Progress in Optics, Volume 38

EDITORIAL ADVISORY BOARD G. S. AGAUWAL,

Ahmedabad, India

T. ASAKURA,

Sapporo, Japan

M.V: BERRY,

Bristol, England

C. COHEN-TANNOUDJI, Paris, France

V. L. GINZBURG,

Moscow, Russia

F. GORI,

Rome, Italy

A. KUJAWSKI,

Warsaw, Poland

J. Rbl.4,

Olomouc, Czech Republic

R. M. SILLITTO,

Edinburgh, Scotland

H. WALTHER,

Garching, Germany

PROGRESS IN OPTICS VOLUME XXXVIII

EDITED BY

E. WOLF

University of Rochester, N. r, l%S.A.

Contributors S. DUTTA GUPTA, P. HELLO, J.L. HORNER, J. JAHNS, B. JAVIDI, A.W. L O H M A " , D. MENDLOVIC, W NAKWASKI, M. OSfiSKI, 2. ZALEVSKY

1998

ELSEVIER AMSTERDAM. LAUSANNE .NEW YORK OXFORD. SHANNON. SINGAPORE. TOKYO

ELSEVIER SCIENCE B.V

SARA BURGERHARTSTRAAT 25 PO. BOX 21 1 1000 AE AMSTERDAM THE NETHERLANDS

Library of Congress Catalog Card Number: 61-19297 ISBN Volume XXXVIII: 0 444 82907 5

0 1998 Elsevier Science B.V. All rights reserved No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without the prior written permission of the publisher, Elsevier Science B.V, Rights & Permissions Department, P.O. Box 521, 1000 AM Amsterdam, The Netherlands. Special regulations for readers in the USA: This publication has been registered with the Copyright Clearance Center Inc. (CCC), 222 Rosewood Drive, Danvers, MA 01923. Information can be obtained from the CCC about conditions under which photocopies of parts of this publication may be made in the USA. All other copyright questions, including photocopying outside of the USA, should be referred to the Publisher, unless otherwise specified. No responsibility is assumed by the publisher for any injury and/or damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions or ideas contained in the material herein. @ The paper used in this publication meets the requirements of ANSI/NISO 239.48-1992 (Permanence of Paper).

PRINTED IN THE NETHERLANDS

PREFACE It is a pleasure to report two happy events. Professor Claude Cohen-Tannoudji, a long-time member of the Editorial Advisory Board of Progress in Optics has been named co-recipient of the 1997 Nobel Prize in Physics, and Professor Michael Berry, whose important contributions to optics and other fields of physics are also well known, has agreed, not long ago, to become a member of the Board. The present volume contains six review articles on a wide range of topics of current research in optics. The first article, by S. Dutta Gupta, deals with various nonlinear optical phenomena in stratified media. It shows that resonances which arise from stratification are of considerable importance for achieving low-threshold nonlinear optical devices. The article also includes a thorough study of optical bistability and harmonic generation in Kerr nonlinear layered media, and various phase matching techniques are discussed. Recent trends involving novel geometries and new materials are outlined. More recent developments concerning gap solitons in periodic structures, weak photon localization in quasi-periodic structures and enhancement of nonlinear susceptibilities in layered composites are also discussed. The second article, by I? Hello, reviews the optical aspects of interferometric gravitational-wave detectors. Different optical configurations are reviewed and their sensitivities are estimated for typical values of the optical parameters. The next article, by W. Nakwaski and M. Osinski, presents a review of temperature-related effects and thermal modeling of vertical-cavity surfaceemitting lasers (VCSELs). The effects of temperature on the characteristics of such devices are discussed, including the temperature dependence of the longitudinal mode spectra, the transverse-mode structure and the output power. The principles of thermal VCSEL modeling are then outlined. Both analytic and numerical approaches are treated. Finally, the most important results obtained by the use of such models are presented. The fourth article, by A.W. Lohmann, D. Mendlovic and Z. Zalevsky, entitled Fractional Transformations in Optics, describes some recent theoretical developments in mathematical techniques that are used in physical optics and in optical information processing. Many of the usual transforms used in these fields V

vi

PREFACE

contain various parameters which are integers. The generalizations reviewed in this article have parameters which take on fractional or even complex values. The article surveys these developments and discusses the use of such generalized transforms in some areas of optics. The article that follows, by B. Javidi and J.L. Homer, discusses a number of Fourier-plane nonlinear filtering techniques for use in image recognition. Such nonlinear filters can be optically implemented by a processor known as a joint transform correlator. Their performance is discussed and the use of nonlinear techniques in the design of distortion-invariant composite filters for image recognition is considered. The use of joint transform correlators for security verification of credit cards, passports and other documents is also discussed. The concluding article, by J. Jahns, presents an overview of the field of optical digital computing and interconnection. Following an outline of the historical development of the subject, the motivation for the use of free-space optics in computing applications is discussed. Computational aspects of nonlinear optical devices and optical interconnections and their implementations are then reviewed. The article concludes with an overview of architectures and systems for free-space optical computing and switching. Emil Wolf Department of Physics and Astronomy University of Rochester Rochester, New York 14627, USA January 1998

E. WOLF, PROGRESS IN OPTICS XXXVIII 0 1998 ELSEVIER SCIENCE B.V ALL RIGHTS RESERVED

I NONLINEAR OPTICS OF STRATIFIED MEDIA BY

S. DUTTA GUPTA School of Physics, University of Hyderabad, Hyderabad 500046, India

I

CONTENTS

PAGE

Q 1. NTRODUCTION

. . . . . . . . . . . . . . . . . . .

9 2. NONLINEAR TRANSMISSION AND OPTICAL BISTABILITY

3

IN LAYERED MEDIA . . . . . . . . . . . . . . . . .

6

Q 3 . HARMONIC GENERATION AND OTHER NONLINEAR PHENOMENA IN LAYERED GEOMETRY . . . . . . . .

50

$ 4. NONLINEAR OPTICAL PROPERTIES OF LAYERED

COMPOSITES . . . . . . . . . . . . . . . . . . . .

66

$ 5 . CONCLUSIONS . . . . . . . . . . . . . . . . . . .

75

ACKNOWLEDGEMENT . . . . . . . . . . . .

76

REFERENCES . . . . . . . . . . . . . . . . . . .

76

2

8

1. Introduction

The past two decades have witnessed an intense development of nonlinear optics of stratified media. Surveying the past, present and fbture of nonlinear optics, Bloembergen [ 19921 commented that nonlinear optics has entered the technology phase. This has been possible due to a tremendous development both in the understanding of the underlying phenomena and in the remarkable growth of technology. Now diode lasers with sub-watt power levels can be used to observe most nonlinear optical effects. As pointed out by Stegeman [1992], two factors, namely the development of new nonlinear optical materials with better characteristics and a manipulation of the sample geometry, have played the most significant role. In the context of the latter, optical fibers (Agrawal [1989]) and stratified layered media (Stegeman [ 19921) have been of great importance. Our review explores the various possibilities of nonlinear optics in layered configuration, even to the extent of fabricating a nonlinear medium with better characteristics. Linear properties of layered media are well documented (see, e.g., Yeh [1988]). It is now well understood that the optical properties of a layered medium can be distinct from those of its bulk constituents. The simplest possible example is the “structural” dispersion, in stratified geometry. Even with a single dielectric slab (Fabry-Perot (FP) cavity) lacking in material dispersion the transmission is frequency selective due to multiple reflections from the interfaces. The effect is more drastic for a multilayered geometry, for example, for a periodic layered medium, where the interference of the forward and backward propagating waves in each slab can lead to frequency stopgaps. Guided waves (see, e.g., Hunsperger [ 19841) with geometry-dependent dispersion is another technologically significant example. The most important feature of layered structures is their ability to support resonances, which are always associated with local field enhancements. It is in this context that low threshold nonlinear phenomena can be realized in such structures. Guided wave structures play an important role from the viewpoint of device applications. The confinement of the electromagnetic field in waveguides can lead to large power densities over a longer length compared with what can be realized in bulk samples. In addition, they offer the possibility of integration of various functions 3

4

NONLINEAR OPTICS OF STRATIFIED MEDIA

[I, 0 1

on the same optical “chip”. In fact, integrated optics and its nonlinear extension have developed to a stage that the realization of all-optical chips is only a matter of time. In the context of nonlinear stratified media the theoretical and experimental achievements have been fascinating. Plausible theories now explain most nonlinear effects, including self-action optical bistability, frequency up-and-down conversion, four-wave mixing, and phase conjugation, self-focusing, and spatial solitons. These effects have been observed experimentally. Predictions have been made of new physical phenomena that may be observed in the future. One example is the prediction of new nonlinearity-induced modes (without any linear counterpart) in Kerr nonlinear waveguides, which need large power levels for excitation. With the availability of high-power lasers the problem lies not in coupling sufficient power to the guided mode, but rather, with the low damage threshold of the nonlinear materials suitable for waveguide fabrication. Thus the search for new waveguide materials with large nonlinearity and high damage threshold continues. Since the early days of nonlinear optics in stratified geometry, Kerr nonlinearity leading to intensity-dependent refractive index has drawn considerable attention. It offers, perhaps, the simplest possible model amenable to satisfactory theoretical analysis. The second factor is the easy availability of Kerr nonlinear materials such as CS2, nitrobenzene, and liquid crystals. As a result, the literature on Kerr nonlinear effects in stratified media, both for normal and oblique incidence (mostly in waveguides), is extensive. We review both the theoretical and experimental achievements, concentrating on the exact theoretical models (Chen and Mills [1987a-c], Leung [1985, 19891) and surveying some recent experiments involving liquid crystal, organic, and semiconductor films. Results pertaining to periodic and quasiperiodic media are discussed separately because of the specific properties of such structures. In fact, prediction of gap solitons (Chen and Mills [1987b]) in nonlinear periodic structures and of weak photon localization in linear quasiperiodic structures (Kohmoto, Sutherland and Iguchi [1987]) was one of the major achievements. We also discuss other approximate and numerical methods. We hghlight the switching and bistability experiments, which hold considerable potential for optical switches and other signal processing and communication applications (for a survey of device potentials of Kerr nonlinear layered media, see, e.g., Assanto [1992]). Harmonic generation, and in particular, second harmonic generation, has remained one of the most pursued branches of nonlinear optics since its inception. In layered configuration it is attractive, since one can have enhancement of

1, § 11

INTRODUCTION

5

the generated harmonic using the resonances of the stratified medium. The development of a general theory for a multilayered medium, albeit without pump depletion, was rather recent (Bethune [1989, 19911, Hashizume, Ohashi, Kondo and Ito [1995]) and we review this theory in great detail. In the context of harmonic generation in waveguides, coupled mode theory has been applied extensively, and the results are well documented (Stegeman and Seaton [1985], Stegeman [1992]). The advantage of guided wave structures in the context of harmonic generation is obvious. In addition to a large power density over a large interaction length, the major advantage stems from the flexibility in options for phase matching. A real technological breakthrough was the realization of quasiphase matching by means of ferroelectric domain reversal in LiNbO3, LiTaO3, and KTiOPO4 (KTP) waveguides (for a detailed treatment of poling techniques in these materials, see, e.g., Fejer [1992], Bierlein [1992]). Along with other mechanisms of phase matching, we describe the major achievements in quasiphase-matched waveguides. We also discuss the case of surface-emitted second harmonic with counterpropagating fundamental waves. Note that some of these second harmonic devices using the infrared (IR) diode lasers can lead to efficient blue light sources that are much needed for the xerography and laser printing industry. In addition to harmonic generation, we review some recent trends using cascaded second-order nonlinearity leading to efficient “third” order processes. We also outline some four-wave mixing experiments along with new theoretical proposals. In the context of better nonlinear optical materials, a new proposal to enhance the effective nonlinear susceptibility exploiting the local field corrections in layered composites was advanced (Boyd and Sipe [1994]) and tested (Fischer, Boyd, Gehr, Jenekhe, Osaheni, Sipe and Weller-Brophy [ 19951). After briefly surveying the properties of nonlinear Maxwell Garnett composites, we discuss the details of the proposal and summarize the experimental observations. Since the field is rather new, the device applications of such composites have not yet been probed. In view of the vast literature available, the scope of the current review is limited. For example, we restrict discussion only to macroscopic nonlinear optical effects in the framework of a classical theory based on Maxwell’s equations. It is clear that for sufficiently thin layers, quantum confinement effects (for reviews, see, e.g., Flytzanis, Hache, Klein, Ricard and Roussignol [1991], Flytzanis [1992]) can become important, leading to the breakdown of the classical description. We thus consider layered media, where each layer can be characterized by its macroscopic parameter like the dielectric function. The other limitation is that we describe only the parametric processes, and do not

6

NONLINEAR OPTICS OF STRATIFIED MEDIA

[I, 0 2

cover important topics such as Raman and multiphoton processes, although we briefly discuss the multiphoton-induced Kerr-like effect. Section 2 is devoted to Kerr nonlinear effects and is divided into four parts, the first three of which examine theoretical developments and the fourth summarizes the experiments. The cases of normal and oblique incidence are covered in the first two sections, and the third discusses the properties of periodic and quasiperiodic media. Section 3 describes harmonic generation and other effects like cascaded second-order processes and four-wave mixing. Section 4 discusses the properties of nonlinear composite materials, especially layered composites.

Q 2. Nonlinear Transmission and Optical Bistability in Layered Media The nonlinear optical effects in layered media (for earlier reviews see, e.g., Stegeman and Seaton [ 19851, Stegeman, Seaton, Hetherington, Boardman and Egan [ 19861, Mihalache, Bertolotti and Sibilia [ 19891, Langbein, Lederer, Peschel, Trutschel and Mihalache [ 19901; see also Ostrowsky and Reinisch [ 19921) can be diverse, depending on the nature of the nonlinearity. An important nonlinear optical effect, optical bistability, originates in the intensity dependence of the real and imaginary parts of the refractive index. Since the real (imaginary) part of the refractive index defines the dispersive (absorptive) properties of the medium, the resulting multivalued response was labeled as dispersive (absorptive) bistability. Bistable response resulted when the nonlinear medium was contained in a Fabry-Perot cavity. Thus, feedback was identified as an important factor needed to have bistable response. The proposal for optical bistability with nonlinear cavities was advanced almost three decades ago by Seidel [1969] and Szoke, Daneu, Goldhar and Kurnit [1969]. Experimental observation of optical bistability (McCall, Gibbs and Venkatesan [ 19751, Gibbs, McCall and Venkatesan [1976]) was delayed because of the problems of fabricating a high finesse cavity. The importance of optical bistable devices in the context of optical signal processing as well as optical computing (see, e.g., Gibbs, Mandel, Peyghambarian and Smith [ 19861) was well understood. Although the initial studies on optical bistability were in Fabry-Perot geometry, later proposals were advanced for having mirrorless bistability (Bowden and Sung [1980, 19811). Nonlinear phenomena other than self-action of the wave were suggested for having bistable output. Winful and Marburger [ 19801, Flytzanis and Tang [ 19801, Agrawal and Flytzanis [1981] and several others demonstrated the possibility of optical bistability in degenerate four-wave mixing and phase conjugation. An exhaustive treatment of bistable and multistable behavior in

I,

9 21

NONLINEAR TRANSMISSION AND OPTICAL BISTABILITY IN LAYERED MEDIA

I

optical systems was presented by Gibbs [1985]. Our review attempts to examine the later developments in the context of layered geometry, beginning with the theoretical developments and followed by the experimental results. We restrict discussion to dispersive bistability because of its tremendous potential for device applications. In the context of layered media one can classify the problems under two broad categories, namely, when the structure is illuminated by radiation incident normally and at an angle. It is clear that in the first case, one concludes with a system of coupled nonlinear and linear Fabry-Perot cavities with no surface excitations, whereas in the second case one has to take into account the possible surface and guided modes of the structure. Note that irrespective of whether the incidence is normal or oblique, the resonances of the structure are responsible for the local field enhancements leading to low threshold nonlinear optical effects. This review addresses the cases of normal and oblique incidence separately. 2.1. NORMAL INCIDENCE

Marburger and Felber [ 19781 first studied dispersive bistability using Fabry-Perot cavity. The cavity was represented by the mirror reflection coefficient R and the intracavity medium was assumed to have a Kerr-type nonlinearity. As pointed out later by Leung [1989], the problem of transmission through a nonlinear slab is more complicated when compared with that of a Fabry-Perot cavity with localized feedback (mirrors at, say, z=O and z = d with given reflectivities). In fact, in the case of a nonlinear layer the reflectivities at the two surfaces are not given, but rather, they must be determined together with the reflectivity of the film in a self-consistent fashion. The difficulties increase substantially when one deals with a layered medium comprising a combination of nonlinear layers. It is thus necessary to have a theory, that can adequately describe the transmission characteristics of a general nonlinear layered media consisting of both linear and nonlinear layers. Approximate and exact methods in the context of Kerr nonlinearity were proposed to solve this complicated problem. We describe the exact method (Chen and Mills [1987a-c]) followed by other numerical and approximate schemes.

2.I . I . Chen-Mills exact solution The exact solutions for a Kerr nonlinear slab for normal incidence of plane polarized wave was first presented by Chen and Mills [1987a]. The general solution was shown to involve four parameters, three of which could be expressed

8

NONLINEAR OPTICS OF STRATIFIED MEDIA

[I,

52

in terms of the fourth, which with proper scaling was bounded between zero and one. Use of appropriate boundary conditions (i.e., continuity of the electric field and its normal derivative) led to the numerical evaluation of these constants, and finally to the power-dependent transmission and reflection coefficients of the structure. A generalization to the case of a nonlinear layered medium consisting of a finite number of nonlinear slabs was presented later (Chen and Mills [ 1987b,c]).We briefly describe details of their analytical and numerical method. Consider first the solution of the nonlinear wave equation for the propagation of a plane y-polarized wave along the z-direction. Measuring the field in terms of the incident field amplitude Eo, the expression for the wave in the nonlinear medium can be written as E(z) = Eo € ( z ) exp[icp(z)].

(2.1)

E(z) given by eq. (2.1) must satisfy the nonlinear wave equation d2E --+k2 dz2

where k

[1+aIE12]E=0,

= (w/c)no

n2 ( 1 ~ 1 ’ )

=

and we have assumed a nonlinearity of the form

+a 1 ~ 1 ~ ) .

(2.3)

In eq. (2.3), no is the low-power limit of the refractive index and a is the nonlinearity constant. Substitution of eq. (2.1) into eq. (2.2) and subsequent integration lead to the following two equations for the phase ~ ( z and ) the amplitude E(z): dcp- w dz E2’

(

$)2

+

+k2E2 + ik2ZUE4 = A .

In eqs. (2.4) and (2.5), W and A are integration constants, and ZU = a (Eo12. The incident field may then be taken to have unit amplitude, and the intensity dependence of the response can be probed by varying 2. Equations (2.4) and (2.5) can be integrated to yield

with Z(z)=E2(z). In writing eqs. (2.6) and (2.7) it was assumed that the nonlinear medium terminates at z = d . It is clear from eqs. (2.6) and (2.7) that the general

I, 5 21

NONLINEAR TRANSMISSION AND OPTICAL BISTABILITY IN LAYERED MEDIA

9

solutions involve four constants, W ,A, Z(d) and q(d), which are to be determined from the boundary conditions. The integral in eq. (2.6) can be expressed in terms of the Jacobian elliptical functions, and results are different for cases when 6 = 0, 6 > 0, and 6 < 0. For 6 = 0 (linear case), one can express Z(z) in terms of elementary functions as follows: 2

A + (A2 - 4k W

2 112

)

. sin f2k(z - d ) + sin-'

(

For a self-focusing nonlinearity (5 > 0), Z(z) is given by

'

(2.8)

where

In eqs. (2.8)-(2.10), Z('), and Z(3) are the roots of the cubic polynomial in the denominator of the integrand of eq. (2.6). These roots are given by

I(" =

-2- (,),I2 36

COS

(8

+ Fl),

1 = 1, 2, 3,

(2.1 1)

and they are arranged such that Z(')>I(') > I ( 3 ) .In eq. (2.1 l), 8 is determined from the relation

(2.12) with 4 2A p = - 3G2 - - + - k26'

16

'=

27&3

4A 2W2 3k2a2 k2a

+7+7.

(2.13) (2.14)

Note that the choice of the sign in eqs. (2.8) and (2.10) is critical to arrive at the correct results. This issue was discussed in detail by Chen and Mills, clearly demonstrating that a proper implementation of the boundary conditions makes the choice unique. The case of defocusing nonlinearity (6 < 0) can be developed

10

[I,

NONLINEAR OPTICS OF STRATIFIED MEDIA

D

2

X

ni f

nt

"j

I

IEt

I

+ Z

~~

YA Zo=O

zz

z,

4-1

Zj

'N-I

N'

Fig. I . Schematic view of a layered medium consisting of N layers hounded on the left (right) by a linear medium with refractive index n, (nt).The j-th layer is characterized by linear refractive index ni and nonlinear coefficient ai.

similarly as for & > 0, but we do not describe it here because of the rather lengthy expressions involved. We now apply the solution for the nonlinear medium to a multilayered system with N nonlinear layers (fig. 1) embedded in vacuum (n,= nt = 1). Let the layers be labeled by integers j ( j = 1,. . . ,N ) . Each layer is characterized by its linear refractive index n, and nonlinearity 6,, electric field amplitude E,, and phase q,.Let the beginning of the structure be at z = 0, and let the boundary between the j t h and j + 1-th layers be at z,. For incidence from the left, the boundary conditions at z = 0, z = z , and z = Z N can be manipulated to yield the following relations: ft1:6lZ:(O)

+ (n:

-

W

A

1)Zi (0) + 4 - 2 - - - = 0, ko k;

ko

=

w

-, C

(2.15)

(2.17)

(2.21) In the units chosen, W is a real number bounded between zero and ko. The numerical procedure treats W as a parameter. A guessed value of W determines

I, 9 21

NONLINEAR TRANSMISSION A N D OPTICAL BlSTABlLITY IN LAYERED MEDIA

11

I N ( z N ) through the relation W = k o Z ( z ~(see ) eqs. 2.4 and 2.21) and the value of the constant A N (eq. 2.20). Since all the constants W , A N , and Z N ( Z N ) for the N-th slab are known (note that the intensity given by eq. (2.6) does not contain the fourth constant C ~ N ( Z N ) )the , solution at the left edge of the N-th slab can be evaluated using eq. (2.9) with eqs. (2.10)-(2.14). One thus knows ZN-I(ZN-~). A N - I is then evaluated using eq. (2.19), with W N= W N - I= W . With the knowledge of all the relevant constants in the ( N - 1)-th layer, the solution can be propagated to the left edge of the ( N - 1)-th layer. The continuation of the procedure leads to Z,(z) for all m. Zl(O), thus evaluated, must satisfy eq. (2.15). This condition chooses the allowed value of the constant W . Once the solution for W is known, the intensity transmission coefficient T can be evaluated using the boundary condition at z = zN. T is given by T

(2.22)

= ZN(ZN).

2.1.2. Other numerical and approximate methods Perhaps the simplest possible approach to the calculation of the reflection and transmission coefficients of a layered medium comprising Kerr nonlinear slabs was the extension of the linear transfer matrix method (see, e.g., Born and Wolf [1989]) to a nonlinear regime. The nonlinear transfer matrix method was first developed by Dutta Gupta and Aganval [1987], who applied the theory to calculate the transmission of a system of single and coupled nonlinear FabryPerot cavities. Optical bistability for both cases was demonstrated, and the role of coupling between the cavities (in the case of coupled Fabry-Perot cavities) was assessed. Later the theory was generalized to multiple layers, and an efficient numerical scheme to handle such systems was presented (Dutta Gupta and Ray [ 19881). We recall the essential steps pertaining to a multilayered medium consisting of N nonlinear slabs (fig. 1) bounded on the left (right) by a linear medium with dielectric constant E, ( E , ) . Let a TE- (or s-) polarized plane wave be incident on the structure from the left. Let the nonlinearity of thej-th layer be given by the nonlinear displacement vector 6NL as follows (Maker, Terhune and Savage [ 19641): (2.23)

x

where E, is the linear dielectric constant, is the constant of nonlinear interaction, A,,BJ are the Kerr and electrostriction nonlinearity constants, and 2,is the electric field vector, all pertaining to the j-th layer. In the j-th slab

12

NONLINEAR OPTICS OF STRATIFIED MEDIA

[I, 0 2

with nonlinearity given by eq. (2.23) the solutions of Marburger and Felber [1978] for the y-component of the electric field in the slowly varying envelope approximation can be written as EJ. = AJ+. e’S+z+ AJ,- e-ikj-2,

(2.24)

with

h&= ko&

(1

u,, = aj IA~*

I

2

+ L!,* , aj

+2U,,)1’2 =

= konj*,

x/ (Aj +Bj).

ko

=

w

-, C

(2.25) (2.26)

In eqs. (2.24)-(2.26), A,, (Aj-), kj+ (kj-), and Uj+ (UJ-) are the constant amplitude, wave vector, and dimensionless intensity, respectively, of the forward (backward) propagating wave. Using eq. (2.24), one can obtain the expression for the tangential component of the magnetic field. Furthermore, one can eliminate A,j+ and A,- from the expressions of the tangential field components at the left and right interface of thej-th slab. This yields the characteristic matrix M j that relates the tangential magnetic and electric field components at the left and the right faces of the j-th slab. The elements of the characteristic matrix mi, are given by m:,

= (a. .I-

e-ikJ+‘1

+ nj .+ eikJ-‘I ) / ( n j + + Cj-1,

(2.27a) (2.27b)

’J+ nJ- elk/-‘/)/(n,++ n,-).

mi2 = (nJ+e-lk/+

(2.27~) (2.27d)

Since the boundary conditions demand the continuity of the tangential components, the application of characteristic matrices to evaluate the tangential field components at any interface becomes straightforward. Henceforth, we assume that all media have the same nonlinearity constant, namely, aJ = a for all j (otherwise one needs to know all a,’s). One starts at the right edge, that is, with the N-th layer, treating the transmitted intensity U, = a IAtI2 ( A , is the transmitted amplitude) as the parameter. Forward and backward wave intensities in the N-th layer can then be expressed as (2.28) where n t = & and 1 . 1 2 implies mod square of the elements of the column matrix. The set of nonlinear algebraic equations (2.28) is solved numerically

I, 8 21

NONLINEAR TRANSMlSSION AND OPTICAL BISTABILITY IN LAYERED MEDIA

13

by any standard technique (e.g., fixed-point iteration) to obtain the values of U N ~A. knowledge of U N yields ~ n N f through eq. (2.25). The evaluation of the characteristic matrix M N using eq. (2.27) for the N-th slab is then straightforward. An analogous procedure is applied to the successive layers. For anyj-th layer ( j 3 1) one thus needs to solve the coupled nonlinear equations

which yields the characteristic matrix Mi for the j-th slab. For a total of N layers the total characteristic matrix is given by (2.30) As in the case of a linear stratified medium, the reflection and transmission coefficients are then given by (2.31) (2.32) Note that if any layer is linear, eq. (2.27) leads to the linear characteristic matrix and solving the coupled nonlinear equations like eq. (2.29) is unnecessary. The generalization of the above theory to include oblique incidence is also straightforward (see 9 2.2.1). We now stress some important aspects of the nonlinear characteristic matrix formalism of Dutta Gupta and Agarwal [ 19871. It may be noted from the solution given by eq. (2.24) that the effective refractive indices experienced by the forward and backward waves, nj+and ni-, respectively, are not the same, leading to the socalled nonreciprocity. In the context of counterpropagating waves with arbitrary polarization, the light-induced nonreciprocity can lead to interesting effects (Kaplan and Meystre [ 19811, Kaplan [ 19831, Kaplan and Law [ 19851). The other important aspect is the nonlinearity of the boundary condition (Agarwal and Dutta Gupta [1987]). The term “nonlinearity of the boundary condtions” is used in the sense that the boundary conditions involve magnetic fields whch are nonlinear functionals of the electric fields in the medium. In other words, the magnetic field amplitudes in thej-th nonlinear medium, as can be seen from

14

NONLfNEAR OPTICS OF STRATIFIED MEDIA

[I, § 2

the derivative of eq. (2.24), are functions of the n,h, which in turn depend on the forward and backward wave amplitudes Aj& of the electric field. Neglect of the nonlinearity of the boundary conditions amounts to ignoring the nonlinearity in the amplitudes of the magnetic field while retaining the same in the phases. Note that the complications associated with the nonlinearity of the boundary conditions does not arise when the Fabry-Perot cavity is characterized by mirrors with given reflection coefficients. This was the approach earlier (McCall [1974], Marburger and Felber [1978], Carmichael and Agrawal [1981], Abraham and Smith [ 19821, Cooperman, Dagenais and Winful [ 19841, Lang and Yariv [ 19861, Nishiyama and Kurita [1986]). Agarwal and Dutta Gupta [1987] discussed the consequences of the neglect of the nonlinearity of the boundary conditions in detail. They considered a nonlinear slab of width d coated on both sides by alternating m low-index and m + 1 high-index linear slabs (fig. 2a), and compared the results with and without the nonlinearity of the boundary conditions. The results for the transmission coefficient for the nonlinear structure are shown in fig. 2b. An increase in the number of periods m of the coating leads to higher finesse of the cavity, resulting in a lower bistability threshold (fig. 2b). It is clear from fig. 2b that a higher bistability threshold leads to higher deviations of the approximate results (with neglect of nonlinearity of boundary conditions) from the exact nonlinear characteristic matrix theory. The corrections are almost insignificant for m = 3 when the threshold (as well as the nonlinear correction to the effective refractive indices) is rather low. Note also that for a given m the deviations are more prominent near the upper bistability threshold, which is due to larger intracavity field intensity at nonlinear resonance. Thus, the full nonlinearity of the boundary conditions is important whenever one deals with relatively large intensities in the nonlinear medium. Finally, we consider the implications of simultaneous neglect of the effects of nonreciprocity and nonlinearity of the boundary conditions. The assumption of the same effective refractive indices for the forward and backward waves together with the neglect of nonlinearity of the boundary conditions amounts to using the standard Fresnel formulas with the linear refractive index nj replaced by the intensity-dependent refractive index nj[l+ (Uj++ Uj-)]"2. In fact, some of the earlier attempts to analyze optical bistability in the context of oblique incidence made use of an analogous simplified approach (Wysin, Simon and Deck [ 19811, Martinot, Lava1 and Koster [1984]). A (computationally) different (although essentially the same) characteristic matrix approach was proposed later by Danckaert, Thienpont, Veretennicoff, Haelterman and Mandel [1989]. A detailed study exploring both the domain of applicability of the matrix method and the validity of the various approximations

I,

5 21

15

NONLINEAR TRANSMISSION AND OPTlCAL BlSTABlLlTY IN LAYERED MEDIA

A, A,

m=3

...

I..

A,

m=2

m= 1

Fig. 2. (a) Schematic view of a nonlinear Fabry-Perot cavity with reflection coatings, composed of m low-index (nb) and rn + 1 high-index (na) k/4 plates. Parameters are chosen as follows: n, = 2.3, q,= 1.3077, n = 1.7149. (b) Transmission coefficient T as a function of the incident intensity Ui : solid (dashed) curves give the results with (without) nonlinearity of the boundary conditions. Different curves are labeled by the values of ni. The parameters have been chosen as follows: kond = (2 - 2 A ) z , A=0.113 for m = l , A=0.04 for m = 2 , and A=O.O18 for m = 3 . A gives the half width at half maximum of the linear transmission resonances (Aganval and Dutta Gupta [1987]).

was carried out by Danckaert, Fobeles, Veretennicoff, Vitrant and Reinisch [1991]. The basic difference between the method of Dutta Gupta and Aganval [1987] and that of Danckaert, Thienpont, Veretennicoff, Haelterman and Mandel [I9891 is that the latter group uses matrices that relate the constant amplitudes of the forward and backward waves in adjacent layers, whereas the former group uses tangential components of the electromagnetic fields. The advantage of the earlier approach is that it can directly yield the electric and magnetic fields at any point of the layered medium, whereas the other gives the reflection and transmission coefficients in a straightforward way. Moreover, in the latter

16

NONLINEAR OPTICS OF STRATIFIED MEDIA

[I, § 2

approach solving coupled nonlinear equations is unnecessary, which, although simple, can be slightly more time consuming. With respect to the main results of Danckaert, Fobeles, Veretennicoff, Vitrant and Reinisch [ 19911 regarding the validity of the nonlinear characteristic matrix formalism, conclusions were reached by comparing the results of the characteristic matrix formalism with the exact numerical solution of the nonlinear wave equation pertaining to the layered medium. The major approximations essential for developing the formalism can be listed as follows: (i) neglect of the spatial third harmonics (terms like efi3!7; (ii) slowly varying envelope approximation (SVEA). The motivation for the neglect of the spatial third harmonic is obvious. For a nonlinear layer with thickness d > L , an averaging over the high frequency components leads to a smearing of the contributions from such terms. However, for d A, as was pointed out by Biran [ 19901, this can be a poor approximation. The validity of the slowly varying envelope approximation was tested on the basis of exact and approximate calculations for a superlattice with alternate linear (with comparatively larger linear refractive index) and nonlinear layers. Results were obtained for several structures, in particular, hundred-period (highindedlow-index) and a five-period (8 high-inded8 low-index) structures. In both cases approximately the same peak strength of the fields were noted. SVEA was shown to be a good approximation for the five period superlattice, but it failed miserably for the hundred period structure. This was explained in terms of accumulation of errors (due to SVEA) through successive applications of the boundary conditions. T h s explanation seems incomplete, since the authors do not take into account the specifics of the field distribution in the distributed feedback structure. Note that in case of the hundred period superlattice one has the formation of the stationary soliton-like distribution along the length of the superlattice (see 6 2.3.1), whereas they do not emerge in the structure with lower periods. To summarize, the major conclusion was that SVEA holds in the case of nonlinear layers with widths larger than the wavelengths. Nonlinear characteristic matrix formalism and its simplified versions (with the neglect of nonreciprocity and/or nonlinearity of boundary conditions), because of its elegance and simplicity, have found many applications. Structures involving a few layers as well as periodic and quasiperiodic layered media have been studied. Because of the special properties of periodic and quasiperiodic structures, the results pertaining to such systems will be addressed separately. We present here the results for systems laclung periodicity or quasiperiodicity along the direction of stratification. An interesting effect that appears in the optical response of asymmetrical nonlinear layered media is the nonreciprocity of the overall structure. To be more

<

I,

5 21

NONLINEAR TRANSMISSION AND OPTICAL BISTABILITY IN LAYERED MEDIA

17

specific, the structure does not respond in an identical fashion for light incident from the left or from the right. Note that in linear systems (even with asymmetry) the response is symmetrical. The nonreciprocity in the overall response was noted by Chen and Mills [1987c] in the context of exact solutions for a bilayer. Danckaert, Fobeles, Veretennicoff, Vitrant and Reinisch [ 19911 showed that the simple characteristic matrix approach can capture the nonequivalence of incidence of light from left or right. Obviously this property can be used efficiently for optical diode action. A detailed proposal for an optical diode using a ramped periodic structure was recently given by Scalora, Dowling, Bowden and Bloemer [1994a]. They demonstrated that a structure with alternating high and low indices of refraction with a superimposed ramp could exhibit nearly two orders of magnitude more transmittance in one direction than in the opposite direction. A beam propagation method (Scalora and Crenshaw [ 19941, Dowling, Scalora, Bloemer and Bowden [ 19941, Scalora, Dowling, Bowden and Bloemer [ 1994b1) that incorporated reflection was used to study the systems. More recently, using the matrix method of He and Cada [1992] (for a description of the method see below), Tocci, Bloemer, Scalora, Dowling and Bowden 119951 overcame the drawbacks of the results of previous calculations, namely, the use of somewhat unrealistic parameters and neglect of absorption. The ramp in their periodic structure consisted of a slight systematic variation in the refractive index from period to period. Polydyacetylene and rutile were chosen as the constituent media. The particular nonlinear lattice was shown to exhibit more than five times as much transmittance in one direction as in the opposite direction. The overall structure had a small width (-2 pm) and was also polarization insensitive. In most investigations involving nonlinear layered media or coupled nonlinear Fabry-Perot cavities, the stress was on the relationship between the input and output powers to reveal the parametric domain where the system leads to bistability or multistability. Little has been done with respect to the frequency response and modes of such structures (Arlot, Pic, Reinisch and Vitrant [ 19861). The concept of modes of a nonlinear structure may prove to be of little use in the context of mathematical applications, since the linear superposition principle is no longer valid for such systems. Therefore modal expansion and related simplifications are not permitted in nonlinear systems. However, the knowledge of the power-dependent modes can offer the key to many practical questions (especially in the case of nonlinear waveguides). For example, for a given power level, one knows where to expect the resonances and with what characteristics. It is well understood that the modes have their signature on the frequency response of the structure. The frequency response of a single nonlinear Fabry-Perot cavity exhibits bends of the Airy resonances towards left or right (Arlot, Pic, Reinisch

18

NONLINEAR OPTICS OF STRATIFIED MEDIA

[I, 9: 2

and Vitrant [1986]), depending on the sign of the nonlinearity. Bending of the resonance curves eventually may lead to hysteresis loops in the frequency response. In light of the preceding, a system of coupled nonlinear Fabry-Perot cavities was recently investigated (Pande, Singh and Dutta Gupta [ 19931). The coupled cavity system was shown to exhibit a rich frequency response when the constituent cavities had intracavity nonlinear medium with opposite signs of nonlinearity (i.e., they were of self-focusing and defocusing nature). The interplay between the positive andor negative drag of the mode frequencies led to interesting effects. The system was studied in the framework of SVEA with Fabry-Perot cavities with given mirror transmission. Coupling between the two cavities, as expected, played a crucial role in the overall response. Powerdependent modes (treating the emitted intensity as a parameter) of the structure were studied. It was shown that for specific choices of the cavity parameter with opposite types of nonlinearity, the mode dispersion curve may twist in such a way that for a particular branch, the same emitted intensity corresponds to more than one mode frequency. Since the bending and twisting are a consequence of the nonlinearity, the generation of these was attributed to the nonlinearity of the system (such nonlinearity-induced modes were discussed also by Leung [1989]). The larger the emitted power, the larger is the number of such modes. These modes left their imprint on the power dependence of the transmission coefficient, leading to additional bistability loops. It is worth mentioning that numerical methods have used the transfer matrices to study Kerr nonlinear layered media. One such method, mentioned earlier, is due to He and Cada [1992]. Each nonlinear layer was divided into a large number of sublayers with thicknesses much less than the wavelengths, such that the forward and backward wave amplitudes in each sublayer could be considered as constants. Application of the boundary conditions in the subsequent sublayers led to the matrix approach. The above method, as can easily be noted, does not use the SVEA, and can lead to exact results if the thickness of the sublayers is extremely small. Note that in the linear limit the solutions are exact irrespective of the number of subdivisions of each layer. The numerical method was applied to study a combined distributed feedback (DFB) and Fabry-Perot structure with an additional phase matching layer. This combined structure and layer were placed in a Fabry-Perot cavity with Bragg mirrors (another pair of DFB structures). It was shown that the combined DFB-Fabry-Perot structure can lead to a much lower bistability threshold compared with an ordinary DFB structure with comparable length. The reduction of the bistability threshold is a consequence of enhanced local fields due to a narrowing of the resonances at the edge of the band gap of the DFB structure in the presence of the Fabry-Perot cavity. In fact,

I,

5 21

NONLINEAR TRANSMISSION AND OPTICAL BISTABILITY IN LAYERED MEDIA

19

the modes of a combined DFB-Fabry-Perot structure (with a Fabry-Perot cavity formed by mirrors) was studied in detail by Dutta Gupta and Agarwal [ 19931 in the context of cavity QED application. It was shown that in the context of certain modes, even a modulation of 5% in the refractive index of the medium contained in the Fabry-Perot cavity with mirror reflectivity R = 0.7 is equivalent to a FabryPerot cavity with R M 0.9 without modulation of the intracavity medium. Thus, selective to specific modes, the quality factors can be enhanced. Optical bistability with such line-narrowed modes was also shown by Dutta Gupta [1994]. 2.2. OBLIQUE INCIDENCE

The case of oblique incidence for nonlinear layered media is complicated because of the essentially different behavior of the s- and p-polarizations. Note that for normal incidence one need not distinguish between the different polarizations. Another complication that arises is because waves can now become evanescent, depending on the direction (with respect to the direction of stratification) of propagation. In fact, this property has been exploited to generate various waveguide structures. It is clear that even for a single nonlinear slab with the simplest kind of nonlinearity (namely, Kerr-type nonlinearity) the general solutions for the electromagnetic waves in the layer can be extremely complicated. In the following section we restrict ourselves to Kerr-type nonlinear media and review the achievements separately for s- and p-polarizations. Most of these investigations are carried out keeping in mind the nonlinear guided and surface modes of the structure. 2.2.1. TE- or s-polarized waves Exact solutions for the scattering of a s-polarized wave from a linear-nonlinear interface was first worked out by Kaplan [1976, 1977, 19811. Some earlier studies on nonlinear layered media relied heavily on the solution of Kaplan, even for the case of p-polarized waves. Note that Kaplan’s solution holds only for semiinfinite nonlinear media. The exact solutions for a nonlinear slab for the general case of oblique incidence were given by Leung [1988, 19891. The exact results for the simpler case of normal incidence (Chen and Mills [ 1987a1)was discussed in 6 2.1.1. The method applied by Leung to reduce the problem to quadratures is analogous to that used by Chen and Mills [1987a,c], although Leung used a more general form of the intensity-dependent refractive index given by (2.33)

20

NONLINEAR OPTICS OF STRATIFIED MEDIA

with the condition that Cr

(Iz12) -+

1- 1

0 as E

-+

[I, § 2

0. Both approaches assume that

the structures are lossless, so that the conservation of flux along the z-direction can be exploited. Assuming the x-dependence to be of the form eikTxwith xz as the plane of incidence and writing the solution (for the y-component of the electric field in the nonlinear layer) as (2.34)

where g = koz, E and q are real functions and v is a real constant, the equation for E(E) and q(5) were obtained. Integration of these equations led to the generalization of eqs. (2.4) and (2.5) to the case of oblique incidence as follows: (2.35)

(E’>2 + V(E) = A.

(2.36) 2 In eqs. (2.35) and (2.36) the prime denotes the derivative with respect to E; and 2 are integration constants. Equation (2.36) can be interpreted as the energy

w

integral, where the potential function V ( E ) is given by (2.37)

In eq. (2.37), I? = ni - n?, n,=k,lko. Note that the sign of I? determines whether the waves are propagating or evanescent in the nonlinear layer for vanishing intensities and the solutions take different forms for these two cases. Equation (2.36) after integration leads to the quadrature determining E ( 5 ) in an implicit form: (2.38)

50 is another integration constant. For Kerr-type nonlinearity, 1812)= a one can defme a dimensionless intensity as

where

(

1l?I2

z(E>= n&x E2/n,2, R = f n : ,

where

(2.39)

and carry out integration of eq. (2.38) to obtain the solution in terms of Jacobian elliptical functions. The behavior of the solutions is then determined (as in the

I,

0 21

NONLINEAR TRANSMISSION AND OPTICAL BISTABILITY IN LAYERED MEDIA

21

case of normal incidence) by the cubic dependence of the potential V(Z) given bY

V(Z)= z3 2z2 - 471,z + 271.;

(2.40)

with -

ul = n,a-

2 ‘ 4

n4

-

and

712 =

W nr3

nia-.

(2.41)

The major difference between the two approaches lies in the implementation of the boundary conditions. Whereas Chen and Mills [ 1987~1treated the global constant as the free parameter, Leung [1989] used Z(2) (the intermediate root of the equation V(Z)=O) as the parameter. In the previous approach the equivalent constant W had to be scanned through a fixed range to find particular values that yielded a solution consistent with the boundary conditions. This amounts to solving for the roots of a single but highly complicated algebraic equation. The problem becomes further complicated since there can be several roots in the multivalued domain. In contrast, Leung’s approach avoids this problem, since no equation needs to be solved numerically. However, the method of Leung has not been generalized to a multilayered system. Using the exact solutions, Leung calculated the reflection coefficient from a nonlinear layer with a linear dielectric constant larger than that of the substrate and cladding, and demonstrated multivalued output in reflection. Moreover, Leung [ 19891 also reported nonlinearity-induced modes in the structures. In nonlinear structures, because of the dependence of the dielectric function on the intensity, the nonlinearity-induced increase in the optical width of the nonlinear layer may lead to additional resonances. Leung [1989] explored the origin of such resonances, and calculated the value of the incident intensity for which such resonances occurred in a nonlinear film of given thickness. Using the exact solutions for the nonlinear layers, Langbein, Lederer, Peschel, Trutschel and Mihalache [1990] developed a “matrix” method to handle the transmission and reflection coefficient for oblique incidence. Obviously, the method is not as simple as the nonlinear characteristic matrix approach, where the matrix for a N-layered nonlinear system can be obtained by a direct multiplication of the matrices in proper order. The difficulty arises because these authors deal with the intensities and their derivatives (the solutions for which are known in terms of Jacobian elliptical functions) rather than the complex amplitudes of the fields. The other constraint is the assumption of lossless media, which enables the use of flux conservation.

w

22

NONLINEAR OPTICS OF STRATIFIED MEDIA

[I, § 2

Applying the matrix method, these authors explored the transmission resonances of various structures. Note that a transmission resonance is defined by the zero of the intensity reflection coefficient R . Thus, under the condition of a transmission resonance and in the absence of losses, all the incident energy is transmitted by the structure. They studied the power-dependent evolution of the transmission resonances. The relation between these transmission resonances and all-optical switching was also demonstrated. Transmission resonances without a linear analog (i.e., nonlinearity-induced modes) were another offshoot of their studies. Since other reviews have detailed the earlier studies on s-polarized nonlinear guided and surface waves (see, e.g., Stegeman and Seaton [1985], Mihalache, Bertolotti and Sibilia [1989], Stegeman [1992]), we do not discuss them here. Some major findings included the prediction of new nonlinearity-induced guided and surface modes, which do not have any linear counterpart. Another interesting observation was the possibility of s-polarized surface plasmon polaritons. The existence of s-polarized surface plasmon polaritons in thin metal films bounded on both sides by self-focusing media was predicted by Stegeman, Valera, Seaton, Sipe and Maradudin [1984]. Obviously, these kinds of surface plasmon polaritons do not have any linear analog, since surface plasmons in linear structures are essentially p-polarized. The dispersion relation for the s-polarized nonlinear surface plasmon polaritons guided by a metal film sandwiched between a linear substrate and nonlinear cladding was studied by Lederer and Mihalache [ 19861 and Mihalache, Mazilu and Lederer [1986]. The dispersion curves revealed a local power minimum and a restricted region for the permitted propagation constants. Along with the study of the power-dependent nonlinear surface and guided modes, optical bistability mediated by such modes has drawn considerable attention. A convenient structure to study optical bistability with such modes was the attenuated total reflection (ATR) (sometimes referred to as frustrated total reflection) geometry, where a high-index prism with or without a lowindex spacer layer loaded on top of the guiding interfacellayer is used to couple the incident radiation to the surfacefguided mode. Reflectivity of the structure is monitored as a hnction of the angle of incidence. A change in this angle changes the surface component of the wave vector, thereby matching the propagation constant of the surface/guided mode. In such a case one observes a dip in the reflection coefficient, which otherwise is close to unity because of total internal reflection at the prism spacer layer interface. In the context of nonlinear ATR configuration, to derive a bistable response, one first obtains the angle of incidence when the surfacelguided modes are

I,

3 21

NONLINEAR TRANSMISSION AND OPTICAL BISTABILITY IN LAYERED MEDIA

23

excited at low power levels. Keeping the angle of incidence slightly detuned in the proper direction (determined by the sign of nonlinearity) from the linear (low-power) resonance, one can sweep through the resonance by increasing the incident power. The sweeping is possible because of the dependence of the optical path in the nonlinear layers on intensity. The overall effect of the nonlinearity is a “bending” of the resonances, which eventually leads to the hysteretic response. Several authors exploited ATR geometry to demonstrate optical bistability with guided modes (Stegeman [ 19821, Reinisch, Arlot, Vitrant and Pic [1985]). Modifications of ATR geometry to lower the threshold were also proposed (Haelterman [ 19881). In his proposal Haelterman exploited the intensity-dependent phase jump near the guided wave resonance to achieve low threshold bistability. As mentioned in tj 2. I .2, in the context of normal incidence on a multilayered medium with an arbitrary number of nonlinear layers, the characteristic matrix approach can be generalized to include the case of oblique incidence. The general expression for the nonlinear characteristic matrix given by eq. (2.27) holds, except that now the effective refractive indices for the forward and backward waves (i.e., n j t ) are to be replaced by nz,if,where n,jh are given by (2.42) with

nli

= ~j

-n, 2 > 0, n,

= k.Jko,

(2.43)

where k, is the x-component of the wave vector determined by the angle of incidence, and nzif give the nonlinearity-modified z-component of the propagation constants scaled by the vacuum wave vector k,; n, gives the propagation constant (in units of ko) or the effective index of the guided mode along x. Note that n, is continuous across the interface. The matrix approach for oblique incidence, however, has a restricted domain of application; that is, the nil in each nonlinear layer has to be real. This implies that the method is applicable only when the waves are propagating in the nonlinear layers. Thus, it fails in the case of waveguides with nonlinear substrate or cladding, where the fields are generally evanescent. A much more accurate matrix method using Jacobian elliptical functions as the solution in the nonlinear slab, was proposed by Trutschel, Lederer and Golz [1989]. They considered a system of an arbitrary number of unit cells sandwiched between linear substrate and

24

NONLINEAR OPTICS OF STRATIFIED MEDIA

11, § 2

cladding. Each unit cell consisted of a linear layer between two Kerr nonlinear slabs. The method was applied to a GaAdAlGaAs multilayer. They reported both symmetrical and asymmetrical TE guided modes and presented a detailed study of the corresponding dispersion characteristics. Recently, the optical properties of a Kerr nonlinear layer near a phase conjugate mirror (PCM) was studied using the characteristic matrix approach. Some studies in linear systems with PCM (Agarwal and Dutta Gupta [1995], Dutta Gupta and Jose [1996]) demonstrating the major role of evanescent waves served as a stimulus for such studies. Recent experimental observation of evanescent waves and their phase conjugation (Bozhevolnyi, Keller and Smolyaninov [ 19941, Bozhevolnyi, Vohnsen, Smolyaninov and Zayats [ 19951, Bozhevolnyi, Keller and Smolyaninov [ 19951) was another motivation for the theoretical studies. Dutta Gupta and Jose [ 19961 probed the guided and surface wave structures near a PCM (see inset to fig. 3) for the signature of the interaction of guided and surface modes with a PCM. Phase conjugation of guided and surface modes leading to enhanced back scattering was reported (fig. 3). The studies on the nonlinear counterpart (Jose and Dutta Gupta [1998]) focused on two issues: (a) to probe the well-known distortion correction properties of PCM (see, e.g., Aganval and Wolf [1982], Aganval, Friberg and Wolf [1982a,b, 19831, Friberg and Drummond [1983]) in the context of a nonlinear layer; and (b) to look for the signature of nonlinearity in ordinary and phase conjugated reflectivity. In the domain where the waves were propagating in all the layers, it was shown that a PCM with reflectivity ,u (I,ul= 1) can completely correct for the distortions introduced by a Kerr nonlinear slab. This was an explicit verification of an earlier general theorem of Agarwal [1983] that was applicable to a broad class of nonlinear media. In the domain where guided modes are excited, it was shown that the presence of the PCM can lead to optical multistability in both specular and back scattering directions (fig. 4). 2.2.2. TM- or p-polarized waves An exact solution for the scattering of p-polarized waves is known (Leung

[1985]) only for a single linear-nonlinear interface (i.e., for a semi-infinite nonlinear medium). To this date, to our knowledge, the general solution for a nonlinear slab and in general a layered medium, has not been worked out. Here we briefly recall Leung’s exact solution pertaining to a single interface. Consider a semi-infinite isotropic nonlinear Kerr medium with dielectric function Et = 1 + a1E12)occupying the space z < 0. Assuming the x-dependence of

I, fj 21

NONLINEAR TRANSMISSION AND OPTICAL BISTABILITY IN LAYERED MEDIA

U

1.0

-

0.9

-

0.8

-

0.7

-

0.6

-

25

L. E)

0.5 -

0.4

E2

I ,

-

--lo* lo* lo* -

e [degrees]

10'

100 10.'

_z

102

104

104 10.'

10-Q 10-10

-lo." -10."

10'2

10'3

10.'~ 10-'8

lo-" 1v'B

-

I

10-n

3

I

I

I

20

40

60

I

so

\

.

I

100

e [degrees]

Fig. 3. Linear results for the reflection coefficient in (a) specular direction R and (b) back scattering direction R, as functions of the angle of incidence 0 for p = 1.0. The various dips are labeled by their corresponding mode numbers. The inset shows the layered medium on top of the PCM withthe followingparametervalues: dl = I p m , dZ=5.5pm, d3=0.12pm, 1=0.82pm, E I =3.1329, €2 = 12.95996+0.0453i, ~3 = 1, €4 =6.145 (Jose and Dutta Gupta [1998]).

26

NONLINEAR OPTICS OF STRATIFIED MEDIA

[I, 9 2

1 .o

0.9

0.8

U 0.7

0.6

0.5

I

I

0.1

1 u4

0.01 '

I

' ' ' "

1

0.1

u4

Fig. 4. Reflection coefficient in (a) specula direction R and (b) back scattering direction R, as functions of input intensity U4 for #=47.69" (corresponding to point P in fig. 3a). Other parameters are as in fig. 3 (Jose and Dutta Gupta [1998]).

I,

4

21

27

NONLINEAR TRANSMISSION AND OPTICAL BISTABILITY IN LAYERED MEDIA

the fields to be -eikxxand with no variation along y , the Maxwell equations for p-polarized waves can be written in the form (2.44) (2.45) (2.46) where 5 = koz, and n, = k.Jk0. The set of equations (2.44X2.46) can be reduced to a second-order differential equation of the form

(z)‘(5 =

-

I ) By,

(2.47)

which can be solved for B, and B.L. The solutions for B, and Bh are given by (2.48)

where (2.50) Making use of these exact solutions, Leung [ 19851 found new waves with no linear counterpart. The dispersion relation for the surface modes was obtained without having to solve for the field profiles. Optical bistability mediated by nonlinear p-polarized waves has been an intense field of investigation. Surface plasmons (see, e.g., Raether [ 19771, Kovacs [ 19821) at a metal-dielectric interface played an important role in such studies. These modes are localized near the surface in the sense that their fields decay exponentially away from the surface. Surface plasmons are excited by p-polarized waves in ATR configuration or by surface inhomogeneities (like grating or surface roughness). There can be variations of the ATR geometry (fig. 5). The Otto geometry (Otto [1968]) has a low index spacer layer between the high index prism and the metal film, whereas, in the Kretschmann configuration (Kretschmann [1971]) the metal film is deposited on the base of the high index prism. A different geometry, which can support coupled surface plasmons in very thin metal films, was suggested by Sarid [1981]. In

28

NONLINEAR OPTICS OF STRATIFIED MEDIA

Otto

Kretschmann

Sarid ,A

1 dielectric I

dielectric metal ~

dielectric

~

Fig. 5 . (a) Otto, (b) Kretschmann and (c) Sand geometries for the excitation of surface plasmons. Arrows indicate the interface near whlch the surface excitation is localized. Note the possibility of the excitation of coupled surface plasmons in Sarid geometry

the Sarid geometry one can excite both the symmetrical short-range (SR) and the antisymmetrical long-range (LR) surface plasmons. The long-range surface plasmons (LRSP) have the added advantage of large local field enhancements associated with them (Sarid, Deck, Craig, Hickernell, Jameson and Fasano [ 19821, Agarwal [ 19851). Various nonlinear optical phenomena exploiting this extra enhancement were demonstrated by Sarid, Deck and Fasano [1982], Deck and Sarid [1982], and Quail, Rako, Simon and Deck [1983]. Optical bistability with surface plasmons at a metal-nonlinear dielectric interface was demonstrated by Wysin, Simon and Deck [1981]. Martinot, Lava1 and Koster [I9841 used the ATR configuration with a nonlinear prism loaded on top of the metal film, The electric field intensity in the prism (i.e., &, and incident and reflected plane wavefields, respectively) was approximated by the sum of the intensities of the incident and reflected waves. Moreover, they assumed plane wave solutions in the nonlinear medium with the linear refractive index replaced by its nonlinear counterpart. Dutta Gupta and Agarwal [ 19861 investigated optical bistability in the prism-metal film-nonlinear substrate configuration without the assumption of a plane wave solution for the nonlinear dielectric. They used Kaplan's solutions (Kaplan [1981]) for an approximate wave equation suitable for p-polarized waves. Hickernell and Sarid [I9861 demonstrated the advantages of LRSPs in the context of lowering the bistability threshold. They showed that the power threshold for switching between the two bistable states for LRSP can be two orders of magnitude less than that required for a single interface surface plasmon.

Izi+z~I*,

z~

I,

21

NONLINEAR TRANSMISSION AND OPTICAL BISTABILITY IN LAYERED MEDIA

29

Exact results for optical bistability with surface plasmons in a layered structure on a nonlinear substrate were presented by Aganval and Dutta Gupta [1986] using the solutions of Leung [1985]. Aganval and Dutta Gupta [ 19861 considered a linear stratified medium consisting of N layers occupying a region -ZN < z < 0 on a Kerr nonlinear substrate filling the space z 2 0 (note the location of the nonlinear interface at z=O for obvious convenience). For the incidence of a p-polarized light from the left at an angle 8 the boundary conditions were written as

In eq. (2.51), Bi (B,) is the magnetic field amplitude of the incident (reflected) wave, n,i = cos 8/& ( ~ i dielectric , constant of the medium of incidence), A4 is the characteristic matrix of the stratified medium (Born and Wolf [1989]) occupying -ZN < z < 0, and BY,*l(O)and Ex,n,(0)are the tangential components of the magnetic induction and the electric field, respectively, at z=O+. B,,,l(O) and Ex,nl(0)are evaluated using eqs. (2.48)-(2.50) and taking the limit 5 + O+. They are given by (2.52)

(2.53) with (2.54) Applying eq. (2.51) and treating I ( 0 ) as the free parameter enabled the straightforward calculation of the reflection coefficient of the structure. The exact results thus obtained were compared with the approximate results of Wysin, Simon and Deck [1981] and Dutta Gupta and Aganval [1986]. The set of parameters applied by Wysin, Simon and Deck [1981] was used for calculation. It was shown that the results of Wysin, Simon and Deck underestimated, whereas those of Dutta Gupta and Agarwal overestimated the switching thresholds. Calculations for the Sarid geometry supporting LRSP as well as SRSP were also performed. The former were shown to

30

NONLiNEAR OPTICS OF STRATIFIED MEDIA

[I, § 2

lead to a switching threshold at least one order of magnitude less when compared to a structure supporting ordinary (single interface) surface plasmons. Exact numerical solutions for a nonlinear dielectric slab bounded symmetrically by thin metallic layers was studied by Pande and Dutta Gupta [1990]. The linear equivalent of the structure was proposed by Welford and Sambles [1988]. Pande and Dutta Gupta [1992] made a detailed study of the linear reflection and transmission when the dielectric layer possessed material absorption and dispersion. The metal bound dielectric structure has an advantage that both the asymmetrical and symmetrical coupled modes (i.e., LRSP and SRSP) have comparable damping, unlike the case of Sarid geometry where the SRSP has a large decay compared with the LRSP. Since both the long-range and short-range surface plasmons have comparable decay, both (for proper operating conditions) can be affected by the nonlinearity to the same degree. Note that it is difficult to have optical bistability with the short-range modes in the other conventional scheme (Sarid configuration) where the coupling is by means of the metallic layer. In contrast, metal bound dielectric film can exhibit optical bistability with both LRSP and SRSP (Pande and Dutta Gupta [1990]). Pande and Dutta Gupta [1991] also considered the case of the saturation type of dispersive nonlinearity (Peschel, Dannberg, Langbein and Lederer [ 19881) with the exact numerical method, and studied the effect of saturation on the bistable response. Nonlinearity-induced modes for both the cases of Kerr- and saturation-type nonlinearities were also reported. A more general form of nonlinearity allowing for the nonlinearity-induced anisotropy was considered by Boardman, Maradudin, Stegeman, Twardowski and Wright [1987] and Boardman and Twardowski [1989]. It was shown from first principles that for a macroscopically isotropic nonlinear material the induced polarization at frequency o (= w + o - o)can be calculated using the dielectric tensor which is given by

x,

Ern

i=(

+ CC(E.:+ PE,Z + YE:) 0 . 0

0

&to

+ a@E; + E,' + BE;) 0

0 0 &to + cr(yE,2 + PE;! + E-?)

(2.55) In writing eq. (2.55), a planar guiding structure with the interface at z=O supporting waves with wave vector along x was assumed. The values of the constants fi and y are determined by the nature of the nonlinearity. For example, y = 113, -1/2 and 1, and fi=2/3, 1/4 and 1 for electronic distortion,

I, 5 21

NONLINEAR TRANSMISSION AND OPTICAL BISTABILJTY IN LAYERED MEDJA

31

molecular orientation, and thermal nonlinear mechanisms, respectively. The case of pure p-polarized waves on a semi-infinite nonlinear medium (with E , = 0) was considered for both metal and dielectric bounding media (Boardman, Maradudin, Stegeman, Twardowski and Wright [ 19871). The exact equations of motion were integrated numerically (using a finite element code) to study the power flow as a function of the effective index of the guided mode. For metallic bounding media an extremal behavior (maximum) was predicted. However, access to this maximum requires a large change in the refractive index, which cannot be achieved at moderate power levels, Boardman and Twardowski [1989] studied single interface as well as linear film on a nonlinear substrate and focused on the mixed p- and s-polarized modes. Note that mixing is possible because of the presence of all field components in the expression for 2. Numerical studies were simplified because some of the first integrals could be found. It is clear that in contrast to the case of s-polarization, the study of p-polarized waves in nonlinear layered media poses a much more challenging problem. For s-polarized waves propagating along x-direction with z-axis giving the direction of stratification, one needs to consider the only nonzero component of the electric field E , and the intensity-dependent nonlinearity can be expressed as a function of local intensity IEYl2. In contrast, for p-polarized waves the local intensity becomes a hnction of 1E,j2 + J E z1 assuming an isotropic nonlinearity. Because of the complexity of solving the Maxwell’s equations with both the components of the fields present in the nonlinearity, the earlier approaches resorted to various approximations for IE,I and IE,J. One of the first attempts to model the p-polarized waves (Lederer, Langbein and Ponath [19831) was based on the so-called longitudinal uniaxial approximation for which ( E ,I >> IE, I. Obviously, the approximation proved to be poor for guided waves (Seaton, Valera, Svenson and Stegeman [1985], Stegeman, Seaton and Ariyasu [1985], Langbein, Lederer, Mihalache and Mazilu [ 19871, Boardman, Maradudin, Stegeman, Twardowski and Wright [1987]). This led to the other approach known as the transverse uniaxial approximation which resorted to the other extreme, namely, IE,(<<( E ,1 (Seaton, Valera, Svenson and Stegeman [1985], Stegeman, Seaton and Ariyasu [198S]). In both preceding cases the nonlinear susceptibility becomes the function of any of I E, I or I E, I 2, and analytical techniques similar to those in the case of s-polarization can be applied. As mentioned above, the general analytical solution for p-polarized waves in a nonlinear slab is still missing. Analytical solutions can be obtained only in special cases. One such solution refers to the case of the so-called balanced modes for which the guided electromagnetic fields have two electric components

32

NONLINEAR OPTICS OF STRATIFIED MEDIA

[I, § 2

of equal magnitude (i.e., lE,l=lEzl).This case was studied by Jasinski [1995], who obtained the analytical expression for the modes and calculated the power flow for the nonlinear film with linear substrate and cladding. Analytical but approximate solutions were also derived for the case of quasibalanced modes, when fields satisfy the relation IE, I M IE, 1. The results were compared with the exact numerical calculations. Detailed reviews of the work done up to the late 1980s were conducted by Stegeman and Seaton [ 19851 and Mihalache, Bertolotti and Sibilia [ 19891. Recently, Yokota [ 19931 obtained analytical expressions for the dispersion relations for p-polarized guided waves in a weakly Kerr nonlinear film bounded by linear dielectrics through a perturbation technique. Exact dispersion relations for p-polarized waves guided by a thin dielectric film bounded by nonlinear media were presented by Chen and Wang [ 19931 applying the method of Mihalache, Stegeman, Seaton, Wright, Zanoni, Boardman and Twardowski [1987]. 2.3. PERIODIC AND QUASIPERIODIC LAYERED MEDIA

2.3.1. Periodic layered media: optical bistability and gap solitons Wave propagation in periodic structures has many interesting and potentially useful applications (see, e.g., Yariv and Yeh [1984], Yeh [1988]). We discuss examples of periodic structures that can be diverse. For example, the system can be composed of a periodic arrangement of dielectric slabs with given optical properties and widths or can be a medium with periodic variation of, perhaps, the refractive index. It can also be a layered medium with one or more interfaces with a periodic profile. Irrespective of the details, these structures possess forbidden frequency bands or stopgaps as a direct consequence of the Floquet-Bloch theory. These gaps are located around the Bragg frequencies, which are determined basically by the modulation period and the average refractive index. Any wave (propagating along the direction of modulation) with frequency in the stopgap is reflected by the structure, and the stopgaps are “perfect” (i.e., with infinite slope at the gap edges) for periodic media with infinite extent. However, for finite structures sharp transmission resonances exist at the gap edges. With the assessment that the location of the gap and such resonances depend critically on the refractive index, the importance and prospects of dispersive nonlinearity in the context of such structures are immediately understood. In a nonlinear medium, where the refractive index depends on the local intensity of the wave, one expects that both the transmission resonances and gaps of the structure can be

I,

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NONLINEAR TRANSMISSION AND OPTICAL BISTABILITY IN LAYERED MEDIA

33

significantly tailored. This suggests tremendous and diverse potentials for alloptical devices. In an earlier review Winful and Stegeman [1984] summarized some potential applications of nonlinear periodic structures, including optical bistability, nonlinear coupling and tuning of guided and surface modes, tunable filters and pulse compression. Recently, other important applications, such as power limiting (Herbert, Capinski and Malcuit [ 1992]), all-optical logic gating (Cada, He, Acklin, Proctor, Martin, Morier-Genoud, Dupertuis and Glinski [ 19921, and realization of VLSI (very large scale integrated) compatible switching devices (Bieber, Prelewitz, Brown and Tiberio [ 1996]), were also demonstrated. The analytical treatment of the nonlinear periodic structure can be complicated, especially when the modulation is strong. The difficulty in this case is associated with multiple scattering by the grating leading to many spatial harmonics. Under suitable conditions, however, various approximate approaches, such as coupled mode theory or method of envelope functions, can be applied. Despite their approximate nature, these approaches shed much light on the physics of nonlinear periodic media. The coupled mode theory is examined, followed by the envelope function approach. Coupled mode theory for periodic structures was discussed in detail by Kogelnik and Shank [ 19721 in the context of distributed feedback lasers, and was successfully applied to nonlinear periodic structures by Winful, Marburger and Garmire [ 19791 to show bistability for continuous wave excitation. Henceforth, coupled mode theory applied to the latter structures has been used extensively (Mills and Trullinger [1987], Chstodoulides and Joseph [1989], de Sterke and Sipe [1990b,c], He and Cada [1991]). Here, applying a model system, we recall the essential steps and approximations involved in coupled mode theory. Consider a medium with an index of refraction given by

I -I2 ,

n(z) = no + n l cos(Kz)+n2 E

(2.56)

where no is the background refractive index, n1 is the modulation amplitude, n2 gives the nonlinearity parameter (nonlinear refractive index coefficient), and K (=2x/A, with A the modulation period) is the modulation wave number; n2 can be related easily to a of eq. (2.3). Note that eq. (2.56) describes a stratified medium with the simplest law of harmonic modulation. It can also describe periodically corrugated waveguides or fibers where n is to be interpreted as the distribution law for the effective mode index. Coupled mode formalism can be applied to the case of weak modulation (nl <<no), when one ignores

34

NONLINEAR OPTICS OF STRATIFIED MEDIA

[I, 0 2

the contributions of higher-order spatial harmonics and writes the field as a superposition of the forward and backward waves E

=

(A+(z,t )elkr+ ~ ( zt )e-I") , e8''

+ C.C.

(2.57)

The other requirement is the narrow width of the wave number distribution, namely, detuning 6 defined by 6 = k - Kl2 ( k = wno/c) must satisfy 6 << Kl2 (de Sterke and Sipe [1990a]). Modification of eq. (2.57) in the case of guiding structures is straightforward: one has to multiply the righthand side of eq. (2.57) with the transverse distribution function (e.g., f ( x ) ) of the corresponding mode. Vector fields ?I can be accounted for by the suitably defined vector mode functionsT(x) (de Sterke and Sipe [199Ob]). Substitution of eq. (2.57) in the wave equation with the refractive index given by eq. (2.56) leads to the following two coupled equations for A+ and A - :

where

(2.60)

In eqs. (2.58) and (2.59) we have applied the fact that both A+ and A- are slowly varying functions of z and t . Equations (2.58) and (2.59) were derived by Christodoulides and Joseph [ 19891. Analogous equations were obtained and analyzed numerically by de Sterke and Sipe [ 1990bl in the context of switching and temporal dynamics. We first recall the linear properties (by setting y=O) of the system. For simplicity, assuming solutions with temporal dynamics given solely by e-I"', we remove the time derivatives in eqs. (2.58) and (2.59). Introducing a new pair of amplitudes a+@) as

(i:)

=

V(z)

)I:(

, V ( z )= Diag(e-'", eihZ),

(2.61)

I,

5 21

NONLINEAR TRANSMISSION AND OPTICAL BISTABILITY IN LAYERED MEDIA

35

the coupled equations (2.58) and (2.59) can be written as

5 dz ("+) a= isM

(z:),

(2.62)

with (2.63) where

Equation (2.64) gives the dispersion relation of the structure, which shows that s becomes imaginary whenever 161
2

ho

2

(2.65)

with (2.66) Thus, ho gives the width of the forbidden gap. In the gap the forward wave will transfer the energy to the backward wave and will therefore decay, resulting in the Bragg reflection. Noting that M 2 = I (Z is the unit matrix) the following relation holds (Dutta Gupta and Agarwal [ 19931): exp(iA4sz) = Z cos(sz) + i M sin(sz).

(2.67)

Applying eq. (2.67), the solution for the forward and backward waves are written as

Thus, the matrix U gives the spatial evolution of the fields inside the medium. Naturally one has to incorporate proper boundary conditions for a finite stack to evaluate linear response of the structure.

36

NONLINEAR OPTICS OF STRATIFIED MEDIA

[I? 0 2

It is clear from the structure of eqs. (2.58) and (2.59) that a general analytical solution is extremely complicated. These equations were solved for 6 = 0 (i.e., when the frequency of the wave is resonant with the Bragg frequency) (Chnstodoulides and Joseph [ 19891). Similar equations with the omission of temporal dynamics were investigated by Mills and Trullinger [ 19871. The major outcome of these studies was the confirmation of solitary wave profiles with the frequency in the stopgap of the structure. The static (immobile) profiles, termed as “gap solitons” (for a review, see, e.g., Sipe [1992], de Sterke and Sipe [ 1994]), were first discovered in numerical experiments carried out by Chen and Mills [1987b,c]. The exact method (see 52.1.1) was applied to calculate the transmission from a finite superlattice with alternating linear and nonlinear slabs. The superlattice was illuminated with normally incident plane waves with frequency in the stopgap but close to the edge. Bistability in transmission was observed. The intensity distribution corresponding to the total transmission ( T = 1) state exhibited the solitary wave profile. Calculations for a large (“infinite”) system showed that these dstributions could be accurately fitted to f ( z ) = EM sech(Dz) with no contribution of nonlinearity in fi. The maximum field EM obeyed the relation aE$ = constant [recall that a defines the nonlinearity by means of eq. (2.3)]. Thus, it was interpreted that total transmission of the nonlinear superlattice is mediated by the excitation of these gap solitons. Existence of such solitons in systems with weak harmonic modulation was demonstrated by Mills and Trullinger [ 19871, who also obtained the analytical expressions for the solutions in the framework of a coupled mode theory, In the particular case of stationary solutions (with no energy transport) satisfying ]A+I - IA- l 2 = 0, gap solitons with an approximate “sech” profile were demonstrated. In contrast to the gap solitons that transport no energy, the soliton solutions of partial differential equations (Christodoulides and Joseph [ 19891) could propagate undisturbed even though their power spectra lie well within the stopgap and the carrier frequency is in resonance with the Bragg frequency (at the center of the stopgap). Although the existence of soliton solutions away from the Bragg frequency was mentioned, they were not reported by these authors. An alternate approximate approach to the class of problems involving nonlinear lossless periodic layered media (not restricted to low modulation depths) was the envelope function formalism proposed by Sipe and Winful [1988] and developed in a series of papers by de Sterke and Sipe [1988, 1989a,b, 1990~1.The essence of the method lies in exploiting the periodicity of the relevant parameter (namely, dielectric function), making full use of the Bloch theorem. The solutions of the linear wave equation are analyzed first, using

I,

5

21

NONLINEAR TRANSMISSION AND OPTICAL BISTABILITY IN LAYERED MEDIA

the mode functions qm(z)with temporal dependence exp(-iw,t), equation

37

satisfying the

(2.69) where E ( Z ) is real and periodic with period A. The eigenfunctions q,Jz) are orthogonal in the sense (2.70) Next, the method of multiple scales is applied by writing the solution E of the nonlinear wave equation as

‘

and introducing different spatial and temporal scales as z(‘) = p‘z and t(‘)= P t ( p << 1, I = 0, 1, 2, . . . ). Use of expansion (2.7 1) in the wave equation, together with the relations (2.72) (2.73) leads to the equations for various e(‘)’s.The lowest order equation is satisfied by a solution of the form

In eq. (2.74) the envelope function a depending only on the slow variables was introduced. d2)is then sought as a superposition of the mode functions qnl(zco)) and the envelopes b,(z(’), z ( ~ ). ,. . , t ( 2 ) ., . . ). Using the orthogonality of the mode functions, b,’s can be eliminated and e(’) can be expressed in terms of envelope a, ~ ( Z ( ~ ) ) ’and S matrix elements of the operator SZ (= -ic&). Knowledge of e(l)and d2)in the third-order equation then leads to the equation for the envelope function a . This technique was applied to the case of a periodic medium with Kerr nonlinearity, and it was shown that the envelope satisfies a nonlinear Schrodinger

38

NONLINEAR OPTICS OF STRATIFIED MEDIA

[I, § 2

equation (NLS) (Sipe and Winful [1988]) with the spatial coordinate redefined as (2.75) w;! = c w;I (mlQlm),

(2.76)

thus making the envelope a function of 6 and r(*); w; can be identified as the group velocity. The stationary solution of the NLS equation describes a soliton propagating with this group velocity. A detailed description and physical interpretation was given by de Sterke and Sipe [1988], who also looked at the limiting case of the gap solitons with no energy transfer. Their subsequent paper (de Sterke and Sipe [1989b]) established connections of the envelope function approach with effective mass theory in condensed matter. Moreover, effects of third harmonic generation by the cubic nonlinearity were also incorporated by means of Hamiltonian methods. Envelope function approaches were used by de Sterke and Sipe [199Oc] to study equations such as (2.58) and (2.59) in the context of copropagating and counterpropagating waves in nonlinear periodic media. In both cases the NLS equation was obtained under suitable conditions. de Sterke and Sipe [ 1989al proposed corrugated nonlinear waveguide structures as suitable candidates for observing gap solitons. Expansion of the field in terms of the waveguide modes was used to arrive at a one-dimensional wave equation with a nonlinear source term. The envelope function approach then led to the gap solitons. Bistability in transmission was also demonstrated. The numerical study of the temporal dynamics of the system described by eqs. (2.58) and (2.59) was conducted by de Sterke and Sipe [1990b]. It was shown that for low powers, transitions from low to high transmission occurred, leading to the excitation of gap solitons. At higher intensities, however, periodic self-oscillations occurred. For even larger input powers the dynamics were chaotic. Investigators suggested various arrangements for the formation and observation of gap solitons. They were reported for oblique incidence of s-polarized light in a finite periodic structure by Dutta Gupta [1989a], using the tunneling resonances (Yeh [1987]). Nabiev, Yeh and Botez [1993] suggested the use of periodic leaky wave coupled and evanescent wave coupled array structures to this end. Feng [ 19931 proposed the use of two intense continuous wave beams to excite gap solitons with two spatial dimensions. Lee and Ho [ 19931 showed that two orthogonally polarized pulses can copropagate through a nonlinear periodic medium as a coupled gap soliton while each pulse separately gets reflected due to Bragg reflection. Based on this effect a new, all-optical switching scheme was proposed.

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NONLINEAR TRANSMISSION AND OPTICAL BISTABILITY IN LAYERED MEDIA

39

2.3.2. Quasiperiodic layered media: weak photon localization Quasiperiodic structures, with two or more incommensurate periods, are intermediate between periodic and random media. The interest in quasiperiodic layered media originated in studies on analogous systems in solid state physics. Localization in solid state physics (for a review see, e.g., Merlin [ 19881, Gradeskul and Freilikher [ 19901) involving one or higher dimensional quasiperiodic systems has been studied extensively (Kohmoto, Kadanoff and Tang [1983], Kohmoto and Banavar [1986], Hawrylak and Quinn [1986], Kohmoto, Sutherland and Tang [ 19871). The theoretical studies received an impetus with the realization of quasiperiodic superlattice structure by Merlin, Bajema, Clarke, Juang and Bhattacharya [1986]. The optical realization of a one dimensional quasiperiodic structure involving a stack of dielectric layers arranged in a Fibonacci sequence was proposed and studied theoretically by Kohmoto, Sutherland and Iguchi [ 19871. The Fibonacci multilayer is constructed recursively by dielectric slabs A and B (with linear refractive indices n, and n b and widths d , and db, respectively) as S,+l=Sj-lSj with So=(B) and SI=(A). Thus, S2 = (BA), S3 = (ABA), S4 = (BAABA), etc. The optical system has several advantages over its solid state counterpart. In solid state physics electron-electron and electron-phonon interactions are inevitable. This has been a real threat for direct tests of the theoretical predictions, On the other hand, optical experiments are more pure, since photons are noninteracting. Moreover, polarization of light beams adds a new feature to the localization problem. We now summarize the results pertaining to a linear Fibonacci multilayer (Kohmoto, Sutherland and Iguchi [ 1987]),where the constituent layers were assumed to have the same optical width. It was shown that the system can be described by a (2 x 2 ) matrix (belonging to an SL(2, R ) ) map, which can be reduced to a trace map. The most interesting consequence was the self-similarity of the transmission coefficient as a function of the optical width for various generations. Moreover, it was shown that the allowed states form a Cantor set with Lebesgue measure zero. Later, Sendler and Steel [ 19881 presented a detailed treatment of the same structure and compared the results with those for a periodic stack. The Landauer resistivity RL = R/( 1 - R ) (R, reflection coefficient) was studied together with the transmission. It is clear that RL becomes exponential or power law bounded (Landauer [ 19701) as a function of system length for localized and critical states, respectively, For an extended state the transmission is given by a constant or a bounded function of material length. The nature of the field distribution for various values of the optical width of the slabs was also investigated. The major results can be summarized as follows: in the limit when the number of layers

40

NONLINEAR OPTICS OF STRATIFIED MEDIA

[I, § 2

becomes large all the observable states are exponentially bounded surface states. For a smaller number of layers, however, some states appear to be critical but cross over to exponentially localized states as the number of layers increases. The exponential (critical) character of the states was tested by looking directly at the field distribution or by inspecting the linearity of the dependence of log(RL) on the number of layersj (on log(j)). The scaling behavior and the self-similarity in a one-dimensional Fibonacci multilayer consisting of quarter wave plates of SiO2 and Ti02 was recently observed experimentally (Gellermann, Kohmoto, Sutherland and Taylor [ 19941). The theory for a nonlinear Fibonacci multilayer (Dutta Gupta and Ray [ 1988-1 9901) was developed using the nonlinear characteristic matrix formalism described in 9 2.1.2. The nonlinearity was assumed to be given by eq. (2.23). It was shown that nonlinearity can lead to a rich multivalued character with respect to the power dependence of the transmission (Dutta Gupta and Ray [ 19881). States corresponding to the total transmission of the nonlinear structure were shown to correspond to bulk localized states described by a "sech" distribution. The emergence of these structures has the same origin as that with nonlinear periodic stacks. These bulk localized states for various generations were shown to be self-similar (Dutta Gupta and Ray [1989]). A detailed numerical study of the effects of nonlinearity on localized, extended, and critical states revealed (Dutta Gupta and Ray [1990]) strong surface localized states, which are observed for forbidden states ( R z 1) in linear theory and which become increasingly extended, until1 they spread over the whole structure as the power is increased. Thus, nonlinearity leads to delocalization. However, the extended states corresponding to the allowed regions (with R % 0) in the linear theory remain almost unaffected by nonlinearity (except for a scaling of the amplitude). The evidence of critical-like states in the nonlinear structure was also demonstrated. The most important result was the evidence of bulk localization, which persisted even with an increase in system size. This is in sharp contrast with the linear theory and is an indication of the delicate interplay between nonlinearity and dispersion due to quasiperiodicity. Studies were conducted on systems with quasiperiodicity other than that of a Fibonacci sequence, for example, a self-similar Fabry-Perot resonator consisting of alternating dielectric layers of different refractive indices with the high index layers belonging to a triadic Cantor set (Bertolotti, Masciulli and Sibilia [ 19941). Bertolotti, Masciulli, Sibilia, Wijnands and Hoestra [ 19961 proposed the waveguide realization (i.e., a waveguide with Cantor corrugated surface). Bertolotti, Masciulli, Ranieri and Sibilia [ 19961 studied the nonlinear Cantor corrugated waveguide to show bistability with the narrow resonances

I,

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NONLINEAR TRANSMISSION AND OPTICAL BISTABlLlTY IN LAYERED MEDIA

41

in the stopgap. The results were compared with those of a periodic waveguide with equivalent optical length. However, these authors do not address the most significant aspect of quasiperiodic structures, namely, the possibility of weak localization of photons in them. 2.4. EXPERIMENTS ON NONLINEAR TRANSMISSION IN LAYERED MEDIA

The spectrum of experimental studies involving nonlinear layered media is extremely broad. An excellent description of the achievements until1 the mid1980s is given by Gibbs [1985]. Experimental results were classified under different types of nonlinearities. Experiments conducted with widely used nonlinear materials (e.g., Kerr media, such as CS2, nitrobenzene, liquid crystals, semiconductor etalons, semiconductor quantum wells, interference filters) were reviewed separately. Waveguides and distributed feedback structures were also described. Since the literature and available experimental data are extensive, we shall limit our focus to a few recent experiments describing some important issues. Liquid crystals played an important role ever since the early demonstration of optical bistability in Fabry-Perot etalons (Khoo [1982], Cheung, Durbin and Shen [1983], Khoo, Hou, Normandin and So [1983]) and in waveguide configuration (Vach, Seaton, Stegeman and Khoo [ 19841). Liquid crystals in the nematic phase have a large nonlinearity due to optical field-induced collective reorientation of molecules. Durbin, Arakelian, Cheung and Shen [1982] demonstrated that a continuous wave laser of 100 W/cm2 power was sufficient to lead to a significant change in the refractive index. The other mechanism that can be relevant in liquid crystals is the thermally induced nonlinearity. The refractive index change as a function of temperature has a negative gradient (i.e., dn/dT
42

NONLlNEAR OPTICS OF STRATIFIED MEDlA

[I, § 2

nonlinearity (with dnldT = K-') as the dominant mechanism. The sample was irradiated by an Arf laser at an angle, and the transmitted power was monitored as a function of the input power. For lower refractive index contrast, switching as well as bistability was observed at power levels of 30-100mW, whereas for the larger contrast almost fourfold larger switching and bistability thresholds were demonstrated. Switching (bistability) was observed for the angle of incidence close to (larger than) the TIR angle. The observations were supported qualitatively by theoretical calculations for s-polarized waves. Recently, organic and polymer materials have drawn considerable attention because of their large and fast nonlinearities (Carter, Thakur, Chen and Hryniewicz [ 19851, Wu, Heflin, Nonvood, Wong, Zamani-Khamiri, Garito, Kalyanaraman and Sounik [ 19891, Rochford, Zanoni, Stegeman, Krug, Miao and Beranek [1991]). The nonlinearity of such materials, though large, is not sufficient to lead to observable nonlinear effects in bulk samples at moderate power levels. Thus, the optimal choice for their use is in waveguiding geometries (for a review, see, e.g., Kajzar [1992]), where large power densities can be achieved. Some prospective candidates, which can be easily manipulated to yield high-quality cubic nonlinear films, belong to the group of phthalocyanines. These materials have several advantages. They have low absorption in near IR, are easily processable, and can be made into channel waveguides. It is not surprising that in the past decade the phthalocyanines have emerged as one of the most widely used materials for third-order effects in waveguide geometries. Polymer waveguides have been shown to lead to optical bistability (Singh and Prasad [1988], Sasaki, Fujii, Tomioka and Kinoshita [ 19881). Polydyacetylene was used in these experiments. Low power optical bistability with low response times was demonstrated by Wang, Ye, Qiu, Fu and Shen [1995]. The same group (Si, Wang, Zhao, Zou, Ye, Qiu, Shen, Cai and Zhou [1996]) demonstrated optical bistability with ultrafast switching using ODVPC (2,9,16,23-tetraoctadecanoylamido-substituted vanadyl phtha1ocyanine)-dopedpolystyrene film. The switch up (down) time reported was less than 10 (20)ps. Their typical experimental arrangement is shown in fig. 6 . A frequency doubled mode-locked Nd-YAG laser pulse (with 60ps pulse width and 1 Hz repetition rate) was split into two. One part was used as a reference, and the other was coupled to the waveguide by means of a ZF, glass prism. A similar prism was used for outcoupling. The waveguide with 3 pm width was deposited on a fused quartz substrate. The pulse shapes of the output and reference were monitored by a streak camera. Excitation of several modes at low power levels was observed, although the measurements were performed on the TMo mode. The optimum coupling angle for this mode was about 31.5". At 532nm the polymer had low

I,

9 21

NONLINEAR TRANSMISSION AND OPTICAL BlSTABlLlTY IN LAYERED MEDIA

, -

.R

43

Aperture

Fig. 6. Schematic view of the experimental set up for optical bistability in ODVPC-doped polystyrene waveguides. Symbols: M, mirrors; R, rotation stage (Si, Wang, Zhao, Zou, Ye, Qiu, Shen, Cai and Zhou [ 19961).

losses, which was inferred from the linear absorption measurements. Output power from the waveguide was monitored as a function of input power. For positive detunings of the angle of incidence ha, namely, 6u=96‘ and 156’, bistability was observed (figs. 7a,b, respectively), whereas for negative detunings 6u =-135’, bistability was absent (fig. 7c). The observed effect of bistability only for positive detunings confirmed the self-focusing character of the nonlinearity and ruled out the possibility of thermal origin. Note that thermal nonlinearity in ODVPC would have dnldT < 0, which was confirmed by an independent set of z-scan measurements. From the bistability loops and the observed pulse shapes for 6u=96’, the switch up and down times were estimated to be 10 and 20ps, respectively. This is the fastest all-optical switching achieved with organic films. Third-order effects in semiconductor waveguides and multiple quantum well structures (Robson [ 19921) were extensively studied. The measurement of nonlinear phase is an important class of experimental studies on nonlinear optical properties of semiconductors. Various methods involving a single pass of the beam in the sample were developed. For example, interferometric techmques (Bliss, Speck and Simmons [ 1974]), degenerate wave mixing (Miller, Chemla, Eilenberger, Smith, Gossard and Wiegmann [ 19831, Miller, Manning, Milsom, Hutchings and Crust [ 19891) and self-focusing and defocusing (Miller, Seaton, Prise and Smith [1981], Sheik-Bahae, Said and Van Stryland [1989]) have been exploited to this end. In the layered configuration (e.g., in Fabry-Perot cavities) the estimation of the nonlinear phase shift or nonlinear index change

80

?

60

?

v

a"

E 6 0

40

a" 2

I

I

a

a

$

0

40

1

8

20

o

t

1

20

40

60

Input Power (X7.5kW)

(4

80

20 n

0

20

40

60

80

Input Power (X7.5kW)

(b)

100

120

0

40 60 80 100 Input Power (X7.5kW)

20

120

(c)

Fig. 7. Measured input-output characteristics (Si, Wang, Zhao, Zou, Ye, Qiu, Shen, Cai and Zhou [1996]) for different detuning angles: (a) 6a=96', (h) 6a= 156', (c) 6a=-l35'. Triangles and circles indxate data for the upper and lower sides of the input pulses. Note the absence of bistability for negative detuning, indicating the predominantly electronic character of the nonlinearity.

I,

5 21

NONLINEAR TRANSMISSION A N D OPTICAL BISTABILITY IN LAYERED MEDIA

45

becomes more complicated because of the multiple reflections at the mirrors of the cavity. The situation is the same with semiconductor bistable devices with integrated Bragg reflectors such as in the early experiment of Gibbs, McCall, Venkatesan, Gossard, Passner and Wiegmann [ 19791. Recently, a monolithic multiple quantum well GaAs/AlGaAs etalon with integrated AlAs/AlGaAs Bragg reflectors was studied by Sfez, Oudar, Michel, Kuszelewicz and Azoulay [ 19901 and Sfez, Kuszelewicz and Oudar [1991] to show bistability in reflection. In contrast to an earlier study on nonlinear refraction in Fabry-Perot cavity, where pump and probe beams at different wavelengths were used (Lee, ChavezPirson, Rhee, Gibbs, Gossard and Wiegmann [ 1986]), these authors used the same beam for inducing the change and probing it, using a sample with a high finesse F (=38) and good contrast (30) between the high and low reflectivity states. To measure the nonlinearity-induced refractive index change, Sfez, Kuszelewicz and Oudar [ 1991 ] applied rectangular pulses at various power levels to obtain the quasi-steady state response. The shift in the reflectivity dip was monitored as a function of wavelength for different input powers. The change in the refractive index 6n was estimated by measured shift 6A in the reflectivity dip in the scale of the wavelength. 6n as a function of 6A is given by

6n =

?,

2 AA d6" ~

(2.77)

where AA = T C (cp~ is the half roundtrip phase) is the local free spectral range and d is the width of the nonlinear layer; A?, for the sample with d = 2.6 pm was estimated to be 23 nm at A= 838 nm. The intracavity power P(cp) was estimated by using the formula (2.78) where Pi, is the incident beam power, and R ( q ) was approximated by the standard Fabry-Perot expression (2.79)

R,,, in eqs. (2.78) and (2.79) is the minimum reflectivity (evaluated from linear spectra at low power levels). y for the sample (for which Rmin<< 1, 3>> 1) in eq. (2.78) is approximately given by 2F/n. Bistability in input-output power characteristics in reflection was reported with 1 ps long triangular pulses with 10 kHz repetition rate. The measurement of the reflection coefficient as a function

46

NONLINEAR OPTICS OF STRATIFIED MEDIA

[I, § 2

of wavelength at various power levels yielded the dependence of the nonlinear index change as a function of input power. A saturation of the index change with increasing power was reported. The saturating value of the index change was estimated to be an, = - 2 . 4 ~ with power P, = 202 mW needed to reach half the saturating value. Note that an a priori model of the nonlinearity was not used in the preceding study. A clever application of a spike (with duration 20 ps) over the rectangular pulse profile enabled the probing of the various branches of the hysteresis loop. Cada, He, Acklin, Proctor, Martin, Morier-Genoud, Dupertuis and Glinski [ 19921 studied the reflection characteristics of a distributed feedback structure without the Fabry-Perot cavity (i.e., without the additional Bragg reflectors). Their sample consisted of 30 periods of GaAs/AlAs alternating layers on a GaAs substrate. The structure was grown by molecular beam epitaxy (MBE), and the widths of GaAs and AlAs layers were 58.8 and 67.7nn-1, respectively. These design parameters were chosen such that a sharp reflectivity minimum occurred at A = 885 nm, which is just below the electronic band edge of GaAs leading to low absorption but still with a large nonlinearity. Two sets of measurements were performed. In the first set the sample was irradiated normally by a focused argon-pumped dye laser giving out 1 ps rectangular pulse at 10 lcHz repetition rate. Choice of the short pulses led to reduced thermal nonlinearity. Reflectivity as a function of the wavelength of the laser was measured and compared with theoretical calculations. A shift of the reflectivity dip to shorter wavelengths with increasing power as well as more pronounced asymmetry were observed. However, multiple valued response was not observed, which was ascribed to a nonuniform illumination of the sample, negligible rise and fall times of the pulses, and the lateral inhomogeneity of the sample. In the second set the laser beam was modulated as triangular pulses at a repetition rate of 50kHz and focused on the sample with a spot size of 10 pm. Bistability in the input-output power dependence was observed. Shorter pulses (with FWHM loons) led to larger bistability loops. For pulses larger than 1 ps, the loop changes its direction in the sense that switch-on intensity becomes lower than switch-off level. This was interpreted by means of competing mechanisms of electronic and thermal nonlinearity with different response times. Finally, the basic logic operations, such as AND, OR, and NOR, were demonstrated when the incident intensity simulated the superposition of two binary inputs. From an experimental angle one of the most widely used periodic structures for nonlinear optical applications is the corrugated waveguide (Sankey, Prelewitz and Brown [ 19921, Prelewitz and Brown [ 19941, Sankey and Brown [ 19951, Bieber, Prelewitz, Brown and Tiberio [ 19961). Sankey, Prelewitz and Brown

1, § 21

NONLINEAR TRANSMISSION AND OPTICAL BISTABILTTY IN LAYERED MEDIA

47

[ 19921 studied a silicon-on-insulator structure with the top surface of the Si layer corrugated with a period A = 162.51~11,designed to be resonant with the TEo mode of the waveguide. Transmitted and reflected signals for an incident Nd-YAG beam at 1.06 ym (in both continuous wave and pulsed regime). The continuous wave laser was used to measure the effective index of the guided TEo mode, whereas actual measurements for switching carried out with the pulsed laser with widths ranging from 25 to 50ns and energies ranging from 0.78 to 18.75 yJ/pulse. Pulse profiles of the reflected, transmitted and reference (input) signals at various pulse energies were obtained. When the TEo mode was approximately resonant with the center of the stopgap of the first-order Bragg resonance (defined by A = 2 n,ff A, with n,r the effective index of the mode), switching was not seen. An increase in detuning, defined by

(2.80) was introduced by lowering the effective index n , R by two oxidation steps to thin the guiding Si layer. Once the operating point was moved to the left edge of the gap, switching was observed in the pulse shapes. Lowering of the index was necessary since the Si nonlinearity is dominantly selfdefocusing in character. At lower energies (2.86 yJlpulse) the signal exhibited a sharp switch down to a low reflectivity state and showed small oscillations, whereas at higher intensities (1 8.75 pJ/pulse) signs of instabilities occurred, which is consistent with theoretical predictions (see, e.g., de Sterke and Sipe [1990b]). A somewhat analogous setup with silicon on an insulator substrate with aluminum interdigitated fingers deposited on the surface and overcoated with amorphous silicon (fig. 8), was studied recently to demonstrate all-optical switching (Bieber, Prelewitz, Brown and Tiberio [ 19961). These structures are studied in order to realize VLSI-compatible optical switches. The merits of using the metal grating were discussed in detail. Figure 9 shows the typical incident and reflected pulse shapes. At the leading edge of the pulse, the reflected signal mimics the incident signal. With an increase in the incident energy, the shift of the stopgap can be sufficient for the incident field to tune itself out of the stopgap, which leads to a sudden drop of the reflection and corresponding decrease in the peak amplitude of the reflected signal. As the incident intensity decreases, a sudden rise occurs in the reflected amplitude, after which there is a steady decrease following the incident pulse profile. The switching dynamics had a crucial dependence on temperature-dependent detuning, which was studied in detail.

48

NONLINEAR OPTICS OF STRATIFIED MEDIA

output

Coupler A=600 nm

,*

\

Input

/’

\~~

:/I’

Bragg Grating A=167.5 nm ^.

0 055 iim

e

‘\,

‘a

Coupler A=620 nm LTILJ

-

Fig. 8. Schematic view of the metal-semiconductor-metal waveguide structure with 55 nm thick a-Si layer. The spacing between the couplers and the Bragg grating is 0.5 mm. The interdigitated aluminium fingers are 50nm thick. Second-order backward coupling is used to excite the TMo guided mode (Bieber, Prelewitz, Brown and Tiberio [1996]).

Together with several other applications (see, e.g., Assanto [ 1992]), power limiting was a major goal. Power limiting in nonlinear periodic structures was demonstrated using dye-doped colloidal crystals (Herbert, Capinslu and Malcuit [ 19921) and in semiconductor corrugated waveguides (Prelewitz and Brown [ 19941). The origin of power limiting in distributed feedback structures is simple. The intensity-dependent refractive index has two major effects on the stopgap of the structure. For a defocusing nonlinearity in a layered configuration, the gap shifts to the lower wavelengths because of a decrease in the average refractive index, and at the same time a broadening of the gap occurs due to an increase in the refractive index contrast (assuming the nonlinear layers have the lower linear refractive index). For the red edge of the gap a competition occurs between these two effects. However, for the blue edge a decrease in transmission will occur, leading to power limiting. Herbert, Capinski and Malcuit [ 19921 reported a limiting intensity near 10 kW/cm2 with light close to the blue edge of the gap using a colloidal solution doped with M Kiton Red dye. It was demonstrated experimentally that Kerr-like nonlinearity can be induced by two (or multi-) photon processes in semiconductor waveguides operated near (or below) half the bandgap energy (see, e.g., Yang, Villeneuve, Stegeman, Lin and Lin [1993a,b], Villeneuve, Yang, Stegeman, Ironside, Scelsi and Osgood [ 19941, Al-hemyari, Villeneuve, Kang, Aitchison, Ironside and Stegeman [1993]). An important feature of such structures is the strong anisotropy, which leads to different nonlinear index coefficients for the TE- and TM-modes.

I, $ 21

NONLINEAR TRANSMISSION AND OPTlCAL BlSTABlLITY IN LAYERED MEDIA

49

Reflected pulse

-20

0

20

40

60

80

Time (ns)

Fig. 9. Incident and reflected pulse profiles for the structure of fig. 8 (Bieber, Prelewitz, Brown and Tiberio [1996]).

Another characteristic feature is the strong frequency dependence of both the nonlinear refractive index n2 and the two-photon absorption coefficient, which manifests itself in a pronounced asymmetry in the spectral profile of the output pulse (Yang, Villeneuve, Stegeman, Lin and Lin [1993b]). The induced Kerrlike effect was exploited to demonstrate ultrafast optical switching in AlGaAs directional couplers in a 1.55 pm spectral region (Aitchison, Kean, Ironside, Villeneuve and Stegeman [ 19911). Three photon processes in AlGaAs operated at below half the bandgap were also exploited for low threshold switching and nonlinear phase shift (Al-hemyari, Villeneuve, Kang, Aitchison, Ironside and Stegeman [ 19931). Nonlinear phase shift of 6n without significant nonlinear absorption (mostly 3 photon) was reported. The index change causing the nonlinear phase shift corresponds to a large value of the nonlinear coefficient n2, which was exploited to generate spatial solitons at lower intensities than in comparable experiments with glass or CS2. Ultrafast saturable absorption using

50

NONLINEAR OPTICS OF

STRATIFIED MEDIA

[I, § 3

these spatial solitons was demonstrated by Villeneuve, Aitchison, Kang, Wigley and Stegeman [ 19941.

Q 3. Harmonic Generation and other Nonlinear Phenomena in Layered Geometry 3.1. HARMONIC GENERATION

This section examines harmonic generation in stratified media. We distinguish between two geometries, namely, layered media without the excitation of the guided modes and those where the fundamental and/or the generated harmonic constitute the guided or surface modes of the structure. 3.1.1. Harmonic generation in reflection and transmission

Harmonic generation and wave mixing in a layered configuration was first considered by Bloembergen and Pershan [ 19621. The output harmonic field was expressed as a linear combination of a general solution of the homogeneous equation and a particular solution of the inhomogeneous equation. The unknown coefficients of the general solution were determined from the boundary conditions. The interest in harmonic generation in layered geometry was revived by recent studies on surface-emitting second harmonic (SH) generators and SH generators that exploit the cavity-induced enhancement (Normandin, Letourneau, Chatenoud and Williams [ 19911, Vakhshoori, Fischer, Hong, Sivco, Zydzik, Chu and Cho [1991], Takahashi, Ohashi, Kondo, Ogasawara, Shiraki and Ito [1994]) of the fundamental field. The concept of quasiphase matching (see, e.g., Fejer, Magel, Jundt and Byer [1992]) in such devices played an important role. In type 111-V semiconductor multilayers with large but nonphasematching nonlinearities, the phase mismatch between the fundamental and generated harmonic was compensated for by periodic modulation of linear andlor nonlinear parameters of the system. A systematic treatment of the problem of harmonic generation for a general multilayered media was presented for both isotropic (Bethune [ 19891) and anisotropic Bethune [ 19911 layered structures. A matrix method incorporating the contribution of source polarization and suitable for computation was developed under undepleted pump approximation. The method was applied to show enhanced third harmonic generation in a periodic structure with alternating linear and nonlinear layers by exploiting the enhancement of the fundamental field due to resonances at the edge of the

I,

5

HARMONIC GENERATION AND OTHER NONLINEAR PHENOMENA IN LAYERED MEDIA

31

51

stopgap (Mahalakshmi, Jose and Dutta Gupta [ 19961). Recently, Hashizume, Ohashi, Kondo and Ito 119951 proposed an alternate formalism. In contrast to the method of Bethune, who used the solutions of Bloembergen and Pershan [ 19621, these authors combined the Green’s function approach of Sipe [1987] for the solutions with the matrix formalism of Bethune. An analogous Green’s function technique was developed earlier in 1975 by Aganval, and was applied to various problems in a different context (Agarwal [1975a-c]). The result, we believe, is a clearer physical picture of the mathematical expressions. Note that analogous methods have been used recently but in the context of specific systems (Vakhshoori [ 19911, Whitbread and Robson [ 19941). We examined Hashizume, Ohashi, Kondo and Ito [1995] to recall the essential steps and results of the matrix formalism, which was developed under the same approximation of an undepleted pump, and we largely conform to their ideas. Consider the structure shown in fig. 1. Let the j-th layer be nonlinear. The key step in the derivation is to express the harmonic field in the j-th layer as

E,

Etotal = ’self J

+iydd,

(3.1)

where i?7lfis the harmonic field generated in region j , which would be present in the absence of any interface. 2Flf is calculated by means of Green’s function approach and draws its origin in the source polarization due to the fundamental distribution; is the harmonic field in region j , reflected by the boundaries at ~ ~ - 1 z,, plus those coming from other regions (fig. 10 for the meaning of i?fdd).Green’s function technique is then used to calculate the self-field, 1 the z-dependence of which can be written as

i?=dd

where

52

NONLINEAR OPTICS OF STRATIFIED MEDIA

- 1

Fig. 10. Self and additional harmonic waves, E7lf and EJdd,respectively. Eylf is generated within the slab and would be there if no reflections were to take place at the boundaries. EJaddgives the harmonic field generated in the slab and reflected by the boundaries plus those coming from other regions.

where pj(z) gives the z-dependence of the nonlinear polarization at harmonic frequency. Unit vectors 9 and $*I arise from a decomposition of the field into s- and p-polarizations. They are defined as

The x-dependence was universally assumed to be of the form exp(ik,x). Gaussian units were used to write eqs. (3.3) and (3.4). It is clear from eqs. (3.3) and (3.4) that I?:r) is a superposition of forward and backward propagating waves, whereas I?; is bound to the j-th layer. Both the forward and backward propagating waves, when reflected by the boundaries at Z,-I and z, along with incoming waves from other layers, constitute Thus, I?;dd should be calculated using the boundary conditions, to which I?; do not contribute. The forward and backward propagating components of the field I?,n(z) given by

are then arranged in a two-component column matrix

as follows:

I, $ 31

HARMONIC GENERATION AND OTHER NONLINEAR PHENOMENA IN LAYERED MEDIA

where

iJ(=Jj*),

jj*), s,

-

53

P(k)-*-s) defines the polarization and E:f’ (E;;)) gives

the forward (backward) wave amplitude. Transfer matrix theory is then invoked to write the boundary conditions at z =zI, F,+l,A(’J)

= Ytl,,&A(ZJ>,

where M,”,l,Iis the interface transfer matrix given by

(3.8)

and r& and t& are the usual interface reflection and transmission coefficients for

the specific polarization A. Propagation from z, to z h in the j-th layer is accounted for by propagation matrix Qj as follows: 4d(zu)

~ ( Z -zb)&A(zb), U

zj-I

6 z h < za

< zj,

(3.10)

with Oj(z) = Diag (eik;lZ,e-ikz,Z),

(3.11)

Inverse propagation is given by q(-z)= j(z)=9:’(z). Applying the boundary conditions at z=zi and z=zi-l and using eq. (3.10), the following relation is obtained: = MJ+l,j

[qMj,j-, F i ” _ 4 Z j - l )

+Fi”],

(3.12)

where

qy=F,’(zj)-drjFi’(Z/-l),

(3.13) j = 1) . . . )N Recall that superscript “r” refers to the self-field given by eq. (3.3). The field amplitudes outside the layered portion, namely, Ftaddand Fiadd,can then be evaluated by successive application of propagation and interface transfer matrices. This constitutes the contribution of the j-th layer to the forward and backward scattered harmonic wave. Since, under the approximation of the undepleted pump, the contributions from the nonlinear layers are additive, the total forward scattered harmonic amplitudes can be written as (3.14) (3.15)

Ta/i =

M - I ~ (. ~. . -@p+l ~ Mp+l,p,P < a, Q ~ +MIa + ~ , n + 2 . . . @ p i M(j-1,~~P > a.

M a , a - ~%I Ma,a+i

(3.16)

Equation (3.14) can be inverted to yield F i a d d ( z 0 ) in terms of F p d d ( z N ) . Note that one component in each Ftadd(zN) or Frdd(zo)is zero when no additional harmonic

54

NONLINEAR OPTICS OF STRATIFIED MEDIA

1 --

.g

251

5 -

20

L

I v)

a,

(a) 7 nm cap layer

15

2

10

2

5

B

Calculation

11, § 3

0

600

800

1000

1200

Film Thickness (nm)

600

800

1000

1200

Film Thickness (nm)

Fig. 1 lab. Comparison of the theoretical and experimental results for the relative SH intensity with (a) 7 nm and (b) 28nm GaP cap layer for the structure shown in fig. 1 Ic. The open circles give the experimental results while the solid line represents the theoretical calculations (Hashizume, Ohashi, Kondo and It0 [1995]).

waves are incident on the structure from left or right. Thus, eq. (3.14) represents add(+) add(-) two simultaneous equations for two unknowns Et ( Z W ) and E, (zo). The correspondence of the theoretical predictions and experiment was tested on a sample of an air-Gap-Alp-Gap multilayered structure (Hashizume, Ohashi, Kondo and Ito [1995]). Crystal orientation in the various layers was chosen to maximize the nonlinear polarization. An s-polarized fundamental at o was incident from air at 45". Thus, for all the regions. The field distribution of the fundamental in each layer was determined using the transfer matrix technique. For a given input field amplitude EE), taking into account the crystal symmetry, the resulting nonlinear polarization was written as

k:=h

pfW(fi = -;pJ(z)

(3.17)

where PJ(z)is determined by the forward and backward fundamental amplitudes

I, 0 31

HARMONIC GENERATION AND OTHER NONLINEAR PHENOMENA IN LAYERED MEDIA

w

55

X

Fig. 1 lc. Schematic view of the layered medium with a GaP cap layer.

in the j-th layer, the nonlinear d-coefficient, and the z-component of the wave vector. The second harmonic is generated with p-polarization, since the source polarization is oriented along the z-axis. The reflected SH emerges at an angle 45" because of the relation k,"' = 2k,W = % cos45". The experimental samples with two different thicknesses of the GaP cap layer (see fig. 1lc), namely, 7 and 28nm, were grown by the MBE technique. The sample was excited by a fundamental at 1.064 pm obtained from a Q-switched Nd-YAG laser. Relative SH intensity was monitored as a function of the A1P film thickness. Excellent agreement between theory and experiment was shown (fig. 11). A comparative study of the second-order nonlinearities of GaP and A1P was also carried out; IdAp/dGapJwas found to be 0.33f0.06. A major achievement from both fundamental and applied angles was the observation of SH generation in quasiphase-matched (for a description of quasiphase-matching see Q 3.1.2), quasiperiodic structures (Zhu, Zhu, Qin, Wang, Ge and Ming [1997]). In a quasiperiodic structure the condition of quasiphase-matching can be written as Ak = k2w - k, - K,,,n = 0 , where m, n are integers, k,, k2, are the wavevectors of the fundamental.andthe SH, respectively, and K,, is the reciprocal vector. The reciprocal vector requires two integers for labeling because of the two incommensurate periods of the quasiperiodic structure. Note that for a periodic structure the reciprocal vector K,,, (=mK, K = 2 d A , A-period) is indexed with only one integer. Thus, because of the wider options for K,,n, quasiphase-matched SH generation in quasiperiodic structures can cover a large number of harmonic frequencies. The quasiperiodic structure in the experiment of Zhu, Zhu, Qin, Wang, Ge and Ming [1997] was fabricated using the pulse field poling technique in a LiTaO3 wafer at room temperature. Each of the building blocks A and B of the structure consisted of one positive and one negative ferroelectric domain and were arranged as Sj = Sj-1 Sj-2, j 2 3,

56

NONLMEAR OPTICS OF STRATlFlED MEDIA

[I, § 3

with S1= A , S2=AB. A weakly focused pulsed beam from a tunable optical parametric oscillator was used as the pump to measure the spectrum of generated SH. Quasiphase-matchedoutput with an efficiency ranging from 5% to 20% was recorded when the fundamental wavelength was tuned to 0.9726, 1.0846, 1.2834, 1.3650, and 1.5699ym. Self-similarity features were shown to be destroyed due to dispersion in the constituent layers. Theoretical calculations were carried out, and they showed excellent agreement with the experimental observations.

3.1.2. Harmonic generation in guided wave conjiguvation Harmonic generation in guided wave geometries (see, e.g., Stegeman [ 19921) has been studied intensively in view of the potential applications in integrated optical devices. Frequency doubling of GaAs-based lasers for data storage and xerography applications are just a few examples of the scope of such devices. The advantages of guided wave structures over their bulk counterparts are now well understood. We note only the salient features. The guided modes are characterized by typical transverse field distributions. The spatial overlap of these mode functions (which are generally orthogonal and normalized for a particular set of modes) plays a crucial role in determining the efficiency of the nonlinear interaction. The larger the overlap integral, the larger is the efficiency. The other important feature to note is that in guided wave geometries the modes are characterized by their corresponding effective indices of refraction n , f . Thus, the phase-matching condition has to be expressed in terms of the effective indices of the interacting modes. The major advantage of wave mixing in guided wave geometries is in the additional flexibility in satisfying the phase-matching condition. To illustrate these features, we consider the case of second harmonic generation in a slab waveguide, where both the fundamental and the second harmonic copropagate. The overlap integral is maximized when both the fundamental and second harmonic are of the lowest order, since no nodes of the distribution occur in the nonlinear guiding layer. In the context of phase matching, the options in the bulk are restricted. The usual techniques such as temperature tuning and angle tuning depend on the birefringence of the second-order nonlinear optical materials. The dispersion in the fundamental and second harmonic is compensated by choosing mutually orthogonal polarizations for the interacting waves. In guided wave geometries over and above the use of birefringence (see, e.g., Stegeman and Seaton [ 1985]), other phase-matching techniques like quasiphase matching (QPM) (Armstrong, Bloembergen, Ducuing and Pershan [ 19621, Lim, Fejer and Byer [ 19891, Fejer, Magel, Jundt and Byer [1992]) and kerenkov configuration (see, e.g., De Micheli

I,

5

31

HARMONIC GENERATION AND OTHER NONLINEAR PHENOMENA IN LAYERED MEDIA

57

[ 19921) can be applied. For phase matching the three techniques are explained as follows: (1) Birefringent phase matching: for birefiingent phase matching the fundamental (SH) is chosen to be of TE (TM) polarization. However, for this the off-diagonal nonlinear tensor elements must be nonvanishing for the mixing to take place. (ii) Quasiphase matching: for this technique any relevant waveguide parameter (e.g., refractive index or nonlinear coefficient) is spatially modulated with modulation vector K (=2n/A, with A the period). The second harmonic effective index n:$ can then be matched with a Bragg scattered fundamental effective index as follows

m integer.

(3.18)

(iii) cerenkov phase matching: this technique needs the fulfillment of the inequality n2w>nrE such that the SH is no longer the guided mode and it is radiated in the substrate at an angle 8 (= cos-'(n~R/n2w)). The radiated SH power in this case is proportional to the interaction length, in contrast to the quadratic dependence for guided SH. The literature on harmonic generation, especially second harmonic generation in guided wave structures, is extensive, and it is beyond the scope of this review to describe all the achievements. We discuss some recent trends and report on a few experimental studies. Although the concept of QPM was understood as early as 1962, its efficient implementation for parametric up- and down-conversion was achieved only since about the early 1990s. The real technological breakthrough was the domain reversal technique in ferroelectric materials like LiNbO3, LiTaO3, and KTP (see, e.g., Fejer [1992], Bierlein [1992]). These crystals are some of the most widely used second-order nonlinear materials since the early days of nonlinear optics. Obviously the purpose of domain reversal is to have a periodic segmented waveguide with neighboring segments that have the opposite signs of nonlinearity. Before we go into the fabrication details, following Bierlein [ 19921 we analyze the phase-matching condition (eq. 3.18) in more detail, considering the individual segments in each period. The output SH power P of a segmented waveguide, normalized to the output power PO of a phase-matched uniform waveguide of equal length, can be written as (3.19)

58

NONLINEAR OPTICS OF STRATlFlED MEDIA

[I, 0 3

where the grating function G is given by G = - 1 sin(Nq/2) N sin(q/2) ’

(3.20)

q = Akl 11+ Ak2 12.

(3.21)

In eqs. (3.19-21) N is the number of periods, and each period is assumed to consist of two segments of lengths 1 1 and 1 2 , with phase mismatches Akl and Ak2, respectively. The coherence length of the i-th segment is lic = 2n/Aki. At phase matching (i.e., q =2mn, m integer), when the function G is maximized, the quantity I in eq. (3.19), which describes the effects of interference of the second harmonic fields in the neighboring segments of a period, can be expressed as (3.22) where the “positive” sign corresponds to adjacent segments with domain reversal, and the “negative” sign refers to the case of a nondomain-inverted refractive index grating. In the context of the phase-matching condition q = 2mn, one can distinguish the case of balanced phase matching (m=O) from QPM (m # 0). For balanced phase matching, the phase mismatch acquired in one segment is balanced by that of the other segment. For a waveguide with a noninverted domain structure, for sufficiently small periods, the ratio PiPo can approach the value of unity. For QPM with domain inversion when 1l c = 1lc, the maximum 2 SH power achievable is given by P = PO(&) , which for m = 1 occurs for 11 =1,,/2, 12 =12,/2. The method for patterning the nonlinear susceptibility (or domain reversal) in ferroelectric crystals depends on the specifics of the crystals, especially their Curie temperature (Fejer [1992]). Since the Curie temperature is very high for LiNbO3 (1 15OoC), attempts to pattern the domains at an elevated temperature with periodic electric field by means of interdigital electrodes were unsuccessful due to diffusion of the electrodes. The same can be applied to LiTaO3 with a much lower Curie temperature (610°C). The most feasible method was the patterned dopant technique involving patterned in- or out-diffusion of certain dopants leading to a corresponding distribution of reversed domains. It was shown that periodic domain reversal in a LiNbO3 waveguide can be achieved by indifision of a Ti film patterned into a grating of a desirable period by standard photolithographic techniques at temperatures close to the Curie temperature (Lim, Fejer and Byer [1989]). In some of these devices the fabrication of the quasiphase-matched LiNbO3 waveguide is a two-step process, namely, the

I,

5

31

HARMONIC GENERATION AND OTHER NONLINEAR PHENOMENA IN LAYERED MEDIA

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high temperature process of Ti indiffusion for patterning, followed by a low temperature process, such as an annealed proton exchange for forming the waveguide (Lim, Fejer, Byer and Kozlovsky [1989]). Annealing was identified as a crucial step, since waveguides made by proton exchange exhibited reduced nonlinear susceptibilities before annealing. The patterned dopant technique was successfully applied toward domain reversal in LiTaO3 (Nakamura and Shimizu [1990]) and in KTP (van der Poel, Bierlein, Brown and Colak [1990]). Laser diode-based SH generation of blue light holds much potential for use in optical storage and xerography because of the compact nature of such devices. Blue light generation with diode sources was demonstrated using QPM in LiTaO3 domain inverted channel waveguides (Yamamoto, Mizuuchi and Taniuchi [1991], Yamamoto, Mizuuchi, Kitaoka and Kato [1995]). QPM has been an attractive method to obtain blue light sources since phase matching of an arbitrary wavelength can be achieved with appropriate patterning of the nonlinear polarization. QPM was used earlier in LiNbO3 (Feng, Ming, Hong, Yang, Zhu, Yang and Wang [1980], Lim, Fejer, Byer and Kozlovsky [1989]) and in KTP (van der Poel, Bierlein, Brown and Colak [1990]) to obtain such blue sources. A domain-inverted LiTaO3 channel waveguide (Yamamoto, Mizuuchi and Taniuchi [1991]) was shown to lead to a 2.4mW of SH at 424nm wavelength. In a similar structure higher efficiency was reached by enhancing the peak power of the laser diode by gain switching and locking the oscillation wavelength by grating feedback (Yamamoto, Mizuuchi, Kitaoka and Kato [1995]). A 4.5 mW of average blue light power with 13% conversion efficiency was achieved. Other efficient designs have been used for quasiphase matching. For example, a LiNb03 waveguide consisting of an array of channels and a ferroelectric fan patterned domain-inverted grating for QPM was studied to demonstrate 17% SHG efficiency using a continuous wave Nd-YAG laser (Ishigame, Suhara and Nishihara [ 19911). In the context of second harmonic generation an interesting geometry corresponds to the case of counterpropagating fundamental waves that belong to the same mode. Since in this case the resulting nonlinear polarization at SH frequency does not have any surface component of the wave vector, the generated second harmonic is radiated normal to the surface in both the cladding and substrate. Ever since the first demonstration of SHG with counterpropagating fundamental in a Ti-indiffused LiNbO3 waveguide (Normandin and Stegeman [ 1979]), the device potential of this phenomenon has been well understood (Normandin, Letourneau, Chatenoud and Williams [ 19911, Dai, Jam, Normandin, Williams and Dion [ 19921). Various different waveguide structures have been exploited to this end. GaAs-based waveguides (see, e.g., Vakhshoori, Fischer,

60

NONLINEAR OPTICS OF STRATIFIED MEDIA

[I, § 3

Hong, Sivco, Zydzik, Chu and Cho [ 19911, Normandin, Letourneau, Chatenoud and Williams [1991], Normandin, Dai, Jam, Delage, Brown and Chatenoud [1993]) show promise in cases where some form of QPM in multilayered geometry could be used to improve the efficiency. The first observation of the effect in polymer waveguides was reported by Otomo, Mittler-Neher, Bosshard, Stegeman, Horsthuis and Mohlmann [ 19931, who used a channel waveguide made of 4-dimethylamino-4'-nitrostilbene(DANS) side chain polymers. The measured nonlinear cross section was shown to be comparable with a single film GaAs waveguide. The output SH power was 17 mW for a 32 W input power from a Nd-YAG laser. The effect was recently demonstrated in LangmuirBlodgett (LB) films (Bosshard, Otomo, Stegeman, Kupfer, Florsheimer and Gunter [1994]). LB films of organic materials are attractive for nonlinear optical applications because they can be deposited one monolayer at a time with precise second-order susceptibility and thickness. The control of thickness provides a means to influence the effective index of the guided mode, and thus, on the phase-matching condition (recall that effective index depends on the film thickness). The flexibility in the deposition technique adds the freedom to reverse the direction of the nonlinearity within a film, resulting in the possibility of quasiphase matching. Fabrication of low loss LB films has led to efficient collinear SH generation (Decher, Tieke, Bosshard and Giinter [19891, Bosshard, Kupfer, Giinter, Pasquier, Zahir and Seifert [ 19901, Bosshard, Florsheimer, Kupfer and Giinter [1991], Clays, Armstrong and Penner [ 19931). Counterpropagating fundamental wave geometry in the context of organic LB films has an added advantage. Since the output signal traverses only a few wavelengths in the sample, the large resonant nonlinearities of organic materials can be exploited. The experiment of Bosshard, Otomo, Stegeman, Kupfer, Florsheimer and Gunter [ 19941 was performed in a four-layer geometry, including an LB film made of 2-docosylamino-5-nitropyridine (DCANP). The LB film was deposited on a TiOdSi02 waveguide over a glass AF45 substrate. Theoretical estimates (based on coupled-mode approach) for the parameters optimizing the nonlinear cross section were performed. Equally split pulses (with width loops) from the same source (Q-switched and mode-locked Nd-YAG) were coupled into the TEo modes through the two ends of the waveguide by means of grating couplers separated by 5.5 mm with coupling efficiency of 11% and 8%, respectively. The generated SH light could be seen with bare eyes perpendicular to the surface. The measured output peak power was 12.6mW with an angular divergence of 0.036rad. The measured value of the nonlinear cross section was 1.3x W-', which agreed well with the theoretical calculations. Unusual suggestions have been made for phase matching in periodically

I,

3

31

HARMONIC GENERATION AND OTHER NONLINEAR PHENOMENA IN LAYERED MEDIA

61

corrugated waveguides for surface emitted SH. One such proposal involved the use of higher-order spatial harmonics for both the fundamental and the SH in the guiding layer. A detailed theory was worked out by Popov and Neviere [ 19941 and Reinisch, Neviere, Popov and Akhouayri [ 19941. Recently, the proposal was tested using a corrugated poly (methyl methacrylate)/ para-nitroaniline side chain polymer waveguide on a glass substrate leading to counterpropagating spatial harmonics of the fundamental and SH guided modes (Blau, Popov, Kajzar, Raimond, Roux and Coutaz [1995]). The reason for the observed low SH efficiency (0.18 x 10-2 m2/W) was attributed to the large spectral width of the pump beam used to excite the guided modes by means of the same grating. Theoretical proposals have been advanced for combining counterpropagating fundamental geometry with quasiphase matching (Ding and Khurgin [ 19961). Coupled mode theory was applied to arrive at the expression for the conversion efficiency as a function of the pump power. The cases of copropagating and counterpropagating fundamental waves were compared. The latter exhibited a maximum for certain critical value of the pump, whereas the former showed a monotonic increase in the efficiency with increasing pump power. Thus, for a broad range of input pump powers below this critical value the counterpropagating geometry appears to be more advantageous. In contrast to the birefnngence and quasiphase-matchingschemes, in Cerenkov configuration the phase matching is automatic since the radiated SH can be chosen from a continuum of modes. Because of the oscillatory nature of these radiation modes, however, the overlap between the fundamental and the SH drops, leading to low efficiencies. Several recent theoretical studies have explored the possibility of enhancing the efficiency in the eerenkov configuration. A multilayered planar waveguide loaded by an organic nonlinear film was proposed by Hayata, Yanagawa and Koshita [1990]. The surface of the organic film was bounded by a thick transparent dielectric cover. For a cover refractive index at SH frequency larger than the refractive index at the pump frequency, a larger output was predicted compared with that in Ti-indiffused L i m o 3 waveguides. Calix[4]arene waveguides were proposed, analyzed and studied experimentally by Noordman, van Hulst and Bolger [ 19951. A maximum efficiency of 0.23% for a fundamental wavelength of 820nm with power densities of 100 MW/cm2 in the waveguide was reported. One of the major difficulties of the Cerenkov configuration, from the viewpoint of device applications, is the crescent shaped beam in a channel waveguide. A parabolic mirror optics was proposed recently to collimate the crescentshaped beam (Wang, Wada and Koga [ 19961).The convergence of the collimated beam was reported to be less than 1.6mrad, and by means of an objective

62

NONLINEAR OPTICS OF STRATIFIED MEDIA

[I, § 3

lens the beam could be focused to a diameter, that was just 1.27 times the diffraction-limited focusing point. The evolution of the intensity profile of the SH radiation with propagation distance in a planar z-cut LiNbO3 waveguide was studied both theoretically and experimentally by M a h a l a k s h , Shenoy and Thyagarajan [ 19961, who reported good agreement between theory and experiment. Effects of losses due to propagation and pump depletion on the efficiency of conversion was calculated using the coupled mode approach (Shenoy, Mahalakshmi and Thyagarajan [ 19931). The SH generation efficiency was shown to exhibit an exponential behavior in contrast to a linear relation for an undepleted pump. The theoretical predictions were tested on LiNbO3 waveguides with varying losses. Another perspective suggestion for enhancing the efficiency in the Cerenkov configuration was the QPM Cerenkov geometry (Tamada [1991]). In this configuration a periodic variation of the nonlinear properties along the propagation direction was used to phase match to a radiation mode at small angles with the waveguide axis. Again, phase matching is automatic with improved efficiencies due to a better overlap. A comparative study of various configurations (namely, Cerenkov, guided-to-guided using QPM, and QPM Cerenkov) was carried out by Thyagarajan, Mahalakshmi and Shenoy [1993]. A comparison of the performances of these configurations in terms of the maximum achievable conversion efficiency and their tolerance to variations in the waveguide parameters revealed that SH generation using QPM Cerenkov has certain advantages over both Cerenkov and QPM configurations. An interesting suggestion to enhance the SH output in the QPM cerenkov scheme by using a chirped nonlinear susceptibility grating is due to Sato, Azumai and Nozawa [ 19941). With a VDCNNAc copolymer waveguide, they demonstrated a significant enhancement compared with a uniformly modulated waveguide. Considerable research has been conducted on other second-order wave-mixing processes in guided wave geometries. With pump wavelength in the visible range, parametric down conversion and oscillation in the L i m o 3 waveguide were reported more than a decade ago (Sohler and Suche [1981], Hampel and Sohler [1986]). Phase matching in these experiments was achieved by exploiting the birefringence. In contrast to these experiments where d3 I played the dominant role, the QPM scheme in the L i m o 3 channel waveguide applied the largest nonlinear coefficient in LiNb03, namely, d 3 3 (Baldi, Nouh, De Micheli, Ostrowsky, Delacourt, Banti and Papuchon [ 19931, Baldi, Aschieri, Nouh, De Micheli, Ostrowsky, Delacourt and Papuchon [ 19951). Others have also reported about parametric amplification and oscillation (Bortz, Arbore and Fejer [1995]). In all these QPM experiments the sign of nonlinearity was

I,

8

31

HARMONIC GENERATION AND OTHER NONLINEAR PHENOMENA IN LAYERED MEDIA

63

periodlcally modulated by ferroelectric domain reversal. Thyagarajan, Rastogi, Shenoy, Ostrowsky, De Micheli and Baldi [1996] proposed a QPM kerenkov scheme to have a large bandwidth for the parametric amplification process. In this scheme the pump and signal formed the waveguide modes, and the idler was a radiation mode. This Cerenkov-idler configuration provided a large signal gain bandwidth and was more tolerant to variations in the pump frequency. 3.2. OTHER MISCELLANEOUS EFFECTS

3.2.I . Cascaded second-order nonlinear effects

Cascaded second-order nonlinearity has received considerable attention during the past decade. A major direction in these studies was the intensity-dependent phase shift, much like that in third-order nonlinear materials. The principle behind the intensity-dependent phase shift in cascaded second-order nonlinear processes is rather simple. A fundamental wave is up-converted to, say, the second harmonic, which is then down-converted back to the fundamental frequency. The resulting wave at the fundamental frequency can have a phase, which is distinct from the original fundamental wave. Since second-order nonlinear effects are much larger than third-order effects, the resulting “thirdorder” effect due to the cascaded process can be orders of magnitude larger than conventional third-order effects. Effective third-order nonlinearity due to cascaded second-order processes was first proposed by Ostrovskii [ 19671 and detailed study was carried out by Kaplan [1993]. This effect was confirmed experimentally by (Belashenkov, Garanskii and Inochkin [ 19891 and DeSalvo, Hagan, Sheik-Bahae, Stegeman, Van Stryland and Vanhertzeele [ 19921). The first experimental demonstration of cascading in guided wave geometry was given by Sundheimer, Bosshard, Van Stryland, Stegeman and Bierlein [ 19931 who used a quasiphase-matched KTP waveguide to measure the selfphase modulation. Absolute value of the phase was inferred from the measured spectral broadening. A direct interferometric measurement was performed recently by Schiek, Sundheimer, Kim, Baek, Stegeman, Seibert and Sohler [ 19941, where a Ti-indifised LiNbO3 channel waveguide with propagation along x of a y-cut crystal was used. Temperature tuning of the wavevector mismatch was applied to control the sign and magnitude of the nonlinear phase shift. The phase shift was measured by a Mach-Zender interferometer. Both the phase shift and fundamental depletion were measured as a function of temperature. A large phase shift was reported for a broad range of phase mismatches where fundamental depletion was insignificant. An all-optical switch with a contrast

64

NONLINEAR OPTICS OF STRATIFIED MEDIA

[I, § 3

of 7: 1 in a nonlinear hybrid Mach-Zender interferometer using a nonuniformly temperature-tuned LiNb03 channel waveguide was also demonstrated (Baek, Schiek and Stegeman [ 19951). The eerenkov configuration of SH generation was also studied for obtaining a nonlinear phase shift (Krijnen, Toruellas, Stegeman, Hoekstra and Lambeck [1996]). A beam propagation method was used to calculate the phase shift and conditions for its optimization. Initial reports have been published on calculation using a coupled mode approach for QPM eerenkov configuration (Thyagarajan, Vaya and Kumar [ 19971). The possibility of optimizing the phase shift by a proper choice of the grating period was demonstrated. A proposal for an unusual type of frequency shifter, based on the cascaded second-order process, was presented by Gorbounova, Ding, Khurgin, Lee and Craig [1996]. It involved SH generation in a waveguide by a counterpropagating fundamental frequency, followed by difference frequency generation in a second waveguide on top of it with an input probe wave at w,. Recall that the counterpropagating geometry leads to the SH (at 2 0 ) emitted normal to the surface of the waveguide. This SH can then interact with the probe of the second waveguide to lead to an idler at frequency wt = 2w - us,which propagates in the direction opposite to the signal. The use of a multilayered structure with domain-inverted second-order susceptibility for the waveguides was proposed for quasiphase matching. The use of mirrors at the top and bottom (vertical cavity) and also at the side faces of the structure (vertical-horizontal cavity) led to enhanced efficiency of the frequency shifters. A comparison of frequency shifters based on four-wave mixing (using x(3)nonlinearity) revealed that the efficiency in the vertical cavity configuration can be Q,' (Q, is the quality factor of the vertical cavity at 2w) times larger. This can be eight orders of magnitude for typical waveguides made of GaAs/AIGaAs multilayers. 3.2.2. Four-wave mixing

Four-wave mixing (in particular, degenerate four-wave mixing) leading to phase conjugation (see, e.g., Fisher [1983]) in optical waveguides, was first reported more than a decade ago (Karaguleff, Stegeman, Fontenberry, Zanoni and Seaton [ 19851). Together with many vital applications in signal processing, four-wave mixing remains a reliable technique for determining the nonlinear susceptibility x ' ~ )especially , in materials that can be fabricated into waveguides. The knowledge of R e ( ~ ( ~ 1is) important for assessing the nonlinear phase shift, whereas I m ( ~ ( ~plays j ) a crucial role in optical energy loss leading to loss of energy efficiency in the switching process. The estimation of

I,

8 31

HARMONIC GENERATION AND OTHER NONLINEAR PHENOMENA I N LAYERED MEDIA

65

both R ~ ( x ( ~and ) ) I ~ ( x ( ~by ) ) means of the measurement of I ~ ( x ( ~and )) 1 x(3)1 using nondegenerate four-wave mixing was proposed and demonstrated experimentally by Le, Goodhue and Rauschenbach [1990]. They used a AlGaAsIGaAs quantum well waveguide excited by a pump (at 01) and a probe (at y?). Both the pump-induced probe power loss, and the efficiency of the Stokes wave generation were monitored. These two quantities are determined by two different x ( ~processes. ) The first is characterized by I m ( ~ ( ~ )= (m u1 + y? - UI)), and = 01 + 01 - y ?)l determines the efficiency of the generated Stokes wave. Note that both these quantities are related, provided x ( ~ ) ( = @w1 + m - U I ) + X ( ~ ) ( U=~u1 + ul- y?) for y?+w which is true in the absence of strong narrow band resonances at U If u1 or U I f @. For Aul M hy? = 1.43eV, it was found that Im(~(~))=(6.1f0.5)xlO-~~esu, = (7.6f1.7)~ lo-" esu, which clearly indicates the dominance of twophoton absorption. Another interesting study in waveguide geometry included self-pumped phase conjugation in planar waveguide structures in a BaTi03 single crystal (James, Youden, Jeffrey, Eason, Chandler, Zhang and Townsend [19931). A phase conjugate reflectivity larger than 20% was reported. A comparison with the experiment in a bulk sample and in waveguide geometry revealed an order of magnitude reduction in the response time in the latter. Several new theoretical proposals were made involving four-wave mixing. For example, continuous wave parametric amplification in Bragg gratings was investigated theoretically (Steel and de Sterke [ 19951). The physical process corresponds to the case in which two pump photons with frequency upbecome decomposed into a pair of signal and idler with frequencies U, and u,, respectively (i.e., q,+ up = us+ ui).Grating-enhanced amplifications with a strong frequency dependence favoring one side of the bandgap was predicted. A planar single mode Kerr nonlinear waveguide excited by a strong pump (with uniform transverse profile) normally incident on the surface was studied by Robert and Sipe [ 1990-19921. The equations describing the spatiotemporal evolution of the counterpropagating waveguide modes were reminiscent of the corresponding equations for phase conjugation involving the signal and phase conjugated wave. In fact, the right propagating mode amplitude was shown to have a source term proportional to the complex conjugate of the left propagating amplitude. Except for the degree of nonlinearity, the situation is analogous to that investigated by Ding, Lee and Khurgin El9951 for transversely pumped parametric oscillation and amplification. An instability in the waveguide fields leading to the formation of coherent periodic patterns was predicted (Robert and Sipe [ 19901). A detailed analysis of the spatiotemporal dynamics of the system was performed to show the evolution of the instability to the steady state.

Ix(~)(u~

Ix'~'~

66

NONLINEAR OPTICS OF STRATIFIED MEDIA

[I, § 4

In addition to the preceding effects, numerous other studies were published related to pulse transmission (Paye and Hulin [ 19931, Scalora, Dowling, Bowden and Bloemer [ 1994a,b]), pulse compression (Steel, Jackson and de Sterke [1994]), parametric amplification of pulses (Steel and de Sterke [ 1996]), counterpropagating wave interaction (Afanasev, Samson and Tolkacheva [ 1995]), spatiotemporal dynamics (Aceves and De Angelis [ 19931, Aceves, Capobianco, Constantini, De Angelis and Nalesso [1994], Ryan and Agrawal [ 1995]), nonuniform grating (Broderick, de Sterke and Sipe [ 1994]), surfaceenhanced nonlinear effects (Naviere, Popov and Reinisch [ 1995]), nonlinear directional couplers (Yasumoto, Mitsunaga and Maeda [ 1996]), nonlocal effects (Assanto and Stegeman [ 1990]), discrete periodic systems (Christodoulides and Joseph [ 19881, Coste and Peyraud [ 1989]), spatial solitons (Aitchison, Weiner, Silverberg, Oliver, Jackel, Leaird, Vogel and Smith [ 19901, Moloney [ 19921, Kang, Stegeman and Aitchison [ 19961).

0

4. Nonlinear Optical Properties of Layered Composites

In nonlinear optics the search for new nonlinear optical materials with higher and higher nonlinearities has been a continuous process. Much effort has been devoted to synthesize new optical materials with desirable properties, such as high nonlinear response, high breakdown threshold, and short response times. An alternate scheme to enhance the nonlinear susceptibilities with existing materials was proposed recently by Sipe and Boyd [1992] that exploits the local field effects in composite materials. By definition, a composite consists of two or more constituents as “grains”, with a grain size or linear dimension of each constituent such that they may be represented by bulk properties. At the same time the intergrain spacing and the grain dimension must be smaller than a wavelength, so that an effective medium description holds (Gehr and Boyd [1996]). We summarize here some earlier results pertaining to the linear and nonlinear optical properties of composites and their physical origin, and explore the proposal of Boyd and Sipe [ 19941 in the context of layered composites. The study of composite materials was initiated by the pioneering work of Maxwell Garnett [1904, 19061 to explain the color of metal-doped glasses. A model of spherical inclusions with diameter a much smaller than A (wavelength) in a dielectric host was proposed and the effective medium parameters were calculated. Note that the condition a << A enables one to use the quasistatic approximation, which greatly simplifies the mathematical description. It is now well understood that local field effects play a crucial role in determining the

NONLINEAR OPTICAL PROPERTIES OF LAYERED COMPOSITES

67

macroscopic parameters of the composite system. Let the composite consist of spherical inclusions with dielectric constant E , embedded in a host of dielectric constant &h.A single spherical inclusion in the composite does not experience the applied field I!?, but rather, is driven by the local field I!?~,,, which is given by the expression +

El,,

-+

=E

4n+ -€' . 3Eh

(4.1)

The field I!?loc,, inside the spherical inclusion is given by

The importance of the local field correction can be immediately understood, since the possibility of resonant enhancement now exists. In fact, if the inclusion is made of metal with Re(&i)< 0, one may choose the host and the inclusion dielectric constants such that Re(&i+ 2 4 ) = 0, which corresponds to the plasmon resonance. Thus, the inclusion can experience an enhanced local field due to the excitation of the localized plasmons. In the case of several such inclusions randomly dispersed in the host, Maxwell Garnett calculations lead to the following relation for the effective dielectric constant E , R :

where f is the volume fraction of the inclusion material. It is clear from eq. (4.3) that for metallic inclusions and for the specific choice of € i and &h, an enhancement in occurs that will drastically affect the effective dielectric constant. Thus, in the context of the Maxwell Garnett model, an efficient handle (choice of constituents and their volume fractions) is available to tailor the effective medium parameter, namely, the dielectric constant. A great deal of work has been done on linear composites. The Maxwell Garnett theory was extended to ellipsoidal inclusions by Cohen, Cody, Coutts and Abeles [1973]. The shortcoming of the Maxwell Garnett theory was overcome by Bruggeman [ 19351, who proposed a theory that treated the constituents symmetrically. The literature on linear composites is vast, and we mentioned only a few important papers. Progress in this field was reviewed earlier by Rouard and Meessen [1977] in the context of granular thin films. A recent review by Nan [ 19931 summarized the various models, both theoretical approaches and experimental data.

68

NONLINEAR OPTICS OF STRATIFIED MEDIA

[I, 9: 4

Although the possibility of control of effective properties was understood in the context of linear structures, it was not until the 1980s that an equivalent nonlinear problem was studied. The first experimental measurement of nonlinear susceptibility in composites was conducted by Jain and Lind [ 19831, using CdSSe crystallites in a glass matrix. They reported a large ( lopxesu) nonlinearity and subnanosecond relaxation times. Rustagi and Flytzanis [ 19841, Ricard, Roussignol and Flytzanis [ 19851, Roussignol, Ricard, Lukasik and Flytzanis [ 19871 studied the nonlinear mixing signal produced in metal doped glasses for the signature of enhanced nonlinear optical response. Ricard, Roussignol and Flytzanis [ 19851 developed a phenomenological theory for the effective thirdorder nonlinearity. They used a Taylor series expansion of the effective dielectric function as follows:

In writing eq. (4.4) an expansion of eq. (4.3) in the lowest order of volume fraction f was used. Furthermore, treating the inclusion material to be of cubic nonlinearity (given by the nonlinear susceptibility x ( ~ )the ) , change in the inclusion dielectric constant 66 was expressed as (Rustagi and Flytzanis [ 19841, Ricard, Roussignol and Flytzanis [ 19851)

where ,!?lot,, is determined by eq. (4.2) with The final result for ~ E can ~ R be written as

&c,

replaced by the applied field.

(4.6) Note that the local field correction factor x enters eq. (4.6) in the fourth power, thus bringing about the possibility of strong resonance enhancement. The theoretical predictions were tested in degenerate four-wave mixing experiments leading to phase conjugation (Ricard, Roussignol and Flytzanis [ 19851). Samples were made of gold and silver colloids in water. For both samples they observed a resonant enhancement of phase conjugate reflectivities that was several thousand' times stronger than for the off-resonant case. A rigorous treatment (although ignoring the tensor nature of the susceptibilities) was undertaken by Agarwal and Dutta Gupta [1988]. They generalized the T-matrix approach of Gubernatis [1978] to the case of composite media with

1 , s 41

NONLINEAR OPTICAL PROPERTIES OF LAYERED COMPOSITES

69

nonlinear inclusions in a linear host. The nonlinearity was assumed to be of the Kerr type, given by the relation Z i ( 0 ) = G ( W ) + ~2

I. 2

lgloc,i

(4.7)

It was shown that a cubic nonlinearity in the inclusion can lead to odd higher-order nonlinearities in the effective medium. In the limit of vanishing nonlinearity their results reduced to the Maxwell Garnett expression given by eq. (4.3). The result, for example, for effective cubic susceptibility xL;F’, is given bY 4nx;;

=f &,2

lXl2

x2

ll+f(x-1)12(1+f(X-l))2’

(4.8)

where x is defined by eq. (4.2). Thus the T-matrix approach of Agarwal and Dutta Gupta exposed the dependence of the resonant denominators on the volume fraction f that was missing in earlier work (Ricard, Roussignol and Flytzanis [1985]). Other approaches and generalizations have been proposed with more complicated geometries (Stroud and Hui [1988], Zeng, Bergman, Hui and Stroud [1988], Haus, Inguva and Bowden [ 19891, Haus, Kalyaniwalla, Inguva, Bloemer and Bowden [1989], Stroud and Wood [1989], Neeves and Birnboim [1988, 19891, Zhang and Stroud [1994]). One possible application of nonlinear composites for cavityless optical bistability was pointed out by Leung [ 19861, Chemla and Miller [1986], and Schmitt-Rink, Miller and Chemla [1987]. The substantial progress in the field of nonlinear composites was reviewed by Flytzanis, Hache, Klein, Ricard and Roussignol [ 19911. A recent review by Gehr and Boyd [1996] summarizes the latest developments. Although the general case of nonlinear inclusions in a nonlinear host was considered earlier by (Haus, Inguva and Bowden [1989], and Zeng, Bergman, Hui and Stroud [1988]), not until the work of Sipe and Boyd [1992] was the potential of the composite media to enhance the nonlinear optical susceptibilities understood. Sipe and Boyd [1992] considered a composite in the Maxwell Garnett geometry allowing for the nonlinearity of both the inclusion and the host. They considered the case of cubic nonlinearity, and under the approximation that each constituent and the composite were macroscopically isotropic, they could incorporate the full tensor nature of the nonlinear interaction. The key result from Sipe and Boyd [I9921 was that under special circumstances the composite could exhibit an effective nonlinear susceptibility, that was larger

70

NONLINEAR OPTICS OF STRATIFIED MEDIA

[I, § 4

than those of the constituents. Note that this enhancement of the effective medium characteristics does not have its origin in enhanced local field due to the excitation of the localized plasmons in metal composites. The enhancement, although resulting from the local field effects, can be present in composites even with only dielectric constituents. The other major results from Sipe and Boyd [1992] can be summarized as follows: (i) Nonlinear inclusion in a linear host: No enhancement of effective x(3)is possible. An increase in the volume fraction of the inclusion from zero to one leads to a monotonic increase of the effective nonlinearity from zero to that of the inclusion. (ii) Linear inclusion in a nonlinear host: Enhancement in the effective nonlinearity is possible when the inclusions have a much larger linear index of refraction than the host. The origin of the enhancement can be traced to the increase in the local field in the host due to the dipole field of the inclusion, which forces the electric field to concentrate in the host. In this case an increase in the volume fraction leads to an increase in the effective nonlinearity (normalized to host nonlinearity) from unity to larger values. Another interesting observation was that the tensor nature of the nonlinear susceptibility of the composite is a function of the parameters of the composite and can be considerably different from that of the host nonlinearity. (iii) Nonlinear inclusion in a nonlinear host: In this case the result is the sum of the preceding two cases. The predictions of Sipe and Boyd [ 19921 in the context of nonlinear Maxwell Garnett composite have not yet been tested experimentally because of the enormous difficulties in fabricating such a composite. As stated earlier, the major requirements are a large contrast of the dielectric constants of the inclusion and the host, and a comparatively large volume fraction of the inclusion. It is a formidable task to meet both these requirements. Keeping these in mind, an alternate proposal was to look at the effective medium parameters of nonlinear layered composites (Boyd and Sipe [ 1994]), comprising alternating layers of two or more constituents. Each layer is assumed to have a width that is much less than the wavelength. In contrast to the Maxwell Garnett medium, such layered composites are essentially anisotropic in character. The linear properties of layered composites are well understood (see, e.g., Yeh [ 19881). The layered composite resembles a uniaxial crystal with the optic axis oriented normal to the layers. In view of the anisotropic character of the layered composites, it is necessary to distinguish between the two polarizations of the applied electromagnetic field, namely, the field parallel and perpendicular to the

1,

Q 41

NONLINEAR OPTICAL PROPERTIES OF LAYERED COMPOSITES

71

layers. For the electric field parallel to the layers (and, hence, continuous across the layers), the effective dielectric constant ~ 1 is 1 given by the weighted average of the two constituents

In eq. (4.9),fi and E , are the volume fraction and linear dielectric constant of the i-th (i= I, 2) constituent. For the electric field polarized perpendicular to the layers, boundary conditions impose the continuity of the displacement vectors across the interface, and hence, the problem is more complicated. For a linear composite medium the effective dielectric constant EL can be expressed as (4.10) It is clear that the theoretical basis of the derivation of the effective medium parameters such as E ~ Iand E L was the long wavelength approximation (layer width<
12

NONLINEAR OPTICS OF STRATIFIED MEDIA

> c .-

n

a 0 U 3 J

2.5 2.0

'

1.5

'

1.0

'

UJ

wa

-0

.-

0.5 C

0.0

-0.5

I

I

I

I

I

I

0.0

0.2

0.4

0.6

0.8

1.o

I

volume fraction f,

~ 1 ~ '

Fig. 12. Effective second-order nonlinear susceptibility x$! normalized to as a function of the volume fraction of the second componentfz. Various curves are labeled by the corresponding values of the ratio of the linear dielectric constants &*/&I (Boyd and Sipe [1994]).

as in the case of eq. (4.9). The following expression can be used for the effective nonlinear susceptibility

x:; =fix? (4.11) where xi"'is the nonlinear susceptibility of the i-th constituent. In the case of an

electric field polarized perpendicular to the layers, the specifics of the nonlinear interaction have to be retained. Boyd and Sipe 119941 considered the case of second harmonic generation, linear electro-optic effect, and nonlinear refraction, and they presented explicit results for these processes that are summarized below. For second harmonic generation the effective second-order nonlinearity & ) ' i s given by

where

EL

is given by eq. (4.10). Note that the local field correction E I / E I and absence of dispersion, i.e., E,(u)= &i(2w),i = 1, 2) enter eq. (4.12)

E L / E ~ (in the

1,

P 41

73

NONLINEAR OPTICAL PROPERTIES OF LAYERED COMPOSITES

EJE.

I

I

=4.0

\

3.0

I

I

I

I

I

I

0.0

0.2

0.4

0.6

0.8

1 .o

volume fraction fl Fig. 13. Effective third-order susceptibility x r i normalized to xi3’ as a function of the volume fraction of the first component f l . Various curves are labeled by the corresponding values of the ratio of the linear dielectric constants E ~ / E I(Fischer, Boyd, Gehr, Jenekhe, Osaheni, Sipe and WellerBrophy [ 19951).

in their third power. The results for x:2 (normalized to ~ 1 ~as’ a) function off2 for nonlinearity in only one component (x;” # 0, xi” = 0) is shown in fig. 12. Dispersion in the constituents was neglected in fig. 12. It is clear from fig. 12 that for an optimal choice of the parameters of the composite an enhancement by a factor of three can be achieved. The results for linear electro-optic or Pockels effect are analogous to those for second harmonic generation, except that ~ i ( 2 0 and ) ~ ~ ( 2 in0 eq. ) (4.12) are now replaced by their corresponding low frequency limits (i.e., ~ i ( 0 ) and ~ l ( 0 ) ) Neglect . of dspersion (~i(w)=&i(O))in this case can be a poor approximation, since the dc dielectric constants span a larger range. A much larger enhancement factor (-14) is predicted for a choice of composite with nonlinearity in only component “1” with parameters satisfying E Z ( ~ ) / E(0) I = 1000 and ~2 (W ) / E1 (0) = 4.0. Perhaps the most interesting and experimentally realizable case occurs when

74

NONLINEAR OPTICS OF STRATIFIED MEDIA

0.0

-I

0

----I..------.

without enhancement 1

1

I

45

1

Angle of Incidence 8

1

1

90

Fig. 14. Measured nonlinear phase shift for a PBZT/TiOZ composite for s- and p-polarized light. The solid curve gives the theoretical results, while the dashed curve shows the behavior without the local field effects (Fischer, Boyd, Gehr, Jenekhe, Osaheni, Sipe and Weller-Brophy [1995]).

the constituents possess a third-order nonlinearity. The result for the nonlinear refractive index exhibited by such a composite is given by

As in the case of the cubic nonlinear Maxwell Garnett composite, the fourthpower dependence on the local field correction is again present. The results for X ; $ / X / ~ ' as a fimction of volume fraction f l are shown in fig. 13. Figure 13 clearly demonstrates about an order of magnitude enhancement for an optimal choice of parameters. Fischer, Boyd, Gehr, Jenekhe, Osaheni, Sipe and Weller-Brophy [ 19951 verified the predictions of Boyd and Sipe [ 19941 with respect to the enhancement in the nonlinear refractive index. They used a z-scan technique to measure the nonlinear phase shift in the composite. The sample composite consisted of alternating layers of TiOz and conjugated polymer poly(p-phenylene-benzobisthiazole) or PBZT, which was deposited by spin coating. The layer thickness

1,

P 51

CONCLUSIONS

75

of Ti02 (PBZT) was 500 A (400 A), which corresponded to a volume fraction almost ideal for having the peak enhancement. Treating the Ti02 films as being linear with nonlinearity only in PBZT, an enhancement of 35% for the parameters was theoretically predicted. In the experimental z-scan setup the flexibility to control the relative angle between the beam and the uni-axis of the sample was retained. This enabled the control of the normal to the surface component of the incident p-polarized radiation to be maintained. Measurements were carried out for p- and s-polarized light. A detailed analysis for the nonlinear phase shift in the context of the experimental setup was given by Gehr, Fischer, Boyd and Sipe [1996], based on the calculation of the nonlinear phase shift experienced by a plane wave propagating through a finite sample of effective anisotropic nonlinear medium. Figure 14 shows experimental results together with the theoretical predictions. It is clear from fig. 14 that predictions of the effective medium theory match the experimental results well, although a minor adjustment of the refractive indices n(Ti02)ln(PBZT) from 1.22 (experimentally observed) to I .33 was needed for the fit.

Q 5. Conclusions This review has demonstrated that nonlinear optical effects in stratified media can offer much more when compared to the conventional nonlinear optics in bulk homogeneous samples. The resonances arising from the stratification and associated local field enhancement make it possible to have efficient nonlinear interaction at even subwatt power levels. Waveguide structures, maintaining large power densities over long propagation distances, were shown to hold the key for efficient nonlinear optical effects. Many of these effects, such as all-optical switching, optical bistability, power limiting, harmonic generation, and frequency up- and down-conversion, have been demonstrated experimentally, and their commercial implementation is not far off. However, the major problem is still the lack of nonlinear optical materials with high nonlinearity, low absorption, high damage threshold, and sufficiently fast response. These materials obviously must be compatible with the technology of fabricating the stratified media. Unusual proposals, such as that of layered composites, have been suggested and tested for enhancing the nonlinear susceptibility. Much research is needed to explore the device potentials of such composites, and several theoretical predictions have yet to be tested. We look forward to the realization of nonlinear materials that can cope with the demands of the scientific community and to new results in the exciting field of nonlinear optics of stratified media.

76

NONLINEAR OPTICS OF STRATIFIED MEDIA

Acknowledgement It is a pleasure to thank G.S. Aganval for his continued support and valuable suggestions throughout the writing of this review. I also wish to thank J. Jose for her help in the preparation of this manuscript. Financial support from the Department of Science and Technology and National Laser Programme, Government of India, is gratefully acknowledged.

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E. WOLF, PROGRESS IN OPTICS XXXVIII 0 1998 ELSEVIER SCIENCE B.V ALL RIGHTS RESERVED

I1 OPTICAL ASPECTS OF INTERFEROMETFUC GRAVITATIONAL-WAVE DETECTORS BY

PATRICE HELLO Laboratoire de I 'Acceldrateur Lindaire, Groupe VIRGO, Bbtiment 208, Uniuersite de Paris-Sud, 91405 Orsay, France

85

CONTENTS

PAGE

5 1.

INTRODUCTION . . . . . . . . . . . . . . . . . . .

87

0 2.

PRINCIPLE AND PRACTICE OF THE INTERFEROMETRIC DETECTION . . . . . . . . . . . . . . . . . . . . .

89

5 3.

COUPLEVG WITH GRAVITATIONAL WAVES . . . . . . .

99

5 4.

OPTICAL COUPLINGS . . . . . . . . . . . . . . . . .

121

5 5.

THERMO-OPTICAL COUPLING . . . . . . . . . . . . .

141

ACKNOWLEDGEMENTS . . . . . . . . . . . . . . . . . .

160

REFERENCES . . . . . . . . . . . . . . . . . .

160

86

8 1.

Introduction

Until recently, the science known as Astronomy was only the astronomy of photons, since all astrophysical information about the near or the far universe was obtained by the observation of light in almost its entire spectrum, from radiowaves to y rays. It is likely that by the turn of this century a new astronomy will be born; viz., the astronomy of gravitational waves or gravitational astronomy. This will be a radically different astronomy since gravitational waves bring different information than those of optical astronomy. Indeed, according to General Relativity, gravitational waves are ripples in space-time generated by accelerated matter. However, the coupling of gravitational radiation to matter is so weak that only events among the more violent in the universe may give gravitational waves detectable on Earth. The more promising events are the gravitational collapse of massive stars (supernovae) and coalescences of compact binaries (see Thorne [1987], Blair [1991], Bonazzola and Marck [1994] and Thorne [ 19951 for reviews). Since the pioneering work of Joseph Weber in the early sixties, gravitationalwave detectors have been continuously improved in two distinct directions: as resonant detectors and as interferometric detectors. Weber’s first detector belonged to the first class. It was an aluminium cylindrical bar at ambient temperature and with a crude seismic isolation (Weber [1960]). Second generation “Weber’s bars” were built in the seventies, together with the first interferometric prototypes. This “boom” seems linked to Weber’s claim to have discovered strong gravitational-wave signals from the center of the Galaxy (Weber [ 19691); this false discovery has paradoxically given a real impulse to gravitational-wave physics and a number of groups all around the world have begun to study gravitational-wave detectors of all kinds. The idea of interferometric detection of gravitational waves is due independently to Weber and to two Russian physicists, Gertsenshtein and Pustovoit [1963]. The first serious work on feasibility, study of noise sources, etc., was done by Weiss [1972], while Forward at Malibu was building the first prototype (Moss, Miller and Forward [1971], Forward [1978]). This was a Michelson interferometer under vacuum with effective 8.5m long arms and a good sensitivity in the kHz region. After the construction of various prototypes 87

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OPTICAL ASPECTS OF INTERFEROMETRIC GRAVITATIONAL-WAVE DETECTORS

[II,

9

1

with Fabry-Perot cavities as arms at Glasgow (10m long arms) and at Caltech (40m long arms), or with delay lines as arms at Garching (30m long arms; Shoemaker, Schilling, Schnupp, Winkler, Maischberger and Rudiger [ 1988]), several big projects have been conceived and are currently under construction, with the real aim of a first direct detection of gravitational waves in the frequency window from a few Hz to a few kJlz where astrophysical events may occur. These projects are the FrencWItalian VIRGO collaboration (Bradaschia, Del Fabbro, Di Virgilio, Giazotto, Kautzky, Montelatici, Pasuello, Brillet, Cregut, Hello, Man, Manh, Marraud, Shoemaker, Vinet, Barone, Di Fiore, Milano, Russo, Aguirregabiria, Bel, Duruisseau, Le Denmat, Tourrenc, Capozzi, Longo, Lops, Pinto, Rotoli, Damour, Bonazzola, Marck, Gourgoulhon, Holloway, Fuligni, Iafolla and Natale [1990]), for the building of a 3 km interferometer in Cascina near Pisa (Italy), the American LIGO collaboration, Abramovici, Althouse, Drever, Gursel, Kawamura, Raab, Shoemaker, Severs, Spero, Thorne, Vogt, Weiss, Whitcomb and Zucker [1992], for the building of two 4km interferometers, one in Hanford (Washington), one in Livingston Parish (Louisiana) and the EnglisWGerman GE0600 collaboration for the building of a 600 m interferometer in Hannover, Germany, (Danzmann, Luck, Rudiger, Schilling, Schrempel, Winkler, Hough, Newton, Robertson, Ward, Campbell, Logan, Robertson, Strain, Bennett, Kose, Kuhne, Schutz, Nicholson, Shuttleworth, Welling, Aufmuth, Rinkleff, Tunnermann and Willke [ 19951). A Japanese 300 m interferometer prototype, named TAMA-300, is also under development in Tokyo (Tsubono [1995]). The LIGO and VIRGO detectors basically share the same philosophy. These are power recycled broadband detectors with Fabry-Perot cavities as arms, but with a better sensitivity for VIRGO at low frequencies (around 10Hz) due to a more sophisticated seismic isolation (Virgo Collaboration [ 19921). In contrast, the GE0600 interferometer is a detector designed for narrow-band operation, because of the use of the dual recycling techtuque. Parallel to this, several second-generation bar detectors have been in operation since the end of the seventies. These bars are made from high-Q materials, aluminium 5056 for all the bars except the Australian one which is made of niobium, and are cooled to ultra-low temperature (from the liquid He4 temperature to as low as l00mK) in order to decrease the thermal noise (see Blair [1991] for example). Due mainly to the noise in electronics, these detectors are in fact narrow-banded around their resonance frequency, and are designed to be in the range typical of supernova events (about 1 kHz). Because of this narrow-band sensitivity, resonant-bar detectors actually miss a large part of the gravitational-wave spectrum during a possible detection. This implies a loss in

PRINCIPLE AND PRACTICE OF THE INTERFEROMETRIC DETECTION

11, § 21

89

astrophysical informations presented by gravitational waves. In contrast, large interferometers like LIGO or VIRGO are intrinsically broad-band detectors. The situation for gravitational-wave detectors at the beginning of the next millennium is likely to be the following: three broad-band interferometric detectors provided by LIGO and VIRGO, and some narrow-band but sensitive detectors like the GE0600 interferometer and “third-generation” resonant detectors. In this review we discuss some of the aspects of interest for the optical design of interferometric detectors, in their first generation as well as in the next (advanced) generations. This article is organized as follows. In $2, we recall the principles of the interferometric detection of gravitational waves and we list the main noise sources, the ones whch effectively limit the LIGONIRGO sensitivities, in their present design or in their future advanced configuration as well. In $ 3 , we review the different optical configurations proposed for interferometric detectors, broadband configurations as well as narrow-band operations. In 0 4, we give the optical specifications for a shot-noise limited interferometer such as VIRGO and recall that the use of the dual recycling technique can relax these (severe) specifications. The coupling of the laser geometry fluctuations to the interferometer misalignments is also discussed. In the last chapter, we study a particular kind of optical problem: the nonlinear thermal-optical coupling in Fabry-Perot cavities, which is an extra source of losses in gravitational-wave detectors and may in general lead to stability problems.

8 2.

Principle and Practice of the Interferometric Detection

2.1. PHYSICAL EFFECT INDUCED BY A GRAVITATIONAL WAVE

Gravitational waves can be defined as perturbations, hap, of the space-time propagating at the speed of light. The observational effect of a gravitational wave propagating in the neighborhood of two testmasses is to change slightly the distance between them. Here “distance” has to be understood in its operational definition; i.e., time of flight for the light between the two masses.IfA and B are the space-time locations of the two testmasses and if we set AB = (x“), the vector (x“) obeys the equation of geodesic deviation in weak field (Misner, Thorne and Wheeler [ 19731):

90

OPTICAL ASPECTS OF INTERFEROMETRIC GRAVITATIONAL-WAVE DETECTORS

[II, 0 2

where TT denotes the transverse traceless gauge. The maximum change in the measure of the distance AB is then

h 6L = -L, 2

(2.2)

where h is the dimensionless gravitational-wave amplitude and L is the distance AB at rest. Amplitude h can then be interpreted as a rate of deformation of space. According to eq. 2.2), the effect of an incident gravitational wave is to change the relative distance between the two testmasses. Moreover, this effect is differential; the relative changes have opposite signs in two orthogonal directions (like the arms of a Michelson interferometer), recalling the quadrupolar nature of gravitational waves. A supernova collapse in our galaxy could produce on Earth gravitational waves of amplitudes as high as h = lo-'*. Unfortunately, such a phenomenon is believed to occur only once or twice per century. In order to detect gravitational waves with a sufficient rate (say at least some detections per year), gravitationalwave detectors should be sensitive up to the VIRGO cluster distance (15 Mpc), giving a sample of about 2500 galaxies to observe. Since gravitational-wave amplitudes from the VIRGO cluster are expected to be about h M an Earth-based detector should then be sensitive to relative changes of length of the same order of magnitude, SL/L M The goal of LIGONIRGO is consequently to achieve a sensitivity (spectral density) about h M to detect at least a few events per year from the nearby universe (<10-20Mpc). 2.2. THE DC MICHELSON: MINIMAL PHASE SHIFT DETECTABLE

Let us consider a simple Michelson interferometer such as the one shown in fig. 1, and a gravitational wave incident on it with optimal polarization. The apparent effect of the gravitational wave can be described as a &fferential change in the arm lengths, i.e., one arm seems to be longer and the other shorter and vice versa half a gravitational-wave period later. Thus, there is then a phase shift between the two beams circulating in the two arms and the interfering pattern is there modified, giving a signal at the photodetector. The interferometer, which is assumed to be perfect (contrast C = l), is illuminated by a laser source of power Pi,.The phase shift between the two arms is the superposition of a DC offset, a, which results from the tuning of the interferometer, and of the phase shift due to the gravitational wave, QgW. It can then be written as

11,

0 21

PRINCIPLE AND PRACTICE OF THE INTERFEROMETRIC DETECTION

91

Photodiode

Fig. 1. Sketch of the Michelson interferometer.

If a = 0, and without incident gravitational waves, 64 = JG, and the interferometer is tuned at a dark fringe. The output power, Pout,transmitted by the interferometer is related to the input power and the phase shift between the two arms: pout= pi, sin2 and, since

Qgw

(“+2”.-).

(2.4)

~

<< 1, it becomes:

The signal to noise ratio can then be computed from this equation, assuming that the only noise source is the shot-noise. The first (DC) term corresponds to the noise and the second to the signal to be detected. Without gravitational waves, the number of photons, N , detected by the photodetector (assumed to have perfect quantum efficiency) within the time interval At is related to Pi, and

a: Nzlw = Pi, sin2

(4)

At,

where zl is the reduced Planck constant and o frequency, so

(2.6) =

2nv is the light angular

This number of photons detected per unit time obeys Poissonian statistics and thus fluctuates as v%. The fluctuation in detected power is then 6PAt = fizlw,

(2.8)

92

OPTICAL ASPECTS OF INTERFEROMETRIC GRAVITATIONAL-WAVE DETECTORS

[II,

52

and finally

The gravitational wave signal induces the variation of power, P,: P,

= f ~ i sin(a)QggW, ,

that exactly reproduces the variations of S/N = PS/6P,is then

(2.10) Qgw.

The signal to noise ratio,

(2.1 1)

The signal to noise ratio is then maximum for a = 0; i.e., when the interferometer is tuned to a dark fringe. In fact this result is slightly different when the contrast is not perfect (realistic case : C < 1). In the latter case, the optimum is obtained when the interferometer is detuned a little from the dark fringe, depending on the contrast defect (Virgo Collaboration [1989]): a P In the perfect contrast case, the minimal phase shift that the interferometer can detect corresponds to S/N = 1 and is

d , .

(2.12)

For a laser power Pi, = l o w , and a laser wavelength A = C / Y = 1.064pm, we get QF$' M lo-'' r a u f i . Furthermore, a monochromatic gravitational wave of amplitude h (with optimal polarization) produces a phase shift (Giazotto [ 19891) 4nL

OggW = -h,

A

(2.13)

where L is the arm length, so that the minimal gravitational-wave amplitude the Michelson interferometer can detect is (2.14)

With the previous numbers and for L = 1 m, we get h,i, M lo-''/&, while for L = 3km, we get h,,, M 4~10-~'/&. There are at least two orders of which magnitude to achieve a shot-noise limited sensitivity about 1O - 2 3 / f i

11, 0 21

PRINCIPLE AND PRACTICE OF THE INTERFEROMETRIC DETECTION

93

is the goal of LIGONIRGO, as set by astrophysical constraints. We see in eq. (2.14) that the only way to increase the sensitivity is to increase the arm length and/or to increase the input power. This is achieved in LIGO and VIRGO with the use of kilometric Fabry-Perot cavities as arms, together with the use of the power recycling technique (see $3). 2.3. REALISTIC DETECTION SCHEME

The DC detection scheme is actually very sensitive to power fluctuations of the laser source and it is necessary to perform the detection at higher frequencies, typically in the RF range (IOMHz), where the laser power fluctuations are negligible. This is achieved by a coherent (homodyne) detection scheme. The classical internal-modulation technique is not suitable for gravitational-wave detectors, since the phase modulators would be located inside the arms and can produce prohibitive beam distortions. Two other modulation schemes have been proposed within the framework of gravitational-wave detectors. The first, named external modulation (Man, Shoemaker, Pham Tu and Dewey [1990], Gray, Stevenson, Harb, Bachor and McClelland [1996]), is a scheme where some light (reference beam) is taken from the incident laser, then modulated by a phase modulator and finally recombined with the output beam leaving the interferometer; the resulting signal on the output photodetector is then coherently demodulated. The drawback of this modulation technique is mainly the need for extra components that must be implemented and controlled. There can be another problem in kilometric gravitational-wave detectors due to the finite light spatial coherence and to the fact that the main beam in the interferometer propagates over effective distances of tens of kilometers, due to the amplifying effect of Fabry-Perot arms (see $ 3.1), while the reference beam typically propagates only over some meters. But this can be avoided if the reference beam is taken as the reflected beam by the AR coated side of the Michelson beamsplitter (Man, Shoemaker, Pham Tu and Dewey [ 19901); this beam is a tiny fraction of the beam coming from one of the Fabry-Perot arms, and so has endured a propagation distance (relative to the laser source) as long as the main beam. The other modulation technique, named frontal modulation, has been first proposed by Lise Schnupp [1986], and is to be implemented in LIGO and VIRGO. The principle is simple: the incident light is modulated just before entering the interferometer and the output signal on the photodiode is coherently demodulated. The advantage here is that the same modulation can also be used for the locking of the interferometer (Regehr, Raab and Whitcomb [1995], Flaminio and Heitmann [1996]). The drawback is that a length asymmetry

94

OPTICAL ASPECTS OF INTERFEROMETRIC GRAVlTATIONAL-WAVE DETECTORS

[II, $ 2

mirror

Fig. 2. Simplified optical layout of VIRGO.

must be introduced in the Michelson interferometer. Indeed, in the perfectly symmetrical case, the interferometer is tuned at a dark fringe for the laser frequency but also for any other frequency, in particular for the sideband frequencies. If a gravitational wave now slightly detunes the interferometer, the same phase shift is endured by the sidebands and the carrier, and the final demodulated signal, resulting from the beats between the carrier and the sidebands, vanishes. This does not occur if the sidebands are partly transmitted to the output photodiode and so some length asymmetry is needed. The length asymmetry can be optimized so that the sideband power transmitted to the photodiode equals the power lost inside the Michelson interferometer (Flaminio and Heitmann [1996]). In the case of LIGONIRGO, the required length asymmetry is typically 0.5-1 m, and so is only a small fraction of the arm length. 2.4. OPTICAL LAYOUT OF A GRAVITATIONAL-WAVE INTERFEROMETRIC DETECTOR

A simplified scheme of an interferometric gravitational-wave detector such as VIRGO is drawn in fig. 2. We can follow the laser beam from the laser bench to the detection bench. The laser bench carries the laser itself and the frequency pre-stabilization stage (Shoemaker, Brillet, Man, Cregut and Kerr [ 19891, Bondu, Fritschel, Man and Brillet [ 19961).The beam then passes through a mode-cleaner cavity ( L = 144m, finesse -1000 for VIRGO) which acts as a low pass filter (see Q 3.1) for the laser frequency and amplitude fluctuations as well as for its geometry fluctuations like angular jitters. The attenuation factor of thls filter is

11, 5 21

PRINCIPLE AND PRACTICE OF THE INTERFEROMETRIC DETECTION

95

expected to be about 2 x in the low-frequency range (Virgo Collaboration [1995]). The cleaned beam is then an almost perfect TEMoo Gaussian mode that can couple to the recycled interferometer. The details of the interferometer (Fabry-Perot arms and recycling cavity) will be discussed in $ 3 . The beam transmitted to the detection bench eventually carries the gravitational wave signal. The detection bench also holds a mode-cleaner cavity which filters the higher-order modes generated by the interferometer defects. The effect is to improve the contrast and thus the sensitivity (see $ 4). 2.5. MAIN NOISE SOURCES

We will discuss the main noise sources in interferometric gravitational-wave detectors except shot-noise which depends on the optical configuration and which will be discussed in detail in the next section. 2.5.I. Seismic noise

The displacement spectral density due to seismic activity has been measured at the VIRGO site in Cascina (Virgo Collaboration [1992, 19951) as: (2.15)

and is typical of quiet regions. Such a seismic noise gives an equivalent gravitational-wave signal h M 7 x 1 0 - 1 0 / ~ 2 and / ~ , so must be damped by more than 10 orders of magnitude in order to achieve a sufficient sensitivity at 10 Hz. The seismic isolation is based on the fact that an harmonic oscillator, like a spring, acts as a filter for frequencies above its natural resonance frequency YO. The transfer function between the seismic displacement spectral density and the displacement spectral density after filtering is (Giazotto [19891): (2.16) for Y >> VO. This shows effectively an attenuation for frequencies above the resonance frequency. If n identical resonators are hung in series, the total transfer function is then: (2.17)

As an example, 5 springs of frequency vo = 1 Hz give an attenuation factor of -10'' at 10Hz. The seismic isolators developed for VIRGO (the so-called

96

OPTICAL ASPECTS OF INTERFEROMETRIC GRAVITATIONAL-WAVE DETECTORS

[II, $' 2

Super-Attenuator;Virgo Collaboration [ 1989, 1992, 1995]), follow this principle. The Super-Attenuator is a chain of 5 isolation stages, each stage being built using vertical blade springs. The performance of the Super-Attenuator is so good that the sensitivity of VIRGO should not be limited by the seismic noise above about 4 Hz. 2.5.2. Thermal noise An interferometric gravitational-wave detector can be seen as a collection of

mechanical resonators, each degree of freedom being in equilibrium with the external environment and having an energy i k T . These resonators are the pendula made by the suspended mirrors, the mirror substrates, the suspension wires that resonate like violin wires, etc. Since the quality factor of each of these resonators is finite, the thermal noise will excite the resonance of each one, according to the fluctuation-dissipation theorem (Callen and Welton [ 19511). The important parameters for deriving the displacement spectral density are temperature T , resonance frequency of the resonator YO = W O / ~ J Gits , quality factor Q and its effective mass M . In case of internal damping, which seems the relevant one for gravitational-wave detectors (Saulson [ 1990]), the displacement spectral density is (2.18) where o = 2xv. In the current design of interferometric detectors, the pendulum thermal noise should limit the sensitivity at low frequency (
11, § 21

i

t t 1 lo

13'

97

PRMCIPLE AND PRACTICE OF THE INTERFEROMETRIC DETECTION

Vlt

GO 39

iC7S

---if-- 1 / I

1 1

I 1 1 E t

Fig. 3. Planned sensitivity for the VIRGO interferometer vs the frequency (in Hz). The seismic noise does not appear here, since it is planned to be strongly attenuated in the displayed frequency window. However, it would produce an almost vertical barrier around 4 Hz.

and is then an huge noise source at low-frequency, but fortunately decreases rapidly as Y ~ ' ~Above . this, the mirror thermal noise, which is roughly (2.20) dominates up to 900Hz and finally the shot noise (filtered by the arm FabryPerots) dominates above 1 kHz.The vertical peaks are due to the resonances of the suspension wires (violin modes). 2.5.3. Quantum limit

Shot noise is in fact only one aspect of the quantum noise in an optical measurement of the position of a test mass (a mirror). The uncertainty in position measurement due to shot noise can be computed as in 6 2.2, or equivalently from the uncertainty principle for phase measurement: (2.21)

98

OPTICAL ASPECTS OF INTERFEROMETRIC GRAVITATIONAL-WAVE DETECTORS

[II,

52

so the position uncertainty is

(2.22) with similar notations as in 02.2, and where w is the light angular frequency. In order to decrease this uncertainty, it is then necessary to increase the number of photons N or equivalently the circulating power P . But it is not so simple since increasing the power makes the mirror more and more sensitive to radiation pressure fluctuations which induce fluctuations in impulsion of the mirror (Caves [ 19801):

c c The corresponding uncertainty in mirror position is roughly

-

(2.23)

a.

6pAt - 2hoAt (2.24) axxrp __ - m rnc The total uncertainty in the mirror position is the quadratic mean of the two contributions, if they are independent, as is the case for classical interferometers: ax2 = ax;" +

=

(2.25)

This uncertainty is minimal for a certain value of N (for a given At):

hmc (2.26) 8zhwAt In the optimal case, shot-noise and radiation-pressure-noise contributions are the same and the minimal error in the mirror position is then: No

=

~

'

(2.27) The quantum noise in an interferometer like VIRGO or LIGO can thus be computed by incoherently summing the four contributions from the four mirrors in the arms:

(2.28) and depends only on the mass of the mirrors and on the arm length. For testmasses rn = 30 kg and an arm-length L = 3 km,we obtain

(2.29)

11, § 31

COUPLING WITH GRAVITATIONAL WAVES

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The planned interferometers for gravitational-wave detection will have sensitivities above the quantum limit. However, this limit is not so low. T h s is especially true if the thermal noise improves, in which case the sensitivity should be rather close to the quantum noise at low frequency. Note finally that this quantum limit is not an absolute one and techniques such as “squeezing of light” exist in order to beat this limit (Jaekel and Reynaud [1990]).

5 3.

Coupling with Gravitational Waves

The goal of this chapter is to show that it is possible with current technology to achieve sensitivities up to h for suitably designed shot-noiselimited Michelson interferometers. Such sensitivity can only be achieved by including some refinements with respect to the usual Michelson configuration. Since these refinements refer widely to the basic properties of Fabry-Perot cavities (e.g., reflectivity and power gain), we first recall all the useful properties of reflective cavities. We then derive and discuss the standard recycling configuration (the one to be used in LIGO and VIRGO), the dual recycling configuration (planed to be used in GEO600), and finally we discuss briefly the other proposed configurations (not planned to be implemented in any project); e.g., resonant recycling, detuned recycling and resonant sideband extraction.

-

3.1. OPTICAL PROPERTIES OF REFLECTIVE FABRY-PEROT RESONATORS

Consider a Fabry-Perot cavity used in reflection (see fig. 4). Let rl and tl be the amplitude reflectivity and transmittivity of the input mirror, with r: + t: = 1 - A I , A I representing the power losses of this mirror, and r2 the amplitude reflectivity

Fig. 4. Optical cavity: notations.

100

OPTICAL ASPECTS OF INTERFEROMETRIC GRAVITATIONAL-WAVE DETECTORS

[II,

3

of the end mirror, which is assumed to be close to 1 (1 - ri << 1). The light field amplitudes as defined in fig. 4 obey the following set of equations:

v 4

=

Vref =

.XP(-)

2invL

(3.4)

V3,

t~V4 + 171Vinin,

(3.5)

where L is the length of the cavity and v is the light frequency. Note that here we use the convention of a phase shift of n/2 for any reflection and no phase shift for transmission. We easily find the intra-cavity field V1 and the reflected field Vref as a function of the input field: Vl

=

Vref =

tl

1 + rl r2 exp (4invLIc) rl + (1 -Al)r2 exp(4invLk) i Vin 1 + r1r2 exp (4invL/c) Vin

,

Already we see that if 4nvL/c = n mod(2n), the cavity is resonant for the frequency v (i.e., the intra-cavity power is maximal) and the power gain is G = t;/(l - r1t-2)~N (1 + r1)/(1 - rl) for a reflective cavity with low losses (r2 z 1 andAl << 1). For a given length L, the interval between two successive resonances is

z, C

A ~ F S= R

(3.8)

and is called the free spectral range (FSR). For the very long cavities used in gravitational-wave detectors, the FSR is typically tens of kHz. The Jinesse of the cavity is defined as the ratio of the FSR to the frequency full-width at half maximum of a resonance peak (Airy peak). This spectroscopic definition matches the photometric one (Saleh and Teich [1991]):

Now let us define the reflectivity R of the cavity by Vref = iRV,,,, and let the frequency be v = vo + Av, where vo is a resonant frequency, verifying 4nvoL/c = n.We then have:

R=

rI

( 1 - A I )r2 exp (4inAvLIc) ' 1 - rl r2 exp (4inAvLk)

-

(3.10)

11,

P 31

As

r2

COUPLING WITH GRAVITATIONAL WAVES

101

is very close to 1, we can write

R-.

rlr2 - (1 - A )r,’ exp (4inAvL/c) 1 - rl r2 exp (4inAvUc)

(3.1 1)

Let us define L as the total losses of the cavity: 1- L

and, assuming that r~r2 is close to 1, so rIr2 R-.

(3.12)

= (1 -Al)r2, 2

1 - n/F - (1 - L ) exp (4inAvL/c)

1 - (1 - n / 3 ) exp (4inAvUc)

N

1 - n / 3 , the reflectivity becomes

.

(3.13)

Assuming finally that the deviation from resonance is weak, that corresponds to Av << AVFSR,so Av is typically less than a few kHz, and after neglecting second-order terms we find that RE-

1 -p+ipAf 1 - ipAf ’

(3.14)

where p = C F / n and Af = (43L/pc)Av is the (dimensionless) reduced frequency. Similarly, we can write the cavity power gain with the help of these reduced parameters. From eq. (3.6) we can derive G=

t: 1 (1 - rI rd2 1 + ( 2 3 / ~ sin2 ) ~ (2nLAv/c) ’

(3.15)

so it becomes G

P(2-P) Y P

L

1 1+(pAf)2’

(3.16)

From the expressions for G and R, the Fabry-Perot cavity can be considered as a low-pass filter with a cut-offfrequency defined as A& = l/p in reduced units, or, returning to normal units:

vc =

C

43L ~

(2vc is also the full width of the resonance curve at half maximum).

(3.17)

102

OPTICAL ASPECTS OF INTERFEROMETRIC GRAVITATIONAL-WAVE DETECTORS

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93

When the cavity is resonant (Af = 0), the reflectivity and the power gain of the cavity are simply related to p and L (their combination perfectly defines the properties of the cavity): (3.18) The parameter p is characteristic of the coupling of the cavity (and can also be interpreted as the defect of reflectivity at resonance). Since r1 < d m , we have rlr2 < d m r 2 , the second term being nothing but we then get 1 - x / 3 < 1 - L/2, and finally L F / n < 2. The variations of p are thus:

m;

o
(3.19)

If 0 < p < 1, the cavity is undercoupled, if p = 1, the cavity is optimally coupled and if p > 1, the cavity is overcoupled. In particular, in the case of optimal coupling, p = 1, we see that R = 0 and G is maximal with respect to p and is simply the inverse of the total losses: G = -1

(3.20)

L’

In this case, at resonance, all the light that enters the cavity is dissipated internally (the power gain compensates exactly for the losses) and nothing is reflected. This is a nice, although uncommon, example of impedance matching. This optimal coupling, which maximizes the power gain, will be used by the power recycling technique in gravitational-wave interferometric detectors (see 9 3.2). Now let us look more closely at the cavity reflectivity R. From eq. (3.14), we derive the power reflectivity as the squared modulus of R: (3.21) and the reflectivity phase, arg(R), depends on the coupling case: ifp < 1 :

arg(R)

= n+arctan

if p = 1 :

arg(R)

=

if p > 1 :

arg(R)

= arctan

(3.22)

+ arctan(pAf) x/2 + arctan(pAf)

-x/2

();!r

-

if Af > 0 if Af < 0 ’

+ arctan( pAf ).

(3.23) (3.24)

The squared modulus and the phase of the cavity reflectivity are plotted in fig. 5 and fig. 6 for different values of p. With a low-losses cavity, say C = 20 ppm,

11, P 31

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COUPLING WITH GRAVITATIONAL WAVES

0.6

0.4

0.2

l-l/(l

+X**2)

Reduced Frequency

J

Fig. 5. Power reflectivity of a Fabry-Perot cavity as a function of the reduced frequency Af (detuning of the cavity), for different values of p.

the values of p chosen for the illustration correspond to high-finesse cavities. With p = 0.25, the cavity has a finesse 3 M 40000, while for the optimal coupling p = 1, its finesse is then 3 M 160000. The main difference between undercoupled and overcoupled cavities appears through the phase behavior. In the case of undercoupled cavity, p < 1, the phase varies linearly over a large frequency range around the resonance point. The slope of the linear variation is derived from the first-order expansion of arg(R): wdR\

-

P(2 -hf - P )

(3.25)

TTTC

(3.26) . . . . c ,n, ., For the overcoupled cavity, there is also a linear benavior of arg(rC1 arouna me resonance, but over a much more limited range. -~

.

I

1.

1

104

OPTICAL ASPECTS OF INTERFEROMETRIC GRAVITATIONAL-WAVE DETECTORS

-1

1'''

" " ' 1 " " 1 " " 1 " " 1 " " 1 " '

111,

5

3

1 " " 1 " ' " " '

Reduced Frequency

Fig. 6 . Reflectivity phase of a Fabry-Perot cavity as a function of the reduced frequency Af (detuning of the cavity), for different values of p.

In the case of very small p (i.e., p << l), as it is the case for long arm cavities in gravitational-wave interferometric detectors, we have simply: (3.27) In terms of dephasing of the cavity, cp = 4 n L h / c , we have a very simple relation between the finesse of the cavity and the slope of arg(R) in the neighborhood of the resonance: (3.28) This means that a detuning of the cavity by an amount 6cp (e.g., induced by a gravitational wave), produces a change in the phase of the reflecting beam which is amplified by a factor about 2F/n, hence the interest in undercoupled FabryPerot cavities as gravitational-wave transducers. This also forbids the use of

11, § 31

COUPLING WITH GRAVITATIONAL WAVES

105

overcoupled cavities since in this case, the reflectivity phase varies only slightly around 0 with respect to 6cp. Finally, we present some data concerning the long cavities used as arms in interferometric detectors of gravitational waves. The VIRGO cavities are 3 km long and have a finesse 3 M 50; those of LIGO are 4km long and have a finesse .F M 200. The free spectral ranges are, respectively, 50kHz for VIRGO and 37.5 kHz for LIGO, and the cut-off frequencies are vc = 500 Hz for VIRGO and vC = 100Hz for LIGO. For reasonable total losses C M 2 x we then for LIGO. In both detectors the have p M 3 x for VIRGO and p M arm cavities are well undercoupled (the reason why neither LIGO nor VIRGO are planned to use more optimally coupled cavities will appear in the study of recycled interferometers). The properties of reflective Fabry-Perot cavities will be used in the following sections dealing with the different optical configurations of gravitational-wave interferometric detectors and their sensitivities. In particular, some notions are useful to keep in mind: optimal coupling in the study of the power recycling and undercoupling for long arm cavities (gravitational-wave transducers). 3.2. POWER RECYCLING IN INTERFEROMETRIC DETECTORS

3.2.1. Improvement of the basic Michelson interferometer We have seen in 0 2 that the Michelson interferometer shot-noise-limited sensitivity to small displacements can be improved by only two means: increasing the arm length or increasing the laser power. Firstly, the physical arm length can be increased up to a few kilometres for ground-based detectors (4 km for LIGO, 3 km for VIRGO); it is not easy to go further because, of course, of the cost and also because complications arise particularly due to the curvature of the Earth. Nevertheless, the optical path in the arms can be enhanced by the use of reflective cavities. As was noted in 9 3.1, the reflected beam from a cavity which suffers some detuning 6cp, gets a dephasing 2F/n x 6cp, so if this detuning is due to some displacement 6x (6cp = 4n6x/A), the relative length variation is 6x/L,,, instead of 6xlL for a simple Michelson interferometer, where the equivalent arm length is defined as L,,

=

.FL

-.

JG

(3.29)

The property of amplifying the dephasing can then be also interpreted as an increase in the arm length by a factor 3/n. For example, in the case of

106

OPTICAL ASPECTS OF NTERFEROMETRIC GRAVITATIONAL-WAVE DETECTORS

111,

5

3

n

6

Recycling mirror

Input mirror

mlrrOr

laser

y

Photodiode

Fig. 7 . Sketch of a long-baseline recycled interferometer.

VIRGO, this enhancement factor is about 16 and the equivalent arm length of the interferometer is then L,, M 50 km. This factor of 16 is also roughly the net gain in sensitivity resulting from the use of such Fabry-Perot cavities as arms. The other way to increase the sensitivity (if shot-noise limited) is to increase the input laser power. But this cannot be indefinitely improved because of technological limitations. As a continuous and very stable laser source is required, the current technology sets the laser power limit at 10-20 W. However, the circulating power in the interferometer, say the power at the beamsplitter, can be increased by the power recycling technique (Drever [1983]). Indeed, it has been shown in 0 3.1 that the maximum sensitivity of the Michelson interferometer is achieved when the interferometer is tuned at a dark fringe. All the light power is then reflected back towards the source, and the interferometer behaves like a highly reflective mirror. The addition of an extra mirror located between the laser source and the beamsplitter (see fig. 7) allows one to “recycle” the reflected light, by building up another Fabry-Perot cavity (recycling mirror + equivalent interferometer mirror). If this cavity is resonant for the input laser beam, there is a power enhancement according to the previous section, which is maximal in case of optimal coupling. From the point of view of the beamsplitter, there is now an apparent laser source much more powerful than the actual one. In the case of VIRGO, a recycling power gain of about 100 is planned. This leads to a circulating power of about 1 kW and a gain in the shot-noise limited sensitivity of one order of magnitude. The combination of multi-kilometer Fabry-Perot arms (with moderate finesse) with the recycling technique allows us to finally reach a shot-noise limited sensitivity of h M a few It has to be realized that the Fabry-

11,

5

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COUPLING WITH GRAVITATIONAL WAVES

107

Perot cavity arms are necessary undercoupled, so that they can have a high reflectivity (at resonance) and the recycling can be efficient; otherwise, as the coupling approaches the optimal case, their reflectivity goes lower and lower and consequently the recycling cavity exhibits lower and lower power gain, even if optimally coupled. 3.2.2. Sensitivity of the power-recycled Fabry-Perot arms interferometer Consider a monochromatic gravitational wave of frequency vgw and with amplitude h, incident on the detector with optimal polarization along the arms of the interferometer. Depending on the description of the interaction of the gravitational wave with the (free-falling) mirrors (Del Fabbro and Montelatici [1995], Tourrenc [1996]), the effect of the gravitational wave is either to change the vacuum index between test masses, without moving them (Einstein coordinates description) or to move the mirrors, without changing the vacuum properties (Fermi coordinates description). Regardless of the description, the net effect from the point of view of the light, is to (weakly) modulate the frequency of the circulating light inside the interferometer at the (low) gravitational-wave frequency. Everywhere in the interferometer, the light field can be resolved into three frequencies: the carrier at the laser frequency YO and 2 sidebands at frequencies YO i vgw(to first order). The complex optical amplitude at any point in the interferometer can thus be written generally as A(t) = A" exp(2invot)

+ i h A l exp[2in(vo + vgw)t]+ fhAz exp[2in(vo - vgw)t].

(3.30)

This amplitude can then be represented by a three-dimensional (3-D) complex vector:

The action of any optical element (including free propagation) can then be represented by a 3 x 3 matrix operator acting upon the amplitude vector A . This matrix has the general form (Vinet [1986])

M=

Moo 0 M," M,1 M20 0

(

:).

(3.32)

M22

The non-diagonal matrix elements MI" and M.0 represent the possible power transfer from the carrier to the sidebands, due to the action of the gravitational wave.

108

OPTICAL ASPECTS OF MTERFEROMETRIC GRAVITATIONAL-WAVE DETECTORS

[II, $ 3

Now consider a reflective Fabry-Perot cavity; the reflected amplitude vector A,f is deduced from the input amplitude vector A,,, by A,f = F ’ Ai,, where F is the Fabry-Perot matrix operator: (3.33) The diagonal elements Ro, R, and R- are the ordinary amplitude reflectivities, for the frequencies vo, YO+ vgwand YO- vgw,respectively, and can be computed for each frequency according to eq. (3.14). The sensitivity to the gravitational wave appears through the non-diagonal elements which have the form (Vinet, Meers, Man and Brillet [ 19881)

With the notations of

5 3.1, it is then straightforward to derive IF* I: (3.35)

where Q = 2nvoL/Cc is dimensionless (analogous to a quality factor), Af stands for the detuning of the carrier from resonance as in $3.1, and f = 4nLvg,/Cc is the reduced gravitational-wave frequency; in the case of VIRGO, YO = 2.8xl0l4Hz, L = 3km and for total losses C = 20ppm, we have numerically Q M 10”. If the cavity is tuned at resonance for the laser frequency (Af = 0), we get simply (3.36) which shows that a resonant cavity exhibits the same behavior for the two sidebands. Now let us consider a complete Michelson interferometer with FabryPerot arms. We can express the total transfer matrix from the input beam, 0, 0), to the transmitted beam to the output photodetector, A,,, = (&, A,,, = T .Ai,. In computing IAout(2,it is easy to realize that the gravitational wave gives a signal on the photodetector resulting from the beat between the carrier and the two sidebands and is proportional to IT+l + IT-1, so that the minimum gravitational-wave amplitude h detectable with the shot-noise

11, 0 31

COUPLING WITH GRAVITATIONAL WAVES

I09

limited interferometer (meaning SNR = 1) characterized by the matrix T is given by (assuming a perfect photodetector): (3.37) where hp is the Planck constant, Pi, = IA,,12 is the input laser power and where we define the gravitational-wave transfer function: (3.38) Assuming identical arms, the calculation of T5 from the analysis of the recycling cavity is straightforward: (3.39) where A , denotes the beamsplitter losses, t, and rr are the amplitude transmittivity and reflectivity of the recycling mirror, Ro is the Fabry-Perot reflectivity for the laser frequency and IF* I are the gravitational-wave transfer functions of a Fabry-Perot cavity. Assuming that the Fabry-Perot arms are resonant with the laser frequency, giving lRol = 1 - p and IF+I = IF-1, the interferometer gravitational-wave transfer function is finally: (3.40) The second fraction corresponds to the use of Fabry-Perot cavities as arms, while the first one corresponds to the light power enhancement due to the recycling cavity (the gain in power is actually the square of this term). This enhancement is, of course, maximized if the recycling cavity is coupled optimally to the input laser. Here, the recycling mirror reflectivity that maximizes the sensitivity is (maximizing the first fraction of G(vgVgw)): r:

= (1 -AN1

- A d 1 -PI,

(3.41)

where A , is the recycling mirror losses. In the case of optimal recycling, the gravitational-wave transfer function becomes: (3.42) If we set C,,f = A , + 2A, + 2p, i.e., the total losses of the interferometer: (3.43) We again find the features of the optimal coupling: the optimal power gain corresponds to the inverse of the total losses. As for an optimally coupled single

110

OPTICAL ASPECTS OF INTERFEROMETRIC GRAVITATIONAL-WAVE DETECTORS

[II,

83

Fig. 8. Sensitivity of a recycled interferometer in the optimal recycling configuration. The length of the arm cavities is 3 km,the input power is 20 W and the losses have been set to: A , = 10 ppm, A,y= 100ppm and cavity losses C , = 20 ppm.

cavity, the light reflected back by the interferometer with optimal recycling cancels out, and all the laser light is dissipated in the interferometer. Figure 8 shows the shot-noise sensitivity of an optimally recycled interferometer with Fabry-Perot arms of different finesse. For a real interferometer, with Fabry-Perot arms having gaussian eigenmodes (see for example Kogelnik and Li [1966] for the basics of gaussian optics), and matched to the fundamental one, any asymmetry (caused by some defect) does excite high-order modes, and since the dark-fringe condition is fulfilled by only one frequency, the fundamental mode, these high-order modes leave the interferometer through the output port. In the case of optimal recycling, the input TEMoo laser beam, assumed to be matched to the interferometer cannot reflect back and is then dissipated through the mirror losses and above all converted in high-order modes that are transmitted to the output photodetector. The final result is that an optimally recycled interferometer with low loss mirrors converts a nice TEMoo laser beam into a very “nasty” beam due to all the internal defects (mirror shapes, misalignments,

11, P 31

111

COUPLING WITH GRAVITATIONAL WAVES

-

5

1

L

08 0 7

06 05 0 4

0 3

02 01

00

Rrec

Fig. 9. Simulation of an imperfect recycled interferometer: behavior of the reflected and transmitted powers as functions of the recycling mirror power reflectivity The optimal coupling is reached for a reflectivity about 0.993 (see text).

etc.). Figure 9 illustrates this phenomenon; it is the result of a beam propagation code (described in $4) simulating a complete interferometer such as VIRGO (arm cavities+recycling). Here the end mirror of one of the 3km-long arm cavities (finesse = 50) is tilted by an angle of 0.5 prad, the mirror losses are set to Sppm (input mirrors) and lOppm (end mirrors), and the input laser, normalized to 1 W, is a TEMoo perfectly matched to the ideal 'interferometer. The figure shows the output power (the power at the output photodetector) and the reflected power as functions of the recycling mirror reflectivity. We see that the optimal recycling is obtained with R, M 0.993, and we check in fig. 10 that the power circulating in the recycling cavity is then maximal too. Note that in this example, the output beam is mainly a TEMlo mode. Here the perfect input TEMoo is dissipated through the mirror losses and converted mainly into a TEMlo mode. This shows the need for an output mode-cleaner matched to the input TEMoo beam, in order to improve the contrast (and so the sensitivity) and to dramatically lower the power reaching the photodetector (typically 100 mW instead of Watts in the case of VIRGO).

112

OPTICAL ASPECTS OF INTERFEROMETRIC GRAVITATIONAL-WAVE DETECTORS

0 95

0 96

0.97

0.98

0.99

[II, 9: 3

1

Rrec

Fig. 10. Simulation of an imperfect recycled interferometer: the power stored in the recycling cavity as a h c t i o n of the recycling mirror power reflectivity. The circulating power is maximal when the optimal recycling is achieved.

3.3. DUAL RECYCLING IN INTERFEROMETERS

The idea of dual recycling was first proposed by Meers [1988, 19891. The principle is to add to the power recycling technique (recycling of the carrier at laser frequency), the recycling of the sidebands induced by a gravitational wave. Since the Michelson interferometer is tuned at a dark fringe for the carrier, these sidebands are transmitted to the output photodetector. A partially reflecting mirror located between the beamsplitter and the photodetector (see fig. 11) can then reflect them back into the interferometer so that they again enter into long Fabry-Perot arms to experience coherently several times the phase shift due to the gravitational wave. This mirror builds up a new Fabry-Perot cavity with the rest of the interferometer, the so-called signal recycling cavity. This cavity can be tuned either at the carrier frequency (“broadband dual recycling”) or at some particular frequency in the detector frequency window (“tuned dual recycling”). Generally, signal-recycled interferometers are narrow-banded (despite the use of the term “broadband” for the first configuration mentioned above) in contrast

11,

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COUPLLVG WITH GRAVITATIONAL WAVES

n

Recycling mirror

Input mirror

n

JW

mirror

4splitter

y

Signal recycling mirror Photodiode

Fig. 1I. The dual recycling configuration.We note the extra mirror (signal recycling mirror), building up with the Michelson interferometer the signal recycling cavity where the sidebands induced by a gravitational wave can be resonant.

to power recycled interferometers. From the general properties of a FabryPerot cavity applied to the signal recycling cavity, we can easily compute the gravitational-wave transfer function of a dual recycling interferometer: Gd(vgw) =

where ID*

(3.44)

ID+/-t ID-\,

1 is, in case of optimal power recycling: IF+I

X

J1

+ (%)2sin2(+(vgw

vd)+ i(@(vgw)r

,

(3.45)

@(vd)))

where F* (defined by eq. 3.34) are the transfer functions of one Fabry-Perot arm for the sidebands, L d is the mean length of the signal recycling cavity, Yd is the resonant frequency of the signal recycling cavity, @( Y ) is the phase of the reflectivity of one Fabry-Perot arm (defined by eq. 3.22), Y: = (1 -As)rd is the amplitude reflectivity of the signal recycling mirror corrected for the beamsplitter losses, td is the amplitude transmittivity of this mirror and 3 d is the equivalent finesse of the signal recycling cavity: (3.46) In case of broadband dual recycling, the signal recycling cavity is tuned at the zero gravitational-wave frequency (i.e., at the laser frequency), so we get

114

OPTICAL ASPECTS OF INTERFEROMETRIC GRAVITATIONAL-WAVE DETECTORS

1

10

1 o2

[II,

3

I o3

Frequency (Hz)

Fig. 12. Sensitivity of a dual recycled interferometer, with optimal power recycling, for various reflectivities of the signal recycling mirror. The length of the arm cavities is 3km, their finesse is 50, the input power is 1OW and the losses have been set at A , = 100 ppm, A , = 10 ppm and L = 5 x w4;the signal recycling cavity is 6 m long and is tuned at vd = 100 Hz.

ID+I = 1D-l; the response is the same for the two induced sidebands and both can be enhanced, provided the two sidebands are within the bandwidth of the cavity (well inside an Airy peak). In case of tuned dual recycling, the signal recycling cavity is tuned at a particular sideband frequency, and the other (of opposite sign) is then anti-resonant: the response for one sideband is greatly enhanced whle the response for the other practically vanishes. Fortunately, the loss of the nonresonant sideband is more than compensated by the gain in the resonant one (see fig. 12). Not surprisingly, we can note that the new term in the transfer function due to the signal recycling mirror has the form of a Fabry-Perot transmission function. This shows that there is effectively a gain at the resonant frequency but to the detriment of the bandwidth - what is gained somewhere is lost elsewhere, in accordance with Mizuno’s sensitivity theorem (Mizuno [ 19951). Among all the projects of interferometric detectors of gravitational waves under construction, GE0600 is the only one planning to implement the signal recycling

11, § 31

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115

technique, with a moderate signal recycling mirror reflectivity in order to not limit the bandwidth too much. The tuning frequency could be around 346Hz, which is twice the rotating frequency of the nearest pulsar discovered (at about 100pc; Johnston, Lorimer, Harrison, Bailes, Lyne, Bell, Kaspi, Manchester, D'Amico, Nicastrol and Shengzhen [ 19931. The main practical problem with the signal recycling technique is the control of the signal recycling cavity, which is difficult, since in principle no light can exit the interferometer during the current operation, except high-order modes generated by the interferometer defects, or modulation sidebands in case of frontal modulation. However, Strain and Meers have demonstrated the feasibility of such a configuration in a table-top experiment by using a fraction of the laser source shifted by some frequency in order to lock the signal recycling cavity at the desired frequency (Strain and Meers [ 19911). Note that only the broadband dual recycling operation has been tested. 3.4. OTHER CONFIGURATIONS

Various other configurations have been proposed, mainly to increase the gravitational wave signal in some bandwidth. The ideas of synchronous recycling, detuned recycling and resonant sideband extraction, like the dual recycling concept, are based on the same idea of making (at least) one of the gravitational-wave induced sidebands resonant in the interferometer. 3.4.1. Synchronous recycling

Synchronous recycling (or resonant recycling) is an idea of Drever (Drever [ 19831).The optical configuration is basically different from the others, since the beamsplitter is rotated by 90" with respect to the usual Michelson configuration, making the optical system rather like a ring cavity (see fig. 13). This ring cavity is formed by three main mirrors - the recycling mirror and the two reflective Fabry-Perot arms seen as two highly reflecting mirrors. In this respect, this configuration is closer to a Fabry-Perot cavity than to a Michelson interferometer. If the storage time of the arm cavities corresponds to half the period of the incident gravitational wave, one of the induced sidebands can grow coherently after several round-trips in the arms, because it sees pass after pass from one arm to the other the same net dephasing due to the gravitational wave (because of its quadrupolar nature). The frequency of the peak sensitivity can be adjusted by tuning the central cavity. The in-depth study of synchronous recycling demonstrates that its sensitivity is very similar to the tuned dual recycling configuration (Vinet [1986], Vinet, Meers, Man and Brillet [1988], Meers [1988]).

116

OPTICAL ASPECTS OF INTERFEROMETRIC GRAVITATIONAL-WAVE DETECTORS

Input mirror

[It,

5

3

mirror

Photodiode

Fig. 13. The synchronous (or resonant) recycling configuration.

3.4.2. Detuned recycling The detuned recycling configuration (Vinet, Meers, Man and Brillet [ 19881) looks like that for standard power recycling, except that the arm cavities are no longer tuned at the laser frequency, but are slightly detuned to the frequency of the gravitational wave which we want to optimize. Consequently, when a gravitational wave of the correct frequency passes through the interferometer, one of the induced sidebands will be resonant in the arm cavities, but only one, as in the resonant recycling or the tuned dual recycling. Since the arm cavities are detuned from the laser frequency, their reflectivity at this frequency is larger than at resonance. Consequently, the power recycling will be more efficient than in the resonant case. We see from eq. (3.21) the reflectivity of the arm cavities out of resonance, (3.47) where Af stands for the detuning. Then from the Fabry-Perot transfer function, eq. (3.39, we can write the gravitational-wave transfer function:

This transfer function can also be optimized with respect to the interferometer losses (optimal coupling) by setting Li,f = A, + 2A, + [ p(2 - p)]/[ 1 + (pAf)*];

11, § 31

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COUPLING WITH GRAVITATIONAL WAVES

1

10

1 o2

1 o3

Frequency (Hz)

Fig. 14. Sensitivity of the detuned recycling configuration for different tuning frequencies. The following parameters have been used: arm cavity finesse = 50, laser power = 10 W, beamsplitter losses = 100ppm, recycling mirror losses = 10ppm, arm cavity losses = 500 ppm.

i.e., the total losses in case of detuned arm cavities, we have the maximal value for the transfer function:

We see that the total losses here are less than in the standard power recycling configuration, since [p(2 - p)]/[l + PA^)^] < 2p, and that the response is peaked at the frequency Af. An example of sensitivity is shown in fig. 14. There is an immediate drawback for this particular configuration. As the power recycling is to be more efficient in this case, there would be a high power hitting the splitter. The sensitivity would then be limited mainly by the beamsplitter losses, and especially by its thermal absorption.

118

[II,

OPTICAL ASPECTS OF INTERFEROMETRIC GRAVITATIONAL-WAVEDETECTORS

53

3.4.3. Resonant sideband extraction

The resonant sideband extraction configuration (Mizuno, Strain, Nelson, Chen, Schilling, Rudiger, Winkler and Danzmann [ 19931, Mizuno [ 19951) looks like the dual recycling one but is fundamentally different in principle. The idea is to use high finesse arm cavities together with the signal extraction cavity, a cavity formed by the interferometer and a mirror located between the beamsplitter and the output photodetector. This mirror must not be confused with the signal recycling mirror in the dual recycling configuration. Here, the role of this mirror is to build up a cavity able to extract the gravitational sidebands from the high-finesse arm cavities. Indeed, we note that in the power recycling configuration (see fig. 7), the arm cavity finesse determines the bandwidth of the detector for given losses, the lower the finesse the broader the response. In particular, if high finesse cavities are used, gravitational-wave frequencies above the knee frequency c/4FL (which is of the order of 25 Hz for a finesse 1000) are cut off since the signal is jammed by a number of incoherent round-trips in the arm cavities. This is no longer true, however, if the signal extraction cavity is implemented. This cavity is in fact equivalent to a compound mirror acting as an input mirror for the arm cavities at the frequency of the sidebands. As we have seen in 6 3.1, the global reflectivity of this mirror can be lower than that of the actual input mirror. The apparent finesse for the gravitational signal sidebands of the 3-mirror-coupled cavity is then lower than the arm cavity finesse. The sensitivity of this configuration is given by Gd(vgw)= IS+l + IS-[, where I& are given, in the case of optimal power recycling, by

-

(3.50) where to is the transmittivity of the signal extraction mirror, r; = (1 - A,)ro is its reflectivity corrected for the splitter losses, L, is the length of the signal extraction cavity, 6 = 2nL,,,vo/c corresponds to the detuning of the laser frequency from resonance in the signal extraction cavity, and IRI and 0 are the modulus and phase of the arm cavities reflectivity (see eqs. 3.21 and 3.22). The behavior of the resonant sideband extraction is fixed mainly by three parameters: (i) detuning of the signal extraction cavity from the laser frequency 6, (ii) length of the signal extraction cavity L,,,, and (iii) losses of the signal extraction cavity

COUPLNG WITH GRAVITATIONAL WAVES

1

10

1 O'

119

1 OJ Frequency ( H r )

Fig. 15. Sensitivity of the resonant signal extraction configuration in the case 6 = 0 (symmetric extraction of both sidebands) for different cavity lengths. The following parameters have been used: arm cavities finesse = 1000, t-0 = 0.95, laser power = 10W, beamsplitter losses = 100 ppm, recycling mirror losses = 10 ppm, arm cavity losses = 500 ppm.

(losses of the arm cavities +reflectivity defect of the signal extraction mirror) that change the range of variations of the signal extraction cavity reflectivity. The sensitivity is shown in figs. 15 and 16 for different configurations. We see clearly that the response can be rather broadband despite of the arm cavity finesse (set to 1000) which would give a cut-off frequency of about 25 Hz in the power recycling scheme. The different losses in these examples are the same as for the sensitivity calculations for the other configurations. Here, because of the large finesse, the total interferometer losses are in fact very high (about 0.32) and the sensitivity gain obtained because of the optimal power recycling (used here) is very poor; viz., about l / m M 1.7 (instead of some tens in the single power recycling configuration). This also means that the circulating power is not as high as in the standard power recycling, typically 20W in the examples shown, instead of 1kW. Nevertheless, the peak sensitivity as well as the bandwidth of the resonant sideband extraction can be compared with the standard power recycling scheme

120

OPTICAL ASPECTS OF INTERFEROMETRIC GRAVITATIONAL-WAVE DETECTORS

1

10

10'

[II,

63

o3

1 Frequency (Hz)

Fig. 16. Sensitivity of the resonant signal extraction configuration for various cavity detunings 6 (given in radan). The following parameters have been used: signal extraction cavity length= loom, arm cavity finesse= 1000, YO = 0.95, laser power= 10W, beamsplitter losses= 100ppm, recycling mirror losses = 10 ppm, arm cavity losses = 500 ppm.

(the case of low signal extraction cavity lengths). For example, see the curve labeled Lsec= lOOm in fig. 15, where the sensitivity is of the order of 4 x 1O-23/&ii within a bandwidth about 1 kHz. Finally, such a configuration can be helpful if, for example, the sensitivity is actually limited by heating effects in the substrates of the beamsplitter and of the input mirrors of arm cavities, since the circulating power in the central zone can be then reduced to a few tens of Watts without loss of bandwidth and without too much loss of peak sensitivity. 3.5. CONCLUSION

In this section we have analyzed, using the Fabry-Perot cavity properties, the different interferometer configurationsproposed for gravitational-wave detection. It has been recalled that each of these is able to achieve shot-noise-limited sensitivity, in some bandwidth high enough to be able to detect gravitational waves. The standard power recycling configuration can be seen as a general

11,

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OPTICAL COUPLINGS

121

purpose configuration, since it has basically a broadband sensitivity (from DC to a few kHz). All the other configurations are able to reach better sensitivities than the standard scheme, in some reduced bandwidth. These narrow-band configurations can be useful if optimization around a certain frequency is required, as in the search for known pulsars (known frequency, known position).

9 4.

Optical Couplings

The previous shot-noise-limited sensitivities can be achieved, of course, if the circulating power in the interferometer is effectively close to ideal, i.e., the power stored in the ideal interferometer. However, in the real situation there are many sources of losses, such as mirror defects, whch generate high-order modes that interfere poorly and are responsible for the imperfect contrast. This spurious light (internally generated) exits the interferometer via the output port and the corresponding power is then lost and can not be recycled. Another loss of power is related to the matching of the laser to the interferometer. The incident light that can not couple to the interferometer is reflected back, the recycled interferometer playing here the role of a mode-cleaner. In the following we will refer to these problems as the DC optical coupling problem. The AC optical coupling is actually a different problem - how to limit extranoises due to light source geometry fluctuations, which can simulate an event, through the interferometer asymmetries. 4.1. MATCHING THE LASER INPUT BEAM TO A REAL INTERFEROMETER

In this section, we consider the coupling between a perfect input laser beam to a real interferometer, with aberrations. In particular, as the arm cavity mirrors are not perfect, the input laser beam (which is here supposed to be perfect) can not couple perfectly to each of these. As a result, the interference between the two reflected beams is not perfect and some light leaves the recycling cavity through the output port, and the contrast is poorer. As a consequence of this loss, the recycling power decreases and the shot-noise level increases. Therefore, since the input laser power cannot exceed say 10 or 20 W with current technology and since a shot-noise level of about 3 x 1 0 - 2 3 / (corresponding ~ to 1 kW of recycled power) is demanded in the case of VIRGO, some severe constraints must be put on the mirrors.

122

OPTICAL ASPECTS OF INTERFEROMETRIC GRAVITATIONAL-WAVE DETECTORS

[II, $ 4

4.1.1. Optical speciJicationsfor the VIRGO interferometer

In this section we describe the numerical code we have constructed and used to set the mirror specifications for VIRGO. Such a code must be as general as possible in order to be able to treat all defects that can occur; e.g., surface defects, substrate index inhomogeneities, curvature radius mismatch, misalignments, etc. It is then convenient to develop an operator formalism. In our model, all the distortions undergone by the light beam when propagating, reflecting on a mirror or crossing through a mirror, are represented by a linear operator acting on the complex amplitude of the beam V(x,y). Resolution of the Helmholtz equation, within the paraxial approximation,

leads to the following free propagation operator over a distance L (Sziklas and Siegman [ 19751):

P = F'exp(ikl) exp

(4.2)

where F represents the two-dimensional Fourier transform operator defined by

while the inverse two-dimensional Fourier transform operator is V(x7.Y) = [F' $l(X,Y)

=

1

4JG2

//

exp(ipx) exp(iqy) $(I& 4 ) dP d4.

(4.4)

Implementation of this method for propagating Gaussian beams (or any beam having a practically finite transversal size) is numerically very efficient, since it allows for the use of Fast Fourier Transforms, available in most computer libraries. Now let us consider the transmissiodreflection operators of the cavity mirrors. First, a reflection operator is associated with the cavity spherical end mirror, within the parabolic approximation (which is valid here, because of the large curvature radii): Rend(x,y)= ire e x p ( - i k x y )

,

(4.5)

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OPTICAL COUPLINGS

123

Fig. 17. Simulation of a cavity: notations.

where R is the curvature radius of the mirror and re is its amplitude reflectivity. For an aberrated mirror, iff(x, y ) represents the deviation from the ideal shape, the reflection operator becomes:

In the case of flat cavity input mirrors with aberrated reflective surfaces, we have the transmission and internal and external reflection operators (Vinet, Hello, Man and Brillet [ 19921):

where f(x,y) is the deviation from the ideal (plane) reflective face, n is the optical index of the substrate materials, and ri (6) is the amplitude reflectivity (transmittivity) of the input mirror. Note that the “aberration” function may include surface defects or mirror tilts as well. Now, let us turn our attention to the simulation of a cavity, such as that shown in fig. 17. From the recombination relations between the beam amplitudes in the cavity, (4.10) (4.11) (4.12) (4.13) (4.14)

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OPTICAL ASPECTS OF INTERFEROMETRIC GRAVITATIONAL-WAVE DETECTORS

we can derive the implicit equation for the intra-cavity field, say

[II,

54

v1 (4.15)

To find the stationary intra-cavity field I/& numerically, we solve eq. 4.15 by iteration. We make as a first guess, for example, the ideal TEMoo beam perfectly matched to the ideal cavity and then iterate eq. (4.15) until the stationary state is reached (meaning that the standard deviation between the two last iterations is less than some threshold). The key point of the numerical method is the h e tuning of the resonance. Since small deviations from the ideal cavity are assumed (e.g., small mirror aberrations, small misalignments), a perturbation calculus is justified; the fundamental eigenmode of the ideal cavity is the TEMoo beam of amplitude Yoo(x,y), and, similarly to first-order perturbation theory in Quantum Mechanics, we find that the round-trip phase that tunes the cavity at resonance is, to first order (Vinet, Hello, Man and Brillet [1992]), (4.16) where C = Ry,$,P(L)Rend?(L) is the round-trip operator and (AIB) denotes the scalar product between the wave functions A ( x , y ) and B ( x , y ) . The phase 6 q can then be added to the round-trip operator phase, or equivalently, half of 6cp can be added to the propagator phase. This method requires only one round-trip simulation (one loop) and is much faster than maximizing the power stored in the cavity by successive iterations (dichotomy) like in similar simulations (Tridgell, McClelland and Savage [1991]). Note, however, that this fast fine-tuning method fails in the case of tilted mirrors, because the change in length of the cavity 6L due to tilts of the mirrors is (Kawamura and Zucker [ 19941) (4.17) which is thus a second-order expression with respect to the tilt angles of the input and end mirrors 19iand 4, respectively. The phase of the propagator must then be corrected for the corresponding one-way dephasing 2n6L/A. The optical specifications in the case of the VIRGO interferometer have been established by the simulation of a Michelson interferometer having Fabry-Perot cavities as arms, but power recycling was actually not considered. Indeed, the specification criterion was to get a minimum recycling factor (power gain in the recycling cavity) of about g N 50 in order to reach the shot-noise-limited sensitivity of h = 3 x 1 0 - 2 3 / a at 100 Hz. This corresponds, for an optimally The optical coupled recycling interferometer,to total losses Citf = l/g M 2 x

11, P 41

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OPTICAL COUPLINGS

Table 1 Optical specifications Input mirrors

End mirrors

Curvature radius (m)

00

3450f 100

Surface deformation (nm)

10 on @ = 4 0 m m 35 on @ =lOOmm

10 on $= 120mm 35 on @=300mm

Substrate quality (nm)

40 on @=40mm 120 on @ = l o o m

Transversal misalignment (mm)

10

1

Tilt angle ( p a d )

1

0.2

specifications have then been set by limiting the total Michelson losses to be less than 2%. The losses have been computed simply by calculating the maximal coupling of the reflected beam by the interferometer (tuned at a dark fringe), Yref(x,y),to the ideal input TEMoo beam Y o o ( x , ~ ) : (4.18)

d m

is the norm of the wave function Y ( x , y )(the squared of where 11 Y 11 = the norm is simply the power in the beam). Of course, in the case of a lossless interferometer, the reflected beam is found to couple perfectly with the input TEMoo beam and Cltf = 0. The main optical specifications for the VIRGO interferometer are reported in table 1 (Hello [1990], Virgo Collaboration [1989, 1992, 19951). Only the specifications for the arm-cavities mirrors are given since these are the most critical. In table 1, the specifications for the surface deformation (peak-to-valley) are split into two different categories, one for the beam zone (r < w , w being the beam size), the other for the whole coated surface. This distinction is in fact related to the spatial scale of the defects, as the corresponding cavity losses (i.e., the generation of high-order modes) depend on the ratio between the beam size on the mirror and the typical scale of the defects (Winkler, Schilling, Danzmann, Mizuno, Riidiger and Strain [ 19941). The substrate quality specification refers to the maximal deformation of the wave-surface tolerable when crossing the coated mirror (it includes the substrate inhomogeneities and the mirror surface defects). Once again, this specification is split into the two categories above. Every surface deformation is given in terms of peak-to-valley values. We note

126

OPTICAL ASPECTS OF INTERFEROMETRIC GRAVITATIONAL-WAVE DETECTORS

[II,

54

that the specifications are rather severe, especially for the accuracy of the mirror shape (e.g., 3% of accuracy for the curvature radii, or lOnm for the peakto-valley maximal deformation inside the beam zone), but they seem to be achievable within current technology. Finally, the main reason why a power recycled interferometer is so sensitive to arm mirror defects is that the hgh-order modes, generated by these defects, are transmitted immediately to the output photodiode, since the dark ftinge condition holds only for the fundamental mode, and the corresponding light power is lost irretrievably. The situation is different when signal recycling is implemented, since the extra signal recycling mirror can partially reflect the “bad” light and gives it a chance to re-couple to the interferometer. 4.1.2. Optical specijkations for dual recycling interferometers

As first pointed out by Meers and Strain [1991], the dual recycling configuration is much less sensitive to mirror defects than that for single power recycling. The reason, as mentioned above, is that the high-order modes generated by imperfections see a highly transmittive interferometer and are directly lost in the case of power recycling but see a much more reflective one in case of dual recycling. This is especially true if these high-order modes are not resonant in the signal recycling cavity, since they only see a simple mirror of low transmission, when they are ejected from the power recycling cavity. The light is therefore reflected back and re-enters the interferometer. The spurious light can then be recycled with the help of the signal recycling mirror. If this recycled light can still couple into the power recycling cavity, the result is that the power circulating in the interferometer is maintained with respect to the ideal case, and finally that the sensitivity to gravitational waves is not degraded. In order to check this rather intuitive demonstration, we need to numerically investigate the behavior of a dual recycled interferometer. Simple configurations have been studed (power + signal recycling, no cavities in the arms; Tridgell, McClelland and Savage [ 19911, McClelland, Savage, Tridgell and Mavaddat [ 19931, Mavaddat, McClelland, Hello and Vinet [ 1995]), and show a circulating power recovery in case of mirror tilts or mirror curvature mismatches, resulting in an improvement in the shot-noise level and thus in the sensitivity. Finally, this proficiency in repairing imperfect interferometers can relax the optical specifications of dual recycling configurations with respect to the single power recycling configurations. A similar power recovery is expected to occur in the resonant sideband extraction configuration, but is of less importance since the key point of the

11,

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127

OPTICAL COUPLINGS

sensitivity here is not the power stored in the power recycling cavity but rather the power stored in the arm cavities. Note however that this latter configuration is less sensitive to beamsplitter imperfections (especially thermal effects; Mizuno [ 1995]), and should be a good configuration for future detectors, if the specifications of the beamsplitter become crucial, since it is likely to be the main source of asymmetry in the interferometer. 4.2. COUPLING THE REAL LASER INPUT BEAM TO A PERFECT INTERFEROMETER

4.2.1. Power losses due to the laser beam distortions

Let us now consider the orthogonal problem of matching a real laser beam to an ideal interferometer. In general, if the geometry of the input beam is distorted, power losses occur in the cavities due to couplings between the fundamental mode and high-order modes (Anderson [19841, Fattacioli, Boulharts, Brillet and Man [1986]). In the following, “ideal interferometer” will mean a symmetric interferometer with identical perfect arm cavities, thus having the same HermiteGaussian TEM,, eigenmodes, of amplitudes at the waist position (located at the input cavity mirrors) Ym,(x,y). The cavities are tuned at resonance for their common TEMoo mode, the interferometer is tuned perfectly at a dark fringe (no light goes out from it to the output photodiode) and the power recycling cavity is also tuned at resonance for the TEMoo mode. The recycling mirror is assumed to be flat so that the recycling cavity is nondegenerate only for the TEMoo mode and infinitely degenerate for all the other modes. This comes from the fact that the reflectivity of the arm cavities suffers a phase shft of x at anti-resonance relatively to the resonance case. In the case of LIGONIRGO, the Rayleigh range of the resonant beam is typically of the order of 1 km,and is therefore much larger than the recycling cavity length (12m in VIRGO). Consequently, we assume the TEMoo mode of the arm cavities to also be the fundamental mode of the recycling cavity (this amounts to neglecting diffraction inside the recycling cavity). Let us consider an imperfect input laser beam, Yin(x,y),which can be written as a combination of the Y,,(x,y), as they build up a complete orthonormal basis for the space of light wave amplitudes (Siegman [ 19861): (4.19) The ideal interferometer tuned at resonance for the TEMoo mode can be seen as a single (compound) cavity, where the first mirror is the recycling mirror,

128

OPTICAL ASPECTS OF INTERFEROMETRIC GRAVITATIONAL-WAVE DETECTORS

[II, $ 4

with reflectivity r, and transmission t,, and the second one is the Michelson interferometer (equivalent to a single arm cavity, since it is assumed to be perfectly symmetrical) of reflectivity Roo = -( 1 - p ) for the TEMoo mode and R,, E 1 for the other modes, since the TEMoo mode is resonant in the arm cavities while all the high-order modes are not. As in Q 3.1, p denotes the reduced losses of the arm cavities. The stationary beam Y ( x , y )in the recycling cavity can then be written as

The higher-order modes not resonant in the interferometer are practically completely reflected back by the recycling cavity. Clearly, the circulating power in the interferometer [norm of Y ( x , y ) ] is lowered due to the input laser distortions, and is roughly (4.21) The distortions of the input laser beam thus induce power losses in the interferometer. However, the main specifications are not very stringent (Hello [1993]) - about 0.5mm for transversal misalignments of the laser beam axis, 0.5 p a d for tilts of the laser beam axis, a waist mismatch SWO/WON 5% and a waist location error J6zJE 180m (the latter is not surprising because of the very large Rayleigh range of the input beam). In any case, it is always possible to increase a little the power delivered by the laser source, in order to improve the coupling factor 1 ~ 0 0 1and ~ so to compensate for this kind of power losses, but the feedback problems due to the reflected high-order modes will become accordingly worse. 4.2.2. Low-loss beam expander design The laser beam must be matched to the ideal recycled interferometer discussed above, meaning that the waist must be located, say at the input mirrors of arm cavities (or somewhere else in the recycling cavity. It does not matter because of the very large Rayleigh range as discussed above), with the matched waist size; in the case of VIRGO, with arm cavity lengths of 3km and endmirror curvature radius of 3.45km, the theoretical waist size is wo = 1.98cm and the corresponding Rayleigh range is n w i / h N 1162m. The matching typically requires two steps: first matching the laser to the input mode-cleaner

11,

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OPTICAL COUPLINGS

I29

(see 5 l), and second matching the output beam from the mode-cleaner to the interferometer. For this purpose low-loss beam expanders must be designed in order to minimize the midway power losses between the laser itself and the interferometer due to absorption as well as to imperfect coupling of the distorted laser beam to the interferometer. First of all, reflective optics must be preferred to refractive optics (lenses) in order to avoid light feedback into the laser itself, due to imperfect anti-reflection coatings borne by the lenses, and avoid distortions due to substrate imperfections. Let us then consider a 2-mirror, off-axis telescope such as that shown in fig. 18 and a laser beam, supposed to be a perfect Gaussian

Fig. 18. Two-mirrors off-axis beam expander.

beam, incident on it. This beam is characterized completely by its so-called qparameter (Kogelnik and Li [ 19661): q1

= Z I +ibl,

(4.22)

where z1 is the beam waist position and bl = xwi/d is the Rayleigh range (wg is the beam waist size). We first consider perfect mirrors, derive the ABCD matrix of the telescope and show that the resulting astigmatism (aberration of lowest order) of the output beam can be removed by a proper choice of the incident angles on the mirrors. We will then extend the discussion to the case of real (spherical) mirrors. The off-axis telescope can then be described by a pair of ABCD matrices: one for the tangential plane, say Oxz, the other for the sagittal

130

OPTICAL ASPECTS OF INTERFEROMETRIC GRAVITATIONAL-WAVE DETECTORS

[II,

54

plane, say Oyz. The telescope ABCD matrices in both planes can be computed easily (Kogelnik and Li [ 19661):

and using a similar expression for My.Herefi, andfi” denote the focal lengths of the first mirror in the tangential and sagittal planes, respectively, and the same notation applies for the second mirror. These focal lengths are related to the angle of incidence by (Jenkins and White [1957]): (4.24) where fi = + R I ,and similarly for the second mirror. The q-parameters, in tangential and sagittal planes, of the output beam are given by (4.25) and a similar expression for 42,“. The condition for no astigmatism for the optical system is simply that z k = zzV, and we obtain, with eq. (4.25) and the analogous expression for 4zy: (A,zI + B.T)(Gzi + 0,) + A,C,b: (Gzi + +(C,~I)~

=

(A,zI + BY)(C+i + D.y) + A.”C& .
(4.26)

Solving eq. (4.26) in the general case is quite difficult, but it can be discussed more easily in the two cases of interest: bl << z1 (small Rayleigh range of the input beam) and bl >> zl (large Rayleigh range or waist located close to the input mirror of the telescope). In the first case (bl << z l ) , eq. (4.26) becomes, after replacing A, . . . by their expressions, and within the small angle approximation (Hello and Man [ 19961): (4.27) while in the second case (bl >> z l ) , eq. (4.26) yields: (4.28) Hence it is possible in both cases to compensate for the astigmatism induced by an off-axis telescope, provided a proper set of incident angles on both mirrors

11, P 41

OPTICAL COUPLINGS

131

is chosen according to eq. (4.27) or eq. (4.28). Note in this case that the output beam is not parallel with the input beam, in contrast to the usual implementation. But the astigmatism cancellation is only possible with mirrors with two focal lengths of opposite signs, as indicated by the square root in the two expressions; i.e., the telescope must be implemented with a concave and a convex mirror. Also note that both eq. (4.27) and eq. (4.28) can be simplified if we can also neglect the length of the telescope, i.e., d << z1 and d <
(4.29) In order to get a low-loss telescope, the second step is to choose mirrors with curvature radii high enough to avoid distortions caused by spherical aberration. Indeed, when reflecting on a spherical mirror of curvature R, the beam phase changes by an amount of 2kz(x,y), where k = 2 n / A and z(x,y) = R - JR2 - (x2+ y 2 )is the mirror surface equation. After second-order expansion, the latter becomes (4.30) The first fraction represents the usual paraboloidal approximation, the one used in eq. (4.5) for the reflection operator of a curved mirror, whereas the second (in parentheses) represents the first correction which is due to the actual spherical shape of the mirror. It is clear that this correction will be negligible if w2/R2 << 1, where w is the typical beam size on the mirror. In the low-loss telescope design, we can then proceed as follows. First we chose a first mirror with a curvature radius R I such as w:/R: << 1. Then we chose the curvature radius of the second mirror and the beam-expander length in order to fulfil the three conditions: (i) get the right magnification, (ii) get low spherical aberration for the second mirror, and (iii) avoid congestion problems on the optical bench. The two angles of incidence on the two mirrors are finally adjusted in order to cancel the astigmatism, without forgetting congestion problems. For example, let us consider the matching telescope of the mode-cleaner cavity in VIRGO. This later has to magnify a beam of waist size w1 N 0.96mm to the matched waist of the mode-cleaner cavity which is w2 N 4.9mm. Based on the above considerations about the spherical aberration, and giving the right magnification, a good choice for the length of the telescope is: d = 402 mm, and for the curvature radii: R1 5 -200 mm and R2 N 1000mm. The incidence angle

132

OPTICAL ASPECTS OF INTERFEROMETRIC GRAVITATIONAL-WAVE DETECTORS

[II, 8 4

on the second mirror, which is tolerable with the allowed space, is 8 2 N 2.5", and the condition for astigmatism cancellation gives the first angle: 81 N 5.6". The configuration has been further checked with an off-axis beam propagation code (Hello and Vinet [1996]), which is able to propagate a TEMoo beam through the system and to compute the losses at the telescope output. The losses C are computed as the maximal coupling of the output beam Y,,,(x,y)with the nearest perfect TEMoo mode Yoo(x,y) (see eq. 4.18); the nearest TEMoo beam is simply given by the TEMoo mode having a beam size w and a curvature radius R computed as the harmonic mean of the sizes and curvatures, respectively, of the output beam in the tangential (w,,R,) and sagittal planes (w,,, . R,,): . (4.3 1) The telescope losses present a minimum as a function of the tilt angle 81 of the first mirror. As predicted above, the angle that minimizes the losses is 81 N 5.6"; the corresponding minimum losses are C N 7 . 4 ~ 1 0 - which ~, is acceptable in the VIRGO context. The remaining losses are due to the spherical aberration (rather low here because of the choice of the curvature radii) and higher-order aberrations. It is thus possible to design low-loss telescopes by simply removing the lowestorder aberration (astigmatism) by a proper choice of incident angles on the mirrors, and by taking care of the spherical aberration which can be lowered by a proper choice of the mirror curvature radii. 4.3. EFFECT OF LASER GEOMETRY FLUCTUATIONS

As first pointed out by Rudiger, Schilling, Schnupp, Winkler, Billing and Maischberger [ 19811, the geometry fluctuations of the input laser can induce an extra-phase noise in the interferometric detection, through couplings to the geometrical asymmetries of the interferometer. This effect is similar to the noise generated by the frequency fluctuations of the laser that can couple to the arm length asymmetry of the interferometer I . Particular care has been devoted to this in the VIRGO experiment (Bondu, Fritschel, Man and Brillet [ 1996]), since, starting from the natural laser frequency fluctuations, a gain of many orders of magnitude was needed to decrease this noise under the shot-noise level,

'

Here the arm length means the equivalent arm length F L / n and the asymmetry is in fact mainly due to the difference of finesse between the two arm cavities.

11, Q 41

OPTICAL COUPLINGS

I33

assuming a typical interferometer arm length asymmetry of a few percent. We will show below that laser geometry fluctuations may impose severe constraints, in particular for the alignment of the interferometer (Barone, Calloni, DiFiore, Grado, Hello, Milano and Russo [ 19961). 4.3. I . Modelling of laser jitters and cavities misalignments

As in $4.2, we will note Ymn the amplitudes of the Hermite-Gaussian modes of the perfect interferometer, and aPplthe amplitudes of the Laguerre-Gaussian modes. The first will be useful to describe misalignments of the cavities, the second will be useful to describe waist size and location mismatches. Let us note that the fundamental mode of the perfect cavities is the same in the LaguerreGauss description as in the Hermite-Gauss description: Yo0 = @opoo. When no jitter is present, the input laser beam of amplitude Y,,(x,y) is assumed to be perfectly matched to the perfectly aligned interferometer, so W n ( x , y ) = Ai,Yoo, where IA,,1* is the total input power. In case of angular jitter O(t) (say in the direction Ox),the input beam couples to first order to the first-order Hermite-Gauss mode (Anderson [ 19841):

(4.32) where 0, = A/?CWO is the beam divergence (Od N 17 prad in the case of VIRGO). Likewise, in case of lateral jitter a(t), the input beam becomes (4.33) Similarly, in case of waist size or location fluctuations, the laser beam couples to the first-order (mi-symmetrical) Laguerre-Gauss mode @lo (Anderson [ 19841). For waist size fluctuations w(t), we then have: (4.34) and for waist location fluctuations z(t): (4.35)

134

c,

OPTlCAL ASPECTS OF INTERFEROMETRIC GRAVITATIONAL-WAVE DETECTORS

\

[II,

44

,

Fig. 19. A slightly misaligned interferometer. The dashed lines refer to the axis of the perfectly aligned interferometer. Here one of the arm cavities is tilted by an angle 6,the other is translated by 6 x .

In exactly the same way, it is straightforward to express the new fundamental mode and the new first-order mode of a slightly misaligned or mismatched cavity (see fig. 19). In the case of misalignment we get (4.36) (4.37) where E = Gx/wo in the case of lateral displacement of the cavity axis (meaning for a flat-curve cavity like these of LIGONIRGO a displacement of the end curved mirror Gx), or E = iWOd in the case of tilt of the cavity axis by an angle 0. In the case of mismatch, we get: (4.38) where E = GW/WO in case of waist size mismatch 6w, and 6 = iOdSd2wo in case of waist location mismatch 62. 4.3.2. Coupling of laser angular jitter and misaligned cavities As an example, and without loss of generality, let us first consider the case of an input laser beam having an angular jitter at some low frequency v and in some

11,

0 41

OPTICAL COUPLINGS

135

direction, say Ox. The input beam amplitude is then given by eq. (4.32) with, for example,

The input beam amplitude is therefore: (4.40) The input beam is then the superposition of a carrier at laser frequency Va, which has components only on the TEMoo mode, and two sidebands at the frequencies YO f Y, which have components only on the TEMlo mode. The misaligned interferometer is going to mix the different spatial modes so that at each frequency, the amplitude Y of the beam anywhere in the interferometer is going to be the superposition of two components (to first order): one on the TEMoo mode, the other on the TEMlo mode. The amplitude can then be conveniently represented by a 2-D vector, denoted Y = [A;B], whose components are the spatial components on each of the TEMooand TEMlo modes: [A;B ] = A Y o 0 + B Yio. For instance, the input beam can also be represented + Yin(-v>, where the different vectors, in this way: Yi, = Yin(0)+ Y~,(Y) corresponding to the different frequencies, are defined as

(4.42) The light amplitude at the location of the recycling mirror, Yre, = Yrec(0)+ Yrec(v)+ Yry(-v) can then be deduced from the input beam amplitude, using the transfer matrices, one for each frequency: (4.43) (4.44) Assuming that the cavities are resonant for the TEMoo modes and that they are translated by an amount 6x1 for the first and ax2 for the second, and in

136

OPTICAL ASPECTS OF INTERFEROMETRIC GRAVITATIONAL-WAVE DETECTORS

[II,

84

the quasi-static approximation (the computed amplitudes are stationary), it is straightforward to compute the three transfer matrices. For the carrier frequency, we find:

&(&-A))

1 -trr,ro

M(o)=

(2(~ t, 2wo

_

_

1 +trY,YI

1 -?-,Yo

)

_

-tr

+

1 rrq

(4.45)

where 6x = 6x1 + 6x2, t, and r, are the amplitude transmission and reflectivity of the recycling mirror, and YO and rl are the moduli of the reflectivity of the arm cavities for the TEMoo and TEMlo modes, respectively. With the notation of Q 3.1, we have YO = 1 - p and 1-1 = 1, assuming low-loss arm cavities. The fact that the arm cavities are resonant for the TEMoo and not for the TEMlo mode appears simply through the signs in the denominators of the matrix elements. For the sideband frequencies, we find analogously: tr

M(fv) =

1 - rrrh -tr

1 + rrrl

(4.46) where Y+ are the arm cavities reflectivity for the sideband (low) frequencies. With the notation of Q 3.1, we have Y+ = ~ ( v of v)exp[icp(vo f v)], with Y the reflectivity modulus (see eq. 3.21), and n + cp the reflectivity phase (see eq. 3.22); at resonance cp = 0. Because the two sidebands are symmetrical on either side of a resonant frequency, we have hrthermore I = Ir- I and the phases are opposite, q(vo - v) = -cp(vg + v). The two spatial components of the light amplitude in the recycling cavity are then given by

IY+

y r e c = M(O)[Ain; 01

+ M ( v)[O; i Ain 1- 60 od

+ M(-v)[O; i A-1 in 60 od

exp(2invt)

(4.47)

exp(-2invt).

It is easy to check that if there is no jitter (00 = 0 and the input laser beam is Yin = A,,lyoo and is represented by Yin = [Ain;0]), and if the cavities are perfectly aligned (6x = O), then the recycling beam is perfect: Yrec= YOO.

&

11, § 41

137

OPTICAL COUPLINGS

In the general case, the beam amplitude in the recycling cavity is computed using eq. (4.47):

+(&[--tr

(4.48)

We can already see the coupling between the laser jitter and the cavity misalignment that appears in the TEMoo components of Yrecat the sideband frequencies. 4.3.3. Phase noise induced by the laser geometrical ftuctuations

The phase noise at frequency Y due to the coupling can then be computed as the phase difference between the two interfering beams at the beamsplitter. Assuming a perfect beamsplitter @erfectlysymmetric with identical transmission and reflection coefficients) and forgetting the phase terms due to the propagation in the recycling cavity), the two interfering beams, coming from the two arm cavities, are simply: (4.49) (4.50)

where t, and r, are the amplitude transmission and reflectivity of the splitter (ts = r, = l / d ) , and R I and R2 are the reflectivities of the arm cavities (different for each of the spatial modes and for each of the frequencies). The phase difference between these two beams is, in fact, amply dominated by the relative phase of the TEMoo components. The TEMoo components of the interfering beams are given by:

where cp = YO + Y), and the analogous expression for only 6x1 by 6x2).

(Y2IYoo) (replacing

138

OPTICAL ASPECTS OF INTERFEROMETRIC GRAVITATIONAL-WAVE DETECTORS

[II,

54

We derive then the phase noise, as the phase difference at frequency Y: (4.52) The phase noise is then proportional to the amplitude of the angular jitter of the laser I90 and to the asymmetry of the interferometer cavity arms 6x1 - 6x2. In particular, if both cavities are translated, but by the same amount (6x1 = 6x2), there is no phase noise, only a power loss due to imperfect matching. We note also that the noise is in fact due to the coupling of the TEMlo component produced by the interferometer asymmetry to the quadraturecomponent of the input laser beam on the same TEMlo mode. Accordingly, we find there is no noise (only a power loss) when coupling the angular jitter to the cavity axis tilts or coupling the lateral jitter to the cavities axis lateral translations, since the induced TEMlo components for the input beam and the cavity modes are in-phase. To the contrary, considering the coupling of the lateral jitter a(t) = a ~ c o s ( 2 m t )to the cavity axis tilts 81 and 192, we find the phase noise (same derivation as above, but with &/@d being replaced by ao/wo and 6xk ( k = 1,2) being replaced by iOk/Sd): (4.53) Considering now the coupling of the laser waist size and location fluctuations to the interferometer asymmetry in waist location and size, a similar derivation can be done, the Laguerre-Gauss modes 0 0 0 and 0 1 0 replacing the HermiteGauss modes 'yo0 and Y,o. In the case of laser waist size fluctuations w(t) = wo+ 6wo cos(2nvt), we find that the phase noise (4.54)

appears through the asymmetry in waist location of the arm cavities. Here r1 denotes the cavity reflectivity for the Ol0 Laguerre-Gauss mode (rI N 1 in our case). Likewise, the laser waist location fluctuation z(t) = z0 cos(2nvt) produces the phase noise (4.55) through the asymmetry in waist size of the arm cavities.

11, § 41

OPTICAL COUPLINGS

139

The common features of the different phase noises above are first the filtering effect of the recycling cavity, that appears through the fraction (1 - rrrO)/( 1 + r,r1). Not surprisingly, the recycling cavity acts like a modecleaner for the low-frequency fluctuations considered here. Secondly, the dependence on frequency appears through the term YI + IT+/ cos(q). This term is at a maximum for the resonant frequencies (q = 0), for example in the DC limit, and it shows a lowering of the noise for antiresonant frequencies (q = n). In order to give orders of magnitude, the last fraction of the noise expressions can be over-estimated to be the DC value which is, for low-loss cavities, (4.56) In the first case, laser angular jitter and translations of the cavity axis, replacing e d by its expression, we then find (4.57) which gives the same result as Rudiger, Schilling, Schnupp, Winkler, Billing and Maischberger [1981] obtained in the case of no recycling (r, = 0). The typical jitter of the VIRGO laser will be 00 M 10-9rad/&, and after mode-cleaning, 60 FZ lo-'' rad/& (Man [1995/96]). In the case of VIRGO, the phase noise due to the coupling of laser angular jitter and translations of cavity is then

64 N_ 5 . 9 ~

(6x1 - 6 ~ 2 ) r a d / J H z .

(4.58)

This has to be compared to the shot-noise-limited phase sensitivity, which is (4.59) where P is the power at the beamsplitter, of the order of 1kW for VIRGO. The shot-noise-limited phase sensitivity is then N 1 . 4 lo-" ~ rad/&. The phase noise due to the laser angular jitter will then be dominated by the shotnoise if the position of the axis of the two cavities can be controlled in such a way that (6x1 - 6 ~ 2 ) < 2.4xlO-'m.

(4.60)

This imposes a very stringent limit for the alignment of the mirrors. Indeed, considering that the position of the cavity axis is controlled by orienting the

140

OPTICAL ASPECTS OF INTERFEROMETRIC GRAVITATIONAL-WAVE DETECTORS

[II, $ 4

curved end mirror, which has a curvature radius R % 3500m, and since the tilt of the mirror 8 is related to the axis lateral translation 6x by 6x = R8, we find that this mirror must be aligned with an accuracy better than

8 < 7x

rad.

(4.61)

This is a much more severe limit than the DC specification obtained in 5 4.1. The relative lateral jitter is expected to be tiny in the case of VIRGO, and so it does not impose particular requirement on the angular alignment of the cavities. Considering now the typical laser waist fluctuations (after mode-cleaning) GWO/WO N BdzO/wO M rad/&, and considering typical interferometer asymmetries ( w l - WZ)/WO E 8d(z2 - zl)/wg M we find a typical phase noise induced by waist size and location fluctuations of the laser of the order of 64 M 2x rad/&, which is well below the shot-noise limit, and therefore negligible. It must be noted also that there exist couplings between angular jitters of the laser and waist size asymmetry of the arm cavities, or between lateral jitters and waist location asymmetry, but these couplings produce lower-order effects than for the case of coupling to the cavities misalignments, and the corresponding noise is then lower than the ones above.

4.4. CONCLUSION

In this section we have discussed some aspects of the optical couplings in interferometric gravitational-wave detectors such as LIGO or VIRGO. The DC coupling between an ideal laser beam and an imperfect interferometer imposes rather severe tolerances on the mirror defects in order not to degrade the shot-noise level too much. A particular class of aberrations, due to heating effects, will be studied in the next chapter. On the other hand, the losses induced by the coupling of an imperfect input laser to the interferometer can be made negligible by proper design of the matching optics. The AC coupling between the laser geometrical fluctuations and the geometrical asymmetry of the interferometer produces a noise which is negligeable for each type of coupling except the one between the laser angular jitter and the translations of the axis of the arm cavities. This imposes a tolerance in the DC pointing of the arm end mirrors less than rad/-, which is difficult to achieve.

11, P 51

THERMO-OPTICAL COUPLING

0 5.

141

Thermo-optical Coupling

5.1. INTRODUCTION

We have seen in Q 3 that it is necessary to increase the power circulating in interferometric gravitational-wave detectors in order to reduce the shot-noise level. The use of a 10-20 W laser source together with the recycling technique, leading to an equivalent 1 kW laser source, is sufficient for the first generation VIRGO detector. With such a configuration there will be about 15 kW of stored power in the arm cavities (of finesse about 50). The cavity mirrors, the sensitive part of the detector, may be heated by absorption of a little amount of this power. Moreover, the heating is nonuniform, because of the (nearly) gaussian geometry of the light beams. This may cause temperature gradients and as a consequence, transversal index gradients (“thermal lensing”) and induce thermoelastic distortions of the mirrors, or in other words: thermal aberrations. These aberrations can in turn cause a loss of power circulating in the interferometer and a loss of contrast, and then a deterioration of the detector performance (Hello [ 19901, Winkler, Danzmann, Riidiger and Schilling [ 19911). Moreover, the problem is non-linear, due to the coupling of the optical system on one hand and of the thermal system on the other hand. The optical system gives the optical field stored in the cavities and thus the intensities hitting or crossing the mirrors, which are responsible for temperature gradients. The thermal system gives the thermal aberrations and then reacts on the optical system. In this section, we first recall the derivation of transient thermal aberrations of large radiatively cooled mirrors heated by absorption of light power either in the coating or in the substrate. We then couple this thermal model to an optical dynamical simulation of the lulometric cavities and of the whole interferometric detector. 5.2. THERMAL ABERRATIONS OF A MIRROR HEATED BY A HIGH POWER LASER SOURCE

5.2.1. The heating problem and the temperature distribution

Let us consider a cylindrical mirror (radius a and width h ) illuminated by a gaussian beam of size wo (see fig. 20). The mirror can be heated by absorption of light power either via its substrate or via its coating. The coating is approximated by a very thin layer located at z = -h/2 and absorption in the coating will be treated as a boundary condition. The coating absorption coefficient E is typically of the order of some ppm for good optics. People have measured coating

142

OPTICAL ASPECTS OF INTERFEROMETRIC GRAVITATIONAL-WAVE DETECTORS

[II,

5

5

Coating

\ Radius a

--+

10-20 kW -

-*

z

Z=

Fig. 20. Heating of a VIRGO mirror by a laser beam. The mirror can absorb some laser power in a coating or in its substrate and heat can escape only by thermal radiation.

absorptions at 1064 nm (the wavelength of the LIGO and VIRGO lasers) at the level of 1 ppm (Rempe, Thompson, Kimble and Lalezari [1992], Uehara, Ueda, Ueda, Sekigushi, Mitake, Nakamura, Kitajima and Kataoka [ 19951, Mackowski [1996] and this should be the right order of magnitude for LIGONIRGO first generation interferometers. The substrate absorption coefficient per unit of length a is typically of the order of some ppm/cm to tens of p p d c m for silica substrates and for the best sapphire substrates. The best “standard” silica substrates are Suprasil 3 11 and 3 12 from Heraeus with absorptions measured in the range 4-6ppdcm (Baures and Man [1993]), while the best sapphire substrates, from Crystal Systems Inc., have absorptions about 3 4 ppmicm, (Blair, Cleva and Man [1996]). Heraeus has also developed a special version of silica, Suprasil 31 1 S (for the beamsplitter) and 312 S (for arm input mirrors) for VIRGO, with a drastic reduction in OH radicals which are the main cause for light absorption at 1064 nm. These special silicas have very low absorption; viz, about 1 p p d c m . For such coating and substrate absorptions, the heating can be assumed a priori to be small. The mirror we consider is suspended by thin wires in a vacuum tank, so the heat losses are due only to thermal radiation. The surrounding tank is assumed to be at uniform temperature To. Furthermore, we make the assumption of axial symmetry; i.e., we assume that the absorption coefficients are uniform and that the incoming laser beam is itself axi-symmetrical. Thus, the mirror temperature field written in cylindncal coordinates T ( r ,z , t )

11,

0 51

143

THERMO-OPTICAL COUPLING

(defined modulo TO;i.e., the actual temperature is TO+ T ( r ,z, t ) ) obeys the Heat Equation:

dT pC- - KAT

dt

=

aZ(r),

(5.1)

where p is the substrate material density, C its specific heat and K its thermal conductivity. In the Heat Equation, I ( r ) denotes the incident gaussian beam intensity (W m-2), which is assumed to be constant in time and not to depend on z. This last condition also means that we neglect attenuation of the beam intensity during its travel through the substrate. This approximation is obviously valid for the low-loss mirrors we consider. The temperature field must also satisfy the boundary conditions (equality of heat fluxes) at each surface of the mirror:

dT -K--(r ay

= a,z,t)

=

4eaTo3T(r = u,z,t),

(5.2) (5.3)

dT -K-(r,z

= -$A,

az

t ) = d ( r ) - 4eaTo3T(r,z = -;A, t),

(5.4)

where 5 is the Stefan constant, e the emissitivity of the materials, and the radiative heat flux oe((T0 + T)4 - To4) has been linearized, according to the low heating assumption, T << TO.The linearization of thermal equations implies that we can study separately the two sources of heating and then add them together to get the general solution. The general axi-symmetrical solution for the temperature field can be expressed as a (numerically rapidly converging) Fourier-Dini series (Hello and Vinet [ 1990al):

T(Y,z,t ) =

C

(Smp(t)

cos(u,z/a)

+ Anlp(t)sin(u,z/u))

Jo(Cmr/a),

(5.5)

W P

where

Cm is the rn-th solution of:

where -t is the reduced radiation constant: z = 4eoTia/K, and up and u,, are, respectively, the p-th solution of the dispersion equations: u

=

u

= --t

zcot(uh/2a), tan(uh/2a).

144

OPTICAL ASPECTS OF INTERFEROMETRIC GRAVITATIONAL-WAVE DETECTORS

[II,

5

5

The coefficients of the series depend both on initial conditions and on the heating mechanism and can be written generally as: (5.9) (5.10)

with the general result AKbstrate(t)= 0, since this term corresponds to an even solution in z and the substrate absorption is supposed to be uniform all along the mirror. In the case of initial thermal equilibrium for the mirror, T ( r , z ,t = 0) = 0. Switching the laser on at t = 0, we get the solution:

(5.13)

where the time coefficients are given by: (5.14) (5.15)

and pmis the p-th term of the Dini expansion of the beam intensity Z(r): (5.16) with pmbeing obtained from the normalized scalar product: (5.17) The pmcoefficients are in general proportional to the beam power P , and the temperature field is thus itself linear with respect to P.

11,

P

THERMO-OPTICAL COUPLING

51

145

........................................ h

Y

-12

2

Y

$10

i

I-

8

............................... ..........................

Fig. 2 1. Temperature distribution in a VIRGO-like silica mirror ( a= 20 cm, h = 10 cm), absorbing 1 W of light power in the coating.

An example of a computation of a temperature distribution is given in fig. 2 1. It shows the steady-state ( t -+ 00) temperature field of a mirror illuminated by a perfect TEMoo laser beam of waist size wg = 2cm and with I Watt power absorbed in the coating. We see that the maximum temperature rise is obtained at the center of the coating and is about 12K for one absorbed Watt. As the VIRGO mirrors, for instance, will absorb much less, say about ePcavM 1OP6x10 kW ~ l O r n W the , expected maximal temperature raise due to coating absorption is of the order of 0.1 K, that gives a temperature transversal gradient of the order of 5mWcm for a mirror of radius a = 20cm. An analogous calculation for substrate absorption gives a maximal temperature (at the center of the mirror) of about 2 K per Watt absorbed, so in the case of VIRGO mirrors, the power dissipated in the substrate will be of the order of aPr,,h M lop6cm-' x 1 kW x 10 cm M 10 mW again; the maximal temperature rise in the case of substrate absorption is then of the order of 0.02K (5 times less than in the case of coating absorption), that gives a maximum temperature transversal gradient of about 1 mWcm.

146

OPTICAL ASPECTS OF INTERFEROMETRIC GRAVITATIONAL-WAVE DETECTORS

[II, § 5

5.2.2. Thermal lensing in the mirror

The absorption of optical power induces a temperature gradient inside the mirror substrates and thus an index gradient, since the index depends in general on the temperature. The mirror then behaves like a lens and the beam crossing it is distorted. This is particularly harmful if the mirror is the input mirror of a Fabry-Perot cavity, as the beam is then mismatched to the cavity, due to this thermal lensing. The temperature distribution is generally not parabolic, so the thermal lens is aberrated, and the mismatch can not be corrected for by using matching optics. The total optical path distortion Y ( r ,t ) due to thermal lensing can be simply calculated from the eikonal equation:

Y ( r , t )= dn dT

-h/2

T(r,z, t ) dz,

(5.18)

K-' for silica as well where dn/dT is the thermal coefficient of the index as for sapphire), and is then computed directly from eq. (5.5). Numerical results give roughly the same thermal lensing for coating and substrate absorptions, although the temperature rise is larger locally for the coating absorption than for the substrate absorption, for the same amount of absorbed light power. In cases of the VIRGO cavity input mirrors, both substrate and coating absorptions provide an optical path gradient of about 0.2 pm cm-' for each Watt absorbed with a sagitta about 1.5 pm per absorbed Watt. This gives a focal length of about a few hundred meters per absorbed Watt. 5.2.3. Thermoelastic deformation

The dynamic thermoelastic problem is generally a very difficult one. Starting from the early work of Cutolo, Gay and Solimeno [1980], in the case of axial symmetry and with the assumption of large mirrors (a >> W O , a being the mirror radius and wo the incoming beam size), an approximate solution for the components of the displacement vector can be written as a Dini series (Hello and Vinet [ 1990bl): (5.19) (5.20) (obviously ug = 0), with the same Crn as before, and where the details of the function UnJand V, depend directly on the Fourier-Dini expansion of the

11, P 51

147

THERMO-OPTICAL COUPLING

temperature distribution that has been computed previously, talung into account the amount of power dissipated either in the substrate or in the coating. The deformation S ( t , r ) of the reflective surface located at z = -:h can now be computed as:

C

(5.21)

S(t,r) = ~ ~ ( r , ~ = - $ h , t ) =Um(-ih,t)Jo(cnzr/~). m

The coefficients Umof this Dini series can be expressed as functions of Sp, and Apm,the coefficients of the Fourier-Dini expansion for the temperature field:

with

D(U)

=

cosh(

g)

[up sinh

(g) (g) cos

-

cnlcosh

(g)

(5.23)

sin(

)]

Pm

(5.24)

Iand p are the Lam6 coefficients of the substrate and Y is the thermal where , stress modulus, related to the usual thermal expansion coefficient a,,, by Y = aex,,(3A + 2p); the Lam6 coefficients are related to the Young modulus E and the Poisson ratio a by A = aE/(l - 2a)(l + a) and p = E/2(1 + a). Numerical results for (silica) VIRGO mirrors give sagittas of about 0.2 pm (h/5) for 1 Watt dissipated in the coating (see fig. 22) and about 0.03 pm (A/30) for 1Watt dissipated in the substrate (see fig. 23). The thermoelastic effect seems weaker than the thermal lensing effect, for the same power dissipated; in the case of the VIRGO mirrors ( E = 1 ppm and a = 1 p p d c m expected), about lOmW is absorbed in the substrate, inducing a thermal lens sagitta of about I5 nm and a deformation sagitta of about 0.3 nm, and lOmW is absorbed in the coating, inducing a thermal lens sagitta of about 15 nm and a deformation sagitta of about 2 nm. In order to see the relative importance of each effect, however, it is necessary to study first a complete

148

OPTICAL ASPECTS OF WTERFEROMETRIC GRAVITATIONAL-WAVE DETECTORS

[II,

5

5

0.08 -

0.06

0.04

~

-

0.02 -

radius (n

15

Fig. 22. Distortion of the reflective surface of a VIRGO mirror for 1 W absorbed in the coating.

35

Fig. 23. Distortion of the reflective surface of a mirror for 1 W absorbed in the substrate.

11,

o 51

THERMO-OPTICAL COUPLING

149

VIRGO cavity, taking into account the non-linear coupling between the optical properties of the cavity. 5.2.4. Thermal birefringence The non-uniform stresses in the mirror due to heating not only produce thermoelastic deformations of the reflecting surfaces, but also some birefringence. Birefringence can lead to depolarization losses when the light crosses the mirror and then deteriorates the interferometer performance. Let us first note that the natural birefringence amplitude of fused silica is in general very low, i.e., of the order of a few mradcm, as measured by Logan [ 19931 and Boccara [ 19971 for silicas from Corning and Heraeus (in any case much lower than the birefringence quoted by the manufacturers) and should not limit the sensitivity of gravitationalwave detectors. The thermally induced birefringence has been studied carefully by Wmkler, Rudiger, Schilling, Strain and Danzmann [1994]. They find that the depolarization power losses, L b i r , for absorption in the substrate can be compared with the power losses due to thermal lensing, C,,,,,through: (5.25) where CB is the coefficient of birefringence (C, M for silica). The ratio is about for fused silica, which shows clearly that thermal lensing is the dominant effect. The same result occurs for absorption in the coating. In the following, we will therefore neglect the thermal birefnngence and consider only the thermal lensing and thermoelastic deformations as sources of loss. 5.3. HEATING EFFECTS IN THE VIRGO INTERFEROMETER

5.3.1. Heating effects in a VIRGO-like cavity This section describes the numerical model that we built which is able to simulate, for instance, the evolution of the stored power in a cavity in presence of power dissipation in the mirrors. Let us then consider a 3 km VIRGO-like cavity. Both mirrors, the input mirror being flat and the end mirror being curved, can in principle absorb some light power. However, we may neglect thermal effects in the lughly reflective end mirror, because this mirror, which is not crossed by the beam, can be manufactured from materials other than homogeneous fused silica, especially from a material with a very low thermal expansion coefficient

150

OPTICAL ASPECTS OF NTERFEROMETRIC GRAVITATIONAL-WAVE DETECTORS

[II,

85

like U.L.E., so the thermoelastic effect can be neglected (thermal lensing does not need to be considered for this mirror). The input mirror has coatings on each face, an A.R. coating on the outside face and the reflective coating on the cavity-side face, and each is able to absorb some optical power. We can neglect the absorption in the A.R. coating since it is dominated by the absorption in the intra-cavity coating by a factor of the order of the cavity power gain (-30 for VIRGO). The basis of the simulation is a fast optical propagation code, describing the optics in a cavity, mixed with the previous analytical formulas for the thermal aberrations which are used to modify the phases of the mirror transmission and reflection operators, as explained in 9 4.1. Rather than use the FFT propagation method, it is better here to take advantage of the axial symmetry and to use the Hankel (or radial Fourier) transform (Siegman [1977]). Indeed the propagation of axi-symmetrical beams by the Hankel transform allow an explicit matrix expression for the propagation operator (Vicari and Bloisi [1989], Bloisi and Vicari [1992]) as well as for the cavity operators (Vinet and Hello [1993]). The Hankel transform allows a high gain in computer time with respect to the FFT-based codes (a factor of about lo3 in our case). The coupling between the optical part and the thermal part follows the sequential modelling described in Hello and Vinet [ 1993a).At each time step, the beams able to heat the mirrors are computed (after tuning the resonance of the cavity, simulating an ideal servo loop) and the intensities absorbed in substrates and coatings are derived. Afterwards, the thermal aberrations are deduced and accordingly the phases of the optical operators are finally updated in order to be used in the next iteration. We must be aware of the time increment used in the simulation, because it represents the temporal accuracy of the code. However, since we are dealing with massive mirrors (about 30 kg for the VIRGO cavity mirrors), the thermal time constants are in fact very long (some hours), and are so much larger than the optical time constants, like the cavity storage time (a fraction of ms), that a quasi-static approximation is thus valid. Of course, this approximation forbids the study of any phenomenon with a short time scale; i.e., with a characteristic time of the order of the cavity storage time, such as fast (high frequency) oscillations or thermal bistability. Possible dynamical effects will be discussed in the conclusion. Some results of the simulation of a VIRGO cavity are shown in fig. 24 (coating absorption) and fig. 25 (substrate absorption). The parameters used for these simulations are: cavity finesse = 50, input power = 500 W (corresponding in the whole VIRGO case to a 10 W laser source and a recycling power again about loo), external temperature TO= 300K, and the mirrors are “standard” VIRGO mirrors made of hsed silica (see table 2). The required CPU time for each of

151

THERMO-OPTICAL COUPLING

...................................................

14.8 14.6 14.4 14.2 00

Fig. 24. The evolution of the intra-cavity power for various coating absorptions.

-2

16

5 5.9 b,.;: . :.:._

v

...................................................................................

..o=Tppm~cm 9.%!PP.rnf.c.rn .... .......

0

a

....

.......................................

15.8 15.7 15.6

0=1 Opprn/cm

15.5

15.2

15

1t '

'

;000 ' 2000

'

fd00

'

4000 ' 5000

'

6doO ' $00

'

; 30

time (!

Fig. 25. The evolution of the intra-cavity power for various substrate absorptions.

152

OPTICAL ASPECTS OF INTERFEROMETRIC GRAVITATIONAL-WAVE DETECTORS

[II,

5

5

these simulations is of the order of a few minutes on a DEC Alpha workstation. We see that for low dissipation rates, less than 2 ppm in the coating and 2 p p d c m in the substrate, the power loss inside the VIRGO cavity is less than 1%, which seems reasonable for good operation of the whole detector. The power loss is in fact mainly due to a varying mismatch of the input gaussian beam. During each step of the simulation, this beam is assumed to be a perfect TEMoo matched to the ideal VIRGO cavity. It is then possible to avoid this effect with an adaptive input beam which could be matched to the cavity at any time. Obviously higher-order aberrations (beyond the parabolic order for thermal lensing and deformation) would remain. Unfortunately, there are two cavities in an interferometric detector and they are not exactly identical and there is no (technological) reason for them to have exactly the same coating and substrate absorptions. It is thus impossible in practice to match the input laser beam to each cavity. In fact, as shown in the following, what is important is the dissipation asymmetry between the two cavities of the detector, rather than the absolute dissipation level itself. 5.3.2. Heating effects in the VIRGO interferometer We describe here the simulation of heating effects in the VIRGO interferometer, taking into account the power recycling configuration, and we give the main results for the absorption specifications. The complete simulation involves the two 3 km long cavities coupled together by the recycling mirror (Hello and Vinet [1993b]). Just as in the code described in the previous section, the simulation computes at each iteration the stored powers and the beam intensities in the different cavities (arm cavities and recycling cavity), in order to compute the thermal aberrations for the next iteration. The simulation is then able to follow the evolution of the beams in the interferometer, until a steady-state (i.e., a stable working point is reached). We can therefore define a figure of merit as an equivalent signal-to-noise ratio (SNR): (5.26)

where P,, is the stored power in the recycling cavity and P,,, is the lower power stored in the two arm cavities. This estimate of the signal-to-noise ratio assumes a shot-noise-limited sensitivity, the shot-noise level being inversely proportional to G.On the other hand, we assume SNR to be proportional to PCav, because the gravitational-wave transfer function is directly proportional to the arm cavity power gain (Vinet, Meers, Man and Brillet [ 19881). In all of the following numerical simulations, we have assumed a recycling power gain of about 50-100, as planned in the VIRGO FCD, for the input

11,

P

51

153

THERMO-OPTICAL COUPLING

Table 2 Optical and thermal parameters Parameter

Fused silica

Sapphire

Index at 1064 nm

1.45

1.75

dnldT (K-')

1

o-~

1.3x 1O@

Thermal conductivity (W m-' K-' )

1.38

23.0

Specific heat (Jkg-' R ' )

145

756

Thermal expansion (K-'

0.54~ lo@

8.4~ 10@

Density (kg m-3)

2202

3980

Mirror radius (cm)

20

20

Mirror thickness (cm)

10

I0

mirrors of the arm cavities being made of fused silica (FSi) or sapphire (Sa) substrates. FSi/Sa optical and thermal parameters are given in table 2. We note in particular the high thermal conductivity of sapphire, which makes sapphire a very interesting material from the thermal point of view. In the following results, we have set the (arbitrary) tolerance thresholds to 5% and 10% SNR loss. This choice seems indeed reasonable and does not limit detector sensitivity too much. First results show that if the two arms are perfectly symmetrical, we can tolerate rather high absorption rates, say 20ppm in the coatings and 25 p p d c m in the substrates for 5% SNR loss. The result is quite different for the more realistic case of different absorptions in both arms. Indeed, any asymmetry between the power absorptions in the two arms induces two different gaussian beams in the arm cavities (since the thermal aberrations are different), and then a contrast defect (power leak) at the interferometer output. This does not arise when symmetry is maintained, since the two cavity modes are identical and the interference is perfect. Figure 26 shows that with current FSi absorptions (5 p p d c m ) a SNR loss of 5% is obtained with an absorption asymmetry about 6%. Unfortunately, silica manufacturers can not guarantee such a reproducibility (less than 10%). The solution could be to ask for lower absorptions, say 2 p p d c m , for which the 5% SNR loss threshold is reached with a 15% asymmetry. This does not seem to be a difficulty for the Heraeus company which is expected to provide to VIRGO fused silica substrates with absorption less than 1 p p d c m . In the case of absorption in the coating (see fig. 27), similar considerations mean asking for about 2 ppm ilo%, which seems feasible, the absorption in the coating being

154

OPTICAL ASPECTS OF INTEWEROMETRIC GRAVITATIONAL-WAVE DETECTORS

Fig. 26. SNR loss for asymmetrical absorptions in the substrate (silica mirrors)

Fig. 27. S N R loss for asymmetrical absorptions in the coating (silica mirrors).

[II, 0 5

11, § 51

THERMO-OPTICAL COUPLING

155

14

12

Fig. 28. SNR loss for asymmetrical absorptions in the substrate (sapphire mirrors)

expected to be less than 1ppm. The results for sapphire substrates are shown in fig. 28 for the case of an absorption of 12ppdcm. The 5% SNR loss level is reached with an asymmetry of about lo%, but if the absorption can be as low as 5 p p d c m (the case for the best Sa samples; Blair, Cleva and Man [1996]), the same threshold is obtained with an asymmetry of about 20%, which is more feasible. Thus it appears that sapphire would be a good material, not necessarily for first-generation detectors for which fused silica seems to be sufficient, but maybe for the next generation, when higher powers are required to improve the sensitivity. The intensive exploitation of the simulation code allows us to find, for each mean value of the absorption coefficients, the asymmetry required to match the 5% (resp. 10%) SNR loss threshold. The results can be seen in figs. 29 (absorption in the coating) and 30 (absorption in the substrate). These figures work as follows (look for example at fig. 30 for the asymmetry of substrates): if the total substrate absorption in arm 1 is, say, 2 p p d c m , then let us plot a vertical straight line at abscissa 2 p p d c m and the intersection points with the abacus give the interval of allowed absorption in the second arm, which is roughly [ 1.7,2.3]p p d c m for a SNR loss of 5%. More generally, we may note that the tolerance zones are roughly defined by straight lines, for

156

OPTICAL ASPECTS OF WTERFEROMETRIC GRAVITATIONAL-WAVE DETECTORS

1

2

[II, 0 5

3

Coating absorption arm 1 (pprn)

Fig. 29. Tolerances for absorption in the coatings. The lines delimit the zone of tolerable asymmetry between the two arms for a loss of 5% in SNR (see text).

absorptions either in coatings or in substrates. The important consequence is therefore that we may deal with specifications expressed as absolute deviations between the two arms. Whatever the mean value of the absorption coefficient is, the tolerable deviation is constant in the limit of low dissipation (<5 ppm for coatings and <5ppm/cm for substrates). Thus, the specifications for the first generation VIRGO interferometer can be given as: - absorption in the coatings: A€ < 0.20 (0.35)ppm, - absorption in the substrates: ha < 0.25 (0.40)ppm/cm, where the first number refers to the 5% of SNR loss level, while the second refers to the 10% of SNR loss level. Let us finally address the problem of the beamsplitter. The splitter is an important source of asymmetry, since it is crossed by only one of the beams (say the beam entering cavity 1) incident on it. The thermal effects in the beamsplitter are not directly computed in our model, mainly because it breaks the axial symmetry. However, we can very roughly take it into account by considering it to suffer the same thermal lensing that the substrates of the input mirrors of the arm cavities suffer; let us also say that the absorption in the substrate is the only one relevant for the heating of the beamsplitter, since it is crossed by the

THERMO-OPTICAL COUPLING

157

Fig. 30. Tolerances for absorption in the substrates. The lines delimit the zone of tolerable asymmetry between the two arms for a loss of 5% in SNR (see text).

same amount of power (1 kW) as the input mirror substrates and since the power incident on the coatings is also 1 kW, about 10 times less than for the intracavity coatings. Considering the beamsplitter, the absorption balance condition between the 2 arms becomes CYI h

+ ase = a2h,

(5.27)

where h is the thickness of the input mirrors, a1 and a2 are the absorptions in the substrates of the input mirrors of cavity 1 and of cavity 2, respectively, a, is the absorption in the beamsplitter substrate, and e is the effective thickness of the beamsplitter, e = h,/cos(8), where h, is the thickness of the splitter (5.5 cm in VIRGO). The refracted angle 8 is deduced from the Snell-Descartes Law for refraction: nsilicasin(8) = sin(45"). In the case of VIRGO, h = 10 cm, h, = 5.5 cm and nsilica = 1.45, the balance is obtained if:

al + 0.63a, = q ,

(5.28)

which must be satisfied with a f0.25(0.40)ppm/cm accuracy, according to the 5( 10)?40 SNR loss threshold specification.

158

OPTICAL ASPECTS OF INTERFEROMETRIC GRAVITATIONAL-WAVE DETECTORS

[II, 0 5

This, in fact, shows that the beamsplitter can play the role of a compensator, with the less absorbing input mirror located in arm 1 and the more absorbing one in arm 2. According to the Heraeus Co., which will provide the mirror substrates for VIRGO, and to some preliminary measurements made in Orsay, the absorption in the beamsplitter of VIRGO should be about l p p d c m , and the balance equation above could be satisfied (within the f 0 . 2 5 p p d c m accuracy) even with input mirror substrates with absorptions of, for example, a1 M 1 p p d c m and a2 M 2 p p d cm . Therefore, the heating problem does not seem a critical one for first generation VIRGO and also for LIGO, where optical powers are similar. But do not forget that the shot noise in these interferometers depends on the square root of the circulating power, so if we want to gain one order of magnitude for the shot noise in future detector generations, we have to increase by two orders of magnitude the recycling power, thus increasing by a factor 100 the absorbed powers in the intra-cavity coatings and in the substrates. To conclude, let us look at fig. 31. It shows the circulating power in VIRGO

0

A'

' '

50''

'

100'

'

i 5 0 ' ' 200.' 250 ' ' 3 0 0 ' ' 3 4 0 ' ' 400 Input Power (W)

Fig. 31. Powers stored in the recycling cavity as a function of the input power for various configurations. Curve (1): mirrors made of fused silica, a1 = 1.1p p d c m , € 1 = 1.1ppm, a2 = I.Oppm/cm, € 2 = 1.0ppm. Curve (2): mirrors made of fused silica, at = l.lppm/cm, € 1 = l.Oppm, a2 = l.Oppm/cm, €2 = 1.lppm. Curve (3): mirrors made of sapphire, a~=3.3ppm/cm,~~=l.0ppm,a~=3.0ppm/cm,~~=l.lppm.

11,

8 51

159

THERMO-OPTICAL COWLING

as a function of the input laser power for various configurations. We see that for each configuration, there is a maximum circulating power achevable. 5.4. CONCLUSION

In this section we have studied the non-linear thermal effects that arise in interferometric detectors of gravitational waves. The main results are first: there does not seem to be any problem for the first generation of detectors such as VIRGO/LIGO. The loss of circulating power induced by thermal aberrations (thermal lensing and thermal deformation) is indeed weak and consequently, the (shot-noise limited) sensitivity deterioration is acceptable. Similarly, a first approach of the study of the dynamical thermo-optical coupling (Dhurandhar, Hello, Sathyaprakash and Vinet [ 1997]), has shown that no instability occurs in the LIGONIRGO cavities, at least at the lowest (longitudinal) order and even for 1 MW of stored power. In fact, the optical cavity coupled to the thermal system behaves just like a capacitor, up to this power. There is actually a non-linear coupling, but this coupling is weak for the first generation of interferometric detectors. However, for next generations, if an increase of circulating power is required, for instance, in order to reduce the shot-noise level, coupling will be stronger and several phenomena may occur. First, the power fluctuations of the laser source can couple to the asymmetry of absorption in the interferometer arms and induce an extra phase noise (Hello and Vinet [ 19971): &(Y) M

3~10-~~/Y

( P,,, ) ( 1 kW

61'/P 10-8/,&

)(

Aa 0.2ppdcm

)

'

where P,,, is the power stored in the recycling cavity, 61'/P are the laser relative power fluctuations and A a is the absorption asymmetry of the arm input mirror substrates. This noise is negligible for the first generation of detectors but can be of the order of the shot-noise in the observational frequency window for next generation detectors, unless the asymmetry in absorption between the two arms is lowered (well in the sub-ppdcm range) or unless the amplitude stability of the laser source is improved. Secondly, in the strong coupling case, multistability may arise, just as in the Kerr effect or radiation pressure effects, and particular attention must be payed to the working point of the servoed cavities; especially the usual resonant points (tops of the Airy peaks) can be unstable ones and be very sensitive to power fluctuations, whatever they are. All these expectations about the next generation of detectors have been made with silica mirrors, the same that are to be used in the first VIRGO/LIGO. But,

160

OPTICAL ASPECTS OF INTERFEROMETRIC GRAVITATIONAL-WAVE DETECTORS

as already outlined in 9 5.3, silica may not be the best material to use for high power cavities. Sapphire, especially, has very good thermal properties. Its thermal conductivity is 16 times larger than silica and its density is about double that of silica. On the other hand, its thermal expansion is worse by more than an order of magnitude with respect to silica. However, the latter is somewhat compensated for by larger Lam6 coefficients. Moreover, sapphire is not only interesting for its thermal properties but also for its mechanical ones. It should be possible to gain about an order of magnitude for the mirrors’ thermal noise (Ju, Notcutt, Blair, Bondu and Zhao [ 19961). In particular, sapphire substrates are expected to have quality factors up to Q M lo7 (Braginsky, Mitrofanov and Panov [1985]), while silica substrate quality factors are typically Q M lo6 for the best samples. In summary, sapphire appears to be a very good candidate to replace silica mirrors in future generation interferometers, from the point of view of heating effects and of thermal noise, as well. Acknowledgements I am deeply grateful to Prof. Claude Fabre, Dr. Jean-Pierre Lasota, Dr. Matthew Taubman and Prof. Philippe Tourrenc for their careful reading of this manuscript, and to Drs. Jean-Yves Vinet, Nary Man, Frangois Bondu, Alain Brillet, Claude Boccara, Enrico Calloni, David McClelland and Sanjeev Dhurandhar for a number of fruitful discussions. References Abramovici, A., WE. Althouse, R.W.P. Drever, Y. Gursel, S. Kawamura, F.J. Raab, D. Shoemaker, L. Sievers, R.E. Spero, K.S. Thorne, R.E. Vogt, R. Weiss, S.E. Whitcomb and M.E. Zucker, 1992, The laser interferometer gravitational-wave observatory, Science 256, 325. Anderson, D.Z., 1984, Alignment of resonant optical cavities, Appl. Opt. 23, 2944. Barone, F., E. Calloni, L. DiFiore, A. Grado, F? Hello, L. Milano and G. Russo, 1996, Effects of misalignments and beam jitters in interferometric gravitational-wave detectors, Phys. Lett. A 217, 97. Bauris, P.-Y., and C.N. Man, 1993, Measurements of optical absorptions at 1.06 pm in low-loss materials, Optic. Mater. 2, 241. Blair, D.G., ed., 1991, The Detection of Gravitational Waves (Cambridge University Press). Blair, D.G., F. Cleva and C.N. Man, 1996, Optical absorption measurements in monocrystalline sapphire at 1 micron, VIRGO note NTS96-020. Bloisi, F., and L. Vicari, 1992, Vector formalism for circularly symmetric laser beams, Appl. Opt. 31, 2714. Boccara, C., 1997, Unpublished measurements from ESPCI, reported in Virgo note VIR-NOT-PCI1390.077.

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OPTICAL ASPECTS OF MTERFEROMETRIC GRAVITATIONAL-WAVE DETECTORS

[I1

Hello, P, and C.N. Man, 1996, Design of a low-loss off-axis beam expander, Appl. Opt. 35, 2534. Hello, P., and J.-Y. Vinet, 1990a, Analytical models of thermal aberrations in massive mirrors heated by high power laser beams, J. Phys. (Pans) 51, 1267. Hello, P., and J.-Y. Vinet, 1990b, Analytical models of transient thermoelastic deformations of mirrors heated by high power cw laser beams, J. Phys. (Pans) 51, 2243. Hello, P.,and J.-Y. Vinet, 1993a, Numerical model of transient thermal effects in high power optical resonators, J. Phys. France 13, 717. Hello, P., and J.-Y. Vinet, I993b, Simulation of thermal effects in interferometric gravitational-wave detectors, Phys. Lett. A 178, 351. Hello, P., and J.-Y. Vinet, 1996, Simulation of beam propagation in off-axis optical systems, J. Opt. (Pans) 27, 265. Hello, P, and J.-Y. Vinet, 1997, Noise induced by laser power fluctuations via absorption asymmetry in gravitational-wave interferometric detectors, Phys. Lett. A 230, 12. Jaekel, M.T., and S. Reynaud, 1990, Quantum limits in interferometric measurements, Eur. Phys. Lett. 13, 301. Jenkins, F.A., and H.E. White, 1957, Fundamentals of Optics (MacGraw-Hill, New York). Johnston, S., D.R. Lonmer, P.A. Harrison, M. Bailes, A.G. Lyne, J.F. Bell, VM. Kaspi, R.N. Manchester, N. D’Amico, L. Nicastrol and J. Shengzhen, 1993, Discovery of a very bright, nearby binary millisecond pulsar, Nature 361, 613. Ju, L., M. Notcutt, D.G. Blair, F. Bondu and C.N. Zhao, 1996, Sapphire beamsplitters and test masses for advanced laser interferometer gravitational wave detectors, Phys. Lett. A 218, 197. Kawamura, S., and M.E. Zucker, 1994, Mirror-onentation noise in a Fabry-Perot interferometer gravitational-wave detector, Appl. Opt. 33, 3912. Kogelnik, H., and T. Li, 1966, Laser beams and resonators, Appl. Opt. 5, 1550. Logan, J.E., 1993, Ph.D. Thesis (University of Glasgow). Logan, J.E., N.A. Robertson and J. Hough, 1992, An investigation of limitations to quality factor measurements of suspended masses due to resonances in the suspension wires, Phys. Lett. A 170, 352. Mackowski, J.-M., 1996, IPN-Lyon, France, unpublished measurements. Man, C.N., 1995196, personal communications. Man, C.N., D. Shoemaker, M. Pham Tu and D. Dewey, 1990, External modulation technique for sensitive interferometric detection of displacements, Phys. Lett. A 148, 8. Mavaddat, R., D.E. McClelland, P. Hello and J.-Y. Vinet, 1995, Dual recycling laser interferometer gravitational wave detectors: simulating the performance with imperfect mirrors, J. Opt. (Pans) 26, 145. McClelland, D.E., C.M. Savage, A.J. Tndgell and R. Mavaddat, 1993, Tolerance of dual recycling laser interferometric gravitational wave detectors to mirror tilt and curvature errors, Phys. Rev. D 48, 5475. Meers, B.J., 1988, Recycling in laser-interferometric gravitational-wave detectors, Phys. Rev. D 38, 23 17. Meers, B.J., 1989, The frequency response of interferometric gravitational-wave detectors, Phys. Lett. A 142, 465. Meers, B.J., and K.A. Strain, 1991, Wave-front distortion in laser-interferometric gravitational-wave detectors, Phys. Rev. D 43, 31 17. Misner, C.W., K.S. Thorne and J.A. Wheeler, 1973, Gravitation (Freeman, San Francisco, CA). Mizuno, J., 1995, PhD Thesis (Max Planck Institut fiir Quantenoptik, Garching). Mizuno, J., K.A. Strain, PG. Nelson, J.M. Chen, R. Schilling, A. Rudiger, W. Winkler and K. Danzmann, 1993, Resonant sideband extraction: a new configuration for interferometric gravitational wave detectors, Phys. Lett. A 175, 273.

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Moss, G.E., L.R. Miller and R.L. Forward, 1971, Photon-noise-limited laser transducer for gravitational antenna, Appl. Opt. 10, 2495. Regehr, M.W., F.J. Raab and S.E. Whitcomb, 1995, Demonstration of a power recycled Michelson interferometer with Fabry-Perot arms by frontal modulation, Opt. Lett. 20, 1507. Rempe, G., R.J. Thompson, H.J. Kimble and R. Lalezari, 1992, Measurements of ultralow losses in an optical interferometer, Opt. Lett. 17, 363. Rudiger, A., R. Schilling, L. Schnupp, W. Winkler, H. Billing and K. Maischberger, 1981, A mode selector to suppress fluctuations in laser beam geometry, Opt. Acta 28, 641. Saleh, B.E.A., and M.C. Teich, 1991, Fundamentals of Photonics (Wiley Interscience, New York). Saulson, P.R., 1990, Thermal noise in mechanical experiments, Phys. Rev. D 42, 2437. Schnupp, L., 1986, unpublished. Shoemaker, D., A. Brillet, C.N. Man, 0. Cregut and G. Kerr, 1989, Frequency-stabilised laser-diodepumped Nd:YAG laser, Opt. Lett. 14, 609. Shoemaker, D., R. Schilling, L. Schnupp, W. Winkler, K. Maischberger and A. Rudiger, 1988, Noise behavior of the Garching 30 meter prototype gravitational wave detector, Phys. Rev. D 38, 423. Siegman, A.E., 1977, Quasi-fast Hankel transform, Opt. Lett. 1, 13. Siegman, A.E., 1986, Lasers (University Science Books, Mill Valley). Strain, K.A., and B.J. Meers, 1991, Experimental demonstration of dual recycling for interferometric gravitational-wave detectors, Phys. Rev. Lett. 66, 1391. Sziklas, E.A., and A.E. Siegman, 1975, Mode calculations in unstable resonators with flowing saturable gain. 2: Fast Fourier transform method, Appl. Opt. 14, 1874. Thorne, K.S., 1987, Gravitational radiation, in: 300 Years of Gravitation, eds S.W. Hawking and W. Israel (Cambridge University Press, Cambridge) p. 330. Thorne, K.S., 1995, Gravitational waves, in: Proc. 1994 Snowmass Summer Study: Particle and Nuclear Astrophysics and Cosmology in the Next Millenium, eds E.W. Kolb and R.D. Peccei (World Scientific, Singapore) p. 160. Tourrenc, ,!F 1996, Is something moving in the Virgo experiment? preprint. Tridgell, A.J., D.E. McClelland and C.M. Savage, 1991, Numerically modelling a dual recycling interferometric gravitational-wave detector, in: E. and F. White Research Conference on Gravitational Astronomy: Instrument Design and Astrophysical Prospects, eds D.E. McClelland and H.-A. Bachor(Wor1d Scientific, Singapore) p. 223. Tsubono, K., 1995, 300-m laser interferometer gravitational detector (TAMA300) in Japan, in: Gravitational Wave Experiments, eds E. Coccia, G. Pizella and F, Ronga (World Scientific, Singapore) p. 112. Uehara, N., A. Ueda, K. Ueda, H. Sekigushi, T. Mitake, K. Nakamura, N. Kitajima and I. Kataoka, 1995, Ultralow-loss mirror of the parts-in-10-6 level at 1064 nm, Opt. Lett. 20, 530. Vicari, L., and F. Bloisi, 1989, Matrix representation of axisymmetric optical systems including spatial filters, Appl. Opt. 28, 4682. Vinet, J.-Y., 1986, Recycling interferometric antennas for periodic gravitational waves, J. Phys. (Paris) 47, 639. Vinet, J.-Y., and P. Hello, 1993, Matrix simulations of optical cavities, J. Mod. Opt. 40, 1981. Vinet, J.-Y., P. Hello, C.N. Man and A. Brillet, 1992, A high accuracy method for the simulation of non-ideal optical cavities, J. Phys. France 1 2 , 1287. Vinet, J.-Y., B.J. Meers, C.N. Man and A. Brillet, 1988, Optimization of log-baseline interferometers for gravitational-wave detection, Phys. Rev. D 38, 433. Virgo Collaboration, 1989, Virgo proposal to the CNRS and INFN, unpublished report. Virgo Collaboration, 1992, Virgo Final Concept Design, unpublished report. Virgo Collaboration, 1995, Virgo Final Design, Version 0, unpublished report. Weber, J., 1960, Detection and generation of gravitational waves, Phys. Rev. 117, 306.

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E. WOLF, PROGRESS IN OPTICS XXXVIII 0 1998 ELSEVIER SCIENCE B.V. ALL RIGHTS RESERVED

I11 THERMAL PROPERTIES OF VERTICAL-CAVITY SURFACE-EMITTING SEMICONDUCTOR LASERS BY

WLODZIMIERZ NAKWASKI * Institute of Physics, Technical University of Eddi, ul. Wblczarisku 219, 93005 tddi.Poland

MAREKO S I ~ K ** I Center for High Technology Materials. University of New Mexico, 1313 Goddurd SE, Albuquerque, NM 87131, USA

* Also with the Department of Electrical and Computer Engineering, University of New Mexico, Albuquerque, NM 8713 1, USA. ** Also with the Department of Electrical and Computer Engineering, and the Department of Physics and Astronomy, University of New Mexico, Albuquerque, NM 87 13 1-6081, USA. 165

CONTENTS

PAGE

9 1. INTRODUCTION . . . . . . . . . . . . . . . .

,

. .

167

9 2. COMPARISON OF VERTICAL-CAVITY SURFACE-EMITTING AND EDGE-EMITTING DIODE LASERS. . . . . . . . . 168

9 3. EFFECTS OF TEMPERATURE ON VCSEL OPERATION

.

182

FUNDAMENTALS OF THERMAL MODELING OF VCSELs .

205

COMPREHENSIVE THERMAL MODELS OF VCSELs . . .

218

CONCLUSIONS . . . . . . . . . . . . . . . . . . . .

254

ACKNOWLEDGMENTS . . . . . . . . . . . . . . . . . . .

255

REFERENCES.. . . . . . . . . . . . . . . . . . . . . .

256

9 4. 9 5. 6 6.

166

,

Q 1. Introduction Vertical-cavity surface-emitting lasers (VCSELs) generate considerable interest due to their unique features which distinguish them from conventional edgeemitting lasers (EELs): narrow low-divergence circular non-astigmatic output beam, inherent dynamic single-longitudinal-mode operation, geometry suitable for integration into two-dimensional (2D) arrays or for monolithic integration with electronic devices, compatibility with vertical-stachg architectures, and so on. Rarely, the paramount role of thermal effects in VCSELs is listed among those features, perhaps because it usually represents an obstacle rather than an advantage. Ever since the inception of VCSELs, thermal problems have plagued their development (Iga, Koyama and Kinoshita [1988], Geels and Coldren [1991], Hasnain, Tai, Dutta, Wang, Wynn, Weir and Cho [1991]), and to this date they represent a major hurdle that must be overcome if massive integration into 2D arrays is ever to be realized (Osinsh and Nakwaski [1995a]). Compared to EELs, VCSELs have a number of unique features which make them potentially more immune to damage caused or accelerated by external ambient. These include complete isolation of the active region from the external medium, a very small active-region volume, the potential for temperatureinsensitive operation, and high internal optical power density (low radiative carrier lifetime). These properties make VCSELs attractive for applications in harsh environments. In spite of the remarkable progress achieved over the last several years (e.g., Iga [1992a,b], Coldren, Geels, Corzine and Scott [1992], Iga and Koyama [ 19931, Huffaker, Deppe, Kumar and Rogers [ 19941, Morgan [ 19941, ChangHasnain [ 19941, Morgan, Lehman, Liu, Hibbs-Brenner and Bristow [ 19971, Coldren, Hegblom, Strzelecka, KO, Akulova and Thibeault [ 19971, Morgan [ 19971, the continuous wave (CW) performance and integration scale of VCSELs are still seriously limited by their thermal behavior. To this day, closely packed VCSEL arrays can be operated only if the array elements are excited sequentially, one at a time. Average heat-flux densities generated inside the active regions of VCSELs are often extremely high (-300 W/mm2 in 10-ym diameter etchedwell VCSELs (Nakwaski and Osinski [1991a]), compared to -30 W/mm2 in 15-pm wide stripe-geometry edge-emitting lasers (Nakwaslu [ 1984]), both 167

168

THERMAL PROPERTIES OF VCSELs

[IK

82

evaluated at CW currents 2 times higher than the pulsed threshold). This leads to a substantial increase in temperature, accompanied by a corresponding increase in the threshold current density, a shift in the emission wavelength, and a reduction of the optical output power. Efficient heat dissipation, along with an ultra-low threshold, is therefore critical for large-scale integration into 2D arrays. In addition, since the operating lifetime of semiconductor lasers usually decreases exponentially with temperature, it is essential to design lasers with consistently low self-heating. In this chapter, we give a comprehensive review of temperature-dependent phenomena in VCSELs, followed by a discussion of various comprehensive approaches used so far to model thermal properties of VCSELs. The chapter is organized as follows. In $ 2, the main differences between properties of VCSELs and EELs are highlighted. In $ 3, we discuss in more detail the experimental data on various device characteristics affected by temperature. The basics of thermal VCSEL modeling are formulated in $4. In $5, we describe comprehensive analytical and numerical thermal models of VCSELs. Such models are especially important if device design needs to be optimized.

tj 2. Comparison of Vertical-Cavity Surface-Emitting and Edge-Emitting Diode Lasers

The main structure difference between edge-emitting diode lasers (EELs) and vertical-cavity surface-emitting diode lasers (VCSELs), reflected in their very names, is illustrated in fig. 1. In EELs, stimulated radiation traveling between resonator mirrors propagates in the plane parallel to the p-n junction, remaining all the time within the active region. In VCSELs, the lasing radiation travels in the direction perpendicular to the p-n junction plane and is amplified inside the active region only during a small fraction of each round trip. Several important consequences follow from the vertical-cavity configuration. First of all, the coupling between the optical field and the gain medium is much weaker in VCSELs than in EELs, therefore a much higher gain would normally be required in VCSELs to achieve a lasing threshold comparable to that of EELs. The high gain requirement can be mitigated to some extent by reducing the radiation losses, which is accomplished mainly by utilizing VCSEL resonator mirrors of much higher reflectivity (very close to unity) than in standard EELs. This, however, causes an increase in densities of both internal stimulated and spontaneous radiation within the VCSEL resonator and a decrease in the differential quantum efficiency. Consequently, the design of high performance

111, § 21

COMPARISON OF VCSELs and EELs

Edge-emitting diode laser

-

169

Vertical-cavity surface-emitting diode laser Radiation

Radiation

-

Radiation

v Radiation

Resonator mirror -0

...

p-n junction Fig. 1 . Schematic configuration of edge-emitting diode lasers (EELs) and vertical-cavity surfaceemitting diode lasers (VCSELs).

VCSELs involves a compromise between low-threshold and high-efficiency requirements. Low-threshold VCSELs should have smaller active regions, fewer quantum wells, and mirrors of reflectivities as high as possible, whereas highly efficient VCSELs should have larger emitting areas, more quantum wells, and lower-reflectivity mirrors (cf. Scott, Geels, Corzine and Coldren [ 19931). Another important consequence of vertical-cavity configuration is the very small magnitude of the cavity length L. While in EELs L is typioally between 100 pm and 500 pm, the effective length of VCSEL resonators (cf. eq. 7, below) is usually in the range of as little as about 1 pm (Hasnain, Tai, Yang, Wang, Fischer, Wynn, Weir, Dutta and Cho [1991], Taylor and Evaldson [1994]) to several micrometers (Iga, Koyama and Kinoshita [ 19881). The longitudinal-mode spacing AAFPfor the Fabry-Perot (FP) resonator can be approximately written as (Young [1993]):

where nG is the group refractive index (eq. 4) and A is the radiation wavelength. Taking A = 0.85 pm and nc =4.3 for GaAs (see table 3, p. 183) gives AAFp =4.2 A for EELs with cavity length L = 200 pm, compared to AAFPas large as 840 A for VCSELs with effective cavity length L,ff = 1 pm. For that reason, usually several longitudinal modes are close to the gain peak in EELs, resulting in multimode

170

THERMAL PROPERTIES OF VCSELs

t

~~

§2

wavelength

I

wavelength

Fig. 2. Illustration of different reactions of output spectra from (a) EELS and (b) VCSELs to an increase in their active-region temperature.

111,

8 21

171

COMPARISON OF VCSELs and EELS

I

oDF-B

il-

TBEML

!J7J ...................

I

.......

I

DMEWL

SMEWL

semiconductormirror dielectric mirror proton-implanted region oxidized region

oxide I Icontact = n = *=

-- --

reverse- olarized p-n j u d i o n active region

polymide

Fig. 3a. Schematic illustration of bottom-emitting VCSELs. See table 1 for abbreviations.

spectral Characteristics (see fig. 2a). In contrast, only one (if any) longitudinal mode is within the spectral gain bandwidth in VCSELs (fig. 2b); therefore, they always lase a single longitudinal mode. Inherently, VCSELs operate in a single longitudinal mode even under modulation conditions, which is one of their most important advantages. Over the last few years, numerous VCSEL structures of different designs were studied. Figure 3 presents schematic structures of some of these devices

172

THERMAL PROPERTIES OF VCSELs

0

0

.....

I

PITSEL

0

-

I

PITSEL-U

BCCL

HMSCL-u

HMIDL

.....

HMSCL

0 I

.....

HMUML

I

HMCCL

TEML

TEML-I1

Fig. 3b. Schematic illustration of top-emitting VCSELs

referred to in this chapter, together with designated abbreviations by which they are identified. Table 1 contains a summary of key features of these devices.

111,

8 21

173

COMPARISON OF VCSELs and EELS

0

..:.;.;.... .:. , ;..:;;;. ......... .;, ;y...:.,'..:

..:..: . .

...

.... , ,.,:.;:

TEOML

TEOML-II

,

TEOML-I11

i---. ....-I HMOL

0

AGTL

HMOL-II

-0"""

Fig. 3c. Schematic illustration of oxidized VCSELs.

DCCOL

Table 1 Vertical-cavity surface-emitting laser structures ~

Abbreviaaon

Active region

Spacer cavity

Bottom reflector

PITSEL-AIT-A

4 x 10IUII GaAs/Alo 2 G q gAs MQW

GRIN Al,Gal_,As 03<x<06 2 waves

Hasnain, Tai, Dutta, Wang, Wynn, Weir 30.5 periods 20 periods AlAs/Alo,15Ga0,85As A I A s / A ~ ~ . I ~ G % ,and ~ ~ Cho A s [1991]; Hasnain, Tai, Yang, Wang, Fischer, Wynn, Weir, Dutta and stair-case barriers stair-case barriers Cho [I9911

PITSEL-AIT-B

4 x 10IUTI GaAs/Alo 3Gao 7As MQW

GRIN Al,GaI-,As 0 3 QxQO 6 2 waves

Tu, Wang, Schubert, Weir, Zydvk and 29.5 periods 20 periods AlAs/Alo 14Ga0.86As AlAdAlo 14Gq.86AS Cho [1991] stair-case barriers stair-case barriers

Top reflector

Reference

4 x 8 ~l~ll GRIN GaAs/Alo l 5 G q 85As Al,Gal,As 015<x<10 MQW I wave

43.5 periods 24 periods Zhou, Cheng, Schaus, Sun, Zheng, AlAVAlo I 5 (3% 85 AS AlAs/Alo I 5 G% 85 AS Armour, Hains, Hsin, Myers and Vawter graded barriers graded barriers [I9911

PITSEL-SNL

3 x 8 ~III A10 SInO s p G% 461% 54p/ 2 x 3 waves (A10 5Gao 510 5Ino5p MQW

55.5 periods G q .5As AlA~/Alo.5 parabolic-graded barriers

36 periods AlAs/Alo.sGq,sAs parabolic-graded barriers

Schneider, Choquette, Lott, Lea, Figiel and Malloy [1994]

30.5 periods Alo.7G%.3AdGaAs

28 periods Alo.7 Gq.3AsIGaAs graded and &doped barriers

Zeeb, Reiner, Ries, Moller and Ebeling [ 19951

AS

3 InGaAdGaAs MQW

A10 3 G q 1 wave

HMlDL-SFIT

1 pm GaAs DH

Alo,15Gao,ssAs >1 pm

3

E $

PITSEL-UNM

PITSEL-11-UU

i

20.5 periods 7 periods AlAs/Alo,15Gq.85As SiOz/TiOz

3

5

9

Qm

6:

Wiitbrich, James, Ganiire and Reinhart [ 19901 continued on next page

@

Kn

N

H

H c (

M

Table 1, continued

N

Y

~~

Abbreviation

Active region

HMUML-UCSB

0.3pm InGaAsP DH InPiInGaAsP (a, = I .3km) 1.4pm

HMSCL-RIT

9X7nm InGaA s P h G aASP MQW

HMSCL-II-cu

HMF'IL-HTC

HMCCL-BTL

BCCL-TUM

3Ox6nm InGaAsP5nGaAs strain-compensated MQW 3x7nm GaAs MQW

1.55pm InGaAsP (kg= 1.4pm)

Spacer cavity

InPhGaAsP 5 waves

Bottom reflector

Top reflector

Reference

27 periods AlAsIGaAs digital alloy grading + 1GaAs

4 periods Si/SiO2

Dudley, Babic, Muin, Yang, Miller, Ram, Reynolds, Hu and Bowers [1994]

50 periods InGaAsPfLnF'

5 periods Si/SiO2

Streubel, Rapp, Andrt and Wallin [19961

1.42pm InF' + 0.44pm SiO2lAl InGaAsP

n 0

z

9

4 periods Si/SiO2

Chua, Zhu, Lo, Bhat and Hong [1995]

$

z

$ < A10.6G%.4As 1 wave

30.5 periods 8 periods Morgan, Hibbs-Brenner, Lehman, AlAslAlo,I ~ G W , ~ AlAslAlo, ~AS (6GW.84AS Kalweit, Walterson, Marta and graded barriers graded barriers + 5.5 Akinwande [1995] periods SiOziTi02

InP

45 periods InF'/In GaAsP (ag= 1.4pm)

4 periods Si/Al2O3

Fisher, Huang, Dann, Elton, Harlow, Perrin, Reed, Reid and A d a m [I9951

21 periods AIAsIGaAs

18 periods AIAsIGaAs

Rochus, Hauser, Rob, Kratzer, Boehm, Klein, Traenke and Weimam [1995]

1 wave

3 ~ 8 m 1% 35G%.65AS Ing,~8Ga".g2As/GaAs 860nm MQW

continued on next page

G:rn tT Iy

a a

rn m

c;

Table 1, continued Abbreviation

Active region

Spacer cavity

Bottom reflector

Top reflector

Reference

TEML-UCB

3 x 8 nm In02Ga0gAslGaAs MQW

AIO 33G9 67AS 2xI500A

33.5 periods Alo,gGao 1 As/GaAs superlattice barriers

7 periods Alo.gG%,1 AsIGaAs superlattice barriers +external cavity

Hadley, Wilson, Lau and Smith [1993]

TEML-11-NTT

6 GaAsIAlo 2Gao MQW

A10 3Gao 7As 1 wave

34.5 periods 20 periods Ahl5Ga0 X5AsfMAs MO.l5G%.85As/ Al0.5Gag,jAsIAlAs

Ohiso, Kohama and Kurokawa [1995]

BEML-UCSB

3x 8 InO.ZGa0 8 AsIGaAs MQW

Alo.2Gao 8AS 1 wave

18.5 periods AlAslGaAs graded barriers + Au contact

16 periods PJAsIGaAs graded barriers

Geels, Thibeault, Corzine, Scott and Coldren [I9931

BEML-11-TIT

3 x 8 nm A10.4Ga0.6As In0,2G~,8As/GaAs 1 wave MQW

22 periods A1AslGaAs graded barriers

25 periods A1AsIGaAs graded barriers i rnetalization

Mukaihara, Hayash, Hatori, Ohnoki, Matsutani, Koyama and Iga [1995]

25.5 periods

24 periods AlAsIGaAs graded barriers

Babic, Dudley, Streubel, Mirin, Bowers and Hu [1995]; Babic, Streubel, Mirin, Margalit, Bowers and Hu [1995]

20 periods AMsIGaAs stair-case barriers + metalization

Wipiejewski, Panzlaff, Zeeb and Ebeling [ 19941

AS

DF-BEML-UCSB 7 x 7 nm InGaAsP MQW

InGaAsP 5.2 waves

AlAslGaAs

graded barriers TBEML-W

~X~IXXI

Ino.zGa",8AsIGaAs MQW

M0.4Ga0.6As 1 wave

24.5 periods AlAsIGaAs stair-case barriers

continued on next page N

Table 1, continued Abbreviation

Active region _

Spacer cavity _

_

_

~

Bottom reflector

Top reflector

Reference

~

2 x SiOx/SiNxHl,/SiOx Sugihwo, Larson and Harris [ 19971 + A14 GaAs + metalization + air gap + 7 periods InPAnGaAsP + 5 periods SiOz/TiO2

AGTL-SU

2x6nm AlAdAl0.31Ga0,69As/ 22.5 periods IIQJ~ Gao,79As/GaAs GaAs GaAsIAlAs MQW 2 waves

AGTL-11-UCB

3 GaAs MQW

AlAs 1 wave

24 periods

4.5 periods + air gap Vail, Li, Yuen and Chang-Hasnain +22.5 periods+Au [1996] metalization

DMEWL-TIT-A

2 pm InGaAsP DH ( k g= 1.3 pm)

InP 5.8 pm

SiO2/AdZn/Au

k/4-SiOz/Au

DMJiWL-TIT-B

2.5 pm GaAs DH

A10.3Gq.7As (2.7 pm) 5 periods Alo.1Gq,gAs (0.3 pm) SiOz/TiO2

AdSiO~/TiO2/SiO~Koyama, Kinoshita and Iga [I9891

DMEWL-OFL

12x41~11 InGaAsPhGaAsP (kg=l.lpm) MQW

InP

8.5 periods silA1203 + Au metalization

6 periods SilSiOz

Uchiyama and Kashiwa [1995]; Uchiyama, Yokouch and Ninomiya [ 19971

0.5 pm GaAs DH

Al0.3Gq.7As 300 nm

5 periods Alo.lGao.9As’ AlO.7G%.3As

22 periods AIAsIAlO,1 G%,gAs

Tai, Fischer, Seabury, Olsson, Huo, Ota and Cho [ 19891

SMEWL-ATT

5 pm

Uchiyama and Iga [I9841

continued on next page

5: c

0 w rn

s

Table 1, continued Bottom reflector

Top reflector

Reference

0.5 km InGaAsP DH InP ( k g = 1.3 p) 4w

5 periods Si3NdiSi

5 periods Si3N4iSi

Wada, Babic, Crawford, Reynolds, Dudley, Bowers, Hu, Merz, Miller, Koren and Young [ 19911

HMML-NTT

9 x 1%-compressibly InGaAsP (kg= 1.2 km) strained InGaAsPiInGaAsP MQW

25 periods GaAsiAlAs + 10.5 periods MIInGaAsP

7 periods InPhGaAsP + 5 periods Si02lTiOz

Ohiso, Amano, Itoh, Tateno, Tadokoro, Takenouchi and Kurokawa [1996]

PIBEL-ATT

3 x 8 nm In0,2G%.gAs/GaAs MQW

20 periods AlAsIGaAs

14 periods AMsIGaAs

Jewell, Scherer, McCall, Lee, Walker, Harbison and Florez [1989]

BEOML-TIT

3 x 8 nm In0 zGao,8As/GaAs MQW

22.5 periods

20 periods AlAslGaAs graded barriers metalization

Hayashi, Mukaihara, Hatori, Ohnoki, Matsutani, Koyama and Iga [ 19951

Abbreviation

Active region

UMEWL-UCSB

Spacer cavity

GaAs 1 wave

AMsIGaAs

+

Thibeault, Bertilsson, Hegblom, Strzelecka, Floyd, Naone and Coldren [ 19971

BEOML-11-UCSB 3 x 8 nm Alo 3Gao 7As Iq,17Gwg3AsiGaAs 1 wave MQW

18.5 periods AlAsiGaAs digital alloy grading

21 periods Alo,9Gq,I AsiGaAs Au metalization

BEOML-111UCSB

18.5 periods AIAsiGaAs graded barriers

Floyd, Thibeault, Coldren and Merz 26 periods Alo,~~Ga02sAs/GaAs [1996] + 1 period GaAslAlAs graded and &doped barriers

3 x8 nm

In0 174Gao82&/

GaAs MQW

SGaO SAs

0.24 km

continued on next page

3 H

M

Table 1, continued Abbreviation

Active region

TEOML-SNL-A

Spacer cavity

t4

Y

Bottom reflector

Top reflector

4 x 6 nm AlGaInP h.56Ga0 44p/ 1 wave (Al0.5G% 5 )0.5I"0.5p MQW

55.5 periods AlAs/Al0,5G%,5As parabolically graded barriers

34 periods Crawford and Schneider [1995]; AIAs/Alo,5G~,5As Choquette, Schneider, Crawford, Geib parabolically graded and Figiel [ 19951 barriers

TEOML-SNL-B

3 x InGaAs MQW

38 periods 25 periods A10,96Ga0,04As/GaAs &,96G%,o4As/GaAs uniparabolically uniparabolically graded barriers graded barriers

Lear, Schneider, Choquette, Kilcoyne, Figiel and Zolper [ 19941; Choquette, Schneider, Lear and Geib [ 19941; Lear, Choquette, Schneider and Kilcoyne [ 19951

TEOML-11-USC

3 x 6 nm A10 5% In0 l 5 G q 85As/GaAs 1 wave MQW

30 periods AIAs/GaAs

4 periods AlAs oxide/GaAs

MacDougal, Dapkus, hdikov, Zhao and Yang [1995]; Yang, MacDougal and Dapkus [1995]

8 Ba

AIGaAs 1 wave

54s

Reference

TEOML-111KAIST

4x8~11 A l o 1 I Gao 89As/ A l O 3 GaO 7 As MQW

AlGaAs 1 wave

40 periods A10.3G%.7As/ Al0.9G%.I As linearly graded barriers

26 periods M0.3G%.7ASI A0.9G%.1 As linearly graded bamers

Shin, Ju, Shin, Ser, Kim, Lee, Kim and Lee [1996]

TEOML-111-UU

3 x8nm GaAdAlo 2 Gao,8 As MQW

A10.5Ga0.gAS 1 wave

30.5 periods AlAs/Alo,;?GaoAS

26 periods Alo,9G~.lAs/ Al0.2G%.8AS step-graded and b-doped bamers

Schnitzer, Fiedler, Grabherr, Jung, Reiner, Zick and Ebeling [ 19961; Jager, Grabherr, Jung, Michalzik, Reiner, Weigl and Ebeling [1997] continued on next page

<

m 8

t

m m r

e

0 m

Table 1, continued Abbreviation

Active region

Spacer cavity

Bottom reflector

Top reflector

Reference

HMOL-UTA

3x6m In02Ga0,8As/GaAs MQW

GaAs 1 wave

26 periods AlAs/GaAs

1 period AlAs/GaAs + 4-6 periods ZnSe/CaFz

Huffaker, Deppe and Rogers [1994]; Huffaker, Shin,Deng, Lin, Deppe and Streetman [19941

HMOL-II-UTA

8nm Ing2Gag,8As/ GaAs SQW

26 periods AlAdGaAs

5-1 periods MgFIZnSe

Huffaker, Graham, Deng and Deppe [19961

DCCOL-SNL

DO-DBRL-USC

Alo.96Ga004As+ 75 GaO 25 As/ GaAs superlattice + 1/2 wave GRIN AlGaAs 3 ~ 8 h0,2G%,8AS m GRIN AlGaAs MQW 1 wave 2x6.5nm h 0 . 2 GW.8As (h, = 0.99 pm) GaAs MQW

AlGaAs-GaAshGaAs 1 wave

~

E $

z

38 periods 18 periods Lear, Mar, Choquette, Kilcoyne, GaAs/Alo,92Gao,o8As GaAdAl0.92Ga0.08As Schneider and Geib [1996]; Choquette, uniparabolically uniparabolically Chow, Crawford, Geib and Schneider graded barriers graded barriers [19961 7 periods

5 periods

AlAs oxide/GaAs

AlAs oxide/GaAs

MacDougal, Yang, Bond Lin, Tishinin and Dapkus [I9961

continued on next page

8m

'i E

% e

r

-

Table 1, notes

M

Abbreviations PITSEL, proton-iniplanted top-surface-emitting laser HMIDL, hybrid-mirrors in-diffused laser HMUML, hybrid-mirrors undercut-mesa laser HMSCL, hybrid-mirrors strain-compensated laser HMPIL, hybrid-mirrors proton-implanted laser HMCCL. hybrid-mirrors current-confinement laser BCCL, buried current-confinement laser TEML, top-emitting mesa laser BEML, bottom-emitting mesa laser DF, double-fused TBEML, tunable bottom-emitting mesa laser AGTL. air-gap tunable laser DMEWL, dielectric-mirrors etched-well laser SMEWL, semiconducting-mirrors etched-well laser UMEWL, undercut-mesa etched-well laser HMML, hybrid-mirrors mesa laser PIBEL, proton-implanted bottom-emitting laser BEOML, bottom-emitting oxidized mesa laser TEOML, top-emitting oxidized mesa laser HMOL, hybrid-mirron oxidized laser DCCOL, double-current-confinement oxidized laser DO-DBRL, double-oxide DBR laser

N

ATT, AT&T Bell Laboratories UNM, University of New Mexico SNL, Sandia National Laboratories UU, University of Ulm CU, Cornell University SFIT, Swiss Federal Institute of Technology UCSB, University of California at Santa Barbara RIT, Royal Institute of Technology HTC, Honeywell Technology Center BTL, BT Laboratories TUM. Technische Universitat, Munchen UCB, University of California at Berkeley NTT, NTT Opto-electronics Laboratories TIT, Tokyo Institute of Technology SU, Stanford University OFL, Optoelectronics Furukawa Laboratory USC, University of Southern California UTA, University of Texas at Austin KAIST, Korean Advanced Institute of Science and Technology

MQW, multiple quantum well SQW, single quantum well DH, double heterostructure GRIN, graded index A g , bandgap wavelength

Y

B

<

a

M

r

ro

a W r V

182

THERMAL PROPERTIES OF VCSELs

6

3. Effects of Temperature on VCSEL Operation

In this section, temperature-related effects on VCSEL performance are presented, including the temperature dependence of the longitudinal mode spectra, the transverse-mode structure, the threshold current, and the output power. 3.1. TEMPERATURE DEPENDENCE OF THE LONGITUDINAL MODE SPECTRA

In a resonant cavity of length L, filled with a homogeneous material of refractive index n R , the Wavelength &I of a longitudinal mode A4 is given approximately by (neglecting phase shifts at the cavity boundaries):

The rate of longitudinal mode wavelength change with temperature is determined by the temperature sensitivity of the refractive index and by the thermal expansion of the cavity length (Mroziewicz, Bugajski and Nakwaski [ 19911, p. 151):

where aT is the linear thermal expansion coefficient, and )2G is the group index,

The partial derivatives ( d n ~ / a T ) Iand ~ (anR/ak)l, in eqs. (3) and (4) are to be evaluated at constant wavelength and temperature, respectively. The refractive index in semiconducting materials changes approximately linearly with temperature. Near the lasing wavelength of 0.85 pm, a value of ( & Z R / ~ T=)4~x~ K-' can be extracted from the data of Marple [ 19641 for GaAs, and 1x K-' can be extracted from the data of Grimmeiss and Monemar [1971] for AlAs. For InP, a similar value of 3 x K-' near the bandgap can be obtained from the data of Pettit and Turner [1965]. The latter value has been also assumed by Bissessur, Ettinger, Fernandez and Davies [ 19931 for the quaternary InGaAsP materials. These values are about an order of magnitude higher than the product of the group index and the linear expansion coefficient, listed in tables 2 and 3. Hence, the longitudinal mode wavelength shift is determined primarily by the temperature dependence of the refractive index.

111,

4 31

EFFECTS OF TEMPERATURE ON VCSEL OPERATION

183

Table 2 Thermal expansion coefficient a T near 300 K aT [ 10-6PC]

Material

Reference(s)

GaAs

6.449

Swaminathan and Macrander [1991]; Adachi [1985]

AlAs

4.9-5.2

Swaminathan and Macrander [1991]; Adachi [1985]

as

6.4- 1 . 2 ~

Adachi [1985]

InP

4.56

Adachi [1992]

Inl-xGa,As,,Pl,a

4.56+ 1 . 1 8 ~

Adachi [1992]

A,! Ga 1

a

Lattice-matched to InP, with x = 0 . 4 5 2 7 ~ 41 4 . 0 3 11y).

Table 3 Group index nc near 300 K Material

Wavelength nC

Reference

0.85 pm

4.3

Buus [1983]

0.85 pm

4.2

Determined from data given by Casey and Panish [1978]

In0.73Ga0 27A~0.6P0.4~ 1.30pm

4.3

Buus and Adams [1979]

In0.6Ga0.4As0s5P0 15 1.55 pm

4.6

Buus and Adams [1979]

GaAs A1O

a

2Ga0.8As

Bandgap wavelength A,

= 1.3 pm.

Bandgap wavelength 1, = 1.55 pm.

The diode laser cavity is not homogeneous; hence, except for devices where the optical field is very well confined within the active region, eqs. ( 2 ) and (3) only approximately describe the actual situation. For EELS, the refractive index n R should be replaced with the effective index n,R, which averages the refractive indices of the active region and the claddings with proper weights determined by field penetration into the claddings (Thompson [ 19801, p. 1 18). Rather than employing discrete mirrors, most VCSEL designs incorporate distributed Bragg reflectors (DBRs). For a DBR consisting of alternating quarterwave layers of thicknesses and refractive indices dL, nL, and dH, nH, respectively, the center wavelength of its reflectivity spectrum is determined by the optical thickness of the DBR period:

184

THERMAL PROPERTIES OF VCSELs

LS

.

.

"Et

"H

"Et

Fig. 4.Schematic structure of a typical VCSEL cavity, showing the refractive indices of the bottom exit medium (nEb), high- and low-index layers of the bottom DBR (nHb, nLb), spacer medium (n,), low- and high-index layers of the top DBR ( n ~ nLt), ~ , and the top exit medium (nEt), as well as bottom and top phase penetration depths (Lpb, Lpt) and the spacer thckness L,.

The temperature variation of the Bragg wavelength, ADBR, is then described by a formula analogous to eq. (3) (Dudley, Crawford and Bowers [1992]):

where a T 3 L and aT,H are thermal expansion coefficients of the low- and highindex material, respectively. Again, the terms containing ~ T , Land aT,H are much smaller than those containing the variation of the refractive index with temperature, and the dispersion terms are much smaller than Experimentally, dADBR/dT was determined to be 0 . 8 7 U C for a GaAdAlAs mirror centered at 0.98 pm, and 1.1h C for an InFVInGaAsP (bandgap wavelength Ag = 1.15 pm) mirror centered at 1.32 pm (Dudley, Crawford and Bowers [1992]). A typical VCSEL cavity, shown schematically in fig. 4, consists of two DBRs separated by a spacer region which incorporates the active region. The spacer thickness L, (including the optically equivalent active-region thickness) and the phase penetration depths of the top and bottom DBRs Lpt, Lpb, respectively, combine into an effective VCSEL cavity length L e ~ :

k.

Leff

= Ls

+ Lpt + Lpb,

(7)

111,

P 31

EFFECTS OF TEMPERATURE ON VCSEL OPERATION

185

where (Babic and Corzine [ 19921)

with

n

sa = -.

~

nEHa

~

a

(1 1)

Here, A D B R , ~ ( a = t , b ) is the center wavelength of the top (t) or bottom (b) DBR, N , is the total number of quarter-wave layers, n, is the refractive index of the spacer region (assumed to be homogeneous), and qu, pa, and sa are, respectively, the ratios of the lower to higher refractive indices at the interfaces of the incident (I) medium and the DBR, the Bragg mirror layers, and the DBWexit (E) medium. For a top-emitting VCSEL, the exit medium for the top reflector is normally the air, and for the bottom reflector it is the substrate. For a bottom-emitting VCSEL, the top exit medium is usually the metallic contact, and the bottom exit medium is the air. The longitudinal modes in a VCSEL cavity can be found by considering the round-trip phase condition that takes into account propagation in the cavity of the effective length L e given ~ by eqs. ( 7 t ( 11) and the detuning between the mode wavelength, AM, and the center wavelengths of the two DBRs, dDBR,t and kDBR,b. Assuming a homogeneous spacer medium, we obtain the following longitudinal mode resonant condition:

In an ideal VCSEL cavity with a thin (quantum well) active layer and two Bragg mirrors of identical center wavelength, ADBR, the thickness of the spacer region should be chosen such as to select AM =ADBR. In this case, eq. (12) simplifies to an expression hlly analogous to eq. (2):

Note that because of the A M = ADBR condition, only one longitudinal mode can be supported by an ideal VCSEL cavity. In addition, the effective cavity length,

186

THERMAL PROPERTIES OF VCSELs

WI,§ 3

L,R, no longer appears in the resonance condition (13). We see that the optical thickness of the spacer must be an integer multiple of the half-wave &)BR/2. In VCSELs with quantum-well active regions, the additional requirement for the maximum of the standing wave intensity pattern to occur at the center of the spacer limits A4 to even numbers; i.e., nsLs must be equal to an integer multiple of ADBR.The control of the spacer thickness is less critical in VCSELs with bulk active regions (over 400 thick), since the standing-wave intensity maximum will always be located somewhere within the active layer. The mode wavelength shift with temperature in an ideal VCSEL cavity is still given by eq. (3) with the spacer index n,, provided that the spacer material satisfies the following thermal dispersion matching condition:

A

Equation (14) was derived by neglecting the thermal expansion and wavelength dispersion terms in eqs. (3) and (6). Measurements of the temperature variation of the lasing wavelength in GaAdAlGaAs-based VCSELs give the mode sh@s near room temperature in the range of 0.56-0.90 h C (e.g., Van der Ziel, Deppe, Chand, Zydzik and Chu [ 19901, Hasnain, Tai, Yang, Wang, Fischer, Wynn, Weir, Dutta and Cho [ 19911, Tell, Leibenguth, Brown-Goebeler and Livescu [19921, Wipiejewsh, Panzlaff, Zeeb and Ebeling [ 19931, Catchmark, Morgan, Kojima, Leibenguth, Asom, Guth, Focht, Luther, Przybylek, Mullay and Christodoulides [ 19931, Geels, Thibeault, Corzine, Scott and Coldren [1993], Norris, Chen and Tien [1994]) for both pulsed and CW operation. Most of the reported values are very close to the measured center wavelength shift of 0 . 8 7 h C in the GaAs/AlAs DBR mirrors (Dudley, Crawford and Bowers [ 1992]), indicating that condition (14) seems to be satisfied in GaAs/AlGaAs devices. They are also comparable to mode shift rates in GaAslAlGaAs Fabry-Perot EELS, which are typically about 0 . 5 O b C (Mroziewicz, Bugajski and Nakwaski [1991], p. 151). Because of the higher thermal resistance of VCSELs (see table 4), the VCSEL active region heats up with the ambient temperature at a faster rate than in edge emitting lasers. Hence, the observed wavelength shft should be somewhat larger than in edge-emitters. This is indeed the case, whch confirms that eq. (3) describes well the mode wavelength shift in VCSELs. Figure 5 illustrates the lasing wavelength shift with temperature under both pulsed and CW conditions for two PITSELs with the active-region diameter of 16 pm and with different room-temperature values of dM, It is clear that under pulsed conditions, AM( T ) dependence can remain linear over a wide temperature

111,

5 31

187

EFFECTS OF TEMPERATURE ON VCSEL OPERATION

Table 4 Thermal resistance RTH

RTH (WW)

Size (pm)

Structurea

Reference

300

6x6

HMIDL-SFIT

Wiithrich, James, Ganiere and Reinhart [19901

1788

150

PITSEL-ATT-A

Hasnain, Tai, Yang, Wang, Fischer, Wynn, Weir, Dutta and Cho [1991]

1250

l o x 10

deep-etched BEML-UCSB

Coldren, Geels, Corzine and Scott [1992]

1000

l o x 10

shallow-etched BEML-UCSB

Geels, Thibeault, Corzine, Scott and Coldren [I9931

800

150

PITSEL- ATT-A

Wu, Tai and Huang [1993]

920

160

PITSEL-UNM

Lu, Zhou, Cheng and Malloy [1994]

2100

160

PITSEL-UNM

Lu, Zhou, Cheng, Malloy and Zolper [ 19941

3000

11 0

wafer-hsed HMUML-UCSB

Dudley, Babic, Mirin, Yang, Miller, Ram, Reynolds, Hu and Bowers [1994]

233-345

60 0

TEML-UCB

Norris, Chen and Tien [1994]

27&384'

60 0

TEML-UCB

Chen, Hadley and Smith [I9941

2500

7x7

TEOML-SNL

Choquette, Schneider, Lear and Geib [1994]

613

21 0

Au-plated BEML-UCSB

Wipiejewski, Young, Peters, Thibeault and Coldren [ 19951

422 1

4.5 0

Au-plated BEML-UCSB

Wipiejewski, Young, Peters, Thibeault and Coldren [ 19951

800

20 0

BEML-11-TIT

Mukaihara, Hayashi, Hatori, Ohnoki, Matsutani, Koyama and Iga [1995]

1300

100

BEML-11-TIT

Mukaihara, Hayashi, Hatori, Ohnoki, Matsutani, Koyama and Iga [ 19951

3000

60

BEML-11-TIT

Mukaihara, Hayashi, Hatori, Ohnoki, Matsutani, Koyama and Iga [1995]

800

20 0

PITSEL-ATT-A

Morgan, Hibbs-Brenner, Marta, Walterson, Bounnak, Kalweit and Lehman [ 19951

1210

160

TEML-11-NTT

Ohiso, Tateno, Kohama, Wakatsuki, Tsunetsugu and Kurokawa [1996]

660

26 0

TEML-11-NTT

Ohiso, Tateno, Kohama, Wakatsuki, Tsunetsugu and Kurokawa [1996]

2550

2x2

HMOL-UTA with SQW

Huffaker and Deppe [1996]

2280

3.5 x 3.5

HMOL-UTA with SQW

Huffaker and Deppe [1996] continued on next page

188

WI, 0 3

THERMAL PROPERTIES OF VCSELs

Table 4, continued RTH

1615

a

(KMI)

Size (p.m)

Structurea

Reference

5.4 0

BEOML-11-UCSB with SQW

Akulova, Thibeault, KO and Coldren [ 19971

For an explanation of the abbreviations see table 1. Pulsed operation. For CUITent range from Ith,cw to 31th,,,, where Ith,,-, is the threshold current for cw operation

range, although different regimes characterized by different values of dAMIdT may exist. For example, the slope of AM(T)changes at 300 K for the “865 nm mode” device. Under the CW conditions, AM(T)deviates significantly from a linear behavior, mostly because of varying degree of heating occurring at the CW threshold current, at which the data were taken (cf. 0 3.2). Knowledge of the wavelength shift rate with temperature can be used 875 870 865 860

855 850 845

840

100

200

300

400

500

600

SUBSTRATE TEMPERATURE (K) Fig. 5. Wavelength shift with temperature for two 16p.m PITSELs with different lasing mode positions relative to a common gain peak (847nm at 300K), measured under pulsed (loons, 0.1% duty cycle) and CW conditions (after Lu, Zhou, Cheng, Malloy and Zolper [1994]). The devices were placed on a temperature-controlled stage and the substrate temperature is assumed to be equal to the stage temperature.

111, § 31

EFFECTS OF TEMPERATURE ON VCSEL OPERATION

189

conveniently to estimate the average active-region temperature, necessary to evaluate the thermal resistance of the device (defined by eq. 49, below). This approach has been employed widely, giving the estimates of thermal resistance in various VCSELs between 233 and 4221 WW (see table 4). These values are much higher than typical thermal resistances of EELS, ranging between 20 and 50KIW (e.g., Joyce and Dixon [1975], Manning [1981], Hughes, Gilbert and Hawrylo [ 1985]>. 3.2. TEMPERATURE DEPENDENCE OF THE THRESHOLD CURRENT

The lasing threshold in all diode lasers is determined by the balance between optical gain and losses. With increasing temperature, two main effects take place: the energy gap in 111-V semiconductors shrinks (see table 5), and the carrier density distribution within each band broadens, with its peak shifting further into the band. The bandgap shrinkage is dominant, and the net result of these two opposing effects is that the gain-peak shifts towards the longer wavelengths. In addition, the gain peak is lowered at a given carrier concentration. Thus, even if the optical losses were to remain unchanged, the threshold current would have increased with temperature, since higher current density is required to maintain the same gain level. In addition, optical losses do increase with temperature, since the higher density of carriers necessary to maintain the required gain level results in increased free-carrier absorption in the active region. Somewhat less important, at least at room temperature and above, is an increase in free-carrier absorption that can occur in passive layers, caused by temperature-dependent impurity ionization. These considerations assume implicitly that all current flowing through the diode laser results in radiative transitions. However, only a fraction of electron-hole pairs recombines radiatively. Nonradiative processes can also be temperature dependent, either directly, or indirectly via increased carrier density necessary to balance the optical loss. An example of such a process is Auger recombination that increases rapidly with carrier density. Another mechanism of carrier loss is leakage along a shunt path away from the active region or straight over the quantum-well active region. The current density associated with the latter process can be described using a simple expression, analogous to the standard current-voltage equation for a p-n junction (Scott, Corzine, Young and Coldren [ 19931):

190

THERMAL PROPERTIES OF VCSELs

Table 5 Temperature dependence of bandgap E , and peak gain g,,,

3.95+1.15xf 3.34.0 h,i InO.73Ga0 27AS0.6P0.4

3.25-3.82 iJ

1.48'

InO.6G%.4As0.85PO15

3.75'

1.22

In1-xGaxAs,P

4.w.3yi

1

The values of aE,/dT are not independent of temperature. Using the results of Thurmond [ 19751 T ( T + 408)/(T+204)2 eVPC. for GaAs, we obtain dEg/aT=-5.405x Bandgap wavelength I , = 1.3pm. Bandgap wavelength I , = 1.55 pm. Lattice-matched to InP, with x=0.4527/(1-0.031 ly). Swaminathan and Macrander [1991], p. 16. Adachi [1985]. g Yan and Coldren [ 19901. Lautenschlager, Garriga and Cardona [1987]. ' Adachi [1992]. J Dutta and Nelson 119821. Extracted from theoretical curves reported by Stem [1973]. Extracted from theoretical curves reported by Dutta and Nelson [1982]. a

'

where E g is~the local bandgap energy in the cladding (barrier) region (dependent on temperature), AF,, is the local separation of quasi-Femi levels in the active region (dependent on both carrier concentration and temperature), ks is the Boltzmann constant, and T, is the local active-region temperature. For indexguided bottom-emitting mesa lasers, deeply etched just through the active region, the parameterjo was fitted to 8x lo3kA/cm2 (Scott, Corzine, Young and Coldren [ 19931). Another implicit assumption is that the laser operates at the gain-peak wavelength, and that the lasing wavelength follows the gain-peak shifts with temperature. This is true only when the spacing between the longitudinal modes in the lasing cavity is small, as in conventional Fabry-PCrot lasers. If the spacing is large or if some additional frequency-selective elements are used, the gainpeak wavelength will not coincide with the lasing wavelength. This situation occurs in distributed-feedback (DFB) and distributed-Bragg-reflector (DBR) EELS, and is characteristic of all VCSELs. Depending then on the sign of the

111,

5

31

EFFECTS OF TEMPERATURE ON VCSEL OPERATION

191

initial detuning from the gain peak at room temperature, the lasing wavelength can approach the gain peak or depart from it. In the former case, which takes place when the room-temperature lasing wavelength is offset from the gain peak towards the longer wavelengths, the increase in threshold current described in the first paragraph is partially compensated by the simultaneous shift of the lasing wavelength towards the gain peak. Conversely, if the room-temperature lasing wavelength is on the short-wavelength side of the gain peak, temperature variation of the threshold current will be accelerated. According to Chow, Corzine, Young and Coldren [ 19951, many-body Coulomb interactions between carriers are not canceled completely by plasma screening. This leads to a decrease in the wavelength dependence of the threshold carrier concentration and consequently, a greater tolerance to the changes in the resonance/gain overlap with temperature on the low-temperature side of the threshold minimum. It is clear that the complex interplay between all these mechanisms can result in a variety of different patterns of threshold current evolution with temperature. Yet, it is a common practice to describe the temperature dependence of the threshold current using the Arrhenius-type relation, Ith(T) = Ith(300 K) exp

(

-

O;

")

f

with T in Kelvin and the characteristic temperature To used as a measure of temperature sensitivity of the threshold current. With TO constant, eq. (16) usually approximates the actual threshold variation within a certain temperature interval. More generally, TO is itself a function of temperature, and for an arbitrary Ith(T) dependence it can be defined simply as

In spite of its limited applicability to VCSELs, eq. (16) is the only analytical form in which the temperature dependence of the VCSEL threshold current was reported. Not surprisingly, measured values of To for VCSELs are scattered widely and sensitive to device structure (see tables 6a,b). They range from very high (practically infinite) values in devices where the threshold current remains practically constant within a certain temperature range (cf. Geels, Thibeault, Corzine, Scott and Coldren [1993]), through moderate values of 130-150K, (see, e.g., Uchiyama, Ohmae, Shmizu and Iga [1986] and Iga, Koyama and

192

THERMAL PROPERTIES OF VCSELs

[IK 0 3

Table 6a Characteristic temperature To for pulsed operation

To (K) Range (K)

Size (Fm)

Structure

Reference

200

100-220

20-30 0

DMEWL-TIT-A

Uchiyama, Ohmae, Shimizu and Iga [ 19861

70

220-263

20-30 0

DMEWL-TIT-A

Uchiyama, Ohmae, Shimizu and Iga [ 19861

210

283-363

15 0

PITSEL-ATT-A

Hasnain, Tai, Yang, Wang, Fischer, Wynn, Weir, Dutta and Cho [1991]

47.5

223-253

8x8

UMEWL-UCSB

Wada, Babic, Crawford, Reynolds, Dudley, Bowers, Hu, Merz, Miller, Koren and Young [1991]

26.8

253-339

8x8

UMEWL-UCSB

Wada, Bahic, Crawford, Reynolds, Dudley, Bowers, Hu, Merz, Miller, Koren and Young [1991]

24

203-253

20 x 20

wafer-fused HMUML-UCSB

Wada, Bahic, Ishikawa and Bowers [ 19921

47

203-298

8x8

wafer-fused HMUML-UCSB

Wada, Babic, Ishikawa and Bowers [1992]

67

200-300

11 0

wafer-fused HMUML-UCSB

Dudley, Babic, Mirin, Yang, Miller, Ram, Reynolds, Hu and Bowers [I9941

Kinoshita [ 1988]), to negative values in devices that are detuned towards the longer wavelengths (cf. fig. 8). Thus, in stark contrast to Fabry-Perot-type EELS, the To parameter in VCSELs becomes more of a design parameter (Tell, Brown-Goebeler, Leibenguth, Baez and Lee [ 1992]), than a material- or structure-related characteristic. Since the arbitrary temperature sensitivity of a VCSEL threshold current can be obtained in principle, this opens up a possibility of designing temperatureinsensitive VCSELs, with infinitely large To. Some interesting examples of such constructions were demonstrated not only for ambient room temperatures (Young, Scott, F.H. Peters, Thibeault, Corzine, M.G. Peters, Lee and Coldren [ 19931, Kajita, Kawakami, Nido, Kimura, Yoshikawa, Kurihara, Sugimoto and Kasahara [ 1995]), but also for cryogenic conditions (Lu, Luo, Hains, Cheng, Schneider, Choquette, Lear, Kilcoyne and Zolper [ 19951, Ortiz, Hains, Lu, Sun, Cheng and Zolper [1996], Goncher, Lu, Luo, Cheng, Hersee, Sun, Schneider and Zolper [ 1996]), and elevated temperatures (Dudley, Ishikawa, Babic, Miller, Mirin, Jiang, Bowers and Hu [ 19931, Catchmark, Morgan, Kojima, Leibenguth,

111,

D

31

EFFECTS OF TEMPERATURE ON VCSEL OPERATION

193

Table 6b Characteristic temperature 7'0 for CW operation

To (K)

Range (K)

Size (pm)

Structure

Reference

115

288-323

15 0

SMEWL-ATT

Tai, Fischer, Seabury, Olsson, Huo, Ota and Cho [1989]

120

25&300

100

DMEWL-TIT-B

Extracted from data reported by Koyama, Kinoshita and Iga [1989]

210

293-363

150

PITSEL-ATT-A

Hasnain, Tai, Dutta, Wang, Wynn, Weir and Cho [1991]

130

283-323

150

PITSEL-ATT-A

Hasnain, Tai, Yang, Wang, Fischer, Wynn, Weir, Dutta and Cho [1991]

330

213-298

100

PITSEL-ATT-B

Tu, Wang, Schubert, Weir, Zydzik and Cho [1991]

80

328-348

100

PITSEL-ATT-B

Tu, Wang, Schubert, Weir, Zydzik and Cho [I9911

40

283-353

20 x 20

PIBEL-ATT

Von Lehmen, Banwell, Carrion, Stoffel, Florez and Harbison [1992]

60 a

328-393

10x10

shallow-etched BEML-UCSB

Geels, Thibeault, Corzine, Scott and Coldren [1993]

156

293-3 18

10 0

PITSEL-SNL

Schneider, Choquette, Lott, Lear, Figiel and Malloy [I9941

a

Determined using the active-region temperature.

Asom, Guth, Focht, Luther, Przybylek, Mullay and Chnstodoulides [ 19931, Shoji, Otsubo, Matsuda and Ishikawa [1994], Lu, Zhou, Cheng, Malloy and Zolper [ 19941, Morgan, Hibbs-Brenner, Marta, Walterson, Bounnak, Kalweit and Lehman [ 19951, Ohiso, Tateno, Kohama, Wakatsuki, Tsunetsugu and Kurokawa [1996]). It should be emphasized that the temperature T usually used in eq. (17) in experimental determination of TO,is the ambient (stage or heat sink) temperature (cf. 0 4.2). Under low-duty-cycle pulsed conditions, it coincides with the activeregion temperature. The pulsed and CW values of To should be very similar if the active-region temperature is used instead of the ambient temperature, except for weakly guiding or weakly antiguiding VCSEL structures (cf. 0 3.3) in which lateral nonuniformity of CW temperature profiles plays an important role. It should be noted that the temperature sensitivity of the threshold current depends on the size of the active region. Larger devices usually exhibit lower values of the characteristic temperature TO,which results from poorer overlap

194

THERMAL PROPERTIES OF VCSELs

0.875

0.870

0.865

0.850 0.845

0.mo

Fig. 6 . Temperature effects on the gain-peak wavelength Amax and the vertical-cavity mode = ADBR point corresponds to the minimum threshold wavelength ADBR in a PITSEL. The A,, current. TO is negative in the region when ADBR is offset towards longer wavelengths relative to I,,,. After Tell, Brown-Goebeler, Leibenguth, Baez and Lee [ 19921.

between the gain and photon density profiles (Wada, Babic, Ishikawa and Bowers [ 19921) as well as from worsening thermal properties, with increasingly onedimensional heat flow. The shft of the gain spectrum in VCSEL structures can be determined experimentally by fabricating Fabry-Perot EELs from VCSEL wafers and measuring the lasing wavelength shift with temperature. Typical measured values of dA,,,/dT for GaAdAlGaAs VCSELs are (3.2-3.4) &‘C (e.g., Geels, Thibeault, Corzine, Scott and Coldren [ 19931, Tell, Brown-Goebeler, Leibenguth, Baez and Lee [1992], Scott, Corzine, Young and Coldren [1993]), which is greater than the analogous value of - 2 . 5 b C in conventional EELs. The accelerated shift of gain-peak wavelength is probably caused by heating associated with higher series resistance of multilayer EELS incorporating horizontal Bragg reflectors and by reabsorption of amplified spontaneous emission in the vertical direction, enhanced by high-reflectivity horizontal Bragg

111, § 31

195

EFFECTS OF TEMPERATURE ON VCSEL OPERATION

100

150

200

250

300

350

400

450

SUBSTRATE TEMPERATURE (K) Fig. 7. Temperature dependence of the CW lasing threshold currents for three 16 pm PITSELs with different lasing mode positions relative to a common gain peak (847nm at 300K). The minimum threshold current occurs close to the temperature where the gain peak and lasing mode wavelengths coincide (after Lu, Zhou, Cheng, Malloy and Zolper [1994]).

mirrors. On the other hand, as illustrated in fig. 6 , the observed mode wavelength shift (cf. 9 3.1) is 4-5 times slower than the gain-peak wavelength shift. For each VCSEL design there exists an optimal temperature, for which an ideal alignment between the gain spectrum and the vertical-cavity resonant mode takes place. This usually coincides with the condition for minimum I,h(T), provided optical and electrical losses are not changing drastically around this temperature. Figure 7 presents typical I t h ( T ) curves for three PITSELs with different lasing mode positions at 300K (Lu, Zhou, Cheng, Malloy and Zolper [1994]). The larger the room-temperature detuning of ADBR towards the longer wavelengths, the higher the temperature at which the threshold current reaches minimum. Figure 8 shows the temperature dependence of TO extracted from the data of fig. 7 using eq. (17). It is clear that TO can be considered constant only over a very limited range of temperatures, away from the vertical asymptote. The asymptotes in fig. 8 correspond to the minima of Ith(T) curves in fig. 7. The temperature dependence of the threshold current in VCSELs, with a minimum occurring near the temperature at which ADBR and A,, are aligned, resembles that of frequency-selectiveEELS, such as DFB or DBR lasers. In edgeemitting DFB lasers, however, Ith(T) characteristics may be more complicated,

196

THERMAL PROPERTIES OF VCSELs

Heat-sink

temperature, THS [K]

Fig. 8. Temperature dependence of the characteristic temperature TOextracted from the data of fig. 7.

with multiple minima corresponding to various transverse modes (cf. Aiki, Nakamura and Umeda [ 19761). 3.3. TEMPERATURE DEPENDENCE OF TRANSVERSE-MODE PROPERTIES

Compared to EELS, transverse-mode properties of VCSELs are considerably more complex. Transverse modes in VCSELs are determined by an intricate interplay of gain profile, absorption, diffraction, reflection, spatial filtering, builtin index waveguiding, and temperature distnbution (Scott, Young, Thibeault, Peters and Coldren [1995], Michalzik and Ebeling [1995]). Depending on the particular VCSEL structure, some of these effects can be more important than others, but rarely can a single mechanism be identified as the dominant one. In addition, VCSEL cavities can usually support many transverse modes (Valle, Sarma and Shore [ 1995a]), especially in large-diameter ( 2 20 pm) devices or in strongly index-guided structures. Hence, mode competition and singletransverse-mode control are important problems, even though the device operates in a single longitudinal mode. The difficulties with transverse mode control are best illustrated by the small value (4.4mW) of the fundamental-mode CW output power achieved so far (Lear, Schneider, Choquette, Kilcoyne, Figiel and Zolper [1994]). Coupling to an

111,

P

31

EFFECTS OF TEMPERATURE ON VCSEL OPERATION

197

external cavity, which is a proven technique for increasing the single-mode output power in EELS, has until now resulted in single-mode CW powers of only 22.4mW (Hadley, Wilson, Lau and Smith [1993], Wilson, Hadley, Smith and Lau [ 19931). Spatial filtering with the aid of a graded-index lens has recently extended this value to only 4.5 mW (Koch, Leger, Gopinath, Wang and Morgan [ 19971). This contrasts with significantly larger single-mode powers of 36-1 00 mW in the same external-cavity VCSELs injected with 100 ns pulses at 0.1% duty cycle (Hadley, Wilson, Lau and Smith [1993], Wilson, Hadley, Smith and Lau [ 19931). The large difference between the pulsed and CW results clearly illustrates the strong effect of heating on the transverse mode structure. One of distinct features of VCSELs is that their threshold currents for pulsed and CW operations are often very similar to each other and that the CW threshold can sometimes be even lower than the pulsed one (e.g., Hasnain, Tai, Yang, Wang, Fischer, Wynn, Weir, Dutta and Cho [1991]). This is caused by the socalled thermal lensing effect, which focuses the radiation in regions of higher temperature because of temperature-dependent refractive index. Thermal lensing can strongly influence the transverse mode structure in socalled gain-guided (or carrier-guided) diode lasers without a built-in waveguide in the p n junction plane, where lateral confinement of the optical field occurs via a combination of gain guiding and index antiguiding (e.g., Nash [1973], Cook and Nash [1975], Thompson [1980] (chapter 6.4.1.), Hadley, Hohimer and Owyoung [1987], Cherng and Osinski [1991]). A similar situation takes place in VCSELs with no built-in lateral waveguide, for example in PITSELs (e.g., Hasnain, Tai, Yang, Wang, Fischer, Wynn, Weir, Dutta and Cho [ 19911, Chang-Hasnain, Harbison, Hasnain, Von Lehmen, Florez and Stoffel [ 19911, Zeeb, Moller, Reiner, Ries, Hackbarth and Ebeling [ 1995]), and, partially, in TBEMLs (Michalzik and Ebeling [19931). In low-duty-cycle short-pulse operation, when thermal effects are negligible, the confinement of the optical field in the radial direction occurs via a combination of gain guiding, carrier antiguiding, absorption, diffraction (Babic, Chung, Dagli and Bowers [ 19931, Jansen van Doorn, van Exter and Woerdman [ 1995]), and spatial filtering at the top contact. Carrier antiguiding tends to defocus the optical field, which leads to large diffraction losses (Hasnain, Tai, Yang, Wang, Fischer, Wynn, Weir, Dutta and Cho [ 19911, Dutta, Tu, Hasnain, Zydzik, Wang and Cho [ 19911). Under CW conditions, the active-region heating results in a non-uniform, bell-shaped temperature distribution which peaks in the active region and falls off in the radial direction (cf. fig. 16 in $5.1.1). Since (dnR/dT)/n is positive (see §3.1), the thermal contribution to the refractive index also peaks in the active region, causing the thermal lensing effect. Nonuniformity of the temperature distribution

198

THERMAL PROPERTIES OF VCSELs

[IK § 3

becomes more pronounced with increasing pumping current, to the point where real-index guiding associated with temperature profile may become dominant, resulting in tighter focusing of the optical field. Experimental observations of narrowing near-field patterns of the bell-shaped fundamental transverse mode with increasing current in PITSELs (Chang-Hasnain, Harbison, Florez and Stoffel [ 19911, Chang-Hasnain, Harbison, Hasnain, Von Lehmen, Florez and Stoffel [1991]) have been confirmed by the calculations of Michalzik and Ebeling [ 19931. Thermally-induced waveguiding improves the overlap between the optical field and the gain region and reduces the diffraction loss. In the intermediate regime of relatively long pulses (over 100ns long), the build-up of thermal waveguide leads sometimes to anomalously long time delays in lasing. When the pulse amplitude is only slightly larger than the CW threshold current, the time delay before the onset of lasing can be as long as several ps (Hasnain, Tai, Yang, Wang, Fischer, Wynn, Weir, Dutta and Cho [ 19911). The time delay rapidly decreases with increasing current and reaches the “normal” level of 25 ns when the pumping current amplitude exceeds the pulsed threshold value. A quantitative analysis of the thermal lensing and its effects on the time delay in PITSELs is given by Dutta, Tu, Hasnain, Zydzik, Wang and Cho [1991]. At the beginning of a low-amplitude pulse, threshold losses are higher than the modal gain, and therefore lasing action cannot start. As the device starts to heat up, thermal lensing begins to play a more and more important role, steadily reducing diffraction losses. The observed time delay is simply equal to the time necessary to create a sufficiently strong thermally-induced waveguide. A similar phenomenon has been observed by Prince, Patel, Kasemset and Hong [ 19831 in carrier-guided stripe-geometry EELS and was explained in terms of thermallycontrolled dynamic evolution of waveguide properties. While a thermally-induced waveguide is beneficial from the point of view of lowering the CW threshold current, it can at the same time facilitate excitation of higher-order transverse modes. At higher currents, a stronger real-index thermal waveguide supports a larger number of high-order modes which can then compete with the fundamental mode. Therefore, the dynamic switch-on response of VCSELs initially contains sometimes a single-lobe profile (the fundamental transverse mode), and after the time (dependent on pumping conditions) necessary for the thermal waveguide to build up transforms into a double-lobe profile (the first-order transverse mode) (Yu and Lo [1996], Buccafusca, Chlla, Rocca, Feld, Wilmsen, Morozov and Leibenguth [ 19961). Once the thermal waveguide is established, the main mode competition mechanism switches to spatial hole burning (e.g., Vakhshoori, Wynn, Zydzik, Leibenguth, Asom,

111, D 31

EFFECTS OF TEMPERATURE ON VCSEL OPERATION

199

Kojima and Morgan [1993], Scott, Geels, Corzine and Coldren [1993], Scott, Young, Thbeault, Peters and Coldren [1995], Valle, Sarma and Shore [1995b], Law and Agrawal [1997]). The fimdamental transverse mode is localized in the central part of the active region, therefore the stimulated recombination associated with this mode takes place primarily in this area. This depresses the local carrier density and the gain in the central part of the laser cavity, reducing the modal gain for the fimdamental mode, while allowing the carriers to build-up near the edges of the active region and increasing the modal gain of higher-order doughnut-shaped transverse modes. Eventually, the laser ends up operating in multiple transverse modes. In VCSELs with no built-in lateral waveguide or with weak index-guiding, spatial hole burning can cause a positive-feedback phenomenon known as self focusing, which further reinforces the real-index thermal waveguide (see Wilson, Kuchta, Walker and Smith [ 19941). A depression in the carrier concentration produces a local increase in the refractive index which can further intensify the stimulated emission, locally reducing the carrier concentration, and so on. A similar effect can also arise from thermal lensing via absorption of the emitted light within the core of thermal waveguide. A depression in carrier density, similar to that caused by the spatial hole burning, can also be caused by nonuniformity of current injection in devices with annular contacts (see Osinski, Nakwaski and Varangis [1994]). The two effects can be distinguished by observing the spontaneous emission profile, which is proportional to carrier density distribution. Nonuniformity due to current spreading should also manifest itself below the lasing threshold, while spatial hole burning can occur only above threshold. The only experiments reported so far by Vakhshoori, Wynn, Zydzik, Leibenguth, Asom, Kojima and Morgan [ 19931 and by Wilson, Kuchta, Walker and Smith [ 19941, involving measurements of the spontaneous emission profile above threshold, indicate that at moderate currents the spontaneous emission profile has a doughnut shape. Further above threshold, when higher-order transverse modes become excited, the carrier density profile is sensitive to details of the laser structure. For example, smooth profiles were observed by Wilson, Kuchta, Walker and Smith [ 19941 in their bottom-emitting VCSELs with circular top contacts, indicating that spatial hole burning was the dominant effect. In contrast, Vakhshoori, Wynn, Zydzik, Leibenguth, Asom, Kojima and Morgan [ 19931 observed dark spot near the center of the spontaneous emission profile even high above threshold, which suggests that nonuniform injection was the main effect in their top-emitting devices with annular contacts. Nonuniform current injection, with current crowding near the edges of the

200

Fig. 9. Current density profiles in the p n junction plane for a 16 vm etched-well GaAsiAlGaAs VCSEL (structure DMEWL-TIT-B) with parameters given by Nakwaski and Osinski [ 19931.

active region in VCSELs with annular contacts (e.g., Nakwaski and Osinski [ 199lb], Nakwaski, Osinski and Cheng [ 19921, Wada, Babic, Ishikawa and Bowers [ 19921, Scott, Geels, Corzine and Coldren [ 1993]), also favors excitation of higher-order transverse modes. To some extent, the nonuniform injection is counterbalanced by the ambipolar radial diffusion of carriers prior to their recombination (Sarzala and Nakwaski [ 1997]), which makes the local gain distribution more uniform than the current-density distribution (e.g., Wada, Babic, Ishikawa and Bowers [1992], Chong and Sarma [1993], Sarzala, Nakwaski and Osinski [1995]). Nevertheless, the gain profile still has an onaxis minimum and is better matched to the lugher-order transverse modes than to the fundamental one. This effect is usually not strong enough to suppress the fundamental mode near threshold, but gains in importance with increasing pumping level, as the current crowding becomes more and more intense (see fig. 9). The better overlap of the gain profile with the optical field of the higherorder modes may then become sufficient to overcome the higher difiaction loss suffered by these modes. Nonuniformity of the current density in devices with annular contacts can be largely leveled out if the heterointerfaces between the alternating layers of

111, 5 31

EFFECTS OF TEMPERATURE ON VCSEL OPERATION

20 1

Bragg mirrors are not graded (Michalzik and Ebeling [ 19931). This, however, increases the series resistance (the specific heteroresistance between p-GaAs and p-AlAs layers can be as high as 2.5 x Qcm’) and results in more intense Joule heating. Built-in index antiguiding can be used as a mechanism for extending the single-transverse mode operation range, since the higher-order modes suffer a higher diffraction loss penalty than the fundamental mode (e.g., Chang-Hasnain, Wu, Li, Hasnain, Choquette, Caneau and Florez [1993], Wu, Chang-Hasnain and Nabiev [ 19941, Wu, Li, Nabiev, Choquette, Caneau and Chang-Hasnain [ 19951, Yoo, Chu, Park, Park and Lee [1996]). The negative index step between an equivalent index of the DBR reflector and the surrounding high-index medium can be made as large as 0.18 (Wu, Chang-Hasnain and Nabiev [1993]), hence the antiguide cannot be affected significantly by the much smaller (one-two orders of magnitude) positive index step due to radial temperature profile. So far, however, this approach has had only limited success. While the near-field intensity profiles in bottom-emitting passive-antiguide-region InGaAdAlGaAs VCSELs show no symptoms of thermal lensing, spatial hole burning or self-focusing, the maximum single-transverse-mode power is still limited to only 1.2 mW (Wu, Chang-Hasnain and Nabiev [ 19931, Wu, Li, Nabiev, Choquette, Caneau and Chang-Hasnain [ 19951). Introducing higher doping at the active region perimeter to increase free-carrier losses and using low-reflectivity ring contacts on the top VCSEL reflector were other mode selection methods postulated by Morgan, Guth, Focht, Asom, Kojima, Rogers and Callis [1993]. Another class of temperature-insensitive waveguide involves strong index guiding (Jewell, Scherer, McCall, Lee, Walker, Harbison and Florez [1989], Geels, Corzine, Scott, Young and Coldren [1990], Geels and Coldren [1990,1991], Shimizu, Babic, Dudley, Jiang and Bowers [1993], Dudley, Babic, Mirin, Yang, Miller, Ram, Reynolds, Hu and Bowers [1994], Young, Kapila, Scott, Malhotra and Coldren [1994], Yoffe, van der Vleuten, Leys, Karouta and Wolter [1994], Yoo, Park and Lee [1994]). Compared to strongly antiguiding VCSELs, index-guiding structures have a serious disadvantage of lowering the threshold of higher-order-mode excitation (Chang-Hasnain, Orenstein, Von Lehmen, Florez, Harbison and Stoffel [ 19901, Schroder, Grothe and Harth [ 19961). Consequently, fundamental-transverse-mode operation can be maintained only over a very limited current range near threshold. 3.4. TEMPERATURE DEPENDENCE OF THE OUTPUT POWER

Because of the thermal lensing effect (see §3.3), the threshold current for

202

THERMAL PROPERTIES OF VCSELs

[III, § 3

the CW operation in PITSELs is often distinctly lower than for the pulsed one. The external differential quantum efficiency, which is the laser parameter proportional to the slope of the light-current characteristic above the threshold current, is, however, much higher for the pulsed operation (Hasnain, Tai, Yang, Wang, Fischer, Wynn, Weir, Dutta and Cho [ 19911). Similarly, maximum available output power and the operating current range are enhanced under pulsed conditions. From fig. 7, we may conclude that in order to obtain efficient CW highpower operation of VCSELs at room temperature, their cavity-mode positions at this temperature should be on the long-wavelength side of the gain spectrum. Although such lasers may have higher threshold currents for pulsed operation than those with aligned cavity-mode and gain-peak positions, nevertheless their CW threshold currents will be lower because of the active-region heating (provided the cavity-mode and gain-peak wavelengths are matched at the activeregion temperature). However, since the active-region temperature depends on the driving current, the conditions for minimum threshold current would in general be different from the conditions for maximum output power. This is illustrated in fig. 10, showing the temperature dependence of lightsurrent (LI ) characteristics of a PITSEL device with a room-temperature detuning of the cavity mode by 18nm towards the longer wavelengths. The CW lasing threshold for this device, shown also in fig. 7, has a minimum at 350 K. All LI characteristics display a typical thermal roll-off behavior, indicating that over the wide temperature range of 9 0 K 4 0 0 K , the output power Po,, is thermally limited. The maximum output power is determined primarily by the temperature variation of the peak gain (see table 5) and by changes in the external dierential quantum eflciency Qd. The latter can be extracted from fig. 10 using the following formula (Agrawal and Dutta [ 19931):

where e is the electron charge, h is Planck’s constant, and c is the speed of light. Equation (18) implies that all output power from a top-emitting VCSEL is collected through the top mirror. Figure 11 shows the temperature dependence of v d , calculated by applying eq. (18) for the device of fig. 10. The raising part of L-I curves, not too far above threshold, is used to determine T]d. A(T) is obtained from the data of fig. 5, taking the CW lasing wavelength for the “865 nm mode” and extrapolating down to 90 K. In any case, the wavelength variation represents only a very small correction to v d determined from the slope efficiency dPoUtldlwith a constant

203

EFFECTS OF TEMPERATURE ON VCSEL OPERATION

15

10

5

0

0

10

20

30

40

50

I(-) Fig. 10. Temperature evolution of the light-current characteristics for a 16pm PITSEL shown in fig. 7 as having minimum CW threshold at 350 K (mode wavelength 865 nm at 300 K) (after Lu, Zhou, Cheng and Malloy [1994]).

Temperature, T

[K]

Fig. 11. Temperature dependence of the differential quantum efficiency q d for the device in fig. 10, using either the stage temperature THS(dotted line) or the active-region temperature TA (solid line) as the argument in vd(T).

204

THERMAL PROPERTIES OF VCSELs

[III,

p3

wavelength. The logarithmic scale in fig. 11 is chosen to verify whether an exponential formula analogous to eq. (16),

with a constant characteristic temperature T,, would hold for VCSELs, as it does for EELs (e.g., Papannareddy, Ferguson and Butler [1987]). Note that sometimes a simpler approximation is used (Wipiejewski, Peters, Thibeault, Young and Coldren [1996]): T]d(T) = vmax(l- AT/Tmax),where Tma, is the characteristic roll-over temperature of the laser. The two curves in fig. 11 correspond to results obtained using either the stage temperature THS (dotted line) or the active-region temperature TA (solid line) as the argument in qd(T). TA is estimated using the wavelength shft between the CW and pulsed operation shown in fig. 5, again extrapolating down to 90 K. Under pulsed conditions, the “865 nm mode” device has two clear regimes of linear wavelength shift, characterized by d&/dT = 0.41 h C for 90 < THS< 300 K, and d&/dT = 0.59 &T for 300 < THS<400K. The deviation of the CW wavelength from the pulsed one gives then the following estimates of the active-region temperature at cw threshold: T ~ = 1 9 8 Kat T H S = ~ O KT, ~ = 2 3 9 Kat THS=~OOK,T ~ = 3 1 0 K at THS=~OOK, T ~ = 3 5 4 Kat T H S = ~ ~ Oand K , T ~ = 4 0 3 Kat T ~ s = 4 0 0 K . If the stage temperature THS is used (dotted curve in fig. l l ) , T, decreases steadily with THS (T, = 241 K for 90 < THS< 200 K, T, = 129 K for 200 < THS< 300 K, T, = 104 K for 300 THS< 350 K, and T, = 63 K for 350 < THS < 400K). Hence, it is clear that if THSis used as an argument in T]d(T), eq. (19) can be used only over a very small temperature range. Conversely, using the active-region temperature TA (solid curve in fig. 1l), produces two regimes of behavior (T,=91 K for 198 < TA <310K, and T,=62K for 310 < TA <403K) that coincide with the two regions of constant pulsed-wavelength shift rate (cf. fig. 5). It is interesting to note that the value of T, = 91 K agrees very well with Tsp= 93 K (valid for 300 K TA< 440 K) determined experimentally by Chen, Hadley and Smith [1994], where Tsp is the characteristic temperature for the spontaneous emission efficiency variation with temperature. This indicates that reduction of internal quantum efficiency, rather than changes in optical losses, is the main effect responsible for the temperature variation of the external quantum efficiency qd. The increase in slope of d&/dT and the reduction in T, from 91K to 62K around T ~ = 3 1 0 Kis most likely caused by the onset of carrier leakage over the heterobarriers (cf. eq. 15). In EELs, opening of an additional current

<

<

111, P 41

FUNDAMENTALS OF THERMAL MODELING OF VCSELs

205

leakage path would have unavoidably reduced the value of TO.In VCSELs, this process is obscured by strong variations of Ith(T) caused by the detuning between the gain peak and the cavity mode (cf. G3.2). Nevertheless, fig. 7 does display an asymmetry between the low- and high-temperature branches of I t h ( T ) , indicating a faster growth at high temperatures. Additional corroborating evidence is provided by fig. 8, where, compared with low-temperature behavior, TOreaches lower values on the high-temperature side of the asymptotes. While TO for the “865 nm mode” device (curve 3 in fig. 8) continues to fall at 400 K, the “852 nm mode” device (curve 2 in fig. 8) reaches a floor of TO= 62-65 K. Although more studies of t h s subject are needed to eliminate the possibility of a serendipitous coincidence, the close matching of the high-temperature values of TO and T,, for the same device is remarkable. Moreover, a T,, of 62 K is very close to the high-temperature value o f To = 60 K reported by Geels, Thibeault, Corzine, Scott and Coldren [1993] for a BEML, indicating that the phenomenon responsible for the low To values at high temperatures may be the same in different VCSEL structures. Carrier leakage over heterobarriers could indeed represent such common mechanism, since both PITSELs and BEMLs incorporate quantum-well active-regions.

0

4. Fundamentals of Thermal Modeling of VCSELs

Thermal modeling of VCSELs is a very involved task because of: (i) a multilayer structure (sometimes containing over a hundred layers), often of nonplanar or buried-type designs, with many heterojunctions, graded layers, strained layers, single or multiple quantum wells, superlattices, oxide and oxidized layers, barriers, etched wells or mesas, etc. (cf. fig. 3); and (ii) the necessity of taking into account many mutual nonlinear interactions between particular physical phenomena; i.e., thermal, electrical, optical and sometimes also mechanical and photochemical processes, taking place during laser operation (cf. fig. 15, below). In analytical modeling of VCSELs, their complex structure imposes the necessity of solving the equations describing their operation separately for each homogeneous region. Mutual correlation of those partial solutions, with the aid of proper boundary conditions (see, e.g., Nakwaski and Osinsh [1994]), should then be carried out. This method is very time-consuming, and therefore simplified approaches are sometimes adopted, transforming the inhomogeneous laser structure made of isotropic materials into an equivalent homogeneous, but anisotropic medium (see, e.g., Osinski and Nakwaski [ 1993a1,Osinski, Nakwaski

206

THERMAL PROPERTIES OF VCSELs

[IK 0 4

and Varangis [ 19941, Osinski, Nakwaski and Leal [ 19941, Sarzala, Nakwaski and Osinski [ 19951). The second of the above mentioned factors, i.e., mutual interactions between individual physical processes occurring during a diode laser operation, whose incomplete review will be presented in fig. 15, obliges us to use a self-consistent approach. A flow chart of a typical self-consistent diode laser simulation will be presented in fig. 14. 4.1. HEAT CONDUCTION EQUATION

For the transient-state condition, the heat conduction equation in a medium of cylindrical symmetry is of the following form (Oezisik [ 19801):

which for a uniform medium with temperature-independent k, K , and g may be reduced to

d2T 1 d T d2T 1 dT + -- + -= -dr2 r 6’r &* K dt

-

-

1 -g(r,z). k

In the above equations, k and K stand for the thermal conductivity and the thermal diffusivity, respectively, and g ( r 7 ) is the distribution of the volume density of heat generation (in W/cm3). In the steady-state case, the first term of the right-hand-side of eqs. (20) and (21) vanishes. Examples of roomtemperature values of thermal conductivities for several semiconductor materials commonly used in VCSEL structures are listed in table 7. To solve the thermal conduction equation (20) (or 21), it is necessary to formulate an appropriate set of boundary conditions. VCSEL structures are usually axially symmetric, so the first boundary condition is written as follows (cf. fig. 12):

In a typical packaging scheme, the outer surfaces of VCSELs are exposed to interaction with an external ambient medium (usually air). In principle, this may result in heat transfer to the external ambient medium via the direct contact of air particles with the laser walls and subsequent diffusion and convection processes. In natural convection, the flux of heat from the surface

111,

4 41

FUNDAMENTALS OF THERMAL MODELING OF VCSELs

207

Table 7 Thermal conductivity k Material

k (W/mK)

Reference

GaAs

44

Adachi [1985]

AlAs

91

Adachi [1985]

Al, Gal -x As

100/(2.27+28.83~-3 0 . 0 ~ ~ ) Adachi [1985]

InP

68

1n0.73Gao,27As0.6P0.4 a

5.5

Nakwaski [ 19881

In0.63G%.37As0.79P0.21

5.1

Nakwaski [I9881

1n0.6G%.4As0.85P015

5.1

Nakwaski [ 19881

a

Bandgap wavelength I, = 1.3 p. Bandgap wavelength I, = 1.45 km.

Kudman and Steigmeier [I9641

Bandgap wavelength I, = 1.55 km.

Laser crystal Contact and solder

G Fig. 12. Boundary conditions of the thermal conduction equation for VCSELs.

is approximately proportional to the 514 power of the difference between the surface temperature and the ambient temperature (Carslaw and Jaeger [ 19881, p. 21). In VCSELs, however, considerable temperature increase occurs only in

208

THERMAL PROPERTIES OF VCSELs

[In § 4

the vicinity of the active region, buried deep inside the device. The temperature at the outer surfaces is usually only slightly higher than that of the surrounding medium. Thus, heat transfer through the surfaces is very small compared to a very efficient heat conduction through the laser heat sink and can be neglected completely. Consequently, the outer surfaces of the VCSEL chip can be assumed to be adiabatic (cf. fig. 12):

El ay

=o, side walls

T( a

=o,

upper walls

Heat flux conduction through the VCSEL base (see fig. 12) produces a temperature increase inside the contact and solder layers (ATc) as well as inside the heat sink (ATHs). The corresponding boundary condition can be written as follows: Tibottomwalls

=

Tamb + ATHS iATC,

(25)

where Tambstands for the temperature of the ambient and AT,, and ATc are to be determined and will be analyzed in the next section. 4.2. HEAT-SINK AND CONTACT/SOLDER-LAYER TEMPERATURE INCREASE

Exact determination of the temperature increase ATHSin the heat sink usually requires an involved three-dimensional thermal analysis. An estimate of A T H ~ can, however, be made using the concept of the effective diameter, D E , for the heat flux entering the heat sink from the device, as proposed by Nakwaski and Kontkiewicz [ 19851 for cylindrically symmetric devices. This approach is based on an assumption that a uniform heat flux entering the heat sink through an effective circular thermal contact of diameter D E will cause exactly the same average temperature increase AT,, in the heat sink as the actual nonuniform heat flow. D E can be determined using the following expression: -1/2

DE = 2 lim ( x k a ~d d , T) Ad,,1 0

H

,

where RTH is the laser thermal resistance (defined by eq. 49) and Ada is the thickness of a hypothetical additional layer (of a thermal conductivity k,) between the heat sink and the device.

111, P 41

FUNDAMENTALS OF THERMAL MODELING OF VCSELs

209

Once D E is determined, the average temperature increase ATHSinside the heat sink can be found using the following formula (Carslaw and Jaeger [1988], p. 216):

where QT stands for the total power of the heat flux entering the heat sink, and k H S is the thermal conductivity of its material. In the derivation of eq. (27), the heat sink is assumed to be much larger than the laser chip. To determine ATc, the thicknesses of all the intermediate layers between the laser chip and the heat sink (i.e., contact and solder layers, and for some VCSEL structures also oxide layers) should be averaged over the effective diameter DE. The average temperature increase within these layers ATc can then be expressed as

where the summation should be carried out over the intermediate layers, d,,a,are their averaged thicknesses and ki are their thermal conductivities. 4.3. HEAT SOURCES

Diode lasers are supplied with the electric power which, in processes of current spreading, carrier recombination as well as emission and absorption of radiation, is transformed into output power and heat in a way illustrated in fig. 13. In VCSELs, the following heat generation mechanisms should be taken into consideration: volume and barrier Joule heating, nonradiative recombination of carriers, absorption of spontaneous radiation, and absorption of laser radiation. Distributions of these heat sources are directly associated with distributions of carrier concentration, current density, and spontaneous and stimulated radiation intensities, hence they are influenced by current spreading, carrier difision and waveguiding mechanisms. Their magnitudes are influenced in turn by such device and material parameters as internal quantum efficiency, threshold current density, threshold carrier concentration, absorption coefficients, electrical resistivities - all nonlinearly dependent on temperature. The power balance in a semiconductor laser may be presented in the following form: P

=

U I = (Upn+IRs)I

210

THERMAL PROPERTIES OF VCSELs

=

Z2Rs

Joule heating

+UpnIth(l - %p)

nonradiative recombination at threshold

+UpnIthr]sp(l-f

+UpnIth %pf

T)

T

absorption of spontaneous radiation inside the active region at threshold escape spontaneous radiation at threshold

+upn(f -zth)(1 - %> (1 - Vsp)

nonradiative recombination for I > I t h

+Upn(I - Ith)( 1 - 7i)Qsp

absorption of spontaneous radiation inside the active region for I >Jth

-f T)

w,§ 4

(29)

+Upn(I - Ith)( 1 - ri)qspf T escape spontaneous radiation for I > f t h +Upn(I - Ith)(qi - Qd)

absorption of stimulated radiation outside the active region

laser output power. +Upn(I - Ith)vd In the above equations, P is the electric power supplied to the laser, U is the total voltage drop inside the device, Up, is the voltage drop at the p-n junction, I and I t h are the operation current and the threshold current, respectively, vSpr Q,, and v d are the quantum efficiencies: internal for spontaneous emission and for lasing radiation as well as differential external for lasing, respectively, and f T is the radiative transfer coefficient for spontaneous emission (cf. 5 4.3.1). All heat sources in semiconductor lasers are associated with current densities, carrier concentrations or radiation intensities, which are nonuniform. Therefore these heat generation processes are nonuniform not only along the z axis (because of different structure layers of different heat yields) but also along the r axis inside all homogeneous layers. With the exception of the Joule barrier heating, all heat generation processes represent volume heat sources. They are, however, often treated as flat (planar) heat sources located in the middle of their generation regions. Such approximation can deteriorate the accuracy of the calculated temperature profiles, especially in the close vicinity of the heat source, so it may be used only when the heat sources are relatively thin.

111, § 41

FUNDAMENTALS OF THERMAL MODELING OF VCSELS

r-l

21 I

Supply Power

Current Spreading

Recombination

Joule Heating

7

J \

Stimulated

Spontaneous

output

Heat

Fig. 13. Power balance in diode lasers.

4.3.1. Active-region heating

Nonradiative recombination of carriers takes place within the active region, where carriers of both kinds (electrons and holes) can meet each other. This process is associated with the eficiency of radiative recombination, which for an ideal injection is simply equal to the internal quantum eflciency. Below the threshold, i.e., for j < j t h , where j and j,, stand for the operation current density and the threshold current density, respectively,most of the electric power supplied to the active region is transformed into heat because the internal quantum efficiency for the spontaneous emission qsp is much less than unity. Then the volume density g A (in W/cm3) of active-region heat generation may be expressed as

212

[IK § 4

THERMAL PROPERTIES OF VCSELs

where d A stands for the active region thickness (cumulative thickness in the case of multi-quantum-well active regions). This expression should be modified slightly if we take into account absorption of some part of spontaneous radiation in the active region and radiative transfer of the remaining part out of this region:

where f T stands for the radiative transfer coefficient (Kobayashi and Furukawa [1975], Nakwaski [1979]):

with (33)

acr= arctan (d&)

and where nR1 and nm are the refractive indices of the cladding and the activeregion materials, respectively. Above the laser threshold, an analogous equation reads as follows (Kobayashi and Furukawa [ 19751, Nakwaski [ 19791): gA =

Upn(1 - r ~ s p f[~j t )h + ( j - j t h ) ( l - ail1 dA

>j t h ,

(34)

where qi is the internal quantum efficiency of the stimulated emission. Whenever intense heating takes place, it is important to remember that the threshold current density j t h is not a constant device parameter, but is itself temperature dependent. As the pumping current density j increases, so does the active-region temperature, and therefore j t h is also current dependent. To emphasize this, Scott, Geels, Corzine and Coldren [1993] have introduced the concept of a current-dependent effective threshold current density jth,e 3j t h ( j ) . Alternatively, we could write j t h = j t h ( TA),where TA is the average active-region temperature. For high reliability, the quality of semiconductor laser materials must be very good. Consequently, in most cases the internal quantum efficiency for stimulated emission q, is very close to unity (Petermann [1991]). Thus, eq. (34) reduces to:

111, P 41

FUNDAMENTALS OF THERMAL MODELING OF VCSELs

213

In the case of proton-bombarded VCSELs, e.g., in PITSELs, this part of the spontaneous radiation, which is leaving the active region, is mainly absorbed in the closest vicinity of the active region; i.e., in surrounding it highly absorbing (of high absorption coefficient a ) areas exposed during their fabrication to a stream of protons. Thicknesses (= a-') of these new heat sources are very low. Therefore, it is quite a good approximation to assume that these absorption events take place also inside the active region. Then, the radiative transfer coefficientf T should be put equal to zero in all the above expressions. Saturation of the voltage drop Upn(r) at the p-n junction above the lasing threshold (e.g., Sommers [1971], Paoli [1973]) should also be taken into account. It does not simply mean that Upn(r)is taken as a constant distribution for all currents above the threshold, because an increase in the pumping is followed by an increase in the active-region temperature, which results in an increase in the threshold current. Therefore, for a given value of the pumping current, the saturated profile of the voltage drop at the p-n junction should correspond to an actual active-region temperature increase. In laser structures, where diffusion of minority carriers within the active region before their recombination (radiative or nonradiative) plays an important role, i.e., in lasers without radial carrier confinement mechanisms, it is more justified to associate the above heat generation with carrier concentration distribution rather than with a current density profile. Each act of nonradiative recombination is followed by heat generation of energy equal to about hv, where h is the Planck constant, and v is the laser radiation frequency. Generally, especially in lasers with quantum-well active regions, this energy may be different than the energy eUpn,where e is the unit charge. Then this heat generation consists of two processes - carrier thermalization and carrier recombination, whose sum must give the supply energy eUpn.Even if they are separated in space, they both occur inside or very close to the active region. Therefore we may neglect their separation. Equation (35) will be then modified to the following form:

where PA stands for the total effective threshold power generated (mainly nonradiatively) inside the active region, defined as

Jo

214

THERMAL PROPERTIES OF VCSELs

[IK 0 4

and NA,, is the total carrier number composing the effective threshold within the active region:

with rs the structure radius and &h,e the threshold effective carrier concentration (associated withj,h,,). In the above, we assume that all the heat generation inside the active region is distributed uniformly over NA,, recombining carriers. 4.3.2. Absorption of laser radiation Absorption of laser radiation is associated with generation of heat of a volume density gabs:

where a is the absorption coefficient (different in various layers) for the laser radiation and pintis its internal density inside the resonator:

Note that according to the suggestions of Petermann [1991], the internal quantum efficiency for stimulated emission is taken equal to unity in the above equations. 4.3.3. Absorption of spontaneous radiation In contrast to a stimulated radiation, spontaneous radiation is always emitted isotropically in all directions. Some part of its vertical component is reflected at boundaries between the active region and the cladding layers as well as from the resonator mirrors and is effectively absorbed within the active region, which was already taken into account in 4 4.3.1. The in-plane emission, on the other hand, can be amplified significantly by stimulated processes within the active region (Onischenko and Sarma [ 19971). Spontaneous radiation reaches sometimes distant regions of the laser. Its absorption may, therefore, occur in many different places. For that reason, a distribution of heat generation associated with this absorption is usually difficult to determine, unless the active region is surrounded by highly absorptive areas, as in PITSELs (8 4.3.1).

111, P 41

FUNDAMENTALS OF THERMAL MODELING OF VCSELs

215

4.3.4. Joule heating

In all layers, a current flow is followed by generation of the volume Joule heating gJ: gJ =j 2 p

[w/cm31,

(41)

where p stands for the electrical resistivity (in Qcm). A current flow through a potential barrier as contacts and heterobarriers is in turn followed by a generation of the Joule heat of a surface density qB:

where R B is the specific contact resistance (in Qcm2) of the barrier. 4.4. SELF-CONSISTENT APPROACHES

The thermal conductivity, k, of a semiconductor material is a temperaturedependent parameter. This dependence is especially important for relatively high temperature increases because of its strong nonlinear behavior. It may be easily taken into account with the aid of the Kirchhoff transformation (Carslaw and Jaeger [1988], p. 11):

Then all of the calculations are carried out for the transformed temperature 0; i.e., as if the thermal conductivity were constant. These temperature profiles should afterwards be recalculated for the temperature-dependent thermal conductivity case, using the inverse transformation. In eq. (43), TR stands for the reference temperature. Usually we assume it to be equal to the lowest temperature inside a semiconductor medium; i.e.,

The detailed form of the reverse transformation depends on a functional dependence k( T ) in a temperature range of interest. At temperatures around and

216

[IIL ii 4

THERMAL PROPERTIES OF VCSELs

over room temperature, for example, the thermal conductivity of GaAs may be expressed as (Amith, Kudman and Steigmeier [ 19651): kGaAs(T)

=

0.44 . (300/T)1.25

[W/cm K],

(45)

and that of InP as klnP(T) =

[

1.47 + ( T - 30O)l-I 117

[W/cm K].

Equation (46) was obtained on the basis of fig. 1, published by Kudman and Steigmeier [1964]. Introducing successively eqs. (45) and (46) to eq. (43), we get the inverse transformation formula for GaAs in the following form:

[

T = T , 1-

kGaAs(TR)@

528

and that of InP as T

=

(&s]

=

[

T, 1 -

]

0(5) 11/4

1200 300

,

(47)

[

128 + (TR- 128)exp k I y R / O ]

In all the above equations, temperature should be put in Kelvin. Thermal difhsivities K of semiconductor materials are also dependent on temperature. It is not, however, possible to take into consideration at the same time the temperature dependencies of both these thermal material parameters; i.e., k(T) and K ( T ) ,using a transformation analogous to that presented above. Therefore, in detailed analytical transient thermal analyses, i.e., when both the above parameters should be included, another method of calculation, namely the so-called staircase approach, is recommended. For each time step, At, temperature profiles are determined using values of k and K found in a previous calculation step, starting from an initial temperature of the entire structure equal to that of the ambient (Tamb). There is still another temperature-dependent term in the thermal conduction equation (20) - including the volume power density of a heat generation; i.e., g . This is because many material parameters (such as electrical resistivities, refractive indices, absorption coefficients) and device parameters (such as threshold current and quantum efficiencies),which influence the heat generation, are strongly dependent on temperature. The above may be included in the model using the self-consistent approach, when in successive iteration loops of the

111, P 41

FUNDAMENTALS OF THERMAL MODELING OF VCSELs

217

Q START

temperature

Determine all temperaturedependent New average temperatures spreading

No Carrier diffusion

profiles

sources

Kirchhoff transformation

Linear

Determine reference temperature

conduction

*

STOP

Fig. 14. Flow chart of the thermal-electrical self-consistent calculations in VCSELs.

calculation values of the above parameters determined in the previous loop are used. Self-consistency is assumed to be reached when differences between results of calculations obtained in two consecutive loops are below given limits. Strictly speaking, not only material and device parameters but also distributions of current densities and carrier concentrations within the whole laser structure are dependent on current temperature profiles. This is because the current spreading and the carrier diffusion phenomena are temperaturedependent processes. Therefore in more exact thermal analyses of VCSELs, the thermal-electrical self-consistent procedure is recommended (fig. 14), in which mutual interactions between thermal and electrical processes in the laser are included. Even more exact is the thermal-electrical-optical self-consistent approach in which optical processes, with their mutual interactions with both the thermal and electrical processes, are also taken into consideration. The full picture of mutual

218

THERMAL PROPERTIES OF VCSELs

Fig. 15. Mutual interactions between thermal, electrical, and optical processes in semiconductor lasers.

interactions between all these processes is shown in fig. 15. In VCSELs with strained active regions, additionally mechanical processes should be included.

5

5. Comprehensive Thermal Models of VCSELs

The key parameter used in all simplified treatments of steady-state thermal problems in diode lasers is the thermal resistance RTH (in WW), defined as to the total the ratio of the average active-region temperature increase dissipated thermal power QT (Joyce and Dixon [1975], Manning [1981]):

It should be noted that although the thermal resistance is a very useful parameter to compare the thermal properties of various devices, it may sometimes give

111,

o 51

COMPREHENSIVE THERMAL MODELS OF VCSELs

219

misleading information. Consider, for example, a device with a very poor electrical contact between the device chip and the heat sink. The resultant heat, generated at the laserheat sink interface, would be very efficiently extracted by the heat sink (assuming it is made of a high thermal conductivity material), so its influence on the active-region heating would be relatively small. However, the heat generated near the heat sink would still contribute to the total heat power QT. Therefore, when eq. (49) is used to determine R T H , such a device would have lower thermal resistance than a well mounted laser with low-electrical resistance contact. Thermal-electrical behavior of VCSELs is described by a coupled system of partial differential equations with complicated boundary conditions. The approaches towards solving these equations can be classified into two major types: analytical and numerical models. In analytical models, the solution is written in the form of an analytical expression, usually at the expense of some approximations imposed by the postulated form of the solution. In numerical models, no functional form of the solution is sought, hence details of the device structure can be rendered more faithfully. However, in contrast to analytical models where the accuracy of the solution can be easily controlled, it is more difficult to verify that the purely numerical solution does not contain significant errors. Details of hitherto known analytical and numerical comprehensive thermal VCSEL models are compared in table 8a and table 8b, respectively. 5.1. COMPREHENSIVE ANALYTICAL MODELS

5.1.I . Multiluyer rudiulEy uniform structures

The first comprehensive approach to the thermal properties of VCSELs was developed by Kinoshita, Koyama and Iga [1987], Iga, Koyama and Kinoshita [1988], and Iga and Koyama [1990], who assumed only a single flat-disk heat source located in the center of the active region, but considered the influence of multilayer device structure on the 2D heat-flux spreading. The heat exchange with the exterior is assumed to take place only through the heat sink, with adiabatic boundary conditions for all remaining surfaces defining the device. For each layer, assumed to be radially uniform, 2D azimuthally symmetric temperature profiles are expressed in terms of infinite series containing the Bessel and hyperbolic functions. The expansion coefficients are found by imposing the boundary conditions of continuity of the temperature and heat flux profiles across the interfaces between the layers. The method is analogous to that proposed originally by Joyce and Dixon [1975] for edge-emitting lasers. The main

Table 8a Analytical comprehensive thermal VCSEL models Ref.

Year

Structure

Method

1

1987

DMEWL

Fourier

2

1991

DMEWL

Green

3

1992

PITSEL

Fourier

4

1995

DMEWL

Green

5

1995

PITSEL

Green

Current spreading

Carrier diffusion

-

-

crude

-

good

-

exact

-

good

-

fair

+ + + +

Structure modeling

Heat sourcesa

NR

SP

ST

+

-

-

+ + + +

+ + + +

+ + -

-

+ + + +

Self-consistency

BJ

VJ -

CJ -

- + + - + - -

k(T) -

+ + + -

Th-El Th-Op El-Op -

-

+ + +

+c

-

+c

-

+C

-

-

-

-

?

-?

i;;

Abbreviations: NR,nonradiative recombination; SP, absorption of spontaneous radiation; ST, absorption of stimulated radiation; VJ, volume Joule heating; BJ, barrier Joule heating at heterojunctions; CJ, barrier Joule heating at the p-side contact. Abbreviations: k(T), temperature-dependent thermal conductivity; Th-El, thermal-elecbical; Th-Op, thermal-optical; El-Op, electrical+ptical. Partly.

a

References (1) Kinoshita, Koyama and Iga [1987] (2) Nakwaski and Osinski [1991a,b, 19931 (3) Nakwaski and Osihslu [1992c, 19941

*I:

(4) Osihski and Nakwaski [1995b] (5) Zhao and McInemey [1995]

2

2

E

Table 8b

e e

Numerical comprehensive thermal VCSEL models

M VI

Y

Ref.

Year

Structure

Methoda

Current spreading

Camer diffusion

-

-

1

1993

UMEWL

FDM

2

1993

TBEML

FEM

3

1994

HMML

FEM

4

1994

TEML

CVM

5

1994

PITSEL

FEM

6

1995

DMEWL

FEM

+ + + + +

7

1995

DMEWL

FEM

8 9

1995

PITSEL

FEM

1996

PITSEL

FEM

a

Structure modeling

Heat sources

Self-consistency

NR

SP

ST

VJ

BJ

fair

+

?

-

-

-

+

-

exact

+

+

+

-

fair

+

+

-

-

fair

-

fair

-

exact

+ + +

+ + +

+ + -

-

-

fair

+

+

-

-

+ +

+ +

exact

+ +

+ +

+ +

+ +

fair

-

CJ -

+ -

+ + +

-

+ + -

+ +

k(T)

Th-El Th-Op El-@

-

-

-

-

+

-

-

-

-

-

-

-

-

-

-

+ +

+

-

+

-

-

-

-

-

+ + - -

+

+

-

-

+ +

Abbreviations: FDM, hte-difference method; FEM, finite-element method; CVM, control-volume method. See table 8a.

-

3 5 %

I

rn

i5

rn ~

-

+

1

270

I

I

xr 0

(fl

%

References

(1) S h m h , Babic, Dudley, Jiang and Bowers [1993] (2) Michalzik and Ebeling [1993] (3) Piprek and Yo0 [1994] (4) Noms, Chen and Tien [1994], Chen, Hadley and Smith [1994], and Chen [1995] ( 5 ) Piprek, Wenzel and Szteflca [1994], Piprek, Wenzel, Wiinsche, Braun and Henneberger [ 19951

(6) Rahman, Lepkowski and Grattan [1995] ( 7 ) Baba, Kondoh, Koyama and Iga [1995a] (8) Sarzaka, Nakwaslu and Osibski [1995] (9) Hadley, Lear, Warren, Choquette, Scott and Corzine [1996]

N

N

222

THERMAL PROPERTIES OF VCSELs

[III,

5

5

limitation of this approach is that it neglects any structural nonuniformity in the radial direction. Consequently, in the case of buried-heterostructure DMEWLs (e.g., Koyama, Kinoshita and Iga [1989], see also fig. 3 and table I), to which it was applied, neither the lateral confining layers nor the dielectric mirrors on the heat-sink side could be accounted for. The model of Kinoshita, Koyama and Iga [1987], Iga, Koyama and Kinosbta [1988], and Iga and Koyama [1990] is not self-consistent; i.e., the effect of calculated temperature profiles on material parameters and heat source efficiencies was not considered. In VCSELs, where heating is much more intense than in EELS, non-self-consistent models can underestimate the severity of the thermal problems. The first self-consistent treatment of thermal problems in VCSELs (Nakwaski and Osinski [ 1991a,b]) was applied to buried-heterostructure DMEWL devices and is discussed in Q 5.1.2. As a matter of fact, this was the very first self-consistent thermal-electrical model applied to any semiconductor laser, including the edge emitters and high-power laser arrays. While the radially nonuniform DMEWL structure is too complex for the model of Kinoshita, Koyama and Iga [1987], Iga, Koyama and Kinoshita [1988], and Iga and Koyama [1990] to give accurate results, the same model can be applied to radially uniform structures, such as PITSELs. The thermal conductivity of highly electrically-resistive regions which funnel the injected current into the active region is practically unaffected by the implantation process (Vook [1964]), which combined with the planarity of the PITSEL structure, makes it particularly suitable to be modeled analytically. We have incorporated an analytical approach similar to that of Kinoshita, Koyama and Iga [1987] in the analysis of PITSELs, as a portion of our comprehensive, thermal-electrical self-consistent model (Nakwaski and Osinski [1992a,c, 1994]), featuring the temperature-dependent distribution of multiple heat sources, and the temperature dependence of material and device parameters. In the analysis, all important heat-generation mechanisms are taken into account, including nonradiative recombination, reabsorption of spontaneous radiation in the active region, free-carrier absorption of laser radiation, volume Joule heating and absorption of stimulated radiation in all the layers, and barrier Joule heating at heterojunctions. These dstributed heat-generation processes are lumped into three uniform flat-disk heat sources, each of the active-region diameter DA= 2 r ~ , located in the centers of the active region and two Bragg mirrors. An analytical solution is obtained for the entire structure separately for each heat source. Using the superposition principle, a cumulative temperature distribution in the whole volume of the device is determined by adding together contributions from all heat sources. Subsequently, a self-consistent solution is found with the

111, Q 51

COMPREHENSIVE THERMAL MODELS OF VCSELs

223

aid of an iteration procedure, taking into account the temperature dependencies of material and device parameters, including thermal conductivities, threshold current, electrical resistivities, voltage drop at the p n junction, free-carrier absorption as well as internal and external differential quantum efficiencies. The flow chart of numerical calculations of this type is shown in fig. 14, where, however, carrier diffusion is included additionally. Note that large temperature variations in VCSELs affects substantially their lasing characteristics,because of strongly nonlinear thermal-electrical interactions, eventually leading to thermal runaway. For each flat heat source situated between the j th and the (J + 1)th layers and for each ith layer, we are looking for the transformed temperature distribution in the following form:

where rs is the structure radius, z; is the coordinate of the top boundary of the ith layer, zi-1
(51)

JO and JI are the zeroth-order and the first-order Bessel functions of the first kind, while aj,,,,,and /?J,i,n are the expansion coefficients to be determined below. Cumulative profiles of the transformed temperature O(r,z)should be recalculated using the reverse transformation (cf. 5 4.4). The aj,i,nand pj,i,, coefficients are determined from the boundary conditions of temperature profiles and heat fluxes at all the boundaries between various layers. For temperature profiles, we obtain from eq. (SO):

where di is the thickness of the ith layer, whereas for heat fluxes we have:

224

THERMAL PROPERTIES OF VCSELs

ki [a,,i,n sinh(K,d;)

+ /?;,;,ncosh(~ndj)]= for i # jA n 2 1,

(55)

where ki is the ith layer thermal conductivity and qi stands for the density of a given heat flux generated at the boundary between the j t h and the ( j + 1)th layers. From the boundary conditions (24) and (25), we have:

a,,l,n= 0 for n

0,

(56)

and

Using eq. (54), we can write

where N is the total number of structure layers and hence, from eq. (52), we get

We now introduce the r,,i,ncoefficients, defined as

Then, taking eqs. (53) and (55) for i f j , we obtain

111, P 51

225

COMPREHENSIVE THERMAL MODELS OF VCSELs

and from eq. (58) we have: r j , N , n = -tanh(K,dN)

for n

> 1.

(63)

Again using eqs. (53) and ( 5 5 ) for i f j , but now for i = 1, and taking advantage of eq. (56), we get:

and

Now we can determine for 2 6 i 6 N all rj,i,nworking inward from r j , ~and ,~ rj,N,,,. To determine all ~ ~ i , i ,we ~ , once more return to eqs. (53) and ( 5 9 , but this time for i = j . After some mathematical manipulation, we find:

with

The remaining aj,i,ncoefficients can be determined from eq. (53), which after using the rj,i,ncoefficients can be rewritten in the following form:

q i + ~=,aj,i,n n [cosh(Kndi)+ rj,i,nsinh(~ndi)] for n

> 1.

(69)

All the P,,i,,coefficients can then be found from eq. (61). Figure 16 shows the pumping current dependence of radial temperature profiles in the midplane of the active region calculated for a 35-pm PITSEL (Zhou, Cheng, Schaus, Sun, Zheng, Armour, Hains, Hsin, Myers and Vawter [1991]). The total number of layers in the simulation, from the highly doped cap layer on the p-type DBR side to the solder contact below the substrate, including the linearly graded interfaces, is 284. Note that the CW threshold current for this device is 10.2mA, hence the lowest profile in fig. 16 shows the temperature profile just above threshold. Superlinear increase in the temperature at the center of the active region (r=O) in response to increased pumping

226

THERMAL PROPERTIES OF VCSELs

. r = DA/2 60

DA = 35 p m

c

40

1: 10.5 mA

30 20 10

0 0

50

100

150

200

250

Radial distance, r [pm] Fig. 16. Radial temperature profiles in the plane containing the active region of a 35pm GaAsiAlGaAsiAlAs PITSEL for various CW pumping currents, calculated using self-consistent thermal-electrical model. Chip diameter DS is 500 pm.

current can be seen clearly. Consequently, the temperature profile becomes increasingly inhomogeneous, with a large temperature step between the center and the edge (r = r A ) of the active region. This results in the creation of a strong thermal waveguide (cf. 5 3.3), with the refractive index step as large as 1 . 4 ~ at I = 50 mA, which corresponds to an index step that would have been obtained if the active region were surrounded by Alo.o2Gao98As rather than GaAs. On the other hand, since the slope dTldr is a measure of the lateral heat flow, it is evident that the importance of 2D heat flow increases with the pumping current. Pumping-current evolution of the relative share of three major heat sources in the same PITSEL device is illustrated in fig. 17. The active-region heating is a dominant heat source near threshold, but gradually the p-type mirror heating takes over, due to its roughly quadratic dependence on the pumping current. The situation reverses again near the thermal runaway limit, where we observe an accelerated increase in relative importance of the active-region heating, caused primarily by nonlinear processes intensifying the nonradiative recombination. Figure 18 illustrates the current dependence of the thermal resistance, RTH, as defined in eq. (49), for PITSELs of the same vertical structure as in figs. 16 and

227

COMPREHENSIVE THERMAL MODELS OF VCSELs

500

400n U

L

0)

3

0

a

I

-

I

I

I

I

I

1

1

1

1

I

-

1: active region 2: P-type mirrors 3: N-type mirrors

-

300-

-

200-

-

100-

-

-

-

-

00

20

40

60

80

100

120

Current [mA] Fig. 17. Yields of three major heat sources in the 35 pm-diameter PITSEL of fig. 16 shown as a function of the pumping current.

L Q,

r !-

-

. . . . . . uniform cylinder model

-

Fig. 18. Pumping-current dependence of thermal resistance RTH for 35 pm-diameter PiTSELs with various electrical series resistances. Curve 2 corresponds to the device simulated in figs. 16 and 17 and reported by Zhou, Cheng, Schaus, Sun,Zheng, Armour, Hains, Hsin, Myers and Vawter [1991].

22 8

THERMAL PROPERTIES OF VCSELs

[IIL 0 5

17. In addition to the experimentally realized device with the series resistance Rs,O= 33 Q (curve 2), we also consider hypothetical devices with lower (curve 1) or larger (curves 3 and 4) series resistances. Corresponding threshold voltages are readjusted using the following equation:

where U p , is the contribution of the p n junction to the threshold voltage, assumed to be independent of the series resistance R,, and determined from the IV characteristic of the 33 Q device. No variation of pulsed threshold current with R, is assumed to take place. The CW threshold, however, does depend on R, due to changing temperature of the active region, although for the devices considered here these changes are very small, primarily due to the high value of TO= 2 10 K (Hasnain, Tai, Dutta, Wang, Wynn, Weir and Cho [1991]) (cf. table 6b, p. 193) assumed in the calculations. Although RTH is usually treated as a constant parameter with a value characteristic of any particular device, (cf. table 4, p. 187), it is clear that due to nonlinear processes it varies substantially with the pumping current (see Nakwaski and Osinski [1992b]). The relatively high values of RTH displayed in fig. 18 are caused primarily by the “junction-up” configuration of PITSELs. The horizontal dotted line represents the thermal resistance calculated using the simplified uniform cylinder model (Nakwaski and Osinski [1992d]). It is clear that this model represents a reasonable approximation only in the linear regime, near the lasing threshold. The effect of the series electrical resistance on the average temperature increase of the active region AT*+”, used in calculation of the thermal resistance RTH,is illustrated in fig. 19. Due to nonlinear processes, the penalty for a too high series resistance of the device increases rapidly with the pumping current. The operating current range of the 100 Q device is nearly half that of the low-series-resistance (20 Q) device. Zhao and McInerney [ 19951 have recently reported an analytical solution of the thermal conduction equation for a GaAdAlGaAs PITSEL volume using the Green’s function approach proposed for surface-emitting LEDs by Nakwaski and Kontkiewicz [1985]. In the model, the complex multilayer VCSEL structure seems to be replaced with an equivalent uniform structure, although the authors did not mention anything about it: in the solution, average (?) values of thermal conductivity and difisivity are used for the entire VCSEL volume. The model would be exact if Green’s function solutions (with unknown expansion coefficients) were assumed separately for each uniform structure layer. Then

229

COMPREHENSIVE THERMAL MODELS OF VCSELs n

Y

Y

.-

E3

e

Y

W

E" W

Y

e p

P

// /

1004

-3 " i0- '

uniform cylinder I

20

I

I

40

I

60

I

' 80

'

model

100

I

'

120

I

i

140

Current, I [mA]

Fig. 19. Pumping-current dependence of the average active-region temperature increase AT,+,, in a 35 pm-diameter PITSEL. Curve 2 corresponds to the device rcported by Zhou, Cheng, Schaus. Sun, Zheng, Armour, Hains, Hun, Myers and Vawter [1991] and simulated in figs. 16 and 17.

the coefficients would be found from continuity conditions at all layer edges for profiles of both temperature and heat flux, similarly as in the approaches proposed for VCSELs by Nakwaski and Osinski [ 1992~1and earlier for EELS by Joyce and Dixon [1975]. As heat sources, Zhao and McInerney considered nonradiative recombination and absorption of spontaneous emission in the active region and volume Joule heating in layers of current spreading. Unfortunately, they did not solve exactly the current-flow process, using a simplified approach with two adjusting parameters of values difficult to estimate. The temperature dependence of all model parameters was neglected. Because of all the above facts, the exactness of the model seems to be very limited. Nevertheless, the model was used later in an interesting analysis of transverse modes in VCSELs (Zhao and McInerney [ 19961). 5.1.2. Multiltryrr rudiutiv nonunifi,rm structures

Most VCSEL structures are either nonplanar, or contain laterally nonuniform layers confining the carriers, defining the waveguide/antiguide, or acting as reflectors. The analytical approach of 4 5.1.1

5.1.2.1. GaAs/AICuAs lusrrs.

230

THERMAL PROPERTIES OF VCSELs

SII, § 5

may only be used for such structures for which the radially nonuniform layers can be replaced with thermally equivalent uniform layers. An alternative analytical approach that takes into consideration lateral nonuniformity without requiring thermal equivalencies in the radial direction has been developed by Nakwaski and Osinski [1991a,b, 19931 and applied to buried-heterostructure . DMEWLs (Koyama, Kinoshita and Iga [ 19891) (see fig. 3 and table 1). First, current spreading between the etched-well substrate and the heat sink is found using approximate analytical formulae (Bugajski and Kontluewicz [ 19821, Nakwaski and Osinslu [19931). Realistic, radially nonuniform, multiple heat sources associated with different layers of the device are considered, each with axially uniform distribution across the layer thickness. The following heat sources are included: the active region, the N-type and the P-type cladding layers and the p-side contact resistance. The device is then divided into two concentric cylinders (internal with 0 < r < D ~ l 2and external with DA/2< r 6 DsI2) such that within each cylinder all layers are radially uniform. While the dividing wall at r=DA/2 is considered to be thermally insulating, prior to finding the solution of the heat spreading problem the heat generated by each source is redistributed between the two cylinders using an electrical analog model (Nakwaski and Osinslu [ 1991al). Due to the smaller size of the inner cylinder, the redistribution of heat within that cylinder, containing the active region, is considered to be more accurate. For each cylinder, the multilayer structure is replaced with a thermally equivalent medium and an analytical solution for the temperature profiles is found for each ith heat source using the Green’s function method in the following form: Region I :

Region I1 :

In the above equations, T R stands for the reference temperature equal to the temperature at the bottom edge of the laser crystal, r A and rs are the radii of the active region and of the laser structure, respectively, j ~ (n,= 1,2,3,. ~ . . ) is the nth zero of the first-order Bessel function of the first kind, cm= n ( m ( m = I, 2,3,. . . ) is the (m + 1)-st zero of the cosine function, z , ~denotes , ~ the z coordinate after the space transformation, and deq,ais its value for the bottom of

i)

111, § 51

COMPREHENSIVE THERMAL MODELS OF VCSELs

23 1

the etched well (the n-GaAsN-AlGaAs interface), both for Region a ( a =I, 11). The coefficients Anm,i,rand Anm,i,~l are calculated using the following formulae:

where keq,a stands for the equivalent thermal conductivity of Region a, is the transformed coordinate of the top of the ith layer in Region a, and gi,eq,js is the equivalent distribution of the ith heat source after its redistribution. In each loop of the self-consistent calculations, the cumulative profiles of transformed temperature are found using the superposition principle by adding together contributions from all heat sources. Then, the actual temperature profiles are determined by performing the inverse Kirchhoff transformation (cf. § 4.4). At the boundary between the two cylinders, the temperature profiles are matched by adjusting the profile of the outer cylinder to the level of the inner cylinder. The adjustment only affects the near vicinity of the boundary, on the side of r >DA/~. A very important feature of the model is its self-consistency (cf. 4.4), which accounts for mutual interactions between thermal and electrical phenomena. In an iterative loop, temperature dependencies of many material and device parameters are considered, including thermal conductivity, electrical resistivity, threshold current, quantum efficiencies, and voltage drop at the p n junction. Also, the temperature dependence of all important heat generation mechanisms is taken into account, including nonradiative recombination, absorption of spontaneous emission, as well as the Joule heating in all layers. The model described here represents the first application of a self-consistent approach to thermal problems in any semiconductor laser, including the edge emitters. We

232

THERMAL PROPERTIES OF VCSELs

n

- self-consistent . . . . . non-self-consistent

-

-

0

10

20

30

40

Radial distance, r [ p m ] Fig. 20. Radial temperature profiles in the plane containing the active region of a 16km buriedheterostructure GaAs/AIGaAs DMEWL for the pumping current I = 5 1 t h ~(1th.p = 38.4 mA), where Ith,P stands for the room-temperature pulsed threshold current, calculated using self-consistent thermal-electrical model (solid line) or taking the output of the first loop of the iterative process as a non-self-consistent solution (dotted line). Chip diameter Ds is 500 Fm.The kink near r = 23 km corresponds to the edge of the outer oxide layer.

have subsequently used the self-consistent approach in all of our comprehensive thermal modeling, including the model described in 6 5.1.1. The importance of self-consistency is illustrated in fig. 20, comparing radial temperature profiles in the active-region of a 16 pm GaAs/AlGaAs DMEWL, obtained using the self-consistent solution (solid line) and taking the output of the first loop of the iterative process as a non-self-consistent solution (dotted line). The device structure is similar to that of Koyama, Kinoshita and Iga [1989], except for an enhanced P-AlGaAs-cladding doping level of 2 x 10l8cm-3 which significantly improves device thermal properties (see Nakwaski and Osinski [ 1991a,c]).A pumping current of 192 mA was assumed, corresponding to 5 times the room-temperature pulsed threshold current. Clearly, at currents significantly above the threshold, the non-self-consistent solution grossly underestimates the active-region temperature increase. Another interesting feature displayed in fig. 20 is the on-axis dip in the temperature profile, which is a direct consequence of nonuniform current injection. Associated with the dip is a thermally induced antiguide. Selfconsistent analysis reported by Nakwaski and Osinski [ 1991a,c,d] and Osinski

233

COMPREHENSIVE THERMAL MODELS OF VCSELs

1-1

3

'=.

U

500

400

DA:

300

1: 5 pm 2: 10 prn 3: 16 prn 4: 20 prn 5: 30 prn 6: 40 prn

100

0

2

4

6

8

10

12

14

16

18

Relative pumping current, Fig. 21. Current-dependence of thermal resistance for buried-heterostructe GaAdAIGaAs DMEWLs of various active-region diameters D A .

and Nakwaski [1992] reveals that the sign of thermal waveguiding in the active region can be controlled by the N-AlGaAs doping level. Increasing the N-AlGaAs doping level beyond the value of N = 7 x 1017cmP3 used in fig. 20, results in improved uniformity of the injected current density. This is manifested initially by flattening of the active-region temperature profile, and eventually by occurrence of a maximum at r=O for N=7xlOI8 ~ m - Freedom ~. to engineer thermal waveguide in the active region is a characteristic feature of all etchedwell VCSELs. Depending on the application, it might be more beneficial to focus the output light into a narrower spot or to spread it over a wider area without changing the active-region diameter. A thermal antiguide can also enhance single-transverse-mode operation (cf. 9 3.3). Figure 21 displays the current dependence of thermal resistance RTH for DMEWL devices with various active-region diameters. Except for their lateral dimensions, the devices have the same structure as the device of fig. 20. Comparison with fig. 18 reveals a device-type-dependent variation of RTH with current. While the RTH(Z)curves increase monotonically in PITSELs, they have distinct minima in DMEWL, particularly for small-size emitters that can operate in CW mode far above threshold (curves 1 and 2 in fig. 21). These seemingly contradictory results can be understood by considering various factors that can influence the evolution of RTH with current.

234

THERMAL PROPERTIES OF VCSELs

WI, § 5

Figures 18 and 2 1 indicate that the thermal resistance in VCSELs is governed by a number of mechanisms that may affect the RTH(I)dependence in opposite ways. Variation of VCSEL thermal resistances with a pumping current is caused by a temperature dependence of thermal conductivities of constituent materials and by a change of intensities of various heat generation processes located in different places of a laser. The former mechanism always increases the value of R T H whereas , the latter one may increase or decrease RTHdepending on the laser structure. This is a reason for a different R T H ( I shown ) in figs. 18 and 21. An increase in the pumping current invariably heats up the device, which in turn reduces the thermal conductivity and increases the thermal resistance RFA associated with every heat source a. This effect is nearly negligible at low currents, but steadily becomes more and more significant at higher currents, as evidenced in fig. 21 by a sudden increase in RTH near the thermal runaway conditions. A more subtle effect is that of the heat source distribution. The thermal resistance of the device is obtained by summing together fractional resistances RFd with weights QJQT determined by the relative shares of corresponding heat sources. If the relative share of heat sources with high fractional resistances RFA increases, the total thermal resistance will have a tendency to increase. As shown in fig. 18, this is obviously the case of PITSELs. However, if the relative share of heat sources with high fractional resistances RFA decreases, the variation of total RTHwiIl depend on which of the two opposite mechanisms prevails: the increase in thermal conductivity or the lower average fractional resistance. It follows from fig. 21 that this more complex behavior is the case for DMEWLs. In a typical VCSEL configuration, where all heat flux is directed towards the heat sink located on the side opposite to the output mirror, the fractional resistances of all heat sources of the same diameter are determined primarily by their distance from the heat sink. Hence, in a junction-up configuration of PITSELs, the P-type Bragg mirror has the largest fractional resistance, while in DMEWLs mounted junction-down the P-AlGaAs cladding layer has the lowest fractional resistance. The Joule heating associated with DBRs or cladding layers is roughly proportional to I* (this dependence would have been exact if current spreading mechanisms and electrical resistivities were independent of temperature) and, at least well below the thermal runaway, tends to grow faster than the active-region heating. Due to relatively high electrical resistivity of p-type semiconductors, the p-type Joule heat sources are the most important ones to consider since they end up having higher weights &/&. From the above considerations it follows that the p-type Joule heat sources in PITSELs, being on the high end of fractional thermal resistances, cause a further increase in RTH

235

COMPREHENSIVE THERMAL MODELS OF VCSELs

..... I

n

t-<

W

400

2

= I

th,cw (TA)

-

U

+- 300c 0) I

3

0

200

-

L

v

G

0

100

200

300

400

500

Pumping current, I [mA] Fig. 22. Graph of effective CW threshold current I,h,,,(TA) variation with pumping current for buried-heterostructure GaAdAIGaAs DMEWLs of various active-region diameters D A . The ambient temperature is 300 K. Device parameters are the same as in fig. 21.

with current. In contrast, the increasing share of low-thermal-resistance P-type Joule heat sources in DMEWLs tends to reduce the total RTHwith current. That is why fig. 21 shows a reduction of RTH, as long as the device is well below the thermal runaway regime where nonlinear effects cause a fast increase in the share of the active-region heating. When no direct current flows through the laser (and, consequently, the activeregion temperature is equal to ambient temperature), the CW threshold can be considered to coincide with the pulsed threshold. With increasing CW pumping current, the active-region temperature rises, hence the CW threshold, I t h , c w ( T A ) , also increases. In order to avoid conhsion with the CW threshold current, Ith,cw, corresponding to the onset of CW lasing action, the current-dependent CW threshold has been termed the effective CW threshold, after Scott, Geels, Corzine and Coldren [1993]. Figure 22 shows the evolution of the effective CW threshold current with pumping current for DMEWLs of various activeregion diameters. If the thermally-induced increase in the effective CW threshold Ith,cw( T A ) is slower than the increase in the pumping current I , at some pumping level the condition Z = I t h . c w ( T ~ ) = I t h , c w will be met and the laser will start CW operation. The onset of CW lasing corresponds to the lower intersection of the I t h , c w ( T A ) curve (solid lines in fig. 22) with the I = I t h , $ w ( T A ) line (dotted line).

236

THERMAL PROPERTIES OF VCSELs

I L I L I I I I I L

100

D,: 80

be

Y

r= Y

W

L %

5 pm

1:

2: 10 p m 3: 16 p m 4: 20 p m

n

60

5: 30 p m 6: 40 p m

40

5

Y

20

0

2

4

6

8

10

12

14

16

18

Relative current, Fig. 23. Fraction of the total pumping current taken by the effective CW lasing threshold Ith,cw(rA) as a function of the total pumping current expressed in pulsed-threshold units, calculated for various values of the active-region diameter DA. Crosses represent pumping conditions at which [I - Ith,cw( TA)] reaches a maximum. The ambient temperature is 300 K. Device parameters are the same as in fig. 2 1.

However, as the pumping current is increased further, Ith,cw(T . ) starts increasing superlinearly, and eventually thermal runaway takes place. The CW lasing action is no longer possible when the second intersection of the Ith.cw(TA)curve with the Z=Ith,cw(T~) line is reached. Thus, fig. 22 shows the entire CW operating range for each device. Within this range, the optical output power is roughly proportional to the excess pumping current [I -Ith,cw( TA)]. For any particular device size, there exists an optimal pumping level such that [I -Ith,cw(TA)] reaches a maximum. Another representation of the operating range (between the onset of CW lasing and the thermal runaway) for room-temperature CW excitation of etched-well VCSELs is illustrated in fig. 23, The value of lth,cw(TA)/I can be interpreted as a measure of the efficiency of converting the supplied electrical energy into the energy dissipated into heat. It can be seen that from the point of view of their efficiency, the lowest-diameter devices display the best behavior. For 5 pm devices, only 10-20% of the pumping power is used to reach the threshold in the pumping current range of 4 1 t h ,through ~ 16Zth,p.Also, both the operating range, expressed in terms of the pulsed threshold current Ith,p, and the relative surplus

111, P 51

COMPREHENSIVE THERMAL MODELS OF VCSELs

23 I

current available over and above the effective CW threshold, are the largest for small-diameter devices. With an increasing active-region diameter, the operating range shrinks systematically, while an ever increasing portion of the pumping current must be consumed to support the CW lasing action. When DA= 40 pm, at least 90% of the pumping power is used up just to attain the threshold. As illustrated by fig. 23, the reduced operating range with increasing active-region diameter is directly associated with intense heating. However, even though the efficiency characteristics are superior for devices with the smallest D A ,their total output power remains small. The points marked by crosses in fig. 23 indicate the pumping conditions such that [Z- Ith,cw(T,)](and, approximately, the optical power) reaches a maximum. It is worthwhile noting that optimal conditions for optical power and overall efficiency do not coincide, and the maximum output power is always reached at a pumping level higher than that yielding a maximum overall efficiency. The self-consistent analytical model described in this section has also been used to optimize the DMEWL device design for high-power low-thermalresistance operation (Osinski and Nakwaski [ 19921). The tradeoff between an increasing pumping power, increasing Ith,cw(T A ) , and a shnnlung operating range as the active region becomes wider, leads to an optimal value of DA= 16 pm. The devices of this size offer the largest surplus current [ Z - I t t , c w ( T A ) ] and, consequently, the largest total output power. Further optimization of the inner structure of 16 pm devices results in the thermal resistance RTH of only 188 K/W at the pumping level of Z = 3Zth,p.The average active-region temperature increase at that pumping level is less than 49 "C. Note that this value of RTHis significantly lower than the lowest limit in table 4 for TEML devices with 60 pm diameters (cf. Norris, Chen and Tien [ 19941, Chen, Hadley and Smith [ 1994]), especially if one recognizes that the rough scaling of RTHwith the active-region area indicates that the thermal resistance of 16 pm devices should be -14 times higher than that of 60 pm devices. 5.1.2.2. ZnGaAsP/ZnP lasers. Compared with short-wavelength devices, thermal problems in all-semiconductor long-wavelength InGaAsP/InP VCSELs are exacerbated by a smaller refractive index step between Id' and InGaAsP (0.2-0.3 vs. -0.5 for AlGaAdAlAs), which requires about a twice larger number of quarter-wave layers in distributed Bragg reflectors (DBRs) to achieve high reflectivities indispensable for low threshold current density. In addition, due to longer wavelength, all layers in the DBRs must be thicker. Thus, it can be expected that thermal behavior of long-wavelength VCSELs may be even more critical for device operation than is the case for GaAs-based VCSELs.

23 8

THERMAL PROPERTIES OF VCSELs

[IK

05

In view of these difficulties, dielectric mirrors are often used in longwavelength VCSELs, in place of semiconducting Bragg reflectors (e.g., Wada, Babic, Crawford, Reynolds, Dudley, Bowers, Hu, Merz, Miller, Koren and Young [ 19911, Uchida, Miyamoto, Yokouchi, Inaba, Koyama and Iga [ 19921, Tadokoro, Okamoto, Kohama, Kawakami and Kurokawa [ 19921, Miyamoto, Uchida, Yokouchi, Inaba, Mori, Koyama and Iga [1993], Baba, Suzuki, Yogo, Iga and Koyama [1993a,b], Baba, Yogo, Suzulu, Koyama and Iga [1993, 19941). This, however, imposes further limitations on the current path, and consequently on the electrical series resistance of the device. Whlle roomtemperature pulsed operation of electrically pumped VCSELs with dielectric mirrors was obtained both at 1.3 pm (Wada, Babic, Crawford, Reynolds, Dudley, Bowers, Hu, Merz, Miller, Koren and Young [1991]) and 1.5pm (Uchida, Miyamoto, Yokouchi, Inaba, Koyama and Iga [ 1992]), so far room-temperature (up to 36°C) CW operation has been very recently achieved only at 1.3 pm (Uchiyama, Yokouchi and Ninomiya [ 19971). An alternative approach was proposed by Dudley, Ishikawa, Babic, Miller, Mirin, Jiang, Bowers and Hu [1993]. They used wafer fusion to integrate GaAs/AlAs mirrors with InPhGaAsP double heterostructures. These lasers did not show any symptoms of device degradation despite a 3.7% lattice mismatch between the wafers. VCSELs fabricated using wafer fusion have demonstrated room-temperaturepulse operation at 1.3 pm (Dudley, Babic, Mirin, Yang, Miller, Ram, Reynolds, Hu and Bowers [1994]) and at 1.5 pm (Babic, Dudley, Streubel, Mirin, Bowers and Hu [1995]), and recently also roomtemperature CW operation at 1.5 pm (Babic, Streubel, Mirin, Margalit, Bowers, Hu, Mars, Yang and Carey [ 19951). The first comprehensive, self-consistent thermal-electrical analytical model of dielectric-mirror etched-well laser operation at long-wavelengths has been reported by Osinski and Nakwaski [1995b]. The model features a realistic, current-dependent distribution of heat sources and incorporates calculation of lasing threshold. The circular planar buried heterostructure VCSELs modeled in this section (see the DMEWL structure in fig. 3a) are similar to those developed by Baba, Suzuki, Yogo, Iga and Koyama [1993a,b] and Baba, Yogo, Suzulu, Koyama and Iga [1993, 19941. The reported external differential efficiency is very low (T]d = 0.16%), and consequently the output power is very small. It is therefore worthwhile to examine the design of these lasers with the goal of identifying the main factors which determine their performance. Nearly all semiconductor layers of the laser are made of InP (cf. fig. 24). Only the active region is manufactured from InGaAsP (Ag= 1.37 pm, where Ag

239

COMPREHENSIVE THERMAL MODELS OF VCSELs

I

diamond heat sink

I

i

~~~~~~~

Ga solder

AulZnlAuITi

I

I-

I

~

/

(SiO,/Si),

air

'

n-InGaAsP

2

N-lnP

]substrate

rW

I Fig. 24. Schematic structure of an InGaAsPRnP circular planar buried-heterostructe dielectncmirrors etched-well laser (after Baba, Yogo, Suzuki, Koyama and Iga 11993, 19941).

stands for the bandgap wavelength) and the cap layer is produced from InGaAs. (The thin n-type InGaAsP layer reduces heat flux penetration of the substrate layer, which is negligible and is neglected in the model). Both the above layers are relatively thin. Their thermal conductivities, however, are many times lower than that of In€', so they greatly influence the heat-spreading process. Heat fluxes penetrate poorly low-thermal-conductivity areas, and when they do penetrate them, they go across these layers along the shortest possible path inside them. Therefore we may assume approximately a one-dimensional (ID) heat flow through both these layers, perpendicular to their long edges. The space transformation was proposed by Nakwaslu [19831 to solve similar problems. The transformation is fully justified for 1D heat flow. Therefore we may apply it to the InGaAsP active region, to the InGaAs cap layer, and also to the dielectric layers of the output mirror (of even lower thermal conductivity), replacing them by appropriate thermally equivalent layers of InP (see fig. 25).

240

THERMAL PROPERTIES OF VCSELs

Transformation of semiconducting layers

3 n

InP InGaAsP

Fig. 25. Thermal equivalent of the central part of the circular buried heterostructure VCSEL under consideration.

After the transformation, the entire bulk of the device can be treated as a homogeneous InP cylinder. The following heat generation mechanisms are included in the model: - nonradiative recombination and absorption of the spontaneous radiation within the active region, - absorption of the laser radiation in all layers of the resonator, - volume Joule heating in all layers of the current path, and - barrier Joule heating at the p-side contact. For each of the heat generation areas, the transformed temperature profile (after the Kirchhoff transformation, cf. $4.4) is calculated for the entire laser structure, and next the cumulative transformed temperature profile is found with the aid of the superposition principle, using the approach similar to that reported for GaAs/AlGaAs lasers by Nakwaski and Osinski [1991a, 19931. The threshold carrier concentration is determined by considering the balance of gain and losses, with the gain coefficient changing linearly with the carrier density. The following loss mechanisms are included: free-carrier absorption in n-InP, intervalence band absorption in p-InP, active-region loss, diffraction loss, as well as absorption and scattering losses in dielectric mirrors. Threshold current is then found, taking account of Auger recombination, carrier loss due

111, § 51

COMPREHENSIVE THERMAL MODELS OF VCSELs

24 1

to trap and interface recombination, and carrier leakage over the heterobarrier. Analytical approximations are used in the model to describe the current spreading in the device. The main goal of the analysis was to determine which parameters have significant influence on device characteristics and to identify means of improving the device performance. In the following, we present a systematic discussion of the role of the most crucial device parameters. Large differences between theoretical and experimental values of front and back mirror reflectivities indicate that the mirrors suffer from scattering loss, absorption, or poor control of layer thicknesses. While improvement of mirror quality may not be easy in practice, it is interesting to see what advantage could be expected from increased mirror reflectivity. From our calculations, we see that improvement in mirror quality leads to a dramatic reduction of the threshold current. This in turn reduces considerably the Joule heating and the Auger recombination, resulting in a reduction of the total generated heat power, the active region temperature increase, and the thermal resistance. At a fixed pumping current, the output power increases substantially. Contrary to experimental results reported by Baba, Suzuki, Yogo, Iga and Koyama [ 1993b], our analysis indicates that the laser would operate in CW mode at an ambient temperature of 5"C, but not at 14°C. This may indicate that the particular device for which near-room-temperature CW operation was achieved may have had superior properties, such as very low scattering loss in the dielectric mirrors. In order for a 12pm device to lase at 14"C, the mirror reflectivities must be increased to at least 99.8% and 99.9% for the front and the rear mirrors, respectively. The device with only a slightly lower front-mirror reflectivity of 99.7% would not lase in CW mode at that temperature. Mirror quality in the device reported by Baba, Suzuki, Yogo, Iga and Koyama [1993b] was therefore most probably higher than that measured typically for otherwise identical devices or Bragg mirrors. In the above VCSELs, dielectric SiO2/Si DBR structures were used as the front mirrors. It is worthwhile to consider how much improvement can be expected by replacing thermally insulating SiOz layers with highly conducting MgO layers, which are almost as good thermal conductors as Inl? In order to account for a lower refractive index contrast between Si and MgO, compared to SiO2/Si system, the number of periods in the front reflector is increased from 6 to 7. It is seen from our results that the above improvement is sufficient, for example, to make the difference between lasing and non-lasing at 14°C in 14 pm devices. Let us now investigate the influence of the active-region diameter on device characteristics. Figure 26 shows that the original 12 pm diameter (Baba, Yogo,

242

THERMAL PROPERTIES OF VCSELs

0

10

20

30

40

Current [mA]

Fig. 26. Light-current characteristicscalculated for etched-well 1.3 Krn InGaAsP/InF' lasers of various active-region diameters D A of 7 periods of MgO/Si quarter-wave dielectric layers.

Suzuki, Koyama and Iga [1993]) is close to optimal. A slight improvement in maximum output power is expected for devices with 10 pm diameter. Altogether, however, the range of the active-region diameter for which the device would lase CW at 14°C is remarkably narrow. Figure 26 also illustrates the importance of nonlinear thermal effects in long-wavelength VCSELs. Thermally reduced roll-off of the light+urrent characteristics is much rapid than that calculated for GaAsiAlGaAs etched-well VCSELs by Osinski and Nakwaski [1995al. In the range of the active-region diameters DA between 8 p m and 14ym, the thermal resistance, RTH,reduces with decreasing DA (e.g., at I = 20 mA, R T H = 520WW for DA= 14 pm, and R T H E 3 5 0 W for DA = 8 pm). This variation is, however, much smaller than the dynamic range of RTH for each device, caused by an increasing share (with increasing current) of the P-InP Joule heat source with low fractional thermal resistance. For example, for DA = 10 pm, RTHN 103OWW near threshold (Z=9mA), and drops down to -18OWW near thermal runaway ( I = 35 mA). Known comprehensive analytical thermal VCSEL models are compared in table 8a (p. 220) taking into account the method used, accuracy of the structure modeling achieved, heat sources included and self-consistencies taken into consideration.

111, 4 51

COMPREHENSIVE THERMAL MODELS OF VCSELs

243

5.2. COMPREHENSIVE NUMERICAL MODELS

The comprehensive analytical approaches discussed in 8 5.1 require extensive computations in order to determine the series expansion coefficients in temperature profiles. In addition, they can only be applied to those VCSELs whose structures are consistent with the formulation of the method. A purely numerical approach to heat conduction problems offers more flexibility in terms of defining the device structure, and is often easier to implement since existing generalpurpose software packages can be utilized. In the following, we discuss the reported numerical models of VCSELs, focusing on essential features defining the model and addressing various assumptions and approximations made by their authors. One potential problem with the application of multi-purpose finite-element packages is a difficulty in obtaining self-consistent solutions of nonlinear thermal problems. Accurate treatment would normally require an iterative approach. T h s can easily become a very complex task, since for each element, local values of device and material parameters would have to be determined using the average local temperatures. Besides, most authors choose to accept a linear solution of the problem, neglecting the temperature dependence of thermal conductivity, which can easily lead to significant errors. Therefore, it is usually recommended to prepare one’s own numerical code for a strictly specified VCSEL structure, enabling an iterative self-consistent approach. The finite difference method (FDM) was used to model thermal behavior of UMEWLs (cf. table 1 and fig. 3) by Shimizu, Babic, Dudley, Jiang and Bowers [1993]. The authors lumped all heat generation processes into the active region and neglected the temperature dependence of both heat generation mechanisms and thermal conductivities, which confines applicability of the model to relatively low operation currents. They also considered the heat sink contribution to the thermal resistance, assuming the heat flux at the laserheat-sink interface to be uniform. A dramatic reduction in heat-sink thermal resistance was obtained for diamond heat sinks compared to the standard copper ones. A similar conclusion was reached by Osinski and Nakwaski [1993b], using the analytical model of PITSELs described in 4 5.1.1. Thode, Csanak, Hotchkiss, Snell and Campbell [ 19951 have reported on the FDM, time-dependent VCSEL model, which requires extensive mainframe computer resources to implement. To shorten the CPU time, the simulation region is limited to a cylinder of 12 pm radius and 6.75 pm height. A uniform mesh is used, therefore 80 uniform regions of average (?) material parameters are defined. The PISCES code was applied to approximate temperature

244

THERMAL PROPERTIES OF VCSELs

[III,

55

profiles. No details about heat sources and modeling thermal structure are given. Piprek and Yo0 19941 used the kite-element method (FEM) to model operation characteristics of the 1.55 bm HMMLs. A single, uniform heat source located at the active region was assumed. For simplicity, each DBR mirror was replaced with an equivalent, uniform but thermally anisotropic medium, whose radial and axial thermal resistivities are calculated using formulae given by Osinski and Nakwaski [ 1993al. This approximation, appropriate when simple analytical formulae are applied, is questionable when a fully numerical FEM solution is sought, as it inevitably deteriorates the accuracy of the numerical solution. This is especially important when layers of very different thermal conductivities are located close to small heat sources, as in the case of VCSELs. Since the FEM modeling can easily handle multilayer structures, it is better to avoid such approximations. More advanced was the model of Piprek, Wenzel and Sztefka [ 19941, prepared for the 0.98 ym InGaAsiAlGaAs PITSELs, where additionally the current spreading effect was considered and all important heat sources were taken into account, including active-region heating, laser light absorption as well as volume, contact and barrier Joule heating and some thermal-electrical-optical interactions. The model was generalized succesfully by Piprek, Wenzel, Wiinsche, Braun and Henneberger [ 19951 for long-wavelength VCSELs. Surprisingly, the very important temperature dependence of the thermal conductivity seems not to have been taken into consideration in these otherwise quite sophisticated models. Nevertheless, this simulation was used to model long-wavelength VCSEL performance characteristics (Piprek, Babic and Bowers [ 19961). FEM was also used by Michalzik and Ebeling [ 19931 in their comprehensive self-consistent thermal model of TBEMLs (cf. table 1 and fig. 3a), which includes the temperature dependence of thermal conductivity. The main emphasis is laid on realistic modeling of current spreading in a rather complicated TBEML structure, achieved with the aid of a 2D resistance network model. The electrical conductivity profile in the proton-implanted region is assumed to have Gaussian tails with different decay constants in the radial and axial directions. Multiple heat sources are considered, including distributed Joule heating (with an exception of the p-type contact Joule heating), heterobarrier heating, absorption of stimulated radiation within the laser cavity, and absorption of spontaneous radiation within the active region. The calculated temperature distribution is then used to define a thermally induced, axially nonuniform waveguide. VCSEL cavity modes are found by applying a one-dimensional transfer matrix method, similar to that developed for graded-index optical fibers

111%§ 51

COMPREHENSIVE THERMAL MODELS OF VCSELs

245

by Morishita [1991]. Carrier diffusion and spatial hole burning effects are neglected. Transverse profiles of cavity modes are found to be sensitive to driving current via the thermal lensing effect (cf. 9 3.3). Calculated modal gains are then used to determine threshold gain, threshold carrier density, and threshold current density, including the effects of detuning between the mode and the gain peak wavelengths (cf. 6 3.2.). The model does not include any feedback between the temperature distribution in the device and the electrical properties (including redistribution of the heat sources), nor between the calculated optical fields and the thermal-electrical phenomena. It was used later by Michalzik and Ebeling [ 19951 to determine CW performance characteristics of PITSELs. Baba, Kondoh, Koyama and Iga [1995a] reported the FEM thermal model of InGaAsP/InP DMEWLs. The model is rather approximate, without any selfconsistency. Its boundary conditions are artificial: without any justification, constant temperature is simply assumed to be maintained for all walls of the cylinder of 50pm radius and 50pm height. Only one heat source within the active region (nonradiative recombination and absorption of spontaneous emission) is considered. With the aid of the model, it was found that thermal resistances RTH of 1.3 pm DMEWLs using MgO/Si mirrors are nearly half of those using SiOz/Si ones. Similar simulation of BEMLs (Baba, Kondoh, Koyama and Iga [1995b]) revealed that RTH of devices with GaAs/AlAs DBRs fused epitaxially to InGaAsP/InP emitting layers is 1/3 of that with InGaAsP/InP DBRs. Another comprehensive thermal model of VCSELs was proposed by Norris, Chen and Tien [1994] and Chen, Hadley and Smith [1994] for p-substrate GaAdAlGaAs TEMLs (cf. table 1 and fig. 3 ) with short-period-superlattice DBRs. It should be emphasized that these devices will in general have much poorer electrical and thermal properties than other AIGaAs-based VCSELs, since thermal and electrical conductivities in very thin layers are much higher than in their bulk counterparts, especially along the axial direction. In the model, a very careful evaluation of heat generation mechanisms is carried out including radiative transfer of spontaneous emission from the active region, nonradiative recombination, absorption of spontaneous and stimulated radiation, and all barrier and volume Joule heating processes except for the contact heating. Detailed analysis of the current flow is performed, including the thermionic and tunneling currents through heteroboundaries, the effect of band-gap discontinuity on the heterointerface resistance, and the anisotropy in the electrical conductivities caused by quantum effects in layered structures. However, carrier diffusion effects are neglected. Somewhat surprisingly, the temperature dependencies of material parameters and heat generation processes

246

THERMAL PROPERTIES OF VCSELs

tIII,

8

5

are not included in this otherwise quite sophisticated model, which compromises its accuracy and limits its applicability to the linear regime well below thermal runaway. In addition, the entire VCSEL cavity, comprising the active region and the short-period-superlattice GaAdAlGaAs DBR mirrors, is replaced with a uniform thermally equivalent medium, characterized by the anisotropic thermal conductivity (k,= 12 W/mK and k, = 10 W/mK), which is a rather crude approach for purely numerical modeling. The same control volume method was used by Chen [1995] to compare thermal characteristics of various VCSEL structures. In the analysis, the microscale electrical and thermal conduction within VCSELs multi-layered volumes were taken into account, including anisotropy of both electrical resistivity and thermal conductivity. The study reveals the interesting fact that the p-type substrate VCSELs (e.g., Hadley, Wilson, Lau and Smith [ 19931) are characterized by lower thermal resistances than the n-type substrate VCSELs (e.g., Young, Scott, F.H. Peters, M.G. Peters, Majewski, Thibeault, Corzine and Coldren [1993], Hasnain, Tai, Yang, Wang, Fischer, Wynn, Weir, Dutta and Cho [1991]). For all the VCSEL structures under consideration, the calculated maximum temperature rise occurs along the optical axis. This is in contrast to the results reported by Nakwaski and Osiriski [ 1991a, 19931 for the etched-well VCSEL, where the temperature profiles peak usually (but not always, depending on the VCSEL structure) close to the edge of the active region. It is, however, very difficult to compare the results of the very sophisticated, self-consistent model given by Nakwaski and Osiriski [1991a, 19931, which is very sensitive to modeling details (compare discussion of fig. 20), with those presented by Chen [1995]. The heat generation profiles assumed by Chen [1995] are, for example, essentially different from those determined by Nakwaski and Osinski (cf. Nakwaslu and Osinski [1991e]). Rahman, Lepkowslu and Grattan [ 19951 have used FEM to model the thermal properties of GaAdAlGaAs DMEWLs. With the exception of the numerical calculation procedure, the model is almost identical to the analytical one reported earlier by Nakwaski and Osinski [ 199la]. All important heat sources are included in this simulation. Details of the complex multilayered buried-heterostructure are taken into account. Full thermal-electrical self-consistency is achieved. Surprisingly, the algorithm of this ambitious simulation is convergent very quickly. The solution is given in the form of temperature and heat-flux profiles as well as isotherm contour maps. Sarzala, Nakwaski and Osinski [ 19951 developed a comprehensive, hlly selfconsistent thermal-electrical finite-element model to investigate the thermal properties of GaAdAlGaAs PITSELs (cf. table 1 and fig. 3). In the model,

247

COMPREHENSIVE THERMAL MODELS OF VCSELs

0 t

‘A I

rC I

r?3 I

)

-Top contact ---Radial current flow region

----Bottom contact Fig. 27. Schematic illustration of current flow in a PITSEL. YA, active region radius; YC, inner radius of the annular contact; r s , structure (chip) radius; d, thickness of the current spreading (radial current flow) region; d p , thickness of the high-resistance proton-implanted region; U , voltage drop between radial current the top contact and the active region; Up,, voltage drop across the p-n junction; I(Y), in the current spreading region; jo(r), axial current density in the current confinement region.

special attention is paid to radial carrier diffusion (Sarzala and Nakwaski [ 19971) within the layer containing the active region and to its influence on temperature profiles. The radial diffusion is enhanced in structures of cylindrical geometry, so this effect is expected to play a significant role in the heat generation as well as in the heat spreading processes in PITSELs. Knowledge of the local current density in the entire volume of the device is essential for proper treatment of the Joule heating source. The main factors affecting the current density distribution are: the cylindncal symmetry of the device, the annular shape of the top contact, and a very high electrical resistivity of the region exposed to the proton bombardment. Figure 27 shows schematically the current flow in a PITSEL structure. The top section, labeled as the radialcurrent-flow region, plays the dominant role in determining the radial distribution of the current injected into the active region. Inside the p-GaAs contact layer and an upper part of the P-type DBR mirror, not affected by the proton bombardment, an approximately two-dimensional (2D) radial flow of current Z(r) takes place. After that, in an inner part of the P-type DBR mirror, which was not exposed to

248

THERMAL PROPERTIES OF VCSELs

Layer #

Thickness

4

d,

N

.. .

dN

tIK § 5

Conductivity

P4

.

0 0

PN

Fig. 28. Electrically equivalent anisotropic medium concept for a multilayer medium.

proton bombardment (axial current flow region), an approximately 1D current flow of a density jo(r) occurs, directed downwards along the main laser axis. Afterwards, a three-dimensional (3D) current spreading towards the bottom contact takes place below the p n junction in the N-type DBR mirror and the n-type GaAs substrate. Note that due to the azimuthal symmetry, 2D and 3D current flows actually reduce to 1D (radius-dependent) and 2D problems, respectively. In order to account for the multilayer composition of the DBR regions, we introduce an equivalent anisotropic medium illustrated in fig. 28, with radial and axial electrical resistivities pr and p z , calculated according to the following expressions (Osinski, Nakwaslu and Cheng [19921):

where d ; and pi stand for the thickness and the resistivity of the ith layer, N is the total number of layers contained in the region, and dtOtis their cumulative thickness. The basic equations describing the current and voltage distributions in the top section of the device, comprising the radial and axial current flow regions, were given by Osinski, Nakwaski and Varangis [ 19941 (refer to fig. 27 for explanation of some of the symbols used):

111,

5 51

COMPREHENSIVE THERMAL MODELS OF VCSELs

249

where j , is the reverse saturation current density at the p n junction, /? is the diode parameter, pr is the effective resistivity in the lateral direction in the radial current flow region, and pzp is the effective resistivity in the vertical direction in the axial current flow region. Approximate analytical solution to the above set of equations is given by Nakwaski [1995]. Before the carriers injected into the active region recombine radiatively or nonradiatively, they diffuse in the radial direction. Note that since the injected current density peaks near the edges of the active region, diffusion can occur in two directions: towards the center and away from the active region. In a cylindncally symmetric structure, the carrier diffusion is governed by the following equation:

where n(r) is the carrier concentration, dA is the active region thickness, D, is the ambipolar diffusion constant, B is the radiative bimolecular recombination constant, ,z, is the carrier lifetime with respect to the nonradiative recombination, j o ( r ) is the pumping current density, and e is the electron charge. In our calculations, we adopt the following values of the above parameters, reported by Lengyel, Meissner, Patzak and Zschauer [ 19821: D,= 40 cm2/s, B=9.7x10-" cm3/s, and r , = 1 . 8 ~ 1 0 - ~ s . For an ideal laser structure, i.e., with perfect grading of all heterobarriers, its total series electrical resistance, Rid, is determined theoretically from known values of electrical resistivities and thicknesses of all the layers. For any particular device, the residual heterobarrier electrical resistance, RHB,is then found from the measured value of resistance R and the calculated value of Rid :

In the model, the R H Bresistance is assumed to be distributed equally among all gradient layers in the p-type DBR mirror. Operation-current dependence of the external quantum efficiency T]d was deduced from an experimental light+urrent characteristic reported by Zhou,

250

THERMAL PROPERTIES OF VCSELs

[III,

55

Cheng, Schaus, Sun, Zheng, Armour, Hains, Hsin, Myers and Vawter [1991]. A self-consistent solution is found numerically in the entire volume of the laser with the aid of an iteration procedure, taking into account the localtemperature dependencies of material and device parameters, including thermal conductivities, electrical resistivities, reverse saturation current density, freecarrier concentrations, absorption coefficients, threshold current, as well as internal and external differential quantum efficiencies. The flow chart of numerical calculations is illustrated in fig. 14, without, however, determination of a threshold current. Note that strongly nonlinear thermaklectrical interactions, enhanced by large temperature variations in VCSELs, affect substantially their lasing characteristics, eventually leading to thermal runaway. Multiple nonhomogeneous heat sources are considered, including the carrierconcentration-dependent active-region heating (nonradiative recombination and absorption of spontaneous radiation), current-density-dependent volume and barrier (p-side contact and heterobarriers) Joule heat generations as well as internal-radiation-density-dependentabsorption of laser radiation. More details may be found in the paper of Sarzala, Nakwaski and Osinski [1995]. In terms of thermal properties, the main effect of carrier diffusion is a reduction of the active-region temperature, compared with the temperature obtained at the same pumping current without diffusion. According to eq. (36), the local density of heat generated in the active-region plane is proportional to the local carrier density. Without diffusion, the active-region heat source would be contained within the circle Y < ?-A, and its spatial profile would be similar to that of current density. Consequently, intense heat generation would take place mostly around the edges of the active region. As a result of diffusion, the local carrier density decreases, especially at the edges of the active region, while the area occupied by the heat source becomes larger. The heat generated outside the circle r 2 r A is conducted away much more easily than the heat generated inside the active region. This leads to a significant reduction in the active-region temperature increase, as illustrated in fig. 29. Figure 29 can be regarded as representing the key result of a paper presented by Sarzala, Nakwaski and Osinski [1995]. In addition to a remarkable lowering of the active-region temperature, dramatic changes take place in the radial temperature profiles. Without diffusion, the temperature profiles essentially reflect the nonuniform current injection, with large maxima at the edges of the active region. Inclusion of the diffusion results in nearly complete disappearance of these maxima. Only at very high pumping levels one can discern a slight positive slope of the T , ( Y curve. ) The uniformity of temperature profiles within the active region is truly astonishing, bearing in mind that it was obtained with

o

111, 51

COMPREHENSIVE THERMAL MODELS OF VCSELs

25 1

Fig. 29. Comparison of radial temperature profiles in the active-region plane, calculated with diffusion (solid lines) and without diffusion (broken lines) for indicated pumping currents.

highly inhomogeneous current injection. In fact, if a uniform active-region heat source is assumed in PITSELs, as by Nakwaski and Osinski [1994], the radial temperature profile has a bell shape. Hence, a slightly raised carrier density near the active-region edges turns out to be optimal for getting a uniform temperature distribution inside the entire active region. It should be emphasized, however, that the radial temperature profiles may well depend on the activeregion diameter DA. For devices with larger values of DA, we expect a greater nonuniformity of injected current profiles, carrier density profiles, and activeregion temperature profiles. The analysis demonstrates that the carrier diffusion influences strongly the distribution of the main heat source located in the active region. As a result, both current- and heat-flux distributions are modified and a temperature spike,

252

THERMAL PROPERTIES OF VCSELs

[III, § 5

Fig. 30. Isotherm profiles in the vicinity of the active region in an 11 pm PITSEL dnven at 6 mA, calculated by neglecting diffusion effects. The isotherms are drawn at intervals of -0.5"C. Note characteristic refraction effects at the interfaces between high- and low-thermal conductivity layers.

which appears at the edge of the active region when carrier diffusion is neglected, smooths out and practically disappears. This redistribution of carrier density results in a smaller number of carriers near the edges of the active region, which reduces considerably the modal gain for high-order modes and favors the excitation of the fundamental transverse mode. Figures 30 and 31 display the isotherm profiles in a section of the device comprising the active region, obtained without and with diffusion, respectively. The two flat maxima in fig. 30, with the temperature of 310.9K, are located in the active-region plane and correspond to the maximum of the 6 m A dashed line in fig. 29. They disappear entirely in fig. 31, and the active region has a remarkably flat temperature profile, again in accordance with the 6 mA solid line in fig. 29. Instead a lower-temperature maximum, just above 308.3 K, emerges in the central part of the P-type mirror. Recently, Hadley, Lear, Warren, Choquette, Scott and Corzine [ 19961 have presented the results of their comprehensive (and very sophisticated in its optical part) full thermal-electrical-optical numerical modeling of PITSELs using a finite-difference technique. The model considers the following major physical processes (i) the ohmic transport of carriers through the cladding layers to an active region, (ii) the heat transport from multiple heat sources toward a heat

111, § 51

253

COMPREHENSIVE THERMAL MODELS OF VCSELs

I=6mA

I

I

1

0

106.9pm

6.5

1

>

r[w1

Fig. 3 1. Isotherm profiles in the vicinity of the active region in an 11 pm PITSEL driven at 6 mA, with diffusion effects taken into account. The isotherms are drawn at intervals of -0.3"C.

sink, (iii) the radial diffusion of carriers inside the active region, and (iv) the multimode optical field. In the model, many important features are taken into consideration, including carrier leakage effects, interaction of transverse modes with carriers, effects arising from the strained band structure usually present in the quantum wells, and so on. The model also contains some thermaloptical (thermal lensing effect) and electrical-optical (spatial holeburning effect) self-consistency procedures. Three main heat generation mechanisms are taken into account, namely nonradiative recombination, reabsorption of radiation, and ohmic dissipation. A temperature increase inside the contact layers and the heat sink does not seem to be taken into account. The authors claimed to successfully predict the threshold current, output powers, and transverse-mode behavior of gain-guided VCSELs, including thermal rollover effects at high injection currents. At least the last prediction is a real surprise: the validity of the above approach seems to be confined to only relatively low operation currents because both the heat and the carrier transports are assumed to be linear and the model does not contain the thermal-electrical self-consistent procedure. As in the model of Piprek and Yo0 [ 19941, multi-layered DBR mirrors are replaced with an equivalent uniform and anisotropic medium whose composite thermal conductivities (and also some other model parameters) are additionally modified to reproduce the measured device temperature rise. Thus, the agreement between theory and experiment can hardly be regarded as a confirmation of the models

254

THERMAL PROPERTIES OF VCSELs

UII,

P6

validity. Nevertheless, the above model seems to be now the most advanced thermal-electrical-optical simulation of a VCSEL operation. An intricate thermal VCSEL analysis has been recently reported by Ning, Indik and Moloney [1995], Ning, Indik, Moloney and Koch [1995], and Ning and Moloney [1995]. They have introduced plasma and lattice temperatures as two independent variables described by kinetic equations coupled to the conventional laser equations for carrier density and field amplitude. According to this approach, lattice and plasma each absorb heat from their respective heat sources and dissipate heat to their heat sinks. In the CW region, lattice and plasma can each hold their individual temperatures and maintain a constant temperature difference because of the sustained pumping as well as the heat exchange and dissipation. New achievements of this approach seem, however, to be too subtle to have practical meaning now. Besides, there are also some essential drawbacks of thls analysis. First of all, it considers important laser variables (e.g., temperature, carrier concentration and current density) as position independent, average quantities whereas their three-dimensional profiles are very important to model correctly VCSEL operation. Furthermore, most of decay rates in the hnetic equations are used as adjustable parameters. All construction and material details are hidden in these parameters. Their values, as used in example numerical calculations, are not justified or evaluated by any means. Finally, the above parameters depend, for example, on VCSEL thermal and electrical resistances, whose exact determination now seems to be more essential for modeling of VCSEL operation, than a subtle distinction between the temperatures of lattice and plasma. Thus, the approach based on this distinction seems to be premature at this moment, but it should not be neglected. It should serve as an important direction in which present VCSEL 3-D models can be extended and should be included into self-consistent comprehensive VCSEL analysis in future. Numerical comprehensive thermal VCSEL models are compared in table 8b (p. 221). In the course of time, they become more and more involved, but also more and more exact.

8

6. Conclusions

Although vertical-cavity surface-emitting lasers (VCSELs) are generally considered to be very promising, their relatively poor thermal properties still represent the main obstacle on the way towards their wide application, and are especially pressing when integration into densely-packed two-dimensional

1111

ACKNOWLEDGMENTS

255

arrays is contemplated. To overcome these difficulties, a good understanding of the thermal problems must be reached. In particular, detailed modeling of heat generation processes, heat-flux spreading, and mutual interactions between thermal, electrical, and optical phenomena is desirable to design thermally optimized devices. As long as the optical output power in VCSELs remains small, the coupling between optical and thermal effects can be ignored in determining the temperature distribution inside the device. On the other hand, thermal-electrical interactions are very important, and thermal4ectrical selfconsistency is essential for realistic modeling of thermal effects. It is also important to include a realistic distribution of heat sources in the model. The current-induced variation of thermal resistance is very sensitive to VCSEL structure, and in particular to the relative distribution of heat sources and their location with respect to the heat smk. This chapter summarizes the present understanding of thermal effects in VCSELs. It is clear that these effects are preponderant and affect almost all device characteristics. As a rule, elevated temperature has a detrimental effect on these characteristics and should be minimized. There are, however, exceptions to this rule. Some aspects of thermal behavior of VCSELs may actually be beneficial. For example, thermally-induced optical waveguide in devices with no built-in lateral guiding helps to reduce losses and lower the threshold current. Unique opportunities for thermal-waveguideengineering exist in etchedwell VCSELs. Elevated temperature may be also beneficial in VCSELs with transparent electrodes [Chua, Thornton, Treat, Yang and Dunnrowicz [ 19971). Finally, non-monotonous variation of threshold current with temperature has been exploited to design temperature-insensitive devices.

Acknowledgments Much of our work reported here would not have been possible without the active contribution and support of many of our colleagues. In particular, we would like to acknowledge numerous fruitful discussions and advice from Julian Cheng of the University of New Mexico, Albuquerque, USA, and G. Ronald Hadley of the Sandia National Laboratories, Albuquerque, USA, as well as from Maciej Bugajski and Bohdan Mroziewicz of the Institute of Electron Technology, Warsaw, Poland. We are grateful to Bo Lu and Ping Zhou of the University of New Mexico, USA, for technical discussions and for providing their data on the temperature dependence of the threshold current. We acknowledge valuable contributions from our students, Robert P. Sarzala, Jacek Wilk, Georgi K.

256

THERMAL PROPERTIES OF VCSELs

[111

Yanakiev, Petros M. Varangis, Antonio Leal, Jonathan Stohs, Gilles du Crest, and Joachim Kastner. We are also thankful to Mrs. Katarzyna A. Steele (Jezierska) and Mr. Bill Johnson for their apt technical assistance. This work was supported by the Polish State Committee for Scientific Research (KBN) (Grant # 8-T11B0 18-12), the National Science Foundation (NSF), the Advanced Research Projects Agency (AREA), the Air Force Office for Scientific Research, and the New Energy and Industrial Technology Development Organization WEDO) of Japan. References Adachi, S., 1985, J. Appl. Phys. 58, RI. Adachi, S., 1992, Physical Properties of 111-V Semiconductor Compounds: InP, InAs, GaAs, Gap, InGaAs, and InGaAsP (Wiley, New York). Agrawal, G.P., and N.K. Dutta, 1993, Semiconductor Lasers, 2nd Ed. (Van Nostrand Reinhold, New York) p. 59. Aiki, K., M. Nakamura and J.-I. Umeda, 1976, E E E J. Quantum Electron. QE-12, 597. Akulova, Y.A., B.J. Thibeault, J. KO and L.A. Coldren, 1997, IEEE Photon. Technol. Lett. 9, 277. Amith, A,, I. Kudman and E.F. Steigmeier, 1965, Phys. Rev. 138, A1270. Baba, T., T. Kondoh, F. Koyama and K. Iga, 1995a, Opt. Rev. 2, 123. Baba, T., T. Kondoh, F. Koyama and K. Iga, 1995b, Opt. Rev. 2, 323. Baba, T., K. Suzuki, Y. Yogo, K. Iga and F. Koyama, 1993a, Electron. Lett. 29, 331. Baba, T., K. S d , Y. Yogo, K. Iga and F. Koyama, 1993b, IEEE Photon. Tech. Lett. 5, 744. Baba, T., Y. Yogo, K. Suzuki, F. Koyama and K. Iga, 1993, Electron. Lett. 29, 913. Baba, T., Y. Yogo, K. Suzuki, F. Koyama and K. Iga, 1994, Jpn. J. Appl. Phys. Pt. 133, 1905. Babic, D.I., Y. Chung, N. Dagli and J.E. Bowers, 1993, IEEE J. Quantum Electron. QE-29, 1950. Babic, D.I., and S.W. Corzine, 1992, IEEE J. Quantum Electron. QE-28, 5 14. Babic, D.I., J.J. Dudley, K. Streubel, R.P. Mirin, J.E. Bowers and E.L. Hu, 1995, Appl. Phys. Lett. 66, 1030. Babic, D.I., K. Streubel, R.P. Mirin, N.M. Margalit, J.E. Bowers and E.L. Hu, 1995, Electron. Lett. 31, 653. Babic, D.I., K. Streubel, R.P. Mirin, N.M. Margalit, J.E. Bowers, E.L. Hu, D.E. Mars, L. Yang and K. Carey, 1995, IEEE Photon. Technol. Lett. 7, 1225. Bissessur, H., R.D. Ettinger, F.A. Fernandez and J.B. Davies, 1993, IEEE Photon. Technol. Lett. 5, 764. Buccafusca, O., J.L.A. Chilla, J.J. Rocca, S. Feld, C. Wilmsen, V. Morozov and R. Leibenguth, 1996, Appl. Phys. Lett. 68, 590. Bugajski, M., and A.M. Kontlaewicz, 1982, Electron Technol. 13(4), 63. Buus, J., 1983, IEEE J. Quantum Electron. QE-19, 953. Buus, J., and M.J. Adams, 1979, E E J. Solid State Electron Dev. 3, 189. Carslaw, H.S., and J.C. Jaeger, 1988, Conduction of Heat in Solids (Clarendon Press, Oxford). Casey Jr, H.C., and M.B. Panish, 1978, Heterostructure Lasers, Part A: Fundamental Principles (Academic Press, New York) p. 44. Catchmark, J.M., R.A. Morgan, K. Kojima, R.E. Leibenguth, M.T. Asom, G.D. Guth, M.W. Focht, L.C. Luther, G.P. Przybylek, T. Mullay and D.N. Christodoulides, 1993, Appl. Phys. Lett. 63, 3122.

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IV FRACTIONAL TRANSFORMATIONS IN OPTICS BY

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DAVIDMENDLOVIC Tel-Auiv University, Faculty of Engineering, 69978 Tel-Auiu, Israel

AND

ZEEVZALEVSKY Tel-Auiu University, Faculty of Engineering, 69978 Tel-Auiu, Israel

263

CONTENTS

PAGE

0 1.

INTRODUCTION . . . . . . . . . . . . . . . . . . .

265

0 2.

THE FRACTIONAL FOURIER TRANSFORMATION (FRT) .

268

9 3.

THE FRACTIONAL HILBERT TRANSFORMATION (FHiT) .

276

0 4.

THE FRACTIONAL ZERNIKE TRANSFORMATION (FZT) .

284

0 5. 0 6.

THE HARMONIC REAL TRANSFORMATIONS. . . . . .

286

OTHER FRACTIONAL TRANSFORMATIONS . . . . . .

289

9 7. 3 8.

FRACTIONAL FILTERING. . . . . . . . . . . . . . .

302

OTHER ASPECTS OF FRACTIONALIZATION . . . . . .

324

0 9.

FRACTIONALIZATION AND GROUP THEORY. . . . . .

326

0 10. CONCLUSIONS . . . . . . . . . . . . . . . . . . .

333

ACKNOWLEDGMENTS . . . . . . . . . . . . . . . . . . .

334

Appendix A . ABOUT THE WIGNERDISTRIBUTION . . . . . .

334

REFERENCES . . . . . . . . . . . . . . . . . . . . . . .

339

264

0 1.

Introduction

1.1, WHAT IS MEANT BY “FRACTIONAL’?

The term “fractional” has been used in mathematics for many decades, but has been used in optics for only a few years. The process of “fractionalization” in optics corresponds strictly to its mathematical counterpart. The most prominent example is the fractional Fourier transformation, which will be treated later on in some detail. A simple mathematical example will illustrate the process of “fractionalization”. We start with a certain set of rational numbers g,,, which are identified by the integer index m: {. . . , $, 1, 3, 9, 37, . . .}. It is apparent that this set is defined by the law

A, 3,

(1)

g(m) = 3“. Now we generalize the integers m into the real numbers P: g(P) = 3p.

(2)

The function g(P) is the result of the “fractionalization” of the set of rational numbers with integer indices g(m). Notice that the function g(P) assumes the value g(m) if P = m. Hence, “fractionalization” is a lund of interpolation. We must remember that interpolations are not unique processes, unless certain additional properties are specified, such as smoothness or finite bandwidth of g(P). 1.2. SIMPLE EXAMPLES, BASED ON A FINITE SET OF INTEGERS

Suppose we wish to fractionalize the following set of four images: go = u(x,y), gI = u(y,-x), g2 = u(-x,-y), g3 = u(-y,x). These four images are apparently alike, apart from successive rotations by 90”. Now we generalize from a quantized rotation to a continuous rotation and obtain g(x,y , P ) = u(x cos $ + y sin #, -x sin $ + y cos $).

(3)

The angle $ is related to the real number P by $ = Pn/2. The interpolation of g, = g ( m ) to g(P) is a case of “fractionalization”, where, in this example, the 265

266

FRACTIONAL TRANSFORMATIONS IN OPTICS

“0

1

outcome may be expanded periodically in P. That is typical if the set g,,, (here with m = 0,1,2,3) has a finite number of members, four in this case. The second example relates to spatial filtering, which will be treated later on in more detail. Here, we concentrate on only as many of the essentials as are needed for illustrating the process of fractionalization. The original spatial filter may be binary, and phase-only:

E(v) =

exp[-in/2] for region R , exp[in/2] for region R2

(4)

The generalization is now obvious:

Ep(v) =

exp[-iPn/2] for region R , exp[iPn/2] for region R2

Here again, the fractional index P is related to an angle 9 = Pn/2. For the experimentalist, the fractionalization of the classical ; ~ dphase shift filter E into the fractional filter means that he has to vary the phase between the two regions R1 and R2 continuously. Various additional examples demonstrate the idea of fractionalization or continuous extensions. A prominent example is fuzzy logic, which may be considered as fractional Boolean logic. The Boolean logic has been used widely and for a long time in the context of transmission and processing of information. Various devices such as computers were constructed to implement this logic format. Recently it has been discovered that fuzzy logic may be a useful tool for controlling unstable systems. In this approach, the control signals are not only “yes” and “no”, but also a gray zone is allowed. A simple optical implementation has been achieved by Itoh, Mukai and Yajima [ 19941. Similar continuous extension occurred in the fractal field. The definition of the conventional dimensions (dot is one-dimensional, line is two-dimensional and volume is three-dimensional) has been extended to shapes which have items repeated periodically while their sizes are being decreased monotonically (termed fractals) (Tricot [ 19921, Fontana and Viliani [1992]). Thus, instead of integer numbers, the dimension may also assume real values like 2.23. Another important continuous generalization is the expansion of the regular calculus into fractional calculus, where a fractional derivative and fractional integral are defined, by Mcbride and Roach [1985].

rv, §

11

INTRODUCTION

267

1.3. AN EXAMPLE BASED ON AN INFINITE SET OF INTEGERS

The set of differential operators D,, characterized by the integers m = 0, 1,2,. . ., is given as (6) If u(x) is replaced by its Fourier integral, eq. (6) becomes

I, Dc)

D~[u(x= ) ]u(x) = M

1(v)exp(2nivx)dv, (7)

D,rl[u(x)] = Lm(2niv)“ O(v)exp(2nivx) dv. Fractionalization means that one replaces the integer rn by a real-valued p : D,,[u(x)]= L I ( 2 s i v ) ” O(v)exp(2sivx) dv. This case of fractionalization has already appeared in textbooks such as Bracewell’s [1978]. It is supposedly useful in the theoretical mechanics of soft matter. So far we have used only the nonnegative integers (rn = 0, 1,2,. . .). Negative integers correspond to integration. In that case, a singularity at zero frequency must be handled with care. We mention the extension to negative integers because it implies the existence of inverse operations. That aspect will be significant in $9, where fractionalization will be described in terms of group theory. 1.4. MOTIVATION

The number of papers on “fractional optics” is approaching 150, despite the fact that only four years have elapsed since the first publication appeared. We hope to review here, in a systematic manner, the existing results. Such a review might identify some flaws of the fundamentals. It might stimulate new applications of existing fractional transformations. In addition, it might animate someone to fractionalize some other classical transformations. We are quite optimistic about those opportunities.

268

FRACTIONAL TRANSFORMATIONS IN OPTICS

1.5. OUTLINE

The remaining sections of this chapter belong to four categories. First, in $ 2 we present the fundamentals of the fractional Fourier transformation (abbreviated as FRT). We do this in some detail not only because the FRT is the earliest case of fractionalization in optics, but also for making it easier later on to understand some other fractional transformations ($0 3-6). Section 7 deals with applications of several fractional transformations. Section 8 provides some other aspects of the fractionalization techniques. These applications are “filtering processes”. In $ 9 we attempt to look at fractional transformations from a higher, more abstract level by involung group theory. Some historical remarks are part of the conclusions. An appendix serves as introduction to the Wigner distribution functions.

9 2.

The Fractional Fourier Transformation (FRT)

2.1. FUNDAMENTALS

The fractional Fourier transformation (FRT) has been known in mathematics for 70 years (see Q 9). Unaware of that history, optical scientists have reinvented the FRT twice. The first re-invention was related to the propagation of light within a medium with continuously varying refractive index (Mendlovic and Ozaktas [ 19931, Ozaktas and Mendlovic [1993a,b]). The other re-invention (Lohmann [1993]) was a natural consequence of the desire to describe physical optics by means of the Wigner distribution function (abbreviated as WDF; a brief introduction to “Wigner optics” is presented as an appendix). Soon the equivalence of these two re-inventions was proven (Mendlovic, Ozaktas and Lohmann [1994a,b]). The following subsections contain the two optical FRT definitions and some basic FRT properties. 2. I . I . Dejnition based on light propagation in graded index mediu

The first FRT definition (Mendlovic and Ozaktas [1993], Ozaktas and Mendlovic [1993a,b]) is based on the field propagating along a quadratic graded index (GRIN) medium (Kogelnik [ 19651) having a length proportional to p ( p being the FRT order). The eigenmodes of quadratic GRIN media are the Hermite-

Iv, § 21

269

THE FRACTIONAL FOURIER TRANSFORMATION (FRT)

2f

4f

6f

8f

Fig. 1. Light propagation through a bulk optical setup.

Gaussian (HG) functions, which form an orthogonal and complete basis set. The mth member of this set is expressed as

where H , is a Hermite polynomial of order m and w is a constant associated with the GRIN medium parameters nl and n2, as they appear in

n2(x,y)= n: -nln2(x2 + y 2 ) ,

(10)

where n(x,y ) is GRIN'S index of refraction. An extension to two lateral coordinates x and y is straightforward, with Y,,,(x)'Y,(y) as elementary functions. The propagation constant for each HG mode is given by

with k

= 2n/A.

The HG set is used to decompose any arbitrary distribution u(x),

m

where the coefficient A,, of each mode Y,(x) is given by

I, M

A*

=

u(x)Y&)/~,, h,

(13)

with h,, = 2"'m!&w/&. Using the above decomposition, the FRT of order p is defined as

L = (n/2)is the GRIN length that results in the conventional Fourier transform. It was shown (Mendlovic and Ozaktas [1993]) that this definition agrees well with the classical Fourier transform definition when p = 1. Figure 1 serves to make it plausible that at 2f intervals, parallel rays change into converging rays, and vice versa. The same occurs within the GRIN fiber

270

[IV,

FRACTIONAL TRANSFORMATIONS IN OPTICS

4f

2f

6f

02

8f

Fig. 2. Light propagation through a GRIN fiber

(fig. 2). Expressed in wave-optical terms, every 2f subsystem performs a Fourier transformation. Hence, an upside-down image is generated at 4f, and an upright image at Sf. The only difference between the two setups is that the deflection of the rays occurs abruptly when classical lenses are used. Within the GRIN fiber the deflection is smooth and continuous. 2.1.2. Dejinition based on the Wigner distribution function A complete signal description, simultaneously displaying space and frequency information, can be achieved by the space-frequency Wigner distribution hnction (WDF) (Wigner [1932]). The Wigner distribution is provided in the appendix. An optical Fourier transformation (Fraunhofer diffraction) corresponds to a 90" rotation in the Wigner domain. An upside-down image corresponds to a 180" rotation in Wigner space. Based on this background, it was quite natural to define the FRT as what happens to the signal u(x) while the WDF is rotated by an angle 4 = pM2. Note that the WDF of a 1-D function is a 2-D function and the rotation interpretation is easily displayed. Lohmann [ 19931 generalized the same rotation strategy to 2-D signals, i.e. images, whose WDFs are 4-D distributions. The WDF of a function can be rotated with bulk optics. It was suggested by Lohmann [1993] that the optical system illustrated in fig. 3 be used for Input

lens

output

lens

lens

-

+Z

= Rf,+z=

(4

'

f =f, / Q

f =f, / Q

Rf,

f =f,

z=Rf, (b)

Fig. 3. The two possible optical setups for obtaining the FRT.

4

/Q

n! § 21

THE FRACTIONAL FOURIER TRANSFORMATION (FRT)

27 1

implementing the FRT operator. The second interpretation may be illustrated by rotating the Wigner distribution and projecting it onto the x-axis of the twodimensional WDF: y)}, IFP[uI(x)I2= R-@{W(X>

(15)

where Reg{ W ( x ,v ) } is the Radon transform at angle -9 of the WDF W ( x ,v ) (Lohmann and Soffer [1994]). More details regarding the Radon transform are given in 4 7.2. These optical setups represent three shearing operations in WDF space: x, v,x-shearing or v , x , v-shearing; v is the spatial frequency and x is the space coordinate. x-shearing is performed by free-space propagation, then a lens performs v-shearing, then x-shearing is again performed by free-space propagation. In his paper, Lohmann [ 19931 characterized this optical system using two parameters, Q and R:

where5 is an arbitrary but fixed length,f is the focal length of the lens, and z is the distance between the lens and the input (or output) plane. Lohmann [1993] showed that for an FRT of order p , R and Q should be chosen as

R = tan ($/2),

Q = sin (9)

(17)

for the configuration in fig. 3a, and as R = sin (g),

Q = tan ( @ / 2 )

(18)

for the configuration in fig. 3b. Note that $ =p(n/2). By analyzing the optical configuration of fig. 3, Lohmann [1993] obtained

with

This last equation defines the FRT for one-dimensional (1-D) functions with A as a wavelength. Generalization to two-dimensional hnctions is straightforward. Note that A2 = Aji is also termed the scaling factor.

272

[IY § 2

FRACTIONAL TRANSFORMATIONS IN OPTICS

The two interpretations of the FRT operation can be described by the same transformation kernel Bp, as shown by Ozaktas, Barshan, Mendlovic and Onural [1994]:

where B,(x,x’) is the kernel of the transformation and p is the fractional order. The kernel has two optical interpretations, one as a propagation through GRIN medium (Mendlovic and Ozaktas [ 1993]),

B,,(x, x’) =

Jzexp [

-( x 2

y”’]2

&H,,

(2x)

Hn ($XI)

,

(22)

n=O

and the second as a rotation operation in the Wigner plane (Lohmann [ 1993]),

Note that w is the coefficient that connects the two interpretations:

Notice further that the kernel is usually not simply a function of a coordinate difference, as would be the case for a convolution. 2.1.3. Properties of the FRT 9

Linearity: The FRT of a linear combination of two input functions uI and u2 behaves according to the definition of linear systems; CI and c2 are constants:

Continuity: Two FRTs with different orders p1 and p2 yield the following theorem

This feature is sometimes called “index additivity”.

n! § 21 9

273

THE FRACTIONAL FOURIER TRANSFORMATION (FRT)

Parseoal h theorem:

J-Ca

J-‘33

or more generally,

Shqt theorem: If the input object is shifted by the amount of a, then its FRT yields (Lohmann, Mendlovic and Zalevsky [ 19951): uP(x;a) = {FP[uo(xo- a ) ] } = exp[ina sin 4(2x - a cos @)]uP(x- a cos 4).

(29) In other words, a shift of the input will not simply cause an equivalent shift of the output. This feature will be referred to as “shift variance of the FRY. Scaling theorem: If the input object is scaled by the factor of a, then its FRT yields (Lohmann, Mendlovic and Zalevsky [ 19951):

where

j= 2 tan-’ (a2 tan(-)) EP n 2 Y = e x p [ ixx2(a4- 1) a4tan4+

&

1.

These equations reduce to the well-known Fourier theorem if p

=

1

2.2. ANAMOWHIC FRT

The FRT concept has been extended to the anamorphic case (Sahin, Ozaktas and Mendlovic [ 19951, Mendlovic, Bitran, Dorsch, Ferreira, Garcia and Ozaktas [1995]). This modification permits the use of different fractional orders for two orthogonal axes of a two-dimensional image. The main advantage of this extension is the possibility of varying the shift variance of the processor according to the characteristics of the input image. A particular example

274

FRACTIONAL TRANSFORMATIONS IN OPTICS

[IY 8 2

occurs when the detection of objects along a row is needed. In the direction perpendicular to the row, there is no need to keep the shift invariance of a conventional Fourier transform processor. Depending on the characteristics of the object to be detected, the decrease of shifi invariance may result in a gain in the performance of the correlator, mainly in peak sharpness and SNR (Mendlovic, Ozaktas and Lohmann [ 19951, Bitran, Zalevsky, Mendlovic and Dorsch [ 19961). The shift variance can also help in locating the object, because a detection peak will only be produced when the input object lies on the line along which the shift invariance is maintained. The anamorphic FRT is defined as

J-,

with

T, = Ahxtan A,

sl, = A h r sin gX,

A = p X~ d 2 ,

Tl,= Ah! tan q&,

S,, = Ah,,. sin @,,,

9" = p!,x/2,

(34)

where the subscripts x and y indicate the horizontal and vertical directions of the system, respectively. Due to the index additivity of the FRT, the anamorphic FRT can be implemented by cascading two systems, each one performing the appropriate transform in one of the main axes and imaging in the other. Other setups providing greater compactness or flexibility can be used, as well (Sahin, Ozaktas and Mendlovic [ 19951, Mendlovic, Bitran, Dorsch, Ferreira, Garcia and Ozaktas [ 19951). The optical setup for obtaining an anamorphic FRT can be built as a cascade of an amorphic setup, which performs the FRT with the order which is the lower of p x and pl,,and an anamorphic system which renders an FRT in one axis and imaging in the perpendicular one (Mendlovic, Bitran, Dorsch, Ferreira, Garcia and Ozaktas [1995]), as seen in fig. 4. However, if the transform module is the first stage of a correlator, an inexact FRT (without the final quadratic phase factor) may be obtained with a setup analogous to that described by Garcia, Dorsch, Lohmann, Ferreira and Zalevsky [ 19971. In the case of an anamorphic FRT the convergence of the beam at the output of the filter plane will be different in the two main axes. An anamorphic system will be needed to focus the correlation.

Iv, § 21

THE FRACTIONAL FOURIER TRANSFORMATION (FRT)

215

Fig. 4.Experimental setup for obtaining the anamorphic FRT

2.3. SOME APPLICATIONS

Besides the ability to change easily the space variance of the optical system, the FRT has been shown to be very useful for many other applications in signal processing. The main application is related to chirp noise removal. This application is based on the fact that if a chirp-type noise of exp(-iJcax2) is fractionally Fourier-transformed with order p = $tan-'( l/Ah a), the result is a delta function. Thus, in order to remove the noise a simple notch filter may be placed at the proper location in the proper FRT plane. Since the filter is a notch, the amount of information loss in the signal is minimal. Another important application of the FRT is related to the fact that the FRT corresponds to rotation of the Wigner chart by an angle of p n / 2 . Thus, assuming that the Wigner chart for the signal and the noise distributions is as illustrated by fig. 5, one may see that filtering either in the Fourier plane (corresponding to projection of the Wigner chart on the Y-axis) or filtering in the spatial plane (corresponding

4 f.

Fig. 5. A Wigner chart of a signal and a noise where the FRT filtering is very applicable.

276

FRACTIONAL TRANSFORMATIONS IN OPTICS

[IY

53

Fig. 6. A Wigner chart of a signal which may be more efficiently multiplexed via an FRT.

to projection of the Wigner chart on the x-axis) will result in partial loss of information in the signal (see appendix). However, filtering in the proper FRT plane (the angle in the Wigner chart at which there is full separation between the projections of the signal and the noise) may result in a perfect reconstruction of the signal (Dorsch, Lohmann, Bitran, Mendlovic and Ozaktas [1994]). Another important application of the FRT is related to signal multiplexing. Due to the ability of the FRT to rotate the Wigner chart, the Wigner distribution of a signal may be arranged in a more efficient manner (Ozaktas, Barshan, Mendlovic and Onural [ 19941) as seen in fig. 6. That efficient arrangement saves additional bandwidth that may be needed for the transmission of the signal. Note that the Wigner chart of fig. 6 does not contain any cross terms due to our assumption that the different signals which appear in this Wigner chart are mutually incoherent. Other promising applications will be mentioned in the relevant sections of this chapter.

9 3.

The Fractional Hilbert Transformation (FHiT)

3.1. FUNDAMENTALS

The optical implementation of the Hilbert transformation (HT) has been invented twice in 1950. Kastler [ 19501 used it for image processing, especially for edge enhancement, since the HT emphasizes the derivatives of the image. Wolter [ 19501used it for spectroscopy. Later on, the HT processor was made achromatic (Hauk and Lohmann [ 1958]), two-dimensional (Lowenthal and Belvaux [ 1967]), and angular isotopic (Eu and Lohmann [1973]). The achromatic version is

1%

Q

31

THE FRACTIONAL HILBERT TRANSFORMATION (FHIT)

X

277

V

Fig. 7. An optical setup for performing the conventional HT

based on the “detour phase” concept, which later became the root of synthetic complex spatial filters (Brown and Lohmann [ 19661) and of computer-generated holograms (Lohmann and Paris [ 19671). All of these physical applications made the HT an important transformation used variously in different scientific fields, including solid-state physics and signal processing. The Hilbert transformation (HT) has been generalized by defining the fractional Hilbert transformation (FHiT) operation. In the first stage, two different approaches for defining the FHiT are suggested. One is based on modifying only the spatial filter, and the other proposes to use the fractional Fourier plane for filtering. In the second stage, the two definitions have been combined into a FHiT, which is characterized by two parameters (Lohmann, Mendlovic and Zalevsky [1996]). We will now generalize the classical HT into the FHiT, which can be implemented optically quite easily as a spatial filtering setup. Two alternative ways for fractionalizing the HT are proposed. The two definitions are not equivalent, and they introduce different improvements in the image processing. The first way is a modification of the spatial filter with a fractional parameter P. The second approach is based on the FRT. Here, the spatial filter is exactly the same as in the conventional HT case, but instead of using a conventional Fourier filtering, a symmetrical fractional correlation scheme with a fractional order Q, is used (Mendlovic, Ozaktas and Lohmann [1995]). Both FHiT approaches were invented according to the following postulate: the first order FHiT should be consistent with the conventional Hilbert transform. The two alternatives are merged into a more general definition. T h s general definition has two parameters P and Q. For Q = 1 and P variable, this general definition coincide with the first way of fractionalizing. For P = 1 and Q variable, the general definition matches the second way of fractionalizing. Figure 7 shows the optical setup used by Kastler and Wolter. It is a

278

FRACTIONAL TRANSFORMATIONS IN OPTICS

tIY

03

conventional optical spatial filter with a filter function consisting of a glass plate, covered halfway by a n-phase shifting layer:

when S ( Y ) is a step function. The first fractional generalization contains the filter:

where

n

Q=P2.

(37)

This can be conveniently re-written as

For an input function U ( x ) with the spatial spectrum

U(Y),the output image

is 03

~(~)I?p(v)exp(2nivx)dv.

Vp(x) =

(39)

J-W

On the basis of eq. (38) we conclude that Vp(x) is apparently a superposition of the ordinary bright field kind, VO,and the classical HT, V I ,in the following form: Vp(x) = cos Q

Vo(x)+ sin Q VI (x).

(40)

The weighting parameters cos Q and sin Q can be vaned easily if the object is linearly polarized at 45", if the filter consists of two pieces of a quarter wave plate (oriented at 0' for Y 3 0 and at 90" otherwise). The output polarizer is oriented at the angle 6. An alternate approach for defining the FHiT can be understood by means of the flow chart presented in fig. 8. This configuration has been called a symmetrical fractional correlation system with order Q (Lohmann, Mendlovic

Fig. 8. Block diagram for performing the fractional Hilbert transform based on the FRT operation.

n! o

31

THE FRACTIONAL HILBERT TRANSFORMATION (FHiT)

279

and Zalevsky [ 19961). This scheme perfectly matches the conventional Fourier correlator except that instead of conventional Fourier and inverse Fourier transforms two FRTs with fractional orders of Q and -Q are performed, respectively. The input Uo(x) is at first transformed by means of the FRT of degree Q, yielding ~ Q ( Y )which , is then multiplied by Z?I(Y) (see eq. 35). Finally, the signal is back-transformed, as indicated in fig. 8 by the box with (-Q). The mathematical expression for this symmetrical fractional correlator system is given by

where FQIUo(x)]is the FRT of Uo(x) with the fractional order Q. Both approaches for defining the FHiT mentioned above satisfy the reasonable requirement that the classical HT is obtained if P = Q = 1.

U

U

U

Fig. 9. Block diagram that demonstrates a combination of the two fractional Hilbert definitions with the two free parameters P and Q.

The two definitions given above are not equivalent; rather, they are different ways to improve the image edge enhancement or the image compression. The combination of the two definitions provides two degrees of freedom (P,Q) which can control and achieve the desired image processing. Figure 9 describes the flow chart of such a combination. Here, for P f 1 and Q # 1 we have a two-parameter generalized FHiT, i.e.:

The generalized FHiT shown in fig. 9 has been illustrated by several computer simulations. As an input signal, a rect function with a width of 40 out of 256 pixels was used. Figure 10 shows the FHiT output for Q = 1 and various values of P. One can notice that for 0 < P < 1 the negative derivative of the input is emphasized, while for 1 < P < 2 the positive derivative gets higher values. Several computer simulations were carried out to illustrate the FHiT for different values of P and Q. Figures 11 and 12 show the FHiT for various values of P,

280

FRACTIONAL TRANSFORMATIONS IN OPTICS

P=O

P=0.2

P-0.5

:I,l~~~~ 0

100

0

200

0

100

200

0

P=l

P=O.8

100

200

P=i .2

Lmxml

:m;m:bl

00

100

200

0

P11.5

0

0

100

100

200

0

P.1.8

200

0

100

100

200

P=2

200

OO

100

200

Fig. 10. Computer simulation of the case when Q = 1.

LcIIrn!rn P.0

00

100

P.0.2

200

0

P-0.8

0

100

100

P.0.5

200

0

P=l

200

0

100

100

200

Psl.2

200

0

100

200

D!JIrnILn_ P=l.5

00

100

P-1 .I

200

0

100

200

P.2

0

100

200

Fig. 11. Computer simulation of the case when Q = 0.8.

IY

P

31

THE FRACTIONAL HILBERT TRANSFORMATION (FHiT) P.0

P.0.2

I

I

P-0.5

I

I

I

1 .

lu:u:w 0

100

200

0

P-0.8

-

28 1

100

200

2

100

P=1.5

200

100

200

P.1.2

- 2

- 2

00

0

P.1

0

200

100

P.1

.a

0

100

200

P.2

Fig. 12. Computer simulation of the case when Q = 0.5

with Q = 0.8 and Q = 0.5, respectively. Note that decreasing Q to 0.8 changes the results and gives more emphasis to the positive derivative. For Q = 0.5 there is no preference for the negative or for the positive derivatives. Several comments about the two FHiT definitions are in order here: For P = 0, the first definition provides exactly the input. According to the second definition, the output is the input, directly multiplied by the filter Hp. Remember, an FRT with Q = 0 is the identity operator. Both definitions are periodic in P. The period is 4 for a general input signal. Only the first definition is index-additive, i.e., performing FHiT with order P I after FHiT with P2 means an FHiT with order P I + P2. For the second definition, and also for the composed definition, the following generalization is possible: According to Mendlovic, Ozaktas and Lohmann [ 19951, instead of using a symmetrical fractiona1correlator configuration, one can use an asymmetrical one. This will lead to an additional free parameter.

3.2. ONE-DIMENSIONAL IMPLEMENTATION

In this subsection we will describe the optical implementation of the FHiT for

282

[n!§ 3

FRACTIONAL TRANSFORMATIONS IN OPTICS

t v0

I Image Fig. 13. The optical setup for implementing the I-D FHiT.

I

r

AY

L I

Fig. 14. The filter used in the setup of fig. 13

one-dimensional objects (Lohmann, Ojeda-Castaneda and Diaz-Santana [ 19961); the setup is shown in fig. 13. The essence is a slit filter, a mask (fig. 14):

In our case, G(v) = -b rect

(-)

Y-

iAV

$AV

+

(-)

rect Y + ~ A v ;AY

(44)

rv, 5 31

283

THE FRACTIONAL HILBERT TRANSFORMATION (FHiT)

A one-dimensional object u(x~)b( yo) is now placed in the input plane. Propagation of this input through the system results in

[

x exp 2nibyrect

which equals v(x, y) =

(',d,"' ~

1: [

~ ( v )exp(-2niby) rect

+ exp(2niby) rect

('i;:')] ~

exp(2nivx) dv.

(46) The expression in square brackets is equivalent with the FHiT (eqs. 36-39), if one sets P

=

4by.

(47)

In other words, the output signal V(x,y) represents the FHiT with a running index P along they direction. 3.3. TWO-DIMENSIONAL IMPLEMENTATION

The FHiT definition (eq. 39) may be extended to two dimensions (Lohmann, Tepechin and Ramirez [ 19971):

where e ( v , p ) is the spatial spectrum of the input function U(x,y) and

Rp(v,,u)is the two-dimensional extension of eq. 36. In order to implement optically the two-dimensional FHiT one may use the setup shown in fig. 15. As seen in fig. 15, the object is illuminated by a monochromatic plane wave. A polarizer at 45" orientation ensures that the two polarization components (in the x - and y-directions) are equally strong. A Wollaston prism W splits the two polarization components so that two spatial frequency spectra v,p + b)and

e(

284

FRACTIONAL TRANSFORMATIONS IN OPTICS

[IY 0 4

Y

*

Fig. 15. The optical setup used for the two-dimensional implementation. tP

Fig. 16. The structure of the filter used in fig. 15

O(Y,p - b)will appear one above the other in the filter plane. There (see fig. 16) one of the spectra loses its positive frequencies (Y > 0) and the other loses its negative frequencies. This is achieved by two properly located razor blades as "knife edges". A second Wollaston prism W placed shortly before the image plane of fig. 15 re-aligns the two linearly polarized components. A quarter-wave plate Q (at 45" orientation) converts the two orthogonal linear polarization components into two circularly polarized components, with opposite circular directions. The final polarizer, at an orientation of 45" + 4, produces phase shfts *$J for the two polarization components, which have traveled through the upper and lower part of the filter of fig. 16, respectively. In other words, that polarizer varies the phase 4, as it appeared in the FHiT filter of eq. (48).

0 4.

The Fractional Zernike Transformation (FZT)

In this section we explore the fractionalization procedure of Zernike's phase contrast microscope. This method is a spatial filtering process somewhat similar

nr, P 41

285

THE FRACTIONAL ZERNIKE TRANSFORMATION (FZT)

Fig. 17. A schematic illustration of the Zernike transform.

to the optical Hilbert transformation. Hence, it is easy to show how one should define and design the fractional Zernike transform of complex order (FZT). Note that similar transformations already exist as variable phase contrast (Lohmann, Schwider, Strebl and Thomas [1988], Lohmann [1954, 19921, and as an array illuminator (Osterberg [ 1947, 19501). The FHiT has been defined by eqs. (39) and (36). In those equations, P is the fractional order. This definition can be extended to complex orders: P P + iQ. An optical implementation of the FHiT with complex order may be performed easily either for one-dimensional objects or for two-dimensional objects similarly to the implementation suggested in $0 3.1, 3.2 and 3.3. The known definition for the Zernike transformation is similar to the definition of the Hilbert transformation. However, in the filter plane the mask is .--)

this is illustrated schematically in fig. 17. It is apparent that a 90" phase difference exists between the two zones, and a relative amplitude of A . We generalize this transform into the FZT as follows: The mask placed in the filter plane is now

By choosing P = 0.5 and Q = (In A ) / x , the FZT converges to the conventional Zernike transform. Note that if P = 1 a positive phase contrast is achieved, while for P = -1 a negative contrast is obtained. In the optical implementation of the two-dimensional FZT, a mask with two complementary parts (fig. 18) is now used instead of the two-quadrant mask (fig. 16) for the two-dimensional FHiT.

286

FRACTIONAL TRANSFORMATIONS IN OPTICS

tIv,

55

Fig. 18. The structure of the filter used in fig. 17.

5 5.

The Harmonic Real Transformations

5.1. THE FRACTIONAL SINE AND COSINE TRANSFORMATIONS

The cosine transform is defined like the conventional Fourier transform, but it uses only the real part of the Fourier transform kernel. Its definition is:

1,

co

{ C ’ u } ( x )=

u(x’)cos(2xxx’) dr’

The cosine transform for odd objects is zero, due to the symmetry of the cosine. Note that for real inputs, the cosine transform is exactly the real part of the conventional Fourier transform. In the same manner, one may define a transform whlch uses the real part of the kernel of the FRT. Such a transform may be termed the “fractional cosine transform” with fractional order p :

s,

03

{C”u)(x) =

{ w 4 4 x , x ’ ) l } u ( ~ ‘ )b’,

(52)

where BJx, x’) is the FRT kernel defined in eqs. (22) and (23), and RE is the reaI operator. Obviously for fractional order p = 1, the fractional cosine transform is exactly the conventional cosine transform. For fractional order p = 4 the original object is obtained:

{c4“u1}= w

~

~ = u(x) (

4

(53)

The transform that uses the imaginary part of the Fourier transform kernel may be termed the sine transform. Its definition is:

The sine transform for an even object is zero, due to the symmetry of the sine (Lohmann, Mendlovic, Zalevsky and Dorsch [1996]). For real inputs the sine

n! Q 51

THE HARMONIC REAL TRANSFORMATIONS

287

transform is exactly the imaginary part of the Fourier transform. As before, one can define the “fractional sine transform” as a transform that uses the imaginary part of the kernel of the FRT:

where B,(x,x’) is the FRT kernel defined in eqs. (22) and (23), and IM is the imaginary operator. Once again, for fractional order p = 1, the fractional sine transform is exactly the sine transform. In this case the sine transform of order p = 4 will be zero for real input objects: {S4[u]}(x) = IM{p[u]}(x)

= IM{u(x)} = 0.

(56)

5.2. THE FRACTIONAL HARTLEY TRANSFORMATION

The Hartley transformation has a real-valued kernel. It generalizes both the cosine and the sine transforms (Hartley [1942]). A possible application of t h s transformation may be data compression and digital image processing, especially since the invention of the fast Hartley transform (Bracewell [1984]). This transformation can be obtained optically as well (Bracewell, Bartelt, Lohmann and Streibl [1985]). The definition of this transformation is

Generalizing this transformation for any fractional order could be significant in image processing since it provides the advantages of the FRT (a shift variant transformation used for pattern recognition and efficient noise filtering). It is a real transformation, and thus its computing complexity is smaller if the input is real and it can be implemented experimentally with incoherent illumination, which by itself has several major advantages (such as the lack of speckle effect) (Bartelt, Case and Hauck [1982]). Moreover, while the cosine (or sine) transformation takes into account only the symmetrical (or asymmetrical) parts of the input object, the Hartley transform processes the whole input information

288

FRACTIONAL TRANSFORMATIONS IN OPTICS

tw 5 5

(symmetrical and asymmetrical). The generalized transform is termed here the “fractional Hartley transform” (FHT). We define this transformation as

{ X P u } ( x= ) / ~ { R E [ B , , ( x , x ’ ) ] } u ( x ’dx’ ) +

I,

{IM[B,(x,x’)]}u(x’) dx’.

Taking into account eqs. (52) and (55), one obtains

{XPu}(x)= {C”}(x>

+ {SPu}(x).

(58)

(59)

According to this definition, the FHT is a real transformation, but not indexadditive:

Note that the fractional sine and cosine transforms are also not index-additive:

{ CP’+P2 u } ( x ) f {CP2 { P’u } }(x),

{ SPI +PZu } ( x ) # {S”2 { SP’u } } (x). (61)

For example, if a real object is used: {C”}(X)

=

u(-x)

f {C’+IU}(X)

= +(X)

+ u(-x)].

(62)

Index-additivity is a desirable convenient feature. It is also relevant for the grouptheoretical classification (§ 8). 5.3. IMPLEMENTATIONS

Let us now examine several additional important properties of the FHT. According to eq. (58), one can easily obtain

(Xl{X’u}}(x) = u(x).

(63)

Thus, the periodicity of the transformation is 2 and not 4 as for the FRT. This property is important and could be useful for signal compression and representation. As stated before, the FHT is real for a real input object. Thus, this transformation can be performed optically with spatially incoherent illumination since no phase information is included. Using the optical implementation based

n! §

61

OTHER FRACTIONAL TRANSFORMATIONS

289

on the shearing interferometer (Mendlovic, Zalevsky, Konforti, Dorsch and Lohmann [ 19951) the output intensity obtained is

4E,rl)=

(IBIl2+ P 2 I 2 ) /m/mhdYll(x,Y)12

/m

-m

+ 2IBlB2l

-m

-cm

--M

(64)

h d Y It(w>12cos(V),

where B I and B2 are constants, t ( x , y ) is the input transparency, and )I is defined bY

with dl and d2 the path lengths of the shearing interferometer, and k = 2n/A (A is the wavelength). In most applications of the Hartley transform (Bracewell [ 1984]), a generalization to the fractional domain is possible. In addition, a fractional correlator (Mendlovic, Ozaktas and Lohmann [ 19951) with incoherent output could be obtained using first a conventional Fourier transforming system and then placing a rotating diffuser and a shearing interferometer for performing the FHT (Mendlovic, Zalevsky, Konforti, Dorsch and Lohmann [ 19951). Actually, t h s work of Mendlovic, Zalevsky, Konforti, Dorsch and Lohmann [1995] is an important motivation for defining this transform despite the fact that it is not index-additive. Owing to the fact that it is real (the complexity of the digital calculation is decreased) and that the different fractional orders contain all the information about the object, it can be used widely in digital signal processing as well while variation of the fractional order determines whether the odd or the even part of the object is given greater emphasis. Q 6. Other Fractional Transformations 6.1. THE ABCD-BESSEL TRANSFORMATION

The so-called ABCD transformation comprises several important special cases, such as the Fourier transformation, the Fresnel transformation, and the FRT. Usually, these transformations operate with Cartesian coordinates. Here, however, we

290

[IV, 0 6

FRACTIONAL TRANSFORMATIONS IN OPTICS

use polar coordinates, which is sensible if the input is known to be rotationally symmetric, or if the angular features of the object are not significant. The ABCD-Bessel transformation has more degrees of freedom than the transformations discussed above, and yet it may be easy to implement and efficient to compute. The additional new degrees of freedom may contribute to improved filtering and pattern recognition abilities, improved image-processing performance, and to presenting a wider and more physically correct point of view for objects with angular symmetry transmitted through lossless optical systems. Note that the ABCD transformation is also known as a canonical transformation or as a generalized Huygens integral. 6.1.I . The ABCD transformation family

The general ABCD transformation can be related to the ABCD matrix of ray optics. If one uses the notations of ri, 8, for the position and the direction of the ray in the entrance to an optical system, and r,, 8, for the ray’s position and direction in the output of the system, the mathematical relation between the entrance characteristics and the output is

[el

=

:[ ]:

[;i]

The ABCD transformation of paraxial wave optics can be expressed using the following formula (Baues [1969], Collins [1970], Abe and Sheridan [ 1995a,b]):

where C‘ is a constant such that J

-00

A is the wavelength, and the coefficients A , B and D are the components of the ABCD matrix. Note that for a lossless system the determinant of the ABCD matrix must be unity, i.e., A D + B C = 1.

(69)

The special transformations with relevance to optics are named the Fresnel, the Fourier and the FRT. The special parameters to these transformations are: Fresnel D = 1, B=z, A = 1; Fourier D = 0, B=f, A Fractional Fourier D = cos( i p x ) , B = f sin( i p x ) , A here p is the fractional order.

= 0; = cos( i p x ) ;

(70)

n! 0 61

29 1

OTHER FRACTIONAL TRANSFORMATIONS

6.1.2. From Cartesian to polar coordinates

Input and output may now depend on two variables:

1

+y;) + D(x2 +y 2 )- ~ ( X O+Xyoy)] Note that an anamorphic generalization is conceivable where A, D., f D,. We will not pursue the anamorphic option here. If polar variables are introduced, xg

= rcos 8,

x = pcos

v,

yo = r sin 0,

y

= psin

f

d x o dyo.

(71) A,> and

111,

(72)

the ABCD integral transform assumes the form U( p,

v)= C ' l

00

1

2n

uo(r, 0) exp

+ Dp2 - 2pr cos(0 - v)]

(73) The three special transformations can be deduced immediately by invoking the three cases according to eq. (70). 6.1.3. The Bessel series The exponential function in eq. (73) contains a term which may be replaced by a Bessel series:

The 8 integral can be understood as a decomposition into angular harmonic components:

I'"

uo(r, 8 ) exp(-im8) d 8 = 2xum(r).

(75)

This result changes eq. (73) to: u(p, v )= 2 x c '

C exp[im(t/ - in)] m

x

lm (s) [ u,,(r)J,

1

exp g ( A r 2 + Dp2) r dr.

(76)

292

tIY § 6

FRACTIONAL TRANSFORMATIONS IN OPTICS

6.1.4. Rotationally symmetric input

A rotationally symmetric input is an important special case which benefits from conversion of the coordinates from Cartesian to polar. The angular integral is

I, 03

exp(im6’)d6’ =

2n

for m

0

otherwise.

=

0,

(77)

Thus, the resultant transform, which we term the ABCD Bessel transform, becomes: u( p ) = 2nC’

For A

=D =

=

D

=

uo(r)Jo

2nrp

1

exp g ( A r 2 + Dp2) r dr.

1 and B = z , one may obtain the Fresnel-Bessel transform:

U( p) = 2nC’

for A

lm (x) [ lm (x)[ uo(r)Jo 2 n r p

exp z ( r 2 + p’)] r dr;

(79)

0 and B =f one obtains the Fourier-Bessel transform:

u( p ) = 2nC’

lm (f ) uo(r)Jo

2nrp

Choosing A = D = cos i n p and B Fourier-Bessel transform:

r dr

= f sin

inp in eq. (78) yields the fractional

The possible advantages of the suggested transform may be summarized as follows: - Superior performance of ABCD-based optimal filtering in restoring signals degraded by linear time or space variant distortions and nonstationary noise, compared to standard Fourier based optimal filtering (Barshan, Kutay and Ozaktas [ 19971). This improvement comes at no additional computational cost. - Easier synthesis of rotationally symmetric optical systems. One has more degrees of freedom (the ABCD parameters) compared to the conventional Fourier-Bessel transform. - Simple optical implementation based on cascading a combination of freespace distances and lenses according to the desired ABCD coefficients.

n! § 61

OTHER FRACTIONAL TRANSFORMATIONS

293

- Optimal filtering for objects of rotational symmetry. The optimal Wiener filter

-

-

may be derived and applied in the ABCD domain for rotationally symmetric objects. Developing image processing algorithms having more degrees of freedom which may thus obtain improved processing operations. Existence of fast numerical algorithms based on the fast Fourier transform (FFT) method.

6.2. THE FRACTIONAL BESSEL TRANSFORMATION

As seen in eq. (70), by choosing A = cos(ippn), B = f sin(lpn) and = cos(ipp7~)one may obtain the fractional Bessel transformation by eq. (81). However, another way may exist for its fractionalization. The Bessel transformation may be defined as:

D

where Jn are the Bessel functions with integer index n, and the w,, are weighting functions. For example, eq. (76) belongs to this category. Now we go one step further by replacing the integer index n by a real-valued index p. This process of fractionalization is straightforward since Bessel functions with non-integer indexes ( p instead of n) are well known. A two-dimensional display of VJx), where the p-axis is orthogonal to the x-axis, can be obtained by means of a fairly simple optical experiment (Lohmann, Ojeda-Castaneda and Heredia [1994]). Their approach can be modified such that Airy functions or Leguerre polynomials replace the Bessel functions (Lohmann, Ojeda-Castaneda and Heredia [1996]). We do not pursue this kind of fractional Bessel transformation any further because there seem to be no applications in sight. 6.3. THE FRESNEL TRANSFORMATION AND THE FRACTIONAL TALBOT EFFECT

Fresnel diffraction is described in paraxial approximation by an integral that has been called occasionally a “Fresnel transformation” (Mertz [ 19651):

This transformation is already “fractional” since the index 2 is not confined to integer multiples of a basic 21 value. The 2 is the axial coordinate.

294

FRACTIONAL TRANSFORMATIONS IN OPTICS

tn! § 6

If the input f(x0) is periodical,

the output will assume identical values at particular depth locations (Talbot [ 18361):

Z,,, =ZO+rnZl,

2T2 21= -,

A

where T is the basic period in the input plane, A is the wavelength and m is an integer. The starting location 2, is basically arbitrary. However, the ~ ( x2 ;0 ) is especially interesting if 2 0 is a simple fraction of 21, i.e., zO/Z~ = 20/2,= . . . (Winthrop and Worthington [1965]). Those special outputs are sometimes called “Fresnel images” and sometimes “fractional Talbot images”. The associated longitudinal locations are sometimes called “fractional Talbot planes”. The formation of such fractional Talbot images may be referred to as “fractional Talbot transformation”. Fractional Talbot images are used as socalled array illuminators, which provide an array of bright spots (Lohmann [1988], Lohmann and Thomas [1990], Arrizon, Ibarra and Lohmann [1996]). Montgomery [ 19671 has shown that “self imaging” at integer multiples of Z1 occurs not only for periodic objects (eq. 84) but also for quasi-periodic objects:

3,

i,

The variety of such quasi-periodic objects is much richer than that of periodic objects. Another way to define the fractional Talbot effect (FTE) is to investigate the periodic phenomenon of light propagating through a graded-index (GRIN) medium. This medium is useful for different optical implementations related to optical communication and information transmission (Yariv [ 19851); see also 9 2.1.1. Such a medium can be regarded as consisting of infinitesimal layers in whch focusing and propagation take place simultaneously. This property is apparent on examination of the refractive-index distribution of a parabolic profile,

where r2 = x2+ y2 is the radial distance from the optical axis and nl , 122 are the parameters of the GRIN medium. As is commonly assumed, the field distribution

n! § 61

OTHER FRACTIONAL TRANSFORMATIONS

295

of interest is confined to a neighborhood of the optical axis such that the value of n(r) dictated by eq. (87) is greater than 1. By solving the ray equation one can show that a parallel bundle of rays will be focused at a distance L from the input plane, where

Thus, light propagation through a parabolic GRIN fiber along a distance of L yields a Fourier transformation. In other words, the focusing property of a GRIN medium is confirmed from a wave-optics viewpoint as well. The light distribution observed in any fraction of this length L ,

z =pL,

(89)

is by definition the p-order fractional Fourier transform (FRT) of the input light distribution. An optical implementation of this FRT was obtained using the GRIN medium fiber (Mendlovic and Ozaktas [1993], Ozaktas and Mendlovic [ 1993a,b]). Now we suggest plotting a chart that contains a continuous representation of the FRT of a signal as a function of the fractional Fourier order. This chart is termed the ( x , p ) chart (Mendlovic, Zalevsky, Dorsch, Bitran, Lohmann and Ozaktas [ 19951, Mendlovic, Dorsch, Lohmann, Zalevsky and Ferreira [ 1996]), and may be useful for various applications in optics. Using this chart, optical systems can be designed and analyzed. The spatial resolution and spacebandwidth product can be estimated. This chart actually shows explicitly the propagation of a signal through a graded-index medium. For a 1-D object, this plot contains two axes: the space and the FRT orderp. The vertical axis x is the spatial coordinate of up@).The horizontal axis is the fractional Fourier transform order, p. More explicitly, one may write W , P ) = up(x>,

(90)

where F is the chart function and up is a fractional Fourier transform of order p . As a result, in t h s plot all of the fractional Fourier orders of the original function u&) are calculated and displayed simultaneously. A diffractive optical implementation for obtaining this chart was suggested by Mendlovic, Dorsch, Lohmann, Zalevsky and Ferreira [1996]. The optical setup that produces the (x,p) chart is illustrated in fig. 19. Briefly, the input 1-D object is converted to a 2-D object using a cylindrical lens. Then follows a setup that consists of a sandwich of 2 filters separated by

296

FRACTIONAL TRANSFORMATIONS IN OPTICS

The flrst mask

‘1

The second

f\

mask

The Output plane

Fig. 19. The optical setup for obtaining the ( x , p ) chart.

2 free-space propagations. Each filter consists of many strips, each strip being a Fresnel zone plate with different focal length that is selected for obtaining the different fractional order p , and eventually the 2-D output will be exactly the (x,p)chart of the 1-D input function. The output obtained in such a chart gives the light distribution in the GRIN fiber, as a function of the distance Z :

F(x,Z/L) = u(x,Z),

(91)

where u(x,Z) is the light propagating in GRIN media. Figure 20 illustrates a

Fractional order p

Fig. 20. The computer simulation results for the ( x , p ) chart for an input of a Ronchi grating.

IY § 61

297

OTHER FRACTIONAL TRANSFORMATIONS

typical ( x , p ) chart plotted for a Ronchi grating. In this figure, one may notice a longitudinal periodicity. This effect was termed by us the fractional Talbot effect (FTE; Mendlovic, Dorsch, Lohmann, Zalevsky and Ferreira [ 19961, Zalevsky and Mendlovic [1997]). In order to analyze this phenomenon let us decompose our input object into Fourier series (eq. 84) in which T is the basic period of the object. Substituting eq. (84) into the integral definition of the FRT (eq. 19) yields

After changing the order between the integration and the summation, one obtains

Calculating the last integral,

" x i ) exp (

Ah tan@

=

exp [-2nix0

(&

-

a)]

b o

xn + 2ni- T cos - niAfi tan @

Czexp

(94)

where C2 is a constant. After eliminating the phase factors (in any case, one observes only the intensity in the ( x , p ) chart), one obtains

where C is a constant. For a reconstruction of the input we require that all Fourier terms have the same phase factors, thus

Ah tan $ = m,

2T2 where m is an integer. Substituting eq. (89) into the last equation gives:

z=2L n

2mT2

(96)

(97)

298

[N§ 6

FRACTIONAL TRANSFORMATIONS IN OPTICS

Using L = f i d 2 from eq. (24), the previous relation may be rewritten as:

L)

2L n mT2 z= tan-’ ( n

.

In the limit of free-space propagation, L goes to for x -+ 00, in eq. (98), yields

(T )

2L nmT2 z ; %

=

00. Using

tan-’( l/x)

= (l/x)

2mT2 ~

A ’

which corresponds to the classical Talbot formula (eq. 85, with ZO= 0). A comparison of eq. (95) with eq. (84) shows that a reconstruction of the input is obtained, but with a change of the lateral period by a factor of C O S ~ , and the longitudinal periodicity is obtained in planes placed according to a tan-’ relationship. This result is clearly visible in fig. 20, where a Ronchi grating was used as the input. The quasi Talbot images at longitudinal locations according to tan-’ show decreasing lateral periods. 6.4. THE FRACTIONAL LEGENDRE TRANSFORMATION

The Legendre transformation is useful in optics if one wants to compute the point spread function P(x’). The wave aberration w(x) and the amplitude transmission A(x) of the pupil are assumed to be known. Mathematically, the Legendre transformation consists of two steps:

In the first step one composes the pupil function P(x), which describes the complex light amplitude if the object is a point source on axis. The second step, which describes the light propagation from the pupil plane to the image plane, is apparently an optical Fourier transformation. The integral is solved in approximation by the method of stationary phase (Felsen and Marcuvitz [ 19731, Bryngdahl [1974]). The phase is stationary where its derivative is zero: d[kw(x) - 2n~x’/(Af)]

dx

= 0.

(100)

This equation says in essence that the light at a particular output location x’ comes from a point xo in the pupil plane as defined by

IV, § 61

OTHER FRACTIONAL TRANSFORMATIONS

299

The amplitude at that point depends on the pupil amplitude and on the wavefront curvature, which is proportional to the second derivative of the wave aberration:

Recently, Alonso and Forbes [ 19961 have generalized the Legendre transformation by modifying the second step of eq. (99). They replaced the ordinary Fourier transformation by a fractional Fourier transformation. Hence, the result will be the light distribution within a GRIN fiber at a distance z =pL, where L is a so-called quarter pitch length. (The GRIN fiber produces a classical Fourier transformation at z = L). 6.5. ALTERNATE FRACTIONAL FOURIER TRANSFORMATION

This subsection serves to demonstrate that the process of fractionalization is not unique. We present an alternate fractionalization procedure of the Fourier transformation. We also compare the traditional FRT with the alternate FRT.

6.5.1. The alternate concept Our task is to define a function V ( x ,4) such that it coincides with the input u&) if 4 = 0, and with its standard Fourier transform il&) if 4 = n/2:

The phase 4 is related in the usual manner to the fractional indexp as 4 = p d 2 . We use here reduced coordinates in order to simplify the derivations x + x/A and A2 = ,If. One ought to remember that scaling is not a trivial issue for the FRT (Dorsch, Zalevsky, Mendlovic, Bitran and Lohmann [ 19951). Equation (103) implies that:

The traditional FRT was based on two different procedures, which yielded the same results. The first procedure related the fractional index p to the length of a GRIN fiber. The second procedure did relate to a rotation in Wigner space by an angle Q = p n / 2 . Here, we are influenced by the fractionalization of the Hilbert transformation. The FHiT output turned out to be a linear superposition of the input and the

300

FRACTlONAL TRANSFORMATIONS IN OPTICS

[IV, § 6

classical Hilbert output (40). Now in a similar manner we define the alternate FRT as: V(x, $) = uo(x) cos $ + &(x) sin $.

(105)

The two conditions of eq. (103) are clearly satisfied. 6.5.2. Comparison of the two FRTs

We will compare the traditional FRT with the alternate FRT in two ways. First we show the two ( x , p ) charts (Mendlovic, Dorsch, Lohmann, Zalevsky and Ferreira [1996]) and then we show what happens in Wigner space. We use a “Gabor object” as input:

For the computer simulation we have chosen a = 30 pixels, X = 20 pixels and V = 10 pixels. The (x,p) chart for the traditional FRT in fig. 21 shows qualitatively how light propagates in a GRIN fiber. The ( x , p ) chart for the alternate FRT consists of separated islands, as seen in fig. 22. Figure 23a shows the WDF display of the input (eq. 106); this WDF display is valid both for u(x,p = 0) and for V(x,$ = 0). In fig. 23b, one sees the WDF of u(x, in),which corresponds to a rotation by in.The WDF of V ( x , in)in fig. 23c consists of three parts: the WDF for $ = 0, the WDF for $ = in,and an interference term midway between the two other islands.

Fig. 21. The (x,p) chart of a Gabor object.

IV, § 61

OTHER FRACTIONAL TRANSFORMATIONS

301

40

6’3 ??

2

4

6

8

10

12

14

16

18

20

Fig. 22. The ( x , p ) chart using the alternate defirution.

6.5.3. From two to four contacts of the two FRTs

The alternate fractionalization procedure required two contacts between V(x,9) and the classical Fourier transformation (eq. 103). Unfortunately such a contact does not exist any longer at 9 = n and at $I = +n: V ( X , n)= -uo(x),

V(X,

(107)

~ T C= ) -iio(x).

So far, the alternate FRT had only two contacts with the classical Fourier transformation, at 9 = 0 and at 0 = in.Now we want contacts at four index phases: 9 = 0, :n,fn,in.This is achieved with a somewhat different fractionalization procedure: V,(x, 9) = Vo(x) +A(x) cos 9 + B(x) sin I$+ so that V2(x,n)= uo(-x),

V,(x, Sn)= iio(-x),

C(X)

cos 29,

(108)

3 02

FRACTIONAL TRANSFORMATIONS IN OPTICS

which yields 4vo(X)

= U o ( X ) iUo(-X)

4C(x) = uo(x)+ uo(-x)

+ &(X) -

-b fiO(--X),

iio(x)- iio(-x),

2 4 ) = UO(X) - uo(-x),

2B(x) = iio(x) - Go(-x).

The four contacts at 4 = 0, in,n,in are not the only condition one might impose upon fractionalization. One might require, for example, additional contacts with the FRT at 4 values of in,in,:n and :n.Functional values at other 4 values may be defined, for example, by the periodic version of the sampling theorem. Interpolation based on the sampling theorem corresponds to developing V(x,Cp)in a Fourier series in Cp with kernels exp(2nim/N) and with 2N + 1 x-dependent coefficients. The essence of this subsection is that the process of fractionalization can also be considered as an interpolation procedure. Just as there exist many different interpolation procedures, there also exist many different kinds of fractionalization. Each kind is “optimal” according to some criterion, e.g., experimental convenience, algorithmic efficiency, or relevance for a particular application.

9 7.

Fractional Filtering

7.1. FRACTIONAL CORRELATION, CONVOLUTION

The conventional correlation, mostly implemented by means of the Fourier transform, has been extended to the so-called ‘fractional correlation’ (Mendlovic, Ozaktas and Lohmann [1995]). The algorithm for performing a fi-actional correlation and the optical configuration are shown in fig. 24. It consists of the multiplication of the fractional transforms of the two signals to be correlated

Fig. 24. Algorithm for obtaining a generalized fractional correlation and its optical configuration.

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303

followed by a third FRT. Analytically, the operation of fractional correlation of an input function, f ( x ) , with a reference pattern, g(x), is defined as follows: q7,.p*,,&’)

= FP3{FP‘ {f(x))

FP2{g(.>>>.

(111)

The parameters P I ,p2 and p3 are the orders of the FRTs which are in principle arbitrary. For simplieing the exponent of the transformation integral that describes the output, detailed by Mendlovic, Ozaktas and Lohmann [ 19951, the most obvious choice is: PI = p ,

p2=-p,

P3 = - I ,

(1 12)

with P ranging from 0 to 1. In this case, if input and reference are identical and real-valued, a perfect phase matching between object and reference FRTs in the fractional domain is obtained. The inverse Fourier transform will just focus the resulting plane wave. In order to build optically a fractional correlator, instead of preparing a full setup containing two lenses and free propagations, the object is illuminated with a converging beam (Garcia, Dorsch, Lohmann, Ferreira and Zalevsky [ 19971). This permits the change of the convergence phase factor, multiplying the object, by displacing it along the optical axis. The matching between the object-filter distance and the convergence of the beam may produce any desired order and scaling factor. Hence, this approach is more convenient for the experimentalist, as the exact sizes of the input and filter transparencies are often not determined precisely. This is especially important for the case of using spatial light modulators (SLMs) for implementing the filter. As the FRT is not exact there will be a quadratic phase factor multiplying the output plane. This means that the correlation plane will be displaced along the optical axis. The definition of the anamorphic fractional correlation is a straightforward extension of eq. (1 11). Analogously to the standard case, it is obtained by inverse Fourier transformation of the product between the anamorphic FRT of the target and the input image. A possible setup for performing the anamorphic fractional correlation is depicted in fig. 25. The proper adjustment of distances a, and a, will provide the FRT of the desired order in the filter plane. The scale factor of the FRT is variable, as a parameter independent of the order. Due to their relatively high price, limited availability, and poor performance, we expended some effort to reduce the number of cylindrical lenses used in the optical setup. In the chosen configuration, only three cylindrical lenses and one spherical lens are used. The price to be paid for this simple setup is that the aspect ratio of the FRT (quotient

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FRACTIONAL TRANSFORMATIONS IN OPTICS

f”

f”

Input

FRT

output

Fig. 25. Experimental setup for obtaining the anamorphic fractional correlation

between the x and y scale ratios) cannot be adjusted. A modification can be made to avoid this problem. It consists of inserting an additional anamorphic imageforming system (that provides different magnifications in both axes) creating a stretched image of the FRT plane. The output of this imaging system is then taken as the input for the inverse transforming subsystem. Nevertheless, this additional complexity can be avoided in most practical cases. According to figs. 25 and 3, since the Z distance is equal for both axes, one may write

Thus the aspect ratio (A.R.) between the two axes is

7.2. THE FRACTIONAL RADON TRANSFORMATION

The Radon transform is a mathematical tool that assists in obtaining the structure of a 3-D object from its projections. As such, it became important and useful in tomography (Wood and Barry [1994a]). Over the years, the Radon transform revealed several essential properties that might be useful in pattern recognition and signal processing (Wood and Barry [1992, 1994b]), and especially in optical signal processing systems (Clack and Defrise [ 19941, Easton, Ticknor and Barrett [ 19841, Woolven, Ristic and Chevrette [1993], Barrett [1982], Feng and Fainman [ 19921). In this subsection, a novel fractional transformation that we term the fractional Radon transform is defined (Zalevsky and Mendlovic [ 1996a1). This transform generalizes the Radon transform and combines it with the FRT. Both

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Fig. 26. A central slice at angle 6 of the 2-D Fourier transform of an input object.

transformations are useful tools for invariant pattern recognition, tomography and signal processing. Some of the properties of the new transformation, as well as further directions for investigation, are presented. The motivation for this new transformation is based on the famous “central slice theorem”, which reveals a fundamental property of the conventional Radon transform: If one denotes by g(s,8) the Radon transform of a certain twodimensional function, and performs a 1-D Fourier transform with respect to the parameter s, the function obtained equals the central slice, at angle 8, of the 2-D Fourier transform of the input object. This means the 2-D Fourier transform in polar coordinates equals the 1-D Fourier transform of the Radon transform. The 1-D Fourier transform is performed on the radial axis s and corresponds to a certain angle 8. This property is illustrated in fig. 26. The same property, when defined with the FRT, will provide the fractional Radon transform. Thus, if one denotes by gp(s,8) the fractional Radon transform of a certain function, and performs a fractional Fourier transform with respect to the parameter s, the function obtained equals the central slice at angle 8 of the 2-D fractional Fourier transform of the object. Now, let us derive the above arguments mathematically. The mathematical definition of the conventional Radon transform is

g(s, 0) = Rf =

1:

/ I f ( x , y ) 6(x cos 8 + y sin 8 - s) dx dy,

(1 15)

where f ( x , y ) is the input object, and the 6(. . .) is the Dirac delta function. An equivalent definition is DC1

g(s, 8) = L m f ( s cos 8 - u sin 8, s sin 8 + u cos 8)du,

(116)

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[Iv,

5

7

Fig. 27. The Radon transformation.

where s = x cos 8 + y sin 8,

u = -x sin 6 + y co s 6.

(117)

Thus the Radon transform is a projection of a 2-D object at certain angles, as illustrated in fig. 27. Here are some important properties of the conventional Radon transform: (1) Linearity: R { f + g } = Rf + R g .

(118)

(2) Rotation of the input object by an angle 80 will rotate the Radon transform by the same angle 80, i.e., the Radon transform is rotationally invariant. (3) Shift property: Rf(x-xO,y-yo)

=g(s-xocos&y0sin8,6),

(1 19)

where g(s, 0) is the Radon transform off(x,y). (4) The I-D Fourier transform with respect to s of the Radon transform of an object equals the central slice, at angle 60, of the 2-D Fourier transform of this object:

where G(E,8) is the 1-D Fourier transform according to the s-axis of the Radon transform of f(x,y) and F ( c , 0) is the 2-D Fourier transform of f(x,y), at polar coordinates.

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Since the last property is fundamental for the definition of the fractional Radon transform that follows, we will prove it. By defimtion,

G(E,0) =

g(s, 0) exp(-i2@s) ds. -a

After substitution of the definition of g(s, 0) one obtains f m f ( s cos 8 - u sin 0, s sin 0 + u cos 0) exp(-i2nEs) ds du.

G(E,0)=

Since in angular rotation transformation the Jacobian is unity, we may change the variables and obtain

G(E,0) =

/

rcu

F00

/-,f(x,

-03

= F(E cos 0,

y) exp[-i2x(xE cos 0 + yE sin 0)] dx dy

(123)

E sin 8),

and the argument of eq. (120) is proven. Now we can define and analyze the fractional Radon transform. Using the definition of Lohmann [ 19931 for the FRT, we obtain

where @ = i p n and ( u , u ) are the Cartesian coordinate set. F,(u,u) is the FRT off(x, y) expressed in Cartesian coordinates. In polar coordinates this equation may be written as

-2xi -(xs sin @ where (E,t9) is the polar coordinate set according to u=Ecos8,

u=Esin0

1

cos 0 + yE sin 0) dx dy, (125) (126)

FJE, 0) is the fractional Fourier transform of f(x,y) expressed in polar coordinates. After defining x = s cos 0 - u sin 0,

y

= s sin 0 + ucos 19,

(127)

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FRACTIONAL TRANSFORMATIONS IN OPTICS

we obtain

F ~ ( Ecos 8, E sin 8 ) =

[I 1:

f(scos 8 - u sin 8,s sin 8 + u cos 8)

and by definition,

Fp(Ecos 8, E sin 8 ) = Gp(E,O),

(129)

where GJE, 8) is the fractional Fourier transform of the fractional Radon transform. Thus after performing the -p order FRT with respect to the E parameter, we obtain

where g&, 8) is the fractional Radon transform of f(x,y). One may notice that for fractional order p = 1 the conventional Radon transform is obtained. According to eq. (1 15) the previous equation may also be written as

s,I , 3

g&, 6) = Rpf =

0

,

f ( x ,y ) 6(x cos 8 + y sin 8 - s)

1

cos 8 - x sin 8)2 dx dy. (131) Here we summarize some of the mathematical properties of the fractional Radon transform which we denote by Rp (1) Linearity:

(2) Rotation of the input object by an angle of 60 will rotate the fractional Radon transform by the same angle 60. Thus, the fractional Radon transform is rotationally invariant. (3) Shift property. In contrast to the conventional Radon transform, in the fractional Radon there is no relation as in eq. (119). The amount of shift variance is controlled by the fractional order p .

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(4) The FRT, with respect to s, of the fractional Radon transform of an object at angle 6, is the central slice at angle 6,)of the 2-D fractional Fourier transform of this object:

where G,,(E,6) is the FRT, with respect to s, of the fractional Radon transform of f ( x , y ) , and Fp(€j,8) is the 2-D FRT of f ( x , y ) in polar coordinates. An example of a conceivable application is given by Zalevsky and Mendlovic [1996a]. They show that minimizing of the mean square error obtained after filtering nonstationary signals, is related directly to the fractional Radon transform. Another possible application for the fractional Radon transform is in pattern recognition. Due to its mathematical properties, which are illustrated in this subsection, the transform can perform a rotationally invariant pattern recognition with controllable degree of the shift property. The actual implementation of the fractional Radon transform is feasible with optical hardware. Taking into account the extensive calculations required for digital computation of the fractional Radon transform, the optical option seems to be very attractive. It could be based on setup similar to that suggested and tested for performing the conventional Radon transform (see for example Woolven, Ristic and Chevrette [1993]). 7.3. THE FRACTIONAL WIENER FILTER (FWF)

In this subsection a nonstationary Wiener filter [termed the fractional Wiener filter (FWF)] is suggested for restoring degraded reference objects in a fractional correlation systems (Zalevsky and Mendlovic [1996b]). Our goal is to get an output as similar as possible to the input. The performance of the new filter is sometimes superior to that of the conventional Wiener filter. The suggested optical setup includes the conventional fractional correlation scheme of fig. 24 for the specific orders PI = p, p2 = p, p3 = -p. This selection o f p l , p2 andp3 is the most appropriate for noise removal (see 5 2.3). The configuration detailed by Garcia, Dorsch, Lohmann, Ferreira and Zalevsky [1997} may be used in experiment, with the FWF filter placed in the FRT filter plane. Since the FRT is a shift-variant operation, a stationary input signal becomes nonstationary after passing through the filter. Thus, the correlation expression

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no longer depends on the space (or time) difference lj, but will depend on the specific spatial location (or the specific time):

where @ denotes the fractional convolution, h(x) is the suggested filter, E { . . .} is the ensemble average, v is the input signal being embedded with noise, and 2 is the obtained reconstruction of the original reference signal u. The notations XI and x2 denote different spatial locations, or different times if temporal signal processing is involved. The fractional convolution (according to eq. (1 9) and fig. 24 with PI = p2 = p , p3 = -p) might be written as

-2ni -v(a+@-xX) sin Q

!

dadodv,

(1 35) where Q = ; np and p is the fractional order. Note that the coordinates used (a, /?, Y and x ) were normalized by Using eqs. (135) and (1 34), one obtains

m).

In the same manner, using eq. (135) one obtains

and

where R;, is the cross-correlation between the reconstructed signal and the original signal u. The mean square error after the filtering stage is defined as:

N,

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7.3.1. Fractional spectral densify

In order to handle stationary signals, a power density is used. Conventionally, power density is the Fourier transform of the normalized autocorrelation. The transformation is done according to the space (or time) difference parameter E. The above treatment is good for stationary signals, but it fails for nonstationary ones, since the correlation of those signals depends on the space (or time) values themselves, and not only on their coordinate difference. Let us define the fractional spectral density (FSD) as the 2-D FRT of the correlation expression. Note that for a one-dimensional nonstationary signal the FSD is two-dimensional. At this point, the FSD is similar to the Wigner transform. Thus, the FSD of R ; ( x I , x ~is)

where (Y I ,v2) are the spectral coordinates, Su,,(Y I ,19) is the FSD of &(XI, X Z ) , p is the fractional order and H is the FRT of the filter signal. In the same manner, one obtains

7.3.2. Mean square error minimization The mean square error is a 2-D function since it depends on ( x I , x ~ )If. one examines the FSD SeP(vl, 212) at Y I = -v2 = Y , one obtains

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[IV,

a7

The minimum as a function of H J Y ) for fixed Y is found at the zero value of the derivative:

Note that in 5 7.3.4 we will derive a similar relation using a different approach (the calculus-of-variation method). There, a more rigorous mathematical handling is provided to derive eq. (145). The expression given in eq. (145) is the FWF that should be placed in the fractional domain. Thus the filter signal itself is the inverse FRT of H J Y ) :

Talung into account that

one obtains

According to the definition of the fractional Radon transform presented in eq. (1 30), one may write

where R,,(O,-in)is the fractional Radon transform at coordinate s = 0 and angle -in.The fractional Radon is examined at -in since the axis of vI = -v? = Y lies at -$nin the 2-D fractional domain. In the above differentiation, the FSD of the mean square error was minimized. Thus, according to eq. (149), R,(O,-in) was actually minimized as well. What is really the meaning of minimizing the fractional Radon transform at coordinate 0 and angle -in? The answer is illustrated schematically in

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\

Fig. 28. Schematic illustration o f the fractional Radon transform.

fig. 28. The two-dimensional function drawn in this figure is the mean square error Re(x1,x2). Since the fractional Radon transform is the projection of the 2-D function at the proper angle (as examined in 9 7.2), therefore RJO, -in) is the projection of the 2-D function along the line tilted at 45" in fig. 28. Thus, the projections (the summed values of the 2-D functions) along this line have been minimized. In the stationary case of p = 1, the criterion for the minimizing procedure was R,(O), which is a certain dot along the line tilted at 45" in fig. 28. Since in the stationary case all the dots along this line are equal (this line corresponds to = 0), it does not matter which dot exactly is being minimized (minimizing one dot will retroactively minimize all the projections along this line). In the nonstationary case, the components of the 2-D function R,(xl ,x2) are not uniform along this line, and minimizing only a certain value is not general enough. It is more appropriate to minimize the whole sum of the values along this line (R,(x,x)); this is actually minimizing R,,(O,-in). 7.3.3. Computer simulation

The aim of this computer simulation is to calculate the error expression given in eq. (149) as a function of the fractional order. The purpose of the fractional filter demonstrated in this simulation is to restore a signal that has been distorted by additive noise. Figure 29 illustrates the autocorrelation of the input signal R,(c),

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FRACTIONAL TRANSFORMATIONS IN OPTICS

Fig. 29. The autocorrelation of the signal RL,(E),

3

Fig. 30. The autocorrelation of the noise I?,!($).

which was chosen to be a low-pass, i.e., the correlation decreases with increasing distance difference. Figure 30 illustrates the autocorrelation of the noise Rn(E), as a weak chirp. Due to the high frequencies existing in the noise the correlation changes rapidly with increasing distance difference. Note that for a noise with the given correlation function, an improvement is expected. Since the FRT of a chirp can be made to be a delta function if an FRT with the proper fractional order is performed (Dorsch, Lohmann, Bitran, Mendlovic and Ozaktas [ 1994]), chirp noise can be filtered easily. On the other

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‘.n 0.5

0.351

-52.0 0.5

1-

P

Fig. 3 1. The error expression as a function of the fractional order p

hand, in the Fourier domain the chirp signal can not be filtered without also destroying parts from the signal’s spectrum. We assumed that the noise is additive and not correlated with the signal: Ruo(E)= Rl,(E) and R,(Q = R,,(E) + Rn(E).Thus the following expressions were used:

Figure 3 1 illustrates the error expression as a fknction of the fractional order p . The minimal error is seen to be obtained for p = 0.86. As a result, the optical setup should be designed as a fractional convolver of fractional order p = 0.86. The ratio of the errors of the FWF (for p = 0.86) and the conventional Wiener filter (for p = 1) is 0.77. Thus the FWF improves the restoration of a noisy signal by 23%, which is a good result. Thus in the case demonstrated, the shift invariance property was traded for improved restoration ability of the suggested filter. 7.3.4. An alternative way to derive the FWF

In this subsubsection the expression for the optimal filter is derived using the calculus of variations method. Let us assume that the filter function h ( x ) can be expressed as

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where h,(x) is the optimal filter and ~ ( xis) a functional perturbation. We will assume that E ( X ) is real and thus

q v ) = E:@),

(152)

where C J Y ) is the FRT, with fractional order p , of E(x).The FRT of h,(x) will be denoted by Hop(v). We now introduce eqs. (151) and (152) into the FSD expression (144): Sep(vl,Y ) = M(v1, Y ) + M , ( Y I , v21,

M ( v1, v2) = Sop (YI , Y W U P (YI >q;", (v2)

+ Su,,(vl?v2)- Suu, (vl

9

v2>H~*_, (v2) - Surc,,(vl

9

v2)Hup(vI

1 3

(153)

M,(YI, Y )=S"/,(vl,v2) [HOP(YI)CP(V2) + H,*F(v2)€p(vI)]

+ Suu/,(vl,v2Np(v2)- ~ u u , , @ I , v2154v1). Here we assumed that the variations ep(v)are small. Rewriting the previous equation in the spacehime domain yields

T,(x I ,x2) =

sWsW

Me(v1, ~

-03

2 &)( X I

,x2, YI ,v2)dvl d ~ 2 ,

-02

where B,,(xl, x2, YI, v2) is the two-dimensional FRT kernel, defined in eqs. (22) and (23). Our aim is to reduce T, to zero. After a change of variables, and using the relation B p ( X I , X Z , YI, v2) = Bp(xl,x2,v2, YI),

(155)

eq. (154) may be rewritten as:

1, w

T,

=

c13

dvl c.(vI)/

[ Su~,(Y,v2)Ho*,(v2)+Su,,(Y, v1)KIp(v2)

-02

In order to obtain optimal filtering one should force T, to be zero independent of f p ( YI >. This condition requires Su,J(vi,v2)H,*_,(v2)+SUp(~2, Yi)K+,(v2) = SuU,,(vi, v2)+Suu,,(v2, VI).

(157)

v2) = So,,(v2,YI)and Suu,(v~, 1/21 = S,,,,(Y, v1). In For real inputs it is Su,(v~, addition, iff(x) is a real function then Fp(v)= Y P ( v )(where F J v ) is the FRT

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off(x) with fractional orderp). Thus for YI= -15 = --Y the optimal filter H,,(Y) obtained according to eq. (157) corresponds exactly to that obtained in eq. (145). 7.4. THE FRACTIONAL WAVELET TRANSFORMATION

The wavelet transformation is a tool successfully used in dealing with transient signals, data compression, bandwidth reduction (Kaiser [ 19941) and timedependent frequency analysis of short transient signals (Caulfield and Szu [ 1992]), optical correlators (Szu, Sheng and Chen [1992], Lu, Katz, Kanterakis and Caviris [ 1993]), sound analysis (Martinet, Morlet and Grossmann [ 1987]), representation of the human retina, and representation of fractal aggregates (Freysz, Pouligny, Argoul and Arneodo [ 19901). The different wavelet components are scaled and shifted versions of the mother wavelet. Mathematically, the wavelet operation is equivalent to performing a Fourier transform of the input function, multiplying it by a differently scaled Fourier transform of the wavelet mother function, and eventually performing an inverse Fourier transform (Sm, Sheng and Chen [ 19921). Commonly, the mother wavelet function h(x) is a typical window function multiplied by a modulation term. The scaled and shifted versions of h(x) are the daughter wavelets h,h(x):

where b is the amount of shift, a is the scale parameter and J;; is the normalization factor. A typical wavelet mother fimction is the Morlet wavelet function (Martinet, Morlet and Grossmann [ 19871). The definition of this function is h ( x ) = 2 cos(2nh.x) exp ( - ; x 2 ) ,

(159)

and its Fourier transform is ~ ( u= )2 x { exp[-2n2(u

--f012]+ exp[-2n2(u

+ f ~ ) ~.] }

(1 60)

This function is real and non-negative. A one-dimensional wavelet transform of a signalf(x) is defined as (Daubechies [19901)

Note that eq. (161) has a form of correlation between the input signalf(x) and the scaled and shifted mother wavelet function ha&). This fact is the basis for

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[IV, § 7

the optical implementation of this transformation. Since each wavelet component is actually a differently scaled band pass filter (BPF), the wavelet transform is a localized transformation and thus efficient in the processing of transient signals. If the input is decomposed into several wavelet components and reconstructed back, the mean square error between the original and the reconstructed images is not too high, even when a restricted number of wavelet components is used. This property was implemented successfully for digital image compression and transmission. The reconstruction of an image from its wavelet decomposition is done by

C should be h t e (admissible condition) in order to make the transformation reversible. Although extensive calculation is needed for performing those operations, acceleration of the computation may be achieved by using optics. In optical implementation, it is quite easy to obtain a continuous shift parameter. However, the scaling parameter a can be varied only in discrete steps. The wavelet transformation with the continuous b and discrete variable a is termed the hybrid wavelet transformation. Using the hybrid transform, a more convenient definition of the wavelet transformation for optical implementations (see 0 7.4.3) is commonly used. The reconstruction formula of the hybrid case 1s:

Note that the scale factors a were chosen as powers of 2 in order to obtain a fast computing algorithm. In the one-dimensional hybrid wavelet transform a filter with a strip structure is used, with each strip corresponding to a different scaling parameter of the mother wavelet (Zhang, Li, Kanterakis, Katz, Lu, Tolimieri and Caviris [1992], Szu, Sheng and Chen [1992], Sheng, Roberge and Szu [1992]). When a twodimensional transform is wanted, a problem arises regarding where to obtain the result corresponding to each scaling factor. To solve this problem a multireference approach was used by Mendlovic and Konforti [1993], i.e., the Fourier

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domain was separated into rings corresponding to the different scales of wavelet function, while each ring contained a different grating that aimed the transform’s result at different spatial positions in the output plane. The problem is that the different scale versions of the mother wavelets overlap, and thus in order to create the rings of the Fourier plane, the scaled mother wavelets are approximated by rings with a rectangular profile. This approximation is not too exact and thus a different approach based on a replication of the input was suggested (Mendlovic, Ouzieli, Kiryuschev and Marom [ 19951). Here, the spectrum of the input function is replicated using a grating and then filtered by the differently scaled mother wavelets, which are located in different spatial positions according to the replications of the input’s spectrum. The disadvantage of this approach is that a large spatial region is needed in order to obtain the wavelet transform for several scales of the mother wavelet simultaneously. A different approach for implementing the two dimensional hybrid wavelet transform is to multiplex the different scales of the mother wavelet by different wavelengths (Garcia, Zalevsky and Mendlovic [ 19961); this requires a spatially coherent illumination that contains several wavelengths. In this subsection, a novel approach for processing transient signals and image compression is suggested. The new approach generalizes the conventional wavelet transform by using the FRT instead of the standard Fourier transformation. Inspection of eqs. (21) and (23) reveals that the FRT is a localized transformation, similarly to the wavelet transform. Here, we define a new transformation, termed the fractional wavelet transform (FWT) (Mendlovic, Zalevsky, Mas, Garcia and Ferreira [1997]). The FWT adapts the localization of the signal, using the FRT, to the localization needed by the wavelet transformation. When one says ‘localized’ in t h s context, it means that the input function is actually multiplied by a space window as is done in the Gabor transform (Gabor [1946]). In the FRT case the space window is a phase window (the chirp phase function exp[in(x2 + x’*)/(AJ tan@)]) and not an amplitude window as in the original Gabor transform. In a chirp function the local frequency increases with the distance from the center. The fast oscillating contributions from far away are wiped out during the integration. Hence, the phase window is equivalent to an amplitude window. In this way by controlling the amount of localization in the FWT, during the reconstruction (FWT followed by inverse FWT) may reduce the mean square error. Thus, less wavelet components need to be stored in order to achieve the same reconstruction error.

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7.4.I . FWT - Mathematical definition In order to adapt the localization existing in the FRT to the localization existing in the wavelet components, we suggest the following definition for the fractional wavelet transform: Performing an FRT with the optimal fractional order p over the entire input signal and then performing the conventional wavelet decomposition. For reconstruction, one should use the conventional inverse wavelet transform and then an FRT with the fractional order of -p in order to return back to the plane of the input function. A schematic chart of the FWT is presented in fig. 32.

El-EHz

T LLFl

"-order ;RT

(inverse'

Wavelet

transrorm

''ore

transtorm

-

Fig. 32. Schematic illustration of the FWT.

Mathematically, the FWT may be formulated as following:

where W(P)(a,b) is the FWT and B,, is defined by eqs. (22) and (23). Note that for p = 0 the FWT becomes the conventional wavelet transform. The formula for back reconstructing the input is

s,1" I ,

1 " O "

f(x)=

c

m l

- W"'(a,b)B,(x,x')ha,b(X')dadbdx',

a2

(166)

where C is given in eq. (163). The hybrid FWT will be f(x)=

1

n=m

lm 1, 9

2 n = - m -m

" 1

W(,,)(2",b) B-,(x,x') hZn,b(x')dx' db.

(167)

7.4.2. Computer simulations Several computer simulations were carried out in order to demonstrate the performance of the new transformation. In fig. 33a an input transient signal is illustrated. This signal contains a chirp structure which is complicated to compress conventionally. We used one daughter wavelet function for the decomposition of the signal (only with the scaling factor

I

I 1

I

1I

\

1 140

en

.

(d)

15,

\

-

.

100

120

1

(e)

Fig. 33. (a) Input signal; (b) wavelet daughter function used; (c) reconstruction usmg FWT with one daughter function; (d) reconstruction using conventional wavelet with one daughter function; (e) reconstruction using conventional wavelet with five daughter functions.

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[Iv, § 7

a = I), and reconstructed it back. Figure 33b shows the wavelet daughter function (a Morlet function). For reconstruction, the hybrid wavelet transform was used. After reconstruction, the error was calculated according to eq. (168). With the conventional wavelet, when only one daughter function is used, the unnormalized reconstruction error defined by eq. (168) is 0.92. For obtaining the FWT, the optimization of the selected FRT order was done using a trial-and-error algorithm. The fractional order p of the FWT is selected such that the mean square error between the original input and the reconstructed input is minimal. Indeed, this optimization step may require extensive calculations. However, this step needs be done only once. This led to an FRT order of 0.5 which finally provided a FWT reconstruction error of 0.68, an error reduction of about 27%. Note that this fractional order was the optimal order, in the sense of minimal mean square reconstruction error, for this input signal. This error is defined as

wheref(x) is the input signal andf'(x) is the reconstructed signal. Figure 33c illustrates the reconstruction obtained with the FWT with only one wavelet component (one scaling factor). Figure 33d shows the reconstruction obtained with the conventional wavelet with only one component. In Figure 33e, one may see the reconstruction obtained with the conventional wavelet transform with five scaling factors (five daughter functions). Note that even then the obtained reconstruction error is 0.78 (bigger than the error obtained with a single scaling factor of FWT). 7.4.3. Optical implementation As indicated in fig. 32, the amount of calculation required for performing the FWT with a reasonable number of pixels is large; this is mainly due to the optimization stage for finding the optimal fractional order. A significant acceleration of the FWT calculation can be achieved by performing some or all of the stages using optics. The suggested optical implementation is illustrated in fig. 34. This figure contain two parts. In the first part the temporal signal is fed into an acoustic optical cell which converts the temporal signal into a one-dimensional spatial signal. A one-dimensional FRT of the input pattern is performed according to the setup suggested by Lohmann [1993]. Note that the FRT is obtained by bulk optics implementation. However, a graded-index fiber may be used instead (Mendlovic and Ozaktas [ 19931). The

IV, P 71

323

FRACTIONAL FILTERING

WalO H(aZ r) Wa3 r)

Acousto

I ,

_-

TheFRT

output

Wavelet match filter

_-

2

f,

f.

f.

Fig. 34. Optical implementation of the FWT.

second part of the setup performs optically the wavelet transform of the onedimensional signal. This is done by Fourier-transforming the spatial information, multiplying by a wavelet matched filter for the one-dimensional signal, and then performing an inverse Fourier transform (according to the configuration suggested by Zhang, Li, Kanterakis, Katz, Lu, Tolimieri and Caviris [ 19921). The Fourier transform is done using cylindrical lenses since the information is one-dimensional. Spherical lenses are used in order to achieve imaging in the other spatial axis. The wavelet matched filter contains several strips. Every stripe represents a one-dimensional Fourier transform of the scaled mother wavelets, but each strip corresponds to a different scaling of the mother wavelet. The scale parameter a varies along the vertical axis as is defined by the bank of the strip filters. In the output plane one obtains a two-dimensional representation of a one-dimensional wavelet, where the horizontal axis represents the continuous shift parameter b and the vertical axis the discrete scale parameter a (Zhang, Li, Kanterakis, Katz, Lu, Tolimieri and Caviris [1992], Szu, Sheng and Chen [1992], Sheng, Roberge and Szu [1992]). Note that for two-dimensional input signals the multiplexing approach may be applied in order to implement the two-dimensional FWT. The multiplexing may be spatial via Dammann gratings (Mendlovic, Ouzieli, Kiryuschev and Marom [ 19951) or spectral via wavelength multiplexing (Garcia, Zalevsky and Mendlovic [ 19961). In a similar manner, the inverse FWT may be implemented optically. This time the optical setup will contain the inverse wavelet transform first and then the FRT with fkactional order -p (Mendlovic and Ozaktas [ 19931). Experimentally, one would use fractional order 4 - p instead of -p.

324

tIY § 8

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€8. j Other Aspects of Fractionalization

This section contains a collection of topics which do not belong to the main stream of the whole chapter. That does not diminish their importance. On the contrary, lively activity related to some of these topics may be expected in the future. 8.1. FRACTIONALIZATION AS "TERF'OLATION

The extension of the various operators F(n)mentioned earlier (with n = 0,1,2,3; mod 4) to the fractional domain operator F ( p ) (with p real) can be thought of as an interpolation. In fact, that was the idea o f the Gedanken experiment by which a GRIN fiber of Fourier length was broken (fractionalized) into pieces. Similarly, the rotation of the Wigner function, which so far was done in 90" steps, was now performed in arbitrary fractions of 90". The angle $ of rotation was related to the fractional degree p as Q = i p n . The old and the new operators are required to coincide i f p is an integer, especially for p = 0 and for p = 1. This requirement may be satisfied, for example, by

This linear interpolation becomes inappropriate if the range of p extends beyond ( 0 , l ) . Somewhat better is a trigonometric interpolation because it obeys the modulo (Ap = 4; A$ = 2x) property: F ( p ) = cos $F(O)+ sin $F( 1).

(170)

However, this definition is unsatisfactory at the other contact values ( p = 2 and p = 3): F(2) = -F(O),

F(2)[u(x)] = -u(x),

F(3) = -F(l),

F(3)[u(x)] = -S(v). (171)

The proper results should be F(2)[u(x)] = u(-x),

F(3)[u(x)]= S(-v).

(172)

Contacts at all four critical parameters ( p = 0,1,2,3) is possible as was shown in 5 6.5.3. That result may be written as F ( p ) = G + cos QA + sin QB + cos 2$C.

(173)

Iv,

D 81

325

OrHER ASPECTS OF FRACTIONALIZATION

The operations G, A , B and C are linear combinations of the four basic operations F(n). For example, 4C = F ( 0 ) + F(2) - F ( 1) - F ( 3 ) .

(174) Another interpolation strategy is based on some symmetry properties of the FRT: u(x,p + 2) = u(-x,p), and if the input is real-valued, u(x, 0) = u*(x, 0), then

(175)

-PI = u*(x,p>. (176) Hence, knowledge of u ( x , p ) within the p-range of (0, 1) is sufficient. We may now base the u(x,p) interpolation on the samples at p = 0, 1, 4x9

u(x, 4) = A(x) cos @ + B(x) sin 9

i, i:

+ C(x) cos 34 + D(x) sin 34.

(177)

The functions A , B , C and D are linear superpositions of u(x,mx/4) for m = 0, 1,2,3. Interpolations of this kind can be useful if one wishes to compute the so-called ( x , p ) chart (a two-dimensional display of the u(x,p)).The sampling step dp depends on the particular input u(x,O). The viewpoint of eigenmodes within a GRIN fiber is appropriate here. Dickinson and Stieglitz [ 19821 investigated some properties of the digital Fourier transformation in a quite different context. Their tools included somewhat similar interpolation procedures. 8.2. THE FRACTIONAL FOURIER-KRAVCHUK TRANSFORMATION

In the classical Fourier analysis the variables (time, space, . . . ) are considered to be continuous, whereas in digital Fourier analysis the variables are discrete. Neither of these approaches fits the experimental reality directly. Sources, spatial light modulators and CCD detectors consist of arrays of finite components. Nearly 70 years ago, Kravchuk [ 19291 modified Fourier analysis for such a finitedicrete situation. Recently, Atakishiyev and Wolf [ 19961 combined the Kravchuk concept with the fractional Fourier concept. 8.3. COMPLEX FRACTIONAL INDEX

Bernard0 and Soares [1996] modified the FRT by letting the fractional index become complex: p

= p’

+ ip”.

(178) The consequences resemble “apodization”. Lohmann, Mendlovic and OjedaCastaneda [1997] studied the impact of a complex degree on the fractional

326

FRACTIONAL TRANSFORMATIONS IN OPTICS

[Iv,

59

Hilbert transformation. That approach places the Hilbert transform filtering and the knife-edge filtering as special cases under a common roof. 8.4. THE SIGNIFICANCE OF PHASE AND AMPLITUDE IN THE CONTEXT OF THE FRT

It is widely believed that the Fourier phase is much more significant that the Fourier amplitude (Oppenheim and Lim [1981]). This belief is far from being universally true. Some counterexamples, but not all of them, occur if the Fourier input is no longer real and non-negative. A recent study about phase and amplitudes, in the context of the FRT, provided some insights and suggested that one must be cautious when making guesses about the relative significance of phases and amplitudes (Lohmann, Mendlovic and Shabtay [ 19971). 8.5. ESTIMATION OF THE FRT DEGREE p

Suppose an FRT operation has been performed by means of a piece of GRIN fiber, whose parameters, including p , are not known to the observer. Abe and Sheridan [ 19961 provided a tool for estimatingp from the input and output data. 8.6. FRACTIONAL, AND FRACTAL,

These two terms clearly share the linguistic root “to break”. Hence, it is not surprising that several authors searched for connections. Two research teams (Berry and Klein [ 19961, Hamam and de Brouguenet de la Tocnaye [ 19941) studied the fractal images produced at fractional Talbot distances, thus continuing earlier work by Bryngdahl [1973]. The original grating cell is scaled by 1/24 (A4is integer) and then replicated 24 times withm the original cell area. Alieva [1996] used the FRT as a tool for the investigation of fractal objects.

8 9. Fractionalization and Group Theory 9.1. MOTIVATION

Group theory has been very valuable for the evaluation of quantum mechanics and of elementary particle physics. Perhaps group theory may become similarly valuable for optics? So far the group theory has been used only seldom in optics. Lenz [ 19891 did apply group theory for pattern recognition. Abe and Sheridan [ 1994a,b] put the

IY § 91

FRACTIONALIZATION AND GROUP THEORY

327

FRT and some other optical transformations into the framework of group theory. We will follow their lead, partially. A very small amount of group theory will suffice to understand how the FRT is related to the canonical ABCD transformation. We will recognize the equivalence of the ABCD transformation and “Wigner algebra”. On that base we can clarify the difference between the FRT and the Fresnel transformation. Of particular value to us is the concept of isomorphism. It means that two different groups are related to each other such that any process within group A has as counterpart a particular process within group B. “Isomorphsm” implies unique correspondence in both directions. A very famous isomorphism is due to Descartes, who discovered the correspondence between linear algebra and analytic geometry. For example, two linear equations with three unknowns correspond to two planes, which intersect along a straight line in 3D space. The benefit of such an isomorphism is that a certain problem, which is difficult within group A , might be easy to solve within group B. 9.2. ELEMENTARY GROUP THEORY

A group G consists of elements g,,, (m = 1,2,. . . , N ) . N is the ordel; which might be finite or infinite. The elements of G may be numbers or functions, or transformations, operators, or elementary particles or states. A collection of such elements is called a group if these elements obey certain rules. First, there should exist a procedure called “multiplication” (actually, the “multiplication” could be an ordinary arithmetic addition or a convolution). The first rule requires that the product gk of two element is again a member of that group: gm gn ‘

= gk.

The product should be unique: g , . gn = unique.

The “multiplication” should be associative: gm ’ (8, ’ gk) = (gm gn) .gk. ’

There should exist an identity element E Egm

gmE

c G:

= gm.

Every element g, should have an inverse element g;’ c G such that: -1

gm . gm

= gm

.gkl

= E.

328

FRACTIONAL TRANSFORMATIONS IN OPTICS

[IV, 9: 9

Pairs of elements may, or may not, “commute”: g,, .g,, 7 2 g,, .gn7.

(1 84)

If they do, the group is called “Abelian”. It is an essential feature of quantum mechanics that many pairs of operators do not commute. So far, the index m,which identifies a particular element of the group, has been an integer. We may replace the integer index m by a real-valued index p . That is what happened when we generalized the ordinary Fourier transformation to the fractional Fourier transformation. Hence, the meaning of “fractionalization” may be described as gm

(185)

+gpy

where m is an integer and p is a real number. The range (or “support”) of m or p may be finite or infinite. It is often convenient to relate a finite range to an angular coordinate @, which leads to a periodic or cyclic group: =

tpn,

0 < p < 4,

0 < @ < 2n.

( 186)

Further on, we will encounter the important term “subgroup”. For example, the group of integers can be split into two subgroups, the even integers and the odd integers. Both subgroups obey the ordinary law of “multiplication”. Both subgroups are groups of their own, because they satisfy eqs. (179)-(183). Both have “unity” as identity element, eq. (1 83). Finally, we introduce the term “isomorphism”. Two groups g,, and h,, are isomorphic if there exists a one-to-one unique correspondence between pairs of elements. The pairing is regulated by a procedure Q and its inverse Q-’: Q[g,n]

Am,

Q-‘ fhnJl

=gm.

(187)

Such an isomorphism might be quite beneficial if, for example, the elements g,,, are closely related to an experiment, while the elements h,, are more abstract. However, the multiplication of h, elements might be much simpler than the corresponding multiplication of g,, elements. 9.3. THE CANONICAL ABCD TRANSFORMATION AS A GROUP

Here, the element of the group is an integral operator and not the function which is operated upon:

IY 5 91

329

FRACTIONALIZATION A N D GROUP THEORY

The constant factor C‘ is fixed by the requirement of “energy conservation”:

J-00

J-CG

“Multiplication” means in this case to perform this transform integral twice in a row, first with the parameters A I , B I Cl,Dl, , and then with the parameters A2, B2, Ci, D2. The outcome is again an integral of the same structure, but with different parameters A , B, C‘, D. The identity operator (182) is the special case of A = 1, D = 1 and B going asymptotically to zero. In that case the integral kernel is a particular version of the Dirac delta. The inversion operator, which compensates the ABCD transformation (188) such that u(x0) is retrieved, can be deduced from the ‘‘multiplication’’. The parameters A l , Bl and Dl are supposed to be known. The parameters A2, B2 and DZ have to be adjusted such that the parameters A , B and D of the joint transformation are the identity parameters A = D = 1, and B goes asympotically to zero. 9.4. ISOMORPHISM OF THE ABCD TRANSFORMATION AND THE “WIGNER ALGEBRA”

The procedures Q and Q-’ which relate isomorphic pairs of elements are in this case the Wigner transformation and its inverse: ~ ( xv) , =

f

00

J -00

u(x + ;x’)u*(x

-

ix’) exp(-2nivx’)

dx’,

W ( $, Y)exp(2nivx) dv.

The W is the Wigner distribution function (WDF), which is described in more details in the appendix. We insert the ABCD integral (188) into the WDF definition and obtain W ( X Y) ,

+

W(DX- B Y ,AY +x( 1 - AD)/B).

(191)

What the ABCD transformation does to a function u(x) is clearly equivalent (“isomorph”) to a simple change of coordinates in Wigner space:

The parameters A , B and D in this matrix are the same as in the integral transformation (188). Here, in the group of coordinate transformations in

330

FRACTIONAL TRANSFORMATIONS IN OPTICS

Ҥ 9

Wigner space, these parameters cause such processes as rotating, shearing, squeezing, etc. The Wigner space may be somewhat abstract, but it is certainly a convenient arena for ABCD transformations. It replaces the integral transformation by a simple vector-matrix operation. The determinant of this matrix happens to be unity. 9.5. SUBGROUPS

IN BOTH ISOMORPHIC DOMAINS

We continue to ignore the parameter C’, which is determined firmly by the concept of energy conservation (eq. 189). The canonical, or ABCD, transformation is a three-parameter group, with the parameters A , B, D. We will list below several subgroups, with 0, 1 or 2 free parameters. The first three subgroups, called “identity”, “magnification” and “coordinate inversion” (u(x) -+ u(-x)) rely on the particular version of the Dirac delta that we encountered earlier in 9 9.3. The next four subgroups are called “Fourier”, “Fresnel”, “lens” and “FRT”. These four subgroups are termed by us “mathematical transformations”. They have counterparts (numbered 8-1 1) which we call “optical transformations”. This implies, for example, that in the case of the optical Fourier transformation df is considered as a free parameter, which is relevant for any optical experiment such as described by:

s, ‘x

u(x’) exp(-2nix’x/df) dx’ = fi(x/df).

The “mathematical” Fourier transformation is a subgroup of the “optical” Fourier transformation because it is the special case where the experimental parameter df is fixed to be unity. Table 1 surveys eleven subgroups by defining their ABD values and their numbers of free parameters F . The occurrence of B = 0 is meant asymptotically such that the particular version of the Dirac delta emerges. There has been some confusion in the literature about the question of whether the FRT and the Fresnel transformation are one and the same thing. The FRT has apparently one more free parameter, whether we compare the mathematical versions (numbers 5,7) or the optical versions (numbers 9,ll). In other words, the two transformations in question are distinctively different groups. An equivalent statement can be based on the different Wigner matrices which are the isomorphic counter parts to what appears in the list. The Wigner matrix that corresponds to the ABCD transformation has been derived as eq. (192). The 11 particular matrices whch correspond to the 1 1

IV, § 91

33 1

FRACTIONALIZATION A N D GROUP THEORY

Table 1 Subgroups in both isomorphic domains Group number

Characteristics

Number of free parameters

1

Identity A = D = 1, B=O

0

2

Magnification A = M , D = 1/M, B+O

1

3

Coordinate inversion A = D = -1, B+O

0

4

Fourier (mathematical) A=D=O,B=l

0

5

Fresnel (mathematical) A = D = 1,B= 1

0

6

Lens (mathematical) A = 1, D = 1 - B , B-0

0

7

FRT (mathematical) A = D = cos 4, B = sin Q

8

Free parameters

Expression

u(x)

M

u(x/M)

1

Q

u,(x)

Fourier (optical) A = D = O ,B = L f

1

Y

iqX/af)

9

Fresnel (optical) A=D=l,B=Lz

1

az

&(x; az)

10

Lens (optical) A = 1, D = 1 - B / L f , B-0

1

Af

u ( x ) exp(-im2/kf)

11

FRT (optical) A = D = cos Q, B =

2

$3

up(x;

sin Q

Ah)

particular subgroups are shown in the following equations; they are obtained by inserting the ABCD parameters from table 1 into eq. (192): 1 0

[o

11’

[

1/M 0 0 MI’

[

-1 0

0 -11’

[;;], [:, 3 [; ;I, [;;;; [ -f]’[i [ 1A ; -j

7I‘:-

1

3

[si:iAj-

-Af

sin @ cos4

I.

(195) (196)

Notice that the Fresnel matrix describes shearing, and the FRT matrix means rotation. This underlines the distinction between the Fresnel transformation and

332

FRACTIONAL TRANSFORMATIONS IN OPTICS

“v, 0 9

the FRT. More details about the geometrical deformations will be shown in the appendix. 9.6. THE INHOMOGENEOUS CANONICAL TRANSFORMATION

Two simple but important transformations are the “shift” and the “tilt”: u(x)

---f

u(x - E ) ,

u(x) -+ u(x) exp(2nixF).

(197)

We call it “tilt” because the last equation describes what happens to the object amplitude if the illuminating plane wave is tilted. These two transformations can be incorporated into the ABCD integral transformation and also into the isomorphic Wigner formalism, where E and F appear as an additional vector:

The explicit incorporation into the integral transformation can be found, for example, in the book by Wolf [1979], and in the paper by Abe and Sheridan [ 1995a,b]. We conclude this subsection by relating an (almost) arbitrary Wigner matrix to the particular matrix that is the isomorphic counter part to the ABCD transformation:

-”I

[(I-~D)/B A

=

[::]

( 199)

Suppose a, b, d are known and c is determined by requiring the determinant to be unity; then the transform parameters are:

A

= d,

B

-b,

D

= a,

c = - 1-AD

B

’

Equations (199) and (200) are the “dictionary” that ties together the two isomorphic groups, called ABCD transformation, and “Wigner algebra”. 9.7. SOME OTHER INTEGRAL TRANSFORMATIONS AND THEIR GROUPS

The fractional Hilbert transformation (4 3 ) is clearly a one-parameter cyclic group. According to its first definition, this parameter is the optical phase 9 of the spatial filter. In physical terms, the process of multiplication means the cascade

IY § 101

CONCLUSIONS

333

of two FHiT systems, with phases $, and $ 2 . The first filter is imaged on the second filter. Hence, the filter phase of the total system is simply @ I + $2. We may as well define “multiplication” as addition of filter phases, modulo 2n. All other features of a group discussed in this section can be verified easily. Other parts of the present chapter on fractional transformations may also be cast into the framework of group theory. But we stop here by repeating: Fractionalization is the conversion of a group with integer indices into a group with real-valued indices. The former group is a subgroup of the latter and the latter group may evolve from the former group by interpolating real-valued indices between the integer-valued indices.

Q 10. Conclusions

As we have seen, “fractionalization” occurs if an integer index m is replaced by a real-valued index p . The index identifies a specific member of a “family” of mathematical elements, such as a linear transformation. In most cases, the “family” obeys the criteria of a “group”. The mathematical procedure of fractionalization has been motivated by demands in theoretical physics, in signal processing, and in optics. According to Hida [1980] and Kolzow [1996], it was Norbert Wiener [1927] 70 years ago, who fractionalized the Fourier transformation in support of Weyl’s [ 19271 studies on group theory in quantum mechanics. Ten years later Condon [1937] rediscovered the FRT, as did Bargmann [1961] 25 years later. In the 1980s Namias [1980], Mcbride and Kerr [1987] and Kerr [ 19881 took up the issue again for the benefit of quantum mechanics. The FRT was rediscovered again in the signal processing community (Almeida [ 19941) and independently in the optics community (Ozaktas and Mendlovic [ 1993a,b], Mendlovic and Ozaktas [ 19931, Lohmann [1993]). Two different optical approaches were found to be equivalent in mathematical terms (Mendlovic, Ozaktas and Lohmann [ 1994a,b]). Bernard0 [1996] went one step further by allowing the fractional index p to be complex. More recently, fractionalization has been studied not only in the Fourier context but also for several other transformations with relevance for optics. In some instances, existing groups of transformations were identified as fractionalizations of some integer-index groups. On other occasions, new transformations were invented by fractionalization of existing groups. Both approaches created new insights and new opportunities. We hope that our article will stimulate the generation of more opportunities and insights.

334

FRACTIONAL TRANSFORMATIONS IN OPTICS

[IY APP. A

Acknowledgments The authors apologize to all those whose publications may not have been mentioned properly in this article with a finite length. We would like to acknowledge many fmitful interactions with S. Abe, M. Alonso, P. Andres, H.H. Barrett, H.J. Caulfield, R.G. Dorsch, C. Ferreira, J. Garcia, A. Kutay, K. Leeb, E. Marom, J. Ojeda-Castaneda, J.T. Sheridan, B.H. Soffer and E. Tepechin. Haldun Ozaktas deserves a special appreciation for his contribution as a member of the trio that found in 1991 the fractional Fourier optics. Gal Shabtay was very helpful in various ways. A.W. Lohmann also acknowledges support by the DFG.

Appendix A. About the Wigner Distribution One of the most commonly implemented phase space representations is the Wigner distribution function (WDF) (Wigner [1932]). The WDF may be considered as a wave generalization of the “Delano diagram” which is also known as the Yo representation. The Y O diagram is a ray model in which the Y-axis represents the lateral spatial location and the o-axis represents the direction of the ray (the derivative of the first coordinate). The WDF is useful in many fields, such as dual time-frequency processing (Wood and Barry [1994c]) and data compression (Lohmann, Dorsch, Mendlovic, Zalevsky and Ferreira [ 19961). A special importance of the WDF is related to optics, since it is a powerful tool for designing and analyzing optical systems (Mendlovic, Zalevsky, Dorsch, Bitran, Lohmann and Ozaktas [ 19951, Lohmann [1995]). A nice example for an introduction to the WDF comes from the music area (Bartelt, Brenner and Lohmann [ 19801). Neither the representation of music as a function of time nor the representation as a function of frequency is suitable for a musician. Music is displayed as a function of time andfrequency (logarithmically). The pianist knows at every instant of time what note he has to press. Bastiaans [1978] deserves the credit of having introduced the WDF into the area of optical information processing. A. 1. DEFINITION

In its 1-D version, the WDF is a mathematical operation applied to the input field distribution u(x):

IV, APP. A1

ABOUT THE WIGNER DISTRIBUTION

335

The symbol W denotes the WDF operator and W,(x,f,) is the Wigner chart, a two-dimensional display. Since this transform represents simultaneously spatial and spectral information of the function, it takes into account diffraction phenomena as well. For a onedimensional input signal the WDF is a two-dimensional chart presenting the spatial and the spatial-spectrum information of the input (termed the spatial Wigner distribution function SWDF). The Wigner representation is not linear but bilinear, i.e., W{alul(x) + a2u2(x>> =

la112W{uI(x)}+ Ja212W{u2(X)}

+ 2 J,RE [aI a; u I (x + ix’)u; (x jx’)] exp(-2ni~x’)dx’ M

-

f

la1 I2W{UI(X))

+ la212W{u2(x)},

(202) where RE is the real operator. The reconstruction of a function from its Wigner chart can be performed based on: l

u(x) = __

u*(O)

1 w(+~,L)

o

o

e x p ( 2 n i ~ xd) f ~ .

(203)

-02

Note that there is an unknown constant phase factor when performing the inverse WDF. The magnitude of u(O)* is known if x = 0 is inserted into eq. (203). The WDF is also useful for handling optical temporal signals. Based on similar considerations as for the SWDF, the “temporal WDF” (TWDF) is defined as

1, m

Wr(t,J) =

u(t + it’) u*(t - it’) exp(-2niht’) dt’,

where u(t) is the temporal input signal andh is the temporal spectrum coordinate. The inverse TWDF is defined as u(t) =

l ~

u*(O)

/

o

o

-M

it,^) exp(2ni~;t)d ~ .

(205)

Present technology offers many possibilities for temporal optical signal processing, especially in communication applications. Several examples are: demultiplexing of incoming data (Nuss, Li, Chiu, Weiner and Oartovi [ 1994]), femtosecond pulse shaping (Weiner, Heritage and Kirschner [ 19881) and image compression (Nuss and Morrison [ 19951). Recently, space-time devices such as pairs of gratings, time lenses and dispersive media were employed to design

336

FRACTIONAL TRANSFORMATIONS IN OPTICS

[IY APP. A

temporal signal processing systems (Akhmanov, Chirkin, Drabovich, Kovrigin, Khokhlov and Sukhorukov [ 19681, Lohmann and Mendlovic [ 1992a,b], Kolner and Nazarathy [1989]). The TWDF may be an attractive tool for handling such systems. A.2. PROPERTIES

Let us assume the SWDF definition given in eq. (201), and develop some important mathematical properties of the Wigner distribution function. Note that a complete summary reference of the WDF properties is not available although some of the following properties have been mentioned in previous publications (Ozaktas and Mendlovic [ 1993a,b]). A.2.1. Fourier representation If a Fourier representation of a signal is inserted, m

u(x) = one obtains

I,W;)

exp(2nihx) dL,

A.2.2. Projection properties The projections of the Wigner distribution function onto the principal axes have well-known physical meanings:

A.2.3. Shljting the object Shifting the input object, u(x)

+

u(x - xo),

IV, APP. A1

ABOUT THE WlGNER DISTRIBUTION

I

331

Y-shearing X-shearing

t

Fig. 35. The X- and the Y- shearing operations over a rectangle

results in a shifted Wigner distribution: W$(X?fx)= WAX -xo,fx), where Wj(x,fx) is the WDF of the shifted object. A.2.4. Tilting the wavefont Tilting the wavefront of the input signal by ug(x) = u(x) exp(-2xifox), wherefo is related to the tilt direction, yields W;(x,fx>

=

Wx(x,f* +All>

where W; is the WDF of ug, A.2.5. Lens operation

The lens operation is a multiplication with a chirp function

where f is the focal length of the lens. Using the definition of eq. (201) it can be easily seen that

where W i is the WDF after the lens operation. Note that this operation is the shearing operation along thefx-direction as seen in fig. 35.

338

[IV, A m . A

FRACTIONAL TRANSFORMATIONS IN OPTICS

A.2.6. Free-space propagation

The fi-ee-space propagation (FSP) along a distance z can be expressed as a multiplication of the spatial spectrum of the signal by a chirp function:

iiFSPvx) ii(h> exp(-idzA22>.

(217)

=

Using the definition of eq. (207) it is easily seen that GSP(X,h> = WAX

- W,h).

(2 18)

Note that WJsp is the WDF after the free-space propagation. This operation is the shearing operation along the x-direction as seen in fig. 35. A.2.7. Fractional Fourier transformation Performing an FRT of a function with fractional order p causes a counterclockwise rotation of its Wigner chart by an angle i p n . A.3.

A LOOK BACK

We have seen that the basic process of physical optics (free-space propagation, passage through a lens, Fraunhofer difiaction, and so on) can be described nicely as simple deformation of the Wigner distribution function. Figure 36 is a graphic summary of those WDF deformations and their corresponding signal processing operations.

+xgEkypx V

T

V

?

a,.

r(

+$LX

V

?

4FT b

+bx

Fig. 36. Summary of the WDF deformations.

IVI

REFERENCES

339

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E.WOLF, PROGRESS IN OPTICS XXXVIII 0 1998 ELSEVIER SCIENCE B,V ALL RIGHTS RESERVED

V PATTERN RECOGNITION WITH NONLINEAR TECHNIQUES IN THE FOURIER DOMAIN BY

BAHRAM JAVIDI' AND JOSEPH L. HORNER'.~

' University of Connecticut, Department of Electrical and Systems Engineering,

260 Glenbrook Rd., U-I57 Storrs. CT 06269-2157, USA 2Air Force Research Laboratory, Hanscom Air Force Base, Bedford, M A 01 731-5000, USA

343

CONTENTS

PAGE

§ 1. INTRODUCTION

3 2.

. . . . . . . . . . . . . . . . . . . 345

NONLINEAR JOINT TRANSFORM CORRELATORS . . . .

§ 3 . MULTIOBJECT DETECTION USING BINARY JOINT

TRANSFORM CORRELATORS . . . . . . . . . . . . .

347 353

§ 4 . COMPOSITE FOURIER-PLANE NONLINEAR FILTERS . . .

359

§ 5.

ILLUMINATION DEPENDENCE OF BNARY NONLINEAR JOINT TRANSFORM CORRELATOR . . . . . . . . . .

§ 6.

CHIRP-ENCODED JOINT TRANSFORM CORRELATORS .

§ 7.

RANDOM PHASE ENCODED JOINT TRANSFORM CORRELATOR . . . . . . . . . . . . . . . . . . . .

390

§ 8.

SECURITY VALIDATION AND SECURITY VERIFICATION .

398

9

370 .

9 . SUMMARY . . . . . . . . . . . . . . . . . . . . . .

375

404

ACKNOWLEDGEMENT . . . . . . . . . . . . . . . . . . .

405

LIST OF SYMBOLS AND ABBREVIATIONS . . . . . . . . . .

405

Appendix A . PERFORMANCE METRICS . . . . . . . . . . . .

409

Appendix B. FREQUENCY-DEPENDENT THRESHOLD FUNCTION METHODS . . . . . . . . . . . . . . . . . . .

411

REFERENCES . . . . . . . . . . . . . . . . . . . . . . .

416

344

8

1. Introduction

Most optical pattern recognition systems are either spatial filter based processors (Vander Lugt [1964], Goodman [1996], Vander Lugt and Rotz [1970], Kozma [1966], Flannery and Horner [1989]) or joint transform correlators (JTC) (Weaver and Goodman [1966]). We will show in this chapter that an important advantage of the JTC over filter-based processors is the ability to perform nonlinear transformations on the Fourier magnitude of the input scene and the reference signal (Javidi [ 1989a,b], Javidi and Horner[ 1989a1). The disadvantage of a conventional JTC is a reduction in the effective space-bandwidth product of the system due to the display of the reference fimction in the input plane. However, space variant thresholding in the Fourier plane (Javidi, Wang and Tang [1991]), chirp encoding techniques (Tang and Javidi [1993a]), andor subtraction techniques (Javidi, Wang and Tang [ 19911) can remedy this problem. The correlation function at the output of a JTC is obtained by applying a Fourier transform operation to the joint power spectrum. The joint power spectrum, which is the intensity of the Fourier transform of the input scene displayed alongside the reference function, can be obtained by using an intensity device, such as an optically addressed SLM or an optoelectronic device such as a CCD detector. We will show in this chapter that nonlinear transformation of the joint power spectrum, such as binarization, results in very good correlation performance. We call JTCs with nonlinear transformations in the Fourier-plane nonlinear JTCs (Javidi [ 1989a,b], Javidi and Horner [ 1989a1, Javidi, Wang and Tang [1991]). We will show that for many typical images, Fourier-plane nonlinear transformations are necessary to produce a correlation peak larger than the output noise sidelobes (Javidi, Wang and Tang [1991]). Compared with the conventional (linear) JTC (Weaver and Goodman [ 1966]), the Fourier-plane nonlinear transformations improve the correlation performance in terms of light efficiency, correlation peak height, and sensitivity against similar objects. An additional advantage of the nonlinear JTC is hardware realization. Binary JTCs can be implemented easily using binary or high-contrast optical devices, which are readily available commercially. For the first-order harmonic term of the nonlinearly transformed joint power 345

346

NONLINEAR PATTERN RECOGNITION TECHNIQUES IN THE FOURIER DOMAIN

[v, $'

1

spectrum, the Fourier phases of the reference function and the input scene are recovered. Nonlinear transformations modify the Fourier magnitudes of the reference function and the input scene. Correlation-like functions with a variety of characteristics can be generated in the output plane. For example, using certain nonlinear transformations, it is possible to remove the Fourier magnitudes of both the input scene and the reference function (Javidi, Wang and Tang [1991], Refiegier, Laude and Javidi [1995]). In this case, the nonlinear JTC processes the input scene and the reference function based on their Fourier phases only. This is different from the phase-only filter (Horner and Gianino [1984]) where only the Fourier magnitude of the reference function is removed. We will show that the ability of the nonlinear JTC to perform nonlinear transformations on the Fourier magnitudes of the input signal and the reference signal can significantly improve the detection process. In this chapter, we modify the classical composite filters designed for distortion-invariant image recognition to be used in the nonlinear JTC. We find that the performances of these composite filters are improved substantially by applying the Fourier-plane nonlinear techniques. We investigate the correlation performance of binary joint transform correlators with unknown input-image light illumination. The correlation performance of binary joint transform correlators with unknown input-image light illumination is investigated for different thresholding methods used in the Fourier plane. We show that these techniques can provide tolerance to variation in input illumination. A disadvantage of the conventional JTC is the reduction in the spacebandwidth product of the input because both the input image and the filter function are presented at the input plane. In this chapter, we show that using a chirp-encoded JTC technique we can remedy this problem. This technique produces the self-correlation terms and the cross-correlation terms in separate output planes along the optical axis. For multiple-object detection, the intercorrelation terms between the similar targets in the input scene are generated in one output plane, The desired cross-correlation terms are generated in a different output plane. This technique reduces the effect of the on-axis autocorrelation terms on the desired cross-correlation terms. As a result, a better peak-tosidelobe ratio is produced. A technique that produces similar results uses random phase encoding at the Fourier domain for joint transform correlators. A random phase encoded JTC can reduce the undesired self-correlation terms, the redundant cross-correlations, and the high-order harmonic terms. These terms reduce the system space-bandwidth product. Finally, we show an application of all the above technology: the use of

v, § 21

NONLINEAR JOMT TRANSFORM CORRELATORS

341

nonlinear JTCs for security verification of credit cards, passports, and other IDS so that they cannot be easily reproduced. In 9 2, we review the linear (conventional)JTC (Weaver and Goodman [1966]), the nonlinear JTC, kth-law nonlinear JTCs, and binary nonlinear JTCs (Javidi [ 1989a,b], Javidi, Tang, Gregory and Huson [ 19911, Tang and Javidi [1992], Javidi and Wang [1991]). In 9 3, we investigate multiobject detection using binary joint transform correlators (Javidi, Wang and Tang [1991]). In 3 4, the composite Fourier-plane nonlinear filters for distortion-invariant pattern recognition are investigated (Javidi and Painchaud [ 19961, Javidi, Wang and Zhang [1997]). In 9 5, the illumination dependence of binary nonlinear JTCs using different thresholding methods are discussed (Javidi, Li, Fazlollahi and Horner [1995]). In 9 6, we investigate chirp-encoded joint transform correlators (Tang and Javidi [1993a]). In 9 7, we investigate random phase encoded joint transform correlators (Zhang and Javidi [1993], Zhang [1995]). Section 8 explores the idea of a system based on nonlinear JTCs for security verification of credit cards, passports, and other IDS (Javidi and Horner [1994a], Javidi [ 1997a,b]).

0

2. Nonlinear Joint Transform Correlators

In this section, we discuss the effects of nonlinear transformations in the Fourier plane of a JTC (Javidi [1989b]). We begin with a brief review of the linear (conventional) JTC (Weaver and Goodman [ 19661). The discussion of general types of Fourier-plane nonlinear transformations and kth-law nonlinear transformations in a JTC will follow. 2.1. LINEAR JOINT TRANSFORM CORRELATORS

The input plane contains the reference signal r(x-xo,y) and the input signal +xg, y ) which are displayed side-by-side, The joint power spectrum of the two input images is the intensity (absolute value squared) of the Fourier transforms of the two inputs:

s(x

E(a,B>= IFT{r(n:-xo,y)+~(x+xo,y)}12

P>+ R 2 @ ,0) (2.1) + S(a,B) exp[-j@S(a,PI1 R(a, exp[+j@R(a, exp(-jbOa) + S(a,P) ex~[+j@s(a, PI1 R(a,P) e x ~ [ - j @ ~ (B>1 a , exp(jLoa), (a,B) are the spatial frequency coordinates, and S(a,In) exp[j@s(a,P)] = S2@,

where and R(a,/3) exp[j@R(a,P)] correspond to the Fourier transforms of the input

348

NONLNEAR PATTERN RECOGNITION TECHNIQUES IN THE FOURIER DOMAIN

[y 5 2

and reference signals s(x,y) and r(x,y),respectively. In the conventional case, the inverse Fourier transform of eq. (2.1) can produce the following correlation signals at the output plane:

where

In eq. (2.2), the first two terms are the on-axis autocorrelation terms. The terms of interest are the third and fourth terms, which are the cross-correlations of the reference signal with the input signal. 2.2. ANALYSIS OF NONLINEAR JOINT TRANSFORM CORRELATORS

The nonlinear JTC uses exponential techniques in the Fourier plane to transform the joint power spectrum. Its optoelectronic hybrid implementation and alloptical implementation are shown in fig. 1. The nonlinear characteristic of the device is denoted by g(E) where E is the joint power spectrum. An expression for the nonlinearly transformed joint power spectrum can be obtained using the transform method of communication theory (Davenport and Root [1958], Papoulis [1984]). Let the Fourier transform of the nonlinear characteristic of the device be defined by (Goodman [1996])

G(w)= T g ( E ) exp(-jwE) dE. -M

The output of the nonlinear system is given by the inverse Fourier transform relationship: g(E) =

1

/

G(w)exp(jwE) dw.

(2.5)

v, 0 21

349

NONLINEAR JOINT TRANSFORM CORRELATORS

Reference Input image r(x,y) SC ne S(W) Fourier

Frequency Plane

7

source

Reference Input Sky) image r(x,y) I Fourier Coherent light point source

Frequency plane

I

I

Collimating lens

/I

Beam

Fourier Transform Lens

I

Correlation output

I),

\I-

-1

Input plane using electronically or optically addressed spatial light modulator

modulator with nonlinear characteristics

Fig. 1. (a) Optoelectronic and (b) all-optical implementation of a nonlinear joint transform correlator.

350

NONLINEAR PATTERN RECOGNITION TECHNIQUES IN THE FOURIER DOMAIN

[y 9

2

The nonlinearly transformed joint power spectrum can be obtained by substituting in eq. (2.5) for E , given by eq. (2.1). The output of the nonlinear device can be written as (Javidi [1989b])

Here, Eu=

{

1 for u = 0 2 for u < 0

and J, is a Bessel fimction of the first kind, order u. It can be seen that for u= 1, the nonlinear system ..as preserved ‘---e phase of the cross-correlation term [$s(a,B) - $p,(a,/?)I and only the amplitude is affected. This results in the good correlation properties of the first-order correlation signal at the output plane. Varying the severity of the nonlinearity will produce correlation signals with different characteristics. For highly nonlinear transformations, the lugh spatial frequencies are emphasized and the nonlinear JTC becomes more sensitive in discrimination against similar targets. We now discuss the effects of using a kth-law nonlinearity in the Fourier plane of the JTC (Javidi [1989b]). For a full-wave odd kth-law device, g ( E ) = lElk sgn(E), where E is the joint power spectrum and sgn is the signum function. The linear JTC and the binary JTC correspond to the cases of k = 1 and k = 0, respectively. Other types of nonlinear JTCs are obtained by selecting the value of k and varying the nonlinearity. Given that the nonlinearity operates on the cross product terms of the joint power spectrum only, the output of the nonlinear device is given by

where c u k is a constant, and k is the severity of the nonlinearity. It can be seen from the above equation that each harmonic term is phase-modulated by

v, 8 21

NONLINEAR JOINT TRANSFORM CORRELATORS

35 1

u times the phase difference of the input signal and the reference signal Fourier transforms, and the higher-order correlation signals are diffracted to 2x0. The correct phase information of the joint power spectrum is obtained for the firstorder harmonic term (u = 1):

glk(a,

6 )S(a,P)lk cos[hOa + fbS(a,P)- f b R ( a , 611.

= Cuk[R(a,

(2.8)

Varying the severity of the nonlinearity k will produce correlation signals with different characteristics. For highly nonlinear transformations (small k ) , the high spatial frequencies are emphasized. 2.3. BINARY NONLINEAR JOINT TRANSFORM CORRELATORS

In the binary nonlinear JTC, the Fourier transform interference intensity is thresholded according to a threshold value VT before the inverse Fourier transform operation is applied. The binarized joint power spectrum can be written as (Javidi [ 1989b], Javidi and Horner [1989a,b], Javidi, Wang and Tang [1991], Javidi and Wang [1991], Javidi and Kuo [1988])

Given that [R2(a,B) + S2(a,P)I, [ ~ ( P) a ,S(a,P)I, $~s(a, P) and fb~(a, P) are slowly varying compared to cos(2xoa), and that V T is a predetermined threshold value, the binarizing action of the thresholding device converts the amplitude-modulated interference intensity to a pulse-width-modulated and a pulse-position-modulated signal by affecting the width and the position of the transmittance pulses. The Fourier phase difference affects the width and position of the transmittance pulses at the Fourier plane. The function g(a,P) may be considered as a periodic function in the a direction with a period 2nl2xo such that g(a,B) is expanded in a Fourier series. The relationship between the pulse width d and the amplitude spectra is

R ~ ( ~ , B ) + S * ( ~ , P ) + ~ R ( ~ , P ) cos[2~od/2] S ( ~ , B ) = vT.

(2.10)

The function g(a,p) may be considered as a periodic function in the a direction and is expanded in the Fourier series

otherwise,

(2.1 1)

352

NONLINEAR PATTERN RECOGNITION TECHNIQUES IN THE FOURIER DOMAIN

[y 5

2

where K

=

2x0 2n

/

ZJC/~XU

g(a)exp(-j2nua)da.

(2.12)

-2n/4~0

The thresholded interference intensity can be considered as an infinite sum of harmonic terms: M

for u = 0,

Clearly each harmonic term is amplitude modulated due to the nonlinear characteristics of the thresholding technique and phase modulated by u times the phase modulation of the joint power spectrum, where u is an integer. The Fourier component go(a,P) generates a zero-order on-axis signal at the output plane which is referred to as a DC term. The Fourier component of transmittance g l ( a, P) generates the first-order cross-correlation signal for u = 1, which can be written as (2.15) x cos[hOa + # S ( a ,

- #R(a,

In spite of the modified amplitude modulation, gl( a ,p) contains the correct phase correlation information between the reference and the input signal. It generates the first-order correlation signal of interest at the expected location (42x0)at the output plane. The Fourier components gu(a,B) (u < 1) generate higher-order correlation signals at the output plane. Equations (2.13) and (2.14) show that the factor l/u

8 31

MULTIOBECT DETECTION USING BINARY JOINT TRANSFORM CORRELATORS

353

in the amplitude modulation decreases as the order increases. Thus, at the output plane, the high-order harmonic correlation terms are weighted less. Furthermore, when the reference scene and the input scene are placed sufficiently far from each other in the input plane, the high-order harmonic correlation terms and the zeroorder DC term are diffracted far from the first-order correlation terms. Thus, their effects at the location (312x0) in the output plane are negligible. Therefore, the correct phase relation between the reference and the input signal can be achieved at ( f 2 x o ) in the output plane where the first-order correlation is located. Investigation based on a number of tests has shown that the binary JTC outperforms the linear JTC (Javidi [1989b], Javidi and Horner [1989a,b], Javidi, Wang and Tang [1991], Javidi and Wang [1991], Rogers, Kline, Kabrisky and Mills [1990], Javidi and Kuo [1988]). The analysis, based on the low-pass signal and noise models, also shows that the binary JTC exhibits the best correlation performance among the class of nonlinear JTCs in terms of the output peak-tonoise ratio when the noise bandwidth is smaller than the target signal. Thus we have seen that the thresholded interference intensity can be considered as a sum of infinite harmonic terms. Each harmonic term is amplitude modulated due to the nonlinear characteristics of the device, and phase modulated by u times the phase modulation of the nonthresholded joint power spectrum, where u is an integer. Thus, the correct phase information can be recovered for the first-order harmonic term, and as Oppenheim and Lim [ 198I ] have shown, the phase information is much more important than the amplitude information. The higher-order correlation signals are diffracted to 20x0.

5

3. Multiobject Detection Using Binary Joint Transform Correlators

When the input plane of a linear JTC contains multiple signals, the output plane will contain correlation between these multiple signals. This is due to the absolute value square operation in the Fourier plane to obtain the joint power spectrum. The Fourier transforms of these signals are multiplied with the corresponding complex conjugates to form the power spectrum which will produce these additional output terms. To avoid overlap between the correlation terms between the targets in the input scene and the correlation terms between the input signals and the reference signal, a number of methods can be used. It is possible to adjust the separation requirement between the input image and the reference image to avoid the overlap, as we will discuss later. For nonlinear JTCs, it is possible to adjust the threshold function to eliminate the output correlation between the identical targets in the scene.

354

NONLINEAR PATTERN RECOGNITION TECHNIQUES IN THE FOURIER DOMAIN

[\!

53

In this section, we will describe the performance of the JTCs in the presence of multiple targets in the scene (Javidi, Wang and Tang [1991]). The separation requirements between the input image and the reference signal will be discussed for each method. Analytical expressions are provided to describe the effects of multiple input targets on the nonlinear JTC output. Computer simulations are provided to evaluate the performance of the binary JTC for each thresholding method in the presence of multiobject input scenes. The results indicate that binary JTCs perform well and produce well-defined correlation peaks and small sidelobes. The best performance is obtained for the binary JTC using the spatialfrequency dependent threshold function. It produces a maximum correlation peak intensity for the first-order correlation term and it eliminates the evenorder harmonic terms at the output plane. Furthermore, the correlation functions between the different targets in the scene are eliminated. Also, it allows a better utilization of the available space-bandwidth product of the system by eliminating the on-axis DC terms at the output plane. 3.1. SEPARATION REQUIREMENTS OF THE JOINT TRANSFORM CORRELATOR FOR

MULTIOBJECTS DETECTION

Let us assume that the reference image is r(x-xo,y-yo) and an input scene s(x,y) contains multiple targets [SI(x-XI ,y - y l ) , SZ(X-xz,y -y2), . . . , SN(X-xN,Y-YN)I: N

i= I

N

N

N

i=l k=l i>k

v, 5 31

MULTIOBJECT DETECTION USING BINARY JOINT TRANSFORM CORRELATORS

355

i= 1 N

i= I N

i= I

i = I ni= 1 izni

N I V i = l m=1 i>m

It is evident from eqs. (3.2) and (3.3) that when multiple targets are present in input scene, the linear JTC output contains the following terms: The autocorrelation of the reference image [ R ~ ( xy ,) ] and the autocorreiations of the input targets [Rs,s,(x,y),i = 1, 2 , . . . , N ] ; The desired cross-correlations between the reference image and the input targets [ R R s , ( x , ~i) = , 1, 2, . . . , N ] ; The cross-correlations between the different targets [Rs,s,(x,y), i + m, ( i , m = 1, 2, ..., N ) ] . The autocorrelation terms in eq. (3.3) are diffracted on the optical axis at the output plane. The desired correlation functions between the reference image and the input targets [RRs,(x,Y), i = 1, 2 , . . . , N ] and the undesired correlation functions between the different targets [Rs,s,(x,y), i # m, (i,m = 1, 2, . . . , N ) ] may overlap unless the input scene is placed sufficiently far from the reference image. The required separation between the input scene and the reference image can be expressed as max(Ds,sm)< min(Ds,R),

i = 1, 2, . . . , N ,

(3.4)

356

NONLINEAR PATTERN RECOGNITION TECHNIQUES IN THE FOURIER DOMAIN

[v 0 3

where DSJ, is the distance between any two of the targets si and s, and DS,R is the distance between the reference image and any one of the targets si in the input scene. Linear JTCs satisfying the separation condition in eq. (3.4) may require a large space-bandwidth product when multiple targets are present in the scene. 3.2. MULTIPLE INPUT OBJECTS DETECTION WITH A BINARY JTC USING THRESHOLD FUNCTIONS

In this section we discuss the effects of multiple input targets on the binary JTC output. For the binary JTC that uses the threshold function to binarize the joint power spectrum (see eq. 3.1), the Fourier component of transmittance that generates the first-order cross-correlation signal is given by (Javidi, Wang and Tang [ 19911)

where

I

N

N

(3.7) In eq. (3.6), glci(a,@ is the transmittance component of the binarized joint power spectrum that generates the first-order correlation between the reference image and the ith target s(x-x;,y-yl) in the scene. Here, glci(a,B)is weighted by the function l/S(a,B) and the Fourier amplitude of the target S;(a,B). It is evident from eqs. (3.6) and (3.7) that the first-order correlation signal is the correlation between the Fourier phase of the reference image and the Fourier phase of the ith input target, convolved with the inverse Fourier transform of l/S(a, B). The phase-only reference image {exp[j@R(a,B)) is equivalent to an image with unity Fourier amplitude and phase @ ~ ( a , @ The ) . correlation signals are affected by the multiplicative function l/S(a,B) in the Fourier plane. The function S(a,p) is the Fourier amplitude of the input scene s(x,y). It contains

5

31

MULTIOBJECT DETECTION USING BINARY JOINT TRANSFORM CORRELATORS

357

Fig. 2. An image used in tests: reference image and three targets in the scene.

the summation of the linear cross-product terms of the power spectra between the input targets. In general, the function l/S(a,P) is object dependent, and its effect on the binary JTC output depends on the input images. For typical images, S(a,B) may have a base-band characteristic. Thus, l/S(a, P) contains considerably higher spatial frequency content than S(a,P). For images with these characteristics, the Fourier transform of l/S(a,P)may have narrow width and small sidelobes. Using the threshold function (3. l), the amplitude of the first-order term of the binarized joint power spectrum is 2 / n (see eq. 2.15) for both a single object and multiple objects in the input scene. In eqs. ( 3 . 6 ) and (3.7), the summation of all the various components (glci’s)of the first-order terms will produce a constant amplitude of 2 / n . Thus, the zeros of S(a,P) will not cause any problem in the formation of the output correlation signals. Computer simulations of the linear JTC and the binary JTC were performed for the input scene containing multiobjects as shown in fig. 2. The tank image is the reference signal. Figure 2 contains a reference signal and three target tanks in the input scene. For the binary JTC, median thresholding and the threshold function (3.1) were used. The threshold function is the sum of the power spectrum of the reference image and the power spectrum of the input scene. Figures 3a and 3b illustrate the computer simulation results of the binary JTC for three targets in the scene noise [see fig. 21. Median thresholding was used in fig. 3a, while the variable threshold function [ V f ( a ,@) = R2(a, 6) + S2(a,@)I was used in fig. 3b. It is evident from fig. 3 that the binary JTC performs well with both thresholding techniques. The binary JTC with median thresholding contains

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NONLINEAR PATTERN RECOGNITION TECHNIQUES IN THE FOURIER DOMAIN

[y 5 3

DC term

Fig. 3. Computer simulation results of the binary JTC for three targets in the scene noise (shown in fig. 2): (a) using median thresholding; (b) using the threshold function v T ( a , p ) = R2(a,p) + @(a,@.

the output correlations between the targets in the input scene. The binary JTC using the threshold function has the best correlation performance with the highest peak-to-sidelobe ratio. In addition, the output correlations between the targets in the input scene are eliminated. For linear JTC and the binary JTC using median thresholding, the separation between the nearest target and the reference image must be adjusted to avoid the first-order overlap of the correlation between the different targets and the correlation between the targets and the reference image. As the number of targets in the scene increases, more stringent separation requirements between the reference image and the input image must be met. The use of the threshold function (3.1) will eliminate these problems.

v, Q 41

COMPOSITE FOURIER-PLANE NONLINEAR FILTERS

0

359

4. Composite Fourier-Plane Nonlinear Filters

In pattern recognition, it is desirable to recognize distorted patterns or targets. Generally, distortion-invariant pattern recognition uses a composite filter which is synthesized from many orientations of the target to be recognized. The classic equal-correlation-peak (ECP) composite filter (Hester and Casasent [ 19801) is designed so that the output correlation peak is the same for the true-class training images. However, there are some practical problems in using this filter for optical pattern recognition. Various composite filters based on different design criteria have been proposed to optimize some criteria (Javidi and Painchaud [1996], Javidi and Wang [1995], Refregier [1990, 1991a,b], Schils and Sweeney [1988], Mahalanobis, Kumar and Casasent [1987], Hsu and Arsenault [ 19821) or for compromise between criteria (Refregier [ 1991a,b]). For example, the optimal-trade-off composite filter shows the best trade-off among three important performance measures (Refregier [ 1990, 1991a,b]): the signal-to-noise ratio, the peak-to-correlation energy, and the Horner efficiency. A technique that uses nonlinear transformation in the Fourier plane to modify these composite filter designs has been proposed to enable the use of these filter designs in the nonlinear joint transform correlator (Javidi and Painchaud [19961). This technique is an extension of the nonlinear joint transform correlator to detect distorted objects. We note that this technique is not a method for the design of a specific composite filter, but rather is a general technique which can be used in all the specific composite filter designs mentioned above. Our work here focuses on how the Fourier-plane nonlinear filtering affects the correlation performance of a composite filter (Javidi, Wang and Zhang [1997]). We choose the ECP composite filter as the prototype to show the benefits of the nonlinear techruques, because the ECP composite filter is computationally simpler than a number of other composite filters. Let sl(x,y), s ~ ( x , y ) ., . . , s ~ ( x , y represent ) M training images. By sampling these images we can obtain matrices with N pixels each. We then rearrange each matrix into an N-element column vector si,by cascading the rows of the matrix. This image scanning is named lexicographic scanning. It is performed from left to right and from top to bottom. A training data matrix S has the vector si as its ith column. Therefore, S is an N x M matrix. We also use an N-element column vector h to denote the composite image h(x,y). Using this notation, the equal correlation peak composite filter can be expressed as (Hester and Casasent [19801) h = s(S+S)-'c*,

(4.1)

360

NONLINEAR PATTERN RECOGNITION TECHNIQUES IN THE FOURIER DOMAIN

[v $ 4

where Stis the complex-conjugate transpose of S, and (.)-I denotes the matrix inverse. The vector c contains the desired cross-correlation peak value for each training image, and c* represents the complex conjugate of c. Our work here focuses on how the Fourier-plane nonlinear filtering affects the correlation performance of a composite filter. These nonlinear techniques can be used to implement various composite filters. We choose the equal correlation peak composite filter as an example because: (1) it is computationally simpler than many other composite filters, and (2) as we shall see, substantial improvements in terms of discrimination against similar non-target objects, and output peak-to-sidelobe ratio, are obtained by applying nonlinear techniques in the Fourier domain. In order to improve the performance of composite filters in terms of discrimination against objects similar to the target, correlation-peak sharpness, and correlation noise robustness, we apply nonlinear filtering techniques to composite filters (Javidi and Painchaud [1996], Javidi, Wang and Zhang [1997]). First we apply a nonlinearity such as the kth-law nonlinear transform to each of the Fourier transforms of the training images. Then we use these kth-law nonlinear filter functions to form a kth-law nonlinear composite filter. We use the composite filter in eq. (4. l ) as an example to show how to design a nonlinear composite filter. In the frequency domain, the composite filter (4.1) can be expressed as

h = S(j.+S)-lc * , where h is the frequency-domainrepresentation of h and the hat symbol

(4.2)

denotes the Fourier transform. We define the kth-law nonlinear operation on an arbitrary matrix A as follows: h

Ak=[

IA121

k

I

exp(j@All) 1 ~ 2 1 1 ~ e x ~ ( j @. ,. 4. ~ ,I )A M ~exp(j@AA,,) I ~ exp(j@'4,2) . IL4221k exP(j@A*,) ' . IAM21k exp(j@,4A,,>

1~4111

'

exp(j@,4,,,> IA2Nlk exP(j#A*,) . ' .

k

exP(j@A,,*) (4.3) where A,, exp(j#A,,) is the element of ith row andjth column of A. By applying the nonlinear operation to the Fourier transforms of all the training images, we obtain IPlNl

p

==

(@$ . . . &).

Replacing?!, by Sk in the expression for composite Fourier-plane nonlinear filter: hk =

IAMNI

gk(($k)+Sk)-l

h in eq. (4.2), we obtain the kth-law

(4.4) Other composite Fourier-plane nonlinear filters can be obtained from their linear versions in a similar fashion. C*.

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361

In the simulations the size of the training images is 64 x 64 pixels and the size of the input image is 256x256 pixels. The training images are fist normalized to unity and then zero padded to 256x256 to compute the composite filters. These normalized training images are then Fourier transformed and a kth-law nonlinearity is applied to them. The amplitude of the Fourier transforms of the training images is raised to a power of k as shown in eq. (4.3). These nonlinearly modified Fourier transforms are used in computing the composite filters (see eq. 4.4). The input signal is also Fourier transformed and then altered nonlinearly. This nonlinearly transformed spectrum of the input signal is then multiplied by the complex conjugate of the nonlinear composite filter. The nonlinear correlation output is obtained by taking the inverse Fourier transform of the product. The performance metrics for evaluating the composite Fourierplane nonlinear filters are defined in Appendix A. In our simulations, a Mig29 airplane is used as the training target to be recognized. The composite Fourier-plane nonlinear filters are designed to tolerate the target distortions from 0" to 90" of in-plane and out-of-plane rotations. Another type of airplane, an F15, is used as the non-target object to test the discrimination of the composite Fourier-plane nonlinear filters. Mig29 and F 15 fighters are chosen because of their similarity. These similar but different images provide a good discrimination test of the filter designs for both the in-plane and out-of-plane distortions. To simulate in-plane and out-of-plane rotations of the target, the images are created by putting the aircraft on a rotating table. A set of 19 training images with in-plane rotation is created by rotating the Mig29 from 0" to 90" in 5" increments. Two members of the set of training images (0" and 45") are shown in figs. 4a,b. This image set is used to create the composite Fourier-plane linear and nonlinear filters. An additional set of 3 1 Mig29 in-plane rotation images is created in 1" increments of rotation between 20" and 50".This set is used to test the distortion tolerance and other performance of the composite Fourier-plane nonlinear filters in terms of the metrics defined in Appendix A. Figure 4c shows a 36" image from this set embedded in non-overlapping background color noise with mean m, = 0.1 and standard deviation uc= 0.15, corrupted by a zero-mean additive white noise with standard deviation ua= 0.1. For both in-plane and out-of-plane rotations, a set of 31 images of an F15 is created with 1" increments of rotation between 20" and 50". This image set is used to test the discrimination ratio of the various filters; it is also used as the non-target signal image in the simulations. One member of this non-target image set (36" in-plane rotation) is shown in fig. 4d. The out-of-plane distortions of the training images and signal images are shown in fig. 5. These images represent different view angles of the Mig29 and

362

NONLINEAR PATTERN RECOGNITION TECHNIQUES IN THE FOURIER DOMAIN

[y 9 4

Fig. 4. Mig29 in-plane rotation training image set member at: (a) 0" rotation, (b) 45" rotation, (c) 36" rotation embedded in non-overlapping color background noise with rn, = 0.1 and U, = 0.15 and further corrupted by zero-mean additive white noise with u,=O.l. (d) F15 in-plane rotation input-signal set member at 36" rotation.

F15. A set of 19 images of the Mig29 out-of-plane rotation from 0" to 90" in 5" increments is created as the training image set. Two members of this training image set (0" and 45") are shown in figs. 5a and 5b. This image set is used to create the composite Fourier-plane linear and nonlinear filters for out-of-plane distortion. The target image set of 31 images of the Mig29 and the non-target image set of 31 images of the F15 are also created in 1" increment of rotation between 20" and 50" for out-of-plane rotation. Figure 5c shows an example of a 36" image from the target image set embedded in non-overlapping color background noise with mean m, = 0.1 and standard deviation 0,= 0.15, corrupted by a zero-mean additive white noise with standard deviation oa= 0.1. One image of the F15 out-plane rotation (36") is shown in fig. 5d.

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363

Fig. 5. Mig29 out-of-plane rotation training image set member at: (a) 0" rotation, (b) 45" rotation, (c) 36" rotation embedded in non-overlapping background color noise with m, = 0.1 and u, = 0.15 and corrupted by zero-mean additive white noise with u, = 0.1. (d) F15 out-of-plane rotation input-signal set member at 36" rotation.

To compute the statistical performance metrics (see appendix A) in the presence of noise, 30 runs are conducted for each simulation, and the statistical averages are used to compute the performance metrics. The noise used in the simulations includes non-overlapping background colored Gaussian noise and additive white noise. Real scene background clutter noise is also used. Different types of noise are used to investigate the robustness of the composite Fourierplane nonlinear filter. The distortions to be considered here are the in-plane and out-of-plane rotation of the target, a Mig29 airplane. In the implementation of the composite Fourier-plane nonlinear filters (see eq. 4.4), we use the images of a Mig29 as the training images. We test

364

NONLINEAR PATTERN RECOGNITION TECHNIQUES IN THE FOURIER DOMAIN

I.

. . . . ......

[y 0 4

k = 0.6

. * . . . * . * ‘

2000.”.’...

500 20

POE

70

25

35

30

40

Signal rotation angle

* .

so

45

.

60 -

k = 0.6 50

.

.

.

.

.

.

.

.

*

.

.

.

.

.

.

.

*

.... . . . . .

a

40 -

10

20

:* 3

25

30

35

40

Signal rotation angle

45

SO

Fig. 6 . Mig29 in-plane composite filter performance for the input signal with non-overlapping color background noise (m,=0.1, u,=O.lS) and zero-mean additive white noise (u,=O.l).

the performance of the composite filter when the input signal is buried in non-overlapping background color noise (m,= 0.1, ac= 0.15) and further corrupted by zero-mean additive white noise (a, = 0.1). The simulation results corresponding to the in-plane distortion are presented in fig. 6. In figs. 6a-d, the solid line, the dotted line, and the dash-dotted line denote the performance measures of the filters for k = 0.4, k = 0.6, and k = 1, respectively. By comparing the performance of the nonlinear composite filter with that of the linear composite filter, we see that the nonlinear filtering technique significantly increases the peak intensity, the peak-to-output energy ratio (POE), the peakto-sidelobe ratio (PSR), and the signal-to-noise ratio (SNR) of the correlation peak for the in-plane distortions target images, (For definitions and discussion of these metrics see Appendix A). For k = 0.4, the peak intensity is about 3 times higher than that of the linear case. The POE is 4 times higher, the PSR is about

v, P 41

365

COMPOSITE FOURIER-PLANE NONLINEAR FILTERS

35r---

PSR 25

1

h=04

-~

-

---I

h=04

k = 0.6

. . . . . . .

,

...

1

(4 h= 1 .

25

20

.

30

~

35

~

40

Signal rotation angle

,

45

1

50

3~ DR 25

7-

15

Fig. 6 (continued)

3 times higher, and the SNR is about 2 times higher than that of the linear case. For k = 0.6, there are also improvements in these metrics over the linear case ( k = 1). The change of the discrimination ratio with the nonlinearity of the

366

NONLINEAR PATTERN RECOGNITION TECHNIQUES IN THE FOURIER DOMAIN

[v, 9 4

600 500

(4

400 300

20

k= 1 25

30

35

40

Signal rotation angle

45

50

L1 35 40

"20

25

30

Signal rotation angle

50

Fig. 7. Mig29 out-of-plane composite filter performance for the input signal with non-overlapping color background noise (m,= 0.1, a, = 0.15) and zero-mean additive white noise (aa= 0.1).

composite filter is shown in fig. 6e, where the average discrimination ratio over the whole signal rotation angle (20" to SO') is used in order to show the overall performance of the discrimination ratio with respect to the nonlinearity. The discrimination ratio decreases monotonically as k increases (see fig. 6e). Our test results show that, for the images in the presence of non-overlapping color background and additive white noise, the kth-law nonlinear composite filter has better performance to distortion compared with a linear composite filter. In the in-plane rotation case, the images of the target all have the same content; that is, each of them is a rotated version of the others. However, in the out-ofplane rotation case, the images of the target will generally have different content, because they are obtained by rotating the target in a 3-dimensional space. This makes the recognition of a target with out-of-plane distortion more difficult. The simulation results corresponding to out-of-plane distortion of the input

Y

o 41

367

COMPOSITE FOURIER-PLANE NONLINEAR FILTERS

PSR

..

I .4

I .2 20

6.5

.

~

25

30

35

40

Signal rotation angle

50

45

1 k=0.6

............................. 4.5

k= 1

4 ~

3.5

20

25

~

30

35

40

Signal rotation angle

45

50

7

I'

0

I 0.2

0.4

0.6

Nonlinearity, K

0.8

I

Fig. 7 (continued).

signal with non-overlapping color noise (m,= 0.1, u, = 0.15) and zero-mean additive white noise (ua= 0.1) are presented in fig. 7. As in fig. 6 , the solid line, dotted line, and the dash-dotted line represent the performance metrics of

368

NONLINEAR PATTERN RECOGN[TION TECHNIQUES IN THE FOURIER DOMAIN

[y 5 4

Fig. 8. A scene of an airport with two versions of the out-of-plane rotated target, Mig29 at 60" and 7 5 O , and other false objects. Further, the scene was buried in color background noise (mc = 0, u, =0.25) and zero-mean additive white noise (a, =0.15).

the filters for k = 0.4, k = 0.6 and k = 1, respectively. From the results shown in figs. 7 a 4 , we see that the peak intensity, the correlation peak-to-output energy ratio, the correlation peak-to-sidelobe ratio, and the signal-to-noise ratio of the composite filter are greatly improved after applying the nonlinear techniques. The average discrimination ratio of the nonlinear composite filter over the rotation ranging from 20" to 50" with respect to the nonlinearity is shown in fig. 7e. The discrimination ratio decreases monotonically as k increases. For k = 0, the discrimination ratio is about 4 times higher than that of k = 1. We also examined a scene which contains two rotated views of the target Mig29 (60" and 75"), buried in color background noise (m,= 0, uc= 0.25) and zero-mean additive white noise (0,=0.15), see fig. 8. Figures 9a-c show the plots of the correlation outputs of the composite filters for k=O, k=0.2 and

v; 5 41

COMPOSITE FOURIER-PLANE NONLINEAR FILTERS

369

Fig. 9. Discrimination test of the composite Fourier-plane nonlinear filter to the scene shown in fig. 8: (a) k = O ; (b) k = 0 . 2 ; (c) k = 1 .

370

NONLINEAR PATTERN RECOGNITION TECHNIQUES M THE FOURIER DOMAIN

[v, 5 5

k = 1, respectively. The nonlinear composite filters can still successfully detect the targets in the noisy scene, while the linear composite filter cannot. In summary, the nonlinear composite filter design technique can greatly improve the performance of linear composite filters in terms of peak intensity, peak-to-output energy ratio, peak-to-sidelobe ratio, signal-to-noise ratio, and discrimination ability against undesired objects.

9 5.

Illumination Dependence of Binary Nonlinear Joint Transform Correlator

We investigate the performance of the binary JTC for three different thresholding methods of the joint power spectrum when the illumination over the input image is unknown. We show that proper selection of the threshold value can improve the performance of the processor. The spatial-frequency dependent threshold function method (Javidi, Wang and Tang [ 1991]), the constant median thresholding method (Javidi and Horner [1989a1, Javidi, Wang and Tang [1991], Javidi [1990], Javidi and Wang [1991], Javidi and Kuo [1988]), and a sliding-window local-median thresholding method (Javidi and Wang [ 19911, Hahn and Flannery [1992]) are investigated. Computer simulations and optical experiments are presented. We will show that by using spatial-frequency dependent thresholding, the binary JTCs can perform well in the presence of unknown input-image illumination. The median thresholding does not perform as well in the presence of unknown input-image illumination. The frequencydependent threshold function methods are described in Appendix B. Computer simulations were performed to investigate the performance of the binary JTC when the input image illumination is unknown. The input image illumination “a” was varied uniformly over the entire input image. In the computer simulations two cases of detecting a target without background scene noise, and detecting a target in the presence of background scene noise were investigated. For the latter case, Gaussian noise was placed around a deterministic reference image to simulate the noisy input image. The input image noise (the so-called background noise) was not overlapping with the target (Javidi and Wang [1992], Javidi, Wang and Fazlollahi [1994]). Both white noise and colored noise were investigated in the simulations. For both cases, the mean value and the standard deviation of the scene noise were 0.5 and 0.2, respectively. The maximum value of the reference image in the input plane was normalized to unity. The average value of the reference image was about 0.3. To obtain the colored noise, we passed a white Gaussian noise through a lowpass linear filter with a Gaussian transfer function. The bandwidth of the filter for

Y

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ILLUMINATION DEPENDENCE OF BMARY NONLINEAR JTC

371

Fig. 10. Target used in the computer simulations for illumination tests.

generating the colored noise was fixed to be 30 pixels in both a and p directions. A 256x256 FFT was used to obtain the colored noise. A bandwidth of 30 pixels is close to the bandwidth of the reference image (target). The reference image used in the simulations is shown in fig. 10. Figures lla,b show the target in colored background noise and white background noise, respectively. The size of the target is 15 x26 pixels. The size of the input image is 90x 100 pixels. The input image and the target are placed in an input array of 512x256 pixels. The size of the input image is chosen to be large enough to change the distance between the reference image and the target placed in different positions inside the input image. The target position in the input image is changed to observe the effect of the sliding-window local-median thresholding with a fixed window size on the output performance. Three types of thresholding schemes were used: (1) sliding-window local-median threshold, (2) global median threshold, and (3) spatial-frequency dependent threshold fwnction (eq. 3.1). For the sliding-window local-median thresholding technique (Javidi, Li, Fazlollahi and Horner [1995]), the size of the window around each pixel of the joint power spectrum can be approximated using the separation between the reference image and the target in the input image. When the reference image and the target in the input image are separated in the x direction, the size of this window, wa,in the a direction (the axis parallel to x in the Fourier plane) should be large enough to include at least one period of the cosine h n g e s of the joint power spectrum, and small enough to guarantee that the self-product terms of the joint power spectrum, R2(u,/?)and a2S2(u,/?), remain almost constant within the window. A reasonable size of the window in the a direction is one period of the cosine hnges. Our simulation results support this theory. When the target inside the input image moves away from the reference image, the frequency of the cosine fringes in the joint power spectrum increases and consequently the fringe period decreases. When the target inside the input image moves toward the reference image, the period of the cosine fringes increases. If the array size in the x direction is N pixels and the separation between the reference image and the target is x' pixels, one period of the cosine function in the u direction will be T =NIX' pixels. In the

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[y 0 5

Fig. 11. (a) Target in colored scene noise; (b) target in white scene noise. The noise mean value is 0.5 and noise standard deviation is 0.2. Maximum value of the target is 1.

simulations, N is 5 12 pixels. When the target is at the center of the input image, x’ is 120 pixels and T =NIX’ = 4.3 pixels. When the target is in the uppermost or the lowermost position, the period of the cosine fringes will be T = 3.3 pixels or T = 6.2pixels, respectively. Because the variation of T is around 5 pixels, the window size chosen for the sliding-window local-median thresholding method is 5 x5 pixels. Window sizes of 5 x5, 5 x 1 and 7x 1 pixels provided fairly similar performances. The window size in the @ direction, wo,depends on the input-image noise power spectral density, the target spectrum and the separation between the reference and the target in the y direction in the input plane. Different performance measures have been introduced to evaluate the perfor-

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mance of the optical correlators (Horner [1992]). In our tests with input image noise, we measured the output Peak-to-Noise Ratio (PNR, see Appendix A). The area where the correlation results were measured was 104x 125 pixels, which is the correlation width for the linear correlator. We iterated the tests 30 times for the threshold function method, each time with a new noise sample. Figure 12 shows plots of the normalized correlation peak intensity of the binary JTC versus the illumination coefficient “a” for different thresholding methods when the input image contains a noiseless target. It is obvious from this figure that for a noiseless input image, the sliding-window local-median thresholding technique is less sensitive to illumination variation than is the fixed median thresholding method. In thls case, for all three methods the PNR is very high and the peak-to-sidelobe ratio is much greater than unity (Fielding and Horner [1990], Marom [1993], Javidi and Wang [1991]). In the tests with input image noise, the target is placed in three different positions in the image. Figures 13a-15a and 13b-15b show the output PNR of the binary JTC using different thresholding techniques for colored input image noise and white input image noise, respectively. The PNR results shown in figs. 1315 correspond to the target having been placed at the center, upper corner, and lower corner of the input image, respectively. The target is placed in different positions to test the sensitivity of the window size to target position. For all three methods, the PNR is hlgh for a large range of variations in the input image illumination “a”. It can be seen from figs. 13-15 that for a colored noisy input image, the sliding-window local-median thresholding technique outperforms the spatialfrequency dependent threshold function method in terms of the PNR. The sliding-window local-median thresholding method for the window size used here is not very sensitive to target position. When the input illumination is varied, the PNR of the binary JTC using median

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thresholding is not as large as the binary JTCs using the two spatial-frequency dependent methods. For median thresholding, we have shown mathematically how the correlation peak changes with illumination “a”, and that the correlation peak has its largest value for a = 1. Also, we have shown that the binary JTC employing the spatial-frequency dependent threshold function (3.1) can provide input image illumination invariance.

0

6. Chirp-Encoded Joint Transform Correlators

In a conventional JTC, the input scene and the reference signal are displayed in the same plane. The output of the JTC contains three terms. One term is the sum of the autocorrelation of the input scene and the autocorrelation of the reference signal that is formed on the optical axis (DC terms). The other two terms are the cross-correlation functions between the input signal and the reference signal which are diffracted off-axis. These two cross-correlation terms are positioned symmetrically from the optical axis and they will indicate the presence and locations of the input targets. In this section, we describe a JTC that produces the three output correlation functions in different planes (Tang and Javidi [ 1993a,b],Javidi and Tang [ 19941). In this system, the reference signal and the input scene are placed in different

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[y 4 6

input planes. We show that the effect of having two input planes is to encode the joint power spectrum with a different quadratic phase function (chirp function) for each correlation term. As a result, the JTC will have three output planes. The autocorrelation functions on the optical axis (DC terms) are focused onto one of the output planes, and the off-axis cross-correlation functions are produced in two separate output planes. Because of the formation of this chirp modulation, we call this technique chirp-encoded JTC. This technique can be applied to both linear and nonlinear JTCs. We show that for the nonlinear JTC, the higherorder correlation terms are produced in other output planes than the first-order correlation terms. We begin by describing the basic concept of this chirp-encoded JTC. The nonlinear JTC using the chirp-encoding technique is discussed and the advantage of chirp encoding for multiple-object detection is described. Computer simulations are provided to verify the theory and illustrate the system performance and optical experimental results are presented. 6.1. ANALYSIS OF THE CHIRP-ENCODED JOINT TRANSFORM CORRELATOR

In the conventional single input plane-single output plane JTC, both the reference signal and the input signal are located in one input plane. For a reference signal r(x + x ~ , yand ) an input signal ~ ( x - x o , ~ ) ,

are the optical Fourier transforms of the input signal s(x,y) and the reference signal r(x,y), respectively:

where a and B are the Fourier-plane coordinates, A is the wavelength of the illuminating coherent light, andf is the focal length of the transform lens. The cross-correlation functions are separated by 2x0 from the optical axis. The implementation of the chirp-encoded JTC with an optically addressed spatial light modulator (SLM) at the Fourier plane is shown in fig. 16. In this system, the reference signal and the input scene are placed in different input planes. Plane PI is the reference plane that contains the reference signal

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r(x+xo,y) and is positioned a distance d, from the Fourier-transform lens FTL1. Plane P2 is the input plane that contains the input signal s(x -xo,yj and is positioned a distance d , from the transform lens FTLl . The images are then Fourier transformed by transform lens FTLl and the interference between the Fourier transforms of the input signal and the reference signal is produced at the input of an intensity device such as the liquid-crystal light valve (LCLVj in plane P3:

where k is the corresponding wave number. Here S(a,B) and R(a,P) are the Fourier transforms of Y ( X , y ) and s(x,y ) , respectively. The Fourier-transform interference intensity distribution is obtained from the liquid-crystal light valve output which can be written as

where * indicates complex conjugation. The difference between the joint power spectrum of the conventional JTC (eq. 2.1) and the joint power spectrum in eq. (6.4) is the quadratic phase function

which modulates the cross-power spectra of the reference signal and the input signal. The quadratic phase has the form of a chirp signal. Hence, we call this phase modulation chirp encoding.

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Now, we show that due to the chirp encoding, when a Fourier transform lens is used to obtain the Fourier transform of eq. (6.4), the correlation signals will be focused in different output planes. The transform lens also introduces a quadratic phase function. The Fresnel diffraction introduces additional quadratic phase functions which are dependent on the distance from the lens. The output correlation functions are formed in the output planes where the quadratic phase functions become zero. This is illustrated in the following analysis. The Fourier transform lens FTL2 produces the Fourier transform of E(a,P) at its back focal plane. For the general case, at plane P, which is positioned a distance z from the transform lens FTL2, the light distribution is given by

(6.6) where FT indicates the Fourier transform operator, and (x’,y’) are the coordinates of the plane P,. Here the Fourier transform is computed at frequencies scaled by llkz. Using the expression for the joint power spectrum (eq. 6.4), the output light distribution can be written as

(6.7) where the Fourier transform is evaluated at x’lAz, y’lilz. The first term of eq. (6.7) will produce the autocorrelation terms s(x’,y’) @s*(x’,y’) and

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NONLINEAR PATTERN RECOGNITION TECHNIQUES IN THE FOURIER DOMAIN

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r(x’,y’)@ r*(x’,y’),where @I denotes correlation. The sum of the autocorrelation of the input signal and the autocorrelation of the reference signal will appear in the plane behind the second Fourier-transform lens where the quadratic phase becomes zero; i.e., zo=f or a distance f away from the lens FTL2. For real signals, the complex conjugate of the signal is equal to the signal. The second and third terms of eq. (6.7) produce the cross-correlation terms between the input signal and the reference signal; i.e., s(x’,y’) @ r*(x’,y’) and r(x’,y’) @Is*(x’, y’). The cross-correlation terms are produced at different planes according to their corresponding quadratic phase modulations. To determine the plane in which each cross-correlation term appears, we set the corresponding quadratic phase term equal to zero. For the second term of eq. (6.7), the zero quadratic phase modulation is produced at

The first cross-correlation term is produced at the plane zI away from the transform lens FTL2:

The separation between the correlation function and the optical axis is given by (6.10) where X I ’ and Z I indicate the position of the correlation term s(x‘,y’)8 ~ * ( x ’ , y ’ ) from the optical axis and the transform lens FTL2, respectively. Similarly, the other cross-correlation term, s*(x’,y’) 18r(x’,y’), is produced behind the lens FTL2 in the plane where the quadratic phase modulation becomes zero: (6.11) In this case, the separation between the correlation function and the optical axis is given by (6.12) Assuming that d, > d, and f > d, - d,, the correlation planes are located at planes ~2,f,andzl,wherezz
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When d, = d,, the single input plane-single output plane conventional JTC is obtained. The correlation planes coincide and all the outputs are formed in one plane; i.e., z2 =zl=f.In this case, the separation of the correlation signals from the optical axis becomes equal to 2x0. When d, is not equal to d,, the various terms of the joint power spectrum produce their corresponding correlation signals at different planes along the optical axis. The autocorrelation functions produced by the self-product terms of the joint power spectrum, s(x’,y’) @ r*(x’,y’) + s*(x’,y’)@ r(x’,y’), are separated along the optical axis from the cross-correlation functions produced by the cross-product terms of the joint power spectrum, s*(x‘,y’) @r(x’,y’)and s(x’,y’) @ r*(x’,y’). At the output plane Pz, positioned a distance Z = Z I (see eq. 6.9) from the lens FTL2, the correlation signal s*(x’,y’) @ r(x’,y’)is obtained. The other terms, s*(x’,y’)@s(x‘,y’),r*(x’,y’)@ r(x’,y’) and s*(x’,y’)@ r(x’,y’),are out of focus on this plane due to the non-zero quadratic phase modulation of the spectrum. Their corresponding correlation signal intensities are relatively small compared with that of the desired correlation signal, s*(x’,y’) @ r(x’,y’). 6.2. CHIRP-ENCODED NONLINEAR JOINT TRANSFORM CORRELATOR

We now apply the nonlinear techniques of 5 2 to a chirp-encoded JTC. For the chirp-encoded nonlinear JTC, each term of the joint power spectrum produces correlation functions in different output planes. The nonlinear transformation of the joint power spectrum results in an improved correlation behavior in each output plane. An expression for the nonlinearly transformed interference intensity is (Javidi [1989b], Tang and Javidi [1993a])

Note that eq. (6.13) is similar to the expression for the output of the conventional single input plane-single output plane nonlinear JTC, except that a quadratic phase term

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is introduced in the argument of the cosine function. When the separation of the two input planes becomes zero, eq. (6.13) is the same as the expression for the nonlinear JTC when the input images are in the same plane. In both cases, for u = 1, the nonlinear system can preserve the correct phase of the cross-correlation term and only the amplitude is affected. For the conventional nonlinear JTC, two symmetric correlation signals will appear at the same plane in the back focal plane of the Fourier transform lens. However, for the chirp-encoded nonlinear JTC, the correlation signals are located at different planes due to the phase modulation

The locations of the correlation peaks are given by eqs. (6.9)-(6.12). For the conventional nonlinear JTC, each harmonic term is phase-modulated by u times the Fourier phase difference of the input signal and the reference signal. The higher-order correlation signals are also diffracted, and they are at the same output plane. However, for the chirp-encoded nonlinear JTC, the various higherorder correlation signals are produced at the different output planes due to the different quadratic phase modulations applied to each harmonic term. We can derive the position of the higher-order harmonic correlation signals by substituting g(E) and letting the corresponding quadratic phase terms go to zero. To find the uth-order correlation signal for the s(x’,y’) @ r*(x’,y’)term, we have:

k -k =0 u--k d r - d s + 2f

f

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(6.14)

The various orders of the correlation terms are produced at the plane z1 away from the transform lens FTL2: r2

Note that zl is dependent on the harmonic order u. The separation of the correlation function from the optical axis is given by (6.16) Similarly, the uth-order correlation term s* (x‘,y’) 8 r(x’,y’)is produced behind the lens FTL2 in the plane where the quadratic phase modulation becomes zero: (6.17)

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In this case, the separation between the correlation function and the optical axis is given by (6.18) For the first-order harmonic term, u = 1 in eqs. (6.15H6.18), the same results as in eqs. (6.9)-(6.12) are obtained. Thus, the location of the first-order correlation signal for the chirp-encoded nonlinear JTC is the same as that of the linear case. The nonlinear transformation of the joint power spectrum sharpens the correlation peaks of interest, and diffuses the unwanted correlation signal functions. The computer simulations and experimental results will indicate that applying nonlinearity to the joint power spectrum improves the performance of the chirp-encoded JTC. 6.3. ON-AXIS CHIRP-ENCODED JOINT TRANSFORM CORRELATOR

Another archtecture for implementing the JTC is the on-axis chirp-encoded JTC (Javidi, Tang, Zhang and Parchekani [ 19941).The reference images and the input image propagate on-axis in the same input plane, as shown in fig. 17. However, their distance from the Fourier-transform lens is different. In the Fourier plane, the spatial carrier frequency of the joint power spectrum is zero, that is, x0 = 0 in eq. (2.1). Suppose that the reference images are denoted by r(x,y), and the input image is denoted by s(x,y). The effective distance for the chrp-encoding between the reference image and the input image is given by A, = d , - d,, where d, and d , are the effective distances of the input and the reference image from the Fourier transform lens, respectively. The reference image and the input image are effectively overlapped and propagate on-axis. Thus, this architecture is called an on-axis chirp-encoded JTC. Using the equations of 9 6.1, we can find the crosscorrelations between the reference image and the input scene that are produced at the plane located a distance z1 or z2 away from the second transform lens. Since the carrier frequency is equal to zero, the joint power spectrum consists of h n g e s that have larger spacing than those produced with carrier frequencies resulting from off-axis spatial shfting of the reference images and the input image. Hence, the joint power spectrum can be recorded by a device with a smaller spacebandwidth product. The on-axis joint transform correlator can handle input images with a larger space-bandwidth product than the conventional joint transform correlator in which the images must be located off-axis. However, since the effective separations in the x and y directions between the input and the references are zero, all the correlation functions

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are overlapped and located on the optical axis at the output plane. The chirpencoding technique and nonIinear transformation of the joint power spectrum diffuse the undesired correlation functions. Javidi, Tang, Zhang and Parchekani [1994], have shown that the diffused terms are small compared with the desired correlation signal.

6.4. COMPUTER SIMULATION OF THE CHIRP-ENCODED JOINT TRANSFORM

CORRELATOR

A computer simulation of the optical correlator is provided to study the effects of the joint power spectrum chirp encoding on the correlation signals. The correlation tests are performed for the tank images shown in fig. 18. Figure 18a shows the tank in the input scene used as the input signal for correlation tests; the input scene has a size of 30x32 pixels. Figure 18b shows the tank image used as the reference signal; this has a size of 15x22 pixels. The chirpencoded joint power spectrum of the input scene and the reference function is produced according to eq. (6.4). The Fresnel diffraction pattern of the joint power spectrum is computed in different correlation planes according to eq. (6.6). For the conventional single input plane-single output plane JTC, the input signal and the reference signal are displayed side by side in the same plane.

(b) Fig. 18. Image used in correlation tests: (a) input image in noisy scene; (b) reference image.

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Fig. 19. Binary JTC results for the tank shown in fig. 18a: (a) chup-encoded binary JTC at plane z = zl ; (b) conventional single input planesingle output binary JTC.

Figure 19 illustrates the computer simulation results for the tank image (see fig. 18b) when a chirp-encoded binary JTC is used. The input signal and the reference signal are the tank image in fig. 18. We concentrate on plane z1, which is the output plane for the cross-correlation term s(x’,y’) C3 r*(x’,y’).Figure 19a is the intensity of the correlation function x(x’,y’) 8 r*(x’,y’)that is produced a distance Z I behind the Fourier transform lens FTL2. The distribution of the on-axis autocorrelation terms (DC terms) and the other cross-correlation term s*(x’, y’) @ r(x’,y’) will not be formed in this plane due to the chirp encoding of their spectra. Figure 19b illustrates the conventional single input plane-single output plane binary JTC for the same images (see fig. 18b). For the conventional binary JTC, the on-axis autocorrelation terms (DC terms) and the two cross-correlation functions are produced in the same output plane. We can also see that for the

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conventional single input plane-single output plane binary JTC, the higher-order correlation terms appear in the same output plane as do the first-order correlation terms. For the chirp-encoded binary JTC, only the first-order term is present at the output plane z1. This may result in an increase in the correlation peak-tosidelobe ratio. The higher-order harmonic terms are produced in the different planes according to eqs. (6.15)-(6.18). It is evident from fig. 19 that using the chirp-encoded nonlinear JTC, the on-axis autocorrelation terms (DC terms) and the cross-correlation terms are produced in different output planes. The formation of these terms in different output planes reduces the overlap of the on-axis autocorrelation (DC terms) and the off-axis cross-correlation terms. One of the important advantages of this technique is in multiple-object detection using a JTC. In a conventional JTC, given an input scene with multiple targets, the output plane contains the desired cross-correlation between the reference signal and the input scene as well as the undesired correlations between the targets in the input scene. Using the chirp-encoded technique however, the desired correlation signals between the reference signal and the input targets, and the undesired correlations between the targets in the input scene are produced in separate output planes. In the desired output plane, the cross-correlations between the reference image and the input targets are generated. In this plane, the cross-correlation terms between the different targets in the input scene, and the autocorrelation terms for different targets, are out of focus due to the quadratic phase modulation in the Fourier plane. As a result, the effect of the intermodulation of the similar targets in the input scene is reduced. In addition, the separation requirement between the input scene and the reference signal may be relaxed. Consider the images shown in fig. 20a. The car is used as the reference object and is placed on the lower part of the input plane. The two tanks are placed side by side to simulate the input scene and are placed on the upper part of the input plane. Assume that the distance between them is the same (D= 20 pixels). For a conventional binary JTC, the cross-correlations of the reference car and the tanks, and the correlation of the two tanks in the scene, are formed in the same plane. Figure 20b shows the conventional output correlation plane. According to the separation condition, an area of 40x40 pixels around the optical axis is considered as the DC terms and should be ignored. The peaks represent the correlation between the two tanks in the scene and fall within the DC area. Figure 20c shows the same test using the chirp-encoded binary JTC, where the correlation peaks between the two tanks in the input scene are not observed in the output plane of interest. In fig. 20c, the correlation peak intensity between

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DC,tem

Fig. 21. Correlation for a scene containing multiple objects (case 2): (a) correlation result using the conventional binary JTC; (b) correlation result using the chirp-encoded binary JTC.

the two tanks in the input scene is 230 times smaller than the same peak intensity shown in fig. 20b. In the second multiple-target detection test, multiple tanks are placed in the input scene as shown in fig. 21. The sizes of the images are 15x22 pixels for the reference tank, and 80x 128 pixels for the input scene. The conventional binary JTC output is shown in fig. 21a, where all the correlations are mixed up. The correlation signals using the chirp-encoded technique are shown in fig. 21b. It is evident from fig. 21b that the DC terms and the unwanted correlation signals between the input targets have been reduced. The experimental results using a chirp-encoded nonlinear JTC are presented elsewhere (Tang and Javidi [ 1991, 1993al). Fresnel encoding has been used in holography and other applications.

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7

7. Random Phase Encoded Joint Transform Correlator

7.1. IMF'LEMENTATION OF RANDOM PHASE ENCODED JOINT TRANSFORM CORRELATOR

In a conventional joint transform correlator (JTC), as shown in fig. 1, the input scene and the reference signal are displayed in the same plane. The output of the joint transform correlator contains three terms. The first is the sum of the autocorrelation of the input scene and the autocorrelation of the reference signal; these are on the optical axis (DC terms). The other two are the cross-correlation signals between the input signal and the reference signal, which are diffracted off-axis. The off-axis correlation terms indicate the presence and the location of the input targets. The separation of the input object and the reference object in the input plane must be large enough to avoid overlap of the above three terms in the output plane. It must be larger than two times the sum of the size of the reference and input objects (separation condition). When the size of the input increases, the separation of the images must increase accordingly to avoid overlap. However, when the separation increases, the carrier frequency of the joint power spectrum increases and the correlation quality suffers due to the limited frequency response of the detector used to detect the joint power spectrum [this can be described by the modulation transfer function (MTF) of the device]. This dilemma can be solved by a modulated architecture of the JTC the random phase encoded JTC (Zhang [1995], Horner, Javidi and Zhang [1994], Zhang and Javidi [ 19931). The random phase encoded JTC can be implemented by the setup shown in fig. 22. In this system, the input signal and the reference signal are placed on two symmetric planes and are Fourier-transformed by two Fourier-transform lenses. At the planes Plf and Pzf, the Fourier transforms of the input and the reference are multiplied by two phase functions (phase modulations), respectively. The modulated joint power spectrum is obtained at P3. It is recorded by a squarelaw device (video camera) and displayed on a spatial light modulator (SLM). It is then multiplied by another phase fimction used as the demodulation function at Pq. We show that by use of the phase modulation and demodulation of the joint power spectrum at the Fourier plane, the correlation signal of the input and the reference can be recovered at the output plane Po, and the undesired signals (the autocorrelation of the input and reference and the complex conjugate of the cross-correlation signal) are phase modulated by some functions. By properly choosing the phase functions, the peaks of the undesired terms can be reduced significantly at the output plane. As a result, the system does not

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Nonlinear device

ykE

Fig. 22. Phase-encoded joint transform correlator. P I is the input plane. P2 is the reference plane. P ~ Jand PY are Fourier planes of PI and P2, respectively, and are imaged onto P3 by a 4-f image system. PO is the correlation output plane.

2>

w m

3 0

392

NONLINEAR PATTERN RECOGNITION TECHNIQUES IN THE FOURIER DOMAIN

[y 9 7

have to meet the separation condition to avoid overlap of these terms, and therefore the requirement on the space-bandwidth product of the system is relaxed. Equivalently, it can handle a larger input image than does a conventional JTC. This is especially useful for multiple-target detection. Suppose that the input signal s(x,y) is put on the plane PI and the reference signal r(x,y) is put on Pz. On the back focal planes of the Fourier-transform lenses 1 and 2, S(a,@)and R(a,P), the Fourier transforms of f ( x , y ) and r ( x , y ) ,respectively, are obtained. On PI,-,S(a,B) is multiplied by the phase-only function of Ml(a,/3).So the light emanating from PI^, denoted byA,I,-(a,P), is the product of S(a,p) and A4 I (a, B), i.e.

For the same reason, on Plf,

By using beam splitter BS and image lenses IL1, IL2, both PI/ and P2/ planes are imaged onto P3. Consequently, on Pj, the interference pattern of Aplf and Ap2f is obtained. The modulated joint transform power spectrum can be obtained by a square-law device such as a video camera. The modulated joint power spectrum (MJPS) can be represented as

where E ( a ,p) is the MJPS that is recorded by a camera. The output of the camera is connected to a nonlinear network to transform the MJPS. Then, the output of the nonlinear network is sent to a SLM to display the nonlinear version of the MJPS. The nonlinearity here can be any type of nonlinear transform, or a linear transform. Since the phase-only mask Md(a, P), which is used as demodulation function, is placed on the SLM, the light emanating from P4 can be represented as

where g { .} denotes the nonlinear transformation and G(a,B) represents the light distribution emanating from P4.Applying another Fourier transform optically, we get the output on the plane Po.

v, P

71

RANDOM PHASE ENCODED JOINT TRANSFORM CORRELATOR

393

7.2. ANALYSIS OF RANDOM PHASE ENCODED LINEAR JOINT TRANSFORM

CORRELATORS

Assume that the demodulation function Md(a,/3) is

It can be seen that the phase-only function Md is used as a demodulation function which recovers the desired correlation signal completely while the other noise terms are still modulated by phase functions. Hence, on the output plane we have g(x,y)=R.s~(x,y)*md(x,y)+R,(x,y) * md@,y)

(7.7)

+ R.v(x,y) + Rr.s(x,y) * md(x,y) * md(x,y),

where * represents the convolution operation. R,,(a,B) and R,.,.(a,p) are the autocorrelation of s(x,y) and v ( x , y ) , respectively. R,y,.(a,p) and R,,(a, B) are cross-correlations between s ( x , y ) and r ( x , y ) [R,s,.(a,/3) = Rz7(a,p)], and

From eq. (7.7) we can see that there are four terms in the output plane Po. The term R,,,.(a,p) is the desired cross-correlation signal. The other terms are considered as noise signals. This is different from conventional JTC. The three noise terms are all convolved with some functions of Md(@,lj).By properly choosing M l ( a , B )and M2(a,b),and therefore Md(a,/3), it is possible to reduce the noise peak intensity so the signal peak-to-sidelobe ratio can be improved. If Ml(a,/3) and Mz(a,/3) are phase-only functions, Md(a,P) is also a phaseonly function, Therefore we have

Is,

1w(a,B)~d(a7/3)l2 dadB =

// Iw(a,P>I2

dadh

(7.9)

90

where W ( a , P ) is an arbitrary function. It can be seen that the total energy of each noise term will not be changed by the demodulation function Md(a,/?) if it is a phase-only function. Only the distribution of the noise in the output plane is changed.

3 94

NONLINEAR PATTERN RECOGNITION TECHNIQUES M THE FOURIER DOMAIN

[v 5 7

From eqs. (7.6) or (7.7) we can see that the noise terms including autocorrelations and one cross-correlation are multiplied by Md(a, P) and Mi(a,P), respectively. Equation (7.5) is the demodulation condition that Md(a,P) needs to satisfy. If Ml(a,P) and M ~ ( c w ,are / ~ )chosen to be random phase fimctions e x p [ j h ( a , B)1 and eXpfiQlm2(a, 8)1, respectively, and Ql,l(a, P) and rP,2(a, P> are two independent white Gaussian processes, then # m d ( Q , P ) = qjm2(a,P) - Qnll (a,B) is also a white Gaussian process. Therefore, Md((x,P) = exp[j&d(a,P)] has the same defising properties. It reduces the noise peaks of the two autocorrelation terms by spreading the energy of the noise terms across the entire output plane. For the same reason M:(a,B) reduces the noise peak of the fourth term in eq. (7.7). We have shown that all the noise terms in a phase-encoded joint transform correlator are diffused by the random phase encoding but the desired correlation term remains intact. 7.3. COMPUTER SIMULATIONS

To study the performance of the random phase encoded JTC, computer simulations were conducted. The car image shown in fig. 23a (21 x 16pixels) was used as the reference and was located at the center of the reference plane in fig. 22. The input image was the same car image located at the input plane and was shifted 36pixels from the origin. Figure 24a shows the correlation output for a linear JTC. We see the strong autocorrelation terms at the center of the output plane and two cross-correlation terms on both sides of the autocorrelation terms. Here we assume a real-valued gray scale SLM that can display positive and negative values. If only positive values can be displayed, a bias has to be used to shift a negative value to a non-negative one. This will result in an extra DC peak at the origin of the output (Flavin and Homer [1989]). In fig. 24a, only one of the cross-correlation peaks is desired. The other crosscorrelation is the redundant term that does not contain any more information. The autocorrelation terms are also considered as noise terms. Figure 24b shows correlation output for the same reference and input images using the random phase encoded JTC shown in fig. 22. It shows that only one cross-correlation peak appears at the output plane. The autocorrelation peaks and the redundant cross-correlation peak are diffised by the random phase encoding. Since the phase encoding at the Fourier plane uniformly distributes the undesired terms at the output plane, the undesired terms contribute to the background noise floor, and the noise peaks are significantly reduced. Note that the random phase encoding and decoding at the Fourier plane recovers the desired correlation signal completely, and that this correlation signal is unchanged. However, since

v, Q

71

RANDOM PHASE ENCODED JOINT TRANSFORM CORRELATOR

395

20 40

60 80 100 120

20

40

60

80

100

120

80

100

120

(a)

20 40

60 80 100 120

20

40

60

(b) Fig. 23. The input and the reference images used for single-target detection using a JTC and a random phase encoded JTC. (a) The reference image (a car image) located at the center. (b) The input image (same car image as the reference) located at (0,36).

the noise terms overlap with the desired correlation signal and are not negligible, the output noise increases slightly. This is due to the limited space-ban dwidth product of the optical system that binds the noise energy to the limited area of the output plane, unlike the ideal case in which we assume that the noise energy

396

NONLINEAR PATTERN RECOGNITION TECHNIQUES IN THE FOURIER DOMAIN

x

Id5

[v $ 7

r( au t o-correlati on

1

0.5 0

0

x , /

cross-correlation

I 2 1 -

60

Flg 24 The correlation outputs of linear JTCs for the reference and input ‘I> ,II< figs 23a and 23b, respectively (a) The correlation output for the lineai TTC. ( b ) 11ie oi output for the random phase encoded linear JTC

I

can be distributed over a sufficiently large area to result in a negligible i n s ~ L Y W to the background noise. We now test the random phase encoded JTC for multiple-target detection. FOLW identical car images used as the targets are placed at the input as shown in fig. 25. The reference image is the same car image placed at the center of the reference plane as in fig. 23. The size of the car images used here is reduced by half to avoid overlapping in the linear JTC. For the linear case, the correlation output for conventional JTC and random phase encoded JTC are shown in figs. 26a and 26b, respectively. Comparing these two plots, we can see that random phase encoding at the Fourier plane eliminates the autocorrelation peaks and the redundant crosscorrelation peaks. These noise terms are distributed over the background noise floor, as expected.

v, P

71

391

RANDOM PHASE ENCODED JOINT TRANSFORM CORRELATOR

50 100 150

200 250

50

100

150

200

250

(a)

50 100 150

200 250 50

100

150

200

250

(b) Fig. 25. The input and reference for multiple-target detection: (a) the input image that contains four identical car images as the targets. (b) The reference car image located at the center of the reference plane.

398

NONLINEAR PATTERN RECOGNITION TECHNIQUES IN THE FOURIER DOMAIN

[\! 0 8

x 4

k

1

0

10” 4,

h

Fig. 26. The correlation outputs of linear JTCs for the reference images shown in fig. 25: (a) the output of the linear JTC;@) the output of the random phase encoded linear JTC.

0

8. Security Validation and Security Verification

We conclude with a practical application for the technology we have been describing so far. Credit card fraud is a serious problem facing many banks, businesses, and consumers. In addition, counterfeit parts such as computer chips, machine tools, etc. are being produced in great numbers. With the rapid advances in computers, CCD technology, image processing hardware and software, printers, scanners, and copiers, it is becoming increasingly simple to reproduce pictures, logos, symbols, money bills, or patterns. Presently, credit cards and passports use holograms for security. The holograms are inspected by human eye. In theory, the hologram cannot be reproduced by an unauthorized person using commercially available optical components. In practice, the holographic pattern can be easily acquired from a credit card (photographed or captured by a CCD camera) and then a new hologram synthesized. Therefore any pattern

V,

8 81

SECURITY VALIDATION AND SECURITY VERIFICATION

399

that can be read by a conventional light source and a CCD camera can be easily reproduced. We propose an idea for security verification of credit cards, passports, and other IDS so that they cannot be easily reproduced (Javidi and Horner [1994a], Javidi [ 1997a,b]). We propose a new scheme of complex phase/amplitude patterns that cannot be seen and cannot be copied by an intensity-sensitive detector such as a CCD camera. The basic idea is to permanently and irretrievably bond a phase mask to a primary identification amplitude pattern such as a fingerprint, a picture of a face, or a signature. Computer simulation results and laboratory tests of the proposed system will be provided to verify that both the phase mask and the primary pattern are identifiable in an optical processor or correlator (Vander Lugt and Rotz [1970], Weaver and Goodman [1966], Javidi [ 1989a,b]). Complex phase/amplitude patterns which cannot be seen and cannot be copied by an intensity-sensitive detector such as the CCD camera are utilized for verification of the authenticity of items bearing the pattern. The phase portion of the pattern consists of a two-dimensional phase mask that is invisible under ordinary light. The complexities of interferometry and the large dimensions of the mask make it extremely difficult to determine the contents of the mask. The code in the mask is known only to the authorized producer of the card. One cannot analyze the mask by looking at the card under a microscope or photographing it, or reading it with a computer scanner in an attempt to reproduce it. Only an elaborate set-up like a Michelson interferometer is capable of deciphering it. The verification system that reads the card could be one of several coherent optical processor architectures. An biometric image such as a fingerprint or primary pattern g(x,y) whose authenticity is to be verified, consisting of an amplitude gray-scale pattern to which a phase mask has been bonded, is placed in the input plane of the processor. Thus the composite input signal is

If there is no primary pattern and the phase mask exp[jM(x,y)] alone is used for verification (as, for example, in product verification or authentication),g(x, y ) will be a constant. If the primary pattern must be verified, the processor will have an a priori knowledge of the primary pattern g(x,y). In that case, the process is repeated for verifying the primary pattern. In fig. 27, the nonlinear joint transform correlator architecture is used to verify the authenticity of the card. The complex mask on the card and an image of a

400

NONLINEAR PATTERN RECOGNITION TECHNIQUES IN THE FOURIER DOMAIN

[V, § 8

CCD

Laser Di

2

r-----------------I If there is a correlation peak, it means that the card is authentic. I

I I If there is no correlation peak, it-means I that the card is not authentic. I- - - - - - - - - - - - - _ _

r]

-1 I I I

---___________

Fig. 27a. Nonlinear single-SLM joint transform correlator for verifying a fingerprint match in addition to verifying the authenticity of the phase mask superposed over the fingerprint on an ID or credit card.

corresponding reference phase mask are placed at the input plane of a JTC. If a SLM is used, the single-SLM version of the JTC (fig. 27a) is attractive because it is a more cost-effective type for this application. Figure 28 illustrates an example of the phase-encoded ID card inserted into the input plane of an optical processor such as the correlator of fig. 27. The processor (fig. 27) used to verify the input mask can also be used to verify the primary pattern such as a fingerprint or a picture. The processor will have an a priori knowledge of the primary pattern g(x,y) and the process is repeated for verifying the primary pattern. For even more security, the primary pattern could itself be phase-encoded. That is, a fingerprint or picture of a face could be written as a phase mask itself and combined with the random phase mask discussed previously. This would have the effect that the combined pattern would be completely invisible to the

v, § 81

40 1

SECURITY VALIDATION AND SECURITY VERIFICATION

CCD

Laser

, 1

I I I I

If there is a correlation peak, it means that the card is authentic. --I

If there is no correlation peak, if means that the card is not authentic.

--__-------------------------

Fig. 27b. Implementation of a verification system with a joint transform correlator without SLM.

eye or to any other detector using conventional light sources. The composite mask could be produced by the same means used to make the random phase mask; e.g., refractive, embossing, or bleaching techniques. This double phaseencoding scheme would have an additional security value, in that anyone wanting to counterfeit the card would not even be able to determine easily what type of a primary pattern they would have to produce on the card. Even if they obtained the primary pattern, they would have to unscramble the random phase code from it. We have verified the idea proposed above with computer simulations. The card-holder’s fingerprint can be verified before the phase mask on the card is verified. The order is not important. A random noise is used for M ( x , y ) to generate the reference phase mask exp[jM(x, y ) ] .This phase mask is multiplied by an image g(x,y) to form the input signal. The input image g(x,y), a segment of a twenty dollar bill, is shown in fig. 29. A phase mask is placed over the image of Andrew Jackson. In the computer simulation tests, the size of the input image is 80x 100 pixels, and the size of the input array (with zero padding) is

402

NONLINEAR PATTERN RECOGNITION TECHNIQUES IN THE FOURIER DOMAIN

Phase Mask \

Front

[y 4

8

\E V E R Y B A N K

VALID FROM 1193

EXPIRATION DATE

L

1/00

An image (Fiigff Rint & Bich%gnatum, ete.) c

24-HOUR CUSTOMER SERVICE. CALL 1-800012-3456

I

'

An image bonded with the phase mask g(x,y)CiM(x.y)

Back AUTHORIZED SIGNATURE

Bic-Signature bonded with thc phase mask

Magnetic Ship Includes Algorithms to determine the sequenceof Codes to be used, PIN No.,etc.

Fig. 28. An example of the phase-encoded ID card inserted into the correlator of fig. 27.

Fig. 29. Part of the input image (a $20 bill) used in the simulations. A phase mask is placed over the image of Andrew Jackson.

v, 8 81

SECURITY VALIDATION AND SECURITY VERIFICATION

403

Fig. 30. Computer correlation simulation with fig. 29 as input: (a) the reference function is the random phase mask; (b) the reference function is a different random phase mask, for example a counterfeit mask.

256~512pixels.The amplitude of g(x,y) in the input plane is normalized to have a maximum value of unity. We have simulated a binary JTC to verify the phase mask. Figure 30 shows the computer simulations of the correlation with the binary JTC. In all cases the input is that shown in fig. 29 - a picture of Andrew Jackson with a random phase code, exp[jM(x,y)], placed over it. In fig. 30a the reference mask is the same random phase pattern. In fig. 30b, the input image has a different phase mask multiplied with the image in fig. 29, that is, exp[jN(x,y)], where N(x,y) is not the authorized code. In this case, the reference image does not match the random phase mask placed on the twenty dollar bill image. It is

404

NONLINEAR PATTERN RECOGNITION TECHNIQUES IN THE FOURIER DOMAIN

[y 5

9

clear from these figures that the processor has verified the authentic phase mask and rejected the unauthorized mask. In conclusion, we have described a phase-encoding scheme that is virtually impossible to copy, and when combined with a biometric image such as an individual fingerprint or photograph, makes a foolproof security entry or identification system. The phase mask can be verified using a variety of system architectures. The phase mask can be made of a light-transmissive embossed plastic film, a bleached photographic film, or etched onto a glass or metallic reflector. The random phase modulating mask could be used alone, as for example on a tag of a manufactured object, for product authentication. We have presented a number of alternative methods to realize the basic idea. The combination used would ultimately be determined by the level of security required.

Q 9. Summary We have discussed a number of Fourier-plane nonlinear techniques that can be used for pattern recognition. We have shown that nonlinear transformation of the joint power spectrum can result in very good correlation performance in terms of correlation peak size, peak to output noise ratio, and sensitivity against similar objects. We investigated the correlation performance of binary joint transform correlators with unknown input-image light illumination for three different thresholding methods of the joint power spectrum. Two types of frequency-dependent threshold hnction have been analyzed: the sliding-window local-median thresholding method and the spatial-frequency dependent threshold method. We have shown that the binary JTC employing the latter method works better than the sliding-window local-median threshold method and can provide input-image illumination invariance. Computer simulations and experimental results show that for varying illumination, the variable threshold function and the sliding-window local-median thresholding perform well by providing large correlation peak intensity, and large peak to noise ratio. We investigated the use of nonlinear techniques in the design of composite filters which enable these filter designs to be used in a nonlinear joint transform correlator. We find that the performance of these composite filters is improved substantially by applying the Fourier-plane nonlinear techniques. The main performance improvements gained from the incorporation of the Fourier-plane nonlinearity in our filter designs are a large increase in the peak intensity, a significant improvement in the peak-to-sidelobe ratio, and a substantial improvement in

VI

LIST OF SYMBOLS AND ABBREVIATIONS

405

the filter's discriminability. Additionally, the nonlinearities improve the SNR of the peak intensity value. We have presented several techniques to reduce the redundant and self-correlation terms of the joint transform correlator. These include the chirp-encoded technique which places the reference and input images in separate planes. The other technique uses random phase encoding. The analysis and computer simulations show that these random techniques can reduce the undesired output signals, including the DC terms, the spurious autocorrelations from multiple targets, and the higher-order harmonic terms. All of these take valuable space-bandwidth product and increase the system size. Since the undesirable output signals are reduced, the input image separation condition for the target and reference is relaxed. As an application we have described a phase-encoding scheme that is very difficult to copy, and when combined with an individual fingerprint or photograph, makes a foolproof security entry or identification system. The phase mask can be verified using a variety of system architectures. The phase mask can be made of a lighttransmissive embossed plastic film, a bleached photographic film, or etched onto a glass or metallic reflector. Also, the random phase modulating phase mask can be used alone on a manufactured item. We have presented a number of alternative methods to realize the basic idea. The combination used would ultimately be determined by the level of security required.

Acknowledgement We acknowledge Dr. Wenlu Wang for his assistance in preparing this chapter. We also acknowledge the USAF Research Laboratory at Hanscom Air Force Base and the National Science Foundation for their support.

List of Symbols and Abbreviations

1.1 ( I-'

Magnitude Matrix inverse

@

Represents correlation An arbitrary matrix

A ~xP(~@ 1A , , ,

a

The element of the ith row andjth column of matrix A Input image illumination coefficient

Cuk

A constant related with u and k

Aij

406 C

NONLINEAR PATTERN RECOGNITION TECHNIQUES IN THE FOURIER DOMAIN

rv

A column vector containing the desired correlation peak for each training image Complex conjugate of c Correlation output at the location of ( x , y ) Side lobe intensity outside the window w(R2) Charge-coupled device Distance between any two of the targets s, and sm Distance between the reference image and any one of the targets si in the input scene

d

Pulse width of the binarized function g(a, /3)

dr

Distance between the reference signal and FTLl

ds DC terms

Distance between the input signal and FTLl The sum of the autocorrelation of the input scene and the autocorrelation of the reference signal that is formed on the optical axis Discrimination ratio Expected value Joint power spectrum A threshold value = median(hist[R2(a)(a2

+ 2a cos(2xoa) + l)]}

=(l/a*)~T,~

A threshold hnction

Equal-correlation-peak Focal length of the transform lens Airplane used as the non-target object to be rejected Fourier transform Fourier transform lens 1 Fourier transform lens 2 Fourier transform of g(E) The nonlinear characteristic of a nonlinear device The output of the nonlinear device with nonlinearity k The first-order harmonic term of gk(a, B)

VI

LIST OF SYMBOLS AND ABBREVIATIONS

407

The amplitude modulation of each harmonic term of the expansion of g(a,0) Composite filter A column vector obtained by lexicographic scanning of the composite filter

I;

Fourier transform of h

hist[.]

Histogram Expected value of the correlation peak intensity

I0

Ju

Identification Image lenses 1 and 2 Bessel function of the first kind, order u

JTC

Joint transform correlator

KO k LCLV

Coefficient of the Fourier series of g(a,B) The severity of the nonlinearity Liquid crystal light valve

M

Number of training images

MkY)

Random code used to generate the reference phase mask Random phase-only function

ID IL1, IL2

Random phase-only function = M;(a, PI M2(a,PI Fourier transform of Md(a,P) Mean of background color noise Median value

Airplane used as the training target to be recognized Modulated joint power spectrum Modulation transfer function Total number of pixels in w(R2)

PNR POE PSR

Random code used to generate the unauthorized phase mask Peak-to-noise ratio Peak-to-output-energy ratio Peak-to-sidelobe ratio

408

NONLWEAR PATTERN RECOGNITION TECHNIQUES IN THE FOURIER DOMAIN

Optical Fourier transforms of the input signal s(x,y ) Autocorrelation of the reference image Cross-correlation of the reference and input targets Cross-correlation between different targets Autocorrelations of the input targets Fourier transform of the reference signal r(x,y), with

@ ~ (B) a ,the corresponding Fourier phase Autocorrelation of the input signal

Autocorrelation of the reference signal Cross-correlation of the input signal and the reference signal Correlation peak intensity located at (x0,yo) Reference signal located at ( X O , 0) in the input plane Reference image located at (x0,yo) in the input plane Training data matrix with si as its ith column Complex-conjugate transpose of S Fourier transform of S Optical Fourier transform of the reference signal r(x,y)

si

Fourier transform of the input signal s(x,y), with @s(a,B) the corresponding Fourier phase Fourier Transform of the input signal si(x,y), with @si( a,@) the corresponding Fourier phase A column vector obtained by lexicographic scanning of the ith training image Input signal put at (-xo, 0) in the input plane The ith target located at (xi,yi)at the input plane of the N targets Spatial light modulator Signal-to-noise ratio Input transmittance function

C: APP. A1

PERFORMANCE METRICS

409

A threshold value for binarizing the joint power spectrum Variance An arbitrary function Neighborhood of the correlation peak in the output plane Distance of term s(x’,y’) @ r*(x’,y’)from the optical axis

VT

Distance of term s*(x’,y’) @ r(x’,y’) from the optical axis

z

Distance between the output plane and FTL2

20

Distance between the DC terms plane to FTL2

21,

z2

Distance between the cross-correlation term plane to FTL2 Point on a-axis where the envelopes (a - 1)2R2(a) reach E

E amax

Point on a-axis where the envelopes ( a + 1)2R2(a) reach E Spatial frequency coordinates A constant related with order LJ Two independent white Gaussian processes = $rn2(a,B)-$mi(a,B)

Wavelength of the illuminating coherent light Standard deviation of additive white noise Standard deviation of background color noise

Appendix A. Performance metrics

To evaluate correlation performance we use a number of criteria. These criteria include: correlation peak intensity, peak-to-output-energy ratio, peak-to-sidelobe ratio, peak-to-noise ratio (Horner [ 1992]), discrimination ratio, and signal-tonoise ratio of the correlation peak intensity (Javidi and Horner [ 1994b], Horner [1992]). In what follows, the notations I . I and E{.} are used to denote the magnitude and the expected value, respectively.

410

NONLINEAR PATTERN RECOGNITION TECHNIQUES

IN THE FOURIER DOMAIN

[v, App. A

The correlation output peak intensity is defined as the maximum peak intensity, as it appears at the output when the input signal contains no noise. That is,

where R2(xo,yo) is the output correlation peak intensity located at (xo,Yo), and c (x , y) is the correlation output at the location of ( x , y ) .Ideally the peak should appear at (xo,yo)=(O,O). The value of c(0,O) is used as a constraint and it is forced to be a constant in the composite filter formulation of eq. (4.1). When the input signal is corrupted by noise, the correlation peak intensity 10 is defined as the expected value of correlation peak intensity at (xo,Yo):

One measure of correlation peak sharpness and correlation efficiency may be evaluated by peak-to-output-energy (POE). POE is defined as the ratio of the expected value of the correlation peak intensity to the expected value of the spatial average of the output energy:

where c ( x , y ) is the correlation output. The overbar symbol in the denominator denotes the normalized integration (spatial averaging) over (x, y ) . A correlation POE of larger than unity does not guarantee a successful detection of the target. The target is detected successfully when the correlation peak is larger than every other pixel in the output plane. PSR, the peak-tosidelobe ratio, also known as peak-to-secondary ratio, is a suitable criterion that shows how well the target can be detected in the presence of noise and/or other objects. The PSR is defined to be the expected value of the ratio of the correlation-peak intensity to the maximum sidelobe intensity in the output plane:

where Ic(x,y)lz, is the sidelobe intensity outside the window w(R2)that represents the neighborhood of the correlation peak in the output plane.

Y APP. Bl

FREQUENCY-DEPENDENT THRESHOLD FUNCTION METHODS

41 1

To evaluate the robustness of the correlation against noises, we need to measure the variability of the correlation peak in response to the noise in the input scene. The signal-to-noise ratio (SNR) of the output correlation peak in the presence of noise in the input scene can be defined as (Javidi and Horner [ 1994b1)

where

is the variance of the output correlation peak intensity due to the input noise. The peak-to-noise ratio (PNR) is defined as the ratio of the correlation peak intensity to the average noise intensity (Horner [ 19921):

where N’ is the total number of pixels in w(R2). Discrimination capability of a filter refers to the filter’s ability to correctly distinguish the true target images against non-target images. For this purpose, we desire to have a strong output correlation peak in response to the target, and small correlation output in response to other signals. This criterion is evaluated by the discrimination ratio (DR). DR is defined as the ratio of the correlation peak intensity of the target to the correlation peak intensity of any non-target: DR

=

10,target

10,non-target

E

-

E

{fi:arget}

{REon.target}

‘

A larger DR means that the composite filter has better discrimination against other non-target objects.

Appendix B. Frequency-dependent Threshold Function Methods In this appendix, we describe three thresholding methods: the global spatialfrequency dependent threshold function method (Javidi, Wang and Tang [199l]),

412

NONLINEAR PATTERN RECOGNITION TECHNIQUES IN THE FOURIER DOMAIN

[\! App. B

the constant median thresholding method, and a sliding-window local-median thresholding method (Javidi and Wang [ 19911). Given unity illumination over the reference image, and uniform illumination ‘‘a” over the input image, the input transmittance fimction is

The joint power spectrum is

E(a,P) = R2@,B) + a2S2(a,P>

+ 2aR(a,

S(a, cos[2xO~-t@da, B) - @R(a,P)],

(B2)

where a and /3 are the angular spatial frequency coordinates, R(a,B) and S(a,B) are the amplitude spectra, and @R(a,P) and @s(a,P)are the phases of the Fourier transforms of r ( x , y ) and s(x,y), respectively. In the binary JTC, the joint power spectrum is thresholded before the inverse Fourier transform operation is applied (Javidi [ 1989133). For a binary nonlinearity with threshold value E T ,the binarized joint power spectrum can be written as 1 for E(a,B) 2 ET 0 for E(a,P) < E T ’ where ET can be a function of (a,B). The thresholded joint power spectrum can be considered as an infinite sum of harmonic terms (Javidi [1989b], Vander Lugt and Rotz [1970]). The first-order harmonic term will generate the firstorder cross-correlation between the reference and the input image. The first-order cross-correlation signal is diffracted to 2x0, and the uth-order cross-correlation signals are diffracted to 20x0, where u is an integer ( u > 1). The Fourier component of transmittance that generates the first-order cross-correlation signal can be written as (Javidi [1989b])

for

where the subscript “1” denotes the first-order component. The Fourier components g,(a, P) (u > 1) generate higher-order correlation signals at the output plane. When the reference and the input scene are placed

V, APP. Bl

FREQUENCY-DEPENDENT THRESHOLD FUNCTION METHODS

413

sufficiently far from each other in the input plane, the high-order harmonic correlation terms and the zero-order DC term are diffracted far from the firstorder correlation terms. The analysis, based on the low-pass signal and noise models, shows that the binary JTC exhibits the best correlation performance among the class of nonlinear JTCs in terms of the output peak to noise ratio when the noise bandwidth is smaller than the target signal (Javidi, Wang and Fazlollahi [ 19941). B. 1. SPATIAL-FREQUENCY DEPENDENT THRESHOLD FUNCTIONS

We will investigate the spatial-frequency dependent threshold function. It is obvious from eq. (84) that the largest correlation peak is obtained when ET is (Javidi [ 1989b1, Javidi, Wang and Tang [ 19911)

When the threshold function Er(a,B) = R2(a,P)+ a2S2(a,P) is used for binarization of the joint power spectrum, the Fourier magnitudes of the reference image and the input image are removed from gl,(a,/?). Thus, the Fourier component of the transmittance function that generates the first order correlation signal is gl.a(a,P)

=

1

j t exp [$S(a9P)1 exp [-j$R(a, PI1exp(jbOa).

(B6)

It can be seen from eq. (B6) that the correlation signal is independent of the illumination “a”. An additional advantage of using the threshold function is that the DC terms and the intermodulation terms in the joint power spectrum are eliminated. This relaxes the requirement on input image separation in the input plane of the system (Javidi, Wang and Tang [1991]). B.2. SLIDING-WINDOW LOCAL-MEDIAN THRESHOLDING

We consider the sliding-window local-median thresholding to investigate the illumination sensitivity of the binary JTC. We will show that this technique can improve the performance of the binary JTC and can reduce its sensitivity to the illumination “a”. As we discussed before, the binary JTC using the frequency-dependent threshold function for binarization is invariant to input-image illumination, but requires computation of ET = R2(a,P) + a2S2(a,P). We show here that

414

NONLINEAR PATTERN RECOGNITION TECHNIQUES IN THE FOURIER DOMAIN

[y App. B

the median thresholding method is sensitive to illumination variations. The sliding-window local-median thresholding technique uses the median of the local histogram of pixel values of the joint power spectrum inside a window around each pixel, to binarize that pixel value. This is a spatial-frequency dependent threshold that adapts to the changes of the amplitude spectrum and provides some illumination invariance; owing to the fact that when t h s method is used, the threshold value obtained mimics the threshold function ET(a,P) = R2(a,/3)+ a2S2(a,P).Implementation of this threshold is simple owing to the small window size. The size of the window in this technique is selected according to the maximum and minimum possible distances between the reference image and the target in the input image. The sliding-window local-median thresholding technique is easier to implement than either the frequency-dependent threshold hnction method or the global median thresholding, and does not require an assessment of “a”, the illumination constant. The computer simulations and the experimental results show that the binary JTC using the sliding-window localmedian thresholding technique performs well in detecting a target with unknown input image illumination. B.3. GLOBAL MEDIAN THRESHOLDMG

We discuss the variation of the correlation peak intensity with changes in inputimage illumination using constant median thresholding for binarization. We show mathematically that the correlation peak in a binary JTC using median thresholding decreases with changing the illumination “a” from unity. We use one-dimensional notation for simplicity. The input transmittance function is t(x) = Y ( X - xo) + as(x + xo). If we assume s(x) = r(n), i.e. autocorrelation, the binarized joint power spectrum that generates the first-order correlation is (see eq. B4)

/

n 1-

gl,u(a>= -

[

I’

(1 + a2)R2(a)- ET,,

2aR2(a)

c0s(2x0a),

where the constant threshold value ET,, is defined as ET,, = median{ hist [R2(a)(a2+ 2a cos(2xoa) + l)] } ,

where “hist” is the histogram of the bracketed quantity. When a becomes

(B8) =

1, eq. (B7)

Y APP. Bl

415

FREQUENCY-DEPENDENT THRESHOLD FUNCTION METHODS

where

ET,, = median{hist[2R2(a)(l + cos(2xoa))]}. Assuming that R2(a)is slowly varying compared to cos(2x0a), the histogram of the joint power spectrum E ( a ) = R2(a)[a2+ 2a cos(2xocr) + 11 can be written as (Javidi, Li, Fazlollahi and Horner [ 19951)

h(E) = -

n

7a

da

d-E2

+ 2(a2+ 1)R2(a)E- (a2- 1)2R4(a)

for any a.

a&"

(B11) In eq. (B1 l), a:," and a;,, are the points on the a-axis where the envelopes (a - 1)2R2(a) and ( a + 1)2R2(a)reach E , respectively. Equation (B11) is the mathematical expression for the histogram of the joint power spectrum when the input image s(x) = ~ ( x has ) illumination a The median E T , is ~ a value that satisfies

/

ET.*

/

Em,,

h(E) dE

=

h(E)d E ,

(B12)

ET,"

0

with Em,, the maximum value of the joint power spectrum E. We have shown elsewhere (Javidi, Li, Fazlollahi and Homer [1995]) how E T , is~ related to R2(a). We have also shown that with some assumptions on R2(a)we can find an upper bound and a lower bound for E T , in ~ terms of E T , as: ~

E T , will ~ lie somewhere between the bounds when "a" changes. Notice that the two bounds are tangent at a = 1. From eq. (B7) and the upper bound in eq. (B13), we have

(g) 2

Since is concave and reaches its minimum at a = 1, we can conclude that the first-order correlation peak reaches its maximum at a = 1 and

416

NONLINEAR PATTERN RECOGNITION TECHNIQUES IN THE FOURIER DOMAIN

[V

decreases by increasing or decreasing “a”. The median values of the joint power spectrum E ( a ) = R2(a)[u2+ 2acos(2xoa) + 11 for a and l/a are related as Er,1Iu = (l/u2)ET,u.Using this in eq. (B7) we can also conclude that the correlation peaks for a and l/u are the same.

References Davenport, W.B.,and W.L. Root, 1958, An Introduction to the Theory of Random Signal and Noise (McGraw-Hill, New York). Fielding, K.H., and J.L. Homer, 1990, I-f binary joint transform correlator, Opt. Eng. 29, 1081. Flannery, D.L., and J.L. Homer, 1989, Fourier optical signal processors, Proc. IEEE 77, I5 11. Flavin, M., and J.L. Homer, 1989, Amplitude encoded phase-only filters, Appl. Opt. 28, 1692. Goodman, J.W., 1996, Introduction to Fourier Optics (McGraw-Hill, New York). Hahn, W.B.,and D.L. Flannery, 1992, Design elements of binary joint transform correlation and selected optimization techniques, Opt. Eng. 31, 896. Hester, C.F., and D. Casasent, 1980, Multivariant technique for multiclass pattern recognition, Appl. Opt. 19, 1758. Homer, J.L., 1992, Metrics for assessing pattern-recognition performance, Appl. Opt. 31, 165. Homer, J.L., and P.D. Gianino, 1984, Phase-only matched filtering, Appl. Opt. 23, 812. Homer, J.L., B. Javidi and G. Zhang, 1994, Analysis of method to eliminate undesired responses in a binary phase-only filter, Opt. Eng. 33, 1774. Hsu, Y.N., and H.H. Arsenault, 1982, Optical character recognition using circular harmonic expansion, Appl. Opt. 21, 4016. Javidi, B., 1989a, Synthetic discriminant function based nonlinear correlation, Appl. Opt. 28, 2490. Javidi, B., 1989b, Nonlinear joint power spectrum based optical correlation, Appl. Opt. 28, 2358. Javidi, B.,1990, Comparison of nonlinear joint transform correlator and nonlinearly transformed matched fiber based correlators for noisy input scenes, Opt. Eng. 29, 1013. Javidi, B., 1997a, Securing information with optical technologies, Phys. Today 50,27. Javidi, B., 1997b, Optical information processing for encryption and security systems, Opt. & Photonics News 8, 28. Javidi, B.,and J.L. Homer, 1989a, Single SLM joint transform optical correlator, Appl. Opt. 28, 1027. Javidi, B.,and J.L. Homer, 1989b, Multifunction nonlinear signal processor: deconvolution and correlation, Opt. Eng. 28, 837. Javidi, B.,and J.L. Homer, 1994a, Optical pattern recognition for validation and security verification, Opt. Eng. 33, 1752. Javidi, B.,and J.L. Homer, 1994b, Real-Time Optical Information Processing (Academic Press, San Diego, CA) p. 29. Javidi, B.,and C. Kuo, 1988, Joint transform image correlation using a binary spatial light modulation at the Fourier plane, Appl. Opt. 27, 663. Javidi, B., J. Li, A.H. Fazlollahi and J.L. Homer, 1995, Binary nonlinear joint transform correlator performance with different thresholding methods under unknown illumination conditions, Appl. Opt. 34, 886. Javidi, B., and D. Painchaud, 1996, Distortion-invariant pattern recognition with Fourier-plane nonlinear filters, Appl. Opt. 35, 3 18. Javidi, B., and Q. Tang, 1994, Single input plane chirped encoded joint transform correlator, Appl. Opt. 33, 227.

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REFERENCES

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Javidi, B., Q. Tang, D. Gregory and T.D. Huson, 1991, Experiments on nonlinear joint transform correlators using an optically addressed SLM in the Fourier plane, Appl. Opt. 30, 1772. Javidi, B., Q. Tang, G. Zhang and F. Parchekani, 1994, Image classification using a chirp encoded optical processor, Appl. Opt. 33, 6219. Javidi, B., and J. Wang, 1991, Binary nonlinear joint transform correlation with median and subset median thresholding, Appl. Opt. 30, 967. Javidi, B., and J. Wang, 1992, Limitation of the classical defitution of the correlation signal-to-noise ratio in optical pattern recognition with disjoint signal and scene noise, Appl. Opt. 31, 6826. Javidi, B., and J. Wang, 1995, Distortion-invariant filter for detecting a noisy distorted target in nonoverlapping background noise, J. Opt. SOC.Am. 12, 2604. Javidi, B., J. Wang and A.H. Fazlollahi, 1994, Performance of the nonlinear joint transform correlator for images with low-pass characteristics, Appl. Opt. 33, 834. Javidi, B., J. Wang and Q. Tang, 1991, Multiple objects binary joint transform correlation using multiple level thresholding crossing, Appl. Opt. 30, 4234. Javidi, B., W. Wang and G. Zhang, 1997, Composite Fourier-plane nonlinear filter for distortion invariant pattern recognition, Opt. Eng. 36, 2690. Kozma, A,, 1966, Photographic recording of spatially modulated coherent light, J. Opt. SOC.Am. 56, 428. Mahalanobis, A,, B.V.K.V. Kumar and D. Casasent, 1987, Minimum average correlation energy filters, Appl. Opt. 26, 2633. Marom, E., 1993, Error d i f h i o n binarization for joint transform correlators, Appl. Opt. 32, 707. Oppenheim, A.V., and J.S. Lim, 1981, The importance of phase in signals, Proc. IEEE 69(5), 529. Papoulis, A., 1984, Probability, Random Variables and Stochastic Process (McGraw-Hill, New York). Refregier, Ph., 1990, Filter design for optical pattern recognition: multicriteria optimization approach, Opt. Lett. 15, 854. Refregier, Ph., 1991a, Optimal trade-off filters for noise robustness, sharpness of the correlation peak, and Horner efficiency, Opt. Lett. 16, 829. Refregier, Ph., 1991b, Optical pattern recognition: Optimal trade-off circular harmonic filters, Opt. Commun. 86, 113. Refregier, Ph., V. Laude and B. Javidi, 1995, Basic properties of nonlinear global filtering techniques and optimal discriminant solutions, Appl. Opt. 34, 3915. Rogers, S.K., J.D. Nine, M. Kabrisky and J.P. Mills, 1990, New binarization techniques for joint transform correlator, Opt. Eng. 29, 1088. Schils, G.F., and D.W. Sweeney, 1988, Optical processor for recognition of three-dimensional targets viewed from any direction, J. Opt. SOC.Am. 5, 1309. Tang, Q., and B. Javidi, 1991, Binary encoding of grayscale nonlinear joint transform correlators, Appl. Opt. 30, 1321. Tang, Q., and B. Javidi, 1992, Sensitivity of the nonlinear joint transform correlators: experimental investigation, Appl. Opt. 31, 4016. Tang, Q., and B. Javidi, 1993a, A technique for reducing the redundant and self correlation terms in joint transform correlators, Appl. Opt. 32, 191 1. Tang, Q., and B. Javidi, 1993b, Multiple object detection with a chirp encoded joint transform correlator, Appl. Opt. 32, 5079. Vander Lugt, A,, 1964, Signal detection by complex filtering, IEEE Trans. Inform. Theory IT-10, 139. Vander Lugt, A., and F.B. Rotz, 1970, The use of film nonlinearities in optical spatial filtering, Appl. Opt. 9, 215. Weaver, C.S., and J.W. Goodman, 1966, Technique for optically convolving two functions, Appl. Opt. 5 , 1248.

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Zhang, G., 1995, Signal detection by optical correlators with phase encoding at Fourier domain, Ph.D. Thesis (The University of Connecticut, Storrs, CT). Zhang, G., and B. Javidi, 1993, Random phase modulation techniques for optical pattern recognition, in: Proc. IEEE Lasers and Electro-Optics SOC.Annu. Meeting, San Jose, 1993 (EEE, Piscataway, NJ) p. 57.

E. WOLF, PROGRESS IN OPTICS XXXVIII 0 1998 ELSEVIER SCIENCE B.V. ALL RIGHTS RESERVED

VI FREE-SPACE OPTICAL DIGITAL COMPUTING AND INTERCONNECTION BY

J ~ G EJAHNS N Fern Uniuersitat Hugen, Optische Nachvichtentechnik, Feithstrasse 140, 58084 Hugen. Germany

419

CONTENTS

PAGE

$ 1. FACETS . . . . . . . . . . . . . . . . . . . . . . .

421

5 2.

SYSTEM MODEL AND COMPUTATIONAL ASPECTS . . .

433

5 3. 9 4. 5 5.

NONLINEAR OPTICAL DEVICES . . . . . . . . . . . .

447

OPTICAL INTERCONNECTIONS . . . . . . . . . . . .

466

ARCHITECTURES AND SYSTEMS . . . . . . . . . . .

486

9 6.

CONCLUSION AND OUTLOOK . . . . . . . . . . . . .

503

ACKNOWLEDGEMENT . . . . . . . . . . . . . . . . . . .

504

REFERENCES . . . . . . . . . . . . . . . . . . . . . . .

504

420

9 1.

Facets

1.1. A LOOK BACK

Since the beginning of the technical age, marked by the invention of the telephone in the middle of the nineteenth century, the three areas of information technology (i.e., communications, processing, and storage) were initially fully covered by electric (or electronic) techniques. In 1876, Alexander Graham Bell received a patent for the telephone. There had been several forerunners, most notably Charles Bourseuil in France (around 1830) and Philipp Reis in Germany (around 1860). Bell also invented an “optical telephone”, called the “photophone”, a couple of years later and patented it in 1880 (Bell [1880]). The photophone consists of a membrane that is excited by the acoustic waves of the speech signal. The membrane modulates a light beam which is reflected off the membrane and then propagates through free space to a detector (fig. 1). Bell considered the invention of the photophone to be even more important than the telephone. However, he could not get it to work at a satisfactory level. One problem was caused by the unguided propagation of the light signal through free-space, which limited the transmission distance. This issue is connected to another problem; namely, the non-availabilityof a suitable light source or in other words, the lack of technology. In his demonstration experiments, Bell used light light source

/*

light signal

I

Se detector

membrane Fig. 1.1. Principle of the photophone invented by A.G. Bell: a light beam is modulated by a membrane. The membrane itself is driven by the sound waves of the speech signal. 42 1

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FREE-SPACE OPTICAL DIGITAL COMPUTING AND INTERCONNECTION

[VI, § 1

rays from the sun. It took around 80 years until, with the laser, a light source was invented that would have helped Bell with the photophone. Nonetheless, one may consider the photophone as the beginning of modern optical communications. The demonstration of the laser (Maiman [1960]) was the key invention for optical information technology, and triggered many activities in areas like fiber optic communications, optical data storage, holography, and analog optical information processing. As is well known, fiber optical communications and optical data storage based on the compact disk technology have since then developed into standard technologies, both making use of digital data formats. The preference for digital over analog techniques has a variety of reasons. Digital signals have a better immunity to noise and therefore provide better quality, in general. Furthermore, digital data formats offer a great deal of flexibility in modifying, exchanging, combining, and storing information. It is less well known that by the early 1960s, scientists started to investigate several nonIinear effects in lasers to demonstrate the feasibility of digital optical logic, as described, e.g., by Basov, Culver and Shah [1972]. A few early considerations on the use of optics for digital computing even date back to the 1950s (Ganzhorn, Schweitzer and Kulcke [1959]). Since those early days, digital optical computing has been going in waves. Comparisons of optical and electronic logic devices put a damper on the field, since it was predicted that thermal problems would be more severe for optics (Keyes and Armstrong [1969]). The digital optical computing field became quiescent until the mid1970s, when the residue number system was discovered for optics (Huang, Tsunoda, Goodman and Ishihara [1979]). The analogy between the cyclical nature of modulo arithmetic and some optical phenomena (e.g., the phase of a light wave) spurred many activities. Most of these came to a gradual end within a few years, since there were still no suitable “optical” devices to work with. However, working hybrid opto-electronic residue processors were demonstrated some ten years later, for example, by Falk, Capps and Houk [ 19881 and Goutzoulis, Malarkey, Davies, Bradley and Beaudet [ 19881. In the late 1970s optical bistability was demonstrated in semiconductors (Miller, Smith and Johnston [ 19791, Gibbs, McCall, Venkatesan, Gossard, Passner and Wiegmann [ 19791). Advances in semiconductor technology allowed the fabrication of practical nonlinear optoelectronic devices; e.g., the SEED device (Miller [1987]). What followed was a phase of intensive research, with major programs around the world including both industrial and academic laboratories. The most comprehensive effort was probably the program at AT&T Bell Laboratories, which covered all areas, including devices, interconnections, systems, and archtectures. Out of those efforts resulted several

VL

5

11

FACETS

423

Table 1.1 Evolution of digital optical computing 1965

Optical logic operations

1975

Optical residue arithmetic

1985

Optical bistability, array optics

1995

Optical interconnections

demonstrations of digital optical processing systems. As a beneficial side effect, a number of significant advances were made in such areas as device technology, micro-optics, and system architectures. Since then, many of these advances have “cross-fertilized” other areas, such as the use of optical interconnections for VLSI systems, which is now the area that is most actively pursued in digital free-space optics. The processing of information, or simply computing, still remains a domain of electronics with optics covering only a few niche applications until now. Nonetheless, there have been significant efforts in both analog and digital optical computing over the past 30 years (table 1.1). What is the potential of optics, in particular in the area of digital computing and what has been achieved? This is the question we shall consider in this chapter. 1.2. WHAT’S IN A NAME?

“Optical computing” or “digital optics” or “optics in computing” - various names have been used; however, the meaning is always the same. Since the early 1960s, there has been interest in the question of whether optical signals could replace or complement electronics in computing. Digital computing is a complex task. It comprises all three areas of information technology: processing, communications, and storage (although other terms are more common in the computer world: logic, interconnections, and memory). The fundamental parts of a digital computer are the processor, the memory, and the bus used for processor-memory communication and datainput/output. The processor typically contains combinational logic to perform arithmetic operations as well as registers to store intermediate results. Data are being exchanged constantly between the processor and the main memory over the bus that connects the various parts of a computer (fig. 1.2). The memory is often divided into two functional units, the main memory and the cache. The main memory holds all the relevant data required for a specific task. Memory

424

FREE-SPACE OPTICAL DIGITAL COMPUTING AND INTERCONNECTION

memory

[VI, 0 1

4

bus

input

f

processor

b

output

u Fig. 1.2. Functional components of a computer: processor, memory, and bus

chips currently hold 16 or 64 megabytes of information and have access times of more than 100ns. The cache is a small, fast memory, located physically on the same chip as the processor to facilitate communications between them. The cache is used to store data which are frequently accessed by the processor. Its capacity is typically less than 100 kilobytes with an access time of less than 10 ns. Since data are often routed repeatedly between storage units and the logic unit, feedback loops are usually an inherent part of computing systems. The performance of a computer depends very strongly on the balance between its three parts. Recently, constant improvements of processor and memory technology have put much pressure on the interconnection capabilities of electronic computers, Therefore, optical solutions are being considered for the various levels of computer communication (i.e., between chips, boards, and racks). More recently, with the development of fiber optic communications, interest arose in the optical routing and switching of optical signals (“photonic switching”).There is a number of differences between computing and switching, but also several similarities, so that both are related areas. A communications network consists of a transmission facility and one or more switching systems (Cloonan [ 19941) (fig. 1.3). The switching system receives signals from multiple transmission links and routes them to the desired output transmission facilities. The switch consists of the following functional parts: the controller, the switching fabric, and inpuuoutput interfaces. The switching fabric again consists of switching nodes which are interconnected by links. Routing information is extracted from the incoming signals and sent to the controller which then sets up the nodes of the switching fabric. Different multiplexing schemes are used based on a spatial, temporal or wavelength-oriented representation of the optical signals. Unlike the data in a computer, the signals in a switching system always flow in a feed-forward fashion.

425

FACETS

VI, § 11

I

=I

control 1

I

L

:I-

ai

input

f.*

f.-

: I

4.2

I

I

switching fabric

output

F

Fig. 1.3. Functional components of a switching system.

The interest in optics is motivated by the large bandwidth of optical signals and - in the case of free-space optics - the possibility of not being bound by wires or waveguides. It turns out that the communications are more and more becoming the weakest link in computing and switching systems. 1.3. LIMITATIONS OF ELECTRONIC SYSTEMS

The processing capabilities of today’s computers are impressive. Supercomputers have been built which can perform several GFLOPs (FLOP: floating point operation). At the chip level, one can still observe enormous increases in the processing power of a chip, with dimensions of the individual devices shrinking to a few tenths of a micrometer. However, the speed of today’s high performance electronic computers is increasingly limited by communication problems like the number and bandwidth of the interconnections and by data storage and retrieval rates rather than by processing power. It is interesting to take a look at the different time scales that exist at the various levels of an electronic system represented by the delay times for a signal (table 1.2). What is strilung about this comparison is the fact that there are several orders of magnitude between the delay times for an individual device and the communications on a systems level. The switching time for a transistor (e.g., a MOSFET transistor) have remained relatively constant over the years: on the order of a picosecond, limited essentially by the channel length, i.e., the lateral dimensions between its electrodes (Meindl [ 19951). Current technology allows one to fabricate transistors with channel lengths of 0.25 pm, while the microelectronics industry is gearing up for the next step to get to 0.1 pm

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FREE-SPACE OPTICAL DIGITAL COMPUTING AND INTERCONNECTION

[VI, § 1

Table 1.2 Delay times at different levels of an electronic computer Transistor

O( 1 ps) ‘1

Gate

O( 10- 100 ps)

Chip

O( 1- 10 ns)

Bus

O( 100 ns)

O(.) denotes the order of magnitude.

structures (Taur, Buchanan, Chen, Frank, Ismail, Lo, Sai-Halasz, Viswanathan, Wann, Wind and Wong [ 19971). The improvements in technology will bring the switching delays of a logic gate down to the range of less than 100 ps and clock frequencies of the processors close to the GHz range (Asai and Wada [ 19971). However, while technology still allows one to improve processing and memory devices, the communications capabilities appear to have reached a final limit. The system bus in a computer has emerged as the bottleneck (Boxer [ 19951). Several communications-related limitations exist for the performance of all-electronic computers: (a) physical limitations such as a limited time-bandwidth product of electric wires due to capacities and inductivities as well as effects like electromigration, (b) topological limitations due to the two-dimensional layout of the wiring that is inherent to electronics, and (c) architectural limitations such as the often-cited “von Neumann bottleneck” that limits the amount of data exchanged between processing unit and memory. It is instructive to take a more detailed look at these issues. 1.3.1. Physical limitations Where does the two orders of magnitude discrepancy in the switching delays for a gate and a chip come from? The explanation lies in the need to communicate information between the various parts of the processor. In a processor, one has short (or “local”) and long (or “global”) interconnects. The distribution of interconnection lengths in a processor shows that there are two peaks, one around 0.1 and one around 0.5&, where A is the chip area (Bakoglu [ 19901) (fig. 1.4). The short interconnections serve the majority of the wiring. There is also the need to carry information between distant parts of the processor via

427

FACETS

Occurrence rate local

====F=

1

Fig. 1.4. Histogram showing the distribution of interconnection lengths L , for intrachipcommunications in an electronic processor. The histrogram shows two distinct peaks for local interconnections and global interconnections. A denotes the chip area.

the global interconnects. With chip areas of typically lOmm x lOmm, local interconnection lengths are 1-2 mm and global interconnects are 5 mm or more. The time delay of an electrical signal traveling over a wire of length L, is given as

r = RL,

(CL

+ iCL,),

(1.1)

where R and C are the resistance and capacitance per unit wire length, respectively, and C, is the capacitance of the load at the end of the wire (Taur, Buchanan, Chen, Frank, Ismail, Lo, Sai-Halasz, Viswanathan, Wann, Wind and Wong [1997]). This delay is referred to as RC delay and must be added to other delay components. The second term in eq. (1.1) describes the part in the delay due solely to the wire itself. Wire capacitance per unit length tends to stay essentially constant at 0.2pF/mm and does not scale with technology. On the other hand, resistance per unit length grows linearly with the scaling factor a if the dimensions of the wire are reduced by that factor. Consequently, long and thin wires cause long delay times. They can be used for local interconnects, but not for global interconnects. The way one deals with that problem is by using a hierarchy of interconnections. While local interconnections are implemented by thin and densely packed wires (e.g., with a width of 0.3 pm), one uses “fat” wires (width 3 1 pm) for global interconnections. The number of wiring planes and the pitch of the wires need to be optimized from case to case. High performance processors based on 0.1 pm technology would require six to eight interconnection layers. Bandwidth limitations are even more apparent for off-chip communications. Long delay times are caused by capacitances and inductivities. Consequently, the

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FREE-SPACE OPTICAL DIGITAL COMPUTING AND INTERCONNECTION

[VI, § 1

speed of chip-to-chip or board-to-board communications is much slower than for intra-chip communications and channel density is much lower. The bandwidth of the system bus is typically lOMb/s. The number of input/output pins of a processor chip used for signal communications is a few hundred. This leads to another problem of electrical interconnects related to their topology. 1.3.2. Topological limitations of 2 - 0 interconnections

With the scaling of device structures to a few tenths of a micrometer, processors now consist of lo6 to lo7 transistors. The processing capabilities of a processor must be supported by an adequate number of interconnections for the processor to work efficiently. This empirical relationship has been quantified by Rent's rule, which says that the number of logic gates Ng and the number of interconnections N, should follow the relationship

N,= aN,', where a and b are empirical constants; b typically lies in the range between 0.5 and 0.7. Often used values are a = 2.5 and b = 0.6. A Pentium processor, for example, has lo5 logic gates. With the values just given, Ni = 2500 inputloutput connections would be required, while in practice only about 500 can be realized. This means that the communications appetite of a processor is not fully satisfied, a trend that will become worse with further improvements of device integration. The fundamental problem is that the number of devices and interconnections scale differently with the chip area (fig. 1S).While the number of devices Ng is proportional to A , the number of interconnects in a two-dimensional geometry scales only with &.

Ng gates, N g = A Ni connections, Ni= A

I12

L12

Fig. 1.5. Two-dimensional interconnection topology: the number of interconnections scales with the square root of the chip area A , while the number of devices grows directly with A .

FACETS

429

1.3.3. Von Neurnann bottleneck Computers are traditionally organized in the von Neumann architecture with processor and memory as separate functional units connected via the systems bus (fig. 1.2). Non-von Neumann architectures are used only in special purpose systems, e.g., systolic arrays. In a von Neumann computer, arithmetic operations like addition, subtraction, etc., are performed by first moving the data from the memory to the processor using a sequential binary addressing mechanism. After the arithmetic operation has been finished, the result is again transferred to the memory. The need to permanently exchange data between processor and memory and the sequential addressing mechanism slows down the operation of a system (Huang [ 19841). The problem is further increased by the bandwidth mismatch between processor and memory. Since 1980, processor speed has increased by a factor of 100, so that currently processors can operate at cycle times of about 5 ns. During the same time period, the clock speed of a DRAM memory chip has fallen only by a factor of 2, from 120ns to 60 ns. This mismatch results in a slowing down of the processor at times when it has to wait for data to be retrieved from main memory. This is true, in particular, for RISC (reduced instruction set computer) architectures that can perform several instructions within one clock cycle. The primary tool for handling the problem is to use a small cache memory which is implemented as a fast SRAM and placed on the same c h p as the processor. The data exchange between process-or and cache does not go over the system bus, so that it can be much faster than the exchange between processor and main memory. By constantly updating, the cache keeps the data which are most likely to be used by the processor, Yet, there are limitations to the improvement that caching offers. These are, again, introduced by the limitations of the system bus and the resulting rate at which the cache can be updated with data from main memory. It was pointed out by Dickinson [ 19901 that increasing the bus-width for processor-memory links from the currently typical 64 channels to 1024 channels would allow one to improve the system performance by a factor of 3 to 4. An even bigger improvement might be expected in multichip architectures where currently electronic crossbar switches are being considered as possible alternatives to the system bus. 1.4. OPTICS AS AN INTERCONNECTION TECHNOLOGY

Basically, one can view optics simply as an interconnection technology. With its

430

rack

FREE-SPACE OPTICAL DIGITAL COMPUTING AND INTERCONNECTION

bus

board

[VI,

01

chip

free-space optics,

waveguide /fiber bundle /free-space optics

- 1 m d

t O . 1 m-

Fig. 1.6. Interconnection hierarchy in a computing system and optics technologies. Typical transmission distances are on the order of meters for rack-to-rack communication, O(O.l m) for board-to-board, and O( 1cm) for chip-to-chip.

large bandwidth and its interconnection capabilities, optics can offer interesting solutions to help alleviate the limitations of all-electronic computer systems. Optics is being considered for the various levels of interconnections in an electronic computing system (fig. 1.6). Depending on the transmission distance and interconnection density, different optics technologies are of interest. Between racks, fiber-optical links have already been used for several years; e.g., as the backplanes of switching systems. For board-to-board communications, polymer waveguides, fiber bundles, and free-space optics can be used. For chip-to-chip interconnections, integrated packaging technologies using either waveguide or free-space optics are being investigated. Optical interconnections behave very much like perfectly matched transmission lines; e.g., the propagation speed in a transmission line with a small permeability and permittivity (i.e., ,ur= 1, E , = 2 4 ) is the same as in an optical fiber. However, for electrical interconnections to achieve transmission line speeds, they must be driven by low impedance drivers and be terminated at the end of the wires in order to avoid reflections. Optical drivers always behave as if they were driven by a low impedance source and optical reflections at detectors are usually not a problem. Optical interconnections also offer the advantage of a bandwidth-independent low absorption and they can support larger fan-outs. Another major advantage is that they lack mutual coupling effects, which is why optical signals can cross through each other without the generation of noise or loss of information. In addition, optical signal transmission can be more energy efficient due to the fact that it is not necessary to charge a wire or a cable in order to transmit a signal. This is the basis of fiber optic

43 1

FACETS

N, connections, N , = A

A

Fig. 1.7 Three-dimensional interconnection topology. the number of interconnections N , scales linearly with the chip area A

communications for long and medium distances. However, even for very short distances optical communications can be of advantage energy-wise as compared with conventional electronic interconnections (Feldman, Esener, Guest and Lee [1988], Miller [1989]). Finally, optics offers various degrees of freedom, such as the wavelength, polarization, and the spatial position of a light beam, which can be used for signal multiplexing to enhance the throughput of a communication channel. In particular, free-space optical propagation is of interest to solve the topological problem of 2-D interconnections. For a 3-D interconnection technology, the number of interconnections scales with A like the number of devices on the chip (fig. 1.7). Free-space optical interconnections with several thousand channels are feasible. From that point of view, free-space would be ideally suited to satisfy the hunger for bandwidth of electronic processors. Yet, the advantages of optical interconnections are not quite clear yet. High performance and reliability at low cost are necessary requirements for any kind of technology to be accepted into the competitive world of electronics. This is dlfficult to achieve, in particular for chip-to-chip interconnections. Here, free space optics offers the largest potential, but also faces a variety of difficulties. Mainly, packaging issues remain to be solved (Jahns and Huang [ 19891). Also, for free-space optical interconnections, the availability and performance of 2-D arrays of light sources and detectors is crucial. Both modulator arrays and arrays of vertical-cavity surface-emitting lasers (VCSELs) are being considered. For large 2-D arrays, addressing mechanisms and thermal issues (Ozaktas and Goodman [ 19921) need to be taken care of. Researchers are actively working to solve the interconnection and packaging problems at many different levels and it is likely that future computing and switching systems will incorporate the results of many of these efforts. Research projects include work on guided wave technology using time-division multiplexing, wavelength division multiplexing and space division multiplexing (Hinton [1993]). Another direction of research is centered around what will

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FREE-SPACE OPTICAL DIGITAL COMPUTING AND INTERCONNECTION

[VI, 0 1

be the focus of this review, viz., the use of free-space optics for digital computing and switching systems. Free-space optics has the potential to enhance the performance of digital electronic system by providing a large number of interconnections. Future computing systems will continue to make use of metallic wires, but the availability of free-space optics will give system designers alternatives for engineering a system. In addition, beyond being merely a high bandwidth replacement of electronic wires, free-space optics offers design possibilities like the spatial position for implementing specific routing schemes or reconfigurability of the interconnections. 1.5. OUTLINE

In this chapter, we will review the progress made in the area of digital free-space optics over the past 10-20 years. Guided-wave technology and analog optical computing are not considered here. An area like digital free-space optics that has recently gone through a phase of rapid development does not lend itself to a clearcut presentation, in particular since it is a field to which many disciplines contribute. Nonetheless, this is an attempt to provide the reader with some oversight. The chapter is organized as follows: in 0 2, we will examine hardwarerelated issues from a computational point of view. As a basis, a model for a digital optical computer is used as it was developed at AT&T Bell Laboratories in the mid- 1980s and was adapted by many other research groups. In 9 3 and 9 4, the physics of nonlinear devices and optical interconnections will be described. We shall concentrate on semiconductor devices which have the potential for high speed. Liquid crystal components which have also been investigated for computing and switching applications shall not be considered here. In Q 5 we describe some of the architectural and systems aspects relevant to free-space optical digital computing and switching. Some concluding remarks will follow in Q 6. In writing this review, is was unavoidable to assume a certain point of view and consequently give more room to some of the work related to digital free-space optics at the expense of other contributions. This is more a question of economics rather than a lack of respect for the work done by many groups working in the field. The author hopes that the reader will find this review a useful source for references. In addition, one may consult other overview articles or books that deal with the subject of digital free-space optics. Review articles include, e.g., Jahns [1980], Sawchuk and Strand [1984], Huang [1984], Goodman, Leonberger, Kung and Athale [1984], Berra, Ghafoor, Guizani, Marcinkowski and Mitkas [ 19891, Streibl, Brenner, Huang, Jahns, Jewell, Lohmann, Miller, Murdocca, Prise and Sizer 119891, Drabik [ 19941, Hinton, Cloonan, McCormick, Lentine

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433

and Tooley [1994]. Books on the subject include Murdocca [1990], McAulay [1991], Hinton [1993], Lalanne and Chavel [1993], Erhard and Fey [1994], and Jahns and Lee [ 19941.

5 2.

System Model and Computational Aspects

Much of the work on optical computing was motivated by the parallelism that optical interconnections offer and the potentially high speed of optical logic devices. In the mid-I980s, the question of how to use that potential of optics was one of the main issues in the optical computing community. Some considerations on that subject were summarized by Prise, Streibl and Downs [ 19881. According to these authors, there are two approaches which one can follow: one is to implement an architecture similar to the architecture of conventional electronic computers where the optical devices are connected by optical waveguides (West [ 19851). This is worthwhile only if the optical switching devices can be operated at much higher speeds than the gates in electronic computers. Alternatively, one can try to take advantage of the interconnectivity of freespace optics. Optical imaging can provide up to lOOOx 1000 channels or more. In order to make use of such a wide interconnect, however, a significant fraction of those data channels must be busy at any instant. This requires computer architectures which are different from the sequential architectures of conventional von Neumann machines. Since interconnections based on imaging imply a certain amount of regularity, it is also necessary to take advantage of the parallelism of the optical interconnections without suffering from the constraint of regularity. Two main types of architectures for free-space digital optics were developed: symbolic substitution (Brenner, Huang and Streibl [ 19861) and the concept of optical programmable logic arrays (Murdocca, Huang, Jahns and Streibl [1988]). Both will be described in 0 5. Figure 2.1 shows the model of a hgital optical computer as used in the work at Bell Labs in the late 1980s. It consists of 2-D arrays of optical switching devices interconnected through free-space. The interconnections may be implemented by lenses, lenslet arrays, beamsplitters, gratings, holograms, etc. The architecture is characterized by a very wide pipeline with thousands of channels. Some problems with an inherent parallelism, such as switching in optical communications, may benefit directly from this architecture. For more general problems, the pipelined architecture shown in the figure is efficient only if the computation can be broken up into sufficiently large chunks of data where registering and storing occur seldom, if at all. In fast electronics, if data are not

434

FREE-SPACE OPTICAL DIGITAL COMPUTING AND INTERCONNECTION

input

4 -

optical interconnect

column processing

__c

logic array

g

2

output

I

-

[VL

row processing

-

Fig. 2.1. Model of a digital optical computer.

registered periodically, signals get out of synchronization due to clock skew and differences in propagation delays of the gates. With optics, since the path lengths can be controlled precisely, constant latency architectures and pulsed logic are feasible. Free-space optical imaging systems can be designed such that the path lengths are constant over the whole field. Tolerances caused by aberrations are on the order of less than a wavelength. This corresponds to transit time differences on the order of femtoseconds. The synchronicity of the architecture is one aspect. The other is the use of logic gates with a constant fan-in and fan-out. The fan-in of the device is the number of inputs, the fan-out is the number of outputs. All devices across the array should have the same computational and physical properties. For any device to be used for implementing logic operations, a minimum fan-in and fan-out of 2 is required. As Prise, Streibl and Downs [I9881 pointed out, this value is optimum because a larger value for the number of inputs and outputs makes the devices less tolerant of any device or system-related variations. Large fan-ins also require high contrast of the devices. Finally, as we shall see below, it turns out that if the number of gates is large, regular interconnects are much easier to implement than random interconnects. It was shown that it is possible to build digital processors using simple regular interconnects with constant fan-in and fan-out. Such a design can also be efficient despite its regularity (Murdocca [ 19871). Regular interconnections based on optical imaging still allow for some degree of space-variance. A variety of proposals for implementing space-variant optical networks, such as the Perfect Shuffle, have been demonstrated and will be reviewed in 9 4. In the remainder of this section, we will discuss the computational aspects of the devices (9 2.1) and interconnections (9 2.2).

VI, § 21

435

SYSTEM MODEL AND COMPUTATIONAL ASPECTS

2.1. COMPUTATIONAL PROPERTIES OF NONLINEAR OPTICAL DEVICES

In this section, we will discuss the properties of optical switching devices merely from a computational point of view. The physics of the devices and some examples will be treated in 4 3. Throughout this section, we will follow the arguments given by Prise, Streibl and Downs [1988]. Figure 2.2 shows the characteristics of optical switching devices, categorized as inverting vs. non-inverting and thresholding vs. bistable. In order to use any of these devices as a logic gate, the signals from the previous stage (fig. 2.1) are fed to the device by the optical interconnect. For an inverting device, the output intensity will be high (we use the abbreviation HI), if the input intensity (or sum of input intensities from several input beams) is less than the switching power Psw. Conversely, the device will be in the low state (LO), if the input intensity is larger than Psw. In an analogous fashion, one can describe the operation of a non-inverting device. An external bias beam Ps can be added to the device to bring it near its bistable

thresholding out

out

non-inverting

4

out

'+

4%' out

inverting

Fig. 2.2. Device characteristics.

436

q-2-p tVI,

FREE-SPACE OPTICAL DIGITAL COMPUTING AND INTERCONNECTION

HI

LO

LO

LO

HI

1 1

LO

HI

LO

HI

02

Fig. 2.3. Critically biased inverting bistable device operated as a logical NOR gate.

switching point. This is essential in the case of devices without inherent gain, in particular for some bistable devices. Bistable devices can be used as logic gates or as latches. If the device is used in a pulsed mode, i.e., if the optical power is reduced below the bistable region before each switching event, the device can be considered as a thresholding device. Otherwise, the device is a latch, i.e., it can keep a state for a certain amount of time. The main difference between a bistable and a thresholding device is that for a bistable device the output power must be supplied by a bias beam. The bias brings the device near the switching point, so that a small additional change in the input power causes the device to switch. Therefore, operation of the device critically depends on the biasing which imposes tight tolerances on the accuracies of the optical power supply and the optical interconnect. Figure 2.3 shows the operation of an inverting bistable device as a logic NOR gate (Hinton [1993]). The bias power PB is chosen to be close to the switching power Psw. The power levels P I and Pz of the two input beams are assumed to be the same. If the bias beam brings the device sufficiently close to its switching point, then any input will exceed the nonlinear portion, thus moving the device from the HI to the LO state. The truth table for the NOR operation is shown in the right of the figure. The output of the device will only be HI if both input beams are LO. The computational parameters of a device are the fan-in (i.e., the number of inputs), the fan-out (number of outputs), and the threshold which is the number of HI inputs required to switch the device. For example, the NOR gate shown in fig. 2.3 has a threshold of 1, whereas an AND gate with a fan-in of 4 would have a threshold of 4. Obviously, the value of the threshold must be larger than zero and less than or equal to the fan-in. For a device to be used as a logic gate in a system, cascadability is important.

VI, § 21

SYSTEM MODEL AND COMPUTATlONAL ASPECTS

437

Fig. 2.4. Characteristics of a thresholding device.

This means, that the output from one device can feed the inputs of another device. Another requirement is that a complete set of logic operations can be implemented. As is well-known from Boolean logic, AND and OR form a complete set, but NOR is also a complete set, for example. For nonlinear devices, tolerances in the input power levels will cause variations in the output powers. Therefore, the slope and position of the characteristic curve of a device play an important role. The device properties are related to the system properties. Both are discussed in the following using the example of a noninverting thresholding device. Figure 2.4 shows the characteristic in detail. A thresholding device is characterized by the following parameters: - P,,, the switching power, - AP,,,the switching window, - Po,, the output power just after switch-on, - Po@,the output power just before switch-on, - THI, the differential transmission of the device at switching point when the device is on, - TLO,the differential transmission of the device at switching point when the device is off. From this, one can define figures of merit characterizing the device: - C,,, the switching contrast:

438

FREE-SPACE OPTICAL DIGITAL COMPUTING AND INTERCONNECTION

-

T,,

-

osw,the relative switching window

[W 0 2

the switching transmission:

as,

AP5w

=psw

Whether a device is operated in transmission or reflection is important. It has been pointed out that reflective operation reduces the switching power, enhances the contrast, and allows one efficient cooling from the back of the device (Wherrett [ 19841). It is important to note that a device can have a gain greater than one. One refers to these devices as having inherent or absolute gain, as opposed to devices with differential gain. The performance of a device in a system is defined not only by its individual parameters as d e h e d above but also by variations in the power of the bias beam supplied to the device. This may vary over an array of devices and therefore variations in the switching behavior may occur. Furthermore, there may be variations in the switching power in time and across the array as well as variations in device transmission (or reflection, respectively). These variations intrinsic to a system can be taken into account by defining an effective switching window oetf:

oeff= a,

+ a,,,

=

a,,

6Pe 8 P T + 6P,, -$---+---. psw

PB

PT

For critically biased devices, nonuniformities of the optical power supply beams pose a serious problem. The illumination of the devices is achieved by array generators (or array illuminators) whch will be discussed below. One of the figures of merit of an array generator is the uniformity over the array of spots. Nonuniform illumination may result in erroneous operation of the devices; i.e., a device might switch due to the h g h intensity of the power supply beam rather than the state of the input beams. That problem can be largely reduced by using the concept of dual-rail logic (von Neumann [ 19631).That term describes a binary coding technique that comprises both intensity and spatial encoding. The LO and HI states of a device are represented by two pixels which are always complementary to each other. A binary 0 would, e.g., be represented by two pixels where the upper one is bright and the lower one is dark. The use of

VI,

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SYSTEM MODEL AND COMPUTATIONAL ASPECTS

439

dual-rail logic means that the logic state of a device depends on the relative intensity levels of the two pixels (or the two light beams illuminating the two pixels, respectively) rather than on the absolute power level of a single beam. This reduces to a large extent the accuracy requirements related to the optical interconnect and the array generator. 2.2. COMPUTATIONAL ASPECTS OF THE OPTICAL INTERCONNECTS

In interconnecting arrays of devices, several basic operations are encountered (fig. 2.5). These are - “copying” of an input array to an output array, - “split-and-shift”, where two or more laterally shifted versions of the same input array are copied to the output requiring the optical implementation of fanning out from a single device, and - “permutation” of the pixel positions. All three types of operations can be implemented by using optical imaging techniques. However, whereas “copy” and “split-and-shift” are space-invariant operations, permutations are, in general, space-variant operations. The former can be implemented by conventional imaging techniques whereas the latter may require the use of microchannel imaging using multifaceted elements. We will discuss the physical implementations of various interconnection schemes in a later section. Here, we will concentrate on the terminology and the computational aspects of optical interconnects. First, we want to describe the terms space-variant and space-invariant, regular and irregular, as well as fixed and dynamic. Space-invariant and space-variant interconnections: Interconnections for discrete input and output arrays can be represented as so-called bipartite graphs (fig. 2.6). Both the input and the output are represented by a regular array of positions which are denoted by characters (e.g., a, b, c, . . . ) or numbers (0, 1, 2, 3, . . . ). Often the number of positions is a power of 2.

Fig. 2.5. Basic interconnection tasks for array logic.

440 input

FREE-SPACE OPTICAL DIGITAL COMPUTING AND INTERCONNECTION

output

input

[VI, § 2 output

El Fig. 2.6. Bipartite graph representing an interconnection network (in this case space-variant).

....

Fig. 2.7. Space-invariant interconnect.

Linear optical systems can be characterized as either space-invariant (si) or space-variant (sv). A system is called space-variant if the array of lines emerging from an input position varies to the next (fig. 2.6). A system is called spaceinvariant if every input position generates the same output pattern (fig. 2.7). In a bipartite graph this means that the arrays of lines emerging from all input positions are the same. For arrays of finite size, space-invariance is not strictly possible since some lines would not connect to a position in the output array. Occasionally, the lines are wrapped around in a cyclical fashion. This case is represented in fig. 2.7 by the dashed line. Regular and irregular networks: These terms are not defined precisely; they are used according to the general understanding of what regular and irregular mean. It is important to note that irregular is not the same as space-variant. Certain space-variant interconnections exhibit a high degree of regularity. Actually, the interconnection pattern for sv multistage networks (see below) can be expressed explicitly in terms of a mathematical mapping of input to output positions. Fixed and dynamic interconnections: In electronics, connections are usually fixed, with a few exceptions such as switch boards. For metallic wires, a change of a connection requires a mechanical displacement. Mechanical switches are

VI, 0 21

44 1

SYSTEM MODEL AND COMPUTATIONAL ASPECTS

object

lens

image

--

_____II-

field (area A )

Fig. 2.8. Imaging setup.

also used in optics. However, since they are slow (milliseconds) their use is limited. Optical connections can also be changed using electro-optic effects (e.g., directional coupler) or the optical elements (dynamic holograms or kinoforms). Some of these effects can be very fast (nanoseconds or less) and they offer the possibility for dynamic routing of light signals. The usefulness of dynamic interconnections, however, is not quite clear. In computing, there appear to be only few interesting applications. Maybe this is because current electronic computers are built on fixed interconnections. In the following, we will consider only fixed interconnections and explain several more terms like fan-out, blocking and non-blocking networks. Conventional imaging systems may be based on single lens (fig. 2.8) or 4f-setups, for example. The interconnectivity or channel capacity is given by the space-bandwidth product (SBP) of the imaging system. The SBP is given as the ratio of the field of the imaging system with an area A and the area of one pixel, 6x2. The resolution 6x is given by the numerical aperture NA of the imaging system with NA = sina. For a circular lens aperture, the pixel size is 6x = 2.44 UNA. The field is the area over which aberrations are not significant. In general, a large numerical aperture yields a smaller optical field. The SBP of optical imaging systems ranges between 1O3 for micro-optical imaging systems to lo* for expensive systems used in lithography, for example. The SBP of an imaging system scales approximately linearly with the diameter of the lens if the NA is kept constant (Lohmann [1989]). A specific aspect of digital optics is that the objects are discrete, i.e., the input object consists of a discrete array of light sources, for example. If the size 6x of the devices is comparable to the pitch Ax, one speaks of a densely packed array (fig. 2.9). (For simplicity, we assume that the device size is the same as the resolution 6x of the imaging system.) If k / 6 x is large, one speaks of dilute

442

FREE-SPACE OPTICAL DIGITAL COMPUTING AND INTERCONNECTION

[VI, § 2

pitch, A x

device size, 6x

densely packed array

dilute array

Fig. 2.9. Visualization of densely packed and dilute arrays.

arrays. The first generations of self-electro-optic effect devices used in earlier systems experiments were densely packed, with typical values 6x = lOpm and Ax = 20pm. Examples of dilute arrays would be arrays of optical monomode fibers (6x < 10 pm and Ax = 125 pm) or smart pixel arrays. It has been pointed out by Lohmann [ 19911 and McCormick, Tooley, Cloonan, Sasian and Hinton [1991] that the use of conventional imaging is not optimal for dilute arrays since the larger part of the SBP would be used for imaging the dead space between the devices represented by the shaded area in fig. 2.9. Furthermore, the imaging of dilute arrays may require optical fields which are significantly larger than those which can be obtained with conventional imaging systems. For example, one might conceive of a situation in which a silicon chip with optical inputs and outputs is interconnected via free space. In that case, the size of the chip might be on the order of 10 mmx 10 mm. To better match the optics to such a situation, a microchannel imaging scheme seems more appropriate. Here, each optical channel is implemented by a pair of microlenses (fig. 2.10a). The advantage of microchannel imaging over conventional imaging is that the array size can be large without affecting the image quality since field size and resolution are independent. A limitation occurs, however, in the longitudinal separation Az of the device arrays. Due to diffraction, light may couple between neighboring channels and thus crosstalk may occur. In order to prevent this, it is necessary that Az D2/21 (Smith, Murdocca and Stone [1994]). Here, D is the effective aperture of each channel. A modified imaging scheme which combines in a hybrid form conventional imaging with the microchannel approach will be described in a later section. Due to the use of faceted array components in the microchannel approach, one can also implement space-variant random interconnections if suitable elements

<

VI, § 21

regular Ax

443

SYSTEM MODEL A N D COMPUTATIONAL ASPECTS

random

6X

Fig. 2.10. Regular and irregular interconnections using microchannel imaging.

for beam deflection are added (fig. 2.10b). These may be either prismatic arrays or diffraction gratings, which in turn may be holographic (Robertson, Restall, Taghizadeh and Walker [ 19911) or lithographic (Jahns and Daschner [ 19901, Sauer, Jahns, Feldblum, Nijander and Townsend [ 19941). For a number of optical interconnection schemes, a split-and-shift operation must be implemented. This means that two or more copies of the input are generated, shifted laterally relative to each other by a well-defined distance, and superimposed in the image plane. Sometimes (e.g., for special logic concepts like symbolic substitution) one may wish to be able to influence the various copies individually. In this case, one wants to completely separate the optical paths using branching and masking (fig. 2.1 1; Hinton [1993]). Another solution to realize a split-and-shift operation is spatial filtering using a suitably designed difiactive phase element in the Fourier plane of a 4F-setup mask

$$$*I

output

Fig. 2 11 Implementation of a space-vanant interconnect using branchmg and maslung

444

[VI, § 2

FREE-SPACE OPTICAL DIGITAL COMPUTING AND INTERCONNECTION

Y object

phase grating, period p

image

Fig. 2.12. 4f-imaging setup with grating in the Fourier plane for implementation of a space-invanant interconnect.

grating

LI4

Az

st

Fig. 2.13. Time skew in microchannel imaging systems

(fig. 2.12). The lateral shift of an image in the output plane is determined by the grating periodp according to AYp, where f is the focal length of the lenses. As mentioned before, synchronicity of the channels is a benefit of free-space optics. It can be guaranteed within a few femtoseconds if conventional imaging systems are used. Pure microchannel imaging systems do not, in general, yield perfect synchronicity due to different path lengths in the different channels (fig. 2.13). The time delay At can be expressed as a function of the deflection angle a and the longitudinal separation hz (Shamir [1987]): At=&( c

1 cosa- 1

).

For deflection angles of about 10" and a longitudinal separation of 10 mm, the maximum delay would be on the order of a picosecond. It is interesting

VI, § 21

445

SYSTEM MODEL AND COMPUTATIONAL ASPECTS

F inputsloutputs Fig. 2.14. Switching node in an interconnection network

input array

output array

switching node

I

0 Y

.$

1

3 4 5

6 7

0 (b)

1

2

3 4 5 output

6

7

Fig. 2.15. Crossbar network. (a) Bipartite graph; (b) 2-D representation with N 2 switching nodes.

to note that while diffraction at a single grating causes pulse dispersion, a tandem configuration of deflection gratings allows one to keep the temporal pulse duration constant (Marathay and Lohmann [ 19891). The subject of free-space optical interconnects for digital optical computing systems is closely related to the field of switching networks, in particular spaceswitching networks. A space-switching network consists of arrays of input and output devices and switching nodes which spatially separate inputs and outputs. They are connected with each other through physical channels such as wires, waveguides, or free space. The specific function of the nodes is not specified at this point. It may vary with the specific application for which the network is used. For a sorting network, e.g., the nodes may be compare-exchange units. Each node may, in fact, be a small network of its own. We may represent it as a “black box” with a certain fan-in and fan-out (fig. 2.14). Figure 2.15a shows an important network, the crossbar, which is used in switching. In a crossbar every input is connected to every output. Another

446

FREE-SPACE OPTICAL DIGITAL COMPUTING AND INTERCONNECTION

IVI,

02

representation of the crossbar is shown in fig. 2.15b, where it becomes obvious that the number of switching nodes (the dark squares) is NM for N input devices and M output devices. The figure shows the special case where N = M . Two quantities to describe an interconnection network are the fan-out of the switching nodes and the diameter of the network. The fan-out F is the number of input positions and output positions of the nodes. The diameter V is the number of switching stages required to link any input to any output device. The crossbar represents an extreme since it consists of only one switching stage, where the devices have a fan-out of F = N . The crossbar allows one to route each of the N inputs to any of the M outputs or, in general, arbitrary combinations of outputs. The total number of interconnection patterns supported by a crossbar is N M (or N N in our special case). The crossbar is a nonbloclung network, i.e., it allows one to connect an input device to an output device - or to any combination of output devices - independent of already existing connections. The crossbar has two disadvantages: one is the high cost in terms of the required number of switching nodes; the second is that the network does not scale very well; i.e., the number of nodes grows with N 2 as the number N of stations grows. Therefore, other types of interconnection networks have been considered for optical computing and switching systems, in particular, multistage networks. Of particular interest are multistage interconnection networks (MINs) that are constructed of 2 x 2 crossbar switches (i.e., F = 2). Permutation networks based on MINs typically consist of log, N stages (i.e., V = log, N ) where each stage has N switching nodes. MINs support all N ! permutations and some, but not all, broadcasting connections. A well-known MIN is the Perfect Shuffle (Stone [1971]) (fig. 2.16). The name is derived from a technique for shuffling a deck of N cards by interleaving the input

output

Fig. 2.16. Perfect Shuffle network for N =8.

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NONLINEAR OPTICAL DEVICES

447

upper and the lower halves with one another. After repeating the same operation log, N times, the initial order of the cards is reestablished. For a 1-D input with eight positions the original order is obtained after three identical stages. In the following two sections, we shall consider physical properties and implementations of optical logic devices and interconnections.

0 3.

Nonlinear Optical Devices

The potential of free-space digital optics is based on the parallelism of the interconnections and on the availability of suitable devices to implement logic operations. A variety of devices has been investigated. In fact, by the early 1960s the implementation of optical logic based on the use of lasers had already been demonstrated (see, e.g., Basov, Culver and Shah [1972]). In the 1970s, the main emphasis was on liquid crystal devices used for both analog and digital processing (Bleha, Lipton, Weiner-Avnear, Grinberg, Rief, Casasent, Brown and Markevitch [ 19781). Optical switching devices must fulfill certain requirements as to speed, power dissipation, size, systems behavior, etc. Various attempts were made to understand the potential and the limits of nonlinear optical devices from fundamental considerations. Keyes [ 19701 came to the conclusion that thermal dissipation would limit the potential of optical logic elements. A similar assessment was made by Landauer [ 19761. Later, more optimistic conclusions were reached by Fork [ 19781 and Smith [ 19821 who emphasized the potential of optics for parallelism and subpicosecond switching using nonresonant effects. In the late 1970s, optical bistability was demonstrated, which stimulated research efforts to build nonlinear optical devices based on semiconductor materials. Since the mid-1980s advances in semiconductor technology (like the development of molecular beam epitaxy) made possible the fabrication of semiconductor devices which can be integrated in large 2-D arrays (Tooley, Lentine and Hinton [ 19931).For research purposes, in particular, the self-electrooptic effect devices (Miller [1987]) became important, since they proved to be useful tools for demonstrations of free-space digital systems. Arrays of active (i.e., light emitting) devices for switching and interconnections are becoming practical since the demonstration of room-temperature vertical-surface-emitting lasers (Jewell, Scherer, McCall, Lee, Walker, Harbison and Florez [ 19891). In this section, we will review the work on switching devices for digital optics. Section 3.1 serves to give a classification of the different devices. Section 3.2 is an overview of the main device types. Finally, in 43.3, several of the most widely used devices will be discussed in detail.

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3.1. N O N L N A R OPTICAL DEVICES

3.1.1. ClassiJication

According to Tooley, Lentine and Hinton [1993], one can classify nonlinear optical devices as either active/passive and intrinsichybrid (table 3.1). Passive devices have no light-generating or -amplifying capabilities. However, they can produce signal gain by transferring the information from a low-power beam of light to a high-power beam. This property is referred to as differential gain as opposed to absolute gain exhibited by an active device. Active devices, such as LEDs or laser diodes can provide real optical gain by converting electrical current into optical power. Table 3.1 Classification of nonlinear optical devices

Intrinsic Hybrid (with electrical feedback)

Passive

Active

Refractive optical bistability; electroabsorptive bistability Modulator/detector systems; Liquid crystal light valve

Laser diode systems Detecuemit systems

Active and passive devices can be further classified into hybrid and intrinsic. In a hybrid device the feedback required for nonlinear optical behavior is provided electrically. This may involve electronic circuitry or can be achieved by the internal electrical characteristic of the device. For switching to occur, some positive feedback mechanism is required. The switching speeds of hybrid devices are limited by the time constants of the (electrical) feedback. Intrinsic devices, on the other hand, are made from optically nonlinear materials like GaAs or InSb, and rely on direct optical feedback or the nature of the nonlinearity itself as in the case of thermal nonlinearities. In this case, switching times are determined by the time constants of the medium, or, in very fast devices, by the optical feedback time.

3.1.2. Physical background In a linear medium the relationship between the electrical polarization P and the electric field E of a light wave is given as

VI,

P 31

NONLINEAR OPTICAL DEVICES

449

Here, EO is the electric permittivity of free space and x is the electric susceptibility of the medium. A nonlinear medium, on the other hand, is characterized by

Here anisotropy and inhomogeneity of the medium are ignored for simplicity. One distinguishes between nonlinearities of second order (or x2 nonlinearities), for which

Second-order nonlinearities play a role in frequency doubling, the electro-optic effect, and three wave mixing. Another important range of nonlinear optical phenomena results from the cubic dependence of the polarization on the electric field as it occurs in media with centrosymmetry. The dominant nonlinearity is then pNL =

E3.

(3.4)

These x3 nonlinearities arise in abundance in semiconductors. They give rise to an intensity-dependent refractive index and absorption coefficient. Thudorder nonlinearities give rise to various effects, such as self-focusing, self phase modulation, four-wave mixing, optical bistability and nonlinear optical switching. These effects are subdivided into two classes called resonant and nonresonant optical nonlinearities. Resonant effects occur when photon energies are close to the fundamental absorption edge of a semiconductor medium and are caused by photogenerated carriers. Nonresonant effects occur at photon energies well below the fundamental absorption edge. As pointed out by Ironside [ 1993b3, with resonant optical nonlinearities, real photocarriers are produced which give rise to the nonlinearity. However, they do not fit well into the theory sketched briefly with the above equations. Therefore, when considering resonant nonlinearities, it is more appropriate to regard them generally as intensitydependent optical properties. Third-order nonresonant effects are more easily described by the equations given above. Resonant effects are slower than nonresonant effects and have response times determined by the relaxation time of the excitation (typically in the range from to lo-'' s). However, the advantage of resonant nonlinearities is that they are proportionally more sensitive by factors of 103-108 (Tooley, Lentine and Hinton [1993]). This made possible the demonstration of a range of optically

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FREE-SPACE OPTICAL DIGITAL COMPUTING AND INTERCONNECTION

[VI, § 3

Table 3.2 n2 values for the optical Kerr effect

Material

n2 (cm2/W) Reference

Si02

3.2~10-l~

(for 1 = 1.5 pm, Ironside [1993a])

Non-linear (metal-doped) glasses

10-'4-10-'s

(for 1= 1.06 pm, Aitchison [1993])

GaAs

4x 2.5~

Baker, Etemad and Kajzar [1987] (theoretical value for 1 = 1.5 km, Sheik-Bahae, Hutchings, Hagan and Van Stryland [1991])

Organic materials e.g., Poly(4BCMU)

2.5~

(for 1 = 1.5 pm, Etemad, Baker and Soos [1994])

InSb at 77K

1o

Miller, Seaton, Prise and Smith [1981]

-~

nonlinear effects, such as optical bistability, at milliwatt power levels. This, in turn, is of importance for the prospects of digital optical computing with large arrays of optical logic gates operating in parallel. The simplest type of resonant optical nonlinearity is the opto-thermo-optic effect. This refers to an optically induced heating of the medium by which the optical constants (absorption and refractive index) are changed. In general, any material with a significant thermo-optic coefficient dn/dT and a reasonable amount of absorption will exhibit a refractive optothermal nonlinearity. Values for 122 ranging from lop4 to 10-6cm2/W can be achieved. Response times, however, are slow (from l o p s to 10ms). They are determined by the time it takes for heat to diffuse away from the illuminated region. x3 nonlinearities can also lead to nonresonant effects such as the optical Kerr effect, where an optical intensity I causes a change in the refractive index n described by

The optical Kerr effect is self-induced, which means that the phase of a lightwave depends on the intensity of the lightwave itself. The magnitude of the effect is expressed by 112. The value of n2 varies over many orders of magnitude, dependmg on the material (table 3.2). As the table shows, the effect is the strongest for some organic materials and under certain conditions for semiconductor materials. However, it is also being used in simple Si02 fibers where, e.g., sufficiently long propagation lengths are used to achieve a phase shift of JG.

VI, § 31

NONLINEAR OPTICAL DEVICES

45 1

Making use of purely dielectric nonlinear effects is relevant to the development of switching devices for digital optics, since one can induce changes in the optical constants of a medium which can be read out at the same wavelength as the inducing light beam. This opens up the possibility for constructing cascadable all-optical logic devices. 3.2. OPTICAL SWITCHING DEVICES

- OVERVIEW

The purpose of this section is to give an overview of the different device types that have been investigated for digital free-space optics (table 3.3). A more complete account was given by Tooley, Lentine and Hinton [1993]. Some devices, namely the nonlinear Fabry-Perot etalons, the self-electro-optic effect devices and the vertical-cavity surface-emitting laser diodes will be described in more detail in $3.3. Optical bistability was predicted by Szoke, Daneu, Goldhar and Kurnit [ 19691 and first demonstrated experimentally in sodium vapor by Gibbs, McCall and Venkatesan [1976]. Since then, a large number of bistable devices has been demonstrated. The key factor for intrinsic bistable devices is a medium with a strong nonlinearity of the refractive index. The intensity-dependent change in the refractive index can be used to detune a Fabry-Perot etalon (Karpushko and Sinitsyn [1982]). Various materials have been investigated such as InSb by Miller, Smith and Johnston [1979]. Bistability in bulk GaAs was reported the same year by Gibbs, McCall, Venkatesan, Gossard, Passner and Wiegmann [1979] and later using multiple quantum well (MQW) structures by Jewell, Rushford and Gibbs [1984] (fig. 3.1). A quantum well consists of a very thin (50-100 A) layer of a semiconductor material, e.g., GaAs sandwiched between Al,Ga,,As which has a higher bandgap. The layer with the lower bandgap is called the well, the material with the larger bandgap is called the barrier. The layers are fabricated by molecular beam epitaxy (MBE) or metal organic chemical vapor deposition (MOCVD). One of the most widely investigated devices for digital free-space optics is the quantum well self-electro-optic effect device (SEED) (Miller [ 19871). The basic SEED device relies on the Franz-Keldysh effect or quantum confined Stark effect (QCSE), a field-induced shift of the absorption peak of GaAs. In bulk material, this effect can be observed only at low temperatures. In quasitwo-dimensional quantum well structures one can also observe the wavelength shift at room temperatures. The shift of the absorption peak can be used, e.g., to modulate a light beam in its intensity. A variety of structures based on the self-electro-optic effect has been reported. The basic structure of the

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FREE-SPACE OPTICAL DIGITAL COMPUTiNNG AND INTERCONNECTION

[VI, 0 3

Table 3.3 Overview of device research for digital optics Device

Reference(s)

Intrinsic devices Passive

Non-linear Fabry-Perot etalon

Karpushko and Sinitsyn [ 19821 Miller, Smith and Johnston [1979] Gibbs, McCall, Venkatesan, Gossard, Passner and Wiegmann [1979] Jewell, Rushford and Gibbs [ 19841

Actiue

VCSEL

Iga, Koyama and Kinoshita [ 19881 Jewell, Scherer, McCall, Lee, Walker, Harbison and Florez [1989] Lee, Tell, Brown-Goebeler, Jewell and Hove [1990]

Hybrid devices Passive

SEED

Miller [ 19871 Lentine, Hinton, Miller, Henry, Cunningham and Chirovsky [1989]

nipi

Kost, Garmire, Darner and Dapkus [1988] Riel, Kiesel, Ennes, Gabler, Kneissl, Bohm, Trankle, Weimann, Gulden, Wu, Smith and Dohler [I9921

DAOS

Sizer, Livescu, Cunningham and Miller [1989]

EARS

Matsuo, Amano and Kurokawa [I9921

Actiue

Thyristor-pnpn DOES

Taylor, Mand, Cho and Simmons [1986]

VSTEP

Kasahara, Tashiro, Hamao, Sugimoto and Yanase [1988] Kara, Kojima, Mitsunanaga and Kyuma [I9891 Heremans, Kuijk, Suda, Vounckx, Hayes and Borghs [I9921

early SEED devices consists of a reverse-biased p-i-n diode with the multiple quantum wells in the intrinsic region (“R-SEED”); see fig. 3.2. When no light is incident on the device, there is no current, and the electric power supply voltage essentially appears across the device. As the optical power is increased, a photocurrent is generated which causes the voltage across the device to decrease. The change of the voltage shifts the absorption peak in the GaAs (see below). If the device is operated at a wavelength for which a decrease in voltage causes an increase in absorption, then this increase in absorption causes an increase of

VI, § 31

453

NONLINEAR OPTICAL DEVICES

GaAs

i150-100

GaAlAs

u

8,

+ energy

I

Fig. 3.1. Multiple quantum well structure in GaAs/GaAlAs and band diagram. A quantum well is a thin layer of GaAs (small bandgap) between “barrier” layers of GaAlAs (large bandgap).

load

input beam

n I

P output beam

Fig. 3.2. The basic structure of a SEED device consists of a reverse biased pi-n-photodiode with a multiple quantum well in the intrinsic region and an external load resistor. Depending on the electric field across the intrinsic MQW region, the absorption of the device can be varied. Its state is read out by an optical beam, here shown in transmission.

the photocurrent. The increase in photocurrent, in turn, causes a larger voltage drop across the resistor, a further reduction across the photodiode, a further increase in absorption, and a further increase in photocurrent. This will continue until the voltage across the photodiode approaches forward bias near OY At this point the quantum efficiency of the photodiode will drop to zero, and the device switches abruptly from a high- to a low-voltage state. Since we assumed increasing absorption with decreasing voltage, the switching of the MQW device from the high- to the low-voltage state corresponds to switching the optical light beam from a h g h to a low optical intensity. Other GaAs devices based on the QCSE are the MQW n-i-p-i devices (Kost, Garmire, Danner and Dapkus [1988], Riel, Kiesel, Ennes, Gabler, Kneissl, Bohm, Trankle, Weimann, Gulden, Wu, Smith and Dohler [ 1992]), the DAOS device (Sizer, Livescu, Cunningham and Miller [1989]), and the EARS device (Matsuo, Amano and Kurokawa [1992]). The basic structure of an n-i-p-i device consists of a small number of quantum wells, sandwiched

454

FREE-SPACE OPTICAL DIGITAL COMPUTING AND R*ITERCONNECTION

[VI, § 3

,

n-GaAlAs i-MQW p-GaAlAs i-MQW’

...

Fig. 3.3. Structure of a multiple quantum well n-i-pi

device.

between a p- and n-type wide gap material (fig. 3.3). This basic structure is repeated several times. As with the SEEDS, the operation is based on electroabsorption. When light is absorbed in the device, electrons and holes are generated. They are separated spatially by the electric field across the MQW structure and move into the p and n regions, thereby reducing the electric field and thus changing the absorption. After a time delay, electrons and holes recombine and the device is reset to its original “high” field state. The speed of the devices is limited by the recombination times of the photocarriers. Because of the spatial separation of the charges in the p- and n-layers, relatively long recombination times in the microsecond regime were achieved. The DAOS (diffusion-assisted optical switch) device is structurally very similar to the SEED. It also consists of an MQW pi-n diode that is reversebiased; however, in contrast to the SEED it uses no load resistor (fig. 3.4). When an optical input beam is applied, photocarriers are generated in the multiple quantum wells which then move vertically through the wells. After

t=O

>

input beam

> t

probe beam

Output

beam

-

Fig. 3.4. Structure of a DAOS device and its operation shown in consecutive stages. At t = 0 the input beam causes charge carriers to he generated in the i-region of the pi-n diode. At tl , a probe beam is used to read out the optical state of the device. Finally, charge carriers are moved out of the device by diffusion.

VL § 31

NONLINEAR OPTICAL DEVICES

455

reaching the p- and n-regions, the carriers screen the external field, thus causing reduced absorption. At this time, an optical probe beam is applied to read out the state of the device. Finally, the carriers diffuse laterally out of the device as a photocurrent. The screening voltage disappears. The advantage of the DAOS device is that the charge packets remain localized. This means that in the ideal case the screening takes place only in a small region defined by the spot size of the probe beam. Therefore, an array of devices can be realized by simply fabricating one large mesa without further need for structuring. However, in the experimental device, diffusion of the carriers in the lateral direction was too fast. This means that the charge packets did not stay in the p- and n-regions long enough to collect all the charge for the probe beam to read them out. The EARS (exciton absorptive reflection switch) device (Matsuo, Amano and Kurokawa [ 19921) is a reflective modulator device consisting of a p-i-n structure like the basic SEED device. However, it consists of a larger number of quantum wells and an integrated heterojunction phototransistor (HPT). The MQW stack consists of up to >200 pairs of layers, resulting in a long optical path (up to 8pm) of the light beam inside the device. Thus the contrast ratio of the device is increased relative to the SEED device. Values of up to 20dB are reported. The transistor is used to maintain the logic state of the device even when the external beam is removed. This memory function can be used to make the device act as a latch or to reshape distorted signals using time-sequential operation. Active (i.e., light-emitting) devices include laser diodes as intrinsic nonlinear devices and hybrid devices. Laser diodes can be implemented as edge emitters or as surface-emitting devices (fig. 3.5). Of particular interest are vertical-cavity surface-emitting laser diodes (VCSELs) in which the Fabry-Perot resonator is built orthogonally to the substrate surface (Iga, Koyama and Kinoshita [1988]). The resonators can be realized either by etching into the substrate (Jewell, Scherer, McCall, Lee, Walker, Harbison and Florez [1989]) or by modifLing the crystal structure inside the GaAs wafer by ion beam bombardment (Lee, Tell, Brown-Goebeler, Jewell and Hove [ 19901). The active layer of the resonator is only on the order of lOnm and is thus much smaller than with edge emitters. The diameter is typically less than 10pm in order to suppress lateral modes. The small volume of the resonator allows one to achieve very small threshold currents of 1mA and less. On the other hand, it requires mirrors with very large reflectivities of more than 99% in order to achieve population inversion. They are realized as stacks of MQW layers. VCSELs are of interest because they can be fabricated as 2-D arrays with electrical or optical addressing. Compared with edge emitters

456

-

FREE-SPACE OPTICAL DIGITAL COMPUTING AND INTERCONNECTION

6-8 pm

/active J

[VI, § 3

Au/Ni contact

resonator

Fig. 3.5. Structure of a vertical-cavity surface-emitting laser diode. Here, a structure is shown, where the resonator is etched into the substrate. High reflectivity mirrors are built up as stacks of GaAs and AlGaAs. In the example shown, light emission takes place through the wafer at a wavelength longer than 870nm which corresponds to the band edge of GaAs.

they exhibit superior beam quality due to the symmetric geometry of the resonator. Several active switching devices based on a thyristor-like structure have been investigated. These are the DOES device (Taylor, Mand, Cho and Simmons [1986]), the VSTEP (Kasahara, Tashiro, Hamao, Sugimoto and Yanase [ 1988]), and - without special acronyms - devices demonstrated by Kara, Kojima, Mitsunanaga and Kyuma [ 19891 and by Heremans, Kuijk, Suda, Vounckx, Hayes and Borghs [ 19921. These devices all use a pnpn-structure in series with a resistor (fig. 3.6). One can understand the operation of the devices using load lines. When no light beam falls upon the device, it will be at point A of the load line which is a region of high electrical impedance and no optical output. As the input light intensity is increased, the device will switch to point B (low impedance, optical output). By momentarily setting the electrical power supply to zero and removing the input beam, the device will return to the state represented by point A in the diagram. Switching can be achieved at high speeds limited by the RC time constants of the device. Light emission has mostly been demonstrated in a LED mode which has dsadvantages in terms of the coupling efficiency to

VI,

P 31

NONLINEAR OPTICAL DEVICES

+U

457

I

Fig. 3.6. pnpn-thyristor optical switching device: structure (left) and load line representation (right). The device switches between two states represented by the letters A and B in the figure. State A has high electrical impedance for no light input when no current flows. State B is at the intersection of the load line 1/R, where R is the resistance of the external load resistor, and the characteristic of the thyristor. Here, impedance is low and a current flows which causes light to be emitted from the device.

the optical system. The required optical switching energy can be low (on the order of 0.01 fJ/pm2) if the device is critically biased just below threshold. 3.3. EXAMPLES OF NONLINEAR OPTICAL DEVICES

3.3.1. Nonlinear Fubry-Perot switching devices A nonlinear Fabry-Perot etalon (NLFP) consists of two high reflectivity mirrors and a nonlinear medium. The properties of a linear Fabry-Perot resonator are well known (Born and Wolf [1987]). We assume a length L of the resonator and the same reflectivities r and transmission factor t for both mirrors. Furthermore, it is assumed that r + t = 1. Then, by multiple reflections, the transmitted wave consists of the superposition of many partial waves. The amplitude Et of the transmitted wave is given as

(3.6)

where Ei is the amplitude of the incident wave. The intensity transmission T is given by the Airy function (fig. 3.7):

458

PI, § 3

FREE-SPACE OPTICAL DIGITAL COMPUTING AND INTERCONNECTION

T

c

\

0

I

I\

I

\

: A

*O

Fig. 3.7. Airy function. The solid line represents the transmission of a resonator with a linear medium, the dotted line represents a shifted curve.

where Ii = IEiI2, It = IEtI2, and R = lrI2. Furthermore, F is the finesse of the resonator, F = 4R (3.8) ( I -R)2’ and 6 is the phase shift between two partial waves: 4nnL (3.9) A ’ ~

For phase shifts of 6 = m 2 n (m = 0,1,2,. . .) the Airy function assumes its maximum value of 1. The higher the finesse of the resonator, the sharper the peaks. We may also express this in terms of the wavelengths A,,, for which maximum transmission occurs for a given resonator length L. If we assume that L >> A (corresponding to m >> l), then 2nL A=-. (3.10) m Now we assume that the index of refraction n of the medium inside the resonator depends on the intensity I of the optical field. Since the Airy function depends on the index of refraction, the resonance curve scales and shifts with 1.The shift is described by 2n2IL A A = -. (3.1 1) m For positive values of n2 the Airy curve shifts towards longer wavelengths. Suppose a wavelength A0 is chosen so that for I = 0 the etalon is reflective.

VI,

o 31

NONLINEAR OPTICAL DEVICES

459

T

t

Fig. 3.8. Input/output characteristic for a nonlinear Fabry-Perot etalon.

Then, by gradually increasing the intensity, the resonant peaks will shift so that the NLFP will eventually become transmissive. The increase will become very rapid beyond the threshold intensity It),. When the intensity is reduced again, the transmission value will eventually decrease. However, the light intensity inside the resonator will initially keep the transmission values up. At a threshold ILh < I t h , T will suddenly drop to small values. The inpub'output characteristic for the device is shown in fig. 3.8. From the curve, one can see how a bias beam is used to bring the device close to switching. One or two signal beams can be used to cause the transmitted intensity to increase strongly as the sum of input intensities goes above threshold. It has been pointed out in $ 2 how a nonlinear characteristic as shown in fig. 3.8 can be used to implement logic operations such as the NOR function. NLFPs were first realized as interference filters (Karpushko and Sinitsyn [1982]). The operation relies on a thermal nonlinearity based on absorption inside the cavity. One of the most studied materials for NLFP devices is InSb (Miller, Smith and Johnston [1979]). The nonlinearity in that material is also based on thermal nonlinearities. The effect is very large (n2 = 1 cm2 kW-2), so that changes of the refractive index in the percent range can be achieved with modest laser powers. Devices operating with less than a milliwatt of input power have been demonstrated. Due to the thermal effect used, however, switching times were modest; i.e., on the order of loons. Faster responses could be achieved with GaAs etalons based on electronic nonlinearity as demonstrated by Gibbs, McCall, Venkatesan, Gossard, Passner and Wiegmann [ 19791. However, high intensity levels in those devices lead to heating and thus detuning of the etalon. This problem can be circumvented by using a pumpiprobe technique

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FREE-SPACE OPTICAL DIGITAL COMPUTING AND INTERCONNECTION

[VI,

03

(Jewell, Rushford and Gibbs [1984]). It uses a high-power probe beam which is not absorbed and reads the state of the device just after it has been set by a low-power pump beam whch is absorbed. 3.3.2. Self-electro-optic effect devices In GaAs, absorption occurs for small fields around 850nm (fig. 3.9). This absorption peak is called the exciton peak, since absorption of a photon leads to the generation of an exciton. An exciton is an electron-hole-pair that remains in a bound state similar to a hydrogen atom. In bulk crystals, excitons are bigger (approximately 300 A) and short-lived, which is why the effect cannot be observed at room temperature. In a thin film, a so-called quantum well, the exciton is confined energetically by the sidewalls of the quantum well and remains stable over a sufficiently long time range. The quantum wells typically have a thickness of 50A. SEEDS make use of the changes in the optical absorption which can be caused by changes in an electric field normal to the quantum well layers of a semiconductormaterial (Miller, Chemla, Damen, Gossard, Wiegmann, Wood and Burrus [ 19841).If the quantum well structure is placed in the intrinsic region of a reverse bias diode, one can change its absorption by variation of the electric field (see 5 3.2). A light beam that is sent onto the device can therefore be modulated by applying an electrical signal to the device. A thickness of the MQW region of a few micrometers is enough to change the transmission or reflection by a factor Absorption Coefficient

bm-']

ov

820

840

Wavelength [nm]

880

Fig. 3.9. Absorption spectrum of the SEED pi-n diode for various applied voltages (after Morgan 119911). One can see how the exciton peak shifts and smears out for increasing voltages.

VI,

P

31

NONLINEAR OPTICAL DEVICES

46 1

Fig. 3.10. Layer structure of a SEED device

of 2 or more. This is sufficient for the fabrication of optical modulators. Usually, SEEDSare operated in reflection, by using a multiple quantum well structure as a reflector (fig. 3.10). The possibility of using the quantum wells in a reverse biased diode allows one to make devices which are very energy-efficient. Since no electrical current flows, except for the photocurrent induced by the incoming light beam, little energy is dissipated. The energy hE, which is necessary to modulate the beam, is given as cv2

(3.12) 2 ’ where C is the capacitance of the device and V is the applied voltage. Switching energies per unit area of a few fYpm2 are typical. This value is comparable to electronic switching devices. The switching speed, however, depends on the power level of the optical input beam. A beam of 1mW of optical power impinging on a device with an area of l o x 10pm2 will switch it within At M loops, which has actually been achieved with single devices (Boyd, Fox, Keller, Miller, Chirovsky, D’Asaro, Kuo, Kopf and Lentine [ 19901). The first SEED devices consisted of a single modulator in series with a resistor or a diode. The latter serves as a load for the modulator (see Q 3.2). For systems experiments in the late 1980s and early 1990s, the symmetric SEED device was used (Lentine, Hinton, Miller, Henry, Cunningham and Chirovsky [ 19891). The S-SEED uses differential inputs and is bistable in the ratio of the two inputs. Thus, one can avoid the problem of critical biasing which occurs in the previously described R-SEED. The S-SEED (fig. 3.11) consists of two p-i-n diodes in series, each containing MQW in the intrinsic region. Each modulator acts as a load diode for the other. Because the switching of the device depends on the ratio of the two optical inputs, the symmetric SEED is insensitive to optical power supply fluctuations if both beams come from the same source.

462

FREE-SPACE OPTICAL DIGITAL COMPUTING AND INTERCONNECTION

n

I

i

[VI, § 3

p

I

Fig. 3.1 1. Schematic diagram of a symmetric SEED consisting of two modulators in series. Two input beams and two output beams which are in complementary states are used to operate the device.

The S-SEED has “time-sequential gain”; i.e., it can be set with a pair of low power beams and remain in that state until it is read out with a pair of highpower beams. Since input and output are separated in time, the device has good input/output isolation. Therefore, one can avoid the problem of “critical biasing” that is common to most bistable optical devices. Operation of the S-SEED works as follows: first, a set of unequal beams Pinl and P i n 2 ; sets the state of the device. Provided the power levels of the two beams are sufficiently large, one can force the device into one of two possible states. A contrast ratio of 2:l is sufficient to achieve this. The second set of beams (Poutland pout^) of equal power is used to read the state of the device. Suppose that diode 1 receives the beam with higher power. It will generate a higher photocurrent than diode 2 so that most of the voltage drop occurs at diode 2. This shifts the absorption in the diodes so that an even higher absorption occurs in diode 1, which in turn causes a higher photocurrent in diode 1. This causes a positive feedback required to force the device into one of two states. Using a suitable combination of preset beams, the S-SEED can be made to operate as a logic gate for a set of input beams. AND, OR, NOR, and OR functions can be implemented as well as latching functions (Lentine [1994]). This was taken advantage of in a variety of systems experiments for computing and switching operation. S-SEEDS evolved rapidly from laboratory prototypes to monolithically integrated device arrays with more than 30 000 devices

VI, § 31

463

NONLINEAR OPTICAL DEVICES

v*

‘bias

n

i

p

I

Fig. 3.12. Transistor-based SEED device: MQW modulator in series with phototransistor

per chip. For systems demonstrations, chips with up to 32x64 devices were used (McCormick, Tooley, Cloonan, Brubaker, Lentine, Morrison, Hinterlong, Herron, Walker and Sasian [ 19921). The S-SEED is an example of a three-terminal device with two inputs and one (or two complementary) outputs. Another example of a three-terminal device is a SEED integrated with a transistor (fig. 3.12). The transistor provides gain between the photodetector and the modulator. Either a heterojunction phototransistor in series with a multiple quantum well modulator can be used as shown in the figure. Alternatively, a FET-transistor can be integrated with a quantum well photodetector and a multiple quantum well modulator (F-SEED). Transistor-based SEED devices can also be operated as logic gates for optical inputs and outputs. More recently, SEED devices were combined with GaAs electronics to implement so-called “smart pixel” devices (D’Asaro, Chirovsky, Laskowski, Pei, Woodward, Lentine, Leibenguth, Focht, Freund, Guth and Smith [ 19931). These will be discussed in 6 3.3.4. 3.3.3. Vertical-cauity surface-emitting laser diodes VCSELs have not been used much for optical computing or photonic switching applications, although attempts have been made to integrate them with transistors (Lee, Song, Yoo, Shin, Scherer and Leibenguth [1993]). However, they are attractive components for optical interconnections for a number of reasons. They can be fabricated in 1-D or 2-D arrays with high density. They require only

464

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FREE-SPACE OPTICAL DIGITAL COMPUTING AND MTERCONNECTION

P-Bragg Reflector

I

~~~~}

I

}

16 Pair

I Pah

GaAlAs Spacer

n-Braw ReflecG;

QW Active Layer

\I

} 20.5 Pair

I

I

I

,,, , 3,

’

i:

~

I

GaAs Substrate

Fig. 3.13. Layer structure of an AlGaAdGaAs VCSEL.

low threshold currents of about 1 mA or less at output power levels of around 1 mW. Important for practical reasons is the fact that they emit beams with nearGaussian profiles, which allows one to efficiently couple them to optical elements like fibers or lenses. Stable operation of VCSELs at room temperature was first achieved by Jewell, Scherer, McCall, Lee, Walker, Harbison and Florez [1989]. The difficulty with surface-emitting laser diodes is to achieve a high carrier confinement so that population inversion happens inside the active medium. Reflectivities of over 99% can be achieved by implementing the mirrors as Bragg reflectors fabricated by MBE or MOCVD. A schematic diagram of a VCSEL is shown in fig. 3.13. The active area typically consists of 10 quantum wells of GaAs (thickness: loOK) interlaced with layers of AlGaAs (thickness: 70A). The mirrors are implemented typically by 10-20 pairs of quarter wave layers of GaAs and AlAs. In fig. 3.13 the top mirror is also the p-contact and the bottom mirror acts as the n-contact for the laser diode. p- and n-doping are achieved by using Si and Be, respectively. The structure must be fabricated such that the active zone is in the maximum of the standing wave between the mirrors. Lateral confinement is provided by index or gain guidance. Index structures are fabricated by etching posts into the layer structure as shown in fig. 3.5. Gain guided structures are fabricated by electrically isolating the active volume from the surroundings. This is achieved by ion beam implantation. Gain guided VCSELs are implemented as truly planar devices (Lee, Tell, Brown-Goebeler, Jewell and Hove [ 19901). Figure 3.14 shows the typical voltage+xrrent and power-current diagrams

VI, § 31

465

NONLINEAR OPTICAL DEVICES

cw o u t ut Power I m b j

Voltage [arb.units]

1 .o 0.9 0.8

20 18 16 14 12 10

0.7

8 6

0.4 0.3 0.2 0.1 -0-

0.6 0.5

2 0

0

Y I I I I

2.0

Current [mA]

4.0

Fig. 3.14. Voltage vs. current (V-I) and optical output power vs. current (L-I) of a VCSEL at room temperature (after Morgan, Chirovsky, Focht, Guth, Asom, Leibenguth, Robinson, Lee and Jewell [1991]).

for VCSELs operating in a continuous wave mode (Morgan, Chirovsky, Focht, Guth, Asom, Leibenguth, Robinson, Lee and Jewell [1991]). In this case, the laser was a gain-guided type fabricated by ion beam implantation. This diagram shows a threshold current of 2 mA, which is typical for current devices, although thresholds well in the submicron regime have been demonstrated, e.g., by Hayashi, Mukaihara, Hatori, Ohnoki, Matsutani, Koyama and Iga [ 19951 and by Yang, McDougal and Dapkus [ 19951. The relatively high threshold voltage of 12 V for the specific lasers shown is a consequence of the high series resistance of the mirror stack. Improved designs lead to significantly lower values of 1-3 V. Due to the small cavity length (less than one wavelength) VCSELs generally emit in a single longitudinal mode, the TEMoo mode. For large currents, however, higher transversal modes are excited as a consequence of saturation effects in the TEMoo mode. These can be seen as additional peaks in the spectrum of the laser radiation. 3.3.4. Smart pixels In the course of device development it became obvious that a combination of electronics and optics would yield a higher functionality and better performance than the basic devices described above. For various device types, combinations of nonlinear optical switching devices with transistors have been demonstrated (Brown, Gardner and Forrest [ 199l]), Ersen, Krishnakumar, Ozguz, Wang, Fan, Esener and Lee [1992], Lee, Song, Yoo, Shin, Scherer and Leibenguth [1993], Goossen, Walker, D’Asaro, Hui, Tseng, Leibenguth, Kossives, Bacon, Dahringer,

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FREE-SPACE OPTICAL DIGITAL COMPUTING AND INTERCONNECTION

[W D

4

chip with optics and electronics

optical I/O devices

Fig. 3.15. Schematic representation of a smart pixel array. Gray areas indicate electronic funcions, bright areas optical input/output positions.

Chirovsky, Lentine and Miller [ 19951). The resulting optoelectronic devices dubbed as “smart pixels” (Hinton [ 19881) combine electronic functionality with optical interconnection capability (fig. 3.15). A resulting benefit is the fact that electric interconnections are short, solving the problem of RC time constants on the chip. From an optical point of view, the chip area becomes considerably larger, therefore requiring alternative interconnection schemes. This is related to the imaging problem sketched in 0 2. We shall address this issue in the following section on optical interconnections.

8 4.

Optical Interconnections

In this section, we will discuss the physical implementation of the operations required for interconnecting 2-D device arrays. Depending on the geometry of the device arrays (dense or dilute), different options for imaging one array to the next exist. As visualized by fig. 4.1, the use of active, light-emitting devices like VCSELs or DOES devices simplifies the optical system, since passive modulator-type devices require optical illumination with an array of bias or power supply beams. This task has been dubbed “array illumination” or “array generation”. This section provides an overview of the techniques for array generation and the implementation of the interconnections. We shall distinguish between simple (space-invariant) imaging and space-variant interconnections as used, e.g., in multistage interconnection networks. Micro-optical implementations for all types of named operations are of interest. We will therefore begin this section with a brief consideration about macro- and micro-optical interconnections and technology.

VI, § 41

, I ,

467

OPTICAL INTERCONNECTIONS

beam splitter ( 1 x N )

2-?interconnect

beam splitterkombiner

quarter-wave plate

array 1

array 2

Fig. 4.1. Interconnecting modulator-based and light-emitting based devices. For modulator-based devices run in reflection as the SEEDS, it is necessary to provide arrays of light beams by a 1x N beam splitter and split and combine to bring the signals on and off the chp. Light-emitting devices do not require those optics marked with the shaded areas and therefore reduce the complexity in the implementation of the interconnections.

4.1. MACRO-OPTICS AND MICRO-OPTICS

Classical optics offers numerous techniques which are useful for digital freespace optics (Lohmann [ 19861). Optomechanical hardware, however, is often bulky and expensive, and therefore in general, not suitable for practical purposes (Jahns and Huang [1989], Kostuk, Kato and Huang [1989]). This is one reason why micro-optic components and systems have become of interest for interconnection applications. Another reason is the increased flexibility that micro-optical elements offer. A specific example is computer-generated gratings as they are used, e.g., for beam splitting in an array generator or lenslet array to implement space-variant interconnections. A variety of lithographic fabrication techniques has been applied to the manufacture of micro-optic elements. It would be beyond the scope of this article to go into any of the details, however, a large number of references is available (for example, Herzig [1997] and Jahns [ 1994al. The use of micro-optic elements offers a variety of new possibilities for implementing optical interconnection in the areas of array generation and imaging. We will make use of some basic considerations by Smith, Murdocca and Stone [1994] and Lohmann [1989]. A specific aspect of the use of microlenses as opposed to large “macro” lenses is their scaling behavior. If the diameter and

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FREE-SPACE OPTICAL DIGITAL COMPUTING AND INTERCONNECTION

[VI,

04

Table 4.1 Scaling properties of various optical parameters as a lens is scaled by a factor a < 1 Diameter

D

+

a . D’

Focal length

f

-

a.f‘

Numerical aperture

NA

+

NA’ =

Resolution

6x

---t

6x’ const.

Wave aberrations

Y - +

a . P

Ray aberrations

5

a . E’

-

& const.

focal length of a lens are reduced by a factor a with 0 < a < 1 by keeping the f-number constant, the lens parameters scale with a as shown in table 4.1. Based on the assumption that the scaling process is such that for different values of a lenses are geometrically similar, the image field can be assumed to scale with a’. Since the resolution remains constant (table 4.1), the number of resolvable spots in the image (the space-bandwidth product) decreases. It was shown by Lohmann [1989], that for a 2-D object this decrease essentially goes with a’. The consequence of this is that micro-optic imaging systems do not have the same capacity in transmitting a certain number of channels. While for high performance imaging systems with large apertures (used, for example, in photometry or lithography), SBPs of 10’ are achieved, micro-optic imaging systems with lens apertures of typically 1 mm or less will only be able to image an SBP of typically lo4. This must be kept in mind for the following considerations on micro-optic interconnections. On the other hand, micro-optics offers a number of new possibilities for implementing imaging and beam splitting operations. 4.2. ARRAY ILLUMINATION

An array illuminator (or array generator) is an optical system that splits a single input beam (usually from a laser source) to a regular 1-D or 2-D array of partial beams. Different physical principles have been suggested and demonstrated for free-space optics (Streibl [ 19941).It should be noted that here we do not consider the generation of widely spaced multiple beams, by using, for example, cascaded beamsplitters; see, e.g., Kubota and Takeda [ 19891 and Walker and Jahns [ 19921. For high-density array illumination, depending on the position of the observation plane one may distinguish between Fourier-type (FOU), Fresnel-type (FRS), and imaging-type (IM) array illuminators (fig. 4.2). As we shall see later, all three types can be subdivided further into different categories. An overview of various

VI,

D 41

469

OPTICAL INTERCONNECTIONS

beamsplitter

FRS

FOU

IM

Fig. 4.2. Classification of array illumination techniques. FRS, array illumination using Fresnel diffraction; FOU, far field diffraction techniques; IM, imaging or wavefront splitting techniques.

Table 4.2 Overview of research on array illumination for digital free-space optics Device

Type

References

FOU

Dammam-type

McCormick [1989] Jahns, Downs, Prise, Streibl and Walker [1989] Krackhardt and Streibl [ 19891 Mait [I9891 Morrison [1992]

kinoform-type

Feldman and Guest [1989] Turunen, Fagerholm, Vasara and Taghizadeh [ 19901 Gale, Rossi, Schuetz, Ehbets and Herzig [I9931 Amzon and Ojeda-Castaneda [ 19941 Sinzinger and Arriz6n [I9971

lenslet-type

Streibl, Jahns, Nolscher and Walker [1991]

FRS

Talbot-type

Lohmann [1988] Leger and Swanson [1990] Anizon and Ojeda-Castaneda [ 19931

IM

phase-contrast

Lohmann, Schwider, Streibl and Thomas El9881

lenslet-type

Walker, Taghizadeh, Mathew, Redmond, Campbell, Smith, Dempsey and Lebreton [1988]

telescope-type

Lohmann and Sauer [I9881 Lohmann, Sinzinger and Stork [I9891

references is given in table 4.2. Overviews of array illumination were given by Streibl [1989, 19941. Design parameters for an array illuminator are - splitting ratio: the number of generated beams,

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FREE-SPACE OPTICAL DIGITAL COMPUTING AND INTERCONNECTION

[VI, 5 4

compression ratio: the ratio of the bright area of a spot and the total area of the elementary cell of the spot pattern. For the I-D case, the compression ratio is the spot diameter w divided by the period d , light eficiency: the amount of light intensity in the generated spot array relative to the intensity of the input beam, uniformity: deviation of the spot intensities from the average value, and modal shape of the beams: depending on the type of array generator, the spot array may consist of focused spots or beamlets with a small diameter. Depending on the application, either case might be favorable. The requirement for a high light efficiency is the reason that phase-only components are used as the beamsplitting element of an array generator. For fabrication, mostly microlithographic techniques have been used. Both refractive and difiactive elements have been demonstrated for implementing lenslet arrays or beamsplitter components. Although certain aspects of the fabrication and technologies play a role in determining the usehlness of different approaches to array generation, a description of micro-optic fabrication would go beyond the scope of this review. The interested reader is referred to the literature, e.g., Herzig [1997] and Jahns and Lee [1994]. 4.2.1. Fourier-type array illuminators

The most widely used type of array generator makes use of far-field diffraction using an optical Fourier transformation (see fig. 4.3). Basically, two types of beamsplitter components can be distinguished: gratings of the Dammann or kinofom type and lenslet arrays. Grating type Fourier array generators: The distinction between Dammann type gratings and kinoform type gratings is somewhat artificial. Both are phasebeamspli tter

lens

g(x)

A

spot array

f Fig. 4.3. Optical setup for generating spot arrays using an optical Fourier transformation. The beamsplitter component is denoted by its complex amplitude transmission g(x). The spot array is generated in the back focal plane of the lens.

m, B 41

OPTICAL INTERCONNECTIONS

1

* ‘\\

,kinoform with redundant spatial resolution

k.

+a!

multilevel Dammann grating

\’.\~\ ,/

\ \ \ ~

*0 8

47 1

k\ ’\\.

degree of spatial encoding

binary Dammann grating, classical computer-generated hologram 1

Fig. 4.4. Phase space representation of difiactive optical elements as used for may generation.

only elements that are computed by numerical optimization techniques after defining a cost (or objective) function (e.g., “make diffraction orders with index -N to +N equally bright”); the distinction between them may be justified by the way information is encoded. In the “classic” Dammann grating, information is encoded in the spatial domain, whereas in the kinoform type grating, information is encoded by the phase with gradual transitions between both types. (This may be represented in a phase-space diagram as shown in fig. 4.4). A Dammann grating is a binary phase grating that assumes the phase values 0 and n (Dammann and Gortler [ 19711). For simplicity, we consider the I-D case. One period of the grating is defined by a set of N transition coordinatesXI, . . , ,XN at which the phase value of the grating changes from 0 to n or vice versa (fig. 4.5). The position of these transition points determines the light distribution in the Fourier plane. Initially, Dammann used a symmetric design to reduce the complexity of the grating design; however, this is not a necessary requirement. For the symmetric case, the transmission function g(x) is then described by

The coordinates of the phase transition points are used to control the intensity of the diffraction orders. With the symmetric design, N transition points allow

412

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FREE-SPACE OPTICAL DIGITAL COMPUTING AND INTERCONNECTION

84

X

I

-x, =-0.5

-x? -x*

-XI

xg

=0

XI

X?

xi

X,

= 0.5

Fig. 4.5. One period of a Dammann grating. g(x) denotes the transmission of the grating. Transmission values are +1 or -1 corresponding to a phase shift of 0 or n. The period is assumed to be 1 in this case.

one to generate arrays with 2N + 1 equal intensity orders. The amplitudes of the diffraction orders are given as: N

A0 =

4 C(-l)""xfl

Am

-

=

L

mn

+ (-l)N+'

Z ( - I ) ~ + sin(2nmxn), '

m

#

0.

n= 1

The design of the grating (i.e., the computation of the transition points) is done by means of optimization techniques. The goal of the optimization is to maximize the diffraction efficiency and to minimize the nonuniformity of the diffraction orders in the generated array. Diffraction efficiencies for symmetric binary Dammann gratings lie in the range between 40-70%. 2-D arrays can be implemented simply by using separable designs. Various modifications of the Dammann concept have been described, including nonseparable designs (Mait [ 1989]), multilevel structures (Walker and Jahns [ 1990]), and special symmetries in the design to eliminate, for example, the zeroth order (Morrison [ 19921). The virtue of the original Dammann approach is its conceptual simplicity. Practical problems arise from the high requirements for the spatial resolution necessary to physically implement the transition points precisely. In particular, for large arrays the limitations of the spatial resolution limit the uniformity of the spot array generated (Jahns, Downs, Prise, Streibl and Walker [1989]). Improvements in the processing power of computers have allowed the realization of computationally more demanding designs, which we denote as kinoform-type array generators, first described by Feldman and Guest [ 19891. For

VI, c3 41

473

OPTICAL INTERCONNECTIONS

period

4

N pixels

pixel

Fig. 4.6. Structure of a kinoform-type Fourier array illuminator. One period consists of N pixels (number of equal intensity orders in the spot array) of fixed size. Information is encoded by the phase value of each pixel, here represented by the shading.

those elements a cell-oriented design is used as in classical computer-generated holograms. This means that one period of the grating consists of a number of cells or pixels. The number of pixels is determined by the number of equalintensity spots to be generated. Each pixel is assigned a specific phase value (fig. 4.6). Ideally, one uses a continuous representation of the phase levels. For practical reasons, this is difficult to achieve by fabrication and one uses a quantized phase structure. By adding more pixels to a period, one can improve the efficiency of the element (Arriz6n and Testorf [ 19971). For both Dammann gratings and kinoform type gratings, various iterative optimization techniques have been used, based on, for example, iterative Fourier algorithms like the Gerchberg-Saxton algorithm (Gallagher and Liu [ 1973]), simulated annealing (Turunen, Vasara and Westerholm [ 1989]), and genetic algorithms (Johnson, Abushagur and Kathman [19931). Lenslet-type Fourier array generators: A lenslet array is a periodic structure and therefore generates a diffraction pattern consisting of discrete spots. In one dimension, the number N of spots generated is given by the diameter of the illuminated area, which is derived from simple geometrical considerations to be p F / f , and the pitch of the spots, AF/p (fig. 4.7):

Here, p is the period of the lenslet array or its pitch, f is the focal length of each lenslet, and F is the focal length of the Fourier lens. The input object g(x) to the Fourier setup is the array of focal spots of the lenslets. Assuming rectangular

474

FREE-SPACE OPTICAL DIGITAL COMPUTING AND INTERCONNECTION

lenslet array, pitch p , focal lengthf

[VI, § 4

Fourier lens, focal length F

Fig. 4.7. Array generation in the far field by using a lenslet array. The lenslet array drawn with shaded lines can be used to compensate the quadratic phase factor which occurs across the focal spots of the lenslets in the first array. The number N of spots in the output array is given as the quotient of the diameter of the illuminated area and the spot separation.

apertures of the lenslets, their focal spots have a profile described by sinc (xp/Af) multiplied with a complex exponential factor. Hence: M/2

(4.4) -M/2

where * denotes the convolution operation and M is the number of lenslets in the array. The spot pattern in the Fourier plane is the interference pattern of several tilted plane waves. The amplitude in the Fourier plane is then given as

=

\

-

c , M/2

X2

[enp(inm)

*rect($)] "

ue(x)

.

exp(2nimz)

(4.5)

-M/2

sin n ( + ~ 11-1X P

[

AF

sin(nz)

'

The term in brackets, u,(x), describes the envelope to the spot array. It is visualized in fig. 4.8. Obviously, a significant nonuniformity occurs across the array. This can, in principle, be removed by placing a second identical lenslet array in the focal plane of the lenslets as indicated in fig. 4.7. In practice, however, this technique is difficult to realize. Even small phase variations across

VI,

o 41

OPTICAL INTERCONNECTIONS

475

Fig. 4.8. Envelope to spot array generated by lenslet-type array generator.

the focal spot array - as they might occur due to fabrication nonuniformities of the lenslets - will again result in a non-uniformity of the output array. 4.2.2. Fresnel-type array generation using Talbot self-imaging

A wave field with a lateral periodicity (in x and y ) also exhibits a longitudinal periodicity along the axis of propagation (z-direction). This effect is well known as Talbot self-imaging (Talbot [1836], Patorski [1989]). Often considered is the case of a 1-D grating with period d . Talbot images of the grating occur at distances TZZT ( n = 1,2,3,. . .), with

Light distributions of high contrast also occur in “fractional Talbot planes” given by z = (n/m)zT, where m and n are both integers. If phase-only gratings with specific phase depths and duty cycles are used in the input plane, high contrast interference patterns occur in those planes. A graphical explanation for the fractional Talbot effect was given by Lohmann [1988]. He used a phasor description to represent the phase values of the grating and the wavefield. This is visualized in fig. 4.9 for a grating with @ = d 2 . The grating is represented by the phase values of f n / 4 indicated by the white and the shaded dot. As the wavefield propagates along the z-axis the two phasors remain in a stable relationship to each other, but rotate together around the origin of the complex plane. Without going into the theoretical details, we would like to mention that this description can be understood in terms of the propagation of the angular spectrum (see, e.g., Goodman [1968]). After the Talbot distance ZT, the two phasors assume their original values, meaning that no amplitude modulation

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FREE-SPACE OPTICAL DIGITAL COMPUTING AND INTERCONNECTION

[VI,

s: 4

6 Fig. 4.9. Array generation using the Talbot effect. In the figure a phase grating of period d is shown. In the lower part of the figure, the conversion from a phase distribution to an amplitude distribution is visualized. The whte and the shaded dot represent the phase values of the &lst diffraction orders.

occurs across the plane. However, in planes z = zT/4 and z = 3z~/4,the field is amplitude modulated. This is described by the situation that one of the two dots does not lie on the unit circle. From further considerations, one can derive a variety of solutions for the design of the grating and the planes in which to observe high contrast amplitude patterns (Suleski [1997]). The design parameters for the grating are the phase depth 4 and the mark-to-space ratio w/d. Two examples for possible solutions are

n 1 d2 -w_- 1 (Ronchi grating) and phase depth : z=zzT=2, d 5

(4.7)

2 1 ld2 and phase depth -n: 3 = -zT 3 = 3h' A slightly different approach for a Talbot array illuminator was described by Leger and Swanson [ 19901 who used multilevel phase gratings with a quadratic phase code.

4.2.3. Array generation using imaging There are two types of array generators in which the spot array is generated in the

VL ij 41

OPTICAL INTERCONNECTIONS

phase grating

phase shifting element

471

image plane

Fig. 4.10. Array illumination using the phase contrast method. After spatial filtering, phase shifted areas in the input grating appear bright in the output plane before a dark background.

image plane. One is based on spatial filtering using the phase contrast method, and the other is based on pupil division using lenslet or prism arrays. Phase contrast array generator: The phase contrast technique after Zernike [1935] relies on spatial filtering principles to convert a phase distribution to an amplitude distribution. This is achieved by putting a phase shifter at the origin of the Fourier plane. This technique was adapted by Lohmann, Schwider, Streibl and Thomas [ 19881 to the problem of array generation. To that purpose, a phase grating with a duty cycle w/d (in one dimension) is used as the input object. The phase shift in the grating is again denoted by @ (fig. 4.10). In order to achieve a high contrast in the output image, one must satisfy certain conditions which relate the phase shift @ and the ratio w/d to each other and to the phase shift a of the phase shifting element. According to Lohmann, Schwider, Streibl and Thomas [ 19881, these relationships are: c o s @ =1 - -

a = -@.

d 2w’

Equation (4.8) leads to a 2-D compression ratio (w/d)2 which lies in the range 6 ( ~ / d6) 1. ~ This means that it is difficult to achieve simultaneously high contrast and a large compression ratio (small spots). A trade-off exists, however, between background brightness and compression ratio. If a moderate background illumination can be tolerated, better compression ratios (i.e., closer to 0) can be achieved. Array generators based on pupil division: Simple setups for generating spot arrays are shown in fig. 4.1 1. One is the simple case in which a light wave illuminates a lenslet array. The second case is the use of telescopelets which

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FREE-SPACE OPTICAL DIGITAL COMPUTING AND INTERCONNECTION

lenslets

[VI, 5 4

telescopelets

Fig. 4.1 1. Spot array generation using pupil division by lenslets and telescopelets.

generate spot arrays by compressing the light into smaller areas (Lohmann and Sauer [1988], Lohmann, Sinzinger and Stork [1989]). The telescopelets are realized by lenslet arrays of the same pitch and numerical apertures but of different focal lengths. 4.2.4. Comparison

The various methods for array generation differ in their properties. A comparison is given in table 4.3 according to the criteria named earlier. The main advantage of the Fourier-type array generators lies in their flexibility and in the fact that good efficiencies and uniformities are achieved. Limitations arise due to fabrication problems and the computational expense of calculating a grating. With “analog” FRS- and IM-type generators, very large spot arrays can be generated. However, compression of the beams is usually worse than Table 4.3 Comparison of the performance of array generation techtnques. The numbers are given for the 1-D case. Limiting factors or trade-offs are given in brackets.

Splitting ratio

FOU

FRS

IM

Grating-type: 6 64 Lenslet-type: 2 100 (fabrication)

Large ( 2 100)

large (2 100)

Limited ( w / d =1/9) (trade-off w. efficiency)

pc: limited (wld= 1/4) (trade-off w. efficiency)

Compression ratio Good (wld < 0.1) (optical setup) Efficiency

Good: 5&90% (design)

High (nonuniformity near edges)

High (nonuniformity near edges)

Uniformity

Good: few percent (fabrication)

Same as illuminating wave

Same as illuminating wave

VI, 5 41

479

OPTICAL INTERCONNECTIONS

for FOU-type generators. Also, the uniformity of the spot array is (essentially) determined by the profile of the illuminating wave. For example, if good uniformity is required, Gaussian beams can not be used to illuminate the beamsplitter; instead, the profile of the laser beam must first be converted to a flat top. 4.3. IMAGING

In order to provide h g h interconnectivity, optical imaging is used. In this section we discuss various techniques for imaging discrete 2-D arrays of spots. The conventional approach would be to use a 4F telecentric imaging system as shown in fig. 4.12. We use the letter F here to denote the focal length of the lenses. Telecentric imaging offers a variety of desirable features as discussed for applications in digital free-space optics by McCormick and Hinton [ 19931. As pointed out by Lohmann [ 19911 and McCormick and Hinton [ 19931, using a conventional imaging setup for imaging discrete spot arrays is wasteful of the space-bandwidth product, since useless information (represented by the dark area in fig. 4.12) is also imaged through the system. An imaging setup that makes better use of the geometry is shown in fig. 4.13. Here, individual optical channels for every device are implemented by a pair of lenslets. It is straightforward to extend this approach to the implementation of space-variant interconnections. A practical problem arises, however, since diffraction at the lenslet apertures may cause crosstalk by light falling onto a neighboring aperture. In order to prevent this, the longitudinal separation Az. of the lenslets must be smaller than d2/2A. Here, d denotes the pitch of the lenslet array assuming a full illumination of the apertures. For example, if A = 0.85 pm input

lens

lens

output

w field

Fig. 4.12. Telecentric setup: for h g h resolution imaging, a large aperture is required, resulting in a small field around the optical axis. F is the focal length of the imaging lenses. On the right, cross-sections indicate the sizes of the lens apertures and the image field.

480

FREE-SPACE OPTICAL DIGITAL COMPUTING AND INTERCONNECTION

n

/ lenslet may

r

L

U

- f

[VI, 0 4

--

--

A2

A

I

,-I

Fig. 4.13.Microchannel imaging of discrete arrays.f is the focal length of the lenslets

and p = 100 pm, one would obtain Az = 6 mm. For practical purposes, however, larger interconnection distances would often be preferred. In order to be able to transmit the signals over large distances, but keep the advantages of microchannel imaging, Lohmann [ 19911 suggested a combination of conventional imaging and the microchannel approach (fig. 4.14). The resulting “hybrid imaging” setup essentially consists of a 4F-system. However, rather than imaging the device arrays directly onto each other, microlenses are used to collimate the light from the input sources and focus it to the output positions. The 4F-setup is merely used to image A, onto A2. The hybrid system splits up the imaging task in a favorable way. The high resolution for forming small spots (e.g., on the order of 10pm in diameter) in the output plane is provided by the microlenses. The large lenses in the 4F-system need only resolve the aperture of the microlenses (e.g., l00pm diameter). Therefore, the requirements for the optical properties of these lenses are greatly reduced as compared with a conventional 4F-setup. Based on simple considerations, one can express this as a reduction of the numerical apertures of the lenses in the 4F-system and the hybrid system (Jahns, Sauer, Tell, BrownGoebeler, Feldblum, Nijander and Townsend [ 19941): (4.10) (4.1 1)

VI,

5 41

48 I

OPTICAL INTERCONNECTIONS

arrayA,

lens

lens

0

0

0

0

0

arrayA2

0

Fig. 4.14. Hybrid imaging system combining 4F-imaging with microchannel imaging. A larger field is achieved than with the conventional 4F-setup.

where d and D are the apertures of the lenslets and the large lenses, respectively, and N is the number of lenslets in the array (I-D). The two equations can be used to show that smaller numerical apertures are required for the hybrid imaging system for achieving the same resolution. Since the aberrations of an imaging system increase strongly as the numerical aperture goes up, the hybrid imaging approach can be very helpful, particularly for micro-optical systems (Jahns and Acklin [1994], Sinzinger and Jahns [1997]). As indicated by the figure, one can achieve systems suitable for imaging discrete device arrays with a larger field than with conventional imaging and without the crosstalk problems of microchannel imaging. It should be noted that the hybrid imaging approach

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appears to be particularly well matched to the task of imaging smart pixel arrays which are “dilute” in the sense that the pitch of the array is considerably larger than the size of an individual optical inputloutput device. 4.4. OPTICAL MULTISTAGE INTERCONNECTION NETWORKS

Optical imaging with its large space-bandwidth product forms the basis for computing and switching purposes. However, for computing and switching purposes, space-invariant interconnections are of limited use. Many applications require the implementation of space-variant interconnections. In addition, fan-out is often required. The implementation of space-variant operations implies, in general, limitations in the achievable space-bandwidth product. An arbitrary space-variant operation may be visualized by the interconnection pattern as visualized in fig. 4.15. This represents the case of a crossbar network interconnection matrix

NxN

Fig. 4.15. Crossbar network for a 2-D input comprising N 2 channels. Only a few channels are shown. In order to provide an arbitrary interconnectivity between input and output, the system has to provide N 4 degrees of freedom. The interconnection pattern is completely space-variant. One may think of the interconnection matrix as a binary mask with apertures whch can be either opaque or transmissive to block or transmit individual light paths.

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for a 2-D input and output. We assume that input and output both consist of N 2 pixels. To be able to realize any arbitrary connection pattern between them, the system must provide N 4 channels or degrees of freedom. This result is merely an extension of the relation for the crossbar for 1-D inputs given in Q 2. The 2-D crossbar may be viewed as a model for an arbitrary space-variant interconnection network. If we denote the number of inputs by SBP, (for spacebandwidth product) and the capacity of the system by SBP,, we can write for the relationship between both for a completely space-variant interconnect: SBP,

= SBP, ”2

(space-variant).

(4.12)

For a given space-bandwidth product of the system, a corresponding size of the input results. As an example, we assume SBP, = N 4 = lo4. In this case, the number of input and output channels would be SBP, = N 2 = 100. For transformations which are separable in x- and y-direction, the operation can be implemented in two sequential steps, each operating only on the rows or columns, respectively. One may think of each step as an N-times parallel matrix-vector multiplier setup for 1-D inputs. Each step then requires a spacebandwidth product of the system of N 3 , N 2 channels for each row or column, respectively. For this case, one has SBP,

= SBP,2’3

(space-uariant, separable).

(4.13)

A space-invariant interconnect is described by a Toeplitz matrix. The number of degrees of freedom in the input and in the matrix is the same, hence

SBP,

= SBP,

(space-invariant).

(4.14)

A space-invariant interconnect can be implemented efficiently by an optical imaging setup. From eq. (4.14) it follows that it is much easier to transmit a large number of data channels. Consequently, it is advantageous if an interconnect can be “constructed” of space-invariant interconnections, as pointed out by Jahns [1982, 19831. As an example of how an sv interconnect can be “synthesized” by si interconnections, we consider fig. 4.16. Figure 4.16a shows one stage of the inverse Perfect Shuffle network (c.f. fig. 2.16). Figure 4.16b shows that the same interconnection pattern can be achieved by the following sequence of operations: - copy: generates two copies of the input, - split-and-sh$t with a shift of one unit distance in one direction,

484

FREE-SPACE OPTICAL DIGITAL COMPUTING AND INTERCONNECTION

COPY

split-and-shift

PI, § 4

undersampling

Fig. 4.16. (a) One stage of the inverse Perfect Shuffle interconnection network for an input with = 4 positions. (b) The space-variant interconnection network can be “synthesized” of two (spaceinvariant) operations (copy and split-and-shift) and subsequent undersampling in the image plane. The dashed lines represent “unused” paths.

N

undersanzpling: only every other position is used. The space-variance of the interconnection pattern is contained in the undersampling operation. Since it can be implemented in the image-plane simply by masking, it does not require an increased space-bandwidth product. Size compatibility of the input and output can be achieved by anamorphic imaging whch compresses the 2-D data field in one dimension by a factor of 2. Fortunately, most interconnection networks of interest for optical computing and photonic switching show some amount of regularity that allows one to implement them efficiently in terms of the space-bandwidth product. The principle described above has been used for various implementations. It has been pointed out by Lohmann [1986] that for implementing the fan-out of 2 interferometer-type setups are of interest. An example is given in fig. 4.17. The split-and-shift operation can be implemented by a number of optical components. For the Perfect Shuffle, the use of Wollaston prisms (Jahns [ 1982]), a pair of prisms (Lohmann, Stork and Stucke [1985]), tilted mirrors in a Michelson-type setup (Brenner and Huang [ 1988]), lenslet arrays (Eichmann and Li [1987], Stirk, Athale and Haney [1988], Sawchuk and Glaser [1988], Kawai [1991]) and a Fresnel double mirror (Sheng [1989]) were suggested. As an example the setup demonstrated by Brenner and Huang [1988] is shown in fig. 4.17. Other multistage interconnection networks equivalent to the Perfect Shuffle are the Banyan network (Goke and Lipovski [1973]) and the Crossover network(Jahns and Murdocca [ 19881). The interconnection diagrams for both are shown in figs. 4.18 and 4.19. Optical implementations based on similar design

-

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485

OPTICAL INTERCONNECTIONS

0 1 2 3 4 5 6 7

mirror

beamsplitter cube

0 1 2 3 4 5 6 7 ' 0 1 2 3 4 5 6 7

~

shifted images

Fig. 4.17. Optical implementation of one stage of the perfect shuffle. For beam splitting and combining a beam splitter cube is used. Two shifted images of the input are generated in the output plane. The shift is controlled by tilted mirrors in the two arms of the setup.

Fig. 4.18. Banyan network.

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FREE-SPACE OPTICAL DIGITAL COMPUTING AND INTERCONNECTION

[VI, § 5

Fig. 4.19. Crossover network.

principles were suggested for the Banyan by Jahns [1990] and Cloonan and Herron [1989], for the Crossover by Jahns and Murdocca [1988]. The Crossover network was invented in particular with a free-space optical implementation in mind. Each stage consists of straight-through connections and crossover connections which implement a spatial inversion of a particular section of the input array. This space-variant inversion can be aclueved optically by a double imaging step with a retroreflector array in the intermediate image plane (fig. 4.20). For the straight-through interconnections, a mirror is used in the intermediate plane.

5 5.

Architectures and Systems

Digital optical computing and switching architectures have been under investigation since the mid-1970s. Depending on the physics, they can be subdivided into guided-wave and free-space systems. Here, we review some of the work on architectural and systems issues using free-space optics. For an overview of photonic switching systems based on guided-wave technology, see Midwinter [1993]. In order to take advantage of the spatial and temporal bandwidth of freespace optics, one must consider parallel processing architectures. In electronics, there have been a number of efforts to make use of parallel processing concepts. A well-known classification of parallel computers goes back to Flynn [1966]. According to the number of control and data streams in a computer, he

VI, § 51

ARCHITECTURES AND SYSTEMS

optical setup

i

487

interconnect

m 01234567

01234567

out

Fig. 4.20. Optical implementation of one stage of the crossover network using a Michelson-type setup. The straight-through and crossover connections are implemented in the two arms of the setup. The spatial inversion is realized by placing a retroreflector array with 90" facets in one of the two intermediate image planes. From one stage to the next, the retroreflector array gets finer to implement the crossover connections on well-defined partitions of the input array as shown in the right half of the figure.

distinguishes between single versus multiple instruction (SI, MI) and single versus multiple data (SD, MD). The classical von Neumann architecture would be a SISD computer. Parallel processing occurs by simultaneously applying the same operation on many data in a SIMD system. Optical models for parallel computing are often based on SIMD structures using regular arrays of logic gates and regular interconnections. As described by Erhard and Fey [ 19941, parallel computers can be categorized into four classes: - pipeline processors, - multiprocessors, - neural networks, and - array processors. Pipeline processors belong to the class of SIMD computers. They are the most common class of computers. They consist of a sequence of processing units each of which performs a specific operation. The data stream flows consecutively through all units. The combination of their operations leads to a total operation of the system. Pipeline structures are important for optical computing. The use of interconnections without skew as offered by free-space imaging systems is essential for the implementation of a pipeline architecture. Optical systolic

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[VI, $ 5

arrays (Caulfield, Rhodes, Foster and Horvitz [1981]) and the concept of optical programmable logic arrays (Murdocca, Huang, Jahns and Streibl [ 19881) are examples of optical pipeline architectures. Systolic arrays consist of simple regular structures and are therefore suitable for an optical implementation. An acousto-optic implementation of an analog optical matrix-vector multiplier using a systolic structure was described by Rhodes and Guilfoyle [ 19841. The importance of systolic arrays for digital fi-ee-space optics was described by Erhard and Fey [1994]. Systolic arrays can be considered to be a subclass of cellular automata which go back to von Neumann [1963]. Proposals for optical cellular automata have been made (see, e.g., Yatagai [1986], Murdocca [1987], and Taboury, Wang, Chavel, Devos and Garda [1988]. Multiprocessor systems are members of the class of MIMD computers. They are built of a number of identical or similar processing units which work jointly on a specific problem. The individual processors work on their subtask independently; this means that at a given moment different commands are executed by different processors. Coupling between the processors is achieved through shared memories or message passing. The role of optics in multiprocessor systems may be to provide fast parallel interconnections between boards or chips. A free-space optical board-to-board interconnect for an electronic multiprocessor with thirty-six boards was demonstrated by Sakano, Matsumoto, Noguchi and Sawabe [ 19911. Neural networks also belong to MIMD systems. Here one follows an architectural model based on the function of the human brain, with a large number of densely packed simple processing units (neurons). Information is passed between the neurons via weighted interconnections. Particularly because of the large number of interconnections that is required, optical solutions have been investigated extensively during the past ten years (Abu-Mustafa and Psaltis [ 19871, Yayla, Krishnamoorthy, Marsden and Esener [ 19941). Array computers form another example for SIMD architectures. They consist of a multidimensional arrangement of processing units, all of which work simultaneously on multidimensionally structured data. In contrast to a multiprocessor, the processing units in an array processor all execute the same operation at a given moment. The interconnect in an array processor often has a grid structure, typically with nearest-neighbor connections from one processor to the north, east, south, and west. An example of more complex interconnection schemes is the hypercube. The requirement for high data throughput and simple functionality of the processing units makes the use of optics interesting for the implementation of array computers. For example, a concept based on symbolic substitution logic was described by Hwang and Louri [1989]. Murdocca [1990]

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described an optical implementation of the Connection Machine (Hillis [ 19851) using programmable logic arrays. 5.1. IMPLEMENTATION OF LOGIC OPERATIONS ON 2-D ARRAYS

Conventional Boolean logic allows one to transform the state of a device according to a set of rules that is described, for example, by a truth table. Free-space optics permits in addition the use of spatial degrees of freedom to represent the logic state by the spatial position. The implementation of parallel logic operations on 2-D arrays relies on the simplicity of 2-D imaging. As pointed out by McAulay [ 19911, the control for implementing “spatial logic” becomes practical if the same operation is performed on the whole array. The technique of computing by updating simultaneously all values in an array, was adapted for free-space digital optics from the theory of cellular automata. In cellular automata, the state of a pixel (cell) is updated based on the states of its nearest neighbors in the array. A simple scheme for implementing logic functions on 2-D arrays based on a shadow-casting technique was described by Ichioka and Tanida [1984]. In some aspects, their technique can be considered to be a forerunner to symbolic substitution, a technique that also relies on the use of space-invariant interconnects between subsequent data planes (Brenner, Huang and Streibl [ 19861).In contrast, the so-called PLA approach (programmable logic arrays) (Murdocca, Huang, Jahns and Streibl [1988]) is based on the use of space-variant multistage interconnections. In the following, we will describe both techniques. 5.I . I . Symbolic substitution

Symbolic substitution relies on the recognition and replacement of a certain pattern in a binary 2-D array. Depending on a suitable choice of the search and replacement patterns, one can implement logic operations with that scheme. A logic function is specified by a substitution rule which combines search and replacement patterns. The principle is shown in fig. 5.1. The upper part of fig. 5.1 shows the input pattern that one searches for and the pattern that it is replaced with. In the lower part, search and replacement are visualized for a simplified situation. From a visual inspection one can recognize the two positions in which the search pattern occurs in the input array. They are found by forming the same number of shifted copies of the input object as there are dark spots in the search pattern. The shift of the copies is implemented according to the reference position in the search pattern which is marked by

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FREE-SPACE OPTICAL DIGITAL COMPUTING AND INTERCONNECTION

[VI,

5

5

substitution rule: search pattern

replacement pattern

W-•

search

replace

input

split-and-shift

NOR

split-and-shift

NOR

Fig. 5.1, Principle of symbolic substition for implementing logic operations on a 2-D binary array. The substition rule shows the pattern to search for in the input array and the replacement pattern. Dots in those patterns mark the reference positions for the split-and-shift operations. Search and replacement operations both consist of a split-and-shift operation and a subsequent logic (in this case, a NOR) operation.

a dot. In the specific example, since we want to superimpose the dark pixel in the upper left with the dark reference pixel, the shifted copy is shifted to the lower right as indicated by the arrow. There are several grey levels in the superimposed pattern; in our case, there are three. By applying either a NOR or an AND operation, one obtains the two positions one is loolung for. At these positions, the two replacement patterns are obtained by another split-and-shift and NOWAND-operation. The optical implementation of the split-and-shift operation can be achieved either by using beamsplitter cubes (Brenner [ 1988]), diffraction gratings (Mait and Brenner [ 19881, Thalmann, Pedrini and Weible [ 1990]), holographic matched filtering (Jeon, Abushagur, Sawchuk and Jenkins [ 1990]), and shadow-casting in combination with Wollaston prisms (Louri [ 19911). The principle of symbolic substitution can be used to realize arithmetic or more complex operations by a spatial encoding of these operations. As an example, we describe the spatial representation of the rules for binary addition in fig. 5.2. The figure shows the two patterns used to represent a binary 0 and 1. Furthermore, the assignment of the spatial position for the two summands A and B in the input array is shown. They appear in the upper right of the 4 x 4 array. The sum bit S and the carry bit C appear in the lower half of the output array. The rules for all four combinations of summands are then described in a spatial representation as shown in fig. 5.2.

49 1

ARCHITECTURES AND SYSTEMS

bit representation

spatial representation input

om

output

1 1 7 A,B - summands S - sum, C - carry bit addition rules

Fig. 5.2. Rules for binary addition of two bits represented by spatial encoding.

The attractive feature of symbolic substitution is that the operations can be implemented in parallel for a 2-D array. In addition, more complex operations than the simple case of a binary addition can be implemented (Eichmann, Kostrzewski, Kim and Li [1990], Alam, Awwal and Karim [1992]). However, practical limits to the complexity of the operations are imposed by the optics and by the accuracy of the logic devices. 5.1.2. Optical programmable logic arrays

In electronics, programmable logic arrays (PLAs) are used to implement arbitrary logic functions. Two matrices are used, one consisting of AND, the other one consisting of OR gates. The logic arrays are programmed by enabling or disabling certain interconnections in these matrices. Thus, any desired logic function can be implemented. This approach is known as sums of products or disjunction of conjunctions (McAulay [ 199I]). Based on this technique, Murdocca, Huang, Jahns and Streibl [ 19881 suggested the implementation of optical digital logic by using 2-D device arrays like the SEEDS interconnected by multistage interconnection networks. The basic idea is to generate all products of two input variables, x and y , in a first step using a network consisting only of AND gates. Different optical paths in the interconnect are enabled or disabled by binary masks in the image planes to allow only certain

492 0 x

[VI, § 5

FREE-SPACE OPTICAL DIGITAL COMPUTING AND INTERCONNECTION

1 -

x

2

3 y

4

.

5

6

7

~

y

AND

0

1

2

3

4

5

6

7

**A?%

UOR

Fig. 5.3. Optical programmable logic arrays. The first array consisting of AND gates shown on the left generates the products of two logic variables, x and y. The array on the right forms the sums of the products. As an interconnect, the Banyan network is used. Solid lines represent enabled interconnections. shaded lines are disabled.

of these products in the output plane. In a second network, sums of the allowed product terms are generated (fig. 5.3). In its basic concept, the problem of the PLA approach is the large number of gates that are required for the design of a specific circuit. Using freespace optics the expense for the interconnections may be quite high, at least, if optomechanical hardware is used. In general, the gate count for performing logic operations is relatively high for the PLA technique. This has led to doubts about the suitability of free-space interconnects at the gate level. Several improvements for reducing the gate count were discussed by Smith, Murdocca and Stone [1994].

5.1.3. Computational origami The motivation behind computational origami (Huang [ 19921) is the assumption that the available optical hardware is simple; i.e., 2-D arrays with relatively few logic gates which are interconnected in a regular way, and therefore often not adequate to directly achieve complex functionality as in VLSI electronics. As a solution to this problem, a complex task is mapped into a sequence of simpler steps which are performed sequentially. The complexity of the task is traded off

493

Fig. 5.4.Principle of computational ongami. Left: conventional optical logic circuit. Right: folded origami network. (After Huang [1992].)

with the number of sequential steps. The resulting time penalty is supposed to be compensated by using very fast optical circuitry. An example is shown in fig. 5.4. On the left, the figure shows a simple conventional logic circuit consisting of eight AND gates. It is redrawn in a folded regular way on the right. An array of logic gates is used with a width of six and a depth of four. Computational origami has similarities with the concept of systolic arrays as described by Erhard and Fey [1994]. It is a way of shaping a computational task into a form such that it becomes suitable for the hardware and communications requirements of an optical processor. It also illustrates how to perform computations with delay lines rather than with random-access memory. The function of memory in a computation is implemented by delaying the data stream until it is needed. Fixed amounts of time are used to accomplish the delays. 5.2. NETWORKS

The idea of switching networks originates from telecommunications with switched telephone connections between arbitrary users. The reason for this is that the problem of designing a permutation network for interconnecting computer parts is related to the problem of designing telephone switching networks. The name “crossbar switch”, for example, was coined in the context of telephone switching. However, the concept of switching has extended to include data traffic in a telecommunicationsnetwork as well as in an individual computer. Examples of localized computer networks are multiprocessor systems which

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FREE-SPACE OPTICAL DIGITAL COMPUTING AND INTERCONNECTION

[VI, § 5

require means for communicating data between the processors and the memory. Special purpose computers with general processors and hardware coprocessors (e.g., to calculate Fast Fourier Transforms) also require computer networks. Consequently, there has been a great deal of interest in data networks for switching and computing. In the following, the term “network” will be used in a broader sense than earlier, where we used it mostly to denote an interconnection such as the Perfect Shuffle or Banyan. In a more general sense, a switching network consists of switching nodes and the interconnections. The actual routing of the data occurs in the switching nodes. The routed traffic is passed to the next stage by the interconnection. 5.2.1. Network properties

Networks are described according to certain properties; these are: blocking vs. non-blocking, rearrangable or fixed, by their latency, fault-tolerance, and complexity. Here, we will not discuss these issues in detail. A thorough discussion was given for photonic switching networks by Cloonan and Hinton [1993]. A more general overview was given by Kuck [1978]. The basic problem of a telephone network is to interconnect any of N inputs with any of N outputs. Mathematically speaking, a switching network must be able to implement N ! permutations. The task can be achieved with a crossbar switch. Since a crossbar consists of N 2 two-state nodes, the number of settings of the switch is 2NZwhich is much larger than the required number of permutations for N >> 1. Therefore a good deal of hardware in a crossbar is wasted. Suppose that one could instead design a network with N log, N two-state switches: The total number of settings of such a switch would be 2”’gN = N N , which is a good approximation to N! % (N/e)N(2~N)1’2.Thus, one can see that only O(N log, N) switching nodes are necessary to perform the N ! required permutations [O(.) denotes the order of magnitude]. This result has led to various networks which have been investigated for switching applications. In order to change from one setting of the switch to another, the network must be rearranged. Therefore, the networks are called rearrangeable networks. A rearrangeable network with O(N log, N) gates is able to perform all possible permutations of N inputs in O(log,N) gate delays. Often, N is chosen to be a power of 2. In this case, the number of switching nodes in the network is minimal. For an optical implementation, the interest in logN-type networks stems additionally from the fact that they are based on the multistage interconnections described in the previous section. As these networks exhibit a certain amount of regularity, it is possible to implement them with a large space-bandwidth

m 5 51

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495

product by using imaging systems. The reduced expense in terms of the spacebandwidth product compared with an optical implementation of the crossbar, for example, is the reason why multistage interconnection networks have been under investigation for free-space photonic switching systems. One of the most important properties is the issue of whether a network is blocking or non-blocking. A non-bloclung network is able to implement any connection (pattern) at any given time. One distinguishes between strictly non-blocking and wide-sense non-blocking networks. A strictly non-blocking network is the Crossbar network. Blocking is usually a result of lacking degrees of freedom in the network. Strictly non-blocking and wide-sense nonblocking networks will require on the order of Nlog,N to N2 switching nodes (N is the number of inputs and outputs). For economic reasons, as described above, one will not, in general, use crossbar networks since the number of required switching nodes (or optically speaking, the required spacebandwidth product) is prohibitive, and will instead use wide-sense non-blocking networks. In telecommunication systems, the Clos network is used widely (Clos [ 19531). For fast applications, advanced self-routing schemes based on multistage interconnection networks are used (Huang and Knauer [1984]).

5.2.2. Switching nodes Switching of a light signal can be achieved using either a routing switch or a logic gate. In a routing switch a beam of light is physically switched between two or more paths. A whole switching fabric consists of an array of routing switches which are set and controlled externally. The speed of the switching devices determines the time it takes to reconfigure the fabric; however, the bandwidth of the transmitted optical signals can be very large. An example of a routing switch is a directional coupler. A free-space optical implementation of a routing switch can be based on polarization optics. The state of polarization of the light beam is controlled by, e.g., a liquid crystal light modulator. A polarizing beam splitter is used to steer the light beam in either of two directions. Such routing switches have been proposed by various groups (Johnson, Surette and Shamir [ 19881). The other switching principle uses logic gates. A switching fabric consists of arrays of such logic gates or more complex logic units like compare-exchange modules. Routing of the light signals is achieved by sorting the addresses contained in the headers of the data packets. Sorting networks are described in the next section. In this approach, the bandwidth of the optical signals that can be transmitted through the switch is limited by the speed of the logic devices.

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FREE-SPACE OPTICAL DIGITAL COMPUTING AND INTERCONNECTION

input 1

+

[VI,

5

5

control

-i.-.-,

output

input 2 Fig 5 5 (2,1,1) node with two inputs, one output and a capacity of one (1 e., at a given point in tune, one signal path can be routed through the node)

x AND

input 1

-w

input 2

-+

OR

output

Fig. 5.6. Optical (2,1,1) node built of AND and OR gates. Solid lines indicate used connections, shaded lines unused connections.

There are different ways to describe the functionality of a switching node. A convenient scheme is the description by a triplet of numbers (a,b, c) where a stands for the number of inputs that the node can receive, b denotes the number of outputs that it can drive, and c is the capacity of the node. Capacity is defined as the number of unique paths which can be routed simultaneously through the node. The simplest example may be the (2, 1 , l ) node shown in fig. 5.5, which one may think of as being implemented in fiber-optical technology. An example of a (2,2,2) node would be a directional coupler. A (2, 1 , l ) node using, e.g., SEED devices and free-space optical connections can be realized using AND-OR logic and customized interconnections - similar to the PLA approach - as shown in fig. 5.6 (Cloonan and Hinton [1993]). It consists of two consecutive planes of logic gates. The first plane contains AND gates, the second OR gates. The two planes are interconnected by simple imaging optics (including a lateral shift). The network is customized by enabling and disabling certain connections. The gates may be implemented as SEED devices (Lentine [ 19941). Complex switching nodes for digital free-space optics have also been implemented by using symbolic substitution (Cloonan and Hinton [ 19931) and the PLA approach (Murdocca and Cloonan [1989]). The sorting node shown in fig. 5.7 is a powerful self-routing switching node that can be used in the STARLITE switching network (Huang and Knauer [1984]).

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ARCHITECTURES AND SYSTEMS

=OR

= NOR

+ +

X

S[)

3.0

S!

s,

x

x

>’

Y

I

uu min max

Fig. 5.7. Sorting node based on the PLA approach (after Murdocca and Cloonan [1989]).

5.2.3. Sorting networks

Sorting is an important operation for computing and switching. Sorting networks date back to the work of Batcher [1968]. The basics of these networks are described by Partridge [ 19941. The realization of optical interconnection networks based on sorting networks was investigated by, e.g., Huang and Knauer [1984] in their STARLITE switch and by Midwinter [1987]. A Batcher sorting network is made up of ( 2 , 2 , 2 )sorting nodes with two inputs and two outputs (fig. 5.8). Two values are entered on the left. The smaller value

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FREE-SPACE OPTICAL DIGITAL COMPUTING AND INTERCONNECTION

a b

1-F

rw 5 5

min(a,b) max(a,b)

Fig. 5.8. Sortmg node in a Batcher network.

is sent to the upper output, and the larger value is sent to the lower output. The sorters contain simple binary comparators and a switch that can be latched to a straight-through or crossover connection. One way of making a sorting network from these sorting elements is to interleave the outputs of the sorters such that the inputs to the next stages are always bitonic sequences. A series of numbers (a1,a2,. . . ,a ~is)bitonic if it can be decomposed into a sequence of numbers with increasing values (a1,. . . ,a k ) and a sequence of numbers with decreasing values ( a k + l , . . . , u N ) . Thus, 1,2,3,4,8,7,6,5 is a bitonic sequence, as is 8,7, 1,2,3,4,5,6. Batcher showed that if one takes a bitonic sequence (ul,. . . , a k , a k + l , .. . , U N ) and computes the sequence (min(al, a k + l ) , min(a2, uk+2),. . . ,min(ak, a2k)) and the sequence (max(a1, a k + l ) , max(a2, ak+2),. . . ,max(ak, alk)), that both of these two sequences are also bitonic and no number in the first sequence is greater than any number in the second sequence. From this observation, it is possible to build a sorter with O(N log, N ) complexity. Figure 5.9 shows a Batcher sorting network for N = 8 inputs. It consists of three (= log, N ) stages with N/2 exchangeibypass nodes. Signals are routed according to the bit sequence in the destination address. An example is shown in fig. 5.9, where data from an arbitrary input port are supposed to be routed to output port 6. The binary address (1 10) is used to set the switches starting inputs

bypass-exchange nodes

outputs

destination address: I10

4

l o g N stages

*

Fig. 5.9. Batcher sorting network for sorting numbers. Here, N = 8 inputs are shown. The address of the destination (here, with the label 6 = ‘110’) is used to route the signal to that output port.

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with the most significant bit in the first stage of the network. For switching applications, it is important to note that the Batcher sorting network can be used as a general permutation network simply by attaching a destination header to each data item and sorting the headers. The data then follow the headers to their proper destinations. In an optical sorting network all permutation stages are realized by optical elements as described in $4. The sorting nodes are implemented using arrays of switching devices. A proposal for an optical sorting network was made by Stirk and Athale [1988]. A hybrid implementation using electronic gate arrays with optical inputs and outputs was described by Zurl and Streibl [19921. Another example of a sorting network is the Omega network (Lawrie [ 19751) which uses the perfect shuffle as the interconnect. For N inputs, it consists of log, N identical shuffle and three bypass-exchange stages. The fact that the interconnection stages are all identical is of advantage, since it may simplify the optical implementation; e.g., by using a feedback setup. Each stage consists of N/2 switching elements. Just as for the Batcher network, every switching element performs a bypass or exchange operation depending on the state of the input variables. However, the state of a single switching element is determined differently. While the Batcher network is controlled internally, the Omega network is controlled externally. Optical implementations of the Omega network were discussed by Cheng and Sawchuk [1992]. 5.3. SYSTEM DEMONSTRATORS

With the availability of switching devices, systems research on digital optical computing and photonic switching became possible. During the past 10 years, several demonstration experiments were reported, some of which we will review here. 5.3.1. Hybrid processors with electronic control Early demonstrations of digital optical processors shared the feature that they were operated under electronic control. We therefore refer to them as hybrid processors. In a hybrid system, typically, the optical output of a particular processing stage was detected electronically and fed back to the optical system through an electronic computer to an optical output stage. Here, we review some examples. Crossbar-type interconnects: The basic idea to realize an optical crossbar for 1-D signals goes back to Goodman, Dias and Woody [1978] who demonstrated

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FREE-SPACE OPTICAL DIGITAL COMPUTMG AND MTERCONNECTION

input

switching node

[VI, § 5

output

Fig. 5.10. Schematic of the optical implementation of a crossbar switch.

a matrix-vector-multiplier.The principle is shown in fig. 5.10. An anamorphic imaging system is used to broadcast light from each pixel, urn,of the input array vertically to a column of modulators or switching nodes in a 2-D matrix, &,,. All light paths from one row are focused onto a single output position, 0,. By controlling the state of the modulators in the array, one can implement arbitrary interconnections. Suitable components to implement the switching matrix would be, e.g., GaAs switching device arrays, liquid crystal, or deformable mirror devices. Further implementations of crossbars were reported, e.g., by Sawchuk, Jenkins, Raghavendra and Varma [1987], Cloonan [1990], and by Fukui and Kitayama [1992]. Fukui and Kitayama discuss the implementation of a crossbar for 2-D objects. In order to achieve a large SBP, they suggested partitioning the optical system by using arrays of laser sources and microlens imaging arrays. A ferroelectric liquid-crystal spatial light modulator is used as a shutter to allow dynamic reconfiguration of the transmission matrix. As a goal they consider the construction of a switch for 20x20 channels, each with 103xlo3 pixels, corresponding to an overall SBP of 4x108. It remains to be seen if such an ambitious system can be realized in practice. In an experiment, a 2 x 2 switch was demonstrated. Guilfoyle [1988] suggested the use of a matrix-vector multiplier based on a crossbar architecture with feedback to build a general purpose optical computer. The basic idea of this proposal is that the control matrix operating on the input data vector produces a complete set of combinatorial functions. A demonstration of an arithmetic logic unit based on an optical crossbar using LEDs and detectors was given by Falk, Capps and Houk [1988]. Fifty million logic operations

VI,

9 51

ARCHITECTURES AND SYSTEMS

50 1

per second were demonstrated, an impressive accomplishment considering the simplicity of the hardware used. Space-variant interconnections were used in a demonstration system by Huang, Sawchuk, Jenkins, Chavel, Wang, Weber, Wang and Glaser [ 19881. They used the concept of a cellular processor realized by an optical gate array. The gate array was emulated by an LED array used as optical input and a video camera for recording the output of the optical part of the system. It consisted of 54 gates. Holographic multifacet elements were used to provide the interconnections. A host computer was used to provide control for the system. 5.3.2. Digital optical processors using PLA logic and split-und-shft interconnect The first digital optical processor to be built was demonstrated by Prise, Craft, Downs, LaMarche, D’Asaro, Chirovsky and Murdocca [1991] at AT&T Bell Laboratories. They used logic arrays with NOR gates implemented by symmetric SEED devices, each 10x20 pm2 in size. Each array consisted of 4 x 8 gates. Four such arrays were operated in a feedback loop as shown schematically in fig. 5.1 1. The devices were illuminated by optical power supply beams generated by a binary Dammann grating. The interconnect consisted of simple split-and-shift operations implemented by imaging and a polarizing beamsplitter. A PLA design as discussed earlier was used to implement various logic operations like a shift device array

mask

interconnect

I1 Fig. 5.1 1. Schematic of the AT&T digital optical processor. Four logic gate arrays are operated in feedback loop.

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FREE-SPACE OPTICAL DIGITAL COMPUTING A N D INTERCONNECTION

IVI, § 5

Fig. 5.12. Photograph of the digital optical processor (courtesy M.E. Prise).

register and a decoder. Binary masks were used to enable or disable the individual connections. The clock cycle time was relatively slow, up to 1.1 MHz. A photograph of the actual system is shown in fig. 5.12. The implementation of the various optical operations requires a significant amount of hardware, wliiiii is here implemented by off-the-shelf opto-mechanic components. Another system based on the use of optical PLAs and split-and-shift jni(:iconnections was presented by Thienpont, Goulet, Baukens, Koczyk, Buczynshi, Nieuborg, Kirk, Heremans, Borghs, Kuijk, Vounckx and Veretennicoff [ 19971. To implement the optical logic gates they used thyristor-type switching devices operated in the dual-rail mode. Since these devices are light-emitting, a much simpler optical setup could be used than for the system described above. A threestage cascaded system was built, and demonstrated various logic operations at a clock frequency of 11 MHz. Higher operation speed of up to 50Mbis was demonstrated for devices on single wafers (Knupfer, Kuijk, Heremans, Vounckx and Borghs [ 19951).

5.3.3. Photonic switching systems using multistage interconnection network System demonstrators for photonic switching were also demonstrated at AT&T

VI, § 61

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Bell Laboratories. The hardware used was similar to that in the digital processor described above. The logic devices were implemented as S-SEEDS. However, a larger number of devices (up to 32x64) was used in the gate arrays to demonstrate a large throughput through the switch (McCormick, Tooley, Sasian, Brubaker, Lentine, Cloonan, Morrison, Walker and Crisci [ 19911). Six stages were demonstrated with fiber optic input and output. As an interconnect, the Crossover was implemented. In later versions, a split-and-shift was also used. Experiments showed that the system was fully functional; however, the system speed was limited to about 50kb/s. Limitations arose, e.g., from the 80mW laser diodes used to supply all of the 2048 devices in one array with optical power, from light losses in the system, and from limited time-constants of the devices. Faster systems (using a smaller number of devices) were demonstrated later, using F-SEEDS operated at 250 Mb/s (Goossen, Lentine, Walker, D’Asaro, Hui, Tseng, Leibenguth, Kossives, Dahringer, Chirovsky and Miller [ 19941). A philosophy similar to that of the AT&T experiments was used in demonstrators described by NTT (Hirabayashi, Yamamoto, Hino, Kohama and Tateno [ 19971). The NTT system utilizes VCSEL devices and microchannel imaging. The specific system was used to demonstrate board-to-board interconnection for electronic switching systems.

0 6.

Conclusion and Outlook

Significant progress has been made in the field of digital free-space optics over the past 10 years. To a large extent this has been made possible by advances in technology. Until the mid- 1980s, only rudimentary hardware existed for experimental work. Now, rather sophisticated smart pixel devices and micro-optic components are available, e.g., through initiatives like the optical computing CO-OP (Athale and Raj [1996]). This will allow hrther systems-related research in the years to come. However, and perhaps even more importantly, there have also been new insights about what the role of optics might be in digital computing. Optics is currently being viewed as an interconnection technology. The potential of free-space optics for building 2-D buses with massive interconnectivity (i.e., 1000 or more parallel channels) at high speed will become more and more realistic with further improvements of the components. It has been argued that the need to convert the informationcarrying signal from electrical to optical and back would be a disadvantage of optical interconnections. However, there is nothing intrinsically inefficient about the signal conversion. It has been pointed out that the integration of the devices

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PI

is the key to energy-efficient interconnections (Miller [ 19891). The smart pixel technology is moving in that direction, and a variety of different components has been demonstrated (see, e.g., Brown, Gardner and Forrest [ 19911, D’Asaro, Chirovsky, Laskowski, Pei, Woodward, Lentine, Leibenguth, Focht, Freund, Guth and Smith [ 19931, Kasahara [1993]). In addition, new packaging schemes for the optical interconnections have been suggested and will further make optics more acceptable to the electronics industry (Drabik [ 19941, Jahns [ 1994b1). The notion of optics as an interconnection technology for VLSI computers relies more on the large temporal bandwidth and to a lesser degree on the spatial bandwidth achievable with free-space. In principle, optical systems can provide much more than a few hundred or thousand interconnections. Systems with a very large number of channels (1 0 000 or more) may be of interest for neural computers (Psaltis and Farhat [ 19851) or special purpose computing systems; e.g., systems designed to do image processing (Craig, Wherrett, Walker, Tooley and Smith [1991], Huang, Kuznia, Jenkins and Sawchuk [ 19941, and Liu, Zhang and Zhang [ 19941). Will optics play a role beyond interconnections? For the time being, the concept of an “all-optical” computer including “optical logic” has moved to the background. An electronic transistor is much smaller than a single optical logic device. Linear dimensions differ by one to two orders of magnitude. Hence, the functionality of optical chips is significantly less than that of today’s electronic processors chips. The rational answer to the above question therefore seems to be “no” at this point. However, understanding of “optical computing” has changed several times and new inventions and ideas might change it again.

Acknowledgement The author gratefully acknowledges fruitful discussions with Stefan Sinzinger, and the diligence and support of Susanne Kinne during the preparation of this manuscript.

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AUTHOR INDEX FOR VOLUME XXXVIII

A Abe, S. 290, 326, 332 Abeles, B. 67 Abraham, E. 14 Abramovici, A. 88 Abu-Mustafa, Y.S. 488 Abushagur, M.A.G. 473,490 Aceves, A.B. 66 Acklin, B. 33,46,481 Adachi, S. 183, 190, 207 Adams, M.J. 175, 183 Afanasev, A.A. 66 Agarwal, G.S. 11, 13-15, 19, 24, 28, 29, 35, 51, 68 Agrawal, G.P. 3, 6, 14, 66, 199, 202 Aguirregabiria, J.M. 88 Aiki, K. 196 Aitchison, J.S. 48-50, 66, 450 Akhmanov, M. 336 Akhouayri, H. 61 Alunwande, T. 175 Akulova, Y.A. 188 Akulova, Y.A.A. 167 Al-hemyari, K. 48, 49 Alam, M.S. 491 Alieva, T. 326 Almeida, L.B. 333 Alonso, M.A. 299 Althouse, WE. 88 Amano, C. 178, 452, 453, 455 Amith, A. 216 Anderson, D.Z. 127, 133 And& J. 175 Arakelian, S.M. 41 Arbore, M.A. 62 Argoul, F. 317 Anyasu, J. 31 Arlot, I? 17, 23

h o u r , E. 174, 225, 227, 229,249 Armstrong, J.A. 56, 422 Armstrong, N.J. 60 Ameodo, A. 3 17 A r r i z h , V. 294,469,413 Arsenault, H.H. 359 Asai, S. 426 Ascheri, €! 62 Asom, M.T. 186, 192, 198, 199, 201, 465 Assanto, G. 4, 48, 66 Atakishiyev, N.M. 325 Athale, R.A. 432, 484, 499, 503 Aufmuth, P. 88 Awwal, A.A.S. 491 Azoulay, R. 45 Azumai, Y. 62 B Baba, T. 221, 238, 239, 241, 245 Babic, D.I. 175, 176, 178, 185, 187, 192, 194, 197, 200, 201, 221, 238, 243, 244 Bachor, H.-A. 93 Bacon, D.D. 465 Baek, Y. 63, 64 Baez, EM. 192, I94 Bailes, M. 115 Bajema, K. 39 Baker, G.L. 450 Bakoglu, H.B. 426 Baldi, P. 62, 63 Banavar, J.R. 39 Banti, X. 62 Banwell, T. 193 Bargmann,V: 333 Barone, F. 88, 133 Barrett, H.H. 304 Barry, D.T. 304, 334 Barshan, B. 272, 276, 292 515

516

AUTHOR tNDEX FOR VOLUME XXXVlIl

Bartelt, H.O. 287, 334 Basov, N.G. 422, 447 Bastiaans, M.J. 334 Batcher, K.E. 497 Baues, P. 290 Baukens, V. 502 BaurBs, P.-Y. 142 Beaudet, PR. 422 Bel, L. 88 Belashenkov, N.R. 63 Bell, A.G. 421 Bell, J.F. 115 Belvaux, Y. 276 Bennett, J.R.J. 88 Beranek, M.W. 42 Bergman, D.J. 69, 71 Bernardo, L.M. 325, 333 Berra, P.B. 432 Berry, M.V. 326 Bertilsson, K. 178 Bertolotti, M. 6, 22, 32, 40 Bethune, D.S. 5, 50 Bhat, R. 175 Bhattacharya, P.K. 39 Bieber, A.E. 33, 4 6 4 9 Bierlein, J.D. 5, 57, 59, 63 Billing, H. 132, 139 Biran, B. 16 Birnboim, M.H. 69 Bissessur, H. 182 Bitran, Y. 273, 274, 276, 295, 299, 314, 334 Blair, D.G. 87, 88, 142, 155, 160 Blau, G. 61 Bleha, W.P. 447 Bliss, E.S. 43 Bloembergen, N. 3, 50, 51, 56 Bloemer, M.J. 17, 66, 69 Bloisi, F. 150 Boardman, A.D. 6, 3&32 Boccara, C. 149 Boehm, G. 175 BOhm, G. 452, 453 Bolger, B. 61 Bonazzola, S. 87, 88 Bond, A.E. 180 Bondu, F. 94, 96, 132, 160 Borghs, G. 452, 456, 502 Born, M. 11,29, 457 Bortz, M.L. 62 Bosshard, C. 60, 63

Botez, D. 38 Boulharts, A. 127 Bounnak, S. 187, 193 Bowden, C.M. 6, 17, 66, 69 Bowers, J.E. 175, 176, 178, 184, 186, 187, 192, 194, 197, 200, 201, 221, 238, 243, 244 Boxer, A. 426 Boyd, G.D. 461 Boyd, R.W. 5, 66, 69-75 Bozhevolnyi, S.I. 24 Bracewell, R.N. 267, 287, 289 Bradaschia, C. 88 Bradley, J.C. 422 Braginsky, VB. 160 Braun, D. 221, 244 Brenner, K.H. 334 Brenner, K.-H. 432, 433, 484, 489, 490 Brillet, A. 88, 94, 108, 115, 116, 123, 124, 127, 132, 152 Bristow, J.P. 167 Broderick, N.G.R. 66 Brown, B.R. 277 Brown, H. 447 Brown, J. 60 Brown, J.B. 59 Brown, J.J. 465, 504 Brown, T.G. 33, 4-9 Brown-Goebeler, K.F. 186, 192, 194, 452, 455, 464, 480 Brubaker, J.L. 463, 503 Bruggeman, D.A.G. 67 Bryngdahl, 0. 298, 326 Buccafusca, 0. 198 Buchanan, D.A. 426, 427 Buczynski, R. 502 Bugajski, M. 182, 186,230 Burrus, C.A. 460 Butler, J.K. 204 Buus, J. 183 Byer, R.L. 50, 56, 58, 59 C Cada, M. 17, 18, 33,46 Cai, Z. 4 2 4 4 Callen, H.B. 96 Callis, S.E. 201 Calloni, E. 133 Campbell, A.M. 88 Campbell, M. 243 Campbell, R.J. 469

AUTHOR INDEX FOR VOLUME XXXVIII

Caneau, C. 201 Capinski, W.S. 33, 48 Capobianco, A.D. 66 Capozzi, M. 88 Capps, C.D. 422, 500 Cardona, M. 190 Carey, K. 238 Carmichael, H.J. 14 Carrion, L. 193 Carslaw, H.S. 207, 209, 215 Carter, G.M. 42 Casasent, D. 359, 447 Case, S.K. 287 Casey Jr, H.C. 183 Catchmark, J.M. 186, 192 Caulfield, H.J. 488 Caulfield, J. 317 Caves, C.M. 98 Caviris, N.P. 317, 318, 323 Chand, N. 186 Chandler, P.J. 65 Chang-Hasnain, C.J. 167, 177, 197, 198, 201 Chatenoud, E 50, 59, 60 Chavel, P. 433, 488, 501 Chavez-Pirson, A. 45 Chemla, D.S. 43, 69, 460 Chen, G. 186, 187, 204, 221, 237, 245, 246 Chen, J. 317, 318, 323 Chen, J.M. 118 Chen, Q. 32 Chen, W. 4, 7, 8, 17, 19, 21, 36, 426, 427 Chen, Y.J. 42 Cheng, J. 174, 187, 188, 192, 193, 195, 200, 203, 225, 227, 229, 248, 249 Cheng, L. 499 Cherng, C.-P. 197 Cheung, M.M. 41 Chevrette, P. 304, 309 Chilla, J.L.A. 198 Chirkin, A S . 336 Chirovsky, L.M.F. 452, 461, 463, 465, 501, 503, 504 Chiu, T.H. 335 Cho, A.Y. 50, 59, 167, 169, 174, 177, 186, 187, 192, 193, 197, 198, 202, 228, 246, 452, 456 Chong, C.H. 200 Choquette, K.D. 174, 179, 180, 187, 193, 196, 201, 221, 252 Choquette, R.P. 192

517

Chow, W.W. 180, 191 Christodoulides, D.N. 33, 34, 36, 66, 186, 192 Chu, G.N.S. 50, 59 Chu, H.Y. 201 Chu, S.N.G. 186 Chua, C.L. 175, 255 Chung, Y. 197 Clack, R. 304 Clarke, R. 39 Clays, K. 60 Cleva, F. 142, 155 Cloonan, T.J. 424, 432, 442, 463, 486, 494, 496, 497, 500, 503 Clos, c. 495 Cody, G.D. 67 Cohen, R.W. 67 Colak, S. 59 Coldren, L.A. 167, 169, 176, 178, 186-194, 196, 199-201, 204, 205, 212, 235, 246 Coldren, L.A.C. 167 Collins, S.A. 290 Condon, E.U. 333 Constantini, B. 66 Cook, D.D. 197 Cooperman, G. 14 Corzine, S.W. 167, 169, 176, 185-187, 189194, 199-201, 205, 212, 221, 235, 246, 252 Coste, J. 66 Coutaz, J.L. 61 Coutts, D. 67 Craft, N.C. 501 Craig, A.E. 28, 64 Craig, R.G.A. 504 Crawford, D.L. 178, 184, 186, 192, 238 Crawford, M.H. 179, 180 Cregut, 0. 88, 94 Crenshaw, M.E. 17 Crisci, R.J. 503 Crust, D.W. 43 Csanak, G. 243 Culver, W.H. 422, 447 Cunningham, J.E. 452, 453, 461 Cutolo. A. 146

D Dagenais, M. 14 Dagli, N. 197 Dahringer, D. 465, 503

518

AUTHOR INDEX FOR VOLUME XXXVIII

Dai, H. 59, 60 Damen, T.C. 460 D’Amico, N. 115 Dammann, H. 471 Damour, T. 88 Danckaert, J. 14-17 Daneu, V 6, 451 Dann, A.J. 175 Dannberg, P. 30, 41 Danner, A. 452, 453 Danzmann, K. 88, 118, 125, 141, 149 Dapkus, P.D. 179, 180, 452, 453, 465 D’Asaro, L.A. 461, 463, 465, 501, 503, 504 Daschner, W. 443 Daubechies, I. 317 Davenport, W.B. 348 Davies, D.K. 422 Davies, J.B. 182 De Angelis, C. 66 de Brouguenet de la Tocnaye, J.L. 326 De Micheli, M.P. 56, 62, 63 de Sterke, C.M. 33, 34, 36, 38, 47, 65, 66 Decher, G. 60 Deck, R.T. 14, 28, 29 Defnse, M. 304 Del Fabbro, R. 88, 107 Delacourt, D. 62 Delage, A. 60 Dempsey, J. 469 Deng, H. 180 Deppe, D.G. 167, 180, 186, 187 DeSalvo, R. 63 Devos, F, 488 Dewey, D. 93 Dhurandhar, S.V. 159 Di Fiore, L. 88 Di Virgilio, A. 88 Dias, A.R. 499 Diaz-Santana, L. 282 Dickinson, A. 429 Dickinson, B.W. 325 DiFiore, L. 133 Ding, Y.J. 61, 64, 65 Dion, M. 59 Dixon, R.W. 189, 218, 219, 229 Dohler, G.H. 452, 453 Dorsch, R.G. 273, 274, 276, 286, 289, 295, 297, 299, 300, 303, 309, 314, 334 Dowling, J.P. 17, 66 Downs, M.M. 433435, 469, 472, 501

Drabik, J.T. 432, 504 Drabovich, K.N. 336 Drever, R.W.P. 88, 106, 115 Dnunmond, P.D. 24 Ducuing, J. 56 Dudley, J.J. 175, 176, 178, 184, 186, 187, 192, 201, 221, 238, 243 Dunnrowicz, C.C. 255 Dupertuis, M.A. 33, 46 Durbin, S.D. 41 Duruisseau, J.P. 88 Dutta, N.K. 167, 169, 174, 186, 187, 190, 192, 193, 197, 198, 202,228, 246 Dutta Gupta, S. 11, 13-15, 18, 19, 24-26, 28-30, 35, 38, 40, 51, 68, 71

E

Eason, R.W. 65 Easton, R.L. 304 Ebeling, K.J. 174, 176, 179, 186, 196-198, 201, 221, 244, 245 Egan,P. 6 Ehbets, P. 469 Eichmann, G. 484, 491 Eilenberger, D.J. 43 Elton, D.J. 175 Ennes, M. 452, 453 Erhard, W. 433, 487, 488,493 Ersen, A. 465 Esener, S.C. 431, 465, 488 Etemad, S. 450 Ettinger, R.D. 182 Eu, J.K.T. 276 Evaldson, P.A. 169 F Fagerholm, J. 469 Fainman, Y. 304 Falk, R.A. 422, 500 Fan, C. 465 Farhat, N. 504 Fasano, J.J. 28 Fattacioli, D. 127 Fazlollahl, A.H. 347, 370, 371, 413, 415 Fejer, M.M. 5, 50, 5G59, 62 Felber, F.S. 7, 12, 14 Feld, S. 198 Feldblum, A.Y. 443, 480 Feldman, M.R. 431, 469, 472

AUTHOR INDEX FOR VOLUME XXXVlll

Felsen, L.B. 298 Feng, D. 59 Feng, J. 38 Feng, L. 304 Ferguson, W. 204 Femandez, F.A. 182 Ferreira, C. 273, 274, 295, 297, 300, 303, 309, 319, 334 Fey, D. 433, 487, 488, 493 Fiedler, U. 179 Fielding, K.H. 373 Figiel, J.J. 174, 179, 193, 196 Fischer, G.L. 5, 73-75 Fischer, R.J. 50, 59, 169, 174, 177, 186, 187, 192, 193, 197, 198, 202, 246 Fisher, M.A. 175 Fisher, R.A. 64 Flaminio, R. 93, 94 Flannery, D.L. 345, 370 Flavin, M. 394 Florez, L. 193 Florez, L.T. 178, 197, 198, 201, 447, 452, 455, 464 Florsheimer, F. 60 Florsheimer, M. 60 Floyd, ED. 178 Floyd, P.D. 178 Flynn, M.J. 486 Flytzanis, C. 5, 6, 68, 69 Fobeles, K. 15-17 Focht, M.W. 186, 192, 201, 463, 465, 504 Fontana, A. 266 Fontenberry, R. 64 Forbes, G.W. 299 Fork, R.L. 447 Forrest, S.R. 465, 504 Forward, R.L. 87 Foster, M.J. 488 Fox, A.M. 461 Frank, D.J. 426, 427 Freilikher, VD. 39 Freund, J.M. 463, 504 Freysz, E. 317 Friberg, A.T. 24 Fntschel, P. 94, 132 Fu, X. 42 Fujii, K. 42 Fukui, M. 500 Fuligni, F. 88 Furukawa, Y. 2 12

519

G Gabler, T. 452, 453 Gabor, D. 319 Gale, M.T. 469 Gallagher, N.C. 473 Ganiere, ID. 174, 187 Ganzhorn, K. 422 Garanski, S.V 63 Garcia, J. 273, 274, 303, 309, 319, 323 Garda, P. 488 Gardner, J.T. 465, 504 Ganto, A.F. 42 Garmire, E. 33, 452, 453 Ganiga, M. 190 Gay, P. 146 Ge, C. 55 Geels, R.S. 167, 169, 176, 186, 187, 191, 193, 194, 199-201, 205, 212, 235 Gehr, R.J. 5, 66, 69, 73-75 Geib, K.M. 179, 180, 187 Gellermann, W 40 Gertsenshtein, M.E. 87 Ghafoor, A. 432 Gianino, P.D. 346 Giazotto, A. 88, 92, 95 Gibbs, H.M. 6, 7,41, 45, 422, 451, 452,459, 460 Gilbert, D.B. 189 Gillespie, A. 96 Glaser, 1. 484, 501 Glinski, J.M. 33, 46 Goke, L. 484 Goldhar, J. 6, 451 Golz, M. 23 Goncher, G. 192 Goodhue, W.D. 65 Goodman, J.W. 345, 347, 348, 399,422,431, 432, 475, 499 Goossen, K.W. 465, 503 Gopinath, A. 197 Gorbounova, 0. 64 Gortler, K. 471 Gossard, A.C. 43, 45, 422, 451, 452, 459, 460 Goulet, A. 502 Gourgoulhon, E. 88 Goutzoulis, A.P. 422 Grabherr, M. 179 Gradeskul, S.A. 39 Grado, A. 133

520

AUTHOR INDEX FOR VOLUME XXXVIll

Graham, L.A. 180 Grattan, K.T.V. 221, 246 Gray, M.B. 93 Gregory, D. 347 Grimmeiss, H.G. 182 Grinberg, J. 447 Grossmann, A. 3 17 Grothe, H. 201 Gubematis, E. 68 Guest, C.C. 431, 469, 472 Guilfoyle, P.S. 488, 500 Guizani, M. 432 Gulden, K.H. 452, 453 Giinter, P. 60 Gursel, Y. 88 Guth, G.D. 186, 192, 201, 463, 465, 504

H Hache, F. 5, 69 Hackbarth, T. 197 Hadley, G.R. 197, 221, 252 Hadley, M.A. 176, 187, 197, 204, 221, 237, 245, 246 Haelterman, M. 14, 15, 23 Hagan, D.J. 63, 450 Hahn, W.B. 370 Hains, C. 174, 192, 225, 227, 229, 249 Hains, C.P. 192 Hamam, H. 326 Hamao, N. 452, 456 Hampel, H. 62 Haney, M.W. 484 Harb, C.C. 93 Harhison, J. 193 Harbison, J.P. 178, 197, 198, 201, 447, 452, 455,464 Harlow, M.J. 175 Harris Jr, J.S. 177 Harrison, P.A. 115 Harth, W. 201 Hartley, R.V.L. 287 Hashizume, N. 5, 5 1, 54 Hasnain, G. 167, 169, 174, 186, 187, 192, 193, 197, 198, 201, 202, 228, 246 Hatori, N. 176, 178, 187, 465 Hauck, R. 287 Hauk, D. 276 Haus, J.W. 69 Hauser, M. 175

Hawrylak, P. 39 Hawrylo, F.Z. 189 Hayashi, Y. 176, 178, 187, 465 Hayata, K. 61 Hayes, R.E. 452, 456 He, J. 17, 18, 33, 46 Heflin, J.R. 42 Hegblom, E.R. 178 Hegblom, E.R.H. 167 Heitmann, H. 93, 94 Hello, P. 88, 123-126, 128, 130, 132, 133, 141, 143, 146, 150, 152, 159 Henneberger, F. 22 1, 244 Henry, J.E. 452, 461 Herbert, C.J. 33, 48 Heredia, S. 293 Heremans, P. 452, 456, 502 Heritage, J.P. 335 Herron, M.J. 463, 486 Hersee, S. 192 Herzig, H.P. 467, 469, 470 Hester, C.F. 359 Hetherington, W.M. 6 Hibbs-Brenner, M.K. 167, 175, 187, 193 Hickernell, R.K. 28 Hida, T. 333 Hillis, W.D. 489 Hino, S. 503 Hinterlong, S.J. 463 Hinton, H.S. 431433, 436, 442, 443, 447449, 451, 452, 461, 466, 479, 494, 496 Hirabayashi, K. 503 Ho, S.T. 38 Hoekstra, J.W. 64 Hoestra, H. 40 Hohimer, J.P. 197 Holloway, L.E. 88 Hong, C.S. 198 Hong, J. 59 Hong, M. 50, 59, 175 Homer, J.L. 345-347, 351, 353, 370, 371, 373, 390, 394, 399, 409, 411, 415 Horsthuis, W.H.G. 60 Horvitz, S. 488 Hotchkiss, R. 243 Hou, J.Y. 41 Hough, J. 88, 96 Houk, T.L. 422, 500 Hove, J.V 452, 455, 464 Hryniewicz, J.V. 42

AUTHOR INDEX FOR VOLUME XXXVlIl

Hsin, W. 174, 225, 227, 229, 249 Hsu, Y.N. 359 Hu, E.L. 175, 176, 178, 187, 192, 201, 238 Huang, A. 422,429,431433,467,484,488; 489,491493,495497 Huang, K.F. 187 Huang, K.S. 501, 504 Huang, Y.T. 461 Huang, Y.-Z. 175 Huffaker, D.L. 167, 180, 187 Hughes, J.J. 189 Hui, P.M. 69, 71 Hui, S.P. 465, 503 Hulin, D. 66 Hunsperger, R.G. 3 HUO,T.-C.D. 177, 193 Huson, T.D. 347 Hutchings, D.C. 43, 450 Hwang, K. 488 I Iafolla, V. 88 barra, J.G. 294 Ichioka, Y. 489 Iga, K. 167, 169, 176-178, 187, 191-193, 219-222, 230,232,238,239,241,245,452, 455, 465 Iguchi, K. 4, 39 Inaba, Y. 238 Indik, R.A. 254 Inguva, R. 69 Inochkin, M.V. 63 Ironside, C.N. 48, 49, 449, 450 Ishigame, Y. 59 Ishihara, S. 422 Ishikawa, H. 193 Ishikawa, M. 192, 194, 200, 238 Ismail, K.E. 426, 427 Ito, R. 5, 50, 51, 54 Itoh, H. 266 Itoh, Y. 178

J Jackel, J.L. 66 Jackson, D.G.A. 66 Jaeger, J.C. 207, 209, 215 Jaekel, M.T. 99 Jager, R. 179 Jahns, J. 431433, 443, 467470, 472, 480, 481, 483, 484, 486, 488, 489, 491, 504

52 1

Jain, R.K. 68 James, J.H. 174, 187 James, S.W. 65 Jameson, R.S. 28 Jansen van Doom, A.K. 197 Jam, S. 59, 60 Jasinsla, J. 32 Javidi, B. 345-347, 350, 351, 353, 354, 356, 359,360, 370,371, 373,375, 381,383,385, 389, 390, 399, 409, 411413, 415 Jeffrey, P.M. 65 Jenekhe, S.A. 5 , 73, 74 Jenkins, B.K. 490, 500, 501, 504 Jenkins, F.A. 130 Jeon, H.I. 490 Jewell, J.L. 178, 201, 432, 447, 451, 452, 455, 460, 464, 465 Jiang, W.B. 192, 201, 221, 238, 243 Johnson, E.G. 473 Johnson, K.M. 495 Johnston, A.M. 422, 451, 452, 459 Johnston, S. 115 Jose, J. 24-26, 51 Joseph, R.I. 33, 34, 36, 66 Joyce, W.B. 189, 218, 219, 229 Ju, L. 160 Ju, Y.-G. 179 Juang, F.T. 39 Jundt, D.H. 50, 56 Jung, C. 179

K Kabrisky, M. 353 Kadanoff, L.P. 39 Kaiser, G. 3 17 Kajita, M. 192 Kajzar, F. 42, 61, 450 Kalweit, E.L. 175, 187, 193 Kalyanaraman, P. 42 Kalyaniwalla, N. 69 Kang, J.U. 48-50, 66 Kanterakis, E.G. 317, 318, 323 Kapila, A. 201 Kaplan, A.E. 13, 19, 28, 63 Kara, K. 452, 456 Karagulef, C. 64 Karim, M.A. 491 Karouta, F. 201 Karpushko, F X 451, 452, 459

522

AUTHOR INDEX FOR VOLUME XXXVIII

Kasahara, K. 192,452, 456, 504 Kasemset, D. 198 Kashiwa, S. 177 Kaspi, VM. 115 Kastler, A. 276 Kataoka, I. 142 Kathman, A. 473 Kato, M. 59, 467 Katz, A. 317, 318, 323 Kautzky, H. 88 Kawai, S. 484 Kawakami, T. 192, 238 Kawamura, S. 88, 124 Kean, A.H. 49 Keller, 0. 24 Keller, U. 461 Ken, F.H. 333 Ken; G. 94 Keyes, R.W. 422, 447 Khokhlov, R.V. 336 Khoo, I.C. 41 Khurgin, J.B. 61, 64, 65 Kiesel, P. 452, 453 Kilcoyne, S.P. 179, 180, 192, 196 Kim, D.H. 491 Kim, D.Y. 63 Kim, I. 179 Kim, T. 179 Kimble, H.J. 142 Kimura, A. 192 Kinoshita, S. 167, 169, 177, 191, 193, 219, 220, 222, 230, 232, 452, 455 Kinoshita, T. 42 Kirk, A. 502 Kirschner, E.M. 335 Kiryuschev, I. 319, 323 Kitajima, N. 142 Kitaoka, Y. 59 Kitayama, K. 500 Klein, M.C. 5 , 69 Klein, S. 326 Klein, W. 175 Kline, J.D. 353 Knauer, S. 495497 Kneissl, M. 452, 453 Knupfer, B. 502 KO, J. 188 KO, J.K. 167 Kobayashi, T. 212 Koch, B.J. 197

Koch, S.W. 254 Koczyk, I? 502 Koga, R. 61 Kogelnik, H. 33, 110, 129, 130, 268 Kohama, Y. 176, 187, 193, 238, 503 Kohmoto, M. 4, 39, 40 Kojima, K. 186, 192, 198, 199, 201, 452, 456 Kolner, B.H. 336 Kolzow, D. 333 Kondo, T. 5 , 50, 5 1 , 54 Kondoh, T. 221, 245 Konforti, N. 289, 318 Kontkmvicz, A.M. 208,228, 230 Kopf, R.F. 461 Koren, U. 178, 192, 238 Kose, V 88 Koshita, M. 61 Kossives, D. 465, 503 Kost, A. 452, 453 Koster, A. 14, 28 Kostrzewski, A. 491 Kostuk, R.K. 467 Kovacs, G . 27 Kovrigin, A.I. 336 Koyama, F. 167, 169, 176178, 187, 191, 193,219-222,230,232,238, 239,241,245, 452, 455, 465 Kozlovsky, W.J. 59 Kozma, A. 345 Krackhardt, U. 469 Kratzer, H. 175 Kravchuk, M. 325 Kretschmann, E. 27 Krijnen, J.M. 64 Krishnakumar, S. 465 Knshnamoorthy, A.V. 488 Krug, W. 42 Kubota, T. 468 Kuchta, D.M. 199 Kuck, D.J. 494 Kudman, I. 207, 216 Kiihne, M. 88 Kuijk, M. 452, 456, 502 Kulcke, W. 422 Kumar, A. 64 Kumar, B.V.K.V. 359 Kumar, K. 167 KUg, S.-Y. 432 Kuo, C. 351, 353, 370 Kuo, J.M. 461

AUTHOR lNDEX FOR VOLUME XXXVIII

Kiipfer, M. 60 Kurihara, K. 192 Kurita, S. 14 Kumit, N.A. 6, 451 Kurokawa, T. 176, 178, 187, 193, 238, 452, 453,455 Kuszelewicz, R. 45 Kutay, M.A. 292 Kumia, C.B. 504 Kyuma, K. 452, 456

L Lalanne, P. 433 Lalezari, R. 142 LaMarche, R.E. 501 Lambeck, P V 64 Landauer, R. 39, 447 Lang, R.J. 14 Langbein, U. 6, 21, 30, 31, 41 Larson, M.C. 177 Laskowski, E.J. 463, 504 Lau, K.Y. 176, 197, 246 Laude, V 346 Lautenschlager, P. 190 Laval, S. 14, 28 Law, C.T. 13 Law, J.Y. 199 Lawrie, D. 499 Le, H.Q. 65 Le Denmat, G. 88 Leaird, D.E. 66 Leal, A. 206 Lear, K.L. 174, 179, 180, 187, 192, 193, 196, 221, 252 Lebreton, G . 469 Lederer, E 6, 21-23, 30, 31, 41 Lee, E.-H. 201 Lee, E.-K. 179 Lee, S. 38 Lee, S.H. 431, 433, 465, 470 Lee, S.J. 64, 65 Lee, S.-L. 192 Lee, Y.H. 45, 178, 192, 194, 201, 447, 452, 455, 463465 Lee, Y.-H. 179 Leger, J.R. 197, 469, 476 Lehman, J.A. 167, 175, 187, 193 Leibenguth, R. 198, 465, 503 Leibenguth, R.E. 186, 192, 194, 198, 199, 463, 465, 504

523

Leibenguth, R.E.L. 465 Lengyel, G. 249 Lentine, A.L. 432, 447449, 451, 452, 461463, 465, 496, 503, 504 Lenz, R. 326 Leonberger, E J . 432 Lepkowski, S.P. 221, 246 Letourneau, S. 50, 59, 60 Leung, K.M. 4, 7, 18, 19, 21, 24, 27, 29, 69 Leys, M.R. 201 Li, G.S. 177, 201 Li, J. 347, 371, 415 Li, M. 335 Li, T. 110, 129, 130 Li, Y. 318, 323, 484, 491 Lim, E.J. 56, 58, 59 Lim, J.S. 326, 353 Lin, C. 48, 49 Lin, C.C. 180 Lin, C.-K. 180 Lin, H. 48, 49 Lind, R.C. 68 Lipovski, G. 484 Lipton, L.T. 447 Liu, B. 473 Liu, L. 504 Liu, Y. 167 Livescu, G. 186, 452, 453 Lo, C.W. 198 LO, S.-H. 426, 427 Lo, Y.H. 175 Logan, J.E. 88, 96, 149 L o h a n n , A.W. 268, 27@-274, 276-278, 281-283, 285-287, 289, 293-295, 297, 299, 300, 302, 303, 307, 309, 314, 322, 325, 326, 333, 334, 336,432,441,442,445,467469, 475, 477480, 484 Longo, M. 88 Lops, M. 88 Lorimer, D.R. 115 Lott, J.A. 174, 193 Louri, A. 488,490 Lowenthal, S. 276 Lu, B. 187, 188, 192, 193, 195, 203 Lu, X.J. 317, 318, 323 Luck, H. 88 Lukasik, J. 68 LUO,W.-L. 192 Luther, L.C. 186, 192 Lyne, A.G. 115

524

AUTHOR INDEX FOR VOLUME XXXVlll

M MacDougal, M.H. 179, 180 Mackowski, J.-M. 142 Macrander, A.T. 183, 190 Maeda, H. 66 Magel, G.A. 50, 56 Mahalakshmi, V. 5 1, 62 Mahalanobis, A. 359 Maiman, T.H. 422 Maischberger, K. 88, 132, 139 Mait, J.N. 469, 472, 490 Majewski, M.L. 246 Maker, P.D. 11 Malarkey, E.C. 422 Malcuit, M.S. 33, 48 Malhotra, V 201 Malloy, K.J. 174, 187, 188, 193, 195, 203 Man, C.N. 88, 93, 94, 108, 115, 116, 123, 124, 127, 130, 132, 139, 142, 152, 155 Manchester, R.N. 1 15 Mand, R.S. 452, 456 Mandel, I? 6, 14, 15 Manh, P.T. 88 Manning, J.S. 189, 218 Manning, R.J. 43 Mar, A. 180 Maradudin, A.A. 22, 30, 31 Marathay, A.S. 445 Marburger, J.H. 6, 7, 12, 14, 33 Marcinkowski, S. 432 Marck, J.A. 87, 88 Marcuvitz, N. 298 Margalit, N.M. 176, 238 Markevitch, B.V. 447 Marom, E. 319, 323, 373 Marple, D.T.F. 182 Marraud, A. 88 Mars, D.E. 238 Marsden, G.C. 488 Marta, T.M. 175, 187, 193 Martin, D. 33, 46 Martinet, R.K. 3 17 Martinot, P. 14, 28 Mas, D. 319 Masciulli, P. 40 Mathew, J.G.H. 469 Matsuda, M. 193 Matsumoto, T. 488 Matsuo, S. 452, 453, 455 Matsutani, A. 176, 178, 187, 465

Mavaddat, R. 126 Maxwell Garnett, J.C. 66 Mazilu, D. 22, 31 McAulay, A.D. 433, 489, 491 Mcbride, A.C. 266, 333 McCall, S.L. 6, 14, 45, 178, 201, 422, 447, 451, 452, 455, 459, 464 McClelland, D.E. 93, 124, 126 McCormick, F.B. 432, 442, 463, 469, 479, 503 McDougal, M.H. 465 McInerney, J.G. 220, 228, 229 Meers, B.J. 108, 112, 115, 116, 126, 152 Meessen, A. 67 Meindl, J.D. 425 Meissner, P. 249 Mendlovic, D. 268, 272-274, 276-278, 281, 286,289,295, 297,299, 300, 302-304, 309, 314, 318, 319, 322, 323, 325, 326, 333, 334, 336 Merlin, R. 39 Mertz, L. 293 Merz, J.L. 178, 192, 238 Meystre, P. 13 Miao, E. 42 Michalzik, R. 179, 196-198, 201, 221, 244, 245 Michel, J.C. 45 Midwinter, J.E. 486, 497 Mihalache, D. 6, 21, 22, 31, 32 Milano, L. 88, 133 Miller, A. 43 Miller, 9.1. 175, 178, 187, 192, 201, 238 Miller, D.A.B. 43, 69, 422, 431, 432, 447, 45@453,459461, 465, 503, 504 Miller, L.R. 87 Mills, D.L. 4, 7, 8, 17, 19, 21, 33, 36 Mills, J.P. 353 Milsom, P.K. 43 Ming, N. 55, 59 Mirin, R.P. 175, 176, 187, 192, 201, 238 Misner, C.W. 89 Mitake, T. 142 Mitkas, P.A. 432 Mitrofanov, VP. 160 Mitsunaga, N. 66 Mitsunanaga, K. 452, 456 Mittler-Neher, S. 60 Miyamoto, T. 238 Mizuno, J. 114, 118, 125, 127

AUTHOR INDEX FOR VOLUME XXXVlll

Mizuuchi, K. 59 Mohlmann, G.R. 60 Moller, B. 174, 197 Moloney, J.V. 66, 254 Monemar, B. 182 Montelatici, V. 88, 107 Montgomery, W.D. 294 Morgan, R.A. 167, 175, 186, 187, 192, 193, 197-199, 201, 460, 465 Mori, K. 238 Morier-Genoud, F. 33, 46 Morishita, K. 245 Morlet, J. 31 7 Morozov, V. 198 Morrison, R.L. 335, 463, 469, 472, 503 Moss, G.E. 87 Mroziewicz, B. 182, 186 Mukai, S. 266 Mukaihara, T. 176, 178, 187, 465 Mullay, T. 186, 192 Murdocca, M.J. 432434,442,467,484,486, 488, 489, 491, 492, 496, 497, 501 Myers, D.R. 174, 225, 227, 229, 249

N Nabiev, R. 201 Nabiev, R.F. 38, 201 Nakamura, K. 59, 142 Nakamura, M. 196 Nakwaski, W. 167, 182, 186, 199,200, 205208, 212, 22CL222, 228-230, 232, 237-240, 242-244, 2 4 6 2 5 1 Nalesso, G.F. 66 Namias, C. 333 Nan, Ce-Wen 67 Naone, R. 178 Nash, F.R. 197 Natale, G. 88 Naviere, M. 66 Nazarathy, M. 336 Neeves, A.E. 69 Nelson, P.G. 118 Nelson, R.J. 190 Neviere, M. 61 Newton, G.P. 88 Nicastrol, L. 1 15 Nicholson, D. 88 Nido, M. 192 Nieuborg, N. 502 Nijander, C.R. 443, 480

525

Ning, C.Z. 254 Ninomiya, T. 177, 238 Nishihara, H. 59 Nishiyama, Y. 14 Noguchi, K. 488 Nolscher, U. 469 Noordman, O.F.J. 61 Normandin, R. 41, 50, 59, 60 Norris, P.M. 186, 187,221,237, 245 Nonvood, R.A. 42 Notcutt, M. 160 Nouh, S. 62 Nozawa, H. 62 Nuss, M.C. 335

0 Oartovi, L. 335 Oezisik, M.N. 206 Ogasawara, N. 50 Ohashi, M. 5, 50, 51, 54 Ohiso, Y 176, 178, 187, 193 Ohmae, Y. 191, 192 Ohnoh, N. 176, 178, 187, 465 Ojeda-Castaneda, J. 282, 293, 325, 469 Okamoto, H. 238 Oliver, M.K. 66 Olsson, N.A. 177, 193 Onischenko, A. 214 Onural, L. 272, 276 Oppenheim, A.V 326, 353 Orenstein, M. 201 Ortiz, G.G. 192 Osaheni, J.A. 5, 73, 74 Osgood, R.M. 48 Osihski, M. 167, 197, 199, 200, 205, 206, 22C222, 228-230, 232, 237, 238, 240, 242-244, 246, 248, 250, 251 Osterberg, H. 285 Ostrovslui, L.A. 63 Ostrowsky, D.B. 6, 62, 63 Ota, Y. 177, 193 Otomo, A. 60 Otsubo, K. 193 Otto, A. 27 Oudar, J.L. 45 Ouzieli, I. 319, 323 Owyoung, A. 197 Ozaktas, H.M. 268, 272-274, 276, 277, 281, 289, 292, 295, 302, 303, 314, 322, 323, 333, 334, 336

526

AUTHOR INDEX FOR VOLUME XXXVIII

Ozaktas, H.W. 431 Ozguz, V 465

P Painchaud, D. 347, 359, 360 Pande, M.B. 18, 30 Panish, M.B. 183 Panov, VI. 160 Panzlaff, K. 176, 186 Paoli, T.L. 213 Papannareddy, R. 204 Papoulis, A. 348 Papuchon, M. 62 Parchekani, F. 383, 385 Pans, D.P. 277 Park, H.-H. 201 Park, M.S. 201 Partridge, C. 497 Pasquier, C. 60 Passner, A. 45, 422, 451, 452, 459 Pasuello, D. 88 Patel, N.B. 198 Patorski, K. 475 Patzak, E. 249 Paye, J. 66 Pednni, G. 490 Pei, S.S. 463, 504 Penner, T.L. 60 Penin, S.D. 175 Pershan, P.S. 50, 51, 56 Peschel, Th. 6, 21, 30, 41 Petermann, K. 212, 214 Peters, F.H. 192, 246 Peters, M.G. 187, 192, 196, 199, 204, 246 Pettit, G.D. 182 Peyghambarian, N. 6 Peyraud, J. 66 Pham Tu,M. 93 Pic, E. 17, 23 Pinto, I. 88 Piprek, J. 221, 244, 253 Ponath, H.E. 31 Popov, E. 6 1, 66 Pouligny, B. 3 17 Prasad, P.N. 42 Prelewitz, D.F. 33, 4 6 4 9 Prince, F.C. 198 Prise, M.E. 43, 432435, 450, 469, 472, 501 Proctor, M. 33, 46 Przybylek, G.P. 186, 192

Psaltis, D. 488, 504 Pudikov, V 179 Pustovoit. VI, 87

Q

Qin, Y. 55 Qiu, L. 4 2 4 Quail, J.C. 28 Quinn, J.J. 39 R Raah, F.J. 88, 93, 96 Raether, H. 27 Raghavendra, C.S. 500 Rahman, B.M.A. 221, 246 Raimond, A. 61 Raj, K. 503 Rako, J G. 28 Ram, R.J. 175, 187, 192, 201, 238 Ramirez, G. 283 Ranieri, P. 40 Rapp, S. 175 Rastogi, V 63 Rauschenbach, K. 65 Ray, D.S. 11, 40 Redmond, I. 469 Reed, J. 175 Refregier, Ph. 346, 359 Regehr, M.W. 93 Reid,I. 175 Reiner, G. 174, 179, 197 Reinhart, F.K. 174, 187 Reinisch, R. 6, 15-17, 23, 61, 66 Rempe, G. 142 Restall, E.J. 443 Reynaud, S. 99 Reynolds, T.E. 175, 178, 187, 192, 201, 238 Rhee, B.K. 45 Rhodes, W.T. 488 Ricard, D. 5, 68, 69 Rief, P.G. 447 Riel, R. 452, 453 Ries, M. 174, 197 Rinkleff, R. 88 Ristic, VM. 304, 309 Roach, G.F. 266 Roberge, D. 318, 323 Robert, B.D. 65 Robertson, B. 443 Robertson, D.I. 88

AUTHOR INDEX FOR VOLUME XXXVIIl

Robertson, N.A. 88, 96 Robinson, K.C. 465 Robson, P.N. 51 Robson, R. 43 Rocca, J.J. 198 Rochford, K. 42 Rochus, S. 175 Rogers, L.E. 201 Rogers, S.K. 353 Rogers, T.J. 167, 180 Rohr, T. 175 Root, W.L. 348 Rossi, M. 469 Rotoli, G. 88 Rotz, F.B. 345, 399, 412 Rouard, P 67 Roussignol. €? 5, 68, 69 Row, J.F. 61 Rudiger, A. 88, 118, 125, 132, 139, 141, 149 Rushford, M.C. 451, 452, 460 Russo, G . 88, 133 Rustagi, K.C. 68 Ryan, A.T. 66 S Sahm, A. 273, 274 Sai-Halasz, G.A. 426, 427 Said, A.A. 43 Sakano,T. 488 Saleh, B.E.A. 100 Sambles, J.R. 30 Samson, B.A. 66 Sankey, N.D. 46 Sarid, D. 27, 28 Sarma, J. 196, 199, 200, 214 Sarzata, R.P. 200, 206, 221, 246, 247, 250 Sasaki, K. 42 Sasian, J.M. 442, 463, 503 Sathyaprakash, B.S. 159 Sato, H. 62 Sauer, F. 443, 469, 478, 480 Saulson, P.R. 96 Savage, C.M. 11, 124, 126 Sawabe, T. 488 Sawchuk, A.A. 432, 484, 490, 499-501, 504 Scalora, M. 17, 66 Scelsi, G. 48 Schaus, C.F. 174, 225, 227, 229, 249 Scherer, A.L. 178, 201, 447, 452, 455, 463465

527

Schiek, R. 63, 64 Schilling, R. 88, 118, 125, 132, 139, 141, 149 Schils, G.F. 359 Schmitt-fink, S. 69 Schneider, R.P. 192 Schneider Jr, R.P. 174, 179, 180, 187, 192, 193, 196 Schnitzer, P. 179 Schnupp, L. 88, 93, 132, 139 Schrempel, M. 88 Schroder, S. 201 Schubert, E.F. 174, 193 Schuetz, H. 469 Schutz, B.F. 88 Schweitzer, P. 422 Schwider, J. 285, 469, 477 Scott, J.W. 167, 169, 176, 186, 187, 189-194, 196, 199-201, 205, 212, 221, 235, 246, 252 Seabury, C.W. 177, 193 Seaton, C.T. 5, 6, 22, 31, 32, 41, 43, 56, 64, 450 Seibert, H. 63 Seidel, H. 6 Seifert, M. 60 Sekigushi, H. 142 Sendler, E. 39 Ser, J.-H. 179 Sfez, B.G. 45 Shabtay, G. 326 Shah, B. 422, 447 Shamir, J. 444, 495 Shank, C.V 33 Sheik-Bahae, M. 43, 63, 450 Shen, Y. 4 2 4 4 Shen, YR. 41 Sheng, Y. 317, 318, 323 Sheng, Y.-L. 484 Shengzhen, J. 115 Shenoy, M.R. 62, 63 Sheridan, J.T. 290, 326, 332 S h i m i q H . 59 Shimizu, M. 201,221, 243 S h i m , S. 191, 192 Shm, H.-E. 179 Shin, J. 180 Shm, J.H. 463, 465 Shin, J.-H. 179 Shiralu,Y 50 Shoemaker, D. 88, 93, 94

528

AUTHOR INDEX FOR VOLUME XXXVIII

Shoji, H. 193 Shore, K.A. 196, 199 Shuttleworth, J. 88 Si, J. 4 2 4 4 Sibilia, C. 6, 22, 32, 40 Siegman, A.E. 122, 127, 150 Sievers, L. 88 Silverberg, Y. 66 Simmons, J.G. 452, 456 Simmons, W.W. 43 Simon, H.J. 14, 28, 29 Singh, B.P. 42 Singh, R. 18 Sinitsyn, G.V. 451, 452, 459 Sinzinger, S. 469, 478, 481 Sipe, J.E. 5, 22, 33, 34, 36, 38, 47, 51, 65, 66, 69-75 Sivco, D.L. 50, 59 Sizer 11, T. 452, 453 Sizer 111, T. 432 Smith, D.E. 442, 467, 492 Smith, J.S. 176, 187, 197, 199, 204, 221, 237, 245, 246, 452, 453 Smith, L.E. 463, 504 Smith, P.W. 43, 447 Smith, P.W.E. 66 Smith, S.D. 6, 14, 43, 422, 450-452, 459, 469, 504 Smolyaninov, 1.1. 24 Snell, C. 243 So, V.C.Y. 41 Soares, O.D.D. 325 Soffer, B.H. 271 Sohler, W. 62, 63 Solimeno, S. 146 Sommers, H.S. 213 Song, J.I. 463, 465 Soos, Z. 450 Sounik. J. 42 Speck, D.R. 43 Spero, R.E. 88 Steel, D.G. 39 Steel, M.J. 65, 66 Stegeman, (3.1. 3, 5, 6, 22, 23, 3@33,41,42, 48-50, 56, 59, 60, 63, 64, 66 Steigmeier, E.F. 207, 216 Stem, E 190 Stevenson, A.J. 93 Stieglitz, K. 325 Stirk, C.W. 484, 499

Stoffel, N. 193 Stoffel, N.G. 197, 198, 201 Stone, H.S. 446 Stone, T.W. 442, 467, 492 Stork, W. 469, 478, 484 Strain, K.A. 88, 115, 118, 125, 126, 149 Strand, T.C. 432 Strebl, N. 285 Streetman, B.G. 180 Streibl, N. 287, 432435, 468, 469, 472, 477, 488,489,491,499 Streubel, K. 175, 176, 238 Stroud, D. 69, 71 Strzelecka, E. 178 Strzelecka, E.M.S. 167 Stucke, G. 484 Suche, H. 62 Suda, D.A. 452, 456 Sugihwo, E 177 Sugimoto, M. 452, 456 Sugimoto, Y. 192 Suhara, T. 59 Sukhorukov, A.P. 336 Suleski, T.J. 476 Sun, S.Z. 174, 192, 225, 227, 229, 249 Sundheimer, M.L. 63 Sung, C.C. 6 Surette, M.R. 495 Sutherland, B. 4, 39, 40 Suzuki, K. 238, 239, 241 Svenson, B. 31 Swaminathan, V 183, 190 Swanson, G.J. 469, 476 Sweeney, D.W. 359 Sziklas, E.A. 122 Szoke, A. 6, 451 Sztefka, G. 221, 244 Szu, H. 317, 318, 323

T Taboury, J. 488 Tadokoro, T. 178, 238 Taghizadeh, M.R. 443,469 Tai, K. 167, 169, 174, 177, 186, 187, 192, 193, 197, 198, 202, 228, 246 Takahashi, H. 50 Takeda, M. 468 Takenouchi, H. 178 Talbot, W.H.F. 294, 475 Tamada,H. 62

AUTHOR INDEX FOR VOLUME XXXVlll

Tang, C. 39 Tang, C.L. 6 Tang, Q. 345-347, 351, 353, 354, 356, 370, 375, 381, 383, 385, 389,411, 413 Tanida, J. 489 Taniuchi, T. 59 Tashiro, Y. 452, 456 Tateno, K. 178, 187, 193, 503 Taw, Y 426, 427 Taylor, G.W. 169, 452, 456 Taylor, P.C. 40 Teich, M.C. 100 Tell, B. 186, 192, 194, 452, 455, 464, 480 Tepechin, E. 283 Terhune, R.W. 11 Testorf, M. 473 Thakur, M. 42 Thalmann, R. 490 Thibeault, B.J. 176, 178, 186-188, 191-194, 196, 199, 204, 205, 246 Thibeault, B.J.T. 167 Thienpont, H. 14, 15, 502 Thode, L. 243 Thomas, J. 469, 477 Thomas, J.A. 285, 294 Thompson, G.H.B. 183, 197 Thompson, R.J. 142 Thorne, K.S. 87-89 Thornton, R.L. 255 Thurmond, C.D. 190 Thyagarajan, K. 62-64 Tiberio, R.C. 33, 4 6 4 9 Ticknor, A.J. 304 Tieke, B. 60 Tien, C.-L. 186, 187, 221, 237, 245 Tishinin, D. 180 Tocci, M.D. 17 Tolimieri, R. 318, 323 Tolkacheva, E.G. 66 Tomioka, T. 42 Tooley, F.A.P. 432, 442, 447449, 451, 463, 503, 504 Toruellas, W. 64 Tourrenc, P. 88, 107 Townsend, P.D. 65 Townsend, W.P. 443, 480 Traenke, G. 175 Trankle, G. 452, 453 Treat, D.W 255 Tricot, C. 266

529

Tridgell, A.J. 124, 126 Trullinger, S.E. 33, 36 Trutschel, U. 6, 21, 23 Tseng, B. 465, 503 Tsubono, K. 88 Tsunetsugu, H. 187, 193 Tsunoda, Y. 422 Tu, L.W. 174, 193, 197, 198 Tiinnermann,A. 88 Turner, W.J. 182 Turunen, J. 469, 473 Twardowski, T. 30-32

U Uchida, T. 238 Uchiyama, S. 177, 191, 192, 238 Ueda, A. 142 Ueda, K. 142 Uehara, N. 142 Umeda, J.-I. 196 V Vach, H. 41 Vail, E.C. 177 Vakhshoori, D. 50, 51, 59, 198, 199 Valera, J.D. 22, 31 Valle, A. 196, 199 van der Poet, C.J. 59 van der Vleuten, W.C. 201 Van der Ziel, J.P. 186 van Exter, M.P. 197 van Hulst, N.F. 61 Van Stryland, E.W 43, 63, 450 Vander Lugt, A. 345, 399, 412 Vanhertzeele, H. 63 Varangis, P.M. 199,205,248 Varma, A. 500 Vasara, A. 469, 473 Vawter, G.A. 174, 225, 227, 229, 249 Vaya, M. 64 Venkatesan, T.N.C. 6, 45, 422, 451, 452, 459 Veretennicoff, I. 14-17, 502 Vicari, L. 150 Viliani, G. 266 Villeneuve, A. 48-50 Vinet, J.-Y. 88, 96, 107, 108, 115, 116, 123, 124, 126, 132, 143, 146, 150, 152, 159 Virgo Collaboration 88, 92, 95, 96, 125 Viswanathan, R.G. 426, 427 Vitrant, G. 15-17, 23 Vogel, E.M. 66

530

AUTHOR INDEX FOR VOLUME XXXVIII

Vogt, R.E. 88 Vohnsen, B. 24 Von Lehmen, A. 193, 201 Von Lehmen, A.C. 197, 198 von Neumann, J. 438, 488 Vook, EL. 222 Vounckx, R. 452, 456, 502

w Wada, H. 178, 192, 194, 200, 238 Wada, 0. 61 Wada, Y. 426 Wakatsula, A. 187, 193 Walker, A.C. 443, 469, 504 Walker, J.A. 465, 503 Walker, J.D. 199 Walker, S. 178, 201, 447, 452, 455, 464 Walker, S.J. 468, 469, 472 Walker, S.L. 463, 503 Wallin, J. 175 Walterson, R.A. 175, 187, 193 Wang, C.H. 501 Wang, H. 42, 55 Wang, J. 345-347, 351, 353, 354, 356, 359, 370, 373, 411-413, 465 Wang, J.M. 488, 501 Wang, M. 61 Wang, W. 347, 359, 360 Wang, Y. 42-44, 59 Wang, Y.H. 167, 169, 174, 186, 187, 192, 193, 197, 198, 202, 228, 246 Wang, Z. 197 Wang, Z.H. 32 Wann, H.-J.C. 426,427 Ward, H. 88 Warren, M.E. 221, 252 Weaver, C.S. 345, 347, 399 Weber, A.G. 501 Weber, J. 87 Weible, K. 490 Weigl, B. 179 Weimann, G. 175, 452, 453 Weiner, A.M. 66, 335 Weiner-Avnear, E. 447 Weir, B. 169, 174, 186, 187, 192, 193, 197, 198, 202, 246 Weir, B.E. 167, 174, 193, 228 Weiss, R. 87, 88 Welford, K.R. 30 Weller-Brophy, L.A. 5, 73, 74

Welling, H. 88 Welton, T.A. 96 Wenzel, H. 22 1, 244 West, L.C. 433 Westerholm, J. 473 Weyl, H. 333 Wheeler, J.A. 89 Wherrett, B.S. 438, 504 Whitbread, N.D. 51 Whitcomb, S.E. 88, 93 White, H.E. 130 Wiegmann, W. 43, 45, 422, 451, 452, 459, 460 Wiener, N. 333 Wigley, P.G. 50 Wigner, E. 270, 334 Wijnands, E 40 Williams, R.L. 50, 59, 60 Willke, B. 88 Wilmsen, C. 198 Wilson, G.C. 176, 197, 199, 246 Wind, S.J. 426, 427 Winful, H.G. 6, 14, 33, 36, 38 Winkler, W. 88, 118, 125, 132, 139, 141, 149 Winthrop, J.T. 294 Wipiejewski, T. 176, 186, 187, 204 Woerdman, J.P. 197 Wolf, E. 11, 24, 29, 457 Wolf, K.B. 325, 332 Wolter, H. 276 Wolter, J.H. 201 Wong, H.-S. 426, 427 Wong, K.Y. 42 Wood, J.C. 304, 334 Wood, T.H. 460 Wood, YE. 69 Woodward, T.K. 463, 504 Woody, L.M. 499 Woolven, S. 304, 309 Worthington, C.R. 294 Wright, E.M. 30-32 Wu, C.C. 187 Wu, J.W. 42 Wu, X.X. 452,453 Wu, Y.A. 201 Wiinsche, H.-J. 221, 244 Wiithrich, C. 174, 187 Wynn, J.D. 167, 169, 174, 186, 187, 192, 193, 197-199,202,228, 246 Wysin, G.M. 14, 28, 29

AUTHOR lNDEX FOR VOLUME XXXVlIl

Y Yajima, H. 266 Yamamoto, K. 59 Yamamoto, 'I: 503 Yan, R.H. 190 Yanagawa, K. 61 Yanase, T. 452, 456 Yang, C.C. 48, 49 Yang, G.M. 179, 180, 465 Yang, L. 169, 174, 175, 186, 187, 192, 193, 197, 198, 201, 202, 238, 246 Yang, VK. 255 Yang, Y. 59 Yang, Z. 59 Yariv, A. 14, 32, 294 Yasumoto, K. 66 Yatagai, T. 488 Yayla, G. 488 Ye, P. 4 2 4 4 Yeh, P. 3, 32, 38, 70 Yoffe, G.W. 201 Yogo, Y. 238, 239, 241 Yokota, M. 32 Yokoucbi, N. 177, 238 Yoo, B.-S. 201 Yoo, J.Y. 463, 465 Yoo, S.J.B. 221, 244, 253 Yoshikawa, T. 192 Youden, K.E. 65 Young, D.B. 187, 189-192, 194, 196, 199, 201,204, 246 Young, M. 169 Young, M.G. 178, 192, 238 Yu, S.F. 198 Yuen, W. 177

53 1

Z

Zahir, S. 60 Zalevsky, Z. 273,274,277,278,286,289,295, 297, 299, 300, 303, 304, 309,319, 323, 334 Zamani-Khamiri, 0. 42 Zanoni, R. 32, 42, 64 Zayats, A.V 24 Zeeb, E. 174, 176, 186, 197 Zeng, X.C. 69, 71 Zemike, F. 477 Zhang, G. 347,359, 360,383,385,390 Zhang, L. 65 Zhang, X. 69, 504 Zhang, Y. 318, 323 Zhang, Z. 504 Zhao, C.N. 160 Zhao, H. 179 Zhao, J. 4 2 4 4 Zhao, Y.-G. 220,228, 229 Zheng, K. 174,225,227,229, 249 Zhou, J. 4 2 4 4 Zhou, P. 174, 187, 188, 193, 195, 203, 225, 227, 229, 249 Zhu, J. 59 Zhu, S. 55 Zhu, Y. 55 Zhu, Z.H. 175 Zick, W. 179 Zolper, J.C. 179, 187, 188, 192, 193, 195, 196 Zou, B. 4 2 4 4 Zschauer, K.-H. 249 Zucker, M.E. 88, 124 Ziirl, K. 499 Zydzik, G.J. 50, 59, 174, 186, 193, 197-199

SUBJECT INDEX FOR VOLUME XXXVIII

A ABCD-Bessel transformation 289-293 Airy function 457, 458 - resonance 17 attenuated total reflection (ATR) 22

B Bessel hnction 223 birefringence 56, 61 -, thermal 149 birefringent phase matching Bloch theorem 36

- _ - resonator 40, 99, 455 - _ - transfer function 108, 116 Fibonacci multilayer 39, 40 fluctuation-dissipation theorem 96 four-wave mixing 4 4 , 64-66, 449 _ _ _ , degenerate 6, 64 Fourier transformation, fast 293, 494 - -, optical 270, 298 fractal 326 fractional Bessel transformation 293 - Boolean logic 266 - convolution 302 - correlation 302 - filtering 302-323 - Fourier transfornation 265, 27C276, 299 - Fourier-Kravchuk transformation 325 - Hartley transformation 287-289 - Hilbert transformation 276-284, 299 - Legendre transformation 298, 299 -optics 267 - Radon transformation 304-309, 312, 313 - spectral density 3 11 - Talbot effect 293 - wavelet transformation 3 17-323 - Wiener filter 309-3 17 - Zernike transformation 284, 285 Franz-Keldysh effect 45 1 Fraunhofer diffraction 270 frequency down-conversion 4 - up-conversion 4 Fresnel diffraction 379 - transformation 293, 330 frustrated total reflection 22 fuzzy logic 266

57

C Cantor corrugated surface 40 -set 39, 40 channel capacity 441 colored noise 366, 368, 370, 372, 374, 375 Curie temperature 58 D Dammann grating 323, 471473 digital computing 432 - optical processor 501 - optics 423, 433, 441, 441 distributed Bragg reflector (DBR) 183, 190 - feedback (DFB) 18, 190

E electro-optic effect 73 evanescent wave 24

F Fabry-Perot (FP) cavity 3, 6, 7, 14, 17-19, 43, 88, 89, 93, 99, 101, 104, 106, 112, 115, 120, 124 _ - _ etalon 41, 451, 457, 459 - - _ interferometer 109

G general relativity 87 graded index (GRIN) medium 268,294,295 533

534

SUBJECT INDEX FOR VOLUME XXXVlll

gravitational wave 87, 90, 118 -, coupling with 99-121 - -, interferometric detection of 87, 89-99 - detector 96 - - transfer function 116 Green’s function 228 group theory 327, 328 -

-

H harmonic generation in guided wave configuration 56 _ _ - layered geometry 50-66 heat conduction equation 206, 207 Helmholtz equation 122 Hennite-Gauss mode 127, 133, 138, 268 Hilbert transformation, optical implementation 276 J joint transform correlator (JTC) 345 _ _ _ - , binary 346, 351 - _ - _ , chirpencoded 375-389 _ _ _ _, linear 347 - - - _ , nonlinear 347-353, 370, 404 - _ _ - , random phase encoded 390-396

K Kern nonlinear film 32 - nonlinearity 4, 7, 24 Kirchhoff transformation 23 1 L L a g u e r r d a u s s mode 133, 138 laser, edge-emitting 167-170 -, GaAsIAlGaAs 229-237 -, infrared diode 5 -, InGaAsPDnP 237-242 -, vertical-cavity surface-emitting, see VCSEL Lebesgue measure 39 LIGO 89, 90, 93, 94, 99, 105, 127, 134, 140, 142, 158, 159 liquid crystals 4, 41, 447 lithography 441 M Mach-Zender interferometer 63 Maxwell Garnett theory 67 Maxwell’s equations 5, 3 1 Michelson interferometer 87, 90, 91, 105, 112, 115, 124, 399

molecular beam epitaxy 45 1 multiple quantum well 45, 451

N neural network 487, 488 nitrobenzene 41 0 optical bistability 4, 19, 28-30, 42, 69, 422, 447, 449 - -, dispersive 7 - -, in layered media 6 computing 423 - fibers 3 logic elements 447 - pattern recognition 345 - switching devices 4 5 1 4 5 7 -

-

P paraxial approximation 122 phase conjugation 4, 6, 24 photonic switching 424 Pockels effect 73 polariton 22 power spectrum 354 programmable logic array, optical 491

Q

quantum limit 97-99 well, multiple 45, 451

-

R Radon transformation Raman process 6 Ronchi grating 297

271

S Sarid configuration 30 Schrodinger equation, nonlinear (NLS) 37, 38 second harmonic generation 57, 59 self-electro-optic effect device (SEED) 45 1, 46&463 - focusing 4, 449 - phase modulation 449 slowly varying envelope approximation (SVEA) 12, 16, 18 soliton, gap 36 spacebandwidth product 345, 346, 441 spatial filter 266, 345 - holeburning effect 253

SUBJECT INDEX FOR VOLUME XXXVIII

Stark effect, quantum confined 45 1 stratified layered media 3 surface plasmon 22, 27-30

T Talbot self-imaging 475 thermal lensing effect 253 thermoelastic deformation 146-1 49 transfer matrix method, linear 11 _ _ _ , nonlinear I 1

V VCSEL (vertical-cavity surface-emitting laser) 165-255, 431,451,455, 463465 ~ _ - - - _ , effect of temperature on 182-205 _ _ ~ _ _, thermal _ modeling of 205-218 _ _ _ _ - _ , - models of 218-254 ~ _ ~ _ ,types - _ of 171-180

535

vertical-cavity surface-emitting laser, see VCSEL VIRGO 89, 90, 93-99, 105, 106, 121-126, 128, 131, 134, 139-142, 145, 147-160 von Neumann architecture 487 - -bottleneck 426, 429 - - machine 433

W wavelet matched filter 323 Weber’s bar 87 white noise 366, 368, 370, 372, 374, 375 Wigner algebra 329 - distribution function (WDF) 270, 271, 276, 324,336339 - space 329, 330 - transformation 329 Wollaston prism 283, 284

CUMULATIVE INDEX - VOLUMES I-XXXVIII

ABELBs,F., Methods for Determining Optical Parameters of Thin Films ABELLA, I.D., Echoes at Optical Frequencies ABITBOL,C.I., see Clair, J.J. Dynamical Instabilities and ABRAHAM, N.B., P. MANDEL,L.M. NARDUCCI, Pulsations in Lasers AGARWAL, G.S., Master Equation Methods in Quantum Optics Crystal Optics with Spatial Dispersion AGRANOVICH, VM., VL. GINZBURG, AGRAWAL, G.P., Single-Longitudinal-Mode Semiconductor Lasers G.P., see Essiambre, R.-J. AGRAWAL, ALLEN,L., D.G.C. JONES,Mode Loclung in Gas Lasers AMMANN, E.O., Synthesis of Optical Birefringent Networks APRESYAN,L.A., see Kravtsov, Yu.A. ARIMONDO, E., Coherent Population Trapping in Laser Spectroscopy ARMSTRONG, J.A., A.W. SMITH,Experimental Studies of Intensity Fluctuations in Lasers ARNALJD, J.A., Hamiltonian Theory of Beam Mode Propagation ASAKURA, T., see Okamoto, T. ASAKURA, T., see Peiponen, K.-E. BALTES,H.P., On the Validity of Kirchhoff’s Law of Heat Ramation for a Body in a Nonequilibrium Environment BARABANENKOV, Yu.N., Yu.A. KRAVTSOV, VD. OZRIN,A.I. SAICHEV, Enhanced Backscattering in Optics BARAKAT, R., The Intensity Distribution and Total Illumination of Aberration-Free Diffraction Images BARRETT,H.H., The Radon Transform and its Applications S., Beam-Foil Spectroscopy BASHKIN, BASSETT,I.M., W.T. WELFORJJ,R. WINSTON,Nonimaging Optics for Flux Concentration P., Scattering of Light by Rough Surfaces BECKMANN, BERAN,M.J., J. OZ-VOGT,Imaging through Turbulence in the Atmosphere BERNARD,J., see Omt, M. Catastrophe Optics: Morphologies of Caustics and their BERRY,M.V., C. UPSTILL, Diffraction Patterns BERTERO, M., C. DE MOL,Super-Resolution by Data Inversion BERTOLOTTI, M.,see Mihalache, D. BERTOLOTTI, M., see Chumash, ?? BEVERLY III, R.E., Light Emission From High-Current Surface-Spark Discharges I., Photon Wave Function BIALYNICKJ-BIRULA, 547

11, 249 VII, 139 XVI, 71

MV, XI, IX, XXVI, XXXVII, IX, IX, XXXVI, XXXV,

1 1 235 163 185 179 123 179 257

VI, XI, XXXIV, XXXVII,

211 247

XIII,

1

183

57

M I X . 65 I, 67 XXI, 217 XII, 287

XXVII, VI, XXXIII, XXXV,

161 53 319 61

XVIII, XXXVI, XXVII, XXXVI, XVI, XXXVI.

257 129 227 1 357 245

548

CUMULATIVE INDEX

-

VOLUMES I-XXXVIII

BJORK,G., see Yamamoto, Y. BLOOM, A.L., Gas Lasers and their Application to Precise Length Measurements BOUMAN, M.A., W.A. VANDE GRIND, P. ZUIDEMA, Quantum Fluctuations in Vision P., see Rouard, P. BOUSQUET, BROWN, G.S., see DeSanto, J.A. BROWN, R., see Omt, M. BRUNNER, W., H. PAUL, Theory of Optical Parametric Amplification and Oscillation BRYNGDAHL, O., Applications of Shearing Interferometry O., Evanescent Waves in Optical Imaging BRYNGDAHL, BRYNGDAHL, O., F. WYROWSKI, Digital Holography Computer-Generated Holograms O., T. SCHEERMESSER, E WYROWSKI, Digital Halftoning: Synthesis of BRYNGDAHL, Binary Images BURCH, J.M., The Metrological Applications of Diffraction Gratings BUTTERWECK, H.J., Principles of Optical Data-Processing Quantum Interference, Superposition States of Light, and BUZEK,V, P.L. KNIGHT, Nonclassical Effects

XXVIII, IX, XXII, IV, XXIII, XXXV.

87 1 77 145 1 61

xv,

1 IV, 37 XI, 167

~

CAGNAC, B., see Giacobino, E. CASASENT, D., D. PsALTis, Deformation Invariant, Space-Variant Optical Pattern Recognition CEGLIO,N.M., D.W. SWEENEY, Zone Plate Coded Imaging: Theory and Applications CHARNOTSKII, M.I., J. GOZANI, VI. TATARSKII, VU. ZAVOROTNY, Wave Propagation Theories in Random Media Based on the Path-Integral Approach Tunneling Times and Superluminality CHIAO,R.Y., A.M. STEINBERG, CHRISTENSEN, J.L., see Rosenblum, W.M. CHRISTOV, I.P., Generation and Propagation of Ultrashort Optical Pulses CHUMASH, V, I. COJOCARU, E. FAZIO,F. MICHELOTTI, M. BERTOLOTTI, Nonlinear Propagation of Strong Laser Pulses in Chalcogenide Glass Films CLAIR,J.J., C.I. ABITBOL, Recent Advances in Phase Profiles Generation CLARRICOATS, P.J.B., Optical Fibre Waveguides - A Review COHEN-TANNOUDJI, c.,A. KASTLER, Optical PLImping COIOCARU, I., see Chumash, V COLE,T.W., Quasi-Optical Techniques of Radio Astronomy COLOMBEAU, B., see Froehly, C. COOK,R.J., Quantum Jumps M. DETAILLE, M. SAME,Some New Optical Designs COURT&,G., P. CRUVELLIER, for Ultra-Violet Bidimensional Detection of Astronomical Objects K., Phase-Measurement Interferometry Techniques CREATH, CREW,A.V, Production of Electron Probes Using a Field Emission Source CRUVELLIER, P., see Courtis, G. CUMMINS, H.Z., H.L. SWINNEY, Light Beating Spectroscopy DAINTY, J.C., The Statistics of Speckle Patterns DANDLIKER, R., Heterodyne Holographic Interferometry A. RENIERI, A. TORRE,Theory of Compton Free DATTOLI,G., L. GIANNESSI, Electron Lasers DE MOL,C., see Bertero, M. DE STERKE, C.M., J.E. SIPE,Gap Solitons

XXVIII.

1

XXXIII, 389 11, 73 XIX. 211 XXXIV.

1

XVII, 85 XVI, 289

XXI. 287 XXXII, XXXVII, XIII, XXIX.

203 345 69 199

XXXVI, XVI, XIV, v, XXXVI, XV, XX, XXVIII,

71 327 1 1 187 63 361

1

1 xx, XXVI, 349 XI, 223 1 VIII, 133

=,

XIV, XVII,

1 1

XXXI, 321 XXXVI, 129 XXXIII, 203

CUMULATIVE INDEX

~

549

VOLUMES I-XXXVIII

DECKERJR, J.A., see Harwit, M. DELANO,E., R.J. PEGIS,Methods of Synthesis for Dielectric Multilayer Filters DEMARIA,A.J., Picosecond Laser Pulses DESANTO,J.A., G.S. BROWN,Analytical Techniques for Multiple Scattering from Rough Surfaces DETAILLE, M., see Court&, G. DEXTER,D.L., see Smith, D.Y. DRAGOMAN, D., The Wigner Distribution Function in Optics and Optoelectronics K.H., Interaction of Light with Monomolecular Dye Layers DREXHAGE, DUGUAY, M.A., The Ultrafast Optical Kerr Shutter DUTTA,N.K., J.R. SWSON,Optical Amplifiers DUTTAGUPTA,S., Nonlinear Optics of Stratified Media EBERLY, J.H., Interaction of Very Intense Light with Free Electrons ENGLUND, J.C., R.R. SNAPP,W.C. SCHIEVE, Fluctuations, Instabilities and Chaos in the Laser-Driven Nonlinear Ring Cavity ENNOS,A.E., Speckle Interferometry ESSIAMBRE,R.-J., G.l? AGRAWAL, Soliton Communication Systems FABELINSKII, I.L., Spectra of Molecular Scattering of Light FABRE,C., see Reynaud, S. FANTE,R.L., Wave Propagation in Random Media: A Systems Approach FAZIO,E., see Chumash, V FIORENTIM, A,, Dynamic Characteristics of Visual Processes FLYTZANIS,C., F. HACHE,M.C. KLEIN,D. RICARD, PH. ROUSSIGNOL, Nonlinear Optics in Composite Materials. 1. Semiconductor and Metal Crystallites in Dielectrics FOCKE,J., Higher Order Aberration Theory FRANCON,M., S. MALLICK,Measurement of the Second Order Degree of Coherence FREILIKHER, VD., S.A. GREDESKLIL, Localization of Waves in Media with OneDimensional Disorder FRIEDEN, B.R., Evaluation, Design and Extrapolation Methods for Optical Signals, Based on Use of the Prolate Functions FROEHLY, C., B. COLOMBEAU, M. VAMPOUILLE, Shaping and Analysis of Picosecond Light Pulses FRY,G.A., The Optical Performance of the Human Eye GABOR,D., Light and Information GAMO,H., Matrix Treatment of Partial Coherence GANDJEAKHCHE, A.H., G.H. WEISS,Random Walk and Diffusion-Lke Models of Photon Migration in Turbid Media GANTSOG, Ts., see Tanai, R. GHATAK, A,, K. THYAGARAJAN, Graded Index Optical Waveguides: A Review A.K., see Sodha, M.S. GHATAK, GIACOBINO, E., B. CAGNAC, Doppler-Free Multiphoton Spectroscopy GIACOBINO, E., see Reynaud, S. GIANNESSI, L., see Dattoli, G. GINZBURG, VL., see Agranovich, VM. GINZBURG, VL., Radiation by Uniformly Moving Sources. Vavilov-Cherenkov effect, Doppler effect in a medium, transition radiation and associated phenomena

XII, 101 VII, 6 1 IX, 31

XXIII, XX, X, XXXVII, XII, XIV, XXXI, XXXVIII,

1 1

165 1

163 161 189 1

VII, 359

XXI, 355 XVI, 233 XXXVII, 185 XXXVII, 95 =, 1 XXII, 341 XXXVI, 1 I, 253

XXIX, 321 1 IV, VI, 71

XXX, 137

IX, 311 XX, 63 VIII, 51 I, 109 111. 187

XXXIV, XXXV, XVIII, XIII, XVII,

333 355 1 169 85

=,

1

XXXI, 321 IX,235

XXXII, 267

550

CUMULATIVE INDEX - VOLUMES I-XXXVlIl

GIOVANELLI, R.G., Diffusion Through Non-Uniform Media GLASEQI., Information Processing with Spatially Incoherent Light GNIADEK, K., J. PETYKIEWICZ, Applications of Optical Methods in the Diffraction Theory of Elastic Waves GOODMAN, J.W., Synthetic-Aperture Optics GOZANI, J., see Charnotskii, M.I. GRAHAM, R., The Phase Transition Concept and Coherence in Atomic Emission GREDESKUL, S.A., see Freilikher, VD. HACHE,F., see Flytzanis, C. HALL,D.G., Optical Waveguide Diffraction Gratings: Coupling between Guided Modes HARIHARAN, P., Colour Holography HARIHARAN, P., Interferometry with Lasers P., B.C. SANDERS, Quantum Phenomena in Optical Interferometry HARIHARAN, HARWIT, M., J.A. DECKER JR, Modulation Techniques in Spectrometry HASEGAWA, A,, see Kodama, Y. HEIDMANN, A., see Reynaud, S. HELLO,P., Optical Aspects of Interferometric Gravitational-Wave Detectors HELSTROM, C.W., Quantum Detection Theory HERRIOT, D.R., Some Applications of Lasers to Interferometry HORNER, J.L., see Javidi, B. HUANG, T.S., Bandwidth Compression of Optical Images IMOTO,N., see Yamamoto, Y. ITOH,K., Interferometric Multispectral Imaging JACOBSSON, R., Light Reflection from Films of Continuously Varying Refractive Index P., B. ROIZEN-DOSSIER, Apodisation JACQLIINOT, JAHNS,J., Free-space Optical Digital Computing and Interconnection JAMROZ, W., B.P. STOICHEFF, Generation of Tunable Coherent Vacuum-Ultraviolet Radiation JAVIDI,B., J.L. HORNER, Pattern Recognition with Nonlinear Techniques in the Fourier Domain JONES,D.G.C., see Allen, L.

KASTLEQ A,, see Cohen-Tannoudji, C. KELLER, O., Local Fields in Linear and Nonlinear Optics of Mesoscopic Systems -00, I.C., Nonlinear Optics of Liquid Crystals KIELICH,S., Multi-Photon Scattering Molecular Spectroscopy KINOSITA, K., Surface Deterioration of Optical Glasses KITAGAWA, M., see Yamamoto, Y. KLEIN,M.C., see Flytzanis, C. KLYATSKIN, VI,, The Imbedding Method in Statistical Boundary-Value Wave Problems KNIGHT, EL., see Buiek, V KODAMA, Y., A. HASEGAWA, Theoretical Foundation of Optical-Soliton Concept in Fibers G., Multiple-Beam Interference and Natural Modes in Open KOPPELMAN, Resonators

11, 109 XXIV, 389 IX, 281 VIII, 1 XXXII, 203 XII, 233 137

xxx,

XXIX. 321 XXIX, XX, XXIV, XXXVI, XII, XXX, xxx, XXXVIII, X, VI, XXXVIII,

1 263 103 49 101 205 1 85 289 171 343 x, 1

XXVIII, 87 XXXV, 145

V, 247 111, 29 XXXVIII, 419

XX. 325 XXXVIII, 343 IX, 179 v, XXXVII, XXVI, XX, IV, XXVIII, XXIX.

1 257 105 155 85 87 321

XXXIII, XXXIV,

1

1

XXX, 205 VII,

1

CUMULATIVE INDEX

~

551

VOLUMES I-XXXVIII

KOTTLER, F., The Elements of Radiative Transfer KOTTLER,F., Diffraction at a Black Screen, Part I: Kirchhoffs Theory KOTTLER, F., Diffiaction at a Black Screen, Part 11: Electromagnetic Theory KRAVTSOV, Yu.A., Rays and Caustics as Physical Objects KRAVTSOV, Yu.A., see Barabanenkov, Yu.N. KRAVTSOV,Yu.A., L.A. APRESYAN, Radiative Transfer: New Aspects of the Old Theory KUBOTA, H., Interference Color LABEYRIE, A,, High-Resolution Techniques in Optical Astronomy LEAN,E.G., Interaction of Light and Acoustic Surface Waves LEE,W.-H., Computer-Generated Holograms: Techniques and Applications E.N., J. UPATNJEKS,Recent Advances in Holography LE~TH, VS., Laser Selective Photophysics and Photochemistry LETOKHOV, LEVI,L., Vision in Communication X-Ray Crystal-Structure Determination as a Branch of LIPSON,H., C.A. TAYLOR, Physical Optics LOHMANN,A.W., D. MENJJLOVIC,Z. ZALEVSKY, Fractional Transformations in Optics LOWS, B., see Omt, M. LUGLATO, L.A., Theory of Optical Bistability LuKS, A., see PeiinovL, V. MACHIDA, S., see Yamamoto, Y. MAINFRAY, G., C. MANUS,Nonlinear Processes in Atoms and in Weakly Relativistic Plasmas MALACARA, D., Optical and Electronic Processing of Medical Images MALACARA, D., see Vlad, V.I. MALLICK, S., see Franpon, M. MANDEL,L., Fluctuations of Light Beams MANDEL,L., The Case For and Against Semiclassical Radiation Theory MANDEL, P., see Abraham, N.B. MANUS,C., see Mainfray, G. MARCHAND, E.W., Gradient Index Lenses Optical Films Produced by Ion-Based Techniques MARTIN,P.J., R.P. NETTER~ELD, MASALOV,A.V, Spectral and Temporal Fluctuations of Broad-Band Laser Radiation MAYSTRE, D., Izlgorous Vector Theories of Diffraction Gratings MEESSEN, A,, see Rouard, P. MEHTA,C.L., Theory of Photoelectron Counting MENDLOVIC, D., see Lohmann, A.W. MEYSTRE,P., Cavity Quantum Optics and the Quantum Measurement Process. MICHELOTI?, F., see Chumash, V. MIHALACHE, D., M. BERTOLOTTI, C. SIBILIA,Nonlinear Wave Propagation in Planar Structures MIKAELIAN, A.L., M.L. TER-MIKAELIAN, Quasi-Classical Theory of Laser Radiation MIKAELIAN,A.L., Self-Focusing Media with Variable Index of ReFraction Surface and Size Effects on the Light Scattering MILLS,D.L., K.R. SUBBASWAMY, Spectra of Solids MILONNI, P.W., B. SUNDARAM, Atoms in Strong Fields: Photoionization and Chaos MIRANOWICZ, A., see Tanal, R.

111,

IV, VI, XXVI, XXIX.

1 281 331 227 65

XXXVI, 179 I, 211 XIV, XI, XVI, VI, XVI, VIII,

47 123 119 1 1 343

V. 287 XXXVIII, XXXV, XXI, XXXIII.

263 61 69 129

XXVIII, 87 =I, 313 XXII, 1 XXXIII, 261 VI, 71 11, 181 XIII, 27

xxv,

1

XXII, XXI, xv, VIII, XXXVIII, XXX, XXXVI,

145

XXXII, 313 XI, 305 XXIII, 113 1

77 373 263 261 1

XXVII, 227 VII, 231 XVII, 279 XIX, 45 XXXI, 1 XXXV, 355

552

CUMULATIVE INDEX - VOLUMES I-XXXVIII

I, 31

MIYAMOTO, K., Wave Optics and Geometrical Optics in Optical Design MOLLOW, B.R., Theory of Intensity Dependent Resonance Light Scattering and Resonance Fluorescence MURATA, K., Jnstruments for the Measuring of Optical Transfer Functions MUSSET,A,, A. THELEN,Multilayer Antireflection Coatings

1 V, 199 VIII, 201

Thermal Properties of Vertical-Cavity SurfaceNAKWASKI, W., M. OSI~SKI, Emitting Semiconductor Lasers NARDUCCI, L.M., see Abraham, N.B. N A ~ T I LK., , see Ohlidal, I. NETTER~~ELD, R.P., see Martin, P.J. NISHMARA, H., T. SUHARA, Micro Fresnel Lenses

XXXVIII, 165 1 XXV, XXXIV, 249 XXIII, 113 XXIV, 1

OmiDAL, I., K. NAVRLTIL,M. OHLfDAL, Scattering of Light from Multilayer Systems with Rough Boundaries OHLLDAL, M., see Ohlidal, I. OHTSU,M., T. TAKO,Coherence in Semiconductor Lasers OKAMOTO, T., T.ASAKURA, The Statistics of Dynamic Speckles OKOSHI, T., Projection-Type Holography OOUE,S., The Photographic Image ORRIT,M., J. BERNARD, R. BROWN,B. LOIJNIS,Optical Spectroscopy of Single Molecules in Solids OS~JSKI, M., see Nakwaski, W. G.V, Yu.1. OSTROVSKY, Holographic Methods of Plasma DiagnosOSTROVSKAYA, tics OSTROVSKY, Yu.L, see Ostrovskaya, G.V. OSTROVSKY, Yu.I., VI? SHCHEPINOV, Correlation Holographic and Speckle Interferometry OUGHSTUN, K.E., Unstable Resonator Modes OZ-VOGT,J., see Beran, M.J. OZNN,VD,, see Barabanenkov, Yu.N. PAL,B.P., Guided-Wave Optics on Silicon: Physics, Technology and Status PAOLETTI. D., G. SCHIRRIPA SPAGNOLO, Interferometric Methods for Artwork Diab-ostics PAToi<sKi,K., The Self-Imaging Phenomenon and Its Applications PAUL,H., see Brunner, W. PEGIS,R.J., The Modem Development of Hamiltonian Optics PEGIS,R.J., see Delano, E. PEPONEN,K.-E., E.M. VARTIAINEN, T. ASAKURA, Dispersion Relations and Phase Retrieval in Optical Spectroscopy PERMA, J., Photocount Statistics of Radiation Propagating Through Random and Nonlinear Media PElir~ovA,V., A. LUKE,Quantum Statistics of Dissipative Nonlinear Oscillators PERSHAN, P.S., Non-Linear Optics PETYKIEWICZ, J., see Gniadek, K. PICHT,J., The Wave of a Moving Classical Electron POPOV,E., Light Diffraction hy Relief Gratings: A Macroscopic and Microscopic View PORTER,R.P., Generalized Holography with Application to Inverse Scattering and Inverse Source Problems

XXXIV, XXXIV, XXV, XXXIV, XV, VII,

249 249 191 183 139 299

XXXV, 61 XXXVIII. 165 XXII, 197

XXII, 197 XXX, XXIV, XXXIII, XXIX,

87 165 319 65

XXXII,

1

XXXV, 197 XXVII, 1 xv, 1 1,

1

VII, 67 XXXVII. 57 XVIII, XXXIII, V, IX, V,

127 129 83 281 351

XXXI, 139 XXVII. 315

553

CUMULATIVE INDEX - VOLUMES I-XXXVIII

PRESNYAKOV, L.P., Wave Propagation in Inhomogeneous Media: Phase-Shift Approach PSALTIS, D., see Casasent, D. PSALTIS, D., Y. QIAO,Adaptive Multilayer Optical Networks QIAO,Y., see Psaltis, D.

RAYMER, M.G., LA. WALMSLEY, The Quantum Coherence Properties of Stimulated Raman Scattering RENIERI,A., see Dattoli, G. REYNAUD, S., A. HEIDMA", E. GIACOBINO, C. FABRE,Quantum Fluctuations in Optical Systems RICARD,D., see Flytzanis, C. RISEBERG, L.A., M.J. WEBER,Relaxation Phenomena in Rare-Earth Luminescence RISKEN,H., Statistical Properties of Laser Light RODDIER,F., The Effects of Atmospheric Turbulence in Optical Astronomy ROIZEN-DOSSIER, B., see Jacquinot, P. RONCHI, L., see Wang Shaomin ROSANOV, N.N., Transverse Patterns in Wide-Aperture Nonlinear Optical Systems ROSENBLW, W.M., J.L. CHRISTENSEN, Objective and Subjective Spherical Aberration Measurements of the Human Eye L., Dephasing-Induced Coherent Phenomena ROTHBERG, ROUARD, P., P. BOUSQIJET, Optical Constants of Thin Films Optical Properties of Thin Metal Films ROUARD,P., A. MEESSEN, ROUSSIGNOL, PH., see Flytzanis, C. RUBINOWICZ, A., The Miyamoto-Wolf Diffraction Wave RUDOLPH, D., see Schmahl, G. SAICHEV, A.I., see Barabanenkov, Yu.N SATSSE,M., see Courtes, G. SAITO,S., see Yarnamoto, Y. SAKAI,H., see Vanasse, G.A. SALEH,B.E.A., see Teich, M.C. SANDERS, B.C., see Hariharan, F? SCHEERMESSER, T., see Bryngdahl, 0. SCHIEVE, W.C., see Englund, J.C. SCHIRRIPA SPAGNOLO, G., see Paoletti, D. G., D. RUDOLPH, Holographic Diffraction Gratings SCHMAHL, SCHLIBERT, M., B. WILHELMI,The Mutual Dependence Between Coherence Properties of Light and Nonlinear Optical Processes Interferometric Testing of Smooth Surfaces SCHULZ,G., J. SCHWIDER, SCHULZ, G., Aspheric Surfaces SCHWIDER, J., see Schulz, G. SCHWIDER, J., Advanced Evaluation Techniques in Interferometry SCULLY, M.O., K.G. WHITNEY,Tools of Theoretical Quantum Optics SENITZKY, I.R., Semiclassical Radiation Theory Within a Quantum-Mechanical Framework SHCHEPINOV, VP., see Ostrovsky, Yu.1. SIBILIA, C., see Mihalache, D. J.R., see Dutta, N.K. SIMPSON, SIPE,J.E., see Van Kranendonk, J.

XXXIV, 159 XVI, 289 XXXI. 227 XXXI, 227 XXVIII, 181 XXXI. 321 XXX, XXIX, XIV, VIII, XIX, 111, XXV, xxxv,

1 321 89 239 281 29 279 1

XIII, 69 XXIV, 39 IV, 145 xv, 77 XXIX, 321 IV, 199 XIV. 195 XXIX, XX, XXVIII, VI, XXVI, XXXVI, XXXIII, XXI, XXXV, XIV.

65 1 87 259 1 49 389 355 197 195

XVII, XIII, xxv, XIII, XXVIII, X,

163 93 349 93 271 89

XVI, XXX, XXVII, XXXI, XV,

413 87 227 189 245

554

CUMULATIVE INDEX - VOLUMES I-XXXVIII

SIPE,J.E., see De Sterke, C.M. SITTIG,E.K., Elastooptic Light Modulation and Deflection SLUSHER, R.E., Self-Induced Transparency SMITH,A.W., see Armstrong, J.A. SMITH,D.Y., D.L. DEXTER,Optical Absorption Strength of Defects in Insulators SMITH,R.W., The Use of Image Tubes as Shutters SNAPP,R.R., see Englund, J.C. Self-Focusing of Laser Beams in SODHA,M.S., A.K. GHATAK,VK. TRIPATHI, Plasmas and Semiconductors SOROKO, L.M., Axicons and Meso-Optical Imaging Devices SPREEUW, R.J.C., J.P. WOERDMAN, Optical Atoms STEEL,W.H., Two-Beam Interferometry STEINBERG, A.M., see Chiao, R.Y. STOICHEFF, B.P., see Jamroz, W. STROHBEHN, J.W., Optical Propagation Through the Turbulent Atmosphere STROKE, G.W., Ruling, Testing and Use of Optical Gratings for High-Resolution Spectroscopy SLTBBASWAMY, K.R., see Mills, D.L. SUHARA,T., see Nishihara, H. SUNDARAM, B., see Milonni, P.W. SVELTO,O., Self-Focusing, Self-Trapping, and Self-phase Modulation of Laser Beams SWEENEY, D.W., see Ceglio, N.M. SwmEY, H.L., see Cummins, H.Z. TAKO,T., see Ohtsu, M. TANAKA,K., Paraxial Theory in Optical Design in Terms of Gaussian Brackets R., A. MIRANOWICZ, Ts. GANTSOG, Quantum Phase Properties of Nonlinear TANA~, Optical Phenomena TANGO,W.J., R.Q. RVISS,Michelson Stellar Interferometry TATARSKII, VI., VU. ZAVOROTNYI, Strong Fluctuations in Light Propagation in a Randomly Inhomogeneous Medium TATARSKII, VL, see Charnotskii, M.I. TAYLOR, C.A., see Lipson, H. TEICH,M.C., B.E.A. SALEH,Photon Bunching and Antibunching TER-MIKAELIAN, M.L., see Mikaelian, A.L. THELEN,A,, see Musset, A. THOMPSON, B.J., Image Formation with Partially Coherent Light THYAGARAJAN, K., see Ghatak, A. TONOMLJRA, A., Electron Holography TORRE,A,, see Dattoli, G. TRIPATHI, VK., see Sodha, M.S. TSWnUCHI, J., Correction of Optical Images by Compensation of Aberrations and by Spatial Frequency Filtering 'MISS, R.Q., see Tango, W.J. UPATNIEKS, J., see Leith, E.N. UPSTILL,C., see Berry, M.V USHIODA, S., Light Scattering Spectroscopy of Surface Electromagnetic Waves in Solids

XXXIII, X, XII, VI, X,

203 229 53 211 165 x, 45 XXI, 355

XIII, XXVII, XXXI, V, XXXVII, XX, IX.

169 109 263 145 345 325 73

11, 1 XIX, 45 XXIV, I

XXXI.

1

XII, 1 XXI, 287 VIII, 133 XXV, 191 XXIII, 63 XXXV, 355 XVII, 239 XVIII, XXXII, V, XXVI, VII, VIII, VII, XVIII, XXIII, XXXI, XIII,

204 203 287 1

231 201 169 1 183 321 169

11, 131 XVII, 239 1 VI, XVIII, 257

XIX, 139

555

CUMULATIVE INDEX - VOLUMES I-XXXVlll

VAMPOUILLE, M., see Froehly, C. VANDE GRIND,W.A., see Bouman, M.A. VANHEEL,A.C.S., Modern Alignment Devices J., J.E. SIPE,Foundations of the Macroscopic Electromagnetic VANKRANENDONK, Theory of Dielectric Media VANASSE, G.A., H. SAKAI,Fourier Spectroscopy VARTIAINEN, E.M., see Peiponen, K.-E. VERNIER, P.J., Photoemission Direct Spatial Reconstruction of Optical Phase from VLAD,VI., D. MALACARA, Phase-Modulated Images WALMSLEY, LA., see Raymer, M.G. WANGSHAOMN, L. RONCHI, Principles and Design of Optical Arrays WEBEQM.J., see Riseberg, L.A. WEIGELT, G., Triple-Correlation Imaging in Optical Astronomy WEISS,G.H., see Gandjbakhche, A.H. WELFORD, W.T., Aberration Theory of Gratings and Grating Mountings WELFORD, W.T., Aplanatism and Isoplanatism WELFORD, W.T., see Bassett, I.M. K.G., see Scully, M.O. WHITNEY, WILHELMI, B., see Schuhert, M. WINSTON, R., see Bassett, I.M. WOERDMAN, J.P., see Spreeuw, R.J.C. WOLTER,H., On Basic Analogies and Principal Differences between Optical and Electronic Information WYNNE, C.G., Field Correctors for Astronomical Telescopes WYROWSKI, F., see Bryngdahl, 0. WYROWSKI, F., see Bryngdahl, 0.

XX, 63 XXII, 77 I, 289 XV, VI, XXXVII, XIV,

245 259 57 245

XXXIII. 261 XXVIII, XXV, XIV, XXIX, XXXIV, IV, XIII, XXVII, X, XVII, XXVII, XXXI,

181 279 89 293 333 241 267 161 89 163 161 263

I, 155 X, 137 XXVIII, 1 XXXIII. 389

YAMAGUCHI, I., Fringe Formations in Deformation and Vibration Measurements using Laser Light XXII, 271 YAMAII,K., Design of Zoom Lenses VI, 105 YAMAMOTO, T., Coherence Theory o f Source-Size Compensation in Interference Microscopy VITI. 295 YAMAMOTO, Y., S. MACHIDA, S. SAITO,N. IMOTO,T. YANAGAWA, M. KITAGAWA, G. BJORK,Quantum Mechanical Limit in Optical Precision Measurement and Communication XXVIII, 87 T., see Yamamoto, Y. YANAGAWA, XXVIII, 87 YAROSLAVSKY, L.P., The Theory of Optimal Methods for Localization of Objects in Pictures XXXII, 145 H., Recent Developments in Far Infrared Spectroscopic Techniques YOSHINAGA, XI, 77 Yu, F.T.S., Principles of Optical Processing with Partially Coherent Light XXIII, 221 Yu, F.T.S., Optical Neural Networks: Architecture, Design and Models XXXII, 61

Z., see Lohmann, A.W. ZALEVSKY, ZAVOROTNY, VU., see Charnotskii, M.I. ZAVOROTNYI, VU., see Tatarskii, VI. ZUIDEMA, P.,see Bouman, M.A.

XXXVIII, 263 XXXII, 203 XVIII, 204 XXII, 77