Photorefractive Materials and Their Applications 3

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Springer Series in

OPTICAL SCIENCES

115

founded by H.K.V. Lotsch

Editor-in-Chief: W.T. Rhodes, Atlanta Editorial Board: A. Adibi, Atlanta T. Asakura, Sapporo T.W. H¨ansch, Garching T. Kamiya, Tokyo F. Krausz, Garching B. Monemar, Link¨oping H. Venghaus, Berlin H. Weber, Berlin H. Weinfurter, M¨unchen

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Springer Series in

OPTICAL SCIENCES The Springer Series in Optical Sciences, under the leadership of Editor-in-Chief William T. Rhodes, Georgia Institute of Technology, USA, provides an expanding selection of research monographs in all major areas of optics: laser and quantum optics, ultrafast phenomena, optical spectroscopy techniques, optoelectronics, quantum information, information optics, applied laser technology, industrial applications, and other topics of contemporary Interest. With this broad coverage of topics, the series is of use to all research scientists and engineers who need up-to-date reference books. The editors encourage prospective authors to correspond with them in advance of submitting a manuscript. Submission of manuscripts should be made to the Editor-in-Chief or one of the Editors. See also http://springeronline.com/series/624

Editor-in-Chief William T. Rhodes Georgia Institute of Technology School of Electrical and Computer Engineering Atlanta, GA 30332-0250, USA E-mail: [email protected]

Editoral Board Ali Adibi

Bo Monemar

School of Electrical and Computer Engineering Van Leer Electrical Engineering Building Georgia Institute of Technology 777 Atlantic Drive NW Atlanta, GA 30332-0250 [email protected]

Department of Physics and Measurement Technology Materials Science Division Link¨oping University 58183 Link¨oping, Sweden E-mail: [email protected]

Toshimitsu Asakura

Herbert Venghaus

Hokkai-Gakuen University Faculty of Engineering 1-1, Minami-26, Nishi 11, Chuo-ku Sapporo, Hokkaido 064-0926, Japan E-mail: [email protected]

Heinrich-Hertz-Institut f¨ur Nachrichtentechnik Berlin GmbH Einsteinufer 37 10587 Berlin, Germany E-mail: [email protected]

Theodor W. H¨ansch

Horst Weber

Max-planck-Institut f¨ur Quantenoptik Hans-Kopfemann-Strasse 1 85748 Garching, Germany Email: [email protected]

Technische Universit¨at Berlin Optisches Institut Strasse des 17. Juni 135 10623 Berlin Germany E-mail: [email protected]

Takeshi Kamiya

Harald Weinfurter

Ministry of Education, Culture, Sports Science and Technology National Institute for Academic Degrees 3-29-1 Otsuka, Bunkyo-ku Tokyo 112-0012, Japan E-mail: [email protected]

Ludwig-Maximilians-Universit¨at M¨unchen Sektion Physik Schellingstrasse 4/III 80799 M¨unchen, Germany E-mail: [email protected]

Ferenc Krausz Max-planck-Institut f¨ur Quantenoptik Hans-Kopfemann-Strasse 1 85748 Garching, Germany E-mail: [email protected]

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Peter G¨unter

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Jean-Pierre Huignard

(Editors)

Photorefractive Materials and Their Applications 3 Applications With 316 Illustrations

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Peter G¨unter Institute of Quantum Electronics Nonlinear Optics Laboratory Swiss Federal Institute of Technology ETH Hoenggerberg HPF E 8 CH-8093 Zurich Switzerland Email: [email protected]

Jean-Pierre Huignard Thales Research and Technology France RD 128 91767 Palaiseau Cedex France Email: [email protected]

With 316 figures. Library of Congress Control Number: 2006925859 ISBN-10: 0-387-34443-8 ISBN-13: 978-0387-34443-0

e-ISBN 0-387-34728-3

 C

Printed on acid-free paper.

2007 Springer Science+Business Media, LLC All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer Science+Business Media, LLC, 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks, and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. 9 8 7 6 5 4 3 2 1 springer.com

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Preface

This is the third and final volume of a three-volume series devoted to photorefractive effects, photorefractive materials, and their applications. Since the publication of our first two Springer books on “Photorefractive Materials and Their Applications” (Topics in Applied Physics, vols. 61 and 62) almost 20 years ago a great deal of research has been done in this area. New and often unexpected effects have been discovered and theoretical models developed, known effects could finally be explained, and novel applications have been proposed. We believe that the field has now reached a high level of maturity, even if research continues in all areas mentioned above and with new discoveries arriving quite regularly. We therefore have decided to invite some of the top experts in the field to put together the state of the art in their respective fields, having been encouraged to do so for more than ten years by the publisher, due to the fact that the former volumes have long been out of print. The first volume is devoted to a description of the basic effects leading to photoinduced refractive index changes in electrooptical materials. In the second volume the status of the most recent developments in the field of photorefractive materials is reviewed and the parameters governing photorefractive nonlinearity are highlighted. This third volume deals with the applications of the photorefractive effects and of materials. Starting about 35 years ago, the attractiveness of the photorefractive effect for data storage, optical metrology, optical signal processing, image correlation, and several others nonlinear optical applications has been recognized. One of the main reasons for this is the large nonlinearity or refractive index change, which can be induced by low light intensities using the photoinduced space-charge fields in electrooptical materials. Many new concepts have been demonstrated in laboratories over all these years. Several of these concepts have also proved useful in other areas of nonlinear optics. Particularly interesting was the observation of a large energy transfer from pump beams to the signal beam in two- and fourwave mixing experiments. These effects lead to coherent amplification of a wave form covering spatial information and to self-pumped optical phase conjugation with applications in the area of wave-front correction of self-induced optical resonators. Also, it is now possible with photorefractive nonlinearities to control optically the group velocity of a modulated signal beam interfering with a continuous beam in the crystal. In these interactions based on dynamic holography, a large group index can be achieved for the demonstration of superluminal and slowdown light propagation of a signal beam carrying temporal information.

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Preface

In this third volume a series of applications of photorefractive nonlinear optics and of optical data storage and processing are presented in several chapters. This and the other two volumes on photorefractive effects, materials, and applications have been prepared mainly for researchers in the field, but also for physics, engineering, and materials science students. Several chapters contain sufficient introductory material for those not so familiar with the topic to obtain a thorough understanding of the photorefractive effect. We hope that for researchers active in the field these books will provide a useful reference source for their work. We would like to thank all the authors who contributed to this work for their excellent contributions and great efforts in presenting attractive overviews of the topics they have developed in this book. We are very much indebted to Mrs. Lotti N¨otzli for her great administrative support. Zürich, Orsay, October 2005

Peter Günter, Zurich Jean-Pierre Huignard, Orsay October 2005

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Contents

1 Introduction ..................................................................... Peter G¨unter and Jean-Pierre Huignard

1

2 Digital Holographic Memories ............................................... Lambertus Hesselink and Sergei S. Orlov 2.1 Introduction ................................................................. 2.2 Optical Architectures ...................................................... 2.3 Holographic Optics ........................................................ 2.4 Multiplexing Techniques .................................................. 2.5 Holographic Recording Materials ........................................ 2.6 Photorefractive Materials .................................................. 2.7 Fixing of Holograms in Photorefractive Media ......................... 2.8 Systems Issues .............................................................. 2.9 Demonstration Platforms ..................................................

7 7 8 11 17 21 22 25 30 32

3 The Transfer Function of Volume Holographic Optical Systems ..... George Barbastathis 3.1 Introduction ................................................................. 3.2 3D Spatial Heterodyning with Volume Holograms .................... 3.3 Derivation of the Optical Response of Volume Holograms ........... 3.4 Examples .................................................................... 3.5 Conclusions and Discussion ..............................................

51

4 Photorefractive Memories for Optical Processing ........................ M. Duelli, G. Montemezzani, M. Zgonik, and P. G¨unter 4.1 Volumetric Optical Data Storage ......................................... 4.2 Optical Pattern Recognition ............................................... 4.3 Holographic Associative Memories ...................................... 4.4 Photorefractive Materials as Volume Storage Media .................. 4.5 Optical Correlators Using Photorefractive Crystals ................... 4.6 All-Optical Nonlinear Associative Memories .......................... 4.7 Summary ....................................................................

77

51 52 56 58 74

79 94 104 109 120 122 128

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Contents

5 True-Time-Delay Adaptive Array Processing Using Photorefractive Crystals ....................................................... G.R. Kriehn and K. Wagner 5.1 Introduction to True-Time-Delay Array Processing ................... 5.2 Output Time Delay and the BEAMTAP Algorithm .................... 5.3 The Traveling-Fringes Detector .......................................... 5.4 Photorefractive Volume Holography Using Polarization-Angle, Read–Write Multiplexing ......................... 5.5 Experimental Implementation of BEAMTAP .......................... 5.6 Conclusion .................................................................. 6

7

8

Photorefractive Mesogenic Composites for Applications to Image Processing ............................................................ Hiroshi Ono, Akira Emoto, and Nobuhiro Kawatsuki 6.1 Overview of Mesogenic Composites for Photorefractive Applications ................................................................. 6.2 Fundamental Properties of Photorefractive Mesogenic Composites for Image Processing ........................................ 6.3 Applications to Image Processing ........................................ Solid-State Lasers with a Photorefractive Phase-Conjugate Mirror ...................................................... T. Omatsu, M.J. Damzen, A. Minassian and K. Kuroda 7.1 Introduction ................................................................. 7.2 Optical Phase Conjugation for Solid-State Lasers ..................... 7.3 Phase-Conjugate Diode-Pumped Nd:YVO4 Laser Amplifiers ....... 7.4 Phase-Conjugate Laser Resonator ........................................ 7.5 Phase-Conjugate Laser Diode ............................................ 7.6 Summary .................................................................... Dynamic Holographic Interferometry: Devices and Applications .... Philippe Lemaire and Marc Georges 8.1 Introduction: Historical Background .................................... 8.2 Requirements for Applicability of HI .................................... 8.3 Potentialities of Photorefractive Crystals for Holographic Interferometry ........................................... 8.4 Holographic Camera with Continuous Laser Illumination for Scattering Objects ...................................... 8.5 Holographic Cameras with Continuous Laser Illumination for Transparent Objects .................................... 8.6 Holographic Camera with Pulsed Lasers ................................ 8.7 Conclusion ..................................................................

135 135 138 141 143 158 166

169

169 172 182

193 193 194 197 209 212 219 223 223 226 229 233 242 244 249

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Self-Organizing Laser Cavities ............................................... Gilles Pauliat, Nicolas Dubreuil, and G´erald Roosen 9.1 Introduction ................................................................. 9.2 Self-Organizing Cavities with Adaptive Fabry-Perot Filters .......... 9.3 Design of Self-Organizing Cavities with 4-Level Laser Amplifiers ............................................................ 9.4 Self-Organizing Cavities with Semiconductor Amplifiers ............ 9.5 Self-Organizing Cavities Operating on the Transverse Structure .... 9.6 Conclusion ..................................................................

253

10 Slow and Fast Lights in Photorefractive Materials ....................... Guoquan Zhang, Fang Bo, and Jingjun Xu 10.1 Dispersive Photorefractive Phase Coupling ........................... 10.2 Phase-Coupling-Induced Slow and Fast Lights ....................... 10.3 Experiments on Slow and Fast Lights in Photorefractive Materials ............................................. 10.4 Slow and Fast Lights with a Stationary Refractive Index Grating ................................................. 10.5 Prospects and Conclusions ..............................................

277

11 Holographic Filters ............................................................. Karsten Buse, Frank Havermeyer, Wenhai Liu, Christophe Moser and Demetri Psaltis 11.1 Introduction ................................................................ 11.2 Telecommunication Networks ........................................... 11.3 Reflection Gratings ....................................................... 11.4 Material Issues ............................................................ 11.5 Holographic Recording .................................................. 11.6 Devices ..................................................................... 11.7 Summary and Outlook ...................................................

295

12 Neutron Physics with Photorefractive Materials ......................... Martin Fally, Christian Pruner, Romano A. Rupp, and Gerhard Krexner 12.1 Basic Concepts ............................................................ 12.2 Materials ................................................................... 12.3 Experiments ............................................................... 12.4 Electro Neutron-Optics ................................................... 12.5 Neutron Holography ...................................................... 12.6 Outlook and Summary ...................................................

321

Index ....................................................................................

355

9

253 254 263 268 273 273

278 280 284 290 292

295 297 301 303 306 309 317

322 326 329 341 345 348

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1 Introduction Peter G¨unter and Jean-Pierre Huignard

The objective of this third volume of the series on photorefractive effects and materials is to highlight the most recent applications of photorefractive nonlinear optics. As already outlined in volumes 1 and 2, which develop the basic physical mechanisms and the material properties, photorefractive effects exhibit a large diversity of very specific properties. At the origin in the 70’s, photorefractive crystals were proposed for high-capacity volume holographic storage using hologram multiplexing. At that time, this ambitious application already stimulated research on materials and system architectures in order to optimize their storage capacity. Nevertheless, if the concepts were demonstrated in the laboratory, no storage system based on holography was developed due the progress of other technologies such as magnetics and read-only bit-by-bit recording with an optical disk. Then it was realized that the recording of dynamic holograms in photorefractives through two-wave or four-wave mixing makes it possible to transfer energy from a pump beam to a probe, thus leading to coherent amplification of a wave front carrying spatial information. These effects, obtained with low incident optical powers, opened a wide range of new applications extending from image amplification to self-organized optical cavities. Photorefractive holography also permits one to demonstrate novel beam interactions leading to control of the group velocity of a light pulse. Beside this, the dynamic aspects of the holographic recording and readout in photorefractive materials enables the real-time implementation of optical processing functions operating in parallel over a large number of pixels. Also, these dynamic photorefractive volume gratings exhibit spectral and spatial band-pass responses leading to unique properties for signal or image filtering as well as pattern recognition by optical correlation methods based on Fourier transform processors. Moreover, there is now a renewed interest in the field of holographic storage applications due to advances in key technologies such as spatial light modulators and detector arrays. Highresolution holograms can now be recorded in some polymer or photorefractive materials after a fixing process for high-density storage of binary two-dimensional data with multigigabit output rates. This exhaustive list of specific characteristics involving the materials’ properties as well as the interactions between the interfering beams is rich in new proposals for applications in photonics. The different chapters included in this third volume address all the domains introduced above. They are written by leading experts in the field, who provide an extended

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Peter G¨unter and Jean-Pierre Huignard

analysis of their recent research achievements in applications of photorefractive nonlinearities. Chapter 2 of this volume gives an overview of digital holographic memories, which have the potential for achieving capacities of one terabyte per disk at a rate of 1 gigabit per second. There is renewed interest in this field, which has benefited from recent progress in key component technologies such as spatial light modulators, compact lasers, and high-speed detector arrays. The storage medium is still a critical issue, but photorefractive crystals like LiNbO3 and photopolymers have already demonstrated their capabilities for hologram multiplexing, leading to an increase in storage density. Optical memory architectures are presented, but it is shown that system optimization is a complex problem with a large number of tradeoffs between several parameters such as the overall system capacity, material dynamic range, hologram efficiency, and signal-to-noise ratio. Consequently, continuing research into improving the characteristics of materials used for data recording in a holographic memory is clearly a key issue for the success of this storage technology. Volume holograms are becoming important optical elements, since they exhibit strong Bragg selectivity in the spatial and spectral domains. The basic properties of phase volume diffraction are used in Chapter 3 for developing a class of applications in which volume holograms are used in optical imaging systems as three-dimensional spatial heterodyning elements for accurate wave-front detection. In these innovative operating principles Bragg selectivity permits one to achieve extremely high positional or color sensitivity, which in turn provide new imaging functions such as high-resolution profilometry, 3D imaging at long distances, and hyper/spectral imaging. Also, the transfer function of an optical system including volume holographic components is analyzed. In particular, the depth selectivity of volume holographic components in imaging systems is outlined, and it is used in several experiments demonstrating new principles of optical slicing of 3D objects. These experiments, which rely on the use of phase volume gratings such as obtained with a fixed grating in a photorefractive crystal or in bulk photopolymer materials, may stimulate new ideas that may extend the limits of classical imaging systems. Parallel information processing is a field of great interest, since dynamic holography can process and manipulate laser beams carrying high-bandwidth spatial and temporal information. In this perspective, Chapter 4, which is reproduced from an earlier Springer Optical Sciences book, reviews the basics of data storage with volume holography and pattern recognition by optical Fourier processing of coherent images. The main system architectures are described and analyzed, and their limitations are discussed. It thus includes hologram multiplexing techniques, limitations on storage capacity due to the optical systems, recording materials and crosstalk between digital pages. Besides data storage, real-time parallel pattern recognition can be performed by a photorefractive correlator involving lenses for Fourier transformation of the complex amplitude of the wave front. The specific properties of analog multichannel optical correlation architectures are reviewed. Then the extension of these techniques to advanced concepts of all

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1. Introduction

3

optical nonlinear associative memories are introduced in this chapter, and possible implementations using volume holographic elements are presented. Chapter 5 highlights that radio-frequency optical signal processing is a domain in which photorefractive-based system architectures can achieve massive parallelism for adaptive true-time delay processing of wide-bandwidth radar signals. This is due to photorefractive material capabilities to record complex gratings and diffract from these gratings with optically modulated beams that carry a wide radio-frequency bandwidth. In this approach, the adaptive weight coefficients of the signals issuing from each antenna module are represented by the amplitude and the phase of the holographic grating, which can be multiplexed in the same crystal volume. These properties permit one to perform antenna beam formation as well as other important analog signal processing operations such as time-integrating correlation or jammer excision. It is thus expected that photorefractive techniques will play a role in microwave photonics, where very wide bandwidth RF signals will be emitted by large phased-array antennas. Optoelectronics allows one to transmit and to process in parallel all the signals issuing from antenna modules operating in emission or receiving mode. These novel concepts have undoubtedly the capability to contribute to new advances in radar using hybrid optical-electronic signal-processing technologies. Another form of optoelectronic parallel information processing is introduced in Chapter 6. The novel approach in this chapter is the use of a reorientational photoinduced effect in dye-doped liquid-crystal molecules. An important advantage of this class of orientational photorefractive effects is that the operating voltage in low and it benefits from the large index modulation achieved with liquid crystal. In early experiments, dye-doped low-molar-mass liquid crystal shows good nonlinear performance but only for large grating periods. Recent improvements using material combining low-molar-mass liquid crystal with polymers have been obtained for applications that require a higher spatial resolution. The control of laser beam quality is now important in many industrial applications involving high-power diode-pumped laser sources. Thermal gradients in the gain media lead to strong aberrations, which severely degrade the beam quality. As reviewed in Chapter 7, nonlinear photorefractive optics can provide elegant solutions to compensate in real time for thermal aberrations of the beam either inside a cavity or in a master-oscillator power-amplifier architecture. A photorefractive nonlinear mirror can efficiently generate a conjugate beam with CW or pulsed lasers, in particular when using a self-pumped geometry. This chapter describes several important aspects of phase conjugation applied to different laser architectures. Moreover, the concept of the phase-conjugate mirror can also be applied to coherently combine the beams emitted by multichannel amplifiers or multielement laser diode arrays. This review, which includes very significant experimental results, outlines the importance of nonlinear materials in performing adaptive correction of laser beams or beam shaping, in particular for the nearinfrared spectral region. They permit a great improvement in laser brightness and bring additional capabilities for power scaling. These applications should stimulate research for improving existing materials’performance in this spectral range

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and for studying new materials with high power-handling capabilities and short response time. An important application area of photorefractive holography that was developed in the early stages using conventional materials is interferometry applied to nondestructive testing or to high-precision measurements. The availability of high-sensitivity photorefractive crystals now permit one to realize dynamic interferometry of large objects. This subject is treated in detail in Chapter 8, which develops several metrological device architectures based on dynamic holography with sillenite crystals. The first system is a holographic camera using continuous laser illumination well suited for the measurement of the displacements of the surface of scattering objects. The second is adapted to interferometry of transparent objects, and the third uses pulsed lasers for studying vibrating structures. Photorefractive interferometry also benefits from the advanced digital processing methods for the detection of the interferogram and for calculation of data corresponding to object deformation. It is now a mature technology, and a few systems are commercially available. Adaptive photorefractive interferometry has a significant potential for metrological applications; it is a flexible tool that adapts to any three-dimensional object structure. There are numerous applications of lasers sources that require singlelongitudinal-mode operation, and this happens in particular in holographic interferometry. As proposed and analysed in Chapter 9, an innovative approach for single-mode oscillation without adjustments of static filters is to exploit a self-organizing process by an intracavity adaptive photorefractive filter. When the crystal is inserted in the cavity, the oscillating modes record a common Bragg grating. For a correctly designed cavity this dynamic grating acts as a spectral and spatial filter, which in turn modifies the relative losses of the modes. This results in a unique-mode laser oscillation corresponding to optimum gain and spectral filtering efficiency of the grating. This intracavity interaction is self-adaptive, and after a rather short time constant it forces the cavity to oscillate on a transverse single and longitudinal mode. These novel types of self-organizing cavities are undoubtedly important for achieving single-mode operation, and it can be easily implemented in diode-pumped solid-state gain media or with extended semiconductor laser cavities. Moreover, it has also been demonstrated that selforganization can also occur on the transverse-mode structure: a laser oscillating on several transverse modes can be forced to oscillate on a single mode due to spatial filtering by the photorefractive volume-grating structure. Chapter 10 deals with a new research topic. It investigates the propagation of a light pulse or of a modulated signal beam inside the volume of a photorefractive grating. It is demonstrated that slow and fast light propagation can be achieved due to the dispersive properties of phase coupling in two-wave mixing interactions. This effect occurs in different types of nonlinear media that exhibit strong dispersive properties, and in particular, dynamic holography in photorefractives can achieve a high value of the group index at low incident intensity. Wave coupling results in changes of both the intensity and phase of the pump and of the modulated signal beams interfering in the material. It results in an

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1. Introduction

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efficient phase transfer from the pump to the signal beam due to photorefractive two-wave mixing. Group-velocity tunability and transition between slow and fast light are demonstrated by tuning the recording parameters such as the pump intensity and the externally applied voltage to the crystal. It is shown that these parameters as well as the frequency detuning between the pump and signal beams contribute mainly to the control of the group time delay inducted by the slope of the photorefractive phase coupling. In Chapter 11, another type of holographic filter is realized with photorefractive crystals like doped lithium niobate or lithium tantalate. These crystals permit one to realize filters that exhibit very low spectral bandwidth, robustness, and low absorption losses in the telecommunication windows. Moreover, the index modulation of the Bragg filter can be permanently fixed with an appropriate thermal process. All these characteristics make the photorefractive filters suitable for dense-wavelength division multiplexing in very high bandwidth communication networks. This chapter details the devices, the selection of photorefractive materials properties, the recording process, and the thermal fixing. All these steps are optimized in order to attain the specific features required for telecommunication applications. The last chapter of this volume opens an original field of investigation and applications: neutron physics with photorefractive materials. Early experiments began with neutron diffraction from a photoinduced grating in polymetacrylate through a photo-neutron-refractive effect. Then, the same type of diffraction was also realized with a conventional photorefractive grating recorded in electro-optic lithium niobate crystal. It is shown in this chapter that these novel experiments are of interest for studying polymerization process and for probing fundamental properties of neutrons. This is the case of neutron diffraction by electrooptic crystals in which neutrons are exposed to the high photoinduced space charge field generated in the crystal volume. Future perspectives of photo-neutron-refractive materials and applications are discussed and neutron holography is also introduced in this last chapter. In conclusion, this collection of chapters provides a broad survey of the most advanced application fields of photorefractive nonlinear optics. The chapters deal with a large variety of components, devices, and systems architectures exploiting the unique characteristics of photorefractive materials. The authors of this volume are leading experts, and their contributions open innovative applications of dynamic holography and nonlinear wave mixing, and in laser optics, optoelectronics, signal processing, and metrology. This volume should serve the needs of the scientific and engineering communities interested in the applied aspects of photorefractive nonlinear optics. The editors of this volume express their warm regards to all the authors for their outstanding contributions and very fruitful cooperation in the preparation of the volume, which should help to stimulate further developments in the field. We also thank Mrs. Lotti N¨otzli for her very valuable secretarial support.

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2 Digital Holographic Memories Lambertus Hesselink and Sergei S. Orlov Stanford University, Stanford, CA 94305-4035 [email protected]

2.1 Introduction Optical data storage is a phenomenal success story. Since its introduction in the early 1980s, optical data storage devices have evolved from being focused primarily on music distribution, to becoming the prevailing data distribution and recording medium. Each year, billions of optical recordable and prerecorded disks are sold worldwide. Almost every computer today is shipped with a CD or DVD drive installed. The roadmap for optical storage, therefore, is of significant interest, since it defines the industry’s view of future products. One such roadmap is shown in Figure 2.1, based on information provided by TDK in 2002. It shows that the end of the DVD roadmap is followed by super-resolution-blue-based technology, which provides up to 100 GB capacity per disk at a transfer rate of 200 Mbps. For larger capacities and transfer rates, a new approach is needed to further extend the roadmap. Near-field optical recording and holographic recording are generally viewed as being the most likely technologies for achieving transfer rates and capacities near 1 TB per disk at a rate of 1 Gbit/sec. In this chapter we explore holographic data storage systems (HDSS) technology, which has a long history of research and development. In the seventies, much fundamental materials and systems work was carried out on HDSS at RCA (Amodei and Staebler, 1971, Staebler et al., 1975), in the (then) Soviet Union [3] and Bell Labs [4], and many other industrial and university laboratories wordwide [5]. New and improved components were demonstrated and incorporated into complex systems. However, there was neither a strong market need, nor a cost competitive system demonstration to drive large-scale commercial development. Research activities subsided. In the mid-1980s, work at the Microelectronics and Computer Corporation and Stanford University, at Northrop, Caltech, and other laboratories signaled a renewed effort in holographic data storage [6]. The primary drivers this time around were new components developed for consumer applications, and a growing commercial need for largecapacity storage devices.

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Holography

Data Rate (Mbps)

1000

Near Field

High-speed blue

Super resolution blue

100

Multilayer blue DVR-blue DVD 10 10

100

1000

F I G U R E 2.1. Optical data storage roadmap (source TDK, 2002).

The recording material, however, was not under development for other consumer applications. This led to the formation of the Photorefractive Information Storage Materials (PRISM) in April 1994, and the Holographic Data Storage Systems (HDSS) NSIC/DARPA/Industry/University consortia in April 1995. The PRISM activities were focused on fundamental materials issues, while new components would be integrated into systems reaching a 100-GB capacity and a raw transfer rate of 10 Gbit/sec [7] through component improvement and system optimization. System optimization, as usual, is a complex problem involving a large number of tradeoffs. For HDSS in particular, the tradeoff between capacity and transfer rate is different from that of other storage systems. Fundamentally, holographic data storage is based on multiplexing many holograms in the same volume of the recording medium. For media with a linear response, this implies that the dynamic range for each hologram is roughly equal to the total dynamic range of the medium divided by N, the number of holograms. Since the diffraction efficiency of each hologram is proportional to the squared index modulation, readout signal strength drops off as 1/N 2 . The larger the capacity of the device, the smaller the readout signal strength and the signal-to-noise ratio (SNR). In turn, small SNRs cause large raw-bit error rates, which above a threshold of 10−3 to 10−4 cannot be further lowered by error-correction schemes. To boost SNR, detector integration times can be increased at the expense of data-transfer rates. High-capacity systems are therefore most easily demonstrated for small transfer rates, and vice versa.

2.2 Optical Architectures Broadly speaking, holographic data storage materials are divided into two classes: thin (a few hundred microns thick) photosensitive organic media and thick (a few millimeters to centimeters) inorganic photorefractive crystals. The thin media

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2. Digital Holographic Memories

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Holographic Engine Laser

SLM

Medium

CCD

Beam Steering Data In

Address ...011001...

0 1 Encode

Data Out

0

... 01100

Control Unit

Decode

F I G U R E 2.2. 90-degree optical architecture, including the electronic control unit.

lend themselves best for transmission-type architectures using a variety of shift or phase multiplexing techniques, while thick media are most suitable for angular multiplexing, typically in the 90-degree geometry. Although phase and wavelength multiplexing techniques have been thoroughly investigated, at present the required lasers and phase masks are not suitably developed to achieve large data densities [6, 8]. As an example, consider the case of the 90-degree geometry shown in Figure 2.2. The data-bearing object beam and the reference beam are incident upon a thick photorefractive medium from orthogonal directions. Intersecting plane signal and reference beams generate an interference pattern with regions of high intensity and low intensity, of the form I = I0 (1 + m cos K x) in which I0 is the intensity, m is the modulation factor, K is the grating vector, and x is transverse to the direction of propagation (Figure 2.2). Photons excite electrons into the conduction band in higher concentrations within the regions of constructive interference than within the regions of destructive interference. Electrons diffuse in the direction of the concentration gradient to the regions of destructive interference, where they relax to empty states in the energy band gap (intrinsic or dopant levels). The second architecture that has been widely investigated is built around a rotating disk, as shown in Figure 2.3. The medium is a photopolymer sandwiched between two glass plates. The photopolymer medium has a thickness of a few hundred microns. Conventional angular and wavelength multiplexing do not allow hundreds of holograms to be superimposed, since their sensitivities (the derivative of the square root of the diffraction efficiency with time normalized by the intensity and thickness of the medium) are inversely proportional to the medium thickness [8]. Shift multiplexing was developed for this purpose by Markov and Darskii and their coworkers, and later extended by the Psaltis group at Caltech. By moving the medium with respect to a fixed spherical reference wave and the object wave, the Bragg condition is violated, allowing a new hologram to be recorded with little crosstalk. Since

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Signal beam

Page Composer

Laser diode

Reference Beam

Imaging lenses Photopolymer Sample Rotating disk

Computer

CCD camera

F I G U R E 2.3. Rotating disk architecture using photopolymer media.

Diffraction Efficiency, cnts/CCD pixel

the spherical reference wave does not provide much selectivity, this multiplexing method does not provide very high recording densities. By modifying the phase front of the reference beam using a random phase mask, excellent selectivity can be achieved. Alignment requirements, however, increase, since the Bragg selectivity curve becomes much narrower compared to the spherical wave case. By engineering the magnitude and the correlation function of the phase mask, the decorrelation distance is modified. Excellent densities can be achieved with this method, as shown in Figure 2.4. The shift multiplexing (mux) selectivity is equal ULSH-500-4 200 micron Polaroid Photopolymer 50 45 40 35 30 25 20 15 10 5 0

0

2

4 6 8 Displacement, microns

10

12

150 μm using a spherical wave shift-mux

F I G U R E 2.4. Spherical wave and random-phase-mask multiplexing results for 200micron-thick photopolymer sample.

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to the autocorrelation length of the speckle field, which is 1.2λ / NA in both the x and y directions where NA is numerical aperture and l is optical wavelength. In 1999, in an effort led by Siros and Stanford, the HDSS team demonstrated shift multiplexing using a correlation multiplexing based on a speckled reference wave, and achieved a density of 20 GB on a disk and a data transfer rate as high as 10 Gbit/sec, as described later in this chapter.

2.3 Holographic Optics Holograms are recorded using a reference beam and a signal beam comprising data. As shown in Figure 2.5(a), the reference beam R and signal beam S interfere to generate a fringe pattern, which is transferred to the medium. In holographic data storage systems, the recorded hologram typically takes the form of an index grating. As shown in Figure 2.5(b), readout is accomplished by illuminating the hologram with the original reference beam R, which diffracts off the recorded hologram to generate an output signal beam S. The electric field amplitude of the propagating reference wave is represented as R = R0 exp(iρ · r ),

(2.1)

where R0 is the complex amplitude, ρ is the wave vector, r is the spatial dimension. The electric field amplitude of the propagating signal wave is similarly represented as S = S0 exp(iσ · r ),

(2.2)

where S0 is the complex amplitude and σ is the wave vector. The interfering signal and reference wave vectors generate a grating with a grating wave vector K given by K = σ − ρ. x z

R

S

R

S

(a)

(b)

F I G U R E 2.5. Optical recording geometry (a) and readout (b).

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Lambertus Hesselink and Sergei S. Orlov F I G U R E 2.6. Wave vector arrangement between object (σ ) and reference beam (ρ), giving rise to grating vector K.

Ky σ K ρ

Kz

This relationship is illustrated in Figure 2.6. The resulting hologram is given by the index grating n(y) = n 0 + Re[δn exp (i K y)],

(2.3)

in which x is the transverse direction of propagation and δn is a complex coefficient given by δn = n 1 R0 S0∗ /I0 ,

(2.4)

where n 1 is the proportionality constant that accounts for the material response to illumination and I0 is the total illumination. The diffraction efficiency η of a recorded hologram is given by η = sinc2 (eR · eS ((π|δn|L)/(λ cos θ ))) exp(−αL),

(2.5)

where L is the thickness of the medium, α is the absorption coefficient of the material, and eR and eS are the polarizations of the reference and signal beams, respectively. Note that the diffraction efficiency can approach 100% for relatively strong holograms. For relatively weak holograms, with negligible absorption, the diffraction efficiency can be represented by η = (eR · eS ((π|δn|L)/(λ cos θ)))2 .

(2.6)

Holograms recorded in a material having a substantial thickness display a property called Bragg selectivity. Bragg selectivity means that holograms recorded with a planar reference wave vector provide the strongest diffraction efficiency when read out with the same reference beam. The original reference beam and the hologram are said to be Bragg-matched. When the wave vector of the read out reference beam is substantially different, the diffraction efficiency is significantly less. The wave vector of the reference beam can be changed by changing the angle of incidence or wavelength of the reference beam. Bragg selectivity, therefore, allows multiple holograms to be recorded in a holographic data storage system because only the hologram Bragg-matched to the readout beam is reconstructed with substantial diffraction efficiency; all of the others are substantially attenuated. When multiple holograms are recorded in a material, the dynamic range of the index grating is divided up among the holograms. This dynamic range is

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characterized by M#, which is defined as 1

/

M# = (ηi 2 ),

(2.7)

in which M is the total number of holograms and ηi , is the diffraction efficiency of the ith hologram. In an ideal material, M# is independent of M for a large number of multiplexed holograms. In this case, M# is given by M = eR · eS ((π|δn|L)/(λ cos θ )).

(2.8)

The diffraction of each hologram in a system of M multiplexed holograms is therefore given by η = ((M#/M))2 .

(2.9)

Another important property of materials used in holographic data storage systems is the recording sensitivity. The material sensitivity, typically measured near t ≈ 0 can be determined using √ S = ∂( η)/∂t/I0 L . (2.10) The material sensitivity depends on material parameters and recording configuration and is generally reported in cm/J. The sample sensitivity can be determined using √ (2.11) S  = ∂( η)/∂t/I0 t, where t is the exposure time. The sample sensitivity depends on the material thickness and is generally reported in cm2 /J. (The type of sensitivity referenced can be generally inferred from the units reported.) Inorganic photorefractive materials such as lithium niobate have sample sensitivity ranging from 0.01 to 0.1 cm2 /J and M# of about 1–10 for 1-cm-thick samples. Polaroid ULSH-500 photopolymers have sample sensitivity ranging from 20 cm2 /J and M# as high as 10 for 0.5-mm-thick samples. Development of stronger, more sensitive, and more stable holographic storage materials is an active area of research. Bragg selectivity is illustrated in Figure 2.7. Signal beam wave vector σ and reference beam wave vector ρi record a hologram grating having wave vector y

ρi

F I G U R E 2.7. Bragg mismatch illustrated on rotation of the reference beam wave vector.

ρj

σp

ΔK

K

K

z

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K. In an isotropic medium, wave vectors σ and ρi have the same wave number β = 2nπ/λ, and therefore reside on the Ewald sphere. Because the original signal and reference wave vector are Bragg-matched, the vector sum lies on the Ewald sphere: σ = K + ρ.

(2.12)

A second reference beam with wave vector ρ j diffracts off the grating with wave vector K, and the resulting vector sum differs from the output signal by K: σ = K + ρ + K.

(2.13)

The second reference beam is therefore not Bragg-matched to the grating and is therefore substantially attenuated. When the wavelength is the same and the angle of the reference beam changes, as is the case for angle multiplexing, the Bragg selectivity is approximated as η = (eR · eS ((π|δn|L)/(λ cos θ)))2 sinc2 ((( K z L)/2)),

(2.14)

where K z is projection of K on the z-axis and sin c(x) = sin(πx)/πx. Thus, Bragg selectivity may be characterized by a minimum separation of grating vectors along the z-axis, given by ( K z )min = 2π/L. The first zero of the sinc function is obtained at this separation vector. The finite lateral extent of the medium puts a similar constraint on the minimum separation of grating vectors on the xand y-axes, so that the corresponding minimum separation vectors are given by ( K x )min = 2n/H and ( K y )min = 2n/W , respectively, for medium height H and width W, respectively.

2.3.1 Fourier Transform Configuration A particularly advantageous configuration for holographic data storage utilizes the Fourier transform properties of a spherical lens. In the Fourier transform configuration, shown in Figure 2.8. the signal path comprises a signal page generated by a spatial light modulator situated at the front focal plane of a lens, situated a σi (a)

f

f

ai

Sj (b) Aj(r)

F I G U R E 2.8. Fourier transform configuration for holographic data storage, shown for (a) a point source and (b) a spatial light modulator.

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focal length f in front of the lens. The holographic recording material is situated near the rear focal plane of the lens, but not necessarily at the focal plane. Figure 2.8 (a) shows that point source ai becomes a plane wave on passing through the lens. More generally, Figure 2.8 (b) shows a spatial light modulator having a spatial distribution given by A j (r ), for position vector r, is transformed to a spectrum of wave vectors according to the relationship  S j (r ) =

dσ exp(iσ · r )A j (2π f σ/λ).

(2.15)

Thus, in this configuration, each pixel corresponds to a plane wave in the holographic storage material. A recorded image of the signal page is imaged onto a detector array using an appropriate lens. In the Fourier transform configuration, the signal page is distributed throughout the holographic storage material and is therefore robust against local defects in the material. This configuration is relatively insensitive to displacements of the holographic storage medium, provided it remains in the same orientation. It is the most widely used optical configuration used in holographic data storage applications. Usually the medium is placed in between two Fourier transform lenses to create an almost symmetric optical system, which has the additional advantage that the magnification is close to one, and under these conditions the optical distortion aberration is close to zero. The medium is not placed exactly at the Fourier plane, since this gives rise to very high incident intensities of the signal beam, which leads to large object-to-referencebeam intensity ratios along the z-axis or thickness of the medium that are ideally kept close to one to optimize the recording efficiency. The medium therefore usually is placed either just before or behind the Fourier plane, breaking the perfect symmetry of the optical setup, yet allowing the aberrations to remain manageably small.

2.3.2 Grating Vectors as Data Channels The data capacity of a holographic data storage system can be estimated using a degrees of freedom model based on resolvable grating vectors [11, 12]. The finite extent of a holographic storage medium generally limits the resolution of grating vectors. In the direction of signal propagation, the grating vector resolution is given by K z = 2π/L; in the directions perpendicular to the direction of signal propagation, grating vectors are given by K y = 2π/W and K x = 2π/H . Here L , W , and H correspond to medium length, width, and height, respectively. In an isotropic medium, the maximum wave vector length is given by K = 4πn/λ, which defines a radius of a grating vector sphere that bounds the space of allowable grating vectors. The volume of this space is given by VK = ((4π)/3)K 3 .

(2.16)

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Defining a data channel as a resolvable grating vector, the total number of accessible data channels is given by Q = VK /( K X K Y K Z ), so that Q = ((32π)/3)((n 3 V )/(λ3 )),

(2.17)

where V is the volume of the holographic recording medium. This quantity allows two data channels in quadrature for each grating orientation, which generally requires coherent demodulation to separate the data channels. For one data channel in each grating direction, the total number of accessible data channels is given by [11, 12] Q = ((16π)/3)((n 3 V )/(λ3 )).

(2.18)

Equation 2.18 indicates that a material with an index of refraction of 2.5, such as lithium niobate, has a channel density of 2 × 1015 resolvable grating vectors. In a practical application, the number of degrees of freedom is reduced by a factor α to 1012 to 1013 due to the limited numerical aperture of a signal beam and limited angular excursion of the reference beam. In a holographic data storage system, the accessible data channels a Q are the product the number of pages N and the number of pixels per page P. Once we have determined the number of data channels, we may then estimate total capacity through Shannon’s law, namely, by multiplying the total number of channels by the capacity of a single channel: C total = α Q log2 (1 + SNR) = NP log2 (1 + SNR),

(2.19)

in which N is the number of pages and P is the number of pixels per page, and SNR denotes the signal-to-noise ratio. For a white Gaussian noise model, given the constraint of constant laser power, and for constant total diffraction efficiency, the SNR decreases with increasing number of pixels per page (SNR < 1/P). At the same time, due to a limited dynamic range, the total diffraction efficiency decreases as 1/N 2 , leading to a corresponding behavior of the SNR (SNR< 1/N 2 ). Thus, there exists a trade off between the SNR and the number of pages that can be superimposed. As a result, given parameters of the system (laser power, optical performance, electronic detection noise, media M#, data rate), there is an optimum number of pages that need to be written in a single location of the volume holographic memory for which the capacity C total is maximized [13]. For realistic parameters of storage in the 90-degree geometry lithium niobate system (P = 106 , M# = 1, laser power 1 watt) the theoretical upper bound on the optimum number of pages is estimated to be <2500, with total raw capacity per storage location of < 2.5 × 109 bits. It is worth noting that the capacity calculated above is merely an informationtheoretic upper bound and does not consider the effects of SNR reduction due to nonideal imaging, and interpixel and interpage crosstalk. The state-of-the-art demonstration of digital hologram recording in 90-degree geometry in lithium niobate [48] experimentally realized a capacity of 109 pixels per storage location (i.e., 1000 superimposed holograms of 1 Mpixel images). The resulting raw channel density of <350 bit/μm2 (calculated as the ratio of capacity and the area

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occupied by the holograms on the entrance face of the storage crystal) represented, however, only ≈1.08% of the theoretical volumetric density limit of 1/λ3 .

2.4 Multiplexing Techniques 2.4.1 Angular Multiplexing Figure 2.9. shows a diagram for angular multiplexing. Data are encoded on a signal beam using a signal-amplitude spatial light modulator (signal ASLM). The appropriate data pattern is loaded onto the signal ASLM through an appropriate electronic interface to a computer or peripheral system. The signal ASLM is situated at the front focal plane of a lens; the modulated signal propagates to the holographic medium in a Fourier transform configuration. Reference beams are plane waves incident on the holographic medium. Each reference vector is limited to a plane of incidence further situated so that the reference beams and an included wave vector in the signal beam lie in the same plane, as shown in Figure 2.9. A separate hologram is recorded with each reference wave, resulting in multiplexed holographic data pages. A data page is read out by illuminating the holographic medium with one of the reference beams. The output data page is focused on the output detector, and then demodulated by the appropriate electronics Throughout this chapter, we are primarily interested in page-oriented multiplexing, in which we organize a large quantity of data recorded as pixels of an image. Each pixel stored by each reference beam may be considered to be a separate data channel. Angular selectivity is maximized for included signal, and reference wave vectors are normally incident to perpendicular faces of the holographic storage medium [16, 17, 18]. For copolarized signal and reference beams, Output Detector

Holographic Medium

Reference Angles Signal ASLM

F I G U R E 2.9. Angle multiplexing architecture.

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the diffraction efficiency is expressed as η = (((π|δn|L)/(λ cos θ)))2 sinc2 (((π L sin(θρ − θσ ))/(λ cos θσ )) θ ), (2.20) where θρ is the mean angle of the first and second reference beams and θσ the angle of incidence of the signal beam. The minimum separation angle is defined as the first extinction of the sinc function, which occurs at

θ = ((λ cos θσ )/(n L sin(θρ − θσ ))).

(2.21)

In this expression, the separation angle refers to the angle inside the material; refraction at the surface of materials having a high index of refraction will substantially increase the minimum separation angle. In a holographic storage material such as lithium niobate having a thickness of 1 cm, the angular selectivity is 50 μrad in free space for recording/readout wavelengths of 500 nm, in the perpendicular geometry. Thus, 10,000 holograms can be accessed with an angular range of the reference beam of about 30 degrees. For moderate nonoverlapping angular spectra of the signal and reference beams we can derive a simple expression to estimate the accessible data channels. A reference wave illuminating the side face of the crystal therefore has a maximum number of degrees of freedom, the number of grating vectors separated by the minimum separation vectors of Q z = 2n L/λ. A signal wave illuminating a face has a maximum number of degrees of freedom Q x = 2n H/λ in the x dimension and Q y = 2nW/λ in the y dimension. In this geometry, the number of accessible data channels is further estimated by taking into account the angular bandwidths of the reference and signal beam to yield Q = ((8V )/(λ3 ))(NA)2 sin(/2),

(2.22)

where NA is the numerical aperture of a signal incident at surface normal and  is the angular range of the reference beam, both in free space outside the material. For a wavelength of 500 nm, signal numerical aperture of 0.25, and reference beam angular range of 30 degrees, the number of data channels is about 1012 cm−3 .

2.4.2 Wavelength Multiplexing Wavelength multiplexing consists of hologram formation in which each reference wave consists of a plane wave of a different wavelength incident at the same angle, such that R1 = ((A1 /2) exp[i(ρ1 · r − ω1 t)] + ((A1 ∗ /2) exp[−i(ρ1 · r − ω1 t)],

(2.23)

where now |ρ1 | and the corresponding angular frequency ω1 vary but the orientation of ρ1 is the same for all reference beams. Only the recording reference waves are Bragg-matched, and any others result in Bragg-mismatched reconstruction. Figure 2.10 shows a diagram for wavelength multiplexing. As with the angular multiplexing case, a Fourier transform configuration is used. The signal and reference beams are incident on opposite sides of the holographic medium, known

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Output Detector

Holographic Medium

Reference Beam

Signal ASLM

F I G U R E 2.10. Generic wavelength-multiplexing architecture.

as a reflection configuration. Wavelength selectivity is maximized for included signal, and reference wave vectors correspond to counterpropagating directions in the holographic storage medium [15–18]. For copolarized signal and reference beams, the diffraction efficiency is expressed as η = (((π|δn|L)/(λ cos θ)))2 sinc2 (((2πnL sin(θ/2))/c) v),

(2.24)

where v is the frequency difference between the recording and readout reference beam, c is the velocity of light in free space, and θ is the angle between a planar signal and reference wave outside the medium. For reflection holograms, the minimum separation frequency is defined as the first extinction of the sine function, which occurs at

v = (c/(2n L sin(θ/2))).

(2.25)

For reflection holograms, the angle between the signal and reference beams is generally substantial. Frequency selectivity is highest for the counterpropagating geometry, in which the signal and reference beams are incident at opposite surfaces of a holographic storage medium and propagate in opposite directions.

2.4.3 Phase-Encoded Multiplexing Phase-encoded multiplexing uses a phase spatial-light modulator to control the reference beam and requires accurate control of the phase modulation and efficient collection of phase patterns to minimize crosstalk. An example architecture for phase-encoded multiplexing is shown in Figure 2.11. Here, a Fourier transform arrangement is used for both the signal and reference beam paths. A phaseencoded reference beam comprises a phase pattern eϕ(x,y) established by a spatial

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Holographic Medium

Reference PSLM Signal ASLM

F I G U R E 2.11. Holographic data storage system using phase-encoding multiplexing.

light modulator. In this example, a one-dimensional spatial light modulator is situated at the front focal plane of the lens, so that the resulting reference beam incident on the material has a wave-vector spectrum in one dimension. By using a set of orthonormal functions and an appropriate optical configuration, multiple holograms may be superimposed without significant crosstalk. When a reference beam is modulated in phase in one dimension, as shown in Figure 2.11, it consists of wave vectors spanning a plane. We choose a spacing of plane wave components such that the discrete spectrum of wave vectors ρ j corresponds to the wave vectors optimized for angular multiplexing. For proper selection of geometries, Bragg-mismatch can be neglected. For this case, Krasnov has shown that the diffracted signal is given by [19] dc ∝ (R j + Rc )Sl ,

(2.26)

in which the proportionality constant depends on the properties of the holographic material. For this special case, (R j · Rc ) defines the inner product between R j and Rc . To minimize crosstalk we require that the reference waves be orthogonal. In a practical application of phase encoding, we will typically have a configuration in which Bragg-mismatch cannot be completely neglected; a more detailed analysis is required [8]. Overall, we expect the crosstalk-limited signal-to-noise ratio of an orthogonal phase-encoded multiplexing system to be comparable to the case of angular multiplexing, and somewhat lower as we deviate from strict orthogonality. Capacity of a phase-encoded system is also expected to be similar to that of an angular multiplexed system. In practice, however, crosstalk between different pages and strict alignment requirements of the reference beam and the rest of the system limits the useful capacity severely. No experimental demonstration of a high-capacity system has been made todate. A simple way to think about phase multiplexing is to consider the system to be an interferometer, since the reference

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and object beams must be stable to within a fraction of a wavelength with respect to each other. This same stability also places very strong mechanical constraints on the system during readout.

2.4.4 Correlation Multiplexing For the holographic disk geometry, shift multiplexing is the most suitable and natural solution, since it doesn’t require any beam steering or wavelength change and is achieved by simply moving the medium with respect to the reference and object beams, as indicated in Figure 2.3. Shift multiplexing can be implemented using a spherical wave [20], or, alternatively, a random or pseudorandom speckle pattern as a reference beam [21]. Shift multiplexing in the spherical-referencebeam case relies on the fact that a spherical-type beam contains multiple angular components, and thus a shift of the media or the reference beam leads to conventional Bragg-mismatch. Multiplexing with a speckle beam is based on the spatial autocorrelation function of the beam used for the recording. In such case the diffraction efficiency of the hologram is determined by the degree of correlation between the speckle beam used for the recording and the speckle beam presented on readout (e.g., a spatially shifted version of the original beam). For random Gaussian speckle, the hologram shift selectivity is determined by the beam autocorrelation function [21]: η(δ) = η(0)|2Jl (kNAδ)/kNAδ|2 ,

(2.27)

where δ is the shift amount, k = 2π/λ, NA is the numerical aperture of the reference beam, and Jl is the first-order Bessel function of the first kind. The speckle selectivity does not depend on the direction of the media shift (Bragg-selective or Bragg-degenerate) and is independent of the media thickness, although the crosstalk buildup is usually much faster in the Bragg-degenerate direction (and in thin media). Since this chapter focuses on the use of thick photorefractive recording media, for details of correlation multiplexing, the reader is referred to Hesselink, 2004.

2.5 Holographic Recording Materials Although different applications require materials of varying properties, an ideal material would have a fast time response (< μsec), be as sensitive as photographic film (μJ/cm2 ), retain information for long time periods (>10 years), have a large spectral sensitivity range extending into the infrared region of the spectrum, and be available in large quantities, at good optical quality, in large size, and at low cost. Such a recording material does not exist today! In general, sensitive materials have small photorefractive response, and vice versa. For example, LiNbO3 is available in large size, of good optical quality, and at relatively low cost, and has a large index variation per absorbed photon. Its response time, however, is rather slow, on the order of milliseconds or longer, depending on the intensity of the recording

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light. Thermal fixing has been demonstrated allowing nondestructive readout for periods of years, but this process typically requires temperatures of at least 100◦ C for fixing. Photopolymer materials usually have much higher sensitivity due to chemical amplification effects and a high dynamic range, but are rather limited in terms of thickness (1 mm or less). Other issues associated with polymers are significantly increased scatter levels compared to single-crystal photorefractives and volumetric shrinkage on photopolymerization. The latter effect can be somewhat mitigated by use of high-molecular-weight monomers and proper choice of the material chemistry. The shrinkage and scatter limits the available usable thickness of photopolymers. Typical minimum material parameters for optimized holographic performance are listed below [22]: Material: – Thickness: L > 0.5 mm; – Sensitivity: S = 500 cm/J (or S L > 20 cm2 /J); – Dynamic range: (M/#)planewave > 5.0 (or, (M/#)image > 2.0); – Shrinkage: < 0.05% (for 500 μm); – Scatter: < 10−5 srad−1 ; – Wavelength response: visible (<532 nm or blue). Media: – Disk wedge <0.5 mrad; – Format: 120 mm to 6.5 diameter disk; – Total wavefront error over <2 mm spot: < λ/10; – Scatter (including substrate) < 10−5 srad−1 . The performance of a holographic data storage system greatly depends on the quality and physical properties of the recording medium. The DARPA PRISM consortium has developed a precision tester [23] for measuring the performance of different holographic media using a common platform. Since 1996 many advanced holographic recording materials have been evaluated and compared using a common comparison framework. The most important parameters include image quality (degree of image distortion), sensitivity, dynamic range, fidelity (spatial resolution), stability, shrinkage, and available thickness. Based on the results published so far [24], the most promising photorefractive crystals remain Fe-doped lithium niobate, while Polaroid CROP materials represent the better option in the photopolymer family. For a comparison among various optical recording properties the reader is referred to Hesselink, 2004.

2.6 Photorefractive Materials Photorefractive materials record holograms through a photoinduced chargetransport (diffusion, drift, and photovoltaic effect), charge-carrier redistribution

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2. Digital Holographic Memories F I G U R E 2.12. Photoexcitation of electrons in the presence of a periodic intensity pattern.

23

I

---

---

---

---

++ ++

++ ++

++ ++

++ ++

Ec

Ev

between the deep traps and buildup of an internal space-charge field. The resulting index distribution is produced due to electrooptic (Pockels) effect Figure 2.12. In general, a variety of trapped and photoexcited states are possible, including electrons and holes. Typically, electrons are trapped at impurity centers and photoexcited into the conduction band. This example is illustrated as follows. Using the example of Figure 2.12, intersecting signal and reference beams generate an interference pattern with regions of high intensity and low intensity, which has the functional form I = I0 (1 + m cos K x),

(2.28)

in which m is the modulation factor that quantifies the contrast of the interference pattern. As shown in Figure 2.12, photons excite electrons into the conduction band in higher concentrations within the regions of constructive interference than within the regions of destructive interference. Electrons diffuse in the direction of the concentration gradient to the regions of destructive interference, where they relax to empty states in the energy band gap. In the presence of a photovoltaic current found in materials such as lithium niobate, the effect is enhanced. A graphical depiction of the relevant physical quantities associated with the photorefractive effect is shown in Figure 2.13 for a diffusion-dominated photorefractive medium. A periodic intensity variation I arises from the in interference of the signal and reference beams. This pattern establishes a Coulombic charge density ρ in the photorefractive material that records the intensity pattern; red indicates positive charge and blue indicates negative charge. At steady state, a periodic potential U associated with the charge density balances diffusion. The separation of electrons from fixed, positive ions establishes an electric field E, typically called a space-charge field, that replicates the functional form of the

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Lambertus Hesselink and Sergei S. Orlov F I G U R E 2.13. Photorefractive effect in diffusion-dominated materials.

ρ

U

E

δn

interference pattern, up to a phase factor ϕ: E = |E SC | cos(K x + ϕ).

(2.29)

This expression is accurate only for small modulation depths, in which E SC = m E 1 and E 1 is the normalized space-charge field, representing the transport properties of the medium. In materials displaying the linear electrooptic (Pockels) effect, for example, the space-charge field generates a periodic index grating δn proportional to the electric field, such that 1 (2.30) δn = − n 3r m E 1 cos(K x + ϕ), 2 where n is the index of refraction and r is the effective Pockels coefficient, which in general depends on the Pockels tensor, light polarization, and crystal orientation [60]. Therefore, the resulting index grating replicates the initial interference pattern up to a phase shift, generating a volume hologram. In a holographic medium that does not display nonlinear beam coupling, the diffraction efficiency is [25]   πmn3 r|E1 | η = sin2 . (2.31) 2λ cos θ Detailed accounts of photorefractive transport can be found in [26–31]. Lithium niobate is the most common photorefractive material used in holographic

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data-storage-system demonstrations. Other inorganic materials of interest include strontium barium niobate and barium titanate. Organic materials include reversible photorefractive polymers that may eventually be candidates for holographic storage systems [61]. Photorefractive polymer materials, however, require large applied electric fields, and good sensitivity as well as longterm stability of recorded holograms have not been demonstrated in the same medium.

2.7 Fixing of Holograms in Photorefractive Media Thermal fixing is the most widely used fixing method in LiNbO3 , followed by two-photon recording. These two primary techniques are described in somewhat more detail below.

2.7.1 Thermal Fixing In a simple photorefractive material with one photoexcitation process and one species of trapped charge, light used to record the hologram will typically erase the same hologram upon readout. Thus, for permanent storage and multiple reads without data degradation, photorefractive materials require a fixing process. Hologram fixing can be accomplished in a number of ways. One method makes use of complementary charge gratings residing in separate subsystems of charge transport that are not sensitive to the readout light. The gratings are complementary in the sense that they have opposite polarity and can screen each other. Figure 2.14 illustrates a generic process that uses complementary gratings to establish stable holograms. Subsystem A comprises a photoexcitable charge species, such as a trapped electron, as shown in Figure 2.14(a). A Coulombic charge density grating ρA is written in subsystem A through the photorefractive effect. A second charge density grating ρB is established in subsystem B using a different process, and ρA

ρB

ρA + ρB

(a) Initial Grating

(b) Complementary Grating

(c) Exposed Grating

F I G U R E 2.14. Graphical depiction of the evolution of complementary gratings.

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may be accomplished by changing a material environment parameter such as temperature, electric field, or type of illumination, as shown in Figure 2.14(b). Optical, holographic, and system considerations put constraints on the use of complementary gratings for nonvolatile readout. For image-bearing beams, an important issue is the expansion of the crystal on heating. If holograms are recorded at an elevated temperature and then cooled, the Bragg condition will be modified, and either the angle of the reference beam or its wavelength or both must be modified to obtain the greatest diffraction efficiency. This effect ultimately puts limits on the image field that may be reconstructed with acceptable fidelity, and will introduce additional crosstalk because not all of the signal can be Braggmatched simultaneously at the lower temperatures. It is a system issue whether simultaneous formation of holograms at high temperature or formation at low temperature and subsequent activation of mobile ions at high temperature give optimal performance. In lithium niobate, for example, subsystem A may establish an initial electronic charge density through the photorefractive effect and subsystem B is established as a complementary charge density through mobile ion transport at elevated temperature [2, 32–34] as shown in Figure 2.15. The ionic conductivity usually obeys an Arrhenius-type dependence on the temperature T: σi = en i μ0 exp(−((E a kB T ))).

(2.32)

where E a is activation energy (approximately 1.2 eV for hydrogen impurity and ≈1.4 eV for interstitial lithium migration), n i is the density of conducting ions, e is the electron charge, μ0 denotes the permeability of free space, and kB is Boltzmann’s constant. Mobile ions are thermally activated and move to screen e

e

e

e Illumination I(x), recording

I(x) ++

++ ni1(t=0) N+d1(t=0)

--

N+d1 (t = 0) grating Elevated Temp., Ionic Transport

ni 1 N+d1 ni 1 N+d 1

Ionic Compensation “Fixing”

Low T, Readout (Developing)

F I G U R E 2.15. Mechanism of hologram fixing via thermally assisted ionic drift.

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out the electronic grating. The resulting hologram becomes screened. However, at lower temperatures, the ions have significantly lower mobility and the grating lifetimes can be considerably longer than for an electronic grating. Thus, at lower temperature, the ionic grating will represent a backbone of the fixed hologram, while upon light illumination, the electronic grating will redistribute itself, partially screening the fixed ionic grating. The ionic grating also eventually decays at a much slower rate (from several months to more than 10 years) depending on the residual conductivity at the storage temperature and the degree of electronic screening. Two alternative approaches exist for creation of fixed ionic gratings in lithium niobate. In the first case, electronic gratings are formed at room temperature. Initial exposure to the signal and reference beams generates an electronic grating, which is followed by ionic compensation at higher temperatures without illumination. At elevated temperatures (above approximately 100◦ C in LiNbO3 [1], ions become active and generate a complementary screening grating that compensates the space charge arising from the electronic grating. These paired electronic and ionic gratings constitute complementary gratings and are derived from charge patterns with opposite polarity. On cooling to room temperature, the ions are significantly less mobile and represent the ionic backbone of the fixed grating. Subsequent light exposure reveals the fixed grating by partially redistributing the electronic charges. In the presence of light exposure, the ionic grating is partially screened, but in the case of LiNbO3 , strong photovoltaic currents significantly reduce the screening. The actual degree of screening depends largely on the grating spacing, photorefractive impurity concentrations (e.g., Fe2+ ), and the reduction state of the material [34, 36]. In the second case, both electronic and ionic gratings are formed simultaneously at an elevated temperature [2]. Simultaneous formation of electronic and ionic gratings typically results in stronger complementary gratings because of mutual screening during formation, in which the magnitudes of both gratings may be substantial, but their difference remains relatively small [32, 33]. The typically used temperature range for this process is 150 to 180 degrees Celsius. Multiple fixed holograms based on ionic gratings have been demonstrated in LiNbO3 [2, 40, 41]. At least 500 holograms may be stored using ionic gratings, for which diffraction efficiencies in the range of 2 to 2.5% have been reported [2, 40, 41]. As part of the PRISM Consortium, Siros Technologies demonstrated fixed ionic gratings by recording over 500 pages of digital data in LiNbO3 [42]. Subsequently, the Psaltis group demonstrated fixing of thousands of holograms [43]. Holographic gratings based on charge redistribution inevitably decay due to ionic and electronic conduction. The lifetimes of fixed ionic holograms are limited by the finite ionic conductivity at low (i.e., room) temperature [34, 36]. Ionic gratings are partially screened by trapped electrons upon readout, which decreases the resulting diffraction efficiency, but proportionally increases the grating lifetime. A significant increase in fixed ionic hologram lifetime is realized in lithium

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Lambertus Hesselink and Sergei S. Orlov Signal (near-IR) GATING BEAM (visible)

Reference (near-IR)

Reference (near-IR)

Reconstructed Signal beam

F I G U R E 2.16. Two-photon recording in LiNbO3 requires a visible incoherent gating beam in addition to two IR recording beams. During readout, only the IR reference beam is present. The light-induced IR sensitivity is about one hundred times higher than without the gating beam, giving rise to prolonged readout.

niobate with low hydrogen impurity content. Fixed hologram lifetimes of about 2 years at room temperature are projected in dehydrated lithium niobate crystals. Due to partial electronic compensation, the actual hologram lifetime is prolonged, but at the expense of reconstruction efficiency. Prolonged recording at elevated temperature (in highly doped lithium niobate) usually results in a nearly fully compensated gratings with no, or little, electrooptic contribution, but with a sufficiently strong index perturbation arising from the photochromic effect due to substantial modulation in the dopant concentrations. Such gratings possess prolonged lifetimes (over 50 years or more) and do not require light for development.

2.7.2 Two-Photon Fixing The difficulty with thermal fixing is that recording occurs at elevated temperatures with reading at room temperature. For thick media, the diffraction efficiency therefore reduces due to Bragg-mismatch. This problem and the associated complications of recording at elevated temperatures can be overcome by using a two-photon recording method. Two-photon recording techniques (Figure 2.16) have been investigated since the early days of research on holographic memories, in particular at Bell Laboratories [4]. The difficulty with the early work was the low sensitivity of the medium and the high power of the lasers required to record data in LiNbO3 . The Stanford group overcame this problem by modifying the composition of LiNbO3 by making it more stoichiometric, adding Fe and other dopants, combined with optimized oxidation and reduction postgrowth treatments [45]. We demonstrated an improvement in recording sensitivity of several orders of magnitude over previous results, making the medium approximately as sensitive as Fe-doped LiNbO3 in the green. Typical sensitivity and index changes as a function of stoichiometry are shown in Figure 2.17. It is interesting to note that the sensitivity exhibits a threshold behavior near 49.5 mol % Li2 O in the crystal. The details of the materials improvement and holographic aspects are described in Hesselink et al., 1998.

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10-2

10-3

S

Δn

10-3

10-4

10-4 48.5

Index change (Δn)

Sensitivity (cm/J)

Sensitivity Index change Δn

10-5 49.0 49.5 Li2O in the crystal (mol%)

50

F I G U R E 2.17. Two-photon sensitivity and index change as a function of stoichiometry in undoped LiNbO3 .

At a Li2 O concentration of 49.9%, the total accumulated grating strength (M# or the sum of the induced refractive index changes of all superimposed holograms) in this material is very high, and we have measured values of M# ≈ 10. This is very significant for data storage applications, since it provides a large dynamic range for superimposing holograms. The single-photon sensitivity in the near IR is very small, thus providing a means for prolonged readout, as shown in Figure 2.18. There is a small initial drop in readout efficiency, but for longer times it proves to be quite stable. When the gating beam is turned on again during readout, information is erased. The Stanford group [46] demonstrated the first digitally recorded holograms using two-photon recording with red Ti:Sapphire light around 800 nm, and gating light from a green argon-ion beam [8]. Besides these two primary fixing

F I G U R E 2.18. Diffraction efficiency in a record-readout-erase cycle using two-photon recording in LiNbO3 .

Diffraction efficiency (%)

80 70

Recording gate on

Readout gate off Erasure gate on

60 50 40 30 20 10 0

0

50

100 150 Time (s)

200

250

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techniques, other techniques for nondestructive readout are also discussed in the reference paper by Hesselink [8].

2.8 Systems Issues 2.8.1 Capacity Versus Transfer Rate Tradeoff The tradeoff between capacity and transfer rates is estimated by computing the number of photons incident on a detector element during page readout. From this calculation we can estimate the maximum number of superimposed pages readable with a certain SNR: 0.5  2 M # θQE Pηopt ηfix tread 2 , (2.33) Number of pages = M N pmin where M # is a variable based on system and materials parameters, θ is the number of photons per watt per second, QE the quantum efficiency of the detector array, ηopt the optical efficiency of the readout system, ηfix is the fixing efficiency, tread is the read time of the recalled data, M and N are the numbers of rows and columns on the data page, pmin denotes the number of photoelectrons needed to get a 20-dB SNR, and P is the power of the readout beam. It is assumed that bit patterns are balanced (although this does not have to be the case) by making half the pixels in a page on and half off. From this equation, the tradeoff between capacity (the product of the stored number of pages, the code rate, and the number of bits per page) and data transfer rate (the number of bits per page times the code rate divided by the readout time) is determined in a photon-limited system. Optical damage and crosstalk limitations are not considered. In practice, however, optical damage often tends to limit capacity, but can be significantly reduced by recording at elevated temperatures. Equation (2.33) provides an upper bound. Alternatively, Odoulov et al. [58] have shown that periodically poled LiNbO3 having thick domains on the order of the fractions of millimeters, the DC photovoltaic damage can be significantly reduced, thereby providing significantly improved storage performance through much higher attainable SNRs [8].

2.8.2 Signal Processing and En/Decoding In a data-storage system, the goal of error-correction coding, data modulation, and signal processing is to reduce the BER to a sufficiently low level while achieving such important figures of merit as high density and high data rate. The distinct feature of digital holographic data storage is that this technology treats each individual pixel of the hologram as a data channel and the entire image as set of data channels. This allows one to employ the modulation coding and signal-processing techniques to improve the system performance in terms of total capacity at an acceptable error rate.

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Input Data

Error Correction Coding

Interleaving

31

Modulation Coding

Equalization Encoding

Holographic Storage Material

Equalization Decoding

Output Data

Error Detection & Correction

De-Interleaving

Demodulation

F I G U R E 2.19. Typical signal path ip a holographic data-storage system.

Coding and signal processing can involve several qualitatively distinct elements. The cycle of user data from input to output can include error-correction coding (ECC), interleaving, modulation encoding, signal preprocessing, data storage in the holographic system, hologram retrieval, analog signal detection and binary digitization (using a CCD or CMOS detector array), signal postprocessing, channel decoding, and ECC decoding of deinterleaved binary data. A simple example of a signal path in a holographic data storage system is shown in Figure 2.19. Input data are processed to generate data blocks with errorcorrection coding. The data are interleaved among blocks to distribute burst errors among multiple blocks. Data are subject to a modulation code that converts binary data into specific modulation formats to be recorded in a holographic material. On readout, data are demodulated and then deinterleaved. The error-correction algorithm is applied to correct and/or identify errors, generating the output data. The purpose of modulation coding is to simplify the detection of the analog signal values from the detector array. Modulation codes generally restrict available patterns for formatting data to a subset that are less likely to be corrupted during recording, storage, and/or readout. This restriction is typically achieved using constraints on the types of data patterns that may occur. A balanced constraint addresses the issue of variation in signal strength across a data page; in a system with binary channels, this issue is addressed by ensuring that a fixed number of OFF and ON pixels (0 and 1, respectively) occur in a block of a given size. A lowpass constraint addresses intersymbol interference arising in high-frequency data

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patterns, typically by eliminating the data patterns most susceptible to intersymbol interference [52]. The simplest detection scheme is global threshold detection, in which a specific threshold value is chosen: Any CCD pixel with an intensity above the threshold is declared a 1, while the rest are declared 0. If all holograms result in equal diffraction efficiencies, a threshold intensity can be chosen to distinguish ON pixels from OFF pixels. Note that we use OFF and ON when referring to SLM or CCD pixels and 0 and 1 when referring to information bits [8, 52]. In the presence of variation in signal strength across a data page, this simple scheme does not perform satisfactorily because the optimal value of the threshold becomes dependent on the position within the page. A modulation code identified as an early alternative to the threshold approach in holographic data storage systems is a balanced constraint known as differential decoding, in which each zero and one of the data are encoded in the form of a group of two adjacent pixels A and B, one ON, one OFF [52]. Upon detection and demodulation, if the intensity of pixel A is lowest, the data bit is declared a 0, and if the intensity of pixel B is lowest, the data bit is declared a 1. Viewed a different way, a 0 1 sequence represents a data bit 0 and a 1 0 sequence represents a data bit 1. This method was applied to holographic data storage by the Stanford group in 1994, when a picture of the Mona Lisa and digital music were recorded and retrieved in the first demonstration of a digital holographic storage system [8, 52]. Other, more optimized, coding techniques were developed subsequent to this demonstration, and have been applied and implemented by a number of researchers worldwide [8, 47–50].

2.9 Demonstration Platforms A number of systems have been designed, built, and tested at Stanford University and Siros Technologies for a variety of purposes. They were built under sponsorship of the Center of Nonlinear Optical Materials, and the PRISM and HDSS consortia. The primary goals are to gain insight into the underlying physics of the recording mechanisms and the system tradeoffs required to achieve specified performance, and to integrate new components into working demonstration platforms. The most-significant demonstrations are briefly described in ensuing sections. Additional demonstrations are discussed in the review paper by Hesselink [8]. We limit ourselves here to the most significant demonstrations using LiNbO3 materials and the Stanford/HDSS final system demonstration, since it still describes the most advanced publicly disclosed digital storage system.

2.9.1 The Stanford University All-Digital System Demonstration (Heanue 1994) In 1994, we demonstrated the first fully digital holographic data storage system [52] by storing digital images and a short soundtrack. The purpose of this

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experiment was to show that a fully functional system could be implemented with off-the-shelf components. Particular emphasis was placed on using digital signal processing techniques to overcome noise issues limiting capacity and transfer rates. A new differential coding scheme was introduced that allowed us to significantly increase capacity while adding extra bits for channel and ECC coding. It allowed us to evaluate the tradeoffs between storage system capacity, bit-error rate, and data-transfer rate for several channel and ECC coding schemes. As described above, the capacity is limited for the most part by the attainable diffraction efficiency of the stored holograms. As the number N of holograms stored in a single stack increases, the diffraction efficiency falls as 1/N 2 , mentioned above. As the strength of the diffracted signal decreases, the signal-to-noise ratio (SNR) decreases because the strength of noise due to scatter and CCD noise is independent of N. The total number of holograms that can be stored is thus determined by the minimum acceptable SNR. A holographic data storage system is also limited by page-to-page fluctuations in signal strength, which can occur due to laser instability or, for example, due to a nonoptimal choice of recording schedule. Intrapage distortion can occur due to imperfections in the optical system or in the SLM or CCD array. These noise sources limit the applicability of direct storage of images in analog form. The majority of previous holographic storage implementations have involved direct storage and retrieval of pictorial information. For example, Mok has demonstrated angular-multiplexed storage of as many as 5000 edge-enhanced analog images in a single crystal [15]. The noise-tolerant nature of digital storage makes it possible to overcome problems associated with the aforementioned noise sources. However, prior to the publication of our Science paper, no fully automated digital holographic data storage system had been implemented and no comprehensive study of the biterror-rate performance of a system operating at reasonable data transfer rates had been carried out. Achievable bit error rates have been projected previously by sampling a small number of digital information bits from a random sampling of 1000 holograms stored in a manually Bragg-tuned system [51]. Using various encoding techniques, we have implemented what to the best of our knowledge is the first fully automated system in which data are written and recalled in digital form. We have used a spatiorotational multiplexing system to store digital representations of color images and compressed video. Critical features of our implementation that allowed us to overcome previous obstacles include location of the Fourier plane outside the crystal volume, use of cylindrical lenses to implement spatial multiplexing, a novel differential encoding technique used to increase error immunity, use of Hamming error-correction codes, and the distribution of consecutive information bits over multiple data pages in order to decrease the probability of burst errors. With these techniques, bit-error rates of 10−6 have been achieved at readout rates of 6.3 × 106 pixels per second. We have used the system to evaluate tradeoffs between bit-error rate and storage capacity at a fixed data-transfer rate.

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A generic diagram of the Stanford system is shown in Figure 2.2 The storage medium is an Fe-doped LiNbO3 crystal cut such that its c-axis is at 45 degrees to the crystal surfaces. The crystal dimensions are 2 cm × 1 cm × 1 cm, of which only a 0.1-cm3 portion is used for data storage. The crystal is mounted on a computer-controlled rotation stage, which is in turn mounted on a computercontrolled translation stage, allowing the crystal to be moved in the vertical direction. The SLM is a 480 × 440 pixel liquid crystal array taken from an InFocus TVT-6000 video projector. It is addressed with an analog video signal produced by a framegrabber board in the computer. The camera is an intensified CCD array. Each stored hologram is read out in 1/30 second. The cylindrical lenses in the reference beam path are used to collimate the reference to an area of approximately 10 mm × 2 mm at the face of the crystal. The combination of the translation stage and the collimation of the reference beam allows holograms to be stored in four different stacks. The signal beam occupies an area of approximately 1.5 mm × 1.5 mm on the front face of the crystal. The Fourier plane is located approximately 3 mm in front of the crystal. A filter is used to select only the central spot of this transform. The filter prevents erasure of previously recorded stacks during writing and eases alignment tolerances at the CCD plane. By recording in a Fresnel plane, rather than the Fourier plane, the modulation depth is sufficiently uniform to eliminate the need for a diffuser [62]. Because previously recorded holograms were erased in this demonstration as additional holograms were recorded, an appropriate recording schedule had to be determined in order to store a large number of pages with equal diffraction efficiency [6]. If all holograms result in equal diffraction efficiencies, a threshold intensity can be chosen to distinguish ON pixels from OFF pixels. Note that we use OFF and ON when referring to SLM or CCD pixels and 0 and 1 when referring to information bits. An incorrect measurement of the time constant can result in a schedule that leads to unequal diffraction efficiencies for each of the stored holograms. A 10 percent error in the assumed time constant can result in more than a factor of 2 variation in diffraction efficiency. In addition, laser fluctuations or anomalous writing behavior can result in unequal diffraction efficiencies. In order to circumvent these problems, we used a differential encoding technique, in which the pixel sequence OFF-ON is written to the SLM to represent a 0 and the pixel sequence ON-OFF is written to represent a 1. In addition to being insensitive to page-to-page intensity fluctuations, the differential encoding technique results in a lower probability of error, assuming a model of additive white noise that is independent from pixel to pixel. For example, the probability of an error given that a 0 is transmitted in a threshold detection system is given by [63, 64]  p≈

∞

aT

 2   r + aL2 r r aL  exp − dr, I 0 σ2 2σ 2 σ2

(2.34)

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where aH and aL are the average amplitudes of ON and OFF pixels, respectively, at the CCD array, σ is the standard deviation of the noise, I0 is the zero-order modified Bessel function of the first kind, and the thresold aT is set to be the mean of the ON and OFF values. The probability of an error given that a 0 is transmitted in a differential detection system is given by    2  ∞ x x aL  x + aL2 dr I exp − p≈ 0 2 2σ 2 σ2 aT σ  2    x r + aH2 r r aH  exp − × dxdr (2.35) I 0 2 2σ 2 σ2 0 σ These equations show that significant improvements can be made using differential encoding over simple thresholding schemes (Heanue, Bashaw, Hesselink, 1994). Another advantage to the differential encoding technique is that it can be used when single pages reconstruct nonuniformly. This nonuniformity can result from poor overlap of signal and reference beams in the crystal, nonuniform illumination of the SLM, or from the introduction of interference fringes due to multiple reflections in the optical system. When a substantial contribution to the variance is due to such large-scale nonuniformities, we expect difference encoding to offer further improvement over simple thresholding techniques. This is indeed the case. Ideally, each pixel on the SLM would be used to represent a single bit of data. The liquid-crystal array used in these experiments exhibited some interpixel crosstalk. In other words, pixels adjacent to an ON (transmitting) pixel exhibit a significant amount of transmission. Also, the SLM and the CCD array are manufactured with different horizontal-to-vertical pixel pitch ratios. Thus, using a simple optical system such as ours, it is impossible to image the SLM onto the CCD array such that each SLM pixel is imaged directly onto a CCD pixel. In practice, a multiple-lens arrangement could be used to achieve the proper anamorphic imaging; however, this greatly increases system complexity. The pitch mismatch results in a systematic geometrical source of errors if too fine a sampling grid is used. In order to circumvent these problems, we used a block of 8 × 8 pixels to represent one bit. The differential encoding technique increases the pixel-to-bit ratio to 8 × 16 pixels per information bit. The data is read out by sampling the CCD output at one pixel for each 8 × 8 pixel block imaged onto it. The difference in intensity between adjacent samples is used to determine whether the information bit is a 0 or a 1. We measured the raw bit error rate of our system to be between 10−3 and 10−4 at video readout rates. In order to improve performance, we implemented a Hamming error-correcting code in which 4 check bits are added to each string of 8 data bits. The Hamming code is capable of correcting a single bit error in the sequence of 12 bits assuming that only one bit error occurs in those 12. In order to reduce the potential of burst errors, each stored data page represented one bit

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plane in a two-byte sequence. Thus, each bit in the 12-bit sequences was stored in a separate page in order to reduce the probability of multiple errors within a given sequence. With the error-correcting codes, we were able to achieve bit error rates of 10−6 , which is adequate for compressed video storage and more than sufficient for uncompressed image storage. We used our system to store both digital color image and low-resolution video data. The total capacity of the system was 1232 pages with 1592 bits per page, resulting in a raw capacity of 245 kB. Taking into account the necessary error-correcting bits, the total useful data capacity is 163 kB. The total pixel capacity of our system is 2.6 108 pixels, with a density of 3 × 109 pixels per cm3 . The transfer rate is 6.3 × 106 pixels per second. The CCD camera outputs one byte per pixel. The total pixel capacity of our system indicates that our storage capacity does not represent a fundamental limit. Rather, the primary limitation in determining the information storage capacity is the necessary oversampling on the SLM. With a higher-quality SLM designed in conjunction with the CCD array, we anticipate that the information capacity and data-transfer rates can be enhanced by several orders of magnitude.

2.9.2 The Siros Fully Automated Video Demonstration (1995) The Stanford demonstration showed that digital encoding techniques could be used to great advantage to improve raw BER, and in combination with errorcorrecting codes, digital images could be retrieved with low error rates. The capacity of this demonstration, however, was rather limited. The next step in the development of digital holographic data-storage systems accomplished storage of compressed high-resolution video data, requiring substantially larger data-storage capacity, and a good bit error rate of about 10−8 . Primarily due to the efforts of John Heanue, Andy Daiber, Ray Snyder, and Jim Colvin at Siros Technologies, the first video movie was stored and retrieved by recording approximately 5 MB of compressed video data. The optical arrangement was similar to the Stanford demonstration, except that the SLM was a Texas Instruments deformable mirror device, with a resolution of 640 × 480 pixels. This device overcame all problems of the previously used liquid-crystal devices. Both video imagery and sound were recorded in LiNbO3 with the MPEG standard using the system of Figure 2.20. Data were encoded using 6:8 channel codes and a Reed Solomon 15:13 error-correction scheme that corrects one byte error per code word. This system was shown on the Discovery Channel in 1995 as part of the Next Step program. In contrast to the Science demonstration, in this experiment the information was permanently stored using ionic fixing. After recording of all data stacks, the crystal was removed from the setup and placed in an oven. It was heated to approximately 120◦ C for 15 minutes, and then slowly cooled down to room temperature. Caused by hydrogen impurities in the crystal, the thermal fixing efficiency was rather high, approximately 50% (i.e., the diffraction efficiency after thermal fixing was approximately 50% of that before the fixing procedure) [42]. Data retrieval in this experiment was implemented in software.

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Illumination Optics Galvanometer Optional Phase mask PM Lens

SLM

Storage medium

Relay Lens

FT Lens

FT Lens

CCD

F I G U R E 2.20. Generic 90-degree recording geometry.

2.9.3 The Siros Fully Automated System with Electronic Readout at Video Rates (PRISM, 1996) All previous demonstrations were under software control with off-line data retrieval. Although it is well recognized that software procedures can often be implemented in electronic hardware, there are many practical issues that make such an implementation far from trivial. At Siros, primarily through the efforts of Bob Okas and Ray Snyder in cooperation with GTE as part of the PRISM consortium effort, a fully electronic readout and control system was implemented in 1996. We believe that this was the first demonstration of a fully automated and electronically controlled system. The block diagram of the electronic layout is shown in Figure 2.21. The electronic architecture is based on a VME bus. The optical system architecture was the same as shown in Figure 2.20. Temperature fixing in LiNbO3 was implemented for nonvolatile readout, and a total capacity

CCD

Display

TCP Frame Grabber Decoding FPGA

Warp

68040 CPU

VME

F I G U R E 2.21. Electronic block diagram for the Siros real-time video readout system.

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Lambertus Hesselink and Sergei S. Orlov F I G U R E 2.22. The Siros smallprototype demonstration device incorporating a 800 × 600 DMD, a 1000 × 1000 SLM, and novel optical arrangement. Capacity is limited to less than 10 GB, with a maximum transfer rate of 10 MB/sec.

of 5 MB of video data was stored and retrieved at video rates. Channel 6:8 coding, ECC RS 13:15 coding, bit shuffling, and data warping were all implemented in electronic hardware, as well as overall system control [53]. A small compact demonstration system using LiNbO3 was built primarily by Mark McDonald at Siros to show that all components could be integrated into an approximately 5.25 form factor depicted in Figure 2.22. This system includes both recording and readout optics and components. Associative data retrieval capabilities were added to the Siros 90-degree demonstration platforms, as described in Hesselink 2004.

2.9.4 The Stanford University and Siros Fully Electronic Data Readout System Achieving 1 Gbit/sec (HDSS, 1999) In previous demonstrations at Stanford and Siros, the total system capacity was limited to tens of Mbytes, with transfer rates on the order of a few MB/sec, far from being competitive with current state-of-the-art optical and magnetic drives. The HDSS consortium was formed to develop the components and the systems to demonstrate 1 Gbit/sec transfer rate and a capacity of 125 GB on a 5.25 disk. This required the development of new components and electronics. A reflective 1000 × 1000 element ferroelectric liquid-crystal device was designed and built by IBM and Display Tech capable of recording 1000 frames per second. A pixelmatched detector array was developed by Kodak, having 1000 × 1000 pixels, and a frame rate of 1000 fps using 8-bit resolution. The output of this camera is divided into 64 digital channels. A pixel-matching very low distortion Fresnel lens was designed and manufactured by Rochester Photonics to achieve less than 0.3 pixel element distortion over the full one million pixels in the field of view. A novel random phase code reference beam was coaxially placed with the imaging lens. These components were assembled in the optical set up of Figure 2.20. In the first

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Alignment shutter Galvo

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PCI-

SLM VME

C7 CCD Camera

Decode x16 Mux 12 streams

ECC

F I G U R E 2.23. Electronic block diagram for the 1 Gbit/sec electronics.

high-speed demonstration, a 90-degree LiNbO3 arrangement was implemented using the electronic architecture of Figure 2.23. A 6:8 channel code and a ReedSolomon 136:156 ECC code for correcting up to ten-byte errors in the sequence are used for this demonstration. A team at Stanford University and Siros consisting primarily of Robert Okas, Sergei Orlov, Xiaochun Li, Eric Bjornson, Fedor Dimov, Bill Phillips, Ray Snyder, Darren Kwan, and Yuzuru Takashima developed the first implementation of a 1 Gbit/sec electronic readout system. Six data files were recorded, five JPEG images, and one counting sequence. The counting sequence was used so that the data could be easily inspected in numerical format. Each source file was divided into two pieces, giving a total of 12 data sequences, corresponding to the 12 output byte streams. Each recorded hologram stored data at a 55% efficiency due to 6:8 encoding, RS encoding, and masking. Thus, each image of 1,048,576 pixels corresponds to 576,000 bits. When this is divided into 12 streams it gives 6 Kbytes per image per output stream. Thus our source data files, each of approximately 48 KB (split into two 24-KB sections) required four images to store. The data were stored as four holograms and retrieved with an integration time of 600 microseconds, consistent with a 1-msec frame rate. Data passed through the entire system, and the 12 parallel byte streams were collected using the full electronics and checked for accuracy. The raw error rate was 8.7 × 10−5 [8].

2.9.5 IBM HDSS Materials and System Testers A research team at IBM Almaden in California has built three holographic digital data-storage test platforms [57]. The PRISM tester [23] shown in Figure 2.24 was designed and built for the purpose of having a stable platform for testing holographic data-storage materials. It was completed in 1996 and has provided an important means for testing a wide variety of materials ranging from photopolymers to inorganic photorefractive

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Fourier lens

Reference beam

θ

Object beam

2θ

F I G U R E 2.24. The primary features of the IBM PRISM holographic materials test apparatus. The SLM is a chrome-on-glass mask, and the detector array is a 1024 × 1024 pixels large CCD camera. A pair of precision rotation stages allow the reference beam to enter the storage medium under test at any horizontal incidence angle.

crystals. Two additional demonstration systems were also built at IBM, the DEMON I and II in 1997, shown in Figures 2.25 and 2.26. These platforms were used to study coding approaches [65] and to study the performance of holographic systems under various conditions . For example, IBM also stored videos similar to the earlier Siros demonstrations and achieved very 640 × 480 CCD LiNbO3 storage Fourier material lens Fourier lens 640 × 480 SLM

Five-element zoom lens Reference beam Object beam Intermediate image

Galvanometrically actuated scanner

F I G U R E 2.25. Salient features of the DEMON I holographic digital data-storage engine. A five-element zoom lens demagnifies the SLM image to an intermediate image plane, which is then imaged onto the CCD detector array with a pair of lenses. This tester was designed for use with LiNbO3 media.

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1024 × 1024 CCD LiNbO3 Shortstorage focalmaterial length Fourier lens

Fourier lens

Polarizing beam splitter

Scan lenses

Reference beam Galvanometrically actuated scanner 1024 × 1024 reflective SLM

Apodizer

Object beam

F I G U R E 2.26. Primary features of the DEMON II holographic digital data-storage engine. This system has demonstrated over 100 bits/μm2 .

high data storage densities in DEMON II using LiNbO3 as a recording medium [48]. Important optical approaches were developed and tested for achieving flat topintensity profiles for the image-bearing signal beam [66] as well as methods for reducing the central spike in the object beam at the Fourier transform plane [57]. Additionally, associative retrieval methods were developed that proved superior performance of holographic-based searches over traditional digital computer searches, as described above.

2.9.6 Rockwell Ultrashort-Access-Time-Testbed Ultrashort data-access times were achieved by a team of researchers at Rockwell International Research Center in Thousand Oaks, CA, as part of the HDSS program. Using acoustooptic beam deflectors for the reference beam, access times of less than 50 microseconds were demonstrated in LiNbO3 in the 90-degree configuration as shown in Figures 2.27 and 2.28 [54, 55]. Short access times combined with associative data retrieval could provide an interesting market opportunity, since other storage devices based on rotating disks have access times of a few milliseconds, and cannot contain as much data under the head. In particular, for searching video databases such capability could be valuable. Besides these milestone demonstration platforms using LiNbO3 as the recording medium, researchers at other laboratories around the world have been investigating holographic storage in polymer-based media that take on the shape of a compact disk, and utilize architectures akin to the one shown in Figure 2.3. For

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Light From Laser xxx

F I G U R E 2.27. Architecture for ultrashort-access-time demonstration.

Reference Arm Arguer xxx ACD

FrequencyCompressing ACD

OCD

Image Arm LOTV

Holographic Recording Medium

example, much pioneering work has been carried out at Caltech and Holoplex, as described earlier in this chapter, as well as at Aprilis, Optware, InPhase Technologies, and others. References to this work can be found in Hesselink 2004. In this chapter we close by discussing the HDSS demonstration system that was built at Stanford University aspart of the DARPA HDSS program. It provides the highest data-transfer-rate demonstration to date, as high as 10 Gbit/sec. Since that demonstration, Aprilis has also demonstrated storage density exceeding 100 bit/μm2 , but at a much reduced data transfer rate closer to 100 Mbit/sec.

2.9.7 The Stanford University and Siros 100-GB Capacity and 1 Gbit/sec Readout System Demonstrator (1999) For thick media like LiNbO3 in the 90-degree geometry, capacity is usually limited by media dynamic range and noise rather than by multiplexing crosstalk. For thin media, such as photopolymers, however, this is not the case, and the number of superimposed holograms in a spatial location is largely determined by the limited number of degrees of freedom available for multiplexing. Angular multiplexing

F I G U R E 2.28. Photograph of Rockwell short-testbed demonstration system.

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Holographic photopolymer disk media

High-speed decoding electronics

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Data out

spindle

lens

to Computer SLM

lens

C7 1000fps CCD

Computer

pulsed laser Sync. electronics

F I G U R E 2.29. Overall system layout for the HDSS WORM demonstration platform.

in the transmission geometry does not allow sufficient data storage density, and other multiplexing techniques such as shift and peristrophic multiplexing are required. Unfortunately, the displacements necessary for achieving low crosstalk in the shift-multiplexing technique are still too large and require the undesirable stop-and-go architecture for the rotating disk when a CW laser is used. To avoid this problem a very sensitive, relatively thick storage medium is required suitable for pulsed laser recording with nsec pulses. Recent developments in the DARPAfunded PRISM consortium have made great steps forward toward achieving such a medium for recording in the green region of the spectrum using a few hundred milliwatts of power. To achieve recording during constant rotation of the disk, a new multiplexing technique had to be implemented based on a phase-modulated reference beam. In such an architecture, performance can be readily analyzed and compared with more conventional technologies. In the system currently under test at Stanford University, densities of over 70 bits/μm2 are expected for a total system capacity of 125 GB and a transfer rate of 1 Gbit/sec using the electronics described above. The overall architecture for this system is shown in Figure 2.29. A 1024 × 1024 pixel-matched (12.8 μm square) 1000 fps Kodak C7 and IBM LC display are used as a detector and page composer respectively. Recording and readout are done using a pulsed doubled Nd:YAG laser (532-nm wavelength, 500-μJ/pulse, pulse width 25 nsec). A rotating holographic photopolymer disk is mounted on a precision air-bearing spindle. Typical rotation rate is 300 RPM. The angular addressing is done using a precision optical shaft encoder (16,384 counts per revolution). With appropriate synchronization electronics, the stability of the disk rotation speed and timing accuracy of the shaft encoder and laser pulses allow us to address different angular positions on the disk with overall positioning repeatability of better than ±0.1 μm. Different radial positions are addressed via moving the spindle with mounted disk using 25-nm resolution Newport PM600 translation stage. The HDSS WORM demonstration platform is shown in Figures 2.30 and 2.31.

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Reference Beam Diffuser SLM Diffuser

CCD FT Lens 1 Object Beam

FT Lens 2 Photo-polymer Disk Media

F I G U R E 2.30. Optical layout for the HDSS WORM demonstration platform.

The areal information density D (raw) for holographic disk storage can be approximated as 2  d × N, (2.36) D= ζ λF where d is the size of the SLM, λ is the wavelength of light, F is the focal length of the imaging optics, ζ is a dimensionless parameter that relates the actual hologram size to the Nyquist limit (typical values used in holographic storage systems are within the range from 1.1 to 1.6 depending on the image quality requirements), N is the number of holograms superimposed in the same storage location using different multiplexing techniques. The number of holograms that can be superimposed is limited by a variety of parameters such as required transfer rate (i.e., photon budget), material dynamic range (i.e., M/#), scatter, and information crosstalk. In order to achieve high storage density, it is therefore important to achieve high data density per hologram, which according to equation (2.36) requires a high NA for the imaging optics. The demonstration platform employs

F I G U R E 2.31. Photograph of the Stanford HDSS 1 Gbit/sec WORM demonstrator.

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F I G U R E 2.32. Sample 1024 × 1024 data page (η ≈ 2 × 10−3 , right) and enlarged 48 × 48 portion of it (left).

a custom-built (by Coastal Optical Systems, Inc.) short-focal-length (17.1 mm) optical system that allows 1-Mpixel images to be relayed from SLM to CCD array with distortions of less then ±1.5 μm over the entire SLM format (13.1 mm × 13.1 mm). The total numerical aperture of 0.75 is divided between the central high-resolution, low-distortion portion (NA = 0.36) used by the SLM and the outer area used by the reference light-generating optical elements. The spot size in the photopolymer medium is ≈2mm2 , which corresponds to raw areal density per hologram of ≈0.5 bits/μm2 . In current implementation, a speckle correlation-shift multiplexing technique is used for hologram multiplexing Figures 2.4., 2.30. A sample multiplexed hologram recorded in rotating disk media and retrieved with a single laser pulse at 1000 fps is shown in Figure 2.32. The raw byte error rate is < 2 × 10−3 , which provides error-free reconstruction of data with currently used 136:156 Reed-Solomon ECC.

2.9.8 The Stanford Ultrahigh >10 Gbits/sec Optical Data Rate Demonstration (2000) In the holographic disk system at 1 Gbit/sec, the hologram signal strength largely exceeds the camera noise. Thus, the holograms can be read at much higher optical data rates, provided that the holograms can be physically transported under the optical readout head at sufficient and sustainable rates. The latter depends on the recording density and the disk linear velocity. In later experiments we achieved sustained optical data rates as high as 10 Gbit/sec [56] by increasing the optical efficiency of the optical delivery system. The storage density in these experiments was approximately 10 bits/μm2 .

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Acknowledgments. This work was partially funded by the DARPA/NSIC/ Industry/University PRISM and HDSS Consortia. The authors acknowledge the technical contributions of and helpful discussions with the many researchers who participated in these consortia. One of us, L.H., wishes to acknowledge all the hard work done by the consortia members, present and past students, postdocs, and visitors to his group at Stanford University.

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3 The Transfer Function of Volume Holographic Optical Systems George Barbastathis Mechanical Engineering Department, Massachusetts Institute of Technology, 77 Massachusetts Ave., Room 3-461c, Cambridge, MA 02139 USA [email protected]

Abstract. We present a theoretical formulation for analysis and design of optical systems that utilize three—dimensional spatial heterodyning implemented by volume holograms of arbitrary shape. Basic volume holographic properties of angle, wavelength, depth selectivity and degeneracy are characterized and the impact of these properties in terms of imaging performance is discussed.

3.1 Introduction Volume holography has been one of the principal applications of the science of photorefractives. A volume hologram is essentially a three-dimensional (3D) grating, or a superposition of 3D gratings, in a “thick” holographic material. Volume holograms diffract in the Bragg regime, which means that (a) there is only one diffracted order, the +1st ; and (b) the nature of the diffracted field is strongly dependent on the probe field that illuminates the hologram, e.g., if the probe field wavelength or angle of incidence is different from the reference beam, even slightly, then the diffracted beam may become very weak or altogether absent. By contrast, “thin,” or Raman—Nath, holograms diffract into multiple orders, and generally diffract in response to any probe field. Because of its two aforementioned key properties, volume holography has been attractive for numerous coherent information processing [1] applications, including interconnects [2], data storage [3, 4], telecommunications [1], and imaging [5]. In interconnects, the Bragg selectivity is used to create large numbers of noninteracting “data paths” [6]. Similarly, Bragg selectivity in holographic memories enables the multiplexing of thousands of data pages. In optical communications, the dispersion properties of volume diffraction are used instead as filter banks for spectral multiplexers or demultiplexers. Our research group at MIT has pursued a different class of applications of volume holograms in optical imaging systems [7] as 3D Spatial Heterodyning (3DSH) elements for accurate wavefront detection. In this case, Bragg selectivity is designed to result in extreme positional or color sensitivity, which in turn enables imaging functions with 51

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demanding requirements, such as high-resolution profilometry [8] and 3D imaging at long working distances [9] with minimal scanning [10], hyperspectral imaging [11, 12], etc. 3DSH belongs to a class of “computational imaging” systems that simultaneously emphasize information extraction from the “raw” optical intensity (measured by a digital camera or detector array) and the transformation of the extracted information into maximum-utility representations via digital postprocessing. For example, we have used the pseudoinverse [13, 14], Viterbi [15], and Radon transform [16] algorithms to “restore” images obtained from 3DSH instruments implemented with volume holograms. This chapter is concerned with the analog or optical part of 3DSH computational imaging systems, in particular the response of optical systems with volume holograms as imaging elements. In Section 3.2 we describe the generic format of 3DSH/volume holographic imaging systems, and the various image acquisition modes. It then becomes evident that a uniform method of characterizing the imaging system, such as a point-spread function (PSF) or transfer function formulation is necessary. This derivation is carried out in Section 3.3. The result, expressed as a 3D Fourier transform constrained on a surface, is general within the limits of weak diffraction and paraxial approximation that are used to derive it. Examples are given in Section 3.4, selected with the intention of building intuition on the new formulation and also explaining the operation of volume holograms as imaging elements in 3DSH systems. The first example, in Section 3.1, is the Bragg-matched response of the volume hologram, which indicates some rather surprising properties that can be interpreted as field-dependent apodization and superresolution. The second example is the rederivation of well-known angle and wavelength selectivity properties of holograms in Section 3.2. The intent is to validate the new formulation, and also to indicate that the detail of the diffracted field can be quite dependent on the 3D shape of the hologram. The final example is the calculation of the volume holographic response to defocus, i.e., displacement of the probe source in the longitudinal direction, which is presented in Section 3.3. The depth selectivity property of volume holograms has been the key for many of our “optical slicing” 3DSH experiments, and it follows rather simply from the formulation. The results are summarized in Section 3.5, where suggestions for future work are also brought up.

3.2 3D Spatial Heterodyning with Volume Holograms The principle of 3D spatial heterodyning is shown in Figure 3.1. The incident or “probe” beam is mixed with a local oscillator signal represented by the volume hologram. The mixing process is not a product, but rather the phenomenon of volume diffraction itself, which includes multiplication, phase conjugation, and low-pass filtering (spatial integration), as can be seen from (3.4) in Section 3.3 ahead. The reason we refer to this process as 3D spatial heterodyning is because

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probe wavefront

volume hologram

53

lens

detector probe field

diffracted field

“local oscillator” wavefront

F I G U R E 3.1. The principle of 3D spatial heterodyning with a volume hologram.

mixing happens along the propagation direction. The 3D feature increases the sensitivity, as we will see shortly. The outcome of the 3DSH process is carried by the field diffracted by the volume hologram. Typically it is convenient to spatially Fourier transform the diffracted field and observe its intensity on a digital camera or measure the spatially integrated intensity on a photoelectric detector. We now describe how the 3DSH method can be used for imaging. The first step is to construct the mixing element, i.e., the volume hologram. The typical recording process is a sequence of exposures of a photosensitive thick holographic medium, such as a photorefractive crystal or photopolymer. Each exposure is the result of interfering two mutually coherent beams inside the holographic medium. Although the holographic recording process is the most convenient, it is not the only one. Alternatives such as 3D interferometric lithography, layered nanofabrication, and the Nanostructured OrigamiTM 3D fabrication and assembly process are likely to become practical in the future as the field of 3D nanomanufacturing advances. One advantage of these techniques is that they can be used to fabricate high-index contrast 3D structures. In this chapter we restrict the discussion to low-index contrast, which is more likely to result from holographic processes. Once recorded or fabricated, the 3DSH/volume holographic element is used as shown in Figure 3.1. Light emitted or scattered by the object of interest is the probe field, with the possible addition of “objective optics” between the object and the volume hologram. The role of objective optics is to shape the probe field to a more convenient form, which can yield advantages such as long working distance and smaller physical size of the elements [8]. It is important to emphasize that the 3DSH imaging process does not involve the recording of a hologram of the object; the role of the hologram is simply that of an optical element in the train, like a “superlens.” The result of spatially heterodyning is to form projections of the object on the camera. For example, a useful form of projection that we have used extensively

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invisible

invisible

focal plane

h1 h1

visible

focal plane

h2 h2

F I G U R E 3.2. Optical slicing with 3DSH. Left: focus at height h 1 above ground. Right: focus at height h 2 above ground.

in our 3DSH experiments is “optical slicing,” which is described in Figure 3.2. The 3DSH element in this case can be particularly simple, such as a volume hologram recorded as the interference between two plane waves or a plane wave and a spherical wave. The 3DSH element and the objective optics are arranged so as to define a “focal plane.” Unlike traditional optical systems, however, in the 3DSH system only a slit-shaped portion of the object that intersects the focal plane is visible; the remainder of the object is invisible. In the simple object depicted in Figure 3.2, setting the focal plane at distance h 1 from the ground plane would result in a slice of the tallest surface becoming visible, as indicated on the left-hand side. Setting the focal plane at h 2 instead makes a slice of the middle-height object visible. The optical slicing principle was first implemented with a confocal microscope. In both cases, the system rejects light originating away from the focal plane. At the same time, the field of view is limited to a point (confocal microscope with pinhole at detector plane) or a slice (3DSH and confocal microscope with slit-shaped pinhole.) The rest of the object is recovered by scanning in three and two dimensions, respectively. In the confocal microscope, extensive scanning is the only option to recover the object in its entirety. 3DSH provides alternative options that reduce or eliminate scanning altogether: (i) Multiplex method [11], where the 3DSH element is multiplexed so that multiple slices from the object are simultaneously imaged on nonoverlapping segments of a large camera. This unusual imaging mode is illustrated in Figure 3.3 and was implemented in collaboration with the optics group at Caltech [11, 12]. Provided that the dynamic range of the hologram and the photon count are sufficiently high, and that the camera has enough pixels available, the entire 3D object shape can be mapped onto the 2D camera without scanning. (ii) Rainbow method [17], where the object is illuminated with a rainbow as shown in Figure 3.4. In this case each color acts as its own slit and forms independently a depth-selective image on the camera. Therefore, scanning in the depth direction only is required to recover the object in its entirety.

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object surface object space (voxels)

camera space (pixels)

F I G U R E 3.3. Real-time 3D imaging in the multiplex 3DSH mode. The volume holographic element defines the map between object voxels and camera pixels according to the dotted arrows in the figure. Each camera pixel appears bright if its corresponding voxel intersects with the object (i.e., if it belongs to the object’s surface in the case of reflective objects, as shown here, or if a source exists in that voxel in the case of self-luminous or fluorescent objects, as in [11, 12]).

The implementation details of these methods are beyond the scope of this work. Instead, in the next section we lay out the fundamental theory that can be used to understand the model of the operation of volume holograms as 3DSH elements for imaging.

red camera

green focal plane

focal plane

h2

h1 h1

green

red camera

h2

F I G U R E 3.4. Image formation with the rainbow 3DSH mode. All color strips are imaged simultaneously, but each is visible if it is reflected by a part of the object surface that is in focus.

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probe plane

x

lens

volume x′′ hologram

image x′ plane

lens

z′′ y

y′

y′′

f1

f1

f2

f2

F I G U R E 3.5. Geometry of the 4-F optical system (telescope) with volume holographic aperture filter.

3.3 Derivation of the Optical Response of Volume Holograms Figure 3.5 shows the geometry that we will be considering in this paper. It consists of a 4F system (telescope) formed by two ideal thin lenses L1, L2 of focal lengths f 1 and f 2 , respectively. At the front focal plane of L1, the light distribution is assumed to be monochromatic and spatially coherent, with wavelength λ and amplitude p(x, y). Our goal is to determine the light amplitude distribution q(x  , y  ) at the back focal plane of L2. This system essentially conforms with the 3DSH paradigm of Figure 3.1 with the addition of a Fourier transforming lens as objective. If a thin transparency (amplitude or phase) is placed at the shared focal plane of the lenses in Figure 3.5, the answer is well known: the transparency (also known as “pupil function”) acts as a Fourier-plane filter. The significant difference considered here is that the Fourier filter is three-dimensional (3D). As a result, the filter behavior changes qualitatively: for example, we will see that the shift-invariance property is destroyed. In return, due to the 3D nature of the amplitude modulation, we obtain more degrees of freedom in determining the filter response. Let ε(x  , y  , z  ) denote the 3D modulation of dielectric index in the vicinity of the shared focal plane. If the modulation has a spatial carrier, i.e., it can be written in the analytic form ε(x  , y  , z  ) = εbb (x  , y  , z  ) × exp {iK · r },

(3.1)

where r ≡ (x  , y  , z  ) is the Cartesian coordinate vector, K = (K x , K y , K z ) is a grating vector, and εbb (x  , y  , z  ) is a baseband (low-frequency) modulation, then we will call ε (x  , y  , z  ) a “volume hologram.” Herein we will be considering volume holographic types of modulations only.

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The field generated from the propagation of p(x, y) through L1 to the vicinity of the shared focal plane is given by      x x  + yy  z     P(x , y , z ) = exp i2π p(x, y) exp −i2π λ λ f1  2 2   (x + y )z × exp −iπ d x d y. (3.2) λ f 12 Here and in the rest of this paper we will be neglecting amplitude terms of the form 1/iλz from the expressions for spherical waves. These terms contribute little to the diffraction integrals, in comparison with the fast-varying exponentials. It is clear that P(x  , y  , 0) reduces to the 2D Fourier transform of p(x, y). We also define the new function g (x  , y  , z  ) = P(x  , y  , z  ) × ε (x  , y  , z  ).

(3.3)

One can think of g(x  , y  , z  ) as the radiation field emitted by an elemental point scatterer inside the volume hologram in response to the Fourier-transformed probe field. We will now show that the output field is     x 2 + y 2 x y 1 √   1− , (3.4) , , q(x , y ) = ηG λ f2 λ f 2 λ 2 f 22 where η  1 is the diffraction efficiency of the volume hologram, and G(u, v, w) is the 3D spatial Fourier transform of g(x  , y  , z  ). From this expression we have deliberately √ neglected the undiffracted portion of the input field itself, which is simply 1 − η × p(− f 2 x /f 1 , − f 2 y /f 1 ). From now on, we will also neglect √ proportionality constants such as η from the field expressions and arbitrarily normalize the maximum (Bragg-matched) diffracted intensity to one. The necessary normalization factors will be implicit in this section’s formulas, and will be explicitly calculated in subsequent sections. We begin by calculating the diffracted field at observation coordinates (xb , yb , z b ) in the vicinity of the hologram. Under the first-order Born and paraxial approximations, the diffracted field is    z b − z  g(x  , y  , z  ) exp i2π e (x b , yb , z b ) = λ    2 (xb − x ) + (yb − y  )2 × exp iπ dx  dy  dz  (3.5) λ(z b − z  ) The most natural selection for zb in the above equation is z b = f 2 , i.e., to the left of and immediately adjacent to L2. This selection, as long as the hologram is confined near the shared focal plane, as shown in Figure 3.5, ensures that the paraxial approximation used in (3.5) for the free-space Green function remains valid. We have also avoided the potential singularity that would occur if z b were allowed to be inside the volume hologram. It should be noted that (3.5) is the basis

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of the 3D spatial heterodyning function of the volume hologram, as we noted in Section 3.2. The system output is the field generated by L2 at its back focal plane when the illumination immediately to the left of L2 is e (xb , yb , f 2 ) according to (3.5). From basic Fourier optics we therefore obtain      x  x b + y  yb x 2 + y 2 q(x  , y  ) = exp iπ exp −i2π λ f2 λ f2 × e (xb , yb , f 2 ) dxb dyb . (3.6) Combining integrals (3.5) and (3.6) and performing the xb , yb integrations first, the result is    x  x  + y  y       g(x , y , z ) exp i2π q(x , y ) = λ f2     x 2 + y 2 z  dx  xy  dz  , × exp −i2π 1 − (3.7) λ 2 f 22 which is consistent with (3.4). Result (3.4) is general, within the limits of validity of our approximations. As with any linear system, of particular interest is the special case p(x, y) = δ(x − x0 ) δ(y − y0 ), which gives us the impulse response h(x  , y  ; x 0 , y0 ) of the system. By direct substitution to (3.2) and (3.4) we obtain      1 y0 1 x0 x y   h(x , y ; x 0 , y0 ) = E , , + + λ f1 f2 λ f1 f2   1 x 02 + y02 x 2 + y 2 , (3.8) − λ 2 f 12 2 f 22 where E (u, v, w) denotes the 3D spatial Fourier transform of the dielectric modulation ε(x  , y  , z  ). Therefore, the system input—output relation can also be written as    p(x, y)h(x  , y  ; x, y) dx dy. (3.9) q(x , y ) = It is evident from the impulse response (3.8) that the 4F system with a volume holographic pupil filter is not shift invariant. It should also be noted that (3.4) is computationally more efficient than (3.9) because the fast Fourier transform algorithm, albeit 3D, is still faster than 2D space-variant superposition.

3.4 Examples 3.4.1 Bragg-Matched Reconstruction We now turn to the first specific use of the general formulation described above. We will calculate the response of the volume hologram under “Bragg-matched”

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probing, which is defined as a probe field that exactly phase-conjugates the reference beam used to record the hologram. Consider a signal hologram recorded as the interference of two mutually coherent reference and fields pf (x, y) and qs (x, y), respectively, formed at the input plane during the recording phase of the hologram. In the vicinity of the Fourier plane, the fields are transformed as     Pf (x , y , z ) = dx dy pf (x, y)     (x 2 + y 2 ) z  x  x + y  y × exp −iπ exp −i2π , (3.10) λ f1 λ f 12  Q s (x  , y  , z  ) = dx dy qs (x, y)     x  x + y  y (x 2 + y 2 ) z  exp −i2π . (3.11) × exp −iπ λ f1 λ f 12 As a result of the exposure, the refractive index of the holographic material is modulated according to |Pf + Q s |2 = |Pf |2 + |Q s |2 + Pf∗ Q s + Pf Q s ∗ .

(3.12)

Only the third term is relevant for Bragg-matched reconstruction. It can be shown that the remaining three terms are Bragg-mismatched, so we will neglect them from now on. Therefore, we write the dielectric modulation as ε(x  , y  , z  ) = Pf ∗ (x  , y  , z  ) Q s (x  , y  , z  ) s (x  , y  , z  ), 



(3.13)



where the “hologram shape” function s(x , y , z ) is defined as  1, if r is inside the hologram,    s(x , y , z ) = 0, otherwise.

(3.14)

In the narrowest definition, Bragg-matching is obtained if the hologram is probed with a field identical to the reference beam, i.e., p(x, y) = pf (x, y). Substituting into (3.9) and carrying out the integrations, we find that  1 dx dy qs (x, y) q(x  , y  ) = V         1 x 1 y 1 x 2 + y 2 x y x 2 + y2 ×S , , , + + − λ f2 f1 λ f2 f1 λ 2 f 22 2 f 12 (3.15) where the normalization factor V equals the hologram volume  V = s(x  , y  , z  ) dx  dy  dz  .

(3.16)

Result (3.15) confirms the action of the volume hologram as a shift-variant

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filter because of its 3D nature, namely, the third argument of the superposition kernel S(., ., .). It is evident that a thin transparency or diffractive element would have a shape function of the form s(x  , y  , z  ) = s⊥ (x  , y  )δ(z  ), and so its Bragg-matched response would reduce to the well-known result        1 x 1 y 1 x y , , + + dx dy qs (x, y) S⊥ q(x  , y  ) = V⊥ λ f2 f1 λ f2 f1 (3.17) where V⊥ denotes the hologram aperture  s⊥ (x  , y  ) dx  dy  . V⊥ =

(3.18)

Returning to the 3D result expressed by (3.15), we define x − x , 2 y − y yˆ = , 2

xˆ =

(3.19) (3.20)

and without loss of generality, we assume that the system has unit magnification, i.e., f1 = f 2 ≡ f . We can then rewrite the superposition integral (3.15) as  1   q(x , y ) = dx dy qs (x, y) V    ˆ  + x) + yˆ (y  + y) x + x y  + y x(x ×S , , . (3.21) λf λf λf 2 We now assume that the superposition kernel assumes significant values only in the vicinity of the Gaussian image (x  , y  ) = −(x, y). Then, the variation of the superposition kernel within that region is primarily due to arguments x  + x and y  + y, rather than xˆ and yˆ . If we approximate xˆ and yˆ as constant over the region of integration, then indeed (3.21) looks like a convolution integral with position-variant PSF   1 x y xˆ x + yˆ y h(x, y) = S , , . (3.22) V λf λf λf 2 The assumptions that led us to this formulation are very similar to the familiar practice of quoting a field-dependent modulation transfer function (MTF) in optical engineering analysis. In the same spirit, we rewrite (3.21) in the spatial frequency domain (u, v) as ˆ yˆ ). Q(u, v) ≈ Q s (u, v) H (u, v; x,

(3.23)

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ˆ yˆ ) is the field-dependent amplitude-transfer function The function H (u, v; x, (ATF), given by    x y xˆ x + yˆ y 1 ˆ yˆ ) = F(x,y) S , , , (3.24) H (u, v; x, V λf λf λf 2 where F(x,y) denotes Fourier transformation with respect to (x, y). A straightforward calculation yields    xˆ z  yˆ z    H (u, v; v, ˆ yˆ ) = dz s −λ f u − , −λ f v − ,z . (3.25) f f ˆ yˆ ) is a projection of the hologram shape function through Therefore, H (u, v; x, the straight line defined by equations xˆ z  , f yˆ z  y  = −λ f v − , f

x  = −λ f u −

(3.26) (3.27)

in the 3D hologram space. To interpret result (3.25), we consider the specific case of a rectangular hologram, which we further simplify by neglecting one lateral dimension. Thus, we take       z x   rect , (3.28) s(x , z ) = rect a L where a is the hologram aperture and L is the hologram thickness. The projection lines (3.26–3.27) for this case are illustrated graphically in Figure 3.6 for two spatial frequencies u 1 , u 2 , and several field locations xˆ1 through xˆ5 . The field-dependent ATF can be calculated explicitly from this picture, or by direct volume hologram

probe x plane ^x1

lens

x′′

H(u1 ; x^5) H(u1 ; x^1)

image plane

lens λf u1 λf u2

x′

z′′

H(u2 ; x^1)

^x5

H(u2 ; x^5) f

f

a L

f

f

F I G U R E 3.6. Illustration of the spatial frequency projections through the hologram, as described by (3.28)–(3.29). This picture also explains the shift variance of the volume holographic optical system in Bragg-matched condition.

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substitution into (3.25). For small fields ⎧ 0, ⎪ ⎪  ⎪ f a 1 ⎪ ⎪ ⎪ + + λf u , ⎪ ⎨2 ˆ 2 L|x| ˆ ≈ 1, H (u; x)  ⎪ ⎪ f a 1 ⎪ ⎪ + + λ f u , ⎪ ⎪ ˆ 2 L|x| ⎪ ⎩2 0,

ˆ < a f we obtain L|x| if u < −u c (1 + α); if − u c (1 + α) < u < −u c (1−α); if −u c (1 − α) < u < u c (1 − α); if − u c (1 − α) < u < u c (1 + α); if u > u c (1 + α), (3.29)

where we used uc =

a 2λ f

(3.30)

for the cutoff frequency of the equivalent thin clear aperture, and α=

ˆ L|x| <1 af

(3.31)

for the correction to the band edge due to the hologram thickness L. We will refer to α as the “field-dependent apodization parameter” for reasons that will become obvious shortly. For large fields α > 1 we obtain ⎧ 0, if u < −u c (1 + α); ⎪ ⎪  ⎪ 1 f a ⎪ ⎪ ⎪ + λ f u , if − u c (1 + α) < u < u c (1 − α); ⎪ + ˆ ⎪ 2 L|x| 2 ⎪ ⎪ ⎪ ⎨ af , if u c (1 − α) < u < −u c (1 − α); ˆ H (u; x) ≈ L|x| ˆ ⎪ ⎪  ⎪ f a ⎪1 ⎪ ⎪ + − λ f u , if − u c (1 − α) < u < u c (1 + α); ⎪ ⎪ ˆ 2 2 L|x| ⎪ ⎪ ⎪ ⎩ 0, if u > u c (1 + α). (3.32) Finally, on-axis equation (3.25) reduces to

 u . H (u; 0) ≈ rect 2u c 

(3.33)

The effect of the third dimension is seen in Figures 3.7 and 3.8. The Braggmatched volume hologram acts as an apodizer that smoothens and reduces the PSF side lobes off-axis. For α > 1 we also observe significant narrowing of the main lobe of the PSF, which is attributed to the considerable broadening of the spatial bandwidth due to the long-haul projections at these points (see Figure 3.6.) This “superresolution” phenomenon has never been reported before to the author’s knowledge.

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H 1 u −uc uc −uc (1+ α) −u uc (1+ α) (1− α) u (1− α) c c (i) H

1/α u uc −uc uc (1− α) −uc (1− α)

−uc (1+ α)

uc (1+ α)

(ii) F I G U R E 3.7. Shift-variant transfer function of Bragg-matched hologram for (i) α < 1; (ii) α > 1. 1 xhat =0 xhat =500λ xhat =1,500λ

0.9 0.8

q(x′) [a.u.]

0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 −10

5

0 x′−xhat [λ]

5

10

F I G U R E 3.8. Shift-variant PSF for a = 103 λ, f = 2.5 × 103 λ, L = 2 × 103 λ, and three ˆ different values of x.

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3.4.2 Angular and Wavelength Detuning We now consider the case of a volume hologram recorded in the system of Figure 3.5 with two mutually coherent reference and signal point sources of wavelength λf located at xf and x s on the input plane. Using the paraxial approximation, it is easy to see that the volume hologram recorded by the two resulting plane waves is then expressed as    x2 − x2 2π xs − xf −x  s(x  , y  , z  ). (3.34) ε(x  , y  , z  ) = exp i + z  f 2 s λf f1 2 f1 We probe this hologram with a point source of wavelength λ located at x on the input plane. For later convenience, we define the normalized wavelength μ=

λ λf

(3.35)

and the lateral magnification of the 4F system m=

f2 . f1

(3.36)

Substituting in (3.3) and Fourier-ransforming, we obtain    x s2 − xf2 + x 2 /μ xs − xf + x/μ 1 − G(u, v, w) = S u − . , v, w − λf f 1 λ 2λf f 12 (3.37) According to (3.4) we obtain the hologram response   (x + μmx s ) + m(x − μxf ) y  √ , , q(x  , y  ) = ηS λ f2 λ f2  (μm 2 xs2 − x 2 ) + m 2 (x 2 − μx f2 ) . 2λ f 22

(3.38)

To interpret (3.38) we first locate the Gaussian image of the probe point source by setting the first two arguments equal to zero: x  = −μmxs − m(x − μxf ), y  = 0.

(3.39) (3.40)

Substituting into (3.38), we find that the field qG at the Gaussian image is   μ(1 − μ)xs2 − μ(1 + μ) xf2 + 2μ (μxs x f − x x s + x x f ) √ qG = ηS 0, 0, . 2λ f 12 (3.41) Since s(x  , y  , z  ) is an index function, its Fourier transform has the property |S (u, v, w)| ≤ |S(0, 0, 0)|

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for |u| + |v| + |w| > 0. It follows that the third argument in (3.41) above expresses Bragg mismatch, in other words, the decrease in diffracted power when xp = xf . We can see this more clearly by examining a few special cases: (a) Angle selectivity of slab-shaped hologram. We select the probe wavelength to equal the recording wavelength, i.e., μ = 1, and set x = xf + x. We also select the hologram shape as          y z x    s (x , y , z ) = rect rect rect . (3.42) a b L Using the same normalization as in (3.15), we find that   (x s − x f )x , qG (x) = sinc λ f 12

(3.43)

which is consistent, in the paraxial approximation, with standard derivations that assume infinite hologram aperture. For example, the angular deviation corresponding to the first Bragg null is given by λf f 1 x = . f1 L xs − xf At arbitrary locations (x  , y  ) in the output plane, the field is given by      y (x  + mx s ) + m(x − xf )   sinc b q(x , y ) = sinc a λ f2 λ f2  2 2 2 2 2 2  (m x s − x ) + m (x − x f ) × sinc L . 2λ f 22

(3.44)

(3.45)

It is evident from (3.45) that the yˆ dimension is completely uncoupled from the Bragg-mismatch effect in this geometry. This is sometimes referred to as out-ofplane Bragg degeneracy.” Figure 3.9 shows the Bragg-matched response plotted against the probe location x and output coordinate x  . As the probe is displaced away from its matched position x = x f , the Gaussian image is displaced in the opposite direction and is attenuated because of Bragg mismatch. The main lobe of the diffracted pattern remains approximately unchanged, but the side lobes can be seen to depend significantly on the amount of Bragg mismatch. This phenomenon has been typically neglected in interpage crosstalk calculations, which have instead approximated the diffraction pattern without taking into account the 3D nature of the pupil function. (b) Wavelength selectivity of slab-shaped hologram. We select the probe location to be identical to the recording reference source, i.e., x = xf , and the probe wavelength mismatch as μ = 1 + μ with |μ|  1. The Gaussian image location can then be identified as x  = −m[xs + μ(x s − x f )], y  = 0.

(3.46) (3.47)

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George Barbastathis +100

x′ [λ]

+50

+0

−50

−100 +904

+952

+1000 x [λ]

+1048

+1096

F I G U R E 3.9. Bragg-mismatched PSF for angular detuning of a slab-shaped hologram with a = 500λ, L = 103 λ, xf = 103 λ, and a telescope with f 1 = f2 = 4 × 103 λ.

The Bragg-mismatched field at the Gaussian image location is  qG (μ) ≈ sinc

μ (xs − xf )2 L 2λf f 12

 .

(3.48)

Therefore, the wavelength selectivity is λf μ = 2 L



f1 xs − xf

2 .

(3.49)

At arbitrary locations (x  , y  ) in the output plane, the field in this case is given by      y (x  + xs (1 + μ)m) + mμx f sinc b q(x , y ) = sinc a λ f2 λ f2   ((1 + μ)m 2 x s2 − x 2 ) − m 2 μxf2 × sinc L . (3.50) 2λ f 22 



Figure 3.10 shows the Bragg-mismatched response plotted against the relative wavelength μ and output coordinate x  . The behavior of the diffracted field is similar to the case of angular detuning. (c) Wavelength degeneracy. We now consider the case μ = 1 and probe also displaced to a new location x = xf + x. The field at the Gaussian image

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+200

x′ [λ]

+100

+0

−100

−200 0.808

0.904

1.000 μ

1.096

1.192

F I G U R E 3.10. Bragg-mismatched PSF for wavelength detuning and optics as in Figure 3.9.

is then qG (μ, x) ≈ sinc



μ(1 − μ) m 2 (x s2 + xf2 ) + 2μ m 2 (x s − xf )x L 2λ f 22

 , (3.51)

from which we find that if the probe displacement satifies x =

(μ − 1)(xs2 + x f2 ) , 2(xf − xs )

(3.52)

then the diffracted power is maximized. This condition is known as “wavelength Bragg degeneracy,” and it has been used for nondestructive readout of holograms at wavelengths where the material sensitivity to photoexposure is very small. (d) Parallelepiped-shaped hologram. Very often the result of recording a volume hologram with limited-size beams is a parallelepiped shape, as shown in Figure 3.11. This is because the hologram is recorded only in the region where the signal and reference beams overlap. The question then arises, does this strange shape impact the performance of the volume hologram as imaging element? It can be seen that the hologram shape in this case can be written as          y z x − θf z     rect rect , (3.53) s(x , y , z ) = rect a B L where θf ≡ xf / f 1 in the paraxial approximation. Substitution into (3.4) and a straightforward integration yield   y   q(x , y ) = sinc (Aα) sinc B sinc (L(θf α + β)) , (3.54) λ f2

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x′′ θs

z′′

a

L

where we have used the shorthand (x  + μmxs ) + m(x − μxf ) , α= λ f2 (μm 2 x s2 − x 2 ) + m 2 (x 2 − μxf2 ) β= . 2λ f 22

(3.55) (3.56)

It is readily shown that the Gaussian image location and Bragg selectivities are identical to those derived earlier for the rectangular hologram. However, the point-spread function is quite different, as can be seen in Figures 3.12 and 3.13, +100

x′ [λ]

+50

+0

−50

−100 +904

+952

+1000 x [λ]

+1048

+1096

F I G U R E 3.12. Bragg-mismatched PSF for angular detuning of a parallelepipe-shaped hologram with all other parameters as in Figure 3.9.

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+200

x′ [λ]

+100

+0

−100

−200 0.808

0.904

1.000 μ

1.192

1.096

F I G U R E 3.13. Bragg-mismatched PSF for wavelength detuning of a parallelepipedshaped hologram with all other parameters as in Figure 3.9.

which describe the effect of angular and wavelength detuning, respectively. Notice in particular the change in side-lobe structure, which makes this situation unappealing for imaging applications.

3.4.3 Longitudinal Detuning (Defocus) Consider again a hologram recorded by two plane waves as in (3.34). In this section, we will consider a cylinder-shaped hologram with radius R and thicknes L, i.e.,  





s(x , y , z ) = circ

x 2 + y 2 R



 rect

z  L

 .

(3.57)

For further simplicity, we assume that the signal beam is on-axis, i.e., x s = 0. We are interested in the case in which the hologram is probed by a point source onaxis displaced by z 0 in the longitudinal direction. This is the case of a defocused probe beam, which is known to lead to the depth selectivity properties of volume holographic imaging (see Section 3.2). It is straightforward to derive the field in the vicinity of the hologram as     z 0 x 2 + y 2 z 0 + z  exp iπ . P(x , y , z ) = exp i2π λ λ z 0 z  + f 12 





(3.58)

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After substitution into (3.4) we arrive at the expression         z x  2 + y 2      q(x , y ) = dz rect dx dy circ L R     × exp iπ A(z  )(x 2 + y 2 ) − i2π Bx x  + B y y  + iπC z  , (3.59) where z0 , λ(z 0 + f 12 ) x xf + , Bx = λ f 2 λf f 1 y By = , λ f2 xf2 x 2 + y 2 C= − . λ f 22 λf f 12 A(z  ) =

z 

(3.60) (3.61) (3.62) (3.63)

The integral in (3.59) can be simplified [5] as       z    q(x , y ) = dz rect F 2π A(z  )R 2 , 2π B R exp {iπC z  }, (3.64) L  where B = Bx2 + B y2 and the function F(., .) is the field distribution near the focus of an ideal quadratic lens [8]. The last equation can be numerically integrated using the asymptotic expansions for F(., .) found in (8.8.17a–b), (8.8.20a–b) of [18]. The effect of defocus on the volume holographic optical system is seen in Figure 3.14 for three successive values of z 0 . For small defocus, the response is the familiar disk shape obtained from geometrical optics with a “blinking spot” in the center and ripples near the edge due to diffraction. As the defocus increases, so does the radius of the disk at the image plane, but simultaneously a second effect becomes evident: the disk is screened by a slitlike function. The slit was explained in [10] as a consequence of the Bragg selectivity of the hologram and was derived approximately using geometrical arguments. The derivation given here is more rigorous, since it takes diffraction effects fully into account within the paraxial approximation. The function of the hologram with defocused input can be explained with the help of Figure 3.15. The longitudinally displaced source produces a small disk Dp of spatially coherent illumination at the entrance plane of the optical system. The disk Dp can be thought of as the equivalent input to which the system is responding, in lieu of the true point source. Without the volume hologram, the telescope would have produced an image Di with radius equal to the radius of Dp multiplied by the lateral magnification m. However, if the defocus is sufficiently

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z0 =64λ (i)

100

y′ (λ)

50

0

−50

−100 −1100

−1050

−1000

−950

−900

−900

−800

x′ (λ) (ii)

200

y′ (λ)

100

0

−100

−200 −1200

−1100

−1000 x′ (λ)

F I G U R E 3.14. Defocused response for a slab-shaped hologram with a = 103 λ, L = 103 λ, x f = 103 λ, and a telescope with f 1 = f 2 = 4 × 103 λ. The values of defocus and maximum intensity are (i) z 0 = 64λ, Imax ≈ 10−2 ; (ii) z 0 = 200λ, Imax ≈ 10−3 ; (iii) z 0 = 400λ, Imax ≈ 5 × 10−4 . The Bragg-matched maximum intensity is normalized to 1.

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200

y′ (λ)

100

0

−100

−200 −1200

−1100

−1000 x′ (λ)

−900

−800

F I G U R E 3.14. (Continued)

large, then part of Dp is outside the Bragg range x that was derived in (3.44) of example (a) in Section 3.4.2 above. The Bragg-mismatched part of the disk is not imaged by the hologram; the remaining part that does get imaged looks like a slit oriented along the degenerate direction yˆ cut out of the disk Di . The effect of wavelength change on the defocused reconstruction is more complicated. Examples with mismatched wavelength are given in Figure 3.16. It can be seen that for sufficiently large defocus, the probe color can become Bragg matched

probe x plane 2Δ x probe D p source

lens

volume x′′ hologram

image x′ plane

lens

z′′ y

Bragg matched

Di y′

y′′

image

Bragg mismatched f1

f1

f2

F I G U R E 3.15. Schematic explanation of the defocused response.

f2

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y′ (λ)

50

0

−50

−100 −1300

(ii)

−1250

−1200 x′ (λ)

−1150

−1100

−1300

−1200 x′ (λ)

−1100

−1000

200

y′ (λ)

100

0

−100

−200 −1400

F I G U R E 3.16. Defocused response as in Figure 3.14 with additional wavelength detuning μ = 1.2 and (i) z 0 = 64λ, Imax ≈ 10−2 ; (ii) z 0 = 200λ, Imax ≈ 2 × 10−3 ; (iii) z 0 = 400λ, Imax ≈ 3 × 10−2 .

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200

y′ (λ)

100

0

−100

−200 −1400

−1300

−1200 x′ (λ)

−1100

−1000

F I G U R E 3.16. (Continued)

due to the color degeneracy effect described in example (c) of Section 3.4.2 above. A complete analysis of this effect is beyond the scope of this discussion, but it suffices to say that the effect is rather weak because only the Bragg-matched fraction of the defocused power is very small.

3.5 Conclusions and Discussion We have presented what is, to our knowledge, the first-ever analysis of a 3D spatial heterodyning imaging system with volume holographic pupil function. We derived the general-purpose formula (3.4), which can be used to compute the diffracted field of arbitrarily shaped holograms in response to arbitrary input fields. We applied the new formulation to rederive well-known angular, wavelength, and depth selectivity results. In the case of depth detuning in particular, we obtained a new expression that fully takes into account diffraction effects and also explains the ringing and wavelength degeneracies that have been observed in experimental implementations. Finally, a surprising effect of shift-variant response and superresolution in the Bragg-matched response was predicted. These observations support the argument that volume holograms are among the most powerful optical elements available to shape the optical response in

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computational imaging systems. This property is particularly promising in the context of 3D imaging, where volume holograms have already proven successful in applications demanding with respect to high resolution. long working distance, and real-time and hyperspectral operation. The possibilities are by no means exhausted. For example, one can use the formulation derived herein to design phase- or amplitude-apodized holograms, chirped holograms, etc. It would also be extremely fruitful to combine the results of the present essentially Fourierbased approach with the photorefractive model and jointly optimize the optical design and material response. Acknowledgments. The author is grateful to Arnab Sinha and Kehan Tian for useful comments and assistance with the simulations. This research was supported by the Air Force Office of Scientific Research, the National Science Foundation Center for Materials Science and Engineering at MIT, and the Army Research Office Institute for Soldier Nanotechnologies at MIT.

References 1. Psaltis, D. Coherent optical information systems. Science 298, 1359–1363 (2002). 2. Psaltis, D., Brady, D., Gu, X.G., and Lin, S. Holography in artificial neural networks. Nature 343(6256), 325–330 (1990). 3. Li, H.-Y.S. and Psaltis, D. Three dimensional holographic disks. Appl. Opt. 33(17), 3764–3774, June (1994). 4. Heanue, J.F., Bashaw, M.C., and Hesselink, L. Volume holographic storage and retrieval of digital data. Science 265(5173), 749–752 (1994). 5. Barbastathis, G. and Brady, D.J. Multidimensional tomographic imaging using volume holography. Proc. IEEE 87(12), 2098–2120 (1999). 6. Lee, H., Gu, X.-G., and Psaltis, D. Volume holographic interconnections with maximal capacity and minimal cross talk. J. Appl. Phys. 65(6), 2191–2194, March (1989). 7. Sinha, A., Liu, W., Psaltis, D., and Barbastathis, G. Imaging with volume holograms. Opt. Eng. 43(9), 1959–1972, (2004). 8. Sinha, A. and Barbastathis, G. Volume holographic imaging for surface metrology at long working distances. Opt. Express 11, 3202–3209 (2003). 9. Sinha, A. and Barbastathis, G. Volume holographic telescope. Opt. Lett. 27, 1690– 1692 (2002). 10. Sinha, A., Sun, W., Shih, T., and Barbastathis, G. Volume holographic imaging in the transmission geometry. Appl. Opt. 43, 1533–1551 (2004). 11. Liu, W., Psaltis, D., and Barbastathis, G. Real time spectral imaging in three spatial dimensions. Opt. Lett. 27, 854–856 (2002). 12. Liu, W., Psaltis, D., and Barbastathis, G. Volume holographic hyperspectral imaging. Appl. Opt. 43(19), 3581–3599, (2004). 13. Sinha, A., Sun. W., Shih. T., and Barbastathis, G. n-ocular holographic 3D imaging. In OSA Annual Meeting (Orlando, FL, 2002). paper WD7.

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14. Barbastathis, G. and Sinha, A. n-ocular volume holographic imaging. In Cooperative Control: Models, Applications and Algorithms, Butenko, S., Murphey, R., and Pardalos, P. M., editors, 1–21. Kluwer Academic (2003). 15. Sun, W., Sinha, A., Barbastathis, G., and Neifeld, M.A. Volume holographic image restoration with the Viterbi algorithm. In Conf. on Lasers and Eletro-Optics (2004). 16. Shih, T. Volume holographic imaging of transparent 3D objects. B.S. thesis, Massachusetts Institute of Technology (2004). 17. Sun, W. and Barbastathis, G. Rainbow volume holographic imaging. In Conf. on Lasers and Eletro–Optics (2004). 18. Born, M. and Wolf, E. Principles of Optics. Pergamon Presss, 7th edition (1998).

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4 Photorefractive Memories for Optical Processing∗ M. Duelli, G. Montemezzani, M. Zgonik, and P. G¨unter Nonlinear Optics Laboratory, Institute of Quantum Electronics, Swiss Federal Institute of Technology, ETH H¨onggerberg, 8093 Z¨urich, Switzerland

The field of three-dimensional optical storage has lately witnessed a tremendous increase of interest from both the scientific and the industrial communities. Volume holographic storage was proposed by van Heerden [1] shortly after the introduction of the laser, and interest in this technology has continued at varying degrees of intensity since. Holography potentially combines the advantages of high storage densities and ease of parallel data readout. The storage of multiple pages of information in the same volume and the parallel readout of many bits promise fast data access times and a high data bandwidth. Furthermore, in holography the information is distributed throughout the storage medium and is therefore less sensitive to local material imperfections. If Fourier holograms are stored in holographic memories, effects of imperfections in the medium are distributed throughout the page, and are not linked to individual pixels; thus valuable bits of data are not lost, and only the signal-to-noise ratio is diminished. This is in contrast to magnetic and conventional optical recording, where an individual bit of information is represented by a highly localized change in some physical property. This advantage may be less important in holographic digital data storage when additional redundancy of information storage and error correction can be easily implemented. Optical data storage currently finds commercial viability mainly in twodimensional low-cost data distribution on CD ROM (optical disk) and in quasiarchival storage on removable disk read–write drives (magnetooptical disk). Both of these technologies were made possible with the development of low-cost diode lasers in the past twenty years. An example of quasi-three-dimensional optical data storage is the new digital versatile disk (DVD) standard, where improvement in storage capacity with respect to CD ROMs is due to smaller bit size (shorter wavelength), reduction of the storage area sacrified for tolerances, and the use of more than one layer as storage plane. Volume holographic storage has been almost ∗ This chapter is reproduced from: P. G¨unter, Ed, Nonlinear Optical Effects and Materials (Springer Series in Optical Sciences 72, Chapter 5), Springer Verlag Berlin (2000).

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completely absent in the data storage market up to now. It has been incorporated only in a limited way into specialized commercial systems. One of the reasons for renewed interest in holographic data storage is improved input/output (I/O) technologies such as liquid-crystal spatial light modulators and integrated detector arrays that are capable of delivering high data bandwidth. The absence of these elements put a strong brake on the development of this technology in the past. The next few years will tell us whether volumetric data storage will make it possible to overcome the current I/O bottleneck in digital computers, where the processor speed far exceeds the ability of conventional rotating disk devices to import or export the data needed for manipulations. Other than for high-speed retrieval of data in a holographic memory system, the high parallelism of optical beams is also very attractive for computationally intensive data processing tasks, such as pattern recognition, edge enhancement, and spatial filtering, where vast amounts of data (a two-dimensional image is considered as a data page) have to be processed. Optical pattern-recognition systems are being developed for many applications such as identification of human pupils, partial or distorted fingerprints, robotic vision, and rapid searching of vast databases. Most of these applications make use of coherent light and the Fourier transform (FT) capability of lenses [2]. Especially, implementations of neural network models [3, 4], which are based on the structure of the human brain, are quite well suited to utilize the natural properties of light. These models of computational devices consist of many simple processing units or “neurons,” which communicate with each other via interconnection weights. Neural networks are successful in solving ill-defined problems, e.g., problems that are partially random, ill posed, or combinatorially complex, so that an algorithm cannot be easily formulated. These problems typically involve processing vast amounts of information, preferably in parallel, and often result in answers that need not be of high numerical accuracy. Communication and parallelism are more important than the computing power of individual neurons. Ideally, the architecture of a neural-network computer should reflect the highly parallel, associative, and nonlinear analog nature of the neural-network models. One approach to achieving such an architecture is to use optics for parallel communication and massive interconnection between a large number of processing units represented by planes of pixels having nonlinear optical properties. A sophisticated aspect of analog optical computing that is based on neural network architectures is that of associative memories (AM) [3]. In contrast to usual computer memories, where a separate unique address points to each stored datum (location addressing), in associative memories the output data is recalled directly using the input data itself (content addressing), without the use of separate addresses. Data can flow through the system, exciting chains of associations until a decision is reached in a global and parallel manner. Associative memories also have error-correction properties in that a complete undistorted set of data can be retrieved using a distorted or partial version of input data. In this chapter we discuss holographic memory systems with a particular emphasis on their applications to optical pattern recognition and associative memories. The problematics of volume digital data storage for computer memories

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is scratched only at the surface, and the interested reader is referred to some recent excellent reviews on this topic [5–9]. Stronger attention is given to implementations of optical processing and associative memory systems that use volume holograms recorded with the photorefractive effect, which is described in Chapter 4 of this book. For lack of space, optical associative memories based on matrix–vector multiplication (see, for instance, [10–13]) or more general optical neural network architectures [14] capable of learning will not be discussed here. The structure of this chapter is as follows. The first part (Sections 4.1 to 4.3) presents a general introduction to holographic data storage, optical pattern recognition, and holographic associative memories, respectively. In the second part (Sections 4.4 to 4.6) the same three topics are discussed by considering in detail implementations based on photorefractive materials. Section 4.1 reviews some recent advances in volumetric data storage, in which the storage capacity is discussed in relation to system architecture and multiplexing schemes. Section 4.2 focuses on optical pattern-recognition systems and discusses specific features of architectures based on volume holograms. Section 4.3 gives a general discussion of linear and nonlinear holographic associative memories. Section 4.4 considers photorefractive materials as volume storage media. We discuss recording schemes, storage capacity, and possibilities of hologram fixing and nondestructive readout. Implementations of optical correlators using photorefractive crystals are discussed in Section 4.5, while photorefractive all-optical associative memories including a deeper discussion of a system developed in our laboratory are presented in the last section. For a detailed discussion of the photorefractive effect itself, the reader is refered to Chapter 4.

4.1 Volumetric Optical Data Storage In this section we discuss system architecture and the storage density limitations of holographic memories. This discussion concentrates on digital storage of information, i.e., each data page is composed of pixels that are either “on” or “off”, which is the most interesting case for very-large-capacity memories. However, most of the conclusions are valid also for the case of analog storage of information, which is necessary for performing optical information processing, as described in Section 4.2.

4.1.1 Light Diffraction by Volume Gratings To analyze the storage capabilities of holographic volume media, we have to know the diffraction properties of volume refractive index or absorption gratings. Consider a thick holographic recording medium in which a grating has been recorded by interference of two plane waves, as shown in Figure 4.1. The grating with grating vector K = k1 − k2 can be read out using a probe beam Ii that may have the same or a different wavelength; the phase-matching conditions for the case of different wavelengths are visualized in Figure 4.1. For

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nω1

I1

Ii

Id

nω2 kd ϑ0 ϑ0 ki

I2

k2 K

ζ

k1

d

F I G U R E 4.1. Schematic of holographic grating recording and readout using a readout beam of longer wavelength than the recording beams. The right side shows the wavevector diagram for isotropic Bragg diffraction. I1 , I2 , Ii , and Id are the intensities of the two recording waves and the incident and diffracted readout waves, respectively. k1 , k2 , ki , and kd are the corresponding wave vectors, K is the grating vector, and ϑ0 is the Bragg angle for readout at frequency ω2 inside the crystal.

thick gratings, one diffraction order can be observed (Bragg diffraction regime), whereas for thin gratings the light is diffracted into many diffraction orders (Raman–Nath regime). The conditions to be fulfilled in order to obtain a thick hologram have been put forward by Gaylord and Moharam [15]. They are Q≡

2πλd >1 n2

(4.1)

and λ2 ≥ 10, (4.2) n2 σ with λ the vacuum wavelength of the reading beam, d the interaction length (thickness of the medium), n the refractive index, and  = 2π/|K| the grating spacing. The quantity σ is proportional to the strength (amplitude) of the grating modulation. For phase gratings one has σ = n, where n is the amplitude of the periodic refractive index modulation, for absorption gratings σ = αλ/2π, where α is the amplitude of the absorption constant modulation. Figure 4.2 depicts the conditions for a thick grating in the case of a phase grating only. The material thickness d as well as the refractive index modulation amplitude n are shown as a function of the grating spacing  for the conditions corresponding to Q = 1 and ρ = 10. In order to consider a grating as thick, the interaction length and the grating modulation amplitude both have to be in the gray areas. Note that with condition (4.2) the maximum amplitude n for which a grating can still be considered as thick is approximately 10−4 already for a grating spacing of 10 μm. In thick gratings, light is diffracted only if the angle of incidence is close to the Bragg angle. This angle is derived from the phase-matching condition kd − ki = K. For the symmetric situation of Figure 4.1, the value of the angle of ρ≡

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400

d [μm]

300 200 100

Q=1

0 100 10–1

Δn

10–2

ρ = 10

10–3 10–4 10–5

0

2

4

6

8

10 12 14 16 18 20

Λ [μm]

F I G U R E 4.2. Material thickness (interaction length) d and refractive index modulation amplitude n as a function of the grating spacing. The gray areas give the regions where conditions (4.1) and (4.2) for a thick grating are fulfilled. Parameters: wavelength λ = 500 nm, refractive index n = 2.2.

incidence inside the crystal for which the Bragg condition is exactly fulfilled is given by   λ . (4.3) ϑ0 = arcsin 2n The most important property of a thick hologram is its diffraction efficiency η, that is, the relative power being diffracted into the first (and single) Bragg order with respect to the total incident power in the material. A general form for η for a fixed transmission hologram with a sinusoidal refractive index modulation has been derived by Kogelnik [16] for isotropic materials using a coupled wave theory and has been extended recently to the anisotropic case by Montemezzani and Zgonik [17]. In this latter case one obtains  Id (d) sin2 ν 2 + ξ 2 −2αd   e , (4.4) η= =  Ii (0) 1 + ξ 2 ν2 where it was assumed that both waves (Id and Ii ) are absorbed equally strongly with an amplitude absorption constant α [17]. The parameter ν is proportional to

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the grating strength and is defined as ν=

π nd , λ cos θ

(4.5)

where the angle θ is obtained from cos θ = (cos θ1 cos θd )1/2 , that is, from a geometric average of the projection cosines of the Poynting vector directions of the incident (θi ) and diffracted (θd ) waves onto the direction of the medium surface normal ζ of Figure 4.1. The parameter ξ describes the angular mismatch of the incident wave with respect to the perfect Bragg angle of (4.3). It is defined as kd , (4.6) 2 where k is the wave-vector mismatch and always points in the direction of the surface normal ζ . For the situation of Figure 4.1, equation (4.6) is approximated by ξ=

ϑ K d , (4.7) 2 and is expressed in terms of the grating vector magnitude K = |K| and the deviation v = v − v0 of the internal angle for the wave vector ki from the exact Bragg angle. In case of perfect Bragg matching (ξ = 0), (4.4) reduces to the well-known formula   π nd 2 e−2αd . (4.8) η = sin λ cos θ ξ=

It should be noted that equations (4.4) through (4.6) as well as (4.8) are valid in general transmission geometries and include also the case of anisotropic Bragg diffraction, provided that the refractive index change amplitude n is calculated in a consistent way for the studied beam interaction [17]. Equations (4.3) and (4.7), in contrast, are specific to the symmetric geometry shown in Figure 4.1. The coupled wave theory leading to the above equations for the diffraction efficiency of transmission phase gratings can be applied also to the case of absorption or mixed absorption and phase gratings, as well as to the case of reflection-type holograms. Expressions describing all these cases for isotropic and anisotropic materials can be found in [16–18].

4.1.2 Hologram-Multiplexing Methods Utilization of the full data storage capacity of volume recording media requires hologram multiplexing. Beside the possibility of storing different holograms at different locations in a volume (spatial multiplexing), three methods have been proposed and tested to store several pages of information in the same region of the holographic medium: angular multiplexing, wavelength multiplexing, and phase multiplexing. In practice, a combination of spatial multiplexing with one of the

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η

(b)

83

η

ksω1 ks

ksω2 K1 kR2

kR1

K2

K2

ζ

K1

ζ

kω2

kω1

F I G U R E 4.3. Wave-vector diagram of (a) angle multiplexing by deflecting the reference beams and (b) wavelength multiplexing.

other multiplexing techniques seems to be ideal. Stacks of pages will be recorded in relatively small regions of the medium, because this approach allows stacks to be recorded and erased without affecting other regions of the storage volume. Angular, phase, and wavelength multiplexing will be discussed shortly. (A) Angular Multiplexing For angular multiplexing, each page of information is addressed by a particular angle at which the reference wave (usually a plane wave) illuminates the holographic material. In the recording step, object pages (all having the same average wave vector kS ) are recorded each using a slightly different wave-vector direction kR for the reference wave (see Figure 4.3a) [1]. To read out a specific stored object, the hologram is illuminated with the reference beam under the angle that was used to create the hologram. The holographic systems discussed in the second part of this chapter are mostly based on this multiplexing technique. The good angle selectivity of angular multiplexed holograms is a direct consequence of using thick gratings; the minimum angle separation between neighboring reference waves can be easily derived from (4.4). Let us consider a holograms with a moderate grating strength (ν  1) as encountered in multiplexing a large number of pages. Then the diffraction efficiency falls to half of its peak value if the mismatch parameter is ξ ≈ 1.39. With (4.7) and K = 4πn sin ϑ0 /λ, one finds that for a symmetric geometry as in Figure 4.1, this value of ξ corresponds to an angular mismatch of ϑ = 2.78λ/(4πnd sin ϑ0 ) and is therefore inversely proportional to the grating thickness d. For an internal incidence angle ϑ0 = 20◦ , λ = 0.5 μm, n = 2.2, and d = 2 mm, one has ϑ ≈ 0.004◦ , so that approximately 1000 holograms can be recorded and independently read out by a total angular displacement on the order of 5◦ inside the medium. It should be noted that most implementations of digital holographic storage by angle multiplexing use an interaction angle of nearly 90◦ between the beams. This has the big advantage of a reduced scattering noise as well as a slightly better angle resolution.

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In addition to the above-described method of changing the reference-beam angle of incidence, angular multiplexing can also be achieved by keeping a fixed crossing angle between the recording beams and rotating the storage medium between consecutive recordings. Each hologram in this orientation multiplexing is again characterized by a differently oriented main grating vector K. The readout address of each hologram is given by the orientation of the probe beam relative to the sample. Peristrophic multiplexing [19], a method whereby the hologram is rotated around the surface normal of the sample, is only one of the many possible variations. Although angular multiplexing requires accurate control of each reference angle for a good reconstruction of the stored objects, it is the most often used and investigated multiplexing technique. The angular multiplexing technique alone was shown by Mok et al. to allow storage of 500 objects or 5000 edge-enhanced objects (320 × 220 pixels) in a photorefractive LiNbO3 crystal [20, 21]. The reference beam direction was adjusted with an acoustooptic deflector in conjunction with a telescope. The holograms showed a diffraction efficiency of approximately 4 · 10−6 . Angular multiplexing together with spatial multiplexing, i.e., the holograms are stored in different regions of the volume, has been used to store 750 (128 × 128 pixels, diffraction efficiency 1 · 10−4 ) [22] and 10,000 images [23] by Tao et al. and Burr et al., respectively. Recently, angle multiplexing has also been used in connection with electrical fixing (see Section 4.5.3) to store 1000 holograms in a photorefractive crystal SBN:75 [24]. Angle multiplexing techniques that use a single laser beam have also been proposed [25]; experimental demonstration was limited to ten multiplexed holograms for the moment. (B) Phase Multiplexing Phase multiplexing uses a more complex reference than the plane-wave-like beams employed in angle multiplexing. The reference wave may be thought of as a linear combination of all possible reference beams used in the angular multiplexing scheme. The addresses of the stored images reside in the adjustable phases of different parts of the reference wave all overlapping with the signal wave in the holographic medium. The phase distortions of the reference beam can be random, introduced using a ground glass plate [26, 27], or deterministic, obtained by using a phase-only spatial light modulator. Multiple holograms may be superimposed without significant crosstalk by choosing a set of orthogonal j functions for the phase pattern, i.e., eiϕ (x,y) , where ϕ j (x, y) describes the phase distribution of the jth reference wave, which is related to the jth image. If the space x,y is divided into N discrete pixels, as obtained for instance by using a phase-only spatial light modulator, the orthogonality condition among the phase patterns is expressed by N  n=1

j

m

eiϕn (x,y) eiϕn (x,y) = 0

if j = m,

(4.9)

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where n is an index related to the position of a given pixel in the phase pattern. For readout, the phase code corresponding to the desired object wave is used, and with efficient code selection only the desired object will significantly contribute to image reconstruction. The fixed alignment geometry allows faster access times as for angular multiplexing. However, the storage capacities of experimental implementations are rather low, partly due to the lack of high-precision phase-only spatial light modulators, which are able to provide complex wave-front structures. Denz et al. [28, 29] presented a deterministic phase-encoding method whereby N objects are stored with N pure and deterministic phase-coded reference beams. Using this method, the image beam simultaneously interferes with all N reference beams inside the storage medium. Each of these references is encoded in phase, and they are angularly spaced by more than the Bragg angle. It has been shown that if N is a power of 2, then N orthogonal phase addresses can be constructed by choosing the phases to be equal to 0 or π and using the Walsh–Hadamard transform [30]. This method of binary phase encoding has been experimentally verified by multiplexing 64 images (128 × 128 pixels) in a photorefractive BaTiO3 crystal [31, 32]. The access time to retrieve any of the stored images was 150 μs. By using a subsampling of phase codes, holograms stored by phase multiplexing can also be used for optical addition, subtraction, and inversion of images [33]. In an earlier experiment, sinusoidally phase-modulated reference beams [34] were also used for the storage of seven different holograms. An alternative multiplexing technique, which may be seen as a special case of phase multiplexing, has recently been proposed [35]. This shift-multiplexing technique is based on a set of reference waves converging to a region of the recording medium. The medium is shifted by a few micrometers between two exposures. In the local frame of the medium, the phase distribution of the reference rays is changed as a result of the shift, thus allowing discrimination of different hologram pages. Shift multiplexing may be implemented using various kinds of reference waves, including spherical waves [36] and highly noncollimated Gaussian waves. It offers the potential advantage of simplicity and of a better compatibility with today’s spinning disk technologies. Multiplexing of a limited number of holograms with a required hologram-to-hologram shift on the order of 5 μm has been reported [35]. (C) Wavelength Multiplexing For wavelength multiplexing, each hologram is recorded using a different laser wavelength. The direction of the reference and signal wave outside the medium remains constant [1]. With respect to angular multiplexing there is the advantage that no readjustment of angles is required. However, a tunable laser source is required. The wave-vector diagram for frequency multiplexing is shown in Figure 4.3(b) for the case in which the writing beams are symmetric with respect to the surface normal of the storage medium. A hologram written by interference of the reference wave with wave vector kω1 and object wave (ks,ω1 ) forms an index

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grating with grating vector K 1 . By illuminating the crystal with light from the same direction but of different frequency ω2 , no diffracted light results, because the vector kω2 is not Bragg-matched. Reflection gratings are often used in wavelength multiplexing because of their efficient use of k-space. The grating vectors are oriented nearly parallel to the reference beam and are of magnitude approximately twice that of the reference beam. Therefore, in this case the grating vectors may span a greater portion of k-space than grating vectors arising from oblique beams [37]. For reflection gratings recorded by counterpropagating beams, the number of holograms N that can be distinguished in a medium of thickness d and within a wavelength range of λ − λ/2 to λ + λ/2 can be easily estimated. As for the case of transmission gratings, the diffraction efficiency of a weak reflection hologram falls to half its maximum value if the mismatch parameter of (4.6) is ξ ≈ 1.39 [17], so that 2πn d λ. (4.10) N∼ = 2 λ 2.78 With this criterion (4.10) tells us that about 400 holograms can be distinguished for a d = 2-mm-thick medium with refractive index n = 2.2 and a total wavelength tuning of ±5 nm around a central wavelength of 500 nm. It has to be remembered that if broad-bandwidth lasers are used, the maximum number of multiplexed pages might be limited by the laser bandwidth and not by the material thickness [38]. In a first experimental demonstration using a tunable solid-state laser diode, 26 images were multiplexed by tuning the wavelength over the range 670 ± 6 nm [39]. More recently, a digital wavelength-multiplexed holographic data storage system based on the same tunable laser diode technology has been demonstrated [40]. A 60-kbyte data file split into 27 wavelength multiplexed pages was stored and retrieved. Instead of a counterpropagating (180◦ ) configuration, this system uses a 90◦ angle between reference and signal beams, which has the important advantage of reduced reflection noise.

4.1.3 System Architecture Holographic storage systems using one of the multiplexing techniques described above may be implemented with the system architecture depicted in Figure 4.4. This schematic architecture is useful for the estimation of the limitations on storage capacity due to the optical system, as will be discussed in Section 4.1.4. The system consists of three distinct optical arms: the reference arm, which provides the address of a data page; the object arm, where the input information is injected during hologram recording; and the output arm, which recovers the output page information upon readout. In the case of angular multiplexing, encoding corresponds to switching on one single pixel at the reference plane (x i , yi ), which, for instance, may be achieved by placing a spatial light modulator (SLM) in that plane. The position of the bright pixel defines the reference beam direction after the transforming lens. For phase

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reference plane ⊗ xi

87

yi

L η

y1

yo L ⊗

ψ

xo z f

object plane

a

M by M input array

f

L ⊗

ξ

d volume holographic medium

⊗ x1

f

f output plane

α

H by H hologram array

M by M output array

F I G U R E 4.4. Schematic diagram of the holographic storage system used to determine the storage capacity and storage density at a given signal-to-noise ratio; L: lenses, f : focal length of the lenses. a pixel size, α hologram size.

multiplexing, in contrast, a certain number of pixels with a mutually well-defined phase relationship is switched on at the reference plane and given the address of the recorded page. Finally, for wavelength multiplexing the wavelengths of both object and reference wave are changed. The electronic data stream representing the information to be stored (one data page) is transformed into a coherent optical signal by the use of an SLM (assumed to have M × M elements) placed at the object plane. The light diffracted from the SLM is collected by a lens and concentrated in a small region at the plane of the storage material as either a Fourier or Fresnel transform of the bit pattern. In the specific case of Figure 4.4, the lens in the object arm provides a Fourier transform relationship [41], between the optical distributions f (x0 , y0 ) in the front and g(ξ, η) in the back focal plane. Random phase shifters can eventually be used in conjunction with the SLM to reduce the dynamic range of the light distribution at the hologram plane. Finally, the output arm provides the transformation of the light diffracted off the hologram upon readout into an appropriate intensity distribution at the output plane, where a charge-coupled device (CCD) sensor array consisting of M × M pixels can be placed in order to obtain an electronic signal of the reconstructed

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page. This signal can now be interfaced to the host processor for further manipulations using electronic signal processing. Using parallel readout of the CCD, high transfer rates are achievable, possibly exceeding gigabits per second [6]. In general, if the holographic medium is large compared to the hologram size α, multiple holograms may be accommodated by spatially multiplexing H × H holograms across the surface. In the simplified scheme of Figure 4.4 it was assumed for simplicity that the pixel distance (pitch) at the object arm SLM is equal to the pixel size a, and that similarly, the hologram size α and the hologram distance are the same. As discussed above, for a volume holographic medium, multiple images may be superimposed onto a stack at the location of a single element of the H × H hologram array. Storage in stacks of holograms is preferred due to the limited superposition of holograms, and it allows stacks of pages to be recorded and erased without affecting other regions of the storage medium.

4.1.4 Storage Capacity of Volume Media The fundamental upper limit for storage capacity of volume media is determined from diffraction considerations [42]. However, in addition to diffraction, other effects such as the optical quality of the holographic medium, the aperture of the optical system, lens aberrations, and crosstalk noise may limit the practically achievable storage capacity and density. Some of these aspects are discussed below. (A) Storage Capacity Limitation Due to Diffraction The limiting storage capacity of volume media may be quickly estimated by extrapolating the results for two-dimensional media. Let us assume that the recording medium used for storage has an area A facing the recording and reconstructing beams. The area used for each hologram is A/H 2 = α 2 , as indicated in Figure 4.4, and elementary considerations of diffraction show that the number of bits that can be stored in such a hologram is on the order of α 2 /λ2 , where λ is the wavelength of the readout beam. Hence, the maximum number of bits that can be stored if the full area of the medium is occupied by thin holograms is about A/λ2 . By making the hologram thick compared to the fringe spacing, we add another dimension. Using angular multiplexing, each hologram can be addressed by choosing the direction of incidence of the readout beam. Diffraction limits the number N of angularly multiplexed holograms to the order of d/λ, where d is the sample thickness. At each stack position the storage capacity in bits is the product of N and storage capacity of thin holograms α/λ2 . The diffraction-limited storage capacity C of the full volume storage medium is therefore given by C max ≈ (α/λ2 )(d/λ)H 2 , which can be rewritten as Cmax ≈

V , λ3

(4.11)

where V = A d is the volume of the storage sample. As expected, this result is similar to what is expected using a nonholographic sequential three-dimensional

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optical recording by “focusing a bit” to a localized volume of λ3 (every cube with an edge of length λ acts as a separate storage cell). Indeed, it has been shown that even under consideration of optical system limitations, holographic and layeredbit recording techniques lead to similar storage capacity in volume media [43], with the holographic technique being slightly better if lenses with an F-number larger than about 1 are used. The result (4.11) can also be derived by considering wavelength multiplexing [1] or by calculating the maximum number of orthogonal wave-vectors that are available within the storage medium [9, 44–46]. For wavelengths in the visible region, (4.11) predicts a tremendous storage density of 1013 bits per cm3 . However, as shown below, this number is too optimistic because of other less-fundamental limitations that lead to lower, but more realistic, storage densities. (B) Storage Capacity Limitation Due to Optical System We first consider the capacity limitations for a thin (two-dimensional) holographic storage medium on the basis of the limits of the optical system. Several authors have considered this problem; specifically, three papers in the 1970s [47–49] reported conceptually similar analyses: they optimize the capacity of the holographic storage system with a proper selection of input, output, and hologram arrays and the aperture of the lenses in the system. Following these analyses, the two-dimensional capacity C (total number of bits that can be stored in a given system) and areal storage density D (number of stored bits per area) can be expressed solely as a function of the parameters of a general optical system (Figure 4.4) [47, 48, 50]: 2  DL C(2-dim) = , (4.12) 8Fn r λ  2 1 1 D(2-dim) = , (4.13) 8 Fn r λ where F is the lens F-number (the focal length f divided by the diameter D L ) and the Rayleigh criterion n r is taken as a measure indicating when two bits can still be resolved. A Rayleigh criterion of n r = 1.22 means that two bits can still be resolved if the center of the Airy pattern generated by one bit falls on the first minimum of the Airy pattern of the neighboring bit. Equations (4.12) and (4.13) are derived under the assumption that the lens in the object arm of Figure 4.4 has an “optimum diameter” D L equal to the sum of the sizes of the input and hologram arrays [48]: √ √ (4.14) D L = 2Ma + 2H α, √ where the factor 2 accounts for measuring along the diagonals. It should be noted that the total storage capacity given by (4.12) can be eventually exceeded by mechanically shifting the holographic medium in the transversal directions to allow for additional spatial multiplexing. In contrast, the density given by (4.13) cannot be exceeded in thin planar gratings. Inserting F = 1, n r = 1.22,

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and λ = 0.5 μm, one obtains an area storage density of D(2-dim) = 34 Mbit/cm2 . This value is rather modest if compared with the density of localized nonholographic two-dimensional storage devices such as CD or DVD. They present densities on the order of (1/nr λ)2 , corresponding to about 270 Mbit/cm2 if the same wavelength and Rayleigh criterion as above are assumed. This reduced density and the fact of having to use rather bulky systems are the price to be paid for parallel random access to the data. The above analysis is valid for thin two-dimensional media. Generalization to the three-dimensional case is easy; the maximum storage density can be expressed as [44]  2 1 1 Nmax (4.15) χ Dmax (3-dim) = 8 Fn r λ d where d is the material thickness, and Nmax is the maximum number of holograms that can be multiplexed at a given location. As will be seen below, Nmax is limited by crosstalk effects due to diffraction or other noise sources and depends on the minimum allowable signal-to-noise ratio (SNR) for detection. In (4.15) the detrimental effect of lens aberration that leads to further crosstalk among bits in the detector plane has been considered through the empirical quantity ⎛ ⎞4 1 ⎠   χ =⎝ (4.16) 1 + α L f nr λ first introduced by Akos et al. [51]. The constant α L is on the order of α L = 10−3 for conventional lenses, where it is assumed that for a fixed Fresnel number, the aberrations are proportional to the focal length of the lens. Recording of multiple hologram stacks may be accomplished most easily by shifting the holographic medium. We can therefore put H = 1 and calculate explicitly the dependence of the storage density on the input and hologram sizes using (4.14) and (4.15):   1 M 1 2 Nmax χ Dmax (3 − dim) = + . (4.17) 4 α a d Here we recall that a and α are the pixel and hologram sizes, respectively, which are approximately related by α = nr λ f /a. Putting this in (4.17) shows that in general, a certain compromise has to be taken between hologram size and pixel size. (C) Interpage Crosstalk and Storage Density The last term in (4.15) and (4.17) contains implicitly the dependence of the storage density on the multiplexing method. To fully evaluate the density Dmax , the maximal number of multiplexed holograms Nmax has to be calculated. This number is limited by the condition that pixels have to be distinguishable in the

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output plane. Assuming that all other noise sources have been reduced to a minimum, the only fundamental noise source remaining is the crosstalk between the stored pages during readout (interpage crosstalk). For angular multiplexing, this crosstalk-limited storage capacity of volume holographic memories has been calculated by Ramberg [50] and Gu et al. [52]. Considering Figure 4.4, it is found that a maximum number of objects can be stored when the angle between signal and reference beam is ψ = 90◦ and the separation between adjacent reference points on the yi -coordinate in the reference plane is chosen as = λ f /d [52]. For this optimum configuration, the worst signal-to-noise ratio (SNR)min in terms of intensity can be expressed as (SNR)min ≈

2d f , λY Nmax

(4.18)

where Nmax is the maximum number of angular multiplexed images, d is the medium thickness, f is the lens focal length, λ is the vacuum wavelength, and Y = 2y1,max is the linear dimension of the output plane (object image). Equation (4.18) shows that the maximum number of image pages depends on the minimum SNR that the detection system can handle. Its derivation assumes that the paraxial approximation is valid and that the transverse dimensions of the medium are much larger than the spatial bandwidth of the objects, i.e., the whole Fourier spectrum (spatial frequency spectrum) of the input object can be recorded. For f /Y = 1 and assuming that an SNR equal to 1 can be tolerated, one obtains from (4.18) Nmax ≈ 2d/λ, in agreement with the estimate made at the beginning of this section. Similar analysis of the interpage crosstalk noise can also be accomplished for the cases of wavelength and phase multiplexing. In the case of wavelength multiplexing, the crosstalk noise is minimal for ψ = 180◦ (antiparallel geometry) and the maximum noise appears for the middle hologram with frequency νm = ν0 , where νm is the frequency of the mth hologram and ν0 is the middle frequency, in the frequency schedule [53]. Assuming that ν0 /νmax ≈ 1, the signal-to-noise ratio for wavelength multiplexing is given by    −1    P 1 − cos 2πmY 2 8 f 2 Y2 2 (SNR)min = π 1 + , (4.19) 8f2 m2 m=1 where N = 2P + 1 is again the number of stored holograms. In Figure 4.5 the SNR is shown as a function of the number of stored holograms following (4.18) and (4.19) for angle and wavelength multiplexing, respectively. The latter has a slightly more favorable crosstalk-limited signal-to-noise ratio than angular multiplexing. By examining a coupled wave theory in the Fourier regime, Bashaw et al. [54] compared the crosstalk for orthogonal phase-encoded and random phaseencoded multiplexing with the crosstalk of angular multiplexing. For an ideal storage medium (with no background scatter, no grating vector dispersion, and

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105

SNR

104 103 100 10 1 0

200

400 600 800 103 Number of holograms

104

105

F I G U R E 4.5. Logarithm of the worst signal-to-noise ratio as a function of the number of stored holograms N for angle and frequency (wavelength) multiplexing. Notice that for N > 1000 a logarithmic scale has been used. The SNR for frequency multiplexing has not yet been calculated for large N . The parameters are material thickness d = 1 cm, focal length f = 30 cm, λ = 500 nm, and output dimension Y = 3 cm.

an infinite lateral extent), they found that orthogonal phase-encoded multiplexing and angular multiplexing have similar crosstalk characteristics. Orthogonal phase encoding exhibits an SNR that is better by a factor of two than angular multiplexing as in (4.18). This improvement occurs because the crosstalk is the page average, rather than the worst-page crosstalk of angular multiplexing. These results partially contradict [55], where the SNR is estimated by a numerical analysis. It is claimed that at least until a capacity of about 500 holograms, the orthogonal Walsh–Hadamard binary phase-encoding scheme [28] produces a much better SNR on reconstruction than angular encoding [55]. With the minimum affordable SNR set, the relationships (4.18) and (4.19) allow one to calculate the limiting storage density Dmax (3-dim) for a volume hologram by means of (4.15) or (4.17). Figure 4.6 shows this theoretical storage density for the most common case of angle multiplexing. Even under consideration of the optical system limitations, if an SNR of unity can be afforded, a storage density of about 1 Tbit/cm3 is theoretically achievable. This value is about one order of magnitude lower than the storage density Dmax ≈ λ−3 (4.11) expected from pure diffraction considerations. However, it should be noted that for such a low SNR, a significant fraction of the stored bits might have to be used for errorcorrection purposes. For SNR = 10 one expects a storage density on the order of 100 Gbit/cm3 for typical array sizes of 1000 × 1000 pixels. If an SNR of 100 is required, the storage density would decrease to the order of 10 Gbit/cm3 , which is in agreement with a statistical analysis by Yi et al. [56] that also takes into account intrapage interpixel crosstalk, which arises from the limited spatial extent of the

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Pixel size: a = 3 μm

1000

a = 30 μm a = 3 μm

100

a = 30 μm

10 102

103

104 105 106 input/output array size

107

108

F I G U R E 4.6. Volume storage density (4.17) for angle multiplexing as a function of input– output parallelism with the input–output resolution as a parameter. The solid curves are for a signal-to-noise ratio SNR = 1, the dashed curves for SNR = 10. Parameters: lens Fnumber F = 1, n r = 1.22, λ = 500 nm, d = 1 cm. The decrease of the density for larger parallelism is due mainly to the effect of lens aberrations characterized by the constant α L = 10−3 in (4.16).

storage material. For both angular and wavelength multiplexing, a storage density between 4 and 20 Gbit/cm3 is predicted depending on the pixel size a for an SNR of 150 or a bit-error rate of 10−9 [56]. The decrease of the density for larger parallelism seen in Figure 4.6 is due mainly to the effect of lens aberrations characterized by the constant αL and is contained in the quantity χ in (4.16). More input pixels requires larger lenses with correspondingly longer focal lengths. Figure 4.6 also shows that the reduction in storage density with increasing parallelism is faster with coarser pixel size a. Thus at any chosen number of pixels, increasing the input–output pixel density also increases the storage density. To summarize, larger input dimensions reduce the storage density, as do also too-large pixels. Instead of storing Fourier holograms, image-plane holograms can also be stored. It has been shown that the SNR is independent of the pixel location at the output plane for wavelength and angle multiplexing and that the worst-case signal-to-noise ratio is slightly better than the worst-case SNR for Fourier transformed holograms recorded with the same parameters [57, 58]. (D) Storage Capacity Limitations Due to Material Quality The crosstalk calculations discussed above show that the storage density of volume media is significantly lower than the ultimate limit given by diffraction. This is particularly true if large signal-to-noise ratios and low bit-error rates of the reconstructed information are required. However, it was assumed that interpage or interpixel crosstalk and lens aberrations are the only noise sources in

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the holographic recording system. In practice, additional noise arises from imperfections of the storage material (such as scattering centers) and decreases the achievable storage density further. Despite the fact that their bad optical quality represents the principal limitation of many holographic media, unfortunately this noise source cannot be easily quantified. Bashaw et al. [54] use the quantity SNR1 to express the ratio between signal and scattered noise if only one hologram is recorded. In photorefractive materials the diffraction intensity decreases as the square of the number of stored holograms N (see Section 4.4.1), and so does the signal-to-noise ratio SNRscatt = SNR1 /N 2 that is due to scattering imperfections. Since the crosstalk signal-to-noise ratio decreases only with the first power of N, in many material samples the scattering noise dominates over crosstalk noise already for a relatively small number of stored holograms. In view of digital holographic storage, the construction of precise experimental test beds for the characterization of the material quality is very important [59]. Furthermore, material research to produce low-noise media is still of crucial importance in order to reach the full potential of volumetric optical data storage [60].

4.2 Optical Pattern Recognition Pattern recognition is one of the most natural signal-processing tasks for which the parallelism of optics provides important advantages. The two common types of optical pattern recognition systems, the joint Fourier transform correlator and the VanderLugt-type correlator, will be discussed in this section. We will focus especially on the performance characteristics of these correlators when volume holograms instead of thin holograms are used. A more detailed discussion on using thin holograms in correlators and the possibilities of electronic pre- and postprocessing can be found, for example, in N. Collings, “Optical pattern recognition” [61] and references therein.

4.2.1 Optical Correlators (A) Joint Fourier Transform Correlator In the joint Fourier transform correlator (JTC), which was first proposed by Weaver and Goodman [62], the object and scene patterns to be correlated are presented at the input plane simultaneously. Both are Fourier transformed by a single lens, and the interference pattern of their Fourier transforms is recorded in a hologram placed in the Fourier plane. The subsequent interrogation of this hologram with a collimated beam yields—after Fourier transforming—the correlation products of the object and the scene patterns. The JTC is especially advantageous when two unknown signals have to be correlated in real time because of its system simplicity. Provided that real-time devices for the input as well as for recording are available, search routines can be performed at the input data rate.

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4. Photorefractive Memories for Optical Processing reading beam

yo xo ⊗ s (xo, yo–h) s1 (xo, yo+ h)

zo

η ξ⊗

L

95

y1 ζ

x1 ⊗

L

z1

h h

input plane

f

f

f holographic medium

f correlation plane

F I G U R E 4.7. Schematic setup of a single-axis joint Fourier transform correlator. L: lenses, f : focal length of the lenses.

Consider a system as sketched in Figure 4.7. A transparency with a real transmittance s1 (x0 , y0 + h), which contains the object set, is positioned below the optical axis with its center at the point (0, −h), and a second transparency with a real transmittance s(x0 , y0 − h) representing the input object is positioned above the optical axis with its center at the point (0, h) at the front focal plane of a converging lens. The two transparencies are illuminated by a collimated beam from a laser and Fourier transformed by the lens. Then in the back focal plane, the light amplitude is F{S1 (x0 , y0 + h) + S(x 0 , y0 − h)}, where F is the Fourier transform operator. The interference of the two input waves creates an intensity distribution at the holographic material that is proportional to the product of the two FTs of the inputs: I (ξ, η) = |F{S1 (x0 , y0 + h) + s(x0 , y0 − h)}|2 = |S1 (ξ, η)|2 + |S(ξ, η)|2 + S1 (ξ, η)S ∗ (ξ, η)e−2iηh + S1∗ (ξ, η)S(ξ, η)e2iηh ,

(4.20)

where S(ξ, η) and S1 (ξ, η) are the Fourier transforms of s(x0 , y0 − h) and s1 (x0 , y0 + h), respectively, and ∗ indicates complex conjugation. This intensity distribution is recorded in a thin holographic medium. The correlation theorem states that the correlation of two functions is equivalent to the FT of the product of their complex field amplitudes in the Fourier plane. Therefore an optical Fourier transformation of the amplitude of the recorded hologram is needed. Illumination of the hologram with a plane wave and Fourier transforming the transmitted amplitude using the second lens in Figure 4.7 accomplishes this task. The output of the joint Fourier transform correlator can be found in the back focal plane of the second lens and is divided into three regions. In the first region, around the coordinate x 1 = y1 = 0, the sum of the autocorrelation of the two inputs is observed. The two side regions around the x1 = 0, y1 = ±2h coordinates correspond to the last two terms of (4.20) and contain the cross-correlation of the object with the scene pattern. The observed intensity distribution is proportional to the absolute square of the amplitude correlation function of the two

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s

correlation plane

s1

F I G U R E 4.8. Input intensity distribution of a 4-channel JTC consisting of the reference set of objects s1 and the unknown input object s. The output plane centered at (x 1 = 0, y1 = 2h) contains the cross-correlation product of the two scenes.

scenes: I (x 1 , y1 ± 2h) = |F{S1 (ξ, η) · S ∗ (ξ, η)}|2 = |S1 (x0 , y0 + h) ⊗ s(x 0 , y0 − h)|2   2   =  S1 (x0 ∓ x1 , y0 + h ± y1 )s(x0 , y0 − h)dx 0 dy0  , (4.21) where denotes the cross-correlation. Complex conjugates in the integral are omitted because the transmittances s and s1 were assumed to be real. In the case of two equivalent images, the cross-correlation has a maximum at (x 1 = 0, y1 ± 2h). In the case of an object set consisting of several images, the cross-correlation shows maxima at positions corresponding to the relative shifts between the similar portions of images in the input plane. This property can be used to identify the position of a smaller image inside a larger image, or to find out whether a certain object is present in a scene (multichannel JTC). If the input object s is similar to one of the objects contained in the reference set s1 , peaks of intensity will occur at the corresponding places (Figure 4.8). The height of the peak is a measure of the degree of similarity of two objects. Thus the identification is performed by detecting the position and relative intensities of the correlation peaks in the output plane. The JTC scheme described above is called a single-axis joint Fourier transform correlator because a single lens is used to perform the optical Fourier tranform of both object and reference image sets in front of the holographic medium. There exists a slightly modified technique (dual-axis JTC correlator) in which each signal is transformed by independent optics, greatly reducing the requirements on the Fourier transform lenses. Due to these practical considerations, most experimental investigations with JTCs have been performed using the latter scheme. Although the dual-axis JTC correlator relieves the stringent requirements of the FT lens, it does introduce the potential problem that the two Fourier planes have a relative tilt with respect to each other. It has been shown that this can lead to a degradation of the sharpness of the correlation peak [63]. JTC correlators have been demonstrated using photographic films in the Fourier plane or optically addressed spatial light modulators [64], as well as with real-time holographic recording media such as thermoplastic plates [63], bacteriorhodopsin

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4. Photorefractive Memories for Optical Processing F I G U R E 4.9. Realization of a matched filter for the frequency-plane correlator; L: lens, f : focal length of the lens.

η reference plane wave ξ ⊗ L

yo xo ⊗

zo

ζ

ψ

s1(xo,yo) f input plane

97

f holographic medium

[65], atomic vapor [66], semi-insulating multiple quantum wells [67], and photorefractive crystals (4.68). The last of these are used as volume holograms, which can significantly alter the performance of this kind of correlator, as we will see in Section 4.2.2. (B) VanderLugt-type Correlator In the frequency-plane correlator (FPC) or VanderLugt-type correlator, the input pattern with a real transmittance s(x0 , y0 ) is compared with a fixed template. The template itself is recorded as a hologram. One method for the construction of the hologram was proposed by VanderLugt [69], and the correlator based on this method is called the VanderLugt correlator. The optical generation of the matched filter that contains the frequency spectrum of the object pattern s1 (x0 , y0 ) is shown in Figure 4.9. The input device is located in the front focal plane of a lens and a holographic recording device (e.g., a photographic film) is placed in the back focal plane. The amplitude and phase of the Fourier spectrum S1 (ξ, η) of s1 (x 0 , y0 ) are recorded by interference with a plane reference wave R(ξ, η) = |R|eicη that is assumed, for simplicity, to propagate in the ηζ -plane. If aberrations are neglected, the observed intensity pattern is given by I (ξ, η) = |R(ξ, η) + S1 (ξ, η)|2 = |R|2 + |S1 (ξ, η)|2 + R ∗ (ξ, η)S1 (ξ, η) + R(ξ, η)S1∗ (ξ, η). (4.22) The holographic medium is developed so that the transmission of the fixed recorded hologram is proportional to I (ξ, η). This filter is now reinserted in the optical system at the back focal plane of the first lens (Figure 4.10). An adjustment precision of a few microns perpendicular to the optical axis and a few tens of microns along the optical axis has to be fulfilled to achieve a proper result. When a transparency s(x0 , y0 ) is present at the input plane and illuminated with a collimated beam, the intensity distribution at the output plane can then be divided into three regions. The first region that appears on the optical axis contains the autocorrelation beam. The observed intensity distribution at the second region centered at (x1 = 0, y1 = f tan ψ) is proportional to the absolute square of the amplitude convolution function of the input and the stored scene. The third region

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yo xo ⊗

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L

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x1 ⊗

z1

ψ

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f fixed template

f output plane

F I G U R E 4.10. 4 f -optical processing system for a frequency plane correlator. The fixed template is inserted at the back focal plane of the first lens.

appears with its center off the optical axis by an amount (x1 = 0, y1 = f tan ψ). The intensity distribution is proportional to the absolute square of the amplitude correlation function of the input object s(x0 , y0 ) and the stored object s1 (x0 , y0 ). As is the case in the JTC correlator, the detection of the correlation peak can be used either to identify the position of a smaller object inside a larger object or to find out whether a certain object is present in a scene. While conventional holographic pattern-recognition systems can identify a target rapidly by matching it to the information stored either in a template (FPC) or a frame store (JTC), they suffer a basic limitation in that the target images must match the template or frame store image exactly, i.e., these types of correlators are not rotation or scale invariant with respect to the input object. Only shifts of the input object within the input plane are permitted due to the fact that a lateral displacement in the input plane will lead to an additional phase factor in the Fourier plane. In this case the correlation pattern will remain the same and will be displaced in the correlation plane. A significant amount of progress has been made in generalizing the pattern-matching capabillity of the FPC using preprocessing operations (coordinate transforms, phase coding), optimized filtering (use of synthetic discriminant functions, circular harmonic filters, multiplexed filters, etc.) and postprocessing operations [61].

4.2.2 Optical Pattern Recognition Using Volume Holograms The above description of optical correlators is valid for a thin nonlinear holographic material placed at the filter plane. If the medium is thick, the Fourier plane of the image scene is just one layer of the hologram, which has important consequences for the correlator’s performance, as described below. (A) Joint Fourier Transform Correlator with Volume Media In using volume holograms such as photorefractive crystals in optical correlators, the Bragg condition for diffraction plays an important role. Nicholson et al. [70] considered the Bragg condition in an optical joint transform correlator using photorefractive Bi12 SiO20 . Their simplified analysis is based on Kogelnik’s results of diffraction from a volume hologram based on a plane-wave grating approximation. Gheen et al. [71] presented a more detailed analysis, which takes

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s1(xo,yo+h) input plane

99

η

zo

ξ ⊗

L

ζ

A1

h h A4 BS output

x1 ⊗

y1

A2 f

ζ

d

A3

θ

holographic medium

F I G U R E 4.11. Optical joint transform correlator using a thick real-time holographic recording medium. The output plane (x 1 , y1 ) can be observed after the beam splitter BS.

into account the structure of the grating. The volume hologram is decomposed into a series of thin slices, and the contributions from the composite thin holograms are added to get the final result. With this method the intensity distribution at the output plane is calculated for the system sketched in Figure 4.11. It is assumed that the beams are not significantly affected through scattering or absorption as they pass through the material. Thus the result is valid only for small diffraction efficiencies. The x0 and ξ dimensions of the system are also ignored to simplify the analysis. The two input signals to be correlated, s(x0 , y0 − h) and s1 (x 0 , y0 + h) are Fourier transformed and illuminate the holographic medium, leading to a phase grating proportional to the intensity distribution. This hologram is read by the third beam, a plane wave that satisfies the Bragg condition for the grating spacing written by the zero-order Fourier components of the two scenes. The diffracted output wave propagates backward through the lens and is observed at the output screen. This geometry is not unique: the readout beam might also differ in wavelength and enter the crystal from the left side at the appropriate Bragg angle. The complex scattered amplitude at a distance ζ from the Fourier plane is calculated in a similar way as for thin holograms. Integrating these amplitudes along the crystal length d and propagating them to the output plane (x 1 , y1 ), one can show that the complex wave amplitude at that plane is    kd(y0 − h)(y1 − h) s(y0 − h) s1 (y0 + h + y1 )sinc s(y1 ) = C dy0 , 2n f 2 aperture

(4.23) where the integration is over the aperture of the input signals, the coordinate x 0 has been ignored, and the waves s and s1 were again taken to be real functions. The constant C depends on material parameters, k is the wave number of the readout beam, and n is the refractive index of the material.

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input object beam

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a b

reference set beam

G1 G2 G3

c d

F I G U R E 4.12. Crosstalk problem. Spatial components along the grating vector lead to crosstalk. The regions G1–G3 are the desired gratings between the spatial components; a, b, c, d are the undesired crosstalk gratings.

d

The expression (4.23) is close to the desired correlation function (4.21). However, the Bragg effect has introduced an extra sinc term. The main effect of the sinc function is to limit the spatial extent of the input object s(x0 , y0 − h) so that s1 (x0 , y0 + h) is being correlated with only a portion of s. This effect varies over the output depending on the position, with the most severe restriction for the y1 values far from the center of the y1 -plane. The width w of the sinc function at the half-intensity level is given by w = 0.9n f 2 λ/ (d(h − y1 )), where λ is the vacuum wavelength of the readout wave. The output correlation is centered at y1 = −h when the coordinates y0 and y1 are chosen as mirror images with respect to BS. As a general guide, the Bragg condition can be neglected if w is larger than the spatial extent of the input object s(x0 , y0 − h). The Bragg condition limits the spread of wave vectors that can be reconstructed and thus the space-bandwidth product of the input scenes. For n = 2.23, f = 300 mm, λ = 500 nm, and a crystal thickness of d = 1 mm, the aperture of the input scenes is limited to y0,max = 7.6 mm. The number of pixels that can be processed in parallel, i.e., the space-bandwidth product, is on the order of the ratio of the input dimension to the smallest feature area in the entrance plane of the nonlinear medium [72]. √ Diffraction limits the smallest feature size to approximately λd. For the above parameters, the minimal feature size is about 22 μm, leading to a space-bandwidth product of about 105 . Using a thin medium (d = 1 μm), the minimum pixel size is around 1 μm, so that a space-bandwidth product of about 108 can be reached for an input aperture of 10 mm. Another serious problem is crosstalk between spatial components directed along the grating vector due to the nonvanishing crystal thickness (Figure 4.12). For spatial components directed orthogonally to the plane of incidence, no crosstalk occurs. A reduction of the recording angle ϕ and the crystal size d diminishes this problem. As we have pointed out, several aspects have to be considered when one is constructing a joint Fourier transform correlator using volume holograms. In order to avoid crosstalk effects, the two beams have to be aligned almost parallel and Fourier transformed by a single FT lens. The resolution limit in the Fourier plane, which depends on the material thickness, determines the pixel size. Efficient reconstruction can be realized only for writing beam components with incidence

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4. Photorefractive Memories for Optical Processing F I G U R E 4.13. Construction of the matched filter using a volume holographic medium situated at the Fourier plane. The scene to be stored is Fourier transformed and interferes with a plane wave incident at an angle ψ. The plane ζ = 0 corresponds to the back focal plane of the lens.

101

η

yo xo ⊗

reference plane wave

ξ⊗

zo

ζ

d ψ

s1(xo,yo) f input plane

f

ζ

holographic medium

angles allowed by the Bragg acceptance which determines the maximum input object size; this leads to problems if wide fields of view are required from both input signals. These two aspects strongly limit the useful space-bandwidth product of such a device. On the other hand, utilization of a volume hologram does not lead to any advantage in such a system. Therefore we can conclude that holographic volume media are not well suited for implementing a JTC [73]. Nevertheless, several real-time correlators using volume holograms, especially photorefractive crystals, have been implemented due to the lack of fast, thin real-time holographic recording media [68, 74–76]. (B) VanderLugt Correlator with Volume Media A calculation similar to that Gheen et al. [71] for the joint transform correlator can also be performed for the VanderLugt-type correlator using the same assumptions. First the phase grating obtained during recording of the matched filter is calculated at a distinct plane in the holographic medium. The intensity distribution at the correlation plane is then obtained by coherently adding the contributions of each thin hologram. In order to simplify the analysis it is assumed that the normal to the front face of the storage medium is aligned parallel to the optical axis of the system. The configuration used for recording the holographic matched filter is shown in Figure 4.13. While for thin media illumination of the recorded hologram with an object wave s(x0 , y0 ) produces three ouput beams (Section 4.2.1.B), for thick media the convolution beam cannot be observed because it is not Bragg-matched. Here we are interested in the correlation beam that propagates in the same direction as the reference beam that was used in the resording step. Neglecting any variations in the x0 or ξ -dimension, the contribution δ E of the hologram layer at the plane ζ to the diffracted amplitude in the correlation beam is expressed as [77] δ E(η, ζ ) = C1 S1∗ (η, ζ )RS(η, ζ ),

(4.24)

where C1 is a complex constant proportional to the refractive-index-change amplitude in the hologram, and S1 is the complex amplitude of the optical field of

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yo xo ⊗ zo

ξ ⊗ d

L

y1

L

s(xo, yo)

ζ

x1 ⊗ ψ

c

ζ input plane

f

f

holographic medium

z1

f

f

correlation plane

F I G U R E 4.14. Correlation of an input pattern s(x 0 , y0 ) with a matched filter in a VanderLugt-type correlator. The reconstructed reference beam is focused in the correlation plane.

the stored scene at the plane ζ given by      kηy0 ky02 ζ dy0 , exp −i s1 (y0 ) exp −i S1 (η, ζ ) = C exp(ikζ ) 2f2 f aperture

(4.25) where C is a complex constant. The amplitude S(η, ζ ) in (4.24) is obtained with the analogous expression to (4.25), and R is the amplitude of the plane-wave reference wave used for storing the matched filter. Expression (4.24) is valid under the assumption that the phase hologram amplitude is linearly proportional to the light intensity. Note that for photorefractive materials this condition is valid only in the small-modulation regime [78]. After Fourier transforming by the second lens in the VanderLugt scheme (Figure 4.14), the complex amplitude K (y1 ) at the output plane can be calculated by integration of the contributions of the different hologram layers: d/2  K (y1 ) = −d/2

   η2 1 exp (ik f ) exp ik F (δ E) dζ, iλ f 2f

(4.26)

where F denotes the Fourier transformation with respect to the coordinate η. Considering that the Fourier transformation of a plane wave is a delta function F(R) = |R|δ(y1 − c) and inserting the expressions for S1∗ (η, ζ ) and S(η, ζ ), the above integration leads to    πdy0 (y1 − c) dy0 K (y1 ) = C3 s1∗ (y0 − y1 + c) s (y0 ) sinc f 2 nλ Aperture

(4.27) where C3 is again a complex constant. The distribution K (y1 ) is centered at y1 = c due to the convolution with the delta function. In (4.27) we accounted for refraction into the crystal, which results in the term n appearing in the denominator

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of the sinc function. The constant c = f tan ψ depends on the angle between the object beam and the reference beam during recording of the matched filter. Equation (4.27) indicates that the Bragg effect, which reduces the shift invariance, introduces an additional sinc function in the correlation plane. In the case of a large crystal size, i.e., the width of the sinc function is much smaller than the input scene dimension, the sinc function vanishes for y1 = c. The intensity distribution, which is the square of the amplitude distribution K (y1 ), is then given by 2         ∗ s1 (y0 ) s (y0 ) dy0  (4.28) lim I (y1 = c) = C3   d→∞   Aperture and in this limit the cross-correlation is obtained only for an exact positioning of the input image s1 (y0 ). The influence of the crystal thickness d on the output intensity distribution can be seen in Figure 4.15. The output intensity, which is given by |K (y1 )|2 , is calculated using (4.27). The parameters are as follows: wavelength λ = 500 nm, refractive index n = 2.23, focal length f = 30 cm, input aperture 2y0,max = 2 cm. In Figure 4.15 the loss of shift-invariance for increasing hologram thickness can be easily recognized. A further increase in crystal thickness narrows the final intensity peak at y1 = c. The lack of shift invariance allows us to distinguish between objects that are shifted in the input plane. An intensity peak is observed only if the input image shows some overlap of its bright areas with the bright areas of the stored scene. While for thick volume holograms the Bragg condition eliminates shift invariance in the y0 -direction, the system retains its shift invariance in the transversal (x 0 ) direction perpendicular to the plane of incidence. Thus the two-dimensional output intensity distribution is given by  2      I (x 1 , y1 ) = I (x 1 , c) = |C|2  s1∗ (x0 − x1 , y0 )s(x0 , y0 )dx0 dy0  .   aperture

(4.29) Therefore, the effect of the thick hologram is to mask the two-dimensional correlation pattern except for one vertical stripe perpendicular to the plane of the beams (x1 ) whose position (c) depends on the angle of the reference beam. If we change the angle of the reference beam and record a different hologram with it, the one-dimensional stripe will be produced at a different location. One can store N different templates and get a system of N correlators with one-dimensional shift invariance. Although the shift invariance is lost in the one dimension, the utilization of volume holograms in the VanderLugt-type correlator is favorable due to the large storage density that can be reached. Instead of storing only one matched filter we can store many templates in one crystal by angular encoding. The shift invariance is lost, but a much greater amount of data can be processed in parallel.

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1-dim. stored scene 1.0

s ( yo)

s1 ( yo)

1.0 0.5

0.0

0.0

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d = 1 μm

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intensity [a.u.]

0.4 0.2 -0.5

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0 -1.0

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d = 10 μm

0.8 intensity [a.u.]

intensity [a.u.]

0.8

0.5

0.6 0.4 0.2 0 -1.0

-0.5

0.0 0.5 y1 - c [cm]

1.0

F I G U R E 4.15. Output intensity distribution as a function of the output dimension. Increasing the crystal thickness d leads to a single intensity peak at y1 = c.

If the loss of shift invariance represents a serious problem for a given system, another type of optical correlator, the reflection-type wavelength-multiplexed correlator, may be used. In volume media the reflection-type correlator has been shown to allow a larger shift invariance than JTC or VanderLugt correlators [79].

4.3 Holographic Associative Memories The holographic storage systems that are discussed in Section 4.1 are locationaddressable memories. To retrieve one document, a mechanical or optoelectronic control is necessary by which a specific reference beam is directed toward the

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storage medium. Thus these systems cannot be called associative memories. If a holographic memory has to implement the associative memory function, then reading must be content-addressable. The desired response must be obtained using a patterned key—either a separate one or a fragment of a stored pattern—without any extra control information. The associative properties of holography have been recognized ever since its invention by Gabor [80]. One can distinguish between linear and nonlinear holographic associative memories, the second kind possessing a nonlinear thresholding function for better discrimination.

4.3.1 Linear Holographic Associative Memories The good match between the parallelism and interconnectivity of optics and the requirements of associative memories was noticed early. Van Heerden [81] predicted in 1963 that a hologram would produce a “ghost image” of a stored image upon illumination of the hologram with a fragment of the original image. These early ghost image experiments were characterized by poor image qualities and low signal-to-noise ratios. The invention of off-axis holography greatly improved the SNR by angularly separating the desired signal term from the undesired noise due to self-interference among scattered waves from the original image. Some of the early experiments in holographic associations were performed by Collier and Pennington [82]. In these experiments a hologram was formed from two object waves. By illumination of the hologram with part of wave front A, a complete version of wave front B was reconstructed. If exposures from several pairs of different patterns (A, B) are recorded (superimposed), an information pattern associated with a particular key pattern can be reconstructed from the stored holograms with reasonable selectivity. The reconstructed wave front is a linear mixture of the images of the B patterns, with relative intensities that depend on the degree of matching of the stored field patterns with the field pattern that is used during reading. Thus one pattern will dominate, whereas the other terms in the mixture will represent superimposed noise. These holographic memories suffered from distortions, poor SNR, and low storage capacity. More recently, Paek et al. [83] proposed a ghost-image-type associative memory using second-order diffraction from a thin hologram to simplify the optical system and improve the output intensity, which usually suffers from the necessity to perform two optical correlation operations in series. As mentioned above, when a content-addressable memory is used to store arbitrary patterns, large storage capacity cannot be achieved by a superposition method because of the crosstalk noise between the patterns. Therefore pageoriented holographic memories were developed that used mechanical or acoustooptic deflection of reference beams to read out one of many spatially separated holograms [84]. This was one of the earliest applications of the optical VanderLugt-type correlator [69] for optical associative memories. In this application, memory data were stored in a large number of spatially multiplexed holograms. The small holograms, corresponding to the various stored entries,

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had to be written using narrow coherent beams confined to them only. During recording, different data planes or “pages” were recorded in a hologram by sequentially shifting a plane-wave reference beam over the entire recording medium. Content-addressable reading, on the other hand, has to be made in an all-parallel fashion, by interrogating all entries simultaneously and in parallel. In the readout phase the light from the input data page illuminated the entire set of holograms. A parallel associative search of all of the stored data could be performed simultaneously. A detector matrix determined the location of the resultant correlation peak, which determined the location of the hologram containing the matching data. This information was used to shift a readout reference to the proper hologram for readout of the associated data. The system could also be used for heteroassociation by shifting the readout beam to a hologram different from the matching one. Associations could be made by processing the correlation plane with lookup tables. Such page-oriented associative holographic memories are capable of large storage capacities, but they handle multiple associations serially because of the mechanical scanning of the readout beam. In response to the need for highly parallel architectures for neural network models, a new class of associative memories has been developed, which is also based on the VanderLugt-type correlator. Unlike page-oriented associative memories, these nonlinear holographic associative memories (NHAM) use nonlinear gain and feedback to implement competition between stored data [85].

4.3.2 Nonlinear Holographic Associative Memories The NHAM is an optical associative memory that combines the fully parallel image-to-image heteroassociative capabilities of ghost-image holography with the high SNR, processing gain, and storage capacity of thresholded VanderLugt correlators. They are potentially superior to linear associative memory approaches, because nonlinearities allow the selection of a particular stored image over all others on the basis of incomplete input data. As a consequence, the storage capacity is increased. Phase conjugation is often used to implement the features of gain, nonlinear feedback, and competition. The concept of NHAM can be also expanded to the optical implementation of neural-network models. A schematic diagram of a representative system is shown in Figure 4.16. The heart of the system is a hologram in which Fourier transforms of objects are recorded sequentially using angularly multiplexed reference beams. For readout of the NHAM, phase-conjugating mirrors or other means of forming retroreflected time-reversed beams are positioned on both sides of the hologram. When a partial or distorted version of an object addresses the hologram via the beam splitter, a set of partially reconstructed reference beams is generated. Each reconstructed reference beam corresponds to the correlation of the input with the stored object associated with that particular reference beam. This part of the system is identical to a matched VanderLugt correlator. The distorted reconstructed reference beams are phase conjugated by the reference-arm phase-conjugating mirror (PCM) and their paths retraced to the hologram. These beams reconstruct

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reference beams

retroreflection

hologram retroreflection + nonlinearity distorted input

stored images

F I G U R E 4.16. Recording and readout of reference-based nonlinear holographic associative memories (redrawn after [85]).

the complete stored objects. The reconstructed objects are phase conjugated by the object-arm PCM, and the process is iterated until the system settles into a self-consistent solution, or eigenmode, assuming that the gain of the PCMs is sufficient for oscillation. The most important common feature of NHAMs is nonlinearity. Without it these associative memories could not “choose” a particular image over all others, and the output would be a linear superposition of multiple recalled images. The nonlinear response and multistable operation allow selections between patterns to be made on the basis of incomplete data, since gain will exceed loss only for the stored pattern with the largest correlation with the input pattern. Nonlinearities also improve the signal-to-noise ratio and storage capacity over ghost-image holography or linear matched-filter correlators. For thin Fourier holograms, the effects of nonlinearities in the reference arm on the SNR and storage capacity of NHAMs has been extensively discussed by Owechko [85]. In this case, since the thresholded reference beam reconstructs not only the object, which has been written with that specific reference beam, centered on the input object, but also partially all other stored objects, an aperture has to be placed over the input plane to block these displaced objects. This aperture prevents ambiguities in the output plane at the cost of a reduced amount of shift invariance. An estimation of the number of stored objects N as a function of the degree of shift invariance leads to N ≈ ( φ f /z)2 [85], where φ is the range of reference-object beam angles for which the hologram has good diffraction

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efficiency, f is the focal length of the Fourier transform field, and z is the amount of shift invariance. For parameter values of f = 200 mm, z = 10 mm, φ = 30◦ , the maximum number of stored images is N = 110. For volume holograms, Bragg selectivity prevents reconstruction of an object written at a different angle than the original. In this case the storage capacity or storage density of an NHAM depends on the storage density of the storage crystal. If we assume that crosstalk during readout with the thresholded phaseconjugated reference beam is the only noise source, i.e., perfect thresholding, we can directly apply the results of Section 4.1.4 for the case of angular multiplexing. With (4.18) the number of images that can be stored and compared in parallel at the same time is N ≈ (2d f /λy(SNR)min ), where Y is the linear dimension of the stored objects and (SNR) min is the minimum tolerable signal-to-noise ratio. For a readout wavelength of λ = 500 nm, an output dimension of Y = 2 cm, a focal length of f = 200 mm, and a minimum signal-to-noise ratio of 100, we can store and associate in parallel 400 images for a 1-mm-thick crystal and 4000 images for a 10-mm-thick crystal. However, in practical NHAM it is often difficult to obtain perfect thresholding, so that the size of the stored image library is limited further. Implementations of nonlinear holographic associative memories can be categorized on the basis of the method used for generating the reference beams used in recording the holograms and on the resonator geometry. All-optical implementations will be discussed in Section 4.6. The basic principles of hybrid opticalelectronic NHAMs are the same as for all-optical ones, but the implementation of the nonlinearity and the input are quite different. Owechko implemented a hybrid NHAM that used a pseudoconjugation system consisting of video detectors and liquid-crystal light valves (LCLV) [86]. This approach allowed the programming of general nonlinear feedback functions. Between the reference loop video detector and the LCLV, the correlation plane is nonlinearly processed in electronic form using digital lookup tables in a PC board level image processor. With this image processor heteroassociations or multilayer optical neural networks can also be programmed [87].

4.3.3 Ring Resonator Associative Memories Another type of associative memory is the ring resonator NHAM that was first described by Anderson [88]. He demonstrated a ring phase conjugate resonator containing the holograms and an externally pumped two-wave mixing configuration to introduce gain to offset the loop loss (Figure 4.17). This system was designed to have two eigenmodes representing orthogonal patterns defined by a hologram prestored in a bleached silver halide emulsion. For a plane-wave input the system would usually resonate with equal strength in each of the two modes. However, if the input object is similar to one of the resonator modes, competition for the pump energy in the gain element will decrease amplification of the second mode below threshold and suppress its oscillation. Therefore, by injection of a portion of one of the original patterns the function of association is performed.

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(a)

(b)

gain

pump

T object wave

109

T injected signal

recording medium

hologram

F I G U R E 4.17. Holographic ring resonator memory. (a) Recording of hologram. (b) Recall by injected signal. Gain is supplied by a pumped photorefractive medium. The general transformation of the original object in the ring is represented by the black-box operator T (redrawn after [88]).

The system locks so that the resonance of the other eigenmode is suppressed [89]. Note that this approach is different from the previously described NHAM architectures. Here during recording of the hologram the reference beam is derived from the object beam itself (Figure 4.17). During readout there is no individual thresholding on the reconstructed reference beam. Instead, a nonlinear gain competition mechanism favors one reconstruction over other possible ones. This results in a simpler design and automatic generation of reference beams during recording, but at the cost of losing storage capacity. These systems also lack some of the discrimination obtainable using separate reference beams, but they do incorporate competition between stored modes using nonlinear gain saturation. In related systems, the sharp thresholding characteristics of resonators and the competition between modes could be used to implement self-learning and different neural network operations on an all-optical basis [90, 91].

4.4 Photorefractive Materials as Volume Storage Media An ideal material for volumetric data storage would have a fast response during writing (submicroseconds), be as sensitive as photographic film (microjoules/cm2 ), retain information for long time periods (>10 years), have a broad sensitivity range extending into the near-infrared region of the spectrum, and be available in large quantities at good optical quality and large sizes. High optical quality and low scatter are required to ensure that the signal-bearing wave front is not adversely distorted and that the noise level from scattered light is manageable. Also, a large refractive index modulation is needed so that there is sufficient dynamic range to multiplex the many holograms. Unfortunately, such a material does not exist today. Photopolymers offer a good sensitivity (mJ/cm2 )

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and a long storage time (years), but the fixing of the holograms leads to film shrinkage, which degrades the quality of the reconstructed pages and shifts the Bragg angle. The main problem with these materials is the limited film thickness of a few tens of micrometers, which limits the number of multiplexed holograms [92, 93]. In a different approach, photochemical hole-burning is used for a large capacity memory system. But high capacities have been demonstrated only for very low temperatures (a few K) [94]. Most promising candidates for volume storage media are photorefractive crystals [60], which have been widely investigated for real-time holography, phase conjugation, and data storage [9]. These crystals offer a large effective optical nonlinearity available at low continuous optical power densities of a few mWcm−2 and long storage times for fixed holograms. It is generally accepted that the photorefractive phase gratings in electrooptic materials arise from optically generated charge carriers that migrate when the crystal is exposed to a spatially varying illumination pattern. A thorough introduction to the photorefractive effect (PE) has been given in Chapter 4 of this book. The effect has been observed in many photoconductive inorganic electrooptic crystals including LiNbO3 , KNbO3 , BaTiO3 , BSO, GaAs, CdTe, as well as in the organic crystals COANP:TCNQ, MNBA, and DAST [49, 95–101]. Besides crystalline materials, photorefractive polymers (see Chapter 4) have also been suggested for holographic data storage. The major limitation in achieving useful device performance in these materials is the thinness (<350 μm) and the large applied fields that are necessary for the observation of the PE. Some of the important system characteristics and experimental techniques that are relevant for applications of photorefractive materials within associative memory systems or as data-storage elements are discussed in this section.

4.4.1 Recording Schemes One potential problem with photorefractive materials when used as storage media is the partial erasure of previously stored holograms while storing a new hologram page. Therefore, holograms stored with equal energies may reconstruct images of unequal brightness, those recorded later being brighter than holograms recorded earlier. To maximize the storage capacity, all the stored holograms should be of almost equal diffraction efficiency. Also, for many applications such as neuralnetwork implementations or associative memories, it is crucial that each hologram be stored with a distinct diffraction efficiency. To ensure a constant hologram strength, two storage schemes have been proposed: sequential recording and incremental recording. Both can be applied to each of the three multiplexing techniques for volume holograms of Section 4.1.2. (A) Sequential Storage Procedure An obvious method used to avoid excessive erasure of the previously stored holograms is to record subsequent holograms for shorter and shorter times. In

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1.0 1st hologram 2nd hologram 3rd hologram 4th hologram 5th hologram

0.8

Δni Δnsat

0.6 0.4 Δnsat, i

0.2

Δnsat

0.0

t1

t2

t3

t4 t5

F I G U R E 4.18. Relative refractive index change of 5 holograms as a function of time. The recording of the next hologram in the sequence is stopped when its refractive index change is equal to that of the previous recorded holograms.

this way, the diffraction efficiency decrease of “old” holograms is reduced, and at the same time the initial diffraction efficiency of “new” holograms diminishes with hologram number. This kind of sequential approach was used already in early times by Burke et al. [102]. However, their sequence of exposure times was such that an absolutely constant diffraction efficiency over 18 hologram pages could not yet be achieved. Accurate formulas for the sequence of recording times needed to reach equal refractive index changes and thus equal diffraction efficiencies over multiple pages may be obtained by taking into account the write and erase characteristics of the material. In first approximation (see Chapter 4), the erasure of the amplitude of the refractive index change n of a hologram recorded for a time t during subsequent recording of another hologram for a time t  can be expressed as      t t n = n sat 1 − exp − exp − , (4.30) τw τe where n sat is the saturation amplitude of the index modulation when only one hologram is recorded, τw is the exponential time for hologram recording, and τe is the exponential erasure time. For equal illumination intensities the time constants τw and τe are usually equal. However, in some materials, such as LiNbO3 , the erasure time τe can be significantly longer than τw . As illustrated in Figure 4.18, in order to obtain a constant diffraction efficiency for all pages, the recording of the ith hologram has to be stopped whenever its diffraction efficiency equals that of the (i − 1)th hologram that is being erased. For a material with equal recording and erasure time constants (τw = τe = τ ) the required recording time for the ith hologram can be calculated making use of

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(4.30), so that one finds the time ti as [103]   1 + (i − 1)β ti = τ ln (i > 1), 1 + (i − 2)β

(4.31)

where β = n 1 / n sat is the ratio of the index of refraction modulation recorded during the first exposure to the saturation index modulation. In Figure 4.18 the saturation parameter β was set to β = 0.8. At the end of the recording schedule all holograms show the same refractive index amplitude change. For full recording of the first hologram (β = 1) [31] reduces to the expression given earlier [42, 104]. It is important to consider how the saturation amplitude of the refractive index change n sat,i depends on the total number of stored holograms N. One obtains n sat,i = n sat,i = n sat

β , 1 + (N − 1)β

(4.32)

which for N  1/β reduces to n sat,i  n sat

1 . N

(4.33)

Therefore the grating amplitude is inversely proportional to the number of stored holograms, and with (4.8), the diffraction efficiency of each single hologram page decreases as the square of the number of hologram pages N. Using the sequential recording schedule given by (4.31), 500 angle-multiplexed holograms with 0.01% diffraction efficiency have been stored in a LiNbO3 crystal [105]. (B) Incremental Storage Procedure Since the above sequential recording schedules are all calculated from the material’s response times and the maximum attainable index modulation, any small error in material characterization can result in highly nonuniform diffraction efficiencies. Making the long initial exposures also introduces problems from beam fanning and coupling between the recording beams, both of which tend to restrict the maximum attainable index modulation below the theoretical value. These problems can be avoided by recording with a series of short exposures, so-called incremental recording. For incremental storage [106–108], each hologram i out of N holograms is recorded with a series of exposures δti , each relatively short compared with the material’s response time. During recording, the object and reference pair are sequentially displayed, repetitively cycling through all N pairs. As this process is repeated, the diffraction efficiencies of all the holograms gradually increase. After many cycles, the recording process approaches saturation, where the increase of the recording increment is exactly matched by the decrease during the (N − 1) erasure steps. This approach puts more severe constraints on the stability of the holographic setup than sequential recording. The interference pattern must be

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Δnsat,5 Δnsat,1 ε 2

Δni

0.20

Δnsat

113

Δnsat,1

0.15 δt

0.10 0.05 0.00

1st hologram 5th hologram

0

1

2

T

3

4

5

t / τw

F I G U R E 4.19. Relative amplitude of the refractive index change as a function of the writing time for the first and the last image (out of five) in the recording cycle. The refractive index change increases until the increase of the recording increment is exactly matched by the decrease during the erasure steps. The maximum difference in the saturation refractive index change for each hologram is given by the parameter ε. Parameters: τe = τw ; δti = δt = 0.1τW ; N = 5; T /τW = 0.5.

stable over all the time needed to record the hologram stack, and not only during recording of a single page. The principal aspects of incremental recording are depicted in Figure 4.19. After many cycles the system asymptotically converges toward a state for which the amplitude of each hologram shows only small variations around a constant average level and recording can be stopped. For short incremental recording time δti as compared to the photorefractive time constants, the asymptotic refractive index change amplitude n sat,i for hologram i is found to be n sat,i = n sat

δti δti + (T − δti )

τw , τe

(4.34)

where τw  and τe are again the writing and erasure time constants of the material, and T = δti is the total cycle time through all holograms. For τw = τe , (4.34) simplifies to   n sat δti n sat,i = n sat = , (4.35) T N in analogy with (4.33), valid for sequential recording. Here the second equality holds only if equal incremental times δt are chosen for every hologram page. It should be remarked that sequential and incremental storage achieve the same final grating modulation level in exactly the same time period [107, 109]. The same holds also for a third technique, parallel storage using N mutually incoherent but pairwise coherent beam pairs [109–111], a technique that has been proposed also

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for hologram copying from an archival storage medium to a secondary storage medium. As seen in Figure 4.19, at equilibrium, for incremental recording the hologram diffraction efficiency ηi shows a small variation ε because holograms incremented more recently are stronger. Of course, this variation can be minimized by shortening the recording increment time δt [106, 108]. If we require that all the holograms satisfy ηsat ≤ ηi ≤ ηsat (1 + ε), then the maximum δt is determined from the erasure during time (N − 1)δt to be τe δt = n sat ln(1 + ε). (4.36) 2(N − 1)

normalized diffraction efficiency

Thus the signal uniformity can be made to satisfy the system requirement by using the single parameter δt. For sake of visualization, in Figure 4.19 a relatively long incremetal time with consequently large variation ε/2 around the average value n sat,i = n sat /5 was chosen. In order to prove the validity of (4.35), five angularly multiplexed holograms have been recorded with a distinct ratio of their diffraction efficiencies in a nominally pure BaTiO3 crystal, where the assumptions of an exponential rise and decay and equal record and decay time constants are well fulfilled. The normalized target diffraction efficiencies ηsat,i /ηsat ∼ = ( n sat,i / n sat )2 of the five holograms have been chosen to be η1 = 1, η2 = 0.5, η3 = 0.8, η4 = 1, η5 = 0.2. The ratio of the writing times δti are equal to the ratio of the square root of the normalized diffraction efficiencies. Each incremetal writing time δti is selected to be short compared with the material’s response time. Setting δt1 = 4 s, the others must be selected as δt2 = 2.8 s, δt3 = 3.6 s, δt4 = 4 s, and δt5 = 1.8 s. The recording procedure was stopped after 50 cycles when saturation was reached. In Figure 4.20 the normalized diffraction efficiency is shown as a function of the angle of the readout beam. 4

1

1.0

3 0.8 0.6

2

0.4

target values η1 = 1 η2 = 0.5 η3 = 0.8 η4 = 1 η5 = 0.2 5

0.2 0.0 9.6

9.7

9.8

9.9 10.0 10.1 10.2 10.3

10.4

θ [deg]

F I G U R E 4.20. Normalized diffraction efficiency ηsat of five angle-multiplexed holograms recorded with different writing times δti in an incremental recording scheme as a function of the readout angle of the reference beam.

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The experimental values of the diffraction efficiencies at the Bragg angle of each of the five holograms are quite close to the predicted ones. It should be noted that these recording schemes assume that the space-charge field of each hologram depends only on the record and erase time and that τw and τe are the same for all holograms, i.e., the dependence of the space-charge field and of the photorefractive response time on the grating spacing or on the wavelength of the incident light is not considered. Crosstalk effects and beam-coupling are also not considered. These effects become important if many holograms are stored using angle or wavelength multiplexing and deviations of the predicted diffraction efficiencies occur that are due to these effects.

4.4.2 Storage Capacity of Photorefractive Holographic Media For the case of photorefractive materials, besides crosstalk and the optical system (see Section 4.1.4), further mechanisms limit the achievable storage capacity. These limitations are due to the finite dynamic range of the photorefractive material [112] and to noise from crystal imperfections and have to be taken into account in determining the storage density of photorefractive crystals. In Section 4.4.1 we have seen that the saturation value of the refractive index change decreases with 1/N , where N is the number of stored holograms. In the case of unequal writing and erasure time constants (τw = τe ) the saturation refractive index change amplitude falls off for large N as [103] n sat,i =

n sat τe , N τw

(4.37)

which is valid for both sequential and incremental recording. Unequal recording and erase time constants can be observed, for instance, in photorefractive LiNbO3 , where the bulk photogalvanic effect is important in the formation of the space-charge field. The refractive index change and diffraction efficiency for a strong volume hologram as a function of the number of stored holograms are shown in Figure 4.21. While the signal-to-noise ratio related to crosstalk noise decreases inversely proportionally to the number of stored pages (Section 4.1.4), the signal-to-noise ratio related to scattering or detector noise decreases more rapidly and essentially quadratically with the number of pages N. This is because the noise amplitude can be considered to be essentially independent of N, while the intensity of the useful signal (proportional to the hologram diffraction efficiency) decreases as 1/N 2 because of (4.37) and (4.8). One may define the minimum allowable signal diffraction efficiency ηmin , which is sufficient to overcome a constant noise level in the apparatus, such as scattering or detector noise. The maximum number of pages Nmax that can be stored can then be easily calculated as   τe φmax −1 , (4.38) Nmax = 1 + τw φmin

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Δnsat, i / Δnsat

(a)

~1/N

0.10

0.01

1

10

100

1.00

F I G U R E 4.21. (a) Normalized refractive index change as a function of the number of stored images. Solid line: symmetric recording/erasure (τW = τe ); dotted line: asymmetric recording/erasure (τe = 5 τW ) from (4.34); dashed line: asymptotic fall-off as 1/N for large N and τe = 5 τW (4.37). (b) Saturation diffraction efficiency as a function of the number of stored images for φmax = 3π/2.

(b)

ηsat,i

τe = 5 τw 0.10

0.01

τe = τw

1

10

100

Number of holograms

where φmin = (ηmin )1/2 and the maximum strength parameter φmax = (π n sat d )/(λ cos θ) is defined by the argument of the sine function in (4.8) and contains the maximum achievable refractive index change n sat and the material thickness d. In Table 4.1 we show an example of the maximum number of storable holograms Nmax and corresponding storage density D. An (arbitrary) minimum diffraction efficiency ηmin of 0.001% (10 mWcm−2 read beam resulting in a 100 nWcm−2 diffracted beam) and a maximum strength parameter φmax = 3π/2 are assumed. The results are summarized for the case of τe = τw and for τe = 5τw . In this example the storage density reaches a few tens of Gbit/cm3 , which is less than T A B L E 4.1. Number of stored holograms Nmax and the storage density D assuming a maximum strength parameter of φmax = 3π/2 and a minimum allowable diffraction efficiency of 0.001% for a d = 1-cm-thick crystal. The storage density is calculated using (4.15) and the parameter values F = 1, f = 50 mm, nr = 1.22, λ = 0.5 μm, and α L = 10−3 φmax = 3π/2

Nmax

D [Gbit/cm3 ]

τe = τw τe = 5τw

1490 7450

18 91

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predicted in Figure 4.6 from limitations due the optical system, diffraction, and crosstalk. This is because the additional limitation due to scattering and detector noise implicitly contained in the value of ηmin further reduces the maximum number of holograms Nmax . Note that the effect of dynamically forming noise gratings [113], arising by interference of a new reference wave with the the Braggmismatched diffraction from older gratings, has been neglected both here and in the treatment of Section 4.1.4. It should be kept in mind that the minimum diffraction efficiency that one can afford in a given system is limited not only by noise, but also by speed and data-rate issues [114]. At the detector a sufficient number of photons has to be integrated for each bit, and thus a too-weak diffracted beam goes at the expense of processing speed. As an example, for the 5000 320 × 220pixel holograms of [21], which were read out with a 50 mW/cm2 beam, and a diffraction efficiency of η = 4 × 10−6 , one can compute a necessary readout time on the order of one ms [9]. Despite the low diffraction efficiency, this readout time still corresponds to the fastest data rates of available CCD cameras, indicating that a fast data rate might represent one of the most important advantages of holographic memories over conventional optical point storage devices. To further increase the storage capacity of a photorefractive memory, conventional error-encoding techniques can be used to reduce the minimum signal-tonoise ratio required to achieve an acceptable bit-error rate. Neifield et al. [115] suggested that a factor-of-2 improvement in capacity can be achieved even when error-correction bits diminish the number of storage bits per page. Thus photorefractive memory systems with a storage capacity of few tens of Gbit/cm3 seem to be feasible. Recent research in academia and industry (primarily in the U.S.) has resulted in a number of impressive laboratory demonstrations, most of which are based on photorefractive materials. Holographic memories seem now to possess the storage density, readout speed, and fidelity required for competitive commercial products. The first fully digital holographic storage system using a photorefractive crystal was demonstrated by Heanue et al. [116]. The total useful capacity of that system based on a LiNbO3 crystal was 163 kBytes. In a later prototype demonstrator constructed by McMichael et al. [117], 20,000 holograms each containing approximately 70 kbits of data were stored in 20 different locations of a LiNbO3 crystal, corresponding to a raw capacity of about 1.4 Gbits. Fully digital holographic storage systems capable of real-time retrieval and electronic decoding of MPEG compressed video have been demonstrated recently [118]. The often cited goal of storage of megabit data pages with gigabit-per-second data rate has been recently demonstrated by an IBM group [119]. Pixel-matched recontruction of 1024 × 1024 arrays of binary pixels with 1 ms optical exposure of the holograms gave a raw bit-error rate as low as 2.4 × 10−6 using globalthreshold detection. A digital holographic storage system incorporating thermal fixing or optical fixing by means of a two-color gating technique has also been demonstrated [120, 121].

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4.4.3 Hologram Fixing and Nondestructive Readout The use of photorefractive crystals in holographic storage data systems, neural networks, and associative memories requires that the stored holograms not be erased during readout. The storage time in the dark is also finite and depends mainly on the dark conductivity of the materials. At least three possibilities are exploited to overcome these problems: the use of long-lived complementary gratings often achieved through heating of the crystal (thermal fixing), nondestructive readout using a wavelength at which the photorefractive material is not photoconductive, and sustained readout using refresh and low-intensity probe techniques. The thermal fixing procedure takes advantage of mobile ions resident in the crystal as a consequence of the growth process, which replicate a hologram or a series of holograms formed by trapped electrons or holes. This process can be described schematically as follows [122–124]: During or after hologram recording, the sample is heated to temperatures in the 100◦ –200◦ C range. At these temperatures the mobile ions move to form another grating and neutralize the electronic grating (fixing stage). On cooling to room temperature, a homogeneous illumination is used to partially erase the electronic grating and bring out the replica (developing stage). The complementary grating remains and cannot be optically erased at lower temperatures. This technique was demonstrated first in LiNbO3 [125, 126] and later in Bi12 SiO20 [127], in KNbO3 [128–130], and in BaTiO3 [131]. With this process an erase-time constant at room temperature for the fixed gratings of several years (LiNbO3 ) or a few days (KNbO3 ) instead of minutes or seconds, respectively, can be achieved even under illumination of the crystal. Another mechanism relies on domain patterns in ferroelectrics. Replication of a space-charge field due to trapped charge is achieved by domain formation. The exact configuration taken by domain walls to compensate the space-charge distribution is not yet known. Three models have been proposed: local switching of the spontaneous polarization wherever the total field (the sum of the applied and the space-charge field) exceeds the coercive field (Figure 4.22a) [132], needle-shaped (a)

(b) P

+ + −− + − + − + − + − + − + −

P + + −− + − + − + − + − + − + −

F I G U R E 4.22. (a) Local switching of domains wherever the applied and the space-charge field are larger than the coercive field. (b) needle-shaped domains nucleate at one electrode and propagate through the crystal until they are stopped by the space-charge field. The charges shown are those redistributed by means of the photorefractive effect. These charges are compensated by the bound surface charges at the boundaries of the domains (charge density σ = 2P · n, where n is the domain wall normal).

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4. Photorefractive Memories for Optical Processing F I G U R E 4.23. The photorefractive space-charge field induces a deformation of domain walls in an unpoled crystal, so that this field is screened.

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P ++− + − + − + − + − + − + − + −

−

domains nucleated at one electrode propagate toward the other electrode until they are stopped by the space-charge distribution (see Figure 4.22b) [133], and deformation of the domain walls by the space-charge field. These domain gratings are not erased by light, and they can be reconverted into photorefractive gratings by applying an electric field. It has been demonstrated that the diffraction efficiency of the revealed photorefractive grating is higher than that of the photorefractive grating that originally generated the domain grating. Deformation of the domain walls by the space-charge field such that this field is screened (Figure 4.23) has been proposed by Cudney et al. [134]. In this case it is possible to record and simultaneously compensate an optically induced spacecharge pattern by depoling the crystal before the grating is created. By repoling the crystal a diffracted beam reappears. Evidence of these effects have been identified in Srx Ba(1−x) Nb2 O6 [135, 136], BaTiO3 [133], and KNbO3 [137]. Recently, storage and fixing of 1000 anglemultiplexed holograms by ferroelectric domain switching has been demonstrated in Sr0.75 Ba0.25 Nb2 O6 [24]. The lifetime of the fixed gratings was on the order of 14 hours. The spectral sensitivity of the photorefractive material can also be used for nondestructive readout. Simple plane-wave holograms written at one wavelength may be read by Bragg-matched light of another wavelength, where the medium is not photoactive. For image-bearing holograms, however, light at the readout wavelength cannot in general be Bragg-matched to all the components of the signal. A nonvolatile memory with different recording and readout wavelengths that uses preformatted data has been demonstrated by Psaltis et al. [138]. This kind of technique is also being explored in connection with shift-multiplexing using spherical reference waves [36]. Recently, two-color illumination has also been attracting considerable interest because of the possibility of obtaining twophoton photorefractive recording at cw power levels. As in previous observations with pulsed light [139, 140], the gating light at the wavelength λ2 populates certain defect levels and increases the absorption at another (usually longer) wavelength λ1 . A hologram can therefore be recorded with the wavelength λ1 , to which the material would be otherwise insensitive. Readout at the wavelength λ1 only does not affect the hologram and is therefore nondestructive [141–143]. While this effect was first attributed to the presence of Pr3+ or other rare-earth ions, the real origin of the involved defect levels is still unknown. The sensitivity

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of the two-color recording effect is enhanced [143] by use of LiNbO3 crystal with a composition close to stoichiometric [144] instead of the more common congruently grown crystals. In addition to these techniques, which rely on the properties of the materials, there are feedback techniques that refresh and sustain holograms during their use [145–148]. In one of these techniques, holograms are periodically copied when their diffraction efficiency falls below a certain level [145]. Holograms may also be circulated between two photorefractive crystals by transferring data from one to the other before the holograms in the first are erased during readout, and then reversing the roles of the crystals [146]. However, the effectiveness of many of these techniques is limited by noise effects, and noisy gratings are copied and amplified as well. As an alternative to all optical approaches, hybrid prototype systems where refreshing is performed using an optoelectronic device acting both as spatial light modulator and detector array have been demonstrated recently [148]. The optoelectronic device is integrated into an optical system with phase-conjugate readout of the photorefractive memory, which makes this approach potentially very interesting for the construction of very compact lensless dynamically refreshed holographic storage systems.

4.4.4 Coherent Erasure and Updating of Holograms High-storage-density optical holographic memories should have the capability of updating pages of information. In photorefractive crystals, erasure of recorded holograms is usually performed either by uniform illumination of the storage crystal at the recording wavelength or by heating the crystal if thermal fixing has been used. These bulk erasure processes apply indiscriminately to all multiplexed holograms in a stack. Selective erasure of a single stored hologram has been demonstrated using a coherent erasure technique [149]. This technique is based on superimposed recording of a π-phase-shifted image. The selective erasure is shown to be much faster than the incoherent erasure owing to the coincidence of the previous charge distribution and the π -phase shifted new light illumination pattern. In another demonstration, incremental recording was used for recording of the initial binary holograms as well as for replacing (selective erasure and updating) a single hologram out of four [150]. This method can not be used only to erase a single hologram but also to erase a single bit or part of a hologram by applying the phase shift not to the reference beam but to the part of the image that is to be erased. Note also that, when after storing a binary image a second image is stored with a π-phase shifted reference beam, some parallel logical operations such as XOR and OR can be implemented [151].

4.5 Optical Correlators Using Photorefractive Crystals In Section 4.2 we have discussed the general features of the joint Fourier transform (JTC) and VanderLugt-type optical correlators and analyzed the consequences of using thick photorefractive materials for the necessary optical processing.

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Despite the fact that the Bragg condition in volume media destroys shift invariance and partly limits the applicability of such systems, recent implementations of optical correlators (see Section 4.2) based on photorefractive nonlinearities are numerous [68, 71, 74–76, 79, 152–165]. While many of these works are laboratory optical-table-size demonstrations, some compact devices have been constructed as well. Rajbenbach et al. [68, 158] have constructed a compact photorefractive JTC system based on a Bi12 SiO20 (BSO) designed for an industrial application, namely, to sort mechanical tools of different basic two-dimensional geometrical shapes. The reference set of objects is presented to a CCD camera, while the unknown object is viewed by a second camera. The two video signals are mixed so as to generate a single video image that is displayed by a liquid-crystal display in order to build a single-axis JTC. To write the grating in the photorefractive BSO crystal a visible green beam from a frequency-doubled Nd:YAG minilaser is used. The real-time phase hologram is read out by a Bragg-matched plane wave from a HeNe laser. A CCD sensor in the correlation plane analyzes the diffracted signal. The response time ranges between 35 and 100 ms for illumination intensities on the order of 10 mW/cm2 . Recognition of tools out of a set of four has been demonstrated, but the system can easily be expanded to more objects. In another demonstration that uses a BSO crystal as well, Yu et al. [75] employed a Galilean telescope to decrease the angles between the object beams in the JTC, which relaxes partly the limitations due to Bragg diffraction. The use of photorefractive crystals as a buffer memory for JTC-type correlators has been suggested by Alves et al. [166]. In order to get the full advantage of using thin holograms in the Fourier plane they proposed to use a photorefractive crystal as a buffer memory for providing a fast data-access time and to store many filters. To achieve correlation at video rate each image has to be compared to a set of filters in less than 40 ms. The Fourier transform of this image is recorded in a nonlinear material by interfering with a plane wave. A third beam carrying the Fourier transform of the filters reads out the hologram. If the whole set of filters can be read out from the photorefractive crystal in less than 40 ms, dead time exists between each video image, which can be used to refresh data. Alves et al. analyze the refreshing scheme and show also erasure and updating of the stored filters in the memory. Nevertheless, fast correlation with such a system has not yet been implemented. If a thin real-time holographic medium is used to record the images that are to be compared, fast image correlation with a large set of filters seems to be possible. A photorefractive crystal for permanent storage of matched filters in a VanderLugt-type hybrid digital/optical correlator has been demonstrated by Duelli et al. [164]. In contrast to most photorefractive crystal-based correlators, where the thick photorefractive material is placed in the filter plane, in this system the optical memory is used as an input device for the reference library and a phase-modulating liquid-crystal television is placed in the filter plane. Thus the correlator is fully shift-invariant. During the recognition process the input scene is Fourier transfomed electronically and displayed on the LCTV, and a sequential search of the input scene within the library of reference objects is performed by

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rotating the storage crystal and reading out the angularly multiplexed matched filters. This correlator shows a high discrimination, sharp correlation peaks, a high light efficiency, and is fully shift-invariant in spite of the photorefractive crystal thickness. In addition to inorganic crystals and organic polymers, another class of thin photorefractive materials, semiconductor multiple quantum well (MQW) devices have been employed for optical correlation in a JTC configuration [67]. In this class of materials the photorefractive effect is mediated by resonant effects such as the Franz–Keldysh effect or the quantum-confined Stark effect [167, 168]. Due to the large carrier mobility these materials exhibit a fast response time, and a correlation operation can be performed in time on the order of 1 μs. On the other hand, the high mobility is detrimental for the achievable resolution, which is limited to a feature size of about 10 μm. All optical correlation has also been demonstrated using infrared wavelength and bulk photorefractive semiconductor materials as real-time holographic media in both types of optical correlators [71, 160].

4.6 All-Optical Nonlinear Associative Memories In this section we mention some reported associative memory systems based on the general architecture described in Section 4.3.2. We distinguish between implementations using thin and thick holograms. An associative memory containing a photochromic saturable absorber as nonlinear threshold element realized recently in our laboratory is treated in greater depth at the end of this section. For optical associative memories based on optical resonators we refer to Section 4.3.3.

4.6.1 Thin Storage Media Implementations All-optical implementations of NHAM can be differentiated by the form and implementations of the nonlinearities and by the storage medium used. The use of thin Fourier transform holograms for storing the pages of information has the advantage of shift-invariance and the capability of heteroassociations by manipulating the correlation plane. A great disadvantage is the lack of Bragg selectivity, which results in a low storage capacity compared to volume holograms. Most implementations use a single-iteration nonresonant configuration, where the object arm phase-conjugating mirror (PCM) is missing (see Figure 4.16). The first all-optical associative memory was demonstrated by Soffer et al. [169, 170]. They used thin thermoplastic Fourier transform holograms as the storage medium and a PCM based on degenerate four-wave mixing (DFWM) in BaTiO3 . The output of the hologram acted as a probe for the DFWM system, generating an amplified phase-conjugate of the correlation plane. A partial version of one of the stored objects was used to address the hologram. The conjugated backward-propagating beam illuminated the hologram, re-creating a complete version of the stored objects. Thresholding was not demonstrated in this system,

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which was basically a linear associative memory. The discrimination of the two stored images was due to the coding of the objects using high-spatial-requency diffusers. Furthermore, Paek and Psaltis demonstrated a single nonresonant associative memory that used a simple mirror to retroreflect the reconstructed reference beams. In their system, four spatially multiplexed objects are recorded simultaneously using a single reference beam. This approach is equivalent to sequentially recording objects centered in the same aperture but with angularly shifted planewave reference beams. During readout the presence of a correlation peak in a particular subregion is a unique label for which the stored object is being recognized. A pinhole array acts as the thresholding element; only the peaks of the correlations pass and the sidelobe noise is suppressed [171]. Using a PCM instead of a simple mirror in connection with a pinhole array association of two stored objects has also been demonstrated [172]. But the discrimination in these implementations was not complete, since faint images of other stored holograms could still be observed in the output plane. Moreover, the quality of the reconstructed images was quite poor. This was attributed to the relative large size of the pinholes. Although a thin hologram was used, which allows shift-invariance, this advantage has been lost due to the fixed pinhole array. Paek and Psaltis discussed approaches for restoring the shift-invariance by eliminating the pinhole array and using quadratic nonlinearities in the correlation plane. A novel method of thresholding was demonstrated by Wang et al. [173]. They used a bistable Fabry–Perot etalon. This nonlinear element was situated in the correlation plane, and holding beams were used to bias the etalon. If the peak of the autocorrelation function was sufficient to switch the etalon, the holding beam at that point was be transmitted. Since the holding beams were aligned to be counterpropagating to the reference beams, after switching, the transmitted holding beam read out the hologram and reconstructed the associative image. The need for a PCM was therefore avoided, but the system’s complexity increased. Although auto- and heteroassociation could be implemented by directing the holding beams to the same or different holograms, the use of a separate holding beam for each stored image is not practical for large data storage.

4.6.2 Volume Storage in Associative Memories The use of volume holograms in associative memories to extend the storage capacity was first pointed out by Yariv et al. [174]. They demonstrated all-optical association between two images stored in a photorefractive BaTiO3 crystal. The reconstructed reference waves from the holographic memory were coupled to a photorefractive bistable ring resonator by means of multimode optical fibers. The ring resonator performed the function of thresholding, in which only one of the two waves survives and illuminates a transparency similar to the one used to record the holographic memory. Xu et al. [175, 176], stored four objects in a photorefractive KNSBN:Co crystal, and liquid-crystal electrooptical switches served as the (optoelectronic) thresholding element. Similar to the bistable

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Fabry–Perot etalon, counterpropagating holding beams have to be used in order to bias the switches and to read out the associated image. Lu et al. [177] presented a multilayer associative memory, which is basically similar to a Hopfield network [4] with nonlinear processing on the inner product between the input and the stored patterns. The hybrid optical–electronic implementation involved storage of volume holograms in two photorefractive crystals while thresholding of the reconstructed reference waves was performed electronically. Reconstruction of one out of four images was demonstrated. Heteroassociation was also demonstrated. Kang et al. [178], Tanaka et al. [179], and Chen et al. [180] used a LiNbO3 crystal as a thick holographic recording medium and a PCM that provided optical feedback, thresholding, and gain. Unfortunately, the thresholding behavior was not analyzed, and associative readout was demonstrated only for a single image or for orthonormal images. Association has also been demonstrated by Ingold et al. [181] using a nonresonant cavity with image-bearing beams. The nonresonant cavity consists of an optical feedback configuration containing a photorefractive crystal and a nematic liquid crystal. The photorefractive KNbO3 crystal is used as an optical amplifier for several beams and for image comparison, and the nematic liquid crystal as a thresholding element. Associative readout, error correction, and image completion are demonstrated for two and three stored black-and-white images. Associative recall of stored images has been demonstrated for one, mostly two, and at maximum five stored images in almost any published all-optical nonlinear holographic associative memories. The maximum number of angularly multiplexed holograms was limited by the high loss through the hologram and other optical components. A further limit was imposed by the low gain of the nonlinear elements. The thresholding capabilities as well as the storage capacity and practical limitations of the proposed or demonstrated systems were mostly not described. In the following we describe in some more detail an all-optical nonlinear holographic associative memory that was developed recently in our laboratory [182]. It is similar to that of Soffer et al. [169] in that it is a single-pass NHAM that uses a phase-conjugating mirror for retroreflection of the reconstructed reference beams. A schematic setup is shown in Figure 4.24. The images are stored in a volume holographic medium consisting of a photorefractive LiNbO3 crystal by angle multiplexing. Since the phase conjugation is realized via degenerate four-wave mixing in a photorefractive KNbO3 crystal, its reflectivity is essentially linear. Therefore an additional nonlinear element has to be added in order to provide thresholding. This is achieved by inserting a saturable absorber with a tunable threshold into the correlation plane. The absorber consists of a photochromiccolorant-doped polymethylmetacrylate (PMMA) and the threshold intensity for absorption saturation can be adjusted by simultaneous illumination with UV incoherent light [183]. As mentioned earlier (Section 4.2.2B), the thickness of the photorefractive crystal makes the holograms Bragg selective, which destroys the shift invariance in the plane of incidence and with it the main advantage of using Fourier holograms.

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reference beams input plane

θ

C

C O

LiNbO3:Ce

L

gain crystal KNbO3:Fe L

f

C

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output plane

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saturable absorber KNbO3 phaseconjugating mirror

F I G U R E 4.24. Schematic of our all-optical associative memory; L : lenses, f : focal length of the lenses.

Furthermore, shift invariance is also destroyed by the presence of a stationary diffuser, which is attached to the liquid-crystal display present in the object arm. A two-dimensional analysis (see (4.29)) shows that for thick media the intensity distribution Iref,i (u) of the reconstructed reference beam i can be approximated by the square of an overlap integral of the transmission functions of the partial input image tinput (x, y) and the stored image ti (x, y) corresponding to that reference beam, i.e., the cross-correlation sampled at the point zero:  2      ∗  Iref,i (u) = Iref,i (u = u i ) ∝  tinput (x, y)ti dx dy  , (4.39)   aperture

where u is the spatial coordinate in the plane of incidence at the saturable absorber, u i is the location of the focus of the reference beam i at the absorber, and x and y are the spatial coordinates at the input plane. The integration runs over the aperture of the input scene. During associative readout the reconstructed reference beams pass through the saturable absorber, are phase conjugated, and retrace their paths to the hologram passing through the saturable absorber again. This double pass through the absorber greatly enhances the intensity difference between the beams, such that only one of the reconstructed reference beams has a nonnegligible intensity by the time it addresses the LiNbO3 crystal from the right side. Due to the strong angular selectivity of the thick hologram, only the stored image that is most similar to the input is reconstructed and observed in the output plane. The number of stored images that can be associated in parallel is increased by amplifying the reference beams. A gain crystal transfers energy from a pump beam to the reconstructed reference beams via photorefractive two-beam coupling. This is necessary since

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Output Intensity Ratio

1000

100

10 0.04 wt.% doped PMMA 0.02 wt.% doped PMMA

1

0

1

2

3 4 5 6 7 Input Intensity Ratio

8

9

10

F I G U R E 4.25. Thresholding characteristic of the saturable absorber. The full squares are measured data for 0.02 wt.% and the open squares are data for 0.04 wt.% colorant-doped PMMA. The curves give the theoretically expected transmission ratios according to a simple model presented in [183].

the diffraction efficiency decreases with the number of images that are stored in the LiNbO3 crystal, and thus the signal-to-noise ratio increases at the output plane. The thresholding effect of the saturable absorber can be seen in Figure 4.25, where the intensity ratio of two reference beams after the thresholding process is shown as a function of the intensity ratio before thresholding. The saturable absorber strongly enhances the intensity ratio and suppresses the weaker reference beam almost completely. An intensity ratio of one to two of the reconstructed reference beams before thresholding is necessary to reach a signal-to-noise ratio of one to ten in the output plane. The readout time of our implementation of an associative memory is limited by the response time of the amplifier and the saturable absorber and is about 1 second for the intensities we used. Ten black-and-white images (letters) with an area overlap of 20–80% have been stored in the LiNbO3 crystal and successfully read out using a partial input. The exposure time per image per cycle during recording was 10 s, and 60 cycles were necessary to reach saturation using an intensity for the object and reference beams of approximately 25 mW/cm2 . Due to the long writing time, the erasure of the recorded holograms during associative readout is negligible. Photographs of the observed output using a partial input are shown in Figure 4.26. The image completion properties can be seen in Figure 4.27. In this case five objects have been stored and the output has been viewed with a CCD camera. This was necessary due to the very small amount of light that was used for the associative readout. Only 80 pixels (out of 85,000 of the entire screen and out of 40,000 of the entire associated image) are necessary to get full reconstruction of the associated image. The small dot that is used as the input has a large overlap

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4. Photorefractive Memories for Optical Processing F I G U R E 4.26. Ten letters have been stored and associative readout using only a partial input image.

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stored images

H M B A K F D N E C H M B A K F D C N E input images

readout images

⇒

⇒ ⇒ ⇒ ⇒ ⇒ ⇒ ⇒ ⇒ ⇒

with the letter A and a smaller overlap with the letter M. Notice that crosstalk is virtually absent. One serious problem encountered by associative memories is to distinguish between strongly correlated images and, in particular, images that are partially or totally enclosed by others. Recently we have presented a storage scheme for amplitude-modulated images that eliminates this problem in nonlinear holographic associative memories [184]. The method is based on controlling the diffraction efficiency of each hologram corresponding to a stored image. It was demonstrated that the weighted storage method allows the discrimination of highly correlated images. The weighted storage can be accomplished by adequately varying the exposure time of each hologram.

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M. Duelli, G. Montemezzani, M. Zgonik, and P. G¨unter F I G U R E 4.27. Reconstruction of 5 stored images. Only 80 pixels are necessary to get full reconstruction of the stored images. The white area was used as input, while the gray letters are shown only in order to indicate the overlap of the small input area with the stored images.

4.7 Summary Holographic data storage has received renewed interest in the last few years. In this chapter we discussed some recent advances in holographic memory systems and given the major emphasis to their applications to optical processing using photorefractive media, such as pattern recognition and associative memories. Several theoretical studies have shown that the storage capacity of a practical optical holographic system is much lower than the ultimate diffraction limit, but it can still reach about 20 Gbit/cm−3 for a bit-error rate on the order of 10−9 . The limitations are introduced by the optical system and crosstalk during readout. Among the most promising materials to implement a holographic storage device are photorefractive inorganic crystals. They offer a large effective optical nonlinearity available at low continuous optical power densities of a few mWcm−2 and long storage times for fixed holograms. Due to erasure of previously recorded holograms, special recording schemes have to be used to achieve

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a constant diffraction efficiency of all stored holograms. Nowadays, the storage capacity in these materials is limited by the intrinsic scattering noise. But still photorefractive storage systems with a storage capacity of few tens of Gbit/cm−3 seem to be feasible. Although this number has not been achieved yet, it has been shown that recording of 1-Mbit data pages with 1-Gbit/s data-transfer rate upon readout are possible in practice. These large storage densities are of great importance if these systems are to be applied to perform image processing. We have discussed the influence of the volume nature of the stored data pages on the performance characteristics of the two common types of optical correlators (joint Fourier transform correlator and VanderLugt-type correlator). The latter is especially attractive in using angularly multiplexed holograms for storage of matched templates and thus for performing several pattern recognitions in parallel. Also, nonlinear holographic associative memories gain from the recent advances in photorefractive storage systems. The latest results for this kind of optical processing task have been summarized, and an associative memory system constructed in our laboratory has been described in deeper detail. Nevertheless, while a few holographic storage devices with limited capacity are present in niche markets, practical commercial systems for mass-storage or associative memories do not yet exist. Material problems remain the principal obstacle. In the case of holographic associative memories, association at TV frame rates with a large set of stored images still appears as a distant goal. Moreover, in order to implement higher-order tasks (such as rotation and scale invariance) these systems have to be incorporated into general neural-network architectures.

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5 True-Time-Delay Adaptive Array Processing Using Photorefractive Crystals G.R. Kriehn1 and K. Wagner2 1

2

California State University, Fresno, Electrical and Computer Engineering Department, 2320 E. San Ramon Ave. M/S EE94, Fresno, CA 93711-8030 [email protected] University of Colorado, Boulder, Optoelectronic Computing Systems Center, Department of Electrical and Computer Engineering, Boulder, CO 80309-0425 [email protected]

5.1 Introduction to True-Time-Delay Array Processing Radio frequency (RF) signal processing has proven to be a fertile application area when using photorefractive-based, optical processing techniques. This is due to a photorefractive material’s capability to record gratings and diffract off these gratings with optically modulated beams that contain a wide RF bandwidth, and include applications such as the bias-free time-integrating correlator [1], adaptive signal processing, and jammer excision, [2, 3, 4]. Photorefractive processing of signals from RF antenna arrays is especially appropriate because of the massive parallelism that is readily achievable in a photorefractive crystal (in which many resolvable beams can be incident on a single crystal simultaneously—each coming from an optical modulator driven by a separate RF antenna element), and because a number of approaches for adaptive array processing using photorefractive crystals have been successfully investigated [5, 6]. In these types of applications, the adaptive weight coefficients are represented by the amplitude and phase of the holographic gratings, and many millions of such adaptive weights can be multiplexed within the volume of a photorefractive crystal. RF modulated optical signals from each array element are diffracted from the adaptively recorded photorefractive gratings (which can be multiplexed either angularly or spatially), and are then coherently combined with the appropriate amplitude weights and phase shifts to effectively steer the angular receptivity pattern of the antenna array toward the desired arriving signal. Likewise, the antenna nulls can also be rotated toward unwanted narrowband jammers for extinction, thereby optimizing the signal-to-interference-plus-noise ratio.

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5.1.1 Narrowband Processing and Limitations Beam steering and jammer nulling can be performed on RF signals from onedimensional (1-D) or two-dimensional (2-D) antenna arrays—which can either be regularly sampled on a uniform grid of antenna element positions, or sparsely or irregularly/randomly sampled in space. For simplicity, we will limit the discussion to a linear, 1-D, equi-spaced antenna array (although the optical BEAMTAP system presented here works equally well for 2-D and even irregularly-spaced arrays). In such a 1-D array, an RF signal at frequency ω arriving from a far-field source becomes a plane wave propagating at an angle θ with respect to the array axis, resulting in RF signals, sn (t) = ei(ωt+φn ) , from the nth antenna element with a phase shift φn = n D(ω/c) cos θ , where D is the element spacing and c is the speed of light. To coherently combine these RF signals, each signal must be multiplied by a conjugate phase factor and summed, which provides a beam-steered  output s(t) = n w n sn (t)—where the complex weights w n = e−iφn |an | may include an additional amplitude taper |an | that can be used to lower the sidelobe level compared to a simple sinc function in the case of a uniformly weighted array. Optically, photorefractive crystals can be used to implement the weighted multiplications by recording appropriate gratings that have the desired strengths and phase shifts. Subsequent diffraction of the single-sideband, optically-modulated signals coming from each antenna element off these gratings and combining the resulting beams with a lens onto a photodetector performs the coherent summation via heterodyne detection with a strong reference beam. The detected output thus produces a current proportional to the desired beam-steered signal. Unfortunately, as the RF bandwidth B of an antenna array, the antenna diameter (N D for an N element 1-D equi-spaced linear array), and the maximum detectable angle from boresight θmax all increase, the propagation delay time across the aperture T = N D cos θmax /c eventually becomes resolvable within the system bandwidth (i.e., when T B > 1). As a result, narrowband techniques are no longer adequate (as is the case when using the simple phased-array beam steering process described above), because these narrowband phased-array beam steering approaches produce unwanted beam squint. This is an undesired rotation of the main beam receptivity angle as a function of frequency, which decreases the signal strength and bandwidth of the final output. In addition, narrowband processing also results in null squint (which is the deleterious rotation of adaptively formed antenna nulls that are pointed toward jammers), so that when the jammers hop to new frequencies, they are no longer nulled, and these interfering signals leak through into the system output.

5.1.2 Wide Fractional Bandwidth Processing To solve this problem, wideband true-time-delay beam-steering techniques become necessary, in which the phases φn = n D(ω/c) cos θ are recognized as being linearly dependent upon the frequency, but are compensated for over a specific bandwidth. This allows for the diffraction of a coherently modulated optical field

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off of a phase-shifted photorefractive grating to remain independent of the RF offset frequency for frequencies as large as the inverse optical transit time through the photorefractive crystal (30 GHz for a 5 mm thick crystal of index 2.0), rather than the much smaller frequency offset in which the phase will change in a resolvable manner due to the inverse RF transit time across the array (10 MHz for a 30 m diameter array). In conventional array processors, tapped-delay lines are used to provide delayed signal histories of every antenna element—but unfortunately, the optical implementation of N wideband tapped-delay lines is problematic for N > 32 when using multi-channel acousto-optic devices [7, 8], or for N > 100 when using ultra-wideband hyperfine optical delay-line technology [9]. Therefore, scaling these optical implementations using traditional time-delay-and-sum architectures [10] for very large antenna arrays using hundreds (or even many thousands) of elements becomes impractical, and the full potential for massive parallelism using photorefractive array processors appears to then be limited. We have, however, solved this problem with the introduction of the optical BEAMTAP (Broadband Efficient Adaptive Method for True-time-delay Array Processing) algorithm [11], in which the linear operation of time delay of the antenna element signals is commuted with the complex weight multiplication. After multiplication, all of the equivalent delays are then combined, resulting in a new sum-and-time-delay algorithm. Although this equivalent algorithm provides no fundamental advantage for digital implementations, it enables the use of a photorefractive crystal (which has little internal time delay), for processing RF signals from large wideband antennas by volume multiplexing the adaptive weights as required by BEAMTAP. The volume multiplexing capability of a photorefractive crystal for up to a billion independently resolvable gratings is sufficient to accommodate an antenna array with up to a million elements using as many time delays as necessary (which, in this case, is roughly a thousand for octave bandwidths).

5.1.3 Chapter Outline Previously, we have developed photorefractive, signal-processing architectures that use an electronic feedback closed-loop adaptive process to record the photorefractive grating with the required amplitude and phase in order to optimally beam steer and jammer null the signals from the array, even if the fiber feed network has unwanted phase errors or arbitrary topology mappings from the antennas to the optical processor [6]. In this chapter, we describe the improved wideband results of an experimental photorefractive implementation using the BEAMTAP algorithm. We first provide a brief overview of the algorithm in addition to a review of the key device required for sum-and-time-delay based processing—an accumulating traveling photodetector known as the traveling-fringes detector—developed by Dolfi et al. at Thales [12, 13]. Next, we describe our technique for simultaneous reading and writing of photorefractive gratings for wideband RF modulated beams from an array of modulators that cover a wide angular bandwidth at the input and output using the parallel-tangents, equal-curvature condition [14, 15]. Finally, we combine these technologies into an operating BEAMTAP system that

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uses an emulated input from an antenna array based on a traveling-wave, acoustooptic device. Results of simultaneous main beam forming and wideband jammer nulling are presented for the first time.

5.2 Output Time Delay and the BEAMTAP Algorithm The conventional broadband time-delay-and-sum approach to adaptive beam forming and jammer nulling requires that a true-time-delay, tapped-delay line be used for every element in a phased-array antenna. Unfortunately, this approach is both inefficient and difficult to implement optically, since a complex spatio-temporal, optical time-delay module (such as an acousto-optic deflector) is required for every element in the array. A solution to this problem can be found if, instead of choosing to delay the signals after they are first detected at the antenna array, the signals are alternatively delayed at the output. This has the tremendous advantage of only requiring a single tapped-delay line for a formed beam, irrespective of the antenna array size. The weight matrix can then be thought of as a structure analogous to a switching matrix, which is used to route signals from each antenna element to an appropriate delay and allows the signals to be time aligned for coherent summation at the output. The number of resolvable delays required to coherently sum these signals will be proportional to the size of the antenna array, but only a single delay line will be required. An example of this is shown in Fig. 5.1. A chirped signal that is incident upon the antenna array is detected at each antenna element, whereby the detected signals are routed through the switching matrix and relayed to their respective time delays for coherent summation. Since the spatio-temporal RF chirp arrives at the top antenna element first, the greatest amount of time delay must be applied to this particular signal before it can be coherently summed with the signals arriving at other antenna elements later in time. Each successive signal arriving at lower antenna elements requires less delay, and since the RF signal arrives at the bottom antenna element last, no time delay is required at this particular element. After leaving the tapped-delay line, the signals are time aligned for coherent summation at the final output. Adaptive beam forming can be integrated into this type of architecture by providing a method by which the switching matrix can be programmed to route the signals to their respective time delays. To accomplish this, a second tapped-delay line is required, giving rise to BEAMTAP. Figure 5.2 illustrates the architecture for the optical implementation of BEAMTAP. As seen in the figure, a single coherent laser is split using two beamsplitters to drive both a fiber feed network and the BEAMTAP processor. The fiber feed network from the phased-array antenna is shown on the left, in which an RF signal impinging upon the array is detected at each antenna element and used to drive an array of electro-optic modulators (EOMs). The modulated light from the EOMs, which contains the RF signal information from each of the antenna elements, is

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F I G U R E 5.1. Using output time delay, only a single accumulating tapped-delay line is required to coherently sum the signals from an antenna array irrespective of the array size. The signals from each of the antenna elements are routed through a switching matrix that relay the signals to their respective time delays so that they can be coherently summed at the output. By applying a feedback signal that contains a desired steering signal to a second input delay line, the appropriate switches can be chosen for the beam forming operation.

then remotely fed via fiber to the BEAMTAP processor and is imaged at the plane z 0 into a photorefractive crystal by the lens system L 0 . Within the processor, the diffracted light from an acousto-optic deflector (AOD) located at the plane x0 interferes with the remotely fed fiber signals sn (t) from the array and forms holographic gratings within the photorefractive crystal (PRC). Diffraction of the phased-array signals off this grating is then detected with a special accumulating, scrolling detector known as the traveling-fringes detector (TFD), which has a carrier velocity matched to the magnified acoustic velocity of the AOD by the lens systems L 1 and L 2 —producing a resonant charge carrier distribution q(x2 , t). The TFD heterodyne detects, delays, and coherently sums the signals from each of the individual antenna elements to form an output signal o(t). The output signal o(t) is then amplified, filtered, and subtracted from a desired signal d(t) to form a feedback signal f (t), which is capable of updating

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F I G U R E 5.2. Optical implementation of BEAMTAP. The main components include electro-optic modulators (EOMs) and the fiber feed network, an acousto-optic deflector (AOD), a photorefractive crystal (PRC), and the traveling-fringes detector (TFD).

the holographic weight matrix formed within the photorefractive crystal to ensure both adaptive beam forming and jammer nulling. In this system, the TFD implements the tapped-delay line used for coherently summing the signals from each of the antenna elements; the PRC acts to spatially route holographically diffracted signals from the phased-array antenna to their appropriate delays on the TFD for coherent summation (which is analogous to the switching matrix); and the AOD serves as the second tapped-delay line necessary for weight adaptation of the holographic gratings. The final output o(t) is given by a correlation-convolution integral, T D /2

o(t) = R κ2 −TD /2

N 

sn (t − tn + τx − tr2 )

n=1

t−T g2 sn∗ (t2 − tn + τx − tr2 ) [d(t2 + (T − td )) − g1 o(t2 )] dt1 dτx , × −∞

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in which the feedback signal [d(t2 + (T − td )) − g1 o(t2 )] correlates against fixed, delayed versions of the signals from the antenna elements sn∗ (t2 − tn + τx − tr2 ) to form the holographic weight matrix within the photorefractive crystal—which is then convolved with the delayed signals from the antenna sn (t − tn + τx − tr2 ), and coherently summed across the N elements by the time aperture TD of the TFD. Although complex, the process is equivalent to the time-domain least-meansquare (LMS) beam former, [16, 17] where the output is given by  t  sn (t − mτ ) sn∗ (t2 − mτ ) [d(t2 ) − o(t2 )] dt2 , (5.1) o(t) = m

n

−∞

which performs the necessary correlation-convolution integral to maximize the signal-to-interference plus noise ratio (SINR). In the case of optical BEAMTAP system, the summation over the M delays within the tapped-delay line is replaced with a continuous accumulation over the time aperture TD of the TFD. The remaining time delays (such as tn , τx , tr 2 , etc.) are simply a consequence of maintaining causality within the system, and are explained in detail in the theoretical development of the BEAMTAP architecture [11]. A critical issue with this system is the need for a high degree of isolation between the reading and writing beams within the photorefractive crystal, while simultaneously maintaining Bragg-matched diffracted readout over a wide angular aperture, since signals arriving at the antenna array may span a large angular spectrum. A high degree of isolation is required because any leakage of the writing beam from the feedback AOD into the output will limit the amount of feedback gain—and therefore the jammer null depth—available to the processor before the system breaks into oscillation [6]. Therefore, it becomes necessary to find an experimental geometry in the photorefractive crystal that satisfies both of these requirements. This can be accomplished with polarization-angle, read-write multiplexing in barium titanate (BaTiO3 ) using the parallel-tangents, equal-curvature condition [14, 15]. After examining the TFD in more detail, discussion of this geometry within BaTiO3 (as it relates to the experimental implementation of BEAMTAP), forms the basis for the rest of the chapter.

5.3 The Traveling-Fringes Detector The traveling-fringes detector (TFD) used in the photonic implementation of the BEAMTAP processor consists of a photo-conducting layer of gallium arsenide (GaAs) that exploits a resonantly-enhanced amplitude of the traveling-wave photo-carrier grating for efficient broadband detection of optically-modulated, high-frequency RF signals as moving interference patterns, and is a key enabling technology for BEAMTAP. Dolfi originally developed this device due to its large conversion efficiency at high (microwave) frequencies and a high optical saturation level owing to its large detection volume [13]. Adapting the TFD to BEAMTAP, the device provides the necessary opto-electronic conversion for

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broadband detection, as well as providing the opto-electronic architecture with the necessary physical time delay required for true-time-delay processing. This section briefly reviews the TFD originally designed for narrow fractional bandwidth detection at a high microwave carrier frequency (1–20 GHz), and shows how to use the TFD in combination with a traveling-wave acousto-optic deflector in the BEAMTAP system for wide fractional bandwidth detection. The TFD uses a photoconductive layer of GaAs to detect RF signals based upon the synchronous drift of photo-generated carriers with a moving interference pattern [12]. Interfering two beams of coherent light tilted by an angle θ with a difference frequency f d , yields a fringe velocity v f = f d λ/ sin(θ ). For broadband operation, in order to keep the fringe velocity constant—even at varying frequencies—the angle of the incident light needs to be scanned in proportion to its frequency, which is exactly the case for light diffracted from a traveling-wave deflector such as an acousto-optic device, as shown schematically in Fig. 5.3. The light that is incident on the photoconductive layer of the detector produces photogenerated charge carriers with a drift velocity v(n, p) = μ(n, p) E a , where μ(n, p) is the mobility of the charge carrier being resonated (μn for electrons and μ p for holes) and E a is the applied field. By varying the applied field transversely across

Z AcoustoOptic Device

Traveling Fringes Detector

Moving Fringes

p(x, t)

X Accumulating Resonant Hole Distribution

RF Input

f

920

L

1

Modulated Signal

L

2

Mixer

Ouput

C

Carrier LNA

f

MHz

920

Signal BW f

250

250

L

920 MHz

R = 50 Ω L

+ V _

MHz

F I G U R E 5.3. Detection of a broadband RF signal using an acousto-optic device imaged onto a velocity matched traveling fringes detector. Traveling fringes produced by the interference of a diffracted beam from the AOD with the DC beam are resonated with the carrier drift velocity by varying the field applied transversely across the photoconductive region of the detector and the magnification.

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the layer, a resonance peak occurs using an electron resonance when the fringe velocity v f equals vn [13]. The accumulation and detection of the intensity pattern in the TFD can be modeled as a drifting diffusion equation in a semiconductor for the motion of photogenerated carriers under the influence of an applied electric field [13, 18], such that  o(t) = R

0 −TD v f



t

−∞

 

I (x, t ) 

1 4π Dn (t −

t )

e

−

[x−vd (t−t  )]2 4Dn (t−t  )

− (t−t τ

)



dt  d x.

(5.2)

Here the length L = TD vd , R is the detector responsivity (A/W), TD is the time window over which the interference pattern is accumulated by the detector, Dn = kB T μn is the diffusion constant of the electrons, and τ is the carrier lifetime. q Integration of Eq. 5.2 yields an exponentially decaying spatial sensitivity away from the electrode, given by exp(−x/τ vd ), that can be partially compensated for by shifting the illuminating Gaussian spot away from the collecting electrode; and a position-dependent, bandwidth-limiting, transfer function due to carrier 2 3 diffusion, √ T (ω, x q ) = exp(−x Dn ω /vd ). This leads to a 3 dB roll-off at 2π f c ≈ vd log2/ Dn τ . For 1:1 imaging of an acousto-optic deflector with a velocity 4.2 mm/μs onto a GaAs traveling-fringes detector with μn = 8500 cm2 /Vs and a lifetime τ = 10 ns, this yields a 3.2 GHz corner frequency, which is as wide as high frequency AODs can achieve. The frequency limit of the broadband TFD can be further increased through the use of lower mobility holes (to 20 GHz), or by magnifying the acousto-optic deflector onto the TFD and increasing the applied voltage correspondingly (thereby increasing v f = Mva and thus f c ). When the TFD is operating in this regime, diffusion is negligible over the detection aperture and will not limit the system bandwidth as the charges drift across the sensitive photoconductive region of the device. The output can be approximated as an ideal accumulating traveling-wave delay line.

5.4 Photorefractive Volume Holography using Polarization-Angle, Read-Write Multiplexing Photorefractive crystals are useful for spatial-temporal information processing of coherent optically-modulated signals due to the real-time, read-write capabilities of the holographic medium. However, when used to write volumetric holographic gratings for real-time applications, it is necessary to isolate the recording, or writing beams, from the diffracted read-out beam. This isolation is an extremely important factor in adaptive feedback systems because it determines the amount of feedback gain, and therefore the jammer null depth, of the overall processor [19]. This section is therefore devoted to exploring the usefulness of read-write multiplexing for the BEAMTAP system, and demonstrates an important holographic application using photorefractive crystals.

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5.4.1 Parallel-Tangents, Equal-Curvature Condition The parallel-tangents, equal-curvature condition [14, 15] is an effective geometry that has been previously used to achieve a high degree of read-write isolation while simultaneously maximizing the Bragg-matched angular aperture between the orthogonally polarized writing and reading beams. The process involves matching the tangents and curvatures of the ordinary and extraordinary momentum surfaces, as shown in Fig. 5.4. The geometry is such that the grating vector K g written between two ordinary beams kw1 and kw2 at angles θw1 and θw2 , respectively, will be Bragg matched to the extraordinary polarized reading beam when read out with a k-vector kr at an angle θr . In this configuration, the ordinary writing beams do not couple due to the small effective electro-optic coefficient r22 , while the readout beam accesses the large coefficient r42 in barium titanate (BaTiO3 ) or r33 in strontium barium niobate (SBN), producing an efficiently diffracted output. The diffracted output angle θeq , with k-vector keq , is chosen so that both the tangents and the curvatures of the ordinary and extraordinary surfaces are matched at θw2 and θeq . For small deviations away from this exact geometry, the Bragg matching condition is satisfied to first and second order so that if one of the writing beams contains a large angular spectrum and multiple grating vectors K g are recorded within the material, the entire angular spectrum can still be Bragg matched and read out using the reading beam kr .

F I G U R E 5.4. Geometry which describes the parallel-tangents, equal-curvature condition. Two ordinary polarized beams, kw1 and kw2 with angles θw1 and θw2 , respectively, write a grating K g in the photorefractive crystal. The grating is read out by an extraordinary polarized beam kr at an angle θr , producing a diffracted beam keq at an angle θeq . Both the writing beam kw2 and the diffracted beam keq can subtend a large angular bandwidth while still maintaining excellent Bragg matching.

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5.4.2 Parallel-Tangents, Equal-Curvature Condition in BaTiO3 Solving for the parallel-tangents, equal-curvature condition in BaTiO3 to determine an experimental setup for polarization-angle, read-write multiplexing in BEAMTAP requires that the equal-curvature condition be derived in a region that simultaneously satisfies the parallel-tangents condition at an angle θeq from the c-axis of the crystal. In the x z plane, the ordinary and extraordinary surfaces map out a circle and ellipse, respectively, where  2 k z2 2π k x2 + 2 = (5.3) 2 no no λ  2 k z2 2π kx2 + 2 = . (5.4) 2 ne no λ Converting to polar coordinates, k x = k sin(θ ) and k z = k cos(θ ), which allows for the tangents of the two surfaces to be solved for as a function of θw2 and θeq —the angles of the second ordinary write beam and the extraordinary read beam, respectively:   dkz = −tan(θw2 ) (5.5) dk x or d    2 dkz no =− tan(θeq ), (5.6) dk x ext ne where setting the two equations equal to each other sets the tangents of the two surfaces equal to each other as well [20]. The extraordinary angle θeq is provided here as a function of the ordinary angle θw2 (both of which are internal to the crystal), giving   ne 2 −1 tan(θw2 ) . (5.7) θeq = tan no Next, to obtain the second-order phase matching condition, a specific set of ordinary and extraordinary angles satisfying the parallel-tangents condition is chosen so that the diffracted read-out beam is at an angle where the ordinary and extraordinary momentum surfaces have equal curvatures in the x z plane. For a function y = f (x) in the Cartesian coordinate system, the curvature ρ is defined as the rate of change of the slope of the tangent of f (x) with respect to the arc length, where



f (x)

ρ= . (5.8) {1 + [ f  (x)]2 }3/2 The curvatures of the ordinary and extraordinary surfaces are thus given by [14] 1 (5.9) ρ(θ)or d = ko (n o /n e )2 ρ(θ )ext =  (5.10)

3/2 , ko 1 + [n e (θ)sinθ ]2 n 2o − n 2e /n 4e

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where the extraordinary index of refraction is a function of the angle θ and is equal to  −1/2 cos2 θ sin2 θ n e (θ) = + 2 . (5.11) n 2o ne Substituting, Eq. 5.11 into the curvature of the extraordinary momentum surface equation in Eq. 5.10, and setting it equal to the curvature of the ordinary momentum surface equation in Eq. 5.9, defines the equal-curvature angle [14], with the elegant although non-intuitive solution cos(2θeq ) =

10/3

− no ne − ne

10/3

+ no ne + ne

no no

2/3 8/3

10/3

2/3 8/3

10/3

.

(5.12)

Finally, when the equal-curvature angle is defined, it can be substituted into the parallel-tangents equation to determine the angle on the ordinary momentum surface that simultaneously satisfies both the parallel-tangents and the equalcurvature condition. Figure 5.5 shows the method by which the four angles defining this geometry are calculated. The ordinary k-circle is first translated at the point kw2 by a vector

R to the parallel-tangents, equal-curvature angle (given by Eq. 5.12) at the point keq , so that the two surfaces are brought into coincidence. Then, a second c-axis Translated Ordinary Momentum Surface

^z

θ eq θ w2 θr θ w1

Parallel ko Tangents ΔR

k e (θ) k eq

k w2

Kg

Kg

kr

ΔR

k w1

^x

Intersection Ordinary Momentum Surface

Extraordinary Momentum Surface

F I G U R E 5.5. Geometric construction by which the angles of the 4 beams are found using the parallel-tangents, equal-curvature condition. The ordinary k-circle is translated by

R to the parallel-tangents, equal-curvature angle on the ellipse, and a second intersection between the translated circle and ellipse determines the angle of the extraordinary read beam θr . Translating from the read beam and parallel-tangents, equal-curvature angle by − R determines the two writing beam angles, θw1 and θw2 , respectively.

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intersection between the circle and ellipse is sought, which determines the angle of the extraordinary-polarized read beam θr . However, the exact parallel-tangents, equal-curvature condition presented by Sarto [14] must be modified, since there is only one additional intersection point for negative uniaxial crystals such as BaTiO3 , which occurs at an inconvenient angle greater than 180◦ . Because the 180◦ angle crossing is difficult to work with experimentally, the symmetry of the system can be broken by slightly deviating away from the exact parallel-tangents, equal-curvature condition by a degree or two. With the deviation included, an additional intersection point at an angle close to θeq arises from the re-introduction of a small parabolic second-order momentum mismatch—and this provides the true representation of Fig. 5.5. Finally, once this intersection  and the point is found, the ordinary k-circle can be translated back by − R, where kr − R touches the original ordinary momentum surface determines the k-vector (and hence, the angle) of the final write beam, kw1 . For BaTiO3 at an operational wavelength of 532 nm, n o = 2.4746 and n e = 2.4118 [21]. Using Eq. 5.12, the equal-curvature angle is thereby found to be θeq = 53.6877◦ , and using the parallel-tangents equation, the corresponding ordinary momentum surface angle θw2 = 55.0827◦ . With the extraordinary, equal-curvature angle and the ordinary, paralleltangents angles determined by the ordinary and extraordinary indexes of refraction, the translation vector R can be defined so that the ordinary momentum surface at the parallel-tangents angle at kw2 can be translated to the paralleltangents, equal-curvature angle on the extraordinary momentum surface. R is given by

R = ke (θe ) − ko (θw2 ) = [ke (θe ) sinθe − ko sinθw2 ] xˆ + [ke (θe ) cosθe − ko cosθw2 ] zˆ . (5.13) The exact equal-curvature condition is broken here by deviating the angle θw2 by 1◦ increments over a ± 10◦ angular range, with the “equal-curvature” angle, θeq , being re-calculated at every deviation. With the symmetry broken, an additional crossing near θeq is introduced. In the translated coordinate frame, the grating vector is given by K g = [kw2 + R] − [kw1 (θ) + R] = keq − [kw 1 (θ) + R], so the momentum mismatch of the read beam kr as a function of θ can be defined as k = (keq − K g ) − kr (θr ) = [kw1 (θw1 ) + R] − kr (θr ). Figure 5.6 shows the momentum mismatch over angles that are near the parallel-tangents, equalcurvature angle—providing a clearer picture of the exact nature of the mismatch. In Fig. 5.6, only curves with momentum mismatch ≤ ±π/L (where L = 2 mm is the length of the crystal), are shown for angles ≤ θeq (θeq , θeq − 1◦ , . . . , θeq − 5◦ ). Several angles > θeq are certainly within the bounds of a ±π/L momentum mismatch as well, and could have just as easily been chosen, but were not considered for the sake of simplicity. The bounds on the momentum mismatch were chosen to lie within ±π/L so that the Bragg efficiency of an extraordinary plane wave diffracting off a grating

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45o +c Axis θw2 Angular Spread where Δk < π/L Crystal Normal

X

Z

x z

F I G U R E 5.6. Momentum mismatch k and k z surrounding the parallel-tangents, equalcurvature angle. Notice that at the exact parallel-tangents, equal-curvature angle (bold) the angular bandwidth over which the momentum mismatch is approximately zero is maximized. As the exact angle is deviated away from its optimum value, the momentummatched angular bandwidth decreases, but an additional crossing where the momentum mismatch equals zero arises, allowing for a convenient read-write multiplexing geometry.

written between two properly oriented ordinary writing beams would only suffer a 4 dB loss when probing a wide angular spectrum in the second writing beam. Although this restriction is somewhat of an arbitrary constraint for determining an experimental geometry, it allows for the momentum mismatch to remain small, minimizes the loss of Bragg-matched angular bandwidth surrounding the paralleltangents, equal-curvature angles, constrains the additional crossing to remain at a convenient angle to work with experimentally, and provides greater experimental tolerances during alignment. Since Fig. 5.6 shows that there are several optimal read-beam angles that can be chosen to satisfy the ±π/L constraint, there is considerable flexibility in determining the exact experimental geometry, depending upon desired location of the new crossing between the ordinary and extraordinary momentum surfaces (i.e., where k = 0). Figure 5.6 also shows that the momentum mismatch has been projected into the z-coordinate frame of the crystal cut since the beams surrounding the parallel-tangents, equal-curvature condition are traveling approximately 45◦ with respect to the +c axis of the crystal, which in a 45◦ cut piece of BaTiO3 , is in the z direction. Once exiting the crystal, the beams then travel (more or less) in the z direction of the laboratory coordinate frame. The difference between the actual momentum mismatch k (dashed lines), and the momentum mismatch projected along the z-axis (solid lines), remains negligible for the purposes of these calculations. Figure 5.7 shows an example of the theoretical Bragg efficiency as the angle of the second writing beam is deviated away from the chosen parallel-tangents, equalcurvature angle, but still satisfies the 4 dB criteria over a broad angular spectrum— shown here to span between 20◦ –25◦ . For a crystal of length L = 2 mm, the

F I G U R E 5.7. Bragg efficiency vs. angle for the chosen geometry of the experimental read-write multiplexing system. The deviation from the optimum condition is chosen to satisfy a 0.5 dB Bragg efficiency criteria.

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Bragg-matching diffraction efficiency is given by ⎡ 2 ⎤

kz L sin( 2 ) ⎦. η = 10 · log10 ⎣

(5.14)

kz L 2

The figure is also representative of the actual geometry chosen for the experimental read-write multiplexing system, where the Bragg efficiency was arbitrarily chosen so that it did not drop more than 0.5 dB below its maximum value surrounding the parallel-tangents, equal-curvature condition. This corresponds to an ordinary momentum-surface angle θw2 = 51.0827◦ [−4◦ from the optimum θw2 (opt)], and defines the extraordinary equal-curvature angle to be θeq = 49.6350◦ . This produces an anticipated theoretical Bragg-matched angular aperture of about 20◦ internal to the crystal and 50◦ external to the crystal. With the chosen geometry, the intersection between the extraordinary and the translated ordinary momentum surface occurs at the extraordinary angle θr = 61.9200◦ , and the final ordinary writing angle θw1 is found by translating the k-vector corresponding to the extraordinary read angle θr back by − R. Specifically, k x (θw1 ) = ke (θr )sinθr − Rx (θ R )

(5.15)

k z (θw1 ) = ke (θr )cosθr − Rz (θ R ),

(5.16)

which forces the final ordinary write-beam angle θw1 to be tanθw1 =

k x (θw1 ) , k z (θw1 )

(5.17)

so that θw1 = 63.1557◦ . Finally, to determine the experimental geometry of the polarization-angle, readwrite multiplexing system, the external angles must be calculated. For a 45◦ cut crystal, the internal and external theoretical angles normal to the surface, as determined by the Anisotropic Snell’s Law, are given in Table 5.1, with the overall geometry of the system shown graphically in Fig. 5.8. Because the second writing beam is shown to contain an angular spectrum θ in Fig. 5.8, upon reconstruction, T A B L E 5.1. Theoretical geometry for the polarization-angle, read–write multiplexing system using the parallel-tangents, equal-curvature condition

θw1 θw2 θr θeq

θ

Internal angles (w.r.t. c axis)

Internal angles (w.r.t. normal)

External angles (w.r.t. normal)

63.1557◦ 51.0827◦ 61.9200◦ 49.6350◦ 20◦

18.1557◦ 6.0827◦ 16.9200◦ 4.6350◦ 20◦

50.4521◦ 15.2017◦ 44.8981◦ 11.3600◦ 50◦

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the entire spectrum will be reconstructed as long as its internal angular aperture remains less than about 20◦ , as alluded to in Fig. 5.7.

5.4.3 Experimental Demonstration of Wide-Angular Aperture Readout Figure 5.9 shows the experimental setup of a system used to demonstrate the concept of the parallel-tangents, equal-curvature condition. Two ordinary beams are input into the crystal at angles θw1 and θw2 , with the θw2 beam focused into the crystal with a lens having an effective numerical aperture (due to space constraints) of NA = 0.5 to test the Bragg-matched angular readout bandwidth, providing an angular aperture of about θext = 53◦ external to the crystal and just over θint = 20◦ internal to the crystal. The third beam, which is extraordinary polarized, is input into the crystal at an angle θr . This read-out beam diffracts off the grating within the photorefractive crystal, is passed through a horizontal polarizer to filter out the writing beams, and imaged onto a charge-coupled device (CCD) camera. Experimentally, the reading beam angle was only deviated by 0.08◦ from its theoretical angle in order to achieve maximum diffraction efficiency over as broad an angular aperture as possible. This provides good confidence in the

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F I G U R E 5.9. Experimental setup using the parallel-tangents, equal-curvature geometry. Two ordinary write beams are used to write a grating in the photorefractive crystal, one of which contains a wide angular aperture. The extraordinary polarized beam efficiently reads out the hologram, reconstructing the wide angular aperture recorded in the BaTiO3 crystal.

material parameters utilized in the parallel-tangents, equal-curvature condition’s theoretical derivation as well as the accuracy of the 45◦ crystal cut. Figure 5.10 shows images of the light that diffracts off the hologram and is captured by the CCD camera when the read beam is rotated about the optimum readout angle θr by 0.05◦ increments. The top left-most picture shows the output at an angle θopt − 0.20◦ ; the bottom right-most picture shows the output at an angle θopt + 0.20◦ ; and the incremental 0.05◦ changes can be seen in the progression of images along each of the three columns. Notice that the middle pictures in the central columns show the optimum diffracted angular bandwidth, spanning 20.2◦ internal to the crystal (denoted by the light present in the horizontal direction). This compares extremely well with the theory just presented, since the internal Bragg-matched angular bandwidth was expected to be on the order of about 20◦ . Furthermore, these results also follow the experimental and theoretical intensity plots first presented by Sarto [14], who performed his experiments at a wavelength of 514 nm. Notice that the angular bandwidth near the parallel-tangents, equalcurvature geometry is actually larger than the angular bandwidth in the Braggdegenerate (vertical) direction. This is because the difference in curvature between

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F I G U R E 5.10. Images showing the diffracted output of a beam using the paralleltangents, equal-curvature condition over a large angular bandwidth. Each picture shows the diffracted output as the read beam is deviated away from the optimum angle by 0.05◦ increments. The middle picture in the central column shows the optimum diffracted angular bandwidth, which occurs at θr .

the sphere and the ellipsoid of revolution changes more quickly in the Braggdegenerate direction than in the plane of the parallel-tangents, equal-curvature condition. This clearly indicates that large z-dimensional angular bandwidth and efficient diffracted readout is possible. The high degree of isolation between the reading and writing beams is verified in Sec. 5.5. The asymmetry between the pictures when the read-beam angle is deviated either negatively or positively from the optimal readout angle occurs as a result of the nature of the Bragg mismatch introduced by the change in curvature of the ellipsoid of revolution after the sub-optimally aligned read beam attempts to diffract off the grating in the crystal. When the read beam is deviated negatively from its optimal angular alignment (Fig. 5.10a–d), the central portion of the beam is still sufficiently Bragg matched to allow a subset of k-vectors to propagate. However, at large angles deviating away from θeq , the diffracted waves no longer

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meet the phase matching requirements necessary to propagate any light. This is because the diffracted k-vectors lie completely inside the ellipsoid of revolution. Conversely, when the read beam is deviated positively from its optimal angular alignment (Fig. 5.10f–i), the central angular components try to launch k-vectors that now lie outside of the ellipsoid of revolution. Again, light will not propagate. However, the sphere and ellipse still intersect at angles that deviate away from θeq , and these angular components are sufficiently Bragg matched to allow propagation within the crystal.

5.4.4 Polarization-Angle, Read-Write Multiplexing with BaTiO3 Once the polarization-angle, read-write multiplexing system was characterized using the parallel-tangents, equal-curvature condition, an experimental open-loop version of the optical BEAMTAP processor was built. Figure 5.11 shows the experimental implementation of the system, including the polarization-angle, read-write multiplexing system used to isolate the writing beams from the reading beam within the photorefractive crystal. Vertically polarized light from a Coherent 5 W Verdi laser is initially passed through a half-wave plate and Glan laser prism to act as a variable attenuator before being spatially filtered and collimated by a 50 mm focal length lens. The beam is then split into 2 beams via an additional wave plate and polarizing beam splitter. One beam serves as a clean reference beam for detection at the traveling fringes detector (TFD), and the other beam is passed through a 3:1 beam compressor. When split by an additional wave plate and

F I G U R E 5.11. Detailed layout of the complete experimental photorefractive BEAMTAP system employing polarization-angle read-write multiplexing for isolation of the ˆ ˆ o-polarized writing beams from the e-polarized reading beam. Key components include the two AODs (acousto-optic deflectors), PRC (photorefractive crystal), and the TFD (traveling-fringes detector).

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polarizing beam splitter, and Fourier transformed into the acousto-optic devices (AODs) using 500 mm focal length lenses, 225 μm × 75 μm spots are produced in each of the wideband AODs (which are well matched to the 75 μm piezo-electric transducer height). The AODs serve as the entrance into either the feedback arm, or the array emulator arm of the processor. Light which enters the emulator arm diffracts off the acoustic gratings of the AOD, which emulates a well-sampled array of signals from an antenna array with an off axis incident RF plane wave. (This yields a traveling-wave signal across the array aperture, just like the modulation produced by the AOD.) The first diffracted order is allowed to pass through an iris while the DC beam is blocked, with the diffracted light then being collimated and split into two orthogonally polarized beams by a Glan Thompson polarizing beam splitter—each with excess of 40 dB ˆ and e-polarized ˆ polarization isolation. The obeams are then focused and imaged (using a single 2f telescopic lens system) into the photorefractive crystal (PRC) using 150 mm and 200 mm focal length lenses, respectively. Finally, as the two ˆ and e-polarized ˆ spots are imaged into the crystal, the obeams are rotated to the write-beam angle θw2 and the parallel-tangents, equal-curvature angle θeq using two pairs of mirrors. The wide angular bandwidth writing angle and the paralleltangents, equal-curvature angle are used for these two beams because the signals arriving from the antenna array may span a large angular bandwidth—in the emulated antenna, the AOD diffraction angles imaged through the photorefractive crystal span an angular range of about 5.0◦ external to the crystal, or about 2.0◦ internally. In the BEAMTAP arm of the processor, the lateral 225 μm spot in the feedback AOD (with acoustic velocity va = 4.2 mm/μsec) creates a tap-delay line with a delay of 54 ns within the 1/e2 Gaussian beam profile. Light that diffracts off ˆ this delay line is focused into the crystal, and because the beam is o-polarized, it ˆ writes a cross-correlation hologram with the o-polarized beam from the emulator arm of the processor. In addition, the angle by which this light is focused into the crystal is set to be centered about the first write beam angle θw1 , with an angular divergence from the AOD of 3.6◦ that is 1:0.75 afocally imaged into the ˆ photorefractive crystal. The e-polarized beam from the emulator is Bragg matched ˆ to the gratings written by the o-polarized beams using the parallel-tangents, equalcurvature condition across the entire angular bandwidth of the emulated antenna array. This reproduces the full angular bandwidth coming from the feedback AOD with only a 3 dB roll-off due to slight deviations from the optimum parallelˆ tangents, equal-curvature geometry. The e-polarized reading beam thus reads out the weight matrix formed by the cross-correlation between the emulated array of signals with the desired signal from the feedback arm of the processor. The ˆ e-polarized light that scatters off this grating diffracts out of the crystal at the read beam angle θr , which is shown in Fig. 5.12 as a top view of the photorefractive crystal. Figure 5.12a shows the overlap of the two ordinary-polarized write beams and the extraordinary-polarized parallel-tangents, equal-curvature beam in the photorefractive crystal. Figure 5.12b shows only the read-out beam, and the light is seen to be clearly diffracting off the hologram to the read beam-angle θr . Note

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F I G U R E 5.12. (a) The overlap of the two ordinary polarized write beams and the extraordinary polarized parallel-tangents, equal-curvature beam in the photorefractive crystal. (b) Only the parallel-tangents, equal-curvature beam, which is used to read out the hologram in the crystal, is shown here. Notice that the beam is clearly diffracted into the read beam angle θr .

that this diffracted beam holographically reproduces the spatio-temporal field profile that was produced by the feedback arm’s traveling-wave AOD (windowed by the time-integrated correlation function with each antenna element from the emulator arm), so that the different RF frequency components are diffracted at an angle proportional to their frequency. This is required for interferometric dispersion-free detection on the TFD, and is illustrated in Fig. 5.3. After the photorefractive crystal, the diffracted beam is adjusted to coincide with the writing beam angle θw1 by a pair of mirrors so that the angle will be properly aligned for coherent, dispersionless, interferometric detection on the TFD with the clean reference beam. The clean reference beam is adjusted to the feedback AOD DC reference beam angle by an additional set of mirrors and is rotated by a half wave plate so that it has the same polarization as the diffracted beam from the photorefractive crystal. The feedback AOD DC reference beam, as well as the original write beam kw1 are blocked by irises so that they will not be detected. This is the key isolation step of the read-write isolation subsystem. Finally, the clean reference beam and the diffracted read beam are focused onto the TFD, where they interfere to produce a set of moving interference fringes. By ensuring that the angular alignment of the diffracted read beam and the clean DC reference beam duplicates the original angular spectrum produced by the feedback AOD, the interferometric detection of each frequency component will produce traveling fringes moving at the same velocity as the imaged acoustic velocity—allowing them to be resonantly detected and accumulated by the TFD. If this condition is not obeyed, then the different frequency components from the desired signal that are detected by the phased-array antenna, diffracted by the photorefractive crystal, and interferometrically detected on the TFD will produce fringes moving at different velocities, and will not be resonantly detected. But with

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the velocity of the photogenerated carriers resonantly matched to the velocity of the interference pattern, the detected output contains the coherent addition of each of the antenna array elements provided by the emulator arm of the BEAMTAP processor. Figure 5.13a shows the detected beam-formed output for a single 1 GHz tone using the polarization-angle read-write multiplexing scheme. The detected power ˆ shown is the e-polarized read beam, since the reading and writing beams are angularly separated from each other, and both writing beams are blocked by irises after the photorefractive crystal. When light from the reading beam is blocked in the emulator arm of the processor, the signal detected at the output of the TFD is minimized—this is because the writing beam is isolated by both polarization and angle from the reading beam. Figure 5.13b shows that the writing beam has been isolated from the reading beam by 45 dB. Furthermore, this isolation remains fixed with time due to the additional degree of orthogonality introduced into the system by angularly separating the two beams. This was verified by taking the data after four hours of waiting once the system was initially setup, indicating that the polarization-angle, read-write multiplexing geometry is extremely robust. This is also in contrast to an alternative read-write isolation scheme that only used polarization multiplexing in crystalographically-cut SBN, in which fluctuations in the isolation were unavoidable due to secondary coupling effects between the writing and reading beams resulting from a buildup of an unwanted space-charge field. It is of interest to note that the RF feedthrough in the system is within 3 dB of the leakage from the writing beam, indicating that a substantial amount of optical isolation has been achieved. Further isolation would not actually improve the system performance unless the RF feedthrough could also be suppressed.

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5.5 Experimental Implementation of BEAMTAP The culmination of the theoretical, analytical, and experimental work presented thus far allows for the integration of various subsystems—most notably an impedance matched TFD capable of detecting large fractional bandwidths and polarization-angle read-write multiplexing—into the overall framework of an experimental demonstration of BEAMTAP. However, this type of integration also requires phase stabilization of the interferometric reference beam with respect to the diffracted light from the photorefractive crystal. As a final step toward this end, a phase-stabilization subsystem was built to stabilize the interferometric reference beam with respect to the diffracted light from the photorefractive crystal.

5.5.1 Phase-Stabilization in BEAMTAP Phase stabilization is required in BEAMTAP because the output signal must have an accurate measurement of the jammer phase if the system is to undergo proper adaptation. If the phase of the interferometric reference beam changes by greater than 90◦ , but the jammer phase remains constant, then after subtraction at the differencing node, a positive jammer feedback signal—instead of a negative one— is applied to the feedback AOD for weight adaptation. Thus a strong correlation grating is written in the photorefractive crystal with the jammer that serves to increase the signal strength of the jammer at the output rather than to null it. The problem does not arise because the system is incapable of handling a dynamic environment where the jamming signal evolves with time, but because phase instabilities in the reference arm induce a false change in the jammer phase that the processor incorrectly “adapts” to. Therefore, it is critical that the optical phase in the interferometric reference arm and the diffracted light from the emulator arm remain constant. Referring back to the experimental setup of BEAMTAP (see Fig. 5.11), the two thicker lines show the path of the interferometer that must undergo phase stabilization—the diffracted light from the emulator arm off the photorefractive crystal, and the interferometric reference arm. (The second interferometer that writes gratings in the photorefractive crystal self stabilizes due BEAMTAP’s adaptive feedback mechanism.) Unfortunately, a simple pilot tone cannot be injected into the system (which would write a photorefractive grating, be subsequently diffracted off the grating and be detected at the TFD—thus enabling it to be used for stabilization), because this would require a connection between the electronics used to generate the feedback signal and the electronics used to generate the phased-array antenna signal for the emulator. Due to electrical reflections and finite isolation in the RF components used to generate these signals, if there is a connection between the feedback arm and the phased-array arm, part of the jammer signal leaks back into the feedback signal and serves as a beam forming source that points a beam (rather than a null) toward the jammer.

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As alluded to Fig. 5.11, the problem can be solved by using the carrier for the signal of interest at 920 MHz as a reference for phase stabilization, as long as the two carriers in the two arms are generated separately and not connected electrically to each other in any way. Using the carrier is convenient, especially since it is the strongest frequency component of the signal of interest, and is always detected with relatively strong efficiency. A 920.01 MHz phase-locked local oscillator is then used to down convert the signal to 10 kHz, where it finally fed into a lock-in amplifier to detect the phase of the signal. The error signal that is produced drives a piezo-electric mirror that compensates for any path length changes in the reference arm of the BEAMTAP processor. Using the carrier frequency does not ensure a problem free solution though, since information content is present with the carrier. It thus becomes necessary to isolate the carrier from the rest of the signal bandwidth to keep any nonlinearities from folding frequency information into the 920 MHz signal. To avoid this problem, a specific modulation scheme was developed to ensure that the carrier could be efficiently isolated from the information bandwidth. This is illustrated in Fig. 5.14. At the output, the detected signal of interest is comprised of a 920 MHz carrier and double-sideband modulation that can potentially span up to 500 MHz of bandwidth. These signals are then mixed down with a 920.01 MHz local oscillator to produce a 10 kHz tone and a number of frequency components that are close to baseband, spanning about 250 MHz in width. However, if the original signal bandwidth is chosen so that it starts at 1 MHz, instead of at DC, then a 1 MHz window is created between the mixed carrier signal and the information

F I G U R E 5.14. A modulation scheme that allows for the unwanted signal bandwidth to be separated from the desired 10 kHz, mixed down carrier for phase modulation. A 1 MHz window separates the carrier from the modulation bandwidth, allowing for a lowpass filter with a 20 kHz cutoff to remove the unwanted frequency bandwidth.

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bandwidth. A sharp, low pass filter with a 20 kHz cutoff frequency can then be used to remove these unwanted frequency components. This produces a clean 10 kHz signal that can be efficiently used for phase stabilization using the lock-in amplifier and piezo-electric mirror.

5.5.2 Experimental Beam Forming and Jammer Nulling Results With an impedance matched TFD, polarization-angle read-write multiplexing system, a phased-array antenna emulator, and a phase stabilization subsystem, the BEAMTAP system is now capable of simultaneous beam forming and jammer nulling. Figure 5.15 shows the electronic RF setup for the full optical system, which includes a jammer signal source and closed loop adaptation. An arbitrary waveform generator and a signal generator are used to create a variable bandwidth jamming signal that is incoherent with the rest of the system. The agile signal generator provides the 920 MHz signal of interest, and the two signals are combined, amplified, and input into the emulator. The synthesized sweeper, which is only phase locked to the agile signal generator, provides the desired steering signal d(t), which, when subtracted from the output o(t), generates the feedback signal f (t). This signal is similarly amplified and applied to the feedback AOD. With the two signals in place, the read beam from the emulator, using the parallel-tangents, equal-curvature condition, diffracts off the photorefractive grating produced by the two writing beams, and is interferometrically detected. The output signal is amplified by 78 dB, and a portion of the signal coupled to a spectrum analyzer to provide the final output to the system. A second portion of the signal is tapped off to the phase stabilization subsystem, and the rest is used to help generate the feedback signal f (t). A variable attenuator is used to allow for changes in the feedback gain, and an RF switch is used to quickly switch between open-loop and closed-loop adaptation. In the phase stabilization subsystem, the stable reference is provided by the carrier of d(t), and the output o(t) provides the signal to be stabilized. Narrowband Adaptive Processing

Figures 5.16 and 5.17 show the first experimental results of simultaneous beam forming and jammer nulling with BEAMTAP. Figure 5.16a shows a 920 MHz tone that was applied to the phased-array antenna as the signal of interest, (b) the open-loop beam-formed output, and (c) the closed-loop signal with negligible loss. The feedback gain in the system is 20 dB lower than the total cascaded gain shown in Fig. 5.15 to provide a reference for comparison against higher feedback gains in later experiments. Figure 5.17 shows the applied phased-array jamming signal, incident at the same angle of arrival (AOA) as the angle of interest (which is one of the most difficult scenarios to account for in adaptation), but at a 820 MHz instead of 920 MHz. The jammer power is 10 dB higher than the signal of interest. But because the jamming signal is not perfectly Bragg matched to the photorefractive grating written at 920 MHz (which is the signal of interest), the open-loop results (b) show that the jammer is detected on a sidelobe

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much longer than 15 minutes—indicating excellent phase stabilization within the system. Next, Fig. 5.18a shows the time evolution of the jammer power over 30 s as the processor is switched between open-loop and closed-loop adaptation. After adapting, the stable closed-loop null depth for the jammer is between 15–20 dB, with the system reaching steady state after about 1 s of adaptation time (once the RF switch is flipped to close the adaptive feedback loop). The adaptation is somewhat slow because the jammer is arriving at the same AOA as the signal of interest. This creates competition between the beam forming and jammer nulling operations, although a nulling rate of over 3 dB per 50 ms is achieved, even in this case. Furthermore, there is enough time delay present in the processor to successfully distinguish between the two signals with only a minimal loss of the signal power. Using more optical power or higher feedback gain should result in much faster nulling dynamics (with sub-millisecond nulling being achievable) when the signal and jammer arrive from different AOAs. Figure 5.18b shows the dynamics of the erasure of the grating components responsible for jammer nulling after the system is switched from closed-loop to open-loop operation. The grating decays away exponentially, as expected, and the jammer power gradually increases to its expected open-loop power level. The grating takes about 1 s to decay back to its open-loop state. After erasure, the jammer power remains constant, which also indicates robust phase stabilization. A second experiment was also performed where the feedback gain was increased by 20 dB and the experiment was repeated. With higher gain, the

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competition between the beam forming and jammer nulling operations was stronger, and the processor had difficulty distinguishing between the two signals. The signal of interest was nulled by about 20 dB, and the jammer by 30 dB. Even though the jammer was suppressed by 40 dB, the actual SINR improvement was only 20 dB, 5 dB worse than the previous case. This occurred, again, because the signal of interest and the jamming signal arrived at the same AOA. When arriving from distinct angles, much better results are to be expected. Broadband Adaptive Processing

After this initial proof of concept demonstrating simultaneous, narrowband adaptive beam forming and jammer nulling, information bandwidth was added to the jammer signal to demonstrate broadband adaptive nulling. As an intermediate experiment, Fig. 5.19 shows results of the (a) phased-array emulator signals (b) open-loop signals, and (c) closed-loop signals of a narrowband, 920 MHz signal and 10 MHz bandwidth jammer centered at 880 MHz. The broad pedestal present 40 dB down on the signal is actually just phase noise that is generated by the signal source, and is not indicative of a broadband signal. The open-loop results demonstrate that both the signal and jammer are detected, but after closed-loop adaptation, the jamming signal is nulled to the level of the noise floor, with the expected (slight) degradation of the signal of interest. The final SINR improves by about 18 dB. However, notice that the total null depth of the carrier frequency of the jammer is about 45 dB. As a final experiment, the jammer informational bandwidth was extended to 100 MHz, while keeping the carrier frequencies of the signal and jammer (920 MHz and 880 MHz, respectively) constant. Now, a portion of the frequency spectra of the jamming signal overlaps with the frequency spectra of the signal of interest at 920 MHz, with both signals being produced by the array emulator as if they were arriving at the same AOA. This, by far, is the most difficult test for the adaptive processor, since the only degree of freedom between the signal and jammer within the narrow bandwidth surrounding the signal tone is a lack of phase coherence between the two signals. Figure 5.20 shows the experimental results of the (a) phased-array emulator signals, (b) open-loop signals, and (c) closed-loop signals. Notice in open-loop adaptation, a large portion of the jammer frequency components have already been nulled. This is a result of the Bragg mismatch of these frequencies with the grating written between the desired signal and signal of interest at 920 MHz. In a sense, the residual frequency spectra of the jamming signal is a plot of the frequency selectivity of the Bragg diffraction in the photorefractive crystal due to the illuminated time aperture in the AOD. The sinc-like pattern (plotted on a log scale) is due to the jammer frequency reading out the signal grating, and appears slightly shifted due to a small misalignment of the read-write multiplexing geometry. Once the RF switch is flipped for closedloop adaptation, the processor still successfully nulls the residual jammer to the level of the noise floor, providing about 45 dB of total null depth for the carrier

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of the jammer—and leaves an attenuated, yet still detected, signal of interest at 920 MHz. Clearly the processor is extremely robust, and is capable of performing simultaneous, broadband adaptive beam forming and jammer nulling under adverse signal conditions (including main-beam jammers), providing the feedback gain is not too strong so as to completely null the signal of interest. Even better performance with deeper jammer nulls, increased SINR improvement, and faster nulling is expected when the signal and jammer arrive from resolvable angles onto the array.

5.6 Conclusion The significance of the work presented in this chapter has centered around an experimental demonstration of a photorefractive RF photonic system that is capable of broadband adaptive signal processing, enabling simultaneous beam forming and jammer nulling operation for large antenna arrays. To that end, this chapter has presented a novel solution using the BEAMTAP algorithm which is based on an output time delay reformulation of adaptive beam forming that enables the use of photorefractive crystals as the adaptive weights. The critical components include the sum-and-time-delay which can be implemented by using a scrolling, accumulating detector known as the traveling-fringes detector for coherent summation at the output; a photorefractive crystal using dynamic holography to store the adaptive weight matrix; and an acousto-optic deflector to provide adaptation for robust, simultaneous beam forming and jammer nulling. In addition, this chapter has provided insight into polarization-angle, read-write multiplexing using the parallel-tangents, equal-curvature condition in 45◦ -cut BaTiO3 . The parallel-tangents, equal-curvature condition is a special condition ˆ that allows for anisotropic Bragg matching of gratings written by two o-polarized ˆ beams and read out by an e-polarized beam over an extremely wide angular bandwidth. This read-write isolation is critical, as it separates the reading beam from the writing beams and prevents the system from breaking into oscillation, and is the determining factor of the amount of jammer null depth available to the processor. Experimental results showed 45 dB of read-write isolation, which was ultimately limited by the RF feedthrough of the system. In conclusion, this chapter has demonstrated the feasibility of the photorefractive implementation of RF photonic adaptive processing using BEAMTAP for the next generation of phased-array antennas. Experimental demonstrations of beam forming and jammer nulling have illustrated the flexibility and robustness of the processor, in addition to verifying the usefulness of the algorithm and the optical architecture for future phased-array antenna applications. We wish to acknowledge the support by Dr. William Miceli of ONR, Dr. Steve Pappert of SPAWAR, and the Office of the Secretary of Defense DDR&E through the MURI program grant no. N00014-97-1-1006. Finally, we are also grateful to Dr. D. Dolfi for providing us with the traveling-fringes detector, and also the

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work Dr. T. Sarto and R. T. Weverka, upon which all of the parallel-tangents, equal-curvature condition work is based.

References 1. D. Psaltis, J. Yu, and J. Hong, “Bias-Free Time-Integrating Optical Correlator Using A Photorefractive Crystal,” Applied Optics 24, 3860–3865 (1985). 2. J. H. Hong and T. Y. Chang, “Frequency-Agile RF Notch Filter That Uses Photorefractive 2-Beam Coupling,” Optics Letters 18, 164–166 (1993). 3. J. H. Hong and T. Y. Chang, “Adaptive RF Notch Filtering Using Photorefractive 2-Beam Coupling,” IEEE Journal Of Quantum Electronics 30, 314–317 (1994). 4. R. M. Montgomery and M. R. Lange, “Photorefractive Adaptive Filter Structure With 40-dB Interference Rejection,” Applied Optics 30, 2844–2849 (1991). 5. J. H. Hong and I. Mcmichael, “Novel Optical Technique For Phased-Array Processing,” Optical Engineering 30, 1976–1980 (1991). 6. R. T. Weverka, K. Wagner, and A. Sarto, “Photorefractive processing for large adaptive phased arrays,” Applied Optics 35, 1344–1366 (1996). 7. L. B. Lambert, M. Arm, and A. Aimette, “Electro-optical signal processing for phasedarray antennas,” in Optical and electro-optic information processing, J. Tippett et al., eds., (MIT press, 1965), p. 715. 8. D. R. Pape, “Multichannel Bragg cells: design, performance, and applications,” Optical Engineering 31, 2148–2158 (1992). 9. T. M. Turpin, F. F. Froehlich, and D. B. Nichols, “Optical tapped delay line,”, US Patent 687029, 2003. 10. R. M. Montgomery, “Acousto-optic/photorefractive processor for adaptive antenna arrays,” In Proc. SPIE, B. M. Hendrickson and G. A. Koepf, eds., Optoelectronic Signal Processing for Phased-Array Antennas II 1217, 207–217 (Bellingham, Wash., USA, 1990). 11. G. Kriehn, A. Kiruluta, P. E. X. Silveira, S. Weaver, S. Kraut, K. Wagner, R. T. Weverka, and L. Griffiths, “Optical BEAMTAP Beam-Forming and Jammer-Nulling System for Broadband Phased-Array Antennas,” Applied Optics 39, 212–230 (2000). 12. D. Dolfi, T. Merlet, A. Mestreau, and J.-P. Huignard, “Photodetector for microwave signals based on the synchronous drift of photogenerated carriers with a moving interference pattern,” Applied Physics Letters 65, 2931–2933 (1994). 13. T. Merlet, D. Dolfi, and J.-P. Huignard, “A Traveling Fringes Photodetector for Microwave Signals,” IEEE Journal of Quantum Electronics 32, 778–783 (1996). 14. A. W. Sarto, K. H. Wagner, R. T. Weverka, S. Weaver, and E. K. Walge, “Wide angular aperture holograms in photorefractive crystals by the use of orthogonally polarized write and read beams,” Applied Optics 35, 5765–5775 (1996). 15. M. Cronin-Golomb and M. P. Tarr, “Applications of birefringent phase matching for photorefractive devices,” Optics Letters 20, 2252– (1995). 16. B. Widrow, P. E. Mantey, L. J. Griffiths, and B. B. Goode, “Adaptive Antenna Systems,” Proceedings of the IEEE 55, 2143–2161 (1967). 17. R. T. Compton, Adaptive Antennas (Prentice Hall, 1988). 18. S. M. Sze, Physics of Semiconductor Devices, 2 ed. (John Wiley, 1981).

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19. A. W. Sarto, Ph.D. thesis, University of Colorado, Boulder, 1996. 20. I. C. Chang, “Noncollinear acousto-optic filter with large angular aperture,” Applied Physics Letters 25, 370–372 (1974). ¨ 21. K. Buse, S. Riehemann, S. Loheide, H. Hesse, F. Mersch, and E. Kratzig, “Refractive Indices of Single Domain BaTiO3 for Different Wavelengths and Temperatures,” Physica Status Solidi A 135, K87–K89 (1993).

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6 Photorefractive Mesogenic Composites for Applications to Image Processing Hiroshi Ono,1 Akira Emoto,1 and Nobuhiro Kawatsuki2 1

Department of Electrical Engineering, Nagaoka University of Technology, 1603-1 Kamitomioka, Nagaoka 940-2188, Japan [email protected] 2 Department of Materials Science and Chemistry, Himeji Institute of Technology, 2167 Shosha, Himeji 671-2201, Japan [email protected]

The industry of optoelectronics, including optical memeory, optical communication, and optical information processing, has developed since the last century. Over the past decade, interest has grown in nonlinear optical materials allowing operations in real time on the phase and on the amplitude of coherent wavefronts. Such nonlinear interaction is a fascinate subject, with great promise for application to image distortion compensation, coherent image amplification, optical image filtering, and novel laser resonator design. Photorefractive materials are by far the most efficient nonlinear optical media for optical wave mixing and phase conjugation, which are indispensable for the realization of most of photonics applications. Among many kinds of photorefractive materials, mesogenic materials show a high diffraction efficiency and relatively fast response under a low operating voltage. The aim of the present chapter is to summarize the development of the photorefractive mesogenic composites and to demonstrate optical information processing using photorefractive mesogenic composites.

6.1 Overview of Mesogenic Composites for Photorefractive Applications Before we discuss the applications to image processing, reviews of material development are in order regarding the photorefractive characteristics of the mesogenic composites. In 1994, the first observation of the orientational photorefractive effect in a mesogenic material utilized dye-doped 4-cyano-4 -pentylbiphenyl (5CB) [1, 2, 3]. Reorientation of liquid-crystalline molecules can occur even

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F I G U R E 6.1. The chemical components of a photorefractive polymer-dispersed liquid crystal (PDLC) fabricated by the solvent-induced phase-separation method. (a) E7 as low-molar-mass nematic liquid crystal, (b) fullerene as photoconductive sensitizer, and (c) polymethylmethacrylate as polymer matrix.

in weak optical and electrical fields, so a space-charge field, which can be orders of magnitude larger than an optical field, should produce impressive photorefractivity. The photorefractivity in mesogenic materials is based on orientattional birefringence, and effects are named “orientational photorefractive effects.” One of the advantages of photorefractive mesogenic systems is that the operating voltage required is much lower than that required for other organic photorefractive materials [4, 5, 6], including polymer composites, organic crystals, and organic glasses. The dye-doped low-molar-mass liquid crystals show highperformance photorefractivities, but unfortunately, good photorefractive performance is observed only for large fringe spacing (>10 μm), which is not suitable for high-resolution applications. Nowadays, a sophisticated method for improving the resolution of photorefractive liquid crystals is to combine the low-molar-mass liquid crystal with polymer [7, 8, 9, 10, 11, 12, 13, 14, 15]. One avenue is to use use polymer-dispersed liquid crystals (PDLCs) [7, 8, 9, 12, 13, 14, 15]. The PDLC films are composed of liquid droplets surrounded by polymer matrix materials. Photorefactive PDLCs, which contain PMMA, E7, and C60 (see Figure 6.1 and are fabricated by means of the solvent-induced phase-separation method, were presented by Ono and Kawatsuki in 1997 [7]. Golemme et al. also reported observation in photorefractive effects in the functionalized PDLCs as described in Figure 6.2, which are fabricated by means of the thermally induced phase separation method [8]. To investigate the photorefractive properties they performed four-wave mixing experiments, and a high diffraction efficiency and a high refractive index modulation were obtained in a 53-μm-thick sample with an applied electric field of 22 V/μm. The resolution of the photorefractive liquid crystal was improved by using the PDLCs, and a Bragg-type grating can be written in the photorefractive PDLCs. However, scattering losses are very large for the diagonal beam incidence because of the phase separation in photorefractive PDLCs.

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F I G U R E 6.2. The chemical components of a photorefractive polymer liquid crystal fabricated by the thermally induced phase-separation method. (a) 2, 4, 7-trinitro-9-fluorenone as photoconductive sensitizer and (b) poly-(9-vinylcarbazole) as func-tionalized polymer matrix.

In 1998, Ono and Kawatsuki presented a new avenue to solve the problems in both photorefractive PDLCs and low-molar-mass liquid crystals [10]. Their presentation is to use a high- and low-molar-mass liquid-crystalline mixture as shown in Figure 6.3. The low-molar-mass liquid crystal can dissolve the polymer liquid crystal and the resulting polymer dissolved liquid crystal composite (PDLCC) exhibits the mesophase without phase separation. The mesogenic groups in the PDLCC film easily align in the homeotropic configuration, and this alignment is stable under no electric field even at room temperature, which is preferable to reducing the scattering loss. The photorefractive properties of the PDLCC described in Figure 6.3 were extensively investigated, and an extreamely high net (b)

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F I G U R E 6.4. The chemical components of a functionalized photorefractive polymer dissolved liquid crystal composite. (a) E7 as low-molar-mass liquid crystals, (b) 2,4,7-trinitro9-fluorenone (TNF) as photoconductive sensitizer, and (c) functionalized polymer for photorefractive PDLCC.

gain coefficient was achieved with low applied electric fields and low scattering loss in the thick-grating regime. Another merit in the phtorefractive PDLCC is to use the high functionalized polymer liquid crystals as shown in Figure 6.4(c) [11, 15]. The functionalized polymers have both mesogenic and photoconductive active groups in polymer side chains. The photorefractive PDLCC containing the functionalized copolymer has been investigated for use in real-time holography [16, 17].

6.2 Fundamental Properties of Photorefractive Mesogenic Composites for Image Processing 6.2.1 Two-Wave Mixing in Photorefractive Mesogenic Composites The nonlocal origin of the photorefractive effect manifests itself in a phase shift (φ) between the recording index and the light fringe pattern that creates it, enabling

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F I G U R E 6.5. A schematic diagram of the two-beam coupling experimental geometry and the resulting self-diffraction in the (a) Raman–Nath and (b) Bragg regimes.

two-beam coupling. Undoubted evidence of photorefractivity is provided by the demonstration of the phase-shifted refractive index grating [18, 19]. Two-beam coupling experiments have been considered the cornerstone of decisive experimental proof of photorefractivity in new materials. Diffraction in a photorefractive grating is treated differently if the grating is thin (Raman–Nath) or thick (Bragg) [20, 21]. For the Raman–Nath regime of optical diffraction, the angular spread of the grating vector is much larger than the Bragg diffraction angle, and multiple orders of diffraction are there fore allowed as schematically described in Figure 6.5(a), while the coupled wave theory is valid in the Bragg regime and the light energy is transferred between the two beams as shown in Figure. 6.5(b). Figure 6.6 shows typical diffraction patterns for the photorefractive PDLCC sample. When the diffraction is of the Raman–Nath type, a multiple diffraction process was effective at certain values of dc voltage applied to the PDLCC sample, and a result, in a far field after the sample, the first- and higher-order diffraction spots appeared as shown in Figure 6.6(a). In the Bragg regime, one has to notice that the intensities of the two transmitted beams are different. The asymmetric diffraction pattern is a result of photorefractive-like nonlocal gratings generated in the PDLCC sample. According to the theory of two-beam coupling

F I G U R E 6.6. Demonstration of diffraction from (a) thin (Raman–Nath) and (b) thick (Bragg) gratings in the photorefractive PDLCC. The chemical components of the PDLCC were presented in Figure 6.4. The value of the parameter Q was set to be about (a) 0.1 and (b) 20. The photorefractive gratings were written using p polarized, mutually coherent, He-Ne laser beams that emit continuous-wave 632.-nm light, with an intensity of 6 mW each. The applied dc field was 0.3 V/μm.

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in photorefractive materials, the intensities of the two beams transmitted in the photorefractive material are [19, 21] 1 + ξ −1 , 1 + ξ −1 ed 1+ξ , I2 (z) = I2 (0) 1 + ξ e−d

I1 (z) = I1 (0)

(6.1) (6.2)

where ξ is the input intensity ratio, d the thickness of the photorefractive grating. The gain coefficient  is given by 2πn 1 sin φ, (6.3) λ cos θ where n 1 is an amplitude of index modulation, λ is wavelength, and 2θ is the incident angle between two coherent beams. Thus diffraction on a photorefractive grating is treated differently if the grating is “thin” or “thick.” The two limiting cases can be decided by the follwing parameter [20]: =

2πλd , (6.4) n2 where n is the refractive index and  is the grating spacing. According to Kogelnik’s coupled-wave model, the most rigorous treatments suggest that Q values of over 10 are required to produce a true volume grating (Bragg regime). Figure 6.7 shows typical asymmetric energy transfer signals in two-beam coupling experiments [22]. The intensity of one beam was monitored while another beam was opened at t = 0 s, and the same experiment was repeated for another beam. The fringe spacing was set to be 2.6 μm and the Bragg condition for diffraction on thick gratings was achieved. No signals were detected in the case Q=

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F I G U R E 6.7. The typical energy transfer signals in the two-beam coupling experiments for photorefractive PDLCC described in Figure 6.3. The composition of polymer liquid crystal and E7 is (a) 0:100 and (b) 0:90. Fullerene was used as photoconductive sensitizer for two composites. The applied dc field was 0.18 V/μm and the fringe spacing was 2.6 μm. The film thickness was 50 μm [22].

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F I G U R E 6.8. Self-diffraction patterns from (a) PDLC and (b) PDLCC samples. The chemical components of photorefractive PDLC and PDLCC are presented in Figures 6.1 and [6.3], respectively. The polymer concentration was 10 wt%, the grating constant 20μm, and the film thickness 10 μm. The dc field applied to the PDLC sample was 6.0 μm and that to the PDLCC sample, 0.4 V/μm [23].

of E7/C60 without polymer as shown in Figure 6.7(a), while a large beam-coupling signal appeared by adding the polymer to E7/C60 as shown in Figure 6.7(b). This means that the resolution of the photorefractive liquid crystals was improved by the addition of the polymer. From the point of view of morphology, we can divide photorefractive liquid crystals containing polymers into two classes: (a) polymer and low-molarmass liquid crystals are phase separated (PDLC) and (b) polymer and low-molarmass liquid crystals are miscible and the resulting mixture exhibits mesophase (PDLCC). The photorefractive properties of the mesogenic composites are expected to be strongly dependent on the morphology. Figure 6.8 shows typical self-diffraction patterns for PDLC and PDLCC samples [23]. Judging from the value of the parameter Q  1.0, we have to deal with a thin grating case, e. g., a Raman–Nath diffraction for which the directions of the mth diffraction orders are given by θm = arcsin(mλ/) [21]. Since the diffraction is of the Raman– Nath type, a self-diffraction process was effective at certain values of dc voltage applied to the PDLC and PDLCC samples, and as a result, in a far field after the sample, the first- and higher-order diffraction spots appeared. The necessary applied dc voltage is strongly dependent on the morphology of the photorefractive mesogenic composites, and the higher dc field is necessary for the PDLC in comparison with the PDLCC. Higher-order diffraction was almost invisible in the PDLC sample, while higher-order diffraction was clearly visible in the PDLCC sample due to the larger refractive index change in the photorefractive PDLCC. The PDLC films strongly scatter the light under no electric field because of the mismatch between the refractive index of the liquid crystals and that of the PMMA binder. Applications of a dc electric field across the PDLC films make the director of the liquid crystalline droplets align parallel to the field. Since the ordinary refractive index of E7 is nearly matched with that of the PMMA binder, the films become transparent for the normally incident light. However, the scattering losses are relatively large for the diagonal beam incidence as shown in Figure 6.8(a). According to PDLC studies for display applications [24], p-polarized light can couple into the extraordinary refractive index of the liquid crystal in the aligned PDLC film. This coupling into the extraordinary refractive index, which is different from the polymer index, provides a refractive index mismatch that scatters

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light. In the case of PDLCC samples, the scattering losses are negligibly small and the clear diffraction spots appear as shown in Figure 6.8(b) because the polymer and low-molar-mass liquid crystals are miscible without phase separation.

6.2.2 Degenerate Four-Wave Mixing in Photorefractive Mesogenic Composites Optical phase conjugation is a fascinating subject, with great promise for application to image distortion compensation, coherent image amplification, optical image filtering, and novel laser resonator design [25]. All optical devices in which the optical phase conjugation phenomenon plays an important role have already been demonstrated in various systems. Phase conjugation in the photorefractive mesogenic composites has been measured with the degenerate four-wave mixing (DFWM) configuration shown schematically in Figure 6.9. Photorefractive mesogenic composites used for DFWM experiments were based on a low-molar-mass nematic mixture, functionalized copolymer, and sensitizer (see Figure 6.4) [26, 27]. Figure 6.10 shows the absorption spectrum of the photorefractive mesogenic composite film. The copolymer and TNF form charge-transfer complexes, and the light in the visible region was absorbed. Experiments of optical phase conjugation were carried out using a frequencydoubled Nd-YAG laser of 532 nm or a He-Ne laser of 633 nm. As shown in Figure 6.10, the electronic transition of the photorefractive mesogenic composite is on-resonant with a wavelength of 532 nm and off-resonant with a wavelength of 633 nm. The output laser beam was divided into three beams that were directed onto the mesogenic composite possessing orientational photorefractive effects. Two of the three beams formed the counterpropagating pump fields E 1 , PBS1

PBS2

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F I G U R E 6.9. Experimental setup for degenerate four-wave mixing with counterpropagating-pump geometry. E 1 and E 2 denote the forward and backward pump beams, respectively. E 4 represents the probe beam and E 3 the phase-conjugate signal beam.

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Absorption (/cm)

F I G U R E 6.10. Absorption spectra of the photorefractive mesogenic composite. The chemical components of the composite were presented in Figure 6.4. The photorefractive mesogenic composite was prepared by mixing the nematic mixture (E7), copolymer, and TNF (20:78:2 wt%). The thickness of the mesogenic composite was 50 μm [26].

177

400 300 532 nm 200 633 nm 100 0 400

500

600

700

800

Wavelength (nm)

E 2 , and another beam was used as the probe beam with electric field amplitude E 4 . Interaction of these beams in the photorefractive polymer composite leads to generation of the fourth beam of field amplitude E 3 , which is phase conjugate to the probe beam E 4 . The polarization directions of the three beams were independently controlled by three half-wave plates. Under the degenerate four-wave mixing configurations, beams 1, 2, 3, and 4 are of the same angular frequency ω E 1 and E 2 are taken to be the counterpropagating pump beams. E 4 is the probe beam and E 3 is the phase-conjugated wave as shown in Figure 6.9. In general, the interference between the beams gives rise to six sets of spatial intensity modulations in the photorefractive mesogenic composite, each with a different fringe spacing. Two sets, between E 1 and E 4 and between E 2 and E 3 , form transmission gratings, while the other four form reflection gratings. In the case of the photorefractive liquid crystals the grating period of the relection grating is shorter than the resolution limit, and the coupling equations including six sets of spatial intensity modulations can be further simplified by assuming that the phase-conjugated response is mostly dependent on transmission gratings. If only transmission gratings are dominant, we arrive at the following equation [18]:    d exp ( + α2 )(r − r  ) dr  ≡ c E 1 (0)/J (r ), (6.5) E 3 = c E 1 (0) ) I (r 0 r where r = z/ cos θ, the z axis is perpendicular to the transmission grating vector, 2θ is the crossing angle between the beams E1 and E 3 , α is the absorption  coefficient, I0 = 4j=1 I j , and c=

E 2 (d)E 4∗ (0) exp(−αd/2)  I1 (0)/J (0) − 1

(6.6)

Taking the square of both sides of equation (6.6), one gets the intensity of the phase-conjugation beam. The phase-conjugate reflectance is defined as the intensity ratio of probe and phase-conjugate beams.

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Reflectance (%)

25 20 15 10 5 0 0.0

0.1 0.2 0.3 0.4 Applied dc field (V/μm)

0.5

F I G U R E 6.11. Phase-conjugate reflectance from the photorefractive mesogenic composite (see Figure 6.4) versus applied dc field. The frequency-doubled Nd-YAG laser (filled circles) or He-Ne laser (open circles) was used as the pump beam. The grating spacing was 3.0 μm. In the case of the experiments using the frequency-doubled Nd-YAG laser, beam powers were I1 = 6.5 mW, I2 = 13.1 mW, and I4 = 6.5 mW. In the case of experiments using the He-Ne laser, beam powers were I1 = 5.9 mW, I2 = 12.7 mW, and I4 = 6.0 mW. Beams 1 and 4 were of the same polarization state (s-polarization) and the polarization direction of beam 3 was perpendicular to that of the other two beams ( p-polarization) [26].

Figure 6.11 shows the typical dependence of the steady-state value of the phase conjugate reflectance (= I3 /I4 × 100) as a function of the electric dc field applied to the photorefractive mesogenic composite. The reflectance increased monotonically as the electric field increased and attained its maximum value of around 20%. A larger external electric field is a necessary condition for observation of the phase-conjugation signal generated by the He-Ne laser beam in comparison with that in the experiment using the frequency-doubled Nd-YAG pump beam. The applied electric field permits charge migration along the grating vector and improves the efficiency of the photocharge generation. We consider that the photocharges are efficiently generated using the frequency-doubled Nd-YAG pump beam because the absorption coefficient at the wavelength of the frequencydoubled Nd-YAG laser beam is larger than that at the wavelength of the He-Ne laser beam as shown in Figure 6.10. The refractive-index-modulation depth n 1 is dependent on the period of the grating . In order to check the influence of this effect on the magnitude of the reflectivity of the phase-conjugated mirror, we performed an experiment enabling us to control the period of the grating K I , which can be carried out by changing the 2θ angle between the E1 and E 4 beams. The phase-conjugate reflectance and the index-modulation depth estimated from the above-mentioned theory are plotted versus the grating period , which is shown in Figure 6.12. The superlinear increase in the index-modulation depth and phase-conjugate reflectance was observed with increasing grating spacing. Although the resolution of our photorefractive mesogenic composite is fairly improved in comparison with the photorefractive low-molar-mass liquid crystal without polymer, for small grating

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Reflectance (%)

3.0 15

2.5 2.0

10

1.5 1.0

5

0.1 0.2 0.3 Grating pitch (mm)

0.5 0.0 0.4

Index modulation depth

(× 10−3) 3.5

20

0 0.0

179

F I G U R E 6.12. Phase-conjugate reflectance in percent and index-modulation depth versus grating spacing. The chemical components of the composite were presented in Figure 6.4. The photorefractive mesogenic composite was prepared by mixing the nematic mixture (E7), copolymer, and TNF (20:78:2 wt%). The thickness of the mesogenic composite was 50 μm. The frequency-doubled Nd-YAG laser was used as the pump beam. The applied electric field was 0.3 V/μm and beam powers were I1 = 6.5 mW, I2 = 13.1 mW, and I4 = 6.5 mW. Open and filled circles represent reflectance and index modulation depth, respectively [26].

spacings the index-modulation depth will be limited by elastic forces when the mesogenic molecules are reoriented. According to equation (6.6), the phase-conjugate reflectance is strongly dependent on both the spatial phase shift (φ) and the index-modulation depth (n 1 ). In Figures 6.13(a) and 6.13(b) the phase-conjugate reflectivities, which can be calculated from equation (6.6), are plotted versus the pump-beam intensity ratio. The experimental data described in Figure 6.13(a) were obtained when beams 1, 2, and 4 were of the same polarization state ( p-polarization), while those in Figure 6.13(b) were obtained when beams 1 and 4 were of the same polarization state (s-polarization) and the polarization direction of beam 2 was perpendicular to that of the other two beams. In the case of the p-polarization state, two-beam coupling between beams 1 and 4 was observed, while the two-beam coupling effects can be ignored when beams 1 and 4 are of the s-polarization state. In the case of the s-polarization state, one may assume that the response is local, and in the first approximation one may put φ = 0 because the two-beam coupling effects can be ignored. In the case of the p-polarization state, the phase-conjugate reflection must be calculated from equation (6.6) including the spatial phase shift obtained from the two-beam coupling experiments. According to equation (6.6), the dependence of phase-conjugate reflectivity on the pump ratio is a manifestation of the same phase that produced the directionality in two-wave mixing. Since the sign of the spatial phase shift is reversed by changing the direction of the applied dc electric field, the photorefractive media with negative and positive phase shift display an optimum pump ratio of over [filled circles in Figure 6.13(a)] and under unity [open circles in Figure 6.13(a)], respectively. In contrast, as shown in

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Reflectance (%)

(a) 15

10

5

0 0.001 0.01

0.1

1

10

100 1000

3.0 (b) Reflectance (%)

2.5 2.0

F I G U R E 6.13. Phase-conjugate reflectance versus pump power ratio. The chemical components of the composite were presented in Figure 6.4. The photorefractive mesogenic composite was prepared by mixing the nematic mixture (E7), copolymer, and TNF (20:78:2 wt%). The thickness of the mesogenic composite was 50 μm. The probe-beam intensity was set to 5.0 mW. The applied dc field was set to 0.4 V/μm (filled circles) and −0.4 V/μm (open circles). (a) Beams 1, 2, and 4 were of the same polarization state ( p-polarization) and (b) beams 1 and 4 were of the same polarization state (s-polarization) and the polarization direction of beam 2 was perpendicular to that of the other two beams. The solid lines denote the theoretical fitting results [27].

1.5 1.0 0.5 0.0 0.001 0.01 0.1 1 10 100 1000 Pump power ratio (I2 / I1)

Figure 6.13(b), in the case of s-polarization state, the photorefractive mesogenic composite displays an optimum pump ratio of around unity because the two-beam coupling effects can be ignored.

6.2.3 Real-Time Holography in Photorefractive Mesogenic Composites Real-time holographic gratings have drawn much attention in the past few years because of their potential application for parallel and real-time optical processing. Photorefractive mesogenic material is promising for real-time holographic applications and has been the subject of considerable interest in real-time holography because it presents high-sensitivity, high-diffraction efficiency and real-time capabilities. Real-time measurements of the diffraction efficiency of the transmission holograms were carried out using the system described in Figure 6.14. The holographic gratings were created by means of a frequency-doubled Nd-YAG laser with a

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F I G U R E 6.14. Schematic of the experimental setup for real-time holographic recording and reading. λ/2: half-wave plate, BS: beam splitter, M1–M4: mirror, and ND: neutral density filter [28].

532-nm wavelength. The total intensity of the writing beams was varied between 1 and 7 mW and a beam ratio of 1/1. The fringe spacing was also varied between 2.2 and 2.8 μm. To enhance the generation efficiency of the photoinduced charges and their redistribution, a dc electric field was applied and the value was varied between 0 and 0.3 V/μm. The sample was tilted 60 degrees from the bisector of the two incident writing beams to provide a projection of the grating wave vector along the direction of the applied dc field. The diffracted beam intensity was monitored with a He-Ne laser with a 633-nm wavelength where our photorefractive PDLCCs (see Figure 6.4) scarcely absorb as shown in Figure 6.15. In order to obtain the intensity of the diffracted beam at each time, a photodiode detector was positioned in the Bragg diffraction direction, and a detector was connected to a digital oscilloscope and a personal computer detected the diffraction beam. The diffraction efficiency is defined as the intensity ratio of the diffracted beam and the incident beam. Figure 6.16 shows the diffraction efficiency versus applied dc fields (top) and writing power (bottom). The diffraction efficiency was increased as the applied dc voltage and TNF concentration were increased. The photoconductivity is increased as the TNF concentration is increased. Improvements in

F I G U R E 6.15. Absorption spectra of the photorefractive PDLCCs. The chemical components of the composite were presented in Figure 6.4. The TNF concentration was described in the figure. A frequency-doubled YAG laser (532 nm) was used as the hologram writing beam, and a He-Ne laser (633 nm) was used as the probe [28, 29].

Absorption (/cm)

500 400 300

523 nm

200

(c)

100

(b)

0 400

633 nm

(a) 500

600

Wavelength (nm)

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1.0

35 30

Normalized eff iciency

Diffraction efficiency (%)

40

(c)

25 20 (b)

15 10

(a)

5 0 0.00

(c)

0.8 0.6

(a)

0.4 (b) 0.2 0.0

0.05

0.10 0.15 0.20 0.25 Applied dc field (V/μm)

0.30

0

2

4

6

8

10

12

14

Writing power (mW/cm2)

F I G U R E 6.16. Hologram recording properties of photorefractive PDLCCs. The chemical components of the composite were presented in Figure 6.4. The concentration ratio by weight (polymer:E7) was 20:80 and TNF concentration was (a) 0.64, (b) 1.00, and (c) 2.00 wt%. The fringe spacing was 2.8 μm. Top: The diffraction efficiencies are plotted versus applied dc field. The writing beam power was 12 mW/cm2 . Bottom: The diffraction efficiencies are plotted versus writing beam power. The applied dc field was 0.3 V/μm [29].

photoconductivity are due to an increase in the quantum efficiency of mobile charge generation. The resultant improvement of the space-charge field may play a role in improving the diffraction efficiency. The sensitivity was improved by increasing the TNF concentration. The grating was generated by irradiating at total power (2.8 mW/cm2 ) of writing beams (1 mW for one writing beam in laser spot half-diameter of 4.8 mm), while maintaining a high grating efficiency of around 15%. These values give the nonlinear coefficient of 0.57 cm2 /W, which is a relatively large magnitude.

6.3 Applications to Image Processing 6.3.1 Image Amplification in Photorefractive Mesogenic Composites Optical image reconstruction, a type of intensity amplification of a twodimensional optical image, was demonstrated using a photorefractive PDLCC described in Figure 6.4. Figure 6.17 shows the experimental setup for reconstruction of a two-dimensional optical image from nonlocal gratings in the photorefractive mesogenic composite. A linearly polarized He-Ne laser beam (λ = 633 nm) was divided into two beams by a polarizing beam splitter (BS), one of the two beams was expanded, and an image was imprinted onto it by a transparent object (signal). This image was Fourier transformed by a lens, and the photorefractive mesogenic film was placed in the Fourier plane. The other beam (pump) was superposed on the same spot and interfered with the signal beam.

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183

F I G U R E 6.17. Schematic of the experimental setup for measuring the image reconstruction from nonlocal gratings in the photorefractive mesogenic composite. The signal and pump beams are p-polarized, where PD represents a photodiode, M1, M2, M3, and M4 mirrors, and BS a polarized beam splitter.

The electric fields of the two light waves can be written as E j = A j exp[i (ωt − k j · r)],

j = 1, 2,

(6.7)

where A1 and A2 are the wave amplitudes of the signal and pump beams, ω is the angular frequency, and k1 , k2 are the wave vectors. The gain coefficients depend on the beam intensity ratio m, and the dependence can be estimated by the experimental observation as shown in Figure 6.18. We now write  A1 = I1 exp(−iψ1 ), (6.8)  A2 = I2 exp(−iψ2 ), (6.9)

F I G U R E 6.18. Gain coefficients of the photorefractive mesogenic composite versus the intensity ratio (m) between the signal and pump beams. The chemical components of the composite were presented in Figure 6.4. The photorefractive mesogenic composite was prepared by mixing the nematic mixture (E7), copolymer, and TNF (20:79:1 wt%). The thickness of the mesogenic composite was 50 μm. The applied dc electric field was 0.2 V/μm.

Gain coefficients (1/cm)

140 120 100 80 60 40 20 0

0

50 100 150 200 250 300 m

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where ψ1 and ψ2 are phases of the complex amplitudes A1 and A2 . By examining the two-beam coupled equations, these phases can be obtained as   1+m β ψ1 (d) − ψ1 (0) = − ln , (6.10)  1 + med   1+m β ψ2 (d) − ψ2 (0) = − ln , (6.11)  1 + me−d where

πn 1 cos φ. (6.12) λ cos θ The two-dimensional optical image was focused by a Fourier-transform lens with f = 30 cm and the signal beam was considered to have an electric field distribution given by [30]:  

x 2 + yi2 1 w(xi , yi , f ) = exp ik f + i · F{u(x, y)}, (6.13) iλ f 2f β=

where u (x, y) is the electric field of the input optical image, and the Fourier transform of a complex function u of two independent variables x and y is represented here by F {u(x, y)}. The second coherent wavefront (pump beam) of known amplitude and phase is added to the signal wavefront described by equation (6.13) and the two wavefronts interfered with each other. The output image-intensity distribution is described by |v(X, Y )|2 : √ v(X, Y ) = F −1 I 2 + iψ2 (d) , (6.14) where the inverse Fourier transform of a function G is represented by F −1 (G). Figure 6.19 shows the comparison between experiments and numerical simulation results, and the experimentally measured image-intensity distributions are in good agreement with the theory.

6.3.2 Edge Enhancement of Optical Image In this section, we expolit the high-performance optical nonlinearity inherent in the photorefractive PDLCC (see Figure 6.4) to demonstrate and explain edge enhancement of a two-dimensional optical image. Figure 6.20 illustrates the the experimental setup for the edge enhancement of the optical image, combining Fourier transform holographic operation and the high-performance photorefractivity in the PDLCC. A laser-diode-pumped Nd:YAG second-harmonic-generating laser beam at λ = 532 nm was divided into two beams by a polarizing beam splitter and one of the two writing beams was expanded and an image was imprinted onto it by a transparent object (signal beam). This image was Fourier transformed by a lens and the photorefractive PDLCC film was placed in the Fourier plane. Another beam (reference beam) of the two writing beams was superposed on the same spot and interfered with the

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6. Photorefractive Mesogenic Composites for Applications to Image Processing 6 (a)

5

Intensity (a.u.)

Intensity (a.u.)

6

185

4 3 2 1 0 0.0

5

(b)

4 3 2 1

0.5

1.0 1.5 X-axis [mm]

2.0

0 0.0

2.5

0.5

1.0 1.5 X-axis [mm]

2.0

2.5

F I G U R E 6.19. (a) Experimentally observed and (b) theoretically calculated reconstructed images from nonlocal gratings in the photorefractive mesogenic composite. Bold solid curves represent the input (signal) images and fine solid curves reconstruct images amplified by the pump beam. The intensity ratio (m) between the signal and pump beam was 50. The photorefractive mesogenic composite was prepared by mixing the nematic mixture (E7), copolymer, and TNF (20:79:1 wt%). The thickness of the mesogenic composite was 50 μm. The applied dc electric field was 0.2 V/μm.

signal beam. The grating constant of the interference light was controlled by the crossing angle of the writing beams and set to be around 2.5 μm. In general, the reconstructed image is a faithful replica of the original image only if the intensity of the signal beam is less than the intensity of the reference beam. If this condition is violated, edge enhancement can be produced in the reconstructed image because the largest index modulation will be produced Input image Lens Object

Sample

Reconstructed image

Mask Probe

Reference

F I G U R E 6.20. Experimental setup for nonlinear intensity filtering by a Fourier transform holographic configuration. To enhance the generation efficiency of the photoinduced charges and their redistribution, a dc electric field of 0.2 V/μm was applied. A laser-diode-pumped Nd:YAG second-harmonic-generating laser beam at λ = 532 nm was used as signal beam and the holographic optical image was reconstructed by a He-Ne laser with a 633-nm wavelength (reading beam) where our photorefractive PDLCCs scarcely absorb. The sample was tilted 45 degrees from the bisector of the two writing beams to provide a projection of the grating vector along the direction of the applied dc field. In this experiment, the signal and reference beams are linearly polarized parallel to each other and are s-polarized in order to prevent strong two-beam coupling. The reading beam is also linearly polarized orthogonally to the writing beams and is p-polarized.

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F I G U R E 6.21. Demonstration of the edge enhancement of two-dimensional optical images. In photograph (a), the intensity of the signal beam is 1/10 the intensity of the reference beam. In photograph (b), it is 20 times the intensity of the reference beam. The bright lines in the images are separated by approximately 0.5 mm [31].

when two optical fields of the same intensity interfere. When the intensity of the refernce beam is much larger than that of the signal beam, the central maximum of the Fourier transform has an intensity greater than the intensity of the reference beam and produces a localized grating with a poor diffraction efficiency. Toward the wings of the diffraction pattern, the intensity of the reference beam, and a highly effiecient grating is formed. The low spatial frequencies of the signal are attenuated while the high spatial frequencies are efficienctly reconstructed, thus giving rise to the edge enhancement in the image, as demostrated in Figure 6.21. Quantitative considerations were performed by means of Fourier transform holograms [31, 32]. Figure 6.22 contains a plot of the calculated and experimentally observed images with the object being a single slit. As shown in Figure 6.22, the enhancement of the edges of the two-dimensional optical images as the intensity ratio is increased and the experimental results are in good agreement with the theoretical expectation.

6.3.3 Spatial Frequency Selective Reconstruction of Optical Images In this section, we demostrate spatial frequency selective reconstruction of two-dimensional optical images using functionalized mesogenic composites possessing real-time holographic capabilities (see Figure 6.4). Figure 6.23 illustrates the experimental setup for spatial frequency selective reconstruction of the two-dimensional optical image, combining a Fourier transform holographic operation and photo-densities in the photorefractive mesogenic composite. The twodimensional optical image was Fourier transformed by a lens, and a part of the

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6. Photorefractive Mesogenic Composites for Applications to Image Processing F I G U R E 6.22. (a) Theoretical and (b) experimentally observed plots of the intensity of the observed image of a single slit 3.0 mm wide when a real image of this object is focused in the photorefractive PDLCC by a 30-mm-focal-length lens. The ratio of the intensity of the signal beam to the intensity of the reference beam is described in the figure [31].

(a)

187

(b)

1/10

2/1

20/1

spatial frequency was brought into interference with the reference beam in the Raman–Nath regime. In order to clarify quantitatively the mechanism of spatial frequency selective reconstruction using photorefractive mesogenic materials, we consider the image plane when the object is a single square as shown in Figure 6.24(a). Under our experimental conditions, the input image is Fourier-transformed by a lens as shown in Figures 6.24(b) (experiment) and 24(c) (calculation). The observed image was in good agreement with the theoretical image. In the Raman-Nath thingrating regime, multiple diffraction orders are generated. Two-beam interference between the Fourier-transformed two-dimensional optical image and the reference PBS

M1

He-Ne laser λ/2

BE

+1

Mask2

BE

−1

L1 M3 Mask1

Sample

L2

M2

F I G U R E 6.23. Schematic of the experimental setup for spatial frequency selective reconstruction using photorefractive mesgenic materials. The Fourier transform operation is performed in the Raman–Nath diffraction regime and the spatial frequency selective reconstruction appears in the self-diffraction beams.

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F I G U R E 6.24. (a) Photograph of test input image, (b) Fourier-transformed image generated by a lens (experiment), and (c) calculated image. Three patterns A, B, and C denote the three kinds of reference beam patterns used in this study.

beam provides holographic gratings. In the back focal plane of the imaging lens, intensity distributions of the spatial frequencies of the image appear, since the Fourier spectrum and the reconstructed images depend on the intensity distribution of the reference beam. Thus the image with favored spatial frequency can be reconstructed by controlling the intensity distribution of the reference beam, and these processes can be applied to optical two-dimensional image processing for the spatial frequency selective reconstruction. As a first example of the frequency selective reconstruction, three kinds of reference beam with different intensity distribution patterns are used for the demostration; the intensity of the signal beam is almost equal to that of the reference beam at the focal plane. The reference beam with pattern A or B is superposed onto the Fourier-transformed image, with high spatial frequencies in the horizontal and vertical directions of the input optical image, respectively, while the reference beam of pattern C is overlapped with the signal image with low spatial frequencies. Figure 6.25 contains a picture of the experimentally observed and calculated images with the object being a single square as shown in Figure 6.4. The experimentally observed images [Figure 6.25(a)] were almost in good agreement with the calculated images [Figure 6.25(b)]. As another case of spatial frequency selective reconstruction using the photorefractive mesogenic composite, we demonstrated spatial frequency selective reconstruction by controlling the intensity of the reference beam. In this case, the diameter of the reference beam was much larger than that of the Fouriertransformed signal beam and assumed to be large enough for the reference beam profile to be homogeneous in comparison with the signal beam profile. Since a highly efficient grating is formed when the intensity of the signal beam is comparable to that of the reference beam, spatial frequency selective reconstruction is expected to be achieved by simply controlling the reference beam intensity, although the cylindrical symmetry is necessary for the spatial frequency of the reconstructed image. Figure 6.26 contains a two-dimensional plot of the experimentally observed and calculated images with the object being a single square on varying the intensity ratio. The observed images are in good agreement with the theoretical expectation.

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6. Photorefractive Mesogenic Composites for Applications to Image Processing F I G U R E 6.25. Comparison of (a) experimentally observed images and (b) numerical simulation results. Patterns A, B, and C are in accordance with the images obtained from three reference beam patterns [33].

F I G U R E 6.26. Comparison of (a) experimentally observed images and (b) numerical simulation results as a function of the intensity ratio between the signal and reference beams. The beam intensity ratio was set to be (1) 0.470, (2) 0.017, and (3) 0.003 [33].

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References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18.

19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29.

E.V. Rudenko and A.V. Sukhov: JETP Lett. 59, 143 (1994). E.V. Rudenko and A.V. Sukhov: JETP 78, 875 (1994). L.C. Khoo, H. Li, and Y. Liang: Opt. Lett. 19, 1723 (1994). W.E. Moerner and S.M. Silence: Chem. Rev. 94, 127 (1994). S.J. Zilker: Chemphyschem, 1, 72 (2000). G.P. Wiederrecht: Annu. Rev. Mater. Res. 31, 139 (2001). H. Ono and N. Kawatsuki: Opt. Lett. 22, 1144 (1997). A. Golemme, B.L. Volodin, B. Kippelen, and N. Peyghambarian: Opt. Lett. 22, 1226 (1997). G.P. Wiederrecht and M.R. Wasielewski: J. Am. Chem. Soc. 120, 3231 (1998). H. Ono, I. Saito, and N. Kawatsuki: Appl. Phys. Lett. 72, 1942 (1998). H. Ono, T. Kawamura, N. Kawatsuki, H. Norisada, and T. Yamamoto: Jpn. J. Appl. Phys. 38, L1258 (1999). Y. Bai, X. Chen, X. Wan, Q. Zhou, H. Li, B. Zhang, and Q. Gong: Appl. Phys. B 73, 35 (2001). Y. Bai, X. Chen, X. Wan, Q. Zhou, H. Li, B. Zhang, and Q. Gong: Appl. Phys. Lett. 80, 10 (2002). N. Kamania, S. Putilin, and D. Stasel’ko: Synth. Met. 127, 129 (2002). H. Ono, H. Shimokawa, A. Emoto, and N. Kawatsuki: J. Appl. Phys. 94, 23 (2003). H. Ono, T. Kawamura, N.M. Frias, K. Kitamura, N. Kawatsuki, H. Norisada, and T. Yamamoto: J. Appl. Phys. 88, 3853 (2000). H. Ono, T. Kawamura, N.M. Frias, K. Kitamura, N. Kawatsuki, and H. Norisada: Adv. Mater. 12, 143 (2000). J. Feinberg and K.R. MacDonald: Phase-conjugate mirrors and resonators with photorefractive materials, in: Photorefractive Materials and Their Applications II, ed. by P. G¨unter and J.-P. Huignard (Springer 1988) pp. 151–198. P. Yeh: Photorefractive effects, in: Introduction to Photorefractive Nonlinear Optics (John Wiley & Sons, Inc. New York 1993) pp. 82–117. H. Kogelnik: Bell Syst. Tech. J. 48, 2909 (1969). P. Yeh: Wave propagation in periodic media, in: Introduction to Photorefractive Nonlinear Optics (John Wiley & Sons, Inc. New York 1993) pp. 118–182. H. Ono and N. Kawatsuki: J. Nonlinear Opt. Phys. & Mater. 8, 329 (1999). H. Ono, H. Shimokawa, A. Emoto, and N. Kawatsuki: J. Appl. Phys. 94, 23 (2003). P.S. Drzaic: Electro-optical effects, in: Liquid Crystal Dispersions (World Scientific, Singapore 1995) pp. 183–352. R. Fisher: Optical Phase Conjugation (Academic Press, Inc., London 1983). H. Ono, K. Kitamura, N. Kawatsuki, and H. Norisada: Jpn. J. Appl. Phys. 40, 1328 (2001). H. Ono, K. Kitamura, and N. Kawatsuki: Liq. Cryst. 29, 1339 (2002). H. Ono, T. Kawamura, N.M. Frias, K. Kitamura, N. Kawatsuki, H. Norisada, and T. Yamamoto: J. Appl. Phys. 88, 3853 (2000). H. Ono, T. Kawamura, N.M. Frias, K. Kitamura, N. Kawatsuki, H. Norisada, and T. Yamamoto: Adv. Mater. 12, 143 (2000).

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30. J.W. Goodman: Fourier transforming and imaging properties of lenses, in: Introduction to Fourier Optics (McGraw-Hill, Inc 1968) pp. 77–100. 31. H. Ono, T. Kawamura, N. Kawatsuki and H. Norisada: Appl. Phys. Lett. 79, 895 (2001). 32. P. Hariharan: Wavefront reconstruction, in: Optical Holography (Cambridge University Press, 1996) pp. 11–24. 33. A. Emoto, H. Ono, and N. Kawatsuki: Liq. Cryst. 30, 1201 (2003).

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7 Solid-State Lasers with a Photorefractive PhaseConjugate Mirror T. Omatsu,1 M.J. Damzen,2 A. Minassian2 and K. Kuroda3 1

Department of Information and Image Sciences, Chiba University, 1-33, Yayoi-cho, Inage-ku, Chiba 263-8522, Japan 2 The Blackett Laboratory, Imperial College, Prince Consort Rd., London SW7 2BW, UK 3 Institute of Industrial Science, University of Tokyo, 4-6-1, Komaba, Meguro-ku, Tokyo, 153-8505, Japan

7.1 Introduction The maintenance of high-spatial-quality radiation in laser devices as they are scaled to high powers is important for many practical applications, including laser material processing, remote sensing, and optical pumping. However, it is increasingly difficult to maintain a fundamental TEM00 high-spatial-quality mode, and a key problem in solid-state lasers is due to the onset of thermally induced refractive-index distortions in the laser amplifier. This is caused by the intensive inversion mechanism that heats the laser medium. Similarly, in high-power semiconductor diode lasers, the beam quality is severely degraded since to achieve high power (e.g., multiwatt), the emitters of the device need to be scaled to large dimensions (broad-stripe emitter) or multiple arrays of emitters must be used. The broad-stripe emitter leads to highly multimode operation and the emitter arrays are further degraded since the output radiation is spatially incoherent. Phase conjugation provides a general technique for the maintenance of high spatial quality. It is interesting to note that soon after the original demonstration of phase conjugation by stimulated Brillouin scattering in 1972 [1], the first application in which phase conjugation was applied was correction of the distortions in a laser amplifier. This was achieved by double-passing the phase-conjugate return back through the distorting amplifier to produce a beam with aberration-corrected wavefront [2]. The resultant laser output radiation was thereby intensified by received two-pass amplification but without degradation in spatial quality. Since this early demonstration much effort has been applied to the application of phase conjugation to laser operation. Many mechanisms have been investigated for phase conjugation including stimulated Brillouin scattering [3], four-wave mixing in a thermal grating media [4] and in resonant amplifier media [5, 6], and via photorefraction [7, 8, 9].

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(a)

(b)

input high- double-pass quality beam aberrationcorrected beam output coupling beam laser mirror

PCM

aberrated beam

PC beam

aberration-free fundamental mode beam splitting/ (c) combining element

coherently combined output beam multichannel amplifier

F I G U R E 7.1. Key architectures in which phase conjugate mirror (PCM) is applied to laser operation: (a) double-phase amplifier with aberration correction; (b) phase-conjugate laser resonator in which one of the laser cavity mirrors is a PCM; and (c) PCM is used to coherently combine the radiation from a multichannel amplifier.

This chapter describes aspects of phase conjugation due to the photorefractive mechanism applied to laser systems and presents experimental results in diode lasers and continuous-wave diode-pumped solid-state lasers. Figure 7.1 illustrates some of the key architectures in which phase conjugation can be applied to laser operation. Figure 7.1a shows the case of a double-pass amplifier with aberration correction, Figure 7.1b shows the case of a phase-conjugate laser resonator in which one of the laser cavity mirrors is a phase-conjugate mirror (PCM), and Figure 7.1c shows where the phase-conjugate mirror is used to coherently combine the radiation from a multichannel amplifier. The latter case of a multielement amplifier can also be related to the case of a multiple emitter semiconductor diode array with phase-conjugate feedback. Results will be described in this chapter of photorefractive phase-conjugate geometries applied to double-pass amplifiers, laser oscillators, and multielement diode arrays.

7.2 Optical Phase Conjugation for Solid-State Lasers Distortion in a high-power solid-state laser is produced by the intense pumping mechanism producing the inversion in the amplifying medium. In addition to the

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phase distortion produced by thermally induced refractive changes, a significant stress-induced birefringence is produced. In a rod geometry, the principal axes of the birefringence follow the cylindrical geometry of the rod and consist of radial and circumferential components. Radiation passing through the rod experiences phase retardation by an amount dependent on the spatial position in the rod. A linearly polarized input beam will be amplified and emerge as a depolarized beam. The latter effect of depolarization is most pronounced in laser crystals with isotropic refractive index (e.g., Nd:YAG), but by using laser crystals with strong naturally birefringent (e.g., Nd:YVO4 ) the depolarization effects can be largely removed. The thermally induced refractive index change in a diode end-pumped geometry leads to a thermal lensing with dioptic power (m−1 ) given by [10] ηh · dn /dT Pa 1 = , f 2κ πω2

(7.1)

where f is the focal length of the thermal lens, Pa is the absorbed inversion pump power, ω is the pump beam radius, ηh is the fraction of the pump power converted to internal heat, dn/dT is the temperature coefficient of the refractive index, and κ is the thermal conductivity. It can be seen that the thermal lensing effect increases with pump power and its strength at any given pump power is increased for laser materials with high thermal coefficient of refractive index and low thermal conductivity. For example, with diode pumping at 808 nm in a neodymium laser operating at 1064 nm, the photon defect leads to a heating efficiency of ηh = 24%. The actual heat generating efficiency may be considerably higher than this for intense pumping due to additional up-conversion effects. For Nd:YAG the thermal conductivity is 13 Wm−1 K−1 and the thermal coefficient is 7.3 × 10−6 K−1 [10], while the equivalent values for Nd:YVO4 are 5 Wm−1 K−1 and 3.0 × 10−6 , respectively. Due to the high absorption at the diode wavelength, Nd:YVO4 tends to have high inversion density and exhibits relatively high up-conversion effects [11]. Strictly, the approximation of the thermal lens theory assumes quadratic distribution of the temperature and refractive index, and this occurs at the center region of the pumping. Near the edges of the pump region the refractive index distribution can be more complex, leading to strong aberration effects for highly pumped laser crystals. When used in a laser resonator, the laser amplifier’s thermally induced distortions lead to changes in the cavity stability due to the pump-dependent thermal lens, and the distortions lead to reduction in beam spatial quality. The ISO standard for quantifying beam quality is the M2 beam propagation parameter, which is closely related to the number of times diffraction-limited that compares the divergence of the radiation to a perfect Gaussian distribution. In lamp-pumped Nd:YAG, the thermal effects become significant at the 10 W optical output level, and above 20 W or more the spatial quality is degraded such that increasing output power is at the expense of increasing M2 spatial factor. For lamp-pumped systems the electrical efficiency of the laser is typically ≈ 1%, corresponding to kilowatt pump powers for optical power ≈ 10 w. Diode-pumped

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solid-state lasers are of considerable interest due to the a significant increase in the laser efficiency. This arises due to a number of factors including the high efficiency of the diode laser but more particularly due to the ability to precisely match its spectral output to an absorption line in the solid-state lasing species, leading to a nearly 100% conversion of pump photons to optical inversion of the laser transition. The photon energy defect between the diode pump and the laser wavelengths can also be much smaller than in the lamp-pumped case, and this results in a lower heating fraction in the laser amplifier. Despite the lower heat input, the diode-pumping geometries employed usually involve small volumes that are intensively pumped and lead to thermally induced lensing that can be even higher than that of lamp-pumped systems. Hence, in end-diode-pumped neodymium lasers, while these systems have much higher efficiency, the influence of thermal lensing causes difficulties due to short focal length and beam-quality reduction at optical output powers near ≈10 W, which is comparable to lamppumped systems. In semiconductor diode lasers, single spatial mode operation can be typically achieved up to 100 mW. For edge-emitting diodes the emitter dimensions may be 1 micron by 3 microns for single-mode operation. The divergence angle (full width at high maximum) of the radiation is ≈40◦ by 10◦ with the high divergence (fast axis) corresponding to the small 1-micron emitter dimension and the smaller divergence (slow axis) due to the wider stripe dimension. The power limitation is primarily due to facet damage due to the very high intensity associated with small emitter size. To obtain higher powers, a broad-stripe geometry is used with stripe dimension up to 500 microns. The output power in the broad stripe can be several watts, but the spatial quality in the slow axis is highly multimode, with M2 ≈ 50 in the slow-axis dimension. Highe-output-power diodes use a line array of broad stripes to form a diode bar with overall emitter width of 10 mm. Output powers of diode bars now exceed 60 W but have M2 ≈ 1000, since the individual emitters are incoherent. Higher powers can be obtained by stacking diode bars to allow output powers exceeding one kilowatt but with high M2 in the fast-axis dimension. High-power-diode lasers have spectral bandwidths of typically 2–3 nm. For solid-state lasers, the wavefront distortion and lensing effects imposed on a wavefront propagating through the inverted laser amplifier may be unraveled by the photorefractive PCM. If the incident beam is a high-quality TEM00 spatially Gaussian mode, this allows the double pass to recover the beam quality. For a phase-conjugate laser, the PCM can recover and maintain the beam quality on each round trip of the resonator. The fundamental Gaussian mode with a uni-phase waist at the plane output cavity mirror will be the self-consistent spatial distribution with minimum diffraction loss. For a perfect PCM, all cavity configurations will be inherently stable with the round-trip ABCD matrix of the cavity being the unit matrix. For semiconductor diode lasers, much investigation has centered on the use of phase-conjugate feedback into the diode amplifier for coherent control of its spatial form and for spectral-line-width narrowing. Interesting self-frequency scanning effects have been observed in early investigations using a self-pumped photorefractive PCM.

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To ensure that the photorefractive phase conjugator operates effectively for laser systems demands that a self-pumped phase-conjugation scheme be utilized (e.g., corner pumped or ring-pumped). The photorefractive material must also have good sensitivity at the laser wavelength and be able to achieve high efficiency (high reflectivity) and ideally fast response time. The photorefractive crystal, barium titanate (BaTiO3 ), has among the highest nonlinearity and displays very efficient self-pumped operation, but its highest sensitivity is in the visible part of the spectrum. Solid-state lasers and high-power semiconductor diode lasers operate predominantly in the near-infrared region. By doping barium titanate with rhodium the photorefractive sensitivity of this material is extended out to longer wavelengths and most significantly to the important wavelength region near 1064 nm, where the main lasing transition of neodymium-doped solid-state lasers occurs. The slow response of the photorefractive effect is a further issue. The most efficient crystals such as barium titanate are among the slowest in their response, and due to decreasing absorption and sensitivity at long wavelengths this can limit applicability in some real-world domains. Since the photorefractive response is intensity-dependent, the operation in high-power laser systems allows improvement in response speed. Ultimately, since the mechanism of photorefraction requires absorption of the radiation for charge excitation and transport, the photorefractive material may itself be excessively heated and cause difficulties unless it is adequately cooled. Despite this issue, demonstrations of high-power solid-state laser operation have been made using photorefractive self-pumped PCM with a low absorption material at the laser wavelength 1064 nm.

7.3 Phase-Conjugate Diode-Pumped Nd:YVO4 Laser Amplifiers Most demonstrations in solid-state lasers have been based on a master-oscillatorpower amplifier (MOPA) scheme with two (or four) passes of a Nd:YAG amplifier with either flashlamp-pumping or quasi-CW diode pumping. Nd3+ -doped YVO4 is a promising laser material for high-average-power diode-pumped operation because it exhibits large stimulated emission cross-section as well as large absorption at laser diode frequency in comparison with Nd:YAG [12, 13, 14]. These factors combine to make Nd:YVO4 suitable for diode-pumping. However, its poor thermal properties such as low thermal conductivity make the power scaling difficult to achieve. Previous reports have presented that the Nd:YVO4 exhibits strong thermal lensing and aberration due to the quantum defect and nonradiative decay including up-conversion excitation. One solution to correct these thermal effects is application of photorefractive phase conjugation to the Nd:YVO4 laser amplifiers. In recent years, Omatsu et al. have demonstrated a multiwatt CW neardiffraction-limited output from a side-pumped Nd:YVO4 double-pass amplifier with a ring self-pumped Rh:BaTiO3 PCM in an external loop cavity geometry

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Amplifier g(z) Ps (0) Ppc (0) z=0

z = L RPs (L) = Ppc (L)

[15]. Furthermore, they extended the phase-conjugate double-pass amplifier to generate 7.4-watt diffraction-limited picosecond pulses [16, 17].

7.3.1 Theory 7.3.1.1 Numerical Simulation of Output Power from the Amplifier

The geometry of the double-pass amplifier is shown schematically in Figure 7.2. A signal beam S travels in the amplifier along the z-axis. The phase conjugate (PC) is produced by the reflection of the amplified signal beam on the right-hand end (Z = L) of the amplifier. The reflectivity of the PCM is R. The signal and phase-conjugate beams experience saturated gain, which is an extension of the steady-state model given by [18], g(z) =

g0 , 1 + (PS (z) + PPC (z)) /(A · Is )

(7.2)

where g0 is the small signal gain, Isat is the saturation intensity, A is the crosssection of the signal and phase-conjugate beams, and Ps and PPC are the powers of the signal and phase-conjugate beams, respectively. A transversely diode-pumped Nd:YVO4 amplifier initially proposed by Alcock et al. has an elliptical gain volume. Recent operation of this scheme with continuous-wave diode-pumping has shown that it exhibits a high small-signal gain exp(g0 L) of typically ≈10, 000[19]. By substituting Is = 1 kW/cm2 , A = 0.2 mm × 1.0 mm, and g0 L = 9.2, equation (8.2) was numerically solved to yield the output power Pout = PPC (0) as a function of the signal input power at various R. Figure 7.3 shows the numerically simulated output power as a function of the incident signal power at various values of R. The threshold incident power for efficient output is less than 5 mW, because of the high gain of the amplifier. Above the threshold, the output power is ultimately limited by the signal beam saturating the amplifier itself. As the reflectivity is increased, the output power increases. The phase-conjugate reflectivity required for obtaining > 6 W is approximately 20%. Above a phase conjugate reflectivity of 30%, the output power is limited to the saturation level. 7.3.1.2 Photorefractive Phase-Conjugate Reflectivity

A photorefractive ring-PCM based on a BaTiO3 crystal allows higher reflectivity and faster temporal response in comparison with a total internal reflection

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R=0.6 R=0.5 R=0.4 R=0.3

Output Power (W)

8 6

199

R=0.2 R=0.1

4

R=0.05 R=0.01

2 0

0

50 100 Input Power (mW)

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F I G U R E 7.3. Output power from a double-pass amplifier as a function of input power at various values of R.

geometry (TIR- or CAT-PCM). Thus, the ring-PCM is a more promising candidate to be applied to a laser amplifier. Reflectivity in the ring-PCM can be numerically simulated on the basis of coupled mode equations given by [20], d A1 dz d A2 dz d A3 dz d A4 dz

1 (A1 A∗2 + A3 A∗4 )A2 =− γ , 2 I0 1 (A∗ A2 + A∗3 A4 )A1 = γ 1 , 2 I0 1 (A1 A∗2 + A3 A∗4 )A2 = γ , 2 I0 1 (A∗ A2 + A∗3 A4 )A3 =− γ 1 , 2 I0

(7.3) (7.4) (7.5) (7.6)

where A1 , A2 , A3 , A4 are the complex amplitudes of the probe, forward-pump, backward-pump, and phase-conjugate waves, γ is the photorefractive two-wave mixing gain, and I0 is the total averaged light intensity, respectively. In the case of the ring-PCM, the boundary conditions can be given by A1 (L) = r A3 (L), A4 (L) = r A2 (L), A2 (0) = 0,    A4 (0) 2  , RPCM =  A (0) 

(7.7) (7.8) (7.9) (7.10)

1

where r is the product of the mirror reflection complex coefficient, L is the crystal length, and RPCM is the phase-conjugate reflectivity, respectively. These equations can be solved to yield phase-conjugate reflectivity RPCM at a given mirror reflectivity parameter |r |2 in the external loop cavity. Figure 7.4 shows theoretical plots of phase-conjugate reflectivity at various |r |2 . The threshold gain product

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F I G U R E 7.4. Simulated phaseconjugate reflectivity of the ringPCM as a function of two-wavemixing gain.

|r|2=0.9 |r|2=0.8

0.8 0.6 0.4 0.2 0 0

2

4 κL

6

8

g0 L for obtaining a value of RPCM of higher than 20% is estimated to be ≈ 2.4 at |r |2 = 0.8. 7.3.1.3 Wavelength Selectivity of a Photorefractive Phase-Conjugate Mirror

An undesirable spectral narrowing phenomenon is frequently observed in laser systems with a photorefractive PCM. This phenomenon reflects the finite twowave-mixing gain-band of refractive index gratings formed in the photorefractive PCM. According to the conventional coupled-mode theory and band-transport model, the two-wave-mixing gain-band of refractive index gratings formed in the PCM is given by [20] λ = (2κ 2 n/π) cos θb , K , κ∝ 2 K + K D2

(7.11) (7.12)

where κ is the two-wave mixing gain coefficient of the BaTiO3 crystal, n is the refractive index of the crystal, θb is Bragg’s angle of the refractive grating, is the grating period, K is the wave-number of the refractive-index grating, and K D is Debye’s wave-number, respectively. These show that the gain-band depends strongly on the grating period. The ring-PCM geometry is preferable in obtaining a broader two-wave-mixing gain-band, because it can prevent the formation of reflection and 2k gratings with the grating period shorter than the laser wavelength.

7.3.2 Experiments 7.3.2.1 CW regime

A schematic diagram of the diode-pumped Nd:YVO4 double-pass amplifier with the ring-PCM is shown in Figure 7.5. The amplifier used is a transversely diodepumped 1.1 at% a-cut Nd:YVO4 slab with a bounce geometry. The amplifier exhibits typically a single-pass gain of > 10, 000. The Nd:YVO4 slab had dimensions

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F I G U R E 7.5. Schematic diagram of CW Nd:YVO4 amplifier with PCM.

of 20 mm × 5 mm × 1 mm. The end surfaces of the crystal were AR-coated for 1064 nm and cut at 2◦ relative to the normal of the pump face to suppress self-lasing within the crystal. The crystal, wrapped in indium foils, was sandwiched between two aluminum blocks, providing thermal conduction through the top and bottom faces. The temperature of the blocks was maintained at ≈ 10◦ C using a water recirculating chiller. The pump diode is a CW single-bar diode array with fast axis collimator and line-focused by a cylindrical lens ( f = 12.7 mm) onto the Nd:YVO4 crystal. The polarization of pump diode was parallel with the c-axis of the Nd:YVO4 crystal, yielding maximum pump absorption (α ≈ 30 cm−1 ). Owing to the strong absorption, 90% of the pump power is absorbed in a depth of 0.7 mm. The focal length of the thermal lens in the Nd:YVO4 due to quantum-defect heating was estimated to be < 0.1 m under maximum pumping on the basis of the heat diffusion equation [21]. A commercial Nd:YVO4 laser with 100 mW of diffraction-limited and singlelongitudinal mode output was used as the signal laser. The signal laser beam is collimated to a φ1 mm spot, and is directed toward the amplifier by mirror M2 with 14% reflection for 1064 nm. The polarization of the signal beam was rotated by a half-wave plate (HWP1 ) to be parallel to the c-axis of the Nd:YVO4 crystal, thereby yielding maximum gain. An optical isolator is placed between the signal laser and the amplifier in order to prevent feedback. The signal laser beam is line-focused in the vertical direction by a cylindrical lens, CL1 , with a focal length of 100 mm to form an ellipsoidal spot in the amplifier with dimensions of ≈ 0.2 × 1 mm2 , so that the injected signal beam can overlap sufficiently well with the elliptical gain volume. The amplified signal beam, emerging from the amplifier, is recollimated by another cylindrical lens, CL2 , with a focal length of 100 mm. The amplified signal beam is directed toward a 45◦ -cut BaTiO3 crystal with a 3200-ppm Rh doping by two turning mirrors, M3 and M4 , with high reflectivity for 1064 nm. The signal beam is relayed to be a 3–4-mm spot onto the BaTiO3 crystal by a spherical lens L1 . The maximum signal intensity on the crystal was ≈ 25 W/cm2 . A half-wave plate HWP2 makes

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the polarization of the signal beam to lie in an extra ordinary plane of the BaTiO3 crystal, yielding maximum two-beam coupling gain. The BaTiO3 crystal measures 10 mm × 9 mm × 2 mm with the entrance and exit surfaces AR-coated for 1 μm. A two-wave-mixing gain of > 15 cm−1 was obtained. The photorefractive crystal is formed into a ring PCM by two mirrors (M5 , M6 ) and two spherical lenses. In the external loop, two spherical lenses ( f = 200 mm) form 4 f -relay-imaging optics, such that the crystal location coincides with the conjugate points of the imaging optics. The coherence length of the signal laser is longer than the external loop length (> 80 cm), leading to formation of reflection gratings that decrease and fluctuate the phase-conjugate reflectivity. The formation of reflection gratings is prevented by the vibration of the M6 mirror driven by an attached stepping motor with a vibration frequency of 5–10 Hz [22]. With this system, the reflectivity of the ring PCM is ≈ 40%, which is consistent with the theoretical value. Figure 7.6 shows the single-pass gain of the amplifier as a function of pump diode power with signal power of 6 mW. At a maximum pumping power of 23.4 W, the amplified signal power was 2.8 W, corresponding to a single-pass gain of 470. The experimental output power of a double-pass amplifier with the ring-PCM is shown in Figure 7.7. The output power increased as a function of the pump diode power, and reached up to 7.8 W at the pump power of 23.4 W. There is good agreement between experimental output power and theoretical output power. Finally, a double-pass gain of as high as 1300 and an optical extraction efficiency of as much as 33% were obtained. In order to compare the system with a conventional double-pass amplifier, the PCM was replaced by a conventional high-reflectivity plane mirror. Open circles in Figure 7.7 show the experimental plots of output of the double-pass amplifier with the conventional mirror. When the pump diode power was comparatively low, the output power was not significantly different from that of the double-pass amplifier with the PCM. When the pump power was 23.4 W, the output power was limited to 75% of that of the amplifier with the PCM. When the pump diode power is high, strong thermal lensing and distortions of the amplifier aberrate the amplified signal wavefront. And thus, the use of the 103

Gain

102

101

100

0

10 20 LD Power (W)

F I G U R E 7.6. Experimental plots of single-pass gain in a CW Nd:YVO4 amplifier.

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PCM Conventional Mirror

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F I G U R E 7.7. Experimental output power from a CW Nd:YVO4 double-pass amplifier with PCM.

6 4 2 0

0

10 20 LD Power (W)

conventional mirror enhances the distortion of the reinjected signal wavefront. Consequently, spatial overlapping between the reinjected signal beam and active region is reduced, whereas the PCM, having a potential to compensate thermal distortion, can allow much more efficient spatial overlapping between the reinjected signal and pumped regions. Therefore, highly efficient output from the double-pass amplifier with the PCM was obtained. To investigate the potential of phase correction in this system, the spatial form of the double-pass amplified beam was measured. The experimental far-field spatial patterns of the amplified signal beam are shown in Figure 7.8. The original signal beam had a near-perfect Gaussian profile, as shown in Figures 7.8(a). Figures 7.8(b) and (c) illustrate the output beam patterns in the double-pass amplifier with the PCM and conventional mirror, respectively. Though the beam profile of the output of the double-pass amplifier with the PCM was fatter along the vertical axis than the original one, the profile along the horizontal axis was a perfect diffractionlimited one. On the other hand, the output with the conventional mirror, instead of PCM, was deeply modulated and showed strong astigmatism. These demonstrate that the double-pass amplifier with the PCM has a good potential for correction of thermal distortion inside the amplifier. Further improvement in the quality of correction of astigmatism in the amplified signal beam has been shown to be achieved by addition of a Dove prism in the loop producing a 90◦ rotation of the beam [23]. Figure 7.9 shows the time evolution of output power of the double-pass amplifier with the PCM at the beginning of buildup. After several seconds (less than 10 seconds) following the amplifier being turned on, the phase-conjugate reflectivity grows to reach a steady-state level. 7.3.2.2 Picosecond Regime

High-average-power picosecond diode-pumped solid-state lasers have received considerable attention in many fields. Nd:YVO4 , which exhibits a broad emission band (≈ 1 nm), is a promising picosecond laser material. To date, Couderc et al. have demonstrated 650-mW and 2.8-ps output from a passive mode-locked diodepumped Nd:YVO4 laser based on a second-order nonlinear process [24].

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F I G U R E 7.8. Beam profiles of (a) the oreginal signal, (b) the output of the double-pass amplifier with the PCM, and (c) the output of the double-pass amplifier with the conventional mirror.

Double-Pass Amplifier Output (a.u.)

The demonstration of the operation of a 7.4-watt picosecond output from a double-pass side-pumped Nd:YVO4 amplifier with a ring self-pumped Rh:BaTiO3 phase-conjugate mirror is presented.

1

0.5

0

0

10

20 Time (s)

30

40

F I G U R E 7.9. Temporal evolution of the output from a CW Nd:YVO4 amplifier with PCM.

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205

Signal (ML-YVO4 laser) Isolator1 Isolator2

Rh:BaTiO3

PBS Output L1

Nd:YVO4 CL3

HWP1

HWP2 CL CL1 2 CW-LD Pump 35W

F I G U R E 7.10. Schematic diagram of a picosecond Nd:YVO4 amplifier with PCM.

Figure (7.10) shows a schematic diagram of the experimental setup, which is nearly identical to that of the CW double-pass amplifier with the PCM. The signal laser used for the amplifier is a commercial diode-pumped CW modelocked Nd:YVO4 laser with a pulse width of 7.2 ps and a repetition rate of 100 MHz. To prevent feedback to the signal laser, an optical isolator (isolator1 in the figure is placed between the signal laser and the amplifier. To eject the amplified output from the system, another optical isolator (isolator2 ) formed by a polarizing beam splitter (PBS), a Faraday rotator (FR), and a half-wave plate (HWP1 ) is placed between isolator1 and the amplifier. The collimated signal beam is directed toward the amplifier by a mirror M1 , with its polarization parallel to the c-axis of the Nd:YVO4 crystal. The crystal was transversely pumped by a water-cooled CW-single-bar diode array. The diode center wavelength was 809 nm with a wavelength width (FWHM) of 1.9 nm. The effective absorption depth is < 1 mm. This diode had a relatively broad spectrum in comparison with the diode used in the CW experiment, thereby producing a deeper active region. Maximum pump diode power was 36 W with a single-pass gain G (= exp(g0 L)) greater than 10,000. The collimated signal beam is directed into the amplifier by a 75-mm vertical cylindrical lens, CL1 , to form a line focus with dimensions ≈ 0.2 mm × 1.0 mm. The amplified signal beam is delivered using cylindrical lenses CL2 , CL3 , and a spherical lens L1 to form a φ2-mm circular spot on the BaTiO3 . The crystal had dimensions of 5 mm × 4 mm × 4 mm. A half-wave plate HWP2 rotated the polarization to be parallel to the extraordinary plane of the BaTiO3 crystal, thereby yielding maximum two-beam coupling gain. Frequency-narrowing effects due to wavelength selectivity of the PCM can broaden output pulses. And thus, an efficient PCM for ultrashort pulses requires broad-band two-wave mixing gain, which makes ultrafast operation difficult to achieve. As stated above, use of a ring-PCM geometry is one solution for obtaining efficient phase conjugation for ultrashort pulses. The BaTiO3 crystal formed the ring-PCM with two turning mirrors and two 4 f -imaging lenses. The angle between probe and forward-pump beams was

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T. Omatsu, M.J. Damzen, A. Minassian and K. Kuroda F I G U R E 7.11. Experimental plots of output power from a picosecond Nd:YVO4 amplifier with PCM as a function of input signal power.

Output Power (W)

8 6 4 2 0

0

50 100 Input Power (mW)

150

≈ 24◦ . The experimental two-wave-mixing gain with this system was > 6 cm−1 . Since the coherence length of the signal laser (< 3 mm) was much shorter than the external loop length (≈ 60 cm), the formation of reflection gratings was negligible even without the vibration of the M6 mirror. By substituting experimental parameters n = 2.5, κ = 6 cm−1 and = 2.5μm into (10), wavelength selectivity λ of the two-wave-mixing gain is estimated to be ≈ 5 nm, which is sufficiently broader than the spectral bandwidth (≈ 0.35 nm) of the signal pulse. And thus, the pulse broadening effect due to wavelength selection by the PCM should be negligible and efficient phase-conjugate reflectivity can be expected. Once the amplified signal laser was injected into the PCM, the PCM output started to build up gradually, and several minutes later its reflectivity reached a steady level. The slow temporal evolution was due to the relatively low concentration of the Rh ions in the BaTiO3 crystal. After the PCM built up sufficiently well, its temporal response to changes in the incident beam was much faster (less than one minute). With this system, the reflectivity of the PCM was typically ≈ 40%. Experimental reflectivity agreed well with the theoretical value. The phases-conjugate beam was fed back into the amplifier, and after passing through the amplifier it was reflected off by the PBS. During the experiments there was no damage to the crystals due to pico second pulses. The output power from the double-pass amplifier was measured by varying the signal power. Figure 7.11 shows the output power from the double-pass amplifier with PCM as a function of signal beam power at pumping power of 36 W. The threshold input power for obtaining efficient output was 2–3 mW. Above threshold the output power eventually saturated, giving a maximum output power of 7.4 W with an input power of 137 mW. This corresponds to an extraction efficiency of > 20%. In Figure 7.12, the simulated curve at the phase-conjugate reflectivity R of 40% is also plotted. There thus was demonstrated good agreement between the experimental and simulated results. Output power as a function of pump diode power was also measured (Figure 7.12). The signal power was fixed to be 137 mW. Output power is proportional to the pump power. A slope efficiency of 22% was obtained. This value was not as high as that in the CW experiment. The diode had a relatively broad spectrum, and a relatively deeper active region was produced, which reduces spatial

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F I G U R E 7.12. Output power as a function of pump diode power.

8 Output Power (W)

207

6 4 2 0

0

10

20 30 Pump Power (W)

40

overlapping between the signal beam and the active region. The output beam exhibited a near-perfect Gaussian profile and its beam propagation factors M x2 in the horizontal direction and M y2 in the vertical direction were <1.3 and 1.1, respectively. In order to compare our system with a conventional double-pass amplifier, we replaced the PCM with a conventional high-reflection plane mirror. The output exhibited strong aberration, and its beam propagation factor M x2 was ≈ 3.1 in the horizontal direction. The output power in this case was limited to ≈ 6 W. An experimental intensity autocorrelation trace of the output pulse, obtained by second harmonic generation in a 5-mm KTP crystal, is shown in Figure 7.13. An FWHM of 13 ps was measured, which corresponded to a 9.2-ps pulse width for a Gaussian-shaped pulse. A finite gain-band of the amplifier can also broaden output pulses. In general, the pulse-broadening result in the amplifier with finite gain-bandwidth is given by [18] 2 = τin2 + τout

(16 ln(2)) ln G D , ωg2

(7.13)

where G D is the small-signal gain of the double-pass amplifier, ωg is the gain-bandwidth, and τin and τout , are pulse widths of input and output pulses, 1 SH intensity (a.u.)

SH intensity (a.u.)

1 0.8 0.6 0.4

0.6 0.4 0.2

0.2 0 −20

0.8

−10

0

10

20

0 −20

−10

0

Time delay (ps)

Time delay (ps)

(a)

(b)

10

20

F I G U R E 7.13. Autocorrelation traces of output pulses from the phase-conjugate amplifier.

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T. Omatsu, M.J. Damzen, A. Minassian and K. Kuroda F I G U R E 7.14. Frequency spectrum of output pulses from a phase-conjugate amplifier.

Power (a.u.)

1 0.8 0.6 0.4 0.2 0

1063

1064

1065

1066

Wavelength (nm)

respectively. In the Nd:YVO4 amplifier with ωg /2π = 300 GHz and high gain G D ≈ 104 , this gives a output pulse width of 9.0 ps. This value is consistent with the experimental one. In addition, as shown in Figure 7.13; the spectral bandwidth (0.19 nm) of the output pulse from the double-pass amplifier was relatively narrower than that of the signal pulse (0.35 nm), while the phase conjugate had a spectral bandwidth of 0.35 nm. The corresponding Fourier-transform-limit was 8.8 ps. Consequently, pulse broadening is due mainly to the finite gain-bandwidth of the amplifier. Such a technique provides attractive features for the realization of highquality, all-solid-state, compact picosecond 1-μm lasers operating at high average power. 7.3.2.3 Nanosecond Regime

Initially, high-quality Q-switched output from a double-pass amplifier with the photorefractive PCM was demonstrated. To date, several researchers have demonstrated the generation of high-quality Q-switched pulses from master-oscillator power amplifier systems with the photorefractive PCM based on Rh:BaTiO3 . Brignon et al. presented a flash-lamp-pumped Nd:YAG four-pass amplifier with a CAT-PCM, and generated a 65-mJ diffraction-limited Q-switched pulse at a pulse repetition frequency of 10 Hz [25]. The CAT-PCM exhibits a small incident aperture compared with the ring-PCM, and thus it is not suitable for generation of high-energy nanosecond pulses. Tei et al. demonstrated 360-mJ near-diffraction-limited Q-switched output from two diode-pumped Nd:YAG zigzag slab amplifiers with the ring-PCM with pulse repetition frequency of 100 Hz [26]. This pulse energy is the highest, to the best of our knowledge, obtained by a phase-conjugate amplifier in the nanosecond regime.

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7.4 Phase-Conjugate Laser Resonator A phase-conjugate resonator (PCR), in which one end mirror is a phase-conjugate mirror (PCM), is another alternative solution for correction of thermal distortions in laser resonators [27]. Self-pumped photorefractive PCM can allow a laser oscillation of the PCR without external pumping beams to the PCM. To date, McFarlane et al. have demonstrated the laser oscillation of a pulsed dye laser using a CAT-PCM and achieved a narrowband output with a line-width of 2 GHz. However, there is no report concerning photorefractive PCR lasers in the 1-μm region. Recently, laser oscillation of a CW diode-pumped Nd:YVO4 phase-conjugate resonator based on a ring self-pumped Rh:BaTiO3 phase-conjugate mirror was demonstrated [28]. In this system, a 7.5-W output with high beam quality was obtained. A schematic diagram of the experimental setup is shown in Figure 7.15(a). A 1.1 at% a-cut Nd:YVO4 slab with dimensions of 20 mm × 5 mm × 1 mm forms the transversely diode-pumped amplifier with a bounce geometry. Two mirrors (M1 , M2 ) with reflectivities of 10% and 4% for 1064 nm form a master resonator. Two cylindrical lenses (CL1 , CL2 ) with focal lengths of 100 mm are placed inside the resonator in order to make the fundamental laser mode overlap sufficiently well with the elliptical gain volume. Mirror M1 acts as the output coupler of the laser system. The laser output emerging from the mirror M1 is directed toward a power meter. The other laser output emerging from the mirror M2 is directed towards a ring-PCM based on a 400-ppm Rh-doped BaTiO3 crystal with dimensions of 5 mm × 4 mm × 4 mm. A spherical lens L1 is used to form a spot size with diameter 2–3 mm on the BaTiO3 crystal in the ring-PCM. The polarization of laser output is rotated by a half-wave plate (HWP1 ) to be parallel to the c axis of the BaTiO3 . The phase-conjugate reflection from the ring-PCM feeds back to the master resonator. This produces a self-injection locked laser by the external phase-conjugate feedback. After the PCM builds up sufficiently well and its reflectivity reaches a steady level, one can make the phase-conjugate resonator self-sustaining without the need of the mirror M2 , as shown in Figure 7.15(b). When the master laser was injected in the PCM, the PCM started to build up gradually. After the PCM built up sufficiently well, the laser output power was enhanced by the feedback of phase conjugation to the master laser. Figure 7.16 shows the experimental laser output power with the ring self-pumped PCM as a function of pump diode power. The laser output power increased nearly as a linear function of the pump diode power, and reached up to 7.5 W at the pump power of 22 W, corresponding to an optical extraction efficiency of 35%. The slope efficiency was 46%. With this system, the phaseconjugate reflectivity was typically 40% over the full range of pump power (8 W–23 W). An experimental far-field pattern of the laser output with coupled ring-PCM is shown in Figure 7.17(a). Even though the far-field pattern had parasitic side

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PCM L3 Rh:BaTiO3 L2

M1 (R~16%)

M2 Nd:YVO4 Amplifier

L1 HWP

CL1

CL2 Diode pump

(a)

PCM L3 Rh:BaTiO3 L2 L1 S

M1 (R~16%)

M2 Nd:YVO4 Amplifier HWP CL1

CL2 Diode pump

(b)

F I G U R E 7.15. Schematic diagrams of (a) phase-conjugate coupled laser resonator, and (b) self-sustaining phase-conjugate laser resonator.

9 8 Output Power (W)

7 6 5 4 3 2 1 0

0

2

4

6

8

10 12 14 16 18 20 22 24 Pump Power (W)

F I G U R E 7.16. Output power as a function of a pump diode power.

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F I G U R E 7.17. Spatial forms of (a) phase-conjugate laser with coupled cavity, (b) selfsustaining phase-conjugate laser, and (c) self-sustaining phase-conjugate laser with a slit.

lobes, it showed a dominantly Guassian profile. The suppression of higher-order modes in the far field is presumed to be induced by spatial-filtering (Gaussian aperture effect) on the PCM itself. In this system, the PCM has a sufficiently high reflectivity for the saturation of the Nd:YVO4 amplifier, and it can act as the end mirror of the laser system. Namely, one can make the resonator self-sustaining without the need of the back mirror M2 . To create the self-sustaining PCR, M2 was removed and the output power of the system reached a steady level, and was unchanged during long-term observation of more than several hours. As shown in Figure 8.17(b), the experimental spatial far-field pattern in the steady state had the side lobes due to higher-order modes. In the PCR, any transverse modes are potentially allowed to oscillate. And thus, higher-order modes as well as the fundamental mode can be generated. To achieve a full suppression of the higher-order transverse modes, a slit (S) was placed at the focal plane of the lens L1 . As shown in Figure 7.17(c), the

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experimental far field had a near-perfect Gaussian profile. There is no degradation of the output power by insertion of the slit. The maximum output power reached up to 7.5 W at the pump power of 22 W. The laser system was quite stable. Its output power and spatial profile remained unchanged during long-term observation of more than one hour. In addition, the spatial quality of the laser output was almost unaffected by changing pump power, thereby demonstrating that the PCR laser system, formed by M1 and the photorefractive PCM, is able to adapt spatially and to dynamically correct strong thermal lensing and distortions of the amplifier.

7.5 Phase-Conjugate Laser Diode 7.5.1 Injection-Locked Laser Diode Using a Mutually Pumped Phase-Conjugate Mirror High-power laser diodes (LD) have a great demand as pumping sources for solidstate lasers. However, a conventional high-power LD, called a broad stripe LD (BS-LD), exhibits poor spatial quality, which makes highly efficient pumping of solid-state lasers difficult to achieve. Free-running BS-LDs generally exhibit an undesirable twin-far-field, which arises from an incoherent superposition of several transverse modes coexisting in the wide active region. External injection of a high-quality laser (master laser) into a BS-LD (slave laser) can generate a single-lobed far-field by the coherent superposition of transverse modes. This technique is called injection-locking [29, 30, 31]. The equivalent spontaneous noise power in free-running BS-LDs can be given by the expression [32] Pnoise =

δgth L P · , 1− R 2π

(7.14)

where P is the intracavity power of the BS-LD, R is the reflectivity of the laser diode facet, gth is the threshold gain coefficient, δ is a factor determined by the active layer structure, and L is the cavity length. Assuming R = 0.1, gth = 48 cm−1 , L = 350 μm, and δ = 0.01, we can find Pnoise = 2 × 10−3 P. The performance of injection-locking is determined by the competition between the spontaneous noise-driven and injection-driven modes, and the injected master laser power must be sufficiently larger than Pnoise . Consequently, the injected power required for a full injection-locking is estimated to be at least 1% of the slave laser output power. Since a slight misalignment reduces the injected power and induces instability of the system, the injection-locking technique generally requires accurate alignment in the injection angle and position of master laser to the slave one. A photorefractive mutually pumped phase conjugate mirror (MPPCM) can diffract the master laser beam, and produce automatically a phase conjugate wavefront of the slave laser, which injects into the slave laser [20]. Therefore,

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213

λ/2 plate

Faraday isolator

Single-mode master laser

43° c

Photodetector

DPCM (BaTiO3) 68°

λ/2 plate Broad-area slave laser

Adjustable slit y Spectrum analyzer, CCD camera z

x

F I G U R E 7.18. Experimental setup for injection-locked laser diode with MPPCM.

MPPCM is capable of improving the reliability and stability of the system, because it can correct any misalignment in the system. A very bright output from the BS-LD injection-locked by a single-mode master laser diode through an MPPCM has been demonstrated [33]. Figure 7.18 illustrates a schematic diagram of the experimental setup. The master laser was a 50-mW single-mode laser diode, and its polarization lay in an extraordinary plane of an undoped BaTiO3 crystal. The master beam passed through an isolator, and was directed toward the BaTiO3 crystal. The incident laser power was approximately 10 mW. The slave laser used was a 1-W broad-stripe laser diode (Sony SLD304XT) with an aperture width of 200 μm and a cavity length of 350 μm. The slave beam was collimated by a high-NA aspherical lens and a spherical lens. The slave laser is initially polarized along the active layer. Its beam quality is near-incoherent along the direction parallel to the polarization vector (incoherent axis) and is partially coherent along the direction perpendicular to the polarization vector (coherent axis). In order to obtain effective phase conjugation using a conventional photorefractive BaTiO3 crystal, the wave vector of the photorefractive grating and the polarization vector of an incident beam should lie in an extraordinary plane of the BaTiO3 crystal. Since the spatial coherence of the BS-LD radiation along its polarization direction (incoherent axis) was very poor, the polarization vector of the radiation was rotated with a half-wave plate (HWP) such that it was parallel to the coherent axis of the BS-LD beam pattern [34]. A slit placed in the far-field plane performed spatial filtering of the slave laser beam. The slit width was ≈ 0.25 mm. The filtered slave beam was directed to the BaTiO3 crystal. The incident power onto the crystal was ≈ 5% of slave output power.

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The external incident angles of the master and slave lasers to the crystal c-axis were 43◦ and 68◦ respectively. The wavelength of the master laser was temperature-tuned to be within the slave laser lasing spectrum. Since, as related above, the presence of reflection and 2k gratings often reduce and fluctuate the phase-conjugate reflectivity, the distance between the slave laser and MPPCM (≈ 4 m) was made as long as possible to prevent the formation of these two gratings in MPPCM. With the buildup of MPPCM, a diffraction efficiency of > 50% was achieved. A portion of the output from the slave laser was picked off by a beam sampler in the optical path of the slave laser and was directed toward a CCD camera. Figure 7.19 shows the lasing spectrum of the slave laser measured by a spectrum analyzer with a resolution of 0.1 nm. The input current was 1.4 times the threshold current, and the slave laser power was ≈ 200 mW. Once the MPPCM was turned on, the lasing spectrum of the slave laser was shrunk to < 10.1 nm (≈ 80 MHz), and its center wavelength was locked to that of the master laser.

F I G U R E 7.19. Lesing spectra of (a) free-running laser diode, and (b) injection-locked laser diode with MPPCM.

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(c) Relative Intesity (arb. units)

Relative Intesity (arb. units)

(a)

0 (b) 0.38°

0 −8 −6 −4 −2 0 2 4 6 Far-Field Angle (degrees)

215

8

0 (d)

0 −8 −6 −4 −2 0 2 4 6 Far-Field Angle (degrees)

8

F I G U R E 7.20. Four-field patterns of (a) free-running BS-LD at 1.9 times threshold current, (b) injection-locked BS-LD at 1.9 times threshold current, (c) free-running BS-LD at 2.1 times threshold current, and (d) injection-locked BS-LD at 2.1 times threshold current.

The spatial form of the output emitted from the slave laser is shown in Figure 7.20. The free-running slave laser diode exhibits a top-hat near-field and a twin-lobed far-field profile with a beam propagation factor M 2 of 61, whereas, the injection-locked slave laser has a single-lobed far-field with a narrow divergence of 1.9 times the diffraction limit, which corresponds to M 2 ≈ 8. When the input current increased to the 2.1-times threshold current, the injection-locked slave laser still exhibited a single-lobed far-field. The beam quality was degraded, and the M 2 factor reached up to ≈ 28, corresponding to 40% of that (M 2 ≈ 86) of the free-running slave laser. These show that the injection-locking technique with the MPPCM has a good potential for improving the brightness of the BS-LD. In this experiment, the output power from the injection-locked BS-LD was limited up to ≈ 500 mW. Further improvement can be expected by use of the cascaded injection-locking technique through the two MPPCMs [35].

7.5.2 Self-Injection-Locked Laser Diode with Phase-Conjugate Feedback A self-injection-locking technique, in which phase conjugate reflection feeds back to a BS-LD laser, is another solution to improve spatial quality of the BS-LD [8, 36, 37, 38]. This technique has attractive features such as self-alignment and no requirement of additional elements except a phase-conjugate mirror. Recently, several researchers have demonstrated high-quality output from a BSLD based on the self-injection-locking technique with phase-conjugate mirror

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as an external feedback mirror. In these current investigations, the CAT-PCM based on a BaTiO3 crystal was used, making the entire optical system simple. However, the required time for the buildup of the CAT-PCM is typically a few minutes even with the use of a Rh-doped BaTiO3 crystal, which has a relatively high sensitivity in the near-infrared region. Self-frequency-scanning and selffrequency-narrowing phenomena are also frequently observed. They are caused by an asymmetric frequency-dependent reflectivity due to wavelength selection of the refractive gratings formed in the BaTiO3 as well as refractive dispersion of the BaTiO3 . Suppression of the self-frequency-scanning by counterbalance of refractive dispersion of BaTiO3 with dispersive elements as a grating has been proposed [36]. However, the dispersive elements give an additional loss to the feedback optics, and degrade the temporal response of the PCM. A self-injection-locked BS-LD based on a ring-PCM has been proposed [39]. The required time for the PCM to build up is expected to be shorter than one minute. Moreover, the ring-PCM itself suppresses the self-induced frequency scanning without the use of dispersive elements, because it exhibits relatively broad wavelength selectivity compared with CAT-PCM. The laser system, then, becomes much simpler. A schematic diagram of the experimental setup is shown in Figure 7.21. A 1-W high-power laser diode (SLD304XT) manufactured by Sony was used, and its output power was 750 mW at a drive current of 1.35 A. The collimating optics for the laser beam are composed of a non-AR-coated rod lens (φ = 8 mm) and a non-AR-coated cylindrical lens ( f = 30 mm). The transmission efficiency of the collimating optics was 80%. As with the injection-locked BS-LD with MPPCM, the polarization vector of the laser beam lies along the

F I G U R E 7.21. Self-injection-locked BS-LD by phase-conjugate feedback.

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217

coherent axis and in the extraordinary plane of the BaTiO3 crystal. Part of the BS-LD beam is out-coupled by a beam sampler, BS, placed in the far-field. Reflectivity of the beam sampler BS was less than 10%. A slit and the cylindrical lens form a spatial filter in the far-field of the laser beam along the incoherent axis to select several transverse modes. The slit width is ≈ 30% of the whole far-field. The filtered laser beam, whose spatial coherence is much superior to that of the whole laser beam, is loosely focused by a 300-mm spherical lens (FL) onto a 45◦ -cut Rh-doped BaTiO3 crystal (3200 ppm). The BaTiO3 crystal measures 9 mm × 10 mm × 2 mm, and forms a ring-PCM. The phase-conjugate beam passes through the slit again and is injected well onto the laser diode. The frequency spectra of the out-coupled beams are shown in Figure 7.22(a). Solid and broken lines show the spectra with and without the phase-conjugate mirror, respectively. The free-running BS-LD had a line width of 1.5 nm at FWHM. On the laser diode turning on, the spectral line width started to narrow. After a

6

Intensity [a.u.]

810

Lasing Wavelength (808.9nm)

809

4

808

3 2

807 Line Width (0.57nm)

1

806

0

805 0

804 805 806 807 808 809 810 811 812 813 814

2

4

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8 10 12 14 16 18 20

Time [min]

(a)

(b) 811

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809

4 Lasing Wavelength

Line Width

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805

Line Width (FWHM) [nm]

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808

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Line Width (FWHM) [nm]

Lasing Wavelength [nm]

811

0 0

2

4

6

8 10 12 14 16 18 20

Time [min]

(c)

F I G U R E 7.22. (a) Lasing spectrum of self-injection-locked BS-LD, temporal evolution of lasing spectra of (b) self-injected BS-LD with ring-PCM, and (c) self-injected BS-LD with CAT-PCM.

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short delay (typically 10–20 s), the phase-conjugate reflectivity reached its steadystate level. And then the spectral line width of ≈ 0.6 nm was measured, and it corresponded to 40% of that of the free-running BS-LD. The lasing spectral width is relatively broad compared with those in previous experiments, in which the CAT-PCM based on the 0◦ -cut BaTiO3 crystal was used. This reflects broad wavelength selectivity of the ring-PCM without the formation of reflection and 2k gratings. The use of the ring-PCM, having much broader wavelength selectivity compared with the CAT-PCM, results in the suppression of the asymmetric feedback to the BS-LD and the stabilization of the lasing spectrum. As shown in Figure 7.22(b), the central frequency of the spectrum was locked at a wavelength of 808.9 nm within 0.1 nm for over 3 h. The ring-PCM was replaced by the CAT-PCM to compare their performance. A 0◦ -cut Rh-doped BaTiO3 (400 ppm) itself formed the CAT-PCM. Its dimensions were 5 mm × 6 mm × 7 mm. The temporal evolution of experimental lasing spectrum is shown in Figure 7.22(c). The lasing spectrum periodically scanned to the red (self-frequency-scanning), and mode-hopping occurred occasionally, consistent with the previous experiments. Figure 7.23 shows the temporal response of the ring-PCM in comparison with that of the cat-PCM. Within 30 seconds from the laser diode turning on, the reflectivity of the ring-PCM reached the steady-state level. In contrast, even several minutes after the BS-LD was turned on, the reflectivity of the cat-PCM was still growing gradually. These experiments show that the ring-PCM has a good potential to improve the temporal response and stabilization of the laser system. As shown in Figure 7.24, the phase-conjugate laser diode generated a singlelobed far-field radiation with a narrow divergence, while the free-running BS-LD generated a top-hat far-field output with a wide divergence. The M 2 factor of

Phase Conjugate Reflectivity

0.15 CAT Ring 0.1

0.05

0 0

2

4

6

Time (min.)

F I G U R E 7.23. Temporal evolution of phase-conjugate reflectivity.

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F I G U R E 7.24. Spatial from of self-injected BS-LD.

the self-injection-locked BS-LD was ≈ 25% of that of the free-running laser diode. We must note that the remaining free-running lobes could be seen around ±5◦ in the experimental far-field pattern. In our experiment, the phase-conjugate reflectivity was at most 5–10%, since the BS-LD was not a fully coherent light source. After taking into account the transmission loss inside the feedback cavity, the phase-conjugate feedback power was estimated as ≈ 0.5% (3 mW) of the total laser power. In these experiments, the phase-conjugate feedback power was not enough for a full self-injection locking. Hence the free-running lobes remained in the experimental patterns.

7.6 Summary In this chapter, we have described the application of photorefractive PCM to high-power solid-state and semiconductor laser systems with high beam quality. Although it has been demonstrated that the photorefractive PCM has many excellent features such as high fidelity, high reflectivity, and the ability to operate at low powers, there were at least two hurdles to overcome before it could be applied to high-power laser system, namely, response rate and damage threshold. The photorefractive response time is so slow that the photorefractive PCM cannot compensate the pulse-to-pulse fluctuations of wavefront aberration in pulsed laser systems. It can only correct the aberration induced by the average temperature distribution in the active media. The second hurdle is caused by the fact that the photorefractive effect is initiated by absorption of laser radiation. Even though the absorption coefficient is small, it limits the highest laser power acceptable for photorefractive materials. Fortunately, these problems are solved positively as described in this chapter. As a result, the beam qualities of Nd:YVO4 laser

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amplifiers can be improved drastically in both CW and pulsed operations. In the CW regime, an output power of up to 7.8 W was obtained in an almostdiffraction-limited beam profile. In pulsed operation, a mode-locked Nd:YVO4 laser with a pulse width of 7.2 ps and a repetition rate of 100 MHz was amplified to an average power of 7.4 W. The amplified beam had remarkable beam propagation factors of Mx2 < 1.3 (horizontal) and M y2 < 1.1 (vertical). Further power scaling of over 10 W is possible by the use of multipass geometry. Even in the Q-switched operation, the photorefractive PCM can compensate wavefront aberration and reproduce nearly diffraction-limited beams. For example, a 360-mJ near-diffraction-limited Q-switched output, corresponding to an average power of 36 W, was demonstrated in a diode-pumped Nd: YAG amplifier with a ring-PCM. This demonstration shows that power scaling of ≈ 100 W in the phase-conjugate laser amplifier can be expected, but the maximum power of the system may be limited by depoling effects observed when 1-μm laser radiation is used. The photorefractive PCM is also utilized in phase-conjugate laser resonators. Recently, a 7.5-W output with an excellent beam profile was demonstrated in a diode-laser-pumped Nd:YVO4 phase-conjugate resonator. The photorefractive PCM can be applied to high-power laser diodes with high beam quality. Injection-locking is one technique used to improve the beam quality of high-power laser diodes. Mutually pumped PCMs (MPPCM) are useful for the conversion of beam profiles between master and slave lasers. Instead of the master–slave lasers scheme, self-injection-locking with an external PCM is also useful for improving beam quality. With these techniques, a single-lobe and high-quality output was successfully obtained from high-power broad-stripe laser diodes.

References 1. B.Y. Zel’dovich, V.I Popovichev, V.V. Ragul’skii, F.S.D. Faizullov. JETP Lett. 15, 109 (1972). 2. O.Y. Nosach, V.I Popovichev, V.V. Ragul’skii, F.S.D Faizullov. JETP Lett. 16, 435 (1972). 3. D.A. Rockwell. IEEE J. Quantum Electron, 24, 1124 (1988). 4. S. Camacho-Lopez, M.J. Damzen. Opt. Lett., 24, 753 (1999). 5. M.J. Damzen, R.P.M. Green, and K.S. Syed. Opt. Lett., 20, 1704 (1995). 6. B.A. Thompson, A. Minassian, M.J. Damzen. J. Opt. Soc. Am. B 20, 857 (2003). 7. A. Brignon, S. Senac, J.L. Ayral, J.P. Huignard. Appl. Opt., 37, 3990 (1998). 8. M. Lobel, P.M. Peterson, P.M. Johansen. Opt. Lett., 23, 825 (1998). 9. N. Huot, J.M.C. Jonathan, G. Pauliat, G. Roosen, A. Brignon, J.P. Huignard. Technical Digest CLEO Europe ’98, Glasgow, September 1998. 10. W. Koechner. Solid-State Laser Engineering, 4th edition (Springer-Verlag, Berlin, 1996). 11. J.L. Blows, T. Omatsu, J. Dawes, H. Pask, M. Tateda. IEEE Photonics Tech. Lett. 12, 1727 (1998). 12. A.W. Tucker, M. Birnbaum, C.L. Fincher, J.W. Erler. J. Appl. Phys. 48, 4907 (1977).

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13. G.C. Bowkett, G.W. Baxter, D.J. Booth, T. Taira, H. Teranishi, T. Kobayashi, Opt. Lett. 19, 957 (1994). 14. D. Shen, A. Liu, J. Song, K. Ueda. Appl. Opt. 37, 7785 (1998). 15. T. Omatsu, M.J. Damzen. Opt. Commun. 198, no. 1–3, 135 (2001). 16. T. Omatsu, T. Imaizumi, M. Amano, Y. Ojima, K. Watanabe, M. Goto. J. Opt. A: Pure Appl. Opt. 5, S467 (2003). 17. T. Imaizumi, M. Goto, Y. Ojima, T. Omatsu. Jpn. J. Appl. Phys. 43, 5A, 2515–2518 (2004). 18. A.E. Siegman. Laser (University Science Books, Mill valley, 1986). 19. J.E. Bernard, A.J. Alcock. Opt. Lett. 18, 968 (1993). 20. P. Yeh. Introduction to Photorefractive Nonlinear Optics (Wiley, New York, 1993). 21. J.C. Bermudez, G.V.J. Pinto-Robledo, A.V. Kir’yanov, M.J. Damzen Opt. Commun. 210, 75–82 (2002). 22. N. Huot, J.M.C. Jonathan, G. Roosen, and D. Rytz. Opt. Commun. 140, 296 (1997). 23. K. Tei, Y. Niwa, M. Kato, Y. Maruyama, T. Arisawa. Jpn. J. Appl. Phys. 38, 5885 (1999). 24. V. Couderc, F. Louradour, A. Barthelemy. Opt. Commun. 166, 103 (1999). 25. A. Brignon, J.-P. Huignard, M.H. Garrett, I. Mnushkina. Opt. Lett. 22, 422 (1997). 26. K. Tei, F. Matsuoka, M. Kato, Y. Maruyama, T. Arisawa. Opt. Lett. 25, 481 (2000). 27. R.A. McFarlane, D.G. Steel. Opt. Lett. 8, 208 (1983). 28. T. Omatsu, A. Minassian, M. Damzen. Jpn. J. Appl. Phys. 41, 2024 (2002). 29. L. Goldberg, H.F. Taylor, J.F. Weller. Appl. Phys. Lett. 46, 236 (1985). 30. L. Goldberg, M.K. Chun. Appl. Phys. Lett., 53, 1900 (1988). 31. J.-M. Verdiell, R. Frey, J.-P. Huignard. IEEE J. Quantum. Electron. 27, 396 (1990). 32. S. Weiss, S. Sternklar, B. Fisher. Opt. Lett. 12, 114 (1987). 33. K. Iida, H. Horiuchi, O. Matoba, T. Omatsu, T. Shimura, K. Kuroda. Opt. Commun. 146, 6–10 (1998). 34. S. MacCormack, J. Feinberg. Opt. Lett., 18, 211 (1993). 35. H. Horiuchi, T. Shimura, T. Omatsu, O. Matoba, K. Kuroda. Appl. Phys. B 68, 1021 (1999). 36. M. Lobel, P.M. Peterson, P.M. Johansen. J. Opt. Soc. Am. B, 15, 2000 (1998). 37. M. Lobel, P.M. Petersen, P.M. Johansen. Appl. Phys. Lett., 72, 1263 (1998). 38. M. Lobel, P.M. Petersen, and P.M. Johansen: J. Opt. Soc. Am. B, 16, 219 (1999). 39. N. Yamada, T. Omatsu, H. Nakano, M. Tateda. Jpn. J. Appl. Phys. 41, 606 (2002).

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8 Dynamic Holographic Interferometry: Devices and Applications Philippe Lemaire and Marc Georges Centre Spatial de Li`ege, Avenue du Pr´e Aily, 4031 Angleur, Belgium [email protected]

This chapter is a review of photorefractive crystals used in holographic interferometry, starting from the basics and early experiments up to devices achievement. We first recall the basics of holographic interferometry and describe the most important pioneering experiments that showed the tremendous potentialities of photorefractive crystals as dynamic recording media for this technique. We then present the main requirements for the development of a holographic interferometer and we analyze the figures of merit and the properties of different photorefractive crystals for that purpose. We emphasize the implementation of the phasequantification techniques because they give access to displacement metrology. The next sections are devoted to the presentation of metrological devices based on dynamic holographic interferometry with sillenite crystals. The first system is a holographic camera with a continuous laser adapted to the study of displacement of scattering objects. We present its main development steps and show that it is highly versatile and can be used in the observation of different types of phenomena. The second range of devices allows the observation of transparent objects: one is especially studied for use in microgravity fluid experiments and the second is an adaptation of the first system for scattering objects to the observation of transparent objects. The third system uses pulsed lasers with applications focused on the study of structure vibrations.

8.1 Introduction: Historical Background In order to have a clear understanding of the different works and approaches of holographic interferometry (HI) with photorefractive crystals (PRCs) carried out by several groups, we will start this section by briefly recalling the different holographic methods. HI [1–3] is a technique working under coherent light that allows one to produce the interference of two wavefronts (or more), at least one of which being 223

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holographically stored. The interference pattern called an interferogram shows the optical path difference (OPD) between the wavefronts as intensity variations (fringe patterns). In most cases, the OPDs come from the same object at two different instants. In the case of the opaque scattering objects, the variations arise from the displacement or deformation of the surface, while in the case of transparent objects, they come from thickness or refractive-index variations. Three main methods of HI exist. In real-time HI (RT-HI) only one hologram is recorded. At the readout step, the object is still illuminated and one observes the interferogram resulting from the superimposition of the wavefronts diffracted by the hologram and the one coming directly from the object (transmitted through the hologram). Each object variation is then observed directly (live fringes). The double-exposure HI (2E-HI) requires the recording of two holograms of the object at different states. A further readout step (without the object beam) shows the superimposition of both stored wavefronts (frozen fringes). For both methods, the interferogram is written at each point (x, y) of an observation plane as I (x, y) = Iaverage (x, y), [1 + m(x, y) cos(φ(x, y))],

(8.1)

with Iaverage (x, y) the average intensity and m(x, y) the contrast. The quantity φ(x, y) is the phase difference between the transmitted and the diffracted wavefronts, and that has to be determined in order to calculate the OPDs. A third technique can be applied to the case of vibrating objects: the timeaverage HI (TA-HI). The hologram is recorded during the modal vibration of the object and over a time longer than the vibration period. Here the phase difference is time dependent and written as φ(x, y, t) = φ0 (x, y) sin(t), where φ0 is the amplitude and  the pulsation of the vibration. The intensity pattern is given by I (x, y) ∝ J02 (φ0 (x, y)), where J0 is the zero-order Bessel function whose maximum is found at the vibration node and whose fringe modulation decreases when φ0 increases. A crucial element of HI is the photosensitive medium used for the hologram recording. In earlier developments of this technique, one generally considers that its principal figures of merit are the energetic sensitivity and the diffraction efficiency. However, with the further evolution of the necessary peripherals (such as lasers, CCD cameras, computers, and frame grabbers), other features such as the self-processing and the erasability/reusability of the medium will appear more important in its practical applicability to HI. Therefore, due to their self-developing in situ and indefinitely reusable properties, the PRCs have been progressively considered interesting alternatives to other recording materials. Moreover, a large number of crystal families and species exist, different charge transport mechanisms can be envisaged (diffusion, drift under external field, photovoltaic effect), different beam arrangements can be used (twowave mixing, four-wave-mixing), different diffraction properties exist (diffraction anisotropy or energy transfer with diffraction isotropy), etc. As a consequence, even if the choice of a particular crystal and its working conditions is not a simple matter, it gives unique and smart optical schemes that cannot be envisaged with other recording media. In the following we will review different experiments as envisaged by pioneer groups in the field as well as more recent studies. These

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works show us the tremendous potentialities of PRCs for application to HI. They allowed us to select some particular combinations that we found the most suitable for the development of advanced tools for industrial and metrological applications. The first laboratory experiments of HI with PRCs were carried out by Huignard and Herriau [4] and employed a two-wave scheme whereby a BSO crystal was used with an Argon laser at 488 nm and under an external electric field. They showed the application of 2E-HI to transparent objects. They pointed out that the recording of the second hologram partially destroys the first one and then the recording time of the second has to be smaller than that of the first. On the basis of the same crystal under an external field, the authors proposed the use of the four-wave-mixing technique [5], yielding a permanent diffracted image. The technique is applied to the study of vibrating objects under the TAHI. The objects considered are transparent or reflecting membranes. The object is continuously monitored during a scan of the sinusoidal excitation frequency. When resonant frequency is reached, mode shapes appear, while they disappear at nonresonant frequencies. Despite the alignment difficulties related to the fourwave-mixing arrangement, the authors note the extreme ease with which mode shapes are visualized. Later they proposed to use the same configuration to study a scattering object [6]. The same group proposed for the first time the use of the coupling effect applied to a BSO crystal [7] when the grating wave vector is aligned along the crystal axis 001. The goal is again the study of vibrating objects with the TA-HI. The interferogram contrast being too weak to be exploited, the authors introduce the technique of the mobile grating in order to reinforce the grating recorded. In 1985, Kamshilin and Petrov [8] showed for the first time the application of diffraction anisotropy in HI. The principle is simple: only two waves are incident on the crystal and simultaneously participate in the recording and the readout of the hologram, ensuring the automatic matching of the Bragg condition. Moreover, when the crystal is used under the diffusion regime and the grating wave vector oriented along the axis 110, they show that for a suitable orientation of the input polarization, the diffracted beam has a linear polarization perpendicular to that of the transmitted beam. This allows one to filter out the diffracted beam from the other by placing an analyzer after the crystal. These authors show convincing results of a vibrating object observed with the TA-HI method with a BSO crystal at 514 nm. Later they used the same configuration with a BTO crystal at 633 nm [9]. Still in 1985, the group of Huignard [10] used benefits of polarization properties to improve interferograms, but instead of considering the energy transfer as in [7], the crystal was cut in order to exhibit diffraction anisotropy. They again applied the mobile grating technique to reinforce the space-charge field. They obtained very good results with a vibrating plate in TA-HI. In 1991, Troth and Dainty [11] considered the diffraction anisotropy configuration and obtained excellent double-exposure and time-averaged interferograms with a BSO crystal at 514 nm. They proposed a deep analysis of the signal-tonoise ratio (SNR) as a function of the ratio R between the beam intensities and found that there is an optimum value of R. These works were the first attempt

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to optimize a holographic interferometer. They were followed by a photometric analysis of the noise characteristics [12]. This approach allowed them to estimate the size of observable objects on the basis of the SNR and the noise characteristics measured as a function of the fringe spacing. They showed that typically, one square-meter objects can be tested with 1-watt laser power. Another optimization study was carried out by Miridonov et al. [13] and was similar to that of Troth and Dainty, although more theoretical. They used a BTO crystal at 633 nm and reached high SNR. These pioneering experiments have paved the way for most of the recent studies targeted at the development of metrological devices. They have all been important for an understanding of different principles that may be applied in more advanced systems. Our developments are based on a tradeoff that we made on the basis of the above existing techniques with the aim of obtaining a device that satisfies some requirements that are mandatory if one wishes to use such an instrument in a wide variety of industrial applications. For that purpose, the next section will review the requirements of such instruments.

8.2 Requirements for Applicability of HI 8.2.1 The Ideal Holographic Measurement Device Even if HI has been demonstrated as an interesting technique that can be applied in full-field displacement metrology, in mode shape visualization or in nondestructive testing, it has often failed to be accepted by industrial end users. The principal reason was that most of the experiments remained at the laboratory level, requiring large cumbersome lasers and specialists to interpret the interferogram. Moreover, the holographic recording media generally necessitate complicated chemical processing (in the case of holoplates) or additional electrical charging and heating/cooling devices (in the case of photothermoplastics). These processes are time-consuming, and the hologram is not usable before some tens of seconds if not minutes. A good example of how an industrial holographic interferometer must work in terms of user friendliness is the Electronic Speckle Pattern Interferometer (ESPI) [2, 3]. These are attractive because the holographic recording is directly performed by the CCD camera. Despite the low resolution of the CCD cameras and the OPD’s result, which shows an important speckle noise (requiring numerical filtering with the risk of information loss), this technique is now well commercially established. The above remarks helped us to understand what the potential user of a full-field truly holographic measurement tool wishes: (1) A compact system. Indeed, some measurement cases need a small lightweight device that can be placed in any position with respect to the object under test. (2) A “cheap” system. This condition is mainly influenced by the laser source but also by the quantity of consumables. The latter is generally important with

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holoplates and photothermoplastics. This also means that the measurement head has to incorporate the least complicated and expensive components possible. A versatile system. Since such a device is an investment for the potential user, it may be useful if it can be adapted or adaptable to different applications. A simple “user-friendly” system. The measurement procedure has to be as simple as possible in order to not require an optics specialist for handling a measurement. This means no adjustment, or at least the fewest possible. The largest observable area. This will have a positive impact in the duration of the inspection of a large object if different successive inspections have to be operated to cover the entire area. Quantified data. This is probably one of the most important factors. The aim is to obtain the object displacement measurement from the interferogram. Due to its importance, we will review some of the possible techniques in the next subsection.

8.2.2 Phase Quantification and Associated Techniques The pioneering experiments presented in Section 8.1 did not consider the application of phase-quantification techniques. Such techniques aim to compute automatically the value of φ(x, y) in expression (8.1), which in turn is used to calculate the displacement of solid objects or refractive index variations in transparent objects. Most automated phase-quantification techniques are based on heterodyning, i.e., on the inclusion of an additional phase term in the argument of the cosine of expression (8.1). Heterodyning can be envisaged temporally or spatially. We briefly present two ranges of methods that are frequently applied and that have been considered in our developments. First, the phase-shifting or phase-stepping (PS) [3,14], which consists in acquiring several interferograms with known phase steps introduced between the acquisitions (here we omit the (x, y) dependence of the variables) : Ik = Iaverage [1 + m cos(φ + βk )],

(8.2)

with k = 1, . . . , N (N an integer and greater than or equal to 3) and the phase step βk . The computation of the phase φ is carried out following one or another algorithm depending on the number N and the value of the additional constant phase βk at each step [3]. Generally, the latter is a fraction of 2π (calibrated phase steps), but we can also use the algorithm of Carr´e (N = 4), which does not require calibrated steps, provided they are equal between consecutive interferograms. Equation (8.3) gives an example of phase computation when four images, phaseshifted by π/2, are captured:   I 4 − I2 . (8.3) φ = arctan I1 − I3 One distinguishes the temporal and the spatial PS. In the temporal version, the interferograms (8.2) are acquired successively, with the phase shifts introduced

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between the acquisitions, by shifting the phase of one of the beams (generally the reference beam). In the spatial version, the interferograms are acquired simultaneously in a multicamera system, with phase shifts introduced optically (generally through different polarization separation elements) [15]. The second technique, referred to as Fourier transform (FT) processing [15], consists in adding a spatial carrier to the phase difference φ; the interferogram is then written I = Iaverage [1 + m cos(φ + 2π f 0 x)],

(8.4)

with f 0 the carrier frequency. The Fourier spectrum of (8.4) shows a central peak with two symmetric sidelobes. The latter contain the information on φ, which is extracted by proper filtering and taking the inverse Fourier transform of the result. Contrarily to the PS, the FT requires only one interferogram for the calculation of φ, but it has to be noted that there are severe constraints on the spectra of Iaverage , m, and φ with respect to the carrier frequency f 0 in order to have no ambiguity in the final result. The FT technique is an alternative method to spatial PS adapted for the analysis of transient events, when it is not possible to have sufficiently stable interferograms to apply temporal PS. Nevertheless, FT generally gives less-accurate results than PS because of the filtering. Following the pioneering experiments presented earlier, different groups developing PRC-based holographic interferometers moved a step further by including one of the above phase-quantification techniques. The group of von Bally has developed a holographic camera that records sequences of double exposures [16, 17]. It is based on a BTO crystal and works with an argon laser at 514 nm. They proposed the application of FT [16] but without addition of a spatial carrier (this way to proceed strongly limits the amplitude of the phase difference to measure). They later showed the application of the PS [17], which is applied by the technique of the double-reference beam [18]. Indeed, in the case of two frozen holograms, the phase between them can be shifted only if different reference beams are used for the recording, the phase being shifted at the readout on one of these beams. This is clearly a disadvantage in terms of simplicity of the procedure. Later they considered the use of pulsed illumination with the FT technique for phase quantification [19]. In this case the spatial carrier is introduced between the two pulses by tilting the reference beam. This process is strongly limited by the important angular selectivity of the crystal which authorizes only a small number of carrier fringes, and therefore the amplitude of φ(x, y) is also limited [15]. Pouet and Krishnaswamy [20] proposed the use of 2E-HI associated with a stroboscopic technique for the visualization of vibration patterns with a BSO at 514 nm. Once again short response times are needed, so relatively small objects are observed. The measurement is operated three times with the same resonant modes but with different phase offsets introduced by an electrooptic modulator (“piston” effect), yielding three phase-shifted interferograms. This instrument gives convincing results but can be applied only in the case of small vibrating objects.

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The group of G. Roosen proposed the use of a special polarization separation technique to obtain, simultaneously, two phase-shifted images of the same object displacement acquired by two separate CCD cameras [21]. They can perform quantitative phase measurement with high accuracy based on a single shot, particularly well adapted to pulse lasers. Their holographic camera is breadboarded and has been successfully used with an argon laser at 514 nm and was further used in pulsed illumination with a ruby laser (694 nm) [22]. In the last case, the wavelength is badly adapted to the sensitivity range of the sillenite crystal. For that reason, a BGO-doped copper crystal has been especially developed to increase the response at these wavelengths. Though the response is weak in these conditions, the quality of the results is acceptable. This is the first use of pulsed illumination with a PRC on an industrial example (turbine blade under vibration). All these works show that it is possible to perform quantified measurements of displacements adapted to a photorefractive crystal-based holographic interferometer, some of the works being adapted to a very specific range of application. With the basic target to achieve a lightweight and portable holographic camera for industrial purposes, we chose, ten years ago, an original approach by combining the use of a sillenite PRC in anisotropy configuration with the RT-HI method. Indeed, with regard to the existing knowledge and also the criteria mentioned above, it appears the most promising, in terms of performance, flexibility, and adaptability, to cover a wide area of applications. The different devices and their applications that will be shown later confirm the opportunity of this approach.

8.3 Potentialities of Photorefractive Crystals for Holographic Interferometry We will give the main figures of merit that are important for HI. Moreover, as pointed out before, crystal optics allow original optical schemes, and therefore we will briefly review different photorefractive properties, configurations, and recording geometries.

8.3.1 Figures of Merit The photorefractive effect is characterized by the variation of the photoinduced refractive index variation n, which is proportional to the local space charge field Esc . The latter is a replica of the incident interference pattern created by the superimposition of the reference and the object beams. In order to find the behavior of E sc with respect to the temporal and spatial characteristics of the incident pattern, one has to solve a system of four equations (charge generation and recombination, conduction, continuity (or charge conservation) and Gauss equations) by injecting a Fourier series of E sc in terms of spatial frequency. The system can be solved analytically if it is linearized in the spatial frequency components, which is the case of small modulation of the incident pattern M(M  1)

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[23, 24]. The more appropriate solution is found in the case of quasi-continuous luminous excitation and with low modulation of the incident pattern, which is generally the case in HI (Iref  Iobj ). For these cases and if we consider the practical case in which we do not use any external electric field (diffusive charge transport), the refractive index variation is given by n = n sat (1 − exp(−t/τ ))

(8.5)

in the case that two beams start to interact at time t = 0. Here n sat is the refractive index at saturation of the process and τ is the response time. The saturation value of n depends on many parameters but mainly on the chargetransport mechanism (presence of external field or not), as well as the electrooptic coefficient (depending on the crystal and its cutting orientation). The first important figure of merit is the diffraction efficiency, which is defined as the ratio between the diffracted beam and the readout beam intensities. Following Kogelnik’s coupled waves theory [25] applied to the case of thick phase holograms, it is given by     π n d αd sin2 , (8.6) η = exp − cos(θ ) λ cos(θ) where α is the absorption, d the crystal thickness, θ the half-angle between beams, and λ the wavelength. The diffraction efficiency depends on different material and experimental parameters. Therefore it is impossible to indicate a value for a given crystal; one generally prefers to give a range of values. An important fact to note for the application to HI is the effect of the ratio R = Iref /Iobj between beams on the recording/readout properties of PRCs. Since the variations n are weak, the efficiency is proportional to n 2 . Since n is proportional to the space-charge field, itself a replica of the light pattern, it transpires that the efficiency is proportional to M 2 . If one expresses it in terms of R, one obtains η ∝ M2 ∝

Iref Iobj R = , 2 (Iref + Iobj ) (1 + R)2

(8.7)

which is inversely proportional to R for R  1. Consequently, the diffracted intensity Idiff , given by the product of η and Iref , is proportional to only the incident object beam intensity Iobj . In contrast to other holographic recording media, PRCs are not affected by a low modulation M, and more practically, we can set the recording time by bringing the major part of light by way of the reference beam, while the object beam is adjusted to have light intensity on the imaging detector after the crystal. Another important figure of merit in practice is the temporal characteristics of hologram recording. Instead of considering the response time, which depends on the intensity of the beams, or the sensitivity (which can be defined in different ways), it is preferable to use the writing fluence given by the product of the

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response time τ and the total intensity of the beams (Iref + Iobj ). It represents the quantity of light necessary to reach (1 − 1/e) of the saturation value of n. Another interesting parameter is the storage time. Indeed, for some applications in which slow phenomena have to be monitored, it could be interesting to keep the hologram for a long time in the dark. At last, the spectral sensitivity is important because it has to be high at wavelengths at which the main commercial lasers emit.

8.3.2 Crystal Families and Choice of the Crystal The three main families of crystals are the ferroelectrics, the sillenites, and the semiconductors. The first two have sensitivities mainly in the visible spectrum, the last one in the near infrared. The main laser sources that are nowadays available with a long coherence length and a high power are the Nd:YAG emitting at 1064 nm and 532 nm after frequency doubling and their pulse version the YAG Q-switch. Gas lasers such as Ar3+ (488, 514 nm), Kr+ (647 nm), and He-Ne (633 nm) and the pulsed ruby laser (694 nm) are progressively disappearing in interferometric applications. Consequently, it is more interesting to consider the choice of a crystal for the two wavelengths of the YAG lasers. The first family is that of the ferroelectric crystals. They generally are considered efficient crystals but with a poor sensitivity. In that family one finds the LiNbO3 , the KNbO3 , and BaTiO3 , among the principal ones. The transport mechanism is generally the photovoltaic regime, but both the diffusion and the drift can be used. The variations of refractive index n sat are on the order of 10−3 to 10−5 , the efficiencies can be ranged from a few percent to 100% under some conditions, and the writing fluences are on the order of 1 to 0.1 J/cm2 [26]. Some species, such as BaTiO3 , have a range extending to the near infrared with suitable dopants [27]. The second family is that of the sillenite crystals. This family includes three compounds: Bi12 SiO20 (BSO), Bi12 GeO20 (BGO), and Bi12 TiO20 (BTO). Their spectral sensitivity is important in the blue-green spectrum but is extended to the red for the BTO and doped species of BGO. Their photoinduced refractive index variations n sat are smaller than in the case of ferroelectrics, typically 10−6 to 5.10−6 [16, 28]; the efficiencies are on the order of 0.05% in diffusive regime [11], whereas they can reach 25% in drift regime [29]. The writing fluences are on the order of a few mJ/cm2 in diffusive regime depending on experimental parameters [28]. The last family is that of the semiconductors (CdTe, AsGa, etc.), whose sensitivities are around 1 micron. They are more sensitive than the sillenites but at the same time are equally or more efficient. Values of writing fluences of 14 μJ/cm2 for CdTe and 110 μJ/cm2 for AsGa have been reported as well as values for n sat larger than 10−6 [30]. For application in holographic interferometry, we have the choice among the different crystals species. It seems obvious to consider the crystals that are the most sensitive. Indeed, even the faster crystals are still less sensitive than classical holographic media with recording energy densities on the order of a few μJ/cm2 .

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Most of the experiments presented in the above sections considered sillenite crystals. A very few examples can be found in the literature that make use of ferroelectrics [31–33]. Even if the quality of the results obtained with the latter is excellent, the writing fluences are quite important, which implies either long response times or a concentration of the beams in order to have fast response times. All the experiments showing good results in LiNbO3 consider transparent objects because there is little loss of light, in contrast to the case of scattering objects. Therefore we consider sillenites as the best candidates for operation in the bluegreen spectrum. The problem with these crystals is their low efficiency. If one considers the RT-HI method, the interferogram observed after the crystal during the readout with the illuminated object has a contrast m given by  Itrans Idiff m=2 , (8.8) Itrans + Idiff where Itrans and Idiff are the intensities of the transmitted and diffracted beams respectively. It is easily understandable that the contrast cannot be good with very small values of the diffraction efficiency. Nevertheless, this problem is overcome if one considers special diffraction properties (beam coupling, anisotropy of diffraction) that are exhibited by the photorefractive crystals, and which are presented in the next subsection.

8.3.3 Photorefractive Configurations “Beam coupling” [34] arises for a crystal cut along 001, 110, and −110 and with the grating wave vector parallel to 001. It can be explained by the fact that the recording beams interact with the grating they are recording inside the material. This modifies the amplitude and phase of the beams inside the crystal. One can show that energy can be totally transferred from the reference beam into the diffracted beam (in the direction of the transmitted object beam), reinforcing the transmitted object beam. The photorefractive gain γ , which is the ratio between the object beam intensities at the crystal output and input, is given by γ = exp[(− − α/ cos θ)d], where is the coupling constant. The latter depends on material parameters, on the value of the external field, and on the orientation of the polarization, among other factors. These beam-coupling properties are used in applications such as image amplification and novelty filtering [26]. It can be used also in HI, with the feature that the diffracted and transmitted beams have the same polarization state [22]. It is also interesting to know the evolution of the contrast as a function of the product d [22, 28]: if the latter is smaller than unity, it is equal to the contrast of the interferograms (m ≈ d) and m = 1 is obtained for d ≈ 1.4. For example, for sillenite crystals having a coupling constant of typically 0.5 cm−1 in the green, the useful crystal thickness is about 2 cm. The “anisotropy of diffraction” or “polarization transfer” arises for the same crystal cut as in the case of beam coupling but for the grating wave vector parallel

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to 110 or −110[8, 11, 35, 36]. It can be explained by the fact that two gratings are recorded at the same time in the crystal. These are phase-shifted by π/2, and after diffraction, the two components of the diffracted wave recombine with this shift. Therefore the crystal acts as a half-wave plate on the diffracted beam, which is rotated with respect to the transmitted object beam. It is sufficient to place an analyzer after the crystal to observe the interference between both waves. Also, due to the fact that the intensity of the diffracted beam is small compared to that of the transmitted beam, a correct orientation of the analyzer axis allows one to obtain an interferogram with contrast equal to unity if no background noise is present. Strictly speaking, the anisotropy of diffraction can be obtained only without applied electric field. Concerning the sillenite crystals, the application of anisotropy of diffraction is somewhat limited by the presence of optical activity. In contrast to other crystals such as the semiconductors AsGa and CdTe, one cannot increase the thickness in order to optimize the diffraction efficiency. For example, in BSO, the presence of natural optical activity yields a first maximum of the efficiency for thicknesses around 2.5 to 3 mm.

8.3.4 Recording Geometries An interesting feature exhibited by PRCs is due to their important thickness, compared to other holographic media, which allows one to consider a reference beam entering by a lateral side of the crystal, whereas the object beam enters through the front window. This 90◦ geometry of beams was first considered by Tontchev et al. in a holographic microscope [37] and can be interesting compared to the classical copropagating geometry, where the two beams enter the crystal from the same window. Indeed, copropagating geometries generally lead to noise entering the CCD cameras and coming from the scattering by dust or scratches located on the crystal optical windows. This is no longer true with the 90◦ geometry. Moreover, short-focal-length objective lenses can be used close to the crystal without being disturbed by the reference.

8.4 Holographic Camera with Continuous Laser Illumination for Scattering Objects 8.4.1 Main Developments and Achievements The different steps in the development of a breadboard holographic camera using a sillenite PRC has already been presented [38–40]. The aim was to build a transportable device that satisfies as largely as possible the criteria of an “ideal” holographic system, as listed in Section 8.2. For that, the RT-HI technique associated with the crystal configuration exhibiting anisotropy of diffraction was chosen, and we showed that this choice best satisfies the system requirements. This choice is motivated by the following facts.

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First, the choice of the RT-HI technique is justified by the fact that it is a priori open to more applications than other techniques (2E-HI and TA-HI). Indeed, static, dynamic, and vibratory displacements can be examined in RT-HI. In the case of vibrations, a stroboscopic technique such as that described by Nakadate et al. can be associated [41]. The 2E-HI technique can also be used in all cases but is more complicated in the case of dynamic (continuously evolving) displacements because sequences of double-exposed holograms must be related one to another, which necessitates multiplexing procedures. Finally, phase-quantification techniques are more complicated to introduce with 2E-HI, the PS process requiring the use of a double-reference scheme [18]. TA-HI is applicable only with vibrating objects, so basically it is much too limited. Also, phase-quantification techniques are generally addressed to sinusoidal fringe patterns, and this is not the case in TA-HI (Bessel function fringe profiles). Second, at the level of the crystal configuration we considered the diffraction anisotropy by self-diffraction, as used by Kamshilin et al. [8] and Troth et al. [11]. Indeed, this technique automatically satisfies the Bragg condition, and once the output polarizer is well oriented, the interferogram contrast doesn’t need to be readjusted. This satisfies the criteria of the ideal system of ease and user friendliness. The study and optimization of this first device included several ways of working. A tradeoff between two optical systems has been carried out in order to optimize the ratio between the available laser power and the object area observed. The responses of several sillenite species were compared and a sample was chosen. Finally, two phase-quantification techniques were considered, the PS and the FT. For both of them, our aim was to analyze the difficulties appearing in applying them with regard to the particularities of PRCs. Figure 8.1 shows the scheme of the first prototype. The gray line surrounds all the elements present on the breadboard, which includes all the components necessary for the holographic interferometer: the laser, the beamsplitter, the reference-beam-forming elements, a mirror mounted on a piezotranslator (to apply the phase-shifting technique), and Camera

M2 SH1 DPSS YAG laser PZT 490 mW

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L1, L2, L3 : lenses M1, M2, M3 : mirrors SH1, SH2 : shutters MO : microscope objective VBS : variable beamsplitter PZT : piezo translator SF : spatial filter SU : stimulation unit OB : object beam RB : reference beam PRC : photorefractive crystal between polarizers

Frame Grabber Driving interface CPU

F I G U R E 8.1. Scheme of the first holographic camera using a sillenite crystal.

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the imaging system, which incorporates a sillenite crystal sandwiched between two polarizers. The imagery is composed of two objective lenses with the crystal in the vicinity of the intermediate image plane. The crystal is a BGO (window size typically 3 × 3 cm2 ); the angle between beams is 50◦ . The working principle is the following. Both recording beams (reference and object) are continuously incident on the crystal. The recording of the hologram takes place under the response time of the photorefractive effect. This response time being the same at the recording and at the readout, it must not be too short in order to allow a proper use of the PS process during interferogram capture. Once the hologram of the first state is recorded, the object undergoes a stimulation that modifies its surface shape or position. When the stimulation is stopped, the readout process starts and gives an interference image between the stored image and the direct one corresponding to the stimulated object. Depending on the time to achieve the stimulation, the laser beam is shut down or not. Once the readout is complete and the first hologram erased, the instrument is ready for a new measurement. The repetition rate of the measurement sequence obviously depends on the response time, which depends on the total intensity incident on the crystal. In the case of scattering objects, most of the light is sent to the object but a small amount comes back to the crystal, so that the ratio R is much larger than unity. Owing to the property described by equation (8.7), the diffracted intensity is directly proportional to the object beam intensity and is unaffected by the reference beam intensity (without considering scattering noise). Consequently, the first operation consists in sending a sufficient quantity of light to the object in order to have a diffracted signal detectable by the CCD camera placed after the second polarizer. The remaining light is used for the reference beam, which can then define the limits of the response time. These are the only adjustments to be made when the object is changed. One can tune the light intensity and distribution between the reference and object beam by a variable beamsplitter and also a half-wave plate working in the reference arm in combination with the polarizer in front of the crystal. The response time to use depends mainly on the external conditions under which the holographic camera is utilized, but generally its value ranges from 5 to 10 seconds if a moderately stable environment is considered (few external vibrations and air turbulence, no need of a vibration-compensated optical table). The size of the observable area depends on the object-illuminating power. We showed that with 340 mW one can observe a 55 × 37 cm2 object coated with removable white powder [40]. Applications of the system were presented: measurement of static displacements (with examples in defect detection), dynamic displacements, and vibration modes. It was clearly observed that the instrument allows one to obtain highquality results on large objects and with a high degree of versatility as a function of the application. In the case of the study of vibrating objects, a stroboscopic technique has been implemented to supplement the basic technique [42]. Despite the good performance of the first prototype, the fact remains that this instrument is not compact and lightweight enough to render it really flexible in use

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Philippe Lemaire and Marc Georges F I G U R E 8.2. Compact holographic camera using a sillenite crystal.

(e.g., it cannot be set to observe an object from the top, it has to lie on a table and observe horizontally). Consequently, we have carried on the development toward a more compact head [43]. Mainly, the laser is removed from the optical head and the light is brought through an optical fiber, the latter acting as a spatial filter. The variable beamsplitter, an electromechanical shutter, and the piezotranslator for the phase-shifting have been reduced and confined in a small compartment attached to the imaging system, the latter remaining unchanged. The result is a cylinder of typically 25-cm length and 8-cm diameter with a typical weight of 1 kg (Figure 8.2). The performance is exactly the same as that of the previous prototype.

8.4.2 Applications 8.4.2.1 Quasi-static Displacements

The first range of applications is that of static displacements, when the object is in a final state that does not evolve (or very slowly) with time. This is the simplest case of applications, and the PS can be easily applied. An interesting example is that of nondestructive testing (defect detection) of aeronautical composite structures with internal defects or surface impacts. The operating mode is the following. First, record the hologram of the object at rest. Second, shut down the laser beam so that the hologram is not erased and the object is adequately stimulated during this blind period. The stimulation must be appropriate to the type of structure studied, as well as to the defects searched; here we heat the structure with a halogen lamp, stop the heating, and let the object relax a few seconds. Afterward, the readout is performed showing an interferogram of the residual deformation, which is sufficiently stable for the PS process. This residual deformation depends on the structure and on the boundary conditions. The defects appear as important local variations in the smoother global deformation. The residual deformation then disappears slowly, and the object slightly returns to its initial state. Figure 8.3(a) shows one of the interferograms of the PS sequence obtained with a 60 × 40 cm2 composite structure with impact defects. Figure 8.3(b) shows the phase interferogram modulo 2π resulting from the phase calculation. These

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F I G U R E 8.3. (a) Interferogram, (b) phase image after phase-shifting, (c) image of the defects after differentiation of (b).

images illustrate the interest in calculating the phase. Indeed, in the interferogram (a), the average intensity and the contrast are weak at the edges of the field, with the consequence that the defects in the bottom of the images are not clearly visible. This problem no longer appears in the phase interferogram. In the rest of the paper we will mostly show phase results. A subsequent phase derivation suppresses the background deformation and puts the defects in evidence (Figure 8.3(c)). Other examples of defect detection were presented in [40, 43, 45]. Figure 8.4(b) shows recent results obtained on an entire car door (Figure 8.4(a)) in order to observe abnormal behavior under mechanical load. Another important application is pure full-field displacement metrology. A first goal was a comparison of measurements with finite-element models of honeycomb structures undergoing thermal loading [44]. Another example, depicted in Figure 8.5(a), is the calibration of piezosheets bonded to the rear side of a metallic plate that return an electric signal proportional to the surface displacement. Figures 8.5(b) and (c) show the interferograms obtained for different mechanical loads applied in the middle of the rear side of the panel. We note the extreme resolution in the fringes, allowing a very large range-displacement measurement.

F I G U R E 8.4. (a) Car door. (b) Phase image obtained after mechanical loading showing abnormal displacement of one part of the door (courtesy of Optrion).

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point where the force is applied

clamping points of the plate (a)

(b)

F I G U R E 8.5. Calibration of piezosheet’s electric signal. (a) Drawing of a clamped metallic plate on which are glued piezosheets. In the center of the plate is located the mechanical loading point. (b) Phase image after mechanical load.

Another example is the determination of the coefficient of thermal expansion (CTE) using the measurement of small displacements between the top of a specimen and a baseplate on which it is standing [43]. 8.4.2.2 Dynamic Displacements

The second range of applications consists in studying phenomena that are not stationary. On the basis of RT-HI, an easy possibility to implement is to perform a sequential readout during the object displacement or deformation. The fact that the object is evolving prevents the use of the phase-shifting technique as it is applied in the above case of quasi-static displacements. Therefore we use a single-frame analysis technique based on Fourier transform filtering as explained in a previous section. Consequently, we need to introduce a spatial carrier in the interferograms. This is done just after the holographic recording, prior to the object displacement and the holographic readout. The classical technique of tilting the reference beam to introduce the carrier cannot be applied due to the high angular selectivity of the thick hologram recorded in the crystal. We have shown the use of a simple technique consisting of displacing transversally the object illumination lens between the recording and the readout [40]. This in fact generates fringes located on hyperboloidal equiphase sheets that are intersected by the object. The carrier is then not an ideal rectilinear fringe pattern but is very close to one. Once the spatial carrier has been introduced, the object can be stimulated and the readout can start. Figure 8.6 shows the complete sequence of a wooden panel undergoing heating. After the processing explained in Section 9.2.2, one obtains phase images such as those of Figures 6(a) to (h). The time after hologram capture is indicated for each image. Between the image-capture instants, the shutter after the laser has been shut in order not to erase the hologram in the crystal. This way,

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F I G U R E 8.6. Phase images obtained after processing by the FT technique on interferograms with carrier fringes. The readout has been performed on the basis of a single reference hologram and with a sequential readout during the continuous heating of a wooden panel. The time indicated above each image is the time of interferogram capture after the heating has been initiated.

one can use the same hologram for all the successive readouts and then follow the evolution of the object. The above technique has its limit: the number of recordable interferograms depends on the response time of the photorefractive effect and the time used to record each separate interferogram. Theoretically, with a perfect synchronization between the shutter and the CCD acquisition, it is possible to obtain 100 interferograms. Finally, the dark conductivity of the crystal gives the ultimate limit of the storage time of the hologram in the crystal. With our BSO and BGO crystal, we observed a few days of dark storage time. 8.4.2.3 Vibratory Displacements

We have applied the stroboscopic technique in combination with the RT-HI for the measurement of vibrations. We were the first to apply it to a PRC-based real-time holographic interferometer [42]. The operating mode is simple: the hologram of the object is recorded at the rest, and when the object is vibrating at a given frequency, a stroboscopic readout is operated in synchrony with the object excitation signal. When the object reaches a resonant frequency, the stroboscope delay is adjusted in order to let the light enter the holographic camera at the instant when the object is at its maximum of modal displacement, say when its speed tends to zero. We have pointed out the experimental particularities related to this technique. In practice, the opening time has to be long enough to obtain a sufficient image intensity at the CCD camera. The duty cycle measures the ratio between

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F I G U R E 8.7. Phase images of different mode shapes of a compressor blade excited by a shaker (courtesy of Optrion).

the opening time and the vibration period. The quantity of light is proportional to this parameter. If the duty cycle is increased, a higher luminous level reaches the CCD, but in contrast, one integrates interferograms of the moving object before and after its maximum displacement. As a consequence, the contrast of the fringes decreases because the object is partly seen moving. We have analyzed the errors coming from the phase calculation as a function of the duty cycle. Also, we have shown an original technique to remove this error from the measurements. After correction, the final accuracy is limited by the external perturbations, as for the static displacement measurements, say λ/40 RMS. In practice, duty cycles of 12 to 16% are used. The first objects to which the technique was applied were academic cases (metallic plates excited by loudspeakers) [42]. With a YAG laser (532 nm) emitting 490 mW, we were able to observe very good interferograms on a 23 × 23 cm2 aluminium plate. Later we showed application to the detection of turbine blade mode shapes [40]. The examples shown in Figure 8.7 are recent results obtained with a holographic camera made available commercially by the OPTRION company [46]. They show some mode shapes obtained for different compressor blades of a new aircraft engine. The aim of the test campaign was to compare the frequencies at which the resonance appears and the shape of the modes to the same information derived from mechanical simulation by the compressor blade conceptor. Another interesting example is searching for abnormal behavior of a microelectronic chip soldered on an electronic board. Figure 8.8 shows the phase images obtained at two resonant frequencies of the electronic board, which show that the chip is not properly attached to the board. Indeed, the fringes are not correctly connected between the chip and the board.

8.4.3 Extension to Tridimensional Displacements The phase difference φ calculated by phase-quantification techniques allow one to compute the displacement L of each point of the surface observed. Their relationship is the following: φ(x, y) = S(x, y) • L(x, y),

(8.9)

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F I G U R E 8.8. Vibration patterns of an electronic board on which is soldered a microelectronic chip. The pattern shows abnormal behavior of the chip caused by bad soldering to the board (courtesy of Optrion).

where S is the sensitivity vector, defined as the difference between the illumination-vector (traveling from the illumination source to a given object point) and the observation vector (traveling from the object point to the observation plane). The vector S can be determined if one knows the geometry of the setup and the object. Equation (8.9) suggests that if one wishes to determine the three components of vector L, three independent measurements of φ have to be performed. In order to do this, one needs different sensitivity vectors, obtained for example by considering different illumination points and a single observation direction. Another possibility is to have one illumination with three directions of visualization (three cameras). Numerous examples exist in the literature, and references [2] and [3] summarize different approaches. In all the applications presented above, we consider only single measurements with one observation direction and one illumination point close to the observation camera. Therefore the sensitivity vector is almost directed along the line of sight of the holographic camera. If the object is observed perpendicularly, one measures mainly the “out-of-plane” displacement component. In order to access the complete vector displacement, we have studied the possibility of recording different holograms in the same crystal with a single reference beam or to use a stack of different crystals [47]. This is the first time to our knowledge that such multiple recording in a sillenite crystal has been used on large objects for interferometric purposes. When the object is displaced, the readout is performed sequentially very fast with one illumination at a time, and the interferograms allow one to compute the displacements with each of the sensitivity vectors. Equation (8.9) is inverted in order to retrieve vector L. This technique has been applied with two point sources illuminating the object laterally and symmetrically located with respect to the holographic camera [48].

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F I G U R E 8.9. Image showing the result of the determination of in-plane and out-of-plane displacement. On the left, the phase images obtained with two different illumination sources; on the right, in-plane (black line) and out-of-plane (gray line) computed from the phase images (courtesy of Optrion).

This arrangement allows one to access the “in-plane” displacement component along one transverse direction, as well as the out-of-plane component. This kind of measurement is of major importance in the determination of strain and stress in materials. Figure 8.9 displays the results of the elongation of a metallic sample. The phase images on the left show the displacements measured with two different illumination points; the diagram on the bottom right shows the in-plane (black line) and the out-of-plane (gray line) displacements, computed on the basis of the phase images.

8.5 Holographic Cameras with Continuous Laser Illumination for Transparent Objects 8.5.1 Study of a Microgravity Fluid-Monitoring Holographic Camera Holographic interferometry allows the measurement of refractive index variations of transparent objects [1]. The basic configuration for fluid holographic measurement is that the object illumination beam is collimated and passes through the complete experimental cell. Since the eighties, this technique has gained the attention of space agencies as a diagnostic tool for microgravity experiments. Most of them use holoplates with liquid bridges or photothermoplastics. Some feasibility studies have been performed in the past with LiNbO3 crystals with long storage times [33]. These crystals are difficult to apply in the space environment because

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F I G U R E 8.10. (a): Sketch of the cell containing silicon oil sandwiched between two heated plates. The figure shows also the convection rolls appearing above a threshold gradient (T2-T1). (b) and (c): Phase images respectively for gradient under and above the threshold.

they require powerful lasers. For that reason we proposed to consider more sensitive crystals, such as the sillenites [49]. We have studied the implementation of a BSO crystal in the future multidiagnostics Fluid Science Laboratory (FSL) of the European Space Agency, to be placed on board the International Space Station (ISS). A first demonstration experiment was to observe convection rolls appearing in a parallelepipedic cell filled with silicon oil and undergoing an increasing temperature gradient (T2-T1) between the bottom and the upper heating plates (Figure 8.10(a)). Figures 8.10(b) and (c) show interferograms arising from refractive index variation between the two plates and integrated along the line of sight. In Figure 8.10(c), the temperature gradient is higher than a threshold above which the convection process is initiated. The final goal of the FSL is to follow the variations over a long time. The procedure to be implemented is to record a single reference hologram, with a response time smaller than 50 ms, and to be able to capture more than 1000 realtime interferograms on the basis of this hologram [50]. The consequence of this is that we have to change strongly the response time between the recording and the readout, e.g., by means of orientable polarization-sensitive elements. The object beam intensity is then reduced for the readout, but this is not critical in the case of transparent objects because there is no light loss from the object. In [50], we have shown some interferograms obtained with such a system. For slowly varying fluid phenomena, the phase-shifting can be applied for quantification purposes. In order to implement such crystals on spaceborne facilities, the change of holographic performance due to the radiation doses aboard ISS has to be measured, mainly with respect to gamma rays and protons [51]. No significant damage or losses were observed that could prevent the use of sillenite crystals aboard ISS.

8.5.2 Extension of the Holographic Camera to Transparent Objects It would be interesting if one could slightly modify the holographic camera in order to allow visualization of fluids instead of scattering objects. The idea is to place an additional device in front of the camera, which produces a collimated beam

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optical fiber

collimated beam

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beamsplitter

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Holographic camera for scattering objects Additional optics for transparent objects

F I G U R E 8.11. Scheme of the adaptation of the holographic camera to transparent objects.

F I G U R E 8.12. Phase images of air variations due to heating by a flame. The flame itself is overexposed in image (a), resulting in a uniform phase.

passing through the object and traveling toward the holographic camera. One possibility is shown in the figure, where a mirror is placed after the transparent cell and reflects the light back to the holographic camera. A beamsplitter is necessary to inject light in the line of sight. The other possibility is to avoid the beamsplitter and directly expand and collimate the beam before two folding mirrors, which allow one to inject the light through the cell. The system depicted in Figure 8.11 has been designed for a test bed in which folding mirrors cannot be placed behind the cell. Figure 8.12 shows two phase interferograms obtained with the RT-HI technique and phase-shifting in the case of air variations around the flame of a candle.

8.6 Holographic Camera with Pulsed Lasers 8.6.1 Early Experiments The Laboratoire Charles Fabry de l’Institut d’Optique (LCFIO) has developed a photorefractive holographic camera operating in the pulse regime [22]. The

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PBS1 M3

PBS2 M2

F I G U R E 8.13. Scheme of the pulse holographic camera adapted to the single-pulse laser. HWPx: half-wave plates, Mx: mirrors, PBSx: polarizing beamsplitter cubes, DL: divergent lenses, CL: convergent lenses, Ox: objective lenses, P: polarizer, Cam x: CCD cameras.

phase-shifting system has been briefly explained in Section 9.2.2; it allows simultaneous acquisition of two phase-shifted images. A preliminary measurement of the average intensity of the images without interference fringes is, however, necessary to calculate the phase. Initially, this camera was designed for vibration analysis using a ruby laser. The main drawback is that the laser wavelength (694 nm) does not match the sensitivity range of BSO and BGO crystals. A copperdoped BGO sample was specially grown to increase the sensitivity at 694 nm. Since then, the Centre Spatial de Li`ege group and LCFIO have carried on this work together to adapt the developed system to a frequency-doubled Qswitched YAG laser (COHERENT Infinity) whose wavelength is naturally adapted to sillenite crystals [52, 53]. For these new experiments, the photorefractive crystal is a nominally undoped BGO sample, cut along the beam-coupling configuration. Figure 8.13 shows a scheme of the complete experiment. Real-time HI is performed: a first pulse records the hologram of the object at some instant; the object is then visualized through the crystal while the hologram is reconstructed by the second pulse. The energy of the incident beams has to be decreased in order to avoid erasure of the hologram and CCD blooming with the object beam during the readout. This forced us to add an energy balance: a half-wave plate (HWP1) defines the ratio between reference and object beam energies for the first pulse, through the use of a first polarizing beamsplitter cube (PBS1). At the second pulse, two Pockels cells are supplied with a half-wave voltage. The beam polarization is rotated in such a way that the object beam energy is reduced, all of the light traveling into the reference beam, but most of it deviated in the beam dump so as not to erase the hologram. The delay line constituted by two moving mirrors (M1, M2) is present to equalize the paths of the reference and object beams (the coherence length depends on the pulse duration, here 90 cm). The third Pockels

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F I G U R E 8.14. (a) Interferogram recorded by one of the two CCDs, (b) corresponding phase image.

cell is used to circularize the reference beam polarization at the readout step, in order to provide the polarization separation through the polarizing beamsplitter PBS3. Here the crystal is orientated in the beam-coupling configuration, so that the polarization state of the diffracted beam is the same as the transmitted one. An optimization experiment has been carried out first in order to evaluate the correct energies for the recording and the readout. Figure 8.14(a) shows the interferogram obtained by one of the cameras, (b) the phase calculated, and (c) the displacement. A method for vibration measurement has been implemented with the pulse system [52, 53]. It is adapted to the sinusoidal vibration. It consists of a first displacement measurement at a given time in the vibration period. A quarter of period later, a new pair of pulses allows the determination of a new displacement. By combining both displacement measurements together with the known vibration frequency and inter pulse delay, one can find the amplitude and phase of vibration. The same procedure is applied over a range of excitation frequencies. At the end, one can obtain the frequency response in all points of the vibrating object. Figure 8.15 shows a series of interferograms appearing when the excitation frequency is progressively increased around a resonance frequency. The object is a metallic plate excited by a loudspeaker. The four-pulse vibration measurement technique described in [52] and [53] allows one to compute the amplitude and phase of the vibration in every pixel.

F I G U R E 8.15. Interferograms of a metallic plate during a frequency scan, passing by a resonance.

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F I G U R E 8.16. Amplitude response of the metallic plate as a function of the excitation frequency.

5 4 Amplitude (μm)

247

3 2 1 0 0

50 100 150 200 250 300 350 400 frequency (Hz)

Figure 8.16 shows the frequency spectrum of the plate above at one pixel. The resonances are clearly seen. The same kind of information can be recovered for all the image points.

8.6.2 Recent Developments 8.6.2.1 Simplifications of the Pulse System

The system above suffers from some limitations or complexities of utilization. First, the laser emits a single pulse, which requires complicated synchronization if one wishes to study vibrations. Second, the energy balance system requires the use of active polarization-changing devices such as a Pockels cell. Third, the technique of phase quantification requires an additional Pockels cell to circularize the polarization at the readout. The first drawback can be eliminated if one uses a double-pulse laser with delay between pulses that can be adjusted. The second can be eliminated if the second pulse of such a laser has a lower energy than the first pulse. Presently the use of such lasers is under investigation. Doing so, we avoid the two first Pockels cells. Concerning the simplification of the two-phase shifting technique, we have found an easy way to do it. Instead of considering the beam-coupling configuration, and consequently the isotropy of diffraction, we consider the anisotropy of diffraction configuration. In that case the diffracted and the transmitted waves have orthogonal polarizations after the crystal. It is sufficient to place a nonpolarizing beamsplitter cube after the latter in order to observe with two cameras. In front of the first camera, we place a polarizer (as with the holographic camera presented in Section 8.4.) and in front of the second camera, we place a quarter-wave plate (QWP) followed by a polarizer. If one of the principal axes of the QWP is aligned along the polarization of, e.g., the diffracted beam, the latter will be phase-shifted by π/2, while the other components will not undergo any phase shift. Both components will then be phase-shifted by π/2, for this camera, and consequently, the interferograms observed by the two cameras will be phase-shifted by the same

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F I G U R E 8.17. (a), (b): Interferograms observed by the each of the two cameras with a π/2 phase-shifting. (c) Phase image obtained with the two former.

quantity one to the other. This is clearly observable in the interferograms shown in Figures 17(a) and (b) obtained with such a system. Figure 17(c) is the phase calculated using the following algorithm. The general expression of phase-shifted images (equation (8.2)) is written here for both images as I1 = Iav1 [1 + m 1 cos(φ)], I2 = Iav2 [1 + m 2 sin(φ)].

(8.10) (8.11)

The computation of phase φ is obtained if one captures preliminary average images (without fringes) Iav1 and Iav2 . Supposing m 1 and m 2 equal (which can be set by correct orientation of the polarizers in each arm), the phase is given by  φ = arctan

I2 Iav2

  I1 −1 −1 . Iav1

(8.12)

8.6.2.2 New Crystals and Geometries

Developments have also been performed on the basis of semiconductor crystals working in the near infrared. Mainly, one uses AsGa crystals, which can be adapted to the pulse regime because they exhibit small storage times. They are more efficient and sensitive in the near infrared than are sillenites in the green (see Section 8.3.2). The 90◦ geometry of the recording beam has also been studied with both the BSO and the AsGa crystals [54]. This can be of great interest in limiting the scattering noise observed in the copropagating geometry (see Section 8.3.4). Due to the presence of optical activity in BSO, one needs to place QWP in the input and output beams, in order to avoid a periodic spatial inhomogeneity of the diffraction efficiency. The quality of polarization in such arrangements limits the quality of the results. This is no longer the case with AsGa crystals, which do not exhibit optical activity. Figure 8.18 shows the interferogram obtained with an AsGa crystal under the 90◦ geometry of beams.

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F I G U R E 8.18. Interferogram obtained with a 90◦ geometry of beams (image by CSL and LCFIO).

8.7 Conclusion In this contribution, we have reviewed the possibilities of photorefractive crystals in the field of dynamic holographic interferometry. For this purpose we made a survey of different possibilities already envisaged by different groups. We have highlighted the criteria that are important for the development of an ideal device in order to justify the choice of the configuration and the method that we considered in our developments. The technique is real-time holographic interferometry with sillenite crystals using self-diffraction in the anisotropy of diffraction configuration. We have then presented the study of different devices adapted to different categories of object and illumination regime that are based on the mentioned technique: continuous illumination with scattering objects, continuous illumination with transparent objects, and finally, pulsed illumination. Acknowledgments. The authors would like to express their thanks to Dr. JeanClaude Launay, of the Institut de Chimie de la Mati`ere Condens´ee of the Universit´e de Bordeaux (France), for his efforts and cooperation in crystal growth; to Dr. Gilles Pauliat and Dr. G´erald Roosen, of the MANOLIA group of the Laboratoire Charles Fabry de l’Institut d’Optique of Orsay (France), for their fruitful discussions and collaboration in common research projects. The results presented have been obtained throughout different research projects funded by the General Directorate of Research, Technology and Energy of Walloon Region of Belgium, by the Research General Directorate of the European Commission, and by the European Space Agency.

References 1. C.M. Vest, Holographic Interferometry, John Wiley & Sons, New York (1979). 2. P.K. Rastogi, ed. Holographic Interferometry: Principles and Methods, Springer Series in Optical Sciences 68, Springer-Verlag, Berlin (1994).

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3. T. Kreis. Holographic Interferometry: Principles and Methods, Akademie Verlag Series in Optical Metrology 1, Akademie Verlag, Berlin (1996). 4. J.P. Huignard, J.P. Herriau. Appl. Opt. 16 (7), pp. 1807–1809 (1977). 5. J.P. Huignard, J.P. Herriau, T. Valentin. Appl. Opt. 16 (2), pp. 2796–2798 (1977). 6. A. Marrakchi, J.P. Huignard, J.P. Herriau. Opt. Comm. 34 (1), pp. 15–18 (1980). 7. J.P. Huignard, A. Marrakchi. Opt. Lett. 6, pp. 622–624 (1981). 8. A.A. Kamshilin, M.P. Petrov. Opt. Comm. 53 (1), pp. 23–26 (1985). 9. A.A. Kamshilin, E.V. Mokrushina, M.P. Petrov. Opt. Eng. 28 (6), pp. 580–585 (1989). 10. J-P. Herriau, J-P. Huignard, A.G. Apostolidis, S. Mallick. Opt. Comm. 56 (3), pp. 141– 144 (1985). 11. R. Troth, J.C. Dainty. Opt. Lett. 16, pp. 53–55 (1991). 12. R.C. Troth, S.L. Sochava, S.I. Stepanov. Appl. Opt. 30 (26), pp. 3756–3761 (1991). 13. S.V. Miridonov, A.A. Kamshilin, E. Barbosa. J. Opt. Soc. Am. A 11 (6), pp. 1780–1788 (1994). 14. K. Creath in reference [2], pp 109–150. 15. M. Kujawinska in Interferogram Analysis: Digital Fringe Pattern Measurement Techniques, D.W. Robinson, G.T. Reid, eds., Institute of Physics Publishing, London (1993). 16. D. Dirksen, G. von Bally. J. Opt. Soc. Am. B 11 (9), pp. 1858–1863 (1994). 17. D. Dirksen, F. Matthes, S. Riehemann, G. von Bally. Opt. Comm. 134, pp. 310–316 (1997). 18. R. D¨andliker, R. Thalmann. Opt. Eng. 24 (5), pp. 824–831 (1985). 19. F. Rickermann, S. Riehemann, G. von Bally. Opt. Comm. 155, pp. 91–98 (1998). 20. B. Pouet, S. Krishnaswamy. Appl. Opt. 35 (5), pp. 787–794 (1996). 21. L. Labrunie, G. Pauliat, G. Roosen, J.C. Launay. Opt. Lett. 20 (15), pp. 1652–1654 (1995). 22. L. Labrunie, G. Pauliat, J-C. Launay, S. Leidenbach, G. Roosen. Opt. Comm. 140, pp. 119–127 (1997). 23. N. Kukhtarev, V. Markov, S. Odulov, M. Soskin, V. Vinetskii. Ferroel. 22, pp. 949–960 (1979). 24. N. Kukhtarev, V. Markov, S. Odulov, M. Soskin, V. Vinetskii. Ferroel. 22, pp. 961–964 (1979). 25. H. Kogelnik. Bell System Tech. J. 48 (9), pp. 2909–2947 (1969). 26. M.P. Petrov, S.I. Stepanov, A.V. Khomenko. Photorefractive Crystals in Coherent Optical Systems, Springer Series in Optical Sciences 59, Springer Verlag, Berlin (1991). 27. N. Huot, J-M. Jonathan, G. Pauliat, D. Rytz, G. Roosen. Opt. Comm. 135, pp. 133–137 (1997). 28. Ph. Delaye, J-M. Jonathan, G. Pauliat, G. Roosen. Pure Appl. Opt. 5, pp. 541–559 (1996). 29. J.P. Huignard, F. Micheron. Appl. Phys. Lett. 29 (9), pp.591–593 (1976). 30. L.A. de Montmorillon, Ph. Delaye, J.C. Launay, G. Roosen. Opt. Mat. 4, pp. 233–236 (1995). 31. A. Hafiz, R. Magnusson, J.S. Bagby, D.R. Wilson, T.D. Black. Appl. Opt. 28 (8), pp. 1521–1524 (1989). 32. X. Wang, R. Magnusson, A. Haji-Sheikh. Appl. Opt. 32 (2), pp. 1983–1986 (1993). 33. J. Mary, Y. Bernard, F. Lefaucheux. J. Opt. Soc. Am. B 7 (3), pp. 2356–2361 (1990). 34. A. Marrakchi, J.P. Huignard, P. G¨unter, Appl. Phys. 24, pp. 131–138 (1991).

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9 Self-Organizing Laser Cavities Gilles Pauliat, Nicolas Dubreuil, and G´erald Roosen

The insertion of a photorefractive crystal in a laser cavity leads to spectacular behaviours. The oscillating modes record a dynamic hologram. For a correctly designed cavity, this hologram acts as a spectral or/and spatial filter, which, in turn, modifies the relative losses of these modes. Through the interaction with the amplifying medium, this modification of losses redistributes the energy between the modes. This dynamic interaction is a self-adavptive process that can, after a short adaptation time, automatically force the laser to oscillate on a single transverse and longitudinal mode without any adjustment. We describe the main features of this adaptive process and its applications in various laser cavities.

9.1 Introduction Any laser oscillation results from a self-organizing process. Each oscillating mode modifies the gain experienced by the other ones through the nonlinear effects taking place inside the laser amplifier: spectral or spatial hole burning, mode couplings through carrier density pulsation, and so on. The steady-state (if any) transverse and longitudinal modal structure depends on the cavity geometry and on the gain medium characteristics. It is, for instance, well known that the competition for the available gain in a homogeneously broadened laser amplifier, set in a unidirectional ring cavity, automatically leads to a steady-state single-longitudinal-mode oscillation. Nevertheless, for most laser cavities, a single-mode steady-state operation cannot be obtained unless spectral filters are inserted inside the cavity. Most often, one uses static filters, for instance Fabry-Perot filters, Lyot filters, or coupled cavities. Mode selection is then obtained by carefully adjusting the filter spectral response to get the maximum transmission for the mode wavelength to be selected. An attractive solution to get the single-mode oscillation, without the adjustments needed by the use of these static filters, is to reinforce the self-organizing process by an adaptive filter. This adaptive filter is created by introducing a dynamic holographic medium inside the laser cavity. Such media spontaneously replicate the illuminating interference pattern under the form of a hologram. An updating of the hologram follows any modification of this pattern. When such a dynamic holographic medium is inserted inside a laser cavity, the oscillating modes record a common dynamic hologram, a hologram that in turn, acts as a 253

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frequency or/and spatial filter for these modes. For a correctly designed system, this mutual adaptation leads to a stationary single-mode oscillation. These filters, being self-adapted, do not require any adjustment. Moreover, they permanently maintain the single-mode oscillation and they adapt to any cavity change resulting, for instance, from a modification of the thermal loading and of the cavity aging. We will demonstrate that the cavity is also much more tolerant to any small misalignment that may result from an imperfect manufacturing. Such cavities have been built with various dynamic holographic media. Recording of thermal holograms in a nonsaturable absorbing medium by the longitudinalmode structure in a Nd:YAG laser has been demonstrated [1]. Recording gain gratings in solid-state amplifiers also induces a modal competition [2]. Similarly, absorption gratings recorded in a saturable absorber cell also lead to selforganization [3]. Among all the possible holographic media, we will hereinafter demonstrate that photorefractive crystals are especially attractive for this application. Although several cavity configurations can be implemented, we restrict this chapter to the most commonly used cavities: linear laser cavities in which the photorefractive crystal is part of an adaptive Fabry-Perot filter. These self-organizing cavities were first demonstrated in [4]. In the following section, we first describe the operation of this Fabry-Perot filter and how to compute its spectral characteristics. We detail an approach to optimize the operation of such a filter in various extended cavities. Because the self-organization that leads to the single-mode oscillation results from the interaction of the adaptive Fabry-Perot with the gain medium through the modal structure, we separately analyzed the operation of this self-organization, first with four level amplifiers (dye, Nd:YVO4 , etc.), and second with semiconductor amplifiers. In all these cavities, the self-organization process acts only on the longitudinal structure; the laser being initially, prior to the self-organization process, oscillating on a single transverse mode (and on a full series of longitudinal modes). In a last section, we demonstrate that this self-organization can also occur on the transverse modal structure to force a laser initially oscillating on several transverse modes to oscillate at steady state on a single transverse and longitudinal mode.

9.2 Self-Organizing Cavities with Adaptive Fabry-Perot Filters 9.2.1 Basic Scheme and Principle of Operation The general scheme for these cavities is depicted in Figure 9.1. Most often, the photorefractive crystal is inserted in the linear cavity between the output coupler and the laser amplifier, although other crystal positions also lead to selforganizations [5]. This crystal should be cut and oriented so that the standing wave pattern induces a photorefractive Bragg grating. One can consider the effect

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L l

d

output cavity mirror

amplifying medium

Photorefractive crystal

Z

0

F I G U R E 9.1. General scheme for a self-organizing cavity based on an adaptive Fabry-Perot filter.

of this dynamic Bragg grating under two equivalent ways: either by considering that it creates two coupled subcavities, or by observing that it acts as the first mirror of an adaptive Fabry-Perot filter, whose second mirror is the output cavity mirror. We choose to describe the operation of the self-organization process using this second approach. We first give below a basic description of the operation of such an adaptive Fabry-Perot filter; a more detailed explanation, including the equations, will be given in Section 9.2.3. Let us first assume that the laser has already reached its single-mode steady state. The photorefractive crystal has been cut and oriented so that the two counterpropagating waves interact by two-beam coupling: the wave from the output mirror is amplified by two-wave mixing at the expense of the wave from the laser amplifier. Because in the selected crystal the photorefractive effect originates from a pure diffusion process, the phase of the amplified wave remains unchanged. The losses of the oscillating mode are thus reduced, while its frequency is not modified. Conversely, the nonoscillating modes are not Bragg-matched with this thick photorefractive grating: the amplitudes of the diffracted waves decrease all the more strongly since the wavelength mismatch between the wavelength of the oscillating mode and their wavelengths is larger. The reflectivity is therefore lower for the nonoscillating modes. This reflectivity is further decreased, since the interference between the two waves is no longer necessarily constructive. The losses for these modes are thus larger than the losses for the oscillating mode. We will also show that their frequencies are slightly modified by the presence of the photorefractive grating. These differences in losses explain why the single-mode oscillation can be a stable state: the other modes are kept under threshold. To understand how the laser is driven in this stable single-mode oscillation, one has to take into account that the buildup time of a photorefractive grating is usually much longer than the other characteristic time constants of the laser cavity. Therefore, just after the laser is switched on, several modes oscillate and contribute to the recording of the same hologram, which is the superimposition of the individual gratings recorded by each mode. These overlapping gratings compete for the same photorefractive refractive index modulation. Indeed, the index

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modulation of each grating is proportional to the ratio of the mode intensity to the total intensity. So, if one mode is more powerful than the others, its contribution to the hologram is larger. In general, the hologram is thus best adapted to the strongest mode. This adaptation decreases the losses for this mode. In turn, this mode saturates the laser amplifier more efficiently. For a homogeneously, or partly homogeneously, broadened laser medium, this saturation decreases the available gain for the other modes. This, in turn, decreases their intensities and decreases the strengths of their corresponding gratings. For a large enough diffraction efficiency, this snowball effect leads to the stable single-mode oscillation. The photorefractive Bragg grating and the output coupler constitute the two mirrors of an adaptive Fabry-Perot filter whose maximum reflectivity is automatically adjusted to the recording wavelength. Its spectral response is thus opposite to that of conventional adjustable Fabry-Perot filters. Indeed, with these conventional adjustable filters being inserted inside the cavity (and not playing the role of the output coupler), their maximum transmission is adjusted to correspond to the wavelength of the oscillating mode. It should be noted that Fabry-Perot spectral filters used in reflection are usually less efficient than when they are used in transmission. Furthermore, the photorefractive Bragg grating just decreases the losses for the oscillating mode. In order to get large loss discriminations between the oscillating mode and the other modes, the output coupler cannot have a too-large reflectivity. When switching on the laser and prior to the self-organizing process occurring, the cavity losses are limited by this output coupler reflectivity. Therefore, a good discrimination at steady state corresponds to relatively large losses at the initial stage. We thus expect this self-organization to be very efficient in lasers with large gains, and to be relatively less efficient in lasers with low gains in which the output coupler reflectivity is high.

9.2.2 Advantages of Using Photorefractive Crystals The photorefractive π/2 phase shift between the interference pattern and the refractive index grating is absolutely required to get a steady-state single-mode oscillation. Nevertheless, one should also note that other reasons make the photorefractive effect especially well adapted to the realization of these adaptive Fabry-Perot filters:

r The steady-state grating strength depends neither on the laser intensity nor on the laser operation, pulsed or continuous. The self-organization process is the same whatever the power delivered by the laser; only the time required to reach this steady state is affected by the output power. r Considering the large optical intensities available in laser cavities, and because adaptation times on the order of a few seconds are enough to compensate most cavity misalignments, crystals with low absorptions are employed. Moreover, the dynamic hologram being a phase grating, the presence of the crystal does not induce significant additional losses in most lasers.

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r In contrast to most other dynamic holographic media, the steady-state photorefractive grating strength is proportional to the modulation ratio of the interference pattern. Therefore the amplified wave has exactly the same transverse structure as the transmitted beam. In other words, the photorefractive hologram does not modify the transverse structure of the oscillating mode. Nevertheless, we will experimentally show that transverse instabilities can be observed during the self-organization, in case of a too-large grating strength. Compared to laser cavities closed by a self-pumped photorefractive phaseconjugate mirror, self-organizing cavities with adaptive Fabry-Perot filters present quite distinct features:

r The mode that is selected by the organizing process is one mode of the original cavity in the absence of the photorefractive grating. Apart from its intensity, its transverse and longitudinal structures are not affected by the presence of the dynamic grating. For a successful self-organizing process, i.e., if the stationary state is single-mode, one never observes any frequency shift, in contrast to the frequency shifts usually observed in phase-conjugate cavities [6]. r The laser already oscillates before any photorefractive grating is recorded. The intensity incident on the crystal is already large at the beginning of the selforganization process. This considerably speeds up the attainment of the steady state. r To implement a self-pumped photorefractive phase-conjugate mirror, the photorefractive two-wave mixing gain must exceed the threshold photorefractive gain (gain interaction-length products l ≥ 4). There is no such threshold with Fabry-Perot self-organizing cavities. For well-designed cavities, as described below, we have obtained successful operations with a product as low as l ≈ 0.12.

9.2.3 Steady-State Spectral Characteristics of the Adaptive Fabry-Perot Filter 9.2.3.1 Introduction

The equations governing the kinetics of the self-organizing process can be obtained by adding the photorefractive wave-mixing equations to the description of the laser amplifier. However, taking into account the complicated modal structure that may oscillate just after switching on the laser is a lengthy process. Indeed, the kinetics depend not only on the system parameters but also on the system history: the photorefractive hologram keeps the memory of past modal structures for a given time. This is observed in experiments. For an initially strongly multimode laser, the temporal evolution leading to the single-mode state is not always reproducible, even if, of course, the single-mode state is always reached. Fortunately, modeling of the laser kinetics is not required to understand the self-organizing process and to predict whether the single-mode oscillation is a

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stable state. The idea underlying this analysis is to assume that this single-mode steady state is reached and then to analyze its stability [5,7,8]. 9.2.3.2 Transfer Matrix Formulation

We rely on Figure 9.1 to derive the Fabry-Perot reflectivity. The axis coordinate is z, and its origin taken at the crystal entrance face holds as the origin of phases. The electric fields E ± of the optical waves inside the photorefractive crystal are expressed in complex notation as    1  ( p) ( p) (9.1) A± (z) exp −i ( ω( p) t ∓ n k ( p) z) + cc , E ± (z) = 2 in which cc represents the complex conjugate, and the subscript + or − denotes a wave propagating toward or away from the output coupler, n is the crystal refractive index, and A the electric field amplitudes. This notation is valid for any mode. The superscript p labels the mode: p=0 refers to the oscillating mode, and p=0 to modes under the oscillation threshold. The optical pulsation ω and wave number in vacuum k also depend on the mode number p. Denoting by rFP the Fabry-Perot reflectivity, and r the output mirror reflectivity, we can express the two boundary conditions at z = 0 and z = l. Adding to these two conditions the linear relation between the amplitudes at z = 0 and z = l, the problem is fully constrained by ( p)

( p)

A− (0) = rFP A+ (0) ,

  ( p) ( p) A− (l) = r A+ (l) exp 2 i k ( p) (n l + d) ,  ( p)    ( p)  A + (l) c11 c12 A+ (0) = . ( p) ( p) c c 21 22 A− (0) A− (l)

(9.2)

From equations (9.2), the derivation of the Fabry-Perot reflectivity rFP is straightforward. Multiplying rFP by the phase term exp(−2 i k ( p) (n l + d)), we get the Fabry-Perot reflectivity r¯FP with the origin of phases taken onto the output mirror: c21 e−2ik (nl+d) − r c11 . r e2ik ( p) (nl+d) c12 − c22 ( p)

r¯FP = rFP e−2ik

( p) (nl+d)

=

(9.3)

The intensity reflectivity is |¯rFP |2 and the phase shift induced by the holograms is the complex argument φFP = Arg[¯rFP ].

(9.4)

The hologram transfer matrix coefficients cmn appearing in equations (9.2) account for the diffraction of any mode # p onto the photorefractive hologram recorded by the oscillating mode p = 0. They are thus functions of the oscillating mode, and of the mode under threshold for which we determine the FabryPerot reflectivity. In the two next subsections, we make explicit how to calculate these coefficients. First, we will use an exact numerical computation based on the coupled wave equations. Then, in a second step, we will give some analytical approximate equations valid in most cases.

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9.2.3.3 Exact Derivation of the Hologram Transfer Matrix Using the Wave-Mixing Formalism

For any mode # p, the evolution of the field amplitudes along the z-axis is governed by the coupled-wave equations ( p)     α ( p)  ∂ A+ ( p) = − M A− exp 2in k ( p) − k (0) z − A+ , ∂z 4 2 ( p)   ( p)    ∗ ( p) α ( p) ∂ A− = − M A+ exp −2in k − k (0) z + A− , ∂z 4 2

(9.5)

in which  is the two-beam coupling gain, α the optical absorption, and M the complex modulation ratio of the recording interference pattern, i.e., corresponding to the oscillating mode p = 0. M is expressed by (0)

(0)

2A+ A− M=



,

(0) 2 (0) 2

A+ + A−

(9.6)

M is a function of coordinate z. Its z-dependence can be easily computed by an iterative relaxation method, taking into account the two boundary conditions (first and second equations (9.2), and the set of coupled wave equations (9.5) applied to p = 0. Once M(z) is determined, the numerical resolution of equations (9.5), for any mode number p = 0, provides the transfer matrix coefficients cmn . In Figure 9.2 we have plotted the modulus √ |¯rFP | of the reflectivity for the same set of parameters ( = 5 cm−1 , r = 0.8, d = 2 cm, l = 0.3 cm, λ(0) = 1064 nm) but for different absorption coefficients: α = 0cm−1 , α = 0.2 cm−1 ,

Reflectivity

0.9

0.8

0.7

−0.10

0.00

0.10

Wavelength mismatch Δλ (nm)

F I G U R E 9.2. Reflectivity of the Fabry-Perot filter versus mismatch λ between the oscillating mode and any other wavelength; top curve for α = 0 cm−1 , middle curve for α = 0.2 cm−1 , bottom curve for α = 0.4 cm−1 .

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and α = 0.4 cm−1 . As anticipated, the absorption decreases the filter reflectivity and also alters the spectral response. In spite of being given more-accurate results in the presence of a nonnegligible absorption, an advantage of this numerical computation is that it can be relatively easily adapted to the study of the kinetics, even in case of a multimode operation. 9.2.3.4 Derivation of the Hologram Transfer Matrix Using Kogelnik’s Formalism

The above numerical approach gives accurate results even in case of a strongly nonuniform hologram, i.e., in case of a large two-wave mixing energy exchange and a nonnegligible optical absorption. Nevertheless, quite often, one may assume a low absorption and a modest energy exchange. Indeed, in laser cavities, the energy density is usually quite large. The photorefractive response time is thus not an issue. Therefore, in order to optimize such an adaptive filter, one usually tries to minimize the optical absorption: this minimizes the crystal laser losses and reduces the thermal load of this crystal. Furthermore, we will demonstrate that a too-large photorefractive gain may lead to instabilities and that such a large gain is usually not desired. In such cases, we can assume, with a very good approximation, that the modulation ratio of the interference pattern inside the crystal is uniform. It is thus given by M≈

2r e−2ik(nl+d) . 1 + |r |2

(9.7)

The hologram transfer matrix is then explicitly given by Kogelnik’s thick grating approach [9]. We get ⎛   −iρl  γl  −iρl ⎞ γ2 l γ1 l ∗ γ2 l  1 γ e − γ e −iκ − e e e e 1 2 1 c11 c12 ⎝ γ1 γ2  γ l ⎠    = c21 c22 γ1 eγ1 l − γ2 eγ2l e−iσ l γ1 − γ2 −i ∗ e 1 − eγ2 l e−iσ l κ (9.8) in which the complex roots γ1,2 depend on the Bragg phase mismatch ϑ and on the complex coupling strength κ as follows:    γ1,2 = 0.5 iϑ ± 4|κ|2 − ϑ 2 , λ , λ(0)2 l κ = −i M , 4

ϑ = −4π n

(9.9)

where λ( p) is the optical wavelength in vacuum for mode # p, λ = λ( p) − λ(0) is the wavelength Bragg mismatch, and ρ and σ are defined by   λ 2πn λ 2πn and σ = − (0) 1 + (0) . (9.10) ρ = (0) 1 − (0) λ λ λ λ

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Phase (Rad)

Reflectivity

0.90 0.85 0.80

0.4 0.0 −0.4

0.75 −0.10

0.00

−0.10

0.10

Wavelength mismatch Δλ (nm)

0.00

0.10

Wavelength mismatch Δλ (nm)

F I G U R E 9.3. Modulus (left) and argument (right) of the amplitude reflectivity of the adaptive Fabry-Perot reflectivity at steady state versus the wavelength mismatch λ.

In the absence of absorption and for small l products (l < 1), the results given by this analytical solution are identical to those obtained with the accurate wavemixing formalism. 9.2.3.5 Discussion

In Figure 9.3 we have plotted the modulus |¯rFP | and the argument Arg[¯rFP ] of the Fabry-Perot reflectivity versus the wavelength Bragg √ mismatch for the following parameters:  = 4 cm−1 , α = 0.2 cm−1 , r = 0.8, d = 2 cm, l = 0.3 cm, λ(0) = 1064 nm. Interpreting such spectral reflectivities is straightforward. The values for λ = 0 correspond to the reflectivity for the oscillating mode. Because of the constructive interference, the reflectivity is at its maximum while the phase shift φFP vanishes: the photorefractive grating does not modify the frequency of the oscillating mode. For | λ| much larger than the Bragg wavelength selectivity λBragg , the optical wave does not “see” the photorefractive grating and the Fabry-Perot reflectivity just originates from the output coupler. It is thus equal to the mirror reflectivity r. The phase shift φFP also vanishes for these large mismatches. As | λ| becomes smaller, the influence of the Bragg photorefractive grating becomes more important and the Fabry-Perot interferometer formed by the grating and the output mirror causes the periodic structure. The frequency spacing between two successive peaks is the free spectral range of this Fabry-Perot. With a very good approximation, the spacing between two peaks is given by 2

λ F P ≈



λ(0)

nl 2 d+ 2

.

(9.11)

For these mismatches, the phase shift φFP differs from 0. For large coupling strengths, it can even reach ±π/2. This induced phase shift modifies the optical length of the cavity and thus shifts the optical frequencies of the modes under threshold. It is possible to demonstrate that this frequency pulling always drags the frequencies of the modes under threshold in the direction of the pits of the

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Reflectivity

0.8

0.6

0.4

−50

50×10−3

0 Wavelength mismatch Δλ (nm)

F I G U R E 9.4. Illustration of the frequency-pulling effect: the frequency position of the successive modes are represented on the reflectivity curve versus the wavelength mismatch λ without the frequency-pulling effect (full triangles) and taking into account this effect (open squares); for the other parameters, see text.

reflectivity curve. This effect is illustrated in Figure 9.4. To enhance this effect, this curve has been plotted for a large “gain interaction length product” with √ the following parameters:  = 10 cm−1 , α = 0 cm−1 , r = 0.6, d = 2 cm, l = 0.3 cm, λ(0) = 1064 nm, and a laser cavity length L = 6.5 cm. This frequency pulling increases the losses for the nonoscillating modes and thus contributes to the stability of the single-mode oscillation. Since l is smaller than d, the curve envelope mimics the Bragg selectivity of the thick grating. The frequency spacing between two successive peaks is the free spectral range of this Fabry-Perot. For a relatively uniform grating (moderate absorption and coupling strength), the width of this envelope is given by 2

λ(0) . (9.12) 2nl Another rule of thumb can be useful in case the coupling strength and the output mirror reflectivity are small. The amplitude of the reflected wave remains small compared with the amplitude of the transmitted wave. The Fabry-Perot spectral reflectivity, to first approximation, just results from the superimposition of the wave reflected by the output mirror (amplitude reflectivity r) with the wave reflected by the grating. For the oscillating mode, we get to first approximation   (9.13) r¯FP ( λ = 0) ≈ r 1 + 0.5 le−αl . λBragg ≈

Although the Fabry-Perot is a self-adaptive filter, the experimenter can adjust three parameters to optimize its operation: the photorefractive two-beam coupling gain, the crystal thickness, and its position inside the cavity. In a following section, we

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will demonstrate how to use these formulae (9.11)–(9.13) to optimize the adaptive Fabry-Perot as a function of the laser cavity and laser amplifier.

9.3 Design of Self-Organizing Cavities with 4-Level Laser Amplifiers The first successful operation of such an adaptive Fabry-Perot filter was reported in a continuous-wave dye laser by Whitten and Ramsey [4]. They reported a dramatic reduction of the number of oscillating modes, down to a single-mode oscillation but just for a very specific position of the photorefractive crystal and very close to the laser threshold. We also applied this principle to various 4-level lasers, but we obtained disparate results. For instance, we demonstrated the automatic reduction of the number of the oscillating modes in a pulsed Ti:sapphire laser systematically down to two longitudinal modes [5, 10]. We also tried with a cw Ti:sapphire laser, but we never succeeded in making it single-mode. As explained in Section 9.2.1, this relatively inefficient operation originates from the low available gain, and thus from the large output coupler reflectivity of this Ti:sapphire laser. We also applied this technique to several continuous-wave Nd:YVO4 lasers [8, 10]. The self-organization process was very robust in these lasers: the single-mode oscillation was systematically reached whatever the photorefractive crystal position. Considering the disparity of these results, we decided to conduct a detailed analysis of the operation of these filters in order to optimize their operation and to predict their performance.

9.3.1 Analyzing Procedure The full modeling of the kinetics of a self-organizing laser is rather complex. Furthermore, as observed experimentally, the evolution leading to the single-mode steady state strongly depends on the initial state. Therefore, to optimize the design of self-organizing cavities, we have chosen to neglect this transient behavior and to focus our attention on the steady state. We thus assume that the laser already oscillates on a single-longitudinal mode. For this mode, we compute the losses for all modes using the formalism described in Section 9.2.3. By definition, the gain for the oscillating mode is clamped to the losses. Then, taking into account the characteristics of the laser amplifier, we compute the gain for the other modes. If their gain is lower than their losses, we conclude that the single-mode oscillation is a stable state; otherwise, we deduce that the single-mode oscillation is not possible. Of course, this simple approach proves only the existence of a single-mode steady state. It cannot certify that the self-organizing process automatically drives the laser into this single-mode state. It could, for instance, be trapped in a stable multimode state. Nevertheless, we have successfully optimized several lasers using this simple approach.

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9.3.2 Modeling 4-Level Lasers In standing-wave 4-level lasers, spatial hole burning shapes the population inversion. Secondary axial modes can take advantage of this nonuniform distribution to oscillate, even if the laser line is totally homogeneously broadened. We solve the rate equation in such a 4-level laser to predict the laser intensity above which a single-mode operation cannot be maintained. We consider a CW standing-wave laser cavity containing a homogeneously broadened gain medium of length la. We idealize this gain by a 4-level system in which the pump populates level 3 from level 0, the laser oscillation occurs between levels 2 and 1, and with ultrafast relaxations that empty levels 3 and 1 respectively into levels 2 and 0. The rate equation for the population densities N0 (z) and N2 (z) of the fundamental and excited levels is  1 σ0 d2 N2 (z) σpump Ipump (z) N0 (z) = + 4I0 sin(k0 z) N2 (z) − D + γ N2 (z)2 , hvpump τsp hv0 dz 2 (9.14) where z is the coordinate along the cavity axis, τsp the spontaneous relaxation time of level 2, hν0 the one emitted photon energy, σ0 the emitting cross section, and k0 the wave number of the oscillating mode. In most cases the Auger upconversion rate γ does not significantly contribute to the dissipation of inversion population and can be neglected. The cavity losses are assumed to be moderate, so that intensity I0 of one traveling wave can be considered as constant over the cavity length. The spectral hole burning is smoothed by the diffusion of energy between the sites [11]. This diffusion process is introduced in (9.14) through the diffusion constant D. The numerical resolution of (9.14) gives the evolutions of population densities N0 (z) and N2 (z) versus pump intensity I pump . The gain experienced by one mode is proportional to the overlap integral between the intensity pattern of this mode and the population inversion N2 (z). Because mode 0 is the only one that oscillates, the gain is lower than the cavity losses for all other modes p. To first approximation, assuming small losses, this condition is expressed by [12]  la σ p 0 N2 (z) sin2 [k p z] dz γ0 , (9.15) Pp > = G p (I pump ) with Pp =  la γp σ0 N2 (z) sin2 [k0 z] dz 0

in which the subscript p refers to the mode number p, and γ p is the cavity loss for mode p. G p and Pp respectively represent the gain and losses of mode p normalized to the gain and losses of the oscillating mode. Gp depends only on the cavity parameters and on the pump level. The photorefractive crystal into the cavity does not change its value. Here in after, the frequency of the oscillating mode is chosen at the maximum of the gain curve. With the photorefractive crystal in the cavity and from the analysis conducted in Section 9.2, we determine Pp , which accounts for the losses induced by the adaptive Fabry-Perot. If the losses are exclusively due to the adaptive filter,

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we get Pp ≈

  2 1 − r¯FP λ p

1 − |¯rFP (λ0 )|2

.

265

(9.16)

Taking a Lorentzian shape for the emitting cross section, one easily calculates the value Gp for each mode using (9.15). In the following part, we compare Gp with Pp to determine whether the single-mode oscillation is stable.

9.3.3 Application to 4-Level Lasers 9.3.3.1 Dye Laser

The first experiment with an intracavity photorefractive element was performed in 1987 in a dye laser [4]. We have reproduced the scheme of the experimental cavity in Figure 9.5. The laser spectrum was highly narrowed by the adaptive Fabry-Perot, but single-mode operation was hardly reached. We introduced the values of the experimental parameters in our model. Energy diffusion is negligible in such a laser and we set D = 0 m2 s−1 . The total length of the cavity was 51 cm, and the distance between the end mirror and the dye jet was dampl. = 5 cm. The photorefractive crystal was a 2-mm-thick BaTiO3 sample. In our simulations, we took the photorefractive gain to be  = 1.2 cm−1 . Assuming that only mode ”0” oscillates, we first plotted, in Figure 9.6 left, the gain and the losses induced by the adaptive Fabry-Perot, with the FP thickness d = 7 cm, and a low pumping level set to 1.5 times the pumping oscillation threshold. One immediately sees that even with such a low pumping level, other modes experience a gain larger than their losses: the single-mode oscillation is not a stable one. In Figure 9.6 right, we plotted the gain and the losses for the same laser, but for a symmetric crystal position d = dampl . One sees that at 1.5 times the threshold (crosses), the single-mode oscillation is a stable state. However, for larger pumping levels (for instance at 5 times the threshold, circles) a single-mode oscillation is impossible. These computations confirm that such a dye laser can hardly be made singlemode by an adaptive Fabry-Perot alone. The spectral selectivity of the adaptive

F I G U R E 9.5. Experimental self-organizing dye laser, redrawn from [6].

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Normalized gain and losses

Gilles Pauliat, Nicolas Dubreuil, and G´erald Roosen Normalized gain and losses

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2.0

1.5

1.0 0

10

20 30 40 Mode number

50

2.5 2.0 1.5 1.0 0

10

20 30 40 Mode number

50

F I G U R E 9.6. Comparison of normalized gains (crosses for 1.5 times the threshold, and circles for 5 times the threshold) and losses (full lines) for two photorefractive crystal positions. Left: asymmetric crystal position. Right: symmetric crystal position d = dampl .

Fabry-Perot is not strong enough. To increase its efficiency, it should be complemented by another coarse filter, a prism for instance, which already reduces the number of oscillating modes, the role of the self-organizing process being just to select one mode among a few. To conclude, a self-adapted Fabry-Perot is not very well suited for such lasers in which:

r the gain frequency bandwidth is large; r the output coupler reflectivity is large; the adapted Fabry-Perot cannot significantly reduce the losses of the oscillating mode;

r the spectral hole burning is too large, due to the small thickness of the amplifying medium and because of the very small value of diffusion coefficient D. 9.3.3.2 Nd:YVO4 Laser

We experimentally studied several CW Nd:YVO4 lasers. In the following we present experimental results and calculations obtained with the cavity depicted in Figure 9.7. The amplifier was a 1-mm-thick, 1% doped Nd:YVO4 sample. In this sample, we measured a large diffusion coefficient, D = 7 10−12 m2 s−1 , and an Auger

F I G U R E 9.7. Scheme of the experimental Nd:YVO4 laser and definition of the notation used in the text.

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upconversion rate equal to γ = 310−21 m3 s−1 [13]. The laser was pumped at 808 nm through the end mirror. The waist of the pump beam onto the Nd:YVO4 sample was 80 μm, slightly smaller than the waist of the oscillating mode. The reflectivity of this mirror was 100% at 1064 nm. The reflectivity of the output coupler was 90% at 1064 nm and its radius of curvature was 10 cm. The cavity length was L = 8.5 cm. The photorefractive crystal is a 3-mm-thick Rh:BaTiO3 crystal grown by FEE provided by D. Rytz (FEE, Idar-Oberstein Germany). It was rhodium doped to increase its sensitivity at 1064 nm. This antireflectioncoated sample was 45◦ cut. We oriented the photorefractive crystal relatively to the amplifying medium so that the beam inside the photorefractive crystal was ordinarily polarized. In this way, the photorefractive gain was relatively modest. We measured  = 120 m−1 . Without the photorefractive crystal, this laser always oscillated on several longitudinal modes. With the photorefractive crystal, the single-mode oscillation was reached less than 2 seconds after switching on the laser. This cavity was specially designed so that distance dampl. and distance d could be varied. In a first series of experiments, we set dampl. = 15 mm, and we varied photorefractive crystal position d. For each value of d, we increased the pump power I pump from its threshold value Ithres. until the laser became multimode. These experimental limits are reported as dots in Figure 9.8. On the same graph, we also reported the same theoretical values for this limit, following the computation exposed in Section 9.3.2. As previously observed with the dye laser, the best crystal position corresponds to the symmetric position d ≈ dampl. − nl/2 = 12 mm. This laser can be pumped at 1.8 times the threshold. However, one should remark that this relatively low pumping level originates from the specific cavity we have studied, in which the amplifying medium is several millimeters away from the end mirror. In case the amplifying medium is onto the end coupler, we

1.8

Pumping rate

1.6 1.4 1.2 1.0 0

10 20 Photorefractive crystal position

30 d (mm)

40

F I G U R E 9.8. Experimental (dots) and theoretical (full line) limits of the pumping rate (Ipump /Ithresh. ) above which the laser becomes multimode.

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1.1 1.0

0.9 0

5

10

15 20 Mode number

25

30

F I G U R E 9.9. Comparison of normalized gains (crosses) and losses (full line) for the Nd:YVO4 laser cavity shown in Figure 9.8. For other parameters, see text.

have demonstrated that the same cavity could be pumped 5 times above threshold. To illustrate this point, we plotted in Figure 9.9 the gain and the losses for this cavity, for dampl. = 0 mm, d = 10 mm, and I pump = 5Ithr esh. For this configuration (amplifier close to the end mirror), the hole-burning effect is not very pronounced: we do not observe a strong modulation of the gain versus the mode number, and this gain decays very quickly versus the mode number because of the finite frequency bandwidth of the amplifier. Therefore, such lasers always operate on a single longitudinal mode, whatever the photorefractive crystal position. For all these experiments, the photorefractive crystal was used with a quite moderate gain,  = 120 m−1 . By rotating the crystal along the cavity axis, one accesses the large extraordinary gain,  = 520 m−1 . Although from the above analysis, one could expect an improved single-mode oscillation, we experimentally observed than the laser never reached a steady state and that the beam profile was no longer Gaussian but very irregular and unstable. We attribute this behavior to the wavefront distortion induced during the recording of the photorefractive hologram: the photorefractive time constant is inversely proportional to the intensity. Therefore, before the steady state is reached, the diffraction efficiency of the hologram is the strongest in the center of the beam: the transverse profile of the diffracted beam differs from the initial Gaussian beam. If the diffracted beam is strong enough, i.e., if the photorefractive gain is too large, this effect significantly alters the beam profile and prevents the laser from reaching a steady state.

9.4 Self-Organizing Cavities with Semiconductor Amplifiers The self-organization process results from an interaction of the photorefractive hologram with the gain medium, through the modal structure. Therefore, this self-organization strongly depends on the gain medium characteristics. We have

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previously seen that in a 4-level amplifying medium, some modes experience more gain than the oscillating mode as a result of the spatial hole burning. To ensure a single-mode oscillation, the losses induced by the photorefractive hologram onto these modes should compensate for the larger available gain. The situation in semiconductor lasers is quite different. Most often, due to the large chargecarrier migration length, the longitudinal spatial hole burning is washed out. In these lasers, multimode operation is favored by other optical nonlinearities: carrier-heating spectral-hole burning and carrier-density pulsation [14–22]. When a mode oscillates it may transfer its energy to other modes through two-wave and four-wave mixings. These nonlinear processes have been extensively studied in the literature. The behavior of a self-organizing cavity with a semiconductor laser can thus be easily analyzed at steady state following the same approach as the one we have just described for 4-level lasers:

r Assume that the single-mode steady state is reached; r Compute the gain for all modes taking into account the wave-mixing processes; r Compare this gain with the losses induced by the photorefractive filter following the calculation of Section 9.2.3;

r Then check the consistency of the single-mode oscillation. In the following, we will just give some examples of self-organizing laser cavities with semiconductor amplifiers.

9.4.1 Nontunable Diffraction-Limited Laser Diode We successfully demonstrated the self-organizing operation with extended cavities using diffraction-limited semiconductor amplifiers at various optical wavelengths: 633 nm, 660 nm, 810 nm, 1550 nm [23–26]. Their design is shown in Figure 9.10. The laser amplifiers are commercially available laser diodes that received on their output facet an antireflection coating. Although we have successfully operated several self-organizing cavities with standard coatings (usually with a reflectivity of about 4%), the deposition of the antireflection coating removes most of the mode hops observed in changing the operating point of the laser (current or temperature). We reported the operation of such a cavity at 810 nm in references [23– 24]. In the following we describe the operation of another cavity operating at 660 nm. The photorefractive crystal is a cobalt-doped photorefractive BaTiO3 Semiconductor diode

Aspherical lens

Photorefractive crystal

Output coupler

HR coating AR coating

F I G U R E 9.10. Self-organizing cavities with diffraction-limited semiconductor amplifier.

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Coherence length (m)

600 500 400 300 200 100

5

6 7 Output power (mW)

8

9

F I G U R E 9.11. Coherence length of the self-organizing cavity versus the output power. The integration time is 5 ms.

crystal. The laser diode received a high-quality antireflection coating. Without the photorefractive crystal, this cavity is sometimes single-mode for very specific operating points (current and temperature). We attribute this behavior to the imperfect antireflection coating of the lens, which creates coupled cavities. With the photorefractive crystal, this cavity is always forced to oscillate on a single longitudinal mode by the self-organizing process. The time required to reach the single-mode steady state is about 1 second. After the thermal stabilization equilibrium is reached (a few tens of seconds), this single-mode operation is maintained. For a temperature-stabilized cavity, we have never seen any mode hopping during the duration of our observations (several tests each lasting 14 hours). The maximum output power of this extended cavity is 10 mW for an injection current of 75 mA. The reflectivity of the Bragg grating induced in the crystal is relatively small, so that it does not significantly modify the intracavity intensity. Therefore, the spectral characteristics of such an extended cavity laser diode are roughly the same as that of a similar cavity without a crystal if it were single-mode. For instance, we expect the coherence length to increase with the intracavity intensity [27] as measured in the experiment (see Figure 9.11). Although this cavity is not frequency stabilized, it is of interest in applications in which the coherence length is an important parameter, but not the exact wavelength, such as two-dimensional metrology (interferometric surface inspection, etc.).

9.4.2 Wavelength-Tunable Diffraction-Limited Laser Diode The previously depicted system is not wavelength-tunable. To force the oscillation on a given wavelength, an intracavity adjustable spectral filter has to be used. We

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F I G U R E 9.12. Sketch (left) and photograph (right) of the grating-tuned extended-cavity laser (from [26]).

demonstrated the operation of such a tunable self-organizing laser cavity using the Littman configuration shown in Figure 9.12 [26]. This cavity, apart from the photorefractive crystal, is similar to those used to test optical communication networks and components around 1550 nm. A multiple-quantum-well InGaAsP laser diode is optically coupled to an extended cavity through its antireflection-coated facet. The opposite facet serves as the output-coupling mirror. The extended cavity consists of a collimating lens and a Littman-mounted grating. Tunable single-mode emission is obtained by means of sending the light back from the extended cavity into the semiconductor chip by use of the Littman grating and retroreflector. The minimum-loss wavelength corresponds to the wavelength that is retroreflected by the grating and is thus optimally back-coupled into the semiconductor amplifier. For other wavelengths, the return beam is shifted from the best back-coupling position. Consequently, the loss increases with the detuning. The oscillating-mode wavelength is thus selected inside a small wavelength window. Current commercial devices based on this principle can be continuously tuned over a range larger than 100 nm, but this requires delicate factory adjustments and a high stability of the extended cavity. Indeed, the wavelength mismatch between the lasing wavelength and the wavelength of minimum loss must be kept inside a restricted range so as to prevent any mode hopping or any multimode operation. We demonstrated that the insertion of a CdTe photorefractive crystal between the Littman grating and the collimating lens creates an adaptive spectral filter that decreases the loss of the lasing mode and thus enhances its stability. This increase of stability extends the available wavelength-mismatch range without mode hopping. The operation of the crystal is similar to the operation of the photorefractive crystals in the previously shown self-adapted cavities. It is oriented so that for the oscillating mode, the light reflected onto the photorefractive Bragg grating interferes constructively with the light reflected by the diehedral mirror and Littman grating. Because the reflectivity of this feedback system (diehedral mirror plus Littman grating) is relatively small (about 30%), the Bragg grating reflects part of the light before it experiences too much loss in this feedback loop.

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multimode (without crystal) single-mode

single-mode limit with CdTe single-mode limit without CdTe multimode limit with CdTe multimode limit without CdTe

0 -50 0

2 4 6 Output Power (mW)

8

F I G U R E 9.13. Single-mode operation domain at 1.6μm with and without CdTe crystal. The data denoted by “single-mode limit” correspond to mode-hops toward another singlemode state or to appearance of multimode operation; the data denoted “multimode limit” correspond to mode-hops from a multimode state toward a single-mode state. From [26].

For other nonoscillating modes, the reflectivity of the adapted Fabry-Perot is not optimized and the losses are larger. This reflectivity can be calculated in a way similar to that detailed in Section 9.2.3. In Figure 9.13 we plotted the measured single-mode operation domain versus both the wavelength mismatch and the optical output power. Without a photorefractive crystal, this domain is limited by the open squares and circles, while with the photorefractive crystal it is limited by the filled squares and circles. For instance, if the cavity is enlarged, up to a square dot in Figure 9.13, then a mode hop occurs and then brings back a new mode inside its stability domain. The photorefractive filter always increases the stability domain, especially at high power when a large multimode operation domain appears without a crystal. In Figure 9.13, this multimode domain is delimited by “x” signs with the crystal, and by “+” signs without the crystal. The stable range always shifts toward positive values when the output power increases. Furthermore, in the case of Figure 9.13, it broadens until the power reaches the value of 2 mW. Then, for higher powers, without the crystal, the multimode operation domain appears and the stable range narrows and eventually closes up. The photorefractive filter efficiently suppresses most of the multimode operations. Moreover, the stable range remains wide open even at high-opticalpower levels. Here, we present only the measurements at 1.6μm, but this kind of improvement of the stable single-mode range is efficient in the full tuning range of the extended cavity, from 1480 nm to 1600 nm. Moreover, we did not notice any degradation of the long-term stability due to the insertion of the crystal. Furthermore, we estimate that for a 1-mW output power, the photorefractive grating builds up within 2 ms. The continuous-tuning speed that could be reached is thus estimated at 10 nm/s.

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9.5 Self-Organizing Cavities Operating on the Transverse Structure All the above-reported self-organizing cavities operate on the longitudinal modal structure only: initially, prior to the self-organization process occurring, they already oscillate on the fundamental single transverse structure, although this transverse structure may correspond to a full comb of longitudinal modes. We do not expect the same self-organization process to efficiently operate in lasers that are initially oscillating on several transverse modes. Indeed, the frequency spacing between these transverse modes is much lower than the frequency spacing between the longitudinal modes of a single transverse structure. Therefore, to induce a single-mode oscillation in such lasers, the self-adapted Fabry-Perot should have a stronger spectral selectivity. Fortunately, we may benefit from another phenomenon to ensure a correct self-organizing operation on the transverse structure. For the sake of simplicity, assume that the self-organization process was successful and that the laser oscillates on a single transverse and longitudinal structure. As explained in Section 9.2.2, and because the photorefractive grating is proportional to the modulation index of the interference pattern, the shape of the recorded hologram is perfectly fitted to the transverse shape of the oscillating mode. In others words, for the oscillating mode, the complex amplitude profile of the beam diffracted onto this hologram is proportional to the recording beam profile. All the diffracted light is thus reinjected inside the mode: no energy is lost. For other transverse modes, if their phase profiles differ from the oscillating mode phase profile, then part of the light diffracted onto the photorefractive grating is not reinjected inside the mode, and may even be diffracted outside the cavity. If these diffraction losses are important enough, the single transverse mode oscillation is a stable state. The transverse phase structure of Gaussian modes being all the same, this process is inefficient in cavities supporting these modes. Conversely, the phase profiles of modes in extended cavities built with broad-area laser diodes considerably differ from one another, making this selforganization process very efficient [28]. We have recently verified this selforganization process in reference [29].

9.6 Conclusion The unique properties of photorefractive crystals make them ideally suited to build self-organizing cavities. We illustrated their use and optimization on very specific examples: a linear Fabry-Perot with 4-level laser amplifiers and semiconductor laser amplifiers. Nevertheless, the same basic principle can be applied to very large varieties of cavities with other self-adaptive photorefractive filters: insert

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a self-adaptive filter inside a laser cavity and do so that its efficiency (spectral or/and spatial selectivity) improves when the spectral or/and spatial content of the oscillating modes reduces. An example of a self-organizing ring cavity is illustrated in [5]. Similarly, a self-adaptive Fabry-Perot filter can also very efficiently reduce the emitted spectrum in self-organizing optical parametric oscillators as demonstrated in [30]. Another idea, not developed in this chapter, but illustrated elsewhere [5,8], is to use this self-adapted filter to implement another function. If the crystal is inserted in the other orientation so that the strongest modes increase their losses, then the emitted spectrum can either become unstable or just enlarged and equalized, according to the specific properties of the amplifying medium. This could be useful for optical coherence tomography, for instance. We have also recently demonstrated [31] that the wavelength of a selforganizing extended cavity can be set by a temporary injection of light from a master laser. The wavelength of the self-organizing cavity is thus optically addressed. Even after the master oscillator is switched off, the self-organizing cavity continues to oscillate on the selected wavelength: the photorefractive grating maintains the oscillation on this wavelength, and the oscillation maintains this photorefractive grating. The potentialities of such a configuration for optical communication is certainly to be considered, since it can be extended to a configuration where several slave lasers share a unique tunable master laser. For all the above reasons, we believe that self-organizing laser cavities will undergo a large development.

References 1. 2. 3. 4. 5.

6. 7. 8.

9. 10. 11. 12. 13. 14.

S. Camacho-Lopez, M.J. Damzen. Opt. Lett. 24, 753 (1999). P. Sillard, A. Brignon, J.P. Huignard. J. Opt. Soc. Am. B 14, 2049 (1997). M. Horowitz, R. Daisy, B. Fisher. Opt. Lett. 21, 299 (1996). W.B. Whitten, J.M. Ramsey. Opt. Lett. 12, 117 (1987). ´ N. Huot, “Etude de BaTiO3 : Rh a` 1,06 μm et application a` la conjugaison de phase pour la correction de fronts d’onde,” PhD dissertation, Universit´e de Paris Sud, Orsay France (1998). M. Lobel, P.M. Petersen, P.M. Johansen. J. Opt. Soc. Am. B 15, 2000 (1998). L. Meilhac, N. Dubreuil, G. Pauliat, G. Roosen. Optical Materials 18, 37 (2001). L. Meilhac, “Cavit´es laser auto-organisables: r´eduction du nombre de modes longitudinaux par un filtre photor´efractif dynamique,” Ph.D. thesis, Universit´e Paris XI, France, October 2001. Kogelnik H. Bell Syst. Tech. Journ. 48 2909 (1969). N. Huot, J.M. Jonathan, G. Pauliat, P. Georges, A. Brun, G. Roosen. Appl. Phys. B 69, 155 (1999). H.G. Danielmeyer. J. Appl. Phys. 42, 3125 (1971). J.J. Zayhowski. IEEE J. Quantum Electron. 6, 2052 (1990). L. Meilhac, G. Pauliat, G. Roosen. Opt. Commun. 203, 341 (2002). M. Yamada. J. Appl. Phys. 66, 81 (1989).

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15. A. Uskov, J. Mørk, J. Mark. IEEE J. Quantum Electron. 30, 1769 (1994). 16. R.F. Kazarinov, C.H. Henry, R.A. Logan, J. Appl. Phys. 53, 4631 (1982). 17. A. D’Ottavi, E. Lannone, A. Mecozzi, S. Scotti, P. Spano, J. Landreau, A. Ougazzaden, J.C. Bouley. Appl. Phys. Lett. 64, 2492 (1994). 18. F. Girardin, G.-H. Duan, P. Gallion, A. Talneau, A. Ougazzaden. Appl. Phys. Lett. 67, 771 (1995). 19. C.-Y. Tsai, C.-Y. Tsai, R.M. Spencer, Y.-H. Lo, L.F. Eastman. IEEE J. Quantum Electron. 32, 201, (1996). 20. C. Doerr, M. Zirngibl, C. Joyner. IEEE Photon. Technol. Lett 7, 962 (1995). 21. A. Godard, G. Pauliat, G. Roosen, P. Graindorge, P. Martin. IEEE J. Quantum Electron. 38, 390 (2002). 22. A. Godard, G. Pauliat, G. Roosen, E. Ducloux. IEEE Journal of Quantum Electronics, 40, 970, (2004). 23. G. Delcourt, F. Seguineau, N. Dubreuil, G. Roosen. CLEO/Europe 2000, Nice, France, Post Deadline paper CPD1.7. 24. A. Godard, G. Pauliat, G. Roosen, P. Graindorge, P. Martin. Opt. Lett., 26, 1955 (2001). 25. S. Maerten, N. Dubreuil, G. Pauliat, G. Roosen, D. Rytz, T. Salva. Opt. Commun. 208, 183 (2002). 26. A. Godard, G. Pauliat, G. Roosen, E. Ducloux. Appl. Opt. 43, 3543 (2004). 27. H. Sun, S. Menhart, A. Adams. Appl. Opt. 33, 4771 (1994). 28. V. Reboud, N. Dubreuil, G. Pauliat, G. Roosen. “Ninth international conference on Photorefractive effect, materials and devices,” La Colle-sur-Loup, France, 17-21 Juin 2003, in:OSA Trends in Optics and Photonics, Advances in photorefractive Materials, Effects, and Devices, 87, pp. 535–540. 29. V. Reboud, N. Dubreuil, G. Pauliat, G. Roosen. Photonics Europe, Strasbourg, 26–29 April 2004. 30. S. Victori, T. L´epine, P. Georges, A. Brun. Advanced solid state laser, Topical Meeting, Qu´ebec, Feb. 2002. 31. N. Dubreuil, G. Pauliat, G. Roosen, 30th European Conference on Optical Communications, Stockolm, Sweden, September 5-9, 2004,ECOC 2004 Proceedings, Vol. 3, Paper We4.P.065, pp. 600–601.

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10 Slow and Fast Lights in Photorefractive Materials Guoquan Zhang,1 Fang Bo,2 and Jingjun Xu3 1 2 3

Photonics Center, College of Physics Science, Nankai University, Tianjin 300071, China [email protected] Photonics Center, College of Physics Science, Nankai University, Tianjin 300071, China [email protected] Photonics Center, College of Physics Science, Nankai University, Tianjin 300071, China [email protected]

Techniques on slow and fast lights are of fundamental interest because they are targeted toward the understanding of physical laws for light-pulse propagation [1] as well as nonlinear optics at the level of single photon [2, 3]. They are also promising for many applications such as controllable optical delay/accelerant lines, optical memories, and devices for quantum information processing [4, 5, 6]. Traditionally slow and fast lights occur when the lights propagate in a dispersive amplifying/absorbing medium [7]. Early works on the demonstration of slow and fast lights were blurred by the pulse deformation due to the amplification or the attenuation of light pulses during propagation [8, 9, 10]. A breakthrough on slow lights was realized by Hau’s group at Harvard University [2], who decelerated a light pulse to as slow as 17 m s−1 based on the electromagnetically induced transparency (EIT) effect [11] in an ultracold gas of sodium atoms, while that on fast lights was followed by the work of Wang et al. [12], who reported the observation of distortionless superluminal light pulses. It is worthy of note that no signal propagates at a velocity faster than the speed of light in vacuum [12, 13]. The photorefractive effect is a versatile nonlinear effect and is able to occur in photorefractive materials even at very low light intensities. The photorefractive wave coupling is inherently a highly dispersive process because it takes time to redistribute the photoexcited charge carriers among different donors and acceptors. Such a photorefractive wave coupling would result in dispersive changes in both the intensity and the phase of the light during the propagation. The dispersive properties of the intensity coupling effect were well discussed in the literature [14, 15, 16, 17, 18] and in the books [19] and [20]. We will pay special attention on the dispersive photorefractive phase-coupling effect and its application to the generation of slow and fast lights.

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10.1 Dispersive Photorefractive Phase Coupling Let us consider a nondegenerate two-beam coupling process, where a weak signal beam with an angular frequency ωs and a strong pump beam with an angular frequency ωp interfere with each other in a photorefractive material. The induced refractive index grating moves along the grating vector but with a delayed space displacement with respect to the interference pattern of two coupling beams. Under the approximation of small modulation depth, the associated light-induced space-charge field E sc can be expressed as [21, 22] E sc = −

E0 − i ED (1 +

ED Eq

+i

E0 ) Eq

− τ ( EE0q − i ττdi − i EEDq )

,

(10.1)

where  = ωs − ωp is the angular frequency difference; E 0 , E D = ekB T /q and E q = eNA (ND − NA )/(q ND 0 ) are the externally applied electric field along the grating vector, the diffusion field, and the limiting space-charge field, respectively; q = 2π/ is the grating vector and  is the grating spacing. ND and NA are the densities of the total dopant and the nonmovable compensation charge that maintains the charge neutrality of the crystal in the dark, respectively; τ is equal to (β + s I0 + γ n 0 )−1 and τdi = 0 /eμn 0 is the dielectric relaxation time; I0 , n 0 , β, s, γ , e, kB , T , and μ are the total incident intensity, the conduction band electron concentration, the thermal excitation rate, the absorption cross section, the recombination coefficient, the magnitude of elementary charge on electrons, the Boltzmann constant, the absolute temperature of the photorefractive material, and the mobility of conduction-band electrons in the photorefractive material, respectively. Note that the photovoltaic effect is not considered here and the electron is assumed to be the only charge carrier. Due to the creation of the space-charge field E sc , the intensities and the phases of the two beams are coupled with each other. The coupling coefficients for the intensity in and the phase ph are

in =

2πn 3breff Im(E sc ) λ

(10.2)

and πn 3breff Re(E sc ) , (10.3) λ respectively, where λ is the operating wavelength; Im(E sc ) and Re(E sc ) represent the imaginary and the real components of the complex space-charge field. It is seen that both in and ph are highly dispersive with respect to the angular frequency difference . This is because the formation of the space-charge field in photorefractive materials involves the transport and redistribution of charge carriers among different donors and acceptors. Such charge-carrier transport and redistribution takes time, which makes the photorefractive two-beam coupling a

ph =

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12

(a)

(b) ∂Γph/∂Ω (m−1s)

10 Γph (cm−1)

5 0 -5 -10

8 4 0

-15 -300 -100

100 300 500 Ω/2π (Hz)

700

900

-300 -100

100 300 500 Ω/2π (Hz)

700

900

F I G U R E 10.1. Typical dispersion curves of ph (a) and ∂ ph /∂ (b) for a Bi12 SiO20 crystal, respectively. The material parameters used to calculate the curves are listed in Table 10.1, other parameters are λ = 514.5 nm, T = 300 K, I0 = 104 W m−2 , E 0 = 106 V m−1 , and  = 30 μm, respectively.

slow-response-rate process. Therefore, the strength and the relative phase (with respect to the interference intensity pattern) of the light-induced refractive index grating depend on the angular frequency difference . As an example, Figure 10.1 shows typical dispersion curves for (a) the phasecoupling coefficient ph and (b) the slope of the phase-coupling coefficient dispersion ∂ ph /∂, respectively, for a bismuth silicon oxide (Bi12 SiO20 ) crystal. The material parameters for a Bi12 SiO20 crystal are shown in Table 10.1. It is evident that the phase-coupling coefficient ph varies rapidly as a function of . A narrow frequency window with a steep positive slope (≈ 10 m s−1 ) with respect to  appears in the middle part of the curve in Figure 10.1a, while the slope ∂ ph /∂ is found to be negative at the two sides of the curves. We will show in the following section that such a dispersive phase-coupling process can be used to generate slow and fast lights in the photorefractive materials. A typical dispersion curve for the intensity-coupling coefficient in is also shown in Figure 10.2. Detailed discussions on the dispersive property of the intensity-coupling coefficient in and its applications can be found in the papers [14, 15, 16, 17, 18] and in the books [19, 20].

T A B L E 10.1. Material parameters for a typical photorefractive Bi12 SiO20 crystal at 514.5 nm. From [22]. NA (m−3 )

ND (m−3 )

 –

nb –

reff (pm V−1 )

μ (m2 V−1 s−1 )

s (m2 J−1 )

γ (m3 s−1 )

β (s−1 )

1022

1025

56

2.62

3.4

10−5

1.06 × 10−5

1.65 × 10−17

0

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60

Γin (cm−1)

50 40 30 20 10 0 -300 -100 100 300 500 Ω/2π (Hz)

700

900

10.2 Phase-Coupling-Induced Slow and Fast Lights 10.2.1 Group Velocity of Light with a Dispersive Phase Coupling Under the pump-undepleted approximation and neglecting the material absorption, the signal beam intensity Is (r ) grows exponentially, Is (r ) = Is (0) exp( inr ), during the propagation, while the phase-coupling-induced phase shift of the signal beam s (r ) − s (0) follows a linear increase with the propagation distance r :

s (r ) − s (0) = phr ,

(10.4)

where Is (0) and s (0) are the initial intensity and phase of the signal beam at the entrance surface of the photorefractive materials. Therefore the total phase shift per unit distance for the signal beam is ph + ks , and ks = 2πn b /λ is the wave vector of the signal beam in the photorefractive materials. By differentiating the phase shift per unit distance with respect to the angular frequency ωs of the signal beam, one obtains the effective group velocity of the signal beam vg propagating in the photorefractive materials [22]: c c vg = ≈ , (10.5) ∂

∂( ph +ks ) n b + c ∂ωphs n b + c ∂ωs where c is the light speed in vacuum, and the slope ∂ ph /∂ωs is expanded to be n 3 reff E D B − E 0 A πn 3b reff τ ∂ ph ∂ ph = b = + ∂ωs ∂ 2c A2 + B 2 λ ⎛   ⎞  ED ED E 02 τdi τdi E0 2(E E + + + B − E A) B( ) − A D 0 ⎜ D τ Eq Eq τ Eq Eq ⎟ ⎟ ⎜ − ×⎜ ⎟, ⎠ ⎝ A2 + B 2 (A2 + B 2 )2 (10.6)

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and the parameters A and B are A =1+

ED E0 − τ Eq Eq

and E0 B= + τ Eq



ED τdi + τ Eq

(10.7)  ,

(10.8)

respectively. In general, the refractive-index dispersion is negligibly small for photorefractive materials and it is ignored in deriving (10.5) and (10.6). In the case that n b  c∂ ph /∂ωs is satisfied, vg can be further simplified to be vg ≈ (∂ ph /∂ωs )−1 . It is evident that subluminal and superluminal light propagation in photorefractive materials are possible with a dispersive phase coupling. Note that the spectral bandwidth of the window having a positive or a negative slope is determined by the response time of the wave-coupling process; slow or fast light is observable only when the light pulse duration is longer than or comparable to the response time of the wave-coupling process. We will discuss this in more detail in Sections 10.2.2 and 10.3.5. Podivilov et al. [23] also showed the ultraslow light propagation in photorefractive materials by solving numerically the dynamic degenerate wave-coupling equations between a Gaussian pulse and a continuous-wave (cw) pump beam under the pump-undepleted approximation, where the time derivative of the spacecharge field was taken into consideration, while those of the amplitudes of the pump and the signal pulse were omitted. The output pulse was verified to experience a considerable time delay with respect to the input pulse when the input pulse duration was larger than the response time of the wave-coupling process. Differences as well as similarities between the EIT nonlinearity and the photorefractive nonlinearity were discussed. The main difference is that the EIT nonlinearity results in a decrease of the light absorption coefficient of the material, whereas the photorefractive nonlinearity leads to an amplification of the signal beam.

10.2.2 Tunability of the Group Velocity of Lights The slope ∂ ph /∂ωs is determined by the response rate and the strength of the photorefractive two-beam coupling. Therefore the group velocity of light in photorefractive material is related to the experimental parameters such as the grating spacing , the incident intensity I0 , and the external field E 0 as well as the material parameters, and it can be tuned to quite a large extent, even from the subluminal to the superluminal, or vice versa. Figure 10.3 shows the influences of the grating spacing  on the phase-coupling coefficient ph and the group velocity vg . Note that the effective spectral window to observe the subluminal light propagation moves with the variation of the grating spacing. This happens also for the superluminal light propagation case, as can be seen from the dispersion curves of ph . The bandwidth of the subluminal spectral window sub increases with the increase of the grating spacing. This is because

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(b)

(a)

10 μm

vg (m s−1)

Γph (cm−1)

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1

0.1

0

100

200 300 Ω/2π (Hz)

10 μm 0

400

100

20 μm

30 μm

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F I G U R E 10.3. Dispersion curves for (a) ph and (b) vg at different grating spacings with E 0 = 106 V m−1 and I0 = 104 W m−2 at 514.5 nm, respectively, in a photorefractive twowave mixing process. The material parameters for a photorefractive Bi12 SiO20 crystal are listed in Table 10.1.

the slope ∂ ph /∂ωs is a function of the term / instead of , according to (10.5)–(10.8). Thus a larger grating spacing would result in a broader effective subluminal spectral window. The limited bandwidths of the subluminal and the superluminal spectral windows put a serious limitation on the spectral bandwidth of the input signal pulse. In order to observe slow or fast lights experimentally, the spectral bandwidth of the signal pulse should be less than that of the subluminal or the superluminal spectral windows. The response rate of the photorefractive two-beam coupling is proportional to the incident light intensity I0 , while the coupling strength is independent of I0 when the photoconductivity is much larger than the dark conductivity of the photorefractive materials. This means that the slope ∂ ph /∂ωs is steeper at lower intensities, which results in a smaller group velocity of light, whereas the bandwidth sub/super is broader at higher intensities (see Figure 10.4). 15

103

104

101

105

100

5

vg (m s−1)

Γph (cm−1)

10

102

0 -5 -10 (a)

-15 100

101

102 Ω/2π (Hz)

103

105 104

10−1 103

10−2 10−3

(b)

102 101

102 Ω/2π (Hz)

103

F I G U R E 10.4. Dispersion curves for (a) ph and (b) vg at different incident intensities I0 = 102 , 103 , 104 , and 105 W m−2 at 514.5 nm, respectively, with E 0 = 106 V m−1 and  = 30 μm. The material parameters for a Bi12 SiO20 crystal are listed in Table 10.1.

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-2

Ω/2π (×103 Hz) 0 2 4 6

8

106

0.01

-0.01

106

-0.02

-5 -10

4

10

-15 0

(a)

105

10

0 104

5

-0.03

106

-0.04

0

200 400 600 800 Ω/2π (Hz)

220

240

260 280 Ω/2π (Hz)

(b)

vg (m s−1)

0

-200

15

0.00

0

5

vg (m s−1)

10 Γph (cm−1)

-4 20

15

Γph (cm−1)

15

Ω/2π (×103 Hz) 0 5 10 -15 -10 -5

283

104

300

F I G U R E 10.5. Dispersion curves for (a) ph and (b) vg with I0 = 104 W m−2 at 514.5 nm and  = 30 μm but at different E 0 = 106 V m−1 (solid curves), 104 V m−1 (dashed curves), and 0 (dotted curves), respectively. The left vertical and the bottom horizontal coordinates are for the case with E 0 = 106 V m−1 , while the right vertical and the top horizontal coordinates are for the cases with E 0 = 104 V m−1 and E 0 = 0. The material parameters for a Bi12 SiO20 crystal are listed in Table 10.1.

The effects of the external field E 0 on the dispersive properties of (a) ph and (b) vg are illustrated in Figure 10.5. At a larger E 0 , the coupling strength is stronger, and the drift length of the photoexcited charge carriers is longer. Therefore the response rate is slower. This results in a very steep slope ∂ ph /∂ωs and therefore an ultraslow light at a high external field. The bandwidths of the subluminal and the superluminal spectral windows also become narrower at a higher external field. Transition between the subluminal and the superluminal light propagation is also possible. This can be achieved by scanning the angular frequency difference  or by tuning the external field E 0 or by tuning the incident intensity I0 . Figure 10.6 shows an example of such transitions between the subluminal and the superluminal light propagation by scanning the angular frequency difference . Several appealing advantages are inherent in this technique: (1) the group velocity of light pulses can be controlled to a great extent at room temperature 30

vg (m s−1)

20 10 0 -10 -20 -30 150

200

250 300 Ω/2π (Hz)

350

400

F I G U R E 10.6. Typical transitions between the subluminal and the superluminal light propagation achieved by scanning the angular frequency difference . Simulation parameters:  = 30 μm, E 0 = 106 V m−1 , and I0 = 104 W m−2 at 514.5 nm. The material parameters for a Bi12 SiO20 crystal are listed in Table 10.1.

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in solids by controlling the phase-coupling coefficient dispersion; (2) both subluminal and the superluminal light propagation are possible by controlling the sign of ∂ ph /∂ωs ; (3) the signal pulses can be amplified by use of unidirectional energy transfer from the coupling beam to the signal pulses; (4) the light source does not have to be a single-line laser, and this technique is effective in a wide spectral range as long as a dispersive phase coupling occurs during a nonlinear wave-mixing process. Moreover, our technique to produce slow and fast lights is valid for all nonlinear wave-mixing processes with a dispersive phase-coupling effect.

10.3 Experiments on Slow and Fast Lights in Photorefractive Materials Experimental demonstrations on slow lights in photorefractive materials have been carried out in several research groups [23, 24, 25, 26, 27, 28]. Group velocities as small as hundreds of micrometers per second were achieved in photorefractive barium titanate (BaTiO3 ) crystals [23]. Soon thereafter, superluminal light propagation with negative group velocities was also demonstrated [29, 30]. We should emphasize here that the observed fast light with negative group velocity is not at odds with causality; it is a result of the negative slope of the phasecoupling coefficient dispersion and the interference among different frequency components of the light pulse. Applications of slow lights such as optical buffers were discussed, although the reported performance was not so attractive because of the pulse deformation problem [25]. In the following, we will review recent experimental progress on slow and fast lights in photorefractive materials.

10.3.1 Measurement of Group Velocity of Light Two techniques are employed frequently to measure the group velocity of light. One is the intensity modulation technique [24, 30]. In this technique, the signal beam is modulated sinusoidally and then coupled with a strong cw pump beam in a photorefractive material. It is evident that the sinusoidally modulated signal beam composes of two components with an angular frequency difference δωs = 2π/T , where T is the periodic time of the signal beam. These two components couple respectively with the same pump beam but of different values of ph because the phase-coupling coefficient ph is highly dispersive in the photorefractive twobeam coupling process. Such a dispersive phase-coupling coefficient results in a phase-delay/phase-advance  s = δ ph L (see (10.4)) of the transmitted signal beam, where δ ph is the phase-coupling coefficient difference between the two components of the signal beam when they are coupled with the pump beam and L is the propagation distance of the signal beam in the photorefractive material. Therefore the group velocity of the signal beam in the photorefractive material

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EO

pump BS1

M2

Ω

EX

PZ

Reference

laser

BS2

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B

D1

D2

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F I G U R E 10.7. Experimental setup scheme to measure the group velocity of lights. M1 and M2: mirrors, EO: electrooptic modulator, BS1 and BS2: beam splitter, EX: beam expander, PZ: piezo-mirror, which is used to generate an angular frequency shift , PRC: photorefractive crystal, B: Blocker, D1 and D2: photodetector, and OS: oscilloscope.

OS

in the presence of a dispersive phase coupling is written as vg = (δ ph /δωs )−1 = L/δt. By measuring the time delay or time advance δt experienced by the signal beam, one can obtain the group velocity of the signal beam. Another technique is to use a single light pulse as the signal beam [23, 26, 28, 29]. By measuring directly the time delay or the time advance δt experienced by the signal pulse with respect to a reference pulse propagating in air, one obtains the group velocity of the light pulse through the relationship vg = L/δt.

10.3.2 Slow Light Slow light was demonstrated in photorefractive materials such as BaTiO3 [23, 25, 26, 27], Bi12 SiO20 [24, 28], Sn2 P2 S6 , and CdTe [23, 27]. The group velocity varies from meters per second to hundreds of micrometers per second, depending on the photorefractive materials and the experimental conditions. Figure 10.8 shows typical steady-state temporal traces for the reference beam and the transmitted signal beam in a Bi12 SiO20 crystal using the intensity modulation technique. A time delay of 110 ms was detected for the transmitted signal beam in a 5.7-mm Bi12 SiO20 crystal, corresponding to an ultraslow group velocity of ≈ 0.05 ms−1 . From the viewpoint of practical applications, it is more interesting to slow down a single pulse while keeping its temporal profile of high fidelity. This was achieved experimentally when the pulse duration was much longer than or comparable to the response time of the two-beam coupling process. For short-duration pulses, serious temporal profile distortion was observed [23, 25, 28], in good accordance with the theoretical prediction. The group velocity of the light pulse was found to depend on the pulse duration [23, 28]. The light pulse decelerated with the increase of the pulse duration but finally leveled off to a minimal group velocity, as shown in Figure 10.9. This is because the photorefractive phase coupling is more effective with a longer pulse duration, especially when the pulse duration is

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F I G U R E 10.8. Steady-state temporal traces of the reference wave (dashed curves) and the transmitted signal wave (solid curves) with the pump beam off (a) and on (b), respectively, during a two-beam coupling in a 5.7-mm Bi12 SiO20 crystal. A 532-nm solid-state laser was used. The experimental conditions were a pump beam intensity Ip = 106 mW cm−2 , the pump to the signal intensity ratio ζ = 1500, an external field E 0 = 8 kV cm−1 , a grating spacing  = 21.3 μm, a periodic time of the signal beam T = 250 ms, and a frequency shift of the pump beam /2π = 50 Hz, respectively. From [24].

much less than the response time of the coupling process, but it finally saturates when the pulse duration is comparable to or much longer than the response time. We also note that the group velocity of a single pulse is larger than that in the quasi-cw case (for example, that measured by the intensity-modulation technique) even under similar experimental conditions [24, 28]. This may be due to the fact that the interaction duration between the signal and the pump in the single-pulse case is determined entirely by the pulse duration, while the photorefractive phasecoupling effect accumulates one pulse after another because of the memory effect of the photorefractive gratings and therefore is enhanced in the quasi-cw case. In addition, the photorefractive phase coupling is more complicated in the singlepulse case than that in the quasi-cw case because the frequency components of a single pulse are more complicated than those of a sinusoidal wave in the quasi-cw case. As a distinguished characteristic of the photorefractive phase-coupling-induced slow lights from those generated by other techniques such as EIT [2] and the

vg (m s−1)

100

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1 0

20 40 pulse duration (ms)

60

F I G U R E 10.9. A typical dependence of the group velocity of light pulses on the pulse duration measured in a Bi12 SiO20 crystal. The values for Ip , ζ , E 0 , , and /2π were set at 152 mW cm−2 , 1400, 8 kV cm−1 , 21.3 μm, and 60 Hz, respectively. From [28].

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1.0 0.0

0

40 80 Ω/2π (Hz)

vg (m s-1)

Γin(cm-1)

3.0

287

0.01 120

F I G U R E 10.10. Dispersion curves for in (connected circles) measured in the cw regime and vg (connected squares) measured using the sinusoidal intensity-modulation technique in a 5.7-mm Bi12 SiO20 crystal with Ip = 106 mW cm−2 at 532 nm,  = 21.3 μm, and E 0 = 8 kV cm−1 . The pump to the signal intensity ratio ζ was set to be 3000 and 1500 for the cases of in and vg , respectively. The reason to use different intensity ratios for the two cases is that the sinusoidally modulated signal beam is composed of two frequency components of equal intensities but with a slight angular frequency difference δωs . The dashed horizontal line indicates the value of the absorption coefficient α at 532 nm for the Bi12 SiO20 crystal. From [24].

quantum population oscillation effect [31], the slow light in photorefractive materials was observed to be amplified because of the unidirectional energy transfer from the pump beam [23, 24, 25, 26, 28]. This is clearly seen in Figure 10.10, which shows the measured dispersion curves of in in the cw regime and vg measured using the sinusoidal intensity-modulation technique under the same experimental conditions except that there is a difference of a factor of 2 in the pump-to-signal-intensity ratio ζ because the sinusoidal signal beam has two components with equal intensities. One might explain the slow light in photorefractive material as a result of the dynamic amplification of the signal pulse in the wave mixing. In fact, this is not the case. Odoulov et al. [27] proved experimentally that it is the strong dispersion of the photorefractive gratings instead of the intensity amplification that is of primary importance for the slow light. They observed a considerable deceleration of light pulses in crystals such as Sn2 P2 S6 and CdTe with nearly compensated space-charge fields due to electron–hole competition, in which the unidirectional photorefractive amplification effect is negligibly small.

10.3.3 Fast Light Fast light was demonstrated by Bo et al. [29, 30] in photorefractive Bi12 SiO20 crystals. A negative group velocity of −2.6 m s−1 was measured for a 90-ms Gaussian pulse propagating in a 5.7-mm Bi12 SiO20 crystal in the presence of a pump beam with an appropriate angular frequency shift  with respect to the signal pulse, as shown in Figure 10.11. The transmitted signal pulse was found to be amplified slightly with the pump beam on as compared with that without the pump beam. This indicates that the fast light is not a result of the energy transfer between the two coupling beams, which requires the attenuation of the

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0.4 40

50

60

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100 150 200

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F I G U R E 10.11. Temporal traces showing fast light propagation in a 5.7-mm photorefractive Bi12 SiO20 crystal. The dashed and the solid curves are for the reference and signal pulses, respectively. A 90-ms Gaussian pulse experienced a time advance of 2.2 ms when a pump beam was on. The pump beam was of 34 mW cm−2 at 532 nm and shifted in frequency by /2π = 2 Hz. The pump-to-signal-intensity ratio was 1400. The external field E 0 and the grating spacing  were 8 kV cm−1 and 21.3 μm, respectively. The insets show the leading and the trailing edges of the signal and the reference pulses. Both the leading and the trailing edges of the pulses were shifted in time by the same amount.

signal pulse. Such a time advance is due to the anomalous dispersion of the phase-coupling effect in the photorefractive two-beam coupling.

10.3.4 Tunability and Transition Between Slow and Fast Lights The group velocity of lights is able to be tuned to quite a large extent in photorefractive materials using the dispersion property of the wave-coupling process. It is evident from Figure 10.1 that the slope ∂ ph /∂ωs is negative first and then becomes positive but finally goes to negative again when one scans the angular frequency shift  of the pump beam. Therefore it is easy to demonstrate slow and fast lights in the same photorefractive material and thereby transitions between the slow and the fast light simply by scanning the angular frequency shift  of the pump beam. Bo et al. [30] demonstrated experimentally such transitions by scanning the pump frequency shift in a photorefractive Bi12 SiO20 crystal, as shown in Figure 10.12. Transitions between slow and fast lights were also demonstrated in photorefractive Bi12 SiO20 crystals at room temperature by tuning the pump intensity Ip or the external field E 0 , as shown in Figure 10.13. This is because the response rate and the coupling strength of the photorefractive two-beam coupling depend on experimental parameters such as the incident intensity and the external field. Therefore the subluminal and the superluminal spectral windows shift when one tunes the pump intensity or the external field, as can be seen clearly from Figures 10.4–10.5 and 10.12. At a specific angular frequency shift of the pump beam, one might observe transitions between slow and fast lights by tuning the pump intensity or the external field.

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F I G U R E 10.12. Group velocity tunability and transitions between slow and fast lights in a 5.7-mm photorefractive Bi12 SiO20 crystal realized by scanning the angular frequency shift of the pump beam. The solid and the empty circles were the results with Ip = 83.0 mW cm−2 and Ip = 19.5 mW cm−2 at 532 nm, respectively. The values of E 0 , Is , , and T for both cases were 8 kV cm−1 , 0.1 mW cm−2 , 21.3 μm, and 30 ms, respectively. The results were measured using the intensity-modulation technique. From [30].

10.3.5 Pulse-Broadening Effect The slow and fast light pulses keep their temporal profiles of high fidelity when the pulse durations are much longer than or comparable to the response time of the photorefractive two-beam coupling. However, the pulse will experience

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F I G U R E 10.13. Transitions between slow and fast lights by tuning the pump intensity Ip or the external field E 0 in a 5.7-mm photorefractive Bi12 SiO20 crystal. (a) Ip -dependencies of vg at /2π = 14 Hz (empty squares) and 62 Hz (solid squares), respectively. An external field E 0 of 8 kV cm−1 was applied cross the crystal. (b) E 0 -dependencies of vg at /2π = 2 Hz (empty squares) and 40 Hz (solid squares), respectively. The pump intensity Ip was set to be 19.5 mW cm−2 . The other parameters Is , , and T for both cases were 0.1 mW cm−2 , 21.3 μm, and 30 ms, respectively. The results were measured using the intensity-modulation technique. From [30].

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serious distortion or a broadening effect when short pulses [23, 28] or pulses with broad spectral bandwidth, for example, a rectangular pulse [25, 26], are used. This is because the spectral bandwidth of the light pulse is much broader than that of the subluminal or the superluminal spectral windows sub/super ; therefore the frequency components of the pulse are not uniformly delayed and attenuated/amplified. Such a pulse-broadening effect will not only degrade the measurement accuracy, but also impede slow and fast lights in photorefractive materials from practical applications such as optical buffers. Several techniques have been proposed to alleviate the pulse-broadening effect. Odoulov et al. [27] delayed a Gaussian pulse without a severe broadening effect using a pump pulse of identical temporal profile as that of the signal pulse. The pulse was delayed through a degenerate photorefractive two-wave mixing with a negligible intensity-coupling effect using electron–hole competition in Sn2 P2 S6 and CdTe crystals. Recently, Deng et al. [26] proposed a novel technique to solve the pulse-broadening effect and to enlarge the delay-time–bandwidth product of slow lights. The key in the technique is to simultaneously slow down all frequency components of the input pulse by the same amount using inhomogeneous broadening. The input pulse is first split into different spectral channels by use of a dispersive element such as a prism or grating. Each spectral channel is composed of a relatively long subpulse with its temporal duration and its center frequency determined by the inverse spectral resolution and the channel number of the dispersive element, respectively. These spatially separated subpulses are then delayed independently by the same amount of time using bandwidth-matched slow-light array elements, and are recombined finally using another dispersive element to produce the output pulse. In this way, a short pulse can be delayed by an amount of time much larger than the pulse duration without temporal profile distortion and broadening effect. As a proof-of-principle experiment, they tried to slow down a rectangular pulse in a photorefractive BaTiO3 :Ce crystal using multiple pump beams, in which each pump beam was tuned at different incident angle and frequency shift with respect to the center frequency of the rectangular pulse. Here the photorefractive crystal simultaneously worked as both a dispersive and a slow-light element. The pulse distortion was demonstrated to be alleviated to some extent, depending on the number of pump beams (spectral channel) used.

10.4 Slow and Fast Lights with a Stationary Refractive Index Grating The phase-coupling-induced slow and fast lights are based on the dynamic and moving refractive index gratings in photorefractive materials. In fact, stationary refractive index gratings can also be used to generate slow and fast lights by using its dispersion of the forbidden bandgap. Yeh et al. [32] demonstrated slow and fast lights using a stationary grating recorded in a photorefractive lithium niobate

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(LiNbO3 ) crystal. In this case, the effective group velocity is written as vg =

c nb

(k/2)2 − κ 2 cosh2 ξ L , sinh ξ L 2 2 (k/2) − κ cosh ξ L ξL

(10.9)

where L is the length of the volume index grating, k = 2n b ωs /c − 2π/ is the phase mismatch, ξ is equal to (κ 2 − (k/2)2 )1/2 , κ = πn 1 /λ is the coupling constant, and n 1 is the index modulation of the stationary grating. At the center of the forbidden gap (where k = 0), one observes fast light with an effective group velocity vg =

c κL . n b tanh κ L

(10.10)

The group velocity is approximately κ Lvg as κ L becomes very large. However, the light transmission is small because of the Bragg-reflection of the volume index grating. At the immediate vicinity of the band edges where ξ L = iπ and the transmission is unity, slow light is observed with an effective group velocity vg =

c π2 . nb κ 2 L 2 + π 2

(10.11)

Yeh et al. [32] obtained a group index n g = 7.5 in a stationary volume index grating with an index modulation n 1 = 2.1 × 10−5 recorded in a 3.5-cm photorefractive LiNbO3 crystal. Gao et al. [33] studied quantitatively the relationship between the group velocity of light and parameters such as the phase mismatch k and the diffraction efficiency η of the gratings in LiNbO3 crystals with different dopants (see Figure 10.14). They found that the effective group velocity of light is roughly proportional to the diffraction efficiency of the grating when the grating amplitude is small and the phase mismatch is fixed. Moreover, the competition between the photovoltaic effect and the diffusion during the grating formation as well as

F I G U R E 10.14. The dependence of the normalized group velocity of light vg /(c/n b ) on the diffraction efficiency η and the phase mismatch k. From [33].

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the competition between the chromatic and the phase gratings will have effects on the effective group velocity of light.

10.5 Prospects and Conclusions We would like to point out that the principle to generate slow and fast lights by dispersive phase coupling in nonlinear wave-mixing processes is of considerable generality. It is not limited to photorefractive process and is also applicable to other nonlinear wave-mixing processes [34]. The physics lies in the dispersive properties of the phase-coupling process is the finite response rate of the wavecoupling process. Therefore any nonlinear wave-coupling process with a finite response rate possesses a dispersive phase-coupling effect, which would lead to slow and fast lights. Research on slow and fast lights in photorefractive materials at the current stage are very elementary. Only simple pulses with quite narrow spectral bandwidths have been decelerated/accelerated while keeping their temporal profiles at the same time. For short pulses, the response time of the coupling process should be shorter than or comparable to the pulse duration. This could be achieved by increasing the incident intensity, but the slope ∂ ph /∂ωs becomes less steep; therefore a tradeoff between the duration and the group velocity of the light pulse should be made. The multiple-pump technique alleviates the pulse distortion effectively, but the system becomes complicated for pulses with broad spectral bandwidths. Another way to solve this dilemma is to use a photorefractive material with a larger coupling strength and a fast response rate. The photorefractive quantum well is a good candidate for short-pulse applications; it is known to possess a strong coupling strength of ≈103 cm−1 and a short response time on the order of microseconds even at the mW cm−2 intensity level in the near infrared [35, 36]. In such material systems pulses of microseconds with a group velocity on the order of centimeters per second is expected. However, for practical applications of slow and fast lights, a large delay-time-bandwidth product or advance-timebandwidth product of pulses may be necessary. Another important topic is the nonlinear optics of slow and fast lights. It is believed that the nonlinear effect is enhanced for slow light because the interaction time between the material and the light increases [2, 3]. Therefore a combination of slow light with other nonlinear effects such as frequency conversion in electrooptic materials may result in novel photonic devices. In conclusion, we have shown that photorefractive phase coupling is a highly dispersive process, which can be used to generate slow and fast lights. The group velocity of lights in photorefractive materials can be tuned to quite a large extent by scanning the angular frequency difference between the signal and the pump beams, or by tuning the experimental conditions such as the incident intensity and the external field. The pulse distortion problem in slow and fast lights has been discussed and techniques to solve this problem have been reviewed. Other

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methods to generate slow and fast lights in photorefractive materials have also been discussed.

References 1. P.W. Milonni: Fast Light, Slow Light and Left-Handed Light (Institute of Physics Publishing, Bristol and Philadelphia 2005) pp. 1–179. 2. L.V. Hau, S.E. Harris, Z. Dutton, C.H. Behroozi: Nature 397, 594 (1999). 3. M.M. Kash, V.A. Sautenkov, A.S. Zibrov, L.Hollberg, G.R. Welch, M.D. Lukin, Y. Rostovtsev, E.S. Fry, M.O. Scully: Phys. Rev. Lett. 82, 5229 (1999). 4. C. Liu, Z. Dutton, C.H. Behroozl, L.V. Hau: Nature 409, 490 (2001). 5. D.F. Phillips, A. Fleischhauer, A. Mair, R.L. Walsworth, M.D. Lukin: Phys. Rev. Lett. 86, 783 (2001). 6. A.V. Turukhin, V.S. Sudarshanam, M.S. Shahriar, J.A. Musser, B.S. Ham, P.R. Hemmer: Phys. Rev. Lett. 88, 023602 (2002). 7. E.L. Bolda, R.Y. Chiao, J.C. Garrison: Phys. Rev. A 48, 3890 (1993). 8. C.G.B. Garrett, D.E. McCumber: Phys. Rev. A 1, 305 (1970). 9. M.D. Crisp: Phys. Rev. A 4, 2104 (1971). 10. S. Chu, S. Wong: Phys. Rev. Lett. 48, 738 (1982). 11. S.E. Harris: Phys. Today 50, 36 (1997). 12. L.J. Wang, A. Kuzmich, A. Dogariu: Nature 406, 277 (2000). 13. M.D. Stenner, D.J. Gauthier, M.A. Neifeld: Nature 425, 695 (2003). 14. G.C. Valley: J. Opt. Soc. Am B 1, 868 (1984). 15. P. R´efr´egier, L. Solymar, H. Rajbenbach, J. -P. Huignard: J. Appl. Phys. 58, 45 (1985). 16. S.L. Sochava, E.V. Mokrushina, V.V. Prokofev, S.I. Stepanov: J. Opt. Soc. Am B 10, 1600 (1993). 17. S.I. Stepanov, V.V. Kulikov, M.P. Petrov: Opt. Commun. 44, 19 (1982). 18. A. Marrakchi, J.P. Huignard, P. G¨unter: Appl. Phys. 24, 131 (1981). 19. S.I. Stepanov, M.P. Petrov: Nonstationary Holographic Recording for Efficient Amplification and Phase Conjugation. In: Photorefractive Materials and Their Applications I, Vol 61, ed. by P. G¨unter, J.-P. Huignard (Springer-Verlag, Berlin Heidelberg New York 1988) pp. 263–289. 20. L. Solymar, D.J. Webb, A. Grunnet-Jepsen: The Physics and Applications of Photorefractive Materials (Clarendon Press, Oxford 1996) pp. 157–169. 21. G. Zhang, R. Dong, J. Xu: Chin. Phys. Lett. 20, 1725 (2003). 22. G. Zhang, R. Dong, F. Bo, J. Xu: Appl. Opt. 43, 1167 (2004). 23. E. Podivilov, B. Sturman, A. Shumelyuk, S. Odoulov: Phys. Rev. Lett. 91, 083902 (2003). 24. G. Zhang, F. Bo, R. Dong, J. Xu: Phys. Rev. Lett. 93, 133903 (2004). 25. Zh. Deng, Ph. R. Hemmer: Proc. SPIE 5362, 81 (2003). 26. Zh. Deng, D. Qing, Ph. Hemmer, C.H. Raymond Ooi, M.S. Zubairy, M.O. Scully: Phys. Rev. Lett. 96, 023602 (2006). 27. A. Shumelyuk, K. Shcherbin, S. Odoulov, B. Sturman, E. Podivilov, K. Buse: Phys. Rev. Lett. 93, 243604 (2004). 28. F. Bo, G. Zhang, J. Xu: Opt. Commun. 261, 349 (2006). 29. F. Bo, G. Zhang, J. Xu: TOPS 99, 386 (2005).

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F. Bo, G. Zhang, J. Xu: Opt. Express 13, 8198 (2005). M.S. Bigelow, N.N. Lepeshkin, R.W. Boyd: Science 301, 200 (2003). S.H. Lin, K.Y. Hsu, P. Yeh: Opt. Lett. 25, 1582 (2001). F. Gao, J. Xu, H. Qiao, Q. Wu, Y. Xu, G. Zhang: Opt. Commun. 257, 185 (2006). Q. Yang, J.T. Seo, B. Tabibi, H. Wang: Phys. Rev. Lett. 95, 063902 (2005). S. Balasubramanian, I. Lahiri, Y. Ding, M.R. Melloch, D.D. Nolte: Appl. Phys. B 68, 863 (1999). 36. M. Dinn, K. Nakagawa, M.R. Melloch, A.M. Weiner, D.D. Nolte: J. Opt. Soc. Am B 17, 1313 (2000). 30. 31. 32. 33. 34. 35.

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11 Holographic Filters Karsten Buse,1 Frank Havermeyer,2 Wenhai Liu,2 Christophe Moser2 and Demetri Psaltis3 1 2 3

University of Bonn, Institute of Physics, Wegelerstr. 8, D–53115 Bonn, Germany Ondax, Inc., 850 East Duarte Road, Monrovia, CA 91016, USA California Institute of Technology, Department of Electrical Engineering, Pasadena, CA 91125, USA

11.1 Introduction Fiber-optic networks are the backbone of Internet and telephone communication lines. Total internal reflection guides light in silica fibers. Low absorption makes transmission over 100 km without amplification possible in single-mode fibers. Distributed-feedback lasers with modulators put up to 10 and soon probably 40 Gbit/s onto a single wavelength. The signal is guided through the fiber, maybe repeated and amplified many times, until it reaches its destination. For about 10 years, wavelength-division multiplexing has been employed: In the transparency window of optical networks many waves of slightly different wavelengths are transmitted simultaneously in order to enhance the transmission capacity of a single fiber. These channels are closely packed. The separation between channels in the third optical window around 1550 nm is 50 or 100 GHz, i.e., approximately 0.4 or 0.8 nm. More than 128 channels in the telecom core networks are used, and bandwidths in single fibers exceed 1 Tbit/s. Erbium-doped fiber amplifiers that boost the energy of many different wavelength channels in a single step have made wavelength-division-multiplexing economically successful. Further improvements to flexible and cost-efficient networks require advanced optical components. Photorefractive crystals can be used to filter and switch light [1]. Thus there exist several potential areas of application in optical telecommunication for photorefractive crystals, as Figure 11.1 reviews. Filtering and separating light, wavelength-selective routing and switching of light, coupling of light in and out of fibers, as well as components that improve the emission characteristics of lasers are promising applications. Low light scattering, robustness, optical manufacturability, and low infrared absorption are desirable properties of photorefractive crystals such as, e.g., ironand copper-doped lithium niobate and lithium tantalate crystals, and such properties make them suitable for optical telecommunication. In this chapter we will focus on optical add-drop multiplexers for telecommunication networks. This became to the best of our knowledge the first application

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F I G U R E 11.1. Application areas of holographic elements in optical telecommunication networks. (A) Filtering: In wavelength-division-multiplexing networks one wavelength might be blocked; other wavelengths might be transmitted. (B) Interconnects: Channels of different wavelengths can be routed by a single holographic element that contains multiplexed holograms. (C) Coupler: Multimode or multicolor light can be coupled into a single mode fiber. (D) Laser feedback: Partial reflection of light with a high angular and wavelength selectivity can lock the wavelength, reduce the spectral bandwidth, and improve the spatial beam profile of diode lasers to a great extent.

of photorefractive crystals with a large sales volume. The idea to use holographic Bragg gratings for filtering of light is not new. Already in 1969, Kogelnik introduced the coupled-wave equations and showed the extraordinary wavelength selectivity of volume phase holograms [2]. In photorefractive crystals this was impressively demonstrated [3, 4, 5] and also proof-of-principle filters for wavelength-division networks were proposed [6, 7]. However, for a real telecommunication product many specifications must be fulfilled that include long lifetime even at enhanced temperature; resistance against humidity, temperature changes, and mechanical shocks; as well as low polarization dependence and small chromatic dispersion. We report in this chapter our efforts to solve these technical problems. First a general introduction about telecommunication networks will be provided. From this it becomes clear where optical filters and especially tunable filters are needed. Then the most important properties of volume reflection gratings will be reviewed, including in particular the effect of apodization. In the following section it is described how photorefractive iron-doped lithium niobate crystals were tailored to make this material suitable for recording of advanced wavelength filters. Here especially improvements in the lifetime of thermally fixed holograms were crucial. Then setups used for holographic recording as

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well as typical results of recording experiments will be presented. Various devices will be shown along with their specifications. Beam combiners and gratings for stabilization of laser diodes will finally be highlighted as emerging technologies.

11.2 Telecommunication Networks All telecommunication networks with speeds beyond 1 Gbit/s and reach beyond a couple of hundred meters generally require some kind of optical transport. Figure 11.2 illustrates the boundaries between technologies for the transport of information. There are many types of optical networks that differentiate themselves by their reach (i.e., distance without regeneration), carrier wavelengths, and the protocol to transfer the information. These different networks can be identified based on their geographical coverage. They can be classified into four categories that will be discussed in this section: 1. 2. 3. 4.

Local Area Networks (LAN) Access Networks Metropolitan Area Networks (MAN) Long-Haul and Ultra-Long-Haul Networks (LHN, ULHN)

Speed

OPTICS

Satellite

Radio

Reach

F I G U R E 11.2. Information transport technologies in terms of speed vs. reach. (MM: multimode, SM: Single mode, VCSEL: vertical cavity surface emitting laser, F-P: Fabry Perot laser, WDM: wavelength division multiplexing)

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F I G U R E 11.3. Example of a local area network with optical links.

After introducing these networks, we will describe one of the key enabling optical components for the emerging agile optical network: the reconfigurable optical add drop multiplexer (ROADM). The network closest to the end users is the local area network (LAN). LANs are networks within enterprises (businesses, university campuses). These types of networks handle mostly data. The Ethernet transmission protocol is used in the vast majority of LANs worldwide. Electrical copper-based Ethernet is used for very short reach (30–50 m) and low-speed (1–100 Mbit/s). Longer reach and higher speed is where optics becomes the preferred solution. A typical LAN in a campus environment shares many media of transport such as electrical wires (twisted pairs), wireless, and fibers, as Figure 11.3 illustrates. With the increasing remote connectivity of large businesses, the distinction between LANs, access, and MANs is becoming more and more blurry. Different optical transport technologies exist in LANs that cover distances from several meters to several kilometers and speeds from 1 Mbit/s to 10 Gbit/s: for example, 850-nm VCSELs (vertical-cavity surface emitting lasers) and multimode fibers can carry gigabit Ethernet traffic up to 550 m (standard IEEE 802.3z). Fabry-Perot 1310-nm lasers with single-mode fibers have a reach of typically 5 km and a speed of up to 10 Gbit/s by combining four wavelengths at the speed 2.5 Gbit/s (LX4) or eight wavelengths at 1.25 Gbit/s. The industry is moving toward low-cost solutions, for

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example by combining single-laser 10-Gbit/s modulation with electronic dispersion compensation (EDC). The access network links the LAN and the MAN. It covers approximately distances from 2 to 40 km. The main transport technology is with single-mode fiber and either 1310-nm or 1550-nm lasers at speeds from 2.5 Gbit/s to 10 Gbit/s. Metropolitan networks (MAN) play the role of LANs for cities: they interconnect city areas through access nodes. Traffic from access networks is interfaced at the nodes and retransmitted in the format for MAN. Transceivers are an important interfacing element. They act as a bridge between networks by performing an optical to electronic conversion of the information from the LANs, which is then retransmitted optically with the format and carrier wavelength used in the MAN. The metro network covers distances from 20 km up to 200–300 km. The optical carrier wavelength is in the range of 1550 nm and propagates in singlemode fibers. The spectral region near 1550 nm has the lowest loss per kilometer in the fiber and thus provides the longest reach for a given launch power. Some networks have implemented a standard called dense wavelength division multiplexing (DWDM) that boosts the transmission capacity by sending in parallel many high-speed channels, each on its own carrier wavelength. On average, 2 to 3 channels are deployed on a single fiber with spacing on a 100-GHz (≈ 0.8 nm) grid in the C-band (1525–1565 nm). The metro WDM system is scalable since it can be readily upgraded to 32–40 channels per fiber. At each node of the network, the wavelengths from each fiber are demultiplexed optically before conversion into electrical signals. Each MAN has a large head-end that connects it to other MAN or long-haul network nodes where 25% to 100% of the traffic may need to be routed (Figure 11.4). The network topology of the MAN today is static.

F I G U R E 11.4. Example of connectivity between the different levels of the network hierarchy.

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The wavelength path is set with optical filters at the time of network installation. Optical reconfiguration is needed to take away the lengthy reconfiguration (3–6 months) of many nodes by manual replacement of optical components. Agile networks are being designed to take advantage of remote provisioning of the optical path, thus removing the need for manual replacement at each node. Long-haul (LH) networks connect large cities and span many thousands of kilometers. The bandwidth per fiber is on the order of 1 Tbit/s. LH networks have over 100 channels, each modulated at 10 Gbit/s, and some are scheduled to be with 40-Gbit/s modulation. The channels are spaced spectrally by 50 GHz (≈ 0.4 nm) in the range from 1525 to 1625 nm. The laser sources are distributed feedback lasers stabilized in frequency with etalons. The range from 1525 to 1565 nm corresponds to the amplification region of erbium-doped optical amplifiers (EDFA). They amplify simultaneously all channels optically. This invention enabled the growth of high-capacity transmission in long-haul networks using DWDM in the late 90s. Ultralong-haul refers to long-haul systems with more than one thousand kilometers of transmission without electrical regeneration. Figure 11.4 illustrates the different network layers from long-haul to access-LAN. The nodes connecting long-haul to metro-core are opportunities to add and drop many wavelengths. Typically, 75% of the traffic goes through the nodes, and 25% is dropped and added. Optical add-drop modules (OADM) play an important role as they establish the wavelength connectivity of the network. Here we discuss an emerging OADM architecture that is being implemented in the metro-core by system integrators and allows from 0 to 100% add and drop capability. The architecture shown in Figure 11.5 is called “broadcast and select” [8, 9]. It can be upgraded for full remote reconfigurability. This architecture requires wavelength blockers, splitters, optical filters, and lasers: a small fraction of the power of the incoming traffic is tapped and split equally by a broadband splitter. All channels are broadcast to a receiver. An optical filter placed in front of the receiver selects the appropriate channel. The wavelength that is dropped can be added with new information modulating a laser at the corresponding carrier wavelength. However, the dropped wavelength must be removed from the main traffic path by a device called a wavelength blocker. The wavelength blocker ensures that there is no collision between dropped and added signals of the same wavelength. The number of add and drop channels can be scaled by simply adding receivers and transmitters. By replacing fixed filters and lasers by their broadly tunable version, every fiber or receiver has remote access to the whole set of wavelengths available. In this architecture, tunable filters and lasers enable full wavelength agility. A different architecture that also requires tunable filters arises in the nodes linking metro-core to access. In this case the (R)OADM handles a fraction of the wavelengths compared to the OADM for metro-core. A low-cost architecture for 4–8 drops consists of cascading hitless tunable 3-port filters. Hitless means that during the tuning process, the express wavelengths are at all times available. The holographic grating devices enable hitless tunable filters, as we will explain later in this chapter.

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F I G U R E 11.5. OADM (optical add-drop multiplexer) architecture that can be upgraded to reconfigurable OADM (ROADM) by exchanging the fixed optical filters and lasers by their tunable correspondent (“2-port filter”: input and drop channels are available; “3-port filter”: input, drop, and through channels are available).

11.3 Reflection Gratings The coupled-wave theory of Kogelnik shows the outstanding angular and wavelength selectivities of thick volume phase holograms that result from the Bragg condition [2]. The properties of such gratings are reviewed and described in detail in the literature [10, 11]. The so-called reflection geometry provides the best wavelength selectivity. Incoming light and diffracted light are counterpropagating. Figure 11.6 shows the refractive index modulation and the resulting diffraction efficiency versus the reading wavelength. The grating has a constant amplitude over the entire thickness. Because of the grating length and the large refractive index modulation, the diffraction efficiency approaches unity over a wavelength range around the Bragg wavelength λB , that is, for perfect antiparallel reflection given by λB = 2 n  ,

(11.1)

where  is the period length of the reflection grating and n the refractive index at the wavelength of the reading light. Please note that the diffraction efficiency is shown on a logarithmic scale. Sidelobes with about 1% diffraction efficiency (−20 dB) appear about 0.8 nm (≈100 GHz) away from the central Bragg wavelength, which is typical for volume holograms. A technique known to improve the

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F I G U R E 11.6. Reflection grating [12]. (A) Refractive index modulation n m versus spatial location. The grating is 20 mm long and the modulation amplitude is n = 2 × 10−4 . The period length of the grating is shown in the figure in a greatly expanded way. The real period length is about  = 0.35 μm. The refractive index of the material is about 2.2. (B) Resulting diffraction efficiency versus wavelength of the reading light for reflection geometry.

spectral profile of such reflection gratings is apodization, i.e., a spatial variation of the modulation amplitude of the refractive index changes [13, 14, 15, 16, 17]. Figure 11.7 shows an example [12]. It can be seen clearly that the sidelobes are strongly suppressed. Such a flat-top profile is highly desired for telecom applications. For about 0.1 nm around the Bragg wavelength the diffraction efficiency is close to unity. In metro networks, where the laser wavelength may drift, the grating will still work, and in long-haul lines where high-bandwidth data are put onto the optical beam the spectrally broadened beam will still be fully reflected. The low sidelobes of the flat-top profile make it possible to place in WDM networks

F I G U R E 11.7. Apodized reflection grating [12]. (A) Refractive index modulation n m versus spatial location. The grating is 20 mm long and the modulation amplitude is n = 2 × 10−4 . The period length of the grating is shown in the figure in a greatly expanded way. The real period length is about  = 0.35 μm. The refractive index of the material is about 2.2. (B) Resulting diffraction efficiency versus wavelength of the reading light for reflection geometry.

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many channels 100 GHz or even 50 or 25 GHz apart. These considerations already show that Bragg gratings are highly suitable for wavelength management in optical networks. Besides the diffraction efficiency, two other parameters must be taken into consideration for application of such gratings in telecommunication: The group-delay dispersion must be weak. At the edges of the flat-top-reflection profiles (Figures 11.6 and 11.7) where the diffraction efficiency is still high, a problem comes up that yields signal distortion: Part of the light is diffracted from the front, from the middle, and from the rear part of the grating. A short pulse, e.g., a single bit in a 10-Gbit/s data stream, will be broadened. For a 20-mmlong grating in a material with a refractive index of 2.2 this may cause a delay that distorts the signal. Apodization and slight chirping of the grating minimizes dispersion. The polarization dependence must be small because telecom fibers are not polarization maintaining. If the holographic gratings are stored in birefringent media, it will be necessary to stay close to the optical axis or to multiplex suitable gratings for ordinary and for extraordinary light polarizations. The polarizationdependent losses (PDL) are a key property of any finalized device.

11.4 Material Issues This section is dedicated to the recording material in which the filters can be fabricated holographically. We developed WDM filters based on photorefractive iron-doped lithium niobate crystals (LiNbO3 :Fe). It has already been shown in the literature that this material is suitable for realization of narrow-band wavelength filters [3, 4, 5, 6, 7]. In this book series the photorefractive mechanism in LiNbO3 is described in detail in the chapter “Photorefractive Effects in LiNbO3 and LiTaO3 ,” Volume:, Chapter:. Any component in telecommunication applications must withstand extended lifetime tests. For the photorefractive wavelength filters this implies that a fixing mechanism is required. In LiNbO3 :Fe the best fixing method that is known so far is “thermal fixing” [18, 19, 20, 21]: Recording at high temperatures (about 180◦ C) or heating of the crystals to high temperatures after recording causes a screening of the electronic space-charge field by ions that become mobile at elevated temperatures. After cooling, the mobility of the ions decreases, and the pattern is persistent against further illumination or against the usual electronic dark conductivity. The important point to understand is that a spatially modulated pattern of filled and empty electron traps, i.e., of Fe2+ and of Fe3+ in LiNbO3 :Fe, is stabilized by an ionic grating. On homogeneous illumination, photoexcitation of electrons from the filled sites yields a spatially modulated current that builds up electronic space-charge fields that modulate the refractive index through the electrooptic effect. Thermal fixing is described in more detail in the above-mentioned chapter. For filter applications, “high-low” recording is much better, since recording at higher temperatures allows one to redistribute many

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F I G U R E 11.8. Amplitude of the refractive index grating normalized to the initial value after recording vs. storage time at 85◦ C in the dark. A monoexponential fit to the data yields a hologram lifetime of about 11 years. Even in the worst case considering all measurement errors the lifetime will be 4 years. The crystal is doped with 0.05 wt. % Fe, and the Fe2+ /Fe3+ concentration ratio is about 0.08. By thermal annealing and spectroscopic control we ensured that the hydrogen concentration is below 4 × 1022 m−3 . The results shown are obtained for four different crystals from three different crystal suppliers. Every single data point shown is the statistical average over 4 to 10 individual gratings on one chip. The period length of the grating is 0.35 μm.

more ions than for heating after recording (“low-high-low” recording). Thus the refractive-index changes are much larger as well. In addition to the information given in the chapter mentioned above, here we will highlight how it was possible to improve the lifetimes of the fixed gratings further. The hologram decay of an optimized sample at 85◦ C is plotted in Figure 11.8. This underlines that thermal fixing can provide a hologram stability that is sufficient for telecommunication applications. The lifetime of thermally fixed holograms is determined by several factors. Under illumination, i.e., in the fully revealed state, the electronic grating follows the decay of the ionic pattern. If the amplitude of the ionic grating becomes smaller, the amplitude of the concentration pattern of filled and empty electron traps will decay as well. Consequently, the modulated currents, space-charge fields, and refractive-index changes will be also reduced. The ionic grating is not perfectly persistent because even at lower temperature some mobility of the ions remains present. They diffuse from the regions of high concentration to the regions of lower concentration. The electrons can follow easily since the crystal is illuminated. Considering diffusion of the ions as the limiting effect, the following equation describes the hologram lifetime [22, 23, 24]: 

 τ = Dion K

2

−1 Hion +1 , Neff

(11.2)

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where Dion is the ion diffusion coefficient, K = 2π/ is the length of the grating vector, Hion is the ion concentration, and Neff is the effective trap concentra−1 −1 −1 2+ tion, i.e., Neff = NFe and Fe3+ concentrations NFe2+ and 2+ + N Fe3+ with the Fe NFe3+ . For an oxidized crystal one can write Neff ≈ NFe2+ . From this it is evident what should be done to improve the hologram lifetime. The length of the grating vector K = 2π/ is determined in our application by the Bragg condition and the telecommunication wavelength region. The concentration Hion should be large enough to allow full compensation of the electronic pattern (Hion > Neff ), but otherwise should not be too large. The most important parameter is the ion diffusion constant Dion . If the crystal is kept in darkness after hologram recording, the decay is further delayed because diffusion of the ions builds space-charge fields up that decelerate further diffusion of ions. Over time, the electronic dark conductivity compensates part of this field and further ions can move. Thus besides the above-mentioned parameters now the electronic dark conductivity is also of relevance. Anyhow, the ion diffusion coefficient is still the most important coefficient that determines the hologram lifetimes. In practice, the filter is embedded into a device and we have to distinguish two operation modes: If the device is turned on, then electrical power is present in order to stabilize the temperature. In this mode light will be present. Besides the infrared light in the telecommunication wavelength region there is need for additional light in the visible spectrum in order to maintain the developed state of the thermally fixed holograms. This is done in our case by placing light-emitting diodes on top of the crystal. However, according to our measurements even if illumination is present, the lifetimes exceed 10 years because in this mode the crystal is close to room temperature. The second mode is that in which the device is turned off and no light is present. In this mode the unit may reach higher temperatures. This can happen during shipment or if the filter is installed in a hot place, but not switched on. Then it must survive temperatures of 85◦ C for more than 3 months. It was necessary to address this point before the gratings became ready for commercial use. As Usually hydrogen ions, H+ , are the species that in thermal fixing experiments compensate the electronic gratings [21, 25]. However, if the crystals are dehydrated by annealing at very high temperatures (around 1000◦ C), the hydrogen ions can be removed completely, and other ions with lower diffusion constant are responsible for the hologram fixing mechanism. In such crystals it is possible to obtain hologram lifetimes that clearly fulfill and even exceed the telecommunication requirements, as shown in Figure 11.8. Figure 11.9 shows the decay time constants of the holograms vs. the temperature. An Arrhenius fit to the data gives an activation energy of (1.31 ± 0.06) eV. Most likely, Li+ ions are then compensating the electronic patterns [26]. Thermal annealing on the one hand should reduce the hydrogen concentration as much as possible and on the other hand should establish a useful Fe2+ /Fe3+ concentration ratio. A multiple-step annealing process makes this possible.

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F I G U R E 11.9. Decay time constant τ of the holograms vs. crystal temperature. The circles show experimental data; the solid line is a fit of the Arrhenius equation to the data.

11.5 Holographic Recording In this section, we describe the holographic recording geometry and method as well as the readout geometry and optical methods associated with fiber coupling. The recording requirements will be analyzed. The recording geometry is shown in Figure 11.10. Two collimated beams crossing at an angle  in air interfere in the material. The interference pattern I (z) in the material can be written    2π I (z) = I0 1 + m cos z , (11.3)  √ where m = 2 I1 I2 /(I1 + I2 ) is the modulation depth of the interference pattern generated by the two beams with intensities I1 and I2 . The period length of the interference pattern is =

λ 2 sin 

(11.4)

in the case of unslanted gratings. Notice that the period is independent of the material’s index of refraction n.

F I G U R E 11.10. Recording is performed in transmission geometry and readout is done in reflection geometry (2: full angle between the recording beams; : angle between the reading beams).

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F I G U R E 11.11. Recording methods: (A) The classical holographic two-beam setup has the advantage that angles between the two beams can be easily changed. (B) The phase mask method generates two beams from the +1 and −1 order of diffraction. The advantage is stability during recording due to the small path distance between the mask and the holographic material.

The two collimated beams can be generated with a beam splitter and two mirrors or, alternatively, with a binary phase mask. Figure 11.11 illustrates the two methods. First-order diffraction from a phase mask with period length p generates two recording beams. Assuming that the incident beam is normal to the mask, equation (11.4) becomes  = p/2, the grating period is one-half the period of the phase mask and does not depend on the recording wavelength or the material index of refraction. In order to reduce phase fluctuations of the interfering beams, the phase mask method is preferable because of the proximity of the recording material to the phase mask. This is especially important with thermal fixing in photorefractive lithium niobate since the recording sample is kept at about 180◦ C (“high-low” thermal fixing) [18, 21]. Several parameters have to be controlled in order to record wavelength filters for WDM applications. We will describe the parameters that influence each specification of the filters: The center wavelength of the WDM filter is specified within ±3 GHz, which corresponds to an accuracy of ±0.024 nm with respect to the filter wavelength. Assuming the symmetric recording geometry shown in Figure 11.11B, the center wavelengths of recorded filters are determined by

r the thermal expansion ac (during recording at high temperature and readout at room temperature),

r the angle  between the recording beams (see Figure 11.10) or equivalently the period of the mask,

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r the angle  between the readout beams (see Figure 11.10), and r the index of refraction n in the readout spectral region. The relationship between the filter center wavelength λB and the parameters above is given by (assuming no slanted grating) λB = n cos(˜ /2)

pH , 1 + ac T

(11.5)

where ˜ is the full angle between the incident and diffracted beams inside the material, pH is the period of the mask at the recording temperature, and T is the temperature difference between recording and readout. The dispersion characteristic of lithium niobate for ordinarily polarized light with wavelengths between 1510 and 1580 nm has been measured. For 30◦ C a linear relation can be used for extrapolation: n(λ) = 2.26326 − 3.31412 × 10−5 λ[nm] .

(11.6)

With today’s phase mask technology, the mask’s period accuracy is ≈ 5 ppm. The absolute recording temperature is 180◦ C and stability during the recording is better than 0.1◦ C. Both, the phase mask and the photorefractive crystal, are kept at that temperature in a custom-made oven. Experimentally, the center wavelength of the finally manufactured fiber-coupled filter falls within the targeted window of ±3 GHz, as Figure 11.12 shows. The isolation of WDM filters needs to be as high as possible in order to minimize the contributions from neighboring channels. Typically an isolation of 25 dB at 87.5 GHz from the center ITU (International Telecommunication

F I G U R E 11.12. Deviation of the center frequencies of 45 volume holographic gratings from the standard frequencies that are specified by the ITU (International Telecommunication Union) versus the absolute center frequencies. The 45 gratings are spatially multiplexed in a single photorefractive crystal chip.

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Union) wavelength is necessary for 100 GHz spacing requirements. To achieve this kind of isolation, the gratings have to be apodized as described above: The grating strength n (amplitude of the refractive index modulation) has to vary along the K-vector direction of the grating. For reflection gratings the grating strength is also the dictating parameter for the spectral bandwidth and chromatic dispersion of the filter. In holographic recording, the strength along the grating can be controlled, and its amplitude as a function of the grating position determines the bandwidth, isolation, and chromatic dispersion of the resulting filter. For thermally fixed gratings in lithium niobate, the strength of the index change can be controlled by the exposure energy. For 0.05 wt. % iron-doped LiNbO3 with a reduction ratio of cFe2+ /cFe3+ = 0.07 the maximum refractive index amplitude is approximately 4 × 10−4 for infrared light in the telecommunication range. The spatial variation of the grating strength can be realized by placing an amplitude mask or movable shutters in front of the phase mask, thereby controlling the exposure energy locally. An alternative method is to build the apodization function in the phase mask by changing the etching depth of the grating, thereby modulating the zero-order (the nondiffracted beam), which in turn changes the modulation depth in equation (11.3). Figure 11.13 shows experimental results (Figure 11.13A, apodization profile; Figure 11.13B, resulting spectral properties). A phase mask with 41 periods has been fabricated, and thermally fixed filters have been recorded. The graphs in Figure 11.14 show the amplitude filter shape.

11.6 Devices Volume holographic gratings (VHGs) compete especially with Bragg gratings that are recorded directly into fibers (“fiber Bragg gratings,” FBGs). However, the volume approach has the advantage that it is possible to smoothly disengage from a particular filter by moving the beam off the grating (and possibly onto another grating), whereas this is impossible in a FBG. Furthermore, utilizing free-space optics allows one more easily to separate or to merge beams. And for applications outside the telecom world, free-beam optics is still very important, e.g., for high-power diode lasers. In what follows we will describe the performance that was measured for four different device types that were fabricated with VHGs. These examples demonstrate the unique advantages of the 3-D construction of VHGs.

11.6.1 Notch Filters A beam collimated by a standard dual-fiber collimator propagates through the grating of Figure 11.15. The light is incident on the grating at the angle /2 with respect to the grating wave vector. The reflected wavelength λB is expressed by equation (11.5). The reflected light makes an angle −/2 with respect to the grating vector. When  is the angle between the beams of the dual fiber collimator, the reflected

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F I G U R E 11.13. (A): Amplitude of the refractive index modulation versus grating length and (B) diffraction efficiency for the reflected light for gratings without and with apodization.

light is coupled in the dual fiber collimator. An example of the notch filter performance is shown in Figure 11.16 A on the left side. The filter has an insertion loss of 1 dB from fiber to fiber. No circulators are used and no waveguides are manufactured. The dual collimator technique is also used with thin-film filters, but is not possible for FBGs. The VHG notch filter exhibits > 50 dB crosstalk suppression

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F I G U R E 11.14. (A): Filter response curves for 41 channels with 100 GHz spacing spanning the C-band and (B) details for the channel centered at 192.9 THz. The shaded regions indicate the 25-GHz-wide pass-band regions for this channel and the two adjacent channels.

at the center of the notch filter, a value that is three times better than that of thinfilm filters. This remarkably high suppression is a consequence of the 3-D nature of the filter. Shown in Figure 11.16B is the filter shape obtained when the transmitted light is directed to a detector instead of being coupled to a fiber. The additional suppression that is measured with coupling into a fiber is attributed to the spatial distortion of the through (express) beam near the Bragg-matched wavelength.

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Bragg grating ε/2

F I G U R E 11.15. Light coming in one fiber is coupled out with the help of a “dual fiber collimator.” A Bragg grating reflects light that matches the Bragg condition. This light is coupled back into another fiber using the same collimator. The full angle between incident and reflected light is .

(A)

(B)

F I G U R E 11.16. Notch filter insertion loss (A) and express light power (B) versus center frequency deviation in a semi-logarithmic plot (dB scale).

11.6.2 Ultrahigh-Isolation Add–Drop Filters The same VHG can be used twice by accessing with a pair of dual collimators as Figure 11.17 illustrates. A more advanced and advantageous double-pass configuration uses a mirror instead of one of the dual-fiber collimators shown in Figure 11.18. In this case the diffracted light travels twice through the same grating area and a circulator is used to separate the drop beam from the input beam. The main advantages of this configuration are: (a) Only one collimator has to be aligned and glued into place. (b) A smaller grating area is needed for each channel. (c) Polarization-dependent losses and polarization-mode dispersion are significantly reduced if a Faraday rotator mirror is used. However, the disadvantages are also obvious: Higher part count and cost as well as higher insertion losses, mainly due to the circulator, are expected. The data in Figure 11.19 A show that this mode reduces the side lobes considerably with little penalty on the bandwidth. Isolation of 25 dB at 50-GHz spacing is routinely achieved. Graphs of the polarization-mode dispersion (PMD) and polarization-dependent loss (PDL) of the filter are shown in Figure 11.19B and C for the double-pass filter.

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F I G U R E 11.17. Double reflection from a Bragg grating increases the isolation. Input

Drop

Circulator Dual-fiber collimator

Bragg grating

Faraday rotator mirror

F I G U R E 11.18. Advanced setup for double reflection from a Bragg grating. A Faraday mirror reduces all polarization-dependent effects.

11.6.3 Integration of Multiple Volume-Bragg Gratings and Tunable Filters Multiple gratings can be fabricated on the same chip as shown in Figure 11.20. The period of each grating can be tailored independently during manufacturing. We have fabricated arrays with up to 41 integrated VHGs with 100-GHz center frequency separations. The maximum deviation of the filter center frequency is <3 GHz or 0.024 nm. Figure 11.14 shows the intensity filter shapes of the array. The graph of Figure 11.14 was measured by sliding a collimator in front of the filter chip as Figure 11.20 illustrates. By positioning the collimator along the side of the chip, any filter can be selected. The grating-vector-angle homogeneity across the chip is better than 0.001 degrees, which allows the collimator to be coupled to any of the VHGs and to maintain alignment with all 41 VHGs. The bandwidth of each individual filter can be varied as well, allowing coarse WDM and DWDM to coexist on the same chip. The device is called a “reconfigurable optical-add-drop multiplexer” (ROADM). For different applications in optical networks so-called two-port, three-port, and four-port devices are required, the functionalities are shown in Figure 11.21. A “two-port device” can separate one wavelength channel, but the other light is

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(B)

(C)

F I G U R E 11.19. (A) Insertion loss versus frequency deviation for single-pass and doublepass filters. (B) Polarization mode dispersion (PMD) and (C) polarization-dependent losses (PDL) versus frequency deviation for a double-pass filter configuration as shown in Figure 11.18. THROUGHPUT Λ1

ADD

Dual-fiber collimator INPUT

DROP Hitless tuning

Lithium-niobate crystal with volume holographic filters

F I G U R E 11.20. Light is coupled out of a fiber with the help of a “dual-fiber collimator” (INPUT). A Bragg grating (period length 1 ) reflects light that matches the Bragg condition. This light is coupled back into another fiber using the same collimator (DROP). If a second dual-fiber collimator is present, the transmitted light can be collected (THROUGHPUT) or light at the Bragg wavelength can be added (ADD). Tuning can be done by moving the collimators to another grating. By movement through homogeneous grating-free material it can be avoided that during tuning some of the channels are blocked (“hitless tuning”).

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F I G U R E 11.21. Functionalities of “two-port,” “three-port,” and “four-port” add-drop wavelength filters.

lost. In a “three-port” device the through light is coupled into another fiber. And in a “four-port” device light at the wavelength of the removed channel can be added with new information onto the throughput fiber. In the “three-port” and “four-port” devices tuning from one channel to another should be performed without interrupting the traffic on the through channel. Any continuously tunable device will hit the wavelength channels in between and block the transmission for a while. In a volume approach we can overcome this problem easily, as is shown in Figure 11.20: The gratings exist only in the top portion of the crystal chip. For hitless tuning to another channel, first the dual-fiber collimators are moved out of the grating region; then they are moved through homogeneous grating-free material to the position below the next selected Bragg-grating, and afterwards the collimators are moved into the active region. Thus during tuning no wavelengths are blocked. Figure 11.22 shows the continuous increase of the

F I G U R E 11.22. Drop losses versus reading wavelength and reading position for a volume holographic filter that exists only in part of the filter material. Such continuous disengagement of a filter is useful for hitless tuning through grating-free material to another filter wavelength.

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drop losses if one moves out of the grating region in a filter where the grating is limited to a section of the filter material. The VHG array can be also coupled to an array of fiber collimators. Fiber collimator arrays have been developed for packaging cross-connects based on micro-electromechanical mirrors (MEMs). The packaging is therefore done in a single step by alignment of two blocks: the first block is an array of integrated VHGs and the second block is an array of packaged collimators. The cost and size per packaged filter is reduced dramatically. The array of fibers can be connected to a 1 × N switch to provide 1 by N wavelength connectivity. This device is a tunable filter, also called a channel selector.

11.6.4 Beam Combiners and Laser Diode Stabilizers VHGs can also find applications in other areas besides the tunable filters highlighted so far. One particularly interesting application is the use of holographic filters in conjunction with semiconductor lasers. VHGs can be multiplexed at the same spot in the material, and each VHG can be recorded with different grating vector orientation. Clearly this is not possible in single-mode fiber gratings and very difficult to accomplish with thin-film filter technology. Beam combiners based on VHGs take advantage of this capability to combine several wavelengths into a monomode or multimode fiber. Figure 11.23 shows eight integrated VHGs to combine the light from eight semiconductor lasers operating at different wavelengths anywhere from 1300 to 1650 nm. Different wavelengths are required to get efficient beam combination because otherwise the light diffracted into the center beam is Bragg-matched to all gratings, and thus the gratings will drain part of the light out of the main beam and send it back to other laser diodes. VHGs are selective mirrors that can be used for frequency-selective optical feedback. Figure 11.24 shows the working principle. A Bragg grating provides a highly wavelength-selective and angular-selective feedback. Reflectivities of maybe 10% already influence greatly the modes inside the diode laser. Figure 11.25 shows an example from the literature that illustrates the spectral and spatial improvements of diode laser light with the help of Bragg-selective feedback [27]. λ1

λ1, . . . , λ n

λn

F I G U R E 11.23. Eight spatially multiplexed gratings combine the output of 8 laser diodes of different wavelengths λ1 , . . . , λn with n = 8 into one diffraction-limited beam.

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(B)

Normalized intensity

(A)

−6 0 6 Far-field angle (mrad)

Normalized intensity

F I G U R E 11.24. Light from a high-power semiconductor laser diode is partially reflected by a Bragg grating. This feedback locks the emission wavelength, reduces the spectral bandwidth, and improves the spatial beam profile.

967 972 977 Wavelength (nm)

F I G U R E 11.25. Normalized laser intensity versus far field-angle (A) and wavelength (B) for a freely running laser diode (dashed lines) and a laser diode with Bragg feedback (solid lines) [27].

This application is of interest for improved pumping of fiber lasers in telecom networks. This is done, e.g., in the near infrared in the range 800 to 950 nm. The grating feedback yields three substantial advantages: 1. The emission wavelength is locked regardless of operating conditions and temperature of the diode laser. Typically, lasers operating in the range 800 to 950 nm can be locked up to 4 nm away from the emission wavelength of the freely running diode. This ensures that the emission is matched with the most effective pump wavelength of the fiber or solid-state laser. 2. Reduction of the spectral bandwidth to typically 0.2 nm. This ensures that most of the power is within the pump absorption band of the fiber or the solid state laser. 3. The spatial beam profile is improved, i.e., the beam divergence is reduced. Thus the light can be focused more effectively onto the fiber or solid-state laser material. This method can be used to improve a diode-laser-pumped solid-state laser system and is thus very general and has a large market volume.

11.7 Summary and Outlook There is strong demand for highly wavelength-selective components in telecommunication as well as in other areas of applied optics. Volume holographic gratings allow us to build free-space optical systems to merge or separate beams. This

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versatility makes them suitable for many technical solutions. Apodization allows one to tailor the grating properties. Based on photorefractive LiNbO3 crystals, especially reconfigurable optical add-drop multilexers have been developed and commercialized. A key issue here was the material improvement, especially with regard to the persistence of the gratings. Other applications include notch filters and laser-beam feedback mirrors. The technology is clearly appreciated by industry. Besides the telecom applications, especially high-power diode-pumped solid-state laser systems can profit from the volume holographic gratings. It is foreseeable that this will become a standard application of Bragg gratings. Photorefractive LiNbO3 will have to compete with other materials, especially inorganic glasses. For high-volume applications the cost is the main driving factor. Furthermore, the strength of photorefractive crystals is their dynamic behavior, i.e., the possibility to erase and rewrite holograms. Fixation of holograms is possible, as we demonstrated here with the method of thermal fixing. One drawback is that light-emitting diodes are required in order to keep the holograms developed. Materials that inherently yield fixed gratings have mid- and long-term advantages over photorefractive crystals for applications where static optical components are required, as is the case for fixed filters. Here inorganic glasses already have shown a very strong performance and will join LiNbO3 in a complementary way for further spread of volume holographic filters in applied optics.

References 1. P. Boffi, D. Piccinin, and M.C. Ubaldi. Infrared Holography for Optical Communications. Springer, Berlin, 2003. 2. H. Kogelnik. “Coupled Wave Theory for Thick Hologram Gratings,” Bell Syst. Tech. J. 48, 2909 (1969). 3. G.A. Rakuljic and V. Leyva. “Volume Holographic Narrow-Band Optical Filter,” Opt. Lett. 18, 459 (1993). 4. V. Leyva, G.A. Rakuljic, and B. O’Conner. “Narrow Bandwidth Volume Holographic Optical Filter Operating at the Kr Transition at 1547.82 nm,” Appl. Phys. Lett. 65, 1079 (1994). 5. R. M¨uller, M.T. Santos, L. Arizmendi, and J.M. Cabrera. “A Narrow-Band Interference Filter with Photorefractive LiNbO3 ,” J. Phys. D: Appl. Phys. 27, 241 (1994). 6. S. Breer and K. Buse. “Wavelength Demultiplexing with Volume Phase Holograms in Photorefractive Lithium Niobate,” Appl. Phys. B 66, 339 (1998). 7. S. Breer, H. Vogt, I. Nee, and K. Buse. “Low-Crosstalk WDM by Bragg Diffraction from Thermally Fixed Reflection Holograms in Lithium Niobate,” Electron. Lett. 34, 2419 (1999). 8. J. Bayne and M. Sharma. “Broadcast-and-Select OADM Enables Low-Cost Transparency,” Lightwave, December 2001. 9. M. Vasilyev, I. Tomkos, M. Mehendale, J.K. Rhee, A. Kobyakow, M. Ajgaonkar, S. Tsuda, and M. Sharma. “Transparent Ultra-Long Haul WDM Networks with “Broadcast-and-Select” AODM/OXC Architecture,” Journal of Light wave Technology 21, 2661 (2003).

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10. J. Goodman. Introduction to Fourier Optics. McGraw-Hill, 2000. 11. A. Yariv. Optical Electronics in Modern Communications. Oxford University Press, 1997. 12. I. Nee. Untersuchung und Optimierung thermisch fixierter Volumenphasenhologramme in photorefraktiven Lithiumniobat-Kristallen f¨ur das Wellenl¨angenmultiplexing. Shaker Verlag, 2001. 13. M. Matsuhara and K.O. Hill. “Optical-Waveguide Band-Rejection Filters: Design,” Appl. Opt. 13, 2886 (1974). 14. P.S. Cross and H. Kogelink. “Sidelobe Suppression in Corrugated-Waveguide Filter,” Opt. Lett. 1, 43 (1977). 15. B. Malo, S. Th´eriault, D.C. Jonson, F. Bilodeau, J. Albert, and K.O. Hill. “Apodizes In-Fiber Bragg Grating Reflectors Photoimprinted using a Phase Mask,” Electron. Lett. 31, 223 (1995). 16. J. Albert, K.O. Hill, B. Malo, S. Th´eriault, F. Bilodeau, D.C. Jonson, and L.E. Erickson. “Apodisation of the Spectral Response of Fibre Bragg Gratings using a Phase Mask with Variable Diffraction Efficiency,” Electron. Lett. 31, 222 (1995). 17. A. Inoue, T. Iwashima, T. Enomoto, S. Ishikawa, and H. Kanamori. “Optimization of Fiber Bragg Grating for Dense WDM Transmission System,” IEICE Trans. Electron E81-C, 1209 (1998). 18. J.J. Amodei and D.L. Staebler. “Holographic Pattern Fixing in Electro-Optic Crystals,” Appl. Phys. Lett. 18, 540 (1971). 19. A. Yariv, S. Orlov, G. Rakuljic, and V. Leyva. “Holographic Fixing, Read-out, and Storage Dynamics in Photorefractive Materials,” Opt. Lett. 20, 1334 (1995). 20. A. Yariv, S.S. Orlov , and G.A. Rakuljic. “Holographic Storage Dynamics in Lithium Niobate: Theory and Experiment,” J. Opt. Soc. Am. B 13, 2513 (1996). 21. K. Buse, S. Breer, K. Peithmann, S. Kapphan, M. Gao, and E. Kr¨atzig. “Origin of Thermal Fixing in Photorefractive Lithium Niobate Crystals,” Phys. Rev. B 56, 1225 (1997). 22. B.I. Sturman, M. Carrascosa, F. Agull´o-L´opez, and J. Limeres. “Theory of HighTemperature Photo refractive Phenomena in LiNbO3 Crystals and Applications to Experiment,” Phys. Rev. B 57, 12792 (1998). 23. L. Arizmendi, E.M. Miguel-Sanz, and M. Carrascosa. “Lifetimes of Thermally Fixed Holograms in LiNbO3 :Fe Crystals,” Opt. Lett. 23, 960 (1998). 24. E.M. Miguel-Sanz, M. Carrascosa, and L. Arizmendi. “Effect of the Oxidation State and Hydrogen Concentration on the Lifetime of Thermally Fixed Holograms in LiNbO3 :Fe,” Phys. Rev. B 65, 165101 (2002). 25. H. Vormann, G. Weber, S. Kapphan, and E. Kr¨atzig. “Hydrogen as Origin of Thermal Fixing in LiNbO3 :Fe,” Solid State Commun. 40, 543 (1981). 26. I. Nee, K. Buse, F. Havermeyer, R.A. Rupp, M. Fally, and R.P. May. “Neutron Diffraction from Thermally Fixed Gratings in Photorefractive Lithium Niobate Crystals,” Phys. Rev. B 60, R9896 (1999). 27. S. Yiou, F. Balembois, P. Georges, and J.-P. Huignard. “Improvement of the Spatial Beam Quality of Laser Sources with an Intracavity Bragg Grating,” Opt. Lett. 28, 242 (2003).

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12 Neutron Physics with Photorefractive Materials Martin Fally, Christian Pruner, Romano A. Rupp, and Gerhard Krexner Fakult¨at f¨ur Physik, Institut f¨ur Experimentalphysik, Universit¨at Wien, ¨ A-1090 Wien, Osterreich http://nlp.exp.univie.ac.at, [email protected], (V: 11th May 2005)

When the subject of photorefractive effects began with the discovery of lightinduced refractive-index inhomogeneities in lithium niobate [1], neutron optics had already been established for more than 20 years [2, 3]. Both of the fields have evolved independently of each other into important branches of science and industry. In 1990 those dynamic areas were linked by an experiment in which cold neutrons were diffracted from a grating created by a spatially inhomogeneous illumination of doped polymethylmethacrylate (PMMA). A typical holographic two-wave mixing setup was used to record a refractive-index pattern, a grating, in PMMA that was reconstructed not only with light, as usual, but also with neutrons [4]. Evidently, the illumination induced refractive-index changes for both light and neutrons! In analogy to light optics this phenomenon is called the photo-neutron-refractive effect. The chapter is organized as follows: Starting with a concise explanation of the relevant concepts in neutron optics, electrooptics, and photorefraction, as well as diffraction phenomena, we introduce PMMA and the electrooptic crystal LiNbO3 as examples of photo-neutron-refractive materials. The main part is concerned with neutron diffraction experiments performed on deuterated PMMA (d-PMMA). It is shown that this type of experiment can be useful in studying the polymerization process itself, when serving simply as a neutron-optical element, or when probing fundamental properties of the neutron. The latter is in particular true of electro neutron-optic LiNbO3 , where the diffracted neutrons are inherently exposed to extremely high electric fields due to the light-induced charge transport. Corresponding experiments are presented and future perspectives of photo-neutron-refractive materials as well as their possible applications are discussed. Finally, atomic-resolution neutron holography is introduced and conducted experiments are presented.

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12.1 Basic Concepts This part will provide the necessary prerequisites in photorefraction, holography, neutron optics, and diffraction physics to be able to understand the experiments and the obtained results. First, we will present the technique for preparing the gratings, introduce the equation of motion for neutron diffraction, define the neutron-optical potential and the neutron-refractive index, and finally discuss the relevant terms of the neutron-optical potential, which is modulated by inhomogeneous illumination with light.

12.1.1 Holographic Gratings Soon after the discovery of photorefraction [1], the technological importance of the effect became clear, e.g., that such materials can be used for information storage and as holographic memories [5]. The big advantage over materials changing their absorption, like standard photographic films, is that intensity losses do not occur, and thus the whole volume rather than the surface may be used for data storage. Important consequences when utilizing thick-volume phase gratings in diffraction experiments are that a sharp Bragg condition must be obeyed and that multiple scattering effects must be taken into account, i.e., dynamical diffraction theory has to be considered. Historically, the latter was originally developed for X-rays by Darwin, Ewald, and von Laue at the beginning of the twentieth century and extended in several review articles (see, e.g., [6]). When lasers became available and the technique of holography had become popular, Kogelnik reinvestigated the effects of coupled waves for light [7]. Finally, when crystals of highest quality and thickness could be produced as a consequence of semiconductor technology, Rauch and Petrascheck performed this task for neutrons [8]. We will summarize the results of this theory in so far as they are necessary to interpret our experiments correctly. As a first step, we will discuss the typical setup for the preparation of lightinduced refractive-index gratings, which is sketched in Figure 12.1. Two coherent plane light waves interfere in the photo-neutron-refractive material (two-wave mixing). In the simplest case with waves of equal intensity and mutually parallel 1/2 I(i) Is

I(i) θ

[e]

2Θ[e] s I0 z

Λ 1/2 I(i) d

x

F I G U R E 12.1. Sketch of the setup for the preparation of light-induced refractiveindex gratings (hologram recording) and the reconstruction with light or neutrons. I (i) denotes the incoming intensity, for further abbreviations, see text.

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polarization states, the resulting modulation of the light pattern is sinusoidal I (x) ∝ cos(K x) with an elementary grating spacing =

2π λ0 = . K 2 sin (θ [e] )

(12.1)

Here 2θ [e] denotes the angle between the interfering beams in air and λ0 the wavelength of light in vacuum. All angles are measured in the medium unless indicated by the superscript [e]. Typical values for the grating spacing are 300 nm <  < 2000 nm. In general, this inhomogeneous illumination of the sample results in a spatially dependent refractive-index change that can be expanded in a Fourier series with a fundamental periodicity K :  n s cos (s K x + φs ). (12.2) n(x) = s

The actual pattern of course depends on the properties of the material and the mechanism of photo-neutron-refraction. Usually in electrooptic crystals a linear response, i.e., n(x) ∝ I (x), n s>1 ≡ 0, is obtained though it may be nonlocal (φ1 = 0). This is not always the case: e.g., the photorefractive effect in PMMA depends on illumination time and intensity and is strongly nonlinear as will be shown in Section 12.3.5. Diffraction from such refractive-index gratings, which represent thick holograms [9], is governed by the basic formulae of dynamical diffraction theory. In our experiments we deal only with nearly lossless dielectric gratings in transmission geometry. To gain information about the material parameters, light or neutrons are diffracted from those gratings in the vicinity of the sth-order [e] Bragg angle s = arcsin[sλ0 /(2)]. In particular, the diffraction efficiency η(θ )s = Is /(Is + I0 ) is measured as a function of the deviation from s , the so-called rocking curve, where Is is the sth diffraction order. In the standard case where only zero-order (forward diffraction) and first-order diffracted waves are present, this reduces to the well-known definition [10]. For a monochromatic plane wave in symmetric transmission geometry the diffraction efficiency is described by [7]   2 2 2 2 νs + ξs (θ) , ηs (θ, ν) = νs sinc (12.3) πn s d , (12.4) λ0 cos (s ) sπ(s − θ)d s K (s − θ)d = . (12.5) ξs (θ ) = 2  The thickness of the grating, which in the ideal case is identical to the sample’s thickness, is denoted by d. The parameters νs (grating strength) and ξs (Off Bragg parameter) contain the relevant material information. In particular, the light-induced refractive-index change n s of order s can be probed and determined νs =

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by such a measurement. If any of those parameters appears without a numerical subscript we assume the case of a linear response, i.e., only the first-order diffraction is present and we simply speak about diffraction efficiency η. Then n 1 , the first-order Fourier coefficient in the expansion in (12.2), is frequently replaced by n in the literature. Another useful measure is the integrated diffraction efficiency (i.e., integrated reflectivity times ν)   ν 2ν d +∞ η(θ, ν) dθ = J0 (u) du, (12.6) I (ν) :=  −∞ 2 0 with J0 (u) the zeroth-order Bessel function. The practical importance of I (ν) lies in the fact that this quantity is independent of the lateral divergence of the beam and that it can be accessed experimentally by performing rocking curves (see Section 12.3.3).

12.1.2 Neutron Optics The equation of motion for a field of nonrelativistic particles, e.g., cold neutrons ˚ < λ < 50 A, ˚ is Schr¨odinger’s equation. We restrict our with a wavelength 5 A considerations to coherent elastic scattering (neutron optics) in condensed matter. Then the coherent wave and the coherent scattering are described by a one-body Schr¨odinger equation with the (time-independent) neutron-optical potential V (x) [11], the energy eigenvalues E, and m the neutron mass: H (x) = E (x), (12.7) h¯ 2 2 ∇ + V (x). (12.8) H=− 2m For a general treatment of neutron optics see, e.g., [11, 12]. Inserting (12.8) in (12.7) leads to a Helmholtz-type equation, with k0 being the magnitude of the vacuum wave vector, if we properly define the refractive index for neutrons nN : [∇ 2 + (n N (x)k0 )2 ] (x) = 0, 

(12.9)

V (x) . (12.10) E It is evident that any change in the potential of matter results in a refractiveindex change for neutrons. In our particular context the main task is to modify V (x) by illumination with light, i.e., to observe a photo-neutron-refractive effect. Therefore it is necessary to discuss the relevant terms of the neutron-optical potential. Interactions between the neutron and condensed matter may be classified pragmatically into three groups according to their magnitude [13, 14]: The strong interaction dominates in nonmagnetic materials, whereas the electromagnetic neutron-atom interaction is at least 2 to 3 orders smaller. The latter, however, is important if high electric fields are applied and thus must be taken into account (see Section 12.4). n N (x) =

1−

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As a consequence of the strong interaction, the scattering amplitude of cold neutrons is proportional to a constant and independent of the scattering angle to first-order approximation, because of the extremely short interaction length. Therefore the nuclear potential is replaced by the Fermi pseudopotential. Dealing with bulk matter, we are interested in the macroscopic optical potential VN (x), which is simply given by a series of Fermi pseudopotentials VN (x) =

2π¯h 2  2π¯h 2 ρ b (x). b j δ(x − y j ) = m m j

(12.11)

Summation is performed over the different nuclei j with scattering length b j at the corresponding sites y j . Here bρ (x) denotes the so-called coherent scattering length density. Assuming that the internal degrees of freedom of the atoms are statistically independent of their positions, (12.11) can finally be simplified VN (x) =

2π¯h 2 ¯ bρ(x). m

(12.12)

Here b¯ is the mean bound scattering length averaged over a unit, e.g., a unit cell in a regular crystal or a polymer unit, and ρ(x) is the number density. The photo-neutron-refractive effect in PMMA is based on the fact that the photopolymer has a higher number density than the monomer MMA. By illuminating the photosensitized sample with a sinusoidal light pattern, we also modulate the number density ρ(x) = ρ¯ + ρ(x) sinusoidally. Thus the neutron-optical potential reads VN (x) = VN + VN (x) =

2π¯h 2 ¯ ρ ¯ (b + bρ(x)), m

(12.13)

with the mean scattering length density per unit volume b¯ ρ . Next we will consider the influence of a (static) electric field E(x) on the neutron-optical potential. From (12.11) it is evident that under the application of an electric field the nuclear contribution of the potential VN can be influenced by changing either at least one of the scattering lengths or at least one of the partial number densities δ(x − y j ) = ρ j (x). Changes of the scattering length could be established by an influence of the electric field on the nuclear polarizability [15]. Here, we discuss only its influence on the number density, which is larger by orders of magnitude. Let us assume an electrooptic crystal, which thus is also piezoelectric by symmetry. Application of an electric field E(x) will hence lead to a strained crystal, i.e., to a density variation ρ(x). Again (12.13) is valid. The magnitude of the density modulation then depends on the symmetry as well as the values of the compliance and the piezoelectric tensor [16]. Moreover, a neutron moving in an electric field E(x) with velocity v gives rise to an additional contribution to the potential (Schwinger term, spin-orbit coupling. Aharonov–Casher effect) due to its magnetic dipole moment μ [17]. If an electric field is applied to

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a crystal, the corresponding term in the potential reads hε ¯ μ · [E(x) × k]. (12.14) mc2 Here, c is the velocity of light in free space and the static dielectric constant ε enhances the Schwinger term. Further terms in the expansion of the potential that h|μ| ¯ are linear in the electric field are the Foldy effect [18, 19] VF (x) = − 2mc 2 [∇ · E(x)], and a tiny contribution due to a possibly existing electric dipole moment d (EDM) of the neutron VEDM = d · E(x). Thus, the total neutron-optical potential in the presence of a static electric field amounts to V = VN + VS + VF + VEDM . Finally, we estimate the neutron-refractive index for the materials that will be discussed in next sections. If we take into account only the leading terms in the potential, i.e., employing (12.11), the neutron-refractive index is  λ2 λ2 ¯ ρ b . n N = 1 − b¯ ρ ≈ 1 − (12.15) π 2π Note the quadratic dependence on the wavelength λ. The refractive indices are ˚ 1 − 6.6 × 10−5 (PMMA). for a typical cold neutron wavelength of λ = 20 A: −4 −4 7 1 − 4 × 10 (d-PMMA), and 1 − 2.6 × 10 ( LiNbO3 ). At this point it is worth emphasizing that in photo-neutron-refractive materials we are dealing with changes of the refractive index. This is quite a challenging task for neutrons. VS (x) = −

12.2 Materials So far, two different types of photo-neutron-refractive effects have been realized: changes of the optical potential V ∝ ρ resulting from chemooptics in photopolymers and V ∝ E resulting from electro (neutron) optics in crystals. Here, we will discuss the basic mechanisms of photo-(neutron) refraction in such materials, represented by PMMA and LiNbO3 respectively.

12.2.1 The Photo-Neutron-Refractive Mechanism in PMMA The polymer PMMA is well known in everyday life as Plexiglas. The basis for the occurrence of a photo-neutron-refractive effect in PMMA is the large difference of number densities for the monomer MMA and the polymer PMMA, ρMMA : ρPMMA = 0.8 : 1. A photoinduced polymerization then allows us to modulate the density and hence the neutron-refractive index by illumination. Moreover, the mechanical and, in particular, the excellent optical properties have made PMMA the favorite candidate not only for fundamental studies but also for potential technical applications. The polymerization process is performed in two steps: a thermal prepolymerization and a light-induced post-polymerization. To polymerize the monomer MMA, a (C = C) double bond must be split. This is established by free radicals, which are created by a thermoinitiator. At elevated temperatures

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radicals are formed that react with MMA and start a chain reaction resulting in the polymer. Termination processes at low temperatures (T ≈ 325 K) leave us with a mixture of residual monomers solved in the PMMA matrix. Those monomers now serve as a reservoir to restart the polymerization by illumination. To sensitize the material, photoinitiator, i.e., a substance [20] that decomposes into free radicals under illumination, had been added prior to prepolymerization. Thus the polymerization is restarted in the bright regions, yielding a density modulation and, according to (12.13), a neutron-refractive-index change. Typical recording intensities were in the range of several hundred W/cm2 . Exposure times between 2 and 60 seconds were employed. This parameter has turned out to be an important quantity for the photorefractive response (see Section 12.3.5). The photosensitivity of the doped PMMA system becomes manifest primarily in light-induced absorption changes α(Q) [20]. Havermeyer et al. proposed a model based on two relevant processes: the decay of the photoinitiator into radicals and inert molecules [20, 21] and a light-induced termination for the radicals. The first mechanism, which is responsible for the polymerization, leads to a permanent change of the refractive index: the photorefractive effect. The solution of the rate equation, which nicely describes the experimentally obtained results, yields for the exposure dependence of the light-induced absorption changes α(Q) = a1 [1 − exp (−k1 Q)] + a2 [exp (−k2 Q) − 1],

(12.16)

with Q = I (i) t the exposure and ai , ki proportionality and rate constants respectively. Via the Kramers–Kronig relations the kinetics of the corresponding refractive-index change n L for light can be obtained. However, it is impossible to discriminate between the contributions of electronic polarizability changes and density changes by employing light. On the other hand, neutron diffraction is sensitive only to density variations, i.e., the refractive-index change n N originates therefrom. Thus, by combining light and neutron diffraction we can unravel the contributions (see Section 12.3.3).

12.2.2 The Photo-Neutron-Refractive Mechanism in LiNbO3 LiNbO3 was the first photorefractive material to be discovered [1, 5]. Its photorefraction is based on the excitation, migration, and trapping of charges when illuminated by coherent light radiation. A space-charge density is building up, which, according to Poisson’s equation, leads to a space-charge field E sc and via the electrooptic effect to a refractive-index change n for light; see, e.g., [10]. When illuminating the photorefractive sample with a sinusoidal light pattern as described in Section 12.1.1, the space-charge field in the stationary state is given by E sc (x) = −E 1 cos (K x + φ).

(12.17)

Here E 1 is the magnitude of the effective electric field, i.e., the first coefficient in a Fourier series, which depends on the recording mechanism. In LiNbO3 :Fe the bulk photovoltaic effect is dominant and thus E 0 ≈ E P V and φ ≈ 0. Employing

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the linear electrooptic effect (Pockels effect), the refractive-index change for light then is given by 1 n L (x) = − n 3L r E sc (x), 2

(12.18)

with n L the refractive index for light and r the effective electrooptic coefficient. It seems worth discussing the linear electrooptic effect more accurately at this point. The electrooptic tensor ri jk evolves from the expansion of the inverse dielectric tensor (ε)i−1 j with a static electric field E k . Considering in addition the elastic degrees of freedom, it is necessary to define which of the thermodynamic variables, stress T or strain S, are to remain constant when we take the derivatives. In an experiment, both of these cases can be realized. Keeping the strain constant results in the clamped linear electrooptic coefficient r S . The applied electric field changes the refractive index directly, i.e., by slightly modifying the electronic configuration. However, when keeping the stress constant, the free linear electrooptic coefficient r T is measured. Here, in addition, a contribution via the piezoelectric coupling (di jk ) in combination with the elastooptic effect ( pi jlm ) must be considered: riTjk = (riSjk + piEjlm dklm ) (see, e.g., [22]). When experimentally realizing the latter case, we arrive at    1  = riTjk E k = riSjk + piEjlm dklm E k (12.19) 2 nL i j for the tensor of the optical indicatrix. In LiNbO3 , rS contributes about 90% of the polarizability to rT . This is plausible, since light is quite sensitive to electronic changes but much less to density variations.

The Electro Neutron-Optic Effect Recalling Section 12.1.2, it becomes evident that in analogy to electrooptics for light we can define the corresponding effect for neutrons: the electro neutronoptic effect. Utilizing (12.10) and neglecting the tensorial character of the effect, the electro neutron-optic effect reads   2m 1  = V = r N E(x). (12.20) 2 (¯hk)2 nN Consequently, we call the proportionality constant r N the electro neutron-optic coefficient (ENOC). By inserting (12.13) ff. into (12.20) and comparing the corresponding terms, we can identify the following relations:   λ 2λ|μ| S + |d|, (12.21) ±ε + i rN = hc ¯ 2 2 λ2 r NT = r NS + b¯ ρ d333 . (12.22) π

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T A B L E 12.1. Contributions to the ENOC for LiNbO3 in [fm/V] for cold neutrons with λ = 2 nm and a grating spacing  = 400 nm. Contribution

r NT − r NS

Schwinger-term

Foldy-term

EDM

[fm/V]

3

±0.02

−2 × 10−6

< 10−9

These equations are valid for LiNbO3 (point group 3m) if the grating vector K is parallel to the trigonal c-axis and the vectors μ, E, k are mutually perpendicular. E ≈ d333 was made. The sign in front of Moreover, the approximation e333 /C3333 the dielectric constant in (12.21) is determined by the direction of the neutron spin μ : + for parallel and − for antiparallel to E(x) × k. In another geometry with μ⊥[E(x) × k] this term even vanishes. The complete expressions are given by equation (12.12) in [16]. In comparison to the electrooptic coefficient for light, the situation is reversed in this case: r NS r NT . This is again reasonable and reflects the corresponding contributions to the neutron-optical potential. We therefore suggest discontinuing the use of terms like “primary” or “true” for r S and “secondary” electrooptic coefficient for rT , which can be found in the literature. An estimation of the various contributions to the electro neutron-optic coefficient for LiNbO3 is summarized in Table 12.1. In analogy to light optics and (12.18), the neutron-refractive-index change induced by a holographically created space-charge field E sc amounts to 1 (12.23) n N (x) = r N E sc (x). 2 Experimental results on measurements of electro neutron-optic coefficients will be presented in Section 12.4.

12.3 Experiments 12.3.1 Neutron Experimental Setup The grating spacings of the holographically produced gratings are on the order of several hundred nanometers. Since cold neutrons are employed for the measurements, the corresponding Bragg angles are a few tenths of a degree. Therefore, small-angle-neutron-scattering facilities (SANS) are used for the experiments. In a typical diffraction experiment, first the photoinduced neutron-refractive gratings are adjusted to obey the Bragg condition for neutrons. Then the diffracted and forward diffracted intensities are measured as a function of time and/or of the deviation from the Bragg-angle by rotating the sample. The diffracted and transmitted neutrons are monitored with the help of a two-dimensional positionsensitive detector. The setup is depicted schematically in Figure 12.2. Aside from the neutron flux, which is defined by the available neutron source, two important experimental parameters are the collimation of the beam and the properties of the velocity selector. The first determines the spread of angles θ impinging on the

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F I G U R E 12.2. Measurement setup for neutron diffraction from light-induced neutronrefractive-index gratings. The sample is placed on a rotation stage. Typical collimation lengths L coll are about 15 to 40 m.

sample, the second the wavelength spread λ. The collimation can be tuned by using slits along the neutron beam path. Typically a rectangular entrance slit of width x n and a slit just in front of the sample xs are used. To ensure a sufficiently collimated beam, the distance L coll between the slits ranges from about 15 m up to 40 m. The wavelength distribution g0 (λ) can also be adjusted to the experimental needs and is typically of a triangular shape around a central wavelength. However, in practice, these parameters are optimized according to minimum demand (coherence properties) on the one hand and convenience (measuring time) on the other. The angular (transverse momentum) distribution gtrans (θ ) forms a trapezoid with a base θb = |x n + xs |/L coll and a top θt = |x n − xs |/L coll . Typical values for the spread are λ/λ ≈ 10% and θ = (θb + θt )/2 < 1 mrad.

12.3.2 The Early Experiments (History) A photo-neutron-refractive material was realized for the first time by Rupp et al. using a slab of PMMA. This PMMA matrix contained residual monomer and a photoinitiator. The grating ( = 362 nm) was prepared as described in Section 12.1.1 with a maximum diffraction efficiency for λ = 1 nm neutrons in the range of η B = 10−3 % [4]. Interpreting (12.4), (12.13), and (12.15), the diffraction efficiency η B at the Bragg condition reads   ¯ 2 2 λd bρ η B = sin (ν) = sin . (12.24) 2 cos  In a follow-up publication the neutron wavelength, the sample thickness, and the grating spacing were varied to reach maximum diffraction efficiencies of about 0.05% [23]. The major limitation of those early attempts was the use of protonated PMMA with its high incoherent scattering cross section. Since then the deuterated analogue d-PMMA has been used instead [24], considerably reducing incoherent scattering and increasing the coherent scattering length density. By optimizing the sample preparation, the polymerization, and the exposure process, maximum diffraction efficiencies of up to 70% for λ = 2 nm neutrons are possible (Figure 12.3).

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ηN(θ) [%]

F I G U R E 12.3. Rocking curve η N (θ ) for neutrons with wavelength λ = 2 nm. Grating spacing  = 380 nm, λ/λ = 0.1, L coll = 17.6 m, x s = 10 mm, xn = 55 mm.

λ=2 nm

60

331

50 40 30 20 10 0 -6

-4

-2

0 2 θ [mrad]

4

6

These first successful experiments have opened up a wide field of potential applications: 1. By following the kinetics of relaxation processes in photopolymers, information on the complex phenomenon of glass-forming processes and polymerization can be obtained (materials scientific aspect; see Sections 12.3.3, 12.3.5). 2. By diffracting the neutrons in the presence of an electric field, fundamental properties of the neutron itself are revealed (pure physics; see Section 12.4). 3. Utilizing knowledge about how to produce gratings for cold neutrons, neutronoptical devices can be designed. Mirrors, beam splitters, lenses, and interferometers are of outstanding technological relevance these days (technological aspect; see Sections 12.3.4, 12.3.6). Those instruments then can again be used in turn to obtain material parameters (e.g., scattering lengths) or insight into the foundations of quantum physics (e.g., EDM).

12.3.3 Temporal Evolution of the Polymerization In order to obtain information about the kinetics of polymerization it is important to systematically improve the production process of light-induced holographic gratings in d-PMMA (see Section 12.2.1). In the photopolymer system d-PMMA/DMDPE the recorded grating itself is used as a sensor to directly follow the glass-forming process over large time scales. Up to the last few years only diffraction experiments with light have been performed, which has turned out to be a favorable technique because of its simplicity. According to the Lorentz–Lorenz relation small photoinduced changes of the refractive index for light n L may be the result of changes either in the density ρ or the polarizability  of the material, n L ∝ (ρ + ρ). Neutron diffraction is a complementary technique in the sense that only density changes ρ are probed. Combining (12.13) and (12.15) yields n N =

¯ 2 bλ ρ. 2π

(12.25)

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10

50 5

0

neutrons λ = 1.1 nm light λ0 = 543 nm

0

10

20

30

40

50

time [hours]

60

70

25 0

ηB;L(t) [%]

ηB;N(t) [%]

15

F I G U R E 12.4. Kinetics of the diffraction efficiency for neutrons η B;N (t) (left scale) and light η B;L (t) (right scale) at the exact Bragg-angle respectively. The measurement was performed after restarting the polymerization by illumination with uv-light for 10 seconds [25]. The reading wavelengths for neutrons were λ = 1.1 nm and λ = 543 nm for light respectively. Sample: d-PMMA.

Therefore light and neutron diffraction from photoinduced refractive-index changes can serve as a tool to clarify several aspects of the polymerization process in PMMA, in particular its kinetics. Since the glass-forming processes are irreversible, soon the demand for a facility was addressed that allowed for the simultaneous performance of light- and neutron-optic experiments. This led to the development and the design of HOLONS, which will be introduced in Section 12.3.4 in detail. The experimentally accessible quantity is the diffraction efficiency, which is related to the physically relevant parameter n in the ideal case by (12.3). Note that a unique inverse function does not exist. The kinetics of the diffraction efficiency η B (t) for light and neutrons that were recorded simultaneously reveal another problem: they do not at all obey a sin2 (ν) dependence. Figure 12.4 shows the temporal evolution of η B;N (t) and η B;L (t) for the first three days after photopolymerization had started. The reasons for the discrepancy between theory and experiment are inhomogeneities along the sample thickness and across the sample area [26]. In the case of neutron diffraction, in addition the wavelength and angular distribution are responsible. Fluctuations in the refractive index change or of the grating spacing lead to a decrease of the contrast between the maxima and minima of the rocking curve. This has a huge influence at the exact Bragg condition, where the extrema in the curve are smeared out until complete disappearance. The fact that the measurements are conducted with a partially coherent neutron beam calls for the complete rocking curve to extract the refractive-index changes unambiguously. However, time-resolved experiments and simultaneous light diffraction still need to be achieved. Therefore the strategy is to measure rocking curves η(θ ) from time to time (1 h for light, 10 h for neutrons) to ensure the correct absolute value of n and to monitor η B in the meantime. Rocking curves for neutrons are shown in Figure 12.5 at certain times after initializing the refractive-index change. When interpreting Figure 12.5, it is striking that the shape of the rocking curve changes significantly during the exposure time of the gratings [27]. Starting with a trapezoidal shape that resembles the angular distribution, we finally end up with a triangular or Lorentzian-shaped curve. To explain this feature we have to account for the partial coherence of the neutrons. Assuming a normalized angular

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12. Neutron Physics with Photorefractive Materials t [h]

25 20 15 10

ηN(θ) [%]

2 15 22 40 47 63 117 240

333

F I G U R E 12.5. Rocking curve η N (θ ) for neutrons at several times after starting the photopolymerization process [25]. λ = 1.1 nm, L coll = 14.4 m, xn = 15 mm, xs = 2 mm, λ/λ = 0.23. Sample: d-PMMA.

5 -1.5

-1.0

-0.5

0.0 0.5 θ [mrad]

1.0

0 1.5

and wavelength distribution g0 (k) and gtrans (θ ) as discussed in Section 12.3.1, the measured rocking curve is the convolution

with η(ν, θ) from

(12.3). The integrated diffraction efficiency I pc (ν) = g0 (k) gtrans (θ )[ η(ν, θ)dθ ]dθ dk, however, is independent of the angular spread. This is important since the integrated diffraction efficiency is accessible experimentally by integrating the measured rocking curve. By Iη we will denote the experimentally determined value multiplied by d/ according to (12.6). Estimating the influence of the wavelength distribution yields I pc (ν) = I (ν) + (νλ/λ)2 [J0 (ν) − ν J1 (2ν)]/6, with J1 (u) being the first-order Bessel function. For the experimental reasons already discussed, it is important to relate η B with I pc (ν), which is not possible in general. In the limit of small ν as well as large ν, approximations can be found [25]. I pc can be solved for ν numerically and thus allows one to evaluate the refractive-index change n N for neutrons. The temporal evolution of the refractive-index changes follows power laws quite well for light and neutrons. In fact, it is astonishing that several hours after illumination with uv-light, which lasts for a few seconds only, the refractive index evolves √ over time spans of weeks. The first hours are governed by an approximate t-dependence of the changes for neutrons and light. The reason for this kinetics is still under debate. Investigations of samples with different grating spacings  lead to similar values for the exponent. This serves as a hint that diffusion does not play a decisive role in the photopolymerization process.

12.3.4 HOLONS The measurements presented in the last section were performed using a novel experimental facility at the Geesthacht Neutron Facility (GeNF), which was designed for time-resolved simultaneous diffraction experiments with light and neutrons. In addition, it allowed us to utilize a complete holographic setup while conducting this type of experiment. The acronym HOLONS stands for Holography and Neutron Scattering [28]. It basically consists of a holographic optical setup including a vibration-isolated optical table, an argon-ion laser (λ0 = 351 nm) for

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Vibration isolated optical table

Collimation line for neutrons: λ=1 - 1.4 nm DPSS-Laser

Rotation device

Metallic plate

→ Detection unit

λ0=473 nm

Argon-ion Laser λ0=351 nm

Air cushion feet

Photodetector

Sample on rotation table

Mirrors, Beamsplitter

F I G U R E 12.6. Sketch of the HOLONS experiment at the SANS-2. The light beams for recording the grating (λ0 = 351 nm), for reconstructing (λ0 = 473 nm), and the neutron beam (λ = 1.1 nm) are indicated by grey lines.

recording the gratings, and a diode-pumped solid-state laser (λ0 = 473 nm) for reading. In Figure 12.6 a sketch of the facility is presented. The optical bench itself is placed on a metallic plate, which can be transferred from the HOLONS cabin to the cold neutron beamline SANS-2 by means of a crane. The HOLONS cabin itself is situated in the guide hall of the GeNF. When performing an experiment, a rotation of the whole optical bench (weight: 1.2 tons) with respect to the incident neutron beam is made possible using air cushion feet and translation stages. Accuracy amounts to about ±0.01 deg over a range of ±2 deg. This technique ensures that the sample can be positioned in the correct Bragg angle for neutrons but keeps the light optical setup unchanged. In other words, neutron rocking curves η N (θ) can be performed while one simultaneously measures the diffraction efficiency η B;L (t) for light! This exactly meets our demand for clarifying the kinetics of photorefraction in doped polymers. The photo-neutron-refractive sample is fixed on another rotation stage with high accuracy (±0.001 deg) in the common center of both rotation devices. Because of the small Bragg angle for cold neutrons (1/10 deg), the neutron beam impinges nearly perpendicularly onto the sample surface. Therefore, the holographic recording geometry must be chosen in asymmetric configuration so that the beam splitter does not block the neutron beam. Thus the sample surface-normal is inclined with respect to the axis beam splitter—sample but still remains the bisector of the recording beams. Figure 12.7 is a photograph of the HOLONS experiment at the SANS-2. Summarizing the benefits of HOLONS, we would like to emphasize its importance for

r improving and controlling the quality of photo-neutronrefractive gratings; r producing diffraction elements for calibration standards, beam splitters, lenses,

and further neutron-optical devices on the basis of photo-neutron-refractive materials;

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F I G U R E 12.7. Picture of the HOLONS experiment. To the left the argon-ion laser for recording the gratings and to the lower right the evacuated tube for the detection system can be seen. (1) marks the end of the neutron collimation system, (2) the beam expansion for the recording beams, (3) the photoneutron-refractive sample (dPMMA). Compare to the beam diagram of the schematic in Figure 12.6. Photograph courtesy of Geesthacht Research Center.

r performing time-resolved simultaneous measurements of the diffraction efficiencies for light and neutrons;

r simultaneously recording holographic gratings and reconstructing them by light and neutrons.

12.3.5 Higher Harmonics Among the noteworthy characteristics of the photopolymerization is its pronounced nonlinear response to light exposure. This can be attributed to the growth and termination of the polymer chains in d-PMMA. Illuminating the photosensitized sample with a sinusoidal light pattern results in a refractive-index change which can be expanded in a Fourier series according to (12.2) with nonvanishing higher Fourier coefficients. Performing diffraction experiments, this means that in addition to the (+1st, −1st) diffraction orders, higher harmonics appear. When the photorefractive effect in PMMA was studied by light, it turned out that the diffraction efficiency of the harmonics can be tuned by the proper choice of exposure. According to the model of the photorefractive effect in doped PMMA (Section 12.2.1, [20]), the absorption changes and thus the refractive-index changes depend exponentially on the exposure Q. Only for very low exposure can the response be approximated to be linear. For comparison of the measured data with the results of the absorption model we used reasonable and appropriate parameters [20] for the rate and proportionality constants in (12.16). Further, we assumed that n ∝ α and I (x) = I (i) (1 + cos (K x)), and expanded I (ν(Q)) into a Fourier series. The exposure dependencies of the first and second Fourier coefficients then were compared to the measured values Iη (Q). The experimentally obtained data and the results of the model are presented in Figure 12.8. The constraint to obey the Bragg condition limits the possibility of observing diffraction from higher harmonics to 2/s > λ0 , where s denotes the diffraction order. Therefore, it was possible to detect the second harmonic only with light of λ0 = 351 nm. This is inconvenient and may even damage the refractive-index

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2.0 1st harmonic (λ0=543 nm) 2nd harmonic (λ0=351 nm)

1.5 1.0 0.5 0.0 0.0

0.5

1.0

1.5

2.0

2.5

3.0

Exposure Q [Ws/cm2]

F I G U R E 12.8. Iη (Q) for a PMMA sample ( = 380 nm) exposed to laser irradiation with λ0 = 351 nm for different time spans (2 s, 4 s, 8 s, 16 s) and an intensity of I (i) = 1700 W/m2 . Rocking curves were measured for the first-order Bragg peak (open triangle) and the second-order Bragg peak (solid triangle) using light of wavelengths λ0 = 543 nm and λ0 = 351 nm respectively. The solid line represents the first Fourier coefficient of I (ν) employing the proposed absorption model.

profile of the sample, since it is photosensitive in the uv region. An elegant way of escaping this problem is to employ neutron diffraction instead. After d-PMMA samples with high diffraction efficiencies for neutrons had become available, Havermeyer et al. detected the second harmonic for three samples with grating spacings  = 400, 250, 204 nm [29]. For the latter two samples it is inherently impossible to detect the second diffraction order by means of light. Nowadays higher harmonics up to the 4th diffraction order have been observed, with a spacing /4 = 135 nm [21, 25]. Figure 12.9 shows the counting rate for four

Counts/Pixel in 5h

106 0K 1K

104

2K 3K 4K

102

100

Θ1 0.0

2.5

Θ2 5.0 θD [mrad]

Θ3

Θ4 7.5

F I G U R E 12.9. Counting rate along a horizontal line (= diagonal line in the inset) of the detector matrix for four angular positions: θ ≈ 1 (squares), θ ≈ 2 (circles), θ ≈ 3 (triangles), θ ≈ 4 (crosses). The full line represents an Off-Bragg measurement at θ ≈ 61 [25]. Bragg peaks up to the fourth order are clearly visible. The inset shows part of the detector matrix with 0K , 1K , 2K , 3K , 4K denoting the corresponding diffraction orders and the sample in the Bragg position for the fourth order (4K ) [21].

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angular positions θ ≈ 1,2,3,4 along a horizontal line of the detector matrix. The controlled setting of the magnitude of higher harmonics by tuning the exposure is important to facilitate the development of neutron-optical elements (see, e.g., Section 12.3.6). Before introducing gratings in d-PMMA based on the photo-neutron-refractive effect as a standard neutron-optical component, the question of lifetime arises. Since the fabrication of gratings with diffraction efficiencies in the range of 10%–70% reached a satisfactory level only a few years ago, the experience has been rather limited. The oldest grating that is available and still can be used was recorded on November 19, 1998. The grating formation process was monitored for one week (see Section 12.3.3). In July 2000 and then in May 2002 rocking curves were measured again. The integrated diffraction efficiency increased for several weeks (followed also by light optical measurements) until it reached a limit Iη ≈ 3.5–4.0, and then decreased to a level of 60% (Iη ≈ 2) after four years [30]. An additional problem might have been that the sample was inhomogeneous and thus the various measurements were not performed on the same area of the sample. We estimate this error to be 20%, which corresponds to Iη = 0.4 [21].

12.3.6 The Neutron Interferometer The extensive studies and experiments performed on the photo-neutron-refractive effect in d-PMMA and described in the previous sections served as a basis to set up a neutron interferometer utilizing holographically produced gratings as beam splitters and mirrors. In our view the interferometer may be regarded as one of the most useful neutron-optical devices, since it provides information about the wave function, i.e., amplitude and phase, in contrast to standard scattering or diffraction techniques, where only the intensity is measured. Three decades ago, the successful development of an interferometer for thermal neutrons [31] led to a boost in neutron optics, opening up completely new experimental possibilities in applied [32–34] and fundamental [35–38] physics, e.g., extremely accurate measurements of scattering lengths [39, 40] or tests of quantum mechanics [41–43]. For an excellent, complete, and recent review of neutron interferometry see [44]. The neutron itself and its behavior in various potentials, e.g., in a magnetic field [45], a gravitational field [46, 47], or such of pure topological nature [48–50], was studied as a model quantum-mechanical system by means of interferometers. Some very recent and amazing results on quantum states of the neutron in the gravitational field [51] or a confinementinduced neutron phase [52] demonstrate the demand for further research on those topics. Coherence and decoherence effects, which are important in any experiment, were investigated [37, 53, 54]. Based on the knowledge about the neutron and being interested in fundamental questions of physics, several groups started to elaborate interferometry with more complex quantum objects like atoms [55–57] and molecules [58–60]. On the other hand, the materials-scientific aspect of neutron interferometry has not yet reached a satisfactory level, i.e., it has not yet become established as a

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standard technique. Thermal neutron interferometers are run at the Institut LaueLangevin (ILL) and the National Institute of Standards and Technology [61], which can be used for precise measurements of neutron scattering-lengths, but are naturally limited in wavelength by the use of perfect silicon crystals as beam splitters. Moreover, high expenditure (stabilization against vibrations and thermal drift) was necessary to ensure phase stability and contrast [62]. In the last few years, investigations on biological materials and soft matter, e.g., complex organic molecules, membranes, and bones, have met with more and more interest, and the life science community has discovered neutron scattering as a nondestructive powerful method [63]. Because of the large-scale structures, small-angle scattering with cold neutrons is employed. The interferometer presented in this section is a flexible and versatile device for the operation at any SANS-facility. The first perfect-crystal neutron interferometer successfully run was designed, by Rauch et al. [31]. It consists of three equally spaced parallel slabs that are produced by cutting two wide grooves in a large, perfect silicon crystal in the so-called LLL geometry (triple Laue case) [44]. Dynamical diffraction from the (220) reflection with a lattice constant of  = 0.19 nm is used to split the incoming neutron beam and finally to recombine the subbeams as sketched in Figure 12.10. Such an interferometer can be properly run with thermal neutrons, e.g., for wavelengths less than λ ≤ 0.6 nm. Interferometers for very cold and ultracold neutrons are based on different techniques: the gratings are created by sputter etching ( ≈ μm range) in the LLL geometry [47, 64, 65], by photolithography ( ≈ 20 μm) in reflection geometry [66], or reflection from multilayers is applied [67]. To close the gap between thermal neutrons and very cold neutrons Schellhorn et al. [68] constructed a prototype interferometer in the LLL geometry built of “artificial” gratings employing the photo-neutron-refractive effect of d-PMMA as described in the previous sections. They succeeded in demonstrating that the arrangement of the three gratings acts as an interferometer. A larger and hence more sensitive interferometer was constructed by Pruner et al. G1

G2

G3 0-beam: ΨIr+ΨIIt

phase-flag ΨI 2Θ incident neutron beam

ΨII Φ

H-beam: ΨIt+ΨIIr L

L

F I G U R E 12.10. Sketch of an LLL interferometer for cold neutrons. It is based on holographically generated density gratings G1, G2, and G3 in slabs of deuterated PMMA. Rotating a phase flag by an angle , a phase difference is generated between beams I and I I . After diffraction from G3 the reflected and transmitted amplitudes rI + tI I and tI + rI I add up to the 0-beam and H-beam, respectively.

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F I G U R E 12.11. Interferometer for cold neutrons based on gratings in photo-neutronrefractive d-PMMA. G1–G3 denote the gratings, L = 150 mm,  = 380 nm.

[69], a photograph of which is shown in Figure 12.11. The crucial point for a successful operation of any LLL interferometer is the extremely accurate mutual alignment of the gratings. Thus perfect silicon crystal interferometers rely on the quality of the silicon crystal. The cutting and etching of a monolithic crystal ensures the desired accuracy over the full distance. For the very cold neutron interferometer, the alignment of the phase gratings is performed dynamically by tilting and translating each grating, which is controlled by operating three auxiliary laser-light interferometers additionally [47]. Thermal and acoustic isolation for both types of interferometers is mandatory. The methodology to overcome such problems and to construct an interferometer for cold neutrons is as follows: (1) Three photosensitive d-PMMA samples are prepared and mounted on a linear translation stage. (2) The first photo-neutron-refractive slab is exposed to the interference pattern of the holographic two-wave mixing setup. (3) Successively the second and third slabs are moved, i.e., nominally translated, to the position of the sinusoidal interference pattern. Prior to exposure the motion is corrected for deviations from the ideal translation (pitch, yaw, and roll). The latter is the decisive step during the production of the interferometer, since demand on accuracy is an absolute requirement. To control the accuracy of the translation and to correct deviations, an optical system was used in combination with piezodriven stages. The roll angle, which is the most critical parameter, is controlled by a polarization optic method that is independent of the distance of translation. Therefore in principle, interferometers of any length may be fabricated using this technique. The advantages of this attempt over other techniques are striking: The grating spacing  is easily tailored to the required value within the range 250 nm <  < 10 μm. Moreover, the sinusoidally modulated light pattern creates—if properly prepared (see Section 12.3.5)—a sinusoidal grating. Therefore only +1st and −1st diffraction orders occur. In addition, the adjustment of the three photopolymer samples is done during the recording of the grating once and for ever. Finally, the whole interferometer has turned out to be very stable, compact, and robust

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despite the high demand on accuracy. Moreover, it can be transferred to any neutron source and then set up ready to run within a few hours. This type of interferometer is useful for many investigations with cold neutrons. The most interesting are determination of the scattering length for various materials in the low-energy regime, probing the coherence properties of cold neutrons, and gaining phase information in large-scale structure investigations. The knowledge of the coherence properties is particularly important for scattering, diffraction, spin echo, and reflection experiments [70–74]. e.g., reflectometry with polarized neutrons where the coherence volume plays an essential role and is a decisive quantity only roughly estimated [75–77] or even more often simply not considered. To access the absolute value of the normalized correlation function  (1) (x − x , t − t ) (coherence function) experimentally, the visibility v of the interference fringes is measured. The phase difference  is thereby varied by rotating a phase flag through both beam paths as sketched in Figure 12.10. The visibility (contrast) is defined as [44] v = (Imax − Imin )/(Imax + Imin ) = m| (1) (x − x , t − t )|, with m being the modulation. Then an interference pattern of the form  I0,H (x) = A0,H ± v cos b¯ ρ λ(x − x E ) − ϕ3 , 

2  (x − x E )λ2 b¯ ρ , v = m exp − lc π

(12.26) (12.27)

is expected if a Gaussian coherence function is assumed. Here x and x E are the geometric path difference due to the rotation of the flag and the initial path difference of the empty interferometer respectively, lc is the coherence length, and ϕ3 the relative phase impressed by G3. The parameters A0.H and m are functions of the diffraction efficiencies η1,2,3 of the gratings. In addition, contrast is reduced because of inhomogeneities of the phase flag, thickness variations, and beam attenuation effects [37, 78]. A typical interference pattern for a wavelength of λ = 2.6 nm and a collimation of xn = 5 mm, xs = 1 mm, L coll = 19 m is presented in Figure 12.12. All three above-mentioned purposes of neutron interferometry can be demonstrated on the basis of this measurement: (1) The coherent scattering length of the phase flag is given by the periodicity of the fringes (real part of the coherence function). (2) The fringe visibility (i.e., the envelope), which is continuously reduced upon larger phase shifts, constitutes the absolute value of the coherence function for the actual neutron beam at the instrument D22 (ILL). (3) The relative shift of the fringes with respect to the envelope comprises the relative phase ϕ3 induced by G3. By replacing the latter by a small angle scattering object, the corresponding phase can be measured. This possibility is of utmost interest for solving phase-sensitive large-scale structure problems [79]. The present approach exhibits several inherent features that will open up new possibilities in neutron physics. The ultimate goal is to create a novel device for neutrons that can easily be implemented and run at any beam line for cold neutrons. The principal future application of such an instrument will be the investigation

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0-beam Neutrons in 1 sec

18 16

ϕ3/bρλ

12 10 8 -20

H-beam -15

-10

-5

0

5

xE 10

15

20

Δx [μm]

F I G U R E 12.12. Interference fringes obtained by rotating a sapphire phase flag around an axis perpendicular to the plane of incidence. Squares and triangles show the measured intensity of the 0-beam and H-beam, respectively, as a function of the geometric path difference x(). At x E (indicated by the solid black vertical line) the envelope reaches its maximum, since any path difference of the beams is compensated by rotation of the phase flag. The maximum of the interference fringes, however, deviates from that value by ϕ3 /λb¯ ρ (dashed gray vertical line).

of mesoscopic structures and their kinetics in the fields of condensed matter physics and engineering, chemistry, and biology. A summary of the cold neutron interferometer’s future perspectives and its possible impact on physics and/or materials science can be found in [30]. It is supposed that the flexibility, the low costs, and the excellent properties of the interferometer composed of gratings created by the photo-neutron-refractive effect will promote the development of novel small angle scattering techniques with cold neutrons. In addition, due to its different design and operating wavelength range, this type of interferometer can contribute constructively to unsolved problems, e.g., about the consistency of the measured gravitational phase shift with theory [47, 80–84].

12.4 Electro Neutron-Optics In this section we will present a standard electrooptic material (LiNbO3 ) exhibiting a photo-neutron-refractive effect of a completely different nature. The corresponding mechanism has already been briefly discussed in Section 12.2.2 and is based on the creation of space-charge fields that modulate the neutron-refractive index. The reason why the scientific literature has not referred to any measurement of ENOCs up to now is that the neutron-refractive-index changes due to an applied electric field are rather tiny (cf. Table 12.1). Therefore, any standard technique

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(refraction, total reflection, interferometry, diffraction from a crystal lattice) must fail in detecting an electro neutron-optic effect. It has been estimated [16] that typical changes are on the order of 10−10 . Hence, a different approach was used to tackle the problem: The photo-neutron-refractive sample was illuminated with a sinusoidal light interference pattern, thus reaching space-charge fields |E(x)| of up to 100 kV/cm, a magnitude that exceeds by far the values usually achieved. As a consequence, the neutron-refractive index is modulated via the neutron electrooptic effect according to (12.23). Then neutrons are diffracted from these holographically recorded gratings. Thus the neutron-refractive-index changes under the influence of (spatially modulated) electric fields with the extraordinary sensitivity required can be detected. The experiment for the determination of the ENOC is performed in three steps: the recording of the grating by a two-wave mixing technique, measuring the rockinge curve for light, and subsequently measuring for neutrons. In the first step the space-charge field is created; then its magnitude E sc is calculated according to (12.18), since the refractive index and the electrooptic coefficient for light are known. Then the sample is transferred to the neutron beam line. From the measured angular dependence of the diffraction efficiency for neutrons η N (θ) the neutron-refractive-index change can be calculated and in turn the ENOC r NT using (12.23). This conceptually simple investigation goes hand in hand with many difficulties, ranging from sample preparation via the narrow Bragg angles to delicate adjustment and the extremely small diffraction efficiencies [30]. The experiment to search for a neutron electrooptic effect was suggested by Rupp. In [85] he gave an overview of tentative standard (light) electrooptic materials including estimations for the magnitudes of their ENOCs. A few years later, the first experimental evidence of the effect was reported [16]. To extract the neutronrefractive-index change (12.6) and finally the ENOC using (12.23), a complete rocking curve was measured in a subsequent experiment using a 7 LiNbO3 :Fe sample [86]. The results are shown in Figure 12.13. Neutrons at each of the angular positions were accumulated for 1 h. Note that the maximum diffraction efficiency η B = 5 × 10−5 is 4 to 5 orders of magnitude lower in contrast to typical ones for d-PMMA samples. Based on the data presented above and the space-charge field determined from light-optical measurements, the ENOC of LiNbO3 is estimated 5x10-3 ηN(θ) [%]

4x10-3 3x10-3 2x10-3 1x10-3 0 -3

-2

-1

1 0 θ [mrad]

2

3

F I G U R E 12.13. Angular dependence of the diffraction efficiency for cold neutrons. The latter are diffracted from a grating induced by the electro neutronoptic effect in 7 LiNbO3 :Fe. +1st and −1st diffraction orders are visible.

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2x10-3

ηB;N(t) [%]

3x10-3

343

F I G U R E 12.14. η B;N (t) during illumination with white light; t = 0 indicates the start of the illumination. Counts were collected for 20 min; the data points are an average over this time span.

1x10-3 0 0

2500 5000 Time [sec]

7500

to be r NT ≈ 3 ± 2 fm/V. To prove that the gratings originate from a light-induced space-charge field and the electro neutron-optic effect, in addition the diffraction efficiency η B;N (t) at the Bragg position was measured during illumination with white light. The resulting time dependence is depicted in Figure 12.14. The decay of the diffraction efficiency gives evidence that the charge carriers composing the space-charge field are electrons and that the observed effect is indeed due to the coupling of the neutron-refractive-index change to that field via electro neutron-optics (cf. (12.23)). Up to this point one might assume that the electro neutron-optic effect is of pure fundamental interest. However, it will be shown that the method proposed can be used, in contrast, to solve applied physical questions as well. One major disadvantage of electrooptic photorefractive materials is that photoinduced changes basically are volatile, i.e., they vanish upon illumination with homogeneous light. Therefore, it is evident that fixing mechanisms have to be developed for long-term applications. Several attempts to solve the problem are reported in the literature [87–90]. Among them the most common technique for LiNbO3 , the material in question, is called thermal fixing (cf. Chapter 12 in the first volume of this series [91]). As discussed in Section 2.2, holographic recording in electrooptic crystals means that a light-induced space-charge density of electrons modulates the refractive index via the electrooptic effect. By increasing the temperature to about 400–450 K, positively charged ions become mobile and neutralize the effect of the electronic space-charge pattern, i.e., the (light) refractive-index grating disappears (E sc = 0). However, density gratings of ions and electrons with equal amplitudes remain. After returning to ambient temperatures, homogeneous illumination partially redistributes the electrons, whereas the ions are insensitive to illumination because of their mass. Thus two gratings, an ionic and an electronic one, appear that slightly differ in their amplitudes. This yields again a space-charge density and a refractive-index grating via the electrooptic effect, which exactly mimics the primary grating: it is fixed. This technologically important process depends significantly on the species of ions, the temperature dependence of their mobility, on their concentration, and various other parameters. Spectros copic investigations have revealed that the ionic

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1.5

0.4

1.0

0.2

0.5

0.0

0.0

-0.2

-3

-2

-1

0 1 θ [mrad]

2

3

0

1 2 θ [mrad]

ηN(θ) [10-5]

ηN(θ) [10-4]

2.0

3

F I G U R E 12.15. Rocking curves performed on the hydrated sample. Diffraction from the combined grating (left) and from the pure ionic density grating (right graph) [94].

density grating can usually be traced back to hydrogen ions [92, 93]. Surprisingly, thermal fixing also works if crystals are dehydrated. In the latter case the question arises which ion will be responsible for the compensation process. Employing the knowledge gained from measuring the ENOC, neutron diffraction from two different LiNbO3 samples (hydrated, dehydrated) in two states (E sc = 0, E sc = 0) was performed to determine the species of ions [94]. The conclusions that can be drawn from the four measurements are the following: If E sc = 0, then the photo-neutron-refractive effect is due only to ionic density gratings, i.e., n ion = −λ2 /(2π)bc ion . For the hydrated sample it was supposed that the whole contribution originates from hydrogen ions, for the dehydrated sample from the unknown ion species. If E sc = 0, the photo-neutron-refractive effect comprises two terms, the ionic density grating as before and the electro neutronoptic (ENO) part according to (12.23), n = n ion + n ENO . Evaluating the neutron-refractive-index changes on the basis of the rocking curves measured and as we know the concentration of ions, the coherent scattering length of the ion species responsible for the diffraction from the density grating can be calculated. In Figure 12.15 the angular dependence of the diffraction efficiency for the hydrated sample is shown. The evaluation for this sample in case of E sc = 0 yields n ion = 9.2 × 10−10 , which yields |bc | = 3.6 fm for the compensating ion. This ties in nicely with the assumption that hydrogen (bc = −3.74 fm) is responsible for the compensation process. Moreover, by adding or subtracting n ion from the total neutron-refractive-index change as evaluated for E sc = 0, the ENOC for LiNbO3 can be estimated as r N = (2.4 − 3.4) ± 0.5 fm/V. Equivalent measurements and evaluations for the dehydrated sample have shown that any other effect is more than an order of magnitude smaller. Therefore, only the upper limit for n ion < 6.5 × 10−10 can be estimated, which in turn yields |bc | ≤ 1.92 fm for the compensating ion. This indicates that thermal fixing in dehydrated LiNbO3 may be attributed to the motion of Li-ions (bc = −1.9 fm).

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12.5 Neutron Holography Holographic techniques are widely applied in the wavelength range of visible light, while Gabor’s original idea, dating back to 1948 [95], aimed at the construction of a lensless microscope with atomic resolution. This goal could be achieved only during the last fifteen years by introducing successively electron, X-ray, and γ -ray holography, largely supported by the increasing availability of synchrotron radiation (for reviews see, e.g., [96, 97]). Neutron holography was not considered feasible for various reasons, and only very recently a detailed analysis showed how to transfer and adapt well-known holographic conceptions to the neutron case [98]. Neutron holography is not specifically related to photorefractive materials. However, it takes a novel approach in linking neutron physics, holography, and structural studies of matter, and therefore a brief introduction to the field appears justified in the present context. Being a holographic imaging technique, neutron holography is based on recording of the interference pattern of two coherent waves emitted by the same source. The first wave reaches the detector directly and serves as the reference wave; the second one is scattered by the object of interest (object wave) and subsequently interferes with the reference wave. Atomic resolution holography is usually done in one of two complementary experimental setups. The first one, known as inside source holography (ISH), puts the source inside the sample while the detector recording the interference pattern is situated at a larger distance. In the second approach, named inside detector holography (IDH), the positions of source and detector are interchanged. Neutron ISH (NISH), can be realized by taking advantage of the extraordinarily large (≈ 80 barns) incoherent elastic neutron scattering cross section of hydrogen nuclei permitting them to serve as pointlike sources of spherical neutron waves within the sample. In an incoherent scattering process a spherical neutron wave is generated at the site of the hydrogen nucleus, which propagates towards the detector either directly, forming the reference wave, or after having been scattered coherently by other nuclei in the sample [98, 99]. We consider a single crystalline sample of a substance containing hydrogen and located in a monochromatic neutron beam of intensity I0 . If the origin of the coordinate system is chosen at the position of a source nucleus, it can be shown that the intensity I observed with a detector at a radius vector R is given by      I0  a 2  aj + I(R) = 2 1 + 2 (12.28) j R with bj aj = exp[i(r j k − r j k)]. (12.29) rj Here, a j stands for the wave amplitude with wave vector k scattered off a nucleus j, while b j and r j denote the neutron scattering length and the position vector

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of the nucleus j, respectively. The summation is taken, in principle. over a large number of nuclei in the sample, yet it is found that under typical experimental conditions only nuclei in the immediate neighborhood of the source will contribute significantly. The first and the third terms in the above equation are due to the reference and the object beams, respectively, while the second term represents the interference between these two. It contains the phase information and depends on the relative orientation of sample and detector (but in no way depends on the incident neutron beam). The amplitude of this intensity modulation is typically ≈ 10−3 of the reference beam, so that the requirements concerning counting statistics and detector stability are quite demanding. A hologram of the arrangement of nuclei forming the local environment of the source nucleus can be recorded by scanning the interference pattern over a sufficiently large solid angle. The real space positions of the nuclei can be reconstructed from the hologram by a proper mathematical procedure based essentially on the Helmholtz–Kirchhoff formula. The first NISH experiment was done by a Canadian group at Chalk River [100] on a sample of a mineral called simpsonite, which contains substantial amounts of hydrogen. Though the precision of the atomic positions obtained was rather limited, this work unambiguously demonstrated the feasibility of the technique. A second NISH study was performed recently at the Laboratoire L´eon Brillouin (LLB) in Saclay [101]. A hologram of a palladium single crystal charged with ≈ 70 at% hydrogen (a well-known metal–hydrogen system) was successfully recorded. The reconstruction of the atomic arrangement around the hydrogen nuclei resulted in the positions of the six Pd atoms forming an octahedron around the hydrogen site. Another paper of the Canadian group, discussing the observation of Kossel and Kikuchi lines in neutron incoherent elastic scattering experiments and their relevance for neutron holography, was published recently [102]. The second approach (neutron IDH = NIDH) can be realized using strongly neutron-absorbing isotopes acting as pointlike detectors in the sample. A single crystalline sample is put in a beam of plane monochromatic neutron waves, which can arrive at the detector nuclei either directly or after having been scattered coherently by other nuclei in the sample. The interference of these two contributions entails a modulation of the probability amplitudes of the neutron waves at the sites of the detector nuclei and, consequently, also the neutron absorption probability. An absorption process in a (properly chosen) detector nucleus creates an excited state that emits γ -radiation upon its transition to the ground state. It is the intensity of this γ -radiation that is actually registered in such an experiment and serves as a measure for the neutron density at the sites of the detector nuclei. Since the interference pattern depends on the orientation of the sample in the beam, a hologram can be recorded by rotating the sample through a sufficient number of orientations. The only NIDH experiment performed so far was done at the ILL in Grenoble on a single crystal of lead alloyed with a small amount of cadmium, a strongly neutron-absorbing element [103]. The γ -rays emitted by the cadmium nuclei were detected by two scintillation detectors, and a hologram was recorded by measuring the intensity of the γ -radiation for about 2000 different orientations of the sample with respect to the detectors. The hologram and the reconstructed

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F I G U R E 12.16. Recorded hologram of Pb99.74 Cd0.26 as a function of the rotation angles χ , φ.

positions of the lead atoms surrounding the cadmium sites are shown in Figures 12.16, 12.17. The holograms recorded in the three above-mentioned experiments contain each on the order of 1000 points with an angular resolution (pixel size) of ≈ 3 × 3 degrees. The neutron wavelength used is about 0.1 nm. Taken together, these studies have demonstrated convincingly that neutron holography can indeed be performed by applying the concepts presented in [98].

12.5.1 Outlook Electron and X-ray holography have undergone a rapid development during the last decade. Likewise, neutron holography offers the potential for a number of interesting applications. However, a number of technical problems, some of which are briefly listed below, have still to be overcome before neutron holography can be performed routinely [104]:

r Presently, the recording of a neutron hologram requires about one week of beam time. For NISH experiments it is obvious that in principle, this time could be reduced to one hour or even less by the use of properly calibrated multidetectors permitting one to record a large number of pixels of the hologram simultaneously. r In contrast to NISH, in NIDH experiments the pixels of the hologram can be measured only one after the other. Nevertheless, the count rates could be increased by one order of magnitude by using larger arrays of γ -detectors.

F I G U R E 12.17. Reconstructed fcc lattice of twelve nearest-neighbor lead nuclei forming a sphere around a cadmium probe nucleus (not shown). The squares indicate different lattice planes.

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r The signal-to-noise ratio in NIDH experiments is highly unfavorable due to a high background of γ -radiation, which originates from various sources that are notoriously difficult to eliminate. The excellent energy resolution of germanium detectors would allow one to single out the characteristic γ -radiation emitted from the detector nuclei and to discriminate background radiation. The use of appropriate arrays of detectors including Compton suppression is under consideration.

Neutron holography is a local technique in the sense that it images primarily the local environment of the incoherently scattering (NISH) or neutron-absorbing (NIDH) probe nuclei. Due to the unique properties of hydrogen, possible applications of NISH are suggested in the investigation of the local structure of hydrogen-containing substances like metal–hydrogen systems [105] and many inorganic as well as organic compounds. In addition, simple estimates show that there are several more elements/isotopes exhibiting sufficiently large incoherent scattering cross sections to be serious candidates for NISH. Likewise, there is quite a number of neutron-absorbing nuclei that could be applied in NIDH. In all, there is also a large class of non-hydrogen-containing materials in the investigation of which neutron holography may serve as a potentially useful tool. It will take a few more years and a number of technical improvements to see whether neutron holography can be established as a standard technique.

12.6 Outlook and Summary Some of the novel aspects opened up by the use of photo-neutron-refractive samples have already been discussed in the relevant previous sections (see, e.g., Section 12.3.6). Here, we will briefly suggest additional future experiments that promise great potential for cold neutron physics and materials science. Moreover, new tentative photo-neutron-refractive materials are suggested. A straightforward experiment to be conducted is an Aharonov–Casher (AC) type diffraction experiment [85]. Using the same experimental configuration as in the determination of the ENOC but employing polarized neutrons, the value of r S according to (12.14) and (12.21) will depend on the mutual orientation of the neutron spin, the space-charge field, and the propagation direction. Using (12.21) and (12.23) the AC contribution of the ENOC to the diffraction efficiency is η(AC) = (|μ|πεEd/ hc2 )2 , and thus independent of the neutron wavelength λ and the grating spacing . Moreover, for further experiments it is noteworthy that the AC effect is suppressed by choosing μE. An interesting aspect of the Foldy contribution (second term in (12.21)) is its appearance as a nonlocal effect. The corresponding neutron-refractive-index modulation is phase-shifted by π/2 with respect to that of the density modulation (12.22). Thus, measuring the phase shift, e.g., by means of an interferometric technique, it might be possible to determine its contribution to the ENOC. Because

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of the small diffraction angles, the Foldy contribution, however, is three to four orders of magnitude smaller than the AC effect. The measurement of the ENOC also provides a possibility for detecting an EDM of the neutron. Novel techniques have been developed, suggested and employed for this purpose [106–109]. Under illumination, huge electric space-charge fields build up in the electro neutron-optic crystal that cannot be reached in vacuum. A combination of the interferometric technique and the use of those electric fields is a tentative method to lower the limit for the existence of an EDM. Further promising photo-neutron-refractive materials are photosensitive polymers, which nowadays are available in large numbers, provided that they exhibit a considerable density ratio ρpolymer /ρmonomer , e.g., [110–114]. Considering materials that are photo-neutron-refractive via the electro neutronoptic effect, one must admit that the effects will be small when compared to the photopolymer systems. However, the advantage of these materials is that they are well established in light optics and respond with linear neutron-refractiveindex changes to illumination. Moreover, the magnitude of the refractive-index changes depends on the ENOC and the space-charge field. Estimations for various common light electrooptic materials are included in [85]. Similar to thermal fixing in LiNbO3 , ionic (=density) gratings may be created in any photorefractive sample with high ionic conductivity. Promising material for this type of effect is LiIO3 , which is known as a quasi one-dimensional ionic conductor at ambient temperatures. Moreover, photorefractive properties have been proven to exist at temperatures below 180 K [115–117]. Since the neutronrefractive-index changes are proportional to the concentration modulation cion of the specific ion, we expect that at lower temperatures the ionic grating must be very efficient in diffracting neutrons. Another very interesting possibility to observe light-induced neutronrefractive-index changes is the use of photomagnetic samples and polarized neutrons, which was suggested by Rupp [118]. The fact that the bound scattering length in general is spin-dependent will be of utmost importance. Provided that the nuclear spins are or can be (re)oriented, extremely efficient diffraction of polarized neutrons from light-induced gratings can be expected. Transparent magnetic borates, e.g., FeBO3 are also highly promising materials. The development of useful devices for neutron optics is based on the simplest optical element: the grating. The combination of gratings and/or the modification of the light-optical setup allows the preparation of new devices. A single grating can be used as a monochromator. By creating more complex neutron-refractiveindex profiles, for example lenses for cold and ultracold neutrons can be designed. Only recently, Oku performed this task for cold neutrons based on compound refractive Fresnel lenses, which consisted of about 50 elements [119, 120]. The fabrication of such a lens by holographic means seams much simpler and cheaper. A range of phenomena related to light-induced changes of the refractive index for cold neutrons was presented. This photo-neutron-refractive effect was defined in terms analogous to those of light optics. We demonstrated that the preparation of gratings based on this effect by light-optical means and the diffraction

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of cold neutrons from that grating is possible. Combined neutron and lightoptical experiments are useful in retrieving information on the photorefractive and photo-neutron-refractive nonlinear mechanism. This knowledge in turn promotes the production of neutron-optical elements, e.g., diffraction gratings, lenses, or even an interferometer. The design, setup, and successful operation of such a neutron interferometer was presented. Employing this device, coherent scattering lengths at cold neutron wavelengths can be precisely determined. Moreover, the coherence properties of the neutrons are revealed by interferometric measurements. Operation of the interferometer with partially coherent waves allows one to retrieve phase and intensity of small-angle diffraction or scattering objects. Fundamental properties of the neutron that are revealed in high electric fields can be probed by diffraction from gratings that originate from a space-charge field through the electro neutron-optic effect. The latter gives rise to a change of the neutron-refractive index under the application of electric fields. Experiments demonstrate the feasibility of this approach and suggest a novel way to search for a possible electric dipole moment of the neutron. Atomic-resolution neutron holography was introduced, the chapter finishing with planned experiments and a discussion of their impact on the foundations of physics. Acknowledgments. We are indebted to Dr. F. Havermeyer for providing us with unpublished data. Measurements presented here have been performed at the Research Center Geesthacht (Germany), the Institut Laue-Langevin (Grenoble, France), the Paul Scherrer Institut (Villigen, Switzerland), and the Laboratoire L´eon Brillouin (Saclay, France). Financial support by the grant FWF P-14614PHY and the Austrian ministry bm:bwk (infrastructure for HOLONS at the GeNF) is acknowledged.

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Index

A acousto-optic deflector, 138, 139, 140, 141, 143, 155 self-organizing cavities oscillation of laser, 257 aerial information density (D), 44 all-digital system demonstration, 32–36 BER. See bit error rates differential coding scheme, 33 differential encoding technique, 34–35, 37 pixel sequence, 34, 35 pixel-to-bit ratio, 35 probability of an error, 34, 35 diffraction efficiency of the stored holograms, 33 Hamming error-correcting code, 35–36 bit error rates, 35, 36 pixel capacity, 36 application to 4-level lasers dye laser, 265–66 Nd:YVO4 laser, 266–68 associative memories (AM), 78, 105, 118, 123, 127 analog optical computing, 78 neural network architectures, 78 B beam combiners and laser diode stabilizers beam combiners, 316 Bragg-selective feedback, 316–17 grating feedback, advantages of, 317 bit error rates, 33, 36, 93 broadband efficient adaptive method for true-time-delay array processing (BEAMTAP) algorithm output time delay, 138 AOD. See acousto-optic deflector electro-optic modulators (EOMs), 138–39

fiber feed network, 138 tapped-delay line, 138 weight matrix, 138 TFD. See traveling-fringes detector C coherent erasure and updating of holograms, 120 superimposed recording of a π-phase-shifted image, 120 D data readout system achieving 1 Gbit/sec 6:8 channel code, 36, 39 pixel-matched detector array, 38 degenerate four-wave mixing, 122, 124 degenerate four-wave mixing in photorefractive mesogenic composites, 176–80 absorption spectrum of the photorefractive mesogenic composite film, 176 phase-conjugate reflectivity, 179–80 spatial intensity modulations, 177 steady-state value of the phase conjugate reflectance, 178 demonstration platforms 100-GB capacity and 1 Gbit/sec readout system demonstrator, 43–45 data readout system achieving 1 Gbit/sec, 38–39 digital system demonstration, 32–36 electronic readout at video rates, 37–38 holographic digital data-storage test platforms, 39–41 ultrahigh >10 Gbits/sec optical data rate demonstration, 45 ultrashort-access-time-testbed, 41–42 video demonstration, 36–37

355

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Index

diffraction efficiency, 110, 181 for hologram pages, 111, 114 for volume hologram, 115 figure of merit, 230 of the gratings in LiNbO3 crystals, 291 for neutrons, 342 digital holographic memories optical data storage devices, 7 dynamic holographic interferometry (HI). See also holographic camera with continuous laser illumination application of 2E-HI to transparent objects, 224 diffraction anisotropy, 224–25 HI [1–3], 223–24 OPD. See optical path difference E edge enhancement of optical image, 184–86 experimental setup Fourier transform holographic operation, 184, 185 grating constant of the interference light, 185 electro neutron-optic coefficient, 328, 329, 341, 342, 344, 348 electronic readout at video rates, 37–38 error-correction coding, 30, 31 experimental implementation of BEAMTAP beam forming and jammer nulling, 159–60 electronic RF setup for the full optical system, 160–61 phase stabilization subsystem, 159 broadband adaptive processing, 164–66 jammer informational bandwidth, 164 impedance matched TFD, 158 narrowband adaptive processing, 160–64 AOA. See angle of arrival feedback gain, 160–161 jammer power, 160, 162, 163 phased-array jamming signal, 160, 162

SINR. See signal-to-interferenceplus-noise ratio phase-stabilization in BEAMTAP, 158–60 jammer phase, 158 reference for phase stabilization, 159 experiments of photo-(neutron) refraction higher harmonics, 335–37 Fourier coefficients, 335 nonlinear response to light exposure, 335 polymer chains in d-PMMA, 335 HOLONS. See holography and neutron scattering neutron experimental setup, 329–30 neutron interferometer, 337–41 interference pattern, 340 perfect-crystal neutron interferometer, 338–39 small-angle scattering with cold neutrons, 338 photo-neutron-refractive material using a slab of PMMA, 330 temporal evolution of polymerization, 331–33 diffraction efficiency, 332, 333 light and neutron diffraction from photoinduced refractive-index, 332 F fast light in photorefractive materials Gaussian pulse propagating in Bi12 SiO20 crystal, 287–288 transmitted signal pulse, 287 filters in telecommunication applications, 303–05 lifetime of thermally fixed holograms, 304 decay of the ionic pattern, 304–05 ion diffusion coefficient, 305 fully automated video demonstration, 36–37 recording of video imagery and sound in LiNbO3 , 36 H hologram fixing and nondestructive readout

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Index space-charge field replication due to trapped charge, 118–19 spectral sensitivity of the photorefractive material, 119–120 two-color illumination, 119–120 thermal fixing procedure, 118 erase-time constant at room temperature, 118 mobile ions, 118 hologram-multiplexing methods. See also holographic data storage systems (HDSS) technology angular multiplexing, 83–84 diffraction efficiency, 83 recording step, 83 reference-beam angle of incidence, 84 phase multiplexing, 84–85 addresses of the stored images, 84 binary phase encoding, 85 phase-only spatial light modulator, 84, 85 wavelength multiplexing, 85–86 diffraction efficiency of a weak reflection, 86 grating vectors, 86 reflection gratings, 86 tunable solid-state laser diode, 86 holographic associative memories, 105–6 “ghost image” of a stored image, 105 content-addressable memory, 105 off-axis holography, 105 page-oriented holographic memories, 106 content-addressable reading, 106 readout reference, 106 reconstructed wave front, 105 storage capacity, 105 holographic camera with continuous laser illumination. See also dynamic holographic interferometry (HI) dynamic displacements, 238–39 single-frame analysis technique, 238 extension to tridimensional displacements, 240–42 phase difference, 240 sensitivity vector, 241

357

quasi-static displacements, 236–38 nondestructive testing of aeronautical composite structures, 236 full-field displacement metrology, 237 RT-HI technique, 234 two phase-quantification techniques, 234–36 behavior of a microelectronic chip, 240–41 response time, 235 stroboscopic technique, 239 vibratory displacements, 239–40 holographic camera with continuous laser illumination, for transparent objects holographic camera to transparent objects, 243–44 microgravity fluid-monitoring holographic camera, 242–43 BSO crystal implementation, 243 holographic camera with pulsed lasers early experiments, 244–47 anisotropy of diffraction configuration, 247–48 camera operating in the pulse regime, 244–45 energies for recording and readout, 246 four-pulse vibration measurement technique, 246–47 vibration measurement method, 246 holographic data storage systems (HDSS) technology. See also Hologrammultiplexing methods angular multiplexing, 17–18 angular selectivity, 17, 18 separation angle, 18 signal-amplitude spatial light modulator (signal ASLM), 17 superimposed pages readable with a certain SNR, 30 correlation multiplexing/ shift multiplexing, 21 autocorrelation function, 21 diffraction efficiency of the hologram, 21 spatial autocorrelation function, 21

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Index

holographic data storage systems (cont.) Fourier transform configuration, 14–15 focal plane, 14, 15 signal page, 15 signal path, 14–15 grating vectors accessible data channels, total number of, 15, 16 maximum wave vector length, 15 signal-to-noise ratio, 16 holographic recording materials, 21 material parameters for optimized holographic performance, 22 parameters of, 22 photopolymer materials, 22 properties of, 21 phase-encoded multiplexing, 19–20 crosstalk, 20 diffracted signal, 20 phase spatial-light modulator, 19 phase-encoded reference beam, 19 photorefractive materials interference pattern, 23 linear electrooptic (Pockels) effect, 24 photoexcited states, 23 photorefractive effect in diffusion-dominated materials, 23–25 signal processing and en/decoding coding and signal processing, elements of, 31 differential decoding, 32 ECC. See error-correction coding modulation coding, 31 signal path in a holographic data storage system, 31 threshold detection, 32 recorded hologram (s), 11–13 SNR. See signal-to-noise ratio photorefractive crystals, 9 angular multiplexing, 9 photosensitive organic media, 8–9 shift or phase multiplexing techniques, 9–11 wavelength multiplexing, 18–19 wavelength selectivity, 18 holographic filters

optical add-drop multiplexers for telecommunication networks, 295–97 photorefractive crystals, 295 properties of, 295 VHGs. See volume holographic gratings wavelength-division multiplexing, 295 holographic recording. See also recorded hologram(s) binary phase mask, 307 collimated beams, 306 thermally fixed gratings in lithium niobate, 309 wavelength filters for WDM applications, 307–9 center wavelength of the WDM filter, 307 isolation of WDM filters, 308 holography and neutron scattering, 333–335 benefits of, 334–335 I image amplification in photorefractive mesogenic composites, 182–84 two-dimensional optical image reconstruction from nonlocal gratings, 182–83 output image-intensity distribution, 184 injection-locked laser diode mutually pumped phase-conjugate mirror, 212–14 broad stripe LD (BS-LD), 212 injection-locking, 212 lasing spectrum of the slave laser, 214 MPPCM. See mutually pumped phase conjugate mirror single-mode master laser diode, 213 slave laser, 213 phase-conjugate feedback, 215–19 collimating optics for the laser beam, 216 filtered laser beam, 217 Rh-doped BaTiO3 crystal, 216

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Index J Joint Fourier transform correlator, 94–97 correlation theorem, 95 dual-axis JTC correlator, 96 intensity distribution, 95 transparency with a real transmittance, 95 L liquid-crystal light valves, 108 local area network, 298, 299 Ethernet transmission protocol, 298 long-haul (LH) networks, 300 amplification region of erbium-doped optical amplifiers (EDFA), 300 M mesogenic composites for photorefractive applications, 169–172 dye-doped low-molar-mass liquid crystals, 170 mesophase without phase separation, 171 orientational photorefractive effects, 170 PDLC. See polymer-dispersed liquid crystals PDLCC. See polymer dissolved liquid crystal composite metropolitan networks, 299–300 interconnecting city areas through access nodes, 299 transceivers, 299 multiplexed hologram, 45 mutually pumped phase conjugate mirror, 212, 213, 214, 215, 216 N narrowband processing and limitations beam steering and jammer nulling, 136 RF signals from one-dimensional (1-D) antenna arrays, 136 narrowband processing, 136 neutron physics with photorefractive materials holographic gratings, 322–24 diffraction efficiency, 323

359

light-induced refractive-index gratings, preparation of, 322 neutron optics, 324 coherent elastic scattering, 324 influence of a (static) electric field E(x) on neutron-optical potential, 325 neutron-refractive index, 326 photo-neutron-refractive effect, 321 PMMA. See polymethylmethacrylate nonlinear holographic associative memories (NHAM) hybrid NHAM LCLV. See liquid-crystal light valves video detectors, 108 PCM. See phase-conjugating mirror perfect thresholding, 108 reconstructed reference beams, 106–7 sequential recording of Fourier transforms of objects, 106 storage capacity, 106 O optical add-drop modules (OADM) OADM architectures, 300–301 wavelength blocker, 300 (R)OADM, 300–301 optical correlators, 120–22 buffer memory for JTC-type correlators, 121 correlation at video rate, 121 compact photorefractive JTC system based on aBi12 SiO20 (BSO), 121 response time, 121 permanent storage of matched filters, 121–22 semiconductor multiple quantum well (MQW) devices, 122 optical data storage, commercial devices, 77–78 optical path difference, 224, 226 optical pattern recognition for optical correlators JTC. See Joint Fourier Transform Correlator VanderLugt-type correlator, 97–98

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Index

optical pattern recognition using volume holograms Joint Fourier transform correlator with volume media, 98–101 Bragg condition, 98–99, 100 complex wave amplitude, 99 crosstalk effects, 100 sinc function, 100 VanderLugt correlator with volume media complex amplitude at the output plane, 102 holographic matched filter, 101–02 shift invariance, 103–04 two-dimensional output intensity distribution, 103 optical phase conjugation for solid-state lasers distortion due to intense pumping mechanism, 194–95 optical phase conjugation in a rod geometry, 195 depolarization, 195 refractive index distribution, 195 thermal lensing with dioptic power, 195 lamp-pumped Nd:YAG, 195–96 photon energy defect, 196 thermal effects, 195 self-pumped phase-conjugation scheme, 197 photorefractive crystal, barium titanate (BaTiO3 ), 197 semiconductor diode lasers, 196 output power, 196 solid-state lasers photorefractive PCM, 196 stress-induced birefringence, 195 optical response of volume holograms angular and wavelength detuning, 64–69 angle selectivity of slab-shaped hologram, 65 hologram response, 64 parallelepiped-shaped hologram, 67–69 volume hologram, 64 wavelength degeneracy, 66–67 wavelength selectivity of slab-shaped hologram, 65–66

Bragg-matched reconstruction “field-dependent apodization parameter”, 62 “hologram shape” function, 59 dielectric modulation, 59 refractive index of the holographic material, 59 shift-variant transfer function, 63 superposition integral, 60 light amplitude distribution at the back focal plane of L2, 56–57 3D modulation of dielectric index, 56 diffracted field, 57 Fourier-plane filter, 56 system input—output relation, 57–58 longitudinal detuning (defocus) Bragg selectivity of the hologram, 70–74 cylinder-shaped hologram, 69 hologram vicinity, 69 successive values of z 0 , 70 P phase-conjugate diode-pumped Nd:YVO4 laser amplifiers cw regime, 200–203 amplified signal beam, 201 BaTiO3 crystal, 202 double-pass amplifier, 202 far-field spatial patterns of the amplified signal beam, 203 signal laser beam, 201 single-pass gain of the amplifier, 202 nanosecond regime, 208 flash-lamp-pumped Nd:YAG four-pass amplifier with a CAT-PCM, 208 numerical simulation of output power, 198, 199 elliptical gain volume, 198 output power, 198 signal and phase-conjugate beams, 198 photorefractive phase-conjugate reflectivity, 198–99 photorefractive ring-PCM based on a BaTiO3 crystal, 198 reflectivity in ring-PCM, 199 picosecond regime, 203–08

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Index amplified signal laser, 206 frequency-narrowing effects, 205 operation of a 7.4-watt picosecond output, 204 output power as a function of pump diode power, 206–07 output power from the double-pass amplifier, 206 picosecond Nd:YVO4 amplifier with PCM, 205–06 second harmonic generation in a 5-mm KTP crystal, 207–08 wavelength selectivity of a photorefractive phase-conjugate mirror, 200 two-wave-mixing gain-band of refractive index gratings, 200 phase-conjugate laser resonator, 209–212 phase-conjugate resonator (PCR), 209, 210, 211, 212 full suppression of the higher-order transverse modes, 211–12 master resonator, 209 phase-conjugate reflection, 209 phase-conjugate mirror, 194, 209, 211, 212, phase-conjugating mirror, 106–07, 122, 123, 124, 194, 198, 202, 203, 205, 206 phase-coupling-induced slow and fast lights, 280 group velocity of light with dispersive phase coupling, 280–81 light propagation in photorefractive materials, 281 tunability of the group velocity of lights, 281–84 limited bandwidths, 282 response rate of the photorefractive two-beam coupling, 282 photo-(neutron) refraction electro neutron-optic effect angular dependence of diffraction efficiency for hydrated sample, 344 ENOC. See electro neutron-optic coefficient photo-neutron-refractive mechanism in LiNbO3 , 327–28

361

linear electrooptic effect, 328 space-charge density, 327 photo-neutron-refractive mechanism in PMMA polymerization process, 326–27 neutron holography, 345–48 atomic resolution holography, 345 lensless microscope with atomic resolution,construction of, 345 neutron-absorbing isotopes acting as pointlike detectors, 346–47 photorefractive crystals for HI choice of the crystal, 231–32 ferroelectric crystals, 231 semiconductors, 231 sillenite crystals, 231 figures of merit n, photoinduced refractive index variation, 239 diffraction efficiency, 230 temporal characteristics of hologram recording, 230–31 photorefractive configurations, 232–33 anisotropy of diffraction, 232–33 beam coupling, 232 recording geometries, 233 90◦ geometry of beams, 233 polarization-angle, read-write multiplexing with BaTiO3 BEAMTAP arm of the processor, 155–57 feedback AOD DC reference beam, 156 tap-delay line, 155 detected beam-formed output for a single 1GHz tone, 157 e-polarized read beam, 157 rf feedthrough, 157 isolating the writing beams from the reading beam, 154 array emulator arm of the processor, 155 oˆ - and eˆ -polarized beams, 155 polymer-dispersed liquid crystals, 170–72, 174 polymer dispersed liquid crystal composite, 171–76, 181–182, 184, 185, 187 Polymethylmethacrylate, 321, 323, 326

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16:47

Index

pulse-broadening effect, 289–90 Gaussian pulse without a severe broadening effect, 290 R rainbow method, 54 read-write multiplexing for the BEAMTAP system, 143–151 parallel-tangents, equal-curvature condition, 144 Bragg matching condition, 144 grating vector (s), 144 parallel-tangents, equal-curvature condition in BaTiO3 , 145–151 Bragg efficiency, 147, 148 Bragg-matching diffraction efficiency, 150 equal-curvature angle, 146, 147 equal-curvature condition, 145, 147 momentum mismatch, 147 θeq, extraordinary angle, 145 real-time holography in photorefractive mesogenic composites, 180–82 diffraction efficiency versus applied dc fields and writing power, 181–82 frequency-doubled Nd-YAG laser with a 532-nm wavelength, 180–81 diffracted beam intensity, 181 diffraction efficiency, 181 reconfigurable optical-add-drop multiplexer, 298, 301, 303 recorded hologram (s). See also holographic recording Bragg selectivity, 12, 13, 14 wave vector, 12 dynamic range of the index grating, 12–13 material sensitivity, 13 electric field amplitude of, propagating reference wave, 11 propagating signalwave, 11 reflection gratings, 301–303 refractive index modulation, 301 apodization, 302 resulting diffraction efficiency versus the reading wavelength, 301 holographic interferometry (HI) ideal holographic measurement device, 226–27

electronic speckle pattern interferometer (ESPI), 226 properties of a holographic measurement tool, 226–27 phase quantification and associated techniques, 227–29 Fourier transform (FT) processing, 227 heterodyning, 227 phase-shifting or phase-stepping (PS), 227 polarization separation technique, 229 ring resonator NHAM, 108–09 eigenmodes representing orthogonal patterns, 108 reference beam (s), 109 ring phase conjugate resonator, 108, 109 ultrashort-access-time-testbed, 41–42 access times, 41 S saturable absorber correlation with associative readout, 124 thresholding effect, 125 enhancing the intensity ratio, 125 self-organizing cavities laser cavities, 253–54 operating on the transverse structure, 273 with 4-level laser amplifiers, 263–65 with adaptive Fabry-Perot filters, 254–56 with semiconductor amplifiers, 267–272 self-organizing cavities operating on the transverse structure, 273 light diffraction, 273 transverse mode oscillation, 273 self-organizing cavities with 4-level laser amplifiers, 263–65 laser oscillation on a single-longitudinal mode, 263 modeling 4-level lasers, 264–65 cw standing-wave laser cavity, 264 frequency of oscillating mode at maximum gain curve, 264–65 gain and losses of the oscillating mode, 264

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16:47

Index self-organizing cavities with adaptive Fabry-Perot filters, 254–56 Fabry-Perot filter, 255 reflectivity, 255 photorefractive Bragg grating, 254–55 stable single-mode oscillation, 255–56 self-organizing laser cavities, 253–54 nonlinear effects taking place inside the laser amplifier, 253 single-mode oscillation, 253–54 thermal holograms in a nonsaturable absorbing medium, 254 self-organizing cavities with semiconductor amplifiers, 267–272 gain medium characteristics, 267–68 laser diode, 269–70 spectral characteristics of an extended cavity laser diode, 270 wavelength-tunable diffraction-limited laser diode, 270 Littman configuration, 271 oscillating-mode wavelength, 271 photorefractive filter, 272 reflectivity of this feedback system, 271 signal-to-interference-plus-noise ratio, 141, 161, 164 signal-to-noise ratio, 8, 91, 93, 126 slow and fast lights in photorefractive materials dispersive photorefractive phase coupling, 278–80 associated light-induced space-charge field E sc , 278 induced refractive index grating, 278 experimental demonstrations, 284–89 fast light in photorefractive materials, 287–88 measurement of group velocity of light, 284–85 pulse-broadening effect, 289–90 slow light in photorefractive materials, 285–87 tunability and transition between slow and fast lights, 288, 289 phase-coupling-induced slow and fast lights, 280–84 photorefractive wave coupling, 277

363

slow and fast lights with a stationary refractive index grating, 290–92 stationary refractive index gratings, 290 slow light in photorefractive materials, 285–87 steady-state temporal traces in a Bi12 SiO20 crystal, 285–87 solid-state lasers with a photorefractive phase-conjugate mirror, 193–94 correction of the distortions in a laser amplifier, 193 spatial frequency selective reconstruction of optical images, 186–88 edge enhancement of the optical image, 184–85 Raman-Nath thin-grating regime, 186–87 reference beam profile, 187 spatial heterodyning, 51–52 spatial light modulator, 2, 19, 86 steady-state spectral characteristics of the adaptive Fabry-Perot Filter, 257 Fabry-Perot reflectivity versus the wavelength Bragg mismatch, 261–63 Fabry-Perot reflectivity, 258 Kogelnik’s formalism, 260–61 transfer matrix formulation, 258–61 wave-mixing formalism, 259–60 storage capacity of photorefractive holographic media, 115–17 conventional error-encoding techniques, 117 N max, maximum number of storable holograms, 116, 117 ηmin , minimum allowable signal diffraction efficiency, 115, 116, 117 storage capacity of volume media interpage crosstalk and storage density, 90–93 (SNR)min , worst signal-to-noise ratio, 91 coupled wave theory in the Fourier regime, 91–92 N max, maximal number of multiplexed holograms, 90, 91 limitation due to optical system, 87, 89–90

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364

16:47

Index

storage capacity of volume media (cont.) maximum storage density, 90 Rayleigh criterion, 89 limitations due to material quality, 93–94 ratio between signal and scattered noise, 94 signal-to-noise ratio, 94 limiting storage capacity, 88–89 D/λ2 , storage capacity of thin holograms, 88 system architecture electronic data stream, 87 encoding in case of angular multiplexing, 86–87 H × H holograms, 88 SLM. See spatial light modulator storage capacity and storage density at a given signal-to-noise ratio, 86–87 transformation of light diffracted off the hologram, 87–88 T telecommunication networks LAN. See local area network long-haul (LH) networks, 300, 301 MAN. See metropolitan networks OADM. See optical add-drop modules types of optical networks, 297 thermal fixing of lithium niobate (LiNbO3 ) creation of fixed ionic gratings in lithium niobate, 27 electronic gratings formed at room temperature, 27 simultaneous formation of electronic and ionic gratings, 27 holographic gratings based on charge redistribution, 27–28 lifetimes of fixed ionic holograms, 27–28 using complementary gratings to establish stable holograms, 25–26 charge density, 25, 26 ionic conductivity, 26 thin storage media implementations bistable Fabry–Perot etalon, 123

mirror to retroreflect the reconstructed reference beams, 123 correlation peak, 123 thin thermoplastic Fourier transform holograms, 122–23 DFWM. See degenerate four-wave mixing transfer function of volume holographic optical systems. See also derivation of optical response of volume holograms 3DSH. See 3D spatial heterodyning Bragg selectivity, 51–52 three-dimensional (3D) grating, 51 traveling-fringes detector, 138–43, 166 opto-electronic conversion for broadband detection, 141–42 drifting diffusion equation, 143 fractional bandwidth detection, 142 photo-conducting layer of gallium arsenide (GaAs), 141 true-time-delay array processing (BEAMTAP) broadband efficient adaptive method for true-timedelay algorithm array processing narrowband processing and limitations, 136 radio frequency (RF) signal processing, 135 RF antenna arrays, 135 wideband true-time-delay beam-steering techniques, 136 tapped-delay lines, 137 two-photon recording techniques, 28–30 sensitivity and index changes as a function of stoichiometry, 28–29 single-photon sensitivity, 29 two-wave mixing in photorefractive mesogenic composites, 173–76 asymmetric energy transfer signals in two-beam coupling experiments, 174–75 diffraction patterns for the photorefractive PDLCC sample, 173 Raman–Nath regime of optical diffraction, 173, 175 theory of two-beam coupling, 173–74

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16:47

Index V VanderLugt-type correlator, 97–98 detection of the correlation peak, 98 observed intensity pattern, 97 volume holographic gratings, 308, 309, 313, 316 beam combiners and laser diode stabilizers, 316–17 integration of multiple volume-Bragg gratings and tunable filters, 313, 314, 315 grating wave vector, 309 ROADM. See reconfigurable optical-add-drop multiplexer packaging cross-connects micro-electromechanical mirrors (MEMs), 316 ultrahigh-isolation add–drop filters, 312, 313, 314 double-pass configuration using a mirror, 312 wavelength filters “four-port” device, 315 “three-port” device, 315 “two-port” device, 313–15 volume storage in associative memories all-optical nonlinear holographic associative memory, 124, 125 photorefractive KNbO3 crystal, 124 nonresonant cavity with image-bearing beams, 124 nematic liquid crystal, 124 photorefractive KNbO3 crystal, 124 thresholding effect of the saturable absorber, 125

365

two images stored in a photorefractive BaTiO3 crystal, 123–24 hybrid optical–electronic implementation, 124 recording the holographic memory, 123, 124 weighted storage method, 127 volumetric optical data storage hologram-multiplexing methods, 82–86 light diffraction by volume gratings, 79–82 angle of incidence, 80–81 angular mismatch of the incident wave, 82 fixed transmission hologram, 81 holographic grating recording and readout, 79–80 refractive index modulation amplitude, 80, 82 system architecture, 86–88 W wide-angular aperture readout, experimental demonstration of Bragg mismatch, 153 charge-coupled device (CCD) camera, 151, 152 concept of the parallel-tangents, equal-curvature condition, 151–52 reading beam angle, 151 light diffracted off hologram and captured by CCD camera, 152–53