Progress in Optics, Volume 39

PROGRESS IN OPTICS VOLUME XXXIX EDITORIAL ADVISORY BOARD G. S. AGARWAL, Ahmedabad, India T. ASAKURA, Sapporo, Japa...

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PROGRESS IN OPTICS VOLUME XXXIX

EDITORIAL ADVISORY BOARD G. S. AGARWAL,

Ahmedabad, India

T. ASAKURA,

Sapporo, Japan

M.V BERRY,

Bristol, England

C . COHEN-TANNOUDJT, Paris, France

v: L. GINZBURG,

Moscow, Russia

F. GORI,

Rome, Italy

A. KUJAWSKI,

Warsaw, Poland

J.

Olomouc, Czech Republic

€"A,

R. M. SILLITTO,

Edinburgh, Scotland

H.

Garching, Germany

WALTHER,

PROGRESS IN OPTICS VOLUME XXXIX

EDITED BY

E. WOLF Uniuersily of Rochesfel: N.Z, US.A.

Contributors L. ALLEN, A.A. ASATRYAN, M. BABMER, G.W. FORBES, &.A. KRAVTSOV; G. LEUCHS, T. OPATFW?, M.J. PADGETT, A. SIZMANN, D.J. SOMEFSORD, S.K. SHARMA, W. VOGEL, D.-G. WELSCH

1999

ELSEVIER AMSTERDAM. LAUSANNE .NEW YO=. OXFORD. SHANNON. SINGAPORE. TOKYO

ELSEVIER SCIENCE B,V S A R A BURGERHARTSTRAAT 25

PO. BOX 21 1 1000 AE AMSTERDAM THE NETHERLANDS Library of Congress Catalog Card Number: 61-19297 ISBN Volume XXXM: 0 444 50104 5

0 1999 Elsevier Science B.V. All rights reserved. This work is protected under copyright by Elsevier Science B.V, and the following terms and conditions apply to its use: Photocopying Single photocopies of single chapters may be made for personal use as allowed by national copyright laws. Permission of the publisher and payment of a fee is required for all other photocopying, including multiple or systematic copying, copying for advertising or promotional purposes, resale, and all forms of document delivery. Special rates are available for educational institutions that wish to make photocopies for non-profit educational classroom use. Permissions may be sought directly from Elsevier Science Rights & Permissions Department, PO Box 800, Oxford OX5 IDX, UK; phone: ( 4 4 ) 1865 843830, fax: (+44)1865 853333, e-mail: [email protected]. You may also contact Rights & Permissions directly through Elsevier’s home page (http://www,elsevier.nl),selecting first ‘Customer Support’, then ‘General Information’, then ‘Permissions Query Form’. In the USA, users may clear permissions and make payments through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA, phone: (978) 7508400, fax: (978) 7504744, and in the UK through the Copyright Licensing Agency Rapid Clearance Service (CLARCS), 90 Tottenham Court Road, London WlP OLP, UK; phone: (4) 171 631 5555; fax: (+44) 171 631 5500. Other countries may have a local reprographic rights agency for payments. Derivative Works Tables of contents may be reproduced for internal circulation, but permission of Elsevier Science is required for external resale or distribution of such material. Permission of the publisher is required for a11 other derivative works, including compilations and translations. Electronic Storage or Usage Permission of the publisher is required to store or use electronically any material contained in this work, including any chapter or part of a chapter. Except as outlined above, no part of thls work may be reproduced, stored in a retrieval system or transmitted in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without prior written permission of the publisher. Address permissions requests to: Elsevier Science Rights & Permissions Department, at the mail, fax and e-mail addresses noted above. Notice No responsibility is assumed by the Publisher for any injury andor damage to persons or property as a matter of products liability, negligence or otherwise, or fiom any use or operation of any methods, products, instructions or ideas contained in the material herein. Because of rapid advances in the medical sciences, in particular, independent verification of diagnoses and drug dosages should be made. @ The paper used in this publication meets the requirements of ANSIlNISO 239.48-1992 (Permanence of Paper). PRINTED IN THE NETHERLANDS

PREFACE In this volume five review articles are presented dealing with topics of current research interests in optics. The fist article, by Yu.A. Kravtsov, G.W. Forbes and A.A. Asatryan, is concerned with the analytic extension of the concept of geometrical optics rays into the complex domain. The extension is intimately related to inhomogeneous (evanescent) waves, which are currently of particular interest in connection with the rapidly developing area of near-field optics. The results are also relevant to investigations of wave attenuation in absorbing media, and to the understanding of light penetration into geometrical shadow regions, excitation of surface waves and propagation of Gaussian beams. The article presents the principles, with special emphasis on the physical significance of complex rays and their applications. The second article, by D.-G. Welsch, W. Vogel and T. Opatrn9, describes recent progress in the general area of quantum-state reconstruction, particularly for extracting information about the quantum state of a given object from a set of measurements. The methods are applicable to the optical field as well as to various matter waves. Some methods of processing the measured data and many of the important experiments in this area are discussed. The next article, by S.K. Sharma and D.J. Somerford, is concerned with the scattering of light in the eikonal approximation. This approximation originated in the theory of high-energy scattering processes and in the broad area of potential scattering. From the well-known analogy between a scattering potential and the distribution of the refractive index, the eikonal approximation was later adapted to the analysis of light scattering by small particles. In this article an account is given of the eikonal approximation in the context of optical scattering, and its domain of validity is discussed. The relationship of this approximation to other approximate techniques as well as some of its possible applications are considered. The fourth article, by L. Allen, M.J. Padgett and M. Babiker, concerns the orbital angular momentum of light. The orbital angular momentum is shown to be an observable quantity which can be profitably used with certain types of light beams. The phenomenological interaction of the beams with matter in bulk is reviewed and the contributions of the orbital angular momentum to the V

vi

PREFACE

dissipative and dipole forces on atoms are calculated in detail. Orbital and spin angular momentum of light are compared and contrasted. The concluding article, by A. Sizmann and G. Leuchs, presents a review of the experimental progress made in recent years in the generation of squeezed light and in quantum nondemolition measurements in optical fibers. The rich nonlinear dynamics of quantum solitons in fibers has led to the discovery of new quantum optical effects, such as intrapulse quantum correlations. The nonlinearity of optical fibers is now used to build passive fiber devices which provide all-optical functions, such as quantum noise reduction, and it is expected that active devices will allow absorption-free measurements of optical signals. The review is also concerned with these and other promising developments in this general area. Emil Wolf Department of Physics and Astronomy University of Rochester Rochestel; N a v York 14627, USA December 1998

CONTENTS I . THEORY AND APPLICATIONS OF COMPLEX RAYS by Yu.A. KRAVTSOV(Moscow. RUSSIAN FEDERATION). (SYDNEY.AUSTRALIA) G.W. FORBESAND A.A. ASATRYAN

5 1. INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1. Pioneering works 1.2. Character of wavefields described by complex geometrical optics . . . . . . . 1.3. Goals of the review . . . . . . . . . . . . . . . . . . . . . . . . . 4 2. BASICEQUATIONS OF GEOMETRICAL OPTICS . . . . . . . . . . . . . . . . . . 2.1. Eikonal, transfer, and ray equations of traditional geometrical optics . . . . . . 2.2. Rays as the skeleton for the wavefield . . . . . . . . . . . . . . . . . . 2.3. Complex form of the geometrical optics method . . . . . . . . . . . . . . 2.4. Alternative approach to phenomena described by complex rays . . . . . . . . 5 3. PROPERTlES OF COMPLEX RAYS . . . . . . . . . . . . . . . . . . . . . . . 3.1. Ray paths in the complex space . . . . . . . . . . . . . . . . . . . . 3.2. Fermat’s principle for complex rays . . . . . . . . . . . . . . . . . . . 3.3. Selection rules for complex rays . . . . . . . . . . . . . . . . . . . . 3.4. Complex rays and the saddle-point method . . . . . . . . . . . . . . . . 3.5. Complex caustics . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6. Spacetime complex rays . . . . . . . . . . . . . . . . . . . . . . . 3.7. Electromagnetic waves and complex rays . . . . . . . . . . . . . . . . . 3.8. Complex rays and uniform asymptotics . . . . . . . . . . . . . . . . . 5 4. COMPLEX RAYSIN PHYSICAL PROBLEMS. . . . . . . . . . . . . . . . . . . . 4.1. Complex rays inside a circular caustic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2. Wave reflection in a layered medium 4.3. Point source in a layered me&um . . . . . . . . . . . . . . . . . . . . 4.4. The vicinity of a caustic cusp in free space . . . . . . . . . . . . . . . . 4.5. Swallow-tail caustic: an example with four ray contributions . . . . . . . . . 4.6. Point source in a parabolic layer . . . . . . . . . . . . . . . . . . . . 4.7. Above-barrier reflection . . . . . . . . . . . . . . . . . . . . . . . . 4.8. Complex rays behind a sinusoidal phase screen . . . . . . . . . . . . . . 4.9. Surface waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.10. Reflection of inhomogeneous waves from an interface 4.1 1. Complex rays in weakly absorbing media . . . . . . . . . . . . . . . . . 4.12. Complex geometrical optics in other wave disciplines . . . . . . . . . . . . 5 5. GAUSSIAN BEAMSAND COMPLEX RAYS . . . . . . . . . . . . . . . . . . . . 5.1. Gaussian beams and complex sources . . . . . . . . . . . . . . . . . . vii

3 3 4 5 5 5 8 10 12 15 15 16 17 19 21 23 24 26 27 27 29 31 32 33 33 36 37 37 38 39 42 43 43

viii

CONTENTS

5.2. Another description of Gaussian beams in terms bf complex rays 5.3. Transformation of Gaussian beams in optical systems . . . . . 5.4. Diffraction of Gaussian beams . . . . . . . . . . . . . . . Q 6. DISTINCTIVE ASPECTS OF COMPLEX G E O ~ T I U COPTICS AL . . . . . . 6.1. Nonlocal properties of complex rays . . . . . . . . . . . . 6.2. Boundaries of applicability of complex geometrical optics . . . Q 7. CONCLUSION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ACKNOWLEDGMENTS REFERENCES. . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . 44 . . . . . . . 47 . . . . . . 49 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

50 50 51 52 53 53

I1. HOMODYNE DETECTION AND QUANTUM-STATE RECONSTRUCTION by D.-G. WELSCH (JENA,GERMANY). W. VWEL (ROSTOCK. CZECHREPUBLIC) GERMANY)AND T. OPATR@(OLOMOUC.

Q 1. INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Q 2. PHASE-SENSITIVE MEASUREMENTS OF LIGHT . . . . . . . . . . . . . . . . . . 2.1. Optical homodyning . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.1. Basic scheme . . . . . . . . . . . . . . . . . . . . . . . . 2.1.2. Quadrature-component statistics . . . . . . . . . . . . . . . . . . 2.1.3. Multimode detection . . . . . . . . . . . . . . . . . . . . . . 2.1.4. Q function . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.5. Probability operator measures . . . . . . . . . . . . . . . . . . . 2.1.6. Positive P function . . . . . . . . . . . . . . . . . . . . . . . 2.1.7. Displaced-photon-number statistics . . . . . . . . . . . . . . . . 2.1.8. Homodyne correlation measurements . . . . . . . . . . . . . . . 2.2. Heterodyne detection . . . . . . . . . . . . . . . . . . . . . . . . . 2.3. Parametric amplification . . . . . . . . . . . . . . . . . . . . . . . . 2.4. Measurement of cavity fields . . . . . . . . . . . . . . . . . . . . . . Q 3. QUANTUM-STATE RECONSTRUCTION . . . . . . . . . . . . . . . . . . . . . . 3.1. Optical homodyne tomography . . . . . . . . . . . . . . . . . . . . . 3.2. Density matrix in quadrature-component bases . . . . . . . . . . . . . . 3.3. Density matrix in the Fock basis . . . . . . . . . . . . . . . . . . . . 3.3.1. Sampling of quadrature-components . . . . . . . . . . . . . . . . 3.3.2. Sampling of the dlsplaced Fock-states on a circle . . . . . . . . . . 3.3.3. Reconstruction from propensities . . . . . . . . . . . . . . . . . 3.4. Multimode density matrices . . . . . . . . . . . . . . . . . . . . . . 3.5. Local reconstruction of P(a; s) . . . . . . . . . . . . . . . . . . . . . 3.6. Reconstruction from test atoms in cavity QED . . . . . . . . . . . . . . 3.6.1. Quantum state endoscopy and related methods . . . . . . . . . . . . 3.6.2. Atomic beam deflection . . . . . . . . . . . . . . . . . . . . . 3.7. Alternative proposals . . . . . . . . . . . . . . . . . . . . . . . . . 3.8. Reconstruction of specific quantities . . . . . . . . . . . . . . . . . . . 3.8.1. Normally ordered photonic moments . . . . . . . . . . . . . . . . 3.8.2. Quantities admitting normal-order expansion . . . . . . . . . . . .

65

69 69 70 72 77 81 86 88 90 92 93 94 95 100 101 106 108 108 115 118 119 122 123 124 128 131 133 134 137

CONTENTS 3.8.3. Canonical phase statistics . . . . . . . . . . . . . . . . . . . . 3.8.4. Hamiltonian and Liouvillian . . . . . . . . . . . . . . . . . . . 3.9. Processing of smeared and incomplete data . . . . . . . . . . . . . . . . 3.9.1. Experimental inaccuracies . . . . . . . . . . . . . . . . . . . . 3.9.2. Least-squares method . . . . . . . . . . . . . . . . . . . . . . 3.9.3. Maximum-entropy principle . . . . . . . . . . . . . . . . . . . 3.9.4. Bayesian inference . . . . . . . . . . . . . . . . . . . . . . . 5 4. QUANTUM STATES OF MATTER SYSTEMS. . . . . . . . . . . . . . . . . . . . 4.1. Molecular vibrations . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 .1. Harmonic regime . . . . . . . . . . . . . . . . . . . . . . . . 4.1.2. Anharmonic vibrations . . . . . . . . . . . . . . . . . . . . . 4.2. Trapped-atom motion . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.1. Quadrature measurement . . . . . . . . . . . . . . . . . . . . . 4.2.2. Measurement of the Jaynes-Cumrnings dynamics . . . . . . . . . . 4.2.3. Entangled vibronic states . . . . . . . . . . . . . . . . . . . . 4.3. Bose-Einstein condensates . . . . . . . . . . . . . . . . . . . . . . . 4.4. Atomic matter waves . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.1. Transverse motion . . . . . . . . . . . . . . . . . . . . . . . 4.4.2. Longitudmal motion . . . . . . . . . . . . . . . . . . . . . . 4.5. Electron motion . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.51. Electronic Rydberg wave packets . . . . . . . . . . . . . . . . . 4.5.2. Cyclotron state of a trapped electron . . . . . . . . . . . . . . . . 4.5.3. Electron beam . . . . . . . . . . . . . . . . . . . . . . . . . 4.6. Spin and angular momentum systems . . . . . . . . . . . . . . . . . . 4.7. Crystal lattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ACKNOWLEDGMENTS APPENDIX A . RADIATION FIELDQUANTIZATION . . . . . . . . . . . . . . . . . . . APPENDIX B. QUANTUM-STATE REPRESENTATIONS . . . . . . . . . . . . . . . . . . B.1.Fockstates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B.2. Quadrature-componentstates . . . . . . . . . . . . . . . . . . . . . . B.3. Coherent states . . . . . . . . . . . . . . . . . . . . . . . . . . . B.4. s-parametrized phase-space functions . . . . . . . . . . . . . . . . . . B.5. Quantum state and quadrature components . . . . . . . . . . . . . . . . APPENDIX C. PHOTODETECTION . . . . . . . . . . . . . . . . . . . . . . . . . APPENDIX D. ELEMENTS OF LEAST-SQUARES INVERSION. . . . . . . . . . . . . . . . REFERENCES

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

ix 139 143 144 145 151 153 155 157 158 159 160 163 163 167 171 173 175 175 178 179 180 182 183 183 185 187 187 189 189 190 191 192 194 195 197 200

IJI. SCATTERING OF LIGHT IN THE EIKONAL APPROXIMATION by S.K.SHARMA (CALCUTTA. INDIA)AND D.J. SOMERFORD (CARDIFF. UK)

8 1. INTRODU~ON . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2. THEEIKONAL APPROXIMATION IN NON-RELATIVISTIC POTENTIAL SCATTERING . . . . .

215 218 218 2.1. Preliminaries of the problem . . . . . . . . . . . . . . . . . . . . . . 2.2. The eikonal approximation . . . . . . . . . . . . . . . . . . . . . . . 219 2.2.1. Approximation from the Schroedinger equation . . . . . . . . . . . 219

X

CONTENTS

2.2.2. Approximation from the integral equation . . . . . . . . . . . . . . . 2.2.3. Propagator approximation . . . . . . . . . . . . . . . . . . . . 2.2.4. Physical picture of propagation in the EA . . . . . . . . . . . . . . 2.3. Scattering amplitude . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.1. Eikonal amplitude . . . . . . . . . . . . . . . . . . . . . . . 2.3.2. Glauber variant of the EA . . . . . . . . . . . . . . . . . . . . 2.4. Relationship with partial wave expansion . . . . . . . . . . . . . . . . . 2.5. Comparison with the Born series . . . . . . . . . . . . . . . . . . . . 2.6. Interpretation of the EA as a long range approximation . . . . . . . . . . . 2.7. Numerical comparisons and potential dependence of the EA . . . . . . . . . 2.8. Modified eikonal approximations: corrections to the EA . . . . . . . . . . . 2.8.1. The eikonal expansion . . . . . . . . . . . . . . . . . . . . . 2.8.2. The eikonal-Born series . . . . . . . . . . . . . . . . . . . . . 2.8.3. The generalized eikonal approximation . . . . . . . . . . . . . . . 2.9. Relationship with Rytov approximation . . . . . . . . . . . . . . . . . 0 3. EIKONAL APPROXIMATION IN OFTICAL SCATTERING . . . . . . . . . . . . . . . . 3.1. Analogy with potential scattering . . . . . . . . . . . . . . . . . . . . 3.2. Validity of scalar scattering approximation . . . . . . . . . . . . . . . . 3.3. Scattering by a homogeneous sphere . . . . . . . . . . . . . . . . . . . 3.3.1. The eikonal approximation . . . . . . . . . . . . . . . . . . . . 3.3.2. Derivation of the EA scattering function from the Mie solutions . . . . 3.3.3. Relationship with the anomalous diffraction approximation . . . . . . . 3.3.4. Corrections to the EA . . . . . . . . . . . . . . . . . . . . . . 3.3.5. Numerical comparisons . . . . . . . . . . . . . . . . . . . . . 3.3.6. One-dimensional models . . . . . . . . . . . . . . . . . . . . . 3.3.7. Backscattering in the EA . . . . . . . . . . . . . . . . . . . . 3.3.8. Vector description . . . . . . . . . . . . . . . . . . . . . . . 3.4. Scattering by an infinitely long cylinder . . . . . . . . . . . . . . . . . 3.4.1. The scattering function for normal incidence . . . . . . . . . . . . 3.4.2. Scattering by a homogeneous cylinder . . . . . . . . . . . . . . . 3.4.3. The EA from exact solutions . . . . . . . . . . . . . . . . . . . 3.4.4. Corrections to the EA . . . . . . . . . . . . . . . . . . . . . . 3.4.5. Numerical comparisons . . . . . . . . . . . . . . . . . . . . . 3.4.6. The EA as (rn - 11--* 0 approximation . . . . . . . . . . . . . . . 3.4.7. Vector formalism . . . . . . . . . . . . . . . . . . . . . . . . 3.4.8. Scattering at oblique incidence . . . . . . . . . . . . . . . . . . 3.4.9. Scattering by an anisotropic cylinder . . . . . . . . . . . . . . . . 3.5. Scattering by a coated sphere . . . . . . . . . . . . . . . . . . . . . 3.6. Scattering by a spheroid . . . . . . . . . . . . . . . . . . . . . . . . 3.7. Scattering of light by neighboring &electric spheres . . . . . . . . . . . . 0 4. APPLICATIONS OF THE EIKONAL. APPROXIMATION . . . . . . . . . . . . . . . . . 4.1. Particle sizing . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 .1. One particle at a time . . . . . . . . . . . . . . . . . . . . . . 4.1.2. Suspension of particles . . . . . . . . . . . . . . . . . . . . . 4.2. Scattering by rough surfaces . . . . . . . . . . . . . . . . . . . . . . 4.3. Plasma density profiling . . . . . . . . . . . . . . . . . . . . . . . .

220 221 222 222 222 223 224 225 226 226 227 227 228 229 229 230 231 232 233 233 236 237 238 241 247 249 251 253 254 255 257 258 259 260 261 263 263 264 266 267 268 268 268 272 273 276

CONTENTS 4.4. Light scattering by.cladded fibers . . . . . . . . . . . . . . . . . . . . 4.5. Diffraction by a volume hologram . . . . . . . . . . . . . . . . . . . . 4.6. Miscellaneous applications . . . . . . . . . . . . . . . . . . . . . . . 5 5 . CONCLUSIONS AND DISCUSSIONS . . . . . . . . . . . . . . . . . . . . . . . REFERENCES. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

xi 278 279 281 282 285

n! THE ORBITAL ANGULAR MOMENTUM OF LIGHT by L. ALLEN(COLCHESTER~ST . ANDREWS, UK). M.J. PADGETT (ST . ANDREWS. a) AND M . BABIKER(COLCHESTER, UK)

5 1. INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2 . THEPARAX~AL APPROXIMATION . . . . . . . . . . . . . . . . . . . . . . . 5 3. NONPARAXIAL LIGHTBEAMS . . . . . . . . . . . . . . . . . . . . . . . . 5 4 . EIGENOPERATOR DESCNPTION OF LASER BEAMS . . . . . . . . . . . . . . . . . 0 5 . GENERATION OF LAGUERRE~AUSSIAN MODES . . . . . . . . . . . . . . . . .

294 296 302 306 309 Q 6. OTHERGAUSSIAN LIGHTBEAMSPOSSESSING ORBITAL ANGULAR MOMENTUM . . . . . 319 5 7. SECOND-HARMONIC GENERATION ANDORBITAL ANGULARMOMENTUM . . . . . . . . 322 0 8. MECHANICAL EQUIVALENCE OF SPIN AND ORBITAL ANGULAR MOMENTUM: OPTICAL SPANNERS 324 Q 9. ROTATIONAL FREQUENCY SHIFT . . . . . . . . . . . . . . . . . . . . . . . 326 $ 10. ATOMSAND THE ORBITAL ANGULAR MOMENTUM OF LIGHT . . . . . . . . . . . . 328 Q 11. ATOMSAND MULTIPLE L A G U EGAUSSIAN R~ BEAMCON~~GLJRATIONS . . . . . . . . 342 5 12. MOTIONOF MG' M MULTIPLE BEAMCONfiGURATlONS . . . . . . . . . . . . . 345 4 13. ATOMSAND CIRCULARLY POLARIZED LIGHT . . . . . . . . . . . . . . . . . . 356 5 14. SPMaRBIT COUPLING OF LIGHT . . . . . . . . . . . . . . . . . . . . . . 363 0 15. CONCLUSIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 366 ACKNOWLEDGEMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 369 REFERENCES. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 369

V. THE OPTICAL KERR EFFECT AND QUANTUM OPTICS IN FIBERS by A. SEMANNAND G. LEUCHS(ERLANGEN. GERMANY)

Q 1. INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Q 2 . HISTORICAL PERSPEC- . . . . . . . . . . . . . . . . . . . . . . . . . . 5 3 . THEOPTICAL KERREFFECT . . . . . . . . . . . . . . . . . . . . . . . . . Q 4 . QUANTUM O m c s TN FIBERS - PRACTICAL CONSIDERATIONS . . . . . . . . . . . . 4.1. Kerr-nonlinearity and power confinement . . . . . . . . . . . . . . . . . 4.2. Optical solitons in fibers . . . . . . . . . . . . . . . . . . . . . . . 4.3. Guided acoustic-wave Brillouin scattering (GAWBS) . . . . . . . . . . . . Q 5 . QUADRATURE SQUEEZING. . . . . . . . . . . . . . . . . . . . . . . . . . 5.1. Properties of Kerr quadrature squeezed states . . . . . . . . . . . . . . .

375 377 380 388 388 389 393 397 397 5.1.1. Single-mode interaction Hamiltonian . . . . . . . . . . . . . . . . 397 5.1.2. Single-mode linearized approach . . . . . . . . . . . . . . . . . 400

xii

CONTENTS

5.1.3. Power enhancement with idtrashort pulses . . . . . . . . . . . . . 5.2. Experiments with continuous-wave laser light . . . . . . . . . . . . . . . 5.3. Experiments with ultrashort pulses . . . . . . . . . . . . . . . . . . . 5.3.1. Ultrashort pulses for GAWBS noise suppression . . . . . . . . . . . 5.3.2. Generation and detection of pulsed quadrature squeezing using a balanced Sagnac loop . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.3. Generation and detection of pulsed quadrature squeezing using a linear configuration . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.4. Experiments with solitons . . . . . . . . . . . . . . . . . . . . 5.3.5. Experiments with non-solitonic ultrashort pulses ( k ” Z 0 ) . . . . . . . 3 6. Q UANT UM NONDEMOLITION MEASUREMENTS . . . . . . . . . . . . . . . . . . 6.1. Concept and realization of a QND measurement of the photon number . . . . . 6.1.1. Cross-phase modulation as a QND interaction . . . . . . . . . . . . 6.1.2. Semiclassical approach . . . . . . . . . . . . . . . . . . . . . 6.1.3. Self-phase modulation noise in the QND measurement . . . . . . . . 6.2. Experiments with continuous-wave laser light . . . . . . . . . . . . . . . 6.3. Experiments with solitons . . . . . . . . . . . . . . . . . . . . . . . 6.3.1. Pulse preparation . . . . . . . . . . . . . . . . . . . . . . . . 6.3.2. Elimination of GAWBS noise in the QND detection . . . . . . . . . 6.3.3. Quantum noise of the probe . . . . . . . . . . . . . . . . . . . 6.3.4. Recent proposals . . . . . . . . . . . . . . . . . . . . . . . . 8 7. PHOTON-NUMBER SQUEEZING. . . . . . . . . . . . . . . . . . . . . . . . 7.1. Spectral filtering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1.1. Amplification and deamplification of quantum noise 7.1.2. Expehental apparatus and results: an overview . . . . . . . . . . . 7.2. Spectral filtering of picosecond pulses . . . . . . . . . . . . . . . . . . 7.3. Spectral filtering of sub-picosecond pulses . . . . . . . . . . . . . . . . 7.3.1. Noise reduction and enhancement as a function of fdter type and cut-off wavelength . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3.2. Noise reduction as a function of fiber length . . . . . . . . . . . . 7.3.3. Intrapulse spectral correlations . . . . . . . . . . . . . . . . . . 7.3.4. Spectral filtering of pulses in the normal group-velocity dispersion regime 7.4. Asymmetric fiber Sagnac interferometer . . . . . . . . . . . . . . . . . 7.4.1. Single-mode analysis of a Kerr-nonlinear interferometer . . . . . . . . 7.4.2. Considerations for pulsed squeezing . . . . . . . . . . . . . . . . 7.4.3. Pulsed photon-number squeezing from an asymmetric Sagnac loop . . . 5 8. FUTUREPROSPECTS. . . . . . . . . . . . . . . . . . . . . . . . . . . . ACKNOWLEDGEMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . REFERENCES. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

404 406 408 408

AUTHOR INDEX . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . SUBJECTINDEX . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . CONTENTS OF PREVIOUS VOLUMES . . . . . . . . . . . . . . . . . . . . . . . . CUMULATIVEINDEX . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

471 487 491 501

409 411 412 415 418 418 418 421 423 428 431 432 433 434 434 435 435 436 440 442 443 444 446 448 449 449 451 452 453 458 460 460

E. WOLF, PROGRESS IN OPTICS xxxD( Q 1999 ELSEVIER SCIENCE B.V ALL RIGHTS RESERVED

I THEORY AND APPLICATIONS OF COMPLEX RAYS BY

Yu.A. KRAVTSOV~, G.W. FORBES~ AND A.A. ASATRYAN’ Space Research Institute, Russian Academy of Sciences, Pvofsoyuznaya Sheet 84/32, Moscow I I781 0, Russian Federation; School of Mathematics. Physics, Computing, and Electronics, Macquarie University, Sydnq, NSW 21 09, Australia

1

CONTENTS

PAGE

0 1 . INTRODUCTION . . . . . . . . . . . . . . . . . . . . 9 2. BASICEQUATIONS OFGEOMETRICALOPTICS . . . . .

3

9 3 . PROPERTIES OF COMPLEX RAYS . . . . . . . . . . . . 9 4. COMPLEX RAYS IN PHYSICAL PROBLEMS . . . . . . .

15

0 5 . GAUSSIAN BEAMS AND COMPLEX RAYS . . . . . . . . 5 6. DISTINCTIVE ASPECTS OF COMPLEX GEOMETRICAL OPTICS . . . . . . . . . . . . . . . . . . . . . . . 9 7 . CONCLUSION . . . . . . . . . . . . . . . . . . . .

5

27 43

50 52

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

53

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

53

2

0

1. Introduction

1.1. PIONEERING WORKS

Complex rays are solutions of the ray equations of traditional geometrical optics, but correspond to extremals in the six-dimensional complex space (x’”’‘,”,y’”z’,z’’), where x’ = Re{x}, x” = Im{x}, etc. These trajectories can be used to derive both the phase and amplitude of the associated wavefield. Complex rays were first considered during the 1930s and 1940s in the theory of radio wave propagation through the lossy ionosphere (Epstein [1930a,b], Booker [1939], Bremmer [1949]), but a more general formalization was not developed until the late 1950s and early 1960s. The decisive step in understanding the analytical nature of complex rays was made by Keller [1958], who introduced the notion of complex rays to treat the area of a caustic shadow. He studied the equations for rays passing through points in the interior of a circular caustic in two dimensions and showed that such rays contact the caustic surface at complex points that lie on its analytic continuation. A year later, Seckler and Keller [19591 studied complex rays in plane-layered media, and Keller and Karal [1960] applied complex rays to the problem of surface wave excitation. Grimshaw [1968] stuhed these surface waves in more detail for particular surfaces (the sphere, the cone, and the plane with inhomogeneous impedance). Babich [ 19611 considered the analytic continuation of the wave function into the caustic shadow, and performed calculations that may be interpreted in terms of complex rays. A similar analytic continuation was also applied by Keller and Rubinow [1960] who studied eigenfunctions in both open and closed optical resonators. Complex trajectories are the quantummechanical analog of complex rays, and were studied by Maslov [1963] in connection with the quasiclassical asymptotics of solutions to the Schrodinger equation. Complex trajectories appeared there as complex solutions of the classical equations of motion in regions that are inaccessible to classical particles (i.e., areas of tunneling). Maslov [ 19641 also pointed out that complex rays may form foci and caustics. The application of complex rays within the theory of radio wave propagation through the lossy ionosphere was resumed by Budden [1961], Sayasov [1962], 3

4

THEORY AND APPLICATIONS OF COMPLEX RAYS

[I, 0 1

Budden and Jull [1964], and Jones [1968, 19701. Generally speaking, in lossy media all rays become complex because the index of refraction, which enters both the eikonal and ray equations, takes complex values. The growing interest in these ideas led to the k s t review paper on complex rays, written by Kravtsov [1967a], in which the analytic nature of complex trajectories was refined and the notion of a complex focus introduced for a Gaussian beam. The idea of using a complex point source to model a Gaussian beam was considered almost simultaneously by Deschamps [1%7, 1968, 19711, Arnaud [1968, 1969b], and Keller and Streifer [1971]. Of note also, in one-dimensional problems the method of phase integrals (see, e.g., Heading [1962]) can be regarded as a precursor to complex geometrical optics; that is, complex ray methods may be viewed as a generalization of the phase integral method for three-dimensional inhomogeneous media. Similarly, the old idea of a complex angle of refraction for the evanescent component in the case of total internal reflection is a clear ancestor of complex geometrical optics. Thus this field has a long hwtory.

1.2. CHARACTER OF WAVEFIELDS DESCRIBED BY COMPLEX GEOMETRICAL OPTICS

In complex geometrical optics the direction of wave propagation is given by the gradient of the real part of the complex phase, and the direction of exponential decay of the field’s magmtude is principally determined by the gradient of the imaginary part. In nonabsorbing media these directions are orthogonal, but in lossy media they are separated by an acute angle. In both cases mhomogeneous waves can enter, and their magnitude changes exponentially on a phase front. In fact, just as homogeneous (propagating) waves are the subject of traditional geometrical optics, inhomogeneous (or evanescent) waves can be regarded as the principal subject of complex geometrical optics (Kravtsov [1967a,b], Choudhary and Felsen [1973], Felsen [1976a]). Like tradtional geometrical optics, complex geometrical optics can involve a multiplicity of rays, so that the total wavefield is then a sum of the waves associated with each ray. As emphasized by Kravtsov [ 1967a,b], when multiple complex rays are present, selection rules are typically required to exclude nonphysical solutions. Another feature of complex rays is that, unlike real rays, they can describe nonlocal (diffraction-like) processes. A clear demonstration of their nonlocal properties is provided by the example of Gaussian beams, see 9 5. It is shown in that section that complex geometrical optics provides a complete description of a Gaussian beam (Kravtsov [1967a,b], Keller and Streifer [ 19711).

4 8 7-1

BASIC EQUATIONS OF GEOMETRICAL OPTICS

5

1.3. GOALS OF THE REVIEW

Significant progress has been made in recent years in new areas of applications for complex geometrical optics - not only in optics itself, but also in other wave disciplines including microwave physics, radio wave propagation in plasmas, and elastic wave propagation. Complex geometrical optics has now been developed into an effective tool for both applied studies and more fundamental research. Although complex rays have been shown to be useful in the analysis of a variety of wave problems, their apparent intangibility has meant that they have sometimes been viewed negatively. Ironically, a moment’s reflection shows that real rays have a similarly tenuous connection to the physical world; both real and complex rays are no more than convenient analytical frameworks for studying wave phenomena. The goal of this review is to present the fundamentals of the method of complex geometrical optics and include a range of applications and examples. Our intention is that this will lift some of the aura that can surround complex rays. Section 2 outlines the basic equations for ordinary and complex geometrical optics. Section 3 is devoted to the properties of complex rays and the selection rules associated with them. These results largely follow from the application of standard asymptotic methods (e.g. stationary phase and saddle-point methods) to the Kirchhoff solutions for wave propagation. Examples of complex trajectories in different optical and physical problems are given in $ 4 . A complex ray analysis of Gaussian beam propagation is presented in more detail in $ 5 . Section 6 touches on certain distinctive aspects of complex geometrical optics, including nonlocality and applicability. These considerations serve to give some measure of the physical significance of a complex ray.

8

2. Basic Equations of Geometrical Optics

2.1. EIKONAL, TRANSFER, Ah?> RAY EQUATIONS OF TRADITIONAL GEOMETRICAL

OPTICS

The equations of geometrical optics may be found both in texts on wave theory (e.g., Born and Wolf [ 19801) and in more specialized books such as Kravtsov and Orlov [1990]. These equations essentially relate to asymptotic forms for a scalar

6


[I, 5 2

monochromatic field (with time dependence e-"" ') that satisfies the Helmholtz equation: Au + k2&(r)u = 0.

(2.1)

Here, k = o / c and &(r) is the medium's permittivity. The refractive index is defined by the relation E(r) = n2(r) and, for a nonabsorbing medium, Im{&(r)} _= 0. An asymptotic solution can be found by writing u(r) as the product of an oscillating factor, exp[ik$~(r)], and a slowly varying amplitude factor, A(r). Expanding the latter into an asymptotic series in negative powers of i k leads to the standard form

Equations for,)I Ao, A I , etc. follow upon substituting this form into eq. (2.1) and equating the coefficients of each power of k to zero. This approach was devised by Debye in 1911 and the details are outlined in the book by Born and Wolf [1980]. Kravtsov and Orlov [1990] also examined t h s material together with an alternative, introduced by Rytov [1938], that is based on expanding A(r) as a series in the small dimensionless parameter ,u = I/(kL), where L is a characteristic scale of the variations in the medium properties or in the wave itself. Of course, these two procedures ultimately lead to equivalent equations, specifically to the eikonal equation,

and to the transport equations for amplitudes Ao, A[, etc.:

The solutions to these equations may be expressed in terms of rays, which appear as the characteristics of the eikonal equation (2.3) or, equivalently as the


Fig. 2.1. Energy conservation in an infinitesimal ray bundle is used to determine the field amplitude. This sketch also illustrates locality principle in traditional geometrical optics.

bicharacteristics of the Helmholtz equation (2.1). It is convenient to represent the ray equations in canonical Hamiltonian form: dr dP - = iVc(r), dz dz where p = V v is the "ray momentum" and z is related to arc length by d r = &/n(r). The propagation of a simple wave then begins with the initial field represented as a function of the curvilinear coordinates 5 and r] on an initial surface, written as Q, in the form -=p,

uo = U0(E, r ] ) = 4%5?rl) exp [iW0 (5, r])]

.

(2.6)

If Q is specified in the parametric form r = r 0 ( L r]),

(2.7)

then the trajectory of the ray through ro(E, r ] ) can be written as r(Z) = &E,

r],

r),

(2.8)

where R(E, q , O ) = ro(5, r]). That is, and r] single out a ray, while z specifies the displacement along that ray. These parameters are the ray coordinates. The initial value of the ray momentum follows upon differentiating v(ro) = lyo(5,r]) with respect to 5 and r]:

The third component follows from the eikonal equation (2.3), i.e., (PO)* = &(ro), together with the prescribed sense of propagation in crossing the surface to determine the branch choice. (See fig. 2.1.) It is now straightforward to use

8


[I,

02

eq. (2.5) to fully determine the ray family of the desired field. More generally, a Green’s function can be used to propagate any field and the asymptotic results from this approach give a clear framework for certain aspects of complex geometrical optics. This is discussed further in 0 3.4. 2.2. RAYS AS THE SKELETON FOR THE WAVEFIELD

Both the phase and amplitude of the wavefield in the geometrical optics approximation follow directly from the rays. If 5 and rj are now taken to select the ray that arrives at r, the eikonal, that is’V(r) of eq. (2.2), is determined by a simple integral along t h s ray: (2.10) The zeroth approximation to the amplitude is then given in terms of A:(g, r) by (2.1 1) Here, (2.12) stands for the divergence of the ray bundle which is determined by its perpendicular cross-section do1, which is shown in fig. 2.1. Alternatively, D ( i ) can be defined as the Jacobian of the transformation from ray coordinates to Cartesian coordinates: (2.13) That is, the wavefield can now be estimated as (2.14) and t h s procedure has been referred to as “sewing the wave flesh onto the ray skeleton” (Kravtsov [ 19681, Berry and Mount [ 19721).

1, 0 21


9

In traditional geometrical optics, therefore, the rays play a dual role. First, they are orthogonal to the phase surfaces v = const, so that the phase may be calculated along the rays according to eq. (2.10). Second, the rays serve as power flow trajectories, since, in the zeroth approximation, the power flux is drected along the ray. To see this, consider

I

1 21k

-(u*Vu- u V u * ) ,

(2.15)

which (in a lossless medium) exactly obeys the conservation law (2.16)

divI= 0

and serves as a measure of power flow, similar to the Pointing vector. When each A, is real-valued in eq. (2.2), the power flux is given by 1 I = A ; p + - [(A?- 240A2)p +A1VA* -AovAI] +. . . . k2

(2.17)

In the zeroth approximation, then, I = A 20 p EIO,

(2.18)

and the power conservation law becomes divIo = div(A: p) = 0.

(2.19)

This is precisely the first of the transfer equations (2.4), since p = V q and div p = A v . According to eq. (2.17), power flow is directed along the rays in loss-free media. That is, I 11 p in both the zeroth and first approximation since I - 10= O(k-*). In a lossy medium, where E is then complex in eq. (2. l), eq. (2.16) becomes d i v l = -k

E”

(uI2.

(2.20)

Here, it is apparent that E” = ImE > 0 is responsible for absorption. A simple approximation that uses the familiar rays follows for the case of weak absorption, that is, when E” 0 (area I1 in fig. 3.2), there is a complex conjugate pair of roots for E. One of the associated values for the eikonal corresponds to a decay away from the caustic, and this should be used, along with the contribution from the real ray, for the wavefield description outside the cusp. 4.5. SWALLOW-TAIL CAUSTIC: AN EXAMPLE WITH FOUR RAY CONTRIBUTIONS

A swallow-tail caustic like that presented in fig. 4.3 arises, for example, when a point source is placed in a homogeneous half-space z < 0 with E = EO that adjoins a layered medium with E ( Z ) = EO - E ~ Z for , z > 0 (Orlov [1966], Kravtsov and Orlov [1990, 19981). There exist four rays in region I, although only one ray is shown in fig. 4.3. Two real rays arrive at each point in region 11, and the other pair of rays prove to be complex. All four rays are complex in region 111. When region I shrinks to zero and ultimately vanishes, only two rays continue to be real. When the displacement of the caustic loop into the complex space is sufficiently small, a localized brightening of the field is exhibited in the real space. This is an example of the manifestation of the effects of complex caustics. tZ

(4

(b)

Fig. 4.3. Point source placed near linear layer (a) forms a caustic of swallow tail type (b) (Orlov [1966]).

4.6. POINT SOURCE IN A PARABOLIC LAYER

The permittivity profile &(Z) = Q

+ &* z2

(4.23)

gives an example where an infinitely large number of rays can reach a given point even for simple initial fields (Kravtsov [1967a]). For a barrier-like layer,

34


I

lo

X

>

Fig. 4.4. (a) A point source placed into a plane layer of parabolic profile forms a real caustic beneath the barrier and a complex caustic above the barrier, and generates an i&te number of complex rays above the real caustic. (b) A point source placed into a potential well of parabolic profile generates an infinite number of real rays, corresponding to waveguide propagation. Just a few of the rays are shown here. EO is taken to be negative and ~2 is positive. If a point source sits at (xo,zo) and zo < z-, where zk = * ( - E o / E ~ ) ' / ~ so that ~ ( q =) 0, the summit of the caustic occurs exactly at z = z- (fig. 4.4a). The ray leaving the source at an angle 8 from the z-axis is described by

x = xo + t no sin 8, z=

cos 8 sinh(EA/2t)+zo cosh(#2t),

(4.24) (4.25)

and these can be used to find t and 8 for the ray that reaches (x,z). Below the caustic, two real solutions exist, but there are none in the shadow area. However, the transcendental equations (4.24H4.25) possess an infimte number of complex solutions at all points. The infinite number of solutions may be appreciated by considering the related problem of a point source placed within a parabolic well where EO > 0 and ~2 < 0. If the point source is located in the zone where E ( Z ) > 0, the trigonometric t)replace the hyperbolic functions in eq. (4.25). functions C O S ( E ~and '~~) It is easy to see in this case that the equation for z possesses an infinite number of real roots for any points within this waveguide, and each corresponds to a real ray. A few such rays are shown in fig. 4.4b. These rays map to complex rays in the case of a potential barrier. Moreover, since the real rays form an

I , $ 41

COMPLEX RAYS IN PHYSICAL PROBLEMS

35

infinite number of real caustics in the waveguide case, it is reasonable to expect an infinite number of complex caustics for the potential barrier. To determine the wavefield at (x,z), it is first necessary to solve eq. (4.25), which is transcendental in t. An analytical solution can only be obtained for x = xo, in which case eq. (4.24) states that 8 = 0. (When x z X O , 0 is complexvalued.) With z > z+ and 8 = 0, eq. (4.25) leads to an infinite set of solutions for t:

zm=z

-i(2m+ 1)x , m= 0 ,1 ,2 ,. . ..

{In z&+

(4.26)

ZO&+&q

This result was also obtained by McLaughlin [1972] through an asymptotic calculation of the path integral. The complex-valued eikonal corresponding to tmis given by (4.27) and the associated Jacobian and amplitude satisfy

D, = -z, A,

=B

d=exp(anirn),

(4.28a,b)

[ z i e ( z ) ~ -exp(-ixm). ~/~

Here, B is the strength of the point source, that is, u x Bexp(ikR)/R, when )r- rol -+ 0. Negative values of m must be excluded so that exp(ik decreases for z > z+, and the total field behind the barrier then takes the form

R

vm)

05

A , exp(ik

u=

vm).

(4.29)

m=O

The ratio of the (m + 1)th term to the mth term is proportional to exp(-2&), where (4.30) A factor of exp(-&) accounts for the tunneling attenuation in a single pass. When & >> 1, therefore, only the leading term is needed in eq. (4.29): u(x0,z) x

B

[T:E(Z)]-'/~

exp(ikv- &),

(4.31)

36


[I, 8 4

where q is just the sum of the optical path between the source and the barrier and from the barrier to the point of observation:

(4.32) More detailed analysis of this problem was performed by Kaloshin and Orlov [1973] who noticed that the caustic behind the barrier proves to be complex. The degree of embedding of this caustic into the complex space depends on the position of the source. When zo + 0, this caustic becomes real. Holford [1981] obtained some new solutions of a reduced wave equation for layered media, and Holford's examples of tunneling phenomena may also be interpreted in terms of complex rays.

4.7. ABOVE-BARRIER REFLECTION

Analysis of above-barrier reflection may ,J performed by using an example like that in Q 4.6, but now % > 0 and &2 > 0 in eq. (4.23). The point source is assumed to be placed at (xo,zo), with zo > 0, and the field on the vertical line x = xo is then given by an expression similar to that in eq. (4.31): u(xo,z)= - i B [r;~(z)]-~'~exp(-61)

where exp(-61) is the attenuation due to above-barrier reflection and (4.34)

The path of integration in eq. (4.34) encloses the two zeros of ~ ( z )ki:-. When & and 61 of eqs. (4.31) and (4.33) are much smaller than unity, many complex rays must be taken into account. Summing the contributions of all these rays by using Poisson's formula was performed by Yakushkin [ 19691 for the case of a source placed inside a parabolic layer with EO < 0.

1,

5 41


31

4.8. COMPLEX RAYS BEHINLI A SINUSOIDAL PHASE SCREEN

Propagation of a normally incident plane wave through a sinusoidal phase screen gives a phase modulation of VO(E)= g sin(KE), (4.35) whch gives another example of a problem with an infhte number of complex rays. Near the screen, only one ray reaches each point and all other rays are complex. Beyond the first caustic cusp, two more real rays appear. As the distance from the screen increases, more of the complex rays become real. Figure 4.5 shows the increase in the number of real rays.

Fig. 4.5. The real caustics generated by a sinusoidal phase screen mean that the number of rays that reach any point depend on how far that point is from the screen. This is mapped schematically here where the Roman numerals @ve the number of real rays in a given area.

4.9. SURFACE WAVES

Keller and Karal [1960] and Grimshaw [1968] were the first to study surface waves on the basis of complex geometrical optics. As an example, suppose that the phase at the plane z = 0 in a space where E 3 1 for z > 0 is given by V O ( B= Y 5, Y > 1. (4.36) The phase velocity, Uph = c / y , is then smaller than c. This initial field readily leads to complex values of 5 (initial value of x-coordinate) and z for the ray that reaches (x, 2): (4.37)

38


The eikonal is now found from eq. (2.10) to be ~ = ~ O + r = y x + i Z ~ ~ , whch corresponds to the surface wave propagating along the x-axis: u = A exp (ik yx -

kzdm).

(4.38)

This is just one example of the result in eq. (2.24), where the complex wavevector is now

(

k = ky, k

J-1

1 - y 2 =(ksin8, kcoso),

with 8 = arcsin y = x/2 + i Arccoshy. Thus, the surface wave has a natural description in terms of complex rays, and this can be applied most simply for the case of total reflection of a plane wave. 4.10. REFLECTION OF INHOMOGENEOUS WAVES FROM AN INTERFACE

Consider the case of an inhomogeneous wave that is reflected at a perfectly conducting surface S. By requiring the sum of the incident and reflected wave to be zero at S, that is

the same relations as for a homogeneous wave result:

These expressions give initial values for Vrefl and Arefl at S, as used by Choudhary and Felsen [1973, 19741 while considering the scattering of a plane inhomogeneous wave and a Gaussian beam from a cylinder. When complex ray optics is used, these boundary conditions are appIied not only on the physical surface, but also on its analytic continuation. Such an approach was proposed by Yakushkin [19701 and developed later by Deschamps [ 19741. Yakushkin studied the 2D problem of a linear source radating near an infinitely long circular cylinder. The analytic extension of this cylinder takes the form of eq. (4.9). Complex rays from the line source can undergo multiple reflection from this extended surface and the resulting wavefield is therefore a sum of the multiply reflected waves. Just as a real ray intersects the cylinder

I , § 41


39

twice and only one of the intersections is physically significant, it is necessary to rule out some of the complex ray intercepts. Deschamps [1974] proposed rules for this purpose that are based on consideration of reflected rays in the limit of a homogeneous plane incident wave. A decomposition of the wavefield into creeping waves results by using Poisson’s formula on this infinite sum and evaluating the result at the real point (x,y). This is another type of diffraction effect that can be modelled by complex geometrical optics. At an interface of two dielectric media, the phase condition (4.40a) is still valid, but a coefficient of reflection must be introduced in eq. (4.40b):

Just as for real rays, the reflection coefficient depends on the angle of incidence, and R(8) is determined by analytic continuation. The complex angle 8 = 8’ + i0“ follows from (4.42) where p = V v is the momentum of the incident ray and n is a normal to the surface S . 4.1 1. COMPLEX RAYS

IN WEAKLY ABSORBING MEDIA

We have already seen that all the rays become complex in absorbing media. Many applications involve weakly absorbing media, however, and as indicated in $2.2, approximate solutions for the rays, eikonals, and wave amplitudes can then be obtained by perturbation of the real geometrical optics results. Bennet [1974, 19781 studied complex and real ray tracing in weakly absorbing media, as did R.M. Jones [1970], D.S. Jones [1978], Censor [1977, 19811, and Sukhy [1981]. Since some studies of complex rays in anisotropic media were already listed in 0 3.7, we consider only the scalar problem here. Different results follow, depending on whether the eikonals or the rays are hndamental to the perturbation. A perturbative solution of the eikonal equation is used in traditional geometrical optics when a mehum is close to a case for which the rays can be found in closed form, for example a weakly inhomogeneous medlum. These methods have been discussed by Kravtsov and Orlov [1990] and directly apply here if we simply take E’ to be the unperturbed permittivity and regard E” as a small

40


perturbation. In the zeroth approximation the unperturbed eikonal (ie., determined by integration along the associated real rays:

[I,

84

wo)

is

t

(4.43)

= JE‘dz, 0

where d t = &/&. It also follows from the results of section 9 of Kravtsov and Orlov [1990], that the first and second corrections are given by T

T

(4.44a,b) 0

0

where the integration is again performed along the unperturbed real ray r ( t ) . Notice that ‘1y1 is purely imaginary and % is purely real. Arsaev and Kinber [1968] derived these results for &, ql and @ directly from eqs. (2.29) by considering E” and V’’ to be small perturbations. The first correction corresponds to the exponential attenuation of the wavefield as given in eq. (2.21). The second-order correction modifies the rays themselves. If the real component of the momentum (i.e., p’ = VV’) is expanded as a series in powers of E”,

then (4.45) This result characterizes the deviation of the phase trajectory from the unperturbed direction p;, and Arsaev and Kinber [1968] used this to deduce that the phase trajectories bend in the direction of the gradient of the relative losses V(E”/E’). The details of the ray bending, however, are complicated by the fact that (VV;)’ in eqs. (4.44) and (4.45) depends not only on E”/E’ but also on the initial form of the wavefield. Since thls approach is based on the point eikonal, it is invalid at caustics and certainly cannot account for any caustics introduced by the perturbation itself. On the other hand, perturbation of the rays is not limited in this way and again proceeds by analogy with the results presented by Kravtsov and Orlov [1990],

1,

o 41

COMPLEX RAYS M PHYSICAL PROBLEMS

41

but now for weakly absorbing media. This application was discussed by Zhu and Chun [1994]. Again, the ray is expanded in powers of E”, so that we have

r( z)

= ro( z)

+ rl (z) + r2( z) + . . . ,

(4.46)

where ro(z) is the real ray. In keeping with the results given above, q(t)proves to be purely imaginary, whereas r2(2) is purely real. For the final point of the trajectory to be real, the complex ray generally originates from a complex point on the initial surface. The associated complex ray parameters 5 and r] of the point of origin follow from eq. (4.46), and the first-order shifts A5 and Au prove to be purely imaginary and proportional to the integral of E”. As a simple example, consider a plane wave incident at an angle 60 to the normal of the interface with a weakly absorbing homogeneous half space (z > 0) with permittivity E = E’ + id’, E’ >> E”. The initial phase on the interface can be written as vo= 5; sin 60, and the solutions of the ray equations take the form x = E+px z,

2

= p z z,

(4.47)

where (4.48) From eqs. (4.47) and (4.48) the initial complex point 5; for the ray that reaches a given real point of observation, say ( x , z ) , is found to be

5; = x - p x z = x - Px - z = E‘+i~’’. Pz

To first order in

el’,

(4.49)

it follows that (4.50)

Since characterizes the familiar real ray for the case of a loss-free medium, A5; = i5;“ gives the shift into complex space generated by the absorption. A focused incident field is another useful example to consider. Notice that this

42


[I,

P4

general method is valid also for the case of media with gain (E” < 0) and is well suited for numerical work since only real ray tracing is required. 4.12. COMPLEX GEOMETRICAL OPTICS IN OTHER WAVE DISCTPLINES

Complex ray methods are widely used outside optics. In fact, problems in microwave physics have driven the development of some of these methods with many valuable contributions made by Felsen and Deschamps, with their respective co-workers. The field contributions associated with complex rays for waves generated by structured antennas and the interaction of inhomogeneous and Gaussian micro-wavefields with various structures have been examined, for example, by Belanger and Couture [1983], Heyman and Felsen [1983], Montrosset and Orta [1983a,b], Ghlone, Montrosset and Felsen [1984], Ghione, Montrosset and Orta [1984], Gao and Felsen [1985], Brown [1987], Einziger, Haramaty and Felsen [ 19871, Ikuno [19871, El-Hewie [ 19881, Ikuno and Felsen [1988a,b], Maciel and Felsen [1989], Hovenac and Lock [1992], Ikuno, Ohmori and Nishimoto [1993], and Goto, Yukutake and Ishihara [1995]. An analysis of the advantages and disadvantages of complex ray methods for microwave antennas is included in the publications by Montrosset and Orta [ 1983a,b]. Another important application is to the propagation of radio waves in a plasma, and we mention a few relevant papers here. Terry [1971, 19781 developed and applied complex ray methods for these lossy systems to study ion cyclotron whtstlers in the ionosphere. Smith [1973] studied the angular diameter of the so-called “Ellis window” in the ionosphere by using a complex ray method. A summary of the techniques for ray tracing in a stratified plasma was published by Budden [ 19891, and similar applications were treated by Andrianov and Sekistov [1978] and Wang [1984]. BravoOrtega and Glasser [ 19911 applied complex geometrical optics to propagation through an inhomogeneous magnetized plasma and presented an innovative numerical treatment of the amplitude equation. A recent paper by Zernov and Lundborg [1996] on high-frequency radio wave propagation in a disturbed ionosphere accounts for diffraction by localized random inhomogeneities in the neighborhood of caustics. Elastic and visco-elastic waves in acoustics and seismology bring distinctive boundary conditions at interfaces. For example, D.S. Jones [ 19781 analyzed the acoustics of splitter plates and Westwood [1989] studied the reflection of acoustic waves from a fluid-fluid interface. Collins and Westwood [1991] compared various results with solutions generated by a complex ray model for long-range acoustic wave propagation in the ocean with large bottom slopes

1,

o 51

GAUSSIAN BEAMS AND COMPLEX RAYS

43

and mhomogeneities in sound speed. Smith and Tew [ 19951 studied ultrasonic surface wave excitation and determined the reach of complex surface rays. Zhu and Chun [1994] described an efficient method for complex ray tracing, and ray perturbation, in inhomogeneous media and provided an accurate description of wave attenuation in a realistic Earth structure. Hearn and Krebes [1990a,b] considered complex rays in viscoelastic media, and emphasized that the angle between the local direction of wave travel and the direction of maximal amplitude attenuation should be taken into account in the processing of seismograms. As mentioned in 6 3, complex classical trajectories (Maslov [1963]) enter quantum-mechanical problems involving molecular collisions (see Miller [19741) as well as barrier penetration and caustic shadows (e.g., see Nussenzveig [1992]). More recently, complex paths appeared in the problem of tunneling between two regular phase space regions that are separated by a chaotic layer (Doron and Frishat [ 19951). Although the wavefunction in quantum mechanics cannot attenuate due to absorption, complex potentials arise, for example, in inelastic neutron scattering (Schiff [1968]) or for resonant atomic tunneling through a laser beam (Zhang and Sanders [1994], Tribe, Zhang and Sanders [1996]). The imaginary part of the potential is introduced in such cases to account for loss to other channels by inelastic scattering.

8 5. Gaussian Beams and Complex Rays 5.1. GAUSSIAN BEAMS AND COMPLEX SOURCES

It is widely recognized that paraxial Gaussian beams correspond to spherical waves (cylindrical in 2D) with their center of symmetry shifted into the complex domain. For example, a point source in 2D at ro = (0,zo) generates a cylindncal wave of the form L

u(r) M -exp(ikR),

A?

where r = (x, z) and R = [(z- ~ the expression in eq. (5.1) to u(x,z) M

0

+)x2]~ 1’2.

ik(z - ZO) +

~

The paraxial approximation converts

2(2 - zo)

44

THEORY AND APPLlCATIONS OF COMPLEX RAYS

With zo = ikw; Gaussian beam: u(x, z ) =

=

[I,

55

i a , R is complex and eq. (5.2) then describes a paraxial

(z - ia)*I2

(5.3)

The beam’s waist falls in the plane z = 0 where the field is proportional to exp(-x2/2w;), and a is its Rayleigh range. For a spherical wave the field from a point source is just L

u(r) = - exp(ikR), R

(5.4)

and, with zo = i a, the paraxial approximation gives an axially symmetric Gaussian beam: (5.5)

This use of complex sources was introduced repeatedly in the late 1960’s. Kravtsov [1967a,b] showed that the 2D Gaussian beam of eq. (5.3) has its focus at ( x , z ) = (0,ikw;). Similarly, Deschamps [1967, 1968, 19711 and Keller and Streifer [1971] identified eqs. (5.3) and (5.5) as the paraxial approximations of fields from complex point sources. Arnaud [1968, 1969bl arrived at the same conclusion from a different approach. The relationship between Gaussian beams and complex point sources was later examined by Couture and Belanger [ 19811, and then in a singularity-free form by Sheppard and Saghafi [1998]. Izmest’ev [1971], Ito [1973], Shin and Felsen [1977aJ, Hashimoto [1987], and Lindell and Nikoskinen [1987] showed that it is also possible to embed multipoles in a complex space to describe higher-order Gaussian modes. It0 [ 19741 described the vectorial Gaussian beam by fields of current sources at a complex location. Wu [1985] further generalized this idea by considering the analytical extension of the Green’s function for an inhomogeneous medium. Finally, we observe that Arnaud [1984], Einziger and Raz [19871, Heyman and FeIsen [1986, 19891, and Schatzberg, Einziger and Raz [ 19881 analyzed pulsed Gaussian beams by using this approach. 5.2. ANOTHER DESCRIPTION OF GAUSSIAN BEAMS IN TERMS OF COMPLEX RAYS

An initial wavefield with a Gaussian profile can be propagated directly by applying complex geometrical optics. This analysis was -first performed by

1,s 51

45

GAUSSIAN BEAMS AND COMPLEX RAYS

Kravtsov [1967a,b] for the 2D case, and by Keller and Streifer [1971] for the 3D case, and clearly demonstrates the capabilities of complex ray methods in the modeling of diffraction effects. Suppose, in the 2D case, that the initial field is given at the plane z = 0 by u0 (5,o)= exp(-$)

=exp(-g),

where wo >> A, so a = k wi >> W O .Equation (2.6) now gives the initial conditions for the complex geometrical optics solution as

This form is appropriate for the case where a is fixed as k becomes large. If wo were to be held fixed instead, the Gaussian profile must then be taken as the initial amplitude and the eikonal vo(5)is therefore identically zero:

Traditional geometrical optics now gives parallel rays, all of the form x = E, and the eikonal corresponds to a plane wave, that is, t ) ~= z. Thus, geometrical optics yields

(

;i;)

uGo(x, z ) = uo(x,Oleikz= exp ikz - -

,

(5.9)

and this result is valid for z eq. (34)]. (After Zucchetti, PDi in the output channels [psdc(D13,023) corresponds to F ' A ~ , , A ~in Vogel and Welsch [1996].)

measurement of a phase-space function P(a;s) of the signal mode (in nonorthogonal coordinates in the phase space):

with

and s = 1 - 2q-I [@ = exp(-in/3)]. The function P(a;s = 1 - 2q-I) can be regarded as a smoothed Q function of the signal mode, which approaches the Q function as the quantum efficiency rj goes to unity. In particular, for perfect detection ( q = 1) the Q function of the signal mode is measured (in non-orthogonal phase-space coordinates).

84

HOMODYNE DETECTION AND QUANTUM-STATE RECONSTRLICTION

PI,

52

Identifying in eq. (B.25) s' with 1 - 2q-', this equation can be regarded as a prescription for obtaining other phase-space functions P ( a ; s ) from the measured one. When s < 1 - 2q-I is valid, then the /3 integration in eq. (B.25) can be performed separately to obtain P(a;s) as a convolution of P ( a ;s = 1 - 2 ~ ~with ' ) a Gaussian, which reveals that all the phase-space functions whch are typically broader than the measured one, can be obtained simply by convolving the measured distribution with a Gaussian. In the opposite case, when s > 1 - 2q-I, then the integration over y must be done fist in eq. (B.25). However, experimental inaccuracies may be exploding via the inverse Gaussian and prevent a stable reconstruction of P(a;s) with reasonable precision. Since the maximum value of s for an always stable reconstruction, i.e., s = 1 - 2q-', tends to minus unity as q goes to unity, the upper boundary of s corresponds to the Q function. As mentioned above, higher than six-port homodyne detectors can also be used in order to measure the signal-mode quantum state in the phase space. Among them, eight-port homodyne detection has been studied widely (Walker and Carroll [1984], Walker [1987], Lai and Haus [1989], Noh, Fougkes and Mandel [1991, 1992a,b], Hradil 119921, Freyberger and Schleich [1993], Freyberger, Vogel and Schleich [1993a,b], Leonhardt and Paul [1993a], Luis and Peiina [1996a]). Let us consider two modes which are superimposed, in accordance with eq. (l), by a 50%:50% beam splitter, and assume that one of the incoming modes, say, the second is in the vacuum. It can then be shown that the joint probability distribution of the two outgoing quadrature components .?i(q)and .?i(q+ n/2) is the (scaled) Q function of the !%st incoming mode 1 2 ,

(Lai and Haus [1989], Leonhardt and Paul [1993a]; for details, see also Leonhardt and Paul [ 19951, Leonhardt [1997c]). Hence, using the two outgoing modes of the beam splitter as incoming signal modes in two separate balanced four-port homodyne detectors and measuring (for a n/2 phase difference) the joint difference-event statistics of the two homodyne detectors, the Q function of the original signal mode is obtained, provided that perfect detection is

Equation (36) can be proved correct, applying the beam splitter transformation (1) and expressing the characteristic function of the two-mode (outgoing) quadrature-component distribution p(x{ ,x;, rp, rp + n/2) in terms of the characteristicfunction of the single-mode (incoming) Q function Qc.1).

85

PHASE-SENSITIVE MEASUREMENTS OF LIGHT

2’

I

rL

fi

’ 0

2-

5

D5

CI

I

3

\

L

\

3 #

X/4

I

1

D3

1 4

Fig. 6. Balanced homodyne eight-port detection scheme for measuring the Q function, using 50%:50% beam splitters and a M4 phase shifter. The signal is fed into port 1 and the strong local oscillator is fed into port 2, and vacuum is fed into the ports 1’ and 2’. The joint differenceis measured in the channels 3 and 4 (Aml) and 5 evenrprobability distribution P L \ ~ , , L \eq. ~ ?(37), , and 6 (Am2). (After Vogel and Welsch [1994].)

accomplished 13. Altogether the setup forms a balanced eight-port homodyne detector, with a signal input, a vacuum input and two local oscillator inputs. Alternatively, the beam splitter and the two four-port homodyne detectors can be combined into an eight-port apparatus with two vacuum inputs and one local oscillator input (fig. 6 ) . A straightforward calculation shows (similar to

l 3 Note that ?(q) and L(rp + d 2 ) play the same role as position and momentum of a harmonic oscillator in quantum mechanics, because of [.?(rp),i(q+ n/2)] = i. Therefore, the notations 4 = i ( q ) and j E 2(rp + n/2) are also frequently used. Equation (36) is an example of (rp) “simultaneous” measurement of a pair of conjugate quantities. Actually, the quantities 4; G i’, and 3 Li(rp + n/2) are measured, which can be regarded as the conjugate quantities il = 21 (rp) and $1 = 21(rp + n/2), respectively, “smoothed” by the introduction of additional (vacuum) noise necessary for a simultaneous (but approximate)measurement of 41 andjl . Accordingly, an additional uncertainty is introduced in the measurement, and it can be shown that the uncertainty product for 4; and?; is twice that of the measurements of 41 a n d j l made individually, Aq’,Api 2 1 (Arthurs and Kelly [1965],for a review, see Stenholm [1992]).

a;

86

HOMODYNE DETECTION AND QUANTUM-STATERECONSTRUCTION

[II, 8 2

the six-port homodyne detector) that measurement of the joint difference-event distribution PA,~,A,,,~ again yields the phase-space function P (a ;s = 1 - 277-I),

(Freyberger and Schleich [19931, Freyberger, Vogel and Schleich [ 1993a,b], Leonhardt and Paul [1993b], Vogel and Welsch [1994], D'Ariano, Macchiavello and Paris [1995], Kochahski and Wodkiewicz [1997]). Compared with the six-port scheme, PA,,,,,A~~ is the (scaled) function P (a ;s = 1 - 2q-I) in an orthogonal basis [cf. eqs. (34) and (37)]. The balanced eight-port homodyne detector was first used by Walker and Carroll in order to demonstrate the feasibility of measuring the components of the complex amplitude of optical signals, extending earlier microwave techniques (Walker and Carroll [ 19841). Finally, it was proposed to measure the Q function by projection synthesis, mixing the signal mode with a reference mode that is prepared in a quantum state such that, for appropriately chosen parameters, the joint-photon-number probabilities in the two output channels of the beam splitter realize the coherentstate projector fi(a)= n-'la)(a1for truncated signal states (Baseia, Moussa and Bagnato [ 19971). The method was first introduced to synthesize (for truncated states) the _phase-state projector fi(@) = I@)(#[ (Barnett and Pegg [1996]; cf. 53.8.3). 2.1.5. Probability operator measures

As mentioned, the reference mode with which a signal mode is mixed must not necessarily be in the vacuum state in order to obtain, in principle, all knowable information on the quantum state of the signal mode. If the reference mode is allowed to be prepared in a quantum state &, then a joint measurement of the (n/2-shifted) quadrature-components of the interfering fields can be regarded as a realization of a complex amplitude measurement (Walker [1987]). Each & implies a probability operator measure (POM) over the complex amplitude which can be characterized by a positive valued Hermitian operator:

with

K § 21

87


The joint probability density prob(a) of obtaining a result a from a measurement described by this POM l4 is prob(a) = ( f i ( a ) ).

=

Re a + i Im a

(40)

Note that from the properties of B(a) it follows that prob(a) 2 0 and J d 2 aprob(a) = 1 . The operational probability density distribution prob(a) in eq. (40), which is also called propensity (Popper [19821), can be given by a convolution of the Wigner function W ( a ) of the signal prepared in a state $ with the Wigner function F V R ( ~ )of the reference mode prepared in a state @R, prob(a) =

J’ d2PWR(P

-

a)W(P)

(41)

(Husimi [1940], Arthurs and Kelly [1965], Kano [1965], Wootters and Zurek [1979], O’Connell and Rajagopal [1982], Rajagopal [1983], Wodkiewicz [1984, 1986, 19881, Takahashi and Sait6 [1985], Walker [1987], Lai and Haus [1989], Hradil [1992], LaloviC, DavidoviC and BijediC [1992], Stenholm [1992], DavidoviC and LaloviC [ 19931, Leonhardt [1993], Leonhardt and Paul [1993b, 19951, Chaturvedi, Agarwal and Srinivasan [1994], Raymer [1994], Buiek, Keitel and Knight [1995a,b], Paris, Chizhov and Steuernagel [ 19971, Wiinsche and Buiek [ 19971). Obviously, the reference mode acts as a filter and smoothes the Wigner function of the signal. A filter system of this type is also called quantum ruler (Aharonov, Albert and Au [ 19811). It is needed in order to resolve the noncommutative quadrature components of the signal. The particular realization of the filter strongly influences the outcome of the measurement. The class of phase-space functions that can be obtained includes the s-parametrized functions with s -1. In particular, detection of the Q function implies that & = lO)(Ol, so that f i ( a ) = n-‘Ia)(al,and hence prob(a) = n-’(al@la)= Q(a). This is the case when vacuum is fed into the reference channels in the homodyne detection schemes in figs. 5 and 6, and the signal is mixed with it. If the signal is mixed with a squeezed vacuum, then the POMs of the form given in eq. (38) with & = 5(E)lO)(Olkt(E) can be realized

-1 the s-parametrized phase-space functions P ( a ; s ) (see Appendix B.4) do not necessarily exist as positive functions, and It should be noted that these POMs can also be realized in heterodyne detection (3 2.2) (Yuen and Shapiro [1980], Yuen [1982]). Further, they can be realized in an unbalanced homodyne scheme with vacuum input but not equal-part signal-beam splitting (Leonhardt [1993, 1997~1). l 6 The ten-port scheme can be regarded as a reduced twelve-port scheme, the latter being suited for measuring simultaneously the Stokes parameters of a two-mode field. l 7 For a direct measurement of the photon-number statistics, see $2.1.7.

11, § 21


89

for s > 0 they are not necessarily well-behaved. The latter also applies to the P function, which is used widely to calculate averages of normally ordered (i.e., measurable) quantities. In order to avoid using singular functions, generalized P representations may be used (Drummond and Gardiner [1980]; see also Gardiner [1983, 1991]), such as the positive P function1*,

The possibility of measuring the single-mode Q function in perfect (sixport or eight-port) homodyne detection also offers the possibility of measuring the quantum state of the mode in terms of the positive P distribution using more involved multiport homodyne detection schemes (Agarwal and Chaturvedi [1994]). Let us again consider a signal mode and a reference mode which are superimposed by a 50%:50% beam Splitter and assume that the reference mode is in the vacuum. It can then be shown that the joint Q function of the two outgoing modes, Q(al ,a2), is related to the Q function of the signal mode, Q(a),as l9

which reveals that Q(a1, a2) is nothing but the (scaled) positive P function of the signal mode,

(

e ( a l , a2) = 4~ a = &al,

p=ha;) .

(47)

Hence, if each of the two output modes of the beam splitter is used as an input mode of a multiport apparatus (such as the six- or eight-port homodyne detector outlined in 0 2.1.4) that measures the Q function, then measurement of the joint Q function of the two modes yields the positive P function of the signalmode under study. Needless to say, for imperfect detection a smeared positive P function is measured. The positive P function is an example of a measurable phase-space function2’ that is defined as a fimction of two complex amplitudes a and p (per mode).

I8

Note that

6 = s d 2 a s d2a’ la)(a’*I ((a’*la))-IP(a,a’) in this representation.

’’Equation (46)can be proved correct, applying the beam splitter transformation (1) and expressing

the characteristic function of the two-mode (outgoing) Q function Q(al,a2) in terms of the characteristic function of the signal-mode (incoming) Q function Q(a).Note that @ a , ,9 )is the two-mode s-parametrized phase-space function P( al ,a2;s = -1). For a method suggested to measure the positive P function of a quantum-mechanical particle, see Braunstein, Caves and Milburn [1991].

’’

90

HOMODYNE DETECTION AND QUANTUM-STATE RECONSTRUCTION

[II, § 2

The concept can also be extended - similar to 9 2.1.5 - to other than vacuum reference inputs (in the beam splitter and/or the multiport homodyne detectors) in order to obtain generalized phase-space functions defined in a four-dimensional phase space. A simple example is the smeared positive P function mentioned above. 2.1.7. Displaced-photon-number statistics Let us return to the four-port homodyne detection scheme (fig. 1) and answer the question of which quantity is measured when a signal mode is mixed with a local oscillator mode by an unbalanced beam splitter and only a single-channel homodyne output is measured. We rewrite eq. (10) as 2;

=

UkI(2-a) = Uklb((X)2bt(a),

(48)

where

and apply the photocounting formula (C.5).The probability of detecting m events in the kth output channel can then be given by

where ;(a) = b ( a ) i b t ( a )is the displaced photon-number operator of the signal mode, and q = I Ukl l2q0 (with rlr, being the quantum efficiency of the detector used). Equation (50) reveals that the observed probability distribution P,,, is nothing but the displaced photon-number distribution of the signal field measured with quantum efficiency r]. In particular, when a signal and a strong local oscillator, laL[+ 00, are mixed by a beam splitter with high transmittance, I UII I = I U22 I + 1, and low reflectance, IU21 I = I U12 I + 0, such that the product I U12I IaLI is h t e , then for high quantum efficiency (q + 1) the displaced photonnumber probability distribution of the signal is measured (Wallentowitz and Vogel [1996a], Banaszek and Wodkiewicz [1996], Paris [1996a]):

P,,

+

p d a ) = (m,alcjlm, a ) ,

( 5 1)

where J m , a ) = b(a)lm) are the displaced photon-number states. It should be noted that for chosen m the quantity p , ( a ) as a function of a can be

11, § 21


91

regarded (apart from the factor m-‘) as a propensity for the signal-mode complex amplitude, which can also be measured in multiport homodyning (see 0 2.1.5). For chosen a it is an ordinary probability for the displaced signal-mode photon number, which can already be obtained from the four-port detector out%ined here. In order to obtain in this scheme p , ( a ) as a function of a, a succession of (ensemble) measurements must be performed. Hence, measurement of the displaced-signal-mode photon-number statistics as a function of the complex parameter a is equivalent to measurement of the signal-mode quantum state, and it is expected that it yields more data than the minimum necessary for reconstructing it (8 3). In contrast to balanced homodyning, measurement of the displaced photonnumber statistics in unbalanced homodyning requires highly efficient photodetectors which can discriminate between n and n + 1 photons in order to resolve the discrete nature of the photon number. Presently, such detectors are not available. Photomultipliers and streak cameras can discriminate between single photons, provided that the field does not contain more than about 10 photons, but the quantum efficiency of about 10-20% is extremely low. Currently available avalanche photodiodes operating in the Geiger regime may reach about 80% quantum efficiency (Kwiat, Steinberg, Chiao, Eberhard and Petroff [ 1993]), but they do not discriminate between single photons. They can only indicate the presence of photons, because of saturation. The problem may be overcome using multichannel coincidence-event measurement techniques, also called photon chopping. In particular, it was proposed to use highly efficient avalanche photodiodes and a beam splitter array to divide the number of readout photons among the photodiodes, so that none is likely to receive more than one photon (Ho, Lane, La Porta, Slusher and Yurke [1990], Song, Caves and Yurke [1990], Paul, TO&, Kiss and Jex [1996a])21. Alternatively, it was proposed to directly defocus the field to be measured onto an array of photodiodes (Wallentowitz and Vogel [1996a]). Let us assume that the mode to be detected enters one of the input ports of a balanced linear 2N-port apparatus (the other input ports being “unused”), and multiple coincidences are measured at the output, placing avalanche photodiodes in the N output channels. If the signal mode contains less than N + 1 photons, then there is a one-to-one correspondence between the measured coincidence-

For detection of squeezing via coincidence-event measurement, see Janszky, Adam and Yushin [1992]; for Fock state detection and preparation, see Paul, Torma, Kiss and Jex [1996b].

21

92


[II, 0 2

event distributionP , ( N ) and the photon-number distributionp n of the signal. To be more specific, it can be shown that 22

n=m

where

and pmln(N)= 0 if n < m (Paul, Torma, Kiss and Jex [1996a]). Here, pmln(N) is the probability of registration of m clicks under the condition that n photons are present. The probabilities pmlq(N)form an upper triangular matrix A,,, = pml,(N)which can be inverted in order to calculate p , from j , ( N ) , N

m=n

2.1.8. Homodyne correlation measurements

As already mentioned, the accuracy with which a (quantum) field can be measured by a homodyne detector is limited by the overall quantum efficiency of the device [see, e.g., eqs (19) and (20)]. To overcome this limitation, one may perform homodyne correlation measurements (Ou, Hong and Mandel [ 1987b1). In contrast to ordinary balanced homodyning, the (time-delayed) intensity correlation between the two outgoing fields is measured. In particular, the information on squeezing is obtained from the time dependence of the measured correlation function. Since the measured coincidence events are proportional to the product of the detection efficiencies of the two detectors, small quantum efficiencies may reduce the measured signal (which could be compensated by longer measurement times), but do not smooth out the desired information. However, a drawback of the method is that the classical noise of the local oscillator is not balanced out; i.e., even small relative fluctuations of the (strong) local oscillator may prevent the quantum noise effects of a weak signal from being measured.

22

Here it is assumed that the balanced 2N-port realizes a unitary transformation UN = U2 @ U N / ~ .

11, § 21


93

It was therefore proposed to use a. weak local oscillator whose intensity is comparable to that of the signal (Vogel [ 1991, 19951). In this case, the classical noise of the (highly stablized) local oscillator may be reduced below the level of the quantum fluctuations of the signal. Moreover, simultaneous measurement of different kinds of correlation hnctions of the signal field is possible. To illustrate this, let us consider the difference between the measured second-order intensity correlation function G(2)(t, t + t) for short and long delay times t, AG'2'(t)= G'2'(t,t ) - r+m lim G'2'(r, t + t),

(55)

and restrict our attention to stationary fields, so that the time argument t can be omitted. Decomposing AG(*) with respect to powers of the local-oscillator amplitude EL, one may observe the following effects. The zeroth-order term yields the normally ordered intensity ( I ) fluctuation of the signal,

The second-order term is related to the normally ordered electric-field variance of the signal,

where k(cp)corresponds to the quadrature-component operator i ( c p ) , q being the phase difference between signal and local oscillator. Eventually, the first-order term represents the correlation between the two signal-field observables,

Note that all these quantum-statistical moments can be separated from each other by using their dependences on the phase shift cp (for the measurement of the corresponding spectral properties, see Vogel [ 19951). 2.2.

HETERODYNE DETECTION

It is worth noting that multiport homodyning for measurement of the complex amplitude is equivalent to (four-port) heterodyne detection (Yuen and Shapiro [1980], Yuen [1982], Yuen and Chan [1983], Shapiro [1985], Shapiro and Wagner [1984], Caves and Drummond [1994]). In the scheme, an optical field is combined, through a beam splitter, on the surface of a photodetector with a

94


[H, 8 2

strong local oscillator whose frequency q is offset by an amount Aw from that of the signal mode in the optical field (Aw 0). Here, rapid variations of the quadrature-component distributions which correspond to frequencies higher than the cut-off frequency z, are suppressed; i.e., the exact distributions are effectively replaced with somewhat smeared ones. In the pioneering experimental demonstration of the method also called optical homodyne tomography (Smithey, Beck, Raymer and Faridani 11 9931, Smithey, Beck, Cooper, Raymer and Faridani [1993]), a pulsed signal field is superimposed with a pulsed coherent-state field much stronger than the signal. The quadrature-component distributions are measured for a squeezed signal field and a vacuum signal field. The squeezed field is generated by using a walk-off compensated, travelling-wave optical parametric amplifier. The generated downconversion signal centered at 1064 nm consists of two orthogonally polarized fields, the signal and idler, and has a bandwidth estimated to be lo4 times that of the laser pump (532 nm, 300 ps, 420 pulses per second). The laser pump and the local oscillator field (1064 nm, 400 ps) are obtained from a common laser source, and each local-oscillatorpulse contains a mean number of photons of about 4 x lo6. The interfering fields are detected with high quantumefficiency (- 85%) photodiodes, and the resulting current pulses are temporally integrated and subtracted. The measurements and reconstructions are performed for a squeezed signal field and for a vacuum signal field, figs. 9 and 10, the field mode detected being selected by the spatial-temporal mode of the localoscillator field. The method of optical homodyne tomography was also extended subsequently to the continuous wave-regime, including squeezed vacuum with a high degree of quantum noise reduction (Breitenbach, Muller, Pereira, Poizat, Schiller and Mlynek [ 19951, Schiller, Breitenbach, Pereira, Muller and Mlynek [ 19961) and bright squeezed light (Breitenbach and Schiller [ 19971, Breitenbach, Schdler and Mlynek [ 19971). As mentioned in 8 2.1.2, the measured quadrature-component distributions do not correspond, in general, to the true signal mode, but they must be regarded as the distributions of a superposition of the signal and an additional noise source

11, § 31

103

QUANTUM-STATE RECONSTRUCTION

h

0.8

**.

X a

**=.

0.6

0.3

'

0.00

0.79

1.57

236

0.6

I 03

3.14

Phase q Fig. 9. (a) In balanced four-port homodyne detection measured quadrature-component distributions at various values of the local-oscillator phase [P&,) corresponds to p(x, rp)]. (b) Variances of quadrature components vs local-oscillator phase: circles, squeezed state; triangles, vacuum state. In the experiment 4000 repeated measurements of the photoelectron difference number at 27 values of the relative phase rp are made. (After Smithey, Beck, Raymer and Faridani 119931.)

[eq. (21)], because of non-perfect detection. Substituting in eq. (70) for p(x, ip) the measured distributions p ( x , ip; q) with q < 1 [eq. (19)] and performing the inverse Radon transform on them, the Wigner function of a noise-assisted signal field is effectively reconstructed. Equivalently, the reconstructed Wigner function can be regarded as an s-parametrized phase-space function of the true signal field, however with s < 0. The characteristic function Y ( z , ip; q) ofp(x, ip; q ) typically measured when q < 1 and the characteristic function @ ( p ; s ) of the phasespace function P(a;s)are related to each other according to eq. (3.27), with Y ( z ,ip; q) in place of Y ( z ,47) and s - 1 + q-' in place of s in the exponential [cf. eq. (23) and footnote 71. Hence, making in the exponentials in eqs. (B.28) and (B.29) for s the substitution s - 1 + 8'yields the relations between P(a;s)

104

tQ § 3


and p(x,cp; q). In particular, eq. (B.29) can be used to obtain any phase-space function P(q,p;s) = 2-‘P[a = 2-”2(q + ip); s] from the measured quadraturecomponent distributions in principle30:

x IzI exp[iz(q cos cp + p sin cp - x)]p(x,cp; q)

1

(74)

.

Obviously, when s = 1 - q-’, then eq. (74) takes the form of eq. (71); i.e., replacing in eq. (70) p(x, cp) with p(x,cp; q), inverse Radon transformation yields the signal-mode phase-space function P(q,p;s = 1- q-‘) in place of the Wigner function,

p(x,cp; q) =

/

dy P(x cos cp - y sin cp, x sin cp + y cos cp; s = 1 - q-I).

(75)

The proposal was made to combine squeezing and balanced homodyning such that generalized quadrature-components

for all real parameters p and Y can be measured, and to reconstruct the quantum state from the corresponding distributions p ( X , ,u,Y) (Mancini, Man’ko and Tombesi [1995], D’Ariano, Mancini, Man’ko and Tombesi [1996], Man’ko [1996], Mancini, Man’ko and Tombesi [1997])31.In fact, i ( p , Y) can be related to i ( c p ) since it represents a scaled quadrature-component,

When s > 1 - q-’, then in eq. (74) an inverse Gaussian occurs which may lead to an artifical enhancement of the inaccuracies of the measured data, so that a stable deconvolution might be impossible and the noise dominates the reconstruction of P ( q , p ; s ) . This effect is not observed when s < 1 - q-’ , and a stable reconstruction with reasonable precision of P(a;s) for s 6 1 - 8’ may therefore be expected to be feasible for any quantum state (see also $3.9). In particular, reconstruction of the Q function is always possible if q > 1/2. When s < 1 - q-’, then in eq. (74) the z integral can be performed first to obtain P ( q , p ; s )in a form suited for application of sampling techniques (0 3.3.1):P ( ~ , Ps); = :J drp J d r K ( q , p , x ,rp; s; V ) P ( X , rp; v), with K(q,p,x, rp;s; 11) being a well-behaved integral kernel. 3 1 For an application of the method (also called symplectic tomography) to trapped-ion quantum state reconstruction, see Mancini, Man’ko and Tombesi [1996], Man’ko [1997]. 30

105


?

2.0

2.0 -

I

1 .o

1.0.

;i

& 0.0 -

-1 .o -

-1 .o

~

-%:o

-1.0

0:o

X

1.0

2.0

-2-90 -1.0

0:o

X

1.0

2.0

Fig. 10. Wigner distributions reconstructed from the measured quadrature-component distributions for (a,h) a squeezed state and (c,d) a vacuum state, viewed in 3D and as contour plots, with equal numbers of constant-height contours [ W ( X ,P ) corresponds to W(q,p)].The reconstruction is performed by using inverse Radon transformation according to eq. (72). (After Smithey, Beck, Raymer and Faridani [ 19931.)

with

(cf

9 2.1.2 and Appendix B), which implies that

Hence, measurement of a particularly scaled quadrature-component distribution by means of an ordinary homodyne detector already yields all scaled quadrature components. Performing the analysis with a variable scaling parameter, it can be

106


[II, 5 3

shown that eqs. (70) and (71), respectively, can be given in terms of p ( x , p, Y ) by32 p ( x , p, Y ) =

‘J J J dk

dq

dp e-ik(x-qlc-pv) W (4 , P ) ,

and

Comparing eq. (81) with eq. (71), we see that in any case a three-fold Fourier transformation is required in order to obtain the Wigner function from the homodyne data. Note that z in eq. (81) can be chosen arbitrarily, which reflects the above mentioned fact of overcomplete data. 3.2. DENSITY MATRIX IN QUADRATURE-COMPONENT BASES

The reconstructed Wigner function can be used in order to calculate the density matrix in a quadrature-component basis (Smithey, Beck, Raymer and Faridani [ 19931, Smithey, Beck, Cooper, Raymer and Faridani [19931). The definition of the Wigner function given in Appendix B.4 can be rewritten, on expanding the density operator in a quadrature-component basis, as

with x = q cos cp + p sin cp,

y

= -q

sin Q, + p cos q,

(83)

which for q = 0 is nothing but the well-known Wigner formula (Wigner [ 19321). Hence the density matrix in a quadrature-component basis can be obtained by Fourier transforming the Wigner function: (x - X I , cpIP[x+ X I , cp)

=

J

dy e-2‘y’J’W(xcos cp - y sin cp,x sin cp +y cos cp).

Note that eq. (84) reduces to eq. (70) for x’

32 For

(84) =

0.

a detailed discussion of the transformation properties, see also Wiinsche [1997].

K 9 31


107

It is worth noting that the reconstryction of the density matrix from the homodyne data can be accomplished with two Fourier integrals, avoiding the detour via the Wigner function (Kiihn, Welsch and Vogel [1994], Vogel and Welsch [ 19941). Writing the density-matrix elements as

‘S

(x-x’, qI@Ix+x’,q) =

dze-irzY(z,x’,q),

(85)

the characteristic function Y(z,x’, cp) can be shown to be the characteristic function of a quadrature-component distribution, Y(z,x’, 9)= Y(Z, @I,

(86)

with 1/2

Z=Z(z,x’)= [ z ~ + ( ~ x ’ ) ~ ] and

@=q-$n+arg(2x’+iz).

(87)

Hence, the density-matrix elements can be obtained from the quadraturecomponent distributionsby means of a ’simple two-fold Fourier tran~formation~~ :

It is worth noting that eq. (88) can be used to obtain the density matrix in different representations, by varying the phase of the quadrature component defining the basis. Further, eq. (88) can also be extended, in principle, to imperfect detection, expressing Y(z, q) in eq. (85) in terms of Y ( z ,q;9) (cf. footnote 7; -for details, see Kiihn, Welsch and Vogel [1994], Vogel and Welsch [1 9941). For the numerical implementation of the reconstruction based on eq. (88) spline-expansion techniques can be used (Zucchetti, Vogel, Tasche and Welsch [19961)34, (x--x’, V I @ I X + X ’ , q) N CKrnn(x,x’, q)p(Zrn+l,@n+l),

(89)

m,n

where Krnn(x,x’,q) = (2n)-1 ~ & & , m [ i ( & + l , x ’ )BA,,,(Sk+l)~b,k(-X), l

(90)

k

with z, = -2x’cot(@, - q) [q...Jk), Fourier transform of B(...)(x)]. As an illustration of this method, in fig. 11 the reconstructed density matrices in the For the two-fold Fourier transformation that relates the density-matrix elements to p ( x , p , v), eq. (SO), see D’Ariano, Mancini, Man’ko and Tombesi [1996]. 34 Choosing a finite set of nodes { x , } , an approximate spline function f , ( x ) of f ( x ) is given by = C , f ( x , + ~ f B ~ ~ , , , ~where ( x ) , gAxn,n(X) = ( x - x,)/Ax, if x , < x < x,+l, B A ~ ~ , , L =~) (x,+2 -x)/Ax,+l if x,+l < x Q x , , ~ , and BA,,,(x) = 0 elsewhere, and Ax, = x,+l - x , ; for mathematical details of spline expansion, see de Boor [1987]. 33

108


[II,

03

Fig. 1 1 . (Real) density matrix (n,rp1&’,rp) of a squeezed state reconstructed from measured quadrature-component distributions in (a) the ‘‘position” basis ( q = 0) and (b) the “momentum” basis (rp = d 2 ) . The homodyne data were obtained by T. Coudreau, A.Z. Khoury and E. Giacobino, using the experimental setup reported by Lambrecht, Coudreau, Steinberg and Giacobino [1996]. In the experiment, the quadrature-compoventswere measured at 48 phases, and at each phase 7812 measurements were performed.

“position” basis, cp = 0, and the “momentum” basis, cp = n/2, of a squeezed vacuum state are shown. The homodyne data were recorded by T. Coudreau, A.Z. Khoury and E. Giacobino. In the underlying experimental scheme the squeezing effect is obtained in a probe beam that interacts with cold atoms in a nearly single-ended cavity (Lambrecht, Coudreau, Steinberg and Giacobino [ 199611. The phases cp = 0 and Q, = n/2 which define the quadrature-component bases in fig. 11 coincide with the phases of minimal and maximal field noise, respectively. 3.3. DENSITY MATRIX IN THE FOCK BASIS

Stimulated by the tomographic reconstruction of the Wip e r function (9 3.1) much effort has been made to obtain the density matrix in the Fock basis from measurable data as direct as possible. Let us again start with analysing balanced four-port homodyning. 3.3.1. Sampling of quadrature-components

The problem of reconstruction of the density matrix in the Fock basis from the quadrature-component distributions can be solved, in principle, by relating the density-matrix elements to derivatives of the Q function (cf. 0 3.3.3) and expressing the Q function in terms of the quadrature-component distributions, using eq. (B.29), with s = -1 (D’Ariano, Macchiavello and Paris [1994a,b]). An equivalent formalism, which is suited for practice and which can also be applied

11,

8 31


109

to the reconstruction of the density matrix in other than the photon-number basis, is based on the expansion of the density operator as given in eq. (B.31). In the photon-number basis, this equation reads as 35

The integral kernel (also called pattern function)

with

is studied in detail in a number of papers (D’Ariano [1995], D’Ariano, Leonhardt and Paul [1995], Leonhardt, Paul and D’Ariano [1995], Leonhardt, Munroe, Kiss, Richter and Raymer [1996], Richter [1996a], Wunsche [1997]). The function fmn(x)(fig. 12) is well behaved, and it is worth noting that it can be given by (Richter [1996a], Leonhardt, Munroe, Kiss, Richter and Raymer [1996])

(x) for En, > En,where Wrn(x) and and qm(x) are the for Em < En, andfmn(x) =Em regular and irregular solutions of the harmonic-oscillator Schrodinger equation for a chosen energy value En,3 6 . Equation (91) reveals that the density matrix in the Fock basis can be sampled directly from the measured quadrature-component statistics, since emncan be regarded as a statistical average of the (bounded) sampling function Kmn(x,q) (D’Ariano, Leonhardt and Paul [1995], Leonhardt, Paul and D’Ariano [1995], Leonhardt, Munroe, Kiss, Richter and Raymer [ 19961, Leonhardt [ 1997~1, D’Ariano [ 1997a1). In an experiment each outcome x of .?(q),with q E [0,n), contributes individually to so that @,n,nn gradually builds up during the data collection. That is to say, emflcan be sampled from a sufficiently large set

35 For the relation between the density-matrix elements and p(x, p, v), eq. (80),see D’Ariano,

Mancini, Man’ko and Tombesi [1996]). 36 Strictly speaking, the irregular ( i t . , not normalizable) function rp,(x) must be chosen such that Qnc& QAcp, = Z/n (for details, see Leonhardt [ 1997~1).Note thatfmn(x)is not determined uniquely.

110


P, § 3

06 01 u2

0 42 -0.4

.O 6 0

2

J

ro

.a.75

2

4

Fig. 12. Pfots of some typical pattern functions fmn(x) (lines) along with the products of regular x) lines). In (a) the diagonal pattern function with n = m = 4 is wave functions v n ( x ) ~ , , , , , ( (dashed depicted. Oscillations between -2/n and +2/nare clearly visible in the classically allowed region. Then the function swings over and decays like x-*. In (b) the off-diagonal pattern function with n = 1, m = 4 is depicted. It is less oscillating than the diagonal pattern h c t i o n and it decays faster in the forbidden zone. In (c) and (d) highly oscillating pattern functions are shown for (c) n = rn = 25 and (d) n = 20, m = 50. (After Leonhardt, Munroe, Kiss, Richter and Raymer [1996].)

of homodyne data in real time, and the mean value obtained from different experiments can be expected to be normal-Gaussian distributed around the true value, because of the central-limit theorem. Moreover, the sampling method can also be used to estimate the statistical error (see also § 3.9.1). Experimentally, the method was applied successfully to the determination of the density matrix of squeezed light generated by a continuous-wave optical parametric amplifier (Schiller, Breitenbach, Pereira, Muller and Mlynek [ 19961, Breitenbach and Schiller [ 19971, Breitenbach, Schiller and Mlynek [ 19971). In the experiment the spectral component of the photocurrent in a small band around a radiofrequency SZ is measured (overall quantum efficiency 82%). In this case the measurement is on a two-mode quadrature-component i ( q ) = eiQR[2(w + Q)e-'P+ 2(w - Q)e'"], 4 R being the phase of the

-

K 8 31


111

Photon number n Fig. 13. Photon-number distribution of a squeezed vacuum and the vacuum state (inset) reconstructed from the quadrature-componentdistributions according to eq. (91). Solid points refer to experimental data, iustograms to theory. The experimentally determined statistical error is 0.03. (After Schiller, Breitenbach, Pereira, Miiller and Mlynek [1996].)

radio-frequency local oscillator, so that the scheme is basically a heterodyne detector. Examples of reconstructed diagonal density-matrix elements are shown in fig. 13 (for reconstructed off-diagonal density-matrix elements, see, e.g., Schiller, Breitenbach, Pereira, Muller and Mlynek [ 19961, Breitenbach and Schiller [ 19971). Further, the method was used successfully to measure the time-resolved photon-number statistics of a 5 ns pulsed field with a sampling time of 330 fs, set by the duration of the local-oscillator pulse (Munroe, Boggavarapu, Anderson and Raymer [1995]). From eqs. (91) and (92) it is easily seen that the photonnumber probability distribution p n = &, can be given by

where p(x) = (2n)-' J dq,p(x, q) is the phase-averaged quadrature-component distribution. Equation (95) reveals that for sampling the photon-number statistics the phase need not be controlled - a situation that is typically realized when the signal and the local oscillator come from different sources. In the experiment, an argon-laser-pumped Tixapphire laser is used in combination with a chirped-pulse regenerative amplifier to generate ultrashort, transform limited local oscillator pulses (330 fs) at a wavelength of 830 nm and a

112

PI, § 3


.I0

-6

0

5

Quadrature amplitude

5

to

:::B, 0 a1

0 00 0

10

0

20

30

Pholon number n

Photon 10 number 20 30 n

Fig. 14. Measured phase-averaged quadrature-component distributions for (a) t = 4.0ns and (b) t = 6.0ns, and the resulting time-resolved photon-number distributions for ( c ) t = 4.011s and (d) t = 6.0ns, obtained from (a) and (b) by using eq. (95). P(E) andp(n), respectively, correspond to p ( x ) andp,. (After Munroe, Boggavarapu, Anderson and Raymer [1995].)

repetition rate of 4 kHz with approximately lo6 photons per pulse. The signal is from a laser diode of wavelength 830 nm and pulse width 5 ns. Figure 14 shows examples of the measured (with 65% overall quantum efficiency) quadrature-componentprobability distributions and the resulting photon-number probability distributions at two times in the signal pulse. In particular, the photonnumber statistics are seen to change from nearly Poissonian statistics (laser above threshold) to thermal-like statistics (laser below threshold) (for ultrafast homodyne detection of two-time photon-number correlations, see 8 3.8.1). So far in the formulas, perfect detection has been assumed. The problem of extending eq. (91) to imperfect detection such that p(x, q) and iU,,(x, cp), respectively, are replaced with p(x, QJ; r ] ) and a sampling function K,,(x, cp; r ] ) that compensates for the losses has also been considered37,

-

~ m n ( x ,QJ; r ] ) =

(4kQJ; r>l.),

(96)

where

37 Equation (96) follows

see footnote 7.

from eqs. (B.31) and (B.32), expressing Y ( z ,QJ) in terms of Y ( z ,rp; 1);

11,

0 31


113

(D’Ariano [1995], D’Ariano, Leonhardt and Paul [1995], Leonhardt, Paul and D’Ariano [1995], D’Ariano [1997a], D’Ariano and Pans [1997a]). It has been shown that Kmn(x,cp; r ] ) is a well-behaved bounded function provided that r] > 1/2. It is worth noting that the reconstruction formula (91) also applies to other than harmonic-oscillator systems (Leonhardt and Raymer [19961, Richter and Wiinsche [ 1996a,b], Krahmer and Leonhardt [ 1997b,c], Leonhardt [ 1997a1, Leonhardt and Schneider [1997]). To be more specific, p(x, cp) in terms of pmn reads as38

where g m n ( x ) = VJ;(X>

(99)

%(XI,

and from eqs. (91) and (98) together with eqs. (92), (94) and (99) it follows that the functions nfmn(x),eq. (94), are orthonormal to products of energy eigenfunctions gmn(x),eq. (99),

n / d ~ j ~ , , ( x ) g ~ + , ~sn,mlsnnf ( x ) = for E , - E , =E,~-E,,~.

(100)

can be the regular and irregular solutions, It can be shown that VJm(x)and qm(X) respectively, which solve a Schrodinger equation,

for chosen energy Em, with U ( x ) being an arbitrary potential39. This offers the possibility of reconstruction of the density matrix (in the energy representation) of a particle in an arbitrary one-dimensional potential U ( x ) from the timedependent position distribution of the particle. The quadrature-component distribution in eq. (91) must be regarded as a time-dependent position distribution p(x,t) = (xl$(t)lx) and the phase integral converts into a time integral,

/ / +T/2

pmn= lim !! T-00

T

-T/2

dt

dx eiYmntfntn(x)p(x, t),

(102)

[pmn = pntn(t)1,-0; vntn,transition frequencies]. Obviously, the position distribution p ( x , t ) can also be used for reconstruction of other quantum-state Note that substituting in eq. (91) for p(x, q) the expression on the right-hand side of eq. (98), carrying out the tp integral and using the orthonormalizationrelation (100) just yields an identity. 39 For a proof, see, e.g., Leonhardt and Schneider [1997], Leonhardt [1997c].

38

114


PI, 5 3

representations, such as tomographic reconstruction of the Wigner function. Note that the time interval T , in which the position distribution can be measured is limited in general (for a data analysis scheme for determining the quantum state of a freely evolving one-dimensional wave packet; see Raymer [1997]; see also Q 4.4.1). In the numerical implementation of eq. (91) [or eq. (102)] the sampling function can be calculated on the basis of the analytical result given in eqs. (92) and (94), using appropriate numerical routines (Leonhardt, Munroe, Kiss, Richter and Raymer [1996]). An alternative way is the direct inversion of the underlying basic eq. (98) that expresses the quadrature-component distributions p(x, cp) in terms of the density-matrix elements pmn (Tan [1997])40. For any physical quantum state the density-matrix elements must eventually decrease to zero with increasing m(n). Therefore it follows that the expression on the right-hand side of eq. (98) can always be approximated to any desired degree of accuracy by setting ennrn M 0 for m(n) > urnax. To obtain the (finite) number of density-matrix elements from the measured quadrature-component distributions, the resulting equation can always be inverted numerically, using standard methods, such as least-squares inversion (Appendix D; see also Q 3.9.2). It should be pointed out that the method also applies when the detection efficiency is less than unity. Recalling eqs. (19) and (20), it is easily seen that p(x, q;q ) is related to en,,, according to eq. (98), with gmn(x)being replaced with

en,,?

gnn~(x;q ) = J’h’viz(x’)vn(x’)p(x

q).

(103)

When the characteristic function Y(z, q) of the quadrature-component distribution p(x, q) can be measured directly (Q 4.2. l), then the density matrix in the Fock basis can be obtained from Y ( z ,q) by replacing in eqs. (91) and (92) the x integral with the z integral over f (z)Y(z, q), where -nin

More explicitly, the result can be given by (Wallentowitz and Vogel [1996b])

en,l+k =

l7 1

rxi

d q eCikT

dz SAk’(z)

Re Y(z, q) if k even, Im Y ( z ,q ) if k odd,

(105)

The underlying basic equation for eq. (102) is given by eq. (98) with p(x. t ) and e?’~~~f*‘ in place of p ( x , q )and e-i(n4-n)$, respectively. Inverting it numerically, the time need not be infinitely large as it might be suggested from the analytical result given in eq. (102) (Opatm9, Welsch and Vogel [ 1997cl).

40

11,

8 31

115


(k 3 0), with

(5)

$qz) =a ,/T ~

J-C

( n + k)!

k+l

~;k)(~2/2) e-z2/4

(-2)"2

if keven,

(-2)(k-1)'2 if k odd, (106)

[ ~ ; ~ ) (Laguerre x), polynomial]. So far, reconstruction of arbitrary quantum states has been considered, which can require measurement of the quadrature components at a large number of phases (see also $ 3.9.1), which reveals that the Pauli problem [i.e., reconstructing a quantum state from p(x, q) and p(x, q~ + n/2)] cannot be uniquely solved in general. However, if there is some a priori information on the quantum state to be reconstructed, then p(x, q) need not be known for all phases within a n interval. In particular when the state is known to be a pure state that is a finite superposition of Fock states,

n=O

then it can be reconstructed from two quadrature-component distributionsp(x, q) and p(x, q +n/2) 4 1 . In this case, the problem reduces to solving blocks of linear equations for the unknown coefficients in the Fock-state expansion of the state (Orlowski and Paul [ 19941)42. 3.3.2. Sampling of the displaced Fock-states on a circle

From $2.1.I we know that in unbalanced homodyning the photon-number distribution of the transmitted signal mode is, under certain conditions, the displaced photon-number distribution of the signal mode, pm(a), the displacement The Pauli problem of determining the quantum state of a particle from the position distribution and the momentum distribution has been studied widely, and it has turned out that it cannot be solved uniquely even if the particle moves in a one-dimensional potential and is prepared in a pure but arbitrary state (for the problem including finite-dimensional spin systems, see Pauli [1933], Feenberg [1933], Kemble [1937], Reichenbach [1946], Gale, Guth and Trammell [1968], Lamb [1969], Trammell [1969], Band and Park [1970, 19711, Park and Band [1971], d'Espagnat [1976], Kreinovitch [1977], Corbett and Hurst [1978], PrugoveEki [1977], Corbett and Hurst [1978], Vogt [1978], Band and Park [1979], Park, Band and Yourgrau [1980], IvanoviC [1981, 19831, Moroz [1983, 19841, Royer [1985, 19891, Friedman [1987], PaviEiC. [1987], Wiesbrock [1987], Busch and Lahti [1989], Wootters andFields [1989], Stulpe and Singer [1990], Bohn [1991], Weigert [1992]). 42 These coefficients can also be obtained from p ( x , cp) and a/dp p(x, q)IT=o Wchter [1996c]). More generally, it can be shown that a pure state can always be determined from p(x,cp) and Warp p(x, cp); i.e., from the position distribution and its time derivative in the case of a particle that moves in a one-dimensional potenial and is prepared in an arbitrary pure state (Feenberg [1933], for the problem, see also Gale, Guth and Trammell [1968], Royer [1989]).

41

116

HOMODYNE DETECTION AND QUANTUM-STATE RECONSTRUCTlON

[II, § 3

parameter a = 1 ale'Qlbeing controlled by the local-oscillator complex amplitude [see eq. (51)]. Expanding the density operator in the Fock basis, p,(a) can be related to the density matrix of the signal mode as

where the expansion coefficients (nlm,a) can be taken from eq. (B.15) or eq. (B. 16). Equation (108) can always be inverted in order to obtain en,,, in terms O f p k ( a ) (Mancini, Tombesi and Man'ko [1997], Mancini, Man'ko and Tombesi [19971)43 :

where

is bounded for s E (-l,O], the 6 operator being given by eq. (B.19). This offers from the displaced Fock-state probability the possibility of direct sampling of pnnl,, distribution pm(a). Since pm(a)as a function of a for chosen m already determines the quantum state ($9 2.1.5 and 3.3.3), it is clear that when m is allowed to be varying, then in contrast to eq. (109) - p m ( a )need not be known for all complex values of a in order to reconstruct the density-matrix elements @kn fromp,(a). In particular, it is sufficient to knowp,(a) for all values of m and all phases 47, la\ being fixed (Leibfried, Meekhof, King, Monroe, Itano and Wineland [1996]44;Opatrn? and Welsch [ 19971, Opatrnl, Welsch, Wallentowitz and Vogel [ 19971). For chosen lal, we regardp,,(a) as a function of cp and introduce the Fourier coefficients,

Equation (109) follows directly from eqs. (B.20) and (129). Here the method was first used for reconstructingexperimentally the density matrix of the centerof-mass motion of a trapped ion (see also § 4.2.2).

43

44

K o 31


117

(s = 0, 1,2,. . .), which are related to the density-matrix elements whose row and column indices differ by s as

where

with J = min(m, n + s) and L = min(m, n). Inverting eq. (1 12) for each value of s yields the density matrix sought. Since there has not been an analytical solution, setting ~ , ern,,M 0 for m(n) > nmax eq. (112) has been inverted n ~ m e r i c a l l y ~ (cf. the last paragraph but two of Q 3.3.1) and using least-squares inversion (Appendix D; Q 3.9.2). In this way, can be given by

where it is assumed thatpn is measured for n = 0,1,2,. . . ,N , with N < nmax,and the matrix F;,,(lal) is calculated numerically. Combining eqs. (1 14) and (1 1 l), can be given in a form suitable for statistical sampling. An extension of eq. (1 12) to nonperfect detection is straightforward. In this case, eq. (50) applies, and the measured probability distribution Pm(a;q) can be related to pnl(a)as shown in Appendix C. From eqs. (1 11) and (C.3), it can be seen that when in eq. (1 12) the Fourier component p;(lal) is replaced with the actually measured one, then the matrix G;,(lal) must be replaced with the matrix

Similarly, multichannel detection of the photon-number distribution can be taken into account. In particular, when a photon-chopping scheme as outlined in

45

For examples, see, e.g., Opatmy, Welsch, Wallentowitz and Vogel [1997].

118


tII, § 3

Q 2.1.7 is used, then the Fourier components of the measured N-coincidenceevent probability distribution can again be related to the density-matrix elements according to eq. (112), but now with Ghn(IaI>V,W

=C P m l k ( W k l l ( V ) k.1

GYn(IaI)

(1 16)

in place of Gkn(la1,v), where Pmlk(N)can be taken from eq. (53). 3.3.3. Reconstruction from propensities As already mentioned, the displaced-Fock-state probability p,(a), eq. (1 08), as a function of a delines for each m a propensity prob(a) = .n-‘pm(a)of the type given in eq. (40). The simplest example is the Q function (m = 0), which can be measured in perfect eight-port balanced homodyne detection, with the vacuum as quantum filter (0 2.1.4). Propensities contain all knowable information on the quantum state, and therefore all relevant properties of it can be obtained from them in principle. However, in order to obtain the “unfiltered” quantum state, the additional noise introduced by the filter must be “removed”, which may be an effort in practice. When the Q function is known, then the density matrix in the photon-number basis can be calculated straightfowardly, using the well-known relation46

which corresponds to

(Wiinsche [1991, 1996a1). Equation (118) can also be extended to other than vacuum filters, in order to obtain the density operator in terms of derivatives of more general propensities of the type given in eq. (40) (Wiinsche and Buiek [1997]). In order to obtain the density matrix from the measured homodyne data, derivatives on them must be carried out, which probably could be done with sufficient accuracy only for states whch contain very few photons. Writing .nela12Q(a) = eIal2(alfila) = Em,, e ~ a ~ ’ ( a ~ m ) ( m ~ ~ and ~ ~ ) recalling ( n ~ a ) , that (nla)= (n!)-1’2ane-la12’2, eq. (117) can be derived easily.

46

11,

o 31


119

In practice, it may be more convenient to handle integrals rather than derivatives. It can be shown that4'

r

-

(119) where m 3 n and s 6 -1, L:(z) being the Laguerre polynomial (Paris [ 1996b1). In particular, when s = 1 -2q-', then P(a;s) is just the smoothed Q function measured in nonperfect detection (cf. 0 2.1.4). Unfortunately, eq. (1 19) is not suitable for statistical sampling, since the integral must be performed first and after that the summation can be carried out. Moreover, the inaccuracies of the measured P(a; s) together with the Laguerre polynomials can give rise to an error explosion in the reconstructed density matrix, so that an exact reconstruction of the density matrix from measured data may be expected to be possible only for states which contain finite (and not too large) numbers of photons. In th~scase, the p sum in eq. (1 19) can be truncated at p = N - rn, where the value N has to be chosen large enough to ensure that ( l i t i ; l j ) = 0 for i , j 2 N . Now the p sum can be performed first, and a (state-dependent) integral kernel for statistical sampling can be calculated (Paris [ 1996b,c]). 3.4. MULTIMODE DENSITY MATRICES

The extension of the methods outlined in 99 3.2 and 3.3 to the reconstruction of multimode density matrices from the corresponding multimode joint quadraturecomponent distributions or multimode joint propensities is straightforward. The situation is somewhat different when combined distributions, i.e., distributions that are related to linear combinations of the modes, are measured. Let us consider the two-mode detection schemes shown in fig. 4 in 0 2.1.3. When the sum quadrature-component distribution of two modes, ps(x, a , cpl , @), is known for all phases q11 and @ within JT intervals and all superposition parameters a E ( 0 , ~ / 2 ) ,then it can be shown, on recalling eq. (30), that the reconstruction of the two-mode density matrix in a quadrature-component basis

(B.22) to k = In)(ml and calculating the c-number fimction F(a;s) associated with fi in chosen order. Note that for s = -1 the integral form (119) corresponds to the differential form (117).

47 Equation (119) can be derived, applying eq.

120


[K 5 3

can be accomplished with a three:fold Fourier integral (Opatrnq, Welsch and Vogel [ 1996, 1997b1):

where

and

(k = 1,2148. The generalization to the N-mode case is straightforward. Suppose that we can measure the probability distribution of a weighted sum of quadratures f = a[})&, where a/ are N - 1 parameters which can be controlled in the experiments, 1 = 1,2,. . . N - 1, and fk are real functions which satisfy C:=,f,2({a[}) = 1 identically for each set of parameters {a/}.Let gr be the inversions of fk, gr[{fk({am})}] = a[ (i.e., the set { a [ }parametrizes the surface of an N-dimensional sphere). From the measured sum quadraturecomponent probability distribution ps(x, {a[},{ qk}) its characteristic function Ys(z,{a,},{qk}) can be calculated as a Fourier transform. The characteristic function of the joint quadrature-component probability distribution can then be calculated as

zr=

(xf=

[z = z ~ ) ~ ” ]from , which the N-mode density matrix can be obtained by an N-fold Fourier transform; the whole N-mode density matrix reconstruction is thus accomplished by an N + 1-fold integration of the measured data.

48

For an extension to imperfect detection, see Opatrn?, Welsch and Vogel [1997b].

11,

P 31


121

In the Fock basis the reconstruction of a two-mode density matrix from the combined quadrature-component distributionps(x, a, 971,@) can be accomplished with a four-fold integration (Raymer, McAlister and Leonhardt [ 19961, McAlister and Raymer [1997b], Richter [1997a]): (ml,mzlGlnI,nz) = S % = d a ~ = d ~ l ~ = d ~ R , : : ~ (cpl@)Ps(x,a, x,a, 971, Qb), ( 124) where the integral kernel R:$;(x, a, 971472) is suitable for application of statistical sampling. It can be given by

where

(126) ( y ) and &I ( y ) are harmonic-oscillator energy eigenfunctions, and Here, yml fm,n,(y) is given by eq. (94). An alternative integral expression for rz:,":(x,a) reads

with f-mn (z) being given by eq. (104). Since f-mn (z) is the kernel fbnction for reconstruction of the single-mode density matrix in the Fock basis from the quadrature-component characteristic function Y ( z , @),from eqs. (105) and (106) it is seen that it can be expressed in terms of the associated Laguerre polynomial. Using t h s in eq. (127), then the integral can be performed to obtain a representation of r:$,":(x,a ) as a finite sum over confluent hypergeometric hnctions (Richter [ 1997a1). Finally, the problem of reconstruction of the quantum state of unpolarized light was studied, by considering the (two-mode) density operator

(Lehner, Leonhardt and Paul [ 19961). Obviously, the probability distributionp n of finding n completely unpolarized photons in the signal field characterizes uniquely the quantum state (128). Two schemes for determiningp n were studied.

122


[I& § 3

First, expressions for the sampling function were derived for the case when the two orthogonal polarization modes are delivered to two balanced homodyne detectors and the joint quadrature-component distributions are measured (Krahmer and Leonhardt [1997a]. Second, it was shown that measurement of the quadrature-component distributions of any linearly polarized component of the signal field is sufficient to determinepn,the corresponding sampling function being closely related to the single-mode hnction (Richter [1997b1). 3.5. LOCAL RECONSTRUCTION OF P(n;s)

The displaced photon-number statistics measurable in unbalanced homodyning (0 2.1.7) can be used for a pointwise reconstruction of s-parametrized phasespace functions (Wallentowitz and Vogel [ 1996a1, Banaszek and W6dkiewicz [1996])49.From eq. (B.21) together with eqs. (B.19) and (B.13) it is easily seen (cf. also Moya-Cessa and Knight [ 19931) that

Hence, all the phase-space functions P( a ; s ) , with s < 1, can be obtained from pol(@for each phase-space point a in a very direct way, without integral transformations. In particular, when s = -1, then eq. (129) reduces to the wellknown result that Q(a) = .n-'pO(a), with po(a) = (alela).Further, choosing s = 0 in eq. (129), we arrive at the Wigner function,

Equation (130) reflects nothing but the well-known fact that the Wigner function is proportional to the expectation value of the displaced parity operator (Royer [19771)5 0 . The method was first applied experimentally to the reconstruction of the Wigner function of the center-of-mass motion of a trapped ion (Leibfned, Meekhof, King, Monroe, Itano and Wineland [1996]; 0 4.2.2). From 0 2.1.7 we know that in unbalanced homodyning the quantum efficiency q = lukl l2qO is always less than unity, even if qD = 1, because of l ~ k I l < 1.

4y 50

For a squeezed coherent local oscillator, see Banaszek and Wodluewicz [1998]. For proposals of measuring the Wigner h c t i o n of a particle using this fact, see Royer [1985,

19891.

11, § 31

Q U A N m - S T A T E RECONSTRUCTION

123

Equation (129) can be extended, to imperfect detection in order to obtain (Wallentowitz and Vogel [ 1996a1, Banaszek and Wodkiewicz [ 1997bl)5 1

[a = -(UkZ/ukI)a~,, eq. (49)], where the measured photon-number distribution P , can be regarded as a smoothed displaced photon-number distributionp,(a; q ) of the signal mode. The method works very well and is we1 suited for statistical sampling if s < 1 - q-’ ;i.e., when the weighting factors improve the convergence of the series52. Hence reconstruction of the Q function with reasonable precision is always possible if q > 1/2 (for computer simulations of measurements, see Wallentowitz and Vogel [ 1996a1, Banaszek and Wodkiewicz [ 1997b1). As already mentioned in Q 2.3, measurement of the photon-number distribution of a linearly amplified signal can also be used - similarly to measurement of the displaced photon-number statistics in unbalanced homodyning - for reconstructing the quantum state of the signal mode. It can be shown that when the idler mode is prepared in a coherent state, then the phase-space funcion P(a;s) of the signal mode can be related to the measured photon-number distribution as 53

(Kim [ 1997a,b]), where a = -[(g - l)/g]”2a;. Note that the gain factor g and the quantum efficiency m of the detector enter separately into eq. (132) [in contrast to q = IUkl12rlrin , eq. (131)l. 3.6. RECONSTRUCTION FROM TEST ATOMS IN CAVITY QED

Let us now turn to the problem of reconstruction of the quantum state of a highQ cavity field from measurable properties of test atoms in detection schemes 5 1 This can be easily proved correct, applying eq. (C.4) and expressingp,(a) in eq. (129) in terms of Pm, eq. (50). 52 The feasibility of reconstruction of the Wigner function of truncated states was also demonstrated for 11 < 1 (Wallentowitz and Vogel [1996a]). 53 From eq. (C.5) it can be found that for a = 0, eq. (129) relates the measured distribution P , to the phase-space function of the detected mode at the origin of the phase space, Pdet(O,S). Equation (132) can then be proved correct, using the unitary tranformation (63) and expressing Pd,(a,s) in terms of phase-space functions of the signal and idler modes by convolution (see also Leonhardt [ 19941, Kim and Imoto [ 19951).

124

H O M O D m DETECTION AND QUANTUM-STATE RECONSTRUCTION

DL § 3

outlined in 0 2.4. Though at a first glance the schemes look quite different from the homodyne detection schemes, there are a number of remarkable similarities between them. 3.6.1. Quantum state endoscopy and related methods Let us first consider a two-level (test) atom that interacts resonantly with a singlemode cavity field according to a k-photon Jaynes-Cummings model, the atomfield interaction Hamiltonian being given by

which for k = 1 reduces to eq. (65). When the atoms are initially prepared in superposition states 1 6 )= 2-1(2(18) fe-iv 1 . ) and the excited-state occupation probabilities P:(t) are measured as functions of time, then the cavity-mode density-matrix elements @,,n+k can be determined (Vogel, Welsch and Leine [1987]). To be more specific, it can be shown that the difference P;(t) - P,'(t) reads as

i.e., the off-diagonal density-matrix elements enn+k can be obtained dlrectly from the coefficients aLk) for two phases I/J, such as I/J = 0 and )I = d 2 . Provided that the interaction time t can be varied in a sufficiently large interval (0, T ) , the Fourier transform of P;(t) - P,'(t) consists of sharp peaks, whose values yield the sought coefficients aik' as 54

If T is not large enough, then the peaks in the Fourier integral contain non-negligible contributions of the tails of the corresponding sinc functions. In this case, the coefficients aLk' can be calculated from a set of linear equations obtained from eq. (134) for different times. 54

11,

o 31

125


enn,

(T -+ m). To measure the diagonal density-matrix elements it is sufficient to prepare the atom in the excited state, P , ( t ) l , = ~= 1, and observe (for arbitrary k) the atomic-state inversion AP = P, - Pg = 2Pe - 1,

en,,

from which can be obtained by Fourier transformations5. In cavity QED the 1-photon Jaynes-Cummings model is typically realized, so that the method - also called quantum state endoscopy - does not yield the off-diagonal density-matrix elements @,,,,+k with k > 1. When the quantum state is known a priori to be a pure state such that is given by = cc ,C , with C,C;+~ $ OVrn56,then eq. (134) [together with eq. (135)] can be taken at a sufficiently large number of time points (and at least at two phases) in order to obtain [after truncating the state at a sufficiently large photon number nmaxaccording to eq. (107)] a system of conditional equations for the expansion coefficients c,, whch can be solved numerically (Bardroff, Mayr and Schleich [ 19951, Bardroff, Mayr, Schleich, Domokos, Brune, Raimond and Haroche [1996]). The reconstruction problem for arbitrary quantum states can be solved by performkg a displacement of the initial state of the cavity field such that 8 is replaced with b t ( a ) $ b ( a )and , hence ( k = 1):

em,,

em,,

(8= 2 ~ K , = id1)), where &(a) = (nlbt(a)$b(a)ln)= (n,al$[n,a). Now Qnn(a) can again be obtained from U ( t ) by Fourier transformation, and fiom &(a) the quantum state of the cavity mode can be obtained, applying, e.g., the methods outlined in $8 3.3.2 and 3.5. Alternatively,the quantum state can also be reconstructed when the interaction time is left h e d and only a = (alei'J'is varied

55 For k = 1, very precise measurements of the Rabi oscillations have been performed recently (Brune, Schmidt-Kaler, Maali, Dreyer, Hagley, Raimond and Haroche [1996]), the peaked structure of the Fourier-transformed data being interpreted as a direct experimental verification of field quantization in a cavity. s6 This condition is not satisfied, e.g., for even and odd coherent states as typical examples of Schr6dinger-cat-like states. For even and odd coherent states, see Dodonov, Malkin and Man'ko [1974]. For a review of Schrodinger cats, see Buiek and Knight [1995].

126


[K 0 3

(Bodendorf, Antesberger, Kim and Walther [ 19981). Regarding the measured atomic occupation probabilities as functions of a = (a(eip,M ( t ) = AP(t,a), and introducing for chosen t and la1 the Fourier coefficients,

ixi2x

hP”(t,la[)= -

d q eiSqAP(t,a),

(139)

= 0,1,2,. . .), it can be shown that they are related to the density-matrix elements by equations of the form

(s

(for explicit expressions for Y;(t, la\), see Bodendorf, Antesberger, Kim and Walther [1998]). Inverting eq. (140), whch resembles, in a sense, eq. (112) in Q 3.3.2, for each value of s then yields the density matrix of the cavity mode. Similarly to eq. (112), the inversion can be carried out numerically; e.g., by means of least-squares inversion (Appendix D), choosing an appropriate set of values of lal. In the two-mode nonlinear atomic homodyne detection scheme (Wilkens and Meystre [ 19911) it is assumed that the signal cavity mode is mixed with a local oscillator cavity mode according to the interaction Hamiltonian

Equation (138) can then be found treating the effect of the local oscillator semiclassically (i.e., replacing the operator 6~ with a c number aL,i i ~ --+ aL).In this case, the scheme is obviously equivalent to an initial displacement a = -aL of the density operator of the cavity mode, so that the atomic-state inversion W ( t )is given exactly by eq. (138). In particular, when laLI is sufficiently large, then AP(t) can be rewritten as

In other words, for I aL I 4 ca the atomic occupation probabilities Pe(g)(t)can be related directly to the characteristic function @(p)of the Wigner function W ( p ) of the cavity mode. Varying the interaction time and the phase q~ of aL, the whole function @(p)can be scanned in principle. Knowing @(p),the

KQ

31


127

Wigner function can then be obtained by Fourier transformation5'. Since for appropriately chosen arguments @( /3) is nothing but the characteristic function of the quadrature-component distribution p(x, 6) for all values of 6 within a n interval (see Appendix B.5), the density matrix in both a quadrature-component basis and the photon number basis can be reconstructed straightforwardly from eq. (142) (08 3.2 and 3.3.1). Later it was found that the semiclassical treatment of the local oscillator restricts the time scale to times less than a vacuum Rabi period, because of the quantum fluctuations in the local oscillator cavity mode (Zaugg, Wilkens and Meystre [ 19931, Dutra, Knight and Moya-Cessa [ 1993]), and it was shown that this difficulty can be overcome when the atoms are coupled weakly to the local oscillator but coupled strongly to the signal (Dutra and Knight [ 19941). A priori knowledge on the quantum state to be measured is required in magnetic tomography (Walser, Cirac and Zoller [1996]). Here, the idea of quantum state mapping between multilevel atoms and cavity modes prepared in truncated states (Parkins, Marte, Zoller, Carnal and Kimble [1995]) is combined with a tomography of atomic angular momentum states by SternGerlach measurements (Newton and Young [1968]). It is assumed that an angular-momentum degenerate two-level atom passes adiabatically through the spatial profile of a classical laser beam [Rabi frequency: Q(t)] and, with a spatioternporal displacement t > 0, through the profile of a quantized cavity mode [atom-cavity coupling: g(t - t)] such that the coupled atom-cavity system evolves according to the time-dependent Hamiltonian

1,Jg,Je

Co,m s ,

being Clebsch-Gordan coefficients.

57 Recall that in the dispersive regime the Wigner function can be measured directly (Lutterbach and Davidovich [1997]; see 5 2.4), without any reconstruction algorithm.

128


PI, § 3

It can be shown that if the time-dependent change of the Hamiltonian during the total interaction time is much less than the characteristic transition energies and if the delay and shape of the pulse sequences are chosen such that

then a coupled atom-cavity-field density operator $AF that can be factorized initially into a pure atomic state and a field state containing less than 2Jg photons will be mapped to a product of atomic ground state superpositions and the cavity vacuum

with

and 58

srn,snbeing possible sign changes. Hence, if the final atomic density matrix (Jg, Jg - rn - 1[$A lJg,Jg - n - 1) is known, then the original cavity-field density matrix (ml$F)Fln)is known. Atomic states of this type can be determined from magnetic dipole measurements using conventional Stern-Gerlach techniques (see 04.5). 3.6.2. Atomic beam defection

When a cavity mode is known to be in a pure state, then the expansion coefficients of the state in the photon-number basis can be inferred from the measured deflection of two-level probe atoms during their passage through the cavity (Freyberger and Herkommer [ 19941; for an application of related schemes to the reconstruction of the transverse motional quantum state of two-level atoms,

5 8 Note that with reverse adiabatic passage, an internal atomic state is prepared uniquely by reading out the cavity state.

a, o 31


129

see 44.4.1). A narrow slit put in front of one node of the standing wave (see fig. 8) transmits the atoms only in a small region Ax n ) pq,k+m-q (bkllolbkl),

( 156)

1 - k = m - n, where Ibkl) is again a state of the type given in eq. (159, so that

When for chosen difference, k - I = n - m, the diagonal elements e k k are - known from a direct photon-number measurement, then the offhagonal elements p k k+m-n, k = 0,1,2,. . ., can be obtained from two (ensemble)

and

132


[II,

03

measurements for different superposition parameters. The whole density matrix can then be obtained by means of a succession of (ensemble) measurements, varying the difference m - n of photon numbers in the reference state (155) from measurement to measurement. The presently unresolved problem, however, consists in building an apparatus that prepares travelling waves in superpositions of two photon-number states Im) and In) for arbitrary difference m--12 in a controllable way. For a radiation-field mode that is known to be prepared in a pure state I Y) and that contains only a finite number of photons, eq. (107), it was proposed to extend the method of photon chopping outlined in 0 2.1.7 to a complete determination of the expansion coefficients c, (Paul, Torma, Kiss and Jex [ 1996a, 1997a1)62. First the photon-number distribution pn = (c,t2 is determined according to eqs. (52)(54). Then the balanced 2N-port apparatus (N 2 nmax)is used for mixing the signal mode with N - 2 reference modes in the vacuum state and one reference mode prepared in a coherent state la) (in place of the N vacuum reference inputs in 6 2.1.7). Using the selected statistics when only photons in the first N/2 output channels are observed, it can be shown that the probability p,(N) of detecting n photons with at most one photon in each channel is given by (Paul, Torma, Kiss and Jex [1996a]) p,(NI=

I

I

I Lm)

- >,

d(n- m)!a"'cn-m .

(158)

m=O

Determining the coincidence probability distribution p,(N) for two coherent states with different phases, the phases cp, of the expansion coefficients c, = Ic,Jel" can be calculated step by step (for known amplitudes Ic,( and up to an overall unimportant phase) from the two measured coincidence-event distributions.More involved formulas are obtained in the case when the complete coincidence statistics are included in the analysis; i.e., photons are allowed to be observed in any output channel and realistic photodetection is considered (Paul, Torma, Kiss and Jex [ 1997a1). It turns out that the reconstruction scheme with two coherent states requires photodetectors which can discriminate between 0,1,2,. , . photons; i.e., detectors which have rather low quantum efficiency. Using avalanche photodiodes, which can have high quantum efficiency, the reconstruction might become rather involved,because additional reference beams

For a proposal to extend the method to a two-mode signal field, see Paul, Torma, Kiss and Jex [ 1997bl.

62

11,

8 31

QUANTUM-STATERECONSTRUCTION

133

with different phase properties Must be used. Similarly, when a mixed quantum state is tried to be reconstructed, then extra reference beams (reference phases) must be used, the maximum number of reference beams being limited by the cutoff in the photon number. It can also be shown (Bialynicka-Birula and Bialyniclu-Birula [ 19941, Vaccaro and Barnett [ 19951) that when a radiation-field mode is prepared in a pure state I Y) that is a finite superposition of photon-number states, eq. (107), then it can be reconstructed from the photon-number distribution p n = /cnl2and the (PeggBarnett) 63 truncated canonical phase distribution Pps(@)= I q(@)12, with64

Taking eq. (159) at 2(n,,, + 1) values of @, such as 01 = h/(nmaX + l), the resulting equations can be regarded as conditional equations for the unknown phases qn,provided that all absolute values I~I(@I)I and Ic,I are known. Apart from the fact that the quantum state must be known to be a pure state, the question remains of how to obtain the phase statistics (see also 5 3.8.3). 3.8. RECONSTRUCTION OF SPECIFIC QUANTITIES

Since the density matrix in any basis contains the full information about the quantum state of the system under consideration, all quantum-statistical properties can be inferred from it. Let fi be an operator whose expectation value,

mn

is desired to be determined. One may be tempted to calculate it fiom the reconstructed density matrix (or another measurable quantum-state representation). However, an experimentally determined density matrix always suffers from various inaccuracies which can propagate (and increase) in the calculation process (cf. 53.9.1). Therefore it may be advantageous to determine directly the quantities of interest from the measured data, without reconstructing the

See Pegg and Barnett [1988]. Note that @(@) = ( @ p ~ ( @ ) l Ywhere ), IY) is given by eq. (107), and the truncated phase state J@ppg($))reads as l @ p ~ ( @ = ) ) (nma, + l)-"* C"""" n = O e&@\n) (cf. 8 3.8.3).

63

134


[II, § 3

ern,,

whole quantum state. In particular, substituting in eq. (160) for the integral representation (91) one can try to obtain an integral representation,

with

suited for direct sampling of (k) from the quadrature component distributions p(x, q) [provided that both the sampling function KF(x,q) and the integral in eq. (161) exist]. 3.8.1. Normally ordered photonic moments It is often sufficient to know some moments of the photon creation and destruction operators of a radiation-field mode rather than its overall quantum state. Let us again consider the measurement of the quadrature-component probability distributions p(x, q) in balanced homodyning (0 2.1.2). Then the normally ordered moments of the creation and destruction operators, (;ltniirn) can be related to p(x, q) as6'

where H,(x) is the Hermite polynomial (Richter [1996b])66.Equation (163) is just of the type given in eq. (161) and offers the possibility of direct sampling of normally ordered moments (;lt";l") from the homodyne data. It is worth noting that knowledge of p(x, q) at all phases within a n interval is not necessary to reconstruct (;lt";l") exactly, and therefore the q integral in eq. (163) can be replaced with a sum. It was shown that any normally ordered moment (;lt";l") can already be obtained from p(x, q) at N = n + m + 1 discrete different phases Qlk (Wiinsche [1996b, 19971).

65 Equation (163) can be proven correct if both sides are expressed in terms of the density matrix in the photon-number basis and standard summation rules for the Hermite polynomials are used. 66 A more involved transformation was suggested by Bialymcka-Birula and Bialymcki-Birula [1995]; see footnote 77.

11,

8 31


135

The method is especially useful, e.g., for a determination of the moments of photon number. Note that for finding the mean number of photons (2) from the Fock-basis density matrix a relatively large number of measured diagonal elements must be included into the calculation, in general, each of which being determined with some error. After calculation of the sum ( 6 ) = Enne,, the error of the result can be too large to be acceptable, and for higher-order moments severe error amplification may be expected. Equation (163) can be extended easily to nonperfect detection (q < l), since replacing p(x, cp) with p(x, cp; q ) simply yields q("+m)'2(iitniim) 67. Provided that the whole manifold of moments (2t"iim)has been determined, then the quantum state is known in principle (Wiinsche [1990, 1996b], Lee [1992], Herzog [1996b])6*. To be more specific, the density operator can be expanded as [cf. eq. (69)]

en,,

M

k.l= 0

where

[{k,Z} = min(k,Z)] provided that (iitkii') exists for all values of k and Z and the series (164) (in chosen basis) exists as well. Since all the moments do not necessarily exist for any quantum state, and the series need not necessarily converge for existing moments, the quantum-state description in terms of density matrices is more universal than that in terms of normally ordered moments69

Recall that an imperfect detector can be regarded as a perfect detector with a beam splitter in front of it, so that the destruction operator 2 of the mode that is originally desired to be detected i,s transformed according to a beam-splitter transformation (4 2.1. l), Cr( 7) = JiiCr + f i b , where b is the photon destruction operator of a reference mode prepared in the vacuum state. This is also true for other than normally ordered moments, such as symmetrically ordered moments (Band and Park [1979], Park, Band and Yourgrau [1980]). 6y An example of nonexisting normally ordered moments is realized by a quantum state whose photon-number distribution behaves like p n n-3 for n + co. Even though it is a normalizable state with finite energy, its moments (itk) do not exist for k 2 2. For a thermal state the relation (iit"~3~) = d,,n! A" is valid ( A , mean photon number), which implies that the series (164) for pmmn does not exist. Note that the problem of nonconvergence of the series (164) may be overcome by analytic continuation.

67

-

136


[K5 3

The extension of the method to the, reconstruction of normally-ordered moments of multimode fields from joint quadrature-component distributions is straightforward. Moreover, normally-ordered moments of multimode fields can also be reconstructed from combined distributions considered in Q 2.1.3 (Opatrn-, Welsch and Vogel [1996, 1997a1, McAlister and Raymer [1997a,b], Richter [1997a]). For simplicity, let us restrict our attention here to two-mode (the subscripts 1 and 2 refer to the two modes) moments (i?pli?pi?yliiy) and assume that the weighted sum i = SZl(q1)cosa + iz(@)sina, eq. (29), is measured in a homodyne detection scheme outlined in 0 2.1.3 (see fig. 4). Measuring the moment (2") for n + 1 values of a, a set of n + 1 linear algebraic equations can be obtained whose solution yields the two-mode quadrahuecomponent moments ( i ; - k ( q l ) i $ ( @(k) )= 0,1,2,. . . ,n). Varying ql and @, the procedure can be repeated to obtain the dependence on q~ and cp2 of the twomode quadrature-component moments. These can then be used to determine step by step the normally ordered moments of the photon creation and destruction can be obtained operators of the two modes. In a closed form, (ii~'i?inzi?yli?~) from the combined quadrature-component distribution ps(x, a, ql,912) as

ei(nl-ml) nl+nz+m1+m2)(~)

x ~n,+n,+m1+m2(~)~~2+m2

e l i(n2-ml)

(166) [C,, = n!m!/(n2("+"Y2(n + m)!)].The functions F f ) ( a )form a biorthonormal system to the fimctions G t ) ( a )= (:) cosl-ka sinks in some Q interval, so that daFf)(a) Gg)(a)= S k p . Note that since for given 1 there is a finite number of functions Gf)(a),the system of functions F f ) ( a )is not determined uniquely and can be chosen in different ways. In particular, when only the phase difference A q = (p2 - ql is controlled and the overall phase ql is averaged out (this is the case when the signal and the local oscillator stem from different sources), then the moments (i?In1i?pi?yiiiT2)can be reconstructed for nl -ml = m2 -n2. If both phases ql and @ are averaged out, then those moments ( i i ~ n 1 i ? ~ n 2 ican i ~ 1still ii~) be obtained, which carry the information about the photon-number correlation in the two modes. The method was used to demonstrate experimentally the determination of the ultrafast two-time photon number correlation of a 4 ns optical pulse (McAlister and Raymer [1997a]). In the experiment, the local oscillator pulses are derived from a Ti:sapphire-based laser system that generates ultrashort, near transformlimited pulses (150 fs) at a wavelength of 830 nm and a repetition rate of

s,

11,

o 31

137


-1 0

-5

0

5

10

7 (PS)

Fig. 15. The second-order coherence, eq. (167),,experimentally determined via balanced four-port homodyne detection (dots) and from the measured optical spectrum (solid line). The value of tl is set to occur near the maximum of the signal pulse. (After McAlister and Raymer [1997a].)

1 kHz. The signal is fi-om a single-spatial-mode superluminescent diode. The broadband emission at 830 nm is filtered spectrally to produce a 4 ns pulse having a 0.22 nm spectral width. For each value of the relative delay At = tl - t2 between two local-oscillator pulses the phase-averaged combined quadrature-component distribution p ~ ( xa) , = ( 2 . 7 ~J) dqp, ~ ~ J dcp2ps(x, a, q1, cp2) is measured for three different values of a: 0, n/2 and x/4. From these the normalized second-order coherence function,

is computed (6, and 62 being the photon destruction operators of the spatialtemporal modes defined by the local-oscillator pulses centered at times tl and t2). This experiment represents an extension of measurements of the HanburyBrown-Twiss correlations70 to a sub-picosecond region. Results are shown in fig. 15. 3.8.2. Quantities admitting normal-order expansion

The basic relation (163) can also be used [similar to eqs. (160)-(162)] to find sampling formulas for the mean values of quantities that can be given in terms 70 See Hanbury Brown and Twiss [1956a,b, 1957a,b]; for more discussion of the Hanbury Brown and Twiss experiments, see Pefina [I9851 and Mandel and Wolf [1995].

138


tII, § 3

of normally ordered moments of photon creation and destruction operators (D'Ariano [1997b]). Let us consider an operator h and assume that it can be given by a normal-order expansion as

n.m = 0

The mean value

(h)can then be obtained from p ( x , q;r ] ) according to 71

where the integral kernel, KF(x, fp; r ] )

p(% 1111

= Tr

%

>

[I?(x, q ;r ] ) being defined in eq. (97)] can be given by

with

(n + m )

-I

00

GF(z,q) =

fnm

(-i.z)"+mei(m-nf9,

n,m = 0

Here it is assumed that the integral kernel K&, q;q) exists and the x integral in eq. (169) exists as well72. The integral kernel (171) exists if for z + 00 the hnction G&, q) grows slower than eqz2'2,so that the integral in eq. (171) converges (for examples, see D'Ariano [ 1997b], D'Ariano and Paris [ 1997b])73. Recall that when the c-number function F ( a ; s )that is associated with an operator k in s order exists for s < -1, then (b)can be obtained from the phase-space function P( a ; s)(measurable, e.g., in balanced eight-port homodyning; see 52.1.4) according to eq. (B.22), provided that the integral exists. 72 Equations (169)-(172) can be obtained by taking the average of eq. (168), substitutingfor (ht"6") the right-hand side (multiplied by q-(n+"y2) of eq. (163), and using the generating function of the Hermite polynomials with the argument (i/m)(d/dz). 73 Since the integral kernel K&, q; q ) does not depend on rp if k is a function of the photonnumber operator, this case can be regarded as the first realization of Helstrom's quantum roulette wheel (D'Ariano and Paris [1997b]).

11. Q 31


139

The integral kernel (17 1) is therefore applicable to such quantum states whose (smeared) quadrature-component distributionsp(x, cp; r ] ) tend to zero sufficiently fast as 1x1 goes to infinity such that eq. (169) converges even if KF(x,cp; q) increases with 1x1. At this point it should be noted that K&, q;r ] ) is determined only up to a function O(x,cp); i.e., K&, q;q) in eq. (169) can by replaced with Kk(x, cp; q) = &(x, cp; r ] ) + O(x,cp) such that

for any normalizable quantum state. Hence, if the integral kernel KF(x,cp; r ] ) that is obtained from eq. (171) is unbounded for 1x1 -+ cm such that the x integral in eq. (169) does not exist for any normalizable quantum state, it cannot be concluded that fi cannot be sampled from the quadrature-component distributions of any normalizable quantum state, since a different, bounded kernel may exist.

3.8.3. Canonical phase statistics The quantum-mechanical description of the phase and its measurement has turned out ta.be troublesome and is still a matter of discussion. Many papers have dealt with the problem and an extensive literature is available (for reviews, see Lukl and Peiinova [1994], Lynch [1995], Royer [1996], Pegg and Barnett [1997]). Here we confine ourselves to the canonical phase that is obtained in the attempt - in close analogy to the classical description - to decompose the photon destruction operator into amplitude and phase such that 2 = hfi. The (nonorthogonal and unnormalizable) phase states,

fl=O

(London [1926, 1927]), which are the right-hand eigenstates of the one-sided unitary operator k , &I$) = ei@l$),can then be used to define - in the sense of a POM - the canonical phase distribution P($) = (A($)),with A($)= I$)($\ 74. Note that the Hermitian operators 2 = (k+k 9 / 2 and 3 =.(A - kt)/(2i) are

For a two-mode orthogonal projector realization in the relative-photon-numberbasis, see Ban [1991a-d, 1992, 19931, Hradil [1993].

74

140


PI, 5 3

the Susskind-Glogower sine and coshe operators (Susskind and Glogower [ 19641)75. Various proposals have been made to measure P ( @ )using homodyne detection or related schemes 76. So the Wigner function or the density matrix reconstructed from the homodyne data could be used to infer the phase statistics in principle (Beck, Smithey and Raymer [1993], Smithey, Beck, Cooper and Raymer [1993], Breitenbach and Schiller [1997]). Further it was proposed to obtain P(@) = (I?(@))from a homodyne measurement of (all) normally ordered moments (ij+m+"ij"),because I?(@) has a series expansion of the type given in eq. (168),

(Bialynicka-Birula and Bialynicki-Birula [ 19951)77. Unfortunately, (I?(@)) cannot be given (even for = 1) by an integral relation of the form (169) suited to statistical sampling, because the integral kernel does not exist (see footnote 79). It was therefore proposed to introduce parametrized phase distributions P(@,E) such that for any E > 0, P(@,E) can be obtained by direct sampling fromp(x, cp), and P ( ~ , E-+) P(@)for E 0 (Dakna, Knoll and Welsch [1997a,b]). The sampling functions can then be obtained as convergent sums of the sampling functions for the density-matrix elements in the Fock-basis (see 5 3.3). Since the value of the parameter E for which P(@,E) M P(@)is determined by the number of photons at which the quantum state under study can be effectively truncated, the method is state-dependent. Further, it was proposed to measure the canonical phase by projection synthesis, mixing the signal mode with a reference mode that is prepared in a quantum state such that, for appropriately chosen parameters, the joint-photon-number probabilities in the two output channels of the beam splitter directly correspond to the canonical phase statistics of the signal mode (Barnett and Pegg [1996], Pegg, Barnett and Phillips [1997]). Apart from the --f

75 For constructing the operators corresponding to classical phase-dependent quantities, see Bergou and Englert [1991]. 76 The proposal was made to regard the quadrature-component distribution at x = 0 as (unnormalized) phase distribution, P(4) p(x = O,$) (Vogel and Schleich [1991]), which yields phase statistics which are quite different from the canonical phase statistics in general (for an improvement, see B&ek and Hillery [1996]). 77 Here it was proposed to obtain (dtm+"dn)from the homodyne data via the (2n + m)th derivative of dq e-imrPe-1z'2(exp[i~(rp- n/2)]).

-

s,"'

K 5 31


141

direct photon-number measurement needed and the fact that the quantum state under study must again be truncated, so that the method is state-dependent, the difficult problem remains to design an apparatus that produces the reciprocal binomial states needed. The problem of homodyne measurement of the canonical phase can be solved when the exponential phase moments yk;[i.e., the Fourier components of P(4)] are considered and not the phase distribution itself,

where 'Yk= (kk)if k > 0, and ' Y k shown that ( k > 0)

= y?k

if k < 0 (YO= 1). Then it can be

(Opatrny, Dakna and Welsch [1997, 19981, Dakna, Opatrny and Welsch [1998]). The integral kernel Kk(x)(fig. 16) can be used for sampling the exponential phase moments from the homodyne data for any normalizable state '*. In particular, Kk (x) rapidly approaches the classical limit 79

K(c/) k

()=

i(-l)(k-')'2k signx

if k odd,

(2n)-'(-l)(kf2Y2klnx if k even

as 1x1 increases, and it differs from the classical limit only in a small interval around the origin. It is worth noting that the method applies to quantum and classical systems in a unified way and bridges the gap between quantum and classical phase. In particular, the integral kernel Ky)(x)as given in eq. (178) is nothing but the integral kernel for determining the radially integrated Wigner function, which reveals that in the classical limit the canonical phase distribution is simply the radially integrated probability density for the complex amplitude a. Note that any radially integrated propensity prob(a), such as the

''

Note that when Kk(x) is calculated according to eq. (171), then it does not apply to all normalizable states (for analytical expressions and properties of &(X), including nonperfect detection, see Dakna, Opatmy and Welsch [1998]). 79 Already from the classical kernel (178) it is seen that eik(q-@)Kk(x)does not exist, and hence P(Q) cannot be obtained from p ( x , rp) by means of an integral transformation of the form given in eq. (177).

xi??-,

142


1.5 -

k=5

0.5 -

Kkb)

0

k=l

-

-0.5

-

-1.5

-

k=3

,

-6

-4

-2

0

2

4

6

X

Fig. 16. The x-dependent part &(x) of the sampling function &(x, rp) = eikpKk(x) for the determination of the exponential moments 'Yk = ( E k ) of the canonical phase from the quadraturecomponent distributions p(x, rp) according to eq. (177) for various (a) odd k and (b) even k. (After Dakna, Opatm? and Welsch [1998].)

11,

8 31

143


radially integrated Q function measurable, e.g., in balanced eight-port homodyne detection, can be used to define an operational phase probability density (Noh, Fougkres and Mandel [1991, 1992a,b, 1993a,b,c]),which in the quantum regime differs from the canonical phase distribution in general (for hrther readings, see Turski [1972], Paul [1974], Shapiro and Wagner [1984], Hradil [1992, 19931, Vogel and Welsch [1994], Leonhardt, Vaccaro, Bohmer and Paul [1995], Leonhardt [ 1997~1). 3.8.4. Hamiltonian and Liouvillian So far, measurement and reconstruction of quantum-state representations and averages of particular quantities at certain time have been considered. The quantum state of an object at chosen time t is, of course, a result of state evolution from an initial time to, the state evolution being governed by the Hamiltonian H of the system or a Liouvillian i in the more general case of the system being open. Since the Liouvillian of an object expresses in most concentrated form the dynamics of the object, knowledge of the Liouvillian is essential for understanding the behavior of the object, and the question may arise of how to experimentally determine its form. To answer the question, it was proposed to appropriately apply quantum-state reconstruction routines, such as direct sampling of the density matrix in balanced homodyning (D'Ariano and Maccone [ 19971). Let us consider a system that is initially prepared in some known state Gin = $(to) and assume that in the further course of time it evolves to a state Gout = g ( t ) at time t ,

where for a system that is homogeneous in time the superoperator 6 = G(t,to) can be given by the exponential of the (time-independent) Liouvillian of the system,

t

The superoperators 6 and i can then be obtained by measuring (reconstructing) Gout for various probe inputs G i n . A typical example may be a radiation-field mode whch (at time to) is fed into nonlinear resonator-like equipment giving rise to amplification and damping. The quantum state of the outgoing mode (at

144


P, 5 3

time t ) can then be related to that ofthe incoming mode according to eq. (179). In the Fock basis, eq. (179) reads as w

Now let us assume that in a succession of measurements the quantum states of the outgoing mode are reconstructed for an appropriately chosen (overcomplete) set of input quantum states Qyn, p = 1,2,. . ., such that for chosen m and n the block of equations, w

can be used as conditional equations for the matrix elements GI,,. Repeating the procedure for all values of rn and n, the G matrix and [by using eq. (180)] the L matrix can be calculated in principle. Apart from a truncation of the Hilbert space in numerical implementations, the main problem that remains in practice is of course the (controlled) initial-state preparation such that all (relevant) matrix elements of the Liouvillian can be probed. In parficular, when the Liouvillian is phase-insensitive, then mixtures of Fock states evolve into mixtures of Fock states. For this case a setup was proposed which uses correlated twin beams produced by a nondegenerate parametric amplifier in combination with conditional photon-number measurement on one beam in order to prepare the other beam in (random) photon-number states. These states can then be used as inital states for probing the Liouvillian. To realize the scheme with available techniques, it should be noted that the reconstruction algorithm can be extented such that the quantum efficiency of the photon-number measurement need not be unity (D' Ariano [ 1997~1). 3.9. PROCESSING OF SMEARED AND INCOMPLETE DATA

In practice, there are always experimental inaccuracies which limit the precision with which the quantum state in a chosen representation can be determined. Typical examples of inaccuracies are data smearing owing to nonperfect

For limits on the measurement of state vectors expressed in terms of channcl capacities for the transmission of information by finite numbers of identical copies of statc vectors, see Jones [1994].

11, § 31

145


detection, finite number of measkement events, and discretization of continuous parameters, such as the quadrature-component phase in balanced homodyning. In particular, the latter is an example of leaving out observables in the expansion (69) of the density operator @.Whereas in balanced homodyning the distance between neighboring phases can be diminished such that the systematic error is reduced, in principle, below any desired level, there are also cases in which a principally incomplete set of observables is available. Then either additional knowledge of the quantum state is necessary to compensate for the lack of observables or other principles must be used to reconstruct the density operator according to the actual observation level.

ai

3.9.1. Experimental inaccuracies

Since in a realistic experiment the quantum efficiency q is always less than unity, the probability distributions of the measured quantities are always more or less smeared, because of losses. The problem of compensating for losses in direct sampling of the quantum state from the quadrature-component distributions measurable in balanced homodyning ( Q 2.1.2) has been studied widely. As mentioned in Q 2.3, active loss compensation may be realized, in principle, by mwns of a squeezer, such as a degenerate parametric amplifier (Leonhardt and Paul [1994a]; see also Leonhardt and Paul [1995]). When the signal mode is preamplified before detection by means of a squeezer [eq. (63) with ei@;liin place of ;[I7 then it can be shown that for appropriately phase matching the (scaled) quadrature-component distribution of the preamplified signal measured with quantum efficiency q reads as8’ P’(&x,

q; rl) =

/

h ’ P ( x ’ , q)P[&(x

-X%

rll,

(183)

(g, amplification factor), with p(x; q) being given by eq. (20). From eq. (183)

together with eq. (20) it is seen that q is effectively replaced with gq/ (1 - g + gg), and hence &p’(&x, Q?; q) tends to p(x7cp) as the amplification becomes sufficiently strongp2.Note that the degree of improving the quantum

’’

Note that squeezing the local oscillator has the same effect as antisqueezing the signal field under the assumption that the coherent component is large (Kim and Sanders [1996]; see also Kim [199-3). 82 For details, see also Vogel and Welsch [1994], and for a discussion of eqs. (183) and (20) in terms of moments of the measured quadrature components and those of i (rp), see Marchiolli, Mizrahi and Dodonov [1997]).

146


[XI, 5 3

efficiency is limited by the realizable mode-matching in degenerate parametric amplification, which deteriorates with increasing pump strength. Without active manipulation of the signal, it has been well established that above a lower bound for q, it is always possible to compensate for detection losses, introducing a modified sampling function that depends on q such that performing the sampling algorithm on the really measurable (i.e., the smeared) quadrature-component distributions p(x, cp; r]) yields the correct quantum state (D’Ariano, Leonhardt and Paul [ 19951). This bound depends on the chosen state representation in general. In particular, reconstruction of the density matrix in the photon-number basis is always possible if q > 1/2. In this case,

where the now q-dependent sampling function Knln(x,cp; q ) is defined by eq. (96) [together with eq. (97)]. Another approach to the problem of loss compensation is that the sampling function is left unchanged [i.e., K,,(x, cp; q) -+ K,,(x, cp) = K,,(x, cp; q = 1) in q ) of the quantum state eq. (184)] and, at the first stage, the density matrix that corresponds to the smeared quadrature-component distributions p(x, q;q) is reconstyucted. After that, at the second stage, the true density matrix = @(; = 1) is calculated from emn(q) (Kiss, Herzog and Leonhardt [1995], Herzog [ 1996a1). Extending the Bernoulli transformation (C.3) and the inverse Bernoulli transformation (C.4) to off-diagonal density-matrix elements, can be calculated from emn(q) asg3

emn(

emmn

entn

It is worth noting that eq. (185) is exact. In other words, when (for precisely given overall detection efficiency) q) is known exactly, then can always be calculated precisely, irrespective of the value of q.

ern(

Replacing in eq. (185) q-‘ with q yields @,,,,,(q) in terms of em,,,which corresponds to a generalization of the Bernoulli transformation (C.3) to off-diagonal density-matrix elements. Since an imperfect detector can be regarded as a perfect detector with a beam splitter in front of it, h(q) and $ can be related to each other applying the beam-splitter transformation (7), h(q) = 2(01e’/’i2$e-iPi2J0)2, with i 2 according to eq. ( 6 ) and c0s2(p/2) = 8 (the mode indices 1 and 2, respectively, refer to the signal and vacuum inputs). It can then be proved easily that $[a) = +$, where i = t ( i t 2 $ +hiti - 22@t), which corresponds to the time evolution of a damped harmonic oscillator at zero temperature (8 -+ e+). 83

11, § 31


147

In practice however, q is not @own precisely in general, and the experimenalways differ from the exact ones. In particular, tally determined values of emn(r) when 11 6 1/2 and the error of does not vanish with increasing m and n, and when there is no a priori information about the quantum state, such as the photon number at which it can be truncated, then cannot be obtained for chosen number of measurements - from the measured @ k / ( v ) , because of error explosion. A typical example of such an error is the statistical error with q) is sampled from the quadrature-component distributions measured which in balanced homodyning. In this case loss compensation (for arbitrary quantum states) is possible only if Q > 1/2 (D’Ariano and Macchiavello [ 19981)84. Clearly when the quantum state to be reconstructed can be truncated at some maximum photon number nmaxrthen the sum in eq. (185) is finite and the density-matrix elements encan also be obtained for q < 1/2, the accuracy being determined by that of Q m n ( V ) * Limitations on variables in real experiments always give rise to systematic errors. This is the case, e.g., when the quadrature components i ( q )are measured at discrete phases Q?k or when only some part of the n interval can be scanned experimentally (for an example, see Q 4.4.1). In particular, in balanced homodyning the quadrature-component distribution p(x, q) is measured at a finite number of phases q k within a n interval and finite x-resolution. When p(x, q) is measured precisely at N equidistant phases qk = ( n N ) k , then the reconstructed density-matrix elements are given by

emn(r])

(186)

in place of eq. (91). A measure of the systematic error is the difference Aemn= emn(N)which reads ass5

em,,,

A e m n = ~ ~ G k m ; e kwith l r k-l=m-nf2jN j=l

k,l

84 When the density matrix at initial time t = 0 of a signal mode that undergoes phase-insensitive damping or amplification is tried to be reconstructed from the quadrature-component distributions p(x, rp, t; TJ) measured at time t > 0, then the additional (phase-insensitive) noise gives rise to a modified overall quantum efficiency q*, so that qs > 1/2 must be valid in order to compensate for the losses (D’Ariano [1997b], D’Ariano and Sterpi [1997]). 85 Equation (187) can be derived by substituting in eq. (186) for p(x, rp) the result of eq. (98) and recalling eqs. (92) and (100). Note that G T = 6,,,k6,,~only holds for k - I = m - n.

148


[K § 3

[G:r = n J dxfm,(x)gk~(x)](Leonhardt,and Munroe [1996], Leonhardt [1997b]). Equation (187) reveals that when the quantum state can be truncated such that

then the density matrix-elements en*,,for 1m - nl < N can be reconstructed precisely from p ( x , q ) at N phases qk according to eq. (186). In particular, one phase is required to reconstruct a completely dephased quantum state that contains only diagonal density-matrix elements. Since any quantum state can be approximated to any desired degree of accuracy by setting emnM 0 for m(n) > nmax,there is always an N M nmax for which the condition (188) can be assumed to be satisfied, so that all (essentially nonvanishing) densitymatrix elements can be reconstructed precisely. If N is not known a priori, the quantum state can be reconstructed in an iterative way, increasing N in a sequence of (ensemble) measurements until I is sufficiently small. Note that any normally ordered moment (6tn6")can be reconstructed exactly from p(x, q) at N = n + m + 1 phases (Wunsche [1996b]; see Q 3.8.1). The methods for quantum-state reconstruction are based on ensemble measurements; i.e., on a sequence of individual measurements carried out on identically prepared systems8 6 . Since the number of individual measurements is principally finite, the measured quantities are always estimates of the true ones. Hence all the quantities which can be inferred from the measured quantities are also estimates, and the statistical error with which the original quantities are measured propagates to the quantities inferred from them. In particular, when the quantities which are desired to be determined can be sampled directly from the measured data, then the statistical error can be estimated straightforwardly by also using the sampling method. The problem of statistical error in quantum-state reconstruction has been studied in a number of papers, with special emphasis on balanced four-port homodyne detection (Leonhardt, Munroe, Kiss, Richter and Raymer [1996], D'Ariano [1997b], D'Ariano and Paris [1997a], D'Ariano, Macchiavello and Sterpi [ 19971, Leonhardt [ 1997~1).Let us consider a quantity and assume that it can be sampled directly from the quadrature-component statistics according to eq. (161). When in an experiment n ( q k ) individual

For the problem of measuring the state of single quantum systems, see Uecla and Kitagawa [1992], Aharonov and Vaidman [1993], Aharonov, Anandan and Vaidman [1993], Imamoglu [1993], Royer [1994, 19951, Alter and Yamamoto 119951, D'Ariano and Yuen [1996]. 86

11, I 31


measurements are performed at phase q k , k estimated as

=

149

0,1,. . . ,N - 1, then (& can be

where x n ( c p k ) is the result of the nth individual measurement at phase cpk. Taking the average over all estimates ($eSt)(N))yields ( E ( N ) ) ;i.e., the desired quantity within the systematic error owing to phase discretization,

where

) terms of the averaged Accordingly, the statistical fluctuation of ( p ( e s t ) ( N )in can be given by, on taking into account estimates of the variance of (k;cest)(N)) that the individual measurements are statistically independent of each other ”,

where

is the variance of the sampling function, and N(cpk) is the number of measurements per phase interval, N(cpk) = n(cpk)N/n. Let us mention that when the kernel KF(x, cp) is a strongly varying function of x in regions where p ( x , cp) is non-negligible, then the first term in eq. (193) is much larger than the second one. In this case the second term can be neglected and the statistical error can be approximated by averaging the square of the kernel. Note that the same result is obtained if one assumes that the numbers of

*’For simplicity, in eq. (192) it is assumed that F is a real quantity.

150


[II, § 3

events yielding the values of x(cpk) in given intervals of x (bins) are independent Poissonian variables (Leonhardt, Munroe, Kiss, Richter and Raymer [19961, Leonhardt [ 1997~1,McAlister and Raymer [ 1997bl). However, these variables are neither strictly independent [their sum is always N(cpk)] nor Poissonian Ethe probability that there is more than N ( cpk) events in a bin is zero]. Therefore, care must be taken before one decides to use the simplified error estimation. Whereas for the Fock-basis density matrix elements the simplified estimation is very good (the kernels are strongly oscillating), for the exponential moments of canonical phase one must take into account both terms in eq. (193), otherwise the error would be overestimated (the kernels are slowly varying functions). From eq. (192) it can be expected that the statistical error depends sensitively on the analytical form of the sampling function. To give an example, let us consider the integral kernels Kmn(x, cp), eq. (92), needed for sampling the densitymatrix elements in the photon-number basis, Qmn. For chosen cp, Kmn(x,cp) is an oscillating function of x, and with increasing distance d = ( m- nl from the diagonal the oscillations become faster and the oscillation range slowly increases. It turns out that with increasing m the diagonal-element variance oimm becomes independent of m ; i.e., saturation of the statistical error for sufficiently large m is observed, oimm< 2/R (R, mean number of measurements per phase interval). On the contrary, the off-diagonal statistical error increases with d, without saturation. The influence of the quantum efficiency q on the statistical error of the density-matrix elements is very strong in general. For chosen m and n the oscillation range of Km,(x, cp; q), eq. (96), increases very rapidly as r] approaches the lower bound r] = 1/2, and the statistical error increases rapidly as well (for numerical examples, see D’Ariano, Macchiavello and Sterpi [ 19971)88. Using the experimentally sampled density-matrix elements for calculating the expectation values of other quantities, such as ( p ) according to eq. (160), the resulting statistical error is determined by the law of error propagation. It is worth noting that error propagation can lead to additional noise which is not observed if the quantities are also directly sampled from the measured data (provided that the sampling method applies).

**

For the error of the exponential phase moments sampled from the quadrature-component distributions in balanced homodyning, see Dakna, Opatrn? and Welsch [1998], and for the error in quantum-statemeasurement via unbalanced homodyning and direct photocounting, see Wallentowik and Vogel [1996a], Banaszek and Wbdkiewicz [1997b], Opatrn?, Welsch, Wall&owitz and Vogel [1997].

11,

5 31


151

3.9.2. Least-squares method In the quantum-state reconstruction problems outlined in the foregoing sections a set of measurable quantities (in the following also referred to as data vector) is related linearly to a set of quantities (state vector) that can be used to characterize the quantum state of the system under study. Both sets of quantities can be discrete or continuous or of mixed type. Typical examples are the relations (70) and (98), respectively, between the Wigner hnction and the density-matrix elements in the Fock basis, and the relations (112) and (140) between the Fourier components of the displaced Fock-state distributions and the atomic-state inversion, respectively, and the density-matrix elements in the Fock basis. A powerful method for inversion of such relations has been leastsquares inversion89. The method has been used for quantum-state reconstruction in balanced optical homodyning (Tan" 1997]), unbalanced homodyning (Opatrny and Welsch [ 19971, Opatrny, Welsch, Wallentowitz and Vogel [ 1997]), cavity QED (Bardroff, Mayr, Schleich, Domokos, Brune, Raimond and Haroche [1996], Bodendorf, Antesberger, Kim and Walther [ 19981) and for orbital electronic motion (Cline, van der Burgt, Westerveld and Risley [1994]). It has been used further to reconstruct the quantum state of the center-of-mass motion of trapped ions (Leibfried, Meekhof, King, Monroe, Itano and Wineland [ 1996]), the quantum state of a particle in an anharmonic potential and the quantum state of a particle that undergoes a damped motion in a harmonic potential (Opatrny, Welsch and Vogel [ 1997~1). An advantage of least-squares inversion is that it is a linear method - the density matrix elements can be reconstructed in real time together with an estimation of the statistical error. Moreover, it allows for an easy incorporation in the reconstruction of various experimental peculiarities, such as nonunity quantum efficiency, finite resolution or discretization of the data, finite observation time, dissipative decay of the system, etc.. These aspects can hardly be treated on the basis of analytically determined (and existing) sampling functions. On the other hand, the method does not guarantee (similarly as any other linear method) that a reconstructed density matrix is exactly positivedefinite (cf. § 3.9.3). To illustrate the method, let us assume that a distributionp(x, cp) of the type of a quadrature-component distribution is measured, and that p(x, cp) can be given

The method of least squares was discovered by Legendre [1805] and Gauss [1809, 18211 for solving the problem of reconstruction of orbits of planetoids from measured data. 89

152

PI, 5 3


en,,!

by a linear combination of all density-matrix elements of the quantum state (x, cp), to be reconstructed, with linearly independent coefficient functions &,,I

where Snn,(x,cp) need not be of the form used in eq. (98)90.Since the density matrix of any physical state can be truncated at some value nmax,the sum in eq. (194) is effectively finite. Direct application of least-squares inversion (Appendix D) yields the reconstructed density-matrix elements @:$)as

where P ( ~ ) ( xcp), is the experimentallymeasured distribution, X and @ being the intervals accessible to measurement. The integral kernel Knnt(x,cp) is given by

and F = G-', with the matrix G being defined by

It can be proved by direct substitution that if the data correspond to the exact quantities p ( x , cp), i.e., P ( ~ ) ( xcp) , = p(x, cp), then the reconstructed density matrix equals the correct one, i.e., = On the other hand, if the experimental data suffer from some inaccuracies, then the reconstructed density matrix has the property that it reproduces the data as truly as possible (in the sense of least squares). In the above given formulas we have assumed that x and ~1 are continuous variables, and that n is discrete. The formulas for other combinations of discrete andor continuous arguments can be obtained in a quite similar way. An essential point of the method is the inversion of the matrix G, which requires the matrix to be sufficiently far from singularity; i.e., the data must carry enough information about all the density-matrix elements which are desired to be reconstructed. Otherwise regularized inversion must be applied (Appendix D). @,,,,I.

90 In particular, when x is the position of a moving particle and p corresponds to the time t , and the particle undergoes damping, then the quantum state evolves according to a master equation whose solution then determines Snn,(x, t ) .

11, § 31

153


Regularized inversion usually decreases the statistical error of the reconstructed density-matrix elements, but on the other hand they are biased. Therefore, in practice such a degree of regularization should be used for which the introduced bias is just below the statistical noise. 3.9.3. Maximum-entropy principle As already mentioned, it is principally impossible to measure the exact expectation values of an infinite number of operators 2;in an expansion of the density operator of the type given in eq. (69), because any realistic experiment can only run for a finite time. So far, the exact formulas have been applied to the analysis of the incomplete measurements including an estimation of the error made. However, the question may arise of how to obtain an optimum result of has been reconstruction of a quantum state when only a finite number of (2;) measured in the experiment. An answer can be given using the Jaynes principle of maximum entropy9' (Buiek, Adam and Drobny [ 1996a,b], Buiek, Drobny, Adam, Derka and Knight [ 19971). of n quantities i = 1,. . . n Let us assume that the expectation values (ji) are determined experimentally9'. The set of measured quantities can be regarded as a measure of the realized observation level. Certainly, there is a number of potential density operators 8, Tr6 = 1, which are compatible with the experimental results; i.e.,

a;,

Tr($;)

=

(&),

i = 1, . . .n.

Among them, that density operator is chosen that maximizes the von Neumann entropy 93 ,S[6]=-Tr(61nQ).

( 199)

See Jaynes [1957a,b]. Probability distributions (or density operators) describe our stage of knowledge about physical systems. If we do not know anythmg, we usually assign uniform distributions to the quantities (or a multiple of the unity operator to the density operators). If we have partial knowledge, we choose such distributions which are as broad as possible and still reflect our stage of knowledge. A suitable measure of the breadth is the entropy; one therefore seeks for such distributions (density operators) which maximize the entropy under the condition that known quantities are reproduced. 92 Note that the determination of already requires an infinite number of individual measurements which cannot be realized during a finite measurement time. 93 See von Neumann [1932]. 9'

(A,)

154


Introducing Lagrange multipliers, the resulting density operator @ the familiar (grand-canonical ensemble) form

tII, = @S

03

takes

where

and represents a partially reconstructed (estimated) density operator on the chosen observation level. Substituting in eq. (198) for @ the density operator @ S from eq. (200), a set of n nonlinear equations is obtained for the calculation of the n Lagrange multipliers A; from the measured expectation values Any incomplete observational level cad be extended to a more complete observational level, in principle, by including additional observables in the scheme, which is usually associated with a decrease of the entropy. However, since rather involved calculations are required to be performed, the method has been studied for reconstructing the quantum state of a radiation-fieldmode on particular (not very high) observational levels (Buiek, Adam and Drobn9 [ 1996a,b], Buiek, Drobnq, Adam, Derka and Knight [ 19971) and/or low-dimensional systems, such as spin states Wuiek, Drobny, Adam, Derka and Knight [ 19971). Since the expectation values cannot be measured with infmite precision, only estimates can be inserted into eq. (198), and therefore it can happen that the solution does not exist; i.e., the non-precisely measured averages are not compatible with any density operator. To overcome this problem, the method can be combined with least-squares minimization (Wiedemann [ 1996194); i.e., the sum of squares of differences

(ai).

(a;)

as a function of the parameters Ai is tried to be minimized, non-precisely measured averages.

(ij)(M) being the

94 In this paper, which is unfortunately unpublished, the operators d are identified with the photon number h and quadrature-componentprojectors 1x1,q k ) (XI, rpk I, where the subscript I labels a finite subset of the continuous quadrature-components at chosen phase rpk. Due to computational limits, 4 phases and 13 values of x at each phase are considered. The partial reconstruction of the quantum state (in phase space) from computer-simulated homodyne data is performed for various states and yields results which reflect typical properties of the states sufficiently well.

n, Q 31

155


3.9.4. Bayesian inference The statistical fluctuations of the data are taken into account in the Bayesian inference scheme (Helstrom [1976], Holevo [1982], Jones [1991, 19941, Derka, Buiek and Adam [1996], Derka, Buiek, Adam and Knight [1996]). Let us assume that the system under consideration is prepared in a pure state that belongs to a continuous manifold of states in a state space 0 and i = 1 , . . . ,n, with consider a repeated N-trial measurement of observables eigenvalues AJ,. The determination of the quantum state of the measured system is then performed in a repeated three-step procedure: (i) As a result of a (single) measurement of 8, a conditional probabilityp(A,, I@) is defined which specifies the result AJ, if the measured system is in state

a,,

0 = IW(W,

where !‘A,, = \Ajt)(Aj,1. (ii) A probability distribution PO(@) defined on the space 0 is specified such that it describes the a priori knowledge of the state to be reconstructed. The joint probability distribution p(AjL,@)is then given by

When no initial information about the measured system is available, then the prior probability distributionPO(@) is chosen to be constant. (iii) Finally, the Bayes rule95 is used to obtain the probability p(@lAj,)of the system being in state @ under the condition that Aj, is measured,

Now the procedure can be repeated, making a second measurement of some observable & , (& = or & # ,&) and using p(@IAji)obtained from the first measurement as the prior probability distribution for the second measurement.

ai

156


tII, § 3

Proceeding in the way outlined, the Ndrial conditional probability distribution ~ ( $ 1 {Aj, 1) is given by

where

is the likelihood function (regarded as a function of 6; the measured values {A,} playing the role of parameters). Note that C(@)is the probability distribution of finding the result { A ] , } in the sequence of N measurements under the condition that the state of the system is 6. The partially reconstructed density operator @LJ is then taken as the average over all the possible states @96,

To apply the method, the state space 9 of the measured system and the corresponding integration measure dS2 must be defined and the prior probability PO(@)must be specified. In particular, the integration measure must be invariant under unitary transformations in the space 9. T h s requirement uniquely determines the form of the measure. It should be pointed out that t h s is no longer valid when S2 is tried to be extended to mixed states. Since a system which is in a mixed state can always be considered as a subsystem of a composite system that is in a pure state, the Bayesian reconstruction can be applied to the composite system and tracing the resulting density operator over the degrees of freedom of the other subsystem (Derka, Buiek and Adam [1996], Derka, Buiek, Adam and Knight [ 19961). As already mentioned, the prior probability PO(@) can be chosen constant if there is no a priori information about the state to be r e c o n ~ t r u c t e d It ~ ~turns . out that with increasing number of measurements the method becomes rather insensitive to the prior probability. In particular,

96 Note that & can correspond to a mixed state even though it is assumed that the system is prepared in a pure state. 97 Note that when the integration measure d B is not defined uniquely, then a prior probability that is constant with respect to a chosen integration measure need not be constant with respect to another one.

11,

D 41

QUANTUM STATES OF MATTER SYSTEMS

157

when the number of measurements approaches infinity, N -+ m, eq. (208) corresponds, on chosen observation level, to the principle of maximum entropy on a microcanonical ensemble98. Contrary to least-squares inversion, Bayesian inference always yields a (partially) reconstructed density operator which is (formally) positive definite and normalized to unity. However, the price to pay is a rather involved procedure that has to be carried out in practice. For this reason, the method has been mostly considered - similar to the method of maximum entropy - for spin systems; i.e., for systems with small dimension of the state space (Jones [1991], Derka, Buiek and Adam [1996], Derka, Buiek, Adam and Knight [ 19961). Another statistical method related closely to the Bayesian reconstruction method is the quantum state estimation based on the maximization of the likelihood function C@), eq. (207). Here, that state 0 in the state space is selected for which C(6) attains its maximum. Because of the difficulty of finding the maximum of L(0)in higher-dimensional state spaces, a procedure was proposed which is based on a sequence of general inequalities satisfied by the likelihood function (Hradil [ 19971).In this way, the problem can be tranformed to that of the diagonalization of an operator given by a linear combination of the projectors where the expansion coefficients must solve a set of nonlinear algebraic equations. The method simplifies the Bayesian treatment but still guarantees the positive definiteness. Significantly, all the solutions based on the deterministic relation (69) between counted data (frequencies) and the desired density matrix are involved. Whenever such solution exists as a positively defined density matrix, then it should maximize the likelihood function as well. The method was applied successfully to the reconstruction of (low-dimensional) density matrices in the photon-number basis of a radiation-field mode from computer-simulated homodyne data, and a comparison with direct inversion of the linear basic relation between the measurable quantities and the density-matrix elements was given (Mogilevtsev, Hradil and PeEna [ 19971).

8 4.

Quantum States of Matter Systems

In the preceding sections we have considered phase sensitive measurements of radiation fields and methods for reconstructing the quantum state of the

98 Note that on a chosen (incomplete)observational level the two methods yield different fluctuations

of the observables in general.

158

HOMODYNE. DETECTION AND QUANTUM-STATE RECONSTRUCTION

“1,

04

fields from the measured data. The problem of quantum-state measurement and reconstruction has also been studied for various matter systems. Different matter systems require, in general, different detection schemes for measuring specific quantities that carry the full information on the quantum state of the system. Although these methods may be, at first glance, quite different from the methods outlined in Q 2 for phase-sensitive measurements of light, there have been a number of analogies between the reconstruction concepts for radiation and matter.

4.1.

MOLECULAR VIBRATIONS

It was shown and demonstrated experimentally that the quantum state of molecular vibrations can be determined using a tomographic method (Dunn, Walmsley and Mukamel [1995]) which resembles the one for a light mode outlined in 9 3.1. The method, called molecular emission tomography, is based on the fact that the time resolved emission spectrum of a molecule allows one to visualise the time dependence of a vibrational wave packet withn the excited electronic state from whch the emission originates (Kowalczyk, Radzewicz, Mostowsh and Walmsley [19901). Alternatively, the desired information on the wave-packet dynamics can be obtained by photoelectron spectroscopy (Assion, Geisler, Helbing, Seyfried and Baumert [1996]). Let us assume that the molecule is prepared in a given vibrational quantum state in the excited electronic state. As can be seen from fig. 17, for appropriately displaced potential energy surfaces of the molecule the position of the vibrational wave packet can be effectively mapped onto the frequency of the emitted light. This fact is used for the tomographic reconstruction of the vibrational wave packet by measuring the time-resolved emission spectrum with a time resolution that is fast compared with the characteristic time period of the molecular state to be studied. The experimental realization has been performed as follows (Dunn, Walmsley and Mukamel [ 19951). A sample of Na2 molecules is illuminated by a 4 k H i train of laser pulses of 60 fs duration and mean wavelength of 630 nm. The laser pulses generate vibrational wave packets in the A’Z;+ state of the sodium dimer, which evolve with a time period of 310 fs. A fraction of the pulses is split off and plays the role of a time-gate shutter. The light emitted from the molecular sample is collected and focused synchronously with the split-off part of the exciting pulse onto a nonlinear crystal. A prism monochromator is used to filter the resulting sum-frequency and the field is recorded by a photon-counting photomultiplier. The resulting temporal resolution of the device is about 65 fs.


159

15,000

10,000

5,000

c

Fig. 17. The vibronic energies for the A'ZG ---t X'Z; transition of Na2 clearly show the possibility to display the vibrational motion (in the excited state) in the time-resolved emission spectrum. (After Kowalczyk, Radzewicz, Mostowski and Walmsley [ 19901.)

4.1.1. Harmonic regime

The first reconstruction of the quantum state of molecular vibrations from a time-resolved emission spectrum was based on the assumption that only low vibrational quantum states are excited such that the relevant potentials can be approximated by harmonic ones. Furthermore, it was assumed that the vibrational frequencies in the two electronic quantum states, which contribute to the emission spectrum, are nearly equal. In this case, the vibronic coupling is caused solely by the displacement of the equilibrium positions of the potentials in the two electronic states. When these approximations are justified, then the time-gated spectrum S(Q, T ) can be related to the s-parametrized phase-space distribution P(q,p;s) 3 2-'P[a = 2-"2(q + ip);s] as (Dunn, Walmsley and Mukamel [ 19951)

S(Q, T ) =

s

dyP[x(Q)cos(vT)+ysin(vT),ycos(vT)-x(Q) sin(vT);s]. (209)

160


111,

54

28 7926 15. 247122 6820 6418 61-

14.544

... .,........

-6.2

-4.6

-3

-1.4

0.2

1.8

3.4

Momentum (Ground state width).' Fig. 18. Tomographic reconstruction of an s-parametrized phase-space function for a vibrational wave packet in sodium from a measured set of emission spectra. The time gate duration was 65 fs, which implies s = -0.8. (After Dunn, Walmsley and Mukamel [199S].)

Here,& and T are the setting frequency of the spectral filter and the setting time of the time gate, respectively ( Y , vibrational frequency). The function x(Q) = (D- K ~ Y ) / ~ Kdescribes V the mapping of the position of the wave packet onto the emitted frequency, K being the ratio of the displacement of the potentials to the width of the vibrational ground state. The ordering parameter s in eq. (209) is related to the temporal width r-' of the time gate as s = -(T/KY)2.

Obviously, the relation (209) between the phase-space distribution P(q,p ; s) of the molecular vibration and the measured spectrum S(Q,T) is very close to the basic relation (70) of optical homodyne tomography. Thus P ( q , p ; s )can be obtained from S(Q, T ) by means of inverse Radon transform (see 9 3.1). A typical experimental result is shown in fig. 18. 4.I .2. Anharmonic vibrations

In general, molecular vibrations are known to be significantly anharmonic when their excitations are not restricted to very small numbers of vibrational quanta. In such cases one cannot apply the approximate reconstruction procedure based on eq. (209). Due to the anharmonicity effects it is no longer possible to reconstruct the quantum state from only one half of a vibrational period.

11,

P 41


161

It was proposed (Shapiro [1995]) to reconstruct the vibrational quantum state from the time-resolved (spectrally integrated) intensity Z(T) = 1d S S ( S , T ) of the light emitted by the molecular sample, which can be related to the densitymatrix elements of the vibrational mode in the excited electronic state as

emn

Here, vmnare the vibrational transition frequencies in the excited electronic state, and k h (o‘;‘:)~ 1(kl.2)2 is determined by the Franck-Condon overlap l ( k l . 2 ) ~of the vibrational wave functions in the two electronic states and the vibronic transition frequency w?!. Equation (210) reveals that, as long as the transition frequencies vmnare nondegenerate, the corresponding density-matrix elements pmncan be obtained, in principle, from an analysis of I ( T) as a function of T. However, the separation of the density-matrix elements from each other may require a rather long time series. Further, the dimension of the set of equations to be inverted can be large, because of the large number of densitymatrix elements that may contribute to the intensity of the emitted light. In the degenerate case, which is observed for the diagonal elements of the density matrix and for some off-diagonal elements due to the anharmonicities, adhtional information is needed. It was proposed to use the stationary spectrum of the light, whose determination requires an additional measurement. Alternatively, the two measurements can be combined such that the density-matrix elements are reconstructed from the time-resolved spectrum (Trippenbach and Band [ 19961)99. It is worth noting that the dimension of the sets of equations that must be inverted numerically can be reduced substantially by employing the 1 1 1 information inherent in the time-resolved spectrum:

-

(Waxer, Walmsley and Vogel [ 19971). In eq. (2 1l), the blurring function g ( w ) = exp(-w2/4r2) is determined by the resolution time r-’ of the time gate (see 54.1.1). In practice, the time-dependent spectrum is available only in a finite time interval of size z, which can be taken into account by multiplying S ( a , T)

99 Trippenbach and Band [1996] also discussed the inclusion of molecular rotations in the reconstruction.

162


[II,

54

by the corresponding samplingswindow function G ( T ,z) in order to obtain S ' ( 0 , T ) = S ( 0 , T ) G ( T ,z). The Fourier transform of S ' ( 0 , T ) with respect to T reads

with G(Y- v,,,,, z) being the Fourier transform of the sampling window. Fixing the frequency Y of the time-series spectrum, the structure of the window function ensures that only some of the density matrix elements contribute at this frequency, depending on the chosen size z of the sampling window. Let us assume that the density matrix elements @,,,(i = 1,2,3,. . . ,N ) contribute to the spectrum at chosen frequency Y. By choosing N frequencies 0 = Qi of the emission spectrum, one gets from eq. (212) a linear set of N equations of rather low dimension that can be inverted numerically loo. In this way, the method allows one to reconstruct rather complicated quantum states on a time scale that can be shorter than the fractional revival time, as was demonstrated in a computer simulation of measurements (Waxer, Walmsley and Vogel [1997]). Note that a reduction of the dimension of the set of equations to be inverted improves the robustness of the method with respect to the noise in the experimental data. It was also proposed to reconstruct the density matrix from the time-dependent position distribution according to eq. (102) (Leonhardt and Rayrner [ 19961, Richter and Wunsche [ 1996a,b], Leonhardt [ 1997a1, Leonhardt and Schneider [1997]). This requires a scheme suitable for measuring either the position distribution of molecular vibrations or another set of quantities that can be mapped onto the position at different times. Note that this is hardly possible in molecular emission tomography in general. Finally it was proposed to use a wave-packet interference technique (Chen and Yeazell [1997], $4.5.1) for reconstructing pure vibrational states in the excited electronic state (Leichtle, Schleich, Averbukh and Shapiro [ 19981).

loo Note that the emission frequencies Sri are determined by the two vibrational potentials involved in the vibronic emission, whereas the degenerate values of the vibrational frequencies v,,,, are determined solely by the vibrational potential in the excited electronic state. The (within the resolution of the sampling window) degenerate density-matrix elements usually contribute to the emission spectrum at distinct frequencies Sri.

11,s 41


163

4.2. TRAPPED-ATOM MOTION

Since the first observation of a single ion in a Paul traplo' (Neuhauser, Hohenstatt, Toschek and Dehmelt [ 1980]), much progress has been achieved with respect to laser manipulation of the quantized motion of single atoms in trap potentials. Such systems are of particular interest since the quantized lowfrequency (- MHz) motion is very stable, and laser manipulations allow one to prepare very interesting nonclassical states. Until now, motional Fock states and squeezed states (Meekhof, Monroe, King, Itano and Wineland [1996]) as well as Schrodinger-cat type superposition states (Monroe, Meekhof, King and Wineland [19961) have been realized experimentally. One might expect that the reconstruction of the quantum state of the centerof-mass motion of a trapped atom may be very similar to the reconstruction of molecular vibrations. However, the vibronic couplings in the two systems are basically different. In the case of a molecule, the vibrating atoms are close together within atomic dimensions and the change of the electronic state substantially alters the potential of the nuclear motion. Ln the case of an atom in a Paul trap, the potential of the center-of-mass motion is given externally by the trap. In this case, electronic transitions can hardly affect the potential. The vibronic interaction in this system is induced by the interaction with radiation. Therefore one may expect that appropriate interactions of a trapped atom with laser fields may open various possibilities for measuring the motional quantum state. 4.2.1. Quadrature measurement

The first proposals to reconstruct the motional quantum state of a trapped atom were based on measuring the quadrature components of the atomic center-ofmass motion in the (harmonic) trap potential. In the scheme in fig. 19 a weak electronic transition 18) = 11) H le) = 12) of the atom is proposed to be driven simultaneously by two (classical) laser beams whose frequencies orand cob, respectively, are tuned to the first motional sidebands, w, = 021 - Y and o.$, = 021 + Y of the electronic transition of frequency ~1 (Wallentowitz and Vogel [1995, 1996b1). Since the linewidth of the transition is very small, the motional sidebands can be well resolved. For a long-lived transition and in the

lo'

For the trap, see Paul, Osberghaus and Fischer [1958].

164


/7

v

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

14 '

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

Fig. 19. Scheme of a trapped ion with a we& transition 11) ++ 12) and a strong transition 11) ++ 13). Two incident lasers of frequencies y = y~- v and = ~1 + v are. detuned from the electronic transition by the vibrational frequency Y to the red and blue respectively. The laser driving the strong transition is used for testing the ground state occupation probability by means of resonance fluorescence. (After Wallentowitz and Vogel [ 19951.)

LamkDicke regime lo2 these interactions are well described by Hamiltonians of the Japes-Cummings (and anti Japes-Cummings) type '03. Assuming that the Rabi'fiequencies of the two (classically treated) laser fields are equal, the two Jaynes-Cummings interactions can be combined to an electron-vibration coupling that in the interaction picture reads as

A' = ;hQLJZ (6- + G+)qq), where QL is the vibronic Rabi frequency of the driven transitions, and q is the phase difference of the two lasers which can be controlled precisely. Assuming

Io2 In this regime, the (one-dimensional) spread of the motional wave packet, Ax, is small compared with the laser wavelength AL over 2n, A x > 1 can be expressed as (Byron, Joachain and Mund [1973]) fex

=h,+ [ki+Lk] A . B + [ki+lk'] C .D + . . . (2.38)

111,

5 21

THE EIKONAL APPROXIMATION M NON-RELATIVISTIC POTENTIAL SCATTERING

229

In contrast, the eikonal series yields:

In the above equations, A and B are of second order in potential strength and C and D are third order in potential strength. A comparison of eq. (2.39)with (2.38) shows that neither & 1 +f ~ 2nor & 1 +fB2 is correct to order l/k2. But because A and B are second order in potential strength while C and D are third order in potential strength,f =fB1 +fB2 is more accurate than the eikonal amplitude for the weak coupling case. The addition of the real part of the second Born term to the Glauber amplitude thus results in impressive improvements for Yukawa type potentials. Two alternative amplitudes corrected in this way are (Byron and Joachain [19771) fEBS

=fBl +fB2 +fE3,

and

f&S

=fE + Ref ~ 2 ,

(2.40a,b)

which are correct up to order (1/k2)and have been referred to in the literature as the eikonal-Born amplitudes. 2.8.3. The generalized eikonal approximation An alternative approach to rectify the defect of the missing Refs2 term in the EA is to write a generalized linear propagator of the form (Chen [ 19841)

1 Gl(r) = -b2(b) 21a

Q ( z )e'p',

(2.41)

where the arbitrary parameters a and are determined in such a way that the dominant real part as well as the dominant imaginary part of the second Born amplitude are correctly reproduced. The resulting formula, termed as the generalized EA(GEA), has been found to work very well for Yukawa and Gaussian potentials even at large angles. 2.9. RELATIONSHIP WITH RYTOV APPROXIMATION

The Rytov approximation (Rytov [1937],Nayfeh [1973])assumes a solution of the Schroedinger equation of the form

230

SCATTERING OF LIGHT IN THE EIKONAL APPROXIMATION

[a8 3

where R(r, p) = C,"=, R,(r) is an expansion in powers of coupling constant (R,(r) is of order 5;"). The scattering amplitude may then be expressed as (Car, Cicuta, Zanon and Riva [19771) f ( q ) = -2k

1

eiq'bd2b [ex 7 Rn - 11 ,

(2.43)

where

The lowest nontrivial order of the Rytov expansion is same as the EA. The first order correction differs from the first order Wallace correction. This is because while the eikonal expansion is an expansion in powers of k-' ,the Rytov expansion is an expansion in powers of potential strength.

5

3. Eikonal Approximation in Optical Scattering

Light scattered by an obstacle is related to its physical properties and hence in principle it is possible to obtain information about the scatterer from an analysis of the scattered IigM. Thus, for many years, the light scattering technique has been used to infer the size, shape and refractive index of particles in various scientific disciplines such as biophysics, colloid physics, molecular physics, optical fibers, plasma diagnostics, atmospheric physics and astrophysics, etc. The fact that it is a non-destructive technique and can be used for atmospheric and astrophysical particles, which are not easily accessible otherwise, makes it a very attractive diagnostic tool. Light scattering experiments generally involve measurements either from one particle at a time or measurements from an ensemble of particles. In either case, a theory for predicting the scattering pattern from a single particle is necessary. Unfortunately, the problems involving the scattering of light are very complex. The exact solutions are unknown except in the simplest and most idealized cases of a sphere (Mie [1908]), an infinitely long cylinder (Wait [ 19551) and a spheroid (Asano and Yamamoto [1975]). Simple inhomogeneous objects such as a concentric sphere (Aden and Kerker [1951]) and an infinitely long cylinder (Tang [1957], Kerker and Matijevi6 [1961]) can also be treated exactly. For particles of other shapes one may resort to numerical procedures. The complexity of numerical solutions has become of less consequence with the advent of faster and faster computers. However, in many applications a numerical

111,

8 31

EIKONAL APPROXIMATION IN OPTICAL SCATTERING

23 1

approach still proves to be tedious, impractical or even impossible, and one must resort to approximation methods. That apart, approximation methods give simple expressions for quick and easy use and also provide a deeper physical and mathematical insight into the general scattering problem. Thus, one is aware of important factors involved in the use of any approximation. A brief review of some analytic approximation methods for spherical particles has been recently given by Kokhanovsky and Zege [ 19971. The EA has proved to be a very good approximation for the analysis of the near forward scattering pattern for particles whose refractive index is close to that of the surrounding medium. It has been applied to the scattering of light by homogeneous spheres, coated spheres, infinitely long cylinders, spheroids and rough particles. A large variety of particles can be modeled using these shapes. It was introduced in the context of optical scattering by drawing an analogy between the Schroedinger equation and the scalar wave equation of optics. Consequently, it is expected to be applicable in situations where the scalar description of the scattering process is sufficiently accurate to allow the vector nature of light to be ignored. Attempts have been made to include the vector character of light in the analysis of the scattering process and a reasonable degree of success has been achieved. 3.1. _ANALOGY WITH POTENTIAL SCATTEFUNG

Consider a scalar wave characterized by the field v(r) propagating through a medium of spatially varying relative refractive index m(r). The field v(r) then satisfies the wave equation V 2 q ( r ) +k2m2(r)v(r) = 0.

(3.1)

A comparison of eq. (3.1) with the Schroedinger eq. (2.1) shows that

U(r) = [1 - m2(r)] k2.

(3 .a

With this relation between potential and refractive index, one can easily write the scattering function S(~)E,(=-i&"(O)E*) as

S(0),, where

= -k2

/

db eiq" [eixo@) - 13 ,

(3.3)

232

[IK8 3


The conditions (2.11) for the validity of the EA in potential scattering now translate to

Im(r)- 11 > 1,

(3.5aYb)

where the dimensionless parameter x = ka is the size parameter and is essentially a measure of the size of the scatterer in terms of the wavelength of the scattering radiation. The notation x , has also been used in the coordinate system r = ( x , y , z ) . However, this notation does not result in any confusion and has been used in most books on the subject. The requirement (3.5a) ensures that at boundaries there is no deviation of the incident ray and that the energy reflected is negligible. The second requirement ensures that the ray travels undeviated through the scatterer as the refractive index varies slowly in a wavelength. The angular range given by,eqs. (2.25a) and (2.25b), now translate to

and

Here, m may be taken as the maximum value of the refractive index. For complex m(r) = n(r) + in’@), the condition (3.5a) is equivalent to In@) - 11 c

d

d

Q,

a lo3 10'

0

10

20

30

40 50 Scattering Angle in degrees

60

Fig. 5. i(0) versus 0 for perpendicularly polarized light by a dielectric sphere with rn = 1.33 + iO.001 and x = 30,60,90. Solid curve, Mie theory; dashed curves, MGEA, dash-dotted curve at x = 30, GEA. (From Chen and Smith 119921.)

be a good approximation at all angles if ~ o ( bis) singular at b = 0 (see Q 2.8). For a homogeneous sphere ~ o ( bis) analytic at b = 0. Hence the approximation is expected to be valid only at small scattering angles. Not many numerical studies of angular variation of i(0)have been performed. Whilst Chen [1988, 19891 and Chen and Smith [1992] have compared the angular scattering patterns of the EA, the ADA and the GEA with exact results, Perrin and Chiappetta [ 19851 and Sharma, Roy and Somerford [1988a] have examined the EP and the EA against exact results. The following conclusions emerge from these studies: (i) The EA works to within 25 percent error in the domain x 3 1, 1 n 6 1.2 and 8 < 10.0. (ii) The GEA method works very well for light scattering by a dielectric sphere. It greatly improves the EA results. More importantly, it appears to work very well for y1 as large as 4.0.However, the success is only for scattering angles up to 5". Its improved variant, MGEA, is found to work well for the scattering of light with perpendicular polarization. It predicts accurately the positions of minima and maxima for 0 up to 60", x 2 5.0, n < 4 and n' < 0.5. Figure 5 shows a comparison of GEA and MGEA with exact i(0) versus 8 curves for x = 30, 60 and 90. The refractive index of the scatterer is m = 1.33 + iO.001. (iii) The simplified version of the GEA, given in eq. (3.43), is found to work as well as the GEA for x > 10. (iv) For a dielectric sphere the EA as well as the EP agree well with the exact results for small angle scattering. In the forward lobe the two approximations are very similar, but the EA has some advantage over the

160°, where the eikonal model predicts oscillations not present in Mie calculations. The degree of polarization is in perfect agreement with the Mie theory predictions for 8 > Om,,. Agreement is better for particles of larger sizes. Bourrely, Lemaire and Chiappetta [ 199I] have also examined the possibility of applying the above formalism to ellipsoidal particles. For a perfectly

n ~o, 31


253

I *

102 101

1 8 (degree)

s

1

.c N ._ L

0

0

a 0.5

r

W

Em W

U

0 0

50

100

150

Fig. 10. (a) The perpendicular component of the scattered intensity and (b) the degree of polarization as a ftmction of scattering angle 0 for a perfect homogeneous sphere of radius a = 150pm and index of refraction rn = 1.3 + iO.01. Here 1 = 6.28 pm. (From Bourrely, Lemaire and Chiappetta [1991I).

homogeneous ellipsoidal particle whose axis of symmetry is parallel to the direction of incident ray, cos z is given by cos z =

bb/a [ 1+ (pi - l)b2/u2]”* ’

where u is the size of the ellipsoid along its minor axis and ,HQ is the ratio of the major axis to the minor axis. This approach, however, is valid when (b- 1) is not very large because the Mie coefficients’are specific to spherical shape. The scattered intensity exhibits a behavior similar to the scalar eikonal solution for a perfect sphere. Numerical comparisons with the exact results have not been performed. 3.4. SCATTERING BY AN INFINITELY LONG CYLJNDER

Scattering of light by an infinitely long homogeneous circular cylinder is another situation where the EA has been examined in detail. This model of scattering

254


§3

lends itself to many practical ,situations and is perhaps the second most widely employed model in light scattering applications. 3.4.1. The scattering function for normal incidence

Any electromagnetic wave incident normally on an infinite long cylinder can be considered a superposition of transverse electric and transverse magnetic waves. If the electric vector is perpendicular to the axis of the cylinder, the wave is termed transverse electric. On the other hand, if the magnetic vector is perpendicular to the cylinder axis the wave is termed transverse magnetic. In either case, both the electric as well as the magnetic vector are perpendicular to

I

HY TEWS

TMWS

Fig. 1 1. Scattering geometry for an infinitely long cylinder.

the direction of propagation. The scattering geometry is shown in fig. 11. The plane ( x , y ) is the scattering plane and the direction of the incident beam is taken to be the z-axis. Since E,(x,y) is independent of z, it can be shown easily that for transverse magnetic wave scattering (TMWS), the Maxwell equations reduce to the following two-dimensional equation:

[V: + k2m2(x,y)]&(x,y)

= 0,

& &

(3.60)

: = + and m ( x , y ) is the relative refractive index inside the where 0 cylinder. For transverse electric wave scattering (TEWS) however, Maxwell's equations take the form eq. (3.60) only if ka >> 1. Using standard techniques, the scattering function for this problem is written as (Alvarez-Estrada, Calvo and Juncos del Egido [1980]) T(ki,/if)=

-ik2 4 ~

J

db e-ik'b[m2(b)-

n ~g, 31


255

where b is a two-dimensional vector in the (x,y) plane. For eikonalization of E,(b), as in the Mie case, a trial solution E;(b) = exp[ikx] $(b) is chosen. This in eq. (3.61) gives

[.:+

$(b) = -k2 [m2(b)- 11 $(b).

Zik;]

(3.62)

Ignoring 0 : in the above equation, the resulting equation with boundary condition ( ~ ( 6=) 1 at x = -00, yields the solution

(3.63) with a as the radius of the cylinder. Theoretically, the inequalities lm2(b)- 1 I > 1 govern the validity domain here, too. The solution (3.63), when substituted in eq. (3.61), gives

where q = ki - kf is the momentum transfer. For small angle scattering, q is nearly perpendicular to the x-axis and one may approximate q . b by qy, where q = -k sin 8. The EA then gives

(3.64) where

(3.65) is the phase shift suffered by the ray in travelling undeviated across the cylinder. 3.4.2. Scattering by a homogeneous cylinder

For a homogeneous cylinder, m(r) = m, and the scattering function becomes

(3.66) Making the change of variable y form,

1

=

a sin y, it can also be cast in an alternative

n/2

T(o>,A = x

dy cos y cos(z sin y ) [I - exp(u* cos y)] ,

(3.67)

where z = x sin 8 and as before U * = (i& - pgAtanB). T ( 6 ) A D A can be obtained from T(~ ) E Aby replacing pEAby piDA.For a thin soft cylinder, such

256

SCATTERMG OF LIGHT IN THE EIKONAL APPROXIMATION

that x/m2- 1I 0 the situation is more complicated. When p > 0, the associated Laguerre polynomial is a function of r, the square of which can no longer be described by a single Laguerre polynomial. In general, the square of a Laguerreaaussian mode with mode indices I and p can be expressed as the sum of LaguerreGaussian modes with indices 21 and p =0,2,. . . 2p (Courtial, Dholakia, Allen and Padgett [1997b]). Although these modes all have the same Rayleigh range, their Gouy phases are not the same. Consequently, the constituent modes interfere to give a beam distribution that changes with propagation. It is only in the far field, where the Gouy phase shifts of the constituent

324

THE ORBITAL ANGULAR MOMENTUM OF LIGHT

Fundamental

I

Second Harmonic

e = i , p=o

e =I, p=i

e=2, p=i

e=i, p=2

Fig. 9. The forked interferograms formed between a beani and its mirror image, for a variety of LG beams and their second harmonic counterparts.

modes differ by multiples of 2n, that the distribution at the beam waist is reproduced.

0

8. Mechanical Equivalence of Spin and Orbital Angular Momentum: Optical Spanners Following the calculations wluch predicted that LG modes should possess an orbital angular momentum along the beam axis, attempts were made to demonstrate the transfer of this orbital angular momentum to matter. Initial experiments attempted unsuccessfully to measure the torque exerted on a cylindrical lens mode converter suspended from a quartz fiber (see Beijersbergen [1996]). Experiments of this kind are extremely challenging, as a slight misalignment between the optical beam and the suspension generates unwanted torque on the lens that may be many times larger than the torque of interest. As an alternative, it was suggested that the physical suspension mechanism could be

Iv, 0 81

OPTICAL SPANNERS

325

removed and the optical beam itself used to “suspend” the object to be rotated (Padgett and Allen [ 1995a1). Any dielectric material falling within an electric field gradient experiences a force towards the region of highest field. In the vicinity of a tightly focused laser beam the gradient force is strong enough to trap a micron-sized dielectric particle in three dimensions (Ashkin, Dziedzic, Bjorkholm and Chu [1986]). This technique is now commonly referred to as optical tweezers and is widely used in many biological applications, such as measuring the compliance of bacterial tails (Block, Blair and Berg [1989]), the measurement of the forces exerted by single muscle fibers (Finer, Simmons and Spudich [1994]) and the stretching of single strands of DNA (Wang, Yin, Landick, Gelles and Block [ 19971). The trapped object is held on the beam axis by use of an LG mode within optical tweezers and any bbserved rotation is due purely to the transfer of orbital angular momentum. The first observation of the transfer of orbital angular momentum to a particle was made using an 1 = 3 helical beam within optical tweezers to trap absorbing ceramic powder suspended in kerosene (He, Friese, Heckenberg and Rubinsztein-Dunlop [ 19951). However, the 100% absorption of those particles resulted in a force in the propagation direction of the laser beam and instead of being held at the beam focus, these particles were trapped against the microscope slide. The rotation of the ceramic particles was attributed to the absorption of the orbital angular momentum from the light beam. With particles which absorb only a few percent of the incident laser light, the gradient force can be sufficient to overcome the force associated with the radiation pressure. For partially absorbing teflon particles suspended in alcohol, it is possible to observe rotation while still forming a genuine 3-dimensional optical trap (Padgett and Allen [ 1995b], Simpson, Dholakia, Allen and Padgett [ 19971). The trapped Teflon particles were observed to rotate at several Hertz; hence the term “optical spanner”. Not only does this eliminate any doubts over the origin of the observed rotation, but it also creates a potentially useful manipulative tool. In principle, the rotation speed could be compared with predictions made on the basis of the estimated absorption of the light and the viscous drag of the surrounding fluid. However, the errors inherent in these measurements make quantification of the orbital angular momentum extremely difficult. However, the orbital angular momentum can be compared directly to the spin angular momentum of h per photon. With a circularly polarized, I = 1, LG mode the handedness of the polarization can be set to give a total angular momentum

326


Orbitalspin

Orbital

,

[n! 5 9

Orbital t spin

,

0 ms

40 ms

80 ms

v

120ms No rotation

Rotation

I

Fast

rotation

Fig. 10. Successive frames of a video showing that the spin and orbital angular momentum terms can be added to give faster rotation, or subtracted to give no rotation, in an optical spanner.

of h+-h=2h or h - h = O per photon. The observed start/stop nature of the rotation, which is shown in fig. 10, confirms that the orbital angular momentum associated with an 1 = 1 LG mode is h per photon (Simpson, Dholakia, Allen and Padgett [1997]). This experiment confirms that both spin and orbital angular momentum of light are transferred to a particle in a mechanically equivalent fashion.

8

9. Rotational Frequency Shift

As discussed in 6 5 , a n/2 mode converter introduces orbital angular momentum into the light beam. This is analogous to a quarter-wave plate introducing spin angular momentum to plane polarized light. Similarly a n mode converter is analogous to a half-wave plate and they reverse the handedness of the orbital and spin angular momentum, respectively. A rotating half-wave plate was shown, some 20 years ago, to shift the frequency of a circularly polarized light beam by twice the rotation frequency of the wave plate (Garetz and Arnold [ 19791). A simple polarization analysis shows

IV, Q 91

ROTATIONAL FREQUENCY S H F T

327

that this is due to a corresponding rotation of the electric field vectors at twice the rotation frequency of the wave plate. If the beam has a circularly symmetric intensity distribution, such a rotation of the electric field vector is equivalent to a rotation of the beam. Consequently, this frequency shift can be considered to arise from a rotation of a light beam which possesses angular momentum and is equal to the rotation frequency of the beam multiplied by the angular momentum per photon in units of fi. Recently, a similar behavior was predicted for a Laguerre-Gaussian beam (Nienhuis [ 19961). Although analyzed in terms of an energy exchange, a rotating n mode converter was shown to introduce a frequency shift of twice the rotation frequency. It has been shown (Courtial, Dholakia, Robertson, Allen and Padgett [ 19981) that both of these effects are examples of the recently highlighted rotational frequency shift (BialynickiABirula and Bialynicka-Birula [19971) or angular Doppler shift (Garetz and Arnold 119791). This shift should not be confused with the translational Doppler shift observed for rotating objects, which is due to the rotation having a linear velocity with respect to the observer. Unlike the translational Doppler shift which is maximal in the plane of rotation, this rotational effect is maximal in the direction of the angular velocity vector, where the linear Doppler shift is zero. The translational Doppler shift is equal to the linear velocity between source and observer multiplied by the linear momentum per photon. By contrast, the rotational Doppler shift is equal to the rotational velocity between source and observer multiplied by the angular momentum per photon. The rotation of a source or detector without the introduction of slight off-axis motion is difficult to achieve. But a rotating Dove prism and half-wave plate can be combined to simultaneously rotate the electric field vector and phase structure of the beam. This provides beam rotation while the source remains stationary. Both the spin and orbital angular momentum of the photon contribute in an additive and interchangeable way to the rotational Doppler shift. For circularly polarized Laguerre-Gaussian modes, the origin of this result is particularly simple to understand by examining the transverse form of the electric field; see fig. 11. In each case the field distribution is (I + a)-fold rotationally symmetric. For circularly polarized light, the electric field rotates at the optical frequency and a rotation of the beam at f2 introduces an additional (I + a ) phase cycles per revolution. The sense of this rotation relative to the circular polarization results in an up-shift or down-shift in frequency of (I + a)B. This shift has been measured directly using a highly accurate frequency counter and a “light” source in the mm-wave region of the spectrum (Courtial, Robertson, Dholakia,

328


a=-I

0=+1

I= 1

1=2

I= 3

Fig. 11. Vector plots of the transverse electric field for circularly polarised LG beams, showing the (I + a)-fold rotational symmetry.

Allen and Padgett [1998]). Any arbitrary field distribution can be expressed as a superposition of circularly polarized Laguerre-Gaussian modes and if rotated will therefore give rise to a frequency spectrum consisting of sidebands about the unshifted frequency. In a similar fashion to the optical spanner, we find that both the spin and orbital components of angular momentum act in an equivalent fashion. $j 10. Atoms and the Orbital Angular Momentum of Light

It is well known that the interaction of conventional laser light with a free

rv, §

101

ATOMS AND THE ORBITAL ANGULAR MOMENTUM OF LIGHT

329

atom can give rise to electromagnetic pressure forces which act on its center of mass. Such forces have been the subject of much investigation in both theory and experiment (Ashkin [1970a,b], Letokhov and Minogin [19871, Kazantsev, Surdutovitch and Yakovlev [1990], Arimondo, Phillips and Strumia [ 19921, Metcalf and van der Straten [1994], Adams and Riis [1997]). The basic features can be understood in terms of a simple model comprising a two-level atom subject to a plane electromagnetic wave which gives rise to two kinds of force. These are a dissipative force arising from the absorption of the light by the atom, followed by its spontaneous emission, and a reactive, or dipole, force arising from the non-uniformity of the field distribution. These forces underpin the manipulation of atoms by lasers in a variety of beam configurations. The dissipative force has been exploited in the Doppler cooling of the atomic motion (Wineland and Dehmelt [1975], Hansch and Schawlow [1975]) and the dipole force used for trapping (Chu, Bjorkholm, Ashkin and Cable [1986]). It appears likely that the interaction of atoms with beams possessing orbital angular momentum should lead to new effects. Theoretical studies of the interaction of LG beams with atoms have been conducted recently (Babiker, Power and Allen [ 19941, Allen, Bablker and Power [ 19941, Power, Allen, Babiker and Lembessis [ 19951, Allen, Lembessis and Babiker [ 19961, Allen, Babiker, Lai and Lembessis [1996], Lai, Babiker and Allen [1997], van Enk [1994], Masalov [1997], Kuga, Torii, Shiokawa and Hirano [1997], Wright, Jessen and Lapeyere [19961) to examine how the main features of Doppler cooling and trapping are modified when a plane wave or a fundamental Gaussian beam is replaced with LG light. It is also desirable to consider the role LG beams are likely to play in the emerging field of atom optics (Dowling and Gea-Banacloche [ 19961). To study the effects of the orbital angular momentum of light on atoms it is necessary to consider the theory of forces due to LG light and their effects on a two-level atom. This has been done by Allen, Babiker, Lai and Lembessis [ 19961, who also extended their investigations on the orbital angular momentum effects to more than one beam, in order to explore more fully the effects of the orbital angular momentum on atomic motion. The theory has been developed in terms of the optical Bloch equations (OBE) (Cook [ 19791, Letokhov and Minogin [ 19871, Dalibard and Cohen-Tannoudji [1985]) which allow the ab initio inclusion of relaxation effects and incorporate saturation phenomena naturally. The solution of the OBE in the adiabatic, or constant-velocity, approximation gave insight into the time evolution of angular momentum effects for an atom in an LG beam. It is useful to outline a derivation of the force acting on a two-level atom in the presence of monochromatic coherent light beam based on the density matrix formalism. The coherent light beam is assumed to have a complex amplitude and

330


[n!§

10

a Laguerre-Gaussian (LG) spatial distribution. The Hamiltonian of the system is

where HA and HF are the zero-order Hamiltonians for the unperturbed atom and field, respectively, and are explicitly given by:

P2 HA = -+ ACIIOJC~JC, 2M HF

= hoata.

(10.2) (10.3)

Here A4 is the atomic mass, P is the momentum of the center-of-mass, and nt and n are the lowering and raising operators of the internal states of the atom with wo the transition frequendy. In eq. (10.3), a and at are the annihilation and creation operator$ of the light of frequency u.The coupling of the atom to the electromagnetic field is given in the electric dipole approximation by:

Hint = -d . E(R),

(10.4)

where E(R) is the electric field evaluated at the position R of the atom. d is the atomic electric dipole moment operator, which may be written as (10.5)

d=D,2(n+d),

with 0 1 2 the dipole matrix element. The electric field for a Laguerre-Gaussian mode propagating along the z-axis can be written as E(R) = i [uZ &kb(R)eioWp@) - h.c.1 ,

(1 0.6)

where 3 is the mode polarization vector and &klp(R)and Okb(R) are, respectively, the mode amplitude function and phase function which are explicitly given by Beijersbergen, Allen, van der Veen and Woerdman [1993] and, fiom eq. (2.19),

k 2 Z

@k[p(R)=

q.2

+

z;)

+ Z@ + (2p + Z + 1) arctan

(10.8)

d p m

Here EkOO is the amplitude for a plane wave of wavevector k;Cb = is a normalization factor; and w(z) is defined in terms of the Rayleigh range ZR

w 6 101

ATOMS AND THE ORBlTAL ANGULAR MOMENTUM OF LIGHT

33 1

by w2(z) = 2(z2 + z i ) / , ~ The . integers 1 and p are indices characterizing the LG mode, as described in 4 5. The time evolution of the system may be determined by transforming to the interaction picture governed by the unperturbed field Hamiltonian hat,. In this picture the field operator a(t) is time-dependent, with dependence given by

a(t) = exp(iwa+at)aexp(-iwatat)

= ae-lWf,

(10.9)

with a similar equation for at(t). In the classical limit in which the field forms a coherent beam, we may replace the field operators by c-numbers: a(t)4 cre-lw';

at(t) --f a*eIwt.

(10.10)

The corresponding interaction Hamiltonian is thus given by

Hint = -d .E(R) = -ifi [$+af(R)

- h.c.1 ,

(10.11)

where we have made use of the rotating-wave approximation and introduced 5 andf(R) by jt = ZelWt

(10.12)

(10.13) We can make use of the semiclassical approximation by replacing the position and momentum operators R and P by their expectation values Ro and PO, respectively, while maintaining a quantum treatment for the internal dynamics of the atom. The validity of the semiclassical approximation requires that the spatial extent of the atomic wavepacket be much smaller than the wavelength of the radiation field and that the uncertainty in the Doppler shift be much smaller than the upper-state linewidth of the atom. This is the case for most atoms (Letokhov and Minogin [19871) if the recoil energy of the atom is much smaller than the upper-state linewidth. Within the semi-classical approximation, the atomic density matrix can be written as

P = W R - Ro)

w - PO)P(t),

(10.14)

where now p(t) contains the internal dynamics of the atom and the evolution of p(t) is in accordance with the well-known relation (10.15)

332


"5

10

where the relaxation dynamics enters via the term Rp. Substituting for H and using eq. (I 0.1 I), we obtain the following equations for the atomic density matrix elements: (10.16)

dP21 - -(r - iAo)P21 + af(Rol6-722 - p1 I ), dt

--

(1 0.1 7)

where A0 = o - wo is the detuning of the field frequency from atomic resonance and P21 = ( 5 ) .Equations (10.16) and (10.17) are the optical Bloch equations for the two-level system interacting with the light. The average radiation force acting on the atom can be shown to be given by

Substitution of eq. (10.1 1) into eq. (10.18) and the use of eq. (1 0.13) shows that the force can be written as the sum of two terms, (F)= (Fdiss) + (Fdipole). Here ( F d i s s ) is the dissipative force, given by (Fdiss) = -h\J@(RO)

and

(pdipole)

{ P12(t) af(RO>+P21(t> a*f*(RO)},

(10.19)

is the dipole force, given by (10.20)

where we have introduced a positionally dependent Rabi frequency as (10.21) In the adiabatic approximation (Cohen-Tannoudji, Dupont-Roc and Grynberg [ 1975]),the velocity of the atom, defined as V = Po/M, is assumed to be constant during the time taken for the dipole moment to relax to its steady-state value. The position Ro of the atom at time t is then given by:

Ro

= ro

+ Vt,

(10.22)

where ro is the initial position of the atom when the beam was switched on. Thus,

N,§ 101

333


-f(rO) exp[iV@(ro) . Vt] ,

(10.24)

where we have assumed that the change in the field amplitude is negligible during the time taken for the dipole moment to relax to its steady-state value. Within the adiabatic approximation, the optical Bloch equations become dP22 dt

-- -- 2 m 2

dfi21 dt

~

-

0lf(ro)fil2-

= -[r-iA(ro,

V)lfi21 +af(ro)(p22 - P I I ) ,

where A is the total detuning, A(r0, V ) = AO - VO(r0) . V , and p21 exp[-itV . VO(ro)].The forces can now be written as

(10.25) (10.26) b21

=

(10.28) For given initial conditions, the solution of the optical Bloch equations (10.25) and (10.26) leads formally to the determination of the forces by direct substitution in eqs. (10.27) and (10.28). Torrey [ 19491 gave detailed solutions of the original Bloch equations. He also recognized that there were three special cases of interest which have relatively simple solutions. These were for strong collisions when the natural lifetime of the state may be replaced by the collision-shortened lifetime, for exact resonance, and for intense external fields. His approach has been applied (see Allen and Eberly [ 19751) to the optical Bloch equations. Consequently, the evolution of the forces from the instant the light beam is switched on can also be examined for a number of special cases. Such effects have been examined in detail for atoms excited by plane-wave light (Al-Hilfy and Loudon [ 19851) where the cases considered were (i) an atom with all relaxation constants equal to zero; (ii) a weak beam; (iii) exact resonance and (iv) steady state achieved by an intense field. This treatment may be readily generalized for Laguerre-Gaussian light. We shall settle simply for the consideration of the steady-state case because the general time dependence of the density matrix elements for arbitrary parameter values can be determined more readily by the numerical solution of the optical Bloch equations (10.25) and (10.26). This enables the evolution of the corresponding forces to be displayed. We display the results for a LaguerreGaussian mode with I = 1 andp = 0 such that Q(r0) = r;A0 = -r and w(0)= 3 5 4

334


0

2

4

6

0

10

6

8

10

rt

0

2

4

rt Fig. 12. Variation with time of (a) the average dissipative force, and (b) the average dipole force for a stationary atom in a single LG beam. The time variation of the corresponding torque would be the same as in (a). The time is in units of r-'.

where L=28Onm is the atomic transition wavelength. The results shown in figs. 12a and b depict the evolutions of the dissipative and dipole forces as given by eqs. (10.27) and (10.28), respectively. These figures show clearly that the dipole moment, and hence force components, relax to their steady-state values within a time of the order of r-'. The steady state corresponds to all time derivatives in the optical Bloch equations being set equal to zero, and coincides with the long time limit. It is straightforward to show that the steady-state solutions to the optical Bloch

n! §

101


335

equations (10.25) and, (10.26) give rise to the following expressions for the steady-state forces: ( F ) = (Fdiss) + (Fdipole)

Y

(10.29)

where (10.30)

where the notation is redefined, so that R now stands for the position of the atom and not ro. It is the proportionality to r that signifies the dissipative nature of the force in eq. (10.30) and its association with spontaneous emission. These results apply to an arbitrary field distribution, including the well-investigated plane wave case. Nevertheless, despite the generality of the results, it is possible to draw some conclusions about the characteristics of the forces. First, it is not difficult to see, by virtue of its proportionality to A and to VSZ, that the dipole force (10.31) would attract the atom to regions of intense field when the laser is tuned below resonance and repel the atom from these regions when tuned above resonance. It is this property of the reactive force that is frequently exploited in atom trapping experiments. The dissipative force in eq. (10.30), on the other hand, contains the factor AVO which corresponds to the momentum imparted by the light to the atom, which then reradiates spontaneously in a random direction. The probability of spontaneously emitting a photon in a given direction is the same as that for the opposite direction. The direction of absorption is well defined, so there is a net momentum change per absorbed photon of magnitude h IVO 1 when averaged over a large number of photons. As the maximum rate at which an atom may spontaneously emit photons is r, the maximum dissipative force on the atom is fir [DO!. It is instructive to consider first the familiar simple case of an atom interacting with a linearly polarized plane wave of wavevector k and polarization z. The expressions entering eqs. (10.1l), (10.13) and (10.21) appropriate for this case are: O(R) = k .R,

(10.32)

and (10.33)

336


[n!5

10

where N is a plane-wave normalization factor. It is then readily shown that the phase factor in eq. (10.24) corresponds to the familiar Doppler shift, (10.34)

6=k.V.

The main influence of this effect is to change the detuning parameter from A O = O - W Oto A, where A = A 0 - 6 . The dipole force defined by eq. (10.31) for a plane wave is identically zero, which follows tivially from the fact that Vst=O. The dissipative force, eq. (10.30), on the other hand, can be written succinctly in the form (10.35) where I is a saturation parameter defined by I

=2

~ ~ / r ~ .

In the saturation limit corresponding to I --t the maximum dissipative force on the atom:

(F&)

= hkT.

(10.36) 00,

we obtain the usual result for

(10.37)

The dissipative force due to a plane wave produces zero torque on the atom about an axis parallel to the direction of propagation. This property stems from the uniformity of the plane wave which precludes the presence of non-axial forces on the atom. By contrast, as we pointed out at the outset and discuss in detail shortly, LG beams have non-trivial influence on the non-axial atomic motion. When explicit reference to a specific Laguerre-Gaussian mode is made, expressions (10.30) and (10.31) lead to the steady-state force on a moving atom due to a single Laguerre-Gaussian beam propagating along the positive z-axis in the form (10.38) where

rv, 8

337


101

and

where R(t) now denotes the current position vector of the atom and V = W d t . The total detuning dklp(R, V ) is both position- and velocity-dependent: dklp(R, V ) = do - I/. V@+(R, V ) .

(10.41)

The evaluation of v@k[p from eq. (10.8) can be carried out in cylindrical coordinates. We find

Substitution for VOkb yields straightforwardly the Doppler shift 6 = VOk, in cylindrical coordinates:

6=

-

(-)+ ( kr2+ krZ z;

22

-

.V

vr-T1v4 vr-7

[ 4 - z 2 ] + ( 2 p + l + 1)zR + k) vz, 2 2 +z; 2(z2 z;) 2 2 + 2;

(10.43)

where V,, V4 and V, are the radial, azimuthal and axial components of the atomic velocity, respectively. The Doppler shift divides naturally into four types of contribution: an axial contribution along the z-direction, a contribution due to the Gouy phase, a contribution due to the beam curvature, and finally an azimuthal contribution. We may write:

6 = &xial+

&ouy

+ &rve + dazimuth.

(10.44)

The axial component simply corresponds to a Doppler shift that would arise from a plane wave traveling along the beam axis, &axial = kVz.

(10.45)

This is normally the dominant shift, provided the atom has a substantial velocity component along the beam axis. The shift caused by the Gouy phase is: (10.46)

338


[n!8

10

It is easily seen that, as typically ZR >> W O ,the Gouy shift is very small for practically all Laguerre-Gaussian beams. The shift arising from the beam curvature is given by ~CU,",

=

(-)

krZ

22

+z;

vr+

(

kr2

2(z2

+ z;)

[

Z 22

q

+ z;

)

v,.

(10.47)

This is a sum of contributions due to the spreading of the beam in the radial and axial directions. These contributions are well understood. They have the same origin as the corresponding shifts in conventional (0,O) mode Gaussian beams and arise from the curvature of the wavefront. They may, in certain circumstances, have observable consequences. Finally, the azimuthal Doppler shift is: (10.48) The important feature of this azimuthal shlft is that it is directly proportional to the orbital angular momentum quantum number 1 of the Laguerre-Gaussian mode; it occurs for motion which is azimuthal to the overall beam propagation. Further insight into the physical meaning of the azimuthal Doppler shift can be gained by equating ZA to the z-component of orbital angular momentum of the LG beam. We have,

lh = ( r x P ) =~ rpb,

(10.49)

where p is the beam linear momentum, formally given by eq. (2.20), and p$ is its azimuthal component, which can be written as

where k@(r)is the local azimuthal component of the wavevector. Equation (10.48) can now be written as (10.51) Thus the azimuthal Doppler shft has the same form as the usual translational Doppler shift, but the shift is now due to motion at right angles to the beam direction. It is the azimuthal component of the spiraling Poynting vector that produces this frequency-independent shift and is an example of the rotational ffequency shift (Bialynicki-Birula and Bialynicka-Birula [ 1997]),

n! §

101


339

discussed W h e r in Q 14. The azimuthal Doppler shift is a potentially observable characteristic of the internal motion of the atom arising from its interaction with the LG beam. We shall see that there is also a light-induced torque associated with the forces acting on the atom due to such beams. This, by contrast, is a measurable property of the gross motion of the atom. A different approach to the distinction between internal and gross motion in atoms due to fields possessing orbital angular momentum is also possible (van Enk and Nienhuis [ 19921). Substitution of V@lp in the expression for the dissipative force shows that, in general, there are non-zero force components in all three directions (?,$,?) of the cylindrical coordinates. In particular, a significant contribution arises in the form of an azimuthal component. This is responsible for a non-vanishing torque around the beam direction, given by (10.52) where FZissis the @-component of the dissipative force. This torque has a magnitude that can be written explicitly in the form (10.53) In the saturation limit I + 00, we obtain l(T)klp/

(10.54)

This result is as remarkably simple as the plane wave saturation force in eq. (10.37). We have seen that for the case of a single Laguerrffiaussian beam the forces are modified relative to the usual case of plane wave linearly polarized light. In particular, an azimuthal component of the dissipative, or radiation pressure, force exists which in the steady state leads to a non-zero torque acting on the atom around the beam axis. In order to elucidate fbrther the nature of the interaction between the LG beam and the atom, we consider analytically the low-velocity limit of the dissipative and dipole forces. In the computational evaluation of the dynamics of the atom, this approximation need not be made. The assumption involved in the lowvelocity limit is that the Doppler shift induced by the motion of the atom is smaller than the atomic width, V 00 1. A geometric phase should still be found to occur as the modes are transformed, but it appears that once again spin and orbital contributions may not behave identically. In the literature of beams with discontinuities in their most general form, the effect of orbital angular momentum has been discussed essentially only by implication. It appears that much of the language of the literature of vortices in light beams should be susceptible to being re-written explicitly in terms of orbital angular momentum. This has not yet been done and this review is not the place to attempt it. It would appear that the overlap is likely to be increasingly significant; the conservation of orbital angular momentum in soliton motion in non-linear media (Firth and Skryabin [ 19971) has been demonstrated theoretically. We have seen that spin and orbital angular momentum are sometimes interchangeable and sometimes not. It might appear that there is an important difference between spin and orbital angular momentum because o is independent

NI

REFERENCES

369

of the axis about which total orbital angular momentum is calculated, while 1 depends upon this axis. However, Berry [1997] has pointed out that it is both possible and reasonable to stipulate a direction z for which the transverse momentum current is exactly zero, so that both 1 and (J are invariant under a shift of axis. As we have shown, 1 depends upon the spatial structure of the light beam and, although (J depends on the state of polarization of the beam, the local polarization density depends upon the gradient of the intensity (2.17) and so is also spatially dependent. Spin and orbital angular momentum thus behave in a remarkably more similar way than seems at first sight likely. The literature of the orbital angular momentum of light beams is only a few years old. The range of its manifestations displayed in this review is already very rich. No doubt a good deal more is yet to be discovered in the years to come concerning its full properties and its relationship to spin.

Acknowledgements It is a pleasure to thank Professor Stephen M. Barnett for reading and commenting on a preliminary version of this review.

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E. WOLF, PROGRESS IN OPTICS XXXIX @ 1999 ELSEVIER SCIENCE B.V ALL RIGHTS RESERVED

V

THE OPTICAL KERR EFFECT AND QUANTUM OPTICS IN FIBERS BY

h R E A S SIZMANN AND

GERDLEUCHS

Lehrstuhl fur Optik, Physikalisches Institut. Universitat Erlangen-Niirnbee, Staudtstr. 7/B2, 91508 Erlangen, Germany email: [email protected]

373

CONTENTS

PAGE

Q 1 . INTRODUCTION

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

375

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

377

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

380

Q 2 . HISTORICAL PERSPECTIVE

Q 3. THE OPTICAL KERR EFFECT

Q 4. QUANTUM OPTICS IN FIBERS PRACTICAL CONSIDERATIONS . .

. . . . . . . . . .

388

Q 5. QUADRATURE SQUEEZING . . . . . . . . . . . . . .

397

Q 6. QUANTUM NONDEMOLITION MEASUREMENTS . . . . .

418

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

435

Q 7. PHOTON-NUMBER SQUEEZING Q 8. FUTURE PROSPECTS

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

458

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

460

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

460

374

0

1. Introduction

If one wished to study nonlinear optical effects, one would not choose silica glass as the nonlinear medium at first sight. Second order effects are not observable because the bulk material is symmetric with respect to inversion. Of all materials, silica glass has about the smallest nonlinear coefficient in third order. Nevertheless, this review deals with the nonlinear optical Kerr effect in precisely this material. It does so for several reasons. Firstly, the nonlinearity of silica optical fibers has increasing importance for optical communication applications. Secondly, the fiber allows for the study of soliton dynamics which is important in many nonlinear wave propagation models. Thirdly, on the fundamental side, it allows for the study of novel quantum optical effects such as intrapulse quantum correlations. Furthermore, the squeezing achieved in fibers, i.e., the reduction of the quantum uncertainty of light, has improved substantially in the last decade. The long term goal to achieve an order of magnitude noise reduction (10 dB) now seems within reach. Furthermore, the devices generating squeezing are becoming more and more practical. This is potentially important for all applications which are limited by the shot noise of a laser beam. The experimental methods which have evolved allow for quantum measurements such as back-action evading or quantum nondemolition detection. The in-depth study of the rich nonlinear dynamics of quantum solitons in fibers and the measurement of the quantum characteristics of optical pulses has only just started. One practical application that bears high economic potential is in optical communication. In the following, this example will be used to present some aspects of quantum optics in fibers. In an optical data transfer fiber channel a minimum number of photons is necessary per bit of information, depending on the maximum allowed bit error rate. With increasing data transmission rates the peak intensity of a pulse carrying a bit is increasing consequently. The small value of the nonlinear coefficient is compensated for by the light power confinement in both the tranverse spatial and the temporal dimensions. The latter occurs if the nonlinear interaction leads to the formation of soliton pulses in the optical fiber. The point has been reached where the fiber nonlinearities can no longer be neglected in the systems which are being presently field tested. On the other hand, the concept of optical solitons makes use of this nonlinearity 375

316

THE OPTICAL KERR EFFECT A N D QUANTUM OPTICS IN FIBERS

[v, §

1

to counter dispersion. Solitons have already been tested in a commercial long distance data transmission line (Robinson, Davis, Fee, Grasso, Franko, Zuccala, Cavaciuti, Macchi, Schiffini, Bonato and Corsioni [ 19981). In addition, solitons in fibers are an interesting and rewarding test-bed for studying the dynamical evolution of the quantum properties of optical pulses in a nonlinear environment. Data transmission is an area where optical technology has taken over from electrical technology because of the higher modulation band width and the much reduced crosstalk between nearby channels. One limitation of optical data transmission today is based on the analog nature of the optical systems. In order to guarantee error-free transmission over long distances, it is eventually necessary to regenerate the signal in a repeater station where the signal must be converted to the electronic level and then back into the optical domain. The finite bandwidth of the electronic circuit is the bottleneck limiting the transmission rate. Working with solitons, passive spectral filters, and erbiumdoped fiber amplifiers, one can overcome this limitation (Essiambre and Agrawal [ 19971). Based on their particle-like stability, solitons build a digital more than an analog system. For networks already installed it is difficult to follow this route because they are not designed to support solitons. An all-optical signal regenerator working much like an amplifying discriminator would boost the achievable data rates enormously. In order to explore this possibilty, existing and new functional devices must be studied with respect to their noise properties. The performance of up-to-date optical transmission lines is limited by the quantum noise introduced by the light generation process in the source, by the attenuation in the fiber, and by optical amplifiers. Therefore, it is especially important to study the mechanisms for the reduction of noise, quantum and classical as well. In particular, this refers to squeezing and to the generation of spectral and intermodal quantum correlations. Along with amplitude noise reduction, the functional behavior of a transmission device may show optical limiting, which is a necessary ingredient towards building an optical discriminator. This article reviews experiments which have been setting the stage for quantum optics in fibers by studying the effect of the nonlinear optical Kerrinteraction and of stimulated Raman scattering on the quantum properties of light in silica fibers. To begin, the historical development is summarized in tj 2. The following two chapters prepare for the main part of the review, starting in 9 3 with an outline of the basic characteristics of the optical Kerr effect. Section 4 reviews the essential properties of optical solitons and thermal noise sources in fibers. The central part of the review begins in tj 5 with a discussion of quadrature squeezing in fibers using self-phase modulation (SPM). It has been the first and so far most intensively researched avenue of fiber squeezing. Recent progress

v, § 21

HISTORICAL PERSPECTIVE

377

includes the use of cross-phase modulation (XPM) for quadrature squeezing. Section 6 is devoted to quantum nondemolition (QND) measurements in fibers, where XPM is used to achieve quantum entanglement of two modes or two pulses. Here the concepts of XPM and SPM are used to diswss progress and perspectives for QND. New squeezing mechanisms are discussed in $ 7 . Only three years ago there emerged the transition from quadrature squeezing and coherent signal extraction using a local oscillator, to directly detectable squeezing. A number of recent experiments which produced photon-number squeezing will be discussed in Q 7. Again, the nonlinear Kerr effect is the basic underlying squeezing mechanism. In Q 8 the perspectives emerging from these experiments, including the potential impact on all-optical signal processing, are discussed.

0

2. Historical Perspective

Nonclassical states of the electromagnetic field, i.e., states which have no classical analog, have been intensively studied experimentally over the past 20 years I . Early experiments showed subpoissonian photon statistics in resonance fluorescence (Teich and Saleh [ 19881). Parametric, phase-sensitive deamplification of noise in a beam of light was first studied in four-wave mixing and was observed in the classical domain (Levenson, Shelby and Perlmutter [19851, Maeda, Kumar and Shapiro [1985], Levenson and Shelby [1985], Slusher, Hollberg, Yurke and Mertz [ 19851).Then, the first true squeezing in the quantum domain was achieved in degenerate four-wave mixing in a Na atomic beam (Slusher, Hollberg, Yurke, Mertz and Valley [ 19851). Next, squeezing through four-wave mixing in glass fibers was observed (Shelby, Levenson, Perlmutter, DeVoe and Walls [1986]) shortly before the generation of squeezing in a x ( ~ ) nonlinear parametric oscillator (Wu, Kimble, Hall and Wu [1986]). The main experimental difficulty in all systems is that the weak nonlinear optical processes must be efficient enough to compete with attenuation, diffraction and material-dependent noise sources. Squeezed state generation

I Squeezed states of light are discussed in a number of textbooks, e.g. Walls and Milbum [1994], Bachor [1998], Scully and Zubairy [1997], and Meystre and Sargent 111 [1990]. In addition, the subject was covered in review articles (Leuchs [1986], Loudon and Knight [1987a], Teich and Saleh [1989], Yamamoto, Machida, Saito, Imoto, Yanagawa, Kitagawa and Bjork [1990], Reynaud, Heidmann, Giacobino and Fabre [1992], and Tanas, Miranowicz and Gantsog [1996]). Several special journal issues were dedicated to squeezed light (Kimble and Walls [1987], Loudon and Knight [1987b], and Giacobino and Fabre [1992]).

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THE OPTICAL KERR EFFECT AND QUANTUM OPTICS IN FIBERS

lX§2

based on the third-order nonlinearity. of single-mode fibers was proposed as a promising system (Levenson, Shelby, Aspect, Reid and Walls [1985]). In contrast to enhanced near-resonant nonlinearities in atomic systems, where noise from the creation of an excited-state population is related to the squeezing interaction itself, the extreme off-resonant nonlinearity in fiber must compete with thermal noise not related to the squeezing interaction. The hope was that the thermal noise could be eliminated whle the power confinement and long interaction length in the highly transparent medium would generate detectable squeezing. After all, traveling-wave geometries offer a larger bandwidth for squeezing and have a natural interface to fiber-optic communication systems, thus making optical fibers attractive for classical and quantum noise reduction. In the early experiments, classical thermal noise unique to fibers was found to be a severe limitation. Elimination of this noise was regarded as a technological (Levenson, Shelby, Aspect, Reid and Walls [1985]) rather than a fundamental physical problem. The development of squeezing experiments in fibers shows that of the many techniques for thermal noise reduction, some were successful, and even fibers with particularly low intrinsic noise were found (Bergman, Haus and Shirasaki [19921). Also, new squeezing mechanisms in fibers which seem to be immune to intrinsic phase noise were discovered recently (Friberg, Machida, Werner, Levanon and Mukai [ 19961). The hstory of fiber squeezing is illustrated in fig. 1. After the first continuous wave (CW) results (Shelby, Levenson, Perlmutter, DeVoe and Walls [ 1986]), the second generation of fiber squeezing experiments used high-peak power pulses and coherent detection (Rosenbluh and Shelby [ 19911, Bergman and Haus [1991]), and a variety of more-or-less successful methods of intrinsic noise reduction were reported. The squeezing results achieved over the years, however, did not show a continuous development towards more noise reduction, as instrinsic noise properties of fibers varied to a great extent. T h s review draws a distinction between photon-number and quadrature squeezing. The former refers to reduced amplitude noise and may be measured by direct detection. The latter refers to measuring a specific field quadrature using a local oscillator. Basically, two classes of pulsed quadrature squeezing experiments were performed in fibers: those using sub-picosecond optical solitons (Rosenbluh and Shelby [1991]) and those using 10 to loops pulses near zero dispersion. The latter have matured experimentallyand have reached the (inferred) 7 dB detection limit for chirped pulses (Bergman and Haus [1991]). In the negative dispersion regime, more than lOdB of noise reduction is predicted for solitons; however, this remains to be demonstrated. With solitons, the need for inducing a large nonlinear phase shift over a short

v, 5 21

HISTORICAL PERSPECTIVE

0

379

cw quadrature squeezing and four-mode squeezing

v$/

pulsed quadrature squeezing pulsed photon-number squeezing (spectral filter)

v m pulsed photon-numbersqueezing (asymmetric lwp) Fig. 1. Progress in squeezing experiments with fibers, quantified by the squeezing ratio R of observed photocurrent noise reduction relative to the shot-noise limit (R=1). Different symbols and shaded areas represent the four generations of squeezing experiments, from the first generation of continuous-wave experiments in cryogenic fibers to the present generation of pulsed squeezing experiments at room temperature. The individual experimental results are reviewed in 5 5 (quadrature squeezing) and 5 7 (photon-number squeezing).

length of fiber had led to the use of sub-picosecond pulses already by 199 1. The limited amount of squeezing observed raised the question of whether stimulated Raman scattering masked the noise reduction. This was not the case, as more complete quantum soliton models showed. Therefore, the goal of more than 10 dB of squeezing from fibers seems feasible. The latest quadrature squeezing experiments use extremely short fibers and short pulses in order to bring out the quantum properties of solitons. There has also been recent progress in novel approaches for quadrature squeezing using the cross-Kerr effect, thus avoiding mode matching problems that have been thought to limit earlier soliton squeezing experiments. The third generation of fiber squeezing experiments produced directly detectable photon-number squeezing via spectral broadening, and will be discussed in 9 7. In these experiments, strong squeezing was observed with spectral filtering (Friberg, Machida, Werner, Levanon and Mukai [19961, Spalter, Burk, StroBner, Sizmann and Leuchs [ 19981). However, according to numerical calculations the spectral filtering technique is not expected to yield more than 8 dE3 of squeezing.

380


[v, 5 3

The fourth generation of experiments makes use of two-beam interference where at least one of the interfering pulses has experienced nonlinear interaction (Schmitt, Konig, Mikulla, Spalter, Sizmann and Leuchs [1998]). In this case, numerical calculations predict more than 10 dB of photon-number squeezing. The stable, directly detectable squeezing combined with the promising predictions will lead to continued experimental effort in this field. Today, the main experimental effort in developing sources of squeezed light is directed towards efficient, stable, and compact devices that are versatile and which can be tailored to specific measurement problems. In this context, the fiber is an attractive system, because it is easy to use and because of its promise for efficient squeezing. Moreover, recent progress in the study of optical quantum solitons and of quantum limits in fiber-optic communication fuels the continued interest in quantum optics in fibers2. One goal is to generate quantum solitons with more than an order-of-magnitude quantum noise reduction (>lo dB). With such a highly nonclassical state, basic paradigms of quantum measurement and nonlinear quantum field theory can be tested. Another perspective is to establish the quantum noise performance in fiber-optic components, such as filters, loop mirrors or switches, in comparison with which actual devices may be evaluated. Before presenting the state of the art of non-classical effects in fibers, we begin by reviewing the basic properties of the nonlinear interaction in fibers.

0 3.

The Optical Kerr Effect

Small perturbations in physical systems close to equilibrium tend to lead to harmonic oscillations around the point of equilibrium. If the deviation from equilibrium is large enough, most systems show anharmonic behavior which leads to a change in the oscillation period and to the appearance of new frequencies and overtones. It is well known that this also happens for atomic valence electrons when excited by an intense light beam. This opens the rich field of nonlinear optics. The response of the medium is described by the nonlinear polarization (Bloembergen [ 19651, Hanna, Yuratich and Cotter [ 19791, Loudon [1983], Ducuing [1977]). With regard to silica fibers, the following discussion is

For a review on quantum optics in fibers see e.g. Milbum, Levenson, Shelby, Perlmutter, DeVoe and Walls [1987], Drummond, Shelby, Friberg and Yamamoto [1993], Haus [1995], Friberg [1996], Spalter, van Loock, Sizmann and Leuchs [1997], Sizmann [1997], and Sizmann, Spalter, Burk, SlroBner, Bohm and Leuchs [1998].

v, 8 31

THE OPTICAL KERR EFFECT

381

restricted to a medium showing inversion symmetry so that the polarization of the medium can be written as an expansion in odd powers of the electric field3:

where E represents the total electric field, which may consist of components having different frequencies and may also be the sum of the electric fields of two beams intersecting spatially. The linear ) term just transfers these frequencies to the polarization of the medium. The nonlinear x ( ~term ) mixes different frequencies, generating new spectral components. Most processes induced by ~ ( ~ such 1 , as third-harmonic generation, require a special effort for matchmg of the phase velocities of all optical fields involved. The only one for which this does not hold is The optical Ken effect which is, therefore, the dominant nonlinear interaction in fibers. When the effect was first discovered by Kerr [ 18751, two of the electric fields were constant and one was an optical field. The effect was used for fast optical switches. It is because of its short time constant that the optical version of the effect generates squeezing in an extremely large frequency band4. If only one optical electromagnetic field is applied, then there is a component of the induced nonlinear polarization, P , which effectively changes the refractive index for the propagating light beam. This can be shown by rewriting eq. (3.1), and is known as the optical Kerr effect (Agrawal [1995]):

x('

with I = iceonoE2 being the intensity of the light beam, c the vacuum speed of light, and no the refractive index given below. The Kerr effect may be described by an effective susceptibility which leads to the following intensity-dependent

A single silicon dioxide molecule does not have this symmetry. The inversion symmetry of silica glass is a property of the bulk material where the molecules are statistically oriented. It is, however, possible to pole a silica fiber so that the second-order term appears in the nonlinear polarization and a second-harmonic wave can be generated (Kazansky,Dong and Russell [1994], Kazansky, Russell and Takebe [1997]). Phase matching may be obtained by using spatially periodic poling. s for a If the nonlinear response is dominated by electrons the time constant is less than nonresonant light field (Owyoung, Hellwarth and George [1972]).

382


[Y 0 3

x(')

refractive index5, assuming that the imaginary parts of and x(3)are negligible and that the intensity-dependent term is much smaller than 1+x('),

or

n = no + nzI.

(3.3)

As a result of the optical Ken- effect, a light field will experience an intensitydependent phase shift. For a light pulse this leads to a phase varying with time, producing a frequency chup across the pulse. If a light beam contains different intensity components, each will be phase shifted differently. This explains the initial evolution of a coherent state of th8 light field in a Kerr medium as will be discussed in more detail below. The intensity-dependent phase shift leads to a wide variety of phenomena and applications such as pulse compression, solitons, optical switclung (Agrawal [ 19951) and quantum measurement (Yamamoto, Machida, Saito, Imoto, Yanagawa, Kitagawa and Bjork [19901). The full richness of the optical Ken- effect is revealed when taking into account field quantization (Loudon [ 19831). The operator for the electric field of one mode is a superposition of the photon annilulation and photon creation operator6 ,

(3.4) The energy density of the field is given by the product of the electric field and the dielectric displacement vector (Bloembergen [19651, Drummond and Walls [1980]):

The contribution of the nonlinear interaction to the Hamilton operator is, therefore, a linear combination of products of four field operators. The order of Here the refractive index is given by n2 EOE= EOE+P V describes the quantization volume. In the case of a fiber, V is given by the cross-section of the fiber core A e f f times the length of the fiber AL over which the photons are distributed. For pulses of width At, V = A , e c A t (Imoto [1989], Imoto, Jeffers and Loudon [1992]).

v, 5 31

383


1

1

2

3

4

5

6

Fig. 2. Schematic energy level diagram for glass. The low- and the high-lying excited states (e) shall correspond to infrared and ultraviolet absorption bands (g = ground state). The arrows pointing up and down represent absorbed and emitted photons. Here all photons have the same frequency. The various groups of arrows describe the four photon processes contributing to the optical Kerr effect and to the nonlinear refractive index.

the field operators in the product depends on the level structure and the electric dipole transition matrix elements of the medium (Bloembergen [ 19651, Hanna, Yuratich and Cotter [1979], Loudon [1983], Ducuing [1977]). Figure 2 shows diagrams of elastic four-photon interactions; i.e., interactions where the state of the medium is not changed. These diagrams contribute to the optical Kerr effect. Some of the diagrams yield non-vanishing contributions to the interaction energy, which leads to divergence when assuming instantaneous response of the medium. This may be solved by renormalization (Abram and Cohen [ 19941). The interaction energy may be calculated in fourth order perturbation theory and is determined by a sum of products of four electric dipole matrix elements and three energy denominators. The processes described by diagrams 1 and 4 may be enhanced by a near-resonant two-photon transition. In the case of silica glass and a wavelength in the near infrared, diagrams 1 and 4 can be neglected with respect to diagrams 2, 3, 5 and 6 which are all enhanced because the initial state is also-one of the intermediate states. Using the commutator relation [a, a t ]= 1, each of these four operator products can be written as a linear combination of fa)^, a two-operator product (ata), and a constant term. The two latter terms

384


(4

[Y 8 3

(b)

Fig. 3. Phase diagram for the state of the light field. The shaded area indicates the region in phase space which the state occupies. The circular region in (a) describes a coherent state and the distorted region in (b) represents a state squeezed by the nonlinear Ken interaction.

produce an overall phase factor which does not change the character of the light field (Imoto, Haus and Yamamoto [ 19853). The quartic term leads to characteristic changes of the quantum uncertainty; i.e., of the distribution function in phase space describing the state (Ritze and Bandilla [1979], Kitagawa and Yamamoto [1986], Gerry and Knight [1997]). This shall be illustrated with the example of a light field which is in an initially coherent state. To this purpose, the electric field may also be written as E(t) = E(0)cos W t +

.

~

0

sin W t .

The phase diagram in fig. 3a shows the area in phase space occupied by a coherent state the size of which is determined by the uncertainty relation. For a coherent state the contour line is a circle. The two-dimensional phase space of one mode of the light field is spanned by the amplitude of the cosine wave, a,=&(O), and the amplitude of the sine wave, a,=d,E(O)lo (Leuchs [1988], Yamamoto, Machida, Saito, Imoto, Yanagawa, Kitagawa and Bjork [ 19901). The mean values of the two conjugate variables a, and a, are the expectation values of (a-at)/2i and (a+at)/2, respectively. In other words, the figure shows the contour of the Wigner hnction characterizing the light field (Teich and Saleh [1990], Walls and Milburn [1994]). The length and angle of the arrow in the phase diagram represent the amplitude and the phase of the field, respectively. As a result of the nonlinear interaction the circular distribution is initially squeezed to an ellipse, as shown in fig. 3b. This is the basis for the quantum measurements reviewed in 5 5 and 5 6. The beginning of this evolution is readily understood based on the intensity-dependent refractive index causing an amplitude-dependent phase shift. The 111 evolution of the

Y 4 31


385

coherent state of the 1ight.field can be calculated by applying the time-evolution operator, u(t)

= exp {-iHff/h},

to the coherent state:

and

Obviously, the time evolution operator does not change a pure Fock state, apart from an overall phase factor. A superposition of Fock states such as a coherent state changes its character, however, because the phase factors introduced by the interaction are different for different Fock states. The discrete nature of the Fock-state basis, In), gives rise to unique quantum features in the absence of dissipation, which shows an interesting formal analogy to the quantum revival in a one-atom-maser (Eberly, Narozhny and Sanchez-Mondragon [ 19801, Knight and Radmore [1982], Yo0 and Eberly [1985], Rempe, Walther and Klein [1987]). The evolution of the coherent state turns out to be periodic (Ritze and Bandilla [1979], Walls and Milburn [1994], Buzek and Knight [1995]). After the time fza given by [ 3 ~ ( ~ ) h u ~ /V)]fzn ( 4 ~ o= 2n, the time evolution takes the light field back to the coherent state. After one quarter of that time, td2 = t2J4, the field is in a superposition of two coherent states lao) and /-ao).Figure 4 shows this evolution. This quantum superposition state is highly sensitive to dissipation (Buzek and Knight [1995], Yao [1997]). The diagrams in fig. 3 differ from those in fig. 4 in the following way. The former &splay contour lines of the Wigner function and the latter show contour lines of the @function,

plotted versus the real and imaginary part of a. The a’s are the eigenvalues of the coherent states onto which the light field is projected. The Q-function is the Wigner function convoluted with the Q-function for coherent states, which is a Gaussian distribution (Walls and Milburn [ 19941). For currently available silica

386

THE OPTICAL KERR EFFECT AND OUANTUM OPTICS IN FIBERS

IV, § 3

T 1

-.

Rea

VIII

IX

Fig. 4.Evo!ytion of an initially coherent light field due to the nonlinear Kerr interaction. Figure 3b shows only the very beginning of this evolution. The diagrams show contour lines of the Q-function as a function of the real and imaginary parts of a. The origin is at the center of each diagram. Starting from a coherent state a. = 2 in the diagram at the top left, the light field evolves as indicated in the following diagrams going from left to right and top to bottom. The interaction time elapsed between two consecutive diagrams is At = t2,/32. In the last diagram at the bottom right the light field has evolved into a superposition of two coherent states. After twice the evolution time (16At) the field evolves back to a coherent state -ao, and after tzn:=32At, back to a0 (Courtesy of N. Korolkova [1998]).

fibers the nonlinearity seems to be too low and the dissipation too large to allow for the observation of this unique quantum evolution7. The dissipative attenuation of about 0 . 2 d B h in a low-loss optical fiber near 1.5 pm is already close to the fundamental limit largely determined by Rayleigh scattering from density fluctuations*. Although the generation of

' For other optical materials see Hilico, Courty, Fabre, Giacobino, Abram and Oudar [1992]. The dissipation in optical fibers due to absorption by and scattering off impurities and host density fluctuations and possibly inhomogeneously oriented silica molecules has been reduced dramatically during the last three decades (Drexhage [1991]). While in 1973 the ultimate loss limit was believed to be 1.2dBkm for fused silica (Pinnow, Rich, Ostetmayer Jr and Di Domenico Jr [ 1973]), 0.2 dskm

Y

o 31


387

quantum superposition states might find applications in quantum communication, this will not be discussed further in this review because such studies are currently lacking experimental viability in the case of optical fibers. When working with sub-picosecond pulses and long fibers, one must also consider stimulated Raman scattering (Yariv [ 1987]), which shifts the spectrum towards longer wavelengths (Stolen and Ippen [19731, Weiner, Heritage and Stolen [1988]) and introduces fluctuations in the blue and red wings of the spectrum which are negatively correlated. Stimulated Raman scattering is an inelastic process that excites optical phonons in the fiber. The interaction Hamiltonian is btala1 + bataz, describing e.g. the process where a photon of frequency w~ is absorbed and a photon of lower frequency w2 and a phonon ( B ) are generated. The frequency of the emitted phonon is equal to the difference of the two optical frequencies, wl - w2 in this case. The experiments reviewed in the following sections deal with the initial elliptical deformation of the area occupied in phase space (fig. 4). This picture uses a single-mode Hamiltonian where the four field operators refer to the same mode (Yamamoto, Machida, Saito, Imoto, Yanagawa, Kitagawa and Bjork [ 19901). Alternatively, one may use a multimode picture (Milburn, Levenson, Shelby, Perlmutter, DeVoe and Walls [1987], Tanis, Miranowicz and Gantsog [19961). As already mentioned, the intensity of a laser pulse which varies across the pulse leads to a phase change varying with time. This in turn corresponds to a frequency shift varying across the pulse. The diagrams of fig. 2 must be modified slightly (fig. 5 ) ; e.g., diagram 2 changes to represent absorption of a photon of frequency w1, emission at w2, absorption again at 01 and emission at 013. The corresponding four-field operator product is aialaial. The requirement on the optical Kerr interaction to be elastic simply means that energy is conserved; i.e., 201 - 02 - w3 = 0. In general, the two pump photons may have different frequencies. This model not only shows that the optical Kerr effect generates new frequency components but also suggests that the fluctuations in the red and blue detuned sidebands are positively correlated because photons are created in pairs. The special case where the interaction is described by the four-field operator product aiazafal is referred to as cross-phase modulation (XPM) because the intensity of one mode changes the phase of the other.

was reached six years later (Miya, Terunuma,Hosaka and Miyashita [1979]). Currently,the minimum loss is approaching 0 . 0 6 d B h (Sakaguchi, Todoroki and Sbibata [1996], Sakaguchi and Todoroki [1997]). With reference to footnote 3 one may wonder whether it is possible to improve upon this fundamental dissipation limit by reducing the orientation inhomogeneity by electric field poling of the silica molecules preferentially during solidification in the fiber pulling process.

388


E

[v, § 4

e

1

1

2

3

4

5

6

Fig. 5 . Schematic energy level hagram for glass as in fig. 2. Here the emitted photons have frequencies different from the absorbed pump photons.

0 4. Quantum Optics in Fibers - Practical Considerations 4.1. KERR-NONLINEARITY AND POWER CONFINEMENT

A nonlinear, i.e., intensity-dependent phase shift produced by the optical Kerr effect was among the first optical nonlinearities proposed for squeezing and quantum nondemolition measurement. Even though the Kerr nonlinearity of silica glass is small compared to that of other nonlinear media, its effect is greatly enhanced in fibers by strong power confinement, chirp-free high-peakpower soliton propagation, and low attenuation at the wavelength of 1.55 pm. The magnitude of nonlinear effects in waveguides can be compared with bulk nonlinear systems using a figure of merit defined as F =I.Ler (Agrawal [1995]). Here Z=P/A,ff- is the intensity and L,ff is the effective interaction length of the nonlinearly coupled modes having an effective cross section of A e r = nwg. In a nonlinear waveguide the effective length L,R is limited by losses only. The lowest attenuation in silica glass fibers is about 0.2dBh-1 around a wavelength of 1.55 pm, with a spectral window approximately 25 THz (200 nm) wide. If attenuation is required to be less than 1% (-0.044dB) for quantum measurements, the corresponding fiber may be up to 220m long (fig. 6). For comparison, a 9pm-diameter focus of a Gaussian beam near the wavelength of 1.5 pm is diffraction-limited to a confocal length of Lee= nnw;/& = 60 pm.

Y 5 41

QUANTUM OPTICS IN FIBERS - PRACTICAL CONSIDERATIONS

389

220 rn

Fig. 6. The dimensions of a 150-fs soliton in a single-mode fiber with a 8-pm core and a 9 y m mode field diameter. The optical high-intensity pulse remains self-confined without experiencing spreading and travels more than 200m with negligible ( 1.3 ym). By adjusting the peak power to the pulse width, the opposite chirps created by GVD and SPM exactly cancel each other. A fundamental soliton may emerge from a variety of envelopes and pulse energies; however N > 0.5 is required for asymptotic soliton formation from a sech input pulse (Zakharov and Shabat [ 19721). Second, the peak power of a fiber soliton scales as Po,sT; = const. as a consequence of LD=LNL.Shorter solitons require more energy, and the nonlinear effects are enhanced dramatically. Next, higher-order solitons are created for integer values of N = d m ,where N is the order of the soliton. The fundamental ( N = 1) soliton propagates without change in pulse shape, but higher-order ( N = 2,3,. . . ) solitons undergo periodic changes in pulse width with a period of zp = ~ J G L D the, so-called soliton period. Finally, the distance a soliton of width To must travel in order to acquire a in nonlinear phase shift is also given by the soliton period, zp, which is proportional to Ti/I k” I and independent of n2, the fiber nonlinearity. The latter determines the energy content of the soliton, which is proportional to I k” I /( y To). The soliton period, which is also the length required for solitonic self-stabilization, depends on the (linear) GVD parameter of the fiber. Therefore, short pulses and strong dispersion are required for fast soliton dynamics in short pieces of fiber. The propagating soliton may experience damping and Raman scattering. Fortunately, the minimum loss window at 1.55 ym lies in the anomalous GVD regime (k” < 0) where solitons can form. As an example, the soliton period of a 1-ps pulse in a standard fiber is zp=25m. This distance is short enough to neglect attenuation (zp 0, 1.06 pm

50 m

[9]

0.31

-5.1

17ps pulse, k” = 0, 1.3 1 pm

90 m

ring ring

4.2

0.40

-4

17ps pulse, k“ = 0, 1.31 pm

90 m

ring

4.2

0.50

-3

0 . 1 5 4 . 2 ~ ssoliton, 1.55pm

8 cm

ring

n.d.

[-21 -3f

phase modulated CW, 1.56pn

7.7km, DC

linear

0 . 1 5 ~ ssoliton, 1.55pm

20 cm

linear

[lo]

[111 [0.61 [12] 0.50f

= 0,

a R is the ratio of observed photocurrent noise power to the shot-noise level. The pulse widths are FWHM values ( Z F ~ = H 1.763To for a secb pulse). The fibers were at room temperature unless otherwise indicated. Noise reduction values in square brackets refer to classical noise reduction, where R is the ratio of observed photocurrent noise power to the classical input noise level.

References [l] Levenson, Shelby and Perlmutter [1985] [2] Shelby, Levenson, Perlmutter, DeVoe and Walls [1986] [3] Schumaker, Perlmutter, Shelby and Levenson [ 19871 [4] Rosenbluh and Shelby [1991] [5] Bergman and Haus [1991] [6] Bergman, Doerr, Haus and Shirasaki [I9931

1.48 pm

2.7

0.3 -1

DS, dispersion-shifted fiber; DC, dispersion-compensating fiber. Four-mode squeezing. n.d., no data. Squeezing based on cross-phase modulation.

[7] Doerr, Lyubomirsky, Lenz, Paye, Haus and Shirasaki [1993] [8] Nishizawa, Kume, Mori, Goto and Miyauchi [1994al [9] Bergman, Haus, Ippen and Shirasaki [1994] [lo] Bergman [I9961 [I 11 Lorattanasane and Kikuchi [1997a] [12] Margalit, Yu,Ippen and Haus [1998]

v, § 51

QUADRATURE SQUEEZING

399

Yamamoto [1986]; see aiso table 1 therein). This limit, however, cannot be reached with fiber squeezing experiments 1 4 . In the practical situation of fibers, where the small nonlinearity requires a large photon number, both descriptions produce quadrature squeezed states; i.e., the quartic and the quadratic interaction Hamiltonian yield equivalent results if the nonlinear self-phase shift of the pump is taken into account in the parametric model” (Milburn, Levenson, Shelby, Perlmutter, DeVoe and Walls [1987], note 6 ; Yamamoto, Machida, Saito, Imoto, Yanagawa, Kitagawa and Bjork [1990], p. 116). Some unique properties of Kerr squeezing can already be understood from the properties of the quartic interaction Hamiltonian. Firstly, Kerr squeezing is a self-squeezing effect of the pump mode. One consequence is that it is not squeezing the vacuum state. The vacuum state is the lowest energy eigenstate of the field and therefore it cannot act as a pump for the nonlinear process. In comparison with a Kerr-squeezed state, a squeezed vacuum is easier to characterize experimentally, because the condition that the local oscillator amplitude should be substantially higher than the signal amplitude can be fulfilled readily without saturating the detectors. A special effort is required to generate a squeezed vacuum via the Kerr effect; e.g., by interference of two bright squeezed states in a balanced fiber loop (Shuasaki and Haus [1990]), as will be discussed in $5.3.2. Another consequence of self-squeezing of the pump is that the four photons involved in the elementary process have nearly the same frequency so that phase matching conditions are greatly relaxed. The large bandwidth of the far off-resonant nonlinear susceptibility in glass fibers

l 4 The creation of a detectable crescent shape of the quasiprobability distribution (QPD) would require a Kerr nonlinearity so strong that a few pump photons would suffice to induce squeezing. In practical situations of fiber squeezing, however, -(n) = lo8 photons are required for detectable squeezing. In order to achleve a detectable curvature or to reach the quadrawe squeezing limit with fibers, more than 27dB= 110 l ~ g [ ( n ) - ” ~ ]of[ photon-number noise reduction below the standard quantum limit is required. It will be difficult to observe more than lOdB of squeezing due to detection efficiency and other limiting factors. To date, the observed squeezing from fibers is still near 5 d8. l 5 In order to correctly predict the observed orientation of the squeezed ellipse, the nonlinear phase shift of the pump, E,, must he taken into account (Levenson, Shelby and Perlmutter [1985]). The

a k a ~ aA , is degenerate four-wave mixing Hamiltonian is then modified to H I c c ~ ( ~ ) [ E ~ I ’ where the amplitude quadrature operator as defined by Milburn, Levenson, Shelby, Perlmutter, DeVoe and Walls [1987, p. 14771, in the rotating frame of the pump wave. As can be seen from this modified Hamiltonian, only amplitude fluctuations of the total field contribute to the nonlinear interaction, and the nonlinear interaction cannot alter amplitude fluctuations because of [ a A , H [ ]= 0 (Milhum, Levenson, Shelby, Perlmutter, DeVoe and Walls [1987]), as is expected for Kerr squeezing.

400


[v, 0 5

thus allows for ultrashort pulses with high peak powers to be used for attaining a strong nonlinear effect in short fibers. Secondly, [a,HI]= 0 implies that the photon number is a constant of motion, and therefore the Poisson distribution of a coherent input state will remain unchanged. This is obvious physically if the glass fiber is seen as a highly transparent medium whch produces only an intensity-dependent phase shift of the beam. Photon-number squeezing from a coherent input beam will not occur in a Kerr interaction unless additional measures are taken to rotate the ellipse. In the CW case, this rotation was demonstrated with strong dispersion; e.g., by reflection from a cavity (Levenson, Shelby and Perlmutter [ 19851) and by highly dispersive fibers (Lorattanasane and Kikuchi [ 1997a1). Interference with a weak coherent beam has also been proposed for this purpose (Kitagawa and Yamamoto [ 19861). In the case of high-power pulses, self-phase modulation (SPM) leads to spectral broadening allowing for other' mechanisms, e.g., spectral filtering, to be used for obtaining photon-number squeezing (see 8 7). Thirdly, the quartic Hamiltonian introduces higher moments into the quasiprobability distribution (QPD) of the field. Numerical calculations performed with small photon number reveal the unique crescent-shaped form of the QPD of Kerr squeezing (fig. 4), which is different from quadrature-squeezed QPDs. This unique feature of Kerr-squeezed states cannot be detected with present technology. The large photon number and small Kerr-induced phase diffusion allow for linearization of the quantum fluctuations. The following linearized semiclassical approach is sufficient to model the detectable squeezing characteristics as a function of the average nonlinear phase shift. 5.1.2. Single-mode linearized approach

The intensity-dependent phase shift provides a physical picture for the squeezing mechanism, which will be presented here in more detail because it will also be useful for the discussion of QND experiments in 0 6 . In the absence of losses, a coherent beam of constant average power PO=IoA,ff acquires a nonlinear phase shift of (see eq. 3.3):

where A,R is the effective cross-sectional area of the mode, y = ( n t / A , ~ko, ) and ko = 27~112.In the following, we use A = a A for the amplitude quadrature and B = a$ for the phase quadrature to avoid complexity in the formal presentation. As is shown in fig. 9a, two different amplitudes A1 ’) . ((An:)’)

K2

In this case, the best measurement strategy is to use an arbitrarily strong probe beam, thus allowing its phase uncertainty to disappear (for a coherent probe, ((A@)2)=1/(4(np)) + 0 for (np) + 00). Then eq. (6.4b) becomes (a=O, (.p)

--$

00):

A$rt

=

KAnr,

(6.10)

and K , the cross-phase modulation (XPM) per photon of such a strong probe, represents a gain that generates a macroscopically observable copy of the quantum signal; i.e., S N Y ‘ = S N F . The strong probe produces a back action onto the phase of the signal beam (Kartner and Haus [ 19931) without degrading the transmitted s i b a l (SNRT’= S N V ) . In optical fibers, however, u f 0 and SPM phase noise increases with probe power (((A~SPM,~)’) = d ( n p ) for a coherent probe with ((An,)’) = (np)). Photonnumber fluctuations of the probe input coherent state feed continuously into phase fluctuations via SPM (fig. 9). The increasing phase uncertainty reduces the XPM sensitivity in a phase measurement (see fig. 17). Now the best strategy is to use a small probe power in order to minimize the nonlinear SPM phase spreading, but still enough power to ensure a small probe phase uncertainty [((A@)’) = l/(4(np))]; i.e., to maximize the signal-to-noise ratio transfer expressed in eq. (6.8) with respect to (np). The optimum probe photon number is found to be (n$‘) = 1420) for a coherent probe, and the transfer becomes S N ~ OoptU=~S~N R ~ 1 1 = SNR; U , 1 1+ 1+ K 2 ((An!)’) W w , / q J ((An!)’) (6.1 1) using ” K / 5= 2 ( 0 , / 0 , ) . Although the choice (o,/op) > 1 is of some advantage, If pulses at different velocities are used for the QND measurement, the effective propagation distance z for SPM and XPM is not the same in eqs. (6.6) and (6.7). The ratio KIU is diminished by the relatively small length L X ~ of M pulse overlap compared to the total length L s p ~over which the probe is self-phase modulated, K / U = 2(o,/op)(LXpM/LSpM). 22

426


Signal

Probe

I

-I I I I I I

Fig. 17. Self-phasemodulation (SPM) noise in a QND measurement: (a) The probe beam experiences both self-phase and cross-phase modulation. Two different signal amplitudes As.1 AS,^ can be resolved in the probe phase when its uncertainty is small enough. Phase difksion due to SPM of the probe becomes a limitation to the QND measurement. (b) A measurement of the probe phase quadrature B is contaminated with phase diffusion noise. A better signal-to-noise ratio of the QND readout is obtained in a measurement of the X(+) quadrature. The combined amplitudephase measurement of the probe cancels SPM noise with correlated amplitude noise. (The figure is not drawn to scale. The uncertainty region should be lo4 of the amplitude.)

a further improvement to the signal transfer to the probe beam requires a correspondingly larger QND gain K to dominate over SPM noise of the probe. The key to a noiseless QND experiment in fibers is the measurement of a linear combination of phase and amplitude quadratures (see fig. 17) (Levenson, Shelby, Reid and Walls [1986], Levenson and Shelby [1987], Imoto [1990], Yamamoto, Machida, Saito, Imoto, Yanagawa, Kitagawa and Bjork [ 1990]), as was already implemented in the first continuous-wave fiber QND experiments (Levenson, Shelby, Reid and Walls [1986], Levenson and Shelby [1987], Bachor, Levenson, Walls, Perlmutter and Shelby [ 19881). A similar improvement towards

v, § 61

427

QUANTUM NONDEMOLITION MEASUREMENTS

noiseless quantum measurement was also predicted (Drummond, Breslin and Shelby [1994], Courty, Spalter, Konig, Sizmann and Leuchs [1998]) for soliton collision QND experiments (Friberg, Machida and Yamamoto [ 19921, Friberg, Machida, Imoto, Watanabe and Mukai [1996]). Since the optimized detection scheme requires a quadrature measurement, the input-output relations (6.4a,b) are transformed into an amplitude (AGUA)and phase ( B = q ) quadrature representation as used in 9 6.1.2,

AAF'

= AAF,

(6.12a)

AB;~= AB$ + 2 a (n,) AA;

+2 4 n , )

(6.12b)

(n,)AA:,

where AA=U(2&$)An and A B = m A @was used. The SPM and XPM coefficients are related to the nonlinear phase shifts (6.6) and (6.7) by 2a(n,) = 2@SPM,pand 2 K d m = 2 @ ~ p ~ , , respectively. Now the probe output fluctuations can be measured in the quadrature X that is rotated by #I with respect to the probe amplitude A; i.e., = AA;'COS(W) + hB;%n(v)

d m ,

xPt(v)

=

{AA;

[cot(W) + 2 a (n,)] +

B; + 2K

(n,) (n,)AA:

.I

sin(v).

(6.13) The self-phase modulation noise term, 2a(n,)AAr, can be eliminated by choosing [cot(?+!~)+2o(la,)]=O in eq. (6.13); i.e., 1 ~ ,= -arccot(2a (n,))

= -arccot(2@spM,,>.

(6.14)

The probe output quadrature, X$'( W), then contains only the probe input phase quadrature noise23 and the QND copy of the signal, similar to the SPM-free (a= 0) interaction discussed above. Therefore, it is not surprising that the transfer of the signal-to-noise ratio from the signal beam to the probe beam is (6.15)

which is identical to eq. (6.9) if quadratures are converted to phase and photonnumber representation. This semiclassical analysis shows that SPM-noise-free QND is indeed possible with the combined amplitude-phase measurement. The local oscillator phase I)is different from the squeezing phase 0 of the probe (see eq. 5.5 in §5.1.2), but the difference vanishes in the limit of strong squeezing. Therefore, in schematic graphcal representations (Yamamoto, Machida, Saito, Imoto, Yanagawa, Kitagawa and Bjork [ 19901, Sizmann [1997], Courty, Spalter, Konig, Sizmann and Leuchs [1998], and fig. 17), I)is shown as the squeezing angle. 23

428


lY§6

6.2. EXPERIMENTS WITH CONTINUOUS-WAVE LASER LIGHT

The first quantum-nondemolition detection of photon-number was performed by the IBM group (Levenson, Shelby, Reid and Walls [1986]) using a modification of the first fiber squeezing experiment (Shelby, Levenson, Perlmutter, DeVoe and Walls [1986]), described in 6 5.2. Prisms were used for combining and separating the probe (60mW, 676nm) and signal (130mW, 647nm) beams copropagating in the cryogenic fiber (1 14m at 2 K). Additional detectors and electronics were used to analyze the individual and combined photocurrents of signal and probe beams. The probe beam was reflected off a phase-shifting cavity, as in the squeezing experiment, in order to detect a low-noise (SPM-suppressed) superposition of amplitude and phase quadratures (Levenson and Shelby [ 19871). The signal intensity was measured and the sum and difference of signal and probe photocurrents were compared. Approximately 37% of the rms fluctuations of the probe were caused by shot noise (quantum noise) of the signal. A correlation dip of approximately 1 dB below the average combined noise was observed, similar to the “four-mode squeezing” experiment with almost identical setup and parameters (Schumaker, Perlmutter, Shelby and Levenson [19871). GAWBS and polarization-decorrelation in the low-birefringence fiber limited the amount of detected quantum correlation. In terms of QND parameters introduced later by Grangier and coworkers (Grangier, Courty and Reynaud [1992], Poizat, Roch and Grangier [1994]), a quantum correlation was measured for the first time in the history of optical QND experiments, although the transfer of signal-to-noise ratios as measured in the photocurrents was too small to put the experiment into the QND quadrant. The detected quantum correlation led to a conditional variance, V , 1 = 0.95, or a 5% reduction of noise below the standard quantum level of the signal (Levenson and Shelby [1987]) when the ratio of signal and meter gains was optimized. Simultaneously, the NTT group (Imoto, Watkins and Sasaki [ 19871, Saito and Imoto [1988], Imoto [1990]) reported on a different test apparatus preparing for a QND measurement. A classical modulation transfer from an on-off modulated signal beam to a probe beam was detected in three different setups with long fibers. In a first setup (Imoto, Watkins and Sasaki [1987]), a nonlinear Mach-Zehnder interferometer with 500 m polarization-maintaining fibers in both arms was used. The signal was co-propagating with the probe in one arm. The modulation transfer from signal power to probe phase was used to measure the optical Kerr coefficient for the nonlinear coupling of both waves. For detection of quantum fluctuations of the signal, the measurement

V,

8 61


429

accuracy was too low by a factor of lo3. Greater stability was achieved with a Sagnac ring interferometer made of 10 km of fiber (Imoto, Watkins and Sasaki [1987], Imoto [1990]). The probe and reference beams were split off the same input beam and counterpropagated through the ring where their average phase difference was self-stabilized. By duect power detection of the probe at the transmission output port of the interferometer, the modulation transfer from the signal to the co-propagating probe beam was clearly observed because the counter-propagating reference beam only sensed the average signal photon number. In the third experiment, the 10-km fiber was cut in half, rotated by 90 degrees and spliced in order to cancel polarization group dispersion (Saito and Imoto [1988]). Yet, the observed signal-to-noise ratio was not sufficient for detecting a quantum noise transfer to the probe. Following the experiments, the criteria and regimes of operation of a QND measurement in a lossy Kerr medium were investigated theoretically for an optimized experimental QND setup. T h s analysis resulted in the first formulation of practical criteria for characterizing a nonideal QND experiment (Imoto and Saito [1989]), as discussed in $ 6.1.2. These first measurements of the transfer of quantum noise and of classical modulation suggested that the long interaction length and high power density of optical fibers enables strong coupling between signal and probe. GAWBS noise was observed in experiments of both groups. The IBM group worked in a GAWBS-noise measurement window with a cryogenic fiber (Levenson, Shelby, Reid and Walls [1986]). The NTT group used a Sagnac fiber ring. It was a fiber configuration that was later proposed (Shxasaki and Haus [1990]) and successfully implemented for pulsed squeezing (see $5.3). It was also proposed for QND measurements with solitons (Yu and Lai [ 1996]), combining the advantage of pulses (no stimulated Brillouin scattering) with the advantage of detecting the optimized SPM-free probe quadrature. A key to the success of the IBM group was the detection of a superposition of the phase and amplitude quadrature of the probe beam, which suppresses SPM noise and optimizes the QND readout (Levenson and Shelby [1987], Yamamoto, Machida, Saito, Imoto, Yanagawa, Kitagawa and Bjork [1990]). A third, theoretically more promising approach in terms of QND gain employed a fiber ring cavity (Bachor, Levenson, Walls, Perlmutter and Shelby [ 19881). A theoretical treatment and preliminary measurements (Shelby, Levenson and Perlmutter [ 19881, Shelby, Levenson, Walls and Aspect [ 19861) showed that QND measurements using a fiber ring cavity to enhance the nonlinear coupling were advantageous. Compared with the traveling-wave scheme (Levenson, Shelby, Reid and Walls [1986]), an enhancement fac-

430


1% § 6

tor of 400 for the QND gain was expected with a resonator finesse of 10 for both modes. The added difficulty in the experimental realization (Bachor, Levenson, Walls, Perlmutter and Shelby [1988]) of the fiber ring resonator, compared with a traveling-wave scheme, is the boundary condition of the resonator. It causes phase shifts which can mix amplitude and phase quadratures. If signal phase noise from SPM and from classical phase noise sources is mixed with amplitude noise, the back-action evasion property is lost and added amplitude noise may appear in the signal. The signal frequency therefore needed to be locked exactly to the center of the resonator mode. Furthermore, low-frequency depolarized light scattering in conjunction with polarization-selective optical elements in the resonator generated amplitude fluctuations, and therefore classical probesignal correlations through the QND interaction, which masked a quantum effect. The polarization conditions varied over a time scale of minutes, so that QND measurements were possible only under temporarily optimum QND conditions. The best quantum correlation coefficient achieved was C = 0.26, exceeding the classical noise correlation limit of C, = 0.14 due to excess noise. The experimental results were obtained with two krypton laser beams at 647 nm (probe) and 676 nm (signal), both resonantly enhanced simultaneously in a 13-m fiber resonator of low birefringence, with a finesse of 7 and 12, respectively. This resulted in an effective interaction length of approximately 100m, and in a circulating power of 112 and 240 mW for probe and signal. In order to optimize the QND readout (Levenson and Shelby [1987]), the probe was phase-shifted by reflection from an external cavity before detection, as in the traveling-wave squeezing (Shelby, Levenson, Perlmutter, DeVoe and Walls [19861, Schumaker, Perlmutter, Shelby and Levenson [ 19871) and QND schemes (Levenson, Shelby, Reid and Walls [1986]). Intrinsic excess noise, locking error and depolarization fluctuations limited the performance of the system. Excess noise in the fiber ring resonator was generated through thermal light scattering processes. GAWBS reduction by cooling to liquid helium temperature and by removing the polymer jacket was not feasible because of the complexity of the setup. The experimental problems of QND detection and of squeezing are related. Therefore, it is not surprising that elimination of GAWBS noise and a major improvement in signal-to-noise ratios was expected from using short soliton pulses (Haus, Watanabe and Yamamoto [ 19891, Sakai, Hawkins and Friberg [1990]) instead of CW laser light. The following fiber QND detection experiments employed picosecond solitons and a differential dual-pulse method for elimination of GAWBS noise at room temperature.

v, 0 61

43 1


6.3. EXPERLMENTS WITH SOLITONS

Interacting solitons experience a permanent phase and position shift proportional to the intensity of the collision partner (Zakharov and Shabat [1972]). The quantum theory of solitons shows that the soliton collision can be used to perform quantum nondemolition measurements (Haus, Watanabe and Yamamoto [1989]). A quantum nondemolition (QND) measurement of the signal photon number detects the phase shft of the probe soliton, using a reference soliton that does not interact with the signal soliton, as is shown in fig. 18. Three major experimental challenges must be taken up for such a QND measurement: (1) the preparation of packets of three solitons (signal, probe and reference) in the fiber, (2) the elimination of classical phase noise, and (3) the readout of the QND signal with suppressed quantum phase diffusion noise of the probe and reference solitons. After a first proposal for an experimental apparatus based on a nonlinear interferometer (Haus, Watanabe and Yamamoto [ 1989]), a detailed feasibility Negative Dispersion Probe

Reference

From Two-Color Soliton Source

k

/ Signal

w

Grating

Iy I \

A

I\

Soliton Overlap

i j\ r..........

...A

Sianal Soliton Shot-Noise

!

RF Spectrum Analyzer

Delay Line, 30 m

Fig. 18. Outline of a soliton quantum-nondemolition measurement of the photon number. After the probe-signal interaction in the fiber, probe and reference are overlapped in a Mach-Zehnder interferometer with a ~ / relative 2 phase delay in addition to the group delay. The phase difference between probe and reference is a readout of the signal soliton photon number (Friberg, Machida and Yamamoto [19921).

432


!x§6

study of a different soliton-collision interferometer was performed (Sakai, Hawkins and Friberg [19901). With this experimental scheme, Friberg, Machda and Yamamoto [1992] realized the first, and so far only, quantum nondemolition measurements with optical solitons in fibers. Both the experimental demonstration of a single back-action evading (BAE) measurement (Friberg, Machida and Yamamoto [ 19921) and fist steps towards repeated BAE detection were reported (Friberg, Machida, Imoto, Watanabe and Mukai [19941, Friberg, Machida, Imoto, Watanabe and Mukai [ 19961). The experimental apparatus for the single (fig. 18) and repeated BAE measurements were very similar. A different pulse sequence and a longer fiber were used to realize two collisions. The following discussion of the first soliton QND experiment shows how the problems of packet preparation and GAWBS noise elimination were solved ($0 6.3.1 and 6.3.2). The problem of quantum phase diffusion noise (6 6.3.3) was addressed in recent proposals to overcome thls remaining limitation ($ 6.3.4).

6.3.1. Pulse preparation All three pulses involved in the measurement - signal (2.6 ps), probe (3.6 ps), and reference (3.6 ps), with proper spacing and relative velocity - were prepared from a single-SPM-broadenedhigh-power pulse using a novel spectral filter and delay-line technique (Friberg, Machida and Yamamoto [19921). The spectrally separated signal, probe and reference pulses were obtained by using a two-color bandpass filter and by cutting two slices (approximately 0.8 nm wide) out of the 10-nm spectrum of the input pulse. Using delay lines behind the spectral filter, the timing of signal, probe and reference was adjusted to their relative velocity (spectral offset of signal and probe) so that a complete collision between signal and probe occurred in the 400m long QND fiber. For shorter solitons in a shorter QND fiber, a more energy-efficient pulse preparation scheme is needed. Shorter solitons require more bandwidth and energy. A promising source may be a two-color mode-locked chromiumYAG laser. For a titanium-sapphire laser this mode of operation was already demonstrated (de Barros and Becker [1993], Evans, Spenche, Burns and Sibbett [ 19931, Leitenstorfer, Furst and Lauberau [19951). However, when probe and signal are derived from one laser source either by spectral filtering of spectrally broadened pulses or by two-color mode-locking in one laser cavity, unwanted correlations between the pulses and excess noise may appear. This was already observed experimentally. Spectral filtering of a shot-noise-limited SPMbroadened pulse revealed spectral correlations (Spalter, Burk, Konig, Sizmann

Y 9 61


433

and Leuchs [1998]) and added up to 15 dB of excess noise into the transmitted pulses when the nonlinear filter losses were large (Spalter, Burk, StroBner, Sizmann and Leuchs [1998]). 6.3.2. Elimination of GAWBS noise in the QND detection

The second problem, the elimination of GAWBS noise in the probe phase measurement, was solved with the probe soliton closely following the reference soliton so that both pulses pick up essentially the same GAWBS phase fluctuations. In the QND experiment, Friberg, Machida and Yamamoto [1992] placed the reference only 30ps ahead of the probe pulse (fig. 18). This eliminated room-temperature GAWBS noise in the measurement. As was shown by Townsend and Poustie [1995], excellent reduction of GAWBS noise by more than 12 dB is achieved when the two pulses are separated by less than 100ps and detected differentially in the phase measurement. On the detection side, the signal was separated from probe and reference pulses by a grating and the QND signal was extracted using a Mach-Zehnder interferometer (fig. 18). The experiment showed a 0.25 dB noise reduction below the combined noise level due to the correlation between signal intensity and probe phase, where the signal intensity was shot-noise limited (Friberg, Machida and Yamamoto [1992]). The achieved signal-to-noise transfer (see 0 6.1.2) was S N V ' = 0.98SNR: for the signal after the collision with 0.1 dB fiber losses (98% transmission). When output coupling losses and linear losses at the grating are included, 8 1% of S N V were transferred to the signal output. The transfer of S N V to the probe detector was S y t = O . 1 5 S N V , derived from the 0.7 dB (15%) increase in probe phase noise when the shot-noise-limited signal beam was turned on (Friberg, Machida and Yamamoto [1992]). The total transfer, S N V ' + S N Y ' , is larger (quantum domain) or just below S N V (classical domain), depending on whether output coupling losses and grating losses for the signal are included. An advantage of soliton QND compared to other QND detection methods is the straightforward and efficient extension to repeated QND. In the doublecollision experiment (Friberg, Machida, Imoto, Watanabe and Mukai [1996]), the signal soliton stayed in the fiber for both interactions. There is no additional mode matching or coupling loss; only propagation loss (0.3dB) degrades the signal before the second collision. In the experiment it was shown that probe and reference pulses picked up the same quantum cross-phase modulation to within 13 dB. An extension to repeated BAE detection requires individual readouts of the meter pulses of the first and second collision, which can be

434


[v, 0 6

implemented easily in this scheme,with an additional soliton that does not collide with the signal soliton. 6.3.3. Quantum noise of the probe The soliton QND experiments were limited by detection efficiency and by phase diffusion quantum noise of the probe beam. The two limitations were inherent in the Mach-Zehnder and pulse delay detection scheme. Firstly, uncorrelated vacuum fluctuations entered the interferometer via beam splitters, because in this pulse delay technique only 50% of each pulse, probe and reference, overlapped. In the context of all-optical soliton switching, a more efficient QND scheme using orthogonally polarized probe and reference pulses was proposed (Friberg [ 19931). However, GAWBS noise will at best be reduced by 7dB; i.e., not as perfect aswith parallel polarized pulses (Townsend and Poustie [19951). A solution to this probkm will be presented below in connection with quantum phase noise suppression of probe and reference, the third major challenge that must be taken up. Secondly, taking the quantum nature of the probe soliton into account, the optimum probe power is found to be approximately 1/10 of the signal power (Drummond, Breslin and Shelby [ 19941). The best signal-to-noise ratio that can he achieved with a probe phase measurement yields a correlation dip of 20% below the shot-noise limit of the signal. This is the absolute lower bound on the conditional variance with a fiber length of approximately five times the soliton interaction length to allow the solitons to separate after the collision (Drummond, Breslin and Shelby [1994]). Therefore, any improvement of the QND measurement requires a different approach. The key to coming closer to an ideal soliton QND experiment is to detect the probe with an optimum superposition of phase and amplitude quadratures (see fig. 17) as in the continuous-wave fiber QND experiments (Levenson and Shelby [1987]). Drummond, Breslin and Shelby [1994] showed that perfect QND detection with optical solitons requires equal amplitudes of signal and probe and low relative velocities in addition to the combined amplitude-phase measurement. 6.3.4. Recent proposals Practical schemes for SPM noise elimination in a soliton QND measurement have been discussed recently (Yu and Lai [1996], Spalter, van Loock, Sizmann and Leuchs [ 19971). A new detection scheme (Courty, Spalter, Konig, Sizmann

v, 5 71

PHOTON-NUMBER SQUEEZING

435

and Leuchs [ 19981) prqmises a simultaneous reduction of both classical GAWBS noise and quantum SPM noise. It uses the combined phase-amplitude measurement together with a delayed-pulse method. A frequency-selective phaseshifting cavity (Levenson, Shelby and Perlmutter [ 19851, Galatola, Lugiato, Porreca, Tombesi and Leuchs [1991]) allows for a shift of the phase of the field fluctuations relative to the phase of the carrier field. The carrier acts as a local oscillator, turning phase fluctuations into amplitude fluctuations such that the SPM noise cancels out (Levenson and Shelby [1987]). A subsequent direct intensity measurement and electronic subtraction of the two noise signals (probe and reference) eliminates the classical GAWBS noise in the QND measurement. With the detection problems identified and promising solutions at hand, new and more efficient QND detection using solitons in optical fibers can be taken on.

0

7. Photon-Number Squeezing

7.1. SPECTRAL FETEIUNG

In 1995, an unexpected intensity noise reduction mechanism was discovered by NTT researchers Friberg, Machida and Levanon [1995] and was discussed in further detail by Friberg, Machida, Werner, Levanon and Mukai [1996]. The noise reduction mechanism is based on the spectral filtering of pulses after propagation through a fiber over a length of several soliton periods. Depending on pulse width, energy and fiber length, the pulse spectrum will broaden or contract. A bandpass filter centered around the output pulse spectrum introduces intensity-dependent losses and creates a nonlinear input-output transfer function for the pulse energy (fig. 19). From the slope of the energy transfer function, the classical intensity noise transfer behavior can be derived (Leuchs [ 19861). In the quantum domain, spectral filtering may deamplify photon-number fluctuations below the shot-noise level at those input energies where classical noise reduction is also found. In contrast to the quadrature squeezing discussed in 6 5, the combination of spectral broadening and spectral filtering does not conserve the photon number and produces directly detectable squeezing of the photon-number fluctuations. In direct detection, phase noise and frequency chirp are not limitations. Indeed, the new squeezing mechanism seems to be immune to GAWBS noise and is applicable to fundamental solitons as well as to chirped pulses, thus extending the range of pulse propagation regimes for sub-shot-noise measurements. However,

436


[v, § 7

Input energy (n,J

Fig. 19. Schematic diagram of (a) the energy transfer and (b) the noise transfer characteristic for spectrally filtered solitons. Noise reduction is expected to occur at certain input energies where the energy transfer function shows a reduced slope. The transfer of photon-number uncertainty from a coherent input to-the output described by the squeezing ratio R = (An2)/(.), must be derived from a quantum model.

the largest noise reduction predicted so far falls short of the lOdB goal. Nevertheless, it is one of the most significant recent developments in quantum fiber optics as it allows for incoherent sub-shot-noise measurements in a wide range of nonlinear pulse evolution, it has led to the best squeezing with subpicosecond pulses (67.3.2) and it has made spectral quantum correlations of solitons accessible to observation (6 7.3.3). Furthermore, spectral filtering is a key element in high-performance fiber-optic communication, e.g., for reduction of Gordon-Haus timing jitter, thus bringing quantum fiber optics closer to present-day incoherent optical communication (see § 8). 7.1.1. Ampllfication and deamplijcation of quantum noise

The nonlinear transfer function and the squeezing mechanism can be understood in terms of nonlinear pulse evolution in the fiber with subsequent filtering; e.g., at the soliton energy. The classical pulse dynamics in the fiber tend to stabilize the soliton. If the input pulse energy or shape differs from that of the fundamental

v, o 71


Time

437

Frequency

Fig. 20. Photon-number squeezing from spectral filtering of solitons. (a) The photon-number uncertainty An of the coherent input soliton becomes correlated with an uncertainty in spectral width (b) during propagation through the fiber. When the out-lying sidebands are removed through spectral filtering (c), the photon-number uncertainty is reduced (R< 1) below the shot-noise limit in the transmitted pulse (d).

soliton, the self-stabilizationdynamics (Hasegawa and Tappert [ 19731) will cause oscillations in the temporal and spectral width as the soliton propagates. In the long run, a certain fraction of the input energy ends up in a fundamental or higher-order soliton with increased or reduced spectral width24. If a spectral bandpass filter is located at a propagation distance where excess energy leads to spectral broadening, the filter will remove the excess energy (fig. 20) and will allow for increased transmission in the case of lower pulse energies due to spectral narrowing. As a result, the filter acts as an optical limiter, transmitting a constant power for a small range of input powers, thus reducing fluctuations in the transmitted pulse (fig. 19). The classical output noise properties can be derived from the nonlinear transfer function. The maximum noise reduction occurs at the points of zero slope of the input-utput function. In contrast to a reduced slope, an enhanced slope in the nonlinear input-output curve is expected to amplify the input fluctuations (Leuchs [ 19861). A l l l y quantum-mechanical treatment of the input-output noise transfer shows that even fluctuations at the quantum level may be deamplified at certain input energies in close analogy to classical noise reduction. However, there are

24 A soliton amplitude of N > 0.5, i t . at least a quarter of the fundamental soliton energy is required

for asymptotically creating a soliton out of a sech2 input pulse (Zakharov and Shabat [1972]).

438

[Y 8 7


Tab!e 3 Predictions for reduction in photon-number noise of spectrally filtered solitons with and without stimulated Raman scattering (SRQa Reference

Rb

R~ ( d ~ ) pump pulse’

Nd

ce

Without SRS

Werner [1996al

0.22

-6.5

soliton

1.o

3.0

Werner [ 1996bl

0.30

-5.2

soliton

1.o

2.6

Werner [1996b]

0.2 1

-6.8

soliton

1.1

2.0

Friberg, Machida, Werner, Levanon and Mukai [ 19961

0.26

-5.9

C = -0.5f,

1.2

4.5

soliton

Mecozzi and Kumar [1997]

0.22

-6.5

soliton

1.o

3.0

Werner and Friberg [1997a]

0.19

-7.1

soliton

1.2

4.0

Werner [1997a]

0.15

-8.1

soliton

1.3

6.7

Spalter [1998]

0.23

-6.3

soliton

1.o

2.9

Spalter [1998]

0.359

-4.59

soliton

1.o

Mecozzi and Kumar [ 19981

0.41

-3.9

soliton

1.0

Werner [1996b]

0.37

-4.3

1.8 ps soliton

1.o

Werner [1996b]

0.34

-4.7

1.8 ps soliton

1.1

2.1


0.61

-2.1

1.8ps sech’, k” > 0

1.2

14.0


0.33

-4.8

1.8ps soliton

1.o

3.0

3.3 99

SRS (r = 300K) included 2.6

The filter function is a bandpass filter unless otherwise indicated. R is the squeezing ratio. The pulse widths are FWHM values ( ~ F W H M= 1.763To). N is the amplitude in soliton units (soliton order for k” < 0). is the propagation distance in units of soliton periods. C is the chirp parameter as defined by Agrawal [1995]. g The filter function is a high-pass or a low-pass filter. The same noise reduction was found for both filter functions. a

also operating regimes where quantum noise enhancement is found even though a classical noise reduction is expected, demonstrating the different qualities of quantum and classical noise models. Therefore, a quantum analysis of this system is required for a prediction of quantum noise reduction. Measurements of quantum noise reduction and enhancement cannot be emulated by classical noise transfer models. Predictions of quantum noise reduction by the novel filtering technique are summarized in table 3. For fundamental (N = 1) solitons of the nonlinear

439


-g

0

-1

Y

& -2 BQ -3 0

.-$

-4

-ca,

-5

0

-6

0

2

4

6

8 1 0 1 2 1 4

Propagation distance

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Quantum mechanical limit in optical precision measurement and communication, in: Progress in Optics, Vol. XXWI, ed. E. Wolf (Elsevier, Amsterdam) p. 88. Yao, D., 1997, Schrodinger-cat-like states for quantum fundamental solitons in optical fibers, Phys. Rev. A 55, 701. Yariv, A,, 1987, Quantum Electronics (Wiley, New York). Yoo, H.-I., and J.H. Eberly, 1985, Dynarmcal Theory of an atom with two or three levels interacting with quantized cavity fields, Phys. Rep. 118, 239. Young, M., 1993, Optics and Lasers, Including Fibers and Optical Wave Guides (Springer, New York). Yu, S.-S., and Y. Lai, 1995, Impacts of the self-Rainan effect and third-order dispersion on pulsesqueezed state generation using optical fibers, J. Opt. SOC.Am. B 12, 2340. Yu, S.S., and Y. Lai, 1996, Quantum-nondemolition measurement of the photon number with a fiher ring: enhancement of the correlation through multiple soliton collisions, in: Int. Quantum Electronics Conf. (IQEC), Sydney, Australia, Vol. 20 of OSA Technical Digest Series (Optical Society of America, Washington, DC) p. 246. Yuen, H.P., and V.W.S. Chan, 1983, Noise in homodyne and heterodyne detection, Opt. Lett. 8, 177. Yurke, B., 1985, Optical back-action-evading amplifiers, J. Opt. SOC.Am. B 2, 732. Zabusky, N.J., and M.D. Kruskal, 1965, Interaction of “solitons” in a collisionless plasma and the recurrence of initial states, Phys. Rev. Lett. 15, 240. Zakharov, YE., and A.B. Shabat, 1972, Exact theory of two-dimensional self-focusing and onedimensional self-modulation ofwaves in nonlinear media, Sov. Phys.-JETP 34,62. [ 1971,Zh. Eksp. Teor. Fiz. 61, 1181.

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AUTHOR INDEX FOR VOLUME XXXIX

A Abarbanel, H.D.I. 215,218, 224, 227 Abbas, G.L. 73 Abram, I. 383, 386 Abramochkin, M. 3 11 Adam, G. 153-157, 185 Adam, P. 91 Adams, C.S. 175, 329 Aden, A.L. 230 Agarwal, G.S. 87, 89, 193, 194 Agrawal, G.P. 381, 382, 388-391, 405, 418, 419,424,438 Agrawal, I? 376, 390 Aharonov, Y. 87, 148 Akulin, VM. 99 Al-Hilfy, H. 333 Albert, D.Z. 87 Allen, L. 295-297, 300-303, 306, 308, 309, 311, 313-316, 318-323, 325-327,329, 330, 333, 350, 364, 365 Alonso, M.A. 16, 19, 21, 27, 50 Alter, 0. 148 Alvarez-Estrada, R.F. 219, 247, 249, 254 Anandan, J. 148 Anderson, M.E. 74, 79, 111, 112 Anderson, M.H. 173, 357, 368 Andrews, M.R. 173, 174 Andrianov, VA. 42 Antesberger, G. 98, 126, 151 Anyutin, A.P. 23 Apresyan, L.A. 17 Aragon, S.R. 264 Aravind, P.K. 309, 311 Arimondo, E. 329 Ark, J. 313, 318 Amaud, J. 4, 44, 48 Arnaud, J.A. 4, 24, 44, 48, 319 Amold, S. 326, 327

Amold, VI. 26 Amott, W.P 281 Arsaev, I?E. 40 Arthurs, E. 85, 87 Asano, S. 230 Asatryan, A.A. 49, 50, 52 Ashburn, J.R. 185 Ashkin, A. 325,329,356 Aspect, A. 378, 404, 406, 407, 429 Assion, A. 158 Au, C.K. 87 Avdeev, VG. 27, 52 Averbukh, I.Sh. 162, 180

B Babich, VM. 3, 49 Babiker, M. 329, 350, 364, 365 Bachor, H.A. 377,407, 424,426,429, 430 Bacry, H. 191 Bagnato, VS. 86 Bajer, J. 194 Baker, A. 215, 228 Balykin, V 183 Ban, M. 139 Banaszek, K. 76, 90, 122, 123, 150 Band, N? 100, 115, 135 Band, Y.B. 161 Bandilla, A. 94, 384, 385, 450, 451 Banerjee, H. 216, 224, 247 Bardroff, P.J. 97, 125, 151, 170, 171, 175 Barnett, S.M. 73,86, 133, 139, 140,303,306 Baseia, B. 86 Basistiy, 1.V 322 Bassichis, WH. 249 Baumberg, J.J. 73 Baumert, T. 158 Bayer, RN? 393-395, 406 Bayvel, L.P. 234 471

472


Bazhenov, VY. 322 Beck, M. 65, 70, 74, 76, 79, 102, '103, 105, 106, 140, 175 Becker, F!D. 432 Beijersbergen, M.W. 296, 297, 300, 309, 311, 314316,319, 320, 324, 330 Belanger, P.A. 42, 44 Bennet, J.A. 39 Berg, H.C. 325 Bergman, K. 378, 395, 396, 398, 405, 408410, 414417, 456 Bergou, J. 140 Bergquist, J.C. 165 Berlad, G. 241 Bernstein, H.J. 77 Berriman, B.J. 227 Berry, M.V. 8, 369 Berstein, I.B. 23 Bertani, P. 77 Bertoni, H. 48 Bertom, H.L. 50 Bertrand, J. 101 Bertrand, P. 101 Beth, R.A. 294 Bhandari, R. 264 Bialymcka-Birula, Z. 133, 134, 140,327,338 Bialynicki-Birula, I. 133, 134, 140, 327, 338 Biedenharn, L.C. 302, 304 BijediC, N. 87 Bjork, G. 377, 382, 384, 387, 399, 426, 427, 429 Bjorkholm, J. 367 Bjorkholm, J.E. 325, 329 Blair, D. 325 Blankenbecler, R. 227 Blatt, R. 164, 165, 167 Block, S.M. 325 Blockley, C.A. 164 Bloembergen, N. 380, 382, 383 Blow, K.J. 73, 389, 395, 452 Bodendorf, C.T. 98, 126, 151 Boggavarapu, D. 74, 111, 112 Bohm, M. 380, 393, 441, 443445 Bohmer, B. 143 B o b , J. 115, 183 Bohren, C.F. 234, 257 Boivin, L. 412, 414 Bolda, E.L. 173, 174 Bollinger, J.J. 164 Bonato, L. 376, 459

Bonse, U. 186 Booker, H.G. 3 Born, M. 5 , 6, 19, 24, 25,216 Boron, S. 269 Borovoi, A.G. 215, 273, 281 Bouche, D. 20, 52, 53 Bourrely, C. 237, 241, 247, 251-253, 2 7 4 276 Bouwkamp, C.J. 294 Box, M.A. 273 Bradley, C.C. 173 Braginskii, VB. 419 Braginsky, V.B. 98,418, 419 Brand, G.F. 312 Braunstein, S.L. 74, 89 Bravo-Ortega, A. 42 Breitenbach, G. 102, 110, 111, 140 Bremmer, H. 3 Breslin, J. 427, 434 Brevik, I. 294 Brinkmeyer, E. 278 Brown, K. 42 Brown, L.S. 182 Brown, R.H. 137 Brune, M. 96-98, 125, 151 Brunner, W. 70 Brunner, WH. 70 Bruns, H. 216 Bryant, ED. 281 Bucksbaum, PH. 181 Budden, K.G. 3, 4, 16, 25,42 Burk, M. 379, 380, 392, 393, 432, 433, 440-448 Burns, D. 432 Burrows, C.R. 10 Busch, P. 115 Buiek, V 87, 118, 125, 140, 153-157, 185, 194, 385 Byron, F.W. 215, 218, 225-229 C

Cable, A. 329, 367 Cahill, K.E. 191, 193 Cai, B. 48 Calvo, M.L. 219, 247, 249, 254, 278-280 Campos, R.A. 70 Car, R. 230 Carmichael, H.J. 70, 74, 76 Carnal, 0. 127, 175, 183 Carri, J. 75

AUTHOR INDEX FOR VOLUME X M I X

Carroll, J.E. 84, 86 Carruthers, P. 421 Carter, G.M. 459 Carter, S.J. 393, 396, 405,406,408,413,454 Casimir, H.B.G. 294 Cas!aiio,V.M. 216 Castillejo, L. 227 Cavaciuti, A. 376, 459 Caves, C.M. 89, 91, 93, 95 Censor, D. 23, 39 Cerveny, V 49 Champion, J.V 281 Chan, VW.S. 73, 93, 392 Chang, T.N. 215 Chaturvedi, S. 87, 89 Chen, T.W. 229, 239, 240, 243-245, 247, 251,264268,270,272 Chen, X. 162, 180 Cherkashin, Yu.N. 17 Chiao, R.Y. 91 Chiappetta, P. 237, 241, 245, 247, 251-253, 273-276, 283 Chlzhov, A.V 82, 87 Choudhary, S. 4, 13, 38 Chu, S. 325, 329, 356, 367 Chun, K.-Y. 41, 43 Ch+lek, P. 281 Cicuta, G.M. 230 Cirac, J.I. 99, 127, 164, 167 Cline, R. 151, 185 Cline, R.A. 185 Coene, W. 216 Coerwinkel, R.P.C. 314 Cohen,E. 383 Cohen-Tannoudji, C. 183, 305, 329, 332, 3563.59 Collett, M.J. 73, 422 Collins, M.D. 42 Connor, K.A. 23,24, 53 Conover, C.W.S. 181 Cook, R.J. 183, 329 Cooper, J. 70, 74, 76, 102, 106, 140 Corbett, J.V 115 Comell, E.A. 173, 357, 368 Corney, J.F. 459 Corsioni, R. 376, 459 Cotter, D. 380, 383 Cotton, R.W. 281 Coudreau, T. 108 Courtial, J. 295, 318, 320, 321, 323, 327

473

Courtis, J.-Y. 345 Courty, J.M. 386, 421, 422, 427, 428, 434, 448 Couture, M. 42, 44 Cowley, J.M. 216 cummings, EW. 97 D Dabbicco, M. 412, 414 Dakna, M. 140-142, 150 Dalibard, J. 329,356359 Danakis, S. 309, 311 D’Arimo, G.M. 86, 104, 107-109, 113, 138, 143, 144, 146148, 150, 195 Dasgupta, B. 277 Davidovii., D.M. 87 Davidovich, L. 96, 98, 127, 169, 172 Davies, E.B. 87 Davies, K.B. 173 Davis, G. 376, 459 Davis, L.W. 296, 304 de B m s , M.R.X. 432 de Boor, C. 107 de Matos Filho, R.L. 168, 171, 172 Debi, S. 215,241,273 Dehmelt, H. 163, 165, 329 Demin, A.V 27, 52 Demkov, Yu.N. 216 Derka, R. 153-157, 185 Deschamps, G.A. 4, 13, 18, 38, 39, 44,4749, 51 d’Espagnat, B. 115 DeVoe, R.G. 377, 378, 380, 387, 395, 398, 399,404,40&408,428, 430 D’Helon, C. 165, 166 Dhiani, A-A. 179 Dholakia, K. 295, 313, 318-323, 325-327 Di Domenico Jr, M. 386 DiManio,F. 258 Dodonov, VV 125, 145, 184 DMIT, C.R. 396, 398, 409, 411, 412, 415417,454, 457 Domokos, P. 97, 125, 151 Dong, L. 381 Doran, N.J. 451, 452 Doron, E. 43 Dougherty, D.J. 406, 413 Dowling, J.P. 194, 329 Draine, B.T. 281 Draine, B.T.D. 282

474


Drexhage, M. 386 Dreyer, I. 125 Drobn?, G. 153, 154, 185, 194 Drummond, P.D. 73, 89, 93, 380, 382, 392, 393, 396, 397, 405, 406, 408, 413,414, 427, 434,454,459 Ducuing, J. 380, 383 D ~ n nT.J. , 158-160 Dupont-Roc, J. 305, 332 Durfee, D.S. 173, 174 Dutra, S.M. 127 D~tta-Roy,B. 224,247 Dykaar, D.R. 181 Dziedzic, J.M. 325

E Eberbard, PH. 91 Eberly, J.H. 333, 385 Einziger, PD. 24, 42, 44, 53 Ekstrom, C.R. 175 El-Hewie, M.F. 42 Elwenspoek, M. 264 Englert, B.-G. 140 Englert, F. 2 15 Ensher, J.R. 173, 357, 368 Epstein, PS. 3 Epstein, S.T. 192 Essiambre, R.-J. 376, 390 Evans, B.T.N. 282 Evans, J.M. 432

F Fabre, C. 377, 386, 421,459 Farn, LeKien 99 Fano, U. 100, 183 Faridani, A. 65, 102, 103, 105, 106 Fauster, T. 441, 449 Feam,H. 70 Federici, J.F. 181 Fedoriuk, M.V. 21, 26 Fee, J. 376,459 Feenberg, E. 115 Felsen, L.B. 4, 13, 20, 23-25, 38, 42, 44, 48, 50, 53 Felsteiner, J. 186 Feng, G. 48 Feng, F! 367 Fermann, M.E. 451 Feynman, R.P 183 Ficker, J. 453457

Fiddy, M.A. 247 Fields, B.D. 115 Finer, J.T. 325 Firth, W.J. 368 Fischer, D. 178 Fischer, E. 163 Flannery, B.P. 197 Flatau, P.J. 281, 282 Fluck, R. 443 Flugge, S. 248 Fontana, T. 49 Forbes, G.W. 16, 17, 19, 21, 27, 49, 50 Fougkres, A. 84, 143 Fournier, G.R. 282 Fox, A.M. 412, 414 Franco, V 215 Franko, F! 376, 459 Freyberger, M. 84, 86,99, 128, 130, 171, 178 Friberg, S.R. 378-380, 392, 393, 397, 405, 406,414,424,427,430435,43843,446, 449,453, 454 Friedman, C.N. 115 Friese, M.E.J. 325 Frishat, S.D. 43 Fritsch, W. 185 Frolov, VV 53 Fujimoto, J.G. 451 Fukumitsu, 0. 49, 53 Fiirst, C. 432 Fymat, A.L. 273, 282 G Gabrielse, G. 182 Galatola, P. 407, 435 Gale, W. 115, 184 Galetti, D. 194 Gantsog, T. 377, 387 Gao, X.J. 42 Gardiner, C.W. 73, 89 Garetz, B.A. 326, 327, 367 Garibaldi, U. 216 Garrison, J.C. 17 Gate, L.F. 281 Gauss, C.F. 151 Gea-Banacloche, J. 329 Geisler, M. 158 Gelles, J. 325 George, N. 381 Gerjuoy, E. 218 Gerlach, W. 183

AUTHOR INDEX FOR VOLUME XXXLX

Gerry, C.C. 384 Gersten, J.I. 215 Ghione, G. 42 Ghosh, G. 247, 248, 284, 285 Giacobino, E. 108, 377, 386, 421, 459 Gibson, N.D. 185 Gien, T.T. 218 Giese, R.H. 273, 274 Gilmore, R. 26 Gisin,N. 459 Glasser, A.H. 42 Glauber, R.J. 95, 188, 189, 191, 193, 195, 215, 218, 219, 222-224, 226 Glogower, J. 140 Goldman, 1.1. 215 Golovchenko, J.A. 186 Golub, G.H. 197 Gomez, A. 216 Gordon, J.E. 282 Gordon, J.P. 356, 390, 391, 404, 439, 442, 458 Goto, K. 42 Goto, T. 395, 398, 409, 415, 416 Gottfiied, K. 294 Gowan,E. 49 Goy, F! 96 Grabow, J. 74, 75 Gracia-Bondia, J.M. 194 Gradshteyn, I.S. 260 Grangier, P. 73, 422,423, 428 Granovslai, Ya.1. 284 Grasso, G. 376, 459 Greenberg, J.M. 215 Greene, B.I. 181 Grelu, P. 423 Grimshaw, R. 3, 37 Grossmann, A. 191 Gruner, T. 189 Grynberg, G. 305,332,345 Gusein-Zade, S.M. 26 Guth, E. 115, 184 Guttmann, M.J. 294, 300

H Haberl, F. 451 Hagley, E. 125 Hahn, Y. 227 Hall, J.L. 377 Hallett, F.R. 273 Hallett, J. 281

475

Hamilton, W.R. 216 Hammond, T.D. 179 Hanna, D.C. 380, 383 Hansch, T. 329 Haramaty,Y. 42 Hardman, A.D. 406, 413 Hardman, P.J. 395 Hamad, J.P. 216 Haroche, S. 9 6 9 8 , 125, 151 Harris, M. 311, 318 Hart, R.W. 282 Hasegawa, A. 390,437 Hashimoto, M. 44, 53 Haus, H.A. 84, 87, 88, 297, 298, 304, 378, 380, 384, 390, 392, 395, 396, 398, 399, 405,406,408-421,425,429431,451,454,

457459 Havener, C.C. 185 Hawkins, R.J. 430, 432 He, H. 313, 325 Heading, J. 4, 31 Heam, D.J. 43 Heckenberg, N.R. 312, 313, 325 Heethaar, M.R. 281 Heffels, C. 281 Heidmann, A. 377, 421 Heinzen, D.J. 164 Heitzmann, D. 281 Helbing, J. 158 Heller, W. 272 Hellwarth, R.W. 381 Helstrom, C.W. 87, 155 Heritage, J.P. 387 Herkommer, A.M. 99, 100, 128, 130 Herzog, U. 135, 146 Hessel, A. 50 Heurtley, J.C. 49 Heyman, E. 24, 42,44 Hilico, L. 386 Hill, C.A. 311, 318 Hillery, M. 140 Hirano, T. 329, 368 Hirleman, E.D. 281 Ho, S.T. 91 Hodgkinson, R.J. 269 Hoekstra, A.G. 281 Hofer, M. 451 Hoffman, D. 367 Hohenstatt, M. 163 Holevo, AS. 87, 155

476


Holford, R.L. 36 Holland, M.J. 422 Hollberg, I,. 377 Hollberg, L.W. 377, 406 Holt, A.R. 267, 281 Hong, C.K. 70, 92 Hoock, D.W 251 Hosaka, T. 387 Hovenac, E.A. 42 Hradil, 2. 84, 87, 139, 143, 157 Huffman, D.R. 234,257 Hulet, R.G. 165, 173 Hunklinger, S . 393 Hunziker, W. 215 Hurst, C.A. 115 Husimi, K. 87, 192, 193 Huttner, B. 73 I Ikuno, H. 42 Imamoglu, A. 148 Imoto, N. 123, 377, 382, 384, 387, 399, 419422, 424429, 432, 433 Ippen, E.P. 387, 398, 405, 406, 408, 409, 412418, 451, 457 Ishihara, T. 42 Itano, W.M. 116, 122, 151, 163-165, 167170 Ito, M. 44 Itzykson, C. 215, 224, 227 IvanoviC, I.D. 115, 185 Izmest'ev, A.A. 44 J Jackson, J.D. 294, 305 Jacob, J.M. 459 Jain, A. 185 Jakob, J. 48 James, G.L. 52 Janicke, U. 175 Janszky, 3. 91, 167 Jauch, J.M. 294, 304 Javanainen, J. 173 Japes, E.T. 97, 153 Jeffers, J.R. 382 Jennings, B.R. 264 Jessen, P.S. 329 Jex, I. 77, 91, 92, 101, 132 Joachain, C.J. 218, 225-229 Jonas, P.R. 281

Jones, A.R. 234 Jones, D.S. 39, 42 Jones, KR.W 144, 155, 157, 185 Jones, R.M. 4, 16, 39 Jones, R.R. 180, 181 Jull, G.W. 4 Juncos del Egido, F! 219, 254, 278-280 Jung, I.D. 443 Jung, Y.D. 216

K Kaloshin, V.A. 36 Kano, Y. 87 Kaplan, D.R. 186 Karal, F.C. 3, 37 Kirtner, EX. 406, 413, 425 Katriel, J. 282 Kazansky, P. 381 Kazansky, P.G. 381 Kazantsev, A.P. 183, 294, 329 Keitel, C.H. 87 Keith, D.W. 175 Keller, J.B. 3, 4, 27, 37, 44,45, 52 Keller, U. 443 Kelley, P.L. 195 Kelly Jr, J.L. 85, 87 Kemble, E.C. 115 Kerker, M. 230, 234 Kerr, J. 381 Ketterle, W. 173, 174 Khalili, F.Y. 98 Khatri, F.I. 454 Kienle, S.H. 178 Kikuch, K. 398, 400, 404 Kim,C. 78 Kim, M.S. 78, 88, 95, 98, 123, 126, 145, 151 Kimble, H.J. 75, 77, 127, 377 Kinber, B.E. 40 Kincaid, B. 186 Kmg, B.E. 116, 122, 151, 163, 164, 167-170, 172 Kiss, T. 91, 92, 109, 110, 114, 132, 146, 148, 150 Kitagawa, M. 148, 377, 382, 384, 387, 397, 399,400, 420, 421,426,427,429,450,451 Kittel, C. 393 Klauder, J.R. 70, 95, 191 Klein, N. 96, 385 Kleiner, W.H. 195 Klett, J.D. 273,281, 282


Knight, P.L. 87, 97, 122, 125, 127, 153-157, 185, 377, 384, 385 Knoll, L. 140, 189 Kochahski, P. 86, 88 Kodama, Y. 390 Kogelnik, H. 47,48, 319 Kogelnik, H.W. 47, 48 Kokhanovsky, A.A. 231, 281 Kokorowski, D.A. 178, 179 Konig, F. 380, 392, 395, 396, 409, 421, 427, 432,434,441,448450,453457 Kopilevich, Yu.1. 53 Korrnilitsyn, B.T. 32 Korolkova, N. 386, 441, 448 Kouyoumjian, R.G. 52 Kowalczyk, P. 158, 159 Krahmer, D.S. 113, 122 Kravtsov, Yu.A. 4 6 , 8, 17, 18, 22-27, 30, 31, 33, 39, 40, 44, 45, 5G52 Krebes, E.S. 43 Kreinovitch, VJa. 115 Kristensen, M. 314 Krotkov, R.V 215 Kruskal, M.D. 390 Krutikov, VA. 215,273 Krylov, D. 456 Kubota, H. 459 Kudou, T. 48, 53 Kuga, T. 329, 368 Kiihn, H. 77, 79, 107 Kujawaski, E. 227 Kujawski, A. 48 Kumar, P. 78, 3 7 7 , 4 3 8 4 0 , 4 4 6 4 8 Kume, S. 395, 398,409,415,416 Kum, D.M. 173, 174 Kurtsiefer, Ch. 175-177 Kumetsov, VV 281 Kwiat, P.G. 91 L La Porta, A. 91 Lagendijk, A. 180 Lahti, P.J. 115 Lai, W. 329 Lai, W.K. 329, 350 Lai, Y. 84, 87, 88, 392, 405, 406, 414, 415, 429,434,454 LaloviC, D. 87 Lamb, W.E. 115 Lambrecht, A. 108

477

Lamy, P. 251, 274 Landau, L.D. 215 Lan&ck,R. 325 Lane, AS. 91 Lapeyere, G.J. 329 Latimer, l? 281 Lauberau, A. 432 Lax, M. 296 Lee, C.T. 135 Lee, S.W. 49 Lefevre, V 98 Legendre, A.M. 151 Lehner, J. 121 Leibfried, D. 116, 122, 151, 168-170 Leichtle, C. 162, 170, 171, 180 Leighton, R.B. 183 Leine, L. 97, 124 Leitenstorfer, A. 432 Lemaire, T. 237, 241, 247, 251-253, 214, 275 Lembessis, VE. 329, 350, 364 Lenz, G. 398, 409,411, 412, 417, 457 Lebn, 1. 216 Leonhardt, U. 70, 74, 76, 78, 80, 84, 8G88, 95, 100, 101, 109, 110, 113, 114, 121-123, 130, 143, 145, 146, 148, 150, 162, 175, 185, 194, 195 Letokhov, VS. 294, 329, 331 Lett, P.D. 357 Leuchs, G. 377, 379, 380, 384, 392, 393, 395,396,407,409,421,427,432435,437, 44W50, 453457,459 Levandovsky, D. 448 Levanon, A. 378, 379, 392, 435, 438, 4 4 443 Levenson, M.D. 377, 378, 380, 387, 393395,398400,404,406408,419,420,424, 426, 42&430,434,435,448 Levesque, R. 186 Levi, A.C. 216 Levy, B.R. 52 Levy, M. 215,227 Lewenstein, M. 189 Lewis, J.T. 87 Lewis, R.M. 23 Li, T. 48 Liberman, VS. 366 Liebman, A. 459 Lifshitz, E.M. 215 Lin, C.D. 185

478


Lin, F.C. 247 Lindell, 1.V 44 Liu, C. 281 Liu, Y. 281 Lock, J.A. 42 Lohmann, A.W. 176 London, F. 139 Lopatin, VN. 281 Lorattanasane, C. 398, 400, 404 Louck, J.D. 302, 304 Loudon, R. 70, 73, 333, 377, 380, 382, 383 Louisell, W.H. 296, 357 Lu, B. 48 Lu, L.T. 48 Ludwig, D. 26, 52 Lugiato, L.A. 407, 435 Luis, A. 70, 84, 88 Luki, A. 139 Lundborg, B. 42, 53 Lunis, B. 345 Lutterbach, L.G. 98, 127, 169 Lynch,R. 139 Lyubomirsky, I. 398,409,411,412,417,457

M Maali, A. 125 Macchi, M. 376, 459 Macchiavello, C. 86, 108, 147, 148, 150 Maccone, L. 143 Machado Mata, J.A. 69 Machida, S. 377-379, 382, 384, 387, 392, 397, 399,424,426,427,429,431433,435, 438,441443 Maciel, J. 42 Maeda, M.W. 377 Mahood, R.W. 281 Maker, P.D. 451 Makin, LA. 125 Mallik, S. 216 Mamyshev, PV 390 Mwcini, S. 104, 107, 109, 116, 173, 174, 182 Mandel, L. 70,73, 84,92, 137, 143, 195,294 Man’ko, 0.V. 104 Man’ko, VI. 104, 107, 109, 116, 125, 184 Marchiolli, M.A. 145, 194 Marcuse, D. 306 Marcuvitz, N. 20 Margalit, M. 398, 408, 412, 414, 415, 418, 457

Marks, J.R. 177 Marte, €! 127 Martell, E.C. 185 Martienssen, W. 70 Maruta, A. 390 Masalov, A.V 329 Maslov, VP. 3, 21, 22, 26, 29, 43, 52 Maslowska, A. 281 Mast, P.E. 47, 48 Matijevih, E. 230 Matthews, M.R. 173, 357 Mattle, K. 77 Mayr, E. 97, 125, 151 McAlister, D.F. 79, 80, 121, 136, 137, 150, 175 McCall, S.L. 70, 95 McDuff, R. 312 McGloin, D. 316 McKeller, B.H.J. 273 McKnight, W.B. 296 McLaughlin, D.W. 17, 35 Mease, K.D. 282 Mecozzi, A. 438-440, 446, 447 Medeiros, J.A. 215 Meekhof, D.M. 116, 122, 151, 163, 164, 167-170, I72 Meeten, G.H. 281 Meixner, A. 186 Menyuk, C.R. 459 Merk, R. 453, 454 Mertz, J.C. 377,406 Meschede, D. 96 Metcalf, H. 329 Mewes, M.-0. 173, 174 Meystre, P. 98, 126, 127, 377 Michler, M. 77 Mie, G. 230 Miesner, H.-J. 173, 174 Migdal, A.B. 2 15 Mikulla, B. 380, 443, 450, 453455 Milburn, G.J. 89, 165, 166, 377, 380, 384, 385, 387, 399, 408, 419 Miller, W.H. 43 Minogin, VG. 294, 329, 331 Miranowicz, A. 377, 387 Mishchenko, A S . 22 Mitschke, EM. 391, 442 Mittelbrunn, J.R. 216 Mittleman, M.H. 215 Mittra, R. 20, 52, 53


Miya, T. 387 Miyashita, T. 387 Miyauchi, A. 398,409,415,416 Mizrahi, S.S. 145 Mynek, J. 102, 110, 111, 175, 183 Mogilevtsev, D. 157 Moliere, G. 215 Molinet, F. 20, 52, 53 Mollenauer, L.F. 390, 391, 404, 442 Mollow, B.R. 95 Monroe, C. 116, 122, 151, 163, 164, 167170, 172 Montroll, E.W. 215, 282 Montrosset, I. 42 Moodie, A.F. 216 Moon, B.R. 281 Moore, EL. 164 Moore, R.J. 226 Mori, M. 395, 398,409,415,416 Moroz, B.Z. 115 Moms, V.J. 264 Mostowski, J. 158, 159 Mount, K.E. 8 Mourikis, S. 186 Moussa, M.H.Y. 86 Moya-Cessa, H. 122, 127 Mukai, T. 378, 379, 392, 424, 427, 432, 433, 435, 438, 441-443 Mukamel, S. 158-160 Miiller, G. 96 Muller, T. 102, 110, 111 Mund, E.H. 227, 228 Munroe, M. 74, 109-112, 114, 148, 150

N Nagourney, W. 165 Nakagaki, M. 272 Nakazawa, M. 459 Napper, D.H. 281 Naraschewski, M. I73 Narozhny, N.B. 385 Nasalski, W. 53 Nath, N.S.N. 215, 279 Natterer, F. 101 Nayfeh 229 Nelson, L.E. 415 Neuhauser, W. 163, 165 Newton, R.G. 99, 127, 184, 234,236 Nicoletopoulos, B.R. 215

479

Nienhuis, G. 302, 305, 307, 308, 319, 327, 339 Nieto, M.M. 421 Nijhof, E.-J. 281 Nikoskinen, K.I. 44 Nishimoto, M. 42 Nishizawa, N. 395,398,409,415,416 Noel, M.W. 180 Noh, J.W. 84, 143 Noordam, L.D. 180, 181 Norris, A.N. 48, 49 Nussenzveig, H.M. 43, 282

0 Ober, M.H. 451 O’Connell, R.F. 87 Ohmori, T. 42 Ono, T. 273, 274 Opatrny, T. 80, 81, 114, 116, 117, 120, 136, 141, 142, 150, 151, 194 Orenstein, M. 282 Orlov, Yu.1. 5, 6, 18, 22-27, 30, 31, 33, 36, 39, 40, 51, 52 Orlowski, A. 115 Orszag, M. 172 Orta, R. 42 Osberghaus, 0. 163 Osterberg, H. 49 Ostermayer Jr, F.W. 386 Ostrovskii, L.A. 23 Ostrovskii, VN. 216 Otsuka, K. 451 Ottewill, R.H. 281 Ou, Z.Y. 70, 77, 92 Oudar, J.L. 386 Owyoung, A. 381 Ozaki, H.J. 277

P Padgett, M. 3 18 Padgett, M.J. 295, 301, 302, 313, 314, 316, 318-323, 325-327 Pancharatnam, S. 368 Paris, M.G.A. 82, 86, 87, 90, 108, 113, 119, 138, 148 Park, J.L. 100, 115, 135 Parkins, A.S. 127, 164 Pathak, PG. 52

480


Paul, H. 70, 76, 84, 86, 87, 91, 92, 94, 95, 97, 109, 113, 115, 121, 132, 143, 145, 146, 195 Paul, W. 163 Pauli, M. 247 Pauli, W. 66, 100, I15 PaviEiC, M. 115 Pavlova, L.N. 281 Paye, J. 398,409, 411,412,417, 457 Pedersen, M.A. 26 Pegg, D.T. 86, 133, 139, 140 Penndorf, R. 282 Penning, EM. 182 Pereka, L.C.P. 13, 49 Pereira, S.F. 102, 110, 111 Perelman, A.Y. 284, 285 Perelomov, A. 191 Peiina, J. 84, 88, 137, 157, 193 Peiinovi, V 139 Perlmutter, S.H. 377,378,380,387,393-395, 398400, 404,40&-408,424,426, 428430, 435,448 P d , J.M. 241,245,251, 274 Petrich, W. 368 Petroff, M.D. 91 Pfau, T. 175-177, 183 Phillips, L.S. 140 Phillips, W.D. 294, 329, 357 Phoenix, S.J.D. 73 Pinnow, D.A. 386 Platunann, PM. 277 Plebanski, J. 192 Poe, R.T. 215 Poizat, J.-P. 422, 423, 428 Poizat, J.-Ph. 102 Polzik, E.S. 75 Popov, M.M. 49 Popper, K.R. 87 Porreca, M.G. 407, 435 Poston, T. 26 Potasek, M.J. 73 Poustie, A.J. 394-396, 409, 416, 433, 434 Power, W.L. 329, 365 Powers, S.R. 258, 259, 269 Poyatos, J.F. 167 Poynting, J.H. 294 Prasad, S. 70 Prentiss, M.G. 367 Press, W.H. 197 Pritchard, D.E. 175, 178, 179

PrugoveEki, E. 115 Psencik, I. 49 Punina, VA. 282

Q Quigg, C. 218 Quirbs, M. 216

R Ra, J.W. 48 Radmore, P.M. 385 Radon, J. 101 Radzewicz, C. 158, 159 Raimond, J.M. 96-98, 125, 151 Rajagopal, A.K. 87 Raman, c. 181 Raman, C,V 215,279 Ramsey, N.F. 179 Ray,P. 215 Raymer, M.G. 65, 70, 74,76, 79, 80, 87, 102, 103, 105, 106, 109-114, 121, 136, 137, 140, 148, 150, 162, 175, 177 Raz, S. 24, 44 Reading, J.E 249 Reck, M. 77 Reichenbach, H. 115 Reid, M. 378, 404, 406, 408, 419, 420, 424, 426, 428430 Reid, M.D. 396 Rempe, G. 96, 385 Renwick, S.P. 185 Reynaud, S. 183, 356-359, 377, 421, 422, 428 Ribordy, G. 459 Rich, T.C. 386 Richter, G. 70 Richter, Th. 109, 110, 113-115, 121, 122, 134, 136, 148, 150, 162 Rigrod, W.W. 3 11 Riis, E. 329, 356 Risken, H. 65, 77, 101, 164, 194 Risley, J.S. 151, 185 Ritze, H.-H. 384, 385, 450, 451 Eva, F. 230 Robbins, M.P. 186 Roberts, A.D. 179 Robertson, D.A. 295, 314, 318, 327 Robinson, D.J.S. 197 Robinson, N. 376, 459 Roch, J.-F. 422, 423, 428

AUTHOR INDEX FOR VOLUME XXXlX

Rohrl, A. 173 Rohrlich, F. 294, 304 Rohwedder, B. 179 Rolston, S.L. 357 Rose, M.E. 294 Rosenbluh, M. 378, 398,408,410, 413 Rouze, N. 185 Roy, A. 283, 284 Roy, A.K. 257 Roy, T.K. 245,247, 248, 258, 260-262 Royer, A. 115, 122, 139, 148 Rum, Y.Z. 48 Rubenstein, R.A. 179 Rubinow, S.I. 3 Rubinsztein-Dunlop, H. 312, 313, 325 Runge, I. 216 Russell, P.S.J. 381 Ryan, J.F. 73, 412,414 Rytov, S.M. 6, 25, 229 Ryzhik, LM. 260

S Sackett, C.A. 173 Saghafi, S. 44 SaitB, N. 87 Saito, S. 377, 382, 384, 387, 399, 422, 424, 426-429 Sakaguchi, S. 387 Sakai, Y. 430, 432 Saleh, B.E.A. 70, 377, 384 Salomon, C. 345 Sanchez-Mondragon, 5.3. 385 Sbchez-Soto, L.L. 70 Sandberg, J. 165 Sanders, B.C. 43, 78, 88, 145 Sands, M. 183 Sargent 111, M. 377 Sarkar, S. 228 Sasaki, Y. 424,428, 429 Saunders, C.P.R. 281 Sauter, Th. 165 Savage, C.M. 451 Saxon, D.S. 217, 227, 249 Sayasov, Yu. 3, 25 Scarlett, 9. 281 Schatzberg, A. 24,44 Schawlow, A. 329 Schenzle, A. 173 Schiff, L.I. 43, 215, 227, 249 S c h i f i , A. 376,459

48 1

Schiller, S. 102, 110, 111, 140 Schleich, W. 140 Scbleich, W.P. 84, 86, 97, 99, 100, 125, 130, 151, 162, 170, 171, 178, 180, 194 Schmidt, A.J. 451 Schmidt-Kaler, F. 125 Schmitt, S. 380, 450,453-4,, Schneider, S. 100, 113, 130, 162 Schrade, G. 170, 171 Sclrodinger, E. 191 Schubert, M. 190 Schiilke, W. 186 Schulp, W.A. 216 Schumaker, B. 408 Schumaker, B.L. 73, 95, 398,408, 428,430 Scully, M.O. 70, 377 Seckler, B.D. 3 Seifert, N. 185 Sekistov, VN 42 Self, S.A. 48 Seng-Tiong, H. 412,414 Senior, M. 281 Senit&, I.R. 192 Seyfned, V: 158 Shabat, A.B. 390-392, 431, 437 Shapiro, J.H. 69, 73, 88, 93, 143, 377 Shapiro, M. 161, 162, 180 Sharma, S. 238,269 Sharma, S.K. 215, 216, 218, 224, 237, 238, 241,243-245,247,248, 250, 251,257-263, 265, 269, 273, 277, 283-285 Shatalov, B.E. 22 Shatalov, YE. 22 Shelagin, A.V: 216 Shelby, R.M. 377, 378, 380, 387, 392-396, 398400,404408,410,413,414,419,420, 424,426430,434,435,448 Shepherd, J.W. 267, 281 Shepherd, T.J. 73 Sheppard, C.J.R. 44 Shibata, S. 387 Shifiin, K.S. 235, 282 Shmizu,K. 282 Shin, S.Y. 25, 44, 48 Shiokawa, N. 329, 368 Shirasaki, M. 378, 395, 396, 398, 399, 405, 409412,415417,429, 454,457 Shore, B.W. 97 Shumay, I. 441, 449 Sibbett, W. 432

482

AUTHOR lNDEX FOR VOLUME XXXlX

Siegman, A.E. 296 Sigel, M. 175 Simmonds, J.W. 294, 300 Simmons, R.M. 325 Simpson, N. 318 Simpson, N.B. 316, 319, 322, 325, 326 Sinah, A. 423 Singer, M. 115 Sitenko, A.G. 215 Simann, A. 379, 380, 392, 393, 395, 396, 409, 421, 427, 432434, 440450, 453457 Skryabia, D.Y 368 Sleator, T. 183 Slusher, R.E. 73, 91, 377, 406 Smith, B.J. 43 Smith, C.B. 273 Smith, C.P. 312 Smith, E.T. 179 Smith, G.M. 314 Smith, L.W. 49 Smith, M.S. 42 Smith, W.S. 239, 245 Smithey, D.T. 65, 70, 74, 76, 79, 102, 103, 105, 106, 140 Somerford, D.J. 218,238,243,245,250,251, 257-262, 265, 269, 284, 285 Sommerfeld, A. 216 Song, S. 91 Soskin, M.S. 322 Spadacini, R. 216 Spalter, S. 379, 380, 392, 393, 395, 396,409, 421,427,432434, 438,440450,453-455 Speiser, S. 282 Spenche, D.E. 432 Spreeuw, R.J.C. 296, 297, 300, 309 Spudich, J.A. 325 Srinivasan,V 87 Steinberg, A.M. 91, 108 Stenholm, S. 77, 85, 87 Stepanov, A.Y 216 Stepanov, N.S. 23 Stephens, G.L. 256, 263, 281 Stem, 0. 183 Stemin, B.Yu. 22 Sterpi, N. 147, 148, 150 Steuemagel, 0. 82, 87, 131 Stewart, N. 26 Stolen, R.H. 387, 390 Stoler, D. 73, 76, 308 Ston, M. 284

Stone, C.D. 185 Streekstra, G.J. 281 Streifer, W. 4, 44, 45 StroOner, U. 379, 380, 393, 395, 396, 409, 433,440-447 Stroud Jr, C.R. 180 Strumia, F. 329 Stulpe, W. 115 Sucher, J. 215, 227 Sudarshan, E.C.G. 191, 193 Sugar, R.L. 227 Sukenik, C.I. 181 Sukhy, K. 23,25, 39 Surdutovitch, (3.1. 294, 329 Susskind, L. 140 Sutherland, R.A. 282 Suzuki, K. 459 Swift, A.R. 224, 226-228 Szajman, J. 258

T Takahashi, K. 87 Takebe, H. 381 Takenaka, T. 49 Tamm, C. 311, 315 Tamura, K. 415 Tan, S.M. 114, 151, 173, 174 Tanas, R. 377, 387 Tang, C.C.H. 230 Tanguy, C. 183 Tanner, C.E. 357 Tappert, F. 390,437 Tapster, P.R. 3 11, 3 18 Tasche, M. 107 Taylor, J.R. 390 Tegmark, M. 183 Teich, M.C. 70, 377, 384 ten Wolde, A. 180 Tenney, A. 215 Terhune, R.W. 451 Terry, ED. 16, 25,42 Terunuma,Y. 387 Teukolsky, S.A. 197 Tew, R.H. 43 Thomas, B.K. 218 Thorne, K.S. 418, 419 Tinin, M.V. 17, 27, 52 Tittel, W. 459 Titulaer, U.M. 188 Tiwari, S.C. 368

AUTHOR INDEX FOR VOLUME XXXlX

Tobocman, W. 247 Todomki, S. 387 Todoroki, S.4. 387 Tollett, J.J. 173 Tombesi, P. 104, 107, 109, 116, 173, 174, 182,407,435 Tommei, G.E. 216 Toni, Y. 329, 368 Torma, P. 77, 91, 92, 132 Torrksani, B. 276, 283 Torrey, H.C. 333 Toschek, RE. 163, 165 Townsend, C.G. 173, 174 Townsend, P.D. 395, 396,409, 416, 433, 434 Trammell, G.T. 115, 184 Tribe, L. 43 Tnppenbach, M. 161 Truffin, C. 215 Turchette, Q.A. 175 Turnbull, G.A. 314 Turski, L.A. 143 Twiss, R.Q. 137

U Udo, M.K. 412,414 Ueda,M. 148 Uhtsev, P.Ya. 52 Ungar, J. 356 Unger, H.G. 48 Unruh, W.G. 418,419 V v. Schickfus, M. 393 Vaccaro, J.A. 131, 133, 143 Vaidman, L. 148 Valley, J.F. 377, 406 Van de Hulst, H.C. 217, 234, 235, 261, 283 van der Burgt, P.J.M. 151, 185 van der Straten, P. 329 van der Veen, H.E.L.O. 309, 311, 315, 316, 319, 320, 330 van Druten, N.J. 173, 174 Van Dyck, D. 216 van Enk, S.J. 302, 305, 307, 308, 319, 329, 339, 368 van Linden van den Heuvell, H.B. 180 van Loan, C.F. 197 van Loock, P. 380,393,434 Varchenko, A.N. 26 Virilly, J.C. 194

483

Varsimashvili, K.V 216 Vasilyev, M.V 448 Vasnestsov, M.V 322 Vassallo, C. 53 Vaughan, J.M. 311,318 Verkerk, P. 345 Vetterling, WT. 197 Vigneron, K. 423 Vogel, K. 65, 77, 84, 86, 101, 194 Vogel, W. 70, 74, 75, 77-83, 85, 86, 90, 91, 93, 97, 107, 114, 116, 117, 120, 122-124, 136, 140, 143, 145, 150, 151, 161-164, 166, 168, 171, 172, 189, 190, 196,421 Vogt, A. 115 Volostnikov, V 311 von Neumann, J. 65, 153 von Plessen, G. 412, 414 Vorontsov, Y.I. 98, 418, 419

W Wagner, S.S. 93, 143 Wait, J.R. 230 Waldie, B. 269 Walker, N.G. 77, 78, 82, 84, 86-88 Walker, T. 367 Wallace, S.J. 224, 225, 227, 228 Wallach, M.L. 272 Wallentowitz, S. 90, 91, 114, 116, 117. 122, 123, 150, 151, 163, 164, 166,.171, 172 Wallis, H. 173 Walls, D.F. 164, 173,174,377,378,380,382, 384, 385, 387, 395, 397-399,404,406-408, 419, 420, 422, 424,426,42&430 Walmsley, LA. 158-162 Walser, R. 99, 127, 167, 173, 174 Walther, H. 96, 98, 126, 151, 385 Wang, J. 273 Wang, M.D. 325 Wang, W.D. 13, 51 Wang, Z.S. 42 Watanabe, K. 392,405,424,427,430-433 Watkins, S. 424, 428, 429 Watts, R.N. 357 Waxer, L.J. 161, 162 Weaver, W.D. 185 Wegener, M.J. 312 Weigert, S. 115, 185 Weiner, A.M. 387 Weinfurter, H. 77 Weiss, C.O. 311, 315

484


Weiss, D.S. 356 Weiss, K. 273, 274 Weiss, U. 224, 226-228 Weiss-Wrana, K. 276 Welsch, D.-G. 70, 74, 77-83, 85, 86, 97, 107, 114, 116, 117, 120, 124, 136, 140-143, 145, 150, 151, 166, 189, 190, 194, 196, 421

Werner,M. 440 Werner, M.J. 378, 379, 392, 406, 413, 435, 438443,446,447, 449, 453, 454

Westbrook C.I. 357 Westerveld, W.B. 151, 185 Westwood, E.K. 42 Weyl, H. 193 White, D.W. 26 Wiedemann, H. 154 Wieman, C.E. 173, 357 Wiesbrock, H.-W. 115 Wigner, E.P. 106, 184, 193 Wilkens, M. 98, 126, 127, 175 Williams, A.C. 224 Williamson 111, R.S. 367 Wineland, D. 329 Wineland, D.J. 116, 122, 151, 163-165, 167170, 172

Winful, H.G. 451 Wiscombe, W.J. 282 Wodkiewicz, K. 76,86-88,90, 122, 123, 150 Woerdman, J.P. 296, 297, 300, 309, 311,

Yamamoto, Y.

148, 377, 380, 382, 384, 387, 392,397,399,400,405,406,414,419421, 424427, 429433, 450, 451 Yanagawa, T. 377, 382, 384, 387, 399, 426, 427, 429 Yang, L.M. 266, 267 Yao, D. 385 Yariv, A. 387 Yarygin, A.P. 27, 52 Yashin, Yu.Ya. 24 Yates, A.C. 215 Yeazell, J.A. 162, 180 Yee, T.K. 73 Yin, H. 325 Yokota, M. 48, 53 YOO,H.-I. 385 You, D. 181 Young, B. 99, 127, 184 Young, M. 394 Yourgrau, W. 100, 115, 135 Yu, C.X. 398, 408,412, 414, 415,418,457 Yu, S.S. 429, 434 Yu, S.-S. 406, 414 Yuen, H.P. 69, 73, 88, 93, 148, 392 Yukutake, K. 42 Yuratich, M.A. 380, 383 Yurke, B. 70, 73, 76, 91, 95, 377, 406,419 Yushin, Y. 91 Yushin, Y.Y. 167

314-316, 319,320,330

Wolf, E. 5, 6, 19, 24, 25, 137, 193, 216, 294 Wolf, M. 252, 274 Wolf, M. 4 5 3 4 5 7 Wong, W.S. 390 Wood, D. 389, 451,452 Wootters, WK. 87, 115, 194 Wright, E.M. 17, 329 Wu,H. 377 WU, L.-A. 377 Wu, R.S. 44 Wiinsche, A. 87, 106, 109, 113, 118, 134, 135, 148, 162

Y Yakovlev, VP. 178, 294, 329 Yakushkin, I.G. 22, 36, 38 Yamada, E. 459 Yamamoto, G. 230

Z

Zabusky, N.J. 390 Zagury, N. 98, 172 Zak, J. 191 Zakharov, YE. 390-392, 431, 437 Zanon, D. 230 Zaugg, T. 127 Zege, E.P. 231, 281 Zeilinger, A. 77 Zel’dovich, B.Ya. 366 Zemov, N.N. 42, 53 Zerull, R.H. 273, 274 Zhang, G. 443 Zhang, W. 43 Zhang, X. 412, 414 Zhang, Z.L. 48 Zhou, W. 48 Zhu, T. 41, 43 Zoller, P. 99, 127, 164, 167 Zolotov, I.G. 235


Zubairy, M.S. 377 Zuccala, A. 316, 459 Zucchetti, A. 82, 83, 107

Zukowski, M. 77 Zumer, S. 281 Zurek, W.H. 87

485


SUBJECT INDEX FOR VOLUME XXXIX

- - _ optical scattering 230-268

A Airy function 26, 30 anomalous diffraction approximation (ADA) 217, 237,238

-equation 5,6,39,216,217 - -, complex 17 - - in anisotropic medium 24 eikonal-Born series 228,229 evanescent wave 4

B back-action evading measurement 418, 432 balanced homodyning 73, 91, 92, 143, 151 Bernoulli process 196 - transformation 146 Bessel function 29, 223, 256, 260 Born approximation 221, 225,228 -series 225 Bose-Einstein condensate 68, 173 Brillouin scattering, guided acoustic wave 393-397, 404, 406

F Fermat’s principle 16, 17 fiber-optic communication, quantum limits in 380 Fock state 189 four-wave mixing 377 Fraunhofer diffraction 234,260, 268

G Gaussian beam 4, 11, 17,43-50 geometrical optics 3, 9, 10, 20, 314 - -, basic equations of 5-14 - -, complex 4, 5, 11, 19, 26, 42, 50-53 - -, ray equation of 5 - -, transfer equation of 5 - - approximation 2 17 Glauber amplitude 224 -formula 224 Gouy phase shift 315, 338 Green’s function 8, 19,221, 279

C Clebsch-Gordan coefficients 127, 174, 184 coherent state 191, 192 complex rays 3,4, 12 - -, properties of 1544 - -, selection rules for 17, 18 - -, space-time 23, 24 - - in physical problems 27-43 - - - weakly absorbing media 39-42

D displaced-photon-number statistics 90 Doppler cooling 295, 329 - shift 295,327,331,337-339,360,367

H Hamilton-Jacobi equation 17 Haakel function 260 Heisenberg picture 70 helicity 294 Helmholtz equation 6, 7, 303 Hermite-Gaussian beam 48 - - mode 309,366 Hilbert space 65

E eikonal approximation 215, 219-222, 227229 - -, applications of 268-282 - - in non-relativistic potential scattering 218-230 487

488

SUBJECT INDEX FOR VOLUME XXXIX

hologram, computer-generated 3 12, 3 13 -,volume 279 homodyne, balanced 392 - detection 63-100 - detector, eight-port 85, 89 - -, fow-port 70, 80, 85, 90 - -, multiport 89, 90 - -, six-port 84, 89 I ionosphere 3

J Jaynes-Cummings dynamics 167-1 7 1 - - model 97, 124, 164

K Kerr effect, nonlinear 377 - -, optical 375,376, 380-388 Kirchboff integral 46 - solution 5 Kramers-Kronig dispersion relation 249

L Laguerre polynomial 119, 121,323 LaguerreGitlrssian beam 295,296,300,301, 322, 327,338, 339, 342-352, 362-366 - - mode 300,301, 306,309, 322,323, 327, 330, 336, 338,359, 366, 368 - - -, generation of 309-319 LambDicke parameter 168, 170 - - regime 166 Legendre polynomial 236 Liouville equation 357 Liouvillian 143, 144 Lippman-Schwinger equation 2 18, 220, 222 Lorentz gauge 297

M Mach-Zehnder interferometer 318, 323, 416, 428 magnetic tomography 127 maser,one-atom 385 Maslov’s asymptotic theory 26 -method 21,27 Maximum-entropy principle 153, 154 M e coefficients 252, 253 -theory 252

0 optical Bloch equations 329, 333, 334, 362 - homodyne tomography 101, 102, 160 - homodyning 69-100

P P-function, Glauber-Sudarshan 193 - -, positive 88-90 parametric amplification 94 - oscillator, nonlinear 377 paraxial approximation 43, 44, 296302 - wave equation 296, 307 partial wave expansion 224 Penning trap 182 phase-sensitive measurement of light 69-100 photodetection 195-1 97 photon statistics, subpoissonian 377 propagator approximation 221 propensity 87, 91, 118

Q Q-function 66, 81-87, 89,95, 108, 193, 194, 385 -, reconstruction of 123 quadrature-component state 190 - squeezing 392, 397417,457 quantum nondemolition measurement 377, 418435 - - _ of the photon number 418-427 - state endoscopy 124 - - measurement,tomographic 65 --of matter ~ y S t e m s 157-187 - - reconstruction, tomographic 66 - - representation 189-195

R Rabi frequency, vacuum 97 radiation pressure 294 radiative transfer, theory of 17 Radon transform 101, 105, 160, 176, 185 Raman scattering, stimulated 376, 379, 387, 389,391,404,446 Raman-Nath approximation 279 Rayleigh range 44,300, 330 -scattering 386 RayleighGans approximation 234, 256 resonance fluorescence 377 Riccati-Bessel function 236 Riccati-Neumann function 236 rotating-wave approximation 331

SUBJECT MDEX FOR VOLUME XXXIX

Rutherford formula 226 , Rydberg wave packet 180 Rytov approximation 229 S s-parametrized functions 87, 192, 193 saddle-point method 5, 19, 2 1 Sagnac interferometer, asymmetric 4 4 8 4 5 1 Schrodinger equation 3, 113, 218, 219, 229 - -, nonlinear 390,405,439 second-harmonic generation 322 Sitenkmlauber approximation 2 15 soliton, experiments with 431435 -, quantum 380 - in optical fibers 375, 389 spiwrbit coupling of light 363-366 squeezed light 110 - vacuum 399, 410414 squeezing 377 -, amplitude 415 -, Kerr 399 -, photon-number 379, 380, 392, 435-458 -, pulsed 429, 452 -, quadrature 392,397417,457 stationary phase, method of 19

489

Stern-Gerlach apparatus 184 - - measurements 99 Struve function 256 superoperator 143 Susskind-Glogower operator 140 symplectic tomography 104

V vibrational wave packet 158 vibrations, anharmonic 160 von Neumann entropy 153

W Wigner function 67, 79, 87, 101, 108, 122, 131, 151, 169, 170, 172, 176, 177, 185, 193, 194, 385 Wittaker-Shannon sampling theorem 177 WKB approximation 221 -method 22 -phase 217 Y Yukawa potential 228,229


CONTENTS OF PREVIOUS VOLUMES

VOLUME I (1961)

I I1

The Modem Development of Hamiltonian Optics, R.J. PEGIS Wave Optics and Geometrical Optics in Optical Design, K. MIYAMOTO 111 The Intensity Distribution and Total Illumination of Aberration-Free Diffraction Images, R. BARAKAT IV Light and Information, D. GABOR V On Basic Analogies and Principal Differences between Optical and Electronic Information, H. WOLTER VI Interference Color, H. KUBOTA VII Dynamic Characteristics of Visual Processes, A. FIORENTIM VIII Modem Alignment Devices, A.C.S. VANHEEL

1- 29 31- 66 67-108 109- 1 53

1 55-2 10 211-251 253-288 289-329

VOLUME I1 (1963)

I

Ruling, Testing and Use of Optical Gratings for High-Resolution Spectroscopy, G.W. STROKE I1 The Metrological Applications of Diffraction Gratings, J.M. BURCH I11 Diffusion Through Non-Uniform Media, R.G. GIOVANELLI IV Correction of Optical Images by Compensation of Aberrations and by Spatial Frequency Filtering, J. TSUJRTCHI V Fluctuations of Light Beams, L. MANDEL VI Methods for Determining Optical Parameters of Thin Films, F. ABELBS

1- 12 73-108 109-129 13t-180 181-248 249-288

VOLUME rn (1 964) I The Elements of Radiative Transfer, F. KOTTLER B. ROIZEN-DOSSIER I1 Apodisation, P. JACQUINOT, III Matrix Treatment of Partial Coherence, H. GAMO

1- 28 29-186 187-332

VOLUME IV (1965) I Higher Order Aberration Theory, J. FOCKE I1 Applications of Shearing Interferometry, 0. BRYNGDAHL 111 Surface Deterioration of Optical Glasses, K. KINOSITA IV Optical Constants of Thin Films, P. ROUARD, P. BOUSQUET V The Miyamot+Wolf Diffraction Wave, A. RUBINOWICZ VI Aberration Theory of Gratings and Grating Mountings, W.T. WELFORD VII Diffraction at a Black Screen, Part I: Kirchhoffs Theory, F. KOTTLER 49 1

1- 36 37- 83 85-143 145-197 199-240 241-280 281-314

492


VOLUME V (1966) I I1

Optical Pumping, C. CoHEN-TANNOUDn, A. KASTLER 1- 81 Non-Linear Optics, P.S. PERSHAN 83-144 111 Two-Beam Interferometry, W.H. STEEL 145-197 199-245 IV Instruments for the Measuring of Optical Transfer Functions, K. MURATA V Light Reflection from Films of ContinuouslyVarying Refractive Index, R. JACOBSSON247-286 VI X-Ray Crystal-Structure Determination as a Branch of Physical Optics, H. LIPSON, C.A. TAYLOR 287-350 351-370 VII The Wave of a Moving Classical Electron, J. PXCHT VOLUME VI (1967)

1- 52 Recent Advances in Holography, E.N. LEITH,J. UPATNIEKS 53- 69 II Scattering of Light by Rough Surfaces, P.BECKMANN III Measurement of the Second Order Degree of Coherence, M. FRANCON, S. MALLICK 71-104 105-170 IV Design of Zoom Lenses, K. YW 17 1-209 V Some Applications of Lasers to Interferometry, D.R. HERRIOT VI Experimental Studies of Intensity Fluctuations in Lasers, J.A. ARMSTRONG, 21 1-257 A.W. S m 259-330 VII Fourier Spectroscopy, G.A. VANASSE, H. SAKAI 331-377 VIII Diffraction at a Black Screen, Part II: Electromagnetic Theory, F. KOTTLER

I

VOLUME VII (1969) Multiple-Beam Interference and Natural Modes in Open Resonators, G. KOPPELMAN 1- 66 67-137 R.J. PEGIS II Methods of Synthesis for Dielectric Multilayer Filters, E. DELANO, 111 Echoes at Optical Frequencies, LD. ABELLA 139-168 N Image Formation with Partially Coherent Light, B.J. THOMPSON 169-230 231-297 V Quasi-Classical Theory of Laser Radiation, A.L. M I K A E L ~ M.L. , BR-MKAELIAN VI The Photographic Image, S. Oom 299-358 W Interaction of Very Intense Light with Free Electrons, J.H. EBERLY 359415 I

VOLUME VIII (1970) 1- 50 Synthetic-Aperture Optics, J.W. GOODMAN 51-131 The Optical Performance of the Human Eye, G.A. FRY 133-200 III Light Beating Spectroscopy, H.Z. Cuhmms, H.L. SWINNEY 201-237 IV Multilayer Antireflection Coatings, A. MUSSET,A. THELEN 239-294 V Statistical Properties of Laser Light, H. h s m VI Coherence Theory of Source-Size Compensation in Interference Microscopy, 295-341 T. YAMAMOTO 343-372 Vn Vision in Communication, L. LEVI 37340 VIII Theory of Photoelectron Counting, C.L. MEHTA

I I1

VOLUME IX (1971) I Gas Lasers and their Application to Precise Length Measurements, A.L. BLOOM 1 1 Picosecond Laser Pulses, A.J. D E W 111 Optical Propagation Through the Turbulent Atmosphere, J.W. STROHBEHN IV Synthesis of Optical Birefringent Networks, E.O. AMMA"

1- 30 31- 71 73-122 123-177


V Mode Locking in Gas ,Lasers, L. ALLEN,D.G.C. JONES VL. GINZBURG Vl Crystal Optics with Spatial Dispersion, VM. AGRANOVICH, VII Applications of Optical Methods in the Difiaction Theory of Elastic Waves, K. GNIADEK,J. PETYKIEWICZ VIII Evaluation, Design and Extrapolation Methods for Optical Signals, Based 6n Use of the Prolate Functions, B.R FRIEDEN

493 179-234 235-280 281-310 3 11407

VOLUME X (1972) I I1

III IV V

VI VII

Bandwidth Compression of Optical Images, T.S. HUANG The Use of Image Tubes as Shutters, R.W. SMITH Tools of Theoretical Quantum Optics, M.O. SCULLY, K.G. WHITNEY Field Correctors for Astronomical Telescopes, C.G. WYNNE Optical Absorption Strength of Defects in Insulators, D.Y. SMITH,D.L. DFXTBR Elastooptic Light Modulation and Deflection, E.K. SITI-IG Quantum Detection Theory, C.W. HELSTROM

1- 44 45- 87 89-135 137-164 165-228 229-288 28S369

VOLUME XI (1973) I

II 111

IV V

VI VII

Master Equation Methods in Quantum Optics, G.S. AGARWAL Recent Developments in Far Infrared Spectroscopic Techniques, H. YOSHINAGA Interaction of Light and Acoustic Surface Waves, E.G. LEAN Evanescent Waves in Optical Imaging, 0. BRYNGDAHL Production of Electron Probes Using a Field Emission Source, A.V CREWE Hamiltonian Theory of Beam Mode Propagation, J.A. A R N A ~ Gradient Index Lenses, E.W. MARCHAND

1- 76 77-122 123-166 167-22 1 223-246 247-304 305-3 37

VOLUME XI1 (1974) Self-Focusing, Self-Trapping, and Self-phase Modulation of Laser Beams, 0. SVELTO 1- 51 Self-Induced Transparency, R.E. SLUSHER 53-100 111 Modulation Techniques in Spectrometry, M. -WIT, J.A. DECKER JR 101-162 IV Interaction of Light with Monomolecular Dye Layers, K.H. DREXHAGE 163-232 V The Phase Transition Concept and Coherence in Atomic Emission, R. GRAHAM 233-286 VI Beam-Foil Spectroscopy,S. BASHKIN 287-344

I

II

VOLUME XIII (1976) I

On the Validity of Kirchhoff’s Law of Heat Radiation for a Body in a Nonequilibrium

Environment, H.P. BALTES The Case For and Against Semiclassical W a t i o n Theory, L. -EL III Objective and Subjective Spherical Aberration Measurements of the Human Eye, W.M. ROSENBLUM, J.L. CHRISTENSEN IV Interferometric Testing of Smooth Surfaces, G. SCHULZ, J. SCHWLDER V Self-Focusing of Laser Beams in Plasmas and Semiconductors, M.S. SODHA, A.K. GHATAK, V.K. TRIPATHI VI Aplanatism and Isoplanatism, W.T. WELFORD

II

1- 25 27- 68 69- 91 93-167 169-265 267-292

494


VOLUME X N (1976) I I1

The Statistics of Speckle Patterns, J.C. DAINTY High-Resolution Techniques in Optical Astronomy, A. LABEYRIE m Relaxation Phenomena in Rare-Earth Luminescence, L.A. RISEBERG, M.J. WEBER N The Ultrafast Optical Ken Shutter, M.A. DUGUAY V Holographic Diffkaction Gratings, G. SCHMAHL,D. RUDOLPH VI Photoemission, P.J. VERNIER VII Optical Fibre Waveguides - A Review, P.J.B. CLARRICOATS

1- 46 41- 87 89-159 161-193 195-244 245-325 327402

VOLUME XV (1977) I I1 111 N V

Theory of Optical Parametric Amplification and Oscillation, W. BRUNNER, H. PAUL A. MEESSEN Optical Properties of Thin Metal Films, P. ROUARD, Projection-Type Holography, T. OKOSHI Quasi-Optical Techniques of Radio Astronomy, T.W. COLE Foundations of the Macroscopic Electromagnetic Theory of Dielectric Media, 3. VAN KRANENDONK, J.E. SIPE

1- 75 77-137 139-185 187-244 245-350

VOLUME XVI (1978) I

II III

Iv V VI VII

1- 69 Laser Selective Photophysics and Photochemistry, VS. LETOKHOV Recent Advances in Phase Profiles Generation, J.J. CLAJR, C.I. ABITBOL 71-117 119-232 Computer-Generated Holograms: Techniques and Applications, W.-H. LEE Speckle Interferometry, A E . ENNOS 233-288 Deformation Invariant, Space-Variant Optical Pattern Recognition, D. CASASENT, 289-356 D. PSALTIS 3 5 7 41 Light Emission From High-Current Surface-Spark Discharges, R.E. BEVERLY I11 Semiclassical Radiation Theory Within a Quantum-Mechanical Framework, 413-448 I.R. SEMTZKY

VOLUME XVII (1980) I Heterodyne Holographic Interferometry, R. DANDLIKER 1- 84 I1 Doppler-Free Multiphoton Spectroscopy, E. GIACOBINO, B. CAGNAC 85-161 111 The Mutual Dependence Between Coherence Properties of Light and Nonlinear 163-238 B. WILHELMI Optical Processes, M. SCHLJBERT, 239-277 IV Michelson Stellar Interferometry, WJ. TANGO,R.Q. 'Mss 279-345 V Self-Focusing Media with Variable Index of Refraction, A.L. MIKAELIAN

VOLUME XVIII (1980) I I1

Graded Index Optical Waveguides: A Review, A. GHATAK, K. THYAGARAJAN 1-126 Photocount Statistics of Radiation Propagating Through Random and Nonlinear 127-203 Media, J. PERINA 111 Strong Fluctuations in Light Propagation in a Randomly Inhomogeneous Medium, 20&256 VI. TATARSKII, VU. ZAVOROTNYI IV Catastrophe Optics: Morphologies of Caustics and their Diffraction Patterns, 257-346 M.V BERRY,C. UPSTILL


495

VOLUME xu( (1981) I

Theory of Intensity Dependent Resonance Light Scattering and Resonance Fluorescence, B.R. MOLLOW 1- 43 II Surface and Size Effects on the Light Scattering Spectra of Solids, D.L. Mrtts, K.R. SUBBASWAMY 45-137 111 Light Scattering Spectroscopy of Surface Electromagnetic Waves in Solids, S. USHIODA 13%210 IV Principles of Optical Data-Processing,H.J. BUTERWECK 21 1-280 281-376 V The Effects of Atmospheric Turbulence in Optical Astronomy, F. RODDIER

VOLUME XX (1983) I

Some New Optical Designs for Ultra-Violet Bidimensional Detection of AstronomF! ~CRUVELLIER, M. DETAILLE, M. WSSE 1- 61 ical Objects, G. C O U R ~ , II Shaping and Analysis of Picosecond Light Pulses, C. FROEHLY, B. COLOMBEAU, M. VAMPOUILLE 63-153 III Multi-Photon Scattering Molecular Spectroscopy, S. KIELICH 155-26 1 IV Colour Holography, P. HARIHARAN 263-324 V Generation of Tunable Coherent Vacuum-Ultraviolet Radiation, W. JAMROZ, B.P. STOICHEFF 325-380

VOLUME XXI (1984) 1- 67 Rigorous Vector Theories of Diffraction Gratings, D. MAYSTRE Theory of Optical Bistability, L.A. LUGIATO 69-216 111 The Radon Transform and its Applications, H.H. BARRETT 217-286 IV Zone Plate Coded Imaging: Theory and Applications, N.M. CEGLIO, D.W. SWEENEY.287-354 V Fluctuations, Instabilities and Chaos in the Laser-Driven Nonlinear Ring Cavity, J.C. ENGLUND, R.R. SNAPP,W.C. SCHIEVE 355428 I I1

VOLUME XXII (1985) Optical and Electronic Processing of Medical Images, D. MALACARA 1- 76 II Quantum Fluctuations in Vision, M.A. BOL~MAN, W.A. VANDE GRIND,F! ZUIDEMA 77-144 III Spectral and Temporal Fluctuations of Broad-Band Laser Radiation, A.V MASALOV 145-196 IV Holographic Methods of Plasma Diagnostics, G.V OSTROVSKAYA, Yu.1. OSTROVSKY197-270 V Fringe Formations in Deformation and Vibration Measurements using Laser Light, I. YAh4AGUCHI 271-340 VI Wave Propagation in Random Media: A Systems Approach, R.L. FANTE 341-398 I

VOLUME XXIII (1986) I

Analytical Techniques for Multiple Scattering from Rough Surfaces, J.A. DESANTO, G.S. BROWN 1- 62 I1 Paraxial Theory in Optical Design in Terms of Gaussian Brackets, K. TANAKA 63-1 11 III Optical Films Produced by Ion-Based Techniques, PJ. MARTIN,R.P NETTER~~ELD113-182 IV Electron Holography, A. TONOMLJRA 183-220 V Principles of Optical Processing with Partially Coherent Light, F.T.S. Yu 221-275

496


VOLUME XMV (1987) Micro Fresnel Lenses, H. NISHRIARA, T. SUHARA Dephasing-Induced Coherent Phenomena, L. R~THBERF 111 Interferometry with Lasers, F! HAR~ARAN N Unstable Resonator Modes, K.E. ~UFHSTUN V Information Processing with Spatially Incoherent Light, I. GLASER I

I1

1- 37 ,39-101 103-164 165-387 389-509

VOLUME XXV (1988)

P. MANDEL, Dynamical Instabilities and Pulsations in Lasers, N.B. ABRAHAM, L.M. NARDUCCI 1-190 I1 Coherence in Semiconductor Lasers, M. OHTSU,T. TAKO 191-278 III Principles and Design of Optical Arrays, WANGSHAOMIN, 279-348 L. RONCHI N Aspheric Surfaces, G. SCHULZ 349-415 I

VOLUME XXVI (1988)

I I1 111 IV V

Photon Bunching and Antibunching, M.C. BICH, B.E.A. SALEH Nonlinear Optics of Liquid Crystals, I.C. KHOO Single-Longitudinal-Mode Semiconductor Lasers, G.P. AFRAWAL Rays and Caustics as Physical Objects, YuA. KRAVTSOV Phase-Measurement Interferometry Techniques, K. CREATH

1-104 105-161 163-225 227-348 349-393

VOLUME XXVII (1989) The Self-Imaging Phenomenon and Its Applications, K. PATORSKI 1-108 Axicons and Meso-Optical Imaging Devices, L.M. SOROKO 109-160 111 Nonimaging Optics for Flux Concentration, I.M. BASSETT,W.T. WELFORD, R. WINSTON 161-226 N Nonlinear Wave Propagation in Planar Structures, D. MIHALACHE, M. BERTOLOTTI, c. SIBMA 227-3 13 V Generalized Holography with Application to Inverse Scattering and Inverse Source Problems, R.P. PORTER 3 15-397

I I1

VOLUME XXVnI (1990) Digital Holography - Computer-GeneratedHolograms, 0. BRYNGDAHL, F. WYROWSKI 1- 86 Quantum Mechanical Limit in Optical Precision Measurement and Communication, S. WCHIDA, S. SAITO,N. IMOTO,T. YWAGAWA, M. KITAGAWA, Y. YAMAMOTO, G. BJORK 87-179 I11 The Quantum Coherence Properties of Stimulated Raman Scattering, M.G. RAYMER, LA. WALMSLEY 181-270 IV Advanced Evaluation Techniques in Interferometry, J. SCHWIDER 271-359 V Quantum Jumps, R.J. COOK 361-416

I I1


497

VOLUME XXIX (1991) I

Optical Waveguide Diffraction Gratings: Coupling between Guided Modes, 1- 63 D.G. HALL I1 Enhanced Backscattering in Optics, Yu.N. BARABANENKOV, Yu.A. KRAVTSOV, VD. O z m , A.I. SAICHEV 65-197 199-291 111 Generation and Propagation of Ultrashort Optical Pulses, I.P. CHRISTOV N Triple-Correlation Imaging in Optical Astronomy, G. WEIGELT 293-319 V Nonlinear Optics in Composite Materials. 1. Semiconductor and Metal Crystallites in Dielectrics, C. FLYTZANIS, F. HACHE,M.C. KLEIN,D. RICARD, PH. ROUSSIGNOL 32141 1

VOLUME XXX (1 992) I

Quantum Fluctuations in Optical Systems, S . REYNAUD, A. HEIDMANN, E. GIACOBINO, C. FABRE I- 85 I1 Correlation Holographc and Speckle Interferometry,Yu.1. OSTROVSKY, VP. SHCHEPrNov 87-135 111 Localization of Waves in Media with One-Dimensional Disorder, VD. FREKIKHER, S.A. GREDESKUL 137-203 IV Theoretical Foundation of Optical-Soliton Concept in Fibers, Y. KODAMA, A. HASEGAWA 205-259 26 1-355 V Cavity Quantum Optics and the Quantum Measurement Process., P. MEYSTRE

VOLUME XXXI (1993) Atoms in Strong Fields: Photoionization and Chaos, P.W. MILONNI, B. SUNDARAM 1-137 II Light Diffraction by Relief Gratings: A Macroscopic and Microscopic View, E. POPOV 139-1 87 III Optical Amplifiers, N.K. DLITTA, J.R. SIMPSON 189-226 IV Adaptive Multilayer Optical Networks, D. PSALTIS,Y. QIAO 227-261 V Optical Atoms, R.J.C. SPREEW,J.P. WOERDMAN 263-3 I9 VI Theory of Compton Free Electron Lasers, G. DATTOLI, L. GIANNESSI, A. RENIERI, A. TORRE 321412

I

VOLUME XXXII (1993) Guided-Wave Optics on Silicon: Physics, Technology and Status, B.P. PAL 1- 59 61-144 II Optical Neural Networks: Architecture, Design and Models, F.T.S. Yu III The Theory of Optimal Methods for Localization of Objects in Pictures, L.l? YAROSLAVSKY 145-201 IV Wave Propagation Theories in Random Media Based on the Path-Integral Approach, VI. TATARSKII, VU. ZAVOROTNY 203-266 M.I. CHARNOTSKII, J. GOZANI, V Radiation by Uniformly Moving Sources. VavilwXherenkov effect, Doppler effect in a medium, transition radiation and associated phenomena, VL. GINZBURG 267-3 12 VI Nonlinear Processes in Atoms and in Weakly Relativistic Plasmas, G. MANFRAY, C. MANUS 313-361 I

498


VOLUME XXXIII (1994) I

The Imbedding Method in Statistical Boundary-Value Wave Problems, W.KLY-

ATSKRi 1-127 129-202 I1 Quantum Statistics of Dissipative Nonlinear Oscillators, V. PEKLNOVA, A. LIJK~ Ill Gap Solitons, C.M. DE STERKE, J.E. SJPE 203-260 IV Direct Spatial Reconstruction of Optical Phase from Phase-Modulated Images, VI. VLAD, D. W A C A R A 261-317 J. Oz-VOGT 3 19-388 V Imaging through Turbulence in the Atmosphere, M.J. BERAN, VI Digital Halfioning: Synthesis of Binary Images, 0. BRYNGDAHL, T. SCHEERMESSER, F. WYROWSIU 389-463

VOLUME XXXIV (1995) I

II

Quantum Interference, Superposition States of Light, and Nonclassical Effects, V: B u ~ KPL. , KNIGHT Wave Propagation in Inhomogeneous Media:; Phase-Shift Approach, L.F! PRESNYAKOV

III The Statistics of Dynarmc Speckles, T. OWOTO, T. ASAKURA N V

1-158 159-1 8 1 183-248

Scattering of Light from Multilayer Systems with Rough Boundaries, I. OHLfDAL, K. N A ~ T I L M., O H L ~ A L 249-33 1 Random Walk and Diffusion-Like Models of Photon Migration in m b i d Media, A.H. GAND~AKHCHE, G.H. WEISS 333402 VOLUME. XXXV (1996)

1- 60 Transverse Paaerns in Wide-Aperture Nonlinear Optical Systems, N.N. ROSANOV Optical Spectroscopy of Single Molecules in Solids, M. ORRIT,J. BERNARD, R. BROWN,B. Lows 61-144 111 Interferometric Multispectral Imaging, K. ITOH 145-196 N Interferometric Methods for Artwork Diagnostics, D. PAOLETTI,G. SCHIRRIPA SPAGNOLO 197-255 V Coherent Population Trapping in Laser Spectroscopy, E. ARMONDO 257-354 VI Quantum Phase Properties of Nonlinear Optical Phenomena, R. TANAS,A. MIRANOWICZ, Ts. GANTSOG 355-446 1

II

VOLUME XXXVI (1996)

I

Nonlinear Propagation of Strong Laser Pulses in Chalcogenide Glass Films, V CHUMASH, I. COJOCARU, E. FAZIO,F. MICHELOTTI, M. BERTOLOTTI 1- 47 B.C. SANDERS 49-128 11 Quantum Phenomena in Optical Interferometry, F'. HARWARAN, C. DE MOL 129-178 III Super-Resolution by Data Inversion, M. BERTERO, 179-244 N Radiative Transfer: New Aspects of the Old TheoIy, Yu.A. KRAVTSOV,L.A. AFXESYAN V Photon Wave Function, I. BIALYNICKI-BIRULA 245-294 VOLUME XXXVII (1997) The Wigner Distribution Function in Optics and Optoelectronics, D. DRAGOMAN Dispersion Relations and Phase Retrieval in Optical Spectroscopy, K.-E. ~ I P O N E N , E.M. VARTIAI", T. ASAKURA III Spectra of Molecular Scattering of Light, I.L. FABELINSKU

I

1- 56

II

57- 94 95-1 84


G.P. AGRAWAL N Soliton Communication Systems, R.-J. ESSIAMBRE, V Local Fields in Linear and Nonlinear Optics of Mesoscopic Systems, 0. KELLER VI Tunneling Times and Superlmninality,R.Y. CHIAO,A.M. STEINBERG

499 185-256 257-343 345-405

VOLUME XXXVIII (1998) 1- 84 Nonlinear Optics of Stratified Media, S. DLITTAGWTA 85-164 Optical Aspects of Interferometric Gravitational-Wave Detectors, €? HELLO 111 Thermal Properties of Vertical-Cavity Surface-Emitting Semiconductor Lasers, W. NAKWASKI, M. OsrErsia 165-262 D. MENDLOVIC, Z. ZALEVSKY263-342 IV Fractional Transformations in Optics, A.W. LOHMANN, V Pattern Recognition with Nonlinear Techniques in the Fourier Domain, B. JAVIDI, J.L. HORNER 343418 4 19-5 13 VI Free-space Optical Digital Computing and Interconnection, J. JAHNS

I I1


CUMULATIVE INDEX - VOLUMES I-XXXIX

F., Methods for Determining Optical Parameters of Thin Films ABELLA,I.D., Echoes at Optical Frequencies ABITBOL,C.I., see Clair, J.J. ABRAHAM, N.B., P. W m , L.M. NARDUCCI, Dynamical Instabilities and Pulsations in Lasers AGARWAL, G.S., Master Equation Methods in Quantum Optics AGRANOVICH, VM., VL. GINZBURG, Crystal Optics with Spatial Dispersion AGRAWAL, G.P., Single-Longitudinal-Mode Semiconductor Lasers G.P.,see Essiambre, R.-J. AGRAWAL, ALLEN, L., D.G.C. JONES,Mode Locking in Gas Lasers ALLEN,L., M.J. PADGETT, M. BABIKER, The Orbital Angular Momentum of Light AMMA", E.O., Synthesis of Optical Birefringent Networks A~RESYAN, LA., see Kravtsov, Yu.A. ARIMONDO, E., Coherent Population Trapping in Laser Spectroscopy Experimental Studies of Intensity Fluctuations in ARMSTRONG, J.A., A.W. SMITH, Lasers ARNAUD,J.A., Hamiltonian Theory of Beam Mode Propagation ASAKURA, T., see Okamoto, T. ASAKURA, T., see Peiponen, K.-E. ASATRYAN, A.A., see Kravtsov, Yu.A. &EL!%,

BAF~IKER, M., see Allen, L. BALTES,H.P., On the Validity of Kirchhoffs Law of Heat Radiation for a Body in a Nonequilibrium Environment BMANENKOV, Yu.N., W.A. Ikwrsov, W.O m ,A.I. SAICHEV, Enhanced Backscattering in Optics BARAKAT, R.,The Intensity Distribution and Total Illuminationof Aberration-Free Diffraction Images BARREIT,H.H., The Radon Transform and its Applications S., Beam-Foil Spectroscopy BASHKKN, BASSETT,I.M., W.T. WLFORD, R. WINSTON,Nonimaging Optics for Flux Concentration BECKMA",P., Scattering of Light by Rough Surfaces Imaging through Turbulence in the Atmosphere BERAN, M.J., J. OZ-VOGT, BERNARD, J., see Orrit, 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. 501

11, 249 VII, 139 XVI, 71

xxv,

1

XI, 1 M, 235 XXVI, 163 XXXVII, 185 IX, 179 XXXM, 291 IX, 123 XXXVI, 179 XXXV, 257 VI, 211 XI, 247 XXXW, 183 XXXVII, 57

XXxM,

1

xxw(,

291

xm,

1

XXIX, 65

I, 67 XXI, 217 XII, 287 XXW, VI, XXXIII, XXXV,

161 53 319 61

XVIII, 257 XXXVI, 129 XXW, 227

502


BERTOLOTTI, M., see Chumash, V BEVERLY 111, R.E., Light Emission From High-Current Surface-Spark Discharges BIALYNICKI-BIRULA, I., Photon Wave Function BJORK,G., see Yamamoto, Y. BLOOM, A.L., Gas Lasers and their Application to Precise Length Measurements BOW, M.A., W.A. VANDE GRIND,P. ZUIDEMA, Quantum Fluctuations in Vision F!, see Rouard, P. BOUSQUET, BROWN, G.S., see DeSanto, J.A. R., see Omt, M. BROWN, BRUNNER,W., H. PAUL, Theory of Optical Parametric Amplification and Oscillation O., Applications of Shearing Interferometry BRYNGDAHL, BRYNGDAHL, O., Evanescent Waves in Optical Imaging O., F. WYROWSKI, Digital Holography - Computer-Generated BRYNGDAHL, Holograms BRYNGDAHL, O., T. SCHEERMESSER, F. WYROWSKI, Digital Halftoning: Synthesis of Binary Images BURCH, J.M., The Metrological Applicattbns of Diffraction Gratings BUTIERWECK, H.J., Principles of Optical Data-Processing BUZEK,V, P.L. KNIGHT, Quantum Interference, Superposition States of Light, and Nonclassical Effects 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 VI. TATARSKII, VU. ZAVOROTNY, Wave Propagation CHARNOTSKII, M.I., J. GOZANI, Theories in Random Media Based on the Path-Integral Approach CHIAO,R.Y., A.M. STEMBERG, Tunneling Times and Superluminality CHRISTENSEN, J.L., see Rosenblum, W.M. CHRISTOV, I.P., Generation and Propagation of Ultrashort Optical Pulses V., I. COJOCARU, E. FAZIO,F. MICHELOTTI, M. BERTOLO~I, Nonlinear CHUMASH, Propagation of Strong Laser Pulses in Chalcogenide Glass Films CLAIR, J.J., C.I. ABITEOL, Recent Advances in Phase Profiles Generation CLARRICOATS, P.J.B., Optical Fibre Waveguides - A Review Optical Pumping COWN-TANNOUDII, C., A. KASTLER, COIOCARU, I., see Chumash, V. COLE,T.W., Quasi-Optical Techniques of Radio Astronomy COLOMBEAU, B., see Froehly, C. COOK,R.J., Quantum Jumps C O U R ~G., S , P. CRWELLIER, M. DETAILLE, M. SAYSSE,Some New Optical Designs for Ultra-Violet Bidimensional Detection of Astronomical Objects CREATH, K., Phase-Measurement Interferometry Techniques CREWE,A.V, Production of Electron Probes Using a Field Emission Source CRWELLIER, P., see Court&s,G. CUMMINS, H.Z., H.L. SWINNEY, Light Beating Spectroscopy DAINTY, J.C., The Statistics of Speckle Patterns DANDLIKEK R., Heterodyne Holographic Interferometry

=,

1

XVI, MNI, XXVIII, IX, XXII,

357 245 87 1 77 IV, 145 XXIII, 1 XXXV, 61

xv, Iv,

1 37 XI, 167

XXVIII,

1

XXXIII, 389 11, 73 XIX, 211

XVII, 85 XVI, 289 XXI, 287 XXXII, 203 XXXW, 345 XIII, 69 XXIX, 199 XXXVI, 1 XVI, 71 X l V , 327

v,

1

MNI, 1 XV, 187 XX, 63 XXVIII, 361 XX,

1

XXVI,349 XI, 223 XX, 1 VnI, 133

m, XVII,

1 1

503


DATTOLI,G.,L. GIANNESSI,A. RENIERI, A. TORRE,Theory of Compton Free Electron Lasers DE MOL,C., see Bertero, M. DE STEW, C.M., J.E. SPE, Gap Solitons DECKER JR, LA., see Harwit, M. DELANO, E., R.J. PEGIS,Methods of Synthesis for Dielectric Mdtilayer Filters DEMARIA,A.J., Picosecond Laser Pulses Analytical Techniques for Multiple Scattering from DESANTO,J.A., G.S.BROWN, Rough Surfaces DETAILLE, M., see Courtes, G. DEXTER, D.L., see Smith, D.Y. DRAGOMAN, D., The Wigner Distribution Function in Optics and Optoelectronics DREXHAGE, K.H., Interaction of Light with Monomolecular Dye Layers DUGUAY, M.A., The Ultrafast Optical Kerr Shutter DUTTA, N.K., J.R. SIMPSON,Optical Amplifiers DUTTA GUPTA,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 Soliton Communication Systems ESSIAMBRE, R.-J., G.P. AGRAWAL,

XXXI, XXXVI, XXXIII, XII, W,

321 129 203 101 67 M, 31

XXIII,

1 1 165 1 163 161 189 1

=, X,

xxxw, XII, XIV,

XXXI, XXXVIII,

w, 359 X X I , 355 XVI, 233 XXXW, 185

FABELINSKLI, I.L., Spectra of Molecular Scattering of Light FABRE,C., see Reynaud, S. F m , R.L., Wave Propagation in Random Media: A Systems Approach FAZIO,E., see Chumash, V FIORENTINI, A,, Dynamic Characteristics of Visual Processes FLYTZANIS, C., F. HACHE,M.C. KLEIN,D. hem, PH. ROUSSIGNOL, Nonlinear Optics in Composite Materials. 1. Semiconductor and Metal Crystallites in Dielectrics FOCKE,J., Higher Order Aberration Theory FORBES,G.W., see Kravtsov, Yu.A. FRANCON, M., S. WLICK, Measurement of the Second Order Degree of Coherence FREILWIER,VD., S.A. GREDESKUL, 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 Pufses FRY, G.A., The Optical Performance of the Human Eye

XXXw, 95

GABOR, D., Light and Information GAMO, H., Matrix Treatment of Partial Coherence GmmAKHcHE, A.H., G.H. WISS, Random Walk and Diffusion-Like Models of Photon Migration in Turbid Media G~SOG Ts.,,see TanaS, R. Graded Index Optical Waveguides: A Review GHATAK, A., K. THYAGARAJAN, GHATAK, A,&, see Sodha, M.S. GIACOBQJO, E., B. CAGNAC, Doppler-Free Multiphoton Spectroscopy

I, 109 111. 187

=, 1 XXU, 341 XXXVI, I I. 253

VI, 71 XXX, 137

IX,311 XX, 63 VIII, 51

XXXIV, 333 XXXV, 355 XVIII, 1 XIII, 169 XW, 85

504

CUMULATIVE INDEX

~

VOLUMES I-XXXIX

GrAcoBrNo, E., see Reynaud, S. GIANNESSI, L., see Dattoli, G. GINZBURG, V.L., see Agranovich, V.M. GINZBURG, VL., Radiation by Uniformly Moving Sources. Vavilov-Cherenkov effect, Doppler effect in a medium, transition radiation and associated phenomena GIOVANELLI, R.G., Diffusion Through Non-Uniform Media GLASER, I., Information Processing with Spatially Incoherent Light GNIADEK, K., J. PETYKIEWCZ, 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 S.A., see Freilikher, VD. GREDESKUL, F., see Flytzanis, C. HACHE, HALL,D.G., Optical Waveguide Diffraction Gratings: Coupling between Guided Modes P., Colour Holography HARIHARAN, HARIHARAN, P., Interferometry with Lasers HARIHARAN, P., B.C. SANDERS, Quantum Phenomena in Optical Interferometry 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. Ro?m-Dossrm, Apodisation JACQUINOT, 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.

KASTLER,A,, see Cohen-Tannoudji, C. KELLER,O., Local Fields in Linear and Nonlinear Optics of Mesoscopic Systems MOO, LC., 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.

=, 1 XXXI, 321 IX,235 XXXII, 261 11, 109 XXIV. 389 IX, WI, XXXII, XI, XXX,

281 1 203 233 137

XXIX, 321 XXIX, 1 XX, 263 XXIV, 103 m 1 , 49 XII, 101 XXX, 205 XXX, 1 XXXVIII, 85 X, 289 VI, 171 XXXVIII, 343 x, 1 XXVIII, 87 XXXV, 145

V, 247 111, 29

XXXVIII, 419 XX, 325

xxxvm,343 IX,119 v, 1 XXXW, 257 XXVI, 105 XX, 155 IV, 85 XXVIII, 87 XXE, 321

505


KLYATSKIN, VI., The Imbedding Method in Statistical Boundary-Value Wave Problems KNIGHT, RL., see Buiek, V KODAMA,Y., A. HASEGAWA, Theoretical Foundation of Optical-Soliton Concept in Fibers KOPPELMAN, G., Multiple-Beam Interference and Natural Modes in Open Resonators K o m ~F., , The Elements of Radiative Transfer KO= F., Diffraction at a Black Screen, Part I: Kuchhoffs Theory KO=& F., Diffraction at a Black Screen, Part 11: Electromagnetic Theory KRAVTSOV,Yu.A., Rays and Caustics as Physical Objects KRAVTSOV, Yu.A., see Barabanenkov, Yu.N. Radiative Transfer: New Aspects of the Old KRAVTSOV, Yu.A., L.A. APRESYAN, Theory KRAVTSOV, Yu.A., G.W. FORBES,A.A. ASATRYAN, Theory and Applications of Complex Rays H., Interference Color KUBOTA, LABEYRE, 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 S, Advances in Holography LEITH,E.N., J. U P A ~ K Recent VS., Laser Selective Photophysics and Photochemistry LETOKHOV, LEUCHS, G., see Sizmann, A. LEVI,L., Vision in Communication X-Ray Crystal-Structure Determination as a Branch of LIPSON,H., C.A. TAYLOR, Physical Optics LOHMA", A.W., D. M E ~ L O V IZ. C , ZALWSKY,Fractional Transformations in Optics LOWS, B., see Orrit, M. LUGIATO, L.A., Theory of Optical Bistability L u K ~A., , see Peiinova, V MACHIDA,S., see Yamamoto, Y. MAWFRAY,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, VI. MALLICK,S.. see FranGon, M. MANDEL, L., Fluctuations of Light Beams -EL, L., The Case For and Against Semiclassical Radiation Theory MANDFL,P., see Abraham, N.B. MANUS,C., see Mainfray, G. MARE.W., Gradient Index Lenses MARTIN, P.J., R.P. NETTER~~LD, Optical Films Produced by Ion-Based Techniques MASALOV, A.V, Spectral and Temporal Fluctuations of Broad-Band Laser Radiation MAYSTRE, D., Rigorous Vector Theories of Diffraction Gratings IMEESSEN, A., see Rouard, P. MEHTA,C.L., Theory of Photoelectron Counting MENDLOVJC, D., see Lohmann, A.W.

XXX, 205 VII, 111,

1 1

IV, 281 VI, 331 XXVI, 227 XXM, 65

XXXVI, 179

=,

1 I, 211

XTV, XI, XVI, VI, XVI, XXXIX, VIII,

47 123 119 1 1 373 343

V, 287 XXXVIII, 263 XXXV, 61 XXI, 69 XXXIII, 129 XXVIII, 87 XXXII, 313 XXII, 1 XXXIII, 261 VI, 71 11, 181 XIII, 27 XXV, 1 XXXII, 313 XI, 305 XXIII, 113 XXII, 145 XXI, 1 XV, 71 VIII, 373 XXXVIII, 263

506


MEYSTRE, I?, Cavity Quantum Optics and the Quantum Measurement Process. MICHELOTTI, F., see Chumash, V MIHALACHE, D., M. BERTOLOTTI, C. SIBILIA, Nonlinear Wave Propagation in Planar Structures MIKAELIAN, A.L., M.L. RR-MWLIAN, Quasi-Classical Theory of Laser Radiation MIKAELIAN, A.L., Self-Focusing Media with Variable Index of Refraction MILLS,D.L., K.R. SUBBASWAMY, Surface and Size Effects on the Light Scattering Spectra of Solids MEOW, P.W., B. SUNDARAM, Atoms in Strong Fields: Photoionization and Chaos MIRANOWICZ, A., see Tanai, R. 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., Instruments for the Measuring of Optical Transfer Functions MUSSET, A., A. THELEN, Multilayer Antireflection Coatings

I, Properties of Vertical-Cavity SurfaceNAKWASKI, W., M. O S ~ ~ S KThermal Emitting SemiconductorLasers NARDUCCI, L.M., see Abraham, N.B. NA~TIL K.,, see Ohlidal, I. NETTE&LD,R.P., see Martin, P.J. NISHMARA, H., T. SUHARA, Micro Fresnel Lenses I., K. N A ~ T I LM., OHL~DAL, Scattering of Light from Multilayer Systems with Rough Boundaries OHL~AL M., , see Ohlidal, I. OHTSU,M.,-T. TAKO,Coherence in Semiconductor Lasers OILWOTO, T., T. ASAKURA, The Statistics of Dynamic Speckles OKOSHI, T., Projection-Type Holography OOUE,S., The Photographic Image O P A T R T., ~ , see Welsch, D.-G. ORRIT,M., J. BERNARD,R. BROWN, B. L o w s , Optical Spectroscopy of Single Molecules in Solids O S ~ S KM., I , see Nakwaski, W. OSTROVSKAYA, G.V, Yu.1. OSTROVSKY, Holographic Methods of Plasma Diagnostics Yu.I., see Ostrovskaya, G.V OSTROVSKY, Correlation Holographic and Speckle OsTRovsKY, Yu.I., V€! SHCHEPINOV, Interferometry OUGHSTUN, K.E., Unstable Resonator Modes OZ-VOGT,J., see Beran, M.J. O z m , VD., see Barabanenkov, Yu.N.

=,

XXX, 261 1

XXW, 227 VII, 231 XVII, 279

m,

45

=, 1

m,355

I, 31

m,

1 V, 199 VIII, 201

XXXVIII, 165 1 XXXIV, 249 Xxm, 113 =, 1

OHLfDAL,

PADGETT,M.J., see Allen, L. PAL,B.P., Guided-Wave Optics on Silicon: Physics, Technology and Status PAOLETIT, D., G. S C ~ P SPAGNOLO, A Interferometric Methods for Artwork Diagnostics PATORSKI, K., The Self-Imaging Phenomenon and Its Applications PAUL,H., see Brunner, W. PEGIS,R.J.,The Modern Development of Hamiltonian Optics PEGIS,R.J., see Delano, E.

XXXIV, 249 XXXIV, 249 XXV, 191 XXXIV, 183 XV, 139 VII, 299 XXXIX, 63 XXXV, 61 XXXVIII, 165

XXII, 197 XXII, 197 XXX, XXIV, XXXIII, XXIX,

87 165 319 65

XXXIX, 291 XXXU, 1

CUhtULATIVE INDEX

~

VOLUMES I-XXXIX

PEPONEN,K.-E., E.M. VARTIAMEN, T. ASAKURA, Dispersion Relations and Phase Retrieval in Optical Spectroscopy PERINA, J., Photocount Statistics of Radiation Propagating Through Random and Nonlinear Media PEhovA, V,A. L d , 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 by Relief Gratings: A Macroscopic and Microscopic View PORTER, R.P., Generalized Holography with Application to Inverse Scattering and Inverse Source Problems 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. RAYMEII,M.G., I.A. WALMSLEY, The Quantum Coherence Properties of Stimulated Raman Scattering RENIERI,A,, see Dattoli, G. C. FABRE,Quantum Fluctuations in REYNAUD, S., A. HEIDMA”,E. GIACOBMO, Optical Systems &CARD, D., see Flytzanis, C. RISEBERG,L.A., M.J. WEBER,Relaxation Phenomena in Rare-Earth Luminescence RISKEN,H., Statistical Properties of Laser Light F., The Effects of Atmospheric Turbulence in Optical Astronomy RODDIE& ROIZEN-DOSSIER, B., see Jacquinot, P. RONCHI,L., see Wang Shaomin ROSANOV, N.N., Transverse Patterns in Wide-Aperture Nonlinear Optical Systems Objective and Subjective Spherical ROSENBLUM,W.M., J.L. CHRISTENSEN, Aberration Measurements of the Human Eye ROTHBERG, L., Dephasing-Induced Coherent Phenomena ROUARD, P., P. BOUSQLET,Optical Constants of Thin Films ROUARD,P., A. MEESSEN, Optical Properties of Thin Metal Films ROUSSIGNOL, PH., see Flytzanis, C. RUBINOWICZ, A,, The MiyamotwWolf Diffraction Wave RUDOLPH, D., see Schmahl, G. SAICHEV, A.I., see Barabanenkov, Yu.N. SA~SSE, M., see Courtes, G. SAITO,S., see Yamamoto, Y. SAKAI,H., see Vanasse, G.A. SALEH, B.E.A., see Teich, M.C. SANDERS,B.C., see Hariharan, P. SCHEERMESSER, T., see Bryngdahl, 0. SCHEVE,W.C., see Englund, J.C. SCHIRRPASPAGNOLO, G., see Paoletti, D. SCHMAHL, G., D. RUDOLPH, Holographic Diffraction Gratings

507

XXXVII, 51 XVIII, 121 XXXIII, 129 V, 83 IX,281 V, 351

XXXI, 139 XXVII, 315 XXXIV, 159 XVI, 289 XXXI, 221

XXXI. 221

XXVIII, 181 XXXI, 321

=,

1

XXH, 321 XIV, 89

VIII, 239 XIX, 281 111, 29 XXV, 219 xxxv, 1

XIII, 69 XXIV, 39 IV, 145 xv, 71 XXIX, 321 IV, 199 XIV, 195 XXIX, 65 XX, 1 XXVIII, 81 VI, 259 XXVI, 1 XXXVI, 49 XXXIII, 389 XXI, 355 XXXV, 191 XIV, 195

508


SCHUBERT, M., B. WILHELMI, The Mutual Dependence Between Coherence Properties of Light and Nonlinear Optical Processes S c m z , G., J. SCHWIDER, Interferometric Testing of Smooth Surfaces SCHULZ,G., Asphenc Surfaces SCHWIDER, J., see Schulz, G. SCHWIDER, J., Advanced Evaluation Techniques in Interferometry SCULLY, M.O., K.G. W ~ YTools , of Theoretical Quantum Optics SENITZKY,I.R., Semiclassical Radiation Theory Within a Quantum-Mechanical Framework SHARMA, S.K., D.J. SOMERFORD, Scattering of Light in the Eikonal Approximation SHCHEPINOV, VP., see Ostrovsky, Yu.1. SIBILIA,C., see Mihalache, D. SIMPSON, J.R., see Dutta, N.K. SIPE, J.E., see Van Kranendonk, J. SIPE,J.E., see De Sterke, C.M. SITTIG, E.K., Elastooptic Light Modulation and Deflection SIZMANN, A., G. LEUCHS, The Optical Kerr Effect and Quantum Optics in Fibers 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. SODHA, M.S., A.K. GHATAK, VK. TRIPATHI, Self-Focusing of Laser Beams in Plasmas and Semiconductors SOMERFORD, D.J., see Sharma, S.K. SOROKO, L.M., Axicons and Meso-Optical Imaging Devices Optical Atoms SPREEUW, R.J.C., J.P. WOERDMAN, STEEL,W.H., Two-Beam Interferometry STEINBERG, A.M., see Chiao, R.Y. STOICHEFF, B.P., see JattUOZ, w. STROHBEHN, J.W, Optical Propagation Through the Turbulent Atmosphere STROKE,G.W., Ruling, Testing and Use of Optical Gratings for High-Resolution Spectroscopy SUBBASWAMY, 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. S ~ YH.L., , see Cummins, H.Z. TAKO,T., see Ohtsu, M. TANAKA, K., Paraxial Theory in Optical Design in Terms of Gaussian Brackets Quantum Phase Properties of Nonlinear TANAS,R., A. MIRANOWICZ, Ts. GANTSOG, Optical Phenomena TANGO,W.J., R.Q. W ~ s sMichelson , Stellar Interferometry TATARSKII, VI., VU. ZAVOROTNYI, Strong Fluctuations in Light Propagation in a Randomly Inhomogeneous Medium TATARSKII, VI., see Charnotskii, M.I. TAYLOR, C.A., see Lipson, H. TEICH,M.C., B.E.A. SALEH,Photon Bunching and Antibunching

XVII, XIII, XXV, XIII, XXVIII, X,

163 93 349 93 271 89

413 213 87 227 189 XV, 245 XXXIII, 203 X, 229 XXXIX, 373 XII, 53 VI, 211 X, 165 x, 45 XXI, 355

XVI, XXXIX, XXX, XXVII, XXXI,

XIII, 169 XXXIX, 213 XXVII, 109 XXXI, 263 V, 145 m 1 , 345 XX, 325 1x. 73 11, 1 XIX, 45 xm, 1 XXXI. 1

xn, 1 XXI, 287 VIII, 133 XXV, 191 XXIII, 63

xxxv, 355 XVII, 239

XWI, 204 XXXII, 203 V, 287

m,

1

509


%R-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. TONOMURA, A,, Electron Holography TORRE,A., see Dattoli, G. VK., see Sodha, M.S. TRIPATHI, Tsunuc~r,J., Correction of Optical Images by Compensation of Aberrations and by Spatial Frequency Filtering Wss, R.Q., see Tango, W.J.

VII, 231 VIII, 201 Vn, 169 mn, 1 XXIII, 183 XXXI, 321 XIII, 169

J., see Leith, E.N. UPATMEKS, UPSTILL,C., see Berry, M.V S., Light Scattering Spectroscopy of Surface Electromagnetic Waves in USHIODA, Solids

VI, 1 XVIII, 257

VAMPOLJILLE, M., see Froehly, C. VANDE GRIND,W.A., see Bouman, M.A. VANHEEL,A.C.S., Modem Alignment Devices VANKRANENDONK,J., J.E. SIPE,Foundations of the Macroscopic Electromagnetic Theory of Dielectric Media Fourier Spectroscopy VANASSE, G.A., H. SAKAI, 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 VOGEL,W., see Welsch, D.-G.

XX, 63 XXII, 77 I, 289

WALMSLEY, LA., see Raymer, M.G. WANGSHAOMIN, L. RONCHI, Principles and Design of Optical Arrays WEBER,M.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. WELSCH, D.-G., W. VOGEL,T. O P A T R Homodyne ~, Detection and Quantum-State Reconstruction WHITNEY, K.G., see Scully, M.O. WILHELMI, B., see Schubert, 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. YAMAGUCHI, I., Fringe Formations in Deformation and Vibration Measurements using Laser Light YAW, K., Design of Zoom Lenses

11, 131 XVII, 239

XM. 139

XV, 245

VI, 259 XXXVII, 57 XIV, 245 XXXIII, 261 XXXIX. 63 XXVIII, XXV, XIV, XXIX, XXXIV,

181 279 89 293 333 IV,241 XIII, 267 XXVII, 161

XXXIX, X, XVII, XXVII, XXXI,

63 89 163 161 263

I, 155 X, 137 XXVIII, 1 XXXIII, 389

XXII, 271 VI, 105

510


YAMAMOTO, T., Coherence Theory of Source-Size Compensation in Interference Microscopy YAMAMOTO, Y., S. MACHIDA,S. SAITO,N. IMOTO, T. YANAGAWA, M. KITAGAWA, G. BJORK,Quantum Mechanical Limit in Optical Precision Measurement and Communication YANAGAWA, T., see Yamamoto, Y. YAROSLAVSKY, L.P., The Theory of Optimal Methods for Localization of Objects in Pictures YOSHINAGA, H., Recent Developments in Far Infrared Spectroscopic Techniques Yu, F.T.S., Principles of Optical Processing with Partially Coherent Light Yu, F.T.S., Optical Neural Networks: Architecture, Design and Models ZALEVSKY, Z., see

Lohmann, A.W. ZAVOROTNY, VU., see Charnotslui, M.I. VU., see Tatarskii, VI. ZAVOROTNYI, ZUIDEMA,P., see Bouman, M.A.

W I , 295

XXVILI, 87 XXVIII. 87

XXXII, XI, XXIII, XXXII,

145 77 221 61

XXXVIII, 263 XXXII, 203 X W I , 204 XXII, 77

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