Handbook of Nonlinear Optics Second Edition, Revised and Expanded Richard L Sutherland Science Applications International Corporation Dayton, Ohio, U.S.A with contributions by Daniel G McLean Science Applications International Corporation Dayton, Ohio, U.S.A Sean Kirkpatrick Air Force Research Laboratory Wright Patterson Air Force Base, Ohio, U.S.A MARCEL ffi MARCEL DEKKER, INC NEW YORK • BASEL Library of Congress Cataloging-in-Publication Data A catalog record for this book is available from the Library of Congress ISBN: 0-8247-4243-5 This book is printed on acid-free paper Headquarters Marcel Dekker, Inc 270 Madison Avenue, New York, NY 10016 tel 212-696-9000, tax 212-685-4540 Eastern Hemisphere Distribution Marcel Dekker AG Hutgasse 4, Postfath 812, CH-4001 Basel, Switzerland tel 41-61-260-6300 tax 41-61-260-6333 World Wide Web http //www dekker com The publisher offers discounts on this book when ordered in bulk quantities For more information, write to Special Sales/Professional Marketing at the headquarters address above Copyright © 2003 by Marcel Dekker, Inc All Rights Reserved Neither this book nor any part may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, microfilming, and recording, or by any information storage and retrieval system, without permission in writing from the publisher Current printing (last digit) 10 21 PRINTED IN THE UNITED STATES OF AMERICA OPTICAL ENGINEERING Founding Editor Brian J Thompson University of Rochester Rochester, New York Editorial Board Toshimitsu Asakura Hokkai-Gakuen University Sapporo, Hokkaido, Japan Nicholas F Borrelli Corning, Inc Corning, New York Chris Dainty Imperial College of Science, Technology, and Medicine London, England Bahrain Javidi University of Connecticut Storrs, Connecticut Mark Kuzyk Washington State University Pullman, Washington Hiroshi Murata The Furukawa Electric Co , Ltd Yokohama, Japan Edmond J Murphy JDS/Umphase Bloomfield, Connecticut Joseph Shamir Technion-Israel Institute of Technology Hafai, Israel Dennis R Pape Photomc Systems Inc Melbourne, Florida David S Weiss Heidelberg Digital L.L.C Rochester, New York Electron and Ion Microscopy and Microanalysis: Principles and Applications, Lawrence E Murr Acousto-Optic Signal Processing: Theory and Implementation, edited by Norman J Berg and John N Lee Electro-Optic and Acousto-Optic Scanning and Deflection, Milton Gottlieb, Clive L M Ireland, and John Martin Ley Single-Mode Fiber Optics: Principles and Applications, Luc B Jeunhomme Pulse Code Formats for Fiber Optical Data Communication: Basic Principles and Applications, David J Morris Optical Materials: An Introduction to Selection and Application, Solomon Musikant Infrared Methods for Gaseous Measurements: Theory and Practice, edited by Joda Wormhoudt Laser Beam Scanning: Opto-Mechanical Devices, Systems, and Data Storage Optics, edited by Gerald F Marshall Opto-Mechanical Systems Design, Paul R Yoder, Jr 10 Optical Fiber Splices and Connectors: Theory and Methods, Calvin M Miller with Stephen C Mettler and lan A White 11 Laser Spectroscopy and Its Applications, edited by Leon J Radziemski, Richard W Solarz, and Jeffrey A Paisner 12 Infrared Optoelectronics: Devices and Applications, William Nunley and J Scott Bechtel 13 Integrated Optical Circuits and Components: Design and Applications, edited by Lynn D Hutcheson 14 Handbook of Molecular Lasers, edited by Peter K Cheo 15 Handbook of Optical Fibers and Cables, Hiroshi Murata 16 Acousto-Optics, Adrian Korpel 17 Procedures in Applied Optics, John Strong 18 Handbook of Solid-State Lasers, edited by Peter K Cheo 19 Optical Computing: Digital and Symbolic, edited by Raymond Arrathoon 20 Laser Applications in Physical Chemistry, edited by D K Evans 21 Laser-Induced Plasmas and Applications, edited by Leon J Radziemski and David A Cremers 22 Infrared Technology Fundamentals, Irving J Spiro and Monroe Schlessinger 23 Single-Mode Fiber Optics: Principles and Applications, Second Edition, Revised and Expanded, Luc B Jeunhomme 24 Image Analysis Applications, edited by Rangachar Kasturi and Mohan M Trivedi 25 Photoconductivity: Art, Science, and Technology, N V Joshi 26 Principles of Optical Circuit Engineering, Mark A Mentzer 27 Lens Design, Milton Laikin 28 Optical Components, Systems, and Measurement Techniques, Rajpal S Sirohi and M P Kothiyal 29 Electron and Ion Microscopy and Microanalysis: Principles and Applications, Second Edition, Revised and Expanded, Lawrence E Murr 30 Handbook of Infrared Optical Materials, edited by Paul Klocek 31 Optical Scanning, edited by Gerald F Marshall 32 Polymers for Lightwave and Integrated Optics: Technology and Applications, edited by Lawrence A Homak 33 Electro-Optical Displays, edited by Mohammad A Karim 34 Mathematical Morphology in Image Processing, edited by Edward R Dougherty 35 Opto-Mechamcal Systems Design: Second Edition, Revised and Expanded, Paul R Yoder, Jr 36 Polarized Light: Fundamentals and Applications, Edward Colleti 37 Rare Earth Doped Fiber Lasers and Amplifiers, edited by Michel J F Digonnet 38 Speckle Metrology, edited by Rajpal S Sirohi 39 Organic Photoreceptors for Imaging Systems, Paul M Borsenberger and David S Weiss 40 Photonic Switching and Interconnects, edited by Abdellatif Marrakchi 41 Design and Fabrication of Acousto-Optic Devices, edited by Akis P Goutzoulis and Dennis R Pape 42 Digital Image Processing Methods, edited by Edward R Dougherty 43 Visual Science and Engineering: Models and Applications, edited by D, H Kelly 44 Handbook of Lens Design, Daniel Malacara and Zacarias Malacara 45 Photonic Devices and Systems, edited by Robert G Hunsperger 46 Infrared Technology Fundamentals: Second Edition, Revised and Expanded, edited by Monroe Schlessinger 47 Spatial Light Modulator Technology: Materials, Devices, and Applications, edited by Uzi Efron 48 Lens Design: Second Edition, Revised and Expanded, Milton Laikin 49 Thin Films for Optical Systems, edited by Frangois R Flory 50 Tunable Laser Applications, edited by F J Duarte 51 Acousto-Optic Signal Processing: Theory and Implementation, Second Edition, edited by Norman J Berg and John M Pellegrino 52 Handbook of Nonlinear Optics, Richard L Sutherland 53 Handbook of Optical Fibers and Cables: Second Edition, Hiroshi Murata 54 Optical Storage and Retrieval Memory, Neural Networks, and Fractals, edited by Francis T S Yu and Suganda Jutamutia 55 Devices for Optoelectronics, Wallace B Leigh 56 Practical Design and Production of Optical Thin Films, Ronald R, Willey 57 Acousto-Optics: Second Edition, Adrian Korpel 58 Diffraction Gratings and Applications, Erwin G Loewen and Evgeny Popov 59 Organic Photoreceptors for Xerography, Paul M Borsenberger and David S Weiss 60 Characterization Techniques and Tabulations for Organic Nonlinear Optical Materials, edited by Mark Kuzyk and Carl Dirk 61 Interferogram Analysis for Optical Testing, Daniel Malacara, Manuel Servin, and Zacarias Malacara 62 Computational Modeling of Vision: The Role of Combination, William R Uttal, Ramakrishna Kakarala, Sriram Dayanand, Thomas Shepherd, Jagadeesh Kalki, Charles F Lunskis, Jr., and Ning Liu 63 Microoptics Technology: Fabrication and Applications of Lens Arrays and Devices, Nicholas F Borrelli 64 Visual Information Representation, Communication, and Image Processing, Chang Wen Chen and Ya-Qin Zhang 65 Optical Methods of Measurement: Wholefield Techniques, Rajpal S Sirohi and Fook Siong Chau 66 Integrated Optical Circuits and Components: Design and Applications, edited by Edmond J Murphy 67 Adaptive Optics Engineering Handbook, edited by Robert K Tyson 68 Entropy and Information Optics, Francis T S Yu 69 Computational Methods for Electromagnetic and Optical Systems, John M Jarem and Partha P Banerjee 70 Laser Beam Shaping: Theory and Techniques, edited by Fred M Dickey and Scott C Holswade 71 Rare-Earth-Doped Fiber Lasers and Amplifiers: Second Edition, Revised and Expanded, edited by Michel J F Digonnet 72 Lens Design: Third Edition, Revised and Expanded, Milton Laikin 73 Handbook of Optical Engineering, edited by Daniel Malacara and Brian J Thompson 74 Handbook of Imaging Materials, edited by Arthur S Diamond and David S Weiss 75 Handbook of Image Quality: Characterization and Prediction, Brian W Keelan 76 Fiber Optic Sensors, edited by Francis T S Yu and Shizhuo Yin 77 Optical Switching/Networking and Computing for Multimedia Systems, edited by Mohsen Guizani and Abdella Battou 78 Image Recognition and Classification: Algorithms, Systems, and Applications, edited by Bahram Javidi 79 Practical Design and Production of Optical Thin Films: Second Edition, Revised and Expanded, Ronald R Willey 80 Ultrafast Lasers: Technology and Applications, edited by Martin E Fermann, Almantas Galvanauskas, and Gregg Sucha 81 Light Propagation in Periodic Media: Differential Theory and Design, Michel Neviere and Evgeny Popov 82 Handbook of Nonlinear Optics, Second Edition, Revised and Expanded, Richard L Sutherland Additional Volumes in Preparation Optical Remote Sensing: Science and Technology, Walter Egan Preface to Second Edition The science of optics, the branch of physics that deals with the properties and phenomena of visible and invisible light, has generated a wealth of knowledge that makes its use pervasive in other physical sciences, biology, medicine, forensics, agriculture, art, industry, and the military This has spawned a technology called photonics, a name based on the quantum of energy in the electromagnetic field, the photon The domain of photonics extends from energy generation to detection to communications and information processing, and includes all means of generating and harnessing light for useful purposes Both the science and technology aspects of optics have and continue to be vastly influenced by the field of nonlinear optics It is a discipline that has enhanced our understanding of fundamental light-matter interactions as well as provided the means for accomplishing a variety of engineering tasks The purpose of this book is to provide a balanced treatment of second- and third-order nonlinear optics, covering areas useful to the practicing scientist and engineer The intent is to serve as a ready source of information useful to those researchers performing characterization of nonlinear materials, using the methods of nonlinear optics in scientific studies, and exploiting nonlinear optical phenomena in photonics This edition of the Handbook of Nonlinear Optics has been updated and new material has been added It is evident from a perusal of the scientific literature that advances in nonlinear optics continue at a rapid pace For example, frequency conversion in new bulk and quasi-phase-matched materials as well as the development of new optical parametric oscillators are areas in which progress is continuing Ultrafast optics and the sub-picosecond domain of optical characterization offer interesting and challenging avenues for probing the properties of materials and developing new applications Furthermore, new techniques are continually being developed to measure and modify the properties of materials for diverse applications such as optical limiting, nonlinear fluorescent imaging, and two-photon photopolymenzation n Preface to the Second Edition As in the previous edition, selection of topics for inclusion was based on a certain bias for what has been important to me as a general practitioner of nonlinear optics In this regard, I have chosen to add work done in my group that is relevant primarily to the characterization and application of nonlinear materials However, in the interest of properly setting the stage for the bulk of the book, and because it so often seems to be a point of confusion for beginners, I have expanded the first chapter, which deals with elements of nonlinear optical theory Chapter 2, "Frequency Doubling and Mixing," and Chapter 3, "Optical Parametric Generation, Amplification, and Oscillation," so important in the generation of light for other nonlinear optics applications, have been expanded and updated primarily to include new results reported in the literature Chapters ("Nonlinear Index of Refraction") ("Characterization of Nonlinear Refractive Index Materials"), ("Nonlinear Absorption"), and 10 ("Experimental Techniques in Nonlinear Absorption') all incorporate new material Several of the chapters tabulating materials data (Chapters 5, 8, and 13) have also been updated Chapter 13 replaces Chapter 11 in the previous edition Two new chapters (Chapter 11, "Ultrafast Characterization Techniques," and Chapter 12, ' Laser Flash Photolysis") have been added, covering important topics in the expanding characterization requirements of nonlinear materials Finally Chapter 17, "Electro-Optic Effects," has been added because the effect plays such a central role in several devices used in optics, as well as in the photorefractive effect, and because it is arguably a nonlinear effect, depending as it does on the interaction of two or more electric fields This second edition also afforded the opportunity to correct errors and misprints that occurred in the first edition My gratitude goes to those who have graciously pointed these out to me As always, I am indebted to several people who have been of great help in preparing this work Not the least of these is my family, which has stood beside me with patience and support I would also like to thank my employer, SAIC, and the U S Air Force Research Lab (AFRL/MLPJ) for their encouragement of this project Finally, I acknowledge my colleagues for their helpful advice and criticism, especially Scan Kirkpatrick and Daniel G McLean, who authored Chapters 11 and 12, respectively, and Suresh Chandra, who contributed to Chapter Richard L Sutherland Preface to the First Edition Shortly after the demonstration of the first laser in 1960, Peter Frankin and coworkers ushered in nonlinear optics (NLO) with the observation of second harmonic generation in a quartz crystal Since then, NLO has burgeoned into a mature field of science and engineering The scope of this discipline includes all phenomena in which the optical parameters of materials are changed with irradiation by light Generally, this requires high optical intensities, which is the main reason that NLO matured in parallel with laser technology Judging by the growth and continued good health of publications and international conferences on the subject, NLO appears to have a strong future in areas of photonics devices and scientific investigations The impact of NLO on science and technology has been twofold First, it has enhanced our understanding of fundamental light-matter interactions Second, it has been a driving force in the rejuvenation of optical technology for several areas of science and engineering NLO has matured in the sense of being a well-developed and systematic theory as well as providing applications for a vanety of engineering tasks Second and third order phenomena and devices are now at a stage of understanding and development such that a coherent description and summary of these areas forming the core of the subject are now possible and desirable The rapid development of the subject has created the need for a handbook that summarizes technical details concerning core areas impacting several engineering and scientific endeavors The general practitioner of NLO requires information in at least four critical areas: (1) mathematical formulas applicable to a variety of experimental and design situations, (2) examples of ways NLO is applied to specific technical problems, (3) a survey of device and matenals data for comparison purposes and numerical evaluation of formulas, and (4) in-depth descriptions of methods required for characterizing new matenals When seeking this information, novice and expert alike are often 932 Chapter 17 Figure 33 Spectral transmittance of a liquid-crystal Sˇolc fan filter illustrating transmittance peaks lm, free spectral range DlFSR, and FWHM bandwidth DlBW with bandwidth DlBW < 1:6 lm ð2m þ 1ÞN ð210Þ The tuning range is dl ¼ dðDnÞ lm Dn ð211Þ The operation of the fold filter can be understood by the following argument For a peak wavelength lm, each plate is a perfect half-wave plate Light that travels through a half-wave plate, with its polarization at an angle r with respect to the slow axis, has its polarization rotated through an angle 2r This is what occurs at the first plate in the fold filter The light incident on the second stage is thus polarized at an angle 2r with respect to the slow axis of that plate Transmitted light is now rotated through an angle 4r, but in the opposite sense This process progresses at each stage so that the next successive rotations are 6r, 8r, 10r, 12r, etc At the last stage the polarization is rotated parallel to the exit polarizer axis Polarized light at other nearby wavelengths is not so perfectly rotated and is thus attenuated Electro-Optic Effects 933 Tunable Fabry – Perot filters A Fabry – Perot filter is an example of a tunable filter that is not based on birefringence It consists of a dielectric medium sandwiched between two high-reflectance, parallel mirrors as illustrated in Fig 34 Incident light leaks through the first mirror and undergoes multiple reflections in the mirror cavity If the round-trip optical path length of the cavity is an integer multiple of a wavelength l, light at that wavelength will experience a maximum transmittance (theoretically 100%) through the filter The transmittance of the Fabry –Perot filter is given by T¼ 1 þ F sin2 ð2pnL=lÞ ð212Þ 4R ð1 RÞ2 ð213Þ where F¼ is called the fringe contrast, with R the reflectance of each mirror The transmittance depends on the refractive index of the medium, and the peak transmittance wavelengths are lm ¼ 2nL m m ¼ 1; 2; 3; ð214Þ The mode number m for the Fabry –Perot can be quite large The peak transmittance is 1, and the minimum is (1 þ F )21 The fringe contrast can be large for high reflectivity mirrors For example, if R ¼ 0:9; F ¼ 360: The FSR is DlFSR < lm l2 ¼ m m 2nL ð215Þ for m 1; while the bandwidth is DlBW < l2m pffiffiffiffi pnL F ð216Þ for F 1: Figure 34 Electro-optic tunable Fabry – Perot filter using a crystal with 3m symmetry 934 Chapter 17 As an example of a tunable Fabry –Perot filter, consider LiNbO3 with its optic axis normal to the mirrors and a field applied in this direction, as illustrated in Fig 34 Unpolarized light incident normally on the filter will see an index given by n ¼ n8 ðn8Þ3 r 13 E ð217Þ where E ¼ V=L: For L ¼ mm and m ¼ 2800; lm ¼ 1536 nm: For this mode the tuning range is 3.4 £ 1025 nm/V, or 34 nm for MV The FSR would be 0.5 nm, while the bandwidth for R ¼ 0.8 mirrors would be 0.04 nm Now consider the parallel aligned nematic LC cell with E7 in the same Fabry –Perot arrangement, but with a mirror spacing of 10 mm Polarized light is incident on the filter, as shown in Fig 35, with the polarization along the optic (director) axis Thus, with no voltage applied, incident light will see an index n e ¼ l:75 for E7 For m ¼ 50; the transmitted wavelength is 700 nm The FSR with respect to this mode is 14 nm, while the bandwidth is 1.0 nm Note again that modes are more densely packed for thicker filters Since the net index change, for an applied voltage of 4– V, is in this case approximately the birefringence of the LC, the tuning range is dl ¼ ðDn=n e Þlm ¼ 88 nm: The spectral transmittance for an E7 Fabry – Perot filter is shown in Fig 36 F Switchable Filters Whereas tunable filters are adjusted to filter a specific wavelength, or set of wavelengths, sometimes it is desirable to adjust the efficiency of a filter at Figure 35 Electro-optic tunable Fabry– Perot filter using a nematic liquid crystal Electro-Optic Effects 935 Figure 36 Spectral transmittance of a liquid-crystal Fabry– Perot filter illustrating transmittance peaks lm, free spectral range DlFSR, and FWHM bandwidth DlBW a specific wavelength In particular, one may want to turn the filter efficiency on (a maximum) or off (a minimum) This type of device is called a switchable filter Two types of electrically switchable filters will be examined here Both are based on the theory of Bragg diffraction The first is a holographic polymer-dispersed liquid crystal (HPDLC) reflection filter, and the second is a cholesteric liquid crystal (CLC) filter HPDLC switchable reflection filter HPDLCs were discussed in Section III.E They are based on a periodic perturbation of the dielectric tensor, which is a tensor hologram An unslanted reflection hologram is illustrated in Fig 37 Unslanted implies that the grating vector Kg is normal to the surfaces of the hologram Incident light scatters from the grating planes If certain conditions are met (see below), the scattered light from all grating planes interferes constructively to produce a coherent diffracted beam in a specific direction In all other directions the scattered light interferes destructively Since the diffracted beam appears on the same side of the hologram on which the incident light impinges, this hologram is called a reflection grating or reflection hologram The ratio of reflected optical power to incident optical power is called the diffraction efficiency h The type of diffraction described above is called Bragg diffraction To achieve this type of diffraction (single coherent diffracted beam), the grating must be optically thick The criteria for this are established with reference to two 936 Chapter 17 Figure 37 Electro-optic switchable Bragg filter using holographic polymer-dispersed liquid crystals (a) Reflecting state and (b) transmitting state parameters, namely Q ¼ 2p lL nL2 ð218Þ and V¼ l2 nn1 L2 ð219Þ where l is the wavelength of incident light, L is the grating thickness, L ¼ 2p=jK g j is the grating period (also called the pitch, grating spacing, or grating constant), n is the average refractive index, and n1 is the average index modulation amplitude (see below for a further discussion of refractive index parameters for a tensor grating) The optically thick grating regime is determined by the conditions Q and V [26] Generally, this implies that L l and n1 ,, 1: If these conditions are met, then light at a specific wavelength, called the Bragg wavelength lB, will be diffracted with high efficiency For an unslanted grating, the angle of diffraction will equal the angle of incidence u, and the Bragg wavelength lB is determined by the Bragg condition, lB ¼ 2nLcos u ð220Þ Electro-Optic Effects 937 A grating in the optically thick regime (i.e., a Bragg hologram) can be analyzed using coupled-wave theory [27] Under the conditions described in Section III.E, the average dielectric tensor of the medium is given by ð0Þ 1x 0 C B ð0Þ B C ð221Þ ð0Þ ¼ B 1y C A @ ð0Þ 0 1z while the modulation tensor of the medium is ð1Þ 1x 0 B C ð1Þ B C ð1Þ ¼ B 1y C @ A ð1Þ 0 1z ð222Þ and the tensor components are given by the combination of Eqs (146 – 148) Note ð0Þ that by Eq (148), the HPDLC medium is essentially uniaxial with 1ð0Þ x ¼ 1y ¼ ð0Þ e ð1Þ ð1Þ 10 ðn8Þ ; but 1z ¼ 10 ðn Þ and 1x ¼ 1y :Thus, the effective average index of the medium, as used in the discussion above, depends on the polarization of the light and its direction of propagation Note also that both ordinary and extraordinary refractive indices will change with the applied voltage For many applications these indices, however, are approximately equal The average index modulation is related to the components of (1) and will also depend on polarization and direction of propagation The incident and diffracted waves are often called the reference (R) and signal (S) waves, respectively The diffraction efficiency for a beam of wavelength l incident at uR and diffracted at an angle us with respect to the film normal is given by [3,27] hpffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffii ðkLÞ2 ðDkL=2Þ2 k2 sinh2 hpffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffii hpffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffii h¼ ðkLÞ2 ðDkL=2Þ2 þ ðDk=2Þ2 sinh2 ðkLÞ2 ðDkL=2Þ2 ½k2 ðDk=2Þ2 Šcosh2 ð223Þ where k¼ pe^ S ·1 ð1Þ ·^eR pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 210 ngl jcos uS jcos uR is the grating coupling coefficient, with pffiffiffiffiffiffiffiffiffiffi n ¼ nS nR pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi g ¼ cos aS cos aR ð224Þ ð225Þ ð226Þ 938 Chapter 17 In these equations, eˆS and eˆR are the polarization vectors, nS and nR the refractive indices, aS and aR the angles between the wavevectors and Poynting vectors of the respective waves Note that uS ¼ p uR ; so cos uS , 0: The quantity Dk is the phase mismatch or detuning factor from the Bragg wavelength and is given by   2n Dk ¼ 2p ð227Þ l L where n is a function of the applied voltage An example of the filter function for lB ¼ 540 nm and normal incidence, illustrating switching for various values of the applied field, is plotted in Fig 38 The peak efficiency is given by h ðpeakÞ ¼ tanh2 kL ð228Þ and the bandwidth is DlBW < kl2B pn ð229Þ The expression for the bandwidth in Eq (228) is not exactly the FWHM value; it is given by the detuning condition for which the argument of the hyperbolic sine becomes imaginary It gives an approximate measure of the FWHM bandwidth, Figure 38 Spectral transmittance of an HPDLC Bragg filter illustrating reflectance peak lB, bandwidth DlBW, and switching Field values are given in terms of the critical field Ec Electro-Optic Effects 939 as illustrated in Fig 38 Another expression for the bandwidth is given by the wavelengths of the first zeros in the diffraction efficiency This yields qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi l2 ð230Þ Dl 0BW ¼ B k2 þ ðp=LÞ2 pn and is seen to decrease with increasing thickness L The bandwidth limit for large L reduces to Eq (229) Fig 38 illustrates the bandwidth and switching for an HPDLC Bragg filter The coupling coefficient and hence the diffraction efficiency are polarization dependent Note that the reference and signal waves will have the same polarization In an unslanted grating, p uS ; uR ; u: Since the film is overall weakly birefringent, cos aR ¼ cos aS < l: For s-polarized waves, ks ¼ p 1ð1Þ 210 n8l cos u y ð231Þ while for p-polarized waves, kp ¼ p ð1ð1Þ cos2 u 1ð1Þ z sin uÞ 210 n e ðuÞl cos u x ð232Þ Hence, the index modulation n1, referred to above, depends on polarization and angle of incidence It is of magnitude 1ð1Þ =210 n; given in terms of the appropriate components The switching of the filter for off-normal incidence will depend on polarization For an s-polarized wave, the filter will be switched off at an applied field E Ec ; where Ec is given by Eq (137), for which 1dy ! 1p ; with 1dy given by Eq (148) For p-polarization, the switching will be determined by a ð1Þ zero-crossing of kp, i.e., when 1ð1Þ x ¼ 1z tan u: Note that this will occur at some nonzero value of the applied field since 1xð1Þ decreases while 1ð1Þ z increases with increasing field, cf Eqs (147) and (148) For normal incidence, the switching will be independent of polarization CLC switchable reflection filter CLC cells were discussed in Section III.B The azimuth angle of the CLC directors precesses about the z axis, which is normal to the substrates, at a rate q, viz fðzÞ ¼ qz ¼ 2p z p ð233Þ where p is the pitch of the CLC and q ðq , 0Þ corresponds to a right-hand (left-hand) twist Light polarized along the local director sees a refractive index n e, while light polarized perpendicular to this direction sees n8, where Dn ¼ n e n8 is the birefringence For light propagating along the z axis, the normal modes of propagation in this inhomogeneous medium can be computed exactly in terms of Bloch waves [11] The normal modes are elliptically polarized, in 940 Chapter 17 general Four distinct regimes appear in the dispersion relations, depending on wavelength Mauguin regime ðl ,, ð1=2ÞpDnÞ: In this short wavelength regime, the normal modes are almost linearly polarized parallel and perpendicular to the local director Short wavelength circular regime ðð1=2ÞnpDn ,, l ,, pÞ: In this regime, the normal modes are almost circularly polarized with opposite handedness (i.e., right and left circularly polarized modes) Bragg regime ðn8p , l , n e pÞ: Here the wavevector of one mode is purely imaginary, which corresponds to an evanescent wave This mode cannot propagate in the medium The other normal mode is almost circularly polarized with the same handedness as the CLC helix Long wavelength circular regime ðl n e pÞ: In this long wavelength regime, the normal modes are both almost circularly polarized with opposite handedness The interesting regime for filters is the Bragg regime, where the CLC acts as a reflection filter The incident light is not necessarily one of the normal modes If the birefringence is not too large, so that ðn e Þ2 ðn8Þ2 ,, ðn e Þ2 þ ðn8Þ2 ; then the wave propagation can be analyzed using coupled-wave theory [11] The field inside the CLC is taken to be a superposition of incident and reflected waves with wavevectors ^ k given by sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2p ðn e Þ2 þ ðn8Þ2 ð234Þ k¼ l If a left-hand circularly polarized wave is incident on a CLC with a right-hand twist (or, a right-hand circularly polarized wave incident on a left-handed CLC), then the wave is reflected with an efficiency R given by R¼ s2 k sinh2 sL þ ðDk=2Þ2 sinh2 sL cosh2 sL ð235Þ where s ¼ k 2 ðk qÞ2 ð236Þ Dk ¼ 2ðk qÞ ð237Þ Â p1ffiffi p ðn e Þ2 ðn8Þ2 k ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi l ðn e Þ2 þ ðn8Þ2 à ð238Þ Note the similarity of Eq (235) to Eq (223) The filter efficiency has the same Electro-Optic Effects 941 form as shown in Fig 38 for the HPDLC Bragg reflection filter However, for unpolarized incident light, the CLC filter has a peak reflection efficiency of Rpeak ð239Þ unpolarized ¼ kL Thus, to obtain high rejection efficiency for unpolarized light (or for linearly polarized light) two CLC filters of opposite handedness in tandem are required When a voltage V V th is applied, where Vth is given by Eq (130), R is switched to a value near zero G Beam Deflectors and Scanners Two types of beam deflector or scanner are described One is called an analog deflector (or continuous scanner), and the second is a digital deflector Analog deflectors Analog deflectors or scanners can be based on prismatic refraction One example is shown in Fig 39 An electric field is applied as shown At zero field, the prism is adjusted so that ue ¼ ui : When a field is applied, the index of the prism changes from n to n þ Dn: The exit beam is then deflected by an angle ui þ du; where Dn du ¼ tan ui ð240Þ n With tan ui , and Dn , ð1=2Þn r eff E; du , n r eff E: Thus, for an applied field of order kV/cm, the angle scanned is of the order of a few microradians A second example of an analog deflector is shown in Fig 40 In this case, two prisms of KDP are combined, with the optic axes of the two prisms oriented in opposite directions [28] The field is applied along the optic axis Incident light is polarized along either the X or Y axis (i.e., along one of the principal axes that is induced by the applied field) With zero field, the beam is not deflected When Figure 39 Electro-optic analog beam scanner based on a triangular Pockels prism 942 Chapter 17 Figure 40 Electro-optic analog beam scanner based on a pair of Pockels prisms with  42m symmetry a field is applied, the index of the upper prism decreases while the index of the lower prism increases This produces an effective index gradient that deflects the beam by du ¼ ðn8Þ3 r63 E L D ð241Þ where L is the length of the deflector prisms and D is the aperture Note that the deflection for this device is , n(L/D ) larger than that of the first example However, the deflection is still of the order of a few tens of microradians for a field of order kV/cm Thus, analog deflectors/scanners can fine tune the beam deflection, but digital deflectors are required to obtain larger deflections Digital deflectors One example of a digital beam deflector is illustrated in Fig 41 Polarized light is incident on an electro-optic polarization converter (e.g., one of the types discussed earlier in this section), which is followed by a static birefringent beam deflector There may be a variety of these, but the one illustrated in this example consists of two prisms composed of a uniaxial medium with optic axes oriented orthogonally With no voltage applied to the polarization converter, the light incident on the deflector is vertically polarized Thus, the first prism has an index n8 while the second prism has a larger index (, n e if the beam deflection is only a few degrees) In this case the beam is deflected upward (toward the normal to the interface) When a half-wave voltage is applied to the polarization converter, light incident on the beam deflector is now horizontally polarized In this case the first prism has a high index (n e) while the second prism has a low index (n8), so the beam is deflected downward (away from the normal to the interface) For a prism angle a (as illustrated in Fig 41), the exit angles u1 and Electro-Optic Effects 943 Figure 41 Digital beam scanner using an electro-optic polarization converter followed by a static birefringent beam deflector u2 are found by solving the following equations: sin u1 ¼ n e ða u 01 Þsinða u 01 Þ n e ða u 01 Þsinu 01 ¼ n8sin a ð242Þ sin2 ða u 01 Þ cos2 ða u 01 Þ ¼ þ ½n e ða u 01 ފ2 ðn8Þ2 ðn e Þ2 and sin u2 ¼ n8sinðu 02 aÞ n8sin u 02 ¼ n e sin a ð243Þ A second example of a digital deflector is an unslanted HPDLC transmission grating illustrated in Fig 42 For a transmission hologram, the diffracted signal wave is on the opposite side of the hologram from which the incident reference wave impinges, and “unslanted” implies that the grating vector is parallel to the hologram substrates A transmission hologram can theoretically have a diffraction efficiency of 100% Thus, when no voltage is applied to the HPDLC, the exit beam is deflected (diffracted 100%) by an angle equal to twice the Bragg angle With a voltage applied such that the index modulation reduces to zero (zero diffraction efficiency), the exit beam will not be deflected [29] Relatively large deflection angles (ten’s of degrees) can be readily achieved For s-polarized light, the coupling coefficient is proportional to 1dy 1p : Since 1dy < 1p for any voltage, the diffraction efficiency of s-polarized light will always be weak Hence, this beam deflector will only work well for p polarization For this 944 Chapter 17 Figure 42 Digital beam scanner using an HPDLC switchable Bragg transmission grating (a) Transparent state and (b) diffracting state case, the coupling coefficient is proportional to ð1dx 1p Þcos2 uB ð1dz 1p Þsin2 uB ; where uB is the Bragg angle The droplet tensor elements are given in Eqs (149) The diffraction efficiency is given by   e^ S ·1 ð1Þ ·^eR h ¼ sin L ð244Þ 210 nl cos uB where eˆS (eˆR) is the polarization unit vector of the signal (reference) wave, and n ¼ n8 for s polarization and n ¼ n e ðuB Þ for p polarization REFERENCES M Born and E Wolfe, Principles of Optics, 5th ed., Pergamon Press, New York, 1975 V I Smirnov, Linear Algebra and Group Theory, Dover Publications, Inc., New York, 1961 Electro-Optic Effects 10 11 12 13 14 15 16 17 18 19 20 21 22 23 945 A Yariv and P Yeh, Optical Waves in Crystals, John Wiley, New York, 1984 D W Berreman, Optics in stratified and anisotropic media: £ matrix formulation, J Opt Soc Am 62:502 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Characterization of Second Order Nonlinear Optical Materials 241 Properties of Selected Second Order Nonlinear Optical Materials 295 Nonlinear Index of Refraction 337 Characterization of Nonlinear Refractive... Implementation, Second Edition, edited by Norman J Berg and John M Pellegrino 52 Handbook of Nonlinear Optics, Richard L Sutherland 53 Handbook of Optical Fibers and Cables: Second Edition, Hiroshi... definition of Eq (1) since the field E (,) is a real quantity Elements of the Theory of Nonlinear Optics For a large number of problems in linear and nonlinear optics, the field can be assumed to be of