restoration after a double pass through a multimode fiber using a polarization-preserving phase conjugator is demonstrated. Results indicate that the resolution of the restored image is limited by the number of guided modes and that the contrast is restored only when the phase conjugator preserves polarization on reflection.

It is well known that image information, encoded as a spatial intensity pattern, is rapidly scrambled because of mode coupling as it propagates in a multimode fiber. The image information will be completely scrambled (energy leaving the fiber is randomly distributed among all the spatial modes as well as the polarization modes) when the beam diffraction is large enough to result in many reflections off the fiber core walls (i.e., >.L/d2 » l; Lis the fiber length, dis the fiber diameter, and>. is the wavelength). 1 The concept of using phase conjugation to restore the original image is not new. 2-4 An image is sent through a multimode fiber, and the scrambled output is reflected from a phase conjugator. After traversing a second identical fiber or retraversing the same fiber, the output image is proportional to the input image. In principle, the first case involving two identical fibers can be used to transmit images through fibers. However, in practice it is unlikely that the mode coupling of two fibers will be identical, and to the best of our knowledge such a case has never been demonstrated. The latter case of image restoration involving a double pass through a

single fiber has been demonstrated by using phase conjugators that do not preserve polarization. 5•6 In a recent paper 1 we demonstrated that in multimodefiber systems the recovery of the original spatial and polarization modes takes place only when the phase conjugator preserves polarization on reflection.' In this Letter we report on image restoration after a double pass through a multimode fiber using a polarization-preserving phase conjugator. We present results indicating that the spatial resolution of the restored image is limited by the number of guided spatial modes and is therefore independent of whether the phase conjugator preserves polarization. However, we also present results indicating that the contrast of the original image is restored only when the phase conjugator preserves polarization on reflection. In order to study image restoration after a double pass through a multimode fiber using phase conjugation, we set up the experiment shown in Fig. 1. Linearly polarized light from an argon-ion laser operating in a single longitudinal mode on the 514.5-nm transition is expanded by lens L 1 (focal length f = -2 cm)

ARGON LASER

BaTi03

Fig. l. Experimental setup used to demonstrate image restoration after a double pass through a multimode fiber by polarization-preserv ing phase conjugation. The image impressed upon the expanded argon-ion laser beam by transparency T is focused into a multimode fiber (MMF). Light leaving the MMF is reflected from the PPPC formed by the optical elements PBS, M3-M6 , A/2, and the BaTi0:1 crystal. The effect of conjugating only one polarization component can be observed by blocking one of the output beams from the polarizing beam splitter PBS. After propagating back through the fiber, the restored image is sampled by beam splitter BS, analyzed by polarizer P, and photographed at the image plane.

0146-9592/87 /070510-03$2.00/0

-© 1987, Optical Society of America

481

July 1987 I Vol. 12. No. 7 I OPTICS LETIERS RESOLUTION CHART

GRAY-SCALE GRID

INPUT IMAGE PATTERN

SINGLE-PASS FIBER OUTPUT

POLARIZATION PRESERVING PHASE-CONJUGATE MIRROR

DOUBLE-PASS FIBER OUTPUT

PHASE-CONJUGATE MIRROR INONPOLARIZATION PRESERVING)

Fig. 2. Photograp hs demonstra ting image restoration after a double pass through a multimode fiber by polarizationpreserving phase conjugation. The first row of photos shows the input image patterns (a resolution chart and a gray-scale grid) impressed upon the laser beam, and the second row of photos shows the scrambled single-pass fiber output. The third and fourth rows show photographs, taken at the image plane in Fig. l, of the image reconstructed by phase conjugation after a double pass through the multimode fiber. For the third row, the PPPC is at the fiber end, and for the fourth row, one of the outputs from the polarizing beam splitter in Fig. 1 is blocked to illustrate the effect of conjugatin g only one of the polarizatio n componen ts (NPPPC). Although the resolution is comparable for the two cases, the contrast is severely degraded in the case of the NPPPC.

and collimated by lens L2 (/ = 15 cm). After an image is impressed upon the expanded beam by transparency T, it is focused into a step-index multimode fiber MMF (core diameter d = 100 µm, numerical aperture N.A. = 0.3, length L = 100 m) by lens L 3 (/ = 5 cm, D = 4 cm). The quarter-wave retarder >../4 is used in conjunction with the polarizer P to eliminate the reflection from the air-fiber interface at the fiber input end. The light leaving the fiber is collimated by a lOX microscope objective L4 , then split into its polarization

511

components by the polarizing beam splitter PBS; one of the polarization components is rotated 90° by the half-wave retarder X/2, and both components are focused into a crystal of barium titanate (BaTi0 3) by lens L5 (/ = 0.5 m) and lens Ls(/= 5 cm) such that selfpumped phase conjugation takes place. Power levels incident upon the crystal are of the order of l mW, with half of the power in each of the two components. The arrangement of optical components PBS, mirrors M:1 -~, X/2, L 5 , Ls. and the BaTi0 3 crystal, forms a polarizat ion-prese rving phase conjugato r PPPC. When the phase-conjugate reflections of the two components recombine at the PBS, they form a phaseconjugate wave that has the same polarization as the incident wave. The effect on the image quality of conjugating only one of the polarization components (nonpola rization- preservin g phase conjugat ion, NPPPC) can be observed by blocking the other component. After retraversing the fiber, the restored image is sampled by the beam splitter BS and photographed at the image plane. With the NPPPC, only one half of the returning light reconstructs the image; the other half is randomly distribute d among all the spatial and polarization modes. 1 The restored image is improved by filtering out one half of the light that does not reconstruct the image (one quarter of the returning light) using P oriented to pass light having the polarization of the image. In order to obtain high phase-conjugate reflectivities (R ~ 40%) from the BaTi0:1 crystal, it was necessary to encase the 100-m fiber in a Styrofoam container. This reduces thermal fluctuations on time scales shorter than the response time of the BaTi03 crystal (which tend to reduce the phase-conjugate signal by washing out the index gratings in the crystal). Figure 2 shows photographs of two images sent into the fiber: a standard U.S. Air Force resolution target in the first column and a gray-scale grid in the second column. The gray-scale grid consists of neutral-density filters having a transmission (starting in the lower left-hand corner and proceeding clockwise) of 1, 10, 50, and 100%. The first row of photos shows the input images. The second row shows the single-pass output from the end of the fiber. The uniformity of the output interference speckle patterns indicates severe image scrambling by the fiber. The graininess of the patterns is indicative of the number of modes supported by the fiber. The last two rows show the doublepass fiber output as sampled by the BS, filtered by P, and recorded at the image plane (see Fig. 1). The third row of photos shows the image restoration with the PPPC; and the fourth row, with the NPPPC (one component in the PPPC blocked). Exposure time is adjusted so that equal energy falls upon the film in each case (PPPC and NPPPC). As expected, the photos of the resolution target show that there is little difference in resolution for the two cases. The measured resolution is somewhere in the range from 1.59 lines/mm (Group 0, Element 5) to 1.78 lines/mm (Group 0, Element 6) and agrees well with the resolution calculated from the number of guided modes in the fiber and the imaging optics, ~1.62 lines/mm.

482

512

OPTICS LETTERS I Vol. 12, No. 7 I July 1987 DOUBLE-PASS FIBER OUTPUT POLARIZATION PRESERVING PHASE-CONJUGATE MIRROR

INPUT IMAGE PATTERN

PHASE-CONJUGATE MIRROR fNONPOLARIZATIOl\J PRESERVING I

LOW

DETAIL IMAGE

HIGH DETAIL IMAGE

Fig. 3. Photographs demonstrating image restoration after a double pass through a multimode fiber by polarization. preserving phase conjugation for a low-detail image and for a high-detail image. The input image and the image reconstructed by phase conjugation after a double pass through the multimode fiber are photographed for both PPPC's and NPPPC's. Degradation of image contrast is so severe for the image reconstructed by the NPPPC that the picture of the high-detail image is barely discernible.

However, the photos of the gray-scale grid show clearly that the restoration of image contrast is much better in the PPPC case than in the NPPPC case (in the NPPPC case, the light areas are darker, and vice versa). Figure 3 shows photographs of the double-pass fiber output for low-detail and high-detail images. The first column shows the image patterns focused into the fiber, and the second and third columns show the double-pass output for the PPPC and the NPPPC cases, respectively. As in Fig. 2, although the resolution is comparable in the two cases, the contrast is greater in the PPPC case. The difference in contrast is best seen with the high-detail image, where the picture is barely discernible in the NPPPC case. We also examined the effect of using unpolarized light to illuminate the image transmitted through the fiber. Unpolarized light was obtained by passing the polarized output of the laser through a 7-m-long, 100µm-diameter step-index multimode fiber before expanding the beam and passing it through the transparency Tin Fig. l. In this case, it was not possible either to improve the restoration with the NPPPC by using the polarizer or to eliminate the reflection from the air-fiber interface at the fiber input by using the V4 retarder. Results similar to the above but with a

483

greater reduction in contrast were obtained for the NPPPC case. In conclusion, we have presented experimental results demonstrating that high-contrast image restoration after a double pass through a multimode fiber requires the use of polarization-preserving phase conjugation. When only one polarization leaving the fiber is phase conjugated, only one half of the returning light reconstructs the image 1; the other hs.lfisrs.ndomly distributed among all the spatial and polarization modes, thereby severely degrading the contrast of the reconstructed image. Reference!l ! . l. McMichael, P. Yeh, and P. Beckwith, Opt. Lett. 12, 507

(1987). 2. A. Yariv, Appl. Phys. Lett. 28, 88 (1976). 3. A. Yariv, J. Opt. Soc. Am. liS, 301 (1976). 4. A. Gover, C. P. Lee, and A. Yariv, J. Opt. Soc. Am. 66,306 (1976). 5. G. J. Dunning and R. C. Lind, Opt. Lett. 7, 558 (1982). 6. B. Fischer and S. Sternklar, Appl. Phys. Lett. 46, 113 (1985). 7. I. McMichael, M. Khoshenevisa n, and P. Yeh, Opt. Lett. 11, 525 ( 1986).

Part VI Applications b. Optical Storage

Theory of Optical Information Storage in Solids P. J. van Heerden

In photography, information is stored in a medium which is essentially two-dimensional.

Threedimensional optical storage is possible in semitransparent colored materials like alkali halides with color

~enters. With the use of coherent light sources, like lasers, large amounts' of information can be stored

m the ~olume: and retrieved with little interference. The storage of information is accomplished by the formation o~ mterference patterns between each two plane parallel waves. This paper develops the theory _of t?1s form of storage. It turns out that the information storage capacity is as if every little .,µ). For different directions Ao' is diffracted by (k,>.,,..) only in a special case.

amplitudes are multiplied with the positive factor: 2bt cos 2 80 ;A 00Aoo*. The only distortion in the reproduction is the term cos 280 , which tends to reduce resolution in the reproduction of details of the order X0•

IV. Simultaneous Storage of Several Pictures The fact that a two-dimensional picture is stored in a three-dimensional medium suggests that the storage space is not used to full capacity. This is actually the case. Many pictures can be stored simultaneously in a crystal, either by illuminating these pictures with light beams of different wavelengths or of different directions. This is most easily seen by representing an interference pattern like (Ao*, A1) as a vector (K, >.., µ.) = (k1 - ko, 11 - lo, m1 - mo) in momentum space (Fig. 4). This vector, added to (k0Zom 0) representing the direction of Ao, gives the vector (k1l1m1) of the diffracted wave A1. When one would illuminate the bleached crystal with light presented by (ko'lo'mo') in the same direction but of different frequency wo', no diffracted wave results, because the vector (ko' + K, lo< + >.., mo' + µ.) must satisfy the condition that the total length is equal to w0 '/c0, meaning that the diffracted wave has the same frequency as the illuminating wave. It can be seen from Fig. 4 that this condition cannot be satisfied. Said in different terms, it means that, although a bleaching plane (Fig. 1) in the crystal will show a reflected wave, the reflected waves from different planes will not add up in phase, since Bragg's relation is not satisfied. Many pictures of different frequencies can be stored, therefore, simultaneously without confusion. The disadvantage is, of course, that the bleaching for every picture becomes weaker, since the total bleaching cannot exceed the original coloring of the crystal. In a similar way, different pictures can be stored by illuminating with light beams Ao and Ao' of the same frequency wo but of different directions [Fig. 4(b)]. It is impossible to draw vectors (K, >..,µ.)and (ic', >..', µ.') from the end of Ao and Ao' which are parallel, of equal length,

the end points of these vectors fall in the plane which reflects Ao into Ao'· Therefore, there is an ambiguity in regenerating the pictures only as far as the e special vectors A1 and A1' are concerned. The over all disturbance in the pictures is minor however, and so many can be stored with different directions Ao' without causing excessive noise. In princip!e it is possible to store time-dependent signals in the three-dimensional medium. The requirement is that the time duration of the signal is less than it takes the light to traverse the crystal. This means that it would be necessary to modulate coherent light sources with frequencies comparable to the frequency of light itself. No physical principle is available, nor is any in sight, by which this could be accomplished. It is still of some interest to discuss this case, because of the storage of slower wave phenomena, which will be mentioned in Section VIII. Imagine, therefore, (Fig. 5) that in 0 every picture element varies its transmission in time independe nt of every other picture element, resulting in a movie of some very fast phenomenon. To be able to reproduce this movie from the bleaching in the storing crystal, there is only one restriction, which is the following: Consider the transmission D,(t) as a function of time in a picture element i, and write D;(t) as a Fourier sum: +n D;(t) =

2:: C,,. exp[illwl).

The restriction is now that, in comparing C'" and C;n in any two picture elements i and j, the ratio C,./C;. must be a real positive number. This reduces the possible information content of the movie by about 50%. When this is satisfied, since a Fourier component D,. gives rise to a light wave with frequency wo + nw, by the modulation of Ao, all wavelets emanating from different picture elements of that frequency will be exactly in phase. It is now the strong wave of frequency wo + nw, parallel to the optical axis, which forms the main interference pattern with all diffracted

Fig. 5. Reproduction of time dependent signals. Amplitude modulating of L with a pulse of a specific shape will reproduce the movie.

396 APPL! ED OPTICS / Vol. 2, No. 4 / April 1963

490

waves of the same frequeney but of different directions. To regenerate the movie from the bleached crystal, it is only necessary to reproduce simultaneously all these waves, of different frequencies but parallel to the optical axis, ,\·ith the proper phase relations. In principle, this can be done as follows. All these waves are focused in the focal point Fo of the imaging Jens. They will add to the amplitude of the original frequency w0 to give a time-dependent signal in F0 • Amplitude modulation of the original illuminating beam of frequency w0 with this signal is therefore identical with generating these different frequencies wo + nw in the proper amplitudeand phase relations. This can be accomplished in principle by an amplitude varying element close to Lo. By merely illuminating the bleached crystal with this pulse of light, the whole movie would be regenerated.

V. The Informatio n Storage Capacity The general considerations determining the storage capacity by bleaching can be given in a simple way. This will lead to an order of magnitude number for this capacity. Let us first consider the number of independent information cells present in a crystal of dimensions a X a X d em 3 (Fig. 2). A wave in a direction (k1l1m 1) emerging from a a X. a cm2 aperture will show, as is well known, a narrow diffraction pattern of a solid angle ,..._,A 2 /a 2 • Therefore, about a2 /A 2 independent directions can be recognized. In the same way, the number of independent wavelengths A that can be stored in the crystal depends on the thickness d. It is known, for instance from the resolving power of echelon gratings, that the number of separately resolved wavelengths is given by ,..._,d/Ao, where Ao is the average wavelength. The total number n of independent storage cells is therefore given by n0 s::; a 2d/Ao 3 = V /Ao 3 • This number is the same as when the information would be stored in specific small volumes, like in the nuclear track plate, instead of in interference patterns. To know the storage capacity, it is necessary to know, besides the number of independent storage cells, the signal-to-noise ratio in each cell. It was shown already in Section III that the noise introduced in the regeneration of the vector A1 by the other bleaching pattern AoA 1 is very small, since it is of the order A0 3/V = l/n0 • The noise introduced in Ao by all n 0 vectors A 1 will only increase this by a factor vii;;. The total noise in A 1 caused by the other bleaching terms is therefore of the order I/vii;;, which is still small. The main fundamental source of noise is the random distribution of color centers in the crystal, in the same way as the random distribution of electrons in a beam gives rise to shot noise. Let there be N 0 color centers per cm 3 in the crystal. In a bleaching pattern (K1A1µ 1) in the crystal, there will be, according to the laws of probability, about vNJT more color centers in the positive part of this pattern than in the negative part,

Fig. 6.

Storage of an image in the crystal with the center beam Ao cut off.

or vice versa. This constitutes the noise N, in the amplitude. To esLimate the signal, there are n independent bleaching patterns, and in a certain point these patterns can be positive or negative; therefore the total bleaching amplitude will be "'vii times the average bleaching amplitude A. of a single pattern. Since there are No V color centers available for bleaching, A, s::; N 0 V/vn. The signal-to-noise ratio S,/N, is therefore given by: S,/N, = vNoV/n. When the storage space is filled to capacity, n = n 0 = V A0 3 , and S,/N, = VNoAo 3 • This result is again as if every cube with side Ao acts as a separate storage cell. The signal is then equal to the number NuAo 3 of color c-enters in that cube, and the noise is the root mean square fluctuation in that number, which is VN0A0 3 • The storage capacity C in bits is equal to the number of independent storage cells times the logarithm to the base two of the signal-to-noise ratio: Cs::; nu log2 S./ N,,. As an example, take a crystal of 1 X 1 X 1 cm 3 , with N 0 = 10 15 color centers per cm 3, which would give about 25%absorpt ion. LetAobelµ . Inthatcase : Cs::;10 12 log2 10 3 s::; 1013 bits.

VI. The Appearance of a Ghost Image In the previous paper 2 describing two-dimensional information storage, it was shown that a ghost image of the whole object will appear in the image plane, when only a small fragment is illuminated in the object plane. The ~ame is true for three-dimensional storage. The important difference is that when the fragment is displaced from its original position, no ghost image will appear. To demonstrate this ghost image, let us consider Fig. 6. To avoid the effect of the central light beam Ao, the parallel light illuminating the transparency is now focused in a point and blocked off. Only the waves diffracted by the object will bleach the crystal. Let these parallel waves originating in 0 again be called A 1 , A 2 , • • • , An. The bleaching is again described as proportional to

t

ij[A,A;*

+ A;*A;).

I

Let, further, the wave amplitude in the ith direction as generated by the small fragment be called A/, and April 1963 I Vol. 2, No. 4 I APPLIED OPTICS 397

491

lj

JHfiLW

Fig. 7. Three-dimensional storage as an associative memory. Exposure of Si in 0 after bleaching will reproduce a ghost image of Li> which in tum will illuminate h

the wave in the same direction be generated by the rest of the object A,". So A, = A/ + A,". Now let the crystal be illuminated by the light from the fragment only, that is, by A1', Az', ... An'. The amplitude of the wave generated in the direction i will now be proportional to I; j[A,A/ + A,*AiJA /. But since Ai =A/+ A/, A;* =A/*+ A/*, this sum contains the part: Since [A/A/* ] is positive, all these terms add up in phase, and the intensity with which A, is generated is simply proportional to I; j[A/A/ *], which means proportional to the energy of the light emitted by the picture fragment. All components of the original object are therefore regenerated in the proper magnitude and phase, and a ghost image will appear in I. This ghost image is therefore analogous to the one in the two-dimensional case, 3 and will require no further discussion. In particular, this is true for the other terms in the bleaching which can be considered as noise. On the other hand, suppose that the fragment is displaced by an amount (XoYoZo) from its original position. Every wave A/, in the second illumination, will have acquired an.extra phase factor exp[iING,.

Fig. 3. Electrooptic characteristic and diffraction efficiency vs reading field in a PLZT (9% La) ceramic; thickness: 350 µm.

index can be observed; when reading, electrons are uniformly photoexcited and redistributed, and the ferroelectric domain pattern is left uncompensated. This pattern is shown to be stable during reading at V = 0 and constitutes the fixed hologram. The concept of the photoinduced ferroelectric domain pattern is supported by the following experiments. After having measured the coercive voltage Ve from the low frequency hysteresis loop (f = 3 X 10-2 Hz) fixing voltage pulses are applied to the crystal with different amplitudes and constant duration (tr = 0.5 sec). Figure 5 clearly shows that the amplitude threshold of the fixing voltage is Vrmin = 700 V. For Vr < 700 V, the hologram is erased in 50 sec or less. For 700 < Vr < 1000 V, the hologram is f"} Ofo

READOUT

demonstrated in Fig. 3, where 11 is plotted versus reading field Er. For Er = 2.75 kV cm- 1 , 11 = 0, which allows us to measure the screening field Es = Er. For Ee = - Ea, the gain in diffraction efficiency is 11( -Ea) /17(Ea) = 15. When the Curie temperature is higher than room temperature, for instance in Sro.1sBao.25Nb20a (Tc ""' 50°C) or BaTi0 3 (Tc = 120°C) single crystals, polarization vs applied field shows the well known hysteretic behavior; we show that stable patterns can be produced that are replica of the original photoinduced patterns. This effect is called electrical fixation of the recorded hologram since it allows permanent readout with the light wavelength used during recording. rn,ia Figure 4 shows typical results of the recording, fixation, and erasure processes in a 8-mm cube shaped SBN crystal, poled at room temperature by applying a pulse voltage with amplitude Vp = 1700 V. Holographic fringes with 10-µm spacing normal to the polar axis are recorded with 1 W /cm2 light power density at the argon laser wavelength A = 488 nm. During recording, the HDE reaches the saturation value 11 = 0.14; optical erasure (one beam suppressed) occurs in 25 sec. The crystal is poled again in order to get the same polarization state as in the previous experiment, and a holographic pattern is recorded; a voltage pulse with amplitude Vr = 1000 V (larger than the coercive voltage Ve = 775 V) and duration tr = 0.5S is applied to the crystal. A strong HDE enhancement occurs during the polarization switching time and is attributed to the increase of electrooptic coefficient near the coercive field. After having applied the fixing pulse, 11 vanishes, then reaches the saturation value 11 = 0.52 during readout at the recording wavelength. The high 11 value remains unchanged during a 10-h readout. These effects may be interpreted as follows: the photoinduced electronic space charge fields are canceled by ionic displacements associated with the polarization switchings; the internal field becomes uniform, therefore, no modulation of the refractive 786

APPLIED OPTICS / Vol. 13, No. 4 / April 1974

501

t

50 ELECTRICAL FIXING---;.

40 ,READOUT I 30 I I WRITING i OPTICAL: WRITING

20

ERASURE I

i i

ELECTRICAL ERASURE -

~

i

OPTICAL

I

ERASURE

I

10

t ' '-~+-~~~:~~~"-""~--+~~~-{-~---="'--+ I

Fig. 4. Holographic diffraction efficiency during recording, fixation, readout, and erasure processes in a cube shaped SBN crystals; edges 8 mm.

=

900

t (5

700

10

20

30

40

50

60

70

80

Fig. 5. Hysteresis loop of a SBN sample, cube shaped, edges 8 mm. Holographic diffraction efficiency vs time for different fixing voltages.

fixed, and the HDE value increases when Vr increases. For Vr > 1000 V, the HDE shows a maximum value, then decreases slowly to the zero value. This latter effect is attributed to an homogeneous polarization reversal, such as the crystal remains single domain but with opposite polarization direction. The minimum fixing time was found to be the polarization switching time. We have recently reportedl7 that photoinduced ferroelectric domain patterns could be obtained by photoinducing in a SBN crystal an electronic pattern at temperature higher than Curie temperature and cooling it through the Curie phase transition. The domain patterns obtained by passing through the Curie phase transition and the coercive field transition have the same properties, i.e., stability and high diffraction efficiency. The electrically fixed hologram can be erased by applying to the crystal a voltage pulse with amplitude high enough to saturate the polarization in the crystal. Figure 4 shows that applying a voltage pulse with amplitude Vp = 1700 V » Ve during 1 sec or more erase almost completely the recorded hologram. The extant hologram can be optically erased and is therefore of photoelectronic origin: electrons are photoexcited during reading in the inhomogeneous internal field and cannot be released from their traps by the erasure field. Electrical controls of diffraction efficiency and storage time in photoferroelectric materials involve nonlinear electrooptic effects, photosensitivity in the visible range, and deep photoelectron traps. Nonlinear electrooptic effects are achieved in materials operating near their phase transitions (structural phase transition at the Curie temperature Tc or polarization reversal at the coercive field Ee). Tc and Ee are easily lowered to practical values by controlling the La/Pb and Sr/Ba ratios. Photosensitivity

in the PLZT and SBN samples we have studied is fortuitous, since up to now no doping has been performed in the host lattices. Photosensitivity must be increased to S "" 103 cm 2 J-1 for practical uses of PLZT and SBN as recording media in high speed read-write optical memories. This work was supported in part by DGRST. References 1. V. M. Fridkin, L. M. Belyaev, A. A. Grekov, and A. I. Rodin, J. Phys. Soc. Japan 28, suppl. 448 (1970). 2. A. Ashkin, G.D. Boyd, J.M. Dziedzic, R. G. Smith, A. A. Balman, and K. Nassau, Appl. Phys. Lett. 9, 72 (1966). 3. F. S. Chen, J. Appl. Phys. 40, 3389 (1969). 4. F. S. Chen, J. T. La Macchia, and D. B. Fraser, Appl. Phys. Lett. 13, 223 (1968). 5. G. Bismuth, Thesis, Orsay (1972). 6. W. Phillips, J. J. Amodei, and D. L. Staebler, RCA Rev. 33, 94 (1972). 7. H. Aronson, Final report on Contract NAS-8-26635. 8. F. Micheron and G. Bismuth, J. Phys., Suppl. to No. 4, 33, C2-147 (1972). 9. A. Ishida, 0. Mikami" S. Miyazawa, and M. Sumi, Appl. Phys. Lett. 21, 192 (1972). 10. J. J. Amodei and D. L. Staebler, RCA Rev. 33, 71 (1972). 11. J.B. Thaxter, Appl. Phys. Lett. 15, 2Hf'(l969). 12. F. Micheron, A. Hermosin, G. Bismuth, and J. Nicolas, C.R. Acad. Sci. Paris 274 B, 361 (1972). 13. G. H. Haertling and C. E. Land, J. Am. Ceram. Soc. 54, 1 (1971). 14. F. Micheron, C. Mayeux, A. Hermosin, and J. Nicolas, Proc. Fall. Meeting of Am. Ceram. Soc., Atlanta, Sept. 1973, to be published in J. Am. Ceram. Soc. 15. F. Micheron and G. Bismuth, Appl. Phys. Lett. 20, 79 (1972). 16. F. Micheron and G. Bismuth, Appl. Phys. Lett. 23, 71 (1973). 17. F. Micheron and J.C. Trotier, in Proc. International Meeting on Ferroelectricity, Edinburgh, Sept. 1973 (to be published in Ferroelectrics).

502

~

Fe-Doped LiNb03 for Read-Write Applications D. L. Staebler and W. Phillips

High erase sensitivity is observed in heavily reduced crystals of Fe-doped LiNb0 3. Only 12 mJ /cm2 of incident 4880-A radiation erases a hologram, nearly 3 orders of magnitude less energy than previously required. However, the maximum diffraction efficiency that can be reached in these crystals is substantially reduced. These results are shown to be consistent with an extremely low density of Fe3+ ioils. The crystals are resistant to optical scattering effects usually observed in Fe-doped LiNb0 3 •

I.

Introduction

Holographic storage media for read-write applications should be capable of being simply and quickly erased. Electrooptic storage materials, which can be optically erased with light of the same wavelength used for recording, 1-3 have always shown promise as storage media. However, most of these materials have not been very sensitive to incident light. In the case of LiNb0 3 , which can be grown as large crystals of high optical quality, Fe impurities were used to enhance the storage sensitivity, 4 •5 but the recorded holograms showed high resistance to optical erasure. 6 This paper is concerned with the use of crystal reduction techniques to improve the erasure sensitivity of Fe-doped LiNb0 3 • A full discussion of the reduction technique is presented elsewhere. 7 Here we describe the storage and erasure behavior of the improved crystals and explain the results with a simple charge transport model. The major conclusion is that Fe-doped LiNb0 3 can be tailored for read-write applications by judicious control of both the concentration and valence state of the impurity ions. A trade-off in diffraction efficiency is discussed. II.

Backg_round

Phase holograms are recorded in Fe-doped LiNbOs through the generation of space charge patterns that set up electric field induced modulations on the refractive index. 1 This model has been established for undoped LiNb0 38 which contains a low but significant concentration of Fe impurities. 5 It is verified for doped crystals by the results presented here. Figure l(a) shows a crystal before storage of a hologram. It contains a uniform distribution of electrons (negative signs) in deep electronic traps (circles). The authors are with RCA Laboratories, Princeton, New Jersey 08540.

Received 10 October 1973. 788

APPLIED OPTICS / Vol. 13. No. 4 / April 1974

503

The empty traps are Fea+ ions, 5 and the occupied traps can be considered as the same ions in a reduced valence state, i.e., 3

Fe +

+

e -

(Fe2 +

Fe2+.

The occupied traps ions) introduce optical absorption due to the photoexcitation of the trapped electron into the conduction band.5,& The amount of coloration for a given crystal depends on the fraction of traps that is occupied which in turn depends on the oxidation/reduction state of the crystal.5-7 .s Figure 2 shows the spectrum of the coloration induced by chemical reduction of Fe impurities in LiNbOa. The corresponding increase in trapped electron density must be accompanied by charge compensation by other ionic species, oxygen vacancies perhaps, but these apparently are not directly involved in the storage process. To record a hologram, the crystal is exposed to two intersecting coherent light beams. These beams form an interference light pattern, shown in Fig. l(b). Photoexcitation of the trapped electrons creates a corresponding pattern of free electrons. The electrons can then migrate due to either (1) drift in a uniform electric field 1 •10 or (2) diffusion caused by periodic gradients in the free carrier density .11 In successive steps of excitation, migration, and retrapping, the electrons are rearranged to produce regions of uncompensated space charge. Figure l(b) shows an extreme case. Nearly all the trapped charge are displaced from the regions of high light intensity into regions of low light intensity. The resulting space charge sets up a periodic electric field pattern that modulates the refractive index via the electrooptic effect. This gives a phase hologram. Once recorded, the hologram can be erased by simply exposing it to a uniform light beam. The light excites all the trapped electrons and allows them to redistribute uniformly, bringing the crystal back to its original state. Erasure by a coherent

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0

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8

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EXPOSURE TO LIGHT LIGHT INTENSITY ELECTRIC ~:~r FIELD AND ~~~~~:~:~~;~~or;'(,~

~,~~,~~

INDEX OF REFRACTION VARIATIONS

Fig. 1. Distribution of electrons (negative signs) in deep photoionizable electron traps (circles), (a) before and (b) after storage in Fe-doped LiNbO,. The traps are Fe 3 + ions when empty and Fe 2 + ions when occupied.

light beam at the proper angle for readout can lead to self-enhancement8 •12 complications associated with interference between the readout beam and the diffracted beam within the crystal. This interference pattern will record a new hologram in the manner described above. The result is a new erasure behavior that depends on (1) the spatial phase of the new hologram relative to the one being erased, (2) the relative sensitivity of the crystal for storage and erasure, and (3) the diffraction efficiency of the hologram to be erased. In some cases, this process can actually increase the net efficiency during readout. 1 2 The rate at which the diffraction efficiency increases (or decreases) during exposure to light of a given intensity is a measure of the crystal's sensitivity. It depends on (1) the diffraction efficiency that corresponds to a given space charge and (2) the rate at which the space charge builds up (or decreases). We are concerned here with the latter and in particular its dependence on the density electronic traps. Clearly, the sensitivity to incident light increases with increasing density of occupied traps, since these absorb the light. This is the basis for adding Fe impurities to improve the sensitivity of LiNb0 3 (Ref. 4) and for the use of reduction techniques for further increasing the material's absorption strength.5· 6 Of course, if too much incident light is absorbed, very little is transmitted through the crystal for readout. An optimum situation is when the net absorption, determined by the occupied trap density and the crystal's thickness, is ~50% at the writing wavelength. In that case, half of the light is used for storage, and half of it is transmitted for readout. The work that we are reporting here concerns the density of empty traps. This determines, in part, how long a free electron remains in the conduction band before it is retrapped, i.e., it determines the lifetime of the free electron. The lower the density, the longer the lifetime, and thus the higher the

quantum efficiency (the charge transport per absorbed photon) for storage. Following a procedure suggested by this model, we investigated crystals with a lower density of empty traps than had been used previously. The lower densities were obtained by reduction treatments that filled nearly all the traps. To maintain the net absorption at a practical level we use crystals with a relatively low Fe concentration ( ~0.0005% to ~0.005%). 111.

Experimental Technique

Poled single crystals of Fe-doped LiNb03 were used. They were grown at RCA Laboratories or purchased from Crystal Technology. They were reduced by annealing in a pure argon atmosphere or in powdered Li2 C03. 8 The crystal thickness was chosen for optimum absorption level (50%) at the writing and readout wavelength (4880 A). Nominal Fe doping levels are listed throughout this paper, and correspond to about twice the actual values. The Fe ion concentrations, and the effect of the reduction treatments on the valence states, were checked by optical absorption measurements and by EPR, the latter done by B. Faughnan of RCA Labs. Holograms were recorded and read out using the apparatus shown in Fig. 3. The polarization vectors and the crystal were aligned for maximum effect. 1 The polarity of the c axis is important and, unless otherwise noted, is aligned as shown, pointing toward the detector side of the apparatus. For storage, the crystal is exposed to both beams and for readout or erasure, only to the reference beam. The

2.0

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-~~ ;::====~===:::::~': al_----,~=::=:::::: 9000 7000 6000 5000 4000 WAVELENGTH (~)

Fig. 2. Optical absorption of occupied traps (Fe 2 + ions) introduced by reduction heat treatments. A fully oxidized Fe-doped LiNbOs crystal contains only empty traps (Fe 3 + ions) which do not contribute appreciably to the absorption. The rising absorption for wavelengths shorter than 4000 A is due to the electronic absorption edge of LiNb0 3 •

Fig. 3. Hologram storage apparatus. The polarization vectors are in the plane of intersection as is the c axis of the crystal. April 1974 / Vol. 13, No. 4 / APPLIED OPTICS

504

789

~-----------~,----

sure with the reference beam. The -c orientation, the one shown in Fig. 3, gives faster erasure than the +c direction, but both record at equal rates. The difference between the two erasure orientations is found to be accentuated at higher diffraction efficiencies. Figure 6 shows the erasure for another lightly doped, highly reduced crystal having high sensitivity. A higher efficiency was obtained by using greater crystal thickness. Both directions completely erase, but the +c orientation requires a much longer exposure. In fact, the efficiency actually increases during the initial erase exposure for this orientation. Such an effect has been observed previously, 12 but without subsequent erasure. Close inspection at low efficiencies of the - c erasure of our crystal revealed a slight oscillatory behavior predicted by Blanc 1 3 from a small signal analysis of the

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INCIDENT EXPOSURE

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Fig. 4. Hologram storage (increase of efficiency) and erasure (decrease of efficiency) for two different LiNb0 3 samples: (a) 0.1% Fe-doped, 0.8-mm thick, partially reduced (argon at 850'C); (b) 0.005% Fe-doped, 1.8-mm thick, almost fully reduced (argon at 1050'C).

diffraction efficiency is given as the ratio of the diffracted beam to the transmitted beam. In these tests, both were plane collimated beams of equal intensity. Total intensities of from 10 mW/cm 2 to 200 mW /cm 2 were used for both storage and erasure. IV.

z

4

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3

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Results

w

Figure 4 shows the write and erasure behavior for two crystals of Fe-doped LiNb03 of comparable thickness. The low doped crystal (0.005%) had been reduced by more than 90%, (EPR measureme nts showed that >90% of the traps were filled.) The more heavily doped crystal (0.1%) was partially reduced (only ~ 15% occupation) . The dopings and reductions were chosen such that both crystals had equal absorption strength. Thus, any differences between the two is caused by a change in the sensitivity to absorbed light. The more heavily doped crystal shows the typical behavior of Fe-doped LiNb0 3 , characterize d by an erase sensitivity much lower than the storage sensitivity. 6 The lightly doped, heavily reduced crystal on the other hand has higher sensitivity, particularly for erasure where there is a nearly 100-fold improveme nt. In addition to having a higher erasure sensitivity, the 0.0053 sample has the added advantage of not exhibiting the problems usually encountered during readout and erasure of holograms in Fe-doped LiNb0 3 namely, (1) self-enhanc ement caused by interference between the readout beam and the object beam within the crystal is discussed in Sec. II and (2) optical scattering caused by large scale optically induced inhomogeneities. 6 These were prevented in the 0.1 % sample by rotating it back and forth during exposure to the reference beam and only periodically reading out the hologram. The 0.005% sample, however, could be quickly and completely erased with the on angle reference or object beams. No optically induced scattering was observed, and the only selfenhanceme nt effect was a small variation in the erase time upon reversal of the crystal direction. Figure 5 shows the effect of c axis polarity on era790

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4 EXPOSURE

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Fig. 6. Erasure during readout with reference beam for a 0.9-cm thick crystal of undoped LiNb0 3 containing -0.0005% of Fe im· purities (measured with EPR). The crystal was reduced in argon at lOOO'C. The -c and +c levels refer to the orientations shown in Fig. 5. Storage energy was -3 Jjcm•.

APPLIED OPTICS / Vol. 13, No. 4 / April 1974

505

10~--,.----.---.--,---..,---:i

z

e

..". .. .... a: 5

0

10

20

30

40

50

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EXPOSURE (mj/cm2J

Fig. 7. Diffraction efficiency during erasure with .the reference beam for holograms in 1.8-mm thick reduced crystals of 0.005% doped LiNb0 3 • In each case, the initial point is the saturation efficiency. The lower initial points correspond to greater degrees of reduction. Optical absorption at 4880 A was -50% in each case.

i

5~

could obtain in the 0.1 % sample used for Fig. 4 which was approximat ely 90%. 15 The erasure rates of these crystals, particularly the bottom two in Fig. 7, show little effect upon c-axis reversal. The nearly straight lines in Fig. 7 show exponential erasure, i.e., the exposure required to erase a hologram by a given fractional amount is independen t of the initial diffraction efficiency. This allowed us to choose a diffraction efficiency for which storage and erasure required roughly the same exposure, as explicitly shown in Fig. 8. Table I summarizes the results of the crystals reported here and gives the efficiency for symmetrica l write-erasure behavior of the crystals used for Fig. 7. The storage sensitivity can be enhanced in our improved crystals by the application of an elect~ic field. Figure 9 shows the storage and erasure with and without a c-axis field applied to the 0.005% sample of Fig. 4. Ten kilovolts were applied with colloidal graphite contacts on opposite c faces of a -1-cm wide crystal. The value of the actual field inside the crystal was not measured. The applied voltage did not appreciably change the erasure rate, and its effect on storage was independen t of polarity. Both results have been previously observed in undoped LiNb0 3 .s.is Upon optical exposure equivalent to a few write-erase cycles, the effect of the applied voltage was greatly decreased, due probably to internal

4~ ~ I

Table I.

Sensitivity of Fe-Doped LiNbO,

>

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Doping

Treatment

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700

800

Fig. 8. Erasure for the 0.005% sample of Fig. 4 for the two different holograms: (a) one close to saturation efficiency; (b) one at a lower efficiency where the write and erase times are approximately equal.

self-enhancement effect. Large oscillations in efficiency during readout have been observed in Rhdoped crystals.14 The erasure sensitivity for a given crystal increases with increasing degree of reduction, but this is accompanied by a decrease in the saturation efficiency. Figure 7 shows this effect for a set of identical samples (same doping, same thickness) reduced to progressively greater degrees. 7 Each reduction was greater than 903 so that all samples absorbed the same amount of light. We see here that as the erasure sensitivity (the slope of the erase line) increases, the saturation efficiency (the beginning point of each erasure line) decreases. These saturation efficiencies are much smaller than the value we

Erasen ('/,) (mJ/ cm 2 )

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[TEMPERATURE]

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FIG. 2, Thermal time constants for erasure of fixed holograms and ionic relaxati9n. The grating spacing depends on the wavelength and the angle between the beams; for/. =4880 A and beam angles of 30° (I= 0. 94 µm) and 90' U= 0. 35 µm).

is observed by cooling the crystal to room temperature and exposing it to light. 5 • 6 The light redistributes the electrons so that the ionic pattern is no longer completely neutralized. The electric field pattern then reappears and the hologram can be read out. Approximate ly 15 J/cm2 of absorbed light was required at high temperature to obtain a fixed hologram with a - 40% diffrac tion efficiency when read out at room temperature . The most intense fixed grating recorded had a refractive 2 index modulation of - 4X10"4, and required -300 J/cm Fe-doped 0.1% of crystal 2-mm-thick a For exposure. LiNb0 3 the modulation amplitude was calculated to be 6 times greater than the value correspondin g to 50% diffraction efficiency by comparing the angular dependence 10 for readout of the grating 9 with Kogelnik's calculation. Figure 2 shows the thermal erase time constants for two crystals as a function of temperature . The erase times were measured with the sample hot by monitoring the decay of a weak absorption hologram associated with the redistributio n of trapped (and absorbing) electrons. The erase of the absorption hologram was accompenied by a proportional decrease of the diffraction efficiency of the resulting fixed phase hologram that would be observed at room temperature . The erasure depends strongly on Fe concentratio n and on beam angle. The latter is shown by the quadratic dependence of erase sensitivity on the grating spacing. Neither the Fe concentration nor the beam angle was observed to strongly affect the fixing time, as is also shown in Fig. 2. From 183

the slope of the lines in Fig. 2 the thermal activation energy for erasure is - 1. 4 eV and for fixing the activation energy is - 1. 1 eV as observed for undoped LiNb0 3 • 6 Thermal erasure is thus minimized during hologram storage with heavily doped crystals and large grating spacings. The procedure used for multiple hologram storage was to sequentially record the holograms, while keeping the crystal at 160 °C and rotating the sample by a slight "mount (-0.1° for a -1-cm-thick crystal) after each recording. After cooling the crystal to room temperature and exposing it to light, the different holograms can be read out by rotating the crystal. The angle for readout of a given hologram differed by a small amount from the angle used for recording due to thermal contraction of the grating between recording and readout. Figure 3 shows the room-tempe rature readout of 100 holograms that were recorded with equal exposures in a O. 02% Fe-doped sample. The gradual decrease of diffraction efficiency from high numbers (last ones recorded) to low numbers (first ones recorded) as a measure of the erasure that occurs. 511 holograms were subsequently stored and fixed in this crystal, attempting to compensate for erasure by progressivel y decreasing the exposure during the storage sequence. The diffrac tion efficiency ranged from 2. 5% for the last hologram recorded to 25% for the first hologram recorded (i.e., we overcompen sated). A total exposure of 1400 j was used. The readout quality of holograms fixed by the technique reported here is as good as or better than previously observed for holograms, fixed or unfixed, in LiNb0 3 • The readout image quality was judged subjectively since detailed signal-to-no ise measuremen ts have not been· made. Image distortion attributable to thermal contraction of the holographic grating between the recording and readout temperature s has not been observed. The good quality arises from the fact that the storage at elevated temperature reduces buildup of the optical scattering effects during storage. Such effects can occur during readout, and lead to a nonpermanent degradation in image quality. Exposure to in60

,. "z

"' difTerentation. This method is compared with that based on angular multiplexing. A reduction in holographic cross talk is predicted. Further cross·talk reduction that is due to sidelobe suppression is observed in experiments by using photorefractive crystals and the proposed orthogonal data storage.

Optical data storage in volume holograms has been an exciting prospect since the early days of holography.' This is due mostly to the theoretical storage capacity of - V/ A3 bits in a volume V. This, as an example, translates to 8 x 10 12 bits in a 1-cm" volume at A. = 0.5 µm. Early attempts at exploiting this potential have been disappointing. 2 • 10 Even though hundreds of holograms have been successfully stored in a single volume in these experiments, the data content of the individual holograms was minimal, and, therefore, so was the overall volumetric information density. This was due partly to material limitations 11 •12 but, more fundamentally, to the intolerably high cross talk resulting from the angular multiplexing method used to record the large number of holograms.•- 7 In this traditional method, many twodimensional images are recorded sequentially, each with its unique angularly multiplexed reference wave. Recently, however, phase-encoded or spatially orthogonal reference beams have been used to multiplex many holograms. 13•14 These approaches, nonetheless, are variations of the basic angular multiplexing technique and differ fundamentally from the orthogonal wavelength-multiplexed storage technique proposed here. To appreciate the nature of the problem we recall that the basic property, and virtue, of holography is the essentially uniform distribution of the recorded information throughout the physical volume of the recording medium. We assert here the self-evident, but heretofore ignored, fact that it is equally important to distribute the information throughout the grating K space as well. Failure to do so increases the cross talk between the reconstructed holograms. Consider a sinusoidal volume grating written by two plane waves E 1 exp{i[wt - k' 11 • r]} and E0 exp{i[wt - k 101 • r]} described by the index-of. refraction distribution

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Such a grating can be identified by the point K in K,, K,., K, space. If the wave E0 is spatially modulated, say by image information, the corresponding K point becomes a surface whose extent (or solid angle) is determined by the smallest feature size of the information. Such a surface can be labeled by the reference wave used to record it and is shown as K,l;!,;ng in Fig. 1 along with the responsible k)!I. The same figure also shows the surface ~~tins that is due to a second image recorded with a second, angle-multiplexed reference wave k~}. Reconstruction of, say, image 2 by illuminating the stored holograms with a wave along ~} will inevitably lead to undesirable parasitic scattering

Fig. 1. K space diagram for volume holography that uses angle-multiplexed storage. © 1992 Optical Society of America

519

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OPTICS LE'ITERS I Vol. 17, No. 20 I October 15, 1992

by :::.w/21T. The increase in the information content of a given image causes a mostly lateral spread of the loci, which allows a more uniform coverage ofK space. The mismatch parameter !!i.k that determines the relative intensity of the cross talk is nearly constant in magnitude so that the level of cross talk is essentially independent of the information content tspatial detail) of the images. This indicates the potential for high data storage capacity by using the proposed storage scheme. The orthogonal nature of the curves in Fig. 2 has led us to call this scheme orthogonal data storage. 16· 17 Experiments were conducted with a 2-mm-thick z-cut and polished lithium niobate (LiNb0 3 ) crystal. Figure 3 shows the wavelength dependence of the diffraction efficiency in the reflection geometry. The hologram was recorded with A= 641.7 nm and the geometry of Fig. 2. The figure also shows a theoretical plot of the expected diffraction efficiency. One major surprise is the smaller cross talk of the actual grating compared with the theoretically predicted values. As an example, at 1 nm from the recording wavelength of 641. 7 nm, the measured value is - 44 dB of the peak value compared with a theoretical value of -37 dB. We believe that the discrepancy is due to an effective apodization whose origin is not fully understood. The holographic cross talk in an orthogonal data storage scheme was investigated by recording two different, high-resolution holograms written with wavelengths 2 nm apart. Integrated circuit masks consisting of 1-5-µm-wide lines were used as the transparency T in the experimental arrangement of Fig. 4. The writing beams were oriented in the counterpropagating geometry of orthogonal data storage. Figure 5 summarizes the experimental results. A holographic cross-talk level of -43 dB is measured at the vacant address at 642 nm, centered 1 nm between the other two infonnationladen holograms. This value is essentially the same as the plane-wave (no information) result of Fig. 3, which thus shows that holographic cross talk does not increase in orthogonal data storage as substantial amounts of information are added to the images. Since !J.k at a given level of cross talk scales as the inverse of the hologram length, the data of Fig. 5 are consistent with a cross-talk level of < -40 dB with

K,

-k,.1

(a)

K,

(b)

Fig. 2. K space diagrams for volume holography that uses orthogonal data storage: (a) single hologram; (bl multiple hologram storage. of the incident wave off the recorded hologram of image 1. The locus of the desired reconstructed k:~ vectors in this case are shown in the figure as well as those of the undesired (cross talk) k!,~;;J. t.aJk vectors due to the scattering off grating 1 by reference beam 2. A relative measure of the cross talk is 15

...

0 z

(1)

where g(z) is the thresholding nonlinearity, Wi is the ith weight, and Xi is the ith element of the input pattern. Such a system can be used to dichotomize a set of patterns into two prescribed classes, and more complex, multiple layered networks can be built up using this as the basic building block. Also, extensions to multicategory pattern classification can be achieved by having a matrix of weights and a multiplicity of output units as is discussed in Sec. III.

The authors are with Rockwell International.Science Center, P.O. Box 1085, Thousand Oaks, California 91360, Received 25 August 1989. 0003-6935/90/203019-07$02.00/0. © 1990 Optical Society of America.

Simple learning algorithms can be characterized by the update equation: w;(p

+ 1) =

w,(p)

+ a(p)x,(p),

where wi(p) is the ith weight at time p, x;(p) is the ith element of the pattern shown at time p, and a(p) is a multiplier that depends on the particular learning algorithm. This includes most deterministic algorithms, where some kind of a descent procedure is involved and incremental changes are made to the set of weights during each iteration. For perceptron learning, 4 Oif output y(p) was correct

a(p)

= { 1 if y(p) = 0 but should have been 1

-1 if y(p) = 1 but should have been 0.

(3)

The threshold bias can be absorbed into the patterns by choosing one element of each pattern to be always equal to a nonzero constant. 4 Note, that both additive and subtractive changes to the weights Wi must be made to implement the algorithm directly. Although multiple layer networks require more complex routines such as Back Propagation, most algorithms share the common requirement of bipolar weight changes. II.

Optical Implementation

The basic components to implement the network described above are an input device to convert the patterns into the appropriate format (e.g., electrical to optical, incoherent to coherent optical), an interconnection device, and a thresholding device for the output unit. The function of the interconnections in this context is to simply compute the inner product between the input pattern Xi and the weights Wi. Volume holograms can be used to implement such functions 5 in a way that is extendable to the multiple category case (i.e., multiple inner products). Consider the arrangement shown in Fig. 2 where a holographic medium is positioned at the Fourier plane of lens LI. The input pattern is displayed in the spatial light modulator (SLM), which is positioned at the front focal plane of 10 July 1990 I Vol. 29, No. 20 I APPLIED OPTICS

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(2)

3019

SLM 1: w, (x,y);

I

2: w,(x,y>

z

1-a:

~~ x, --~0.

Hologram

2 WEIGHTED INTERCONNECTIONS

THRESHOLD GATE

a

J

PHASE CONTROL

~

.

w(x,y)

b

Fig. 3. Multiple output holographic system: (a) recording (point source 1 is activated when w1(x,y) is in SLM and point source 2 is activated when w 2(x,y) is in SLM); (b) read-out (y1 = and Y2 = ).

Fig. 1. One layer neural network.

plane wave

CONTROLLABLE PHASE SHIFTER

LM f(x,y) Recording Fig. 2.

Read-out

Inner product computation using volume hologram. Fig. 4.

the same lens. A hologram is exposed with a pattern w(x,y) in the SLM and a reference plane wave as shown. After development, another pattern f(x,y) is loaded into the SLM. The light passing through the SLM is diffracted by the hologram and the diffracted amplitude can be shown to be the inner product between the two patterns w and f. 5 Clearly, this is an overkill since the same function could have been achieved with a planar hologram. However, in the multiple category case where a number of different inner products need to be computed simultaneously, the added dimension afforded by the volume hologram is necessary; unless one resorts to spatial multiplexing of the planar hologram. 5•6 Multiple category classification is achieved by what is essentially an angular multiplexing of the volume hologram. This is shown in Fig. 3 where multiple holograms are written using the various reference plane waves. For the moment, we focus on the single output case and discuss generalizations for the multiple category problem in Sec. III. By virtue of their dynamic nature, photorefractive crystals are ideal candidates for the holographic medium. In addition, crystals such as LiNb03, BaTi03 and SBN are by far the most efficient holographic media requiring relatively low optical intensity levels (e.g., 1 W/cm 2); the most efficient photorefractive crystals exhibit photosensitivities approaching that of photographic film. The holographic process records both modulus and phase of patterns and, thus, holographic interconnection techniques can be used to store bipolar valued weights and adaptable changes, which are either subtractive or additive, are possible. 3020

Grating phase control using phase shifting device.

The Perceptron implementationreported by Psaltis et al. 2 employed photorefractive crystal for adaptable interconnections but used incoherent erasure to achieve subtractive weight changes. In their system, the photorefractive crystal was placed at the image plane of the SLM and a device (movable piezoelectric mirror) was used to provide either a coherent reference or an incoherent one. For additive changes to the hologram, a coherent reference was provided so that the hologram is strengthened, but for subtractive changes, the reference beam was made to be incoherent with respect to the object beam so that nonuniform incoherent erasure resulted. This is roughly equivalent to providing a weight bias in which erasure will result in subtraction. Such methods involving incoherent erasure do not fully take advantage of the phase sensitive nature of holography. To accurately implement the learning algorithm described by Eq. 2 and to exploit the hologram's coherent capability just described, a method by which both subtractive as well as additive changes to the weights can be made must be devised. Physically, such a method would amount to a multiple exposure hologram in which each exposure results in a grating (index grating in the case of photorefractive crystals) whose phase is either 0 or 7r (relative phase difference is 11"). An obvious method of achieving this is shown in Fig. 4 in which two coherent waves intersect in a medium to create a holographic grating. One of the beams passes through a phase shifting device (e.g., an electrooptic

APPLIED OPTICS I Vol. 29, No. 20 I 10 July 1990

552

light sou,rce 1 light source 2

terferome ter because both input ports are used. The beam from laser source 1 (marked by the dotted line) is split into two paths by the dielectric beam splitter to intersect within the holographic medium to create a grating. Another laser source (which can be derived from laser source 1 or a separate laser with the same nominal wavelength) is positioned so that the directions of the beams transmitte d and reflected by the beam splitter traverse precisely the same paths as those resulting from source 1. The Stokes' principle governs the relationship between the reflection and transmission coefficients seen by source 1 (call them r and t, respectively) and those seen by source 2 (r' and t'). Simple arguments 7 using this principle lead to the result

tj:i---shu tter 1

holograph ic medium Fig. 5. Holographic phase control using Stokes' principle ofreversibility.

t

phase modulator, liquid crystal phase modulating device, piezoelectric movable mirror) to acquire a phase of either 0 or 7r relative to the other writing beam. After writing a hologram with one phase setting, holographic subtractio n can be performed by using the other phase setting in the subsequen t exposure and addition can be performed by using the original phase setting. Such phase shifting devices, however, may suffer from inaccuracies and also performance may degrade with time depending on various device characteristics (e.g., backlash in piezo mirrors, voltage inaccuracies in phase modulatio n devices (electrooptic, liquid crystal). Also, the continuous phase variability of these devices is an overkill in the present application in which only two phase settings, 0 and 71", are required. In this paper, we describe a system that relies on a fundamen tal principle in optics known as the Stokes' principle of reversability7•8 and a pair of shutters for the phase control. The basic principle of operation of the phase control system can be described with the help of Fig. 5. This configuration is known as a double Mach-Zehnder in-

= t' and rt*+ r't* = 0.

(4)

Note that since the amplitude of the hologram due to source 1 is proportion al to rt* and that due to source 2 is proportion al to r't*, the two gratings are mutually 7r out of phase. Of course, the accuracy of this result is dependen t on how well the beams can be aligned. Such a phase shift of 7r has been employed for the parallel subtractio n of images. 7 To quantify the accuracy that can easily be obtained in the laboratory , we performed the experime nt sketched out in Fig. 6. An argon ion laser beam (A = 514.5 nm, coherence length ~I cm) was split into two paths containing shutters. Each of the two beams (serving as the light sources 1and2 of Fig. 5) is directed into another beam splitter whose transmitte d and reflected components are recombined with an output beam splitter. This part of the apparatus is the double Mach-Zeh nder interferom eter mentione d earlier. The combined result is viewed along a line with a linear detector array to give the scanned image intensity distributio ns shown in Fig. 7. When shutter 1 is on (and 2 is off), the distributio n is nearly 7r out of phase with that corresponding to shutter 2 on (and 1 off); we

Argon Ion: 514.5nm

I

path length dilference > coherence length \

Microscope Obj.

Observation Plane

Fig. 6.

Experiment to view holographic fringes.

10 July 1990 I Vol. 29, No. 20 I APPLIED OPTICS

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3021

a

b

Fig. 9. Monitored diffraction efficiency (see text).

c 1 on; (b) Fig. 7. Scanned image of intensity grating: (a) shutter shutter 2 on; and (c) both shutters on.

estimate that the error is to within 3° due to alignment errors and also measurement errors. As expected, when both shutter s are on, the result shown in Fig. 7(c) is a fringe-less distribution with the nonuniformity due to the gaussian nature of the laser beams.

To investigate actual subtra ction of index holograms with the described appara tus, we used photorefractive BaTiOs (6 mm X 6 mm X 2 mm) as the holographic medium in the arrang ement shown in Fig. 8. Again, two beams derived from the same Argon laser served as the two light sources controlled by shutter s 1 and 2. The writing beam sources were of ordinary polarization with respect to 9the crystal to minimize effects due to beam coupling. The crystal was placed at the position where the two writing beams intersect to form a photorefractive grating and the diffraction efficiency of the resulting hologram was monitored with a read beam supplied by a He-Ne laser (A = 633 nm). First, a hologram was exposed with shutte r 1 open and 2 closed. As soon as the hologram reached nearly full strength, shutte r 1 was closed and 2 opened to allow the new hologram to develop. This sequence was repeat ed and the diffracted read beam intensity recorded as shown in Fig. 9. As is appare nt, as soon as shutte r 1 is closed and 2 is opened, the diffracted intensity diminshed to reach a complete null after which it rises again to reach saturation. This phenom-

Argon Ion: 514.5n m

Shutter 1 Shutter 2

BS

c

axis

He-Ne:

---------f-----~~~~~----BaTi03 3022

APPLIED OPTICS I Vol. 29, No. 20 I 10 July 1990

554

Pol.

Fig. 8. Experim ent to view subtraction of hologra ms using a BaTi0 3 crystal.

LIGHT SOURCE.: I I

¢--SHUTTER

1

: I I

READ/WAITE CONTROL SHUTTER

PERCEPTRON ALGORITHM: IF ERROR•1. OPEN SHUTTER 1; IF ERROR•-1, OPEN SHUTTER 2;

IF ERROR•O, GO TO NEXT PATTERN

PHOTOAEFRACTlVE CRYSTAL

ERROR 11.0.-11

DETECTOR

Fig. 10. Optical pattern classifier.

ELECTRONIC THRESHOLD ( 1.01

a:

~

ffi

so

25 iterations

(a)

so

25

ITERATIONS (b)

Fig. 11. Computer simulation learning curves (classification error shown as function of number of iterations): (a) photorefractive system with T/10 exposure time; and (b) ideal Perceptron system.

10 July 1990 I Vol. 29, No. 20 I APPLIED OPTICS

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3023

enon is due entirely to the fact that the newly exposed hologram because shutter 2 is 7r out of phase with respect to the initial hologram, whieh results from shutter 1 being open (the quenching of the hologram is actually aided in part by incoherent erasure). We now describe the overall optical system, shown in Fig. 10, in which an arrangement to allow for multiple exposures of a photorefractive hologram is diagramed. This setup uses the holographic phase control method just described. If the SLM contains a picture whose amplitude distribution is given by a(x',y'), the grating amplitude written in the crystal due to source 1 can be described by f 1(x,y) = K(l - exp(-t/T))

I0

A(x,y) , I A(x,y) 12

where te is the exposure time, so that the old weight distribution is diminished slightly as new holograms are written. This forgetting effect actually may prove to be beneficial in the optical implementation where the amplitude of the hologram has a finite dynamic range, since it helps to somewhat normalize the weight values during learning to keep the crystal from saturating. The output of the photodetector is given by

(5)

+

(8)

y(p) = ll2.

where r is the time constant of the medium (assuming intensity is kept constant for all exposures), ti is the exposure time, A(x,y) is the Fourier transform of a(x',y'), 10 is the reference beam intensity, and K is a constant determined by the characteristics of the particular crystal. If, without changing the picture, source 1 is turned off and source 2 is turned on, the new grating can be shown to be proportional to the first with the opposite sign. In particular, if the second exposure time duration is t2, then f(x,y) = K{(l - exp(-t/r)) exp(-t2 /r) - (1- exp(-t 2/T))j

erasure). This will modify the actual learning algorithm that is used in the following way. The weight update equation in comparison with the ideal rule of Eq. 2 is given by (7) w,(p + 1) = exp(-t/T)w;(p) + [1- exp(-t/r)]a(p)x,( p),

~(x,y)

I 0 + A(x,y) 12

.

(6)

True subtractive weight changes are thus possible without the use of external phase shifters. Note, in Eq. 6 that as the new out of phase hologram is being written, the initial hologram is partially erased due to the presence of the writing beams (incoherent

System simulations were performed using Eqs. 7 and 8 as the model, and the learning curve thus obtained is shown in Fig. ll(a), which verifies that, with the described modifications, the Perceptron training scheme works well. The result of the example shown in Fig. ll(a) was obtained for the task of dichotomizing a set of twelve randomly chosen binary patterns, each with thirty-two elements. The forgetting factor that was used was 0.9 ( = exp(-te/r ), so that an exposure time of ~r/10 was assumed, where r is the photorefractive response time constant. Sequencing through the entire set of patterns once is considered as one iteration of the algorithm. A comparison simulation of the original Perceptron algorithm is sl}own in Fig. ll(b) [same conditions as those in Fig. ll(a)]. Ill.

Multiple Category System

The extension of this concept to the multiple category case, which requires a multiplicity of output lines as opposed to a single output sufficient for dichotomies, is

LIGHT SOURCE 1 I I

¢-SHUTTER1

II I

READ/WAITE CONTROL SHUTTER SHUTTERS

PHOTOREFRACTIVE CRYSTAL

Fig. 12. Multiple category pattern classifier. 3024

APPLIED OPTICS I Vol. 29, No. 20 I 10 July 1990

556

shown in Fig. 12, where the single detector and the reference beam shutter have been replaced by a detector array/shutter combination and an SLM/shutter combination, respectively. Here, the reference for each pattern is no longer a single plane wave but a set of plane waves dictated by the collection of openings in the 1-D SLM. Initially, the crystal contains no holograms and interconnections are built up by simply exposing the hqlogram with light source 1 with the pattern in the 2-D SLM and its associated reference pattern in the 1-D SLM. The process is repeated for each pattern in the training set. After the initialization, the first pattern is loaded into the 2-D SLM and the reference beam shutter is closed to interrogate the system. The reconstructed output pattern is then compared against the desired output pattern to yield the error vector. The algorithm must now be performed in two steps; first, only those portions of the 1D SLM corresponding to the positive portions of the error vector are opened and light source 1 is turned on to strengthen certain interconnections following the Perceptron recipe. Then, only those portions corresponding to negative elements of the error vector are loaded into the 1-D SLM and light source 2 is turned on to weaken the appropriate weights. The buffer needed to store the error result needed for this function is especially simple and can be integrated into the detector array/amplifier assembly. The subtractive capability allows for a more exact implementation of the learning algorithms, even for the multiple category case. Summary We have described a new implementation of a learning machine which implements the Perceptron algorithm for pattern dichotomy. The optical system imIV.

plements the weight storage and update functions using coherent means and, in particular, makes novel use of the Stokes' principle to achieve truly subtractive as well as additive weight changes, precluding the need for biases which are typically used in incoherent implementations as discussed earlier. The preliminary experimental results show that high quality subtractions are possible and the computer simulations will be verified by actual optical system operation in future work. This work is supported, in part, by the Office of Naval Research contract N00014-88-C-0230. References 1. D. Z. Anderson and D. M. Lininger, "Dynamic Optical Interconnects: Volume Holograms as Optical Two-Port Operators," Appl. Opt. 26, 5031-5038 (1987). 2. D. Psaltis, D. Brady, and K. Wagner, "Adaptive Optical Networks Using Photorefractive Crystals," Appl. Opt. 27, 1752-1759 (1988). 3. A. Yariv and S. Kwong, "Associative Memories Based on Message-Bearing Optical Modes in Phase Conjugate Resonators," Opt. Lett. 11, 186-188 (1986). 4. R. 0. Duda and P. E. Hart, Pattern Classification and Scene Analysis (Wiley, New York 1973). 5. D. Psaltis, C. H. Park, and J. Hong, "Higher Order Associative Memories and Their Optical Implementations," Neural Networks l, 149-163 (1988). 6. H.J. Caulfield, "Parallel N 4 Weighted Optical Interconnections," Appl. Opt. 26, 4039-4040 (1987). 7. P. Yeh, T. Y. Chang, and P. H. Beckwith, "Real-time optical image subtraction using dynamic holographic interference in photorefractive media," Opt. Lett. 13, 586-588 (1988). 8. J. H. Hong, P. Yeh and S. Campbell, "Trainable Optical Network for Pattern Recognition," in Technical Digest, Topical Meeting on Optical Computing (Optical Society of America, Washington, DC, 1989), p. 307. 9. See, for example, P. Yeh, "Two-Wave Mixing in Nonlinear Media," IEEE J. Quantum Electron. QE-25, 484-519 (1989).

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3025

a reprint from Optics Letters

1993

Self- orga nizin g phot orefr activ e frequ ency dem ultip lexer Mark Saffma n, Claus Benkert, and Dana Z. Anders on Departme nt of Physics and joint Institute for Laboratory Astrophys ics, University of Colorado, Boulder, Colorado 80309-0440

Received July 8, 1991 We demonstr ate a self-organizing photorefractive circuit that demultiplexes a beam that has two signals imposed on separate optical carrier frequencies into two beams, each containing one of the signals on its carrier. Unlike conventional demultiplexing techniques, this method requires little a priori knowledge about the carrier frequencies. The signal channels must be spatially uncorrela ted, and their frequency separatio n must be more than the photorefr active response bandwidt h (hertz to kilohertz) . The optical circuit uses no local oscillator, and the photorefr active response bandwidt h places no upper bound on the channel bandwidth. Experime ntal results for demultiplexing a beam that has two signals, with a BaTi0 3 circuit, show contrast ratios of better than 40:1 at the outputs.

Consid er the task of demulti plexing a number of signals carried on a multim ode fiber, and assume that we have no a priori informa tion about the carrier frequencies other than that they are differen t. How can the channe ls at the output of the fiber be separat ed? One might use a grating , but then the carriers would need to be well separat ed in wavelength, by at least an angstro m or so, and we do not know ahead of time that they are. One might use interfer ometric techniq ues, but these require additional frequency scan and locking electron ics that must also avoid possible degeneracies in resonan ce conditions. To make things worse, neither approach can handle the high spatial content of the speckle pattern s in a straigh tforwar d manner . Our a priori inform ation about the signals is limited to the fact that they have differen t frequencies, and, in the case of a highly multim ode fiber, that the individual speckle pattern s of the signals will be uncorre lated. Using this informa tion alone, and drawin g on some elemen tary concep ts from neural network s, we have constru cted a photorefractive optical circuit that self-organizes to demultiplex incomin g optical signals accordi ng to their tempora l and spatial differences. An optical circuit that demultiplexes two signal channe ls is shown schema tically in Fig. 1. The self-org anizing behavio r' arises from compet itive interact ions in multim ode photore fractive ring resonators. 2·3 The combin ation of the compet itive interactio n betwee n resonat ors in the gain medium and a local interac tion in each resonat or, which we refer to as reflexiv e gain, leads to a steady- state operating condition in which each resonat or chooses to oscillate at one of the pump frequencies. A portion of the oscillating energy is then coupled out of the resonat ors to give the demultiplexed outputs . Note that the optical circuit is purely signal driven: there is no local oscillat or, and there is no local power source. Argum ents based on steady- state energy transfe r in photore fractive two-bea m coupling will clarify the operatio n of the circuit. We will, for simplicity, con0146-9592/91/241993-03$5:90/0

sider a two-res onator circuit demultiplexing a twosignal input beam. To demultiplex N frequencies we would include N resonat ors, each sharing the same pump volume, and each with a reflexiv e photorefract ive interac tion. The resona tors suppor t many transve rse modes, and to simplify the analysi s we assume that the oscillating-mode superpo sition is always on resonan ce, so pump-d etuning effects need not be considered. When the circuit is above thresho ld the two rings compete for energy in the shared gain medium . Without addition al interactions the rings are at most neutral ly coupled,4 and any mixture of the inciden t frequencies can oscillate in each ring. The key to achievi ng the demultiplexing behavior is a nonline arity that raises the cavity losses when two signals try to oscillate in the same ring. There are two distinct cases to be considered. In the first case the two signals may attemp t to oscillate with identic al spatial- mode superpo sitions. This is a high-loss configu ration since the oscillat ing signals are now spatiall y matched, and each one will scatter off the grating owing to the interfer ence of the other oscillating signal and its pump. Thus one spatialmode superpo sition oscillat ing at two frequen cies is disfavored. In the second case, where the two signals choose orthogo nal spatial- mode superpo sitions, the desired nonline arity is provided by the reflexiv e gain interact ion. A portion of the oscillating energy is taken out of each ring and then coupled back into the same ring by photore fractive two-beam coupling, as is detailed in Fig. 2. The standar d equatio ns for photore fractive two-beam coupling 5•6 can be used to derive an expression for the steadystate transmi ssion T of the resonat or beam through the reflexive gain interact ion. If only one signal oscillates in the resonat or beam we find that T = l/[1 + r exp(-GR )] in the absence of reflecti ve and absorpt ive losses. The pump-t o-signa l ratio incident upon the photore fractive medium is given by r = 8/(1 - 8), where 8 is the intensi ty reflecti on coefficient of the beam splitter , exp(GR) is the smallsignal gain in the photore fractive medium , and © 1991 Optical Society of America

559

1994

OPTICS LETTERS I Vol. 16, No. 24 I December 15, 1991

q=q

Fig. 1. Schematic diagram of the self-organizing circuit.

resonator beam

Fig. 2. Reflexive gain interaction. A fraction /5 of the incident intensity is removed from the resonator beam and then coupled in again by photorefractive two-wave mixing.

GR = rRzR, with rR the photorefractive coupling constant and ZR the interaction length. In the steady state the reflexive interaction simply contributes to the passive resonator losses. 7 When two signals, each with orthogonal transverse-mode superpositions and different carrier frequencies, oscillate in one ring, two orthogonal gratings build up in the reflexive gain crystal. The effective steady-state coupling strength between two waves that have the same frequency when several mutually incoherent beams are present can be written as GR err = GR(Iint/1), where I is the total intensity and I;nt is the portion of the intensity that writes the photorefractive grating. Therefore, when two frequencies are present, the effective coupling for each mode superposition is reduced from the single-frequency case. T is correspondingly reduced, hence multifrequency oscillation is again disfavored. The reflexive gain interaction also serves to ensure a high contrast ratio between the signal intensities in each ring. Even when the input signals are spatially uncorrelated, fanning gratings in the gain medium will couple both input signals to both resonator rings, which tends to reduce the oscillating contrast ratio. Nonetheless, a high steady-state contrast ratio can be achieved because the reflexive gain interaction actively enhances the contrast ratio, counteracting the contrast ratio reduction in the gain medium. The contrast ratio enhancement in the reflexive gain interaction can be calculated as follows. Let the oscillating beam consist of two fields, having different frequencies and occupying orthogonal mode superpositions, with intensities 11 and 12 • The contrast ratio at the input to the reflexive gain interaction is defined as q = 1,/12 • The effective steady-state coupling strengths for the two signals are GRl.eff = GRq/(l + q) and GRz.eff = GRl/(l + q), and the contrast ratio at the output of the reflexive gain interaction is then given by

1 1 + r exp(-GR-- ) 1+q . 1 + r exp(-GR_q_) l+q

(1)

The contrast ratio enhancement follows, because for q > 1, ij > q, and for q < 1, ij < q. We have attempted to provide some heuristic arguments for the observed behavior of the demultiplexer. It is also useful to know the values of circuit parameters that are necessary to ensure the desired behavior. In order to demultiplex N signals there must be sufficient gain for the resonators to be above threshold. The oscillation threshold may be derived from small-signal gain considerations analogous to those for a single-crystal photorefractive resonator. 8 For an N-ring circuit pumped by a beam having N signals of equal intensity we must have (1 - L) l

+

\

r exp -

G /N) exp(Gp/N) > 1, R

(2)

where L is the passive cavity loss. The conditions that ensure stability of the demultiplexing state to small perturbations, and the instability of all other steady-state conditions, have yet to be determined. We have experimentally demonstrated the circuit described above as shown in Fig. 3. The output from an Ar-ion laser operating at 514.5 nm is split into two and then frequency shifted with acoustooptic cells to generate two beams with a frequency separation of 280 MHz. The beams are then coupled into a 1.5-m-long step-index, 100-µ,m-core, multimode fiber. By arranging each beam to have a different angle of incidence on the fiber end face, we obtain uncorrelated output speckle patterns, even with a relatively short fiber. 9 The output of the fiber is then used to pump the photorefractive resonators. By modulating the acousto-optic cells with low-frequency square waves, the carrier content in

,_

+~2.7kHz Wmaai"'l7.8kllz

-t-

Fig. 3. Experimental demonstration of a self-organizing frequency demultiplexer. The circuit configuration is equivalent to that shown in Fig. 1. Each resonator ring has a length of -2 m, of which 1 m is 100-µ,m-core multimode fiber. The two resonator rings are positioned on top of each other with the modes crossing in the pump crystal and occupying separate volumes in the reflexive gain crystal. The pump crystal is 45'-cut BaTi0 3 , and the reflexive gain crystal is O'-cut BaTi0 3 • The resonator parameters are exp(Gp) = 1700, exp(GR) = 230, 1 - L = o.oi, and /5 = 0.96.

560

December 15, 1991 I Vol. 16, No. 24 I OPTICS LETTERS

I,

Ring 1

/

-

~ ~..,,-

I

I

v

.f

I,

1,1/ Ring 2

v

,../

~

r-..

I

I I,

Fig. 4. Transient behavior of the circuit. The curves represent the envelopes of the square waves driving the acousto-optic modulators. Signal I 1 on carrier w 1 oscillates in resonator ring 1, and signal 12 on carrier w 2 oscillates in resonator ring 2.

Fig. 5. Outputs of the two resonator rings. The signal carrier frequencies are separated by 280 MHz. each ring can be monitored by detecting the output intensities with low-speed photodiodes. When the input signals are presented to the circuit it takes a few seconds for the self-organiz ation process to reach steady state. The mapping of the input signals into resonator modes is initially noise driven, so which signal will oscillate in which ring is random. We observe the dynamical evolution by detecting the envelopes of the carrier modulations, as shown in Fig. 4. With a total pump intensity of 10 mW, focused to a 0.58-mm-di ameter spot in the pump crystal, the circuit takes approximate ly 10 s to reach the high-contra st-ratio demultiplex ing state shown in Fig. 5. This operating state is then stable for as long as the pump beam is present. The total steady-state oscillating intensity in the two rings reached 0.4% of the pump intensity. Measuremen ts with a spectrum analyzer show an output contrast ratio of better than 40:1 in each ring. This is consistent with our previous holographic measuremen ts of the orthogonalit y of speckle patterns transmitted over a short length of fiber, 9 where we found bestcase cross-talk levels of 50:1. In order for the circuit to demultiplex the input signals, their intensities must be matched to within approximat ely

1995

10%, otherwise the stronger signal will oscillate in both rings. We are currently investigatin g alternative designs that may relax this constraint. This circuit has potential application to optical communicat ions systems. We showed recently that the spatial-mod e superpositio n can be used as a multiplex parameter to transmit multiple communication channels on a single multimode fiber. 9 In that case it was necessary to record, and periodically refresh, the holograms that demultiplexe d the channels. The circuit described here demultiplexe s spatially distinct channels while continuously adapting to slow variations in the spatial-mod e superposition owing to drifts in the carrier frequency or perturbation s to the fiber. The circuit can also be used with channels that have been transmitted on a single-mode fiber. Taking the output of the single mode fiber and sending it through a sufficiently long length of multimode fiber, before pumping the self-organiz ing circuit, will impress a distinct spatial mode on each channel. 10 In the parlance of neural networks this circuit demultiplexes through a self-organiz ing process. The circuit learns to associate distinct input signals with different resonator modes. The essential characteristics of the input signals that allow them to be separated are their spatial and temporal orthogonality. We have used acousto-optic modulators to generate well-defined input signals. In principle any type of transduction mechanism that provides distinct input frequencies with spatially distinct representations could be used with this circuit. We are grateful for the support of the U.S. Air Force Office of Scientific Research through contract 90-0198. Mark Saffman acknowledg es support provided by a U.S. Air Force Office of Scientific Research laboratory graduate fellowship.

References 1. T. Kohonen, Self-Organi zation and Associative

Memory, 2nd ed. (Springer-Verlag, Berlin, 1989). 2. D. Z. Anderson, C. Benkert, B. Chorbajian, and A. Hermanns, Opt. Lett. 16, 250 (1991). 3. C. Benkert and D. Z. Anderson, Phys. Rev. A 44, 4633 (1991). 4. D. Z. Anderson and R. Saxena, J. Opt. Soc. Am. B 4, 164 (1987). 5. N. V. Kukhtarev, V. B. Markov, S. G. Odulov, M. S. Soskin, and V L. Vinetskii, Ferroelectrics 22, 949 (1979). 6. N. V. Kukhtarev, V. B. Markov, S. G. Odulov, M. S. Soskin, and V. L. Vinetskii, Ferroelectrics 22, 961 (1979). 7. Interestingly, when the pump consists of a single frequency the influence of the reflexive gain interaction on the dynamical evolution of the circuit leads to a flip-flop behavior between the rings, equivalent to that reported in Ref. 2. 8. J. 0. White, M. Cronin-Golomb, B. Fischer, and A. Yariv, Appl. Phys. Lett. 40, 450 (1982). 9. M. Saffman and D. Z. Anderson, Opt. Lett. 16, 300 (1991). 10. M. Koga and T. Matsumoto, IEEE Photon. Technol. Lett. 2, 487 (1990).

561

Reprinted with kind pennission from the SPIE

Resona tor memor ies and optical novelty filters Dana z. Anderson Marie C. Erie University of Colorado Department of Physics and Joint Institute for Laboratory Astrophysics University of ~olorado and National Bureau of Standards Boulder. Colorado 80309-0440

Abstract. Optical resonators having holographic elements are potential candidates for storing information that can be accessed through contentaddressable or associative recall. Closely related to the resonator memory is the optical novelty filter, which can detect the differences between a test object and a set of reference objects. We discuss implementati ons of these devices using continuous optical media such as photorefractiv e materials. The discussion is framed in the context of neural network models. There are both formal and qualitative similarities between the resonatoi memory and optical novelty filter and network models- Mode competition arises in the theory of the resonator memory, much as it does in some network models. We show that the role of the phenomena of "daydreaming" in the real-time programmabl e optical resonator is very much akin to the role of "unlearning" in neural network memories. The theory of programming the real-time memory for a single mode is given in detail. This leads to a discussion of the optical novelty filter. Experimental results for the resonator memory, the real-time programmabl e memory, and the optical tracking novelty filter are reviewed. We also point to several issues that need to be addressed in order to implement more formal models of neural networks. Subject terms: optical information processing; optical resonators; optical memories: holographic optical elements; neural networks; optical novelty filters. Optical Engineering 26(5). 434-444 (May 1987/,

CONTENTS

There is a large overlap between the requtstte features of network models and the natural domains or the optical regime. such as parallel and distributed processing_ That optical signals do not interact except in the presence of a material medium is also a distinction in favor of optical ncuromorphic processors. The majority of network models treat neurons as discrete processors with discrete interconnections between them. TI1cse trnxkls are ol'ten directly translatable into an optical architecture.'· " (iu1 interest here. is to discuss two optical devices whose description in neural network language would require a continuous distribution of (infinitesimal) neurons and a continuous distribution of interconnectio ns. We discuss the storage and recall of information in an optical resonator, a memory that can recall images from partinl images. z. 5 - 9 We also discuss an optical novelty filter. a device that detects the differences between a test image and a set of images that were stored previously. 2 ·' 0 As we show herein. these two optical devices are closely related. The resonator memory and novelty filter treated here musl be considered as prototypes, not merely because they are rather primitive by neural network model standards but also because their relationship to a11y existing neural model has yet to be properly established; in several ways, the relationship is a distant one, at best. Many of the features of these devices are nevertheless strikingly reminiscent of neural models. In the resonator memory, for example, it is appropriate to use the term ··competition" as it is used in some neural models. The competition is described in both cases by the prey/predator Volterra-Lotb equations, except that the equations describe the evolution of different entities in the two cases. Even without true parentage in existing neural models, the devices described here remain of interest in and of themselves. The resonator memory. for instance, exhibits a state wandering that we anthropomorp hicall\

I. Introduction 2. Resonator memory 2. I. Experiments 2.2. Daydreaming 3. Real-time programmable resonators 3. I. Theory or the real-time programmable ring 3.2. Demonstration or image storage in the real-lime ring 4. Comments on resonator memories 4.1. Moving optical resonators closer to neural network models 4.2. Phase-conjugating resonators 5_ Optical novelty filter 6. Conclusion 7. Acknowledgments 8. References

I. INTRODUCTION The recent upsurge of activity in associative optical processors is intimately allied with a blooming interest in neural network models that permeates a diverse assortment of disciplines. The attraction of neural networks lies in what it appears they have to offer for brain-like processing problems such as pattern recognition, language acguisition, motor control, knowledge processing, and so on. r. 2 Neural network models are a dramatic departure from the conventional digital computing '_paradigm. Based on a paradigm inspired by the architecture of the brain, neural network models have been abstracted to many forms. Because they are brain-like in architecture and processing capability, we attach the term "neuromorphi c processors" to implementations of these models. tnvited Paper IP-108 received Feb. 13, 1987; revised manuscript received March 16. 1987; accepted for publication March 16. 1987; received by Managing Editor March 16, 1987. '1'l 1987 Society of Photo-Optical Instrumentation Engineers.

434 I OPTICAL ENGINEERING I May 1987 I VoL 26 No. 5

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RESONATOR MEMORIES AND OPTICAL NOVELTY FILTERS

liken to daydreaming. Nothing akin to this has been described

Input Ollie cl

with a nctwork model. Our hope is thHI these prototypes will

Rncordlna M11tHurn

I,.

lead to more sophisticated continuous-media devices whose processing capahilitics more closely emulate the cognitive funclions

thal neural networks and neuromorphic systems suggest arc possihlc.

2. RESONATOR MEMORY

2 11 15 An exemplary associative memory • - holds in storage many information entities (called objects herein), any one of which may be recalled by addressing the memory with a partial object. The partial object may or may not look like one or more of the stored objects. The memory process should recall the object that looks most like the provided input. Thus, one should be able to recall a cat from a picture of the cat's ear. Furthermore, if insufficient information is provided, the memory might be likely to recall a Siamese cat instead of a tabby cat, but it should be less likely to recall a German shepherd instead of a tabby. In some configuration space sense, objects that look alike should be stored near one another. In other words, an associative memory. to distinguish it from a content-addressable memory. is able to categorize what it stores. The classification abilities of our resonator memory are not well understood at present. although the work ofCohen9 lends insight into this matter. Strictly speak-

Operator

Fig. 1. Ring resonator memory.

the writing beam an eigenmode of the resonator fom1ed hy the recording optics and the hologram. To see this. suppose this beam is initially present just to the right of the hologram in the figure. It then traverses the optics in the same manner as during the writing and is transformed to the same relurn slate. The diffraction grating of the hologram then reconstructs the supposed initial beam from the return beam. Since this optical field repeats itself after one round-lrip, it is an eigcnmode. !11 fae1.

ing. what we describe herein arc the contcnt-addrcssahlc features

depending on the way the hologra111 was recorded, the n:con-

of the resonator. The idea to use an optical resonator as a means of storing and recalling information comes from laser theory. A typical laser using an optical resonator is composed of two spherical mirrors. The resonator supports a set of Hermite-Gaussian eigenmodes-optical fields that, after one round-trip propagation. do not change shape. 16 The eigenmodes of a conventional res17 onator form a complete and orthogonal set. Any optical field within the resonator can be decomposed as a linear superposition of the eigenmodes. When a lasing medium is present within the resonator, light is generated. The output light is described by one eigenmode or a superposition of the eigenmodes of the empty resonator, provided the nonlinearity of the gain medium is not so severe as to distort the resonator modes. The actual form of the eigenmodes of the conventional resonator is due to the choice of spherical mirrors. The conventional resonator may be considered a memory for Hermite-Gaussians. If a different set of modes is desired, then mirrors (or other intraresonator elements) other than spherical must be used. The resonator memory stores .objects as transverse eigenmodes of an optical resonator. A generic resonator memory is 18 19 shown in Fig. I. A holographic recording materia1 · plays the role of a programmable mirror. To program the resonator, an input object is illuminated with monochromatic light. The subsequent information-bearing beam is transformed by an operator labeled T1 in the figure. T 1 might be a lens, another (nonprogrammable) hologram, a fiber bundle, or some other transformation operator. The beam then traverses the recording material and becomes transformed by another operator T 2 before it once again traverses the recording material (the role of T 2 is discussed further in Sec. 3). We let the gain medium be inactive during the recording process, although this is not a requirement. The holographic medium records the grating formed by the interference of the original beam with the return beam. The recording process is somewhat different for a real-time medium than it is for a static medium such as a photographic plate. The former is discussed in Sec. 4. The developed hologram now contains a grating that makes

struction may not be perfect, but for this discussion we assume that it is. Although the eigenmode Is actually a beam nr light with a transverse distribution that varies as it travels arnund the resonator, in this paper we refer to the whole of the heam as the original object itself. By multiply exposing the hologram, several objects can in principle be stored. If the set of stored objects is not an m1hogonal one, a system such as the one described here will lend to orthogonalize the set; therefore, what is "memorized" is not necessarily exactly what has been stored. A lot of work has gone into understanding how various network memory models handle nonorthogonal sets and how resultant errors may be corrected: for further details, the reader is directed to Refs. I and 2. The process of addressing the memory entails illuminating an object or partial object in the same manner as during the writing process. This excites the various modes of the resonator according to how much the input looks like each mode. For the present, we assume that the set of stored objects is an orthogonal set. The partial object If) excites a linear superposition of eigenmodes IOi): (I)

where ai = (Oilf). The approximation to 10 becomes exact if the set of modes IOi) is sufficient to span the space of the resonator. The memory must make a decision as to which stored mode most resembles the input. This is· done through a competiti\'e process in the gain medium. If the gain in the resonator is activated and is sufficiently great, oscillation will commence. spawning one or more of the eigenmodes. The modes compete with one another for gain; in general, the more one mode uses up the available gain, the less there is for another mode. If mode competition is strong, then the presence of one mode can suppress the oscillation of all other modes. When an injected signal is present, the competition is biased in favor of the mode tha1 most resembles the input. OPTICAL ENGINEERING I May 1987 I Vol. 26 No. 5 I

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ANDERSON. ERIE

The nature of the competition depends upon the chosen gain medium. 20 In our cxpcri1ncnts, we used two-beam coupling in barium titanate. 21 - 23 We chose this because it is easy lo implemenl and can provide very large gain cneffkicnts. Unfnrlunately. the physics of two-beam coupling in an optical resonator has not been fully analyzed. One can, however, do a pcrturbati\',c 4 treatment of the evolution of the fields inside the resonator. We present the gist of the theory here and refer the reader in want of detail to Ref. 24. Iii the absence of an injected signal, the equation of motion for the intensity of mode n can be put in the form

11ui11P

Fig. 2. Transient behavior of two-mode oscillation, showing ·competition. Each curve represents a given set of initial conditions on the mode intensities. The time development follows the direction of the arrows until a steady state is reached. The dashed lines represent possible steady-state behavior in the absence of competition.

(2)

where o:n is a linear net gain coefficient. Hnn is a self-saturation coefficient describing how much the presence of a mode reduces the net gain for itself, and 0nm is a cross-saturation coefficient indicating how much the presence of mode m reduces the net gain for mode n. Herein lies the competition. 00 ,,, is proportional to the mode intensity (not amplitude) overlap integral, taken over the gain volume. We designate the amplitude distribution of mode n as Un{r), and the overlap is written as

0,,,,, a

f

IU,,trll'IU,,,(r)l'd'r

l'l11J\11rdn11·11n·

Gain M1·di11111

(3)

~min V(llUlllal by an in1crferorne1er with coupled phase-conjugate reneclOrs." Op1. Lett 10, 282 (1985). 8

or

Optical thresholding and maximum operations Claire Gu, Scott Campbell, John Hong, Q. Byron He, Dapeng Zhang, and Pochi Yeh

Self-oscillations in nonlinear optical four-wave mixing and resonators are considered. Some unique properties of these oscillations can be employed for implementing parallel optical thresholding, comparing, and maximum operations. Both theoretical and experimental results that are obtained by utilizing the photorefractive nonlinearity are presented and discussed.

Introduction

and NLO resonators for parallel optical thresholding and max operations. 4 Both theoretical and experimental results obtained by utilizing photorefractive nonlinearity;-s are presented and discussed.

Thresholding and maximum (max) operations are essential elements in the implementation of neural networks. In a neural network, neurons perform thresholding on the received signals and therefore realize nonlinear dynamics. To implement neurons it is necessary to have devices with a nonlinear thresholding input-output response. In addition, the max operation is also desirable in the implementation of neural networks, such as a winner-take-all type of neural network. Max operation is defined as the operation for finding the maximum value among more than one input variables. Although there have been several optical implementations of neural networks, the thresholding functions have typically been performed electronically. 1-s Optical signals are converted to electronic signals, thresholded, then converted back to optical signals. For example, an optical system can be controlled by an electronic computer that does the thre$holding sequentially. Optical thresholding and max operations have the advantages of parallelism and cascadability without resorting to optoelectronic conversion. Because of the intrinsic high bandwidth, optics can perform thresholding on many input channels simultaneously. The output signals remain in the optics domain and can be used readily for the next stage of processing. Unfortunately, there has been limited work in this area. Here we propose and study the properties of self-oscillation in nonlinear optical CNLO) four-wave mixing (FWM),

Self-Oscillation in Photorefractive Four-Wave Mixing

We refer to Fig. 1 and consider a NLO medium pumped by two counterpropagating plane waves with amplitudes Ai and A 4 • In the configuration of phase conjugation by FWM, along with the two pump beams Ai and A 4 , a probe beam A 2 is incident on the nonlinear medium, and a phase conjugate beam As is generated. Self-oscillation occurs if the two counterpropagating beams A 2 and As are generated without an incident probe, i.e., A 2 (z = 0) = A 20 = 0 (see Refs. 9 and 10). Below we discuss the conditions that are necessary for self-oscillation to occur. For simplicity, we consider FWM with a transmission grating in a photorefractive medium. The coupled mode equations are written as 11 (1)

(2)

(3)

(4)

C. Gu, S. Campbe!I, and J. Hong are with Rockwe!I International Science Center, Thousand Oaks, California 91360. Q. B. He, D. Zhang, and P. Yeh are with the Department of Electrical and Computer Engineering, University of California, Santa Barbara, Santa Barbara, California 93106. Received 14 June 1991. 0003-6935/92/265661-05$05.00/0. © 1992 Optical Society of America.

where r is the complex coupling constant and I 0 = 11 + 12 +ls+ 14 is the total intensity; 11 = IA1 j2 , where j = 1, 2, 3, 4. General solutions to Eqs. (1)-(4) are available. 11 With the boundary condition for phase conjugation, i.e., As(z = L) = AsL = 0, we can calculate the reflectivity p Aso I A20 *, where Aso = As(z = 0). Self-oscillation occurs when p is infinity,

=

10 September 1992 I Vol. 31, No. 26 I APPLIED OPTICS

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We note that, iffo > 0, self-oscillation can occur only when a> 0, i.e., 110 > I4L. Given 110, hi, and f 0 L, Eq. (14) can be solved numerically for fh. Once fh is obtained, we can calculate the self-oscillation intensity l2L, which is given by (15)

z:L

z:O

Photo refractive Crystal

Fig. 1.

FWM by means of a transmission grating.

which leads to (5)

s + a tanh .J, = 0,

where qth (and ~ > 1) stable bidirectional oscillation builds up in the ring, but when q < qth the cavity is inactive. However, when ' > 3 a region near q th in which both trivial and nontrivial solutions are unstable always exists. When q is beyond this region stable oscillation can be maintained again. Within the unstable region, cavity outputs change between zero and nonzero oscillation intensities and are sensitive to any perturbations. Since the transitions from a stable state to instability occur by means of saddle bifurcations, no selfpulsation or other instabilities are expected. Experimental Results

An experimental study was conducted with the setup shown in Fig. 6. An extraordinary polarized input beam at 514.5 nm is split into two paths; one generates the pump beam A1 and the other generates the counterpropagating pump beam ~- The beam for ~ 12

Mirror '-"

10 Slab lo

"' a.

.5

"'0

Q)

ffi

6

Fig. 3. Bidirectional ring resonator.

'

4

2 0.5

Stable

'' '' '' '

... ..

(..)

~

''

0.9.

1.7 1.3 Beam Ratio q

2.1

2.5

Fig. 5. Results of the linear stability analysis.

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Input Output

Variable Attenuator

Fig. 6. Experim ental setup.

Fig. 8.

is sent throug h an acousto-optic _(AO) ~~dulator to impres s a 140-MHz frequency s~ift on it. m o~der to avoid the produc tion of reflection gratmg s m the BaTi0 3 crystal. Then this beam. pas~es. thro~gh a variabl e attenua tor in order to adJust its mtensi ty as needed for the study of thresho lding levels. Beams Ai and Ai are then aligned to couz:terpropagate th:oug h the photore fractive crystal, v.:t~. beam Ai orie~ted such that its fanning seeds the mitial resona nt osci~la­ tion. The resonan t cavity consist s of the BaTi03 crystal as its gain medium, three mirrors , a beam splitter for output cou~lin~, and a pi~hole for. spatially filtering the oscillatmg beam mto a sm_gle transve rse mode. In the absence of beam Ai, the ring resonat or oscillates unidirectionally in a stable transverse mode. Note that the frequency of the oscillating beam which may be different from that of the pump beam' is determ ined by the cavity length as well as by 13 the pump frequency . Bidi~e~tional os_cillati?n is initiate d with the gradual additio n of the mtensi ty of beam Ai. The respective oscillation powers of the two intraca vity beams A2 and A3 are measur ed by power meters placed at the t"."'o outpu~ po_rts of the c~vity's beam splitter . The cavity behavio r is then monito red as the intensi ty of beam Ai is increas ed or decreased. The oscillation outputs , with respect to various beam ratios are shown in Fig. 7. As the intensi ty of beam Ai in~reases to reach and initially surpass the cavi~y thresho ld value, oscillations become unstabl e. In this regime, the oscillations jump from_ one power leve~ to anothe r within the region shown m the plot. During this instability, peak oscillation powers, as well as

,., I

:l

.x·

12'/14 o.30

x;lC.x

Unstable 0 20 ·

,,,

LOscill ations

I

Stable Oscillations

~

-;r.~-;r.

time spent at these peak ~powers, decrease as Ai increases, while minimu m oscillation powers stay at zero. The stronge r Ai becomes the lower the peak oscillation powers become and the more time the cavity spends at its zero oscillation state. Eventually, as Ai increas es still further , oscillations cease altogether and the cavity becomes stable again. Parallel Thresho lding and Max Operation

When an array of beams interac ts at different locations inside a crystal, parallel thresho lding can be achieved. In additio n, by adjusti ng the intensi ty of a referen ce beam max operati on can be implemented. While referrin g to Fig. 8 we consider an array of input beams with differen t intensi ties (which are represented by differe nt line types) and a respective array of referen ce beams with equal intensities. As these respective pairs of input and reference ~earns inte:ac t within the crystal, optical thresho ldmg, described above, can be perform ed in parallel. In addition, ~y adjusti ng the intensi ty of the reference. beam~, it is possible to identify the beam_ with maximui:i mtensity. This is done by increas mg (or decreasmg) the intensi ty of the referen ce beams. When the reference intensi ty is decreas ed osc'illation occurs when the intensi ty of the brighte st input beam _reac~es the regime that allows self-oscillation. At this p~mt, t~e brighte st beam is selected and located. With this technique, the compar ison is done in pa:allel ar:d the maxim um can be found withou t measur mg the mtensities of all the light beams electronically. This approach is extrem ely useful when the numbe r of input beams becomes large. There are some problems with the current system. The most significant problem is the instabil ity near the thresho ld. This makes the distinction betwee n two beams with close intensi ty levels difficult. Further study is needed to overcome this instability. Right now, when doing a max operation, we can easily reject many low intensi ty beams a;nd t?e~eby_ reduc_e the numbe r of remain ing beams with similar mtens1ties.

1·····~~~~~-+-~-1

~~

a.oo 0.00

Conclus ion

We have propos ed and studied, both theoretical~y a~d experimentally, the propert ies of self-os~illat10n m FWM and oscillations in a bidirectional rmg resona-

2.so

11114

Fig. 7. Experim ental results. 5664

NLO Medium Parallel threshold ing and maximum operation.

APPLIED OPTICS I Vol. 31, No. 26 I 10 September 1992

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tor pumped by two counterprop agating beams. These properties can be used to implement optical thresholding and max operation. Similar properties also exist in other NLO media besides photorefrac tive crystals. The authors thank Tallis Chang for technical discussions and a critical review of this paper. This study was supported, in part, by grants from the U.S. Office of Naval Research and the U.S. Air Force Office of Scientific Research. P. Yeh is a Principal Technical Advisor at Rockwell Internation al Science Center. References and Notes 1. J. Hong, S. Campbell, and P. Yeh, "Optical pattern classifier with perceptron learning," Appl. Opt. 29, 3019-3025 (1990). 2. D. Psaltis, D. Brady, and K Wagner, "Adaptive optical networks using photorefractiv e crystals," Appl. Opt. 27, 17521759 (1988). 3. E.G. Paek, J. Wullert, andJ. S. Patel, "Holographic implementation of a learning machine based on a multicategory perceptron algorithm," Opt. Lett. 14, 1303-1305 (1989). 4. C. Gu and P. Yeh, "Optical thresholding and max operation," in Optical Computing, Vol. 6 of OSA 1991 Technical Digest Series (Optical Society of America, Washington, D.C., 1991), pp. 68-71. 5. P. Tayebati and L. H. Domash, "New thresholding device using a double phase conjugate mirror with phase conjugate feedback," in Photorefractiue· Materials, Effects, and Devices, Vol. 14 ofOSA 1991 Technic8.l Digest Series (Optical Society of America, Washington, D.C., 1991), pp. 190-194.

5. M. B. Klein, G. J. Dunning, G. C. Valley, R. C. Lind, and T. R. O'Meara, "Imaging threshold detector using a phase-conjugate resonator in BaTi0 3 ," Opt. Lett. 11, 575-577 (1986). 7. J. Feinberg, "Asymmetric self-defocusing of an optical beam from the photorefractiv e effect," J. Opt. Soc. Am. 72, 46-51 (1982). 8. S.-K. Kwong, M. Cronin-Golomb, and A. Yariv, "Optical bistability and hysteresis with a photorefractive self-pumped phase coajugate mirror," Appl. Phys. Lett. 45, 1016-1018 (1984). 9. A. Yariv and D. M. Pepper, "Amplified reflection, phase coajugation, and oscillation in degenerate four-wave mixing," Opt. Lett. 1, 16-18 (1977). 10. J. F. Lam, "Spectral response of nearly degenerate four-wave mixing in photorefractiv e materials," Appl Phys. Lett. 42, 155-157 (1983). ll. See, for example, M. Cronin-Golomb, B. Fischer, J. 0. White, and A. Yariv, "Theory and applications of four-wave mixing in photorefractiv e media," IEEE J. Quantum Electron. QE-20, 12-30 (1984). 12. C. Gu and P. Yeh, "Reciprocity in photorefractive wave mixing," Opt. Lett. 16, 455-457 (1991). 13. See, for example, P. Yeh, "Two-wave mixing in nonlinear media," IEEE J. Quantum Electron. 25, 484-519 (1989). 14. J. P. Huignard and A. Marrakchi, "Coherent signal beam amplification in two-wave mixing experiments with photorefractive BSO crystals," Opt. Commun. 38, 249-254 (1981). 15. C. Gu and P. Yeh, "Theory ofphotorefract ive phase conjugate ring oscillations," J. Opt. Soc. Am. B. 8, 1428-1432 (1991). 16. See, for example, L.A. Lugiato, "Theory of optical bistability," Prog. Opt. 21, 71-101 (1984).

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OPTICS LETIERS I Vol. 18, No. 2 I January 15, 1993

Matrix-ma trix multiplica tion by using grating degeneracy in photorefra ctive media Claire Gu Department of Electrical and Computer Engineering, The Pennsylvania State Unfrersi(v, University Park, Pennsylvania 16802

Scott Campbell and Pochi Yeh Department of Electrical and Computer Engineering. University of California, Santa Barbara, Santa Barbara. California 93106 Received July 27, 1992 We propose and demonstrate a novel method that utilizes grating degeneracy in photorefractive media in conjunction with an incoherent laser array to implement parallel optical matrix-matrix multiplication. Such a matrix- matrix multiplier is capable of handling large matrices such as those with 1000 x 1000 elements. The principle of operation and preliminary experimental results are presented.

Matrix-matrix multiplication is an important operation in many computational and processing applications including correlation, convolution, Fourier transform of temporal signals, and two-dimensional images. In addition, a large number of signal- and image-processing algorithms can be expressed in terms of matrix operations. Direct matrix-matrix multiplication is often avoided in electronic computers because it is an 0(N3) (where N x N is the number of elements in each matrix) operation that requires a long computation time for serial machines. However, attempts to avoid matrix-matrix multiplications usually lead to other complicated algorithms. Furthermore there are situations in which direct matrix-matrix multiplications are inevitable. Optical computing offers the advantage of parallelism and large capacity. Such capabilities have been successfully demonstrated in parallel vectormatrix multiplication. 1 Although matrix-matrix multiplications can be performed as an extension of vector-matrix multiplication 1 with color multiplexing or time multiplexing, these schemes are complicated by dispersion or time delay. Recently, nonlinear optical techniques have been employed in parallel matrix-matrix multiplication. 2•3 These techniques require complicated alignment and suffer from severe energy loss. In this Letter, we propose and demonstrate a new method that utilizes grating degeneracy' in photorefractive media in conjunction with an incoherent laser array to implement parallel optical matrix-matrix multiplication. Specifically, multiplications are implemented by photoinduced index gratings whose amplitudes are determined by the interference between coherent beams, while summations are implemented by grating degeneracy. Such a matrix-matrix multiplier is capable of handling large matrices such as those with 1000 x 1000 elements. Figure 1 shows the schematic diagram that describes the principle of operation of the matrix-matrix multiplication. Both matrix A (N x N) 0146-9592193/020146- 03$5 .00/0

and matrix B (N x N) are placed at the front focal plane of lens L1 . At the rear focal plane of lens L 1 , a volume holographic medium such as a photorefractive crystal is inserted to record the multiplication of the two matrices. The recorded information is read out by a set of reading beams, which consists of N diagonally aligned point sources placed at the front focal plane of lens L2 • Notice that the recording beams are propagating from left to right while the reading beams are propagating from right to left. The holographic medium is also located at the rear focal plane of lens L2 • The diffracted readout beams are directed by a beam splitter to the output plane that is located at the focal plane of lens L 1 • The matrix elements are arranged in the following order. In the region of matrix A, each line along the y direction represents a row of matrix A. For example, the line shown in the region of matrix A in Fig. 2la) represents matrix elements A11, A1 2 , A13, and A14 • In the region of matrix B, each line along they direction represents a column of matrix B. For example, the line shown in the region of matrix B in Fig. 2(a) represents matrix elements B1,, 8 24 , B:J.1, and 8 44 • To realize matrix-matrix multiplication, the illumination of the matrices A and B is chosen so that all pixels within each line along the x direction

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Fig. l. Schematic diagram for matrix-matrix multiplication. © 1993 Optical Society of America

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Fig. 2. Arrangement for elements during recording. (al At the input plane, pixels with the same shading are coherent, and those with different shadings are mutually incoherent. (bl Degenerate gratings shown in the momentum space (normal surfacel. are mutually coherent while pixels with different y values at the input plane are mutually incoherent. Notice that if the figure is rotated 90', as is shown in the experimental results, we see a transposed matrix A and a matrix B. During the recording, since the input plane is coherently illuminated along each line in the x direction, gratings are formed according to AvB1; •. For example, point Au writes gratings with points Bu. B 12 , B 13 , and B 14 as shown in Fig. 2(a). There are N2 gratings formed within each line along the x direction, and the total number of gratings is N3. Within these N3 gratings, there are only N2 different grating wave vectors, i.e., there are N degenerate gratings for each grating wave vector. For example, within the two degeneracy lines shown in Fig. 2(a), the four gratings formed between elements of matrix A and elements of matrix B have the same grating wave vector and are therefore degenerate. The corresponding degeneracy condition is shown in Fig. 2(b) in the normal surface representation. The index grating representing these four terms is

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diagonally aligned at the reading plane, as shown in Figs. 1 and 3. Each of the reading points reads out N nondegenerate gratings, giving N diffracted points in a line· along the x direction at the output plane . The method described above has been implemented experimentally. Figure 4 shows the experimental setup. Note that the x direction is parallel to the table surface. Th achieve maximum diffraction efficiency and the best angular selectivity, a twooptical-axis system is employed. In our experiment, incoherent laser sources are obtained by delaying a multimode Ar-laser .2L(go-g ,)(1 + U2r2)2

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which has three solutions. The trivial one is Q = 0, which corresponds to an unsplit oscillation. The other roots are given by

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]112 · er [2,,.An, (28) 1 L(go - gi) >. Taking r = 100 msec, L = 30 cm, go - gt = 0.01, l = 1 Jillll, An, = l0-5, A = 0.6328 µ.m, Eq. (28) yields Q0 = 1Q3 sec-1. which corresponds to a frequency split of 160 Hz. Whether the ring gyro will oscillate at the same frequency (Q = O) or with a split !:lo, or both, is a subject of mode stability. This issue is taken up in Sec. IV. In the general case when the external field is present (i.e, the so-called drift case), the phase ¢ 0 is given by7,13 _ _1!Ed(Ed + Ep) + E~ , (29) 1 Uo=:I:- - - l r

, 1 - q,,.,. 1 and If>,, - ¢,.,.2; note that the sign of the reciprocal contribution is the same as before, whereas the nonreciprocal contribution has opposite sign. In the round trip, the reciprocal contributions cancel, and net phase shifts are given by -2¢,.,. 1 and -2¢,.,.2• The phase difference measured by the interference at detectorD, (3) If> = -2(¢,.,., - !/>,.,.) = 41r(R 1L 1 + R.j., 2)fl/A.c,

is proportional to the rotation rate fl and can be used to sense rotation. This configuration has several advantages over our previously reported configuration.5 Here, we can use self-pumped phase conjugation, with the obvious advantage of not having to provide external pump waves that are coherent and form a phase-conjugate pair. In the externally pumped configuration, the pump beam(s) involved in writing the index grating must be coherent with the probe wave to within the response time of the phase conjugator, and the two counterpropagating pump beams must be phase conjugates of each other to produce high-fidelity phase-conjugate reflection. In initial experiments in which an entire externally pumped phase-conjugate gyro was mounted on a rotating table, because of the slow time response of phase conjugation in the barium titanate crystal used, vibrations of the mounts providing the external pumping washed out the gratings involved in the phase conjugation and precluded the measurement of rotation. As an additional advantage of the self-pumped configuration, the sensing fibers Fl and F2 can be made longer (thereby increasing the sensitivity) than the coherence length of the laser, provided that they are equal in length to within the coherence length. Figure 2 shows the experimental setup of the self. pumped phase-conjugate fiber-optic gyro. Instead of using two separate fibers as shown in Fig. 1, we use the two polarization modes of a single polarization-preserving fiber coil. All experiments are done with the argon laser running multilongitudinal mode at 515 nm. The highly reflective beam splitter BSl isolates the laser from retroreflections. The polarization-preserving fiber Fl couples light from the laser to the remaining part of the apparatus that is mounted on a

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Fig. 2. Experimental setup of the self-pumped phase-conjugate fiber-optic gyro. Instead of the two fibers shown in Fig. l, the experimental setup shown here uses the two polarization modes of the polarization-preserving fiber-optic coil. Light from the laser is incident upon polarizing beam splitter PBSl with its polarization at45° to the plane of the page. The components reflected and transmitted by PBSl travel clockwise and counterclockwise, respectively, in the fiber coil. The two beams recombine at PBSl and are then split at PBS2. One of the beams has its polarization rotated by PR, and both beams are incident upon a barium titanate crystal such that self-pumped phase conjugation occurs. The reflected waves retraverse the fiber in an opposite sense, recombine at PBSl, and travel back toward the laser with a phase difference t/>, which is proportional to the rotation ~te. These waves are sampled by the beam splitter BS2, and an additional phase delay of "tr/2 rad is impressed on them when they propagate through the quarter-wave retarder A/4. The half-wave retarder is oriented such that the intensities of the interferences measured by detectors Dl and 02 are proportional to sin tf> and -sin t/>, respectively.

rotating table. The output end of Fl is oriented such that the polarization of light emerging from the fiber is at 45° to the plane of the figure. The component polarized in the plane of the page is transmitted by the polarizing beam splitter PBSl and travels counterclockwise in the fiber coil, whereas the component polarized perpendicular to the page travels clockwise in the fiber coil. The fiber coil is made of approximately 9 m of polarization-preserving fiber coiled in a square of 0.57-m sides and is oriented such that the polarizations of the clockwise and counterclockwise waves are preserved. When the two waves leave the coil they are separated by a Rochon polarizer PBS2. The polarization of the light that travels straight through PBS2 is rotated by the polarization rotator PR such that its polarization becomes identical to that of the light deflected by PBS2. Both beams are incident as extraordinary waves on a barium titanate crys1 tal such that self-pumped phase conjugation occurs. 6 The reflected waves retraverse the fiber in an opposite sense, recombine at PBSl, and travel back toward the laser with a phase difference If> = 8-irRLfl/"Ac. These waves are sampled by the uncoated pellicle beam splitter BS2, and an additional phase delay of 1'/2 rad is impressed on them when they propagate through the

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of using two fibers and the associated complexity of terminating them on the same self-pumped polariza_ tion-preserving phase-conjugate mirror (four beams going into one crystal), it is necessary to ensure that the light waves from the two fibers are coherent to within the response time of the phase conjugator (the change in phase shifts for the two waves due to envi. ronmental effects on the fibers must be slower than the response time of the phase conjugator). The second of the above-mentioned effects can be reduced by wrapping the two fibers together so that they see near. ly the same environment. In conclusion, we have described a new type of phase-conjugate fiber-optic gyro in which self. pumped phase conjugation can be employed to permit the use of sensing fibers that are longer than the coherence length of the laser source. In other, externally pumped, configurations, it is possible to use fibers longer than the coherence length of the laser by using a fiber ~ carry the pumping waves. This, however, complicates the setup and defeats some of the advan. tages of using phase conjugation. We have construct. ed a self-pumped phase-conjugate fiber-optic gyro and demonstrated rotation sensing.

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