Progress in Optics, Vol. 30

PROGRESS IN OPTICS VOLUME XXX EDITORIAL ADVISORY BOARD G. S. AGARWAL, Hyderabad, India C. COHEN-TANNOUDJI, Paris, F...

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PROGRESS IN OPTICS VOLUME XXX

EDITORIAL ADVISORY BOARD G. S. AGARWAL,

Hyderabad, India

C. COHEN-TANNOUDJI, Paris, France

F. GORI,

Rome, Italy

A. KUJAWSKI,

Warsaw, Poland

A. LOHMANN,

Erlangen, Germany

J. PERINA,

Olomouc, Czechoslovakia

M. SCHUBERT,

Jena, Germany

R. M. SILLITTO,

Edinburgh, Scotland

J . TSUJIUCHI,

Chiba, Japan

H . WALTHER,

Garching, Germany

B. ZEL’DOVICH,

Chelyabinsk, Russia

PROGRESS IN OPTICS VOLUME XXX

EDITED BY

E. WOLF University of Rochester, N . Y.. U 3 . A

Contributors C. FABRE, V. D. FREILIKHER, E. GIACOBINO, S. A. GREDESKUL, A. HASEGAWA, A. HEIDMANN, Y. KODAMA, P. MEYSTRE, Yu. I. OSTROVSKY, S. REYNAUD, V. P. SHCHEPINOV

1992

NORTH-HOLLAND AMSTERDAM.LONDON.NEWYORK.TOKY0

@ ELSEVIER SCIENCE PUBLISHERS B.V., 1992

AN righis resewed. No pari of this publication may be reproduced, stored in a retrieval system, or iransmiiied, in any form or by any means. electronic, mechanical, photocopying, recording or oihenvise. withoui the writienpermission of the Publisher,Elsevier Science Publishers B. V.. Copyright & Permissions Department, P.O. Box 521. 1000 A M Amsterdam. The Netherlands. Special regulations for readers in the U.S.A. :This publicaiion has been registered wiih the Copyright Clearance Center Inc. (CCC). Salem, Massachusetts. Information can be obtained from the CCC aboui conditions under which phoiocopies of parts of this publication may be made in ihe U.S.A. All other copyright questions, including phoiocopying ouiside of the U.S.A., should be rejerred to the Publisher, unless otherwise specified. No responsibiliiy is assumed by ihe Publisher for any injury and/or damage to persons or property as a matter of producis liabiliiy, negligence or oiherwise. or from any use or operation of any meihods. products, insiruciions or ideas contained in the material herein.

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CONTENTS OF PREVIOUS VOLUMES

VOLUME I(1961) 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. . . . . . . . . . . . . . . . . . . . . . v1. Interference Color. H . KUBOTA. . . . . . . . . . . . . . . . . . . VII . Dynamic Characteristics of Visual Processes. A. FIORENTINI. . . . . . VIII . Modern Alignment Devices. A. C. S . VAN HEEL . . . . . . . . . . . . I. I1 .

1-29 31- 66 67- 108 109-153 155-210 211-251 253-288 289-329

VOLUME I1 (1963) I. I1 . 111. IV . V. VI .

Ruling. Testing and Use of Optical Gratings for High-resolution Spectroscopy.

G. W . STROKE .

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

The Metrological Applications of Diffraction Gratings. J . M. BURCH . . . Diffusion Through Non-Uniform Media. R. G. GIOVANELLI . . . . . . . Correction of Optical Images by Compensation of Aberrations and by Spatial Frequency Filtering. J . TSUJIUCHI. . . . . . . . . . . . . . . . . . Fluctuations of Light Beams. L. MANDEL. . . . . . . . . . . . . . . Methods for Determining Optical Parameters of Thin Films. F . ABELZS .

1-72 73-108 109-129 131-180 181-248 249-288

V O L U M E I11 ( 1 9 6 4 ) I. I1 .

The Elements of Radiative Transfer. F . KO-ITLER . . . . . . . . . . . B. ROIZEN-DOSSIER . . . . . . . . . . . Apodisation. P. JACQUINOT. Matrix Treatment of Partial Coherence. H . GAMO. . . . . . . . . . .

I.

Higher Order Aberration Theory. J . FOCKE. . . . . . . . . . . . . . Applications of Shearing Interferometry. 0. BRYNGDAHL . . . . . . . . Surface Deterioration of Optical Glasses. K. KINOSITA. . . . . . . . . Optical Constants of Thin Films. P. ROUARD.P. BOUSQUET . . . . . . The Miyamoto-Wolf Diffraction Wave. A. RUBINOWICZ. . . . . . . . Aberration Theory of Gratings and Grating Mountings. W . T . WELFORD . Diffraction at a Black Screen. Part I: Kirchhoffs Theory. F . KO-ITLER . .

111.

1-28 29- 186 187-332

VOLUME I V (1965) 11.

111. IV . V. VI. VII .

V

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


VI

VOLUME V (1966) I. I1. I11. IV V.

Optical hmping. C. COHEN.TANNOUDJ1. A. KASTLER. . . . . . . . . Non-Linear Optics. P . S . PERSHAN . . . . . . . . . . . . . . . . . Two-Beam Interferometry. W. H . STEEL . . . . . . . . . . . . . . . Instruments for the Measuring of Optical Transfer Functions. K . MURATA. Light Reflection from Films of Continuously Varying Refractive Index. R. JACOBSSON . . . . . . . . . . . . . . . . . . . . . . . . . . . . v1. X-Ray Crystal-Structure Determination as a Branch of Physical Optics. H . LIPSON.C. A. TAYLOR. . . . . . . . . . . . . . . . . . . . . . . VII . The Wave of a Moving Classical Electron. J . PICHT . . . . . . . . . .

.

1-81 83-144 145-197 199-245 247-286 287-350 351-370

V O L U M E VI ( 1 9 6 7 ) I. Recent Advances in Holography. E. N . LEITH.J . UPATNIEKS. . . . . . 1- 52 I1. Scattering of Light by Rough Surfaces. P . BECKMANN. . . . . . . . . 53- 69 111. Measurement of the Second Order Degree of Coherence. M. FRANCON. S. MALLICK . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71-104 IV. Design of Zoom Lenses. K. YAMAJI. . . . . . . . . . . . . . . . . 105-170 V . Some Applications of Lasers to Interferometry. D . R. HERRIOTT. . . . . 17 1-209 VI . Experimental Studies of Intensity Fluctuations in Lasers. J . A. ARMSTRONG. A. W. SMITH . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 1-257 VII . Fourier Spectroscopy. G. A. VANASSE.H . SAKAI . . . . . . . . . . . 259-330 VIII. Diffraction at a Black Screen. Part 11: Electromagnetic Theory. F . KOTTLER 331-377

VOLUME VII (1969) I.

Multiple-Beam Interference and Natural Modes in Open Resonators. G . KOPPELMAN. . . . . . . . . . . . . . . . . . . . . . . . . . . I- 66 I1. Methods of Synthesis for Dielectric Multilayer Filters. E.DELANO. R. J . PEGIS 67- 137 I11. Echoes and Optical Frequencies. I . D . ABELLA . . . . . . . . . . . . 139-168 IV. Image Formation with Partially Coherent Light. B . J . THOMPSON . . . . 169-230 V. Quasi-Classical Theory of Laser Radiation. A. L. MIKAELIAN. M. L. TERMlKAELlAN . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 1-297 VI . The Photographic Image. S . OOUE. . . . . . . . . . . . . . . . . . 299-358 VII . Interaction of Very Intense Light with Free Electrons. J . H . EBERLY. . . 359-415

VOLUME VIII (1970)

I. I1. I11. IV . V. VI.

Synthetic-Aperture Optics. J . W. GOODMAN. . . . . . . . . . . . . The Optical Performance of the Human Eye. G. A. FRY . . . . . . . . Light Beating Spectroscopy. H . Z . CUMMINS. H . L. SWINNEY. . . . . . Multilayer Antireflection Coatings. A. MUSSET.A. THELEN. . . . . . . Statistical Properties of Laser Light. H . RISKEN. . . . . . . . . . . . Coherence Theory of Source-Size Compensation in Interference Microscopy. T . YAMAMOTO. . . . . . . . . . . . . . . . . . . . . . . . . . VII . Vision in Communication. H . LEV1 . . . . . . . . . . . . . . . . . . VIII. Theory of Photoelectron Counting. C. L. MEHTA. . . . . . . . . . . .

1-50 51-131 133-200 201-237 239-294 295-341 343-372 373-440


VII

VOLUME I X (1971) Gas Lasers and their Application to Precise Length Measurements. A. L. BLOOM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I1. Picosecond Laser Pulses. A. J . DEMARIA. . . . . . . . . . . . . . . Ill . Optical Propagation Through the Turbulent Atmosphere. J . W . STROHBEHN IV . Synthesis of Optical Birefringent Networks. E. 0. AMMANN. . . . . . . V . Mode Locking in Gas Lasers. L. ALLEN.D . G . C. JONES . . . . . . . . v1. Crystal Optics with Spatial Dispersion. V . M . ACRANOVICH. V . L. GINZBURG VII . Applications of Optical Methods in the Diffraction Theory of Elastic Waves. K . GNIADEK. J . PETYKIEWICZ. . . . . . . . . . . . . . . . . . . VIII . Evaluation. Design and Extrapolation Methods for Optical Signals. Based on Use of the Prolate Functions. B. R. FRIEDEN . . . . . . . . . . . . . I.

1-30 31- 71 73-122 123-177 179-234 235-280 281-3 10 3 1 1-407

VOLUME X (1972) 1. I1.

I11. IV . V.

V1. VII .

1-44 Bandwidth Compression of Optical Images. T . S. HUANG. . . . . . . . The Use of Image Tubes as Shutters. R . W . SMITH . . . . . . . . . . 45- 87 Tools of Theoretical Quantum Optics. M. 0. SCULLY. K . G . WHITNEY. . 89-135 Field Correctors for Astronomical Telescopes. C. G. WYNNE . . . . . . 137-164 Optical Absorption Strength of Defects in Insulators. D . Y . SMITH.D . L. DEXTER . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165-228 Elastooptic Light Modulation and Deflection. E. K . SITTIC . . . . . . . 229-288 Quantum Detection Theory. C. W. HELSTROM . . . . . . . . . . . . 289-369

VOLUME XI (1973) Master Equation Methods in Quantum Optics. G . S . ACARWAL. . . . . Recent Developments in Far Infrared Spectroscopic Techniques. H. YOSHINACA . . . . . . . . . . . . . . . . . . . . . . . . . . . Ill . Interaction of Light and Acoustic Surface Waves. E. G . LEAN . . . . . . . . . . . . . . . IV . Evanescent Waves in Optical Imaging. 0. BRYNCDAHL V . Production of Electron Probes Using a Field Emission Source. A. V . CREWE V1. Hamiltonian Theory of Beam Mode Propagation. J . A. ARNAUD. . . . . VII . Gradient Index Lenses. E . W. MARCHAND. . . . . . . . . . . . . .

I. I1.

1-76 77-122 123-166 167-221 223-246 247-304 305-337

VOLUME XI1 (1974) I. I1. I11. IV . V.

V1.

Self.Focusing. Self.Trapping. and Self-phase Modulation of Laser Beams. 0. SVELTO . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Self-Induced Transparency. R . E. SLUSHER. . . . . . . . . . . . . . Modulation Techniques in Spectrometry. M . HARWIT.J . A. DECKERJR. . Interaction of Light with Monomolecular Dye Layers. K . H . DREXHACE. The Phase Transition Concept and Coherence in Atomic Emission. R . GRAHAM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Beam-Foil Spectroscopy. S . BASHKIN . . . . . . . . . . . . . . . .

1-51 53- 100 101-162 163-232 233-286 287-344

Vlll


V O L U M E XI11 ( 1 9 7 6 )

I. 11. 111.

IV. V. VI.

On the Validity of Kirchhoffs Law of Heat Radiation for a Body in a Non-

1- 25 equilibrium Environment, H. P. BALTES . . . . . . . . . . . . . . . The Case For and Against Semiclassical Radiation Theory, L. MANDEL . 27- 68 Objective and Subjective Spherical Aberration Measurements of the Human J. L. CHRISTENSEN . . . . . . . . . . . . . 69- 91 Eye, W. M. ROSENBLUM, Interferometric Testing of Smooth Surfaces, G. SCHULZ,J. SCHWIDER. . 93-167 Self Focusing of Laser Beams in Plasmas and Semiconductors, M. S. SODHA, A. K. GHATAK,V. K. TRIPATHI. . . . . . . . . . . . . . . . . . . 169-265 Aplanatism and Isoplanatism, W. T. WELFORD . . . . . . . . . . . . 267-292

V O L U M E XIV ( 1 9 7 7 ) The Statistics of Speckle Patterns, J. C. DAINTY. . . . . . . . . . . . High-Resolution Techniques in Optical Astronomy, A. LABEYRIE . . . . 11. 111. Relaxation Phenomena in Rare-Earth Luminescence, L. A. RISEBERG, M. J. WEBER . . . . . . . . . . . . . . . . . . . . . . . . . . . . . IV. The Ultrafast Optical Kerr Shutter, M. A. DUGUAY. . . . . . . . . . V. Holographic Diffraction Gratings, G. SCHMAHL, D. RUDOLPH . . . .. . VI. Photoemission, P. J. VERNIER . . . . . . . . . . . . . . . . . . . . . . . . . . VII. Optical Fibre Waveguides - A Review, P. J. B. CLARRICOATS I.

1- 46 47- 87

89-159 161-193 195-244 245-325 327-402

V O L U M E XV ( 1 9 7 7 ) Theory of Optical Parametric Amplification and Oscillation, W. BRUNNER, H. PAUL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11. Optical Properties of Thin Metal Films, P. ROUARD, A. MEESSEN. . . . 111. Projection-Type Holography, T. OKOSHI . . . . . . . . . . . . . . . IV. Quasi-Optical Techniques of Radio Astronomy, T. W. COLE. . . . . . . Foundations of the Macroscopic Electromagnetic Theory of Dielectric Media, V. J. VAN KRANENDONK. J. E. SIPE . . . . . . . . . . . . . . . . . . I.

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

245-350

V O L U M E XVI ( 1 9 7 8 ) 1- 69 Laser Selective Photophysics and Photochemistry, V. S. LETOKHOV. . . 71-117 Recent Advances in Phase Profiles Generation, J. J. CLAIR,C. I. ABITBOL. 111. Computer-Generated Holograms: Techniques and Applications, W.-H. LEE 119-232 IV. Speckle Interferometry, A. E. ENNOS. . . . . . . . . . . . . . . . . 233-288 Deformation Invariant, Space-Variant Optical Recognition, D. CASASENT, D. V. PSALTIS. . . . . . . . . . . . . . . . . . . . . . . . . . . . , 289-356 VI. Light Emission From High-Current Surface-Spark Discharges, R. E. BEVERLY 357-4 11 111 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . VII. Semiclassical Radiation Theory Within a Quantum-Mechanical Framework, I. R. SENITZKY. . . . . . . . . . . . . . . . . . . . . . . . . . 413-448 1.

11.

IX


VOLUME XVII (1980) Heterodyne Holographic Interferometry, R. DANDLIKER. . . . . . . . I. B. CAGNAC . . 11. Doppler-Free Multiphoton Spectroscopy, E. GIACOBINO, 111. The Mutual Dependence Between Coherence Properties of Light and Nonlinear Optical Processes, M. SCHUBERT, B. WILHELMI. . . . . . . . . . . . IV. Michelson Stellar Interferometry, W. J. TANGO,R. Q.TWISS . . V. Self-Focusing Media with Variable Index of Refraction, A. L. MIKAELIAN .

.

1- 84 85-162 163-238 239-278 279-345

VOLUME XVIII (1980) Graded Index Optical Waveguides:A Review, A. GHATAK, K. THYAGARAJAN 1- 126 Photocount Statistics of Radiation Propagating Through Random and Nonlinear Media, J. PERINA . . . . . . . . . . . . . . . . . . . . . . 127-203 111. Strong Fluctuations in Light Propagation in a Randomly Inhomogeneous V. U. ZAVOROTNYI . . . . . . . . . . . . . 204-256 Medium, V. I. TATARSKII, IV. Catastrophe Optics: Morphologies of Caustics and their Diffraction Patterns, M. V. BERRY,C. UPSTILL . . . . . . . . . . . . . . . . . . . . . 257-346 I. 11.

VOLUME XIX (1981) Theory of Intensity Dependent Resonance Light Scattering and Resonance 1- 43 Fluorescence, B. R. MOLLOW. . . . . . . . . . . . . . . . . . . . 11. Surface and Size Effects on the Light Scattering Spectra of Solids, D. L. . . . . . . . . . . . . . . . . . . . . 45-137 MILLS,K. R. SUBBASWAMY 111. Light Scattering Spectroscopy of Surface Electromagnetic Waves in Solids, S. USHIODA . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139-2 10 IV. Principles of Optical Data-Processing, H. J. BUTTERWECK . . . . . . . 21 1-280 V. The Effects of Atmospheric Turbulence in Optical Astronomy, F. RODDIER 281-376 I.

V O L U M E XX ( 1 9 8 3 ) I.

Some New Optical Designs for Ultra-Violet Bidimensional Detection of Astronomical Objects, G. C O U R T I ~P., CRUVELLIER, M. DETAILLE,M. 1-62 SAYSSE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11. Shaping and Analysis of Picosecond Light Pulses, C. FROEHLY, B. COLOMBEAU, M. VAMPOUILLE . . . . . . . . . . . . . . . . . . . . . . 63-154 111. Multi-Photon Scattering Molecular Spectroscopy, S. KIELICH. . . , . . 155-262 IV. Colour Holography, P. HARIHARAN . . . . . . . . . . . . . . . . . 263-324 V. Generation of Tunable Coherent Vacuum-Ultraviolet Radiation, W. JAMROZ, B. P. STOICHEFF . . . . . . . . . . . . . . . . . . . . . . . . . 325-380 VOLUME X X I ( 1 9 8 4 ) Rigorous Vector Theories of Diffraction Gratings, D. MAYSTRE. . . . . 1. 11. Theory of Optical Bistability, L. A. LUGIATO . . . . . . . . . . . . . 111. The Radon Transform and its Applications, H. H. BARRETT . . . . . . . IV. Zone Plate Coded Imaging: Theory and Applications, N. M. CEGLIO,D. W. . . . . . . . . . . , . SWEENEY. . . . . . . . . . . . . . . V. Fluctuations, Instabilities and Chaos in the Laser-Driven Nonlinear Ring R. R. SNAPP,W. C. SCHIEVE . . . . . . . . . . Cavity, J. C. ENGLUND,

.

1- 68 69-216 217-286

287-354 355-428

X


VOLUME XXII (1985) Optical and Electronic Processing of Medical Images. D . MALACARA . . . Quantum Fluctuations in Vision. M . A. BOUMAN. W.A. VAN DE GRIND.P . ZUIDEMA . . . . . . . . . . . . . . . . . . . . . . . . . . . . I11. Spectral and Temporal Fluctuations of Broad-Band Laser Radiation. A. V . MASALOV. . . . . . . . . . . . . . . . . . . . . . . . . . . . Yu . I . IV . Holographic Methods of Plasma Diagnostics. G . V . OSTROVSKAYA. OSTROVSKY. . . . . . . . . . . . . . . . . . . . . . . . . . . V . Fringe Formations in Deformation and Vibration Measurements using Laser Light. I . YAMAGUCHI. . . . . . . . . . . . . . . . . . . . . . . VI . Wave Propagation in Random Media: A Systems Approach. R. L. FANTE. I. I1.

1- 76 77-144 145-196 197-270 271-340 341-398

VOLUME XXIII (1986) 1.

I1 . Ill . IV . V.

Analytical Techniques for Multiple Scattering from Rough Surfaces. J . A. DESANTO.G . S. BROWN. . . . . . . . . . . . . . . . . . . . . . 1-62 Paraxial Theory in Optical Design in Terms of Gaussian Brackets. K .TANAKA 63-1 12 Optical Films Produced by Ion-Based Techniques. P.J . MARTIN.R. P. NETTERFIELD . . . . . . . . . . . . . . . . . . . . . . . . . . . 113-182 Electron Holography. A. TONOMURA. . . . . . . . . . . . . . . . 183-220 Principles of Optical Processing with Partially Coherent Light. F . T . S. Yu . 221-276

V O L U M E XXIV ( 1 9 8 7 ) 1.

I1. I11. IV. V.

Micro Fresnel Lenses. H . NISHIHARA. T . SUHARA. . . . . . . . . . . Dephasing-Induced Coherent Phenomena. L. ROTHBERG. . . . . . . . . . . . . . . . . . . . . . Interferometry with Lasers. P . HARIHARAN Unstable Resonator Modes. K . E. OUGHSTUN . . . . . . . . . . . . . Information Processing with Spatially Incoherent Light. I . GLASER. . . .

1-38 39-102 103-164 165-388 389-510

V O L U M E XXV ( 1 9 8 8 ) I.

Dynamical Instabilities and Pulsations in Lasers. N . B . ABRAHAM. P. MANDEL.L. M. NARDUCCI. . . . . . . . . . . . . . . . . . . . . I1. Coherence in Semiconductor Lasers. M. OHTSU.T . TAKO . . . . . . . I11. Principles and Design of Optical Arrays. WANCSHAOMIN. L. RONCHI. . IV . Aspheric Surfaces. G. SCHULZ . . . . . . . . . . . . . . . . . . .

1-190 191-278 279-348 349-416

V O L U M E XXVI ( 1 9 8 8 )

I.

Photon Bunching and Antibunching. M. C. TEICH.B . E. A. SALEH. . . . I1. Nonlinear Optics of Liquid Crystals. I . C. KHOO. . . . . . . . . . . . I11 . Single-Longitudinal-Mode Semiconductor Lasers. G . P . AGRAWAL. . . . IV . Rays and Caustics as Physical Objects. Yu. A. KRAVTSOV . . . . . . . V . Phase-Measurement Interferometry Techniques. K . CREATH. . . . . . .

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


XI

VOLUME XXVII (1989) The Self-Imaging Phenomenon and Its Applications, K. PATORSKI I. 11. Axicons and Meso-Optical Imaging Devices, L. M. SOROKO. . . . . . . 111. Nonimaging Optics for Flux Concentration, I. M. BASSETT,W. T. WELFORD, R. WINSTON . . . . . . . . . . . . . . . . . . . . . . . . . . . IV. Nonlinear Wave Propagation in Planar Structures, D. Mihalache, M. BERTOLOTH,C. SIBILIA. . . . . . . . . . . . . . . . . . . . . . . . . Generalized Holography with Application to Inverse Scattering and Inverse V. Source Problems, R. P. PORTER. . . . . . . . . . . . . . . . . . .

1-108 109- 160 161-226 227-313 315-397

VOLUME XXVIII (1990)

-

F. Digital Holography Computer-Generated Holograms, 0. BRYNGDAHL, WYROWSKI 11. Quantum Mechanical Limit in Optical Precision Measurement and CommuniS.MACHIDA, S.SAITO,N. IMOTO,T. YANAGAWA, M. cation, Y. YAMAMOTO, G. B J ~ R K. . . . . . . . . . . . . . . . . . . . . . . KITAGAWA, 111. The Quantum Coherence Properties of Stimulated Raman Scattering, M. G. RAYMER, I. A. WALMSLEY. . . . . . . . . . . . . . . . . . . . . IV. Advanced Evaluation Techniques in Interferometry, J. SCHWIDER. . . . Quantum Jumps, R. J. COOK . . . . . . . . . . . . . . . . . . . . V. I.

1- 86

87-179 181-270 271-359 36 1-4 16

VOLUME XXIX (1991) I.

Optical Waveguide Diffraction Gratings: Coupling between Guided Modes, 1- 63 D.G. HALL . . . . . . . . . . . . . . . . . . . . . . . . . . . 11. Enhanced Backscattering in Optics, Yu. N. BARABANENKOV, Yu. A. 65-197 KRAVTSOV, V. D. OZRIN,and A. I. SAICHEV. . . . . . . . . . . . . 199-29 1 111. Generation and Propagation of Ultrashort Optical Pulses, I. P. CHRISTOV. IV. Triple-Correlation Imaging in Optical Astronomy, G. WEIGELT . . . . . 293-319 Nonlinear Optics in Composite Materials, 1. Semiconductor and Metal V. F. HACHE,M. C. KLEIN,D. RICARD Crystallites in Dielectrics, C. FLYTZANIS, 321-411 AND PH. ROUSSIGNOL . . . . . . . . . . . . . . . . . . . . . . .

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PREFACE This volume presents five review articles by well-known experts, on topics of current interest in optics. The first article concerns quantum fluctuations, a phenomenon encountered directly or indirectly in all optical measurements. Such fluctuations set limits to attainable accuracy with which measurements can be made. In recent years theoretical as well as experimental research has demonstrated that limitations arising from quantum fluctuations can sometimes be circumvented to some extent. These developments are of importance from the standpoint of basic physics as well as in connection with technological applications, such as noise reduction in communication systems. The second article deals with correlation holographic interferometry and speckle photography, paying special attention to the effects of random variation of surface microstructure on the contrast of interference fringes. The article which follows covers an important subject in the broad area of wave propagation in random media, namely wave localization. This phenomenon is a subtle manifestation of interference of multiply scattered waves and provides information about important properties of disordered systems. The article considers mainly localization in one-dimensional systems, which elucidate some of the underlying physics. The fourth article discusses an important nonlinear phenomenon, namely soliton propagation in fibres. Solitons are pulses which can propagate over long distances without change in shape. Because of their considerable stability, they are of particular interest for communication systems. The concluding article presents the theory and describes experiments on elementary quantum systems in the context of cavity quantum optics. Such experiments are providing deeper understanding of the interaction of light with matter and give new insights into the foundations of quantum mechanics. This volume illustrates once again that optics continues to be a highly active field of research where many interesting developments are currently taking place. EMILWOLF Department of Physics and Astronomy University of Rochester Rochester, N Y 14627, USA February 1992


CONTENTS I . QUANTUM FLUCTUATIONS IN OPTICAL SYSTEMS by S . REYNAUD. A . HEIDMANN. E. GIACOBINO and C. FABRE (PARIS.FRANCE)

$ 1. INTRODUCTION. . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1. Field fluctuations . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.1. Vacuum fluctuations . . . . . . . . . . . . . . . . . . . . . . 1.1.2. Squeezed states . . . . . . . . . . . . . . . . . . . . . . . . 1.1.3. Intensity measurements . . . . . . . . . . . . . . . . . . . . 1.2. The beam splitter in quantum optics . . . . . . . . . . . . . . . . . 1.2.1. The beam splitter and photon noise . . . . . . . . . . . . . . . 1.2.2. Homodyne measurements . . . . . . . . . . . . . . . . . . . 1.2.3. Effects of losses . . . . . . . . . . . . . . . . . . . . . . . . $ 2. TECHNIQUES FOR QUANTUM NOISEREDUCTION. . . . . . . . . . . . . . 2.1. Parametric generation . . . . . . . . . . . . . . . . . . . . . . . . 2.2. Bistability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3. Second-harmonic generation . . . . . . . . . . . . . . . . . . . . . 2.4. Non-degenerate parametric generation . . . . . . . . . . . . . . . . . 2.5. Lasers with reduced pump noise . . . . . . . . . . . . . . . . . . . 2.6. Reduction of quantum fluctuations using servo techniques . . . . . . . . 8 3. APPLICATIONS OF SQUEEZED LIGHT . . . . . . . . . . . . . . . . . . . 3.1. Improvement of the sensitivity in optical measurements . . . . . . . . . 3.1. 1. Interferometric measurements . . . . . . . . . . . . . . . . . . 3.1.2. Polarization measurements . . . . . . . . . . . . . . . . . . . 3.1.3. Intensity measurements . . . . . . . . . . . . . . . . . . . . 3.1.4. Symmetric heterodyne scattering . . . . . . . . . . . . . . . . 3.2. Quantum non-demolition (QND) measurements . . . . . . . . . . . . 3.2.1. QND measurements using optical Kerr effe.ct . . . . . . . . . . . 3.2.2. QND measurements using squeezed fields . . . . . . . . . . . . 3.3. Inhibition of phase diffusion . . . . . . . . . . . . . . . . . . . . . 3.4. Interference effects with twin photons . . . . . . . . . . . . . . . . . 3.5. Inhibition of relaxation phenomena . . . . . . . . . . . . . . . . . . 3.5.1. Inhibition of spontaneous emission . . . . . . . . . . . . . . . 3.5.2. Inhibition of tunnelling . . . . . . . . . . . . . . . . . . . . . 3.6. Optical communication techniques . . . . . . . . . . . . . . . . . . 3.6.1. Amplification and noise . . . . . . . . . . . . . . . . . . . . 3.6.2. QND measurements on an optical channel . . . . . . . . . . . . 3.6.3. Decrease of energy cost per bit in an optical channel . . . . . . . . 3.6.4. Communication systems using twin photons . . . . . . . . . . . . $ 4. THEORY OF QUANTUM NOISEIN OPTICS . . . . . . . . . . . . . . . . . 4.1. Quantum field fluctuations . . . . . . . . . . . . . . . . . . . . . . 4.1.1. The quantum field . . . . . . . . . . . . . . . . . . . . . . .

xv

3 4 5 6 6 7

7 8 9 10 10

14 15 16 20 21 22 22 22 24 25 26 21 21 28 29 29 30 31 31 31 32 32 32 32 33 33 33

CONTENTS

XVI

Intensity of the field . . . . . . . . . . . . . . . . . . . . . . Photon noise spectrum . . . . . . . . . . . . . . . . . . . . . Standard shot noise . . . . . . . . . . . . . . . . . . . . . . Limit of small fluctuations . . . . . . . . . . . . . . . . . . . Effect of a reflecting plate . . . . . . . . . . . . . . . . . . . Heterodyne measurements . . . . . . . . . . . . . . . . . . . Imperfect photodetectors . . . . . . . . . . . . . . . . . . . . 4.2. Ideal squeezed states . . . . . . . . . . . . . . . . . . . . . . . . 4.2.1. The generalized Heisenberg inequality . . . . . . . . . . . . . . 4.2.2. Minimum states . . . . . . . . . . . . . . . . . . . . . . . . 4.2.3. Generalized coherent states . . . . . . . . . . . . . . . . . . . 4.2.4. Wigner representation of minimum states . . . . . . . . . . . . . 4.2.5. Frequency multimode squeezing . . . . . . . . . . . . . . . . . 4.3. Squeezed-state generation by ideal parametric interaction . . . . . . . . 4.3.1. Effect of an ideal quadratic Hamiltonian . . . . . . . . . . . . . 4.3.2. Perturbative approach . . . . . . . . . . . . . . . . . . . . . 4.3.3. Parametric amplification and deamplification . . . . . . . . . . . 4.3.4. Semiclassical pendulum . . . . . . . . . . . . . . . . . . . . 5 . SQUEEZED-STATE GENERATION BY PARAMETRIC INTERACTION . . . . . . . . 5.1. Parametric amplification in an optical cavity . . . . . . . . . . . . . . 5.1.1. Resonant case . . . . . . . . . . . . . . . . . . . . . . . . 5.1.2. Non-resonant case . . . . . . . . . . . . . . . . . . . . . . . 5.1.3. Good-cavity limit . . . . . . . . . . . . . . . . . . . . . . . 5.1.4. Case of incoming vacuum field . . . . . . . . . . . . . . . . . 5.2. Degenerate parametric oscillation . . . . . . . . . . . . . . . . . . . 5.2.1. Equations of evolution . . . . . . . . . . . . . . . . . . . . . 5.2.2. Stationary solutions and stability analysis . . . . . . . . . . . . . 5.2.3. Linear analysis of fluctuations . . . . . . . . . . . . . . . . . . 5.2.4. Resonant case . . . . . . . . . . . . . . . . . . . . . . . . 5.2.5. Bistable region . . . . . . . . . . . . . . . . . . . . . . . . 5.2.6. Unstable region . . . . . . . . . . . . . . . . . . . . . . . . 5.2.7. General characteristics of the noise spectra . . . . . . . . . . . . 5.3. Non-degenerate optical parametric oscillator (NDOPO) . . . . . . . . . 5.3.1. Stationary values . . . . . . . . . . . . . . . . . . . . . . . 5.3.2. Linearized equations for the fluctuations . . . . . . . . . . . . . 5.3.3. Quantum fluctuations of the signal field . . . . . . . . . . . . . . 5.3.4. Quantum fluctuations in the difference between the signal and idler fields 5.3.5. Effect of extra losses and imbalance . . . . . . . . . . . . . . . 5.3.6. Effect of phase diffusion . . . . . . . . . . . . . . . . . . . . . SEMICLASSICAL REPRESENTATIONS OF THE FIELD . . . . . . . . . . APPENDIX A .1. Definition of semiclassical representations . . . . . . . . . . . . . . . . A.l.l. Formal definition . . . . . . . . . . . . . . . . . . . . . . . A.1.2. The usual definitions . . . . . . . . . . . . . . . . . . . . . A.1.3. Relation between the representations . . . . . . . . . . . . . . A.2. Semiclassical equations of evolution . . . . . . . . . . . . . . . . . . A.2.1. Hamiltonian evolution in the Wigner representation . . . . . . . . A.2.2. Hamiltonian evolution in the 9- and 2 -representations . . . . . . A.2.3. Quadratic Hamiltonian . . . . . . . . . . . . . . . . . . . . A.2.4. Canonical transformations . . . . . . . . . . . . . . . . . . . A.2.5. Parametric generation . . . . . . . . . . . . . . . . . . . . . ACKNOWLEDGEMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.2. 4.1.3. 4.1.4. 4.1.5. 4.1.6. 4.1.7. 4.1.8.

34 35 36 37 38 39 40 41 41 42 43 45 47 49 49

50 52 53 54 55 55 58 59 60 62 62 62 64 64 65 66 67 67 67 68 69 70 71 72 72 73 73 74 75 76 76 78 78 79 80 81 81

CONTENTS

XVll

I1. CORRELATION HOLOGRAPHIC AND SPECKLE INTERFEROMETRY by Yu . I. OSTROVSKY (ST. PETERSBURG, RUSSIA) and V . P. SHCHEPINOV (Moscow. RUSSIA)

$ I . INTRODUC~ION . . . . . . . . . . . . . . . . . . . . . . . . . . . . $ 2. FRINGECONTRAST I N HOLOGRAPHIC INTERFEROMETRY AND SPECKLE PHOTOGRAPHY AS RELATED TO A CHANGE I N SURFACE MICRORELIEF. . . . . . . 2.1. Fringe contrast in holographic interferometry . . . . . . . . . . . . . . 2.2. Young's fringe contrast in speckle photography . . . . . . . . . . . . . 2.3. Visualization of areas in which the surface microrelief varies . . . . . . . 2.3.1. The use of carrier fringes . . . . . . . . . . . . . . . . . . . . 2.3.2. Image subtraction . . . . . . . . . . . . . . . . . . . . . . . 2.3.3. Simultaneous recording of holograms and speckle photographs . . . $ 3. MECHANICS OF CONTACT INTERACTION . . . . . . . . . . . . . . . . . . 3.1. Determination of the contact area . . . . . . . . . . . . . . . . . . 3.1.1. Evaluation of the plastic component of actual contact surface . . . . 3.1.2. Recording the contact contour surface by correlation holographic interferometry . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.3. Recording the contact countour surface by correlation speckle photography . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2. Contact pressure measurements . . . . . . . . . . . . . . . . . . . 3.2.1. Effect of contact pressure on carrier fringe contrast in holographic interferometry and speckle photography . . . . . . . . . . . . . . . 3.2.2. Contact pressure measurement by holographic interferometry and speckle photography . . . . . . . . . . . . . . . . . . . . . . . . . $ 4. CORROSION, EROSION A N D WEARPROCESSES. . . . . . . . . . . . . . . 4.1. Investigation of chemical corrosion . . . . . . . . . . . . . . . . . . 4.2. Investigation of cavitation-induced erosion . . . . . . . . . . . . . . . 4.3. Investigation of mechanical wear . . . . . . . . . . . . . . . . . . . $ 5. CONCLUSIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

89 90 90 92 96 96 99 101 103 103 103 106 111 113 113

118 121 121 123 130 133 134

111. LOCALIZATION OF WAVES IN MEDIA WITH ONE-DIMENSIONAL DISORDER

. .

by V D FREILIKHER (RAMAT-GAN. ISRAEL) and S . A. GREDESKUL ( B E E R ~ H E VISRAEL) A,

$ 1 . INTRODUCTION. . . . . . . . . . . . . . . . . . . . . . . . . . . . $ 2. STATISTICAL PROPERTIES OF PHYSICAL QUANTITIES I N RANDOMMEDIA . . $ 3. ONE-DIMENSIONAL LOCALIZATION . . . . . . . . . . . . . . . . . . . . 3.1. Lyapunov exponent. Localization length . . . . . . . . . . . . . . . 3.2. Scattering problem . . . . . . . . . . . . . . . . . . . . . . . . . 3.3. Resonance transmission . . . . . . . . . . . . . . . . . . . . . . . 3.4. Intensity of a wave passing through a random layer . . . . . . . . . . 3.5. Some numerical and experimental results . . . . . . . . . . . . . . $ 4. WAVES I N RANDOMLY LAYERED MEDIA . . . . . . . . . . . . . . . . 4.1. Point source in a randomly stratified layer . . . . . . . . . . . . . . 4.2. Fluctuation waveguide . . . . . . . . . . . . . . . . . . . . . . . 4.3. Quasihomogeneous waves . . . . . . . . . . . . . . . . . . . . . . 4.4. Quasistationary states in an open system . . . . . . . . . . . . . . 4.5. Point source in an infinite layered medium . . . . . . . . . . . . . . $ 5 . CONCLUSIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ACKNOWLEDGEMENT . . . . . . . . . . . . . . . . . . . . . . . . . . . . REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . .

. .

139 143 148 148 154 161 164 167 170 170 173 178 185 188 199 200 200

CONTENTS

XVIll

IV. THEORETICAL FOUNDATION O F OPTICAL-SOLITON CONCEPT IN FIBERS by Y . KODAMA (COLUMBUS. OH. USA) and A. HASEGAWA (MURRAY HILL.NJ. USA)

INTRODUCTION. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . J 1. SOLITONS IN OPTICS . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2. Envelope solitons in optical fibers . . . . . . . . . . . . . . . . . . . 1.3. Non-linear Schrbdinger soliton . . . . . . . . . . . . . . . . . . . . 1.4. Modulational instability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. OPTIALSOLITONS I N FIBERS 2.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2. Derivation of the non-linear envelope equation . . . . . . . . . . . . . 2.3. Effects of higher-order terms . . . . . . . . . . . . . . . . . . . . . J 3. GUIDING CENTERSOLITON . . . . . . . . . . . . . . . . . . . . . . . 3.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2. Compensation of fiber loss in long-distance propagation of optical solitons . 3.3. Guiding center equation and the Lie transformation . . . . . . . . . . . 3.4. Properties of the guiding center solitons . . . . . . . . . . . . . . . . ACKNOWLEDGEMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . APPENDIX A. INVERSE SCATTERING TRANSFORM A N D N-SOLITONSOLUTIONS . . . APPENDIX B. CALCULUS O N INFINITE-DIMENSIONAL SPACESOF DIFFERENTIAL POLY-

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

NOMIALS

REFERENCES . .

207 207 207 211 213 220 221 221 222 229 231 231 233 235 243 249 249 253 258

V . CAVITY QUANTUM OPTICS AND THE QUANTUM MEASUREMENT PROCESS

by P . MEYSTRE (TUCSON.AZ. USA)

Q 1. INTRODUCTION. . . . . . . . . . . . . . . . . . . . . . . . . . . . J 2. JAYNES-CUMMINGS MODEL. . . . . . . . . . . . . . . . . . . . . . . 2.1. Eigenstates and eigenvalues . . . . . . . . . . . . . . . . . . . . . 2.2. Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3. Trapping states . . . . . . . . . . . . . . . . . . . . . . . . . . . Q 3. CAVITY QED . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1. Enhanced and inhibited spontaneous emission . . . . . . . . . . . . . 3.2. Collective effects . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3. Coupled-modes approach . . . . . . . . . . . . . . . . . . . . . . 3.4. Vacuum Rabi splitting as a feature of linear dispersion theory . . . . . . . J 4. THEMICROMASER. . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1. Phenomenology . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2. Quantum theory . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3. Fokker-Planck approach and semiclassical limit . . . . . . . . . . . . 4.4. Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . J 5. QUANTUM MEASUREMENTS . . . . . . . . . . . . . . . . . . . . . . . 5.1. Single quantum systems versus ensembles . . . . . . . . . . . . . . . 5.2. Repeated quantum measurements on a single harmonic oscillator . . . . . 5.3. Continuous photodetection . . . . . . . . . . . . . . . . . . . . . . 5.4. Measurement-induced oscillations of the Fano factor . . . . . . . . . . 5.5. Measurement-induced dynamics of the micromaser field . . . . . . . . .

263 265 265 267 270 271 271 277 280 283 286 286 290 293 297 301 301 302 304 307 309

CONTENTS

XIX

$ 6. QUANTUM NON-DEMOLITION MEASUREMENTS . . . . . . . . . . . . . . . 6.1. Back-action evasion . . . . . . . . . . . . . . . . . . . . . . . . . 6.2. QND measurements in optics . . . . . . . . . . . . . . . . . . . . 6.3. QND measurements in micromaser cavities . . . . . . . . . . . . . . 8 7. MACROSCOPIC SUPERPOSITIONS . . . . . . . . . . . . .. . . . . . . . 7.1. Tangent and cotangent states of the electromagnetic field . . . . . . . . 7.2. Effects of dissipation . . . . . . . . . . . . . . . . . . . . . . . . 7.3. Detection: non-linear atomic homodyning . . . . . . . . . . . . . . . $ 8. SEPARATED FIELDS. . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1. Micromaser tests of quantum mechanical complementarity . . . . . . . . 8.2. Quantum superpositions of macroscopically separated cavity fields . . . . $ 9. OUTLOOK: MECHANICAL EFFECTS . . . . . . . . . . . . . . . . . . . . 9.1. Light forces and mechanical motion . . . . . . . . . . . . . . . . . . 9.2. Atomic beam deflection in a quantum field . . . . . . . . . . . . . . . 9.3. Atomic reflection at a micromaser cavity . . . . . . . . . . . . . . . . 9.4. Atomic trapping by the vacuum field in a cavity . . . . . . . . . . . . ACKNOWLEDGEMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

312 312 314 316 321 321 326 334 336 336 338 341 341 343 347 349 351 351

AUTHORINDEX . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . SUBJECT INDEX . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . CUMULATIVE INDEX.VOLUMES I-XXX . . . . . . . . . . . . . . . . . . . .

357 365 369


E. WOLF, PROGRESS IN OPTICS XXX 0 ELSEVIER SCIENCE PUBLISHERS B.V., 1992

I

QUANTUM FLUCTUATIONS IN OPTICAL SYSTEMS BY

SERGEREYNAUD, ANTOINE HEIDMANN FABRE ELISABETH GIACOBINO and CLAUDE Laboratoire de Spectroscopie Hertzienne de I'Ecole Normale Supdrieure et de I'lmiversitk P. et M. Curie, CNRS lJRAOO18, Boite 74 75252 Paris Cedex 05, France

CONTENTS PAGE

$ 1 . INTRODUCTION

. . . . . . . . . . . . . . . . . . . 3

$ 2. TECHNIQUES FOR QUANTUM NOISE REDUCTION . . $ 3 . APPLICATIONS OF SQUEEZED LIGHT

10

. . . . . . . . 22

$ 4 . THEORY OF QUANTUM NOISE IN OPTICS

. . . . . . 33

$ 5 . SQUEEZED STATE GENERATION BY PARAMETRIC INTERACTION . . . . . . . . . . . . . . . . . . . .

54

APPENDIX: SEMICLASSICAL REPRESENTATIONS OF THE FIELD . . . . . . . . . . . . . . . . . . . . . . . .

72

ACKNOWLEDGEMENTS . . . . . . . . . . . . . . . . . . 81 REFERENCES

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

2

81

4

1. Introduction

Quantum fluctuations are present in every measurement. They are often unobservable, however, since they occur at a level which is far below that which can be reached by experiment. It is well known, e.g., that the precision needed to determine the position q and the momentum p of a particle is limited by the Heisenberg inequality, Aq A p 3 A, but the measurements of the position and momentum are usually far from the quantum noise level. In optics, on the other hand, the quantum uncertainty is a real limitation, and high-precision measurements are limited by quantum noise. More recently, this has also become true in the field of microwaves. Optical measurements have three characteristics that enable them to reach the quantum noise level more readily than in other fields of physics: (1) optical signals are naturally immune to external sources of noise; (2) thermal noise at room temperature is negligible in the optical domain, but in the microwave domain, studies on quantum noise must be done at low temperature; and (3) the outstanding qualities of the optical sources and detectors allow a very low level of instrumental noise. In the measurement device quantum fluctuations have an effect similar to that of the instrumental noise or thermal fluctuations. They are of a more fundamental nature, however, and it has long been thought that they were an insuperable barrier to accuracy. In the last decade theoretical studies followed by experimental ones have shown that this limitation could be circumvented. Quantum fluctuations have become a new subject for physicists to study. Experimental demonstrations of quantum noise reduction have increased in recent years, and the degree of noise reduction in several experiments now ranges from 70 to 90%. This expansion in research has also given rise to considerable theoretical developments that can now account for most of the observed phenomena in quantum optics. Applications of quantum noise reduction can be envisaged, including ultrasensitive measurements in physics (the detection of gravitational waves, parity violation in atomic physics, new spectroscopic techniques) and new phenomena (quantum non-demolition measurements, two-photon interference, inhibited relaxation), or even technical uses that go beyond the framework of 3

4

QUANTUM FLUCTUATIONS IN OPTICAL SYSTEMS

[I, B 1

fundamental physics (communication systems, gyrolasers). As is usual in quantum mechanics, several complementary pictures can be used to represent the quantum fluctuations. We shall use two of these. On the one hand, quantum fluctuations are obviously linked to the corpuscular nature of the radiation: any detected signal has a granular character, leading to photon noise, or shot noise. Photon noise is just a consequence of the random distribution of photons in a light beam. On the other hand, one can consider that the photon noise reflects the fluctuations of the various components of the electromagnetic field, amplitude and phase of the field, or quadrature components. These fluctuations can often be emulated with classical fluctuations. Although these representations often give a simpler physical insight into the phenomena than the full quantum calculations, they must be used with caution, since their validity is restricted to certain classes of effects. We shall use these pictorial descriptions of the electromagnetic field for a first introduction to squeezed states in Q 1. In Q 2 and 0 3 we describe the principal experiments that have generated light with reduced quantum fluctuations, and the applications of such light. In these sections a few straightforward calculations involving squeezed fields are presented to explain the experimental results. The squeezed fields are treated in the framework of the ‘semiclassical linear input output theory’, which models the quantum fluctuations by classical random fields. In 0 4 the standard representations of quantum optics are used to treat the quantum fluctuations and properties of the squeezed states, and to study the ideal parametric interaction Hamiltonian in detail. Section 5 discusses the quantum properties of parametric generation in an optical cavity, including the two regimes below and above the oscillation threshold.

1 . 1 . FIELD FLUCTUATIONS

This section will give a simple presentation of the properties of quantum fields and, in particular, of squeezed fields in terms of mean square fluctuations of the components of the fields. A particular mode of the electromagnetic field, corresponding to a plane wave of given frequency, direction, and polarization, is described by E

=

E,cos(wt

The variables E , and

+ 4) = E, coswt + E,

sinwt.

(1.1)

4 are the amplitude and phase of the wave, respectively.

1, t 11

INTRODUCTION

5

E , and E, are the two quadrature components (defined with respect to a given phase reference). In quantum optics the two quadrature components are conjugate variables, such as the position q and the momentum p of an oscillator. Their dispersions in any quantum state obey a Heisenberg inequality AE, AE2 3 E' ,

(1.2)

where E is a constant corresponding to the electric field of a single photon. A similar inequality exists for the product of the fluctuations in the phase $ and in the intensity I: I = E;

=

E:

+ E;,

A I A $ > 4~~ .

(1.3)

(1.4)

1.1.1. Vacuumfluctuations In the case of a particle the Heisenberg inequality concerns two variables q and p that have different physical natures. Subsequently, there is no natural common scale for the dispersions Aq and Ap. In contrast, the two conjugate variables of the electromagnetic field, E , and E,, have the same nature, and most of the usual light sources have no preferred quadrature component. In other words, the fluctuations of the electromagnetic field usually have no natural phase reference. As a result, the dispersions A E , and AE2 are equal, and independently satisfy an inequality,

AE,

=

AE2 3 E

.

(1.5)

The factor E gives a standard scale for the fluctuations of the electromagnetic field and quantum noise. In particular, it gives the scale of the vacuumfluctuations, which correspond to the state with the lowest energy. These fluctuations are equivalent to the zero-point fluctuations for a material oscillator. The corresponding field is a minimal state fulfilling eq. (1.5), since there is no phase reference in the vacuum; hence,

AE,

=

AE2 = E

.

(1.6)

The vacuum fluctuations can often be considered as the source of quantum noise in optical measurements. The coherent state defined by Glauber [ 19651, e.g., is a superposition of a classical field and the vacuum fluctuations. The photon noise can also be interpreted as a manifestation of the vacuum fluctuations in the intensity signal.

6


1.1.2. Squeezed states To surpass the limit set by eq. (lS), it is necessary to break the symmetry between the two quadratures. The Heisenberg inequality does not forbid decreasing A E l below E, provided that AE, is increased. Such states in which the dispersion in one quadrature is smaller than the standard dispersion E are called squeezed states. For example,

AEl c E ,

A E , >/ E ~ ~ A. E ,

(1.7)

One can represent the field fluctuations with a distribution in phase space { E , , E , } , which actually corresponds to the Wigner distribution (see 0 4). The Heisenberg inequality indicates that the surface area must be larger than a minimal area, which is of the order of E’. The fluctuations of the vacuum or of a coherent state correspond to a circle with the minimal area. A squeezed state has a different shape (e.g., elliptical), with a smaller dispersion in one of its components. The fluctuations may be squeezed in one of the quadratures E l or E,, in the intensity I , in the phase $, or in any other component of the field (fig. 1.1). 1.1.3. Intensity measurements The photon noise is associated with the quantum fluctuations in the field intensity. This can be understood qualitatively by considering the field as the

Fig. 1.1. Phase space representation of squeezed states. The vacuum state and coherent states would be represented by circles with the minimal area 118’. Squeezed states correspond to ellipses with the same area. In the figure (a) is a squeezed vacuum, (b) an amplitude-squeezed state, and (c) a phase-squeezed state.

4 5 11

INTRODUCTION

I

sum of a single-mode classical field ( E l ) cos (mot) and of the fluctuations 6El distributed over all the modes of frequencies w. In a linear treatment one obtains the intensity fluctuations 61 = 2 ( E l ) 6El,

(1.8a)

A12 = 4 ( E l ) 2 A E ; .

(1.8b)

The intensity fluctuations are proportional to the fluctuations of the field amplitude, i.e. to the component aligned with the mean field (here El). The standard photon noise corresponds to the case in which the amplitude fluctuations are equal to the vacuum fluctuations A E ; = c2. The photon noise is characterized by a variance proportional to the mean intensity

(A12)st= 4 ( E l )

t2 .

(1.9)

This noise is reduced by squeezing the fluctuations of the E , component of the field. More generally, in every optical measurement it can be shown that the fluctuations correspond to some component of the field, which can be a quadrature component of a field or a superposition of several fields. In all these cases the standard quantum limit or standard shot noise limit is the noise level corresponding to the case where these fluctuations are equal to the vacuum fluctuations. To reduce this noise, one must squeeze the fluctuations of the measured component. To show how the quantum noise affects measurements, we are going to consider the example of a simple system, the beam splitter.

1.2. THE BEAM SPLITTER IN QUANTUM OPTICS

1.2.1. The beam splitter and photon noise When a light beam is sent into the input channel A of a 50/50 beam splitter (fig. 1.2), each photon has a probability of a half of being transmitted or reflected if nothing enters in the other input channel (channel B). The noise in the difference between the intensities in the two output channels is then the standard quantum noise. This results directly from the random character of the transmission or reflection processes, and it is independent of the photon statistics of the input beam. This experiment is a good example of the difficulties associated with the quantum nature of the phenomenon: even if the incident beam has a reduced

8

QUANTUM FLUCTUATIONS IN OPTlCAL SYSTEMS

OUtPUl port

c

T

output

I ID

inpul port A

Fig. 1.2. The input fields at ports A and B are partly transmitted and partly reflected by the beam splitter into the output ports C and D. If the beam splitter has a transmissivity equal to the reflectivity, it acts as a random scatterer for photons arriving at A (probabilities of a half to C or D). When the fluctuations entering port B are squeezed, the scattering statistiscs are modified, and twin beams may be obtained in the output ports.

intensity noise, one retrieves the standard quantum noise after a reflection on a beam splitter. 1.2.2. Homodyne measurements Nevertheless, it is possible to modify the intensity noise after a beam splitter. To understand how this can be done, we have to envision the preceding experiment as a homodyne measurement: the incident beam entering channel A is mixed on the beam splitter with the field entering channel B. Assuming that the beam splitter has amplitude transmission and reflection coefficients t and r (with r 2 = 1 - t 2 ) ,the classical relations give the fields going out of the system in channels C and D (fig. 1.2) as E,

=

tE,

f

ED = - r E ,

rE, ,

(1.10a)

+ tE,,

(1. lob)

where the minus sign in eq. (1. lob) comes from the phase shift of R between the reflections from air off glass and from glass off air. It can also be understood as a consequence of the energy conservation condition, I , + I , = I , + I , . We will now consider an experiment in which r = t = l/,,h. Assuming that the field E , is much more intense than the field E,, the measurement of the intensity difference I = Ic - I , between the two channels gives

I

=

I,

-

I,

=

2 ( E , ) E,, ,

(1.1 la)

where E , is the quadrature component of E , in phase with the mean amplitude

1, B 11

INTRODUCTION

9

( E , ) . The intensity difference I is a perfect homodyne signal of the quadrature component E,, by the local oscillator ( E , ) . This property is also true for the fluctuations, A12 = 4 ( E , ) ’ AE;, .

(1.1 1b)

The photon noise reflects the quantum fluctuations of the quadrature component E,, . In particular, when nothing enters channel B, the standard quantum noise after the beam splitter can be interpreted as a result of the homodyne mixing of the vacuum fluctuations AE; I = E’ with the local oscillator (A12)s,= 4 ( E , )

’

E2

.

(1.12)

It is possible to reduce the noise in the intensity difference after the beam splitter by entering a squeezed field in channel B. One obtains twin-photon beams, i.e., two light beams whose intensities are correlated to better than the standard quantum noise. In this case the squeezed quantity is the difference between the amplitudes of the two beams. As discussed later in 5 2.4, twin beams can also be obtained directly in non-linear optics experiments. 1.2.3. Effects of losses Losses have a drastic effect on all problems related to quantum noise. To model losses that reduce the intensity of a light beam by a factor T , the simplest way is to emulate them with a beam splitter with an amplitude transmission coefficient t, such that t 2 = T. Let us suppose that a squeezed field enters channel A. The vacuum field enters channel B, and eqs. (1.10) imply that (1.13a) (1.13b) The beam splitter mixes the fluctuations of the two incident fields and tends to bring the fluctuations of the squeezed field back to the vacuum fluctuations. This occurs for all kinds of linear optical losses. When the losses are large, the initial squeezing vanishes completely. In a detection system a quantum efficiency smaller than one has the same effect, since a detector with a quantum efficiency T can be considered as a perfect detector preceded by a beam splitter with transmissivity T.

10


8 2.

[I, § 2

Techniques for Quantum Noise Reduction

Several non-linear optical systems have been shown to generate squeezed states. In most cases the phenomena involved are parametric processes. The squeezing of fluctuations can then be understood using the analogy with a parametric resonance of a classical matter oscillator. It is well known that a pendulum of eigenfrequency w submitted to a parametric excitation of frequency 2w has one of its quadrature components amplified exponentially, while the other is exponentially damped. In optics a mode of the electromagnetic field of frequency w plays the role of the pendulum, and a non-linear material generates the parametric effect through the modulation of an effective index of refraction at frequency 2w. When the parametric generator is placed in an optical cavity, the system can be considered as a parametric oscillator. The squeezing of the fluctuations is the largest when the cavity operates close to the oscillation threshold, i.e., when the parametric gain approaches the value of the loss coefficient. This effect is well known: the fluctuations of the amplified component diverge at the threshold, and this phenomenon of critical divergence is associated with a squeezing of the fluctuations of the damped component. The parametric effect can come from a x(’) or x(3)non-linearity. In the former case the pump wave has a frequency w,,,which is very different from the frequencies o1and w2 of the signal waves (in three-wave mixing, energy conservation requires that w,, = w , + w2).In the latter case the pump wave has a frequency close to the frequencies of the signal waves (in four-wave mixing, 20, = o1+ w 2 ) . The systems may be used in the degenerate operation in which o1and w2 are identical or in the non-degenerate operation in which the two signal modes have different frequencies or different polarizations. A different technique has also been used to squeeze the intensity. It does not use a parametric generator, but uses a laser for which the output noise is reduced by modifying the noise of the pumping system.

2.1. PARAMETRIC GENERATION

The first observation of squeezing was achieved by Slusher, Hollberg, Yurke, Mertz and Valley [ 19851 at A T & T Bell Laboratories in an experiment of parametric generation involving four-wave mixing in sodium vapor. The nonlinearity was enhanced by placing the non-linear medium in an optical cavity. One year later, squeezed light was generated in a similar system by Wu, Kimble, Hall and Wu [ 19861, but with three-wave mixing in a non-linear x(2) crystal. In both cases the systems were operated in the degenerate regime.

1.8 21

TECHNIQUES FOR QUANTUM NOISE REDUCTION

11

The optimum value measured by Slusher's group in the four-wave mixing experiment was 24% (Slusher, Yurke, Grangier, Laporta, Walls and Reid [1987]). This technique has the advantage of giving a high degree of nonlinearity, despite the low atomic density of an atomic beam, but it has drawbacks: the atomic fluctuations are significant in the vicinity of the resonance and they tend to hide the squeezing effect, as shown by Reid and Walls [ 19851. We shall describe the type of experiment that was first performed by Wu, Kimble, Hall and Wu [ 19861, which yields the larger amount of squeezing. Let us first briefly discuss the physical process that leads to squeezing. Parametric interaction amplifies input noise, especially here, quantum noise. The effect can then be pictured in a simple way (at least in the limit of low pumping rates): the system processes the vacuum noise input and squeezes it. The output, having a zero mean field, is expected to be the squeezed vacuum (Yuen and Shapiro [ 19791). The parametric medium, which is pumped by a laser, is placed in an optical cavity to enhance the non-linearity. Above some pump power threshold, the system may oscillate, but for the present purpose it is kept below threshold. If we neglect pump depletion, the change of the signal field in one round trip in the cavity can be written as a function of the cavity losses, the parametric gain, and the input field af",which is the vacuum field entering the cavity through the coupling mirror

where z is the cavity round trip time; y is the damping coefficient of the field in the cavity, which is related to the amplitude transmission coefficient t of the output mirror by y = i t 2 ( t is assumed to be small); and q is the parametric gain, which is proportional to the pump field, to the x(2) coefficient, and to the length of the crystal. The parametric gain in a, involves a:, which is characteristic of parametric amplification and is the reason for the phase dependence of the process. Writing eq. (2.1) and its complex conjugate and taking the Fourier transform, we express the solution in terms of the quadratures q , = (aI + a:)/,,h and p I = ( a l - .:)/,,,hi, with similar notations for q p and pf":

On the other hand, the outgoing field a:"' is related to the field inside the

12


[I, I 2

cavity and to the incoming vacuum field by the reflection-transmission relation (1.10):

where the reflection coefficient r has been approximated by one. Using eqs. (2.2) and (2.3), one obtains

Equations (2.4) and (2.5) clearly show that the vacuum field is amplified in quadrature q l , whereas quadrature pI is squeezed and even goes to zero for a = 0 and y = v], i.e., for zero-noise frequency and when the oscillation threshold is approached. The vacuum field is squeezed by the system. A more elaborate treatment of this case will be given in Q 5. The experimental scheme (fig. 2.1) uses a non-linear crystal (LiNbO,), which is pumped by a single-mode doubled Y A G laser. The crystal is placed in an optical cavity that is resonant for both the pump and the subharmonic field. The observation of squeezing in the signal beam is done by homodyne detection as exposed in Q 1.2. The light leaving the parametric amplifier is combined on a beam splitter with the light at the fundamental frequency from the Y A G laser

Fig. 2.1. Experimental scheme used by Wu, Kimble, Hall and Wu [ 19861 to generate a squeezed vacuum in parametric downconversion. The squeezed light coming out of the optical cavity containing the parametric crystal (OPO) is combined with a local oscillator in a balanced homodyne detector formed by the beam splitter and the two photodiodes. The noise in the difference between the two photocurrents is analyzed to demonstrate the squeezing effect.

1 3 8

21


13

(1.06 pm), which acts as a strong local oscillator. The photodetection signal is fed into a spectrum analyzer. The signal at a fixed-noise frequency oscillates around the shot noise level as a function of the phase of the local oscillator. The noise reduction has been verified to be improved when the incident pump power tends to its threshold value. The optimum noise reduction obtained with this technique is 63% (Wu, Min Xiao and Kimble [ 19881). These techniques can be extended to the microwave domain: a noise reduction of 47% has been obtained at 19GHz by Movshovich, Yurke, Kaminsky, Smith, Silver, Simon and Schneider [ 19901 using a Josephson amplifier. The experiment was performed at a very low temperature to avoid thermal noise. In reference to optical frequencies, squeezing has also been observed using forward four-wave mixing in a long silica optical fiber by Shelby, Levenson, Perlmutter, Devoe and Walls [ 19861.In this case there is no optical cavity, and broad-band squeezed states hav.e been generated with a minimum total noise level 20% below the standard quantum limit (Schumaker, Perlmutter, Shelby and Levenson [ 19871). Continuous-wave experiments have been hampered by index fluctuations due to Brillouin scattering. But solitons and short pulses can be squeezed efficiently with this method because their time scale is too fast to be sensitive to Brillouin scattering (Shelby, Drummond and Carter [ 19901, Rosenbluh and Shelby [ 19911, Bergman and Haus [ 19911). Small squeezing effects have been observed in another system without a cavity, using an optically thick sodium vapor as a non-linear medium (Maeda, Kumar and Shapiro [ 19861). Other experiments without an optical cavity use pulsed pumping, which makes it possible to obtain the high parametric gain required for squeezing. Pulsed squeezed light has been generated by pumping a non-linear crystal with picosecond pulses. Broad-band squeezed light has been produced with an optimum squeezing factor of 13% by Slusher, Grangier, Laporta, Yurke and Potasek [ 19871 using KTP (KTiOPO,) and of 24% by Hirano and Matsuoka [ 19901 using BNN (Ba,NaNb,O,,). Finally, the systems that have x(’) non-linearities seem to yield better squeezing than the f ) systems, for two primary reasons: (1) the pump frequency is far from the signal frequencies, which allows easy elimination of the spurious fields due to the pump scattering, and (2) the non-linearity is not resonant in the doubling crystals. Consequently, there are no fluctuations due to the non-linear medium, except for the linear absorption of the crystal or spurious reflections at its ends.

14


2.2. BISTABILITY

One of the earliest proposals for generating squeezed states was to use the properties of a non-linear cavity in the vicinity of the bistability threshold (Lugiato and Strini [ 19821). In the limit where the x(3) non-linearity is purely parametric, the effect has been examined by Collett and Walls [ 19851 and Shelby, Levenson, Walls, Aspect and Milburn [ 19861. It is well known that a critical divergence of the fluctuations occurs near the bistability turning point. This divergence takes place in one quadrature of the field. For the quantum fluctuations the total area of the fluctuation distribution in phase space is conserved, which implies that the other quadrature of the field is squeezed. This can be seen from the form of the semiclassical input-output transformation for the field in a one-ended cavity (Reynaud, Fabre, Giacobino and Heidmann [ 19891). The input and output fields ain and aoutand the field inside the cavity tl satisfy the equations fain = (1 - r e - ' @ ) a , laout

=

(e-i@ -

9

(2.6a) (2.6b)

where r and t are the amplitude reflection and transmission coefficients of the coupling mirror and 9 is the total phase shift around the cavity, including the linear dephasing a0due to the roundtrip in the cavity and the phase shift due to the non-linear medium that has a Kerr coefficient K 9=@o+Kla)2.

(2.7)

The transformation ain+ sou' is a rotation around the origin by an intensitydependent angle. Its result on the Wigner probability distribution is a distortion from a disk shape to an elongated shape having the same area (fig. 2.2). As a result, one quadrature of the output field is squeezed, and this squeezing increases in the vicinity of the bistability turning point. Squeezed-state generation has been observed in a related system, made of two-level atoms placed in a high-finesse cavity. The non-linearity cannot be considered as purely parametric, since the atomic medium is driven close to atomic resonance, and additional effects come into play. A noise reduction of 30% has been observed by Raizen, Orozco, Min Xiao, Boyd and Kimble [ 19871. The squeezing of the vacuum fluctuations in the fluorescence of a collection of two-level atoms has been observed in an indirect way through the study of photon correlations in the same way as the Hanbury-Brown and Twiss effect

1, § 21


15

Fig. 2.2. Phase space representation of the evolution of the field in a non-linear optical cavity containing a x(3) medium. The incoming coherent field undergoes an intensity-dependent phase shift, which leads to a distortion of the probability distribution.

(Grander, Roger, Aspect, Heidmann and Reynaud [ 19861, Heidmann and Reynaud [ 19871).

2.3. SECOND-HARMONIC GENERATION

Second-harmonic generation (SHG) is the reverse process of degenerate parametric generation: the non-linear medium is excited at o,and a field is generated at o,,= 2 0 , . In contrast to parametric generation, SHG produces a field with a finite mean value. The second-harmonic field as well as the pump field at the fundamental frequency can be shown to be squeezed at the output of the non-linear medium. With the non-linear crystal inside a cavity resonant for both the pump and the second-harmonic modes, squeezing is predicted to be the same on the two modes (Lugiato, Strini and De Martini [ 19831, Collett and Walls [ 19851, Kennedy, Anderson and Walls [ 19891). Similarly to the other parametric processes, squeezing in SHG is larger in the vicinity of an instability threshold, which is connected here with the emergence of selfpulsing. Two experiments, both using MgO-doped LiNbO,, have been performed along those lines. The first one, by Pereira, Min Xiao, Kimble and Hall [ 19881, demonstrated 13% squeezing in the fundamental mode. The second one, by Sizmann, Horowicz, Wagner and Leuchs [ 19901, yielded 40% squeezing in the upconverted mode. In the latter case a monolithic cavity, in which the mirror

16


[I, § 2

coatings are deposited directly on the ends of the crystal, was used. The predicted amplitude squeezing is detected by separating the output beam into two parts with a beam splitter. Each of the beams is detected, and the two photocurrents are electronically recombined to form the sum or the difference of their fluctuations. In the sum operation the photon statistics of the original beam is reproduced, whereas the difference operation gives the standard quantum noise level. This technique allows a direct comparison of the intensity noise of the emitted beam with the shot noise level. Both the sum and difference fluctuations are recorded as the cavity is scanned through resonance. Amplitude squeezing shows up at exact cavity resonance for the two fields with the cavity.

2.4. NON-DEGENERATE PARAMETRIC GENERATION

The basic phenomenon in parametric generation in a crystal is the annihilation of one pump photon and the creation of two signal photons at the same time. This pair emission gives specific statistics to the beams emitted by the parametric generator. In the non-degenerate operation the two signal fields differ by either their frequencies or their polarizations and they can be separated. Then the pair emission is responsible for photon correlation and intensity correlations between the two beams (Reynaud, Fabre and Giacobino [ 19871, Reynaud [ 19871). Experiments on these twin beams have been performed in a large range of different conditions, e.g., with low-intensity continuous wave (CW) pump lasers or with high-power pump lasers. In the first case the crystal basically emits parametric fluorescence, and the very low intensity allows a detection by photon counting and analysis of the time correlations. The first experiments of this kind were performed by Burnham and Weinberg [ 19701. Friberg, Hong and Mandel [ 19841 showed that the twin photons were simultaneous to better than 100 ps. Later experiments concentrated on the degree of correlation attainable in such emission. A maximum correlation of 67 % has been obtained in such experiments by Rarity and Tapster [ 1990al. Other experiments use high power pulsed lasers (Abram, Raj, Oudar and Dolique [ 19871). In the case of coherent amplification, intense twin beams are emitted by the crystal, and direct photon counting is impossible. A frequency analysis of the noise in the difference of the intensities of the two beams reveals the correlations. Figures up to 75% reduction when compared with standard quantum noise have been obtained recently (Aytur and Kumar [ 19901).

1.8 21

TECHNIQUES FOR QUANTUM NOISE REDU(JT1ON

17

When the non-linear crystal is placed in an optical resonant cavity and is pumped with a sufficiently high intensity, the system oscillates like a laser. Although squeezing has been predicted in degenerate operation under these conditions, it has not yet been observed. In non-degenerate operation, however, the optical parametric oscillator (OPO) emits twin laser-like beams, with a high degree of quantum correlation. This is because the statistics of the beams leaving the cavity retain part of the properties of the photon pair emission, provided that the measurement is done in a time that is long compared with the storage time of the photons in the cavity. In other words, the noise in the difference between the intensities of the two signal beams is reduced below the standard quantum limit for the frequencies within the pass band of the cavity. To determine the quantum fluctuations theoretically in this case, the semiclassical method used for the OPO below threshold can be extended to the derivation of the preceding threshold case. The depletion of the pump field a. can no longer be neglected, which complicates the problem (Fabre, Giacobino, Heidmann and Reynaud [ 19891). The equations for the signal fields a, and a, are driven by the vacuum fluctuations ap and $L entering the cavity through the coupling mirror

- ya, - xa,*a,

+ tap,

(2.8a)

zit2 = - ya, - xa:a,

+tap,

(2.8b)

zit,

=

where x is the parametric coupling coefficient and the other notation is the same as in eq. (2.1). By linearizing the equations around a particular steady state, one can write the evolution of the fluctuations in the various components of the field. As before, we define the quadrature components qi = (ai + a:)/$ and pi = (ai- a , * ) / d i for i = 0, 1, 2, and we introduce the quantities 4-

= (41

P-

=

-4

(2.9a)

2 Y d

(P1 - P,)/J2

*

(2.9b)

The mean fields ai are assumed to be real. The equation for the fluctuations 6 4 - in the amplitude difference q - is written as (2.10) Taking the Fourier transform and using the input-output relations on the coupling mirror, one obtains (2.11)

18


[I, § 2

Since y = - xqo/& when the OPO operates above threshold it can be seen that the fluctuations in the difference between the amplitudes of the signal fields go to zero for zero-noise frequency. The same is true for the noise in the intensity difference, which is proportional to the noise in the amplitude difference. This does not depend on the noise of the pump beam, whose contribution cancels out in the difference, nor of the pump power; in contrast to quadrature squeezing below threshold, noise reduction in the twin beams takes place even far from the threshold. These characteristics make this property a very robust one and the only processes that may degrade the squeezing are the losses. Losses are not included in the preceding formula. Their effect can be understood by considering that each time a photon is lost inside the cavity, it degrades the correlation between the twin beams. As a result, the predicted squeezing at zero-noise frequency is no longer perfect. The remaining noise is equal to the ratio of the internal losses in the cavity to the total cavity damping coefficient. A thorough discussion of the quantum fluctuations in the OPO is given in 0 5. The experimental set-up used by Heidmann, Horowicz, Reynaud, Giacobino, Fabre and Camy [1987] comprises a non-linear KTP crystal pumped by an Ar laser and placed in an optical cavity (fig. 2.3). The cavity has a high finesse for the signal and idler fields, and a low finesse for the pump. When the pumping power is sufficiently large, the system operates above threshold and emits two laser-likebeams, which have orthogonal polarizations. They are separated at the output of the optical cavity by a polarizing beam splitter. +

Fig. 2.3. Experimental setup used by Heidmann, Horowicz, Reynaud, Giacobino, Fabre and Camy [I9871 to demonstrate the quantum correlation in the two beams going out of an optical parametric oscillator (OPO). The OPO is operated above the oscillation threshold and emits two beams with orthogonal polarizations, which are separated by a polarizing beam splitter. Their intensities are detected, and the noise in the difference of the photocurrents is analyzed. The half-wave plate is used to rotate the polarizations by 45". which transforms the twin beams into uncorrelated beams and yields the shot noise level.

1.8 21

19


The intensities of the two beams are detected by photodiodes, and the photocurrents are subtracted. The resulting signal reflects the fluctuations in the intensity difference. If the beams were perfectly correlated, there would be no noise in the intensity difference. If they are not correlated, the noise in the intensity difference is expected to be the same as the noise of a single beam separated into two parts by a beam splitter; it is the shot noise of a beam that has an intensity equal to the sum of the intensities of the two beams. To check for this property the signal corresponding to the difference between the currents of the two photodiodes is fed into a spectrum analyzer. The noise in the intensity difference is found to be squeezed in a frequency range of the order of the inverse of the cavity storage time (fig. 2.4). Noise reductions of up to 86 % have been observed under such conditions (Debuisschert, Reynaud, Heidmann, Giacobino and Fabre [ 19891, Mertz, Debuisschert, Heidmann, Fabre and Giacobino [ 19911). A noise reduction of 53% has been obtained in a monolithic OPO by Nabors and Shelby [ 19901. The level of the standard quantum noise was determined with a balanced detector by placing a half-wave plate before the polarizing beam splitter, which separated the twin beams, with its axes in such a position that the polarization of each beam is rotated by 45 O . Then the polarizing beam splitter acts like a 50-50 beam splitter for each beam. Since the beams have different wavelengths, they do not interfere, and everything acts as if each beam was alone: the noise detected in the difference of the intensities of the two channels is the

1- . . . . . , ,

I

.

,

. ., , .

. . . . Shot.noise... . . . . . , . ..

.. , .. . . ..

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

. . .. ..

..

.

.

.

01 0 FREQUENCY (MHZ) 2( Fig. 2.4. Noise spectrum in the difference of the intensities of the twin beams, as obtained from the experiment shown in fig. 2.3. The noise is well below the shot noise level on a frequency range of the order of the cavity bandwidth.

20


[I, § 2

Generation of twin photons has also been observed in a parametric oscillator based on four-wave mixing. The experiment performed in the phase conjugation geometry in sodium vapor by Vallet, Pinard and Grynberg [ 19891 has yielded 10% noise reduction. 2.5. LASERS WITH REDUCED PUMP NOISE

It is generally admitted that an ideal laser operating well above threshold generates a coherent state. This can be understood by observing that each elementary emission process is independent of the other ones and is thus subject to the standard photon noise. If one measures its intensity with a photodetector, the photodetection signal I exhibits a variance A12 = 2 e f B ,

(2.12)

where e is the elementary charge, f is the mean value of the current, and B is the bandwidth of the detection channel. On the other hand, the Johnson-Nyquist noise at temperature Tin the load resistor R of the photodetector is A12 = 4(kT/R)B.

(2.13)

The photon noise is dominant when the voltage drop through the load resistance is larger than V,, with Yo = 2kT/e.

(2.14)

( Vo = 52 mV at room temperature.) The fact that the noise in an electron flow can be smaller than the shot noise has been known for decades. This property can be used to reduce the photon noise in the emission. A device that directly and efficiently converts the electrons into photons should be limited only by the electron noise; it can emit a flow of photons that is more regular than a Poisson flow, if the electron flow itself is regular. Quantum noise reduction of this type has been demonstrated in the light emitted in a space-charge-limited Franck-Hertz experiment by Teich and Saleh [1985] and on a light-emitting diode (LED) by Tapster, Rarity and Satchel1 [1987]. The noise reduction is limited by the low efficiency of the electron-to-photon conversion. The same effect can be observed with a laser diode, which has a much better conversion efficiency. An intensity noise reduction of 13% has been measured by Machida and Yamamoto [ 19881. By cooling the laser diode to 4 K, Richardson and Shelby [ 19901 have obtained a 25% noise reduction, and with an improved set-up Richardson, Machida and Yamamoto [ 19911 have reached a noise reduction of about 8 5 % .

1.8 21

21


Other proposals have been made to reduce the noise in the output by reducing the pumping noise in different types of lasers (Golubev and Sokolov [ 19841, Smirnov and Troshin [ 19871, Haake, Tan and Walls [ 19891, Kennedy and Walls [ 19891).

2.6. REDUCTION OF QUANTUM FLUCTUATIONS USING SERVO TECHNIQUES

A common technique to reduce the intensity fluctuations of a laser consists of using an optoelectronic servoloop. A beam splitter reflects off a part of the beam, which is detected with a photodiode. The photocurrent is then processed in an electronic loop to provide an appropriate correction signal, which reacts on the initial intensity. Two schemes are possible; a feedback system, where the correction is applied before the beam splitter, or a feedforward system, where the correction is applied after the beam splitter. However, these schemes cannot be used for the correction of quantum fluctuations because the beam splitter couples the vacuum fluctuations to the beam through its unused port. Large degrees of amplitude squeezing have been obtained in “closed” configurations, where there is no beam splitter and all the beam is lost on detection to provide the correction signal (Machida and Yamamoto [ 19861). It is possible to extract squeezed light by using a twin-beam generator instead of a beam splitter (fig. 2.5). The intensity of one beam is measured by standard techniques and used in a feedback or feedforward configuration to correct the fluctuations of the other. The ultimate intensity noise reduction depends on the amount of initial correlation between the twin beams and on the intensity of the excess noise of each of the beams. Intensity noise reductions of about 20% EOM

twin beam generator

EOM

d

feedback

feedforward

Fig. 2.5. Twin-beam generators allow photon noise reduction in a single beam. The intensity fluctuations of one of the twin beams are measured by the photodiode and are used to monitor an electro-optic modulator (EOM), which corrects for the fluctuations of the other beam, by acting either on the pump beam (feedback) or on the outgoing beam itself (feedforward).

22


[I. § 3

have been observed experimentally with such a set-up, using twin photons emitted in parametric fluorescence (Tapster, Rarity and Satchel1 [ 19881) or intense twin beams generated by an optical parametric oscillator (Mertz, Heidmann, Fabre, Giacobino and Reynaud [ 19901).

8 3.

Applications of Squeezed Light

Since many measurements in optics are limited by quantum noise, it is expected that light squeezed in the appropriate component should improve the signal-to-noise ratio. From another viewpoint, one may want to measure a component of a light field without perturbing it (QND measurement). Studies along these lines have shown that this is possible, either using the properties of non-linear materials or squeezed light fields. Other applications of squeezed light include the inhibition of phase diffusion in lasers, interference with two photons, inhibition of relaxation phenomena, and the improvement of communication techniques.

3.1. IMPROVEMENT OF THE SENSITIVITY IN OPTICAL MEASUREMENTS

Research on photon noise has been greatly stimulated by the projects of interferometric detection of gravitational waves (Meystre and Scully [ 19831). Application of squeezed light to interferometry is still a major challenge. When the large interferometers for gravitational wave detection are in use, squeezed light should improve the signal-to-noise ratio and could facilitate the demonstration of gravitational waves, and later increase the number of detected events. Application of squeezed light to ultrahigh-sensitivity intensity measurements has also been proposed. 3.1.1. Interfrometric measurements

The principle that underlies the use of squeezed light in interferometers can be understood from the properties of the beam splitter discussed in Q 1.2.2 (Caves [ 198 11). The interferometer as a whole behaves like a beam splitter with reflection and transmission coefficients that depend on the operating point. Let us consider the case of a setting at half-maximum of a fringe; any change in the phase difference between the two arms of the interferometer is detected as a change in the difference of the intensities in the two output channels. The

APPLICATIONS OF SQUEEZED LIGHT

23

sensitivity of the phase difference measurement is limited by the standard quantum noise in the output. Reduction of this quantum noise is obtained by injecting a squeezed vacuum in the second input port of the interferometer. The experiment performed by Min Xiao, Ling-An Wu and Kimble [ 19871 employs squeezed light in a Mach-Zehnder interferometer for the detection of the phase modulation in the arms of the interferometer (see fig. 3.1). A coherent field is injected into one of the input ports of the first beam splitter, and the fields that have travelled the two paths of the interferometer are recombined in the second beam splitter. Electro-optic modulators are placed in the two arms of the Mach-Zehnder interferometer, producing a phase shift modulated at 1.6 MHz. Squeezed light with a significant quantum noise reduction at 1.6 MHz is injected in place of the vacuum field into the normally unused port of the first beam splitter. The intensities in the two output ports are detected and subtracted. The difference photocurrent is fed into a spectrum analyzer. The phase modulation signal appears on a background that is only one half of the shot noise level when the squeezed light has the correct phase compared with the strong coherent input field, thus improving the signal-to-noise ratio. In view of the very important problems raised by the detection of gravitational waves, extremely detailed studies have focused on the use of squeezed light in large interferometers (Gea-Banacloche and Leuchs [ 1987a1). In addition to photon counting noise, one must take into account radiation pressure noise, which stems from the random motion of the mirrors due to photon number fluctuations in each arm. Minimizing the sum of the two contributions leads coherent beam

I

t o spectrum

analyser

Fig. 3.1. A phase shift is produced in the arms of a Mach-Zehnder interferometer by electrooptic modulators (EOM).M1 and M2 are highly reflecting mirrors and BSl and BS2 are 50/50 beam splitters. The two beams going out of the interferometer are detected by photodiodes. Injecting intensity-squeezed light into the second input port of the interferometer reduces the background quantum noise and improves the signal-to-noise ratio.

24


[I, 0 3

to the so-called standard quantum limit (SQL), giving the ultimate detectable length variation as (WSQL

=

m.

With coherent light this limit is reached only for very large laser powers, which are not achievable at present. Caves [ 19811 suggested that the SQL could be attained for more reasonable laser powers by entering squeezed light in the second input port of the interferometer, as described earlier. This does not overcome the SQL, however, because the reduction in photon noise is compensated by the increase in radiation pressure noise. The meaning of such a limit has created much controversy (Yuen [ 19831, Caves [ 19851, Ozawa [ 19901). Actually, the photon counting noise and radiation pressure noise are not independent sources of fluctuations, as implicitly assumed in the derivation of the SQL. It is therefore possible to reduce the total noise beyond the SQL by squeezing the appropriate quadrature of the field entering the unused input port (Unruh [ 19831, Jaekel and Reynaud [ 19901). The combination of recycling and of squeezed fluctuations poses specific problems, which have been evaluated by Gea-Banacloche and Leuchs [ 1987bl. The aforementioned demonstration experiment has been performed at the half-maximum of a fringe, but the high-sensitivity interferometers, which use a significant amount of laser power, are preferably operated near a dark fringe. Grangier, Slusher and Laporta [ 19881have studied the enhancement in sensitivity beyond the shot noise limit that can be achieved in the vicinity of a dark fringe when squeezed light is injected into the interferometer. On the other hand, the problem of active interferometers has been treated theoretically by Scully and Gea-Banacloche [ 19861. It seems that the limits in sensitivity associated with the quantum noise are not more favorable than those in passive interferometers (Gea-Banacloche [ 1987a1). 3.1.2. Polarization measurements Polarization measurements are often similar to interferometric measurements. For example, the rotation of a linear polarization can be envisioned as a dephasing between its o+ and o- components. The experiment performed by Grangier, Slusher, Yurke and Laporta [ 19871 (fig. 3.2) uses a polarization interferometer analogous to the Mach-Zehnder interferometer. The first beam splitter is replaced by a polarization-dependent beam splitter. The coherent light and squeezed light have orthogonal polarizations and are directed at the two input ports, so that they copropagate in the interferometer. A half-wave


25

Fig. 3.2. A polarization interferometer is similar to a Mach-Zehnder interferometer in which the two beams travel along the same path but with u+ and u- polarizations. A polarization rotation (PR) due to a Faraday rotator corresponds to a phase shift between the u+ and u- waves. After an overall polarization rotation of 45" achieved by the half-wave plate, the polarization rotation is detected as an imbalance in the homodyne detector. Entering squeezed light into the second input port of the first polarizing beam splitter results in a lower background quantum noise as in fig. 3.1.

plate rotates the polarizations by 45 O and a second polarizer with axes parallel to the first one acts as a 50/50 beam splitter. The two output beams are detected with photodiodes, and the difference photocurrent is measured. When there is no polarization rotation before the second polarizer, and if no squeezed light is entered, the signal is just the shot noise. Any polarization rotation appears as a change in the difference photocurrent. When squeezed light is entered into the second input port, the signal appears on a background that is lower than the shot noise. The improvement demonstrated in the signal-to-noise ratio is of the same order as in the preceding experiment. 3.1.3. Intensity measurements

Squeezed light can also be used to increase the sensitivity in intensity measurements. An experiment by Min Xiao, Ling-An Wu and Kimble [ 19881 has shown an improvement in precision beyond the shot noise limit for the detection of amplitude modulation encoded on a weak signal beam. To perform this experiment, a squeezed field with non-zero mean value is constructed by combining a squeezed vacuum with coherent light on a beam splitter, in the manner shown in fig. 3.3. The field going out in the observed port is the superposition of the reflected part of the squeezed vacuum and of the transmitted part of the coherent field. If the transmissivity of the beam splitter is small, it can be shown, using eqs. (1. lo), that a squeezed field is generated. Then the field is amplitude modulated. The modulation signal appears on a

26

QUANTUM FLUCrUATIONS IN OPTICAL SYSTEMS

[I.

I3

Fig. 3.3. By combining a squeezed vacuum with a coherent beam on the beam splitter BS 1, which has a high reflectivity, one obtains squeezed light with a non-zero mean value. Intensity modulation with an acousto-optic modulator (AOM) can then be detected with a sub-shot-noise precision in the homodyne detector (which comprises a 50/50 beam splitter BS2).

background that is lower than the shot noise in a balanced homodyne detector. On the other hand, twin photons can also be used to increase the sensitivity in intensity measurements. Nabors and Shelby [ 19901 have demonstrated this effect by placing an amplitude modulator in one of the twin beams emitted by an optical parametric oscillator. They then measured the intensity difference between the twin beams. Because of the high quantum correlation of the beams, this technique allows the detection of sub-shot-noise modulation signals. A similar experiment has been performed by Rarity and Tapster [ 1990al to measure the turbidity in a liquid crystal cell. 3.1A. Symmetric heterodyne scattering As proposed by Snyder, Giacobino, Fabre, Heidmann and Ducloy [ 19901, twin beams can also be used to measure scattering or polarization changes due to transparent objects. In the former case, the correlated beams, which have slightly different frequencies, intersect at some angle in the medium under test (fig. 3.4). Some light is scattered from one beam into the other. After intersecting, the two beams are directed onto separate photodiodes, which mix each transmitted beam with the scattered light from the other one. Because the two beams are at different frequencies, the mixing generates a heterodyne signal, modulated at the beat frequency, on each ofthe photodetectors. The beat signals from the two detectors are 180" out of phase. If the two photocurrents are subtracted, the beat signal doubles whereas the common-mode signals cancel. This eliminates not only the classical noise but also the correlated quantum

1, B 31


21

Fig. 3.4. Proposed experimental set-up to detect small scattering effects with twin beams. The twin beams intersect in the medium under test, and some light is scattered from each beam into the other one. Each beam is sent onto a photodiode, which detects a signal at the beat frequency between the twin beams. By subtracting the signals of the two photodiodes, the beat signal appears on a background that is lower than the shot noise if the beat frequency is within the squeezing bandwidth.

fluctuations, provided the beat signal is observed within the frequency range of squeezing. Alternatively, if twin beams with orthogonal polarizations copropagate through a slightly birefringent or polarization rotating medium, a small fraction of each beam is transformed into the polarization of the other beam. They are then mixed by detection after a polarizing beam splitter, the axes of which are parallel to the original polarizations. In the same way as before, the beat signal gives a measure of the perturbation introduced by the medium, which can be detected below the shot-noise level. 3.2. QUANTUM NON-DEMOLITION (QND) MEASUREMENTS

A QND measurement is a measurement that does not modify the measured quantity, neither during the measurement nor after it (Caves, Thorne, Drever, Sandberg and Zimmerman [ 19801). In other words, there is no action of the measurement process back on the measured quantity (Yurke [ 1985a1). This is particularly important in the case of repeated observations (Caves [ 1987a1). Interest in this problem has been stimulated by the existence of quantum limits in position measurements, e.g., of the extremity of a Weber bar, in the case of the mechanical detection of gravitational waves (Braginski and Vorontsov [ 19741). 3.2.1. QND measurement using optical Kerr efect Of course, commonly used optical measurements are not of QND type because a photodetector annihilates the photons monitored. Using non-linear optical effects, one can perform QND measurements on optical fields. For example, a Kerr medium provides a QND measurement for the intensity: when a light beam passes through a Kerr medium, its phase is changed but not its

28


[I, § 3

intensity, if the absorption is negligible. The resulting optical index modification can then be read out by a secondary beam using interferometric techniques (Yamamoto, Imoto and Machida [ 19861). QND experiments have been performed that clearly demonstrate the effect, using optical fibers (Levenson, Shelby, Reid and Walls [1986]), parametric crystals (Laporta, Slusher and Yurke [ 1989]), or two-photon transitions in atomic systems (Grangier, Roch and Roger [ 1991]), although the detection efficiency remains limited for the time being. The efficiency of the measurement can be improved by inserting the Kerr medium in an optical cavity. The resulting device has a bistable behavior, and the efficiency is increased when one gets closer to the bistability threshold (Alsing, Milburn and Walls [ 19881, Grangier, Roch and Reynaud [ 19891). 3.2.2. QND measurements using squeezed fields It is also possible to make a QND measurement by using a field that has squeezed fluctuations (Shapiro [ 19801, Caves [ 19891). The usual way to measure a given light beam A is to use a beam splitter with reflection coefficient r. The intensity of the reflected beam D (called the “read-out field”) is then measured. According to eqs. (1. lo), if r is a small quantity, the mean value and amplitude fluctuations of the transmitted beam C are very close to those of the incoming field A (eqs. (1.13)). Furthermore, the mean value of the D-field amplitude is a measurement of the A-field amplitude. The D-field amplitude fluctuations do not reflect the amplitude fluctuations of the field being measured, however, but instead, the vacuum fluctuations coming from the port B AEkl = r 2 A E i l

+ t 2 AE;,

z AE;,

The knowledge of D-field fluctuations does not provide information about A-field fluctuations, just as a photodetector of poor quantum efficiency does not measure the quantum fluctuations of the incident beam. This conclusion is no longer true when one injects a squeezed vacuum through port B. In the case of perfect squeezing for field B ( A E i 1 = 0), the fluctuations measured on beam D are the perfect replica of the incident beam A fluctuations, and the transmitted beam is less perturbed than previously. Of course, phase fluctuations of beam A have been much degraded in the measurement process. In any kind of QND measurement, the photocurrent resulting from the intensity measurement of the read-out field can be used in a correcting feedback or feedforward device to reduce the intensity fluctuations of the incident field

1. t 31


29

below the shot noise level (Yamamoto, Machida, Imoto, Kitagawa and Bjork [ 19871, Caves [ 1987b]), as has been done using twin beams (8 2.6).

3.3. INHIBITION OF PHASE DIFFUSION

Frequency measurements of laser beams are ultimately limited by the spectral linewidth associated with the phenomenon of phase diffusion (the Schawlow-Townes linewidth). In particular, the accuracy of the beat-note frequency measurement between two laser beams, e.g., in gyrolasers (Chow, Gea-Banacloche, Pedrotti, Sanders, Schleich and Scully [ 1985]), is limited by the diffusion on the difference between the two phases. Phase diffusion in a laser is partly due to the vacuum fluctuations entering the laser cavity through the coupling mirror. It is therefore possible to reduce it by entering squeezed vacuum fluctuations in the laser cavity (Gea-Banacloche [ 1987bI). One can also design oscillators in which the process of phase diffusion does not occur, especially in correlated-emission lasers (Schleich and Scully [ 1988]), to inhibit the diffusion on the phase difference between the two fields emitted by a two-mode laser. This possibility has recently been demonstrated experimentally (Winters, Hall and Toschek [ 19901). In OPOs the phase difference between the twin beams diffuses freely, but not the sum, which is clamped to the pump field phase (Drummond and Reid [ 19881, Reid and Drummond [ 19891, Courtois, Smith, Fabre and Reynaud [ 19911). Propositions exist, based on parametric effects, to make gyrolasers that are no longer limited by the phase diffusion (Yamamoto, Machida, Saito, Imoto, Yanagawa, Kitagawa and Bjork [ 19901). Such propositions must be related to others, which tend to eliminate the blind region of laser gyros by using an optical bias by four-wave mixing techniques (Grynberg and Pinard [ 19881).

3.4. INTERFERENCE EFFECTS WITH TWIN PHOTONS

Generally speaking, interference phenomena appear in situations where the light can take different paths: in quantum optics the quantum amplitudes corresponding to these different paths must be added. Twin photons give rise to new types of two-photon interference effects, where the quantum amplitudes are concerned with the pairs of photons and not with each individual photon as is usual.

30


[I. § 3

Two-photon interference fringes have been observed by Gosh and Mandel [ 19881 and Ou and Mandel [ 1988bl. In these experiments the spatial interference pattern is not observed in the light intensity, which is independent of the detector position, but rather, in the coincidence rate between two different detectors. twin-photon coincidence counting is observed preferentially when the detectors are separated by a distance that is a multiple of the fringe interval, whereas the coincidence rate is independent of the distance when using uncorrelated photons. Pairs of photons produced by parametric splitting have a temporal coherence of less than 100 fs. A time delay between the twins of this order of magnitude is sufficient to wash out the interference pattern. As a result, the use of continuous wave beams of twin photons allows one to perform experiments with the same temporal resolution as with femtosecond pulses of individual photons (Hong, Ou and Mandel [ 19871). A single photon impinging on a semireflecting plate has a 50% chance to be reflected and a 50% chance to be transmitted. If two uncorrelated photons arrive on the plate, one expects a 25% chance of two photons transmitted, a 25% chance of two photons reflected, and a 50% chance of one reflected and one transmitted. If the incident photons are twins, on the other hand, because of the perfect destructive interference between the quantum amplitudes corresponding to the final state consisting of one photon transmitted and one reflected, the two photons are necessarily transmitted or reflected together. This phenomenon has been observed by Ou and Mandel [ 1988al and Rarity and Tapster [ 19881. Twin photons provide quantum-correlated and well-separated objects that are suitable for testing non-locality in quantum mechanics (Reid and Walls [ 19861, Grangier, Potasek and Yurke [ 19881, Ou, Wong and Mandel [ 19891). Violation of the Bell inequality has been observed experimentally in such systems (Ou and Mandel [ 1988a, 19901, Shih and Alley [ 19881, Rarity and Tapster [ 1990bl).

3.5. INHIBITION OF RELAXATION PHENOMENA

Relaxation phenomena arise from the existence of a coupling between the system considered and a reservoir. They therefore depend on the exact state of the reservoir. If the fluctuations of a given component of the reservoir are squeezed, the relaxation of some quantities of the system may be inhibited or even completely suppressed.

1 . 8 31


31

3.5.1. Inhibition of spontaneous emission

The relaxation of a two-level atom is affected by the presence of a squeezed vacuum. One component of the atomic dipole is less relaxed than the other (Gardiner [ 1986]), which modifies the emission and absorption spectra of the atom (Carmichael, Lane and Walls [ 19871, An, Sargent and Walls [ 19881). In the case of atomic excitation by an intense monochromatic field, one obtains a spectacular effect of stabilization of one level of the dressed atom (Courty and Reynaud [ 19891). Such phenomena, of course, are closely related to cavity electrodynamics effects (Jhe, Anderson, Hinds, Meschede, Moi and Haroche [ 19871, Heinzen, Childs, Thomas and Feld [ 19881, Yifu Zu, Lezama, Mossberg and Lewenstein [ 1988]), where the atom is decoupled from the field modes responsible for relaxation by a modification of the mode density distribution. In the present case the same effect is obtained by transferring the fluctuations from the quadrature component responsible for the relaxation to the other component. 3.5.2. Inhibition of tunnelling Squeezed fluctuations are also efficient in reducing the tunnelling rate from a metastable state (Savage and Walls [ 19861). This effect could be used to enhance the sensitivity of detection systems using tunnelling effects, especially in the microwave domain. Here again, this technique and the method used for modifying the spectrum of vacuum fluctuations are complementary (Esteve, Devoret and Martinis [ 19861).

3.6. OPTICAL COMMUNICATION TECHNIQUES

The highest performance optical communication systems to date are close to the limits set by photon noise. Numerous proposals have recently been made to go beyond this limit, and four of these will be discussed. Optical losses always have the effect of destroying the squeezing effect. A general rule, therefore, is that it is useless to put information on a channel with squeezed fluctuations if the channel transmission losses are very large. Future applications of squeezing to communications will more probably involve highflow and short-distance systems, which may have very low losses. The security and error rate of the channel may be also improved (Q 3.6.4).

32


3.6.1. Amplifiation and noise

The first example deals with the amplification of optical signals. Just as absorption losses inevitably introduce noise, it is not possible to amplify an optical signal without adding noise to it. This noise can be transferred from one quadrature component to the other (Takahashi [ 19651). With the help of a parametric amplifier, it is also possible to amplify a given quadrature component with no increase of noise (Caves [ 19821). 3.6.2. QND measurements on an optical channel

In an optical communication network it may be interesting to tap the information from the channel. With standard techniques this perturbs the channel by adding noise to it. Using the QND measurement techniques of 0 3.1, it is possible to perform such optical tapping without any perturbation of either the phase or the intensity of the light, according to the information encoding technique (Shapiro [ 19801). 3.6.3. Decrease of energy cost per bit in an optical channel

The minimum photon number carrying a bit of information is obviously a function of the amount of noise present in the channel and of the desired transmission reliability. To compare different systems one usually takes a bit error rate of even if this value is obviously too large in many applications. For “ordinary” light one then obtains a minimum number of 10.5 photons per bit for amplitude-coded systems, and of 9 photons per bit for phase-coded systems using homodyne or heterodyne mixing techniques. Note that the state-of-the-art systems are not far from this quantum limit (Yamamoto, Machida, Saito, Imoto, Yanagawa, Kitagawa and Bjork [ 19901). If the channel noise is entirely due to optical quantum noise, the use of squeezed states can, in principle, lower the ultimate limit down to approximately 0.3 photons per bit (Yamamoto and Haus [ 19861). 3.6.4. Communication systems using twin photons

Twin-photon pairs can also be used to encode information. The idea is to modulate parallel communication channels made of twin photons and to detect the photon coincidences (Hong, Friberg and Mandel [ 19851, Jakeman and

THEORY OF QUANTUM NOISE IN OPTICS

33

Rarity [ 1986I). The rate of accidental coincidences associated with uncorrelated photons can be made negligible, even in the presence of an important background. Another advantage is that the knowledge of only one channel is insufficient to obtain the information. Such a system thus opens the way to highly secured channels.

8 4. Theory of Quantum Noise in Optics In previous sections we have used classical representations of the field to understand the main properties of quantum fluctuations and photon noise. However, these representations need to be justified from the quantum theory of light. We shall first describe the quantum field fluctuations and derive from the quantum theory of photodetection some statistical measurements, such as the photon noise spectrum (f 4.1). We shall then study ideal squeezed states and describe their fundamental properties (f 4.2). Finally, we shall show that squeezed states can be generated in non-linear processes (f 4.3).

4.1. QUANTUM FIELD FLUCTUATIONS

This section will discuss some aspects of the quantum theory of photodetection (Glauber [ 19651, Loudon [ 19831, Cohen-Tannoudji, Dupont-Roc and Grynberg [ 19871, Cohen-Tannoudji, Dupont-Roc and Grynberg [ 19881). By first examining the case of a perfect detector (quantum efficiency equal to one, instantaneous response, no dark current), we show that photodetection can be considered as a normal quantum measurement, where the observable is the incident photon number. We shall determine the relation between the photon noise and the noise spectrum of the field amplitude. We shall also examine heterodyne measurements where the noise spectrum of any field quadrature can be measured. 4.1.1. The quantum field In the quantum theory of light a monomode field (one direction of propagation and one polarization) is described by an electric field operator E(t), which can be written as

E(t) = E + ( t )+ E - ( t ) ,

(4.1)

34


[I, § 4

where E + and E - are, respectively, the positive- and negative-frequency parts of the electric field. They can be decomposed into Fourier components (4.2a) (4.2b) Each Fourier component corresponds to a frequency mode of the field, associated with the annihilation and creation operators a, and aL, respectively;

B + ( w )= K(w)a,,

(4.3a)

E ( w ) = K(u)u! ,

(4.3b)

where K(w) is a scaling factor proportional to following commutation relations

6. These operators obey the

[a;, UL?] = 0 ,

(4.4a)

[a,, & ] = 271 b ( o - 0 ’ ) .

(4.4b)

[a,, a,.]

=

The operator E(t) contains all the information about the statistical properties of the field. The quantum fluctuations of the electric field are described by the operator 6E(t) = E ( t ) - ( E ( t ) )

9

(4.5)

which has a mean value of zero and non-vanishing correlations functions. For example, in the vacuum state the Fourier transform operator S&o) satisfies the relation (SE(O)SE(W’))

=

( S E + ( W ) S E - ( O ’ ) )= ~ K K ( w ) ’ ~ (+ww ’ ) . (4.6)

4.1.2. Intensity of thefield In the usual quantum theory of photodetection (Glauber [ 1965]), the number m of detections between the times t and t + T is proportional to the energy received by the detector during the delay time T: Zcc

stitldl’ ( Z ( t ’ ) )

(4.7a)

with

I(t) = E-(t)E+(t).

(4.7b)

o

1, 41

35


(The electric field E(t) is normalized so that Z(t) is the instantaneous energy of the field.) A photodetector is sensitive to the number of incident photons, however, rather than to the incident energy (Cook [ 19821, Shapiro [ 19851).The simplest method of obtaining a more realistic model is to modify the normalization of the field operator E(t). Taking K ( w ) = 1 in eqs. (4.3), one obtains E + ( t )=

j2

(4.8a)

awe-'"',

(4.8b)

It follows that the operator M

=

jtifrdtfZ(t')

(4.9)

is the quantum observable associated with the number of photons detected during a time interval T (Shapiro [ 19851, Reynaud [ 19901). In particular, all the statistical properties of the photodetection can be deduced from the various moments of M. The operator Z(t) thus describes the instantaneous intensity of the field, normalized as a photon number. Note that Z(t) is defined as a normally ordered product of operators E - and E' ( E - on the left, E' on the right). This is necessary to preserve the important property that no photon can be detected in the vacuum state.

4.1.3. Photon noise spectrum The photon noise spectrum (Papoulis [ 19811, Gardiner [ 19901) constitutes the best characterization of the statistical properties of the field intensity. Experimentally, it corresponds to the signal at the output of a spectrum analyzer when the instantaneous field intensity is sent to the input. We first define the autocorrelation function C,(t, t ' ) of the intensity C,(t, t ' ) = ( Z ( t ) Z ( t ' ) )- ( Z ( t ) ) ( Z ( t ' ) )

=

( s Z ( t )6 Z ( t ' ) ) ,

(4.10a)

where s q t ) = Z(t) - ( Z ( t ) ) .

For a stationary field C,(t, t ' ) depends only on the delay T

(4. lob) =

t - t' . The photon

36

[I, I 4


noise spectrum S,(o) is then defined as the Fourier transform of CAT)

S,(0) =

s

d z ein7C,(z) ,

(4.1 1a) (4.1 lb)

S , ( 0 ) can also be defined from the Fourier transform sf(62) of the intensity fluctuations sZ(t). Noting that (sf(0)s f ( 0 ’ ) )is the Fourier transform of C,(t, t ‘ ) , one obtains

(si(6-2)s f ( 0 ’ ) )= 272 s(0 + 0’)S , ( 0 ) .

(4.12)

The function s(0+ 0‘)is a consequence of the stationary nature of the field. Finally, the variance AI’ of the intensity can be derived from the noise spectrum, AI’

=

s :: -

~(a)s,(a),

(4.13)

where F ( 0 ) is the function associated with the detection filter used in the measurement (Papoulis [ 19813). 4.1.4. Standard shot noise The photon noise spectrum can be related to other statistical functions of the field, such as the photon correlation function C(t, t ’ ) , which is the probability of detecting one photon at time t and another photon at time c‘. In the quantum theory of photodetection, C(t, t ‘ ) is given by q t , t ’ ) = ( E - ( t ) E - ( t ’ ) E + ( t ’ ) E + ( t ) ).

(4.14)

From the commutation relations of the field operator [eqs. (4.4) and (4.8)], C(t, t ’ ) is related to the autocorrelation function C,(t, t ‘ ) of the intensity C,(t, t ’ ) = ( I ) s(t - t ’ ) + C(t, t ’ ) - ( I ) ’ .

(4.15)

In the particular case of Poissonian photon statistics, photons are randomly distributed in the light beam and there are no temporal correlations between them. C(t, t ’ ) is thus independent of the delay t‘ - t and is equal to ( I ) * . In this case the autocorrelation function and the noise spectrum are given by C1(t,t ’ ) = ( I ) s(t - t ’ ) ,

(4.16a)

s,(m = ( 1 ) .

(4.16b)

1. B 41

THEORY OF QUANTUM NOISE I N OPTICS

31

This is the expression for the standard shot noise, which is independent of frequency and proportional to the mean intensity (Papoulis [ 19811, Gardiner [ 19901). The variance A12 is then given by the usual expression A12 = 2 B ( I ) ,

(4.17a)

where 2B is the bandwidth of the detection filter 2B =

1

! F(0).

(4.17b)

211

In the general case of non-Poissonian photon statistics, it is often convenient to rewrite the noise spectrum as

sAQ) =

( 1 ) [1 + Q(Q)1

3

(4.18)

where Q(0) is the Mandel factor that describes the temporal correlations between photons (Mandel [ 19791, Teich and Saleh [ 19881). This factor gives the increase (Q > 0 for super-Poissonian statistics) or decrease (Q < 0 for sub-Poissonian statistics) of the photon noise compared with the standard shot noise. 4.1.5. Limit of smdljuctuations

In many experiments the mean field is very large compared with the field fluctuations. In this case a linear treatment of the fluctuations can be carried out. The field operator E(t) is decomposed into two parts. The first is the mean field (E(t)) ,which will be assumed to be monochromatic (frequency coo)and with a real amplitude Eo, ( E ( t ) ) = E, eciwn'+ Eo eiwor.

(4.19)

The second part is the quantum fluctuations 6E(t),which are distributed over all frequencies cc). The intensity is then developed linearly with respect to the quantum field fluctuations. One obtains the mean intensity (Z(t)> = P o l 2

(4.20a)

and the intensity fluctuations M(t) =

Eo 6 E , ( t )

(4.20b)

with 6E,(t) = eciW0'6E-(t) + eiwa'6E+(t).

(4.20c)

38


This is the same result as the semiclassical one obtained in the introduction [eq. (1.8a)l. The intensity fluctuations are proportional to the fluctuations of the field amplitude, i.e., the quadrature component in phase with the mean field. They are distributed over all the frequency modes (4.21a) with

s E , ( n ) = s B - ( w , + R ) + 6 E + ( - w 0 + 0).

(4.2 1b)

From eqs. (4.12) and (4.20b) the photon noise spectrum is written as

s,(m= I ~ o 1 2 s E l (=~ () 1 ) sEl(Q) 7

(4.22)

where SEl(62) is the fluctuations spectrum for the amplitude component E , , ( sBl(n)6Bl(Q')) = 2 x 6(R

+ a')S,,(R) .

(4.23)

At each frequency the photon noise reproduces the fluctuations of the amplitude component. More precisely, the noise frequency Rfor the intensity is equal to the difference between the frequency of the field fluctuations and the mean field frequency. The intensity measurement can actually be considered as a heterodyne measurement of the amplitude component fluctuations by the mean field. 4.1.6. Efect of a reflecting plate We consider now the effect of a reflecting plate on the statistical properties of the field. As in 8 1, the two input ports are labelled A and B, and the two output ports C and D (see fig. 1.2). The quantum field operators obey the following relations : E,

=

tE,

+ rE,,

ED = -rEA

+ tE, .

(4.24)

For classical fields these equations are derived from the transmission and reflection relations on the plate, which mix the two input fields linearly. In the quantum theory of light the transformation must also be linear and must preserve the commutation relations of the electric field operators. The change in sign for the reflection coefficient appears as a consequence of the unitary character of the transformation.


39

The intensity operators are then related by

I,

=

ID =

t21A + r21B + rtJA,

+ rtJ,,

,

(4.25a)

r21A + t21B - rtJA, - r t J B A ,

(4.25b)

with (4.25~) JAB = EA EB+ , J,, = E , EL . All the field properties at the output of the reflecting plate can be deduced from these equations. For example, if the field in the input port B is in the vacuum state, the intensity operator I , can be ignored, since for any operator X one obtains (IBX)

=

(XI,)

=

0.

(4.26)

The presence of vacuum fluctuations in the input port B, however, may change the output statistics, since the operators JAB and JBAhave a mean value of zero but non-vanishing correlations (J A B ( t ) JBA(t’)) = ( Ei( t ) EB’ (lE; ) (t’)E,’ ( t ‘ ) = (IA(t))

s(t - t ’ ) .

(4.27)

We have shown that in limit of small fluctuations, the photon noise spectrum is related to the fluctuations of the amplitude component of the field [ eq. (4.22)]. Assuming that the mean field at port B is small compared with the mean field at port A and that the two input fluctuations at ports A and B are independent, one obtains the noise spectrum of the output amplitude components as sC,(fl)

=

t 2 s A l ( f l ) -k r2sBl(a) 9

sDl(fl) = r2sA,(G?)+ t2sBl(fl).

(4.28a) (4.28b)

The reflecting plate thus mixes the fluctuations of the two input ports and redistributes them into the two output ports. In particular, when the port B is in the vacuum state, fluctuations of the two output fields E, and ED are closer to the vacuum fluctuations than those of the input field E A . 4.1 .I. Heterodyne measurements We consider now a semireflectingplate (r = t = 1 / d ) and the measurement of the difference between the two output intensities

I

=

Ic - I D

=

JAB

+ JBA.

(4.29)

40


[I, 8 4

The field at port A is assumed to be an intense local oscillator, of frequency o, and of real amplitude at E,, whereas the mean field at port B is small compared with E,. Taking into account only terms proportional to E,, one obtains I

=

EAEB,

(4.30a)

9

where EBIis the component of the field E, in phase with the mean field E , EBl(t)= e-'""'Ei(t)

+ e'""'E,+(t).

(4.30b)

One obtains results similar to those in the case of direct intensity measurements for intense fields [eqs. (4.20)]. The operator Z corresponds to a heterodyne measurement of the field at port B by the mean field at port A (Yuen and Chan [ 19831, Schumaker [ 19841, Shapiro and Wagner [ 19841, Heidmann, Reynaud and Cohen-Tannoudji [ 19841, Yurke [ 1985b], Collett, Loudon and Gardiner [1987]). The photon noise spectrum S , ( 0 ) then reproduces the of the field component E,, in phase with the local oscilspectrum S,, (0) lator

S,(W = J%s,,

,(a>.

(4.3 1)

4.1.8. Imperfect photodetectors Up to now we have considered perfect detectors. However, a photodetector with a quantum efficiency T can readily be modeled as a reflecting plate with an intensity transmission T (amplitude transmission t = JT) followed by a perfect detector. In this case the measured statistics are no longer equal to the incident photon statistics. From eqs. (4.25) one can show that the measured signals are given by (I)(T)=

T(I) ,

S,(0)'T'= T 2 S I ( 0 )+ (1 - T ) T ( I ) .

(4.32a) (4.32b)

The measured Mandel factor Q ( T )is related to the Mandel factor Q of the incident field [ eq. (4.18)] by Q(0)(')

=

TQ(0).

(4.32~)

The super-Poissonian or sub-Poissonian character of the statistics is always reduced by an imperfect quantum efficiency. The measured noise is closer to the standard shot noise than the real noise (Mandel [ 19791, Reynaud [ 19831, Teich and Saleh [1988]).


1,s 41

41

4.2. IDEAL SQUEEZED STATES

In this section we shall define the ideal squeezed states, using the simplest model of a single field mode, characterized by one direction of propagation, one polarization, and one frequency. We shall first define these states as the minimum states associated with a generalized Heisenberg inequality (Dodonov, Kurmyshev and Man’ko [ 19801, Levy-Leblond [ 19861, Schumaker [ 19861, Luks, Perinova and Perina [ 19881, Simon, Sudarshan and Mukunda [ 19881). We shall then show that this definition is equivalent to that of the two-photon coherent states (Yuen [ 19761). Finally, we shall give a geometric interpretation of squeezed states, based on the Wigner representation of the field. The generalization of these properties to multimode fields does not present any difficulties (Milburn [ 19841, Caves and Schumaker [ 19851, Ekert and Knight [ 19891). 4.2.1. The generalized Heisenberg inequality A single mode of the field is described by the annihilation and creation operators a and at, or by the two quadrature components operators Q and P, which are the analogs of the position and momentum operators of a harmonic oscillator,

Q = (a + a f ) / J Z ,

P = i(at - a)/*.

(4.33)

These operators obey the commutation relations [a,at]= 1 ,

[Q,P]=i.

(4.34)

The field fluctuations can be characterized by the two variances a, /?of Q and P, respectively, and by the symmetrically ordered covariance y between the two operators a = (SQSQ),

(4.35a)

p=

(4.35a)

(6P6P),

Y=;(~P~Q+~Q~P),

(4.35c)

zQ=Q-

(4.35d)

with (Q),

6P=P- (P).

To derive the Heisenberg inequality, we consider the quantity V(A.,p)

42


[I, § 4

(A and p real): V(A, p )

=

(4.36a)

(6At 6 A ) ,

with 6 A = 6 P - ( A tip)BQ.

(4.36b)

V(A,p ) is related to the variances of the field V(A, p )

=

B + (A’

t p2)a - 2 A y - p

,

(4.37)

and is positive for all A and p. This condition leads to the classical statistical inequality ~$3 > y’, when V(A,p ) is considered only as a function of A ( p = 0), whereas one obtains the usual Heisenberg inequality ap > $, when V(A, p ) is considered as a function of p (A = 0). A more general inequality, including the two previous ones, can be obtained when V(A,p ) is minimized both with respect to A and p. The minimum of V(A, p ) is reached for the particular values Amin and pmin: Amin

= Yla 9

1/2a

pmin =

(4.38a)

9

and is equal to V(Amin, pmin) =

B - y2/a - 1/4a

*

(4.38b)

The generalized Heisenberg inequality is then up> y 2 t

a.

(4.39)

4.2.2. Minimum states Ideal squeezed states are defined as the minimum states associated with the generalized Heisenberg inequality [ eq. 4.39)]. They obey the equality a/?= y 2

+:.

(4.40)

All the properties of the squeezed states can be deduced from this definition. In particular, one can show that they are eigenstates of some linear combination of the operators Q and P, and are then identical to the so-called two-photon coherent states (Yuen [ 19761, Schumaker [ 19861). In fact, from eqs. (4.38) the minimum value of V(A,p ) is equal to zero

(6ALin &Amin) = 0

3

(4.41)

where Amin is the operator associated with the parameters Amin and pmin.A

1,s 41

43


minimum state I $) is thus an eigenstate of the operator Amin: (4.42a) (4.42b) Inversely, let us consider an eigenstate I $) of an operator that is a linear combination of P and Q. By a proper choice of multiplicative constant, the operator can be written as

A

=

P

-

+ ip,)Q .

(A,

(4.43)

The minimum value of V(A,p ) is then equal to zero and is reached at A,, pa. From eqs. (4.37) and (4.38a) the variances of the field are

Y = A,/2po

a = Vpo3

>

B = (A', + p3/2po

9

(4.44)

and satisfy the Heisenberg equality [eq. (4.40)]. The state I $) is thus a minimum state. 4.2.3. Generalized coherent states Ideal squeezed states can also be considered as a generalization of the usual coherent states. These states are eigenvectors of the annihilation operator a (Glauber [ 19651). They are particular minimum states, with A = 0 and p = 1, so that the two quadratures have equal variances and are decorrelated, a=p=+,

(4.45)

y=o.

,

We define the operators A as linear combinations of a and a t :

A,

=

a cosh

r - at sinh 5 e2ip

(4.46a)

with

5 = (e2iq.

(4.46b)

The operators A , and A ! have the same commutation relations as a and a":

[A,, A:]

=

cosh2( - sinh2{ = 1 .

(4.47)

We then define the generalized coherent states as the eigenvectors of A , :

A,

IZ,

5)

= z Iz,

5) .

(4.48)

These states are characterized by the two complex numbers z and 5. The usual coherent states correspond to ( = 0. The set of states for a given value of 5 can

44

[I, I 4


be considered as a new set of coherent states, with some similar properties. For example, it forms an overcomplete basis of the field space: the identity operator can be written as (4.49) Other properties concerning the semiclassical representations of the field are described in the appendix. Since A , is a linear combination of the operators Q and P,the generalized coherent states are another definition of the ideal squeezed states. The variances and covariances of the field are related to the parameters 1 and p [eqs. (4.43) and (4.44)] and thus to 5. One finds i(cosh25 + sinh2(cos2rp),

(4.50a)

/I= i(cosh25 - sinh2(cos2rp),

(4.50b)

y = i(sinh25 sin2q).

(4.50~)

a

=

The usual coherent states can be obtained from the vacuum state using a displacement operator. The same property is valid for generalized coherent states, and the state I z, 5 ) is found to be related to the squeezed vacuum 10, 5 ) 12,

5 ) = D,b)

10, 5 )

(4.5 1a)

exp[zA! - z*A,] .

(4.5 1b)

with D,(z)

=

Generalized coherent states can also be related to the usual coherent states. The operators A , and A: are, in fact, obtained from a and at through a unitary transformation

A,

=

U,aU!,

A!

=

Ulat Ul

(4.52a)

with U , = exp[$((a+’ - ( * a ’ ) ] .

(4.52b)

The eigenstates of A , are then related to that of the operator a through the transformation U,: Iz, 5

)

=

u, I Z 9 0 )

(4.53)

*

Finally, from eqs. (4.51a) and (4.53) any ideal squeezed state / z , 5

)

can be

1,s 41

45

THEORY OF QUANTUM NOISE IN OF'TICS

related to the vacuum state 10,O) by the unitary transformation

4.2.4. Wigner representation of minimum states In this section we shall use the Wigner distribution to represent the ideal squeezed states in the phase space {q, p}, which will give a simple geometrical interpretation of the various parameters characterizing the state. We first determine the analytical expression for the Wigner distribution for a minimum state, using some properties of this distribution that are described in more detail in the appendix. As shown in previous sections, a minimum state I $) is an eigenstate of a linear combination of the operators Q and P.The corresponding equation can be written in the { q } representation as

aq ( 4 I $> = [iPo - ( P - i 4 ((1 - 4011 ( 4 I J/>

(4.55)

with PO=

(P>

and q o =

.

Thus ( q 1 $) has the shape of a Gaussian wave packet, and the function flq,

=

).4

( 4 + ;u I $> ( $1 4 -

(4.56)

is a Gaussian function. Since the Wigner distribution W ( p , q ) is obtained from F(q, u) by Fourier transformation (see Appendix), W ( p ,q ) is also a Gaussian function, centered at p = po and q = qo. The exact expression for W ( p , q ) can be deduced from the values of the variances a, 8, and the symmetricallyordered covariance y of the operators Q and P [eqs. (4.35)].One obtains W(P,4) =

= exp { - 2 [ a ( P - Po)2 i- 8 ( 4 - 4 O l 2 i-

2Y(q - 40) (P - Po)l} .

(4.57)

The Wigner distribution of a minimum state is always positive, and can be considered in this case as a classical probability distribution. This will permit the representation of the minimum state in the phase space { q , p } . More precisely, the minimum state is described by an isoprobability curve of the Wigner distribution. The points {q, p} of this curve are such that W ( p ,q ) is equal to K e - '. With this definition the proportion of the distribution outside the curve is equal to e - ',whereas the area enclosed inside the curve represents 1 - e - ' of the total probability. From the analytical expression for the Wigner

46


(178 4

distribution [eq. (4.57)], one finds the equation of the curve, which can be written in polar coordinates centered at the mean value point {qo,p o } : p’[(a

+ 8) - ( a - 8)cos(2+) - 2ysin(2+)] = 1

(4.58a)

with 9 = go + pcos+ and p

= po

+ psin+.

(4.58b)

This is the equation of an ellipse, centered at the point { g o , p o } .The lengths pmin and pmaxof the minor and major axes and the angle dmaX of the major axis can be deduced from this equation. Using the relation between the variances a, 8, and y of the field and the parameters and cp of a generalized coherent state [eqs. (4.50)], one obtains

r

Pmin =

e-l€l ,

(4.59a) (4.59b)

Pmax

-

el€l

+max

=

r~

=

cp + i n otherwise.

if

t>O, (4.59c)

The eccentry of the ellipse is related to the parameter {. In particular, the shape of the isoprobability curve for the usual coherent state (5 = 0) is a circle. The shape is more and more squeezed when is increased. The parameter cp determines the angle of the ellipse. The area of the ellipse is given by

r

a=

nPmin Pmax =

(4.60)

n

and is the same for all minimum states. This appears clearly in fig. 4.1, where the isoprobability curves are plotted for the vacuum state and for a squeezed vacuum (centered at p o = go = 0). Another method to represent a squeezed state is by plotting, as a function of the angle 0, the amount of fluctuations for the quadrature component aligned with 0 (Levenson and Shelby [ 19851 , Loudon [ 19891). The variance V ( 0 )of this quadrature is defined by

V ( 0 ) = 2 ([(9- qo)cos0+ (P - P o ) s i n W )

9

(4.6 1a)

and is found to be

v(e)= eZ‘cos2(0 - cp) + e-’‘

Jvce>

sin’(0 - q ) .

(4.61b)

Plotting the dispersion as a function of 0 in polar coordinates, one obtains a lemniscate (see fig. 4.1). In the case of the usual coherent state (5 = 0), V ( 0 )is equal to one and the curve is a circle. When the parameter (is increased,


41

Fig. 4.1. Various representations of ideal squeezed states in the phase space { q , p } . The gray circle is the isoprobability curve of the Wiper distribution for the vacuum state. The dark-gray ellipse is the isoprobability curve for a squeezed vacuum state (squeezed fluctuations on p ) . The lemniscate represents the variation of the dispersion as a function of 0, for the same state.

the variance V ( 0 ) can be smaller than one. However, V(0) may vary rapidly with 0. When the squeezing factor 5 becomes large, the domain where V(0)is less than one becomes very small. 4.2.5. Frequency multimode squeezing

Up to now, we have considered a single mode of the field, characterized by one frequency. We have shown in f 4.1, however, that the quantum fluctuations are distributed over all the frequency modes. A realistic model of squeezed states must take into account the multimode character of the quantum field. In this case the frequency modes are associated with the annihilation and creation operators a, and UL(f 4.1). The Q and P operators are now time-dependent operators, defined as two conjugate quadrature components of the electric field. More precisely, we shall assume that the mean field ( E ( t ) ) is monochromatic (frequency w,) and with a real amplitude E,, and we define the two operators Q(t) and P(t) as the a.mplitude and phase quadrature components of the field ~+(= t ){ [ ~ ( t+) i ~ ( t ) l / J Z )e-ioo'

(4.62a)

(QW)

(4.62b)

with =

$E,,

(P(0)

=

0.

These operators can be related to the annihilation and creation operators

48


[I, § 4

through their Fourier transform (4.63a)

(4.63b) One obtains:

&a)= (aoo+ fl + aL, "I/* -

P(0)= i @Lo - R - aw0 + ,)I*

,

. (4.64)

The frequency components of Q and P are defined with respect to the carrier frequency oo.They obey the following commutation relations :

re, Q ( W 1 = [P(Q), P(Wl =0

[&a),P(U)]

=

9

2in S(a+ w ) ,

(4.65a) (4.65b)

Compared with the monomode case [eqs. (4.33) and (4.34)], it appears that the operators 0 and P are now associated with two different modes, symmetrically disposed with respect to the frequency oo. We define the spectrum V,, for two quadrature components X and Y as V,,(G)

=

s

dz $ ( 6 X ( t o + z) 6Y(to)+ 6Y(to)6 X ( t o + z)) eiR' . (4.66)

As in § 4.1.5, V,, can be related to the Fourier transform of the field fluctuations

4 (6x(n)6p((af)+ 6p((a')6X(L?))= 2n S ( 0 + Q ' ) V , , ( a ) .

(4.67)

The variances a, fl, and yare then replaced by the fluctuation spectra of the field components VQQ, Vpp, and VQp, respectively. These spectra obey a generalized Heisenberg inequality that can be written as VQQ(")

VPP(n)

a

+ [vQP(a)lz

.

(4.68)

The vacuum fluctuations correspond to the simple case

vQQ(a) = vpp(a)= f,

v Q p (=~0).

(4.69)

Ideal squeezed states are then defined as minimum states of this Heisenberg inequality for each frequency 0.Properties of such multimode squeezed states

1 9 5

41


49

are found to be similar to those of monomode states. In particular, using the Wigner distribution, a squeezed state can be represented in the phase space (4,p}. For each frequency the shape of the isoprobability curve is an ellipse, the parameters (eccentricity, angle) of which are associated with the values of the noise spectra VQe, Vpp,and V Q pat the frequency a.

4.3. SQUEEZED-STATE GENERATION BY IDEAL PARAMETRIC INTERACTION

In this section we shall show that single-mode ideal squeezed states can be obtained by parametric excitation. We shall not describe realistic systems (which are discussed in § 5). We shall show, rather, how squeezed states are generated from coherent states by the action of an ideal quadratic Hamiltonian H

=

i [ f ( t ) Q 2 - g ( t ) (PQ + QP) + h(t)P21,

(4.70)

where f ( t ) , g(t), and h(t) are functions of time. An example of such a Hamiltonian is given by an oscillator where the eigenfrequency is modulated

H = 4 [ P2 + w y t ) e.21 .

(4.7 1)

4.3.1. Effect of an ideal quadratic Hamiltonian General arguments will be used to demonstrate that a quadratic Hamiltonian generates squeezed states. The evolution of the system during a given time t corresponds to a canonical transformation, i.e., a linear transformation for P and Q that preserves the commutation relations. This can be written using a matricial notation

X’ = M X ,

(4.72a)

where X is the vector

x=(;),

(4.72b)

and M is the matrix associated with the evolution M

=(: :).

(4.72~)

To preserve the commutation relations, the determinant of the matrix M must be equal to one.

50

[I, 8 4


It is shown in the appendix that the evolution of the Wigner distribution for a quadratic Hamiltonian is classical. It is then possible to replace the quantum equations of evolution by their classical counterpart in the Wigner representation. More precisely, we consider the classical variables 4 and p associated with the Wigner distribution, which obey the same linear transformation as the operators Q and P XI

=

Mx,

(4.73a)

where the vector x is defined as (4.73b) We are interested in the symmetrically ordered variances a, B, and y of the field, which are the elements of the covariance matrix (4.74) where xT is the transposed vector of x. The evolution of the covariance matrix is found to be V'

=

MVMT.

(4.75)

The determinant of the covariance matrix is det(V) = aB- y 2

(4.76)

and is equal to 0.25 for a minimum state [eq. (4.40)]. From eq. (4.75) it appears that this quantity is preserved during the evolution. A minimum state is thus transformed to another minimum state. However, the canonical transformation may change the individual values of the variances. In particular, if the initial state is a coherent state or the vacuum state, with a=;,

B=' 2 ,

(4.77a)

y=o,

the final state is an ideal squeezed state, characterized by the variances a' = ;(a2 + b 2 ) ,

8'

=

i(c2 + d 2 ) ,

y' = i ( a c t b d ) .

(4.77b)

4.3.2. Perturbative approach We shall now study in more detail the effect of the ideal quadratic Hamiltonian [eq. (4.70)], assuming that a perturbative approach can be used.

1. B 41

THEORY OF QUANTUM NOISE IN OFTlCS

51

We shall show that the action of this Hamiltonian is equivalent to a parametric amplification. The coefficients of the Hamiltonian are developed in the following form: f(t) = YO + fi(t)

A t ) = go + gi(t)

9

9

h(t) = ho + h,(t) ,

(4.78)

where f,,go, and h , are time independent and f l ,g , , and h , are small and have a zero temporal mean value. At the lowest order the Hamiltonian can be written as Ho

= f [ f o Q 2 - goVQ + =

Q P ) + hop2]

+ A,A:],

$w,[A:A,

(4.79a) (4.79b)

where the operators A, and A: are characterized by the parameters ( and cp [eqs. (4.46)] defined by the relations

fo

=

wo(cosh2( - sinh2tcos2cp)

(4.80a)

+ sinh2(cos2cp)

(4.80b)

h o = w,(cosh2( go

=

wo sinh2t sin2cp.

(4.80~)

We have assumed that (4.8 1)

foho - g,2 = wo'> 0 ,

which corresponds to the usual harmonic oscillator. The opposite case of an inverted oscillator can also be studied and will lead to the same conclusions. The evolution of A, and A: due to Ho is

A,

=

ACe-iwot,

A:

=

A: eiwof

(4.82)

where A, and A: are the operators in the interaction representation. At the next order the Hamiltonian can be written as

H

I

-1 - 2 [ f i ( t ) Q 2 - gi(t>(PQ + =

/ ( t ) (A:A,

Q P ) + h1(t)P21

(4.83aI

+ &A:) + m ( t ) e2iwarA:2+ m*(t)e-2i"ofA~,(4.83b)

where / ( t ) and m(t) are linear combinations of fi(t), g,(t), and h,(t). In a perturbative approach only secular terms in H I need to be taken into account, namely, terms that correspond to zero frequency in the interaction representation. From the definition of f l ( t ) ,g,(t), and h,(t) the mean value of l(t) is equal to zero. Only the secular part of m(t) will contribute to H , as follows:

,

H -- 1,(-ixA:'+

ix*A:),

(4.84)

52


[I, 8 4

where x is related to the component of m(t) at frequency 2 0 , : m(t) = - i(x/2) e-2i0d.

(4.85)

Thus we have shown that, in a perturbative approach, only the components of f ( t ) , g(t) and h(t) at frequency 20, will contribute to the evolution. As we shall see, this corresponds to parametric amplification. 4.3.3. Parametric amplijication and deamplijication For the sake of simplicity we consider the case where A, is identified with the operator a ([ = 0). From eqs. (4.80) this corresponds to the particular values f0=ho=w,,

(4.86)

g,=o.

The non-perturbed Hamiltonian then takes the usual form for a harmonic oscillator H,

=

iw,(ata

+ a a t ) = fwo(Q2 + P 2 ) .

Furthermore, we choose the phases so that Hamiltonian can be written as H,

=

x

(4.87) is real. Thus the interaction

- i(x/2) (,t2 - d 2 ) = - (x/2) (PQ + QP) .

(4.88)

(a stands for the operator x in the interaction representation.) This is the usual Hamiltonian of parametric amplification. The evolution operator during a time t is

u = exp [ $ v](dt2 - d 2 ) ]

(4.89a)

q=

(4.89b)

with Xt.

U is an ideal squeezing operator [eq. (4.52b)], which transforms a coherent state into a squeezed state. This can be shown directly, since the evolution of the operators d and dt is given by d(t) = U d ( 0 )U t = d(0) cosh q

- d ( 0 ) sinh q ,

dt(t) = U d t ( 0 )U t = dt(0) cosh q - d(0) sinh v]

(4.90a)

.

(4.90b)

For the operators Q and P one obtains Q(t) = Q(o)e - q ,

P(t) = P(o)eq .

(4.90~)


53

(tip)

The variance is thus squeezed by a factor e-’q, whereas the variance ( t i p 2 ) is increased by a factor of e2q (this ensures that the Heisenberg inequality is still fulfilled). It appears that parametric excitation corresponds to a scale transformation, where one component of the field is amplified and the conjugate component is deamplified. 4.3.4. Semiclassical pendulum This scale transformation is a well-known effect in classical mechanics. For a pendulum parametrically excited at twice its oscillation frequency, the oscillation in phase with the excitation is amplified, while the oscillation in quadrature is deamplified. Figure 4.2 shows the evolution of a semiclassical pendulum. The Hamiltonian of the system corresponds to a harmonic oscillator [eq. (4.71)] parametrically excited by a square wave: the frequency w(t) is shifted between two values w1 and w2 at each quarter of the period. When w is constant, the system evolves as a normal harmonic oscillator. In particular, the effect of the evolution during a quarter of a period is to exchange the variables 4 and p , scaled by w,

a

4(t -t T ) = p ( t ) / w , p ( t + T ) = - 4 ( t ) w .

(4.91)

Fig. 4.2. Evolution of the Wigner distribution for the parametrically excited semiclassical pendulum. The initial state (white circle) is a coherent state. The mode frequency is changed at each quarter of a period. The ellipses corresponding to these times have increasingly darker shading. Parametric excitation produces a .net squeezing of fluctuations for one quadrature component of the field.

54


[I. B 5

Assuming that the first quarter of the period is at frequency wl, the second one at frequency a2,and so on, the net effect for a period T corresponds to the scale transformation (4.92) In fig. 4.2 the Wigner distribution of the semiclassical pendulum is plotted in the phase space ( 4 , p ) . It appears that the area of the distribution is conserved in the evolution because the system stays in a minimum state for a canonical transformation. The shape of the distribution is squeezed by the scale transformation, and the eccentricity of the ellipse is increased at each quarter of a period.

8 5.

Squeezed-state Generation by Parametric Interaction

Parametric interaction between optical waves has been the subject of many investigations since the early days of quantum optics (Louisell, Yariv and Siegman [ 19611, Takahashi [ 19651, Mollow and Glauber [ 19671, Graham and Haken [ 19681, Oshman and Harris [ 19681, Smith and Parker [ 19701, Falk, Yarborough and Ammann [ 19711, Woo and Landauer [ 19711, Mollow [ 19731, Stoler [ 19741, Bjorkholm, Ashkin and Smith [ 19781, McNeil, Drummond and Walls [ 19781, Lugovoi [ 1979a,b], Drummond, McNeil and Walls [ 19811, Savage and Walls [ 19831, Graham [ 19841). In recent years, interest in such a non-linear interaction between three modes has been stimulated by its ability to generate non-classical states of the radiation field (Milburn and Walls [1981], Lugiato and Strini [1982], Yurke [1984], Collett and Walls [1985], Savage and Walls [1987], Reynaud, Fabre and Giacobino [1987]), and, as described in $ 2, significant quantum noise reduction has been observed by various groups using parametric interaction. It is interesting, therefore, to consider this problem in much more detail than in the intuitive approach of $2.1 or in the ideal theoretical case of $ 4.3. This is the purpose of the present section. In particular it will be shown that parametric interaction, although one of the simplest types of interaction, can generate different kinds of squeezed light (squeezed vacuum, fields with reduced phase or intensity fluctuations, intensity-correlated beams), according to the precise device in which it is used. Single-pass parametric generation has been studied in the ideal case in the previous section and is discussed in more detail in the appendix. Unfortunately, with available non-linear crystals and continuous-wave laser sources, such an interaction yields very weak squeezing effects. However, Yurke [ 19841 real-

SQUEEZED-STATE GENERATION BY PARAMETRIC INTERAOION

55

ized that the use of an optical cavity could dramatically enhance the squeezed effects. In this section we shall calculate the quantum fluctuations of the fields leaving an optical cavity containing a f 2 ) parametric medium. We shall use an input-output formalism, related to the approaches of Yurke and Denker [ 19841, Yurke [ 19841, Gardiner and Collett [ 19851, and Gardiner [ 19881, allowing the calculation of the output field fluctuations as a function of the input ones. The method has been called “semiclassical input-output theory” (Reynaud and Heidmann [ 1989]), since it relies on the semiclassical description of field fluctuations, based on the Wigner representation: the quantum operators are replaced by classical stochastic variables, obeying the evolution equations of classical electrodynamics, and the input field fluctuation spectra are identical to the quantum ones (I 4.2.5). As shown in the appendix, this method is rigorously valid for a quadratic Hamiltonian, and more generally for any coupling between light modes as long as linearization techniques are used, reducing the interaction to the quadratic form. In the limit of small fluctuations and the case of pure parametric processes, its formal equivalence with the usual method has been shown by Reynaud and Heidmann [1989] and Fabre, Giacobino, Heidmann, Lugiato, Reynaud, Vadacchino and Wang Kaige [ 19901. 5.1. PARAMETRIC AMPLIFICATION IN AN OPTICAL CAVITY

5.1.1. Resonant case

Figure 5.1 gives the general scheme of an optical parametric amplifier (OPA). The non-linear crystal is inserted in a cavity that we assume to be ring-shaped for the sake of simplicity. Let us first assume that this cavity is resonant for the

Ein

I

E,

+TE4 /-\

I

Fig. 5.1. Scheme of field propagation inside an optical parametric oscillator. E , , E,, E,, and E4 are the intracavity fields located after the coupling mirror, before the non-linear medium, after the non-linear medium, and before the coupling mirror, respectively. Ei, and E,,, are the input and output fields coupled to the inner field through the coupling mirror.

56


[I, 8 5

signal mode and completely transparent for the pump mode. It is then straightforward, using the classical equations of propagation, to determine the output signal field as a function of the input field. Let us denote by indices 1, 2, 3, and 4 the fields inside the cavity located, respectively, after the coupling mirror, before the non-linear medium, after the non-linear medium, and before the coupling mirror, and by the subscripts in and out the input and output fields coupled to the inner field through the coupling mirror of amplitude reflectivity r (see fig. 5.1). When neglecting the pump field variation through the parametric medium, the signal quadrature components q3 and p 3 after the medium are related to the ones before the crystal by eq. (4.90~) q3 = q 2 e - q ,

p3 = p 2 e q ,

(5.1)

where q = Zol/c (aois the pump mean field; I the non-linear medium length). Because of the propagation inside the cavity, the signal field is multiplied by a phase factor equal to eim7,where w is the field frequency and z = L / c is the round-trip time (L is the cavity length). Let be the difference between the actual signal mode frequency w and the degenerate parametric frequency w1= iwo,assumed to coincide with a cavity resonance (w,zis a multiple of 2n). The phase factor is then equal to eiRr,and one has finally

q,(a) = gl(a)e - eiRr, ),(a)= pl(a)eq e i n r .

(5.2)

The outer and inner fields are simply related by the equations on the coupling mirror

@,= ,gin + r g , ,

(5.3a)

-

(5.3b)

go"'

=

p1 = tg'" + rp4 , + tq4, 8""'= - rpin + tp, ,

which leads to the following relations between the inner and outer fields :

gl(Q) (1 - r e - 'I einr) = [#"(a),

(5.4a)

pl(0)(1 - r eq eiR+)= tg'"(n),

(5.4b)

q,(a)(e- q einr - r) = tqo"'(a), p,(a)(eq eiRr- r ) = tpo"'(a),

(5.4c) (5.4d)

and to the final input-output relations, which are a generalization of the simple formulas (2.4) and (2.5):

g""'(a2)= q'"(a)gq(n), p""'(m = 8'"(aEp(a) 9

(5.5a) (5.5b)

1.

o 51

SQUEEZED-STATE GENERATION BY PARAMETRIC INTERACTION

51

where (5.6a)

(5.6b) The cavity acts as a linear filter for input fluctuations, which is different for the two quadrature components of the input fields. If rev is equal to one, there is a divergence for g,,(O), corresponding to the threshold of oscillation: the system turns from a passive amplifier to an active optical parametric oscillator (OPO). Let us consider more extensively the case 62 = 0: (5.7a)

(5.7b) These equations are similar to eqs. (4.91) for the ideal parametric interaction of 0 4.3, but enhanced by the presence of the cavity: whatever the value of q, there is now a possibility of perfect squeezing of the q quadrature component, when rev approaches one. Furthermore, the parametric gain is now frequency dependent. From eqs. (5.6) one deduces the output noise spectra, as defined in 4.2.5: (5.8a) (5.8b)

(5.9a) (5.9b) One obtains an “Airy-like” function, reminiscent of the expression giving the reflection coefficient on a Fabry-Perot interferometer. The enhancement factor of the cavity is thus limited to narrow frequency bands having a width close

58


[I, 8 5

to the cavity bandwidth and centered around the zero frequency and all the multiples of the free spectral range z5.1.2. Non-resonant case In the general case of a detuned cavity, the expressions are more complex. They can be simplified by using a matrix notation. Let us denote by x the column matrix containing the two variables p and q. One can simply show that the matrix describing the roundtrip propagation in the cavity due to parametric coupling and free propagation is

f4(f2)= einTMf,(f2),

(5.10)

where M=l

cosh q sin 6 cosh q cos 6 - sinh q cos q cos 6 + sinh q - cosh q sin 6

(5.11)

and 6 is the differencebetween the roundtrip phase shift and the closest multiple of 2n. The equations on the mirror can also be written in terms of column matrices as follows: f, = tf'"

+ rf4,

(5.12a) (5.12b) (5.12~)

Eliminating the f4and 2, fields between these equations, one finally obtains fout(fl) = T(Q)P"(a),

(5.13a)

with T(l2) = (ein'M - r) (1 - r e i n rM)-

'.

(5.13b)

Such a matrix describes the linear input-output transformation of fluctuations and allows us to calculate the transformation of the different variances. Let us define the covariance matrix by eq. (4.67):

(sf(a)sf+(a')) = 2n 6(Q + a')vxx(a).

(5.14a)

Then the input-output relation takes the simple form

Vz't(0)

=

T(a)V$(a)TT( - a ) .

(5.14b)

1, I 51

SQUEEZED-STATE GENERATION BY PARAMETRlC INTERACTION

59

5.1.3. Good-cavity limit Let us stress that, unlike many other approaches of the OPA problem, this analysis is not restricted to the special case of high-finesse cavities, small detunings, and low noise frequencies. This is the reason why we obtain Airy-like functions for the spectra, instead of Lorentzians centered around zero frequency. It is interesting, nevertheless, to have a closer look at the special case where the reflection coefficient r is close to one: r = 1 - y, with y 4 1 , (5.15) t N (2y)"Z

.

(5.16)

If the noise frequency is also assumed to be small (613 z- l), as well as the parametric gain q and the cavity detuning 6, the equations relating the different fields take a simpler form. In particular, one obtains from eqs. (5.4):

q)$,(a)+ tqin(a),

(5.17a)

iazpl(sl) = ( - y - i 6 + q ) p l ( 0 )+ ~ ~ ' " ( 6 2 ) .

(5.17b)

iazij,(Q)

=

( - y - i6 -

In these equations ql(a)and pl(G?)appear as Fourier transforms of the timedependent quantities q l ( t ) , p l ( t ) , obeying the following differential equations :

+ tq'"(t) ,

(5.18a)

z p l ( t ) = ( - y - i6 + q ) p l ( t ) + tp'"(t),

(5.18b)

zq,(t) = ( - y

- i6 -

q)ql(t)

which, in turn, can be deduced from a single equation for the complex field amplitude a,@)= [q,(t) + ipl(t)]/$: za,(t) = ( - y - i6) a&)

+ qa:(t) + ta'"(t) .

(5.19)

Such an equation has a simple interpretation: it relates the variation of the signal field over one cavity roundtrip za,(t)

N

a,(t

+ z) - a,(t) ,

(5.20)

respectively, to the cavity losses, cavity detuning, variation due to parametric interaction, and input field transmitted through the coupling mirror. Related equations (deduced from an effective Hamiltonian and using the P-representation) were first obiained by Gardiner and Collett [ 19851 and Collett and Walls [ 19851, and have been extensively used in the literature. It is also possible to write a differentialequation relating the inner field to the outgoing field, which is zdr,(t)

= (y -

i6)al(t) + qa:(t)

+ tctout(t).

(5.21)

60


[I, I 5

This is the same equation as eq. (5.19), except that the incoming field is replaced by the outgoing one and that the damping coefficient has changed signs: this equation can also be obtained by time-reversal symmetry arguments (Gardiner and Collett [ 19851). 5.1.4. Case of incoming vacuum @Id In the case where the incoming field is the vacuum, the incoming noise density matrix has the following simple expression [deduced from eqs. (4.69)] :

vE(a)= ; I ,

(5.22)

so that

v;:ya)

=

+ T ( ~ ) T =-( a ) .

(5.23)

Using expression (5.13b) in the limit where q, 6, and a are small quantities, one finally finds for the covariance matrix elements (5.24a) (5.24b) (5.24~) (5.24d) B

=

4yq(R2z2 + q2 + y 2 - b2),

C=8 ~ ~ ~ 6 ,

(5.24e) (5.24f)

and D

=

( 0 ’+ ~ q2~ - y 2 - 62)2 + 4 y 2 S 2 2 ~ 2 .

(5.248)

Similar results can be found in the article by Savage and Walls [ 19871. Note that because B2 + C2 - A’

=

2DA ,

(5.25)

such variances fulfill the minimality condition at any frequency D [ eq. (4.68)] :

vqq(a) v,,,,(a)- vqp(a)2 =a.

(5.26)

For each frequency Sa the domain of fluctuations in phase space is a minimum ellipse, having an eccentricity and axes that are now Qdependent. One can also calculate noise spectra S,(a) (noise variance normalized to vacuum noise)


1,s 51

61

associated with the quadrature component q cos 8 + p sin 8:

s,(a) = 2[ V,,(O) cos28 + V,,(a) sin28 + 2 V,,(a) cos 8 sin 81 , =

[ D + A - B c o s 2 8 + Csin28]/D.

(5.27)

Figure 5.2a gives the variation of S , ( a ) with 8 = 0 (squeezing spectrum of the q-component) as a function of a and of the normalized pump parameter 0=

v/r.

(5.28)

The fluctuations in this component cancel at zero frequency when one approaches the oscillation threshold (0- 1). One therefore obtains a strongly squeezed vacuum when o=! 1 and for noise frequencies lying in the cavity bandwidth y/z. In the neighborhood of such a point, the noise compression is extremely sensitive to any change in the parameters. The eccentricity of the ellipse is then very large, and a tiny change of any parameter leads to a small rotation of

Fig. 5.2. Noise spectrum of the output signal field quadrature component (with 0 = 0) of a degenerate optical parametric amplifier, as a function of the pump amplitude parameter a(a = 1 corresponds to the oscillation threshold): (a) in the resonant case; (b) in the non-resonant case (6 = 0.05 7).

62


[I, § 5

the measured component with respect to the optimum direction and, as already noted in the discussion of § 4.2.4, to a dramatic increase of the fluctuations. Figure 5.2b displays S 0 ( n ) as a function of 0 in the presence of a small cavity detuning 6, as an example of such a sensitivity. 5.2. DEGENERATE PARAMETRIC OSCILLATION

We shall restrict the analysis of the equations of evolution to the simple case of a good cavity, both for the pump and signal modes, with the possibility of cavity detunings. The resonant case has been considered by Collett and Walls [1985], and the general case has been studied in great detail by Fabre, Giacobino, Heidmann, Lugiato, Reynaud, Vadacchino and Wang Kaige [ 19901. We shall outline the main results of this analysis. 5.2.1. Equations of evolution In the good-cavity limit the equations for the pump and signal modes can be written in terms of differential equations analogous to, eq. (5.19) for the time-varying quantities ao(t) and a,(t):

it,

zit,

=

-(yl

= -(yo

+ iS,)a, - XaFa,, + t , a p , + i6,)a0 - x a 3 2 + t,a$ ,

(5.29a) (5.29b)

where x is the parametric coupling coefficient. The optical parametric oscillator corresponds to the case where the system is pumped by an incoming field a$ that has a non-zero mean value, whereas the intracavity frequency doubling is governed by the same equations, but with a pump field a t . We shall not consider this latter configuration, which leads also to interesting squeezing effects (Collett and Walls [ 19851). 5.2.2. Stationary solutions and stability analysis To simplify the equations, we shall use the following reduced notations:

Aj"=~(aj").

Ai=x(ai),

(5.30)

Two different types of stationary solutions can be found (Lugiato, Oldano, Fabre, Giacobino and Horowicz [ 19881): (i) the trivial solutions corresponding to zero mean value for the signal field: A,

=

0,

A,

=

to&/(y0

+ i6,),

(5.3 1)

1. I 51

63


corresponding to the below-threshold case (OPA) studied in Q 5.1.2. (ii) the oscillating solution, where the mean intensities Ii = IAiI2 and I , = IAg 1 are given by the equations (5.32a) I , = y: + 6: ,

’

(5.32b) When 6,6, > yo y,, eq. (5.32b) has two real positive solutions in I , for each value of I,. The stability analysis of such solutions leads to the definition of three regions in the (b,, 6,) plane. (1) A “monostable” region, when 6,6, < y o y , and 6,(6, + 26,) > - yo(yo + 2y,), including the simple resonant cases (6, or 6, equal to zero)), corresponding to a single-valued stable solution: (i) I ,

=

0 for I , < Ithr,

(ii) I , # 0 for I , > Ithr, the value of the oscillation threshold Ithrbeing

(r,’ +

(5.33)

. (2) A “bistable” region, when 6,6, > yo y,, with Ithr =

(7: + 6:)/2

70

(i) one stable solution

I,

=

0 for I , < Ibis,

(ii) two stable solutions

I,

=

0 for I , # 0 for Ibis< I , < I [ h r ,

(iii) one stable solution

I , # 0 for I , > I t h r ,

the value of the bistability threshold being Ibis =

Yl(60?1

+

6l

.

(5.34)

(3) An “unstable” region, when 6,(6, t 26,) < - yo(yo

+ 2y,), with

0 for I p
Ii,,

.

When the pump intensity is increased above the instability threshold I,,,, given by eq. (25b) in the article by Lugiato, Oldano, Fabre, Giacobino and Horowicz [ 19881, the OPO first exhibits undamped self-oscillations (Pettiaux, Ruo-Ding and Mandel [ 1989]), then reaches a chaotic regime after passing through a sequence of period-doubling bifurcations.

64


5.2.3. Linear analysis ofjuctuations As shown previously, in the limit of small fluctuations, the quantum fluctuations of the pump and signal fields can be calculated from the classical equations (5.29) linearized around the stationary solution. We shall therefore write (5.35) a,(t) = ( a , ) + 6a,(t) I

where the quantities ( a,) have been calculated in the previous section. The linearized equations for the fluctuations 6ai are z6a,

= - (7,

~ 6 a= , -(yo

+ i6,) 6a, - A : &a, - A , 6ar + t , 6 a P , + i6,)6a, - A , 6a, + t , 6 a F ,

(5.36a) (5.36b)

The resolution of these equations of straightforward when using the matrix formalism introduced in the previous section, with 4 x 4 matrices acting on column vectors:

(5.37)

Such a calculation can be found in the article by Fabre, Giacobino, Heidmann, Lugiato, Reynaud, Vadacchino and Wang Kaige [ 19901, as well as the analytical expression for the noise spectrum Si,(61)for the &quadrature component of the field ai.We prefer here to focus on the specific features of quantum noise reduction in the different regions of the parameter space. 5.2.4. Resonant case

Let us first consider the resonant case (6, = 6, = 0). One can take real values for the mean fields in this case. Figure 5.3 displays the noise spectrum of the output signal field quadrature component S,,(Q) as a function of the pump intensity. One observes an important quantum noise reduction within the cavity bandwith, even far above the oscillation threshold, except close to 61 = 0. Such a squeezing for the phase of the signal field is the above-threshold analog of the squeezing of the vacuum field below threshold (fig. 5.2a): there is a continuous variation of the field fluctuations on the squeezed component when crossing the oscillation threshold.

1,s 51


65

Fig. 5.3. Squeezing spectrum ofthe phase quadrature component ofthe output signal field in the above-threshold OPO, as a function of the pump intensity I , , in the resonant case (6, = 6, = 0).

The transmitted pump field fluctuations are also strongly modified by the parametric interaction in the cavity. One finds that the phase quadrature component of this field, S,,(R), is given by

~)/[(R*T - ~2 7 0 7 , ~ ) 2+ R 2 ~ 2 ( 2 7+ 1 70)2]2.

S p 0 ( R ) =1 - 87,’7:(0-

(5.38) This phase quadrature is also reduced around the zero frequency, but now by only a factor of two for pump intensities that are far above threshold. 5.2.5. Bistable region It is interesting to look at the distortions of the squeezing spectra when the detunings are sufficiently large to enter the bistable region. Let us first define the “optimum squeezing spectrum” S p ( R ) as the minimum value of Si,(R) with respect to 8 for each value of the noise frequency R (SiB(R)is only related to the eccentricity of the noise ellipse). It can then be shown that S p y 0 ) + Sa’n(0)

=

1,

(5.39)

revealing a kind of complementary behavior of the signal and idler squeezing spectra at zero frequency. Figure 5.4a gives the variation of the signal optimum squeezing spectrum as a function of the pump intensity in the upper branch of the bistability curve. One observes a significant noise reduction for non-zero values of R, even far from the turning point. More important in this case is the choice of the relative values of the cavity finesses for signal and idler modes. The best choice for

66


Fig. 5.4. Optimum squeezing spectrum of the output signal field in the above-threshold OPO, as a function ofthe pump intensity I,. (a) In the bistable region (6, = 6 , = l.l), above the bistability < I , < 51thr);(b) in the unstable region (6, = 2, 6 , = - I), between the threshold ( y , = 5 yo. Ithr oscillation threshold and the instability threshold ( y , = 10 yo).

squeezing on the signal mode corresponds to the situation where the cavity finesse is larger for the pump mode: yI % yo (the reverse condition holds for the pump mode squeezing).

5.2.6. Unstable region Figure 5.4b displays the signal optimum squeezing spectrum SY'"(f2)for pump intensities ranging from the oscillation threshold to the instability threshold. One observes that for intermediate frequencies the squeezing can be important and without much variation between the two thresholds. In the low-frequency domain, on the other hand, the noise increases quickly. This is related to the fact that, at the instability threshold, the system will start oscillating, giving rise to a macroscopic peak in the signal field Fourier spectrum and, therefore, to a divergence of the fluctuations at this frequency. Note that, in the case of fig. 5.4b, the increase in noise is limited to a very narrow region around the self-oscillation frequency.

67


1,s 51

5.2.1. General characteristics of the noise spectra As a general rule, one can say that the best squeezing is encountered close to the oscillation thresholds, either in the monostable or bistable case, either below or above threshold. On the other hand, the vicinity of the instability region is less favorable, even though the increase in noise is limited to a small region of the noise spectrum. In any case, the variation of the spectra is smooth as one approaches the thresholds, and a large degree of squeezing can be found even far from such thresholds. Finally, a good amount of squeezing for the signal (pump) field requires the largest cavity finesse for the pump mode (signal mode). 5.3. NON-DEGENERATE OPTICAL PARAMETRIC OSCILLATOR (NDOPO)

Let us finally consider the case of parametric interaction between three different modes of frequencies wo, wl, and 0 2 ,with wo = w1 + w2 (Reynaud, Fabre and Giacobino [ 19871). The Hamiltonian for such an interaction in the non-linear material is now H,,

=

-ihx(aTa:a,

- aJa,a2).

(5.40)

We shall not consider the most general case of unequal reflection coefficients for the signal and idler modes al and a2 (Fabre, Giacobino, Heidmann and Reynaud [ 1989]), but restrict ourselves to the case of a balanced NDOPO, corresponding to a completely symmetrical device with respect to the signal and idler modes. The field evolution equations are in this case (5.4 1a) nil = - ( y + iSl)a, - xaza, + t a p , 7a2 =

-(y

ni, = - ( y o

+ iS2)a2- xa,c1, + tal;. ,

+ iSo)ao + xa,a2 + t o a t ,

(5.4 1b) (5.41~)

where 1 - y and t are the reflection and transmission coefficients for the signal and idler modes. 5.3.1. Stationary values Using the notations defined in eq. (5.30),the signal and idler stationary values are linked by the following equations:

+ iS,)A, + A;Ao = 0 , ( y - i6,)A; + A I A d = 0 .

(y

(5.42a) (5.42b)

68


[I, 8 5

One finds in this case also the possibility of a trivial solution A , = A , = 0 (below-threshold operation), and of a non-trivial solution (above-threshold operation) when (y

+ i6,) ( y - is,)

-

IA,~,

=

0,

(5.43)

which splits into two real equations y2

= IA,12

6,

=

- 6,6,

(5.44a)

and 6,.

(5.44b)

This last equation has an important consequence, i.e., the NDOPO can oscillate only in the case where the signal and idler detunings are identical. We shall denote 6 as this common value of the detunings. Writing the fields in terms of the intensity and phase a = (I),/,

eip,

(5.45)

one easily shows that the two fields have the same mean intensities ( I , = 1,) and that the sum of the two mean phases p, + p2is locked to the pump phase, whereas the difference 'p, - p, is not determined by the equations, in the same way as the phase of a laser field is not determined by the laser dynamical equations. As a result, the quantity 'p, - 'pz will undergo a phase diffusion process (Graham and Haken [ 19681).We shall neglect this phenomenon in the first approach and take a definite choice of the signal and idler phases. For example, we shall take A,= -y-i6,

(5.46)

so that it is possible, according to eqs. (5.42), to take A , and A , to be real. We shall come back to the problem of phase diffusion in $ 5.3.6.

5.3.2. Linearized equations for the fluctuations The linearized equations are a simple generalization of eqs. (5.36). In the balanced case, when using the symmetrical and antisymmetrical modes, (5.47a) (5.47b)

I , $ 51


69

they can be written in a simpler way as follows:

z 6 i o = -(yo

+ i6,)

6ao + A

+

6a+

z 6 a + = - ( y + i 6 ) 6 a + -Ao6a:

z6i-

= -(y

+ to 6 a F ,

-A: 6ao+t6ai~,

+ is) 6a- + A , &a*_+ t &ai! .

(5.48a) (5.48b) (5.48~)

One observes a complete decoupling between the first two equations, which coincide with eqs. (5.36) of the degenerate case (with 6 a + replacing 6a,) and the last equation. As we have already seen such linearized equations provide the basis for the system stability analysis and for the calculation of quantum fluctuations. As the last equation never leads to diverging fluctuations, the stability analysis is the same for the degenerate OPO and for the balanced NDOPO. The squeezing spectra for the pump fluctuations 6ao and for the quantity 6a+ are also the same as in the previous part. New features specific to the NDOPO will arise only when considering the variable 6a-, or any combination between 6a - and the other variables, such as the signal and idler field fluctuations considered separately.

5.3.3. Quantum fluctuations of the signal field Figure 5.5 displays the output signal field intensity noise in the resonant case as a function of the pump intensity. Close to threshold, one observes large excess noise at low frequencies, which is expected from an active system operating close to its oscillation threshold. At large pump powers the intensity noise decreases and goes below the shot noise. Each beam coming from a

Fig. 5.5. Noise spectrum of the intensity of the signal (or idler) field in a non-degenerate OPO operating above threshold as a function of the pump intensity I,,.

70


[I, § 5

NDOPO operating well above threshold is therefore sub-Poissonian, with a maximum squeezing of 50 2. 5.3.4. Quantum Juctuations in the drerence between the signal and idler jklds According to eqs. (5.46) and (5.48c), the real and imaginary parts 6q- and 6p- of 6cr- obey the following equations: z 6 q - = -2y6qzap-

=

+ t6q'"

(5.49a)

-2i66q- + t 6 p i ? ,

(5.49b)

from which, using the relation (5.3) between the inner and outer fields coupled on the mirror, one can derive the input-output relations for the fluctuations

sqya) =

Spya) =

- inz

iS2z + 2y 2y - iaz iRz

64':

(a),

(5.50a)

spirt(a)-

4y6 6qin(a), flz(idlz + 2 y)

(5.50b)

and the corresponding noise spectra

(5.51b) Let us calculate the noise in the difference between the output intensities of the two beams, I , - I,:

Jz (( cryut) 6q';Ut

& ( I ,- I ~ ) =

= 2 ( cryut) 6qYUt,

-

(crzut)

6qyt)

(5.52) (5.53)

since we have chosen the mean values of the signal and idler fields to be real. It follows from eq. (5.53) that eq. (5.51a) gives the noise spectrum that can be measured in the difference between the two beam intensities. One obtains a perfect noise cancellation on this variable for frequencies well inside the cavity bandwidth. This noise reduction does not depend on the detunings 6, b0 and on the pump intensity and fluctuations due to the decoupling in the equations of motion (5.48). Because of such an insensitivity to external parameters and of large potential squeezing, it seems of particular interest to measure the

1,s 51


71

quantum noise reduction of the intensity difference. On the other hand, in eq. (5.51b) there is an excess noise, and even a divergence at zero frequency, for the quantum fluctuations of the conjugate variable p - . This is related to the phase diffusion of 'p, - (p, of course, which allows for unbound fluctuations over long periods of time. Therefore squeezing on I , - I, and phase diffusion on p, - (pz are closely related phenomena. Actually, there is a simple physical interpretation for the reduction of fluctuations in I , - I,, linked to the fact that, in parametric downconversion, the non-linear medium emits twin photons in the signal and idler modes. As explained in 0 2.4, the intensity fluctuations of the two generated modes are therefore identical, at least for noise frequencies inside the cavity bandwidth.

5.3.5. Efect of extra losses and imbalance The perfect noise reduction in the intensity difference is only valid in the case of an ideal, lossless, perfectly balanced OPO. A complete calculation of the noise spectra of I , - 1, in the presence of various imperfections has been made by Fabre, Giacobino, Heidmann and Reynaud [ 19891. We give the main results of this analysis. In the semiclassical formalism linear losses can be accounted for straightforwardly by adding to the equations the effect of a second coupling mirror in the cavity, with reflection and transmission coefficients r' = 1 - y' and t' . For example, eq. (5.4 1a) becomes za, = - ( y

+ y' + i6,)a, - xa>a, + t a p + t ' a i i n .

(5.54)

Extra losses therefore have the effect of increasing the decay constant ( y + y + 7') and of coupling the considered system to a new source of fluctuations aiin. The calculation of the noise spectrum in this case is straightforward and yields a 2 z 2 + 4y(y + y ' ) s, (a)= a2zz+ 4(y + y'),

(5.55)

The minimum noise is still obtained at zero frequency and is equal to y' / ( y + y ' ) , i.e. to the proportion of photons that are not detected in the output beam. This can be easily understood by interpreting the effect in terms of twin photons (0 2.4). It can be shown that the effect of a slight imbalance between the signal and idler beams (in mirror transmission and losses) is to couple the intensity noise of a single beam (given in fig. 5.5) to the intensity difference signal. If the pump

12


[I

is assumed to be limited by shot noise, one obtains a large rise in noise in the spectrum, but limited to low noise frequencies and to pump intensities close to the oscillation threshold.

5.3.6. Efect of phase dirkion Two different phase diffusion processes may affect the OPO characteristics (i)an external one, affecting the phase of the pump field, since it usually originates from a laser, and (ii) an internal one, on the signal and idler phase difference. Phase diffusion means the divergence of fluctuations at zero frequency, and therefore the breakdown of the linearization method. It is nevertheless possible to perform a calculation in this case, restricting the linearization technique to the fluctuations of the non-diffusing variables and keeping the exact expressions for the others. This calculation has been made in the case of the DOPO by Drummond and Reid [ 19881, Reid and Drummond [ 19891, and Courtois, Smith, Fabre and Reynaud [ 19901. One can show that this process does not affect the squeezing in I , - Z,, as expected from the simple explanation of this phenomenon, but it adds excess noise to the different phase noise spectra. For example, the noise spectrum S,+(O) that gives the noise on the phase sum 'pI + cpz, exhibiting squeezing at low frequency in the blockedphase model, undergoes a divergence at zero frequency due to the pump field phase diffusion. In actual experiments measuring phase noise by means of heterodyne mixing, the local oscillator is itself derived from the pump field, and the measured quantity is v1 + 9, - cpo, which is not subject to phase diffusion. As a result, the noise spectrum of this quantity has no divergence at zero frequency, but it is still sensitive to the excess noise brought by the diffusing pump phase. One can show that the squeezing remains only when the pump laser cavity bandwidth is much smaller than the OPO cavity bandwidth for the signal and idler.

Appendix. Semiclassical Representations of the Field A connection between the moments of classical variables and quantum moments must be established to represent quantum fluctuations by semiclassical fluctuations. It may have various semiclassical representations, associated with different orders ofthe quantum moments. The most commonly used in quantum optics are the P- and %representations (Glauber [ 19651). associated with the normal and antinormal orders, respectively. The Wigner repre-

I1

13

APPENDIX

sentation is another representation that corresponds to the symmetrical order (Wigner [ 19321, Takabayasi [ 19541, De Groot and Suttorp [ 19721, Simon, Sudarshan and Mukunda [ 19871). In this appendix we define and compare these representations.

A . l . DEFINITION OF SEMICLASSICAL REPRESENTATIONS

A. 1.1. Formal dejinition The definition of a semiclassical distribution is associated with a set of coherent states, usually the Glauber coherent states. We shall use a more general definition, associated with a class { I z, i)}= of generalized coherent states (see 5 4.2). The usual semiclassical distributions correspond to the class 5 = 0. A semiclassical distribution 9(z) is such that the classical and quantum moments are equal, for a given order 0 of the operators A ! and A < : -

(O ( A p 4 i ) )

=

Z*kZ'.

(A.1)

The symbols (. . .) and T represent, respectively, the quantum mean value in a state characterized by a matrix density p, and the classical mean value using the distribution 9: (A.2a) (A.2b) (The z integral is over the complex plane.) The distribution 9 ( z ) can be defined by its characteristic function C ( y ) , which is the Fourier transform of 9, C ( y )=

s

d2z 9 ( z ) exp(iy*z

+ iyz*) ,

(A.3a) (A.3b)

C ( y ) can be considered as the classical mean value of the quantity exp(iy*z + iyz*). The distribution is thus defined by C(Y) =

< aexP[iY*Acl exP[iYA:I))

.

(A.4)

2-,and Wigner distributions are associated with the normal, antiThe 9-,

74

[I


normal, and symmetrical orders, respectively. They are defined by their characteristic functions, C ( p ) ,C'Q), and Ow): ~ ( " ( y )= (exp[iyA:] exp[iy*AIl) ,

(A.5a)

c ( Q ' (=~ () e x p [ i y * ~ exp[iyA:]) ~] , ~ ( " ' ( y )= (exp[iy*Ac + iyA:]) .

(A.5b) (ASc)

Note that the normal order for the operators A , and AT is not equivalent to the normal order for the operators a and at. As a result, it may have one representation 9 for each class 5 of generalized coherent states. This is also the case for the %distribution. In contrast, the symmetrical order does not depend on the class 5 of coherent states, and the Wigner distribution is independent of the chosen class. A. 1.2. The usual definitions

s

The $distribution is usually defined from the density matrix p: p=

d2z 9 ( z ) Iz, 5) (z,

51 .

('4.6)

It can be shown that this definition is equivalent to that of eq. (AS): the classical moments are equal to the normally ordered quantum moments

(A:"A;)

=

Tr[A;pA:"]

=

s

d2z B(z)z*"z'.

(A.7)

Similarly, the 2-distribution is related to the density matrix elements

The Wigner distribution is usually considered as a function of the classical variables p and q, associated with the quantum operators P and Q. As noted in the previous section, the Wigner distribution can be defined in the particular class [ = 0 associated with the operators a and at. Using the real variables q, p , u, and u such that z = (q + ip)/Jz,

y

=

(u

+ iu)/*,

(A.9)

one obtains

2n 2n

C w ' ( u , u) exp( - ipu - iqo) ,

CCw'(u,u) = (exp (iPu+ iQu)) .

(A. 10a) (A. lob)

I1

15

APPENDIX

The Wigner distribution can also be related to the elements of the density matrix p. We define the function

m u )=

((I

+ 421 P I4 - 4 2 )

= (41 eiPu12

eiPu12

Iq ) ,

(A. 1 1)

where the states I q ) are eigenstates of the position operator Q and eiPuI2is the translation operator in the { q } representation. We use the Glauber identity: for any operators X and Y that commute with their commutator: (A. 12a)

=

“X, YI, Yl

exp(X + Y)

=

exp( - [X, Y]/2) exp(X) exp(Y),

(A. 12b)

exp(X + Y)

=

exp([X, Y]/2) exp(Y) exp(X).

(A. 12c)

“X, Y1,XI

=

0,

one obtains

From these two equations one can deduce exp(iPu

+ iQu) = exp (iPul2) exp (iQu) exp (iPu/2) .

(A. 13)

From eqs. (A. 10) one obtains the relation between the Wigner distribution and the density matrix elements in the { q } representation (A. 14)

As a result, the integral over p of the Wigner distribution is the usual probability

density of the position q : (A. 15a)

Similarly, it can be shown that the integral over q is the probability density of the momentum p: (PI PIP)

=

s

dq

WP,d.

(A. 15b)

A. 1.3. Relation between the representations

From the Glauber identity (eqs. (A. 12)) one obtains the following relation between the three characteristic functions: C (“’(y)

=

exp ( - i y * y )~ ( “ ( y ),

C ( Q ) ( y= ) exp( - i y * y ) ~ ( ~ ) ( y ) .

(A. 16a) (A. 16b)

16

[I

QUANTUM FLUCTUATlONS IN OPTICAL SYSTEMS

Distributions 9,W, and 9 are then related by convolution product (denoted by @) W ( z )= W&) 63 P ( Z )

(A. 17a)

Y

(A. 17b)

2(z),= W((Z)@W ( Z ) ,

where W,(z) is the Fourier transform of exp( - $y*y) and is a Gaussian function. Since the characteristic function C(')(y) is equal to one for the particular state 10, C), W,(z) can also be considered as the Wigner distribution associated with the squeezed vacuum state 10, 5 ) . The three distributions are then increasingly regular, from 9 to 2. For example, in a squeezed state ( z , 5) the 9-distribution is a delta function, whereas the Wigner and %distributions are Gaussian functions. Furthermore, in any state the &distribution is always positive.

A.2. SEMICLASSICAL EQUATIONS OF EVOLUTION

We shall first examine the evolution equations corresponding to a general Hamiltonian H, and then discuss the effect of a quadratic Hamiltonian. A.2.1. Hamiltonian evolution in the Wigner representation Using the Glauber identity [eqs. (A.l2)], the derivatives of the operator V(u, u) = exp(iPu

+ iQu)

(A.18)

can be written as -i

a,

U(u, u )

= -i =

a, exp ( - iuu/2) exp (iPu) exp (iQu)

(P- u/2) U(u, u) .

(A. 19a) (A. 19b)

One then obtains ( - i a,

+ ~ / 2 U(U, ) U) ,

(A.20a)

W(U, u ) P = ( - i a, - 42) U(U,U) ,

(A.20b)

Q U ( U , U) = ( - i a, - 24/21 W ( U ,

(A.20~)

P V(U,U)

=

and the similar relations

V(u, u)Q = ( - i a,

U) ,

+ 4 2 ) U(u, u) .

(A.20d)

I1

71

APPENDIX

The evolution equation of the characteristic function CCw)(u, u) can be deduced in the following way. First, we write the Hamiltonian H in symmetrical order with respect to the operators P and Q. Second, the evolution of C(w)(u,u) is given by the equation a,C(w)(u, 4 = i ( [ H ( P ,Q), Wu, Third, using eqs. (A.20), the operators expressions. One then obtains

a,

u)

=

41) .

(A.21)

P and Q can be replaced by differential

yW C ( W ) ( u) ~ ,,

(A.22a)

where the differential operator Y is given by

Yw = i[H( - i a, + 42, - ia, - 4 2 ) - H( - i a, - u/2,

-i

a, + u / 2 ) ] . (A.22b)

The evolution equation of the Wigner distribution is obtained by Fourier transform

a, W P ,4) = Y w W P , 4) 9, = i[H(p

+ i a,/2,

(A.23a)

Y

q - i aJ2) - H ( p - i a,/2, q

+ i ap/2)] .

(A.23b)

In the case of a single particle in a potential V(Q): H ( P , Q) = 4 P 2 +

VQ),

(A.24)

the differential operator associated with the Wigner distribution can be written as y w = -Pa,-

cfm(4)a,m+',

(A.25)

where fm(q) are the odd terms of the Taylor expansion of V(q).The first terms ( - p a,) and ( -fo(q) a,) correspond to the classica! equations of motion ar4

=

P

1

alp

=

fo(4) *

(A.26)

The next terms are quantum corrections. Assuming that Aq and A p represent the variation length of the potential and the momentum width of the Wigner distribution, respectively, it appears that the quantum terms are of the order of ( A q A p ) - *"'. Quantum corrections become negligible when (Aq A p ) is large, i.e., in the quasiclassical limit (Wigner [ 19321).

78


A.2.2. Hamiltonian evolution in P- and %representations The evolution equations of the 9-and .%distributions can be deduced in a similar way. Writing the Hamiltonian H(A!, Ac) in symmetrical order with respect to the operators A, and A!, one obtains:

a,p = yPs,

(A.27a)

yP = i[H(z*, z - a,*) - H(Z* - a,, z)] ,

(A.27b)

a t 2 = PQ9,

(A.27~)

yQ= i[H(z* + a,, Z) - H(z*, z

+ a~ .

(A.27d)

To compare with the Wigner distribution, the evolution equation of W can be written using the same notation,

yW = i[H(z* + a ~ 2z, - a,,/2)

-

H(Z* - a,/2, z + a,.p)l.

(~.28)

A.2.3. Quadratic Hamiltonian It is particularly interesting to compare these evolution equations in the case of ideal quadratic Hamiltonians. For a linear Hamiltonian or a quadratic Hamiltonian without squared terms (A: and A12): H

=

kA!

+ k*A, + IAJA, + IAcAI,

(A.29)

the evolution is classical for all distributions

yW = scP = yQ= ik a, - ik* a,.

+ 2i1( - Z* a,. + za,).

(A.30)

In the case of quadratic Hamiltonians with parametric terms proportional to A t and AJ2: H

=

mA12 + m*A:,

(A.3 1)

the evolution for the Wigner distribution is still given by the classical differential operator

ZW= 2 imz* a,

-

2 im*z a,. .

(A.32a)

For the 9-and Z!-distributions, however, the evolution operators contain extra terms

yP = yW - irn at

+ im* a $ ,

yQ= yw+ im a: - im* a:, .

(A.32b) (A.32~)

I1

APPENDIX

19

These terms are second-order derivatives terms, similar to diffusion terms in the usual Fokker-Planck equations (Yuen and Tombesi [ 19861). These quantum diffusion terms are associated with the quantum fluctuations of the field. Their expressions are related to the chosen semiclassical distribution, however, in particular to the order convention of quantum moments. Furthermore, these terms disappear in the Wigner representation, so that this distribution is very useful for obtaining satisfactory semiclassical equivalences (Wigner [ 19321, Schmid [ 19821, Koch, Van Harlingen and Clarke [ 19821, Heidmann, Raimond and Reynaud [ 19851, Ekert and Knight [ 19901). A.2.4. Canonical transformations We shall now consider a linear transformation of the P- and Q-operators that preserves the commutation relations P’

=

aP

+be,

Q‘

=

cP

+ dQ ,

(A.33a)

with ad-bc= 1.

(A.33b)

From the definition of the Wigner distribution and of its characteristic function [eqs. (A. lo)], it appears that the transformed functions are given by (A.34a)

C ’ ( u ’ ,v ’ ) = C(U, v ) ,

W ’ ( P ’ ,4 ’ ) =

W P , 4)

(A.34b)

3

with the transformation of variables u = au’

+ cv‘

p

-

=

dp’

,

bq’ ,

v = bu’

+ dv’ ,

q = -cp’

+ aq’ .

(A.34~) (A.34d)

In other words, the canonical transformation of the Wigner distribution is given by the classical propagators. This should be related to the discussion in the previous section, since such a canonical transformation is associated with a quadratic Hamiltonian. In contrast the canonical transformation of the 9and 2-distributions is classical only if the operators A , and A ! are not mixed in the transformation.

80


A.2.5. Parametric generation Finally, we shall consider the parametric downconversion process, where a signal mode of frequency w, is generated by a pump mode of frequency 20, inside a non-linear medium. Assuming that the medium can be described by an effective non-linear index X , the Hamiltonian is (A.35)

H = ( - i h ~ / 2(af2a, ) - aT;a:),

where a , and a , are the annihilation operators of the signal and pump fields, respectively. Compared with the ideal quadratic Hamiltonian [ eq. (A.3 l)], H takes into account the effect of the pump field, through the operators a , and a:. In this case the Heisenberg equations for operators a, and a , are no longer linear in the field operators (A.36a) (A.36b) The evolution equation for the Wigner distribution is given by

a,w= (2/2)(2a;raoa,, - aF2a,,

-

82,,a,.#)W+

c.c.,

(A.37)

where a, and a, are the classical variables associated with the operators a , and a,. The third term in this equation represents a quantum correction to the classical evolution. The existence of such a term is not surprising, since the Hamiltonian is of the order three with respect to the operators a , and a,. Nevertheless, such a term can be neglected in some situations. The simplest situation is when the pump field is much more intense than the signal field. In this case one can neglect the back reaction of the signal field on the pump field. The operators a , and ad can then be replaced by c-numbers, corresponding to a classical pump field imposed from the outside. The Hamiltonian takes the form of an ideal quadratic Hamiltonian, associated with parametric amplification and deamplification (8 4.3), with the scaling factor

q = Xa,t.

(A.38)

This case is encountered in optical parametric amplifiers pumped by a strong field, where the signal field is assumed to be in the vacuum state at initial time. When the signal field amplitude has a mean value that is comparable to the pump field, such an approach is no longer valid. This situation is encountered, for example, in optical parametric oscillators where the back reaction of the signal field on the pump field ensures the existence of a stationary state.

11

REFERENCES

81

Nevertheless, one can use a semiclassical method (Reynaud and Heidmann [ 1989]), where the quantum terms are neglected. This can be understood using a qualitative argument; namely, in the limit where fluctuations are small compared with the mean fields, the evolution equations for the fluctuations can be linearized. These equations are then associated with an ideal quadratic Hamiltonian, and the evolution of the Wigner distribution is classical. In other words, with such a linear treatment of the fluctuations, the evolution of these fluctuations is described by the classical equations of motion. A more rigorous derivation of the semiclassical method can be achieved using a cumulant expansion of the equations of motion (Van Kampen [ 1974, 19811, Heidmann, Raimond, Reynaud and Zagury [ 19851).

Acknowledgements This work has been supported, in part, by the Direction des Recherches et Etudes Techniques (contract no 87/091) and the European Economic Community (Contract ST2J0278C and ESPRIT Basic Research Action NOROS 3 186). Special thanks are given to J. Y. Courtois, J. M. Courty, T. Debuisschert, L. Hilico and J. Mertz for their helpful contributions to this work.

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E. WOLF,PROGRESS IN OPTICS XXX 0 ELSEVIER SCIENCE PUBLISHERS B.V., 1992

CORRELATION HOLOGRAPHIC AND SPECKLE INTERFEROMETRY BY

YU.I. OSTROVSKY~ loffe Physical-Technical Institute, Academy of Sciences of Russia. 194021 St. Petersburg, Russia

V. P. SHCHEPINOV Physical Engineering Institute 115409 Moscow,Russia

87

CONTENTS PAGE

$1.

INTRODUCTION

. . . . .

,

. . . . . . . . . . . . . 89

8 2. FRINGE CONTRAST

IN HOLOGRAPHIC INTERFEROMETRY AND SPECKLE PHOTOGRAPHY AS RELATED TO A CHANGE IN SURFACE MICRORELIEF

. . . . . $ 4. CORROSION, EROSION, AND WEAR PROCESSES . . $ 5 . CONCLUSIONS . . . . . . . . . . . . . . . . . . . REFERENCES . . . . . . . . . . . . . . . . . . . . . . $3.

MECHANICSOFCONTACTINTERACTION

88

. .

90 103 121

. 133

.

134

8 1. Introduction The contrast (visibility) of interference fringes in holographic interferometry and speckle photography (one of the major methods of speckle interferometry) depends, among other factors, on the degree to which the elements of the microrelief of the surface under study have changed. In the case of holographic interferometry a change in the surface microrelief between two exposures results in the decorrelation of the reconstructed light waves, causing a reduction in the fringe contrast. In speckle photography a change in surface microrelief leads to decorrelation between the two speckle structures of the images of the object and, hence, to a reduced contrast of the fringes (e.g., Young’s fringes). Although these factors, on the one hand, impose constraints on the applicability of the methods of holographic interferometry and speckle photography to measurements of the displacements of objects with diffusely reflecting surfaces, on the other hand, they permit the development of new techniques for studying processes and phenomena that can affect the microrelief of the surface of interest from variations in the fringe contrast. In this article we shall call such methods correlation holographic interferometry and correlation speckle interferometry (correlation speckle photography). This paper consists of two parts. The first derives expressions for the fringe contrast in correlation holographic interferometry and in correlation speckle photography for random variations of the surface microrelief. It evaluates the sensitivities of the two methods, and considers the various optical arrangements that permit a visualization of the surface areas where a variation in the microrelief takes place. These arrangements involve the use of carrier fringes or the optical-image subtraction effect. The second part examines practical applications of these methods. Correlation holographic interferometry and correlation speckle photography can be used in two ways to, solve problems. One way consists in constructing a complete mathematical model of the process (phenomenon) under investigation, including the relations governing the variation of the microrelief elements, whereas the other way entails establishing an empirical correspondence among the physical, mechanical, and other parameters characterizing the process in question and the fringe contrast. 89

90

CORRELATION INTERFEROMETRY

[II, 5 2

Since the second approach is the simplest, we shall first consider its application to problems in the mechanics of contact interaction, such as the evaluation of the plastic component of the true contact surface, measurement of the contact contour surface, and determination of contact pressures. These parameters have to be known when calculating the friction and wear forces or evaluating the contact strength and contact fatigue strength. The first method using correlation holographic interferometry is illustrated by solving problems such as the determination of the chemical corrosion rate and the investigation of the cavitation-induced erosion. The possible study of the process of mechanical wear is also examined.

0 2. Fringe Contrast in Holographic Interferometry and Speckle Photography as Related to a Change in Surface Microrelief 2.1. FRINGE CONTRAST IN HOLOGRAPHIC INTERFEROMETRY

Consider the recording of a holographic interferogram of an object with a rough surface by the two-exposure technique. We assume that after the first exposure that part of the surface of the object undergoes a change in its microrelief caused by a surface process, for instance, by mechanical erosion or chemical corrosion. Denote by u , and u2 the complex amplitudes of the reconstructed light waves reflected from the surface under study before and after the change in its microrelief uI = a , exp(-icp,),

u,=a,exp[-i(cp,

- II/+cpo)l.

(2.1)

Here a, and a2 are the amplitudes of the reconstructed light waves, (p, is the phase of the object wave during the first exposure; cp2 is the phase after the action on the surface; 'p, - cp2 = II/ is the change in the object wave phase caused by the change in the surface microrelief; and cpo is a regular phase variation introduced into the object wave to produce carrier interference fringes localized on the surface of the object. Such a pattern is sometimes called the finite-width fringe interferogram. The interference pattern observed under simultaneous reconstruction of object waves (2.1) recorded on a hologram is described by an intensity distribution I,,

11, B 21

FRINGE CONTRAST AND CHANGE IN MICRORELIEF

91

where the brackets (. . .) denote averaging over an area on the surface of the object corresponding to the limiting resolution of the optical system used to observe the object. We shall assume that the size of this area substantially exceeds the characteristic transverse size of the surface microinhomogeneities, i.e. the optical system is not capable of resolving individual details in the microrelief. If we also assume the amplitudes a , and a,, and phases cp, and cp2, to be statistically independent, eq. (2.2) will transform after some straightforward algebra to IH =

+ 2(ala2)

(a?) +

(cOs(VO +

$1)

*

(2.3)

In the following discussion we shall assume that the random nature of the change in surface microrelief affects only the phase of the reflected wave, i.e. a , = a 2 . With this in mind, eg. (2.3) becomes 1"

-

1+

(COS(cp0

+ $1)

(2.4)

*

Equation (2.4) can be transformed in the following way: I,

=

1 + (cos $) cos cpo - (sin $) sin cpo

= 1

+J < o s

$)2

+ (sin +)

cos [cpo

+ tan-

I

((sin $)/(cos + > ) I

One can see that the contrast 7, of the carrier holographic interference fringes

and for the additional phase shift Acpo caused by the change in the surface macrorelief and affecting the change of the carrier interference fringes, we obtain

Avo = tan- ((sin $)/( cos $)) .

(2.6)

Let us now consider the term (cos $) in eq. (2.5). First, we change from averaging over an area containing a large number of microrelief elements to ensemble averaging. To do this, we introduce a distribution function f(q), where q is the displacement vector of a point on the surface of the object originating from a change in the surface microrelief. The vector q is related to the phase variation through the principal relation of holographic interferometry (Ostrovsky, Butusov and Ostrovskaya [ 19801, Vest [ 19791, Ostrovsky, Shchepinov and Yakovlev [ 19911):

92


PI, 8 2

where e, and el are unit vectors in the illumination and observation directions for the point in question on the surface of the object; 1 is the wavelength of the light. Taking into account the preceding principal relation, the expression for (cos $) takes on the form

s-

a3

(cos $)

=

cos m

(-1

2x

The function f ( q ) is, generally speaking, unknown; its form being determined by the pattern of the irreversible change of the surface microrelief elements caused by the process under study (e.g., erosion and corrosion). In a similar way one can derive the expression (sin$)

= Ja

sin(”

-a

(e, - e , ) * q

A

(2.8)

Thus, according to eq. (2.5) a random change of the surface microrelief results in a decrease of the contrast of the carrier fringes and in a change of their geometry. 2.2. YOUNG‘S FRINGE CONTRAST IN SPECKLE PHOTOGRAPHY

The arrangement to record speckle photographs and the accepted notations of the vectors used in this subsection to analyze the fringe contrast are shown in fig. l a (Osintsev, Ostrovsky, Presnyakov and Shchepinov [ 19921). The vector C lies in the plane of the object, and the vector r, in the plane of the speckle photograph. The microrelief of the surface area under study is assumed to be plane. The complex amplitude u(C) of the light wave reflected from the surface of the object can be written in the form

4t) = 4 C ) exp [i cp(C)l

9

where a ( { ) and q(4) are the amplitude and phase of the wave, respectively. Let the speckle structure of the image of the initial surface state be recorded in the first exposure on a photographic plate. We denote the intensity distribution in the plane of the plate by Yl(r). Now we assume that the process under study produces a random change in the microrelief on part of the surface of the object. After this action the complex amplitude u,(C) of the light wave can be written as =

4 4 ) exp [i $({)I

9

11, I 21

FRINGE CONTRAST A N D CHANGE IN MICRORELIEF

bbject

Speckle-photograph

93

Speckle - photograph

Screen

Fig. 1. Notation and orientation for speckle photography: (a) recording layout for a speckle photograph; (b) observation of Young’s fringes.

where $(C) is a random function describing the change in the phase of the light wave. Note that the random functions q(c) and $({) are uncorrelated, i.e. (dt)$(C)) = 0. The second exposure of the speckle photograph is made after the action on the surface under investigation, and after the photographic plate has been shifted in its plane in the direction specified by the vector d. The intensity distribution recorded in the second exposure will be denoted by J2(r). When scanning the doubly exposed speckle photograph obtained in this way with an unexpanded laser beam (fig. lb), on the screen one will observe Young’s fringe pattern with the same fringe orientation and a period determined by the direction and magnitude of the displacement of the photographic plate. In the regions of the speckle photograph corresponding to the distorted areas of the surface, the fringe contrast will decrease. The ensemble-averaged intensity I,(w) in the screen plane can be written in the form (Dainty [ 19751):

94

[II, § 2


where o is a vector in the screen plane and F(w) is the complex amplitude of the light wave in the screen plane. We rewrite eq. (2.9) in the following way: I,(@)

I d 4 + I Z ( 4 + 1l2(4 + I2l(4

=

(2.10)

9

where

J J J

VI

Id4

=

{exp[io*(r, - r2)I) ( 4 ( r 1 ) 4 ( r 2 ) ) dry

(2.11)

{exp[io.(r, - r2)I) < 4 ( r 1 ) 4 ( r 2 ) ) d r ,

(2.12)

-x. -w

I,(@)

=

--w w

{exp[io.(r, - r2)I) ( 4 ( r , ) 4 ( r z ) > d r ,

IIZ(4 =

(2.13)

-32

121(4 =

I;F2(4.

(2.14)

We assume the optical system in fig. l a to be a linear, spatially invariant system with an impulse response h(r - C). Then the expression for the intensity I l ( r l ) can be transformed to

-m

For the autocorrelation function of the intensity, I,(r), we now have rn

(&(rl)e91(r2))

=

(U(Cl)U*(C2)U(c3)U*(C4))

h(rl - eI)h*(rl - t 2 )

- w

h(rZ - &3)h*(r2- C4) d C l

dC2 d C 3

dC4.

(2. 15)

The fourth-order correlation moment of the complex amplitude of the light wave can be expressed in terms of the second-order correlation moments as follows (Rytov [ 19761): ( U ( C l )U*(C2)U(c3)U *(C4))

=

( '(CI

*({2))

( U(C3)U*(C4))

+ U(c2)u*(C4)) (U(C3)U*(C2)) . (2.16)

For the &correlated complex amplitude u(C), expression (2.16) can be rewritten as

(2.17)

1198 21

FRINGE CONTRAST AND CHANCE IN MICRORELIEF

95

where 6 ( .. .) is the Dirac delta function. Taking into account that the object is illuminated uniformly (( lu(t)I2) = 1) and substituting (2.17) in (2.19, we come to

+

j

O0

h(r,

-

t 1 ) h * ( r 2- t l ) d t l

--oo

sIrnrn h*(rl

-

Cf3)h(r2

-

t3)

dt3.

(2.18)

Assuming that the impulse response is normalized to unity, eq. (2.18) can be transformed to

( 4 ( r 1 ) * 4 ( r 2 ) )= 1 + l r ( r l where

1-

-

%)I2?

(2.19)

00

r ( r , - r2)=

00

h(rl

-

t ) h * ( r 2 - t) dC.

Substituting eq. (2.19) in eq. (2.11) and making the change of variables, rl - r2 = q and r I = r l , we obtain

S-

00

Il(4 =

[ e x p ( i o * r 2 ) [ 1+ I T ( q ) 1 2 ] d q d r , - 6 ( o ) + H ( o ) ,(2.20) m

where H ( o ) is the Fourier transform of the function I r(q)12. Assuming that random changes in the microrelief of a surface do not affect its statistical properties, we obtain

(4(r1)$2(4)

=

(4(r1)4(r2)).

Hence, eq. (2.12) takes on the form I,(@)

-

6(o) + H ( o ) .

(2.21)

In a similar manner one can derive an expression for the intensity crosscorrelation function

( 4 ( r 1 ) 4 ( r 2 ) ) = 1 + I<exp(ill/))l2 IWI

-

r2

- d)I2.

After substituting this expression in eq. (2.13) and changing the variables, rl - r2 = q and r l = r l , we come to

II2(o)-6(0)+ I(exp(ill/)))'[exp(io.d)H(o)

=

I$,(o).

(2.22)

96

[II, Q 2


Substituting eqs. (2.20), (2.21), and (2.22) in eq. (2.10) yields

Is(o)-46(o)

+ 2H(o) + 2H(o) I(exp(i$))I'cos(o.d).

(2.23)

After filtration of the bright spot at the centre of the screen, which is described by the delta function 6(o), eq. (2.23) takes on the form I ~ ( W ) -~

( o 1)+[I ( e ~ p ( i $ ) ) 1 ~ c o s ( o * d ) ] .

(2.24)

Equation (2.24) can be used to derive an expression for the Young's fringe contrast y s : Ys = I s m a x

I S max

I Sm'n + I S min -

'

=

I(exp(i$))I'.

If p ( $ ) is the density function of random-phase variations, then Ys =

CS"

[exP(i$)lP($)d$/2)

= (COS$>'

+ (sin$>*.

(2.25)

-m

A comparison of expressions (2.5) and (2.25) yields Ys

=

YZ

'

(2.26)

Thus the method of correlation speckle photography has a higher sensitivity to changes in the surface microrelief than the correlation holographic interferometry.

2.3. VISUALIZATION OF AREAS IN WHICH THE SURFACE MICRORELIEF VARIES

2.3.1. The use of carrier fringes The areas of arbitrary shape where the surface microrelief has been affected by an external action can be efficiently revealed by holographic interferometry. Irreversible plastic deformation of microrelief elements on the surface of an object results in decorrelation of the light waves reflected from it and, as a consequence, the contrast of the carrier interference fringes decreases down to complete disappearance. Holographic interferograms are best recorded by the two-exposure technique, which is capable of yielding high-contrast interference fringes. One optical arrangement for hologram recording is shown in fig. 2. The carrier interference fringe pattern forms as the mirror M 1 turns through a small

1198 21

FRINGE CONTRAST A N D CHANGE IN MICRORELIEF

91

Fig. 2. Optical arrangement for hologram recording involving a change in the direction of object illumination (LA, laser; BS, beam splitter; K, collimator; 0, object; L, lens; M1,M2, mirrors; H, hologram).

angle 8 between two exposures. The change in the object illumination direction caused by the turn of mirror M1 produces rectilinear equidistant fringes localized on the surface of the object. Strictly speaking, exactly straight and absolutely equidistant fringes can form only if the surface of the object is plane and the illuminating beam is normal to the surface. The orientation and pitch of these fringes depend on the direction and angle of turn of the mirror M 1. Note that a change in the direction of illumination between the exposures also results in decorrelation of the light waves reflected from the surface of the object. This decorrelation, however, is substantially less than that originating from a change in the microrelief of the surface of the object. Carrier interference fringes can also be produced by using the effects associated with a displacement or deformation of an object. Deformation frequently accompanies the change in surface microrelief. In general, however, this method of fringe production is technically more complex. Apart from this, one should bear in mind that, as shown by Ostrovsky, Butusov and Ostrovskaya [ 19801, a displacement of the object relative to the observation point leads in itself to a substantial decorrelation of the interfering fields which exceeds that due to the displacement of the object with respect to the light source. The areas with distorted surface microrelief can also be studied by twoexposure speckle photography. An irreversible change in the surface microrelief results in the decorrelation of two speckle images recorded on the speckle photograph. Just as in the method of holographic interferometry, the microrelief decorrelation areas can be derived from the decorrelation of the image speckle structures using carrier fringes. Figure 3a shows an optical arrangement for recording speckle photographs.

98


s

Screen

Fig. 3. Optical arrangements for speckle photograph recording and interference fringe observation. (a) Recording ofspeckle photographs (W, plane wavefront; 0,object; L, objective lens; SF, speckle photograph). (b) Observation of Young’s fringes (LB, laser beam; S, screen). (c) Spatial filtration outside Fourier plane (A, mask with hole).

The object surface area under study, 0, is illuminated by a plane wave W of coherent laser light. An objective lens L projects a focused image of the object’s surface onto the photographic plate SF. In the first exposure the speckle structure of the image of the surface in the initial state is recorded on the photographic plate. Next, an action (corrosion, erosion, wear, etc.) is applied to the surface of the object, which results in an irreversible change of its microrelief, the plate SF is displaced in its plane by an amount d, and the second exposure is made. Thus, the speckle photograph will record two speckle

KO 21


99

structures of the surface image shifted with respect to one another. Note that the magnitude of d should be somewhat greater than the minimum speckle dimension; i.e. d > 1.21f / D , where f is the focal length of the objective lens, D is the lens aperture, and 1is the wavelength of the light. When any part ofthe doubly exposed speckle photograph S F obtained in this way is illuminated with an unexpanded laser beam LB (fig. 3b), an interference pattern of Young’s-type fringes of the same frequency and orientation will be observed on the screen S . At the points of the image where the surface microrelief has changed, the contrast of the Young’s fringes will be less sharp (relative to the unaffected parts of the surface), or the fringes will not be discernible at all. Thus, by scanning the speckle photograph with an appropriate pitch, one can determine the region where the contrast of the Young’s fringes varies and, hence, where the surface microrelief has undergone a change. Speckle photography also permits one to visualize the entire area of the variation in surface microrelief against the background of carrier fringes (Klimenko, Kvartskheliya, Volkov and Golikova [ 19811). In this case the doubly exposed speckle photograph is fixed in the spatial filtration configuration shown in fig. 3c. A small-diameter circular aperture is placed in plane A located at a distance from the Fourier plane (the focal plane of lens L). Through the aperture one observes a system of parallel straight carrier fringes, their orientation and pitch depending on the aperture position in the filtration plane and the distance from the Fourier plane to the latter. The areas of fringe contrast variation correspond to those where the surface microrelief undergoes a change, as occurs with holographic interferograms. 2.3.2. Image subtraction The optical arrangements discussed in the preceding section permit visualization of the areas of surface microrelief distortion as a set of discrete lines or points. To record the continuous boundary of the surface microrelief distortion area, one can use the so-called image subtraction configurations (Gabor, Stroke, Restrick, Funkhouser and Brumm [ 19651, Collins [ 19681, Metherell, Spinak and Pisa [ 19691). They provide a possibility of introducing a phase shift of R between two exposures throughout the field studied. In this case one obtains a dark-field interferogram, the surface areas with distorted microrelief being visualized as bright spots of different intensity against the dark background. Figure 4 illustrates one of the simplest methods of holographic image subtraction. In the first exposure the hologram of the initial state of the surface of

100


Fig. 4. A hologram recording involving a change in direction of the reference wave direction (LA, laser; BS, beam splitter; K, collimator; M1, M2, mirrors; 0, object; H, hologram).

the object (0)is recorded on a photographic plate H. Next, mirror M2 is turned through a small angle 0, and the surface is subjected to an outside action that changes its microrelief, after which the second exposure is made. A change in the direction of plane-wave incidence between the two exposures will result in the formation of straight equidistant fringes localized on the surface of the hologram H. Reconstruction of the real image of the surface of the object by illuminating the hologram H with an unexpanded laser beam through the center of a dark fringe realizes the case of image subtraction. The image of the surface will be dark everywhere except for the areas with a distorted microrelief, which will be bright. Another method of holographic image subtraction is based on the use of Fourier holograms (Klimenko and Ryabukho [ 19851). Figure 5 shows an optical arrangement for recording a Fourier hologram H of the surface of object 0. The hologram does not record the image of the object but rather a spatial spectrum of the object wave. Lens L, performs a Fourier transform of the object field. A visualization of the area with a distorted surface microrelief by the doubleexposure method is carried out in the following way. In the first exposure the spectrum ofthe object field is recorded on a hologram. Next, a linear phase shift is introduced by turning the reference wave through an angle 8, the surface is acted on, and the object wave spectrum is recorded again on the hologram (the second exposure). The existence of a phase shift between the exposures results in the spatial spectrum of the object field becoming modulated by straight equidistant fringes. If we now illuminate this hologram with an unexpanded laser beam through the center of a dark fringe (fig. 5b), the region with distorted microrelief will be visualized in the form of bright spots against a dark background. In the configuration shown in fig. 5b, lens L, performs an inverse Fourier transformation of the reconstructed field.


101

Fig. 5. Image subtraction by means of Fourier holograms. (a) Fourier hologram recording (W, , W,, plane waves illuminating object 0 and photographic plate; L, ,lens; H, hologram). (b) Light wave reconstruction (L,, lens; S, screen).

Regions with distorted microrelief on objects with a plane surface can be visualized in the image subtraction arrangement, which is based on speckle photography (Debrus, Francon and Grover [ 19711). The doubly exposed speckle photograph S F recorded in the optical arrangement shown in fig. 3a is mounted in the optical configuration for spatial filtration (fig. 6 ) . If the photographic plate is displaced between two exposures by an amount d , the intensity distribution in the focal plane of lens L (Fourier plane) will be modulated by parallel interference fringes. If we now place in the Fourier plane a narrow slit with its axis positioned at the center of a dark fringe, the image seen through it will be dark (the case of image subtraction) except for the areas with a distorted surface microrelief, which will be bright. The brightness results because the speckle structures corresponding to these surface areas are partially or completely decorrelated. 2.3.3. Simultaneous recording of holograms and speckle photographs In some cases the different sensitivities of correlation holographic interferometry and correlation speckle photography require their combined use in

102


Fig. 6. Image subtraction by means of speckle photography (L, lens; SF, speckle photograph; A, mask with slit).

problems associated with changes in surface microrelief (Osintsev, Ostrovsky, Shchepinov and Yakovlev [ 1991, 19921). An optical arrangement permitting the simultaneous recording of a hologram and a speckle photograph of the surface under study is presented in fig. 7. The hologram and speckle photograph are obtained by the two-exposure method in the following order. First, one records on a photographic plate SF the initial speckle structure of the surface image (first exposure of the speckle photograph). Second, a photographic plate H is introduced into the optical arrangement, and a hologram of the initial state of the surface is recorded on it (first exposure of the hologram). During this time the photographic plate S F is screened by an opaque mask A. Third, the surface of the object is subjected to an action affecting its microrelief. Fourth, the mirror M2 is turned through a small angle 0, and a hologram of the distorted surface is recorded on the plate H (second exposure of hologram). Fifth, photographic plate H and mask A are removed, photographic plate SF is displaced in its plane by an amount d, and the speckle structure of the image

Fig. 7. Optical arrangement for simultaneous recording by the double-exposure method of holograms and speckle photographs (LA, laser; BS, beam splitter; M1, M2, M3, M4, mirrors; K, collimator; 0, object; L, lens; A, mask; H, hologram; SF,speckle photograph).

11,

I 31

MECHANICS OF CONTACT INTERACTION

103

of the object after the action is recorded on it (second exposure of the speckle photograph). Note that with the operations performed in this sequence, the directions of the illuminating beam during the first and second exposures of the speckle photograph differ somewhat because of the amount of rotation of the mirror M2 necessary to produce carrier fringes in the holographic interferogram. One could naturally return the mirror M2 into the initial position before carrying out the second exposure of the speckle photograph. As shown by calculations and experiments, however, decorrelation of the speckle structures in this case is negligible, since the tilt angles of the illuminating beam necessary to produce carrier fringes in holographic interferometry do not exceed a few minutes of arc. The surface of the object can be acted on either directly in the optical arrangement or outside it, if special kinematic fixtures (Maklead and Kapur [ 19731, Furse [ 19811) are used. Thus the same change in surface microrelief can be recorded with the preceding arrangement both by correlation holographic interferometry and by correlation speckle photography.

4 3. Mechanics of Contact Interaction 3.1. DETERMINATION OF THE CONTACT AREA

3.1.1. Evaluation of the plastic component of actual contact surface

In the mechanics of the contact interaction of objects with rough surfaces, one deals with the actual contact surface area and the contact contour surface area. The actual, or true, contact surface area represents a sum of the areas of all contacting elements in the surface microrelief of the objects involved. The contact contour surface area is the area of the surface of the object bounded by the contour of the contacting regions. Obviously, the actual contact surface area is less than the contact contour surface area. In its turn, the former may be represented as a sum of the elastic and plactic components. In fact, in the contact interactions of objects with rough surfaces, under loading their contacting microrelief elements undergo elastoplastic interaction. After unloading, the actual contact area decreases because of the removal of the elastic deformation. Atkinson and Lalor [ 19771 were the first to use holographic interferometry to measure the plastic component of the actual contact surface area. Plastic deformation of the surface microrelief elements results in decorrelation of the light waves scattered from the surface before and after the contact interaction,

104

PI, § 3


which leads to a decrease of the carrier holographic fringe contrast y H . Atkinson and Lalor [ 19771 showed that in the first approximation one may assume the change in the carrier fringe contrast yH to be equal to the ratio of the area F of the plastic component of the actual contact area to the area Fo of the contact contour surface Fo - F YH=-FO

F

-1---. FO

The plastic component of the actual contact surface area can also be measured by speckle photography (Osintsev, Presnyakov and Shchepinov [ t9901). We now turn to fig. l a and denote the light intensity distribution in the plane of the speckle photograph in the first exposure by 4(r), and that in the second exposure, by 92(r). Denote by uI(C) and uz(C) the complex light amplitudes at the surface of the object in the first and second exposures, respectively. Then the expressions for the complex amplitudes uI(r) and u2(r) in the plane of the speckle photograph can be written in the form OI(Y) =

s,

u,(C)h(r -

C) d t

7

u2(t)h(r -

C) d t

7

where h(r - C) is the impulse response of the system and Fo is the area of the scanning beam, which, under unit magnification in the stage of speckle photograph recording, is equal to the area of the corresponding region on the surface of the object under study. The expression for u2(r) can now be rewritten as u2(r)

=

IF

uF(C)h(r

= O2F

u20

9

-

C) dC

+

jFo-

u,(C + W ( r -

t) d t (3.2)

where uF is the complex amplitude of the light wave reflected from the plastically deformed surface microrelief elements (it is assumed that ( uFul ) = 0); F is the area where the microrelief has changed. By analogy the expression for uI(r) can be written in the form

I I , 31 ~

105

MECHANICS OF CONTACT INTERACTION

Taking into account eqs. (3.2) and (3.3) for the intensities Il(r) and 12(r) we shall obtain the following expressions:

M2+ IUl0l2+

-ol(r)= Iu,(r)12=

= Iu2(r)I2 = l U2 F 1 2

VIFGtO+ V;rFU,O,

+ Iu2ol2 +

U2FGO

+

UZ*F~20,

(3.4) (3.5)

Obviously, uz0(r) = ul0(r + d), where d is the displacement vector of the photographic plate between the exposures. According to eq. (2.9) the intensity distribution [,(a)in the screen plane with a doubly exposed speckle photograph illuminated by an unexpanded laser beam (fig. lb) will be

j j

co

Is(4 =

{ e x p [ i d r , - r7)Il (I@,) I@,)) dr, dr2

-m 00

=

{exp[ia(r, - r2)l) ( < 4 ( r 1 ) 4 ( r 2 ) ) + < 4 ( r 1 ) 4 ( r 2 ) )

-a

+ 2 = 4[ij:(a2 + b 2 ) 2 ] - ’

By using (2.6), for the additional phase shift Avo we obtain

Avo = tan-

I

[2ab/(a2 - b 2 ) ].

We define the mean corrosion rate V as

where z is the corrosion time. Using the expression for fringe contrast yH, we can derive a formula for the mean corrosion rate

In the study by Petrov and Presnyakov [ 19781 mean corrosion rates varying from 24 to 2.3 A s - ’ corresponded to the fringe contrast range of 0.1 to 0.9 for a corrosion time of z = 150 s. The mean corrosion rate ij, which represents the rate of displacement of the mean microrelief level, can also be determined by conventional holographic interferometry from the change in the interference fringe shape resulting from the additional phase shift Avo. However, the sensitivity of such measurements is considerably lower. Osintsev, Ostrovsky, Presnyakov and Shchepinov [ 19921investigated corrosion of the sample by both correlation holographic interferometry and correlation speckle photography. A sample of the aluminium alloy D16T with an original irregular roughness R , = 1.50 pm, whose edges were acted on by an alkali solution between the exposures, was studied in the optical arrangement shown in fig. 7. Figure 25 is a typical holographic interferogram of the surface of the sample. Although the interferogram reveals a substantial loss of fringe contrast, no fringe bending is observed. In fig. 26 one can see longitudinal distributions of the normalized fringe contrast y / y o (where yo is the fringe contrast outside the corrosion zone). Curve 1 was obtained by holographic

123

CORROSION, EROSION A N D WEAR PROCESSES

Fig. 25. Holographic interferogram of specimen surface aRer corrosion.

interferometry, and curve 2, by speckle photography. As follows from an analysis of these curves, relation (2.26) is well satisfied in this case.

4.2. INVESTIGATION OF CAVITATION-INDUCEDEROSION

The cavitation resistance of samples can be studied in different types of laboratory set-ups. The relative cavitation resistance is determined by comparing either the losses in weight for the same intensity of cavitation impact or the time required to reach the same weight losses. Such tests are generally very long and may take up hundreds of hours. To reduce the time required to determine the zones and intensity of cavitationinduced erosion, one usually employs the method involving easily destroyed

1.0

0.8 0.6 0.4

-

0

20

40

60

X,M M

Fig.26. Change in fringe contrast over specimen surface caused by corrosion, which was obtained by: ( I ) , holographic interferometry; and (2), speckle photography.

124

PI, 5 4


coatings. In this case the degree of erosion is found only qualitatively from the outer appearance of the degraded coating. Since, during the process of erosion, the points of the surface microrelief undergo many displacements (q is the sum of a large number of random terms), we shall assume that the function f ( q ) is Gaussian (Dmitriev, Dreiden, Osintsev, Ostrovsky, Shchepinov, Etinberg and Yakovlev [ 19891):

where qI1and q I are, respectively,the displacement vector components parallel and perpendicular to the object’s surface; ijl is the displacement of the mean microrelief level due to erosion; and oll and oI are the variances of q , l and q1 - q I , respectively. We denote the angle of sample illumination by a,, and the angle of observation by a l . Both angles are measured from the surface normal at the point of interest. Substituting eq. (4.3) in eqs. (2.7) and (2.8), we obtain (cos $) and (sin $), after which eq. (2.5) can be used to determine the holographic carrier fringe contrast yH: (cos a,

+ cos al)’

n2 olf - - (sina, -

A2

sinal)2 (4.4)

The expression for the additional phase shift Aq0 (see eq. (2.6)) caused by the displacement of the mean microrelief level can be written as Acp,

277 (cosa, A

=-

+ cosa,)ij, .

Thus, after determining the change AN in the order of the carrier fringes, one can find -

q1 = I A N (cosa,

+ cosa,)-I.

(4.5)

Equation (4.4) can be conveniently rewritten in the form 1 2712 In - = - [ of (cos a,

+ cos al)’ + olf(sina, - sin al)’] .

(4.6)

YH

Equation (4.6) contains two unknowns, ol and oI1.To find them, one has to record two holograms of the object under study under different observation

I t 8 41


125

angles a, and az. After this we can write two coupled equations In-

1

2n2

= - [a:(cosaS

+ cosa1)2+ alf(sina, - ~ i n a , ) ~ ] ,

(4.7)

+ cos a2)2 + alf(sinas - sin aZ)’] ,

(4.8)

YHI

1 2n2 In - = - [ 0: (cos as YH2

where yH I and yH2 are the carrier fringe contrasts at the surface point of interest reconstructed from holograms at angles of observation of a 1 and a2, respectively. By solving eqs. (4.7) and (4.8), one can find aI and all. The method was checked experimentally on specially fabricated specimens that were subjected to erosive action on a test stand designed for cavitationinduced erosion studies at the Leningrad Metal Machining Factory. The test stand is a flow-through type, with the working part a cavitation nozzle representing a venturi channel of rectangular cross section. The angle at the exit from the slit measuring 4 x 60 mmz is 12 degrees. Specimens 28 x 60 mmz in size fabricated of lX18HlOT-type stainless steel were fixed to the side wall of the nozzle. The maximum flow velocity in the nozzle was 39 m s - I . The double-exposure technique was used to record holographic carrier fringe interferograms (Dmitriev, Dreiden, Osintsev, Ostrovsky, Shchepinov, Etinberg and Yakovlev [ 19891). The specimens were mounted in a special holder per-

Fig. 27. Photograph of specimen used to study cavitation-induced corrosion and the fixture used to reposition it in the optical configuration of the interferometer.

126


Fig. 28. lnterferograms of specimens with various surface roughness R , and for various durations of erosive action P: (a)R, = 2.00 pm, T = 20 min; (b) R , = 2.00 pm, T = 30 min; (c) R , = 2.00 pm, T = 40 min; (d) R , = 0.55 pm, T = 30 min.


127

mitting removal after the first exposure to subject them to cavitation-induced erosion in the test stand with subsequent repositioning. Figure27 shows a photograph of a specimen mounted in the holder. Holographic interferograms of a specimen subjected to erosion for different lengths of time are given in fig. 28. A comparison of the fringe patterns shows that as the duration of the erosive action increases, the carrier fringes become deformed, and their contrast is degraded. This indicates irreversible (plastic) deformation, not only of the elements of the microrelief but also of its central level, which can be determined quantitatively by means of eq. (4.5). Figure 29 shows the optical arrangement of a two-hologram interferometer, in which the specimen under investigation is illuminated perpendicular to its surface (as = 0). Hologram H I provides the direction of observation normal to the specimen's surface (al = 0); hologram H, permits observation at an angle a, = 45". Substituting these values in the coupled equations (4.7) and (4.8), we solve them to obtain

(4.10)

ln--YH2

4@

?HI

Fig. 29. Optical arrangement for recording two holograms (BS, beam splitter; M1, M2, M3, mirrors; L, lens; 0, object; H , , H,, holograms).

128


111, § 4

Fig. 30. Interferograms of a specimen recorded at different observation angles a: (a) a = 0"; (b) a = 45".

Interferograms of a specimen exposed on the erosion test stand for 25 minutes are shown in fig. 30. The interference fringe contrast is determined for the central cross section of the specimen, the values thus obtained being normalized to the contrast in the area not subjected to erosion. Figure 3 1 is the

Fig. 31. Fringe contrast variation over the specimen surface: curve 1 determined from the interferogram of fig. 30a ( y H , ) and curve 2 from fig. 30b (yH2).

CORROSION. EROSION A N D WEAR PROCESSES

129

t

0

10

20

30

40

50

X,MM

Fig. 32. Distribution of variances ul and u,, of displacement vector over the specimen surface.

distribution of fringe contrast over the chosen cross section of the specimen, and fig. 32, the distributions of crl and oll along the specimen length found by means of eqs. (4.9) and (4.10). We see that the variance of the tangential displacement, o,,,is substantially greater than that of the normal displacement. The variance oL grows monotonically and reaches a maximum at I = 45 mm,

Fig. 33. Holographic interferogram of a specimen aRer erosive action for one hour.

130


PI, 8 4

whereas o,, is almost constant over the larger part of the specimen and reveals oscillations only at the surfaces of the specimen. Correlation holographic interferometry and correlation speckle photography offer the possibility of visualizing the areas of cavitation-induced erosion. Figure 33 shows clearly a triangularly shaped erosion zone in the holographic interferogram. Experiments reveal the possibility of determining the zones and extent of specimen erosion by means of correlation holographic interferometry. Compared with the other methods, correlation holographic interferometry permits a quantitative determination of the degree of surface erosion for substantially shorter times of exposure to erosive action, which is an essential factor in reducing the test duration. The method offers the possibility of studying various stages of erosion and the spatial distribution of cavitation-induced erosion over the specimen surface or a constructional part. Thus it can be considered as a non-destructive technique of monitoring the cavitation-induced erosion.

4.3.

INVESTIGATION OF MECHANICAL WEAR

Mutual displacement of joined members involves variation in height and shape of the microrelief elements on both containing surfaces, which is accompanied by the removal of material. This process is called wear. Just as for any other process producing irreversible changes in the surface microrelief, the wear can be studied by correlation holographic interferometry and correlation speckle photography. Note that here, similar to the application of these methods to the mechanics of contact interaction or to studies of erosion, one can both visualize the wear zone contour and measure the amount or intensity of the wear. In the first case it is possible to investigate the wear above the sensitivity threshold of these methods to any extent. In the second case, the amount of wear should not reach the level at which total decorrelation of the light waves or speckle structures is observed to occur. Therefore, the quantitative determination of wear is limited by the relation Ah 4 R , , where Ah is the mean irreversible decrease in the surface microrelief height characterizing the wear. In connection with this, quantitative measurement of wear can be carried out only in its initial stages, in particular, when studying the running-in of work-piece surfaces or wear-resistant members. Wear is obviously a complex process that can occur in the contact of surfaces of different roughnesses, in the presence or absence of lubricants, abrasives, crushed products of wear, and other features. The process may be accom-

11,s 41


131

panied by chemical corrosion, electrolysis, and many other phenomena. Therefore, the development of an adequate model capable of describing a change in the shape of the microrelief elements as wear occurs and the calculation with this model of the degree of decorrelation of light waves or speckle structures are complex problems that can be solved only in specific and simple cases. We shall describe the variation in the height of the microrelief elements as wear takes place by a displacement vector q = ( - q l , 0,O) normal to the macrosurface of the object and assume that the probability density function f ( q l ) is Gaussian, (4.11) where ijl is the mean displacement of the central level of the microrelief and D is the displacement variance. We take a holographic interferometer with the optical arrangement characterized by the following parameters:

e,

=

( - cos 0, - sin 0, 0) ,

el

=

(cos 0, sin 0, 0) ,

where 0 is the angle between the surface normal and the unit vectors of illumination (e,) and observation (el) directions. The expressions for (cos $) and (sin $) found using eqs. (2.7) and (2.8) have the forms (4.12)

(4.13) The carrier fringe contrast yH can be determined by substituting eqs. (4.12) and (4.13) in eq. (2.5): (4.14)

For the additional phase shift By,, we find from eq. (2.6) Ay,, = tan-'((sin$)/(cos$))

411 -

=-

A

q1 c o d .

(4.15)

Thus, with the accepted mathematical model, eq. (4.14) permits a determi-

132


nation of the displacement variance contrast o2 =

A2 In yH/8 n2 cos2 0 ,

c7

[II, $ 4

in the wear from the measured fringe (4.16)

whereas eq. (4.15) makes it possible to find the amount of wear i f l from the interference fringe shift AN: ijl = lAN/2 C O S ~ .

(4.17)

For practical purposes it would be interesting to develop a method capable of establishing an empirical relation between the extent of wear and the change in fringe contrast, which could then be used to determine the extent of wear from the measured drop in the contrast. To construct such a relation, however, one needs an independent and highly sensitive method to measure the extent of wear. No comprehensive investigations in this subject area have been reported. The publication by Osintsev, Ostrovsky, Presnyakov and Shchepinov [ 19921 presents only the preliminary measurements of the wave decorrelation and speckle structures caused by abrasive wear. The investigation was carried out on the specimen discussed in $ 4.1 with the optical arrangement shown in fig. 7. The wear was produced by random translational motion of a plane steel bar over the central part of the specimen surface, with the peripheral parts of the specimen not subjected to wear. An abrasive was introduced into the contact zone. A typical holographic interferogram for this case is shown in fig. 34. Figure 35a presents longitudinal distributions of the normalized fringe contrast y / y o (where yo is the fringe

Fig. 34. Holographic interferogram of a specimen subjected to mechanical wear.

11.8 41

133


I

1

1

20

0 (

*

1

I

60

X,M M

(b’

Y/Yo ‘:) 0.8

40

-

0.6 -

0.4 -

I 0

I

0.2

I

0.4

0.6

0.8

I

*

1.0

(YIY,), Fig. 35. Fringe contrast variation over the specimen surface after wear: (a) obtained by holographic interferometry (curve I), and by speckle photography (curve 2), (b) by a plot of ( y / y o ) i v e w s (Y/Yo)s.

contrast outside the wear zone). Curve 1 was obtained by holographic interferometry, and curve 2 by speckle photography. The same data are presented in another graph in fig. 35b, with ( y / ~ plotted ~ ) ~ along the horizontal axis and ( y / y o ) L along the vertical axis. The straight line thus obtained passes through the origin at 45”, thus demonstrating the validity of expression (2.26) for this case. 5. Conclusions

In addition to the applications described in 0 3 and 0 4, the method of correlation holographic interferometry has already been used to reveal defects

134


PI, § 5

in integrated circuits (Kudreev, Panibratsev, Safronov, Safronova and Titar [ 19791) and to investigate the powder pressing process (Kudrin and Bakhtin [ 19881). It can also be used to study surface processes such as deposition and vaporization. Correlation measurements based on an analysis of the contrast of rectilinear and equidistant interference fringes can readily be automated. Thus, correlation holographic interferometry and speckle photography substantially broaden the extent of the scientific and practical problems in mechanics that can be solved. Their range of applications is likely to increase, not only under laboratory conditions but also in industry, particularly in quality control and in the refinement of various technological processes.

References Ashton, R. A,, D. Slovin and H. J. Gerritsen, 1971, Appl. Opt. 10, 440. Atkinson, J. T., and M. J. Lalor, 1977, in: Proc. Int. Conf. on Application of Holography and Optical Data Proces, eds E. Marom and A. A. Frisem (Pergamon Press, Oxford) p. 289. Bezukhov, N. I., 1968, Fundamentals ofthe Theory of Elasticity, Plasticity and Creep (in Russian) (Vysshaya Shkola, Moscow) p. 268. Collins, L. F., 1968, Appl. Opt. 7,203. Dainty, J. C., ed., 1975, Laser Speckle and Related Phenomena (Springer, Berlin). Debrus, S., M. Francon and C. P. Grover, 1971, Opt. Commun. 4, 172. Dmitriev, A. P., G. V. Dreiden, A. V. Osintsev, Yu. I. Ostrovsky, V. P. Shchepinov, M. I. Etinberg and V. V. Yakovlev, 1989, Zh. Tekh. Fiz. 59, 192. Furse, 1. E., 1981, J. Phys. E 16, 264. Gabor, D., G. W. Stroke, R. R. Restrick, A. Funkhouser and D. Brumm, 1965, Phys. Lett. 18, 116. Klimenko, I. S., and V. P. Ryabukho, 1985, Opt. Spektr. 59, 398. Klimenko, I. S., T. G. Kvartskheliya, I. V. Volkov and N. A. Golikova, 1981, Zh. Tekh. Fiz. 51, 2080. Kudreev, V. N., Yu. A. Panibrattsev, G. S. Safronov, A. I. Safronova and V. I. Titar, 1979, Mikroelektronika, N8, 166. Kudrin, A. B., and V. G. Bakhtin, 1988, Applied Holography, Investigation of Metal Deformation (in Russian) (Metallurgiya, Moscow) p. 153. Maklead, N., and D. N. Kapur, 1973, J. Phys. E 6, 423. Metherell, A. F., S. Spinak and E. J. Pisa, 1969, J. Opt. SOC.Am. 59, 1534. Novikov, S. A., V. P. Shchepinov and V. S. Aistov, 1984, in: Dynamics and Strength of Metallurgical Machines, (in Russian) ed. B. A. Mozorov (VNIIMETMASH, Moscow) p. 47. Osintsev, A. V., Yu. I. Ostrovsky, V. P. Shchepinov and V. V. Yakovlev, 1985, Pis’ma Zh. Tekh. Fiz. 11, 202. Osintsev, A. V., Yu. 1. Ostrovsky, V. P. Shchepinov and V. V. Yakovlev, 1988, Pis’ma Zh. Tekh. Fiz. 58, 1420. Osintsev, A. V., Yu. I. Ostrovsky and V. P. Shchepinov, 1990, Pis’ma Zh. Tekh. Fiz. 16, 33. Osintsev, A. V., Yu. P. Presnyakov and V. P. Shchepinov, 1990, in: Trudy All-Union Symp. Met. Primen. Golograf. Interfer. (KUAT, Kuibyshev) p. 62. Osintsev, A. V., Yu. I. Ostrovsky, V. P. Shchepinov and V. V. Yakovlev, 1991, Zh. Tekh. Fiz. 61(8), 134.

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REFERENCES

135

Osintsev, A. V., Yu. I. Ostrovsky, V. P. Shchepinov and V. V. Yakovlev, 1992, Zh. Tekh. Fiz. 6~41~92. Osintsev, A. V., Yu.I. Ostrovsky, Yu. P. Presnyakov and V. P. Shchepinov, 1992, Zh. Tekh. Fiz. 62(5), 82. Ostrovsky, Yu.I., M. M. Butusov and G. V. Ostrovskaya, 1980, Interferometry by Holography (Springer, Berlin) ch. 4. Ostrovsky, Yu. I., V. P. Shchepinov and V. V. Yakovlev, 1991, Holographic Interferometry in Experimental Mechanics (Springer, Berlin) ch. 2. Petrov, K. N., and Yu. P. Presnyakov, 1978, Opt. Spectrosk. 44,309. Rytov, S. M., 1976, Introduction to Statistical Radiophysics (in Russian) (Nauka, Moscow) p. 361. Shchepinov, V. P., and V. V. Yakovlev, 1979a, Zh. Prikl. Mekh. Tekh. Fiz. 6, 144. Shchepinov, V. P., and V. V. Yakovlev, 1979b, Zh. Tekh. Fiz. 49, 1005. Shchepinov, V. P., B. M. Morozov. S. A. Novikov and V. S. Aistov, 1980,Zh. Tekh. Fiz. SO, 1926. Timoshenko, S. P., and J. N. Goodier, 1970, Theory of Elasticity (McGraw-Hill, New York) p.411. Vest, C. M., 1979, Holographic Interferometry (Wiley, New York) ch. 2.


E. WOLF, PROGRESS IN OPTICS XXX 0 ELSEVIER SCIENCE PUBLISHERS B.V., 1992

LOCALIZATION OF WAVES IN MEDIA WITH ONE-DIMENSIONAL DISORDER BY

V. D. FREILIKHER Department of Physics Bar-Ilan University Ramat-Gan 52900, Israel

S. A. GREDESKUL Department of Physics Ben Gurion University of the Negev P.O. Box 653 Beer-Sheva 84105, Israel

137

CONTENTS PAGE

Q 1 . INTRODUCTION

. . . . . . . . . . . . . . . . . . . 139

Q 2. STATISTICAL PROPERTIES OF PHYSICAL QUANTITIES IN RANDOM MEDIA . . . . . . . . . . . . . . . . . 143 Q 3 . ONE-DIMENSIONAL LOCALIZATION . . . . . . . . . 148 Q 4 . WAVES IN RANDOMLY LAYERED MEDIA . . . . . . . 170

Q 5 . CONCLUSIONS . . . . . . . . . . . . . . . . . . . .

199

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

200

ACKNOWLEDGEMENTS

138

200

6

1. Introduction

Ever since the inception of quantum mechanics as a science, it has widely exploited the mathematical tools borrowed from the classical theory of wave processes. During the subsequent decades of rapid development, however, this branch of physics has acquired incomparably richer and more diversified mathematical techniques of its own. The range of problems considered in quantum mechanics today is so wide that wave mechanics proper forms only one of its sections. The time has come “to pay the debts”, especially because the development of a statistical approach to the solution of wave propagation problems has brought the diffraction theory still closer to quantum mechanics in a way that is inherently random and statistical. At present, some purely quantum mechanical approaches have been efficiently adapted to the solution of radiophysical problems. One example is the Feynman diagram technique, which was initially intended to meet the needs of quantum electrodynamics and is now successfully employed in the theory of wave propagation in random media. In the last decade, new points of contact have appeared due to significant progress in the quantum theory of disordered condensed media. In this field one of the main problems, namely the study of particle motion, is to solve the Schrddinger equation with a random potential. The ideas and methods used to tackle this problem are closely connected with those of the problem of signal propagation in a medium with refractive fluctuations. When these phenomena were investigated, it proved useful to adopt the concept in such a way that the propagation of a wave, a particle, or a quasiparticle in a medium (even a continuous one) is treated as a succession of separate scattering events, making it necessary to sum an infinite series of scattered fields. In this case using the radiation transfer equation (a kinetic equation) can be successful, the solution of which actually leads to the summation of an infinite number of scattering events, and diffraction is taken into account only when a single scattering event is described. As to the rest, the wave nature of the objects under investigation is completely neglected. This approximation has made it possible to explain a number of experimental results in optics (Case and Zweifel [ 1967]), radiophysics (Tsang, Kong and Shin [ 1985]), and solid state 139

140

LOCALIZATION OF WAVES IN MEDIA WITH I D DISORDER

[III, 8 1

physics (Lifshits, Azbel and Kaganov [ 19731). It could not be applied in those cases where the field was primarily formed by backscattering, however, and it was no longer possible to neglect the interference (see, e.g., Apresyan and Kravtsov [ 19731). Therefore, the most important and topical problem of the theory of wave propagation in random media and of the quantum theory of disordered condensed systems is that a consistent allowance be made for the interference of multiply scattered waves. The study of interference has naturally lead to the concept of localization, which reflects the most general properties of disordered systems and stems both from the wave nature of the field scattering and from the random character of the scattering media. The pioneering paper by Anderson [ 19581 argued that states in a three-dimensional disordered system with a sufficiently high degree of randomness are localized (the Anderson localization). In the following decades the localization became the principal concept in the physics of disordered condensed systems. Notions such as the Anderson dielectric, Anderson transition, scale localization theory, and weak localization have appeared in textbooks (Bonch-Bruevich, Zvyagin and Mironov [ 19811, Abrikosov [ 19871) and physical encyclopedias. Over 100 conference proceedings and monographs on various aspects of localization and its consequences have been published (Mott and Davis [ 19791, Shklovskii and Efros [ 19791, Bonch-Bruevich, Zvyagin and Mironov [ 19811, Lifshits, Gredeskul and Pastur [ 19881, Efros and Pollak [ 19851, Ping Sheng [ 19901). The term “localization” has recently appeared in the literature on wave propagation in random media, but less commonly than in the theory of disordered condensed systems. The study of the phenomena connected with localization began in radiophysics as early as the 1950s, however. It was stimulated by attempts to define the range of applicability of the radiation transfer equation and, if possible, go beyond these limits. Gertsenshteyn and Vasilyev [1959a,b] were the first to show that the average transmission coefficient of a plane wave passing through a one-dimensional disordered layer decreases exponentially with thickness. Thus the localization length was actually found to be a function of frequency. The studies of wave propagation in one-dimensional random media continued at a greater pace, which was reflected in the monographs by Klyatskin [1975, 1980, 19861 and Rytov, Kravtsov and Tatarskii [ 19781. It is interesting to note that until recently the localization concept was seldom used in these studies, probably because the principal postulates of the localization theory were formulated and proved for closed systems with self-adjoint boundary conditions. When solving the radiophysical and acoustic problems, it is more common to use open systems with

111, § 11

INTRODUCTION

141

non-self-adjoint boundary conditions such as the radiation condition at infinity. The spectral properties of such systems were first analyzed by Gredeskul and Freilikher [ 19901 and Freilikher and Gredeskul [ 19911. Increasing interest has recently been shown in the localization of various other types of waves and excitations described by the wave equation in any medium without spatial periodicity or homogeneity. Studies are under way about the localization of sound waves in a continuous medium (Papanicolaou [ 19711, Hodges [ 19821, John, Sompolinsky and Stephen [ 19831, Baluni and Willemsen [ 19851, Kirkpatrick [ 19851, Ping Sheng, White, Zhao-Qing Zhang and Papanicolaou [ 19861, Condat and Kirkpatrick [ 19871); of electromagnetic waves in solids and plasmas (Escande and Souillard [ 19841, Ping Sheng, White, Zhao-Qing Zhang and Papanicolaou [ 19861); of gravity waves in a shallow water channel with a rough bottom (Guazzelli, Guyon and Souillard [ 19831, Devillard, Dunlop and Souillard [ 19881, Belzons, Devillard, Dunlop, Guazzelli, Parodi and Souillard [ 19871); of the third and fourth sound in liquid helium films on a randomly inhomogeneous substrate (Cohen and Machta [1985], Condat and Kirkpatrick [1986]); and of surface waves in metals (Farias and Maradudin [ 19831, McGurn and Maradudin [ 1983,19851). Interest in the propagation of short irregular pulses in homogeneous media encouraged investigation of localization properties of the solutions of Dirac-type equations (Bratus’, Gredeskul, Pastur and Schumeiko [ 1988a1, Gredeskul, Pastur and Sheba [ 19901); properties of one-particle excitations in superconductors and semiconductors (Dutyshev, Potapenko and Satanin [ 19851, Bratus’, Gredeskul, Pastur and Schumeiko [ 1988b1); and wave propagation in a layered structure, such as that used in X-ray mirrors (Gaponov 19841). Many papers on three-dimensional weak and strong localization, and backscattering enhancement will not be considered here. References can be found elsewhere (Altshuler, Aronov, Khmelnitskii and Larkin [ 19821, Efros and Pollak [ 19851, Barabanenkov [ 19881, Nieto-Vesperinas and Dainty [ 19901, Barabanenkov, Kravtsov, Ozrin and Saichev [ 19911). This chapter will mainly examine localization and associated effects in random media, whose parameters depend on a single coordinate. This is of considerable practical importance, since such layered one-dimensional structures are widely used in optics, radiophysics, and acoustics as models of the medium of propagation (Brekhovskikh [ 1973j). We believe that theoretical study of such structures is necessary, in particular for a statistical description of wave propagation in natural media (the atmosphere, ionosphere, oceans) in terms of a two-scale model (Vinogradov, Kravtsov and Feizulin [ 19831, Freilikher and Fuks [ 19841). In this model the space spectrum of the refractive index n(R)is divided

142


WI,§ 1

into two statistically independent regions, the influences of which can be considered consistently and, in a sense, independently. One region is turbulent and small scale, and it can be described, e.g., by the Markov process approximation, which applies to relatively short inhomogeneities (Rytov, Kravtsov and Tatarskii [ 19781). The other region is large scale and usually highly anisotropic (because of the interface); therefore, in many cases its variation in the horizontal plane ( x , y ) may be neglected, and it may be described approximately by a function of a single variable z. In this case the solution of the one-dimensional problem is a necessary intermediate stage for solving the three-dimensional problem. One-dimensional problems look attractive because, due to their technical and principal simplicity, they often represent exactly solvable problems that enable the researcher to test sometimes uncontrollable approximations. In addition, the study of one-dimensional models is extremely important for understanding the physics of multiple scattering, since in the case of wave propagation in such media the interference effects are most pronounced. These are the primary mechanisms of field formation, and the associated localization is realized most completely: with an arbitrarily small disorder in a onedimensional random medium, all states are localized, as was shown first by Mott and Twose [ 19611. Section 2 examines the statistical properties of various physical quantities in random media. The characteristics of the media, i.e. the dielectric constant and the random potential, are in most cases spatially homogeneous random functions with decreasing correlations. This creates self-averaging quantities, which become non-random in systems with sufficiently large volumes. The relation between the calculated averages and the experimentally measured quantities is analyzed. The third section focuses on one-dimensional localization. In 3.1 we shall define the principal concepts required, namely, the Lyapunov exponent and the localization length, and describe their behavior in some disordered systems. Section 3.2 will discuss the scattering problem with respect to the behavior of the transmission and reflection coefficients both at an individual realization and on the average. A particular case, resonance transmission, will be considered in § 3.3. In § 3.4 we shall describe qualitatively the coordinate dependence of the wave intensity inside a random layer. Section 3.5 presents a brief review of some of the theoretical and experimental results. Section 4 will address waves in layered media. The first three subsections deal with a stratified layer of afinite thickness bounded on one side by an ideally reflecting surface and forming an efficient fluctuation waveguide. In $ 4 . 1 we

W §21

STATISTICAL PROPERTIES OF PHYSICAL QUANTITIES

143

shall obtain exact dynamic expressions (valid for an individual realization) for the various energy fluxes of the field of a point source in such a layer. In 8 4.2 we show that such a semi-open system has the properties of a waveguide. The most important role is played here by quasihomogeneous waves that result from interference of multiply-scattered waves and the canalizing of the source energy along a layer within distances that are exponentially large compared with its thickness. The quasihomogeneous waves are analogs of the quasistationary states in quantum mechanics. Subsection 4.4studies the quasistationary states in a layer open on both sides. Subsection 4.5 deals with the properties of a randomly layered infinite space, and discusses the expressions for the correlation function, average field, and average intensity of the field of a point source in such a space. In conclusion we shall briefly discuss other promising connections between the theory of wave propagation in random media and the quantum theory of disordered condensed systems. The references are not exhaustive, nor do they demonstrate priority.

6 2. Statistical Properties of Physical Quantities in Random Media A theoretical study of the processes of wave propagation in a real medium usually starts by determining the parameters that describe its electrodynamic (or acoustic) properties. In the macroscopic approach such parameters are the dielectric constant ( 8 ) and the magnetic permeability ( p ) (or, in acoustics, sound velocity and density). For the atmosphere, ionosphere, and oceans these quantities are extremely complicated multiscale functions of coordinates and time, the experimental determination of which generally involves great technical difficulties. Even if some ideal measuring system could determine the values of E ( R )at all the desired points of space, in most cases a theorist would be unable to find the parameters for a signal propagating in space because the field equations can unfortunately only be solved for the simplest model dependences. If another ideal computer solves the appropriate equations, however, the resulting picture will be of little use because it will be overloaded with details typical of only a particular instantaneous state of the medium that is subject to rapid variations in time (the atmosphere) or from one realization to another (in solids). When studying wave propagation processes in natural media, therefore, it is common practice to use the statistical approach in which the parameters of the system in question (e.g., the dielectric constant) are regarded as random

144

LOCALIZATION OF WAVES IN MEDIA WITH ID DISORDER

[III, $ 2

functions of the coordinates and time. A great advantage of this approach is that it permits the most important and interesting averaged characteristics of the corresponding solutions to be found without having to scrutinize the “fine structure”, but using rather approximate statistical information about the parameters (e.g., the average value, fluctuation variance, and correlation radii). The studies in radiophysics and acoustics are concerned with the so-called coherent component (average field), average intensity, energy fluxes, variances of the amplitude, and phase fluctuations. It is necessary to determine, however, what the relationship between the averages calculated over an ensemble and the quantities measured at an individual realization is. This point, although important in terms of the validity of the statistical approach for each case, is ignored in many studies of wave propagation in random media. We shall examine this point in more detail, beginning with the scalar Helmholtz equation:

AM

WL + E(R)u = 0,

(2.1)

C2

where the dielectric constant E ( R )has the form

E(R)= E,

+~E(R).

(2.2)

Here &(R) is a random function of the coordinate R with a mean value of zero. By separating the term E , W ~ / C=~ E, and introducing

WR), U ( R ) = - Eo EO

we can express eq. (2.1) as a stationary SchrOdinger equation:

- A M + U(R)u = E,u .

(2.3)

Let the dielectric constant fluctuations 8~ depend on a single coordinate, for example, z. Then, changing to the Fourier components of the field with respect to the coordinates p (R = (p, z)),

1

u(R) = ( 2 7 ~dKeiK.f’ii(K,z), ~ ~ ~ and i i ( ~z ,) = $(z), we arrive at and putting E, - K’ = E, - E, ~ E ( z ) /=E u(z), the one-dimensional version of eq. (2.3)

-$”

+ u(z)$= E $ .

(2.4)

11198 21


145

The statistical properties of random coefficients u(z) or &(z) should, on the one hand, adequately reflect the properties of real media and, on the other, be consistent with reasonable mathematical models that can be studied theoretically. It is usually assumed in the theory of disordered systems (Pastur [ 19871, Lifshits, Gredeskul and Pastur [1988]) that the medium is spatially homogeneous, on the average, and that correlations in it disappear at infinity. The spatial homogeneity on the average means the invariance of any moments with respect to an arbitrary shift a, (U(R,

+ a). .

*

U(R,

+ a))

=

(U(Rl)** * U ( R , ) ) ,

(2.5)

and the disappearance of correlations means the factorization of the moments U(R) into two groups of points at infinite distances from each other, i.e., lim (U(R, + a ) . . . U ( R , +a)U(R,+,)...U(R,+,)) laI--roo

( U ( R , )*

=

U(Rn)) ( U(Rn + 1 1*

* *

U(Rn + m))

(2.6)

(this last equation is written with allowance taken of the property (2.5)). Hence, in an infinite system, with a probability of unity, all realizations are identical to within the space shift. But then any quantity of the form

f = lim fv, f v = V-" V-00

s

f ( R,,..., R,;[U(.)])dR,...dR,,

(2.7)

where f is a function of the coordinates R,, . ..,R, and a functional of the realization U(.), and the exponent a of the volume V is chosen so that the limit in eq. (2.7) will be finite, is non-random. In other words, lim

v-

w

1

f(R,,. . . , R,; [ U ( - ) ] d) R , . . . dR, r

J Pr. 1 V+m =

iim

(J(R,, . . .

which reflects what is known as the self-averaging of fv in the macroscopic limit. It is clear that this statement and all the following theoretical conclusions are only valid for potentials U(R) that satisfy the conditions of eqs. (2.5) and (2.6). As a rule, the common theoretical models (the Gaussian or Poisson potentials and the telegraph process) meet these requirements. In experiments one should check for compliance with the preceding requirements when studying natural media. In the case of solids the samples should be specially treated with the use of, e.g., neutron doping and the annealing of radiation-

146


WI,8 2

induced defects with a subsequent slow cooling, in order to obtain a statistically uniform distribution of defects in the sample (Zabrodskii [ 19801, Rosenbaum, Andres, Thomas and Bhatt [ 19801. Self-averaging quantities play an extremely important role in the physics of disordered systems. For example, all the specific extensive (proportional to volume) physical quantities, in particular the density of one-particle states, the free-energy density, and thus all the thermodynamic quantities (Lifshits [ 1964]), as well as various kinetic characteristics, such as the electrical conductivity (Pastur [ 1971]), and the transmission decrement of a one-dimensional random medium (Lifshits and Kirpichenkov [ 19791, Anderson, Thouless, Abrahams and Fisher [ 19801). The most substantial property of self-averaging quantities is the coincidence, with the probability of unity, of their values at individual realizations and their mean values in an infinite system. Thus the calculated averages can be treated as quantities that are really observed. Self-averaging quantities are non-random in the macroscopic limit V + 00. Therefore, every such quantity fv has its own characteristic value of the volume Vf in the respective region of the parameters. At V 9 V’, the system becomes macroscopic with respect to this quantity, and the fv distribution is Gaussian, with the mean value (fv) independent of V and the variance

(f:>

- (fv>2--

Vf

V

.

For systems smaller than V ’ (although essentially larger than microscopic ones), the quantity fv is random and sensitive to the details of an individual realization. In this case both the system and the fluctuations in the physical quantities are termed mesoscopic (Kramer, Bergmann and Bruynseraede [ 19841, Altshuler and Khmelnitskii [ 19851, Grinstein and Mazenko [ 19861). For the non-self-averagingquantities that are usually dealt with in the theory of wave propagation in random media, a system of any volume is always mesoscopic, and therefore the mean values of these quantities are generally not useful in describing individual realizations. These meari values can be given a physical meaning and compared with the results of measurements (usually carried out at specific realizations) only if special experiments and suitable data processing techniques are used. One method requires that the quantity under investigation be measured for any one realization for a long enough time, and that the time average be equal to the value calculated for the ensemble average. If the realization varies rapidly with time, the ensemble of realizations obtained during the experiment is fairly

I K § 21


147

rich and justifies this procedure. In the full-scale radiophysical measurements this procedure normally proves to be natural and easy to realize, since the variation required is provided by the time evolution of the parameters of the natural media, e.g., the refractive of the atmospheric index (Kukushkin, Freilikher and Fuks [ 19871) or the shape of the surface of the sea (Braude [ 19621).A vivid example of the use of such ergodicity in laboratory experiments is the backscattering enhancement reported by Estemad, Thompson and Andrejko [ 19861, which was observed on two model systems. One system was a static “fluff’ of submicron SiO, balls in air. To find the intensity peak exactly in the opposite direction, it was necessary to measure 10 realizations and average the result. The other system consisted of polystyrene balls suspended in a liquid. Thermal motion in the liquid resulted in the Brownian vibration of the balls, and a pronounced peak was formed even after measurement on one sample. This picture is essentially the same as that commonly used to explain why the measured values of thermodynamic quantities are non-random. The space analog of the picture was previously used by Lifshits, Gredeskul and Pastur [ 19821 to elucidate the mechanism of self-averaging of the transmission coefficient of a layer with an infinitely large area. Another processing technique can be applied when the result observed depends on some additional parameters. Thus the physical quantities describing the waves in a random medium are usually functions of coordinates and time and also of frequency, transceiver parameters, and other factors. This permits the ensemble averages to be identified with the mean values over one or several parameters. The averaging interval should be sufficiently large compared with the corresponding correlation radius, but the ensemble average should remain almost unchanged within it (Tatarskii [ 19671, Sokolovskii and Cherkashina [ 19711). Thus, when investigating the rough surfaces of solids by the light scattering method, the averaging parameter is the light beam aperture, i.e. the illuminated area of the surface. If the latter is large enough, the averaged indicatrix of light scattering can be obtained on a single sample (Kaganovskii, Makienko and Freilikher [ 19761, Kaganovskii, Freilikher and Yurchenko [ 19841, O’Donnel and Mendez [ 19871, Sant, Dainty and Kim [ 19891). Quantities f v for which such a procedure is inapplicable are physical quantities whose relative fluctuations increase with system volume. Their mean values (f,) provide little information because they are formed by representative realizations of low probability and differ from the values at typical (i.e., most probable) realizations. Additional information concerning the behavior of random functions at typical realizations can be obtained by studying their dependences on the self-averaging quantities. Indeed, if fv can be expressed in

148


[III, B 3

terms of the self-averaging quantity yv and volume V as fv = f(yy, V), then, since at V + co yv tends to the non-random limit y = limv.+m yv, the function f(y, V) ( # (f(yv, V)) !) at large V characterizes the behavior of fy at typical realizations. This type of behavior can be investigated qualitatively. Such an approach is successfully used to analyze the coefficient of wave reflection from a disordered segment, the intensity of a wave passing through it, and the distribution of energy fluxes from a point source in a randomly layered medium (Gredeskul and Freilikher [ 1985, 19881).

8 3. One-dimensional Localization 3.1. LYAPUNOV EXPONENT. LOCALIZATION LENGTH

The principal finding of localization theory is that if the disorder is sufficiently high the initial excitation (a wave or particle) does not propagate in a medium during its evolution in time but stays mainly within a finite region of space. This is how localization was introduced by Anderson [ 19581. Instead of the time evolution, however, one can study the corresponding stationary modes that are attenuated exponentially in space and can be characterized by their localization lengths. The localization arises due to the interference of multiply-scattered waves. In the one-dimensional case, in which only one scattering channel exists, this interference is most pronounced. In this situation all states are localized (Mott and Twose [ 19611, Berezinskii [ 19731, Goldsheidt, Molchanov and Pastur [ 19771). Cases in which the localization is not exponential but instead follows a power law (two-dimensional disordered systems or incommensurable almost-periodic systems) are beyond the scope of this article. As is clear from the preceding section, it is most convenient to use selfaveraging quantities when describing the behavior of random systems, and to try to express the main characteristic of localization, its length, by means of such a quantity. Consider first the simplest case, a closed one-dimensional system described by the SchrOdinger equation (2.4) in the interval [ - L, L]:

- $"

+ u(z)$= E $ ,

E

=

k2

and the self-adjoint (zero-current) boundary conditions $ ( + _ L+) a $ ' ( + _ L = ) 0 , Ima

=

0

(3.1)

imposed on the function $ at the ends of the interval. As a is a real quantity

111.8 31

ONE-DIMENSIONAL LOCALIZATION

149

this means that the segment boundaries are ideally reflecting. In fact, the coefficient of a plane wave with the wavenumber k that is reflected from such a boundary is r = -ika - 1 ika+ 1 ' and Irl = 1 at Ima = 0. The spectrum of such a system is simple, and the solutions to the system can be chosen to be real and written as $(z) = e'sincp,

$'(z)

=

ke'coscp.

Substituting these expressions into eq. (2.4), we can readily arrive at the set of equations

(4

cp' = k - - sin'cp,

k

which are satisfied by the functions cp(z) and ((z) that are responsible for the oscillations in the solution and for the growth of the envelope of the solution. The following true statement can be given (see Lifshits, Gredeskul and Pastur [1988]): the solution r ( z ) of this system with the boundary condition cp(0) = cpo = - tan-lka (the condition for ((0) is insignificant, and we shall assume that ((0) = 0) is such that the ratio ((z)/z at IzI -,cx), with unit probability, tends to a non-random limit that is positive for all k in the case where u(z) satisfies the conditions of space homogeneity on the average and of the disappearance of correlations:

This statement means that the solution to eq. (2.4), which satisfies the self-adjoint boundary condition at a certain point, increases with unit probability exponentially on both sides of this point with a non-random increment (the Lyapunov exponent) y(k') = [21(k2)]- I , where 1(k2) is called the localization length. The meaning of this term becomes clear from the following considerations (Mott and Twose [1961]). Let us construct for some k the solution $- (z) of eq. (2.4) that satisfies the appropriate boundary condition at

150

[III, 8 3

LOCALlZATlON OF WAVES IN MEDlA WITH 1 D DISORDER

the left-hand end z = - L and the solution $+(z) satisfying the condition at the right-hand end z = L. Both solutions with unit probability, grow inside the interval [ - L, L]. If they and their derivatives are matched at some point (and then at all points) of the interval, we shall have the eigenfunction of the problem (2.4), with boundary conditions (3. l), localized within the interval [ - L, L]. As was shown by Goldsheidt, Molchanov and Pastur [1977], the localization property is preserved in infinitely large intervals ;i.e. the Anderson localization of all states takes place in a one-dimensional system. In terms of the wave propagation this means that, e.g., in a waveguide formed by two ideally reflecting planes spaced at a distance H apart, the random stratification of the refractive index n = no + 6n(z)with a sufficientlylarge value of H will lead to the radical rearrangement of the space distribution of the field $,(z) of the waveguide modes over the cross section. In contrast to the homogeneous case of 6 n = 0, where this field oscillates uniformly (t,hn sin m z / H , n being the mode number), in an irregular waveguide the envelopes of the normal waves decrease exponentially on both sides of the randomly situated localization centers z:,

-

The theory is required to calculate the localization length for the given random potential u(z) and spectral parameter k. In the simplest case of a limited potential with a mean value of zero, by proceeding from the definition (3.4) and applying perturbation theory in the parameter u(z)/k2 to the set of equations (3.3), we obtain 1- '(k')

=

(2k2)-

jom

B(z) cos 2kz dz .

This result has, in a sense, a resonance character: only the 2k-Fourier harmonic of the correlation function B(z),

contributes to the formation of the localization length at a particular value of

k. As was shown, e.g., by Tatarskii [ 19671, this harmonic is responsible for the scattering in the strictly backwards direction (with k changing to - k). In the limiting case, where the correlation radius is small compared with the wavelength

r, 4 k -

'

(3.5)

151


the potential can be considered as &correlated B(z) = 2 D S ( z ) , D

-

r,B(O),

(3.6)

and the localization length takes the form of l ( k 2 )= 2 k 2 / D .

(3.7)

In addition, if

r, 4 D -

‘I3 ,

(3.8)

then at IEl 4 r;’

(3.9)

the potential u(z) becomes Gaussian. The localization length behaves as I(E) = 2 E / D ,

D2/3+E4r;2, (3.10)

Using the standard averaging over rapid variables (Klyatskin [ 1980]),in the “high-energy” region, E

=

k2

D2I3,

(3.11)

one can obtain the closed Fokker-Planck equation for the probability density p ( r , z ) of the quantity ( ( z ) , whose solution is

(3.12) with l(k2)from eq. (3.7). Hence, we can see that at distances z larger than the localization length l(k2),the self-averaging ratio t ( z ) / zhas a Gaussian distribution with a variance inversely proportional to the “volume” z, as is typical of self-averaging in the limit of infinitely large volume. The following discussion will address various moments of the solutions to eq. (2.4) and consider the behavior of mean values of the form exp(ar). As follows from eq. (3.12), (eat>

=

exp(- a(a

+ 2 ) zJ

Note that a feature of this equation is the existence of values of

(3.13) c t [~- 2 , 0 ] ,

152


[III, § 3

for which the mean value of the function eat rises exponentially despite its exponential decrease at each realization (i.e. with unit probability). This behavior is characteristic of exponential quantities and means that contributions to the average come frome most unlikely (representative) rather than typical realizations. The average can only be observed experimentally in an exponentially rich ensemble of realizations, and it is more or less informative only in this case. It is also noteworthy that the quantities of the type eat fluctuate exponentially: Their relative fluctuation is proportional to

and does not increase exponentially except for the trivial case of a = 0. It should be pointed out that some consequences of the exponential rise described by eq. (3.4) are well known in statistical radiophysics as stochastic parametric resonance (see Klyatskin [ 1980, 19861); e.g., formulas (1.21) in section 6 of Klyatskin [ 19801 demonstrate an exponential increase in the mean values of the quadratic combinations of the solutions to eq. (2.4) with an increment fully corresponding to eq. (3.13) (at a = 2) and eq. (3.7), and were obtained precisely when satisfying the conditions for eq. (3.8). In our consideration of other examples, we observe first that the preceding results in this subsection fully apply in the case where eq. (2.4) is derived from the Helmholtz equation in a three-dimensional, randomly layered medium. In this case the coefficient D in formulas (3.6) to (3.8) is, in order of magnitude, given by D

N

R;4r,(aJ~0)2,

(3.14)

where A, is the wavelength in a fluctuation-free medium divided by 2n, and a,‘ is the variance of the dielectric constant fluctuations ~ E ( z ) When . the fluctuations are small, a, 4 eO, to replace the potential by the Gaussian white noise, it is sufficient to satisfy the condition r, 4 A,, and the high-energy (shortwavelength) region is bounded by the following inequality (3.15) Similar results were also obtained for systems with two-band spectra. Thus, for the Dirac type of equation -i(hz - b)$‘

+ u(z)*+

Ahx$= k#,

(3.16)

I I I , § 31


153

which describes the dynamics of electrons interacting with a random sound signal in a superconductor (Bratus’ and Schumeiko [ 1985]), where hx, are the Pauli matrices, and the localization length in the short-wavelength region with Pc 1 is

I(k) =

~

1-82 [ k 2 - A2(1 - B2)] 2DA2

(For the conditions for the applicability of this formula see Bratus’, Gredeskul, Pastur and Schumeiko [ 1988a,b].) With /3> 1, all the states in such a problem are delocalized (Bratus’, Gredeskul, Pastur and Schumeiko [ 1988a,b]). For the Zakharov-Shabat system (Zakharov, Manakov, Novikov and Pitaevskii [ 19801) -ii3=$’

+ [u(z)hx+ u(z)&,,]$=

k$,

which arises when solving the non-linear Schrgdinger equation by the inverse scattering transform method, the localization length when both the potentials u(z)and u(z) are Gaussian white noises with the diffusion coefficients D, and D,, respectively, is given by (Gredeskul, Kivshar and Yanovskaya [ 19901)

I-

’ = 2(D, + D,) .

Of special interest is the case of the one-dimensional Helmholtz equation, in which the spectral parameter appears in the “potential” u(z) = - k26e(z)/e,, and L in formulas (3.14) and (3.15) should be identified with I , (k with Then, in the long-wavelength limit L -+ 00 ( k + 0), the conditions that permit the potential to be replaced by the Gaussian white noise and the “energy” to be assumed high are satisfied automatically, and eq. (3.7) for the localization length takes the form

A).

(3.17) Thus, in the limit of infinitely long waves the states are delocalized (the random function u(z) vanishes from the dynamic equation). This result was also obtained by many authors who used other arguments and direct calculations for continuous (Kohler and Papanicolaou [ 19731, John, Sompolinsky and Stephen [ 19831, Azbel [ 1983a,b]) and discrete (Ishii [ 19731, O’Connor and Lebowitz [ 19741) models similar to the one described here. Investigations of the behavior of the localization length in the short-wavelength limit for a somewhat more general situation, described by the equation

154


[III, I 3

(Devillard, Dunlop and Souillard [ 19881, Ping Sheng, White, Zhao-Qing Zhang and Papanicolaou [ 19861)

4 dz

(K(z)2)+

k2p(z)$(z) = 0 ,

(3.18)

showed that when k + l ( k 2 )+ const.

(3.19)

For eq. (3.17), however, the Lyapunov exponent and localization length are defined by a formula different from eq. (3.4), i.e.,

where

It is interesting to note that, in contrast to eq. (3.19), for a special type of randomness in eq. (3.17) the localization length f ( k 2 )increases with k and, therefore, has a minimum for some k = kmin. This minimum was observed experimentally by Belzons, Devillard, Dunlop, Guazzelli, Parodi and Souillard [ 19871 in their study of the effect of waves in shallow water on a vessel with a rough bottom. Recently, reports have appeared of the results of an investigation into localization in non-linear random media (Souillard [ 19861, Devillard and Souillard [ 1986]), but we shall not dwell on them here.

3.2. SCATTERING PROBLEM

The localization of waves or particles is due to the stochasticity of the coefficients of the appropriate dynamic equations (e.g., of the dielectric constant, refractive index, or potential). This property should also manifest itself in the case of randomness in sufficiently long but finite regions of space (remembering that here we mean one-dimensional localization), e.g., in scattering problems. Although these problems deal with open systems, the corresponding quantities within a disordered segment can be expressed in terms of solutions relating to a closed system. Therefore, both the scattering


155

states proper and the transmission ( t ) and reflection (r) coefficients “feel” the presence of localization. Various physical quantities can be expressed by means of scattering characteristics t and r. These quantities include the following formulas: (i) the field of a point source in a randomly layered medium (Brekhovskikh [19731)

x

fl,

2(9, z) sin9 d 9 , z 2 zo ;

(3.20)

(ii) the radiation flux density of a point source in a one-dimensional problem (Gredeskul and Freilikher [ 19901) (3.21) (iii) the intensity of a wave passing through a disordered segment at a point z inside it (Klyatskin [ 19801) (3.22) (iv) the static electric conductance of a one-dimensional disordered system at zero temperature, namely the Landauer [ 19701 formula (3.23) (v) the mean thermal flux through a disordered segment (see, e.g., Keller, Papanicolaou and Weilenmann [ 19781) (3.24) (vi) the power absorbed by a superconductor in the field of a sound signal (Bratus’ and Schumeiko [ 19851). All these formulas contain the disorder-sensitive coefficients of reflection r + ( E ) , r - (z) and transmission ( t ) , so that all the preceding quantities clearly reflect the existence of localization. Let us consider first the one-dimensional scattering problem for eq. (2.4), in which a monochromatic wave with the wavenumber k and the amplitude of

156


[III,

B3

unity is incident on the disordered segment [ 0, L] from the right. The segment transmittance (squared modulus of the transmission coefficient) can be expressed as (Pastur and Feldman [ 19741)

I W I Z = 4{2 + exp[2tc(L)1 + exP[2ts(L)l)-'

9

(3.25)

where t c ( z ) and ( , ( z ) are the solutions of the system(3.3) satisfying the boundary conditions Cp,(O) =

4

,

rP,(O) = 0 ,

tc,,(O)

=

0*

Since zj + 4 it is almost equal to unity. This contribution, #(z0), is distinct from zero only if the localization center of this state is vertically separated from the source by no more than the localization radius and is, in this case, of the order of 4- I . (This follows from the normalization condition, eq. (4.5).) Therefore, the total energy flux @d along the layer, eq. (4.8c), in a thick layer, where L % 1, consists only of a small group of waves for which

176

[III, 3 4


lz, - zjl 5 4 and is of the order of djd

-

1 1

--=

Az 1

(Az)-'.

(4.13)

-

Here, 1 is the localization radius for I E 1 ae(02/c2),and Az is the average distance between neighboring localization centers. The z-distribution of the flux along the layer is highly inhomogeneous: only a narrow strip of the size I z - z, I 4 1 4 L, which is close to the source, carries a distinctly non-zero flux. In the case of eq. (3.6),eqs. (4.12)and (4.13)lead to djd

-

D1I3.

(4.14)

The flux djd(z) along the layer is not a self-averaging quantity and therefore has a mesoscopic fine structure depending on the individual properties of a realization. This structure has just been described, and is shown schematically in fig. 6.It may be observed more simply, however, by studying the derivative of this flux,

which has, in addition to a large peak near z,, the structure of which is also fine, many small peaks near all localization centers (fig. 7). These peaks form mesoscopic oscillations of dji(z), with a characteristic period that is of the same order of magnitude as the distance between the localization centers Az. By changing the position of the source, one can, in principle, locate the localization centers zj of the eigenmodes (i.e. the regions of the inhomogeneous layer that

20

L

-

Fig. 6. The flux (Pd(z) through the side surface of a cylinder with an infinite radius and a height z.

WAVES IN RANDOMLY LAYERED MEDIA

177

Fig. 7. Fine structure of the flux derivative.

are most transparent in the longitudinal direction and thus concentrate in themselves the energy canalization along the layer) and determine the wavefunction amplitudes $(z,) at these centers. The set ofthe quantities zj and Ilr,(z,) unambiguously characterizes a realization (as does, e.g., in mesoscopic conductors, the dependence of the conductivity on the magnetic field, which is known in the literature by the name of magnetofingerprints). The total average flux ( @d ) along the layer is of the same order of magnitude as that at an individual realization. Using eq. (4.8a), we can write the average flux (@d) as

and we observe (see Lifshits, Gredeskul and Pastur [ 1988I) that, when z,,, L - zo 9 r,, the integral on the right-hand side is the average number of discrete levels per unit thickness of the layer, so that

from which in the case of eq. (3.6) the estimate of the relation (4.14) again follows. Thus, as follows from eqs. (4.11) and (4.12), at each realization (except those of exponentially low probability) in a randomly layered medium there is a waveguide type of propagation. It is provided by discrete-spectrum waves with negative values of the parameter E. The rest of the energy is that of the continuous spectrum, the states of which are known to be delocalized; i.e. the field associated with the spectrum does not stay within the layer but propagates to the outside, which makes the system open. We shall show, however, that the openness of these disordered systems, which is responsible for the formation of the continuous spectrum, results in a radical rearrangement of the structure of the part of the field associated with this spectrum. We refer to the generation

178


[III, 8 4

of quasihomogeneous waves (analogs of quasistationary states in quantum mechanics), thus substantially enhancing (when compared with regular structures) the waveguide effect.

4.3. QUASIHOMOGENEOUS WAVES

It can be expected that in a randomly layered medium, continuous spectrum waves will also be canalized. The simple reason for this channelling is that the field of a point source may be represented as an expansion in terms of plane waves; each of these waves (even propagating normal to the layer), as follows from eq. (3.26), is reflected from a sufficiently thick layer with the reflection coefficient, whose modulus differs from unity by an exponentially small value. This increase of backward reflection should result because radiation is partly “locked” along the z-axis and is therefore canalized along the layer. To make sure of this, let us analyze the flux Gc emerging from the layer due to waves with a continuous spectrum. The integrand p(E) = sin2 cp(zo) e - 2 [ < ( L ) - ‘%%)I (4.15) in eq. (4.9), written as (4.16)

is the radiation energy flux per unit interval of the spectral parameter E, i.e. the density of the angular distribution of the “upward” outgoing flux,

e= a r c s i n m ,

jOE0

. . . d E = E,

jo ... nl2

sin2ede.

It can be seen from eq. (4.15) that, since at almost every realization the function r(z) rises primarily in a linear fashion, for L - zo % I the quantity p(E) is exponentially small at nearly all realizations. At first sight this agrees completely with the qualitative arguments described earlier. As was shown in 3 3.2, however, in the case of eq. (3.6) for z, = 0 and r - = 1, the mean value (p(E)) = 1 (as in free space for 6 8 = 0), and is formed at low-probability realizations, where p(E) is exponentially large, p exp (2LII). This average can be found in two situations. In one situation the whole integral of eq. (4.16) is the flux, which is exponentially small at most realizations, and its average is formed at low-probability realizations. In this case N

111, B 41


179

an arbitrary realization with a probability that differs from unity by an exponentially small value has good waveguide properties: the layer emits only a part of the total flux, which is exponentially small in the parameter L/l. In the other situation, since for a fixed value of E the estimate &L)/L cc (21)- is valid for most, but not all, realizations, at each realization there will be E E [0, E o ] , such that the function p ( E ) in the integrand will be exponentially large. As a result, the flux associated with the continuous spectrum is of the same order of magnitude at each realization as that in a homogeneous space:

@c-A;', and exceeds the flux canalized along the layer in (4.14):

To determine which of the two situations is present, we shall use eq. (4. lo), expressing the flux that emerges from the layer by means of the reflection coefficient r ( E ) , according to which +

(4.17) In the region of sufficientlylarge E B 1 u ( , the phase @ + ( E )= argr+(E) of the reflection coefficient behaves, as follows from eq. (3.32), mainly as + + ( E )1: 2LJE. Since at a typical realization 1 - Ir+(E)I

~(exP(-L/O),

for fixed E we generally have p(E)

(4.18)

exp( - L / O

The values En of the parameter E are exceptions, for which

+ + ( E n )= 2 n n .

(4.19)

At these points En N n2n21L2,

the denominator I 1 - r

(4.20) +

I in eq. (4.17) becomes small

11 - r+(En)12-e-2L/',

(4.21 )

180


[IIL 8 4

and p(E) is exponentially large,

p(En) ~ X [PL / l ( E n ) l *

(4.22)

As aresult, p(E,,) represents a sharp function shown by the set of peaks in fig. 8. The distance AE,,= En+ - En between the peaks of p ( E ) (i.e. between the roots of eq. (4.19)) is, according to eq. (4.20), AE,,= 2n2n/L2, and the halfwidth of the peaks 6E,,related to the difference of Ir+ / from unity is

1 (4.23) 2n When calculating aCby eq. (4.16), the function p(E) may be replaced by the smoothed function P(E),which is obtained by averaging over the interval AE, in which E - E const., but including many peaks:

6E,,N - AE,,exp [ - L/l(E,,)] .

P(E) = -

AE

jAE’2

p(E

+ E’)dE’ .

-AEJZ

The calculation using eqs. (4.22) and (4.23) gives P(E) N 1/2n.

(4.24)

Thus, at a typical realization the outgoing flux coincides in the order of magnitude with its average ( G c ) and the value @io) in a homogeneous space. This means that the present system is in a sense characterized by ergodicity in the parameter E: ( p ( E ) ) P(E). This ergodicity explains the difference, described earlier, between the behavior of p(E) at typical and representative realizations for a fixed value of E. Indeed, the probability of p(E) being exponentially small (eq. (4.18)) at a particular realization is equal to the probability of this value of E happening to be outside the interval 6E, which is apparently 6E 1 1 - - = 1 - - exp[ -L/l(E,,)]. AE 2n

-

Fig. 8. Effective density of states p(E).

III.8 41

181


In other words, the measure of realizations that is typical for this value of E differs from unity by an exponentially small value. In contrast, at representative realizations for this E, whose measure of order exp ( - L/l),the function p(E) is exponentially large, as in eq. (4.22). Thus, as is suggested by eqs. (4.16) and (4.24), in this problem we are dealing with the second situation that was mentioned earlier. The foregoing arguments suggest that the total flux emerging from a randomly stratified layer has a strongly inhomogeneous angular distribution. The radiation undergoes a type of focusing near the values 0, = a r c s i n d m corresponding to those values of&, at which the function p ( E )has a maximum. The meaning of the values of E, in eq. (4.20) becomes understandable if the solution of eq. (2.4), with the boundary condition of eq. (3.1) outside the layer z > L, is written as

$(E, z) = 1 - r + ( E ) ]t * ( ~e-ifi(2-L) )

+ [ 1 - r r ( E ) ]t

( ~eifi(z-L), )

(4.25)

where t ( E ) is the transmission coefficient of the disordered segment. In fact, for real E both an incident (from the right) wave and a reflected wave are present. In the case of complex values 8, = E, - b , , - is,,, however, where

r + ( 4 )= 1

(4.26)

3

the incident wave coefficient turns out to be zero, and only the outgoing wave remains :

4 = [ 1 - r: (411 l ( 4 )exp[iJaz - L)1 ( r 3 8 , ) # ( r + ( & n N * = 1 ; ll(4)I2 # 1 - lG)I2). $ ( 4 9

7

(4.27)

In quantum mechanics the wavefunction of eq. (4.27) gives the so-called decay state. The square of its modulus because of the time dependence -e-iSnt = , - i t ( E , - 6 1 ~ ) - t 6 2 ~ ,d ecreases within the time z (8,")- I . In the case

-

of b,, G En - hl,, however, where this time is large compared with the characteristic oscillation period, which is of order (En - b , , ) - I , such a state is quasistationary. These states play an important role in the theory of nuclear reactions and decays (Baz', Zel'dovich and Perelomov [ 19691). In this theory, however, they are due to the specific shape of the potential u ( z ) in eq. (2.4), which represents a potential well separated from its surroundings by a sufficiently wide potential barrier with a height greater than the particle energy. Here, in contrast, we deal with quasistationary states in an "overbarrier" situation, where the particle energy is high when compared with the scattering

182

IIII,§ 4


potential. In addition, the particle “locking” is of a purely quantum character, and results from the interference of waves multiply scattered by potential fluctuations. (In this case, single scattering events can be weak.) In the region of E values for which the localization length I is small compared with the layer thickness, the reflection coefficient modulus differs from unity by an exponentially small value. As a result, the solution of eq. (4.26) becomes (4.28)

gn = En - ib,, ,

where the imaginary part is half the half-width of the respective peak of p ( E ) in eq. (4.17), and is exponentially small in the parameter L/I

,a,

=

$En

1

=-

472

AE, exp( - L/l(E,,))

(4.29)

(The shift b,, of the real part of En is, with the same accuracy, equal to zero.) Thus, the values of En of eq. (4.20), for which the quantity gnhas a maximum, represent real parts of the complex values gn of eqs. (4.28) and (4.26). These values correspond to quasistationary states whose “lifetime” z eLI‘ is exponentially large, and the total flux @= that emerges from the layer results from quasistationary states. Note that the waves $(En, z ) that correspond to E = En are exponentially localized within the layer, although these waves cannot be normalized because of the oscillating tails (eq. (4.25))outside the layer. This result follows from the proportionality of the eigenfunctions $(En, z ) for z inside the layer and r - = 1, to the cosine solutions c(En,z ) (satisfying the conditions c(En,0) = 1 and c‘(En,0) = 0 ) that rise exponentially from the point z = 0 (see 5 3.1), and from the identity

-

c2(En,L ) + E - ’ c ‘ , ( E , , , L ) = ‘ I -r+(E)12-exp[ - L / l ( E n ) ] . 1 - lr+(E)12 It should be emphasized that the characteristic scale of the wavefunction $(En, z ) is the localization length I(E,) < L and not the potential well width, as is the case for standard (“underbarrier”) quasistationary states; this is the difference between disordered and regular systems. The concept of quasistationary states makes it possible not only to understand their role in the formation of the flux @,-, but also to analyze the dependence of the field of the source G on the longitudinal coordinate p. For this purpose we shall use formula (3.20) in which rl is a path of integration in the complex plane 9, k is the wavenumber for the level where the radiation source

111, I 41


183

is situated, and f l , 2(z, 9) are the functions describing the fields in the lower ( z < z o ) and the upper ( z > zo) semispaces on which a unit-amplitude plane wave is incident from a vacuum at the angle 9. Analysis of eq. (3.20) shows that it reduces to the sum of the residues that correspond to the denominator poles and to the integrals on the branching-out sides (Brekhovskikh [ 19731). The sum of residues giving the field within the layer, for I C ~ 1 ~can % be written, with the replacement d = Eo - k2 sin’9, as (4.30) n

Here,

K, =

J E , - dn and the dn are the roots of the dispersion equation

1 - r + ( z o ) r - ( z o )= 0 ,

(4.31)

and r - ( z o ) (r+(zo)) are the coefficients of reflection from the regions [zo, co) ( [ 0 ,z o ] ) ,on which a wave is incident from the left (right). Expressing the r+ - of eq. (3.36) by means of the ratio $’/$and using eq. (4.31), we readily obtain the formula

where the right-hand side at d = dn is zero. Therefore, the solutions ‘8,of eq. (4.31) do not depend on the point zo that was chosen, and thus, in the case of r - = 1, coincide with those of eq. (4.26). As was shown, these solutions include those corresponding to quasistationary states. Since in our case the role of time is played by the distance p between the source and the point of observation in the plane (x, y), such states correspond to quasihomogeneous waves attenuating at large distances 9 from the source that are exponentially large, with a factor L/I, 9 c c ( ( I m ~ , ) - ’a L e L l ‘ .

(4.33)

This attenuation is due not to dissipation but to “upward” emergence of the field (to the region where z > L). When p < 9,,, however, quasihomogeneous waves are locked within the layer. Thus the radical distinction between a randomly stratified layer and a regular dielectric waveguide stems from the essential role of the continuous spectrum. The flux from this spectrum is formed by quasistationary states, the energy of which emerges from the layer but at distances from the radiation source that are exponentially large compared with the layer thickness. The part of the field associated with the continuous spectrum and described by integral terms in the expansion (eq. (4.4))contains the sum over quasistationary states, which corre-

184


[IK8 4

sponds to weakly attenuated quasihomogeneous waves canalizing energy along the layer for enormous distances (Freilikher and Gredeskul [ 19901). The study of the influence of random inhomogeneities on wave propagation in the media with regular refraction, in which the unperturbed refractive index is a regular function of coordinates, is of considerable practical interest (Freilikher and Fuks [ 19841). An example of such a medium is a layer of the atmosphere where the dielectric constant varies with the altitude. The quantum mechanical analog of regular refraction with respect to the problem of electron motion is the external field. For example, a constant field corresponds to the linear dependence of the atmospheric refractive index, which is widely used in the wave propagation theory. Perel and Polyakov [ 19841 investigated the effect of a potential random component on the electron behavior in an electric field. They considered the case of repulsion corresponding to antiwaveguide wave propagation (see Kukushkin, Freilikher and Fuks [ 19871). In the case of ultrashort wave propagation above the ocean, the opposite situation is more common, where the so-called near-water tropospheric waveguide arises due to the decrease in refractive index with the altitude. In the absence of fluctuations in this waveguide it has both a quasidiscrete spectrum corresponding to the modes canalized along the layer with low attenuation to the modes canalized along the layer with low attenuation and a continuous spectrum associated with the outgoing waves. The presence of extended “one-dimensional” inhomogeneities improves canalization by decreasing the waveguide mode attenuation (Freilikher and Fuks [ 1984]), which is evidently a manifestation of the one-dimensional localization just described. In contrast, the wave scattering by isotropic fluctuations results in an energy output from the layer (due to the transformation of discrete spectrum waves into continuous spectrum ones), i.e. in the increase of the attenuation decrement of the waves propagating along the waveguide (Freilikher and Fuks [ 19811). Interesting effects arise when the direction of wave propagation in the waveguide is perpendicular to random layers. Sivan and Saar [1988] demonstrated that in such a geometry, for distances smaller than the localization length I, the contributions to attenuation caused by disorder and scattering to other modes additively enters into the total decrement (the Matthiessen rule). In the opposite limiting case, when the propagation distances are greater than I, the system displays its “locking” properties that can make the medium a kind of fluctuation resonator. In fact, if the dielectric constant fluctuates along the z-axis of a metallic waveguide of an arbitrary cross section, so that for any cross section (z = const.) the value E = E, + ~ E ( zis) the same

111,s41

185


(the layers being perpendicular to the axis), then the variables in the wavefield equation for such a system can be separated. The part $(z) that depends on the longitudinal coordinate z is given by eq. (2.4) with the “quantized” parameter Elm = E,, - q&,. In this case the set of transverse wavenumbers qlm is specified by the dispersion equation following from the boundary conditions at the waveguide walls, e.g., when the cross section is rectangular with the sides a and b, q;,

=

y:( (yy. +

For every realization of the random function &(z) at a sufficiently large waveguide length L , there is, as was shown earlier, a set of En values corresponding to quasistationary states. Since in this case the spectral density p ( E ) is non-zero only if I E - E,, 6En (see fig. 8), there is an efficient excitation of only quasihomogeneous modes with Elm = En. Such a system will work as a resonator whose “Q-factor” is determined by the imaginary part of B“,; i.e. it depends on the fluctuation parameters and waveguide length L . N

4.4. QUASISTATIONARY STATES IN AN OPEN SYSTEM

The results obtained in the preceding subsection apply to a semiopen system that corresponds to ideal reflection. In this system the self-adjoint boundary condition is valid at one of the boundaries z = 0. It is natural to try to investigate the possibility of quasistationary states in an open one-dimensional system (the whole-axis problem) when the dielectric constant fluctuation outside the segment [ - L, L ] is equal to zero. Let us first consider a closed system with self-adjoint boundary conditions imposed on its boundary z = L . Then the normal wave spectrum (the energy spectrum in terms of quantum mechanics) can be found from the equation (Brekhovskikh [ 19731) 1 - r+(E)r-(E)= 0 .

(4.34)

Here, E = k2 is the spectral parameter of eq. (2.4), and r (E) ( r - (E)) are the reflection coefficients for a wave with a length 1 = 2a/k incident on the semisegment [0, L ] ([ - L,’O])from the left (or from the right). (The boundary point can be any point inside the segment [ - L, L ] , but not necessarily its center (see eq. (4.32)) Since the system is closed, i.e. I rlt (E)I = 1, eq. (4.34)has a discrete spectrum of real solutions En. In open systems, since the boundary conditions at the points z = & L are not +

186


[III, 8 4

self-adjoint, no real-valued solutions to eq. (4.34) exist. As was shown in 0 3.2, however, the reflection coefficient moduli I r + ( E )I for typical realizations differ from unity by an exponentially small value. Consequently, eq. (4.34) can have complex-valued solutions 8,,= E,, - is,, with an imaginary part - S,, that is exponentially small in the parameter L/l;these solutions, as well as similar ones discussed in the preceding subsection, correspond to quasistationary states or quasihomogeneous waves. In fact, the coefficients R of reflection from the whole segment [ - L, L ] in such a problem with the real “potential” u ( z ) of eq. (2.4) can be represented by eq. (3.33), where r , , - ( r + ,) are the coefficients of reflection from the semisegment [ - L, 01 ([0, L]), the total transmission coefficient T being +

T=

tl t 2 3

1 - r- r+

where t , and t, are the transmission coefficients for the semisegments [ - L, 01 and [0, L]. For complex-valued &, satisfying eq. (4.34), R * and T become infinite, but

As a result, the two solutions that are linearly independent for real E and correspond to the wave incidence on the segment from the left and right, respectively, merge, at E 4 &, into a single solution. The latter contains only outgoing waves outside the segment and therefore describes a quasistationary (decay) state. As follows from eq. (3.33), the complex roots a,, lie near the real points En such that

arg(r- r + ) I E , = 2nn. Note that for such E, the ratio r*(E,,)/ri(E,,),entering into eq. (3.33), is also real. Consider, in the complex-valued plane 8,the set of curves @,, defined by the equation arg[r+(g)r-(d’)] = 2nn, as well as the curves Q and R defined by the equations I r (8) r - (8) I = 1 and I r + (a)I = 1 r - (8)1, respectively. The points &,q at which @, and Q intersect are the spectral parameter values corresponding to the quasistationary states, whereas the points 8; at which R and @, intersect correspond to the resonance +

111, I 41


187

states. The term “resonance states” implies that for d = 8; the coefficients R of reflection from the whole segment become zero. To find the spectra of quasistationary and resonance states, write the reflection coefficients for semisegments as ~

r,(4

=

exp[ - A , ( € ) + i@,(€)l ,

and write the spectral parameter values 82 and 8 : as

Expanding A + and @, in terms of the, presumably, small corrections

and

62;‘, we find-that

62.‘

( A + ? A - ) ( $ : +@:I . ($’+ + $/ )’ + (A’+ + A / )’ ’

(A+ kA-)(A: ? A : ) ($1+$’-)’+(A’+ ?A’_)’ (4.35) Here, the prime means differentiation with respect to the spectral parameter, all the functions of which are taken at the point En, which is why the subscript n of 6:; is omitted. In a typical situation, according to eq. (3.26), A +- exp( - L / l ) , so that the formulas (4.35) can be simplified as =

-

Hence, the shift of the real part, 6,, is negligibly small in the parameter exp( - L / l ) ;i.e. the curves intersect the real axis nearly at right angles. With respect to the imaginary parts, 6,, by virtue of the exponential character of the A , fluctuations, they coincide in their absolute values to within logarithmic accuracy, and are of the same sign at A + % A - and of the opposite sign for A + 4 A _ . In other words, the spectral parameter values corresponding to quasistationary and resonance states either lie on the same side of the real axis and coincide to within a logarithmic accuracy, or they are on different sides of the real axis at equal (to the same accuracy) distances. In these cases the existence of such states, in contrast to the situation considered in the preceding subsection, does not affect the scattering problem characteristics on the real axis. Here, the reflection coefficients R , take the form of eq. (3.35) and at typical realizations differ from unity by an exponentially small value, whereas for quasistationary states they become infinite, and for resonance states they become zero. Exceptions are the real values of the spectral parameter En, for which A + and A - coincide to within an exponential accuracy. The value 8;

188


[III, 8 4

corresponding to the resonance state is then closer to the real axis than 82, resulting in resonance transparency (see 3 3.3) of the segment [ - L, L] for E = En: 1 - IR ( E n ) ]< 1. We believe that in the limit L -+ co the values gn come closer together and approach the real axis, so as to form a simple dense spectrum of localized states in an infinite system, the existence of which was proved by Goldsheidt, Molchanov and Pastur [ 19771.

4.5. POINT SOURCE IN AN INFINITE LAYERED MEDIUM

In 3 § 4.2 and 4.3 we investigated energy fluxes generated by a point source in a randomly stratified layer. A more detailed characteristic is provided by the space distribution of wavefield intensity I(R, R,) = 1 G(R, R,)J 2, which is expressed by the Fourier transform of the Green function G"( u, z ) as c

J

I(R, R,) = ( 4 ~ ) - ~G"(xI;z, z o ) G"*(K~; z , z,) ei(rl-x2)'d~1 d ~ ; 2

R

=

(P, Z) ; Ro

=

(PO,

zo) ; r = P

(4.36)

- PO .

Because of the isotropy of the medium in the (x, y) plane, the Green function

c" actually depends not on the vector u but on the scalar k 2 = E = E 0 - x2; i.e. c" = G ( k ;z , z,). The average intensity ( I ( R , R , ) ) , as well as the space correlation function

K(Ri

-

Ro, R2

-

Ro) = ( G ( R 1 , Ro) G*(R2, R o ) )

9

is found by simply calculating the correlator K(k19k2;zl - z o , z 2 - ~ 0 ) = (G"(k,;z,,zo)c"*(k2;~2,~,)).

(4.37)

The need to find similar correlators also arises when calculating the intensity of a non-monochromatic pulsed signal that propagates in a random medium (Abramovich and Gurbatov [ 19801). In this case different k correspond to different frequencies o:ki = oi/c. The quantities similar to those of eq. (4.37) also appear in the study of the high-frequency properties of disordered conductors, the conductivity of which is expressed by means of bilinear combinations of one-particle Green functions ofthe Schrodinger equation. In his pioneer paper, Berezinskii [ 19731 developed a diagram technique that made it possible to solve the problem of high-frequency conductivity of one-dimensional metals. This approach is based on the fact that in the one-dimensional case the predominant contribution to conduc-

111, I 41


189

tivity is made not by ladder diagrams (leading to the radiation transfer equation, see Barabanenkov [ 1988]), but by the so-called tightly bound diagrams, in which the rapidly oscillating phase multipliers are compensated. The selection of such diagrams is actually equivalent to averaging over a rapid variable that enables one, e.g., to obtain a closed equation for the distribution function of the modulus of the coefficient of plane wave reflection from a layer of finite thickness (Papanicolaou [ 197 11). Direct application of this procedure to finding the desired quantities in eq. (4.37), however, involves some additional difficulties. We shall briefly describe the procedure and results of the calculation of correlators of the type given by eq. (4.37), using the method based on the ideas of Abrikosov and Ryzhkin [ 1976, 19781, Berezinskii and Gorkov [ 19791, Antsygina, Pastur and Slusarev [ 19811, Kaner and Chebotarev [ 19871, Kaner and Tarasov [ 19881, and modified by Freilikher and Tarasov [ 1989, 1991a,b] for the calculation of radiophysical quantities, namely, the coherent component, average intensity, and energy flux of the point source field in a randomly layered medium. The method consists of approximating the functions G"(kj;zj, zo) to the matrices e(kj; zit zo), so that all correlators of the type

can be expressed in terms of the mean value of the trace of the product of the respective matrices e ( k j ;zj, zo), which can be calculated exactly. Consider eq. (2.4), assuming that inhomogeneities occupy the entire space and c0 -+ E, + i y. The presence of the non-zero imaginary part y of the dielectric constant takes into account the energy dissipation in the medium. This is important both practically (the radio wave absorption in some frequency ranges of real media may prove substantial) and theoretically, since it enables the investigation of the dissipation effect on wave localization in randomly layered media. If the dielectric constant fluctuations are sufficiently small on the average, so that

the scattering may be referred to as weak, i.e. changing k 2 only slightly in a single scattering event. At this time two possibilities exist: incoherent scattering in the initial direction of wave propagation (k+ k) and in the specular direction (k -+ - k). Since scattering by small fluctuations is of the resonance type, it is

190

[III, 8 4


clear that the main contribution to the field comes from the spectral components of tic, with very large spatial periods (forward scattering) and A n/k (backward scattering with the sign of k reversed and K = const.). (In the Born approximation the cross section of scattering from q 1 to q2 is directly proportional to the squared modulus of the spatial harmonic amplitude in the spectrum of fluctuations &, with period A = 2n lql - q21 - (Tatarskii [ 1967]).) This reasoning suggests that in the case of weak scattering the random function &(z) may be approximated only by the sum of resonance harmonics

-

’

&(z) = 6e1(z)

+ ~ E ~ (eZikz z ) + 6 ~ , e*- 2 i k Z .

Here, Ak

&E1(Z) =

(4.38)

s-

Ak

& E ( t ) eirzd t

;

~ E ~ (=z )

61(t + 2k) ei“ d t ,

Ak

(4.39)

where 61(t) is the Fourier transform of the random function ~ E ( z ) whereas , the 6 e l , 2(z) remain almost constant at distances Az k- so that Ak 4 k. On the other hand, we shall assume the interval Ak to be sufficiently large, so that the ~ E ~ , ~could ( z ) vary at distances much smaller than those at which the quantities of interest (e.g., the wavefunction envelope and the coherent component of the field) change. The characteristic space scale of change for these functions is the localization radius 1. Since (Ak)- 4 I, such “slow” quantities can be calculated, assuming that 2(z) are &correlated Gaussian processes. This is only necessary for an interval Ak such that I - 4 Ak 4 r,; i.e. the inequality N

’

1% r,

(4.40)

should be satisfied instead of the more stringent requirement k-I % rc (see eq. (3.5)). In the inequality (3.15), which arises when averaging over the rapid variable, o, is replaced by oe,.* oe:

-=

k r , o ~ , , E , / k 24 1 .

(4.41)

The inequalities are weakened, since the restrictions are imposed only on the resonance harmonic envelopes ticl, of eq. (4.39). In this case the “rest” part of the spectrum of fluctuations &(z) that does not “work” may be fairly arbitrary . Thus if the conditions (4.40) and (4.41) are satisfied, we shall consider 8t1(z) as real, and 8e2(z) as complex Gaussian random processes with the following

111, B 41


191

correlation properties : 0

=

(8E1.2)

3

k t ( S E ~ ( Z &) E ~ ( z ' ) )= 2D16(z - z ' ) , k,

w =

-

C

,

k t ( ~ E , ( z ~) E ; ( z ' ) ) = 2 0 , b(z - z') , 201.2 = k04rC,.2~:,2. (Other pair-wise correlators become zero.) According to the representation of eq. (4.38), let us eliminate the rapid variable in the field $ as well, i.e. search for the solution of eq. (2.4)

+ k,26E(Z)$ = O

$"(z) + k2$(z) in the form of $(z)

=

$l(z) eciqr + $,(z) eiqr,

(4.42)

where q is an intermediate parameter satisfying the inequality Iq2-k21 0.5, two solitons are generated from this initial condition. The characteristic distance for the formation of soliton(s) is given by the dispersion distance z,,. Thus, if zo < y - (i.e. if Y g l), the result of the perturbation method based on the

IV,I 31

GUIDING CENTER SOLITON

235

inverse scattering analysis as presented here holds. If the pulse width is made larger and T approaches unity, however, the distance over which the soliton is formed can be compared to the damping distance, and the result of the perturbed inverse scattering method begins to fail. For example, even if A > 0.5, if by the time that the two solitons are formed the amplitude decays to less than 1.5, two solitons will not be created. This argument leads us to consider another important parameter in the soliton propagation, namely, the amplifier spacing 2, in units of the dispersion distance 2, ( = 1, in the model equation (3.1)). The condition Z , 4 1 indicates that amplifications are applied to a soliton before it adjusts itself to form a soliton from the perturbed initial condition of (1 + A ) sech T. These considerations suggest that a stable soliton transmission can be achieved over an extended distance ( % z,) if a periodic amplification is provided at a distance smaller than z, so that the dispersive waves that are generated from the perturbed initial conditions (1 + A ) sech T have no time to escape from the pulse. Kodama and Hasegawa [ 19821 demonstrated numerically that a pair of solitons with sufficient separation can, in fact, propagate with little distortion even after an amplification of 500 times, by choosing the amplifier spacing to be smaller than the dispersion distance z,; i.e. 2, < 1, for a case with re 1, and A 4 1.

3.3. GUIDING CENTER EQUATION AND THE LIE TRANSFORMATION

The appearance of erbium-doped fiber amplifiers stimulated interest in computer simulations of soliton propagation over an extended distance in fibers with periodic amplifications. It was recognized that a soliton can propagate with little distortion even if its dispersion distance z, is much larger than the loss distance y - ', thus T % 1, provided that the amplifier spacing z, is much less than the dispersion distance z, and the initial amplitude is enhanced by a factor 1 - exp( 2rza -2T2,)

(3.10)

so that the line-averaged intensity is kept to unity (Hasegawa and Kodama [ 19901, Mollenauer, Evangelides and Haus [ 19911). In this situation the pulse amplitude changes from a, to a, exp( - TZ,) in the region between two amplifiers. Since Z , 4 Z,, the dispersive term has no time to respond, hence between two amplifiers the pulse changes its amplitude

236

OPTICAL SOLITONS IN FIBERS

[IV,8 3

but does not change its width. Thus at any point between amplifiers, the pulse does not have the soliton property, in which the amplitude times the width is constant. When the pulse shape is sampled at a distance that is a multiple of Z,, however, the pulse shape remains the same and satisfies the soliton property of width times amplitude being a constant. The behavior of a soliton here is analogous to that of the motion of the center position X of a gyrating charged particle (guiding center motion) with charge e in an inhomogeneous magnetic field. The instantaneous position of the charged particle x is given by x=x+p,

where p is the vector radius of the gyromotion and the equation of motion satisfies

y=p x

w,,

and w, ( = e B / m )is the vector cyclotron frequency. The Hamiltonian and the canonical momentum p at the position x are given by H=-

1

2m

[p-eA(x)]’,

and p = -VH,

B

=

VxA.

The transformation to new canonical variables (X, P) from (x, p ) can be achieved using a generating function

F

=

mw,[$(y - Y)’cOt$-

xY] ,

where $ is the phase angle of the gyromotion. The transformed Hamiltonian H‘ = P + o , has a simple structure, in the new coordinates $ and Y and momenta P+ and mw,X in the limit of small gyroradius p (because o, = w,(X) in H’). But when higher-order corrections in p are obtained, the canonical transformation becomes increasingly complex because the transformed Hamiltonian contains mixed variables of original and transformed coordinates and momenta. Lie transformation avoids this difficulty, and the transformed Hamiltonian can be expressed successively to any order in p terms of only the transformed variables. With this background we consider an application of the Lie transformation to the present problem. (Some mathematical background for an infinitedimensional extension of the Lie transform is provided in appendix B.)

IV,8 31

231


Let us now consider the case where the gain and dispersion, D(Z), of the fiber vary periodically (Hasegawa and Kodama [ 1990, 1991a,b]), as shown in fig. 3.1. The model non-linear Schrddinger equation reads a4 a2s + 1qI2q = - i r q i + ;d(Z) az dT2

+ iG(Z)q,

(3.11)

where d(Z) ( = D(Z)/( D)) is the fractional variation from the average dispersion (D). As before, the time is normalized to the soliton width 2, (ps), and the distance Z is normalized to the dispersion distance z, = - 233.1 ( k ; ) and the amplitude 141 to the soliton amplitude

,/m2.9l3I2 ( ID I ) S/z, , =

where S is the effective fiber cross section in pm2 and z, = 1.762,. Let us first transform q to a new variable u by factorizing out the rapidly changing amplitude a ( Z ) through q(T, Z )

=

(3.12)

a(Z)u(T, Z ) ,

and a ( 2 ) = a, exp

(\Jo[

Z

[ G ( Z )- r ]dZ) .

(3.13)

\

0'

A

228

za

3za

z Fig. 3.1. Typical variations of coefficients of non-linear, a ( Z ) , and dispersive, d ( Z ) , terms.

238


The new amplitude u than satisfies (3.14)

We note that if d(Z) # 0 everywhere in the fiber, its effect can be absorbed into a non-uniform Z coordinate. Thus we first consider the case with no variation in dispersion, d(Z) = 1. We also note that eq. (3.14) can express a case with a periodic variation of the fiber cross section. With a proper normalization the periodically varying coefficient a’ can be written as U*(Z) = 1 + n ( z ) .

(3.15)

The normalization that (a’(Z)) = 1 is achieved by choosing the integration constant a, in eq. (3.13) such that ( a ’ ) = 1, or a, being given by eq. (3.10) for the case where the amplifiers are localized at Z = nZa with n = 1,2, . .. .We shall see that this normalization provides the Lie-transformed equation with a unit coefficient for its non-linear term. In eq. (3.15) the average of the oscillating parts are taken to be zero:

(a)

za

jozad(Z)dZ=O.

(3.16)

With eq. (3.15) the envelope equation u can be put in the form au

-=

az

X [ u , u * ; Z ] = Xo[u, u*] + d(Z)X,,[u,

#*I,

(3.17)

where Xo[u, u * ]

=

XoA[u,u * ]

i a2u -+ ilul’u, 2 aT’

-

=

ilul’u,

(3.18) (3.19)

and [u, u * ] denotes the set of arguments with infinite dimension (u, u*, u T ,u ; , uTT, u ; ~ ,. . .), where uT = aulaT, U? = au*/aT, etc. We now transform the u variable to a new variable u, such that the new variable satisfies the canonical non-linear SchrOdinger equation with constant coefficients to O ( Z z ) ,where the power n will be determined. For this purpose we employ the exponential Lie transformation, which is generalized to a system

IV,8 31

239


with an infinite number of degrees of freedom (Kodama [1985a]; see also appendix B). The transformation reads (Hasegawa and Kodama [ 1991a,b] u

=

e*"u

=

u + $ [ u , u*; Z ]

+ +(Q.V$)

[ u , u*; Z ]

+

* * *

,

(3.20)

where Q = ($, $*) is the generalized Lie-generating function to be determined and the directional derivative 4 . V is defined as in (B.4), i.e. (3.21) with $,= = an$/aTnand unT = anu/aT".In this transformation the variables [u,u*] are expressed in terms of variables [u, u*; Z]. The averaged or guiding center non-linear SchrOdinger equation for u will be obtained in an autonomous form, du/dZ = Y0[u, u*] ,

(3.22)

where Yo[ u, u*] is determined by means of averaging. It should be noted that the total derivative d / d Z in (3.22) operates on the space labeled by (u, u*, uT, u;, ..., Z), i.e. (B.26), dv d a +--.v d Z aZ d Z

(3.23)

Substituting u of eq. (3.20) into eq. (3.17) and using eqs. (3.22) and (3.23), we have du do dv a ($++Q*v$+*.*) +--v($++Q.v$+ dZ dZ dZ az * * a ) + -

a ($+;Q.V$+-) az

=

Yo+Yo.v($++Q.v$+**.)+-

=

X[e""u,

=

x + Q vx + ;Q - V(Q * VX) +

e+"u*; Z ]

=

e""X[u,

u*; Z ] * ' *

.

(3.24)

We now determine $ and Yo in a perturbative way by making the expansions (3.25a) (3.25b) where $, and Yon are shown to have an order of O(Z:). In eq. (3.24) we note that a$,/aZ = O(Z:-') but assume do/dZ = O(1); i.e. we assume that no

240


[IV,B 3

resonance exists between the periodic perturbation and the soliton. The effect of such resonances will be discussed in 0 3.4. From O(1) of eq. (3.24) with the expansions (3.25), we have

(3.26) giving

'*; '1

$I[',

where $lo

=

=

(3.27)

al(z)xOA + $~O['Y u*]

( $, ) and the mean free function d , satisfies

dfil6.

(3.28)

dZ

Equation (3.28) implies a, = O(Z,), so that $, = O(Z,). The function $lo will be determined later from the non-secular condition for $,. From O(Z,) we have

(3.29)

-

where [ $, ,XO]= 9, VXo - X o V$, is the Lie bracket. The non-secular condition for $, requires the average of the right-hand side of eq. (3.29) over Z, to be zero, i.e.

(3.30)

-

*

where $,

=

d , X o , ($,

=

$,

+

Hence, we may choose

Ol0 = Yo, = 0 .

(3.31)

Integrating (3.29), we now have $2[',

'*; '1

=

[XOAYXOI + $2O[u,

'*I

9

(3.32)

where $,o = (I$,) will be determined from the non-secular condition for and the mean free function &(Z) satisfies diiJdZ

=

which gives d,

d, ,

=

O(Z2) and $,

$3,

(3.33) =

O(Z,Z). Going to O(Z,Z), we have

IV, 8 31

24 1


The non-secular condition for $3 (i.e. the average of eq. (3.34) vanishes) gives

(3.35) I

where $, = ii2(Z)[ & A , & ] * eq. (3.35) becomes

+ a20[X0A,

[$209x01

from which $20

=

$20

0

Using eqs. (3.27) and (3.32) for [xOA9XOll

-

y2Cl

=

0

3

and

$2,

(3.36)

and Y2, can be chosen as

(3.37)

9

where a,, = - $ (a&)

(3.39)

.

The non-linear SchrUdinger equation that the transformed (guiding center) variable u satisfies is now obtained from eq. (3.22),

and the solution for u can be expressed in terms of the solution of u of eq. (3.40), as u=u =

+ $1 + $41.V$, + $,

+ O(Z,3)

~ + i i i , ~ o ) ~ u - ~ ~ ~ l u ~ ~ +~ O+ (iZ f: ) , [ ~ ~ ~ , (3.41) ~ ~ ]

with [xOA,xOI

=

-2(u1u,12

+ u*u++ o’vh).

(3.42)

The derivation of eq. (3.42)shows that the equation for u with rapidly varying coefficient a ( 2 ) with a periodicity Z , 4 Z , ( = 1) can in fact be transformed into a non-linear SchrUdinger equation for the guiding center variable u through eq. (3.41), which has constant coefficients to O(Z,2). This remarkable result shows that all the soliton-related properties described by the non-linear SchrUdinger equation are valid to a distance zo(ZO/Za)’,even if the soliton amplitudes oscillate with an amplitude of more than about O( 1).

242


[IV, 8 3

Furthermore, the transformed equation (3.42) can be shown to admit a renormalized solitary wave solution to all orders of Z, in the following form (Kodama [ 1978]), us(T,2) =

Nl

(

Z:yA')sech2"-'qT

1=0 n = l

1

exp(iiq2Z),

(3.43)

where the number of terms NI and the coefficients { yi')}rL of the power of sech qT are determined successively by the equation of order Z i derived from eq. (3.40) with u = us. This implies that the original equation (3.1) can support asymptotically a solitary wave close to the one-soliton solution of the non-linear Schradinger equation, and the level of the radiation generated by the perturbation stays small, provided there is no resonance between soliton and radiation. We now consider the effect of non-uniform dispersion d ( 2 ) in eq. (3.14). As mentioned earlier, if d ( 2 ) # 0 everywhere in the fiber, the effect of d ( 2 ) can be incorporated by using a new coordinate Z', satisfying dZ'/dZ = d(2)

(3.44)

and a new non-linear coefficient a 2 ( 2 ) / d ( Z )(Mollenauer, Evangelides and Gordon [1991]). When the dispersion vanishes in some portion of the fiber, however, this transformation becomes invalid. Thus we apply the Lie transformation directly to eq. (3.14) to obtain the transformed nonlinear Schrgdinger equation for a general case of d(Z) = 1 + d"(Z) with ( 2 ) = 0. The result then reads

Here. XODIU,#*] =

i 8% 2 aT2 '

(3.46)

-

(3.47) with

dil,/dZ = il, and ( Z2)

=

(a,)

d i 1 / d Z= =

d" ,

(2,) = 0. The Lie-generating functions $J1 and $J2 used

IV, § 31


243

in eq. (3.41) are also modified to (3.48) and

(3.49) with

Thus the existence of the guiding center solitons is also seen for the case with periodic variation in the dispersion.

3.4. PROPERTIES OF THE GUIDING CENTER SOLITONS

We have seen in 0 3.3 that a periodic amplification can allow a soliton-like solitary wave (guiding center soliton), which propagates over a distance much larger than the dispersion distance with a proper choice in the initial amplitude eq. (3.10), even if the pulse amplitude oscillates by an order of magnitude. A guiding center soliton is also realized when, in addition to the periodic amplification, a periodic variation of dispersion exists in a fiber. Figures 3.2a,b show the behavior of guiding center solitons that are numerically obtained by solving the non-linear Schradinger equation (3.1) with fiber loss and periodic amplifications, but with a constant dispersion (Hasegawa and Kodama [ 19901). The physical parameters used in these computations are as follows: the soliton width zS = 40 ps, the group dispersion D = 1 ps/nm km, the fiber cross section S = 60 pm2, the loss rate 6 = 0.24 dB/km for which the dispersion distance zp ( = z,2/3.1( - k; )) becomes 41 1 km, and the soliton peak power Po ( = [2.913/2@/z,]2) becomes 1.2 mW, and I'= 12. The difference between figs. 3.2a and 3.2b is the amplifier spacings; fig. 3.2a is for Z , = 50 km and fig. 3.2b is for Z , = 100 km. The initial amplitudes a, required to construct the guiding center solitons are from eq. (3.10), 1.7 and 2.4 for figs. 3.2a and 3.2b, respectively. Shown here is the shape of a pair of pulses in 141 obtained at each distance of multiples of 500 km. From these figures it is clear that the magnitude of the pulse I q I sampled at a distance of multiples of the amplifier spacing retains the initial shape extremely well, even though the amplifiers provide localized gains of 0.24 x 50 = 12 dB and 0.24 x 100 = 24 dB. The small deviation from the initial value of I q I in fig. 3.2b is the result of the second term (and possibly the higher-order terms) in eq. (3.41).

244


(a)

r o

1 . 0 1.5

0.5

2.0

x lo00 km

2.5

Iql

0.4

(b’

2.5 r

I

1

0 0

0.5

I

I

L

4.0 1.5

1

L

2.0

I

2.5

1

L

1

3.0

3.5xi000km

i

J

Fig. 3.2. Magnitude of q for a pair of guiding center solitons shown at multiples of 500 km when they are amplified at distances of (a) 50 km and (b) 100 krn. The enhanced level of I q I at 500 km in (b) compared with that at Z = 0 is considered to be the effect of the higher-order terms in eq. (3.41).

Figure 3.3 shows the result when the unit amplitude is chosen for the initial amplitude in q. Although this choice should provide an exact one-soliton solution for r = 0, this initial condition fails to recover the initial shape here, even if the amplifiers compensate for the fiber loss at each stage. A highly interesting aspect of a guiding center soliton is the effect of resonances between characteristic oscillation frequencies (and their multiples) of the guiding center soliton and the frequency of perturbation due to the periodic amplification (Hasegawa and Kodama [ 1991al). We present the effects of two types of resonances. One type is the one-soliton resonance. The one-soliton solution, q = tj sech ( qT) exp (i q2Z/2),has a periodic variation in phase with frequency q2/2. Hence, when the frequency (or its

245


1

0

0.8

0.6 Iql

0.4

0.2

0

Fig. 3.3. The magnitude of q for a pair of a, = 1 solitons. The pair behaves like hear pulses.

harmonics) of the periodic perturbation 2 m / Z , matches with the one-soliton frequency, i.e. 2 zn (3.50) = - ; n = l , 2 , ...,

iq’

2,

such a resonance occurs, and the guiding center perturbation presented in 3 3.3 breaks down. Figure 3.4 shows the numerical result for one-soliton propagation under this resonance condition. Here 2, = 0.5, r=0.23, a, = 1.06, and q = 5.01 are 5.0-

rl

= 5.01

4.0-

-

3.0-

u -

2.0-

1.0-

0-

d

0

12

4 L

Fig. 3.4. Effect of one-soliton resonance Jq(T)I plotted at a distance Z.At each distance of Z = 0,2, ..., 12, Iq(T)I is shown for -25 Q T < 25. Dispersive waves are periodically emitted away from the soliton until the amplitude is reduced to the stable (non-resonant) range.

246


[IV, 0 3

chosen so that the n = 1 resonance in eq. (3.50) takes place. Note that dispersive waves are emitted from the soliton, and the pulse amplitude decays rapidly to approximately four. Similar calculations show that if the initial amplitudes are chosen between 4 and 5.5 then emission of the dispersive wave and a decay of the amplitude to 4 take place, but for an initial amplitude of less than 4 this does not occur. Thus we can conclude that the one-soliton resonance with the periodic amplification induces an emission of dispersive waves until a new soliton is generated with a reduced amplitude so that it becomes off resonance. Other resonances occur for multisoliton solutions. For example, a twosoliton solution with amplitudes q, and qz has additional resonances at

t ( q : - qi)m = 2 n n / ~ , m, n = 1 , 2 , .. . .

(3.51)

A large number of numerical calculations are performed for various sets of initial eigenvalues q1 and q2 for the initial value of q given by the bound-twosoliton shape

q(T,O) =%(-) D 91 +

[ql cosh(q2T)eie1+ q2cosh(qlT)eie2], tf2

In the absence of the perturbation, the phase Oi evolves according to

(3.53) It was found that the initially bound solitons given by eq. (3.52), separated under resonant conditions, and two solitons with identical amplitudes (as is expected from the conservation law) emerged. This indicates that the twosoliton resonances excited by the periodic perturbation induce a merging of the eigenvalues q, and qz. Figure 3.5 shows the behavior of Iq(T)I at Z = 0,2,4, . . . ,20, of two bound solitons for the initial amplitudes of case (a) (ql = 3.2, q2 = 1.8), case (b) Fig. 3.5. Variation of bound-two-soliton magnitude I q( T)I at distances Z = 0,2, ...,20 with the b initial eigenvalues q I and q2 being (a) (3.2, 1.8), (b) (3.3, l.7), and (c) (3.5, 1.5). Note the abrupt splitting into two solitons in (b), whereas propagation is stable in (c) and marginally stable in (4.

241


5.0

1

(a)

12

8

16

1 1! d

2

Z

s

7 =3.5;7 2 = 1 5

iwwlKwJH

d 16

Z

20

248

[IV, B 3


(ql = 3.3, q2 = 1.7), and case (c) (ql = 3.5, q2 = 1.5) all lying on q1 + q2 = 5. Note that case (b) produces abrupt separation, whereas case (c) is stable and case (a) is marginally stable. Unlike the case for one-soliton resonance, the two-soliton resonance occurs for a continuous choice of q1 and q2 for different combinations of m and n in eq. (3.51). For example, for case (b) with q1 = 3.3 and q2 = 1.7, (q: - &/2 = 4, for case (a) (q: - q3/2 = 7, whereas for case (c) (q: - q3/2 = 10, whereas 2n/Z, = 12.6. Thus, m = 4, n = 1 approximately satisfies the resonance condition for case (b), but it takes combinations of a large value of m and n for cases (a) to (c) to satisfy the resonance condition. All cases have the same total energy q1 t q2 = 5, which is conserved in the perturbed equation (3.1). The splitting occurs when the two amplitudes qI and q2 assume the same value, 2.5, by the resonant perturbation. From the inverse scattering theory the eigenvalues of the associate equation (1.24) are invariant for q, satisfying the unperturbed non-linear Schrddinger equation. The observed splitting into two solitons with q1 = q2 = 2.5 in case (b) clearly shows that the perturbation has destroyed this property; the eigenvalues are moved around and when they acquire the same value, the solitons are forced to split. In this regard, for the bound soliton to split it seems easier if one starts with q1 and q2 having values close to each other.

5

/5 /1

4 72

3

2 1

0

1

2

3

4

5

6

7

71

Fig. 3.6. Stable (crosses) and unstable (circles) regions in the space of the initial amplitudes q , and q2 for two bound solitons. Instability is identified by the separation into two solitons with identical amplitudes ofabout ( q , + q2)/2.Open and solid circles are slow and abrupt separations, respectively, whereas triangles are the cases in which no clear separation occurs.

IVl

249

APPENDIX A

With these considerations, many numerical calculations are performed to study the two-soliton resonance problem by the periodic perturbation with 2, = 0.5. Figure 3.6 gives a summary of the numerical calculations (2 5 20) where cases of abrupt separation (solid circles), slow separation (open circles), and no separation (crosses) are shown in the plane of the initial values of q1 and q2. Triangles represent cases that are not clearly discernible. An abrupt separation is identified as one that took place within a distance of 2 < 10, whereas slow separations are those of within a distance of 2 2 10. The curves show resonance lines of eq. (3.51) for n/m = 1/4, etc., as indicated. An infinite set of combinations of n and m exist that satisfy the resonances, but those with only lower integer values are shown. The solid curve shows the demarcation line between the unstable (separation) and stable (non-separation) regions. There are indications of resonant effects in that the demarcation line runs parallel to the resonant curves.

Acknowledgements The authors would like to thank Prof. E. Wolf for inviting us to contribute this article. The authors appreciate valuable discussions with J. P. Gordon, L. F. Mollenauer, and N. A. Olsson. One of the authors (Y.K.) is partially supported by US NSF grants DMS 8805521 and DMS 9109041.

Appendix A: Inverse Scattering Transform and N-Soliton Solutions This appendix outlines the method of inverse scattering transform for the non-linear SchrOdinger equation (1.15), and gives the formula for N-soliton solution. We note that eq. (1.15) can be obtained from the compatibility condition of the following set of equations, the eigenvalue problem (1.24), and the evolution equation for the eigenfunctions (Zakharov and Shabat [ 19721) (A.la)

(A. lb)

250


and (A.2a)

(A.2b) Here, the compatibility conditions that given by a2lG1/ZiZa T = a2&/a T a Z for I = 1,2 lead to eq. (1.15) with a 5/32 = 0; i.e. the eigenvalue 5 is constant if the potential function q( T, Z ) satisfies (1.15). Let us consider the direct scattering problem of (A.l) and define the scattering data that determine the solution of (1.15). For (A. 1) we define the solutions Y = $2)t and @ = (q1,$2)f with asymptotic values, for real 5 = 5,

a(c 5 ) + y(Y(T;

+

(3 (y)

e-itr,

as T + - co ,

(A.3a)

eitT,

as T+

+ co .

(A.3b)

The pair of solutions Y and Y = ($;, tions, and therefore

- $:)=

forms a complete system of solu64-41

(A.5a) (A.5b) where we use the fact that W(S,g) E fig2 - f i g , ; i.e. the Wronskian of (A.l) for any two solutions f and g does not depend on T. Note that a ( [ ) can be analytically extended to the upper half plane of 5. The points 5 = l,, n = 1,2, . . .,N (Im C > 0) at which a ( ( ) = 0 correspond to the eigenvalues of (A. l), and the eigenfunctions satisfy

@(T;5,)

=

6, WT; C,)

('4.6)

*

The scattering data X(Z = 0) corresponding to the potential function q(T, 0) in (A.l) are defined by the set of these variables, i.e. (Z

=

0) = [ r ( & 0) for 5 real,

{c,,

C,(O)} for n

=

1,2,. . .,N ] , (A.7)

IVJ

25 1

APPENDIX A

where r(5; 0) = b(5; O)/a([;0) and C,(O) = b,(O)/a;(O), (aA(0) = (aa/aC) (C,; 0)). The method of the inverse scattering transform is based on the fact that the correspondence between the scattering data and potential function is one to one (it is sometimes called a non-linear Fourier transform). The inverse problem to find the potential function from the scattering data is achieved as follows: consider the following set of linear integral equations:

(A.8a)

(A.8b) which are the linear equations for $/(T, 5), $/(C Cn) (I = 1,2; n = 1, . . . ,N ) and their complex conjugates. The solutions are completely determined by the scattering data Z = [r(5), {C, C,}]. Then the potential function q ( T ) and the energy function 1q(T)I2 are given by N

1C,*e-iCXT $2*(C

q(T)= -2

Cn)

-7 n1

n= 1

j

r * ( o e-itT$T(T; 5 ) d r ,

-a

(A. 9)

j

lq(T)I2dT’

T

N

=

-2i

C, eiZtzT$l(T; 5,) n= 1

+n

1

co

-00

r ( 5 ) eitT$,(T; 5 ) d r .

(A.lO)

252

[IV


It is interesting to note that from (A.8) and (A.lO) the total energy for N-soliton solutions can be written by

j'

N

00

Iq(T)I*dT=2i

c

N

([n-

0 , =

(A. 14)

7

which is the frequency of the nth soliton (see (1.26)). For the case of N-soliton solutions where r ( r ) = 0, the integral equation (A.8) can be written as a set of algebraic equations for the column vectors F/ = (fil,. . . ,f i N I t defined by fin = $;(Ti is the third cumulant of the photon statistics at time t . This equation shows that the sign of the change in photon variance depends on the second and third cumulants of the original photon statistics. Hence, the onecount process does not change just the mean photon number, but also the whole photon statistics, a result that parallels the findings of 8 5.2. The no-count intervals lead to the change in mean photon number ( n ( t ) ) - + ( n ( t + 0))

=

i a - - - lnTr[p(t)exp(-qatat)].

av

(5.15)

Expanding the right-hand side of this equation in powers of the small parameter q ~ and , taking the limit q ~ +0 leads to (5.16) This shows that during these intervals the average photon number decreases in time at a rate given by the photon number variance multiplied by the inverse

QUANTUM MEASUREMENTS

307

of the expectation value of the waiting times. Therefore, it remains unchanged for number states, but decreases for all other states. In the same limit the time evolution of the photon number variance depends only on the third cumulant, (5.17) From these results one can derive an equation describing the non-unitary evolution of the photon statistics in a photodetection process where we read out all the information concerning registrations of photocounts in real time. One finds (Ueda [ 19891, Ueda, Imoto and Ogawa [ 19901) p m ( z I , 72,.**3z m ; O ,

T)

where T is the time at which the measurement is stopped and the times zi, i = 1,. . ., m, are the times at which the m successive photocounts were registered.

5.4. MEASUREMENT-INDUCED OSCILLATIONS OF THE FAN0 FACTOR

Ogawa, Ueda and Imoto [ 19911 recently applied this formalism to a study of the continuous detection of squeezed states of the electromagnetic field. Squeezed states are generated from a vacuum by the unitary transformation (Stoler [ 1970, 19711)

I a, r>

=

S(r)D(a) 10)

3

(5.19)

where D(a) is the displacement operator ~ ( r= )exp(aat - a*a)

(5.20)

and S ( r ) is the squeezing operator ~ ( r=) exp{ir[a2 - (a+')]},

(5.21)

the squeezing parameter r being taken to be real for simplicity. For large amounts of squeezing and positive r, the squeezed states are strongly super-Poissonian (Walls [ 19831, Schleich and Wheeler [ 19871). Figures 13a-c show the evolution of the average photon number, the photon number variance, and the Fano factor a'calculated using eq. (5.18). This example assumes that

308

CAVITY QUANTUM OPTICS AND THE QUANTUM MEASUREMENT PROCESS

I

I

73

74

[V,4 5

TIME

Fig. 13. Temporal development of(a) the average photon number ( n ( r ) ) , (b) the photon number variance [An(r)I2, and (c) the normalized variance uz = [An(r)l*/(n(r))in a continuous measurement sequence where the photocounts were detected at times z, to .,z. (After Ogawa, Ueda and Imoto [1991].)

one-count processes occur at times T,, z2, z3, and 7,. The figures clearly illustrate how both the mean photon number and the variance decrease monotonically during the no-count intervals. The variance decreases faster than the mean photon number, however, as readily seen from eqs. (5.16) and (5.17). Hence, the Fano factor decreases monotonically during these intervals. The situation is different in the one-count processes, where o2 increases abruptly. Ogawa, Ueda and Imoto [ 19911 refer to this effect as “measurement-induced Fano factor oscillations”. Such oscillations occur generally when both the squeezingfactor r and a are large, and they are a direct consequence of the back action of the measurement on the state of the field. The next subsection discusses a closely related example of measurement-induced dynamics in the

v, 8 51

309


micromaser, where the back action of the measurements on the field leads to complicated dynamics, even though in the absence of measurements the system would be in steady state.

5.5. MEASUREMENT-INDUCED DYNAMICS OF THE MICROMASER FIELD

Returning to the situation of $ 5.2, we now discuss the dynamics induced on a high-Q (but not lossless) micromaser as the state of successive atoms is measured when they leave the cavity. For the sake of illustration we consider a situation where the effective potential (4.26) has two minima of equal depth. The procedure to simulate a typical sequence of measurements is the same as in $ 5.2, except that now the micromaser mode is damped in the time intervals when no atoms are inside the cavity (Meystre and Wright [ 19881). Figure 14 gives the raw results of a typical sequence of measurements for the case N,, = 50, nb = 5, and 0 = 2.116 K, in which the effective potential has two minima of equal depths at vd = 0.18 and v, = 0.68. In contrast to the situation encountered in real experiments, we assume for simplicity that all atoms leaving the resonator are detected, either in their state I a ) or I b ) , with unit quantum efficiency. For clarity the inset in fig. 14a shows a portion of the same results on an expanded horizontal scale. From these raw data and eqs. (5.3) and (5.4), one can infer the cavity mode photon statistics just after the measurement, and determine the corresponding mean photon number ( n ) . This is precisely the strategy that would be followed in the laboratory to extract information about the micromaser field from the clicks at the detector. The results of this reconstruction are shown in fig. 14b, where measurementinduced diffusion between the two wells of the effective potential are readily apparent. Figure 14c gives the Fano factor o2 of the photon statistics for the same sequence of measurements. It exhibits a broad peak during the downswitching between v, and vd. This sequence of measurements illustrates the effect of back action on the dynamics of the micromaser. In the absence of such a back action the mean photon number would evolve monotonically to a steady-state level, as shown in fig. 14d. Another example illustrates particularly well the effect of the measurement back action on the dynamics of the micromaser: at zero temperature (n, = 0) the micromaser photon statistics reduce to (5.22)

310

[V,0 5


...

ATOM -NUMBER

3>12 0 z

Y

0

ATOM -NUMBER

'y 8'5

00: 6

3 z

e W

s

4

I 7 2

W

z0

0 ATOM NUMBER

250

500

ATOM NUMBER

Fig. 14. (a) Raw data from repeated experiments of the state of two-level atoms as they leave the micromaser cavity. The value + 1 corresponds to atoms measured in their upper state, and - 1 corresponds to atoms measured in their lower state. The vertical lines are for visual help only. (b) Average photon number in the field just after a measurement as inferred from the outcome of the measurement. (c) Corresponding Fano factor u2. (d) Conventional ensemble averaged value of ( n ( t ) ) for the same parameters N,, = 50, nb = 5, 8 = 2 . 1 6 6 ~ .Measurement-induced diffusion between the minima of the effective potential (4.26) is clearly shown in (a). (After Meystre and Wright [ 19881.)

A direct consequence of this result is that for values of the parameters Q=

4nJNe,,

(5.23)

where q = 1, 2, 3, . . .,the ensemble-averaged steady-state photon statistics of the micromaser are

(5.24) independently of the initial conditions. That is the cavity mode is in the vacuum state because under conditions (5.23) the vacuum acts precisely as a 2qn pulse for atoms spending a time T inside the resonator. The vacuum state, as any number state, does not exhibit any intensity fluctuations and, hence, can act

v, 8 51


311

as a true 2qn pulse. In most other situations, however, the inherent intensity fluctuations lead to the impossibility of achieving such a “perfect” interaction. This is the case, for instance, if the micromaser has a finite temperature, n b # 0. This situation is shown in fig. 15. Here, condition (5.23) is fulfilled (for N,, = 5 and 0 N 35), but we have nb = lo-’. For the corresponding thermal field the initial atom experiences almost, but not quite, a 1On pulse. Consequently, the probability of measuring it in its upper state at the exit of the resonator is almost unity. Because the initial field density matrix is almost a delta function, there is a nearly exact conservation of the mean number of photons and the field remains almost unchanged. (In the language of § 6 this would be called a nearly quantum-non-demolition measurement.) As further atoms are injected, however, there is a small but finite probability that one of them will eventually be measured in its ground state. This happens first in our example for atom number 3 10 or so. In this case the back action on the cavity mode is particularly drastic: to a very good approximation the average intra-

Fig. 15. (a)Inferred average photon number ( n ) for N,, = 5, 8 = 35 and nb = as a function of the number of atoms injected and measured. (b) Probability for the corresponding atom to be measured in its upper state as it leaves the cavity. (AAer Meystre and Wright [ 19881.)

312


[V,8 6

cavity photon number increases by one, and the field becomes almost, although not quite, the number state 11). For the parameters of this example the probability for the next atom to leave the cavity in its upper state is 0.98, so that it is also very likely to be measured in that state. This is precisely what happens in the subsequent measurements. Hence, the cavity field simply decays back at rate IC towards thermal equilibrium. As further atoms are injected, however, there is a finite probability that eventually another atom will leave in the lower state. This leads to the kind of dynamics shown in fig. 15, which can be understood as measurement-induced relaxation oscillations. As before, the observed dynamics depend sensitively on the outcomes of all the preceding measurements. Since every measurement has an element of randomness attached to it, the dynamics are in the final analysis governed by chance, in stark contrast to the classical case. The micromaser illustrates particularly clearly this irreconcilable difference between classical and quantum physics.

0 6. Quantum Non-demolition Measurements 6.1. BACK-ACTION EVASION

The repeated measurements discussed so far are all characterized by a significant back action of the measurement process on the state of the field just after the measurement. One class of measurements, called back-action evading, or quantum non-demolition (QND) measurements (Braginsky, Vorontsov and Khalili [ 19771, Thorne, Drever, Caves, Zimmerman and Sandberg [ 19781, Unruh [ 19781, Thorne, Caves, Sandberg, Zimmerman and Drever [ 19791, Caves, Thorne, Drever, Sandberg and Zimmerman [ 19801, Caves [ 1983]), does not have such an effect on the system to be measured. Consider an arbitrary quantum mechanical system, with Hamiltonian H,, coupled to a measuring apparatus with Hamiltonian H , by the interaction V, so that the total system-measuring apparatus Hamiltonian is H=H,+H,+V.

Clearly, the interaction Hamiltonian V must depend on both system and measuring apparatus observables, so that information about the system can influence the state of the measuring device. A system observable is said to be a QND observable if it can be measured repeatedly in such a way that the results of each measurement after the first can be predicted with no uncertainty occurring from the result of the preceding measurement.

a

v, 8 61

QUANTUM NON-DEMOLITION MEASUREMENTS

313

The first measurement, at some time to, can be seen as a preparation step that leaves the system in one of the eigenstates A , of A^. This step is characterized by the element of randomness discussed in preceding sections. The result of the second measurement, in contrast, is fully predictable provided that at the time t of the measurement, the system is in an eigenstate of the Heisenberg operator a(t).If this is the case, the measurement outcome will definitely be that eigenvalue, and the measurement will leave the state of the system unchanged. Mathematically, this is expressed by the condition (Caves, Thorne, Drever, Sandberg and Zimmerman [ 19801)

[A'(t),A'(t')]

=

0;

i.e. the operator A^ commutes with itself for all times in the interaction picture (Caves, Thorne, Drever, Sandberg and Zimmerman [1980]). (For a set of Hermitian operators to form a set of QND observables, the condition (6.2) becomes

{a,}

[a;(t), a:(tf)] = 0 ,

(6.3)

but this more general form is unnecessary in the following discussion.) To obtain a physical feel for eq. (6.2), consider a simple interaction Hamiltonian of the form (6.4)

V=KAQ,

where Q is an observable of the measuring apparatus and K the systemmeasuring apparatus coupling constant. To ensure that the instantaneous signal at time twill not contain any contaminant from observables e(t)that do to commute with not commute with a(t),it is necessary and sufficient for the part of the Hamiltonian that describes the interaction of the system with the apparatus. This is guaranteed by the condition (6.2). An alternative way to understand this condition is based on the observation as giving the that physically we can think of the Heisenberg operator aH(t) evolution of the observable when the coupling is turned on, whereas al(t)gives its evolution in the absence of coupling to the measuring apparatus ( K = 0). If aH(t) = A1(t), the evolution of the observable is unaffected by its coupling to the measuring apparatus. Caves, Thorne, Drever, Sandberg and Zimmerman [ 19801 show that condition (6.2) guarantees that the equality holds. It is obvious that any non-degenerate, conserved quantity satisfies eq. (6.2) and is a QND variable. For a simple harmonic oscillator these quantities include the photon number ata as well as the two quadrature phase operators

a(t)

314


[v, $ 6

of the field (real and imaginary parts of the field amplitude)

x

1 -= 2

eiRr +

,t

e-iRt

)

(6.5)

6.2. QND MEASUREMENTS IN OPTICS

Several quantum optical QND measurement schemes have been proposed and demonstrated. Milburn and Walls [ 19831 considered a coupled system-meter model consisting of two harmonic oscillators coupled by a fourwave mixing interaction, so that the total Hamiltonian is H

=

hwaata

+ hwbbtb + hgatabtb,

(6.7)

where a and b are the annihilation operators for the system and detector of frequencies o,and wb, respectively, and g is a coupling constant. They showed that in this case the detector harmonic oscillator can be used to perform QND measurements on the system observable D , ( t ) = cos(gatat) by monitoring the real part of the meter amplitude. A similar proposal to measure the mean photon number of a harmonic oscillator in a scheme where the signal and detector waves interact by means of the optical Kerr effect was proposed by Imoto, Haus and Yamamoto [ 19851. These authors showed that the optical phase of the detector wave is sensitive to the change in refractive index owing to the optical intensity of the signal wave, but does not affect it. The effects of crystal losses in these schemes were analyzed by Imoto and Saito [ 19891, who concluded that losses impose a limit on the device length that can be used and that a given loss rate defines an optimum device length. Yurke [ 19861 discussed a variety of back-action evading measuring devices, including parametric-gain media, parametric amplifiers and frequency converters, beam splitters and degenerate parametric amplifiers, and degenerate fourwave mixers. In the first scheme, for instance, the signal and meter are harmonic oscillators of respective frequencies w, and w,. They interact by means of a non-linear medium characterized by a second-order susceptibility x(') and pumped at both the frequencies w, + w, and o,- 0,. The second scheme consists of a three-stage device consisting of a frequency converter followed by a parametric amplifier and a second frequency converter. The frequency converters are pumped at the difference frequency w, - or, whereas the parametric converter is pumped at the sum frequency w , + w,.

v, 8 61


315

All schemes proposed by Yurke can be described by a set of equations of the form

where uin, uoutrbin, and b,,, are the annihilation operators for the modes entering and leaving the "black box" representing the back-action evading amplifier (Yurke [ 19851). It can be seen that a black box with the input-output transformations described by eqs. (6.8) and (6.9) performs QND measurements by introducing the quadrature component operators for the different field modes

x:($) = $(e-'+u, + ei$uL),

(6.10) (6.11)

x:(+) = $(eci@bin+ ei@bk),

(6.12)

YE(+) = +i(eci@bin - ei*bk),

(6.13)

and similarly for XFt(O+ $), YZut(O+ $), X,""t(O' + +), and Y,O"'(O' + +). From eqs. (6.8) and (6.9) one readily obtains

X,O"'(O + J/) = X F ( $ )

+ 2GXp(+),

ryye + $) = Y:($), x;ye' + = x$(+), Y,""'(O' + +) = Yf(+) - 2GY:($).

(6.14) (6.15) (6.16) (6.17)

Back-action evasion is readily apparent from these relations: from eq. (6.17) one sees that YtUt(O' + +) carries information on 2GY)($); i.e. it measures this component of the incoming signal. However, eq. (6.15) shows that the conjugate component Y,( J/) remains clean. The back-action noise is sent into X,, as seen from eq. (6.14). Shelby and Levenson [ 19871 proposed further QND schemes involving x'') non-linear media, including optical rectification and parametric amplification. In the first scheme a voltage proportional to the intensity of the light, which is unperturbed by the non-linear coupling, appears directly across the non-linear crystal, whereas the second scheme uses frequency-degenerate three-wave parametric mixing with type-I1 phase matching, so that quantum correlations appear between two orthogonally polarized signal and idler fields.

316


[v, $ 6

Experiments demonstrating back-action-evading measurements in schemes using a four-wave mixing interaction (zc3) non-linearity) were carried out by using the Kerr effect in fused silica fibers (Levenson, Shelby, Reid and Walls [ 19861, Imoto, Watkins and Sasaki [ 1987 I). La Porta, Slusher and Yurke [ 19891 demonstrated the scheme proposed by Yurke [ 19861, which performs, as we have seen, a back-action-evading measurement on one phase of the optical field. In the next subsection we discuss an experiment proposed by Haroche to perform QND measurements on the exceedingly low intensity fields that can be generated in micromaser cavities.

6.3. QND MEASUREMENTS IN MICROMASER CAVITIES

The QND measurements performed so far in the optical regime were in conditions far removed from those under which cavity QED effects can be observed. On the other hand, the kinds of measurements discussed so far in cavity QED have a considerable back action. Recently, Brune, Haroche, Lefevre, Raimond and Zagury [ 19901 proposed and analyzed a QND scheme that permits measuring the number n of photons in a micromaser cavity. Their technique relies on the non-resonant coupling of the field to two-level atoms, and infers ( n ) by measuring the phase shift of the atomic wave function at the resonator exit. Because of the strong atom-field couplings that can be achieved with Rydberg atoms, the proposed method has the advantage of being applicable down to ( n ) -+ 0. To understand how this technique works, consider a three-level Rydberg atom with levels l a ) , Ib), and ti), where l a ) and Ib) label the upper and lower levels as before and I i ) labels an intermediate level that can be reached from level I a ) by absorption of one photon. We assume the frequency R of the cavity mode under consideration to be nearly resonant with the transition frequency between the excited and intermediate levels, with a detuning of 6=

Wie -

R.

(6.18)

From eqs. (2.5) and (2.6) we find that for an intracavity field in the Fock state In), the upper level I a ) undergoes a dynamic Stark shift. Subtracting the bare energies of the uncoupled atom-field system from the corresponding dressed energies, we find A(r, n) =

-

6)

=

+[Js2 + 4d2(r)n - 61 ,

(6.19)

where we have slightly generalized the form of the Rabi frequency to allow for

v, 8 61


317

a spatially dependent atom-field coupling constant, which we call d instead of g to avoid confusion between I a ) 4 I b ) and 1 a ) + I i ) transitions. For sufficiently large detunings this expression can be expanded to give d2(r)n d(r, n ) = _ _ .

(6.20)

6

We see, then, that the Stark shift experienced by the excited state with respect to the ground state is proportional to the photon number in the cavity. (The ground state is not notably shifted if R - wab is much larger than 6.) The accumulated phase shift per photon is E = ( d ( r , n = 1)) L,/uo, where the ( ) denotes a spatial average along the atom path through the cavity. Large singlephoton shifts can be obtained by choosing detunings 6 that are relatively small, yet large enough so that no significant absorption takes place. The shift (6.20) alters the probability amplitude for the atom to be in the excited state l a ) relative to that for the ground state. (Note in this context that the absorption probability is much larger for a cavity with a square-shaped mode function than for a mode with a slow spatial variation, whereas the phase shift is of the same order of magnitude, so that such a slow variation is an essential feature of the method proposed by Brune, Haroche, Lefevre, Raimond and Zagury [ 19901 to perform QND measurements on the number of photons in the microwave cavity.) The set-up of the proposed experiment is shown in fig. 16, and is based on Ramsey’s method of separated fields (Ramsey [ 19851). The micromaser cavity is placed between two field zones R , and R,, driving the l a ) - I b ) transition. This transition is then detected behind R , by state-selective field ionization.

L

(4

(b)

Fig. 16. (a)QND set-up for measuring the photon number n in a cavity. The atomic beam prepared in state la) crosses successively the field zone R,,the cavity and the field zone R,, before detection by state-selective field ionization. (b) Diagram of the levels l a ) , I6), and I i). The fields R, and R, induce transitions between the levels l a ) and Ib), whereas the cavity field induces a Stark shift between these levels but no real transition, so that n remains unchanged. (After Brune, Haroche, Lefevre, Raimond and Zagury [1990].)

318

[V, 8 6

CAVITY QUANTUM OPTICS A N D THE QUANTUM MEASUREMENT PROCESS

Consider an atom moving at the nominal velocity uo across the length L, of the cavity. In the absence of signal field the phase shift between the atom and Ramsey field reference is $0

=

(a, - % f ) L , / V O

(6.21)

f

where Or is the frequency of the fields R I and R,. Assuming that each Ramsey field acts as a i n pulse on the atomic transition l a ) - [ b ) for atoms moving at the nominal velocity uo, then the probability that atoms at velocity u leave the second Ramsey field in the excited state is (see, e.g., Meystre and Sargent [ 19901) (6.22)

P,(u, 0) = sin2(nuo/2u)C O S ~ ( $ ~ U , / ~ U ) ,

whereas in the presence of (exactly) n photons in the cavity it becomes (6.23)

P,(u, n) = sin2(nuo/2u)C O S ~ [ (-$ne)uo/2u] ~ ;

i.e. the Ramsey fringes are shifted by an amount nE with respect to their position when the cavity is empty. In practice the atomic response must, of course, be averaged over the atomic velocity distribution 9 ( u ) and the field photon statistics p,,. Figure 17 shows the transition probability from the excited to the ground state plotted against $o for a monokinetic atomic beam and field in a number state In), as well as

I

I

(c)

I

I

(4

+,,

Fig. 17. Probability for the exiting atoms to be in the excited state l a ) versus for E = 271. (a) Monokinetic atomic beam ofnominal velocity u,, and field in a number state In). (b)-(d) Upper state probability averages over the atomic velocity distribution for a cavity mode in: (b) a Fock state, (c) a coherent state, and (d) a thermal field. In all cases the mean photon number is ( n ) = 3. The full vertical scale is from 0 to 1, and the full horizontal scale is 24x. (After Brune, Haroche, Lefevre, Raimond and Zagury [1990].)

v. 8 61


319

after averaging over the atomic velocity distribution for various photon statistics of the field. The shape of the fringe pattern permits one to distinguish a coherent field from a thermal state and a Fock state. In this example the field was assumed to be in the same initial state before the successive atoms were injected inside the cavity. Thus, it was allowed to relax to equilibrium between atoms. We now discuss what happens when this is not the case and repeated measurements are performed on the state of the field with no significant relaxation between atoms. We assume that the field is initially described by the photon statistics p z ) . The analysis of a sequence of experiments can be performed along the exact lines of the discussion in 0 5.2. Depending on whether the first atom is measured to be in the upper or lower state, the field density operator is “reduced” to

(6.24) m

or (6.25) where P(b, u , , n ) is the probability that an atom at velocity u , and interacting with the Fock state In) leaves the cavity in the lower state. Assuming that a sequence of N measurements yields as a result the sequence { s k , U k } , meaning that the atom used for the kth measurement has velocity u& and leaves the second Ramsey field in the state Is) (s = a or b), we find the conditional probability of having n photons in the field as

(6.26) m

k

Figure 18 shows the result of a numerical simulation of such a measurement sequence. This simulation is carried out much the same way as the simulations discussed in 0 5.2 and 0 5.5, except that a supplementary random number must be chosen to select the velocity of the kth atom. Generally, Brune, Haroche, Lefevre, Raimond and Zagury [1990] note that p i N ) converges to the Kronecker delta function representing a Fock state somewhere within the width of the original distribution. This “collapse” requires a certain number of

320


(c)

[V,5 6

(4

Fig. 18. Evolution ofthe photon number distributionpLNN’in a simulation of ameasuring sequence. (a) Initial distribution (coherent state with amean photon number E = 10. (b-d) Photon statistics after 3, 5, and 20 detected atoms. The collapse into a Fock state is clearly observable. (After Brune, Haroche, Lefevre, Raimond and Zagury [ 19901.)

atoms (about 20 for the example of the figure), which these authors call an “elementary measuring sequence”. This shows that a single atom is not sufficient to provide a complete measurement of n, which is “pinned down” to a precise value only by gathering enough information through repeated detections of atoms. Each single detection results in multiplying the photon statistics p n by a function of n presenting peaks and minima, thus decimating efficiently some photon numbers in the distribution, until only one is left. From then on the field statistics can no longer change, and a number state is effectively prepared and can be measured repeatedly. Note that, contrary to the situation encountered in the resonant detection schemes of 0 5, undetected atoms do not change the photon statistics here. The argument leading to eq. (6.26) does not include field dissipation between atoms. For sufficiently weak losses this problem can be treated along the lines of micromaser theory, neglecting dissipation while atoms are inside the cavity (Filipowicz, Javanainen and Meystre [ 1986b1). Brune, Haroche, Lefevre, Raimond and Zagury [ 19901 performed such simulations and demonstrated “quantum jumps” of the field as its energy was dissipated.

MACROSCOPIC SUPERPOSITIONS

32 1

8 7. Macroscopic Superpositions 7.1. TANGENT AND COTANGENT STATES OF THE ELECTROMAGNETIC

FIELD

The generation of macroscopic quantum superpositions is an issue of considerable importance in the study of the relationship between quantum and classical physics (Leggett [ 19801). Although evidence for quantum tunneling has been established (Martinis, Devoret and Clarke [ 1988]), the observation of quantum coherences is more difficult. A major problem is that macroscopic objects are not isolated but are coupled to their environment, which causes quantum coherences to be destroyed on a very fast time scale. Examples showing the influence of dissipation on macroscopic superpositions and the concomitant destruction of quantum-mechanical interference phenomena were discussed by Caldeira and Leggett [ 19851, Walls and Milburn [ 19851 and Savage and Walls [ 19851. Optically, a method to generate a superposition of macroscopically separated quantum states by propagating a coherent state through a Kerr non-linear medium was proposed by Yurke and Stoler [ 19861, but this scheme suffers the same difficulties with dissipation. From this point of view, cavity QED experiments in the microwave regime, with the associated high-Q resonators, provide an interesting alternative. We show in this section that by pumping a micramaser with a polarized beam of two-level atoms, it is possible under appropriate conditions to generate a steady-state field that is almost pure and resembles a macroscopic superposition of quantum states (Slosser, Meystre and Wright [ 19901). Consider a lossless micromaser cavity driven by a stream of polarized atoms, i.e. of atoms prepared in the coherent superposition

I$)

=40)

(7.1)

+816>.

We further assume that the atoms interact with the cavity field for a time z such that the trapping condition (2.26) is fulfilled for some N and some q. Calling pf(0) the initial field density matrix, the field evolution is given by an iterative solution of eq. (4.5) for the initial condition P(0) = Pf(0) c3 I $> ( $1

.

(7.2)

When solving this problem numerically, Slosser, Meystre and Braunstein [ 19891 found that the field alone evolved to a pure (zero-entropy) state if the

integer q appearing in the trapping condition (2.26) was odd and the field

322


[V,8 7

density matrix was initially confined between the vacuum and the state I N ) such that g J m T = qn. A similar result is found (Slosser, Meystre and Braunstein [ 19891, Slosser and Meystre [ 19901) if the initial field density matrix is such that the photon statistics are confined between two adjacent trapping states INd) and IN,) that satisfy eq. (2.26) for two integers qd and qu, such that if qd is even, then q, is odd, and if qd is odd, then q, is even. (By adjacent trapping states we mean that no Fock state between INd) and I N u ) satisfies eq. (2.26), independently of q integer.) Figure 19 illustrates the states reached for Nd = 0, Nu = 25, and q, = 1 and 3, respectively. Guided by these results, it is possible to determine the state reached by the field from a self-consistency argument. We assume that the field is in the pure state

If>

N"

C

=

sn

(7.3)

In> 9

n=Nd

and require that it remain in that same state (to within an overall phase factor) after interacting for a time T with an atom prepared in state (7.1); i.e.

If>@((.lQ>

+PIb))+e'"f)@b'

la>

+P'

(7.4)

Ib)),

+

where 1 a' I + 1p' I = 1 and a ' , 8' as well as the overall phase are independent of n. With eqs. (2.24) and (2.25) we readily find the recurrence relations (7.5) and

. P' exp(i+) - ~ c o s ( g J n + l T ) sn= - 1

a sin (gJn+l T )

9

(7.6)

which must be satisfied simultaneously for all n within the region of Fock space under consideration. These equations are satisfied under the two possible conditions as'

(7.7)

-a'P

=

or a8 =

-

a'p'

.

(7.8)

Assuming, without loss of generality, that a, a ' , P, and /3' are real, one finds ei+ = + 1 -

9

a'= T a ,

P'=

kP

(7.9)


323

G

ffl

0

LJ

(b)

Fig. 19. Moduli of the density matrix elements ( n 1 pf I m ) of the field mode driven by a stream of polarized atoms for I a I = 0.75 and INu) = 125 ). The initial field was in a thermal state with mean photon number nb = 0.1, and the initial distribution was slightly truncated and renormalized to avoid any initial population past the state 125). Part (a) corresponds to qu = 1 and (b) to qu = 3. (After Slosser and Meystre [ 19901.)

324


[v, 8 7

or e i @ =+ I ,

a'= +a,

P'= T P .

(7.10)

No other zero-entropy states are possible under the Jaynes-Cummings dynamics. Setting exp(i$) = 1, eqs. (7.5) and (7.6) yield simple recurrences for the probability amplitudes s,. We find s, =

i(a/P) cot (hg&

z)s,

(7.11)

-

in the first case and

in the second case. The corresponding photon statistics are (7.13) and (7.14) respectively. The boundary conditions sNd- I = 0 and sNu = 0 at the boundaries of the block of Fock space under consideration can only be satisfied if (a) qd is even and q, is odd, in which case the appropriate solution is the so-called cotangent state defined by eq. (7.1 I), or (b) if qd is odd and qu is even, which gives the tangent state (7.12). The vacuum state is a down-trapping state with qd = 0. Since typical initial conditions for the fields, such as thermal fields, have a non-zero vacuum population, cotangent states are therefore particularly relevant here. The properties of the cotangent states were studied by Slosser and Meystre [ 19901 and Slosser, Meystre and Wilkens [ 19911, who found that one-cotangent states (i.e., cotangent states bound between the vacuum and a trapping state with q, = 1) exhibit sub-Poissonian photon statistics and can be squeezed for a broad range of conditions. In contrast, three-cotangent states (i.e. cotangent states bound between the vacuum and an upper trapping state with q, = 3) can exhibit a strongly super-Poissonian character for appropriate choices of a. This character is central to the present discussion, since it is associated with bimodal photon statistics and a state reminiscent of a macroscopic superposition, as shown in fig. 19b. To understand how this bimodal distribution comes about, we note that the +

v, B 71

325


20 16 12 Cot2(X)

8 4

0

0

n/2

7T

3 7 ~ 2/

X

Fig. 20. The cot’x function and the horizontal line at /?/a used in the graphic determination of the photon statistics of the cotangent state. (After Slosser and Meystre [1990].)

recurrence relation (7.11) indicates that Is, I > Is, - ,I, provided that COt(ig&

T) > / ? / a .

(7.15)

The cotangent function cot2x is shown in fig. 20, which has a horizontal line at /?/a.For a given interaction time T the only allowed points on the x axis are such that 4x2/g2z2= n ,

(7.16)

where n is an integer. The preceding discussion indicates that the existence of cotangent states requires that this condition be satisfied at x = (2q, + 1)71/2. Consider first the simple case qu = 1 (1-cotangent state): if the first value of x such that condition (7.16) is satisfied, x = g f i r/2, is also such that cot x < J/a, then we have s, > so and the photon statistics are peaked at some photon number other than zero; otherwise, they are peaked at n = 0. For q, = 3, in contrast, condition (7.16) is satisfied at x = 3 4 2 , but not at n/2. It is clear that the photon statistics of the cotangent state can now become double peaked for appropriate values of b/a. In this case, and for a qu that is sufficiently large, the contangent states acquire a character like macroscopic quantum superpositions.

326


[v,$ 7

7.2. EFFECTS OF DISSIPATION

We have seen that the generation of cotangent states in the micromaser relies in an essential way on the existence of trapping states of the electromagnetic field. In general, however, the isolating effect of these states does not survive in the presence of dissipation (Meystre, Rempe and Walther [ 19881). This can be seen by writing the field master equation (4.4) in component form for the diagonal elements of the field density matrix pn = ( n I p In) . This gives

(7.17) At non-zero temperature (nb # 0) dissipation leads to both upward (n + n + 1) and downward (n -+ n - 1) transitions on the harmonic oscillator ladder. This incoherent mechanism allows the micromaser to jump past the trapping state I Nu), so that thermal fluctuations rapidly wash out its effect. At a very low temperature (nb N 0), however, eq. (7.17) reduces to (7.18)

0 -I

-2

-5 -6 -7 0

2

4

6

8

10

I2

14

LOG ,,(Nex)

Fig. 21. The log-log plot of the entropy of the final micromaser field state as a function of Ncx. The parameters are a = 0.51 and Nu = 10 (solid curve and Nu = 15 (dashed curve). (After Slosser, Meystre and Wright [1990].)

v, I 71

321


and dissipation induces only downward transitions. In this regime the dynamics of the micromaser are still greatly influenced by the existence of trapping states (Meystre, Rempe and Walther [ 19881). This suggests that some remnants of the macroscopic quantum superpositions might survive in this regime. To investigate this possibility, Slosser, Meystre and Wright [ 19901 numerically solved the micromaser field master equation (Filipowicz, Javanainen and Meystre [ 1986b], Bergou, Davidovich, Orszag, Benkert, Hillery and Scully [ 19891)

apf= Lpf + R [ F ( z )- I]pf, at

(7.19)

where I is the identity operator, for atoms initially in the coherent superposition (7.1). The results of this investigation are summarized in figs. 2 1 and 22 for the cases Nd = lo), Nu = 110) or I15), and qu = 3. The initial field is taken to be a thermal field with nb = 0.1, truncated and renormalized so that p , = 0 for n > Nu. Figure 21 shows the von Neumann entropy of the field

S T -kBTr(pflnpf),

(7.20)

where kBis the Boltzmann constant, as a function of the ratio N,, between the atomic injection rate and cavity decay rate. The solid line is for I N u ) = 1 10)

10

8

6 A

t 4 2 0 0

2

4

6 8 LOG ,,(Nex)

10

12

14

Fig. 22. Mean photon number ( n ) (solid curve) and Fano factor uz (dashed curve) as a function of IogN,, for Nu = 15, a = 0.53. (After Slosser, Meystre and Wright [1990].)

328

CAVITY Q U A N T U M OPTICS A N D THE QUANTUM MEASUREMENT PROCESS

[v,8 7

and the dashed line for I Nu) = I 15). In both cases we observe that a broad plateau, where S is roughly constant, is followed by a transition to a region where the entropy decreases as l/Nex.This decrease is clear evidence that the micromaser steady state approaches a pure state. Figure 22 demonstrates that it is not only the off-diagonal elements of the field density matrix that undergo a transition. Rather, the quantitative nature of the solution changes as N,, is increased. This figure shows the mean photon number (solid line) as well as the Fano factor of the field (dashed line) as a function of N,,. We observe a distinct transition between two final states of a completely different nature, the transition region being characterized by a strong peak in the field fluctuations, which is suggestive of a phase-transitionlike phenomenon. Below the transition the field is essentially Poissonian (vacuum field), while it is super-Poissonian above the transition region. (Note that as discussed by Slosser, Meystre and Wright [ 19901, this transition does not correspond to the micromaser incoherent pumping threshold.) The fact that the photon statistics is super-Poissonian and that the entropy decreases above the transition region suggests that in this regime we are generating steady-state macroscopic quantum superpositions. This is further confirmed in fig. 23, which shows the moduli of the field density matrix elements I ( n I pf I m ) I in the low-N,, regime, in the transition region, and in the high-Ne, regime. This figure clearly illustrates the transition to an almost pure “macroscopic superposition”, which is well approximated by the cotangent state. To determine more precisely the nature of the transition, we consider the logarithm of the Q function of the micromaser field density matrix. This is motivated by noting that in equilibrium systems the steady-state density matrix is of the form p=Nexp(-V/k,T),

(7.21)

where .N is a normalization constant, V is the potential, k , is the Boltzmann constant, and T is the temperature. Of course, this expression is not valid for the open system on hand, yet by analogy we expect the logarithm of the density matrix to still yield some kind of an effective potential. The specific choice of the Q function is motivated by the property that, in contrast to other distribution functions, the diagonal expansion of the density matrix on coherent states Q(M:) = ( M: I p I a ) is positive definite and, hence, its logarithm is certain to exist. Figure 24 shows the effective potential V,, = - In Q(a) for various values of N,,, fig, 24e giving the corresponding function in the situation where the offdiagonal elements p,, of the field density matrix have been set arbitrarily equal to zero (Meystre, Slosser and Wilkens [ 19901). Note that the injection of

v, 5 71


329

C

Fig. 23. Moduli of the field density matrix elements ( n I pr I m ) for (a) N,, = 15, (b) N,, = 10'. and (c) N,, = lo9. Here, Nu = 15, a = 0.53. (After Slosser, Meystre and Wright [1990].)

330


[V,$ 7

Fig. 24. Effective potential - InQ(a) at (a) No, = lo5, (b) N,, = lo6, (c) N,, = lo’, and (d) N,, = lo8. Part (e) shows the corresponding function if the off-diagonal elements of the field density matrix are arbitrarily set equal to zero. (After Meystre, Slosser and Wilkens [1990].)


33 1

332


[V,5.7

Fig. 24(e).

polarized atoms breaks the symmetry of the Q function along Ima, a result analogous to the well-known situation of the laser with an injected signal. For a second-order-like phase transition the effective potential would develop its minima above threshold by means of a pitchfork bifurcation from its minimum below threshold. Figure 24 clearly shows that this is not the case here; rather, the transition to a macroscopic superposition resembles a firstorder phase transition and the system is akin to optically bistable systems. Indeed, for large N,, the Q function resembles that of conventional dispersive bistability (Risken, Savage, Haake and Walls [ 19871) but with an essential difference: it was shown by Savage and Cheng [ 19891that conventional optical bistable systems are a mixture of two quantum states localized at the minima of the effective potential. By contrast, the present situation is characterized by a coherent superposition of two such states. The seven maxima in fig. 24c are clear evidence of this fact: in the number state representation the Q function is expressed as

v, 8 71

333


where a = I a I exp ( - i+). Because the sum is truncated at Nu, the phase factors result in positive and negative interference that produce the corona with local maxima and minima in Q(a). A similar corona was found by Milburn and Holmes [ 19861 in the Q function of a lossless anharmonic oscillator. We can understand the main features of this corona by studying the simpler superposition

I$>

=

co 10) + ch4 IM)

(7.23)

9

where 10) is the vacuum state and I M ) is a number state. In this case the Q function becomes

+ 2 COS(M+ + 5 ) l a l M lcoc,l)

Jiz

,

(7.24)

where 5 is the relative phase between co and cM.This function has M zeros for a = I a0l exp(i$o), with laO[and $o given by (7.25) and

&;

i.e. the corona of the Q function In the limit of large Nu one finds I a. I N occurs at a radius a N To relate this result to the micromaser situation, we note that in a cotangent state limited by a 311 trapping state INu), the high-n peak of the photon statistics occurs for a number state I M ) slightly above n = 4n2/g2z2 - 1 (Slosser, Meystre and Braunstein [ 19891, Slosser and Meystre [ 19901). From the definition of Nu this implies that ( N - M ) / M becomes roughly constant for N large. Using this value of M in eq. (7.23), we conclude that the number of peaks on the corona of the Q function scales as N and is located on a circle of diameter so that the angular separation between peaks scales as 1/fi. In the limit N -, 00 the oscillations on the corona average to zero over any finite scale of the a space, and the Q function becomes indistinguishable from the Q function of a mixture. In this limit the Fock space truncation becomes ineffective in producing the “phase quantization” responsible for the corona oscillations of the Q function.

&.

fi,

334


[v,5 7

7.3. DETECTION: NON-LINEAR ATOMIC HOMODYNING

Although several formidable experimental challenges will need to be met to generate macroscopic superpositions in the micromaser, none seems to be fundamental, and the major difficulty might well lie in their detection. This is largely due to the lack of appropriate photocounters in the microwave regime and, hence, to the impossibility of directly measuring the micromaser field. As we have seen, micromaser measurements provide only indirect information about the field, which is inferred from the state of the atoms as they leave the cavity. A problem with such a detection scheme is that the state of the atoms depends only on a highly restricted set of correlation functions of the field, and specifically only on sums of terms of the forms (ata)", (ata).at, and (uta)"a, where n is an integer. Physically, this means that atoms only probe coherences between neighboring Fock states of the field. To detect macroscopic superpositions, however, one needs a measurement scheme sensitive to coherences between vastly different Fock states, i.e. an atomic response sensitive to more general products of field creation and annihilation operators of the generic form (,+)"a". Wilkens and Meystre [ 19911 proposed a new detection scheme that does just that. Their scheme is essentially a non-linear version of a single-atom homodyne detector (Yuen and Shapiro [ 19781, Yurke [ 19851, Mandel [ 19821, Schumaker [ 19841, Yuen and Chan [ 19831, Yuen [ 19821, Braunstein [ 19901). To measure the macroscopic superpositions that have been generated (the "signal"), a second mode is excited (the "local oscillator"). To analyze this scheme further and to avoid conflicts in notation, we call the signal mode a, and the local oscillator a, in the remainder of this section. The excitation of the local oscillator brings the cavity field to the state pf = p,, (8p,,. A test atom is then injected into the cavity, where it interacts with both modes. Its state is measured by state-selective field ionization after it leaves the cavity. The interaction between the test atom and cavity field is given by the two-mode Jaynes-Cummings interaction Hamiltonian

%m

=

hg(a, + a&+

+ adj.,

(7.27)

where we assume for simplicity that the atoms are in resonance with both field modes, with the same coupling strength g. Here, a,, a; and a,, a; are the annihilation and creation operators of the signal and oscillator modes, respectively. This problem is exactly solvable (Quattropani [ 1966]), for instance, by

v, 5 71


335

introducing the composite-mode boson operators A and At, with A

=

1

fi

~

(a,

+ a2),

(7.28)

and the photon number operator K = AtA. This permits the evaluation of the probability that an atom leaves the cavity in the upper state as

where the angular brackets denote the expectation value ( X ) = Tr(p,, p,,X), and pa, o, pb, are the upper and lower state populations of the incident test atom. Assuming that the local oscillator is prepared in a coherent state I/l) sufficiently strong so that K + 1 N K = I , this form reduces to (7.30) where ~ ( pis) the Wigner characteristic function of the signal mode (Louise11 [ 19901) (7.31) p = igz/l/J7 = g z ei(+h+~ 1 2 ,)

(7.32)

and we have introduced the phase $h of the local oscillator. The significance of this result resides in the well-known fact that this characteristic function contains all the possible information about the signal mode. As seen from eq. (7.30), ~ ( pcan ) be fully determined by varying the interaction time zbetween the test atom and the resonator or, more conveniently, the phase of the local oscillator. Hence, the state of the a,th mode of the intracavity can be clearly identified. A number of problems are obviously associated with this scheme, the most important being that the local oscillator mode should not interact with the atoms used to prepare the macroscopic superposition to be detected. This can be achieved by using different atoms in the preparation and measurement stages, so that the “preparation atoms” have selection rules such that they are coupled to the a , th mode only, whereas the “measurement atoms” are coupled to both modes. Another important point is that this scheme requires preparing both the signal and the local oscillator before each measurement and performing ensemble averages in the conventional quantum mechanical fashion. It will be interesting to evaluate the back action of the measurement on the signal mode,

336

CAVlTY QUANTUM OPTlCS AND THE QUANTUM MEASUREMENT PROCESS

[v, $ 8

and to see to what extent repeated measurements can be performed without destroying the quantum coherences of the macroscopic superposition.

4 8. Separated Fields 8.1. MICROMASER TESTS OF QUANTUM MECHANICAL COMPLEMENTARITY

So far, this review has concentrated on the behavior of atoms and fields confined in a single high-Q cavity. Adding a second cavity permits us to address some further questions in quantum measurement theory, as illustrated by the experiment proposed by Scully and Walther [ 19891 (see also Scully, Englert and Walther [ 1 9 9 1 1 ) to test quantum complementarity. The classical example of complementarity, or wave-particle duality, is provided by Young’s double-slit experiment, where it is impossible to tell which slit the light went through and still observe interference. In cavity QED a corresponding effect can be obtained by using a variation on a standard quantum beat experiment (see, e.g., Chow, Scully and Stoner [ 1 9 7 5 1 , Meystre and Sargent [ 19901). Quantum beats can be observed when atoms excited in a coherent superposition of two upper states I a ) and I b ) are allowed to decay to a common lower level Ic). The spontaneously emitted radiation then exhibits beats due to the quantum interference between the two decay channels l a ) -+ Ic) and I b ) Ic). Specifically, if the atom-field system is initially in the state -+

IJ/(O))

=

la) + b(Q) l b ) l @ IIO}>

9

(8.1)

where I { 0) ) stands for the vacuum of the field modes, the state of the system at time t is given by

IJ/(O)

=[a(t)la>

+ b ( ~ ) I ~ ) l @ l { 0 ) +Ic>@~c1(01$1> ) +c2(t)1$2)lr

where the “photon states”

are given by

(8.2)

(8.3) and (8.4)

v, i3 81

331

SEPARATED FIELDS

Here, I l k ) labels the state with one photon in mode k and all other modes in the vacuum, g,, and gbk are the dipole coupling constants between the field mode with wave vector k and the I a ) - I c ) and 1 b ) -1 c ) transitions, respectively ya and yb are the spontaneous decay rates of levels l a ) and Ib), So, = w, - kc and S,, = w, - kc. The first line in eq. (8.2) does not contribute to the photocurrent detected at a distance r from the atom, which is given by

Y(t)cc@(t - r / c ) [ I ~ , l ~ e - ~ ~ ( ‘ - ~ ’ ~ ) +

C,be-(~+iwuh)(‘-r/c)

+a-b].

(8.5)

The explicit form of the coefficients c,, cab, and cb is given, e.g., by Meystre and Sargent [ 19901, y = (yo + yb)/2 and wab = w, - 0,. Clearly, the beat part of the signal originates from the interference between the “photons” I $I, ) and ($2).

The experiment proposed by Scully and Walther [ 19891 to demonstrate complementarity in cavity QED extends these considerations to a set-up consisting of two micromaser cavities, as shown in fig. 25. In this case the atomic level structure is somewhat more complicated than in conventional quantum beats. The atom is initially prepared in a coherent superposition of two upper states I a ) and I b ) . The I a ) -1 a‘ ) transition is resonant with the frequency of the first cavity, while the I b ) - I b‘ ) transition is resonant with the second one. We have already seen how the atom can be made to “see” the micromaser field as a n pulse by properly choosing its transit time through a cavity. Under these conditions, and on passing the first cavity, the atom is “flipped” from I a ) a- P

b- P

Micromaser 1

S

a’

b’

Micromaser 2

Fig. 25. A proposed configuration to perform an experiment to determine which decay path has been followed in a micromaser. In passing through the first cavity the atom undergoes the transition I a ) + I a’ ) ,and in the second cavity it makes the transition I b ) + I b’ ). (After Scully and Walther [1989].)

338


[v, 0 8

to I a’ ), whereas on passing the second cavity it is flipped from 16) to I b‘ ). Afterward, both levels I a‘ ) and I b’ ) decay to the common lower level I c ) . These flips with the concomitant emission of a photon into the cavity, potentially leave information about the decay channel followed by the atom. The micromaser cavities can serve as “which path” detectors, however, only if the extra quantum of energy left by the atom changes the cavity field in a detectable manner. Thus, whether information is available about which path has been followed or not depends on the field states initially prepared in the cavities. A straigthforward extension of the quantum beat calculation shows that in the present situation the cross term in eq. (8.5) is replaced by (Scully and Walther [ 19891)

where I Q i ) is the state of the field in the ith micromaser cavity, and superscripts 0 and f label the initial and final states, respectively. We see, then, that the presence or absence of a beat signal depends in an essential way on the state in which the micromaser cavities are prepared. In particular, quantum beats will be present if the micromasers are initially in coherent states I a i ) ,but not if they are initially prepared in number states I n i ) . This is because in the case of number states the scalar product (@{, CP: I @, @) becomes (n,

+ 1,n21n1,n2+ 1 )

=

0.

(8.7)

In contrast, coherent states are not orthogonal and, hence, the scalar product is non-zero.

8.2. QUANTUM SUPERPOSITION OF MACROSCOPICALLY SEPARATED CAVITY FIELDS

A variation on the separated-fields geometry can also be used to prepare correlated states of macroscopically separated quantum systems. Consider a Gedanken experiment where a two-level atom initially in its excited state I a ) is sent through two macroscopically separated single-mode micromaser cavities. Meystre [ 19901 showed that if the atom leaves the second cavity in its ground state Ib), the field is left in the superposition

where the state I 1 , O ) describes one photon in the first cavity and none in the

v, 8 81

339

SEPARATED FIELDS

second, whereas 10, 1 ) stands for no photon in the fist cavity and one in the second, and I p I + I q I = 1. To clarify, we generalize the Jaynes-Cummings Hamiltonian (2.1) to

x2c= ;ttoa, + ttraa;u, + AG?at,a, + t t [ g , ( t ) a ~ a +- adj.] + A[g,(t)a~a_+ adj.] , where at and aiare the creation and annihilation operators for the ith cavity mode and gi are the corresponding coupling constants. In contrast to the Jaynes-Cummings Hamiltonian, X2,includes an explicit time dependence of the coupling constants to account for the (classical) motion of the atom through the cavities. Specifically, we assume gi(t) = 1 while the atom is in the ith cavity and zero otherwise, an approximation that will be discussed in more detail in § 9. Assuming that the atom is initially in its upper state, while both cavities are in the vacuum state, we have IICl(0)) = Ia,O,O)

(8.10)

*

At the time t , when the atom leaves the first cavity, the state of the composite system then becomes (8.11) I N , ) ) = c, la, 090) + c, 16, 130) where I C, 1 ' + I C, I = 1. The explicit form of the Ciis obtained by solving the 3

Jaynes-Cummings problem for the first cavity, but is irrelevant for the present discussion. Similarly, after the second cavity, and neglecting the effects of dissipation (spontaneous emission and collisions) during the transit time of the atom between the cavities, the state of the system is

),"!I

=D,la,O,O) + D , I b , 1 , 0 ) + D , I b , O , 1 ) ,

(8.12)

'

where ID, I + ID, I + I D31 = 1. The state of the atom can be measured after it leaves the second cavity, for instance, by the method of field ionization. If it is found in its ground state, the field density matrix after measurement is Pf=

=

4TratoI-n 16) ( b I W 2 ) )

(W211

4 [ D 2 I b, 190) + D3 I b, 091 ) 1 [Dz* (b, 190 I + D? (b,O, 1 I 1 (8.13) 9

where 4 is a normalization constant such that Tr, p , = 1 after the measurement. After normalization pf clearly describes a pure state of the form (8.8). Thus, this scheme of preparation using a selective measurement on the atom

340


[V,8 8

leaves the field in the pure coherent superposition of the states with either one photon in the first cavity and none in the second or no photons in the first cavity and one in the second. This superposition can be distinguished from a mixture of the type

Pf = lP12 I 130) ( 1, 01 +

Id210, 1)

(0, 1 I

(8.14)

by a simple population measurement on a test atom. Consider such a test atom, initially in its ground state 1 b), so that the atom-field state before a measurement is

I$@)>

=

lb) @[Pe'@l1,0) + q 10,1>1

(8.15)

where we have generalized the state (8.8) slightly by introducing the phase exp(i$). This permits handling both pure state and mixed state simultaneously, the mixed state being obtained by averaging the final result over the phase $. At the end of the interaction the system is in a coherent superposition of the three states 1 a, 0, 0), I b, 0, 1) ,and I by 1, 0), the first two of which are reached by way of two different channels. For instance, the state la, 0,O) can be reached by the atom absorbing a photon either in the fist or second cavity. Specifically, the final state of the atom-field system is found to be (Meystre [ 19901)

1 JI)

=

(d1ei@+ d2)I a, 0,O)

+ ( 4e'@+ @) I b, 0 , l ) +

16, 1,O) , (8.16)

where the explicit form of the various probability amplitudes is unimportant for the present discussion. Each time such a situation occurs, one can expect the appearance of an interference phenomenon, as already seen in the earlier discussion about quantum beats. Indeed, from this final state the probability for the atom to leave in the upper state is

+

pa = Idl e'@ d2I2,

(8.17)

with an interference term d1e'@@ that is averaged away if the field is in a mixture rather than in a coherent superposition.

v, I 91

OUTLOOK: MECHANICAL EFFECTS

34 1

8 9. Outlook: Mechanical Effects 9.1. LIGHT FORCES AND MECHANICAL MOTION

In this paper we have presented recent developments in cavity QED and demonstrated their impact on our fundamental understanding of the interactions between light and matter in particular and of quantum mechanics in general. Experimental progress is rapid, and many experiments that are proposed and reviewed here will soon be reality. The question, then, is where do we go from here? This section discusses an exciting new direction that combines cavity quantum optics and “atom optics”, and shows considerable promise as a future theoretical and experimental testing ground. Together with cavity QED the manipulation of atomic trajectories by electromagnetic fields is one of the most exciting recent developments in quantum optics and laser spectroscopy. Here, one exploits the fact that every time an atom exchanges energy with the field, the momentum of the absorbed or emitted light must be compensated for by a mechanical motion of the atom. This leads to atomic trapping and cooling, state-selective atomic reflection and diffraction by optical fields, atom interferometry, and other areas of research. In this situations it is usually sufficient to describe the fields classically, whereas spontaneous emission is treated as a stochastic process. In cavity QED, in contrast, the mode structure and quantum nature of the field are essential. A recent emerging effort by several groups attempts to unite these two areas of research (Meystre, Schumacher and Stenholm [ 19891, Englert, Schwinger, Barut and Scully [ 19911, Haroche, Brune and Raimond [ 19911). Questions of particular interest are related to the effects of the internal state of the field (photon statistics) and of the atoms on the mechanical motion, and also to the confinement of an atom inside a high-Q cavity, possibly in a state close to the vacuum state. Before discussing this recent work, we first present a simple description of the effects of light on atomic motion. To proceed, we need to generalize the Jaynes-Cummings Hamiltonian (2.1) to include the atomic kinetic energy due to its center-of-mass motion as well as the spatial dependence of the field. In the semiclassical limit this gives

P2 Xm,= -+ +hamz- [dE,(R, ~ ) a ++ adj.] , 2M

(9.1)

whereM is the mass of the atom and of the center-of-mass momentum operator P,E, is the component of the electric field along the atomic dipole moment d, and R describes the center-of-mass position operator of the atom, with [Pi,Rj] = -ih6,.

342


[V,$ 9

In the Heisenberg picture the center-of-mass motion is governed by the equation of motion dP- i [S,,PI , = d[VE,(R)] [a, dt h

+ adj.] .

(9.3)

Taking the expectation value of this expression over the internal atomic degrees of freedom yields

where the approximate equality stems from the fact that we have factorized VE,(R) outside the expectation value, which is the so-called “independentmotion approximation” (Kazantsev, Surdutovich and Yakovlev [ 19901). For concreteness, consider a travelling wave field of the form

E,(R)

=

~ E , ( x )e’(KZ- ” I )

+ C.C.

,

(9.5)

where E,(X) will later be the cavity spatial mode function. Substituting this form into eq. (9.4) yields, in the rotating-wave approximation,

+ + K ~ $ E , ( x )(io+ei(KZ-Rr)+ adj.) .

(9.6)

Expressing the expectation value over the integral degrees of freedom in terms of the atomic polarization 9 ( Z )finally yields dP = 42

dt

~

dEo(X) Re [ P ( Z ) ] dX

Fg:grad

+ Fscat

.

+ i K i E , ( X ) Im [ S ( Z ) ] (9.7)

We thus find two contributions to the mechanical force exerted on the atom by the electromagnetic field. The first, Fmad, is proportional to the real part of the polarization and depends on the field gradient, which is why it is often called the gradient force. The second force, Fscat,is proportional to the imaginary part of the polarization and is called the scattering or light pressure force. We shall see how a quantized version of the gradient force can be used either to trap atoms inside a micromaser cavity (Haroche, Brune and Raimond [ 19911) or to reflect them off that cavity (Englert, Schwinger, Barut and Scully [ 19911). Remarkably, both effects can occur even when the micromaser field is in a

v, 4 91


343

vacuum. First, however, we discuss in greater detail the scattering force and show how it leads to atomic diffraction in a way that depends in a sensitive way to the photon statistics of the field mode (Meystre, Schumacher and Stenholm [ 19891). 9.2. ATOMIC BEAM DEFLECTION IN A QUANTUM FIELD

To proceed, we generalize the Hamiltonian (9.1) to describe the electromagnetic field as a quantized mode and, for simplicity, neglect the effects of the gradient force. We also treat the motion of the atom in the direction f transverse to the field classically; i.e. we ignore the velocity changes induced by the field in the f direction and describe the evolution of the atom-field system in a reference frame moving at the constant velocity PJM. Furthermore, we neglect the effects of spontaneous emission, an approximation that is well justified in a micromaser-type of situation but not necessarily in the optical regime. The atom-field Hamiltonian is then p;

Jfme,q

=-

2M

+ Ablata t $boa, + Ag(a+a + adj.)cosKZ,

(9.8)

and the Hilbert space of the system is the direct sum of the Hilbert spaces for the center-of-mass motion of the atom, its internal degrees of freedom as well as the field mode. Hence, the general state vector of the system is

where, for simplicity, we have dropped the index on the momentum variable and the 1 P) are the eigenstates of P,. The analysis is considerably simplified by noting that an unusual property of the Jaynes-Cummings model -the state I a, n ) is only coupled to I b, n + 1) - still holds here. Hence, it is sufficient to solve the problem within one such manifold and to sum over the contributions of all manifolds at the end of the calculation. Within each manifold the quantized-field problem is mathematically equivalent to the corresponding classical problem (Cook and Bernhardt [ 19781, Delone, Grinchuk, Kuzmichev, Nagaeva, Kazantsev and Surdutovich [ 19801, Bernhardt and Shore [ 19811, Arimondo, Bambini and Stenholm [1981], Tanguy, Reynaud and Cohen-Tannoudji [ 19841, Kazantsev, Ryabenko, Surdutovich and Yakovlev [ 19851). The quantum nature of the field appears only in the change in strength of the dipole coupling between the field and the atom from one manifold to the next.

344


Expanding the corresponding state vector I $,) tation as

[v, 8 9

in the momentum represen-

and noting that COS(KZ)U,(P,t)

=

i[a,(P

+ Ak, t) + a,(P - hk, t ) ]

(9.11)

yields the equation of motion ih

dt

=

p'

2M

an(P,t)

+ 4Ag-

[b,(P

+ h k ) + b,(P

-

hk)] , (9.12)

and db,(P, t ) P2 ijj--b,(P, t) dt 2M

+ $Ag-

[a,(P

+ Ak) + a , ( P - h k ) ] . (9.13)

For atoms initially in the ground state and with initial transverse momentum Po 3 I$AP, 0))

=

b,(Po, 0) I b, n + 1 )

9

(9.14)

it is convenient to express a,(P, t) and b,(P, t) as the series

(9.15)

This equation can be solved numerically in general and analytically if the kinetic energy term can be neglected (the Raman-Nath or Kapitza-Dirac regime). As an example, fig. 26 shows the probability 9 ( P , t ) for an atom to leave the interaction region with transverse momentum P in the cases where the field is initially in a coherent or thermal state. The three curves on each figure are for three values of the interaction time, all results being in the Raman-Nath regime where the kinetic energy gained by the atoms can be ignored. In this regime a straightforward generalization of the results of Bernhardt and Shore [ 19811 and Arimondo, Bambini and Stenholm [ 19811

345


0.15

0.10

0.05

0.00

-50

0

50

P

i

0.15 -

0

f

0.10 -

0.05

-

0.00

-50

0

50

P Fig. 26. (a) Momentum distribution 9 ( p , f) for an atom to leave the field with momentum p , for a field initially in a coherent state with mean photon number ( n ) = 9, and for three different interaction times. (b) Same as (a) but for a field initially in a thermal state with ( n ) = 9. (ARer Meystre, Schumacher and Stenholm [ 19891.)

346


[v,$ 9

shows that the variance in transverse momentum distribution is given by (Pz)

=

i h 2 k Z( n ) g2tZ,

(9.17)

where ( n ) is the mean photon number in the field mode. Unlike 9 ( P , t), (P z ) does not depend on the photon statistics of the field. So far we have discussed the scattering of an atom by a standing wave. It is worth noting that in the quantum regime there is an essential difference between this process and the scattering of the same atom off a wave consisting of two travelling waves of equal amplitudes and frequencies and with opposite directions of propagation. This difference, which is closely related to the discussion of complementarity in 5 8.1, is that with travelling waves we can know in principle which of the two waves has exchanged a unit of momentum with the atom, e.g., by monitoring the number of quanta in the two counterpropagating waves separately. In contrast, a standing wave is an inseparable quantum unit, the average momentum of which remains zero at all times. This unity is imposed by the fixed mirrors that establish the standing wave (Shore, Meystre and Stenholm [1991]) and that act as infinite sinks or sources of momentum. As such, the scattering of atoms off a standing wave is akin to the light scattering off an infinitely massive grating, whereas the scattering off running waves is like the scattering of light off a grating so light that one can infer the order of scattering from the grating motion. Shore, Meystre and Stenholm [ 19911 show that in the case of running waves the atomic motion is still governed by eqs. (9.12) and (9.13), with the essential difference that the for one of the coupling constant ( h g / 2 ) m is replaced by (hg/2)@ coupling terms and ( h g / 2 ) f i for the other, where - n < v < n. This means that the equations are now truncated after v = f n, which illustrates particularly dramatically the basic difference between standing and travelling waves: for travelling waves, momentum conservation allows one to determine uniquely the number of photons in the two waves, yielding “which way” information about the scattering process. In contrast, atomic photon exchange in a standing wave is always accompanied by symmetrical momentum transfer, yielding “both ways” information. There is no limit to how much momentum can be transferred to the atoms by a standing wave, whereas a travelling wave in a number state In) can give away at most a momentum of n h k momentum before being depleted. Shore, Meystre and Stenholm [1991] show how this difference, which is an atomic scattering illustration of the principle of complementarity, alters the scattering of atoms by light fields.

v, I 91

347


9.3. ATOMIC REFLECTION AT A MICROMASER CAVITY

We now turn to situations where a quantized version of the gradient force, which we call the “correlation force”, is used either to reflect atoms away from a micromaser cavity (Englert, Schwinger, Barut and Scully [ 19911) or to attract them towards the cavity center (Haroche, Brune and Raimond [ 19911). A remarkable aspect of this force is that it acts even when the cavity mode is in a vacuum, thus allowing one to envision “vacuum force atomic traps”. Instead of treating the momentum P, as a constant, we now handle it as a dynamical variable, but for simplification we omit the scattering effects associated with photon emission and absorption, discussed in 0 9.2. In this case the Hamiltonian (9.8) becomes p,’ + hRata + i h o a z + h g U ( X ) (a, a + adj.) . .go, =

(9.18) 2M Consider, first, the simple mode function U(X)= 1 for 0 < X < L and 0 elsewhere, where L is the dimension of the cavity along the atomic direction of propagation f.This situation is similar to the study of transmission through and reflection at square-well potentials, with the important differences that (a) the height of the well is now an operator proportional to the strength of the quantized micromaser field (Englert, Schwinger, Barut and Scully [ 1991 I), and (b) the particle is a spinor. As previously, it is sufficient to solve the problem within one manifold { la, n), I b, n + l ) } of the field and atomic electronic states. The present discussion is limited to the resonant case R = w. Expanding the corresponding state vector I $), in terms of the dressed states (2.8) and (2.9) and working in the coordinate representation, we then have ~

$ A x , t) = (XI =

c1, n

$A))

( X , t) I 1, n ) + c2,

t) I 2, n )

(9.19)

Y

or, by transforming a straightforward phase $n(x, t) =

cJ X ,

- in(a+a+ ~ 2 2 / 2 )

t ) I 1, n ) + c2,JX3 l ) 12, n)

.

(9.20)

The equations of motion for the probability amplitudes Cl,,and C2, are

and

348


[v, $ 9

At resonance these equations are uncoupled, since the dressed states that diagonalize the conventional Jaynes-Cummings Hamiltonian are not coupled by the kinetic energy term in eq. (9.18). They describe the motion of particles in potential barriers of heights V l , n= h g U ( X ) m and ~ 2n .= - h g U ( X n +) 1. r Following the standard treatment of this problem (see, e.g., Cohen-Tannoudji, Diu and Laloi! [ 1977]), we find the transmission and reflection coefficients

and ~ v n(k) , =

i(k?, n - k2)

where

sin (kv, nL) 2kkv, cos(k,, .L) - i(k2 + k?, ,) sin(kv,,L) ' (9.24)

(9.25)

v = 1, 2 and k

=

Mvo/h,

(9.26)

vo being the velocity of the incident atoms. For V , < 0 we have k:, > k2, and the dressed atom encounters an attractive potential. In contrast, for V v , > 0 the atoms encounter either a repulsive potential, for k2 > k:, > 0, or a potential barrier, for k2 > 0 > k:, n , a condition which may be re-expressed in terms of oo as

v;

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