Progress in Optics, Volume 28

PROGRESS IN OPTICS V O L U M E XXVIII EDITORIAL ADVISORY BOARD G. S . AGARWAL, Hyderabad, India C. COHEN-TANNOUDJI...

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PROGRESS IN OPTICS V O L U M E XXVIII

EDITORIAL ADVISORY BOARD

G. S . AGARWAL,

Hyderabad, India

C. COHEN-TANNOUDJI,Paris, France F. GORI,

Rome, Italy

A. KUJAWSKI,

Warsaw, Poland

A. LOHMANN,

Erlangen. F.R.G.

M. SCHUBERT,

Jena, G.D.R.

J . TSUJIUCHI,

Chiba, Japan

H. WALTHER,

Garching, F.R .G .

W. T. WELFORD,

London, England

B. ZEL’DOVICH,

R. Chelyabinsk, U.S.S.

P R O G R E S S IN OPTICS VOLUME XXVIII

EDITED BY

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

Contributors G. BJORK, 0. BRYNGDAHL, R.J. COOK, N. IMOTO, M. KITAGAWA, S. MACHIDA, M.G. RAYMER, S. SAITO, J. SCHWIDER, LA. WALMSLEY, F. WYROWSKI, Y. YAMAMOTO, T. YANAGAWA

1990

NORTH-HOLLAND AMSTERDAM .OXFORD. NEW YORK.TOKY0

@

ELSEVIER SCIENCE PUBLISHERS B.V.,

1990

AN rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted. in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without the prior permission of the Publisher, Ehevier Science Publishers B. V., P.0.Box 211. 1000 AE Amsterdam, The Netherlands. Special regulationsfor readers in the U.S.A. :This publication has been regirrered with the Copyright Clearance Center Inc. (CCC). Salem, Massachusetts. Information can be obtained from the CCC about conditions under which photocopies of parts of this publication may be made in the U.S.A. AN other copyright questions. including photocopying outside of the U.S.A.. should be referred to the Publisher. unless otherwise specijied. No responsibility is assumed by the Publisher for any injury andlor damage to persons or property as a matter of products liability, negligence or otherwise. or from any use or operation of any methods. products. instructions or ideas contained in the material herein. LIBRARY OF CONGRESS CATALOG CARD NUMBER: 61-19297 ISBN: 0444884394

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PRINTED IN THE NETHERLANDS

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I I1.

CONTENTS OF VOLUME 1(1961) THEMODERNDEVELOPMENT OF HAMILTONIAN OPTICS.R . J . PEGIS .

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1-29

WAVE OPTICS AND GEOMETRICAL OPTICS IN OPTICALDESIGN. K. 3 1-66 MIYAMOTO ............................ I11. THEINTENSITY DISTRIBUTION AND TOTAL ILLUMINATION OF ABERRATIONFREEDIFFRACTION IMAGES.R. BARAKAT. . . . . . . . . . . . . . 67-108 I v. LIGHTAND INFORMATION. D . GABOR. . . . . . . . . . . . . . . . 109- 153 v . ON BASICANALOGIESAND PRINCIPAL DIFFERENCES BETWEEN OPTICAL AND ELECTRONIC INFORMATION. H . WOLTER. . . . . . . . . . . . . 155-210 VI . INTERFERENCE COLOR.H . KUBOTA. . . . . . . . . . . . . . . . . 211-251 CHARACTERISTICS OF VISUAL PROCESSES. A. FIORENTINI . . . 253-288 VII. DYNAMIC VIII . MODERN ALIGNMENT DEVICES.A. C. S. VAN HEEL . . . . . . . . . . 289-329

C O N T E N T S O F V O L U M E I1 ( 1 9 6 3 ) I. I1. I11. IV . V. VI .

RULING. TESTINGAND USE OF OPTICAL GRATINGS FOR HIGH-RESOLUTION 1-72 SPECTROSCOPY. G . W. STROKE. . . . . . . . . . . . . . . . . . . THE METROLOGICAL APPLICATIONS OF DIFFRACTION GRATINGS.J.M. 73-108 BURCH . . . . . . . . . . . . . . . . . . . . . . . . . . . . . DIFFUSION THROUGH NON-UNIFORM MEDIA.R. G . GIOVANELLI . . . . 109-129 CORRECTION OF OPTICAL IMAGES BY COMPENSATION OF ABERRATIONS AND BY SPATIAL FREQUENCY FILTERING. J TSUJIUCHI . . . . . . . . 131- 180 FLUCTUATIONS OF LIGHTBEAMS.L MANDEL . . . . . . . . . . . . 18 1-248 METHODSFOR DETERMINING OPTICAL PARAMETERS OF THINFILMS.F . 249-288 ABELBS . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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C O N T E N T S O F V O L U M E I11 ( 1 9 6 4 ) THEELEMENTS OF RADIATIVE TRANSFER. F. KOITLER . . . . . . . . I. P. JACQUINOT. B. ROIZEN-DOSSIER. . . . . . . . . . I1. APODISATION. TREATMENT OF PARTIAL COHERENCE H . GAMO . . . . . . . I11. MATRIX

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C O N T E N T S O F VOLUME I V (1965) HIGHERORDERABERRATION THEORY. J . FOCKE . . . . . . . . . . . I. OF SHEARING INTERFEROMETRY. 0. BRYNGDAHL. . . . I1. APPLICATIONS DETERIORATION OF OPTICAL GLASSES.K. KINOSITA. . . . . 111. SURFACE IV. OPTICAL CONSTANTS OF THINFILMS.P . ROUARD.P. BOUSQUET. . . . V . THEMIYAMOTO-WOLF DIFFRACTION WAVE.A. RUBINOWICZ ...... THEORYOF GRATINGS AND GRATING MOUNTINGS. W. T. VI . ABERRATION WELFORD . . . . . . . . . . . . . . . . . . . . . . . . . . . . VII. DIFFRACTION AT A BLACKSCREEN. PART I: KIRCHHOFF’S THEORY.F . KOITLER . . . . . . . . . . . . . . . . . . . . . . . . . . . . C O N T E N T S OF VOLUME V (1966) OPTICALPUMPING. C. COHEN.TANNOUDJI. A. KASTLER. . . . . . . . I. OPTICS.P. S. PERSHAN . . . . . . . . . . . . . . . . I1. NON-LINEAR INTERFEROMETRY. W. H . STEEL . . . . . . . . . . . . . I11. TWO-BEAM V

1-28 29- 186 187-332

1-36 37-83 85-143 145-197 199-240 24 1-280 28 1-3 14

1-81 83-144 145-197

VI

IV .

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INSTRUMENTS FOR THE MEASURING O F OPTICAL TRANSFER FUNCTIONS. K

MURATA. . . . . . . . . . . . . . . . . . . . . . . . . . . . . FROM FILMS OF CONTINUOUSLY VARYING REFRACTIVE V. LIGHTREFLECTION INDEX.R . JACOBSSON ....................... DETERMINATION AS A BRANCHOF PHYSICAL VI . X-RAYCRYSTAL-STRUCTURE OPTICS.H . LIPSON.C.A . TAYLOR . . . . . . . . . . . . . . . . . . J. PICHT . . . . . . . VII . THEWAVEO F A MOVINGCLASSICAL ELECTRON.

C O N T E N T S O F V O L U M E VI ( 1 9 6 7 ) RECENTADVANCESI N HOLOGRAPHY. E. N . LEITH.J . UPATNIEKS. . . . I. OF LIGHTBY ROUGHSURFACES. P. BECKMANN. . . . . . I1. SCATTERING O F THE SECOND ORDER DEGREEO F COHERENCE. M . I11. MEASUREMENT FRANCON. S. MALLICK . . . . . . . . . . . . . . . . . . . . . . IV . DESIGNOF ZOOM LENSES.K . YAMAJI . . . . . . . . . . . . . . . . . OF LASERS T O INTERFEROMETRY. D . R . HERRIOTT V . SOMEAPPLICATIONS STUDIES O F INTENSITY FLUCTUATIONS IN LASERS.J . A . VI . EXPERIMENTAL ARMSTRONG.A .W . SMITH. . . . . . . . . . . . . . . . . . . . . VII . FOURIER SPECTROSCOPY. G. A . VANASSE.H . SAKAI. . . . . . . . . .

199-245 247-286 287-350 351-370

1-52 53-69 71-104 105-170 171-209 211-257 259-330

VIII . DIFFRACTION AT A BLACKSCREEN. PART 11: ELECTROMAGNETIC THEORY. F. KOTTLER . . . . . . . . . . . . . . . . . . . . . . . . . . . 331-377

C O N T E N T S O F VOLUME VII (1969) I.

MULTIPLE-BEAMINTERFERENCE A N D NATURAL MODES IN OPEN 1-66 RESONATORS. G. KOPPELMAN. . . . . . . . . . . . . . . . . . . I1. METHODS O F SYNTHESIS FOR DIELECTRICMULTILAYERFILTERS.E. DELANO.R.J . PEGIS . . . . . . . . . . . . . . . . . . . . . . . 67-137 FREQUENCIES. I. D. ABELLA . . . . . . . . . . 139-168 I11. ECHOESAND OPTICAL WITH PARTIALLY COHERENT LIGHT.B . J . THOMPSON 169-230 IV. IMAGEFORMATION V. QUASI-CLASSICAL THEORY OF LASERRADIATION. A . L. MIKAELIAN. M . L. TER-MIKAELIAN. . . . . . . . . . . . . . . . . . . . . . . . . 231-297 VI . THEPHOTOGRAPHIC IMAGE. s. O O U E . . . . . . . . . . . . . . . . 299-358 J.H. OF VERY INTENSE LIGHT WITH FREEELECTRONS. VII . INTERACTION 359-415 EBERLY. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

C O N T E N T S O F VOLUME VIII (1970) . . . . . . . . . . THEOPTICAL PERFORMANCE O F THE HUMANEYE.G . A . FRY . . . . . LIGHTBEATING SPECTROSCOPY. H. Z . CUMMINS. H . L. SWINNEY. . . . MULTILAYER ANTIREFLECTION COATINGS. A. MUSSET.A. THELEN. . . V . STATISTICAL PROPERTIES OF LASERLIGHT. H. RISKEN . . . . . . . . OF SOURCE-SIZE COMPENSATION IN INTERFERENCE VI . COHERENCE THEORY MICROSCOPY. T. YAMAMOTO. . . . . . . . . . . . . . . . . . . . H . LEVI . . . . . . . . . . . . . . . . VII . VISION IN COMMUNICATION. OF PHOTOELECTRON COUNTING . c. L. MEHTA . . . . . . . . VIII . THEORY I. I1. I11. IV .

SYNTHETIC-APERTURE OPTICS.J . W. GOODMAN.

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

C O N T E N T S O F VOLUME I X (1971) I.

GAS LASERSAND THEIR APPLICATIONTO MENTS. A . L. BLOOM . . . . . . . . . .

PRECISE

LENGTHMEASURE-

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1-30

VI1

PICOSECOND LASERPULSES,A. J. DEMARIA. . . . . . . . . . . . . OPTICALPROPAGATION THROUGH THE TURBULENT ATMOSPHERE, J. W. STROHBEHN. . . . , . . . . . . . . . . . . . . . . . . . . . . IV. SYNTHESIS OF OPTICAL BIREFRINGENT NETWORKS, E. 0.AMMANN. . . V. MODELOCKINGIN GAS LASERS,L. ALLEN,D. G. C. JONES . . . . . . VI. CRYSTALOPTICS WITH SPATIALDISPERSION, v. M. AGRANOVICH, V. L. GINZBURG. . . . . . . . . . . .. . . . . . . . . . . VII. APPLICATIONS OF OPTICAL METHODSIN THE DIFFRACTION THEORYOF ELASTIC WAVES,K. GNIADEK, J. PETYKIEWICZ . . . .. . . . . . . . VIII. EVALUATION, DESIGNAND EXTRAPOLATION METHODSFOR OPTICAL SIGNALS, BASEDON USE OF THE PROLATE FUNCTIONS, B. R. FRIEDEN . 11. 111.

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CONTENTS O F VOLUME X (1972) OF OPTICAL IMAGES, T. S. HUANG. . . . . BANDWIDTH COMPRESSION THEUSE OF IMAGE TUBESAS SHUITERS, R.w.SMITH . . . . . . . .

I. 11. QUANTUM OPTICS,M. 0. SCULLY, K. G. WHITNEY 111. TOOLSOF THEORETICAL c.G. WYNNE . . IV. FIELDCORRECTORS FOR ASTRONOMICALTELESCOPES, V. OPTICAL ABSORPTIONSTRENGTH OF DEFECTSIN INSULATORS, D. Y. . . . . . . .. . .. SMITH,D. L. DEXTER . . . . . . . . . LIGHTMODULATION AND DEFLECTION, E. K. SIITIG . . . VI. ELASTOOPTIC VII. QUANTUM DETECTION THEORY,C. W. HELSTROM . . . . . , . . . .

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CONTENTS O F VOLUME XI (1973) MASTEREQUATION METHODSIN QUANTUM OPTICS,G. S. ACARWAL. .

RECENTDEVELOPMENTS IN FARINFRARED SPECTROSCOPIC TECHNIQUES, H. YOSHINAGA . . . . . . . . , . . , . . . . . . . . . . . . 111. INTERACTION OF LIGHTAND ACOUSTICSURFACE WAVES,E. G. LEAN . WAVES IN OPTICAL IMAGING, 0.BRYNGDAHL. . . . , IV. EVANESCENT V. PRODUCTION OF ELECTRON PROBESUSINGA FIELDEMISSIONSOURCE, . . . . A.V. CREWE . . . . . , . . . . . . . . . . . . . . . VI. HAMILTONIAN THEORYOF BEAMMODEPROPAGATION, J. A. ARNAUD . VII. GRADIENT INDEXLENSES,E. W. MARCHAND . . . . . . .. .. . , .

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I

31-71 73-122 123-177 179-234 235-280 281-310 3 11-407

1-44 45-87 89- 135 137-164 165-228 229-288 289-369

1-76 77-122 123- 166 167-22 1 223-246 247-304 305-337

CONTENTS OF VOLUME XI1 (1974) I. 11. 111. IV. V.

VI.

SELF-FOCUSING, SELF-TRAPPING, AND SELF-PHASE MODULATION OF LASERBEAMS,0. SVELTO. . . . . . . . . . . . . . . . . . . . SELF-INDUCED TRANSPARENCY, R. E. SLUSHER ... . . . .. . MODULATION TECHNIQUES IN SPECTROMETRY, M. HARWIT,J. A. DECKER JR. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . INTERACTION OF LIGHT WITH MONOMOLECULAR DYE LAYERS,K.H. DREXHAGE . . . . . .. . . . . ... . . . .. . THE PHASE TRANSITION CONCEPT AND COHERENCE IN ATOMICEMISSION, R. GRAHAM . . . . . . . . . . . . . . . . . . . . . , . . BEAM-FOIL SPECTROSCOPY, S. BASHKIN. . . . . . . . . . . . . , .

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

C O N T E N T S O F V O L U M E XI11 ( 1 9 7 6 ) I.

ON THE VALIDITY OF KIRCHHOFF’S LAWOF HEATRADIATION FOR A BODY IN A NONEQUILIBRIUM ENVIRONMENT, H. P. BALTES . . . . . . . .

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1-25

VIII

RADIATIONTHEORY, L. THE CASEFOR AND AGAINSTSEMICLASSICAL MANDEL. . . . . . . . . . . . . . . . . . . . . . . . . . . . . AND SUBJECTIVE SPHERICAL ABERRATION MEASUREMENTS OF 111. OBJECTIVE THE HUMANEYE,W. M. ROSENBLUM, J. L. CHRISTENSEN. . . . . . . TESTINGOF SMOOTH SURFACES,G. SCHULZ,J. IV. INTERFEROMETRIC SCHWIDER. . . . . . . . . . . . . . , . . . . . . . . . . . . . SELF FOCUSING OF LASERBEAMSI N PLASMAS AND SEMICONDUCTORS, V. M. S. SODHA,A. K. GHATAK,V. K. TRIPATHI . . . . . . . . . . . . AND ISOPLANATISM, w. T. WELFORD . . . . .. . VI. APLANATISM 11.

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27-68 69-9 1 93-167 169-265 267-292

C O N T E N T S O F V O L U M E XIV ( 1 9 7 7 ) THE STATISTICS OF SPECKLE PATTERNS,J. c. DAINTY. . . . . . . . . I. 11. HIGH-RESOLUTION TECHNIQUES IN OPTICAL ASTRONOMY, A. LABEYRIE. 111. RELAXATION PHENOMENA IN RARE-EARTH LUMINESCENCE, L. A. RISEBERG, M. J. WEBER . . . . . . . . . . . . . . . . . . . . . . . . OPTICALKERRSHUTTER,M. A. DUGUAY. . . . . . . IV. THEULTRAFAST V. HOLOGRAPHIC DIFFRACTION GRATINGS, G. SCHMAHL, D. RUDOLPH . . VI. PHOTOEMISSION, P. J. VERNIER. . . . . . . . . . . . . . . . . . . VII. OPTICALFIBREWAVEGUIDES - A REVIEW,P. J. B. CLARRICOATS. . .

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1-46 47-87 89- 159 161-193 195-244 245-325 327-402

C O N T E N T S O F V O L U M E XV ( 1 9 7 7 ) THEORY OF OPTICALPARAMETRIC AMPLIFICATION AND OSCILLATION, W. , BRUNNER, H. PAUL . . . . . . . . . . . . . . . . . . . . . . 11. OPTICALPROPERTIES OF THINMETALFILMS,P. ROUARD,A. MEESSEN . HOLOGRAPHY, T. OKOSHI. . . . . . . . . , . . . 111. PROJECTION-TYPE TECHNIQUES OF RADIO ASTRONOMY, T. W. COLE .. IV. QUASI-OPTICAL FOUNDATIONS OF THE MACROSCOPIC ELECTROMAGNETIC THEORYOF V. DIELECTRIC MEDIA,J. VANKRANENDONK, J. E. SIPE . . . . . . . . . I.

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

C O N T E N T S O F V O L U M E XVI ( 1 9 7 8 ) 1-69 LASERSELECTIVE PHOTOPHYSICS AND PHOTOCHEMISTRY, v. s. LETOKHOV J. J. CLAIR,C. I. RECENTADVANCESIN PHASEPROFILESGENERATION, 71-117 ABITBOL. . . . . . . . . . . . . . . . . . . . . . . . . . HOLOGRAMS: TECHNIQUES AND APPLICATIONS, 111. COMPUTER-GENERATED W.-H. LEE . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119-232 A. E. ENNOS . . . . . . . . . . . . . . 233-288 IV. SPECKLEINTERFEROMETRY, DEFORMATION INVARIANT, SPACE-VARIANT OPTICALRECOGNITION, D. V. CASASENT, D. PSALTIS . . . . . . . . . . . . . . . . . . . . . . 289-356 v1. LIGHT EMISSIONFROMHIGH-CURRENT SURFACE-SPARK DISCHARGES, R. E. BEVERLY111 . . . . . . . . . . . , . . . . . . . . . . . 357-41 1 WITHIN A QUANTUM-MECHANICAL VII. SEMICLASSICAL RADIATION THEORY FRAMEWORK, I. R. SENITZKY. . . . . . . . . . . . . . . . . . . . 413-448

I. 11.

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IX

CONTENTS O F VOLUME XVII (1980) HETERODYNE HOLOGRAPHIC INTERFEROMETRY, R. DANDLIKER . . . . 1-84 11. DOPPLER-FREE MULTIPHOTON SPECTROSCOPY, E. GIACOBINO, B. CAGNAC 85- 162 BETWEEN COHERENCE PROPERTIES OF LIGHT 111. THEMUTUALDEPENDENCE AND NONLINEAR OPTICALPROCESSES, M. SCHUBERT, B. WILHELMI . . 163-238 IV. MICHELSONSTELLARINTERFEROMETRY, W. J. TANGO, R. Q. Twrss . . . 239-278 MEDIAWITH VARIABLE INDEX OF REFRACTION,A.L. v. SELF-FOCUSING MIKAELIAN . . . . . . . . . . . . . . . . . . . . . . . . , . 279-345 I.

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CONTENTS OF VOLUME XVIII (1980) GRADEDINDEXOPTICALWAVEGUIDES:A REVIEW,A. GHATAK,K. THYAGARAJAN . . .. . . . . . . ... .. . . .. .., .. . . 1-126 11. PHOTOCOUNT STATISTICS OF RADIATION PROPAGATING THROUGH RANDOMAND NONLINEAR MEDIA,J. PERINA , . . . . . . . . . . 127-203 111. STRONG FLUCTUATIONS IN LIGHTPROPAGATION IN A RANDOMLY INHOMOGENEOUS MEDIUM,v. I. TATARSKII, v. u. ZAVOROTNYI . . . . . . . . 204-256 IV. CATASTROPHE OPTICS: MORPHOLOGIES OF CAUSTICS AND THEIR DIFFRACTION PATTERNS, M. v. BERRY, c. UPSTILL . . . . . . . . . . . . 257-346

I.

CONTENTS OF VOLUME XIX (1981) I.

THEORY OF INTENSITY DEPENDENT RESONANCELIGHTSCATTERING AND RESONANCE FLUORESCENCE, B. R. MOLLOW . . . . . . . . . . . . . 1-43 SPECTRAOF 11. SURFACEAND SIZE EFFECTS ON THE LIGHTSCATTERING SOLIDS,D. L. MILLS,K. R. SUBBASWAMY . . . . . . . . . . . . . . 45-137 111. LIGHT SCATTERING SPECTROSCOPY OF SURFACE ELECTROMAGNETIC

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WAVES IN SOLIDS, USHIODA PRINCIPLES OF OPTICAL DATA-PROCESSING, H. J. BUTTERWECK , THEEFFECTS OF ATMOSPHERIC TURBULENCE IN OPTICAL ASTRONOMY, F.

RODDIER

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139-210 211-280 281-376

CONTENTS O F VOLUME XX (1983) 1.

SOME NEWOPTICAL DESIGNS FOR ULTRA-VIOLET BIDIMENSIONAL DETECASTRONOMICAL OBJECTS, G. COURTBS, P. CRUVELLIER, M.

TION OF

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DETAILLE, M. SAYSSE . . . . . . . . . . . . . . . . . . . . 1-62 SHAPING AND ANALYSIS OF PICOSECOND LIGHTPULSES, c. FROEHLY, B. COLOMBEAU, M. VAMPOUILLE . . . . . . . . . . . . . . . . . 63-154 111. MULTI-PHOTON SCATTERING MOLECULAR SPECTROSCOPY, S. KIELICH . 155-262 HOLOGRAPHY, P. HARIHARAN. . . . . . . . . . . . 263-324 IV. COLOUR GENERATION OF TUNABLE COHERENT VACUUM-ULTRAVIOLET RADIATION, V. W. JAMROZ, B. P. STOICHEFF. . . . . . . . . . . . . . . . . . . 325-380

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C O N T E N T S O F V O L U M E X X I (1984) I. II. 111. Iv. v.

RIGOROUS VECTORTHEORIES OF DIFFRACTION GRATINGS, D.MAYSTRE. THEORY OF OPTICAL BISTABILITY, L. A. LUGIATO. . . . . . . . . . . THERADONTRANSFORM AND ITS APPLICATIONS, H. H. BARRETT . . . ZONE PLATE CODED IMAGING: THEORY AND APPLICATIONS, N. M. CEGLIO, D. W. SWEENEY . . . . . . . . .... . .. ... . .. FLUCTUATIONS, INSTABILITIES AND CHAOS IN THE LASER-DRIVEN NONLINEAR RINGCAVITY, J. C. ENGLUND, R. R. SNAPP,W. C. SCHIEVE. . .

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1-68 69-216 217-286 287-354 355-428

C O N T E N T S O F VOLUME X X I I (1985) OPTICALAND ELECTRONICPROCESSING OF MEDICAL IMAGES, D. . . . . .. .. . . . . . . . . . . . . , . . . .. .. MALACARA 1-76 11. QUANTUM FLUCTUATIONS IN VISION, M. A. BOUMAN, W. A. VAN DE GRIND, . 77-144 P.ZUIDEMA. . . . . . . . . . . . . . . . . . . . . . . . , 111. SPECTRALAND TEMPORALFLUCTUATIONS OF BROAD-BANDLASER RADIATION, A. V. MASALOV. . . . . . . . . . . . . . . . . . . . 145-196 IV. HOLOGRAPHIC METHODSOF PLASMA DIAGNOSTICS, G. v. OSTROVSKAYA, Yu. 1. OSTROVSKY . . . . . . . . . . . . . . . . . . . . . . . . 197-270 V. FRINGE FORMATIONS I N DEFORMATION AND VIBRATIONMEASUREMENTS USING LASERLIGHT,I. YAMAGUCHI . . . . . .. ... . . . . 271-340 IN RANDOM MEDIA:A SYSTEMSAPPROACH, R. L. VI. WAVEPROPAGATION FANTE. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 341-398 I.

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C O N T E N T S O F VOLUME X X I I I (1986) ANALYTICAL TECHNIQUES FOR MULTIPLESCATTERING FROM ROUGH G. S. BROWN. . . . . . , . . . . . . . . SURFACES, J. A. DESANTO, 11. PARAXIAL THEORY IN OPTICAL DESIGN IN TERMS OF GAUSSIAN BRACKETS, K.TANAKA . . . . . . . . . . . . . . . . . . . .. . . . FILMSPRODUCED BY ION-BASED TECHNIQUES, P. J. MARTIN, R. P. 111. OPTICAL NEITERFIELD. . . . . . . . . . . . . . . . . . . . . . . . . . . HOLOGRAPHY, A. TONOMURA. . . . . . . . . . . . . IV. ELECTRON V. PRINCIPLES OF OPTICAL PROCESSING WITH PARTIALLY COHERENT LIGHT, F.T.S.Yu.. . . . . . . . . . . . . . . . . . . . . . . . . . . I.

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1-62 63-1 12 113-182 183-220 221-276

C O N T E N T S O F VOLUME XXIV (1987) I. 11. 111. IV. V.

MICROFRESNEL 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, 1. GLASER. . . . . . . . . . . . . . . . . . . . . . . . . . . .

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

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C O N T E N T S O F V O L U M E XXV ( 1 9 8 8 ) DYNAMICAL INSTABILITIES AND PULSATIONS IN LASERS,N. B. ABRAHAM, . . . . . .. ...... . . 1-190 P. MANDEL,L. M. NARDUCCI . I N SEMICONDUCTOR LASERS,M. OHTSU,T. TAKO . . . . . 191-278 11. COHERENCE AND DESIGNOF OPTICAL ARRAYS, WANGSHAOMIN, L. RONCHI 279-348 111. PRINCIPLES SURFACES, G. SCHULZ. . . . . , . . . . . . . . . . 349-416 IV. ASPHERIC 1.

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C O N T E N T S O F VOLUME XXVI (1988) I. 11. 111. IV. V.

PHOTONBUNCHING AND ANTIBUNCHING, M. C. TEICH,B. E. A. SALEH . NONLINEAR OPTICS OF LIQUIDCRYSTALS, I. c. KHOO. . . . . . . . . SINGLE-LONGITUDINAL-MODE SEMICONDUCTOR LASERS,G. P. AGRAWAL RAYSAND CAUSTICS AS PHYSICAL OBJECTS, YU.A. KRAVTSOV . .. . PHASE-MEASUREMENT INTERFEROMETRY TECHNIQUES, K. CREATH. . .

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

CONTENTS O F VOLUME XXVII (1989) I. THE SELF-IMAGING PHENOMENON AND ITSAPPLICATIONS, K. PATORSKI 11. AXICONS AND MESO-OPTICAL IMAGING DEVICES,L. M. SOROKO. . . . OPTICS FOR FLUXCONCENTRATION, I. M. BASSETT,w. T. 111. NONIMAGING WELFORD,R. WINSTON . . . . . . . . . . . . . . . . . . . . . . IN PLANAR STRUCTURES, D. MIHALACHE, WAVEPROPAGATION IV. NONLINEAR M. BERTOLOTTI, C. SIBILIA.. . . . . . . . . . . . . . . . . . . . GENERALIZED HOLOGRAPHY V. WITH APPLICATION TO INVERSE SCATTERING AND INVERSE SOURCEPROBLEMS, R. P. PORTER . . . . . . . . . .

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PREFACE Just like many of its predecessors, this volume of Progress in Optics presents five review articles describing recent developments in optics. It covers classical as well as quantum optics and theory as well as experiment. The first article describes progress made in the last few years in digital holography. This is a field which differs from conventional holography by utilizing synthetic methods rather than optical processes, at least in some stages of the recording-to-reconstruction process. Digital holography has found useful applications in connection with data processing and data storage, for 3-D displays and in providing new types of optical components, for example, holographic gratings. The second article describes basic investigations concerned with new technologies that may lead to better optical communication systems and improved limits of measurement than are expected from the traditional interpretation of quantum-mechanical measurement theory. Such possibilities have become feasible with the introduction of the concept of squeezed states and quantum non-demolition experiments. The next article presents a review of our current understanding of quantum coherence properties of stimulated Raman scattering. Many interesting developments in this field have become possible by the availability of high-power pulsed lasers which have lead to better understanding of the quantum mechanical evolution of macroscopic systems. The fourth article presents an account of techniques developed in recent years in the field of interferometry, for improvements of high-precision measurements. These improvements have been largely brought about by the progress in photoelectric detection technology, in microelectronic circuitry and in the field of microcomputers. The concluding article deals with the fascinating phenomenon of quantum jumps, which were introduced in the theory of atomic spectra by Niels Bohr in 1913and which until recently were shrouded in mystery. In the last few years ion-trapping technology has lead to the development of single-atom spectroscopy, which has greatly elucidated this phenomenon. The article discusses both theory and observations. The present volume attests once again to the remarkable and rapid progress that continues to be made in many areas of optics. Department of Physics and Astronomy University of Rochester Rochester, NY 14627. USA January 1990 XI11

EMILWOLF

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CONTENTS I. DIGITAL HOLOGRAPHY

. COMPUTER-GENERATED

.

HOLOGRAMS

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by 0. BRYNGDAHL and F. WYROWSKI (ESSEN.FED REP GERMANY) $ 1 . INTRODUCTION. . . . . . . . . . . . . . . . . . . . . . . . . . . . $ 2. EARLYHISTORYAND DEVELOPMENTS. . . . . . . . . . . . . . . . . . 2.1. Modifications of the hologram concept . . . . . . . . . . . . . . . . 2.1.1. Detour phase hologram . . . . . . . . . . . . . . . . . . . . 2.1.2. Delayed sampled hologram . . . . . . . . . . . . . . . . . . . 2.1.3. Kinoform . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2. Simulation of optical holograms . . . . . . . . . . . . . . . . . . . 2.2.1. Carrier type hologram . . . . . . . . . . . . . . . . . . . . . 2.2.2. Computer-generated interferogram . . . . . . . . . . . . . . . . 2.3. 3-D digital holography . . . . . . . . . . . . . . . . . . . . . . . 2.3.1. Synthesis in planes of point objects . . . . . . . . . . . . . . . 2.3.2. Multiple perspective projections . . . . . . . . . . . . . . . . . 2.4. Persistence of early trends . . . . . . . . . . . . . . . . . . . . . . $ 3. MAJORSTEPS IN DIGITAL HOLOGRAPHY. . . . . . . . . . . . . . . . . $ 4. FROM RECONSTRUCTION SPACE TO HOLOGRAM PLANE . . . . . . . . . . . 4.1. 2-D objects . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.1. Fourier transform relationship between object and hologram . . . . 4.1.1.1. Boundary conditions of spectrum and object . . . . . . . . 4.1.1.2. Discrete object: boundary condition consideration . . . . . 4.1.1.3. Continuous object: boundary condition consideration . . . . 4.1.2. Fresnel transform relationship between object and hologram . . . . 4.1.3. Object or image of object in hologram plane . . . . . . . . . . . 4.2. 3-D intensity distributions: model of 3-D object . . . . . . . . . . . . . 4.2.1. Reduction of computation effort: removal of vertical parallax . . . 4.2.2. Reduction of computation effort: utilization of data characteristics . . $ 5. CODING PROCEDURES IN DIGITAL HOLOGRAPHY . . . . . . . . . . . .. . 5.1. Fundamental principles . . . . . . . . . . . . . . . . . . . . . . . 5.2. Coding in Fourier holography . . . . . . . . . . . . . . . . . . . 5.2.1. Amplitude hologram . . . . . . . . . . . . . . . . . . . . . . 5.2.1.1. Point- and cell-oriented coding . . . . . . . . . . . . . . 5.2.1.2. Application of phase freedom: diffuser . . . . . . . . . . . 5.2.1.2.1. Discrete intensity signals . . . . . . . . . . . 5.2.1.2.2. Continuous intensity signals . . . . . . . . . . . 5.2.1.3. Additional application of amplitude freedom . . . . . . . 5.2.1.4. Quantization . . . . . . . . . . . . . . . . . . . . . 5.2.1.4.1. Quantization of point-oriented holograms . . . . . 5.2.1.4.2. Quantization of cell-oriented holograms . . . . . . 5.2.2. Phase hologram . . . . . . . . . . . . . . . . . . . . . . .. 5.2.2.1. Bleached amplitude hologram . . . . . . . . . . . . . . 5.2.2.2. Direct coding in digital phase holography . . . . . . . . . 5.2.3. Quantization of phase hologram . . . . . . . . . . . . . . . . . 5.3. Coding in Fresnel holography . . . . . . . . . . . . . . . . . . . . 5.4. Coding in image holography . . . . . . . . . . . . . . . . . . . . . 8 6. REFLECTION OF MATERIALIZATION AND APPLICATION . . . . . . . . . . REFERENCES .............................. . .

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3 5 5 5 6 6 7 7 7 8 8 8 9 9 10 11 11 12 14 16 19 21 21 24 25 28 28 30 34 36 43 44 46 47 52 52 63 65 66 68 72 76 79 82 84

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CONTENTS

I1. QUANTUM MECHANICAL LIMIT IN OPTICAL PRECISION MEASUREMENT AND COMMUNICATION by Y. YAMAMOTO. S . MACHIDA. S . SAITO.N . IMOTO.T . YANAGAWA. M . KITAGAWA and

G . BJORK(TOKYO.JAPAN)

8 1. INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2. EMERGENCE OF STANDARD QUANTUM LIMITS. . . . . . . . . . . . . . .

Coherent states and quantum noise of laser emission . . . . . . . . . . Optical communication at standard quantum limit . . . . . . . . . . . Gravitational wave detection interferometer at standard quantum limit . . Laser gyroscope at standard quantum limit . . . . . . . . . . . . . . Fluctuation-dissipation theorem and simultaneous measurement of two conjugate observables . . . . . . . . . . . . . . . . . . . . . . . . . 2.6. Minimum noise figure for linear amplifiers . . . . . . . . . . . . . . . 8 3 . NONCLASSICAL LIGHTS . . . . . . . . . . . . . . . . . . . . . . . . . 3.1. Minimum uncertainty states . . . . . . . . . . . . . . . . . . . . . 3.2. Properties of quadrature amplitude squeezed states . . . . . . . . . . . 3.3. Generation of quadrature amplitude squeezed states . . . . . . . . . . 3.4. Properties of number-phase squeezed states . . . . . . . . . . . . . . 3.5. Generation of number-phase squeezed states . . . . . . . . . . . . . . 3.6. Pump-noise-suppressed laser . . . . . . . . . . . . . . . . . . . . . 3.7. Properties of photon twins . . . . . . . . . . . . . . . . . . . . . 3.8. Generation of photon twins . . . . . . . . . . . . . . . . . . . . . 3.9. Generation of quadrature amplitude squeezed states and number-phase squeezed states by photon twins and feedback or feedforward . . . . . . 8 4. QUANTUM NONDEMOLITION (QND) MEASUREMENT . . . . . . . . . . . . 4.1. General quantum measurement and QND . . . . . . . . . . . . . . . 4.2. Contractive state measurement . . . . . . . . . . . . . . . . . . . . 4.3. Projection postulate, first kind measurement, and state reduction . . . . . 4.4. QND measurement for photon number . . . . . . . . . . . . . . . . 4.4.1. Heisenberg picture . . . . . . . . . . . . . . . . . . . . . . 4.4.2. Schredinger picture . . . . . . . . . . . . . . . . . . . . . . 4.5. Effect of self-phase modulation . . . . . . . . . . . . . . . . . . . . 4.6. Effect of loss . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.7. Experimental QND measurement of photon number . . . . . . . . . . 4.8. QND measurement for quadrature amplitude . . . . . . . . . . . . . 4.9. Preamplification function of QND . . . . . . . . . . . . . . . . . . 4.10. Use of squeezed states as a probe wave . . . . . . . . . . . . . . . . 5 5 . QUADRATURE AMPLITUDE AMPLIFIERS AND PHOTONNUMBER AMPLIFIERS. . 5.1. General quantum amplifiers . . . . . . . . . . . . . . . . . . . . . 5.2. Degenerate and nondegenerate parametric amplifiers . . . . . . . . . . 5.3. Phase-locked oscillator . . . . . . . . . . . . . . . . . . . . . . . 5.4. Nondegenerate parametric oscillator with idler measurement-feedback . . . 5.5. Laser oscillator with QND measurement-feedback . . . . . . . . . . . 5.6. Amplification and deamplification for quantum state transformation . . . . 8 6. QUPiNTUM MECHANICAL CHANNEL CAPACITY . . . . . . . . . . . . . . . 6.1. Quantum mechanical channel capacity for narrow-band communication . . 6.2. Minimum energy cost per bit . . . . . . . . . . . . . . . . . . . . 6.3. Broadband communication and time-energy uncertainty relationship . . . 6.4. Precision measurement and time-energy uncertainty relationship . . . . . 2.1 2.2. 2.3. 2.4. 2.5.

89 90 90 94 95 98 99 101

103 103 104 106 111

113 116 120 121 123 125 125 128 129 131 132 133 135 137 139 141 143 145 146 146 147 149 I52 153 156 158 159 164 165 167

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XVII

$ 7. APPLICATIONS ............................. 7.1. Communication breaking the SQL . . . . . . . . . . . . . . . . . . 7.2. Gyroscope breaking the SQL . . . . . . . . . . . . . . . . . . . . 7.3. Gravitational wave detection interferometer breaking the SQL . . . . . . 7.4. Measurement for surpassing the SQL of a free-mass position . . . . . . . 8 8. DISCUSSION AND CONCLUSION . . . . . . . . . . . . . . . . . . . . . ............................ ACKNOWLEDGEMENT ................................ REFERENCES

168 169 169 172 173 174 176 176

I11. THE QUANTUM COHERENCE PROPERTIES OF STIMULATED RAMAN SCATTERING by M.G. RAYMER (EUGENE,OR. USA) and LA. WALMSLEY (ROCHESTER. NY. USA)

$ 1. INTRODUCTION. . . . . . . . . . . . . . . . . . . . . . . . . . . . $ 2. HISTORICAL PERSPECTIVE ........................ $ 3. THEORY OF STIMULATED RAMAN SCATTERING . . . . . . . . . . . . . . . 3.1. Photon rate-equation theory . . . . . . . . . . . . . . . . . . . . . 3.1.1. Quantized field theory of photon scattering . . . . . . . . . . . . 3.1.2. Spontaneous scattering cross-section . . . . . . . . . . . . . . . 3.1.3. Stimulated scattering rate . . . . . . . . . . . . . . . . . . . 3.1.4. Photon rate equations . . . . . . . . . . . . . . . . . . . . . 3.1.5. Molecular polarizability model . . . . . . . . . . . . . . . . . 3.2. Semiclassical propagation theory . . . . . . . . . . . . . . . . . . . 3.2.1. Atomic-density operator equations of motion . . . . . . . . . . . 3.2.2. Wave equation . . . . . . . . . . . . . . . . . . . . . . . . 3.2.3. Solution of semiclassical equations in the linear regime . . . . . . . 3.3. Quantum theory of SRS in a cavity . . . . . . . . . . . . . . . . . . 3.3.1. Equations of motion . . . . . . . . . . . . . . . . . . . . . . 3.3.2. Photon number fluctuations . . . . . . . . . . . . . . . . . . 3.3.3. Equivalent classical random process . . . . . . . . . . . . . . . 3.3.4. Generalization to many Stokes modes . . . . . . . . . . . . . . 3.4. Quantized-field theory of spatial propagation in SRS . . . . . . . . . . 3.4.1. Atomic operator equations of motion . . . . . . . . . . 3.4.2. Operator Maxwell-Bloch equations . . . . . . . . . . . . . . . 3.4.3. One-dimensional propagation . . . . . . . . . . . . . . . . . . 3.4.4. Solution of the linearized SRS equations . . . . . . . . . . . . . 3.4.5. Steady-state power spectrum of SRS . . . . . . . . . . . . . . . 3.4.6. Fluctuations of Stokes pulse energy . . . . . . . . . . . . . . . 3.4.7. Temporal fluctuations of Stokes pulses . . . . . . . . . . . . . . 3.4.8. Spatial fluctuations of Stokes pulses . . . . . . . . . . . . . . . $ 4. EXPERIMENTS ON QUANTUM-STATISTICAL ASPECTSOF STIMULATED RAMAN SCATTERING .................... . . . . . . . . . . 4.1. Stokes-pulse-energy fluctuations . . . . . . . . . . . . . . . . . . . 4.2. Temporal and spatial intensity fluctuations . . . . . . . . . . . . . . . 4.3. Spontaneous generation of Raman solitons . . . . . . . . . . . . . . . 4.4. Cooperative Raman scattering (CRS) . . . . . . . . . . . . . . . . . ACKNOWLEDGEMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . LISTOF SYMBOLS. . . . . . . . . . . . . . . . . . . . . . . . . . . . . REFERENCES ................................

....

183 186 196 197 198 200 200 201 204 206 207 209 211 213 213 214 216 218 219 219 223 225 225 229 231 236 239 245 245 255 259 260 264 265 261

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IV . ADVANCED EVALUATION TECHNIQUES IN INTERFEROMETRY by J . SCHWIDER (BERLIN.FED. REP. GERMANY)

. . . . . . . . . . . . . . . . . . . . . . . . . . . . 273 § 1. INTRODUCTION AND CHARACTERISTICS OF REAL-TIMEINTERFEROMETRIC § 2. CLASSIFICATION METHODS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1. Conventional evaluation techniques . . . . . . . . . . . . . . . . Some forerunners of automatic evaluation techniques . . . . . . . . . 2.2. § 3. FRINGEEVALUATlON (rp = Po + p I X) . . . . . . . . . . . . . . . . . . . 3.1. Phase measurement by interpolation between fringe positions . . . . . 3.2. Assessment of fringe positions . . . . . . . . . . . . . . . . . . . 3.2.1. Low pass filtering and dc subtraction . . . . . . . . . . . . 3.2.2. Parabolic approximation of minimum positions . . . . . . . . 3.2.3. Low pass filtering combined with differentiation . . . . . . . . 3.2.4. Image subtraction and level slicing . . . . . . . . . . . . . 3.2.5. Fringe skeletonizing or thinning operations . . . . . . . . . . 3.3. Analog processing of scanned fringe intensity . . . . . . . . . . . . 3.4. Fringe analysis by means of Fourier transform operations . . . . . . . 3.5. Spatially synchronous fringe analysis . . . . . . . . . . . . . . . . 3.6. Sinusoidal fitting . . . . . . . . . . . . . . . . . . . . . . . . § 4 . PHASEMODULATORS. . . . . . . . . . . . . . . . . . . . . . . . . INTERFEROMETRY (PLI) (cp = a sin wr) . . . . . . . . . . . . § 5 . PHASE-LOCK INTERFEROMETRY (cp = wr) . . . . . . . . . . . . . . . . . § 6 . HETERODYNE INTERFEROMETRY (PSI) (cp = ( r - l)cpo) . . . . . . . . . § 7. PHASESAMPLING 7.1. Phase unwrapping . . . . . . . . . . . . . . . . . . . . . . . . 7.2. Calibration methods for reference phase shifters . . . . . . . . . . . DATAAND MERITFUNCTIONS . . . . . . . . . . . . . . . . § 8. RELEVANT § 9. CALIBRATION METHODSFOR INTERFEROMETERS . . . . . . . . . . . . . 8 10. APPLICATIONS. . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.1. Optical testing and flatness tests . . . . . . . . . . . . . . . . . . 10.2. Microstructure measurements . . . . . . . . . . . . . . . . . . . 10.3. Shape measurements of ground surfaces . . . . . . . . . . . . . . 10.4. Moirt topography . . . . . . . . . . . . . . . . . . . . . . . . 10.5. Measurement of refractive index distributions . . . . . . . . . . . . 10.6. Two-wavelength interferometry . . . . . . . . . . . . . . . . . . 10.7. Holographic interferometry . . . . . . . . . . . . . . . . . . . . 10.8. Wavefront sensors for adaptive optics . . . . . . . . . . . . . . . 10.8.1. Shearing sensors . . . . . . . . . . . . . . . . . . . . . 10.8.2. Point reference sensors . . . . . . . . . . . . . . . . . . 10.9. Speckle interferometry . . . . . . . . . . . . . . . . . . . . . . 10.10. Miscellaneous . . . . . . . . . . . . . . . . . . . . . . . . . . § I 1 . ERRORSOURCESAND MEASURING LIMITATIONS. . . . . . . . . . . . . 11.1 Environmental error sources . . . . . . . . . . . . . . . . . . . 11.1.1. Air turbulence and stratification . . . . . . . . . . . . . . . 11.1.2. Thermal drifts and mechanical relaxation . . . . . . . . . . . 11.1.3. Mechanical strain . . . . . . . . . . . . . . . . . . . . . 11.1.4. Vibrations . . . . . . . . . . . . . . . . . . . . . . . . 11.2. Errors depending on the evaluation method . . . . . . . . . . . . . 11.2.1. Reference phase errors . . . . . . . . . . . . . . . . . . .

276 276 278 278 278 281 283 285 286 287 287 289 290 292 295 296 301 303 308 312 316 320 324 326 326 329 330 332 333 333 334 336 336 336 337 339 339 339 339 340 342 343 343 343

CONTENTS

11.3. Errors typically encountered in interferometry . 11.3.1. Errors due to spurious fringes . . . . . 11.3.2. Detector noise . . . . . . . . . . . . 11.3.3. Quantization noise . . . . . . . . . . 11.3.4. Nonlinearities of the photodetector . . 11.3.5. Coherent noise . . . . . . . . . . . . 11.4. Optical limitations . . . . . . . . . . . . . . 11.4.1. Testing of smooth surfaces . . . . . . 11.4.2. Microstructure testing . . . . . . . . 11.4.3. Limitations in holographic interferometry

XIX

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ACKNOWLEDGEMENTS ........................... REFERENCES ................................

346 346 347 347 348 348 349 349 352 352 353 353

V . QUANTUM JUMPS by R.J. COOK(COLORADO SPRINGS.Co. USA)

8 1. INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0 2. ION TRAPPINGAND COOLING . . . . . . . . . . . . . . . . . . . . . . 0 3. THEORYOF TELEGRAPHIC FLUORESCENCE. . . . . . . . . . . . . . . .

Delay function for two-level atom . . . . . . . . . . . . . . . . . . . Interruption of fluorescence due to shelving . . . . . . . . . . . . . . Delay function for the V-configuration . . . . . . . . . . . . . . . . . Delay function for the A-configuration . . . . . . . . . . . . . . . . . 0 4. THENATUREOF QUANTUM JUMPS . . . . . . . . . . . . . . . . . . . . 4.1. Atomic dynamics during frequent measurements . . . . . . . . . . . . 4.2. Knowledge-induced transitions . . . . . . . . . . . . . . . . . . . . 0 5. OBSERVATION OF QUANTUMJUMPS . . . . . . . . . . . . . . . . . . . ACKNOWLEDGEMENTS ........................... REFERENCES ................................

363 368 377 378 383 387 396 397 398 403 407 413 414

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

417 427 43 1

3.1. 3.2. 3.3. 3.4.

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E. WOLF, PROGRESS IN OPTICS XXVIII 0 ELSEVIER SCIENCE PUBLISHERS B.V., 1990

I DIGITAL HOLOGRAPHY - COMPUTER-GENERATED HOLOGRAMS BY

OLOFBRYNGDAHL and FRANKWYROWSKI Physics Department University of Essen 4300 Essen 1. Fed. Rep. Germany

CONTENTS PAGE

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REFERENCES

0 1. Introduction The 1960s were years when a broadened interest in holography gained momentum. During this decade several milestones for the further development and application of holographic ideas and procedures occurred. The availability of lasers and the adaptation of techniques like carrier recording and diffuse illumination introduced new dimensions. The new branch of optics -holography -suddenly exploded in different directions, giving far-reachingstimulation to other fields. The sixties were also the years when a new generation of digital computers evolved. Their impact in science and technology was fast and overwhelming. The era of punched tape and cards was suddenly gone and replaced by one where the computers were programmed to function in an interactive way. Parallel with the advancement of digital electronic processing, optical information processing was revived by the application of holography to produce complex spatial filters. The two-step character of the holographic process was confronted with the computation power of digital computers as well as their ability to interact with output devices. The experimental holographic recording process could be performed as computer-designed artwork. The realization and functioning of complex-valued filters and masks led the way toward a new field. Digital holography was established as a separate branch of holography from the beginning. Its typical feature is that optical processes are partially or totally replaced by synthetic ones. For example, in contrast to the interferometric recording in optical holography, the construction of a computer-generated hologram (CGH) can be performed in a synthetic step in which optical counterparts frequently do not exist. This is where the strengths and possibilities of digital holography can be found. The reconstruction step from optical holography is generally maintained by means of a diffraction process that connects the recorded information with the reconstructed field. The aim is to produce a CGH in such a way that when this CGH is illuminated, a desired light distribution is formed, which we will call the object. We will also use this designation metaphorically in dealing with computer-generated optical elements, where the object (wavefront) is given in the plane of the hologram. However, it can also be performed by synthetic means. In computer recon3

4

DIGITAL HOLOGRAPHY

- COMPUTER-GENERATED HOLOGRAMS

[I, 8 1

struction of a hologram the object is calculated from the hologram distribution and the result is displayed. The focus here will be on that version of digital holography where the recording step is synthetically performed. In this sense a comparison between optical and digital holography is illustrated in fig. 1. The first step leading to the hologram is completely different in the two cases. In digital holography the amplitude and phase are given in the form of a set of data in the computer. In principle no carrier is necessary for recording the phase, which can be directly implemented in a phase hologram. The nature of the construction process of the CGH gives the manipulations flexibility. The second step, that of reconstruction, is fixed in both cases. Computer-generated holograms have been described in a previous contribution by LEE [ 19781 in Progress in Optics. Our intention is not to repeat the content of that review but to supplement it. In particular the focus will be on developments and trends of coding procedures in later years. Most scientific progress advances in a systematic and programmed way, but sometimes it jumps ahead and inventive factors play a part in the process. In looking back at the introduction and application of digital computers to holography, it should be noted that its history did not start only by applying digital computers to simulate the optical recording process. It began by the introduction of new ideas which had no direct connection to, and no counterpart in, optical holography. The availability of digital computers had a profound impact on holography. In particular, developments of computer hardware and the realization of optical holography

object

+

1

recording process:

+ image

+hologram+ diffraction

interference

digital holography

specfied image -b

(object)

- 1 const r u d ion process inverse diffraction + hologram coding

+ CGH +

reconstruction process.

+ image

diffraction

Fig. 1. Block diagrams indicating the difference between optical and digital holography.

1 9

8 21

EARLY HISTORY A N D DEVELOPMENTS

5

sophisticated software such as the fast Fourier transform (FFT) algorithm made it possible to extend the achievements of holography far beyond those of optical holography. The very important factor in the computational field which influenced the development of digital holography is the Cooley-Tukey algorithm; that is, the FFT algorithm (COOLEYand TUKEY[1965]). The primary role of the Fourier transform is to describe wave propagation and to perform spatial filtering operations.

0 2. Early History and Developments The origin and history of digital holography involve many different factors. A combination of developments, means, needs, and the intriguing possibility of going beyond the existing limits evolved in the mid- 1960s through the work of several scientists independently. Their approaches can be divided into two major categories: formation of the hologram itself and calculations of the complex field distribution to be stored as a hologram. In digital holography, where synthetic ingredients are built into the process, two kinds of ideas can be found in these categories -those which initiated new concepts with no previous counterpart in optics and those which could be associated with existing analog phenomena. Contributions in the first category seem to have had a particularly dominating influence on the early history and further developments of digital holography.

2.1. MODIFICATIONS OF THE HOLOGRAM CONCEPT

2.1.1. Detour phase hologram Lohmann invented a scheme that contained new ideas and factors (BROWN [ 1966]), several of which were difficult to understand at the and LOHMANN time because no direct analog counterpart existed. In fig. 2 the artwork performed for the first computer-generated hologram is shown. It was hand made and served well for a feasibility study. It should be noted that the FFT algorithm was not used to calculate the Fourier transform. The diffraction patterns from the four bars of the letter “ E were calculated separately and added, indicating that CGHs can be produced inexpensively for research purposes. The idea was to sample the Fourier transform of the wanted reconstruction equidistantly and divide the hologram into as many cells as

6

DIGITAL HOLOGRAPHY

- COMPUTER-GENERATED

HOLOGRAMS

[I, B 2

Fig. 2. Byron Brown with the artwork of the first CGH.(Courtesy of Brown and Werlich.)

sampling points. The complex number of each sampling point was coded into a pattern configuration within each cell. Its modulus (amplitude) was coded as the area of an aperture and its argument (phase) as the location of the aperture within the cell. To explain the functioning of this scheme, the expression “detour phase” was introduced to associate it with a distorted grating structure. In their subsequent papers (LOHMANNand PARIS [ 19671, BROWNand LOHMANN[ 19691) the FFT and a computer-driven plotter were used to produce the artwork for the CGH, which was photoreduced to proper size. 2.1.2. Delayed sampled hologram

LEE[ 19701 suggested a hologram encoding scheme, where four real positive functions representing the complex function were sampled with spatial delays. This hologram encoding can be used with implicit carrier and bias.

2.1.3. Kinofom The suggestion of introducing diffuse illumination in optical holography corresponds to superposing a random or quasi-random phase onto the specified object in digital holography. Thus the phase in the hologram has a much greater importance than the amplitude. Based on this fact, LESEM,HIRSCH and JORDAN[ 19671 introduced the idea that a phase mask can be calculated to

1 9

8 21

EARLY HISTORY A N D DEVELOPMENTS

I

function as a hologram without a carrier. They called this recording a kinoform, which is a typical digital holography phenomenon. The phase is not coded indirectly but is directly produced without the introduction of a carrier. The necessary phase distribution can be produced by bleaching a photographic recording of an appropriately computed intensity distribution that is proportional to the phase distribution in the hologram plane. However, the kinoform has no relation to what is commonly called a bleached hologram. The reconstruction from a kinoform does not contain a twin object. Its inherent high efficiency and other characteristics of the fundamental concept are a stimulus to develop this type of digital hologram further.

2.2. SIMULATION OF OPTICAL HOLOGRAMS

2.2.1. Carrier type hologram

The holographic process attracted and stimulated work in optical processing. The carrier hologram version is an attractive way to realize a complex valued filter. Pure amplitude or phase masks were forerunners of a new generation of sophisticated computer-generated complex filters. These filters can be made binary, and they have the appearance of distorted gratings (KOZMAand KELLY [ 19651). The next step was the simulation of conventional hologram structures by using computers. The CGH version has some advantages. BURCH [1967] suggested replacing the conventional holographic bias by a constant one, which has the advantage of reducing the necessary bandwidth of the hologram as well as eliminating autocorrelation noise. Synthetic production provided the freedom to modify the hologram parameters. For example, HUANG and PRASADA[1966] demonstrated how to increase the fringe contrast in the hologram by making the bias proportional to the modulus of the distribution recorded. With an amplitude transmission recording medium, digital carrier type holograms are a natural development. 2.2.2. Computer-generated interferogram

Synthetically produced grating structures evolved early in the 1970s for applications that required specified wavefront shapes (MACGOVERNand WYANT[ 19711, WYANTand BENNETT[ 19721). In the simplest version only phase information was considered.

8

DIGITAL HOLOGRAPHY

- COMPUTER-GENERATED HOLOGRAMS

[I, 8 2

This breed of CGHs is related to image holograms, and in contrast to the other early developments in digital holography no Fourier transform was involved in their conception. In most cases an analytic expression of the phase variation is available. LEE[ 19741 refined the concept of computer-generated interferograms. They form the foundation of what later was called computer-generated holographic optical elements. Many coding schemes to produce CGHs have been suggested over the years. In 5 5 a classification is presented that describes the relationships and differences among the various techniques. We will concentrate on how to relate different object and hologram types to suitable coding techniques from which a proper choice can be made in a particular application situation.

2.3. 3-D DIGITAL HOLOGRAPHY

Interest in 3-D images and proposals to reconstruct them were present from the very beginning of digital holography. However, the immensity of the computational problem was recognized, and simplified schemes were sought. 2.3.1. Synthesis in planes of point objects The strength and convenience of the FFT algorithm together with the quadratic phase factor relation between a Fresnel and Fourier transform encouraged suggestions to decompose the specified object in parallel planes in the third dimension. The contributions arising from the different planes were added to produce the hologram (WATERS[1968], LESEM,HIRSCH and [ 19691). WATERS[ 19661 suggested JORDAN[ 19681, BROWNand LOHMANN decomposing line segments in points and superposing the zone-plate patterns related to them in the hologram plane. In these early realizations no interaction between the planes was considered. The hidden line problem, namely, blocking or scattering in the propagation process from plane to plane, is possible to encounter for calculating a hologram with a realistic reconstruction (ICHIOKA, IZUMIand SUZUKI[ 19711). 2.3.2. Multiple perspective projections A different approach was based on partitioning the hologram in such a way that its separate portions were allocated different perspective views of the

MAJOR STEPS IN DIGITAL HOLOGRAPHY

9

desired image. 2-D perspective views were calculated and transformed into holograms (KING, NOLL and BARRY[1970]). This type of hologram is conceptualized for visual observation, and the result is a stereoscopic image.

2.4. PERSISTENCE OF EARLY TRENDS

The directions set in the early history of digital holography had a surprisingly strong influence on efforts and contributions during the years that followed. A variety of coding schemes were added, and the effects of sampling and quantization were treated. New suggestions on possible applications appeared and the field matured. Specific reviews on digital holography have been written by HUANG[ 19711, LEE [ 19781, YAROSLAVSKII and MERZLYAKOV [ 19801, DALLAS[ 19801, [ 19841, and TRICOLES[ 19871. SCHREIER

8 3. Major Steps in Digital Holography Optical holography is traditionally regarded as a two-step process: (1) the interferometric recording of a wavefield in a hologram and (2) the reconstruction of the wavefield stored in the hologram by diffraction. In general, the technique is used in an imaging situation in which the diffracted field of an object is recorded in the first step, and its image is reconstructed in the second step. The phenomena involved determine the strengths and possibilities as well as the limitations of the procedure. In digital holography, as described here, the hologram recording step is performed synthetically supported by digital computer means, and the reconstruction step remains just the same as in optical holography. Because of the nature of the technique, the procedure begins with a specified object (reconstructed image) and an inverse diffraction is performed to determine the complex amplitude that will be recorded later in the hologram. Thus in digital holography the following steps should be considered: (1) Calculation of the complex amplitude in the hologram plane from specifications of the reconstruction. (2) Generation of the hologram. (3) Optical reconstruction. The strength of digital holography results from the flexibilitiesof the first two synthetic steps, which possess extra freedoms that cannot be exercised by

10

DIGITAL HOLOGRAPHY

- COMPUTER-GENERATED HOLOGRAMS

[I. B 4

optical processes. To circumvent difficulties in the treatment of 3-D light distributions in the first step, models are introduced in digital holography. Frequently holography is incorporated in a larger connection and system aspects prevail. This is evident in the execution of the first two steps by adaptation to specific application needs and physical hardware as well as to existing computer hardware and software. Execution of steps 1 and 2 results in a CGH. Step 1: The starting point is to specify the desired wavefront or data of 1-D, 2-D, or 3-D in the reconstruction space. The statement of the reconstructed wavefield or object has to be in an optical form and the proper style for computer handling. The optical quantities of amplitude and phase (and/or polarization, frequency) can be in a continuous or sampled form. This specification is succeeded by a transformation (image, Fresnel, Fourier), which can be realized by inverse wave propagation methods. As a result, the amplitude and phase are obtained, usually in sampled form, in the hologram plane. Step 2: The complex amplitude to be stored in the hologram is converted to a CGH configuration of analog or quantized values. Practical and physical constraints will guide the choice of procedure for realization. A coding scheme is applied to make the CGH structure conform to requirements of material and recording devices. As a result, the complex amplitude to be stored in the CGH can be encoded in a real and positive valued distribution or in a phase only distribution to adapt to recording media. Step 3: Proper illumination of the CGH results in a reconstruction of the object (data) as a light distribution. It is customary in this process to regard the CGH as a thin hologram. As a result of the specific properties of the CGHs, the reconstructions will show certain characteristics. Sampling and quantization especially demonstrate typical features. Step 1 and 2 will be delt within detail. The specification of the object and hologram type leads to possible coding schemes from which to choose in the actual application.

8 4.

From Reconstruction Space to Hologram Plane

The initial step in the realization of a CGH is to define the desired distribution to be reconstructed, which will be called the object. The object needs to be specified in such a way that the relation, which is an inverse wave propagation relation, between it and the distribution in the hologram plane can be calculated by a digital computer. The transformation from object to hologram is determined by the dimensionality of the object and the distance between the object

o

1, 41

FROM RECONSTRUCTION SPACE TO HOLOGRAM PLANE

11

and the hologram. The object may be described as a 1-D, 2-D or 3-D continuous or discrete distribution of complex amplitudes dependent on the CGH application. The object may be located in the far, near, or even within the field of the hologram, and a Fourier, Fresnel, or no transform is necessary to determine the distribution in the hologram plane.

4.1. 2-D OBJECTS

Fourier, Fresnel, and image plane CGHs correspond to the types of inverse wave propagation and from these resulting transformations between object and hologram.

4.1.1. Fourier transform relationship between object and hologram The aim of Fourier holography is to form a light distribution of a desired reconstructed image (object) in the Fourier plane of the hologram. A distinction is made between complex objects and intensity objects. The distribution of an object is described in the form of a complex amplitude

f ( x ) = If(x)I exp[icp(x)l

(4.1)

for a complex object and in the form of an intensity

If(x) I

=

(4.2)

i(4

for an intensity object, where x = (x, y) and arg [ f ( x ) ] = &). For an intensity object any complex amplitude

with the arbitrary phase &) will lead to eq. (4.2). In this sense q(x) is free to choose; that is, f ( x ) is not fixed. In both eq. (4.1) and eq. (4.2), f ( x ) determines the object. The aim is to calculate the complex amplitude F(u) in the hologram plane, which is transformed into the complex amplitudef ( x ) in the reconstruction plane. In Fourier holography this transformation is a Fourier transform (FT) of f ( x ) ; that is,

5-

m

F(u) = FT[ f ( ~ )=]

f ( x ) exp [ - i2 nux] dx dy ,

(4.4)

m

where I = (u, u) and ux = ux + uy. Equation (4.4) describes the relationship between the object f ( x ) and its

12

DIGITAL HOLOGRAPHY

hologram

F(u)

- COMPUTER-GENERATED HOLOGRAMS

dyk

[I, I 4

reconstruction

f(x)

Fig. 3. Notations used to indicate the complex amplitudes in Fourier holography.

spectrum F(u), and the conditions of digital Fourier holography are determined by this interrelationship. Figure 3 shows the notations that are used.

4.1.1.1.Boundary conditions of spectrum and object From the limited extent of the hologram it follows that the size of F(u) must be limited with extent A F = (AFu,AF”); that is, A F < 00. A F = co will, in general, cause disturbances in the reconstruction. AF < 00 leads to restrictions of the object as follows: (1) For a complex object f ( x ) has to be bandlimited. (2) For an intensity object i(x)has to be bandlimited; that is, AZ < 00, where AZindicates the extent ofI(u) = FT[i(x)].Otherwise, nof(x) with A F < co and 1 f ( x )I = i(x) will exist. Furthermore, q ( x ) according to eq. (4.3)cannot, in general, be chosen freely because f ( x ) needs to be bandlimited (5 4.1.1.3). Another constraint is that the Fourier transform needs to be performed by a computer; that is, sampling is required. In the object plane, sampling of f ( x ) is not a problem because A F < 00. A sampled (discrete) version of f ( x ) is introduced as follows: f ( m ) : = f ( m 6 x , n6y) = f ( x ) comb(x, ax), with integers rn

=

comb(x, a)

(4.5)

(m, n), sampling distance 6x = (ax, by), and

-

co

6 ( x - aa) 6 ( y - fib) , a,,9= - a,

where a = (a, b), 6(x) is the Dirac delta function, and proportionality constant is disregarded. As long as 6x < AF;

and

6y < AF;

-

indicates that a

,

(4.7)

the sampling is correct according to the sampling theorem. Conditions (4.7)can be combined to give 6x < A F -

I

.

All vector equations should be interpreted in the form of components.

(4.8)

FROM RECONSTRUCTION SPACE TO HOLOGRAM PLANE

13

Sampling in the hologram plane causes problems. Because a bandlimited object f ( x ) has an unlimited extent A f = 00, aliasing appears in the reconstruction, the consequences of which are discussed later in 5 4.1.1.2 and Q 4.1.1.3. However, it is necessary to sample and in analogy to eq. (4.9, F(k):= F(kbu, 16u) = F(u) comb(u, 6u) ,

(4.9)

where k = (k, I ) and 6u = (6u, 6u). A computer is used to combine f(m) and F(k); that is, the number of points needs to be limited. Because AF < 00, the sampling leads to a finite number of points in the u plane, but in the x plane with Af = co there will generally be an unlimited number of points M = - 00, ..., 00 and n = - co,. .. , 00. Thus a further restriction of the object is necessary, which will require a limited amount of data; that is, m = - fM, . . ., fM - 1 with M = (M, N ) and 1 = (1, I). This limitation is not serious. In general, the range of the object information in digital holography is bounded, causing sinc-oscillations at its boundary (Gibb' s phenomenon). To formulate this further restriction of the object, an additional quantity Ax < 00 is introduced, where Ax is the finite extent of the discrete distribution f(m) in contrast to Af = 00. Thus, considering the constraints AF < 00 and Ax c 00, the sampled versions of F(u) and f ( x ) are connected by the discrete Fourier transform F(k) = f(m) exp [ - i2nkMm] , (4.10) m

with m, k = - ; M y .. ., i M - 1, M = (Ax/Gx,Ay/6y), and kM = (k/M,l / N ) . The connection between the sampling distances 6u and 6x is

(buy6 ~=)( A x - ' , A Y - ' )

=

{(Mb~)-',(N6y)-'}.

(4.11)

In digital holography there is no reason to choose 6x < A F - ' ; that is, to oversample in the x-plane. Hence, the object is sampled at its Nyquist rate 6x = A F - ' . To indicate this case, m, and M, are introduced; that is, f(mf)

= f ( x ) comb(x, AF- I ) ,

(4.12)

with m, = - i M , , ..., iM,- 1, where M, = (Ax/bx, Ay/6y) = ( A x AF,, AY AF"). The discrete Fourier transform (DFT) of f(mf) results in the corresponding discrete spectrum F(k,) with k, = - i M , , ..., iMf - 1. According to eq. (4.11), 6u = A x - ' is valid independent of 6x. Then in regard to Ax < 00 the spectrum F(u) is sampled with the largest possible distance 6u.

14

DIGITAL HOLOGRAPHY

- COMPUTER-GENERATED HOLOGRAMS

[I, I4

However, for various reasons (aliasing in reconstruction, coding, etc.) it is helpful to use bu < Ax- This is realized by embedding f(mf) in a MN data field matrix of zeros. The DFT will result in

'.

F(k) = F(u) comb(u, bu) ,

(4.13)

with k = - ; M y . . . , ; M - 1 and bu = ((M,/M)Ax- I , (N,/N)Ay- I ) ; that is, an oversampling in regard to Ax with M / M , in the u-direction and N/N, in the v-direction. In short, in case (1) Ax < co and (2) AP < co are secured an arbitrarily sampled version of the desired F(u) (that is, F(k)) is obtained by a DFT of f(m,), whereby the DFT is implemented by the FFT algorithm. In addition, Af = co will cause aliasing in the reconstruction. In (i 4.1.1.2and (i 4.1.1.3is shown how (1) and (2) can be fulfilled. In regard to the applications of the CGHs and their theoretical descriptions it is helpful to make a separation between continuous and discrete object distributions. For discrete objects it is sufficient to specify the object information in the samples of f(mf); that is, the rest of f ( x ) is free to choose. 4.1.1.2. Discrete object: boundary condition consideration A discrete object is given as a 2-D set of data in M,N, points. It is interpreted as a discrete version of a complex amplitude f ( x ) ; that is, as f(mf). Thus the complex amplitude f(x) is only defined in the sampling points. A possible choice of f ( x ) is (4.14)

or in abbreviated form

* sinc (x, bx) ,

f ( x ) = f(mf)

(4.15)

with sinc(x, a )

N

sin nx/a sin ny/b ~, nx/a ny/b

~

whereby a = (a, b). This means that f ( x ) is introduced as the sinc interpolation of the values f(m,). From the choice eq. (4.15) it follows that (1) Ax = (Mf6x,N,by) < co, and (2) AF = bx- < co ;that is, both conditions in (i 4.1.1.1are satisfied independently of the type of object. In particular for intensity objects the condition (2) is satisfied for arbitrary phase samples rp(m,) = arg[f(m,)], which as will be shown in 5 5, is important for some coding techniques.

'

1, 41

FROM RECONSTRUCHON SPACE TO HOLOGRAM PLANE

15

The actual procedure is to start from f(mf) to calculate F(k) and then perform the coding, the CGH production, and the optical reconstruction. In considering the effect of aliasing, we disregard the coding and production steps and assume that reconstruction follows directly after F(k). Then, according to eq. (4.13), f ( x ) is not reconstructed, but

FT-'[F(k)] = FT-'[F(u) comb(u, Su)] =f(x)

* comb(x, 6u-I) ,

(4.16)

that is, a repetition occurs. Thus Af = 00 causes a superposition (aliasing). However, in the case Su- = (a Sx, /.? Sy), where a and /.? are integers, the discrete distribution f(mf) is contained undisturbed in f ( x ) because the zeros of the sinc functions of eq. (4.14) are located in the sampling points. In practice, a = M , and /.? = Nf are chosen; that is, Su- = ( M , Sx, Nf Sy), or multiples thereof. The values in the sampling points are then correct and aliasing is not disturbing. For intensity objects, q(m,) is completely free to choose. Introduction of a random phase causes wild fluctuations (speckles)between the sampling points, which in the case of a discrete object, does not disturb the information. A commonly used procedure to suppress the fluctuation between the sampling points is to repeat the CGH with a period equal to A F (LESEM, HIRSCHand JORDAN [ 19681). In this procedure spots corresponding to the sampling points If(mf)1 are reconstructed. In fig. 4 a typical example of such a reconstruction is shown.

'

'

Fig. 4. Optical reconstruction of a 16 x 16 times repeated graylevel Fourier CGH.Only the intensities in the sampling points are reconstructed. Parameters:M = 256; M,= 64; sampling distance in hologram = 5 pm; size of final hologram = (20 mm)'.

16

DIGITAL HOLOGRAPHY

- COMPUTER-GENERATED

HOLOGRAMS

[I, I 4

A further advantage of the hologram repetition is that the space bandwidth product of the hologram will increase without increasing the size of the FFT. Thus for discrete objects a repetition of the CGH is in general advantageous.

4.1.1.3. Continuous object: boundary condition consideration A continuous object is given as a 2-D distribution of data. The condition (1) A x < 00 needs to be fulfilled;that is, the object distribution should possess this property. In digital holography this is not a significant limitation. To fulfill condition (2) A P < 00 for a complex amplitude object, f ( x ) should be bandlimited or be made bandlimited by a low-pass process. For an intensity object i(x) should be bandlimited or be made bandlimited by a low-pass process; that is, AZ < 00. It is not a simple procedure from i(x) to find a bandlimited complex amplitude. For example, is not unconditionally bandlimited (e.g. i(x) = cos’x + f ( x ) = I cos x I ). In this case the relationshipbetween a bandlimited intensity and the correspondingbandlimited I = i(x) is of fundamental complex amplitude satisfying the property )@fI importance. In the following discussion we consider a complex amplitude corresponding to the intensity object. Proceeding from i(x) with AZ< 00, an f ( x ) with If(x)I = i(x) and A F < 00 is wanted. This problem is formulated in optics in general by ONEILLand WALTHER[ 19631 and WALTHER[ 19631. The main interest in this formulation of the problem may be found in connection with treatment of the “phase retrieval” problem (e.g. BRUCK and SODIN[1979], GARDENand BATES [1982], FRIGHTand BATES[1982], BATES [ 19821, DEIGHTON,SCIVIERand FIDDY[ 19851. FERWERDA [ 19861, ROOT[ 19871). The differences between the problem in “phase retrieval” and in digital holography are as follows: (1) In “phase retrieval” the existence of bandlimited f ( x ) is secured, but not in digital holography. (2) Uniqueness of f ( x ) is desired in “phase retrieval”, but not in digital holography. In digital holography the phase is used, for example, to smooth the spectrum IF(u)l’ (I 5.2.1.2.2). These relations were examined in digital holography by WYROWSKI and BRYNGDAHL[ 19881. They found that the existence of a bandlimited f ( x ) is secured in practice and that the bandlimit off(x) is A P = iAZ; that is, the phase carries object information. Figure 5 illustrates how the phase can be used to advantage to increase the information content of the hologram. The relation between f ( x ) and i(x)is ill-conditioned(BARAKAT and NEWSAM

Jlc.>

’

FROM RECONSTRUCTION SPACE TO HOLOGRAM PLANE

17

Fig. 5. Simulations of reconstructed intensities. In (a) an iterated phase and in (b) a constant phase was applied. Although the bandlimit is the same in (a) and (b), the resolutionin (a) is higher because there the phase is also used to carry information.

[ 19851). This ill-conditioning was used by WYROWSKIand BRYNGDAHL [ 19881 to show that to a limited degree the phase q ( x ) can be used as a free parameter (cf. fig. 25 in $ 5.2.1.2.2).

A method to calculate a bandlimited complex amplitudef ( x ) of the intensity object i(x) uses the iterative Fourier transform algorithm (IFTA), which was introduced into digital holography by HIRSCH,JORDANand LESEM[ 19711, and by GERCHBERG and SAXTON[ 19721 in relation to the “phase retrieval” problem. The IFTA consists of a sequence of Fourier transforms with restrictions in the Fourier as well as the space domain. A general version of the IFTA is presented in $ 5.2.1.4.1 (cf. fig. 35). Analysis and modifications of the IFTA can be found for example in publications by LIU and GALLAGHER [ 19741, FIENUP [ 19821, and WYROWSKI [ 1989bl. The IFTA was considered as a projection algorithm, for example, by GUBIN,POLYAKand RAIK [ 19671, YOULA[ 19781 and LEVIand STARK[ 1983, 19841. The IFTA has been widely used in digital holography (see, for example, and LIU [ 19731, FIENUP HIRSCH,JORDANand LESEM[ 19711, GALLAGHER [ 19801, MAITand BRENNER [ 19871, WYROWSKI [ 1990a,b]), and in particular to handle the influence of speckles (ALLEBACH and LIU [ 19751, ALLEBACH, GALLAGHER and LIU [ 19761, WYROWSKI and BRYNGDAHL [ 1988, 19891). The application of the IFTA to calculate a bandlimited complex amplitude f ( x ) in digital holography generally leads to a stagnation of the calculating [ 19881 suggested a method to avoid process. WYROWSKI and BRYNGDAHL stagnation of the iteration. They showed that a bandlimited f ( x ) can be found and it is possible to calculate F(k) by means of the DFT.

18

DIGITAL HOLOGRAPHY

- COMPUTER-GENERATED HOLOGRAMS

[I, I 4

We now turn to the problem of aliasing. In comparison with $4.1.1.2, the complex amplitude in the reconstruction plane is FT-'[F(R)]

=f(x)

* comb(x, 6 u - ' ) ,

(4.16)

and the corresponding intensity distribution is If(x) * comb (x, 6u - I ) I '. Because of Af = 00, aliasing is unavoidable. However, the aliasing can be made unimportant by a sufficient fine sampling in the u-plane. In practice 6u = +Ax- turns out to be sufficient; that is, a four times oversampling of the spectrum. WYROWSKI and BRYNGDAHL [ 19891have presented optical reconstructions of continuous objects. An example of a binary object is reproduced in fig. 6. For comparison the result of the use of a random phase with AF = 00 is also shown. [ 1987al suggested that Furthermore, WYROWSKI, HAUCKand BRYNGDAHL it is possible to obtain large computer holograms of continuous objects. This method consists of a compromise between the use of the iterative method (WYROWSKI and BRYNGDAHL[ 19881) and that used to achieve large holograms by repetition. In fig. 7, a reconstruction is shown using this technique [ 19891). This reconstruction should be com(WYROWSKI and BRYNGDAHL pared with that in fig. 4.

'

Fig. 6. Speckle-free optical reconstruction of a graylevel hologram of a binary object. The result using (a) an iterated phase to obtain AF = :AI is compared with (b) a random phase resulting in AF = co and speckles. Parameters:M = 256;M, = 64; sampling distance in hologram = 5 Fm; hologram size = (1.3 mm)2.

1, I 41

19

FROM RECONSTRUCTION SPACE TO HOLOGRAM PLANE

Fig. 7. Optical reconstructionof graylevel CGH calculated by a repetition and a phase manipulation technique in the hologram plane. The result using (a) an iterated phase is compared with (b) a random phase. Parameters:M = 256; M,= 64; a modified hologram repetition = 16 x 16; sampling distance in hologram = 5 pm; hologram size = (20 mm)'. reconstruction

hologram n

f (X")

+i+f ( x )

Fig. 8. Notations used to indicate the complex amplitudes in Fresnel holography.

4.1.2. Fresnel transform relationship between object and hologram

In this section the aim is to form a light distribution of a desired reconstructed image (object) at a finite distance z from the hologram compared to z = cc in Q 4.1.1. By appropriate illumination of the hologram a desired 2-D light distribution f ( x ) can be formed in the reconstruction plane. The Fresnel conditions are assumed to be fulfilled and, thus, the Fresnel approximations are valid. The complex amplitude in the hologram plane then can be related tof(x) by a Fresnel transform (FRT) for plane wave illumination; that is, Ax,) = FRT[f(x)l3

(4.17)

where ) ( x u ) denotes the complex amplitude and xu = (xu, y,) denotes the coordinates in the hologram plane. In fig. 8 the notations used are shown. In digital holography the wave propagation is simulated and calculated by means of a computer, using the FFT algorithm as described in Q 4.1.1. In several respects the FFT is the essential tool in digital holography. Because the

20

DIGITAL HOLOGRAPHY

- COMPUTER-GENERATED

HOLOGRAMS

[I, 5 4

FFT is such a valuable aid, it is of great interest to apply it in Fresnel holography as well. The FRT then should be expressed as a FT. The relationship is (4.18)

This expression is equivalent to the following steps: superposition of a quadratic phase onto calculation of the Fourier transform, and once more, superposition of a quadratic phase, this time onto the calculated spectrum. These steps are easily accomplished with a computer. However, as discussed in Q 4.1.1, it is necessary to sample in order to calculate the FT by a computer; that is, to perform a DFT. If the function to be sampled is not bandlimited, aliasing will occur. In Fresnel holography the conditions in the object plane x are that the phasefactor exp [i ( z / l z )I x I ’1 is not bandlimited and therefore, in general, the product f ( x ) exp [i(n/lz) 1x1’] is not bandlimited. (For an exception see the discussion below of intensity objects.) Aliasing errors then will appear in the hologram plane xu. The smaller z is, the worse the errors will be; that is, in general, !(xu) cannot be calculated correctly. The conditions in the hologram plane xu are that a sampled version of f (xu) is obtained by computer calculation. Generally this distribution is also not bandlimited (see the discussion above). Aliasing will then appear in the reconstruction plane; that is, the individual diffraction orders will superimpose in the reconstruction plane as in digital Fourier holography (cf. fig. 50c). The effect increases with decreasing z. For intensity objects i(x) aliasing can be avoided in the hologram plane (see the conditions in the object plane above); that is, f ( x ) exp [i(n/lz) I x I 2] can be made bandlimited. Because the phase arg[f(n)] may be chosen freely, so Can

s(~),

(4.19)

In general, this complex amplitude is not bandlimited. However, since f ( x ) is multiplied by exp[i(n/lz) IxI’], only the Fourier transform of exp[icp(x)] has to be calculated, which is identical to the situation discussed in Q 4.1.1 (cf. eq. (4.3)). The result is that the aliasing is eliminated from the hologram plane. Thus the calculation of !(xu) reduces to the

J1(x)

1,s 41

FROM RECONSTRUCXION SPACE TO HOLOGRAM PLANE

21

synthesis of the FRT pair

[n: 1

)(xu) = exp i - Ixu12 FT{Ji(x)exp[icp(x)]}

(4.20a)

(4.20b) Specifically this means that the Fourier transform of exp [icp(x)] should be calculated as described in Q 4.1.1,and then a quadratic phase is superposed onto the resulting spectrum. The result is the complex amplitude ! ( x u ) in the hologram plane. The desired intensity If (x) I = i(x) is then produced in the reconstruction plane. In addition to the elimination of the aliasing in the hologram plane, the procedure described has the advantage that the calculation effort is reduced because it is not necessary to superpose a quadratic phase to the complex amplitude of the x-plane. In short, for intensity objects everything remains as it has been described in Q 4.1.1, except for additional aliasing in the reconstruction plane.

’

4.1.3. Object or image of object in hologram plane In this situation the complex amplitude of the object or the image of the object in the hologram plane is the object itself, and thus no computation is necessary. We will indicate the object in image holography by F(u) in accordance with the nomenclature used in Q 5.4; F(u) may be given as an analytic or sampled function. In case F(u)is given in analytic form, the hologram structure can be analytically calculated (cf. 5 5.4).

4.2. 3-D INTENSITY DISTRIBUTIONS: MODEL OF 3-D OBJECT

Section 4.1 examined 2-D light distributions, which were produced by illuminating a digital hologram. A distinction was made between complex amplitude and intensity distributions. In this section the formation of a 3-D intensity distribution by illumination of a digital hologram is of primary concern. In three dimensionsmany different

22

DIGITAL HOLOGRAPHY

- COMPUTER-GENERATED HOLOGRAMS

[I, B 4

planes z are present in depth, not only z = co or z = 0. Thus a Fourier or image plane hologram will not suffice,but one is necessary which by nature comprises many planes in depth; that is, the Fresnel hologram for 3-D. The Fresnel conditions are assumed to be fulfilled so that the Fresnel approximations are valid. In many synthetic imaging situations, such as computer graphics and computer tomography, where the result is represented as a 3-D set of data in computer form, it is desirable to display this pictorial information optically. This 3-D set of sampled data then should be interpreted as a 3-D intensity distribution i(x, y, z), which can be formed by illuminating a digital hologram. For 2-D intensity distributions the holographic solution has been mastered and treated extensively in theory as well as in practice. Therefore, reducing the 3-D situation to a 2-D situation is one conceivable procedure. It is important that the depth information contained in i(x, y , z) is approximately preserved by this reduction. Thus the 3-D problem is converted to many 2-D problems. Two major methods exist to carry out this reduction. In the first method the depth information is preserved as J calculated perspectives of i(x, y, z). The result is J 2-D intensity distributions i,(x), j = 1, .. .,J . The methods in Q 4.1 can be applied, proceeding from i,(x) to calculate the corresponding hologram. These holograms can be multiplexed. The final hologram with discrete parallax functions as a stereogram. The depth information appears as a purely binocular effect without accommodation information (KING,NOLLand BERRY[ 19701, YATAGAI[ 1974, 19761). In the second method a 3-D intensity distribution is formed. For light propagation the superposition principle is valid, and the 3-D distribution can be decomposed in independent 1-D, 2-D or 3-D distributions. WATERS[ 19661 suggested a direct approach to display points at well-defined locations ( x , y, z) in space. The hologram followed a point-by-point image synthesis. An efficient realization is to decompose the 3-D intensity distribution into J planes in depth, as shown in fig. 9; that is, the simplified intensity

(4.21) can be considered and each 2-D intensity distribution i,(x) can be treated separately (WATERS[ 19681, LESEM,HIRSCHand JORDAN [ 19691, BROWN and LOHMANN[ 19691). The jth plane is located in zj and the 3-D problem is decomposed into J 2-D situations with the intensity $(x) in z = zj.

1,5 41

23

FROM RECONSTRUmION SPACE TO HOLOGRAM PLANE

hologram

30-data

:I z=o

Fig. 9. Illustration of how a 3-D set of data is decomposed into planes, 6r apart, parallel to the hologram. For notations see text.

The method in 4.1.2 for intensity objects can be applied (see eq. (4.20)). The complex amplitude in the hologram plane is (4.22) where (4.23) Everything is satisfactory as long as the conditions discussed in 5 4.1.2 are considered: in the hologram plane the complex amplitude can be calculated according to eqs. (4.22) and (4.23). After performing a coding procedure and the CGH production, a hologram is obtained from which a 3-D intensity distribution i ( x , y , z ) according to eq. (4.21) can be reconstructed. The depth information is retrievable from continuous parallax and accommodation effects. Due to the assumption of the model used, no superposition effects of the light of the different planes in depth and no blocking (hidden lines) are considered within the 3-D distribution. Suggestionshave been made concerning how these effects may be incorporated in the model, for example, the “ping pong” idea (ICHIOKA, IZUMI and SUZUKI[ 19711, DALLAS[ 19801). In general, the method described in eq. (4.2 1) requires an immense computing effort because of the following principal reasons :

24

DIGITAL HOLOGRAPHY

- COMPUTER-GENERATED HOLOGRAMS

[I, 8 4

(1) In order to use the parallax effect, the size of the hologram needs to be relatively large. In digital holography calculation complications usually limit the achievable space bandwidth product of the CGH. In Fourier holography a common solution to obtain a large hologram is to repeat subholograms (0 4.1.1). A hologram distribution with a small space bandwidth product is calculated, and in the production step it is simply repeated to form a large hologram. In Fresnel holography this simple repetition is not possible because the repetition has to be implemented before superposing the quadratic phase (cf. eq. (4.23)). Then the final CGH can be produced. This procedure requires a computer effort that is too high to be of practical value. (2) Even in the case of not-too-large hologram the calculation effort is high because J 2-D FFTs must be calculated. Modifications of the method have been suggested, in which some of the reasons mentioned above (1 and 2) are directly treated (I 4.2.l), and others in which special properties of the set of data were considered (§ 4.2.2). 4.2.1. Reduction of computation efort: removal of vertical parallax As just mentioned, a high calculation effort is necessary to superpose a quadratic phase onto a large hologram. One possibility of reducing this effort is to eliminate the frequently unimportant vertical parallax. Then the hologram needs only to be repeated in the horizontal direction. Because the utilized height of the hologram is too small to take advantage of the vertical parallax (according to assumption), the y,-dependence of the phase factor in eq. (4.23) can be removed; that is, to leave only exp [i(r/Az,) I’,. 1. Thus reduction of the computation effort is possible because the period in the y,-direction is equal to the height of the subhologram. It is only necessary to calculate a hologram stripe of the height of a subhologram. A repetition in the y-direction is performed to obtain a sufficiently extended hologram that allows visual observation. The hologram will consist of stripes (COLLIER,BURCKHARDT and LIN [ 19711, DALLAS[ 19801). An astigmatic error appears in the reconstruction, which is reduced by increasing zi (LESEBERG [ 19891). In fig. 10 optical reconstructions are shown from a hologram produced by this technique. To reduce the computation effort of the 2-D FFT even more, 1-D FFTs can be performed line by line. The parallax in the y-direction is then removed, and a strong astigmatism is introduced. This technique is very similar to the “rainbow” hologram process by BENTON[ 19691. LESEBERG and BRYNGDAHL [ 19841 and LESEBERG [ 19861 demonstrated a synthetic version and found that

1.4 41

FROM RECONSTRUCTION SPACE TO HOLOGRAM PLANE

25

Fig. 10. Optical reconstructions obtained by illuminating an area of 5 mm diameter of a CGH in which (a) and (b) resulted from illuminated areas 160 mm apart. Parameters: data = 2562 samples; height = 256 samples (10 pm sampling distance) and length of hologram = 64 spectra of 512 samples (5 pm sampling distance); ratio between distances of farthest to nearest ring from Fourier plane = 6 : 1. (Courtesy of LESEBERG[1989].)

in visual display situations the disadvantage of removing the vertical parallax is not severe and can even be of some advantage. A grating structure can be incorporated in the hologram and the vertical direction can be used to accommodate its dispersion instead of the vertical parallax. In this way the hologram can be reconstructed using white light. 4.2.2. Reduction of computation effort: utilization of data characteristics When the 3-D set of data exist for planes inclined to each other, for example, a cube, a decomposition in planes no longer parallel to the hologram may be advantageous in some situations (LESEBERG and FRERE[ 19881). The 3-D problem then is reduced to the treatment of 2-D intensity distributions in tilted planes. To calculate the Fresnel transform between planes inclined in respect to each other, it is necessary to perform an additional coordinate transformation (GANCI[ 19811, PATORSKI [ 19831). The complex amplitude in the hologram plane correspondingto an intensity distribution ~Jx)in a plane tilted at an angle y (see fig. 11) can be expressed as

where CT, indicates a coordinate transformation which depends on y (LESE-

26

DIGITAL HOLOGRAPHY

- COMPUTER-GENERATED HOLOGRAMS

[I. B 4

z=o hologram

data

Fig. 11. Illustration of the geometry for calculation of the complex amplitude in the hologram plane from data if(x)located in a plane zf tilted at an angle y.

and FRERE[ 19881). Optical reconstructions in planes inclined to the hologram are shown in fig. 12. Equation (4.24) is theoretically only valid for infinitely extended distributions i,(x). In case the 3-D set of data only consists of line segments inclined to each and other, it may be useful to decompose the data into lines (FRBRE,LESEBERG BRYNGDAHL [ 19861). The Fresnel transformation of the line segment intenBERG

Fig. 12. Optical reconstructions of two surfaces of a cube 90" apart. In (a) y = - 10" and in (b) 80". The reconstructions were recorded by placing photographic films at these angles relative to the hologram. Parameters: data = 128' samples; hologram size = 256' samples repeated 8 x 8 times; sampling distance in hologram = 5 pm. (Courtesy of LESEEERGand F R ~ R E [ 19881.)

I , § 41

FROM RECONSTRUCTION SPACE TO HOLOGRAM PLANE

27

sities can be analytically evaluated using the “saddle point” method, or it can be obtained by means of a wave propagation consideration; that is, as an analytic solution of the wave equation (LESEBERGand FRERE[1987], LESEBERG[ 19871). For each line segment with an inclination y to the hologram plane the complex amplitude $(xu) in the hologram can be analytically expressed in the form (4.25)

where zj is the point of intersection between the line segment and the optical axis. In case the line segment is also displaced from the optical axis, an additional linear phase is introduced.

Fig. 13. Optical reconstructions of a cube from a hologram of its edges (all inclined relative to the hologram). In (a) is the hologram of 12 line foci of finite lengths and in (b) and (c) are reconstructions in two different planes parallel to the hologram. Parameters: hologram size = 4096* samples; sampling distance in hologram = 5 pm. (Courtesy of LESEBERG [1987].)

28

DIGITAL HOLOGRAPHY

- COMPUTER-GENERATED HOLOGRAMS

[I, 8 5

This consideration is only valid for lines of infinite length. The length of the line segment is controlled by the size of its contribution in the hologram and cannot be too short. In fig. 13 a hologram and reconstructions of a cube are shown.

0 5. Coding Procedures in Digital Holography 5.1. FUNDAMENTAL PRINCIPLES

Methods and procedures to determine the complex amplitude F(u) or ! ( x u ) in the hologram plane were described in the previous section. (In the following of $ 5.1, F(u) stands for F(u) as well as ) ( x u ) . ) From this distribu-

tion we continue with the next step toward the realization of a CGH; that is, to decide upon the configuration and structure of the hologram. We will now proceed from the complex amplitude F(u), which is usually given in the discrete form F(k) = F(u)comb(u, bu), to calculate the CGH pattern. We recall that, in general, F(u) is a complex valued function of the independent variables u = (u, u) and possesses the following properties: Its extent is finite; that is, A F < 00, and it can be transformed into the desired reconstruction (object) f ( x ) by T - [F(u)l

= f ( x )9

(5.1)

where T - ' indicates the reconstruction operator, which, for example, is a Fourier or Fresnel transform, or a filtering operation. Thus F(u) contains a transformed version of the information of the object f ( x ) . The aim of the production of the CGH is that the complex amplitude F(u) will be generated by illuminating the hologram; that is, R(u) H(u) = F(u)

9

(5.2)

where R(u)is the illuminatingwave and H(u) is the hologram distribution. H(u) is directly related to the complex refractive index of the hologram material. This is an approximation that is valid for thin holograms. In the following illumination with a plane wave, R(u) = 1 is assumed (for an exception, see the end of 8 5.2.1). Then H(u) = F(u)

9

(5.3)

and thus the complex amplitude F(u)has to be recorded as a hologram. There are very few materials that can directly influence the amplitude as well as the

29

CODING PROCEDURES IN DIGITAL HOLOGRAPHY

calculation of

CGH in hologram

Fig. 14. Block diagram showing the different steps in the realization of a CGH.

phase. Therefore, we rely on a coding procedure, which can transform F(u) into a form G(u) that can be materialized; that is,

F(u)

coding

G(u)

materialization

CGH .

(5.4)

It is necessary to use a coding procedure when F(u) cannot be materialized. In fig. 14 the steps from the desired reconstruction f ( x ) to the final CGH are indicated. A straightforward materialization of a CGH is the referenceless on-axis complex hologram (ROACH) which has a complex-amplitude transmission proportional to F,(u) = F(k)* rect(u, Su) ,

(5.5a)

with rect (u, a)

N

1 , lul
A

(5.5b)

Iul
where (I = (a, b). To control amplitude as well as phase, the ROACH idea is to use a multiemulsion color film (CHU, FIENUPand GOODMAN[ 19731). The different layers of the film are exposed independently, using light of different colors. In reconstruction with monochromatic light, one layer will absorb and the other two layers will be transparent but introduce phase shifts caused by variation of thickness and refractive index. Illuminating the ROACH with a wave R(u) = 1 will, after a Fourier transformation, result in

=

{ f(n)* comb(x, Su- ’)} sinc(x, Su-

I).

(5.6)

It is generally complicated to influence the amplitude as well as the phase of an illumination wave directly. Therefore, the application of techniques and

30

DIGITAL HOLOGRAPHY

- COMPUTER-GENERATED HOLOGRAMS

[I, § 5

materials which do not rely on complex valued quantities is the standard procedure to produce CGHs. We may group these methods according to the parameter of the illuminating wave being influenced: amplitude (amplitude hologram); phase (phase hologram); quantized amplitude or quantized phase. We will now concentrate on the first step of procedure (5.4);that is, the coding, which in digital holography received much attention because of all the possibilities that synthetic procedures have to offer. The purpose of the coding step is to transfer the complex-valued distribution F(u) into a new distribution G(u), the values of which are being limited to real and positive or complex located on the unit circle, and additional quantization in these cases. The transfer from F(u) to G(u)will restrict the set of values of F(u), and thus one has to make sure that the object distribution f ( x ) is included in the distribution T- “G(4l

=g(4

;

(5.7)

that is, the transformed information about the object must be contained in G(u). In order to carry out the coding, there should be some kind of freedom. An accurate coding is not possible under the requirement that F(u) is the unique complex amplitude in the hologram plane which contains the object information. The distribution in the hologram plane should incorporate freedoms ; that is, it should in principle be possible to transfer F(u) into G(u) without destroying the object information. In conclusion we may state that to code the complex amplitude F(u) successfully, it is necessary (1) to find and formulate the existing freedoms, and (2) to develop a coding scheme which is able to use these freedoms to convert F(u) to G(u) without or with only minor deterioration of the original object information. Coding in Fourier holography situations dominates in the literature, which is also reflected in the following section. 5.2. CODING IN FOURIER HOLOGRAPHY

In $ 4we examined procedures to determine the hologram distribution from the desired object f ( x ) , and we now proceed from our knowledge of

and its discrete form

1,s 51

31

CODING PROCEDURES IN DIGITAL HOLOGRAPHY

with k and m = - i M , . ..,SM - 1. F(k) is the sampled version of F(u); that is, F(k)= F(u) comb(u, 6u) with 6u < Ax-'. When 6u = Ax- where Ax = (Ax, Ay) indicates the extent of the discrete version of the object, we use - 1. This is in the indexed vectors m, and k, with mf,k, = - SM,, ..., contrast to the oversampled version (6u c Ax- ') of F(u), where m and k are used. The coding procedure consists of finding a Fourier pair

',

FT

(5.10)

,

where G(u) fulfils the boundary conditions and the object f ( x ) is a portion of g(x). This means that within an object window IF with extent Ax, equal to that of the discrete object; that is, x0 - $AX < x < x0 +

AX,

(5.11)

g(x) and f ( x ) are related as follows:

g(x)

=

af(x - xo) , x E IF

(5.12)

for a complex object and Ig(x)12 =

a21f(x - xo)12,

XEIF

(5.13)

for an intensity object. In fig. 15 the relative sizes and locations of the discrete versions of f ( x ) and g(x) are shown together with the notations used. In eqs.

M6x

Fig. 15. Illustration of introduced fields and their proportions together with notations used. The discrete complex amplitudeg(m) is given within the total field and the discrete objectf(m), within the object window IF.

32

DIGITAL HOLOGRAPHY

- COMPUTER-GENERATED HOLOGRAMS

[I, B 5

(5.12) and (5.13) the object is not completely specified; a scale factor a is free to choose. The diffraction efficiency of the hologram is dependent on a. Furthermore, the relationship between g(x) and f ( x ) is not unique because of the limited extent Ax c co of f ( x ) (eqs. (5.12) and (5.13)) and the undetermined phase of an intensity object (eq. (5.13)). Thus freedoms exist, and these are the desired freedoms of F(u)expressed in the object plane, viz. formulated indirectly as freedoms of g(x). These freedoms include the following: (1) Freedom of amplitude outside the object window. From the limited extent of the object follows that g(x) can be freely chosen for x-values outside IF, as indicated in fig. 16a. (2) Freedom of amplitude in intermediate points inside the object window. When the object is discrete, there is a connection between the object and the complex amplitude g(x) only in the sampling points themselves. Figure 16b illustrates how g(x) can be freely chosen outside the sampling points x within (3) Freedom of phase. For intensity objects the phase can be freely chosen within IF as indicated in fig. 16c; that is, the values q ( x ) = arg[f(x)] can be chosen freely. It may be observed that for continuous intensity signals the phase freedom is restricted, because the phase superposed onto If(x)I must secure the bandlimit condition AF < 00 (5 4.1.1.3). These are the freedoms which can be used to code F(u). To implement them in practice, restrictions will occur depending on the application; for example, the amplitude freedom is restricted in filtering situations to reserve space for correlations. The more the preceding freedoms can be used, the more successfull the coding can be carried out. As soon as the freedoms are reduced, more complicated coding methods are necessary. In the four cases of objects mentioned in 5 4.1.1, the following freedoms are theoretically applicable: discrete complex amplitude: (1) and (2); continuous complex amplitude: (1); discrete intensity: (l), (2) and (3); continuous intensity: (1) and (3) with

Fig. 16. Illustration to indicate different areas (shaded) containing certain freedoms that can be used in the coding process.

1,s 51

33

CODING PROCEDURES IN DIGITAL HOLOGRAPHY

restriction. The scale factor a (cf. eqs. (5.12) and (5.13)) can be chosen freely in any situation. Large values of a are preferable to achieve large diffraction efficiencies. Most possibilities of performing the coding exist for discrete intensity objects because all three types of freedom exist. Examination of this case is also prominent in the literature. Amplitude freedom outside the object can be introduced in all four cases. Thus coding procedures which only use this freedom are the most common. In summary we may repeat that coding in Fourier holography means the synthesis of a Fourier pair (5.10)

where G(u) fulfils the following conditions in amplitude holography: G(u) is real, positive, and possibly quantized, or in phase holography G(u) is complex with values on the unit circle and possibly quantized. Furthermore, g(x) is If ( x - xo)I 2, related to the object by means of g(x) f ( x - xo) or Ig(x) I respectively, with x belonging to an object window IF. A coding can be realized by using the three freedoms listed earlier. There exist numerous possibilities to combine the possible freedoms in the choice of the distribution within and outside the object in the reconstruction plane with the various constraints which exist in and may be imposed on the

-

N

TABLE 1 Amplitude holograms (§ 5.2.1):Coding schemes and methods which utilize certain combinations between freedoms in the object plane and constraints of the hologram.

Freedom applied in object plane

Constraints in hologram Amplitude Coding (5.2.1.1)

Point oriented (5.2.1.1) Cell oriented (5.2.1.1)

Restriction of range (5.2.1.2;5.2.1.3)

Parity sequence (5.2.1.3) Synthetic coeff. (5.2.1.3)

Quantization (5.2.1.4)

Pulse width (5.2.1.4.1(1)) Pulse density (5.2.1.4.1(2) and (3))

Phase

Amplitude and phase

Diffuser: discrete object (5.2.1.2.1); continuous object (5.2.1.2.2) Iterative pulse density: FT algorithm (5.2.1.4.1 (3)); simulatedannealing (5.2.1.4.1(3))

34

DIGITAL HOLOGRAPHY

- COMPUTER-GENERATED

[I. 5 5

HOLOGRAMS

TABLE 2 Phase holograms (5 5.2.2): Coding schemes and methods which utilize certain combinations between freedoms in the object plane and constraints of the hologram. Freedom applied in object plane

Constraints in hologram Amplitude

Phase

Amplitude and phase

Coding

Bleached ampl. hologr.

(5.2.2.1; 5.2.2.2)

(5.2.2.1)

Iterative FT algorithm (5.2.2.2 (B))

(5.2.2.2 (C))

Phase carrier (5.2.2.2 (A)) Complex error diffusion (5.2.2.2 (A)) Quantization (5.2.3)

Object location (5.2.3 (A)) Complex error diffusion (5.2.3 (A))

Iterative FT algorithm

Iterative FT algorithm (5.2.3 (B))

Simulated annealing (5.2.3 (B))

distribution of the hologram. Associated with these combinations different coding schemes have been and can be conceptualized. In tables 1 and 2 schematic overviews are given of most of the coding schemes described in the following of 5 5.2. The coding and quantization restraints are listed separately. In the amplitude freedom is included the amplitude (1) outside as well as (2) in intermediate points inside the object window. 5.2.1. Amplitude hologram

The objective of the coding is to synthesize a Fourier pair, whereby G(u) is real and positive. To achieve this, g w

= g*(

-4

(5.14)

is required. In order for g(x) to contain the object in undisturbed form, g(4

=f(x

- xo) + f*[- (x + x0)I

(5.15)

is introduced, which means that the application of the amplitude freedom (addition of twin image) is used to fulfil the condition of a real G(u).To separate the object and its twin image spatially, xo 3 $Ax and/or yo 3 f A y has to be fulfilled. The corresponding real-valued distribution G(u) is

G(u) = 2 I F(u)[ COS [ 2 auxo - $(u)] ;

(5.16)

1, ~i 51

CODING PROCEDURES IN DIGITAL HOLOGRAPHY

35

Fig. 17. Illustration of a test object f(x) with extent Ax and amplitude IF(u)I of its spectrum.

that is, G(u) is an amplitude { IF(u)I} and phase {$J(u)}modulated cosine carrier (analog to this is the result of the introduction of a reference wave in optical holography). Figures 17 and 18 illustrate this case. In addition, G(u) > 0 is required. This is achieved by the introduction of a bias function B(u), which results in G(u) = 21F(u)l C O S [ ~ ~ M - $J(u)] X , + B(u).

(5.17)

Examples of different bias functions can be found in the literature (BURCH [ 19671), B(u) = max{2IF(u)l cos[2nux - $J(u)]}= const.,

(5.18)

and (HUAIVGand PRASADA[ 1966]), B(u) = 2 I F(u)I

9

(5.19)

Fig. 18. Illustration of the Fourier pair, as stated in eqs. (5.16) and (5.15). To the right is a magnified portion of the area indicated with a rectangle in the spectrum G(u). Parameter: x, = 0.6Ax.

36

DIGITAL HOLOGRAPHY

- COMPUTER-GENERATED

HOLOGRAMS

[I, § 5

and the holographic bias

B(u) = 1 + IF(u)12.

(5.20)

The use of the amplitude freedom in addition to a twin image to introduce the Fourier transform of B(u) to code F(u) leads to the Fourier pair G(u) = 2 I F(u)I cos [2 nUx0 - @ ( U ) ]

(5.21a)

+ B(u)

5 FT g(x) = f ( x - x,)

+ f*[- (x + x0)l + b(x)

*

(5.21b)

-

B(u) should be chosen in such a way that b(x) does not disturb the object. For a constant bias this is trivial because then b(x) 6(x). However, all varying bias functions B(u)result in conditions that should be taken into account when chosing x,; that is, in order to secure spatial separation of the object and b(x), x, cannot be too small. A large x, requires a large amount of data in digital holography. Thus the possibility of using a constant bias in digital holography is attractive because then the amount of data can be kept smaller. In conclusion may be stated that coding in amplitude holography using the amplitude freedom is sufficient and necessary. The described coding is valid because G(u)is equal to the hologram distribution H(u), which is the case for R(u) = 1. The point symmetry of g(x) in the reconstruction is produced by R(u) = 1. On the other hand, for R(u) # 1 eq. (5.2) indicated that G(u) no longer has to be real and positive because only H(u) needs to fulfill the boundary conditions of being real and positive. Thus the point symmetry does not have to exist and the coding can be modified. [ 1983, 19851 used a conical and a NEUGEBAUER, HAUCKand BRYNGDAHL helical wave for R(u). The hologram carrier then is of polar geometry, and no point symmetry exists in the reconstruction. In what follows we assume R(u) = 1. In practice a modification of the Fourier pair (5.21) is necessary. It is generally not possible to use the continuous distribution G(u)of eq. (5.21a) as hologram distribution because it would then be necessary to have infinite storage space and an output device of infinitely high resolving power (an exception is shown in Q 5.4). Thus it is only possible to use the discrete distribution F(k) and not F(u) as a starting point for the coding procedure. 5.2.1.1. Point- and cell-oriented coding Proceeding from the discrete version of F(u); that is, F(k), it is possible to synthesize the discrete Fourier pair analog to the continuous formulation as

1, B 51

CODING PROCEDURES IN DIGITAL HOLOGRAPHY

follows: G ( k ) = 21F(k)I cos[2nkMm0 - 9(k)l + B(k)

31

(5.22a)

with kM = (k/M,l/M)). G ( k ) is stored in memory in MN points. The conversion to a continuous distribution is performed by hardware. Ideally we obtain G,(u) = G(k)* rect [u, Su] = { G(u)comb [u, Su]} * rect [u, Su] . (5.23)

For eq. (5.2) to be a valid approximation for thin holograms Su > 1is required. The dimension of the transversal structure in Q 5 is assumed larger than the wave length 1of the light. From eq. (5.23) it follows that the reconstruction is not g(x) given by eq. (5.21b), but gr(x)= [g(x) * comb(x, Su-

I)]

sinc(x, Su- ')

.

(5.24)

This means that the modification results in a reconstruction g(x) which is repeated and globally modulated by a sinc function. From this it follows that, in addition to the condition xo 3 ;Ax

or/and yo 2 4Ay ,

(5.25)

we must require that Su-'22x0+Ax,

(5.26)

so that the repetition does not introduce overlap. This case is illustrated in fig. 19.The sinc-modulation in eq. (5.24) can be compensated for in advance. We may conclude that for a constant bias the smallest choice of x,; that is, x, = (0, fAy) or x, = (;Ax, 0), results in Su- 3 (Ax, 2Ay) or Su- 3 AX, Ay). Thus the spectrum should be oversampled by a factor of 2 in at least one direction. This type of coding is called point oriented (DALLAS[1980]) and is characterized by

'

(5.27) that is, a coding step resulting in a discrete hologram distribution which is convolved with a rect function (the pixels of the CGH). The actual procedure used to code a point-oriented hologram is as follows:

-

38

[I, B 5

DIGITAL HOLOGRAPHY COMPUTER-GENERATED HOLOGRAMS

Fig. 19. Illustration of the Fourier pair G,(u)-g,(x) for the point-oriented coding, as stated in eqs. (5.23) and (5.24) (without bias) using the test object in fig. 17. The cutout shown to the right is a magnification of the corresponding area indicated, with a rectangle in the spectrum G,(u) covering exactly four cells. Parameters: x,, = 0.6Ax; repetition period Su- = 4Ax.

'

(1) To embed the discrete object distribution f(mJ with an offset mo and m - - 1, .~. .,iMf - 1, in an MN data field matrix of zeros. Thus, g(m) = f(m - mo) , ( m k n b ) E

,

(5.28) (5.29)

These discrete versions have to satisfy conditions (5.25) and (5.26). From this follows m 0 > i M f and/or n , > $ N ,

(5.30)

and

M

> 2m0 + M , .

(5.31)

(2) To calculate the discrete Fourier transform of g(m). The result is F(k) exp [i2nkMmo]with k = - $ M , . . . ,fM - 1. The spectrum F(k) is M / M f times in the u - and N / N , times oversampled in the u-direction. (3) To add an appropriate bias distribution B(k) onto the real part of F(k) exp[i2nkMmo].The result is G ( k ) = IF(k)I cos[2nkMmo- +(k)] + B ( k ) . (4) To convert G ( k ) by hardware into the continuous distribution G,(u).

o

1, 51

39

CODING PROCEDURES IN DIGITAL HOLOGRAPHY

In step 1 m, and in step 3 B(k) were explicitly introduced. Furthermore, in step 2 the oversampling by M/Mf and N/Nf was also explicitly introduced. In considering how many real values are necessary to code a complex number, two suggestions have been presented by LEE [ 19701 and BURCKHARDT [ 19701. These point-oriented CGHs are characterized by the choice of M,m,, and B(k), where m, and B(k) are implicitly specified; viz., for (1) M: M = (/3Mf,Nf) with /3 a positive integer. LEE [ 19701 used /3 = 4, BURCKHARDT[ 19701 used /3 = 3, and GALLAGHER and BUCKLEW [ 19801 used an arbitrary /3 value. (2) m,: m, = (Mf, 0), which will result in a cosine period of /3 pixels. (3) B(k): B(k) = IF(k)I Icos[2nkMf/M - $(k)] I was shown by CAMPBELL, WECKSUNGand MANSFIELD [ 19741. The implicit introduction of the offset mo and the bias B(k)was accomplished in the following way. F(k)was calculated with an oversampling factor of /3 in the u-direction; that is, M = (/3Mf,Nf), and the complex amplitude F(k)interpreted as the vector I F(k)I (cos $, i sin $) in the complex plane was projected row by row onto the vectors e(k) = (cos [2 nk//3], i sin [2 nk//3]). The values of these projections form a hologram distribution with implicit introduction of a carrier corresponding to nz, = (Mf, 0). All negative-valued projections were simply eliminated; that is, set equal to zero, which determines the bias B(k). The specific point-oriented CGH, which results from this procedure, is given explicitly by G(k) = IF(k)I { c o s [ ~ ~ ~ M - ~$(k)I / M + I c o s [ ~ ~ -~ $(k)Il} ~ / M (5.32) 9

withk=(-i/3M f , - l2 N f) * * * @M,, $Nf)- 1. It remains to be checked that the conditions (5.30)and (5.31) are satisfied using the point-oriented procedure above: From the implicit introduction of m, = (Mf, 0), it follows that the condition m, 2 iMf is fulfilled and that the condition M > 2m, + M, becomes M > (3Mf, Nf). Thus /3> 3 ensures an error-free coding apart from the influence of the bias term b(m). The condition /3> 3 is necessary because mo = (Mf, 0) is implicitly introduced. In case mo = (;Mf, 0) is explicitly chosen, also /3= 2 is possible (also apart from the influence of the bias). This indicates that it is necessary to oversample by a factor of 2 in at least one direction to realize in principle the point-oriented coding correctly. Thus two real values are sufficient to code a complex number; that is, its real and imaginary part. The suggested procedures by LEE [ 19701, BURCKHARDT[ 19701, and GALLAGHER and BUCKLEW[1980] were presented in binary form (see 5 5.2.1.4.1 (1)). 9

9

40

DIGITAL HOLOGRAPHY

- COMPUTER-GENERATED

HOLOGRAMS

[I, 0 5

An alternative to the point-oriented coding characterized by (5.27) can be expressed as

In this case the conversion to the continuous distribution is performed before the coding. This kind of procedure was the very first scheme to be applied in digital holography by BROWN and LOHMANN[ 19661 and has been called cell oriented (DALLAS [ 19801). Process (5.33) can be given in the explicit form F(k) = F(u)comb [u, Su]

(5.34a)

* rect Fr(u) = IF&)/ exp[iq&(u)]= {F(u)comb[u, Su]} *rect[u, Su] (5.34b)

1 coding (cosine carrier and bias) G,(u) = 21Fr(u)l C O S [ ~ I C-U&(u)] X~

+ B(u).

(5.34c)

Within the cells rect[u, Su] the linear phase 2nux0 and possibly also B(u) varies. In point-oriented holograms, on the other hand, G,(u) according to eq. (5.23) is constant in the cells rect[u, Su]. The reconstruction given by FT- '[ G,(u)] results in the distribution gr(x) = fr(x - xo) + L*[- (x + xO)l+ b(x)

(5.34d)

9

with

fr(x) = {f(x)*comb[x, 6u-']} sinc[x, Su-'1.

(5.34e)

Thus in the reconstruction the object and its twin image are separately repeated and separately modulated globally with a sinc-function. This is in contrast to f, ( x

- $1

md

L6"-1 -I

hrr

u

0

Fig. 20. Illustration of a reconstructed object f,(x - x,J of a cell-oriented hologram (cf. eq. (5.34e)). Multiplication by the sinc function causes the higher orders to be very weak. Parameters:x, = 0.6Ax; hu- = 4 A x .

CODING PROCEDURES IN DIGITAL HOLOGRAPHY

41

6U

Fig. 21. Illustration of the Fourier pair G,(u).-rg,(x) for the cell-oriented coding, as stated in eqs. (5.34~)and (5.34d) (without bias). The cutout shown to the right is a magnification of the corresponding area indicated, with a rectangle in the spectrum G,(u) covering exactly four cells. Parameters: x, = 0.6Ax; repetition period Su- = 4Ax.

the point-oriented hologram, where they are influenced jointly. Equation (5.34e) indicates that the zeros of the sinc-function are located in the centra of the higher orders of f,(x). Consequently, the higher orders are considerably weaker than the zeroth order. These conditions are illustrated in figs. 20 and 21. Some conditions concerning x, and 6u need to be fulfilled in principle to reconstruct the object undisturbed. To allowf,(x - x,) and f,*[ - (x + x,)] not to superpose, the condition(s)

6u- 2 AX, Ay) and/or 6u- 2 (Ax,2Ay)

(5.35)

should be fulfilled. Conditions (5.35) ensure that gaps will be created between the repeated f ( x ) in which the repeated twin terms f*(- x ) can be located. To ensure that the terms f*(- x) are located in the gaps, the lower limit that x, must fulfill is x, 2 (+Ax,0) and/or

x, 2 (0, $by)

(5.36a)

and the higher limit is X,

6 i(6u-I - AX),

(5.36b)

which is equivalent to 6u-' 2 2x0 + Ax; that is, the same as was obtained in the point-oriented case (cf. eq. (5.26)). Thus for the smallest choice of 6u- (cf.

42

DIGITAL HOLOGRAPHY

- COMPUTER-GENERATED HOLOGRAMS

[I, 8 5

eq. (5.35)), for example,

6u- I

=

A AX, Ay) ,

(5.37a)

x, should be X, =

(+AX, 0 ) .

(5.37b)

Because C,(u) will vary within each cell of the cell-oriented hologram (see fig. 21), the distribution G,(u) in the hologram plane cannot be explicitly used for hologram production with the same explanation as that used at the end of 4 5.2.1. Thus it is necessary to modify the distribution C,(u) further in the case of a cell-oriented coding. A fundamental modification leads to the “detour phase” coding scheme by BROWNand LOHMANN [ 19661. Then, (5.38) GAu) = 21Fr(u) I cos[2nuxo - @ r ( ~ ) l is chosen. No explicit bias is introduced. Furthermore, the “detour phase” coding is based on the choice x,

=

(6u-’,O),

(5.39a)

which results in one cosine period per cell in the u-direction, and, in general, the choice

6u-I = A X . (5.39b) The combination of eqs. (5.39a) and (5.39b) is chosen, that is, no oversampling of F(u),results in x, = (Ax, 0). These choices violate the conditions according to eqs. (5.35) and (5.36b), which means that superpositions of f,(x) and f,*( - x) occur. In addition, in the case of oversampling; that is, 6u- is chosen according to eq. (5.39, it follows that with the choicex, = (6u- I , 0) the eq. (5.36b) is still violated (fig. 22). Thus x, should be changed. However, x, = ( & - I , 0) is essential for the modification “detour phase” because a cosine period per cell is necessary. For the choice of parameters given by eq. (5.39a), G,(u) will take the shape shown in fig. 22. This distribution has the properties of a cosine of one period per cell, and the phase of the cosine is determined by @,(u). BROWNand LOHMANN [ 19661 suggested replacing the graytone distribution per cell by a binary pulse centered around the maximum of the cosine and with an area equal to the graytone value IF,(u)( at the maximum; that is, IF(k)l in the cell with index k. A binary hologram is the result. The influence of quantization with additional noise and bias is visible in fig. 23. Analyses and modifications of the

1, B 51

CODING PROCEDURES IN DIGITAL HOLOGRAPHY

43

Fig. 22. Illustration of the Fourier pair G,(u)ttg,(x) for the cell-oriented coding for the special case x,, = Su- I , using the test object in fig. 17. Then, a cosine period per cell is formed. The cutout shown to the right is a magnification of the corresponding area indicated, with a rectangle in the spectrum G,(u) covering exactly four cells. Parameter: repetition period Su- = 4Ax.

“detour phase” coding procedure can be found in the literature (LOHMANN and PARIS[ 19671, GABELand LIU [ 19701). The actual steps needed to produce a cell-oriented CGH using the “detour phase’’ coding scheme are as follows: calculation of F(k) with the discrete Fourier transform, usually with Su = Ax-’ (no oversampling), coding the complex amplitude F(k) by the “detour phase” technique, and producing the CGH using, for example, a plotter.

5.2.1.2. Application of phase freedom: diruser It is possible to realize the coding of amplitude holograms, as described in § 5.2.1.1,using the amplitude freedom in form of essentially the twin image and the transformed bias b(x).Thus the coding conditions eqs. (5.25), (5.26), (5.39, and (5.36) result from the requirement that superposition of the separate terms of g,(x) is prevented. In order to increase the diffraction efficiency, to manipulate the effects of disturbances in the hologram, and to decrease quantization errors, further modification of the hologram distribution G,(u) is desired. In point- and cell-oriented codings, because IF@) I determines G,(u) to a large degree, most manipulations found in the literature refer to IF&)). To reach the goals mentioned, I F(k)I is smoothed up to the value I F(k)I = 1. This latter case is also important in phase holography (8 5.2.2.2 (B)).

’

’

44

DIGITAL HOLOGRAPHY

- COMPUTER-GENERATED HOLOGRAMS

[I, 0 5

Fig. 23. Cell-oriented holograms and their simulated reconstructions, in which (a) and (b) illustrate the case shown in fig. 22. In (c) “detourphase” has been applied to bmarize the analog hologram in (a). Quantization noise superposes the corresponding reconstruction in (d). Parameters: x, = (6u- I , 0); Su- I = 2Ax. (Courtesy of Weissbach.)

Section 5.2.1.2 describes how phase freedom can be used to achieve the desired manipulations for intensity objects, and § 5.2.1.3 shows how, in addition, amplitude freedom can be applied to achieve the same goals. WYROWSKI[ 1990al has treated the effect that different manipulations of amplitude holograms have on their diffraction efficiency. Several suggestions have been made which use the phase freedom to modify the hologram distribution G,(u). Discrete and continuous intensity signals are discussed separately in the following subsections. 5.2.1.2.1. Discrete intensity signals. The goal is to restructure F(kf) of the discrete Fourier pair (5.40)

1.8 51

CODING PROCEDURES IN DIGITAL HOLOGRAPHY

45

where i(m,) is the discrete signal with 6u = Ax- ' ; that is, the spectrum is not oversampled. The phase cp(m,) is used for the redistribution. The smoothing of IF(k,)I2is a goal of special importance. Some examples are mentioned below. Random phase. In digital holography the introduction of a random phase distribution - II < cp(m,) < K corresponds to diffuse illumination in optical holography. A smoothing of the discrete JF(k,)lvalues is obtained. Furthermore, the object information is distributed over the hologram, which results in a typical behavior of Fourier holograms against disturbances. A blocking of a part of the hologram will introduce noise (reduction of resolution) but no macroscopic signal damage (WYROWSKI, HAUCKand BRYNGDAHL [ 19861). In three dimensions, the distribution of information is of fundamental importance to connect all parts of the object with the hologram (LEITH and UPATNIEKS [ 19641). In addition to the advantages, there are also some disadvantages. Using a random phase will cause wild fluctuations (speckles) between the reconstructed samples i(mf).These fluctuations can be suppressed by repetition of the hologram distribution. This is suited for discrete signals, but for continuous signals a random phase is unsuitable (see 3 4.1.1.3 and the end of this section). A compromise between spectrum smoothing and speckle introduction is the use of a pseudo-random diffuser (KARL [1972], NAKAYAMA and KATO[ 1979, 1981, 19821). Deterministic phase. DALLAS [ 19731 examined the phases cp(m,) that will lead to the discrete Fourier pair

that is, the phase cp(m,) generates a discrete spectrum of constant amplitude after superposition onto a constant amplitude. Such a phase will completely eliminate the dc peak in the spectrum of an arbitrary distribution If(mf) 1. These cp(rn,) distributions are called object-independent deterministic diffusers. DALLAS [ 19731 introduced a whole class of them. WYROWSKI, HAUCKand BRYNGDAHL [ 1987bl generalized the idea and showed that for some particular periodic objects a phase can be found to achieve IF(k,)I = 1. A comparison [ 19731, between deterministic and random phases was performed by AKAHORI who found that deterministic diffusers often are preferable to smooth the spectrum. Thus deterministic phases are an alternative to a random diffuser in several applications. A combination of pseudo-random and deterministic diffusers was made by TORII[ 19781. Zteratedphase. HIRSCH, JORDAN and LESEM[ 19711 and GALLAGHER and

46

DIGITAL HOLOGRAPHY

- COMPUTER-GENERATED HOLOGRAMS

[I, 8 5

LIU [ 19731 have developed an iterative method to calculate object dependent diffusers based on the iterative Fourier transform algorithm (Q 5.2.2.2 (B)). The aim of their method is the synthesis of the discrete Fourier pair

f h ) = If(’%) I exp [icp(mf)l

DFT

F(kf) = F0 exp [i $(kf)] , (5.42)

that is, a complex amplitude with a constant power spectrum

I F(kf)I ’= F t .

(5.43)

’

The method leads over eq. (5.43) to smoothed amplitude samples I F(k)I but not to I F(k)I = const. Equation (5.43) is the starting point to realize kinoforms (Q 5.2.2.2 (B)). In fig. 24 the influence of different phases on the calculated graylevel distribution in amplitude holograms is shown.

’

5.2.1.2.2. Continuous intensity signals. Section 4.1.1.3 described how, for a = can serve to continuous signal i(x), the phase cp(x) added to [)@fI limit the bandwidth of the complex amplitudef(x) = If(x)I exp [icp(x)] ;that is, to realize AF < co.However, in this case the phase freedom is reduced; for example, a random phase is impossibleto use. When iterative methods are used

Jlc.>

0

1

0

1

0

1

Fig. 24. Illustration of the smoothing effect using the phase freedom. Histograms of graylevels in calculated amplitude holograms when (a) a constant, (b) a random, (c) a deterministic, and (c) an iterated phase is utilized.

CODING PROCEDURES IN DIGITAL HOLOGRAPHY

41

Fig. 25. Moduli of typical spectra after the introduction of (a) a constant. (b) a random, and (c) an iterated phase.

to calculate a bandlimited f ( x ) , a smoothing condition may also be impleand BRYNGDAHL[ 19881). A smoothing of IF(&,)[ commented (WYROWSKI parable to that obtained with a random phase, is then achieved. This is illustrated in fig. 25. 5.2.1.3. Additional application of amplitude freedom

In (j 5.2.1.2 it was described how the application of phase freedom can be used to smooth the power spectrum IF(&,)I2 of the object. It is even possible to achieve IF(k,)12 m 1. Another way of reaching this result exists by using amplitude freedom. In general, the amplitude freedom is utilized to introduce a twin image and bias necessary for the coding. However, in addition, the amplitude freedom can be employed to shape F(u),for example, to make the modulus of the spectrum constant. Here this has no influence on the diffraction efficiency, but errors introduced by further quantization can be suppressed. In the following procedure a continuous distribution is used to demonstrate that the methods are independent of Su. Thus, starting from F(u),one is looking for a modified distribution F(u), which fulfils certain boundary conditions, for example, IF(u)l’ = 1. This may also be formulated as follows: Starting from F(u),an additional term P(u) is introduced in such a way that

F(u) = F(u) + P(u)

(5.44)

fulfils the boundary conditions. For P(u) it is required, as for the bias term in the coding of an amplitude hologram, that the corresponding p ( x ) and the object f ( x ) are spatially separated in the reconstruction plane; that is, in

7<4= f W + P ( 4

(5.45)

p ( x ) is not allowed to disturb f ( x ) . In general, p ( x ) and f ( x ) do overlap. One formal way to separate p ( x ) from f ( x ) is to introduce a linear phase

48

DIGITAL HOLOGRAPHY

- COMPUTER-GENERATED HOLOGRAMS

(1, § 5

2nuxp. Thus we proceed to the new Fourier pair

F(u) = ~ ( u+) P(U)exp [i2 n u p1

(5.46aI

1 FT

7(4= fW

+ P(X

(5.46b)

- xp)

3

whereby the following requirements have to be fulfilled: xp has to lead to a spatial separation, and in spite of the linear phase, the term P(u) = P(u) x exp[i2nuxp] needs to satisfy the desired boundary conditions of F(u), for example, IF(u)12 = 1. It is possible to separate&) spatially from the object, if the object is discrete (sampling raster ax). For a discrete object F(u) is periodic; that is, F(u) = F [ u + (AF,, O)] = F [ u + (0, AF,)] = F(u + AF) with Sx = AF-'. Therefore, with a periodic P(u) (period AF), p ( x ) is discrete and with the choice xp = $ 6 separated ~ from the object. Thus a separation is possible for -

F(u) = F(u) + P(u) exp[inuSx]

.

(5.47)

As mentioned earlier, it is also necessary that ?i(u) = P(u) exp [inuSx] fulfils the desired boundary condition of F(u).To this end, at the location u, the value P(u,) = F(u,) - F(u,) can be defined. In this way Pi(#,) can be found as long as u, is located within a period of F(u), but when the same value F(u,) appears again, that is, after a period such as (AF,, 0), then F[uo + (AF,, 011 = F[uo +

011

+ P [ u , + (AF,,

O)] exp[inu,Sx] exp[inAFU6x], (5.48a)

that is,

-

F [ u , + (AF,, O)] = F(u,)

Because Sx = AF;

+ ?i(uo)exp[inAFU6x].

(5.48b)

+ P(uo)exp [in] .

(5.48~)

', we obtain

F [ u , + (AF,, O)] = F(u,)

Thus, both F(u,) and F [ u, + (AF,, O)] satisfy the desired boundary conditions ),@fi together with B(u,) as well as P(u,) exp [in] can be made to fulfil them. The same explanation follows for the period (0, AFJ. Other multiples of periods do not lead to new conditions concerning P(u,). From this we may conclude that types of boundary conditions which can be satisfied with this kind of consideration are limited. CHU and GOODMAN [ 19721 have investigated two cases (two boundary

1, B 51

CODING PROCEDURES IN DIGITAL HOLOGRAPHY

-1

49

I

Fig. 26. Illustration of how a value Fof the spectrum by addition of and P , can be decomposed in P and F , of equal amplitude.

conditions) under which the foregoing procedure is applicable. One is mentioned here and the other case is discussed in § 5.2.1.4.2. [ 19721 fulfilled the boundary condition IF(u)I * = 1 by CHUand GOODMAN introducing a suitable p(u).Chu and Goodman used the diagram in fig. 26 to describe their encoding method. The vector p or p, = p exp[in] is added orthogonal to F, which represents one value of F(u). The vector sums F = F + P and F , = F + P, both have the amplitude 1. F = P(uo) and F , = F(uo) exp [in] are encoded as hologram elements. Figure 27 shows how a periodic distribution F is transformed into values with IF1 = 1 by the addition of p and P,. From the preceding considerations it is evident that P(u) may be introduced

F

Fig. 27. Illustration of a periodic distribution of four different sampling values showing where to add P and P , in order to transform the distribution to pure phase values.

50

DIGITAL HOLOGRAPHY

- COMPUTER-GENERATED HOLOGRAMS

[I, 8 5

so that p ( x ) and f ( x ) are spatially separated and, in addition, that I F(u)I * = 1. Discretep(x) values are then located between the discretef(x) values. CHUand GOODMAN [ 19721 call p ( x ) a parity sequence. The method was originally suggested for the kinoform, but, of course, it is also suited for amplitude holograms. Another method is based on the following reasoning: In case a p ( x ) (cf. eq. (5.45)) offinite extent is found; that is, Ap < a, then xp > Ap (for A p > Ax) would be sufficient to separate p ( x ) and the object spatially. CHU and FIENUP[ 19741, and later modified by HSUEHand SAWCHUK [ 19781, developed a method to satisfy I F(u)I = 1, in which p ( x ) essentially is the object f ( x ) itself. When this is the case, the choice of xp = Ax should be sufficient to realize approximately a spatial separation (CHU and FIENUP [ 19741). HSUEHand SAWCHUK[ 19781 introduced xp = nAx to reduce the remaining disturbances in the reconstruction further. The formulation of the method does not start from F(u) but from F,(u), where

'

F,(u) = F(k) * rect [u, Su] ,

(5.49)

with Su = Ax-'. Instead of eq. (5.46), one now obtains -

F,(u) = F,(u) + P,(u) exp [i2 nuxP]

r

(5.50a)

FT (5.50b)

with xp = ndx. The index r of P,(u) also indicates that the term P,(u) is chosen to be constant in each cell rect [u, Su]. This is important because it enables&) to be small in size. However, P,(u) = const. will cause problems because the linear phase goes through n times 2 n in each cell, which means that it is impossible within a cell with P,(u) exp [i2nuxp] to satisfy the boundary conditions. An exception was mentioned earlier (after eq. (5.48)), that for two particular phase values, namely, 0 and II,boundary conditions exist which can be satisfied by the addition of P,(u,) and P,(u,) exp [in]. Thus an adjustment is possible on a raster with Su, = (2xp)- = Su/2n; that is, it is 2n times finer than the extent of the cell. Then the discrete version follows:

'

Fr(ks)= F,(u) comb [ u, Su,] =

F r ( k ) +Pr(ks) ~ X[ Pi ~ ( k s+ L)l,

(5.51a)

with k, = (k,, l,) = - n M , ...,nM - 1. From this it follows that only the phase values 0 and n are present and that, with the addition of P,(k,) x exp[in(k, + ls)], it is possible to satisfy special kinds of boundary conditions.

1,s 51

CODING PROCEDURES IN DIGITAL HOLOGRAPHY

51

addition

Fig. 28. Illustration of how a dilated distribution of four sampling values is transformed by the addition of P and P, into a distribution of pure phase values.

’

For example, for the condition I Fr(ks)I = 1 to be fulfilled, P and P, should be chosen as and P , in fig. 26. The result is a discrete version of a distribution

F,(u) = F,(u) + P,(u) exp [i2nnuAx] ,

(5.51b)

namely, F,(u) comb@, 6us)= F,(k,), which fulfils lFr(ks)12= 1. As a consequence, z(x) = f,(x) + p,(x - nAx); that is, the contribution from the additional term P(u), necessary to satisfy the boundary conditions, appears in the nth diffraction order. CHUand FIENUP [ 19741 in particular chose n = 1 and produced a kinoform ($ 5.2.2.2. (B)) from F,(k,), this synthetic coefficient method is illustrated in fig. 28. HSUEHand SAWCHUK [ 19781 coded F,(k,) by means of the “detour phase” scheme into an amplitude hologram (called double-phase coding), which is shown in fig. 29. They used a 1-D displacement xp = (nAx, 0) resulting in the sampling distance 6us = (6u/2n, 60);that is, 2n times oversampling in the x-direction. The result ?;;(ks)= Fr
F,(u) = {F,(U)comb[u, bus]}* rect [u, Su,]

.

(5.52)

detour phase’

Fig. 29. Illustration of how a 1-D distribution of two sampling values is transformed by the addition ofP and P, into a distribution of pure phase values (cf. fig. 26). Application of the “detour phase” coding results in the hologram configuration to the far right.

52

DIGITAL HOLOGRAPHY

- COMPUTER-GENERATED HOLOGRAMS

The coding of Fs(u) followed according to the description in

11.8 5

3 5.2.1.1,An

xo = (0, Ay) was implicitly introduced; that is, “detour phase” coding in the y-direction. A typical structure of G,(u) is sketched in fig. 29 for n = 1.

5.2.1.4. Quantization Section 5.2.1.1 described the coding procedure used to construct a CGH pattern, and two cases could be distinguished, namely, either the computer output device is used to convert the discrete distribution G(k)into the hologram distribution G,(u), a point-oriented hologram, or the output device converts the complex amplitude F(k) into a binary “detour phase” hologram, a cell-oriented hologram. In this process the following values are quantized: (1) G(k) or G,(u) is quantized because the output device and/or the holographic recording medium is only capable of generating quantized positive real values. (2) The phase $(k) and the amplitude IF(&)I of F(k) are quantized because the position and the area of the aperture in each cell, which controls the phase and the amplitude, can only be varied by discrete amounts. The following subsections examine the quantization of G(k) or G,(u) for point-oriented holograms and the quantization of F(k) for “detour phase” holograms. 5.2.1.4.1.Quantization of point-oriented holograms. The coding procedure results in the discrete Fourier pair G(k) = 2lF(k)l cos[2nkMm0- $(&)I $ DFT g(m) = f

+ B(k)

h - mo) + f *[ - (m + moll + b(m)

(5.53a) 9

(5.53b)

with the positive and real valued spectrum G(k).Two possibilities to obtain a quantized continuous hologram distribution cr(u) exist (- indicates quantized quantity) as follows: (1) Procedures based on pulse (width) area modulation. At the conversion of G(k) into G,(u) using an output device, the graytone value G(k) is coded by means of a pulse width (height) modulation (BASTIAANS[ 19871); that is, G,(u) = G ( k )* rect [u, 6u]

(5.54)

is not formed, but -

G,(u) = comb(u, Su) * rect [u, w(k)]

(5.55)

with the varying pulse width (height) w(k),which is a suitable function of G(k),

CODING PROCEDURES IN DIGITAL HOLOGRAPHY

53

for example, w(k) = { G(k)Su, 6 u ) or w(k) = { Su, G(k)Su} ,

(5.56)

whereby G(k)is assumed to be normalized. The resulting distribution cr(u) is binary. Bastiaans has investigated pulse width modulation analytically. This procedure was suggested by BURCH[ 19671, and, of course, it can be applied to realize any point-oriented hologram (LEE [ 19781). The pulses are here centered around points of an equidistant sampling raster, which indicates that the coding is point oriented. The procedure can be summarized as (5.57)

This procedure is attractive using a high-resolution binary output device, for example e-beam, because it is then possible to utilize the high resolution with only a small (MN)amount of data. Comparative analyses between this [ 19751, procedure and the binary “detour phase” method appear in GABEL BUCKLEWand GALLAGHER [ 19791, ALLEBACH[ 19811, and FARHOOSH, FELDMAN, LEE, GUEST,FAINMAN and ESCHBACH [ 19871. (2) Procedures based on pulse density modulation. Another procedure is to introduce a quantization of G(k) directly with the result E(k) and then to convolve G(k) with a rect[u, 6u], which results in a quantized Gr(r).This method of quantization in connection with point-oriented CGHs may be summarized as (5.58)

-

where g(m) f ( m - mo) should hold within the object window IF. This conversion of G(k)to which is discussed below, leads to a pulse density modulation with pulses of extent Sr in contrast to the preceding pulse (width) area modulation with varying pulse area. The method to quantize point-oriented holograms summarized in (5.58) is based on consideration of a discrete Fourier pair, which comprises the object; that is, g(m) = f ( m - m,) -I-

c,(u),

f*[- (m + moll -I- B 6(m)

(5.59)

with a chosen constant bias. Starting from the pair (5.60)

54

DIGITAL HOLOGRAPHY

- COMPUTER-GENERATED HOLOGRAMS

[I,

I5

with G(k)< 1, a new discrete Fourier pair

(5.61) is synthesized. G(k)is here quantized in discrete levels 0,2A,4A, ... , 1, where

2A

=

(Z - l ) - '

(5.62)

is the separation between the equidistant levels; Z is called the number of quantization levels. Within the object window 5,g(m) f ( m - m,) is required. A direct way to quantize the distribution G ( k ) in Z graylevels is to use the quantization operator N

G ( k )< A ,

0,

-

I

G ( k ) = g [ G ( k ) ]= 2(2A), (22 - 1)A < G ( k ) < (22 + l)A , 1,

( 2 2 - 3)A < G ( k ) ,

(5.63) withz=O, ..., Z - 1. In fig. 30 L Z is illustrated for Z = 2; the result of this hard-clip operator is a binarization. According to eq. (5.63) G(k) undoubtedly satisfies the required boundary conditions in the hologram plane. To examine when and how g(m) f ( m - m,) is satisfied within the object window 5, the following can be considered: The term N

Q ( k ) = G(k)- G(k) = @[ G ( k ) ] - G ( k )

Fig. 30. Illustration of the hard-clip operator p2.

(5.64)

1, B 51

55

CODING PROCEDURES IN DIGITAL HOLOGRAPHY

is introduced. It describes the change in the hologram plane that is caused by L Z . The discrete Fourier pair produced by L Z is then -

G(k)

=

G(k) + QW)

(5.65a)

5 DFT a m ) = g(m) + q(m)

(5.65b)

9

where q(m) is an expression for the quantization noise produced by L Z . As soon as q(m) is not equal to 0 within IF, the object generally is disturbed; that is, the addition of q(m) to eq. (5.59), a m ) = f ( m - m,) + f*[- (m + moll + dm1 + B

9

(5.66)

implies that g(m) is not proportional to f(m - m,) within IF. An exception is will a point f ( m - m,) = 6(m - m,) as object. For example, the operator change a cosine grating into a Ronchi grating, and in this case q(m,) # 0 is not disturbing because a point object cannot be disturbed and q(mo)will cause the diffraction efficiency to increase from 6.25% to 10.13%. The following conclusions may be drawn to characterize different quantization methods: (A) In case only amplitude freedom can be used, q(m) = 0 should be realized in IF; that is, the quantization noise has to be separated from the object. To achieve this, two possibilities exist: (1) to modify: (a) the initial discrete Fourier pair g(m) DFT G(k), for example, the bias B(k); (b) the quantization operator; (2) to apply an iterative method. (B) In case the object is an intensity distribution, phase freedom is also applicable. Then g(m) needs to fulfil the condition

-

IE(m)12 = a21f(m - moll2

within IF. Thus q(m)# 0 is allowed within IF, due to a and the phase freedom. An example of the phase freedom is illustrated in fig. 3 1. It is possible here to apply iterative methods. Various quantization methods start from the discrete Fourier pair DFT g(m) G(k) and end at the discrete Fourier pair g(m) = g(m) + DFT q(m) G(k), whereby q(m) = 0 within IF. The diffraction efficiency i j of this hologram is not changed compared with q of the corresponding analog one; that is, i j = q, because the additional light introduced by the quantization

-

56

DIGITAL HOLOGRAPHY

- COMPUTER-GENERATED HOLOGRAMS

[I, 0 5

Fig. 31. Illustration of how the phase freedom is used to add a q to change the amplitude g into g without changing its modulus.

is separated from the object. To increase the diffraction efficiency in the quantization step the amplitude freedom has to be combined with the scale factor freedom ( a > 1). An additional use of phase freedom simplifies the realization of this aim (WYROWSKI[ 1990al). In short, two methods exist to realize quantized point-oriented holograms, namely, pulse width/area modulation and pulse density modulation. In the case of pulse density modulation, methods exist which only use amplitude and others which use amplitude and phase freedom. ( 3 ) Examples of binarization procedures based on pulse density modulation. (A) Amplitude freedom only. These procedures are applicable to all kinds of objects. Here the aim is to introduce a quantization method wich leads to q(m) w 0 in IF. The halftone technique can be used, which is the standard procedure in printing to transfer a graytone picture into a binary version; that is, 2 = 2. This technique can be introduced as a carrier method whereby the clipping of a modulated carrier will result in a binary distribution. In holography, where a cosine carrier is already present, the carrier method can be applied directly (LEE [ 19781). To secure the object undisturbed in the first diffraction order (within the object window IF), the constant bias should be modified to B(k) = sin{arccos[ I F ( k ) ( ] }+ 1,

(5.67)

whereby (F(k)lshould be normalized. In this way a single carrier is used to code the amplitude as well as the phase information. However, the coding of

I , § 51

CODING PROCEDURES IN DIGITAL HOLOGRAPHY

51

amplitude and phase can be performed using separate carriers in orthogonal directions. This is achieved by introducing a separate carrier for the binarization (LEE [ 19781,JUST,HAUCKand BRYNGDAHL[ 19851). It may be advantageous to orient the binarization carrier in the direction of slowest hologram variations in order to minimize m, and to equalize the number of sampling points needed in both spatial directions. The extent of b(m) in the reconstruction due to the bias should be considered by the choice of m,. It has been found empirically that the choice M > 8M, (the rate by which F(u) is oversampled) is sufficient. Furthermore, the distributions are discrete, which will cause aliasing due to disturbances from higher diffraction orders of the carrier and result in additional restrictions on m, (JUST,HAUCKand BRYNGDAHL[ 19861, JUST and BRYNGDAHL[ 19871). The halftone processes in the preceding description are passive; that is, only a pointwise comparison between the analog function and a threshold function is performed. The process can be made active in the sense that computational means are added to the process (HAUCK and BRYNGDAHL [1984], WEISSBACH,WYROWSKIand BRYNGDAHL [ 19881, BARNARD[ 19881). The quantization operator is sequentially applied. However, the error due to the binarization at the location k,, that is, Q(k,), is distributed with certain weights to the not-yet-binarized neighbor points before &2 is executed. Therefore, the error correction algorithm illustrated in fig. 32 is usually called the “error diffusion” algorithm. The result will be a modified distribution Q(k) in

X

X

X

d X

X

0

0

0

0

a

.

b

o

c

o 0

Fig. 32. The error correction and diffusion algorithm processes the sampled values sequentially. The elements indicated by crosses have already been processed and the solid point is to be processed. The difference between the binary and desired value, the effect of binarization, is divided and distributed with weights a, b, c, and d to the the unprocessed neighbors before proceeding to the next point and repeating the process.

58

DIGITAL HOLOGRAPHY

- COMPUTER-GENERATED HOLOGRAMS

[It

85

contrast to the result of @ alone. It is possible to affect the corresponding quantization noise q(m) to a large extent, due to the adaptive character of the process. In particular, depending on the diffusion weights, areas occur with low noise. Thus for an optimum combination of mo and weights, it is possible to obtain q(m) m 0 in IF. Specifically, when the directions of error diffusion and WYROWSKI of carrier are orthogonal, q(m) will be small within IF (WEISSBACH, and BRYNGDAHL [ 1988]), which is outlined in fig. 33. As before, M I M , 2 8 has been found empirically to obtain a satisfactory separation. The error diffusion procedure is not limited to Z = 2 but can just as well be applied to 2 > 2. Figure 34 shows how the reconstruction quality of a hologram binarized by the error correction and diffusion algorithm can be drastically improved when the diffraction efficiency of the original graylevel hologram is increased using an iterative procedure (WYROWSKI [ 1990al). The diffraction efficiency of the binary version of this graylevel hologram is increased by the same amount. SELDOWITZ, ALLEBACH and SWEENEY[ 19871 presented another type of binarization procedure, which they called “direct binary search” and which is related to the simulated annealing algorithm (see, for example, VAN LAARHOVEN and AARTS[ 19871). It is based on iteration and confined to 2 = 2. A binary distribution is looked for directly, and it does not start from a graytone hologram distribution G(k).The aim is to find a discrete Fourier pair (5.68)

-

where G(k)is binary and g(m) f ( m - m0) within the object window IF. hologram

r econ struc t ion

Fig. 33. Illustration showing how an error diffusion direction D along the hologram fringes will cause the quantization noise in the reconstruction(shaded areas) to be dislocated orthogonally to the direction Tin which the reconstruction appears.

I , § 51

59

CODING PROCEDURES IN DIGITAL HOLOGRAPHY

Fig. 34. Simulated reconstructed intensities with clipped dc-peak of amplitude holograms using (a)a random phase and (b)an iterative procedure and the error correction and diffusion algorithm for binarization. The calculated diffraction efficiency in (a) q rz 0.5% and in (b)q%3.5%.Parameters:M= 512;M,=64.

DIT

The method is purely iterative and starts from a pair g(m) G ( k ) with a random binary distribution G(k).For each point of G ( k )it is checked whether or not the change of the binary value in that point results in a less disturbed object. When this is the case, the binary value in that point is changed. After DFT g(m) is obtained with several iteration cycles the pair -d(k) g(m) = af(m - m,) in IF. In general, a > 1 is obtained which results in ?j> q. “Direct binary search” has been compared with noniterative methods by JENNISON, ALLEBACH and SWEENEY[ 19891. (B) Amplitude and phase freedom. In the case of discrete intensity objects total phase freedom exists. Phase freedom may be used in addition to amplitude freedom. The aim is to adjust the quantization noise q(m) within IF caused by the quantization procedure to the object by means of the phase freedom. This is done in such a way that in the object window IF

-

Idm) + g(m)12 = Is(m)12 = a21f(m - moll2

(5.69)

is obtained. Iterative methods that use the phase freedom can be applied in this situation. WYROWSKI [ 1989bl presented an iterative quantization method based on the iterative Fourier transform algorithm (IFTA).

60

DIGITAL HOLOGRAPHY

- COMPUTER-GENERATED HOLOGRAMS

[I. 8 5

choose initial amplitude

constraints:

E, I ul = YI G, I ul I i .I( x ) = g j i x i + qI. i x )

4

E~Iu)=G,(~I+Q~(~)

FT-1

interrupt:

-

error criterion number of iterations

E,

lul

Fig. 35. Illustration of the iterative Fourier transform algorithm. The constraints are executed by the operator in the hologram plane u and _X in the object plane x.

This procedure starts with the Fourier pair go(m) =

Ifh - moll exp[irp,(m - mO)l+ B 6(m)

(5.70a)

1 DFT Go(k) = 21F(k)I ~0~[2nk,mO- $(k)] + B ,

(5.70b)

with the random phase rp,(m). The IFTA is illustrated in fig. 35. The operator X - is X[gi(m)l = gi+ I(m)

(5.72)

I,! 51

61

CODING PROCEDURES IN DIGITAL HOLOGRAPHY

This iteration stagnates for U = L Z . The stagnation can be avoided, if Qz is not directly introduced, but in p = 1, . . .,P (e.g. 10) steps. For example, for 2 = 2, each step is iterated with another operator U = g p . The operator is given by 0,

I

g P [ G j ( k ) ]= Gj(k) = 1,

G,(k) < dP), 1 - 8‘’) < G,(k) ,

(5.73)

Gj(k), otherwise ,

with 0.5 = dP)> d P -’)> * * > d’)> 0 (WYROWSKI[ 1989b], BROJA, WYROWSKI and BRYNGDAHL [ 19891). The operator GJ’ is shown in fig. 36. In each step, for example, 5 iteration cycles are performed as illustrated in DFT fig. 37. In this way the desired discrete Fourier pair g(m) G(k), with Ig(m)l’ = a21f(m- mo)12 is obtained with a> 1. With this procedure it is possible to obtain a high diffraction efficiency i j . For example, the corresponding binary hologram of a test object (see fig. 38a) has an efficiency 7 FZ 7 % , which is possible because q(m)# 0, as is clear from fig. 38b. In the case of continuous intensity objects the phase freedom is severely limited and iterative quantization methods seem not to be appropriate. Then methods can be used that only apply amplitude freedom, for example, the error diffusion algorithm. In fig. 39 some types of binary hologram distributions are shown for comparison and to indicate how it is possible to explain different results. Figure39a shows the hard-clipped version of a graytone hologram, The 9

-

Fig. 36. Illustration of the

E poperator, which is used to avoid stagnation of the iterative procedure.

62

DIGITAL HOLOGRAPHY

- COMPUTER-GENERATED HOLOGRAMS step 10.

5 cycles

Fig. 37 Illustration of how the g

5 cycles

p

[L § 5

1

iL

operator is changed stepwise during the iteration.

Fig. 38. Simulated reconstructed intensity, with clipped dc-peak of the hologram binarized by iteration, is shown in (a); in (b) the amplitude of the noise term q(m) produced by the iteration is shown. Although q(m) superposes f ( m - mo), no disturbance appears in the object because of the use of the phase freedom. Parameters: M = 256; M, = 64; i j % 7%.

hard-clipped modulated cosine results in a distinct fringe structure with a preferred direction. When only the amplitude freedom is used, as in fig. 39b where the error correction and diffusion algorithm was used, it is possible to arrange for the quantization noise to be shifted in another direction from the reconstructed object. This is reflected in the breakdown of the fringe pattern. The destruction of the preferred direction explains the different displacement directions of reconstruction and noise. Using the phase freedom in an iterative method, as in fig. 39c, it is clear that the preferred fringe direction is preserved. Thus much light is diffracted into the object window resulting in a high diffraction efficiency and in comparison with fig. 39a the small structure correction enables the elimination of the disturbing noise within the reconstructed object.

1, B 51

CODING PROCEDURES IN DIGITAL HOLOGRAPHY

63

Fig. 39. Illustration of typical binary hologram distributions.A graylevel hologram was binarized by (a) hardclip, (b) the error correction and diffusion algorithm, and (c) an IFTA procedure. Reconstruction of the hologram in (b) is shown in fig. 34a, and that of (c) in fig. 38a. Parameters: M = 256; M,= 64.

Optical reconstructions from holograms quantized with the IFTA method in 2 and 3 levels are shown in fig. 40. FELDMAN and GUEST[ 19891 have suggested a binarization method based on simulated annealing and similar to “direct binary search”. 5.2.1.4.2. Quantization of cell-oriented holograms. The aim in the quantization of cell-oriented holograms is to quantize F(k), that is, IF(k)I and @(k) = arg[F(k)],using as few values as possible. For example, in a cell of 4 x 4 pixels, 4 amplitude and 4 phase values in addition to the zero value are possible.

64

DIGITAL HOLOGRAPHY

- COMPUTER-GENERATED HOLOGRAMS

[I, B 5

Fig. 40. Optical reconstructions from holograms produced by an IFTA method in which (a) and (c) are quantized with 2 = 3 and (b) and (c) with Z = 2. Parameters: M = 256; M,= 64; hologram repetition = 8 x 8; sampling distance in hologram = 5 pm; hologram size = (10 nun2).

(1) Amplitude quantization. The methods that are attractive and have been used are, in particular, those where IF@)/ = 1 (see $ 5.2.1.2 and $ 5.2.1.3). (2) Phase quantization. With the methods of CHU and GOODMAN [ 19721 and CHU and FIENUP [ 19741 all boundary conditions can be satisfied which have the property that a particular term of value F(k,,)can be corrected with P(ko) and also with P(ko)exp [in]. The method may be used to quantize the phase as shown in fig. 41. It is possible to realize an arbitrary quantization. However, then F(k) can no longer be kept constant. ( 3 ) Amplitude and phase quantization. For double-phase coding (HSUEH and SAWCHUK [ 19781, the amplitude is made constant by using the amplitude freedom, and only the phase is left analog. In order to avoid phase quantization

1, B 51

CODING PROCEDURES IN DIGITAL HOLOGRAPHY

-1

65

I

Fig. 41. Illustration of how a value F of the spectrum can be decomposed in F and i?, with quantized phase values by the addition of and pn.The phase quantization levels are 0, 4 2 , x, and 3x12.

errors a prequantization method has been applied by RINESand GALLAGHER [ 19811. MAITand BRENNER[ 19871 used an IFTA algorithm to realize the phase quantization; that is, phase freedom was used in addition to amplitude freedom. FIENUP[ 19801 treated with a modified IFTA algorithm the simultaneous quantization of IF(k)l and $(k). Specifically for 4 x 4 cells it was shown how to quantize into 4 amplitude and 4 phase levels in addition to the zero level and to use the result in a “detour phase” coding. Here, too, the iterative technique can be improved by a stepwise introduction of the quantization to avoid stagnation (cf. eq. (5.73)). 5.2.2. Phase hologram

The nomenclature introduced in the beginning of $ 5.2 is maintained in this discussion. In the coding procedure of phase holograms the starting point is the discrete Fourier pair of eq. (5.9)

Proceeding from this pair, a new discrete Fourier pair

66

DIGITAL HOLOGRAPHY

- COMPUTER-GENERATED HOLOGRAMS

[I. 8 5

is looked for which satisfies the conditions IG(k)I

=

1,

(5.74)

that is, a pure phase function G(k) = exp [iF(k)], and a g(m) which contains the object; that is, (5.75a) (5.7 5b) The continuous phase hologram distribution is formed (materialization process) by the convolution G,(u) = G(k) * rect [u, Su] .

(5.76)

The reconstruction g,(x) = { g ( x )* comb[x, Su-'I} sinc(x, Su-I)

(5.77)

is then obtained, which indicates total analogy to the point-oriented coding procedure of amplitude holograms. Thus the essential problem is to synthesize the discrete Fourier pair DFT g ( m ) with the requirements (5.74) and (5.75a or 5.75b). To this G(k) end, some coding procedures are described below.

-

5.2.2.1. Bleached amplitude hologram In optical holography the phase of a wave is recorded by the introduction of a reference wave, resulting in an amplitude hologram. The phase hologram is then obtained by transforming the amplitude G,(u) into a phase hologram G,(u). A common technique is to bleach the amplitude hologram. As an ideal result, a phase hologram distribution GJu) is achieved, the phase values r(u) of which are proportional to the values of the amplitude hologram G,(u) within the interval [0, n];that is,

r(u)= nGa(u) ,

(5.78)

with r(u) = arg[ G,(u)] and 0 < GJu) < 1. In digital holography this method may also be applied. The discrete Fourier pair G,(k)

=

exp [ir(k)] = exp [i zG,(k)]

(5.79a)

DFT g,(m)

=

DFT-'[G,(k)l ,

(5.79b)

CODING PROCEDURES IN DIGITAL HOLOGRAPHY

67

or in an equivalent form

G,(k)

=

cos [ nG,(k)] t i sin [ nG,(k)]

(5.80a)

$ DFT

'

g,(m) = DFT- {cos [ nG,(k)]} t i DFT- {sin [ nG,(k)]}

(5.80b)

is synthesized. From eq. (5.80b) it follows that the resulting distribution g,(m) contains a distorted object. This occurs because the real part of G,(k) contains an amplitude distribution disturbed by cos [ nG,(k)]; that is, DFT - {cos[ nG,(k)]}, in general, leads to a distorted object. Furthermore, an additional disturbance occurs caused by DFT- {sin [ nG,(k)]}. However, in two special cases the object is not disturbed by the described transition G,(k) -+ G,(k). (It should be observed that the reconstruction of G,(k) may already be disturbed by quantization, for example.): (1) For an intensity signal,where a random phase has been superposed onto the object, the value of { G,(k) - 0.5) is small (the bias is subtracted); that is, most hologram values of G,(k) are close to 0.5 (cf. fig. 24b). In case no phase is superposed onto the object, this consideration is also valid (cf. fig. 24a). The development of G,(k) in terms of K[ G,(k) - 0.51, including the second order, results in G,(k)

-

G,(k) - i 0.5 { G,(k)}* + const.

(5.81)

From eq. (5.81) follows the reconstruction g,(m)

N

g,(m) - i 0.5ga(m)@ga(m) + const. 6(m) .

(5.82)

Thus, in addition to the reconstruction g,(m) of the corresponding amplitude hologram, an autocorrelation indicated by @ appears, as illustrated in fig. 42. The use of the amplitude freedom with an appropriate choice of m, then makes the separation of the object and the autocorrelation possible. A large m, means a large set of data; that is, a large space bandwidth product, which is not a small achievement in digital holography in contrast to optical holography, where an inclined reference wave is used. (2) For binary amplitude holograms G,(k) = 0 or 1 so that sin [ nG,(k)] = 0 and cos [ nG,(k)] = G,(k) with values - 1or 1. Thus in this case the distribution of the phase hologram is identical to the distribution of the amplitude hologram without bias. The reconstruction is now undisturbed; that is, not more disturbed then the reconstruction of the amplitude hologram. This case is of special importance because binary phase holograms can be considered as bleached

68

DIGITAL HOLOGRAPHY

- COMPUTER-GENERATED HOLOGRAMS

[I, B 5

Fig. 42. Simulated reconstructed amplitudes with clipped dc-peak of bleached amplitude holograms (no random phase was superposed onto the object). In (a) only the g,(m)Qg,(m) term is shown, and in (b) the entire reconstruction is shown consisting of the undisturbed reconstruction superposed by the distribution in (a). Different normalizations are used in (a) and (b).

versions of binary amplitude holograms. Thus, binarization procedures in amplitude holography (see 8 5.2.1.4)can also be applied to obtain binary phase holograms, and vice versa (DAMMANN and GURTLER [ 19711, DAMMANN and KLOTZ[ 19771, KRACKHARDT and STREIBL [ 19891, TURUNEN, VASARAand WESTERHOLM [ 1989b1). In the preceding two cases qp = 4 q, holds for the diffraction efficiencies qp and q,, whereas in the first case it is assumed that the autocorrelation and the object are separated. 5.2.2.2. Direct coding in digital phase holography In digital phase holography it is not necessary to start from an amplitude hologram. It is possible to look directly for a discrete Fourier pair which satisfies the coding requirements of a phase hologram; that is, the discrete Fourier pair g(m)

=mmo) -

=

If(m - moll exp[icp(m - moll

9

( m kn M E F (5.83a)

5 DFT G ( k ) = exp[ir(k)].

(5.83b)

This Fourier pair is illustrated in fig. 43. The shaded area indicates that outside the object window IF the introduction of amplitudefreedom is permitted.

1,s 51

CODING PROCEDURES IN DIGITAL HOLOGRAPHY

69

Fig. 43. Illustration of the conditions in digital phase holography. The shaded area indicates the possibility of using amplitude freedom.

The coding problem consists of the use of amplitude freedom, phase freedom, or amplitude and phase freedom of g(m) to satisfy the condition 1 G(k)I = 1. A direct solution exists forf(m - m,) = 6(m - mo);that is, a point object, then G(k) = exp[i2xk,m0] and I G(k)l = 1 are always fulfilled without using amplitude and phase freedom and there is nothing to code. This phase distribution G(k)describes a discrete version of a blazed grating, which is easy to conceive because the linear phase is transformed in periods of 2 R of G(k) into a sawtooth profile, as shown in fig. 44. Thus the blazed grating is a special case of the desired phase hologram. For a general object it is usual that I G(k)I # constant and existing freedoms have to be used. (A) Application of amplitude freedom. CHU and GOODMAN[ 19721 have described how it is possible for discrete objects to secure a constant amplitude of the spectrum if a certain distribution is interlaced between the sampling values g(x) comb [x, 6x1 = g(m).CHUand FIENUP [ 19741introduced a second

Fig. 44. Illustration of blazed grating with sawtooth profile.

70

DIGITAL HOLOGRAPHY

- COMPUTER-GENERATED HOLOGRAMS

[I, B 5

method for continuous objects. These methods show (see 5 5.2.1.3) that it is possible to realize IG(k)I = 1 with amplitude freedom, although only the on-axis case was examined. By using the amplitude freedom as little as possible to minimize the energy of the additional term (to maximize q), an iterated phase can be introduced to presmooth I G(k)I (CHUand GOODMAN[ 19721). KIRK and JONES[ 19711 made a suggestion whereby they introduced a carrier modulated by a function of IF(kf)I. With this method it is also possible to fulfil I G(u)l = 1, using only [ 19891 the amplitude freedom. WEISSBACH, WYROWSKI and BRYNGDAHL presented another alternative to use the amplitude freedom based on a complex error correction and diffusion concept. The IFTA is also applicable to satisfy I G(u)l = 1 using only the amplitude freedom (WYROWSKI [ 1990~1). (B) Application of phase freedom: kinofom. When only the phase freedom is used to secure I G(k)l = 1 for an intensity signal, we look for the discrete Fourier pair g(m) = If(m

- moll exp[idm - moll

(5.84a)

$ DFT

G(k) = IF(k)I exP[QKkA4mo - @(k)l,

(5.84b)

with I F(k)I = 1. The conditions in this case are illustrated in fig. 45. For g(m) = S(m - mo) no complications occur. However, for objects of [ 1990dl has shown that for M 3 2Mf no solution general shape WYROWSKI

zero

Fig. 45. Illustration of boundary conditions for a kinoform. No amplitude freedom is permitted outside the object window F. There should be no light outside F.

1,s 51

CODING PROCEDURES IN DIGITAL HOLOGRAPHY

71

exists, for M < 2M, a solution may exist which in general occurs for M = M,; that is, m, = 0, and g(m,) = f(mf) and G(&,) = F(&,). LESEM,HIRSCHand JORDAN [ 19691 did suggest this case, and the hologram is called a kinoform. Their proposal was to introduce

G ( W = exp [i$J(kf)l

(5.85)

by simply introducing IF(&,)[ = 1, even though IF(&,)[ # const. in general; that is, the amplitude information is lost. The corresponding reconstruction suffers from disturbances (KERMISCH[ 19701). Another proposal by HIRSCH,JORDAN and LESEM[ 19711 and GALLAGHER and LIU [ 19731 was to apply a specific version of the IFTA to determine a phase rp(m,), which allows a good approximation IF(&,)( x 1. Thus, using the IFTA, it is possible to find the discrete Fourier pair

JACOBSSON, HARDand BOLLE[ 19871used an IFTA to secure IF(&,)1 x 1 and simultaneously improve the defect resistance of the kinoform. Starting from the discrete version (eq. (5.86)), the continuous kinoform distribution is F,.(u) = F(k,) * rect [u, 6u] .

(5.87)

This kinoform distribution results in f,(x)

=

{ f ( x ) * comb(x, Ax)} sinc [x, Ax]

(5.88)

in the reconstruction plane; that is, the object is periodical with its own extent. The diffraction efficiency is considerably reduced because of the higher diffraction orders. Should additional quantization be introduced, no space is available for the quantization noise; that is, no amplitude freedom exists. Therefore, it would be interesting to realize a coding for M > M,. However, amplitude freedom then should be allowed. ( C ) Application of amplitude and phase freedom: blazed phase hologram. When amplitude and phase freedom are permitted, the discrete Fourier pair of fig. 43 is synthesized. In this case, too, the IFTA is useful (WYROWSKI [ 1989a, 1990b]), since in the x-plane the operator X is given in the same form as in the quantization of amplitude holograms (cf. e& (5.7 1)); that is, amplitude freedom is allowed. Then I G(&)I x 1 can easily be fulfilled with high accuracy. Thus it is not only attractive to use the amplitude freedom to achieve M > M,, but in addition, to use the amplitude freedom to improve the accuracy of

72

DIGITAL HOLOGRAPHY

- COMPUTER-GENERATED

HOLOGRAMS

[I, 5 5

Fig. 46. Simulatedreconstructedamplitudesofblazed analogphase holograms.In (a) the on-axis and (b) the off-axis reconstruction, the efficiency q = 87% in both cases. As shown, a small amount of amplitude freedom was sufficient. Parameters:M = 256; M,= 64.

I G(k)I z 1, compared with M = M , using only the phase freedom (AKAHORI [ 19861). Examples are shown in fig. 46. From the example in fig. 46, it follows that little use is made of the amplitude freedom, and the efficiency is only moderately reduced to q z 90%. It is impossibleto achieve 100% efficiency because amplitude freedom is necessary. Thus it may be concluded that with amplitude and phase freedom it is possible to use the IFTA to synthesize a discrete Fourier pair, which in the object window IF contains the object; that is, Ig(m)lz If ( m - mo)lzin IF and in addition has a constant spectrum I G(k)l = 1 for an arbitrary ratio M / M , . The result is a blazed phase hologram. TURUNEN, VASARAand WESTERHOLM [ 1989al have presented a method for calculate blazed phase holograms with separable distributions based on the simulated annealing algorithm. WYROWSKI[ 1990~1has derived the theoretical limit of the diffraction efficiency of phase holograms dependent on the freedoms available. The limit takes the shape of the object into consideration and is independent of the hologram calculation method used.

-

5.2.3. Quantization of phase hologram

Quantized phase holograms are of interest because of available microlithographic techniques (DAMMANN [ 19691, GALLAGHER, ANGUS,COFFIELD, EDWARDSand MANN[ 19771). It is important to be able to realize phase

o

1, 51

13

CODING PROCEDURES IN DIGITAL HOLOGRAPHY

holograms using only a few quantization steps. Then, G(k) = exp [ir(k)]

quantization

-

G ( k ) = exp[iF(k)]

(5.89)

is desired, and the question arises concerning how in the different coding procedures an additional quantization may be introduced. The binary case was discussed at the end of Q 5.2.2.1 and we will now turn to multilevel (z > 2) quantization. For bleached amplitude holograms, quantized phase values simultaneously imply quantized graylevels in the amplitude hologram. It is important to pay attention to the disturbances due to the bleaching. For a kinoform a quantization always results in large disturbances of the reconstructed object. Because of M = Mf, no amplitude freedom exists, and for quantization noise there is no space available. For blazed phase holograms the situation is different, namely, amplitude freedom is introduced. As with the amplitude hologram, the objective is to proceed from the analog phase distribution r ( k ) ( - II Q r ( k )Q II) of a blazed phase hologram to find a quantized T(k), so that the resulting distribution C(k)= exp [iF(k)] leads to g(m) in the n-plane with Ijj(m)l* If(m - mo)I2 within the object window F. A direct quantization rule is N

-

G(k) = 433[G(k)I

( 2 2 - 1)A < r ( k ) +

11,

(5.90)

with A = n/Z and z = 0,. .., Z. The operation QPZ quantizes the analog phase into Z different values. When the difference between -d(k) i d G(k) is indicated by Q(k) = C(k)- G(k), then

(5.91)

a m ) = g(m) + d m )

(5.92)

is the result in the spatial domain. It is possible to give q(m) in an explicit form, and SILVESTRI [ 19701 using the results from an investigation by GOODMAN and DALLAS[ 19711: q(m) = g(m) [sinc(l/Z) - 11 + q’(m) sinc(l/Z),

with

(5.93a)

74

DIGITAL HOLOGRAPHY a,

q’(m) =

C

[( - l)j/(jZ

- COMPUTER-GENERATED HOLOGRAMS

[I, 5 5

+ I)] {g(m)* * - - j Ztimes * * g(m)} * *

j = 1

-1

+ C

[(- l)j/(jz+ l)] { g * ( - m ) * * * * I j Z + 2 1times..**g*(-m)}

-a

I

(5.93b)

Thus the light in the object window IF is reduced by g(m) [ 1 - sinc (l/Z)], and false images will appear, namely, the sum q’(m) of convolution terms. In order to keep the disturbances due to q’(m) small, the amplitude freedom or the amplitude and phase freedom should be applied to separate q‘(m) spatially from the object, or to further apply the phase freedom to adapt q’(m) to the object. [ 19701 (A) Application of amplitude freedom. GOODMANand SILVESTRI have shown that with the choice of the offset m,, it is possible to separate the center of gravity of q’(m) from the object. This is illustrated in fig. 47. The reconstruction should be placed off-axis with m, # 0, and then it is possibleto achieveq’(m) z Owithin IF. From eq. (5.93a)it is possible to conclude that under the assumption that q’(m) z 0 in IF i j z [sinc(l/Z)l2q

(5.94)

is obtained, where i j is the diffraction efficiency of the quantized phase hologram and q is the diffraction efficiency of the analog phase hologram. Generally q < 100% (an exception is a point object). Equation (5.94) was [ 19701 for the special case of a quantized blazed grating, found by DAMMANN

Fig. 47. Demonstration of the use of amplitude freedom in quantization (Z= 3) of blazed analog phase holograms. In (a) m, = 0 and in (b) m, = ( - 33,33) were chosen. The false image due to j = - 1 is displaced because of the offset. Parameters:M = 256; M,= 64.

I,! 51

CODING PROCEDURES IN DIGITAL HOLOGRAPHY

75

TABLE 3 Comparison between diffraction efficiencies of quantized versions of a blazed grating and the blazed phase hologram of fig. 46; 11 = 100% and 87% are valid for an analog phase variation. Z is the number of phase quantization levels.

where q’(m) = 0 is valid exactly in F (7 = [sinc(l/Z)12 x 100%). In table 3 some q-values are listed. In short, using the amplitude freedom and an appropriate choice of mo, for practical purposes, it is possible to separate the object from false images. The complex error correction and diffusion concept can be used to spatially separate the object from quantization noise, independent of the object position (WEISSBACH, WYROWSKI and BRYNGDAHL[ 19891). However, in both cases noise may remain because higher order terms are superposed onto the object. MAIT[ 19891 has suggested a four level ( z = 4) quantization method based and G ~ R T L E [ 19711 R on a generalization of the grating concept (DAMMANN and DAMMANN and KLOTZ[ 19761). (B) Application of amplitude and phase freedom. WYROWSKI[ 1989a, 1990bl has used an IFTA to achieve quantization of blazed phase holograms of intensity signals by a method corresponding to the one used for amplitude holograms. The operator QZp modified to suit phase values was applied,

Fig. 48. Illustration of how the operator E P is changed between successive steps for Z = 3. The shaded areas indicate the regions not yet considered. These regions decrease with the iteration, which is analog to the case shown in fig. 37.

16

DIGITAL HOLOGRAPHY

- COMPUTER-GENERATED

HOLOGRAMS

[I, 8 5

Fig. 49. Illustration of a quantized blazed hologram in which (a) shows a blazed phase hologram quantized in 3 levels produced by iteration, and (b) shows its simulated reconstructed intensity. Parameters: M = 256; M,= 64; m, = ( - 33,33); Tj = 60%.

namely, U = QpZp,where Q p Z p [G(k)]

=

r(k)+

A

< (1 - &("))A,

(22 - 1 + dp))A < r ( k ) +

A

< (22 + 1 - dp))A,

( 2 2 - 1 + &('))A < r(k) +

A,

I exp[ - i n ] ,

I

exp [ - i n

+ iz(2A)I ,

otherwise, (5.95)

and 0 = d P )< dP- < . ..dp) ... < < 1. With this operator an iteration can be achieved as in the procedure described for amplitude holograms and indicated in fig. 48. The inverse DFT of the analog blazed hologram distribution is used as the initial amplitude. Quantized blazed phase holograms were realized and an example is shown in fig. 49 for 2 = 3.

5.3. CODING IN FRESNEL HOLOGRAPHY

In 5 4.3.1 methods to calculate the Fresnel hologram distribution !(xu) were described. The distribution ) ( x u ) is related to the object by means of a

1, t 51

CODING PROCEDURES IN DIGITAL HOLOGRAPHY

I1

Fresnel transformation; that is, f ( x ) = FRT-

"h U > l

(5.96)

9

where f(x) can be a 3-D object with x = (x, y, z). The aspects of the coding in the case of a Fresnel hologram given in this section are confined to 2-D objects. A generalization of these aspects to 3-D objects is possible. The objective of the coding in the Fresnel holography consists of synthesizing a Fresnel transform pair

(5.97) proceeding from )(xu). It is necessary for &,) to satisfy the boundary conditions in the hologram plane and forg(x) to contain the object distribution; I f ( x - x,,) 1 in the object window (F. that is, g(x) f ( x - no) or Ig(x) I The objective is analog to the Fourier pair synthesis in coding of Fourier holograms with corresponding freedoms of g(x), that is, amplitude and phase freedom, described in the beginning of $ 5.2. There appears to be no investigation of the coding of Fresnel holograms that is comparable in depth to those studies undertaken in Fourier holography. In Fresnel holography the coding was usually performed as in the Fourier case, that is, )(xu) has been coded as would it have been F(u). The consequences of such a procedure are mentioned here for intensity objects in Fresnel holography* As shown in $ 4 for an intensity object i(x) = If(x)12, it is possible to relate the distribution ) ( x u ) to the Fourier transform of f ( x ) , namely, F(u) = F(x,/Az) by means of a quadratic phase factor:

-

-

(5.98)

1

When ( x u )is coded in the same way as F(u) alone, no unpleasant phenomenon occurs as long as the distribution (5.99)

in addition to G(x,/Lz) (where G(x,/Lz) is the coded distribution F(x,/Lz)), both satisfy the required boundary conditions in the hologram plane. Then g(xu) is a satisfactorily coded hologram distribution in Fresnel holography.

78

DIGITAL HOLOGRAPHY

- COMPUTER-GENERATED HOLOGRAMS

11, B 5

Unfortunately,

is only valid for analog phase holograms; that is, the kinoform and analog blazed phase hologram. (FH stands here for Fourier hologram.) Especially for an amplitude hologram G ( x , / l z ) , it follows that g(x,) according to eq. (5.99), is not a real-valued distribution and is therefore completely unsuitable. In the case of a quantized phase hologram G(x,/Az), the introduction of the quadratic phase factor makes &,) analog, which is unsuitable.

Fig, 50. Illustrationsof simulated reconstructionsof Fresnel holograms.The value of z is smaller in (b) than in (a). With decreasingz the spread of the dc-peak and the twin image increases. In the larger portion of the reconstructionplane in (c), aliasing by unavoidable undersampling in the hologram plane is visible. (Courtesy of Bernhardt.)

1. Q 51

CODING PROCEDURES IN DIGITAL HOLOGRAPHY

19

Then, in general, the quadratic phase factor has to be introduced before performing the coding; that is, (5.101)

The Fourier hologram coding in this procedure results in effects different from those in the Fourier holography, in particular depth effects; that is, the terms introduced by the application of amplitude freedom in the Fourier hologram coding, for example, dc-peak and twin image in the amplitude holography, do not appear in the reconstruction plane in the Fresnel holography but occur in other planes in depth. A well-known example is the appearance of the reconstructed object in front of, the dc-peak in, and the twin image behind the Fourier plane of the hologram. Because of phase quantization, false images appear in different planes in depth (DALLASand LOHMANN [ 19721). When depth effects occur, it follows that the terms due to the amplitude freedom may be extensively spread out in the reconstruction plane of the Fresnel hologram. Overlap will appear with the object if parameters like mo and 6u are left unchanged compared with Fourier holography. A modification of the Fourier pair synthesis is necessary if Fresnel pair synthesis will be pursued (BERNHARDT, WYROWSKI and BRYNGDAHL [ 19901). However, in general, these depth effect terms have considerably lower intensity in the reconstruction plane because of their spread. Thus their superposition onto the reconstruction is not necessarily too disturbing. For example, in the Fourier amplitude holography the twin image should be separated from the reconstructed object; that is, an off-axis hologram is necessary. On the other hand, in Fresnel holography it is possible to use an on-axis hologram (Wu and CHAVEL[ 19841). Figure 50 illustrates simulated reconstructions of digital Fresnel holograms.

5.4. CODING IN IMAGE HOLOGRAPHY

In image holography we proceed from a distribution F(u). Of particular interest here is a wavefront F(u) = exp [i$(u)]. Since F(u) is generally complex, a coding procedure is necessary to secure the use of holographic materials that possess a narrow value range. To code and use the digital output for the production of a CGH here as in 5.2, a discrete version F(k) = F(u)comb [u, 6u]

(5.102)

80

DIGITAL HOLOGRAPHY

hdogram

- COMPUTER-GENERATED HOLOGRAMS

filter

(1, s 5

reconstruction

Fig. 51. Illustration of a model of image holography. The desired reconstruction is selected by the window F in the filter plane.

serves as a starting point. However, this is not necessarily the situation where the wavefront F(u) can be given in the form of a function. In the case of IF(k)I = 1, a phase hologram F,(u) = F(k) * rect [u, Su]

(5.103)

can be produced directly. Let us now turn to the question of how to code in image holography. According to the discussion in 5 5.1, in digital image holography it is also necessary to find and formulate the freedoms of F(u). In Fourier as well as Fresnel holography the freedoms were formulated in the reconstruction plane as amplitude and phase freedoms of the distribution g(n). In order to formulate freedoms of F(u) in image holography, we turn to a model of the reconstruction of an image hologram, shown in fig. 51. The reconstruction in image holography may be given in the form of a filtering process. It is convenient to formulate the freedoms of the distribution F(u) in the filtering plane. The coordinates of the filtering plane are given by x = (x, y), and the relation between the object and the filter plane is FT[F(u)] = f ( x ) . This model indicates that even in image holography the coding can be based on the FT synthesis of a Fourier pair F(u) f ( x ) , where f ( x ) is the distribution in the filtering plane. Because of the filtering process, the total distribution f ( x ) does not need to be defined everywhere in the filtering plane. In the case where a filter function a(x) is defined as

-

a(x) =

1,

X€lF

0 , otherwise,

(5.104)

1,s 51

81

REFLECTION OF MATERIALIZATION AND APPLICATION

then the contributions to f(x) outside IF are blocked. Thus they are free to choose, which means that amplitude freedom exists outside IF. The coding in image holography therefore proceeds from the Fourier pair (5.105)

to the synthesis of a new Fourier pair

G(u)-'

FT

(5.106)

where C(u)satisfies the desired boundary conditions and g(x)a(x) f ( x ) a ( x ) is valid. This latter expression is obviously almost identical to the coding problem in Fourier holography. An important difl'erence is that here phase freedom does not exist. Thus all methods that do not rely on phase freedom and are applicable in Fourier holography can be used here too. In the past the following case attracted particular interest: 1 F(u)I = 1;that is, a wavefront exp[i$(u)] is specified in the hologram plane. A coding is performed by introducing a cosine carrier and a bias. This is only one of many possible ways to perform the coding. If a binary hologram is desired, the @ operator (cf. eq. (5.63)) is applied and the result is a binary interferogram. For IF(u)I = 1 the bias B(u) according to eq. (5.67) results in B(u) = 1; that is, a constant amplitude results in a constant bias. Thus when F(u) = exp[i$(u)] is coded in real values, the normalized hologram distribution N

C(U)= 0.5 { C O S [ ~ ~ U $(u)] X,

+ l}

(5.107)

is obtained, and binarization by @ results in a binary hologram; that is, a phase-modulated binary grating. Frequently the special case is of interest where the wavefront is known analytically; then the hologram distributioncan also be analytically determined. LEE[ 19741introduced a method for this case in which a binary interferogram structure can be analytically calculated; that is, the analog distribution G(u) is transformed into c(u). The method consists of a procedure to find the maxima of the interference fringes; that is, to solve 2nux0 - $(u) = 2nn

(5.108)

and then plot the result as fringes. Thus when F(u) is an analytically known function, it is possible to realize the coding continuously.

82

DIGITAL HOLOGRAPHY

- COMPUTER-GENERATED HOLOGRAMS

[I, 8 6

LEE [1979] compared this with other types of binary holograms and suggested how this method can be applied to a discrete F(u);that is, F(k), to solve eq. (5.108).

8 6. Reflection of Materialization and Application The final step in the production of a CGH is its materialization, which is realized by choosing and employing an appropriate material and computer output device. This requires mostly practical considerations, which are largely determined by, and dependent on, the CGH application and existing hardware. In general, CGHs function as diffractiveelements in optical systems, and the desired requirements are much the same as in the IC-mask and pattern production field. The advantages in that field are directly applicable to the materialization of CGHs. CGH structure implementation by using ablating processes, such as etching, embossing or burning, on surfaces of transparent and reflective materials have attractive and desirable features. Some of these techniques are combined with an accessory process, such as coating with a light-sensitive film (photo-lac, photo-resist), as is common in lithography. In photographic materials this coupling is present in a more integrated form, and different processed modes have been used in holography. An abundant number of photosensitivematerials are available (SMITH[ 19771). The photographic films and plates available for optical holography are also convenient and efficient to use in digital holography. In the early days of digital holography the CGH structure was produced by a plotter or printer, or it was displayed on a CRT and then photoreduced to a proper scale. With the increasing demand of holograms with larger space bandwidth products and structure detail dimensions in the neighborhood of the wavelength used in reconstruction, it is necessary to turn to other techniques. To master large space bandwidth products, scanning systems are used. Drum and flat bed laser scanners are used to immediately create exposure patterns onto photosensitive materials in the right scale. Many scanning systems function differently in different directions; for example, they may work in a continuous fashion in one dimension and in a discrete fashion in the other dimension. Furthermore, their precision in positioning is frequently better than the size of the finest spot. To achieve the required resolution, high demands are made on the optical and mechanical systems. The dimensions required compare with those in IC-patterns, and the same or a similar system may be used to produce CGHs. However, the configurations of the patterns in these cases

1 9 8

61

REFLECTION OF MATERIALIZATION A N D APPLICATION

83

generally possess different features. This sometimes causes software problems when using commercially available electron-beam lithography systems and ion exchange techniques to fabricate CGHs. No fundamental limitation exists, and the future looks promising in respect to materialization of high-quality CGH patterns. The required specifications are determined from the specific application of the CGH. These will guide the choice of material and pattern generator. The situations in which digital holography and computer-generated holograms are applied are numerous and farreaching. The hybrid nature of digital holography suggests that its fundamental characteristic of a two-step process with intermediate storage of data can be applied to advantage in computational and holographic schemes. Frequently CGHs are applied as components or devices of larger systems. It is important to be able to optimize their performance and properties. 3-D display, data storage, data processing, interferometry, and scanning have been and remain application fields of CGHs, in addition to their use as optical components (LEE [ 19781). Diffractive optical elements and digital holograms are breeds of optical components the potentialities of which have profited from the same factors. With the advances in microlithographic technology it is now possible to realize the required sizes and precision of the diffracthe structures which are necessary to be met for their utilizations in practical technical situations. This special branch is known as diffractive optics and deals with quantized phase structures to shape wavefields. A close relationship and connection between these elements and quantized phase holograms exist. So far their proposed applications have been slightly different, but no doubt developments in these related fields will benefit from one another. In recent years it has become possible to manufacture structures with every finer details in a controllable fashion. With decreasing dimensions approaching the wavelength of light, eq. (5.2) on which the coding schemes presented here are based becomes a nonacceptable approximation. In the development of methods to treat the optical phenomena and the search for new appropriate coding schemes in a micron and submicron world, the tools and ideas available in digital holography as described here may be valuable and helpful. In short, the particular application of digital holography and a computergenerated hologram narrows down the choice of material and pattern generation scheme. They, in turn, will confine the choice of coding technique, and its optimization will depend on the kinds of parameter freedom that exist. Thus the coding serves to adapt the data to the existing hardware requirements.

84

DIGITAL HOLOGRAPHY

- COMPUTER-GENERATED HOLOGRAMS References

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-

E. WOLF, PROGRESS IN OPTICS XXVIII 0 ELSEVIER SCIENCE PUBLISHERS B.V., 1990

I1

QUANTUM MECHANICAL LIMIT IN OPTICAL PRECISION MEASUREMENT AND COMMUNICATION BY

Y. YAMAMOTO, S. MACHIDA, S. SAITO, N. IMOTO, T. YANAGAWA, M. KITAGAWA and G . BJoRK* Basic Research Laboratories, Nippon Telegraph and Telephone Corporation, Musashino-shi, Tokyo 180, Japan

*

G. Bj6rk is on leave from the Royal Institute of Technology, Stockholm, Sweden.

CONTENTS PAGE

. . . . . . . . . . . . . . . . . . . EMERGENCE OF STANDARD QUANTUM LIMITS . . . NONCLASSICAL LIGHTS . . . . . . . . . . . . . . .

$ 1 . INTRODUCTION

89

$ 2.

90

$ 3.

103

$ 4 . QUANTUM NONDEMOLITION (QND) MEASUREMENT 125

$ 5 . QUADRATURE AMPLITUDE AMPLIFIERS AND PHOTON NUMBER AMPLIFIERS . . . . . . .

. . $ 6 . QUANTUM MECHANICAL CHANNEL CAPACITY . $ 7. APPLICATIONS . . . . . . . . . . . . . . . . . . $ 8. DISCUSSION AND CONCLUSION . . . . . . . . . ACKNOWLEDGEMENT . . . . . . . . . . . . . , . . . REFERENCES . . . . . . . . . . . . . . . . . . . . .

. . . . . .

. . . .

146 158

168 174

. 176

.

176

0 1. Introduction The quantum mechanical limit of optical communication has been an interesting theoretical subject for many years. The quantum mechanical limit of precision measurement for a weak force, such as gravity wave detection, has also been an active theoretical field. The effect of the “granular nature” of electromagnetic waves on Shannon’s channel capacity has been discussed by many authors since the advent of lasers (STERN[ 19601, GORDON[ 19621, LEVEDEV and LEVITAN[ 19631, TAKAHASHI [ 19651, SHE[ 19681, HELSTROM [ 19761, BRAGINSKY and KHALILI[ 19831, YAMAMOTO and HAUS [ 19861). The notion of “squeezed states” has been studied for the reduction of the quantum limit in measuring one observable by sacrificing the information extraction from the conjugate observable (TAKAHASHI [ 19651, STOLER[ 1970, [ 19801, WALLS[ 19831, CAVESand 19711, YUEN[ 19761, YUENand SHAPIRO SCHUMAKER [ 19851). The new strategy “quantum nondemoliton (QND) measurement” has been proposed to exceed the quantum limited sensitivity in repeated measurements (BRAGINSKY,CAVES and THORNE [ 19771, BRAGINSKY, VORONTSOVand THORNE[ 19801, UNRUH [ 19781, CAVES, SANDBERG and ZIMMERMANN [ 19801). THORNE,DREVER, Recently, the most advanced optical communication technologies, such as coherent communication (YAMAMOTO [ 1980a]), laser amplifier communication (YAMAMOTO [ 1980b]), and photon counting communication (PIERCE, [ 1981]), are now approaching the standard quantum POSNERand RODEMICH limit (SQL), the so-called shot noise limit. The optical precision measurement technologies such as a passive interferometer gyroscope (EZEKIEL,COLE, and SANDERS[ 1978]), active laser gyroscope (DORSCHNER, HARRISON HAUS, HOLZ, SMITH and STATZ [ 19801, CHOW, GEA-BANACLOCHE, PEDROTTI,SANDERS,SCHLEICH and SCULLY[ 19851) and a gravitational wave detection interferometer (HOUGH [ 19831, BILLING, WINKLER, SCHILLING, RUDIGER,MAISCHBERGER and SCHNUPP[ 19831) are also close to the SQL. The purpose of this paper is to study the feasibility for circumventing the SQLs in optical precision measurement and communication by new strategies such as squeezed states and QND measurement. The paper is organized as follows. Section 2 describes how the SQLs in 89

90

QUANTUM MECHANICAL LIMlT IN OPTICS

[II, 0 2

optical precision measurement and communication emerge. It is pointed out that the three different SQLs imposed by the Heisenberg uncertainty principles on photon generation, detection, and amplification are now being approached by the existing technologies. In 0 3 the two kinds of “nonclassical light”, such as “quadrature amplitude squeezed states” and “number-phase squeezed states (number states)”, are demonstrated to exceed the SQL on photon generation. The principles of and experiments in generating these nonclassical lights are reviewed. The other approach of generating a “correlated photonpair” is also discussed. In § 4 new detection schemes, such as “QND” (or “back action evading”) measurements, are shown to overcome the SQL on photon detection. The physical realization and experiment of QND measurement for photon number and quadrature amplitude are reviewed. The other approach of “contractive state measurement” (YUEN [ 1983b], WODKIEWICZ[ 19841, LYNCH [ 19841, CAVES[ 19851, NI [ 19861) is also delineated. Section 5 discusses how to surpass the SQL on photon amplification and introduces “quadrature component amplifiers” and “photon nylmber amplifiers”. In fi 6 the ultimate performance of quantum precision measurement and communication is evaluated. Quantum mechanical channel capacity and Bohr’s time-energy uncertainty relationship are discussed for this purpose. Finally, we will describe some applications in $7.

0 2. Emergence of Standard Quantum Limits In this section we will discuss the SQLs imposed on generation, detection, and amplification of electromagnetic fields. The present states of the art in optical communication and precision measurement are already at the point where these SQLs become observable.

2.1. COHERENT STATES AND QUANTUM NOISE OF LASER EMISSION

A classical electromagneticfield is described by two quadrature phase amplitudes a , and a,:

E(t) = I ( a , cos wt

+ a, sin wt) ,

(1)

where I is a constant, The values a , and a, can be assigned to certain values simultaneously in a phase space as shown in fig. la. Quantum mechanically, however, d , = (d + dt)/2 and d, = (d - dt)/2i are conjugate observables and

1 1 3 8 21

EMERGENCE OF STANDARD QUANTUM LIMITS

Fig. 1. Light waves in a ,

91

- a 2 phase space and in time domain.(a) Classical field, (b) coherent state field, and (c) ideal laser field.

must obey the commutation relation [d, d + ] = 1

=.

(2) Here a Hilbert space operator is denoted by a caret “ ”. It is easily derived from the Schwartz inequality that the uncertainty product of d , and d 2 should be larger than the minimum allowable value (MESSIAH[ 19611): (Ad:)

(Ad:)

[ d l yd2] = t i

3

I ( [ d , , d,] ) 1’

=

&.

(3)

The state that satisfiesthe equality in eq. (3) is called “minimum uncertainty state”. One of such a minimum uncertainty states is a coherent state 1 a ) ,which [1963], is an eigenstate of a photon annihilation operator (GLAUBER SUDARSHAN [ 19631, KLAUDERand SKAGERSTAM [ 1987]), dla) =ala).

(4)

The coherent state satisfies the minimum uncertainty product by sharing equal amounts of noise in the two quadratures (Ad:)

(Ad:)

=$ =$

sl (minimum uncertainty product) , d

as shown in fig. lb.

(5)

92

QUANTUM MECHANICAL LIMIT IN OPTICS

16‘

I

Id

100

Normalized Frequency

n/#)

Fig. 2. Amplitude (solid line) and normalized phase (dashed line) noise spectra of a laser field. r = P/P, - 1 is a normalized pump rate.

Glauber has shown that a classical current source generates a coherent state (GLAUBER [ 19651). In a more realistic context the oscillating dipole moment (“dipole current”) in a laser medium far above its threshold generates a sequence of states that are not too far from a coherent state (have little excess noise above the quantum noise). The amplitude and phase fluctuation spectra of a laser oscillator output field are shown in fig. 2 as a function of the pump rate (YAMAMOTO, IMOTO and MACHIDA [ 19861). The amplitude noise spectra approach the standard quantum limit of a broadband coherent state as the pump rate increases: sAr(a)

4

(p/pth

% l) *

(6)

The phase noise spectrum normalized by the output power, on the other hand, does not depend on the pump rate and features the excess noise proportional in a lower frequency region below the cavity bandwidth: to

The first term corresponds to the intrinsic quantum noise and the second term to a phase diffusion process, that is, a Schawlow-Townes linewidth of a laser emission. The phase of the successive states thus goes through a “random walk”, but it can be suppressed if part of the same laser emission is used as

KO 21

EMERGENCE OF STANDARD QUANTUM LIMITS

93

a phase referencewave, or it can be phase locked to a “local oscillator” master laser. Thus one may say that the generation of coherent states for optical precision measurement and communication is approximately realizable by a laser operating far above threshold. An ideal laser field is schematically shown in fig. lc. The commutation relation and the uncertainty relation for photo number and phase are often written as

[A,$] = i

=.

(AA~)

(AP)a $ .

(8)

Quantum mechanically this is “not” a correct expression. It can be easily proved that a Hermitian phase operator does not exist (SUSSKINDand GLOGWER [ 19641). The proper expression is

e

and NEITO where and are the sine and cosine operators (CARRUTHERS [ 19681). To be exact, a coherent state does not satisfy the equality in (9), so that it is not a “number-phase minimum uncertainty state”. Nevertheless, when the averagephoton number is much greater than unity, a coherent state satisfies the “approximateyyminimum uncertainty relationship (Ah2)

=

(A) %

(minimum uncertainty product) .

Equations (5) and (10) are referred to as standard quantum limits (SQLs) for quadrature amplitude, photon number, and phase. When a laser output field is measured by a photon counting detector, Poissonian photon statistics are obtained. This is a result of the photon number noise of coherent states,

where (A, ) is the average signal photon number. When a laser output field is measured by an optical homodyne detector, Gaussian distributions are obtained. This results from the quadrature amplitude noise of coherent states, 1 Pr(a,)= I(% Ia>I2=-exp[-(a, - (a,>)21, II

(12)

94

QUANTUM MECHANICAL LIMIT IN OPTICS

“off”

4

Pr (01,) = exp[-(a, - (%))‘I “O-phOSe” (Gaussian)

-(al)

10

(al)

>

af

j L P r (a,)dal = “error” (b)

Fig. 3. Errors in (a) photon counting detection and (b) homodyne detection for a coherent state field.

where ( B ) is the average quadrature amplitude. These statistical properties of a laser emission, shown in fig. 3, are the simple manifestations of the quantum noise of coherent states.

2.2. OPTICAL COMMUNICATION AT STANDARD QUANTUM LIMIT

Suppose an on-off intensity modulated signal must be received with a bit error rate of lo-’, the average signal photon number for the “on” pulse should be greater than 21 due to the Poissonian distribution (see fig. 3a), P,

= Pr(n = O/(A,))

=

exp( - (A,))

< lo-’*

(A,) k 21.

(13)

When “on” and “off pulses are sent with equal probabilities, the average signal photon number must be greater than 10.5. When a 0-n phase modulated signal must be received with the same bit error rate, the signal photon number should

11,s 21

95

EMERGENCE OF STANDARD QUANTUM LIMITS

be greater than 9 because of the Gaussian distribution (see fig. 3b), P,

= J[

0

Pr(a,) da,

=

e r f c ( d m )<

*

(tis) 3 9.

(14)

-03

These limits imposed on “photon counting communication” and “coherent communication” originate from the uncertainty relation (10) and (5), respectively. These minimum required signal photon numbers place a limit on the achievable receiver sensivity in optical communication (YAMAMOTO [ 1980al). How close is the current state of the art in optical communication technologies to such a SQL? To date, the best receiver sensitivity obtained for coherent communication has been 45 photonsbit (LINKE[ 19871). The best receiver sensitivity obtained for photon counting communication has been 50 photons for an on pulse (LEVINEand BETHEA[ 19841). The above SQL is now being approached by the existing technologies.

2.3. GRAVITATIONAL WAVE DETECTION INTERFEROMETER AT STANDARD QUANTUM LIMIT

Let us consider the two gravitational wave detection systems shown in fig. 4. Suppose a gravitational wave is propagating in the x direction. The correction to the metric space-time from a flat one modifies the Maxwell equations, and the resulting dispersion relations are (SCULLY and GEA-BANACLOCHE [ 1986]),

and

k$ =

(2)

2*

Here h, is the strength of the gravitational wave, W , is its frequency, and k , is its wave number. In the case of a passive interferometerdriven by an external light source with fixed frequency, the above dispersion relations are considered as a change in wave number,

k, - k ,

=

(E)

h , cos(o,t - k,x)

.

96

QUANTUM MECHANICAL LIMIT IN OPTICS

Fig. 4. Gravitational wave detection interferometer. (a) Passive interferometer and (b) active laser interferometer.

The phase shift due to a gravitational wave is given by multiplying this change in wave number with the effective path length, which is the length 1 of the arm times the number b of bounces, A+= = (w/co)blho.The phase measurement error is determined not by the phase noise (Ad; ) /A; of a laser beam itself but by the quadrature amplitude fluctuation ( A t ; ) of a vacuum field incident on a beam splitter from an open port (EDELSTEIN, HOUGH,PUGHand MARTIN [ 19781, CAVES[ 19801, LOUDON[ 19801). The rms measurement error of a phase shift is (A+:)'/' = l/(As)'12 = (ho/Pz)'12. Here (A,) = A : is the total average signal photon number, P is the total average signal wave power, and z is the measurement time. The minimum detectable gravitational wave amplitude, defined by the signal-to-noise ratio, +,2/(A+:) = 1, is given by (passive, vacuum fluctuation limit) . Here y1 is co/bl is the effective photon decay rate of each arm.Note that the

K8 21

97

EMERGENCE OF STANDARD QUANTUM LIMITS

phase diffusion noise of a laser does not enter the quantum limit of a passive interferometer. In the case of an active system consisting of two independentlaser oscillators, the dispersion relations are considered as a change in frequency, w1 - w2 = - zcokho 1 cos(o,t - k,x)

.

(19)

The beat frequency measurement error is determined by the phase diffusion noises (Schawlow-Townes linewidths)of the two lasers. The rms measurement Here y2 = w / Q is the error of a frequency offset is (Am: ) = 2y,(h~/Pz)'/~. photon decay rate of the laser cavity in this case. The minimum detectable gravitational wave amplitude is given by (active, phase diffusion noise limit) . (e) Pz 1/ 2

h(") mm = ?! 0

The two systems have an identical sensitivity, if y1 = y2. According to (17) and (20), the measurement accuracy improves with increasing the laser power. However, this is not generally the case but, rather, the best measurement accuracy is obtained at optimum laser power. This is because the detection noise is offset by radiation-pressure-induced fluctuations that increase as the power increases. The free mass M attached to the mirror is kicked by the random momentum of the signal photons, (Ap2 ) = (2bhw/~,)~ (A2 ) . The radiation pressure noise in a passive system is not determined by the photon number noise of the laser beam itself but by the in-phase fluctuation (Ak: ) of a vacuum fluctuation. If the measurement is made discretely rather than continuously over a time interval z, then the measurement accuracy of the free mass position is (CAVES[ 19801, LOUDON[ 19801)

at the optimum signal photon number (As)opt = Mc2/8h02b2z. For reasonable numerical parameters such as M = 102kg and z=2x sec, w = 4 x l O I 5 rad/sec and b = 200, the required laser power to reach the best measurement accuracy, eq. (21), is of the order of 10 kW, which is far beyond the present state of the art in a single-frequencyCW laser oscillator. Moreover, the measurement accuracy, eq. (21), itself is often not sufficient to detect a typical gravitational wave (CAVES,THORNE,DREVER, [ 19801).Thus it is fair to say that these SQLs SANDBERG and ZIMMERMANN prohibit the interferometric detection of a gravitational wave in two ways.

-

98

QUANTUM MECHANICAL LIMIT IN OPTICS

2.4. LASER GYROSCOPE AT STANDARD QUANTUM LIMIT

The last example to be discussed is a ring laser gyroscope, shown in fig. 5. The idea of using a ring interferometer as a rotation rate sensor is based on the Sagnac effect (SAGNAC[ 19131). There are two types of ring laser gyroscopes, that is, an active one and a passive one. In an active ring laser gyroscope, shown in fig. 5a, a laser medium is put inside a ring cavity, and the rotation rate can be measured by means of the oscillation frequency difference for a clockwise and counterclockwise wave. The measurement accuracy 6nof the rotation rate is determined by the Schawlow-Townes linewidth of the laser emission. The two counter-propagating waves experience uncorrelated phase diffusion processes, so that the resulting finite linewidths impose the SQL on the rotation rate measurement accuracy given by (DORSCHNER, HAUS, HOLZ, SMITHand STATZ[ 19801; HAMMONSand ASHBY[ 19811, CHOW,GEA-BANACLOCHE, PEDROTTI,SANDERS,SCHLEICH and SCULLY[ 19851)

Here S = 4A/LL is the Sagnac constant, 3, is the oscillation wavelength, L is

(b)

Fig. 5. Ring laser gyroscope. (a) Active type and (b) passive type.

11, B 21

EMERGENCE OF STANDARD QUANTUM LIMITS

99

the cavity length (perimeter),A is the area of the cavity, y = w / Q is the photon decay rate, P is the laser power, and z is the measurement time interval. In a passive ring cavity gyroscope, shown in fig. 5b, a laser beam enters an empty ring cavity in order to sense the resonant frequency difference for a clockwise and counterclockwise wave. The random-walk phase diffusion noise can be suppressed in this case if the two beams are prepared from the identical laser oscillator. When the two laser beams are prepared by a beam splitter, as shown in fig. 5b, the vacuum field incident on the beam splitter from an open port introduces this quantum phase noise in the same manner as a passive interferometric gravitational wave detector. If the photon decay rate y is interpreted as that of the cavity, the measurement accuracy for a passive ring cavity gyroscope is identical to that for an active ring laser gyroscope (eq. (22)), even though the origins for the SQL are different (LIN and GIALLORENZI [ 19791, DAVISand EZEKIEL[1978], EZEKIEL, COLE, HARRISONand SANDERS [ 19781). The experimental measurement accuracies of the rotation rate are close to the preceding S Q L , both by the active four frequency laser gyroscope (DORSCHNER, HAUS,HOLZ, SMITHand STATZ[ 19801) and by the passive and SANDERS[ 19781). ring cavity gyroscope (EZEKIEL, COLE, HARRISON Hence the present laser gyroscope technology is already at the point where the SQL becomes a real technological problem.

2.5. FLUCTUATION-DISSIPATION THEOREM AND SIMULTANEOUS MEASUREMENT OF TWO CONJUGATE OBSERVABLES

When the information carried by an optical signal wave is extracted by a waveguide tap, part of the signal power must be taken from the transmission line as shown in fig. 6a. Since an output mode as well as an input mode must preserve a proper commutator bracket, [&in,

bit,]

= [do,,,

d&tI

=

1

9

(23)

the attenuated output mode accompanies a newly added vacuum field fluctuation, do,, = 8, + J1-r E . (24)

JT

Here T is a power transmission constant, and 2 is the vacuum field operator and satisfies [t,Et]=l

and

(E)=O.

(25)

100

QUANTUM MECHANICAL LIMIT IN OPTICS

e

readout (0)

to*

60°C

Fig. 6. Standard quantum umit on photon measurement. (a) Waveguide tap and (b) quantum noises of input and output waves.

This is the quantum mechanical fluctuation-dissipation theorem (SENITZKY [ 19701). As a result of the added vacuum fluctuation, quantum noise is not decreased even though coherent excitation (“signal power”) is attenuated as shown in fig. 6b. Thus the signal-to-noise (S/N) ratio is degraded. It is expressed, for instance, for coherent homodyne detection by (S/N)OUt =

(a0lIt.J’

w:ut.

1

)

-

>’

T(ain, 1 T ( A G 1 ) + (1 - T ) ( A m

(26)

When the input signal is in a coherent state, its quadrature fluctuation (Ah:, ) is equal to the vacuum field quadrature fluctuation ( A 2 : ) = :. Therefore the SIN ratio is degraded by the attenuation, (S/N)out+ T(S/N),, (coherent state input) .

(27)

The S/Nratio of the readout signal, on the other hand, increases with decreasing T,

11,s 21

EMERGENCE OF STANDARD QUANTUM LIMITS

101

Thus the SIN ratio is divided into the readout SIN ratio and throughput SIN ratio. The maximum number of waveguide taps in an optical communication loop or the maximum allowable repeated measurements in an optical precision measurement is thus determined by the preceding SIN ratio degradation. This is one form of the SQL imposed on photon detection. Suppose the information of d 1 and d 2 is measured simultaneouslyby a 50-50 beam splitter followed by two optical homodyne detectors. The measurement accuracy (Ad: )meas of h is not only determined by the quadrature amplitude quantum noise ( A d : ) of the signal wave but also by the vacuum field quantum noise (Ae:),

and, similarly, we have (AhzZ)rneas

=

(AdzZ) + (At,’)

*

(30)

The product of the measurement accuracies is thus given by (Ah2)1/2 (Ad2)1/2 1 meas 2 meas

-1 -29

(31)

which is twice as large as the Heisenberg minimum uncertainty product. This doubling of the uncertainty product in the simultaneous measurement of 6 and 6, occurs for a variety of detection schemes (YAMAMOTO and HAUS[ 19861). It is the so-called generalized uncertainty principle for simultaneous measurement of two conjugate observables (ARTHURSand KELLY[ 19651, SHE and HEFFNER[ 19661, GORDON and LOUISELL [ 19661) and is the other form of the SQL imposed on photon detection. The foregoing discussion clearly demonstrates that the uncertainty principle is twofold; that is, one uncertainty principle emerges when a photon is generated and the other one is required independently when a photon is detected. In the next section we will see the third uncertainty principle imposed on photon amplification.

2.6. MINIMUM NOISE FIGURE FOR LINEAR AMPLIFIERS

As far as a signal power is concerned, “loss” could be compensated for by a linear ampliiier. Commutator bracket conservation (20), however, requires that an additional fluctuation be added to the amplified signal,

do, = @din

+J

z z 27,

(32)

102

QUANTUM MECHANICAL LIMIT IN OPTICS

[II, 8 2

where the added noise operator satisfies [ d " , d t ] = l and

(33)

(d")=O.

The origins of the added noise d^ depend on the physical mechanisms of the amplifiers. They are, for instance, dipole moment fluctuations for laser amplifiers (YAMAMOTO,MACHIDAand NILSSON[ 1986]), phonon mode fluctuations in Raman and Brillouin amplifiers (LOUISELL,YARIV and SIEGMAN[ 1961]), and zero-point field fluctuation at idler frequency for parametric amplifiers (CAVES[ 19821). Under ideal operating conditions the noise levels ( A d" f ) (i = 1 or 2) are all equal to the vacuum field fluctuation level. As a result of this additional fluctuation, shown in fig. 7, the SIN ratio is degraded by

When the input signal is in a coherent state and the amplifier additional fluctuation is its minimum level, the SIN ratio degradation is (S/N)out+ ; ( S / N ) ,

(coherent state input + ideal amplifier) .

(35)

This is the amplifier uncertainty principe (HAUSand MULLEN[ 19621). The minimum noise figure F = (S/N)in/(S/N)outfor linear amplifiersis not 0 dB but 3 dB.

I

I

ci

Fig. 7. Standard quantum limit on photon amplification. (a) Photon amplifier and (b) quantum noises of input and output waves.

11,s 31

NONCLASSICAL LIGHTS

103

The best noise figure obtained for semiconductor laser amplifiers is 5.2 dB, which is only 2.2 dB above the quantum limit (SAITOHand MUKAI[ 19881).

0 3. Nonclassical lights In this section we will discuss the properties, generation schemes, and applications of the three nonclassical lights. They can circumvent the SQL imposed on photon generation. 3.1. MINIMUM UNCERTAINTY STATES

The basic concept for surpassing the SQL on photon generation is a rather old notion of quantum mechanics, “generalized minimum uncertainty state” (MESSIAH[ 19611). The generalized Heisenberg uncertainty principle for two conjugate observables 8, and 6,is expressed as

[6,,6,]= i d , (Ad:> ( A & > 241(8,>12. (36) Here Ad, = 6, - (6,>,i = 1 or 2. A minimum uncertainty state that exactly satisfies the equality in (36) is mathematically defined as an eigenstate of the operator

8 ( r ) = er8, + ieTr6,,

(37)

where r is a “squeezing” parameter. The squeezing parameter determines the distribution of quantum noise between 8, and 6,:

(Ad:> = 4<8,>e-,‘ %A

( A @ > = 4(8,>e2’

(minimum uncertainty product).

(38)

d

Quantum mechanics does not necessarily require that the uncertainty product be shared equally by the two noncommuting observables. In optical precision measurement and communication only one quadrature amplitude d or only one photon number A is generally used to extract information; the conjugate observable is not utilized. For such usages it is helpful and possible to suppress one quadrature amplitude noise by sacrificing the other quadrature amplitude noise or to decrease photon number noise by enhancing phase noise. The former is a quadrature amplitude squeezed state and the latter is a number-phase squeezed state.

104

QUANTUM MECHANICAL LIMIT IN OPTICS

PI, § 3

3.2. PROPERTIES OF QUADRATURE AMPLITUDE SQUEEZED STATES

A quadrature amplitude squeezed state of the electromagneticfield is one of

minimum uncertainty states, in which 6 ,and 6, correspond to the two quadrature amplitudes d , and d,, and 6, is then a c-number ( = i).This squeezed state features reduced quantum noise in one quadrature amplitude and enhanced quantum noise in the other quadrature amplitude (TAKAHASHI [ 19651, STOLER[ 1970, 19711, YUEN[ 19761). The quasi-probability density (Q-representation) defined by Q(4=

(4

8,la)

,

(39)

for the squeezed state is shown in fig. 8. In the special case of r = 0, the two quadrature amplitudes share the same amount of quantum noise, (Ad:) = (Ad:) = that is, a coherent state is a special case of squeezed states. The conventional diagonal coherent state expansion (P(cr)-represen-

a;

Fig. 8. Quadrature amplitude squeezed states in a I - a2 phase space and in time domain.

11,s 31

105

NONCLASSICAL LIGHTS

tation), defined by

does not exist for the squeezed states (WALLS[ 19831);that is, a squeezed state cannot be described by a classical mixture of coherent states, which is why a squeezed state is called a “nonclassical light”. Quadrature amplitude squeezed states feature Gaussian distribution for coherent homodyne detection, and the variance in one quadrature is below that of a coherent state, (A4: ) = ,ec2‘, and the other is above that, (Ad: ) = 4e2’ (r > 0). In order to maximize the SIN ratio for a given average photon number, the squeezing parameter should assume its optimum value (YUENand SHAPIRO[ 1980]), ropt =

W

d

m

)*

(41)

The S/N ratio for such an optimum squeezed state is far better than that for a coherent state: (S/NL = 4 (A, )

-+

(SIN),, = 4 ( % ) ( ( A S )

+ 1)

(homodyning) *

(42) The difference is large when (A, ) is much greater than one, as in the case of optical precision measurement. The improvement is modest, however, for optical communication application. The required average signal photon number for a bit error rate of 10- for a 0-n phase modulation optical communication is decreased from nine for coherent states to three for such an optimum squeezed state. The reason why the optimum squeezing parameter is not infinite but, instead, is given by (41) is that the enhanced quadrature noise “wastes” part of the signal photon number. This can be seen by the fact that the average signal photon number for the squeezed states is given by ~

(fi)ss = ( d , ) 2

+ ( S 2 ) 2 + sinh2(r).

(43)

There is a trade-off relation between quantum noise reduction and available coherent excitation ( d , ) ’. It is obvious from eq. (43) that a quadrature amplitude eigenstate with zero quantum noise (Ad:) + 0 (r-l a)is realized only when the signal photon number becomes infinite. Quadrature amplitude squeezed states feature either sub-Poissonian or super-Poissonian distributions for photon counting detection, depending on

106

QUANTUM MECHANICAL LIMIT IN OPTICS

PI, 8 3

the squeezing direction and strength. The rather complex expression for the photon number distribution of a squeezed state has been given in the literature (STOLER[ 1970, 19711, YUEN[ 19761). Squeezing in the quadrature amplitude carrying a coherent excitation results in a sub-Poissonian distribution, but excessively stronger squeezing rebroadens the photon number distribution to super-Poissonian, The minimum variance of photon number distribution is approximately (AA2 )ss N ( A ) 2/3, For stronger squeezing the photon distribution starts to oscillate because of the coherent interference effect between a “signal‘s squeezed state” and “detector’s number state” in a phase space (SCHLEICH and WHEELER [ 19871). Squeezing in the other quadrature always results in a super-Poissonian distribution.

3.3. GENERATION OF QUADRATURE AMPLITUDE SQUEEZED STATES

Quadrature amplitude squeezed states can be generated by unitary evolution from a coherent state. The interaction Hamiltonian required for realizing this unitary evolution is known to be quadratic in photon annihilation and creation operators (STOLER[ 1970, 19711, YUEN [ 19761) H,

= ih(Xdt2

+ x *P).

(44)

The Hamiltonian function expresses simultaneous two-photon generation and absorption processes, which can be realized by the second- or third-order nonlinear processes. The nonlinear interaction parameter x is expressed as (WALLS[ 19831)

x={

x(’)E, (degenerate parametric down-conversion) , f3)E,Z (degenerate four-wave mixing) .

(45)

Here E , is the pump field amplitude (c-number), and xC2)and x(3) are the second and the third order susceptibilities. The unitary operator 0 due to such “two-photon interaction” is given by

where c is the light velocity and L is the interaction length. The Heisenberg operator for the output mode is written as dOut= iYdinOt = cosh(r)d, - e”sinh(r)@, ,

(47)

11,s31

107

NONCLASSICAL LIGHTS

where x(L/c) = ire” ( I is a real number) and the phase 8 depends on the phase of the pump wave. When the pump phase is adjusted to realize 8 = 0 and the input mode din is in a coherent state, the variances in the output mode become {Ad:,,,,,

,) = i e - 2 r ,

=

2)

:e2*.

(48)

This is the principle of quadrature amplitude squeezed state generation. Physical realization of the interaction Hamiltonian (44)is either a degenerate parametric amplifier or a degenerate four-wave mixer, as depicted in fig. 9. In a degenerate parametric amplifier a pump photon of frequency wpis converted to two nearly degenerate photons of frequencies w, and wi (w, x wi) where w, = w, t mi. The effective nonlinear interaction requires the phase matching condition, k, = k, t ki.Usable nonlinear crystals and possible choices of pump and signal frequencies are limited so as to satisfy the phase matching condition (BYER [ 19731). In order to achieve effective nonlinear interaction with a relatively small x(’) coefficient, a high-Q cavity is usually employed, as shown in fig. 9. In a high-Q cavity, which is a degenerate parametric amplifier and oscillator, the squeezed states generated have limited bandwidth. The normalized quadrature amplitude noise spectra for a degenerate parametric amplifier and oscilla-

kP

’ -6-7-

R,<1

Rs.1

kr

‘

ki

’

Rp = 1

RE<^

state

WP

Fig. 9. A degenerate parametric amplifier and four-wave mixer.

108

[II, 5 3

QUANTUM MECHANICAL LIMIT IN OPTICS

tor are given by (COLLETT and WALLS [1985], BJORK and YAMAMOTO [ 1988a-c]) 4Q2)' + 46-22

(

4s2+ 1 - S 2 ? P + Sab, = 2

(1

(5); -

2

Y s Yp

-)

-)Y.

+ s2Tp - 4 n 2 + 46-22 (i 1+1TP2

(49)

Yp

Here

2 ic (A, ) s=

P=

fi 2 ic (A, ) '12

(normalized signal wave amplitude),

(normalized pump wave amplitude),

Ys

where ys is the signal photon decay rate, y, is the pump photon decay rate, 6-2 is the fluctuation frequency, and K = f2)/2i. The pump quantum noise, which is neglected in eqs. (44)and (49, is included in eq. (49). Figure 10 shows the normalized quadrature amplitude noise spectra as a function of normalized pump rate t = 4 ic (flp) 'I2/( ys yp)''2. Here (flp) is the average incident pump photon flux (not the average internal pump photon number (A, ) ); t = 1 corresponds to the oscillation threshold. The noise level

962

1oo

lo2

Normalized Frequency n/Ts

Fig. 10. In-phase and quadrature phase amplitude spectra of a degenerate parametric amplifier. t = P/P,,is a normalized pump rate.

11,s 31

NONCLASSICAL LIGHTS

109

below unity indicates the “squeezing” of quadrature amplitude noise. The maximum noise reduction is obtained at zero frequency D = 0 and at the pump level of the oscillation threshold. The bandwidth for squeezing at t = 1 is determined by y,. Decreasing y, will decrease the threshold pump power but also decreases the effective bandwidth for squeezing. A travelling-wave type of degenerate parametric amplification promises broadband squeezed state generation. Above the oscillation threshold the squeezing at near dc decreases with the pump rate. The maximum squeezing occurs at a so-called relaxation oscillation frequency, where the other quadrature carrying a coherent excitation features the noise peak resulting from the resonant coupling effect of the pump photon and signal photon. Quadrature amplitude squeezed state generation in a degenerate parametric amplifier has been demonstrated by Kimble and his collaborators (Wu, KIMBLE,HALLand Wu [ 19861). A MgO : LiNbO, crystal (a-axis) put inside a high-Q cavity was pumped by the second harmonic generation (SHG) of a 1.06 pm Nd : YAG laser. The noise level measured by a balanced homodyne detector was 60% ( - 4 dB) below the SQL. This is the most effective squeezing observed so far. They also confirmed the minimum uncertainty product of the two quadrature amplitudes. If the effect caused by a detection circuit quantum efficiency and cavity internal loss was corrected, the noise level was 90% ( - 10 dB) below the SQL. In a degenerate four-wave mixer two pump photons of frequency opare converted to two nearly degenerate signal photons of frequencies a,and oi (o,x wi), where 20, = o,+ oi. The phase matching condition 2k, = k, + ki, can be easily realized by choosing an appropriate angle between the pump, signal, and idler waves, because the four photons have nearly the same frequency. Thus the severe limitation on pump and signal wavelengths and usable crystals can be lifted in a degenerate four-wave mixer. Two kinds off3) materials have been used for the purpose. One scheme uses the resonant x(3) process, in which the laser frequency is close to the atomic (or molecular) resonance. The x(3) coefficient can be enhanced by the resonance, but “incoherent” processes such as absorption and spontaneous emission generally compete with the “coherent” four-photon process. The first experimental observation of quadrature amplitude squeezed states (SLUSHER, HOLLBERG, YURKE,MERTZand VALLEY[ 19851) utilized the resonant xC3) process in Na atomic beam for the four-wave mixing process. The D, line of Na atomic beam in a high-Q cavity is pumped by a slightly detuned single frequency 0.5145 pm Ar laser. The observed noise level was 16% below the SQL. The residual noise mainly stems from an incoherent spontaneous

I10

QUANTUM MECHANICAL LIMIT IN OpTlCS

[II, § 3

emission. Squeezed state generation in a completely resonant condition has also been reported using a traveling-wave type of interaction in a Na vapor (MAEDA,KUMARand SHAPIRO[ 19871). The other scheme utilizes a nonresonant f 3 ) process, in which the laser frequency is detuned far from the resonance. The xc3) coefficient is then generally smaller, but the incoherent processes do not contaminatethe coherent four-wave mixing process. Squeezed state generation has been reported using a travelling-wave type of four-wave mixing in a single-mode, silica-rich fiber (SHELBY,LEVENSON,PERLMUTTER, DEVOEand WALLS[ 19861). The fiber was cooled to 1.4 K to suppress the phonon scattering noise and was pumped by a phase-modulated single-frequency0.65 pm Kr laser. The observed noise level was 17% below the SQL. The residual noise mainly stems from the phase noise due to a guided acoustic wave Brillouin scattering (GAWBS) process (SHELBY,LEVENSONand BYER[ 19851). In the afore-mentioned generation experiments the relaxation rate of the nonlinear medium is large compared with the cavity decay rate. The nonlinear dipole moment then adiabatically follows variations in the cavity field, which is usually the case for laser physics and nonlinear optics. In the microwave domain, however, the relaxation rate of Rydberg atoms is relatively low, and the coupled atom-field system features the eigenmode structure with the vacuum field Rabi splitting g@ ( U L U Z W , GOY,GROSS,RAIMONDand HAROCHE[ 19831, MESCHEDE,WALTHERand MULLER[ 19851, REMPE, WALTHER and KLEIN[ 19871). Heregis the field-atom coupling coefficient and N is the number of atoms. This means that there is a coherent exchange of excitation between the sentive atoms and field, which leads to phase-dependent amplification and deamplification required to generate the squeezed states (HEIDMANN,RAIMONDand REYNAUD[ 19851). The interaction Hamiltonian for this process reads =

hg(La+

+ $+a),

where .f* is the collective atomic operator for the N atoms that lowers and raises the atomic excitation. This mode of generating the squeezed states has been demonstrated even in the optical frequency domain (RAIZEN,OROZCO,XIAO, BOYDand KIMBLE [ 19871, OROZCO,RAIZEN,XIAO,BRECHAand KIMBLE[ 19871). This experiment realizes the preceding dynamics of the coupled atom-field system by satisfying the condition,

11, S 31

111

NONCLASSICAL LIGHTS

where K is the cavity decay rate, yl is the atomic dipole decay rate, and y is the atomic inversion decay rate. The noise was reduced by 30% ( - 1.55 dB) below the SQL at the Rabi flopping frequency $2 N g f i .

3.4. PROPERTIES OF NUMBER-PHASE SQUEEZED STATES

A number-phase squeezed state of the electromagnetic field is the other kind of minimum uncertainty states discussed in 3 3.1. The observables 6 , and 6, correspond to the photon number and sine operators, and 6, is then the cosine operator defined by

8, = fi = 6 t h ,

(54)

1

6, = S = 7 [(A + 1)21

8, = e = $[(A + 1)-

1/2d

+

- dt(A + 1)-

1/2d d+(A

1/21

+ 1)-’/2]

,

,

(55) (56)

The normalizable number-phase minimum uncertainty state can be constructed mathematicallywhen ( 3 ) = 0 (JACKIW[ 19681). However, the average photon number ( A ) and the ratio of photon number noise to sine operator noise ( AA2 ) / (ASZ) cannot be chosen independently but, rather, should satisfy the constraint I-1-[(-)1i2]=o* (AA2)

(AS,)

(57)

where I J x ) is a modified Bessel function of the fist kind of order p. From the constraint in eq. (57), ( A ) must be chosen as ( A ) E [2k, 2k + 11 (k = 0, 1,2, 3 , . ..). The photon number noise (AA’) goes to zero at ( A ) = 0, 1, 2, .... This is a photon number state, even though it does not satisfy the number-phase minimum uncertainty product in an exact sense. The mathematical foundation of a Jackiw state is concrete, but physically realizable states, which will be discussed later on, do not necessarily satisfy the foregoing constraint (57). A number-phase squeezed state features the squeezed quantum noise between the photon number and sine operators, as shown in fig. 11. When the squeezing parameter ris equal to - In (2 ( A ) ) and the average photon number ( A ) is much greater than unity, the photon number and sine operator noise are reduced to (Ah2) = ( A ) and (AS2) N 1/4( A), respectively. This state is “close” to a coherent state, although a coherent state does not satisfy a

112

QUANTUM MECHANICAL LIMIT IN OPTICS

Fig. 11. Light waves in a,

- a2 phase space and in time domain. (a) Number state and (b) number-phase squeezed state.

number-phase minimum uncertainty product in an exact mathematical sense (CARRUTHERS and NEITO[ 19681). We will also refer to the preceding noise levels as the SQL, since they are equal to the noise levels of coherent states. A number-phase squeezed state with r greater than - f In (2 (A)) has a smaller photon number noise and a larger sine operator (approximate phase) noise than the SQL:

:<e)eP2’< ( A ) , (As2)=f(e)ezr<-. 1 (AA2)

=

(58) (59)

4(A)

In the limit of r + co the photon number noise can be reduced to zero, for which the number-phase minimum uncertainty product (1 1) is approximately pre+ 0. This is “close” to a photon number state. served by The SIN ratio for photon counting detection of a photon number state is improved infinitely when compared with that for a coherent state:

(e)

(SIN),, = (A, ) -+ (SIN),,

=

co (photon counting) .

(60)

The required average signal photon number for a bit error rate of for an on-off intensity modulation optical communication is decreased from twentyone for coherent states to one for number states. The improvement is larger than that for a quadrature amplitude squeezed state.

11,s 31

NONCLASSICAL LIGHTS

113

The unique feature of number-phase squeezed states and photon number states is that the enhanced “phase” noise does not waste the signal photon number and thus it is free from the trade-off relation between quantum noise reduction and available coherent excitation. A photon number state with zero quantum noise is realizable irrespective of the signal photon number.

3.5. GENERATION OF NUMBER-PHASE SQUEEZED STATES

Number-phase minimum uncertainty states can be generated by two successive unitary evolutions from a coherent state. The interaction Hamiltonian required for realizing the first unitary evolution is known to be quartic in photon [ 1986]), annihilation and creation operators (KITAGAWA and YAMAMOTO HSpM =

(61)

~ x B ~ ~ & ~ .

The other is a linear translation Hamiltonian HT = h(lcBt

+~*ri).

The Hamiltonian function (61) expresses “self-phase modulation”, and neither photon absorption nor emission is involved. This can be realized by the third order nonlinear process (Kerr effect). The unitary operator Us,, resulting from this “four-photon interaction” is given by

[:

1

USPM = exp iX - A(fi - 1)

.

The unitary operator U, resulting from the translation Hamiltonian function (62) is

U, = exp ((dt - (*a),

(64)

where ( = ilcL’/c and L‘ is the interaction length. The Heisenberg operator for the output mode is ,,,ci

= U, ~ s p M ~ j i , , ~ i U; p M = ei2r(L/C)* 6, + ( .

(65)

Equation (65) indicates that the input coherent state first experiences “selfphase modulation’’ and then is translated by c-number coherent excitation (. Physical realization of this unitary evolution is shown in fig. 12. The unitary operations (63) and (64) are realized in an optical Kerr medium and approximately by a high reflection mirror. As shown in fig. 13, a quantum mechanical

114

QUANTUM MECHANICAL LIMIT IN OPTICS

Fig. 12. Kerr nonlinear interferometer for number-phase squeezed state generation.

Ima

00

Irna

Fig. 13. (a) Self-phasemodulationin Kerr medium and (b) interference at high reflectivitymirror; y is a nonlinear interaction parameter.

11, I 31

115

NONCLASSICAL LIGHTS

correlation is established between the photon number and phase in an optical Kerr medium, but the photon number noise is not yet reduced at this stage. In order to reduce the photon number noise, the direction of coherent excitation should be perpendicular to the stretching direction of the phase noise. For this purpose 5 must be f n out of phase from the coherent excitation of the Kerr medium output wave, as shown in fig. 13. Figure 14 shows the normalized photon number noise, (AA’ ) / ( A ) ,and the number-phase uncertainty product as a function of the nonlinear strength y = 2xL/c. For a given y value there is an optimum (value to minimize (Ad2 ) and the minimum (AA’) for the optimum ( value are plotted in fig. 14. The photon number noise cannot be reduced to zero, but the minimum achievable photon number noise is bounded by

(AA’),,

=

(at y , , t ~ ~ ( A ) - 2 / 3 ) .

(66)

Note that the minimum photon number variance (66) is smaller than the minimum photon number variance for a quadrature amplitude squeezed state. The normalized phase noise, (A$’) (A), is also plotted in fig. 14. The

r Fig. 14. Number-phaseminimum uncertainty relation in Ken nonlinear interferometer.

116

QUANTUM MECHANICAL LIMIT IN OPTICS

111, § 3

minimum uncertainty product, (Ad2) ( A $ 2 ) = i,is preserved for y values below the optimum value yopt. The forward four-photon mixing for squeezed state generation (SHELBY, LEVENSON, PERLMUTTER, DEVOEand WALLS[ 19861, MAEDA,KUMARand SHAPIRO [ 19871)is essentially the same as the present scheme for the numberphase squeezed state. The difference between quadrature amplitude squeezed states and number-phase squeezed states submerges when the nonlinear interaction strength is weak. 3.6. PUMP-NOISE-SUPPRESSEDLASER

Number-phase squeezed states can also be generated by a laser oscillator, in which the pump-amplitude-fluctuation is suppressed (YAMAMOTO, MACHIDAand NILSSON[ 19861). The amplitude noise spectra of a conventional laser output field (not a cavity internal field) approach the SQL at a high pump level, as shown in fig. 2 (YAMAMOTO, MACHIDAand NILSSON [ 19861). This SQL stems from shot-noise-limited “pump amplitude fluctuation,, in the low-frequency region below the cavity bandwidth, f2 < o/Q(near cavity resonance). The SQL in the high-frequency region above the cavity bandwidth f2 > o/Q(off cavity resonance) originates from vacuum field fluctuation incident on and reflected back from the mirror. The well-known white spectrum (SQL) of an ideal laser emission is the result of an incidental balance between the system’s ordering force (gain saturation) and these two fluctuation forces from reservoirs. The shot-noise-limited pump amplitude fluctuation apparently occurs because coherent or incoherent pump light has the shot-noise character (Poisson limit) and any random deletion processes in a pumping stage do not change the shot-noisecharacter. This is true for an optically pumped laser, but it is not necessarily the case for other pumping schemes. The pump amplitude fluctuation can be substantially reduced below the shotnoise level in an injection current pumped semiconductor laser. This is because the junction voltage modulation in a depletion layer, which is caused by thevoltagedrop acrossthe sourceresistanceR,,regulatesthe minority carrier injection process into the active layer. The microscopic Langevin theory (YAMAMOTOand MACHIDA[ 19871) clarifies the fact that pump current fluctuation in a semiconductor laser is actually thermal noise generated in a source resistance. It is given by

11, B 31

NONCLASSICAL LIGHTS

117

where V , = k, T,/q is the thermal voltage, q is an electron charge, and k, is the Boltzman’s constant. When R, is larger than 2R, where R is a diode’s differential resistance (SZE [ 1969]), eq. (67) is reduced below the shot noise level, SiP(62) < 2qZ (R, > 2 R ) . (68) Since the diode’s differential resistance monotonically decreases with increasing pump level, eq. (68) will eventually be satisfied at a high pump level for whatever small R, value. When the pump amplitude fluctuation is thus eliminated, the photon field amplitude noise spectrum at 62 < o/Qis also reduced to below the SQL, as shown in fig. 15. The amplitude fluctuation spectrum is given by (YAMAMOTO and MACHIDA[ 19871)

Fig. 15. Amplitude (solid lines) and normalized phase (dashed line) noise spectra of a pumpnoise-suppressed laser.

I18

QUANTUM MECHANICAL LIMIT IN OPTICS

Here,

where ye and yo are the p-oton Lxay rates due to output coupling .AS and cavity internal loss, y = ye + yo, and N , and A , are the average carrier number and photon field amplitude; E,, and E,, are the stimulated emission and absorption rates. As also shown in fig. 15, the photon field phase noise spectrum at &? < y = o/Qincreases in proportion to &?- due to the so-called "phase random walk". The uncertainty relationship between amplitude and phase noise spectra is still preserved by the phase diffusion noise, and the uncertainty product is actually twice as large as the minimum value. An injection current pumped semiconductor laser thus produces a number-phase squeezed state at &?y. Number-phase squeezed state generation by a semiconductor laser has been demonstrated by Yamamoto and his collaborators (MACHIDA,YAMAMOTO and ITAYA [ 19871, MACHIDAand YAMAMOTO[ 1988, 19891). The photocurrent spectrum for I/Ithg 10 is compared with the SQL in fig. 16. The source resistance is 1 kR, which is high enough to suppress the pump amplitude fluctuation far below shot noise level. The photocurrent spectrum from near dc to 1.1 GHz was reduced to below the SQL. The noise reduction was - 0.4 to - 0.9 dB (9% to 19%) depending on the frequency, which corresponds to amplitude squeezing of - 1.7 dB (32%)below the SQL of the laser output field. The difference is caused by nonunity detection quantum efficiency. l

I

"

"

<

0.2

"

/ I

"

'

a

1

'

I

I

06 a8 Frequency (GHz)

0.4

I

a

1.0

I

Fig. 16. Observed intensity noise spectra for a constant current-driven semiconductor laser. A is test laser single detection, B is reference laser balanced-mixer detection, C is reference laser single detection, and D is test laser balanced mixer detection.

11, B 31

NONCLASSICAL LIGHTS

119

The amplitude noise versus pump levels for the 0.85 pm GaAs laser and the 1.5 pm InGaAsP laser were in good agreement with the theoretical prediction, as shown in fig. 17. The theoretical amplitude noise for a conventional laser with shot-noise-limitedpump amplitude fluctuation is also plotted for comparison. The advantages of the present scheme are as follows: 1. Amplitude noise reduction is not limited by nonlinear optical material constants but is determined only by laser quantum efficiency. Photon number squeezing of more than 10 dB is achievable, since a quantum efficiency higher than 90% is not difficult to realize. 2. The squeezing bandwidth is ultimately determined by a photon lifetime on the order of 1 psec. A squeezing bandwidth higher than 10-100 GHz is achievable at reasonable pump level. 3. Photon number squeezed light is available at any wavelength between 0.7 pm and 10 pm by choosing an appropriate semiconductor material system.

Normalized Pump Rate l / l t h - l

(b)

Normalized Pump Rate I/lth-1 (a)

Fig. 17. Normalizedintensity noise level of (a) GaAs laser and (b) InGaAsP laser. Solid line and dashed line correspond to theoretical curves without and with pump fluctuation, respectively.

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QUANTUM MECHANICAL LIMIT IN OPTICS

3.7. PROPERTIES OF PHOTON TWINS

A nondegenerate parametric amplifier, in which signal and idler frequencies are separated by a large amount, establishes a quantum mechanical correlation between the signal and idler waves. The interaction Hamiltonian function for a nondegenerate parametric process is given by

HI - 1,h(Xd!dt

+ X*d,di).

(73)

The input and output relations for Heisenberg operators for the signal and idler modes are written as

hs=fis,+JG-1bt,

(74)

ht = J c d t + JG-1CiS.

(75)

and

From eqs. (74) and (75) we have the operator Manley-Rowe relation (HONG and MANDEL[ 1986]),

A&)

- A&)

= AS(O)

- Ai(0) ,

(76)

where ri,(O) = dJd, and Ai(0) =Adtdi are the input signal and idler photon h A h number operators, and A,(t) E b J b , and Ai(t) E b t b , are the output ones. It suggests that the photon number correlation is perfect when both the signal and idler waves are initially in vacuum states or the amplifier gain goes to infinity. When the idler output photon number fluctuation is measured and the measurement result is used to manipulate the signal photon number, the signal output photon number fluctuation can be reduced. This can be considered as number-phase squeezed state (or photon number state) generation. From eqs. (74) and (75) the quadrature components of the output modes satisfy

b,,

=

hi, + (JG- JZz)(dsl - d i l ) ,

b,,

=

- hi,

(77)

and

+ (Jc- JG-I)(4,+ 4,)

I

These relations suggest that the in-phase components become positively correlated (h,, N hi$ and the out-of-phase components become negatively correlated (h,, N - biz) when the amplifier gain G goes to infinity. When the idler output quadrature amplitude is measured and the measurement result is used to manipulate the signal wave quadrature amplitude, the signal quadrature

121

NONCLASSICAL LIGHTS

1

Fig. 18. Input and output, signal and idler waves of a nondegenerate parametric amplifier in a , - a2 phase space.

amplitude fluctuation can be eliminated. This can be considered as quadrature amplitude squeezed state generation. The signal and idler output waves of a high-gain nondegenerate parametric amplifier can be used as either number-phase squeezed state or quadrature amplitude squeezed state generation. When the amplifier gain is very large, the signal and idler output waves are approximately in a conjugate relation,

s, = 8: JG (8, + lit).

(79)

N

This “photon twin” generation in a nondegenerate parametric amplifier is schematically shown in fig. 18.

3.8. GENERATION OF PHOTON TWINS

As discussed earlier, a photon twin (8,N 8’) can be generated by a nondegenerate parametric amplifier. A high-Q cavity can be also used for an efficient nonlinear interaction in this case. The total Hamiltonian of a triply resonant nondegenerate parametric amplifier is expressed as (BJORK and YAMAMOTO [ 1988a]),

H

=

+

h ~ , 8 , t 8+ , h ~ , ~ i t hUpa,tap s~

+ +itrf2)(@ritci,

- d,a,(i,’)

+ ihEp(@ e-imp‘ - 8 e’ap‘ P

- i h ( J y , (i,f,+

)

+ f i ci,=j+ +

dpfpt - H.c.)

.

(80)

122

[II, 0 3

QUANTUM MECHANICAL LIMIT IN OPTICS

Here d,, di, and d, are the cavity internal field annihilation operators for signal, idler, and pump waves; x(’) is the second order nonlinear coefficient responsible for the parametric process; E, is the pump rate; ys, yi, and y are the cavity decay rates for signal, idler, and pump photons; and and are operators for the three input waves. The correlation spectra for an out-of-phase amplitude fluctuation are given by ( B J ~ R and K YAMAMOTO [ 1988b1)

ls,l, &,

+

(

(l-p)(l-p-s2(1+3p))+4n2 1

+ p2s2 Yi

(

ys

%)>’

(1 - p)2 Yp

YsYiYp

+((1 - p ) ’ ( 1 +s2)+462’(yi+ Y sYYi pYp- x ) ) 1

+ 462’

(=+ Ys

p’s2 - 1

(1 - p)’

Yi

Yp

YsYiYp

The normalized correlation spectral desSity (81) is plotted as a function of normalized pump rate t in fig. 19. Even though the quadrature amplitude * , S A ~feature , 2 an 62-’ dependence (random phase walk) at spectra S A ~and frequencies below the cavity bandwidth (62 d ys), the added quadrature amplitude S A ~ s 2 +isAreduced ~ , 2 below the SQL. The noise level below the SQL indicates the “quantum mechanical correlation” of the phases of two waves. The ideal quantum correlation is obtained at zero frequency and at the oscillation threshold. Above the oscillation threshold the quantum correlation at near dc decreases with increasing a pump rate. However, the good correlation is still obtained near a relaxation oscillation frequency at above the oscillation threshold. These results are completely analogous to the quadrature amplitude squeezing for a degenerate parametric amplifier and oscillator (see fig. 10). Quantum mechanical photon number correlation for a nondegenerate parametric fluorescence has been observed (BURNHAM and WEINBERG [1970]). The very short time delay (less than a few hundred picoseconds) between a signal and idler photon has been demonstrated (FRIBERG,HONG and MANDEL [ 19851). Quantum mechanical photon number correlation

123

NONCLASSICAL LIGHTS

.-r,

102

4-

C In

f ’10’ e

c

g

100

v

; 0

1CT2

10-2 lo-’ Normalized frequency

loo

(n/‘Ys)

Fig. 19. Quadrature-phase amplitude noise spectra (dashed-dotted lines) m - correlated noise spectra (solid lines) of photon twins; t = P/P,,.

between signal and idler waves for a nondegenerate parametric oscillator has been demonstrated. A KTP crystal in a high-Q cavity was pumped by a 0.5145 pm Ar laser, and type I1 interaction was employed in the experiment (HEIDMANN, HOROWICZ, REYNAUD, GIACOBINO, FABREand CAMY [ 19871).

3.9. GENERATION OF QUADRATURE AMPLITUDE SQUEEZED STATES AND NUMBER-PHASE SQUEEZED STATES BY PHOTON TWINS AND FEEDBACK OR FEEDFORWARD

If the photon number fluctuation AA, or the quadrature component fluctuation Ad,, (Adi2) of the idler wave is measured and that of a signal wave is counteracted by the measurement result, the quantum noise of the signal wave may be suppressed because of the quantum correlation of the photon twins. This can also be considered as generation schemes of squeezed states (table 1). The feedback and feedforward schemes incorporating idler photon counting detection are shown in fig. 20. It is theoretically and experimentally demonstrated that a feedback loop can suppress the photon number quantum noise completely by sufficiently high loop gain (MACHIDAand YAMAMOTO [ 19861, YAMAMOTO, IMOTO and MACHIDA[ 19861, HAUSand YAMAMOTO [ 19861). A low-intensity sub-Poissonian light has actually been generated by the combination of spontaneous parametric down-conversion and optical attenuator

124

[II, $ 3

QUANTUM MECHANICAL LIMIT IN OPTICS

TABLE1 Properties and generation schemes for quadrature amplitude squeezed states, number-phase squeezed states, and photon twins (0) Quadrature amplitude squeezed states (1) ( A d : ) ( A d : )

=&

Number-phase squeezed states

(AA2)

(As2) =:(c’)

Photon twins

(Ad:) (AA’)

(86:)

=&

(As’) = :(c2)

H , = fh(xdJd> + x*dSdi)

(2) H , , = ?jh(xdt2 + x*d2)

H,,,

(3) elliptic

crescent

(correlated pair)

(4) 4N,(N* + 1) (homodyne)

03

(photon counting)

4N,(N, + 1) (homodyne) a J (photon counting)

(5)

= h~d~’d’

- degenerate OPO/OPA - degenerate four-wave mixer - nondegenerate OPO/OPA - pump-noise-suppressedlaser - feedback/feedforward with photon counting

(0) Nonclassical lights (1) Minimum uncertainty relation (2) Interaction Hamiltonian (3) Quasi-probabilitydensity (4) Signal-to-noiseratio (5) Generation schemes

(WALKERand JAKEMAN [ 19851, RARITY, TAPSTER and JAKEMAN [ 19871). However, the finite delay time inside the feedback loop places severe limits on both the degree and bandwidth of squeezing (SHAPIRO,SAPLAKOGLU, Ho, KUMAR,SALEHand TEICH[ 19871). The feedforward scheme is, on the other hand, not subject to the loop delay problem. A high-intensity signal wave can be counteracted by an attenuator or amplifier or their combination, as shown in fig. 20. The vacuum field fluctuation is inevitably introduced in such a

Fig. 20. Number-phase squeezed state generation by a nondegenerate parametric amplifer and feedforward.

11,s 41

QUANTUM NONDEMOLITION MEASUREMENT

Phase modulator

125

#

Fig. 21. Quadrature amplitude squeezed state generation by a nondegenerate parametric amplifier and feedforward.

feedforward scheme, and the minimum photon number variance is bounded by (Ari’) ( A ) ’/’ ( B J ~ R Kand YAMAMOTO[ 1988~1). A nondegenerate parametric oscillator is suitable for this purpose because the initial photon number distribution is close to Poisson limit at well above the oscillation threshold. The feedforward scheme incorporating idler out-of-phase quadrature component homodyne measurement is shown in fig. 21. The counteraction can be realized by a lossless phase modulator, and quadrature amplitude squeezed states can be generated in this case. The limit imposed on the degree of squeezing is only determined by the degree of quantum correlation established in a parametric process in this case. A nondegenerate parametric amplifier is preferable for this purpose because the out-of-phase quadrature amplitude correlation is degraded at above the oscillation threshold. N

4 4. Quantum Nondemolition (QND) Measurement 4.1. GENERAL QUANTUM MEASUREMENT AND QND

In a general quantum measurement the observable of the signal system, A,, is measured by means of the change in the observable of the probe system, Ap, after the proper interaction between the signal and probe systems, as shown in fig. 22. The Heisenberg equations of motion for A, and Ap are written as d dt

-ih

- A, =

-ih

-

[A,,A,]+[@,,A,],

and d dt

Ap = [@,,A,]+ [@,,Ap],

(83)

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QUANTUM MECHANICAL LIMIT IN OPTICS

dissipation

I

> fluctuation

Probe system

Signal system

{

external DOFs (mo~~gwic)

Fig. 22. QFD measurement model with a probe system and a macroscopic meter. Signal observable A s Ais nondemolitionally measured through demolitional measurement of probe observable A,, whose motion is affected by d, by means of interaction Hamiitonian I?,. If I?, does not commute with A,, the motion of A, is kicked by I?, (back action I). If I?, contains 8,. the motion of d, is also kicked by the uncertainty principle between A, and 8, (back action 11).

where A, is the unperturbed Hamiltonian of the signal system, '-ipis that of the probe system, and HI is the interaction Hamiltonian between the two. The first commutators in eqs. (82) and (83) represent the free motion of A, and Ap. The second commutators represent the contribution of the interaction between the signal and probe systems. In order to measure A, using [A,,Ap] in eq. (83) should not be zero and HI should be a function ofA,. These are the requirements for a general quantum measurement: 1. A, is a function of A,. 2. [A,,A,] # 0. In general, a measurement of a, affects the motion of A, itself in two ways (measurement back action). One way is the change of A, due to [HI,A,] in eq. (82). Such an example is a photoelectron emission in a photodetector. When a signal photon number is measured by a conventional photodetector, the photon absorption process naturally changes the signal photon number. The electric dipole interaction,

ap,

H,=hg(dSt+Bt&),

(84)

is switched on between the signal system (photon field) and probe system (collective two-level atoms). In this equation g is the coupling constant, and S and St are Pauli's pseudo-spin operators that represent the atom's down and up transitions (SARGENT,SCULLY and LAMB[ 19741). The observable

11, B 41

QUANTUM NONDEMOLITION MEASUREMENT

127

A,= R = Btd does not commute with A,; that is, [A,,a,]

# 0. Such a back action is shown by the dashed line in fig. 22. If the interaction Hamiltonian HI commutes with

a,,

[HI,A,l

=

(85)

0,

HI is called a “back action evading” type, and this first source of back action can be eliminated. The other way of changing is through the uncertainty that is introduced by the measurement. The Heisenberg on the conjugate observable of uncertainty principle dictates that when the observable is measured with accuracy ) , the uncertainty imposed on the conjugate observable B, should be larger than (BOHM[ 19511)

a,

(a:

a,

a,

If the unperturbed Hamiltonian H, contains the conjugate observable B,, the motion of A,is affected by this uncertainty of B,. An example is the position measurement of a free particle. When a free particle’s position is measured by a y-ray microscope (HEISENBERG [ 1930]), the measurement accuracy is proportional to the photon wavelength, Aq N 1. A photon scattered by a free particle changes the particle’s momentum. This back action imposed on the particle’s momentum is of the order of the photon’s momentum itself, Ap N h/1. The kicked momentum then perturbs the free motion of the particle’s position according to the following relation:

Here d(0) and p ( 0 ) are the position and momentum just after the first measurement. This back action is shown by the dotted line in fig. 22. The SQL (eq. (22)) on a free-mass position measurement stems from this type of back action noise. If a,(O) commutes with a&), [ ~ , ( O ) , ~ , ( t )=l 0

Y

(88)

a, is called a “QND observable” and the second source of back action can be eliminated. Equations (87) and (88) are the conditions for a QND measurement (CAVES,THORNE, DREVER,SANDBERG and ZIMMERMANN [ 19801). In a QND measurement the observable A,is not affected at all, even though the measurement accuracy of a, is increased arbitrarily. In optical communica-

128

QUANTUM MECHANICAL LIMIT IN OPTICS

[II, 5 4

tion this means that the information carried by a signal wave is fully extracted at one receiver but a signal wave still can transport exactly the same information to the other receivers. Thus the SQL imposed on photon detection, discussed in 5 2, can be overcome.

4.2. CONTRACTIVE STATE MEASUREMENT

As indicated by eq. (87), the free-mass position 4 is not a QND observable in the sense of eq. (88). There is, however, the other way of evading the back action noise in repeated measurements. The position uncertainty at the time t after the first measurement is calculated from eq. (87) as .2

Suppose the third term of eq. (89) is zero or positive, the measurement uncertaintyis constrained by the so-called SQL on free-mass position measureSANDBERGand ZIMMERMANN [ 1980]), ment (CAVES,THORNE,DREVER, (Ad2)

sQL

2

t

(Ad’(0))

”*

(A$’(O))

‘1’

ht 2-. 2M

But if the third term of eq. (89) is negative, that is, the position and momentum just after the first measurement are negatively correlated, the measurement

uncertainty is not necessarily constrained by eq. (90) (YUEN [ 1983b1). The position uncertainty after the first measurement can decrease during the free motion, and it becomes minimum at the instant of the second measurement. There have been intensive arguments concerning whether such a contractive state can be realized by an infinitelyhigh precision measurement (WODKIEWICZ [1984], LYNCH [1984, 19851, CAVES [1985]). The two interaction Hamiltonians realizing the contractive state measurement have been proposed (NI [ 19861, OZAWA[ 1988]),even though their physical realizability is still open to question.

11, B 41

129

QUANTUM NONDEMOLITION MEASUREMENT

4.3. PROJECTION POSTULATE, FIRST KIND MEASUREMENT, AND STATE REDUCTION

The SQL in measurements that were discussed in 5 2.5 can be circumvented by QND measurement and (QND-like)contractive state measurement. In this section some generalmathematical backgrounds of quantum measurements are presented as preparation for the detailed discussions of these measurement strategies. Characterization of general quantum measurements requires specifications for the possible readout, for the probabilities of obtaining each readout, and for the system state after measurement for each readout. The present framework of quantum mechanics can answer all these questions except the interpretation issue (WHEELERand ZUREK[ 19831). In the standard reply (VONNEUMANN [ 19551) the readout is the eigenvalues a, of the self-adjoint probe operator A,. The a, provide information on the signal operator A,by means of the unitary evolution governed by the interaction Hamiltonian H I . The probability of obtaining an is P(an)=,(anl

A (red) Pp Ian>p.

(91)

asp)

Here p y ) = Tr,( = Tr,@),(O)@ as(0)ot) is the reduced density operator for the probe system and 0 is the unitary transformation realized by H I . The initial density operators for the signal and probe systems, are p,(O) = I$), ,( $1 and p,(O) = p ( $1. Equation (91) can be rewritten as

where &a,)

=

[,(a,I

0 I$>,lt [p(a,I 0 I$>, ]

-

(93)

f(cr,)is a generalized projection operator that includes measurements of a finite error. When eq. (93) is a continuous projector rather than a discrete projector or a nonorthogonal projector for an overcomplete set of states, the probability can still be obtained by (DAVIES[ 19761, HELSTROM [ 19761)

P(a) = Tr,[ B,(O)ff(a)l

9

(94)

where ff(a) is an “operator valued measure” and satisfies

s

f ( a ) d a = I,.

(95)

130

QUANTUM MECHANICAL LIMIT IN OPTICS

Three typical measurements for a photon field are given by

I

In) ( n I

(photon counting),

d = l a l ) (all (homodyning), I a ) (a1 (heterodyning) .

(96)

As far as the system state after measurement is concerned, the projection postulate gives it as follows (VONNEUMANN [ 19551): /S(meas, a,) =

Trp( I a n

>

p p( an

I @~sp)

(97)

*

It should be pointed out here that the projection postulate (97) can be applied to alimited class of quantum measurements, which is often called the “first kind measurement” (PAULI[ 19581). In most of the measurements the system state after the readout is totally unpredictable, and these “dirty measurements” are called the “second kind measurement”. a n ) after measurement is not linked to the In general the system state pcmeas, initial state p,(O) by means of unitary evolution. The remarkable characteristic of such a nonunitary process is that the initial pure state p,(O) = I $), s ( $1 loses its quantum coherence, at least partially, and becomes transformed into the statistical mixture state. For instance, suppose the unitary evolution is designed so that the readout an of the probe operator corresponds to the eigenstate ISn) of the signal operator. The unitary evolution establishes the quantum correlation

<

The Von Neumann postulate informs us that this pure-state density matrix decays into the mixed-state density matrix after the “second stage of the measurement” (readout):

-+

Bmix

=

C IU’ISn>ss(SnI

@ Ian)pp
*

(99)

n

The physical interpretation of this “state reduction” has been a central issue for establishing the foundation of quantum mechanics. The most popular approach is schematically shown in fig. 22. The probe system is coupled to many degrees of freedom that supposedly constitute a macroscopic meter. During the probe’s dissipation process into them, the quantum coherence of the

I L 8 41

QUANTUM NONDEMOLITION MEASUREMENT

131

probe and, subsequently, the signal systems are lost. Here we cite only limited references about such models (MACHIDAand NAMIKI[ 1980a,b], ZUREK [ 1981, 19821).

4.4. QND MEASUREMENT FOR PHOTON NUMBER

When the observable of the signal wave is a photon number a, = A,, condition (88) is satisfied, since the system-free Hamiltonian is given by H , = ho,(A, + i) and [H,, A,] = 0; that is, a photon number is a “QND observable”. Let us consider the configuration shown in fig.23 (IMOTO, HAUS and YAMAMOTO[ 19851). The signal wave at frequency o,propagates along an optical Kerr medium with a probe wave at frequency upin which the phase of the probe wave is modulated by the photon number of the signal wave. The phase modulation of the probe wave is detected by a homodyne receiver, and thus the information about the signal photon number can be extracted. If the signal wave is not attenuated, the signal photon number is preserved. The readout observable of the probe wave is a quadrature component = Lip, = (Lip - hJ)/2i. The interaction Hamiltonian for the measurement is that for the interphase modulation between the signal and probe waves:

ap

HI= hxA,Ap.

(100)

The constant x is equal to (h/2 V ~ ~ ) o ~ o ,where x ( ~ Vis ) , the mode volume and x(3) is the third order susceptibility; E is the dielectric constant of the medium. It is easily verified that the interaction Hamiltonian (100) satisfies the general quantum measurement conditions 1 and 2 in 8 4.1. The QND condition bock-act ion Signal wave

US

Kerr medium / I ns /eS ~

nP

1-

s [+p

-

>

-

measurement

Probe wove

homodyne receiver

Fig. 23. Schematical view of a QND measurement scheme with an optical Kerr medium. The signal photon number is nondemolitionallymeasured through the homodyne detection of the probe phase, which carries the signal photon-number information by means of the optical Kerr effect.

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QUANTUM MECHANICAL LIMIT IN OPTICS

[II, 0 4

(eq. (85)) is also satisfied, and thus the interaction Hamiltonian (100) is a “back action evading” type. 4.4.1. Heisenberg picture The unitary evolution for the signal and probe wave annihilation operators resulting from the onset of eq. (100) is expressed by d,(L) = exp(i f i A , ) d , ( O )

(101)

Y

and d , ( ~ ) = exp (i

JFA,)B,(o).

(102)

fi

Here = xL/c,L is the length of the Kerr medium, and c is the light velocity. The optical homodyne receiver with a proper local oscillator phase (90 degrees out of phase for the coherent excitation of the probe wave) measures the quadrature phase component of up(L),

Since the probe wave has the coherent excitation only along d,,(O), that is, (dPl(O)) # 0 and (dp2(O)) = 0, and the phase modulation JFA, is a small quantity, eq. (103) is reduced to 6,2(L) k 2

@,do))

J F A , + A6,2(0)

*

(104)

The signal photon number to be measured is expressed by

Taking the expectation value and variance of eq. (105), we obtain (fipbs) ) = (A,)

9

(106)

Thesecond termofeq. (107), (AAL2) = ((Ad,2(0)2)/F(A,),represents the measurement error, which is determined by the probe wave phase noise ( A @ ) k2 (Ad,,(0)2 )/(A, ) and the nonlinear interaction strength It

fi.

11, B 41

QUANTUM NONDEMOLITION MEASUREMENT

133

fi

can be decreased arbitrarily by increasing and/or decreasing ( A e ) . This QND measurement is ideal, since the expectation value and variance, that is, the statistics of measurement results, are equal to those of the signal wave itself. When the probe wave is in a coherent state, ( A I&) is given by 1/4(dP). When the probe wave is in an optimum squeezed state, ( A @ ) can be reduced to 1/(4(dp) ( ( A , ) + 1)) (YUEN[ 19761). The phase noise of the signal wave is increased by the other interaction represented by eq. (101). The quadrature phase component of the signal wave is similarly calculated as (ciSl(o))

f i A p

+ “S2(’)

’

(108)

The phase noise is thus given by

Here ( A $,(O)’ ) = (Ad,2(0)2 ) / (ci, (0))’ is the initial phase noise of the signal wave. The second term, ( A $:’) = F( Ad: ), is the back action noise added by the measurement. By decreasing the measurement error, the back action noise increases. When the probe wave satisfiesthe minimum uncertainty product, (Aci,,(0)2) (Ad,,(O)’) = $, the measurement error and the back action noise satisfy the Heisenberg minimum uncertainty product:

(Ad:’)

(A$:’)

= $.

(110)

The preceding argument concerning the measurement error and back action noise suggests that the signal wave after QND measured is “reduced” to a number-phase squeezed state. This can be easily shown by using the projection postulate. 4.4.2. Schr6dinger picture Suppose the signal and probe waves are initially in coherent states I a ) , and 1 B ) ,. The density operator after the interphase modulation is expressed by the unitary operator, Bsp

=

0 = exp(i ,/FA,A,),

0 l ~ o ) s s ( ~ o 60 I IBo>,,(Bol

0+.

(1 11)

The density operator after the measurement of the quadrature amplitude of the probe wave is given by (for the specific readout 8’) p p e a s . 8 ~ )=

Trp(lS2)p p(B’l63

Bsp)

(1 12)

134

PI, 5 4

QUANTUM MECHANICAL LIMIT IN OPTICS

(66)

-6)

(-6;6 ) Fig. 24. Quasi-probability density of the state after the measurement.

The quasi-probability density ( a ( )imeas,flz) I a ) is shown in fig. 24 (KITAGAWA,IMOTO and YAMAMOTO[1987]). The variances for photon number and sine operators (&j2 ) = Tr,[ );meas. 8 2 ) (A - (W21 N

(& W P l 2 )

-1

(1 13)

+

and (A$2)

=

Tr,[

);meas*f12)

(3 - (S>)*I

1.0

.8 .6 .4 .2 0

0

.05

.10 .15 Nonlineority

.20

.25

fi

Fig. 25. Uncertainties (Afi2), (A?) anduncertaintyproductP,, = (AW2) the state after the measurement; ( 1 a0( = lSol = 4, S2 = 0).

(AS2)/( C)20f

11, B 41

135

QUANTUM NONDEMOLITION MEASUREMENT

satisfy the number-phase minimum uncertainty product, (Ah’ ) x = :( as shown in fig. 25 (KITAGAWA, IMOTO and YAMAMOTO [ 19871). The initial coherent state is transformed to a number-phase squeezed state by the nonunitary state reduction. QND measurement is considered as the generation process of a nonclassical light (YAMAMOTO, MACHIDAand IMOTO [ 19861). Two similar but slightly different configurations for QND measurement of photon number have been proposed. One utilizes a microwave photon in a high Q-superconductor cavity as a probe system (BRAGINSKYand VYATCHANIN [ 19811). The other employs a nonlinear coupler based on four-wave mixing (MILBURNand WALLS[ 19831).

(As’)

e)’,

4.5. EFFECT OF SELF-PHASE MODULATION

For a nonresonant Kerr medium the interaction Hamiltonian H I usually accompanies the additional terms for self-phase modulation, H I = h ~ f i , f i +, i h ~ f i +; i h ~ f i i .

(115)

The unitary evolution for the signal and probe waves due to the simultaneous interphase and self-phase modulations is expressed by &,(L)= exp [i

JF(fi,t ~A,)IB,(o),

(1 16)

In such a case the probe wave is squeezed as discussed in § 3.5 due to the self-phase modulation effect. The quasi-probability densities of the probe waves after the interaction given by eq. (102) or (1 17) are schematically compared in fig. 26. The optical homodyne receiver with a proper local oscillator phase (90 degrees out of phase) for the “squeezing direction”, as shown in fig. 26b measures (ip0

= &,,(L)cos 6 + &,&) sin 6 = &,,(o)[COS JF(fi,+ $yip> cos e + sinJF(fi, + ifi,) sin el + c,i(o) [cosJF(fi, + ;A,) sin e - sinJF(fi, + $tip) cos el . (118)

When the probe wave has the coherent excitation only along &,,(O)

and the

136

QUANTUM MECHANICAL LIMIT IN OPTICS

T

Y

'%elf phase modulation"

Fig. 26. Suitable local-oscillator (LO) phase for the Kerr-QND measurement scheme. (a) Free from the self-phasemodulation (SPM)effect and (b) with SMP.The S/N ratio is kept undegraded by choosing the proper LO phase.

11,s 41

137

QUANTUM NONDEMOLITION MEASUREMENT

When the local oscillator phase 0 satisfies

the second and third terms of eq. (120) cancel each other, and eq. (120) is reduced to eq. (-105). Thus the measurement error is not affected by the selfphase modulation effect if the local oscillator phase is properly adjusted (SHELBY, LEVENSON, PERLMUTTER,’ DEVOEand WALLS[ 19861).

4.5. EFFECT OF LOSS

When the interphase modulation process is resonantly enhanced by the atomic transition level (due to coherent and virtual excitation of electrons), the signal and probe waves also experience attenuationresulting from the inevitable incoherent and real excitation of electrons. For such a lossy Kerr medium the interaction Hamiltonian is expressed as

ij

Here ys and yp are the decay rates for the signal and probe waves, and and jpj are the loss oscillators for signal and probe waves. The signal photon number is no more preserved in such a lossy Kerr medium, and so this is not a QND measurement in an exact sense. However, when we recall that the SQL on photon detection for coherent states is expressed by (SIN), = (S/N)out+ (S/N)readout, we can define a quasi-QND measurement by the measurement that overcomes the preceding SQL. Thus an ideal QND measurement and quasi-QND measurement for coherent states satisfy, respectively, (S/N)readout

=

(S/N)readout

> (S/N)in - (s/N)out

(S/Wout = (S/Win

= (As ) : = (l =

-

ideal QND (S/Nhn

(1 - E ) (A,):

quasi-QND, (123)

where E is the insertion loss (0 < E 6 1) of the measurement scheme. This quasi-QND condition is expressed in terms of loss-error function A(&). The normalized photon-number measurement error A is defined as the added photon-number variance divided by the mean photon number:

138

QUANTUM MECHANICAL LIMIT IN OPTICS

[II, 8 4

( (AA,,.J2 ) / (ti, ) . The quasi-QND condition is expressed as A < (1 - E)/E, which is shown in fig.27a. For the Kerr-QND measurement scheme the preceding condition for quasi-QND measurement is written as (IMOTO and SAITO[ 19891)

Figure 27b shows the region satisfying eq. (124) for the F ( A , ) (ti,) value versus the total loss E. When the total loss E is larger than 0.715 ( - 5.5 dB), the

INSERTION LOSS E

"0

0.2 0.4

0.6

0.8

1.0

I N S E R T I O N LOSS E

Fig. 27. (a) A quasi-QND criterion for measurement error and insertion loss. (b) Required magnitude of the Kerr effect versus total insertion loss for QND measurement of a photon number. A loss value below 0.715 is required for QND measurement.

11, § 41

139

QUANTUM NONDEMOLITION MEASUREMENT

PO larizotion

maintaining fiber

I pI'==&e A

(balanced receiver) readout

11

~ T

( 1.06um)

(1.32um) output

Fig. 28. Experimental configuration for QND measurement of photon number.

quasi-QND measurement cannot be realized by any large G(A,) (A,) values.

4.7. EXPERIMENTAL QND MEASUREMENT OF PHOTON NUMBER

Experimental efforts to demonstrate QND measurement of a photon number MACHIDAand IMOTO are now underway at several laboratories (YAMAMOTO, [ 19861, IMOTO, WATKINS and SASAKI [ 19871, SHELBY, LEVENSON, P E R L M ~ E RDEVOE , and WALLS[ 19861). The experiments use a high-silica fiber as a Kerr medium because its small loss realizes enormous interaction length, its small cross-section achieves high power density, and its fast response makes possible the use of very short pulses. The experimental configuration for QND measurement is shown in fig. 28 (IMOTO, WATKINSand SASAKI[ 19871). Two single mode/single polarization (PANDA) fibers, 5 km long, are joined with a 90-degree axis rotation, as shown in fig. 28. Signal, probe, and reference probe waves are all confined in a small core area A,, of 10 pm diameter, and the total interaction length is as long as 10 km. A PANDA fiber maintains the polarization with a cross talk to the orthogonal polarization less than - 20 dB after the propagation of 10 km. The losses for the signal wave from a 1.06-pm YAG laser and the probe wave from a 1.32-pm YAG laser are 1.5 dB/km and 1.0 dB/km, respectively. A ring interferometer configuration is formed only for the probe wavelength,

140

QUANTUM MECHANICAL LIMIT IN OPTICS

-3

I

I

N

Required loser power for the fiber used in the ex-

I

I

Fig. 29. Required laser power product versus silica-fiber length for QND measurement.

and it can eliminate various fluctuation sources. Since the two counter-propagating probe and reference probe waves pass through an exactly equal optical path length in a main ring interferometer, the probe laser frequency noise and the mechanical, acoustic, and thermal vibration of fiber length/refractive index can be canceled out. The signal wave is coupled in and out by the wavelength selective couplers and propagates in one direction. The probe wave, co-propagating with the signal wave, interacts with the same signal wave packet and accumulates the signal photon number information due to the interphase modulation, while the reference probe wave, counter-propagating with the signal wave, interacts with many different signal wave packets and obtains only the information of average signal photon number. Thus the latter can be used as a reference wave. Figure 29 shows the condition of quasi-QND measurement mentioned earlier for power product P,P, and fiber length 1. The present experimentalpoint is compared with the theoretical criterion. In order to satisfy the quasi-QND condition, the laser wavelength must be shifted to 1.55 pm, where the fiber loss is in its minimum of less than 0.2 dB/km. The quasi-QND condition for this case is shown by a dashed line. The laser power must also be increased. Unfortunately, the probe phase is modulated not only by the signal photon number but also by the guided acoustic wave Brillouin scattering (GAWBS) (SHELBY,LEVENSONand BYER[1985]). The GAWBS noise can be suppressed by cooling a fiber down to a cryogenic temperature (SHELBY,

11, B 41

QUANTUM NONDEMOLITION MEASUREMENT

141

LEVENSON, PERLMUTTER, DEVOEand WALLS[ 19861). The use of soliton collisions in fibers is proposed to increase the peak power and to suppress the GAWBS noise for QND measurements (HAUS,WATANABE and YAMAMOTO [ 19891). It has been shown that self-induced transparency solitons feature a more efficient QND measurement scheme (WATANABE, NAKANO,HONOLD and YAMAMOTO[ 19891). 4.8. QND MEASUREMENT FOR QUADRATURE AMPLITUDE

When the observable of the single wave is a quadrature amplitude A, = dsl (or ds2), eq. (88) is still satisfied;that is, a quadrature amplitude is also a “QND observable”. Here it should be noted that a,, and 4, are not rapidly varying operators but are quadrature components with respect to the reference wave. The rapidly varying quadrature components are not QND observables but VORONTSOV and THORNE stroboscopic QND observables (BRAGINSKY, [ 19801). Let us consider the configuration shown in fig. 30. The incident signal wave 8, is partly transmitted through a high reflection mirror M,and the transmitted wave 8 is amplified by a travelling wave degenerate parametric amplifier. The amplified quadrature amplitude E l of the signal wave E is measured. The photodetector surface emits a vacuum fluctuation 2, which is also amplified or deamplified by the travelling wave degenerate parametric amplifier. The amplified/deamplified vacuum fluctuation b is partially reflected back to form the wave 8 and is partially transmitted to form the output wave a,,. The phase of the pump wave E , can be adjusted to realize the relation of 8 and E expressed by E=

Jc b + JG-I8 7 .

(125)

Here G is the parametric amplifier gain. It is also possible to relate b with 2 by the proper phase choice of the pump wave E,, b=

Jc 2 - J z T

&.

(126) \

M

Fig. 30.

El

hornodyne

\ ‘

F-

Configuration for QND measurement of quadrature amplitude.

142

QUANTUM MECHANICAL LIMIT IN OPTlCS

The scattering matrix of the mirror M is expressed by

From eqs. (129, (126), and (127) we obtain

+ J Z T d i t , ) + J1-.6

E = &(*din

The quadrature amplitude of the signal wave to be measured from the E l measurement is expressed by Adl d(ObS) = El

-

JE(JG+ $ 3 7 )

=

din, 1 +

J1-.

JE(JG+J=)

Here t r 0 (high-reflectivity limit) is assumed. Taking the expectation value and variance of eq. (129), we obtain (d,c,q”B’>

=

(d$b:))2)

(130)

(din, 1 ) 7 =

(Ad;,

)

+

1

4&(&

+JG-1)2

The second term of eq. (13 1) represents the measurement error (Au;; ) , which is determined by the transmission coefficient E of the mirror and the degenerate parametric amplifier gain. It can be decreased arbitrarily by increasing G, even though E is very small. The measurement is thus ideal because the measurement statistics are identical to the statistics of the signal wave. The output wave dout is also obtained from eqs. (126) and (127) as

do,,

=

-din

+ &[JG 6 - JG-I 6 + ] .

(132)

The quadrature amplitude bout, is reduced to

+JE(JG-JEZ)A~~+,,~, (133)

B,,,,~

when E z 0 and G >> 1 are assumed. Thus the “measured” quadrature amplitude is not disturbed and the QND measurement of 6 , is realized. The other quadrature amplitude is dOut,

=

J1-.din, + JE(JG+ JG-1)A & .

(134)

I I , § 41

QUANTUM NONDEMOLITION MEASUREMENT

143

The measurement back action noise imposed on the other quadrature amplitude is thus given by

(A4’Z2) =be(*

+ Jm)”

(135)

The measurement error and the back action noise satisfy the Heisenberg minimum uncertainty product

The essential part of the QND detector for the quadrature amplitude discussed here is a beam splitter M with infinitesimal coupling loss and the preparation of a squeezed vacuum &. In an ordinary beam splitter the vacuum fluctuation 2 prevents the measurement error from being decreased arbitrarily. In order to resolve this difficulty, the vacuum fluctuation can be squeezed so that the measurement will not be disturbed. In this respect the configuration shown in fig. 30 is similar to the schemes proposed independently by SHAPIRO[1980], YURKE [1984], SHELBY and LEVENSON[1987] and SLUSHER, LAPORTAand YURKE[ 19841). One of the important differences between the QND measurements for the photon number and for the quadrature amplitude is that the latter should always involve the signal amplification process. This is because the back action noise imposed on the other quadrature amplitude accompanies the increase in signal photon number. In order to decrease the measurement for the photon number, on the other hand, the back action noise imposed on the phase does not require the increase in the signal photon number, and therefore it can be realized by a passive device. This fact is analogous to the finding that a photon number state is realized by a finite photon number but a quadrature amplitude state is realized only by an infinite photon number.

4.9. PREAMPLIFICATION FUNCTION OF QND

The signal-to-noise (SIN) ratio achieved by a photodetector with quantum efficiency q is given by

144

QUANTUM MECHANICAL LIMIT IN OPTICS

[II, 8 4

and (s/N)NPS,

PD

=

(AS)

1-q

f+-+-

v

(An;) v2(As)

for number-phase squeezed states, (138)

Here f 3 ( AA? )/(As ) is the Fano factor and (An&) is the thermal noise electron counts of electronic amplifiers. The SIN ratios are degraded by nonunity detector quantum efficiency and electronic amplifier thermal noise. For a Fano factor f = 10and the required quantum efficiency for suppressing the SIN ratio degradation within 3 dB is 0.91,0.99, and 0.999, respectively. Such a high quantum efficiency is difficult to achieve. Moreover, the quantum noise-limited S / N ratios are realized only when the electronic amplifier thermal noise can be negligibly small. This is not the case for a reasonable number of signal photon number ( A s ) . A QND detector transfers the information of a (weak) signal wave to an intense probe wave that is more robust to nonunity detector quantum efficiency and electronic amplifier thermal noise. In general, if a weak signal is ideally (noise free) amplified to a macroscopic level before detection, the S / N ratio degradation due to nonunity detector quantum efficiency and electronic amplifier thermal noise can be suppressed (YUEN[ 19861). A QND detector possesses this ideal preamplification function. The S / N ratios are given by

',

The second terms of the denominators in eqs. (139) and (140) express the S/N ratio degradation due to the phase noise of a probe wave. For large nonlinear coupling strength F and probe photon number, the ideal S/N ratios are realized irrespective of nonunity detector quantum efficiency and electronic amplifier thermal noise.

11,s 41

QUANTUM NONDEMOLITION MEASUREMENT

145

4.10. USE OF SQUEEZED STATES AS A PROBE WAVE

For the signal wave photon number (A,) = 100, the second terms of the denominators in eqs. (139)and (140)can be reduced to less than one at the probe laser power as much as lo4 W! The numerical parameters for a silica mKs, L (interaction length) = 10 km, r fiber system such as f 3 ) = (mode radius) ~2 pm, and ‘c (pulse duration) of 10 psec are assumed. If the quantum phase noise of a coherent state probe laser is suppressed by a squeezed state, the required probe laser power can be reduced to a reasonable level. The two schemes for suppressing the quantum phase noise of a probe laser are shown in fig. 31. In a Mach-Zehndar interferometer configuration, shown in fig. 3 la, the origin of the noise is the vacuum field fluctuation incident on the beam splitter (IMOTO, HAUSand YAMAMOTO [ 19851). This noise can be minimized when a squeezed vacuum field is fed into the beam splitter. For this case the second term of the denominators in eqs. (139)and (140)becomes

1

x‘”

,

DM signal WS -.-.: ......*: ,............,. r:.........*...&

DM

I

I

(b)

Fig. 3 1. Improved Kerr-QNDmeasurementscheme using squeezed states. (a) Use of a squeezed vacuum and (b) use of correlated probe and reference beams.

146

QUANTUM MECHANICAL LIMIT IN OPTICS

[II, 0 5

This term is reduced to less than unity even when the probe laser power is as small as 4 mW for the same numerical parameters. In a PM-to-AM conversion scheme using a Fabry-PBrot interferometer, shown in fig. 31b, the quantum phase noise of a probing signal wave can still be suppressed by referring the quantum phase noise of an idler wave. As discussed in $ 3.7, the photon pair (signal and idler waves) from a nondegenerate OPA features the complete phase correlation so that the quantum phase noise of a probing signal wave is counteracted by measuring that of a reference idler wave. For such a case the same suppression as that in eq. (141) is realized (BJORKand YAMAMOTO [ 1988b1).

0 5. Quadrature Amplitude Amplifiers and Photon Number Amplifiers 5.1. GENERAL QUANTUM AMPLIFIERS

The SQL on photon amplification, discussed in $2.4, stems from the fact that an ordinary linear amplifier amplifies the two conjugate observables simultaneously. After amplification of the signal to a classical power level, the signal can be measured with no additional uncertainty. Any simultaneous measurements of two conjugate observables inevitably suffer from additional noise (AUTHERSand KELLY[1965], GORDON and LOUISELL[1966], SHE and HEFFNER[ 19661, YUEN[ 1983a1). The uncertainty product is at least twice as large as the intrinsic uncertainty product because of this additional noise. This generalized uncertainty principle also must be obeyed when two conjugate observables are simultaneously amplified and then measured. Additional noise in such a simultaneous measurement are attributed to amplifier internal noise (YAMAMOTO and HAUS[ 19861). If only one observable is amplified and the conjugate observable is deamplified (attenuated), the amplified signal does not necessarily suffer from excess noise. Suppose the two quadrature components are amplified with different gain constants G, and G,,

6, = &d,

+El

(142)

y

h2=JG2d2+E2.

(143)

Here bl and 6, are the two quadrature amplitudes of the output mode, and d and d 2 are those of the input mode. f1and f 2 are the internally generated fluctuation operators. To satisfy the commutation relation h

A

[ b , , b,l = [ d , ,

4 1 = 1.

5 ' 9

(144)

11.8 51

147

QUANTUM AMPLIFIERS

one finds

[El,E,] = $i(l - ,/-I.

(145)

Here it is assumed that the input mode and internal fluctuation operators are quantum mechanically independent; that is, [ d , , P,] = [El, (i,] = 0. The uncertainty product of the input equivalent noise operators, defined by P , / a and f2/&, results from eq. (145),

,

In the special case of ,/G G, = 1 the uncertainty relationship (146) disappears, and the amplifier does not necessarily add noise to the amplified signal (CAVES [ 198 11). In this case, however, only one quadrature is amplified (G, > 1) so that the information extraction from the other quadrature becomes impossible. Even though the two gain constants are both greater than unity, B 1, it is still possible to suppress one quadrature added noise, for instance (AF:)/G, < $, by enhancing the other quadrature noise AF:)/G2 B $. Although the two quadrature components are simultaneously amplified, the information extraction from the quadrature d, becomes impossible because of the large internal noise. The SIN ratio for one quadrature is preserved in the afore-mentioned two amplifiers,

,/a

<

Thus the SQL on photon amplification can be overcome.

5.2. DEGENERATE AND NONDEGENERATE PARAMETRIC AMPLIFIERS

A degenerate parametric amplifier, shown in fig. 32a is an example of the amplifier that satisfies G , G, = 1. The evolution equation is written as

a f iJm=

= + Here satisfy the relation indicated by eq. (148).

> 1 and

=

f i - Jm< 1 exactly

1, and the amplified output is free from noise as

148

QUANTUM MECHANICAL LIMIT IN OPTICS

inpu

DM

DPA

HM

Fig. 32. Degenerate parametric ampliier for noiseless in-phase amplitude amplification. (a) Schematic setup and (b) input and output states in phase space.

As discussed in 3 3.3, a degenerate parametric amplifierproduces a squeezed vacuum state. The major difference between the squeezed vacuum state generation and the one-quadrature amplification is that the phase of a pump wave must be locked to that of an input signal wave in order to amplify the quadrature component which carries some useful information. A phase-locked loop incorporated with an optical homodyne receiver can be used for this purpose. A nondegenerate parametric amplifier with a squeezed vacuum state input at an idler channel, shown in fig. 33a, is an example of the amplifiers that have squeezed internal noise sources. The signal output mode is written as

Suppose the input state at an idler channel is squeezed such that ( A d : ) 4 i, the amplified output bSl is almost free from additional noise. On the other hand, the amplified output 6,, is subject to additional noise because of (Ad&) 9 i. The input and output relations for the two amplifiers are schematically shown in fig. 32b and fig. 33b, respectively.

QUANTUM AMPLIFIERS

149

Fig. 33. Nondegenerate parametric amplifier with squeezed idler input for quasi-noiseless in-phase amplitude amplification. (a) Schematic setup and (b) input and output states in phase space.

5.3. PHASE-LOCKED OSCILLATOR

Let us consider a phase-locked oscillator with an optical homodyne receiver and feedback loop, shown in fig.34a. Suppose the coherent excitations of an input signal and local oscillator waves are in quadrature phase (90 degrees out of phase), as shown in fig. 34b. This phase locking between the input signal and local oscillator waves is realized by feedback stabilization. The field incident

150

QUANTUM MECHANICAL LIMIT IN OPTICS

0output signol

(0)

input signol local oscillator

(b)

Fig. 34. Beam splitter with correlated oscillator for quasi-noiselessquadrature amplitude amplification. (a) Schematic setup and (b) input and output states in phase space.

on the photodetector is then written as

8,

= JEd, =

+J

G d ,

& ( (dsI ) + Ad,, + iAd,,) + J1-E[A d , , + i( (d,, ) + Ad,,)]

Here E is a power transmission coefficient of a beam splitter; Ad,, and Ad,, are the quadrature fluctuation operators for the signal wave, and Ad,, and Add2 are those for the local oscillator wave. Note that Ad,, includes the phase modulation term driven by the error signal extracted from the homodyne receiver. The photodetector with unity quantum efficiency measures the operator AD,

+ 2JEU - 4 [ (

4 1

)Ad,, + ( ~ , , > A ~ , , I*

(151)

The products of small fluctuation terms are neglected here. The average photon number E ( d,, )’ + (1 - E ) (d,,)’ can be compensated for by the dc offset,

II,O 51

QUANTUM AMPLIFIERS

151

as shown in fig. 34a. The error signal (current) then is proportional to the photon number fluctuation operator AADy AAD

AD

=

-

(AD)

2~(d,,)Ad,,

+ 2% J3-ca,s

+ 2(1 - E)

(d,2)AdI2

)Ad,, + ( d , 2 ) A 4 2 ) .

(152)

As mentioned earlier, the phase modulation to the local oscillator wave is realized to make the average value for AAD zero; that is, the coherent excitations of the signal and local oscillator waves are 90 degrees out of phase. Moreover, if the phase-locked loop has sufficient gain and bandwidth covering the signal modulation bandwidth, the fluctuation term AAD itself can also be reduced to zero. Thus the modulated Ad4, term becomes quantum mechanically correlated with the other fluctuation terms,

If the local oscillator wave is much more intense than the input signal wave (d,, ) % (d,, ) and the power transmission coefficient of the beam splitter is close to unity E E 1, the second and third terms can be neglected when compared with the first term. Under such conditions Ad,, is negatively correlated with Ad,,. The output field ,& ,, from the beam splitter is

E,,,

=

-J1-Eds

+ JEd,=d,,

( 154)

where the relation E N 1 is used to derive the second equality. From eqs. (153) and (154) it is obvious that the output wave phase is a complete replica of the input signal phase

This is considered to be an ideal amplification process of the signal wave phase information. The performance of the phase-locked oscillator is similar to that of a degenerate parametric amplifier, but there is a subtle difference. A degenerate parametric amplifier amplifies one quadrature amplitude, but a phase-locked oscillator responds to the signal phase. The difference, however, is not clear in the linearized analysis just described. Anothcr difference is that the other

152

QUANTUM MECHANICAL LIMIT IN OPTICS

[II, 8 5

quadrature amplitude is deamplified, that is, squeezed in a degenerate parametric amplifier, but the photon number which is a conjugate observable to the phase is not squeezed in a phase-locked oscillator.

5.4. NONDEGENERATE PARAMETRIC OSCILLATOR WITH IDLER MEASUREMENT-FEEDBACK

A nondegenerate parametric oscillator with idler measurement-feedback, shown in fig. 35, works as an ideal amplifier of a signal photon number. As discussed in 3 3.8, the output photon numbers of the signal and idler waves in a nondegenerate parametric oscillator are quantum mechanically correlated when the counting time interval is longer than the photon lifetime. The quantum correlation is perfect when the cavity internal loss is negligible and there are no input signal and idler waves. When the photodetector output current for the idler photon number Ai(out) is subtracted by the G-times of the photodetector output current for the input signal photon number A!'"), the error signal for feedback stabilization to the pump wave intensity is obtained. With sufficient feedback gain and bandwidth, A?) becomes equal to GAP). Because of the quantum correlation mentioned earlier, the output signal photon number is then the amplified replica of an input signal wave photon number. In this system the phase information of the input signal is completely lost. The feedback scheme shown in fig. 35 suffers from the limited bandwidth due to the finite loop delay. To overcome this problem, we can employ the feedforward technique shown in fig. 36. According to the error signal, the output signal wave from the OPO is either attenuated or amplified. By adjusting the two delay times for the output signal wave and the control electrical signal so that they are exactly equal, the bandwidth limitation problem can be lifted. However, any linear attenuator and amplifier impose an additional fluctuation on the signal wave. As already stated in 3 3.8, the residual photon

-*qz

signal input

pump wave amplitude modulator

Fig. 35. Nondegenerate parametric oscillator with idler photon counting and feedback for photon-number amplification.

QUANTUM AMPLlFIERS

153

4

pump wave UP

Fig. 36. Nondegenerate parametric oscillator with idler photon counting and feedforward for photon number ampliflcation.

-

number fluctuation caused by such additional noise is given by (AA2 ) ( A ) 'I2 ( B J ~ R and K YAMAMOTO [ 1987~1).If the input signal has the average photon number (A, ) and the variance (AAL ),and the amplification factor is G,the amplified signal photon number and variance are G (A, ) and G2( AAi ) .Thus ideal (noise-free) amplification is realized when the following condition is satisfied: (AA2)res

E

(G(Ain))'/2 4 G2(AA,',)

.

Apparently, ideal amplification for a pure photon number state cannot be realized because of (Ad,', ) = 0. But for the input signal wave with a finite photon number noise, (Ad: ) # 0, the amplification factor G can be always made large enough to satisfy condition (156).

5.5. LASER OSCILLATOR WITH QND MEASUREMENT-FEEDBACK

Let us consider a laser oscillator with a QND measurement-feedback loop, as shown in fig. 37. A Mach-Zehnder interferometer is constructed for a probe wavelength, in which the two Kerr media are placed in each arm and are driven by the input signal wave and by an independent laser oscillator output wave. The nonlinear interaction strength for the Kerr medium 1, = X \ ~ ) L ~ / A',, is ~ ,made to be G-times that of that for the Kerr medium 2, = x$~)L,/A,,2. Here L, and A , , are the third order nonlinear coefficient, crystal length, and beam cross-section of the medium (i = 1 or 2). Therefore, when the oscillator output photon number is exactly G-times that of the incident signal photon number, the probe waves in the two arms experience the same phase shift and the dual detector output is nullified. The error signal extracted from the dual detector thus represents the deviation of the from the G-times that of the incident oscillator output photon number A?)

fi

,

154

QUANTUM MECHANICAL LIMIT IN OPTICS

vacuum input

Kerr medium 1

signal

probe

c

err medium 2

oscillator

Fig. 37. Photon-number amplifier by differential quantum nondemolition measurement of photon numbers and negative feedback to an internal oscillator.

signal photon number GA$"),

fiy) - GAY).

(157)

In the limit of large feedback gain the error signal can be suppressed to zero, which indicates that the oscillator output photon number exactly equals G-times that of the input signal photon number. The unitary evolution for the probe wave bin in the Kerr medium 1 is given by

h,,

=

exp(i JF,A!'"))

hi, ,

(158)

where 1 hi" = (a + e) .

(159)

Jz

In eq. (159), d is the probe wave incident on the dichroic half-mirror DM,, and c^ is the vacuum fluctuation at the probe wavelength incident on DM,. DM, and DM, are 50-50 beam splitters for the probe wavelength but are completely transparent for the signal wavelength. The dichroic mirrors DM,, DM,, and DM, are completely transparent for the signal wavelength and completely reflective for the probe wavelength. The unitary evolution for the probe wave &, in the Kerr medium 2 is written as

dut= exp(i

A?))

ain,

(160)

11, I51

QUANTUM AMPLIFIERS

155

where

The output waves C and bout and dout:

f

from DM, are the interference outputs of

and A

1

A

f = - ( - bout + iS,J

Jz

*

(163)

The dual detector output measures the following operator: &+P

-

f +f = i[8,tUt&,, - S , ~ t 8 0 u t ~

where the probe wave is assumed to be linearized as follows: 6 = (6,)

+ Adl + iAa2 .

(165)

The feedback loop suppressesthe error signal (164), and thus the output signal photon number is

where G , = JFl/F2 is used. The average and variance of eq. (166) are

(A?))

=

G,(rf$")),

and

Here ( A p ) = ( 6 ) is the average probe wave photon number. Ideal amplication is realized when the second term of eq. (168) is negligibly small when

156

QUANTUM MECHANICAL LIMIT IN OFTICS

compared with the fist term. The input equivalent noise is

The signal wave at the output of the Kerr medium still possesses the phase information, even though it is contaminated by the back action noise caused by the photon number noise of the probe wave bin, A$?)

=

A$.s(~")+

f i (ci, ) (

~ +d~~

q .

(170)

If we measure (170) by an optical homodyne receiver and modulate the output signal wave by (170), the phase information is carried over to the output signal wave but the noise is added at the same time. The output signal wave is further added by the noise caused by the photon number noise of the probe wave din A

9

JF2 (41)

(Ad1 - Ael).

(171)

f i 6.

4 This is much smaller than the second term of eq. (170) because Since the phase information is not amplified, G, = 1, the input equivalent noise is

(172) The uncertainty product of the amplifier internal noise is just double the minimum required value:

(A@:) G,Z

(AP;) -- 12 ' G,'

(173)

It is also possible to employ a linear feedforward scheme to overcome the loop delay problem in this case.

5.6. AMPLIFICATION AND DEAMPLIFICATION FOR QUANTUM STATE TRANSFORMATION

Both quadrature amplitude squeezed states and number-phase squeezed states lose their nonclassical natures when they encounter a loss. This is an inevitable phenomenon because the quantum noise of these nonclassical lights is smaller than that of a vacuum field and the vacuum fluctuation is coupled

I I , § 51

157

QUANTUM AMPLIFIERS

to the signal wave because of the quantum mechanical fluctuation-dissipation theorem. Suppose a squeezed state is amplified so that the quadrature amplitude noise is well above the SQL while preserving the SIN ratio, as shown in fig. 38. After amplification followed by attenuation, the SIN ratio is given by SIN =

GT(d1)2 GT(Ad:) +a(l - T)

(174) '

fi

Here is the amplitude gain of an amplifier and JT is the amplitude transmission coefficient of a lossy medium. The second term of the denominator represents the vacuum fluctuation coupled from reservoirs. Therefore the S / N ratio degradation is decreased if the amplifier gain satisfies the constraint

Here the second equality assumes the optimum squeezing (Ad:),,,,t = 1/[4(2(fi,) + l)] thatmdmizestheSINratiofor agiven ( A , ) value. At this time the other quadrature amplitude experiences deamplification and dissipation, and its variance becomes that of a vacuum field,

amplified signal

deamolified

a1

I

Fig. 38. Changes of field in phase space through amplification,attenuation, and deamplification.

158

QUANTUM MECHANICAL LIMIT IN OPTICS

T (Ad:) G

-

+ $(l - ?‘)-+:

PI, 0 6

(?‘a1).

The attenuated signal is no longer in a minimum uncertainty state, but the SIN ratio is preserved at least for one quadrature amplitude. When the amplifier gain satisfies condition (176), the signal photon number is much larger than that of an input signal. Often a photodetector and other elements cannot respond properly to high intensity because of saturation characteristics and nonlinear effects. For this case we can deamplify the signal wave so that the photon number is decreased while preserving the S/N ratio. When the deamplification factor is equal to GL and eq. (176) is satisfied, the SIN ratio is eventually given by

As shown in fig. 38, the variance of the other quadrature is much larger than that of an input state so that the photon number becomes higher. Similar amplification and deamplification to preserve the SIN ratio are possible for a number-phase squeezed state. In this case, however, the photon number is preserved because the enhanced phase noise does not waste the signal photon number. The importance of a quadrature amplitude amplifier and photon number amplifier is that the S/N ratio degradation due to dissipation processes can be suppressed only by them. It is equally useful to deamplify the signal power level without degrading the SIN ratio, which is also realized by these amplifiers.

8 6.

Quantum Mechanical Channel Capacity

We have shown that the SQL in optical precision measurement and communication is not an intrinsic one and can be, in principle, exceeded by the nonclassical lights, QND measurements, and single observable amplifiers. In this section we will discuss the intrinsic quantum limit, which ultimately determines the information extraction from a light wave. It emerges in the form of quantum mechanical channel capacity and Bohr’s time-energy uncertainty principle.

11, B 61

159

QUANTUM MECHANICAL CHANNEL CAPACITY

6.1. QUANTUM MECHANICAL CHANNEL CAPACITY FOR NARROW-BAND COMMUNICATION

The intrinsic quantum limit (IQL) in optical precision measurement and communication emerges in a very simple manner if the classical Shannon's channel capacity is quantized (STERN[ 19601, GORDON [ 19621, LEVEDEVand LEVITAN[ 19631, TAKAHASHI [ 19651, HELSTROM [ 19761, YAMAMOTO and HAUS [ 19861). The quantum mechanical channel capacity for narrow-band communication is derived from the negentropy principle of information (BRILLOUIN [ 19651) and the sampling theorem (NYQUIST[ 19281, SHANNON [ 19481). Suppose each degree of freedom (DOF) that corresponds to a Nyquist mode for a band-limited electromagnetic wave traveling along the transmission line has the average number of photons, (A)=

( A s )

+

(Ath)

(178)

*

The maximum entropy per mode is given by the thermodynamic entropy for a bosonic system (LANDAUand LIFSHITZ[ 19591, KUBO[ 19651) H,,,

= (A)

( + (1)) + l n ( l +

In 1

-

(A))

(179)

Here ( A , ) is the average signal photon number and ( & ) = 1/ (exp ( h u l k , 0) - 1) is the average background thermal noise photon number. The maximum entropy (179) is realized when the total photon number distribution per mode obeys the thermal (geometrical) distribution, P(n) =

(A)" (1 + ( A ) ) " "

However, the thermal distribution of the signal photon number does not realize eqs. (179) and (180). Equation (179) is an upper bound of the entropy, which is difficult to realize. According to the negentropy principle of information, the maximum amount of information Z that can be extracted from each DOF is equal to the difference between the total entropy (179) and the residual (noise) entropy. The noise entropy is calculated for a thermal equilibrium field and is given by

160

QUANTUM MECHANICAL LIMIT IN OPTICS

PI, 8 6

Taking the difference between eqs. (179) and (181), one obtains

-

(fith)

(

In 1 +-(A:h

)) (natural digits) .

This thermodynamic negentropy principle of information is also valid quantum mechanically and can be interpreted as follows. It is implicitly assumed that to derive eq. (182), photon number states and photon counters are employed as a quantum state of the electromagneticfield and a detection scheme. Before the ideal photon counting measurement the received signal plus noise wave is assumed to be in a statistical mixture of photon number states represented by the density operator 8. Suppose the diagonal matrix element p,, = ( n 1 1 n ) is given by eq. (180). The entropy for such a wave before the measurement is then given by

which is equal to eq. (179) when pnn is given by eq. (180). According to the projection postulate (VON NEUMANN[ 1955]), after the ideal photon counting measurement the density operator is reduced to that of the photon number eigenstate lacmeas) = I n o ) (no I, where no is the measurement output. However, this does not indicate that the received signal photon number n, is exactly equal to no - (fit,, ) ,because the thermal photon number also features the geometrical distribution

Thus the density operator p,(meas)of the signal wave is not that of a photon number eigenstate but is still in a mixed state. The diagonal matrix element p:fn,en) is written as

Therefore the residual entropy of the signal wave is written as

H,,, - Tr( /3,1meas)

=

-

pifnnen89)In p,(fn,enas),

( 186)

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QUANTUM MECHANICAL CHANNEL CAPACITY

161

Fig. 39. Quasi-probability density Q(a) = (a1 1 I a ) and photon statistics p(n) = ( n I 1 In) for the density operator before and after measurement.

which is equal to (181). The statistical properties of the wave before and after the measurement are schematically shown in fig. 39. Suppose a signal wave is centered at an angular frequency wo and has a bandwidth of 2nB. The Fourier transform of the flat spectrum extending from oo- nB to oo+ nB is proportional to the Nyquist function, sin (nBt)

( a , COSOt

+ a, sinot).

(nB0 A sequence of Nyquist functions displaced by z = 1/B is orthogonal with each other and reproduces fully any bandwidth-limited function confined to wo - nB and wo + nB. Each Nyquist mode has two degrees of freedom, that is, two quadrature amplitudes a , and a, or photon number n and phase $I. As we have seen already in § 3 and $4, the complete preparation and measurement of one observable is possible only when the information extraction from the conjugate observable is totally sacrificed. In the preceding discussion the signal wave consists of a sequence of photon number states, and thus the phase is completely random for this wave. Therefore we can conclude that the arrival rate of the degree of freedom through a channel of bandwidth B is not 2B but is B ; that is, it is equal to the arrival rate of Nyquist modes. The channel capacity is then written as

162

QUANTUM MECHANICAL LIMIT IN OPTICS

(a 1

21 mlo[ number state

-

1

1 6 ~1d

6

1 10 lo2 Average Photon Number Ns

6

lo3

Averoge Photon Number Ns

Fig. 40. (a) Normalized channel capacity C/B versus average photon number n , for the three cases: number state and photon counter, squeezed state and homodyne receiver, and coherent state and heterodyne receiver. (b) Normalized energy cost per bit E/(k, T h2)versus average photon number for the case with number state and photon counter; nth is the thermal photon number.

The channel capacity C/B normalized by the channel bandwidth versus (A, ) is plotted in fig. 40a. Note that so far we do no mention any specific modulation-demodulation scheme in the derivation of eq. (188). In fact, no signal modulation scheme realizes eq. (188) when (nth) # 0. The limit given by eq. (188) is only approximately realized. Here we describe the two simple examples, a pulse position modulation (PPM) and multiple frequency shift [ 19811). For a PPM keying (MFSK) signals (PIERCE,POSNERand RODEMICH signal each degree of freedom corresponds to each slot that constitutes one word, as shown in fig. 41. Suppose one word consists of M slots of duration z (A4= T/z)and one photon number state I 1) is assigned to one of the M slots with all the other slots unexcited (vacuum states 10)). This single photon carries the information log, M bits per time interval T. Note that the photon number distribution of each slot approaches eq. (180) when M goes to infinity

11, 8 61

163

QUANTUM MECHANICAL CHANNEL CAPACITY

Symbol

B

A

D

A

C

F

B

MFSK

Fig. 41. Pulse position modulation (PPM) signal and multiply frequency shift keying (MFSK) signal.

-

and the channel bandwidth is in the order of B 117, that is, the arrival rate of the degree of freedom. For a MFSK signal each degree of freedom is the carrier frequency (color of photon) assigned to the word interval T, as shown in fig. 41. Since the spectrum of this pulse of duration T spreads over Am 2n/T centered at each carrier frequency, the number of independent colors that can be transmitted within the channel bandwidth B is given by M = 2nB/Aw = T/z, where z = 1/B. The different color in MFSK signals corresponds to the different position in PPM signals, and both are equivalent. When the average signal photon number (A,) is much smaller than one, the channel capacity of these two orthogonal modulation schemes are close to the limit (188). The channel capacity of (188) is that for the combination of photon number state and photon counting detection. It is independent of the choice of degrees of freedom (position or color), as just mentioned. The channel capacities for various combinations of different quantum states and detection schemes are and HAUS[ 19861). The combisummarized in a previous paper (YAMAMOTO nation of photon number state and photon counting detection is optimum from the viewpoint of channel capacity for a given (A, ) value. The channel capacity for the combination of a squeezed state and its optimum receiver, the homodyne detector, is given by

-

C = Bln(1

+ 2(A,))

(S.S.

+ homodyne).

(189)

The channel capacity for the combination of a coherent state and its optimum receiver, the heterodyne detector, is given by C = B In (1 + (A, ) ) (C.S.+ heterodyne)

.

( 190)

As shown in fig. 40a, these channel capacities drop off rapidly when (A,) becomes smaller than one. This characteristic is not because a coherent state and squeezed state feature finite quantum noise but mainly because optical

164

QUANTUM MECHANICAL LIMIT IN OPTICS

[II, 8 6

homodyne and heterodyne receivers extract only the wave entropy, the second term of eq. (179), which drops rapidly as (A, ) + 0. The photon entropy, the first term of eq. (179), is dominant for small (A,) values, and it can be extracted only by a photon counter (YAMAMOTO and HAUS[ 19861).

6.2. MINIMUM ENERGY

COST PER BIT

The information that can be carried by a single photon is defined by

In the limit of the small signal photon number (A, ) G 1, this is reduced to the following two cases:

The information that can be transmitted by a single photon goes to infinity at 0 = 0 (no thermal background noise). As far as the minimum photon number per bit is concerned, there is no quantum mechanical limit. However, at a finite temperature there exists a lower limit for the minimum photon number per bit determined by the thermal background noise. This number is 32 nats/photon (46 bits/photon) for a wavelength of 1.5 pm and a temperature of 300 K. As shown in fig. 40b, the minimum energy per bit approaches k, T In 2 as (A, ) becomes much smaller than one. This minimum energy required to make

11,s 61

QUANTUM MECHANICAL CHANNEL CAPACITY

165

a “measurement”was first discussed by Szilardto exorcise a “Maxwell demon” [ 18751, SZILARD and to defend the second law of thermodynamics (MAXWELL [ 19291). It has been believed for many years that this is the minimum energy “dissipation” to measure one bit of information (BRILLOUIN[ 19561, GABOR [ 19611). The two independent inventions of quantum nondemolition measurement for gravitation wave detection (BRAGINSKY, VORONTSOVand THORNE [ 19801, CAVES, THORNE, DREVER, SANDBERG and ZIMMERMANN [ 19801) and a reversible logic in quantum mechanical computers (FREDKINand TOFFOLI [ 19821)incidentally verified that the information can be extracted and processed without any energy dissipation. The minimum energy k, T ln2 is the energy that the system must possess to carry one bit of information. A quantum mechanical limit on the minimum energy cost per bit emerges if an optical homodyne or heterodyne receiver is used instead of a photon counter. The maximum amount of information that can be extracted from a singlephoton is 1.44 bits for a heterodyne detector and 2.88 bits for a homodyne detector, respectively (YAMAMOTO and HAUS [ 19861). This ultimate information efficiency is achievable only by an enormous sacrifice of the channel efficiency C/B, as shown in fig. 40b.

6.3. BROADBAND COMMUNICATION AND TIME-ENERGY UNCERTAINTY RELATIONSHIP

Even though the quantum mechanical limit on minimum energy cost per bit, hw,B (A, ) ln2/C, is lifted by a photon number state and photon counter, there is still a trade-off relationship between energy cost per bit and time interval per bit, ln2/C, as mentioned earlier. It reminds us of the Bohr’s time-energy uncertainty relationship. To study this problem, we need to clarify the ultimate bandwidth available for a channel, that is, the narrow-band analysis must be extended to a broad-band system. Suppose the signal spectrum is centered at an angular frequency wo and extended to f w,/2. The arrival rate of Nyquist’s modes is given by 0,/2 R in this case. This rate corresponds to the sequenceof signal pulses with a duration of one optical cycle z = 2n/u0. The localization of photons in space-time and its relation to the frequency spectrum have been extensively discussed in the literature (HAN, KIM and Noz [ 19871 and references therein). Putting aside the fuzzy relationship between pulse shape and frequency spread of a weakly localized light wave, the maximum channel bandwidth for a photon of center angular frequency wo is of the order of B = w0/2n.

166

PI,$ 6

QUANTUM MECHANICAL LIMIT IN OPTICS

The signal energy E required to transmit one bit of information is given by

hw, ln2

E = h ~ ~ ~ B ( hl i2 ,/ C ) =

.

(192)

The time interval T required to transmit one bit of information is t = -In=2

27rln2 w,[Ln(l+ (A,))+

(ti,) In

(193)

1 +-

(

Here the thermal background noise is neglected. The normalized product Et/(h/2) versus (A,) is plotted as a function of ( A t h ) in fig. 42. The time-energy product Et is minimum at (A, ) = 1 and is close to i h . A small factor for the deviation of Et from $h is not important because our deviation of the channel bandwidth B = w0/27t is only an approximate one. When ( A , ) is greater or smaller than unity, the time-energy product is higher than i h . Bohr’s time-energy uncertainty product, AE At

ih,

( 194)

and has been a somewhat provoking subject for many years (MANDELSTAMM TAMM[ 19451, FOCKand KRYLOV[ 19471, AHARANOVand BOHM[ 19611, PARTOVI[ 19861). Its rigorous interpretation is that the product of the dispersion in the energy of a system and the time duration over which the expec-

Averoge Photon Number

Ns

Fig. 42. Normalized energy-time product per bit ET/(h/2)versus average photon number Nsfor the case with number state and photon counter.

ILO 61

QUANTUM MECHANICAL CHANNEL CAPACITY

167

tation value of a system observable varies is greater than $h. However, the preceding argument suggests that the time-energy uncertainty relation is extended to the energy of a system times the duration over which it is measured must be larger than t h . This point is also discussed in the literature (WIGNER [ 19721). A similar value for the time-energy uncertainty product is obtained in the broadband channel formula using the thermal equilibrium blackbody radiation at an elevated temperature (LEVEDEV and LEVITAN [ 19631, PENDRY[ 19831). It is important to note that the energy constrained by this relation is not an energy which must be dissipated in the measurement process but the energy which a system must have. The fact that it is possible to have information extraction without the system dissipating energy is obviously demonstrated by the quantum nondemolition measurement discussed in § 4 and also by the proposed (quantum mechanical)reversible computers (BENNETT [ 1973, 19871, BENIOFF [ 19821, FREDKINand TOFFOLI [ 19821, FEYNMAN [ 19851, LANDAUER [ 19871). 6.4. PRECISION MEASUREMENT AND TIME-ENERGY UNCERTAINTY

RELATIONSHIP

In communication (the transfer of information between a transmitter and receiver) the signal arrive1 time is anticipated. They are called “heralded signals”, in which the system clock is predetermined. The absence of a photon, a vacuum state, thus carries the information so that a single photon can transmit an infinite amount of information, in principle. In precision measurement (the transfer of information from the state of a natural system to the observer), however, the signal arrival time is not anticipated. They are called “self-heralded signals”. In recording the gravitational waves from a supernova, the astronomer is unaware of the arrival of the gravitationalwave until he or she records it. The absence of a photon cannot carry the information in this case. Some physical phenomena, such as spontaneous decay of an excited atom, Rabi flopping and energy splittingof an atom in an empty cavity, and the Casimir effect are the manifestation of the existence of vacuum fields. They might be utilized to detect the existence and nonexistence of vacuum fields and be relevant to information extraction from vacuum states if some clever detection scheme is invented. But these effects appear in well-defined boundaries for the field, which is unnatural for the previously mentioned applications. Thus we have to exclude the use of vacuum states for precision measurement application.

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QUANTUM MECHANICAL LIMIT IN OYl'ICS

[II, 5 7

Suppose there is a signal pulse of duration T consisting of many carrier frequencies (MFSK), in which each independent signal state is represented by l a ) = {n,; j = 1,2, . . . ,M } and n, is the photon number of carrier frequency w,. The independent carrier frequencies must be separated by Aw = 2 4 T , as mentioned already. If the probability of sending the state ( a ) is Pa,the information carried by the signal pulse is given by

I=

-1PalnPa,

(195)

a

where Xu P, = 1. Since the energy of the state I a ) is Z,hon,, the average signal energy is given by

E

=

c Pa c n,hw,. a

i

The maximization of eq. (195) under the constraints (196) and X u Pa = 1 is realized by the variational principle. The vacuum state is excluded here. The channel capacity is bounded by (BEKENSTEIN[ 19871) E h

C < - x 0.2279 (bits/s) .

(197)

This corresponds to a minimum time-energy product Et N 4.388h, which is about ten times worse than that for communication.

9 7. Applications In this section we discuss applications of nonclassical lights, QND measurements, and single-observable amplifiers. The first application is optical communication, for which QND measurement and single-observable amplifiers play central roles in breaking the SQL.Nonclassical lights find limited applications, only in case where fiber loss in negligibly small, that is, in a short-distance system, since vacuum fluctuations coupled by means of loss destroy any nonclassical lights. The second application is the laser gyro, for which nonclassical lights play central roles in breaking the SQL. Photon twins and squeezed vacuum states will find applications in an active and passive laser gyro, respectively. The third application is an interferometer breaking the SQL on gravitational wave detection. QND-like measurements (including a contractive state measurement) of a free mass position play a central role.

11, II 71

APPLICATIONS

169

7.1. COMMUNICATION BREAKING THE SQL

In short-distance optical communications, such as a local area network and a data bus loop for connecting computers and terminals, signal attenuation due to fiber loss is negligibly small. The minimum loss of a silica fiber is 0.16 dB/km at 1.55 pm wavelength. The main source for degrading the S/Nratio of a signal wave in such a system is an insertion loss of devices for information tapping and sending. QND measurements of photon number and quadrature amplitude, which were discussed in 4,make it possible to read out the information of a signal wave with smaller insertion loss than that required for a usual waveguide tap. A photon number amplifier and quadrature amplitude amplifier followed by a usual waveguide tap can also be utilized for the same purpose, as discussed in $ 5.6. Here the loss T is an information tapping loss, and the SIN ratio degradation is suppressed if the gain G is much greater than the loss. The preamplification function of these two measurement schemes becomes important as the system bit rate increases. This is because the electronic amplifier thermal noise increases explosively as a bit rate exceeds a few Gbit/s. The present linear signal detection schemes, such as an avalanche photodiode and optical heterodyne and homodyne receivers, do not possess a sufficiently large gain-bandwidth product to overcome this problem. QND measurement of the photon number is a nonlinear signal detection scheme, for which the thermal noise is more efficiently suppressed as the bit rate increases (as the pulse duration decreases). Note that the nonlinear interaction parameter JF in eqs. (139) and (140) suppressing the thermal noise is inversely proportional to a mode volume V, that is, a pulse duration. A number-phase squeezed state and quadrature amplitude squeezed state improve the minimum required signal photon number in a small-loss system, as discussed in $3.2 and 5 3.4. However, the real importance of a squeezed state in optical communication is its capability to realize such QND measurements and single observable amplifiers with a modest optical power, as discussed in $ 4.10 and $ 5.5, rather than its use as a signal wave itself.

7.2. GYROSCOPE BREAKING THE SQL

In $ 2 . 4we have discussed the fact that the SQL of an active laser gyroscope is imposed by the phase diffusion noise, that is, Schawlow-Townes linewidth of a laser. Let us consider a ring cavity containing two independent, clockwise

170

QUANTUM MECHANICAL LIMIT IN OPTICS

ider

[II, 5 7

signal

( b) (left circularly polarized) (right circularly pobrized)

Fig. 43. Four-frequency ring gyroscope based on two nondegenerate parametric oscillators.

(CW) and counter-clockwise (CCW), nondegenerate optical parametric oscillators (NOPO), as shown in fig. 43. Suppose that signal and idler waves are in orthogonal polarizations (type-I1 interaction). If we put a Faraday rotator inside a ring cavity, the frequencies of the CW and CCW signal waves in right circular polarization are offset by = a. At this time the frequency of the CW and CCW idler wave left circular polarization are offset by = - a. With an identical amount but with opposite direction, q(cw) the rotation rate both signal and idler waves of CW experience the same frequency shift by + $a, where S is the Sagnac constant given in $2.4. The signal and idler waves of CCW, on the other hand, experience the opposite frequency shift of - iSn. To realize these operations of the two NOPOs under the rotation rate n,the energy conservation o,+ oi= wp must be satisfied and the pump frequency upmust be modulated according to the rotation rate for both NOPOs simultaneously: WyJ)

+ oi(CW) = (&) + f a + fsn+0) - fa + $a =

and

do) P + sn,

(198)

11, s 71

171

APPLICATIONS

Here a : ' ) , a ! ' ) and a ; ) are the signal, idler, and pump frequencies without a Faraday rotator and at rest (a= 0). The rotation rate is measured by beating the two beat notes of the photodetectors 1 and 2:

=

2sn.

(200)

Note that the rotation rate readout is independent of the absolute signal and idler frequencies and is free from the fluctuation of a Faraday bias a. The energy conservations (198) and (199) are satisfied by the closed feedback stabilization to the pump wave. A conventional phase-locked-loop technique, which is not shown in fig. 43, can be used for this purpose. Thus the two NOPOs operate in a self-consistent manner, and the rotation rate 62 can be extracted from the feedback error signal of the phase-locked loop (PLL). As mentioned in 3.8 and shown in fig. 19, the phase of signal and idler waves of an NOPO is negatively correlated. In a pump level not far from the oscillation threshold, this negative correlation is established not only for phase diffusion noise but also for intrinsic quantum phase noise. The time derivative of the negative phase correlation results in the following relations for the frequency noises between signal and idler waves:

+A

AO:") Aa:ccw)

o ~ ( =~ Am ) P'

+ Aaiccw)

=

(20 1) P'

(202)

Here wpis the pump frequency noise due to a finite linewidth of a pump laser that is common for the two NOPOs because the two pump waves are taken from the same laser. Including the noise terms in (200), the rotation rate readout is now modified as [ a:CW) + Aa:CW) - (a:CCW) + Aa$CCW)]

- [oi = (a =

+ A,qi(ccw) - (q(cw) + Aai'Cw))]

+ S a ) - ( a - Sn)+ A a P - 60,

2sn.

(203)

The frequency noise can be canceled out exactly as well as the Faraday bias, and thus the SQL (22) on an active laser gyroscope can be circumvented. The SQL on a passive laser gyroscope can be overcome if a squeezed vacuum state is input on an open port of the beam splitter shown in fig. 5b.

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QUANTUM MECHANICAL LIMIT IN OPTICS

[II, I 7

7.3. GRAVITATIONAL WAVE DETECTION INTERFEROMETER BREAKING THE SQL

In 3 2.3 we have seen that the SQLs of active and passive gravitational wave detection systems are imposed by the phase diffusion noise of a laser and by the vacuum fluctuation incident on a beam splitter, respectively (see fig. 4). Moreover, the trade-off relationship between “photon counting error” and “radiation pressure error” dictates that the measurement error of a free mass position must be greater than the value given by eq. (21). The twofold SQL on gravitational wave detection systems can be exceeded by the following strategies. Let us consider an active gravitational wave detection system, as shown in fig. 44, in which the two lasers in a conventional system, shown in fig. 4b, are replaced by two NOPOs pumped by the same pump laser. NOPO 1 and NOPO 2 satisfy the energy conservation w, + wi= wpwith a slightly detuned frequency 0:’-)0:’) = o,(~) = a. When a gravitational wave arrives at the interferometer, the signal and idler frequencies coil) and o,(’)of NOPO 1 are modulated by the same amount $,kh, and those of NOPO 2 are kept constant.

Fig. 44. Gravitation wave detection interferometer based on two nondegenerate parametric oscillators.

11, I 71

APPLICATIONS

173

The gravitational wave amplitude is extracted by

= cokho.

(204)

Thus the initial frequency bias ct and its fluctuation are canceled out. It is obvious that the phase diffusion noise of the NOPOs and pump laser are suppressed in an identical manner to that of the gyroscope discussed earlier. A correlated spontaneous emission laser (CSEL) has also been discussed for surpassing the SQL (SCULLY[ 19851). In a CSEL the population inversions of the two lasers have a correlation such that the spontaneous emission-induced random phase walk becomes correlated with the two lasers. The difference between the ultimate limits of the NOPO gyroscope and CSEL gyroscope has yet to be explored. With a passive gravitational wave detection interferometer the SQL can be overcome if a squeezed vacuum state is incident on the open port of a beam splitter as shown in fig. 4a (CAVES[ 19811). The required signal photon number to reach the maximum sensitivity (21) is decreased to

Here ( A s )cII,opt is the optimum signal photon number for the case where a vacuum fluctuation is not suppressed. This squeezed vacuum state technique was demonstrated by a laboratory experiment (XIAO, W u and KIMBLE [ 19871). 7.4. MEASUREMENT FOR SURPASSING THE SQL OF A FREE-MASS POSITION

As already discussed in 8 4.1, the SQL of a free-mass position measurement can be suppressed by a contractive state, even though the free-mass position is not a “QND observable”. Figure 45 shows an optical interferometer that surpassesthe SQL of a free mass position measurement (BONDURANT[ 19861). Each optical beam passes through a Kerr medium, in which the phase becomes correlated with the photon number due to the self’-phase modulation effect. If the Kerr media are designed so that the effect of the self-phase modulation is negativewhen compared with the radiation pressure-induced phase modulation noise, the phase modulation noise can be canceled out. Thus the trade-off relationship between the measurement error determined by the phase noise of the probe wave and the radiation pressure-induced back action error determined by the photon number noise of the probe wave can be lifted.

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OUANTUM MECHANICAL LIMIT IN OPTICS

lr

===?La HM

j

Sqieezed

M

Vacuum

Fig. 45. A mass-position-measurement interferometer having lower noise than the SQL by virtue of the Kerr effect.

A similar effect can be realized by using two frequency (mutually correlated) squeezed states (BONDURANT and SHAPIRO [ 19841). These schemes are physical realizations of Yuen’s “contractive state measurement” (YUEN[ 1983a,b], CAVES[ 19851, NI [ 19861, OZAWA[ 19881).

8 8. Discussion and Conclusion This paper has reviewed standard and intrinsic quantum mechanical limits on optical precision measurement and communication. Nonclassical lights such as a quadrature amplitude squeezed state, number-phase squeezed state (number state), and correlated photon pair circumvent the SQL on photon generation. A gravitational wave detection interferometer driven by a squeezed vacuum state at an open port is such an example. The limit on repeated measurement accuracy of a free-mass position can be realized with much smaller optical power than that required for a conventional system without a squeezed vacuum state. The similar improvement is also realized by the use of negative phase correlation of photon pairs generated in a nondegenerate parametric amplifier. The measurement accuracy of a free-mass position is determined by the balance between the “photon counting error” and the “radiation pressure noise” (CAVES[ 19811). At first sight this resembles the general uncertainty relationship for simultaneous measurements of two conjugate observables (phase and photon number). However, the schemes to circumvent the preceding measurement limit were proposed by using two frequency squeezed and SHAPIRO [ 19841) and by using self-phase modulation states (BONDURANT in a Kerr medium (BONDURANT [ 19861). These possibilities are physical

11, 8 81

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175

realizations of Yuen’s “contractive state measurement” (YUEN[ 1983a1, CAVES [ 19851, NI [ 19861, OZAWA[ 19881). A four-frequency active gyroscope using two nondegenerate optical parametric oscillators is the other example. In a conventional laser gyroscope (DORSCHNER, HAUS, Holz, SMITH and STATZ [1980], CHOW, GEABANACLOCHE,PEDROTTI,SANDERS,SCHLEICH and SCULLY[ 19851) the measurement limit on inertial rotation is imposed by the random walk phase diffusion noise (Schawlow-Townes linewidth). The limit is incidentally the same as that of a passive ring gyroscope operating with the same photon number (EZEKIEL, COLE,HARRISON and SANDERS[ 19781). Since the signal and idler waves in a NOPO have a negative phase correlation in their phase, the preceding limit can be surpassed. The scheme resembles the proposed correlated spontaneous emission laser (CSEL) (SCULLY[ 19851). Quantum nondemolition measurement of a photon number, using interphase modulation in a x(3) medium, and that of quadrature amplitude, using degenerate parametric amplification in a x(’) medium, circumvent the SQL on photon detection. In a short-distance local area network the fiber loss can be safely neglected, but instead an information tapping loss is the dominant process that determines the system performance. The lossless information tapping with a QND detector may solve this problem. Since the loss can be negligibly small in such a short-distance system, a photon number state and squeezed state do not lose their advantage over a coherent state. A quadrature amplitude amplifier and photon number amplifier are ideal amplifiers in the sense that the signal can be amplified without any excess noise. The same devices also operate as ideal deamplifiers. The degradation of the SIN ratio for nonclassical lights due to loss can be suppressed by ideal preamplification. The degradation of the SIN ratio due to a nonlinear process or detector saturation can be overcome by ideal predeamplification. The full information of a nonclassical light can be transmitted over a lossy transmission line and can be extracted by a photodetector with power limitation. The intrinsic quantum limit on optical precision measurement and communication emerges when the trade-off relationship between two conjugate observables is under consideration. The quantum limit on simultaneous measurement of two conjugate observables is one example of this. The relationship between the minimum energy and time interval to carry (and extract) one bit of information is another example.

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Acknowledgement The authors wish to thank Professor H. A. Haus and Professor H. P. Yuen for their useful discussions. They also wish to thank Mrs. C.Murata for her careful typewriting.

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E. WOLF, PROGRESS IN OPTICS XXVIII 0 ELSEVIER SCIENCE PUBLISHERS B.V., 1990

III

THE QUANTUM COHERENCE PROPERTIES OF STIMULATED RAMAN SCA'ITERING BY

M. G. RAYMER Department of Physics Universiry of Oregon Eugene, OR 97403, USA

I. A. WALMSLEY The Instimte of Optics Universiry of Rochester Rochester, NY 14627, USA

CONTENTS PAGE

0 1. INTRODUCTION . . . . . . . . . . . . . . . . . . 183 0 2. HISTORICAL PERSPECTIVE . . . . . . . . . . . . . . 186 0 3. THEORY OF STIMULATED RAMAN SCATTERING . . . 196 3 4. EXPERIMENTS ON QUANTUM-STATISTICAL ASPECTS OF STIMULATED RAMAN SCATTERING . . . . . . . 245 ACKNOWLEDGEMENTS

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LIST OF SYMBOLS REFERENCES

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8 1. Introduction Although it has been known since its discovery in 1928 that spontaneous Raman scattering is an intrinsically quantum mechanical process, Raman amplification of a coherent input wave at the Stokes frequency can be well described purely classically. In a Raman generator, where no light at the Stokes-shifted frequency is incident on the Raman-active medium, the Stokes light builds up from nonresonant, spontaneous Raman scattering, which is subject to the same type of quantum statistical behavior as spontaneous emission. The question that naturally arises is - What is the impact of this underlying behavior of the spontaneous scattering on the resulting macroscopic Stokes field? The answer is that the fluctuations associated with spontaneous scattering persist, even for Stokes fields containing large numbers of photons. Although this fact was known fairly soon after the discovery of stimulated Raman scattering (SRS) in 1962, its implications were not realized until recently. For example, since high-gain SRS is always carried out using pulsed lasers, the amplified Stokes light is emitted as a pulse whose energy and pulse shape can undergo large-scale variations from pulse to pulse. Furthermore, in the case of the Raman generator, the emitted Stokes beam profile can have random modulation, or speckle, on it, which changes on each pulse. Thus quantum noise affects both the temporal and spatial characteristics of the Stokes light. Even in a Raman amplifier, where a coherent Stokes field is incident on the medium, the spontaneous Raman scattering acts as a potentially strong noise background. These and other manifestations of quantum noise in the SRS process were first studied experimentally in the 1980s. The observation and detailed explanation of these characteristics of SRS have been made possible recently by two complementary developments. Firstly, high-power, pulsed lasers have become available with highly stable, singletransverse and longitudinal-mode output, allowing the intrinsic quantum fluctuations to be distinguished from those induced by fluctuations in the laser. Secondly, the quantum theory of SRS has been developed sufficiently to allow accurate predictions and quantitative comparisons with experiments to be made. The purpose of this review is to acquaint researchers and students with the 183

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[IK 8 1

current understanding of the quantum coherence properties of stimulated Raman scattering through a discussion of both theory and experiment. Emphasis will be placed on the generation of Stokes light, with frequency ws = wL - wR, by stimulated scattering of laser light, with frequency oL,in a gaseous, Raman-active medium with Stokes shift wR. Coupling to higher order Stokes light at wL - n o R (where n = 2,3,4, ...) and to anti-Stokes light at wL + n o R (where n = 1,2,3, ...) will not be discussed. The historical and theoretical review will trace only those developments necessary to gain an understanding of the quantum coherence properties of SRS. No attempt will be made to survey all areas of phenomenology or application of SRS. The literature in this area is truly prodigious, and several excellent reviews already exist that summarize much of the work done before 1980 (BLOEMBERGEN [1967], KAISER and MAIER [1972], WANG [1975], GRASWK [1976], and KAISER[ 19791). It is important to point out PENZKOFER, LAUBEREAU that the understanding of some fundamental aspects of SRS is still incomplete, and that development of new applications is still continuing at a fast pace. It is hoped that the present article will provide researchers with a foundation in some of the fundamental principles that govern the SRS process and will thereby contribute to further development. Stimulated Raman scatteringis interesting from a basic physics point of view for several reasons. From the viewpoint of quantum-measurement theory SRS provides one of the few examples of a macroscopic system that begins in a pure state and through Hamiltonian quantum evolution ends up having highly unpredictablevalues of its physical observables. The Stokes field that is emitted by a high-gain Raman generator pumped by a sufficiently short laser pulse (transient SRS) is essentially a transform-limited pulse, and it can be described as a macroscopic, classical field. But its phase and its energy are completely unpredictable on a given laser pulse. In simple terms one can say that they are determined by the first few spontaneously scattered Stokes photons which happen to travel in the proper direction for strong amplification. In a sense the problem of understanding the quantum properties of the Stokes field is like understanding the famous cat paradox of Schrodinger: If a radioactive particle decays, it triggers an amplification process that leads to the death of a (gedanken) cat. What quantum state is the cat in before it is observed? GLAUBER [ 19861 carried out an elegant treatment of this class of problem that is valid, at least, for optical amplifiers, such as the Raman amplifier. He showed that the density matrix of the amplified field becomes more nearly diagonal as amplification proceeds, so that in the macroscopic limit the final state becomes essentially a classical mixture: The cat is either dead or alive, with no

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measurable coherence between these two possible outcomes. Thus the quantum amplitudes for generating a macroscopic Stokes field with different amounts of energy are phase-uncorrelated with each other. We might expect, then, from this argument, that a classical interpretation of the Stokes energy fluctuations can be given. The careful quantum treatment of the problem, reviewed in 5 3.4.2, confirms this expectation. From another viewpoint the study of SRS illuminates some of the same issues that are involved in understanding the distinction between amplified spontaneous emission (ASE) and superfluorescence. Either of these processes can arise, in the proper circumstances, when a collection of two-level systems are all put into their excited states and left to decay by emitting light. As in SRS, a few spontaneously emitted photons can trigger the entire collection to decay rapidly. The question is - Are all the emitted photons coherent with respect to each other? Or, put classically, is the emitted field temporally coherent? As will be discussed in 5 4.3, the Raman problem is mathematically equivalent to the two-level problem in some regimes, and so insight that is gained concerning one can be useful in understanding the other. The resolution of the posed question depends largely on understanding the influence of dephasing the active molecules by collisions with any background gas. If the dephasing rate is sufficiently high to overcome the tendency of the molecules to emit coherently, the emission is incoherent and ASE-like; otherwise it is coherent and superfluorescence-like. The analogue of two-level superfluorescence has been observed in Raman scattering and will be discussed in 5 4.4. From a quantum-optics-theory point of view the SRS problem is interesting because of the necessity to treat in a time-dependent manner spatial propagation in the presence of spontaneous quantum noise and collisional dephasing noise. Earlier quantum treatments of SRS dealt with only a few modes of the radiation field, limiting the validity of those treatments to cavity situations, whereas here we are interested in free-space propagation. The more recent treatments use the quantized Maxwell’s equations for the Stokes electric field operator in the Heisenberg picture. This is a rigorous approach that has the advantage of making the equations of motion essentially identical in form to those of the semiclassical theory. The quantum noise then arises naturally from the commutation relations of the field operators. The collisional dephasing noise is introduced by means of quantum Langevin noise operators in the way that is well established in laser theory (HAKEN[ 19701, LOUISELL[ 19731, SARGENT,SCULLYand LAMB[ 19741). Since the foregoing discussion of Schrodinger’s cat indicated that we may be able to find a classical statistical interpretation of the fluctuations of the Stokes field, it is especially appropriate

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to use the Heisenberg picture, in which the equations of motion are for the physical variables themselves (e.g. Stokes field), rather than for the density operator. The three-dimensional nature of the propagation can exhibit interference between the light in differential spatial modes, leading to the formation of a speckle pattern on the generated Stokes beam. The spontaneous scattering excites many radiation modes with random phases, whereas the propagation through the pencil-shaped, pumped volume acts to filter the light spatially, amplifying only those modes within some small solid angle. For sufficiently small solid angles only a single transverse spatial mode can be strongly amplified; then the Stokes field becomes spatially coherent. For larger solid angles several or many modes are amplified and propagate out the end of the medium, giving rise to an interference, or speckle pattern. It is interesting to note that the quantitative analysis of this aspect of SRS follows closely the classical treatment of speckle. In the following chapters the historical development of the subject of SRS is reviewed, followed by more detailed treatments of the theoretical and experimental aspects. Our aim in this article is to emphasize that, although SRS has been extensively studied for a long time and plays a role in many applications, the subject still remains an active area for research into fundamental phenomenology.

8 2. Historical Perspective Since several excellent reviews of the subject already exist, a comprehensive overview of the historical development of SRS will not be attempted. Rather, we concentrate on the work that has special relevance to an understanding of the quantum statistical properties of the process. Note that this does not include the effects of the statistical properties of the pump laser on the scattered radiation, but only the statistics which arise from the inherently quantum mechanical nature of the amplified spontaneous scattering. This chapter is not necessary for an understanding of the subsequent chapters and may be skipped on a first reading. The occurrence of inelastic scattering of light was first predicted by SMEKAL [ 19231 in the early days of quantum mechanics, using arguments based on conservation of energy. This idea was developed further by KRAMERSand HEISENBERG [ 19251, using correspondence-principle arguments to arrive at the correct quantum-mechanical formula for the inelastic scattering crosssection. KLEIN[ 19271 outlined a semiclassical approach in which an expres-

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sion for the time-dependent dipole-matrix element at the Stokes frequency was calculated using (non-adiabatic) perturbation theory in the Schrddinger picture. This expression was substituted into the classical formula for the energy radiated by a dipole to give the scattered intensity. PLACZEK[ 19341 summarized these theories in his classic treatise on Raman scattering, which also fist presented a discussion of the Raman process in terms of a molecular nonlinear polarizability. The quantization of the electromagnetic field by DIRAC[ 19271, and his rigorous derivation of the Kramers-Heisenberg dispersion formula, was followed by GUPPERT-MAYER’S theoretical discovery [ 193 11 of stimulated two-photon processes. It might be said that this work was the first to recognize the possibility of stimulated Raman scattering. Although PLACZEK [ 19341 mentioned this work in his account of spontaneous scattering, which postdates that of Gdppert-Mayer, inelastic scattering was still referred to as incoherent. The possibility of obtaining stimulated scattering was ignored! [ 19281 made the fist experimental observations of RAMANand KRISHNAN the phenomenon of “modified radiation”, or spontaneous Raman scattering. Of course, the pump source (sunlight) was much too weak to produce any gain at the Stokes frequency. Almost simultaneously LANDSBERGand MANDEL’STAMM [ 19281 discovered a similar scattering from solids. Within two years the effect had been catalogued in 60 different liquids and gases. These experiments were recognized at the time as providing support for the correctness of the quantum theory as embodied in the Kramers-Heisenberg formula. For example, R. W. Wood, as quoted by JAYARAMAN and RAMDAS[ 19881, stated, “It appears to me that this very beautiful discovery .. . is one of the most convincing proofs of the quantum theory which we have at the present time.” It is interesting, for our purposes, that RAMANand KRISHNAN[1928] described their results as being due to “the effect of fluctuations (of the atoms) from their normal state.” However, the fluctuations to which they referred were those associated with thermal excitation of the phonon mode, rather than vacuum fluctuations. Between 1927 and the late 1950s Raman scattering was developed as an important spectroscopic tool, but little attention was paid to the coherence properties of the scattering or to the possibility of stimulated scattering. This situation was altered by the discovery of nonlinear optics in the 1960s. The development of high-field-strength light beams meant that scattered fields containing many photons per mode could be produced. This feature enhanced the usefulness of the Raman process as a spectroscopic tool by introducing the possibility of coherent, stimulated scattering. Stimulated scattering was first observed by WOODBURY and NG [1962] and was identified as such by

188

QUANTUM COHERENCE PROPERTIES OF RAMAN SCATTERING

[IKB 2

ECKHARDT, HELLWARTH, MCCLUNG,SCHWARZ, WEINERand WOODBURY [ 19621. Soon thereafter, two theoretical approaches to explain this result were developed. The first was a rate-equation approach outlined by HELLWARTH [ 19631, in which one takes a transition cross-section calculated using the Kramers-Heisenberg formula and finds the rate of change of the number of photons in the scattered beam by considering the flow of energy through a small volume containing a Raman-active medium. Although this approach describes both spontaneous and stimulated scattering, it cannot be used to calculate the statistics of the scattered radiation, because equations are derived only for the average photon number. In the second approach classical or semiclassical equations of motion for both the matter and field amplitudes are used. Because Maxwell’s equations are used to determine the scattered field amplitude in this approach, amplification during propagation through an extended (threedimensional) medium is described in a more rigorous manner than in the phenomenological description by rate equations. GARMIRE,PANDARESE and TOWNES[1963] calculated the rate of generation of Stokes light, using Placzek’s polarizability ansatz for molecules, and discussed the importance of phase matching in higher order wave-mixing processes. LOUDON[ 19631 pointed out that lattice vibrations in crystals could also give rise to stimulated scattering. BLOEMBERGEN and SHEN [ 1964a,b] and PLATONENKO and KHOKHLOV [ 19641 derived coupled differential equations for the slowly varying amplitudes of the Stokes field and the vibrational wave. The connection between Placzek’s classical polarizability model and the semiclassical nonlinear susceptibilitymodel was explained by SHENand BLOEMBERGEN [ 19651. This connection was explored in more detail by WANG[ 19691, who calculated an expression for the polarizability of a molecule using the Born-Oppenheimer approximation. MAKERand TERHUNE[ 19651 further developed the semiclassical theory of ARMSTRONG, BLOEMBERGEN, DUCUINGand PERSHAN [ 19621 to account for scattering near an atomic resonance by including damping in the nonlinear Raman susceptibility. The concept of a phonon mode interacting with laser and Stokes photon modes leads naturally to the idea of SRS as parametric process in which the signal and idler mode build up at the expense of the pump mode. This idea was developed by GIORDMAINE and KAISER[ 19661, who used the Schrbdinger equation in a two-level approximation to derive the dynamical equations for the molecular vibrational coordinate and the population inversion. This allows a treatment of saturation effects in the Raman transition. When the Schrbdinger equations are coupled to the electromagnetic field equations, this formalism is referred to as the Maxwell-Bloch theory. A phenomenological damping con-

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stant for the molecular vibrational coordinate was included. Kaiser and Giordmaine obtained solutions to these equations only in the rate-equation approximation. Solutions to the full Maxwell-Bloch equations were found by [ 19681, TAKATSUJI [ 19751 and GRISCHKOWSKY, LOYand LIAO HARTMANN [ 19751. These authors predicted coherent transient effects analogous to those known in two-level systems interacting with a single electromagnetic field. It is necessary to retain the full set of Maxwell-Bloch equations when studying the interaction on time scales comparable to the damping time of the atomic or molecular coherence. In general, the intensity of light amplified by stimulated scattering processes does not grow exponentially in this regime. In fact, HAGENLOCKER, MINCKand RADO [ 19671 experimentally observed reduced gain in Raman amplifiers when pump pulses shorter than the phonon damping time were used. WANG[ 19691 solved the linearized Maxwell-Bloch equations for the Raman amplifier and confirmed that the transient gain coefficient was related to the square root of the steady-state gain coefficient. His method was based on the work of KROLL[ 19651 in stimulated Brillouin scattering,but the results were valid for arbitrarypump pulse shapes. The shape and delay of the Stokes pulse in the transient regime were studied in detail theoretically by CARMAN, SHIMIZU,WANGand BLOEMBERGEN[ 19701 and experimentally by CARMANand MACK [1972]. The existence of a delay between the peak of the Stokes pulse and pump pulse was confirmed, and “compression” of Stokes pulses in the transient regime was observed. The transient regime has bounds, and it is important to define them. Not only the total intensity of scattered light, but also its statistical properties are determined to some extent by the “transiency” of the scattering. The fist definition BRET, of the transient regime in SRS was proposed by BLOEMBERGEN, LALLEMAND, PINEand SIMOVA [ 19671, who compared the coherence time of the pump pulse with the coherence time of the Stokes pulse. If the ratio of these two times is much less than unity, the experiment is said to be in the transient regime. A more commonly used definition of the transient regime is that the laser pulse duration, for a transform-limited pulse, is smaller than the inverse of the spontaneous Raman linewidth, but this is unnecessarily restrictive. This is because in calculating the coherence time of the Stokes pulse, it is necessary to include the effects of the gain narrowing of the Stokes line, as fist pointed out by TANG[ 19661 for stimulated Brillouin scattering. A more complete definition of the transient regime, which is applicable to a general nonlinear optical process, was proposed by KROLLand KELLEY[ 19711. This definition was based on dispersion relations for the pump and Stokes fields derived from the three-dimensional Maxwell equations with a nonlinear source term.

190

QUANTUM COHERENCE PROPERTIES OF RAMAN SCAlTERING

WI,B 2

In the semiclassical theory, because the electric fields are treated explicitly, the effect of different pump-field statistics on the amplification and generation of Stokes radiation may be determined. This was done first by BLOEMBERGEN and SHEN[ 1964~1,who predicted an increased gain for a chaotic pump when compared with a coherent pump. The effects of multimode pumps in SRS are still not completely understood, despite extensive studies. (See, for example, the review articles by VALLEY[1986] and by PENZKOFER,LAUBEREAUand KAISER[ 19791.) The discovery of stimulated scattering also renewed the interest in understanding the fundamental issues related to the quantum statistical properties of Raman scattering. Whereas spontaneous scattering from a collection of molecules has a low degree of coherence, stimulated scattering might be expected to lead to increased coherence. In this article we are primarily interested in how the presence of stimulated scattering affects the coherence properties of the generated Stokes light. The work discussed above concentrated on understanding the behavior of the average Stokes intensity. The earliest paper to mention the possibility of intrinsic fluctuation of generated Stokes light intensity was that by ZEIGER,TANNENWALD,KERN and HERENDEEN [ 19631, who noticed fluctuations in their experiment but pointed out that they were probably dominated by pump laser fluctuations. To treat correctly the fluctuations of the Stokes pulses, it is necessary to use a quantized-field theory of light scattering. It is then possible to treat spontaneous and stimulated scattering consistently, as was done by FAIN and YASCHIN[ 19641, who were able to relate the spectral intensities of the spontaneous and stimulated scattering. GROB[ 19651 was the first to find steadystate solutions for the Heisenberg field-operator equations, and he calculated the Stokes field intensity from a quantum-electrodynamic theory. Theoretical studies of the quantum statistical properties of nonlinear optical processes were made by SHEN[ 1967J. He derived the equations of motion of the density operator for single and multimode quantized Stokes and laser fields. In his approach the atomic variables are eliminated, leaving equations for the moments of the Stokes field creation and annihilation operators. This treatment was limited to scattering in a cavity, since the system was closed, and neither propagation nor external losses were included. It was demonstrated that, for nonlinear interactions, the statistics of the scattered radiation are dependent on the pump-field statistics. For example, he concluded that a chaotic field is a more efficient pump than is a coherent field. This paper contains the first attempts to understand the statistical nature of the Stokes radiation as determined by spontaneous scattering. Using the Glauber-

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191

Sudarshan quasi-probability distribution for the field, Shen showed that, for a single Stokes mode generated by an undepleted pump field, the statistics are essentially those of a thermally excited harmonic oscillator. In addition, he pointed out that this corresponds to a Stokes field with a Gaussian probability density. MISHKINand WALLS[ 19691used a similar approach to study the statistical properties of the Stokes light. They examined the interaction of three field modes (Stokes, phonon, and anti-Stokes), which are coupled together by means of an undepleted classical pump field. No damping was included and implicit in the few-mode approximation is the absence of spatial propagation. The statistics of the field scattered into these modes were calculated by determining the time development of a quasi-probability distribution for each mode. In this way they showed that spontaneous emission adds noise to the scattered radiation, as expected. Their formalism includes the possibility of an initial vacuum state for each of the modes, resulting in Gaussian probability distributions for the generated fields. It is also shown that the quasi-probability density for each of the fields remains positive definite at all times, implying the existence of an equivalent classical random field. Similar calculations were made by MOLLOWand GLAUBER[ 19671. WALLS[ 19701 also developed a quantum theory of higher order Raman scattering from phonons, including quantum mechanical pump depletion, and calculated time-dependent probabilities of photon scattering. A master-equation approach to the statistical problem was proposed by WALLS[ 19731 and MCNEILand WALLS[ 19741 for calculating the moments of the photon number distribution for two-photon interactions. This theory was quantum mechanical, and the equations of motion of the field density matrix were derived. The master equation for the scattered-field density operator was derived in the usual way from the Hamiltonian for the quantized radiation field and an ensemble of two-level atoms. The interaction was described by the general n-photon Hamiltonian first introduced by SHEN [ 19671. This technique is suitable for describing a variety of nonlinear optical phenomena; however, no account is taken of damping other than spontaneous emission. Since it is not known how to solve the resulting nonlinear master equation in general, McNeil and Walls approached the problem by finding the moments of the field distribution; that is, ( (dz dk)* ) where d z is a creation operator for the kth field mode. In principle one could calculate the moments of the field to arbitrary order using this method. The scattered field intensity, that is, the second moment, was calculated in terms of the initial distribution of photons and the Stokes field shows exponential growth in time. However, their result differs in

192

QUANTUM COHERENCE PROPERTIES OF RAMAN SCATrERING

[III, B 2

form from the classical result because of the explicit inclusion of spontaneous scattering. A similar approach was used by SIMAANand LOUDON[1975a,b] and SIMAAN[ 19751. The master equations in their treatments are derived phenomenologically by considering the balance between the numbers of photons in the incident and scattered fields. The probabilities of these numbers were coupled using transition probabilities derived from the Kramers-Heisenberg formula. The results of this approach are equations of the same form as those of SHEN[ 19671 and MCNEILand WALLS[ 19741. SIMAAN[ 19751 found the probability density for the number of photons in the Stokes field by explicitly solving these equations for arbitrary initial distributions. The main conclusion is that a chaotic pump beam produces a higher transient gain than does a coherent beam. This work represents the most rigorous application of master equations to the Raman problem, but its description of real experiments is limited in that it does not adequately describe spatial propagation. VON FOERSTER and GLAUBER[ 19711 provided the first fully quantummechanical treatment of the SRS problem that included spatial propagation and phonon damping by obtaining a solution of the coupled Maxwell and phonon amplitude operator equations in the unsaturated-gain limit. These Heisenberg-picture operator equations were derived from first principles using Placzek's form of the Hamiltonian density and an assumption of onedimensional propagation. The source polarization was found in terms of the phonon dynamics, which included, for the first time in the SRS problem, a quantum-mechanical Langevin operator to satisfy the requirements of the fluctuation-dissipation theorem. Von Foerster and Glauber demonstrated that spontaneous scattering may be viewed as arising from the zero-point fluctuations of the phonon mode, and they showed that fluctuations are to be expected in the scattered Stokes light. The quasi-probability density calculated in this paper confirms the results of SHEN[ 19671 and MISHKINand WALLS [1969] and introduces the idea that randomness arising from spontaneous noise may also be apparent in light that is scattered and amplified as it propagates in an extended medium. In this case, also, the quasi-probability density of the Stokes field is always positive definite, confirming the existence of an equivalentclassical description. It should be pointed out that AKHMANOV, DRABOVICH, SOKHORUKOV and CHIRKIN[ 19711 solved a set of equations, similar to those of Von Foerster and Glauber, which also included a random driving term (Langevin force) for the phonon amplitude. However, since the field was taken to be classical, the role of the Langevin force was to model spontaneous scattering in a phenomenological way. This method is in contrast

111, § 21

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193

to that of Von Foerster and Glauber, who took a more fundamental approach. IWAZAWA [ 19761 has also derived a set of Langevin equations for the SRS problem from quantum field theory and has discussed the onset of SRS as a phase transition, analogous to that of a laser at threshold. The one-dimensional, linearized, Maxwell-Bloch operator equations were derived in the Heisenberg picture using transition-projection operators by EMEL’YANOV and SEMINOGOV [ 19791, MOSTOWSKI and RAYMER[ 19811, and RAYMERand MOSTOWSKI [ 19811. The electric-dipole Hamiltonian was used, and an explicit connection to the Kramers-Heisenberg formula was made. Using this theory, RAYMER, RZ~ZEWSKI and MOSTOWSKI[1982] predicted that, under transient conditions, it would be possible to observe that the energy of an intense Stokes pulse would be a random quantity, which fluctuates from pulse to pulse in a way reflecting the underlying spontaneous initiation. Since the generated field has Gaussian statistics, as discussed earlier, the pulse energy obeys a negative-exponential probability distribution, at least in the transient regime. The origin of this behavior has been clarified by HAAKE [ 19821. The statistical character of the pulse-energy fluctuations is altered by the damping of the phonon mode and by spatial propagation, the latter being especially important when the gain medium has a finite transverse extent. The effects of including transverse spatial modes, which are not included in a one-dimensional theory, have been examined by MOSTOWSKIand SOBOLEWSKA [ 19841, RAYMER,WALMSLEY, MOSTOWSKI and SOBOLEWSKA [1985], HAZAK and BAR-SHALOM[1988], and STRAUSS, OREG and BAR-SHALOM [ 19881. The full-scale (that is, macroscopic) fluctuation of the Stokes-pulse energy and RAYMER[ 19831. In this experiwas first observed clearly by WALMSLEY ment the laser pulse duration was comparable to the inverse Raman linewidth. Observations of the fluctuations in the transient regime were made by FABRICIUS, NATTERMANN and VON DER LINDE[ 19841 and by RAYMERand WALMSLEY [ 19841, who verified the predicted negative-exponential probability distribution. Systematic experimental studies of the influences of line broadening and scattering volume geometry were made by WALMSLEY and RAYMER [ 19861 and NATTERMANN, FABRICIUSand VON DER LINDE [ 19861. They found that if either the pulse length or the beam cross-sectional area became large, the pulse-energy distribution approached a Gaussian function peaked at the mean energy. In explaining the modifications to the pulse energy that arise from the damping of the phonon mode and from propagation, it is useful to define a set of spatially and temporally coherent modes for the problem, by analogy to classical coherence theory.

194

QUANTUM COHERENCE PROPERTIES OF RAMAN SCAlTERING

[III, B 2

It is to be expected that there will also be fluctuations in the spectrum of Stokes light just as there are in intensity. Such fluctuations have been observed SWANSONand CARLSTEN[ 1988al. In addition, the flucby MACPHERSON, tuation of the occupation numbers of the spatially coherent modes can lead to the formation of speckle patterns on the Stokes beam from a Raman generator. The speckles change from one pulse to the next, even when every pump pulse is identical, as seen by HENESIAN,SWIFTand MURRAY[ 19851 and by Kuo, and RAYMER[ 19871. The speckle pattern changes during each RADZEWICZ coherence time of the Stokes pulse, however, so that the random spatial modes may only be seen in the transient regime. The statistical properties of the scattered radiation alter drastically in the saturated-gain regime. This regime arises when the Stokes gain is sufficiently high to remove a sizeable fraction of photons from the pump beam or a sizeable fraction of molecules from their ground states. In the case where saturation occurs because of pump depletion, LEWENSTEIN[ 19841 solved the Maxwell-Bloch equations numerically and found that the fluctuations in the Stokes-pulse energy are dramatically reduced. WALMSLEY, RAYMER,SIZER, DULINGand KAFKA[ 19851 measured experimentally the pulse-energy distribution in the saturated regime and found good agreement with the theoretical results. TRIPPENBACH and RZ~ZEWSKI [ 19851 calculated the probability distributions of Stokes-pulse energy in the saturated regime excluding propagation, and similar distributions were measured by GRABCHIKOV, KILIN, KOZICHand IODO[ 19861. When the pump pulses are very long and highly depleted (about 80% of the light has been scattered), solitons have been observed by MACPHERSON, SWANSONand CARLSTEN[1988b, 19891 to appear spontaneously in the depleted region of the pump pulse. It has been suggested by ENGLUNDand BOWDEN[ 19861 that the soliton formation is due to an instability which is initiated by quantum noise. A second way in which the Raman process can become saturated is when the ground state of the gain medium is depleted while the pump field remains constant. In this case the Raman gain coefficient remains constant during the scattering, and the situation is analogous to superfluorescence from a system of inverted two-level atoms. A cooperative dipole moment builds up at the Stokes frequency, starting from a few spontaneously scattered photons, leading to cooperative Raman scattering (CRS) (SHIMODA[ 19711, WALLS[ 19711). The statistical nature of the initiation is manifested in the delay time of the cooperative pulse and in the pulse shape. The problem of CRS was taken up by FLUSBERG[ 19751 and by RAUTIANand CHERNOBROD [ 19771, following

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195

earlier experimental work (KIRIN, POPOV, RAUTIAN, SAFONOVand CHERNOBROD [ 19741). They proposed a one-dimensional propagation model analogous to that of superfluorescence in two-level atoms, which predicted identical behavior to two-level superfluorescence in the absence of anti-Stokes generation. More generally, pulsing between Stokes and anti-Stokes emission was predicted. A more complete theory, allowing for three-dimensional propagation and pump depletion, was also developed (EMEL’YANOV and SEMINOGOV [ 19791). CHERNOBROD [ 19791 derived a semiclassical theory of pulse propagation in an extended, but one-dimensional, Raman medium. The quantum initiation of CRS was described by a random initial polarization throughout the medium. CRS was first convincingly demonstrated by PIVTSOV, RAUTIAN, SAFONOV, FOLIN and CHERNOBROD [ 198 11, who found the emitted Stokes pulses to have the characteristic properties of superfluorescence; that is, the pulse duration and delay depend on the number of cooperating molecules. As yet there have been no measurements of the statistical properties of the scattered radiation (for example, the random delay time), even though this has been an important topic in the study of two-level superfluorescent emission. A theoretical study of the cooperative Raman effect in small samples using group theoretical arguments was proposed by Hu and HUANG[ 19821, but this was, strictly speaking, a consideration of superradiance and, therefore, did not include the problem of spontaneous initiation. More recent experiments (ZABOLOTSKII, RAUTIAN,SAFONOV and CHERNOBROD [ 19841) have studied the scattered pulse envelope modulation due to the degeneracy of the final states. Because of the difference in polarizability of each of the degenerate levels, there is a different cooperation time for each of the transitions. This leads to a cooperative pulse, which is a sum of the cooperative emissions from each of the possible degenerate transitions. In summary, it is only recently that clear manifestations of the quantum fluctuations of SRS have been observed. In the experimental studies both damping and propagation are important. In this case the calculation of the statistical properties of SRS is best treated by means of the Heisenberg-picture operator equations. It is worth emphasizing that in all cases discussed here the Glauber- Sudarshan quasi-probability density remains positive definite at all times. Thus, although quantum mechanics plays an important role in initiating the scattering, there is an equivalent classical random process or the Stokes field. However, it has been pointed out that the joint quasi-probability distribution for the laser and Stokes field amplitudes is not always positive definite, and therefore nonclassical correlations may arise between these fields. This development shows that it is impossible to find, in general, a classical description

196

QUANTUM COHERENCE PROPERTIES OF RAMAN SCATTERING

[III, s 3

of the Raman process (MISHKINand WALLS[ 19691, SZLACHETKA, KIELICH, and PE~~IN A PERINOVA[ 19791).

4 3. Theory of Stimulated Raman Scattering In this chapter four major theoretical approaches to treating SRS will be reviewed: the photon rate-equation theory, the semiclassical propagation theory, the quantum theory for modes in a cavity, and the quantum-field propagation theory. One of the main goals of this section will be to elucidate the relations between the different approaches and the circumstances in which each is most appropriate for use. A tutorial approach will be taken, with the given references chosen for their clarity and direct relevance, rather than for historical significance. Since our goal is to treat spatial propagation, we will use the Heisenberg picture where possible. Master-equation techniques will not be discussed in detail, although they were among the first techniques to be applied to the Raman problem. This was done by BLOEMBERGEN and SHEN[ 1964a1, who postulated a general nonlinear optical interaction Hamiltonian in which two or more modes of the field are coupled. Their ideas lead naturally to two distinct ways of dealing with the statistical properties of the scattered radiation - either by specifying the moments of the field or by specifying the joint probability densities of the photon numbers. In the first case a natural extension is to calculate quasi-probability densities for the fields using the well-known formalism developed by GLAUBER[1963] and SUDARSHAN[1963]. The second approach lends itself to a phenomenological derivation, which illustrates well the physics involved in the interchange of photons between various modes of the field. Its main advantage over the density-operator method is that it is straightforward to include saturation of the Stokes gain. The disadvantage of both of these methods is that amplification by spatial propagation is not easy to include. An alternative method is to solve explicitly the Heisenberg equations of motion for the field-mode creation and annihilation operators. From these solutions the characteristic functions of the fields may be developed, and thereby the full density operator of the field. This method, of course, allows one to determine arbitrary expectation values of the field operators and has proved useful in application to optical parametric amplifiers and similar devices (TANG [ 19691). The major drawback of this technique is that it does not lend itself to modeling propagation of a pulse through an amplifying medium, which may only be achieved by using tricks developed for intracavity fields. Most experi-

111,s 31

THEORY OF STIMULATED RAMAN SCATTERING

197

ments involve using finite-duration pump pulses propagating through an extended interaction volume, and it would be more satisfactory to have a fistprinciples theory to model them. Herein lies the use of the more recent theory of SRS,which is also formulated in the Heisenberg picture but deals with the dynamics of the total field, without using a mode decomposition. The equations for the macroscopic field operators have a similar form to the classical equations and can be solved by the same techniques. In addition, because one deals directly with operators rather than with expectations, damping may be included easily using the operator form of the fluctuation-dissipation theorem. In the Heisenberg picture it is straightforward to deal with pulse propagation and transverse effects, and much of the machinery of classical coherence theory may be applied. This theory has been most successful in predicting observable manifestations of the quantum initiation of stimulated scattering. Similar theories have been successfully applied to the problem of supertluorescence,in which quantum noise plays a role parallel to that which it does in SRS.

3.1. PHOTON RATE-EQUATION THEORY

If the spectral width of the Raman-active molecular transition is sufficiently large, and if one is primarily interested in the average rate of energy flow from the incident to the scattered fields, then the photon rate-equation theory is useful. This approach uses time-dependent perturbation theory in the quantum theory of light to calculate the spontaneous and stimulated photon scattering X

2a

z=o

I I

Us

Z=L

Fig. 3.1. A schematic illustration of the geometry of Raman scattering. The interaction volume, containing a Raman-activemedium, is a uniformly pumped cylinder of length L and radius a. A laser pulse of mean frequency wL is incident on the left-hand end face of the cylinder. Radiation at the Stokes frequency usis generated by spontaneous scattering and builds up by means of stimulated scattering on propagation through the interaction region. T h e Stokes pulse exits the gain region through the right-hand face at z = L. The coordinate system in this face is illustrated on the right side of the figure. The Fresnel number of the interaction volume is defmed as F = na2/ASL.

198

QUANTUM COHERENCE PROPERTIES OF RAMAN SCATTERING

[III, 8 3

rates, and then it uses a phenomenological argument to obtain propagation equations for photon fluxes. Useful reviews of various aspects of this theory have been given by SAKURAI [ 19671, LOUDON[ 19731, WYNNEand SOROKIN [ 19771, and PENZKOFER, LAUBEREAU and KAISER[ 19791. Consider a cylindrical laser beam with cross-sectional area A and angular frequency oL,propagating in the z-direction and incident on a uniform gaseous medium with length L and average number density N,as shown in fig. 3.1. The laser intensity, in photons cm-2 s - l , is I, = (c/2nhaL) IELIZ,where IE,( is the amplitude of the electric field (in Gaussian units) EL(z, t) = E,(z,

+

t ) ei(WLr-kLz) C.C.

(3.1.1)

In terms of a photon number nL, the laser intensity can be written

I,

C

=-

V

nL,

(3.1.2)

where V is some quantization volume. 3.1.1. Quantizedfild theory of photon scattering

With reference to fig. 3.2 each molecule begins in its ground state I 1) and can scatter a Stokes photon with frequency near as = aL- w31,while a laser

Fig. 3.2. Energy level scheme for stimulated Raman scattering from, for example, the Q o l ( l ) vibrational transition in hydrogen. The molecule is initially in the ground state I 1 ) when an intense laser pulse, with frequency oL,excites it to a virtual level with large detuning from the manifold of states I m). Scattered photons with frequency osare generated, leaving the molecule in the excited state 13 ).

111, I31

THEORY OF STIMULATED RAMAN SCATTERING

199

photon is destroyed, leaving the molecule in excited state ) 3 ) , named such because it is the third state in which the molecule finds itself after going through a virtual state I m ) . Two time orderings contribute to the scattering, as shown. The Stokes photon is emitted either after or before the destruction of the laser photon. These two processes will be identifiable in the final formula for the scattering cross section. A second-order generalization of Fermi's Golden Rule for quantum mechanical transitions says that if the density of final states is continuous and broad, the rate of transitions from initial state l i ) with energy h a i to final state I f ) with energy h a , is given by (LOUDON [ 19731, p. 195)

(3.1.3) where the intermediate (virtual) states are denoted by l j ) , and it is understood that the delta function is to be integrated over the density of final states. P i s the interaction Hamiltonian operator. We consider cases in which the electric dipole interaction is suitable to describe the interaction, so

P= - i.8,

(3.1.4)

where d is the dipole operator and the electric field operator is given in terms of a sum over modes with frequency w,, polarization vector u,, and creation (annihilation) operator d f ( d r ) :

B=ic r

(?)2

R f t 0, ' I 2

e,(dr - df)

.

(3.1.5)

We are interested in scatteringfrom the initial state I i ) having molecule in state I 1), nL photons in the laser mode, and nS photons in some other mode with having molecule in state 13), nL - 1 laser frequency wB, to final state I f ) photons, and nS + 1 photons in the scattered field mode; that is,

li)

=

lj)

= Im>

If)

=

11) Inp nL>

tn;,nt>

w,= w1+ nSwB+ nLwL,

3

9

13) InS + 1, nL - 1 ) ,

+ n;wB + &aL, (3.1.6) Of = w3 + (n, + 1)0b + (nL - l ) ~ , .

O, = W,

Modes not indicated are assumed to contain no photons. Two types of intermediate state will contribute to the sum in eq. (3.1.3):

lj)

= Im>

InpnL- 1 ) and Im) In,+

1,nL)

200

QUANTUM COHERENCE PROPERTIES OF RAMAN SCATTERING

[III, 8 3

Using standard rules for harmonic oscillators, the sum in eq. (3.1.3)is evaluated to give for the rate of scattering photons from mode L to mode /?

(3.1.7) where

(3.1.8) and wm1= om- wl, d3m= (31

6 Im), etc.

3.1.2. Spontaneous scattering cross-section If the mode fl initially has zero photons ng = 0, a photon can appear in this mode by spontaneous scattering. The number of modes of a single polarization 8@,with frequency between ws and og+ d o g and in a small solid angle element ddl is equal to [ V / ( ~ Z C ) ~ dogdQ. ]W$ Thus the total scattering rate per unit solid angle is given by dR - V dG!

jOm

R,w$ d o s .

(3.1.9)

(27~)~

This rate can be evaluated and written as do -dR =IL ddl dQ ’

(3.1.10)

where I, = cnL/V is the laser photon flux and do/dQ is the differential crosssection for scattering laser photons into a mode with polarization eg and frequency near wL - ojl,and is given by

(3.1.11) This is the well-known Kramers-Heisenberg formula. The two terms in the sum correspond to the two time orderings of scattering shown in fig. 3.2. 3.1.3. Stimulated scattering rate

If the mode /? initially has nS photons, stimulated scattering can lead to the increase in the number to ns + 1. We will assume that the molecular transition

I K 8 31

THEORY OF STIMULATED RAMAN SCA'ITERINO

20 1

I 1 ) -P 13) centered at frequency 0131 is broadened by some mechanism (e.g., collisions) and has a Lorentzian line-shape function L ( w ; , ) which peaks at =

"31

(3.1.12) Then the rate of scattering a photon into mode jand having the molecule end up in state 1 3 ) is (3,l. 13) which, on inserting eq. (3.1.7) and using Is = cns/V for the intensity (photons cm - s - ' )of mode /3, becomes (3.1.14) where aswas replaced by as(= wL - 0 3 1 ) ~ since L(w, - as)is strongly peaked there. 3.1.4. Photon rate equations The preceding discussion has applied to scattering from a single molecule. If molecules are present with a number density N, then, as the photons in mode /3 propagate in the z-direction with velocity c, a phenomenologicalconsideration of energy flow suggests that the spatial rate of change of intensity is given by (3.1.15) Using eq. (3.1.14), this becomes d dz where the gain coefficient (cm-

(3.1.16) I)

is given by (3.1.17)

and the second term is the spontaneous source. We will assume that the laser intensity I, does not decrease during propagation.

202

QUANTUM COHERENCE PROPERTIES OF RAMAN SCATTERING

[III, 8 3

The solution of eq. (3.1.16), with initial condition I& = 0) = I(O), is n

~ , ( z )= I ( O )eg(ma)z

+4 V

[eg(w+ - 11 .

(3.1.18)

The first term describes Raman amplification and the second describes Raman generation. Since we are interested primarily in generation, we will set I(0) = 0. This result is for a single mode. In experiments, however, Stokes light is generated in many modes, both in angle and frequency. To calculate the total Stokes intensity I,, an integral should be performed over the frequencies and solid angles for which there is substantial gain I,

=

-?!-

(2 nc)3

jn joT w i dl;,

dw,

I,(z)

,

(3.1.19)

where again the density of modes was used. With reference to fig. 3.1 the gain is highest for those photons originating near z = 0 and traveling through the pumped end face with area A at z = L. Thus the effective solid angle is approximately dl;, = A/Lz. Thus, inserting eq. (3.1.18) into (3.1.19) gives, at z = L, with I(0) = 0, (3.1.20) In the low-gain limit, where g(w,)L da dl;,

I s = NLI, - d n ,

< 1, this integral can be carried out to give (3.1.21)

where eqs. (3.1.8) and (3.1.1 1) were used. Thus the propagation theory reduces in this limit to the single-molecule spontaneous scattering result. For arbitrary gain the integral in eq. (3.1.20) can be carried out approximately by replacing g(w,) with its maximum value g, given by (3.1.22) for frequencies within a bandwidth n r around a, and by assuming g(w,) = 0 otherwise. This replacement preserves the normalization of g(w,). Then we fhd for Stokes intensity I,

N

w i dl;, n r [egL - 1 1 . (2x)3c*

(3.1.23)

111, § 31

THEORY OF STIMULATED RAMAN SCATTERING

203

This is often quoted as the result of the photon rate-equation treatment. Note that the prefactor in eq. (3.1.23) is just c times the number of modes per unit volume in a solid angle d i l and frequency range Arcentered at us.This means that the effective source term for the Raman generator is one photon per mode. A useful measure of the shape of the interaction volume is the Fresnel number F, defined here to be (3.1.24) (some authors use A/& L), where As = 2nc/u, is the Stokes wavelength. In terms of F, and using d62 = A/L2, eq. (3.1.23) is written as I,z+A--'F2I'[egL - 11.

(3.1.25)

For a fixed wavelength A s , F 2 is the number of spatial modes of the interaction volume AL. If F = 1, this means that &/A'/' = A'12/L; that is, the diffraction angle of the light equals the geometrical angle of the interaction volume; thus the Stokes light is diffraction limited, or single transverse mode. It is interesting to note that the integral in eq. (3.1.20) can actually be carried out exactly, by comparing eqs. (3.4.49) and (3.4.43), to yield 1 -- 1,A - 'F'TgL [Z0(gL/2)- Zl(gL/2)] egL12,

(3.1.26)

where Zo(x) and Zl(x) are modified Bessel functions and again g is given by eq. (3.1.22). This result has the asymptotic form in the high-gain limit (gL % 1) 1 r Z , = A - ~ F ~--egL. 2n'/2 (gL)'/'

(3.1.27)

This result is similar to the less accurate eq. (3.1.25) in the high-gain limit, except the effective source for spontaneous generation is seen to be lowered by a factor (ItgL)-'12. The origin of this factor can be seen by examining eq. (3.1.18), which gives the spectral distribution of generated Stokes photons. Using eqs. (3.1.22), (3.1.17),and (3.1.12),this becomes, at z = L, with Z(0) = 0, (3.1.28) In the high-gain limit (gL % l), this can be approximated by C

laN - egL exp [ - gL(o, - ~ ~ ) ~, / r ~ ] V

(3.1.29)

204

QUANTUM COHERENCE PROPERTIES OF RAMAN SCAnERING

6-

-

'

-

L

=:

s

4-

-

[III, 9 3

t -

Fig. 3.3. Steady-state Stokes intensityas a functionofthe Raman gaing, for F = 1 andA = 1 cm. Curve (a) is the exact result, given by eq. (3.1.26), and curve (b) is the usual photon rate equation result given by eq. (3.1.25). The curves show the transition from linear spontaneous growth to [198 11.) quasi-exponential,stimulated growth. (From RAYMERand MOSTOWSKI

which is valid in the region I as- w& < r. We see that the effect of amplification is to narrow spectrally the generated light from a half width at half maximum of r in the spontaneous regime, to a l/e width of r(gL)- ' I 2 in the high-gain regime. Thus the effective region of integration in eq. (3.1.20) is decreased by a factor of order (gL)- lI2, which yields essentially eq. (3.1.27). For comparison eqs. (3.1.25) and (3.1.26) are plotted in fig. 3.3. We see that they have similar behavior. 3.1.5. Molecular polarizability model The Kramers-Heisenberg formula (3.1.11) for the differential cross-section is written in terms of electric dipole matrix elements, which are not easy to calculate for molecules. A model involving the variation of electronic polarizability tensor ~1 with internuclear coordinate q was introduced by PLACZEK [ 19341. It is possible to establish an explicit connection between these two formalisms, as will be shown here. For a related discussion see WANG[ 19751.

IIL8 31

THEORY OF STIMULATED RAMAN SCATTERING

205

The results of this section are not needed for understanding the quantum statistics to be discussed later but are included for clarity. The differential cross-section d a / d n i s proportional to I S I 2 , where S is given by eq. (3.1.8), and can be written as (nu‘ 1

.=z[

2s

n“ u”

) (n’fu’’I

ln’~ujf

2~ [ n u >

on.,- oL

+ (nu”

2L

In”u”) (n”u”I a q n u )

+

on,,n W L

1,

(3.1.30)

where 2.’) = 2 escL) and the state I nu) is the ground molecular state with electronic quantum number n and vibrational quantum number u. The intermediate state is In’’U“ ), where n” # n, and the final state is lnu’ ), where u’ # u; that is, a vibrational excitation has taken place. Rotational states will be ignored. In eq. (3.1.30) the relatively small contributions of the vibrational energies to the frequency denominators have been neglected. In the Born-Oppenheimer approximation the state I nu) corresponds to the product wavefunction Qn(x,q ) Xnu(q), where x is the electronic coordinate(s) and q is the nuclear coordinate (consider for simplicity a diatomic molecule). A typical matrix element can be written

(no’ I

d pI ~ ” U ”

)

=

ss dx

dq @:(x, q)x,*,(q)dS(x, q)Qn..(x,q)Xn.,.(q). (3.1.31)

Using this form for each element in eq. (3.1.30), and using the closure relation for the nuclear wavefunctions,

C xn,,u,,(q)X3,u,,(q’)

= Nq

(3.1.32)

- 4’)

u”

leads to, for example,

C ( n u ’ / 2 s I n f ‘ u ” ) (n”u‘’1 U”

where

d!L$)(q) =

s

d x @:(x,

[nu) =

s

dqX:u,(q)d,B,,,(q)d,L;.n(q)Xnu(q),

q)dScL)(x,q ) @np,.(~, q)

(3.1.33)

(3.1.34)

is the electronic dipole matrix element between initial and intermediate states. Using these definitions, eq. (3.1.30) can be written as a matrix element, between

206

QUANTUM COHERENCE PROPERTIES OF RAMAN SCATTERING

B3

initial and final vibrational states, of the quantity a(q), S = ft

s

dq x&,(q)a(q)Xnu(q)

9

(3.1.35)

where a(q) is the /3, L component of the frequency-dependent electronic polarizability tensor of the molecule

(3.1.36) evaluated for a fixed value of internuclear coordinate q (see, for example, [ 19861). Since a(q) is a function of q, a Taylor series SCHUBERT and WILHELMI expansion can be made around the equilibrium value, q = 0, a(q) = a(0)

+

(30 -

q

+

a

*

*

.

(3.1.37)

The a(0) term gives zero in eq. (3.1.35), since u’ # u. The second term can be evaluated by the usual rules for a harmonic oscillator with reduced mass m and frequency 031, assuming u = 0 and u’ = 1, to give

(3.1.38) For u‘ # 1, S is zero. Using this formula in eq. (3.1.1 l), and assuming that the Stokes shift 031 is small compared with wL,gives for the scattering crosssection

(3.1.39) This result confirms the usual result of the Placzek model (see PENZKOFER, LAUBEREAU and KAISER[ 19791).

3.2.

SEMICLASSICAL PROPAGATION THEORY

One of the main limitations of the photon rate-equation theory is its inability to treat the coherent interaction between field and molecules. This limitation becomes especially important when the duration of the laser pulse is comparable with, or shorter than, the collisional dephasing time r- of the Raman

111, B 31

207

THEORY OF STIMULATED RAMAN SCATTERIN0

transition. In the case where a strong Stokes field is injected into the Raman medium along with the laser field, the semiclassical propagation theory is useful. Quantum fluctuations are not dominant in this case, so that the electromagnetic field may be treated approximately as a classical quantity. The atoms or molecules are treated quantum mechanically by means of the density operator. In f 3.4 it will be shown how to unify the two different aspects of SRS treated in 9 3.1 and f 3.2: quantum initiation and coherent propagation. 3.2.1. Atomic-density operator equations of motion

The density operator ) ( t ) for an atom or molecule evolves in time in the Schrddinger picture according to d 1 (3.2.1) - I = - [),A], dt h where the Hamiltonian is, in the dipole approximation,

A=A,- d * E , (3.2.2) where A, is the atomic Hamiltonian, d is the electric dipole operator, and E is the classical field, assumed to be linearly polarized along the 8 direction. With reference to the energy-level structure of fig. 3.2, the equations of motion for the relevant density matrix elements ),b = (a1 ) Ib), where A, l a ) = hw, l a ) , are d dt

- Plm

i dlm(pll - pmmlE - - d3mp13E h h 1

(3.2.3)

= iwmlPlm - -

d dt

- ~ m = 3

i03m ~

-

i dm3(~mm- ~ 3 3 ) E+ h

m 3-

1

dm, ~ 1 3 E 9

(3.2.4) (3.2.5)

(3.2.6)

where dab = (a1 8 - 8 Ib), d13 = 0 = d,, and Eis the scalar amplitude ofthe field. Also, terms that correspond to excitation between the weakly excited intermediate states Im) have been neglected. Equations (3.2.3 to 3.2.6) are the same as in, for example, RAYMER and MOSTOWSKI[1981], wherein Pab =

( bba ) *

208

QUANTUM COHERENCE PROPERTIES OF RAMAN SCAITERING

WI, 8 3

Since the intermediate levels 1 rn ) are driven by the field far-off resonance, they adiabatically follow the field and can thus be eliminated from appearing explicitly in the equations. To do this, it is necessary to integrate formally eq. (3.2.3):

X

{dlm[Pll(t') - P m m ( t ' ) I E ( t ' ) + d3mP13(l1)E(t')}*

(3.2.7)

The total field is given by E(t) = EL(z, t ) + Es(z, t), where the laser field E,(z, t) and the Stokes field Es(z, t) are given by E P (2,t) = EJz, t ) ei(w@r-kfiz) + C.C. ,

(3.2.8)

where p = L, S, with z being the location of the atom and EL and E, being slowly varying compared with the detunings from all intermediate states. To avoid explicit z-dependence in the equations, it is necessary to define the slowly varying variable Q(t) by Q(') = p13(t), - i ( w , - w s ) r + i ( k ~ - k s ) z

(3.2.9)

Also, exact Raman resonance will be assumed; that is, wL - us= 031. When eq. (3.2.7), and a similar one forpm3(t),are substituted into eq. (3.2.5), and only terms oscillating near the frequency 031 are retained, the following equation results

(3.2.10)

(3.2.11)

and 6 is the ac Stark shift, given by 6 = 6,

bi=G 1

c ~ d i m ~ ~ [ t E L ~ 2 (1 m

"mi

+

+ lEs12( "mi

- 6,,

+ "L

1 + "S

"mi

where

- "L

)].

+ "mi

- US

(3.2.12)

I K § 31

THEORY OF STIMULATED RAMAN SCAITERING

209

In the following discussion the ac Stark shift will be assumed to be negligible. In terms of Q(t) and the population inversion w(t) = p3Jt) - p l l ( t ) , the adiabatic equations of motion are

a

at

a

at

Q=

- r Q + itc:E,Eaw

(3.2.13)

w

2iic,EtE,Q

(3.2.14)

=

-

2ilc:E,EgQ*,

where a phenomenological dephasing term - r Q has been added, and the time derivatives are written as partial derivatives, since Q and w can also depend on position z. These equations are of the same form as the Bloch equations for a two-level system. The “two-photon Rabi frequency” is given by 2 tcl E,E, (TAKATSUJI [ 19751 and GRISCHKOWSKY, Lou and LIAO[ 19751). 3.2.2. Wave equation

For plane-wave amplification the Stokes field E,(z, t ) obeys the onedimensional wave equation

(9-f

$ ) E d z , t ) = - 411 a2 P(z,t), cz at2

(3.2.15)

where us = c/n is the phase velocity of Stokes light, and we have neglected group-velocity dispersion. P(z, t ) is the magnitude of the nonlinear polarization at the Stokes frequency. In the semiclassical theory this is given by the expectation value, or trace with the density operator, of the electric dipole operator

P = NTr()d-8) (3.2.16) which can be shown in the adiabatic approximation to be P(z, t) = N h rc:E,(z,

t)Q*(z, t ) ei(wsr-kss)+ C.C.,

(3.2.17)

where N is the molecular number density and Q has been allowed to depend on z . Assuming that the envelope E,(z, t ) is slowly varying compared with as, the wave equation becomes (3.2.18)

210

QUANTUM COHERENCE PROPERTIES OF RAMAN SCATTERING

[III, 0 3

where K~ = 27tNho, rc:us/c2. A similar equation holds for the laser field envelope

where vL is the phase velocity at oL. Equations (3.2.13), (3.2.14), (3.2.18), and (3.2.19) are the basic equations describing the SRS process. They were first presented fully by GIORDMAINE and KAISER[ 19661, using the molecular polarizability model of Placzek. (See LAUBEREAU also KAISERand MAIER[ 19721, WANG [ 19751, PENZKOFER, and KAISER[ 19791.) In the present derivation we have used the electric dipole moments dii rather than the molecular polarizability. (See also EMEL’YANOV and SEMINOGOV[ 19791, RAYMER,MOSTOWSKIand CARLSTEN[ 19791, RAYMERand MOSTOWSKI [ 19811.) To make the connection between these two formalisms, we can write eq. (3.2.11) for the coupling constant as K , = S / h 2 ,where S is given in eq. (3.1.8). Thus, from the discussion in 3.1.5 we can rewrite K , as (3.2.20) Then, to find the connection between Q, which is a dimensionless densitymatrix element, and q, which is the internuclear separation, it should be noted that the expectation value of the internuclear operator 4 is given by a trace with the density operator (3.2.21) Here the states I k ) are the electronic vibrational states of the molecule. In the present case population resides primarily in the ground state I 1 ) = [nu), where the electronic quantum number is n and the vibrational quantum number is u = 0, and in the excited state 13) = I nu) ,where u’ = 1. Thus in eq. (3.2.21) k and k‘ vary only over 1 and 3. The relevant matrix elements are then (11411) = ( 3 1 6 1 3 ) = O m d

where, again, the usual rules for a harmonic oscillator were used. Thus (4)

THEORY OF STIMULATED RAMAN SCAITERING

21 1

becomes (3.2.23) or, using eq. (3.2.9),

(4)

= (+I2

h

~,i(wL-os)t-i(kL-ks)r

+

c.c.

(3.2.24)

2mw31

If we define a slowly varying nuclear coordinate 4 5 (h/2mw3,)’/’Q, and use eq. (3.2.20) for rcl, then eqs. (3.2.13) and (3.2.18) become (3.2.25)

(3.2.26) These are the usual SRS equations in the molecular polarizability model (see, LAUBEREAU and KAISER[ 19791). for example, eq. (3.1.44) in PENZKOFER, 3.2.3. Solution of semiclassical equations in the linear regime If the amplification of the Stokes field E,(O, I ) incident on the input face of the medium is not too large, the laser field E,(z, t) can be taken to be a prescribed function of space and time that depends only on the local time variable z = t - z/vL. In addition, the population difference can be well approximated by w = - 1, since most of the molecules remain in their ground states. If, furthermore, the velocities us and uL are equal, eqs. (3.2.13) and (3.2.18) can be written as

a

-

az

Q*@, 7)

=

-rQ*(z, z)+

E,(z, z)

=

-iic2EL(z)Q*(z,

iic,E?(z)E,(z,z),

(3.2.27)

II

U

-

aZ

7).

(3.2.28)

Equations (3.2.27)and (3.2.28)were first solved by KROLL[ 19651 in the case ofconstant EL(z)and later by CARMAN,SHIMIZU, WANGand BLOEMBERGEN [ 19701 for the general case, using Riemann’s method, A method using Laplace

212

QUANTUM COHERENCE PROPERTlES OF RAMAN SCATTERING

[III, 8 3

transforms is given in RAYMER and MOSTOWSKI[1981]. Assuming Q(z, 0) = 0, the solution for the amplified Stokes field is Es(z, z) = Es(O, 7)

+

where

lo7

G(z, 7, z’)E,(O,

7 ’ ) dz‘

,

(3.2.29)

(3.2.30) and p(z) =

lo7

IEL(z”)12dz’’,

(3.2.31)

and I,(x) is the modified Bessel function (ABFUMOWITZand STEGUN[ 19641). In the case of constant laser field EL and input Stokes field Eso, and long times, rz$- 1, the upper integration limit in eq. (3.2.29) is taken to infinity, leading to the following steady-state Stokes intensity lEs(z,

4l2 = IEsoIZegZ,

(3.2.32)

where the gain-coefficient g is given by (3.2.33)

g = 2~~ K~ lEL12/I‘.

This result for g is identical to that found in the photon rate-equation treatment (eq. (3.1.22)). The criterion for the solution to be well approximated by eq. (3.2.32) can be stated more accurately as r z > gz. In the transient limit, r z 4 1, the factor exp [ - r ( z - z’)] can be dropped in eq. (3.2.30). Then, if both the laser and Stokes fields are turned on suddenly at z = 0 and then held constant, the integral can be evaluated to give for the Stokes intensity IEs(z, z)I2

= IEso12Zo({2gzr~}’/2).

(3.2.34)

In the high-gain limit, g z r z % 1, the asymptotic form for the modified Bessel function can be used, Z,(x) (271~)-ll2 ex, for x B 1, to give N

(3.2.35) It should be noted that the product of g r is independent of r, and that if the

W 8 31

THEORY OF STIMULATED RAMAN SCATTERING

213

input Stokes field E,, is zero, the output Stokes field is also zero. That is, there is no spontaneous initiation of Stokes light in the semiclassical theory, as expected. The case of transient amplification pumped by a smooth, but arbitrarily shaped, laser pulse EL(z) cannot be easily treated analytically, and numerical SHIMIZU,WANG studies have revealed some interesting features (CARMAN, and BLOEMBERGEN[ 19701). The peak of the amplified Stokes pulse is delayed with respect to the peak of the laser pulse. In addition, for rapidly falling laser pulse shapes (such as a Gaussian), the Stokes pulse is shorter in duration than the laser pulse, although for laser pulse shapes such as a Lorentzian the amplified Stokes pulse can be longer than the laser pulse. Another finding is that the net transient gain is relatively insensitive to the shape of the laser pulse and depends strongly on its energy. The transient gain is smaller than the steady-state gain, because of the square root nature of the growth, as illustrated in eq. (3.2.35).

3.3. QUANTUM THEORY OF SRS IN A CAVITY

In an attempt to treat both temporal coherence and spontaneous initiation of the Stokes scattering, a number of quantum treatments were developed in which only a single mode (or several modes) of the Stokes field was allowed and GLAUBER [ 19671, MISHKINand WALLS[ 19691, to be excited, (MOLLOW WALLS[ 1970, 1971, 19731, SIMAAN[ 19751). This approach is most suited to interactions inside a high-Q cavity, where the field amplitudes are uniform (mean-field theory). Only the simplest form of this theory will be reviewed here to illustrate several important concepts that will be useful in 8 3.4 and 8 3.5. 3.3.1. Equations of motion

The simplest single-mode treatment begins with a Hamiltonian with the postulated form

A = ho,citci

+ ho,,hth + h~ [e-'mLrci+ht + e ' ~ ~ ~ h,c i 1

(3.3.1)

where cit(ht) is the boson creation operator for the Stokes (phonon) mode withenergyho,(ho,,), icisacouplingcoefficient,andagainw,( = w, + is the frequency of the laser field, which is here treated classically (ICcontains a factor EL).Damping has been neglected, and only a single phonon mode has been considered. Therefore, the treatment is most suited to describing SRS in

214

QUANTUM COHERENCE PROPERTIES OF RAMAN SCATTERING

[III, B 3

the transient regime. Coupling to anti-Stokes light has also been ignored. The operator d(t) evolves in time according to the Heisenberg equation ih

d dt

- i ( t ) = [d(t), A] ,

(3.3.2)

and similarly for h(t). Changing variables to a(t)= d(t)exp(io,t) and = 6 ( t ) exp(io,,t), the equations of motion are

$(t)

d dt

-A=

-id?,

(3.3.3)

The solution of these equations is

+ i sinh(Ict)$(O) , - i sinh(Ict)at(0) + cosh(Ict)$(O) ,

at(t)= cosh(Kt)at(0) $(t)

=

(3.3.4)

where a(0)and $(O) are the SchrOdinger-pictureoperators. The operator for the number of Stokes photons is A(t) = at(t)a(t)and its expectation value ( A ( t ) ) = ( Y(0)I A ( t ) I Y ( 0 ) ) can be calculated using eq. (3.3.4). If the initial state I Y ( 0 ) ) is the vacuum state for both Stokes and phonon modes, by using a(0)I "(0)) = 0, we find that ( A ( t ) ) = (Y(0)I d(0)Bt(O) I Y ( 0 ) ) sinh2(Ict) =

sinhZ(x t ) .

(3.3.5)

It should be noted that this result arises from the d(0) term in the solution (3.3.4); one can say that the laser field scatters off the quantum mechanical zero-point motion of the phonon mode. This effect is, of course, absent in the semiclassical theory. 3.3.2. Photon number fluctuations

Although the mean Stokes photon number ( n ) grows as in eq. ( 3 . 3 3 , on any given trial of the experiment a different number will be observed; that is, there are quantum fluctuations in the Stokes photon number n. In this section a simple, direct method will be presented for obtaining p(n), the probability of observing the value n. According to radiation detection theory (KELLEY and

111, P, 31

215

THEORY OF STIMULATED RAMAN SCATTERING

KLEINER[ 19641, GLAUBER [ 19641, MANDELand WOLF[ 1965]), p(n) is given, for a unity quantum-efficiency detector, by (3.3.6) where P( W) is a quasi-classical distribution function for the Stokes field energy W (measured in units of ho,),and is given by P ( W ) = ( : 6(W - A): ) ,

(3.3.7)

where the brackets ( ) indicate an expectation value in the vacuum state and the double colon indicates normal ordering; that is, d 2 s go to the right of all at’s.In the limit of large Stokes gain the photon number becomes so large that it behaves like a quasi-continuous variable. Then W can be considered to be the measured quantity, rather than n, and P(W) is an excellent approximation to the measured distribution function. In this connection it should be noted that eq. (3.3.7) is similar to the usual expression P( W) = ((6( W - n(t)))) found in classical statistics theory, where the double brackets indicate an ensemble average over the random process n(t) (GOODMAN[ 19851). To evaluate P(W), it should be written as a Fourier transform of the characteristic function C,(t) (3.3.8) where C,(t)

=

( : exp(i@) : )

= (1

+ itP(t)A(t) + i(it)’P2(t)A2(t) +

).

(3.3.9)

When the solution for A(t) in eq. (3.3.4) is inserted into (3.3.9), the terms containing A(0) all go to zero when acting on the initial vacuum state. Thus C,(t) is given by ~ i n h ~ ( i c t ) B ~ ( O ) 8+~*~. -( 0))

C , ( t ) = ( 1 + i t sinh2(ict)B(0)Bt(O) +:(it)’ =

1 + i t sinh2(ict) + [ i t sinh’(ict)]’

= 1/[ 1

- i t sinh’(ict)] ,

+

*

(3.3.10)

where the usual boson rules Bt(0) I m ) = (rn + l)l/z I m + 1 ), etc., were used. Identifying sinh2(rct) as the mean photon number (A), and carrying out the

216

QUANTUM COHERENCE PROPERTIES OF RAMAN SCAnERING

[III, s 3

Fourier transform in eq. (3.3.8) gives (3.3.11) Thus the most probable value of Stokes energy W is predicted to be zero, whereas the mean value is (( W)) = ( A ) , which can be a macroscopic quantity, say 10l6 photons, correspondingto 1 mJ at a wavelength 0.6 pm. This is a remarkable result because it demonstrates a system that begins in a pure state and evolves by Hamiltonian dynamics into a state with a macroscopic, but stronglyfluctuatingvalue of energy. This result can be called a macroscopic quantum fluctuation, and it is interesting because it can be observed with the unaided eye, rather than requiring electronic detectors and amplifiers. To obtain a deeper understanding of the source of these macroscopic fluctuations, we use eqs. (3.3.11) and (3.3.6) to calculate the photon number distribution, which gives (3.3.12) that is, a Bose-Einstein distribution. Thus the Stokes field has the same statistics as a single mode of a thermal, or blackbody, field because the source is spontaneous scattering. Since the amplification process is unsaturated, it is linear, and the statistics of the field remain Bose-Einstein as the mean number becomes amplified to macroscopic levels. 3.3.3. Equivalent classical random process Another perspective from which one can view the result eq. (3.3.11) for Stokes energy fluctuations is by finding an equivalent classical stochastic process, which represents the Stokes field and from which eq. (3.3.11) can be derived by classical means. This can be done in a simple operational way, and HAAKE[ 19781. It should be noted that following the idea of GLAUBER when the normally ordered expectation value in eq. (3.3.9)was evaluated, terms of the following form arose ({B1(O)B+(O)}m)

= m!

.

(3.3.13)

But we should note that this result can be equally well obtained by formally replacing 8(0) by a complex, classical random variable B with zero mean and

I K 5 31

THEORY OF STIMULATED RAMAN SCAlTERlNG

217

probability distribution (3.3.14) where in this case (( lB12)) average. Then we find (({BE*}"))

=

=

1 and the brackets now mean an ensemble

{ B B * } m P ( B )d2B = rn! ,

(3.3.15)

which is exactly the same as eq. (3.3.13). This is a form of the Gaussian moment theorem. Thus any expectation value of a normally ordered function of the operator B(0) can be evaluated by computing a classical average, ( :f(B(0),Bt(O)): )

=

((f(B,B*))) =

s

f(B,B*)P(B)d'B.

(3.3.16)

In particular, the characteristic function can be easily calculated by this method, by replacing (t) by the classical variable A *(t) = i sinh (K ~ ) and B integrating,

at

C W ( 8 = ((exp =

=

1

*(t)A(t)l>>

exp[i(sinh2(rct)B*B]P(B)d2B

I/[ (1 - i t sinh2(ict)] ,

(3.3.17)

which is identical to the result in eq. (3.3.10). Looking at the problem in this way allows us to associate with the quantized Stokes field At(t) an equivalent classical random process A *(t), which has Gaussian statistics, with distribution (3.3.18) where (( [ A1)' = sinh2(Kt) = ( A ) . Then, to find the probability distribution for the Stokes energy, the classical energy variable is defined as W = [ AI '. Using standard change-of-variabletechniques, eq. (3.3.18) immediately implies (3.3.19) which is exactly the same as eq. (3.3.11).

218

QUANTUM COHERENCE PROPERTIES OF RAMAN SCAITERING

[III, 0 3

It has been assumed here that the measurement of energy is instantaneous. However, ifwe allow the detector to have a finite measurement time, the results are not substantiallyaffected. This is because in this model damping is neglected and so the field is temporally coherent. The effects of damping will be discussed in $ 3.5. It should be mentioned that this method is equivalent to that of GLAUBER [ 19631 and SUDARSHAN [ 19631, in which A is associated with a coherent state ( A) which obeys &) [ A ) = A(t) [ A ) . Then P(A) is a diagonal representation of the density operator ),

fi =

, A ) ( A [ P(A)d2A.

(3.3.20)

Then P(A)in eq. (3.3.18) is equal to ( : a2(A - A): ),which is of the same form as eq. (3.3.7). It should be noted that for the case of Stokes generation, P(A) is positive definite, implying that an equivalent classical process exists.

3.3.4. Generalization to many Stokes modes If there are N Stokes modes in the cavity that become excited independently, and the ith mode contains energy W,, the total energy is

w=

c wi. N

(3.3.2 1)

i= 1

Each statistically-independentrandom variable Wi obeys a negative-exponential distribution of the form of eq. (3.3.19). If the mean of Wi is denoted A,, and no two A’s are equal, the usual rules of statistics imply that W obeys the following distribution (see, for example, SALEH[ 19781)

(3.3.22)

In the special case that all the mean values 1,are equal to A, P( W) is the gamma distribution P(W) =

1

wN-’

-e - WIA . (N - l)! AN

~

(3.3.23)

For large N this function is narrow and peaks near W = NA, which is equal to the mean value of W. Thus, as more independent modes are added together,

111, B 31

THEORY OF STIMULATED RAMAN SCATTERING

219

the total energy W fluctuates less. For N + 00, P( W) converges to a Gaussian distribution, which is an example of the central limit theorem.

3.4. QUANTIZED-FIELD THEORY OF SPATIAL PROPAGATION IN SRS

Although the single- or several-mode theory described in the previous section is illustrative in that it correctly predicts the existence of macroscopic quantum fluctuations of Stokes pulse energy, it is not appropriate for describing the most common experimental arrangement for studying SRS : A laser pulse propagates through a Raman-active medium and a Stokes pulse builds up and propagates out of the medium. No cavity mirrors are present. In principle, this situation can be treated by solving for the time evolutions of an infinite number of field-mode creation operators, and then summing them together to find the total, spatially dependent field. Since this procedure would be equivalent to solving Maxwell's equations for the total field, it is convenient and instructive to formulate the equations of motions as a wave equation for the total Stokes field operator E,(v, t). This method allows the connection between the semiclassical and quantum theories of propagation to be seen clearly. As in the semiclassical theory, one can approach the derivation, either by using the molecular polarizability ansatz of Placzek or by using the electricdipole matrix elements of the multilevel system. The former method was used and GLAUBER [ 19711, whereas the latter method was used by VON FOERSTER by MOSTOWSKI and RAYMER [ 19811 and RAYMER and MOSTOWSKI [ 19811, and SEMINOGOV [ 19791. The latter paper includes as well as by EMEL'YANOV the effects of significant population building up in the upper Raman level which under certain conditions can lead to cooperative scattering; this process will be discussed in 3 4.4. 3.4.1. Atomic operator equations of motion The derivation to be outlined follows that of RAYMER and MOSTOWSKI [ 19811 and closely parallels the semiclassical derivation in 5 3.2. Here we utilize the atomic (or molecular) projection-transition operators, denoted by h&), which at the initial time t = 0 are given by hU(O)= li) ( j l , where l i ) is an energy eigenstate of the atom. The hu obey the following commutation relation, (3.4.1)

220

QUANTUM COHERENCE PROPERTIES OF RAMAN SCATTERING

“11, § 3

For i # j , b,, acts like an atomic raising (or lowering) operator. Any atomic operator a can be represented in terms of the b,, by means of

a = C ( i l 2 I j ) biJ.

(3.4.2)

ij

Thus the evolution of the b;s in the Heisenberg picture fully specifies the state of the atomic system. A simple relation exists between the averages of the b, operators and the elements of the density operator that is,

a;

(4j)=Tr(bb,,)= (jl

bl0

=p,,,

(3.4.3)

where eq. (3.4.2) was used to represent 8. Thus the results of the semiclassical theory of § 3.2 should occur as a special case of the present calculation. The time evolution of each b,, is determined in the Heisenberg picture by d ih - b,,(t) dt

=

[ b,(t),

A] ,

(3.4.4)

where the Hamiltonian is, in the dipole approximation,

A = A,

-

J(t) eE(t) (3.4.5)

where h a i is the energy of the atomic state l i ) and @(t) is the electric-field operator at the location of the atom. As in 3.2, we assume, referring to fig. 3.2, that d,, = d, = 0 and that the field is linearly polarized along the direction 8. Then eq. (3.4.4) leads to -

d 1 1 bml = iwmlbml - - dlm(bll - bmm)E- - d3mb318y h h dt

(3.4.6)

d 1 1 b3, = iw3mb3m- - dm3(bmm- b33)E+ - dmlS 3 1 E , dt h h

(3.4.7)

-

(3.4.8)

(3.4.9) where, again, we have neglected terms corresponding to excitation between the weakly excited intermediate states I m ) .

III,! 31

221

THEORY OF STIMULATED RAMAN SCATTERING

Since eqs. (3.4.6)-(3.4.9) are identical in form to eqs. (3.2.3)-(3.2.6),with pI/ replaced by b,i, the same adiabatic elimination of intermediate states described in $ 3.2 can be applied here. This leads to (see eq. (3.2.10)) d dt

-

a 3 1 ( t ) 1:

i03,&31(t) - iicT&)(r,

t)BS+)(r, t )

e i [ w 3 ~ r - ( k ~ - k s ) 4[ bi i(O

-

bdOI

9

(3.4.10)

where icl is given by eq. (3.2.11) and the ac Stark shift was neglected. The total field operator was taken to be &r,

t) =

&(r,

t)

+ ,Gs(r, t )

- E(-)(r, t ) ei("Lf-kLZ) + a (S- ) ( , t ) ei(wsf-kss)+ H.a., (3.4.11) L and r is the location of the atom. Defining a slowly varying operator Q(t) = b3,(t), - i ( ~ ~ - ~ s ) f + i ( k ~ - k s ) ~

& by (3.4.12)

in analogy with eq. (3.2.9),leads to d dt

- &(t) =

itcrEi-)(r, t)E".)(r,

t)$(t),

(3.4.13)

where E','" is the adjoint of EL-) and $(t) = b33(t)- &,,(t) is the population inversion operator. The incorporation of collisional dephasing of the Raman coordinate &(t) requires some discussion. In eq. (3.2.13)for the density-matrix element p13, a damping term -rQ was simply added, in the spirit of the optical Bloch equations. It is known, however, that for operators this is not sufficient. In addition to the damping term, a so-called Langevin operator f ( t ) must be added. Thus eq. (3.4.13)becomes

In a classical theory the Langevin "force" F(t) would be responsible for causing stochastic fluctuations of the oscillator coordinate due to sudden, random collisional impacts. Here, however, its meaning is more subtle - it is related to quantum noise. The form of the Langevin operator P(t)can be derived using quantum reservoir theory (for review see HAKEN [1970] and LOUISELL [ 1973]), and it can be shown to have the properties (E(t))

= (ft(t)) =

0,

(3.4.15)

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QUANTUM COHERENCE PROPERTIES OF RAMAN SCATTERING

[III, 5 3

( f t ( t ) f ( t ) )= 2 r ( a l l ( t ) )s(t - t ’ ) ,

(3.4.16)

(f ( t ) f t ( t ’ ) ) = 2 r ( a,,(t)) s(t - t ’ ) ,

(3.4.17)

where the brackets indicate an expectation value over the dephasing reservoir variables. Thus (a,) is still an atomic operator. In specifying the factors multiplying the delta functions, we have assumed that there is negligible spontaneous decay from state 13) to state I 1 ) . It is important to note that the presence of the Langevin operator maintains the expectation values of the commutation relations of the & operator. For example, we should have at all times = b,,b3, = all, To illustrate that this relation is maintained in the simple case of no coupling between the atom and the laser field, eq. (3.4.14) is integrated to give

ot&

(3.4.18) Then calculate

<

( Qt(t)Q(t)>= &t(to)&(to)>e+

It: It: dt‘

2r(r-to)

dt’f e - r ( z f - t ’

-t”)

(P(t’)f(t’’)) , (3.4.19)

where ( & t ( t o ) f ( t ’ ) ) = 0 for t’ = to was assumed. It should be noted that (&t(tO)&(tO)> = ( a13(t0)a31(20)> = ( a l l ( t O ) ) * using eq*(3.4.16), and noting that in this simple example ( b l l ( t ) ) is time independent since there is no spontaneous population decay, eq. (3.4.19) can be easily integrated to give ( & t ( t ) $ ( t ) ) = ( all(to)) e-2r(r-ro) = (~I,(O)

*

+ ($,,(t))

[ 1 - e-2r(t-fo)1

(3.4.20)

Thus the commutation relation is maintained, at least under the reservoir average, which means that there is an intimate connection between the fluctuations f ( t ) associated with collisional damping and the zero-point motion of the oscillator @(to). Even though the damping would appear to cause the zero-point fluctuations to decay exponentially (eq. (3.4.18)), the Langevin operator continually reintroduces fluctuations of the proper magnitude. For example, one could set the initial time to equal to minus infinity; then the solution would contain only f ( t ) and not &(to), illustrating that &(to) in eq. (3.4.18) can be thought of as being the integral of exp[ - r(to- t ’ ) ] & ’ ) from - to to.

111, s 31

223

THEORY OF STIMULATED RAMAN SCATTERING

In the more general case in which the atom and field are coupled and population is allowed to decay spontaneously, the commutation relations can still be maintained by using suitable Langevin operators. 3.4.2. Operator Maxwell-Bloch equations The Heisenberg equation of motion for the Stokes field operator I?&, implies the operator wave equation (in Gaussian units)

r)

(3.4.21) where P(r, r) is the component of the macroscopic polarization operator, or dipole moment per unit volume, which oscillates near frequency as,

1 =-

AV

C C [ d m , a z , ( t ) + dm3az3(t)l

+ H.a.9

(3.4.22)

{a), m

where the sum is over atoms, labelled by a, located within a small volume A V centered at position r. Using the adiabatic elimination procedure that led to eq. (3.4.10) gives for the polarization

P(r, t ) N N h ~ : @ - ) ( r , t)Qt(r, t ) ei(ost-kss)+ H.a. , where N is the average atomic number density and atomic operator, defined as

(3.4.23)

&, t ) is the collective (3.4.24)

where NA V is the average number of atoms per volume V. Each &“(t) obeys an equation of the form of eq. (3.4.13). Therefore, Q(r, t ) obeys

a

at

&(t) =

- r&(r, t ) + iK:&L-)(r, t)EL+)(r, t)&(r, t) + f ( r , t ) , (3.4.25)

where the collective Langevin operator is defined as (3.4.26)

224

QUANTUM COHERENCE PROPERTIES OF RAMAN SCATTERING

[III, 8 3

and ~ ( rt,) is defined analogously. Since each pa(t)is statistically independent, t ) has the following correlation functions, in the continuum limit,

&,

(Pt(r, t)&’, t ’ ) ) = 2rN-I ( aI1(r,t ) ) b3(r - r’)b(t - t ’ ) , (3.4.27) ($(r, t)Pt(rf, t ’ ) )

=

2I’N-I ( b33(ry2 ) ) b3(r - r’)b(t - 2 ’ ) .

(3.4.28)

Finally, putting eq. (3.4.23) into eq. (3.4.21), and assuming that the positiveand negative-frequency parts of the field are not coupled, gives for the negativefrequency part

(3.4.29) where u2 = 2nNhus ictvs/c2. We will assume that this equation describes a wave travelling more or less in the + z direction and that coupling to the wave travelling in the opposite direction can be neglected. This condition should be valid as long as laser depletion effects are not strong. Equations (3.4.29) and (3.4.25), coupled with an equation for the population inversion operator,

a

at

G(r, t ) = - i x1EL- )(r, t ) Ep )(r, t ) Q(r, t )

- iK,

Er )(r, t)E$- )(r,

t ) @(r, t ) ,

(3.4.30)

describe the spontaneous initiation and spatial propagation of SRS. An important property of the operator Q(r, t ) is easily shown: If the atoms all begin in their ground states 11) , the expectation value of Qt& at t = 0 is equal to (Qt(r,O)Q(rf,O))

=

~ - ‘ b ~ -( r ’ ) ,

(3.4.31)

in the continuum limit. This equation can be interpreted as showing that although the mean (Q(r, 0)) of the initial molecular coordinate is equal to zero, its variance is not. This is due to the quantum mechanical zero-point motion of the oscillator. We will see later that this dipole noise can be thought of as initiating the Raman scattering.

111, ti 31

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THEORY OF STIMULATED RAMAN SCATTERING

3.4.3. One-dimensional propagation Assume that the laser pulse has a cross-sectional area A and travels through a Raman-active medium with length L. Assume further that L B A’/’, that is, a long, thin, pencil-shaped volume of molecules is pumped. The Fresnel number is defined as in eq. (3.1.24), F = A / I , L . In the case where F i s of the order of unity, the diffraction-limited angular divergence angle (-A,/A’/’) is of the order of the geometrical angle subtended by the pencil ( -A”’/L), and therefore only a single transverse spatial mode contributes strongly to emission along the pencil axis. One may think of the two ends of the pencil-shaped, amplifying medium as forming an active spatial filter that transmits only diffraction-limited light. In this case (F- 1) the volumes in eq. (3.4.22) can be taken to be thin transverse slices with volume A V = AAz, where Az is the thickness of a slice located at position z along the axis. Assuming the field varies little in the transverse directions, the Laplacian in eq. (3.4.29) is replaced by a2/az2.Then the slowly varying envelope approximation is applied, in which it is assumed that the variables E$), EL-), and Qt vary little in a time wg or in a distance

’

k,

I.

tZ 3-

+--

EL-)(z, t) = -ix2EL-)(z, t)&+(z,t).

u:

(3.4.32)

This equation is identical in form to the semiclassical eq. (3.2.18). Approximate numerical methods for solving the coupled, nonlinear operator eqs. (3.4.32)and (3.4.25) exist (LEWENSTEIN [ 19841, ENGLUNDand BOWDEN [ 19861). Here, though, analytic solutions to the linearized equations alone will be discussed.

3.4.4. Solution of the linearized S R S equations In the case in which the Raman gain-length product is not too large (<20), the molecules will essentially all stay in their ground states (Is N - i) and the laser field will remain undepleted. Thus, if the laser field is in a coherent state, it is valid to replace the operator fiL- )(z, t ) by a prescribed c-number function EL(z, t). If, fwthermore, we assume that us and uL are equal, eqs. (3.4.25) and (3.4.32) can be written as

a

aZ

@(z, t) = - rQt(z,

Z)

+ iicIEt(Z)f&-)(z, t ) + ft(z, z),

(3.4.33)

226

QUANTUM COHERENCE PROPERTIES OF RAMAN SCATTERING

a

-

EL-)(,, z) = - iic2EL(z)@(z, z),

1111, B 3

(3.4.34)

aZ

where z = t - z/uL.These equations are the same as eqs. (3.2.27) and (3.2.28), except for the operator nature of the variables, and the initial properties (from eqs. (3.4.31), (3.4.27), and (3.4.28)) ($t(Z, z = O)&(Z’, z = 0)) = (AN)-’b(z (Et(Z, z)P(z’, z’)) (P(2,

z)P(z‘,

=

7’)) =

- z’),

(3.4.35)

2 r ( A N ) - ’ b ( z - z’)S(z - z’),

(3.4.36)

0,

(3.4.37)

where AN is the linear density of atoms along z. The initial condition for the Stokes field is that, at the input face of the medium (z = 0), Ei-)(O, z) is equal to the free-field operator (the operator in the absence of any interaction with atoms). Equations (3.4.33) and (3.4.34) can be solved (RAYMERand MOSTOWSKI [ 19811) to yield

+

joz

dz’ H(z, z’, 5 0 ) ot(zf, 0)

+ joTdz’j;dd

H(z,z’,

T, zr)Pt(zf,TI),

(3.4.38)

where G(z, z, z‘) is the same function that appears in the semiclassical solution (3.2.29), and H(z,z’, z, z’)

=

-i~~EJz)e-~(~-~’) x Z0({4K’ K2(Z - z’) [ p ( z ) - p(z’)]}’’2),

(3.4.39)

and again p(z) =

JOT

IEL(~”)12 dz” .

(3.4.40)

We observe first that if an expectation value is taken of eq. (3.4.38), noting that (@(z’, 0)) = (Et(z’, z’)) = 0, and if (E$-)(z, z)) is identified as the semiclassical field E,(z, z), the quantum result exactly reproduces the semiclassical amplifier result eq. (3.2.29), as expected.

111, I 31

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THEORY OF STIMULATED RAMAN SCATTERING

Now let us consider the generation of SRS when there are no Stokes photons incident on the medium; that is, the initial state I Y ( 0 ) ) of the system has all molecules in their ground states, which gives eq. (3.4.39, and the Stokes field in the vacuum state. Then Ev )(O, z) I Y ( 0 ) ) = 0, since the negative-frequency part of the free field is made up of annihilation operators. The mean Stokes intensity I,(z) is defined as the mean number of photons emitted per second and per area, through the end face (z = L ) of the pencil-shaped pumped region, and into the solid angleA/L*defined by the geometry of the region. I,(z) is given by (3.4.41) I s ( z ) = (u,/2xhos) (E(,-)(L,z)Ec,.)(L, 7 ) ) . Using eqs. (3.4.35)-( 3.4.38), this becomes

+ 2rjo‘dz’ In steady state (z+

joz

dz’ IH(z, z’,

7,

z’)I2

).

(3.4.42)

these integrals can be carried out exactly to give

00)

(3.4.43) where the gain-coefficient g is the same as in the semiclassical treatment (eq. 3.2.33)) or the photon rate-equation treatment (eq. (3.1.22)). It may seem remarkable that this result for the Stokes intensity I , ( ~ o )is exactly the same as that found in the photon rate-equation treatment, eq. (3.1.26), with F = 1. However, this result just indicates that the rate-equation theory, when executed carefully, is perfectly valid for calculating average intensities in steady state. In the limit of low gain (gL 4 1) eq. (3.4.43) becomes I,(co) = ; A - ‘ r g L .

(3.4.44)

This can be seen, using eqs. (3.1.22) and (3.1.11), to be equal to I,(oo)= F-2NLI,

do dfl

- dfl,

(3.4.45)

which, since F = 1, is identical to the result (eq. (3.1.21)) found earlier for the spontaneous photon scattering intensity. Thus the present theory is seen to encompass both the semiclassical amplifier treatment and the spontaneous photon scattering treatment. A result not obtainable using the previous treatments is that of transient

228

IIK8 3

QUANTUM COHERENCE PROPERTIES OF RAMAN SCAlTERlNG

2.0E13 v)

B 1.5E13

Bg R

5 1.OE13

2

c

v

03213

O.OE13 -2

0 Time (units Y 1 )

2

Fig. 3.4. The transient Stokes intensity generatedby a Gaussian-shapedpump pulse versus time, calculated by using eq. (3.4.46). (a) Pump pulse, which has peak intensity I,; (b)-(d) Stokes intensity for several values ofthe Raman gain,& = 16 K, KZIOL/f, where the symbols are defined in the text; (b)gL = 150; (c)gL = 125; (d)gL = 100.

Raman generation. In this regime both molecular coherence effects and spontaneous scattering are important. In the transient (rz < g L ) and high-gain ( g L r z % 1) limits, eq. (3.4.42) becomes approximately (3.4.46) Plots of the transient intensity as a function of time are shown in fig. 3.4 for various gains, assuming a Gaussian-shaped laser pulse EL(z). To illustrate the transition from steady-state spontaneous scattering to steady-state stimulated scattering, eq. (3.4.42) is evaluated numerically for a constant laser intensity, switched on suddenly at z = 0. Figure 3.5 shows I,(z) versus z for a number of values of gL. It is seen that for g L > 1 the scattering changes from spontaneous to transient at roughly z = l / r g L , and then to steady state at roughly z = g L / r .

111, B 31

THEORY OF STIMULATED RAMAN SCA'ITERING

229

Fig. 3.5. Instantaneous Stokes intensity as a function of time after a constant laser intensity is turned on, for a number of different values of gz (number of steady-state gain lengths). Solid curves are obtained by numerical evaluation of eq. (3.4.42) with A = 1 cm2. Broken curves are the analytical approximations given by eq. (3.4.46) for small times and eq. (3.4.43) for long times. For very small gains (gz < 0.1) and/or very small times (log,,(Ti) < - 2) the Stokes scattering is spontaneous and agrees with eq. (3.4.45). For larger gains (gz 3 1) the scattering is seen to increase rapidly at a time r u (I/rgL) and then reach a steady-state stimulated intensity at a time T = (gz/T). (From RAYMER and MOSTOWSKI [1981].)

3.4.5. Steady-state power spectrum of SRS

The solution (3.4.38) of the linearized equations can be used to calculate the power spectrum of SRS in steady state, defined as

x (EL-)& z + s)Ec)(L,z)) ds , (3.4.47)

where the autocorrelation function is found using eqs. (3.4.35)-(3.4.37), (3.4.38) to be, for z+ co,

230

[IK8 3

QUANTUM COHERENCE PROPERTIES OF RAMAN SCATTERING

These integrals can be evaluated exactly to give (3.4.39)

The spectrum is normalized so that its integral over w is equal to Zs(co) in eq. (3.4.43). This result is identical in form to that found in the photon rateequation treatment (eq. (3.1.28)), again showing the validity of that treatment for describing steady-state scattering. At low gain the spectrum is Lorentzian with width r, whereas at high gains the spectrum becomes Gaussian, with a gain-narrowed linewidth of about T/(gL)'". Examples of the spectrum are shown in fig. 3.6. Note that the power spectrum is an ensemble-averaged quantity, defined for a stationary process. For short pulses of Stokes light the energy spectrum is a more appropriate quantity. The shape of the energy spectrum is found to fluctuate from pulse to pulse (MACPHERSON,SWANSONand CARLSTEN [ 1988a1).

-2

-1

0

1

2

(W-Ws)/T

Fig. 3.6. The steady-state Stokes power spectrum, showing gain-narrowing of the Stokes line. The curves are calculated from eq. (3.4.49), using the following parameters: (a)gL = (b)gL = 3; (c)gL = 15. As the gain-coefficientis increased, not only does the spectrum become narrower but also its shape changes from a Lorentzian to a Gaussian.

I I L 8 31

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THEORY OF STIMULATED RAMAN SCATTERING

3.4.6. Fluctuations of Stokes pulse energy As discussed in 3 3.3.2, the energy content of each Stokes pulse is expected to be a random quantity because of quantum noise. The present treatment provides a method to calculate accurately the probability distribution P( W) for pulse energy W, in the linear regime. Let us first discuss the probability distribution P ( I ) for Stokes intensity I, which also fluctuates. As in § 3.3.3, it is useful to associate with the Stokes field operator E&-)(z, z) an equivalent classical random process E(z, z), which has Gaussian statistics. The present case is more complicated because of the integrals over z and z in eq. (3.4.38) for the Stokes field operator. In the case of Stokes generation the first two terms in this equation do not contribute to the intensity, because they represent free (vacuum) fields. The third term can be written in discrete form Joz

dz’ H ( z , z’, z, O)&+(z’,0) N

c Hi#,

(3.4.50)

i

where Hi = (Az/AN)’/’H(z, zi,z, 0) ,

(3.4.51)

and the discretized atomic operator is (3.4.52)

47 E (ANAZ)”’&~(Z~, 0) , which has been normalized, so that, from eq. (3.4.35),

=

6,.

(3.4.53)

Here Az is the thickness of each spatial slice. The point of defining the 47 in this manner is that they can be shown to satisfy the Gaussian-momenttheorem:

(ri?,~?;.

*

* d f n d j j , ( i i 2 . * d j m >=

c P

<4?,4j,>

<4?24jZ> * *

9

(3.4.54) where the summation is over the n ! permutations of the dj variables. This result is discussed by GLAUBER and HAAKE[ 19781 and by LOUISELL [ 19731, p. 184. It is a generalization of eq. (3.3.13) to the case of many variables. Note that here df plays the role of a lowering operator for molecular vibration. We can thus conclude that, for the purpose of calculating normally ordered expectationvalues, the di(like the Bt(0)) can be replaced by equivalent classical random variables 4i,which are Gaussian and statistically independent. A

232

[III, B 3

QUANTUM COHERENCE PROPERTIES OF RAMAN SCARERING

similar argument can be made for the fourth term in eq. (3.4.38), where the Langevin operator f ( z , z) is composed of many reservoir oscillators. Thus we can conclude further that the Stokes field operator Eb-)(z, 2) can be replaced by an equivalent complex, Gaussian, random process E(z, 2). From this we can immediatelywrite down the probability distributionfor the complex field, 1 P(E, Z) = IE1* (3.4.55) ?I (Ek- ’(2)EV’ ( 2 ) ) (e&-),(z)E&+)(2))

),

where ( e(,-)(r)E(,.)(~)) is proportional to the mean intensity Z,(z) as given in eq. (3.4.41). Changing the variable to Z = (oS/2zho,) I El 2, we can write the distribution for intensity, (3.4.56) This result is similar to that found in the single-mode treatment in 5 3.3.3. We may now examine the more difficult task of finding the probability distribution P( W) for Stokes pulse energy W. Again the distribution P( W) is the Fourier transform of the characteristic function C,(<), given by

~ , ( t =) ( : exp(i<@): ) . This is similar to eq. (3.3.9), except that the single-mode photon number operator A has been replaced by the general pulse-energy operation @, defined by

1

m

fi = (A~,/2?1ho,)

dr&-)(L, z)&‘,+)(L,2 ) .

(3.4.57)

-m

The evaluation of the characteristic function C,(<) is complicated by the presence of the time integration in eq. (3.4.57). The calculation can be carried out by functional integration techniques (RAYMER,RZAZEWSKI and MOSTOWSKI[ 19821, RZABEWSKI, LEWENSTEIN and RAYMER[ 19821) or, equivalently, by using a Karhunen-Loeve expansion (RAYMER,WALMSLEY, MOSTOWSKIand SOBOLEWSKA [1985]). Here we will follow the latter approach, in which the electric field operator is expanded in a type of generalized Fourier expansion over a Set of mode functions Fk(7): (3.4.58) Since the mode functions are assumed to be orthogonal, the time-independent

111, B 31

233

THEORY OF STIMULATED RAMAN SCAlTERING

amplitude operators

h l are defined by (3.4.59) J - m

Then the energy operator is simply (3.4.60) k

The ‘y,(r) are made unique by requiring that the 6 , operators be statistically uncorrelated; that is, ( 6l S,) = I , h,, where 1, = ( 8,) is the average number of photons emitted into the “temporally coherent mode” Y,(z). Note that the field autocorrelation function can be expressed as

Multiplying this equation by Y,(z’) and integrating over z’ gives 00

21lhos

(E$-)(L, z)E‘,+)(L, 7’)) Y,(z’) dz’

=

I , Y l ( z ) . (3.4.62)

-a

This is an eigenvalue problem that uniquely determines the I , and Y,(z), of which there are an infinite number. The autocorrelation function appearing in the integral can be determined explicitly from the solution eq. (3.4.38) and the properties eq. (3.4.35)-(3.4.37). The special case of steady state is given in eq. (3.4.48). Since the autocorrelation function equals exactly zero outside the laser pulse, the integration limits in eq. (3.4.62) only need to be large enough to include the entire laser pulse. Some examples of the temporally coherent modes are shown in fig. 3.7. We will now evaluate Cw([),which can be written

c,(<)

=

< : n exp(itS18,) k

:) .

(3.4.63)

We just have shown that the Stokes field operator can be replaced by a Gaussian random process, which implies that the 6 , can be replaced by equivalent Gaussian random variables b,. Furthermore, since we know that the b, are uncorrelated; that is, ( b z b , ) = I,$&,, their Gaussian nature implies they are statistically independent. Thus the distribution for the set of b;s is (3.4.64)

234

QUANTUM COHERENCE PROPERTIES OF RAMAN SCAmERING

[IK5 3

:om .-c

g

I

0

+

iJ

1 , . . . , . . . I -20 0 20 40 -1 0 1 Time (units Fig. 3.7. Examples of the lowest order temporal coherence modes !Fk(~).These were calculated using eqs. (3.4.62) and (3.4.38), using a Gaussian-shaped pump pulse of duration sL, centered at z = 0. The conditions are: (a)-(d) rsL = 80,gL = 30; (e)-(h) rr, = 1,gL = 30. The eigenvalues, which are the average numbers of photons scattered into the corresponding mode, are: (a)0.471 x 10”; (b)0.152 x lOI3; (c)0.515 x 10”; (d)0.185 x 10”; (e)0.373 x lo6; (f) 0.103 x lo4; (g) 0.182 x 10’; (h) 120. ( Mer LI,WALMSLEY and RAYMER.)

r-’)

Now the task is easy. Since eq. (3.4.63) is in normal-ordered form, it can be evaluated simply as a classical average over the his, to give cW(8 =

1n

eXP(itbk*bk)P({bk})d2{bk)

k

=

n

1/( 1 - iyn,).

(3.4.65)

k

This is a generalization of eq. (3.3.17). The probability distribution is then found by taking the Fourier transform of eq. (3.4.65) to give

(3.4.66) j#k

235

THEORY OF STIMULATED RAMAN SCATI’ERING

1

-+

0.1

I .01 z .001

0

2

W/<W>

4

6 0

2 4 W/CW>

6

Fig. 3.8. Theoretical Stokes pulse energy distributions, calculated using the one-dimensional theory from eq. (3.4.66). The parameters are: (a) rr, = 10 and gL = 18; (b) Tr, = 30 and gL = 15. (From RAYMER and WALMSLEY[1984].)

The integer K is the number of discretization points used to convert eq. (3.14.62) into amatrix equation, which is then solved by standard numerical methods, to yield the I,. K is made sufficientlylarge (50 - 100) for convergence. It should be noted that this result has the same form as eq. (3.3.32). It also should be pointed out that the Karhunen-Loeve expansion method used here is identical in form to that used in classical coherence.theory(SALEH[ 19781). Figure 3.8 shows examples of P( W) calculated for different values of rzL. At very small values of T z L ( 4 g L )it is found numerically that one eigenvalue I is dominant. Therefore the average energy is ( F@) 11 I,,and P( W )is given approximately by (3.4.67) Thus we see that for small values of I‘z, (transient regime), the single-mode result (eq. (3.3.19)) is recovered. This is because in this case the field is temporally as well as spatially, coherent. HAAKE[ 19821 has observed that in this case the solution for the Stokes field eq. (3.4.38) can be approximated as

where the random field Q*(z,0) has been replaced by a single Gaussian random variable Q*(O) with zero mean and unit variance. This outcome implies that in the transient regime each Stokes pulse has the same temporal shape but a

236

QUANTUM COHERENCE PROPERTIES OF RAMAN SCAlTERING

m,B 3

random energy and phase. This representation leads immediately to the negative-exponential energy distribution (eq. (3.4.67)). On the other hand, in the steady-state regime I'zL becomes large, and many eigenvalues 1, become comparable. Then P( W) narrows and its peak moves toward the mean value ( W ) . This behavior can be understood in terms of the central-limit theorem of classical statistics, which states that when many independent random variables are added together the resulting variable has approximately a Gaussian distribution (see 3 3.3.4). In the present case the independent variables are, roughly speaking, the energies emitted during different coherence time intervals of length (gL)'I2/I'.Again, this type of behavior is well known in classical coherence theory (MANDELand WOLF [1965], MEHTA[ 19701, SALEH[ 1978]), strengthening the identification of the quantum process with an equivalent classical random process. In that context it is known that in the special case in which there are a certain number of nonzero eigenvalues A, which are all equal, P( W) in eq. (3.4.66) becomes the gamma distribution. This provides a simple analytical approximation for the actual P(W) (B~DARD, CHANGand MANDEL[ 19671, SALEH[ 19781, p. 193). 3.4.7. Temporalfluctuations of Stokes pulses In addition to fluctuations of the total energy of the Stokes pulse, the results of the previous section also imply pulse-to-pulse fluctuations of the temporal pulse shapes. This feature is most directly seen by replacing the field operator E$-)(L, z) in eq. (3.4.58) by the equivalent classical random process (3.4.69) E(L, 2) = ( ~ ~ ~ C O , / A U C, bz ) ' Y,(z), /~ k

where the mode amplitudes b, are independent, classical random variables with Gaussian distribution given in eq. (3.4.64). For each pulse the temporal modes Y,(z) are excited with a different set of random amplitudes { b z } which leads to a different and unpredictable pulse shape IE(L, z)I2 for each pulse. Using a random-number generator to simulate the b,, and using the mode functions Yk(z)from fig. 3.7, we have constructed some typical Stokes pulses, shown in fig. 3.9. The types of pulses that occur depend on the spectrum of the eigenvalues {A,}, since these determine the probabilities for the various temporal modes to be excited (see eq. (3.4.64)). The eigenvalues, in turn, depend on the values of the parameters rzLand gL. For small r z L / g Lthe eigenvalue A1, corresponding to mode Yl (z), is dominant, so that most of the pulses are structureless. As I'zJgL increases, higher order eigenvalues become larger, implying more structure on the resultant pulses.

THEORY OF STIMULATED RAMAN SCATTERING

237

-x; Time (ns) Fig. 3.9. Theoretical realizations of the random Stokes-pulse intensity, calculated from eqs. (3.4.68) and (3.4.64), using the parameters rr, = 80, rL = 5 ns, gL = 30, and L = 50 cm. Six temporally coherent modes, four of which are shown in figs. 3.7a-d, were used, with the eigenvalues in table 3.1 used to generate the random, complex amplitudes 6,. Each mark on the vertical axis represents 10” photons/second. These results should be compared with experimental measurements of the Stokes-pulse intensity shown in fig. 4.10. The single-peaked pulse shown in (a) is the most common, with the probability of observing pulses with multiple peaks decreasing as the number of peaks increases. (Atter RAYMER,LI and WALMSLEY[1989].)

It is important to understand that even when there are comparable probabilities for exciting several modes simultaneously, there is a nonzero chance to obtain a nearly pure excitation of one of the modes by itself. Using the statistical independence of the bk variables, it is easy to show that the probability to produce a Stokes pulse in which mode YJz) contains a fraction f or more of the pulse’s total energy W is given by Af-3

P(W, z f W ) = lim K+m

[-&+ 1

(1 - f )A,

K

C k#n

K

n

/#k, n

- Aj)

(3.4.70)

238

[III, B 3

QUANTUM COHERENCE PROPERTIES OF RAMAN SCATTERING

TABLE3.1 Temporal-mode eigenvalues, A,, corresponding to the mean number of photons in each of the temporally coherent modes Y,,(r), determined from eqs. (3.4.62) and (3.4.38). The number of significantly excited modes increases as rr JgL increases

r?L

gL

rrJ g L

1

50

0.02

0.374 (6)* 0.103 (4) 0.181 (2) 0.200 (1)

7

7

1

0.250 (3) 0.431 (2) 0.137 (2) 0.628 (1) 0.357 (1) 0.232 (1) 0.164 (1)

80

30

2.67

n

1 2 3 4 5 6 7 8 9 10

An

0.471 (13) 0.152(13) 0.515 (12) 0.185 (12) 0.697 (1 1) 0.275 (1 1) 0.114(11) 0.490 (10) 0.219 (10) 0.101 (10)

* The number in parentheses indicates the exponent of a multiplicative factor of 10.

Using the values of the eigenvalue 1, from table 3.1 for the case rzL = 80 and g L = 30, we find that the probability for the lowest order mode Y, (z) to contain at least 75 % of the pulse energy is 0.24,and the probability for the same fraction in the next mode !P2(z) is 0.017.Thus we see that there can be a significant probability to observe a Stokes pulse whose shape is almost entirely determined by a single temporal mode. The usefulness of the temporally coherent mode description is that it reduces the number of random variables which must be generated to simulate typical Stokes pulses. Whereas a simulation treating pt(z’,z) in eq. (3.4.38)as a random field would require a very large number of random values, the modes description requires only random complex amplitudes b, for those few lowest order modes that are significantly excited.

111,s 31

THEORY OF STIMULATED RAMAN SCATTERING

239

Another consequence of the random generated field eq. (3.4.69) is the pulseto-pulse variation of the pulse-energy spectrum, defined as the random function SW(4=

ljm

2

E(L, r)e-”dr/

.

(3.4.7 1)

-co

This can be written as (3.4.72) are the Fourier transforms of the temporal modes ‘Pk( r) and where the @k(o) can be called spectral modes. Experimental study of the spectral modes has been carried out by MACPHERSON,SWANSONand CARLSTEN[ 1988al. 3.4.8. Spatial fluctuations of Stokes pulses In the previous sections it was assumed that a one-dimensional wave equation is sufficient to describe the Stokes field when the Fresnel number, F = A/IZsL, is comparable to unity. It was argued qualitatively that in this case the generated field is approximately diffraction limited, and thus spatially coherent. Experiments showed, however, that the 1-D theory is inadequate for an accurate quantitative explanation of the energy distribution P(W) (RAYMER and WALMSLEY[19841). Deviation from a purely negative exponential (eq. (3.4.67)) was seen for energies W less than one tenth of the mean, as discussed in 5 4.1. It was found that a three-dimensional treatment of the problem was in better agreement with the experiments and that the Stokes field was predicted not to be fully spatially coherent even for a small Fresnel number (WALMSLEYand RAYMER[ 19861). The 3-D theory of Stokes generation starts from eqs. (3.4.29) and (3.4.25) for the operators &-)(r, t ) and &r, t). Again, the equations are linearized by setting the population inversion ~ ( rt), equal to - 9. Then the equations have the followingsolution in the paraxial, high-gainlimit, where r is some point in the output face (see fig. 3.1) at z = L (RAYMER,WALMSLEY, MOSTOWSKI and SOBOLEWSKA [ 1985]),

EL-)(, 2) =

j + j S,”

d3r’ K(r, r’, T, O)&t(r’, 0) d3r’

K(r, r’, r, z’)ft(r’, r’) ,

(3.4.73)

240

QUANTUM COHERENCE PROPERTIES OF RAMAN SCATTERING

[III, B 3

where the free-field terms have been dropped, since they do not contribute, and the r’ integral is over the uniformly pumped, cylinder-shaped interaction volume, with

K(r, r ’ , z,

ik,

7’)= - H ( L , z’, 2Z L

z,

2’) exp

I

(3.4.74)

where H(L, z ’ , z, 7’)is given in eq. (3.4.39).The exponential factor depends on the radius vector p = ( x , y ) in the output face and is the paraxial approximation to a spherical wave originating at z = 0. Thus the solution is the same as in the 1-D case, except that the source Qt(r’, 0) is now distributed throughout the circular input face at z = 0, and each point on the input face emits a spherical wave. In this approximation the gain medium acts like an active spatial filter; only rays that exit the output plane (z = L) within the circular output face, with area A, will experience high gain and will dominate the emission. This solution is only valid for F 3 1, since it neglects reflections from the cylinder sides, which can cause waveguiding effects for a smaller Fresnel number (MOSTOWSKI and SOBOLEWSKA [ 19841). In this approximation the mean intensity of the Stokes field at the output face is found to be Z,(P, z ) = F21,(z)

(3.4.75)

Y

where Z,(z) is the mean intensity in the I-D theory (eq. (3.4.42)). This result is as expected, since F2 is approximately the number of spatial modes at wavelength I, that fit into the cylindrically shaped volume and propagate within the forward solid angle A/Lz. To study the spatial coherence properties of the Stokes field, we will first calculate the two-point, two-time correlation function at the output face, defined as

2 Gz(z1,

72) GJPl P2) 9 Y

(3.4.76)

where the factorization into time and space parts results from using eqs. (3.4.73), (3.4.31), and (3.4.27) in the high-gain limit. The temporal correlation function G,(z,, z2) is found to be just F2 times the I-D correlation function, whereas the spatial correlation function is found to be

(3.4.77)

111, s 31

THEORY OF STIMULATED RAMAN SCATTERING

24 1

where J , ( x ) is the Bessel function of order 1 and a is the radius of the cylinder; that is, A = nu2. This result is the well-known measure of spatial coherence of light emitted incoherently from a disk with radius a in the plane z = 0 and observed in the plane z = L (BORNand WOLF[ 19751, 5 10.3 and 3 10.4). It is familiar in the context of starlight, which is incoherent at the source and is partially coherent at the Earth; that is the reason why stars twinkle. The coherence area of the light at z = L can be seen from this result to be approximately na2/F2,or (I,L)2/xaZ,which shows that the farther the light travels from its source the more spatially coherent it becomes. Thus F Z is the number of coherence areas that fall within the output face at z = L. If F 1: 1, approximately one spatially coherent mode can pass through the output face and be amplified strongly. If F 9 1, many coherent modes can pass through the output face and the light will not be spatially coherent. An interesting question that naturally arises is: Can one think of the emitted field as being composed of a sum of coherent but statistically independent modes? The answer, within this linearized, high-gain theory is “yes”. Photons within a given coherent mode are added with the same phase, but the phases of the modes are random and independent. The coherent modes turn out not to be circular regions with area na2/F2,but they each cover the entire output face. They are found using a Karhunen-Loeve expansion analogous to that MOSTOWSKI and employed in 3 3.4.6. (For details see RAYMER,WALMSLEY, SOBOLEWSKA [ 19851.) The Stokes field operator (eq. (3.4.73)) is expanded in the z = L plane as (3.4.78) where the spatially coherent modes Gn(p)are assumed to be orthonormal in p over the domain 0 < IpI < a. The creation operators ci,?(z) are assumed to be statistically independent, and this leads to

where fin is the eigenvalue of the following equation r

(3.4.80) which is analogous to eq. (3.4.62). The eigenvalue fin is equal to the fraction of light emitted into mode an(p). The eigenfunctionscan be labeled by two integers

242

QUANTUM COHERENCE PROPERTIES OF RAMAN SCATTERING

[III, 8 3

Fig. 3.10. The absolute value of the radial eigenfunctionsgik)(p) for the case F = 1, calculated by using eqs. (3.4.80) and (3.4.81). Each function is plotted so that its maximum value is unity. (From RAYMER, WALMSLEY, MosTOWSKI and SOBOLEWSKA [ 19851.)

k and 1, and can be written @ik)(p)= eikegjk)(p)exp [ - iFp2/a2],

(3.4.81) where p and 0 are polar coordinates, and k = 0, f 1, f 2, .. . and 1 = 1,2,3, .. .. Examples of the radial eigenfunctiongjk)(p)are shown in fig. 3.10 for Fresnel number F = 1. In this case we find pio) = 0.630, jI{*') = 0.161, and 8; * 2, = 0.019; the rest are smaller. Thus 63 % of the emission goes into the single mode g'p)(p), which means that the light is fairly spatially coherent, as expected for F = 1. It is noteworthy, however, that 37% of the light goes into other modes, meaning that the l-D theory is not quantitatively accurate, even for F = 1. Figure 3.1 1 shows examples of the radial eigenfunction for F = 3. In this case we find B { O ) = 0.11 1, p{ * ')= 0.109, @{* 2, = 0.099, pio) = 0.092, fif * 3, = 0.07 1, /3$ * ')= 0.048, /I*{4, = 0.035; the rest are smaller. It should be noted that for F = 3 there is significant excitation of modes with radial nodes.

111,s 31

THEORY OF STIMULATED RAMAN SCATTERING

243

Fig. 3.11. The absolute value of the radial eigenfunctions gik)(p) for the case F = 3. (From RAYMER, WALMSLEY, MOSTOWSKI and SOBOLEWSKA [ 19851.)

When many modes are excited, with random phases, in a single pulse, a randomly modulated pattern results from the interference of the modes. Whether or not this speckle pattern can be observed when the Stokes intensity is temporally integrated over the pulse depends on the relative temporal coherence of the different spatial modes. If the pulse is sufficiently short (zL < gL/I'), the phase of each spatial mode will not vary during the pulse, although it will have a random value. In this case the speckle pattern will be stationary during the pulse, with a speckle size of the order of naZ/F2.On the other hand, if the pulse is longer than gL/I', the phase of each mode will fluctuate randomly during the pulse and the speckle pattern will change rapidly and wash out. It is now easy to see how the presence of spatial structure on the Stokes pulse affects the fluctuations of the total pulse energy W. A calculation similar to that leading to eq. (3.4.66) in the 1-D case leads in the 3-D case to K

P( W ) = lim K*N-rm

N

C C c ~ K, T exp( - w/fin&), k=l

(3.4.82)

where (3.4.83) m#n

Ifk

244

QUANTUM COHERENCE PROPERTIES OF RAMAN SCATTERING

[III, 0 3

{

.i ...

.i.i. Ill

Ill

I w/<w>

2

OO

I

w/<w>

2

OO

I w/<w>

Fig. 3.12. The normalized probability density function P ( W / ( W)), of the Stokes pulse energy W,for various Fresnel numbers F,collisional dephasing rates r,and Raman gain-coefficientsgL. The ratio TtJgL, where zL is the pump pulse duration, is also given. This quantity provides a rough measure of whether the scattering occurred under transient conditions. As either F or TzJgL increases, P ( W / ( W)) narrows and peaks closer to its mean value, indicating that the Stokes pulse energy fluctuations have decreased. This is associated with a decrease of the spatial MOSTOWSKI and and/or temporal coherence of the Stokes light. (From RAYMER,WALMSLEY, SOBOLEWSKA [1985].)

where N is the number of discretization points used to solve eq. (3.4.80) numerically. The I ; are related to the I, in eq. (3.4.66) simply by I ; = F2&. The product /?,,A; is the average amount of energy emitted into the spatialtemporal mode @,(p) Yk(z).In the case of an axially symmetrical pump field, as here, some of the eigenvalues are degenerate. Then eq. (3.4.82) and (3.4.83) must be modified to account for this occurrence (MOSTOWSKIand SOBOLEWSKA [ 19841, WALMSLEY [ 19861). Examples of P( W )are shown in fig. 3.12 for different values of F, I%,, and gL. When F = 1 and rzL/gL < 0.02, a single spatial-temporal mode is dominant. Nevertheless, a strong deviation from negative exponential is seen for W less than one-half the mean because of the weak excitations of other spatial modes (predominantly @{* ')(p)), even for F = 1. This result is significantly different from that predicted by the 1-D theory, shown in fig. 3.8. When F = 1 and Z'zJgL is large, a single spatial mode is dominant and many temporal modes are excited; that is, the pulse is not temporally coherent. Then, as expected, P ( W ) narrows. Again, this can be interpreted in terms of the central-limit theorem. It should be mentioned that the 3-D model presented here is based on a

111, s 41

EXPERIMENTS ON STIMULATED RAMAN SCATTERING

245

rather unrealistic pump laser beam shape - a uniform cylinder. Experiments using a Gaussian beam may be expected to yield qualitatively similar behavior, but to differ in quantitative detail. Results of experiments are discussed in 4.1. Other methods have been developed for treating nonuniform beams. PERRY, RABINOWITZand NEWSTEIN[ 19831 have used a Hermite-Gaussian mode expansion to treat Raman amplification but have not treated quantum fluctuations. This method was generalized to include quantum fluctuations by STRAUSS,OREGand BAR-SHALOM [1988] in the limit of a large Fresnel number. Another treatment, which is valid for very large Fresnel numbers, uses numerical ray tracing to determine the Green’s function for the Maxwell-Bloch equations (HAZAKand BAR-SHALOM [ 19881).

5 4. Experiments on Quantum-Statistical Aspects of Stimulated Raman Scattering As discussed in the previous section, there are several manifestations of the quantum initiation of Stokes scattering. The experimental evidence for these macroscopic effects of quantum noise is presented in this section. A characteristic feature of spontaneous scattering is its randomness. Thus most experiments involve a determination of the statistics of some property of the Stokes pulse, such as its energy, temporal shape, spectrum, or delay with respect to the pump laser pulse, which are easily detected for many-photon pulses. Therefore, although the experiments are in principle straightforward, considerable attention must be paid to making each repeated measurement under the same conditions.

4.1. STOKES-PULSE-ENERGY FLUCTUATIONS

Fluctuations in the Stokes pulse energy Warise because, whereas the Raman gain may be considered constant over some finite duration and spatial extent, the generation of the fist Stokes photon by spontaneous scattering is random, in time and position, and thus the total energy of the Stokes pulse will be random also. These fluctuations are characterized in the theory by the probability density function P ( W ) , and it is this quantity that has been measured experimentally. The pragmatic definition of a probability density in an experiment is the obvious one; that is, we conduct an ensemble of N experiments and measure the number of times, FI w,that the Stokes pulse has an energy within

246

QUANTUM COHERENCE PROPERTIES OF RAMAN SCATTERING

54

A W of W. Then P ( W ) = lim N-.m

n

1 N AW

2 -

.

The ensemble is generated by repetitive experiments in which a reproducible input laser pulse generates a Stokes pulse, whose energy is then measured. The experimental distribution P( W) may be used to estimate the true P( W) by any of the various algorithms well-known to statisticians. The requirement of a well-characterized input pulse is important. Gain fluctuations due to a fluctuating pump pulse energy can easily mask the fluctuations due to the spontaneous scattering. For example, in the steady-state case eqs. (3.1.22) and (3.1.23) show that the relative change of the Stokes intensity is given by

-

For the gains found typically in these experiments (gL 20), a 5 % change in laser intensity produces a 100% change in the Stokes intensity, which would obviously completely mask the quantum fluctuations. Furthermore, the effect of a broad-band, multimode pump on the statistics of the Raman generator has not been examined here, where we consider only cases in which a laser that was close to transform limited was used as a pump. Several experiments satisfying the preceding conditions have been carried out, the earliest being WALMSLEYand RAYMER [1983] and FABRICIUS, NATTERMANN and VON DER LINDE[ 19841. In these experiments a pulse from a frequency-doubled Nd : YAG laser was incident on a cell containing molecular hydrogen. The Qol(1) vibrational transition of this molecule was used as the Raman transition because it has a well-characterized linewidth and gain coefficient and has low dispersion at visible wavelengths. Both the pump pulse energy and transverse beam profile were monitored carefully to ensure that there were no hot spots which might alter the Raman gain, which again would mask the quantum fluctuations. Measurements of the pump pulse spatial profile are also required to determine the Fresnel number of the interaction volume, which must be well defined so that the experimental results will be comparable with theory. and RAYMER A typical experimental setup is shown in fig. 4.1 (WALMSLEY [ 19831). A single-mode, Q-switched Nd : YAG laser that produced pulses with duration zL N 10 nsec was used. At 12 atm pressure the Qo,(l) transition linewidth is about 0.64 GHz (BISCHELand DYER[ 19831) so that rz, N 19,

111, I 41

241

EXPERIMENTS ON STIMULATED RAMAN SCAITERING

SF LASER

PD2 DC

F PDI MC

CLC ~

PMT -

Fig. 4.1. Apparatus for the determination ofthe distribution of Stokes pulse energies, P ( W ) .The output of a frequency-doubled, single-mode Nd : YAG laser was spatially filtered (SF) and focussed by a lens (L,) into a cell containing molecular hydrogen. The generated Stokes light was filtered using a prism (P) and filters (DC, F, MC) and detected by a photomultiplier (PMT). The laser pulse was also detected (PDI) after passing through an optical delay line (OD, L3).The energy of both pulses was measured by an integrator (B)and stored in a computer (1C).The pump-beam spatial profile was also measured using a detector array (R) and a transient digitizer (TD). A discriminator (PD2, PD3, D) was used to suppress data acquisition if the pump pulse exhibited any temporal structure, indicating that it was not single-mode.

>-

t m

a

m

.01

8a

0

2

4

6

6

0

2

4

6

8

1

-as I

>-

0.1

k 4

;a 8a

.01

.001 0 2 4 6 6 STOKES ENERGY-W

0 2 4 6 8 STOKES ENERGY-W

Fig. 4.2. Experimental Stokes-pulse-energy distributions for various laser powers: (a) at the The slopes and positions lowest laser energy WE;(b) at 1.01 Wt;(c) at 1.025 WE;(d) at 1.04 Wf. of the lines are theoretical exponential distributions with the means determined from the first five and RAYMER [1983].) points in the corresponding figure. (From WALMSLEY

248

QUANTUM COHERENCE PROPERTIES OF RAMAN SCATTERING

[III, 8 4

where r is the halfwidth of the Raman transition in radians per second. By increasing the gas pressure, it is possible to make this parameter much larger, and values up to rzLN 85 have been used in subsequent experiments. The gain coefficient was gL N 10, so that this experiment was in the regime intermediate between transient and steady-state scattering. Also, since the Fresnel number was small (F N 0.8), the shape of the Stokes-pulse-energy distribution P ( W ) was found to be approximately exponential, indicating that only a small number of coherent modes were excited. Some results of these early experiments are shown in fig. 4.2. There were some indications in these experiments that the one-dimensional theory was inadequate to describe the distributions, since P ( W ) was found to be smaller than expected for values of the Stokes pulse energy much less than the mean. This shift of the most probable energy away from zero is indicative of the excitation of several coherent modes - a feature not predicted by the one-dimensional theory for the transient regime. Experiments in the extreme transient regime (Z3, I! 0.38 4 g L ) were performed by FABRICIUS,NATTERMANNand VON DER LINDE[ 19841 and by RAYMER and WALMSLEY[ 19841. In these experiments mode-locked Nd : YAG lasers producing pulses of less than 100 ps duration were used as the pump. The pulse-to-pulse energy fluctuations were monitored, and a resolution of at least & 1% was maintained. Figure 4.3 illustrates measurements of

10' 10-2 t.

t 2

m i

3 lo-' lo+ 10-~

0

0.1 0.2 0.3 0 0.1 0.2 0.3 0.4 STOKES ENERGY W, [pJ]

Fig. 4.3. Examples of measured Stokes-pulse-energydistributionsP( W )in the transient regime, Tr, = 0.085, in molecular hydrogen. The shaded regions represent statistical errors. The pump pulse energies are: (a) 187 pJ; (b) 196 pJ; (c) 204 pJ; (d) 214 pJ. The lines are the results of theoretical calculations using the one-dimensional theory. Note the deviation from exponential behavior at small Stokes energies in (d), indicating the inadequacy of the one-dimensionaltheory even at small Fresnel numbers. (From FABRICIUS, NATTERMANNand VON DER LINDE [1984].)

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EXPERIMENTS ON STIMULATED RAMAN SCA'ITERING

249

P ( W ) in this regime for a Fresnel number of about unity. In both sets of experiments, the resulting P( W) showed exponential behavior for energies greater than the mean energy, and also showed deviation from an exponential distribution at small Stokes energies. Since, again, the Fresnel number was small, this result gave further evidence for the inadequacy of the one-dimensional theory. It is important to realize that the fluctuations observed in the extreme transient regime are purely quantum mechanical in origin, and the statistics are not modified in any way by the effects of collisional dephasing of the phonon during the pulse. Furthermore P( W) is approximately exponential for a range of TzLfrom 0.01 to 1. The distribution of Stokes pulse energies is a sensitive test of the dephasing collisions because the most probable energy is a function of the number of temporally coherent modes excited. Since the distribution did not change significantly over a wide range of rzL,these transient regime results are well described by the coherent-mode picture, which predicts that only a single coherent mode will be significantly excited if rzLis such that the scattering is in the transient regime. 1.o

0.5

2-

;:

-a3-z 0.5 \

>

t

i 0.0 a 1.0

P

0.5

0.0 0

1

2 3 4 0 1 2 STOKES ENERGY W + W p

3

4

Fig. 4.4. Examples of measured Stokes-pulse-energydistributionsP( W) in the transient regime, rrL = 0.38, in molecular hydrogen, for several Fresnel numbers, F. The distributionsshow the characteristic narrowing around the mean energy as the Fresnel number increases. This is associated with a decrease in the spatial coherence of the Stokes light. (From NATTERMANN, FABRICIUS and VON DER LINDE[1986].)

250

QUANTUM COHERENCE PROPERTIES OF RAMAN SCA'lTERING

[III, § 4

As the Fresnel number F of the interaction region is increased, with rzL such that the experiment still remains in the transient regime, the pulse-energy distribution tends to become more symmetrical around the mean value. Ultimately the distribution becomes Gaussian, in the limit that F % 1. This has been and VON DER LINDE[ 19841, who demonstrated by FABRICIUS, NAITERMANN found that P( W) was approximately a Gaussian when F was about 22. Results of their experiments are shown in fig.4.4. They concluded that the onedimensional theory is valid for energies greater than the mean, for Fresnel numbers up to about F = 4. At larger Fresnel numbers than this the distributions peak near the mean energy. These authors used a gamma distribution to model their data, as discussed in 8 3.4.8. However, a comparison with the experimental results indicates that the distribution converges to a Gaussian more slowly than expected from this model. The results can be explained in terms of the coherent-mode model by noting that, although there is still only one temporal mode excited, there are more and more spatial modes excited as F increases. In the limit that F % 1 the central limit theorem applies and P( W) converges to a Gaussian no matter what the underlying statistics of each mode

I.o

I

h

$ \

5 0.1 n

0.0I

0

I

2 W/<W>

3

4

Fig. 4.5. Experimental and theoretical Stokes-pulse-energy distributions, showing the effects of the finite transverse dimension of the interaction region. The various symbols are experimental measurements and the solid lines are best-fit theoretical plots calculated using eq. (3.4.82). The experimental parameters were: (i) F = 0.27, rTL = 19, and gL = 15; (ii) F = 1.2, r ~=,13, and gL = 13; (iii)F= 3.8, rTL = 12, and gL = 15. The theoretical plots were calculated using a Gaussian pump intensity pulse shape, and the parameters: (i)F = 0.273, rtL= 18.9, and gL = 25.1; (ii) F = 1.23, rzL= 13.4, and gL. = 22.1 ;(iii) F = 3.5, r ~=,12.3, and gL = 24.6. In all cases the photon conversion was less than lo-' and there were more than lo6 Stokes photons per pulse. (From WALMSLEYand RAYMER[1986].)

111, I 41

EXPERIMENTS ON STIMULATED RAMAN SCATTERING

25 1

are. (In this case each mode individually still retains the exponential distribution of energy.) To make a detailed comparison of the three-dimensional theory of $3.4.8 to experiment, further experiments were performed by WALMSLEYand RAYMER [ 19861 in which the pump pulse duration was 10 ns, the ratio rz,/gl was kept constant at a small value (1.6), and the Fresnel number was varied between 0.3 and 3.8. Results from these experiments are shown in fig. 4.5. The characteristic narrowing of the distribution, the approach of the most probable energy to the mean energy, and the change in shape from exponential to Gaussian are all indications that many spatially coherent modes are excited. An analogous argument applies in the case in which the Fresnel number is kept small and the pump pulse duration is made large. As rz, is increased beyond gL,several temporally coherent modes are excited. Thus P( W )begins to look less exponential, and eventually as rzL % gL,the distribution becomes Gaussian and is peaked at the mean, even when F < 1. This behavior has been confirmed by experiment, as shown in fig. 4.6.In these experimentsthe interactionregion was a pencil-like volume with F N 0.3, so that only a single spatial mode was

.o

I

-

5 \

2 0.1 a

0.0 I 0

I

2 W/W>

3

Fig. 4.6. Experimental and theoretical Stokes-pulse-energy distributions, showing the effects of collisional dephasing. The various symbols are experimental measurements and the solid lines are best-fit theoretical plots calculated using eq. (3.4.82).The experimental parameters were: (i) r7, = 19 and gL = 15;(ii) T7, = 49 and gL = 17;(iii) r7, = 85 and gL = 9.4.In all cases the photon conversion efficiency was less than lo-’ and there were more than lo6 Stokes photons per pulse. The Fresnel number for all experiments was F = 0.27.The theoretical plots were calculated using a Gaussian-shaped pump intensity and the parameters: (i) r7, = 18.9 and gL = 25.1;(i) rtL= 48.7 and gL = 28.8;(iii) r7L= 85.2 and gL = 15.8.For all the theoretical plots the Fresnel number was F = 0.273.(From WALMSLEYand RAYMER [1986].)

252

QUANTUM COHERENCE PROPERTIES OF RAMAN SCATTERING

[IK8 4

significantly excited. As the ratio rz,/gL was increased from 1.6 to 11, the measured distribution changed from an exponential-like shape to a Gaussianlike shape, indicating that many temporally coherent modes were excited. This observation can be explained as follows. The number of temporally coherent modes excited is approximately given by the number of coherence times of the Stokes light that occur during the pump pulse. Thus the ratio rz,/gL is approximately proportional to the number of temporal modes excited. On the other hand, the number of spatial modes excited is given approximately by the number of coherence “areas” (that is, the regions over which the modulus of the complex degree of coherence is not significantlydifferent from unity) within the cross-section of the interaction volume. It may be shown that this number is proportional to F2.Thus the number of spatial modes excited increases by an order of magnitude when F increases by a factor of about three, whereas a comparable increase in the number of temporal modes requires an order of magnitude change in the parameter rz,/gL. A comparison of theory (eq. 3.4.82) to the experiment using no free parameters indicates that the Stokes should be less coherent than was actually observed, in the sense that P(W) should be narrower, more Gaussian in shape, and have a larger most probable energy. Theoretical fits of P ( W ) to the experimental data, using the gain as a free parameter, show that an increase of the gain of 70% over that actually measured is enough to produce good fits. The solid lines in figs. 4.5 and 4.6 indicate fits of the theory to the experimental results. The discrepancy may be partly due to an uncertainty in a precise value of the gain, but it is more likely due to the restricted conditions under which the theory applies; that is, the interaction volume is assumed to be a uniformly pumped cylinder, with a medium which is homogeneously broadened. The use of a beam with a Gaussian spatial profile makes it difficult to calculate the electric field distribution of the spatially coherent modes and their degree of excitation. Some progress has been made in this regard in the limit of large Fresnel numbers (HAZAKand BAR-SHALOM [ 19881, STRAUSS,OREGand BAR-SHALOM [ 19881). Furthermore, there are some subtle questions about the validity of the theory for small Fresnel numbers (MOSTOWSKIand SOBOLEWSKA [ 19841) and the factorization of the field correlation function into spatial and temporal parts, as was done in eq. (3.4.76). Ignoring these effects may distort the shape of the calculated P( W), although this question is not yet resolved. Although in all the experiments described thus far the Stokes pulse has contained a large number of photons (between lo6 and so that the experiments were performed in a regime of stimulated scattering rather than

1 1 ~ 41 8

EXPERIMENTS ON STIMULATED RAMAN SCAITERING

253

spontaneous scattering,the gain was not so large as to be saturated. If the pump energy is increased, however, a regime is reached in which the Stokes gain becomes saturated because of the depletion of the pump. It is well-known that the output of a saturated amplifier has reduced fluctuations compared with its input, and a similar effect has been observed in a saturated Raman generator (WALMSLEY,RAYMER,SIZER,DULING and KAFKA [ 19851). In this case the Stokes-pulse-energy fluctuations were dramatically decreased as the conversion efficiency of the input pump was increased. Measurements were again made using the Qo,(l) transition in hydrogen in the transient regime (rz, N 0. l), and they are illustrated in fig. 4.7. For low pump powers (energy conversion efficiency q 10- ’) P( W )was an exponential, characteristic of the

-

,

I

1

2

w/<w>

Fig. 4.7. Experimental and theoretical Stokes-pulse-energy distributions for a small Fresnel number as a hnction ofenergy conversion e5ciency. (a) Experimental measurement (points) and a simple exponential (dashed line). The mean Stokes energy is about lO-”J, thus the gain is unsaturated. (b) Experimental measurement in the case of saturated gain, where the mean energy is about J. (c) Theoretical prediction of LEWENSTEIN [1984] for the same experimental parameters as those in (b). Both (b) and (c) show the stabilization of Stokes pulse energy in the RAYMER, SIZER,DULING and KAFKA [1985].) nonlinear regime. (From WALMSLEY,

254

QUANTUM COHERENCE PROPERTIES OF RAMAN SCA'ITERING

[III, 8 4

unsaturated transient regime. For high conversion efficiencies ( q N 0. l), however, there was a striking narrowing of the distribution and a shitt of the most probable energy to the mean energy. Note that the shape is not Gaussian, however, in contrast with the changes observed in P(W )for purely unsaturated scattering. The distribution is asymmetrical about the mean energy. The narrowing in the saturated case does not arise from the excitation of many modes; rather, a single spatial-temporal mode is excited. There is good agreement between the measured distribution and a numerical simulation made by LEWENSTEIN [ 19841 for similar experimental parameters. Similar results were found by GRABCHIKOV, KILIN,KOZICHand IODO [ 19861, who made a more extensive study of the transition to the saturated-gain regime. Their results are shown in fig. 4.8. In their experiments the same

... .. Fig. 4.8. Examples of measured Stokes-pulse-energydistributions P( W )for the transient regime, rzJgL N 1, in molecular hydrogen, as a function of the energy conversion efficiency of the pump laser pulses q. These measurements were made using a broadband pump, but because the scattering took place in the transient regime, they show trends similar to those using a narrowband pump (see fig. 4.7); that is, the Stokes pulse energies stabilizerapidly as the nonlinear regime of scattering is approached. The conversion efficiencies are: (a) q = 1%; (b) q = 4%; (c) q = 6%; (d) q = 46%. (From GRABCHIKOV, KILIN,KOZICHand IODO [1986].)

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EXPERIMENTS ON STIMULATED RAMAN SCARERING

255

vibrational transition in hydrogen was excited using both a broad-band and a narrow-band pump laser. The shape of P( W )changed from exponential to a distribution peaked at the mean energy, as the conversion efficiency was increased from q = 0.01 to q = 0.46. The distribution first “flattened out”, so that all energies less than the mean energy had approximately equal probability and then began to “peak” around the mean energy. Although no explicit comparison of these results with theory has been made, there is a qualitative agreement of the measured P( W) With the theoretical results Of TRIPPENBACH and R Z ~ Z E W S K I[ 19851. 4.2.

TEMPORAL AND SPATIAL INTENSITY FLUCTUATIONS

The fluctuations in the total Stokes pulse energy can be viewed as arising from the random distribution of the initially scattered photons into the coherent modes. Since each of the temporal modes has a different time development, this random population of the modes on each successive pump pulse leads also to fluctuating Stokes pulse intensity. A similar argument holds for the random excitation of the spatial modes. If the transverse beam profile of the generated Stokes light is measured in the transient regime, the Stokes beam at the output of the interaction volume exhibits a speckled pattern because of the interference of the electric fields of the different spatial coherence modes. This speckle pattern fluctuates from pulse to pulse in such a way that the ensemble average yields a pulse energy at any small part of the beam which is just the incoherent sum of the energies of each of the modes at that part. By measuring the correlation function of the speckle intensity, the correlation function of the Stokes field may be deduced, assuming it has Gaussian statistics. Of course, if the Fresnel number is small, only one or two modes are excited and there is a high probability of observing a smooth intensity profile each time the laser is fired. The formation of speckle patterns on the Stokes beam has been observed in large-Fresnel-number Raman generation (HENESIAN, SWIFTand MURRAY [ 19851) and has been studied in detail by Kuo, RADZEWICZand RAYMER [ 19871, who have examined the shot-to-shot fluctuations of the Stokes beam spatial profile. Some examples of the Stokes speckle are shown in fig. 4.9. Similar speckle patterns produced in amplified spontaneous emission have been studied by KRAVISand ALLEN[ 19771. Obviously alarge energy is required to pump a large interaction volume, and a mode-locked, Q-switched, and cavity-dumped Nd : YAG laser generating 300 pJ, 100 ps pulses was used as an amplifier seed for producing the high-power pump pulses. The short pulse

256

QUANTUM COHERENCE PROPERTIES OF RAMAN SCATTERING

[IKB 4

Fig. 4.9. Single-shotimages ofthe Stokes beam leaving the hydrogen cell, for two different values of Fresnel number F. (a)-(d)F = 2.5; (e)-(h) F = 6.6. The variations from shot to shot are due to quantum noise. For the larger Fresnel number a random speckle pattern is observed on each shot. (After Kuo, RADZEWICZ and RAYMER [1987].)

duration ensured that the experiments were in the transient regime - an important requirement for observing the speckle. In the case of unsaturated gain the spatially coherent modes are qualitatively similar to those calculated using the theory described in 3.4.8, which assumes a spatially uniform pump intensity distribution. A quantitative description for Gaussian beams has been developed only for large Fresnel numbers, as discussed earlier in relation to the Stokes pulse energy distributions (HAZAKand BAR-SHALOM [ 19881, STRAUSS,OREGand BAR-SHALOM [ 19881). An important difference between the spatially uniform pump and Gaussian profile pump cases is that in the latter the gain at the center of the beam is larger than that at the edge of the beam, which leads to transverse spatial narrowing of the Stokes beam as it is amplified. In the case in which rzLB gL, the speckle is “washed out”, even when F is large. This is because each “coherence time”, defined by (gL)’/’/r,contains, in effect, an entirely independent distribution of temporal-mode amplitudes and phases. Adding up the contribution from independent coherence times effectively adds up several independent speckle patterns, which is like producing a small ensemble of independent experiments within each pump pulse, and thus the speckle pattern is averaged out. In the case of a depleted pump pulse the effective interaction volume becomes shorter than the length of the medium, leading to a larger Fresnel

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EXPERIMENTS ON STIMULATED RAMAN SCATTERING

251

number, and consequently larger Stokes divergence angle (CARLSTEN, RIFKIN and MACPHERSON [ 19861, BOBBSand WARNER[ 19861). The nature of the speckle pattern under conditions of pump depletion is still not well understood. The temporal fluctuations of the Stokes pulse intensity have also been LI and WALMSLEY [ 19891). The pulse shape studied experimentally (RAYMER, varies randomly from pulse to pulse for an interaction volume with small Fresnel number. In general, the shape of the Stokes pulse is found by linear combination of all the temporally coherent modes with random weights, but

0 1 2 2 Time (nsl Fig. 4.10. Examples of the random Stokes pulse shapes observed from a Raman generator. The plots have all been scaled arbitrarily to best illustrate the pulse shapes. The experimental parameters for all the plots are rrL*Y 80, tL = 4.8 ns, gL = 32.6, and L = 1 m. These experimental pulse shapes should be compared with the pulse shapes predicted using the coherent modes theory, which are shown in fig. 3.9 and are selected from a similar statistical sample and arranged to show the similarity. The frequency of occurrence of each of the shapes decreased from (a) to (h), the single-peaked pulse shape being the most common by far. (From RAYMER, LI and WALMSLEY[1989].) 0

1

258

QUANTUM COHERENCE PROPERTIES OF RAMAN SCA’ITERING

[III, 5 4

there is a finite probability that a reasonable fraction of the pulse energy will be in a single mode, so that the intensity of the Stokes pulse is nearly that of a “pure” temporally coherent mode. In these experiments a single-mode, @switched Nd : YAG laser was incident on a cell containing molecular hydrogen at 100 atm, so that rzL = 80, and since gL was about 30, the scattering was in the steady-state regime. The generated Stokes light was recorded by a streak camera and a multichannel analyzer. Some typical Stokes pulse shapes are shown in fig. 4.10. Some of these shapes are particularly simple and similar to the theoretical realizations shown in fig. 3.9. The realizations shown in fig. 3.9 were selected from a sample of about 100 random pulse shapes. The ensemble was therefore of a size similar to that from which the experimentally observed pulse shapes offig. 4.10 were taken. Figures 4.10 and 3.9 are arranged to facilitate comparison between experiment and theory. It should be noted that, although none of the pulses has exactly the shape of the pure mode intensities, they show that it is possible to have a large fraction of the total pulse energy concentrated in a single coherent mode. Furthermore, the experiments showed that this was a reasonably common occurrence. For these experimental conditions the probability that a single mode (say !PI(z)) in the case of fig. 4.10a will be excited predominantly can be calculated using eq. (3.4.70). Taking a “single mode” to mean that more than 70% of the pulse energy is in that mode, the predicted probability of observing a shape similar to that in fig. 4.10a is 0.29, and higher order modes have smaller probabilities. The typical Stokes pulse shape is complicated, and a detailed experimental analysis of the statistics of the occurrence of the “single” modes has not yet been carried out. I

I

I

I

I

I 400 500 600 0 FREQUENCY (MHz) Fig. 4.11. Example of a single-shot Stokes spectrum with several narrow spikes (solid curve).The predicted gain-narrowed line shape (dashed curve) is shown for comparison. (From MACPHERSON, SWANSON and CARLSTEN [ 1988al.)

0

0

100

200

I 300

I

IIkO 41

259

EXPERIMENTS ON STIMULATED RAMAN SCAmERING

A related observation has also been made in the frequency domain. By examining the Stokes-pulse-energy spectrum of successive shots, MACPHERSON, SWANSONand CARLSTEN [ 1988al have observed spectral fluctuations that correspond to the random excitation of temporally coherent modes. As shown in fig. 4.11, the spectrum of a single Stokes pulse was found to have several peaks, each of which was much narrower than the ensembleaverage linewidth (eq. (3.4.49)). The positions and widths of the peaks varied randomly from one Stokes pulse to the next, although the ensemble-averaged spectrum was in agreement with the gain-narrowed spectrum predicted by the theory. The “mode structure” seen in individual spectra depends on the degree of transiency of the scattering. There are more peaks in the spectrum when rzL is large; that is, in the steady-state regime. The multiple-peaked spectrum observed on a single experiment corresponds to the Fourier transform of a linear combination of temporally coherent modes.

4.3. SPONTANEOUS GENERATION OF RAMAN SOLITONS

In the regime of SRS in which the gain is saturated, solitons may arise in the heavily depleted pump pulse because of sudden phase reversal in the Stokes, pump, or phonon fields. Such a phase reversal effectively changes the sign of the Raman gain, so that there is gain at the pump laser frequency and loss at the Stokes frequency. Therefore, energy is transferred from the Stokes pulse

0

10

20

30

40

50

60

70

TIME (ns> Fig. 4.12. A typical spontaneously generated soliton in the output pump pulse. The input pump SWANSON and CARLSTEN [1989].) pulse is shown as a dashed curve. (From MACPHERSON,

260

QUANTUM COHERENCE PROPERTIES OF RAMAN SCATTERING

[III, 8 4

back to the laser, which under certain conditions, leads to the generation of a soliton in the pump pulse. Raman solitons were first observed by DRUHL, WENZELand CARLSTEN [ 19831 in an experiment where the rotational levels of para-hydrogen were excited by Raman scattering of a single-mode carbon dioxide laser. When the pump was highly depleted ( q = 0.6), a short soliton pulse was occasionally formed. An example of such a spontaneously generated soliton is shown in fig. 4.12. The soliton arises in a depleted region of the pump pulse, and a corresponding "dark" soliton arises in the Stokes pulse, although this is not shown in the illustration. It was later demonstrated by the same group (MACPHERSON,CARLSTEN and DRUHL [1987]) that a soliton could be induced in the pump beam by externally imposing a 180-degree phase shift on a Stokes pulse, which is the input to a Raman amplifier. They showed that, if the phase change takes place in a time which is short compared with the correlation time of the Stokes light, a soliton will appear in the depleted pump beam. It has been shown theoretically (ENGLUNDand BOWDEN[ 19861) that quantum noise can be sufficientto provide the phase change which gives rise to the spontaneous solitons, and that the amplitude of the soliton depends on the size of the phase change. The degree of phase change is determined ultimately by the quantum noise inherent in spontaneous scattering. Thus the amplitude of the soliton will be random, with statistics reflecting those of the underlying vacuum fluctuations.

4.4. COOPERATIVE RAMAN SCATTERING (CRS)

Several early theories pointed out the possibility of cooperative effects in SRS (WALLS[ 19711, SHIMODA[ 1971]), which can occur when a significant fraction of the molecules (or atoms) are driven into the excited state ( 13) in fig. 3.2) before the laser pulse is depleted. This saturation of the Raman gain is complementary to that described previously, because here it is the molecular ground state and not the laser pulse which is being depleted. The process is analogous to two-level supertluorescence, in which the atoms begin in an excited state and cooperation develops as the atoms emit spontaneously. In fact, the equations of motion are, in the linearized regime, identical for the two processes if anti-Stokes and higher order scattering are ignored. This characteristic was pointed out by FLUSBERG[1975] and RAUTIAN and CHERNOBROD [ 19771. Thus one can directly use the methods developed for two-level supertluorescence (GLAUBER and HAAKE[ 19781, HAAKE,KING, SCHR~DER, HAUS and GLAUBER [ 19791, POLDER,SCHUURMANS and

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EXPERIMENTS ON STIMULATED RAMAN SCA’ITERING

26 1

VREHEN[ 19791, SCHUURMANS and POLDER [ 19791, SCHUURMANS [ 19801). To make the analogy clear, we define an effective dipole moment d , = h K,E,*, assuming the classical laser field EL*to be constant. Then the 1-D eqs. (3.4.33) and (3.4.34) for SRS can be written

(4.4.1)

(4.4.2) where K, = 2nNd,*os/c. These are exactly the linearized equations for superfluorescence from two-level atoms, if we reinterpret $? as the atomic raising operator (recall that in the SRS theory is the lowering operator). It should be noted that the magnitude of the coupling between molecules and Stokes field can be varied by the experimenter by means of the laser field strength. The mean intensity emitted at the output face is given by eq. (3.4.42), which can be written as

ot

zs(z) = e-2rr&(r)

+2r

S,‘

e-2r7‘zTR(r’)dr’

3

(4.4.3)

I~({4ZZ’/ZRL}”2)dz‘ ,

(4.4.4)

where ITR(z) is the intensity in the transient regime ZTR(z) = (ALzR)-l

where A is the area of the output face and zR is the collective decay time, defined as zR = (8n/3)ze(NLls)-’, where z, = 3hc3/4d,2w,3is the effective natural lifetime (inverse of spontaneous scattering rate). If no dephasing collisions are present, r = 0 and Z,(z) = ZTR(r), This function grows rapidly until the atoms’ initial states begin to become depleted, which happens roughly at the delay time ,,z, defined by (4.4.5) that is, when the number of emitted photons equals one half the number of atoms No = NAL in the volume AL. In the limit of large z/zRthis equation can be approximated to find, for the delay time,

zD = zR [$In (2 X N ~ ). ] ~

(4.4.6)

262

QUANTUM COHERENCE PROPERTIES OF RAMAN SCAITERING

[III, s 4

Similarly, the duration of the pulse can be approximated by zp = 2(2’ - zD), where z’ is the time at which one fourth of the atoms have left their initial states. This gives approximately z, = zR iln (2 ZN,),

(4.4.7)

which is much smaller than zD, since No is large. Thus a short pulse is emitted after a relatively long delay. The peak intensity is roughly No/zp,which scales like Nt/ln(2 do). The Nt-factor is a signature of coherent emission. In the presence of collisions (r# 0) the intensity eq. (4.4.3) reaches a steady state after a time zss = l/z,r2 = 2gL/r. This is the time at which the derivative of exp( - 2rz)ZT,(z) is zero. Thus a superfluorescent pulse will be emitted if zD 4 zss. This condition can also be written in the equivalent form zp 4 l / r , which means that there is little dephasing during the pulse. On the other hand, if zss 4 zD, the output will be incoherent and is called amplified spontaneous emission (ASE). The intensity is given in this case by eq. (3.4.43). The practical obstacle to making clear observations of cooperative Raman scattering (CRS) is the need for a long, constant-intensity laser pulse that is not significantly depleted by Raman scattering, which means that there must be more laser photons than there are molecules. Then all of the molecules are simultaneously raised to an intermediate virtual state from which they may radiate by means of cooperative emission. The emitted Stokes pulse will have a duration that depends inversely on the number of atoms in the excited volume and a random delay with respect to the turn-on of the pump pulse. This random delay is due to the uncertainty in the spatial and temporal origin of the first spontaneously scattered photon. The first experimental evidence that cooperative Raman scattering occurs was found by KIRIN,POPOV,RAUTIAN,SAFONOV and CHERNOBROD [ 19741. They observed that the Stokes light from a Raman generator using the Qo,(l) vibrational transition of hydrogen pumped by a Q-switched ruby laser exhibited a large spike, the peak of which was delayed from that of the pump pulse and had a duration that depended on pump power. However, in this experiment the dephasing time of the Raman transition was too large to see clear CRS behavior. The first convincing observations of CRS were made several years FOLIN and CHERNOBROD [ 1979,198l]), later ( P ~ v ~ s oRAUTIAN, v, SAFONOV, using a similar experimental arrangement. When a sufficientlyhigh pump power density was attained, the molecular ground states were depleted, and short, intense pulses of Stokes radiation were found. Figure 4.13 shows examples of the cooperative Stokes and anti-Stokes pulses, together with a typical pump pulse by which they were generated. The duration of the cooperative pulses was

EXPERIMENTS ON STIMULATED RAMAN SCATTERING

263

e I

0

10

I

.

30

50 t,nsec

Fig. 4.13. Examples of experimental observations of cooperative Raman scattering. A shaped pulse from a Q-switched Ruby laser was scattered using the Qo,(l) vibrational transition in low-pressure molecular hydrogen. For sufficiently large pump pulse energies a collective dipole moment at both the Stokes and anti-Stokes frequency builds up. An example of the pump pulse is shown in (a), together with some typical cooperative Stokes pulses (b) and (c) and anti-Stokes pulses (d) and (e). Note the temporal modulation of the cooperative scattered pulses due to the feedback between the two frequencies, which is provided by the collective excitation of the optical phonon mode in the hydrogen molecules. (From PIVTSOV,RAUTIAN,SAFONOV,FOLINand CHERNOBROD [1981].)

in good agreement with an earlier semiclassical theory (CHERNOBROD [ 1979]),which gave essentially the same results as the quantum theory leading to eq. (4.4.7). That these pulses arose from cooperative scattering was demonstrated by showing that their delay time depended inversely on the laser intensity and the gas pressure. When the pump pulse was turned on slowly compared with r- however, the pulses exhibited no delay, because ordinary stimulated scattering prevailed over the cooperative emission. One major difference of CRS from two-level superfluorescence is the occurrence of anti-Stokes emission, simultaneously with the Stokes emission. This characteristic gives rise to a phenomenon unique to CRS, that is, both Stokes and anti-Stokes pulses exhibit “ringing” due to the dynamic balance of energy between the two frequencies. The experimental observation of this effect confirmed qualitatively the earlier theoretical predictions of CHERNOBR~D [ 19791. It is possible to suppress the anti-Stokes light and isolate just the cooperative Stokes pulse by a suitable arrangement of the polarizations of the pump and Stokes beam. This method was used by ZABOLOTSKII, RAUTIAN, SAFONOV and CHERNOBROD [ 19841, who studied rotational Raman scattering in hydrogen rather than vibrational scattering. In this system, when the pump laser and

’,

264

QUANTUM COHERENCE PROPERTIES OF RAMAN SCATTERING

[III

the Stokes light have opposite handedness circular polarization, there can be no parametric generation of anti-Stokes light. They showed that in para-hydrogen, where there is only one allowed final state, the pulses show “ringing” analogous to the Burnham-Chiao ringing in two-level superfluorescence. This ringing was not observed in ortho-hydrogen, because the triple degeneracy of the final state means that there are three simultaneous cooperative transitions, each with a different delay time; the sum of the fields from each of the cooperative Stokes pulses tends to wash out the modulation expected in the pulse from an individual transition. It is spontaneous scattering that gives rise to the macroscopic polarization of the medium and therefore to the occurrence of CRS. There is no initial coherence at the Stokes frequency in the Raman system, so that CRS is a convincing demonstration of a macroscopic quantum effect. Although it has been demonstrated that the cooperative Stokes pulses have a fluctuating delay with respect to the pump (PIVTSOV,RAUTIAN, SAFONOV,FOLINand CHERNOBROD [ 1979]), measurements of the statistics of this quantity have not been reported. On the other hand, measurements of the delay time in two-level superfluorescence have played an important role in developing an understanding of the way in which propagation and damping influence the dynamical growth of the cooperative emission (VREHENand DER WEDUWE [ 19811).

Acknowledgements It is a pleasure to acknowledge Jan Mostowski and Kazik Rz&ewski for playing key roles in the development of the quantum theory of SRS which has been used extensively in this article. Many others have contributed to our understanding of the subjects treated here, through discussions or collaborations, and we would like to thank especially John Carlsten, Fritz Haake, Shih-Jong Kuo, Zheng-Wu Li, Czeslaw Radzewicz, Bozena Sobolewska, Carlos Stroud, Lynn Westling, and Emil Wolf. The research of the authors relevant to the subjects treated here has been supported by the U.S. Army Research Office, the Joint Services Optics Program, and the University Research Initiative.

1111

LIST OF SYMBOLS

265

List of Symbols

h oi

P t

cross-sectional area of the interaction volume laser field frequency direction of plane-wave propagation spatial coordinate length of interaction region number density of scatterers laser electric field slowly varying laser electric field amplitude laser intensity velocity of light quantization volume number of laser photons Stokes field frequency molecular transition frequency quantum-mechanical states of scattering medium, in Dirac notation energy of state l i ) interaction H amiltonian operator electric field operator electric dipole operator time coordinate local time coordinate defined by eqs. (3.1.8) and (3.1.35) transition rate for scattering between states t i ) and If) unit polarization vector annihilation operator for mode r creation operator for mode r matrix element of the electric dipole operator between states 19 and 1.0 element of solid angle differential scattering cross section Lorentzian function halfwidth at half-maximum of Lorentzian function; collisional damping rate intensity of Stokes field gain coefficient for radiation in mode 6 maximum Raman gain coefficient Fresnel number of interaction region

266

QUANTUM COHERENCE PROPERTIES OF RAMAN SCAlTERING

[III

modified Bessel functions of the first kind electronic polarizability tensor scalar product of electronic dipole matrix element between states In) and In” ), and polarization vector of mode jl fl, L component of molecular electronic polarizability tensor density operator atomic Hamiltonian operator density matrix element transition-projection operator slowly varying density matrix element ac Stark shift coupling constants population inversion phase velocity at frequency oLand os magnitude of nonlinear polarization at the Stokes frequency integrated intensity of laser pulse slowly varying annihilation operator for Stokes field slowly varying annihilation operator for phonon amplitude Stokes photon number operator initial state vector of system probability of observing n photons in the Stokes field Stokes pulse energy probability distribution of Stokes pulse energies characteristic function of Stokes pulse energy distribution classical random variable probability distribution of B classical random process probability distribution of A energy in the ith Stokes field mode mean photon number in the ith Stokes field mode number of Stokes field modes positive (negative) frequency component of Stokes field operator normalized slowly varying phonon annihilation operator Langevin operator macroscopic polarization operator at Stokes frequency label for atoms located within a small volume element small volume element of interaction volume thickness of small volume element in z-direction

1111

REFERENCES

267

Stokes wavelength Stokes wavevector probability distribution of Stokes intensity Stokes pulse energy operator temporally coherent mode function temporally coherent mode lowering operator Kronecker delta symbol laser pulse duration spectral mode energy spectrum of Stokes radiation

Green functions radial coordinate in output face of interaction volume radius of output face of interaction volume second-order Stokes field correlation function time-factored part of G(p,, 2 , ; pzyz2) space-factored part of G(p,, 2,; pZyz2) spatially coherent mode functions fraction of Stokes radiation in the nth spatially coherent mode radial mode function azimuthal angle coordinate of output face of interaction volume laser-to-Stokes energy conversion efficiency

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E. WOLF,PROGRESS IN OPTICS XXVIII 0 ELSEVIER SCIENCE PUBLISHERS B.V., 1990

IV

ADVANCED EVALUATION TECHNIQUES IN INTERFEROMETRY BY

J. SCHWIDER Physikalisches Institut Universitat Erlangen-Niirnberg Staudtstrasse 7 0-8520 Erlangen, Fed. Rep. Germany

CONTENTS PAGE

§ 1. INTRODUCTION

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

§ 2. CLASSIFICATION AND CHARACTERISTICS OF REAL-

. . . . . . . FRINGE EVALUATION (cp = po + pix) . . . . . . . . PHASE MODULATORS . . . . . . . . . . . . . . . PHASE-LOCK INTERFEROMETRY (PLI) (cp = a sinwt) . HETERODYNE INTERFEROMETRY (cp = at) . . . . .

TIME INTERFEROMETRIC METHODS § 3. § 4. § 5.

§ 6. § 7.

PHASE SAMPLING INTERFEROMETRY (PSI) ( q = ( r - l ) c ~ , ). . . . . . . . . . . . . . .

.

§ 8. RELEVANTDATAANDMERITFUNCTIONS

,

.

278

301

. 303 308 320

. 324

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

1 1 . ERROR SOURCES AND MEASURING LIMITATIONS

276

. 296

. . . . . . . . . .

§ 9. CALIBRATION METHODS FOR INTERFEROMETERS

10. APPLICATIONS

. .

273

326

. 339

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

ACKNOWLEDGEMENT.

353

REFERENCES

353

4 1. Introduction The invention of the laser greatly boosted the development of interferometric methods (HARIHARAN [ 19871). Other technological developments in the early 1970s also contributed to a considerable expansion in the field of interferometry. The impact of photoelectrical detector technology and the general application of microelectronic circuitry to signal processing can be felt in all modem branches of interferometry. Today, the microcomputer has found its way to nearly every measurement device, and thus on-line evaluation techniques are a common feature of modem interferometric methods. Together with the technological progress in the field of electronic and optical hardware the need for high-precision measurements has become apparent (e.g. in surface testing, holographic interferometry, and other fields of interferometric measurements). Thus surface features are strongly connected with the fabrication process (HEYNACHER[ 19831 and BECKERand HEYNACHER[ 19871). One modem example is the generation of aspheric surfaces; another example is the diamond [ 19871). turning process of metal mirrors for optical purposes (LANGENBECK Furthermore, the development of imaging devices (LEE,BARTLETTand KANIA [ 19851) for soft X-ray radiation has also improved interferometric test methods in the visible region to enable them to meet the requirements for sufficient image quality at X-ray wavelengths. The improvement of computers, most notably in terms of their speed and memory capacity, has increased the sophistication of methods in digital image processing. Since steps in the evaluation process of single interferograms can be understood as a special case of digital image processing, the widespread use of such methods especially in holographic interferometry has become common practice. Therefore, it seems justified to give a survey of such advanced methods used in interferometry and to attempt some form of classification of the different methods to help newcomers in this field to understand the principles and to guide them through the vast variety of possibilities. Among existing interferometers two-beam devices are the most common. Therefore the major part of the new techniques has been applied to the evaluation of two-beam interference patterns. On the one hand, because of the high degree of coherence of laser light, it is relatively easy to generate a 213

214

ADVANCED EVALUATION TECHNIQUES IN INTERFEROMETRY

SPLITTING

[IV, § 1

COMBINING

L\

SAMPLE

1 EP

I

Fig. 1. Principle of a two-beam interferometer.

two-beam interference pattern. On the other hand, the extraction of the essential phase information is a formidable task. To illuminate the difficulties, a short explanation of the mathematical background of two-beam interference might be useful. A two-beam interferometer has the principal structure outlined in fig. 1. An impinging electromagnetic wave with the field strength E is split coherently into two parts, that is, the test wave with the field strength Epand the reference with the field strength E,. The beam splitter in most widespread usage is a semitransparent mirror. After travelling along the two separate a r m s and accumulating all existingphase lags, the waves Epand E, are superimposed by a second beam splitter, such as a semitransparent mirror or other means. The resulting sum field strength is

E,

=

Ep + E, .

(1)

Since only square law detectors exist in optics, the detected quantity is the intensity I

w, y)

=

E ( x , y ) E * ( x ,y )

9

(2)

where the electrical field strength is assumed to be a complex physical quantity (BORN and WOLF [ 19641)

E ( x , y, t ) = E ( x , y) e - J w r ,

(3)

with a complex spatial amplitude E and a harmonic time dependence. In terms of the eikonal equation the complex amplitude can be written as

E ( x , y ) = A ( x , y )d @(x *-" ),

(34

where A is the modulus of the field strength and Q, represents a phase term directly related to the optical path differenceof the wave arising from refraction, reflection, or simply propagation through space. Substituting eq. (2) into eq. (l), the intensity in the exit pupil coordinate system of the interferometer

IV,I 11

215

INTRODUCHON

can be written as I ( x , Y ) = EsE,*(x, Y ) = (Ep + Er) (Ep*+ E,*)

-

(4)

Since Ep,E, have the same time dependence due to coherent splitting of the incoming field strength, eq. (4) can be expressed by substituting eq. (3a) into (4): Z(x, y ) = A ;

+ A: + U P A ,COS(Gp - Qr),

or with the substitutions I, = A ; + A : for the mean intensity and V ( x , y ) = 2ApA,/(A,2 + A t ) for the visibility after Michelson and djp = @, djr = rp for the phase to be measured and the reference phase, respectively: I ( x , Y ) = I O U + V(x, Y ) cos(@(x,Y ) - d)*

(6)

This is the basic equation for two-beam interferometry. The only measurable quantity is I(x,y). The phase @(x,y) should be extracted from eq. (6), although it is screened by two other spatially dependent functions, that is, the mean intensity and the visibility (see BORNand WOLF [ 19641). Furthermore, the intensity depends periodically on the phase to be measured, which causes two additional problems for the evaluation: 1. Because of periodicity, the phase is only determined mod2x, or equivalently, an addition of 27cm (mwhole number) does not change the intensity pattern. 2. Because of the even character of the cosine function, that is, cos @ = cos( - (o), the sign of d j cannot be extracted from a single measurement of I(x, y ) without prior knowledge. Another essential feature of eq. (6) is the fact that only phase differences can be measured, with the following two consequences : 1. Relative freedom in the choice of the reference phase, enabling different interferometer philosophies. 2. The necessity for calibration, since in the general case the reference phase may be affected by the aberrations of the interferometer components.* One of the consequences of modem photoelectric detection is sampling, which means that instead of a continuously valued function only a selection of values is measured and the reconstruction of the continuous functions is possible only if the assumptions of the sampling theorem are fulfilled. Quanti-

* In most cases pis consideredas constant over the interferometer apertureor space invariant. But the space-variantpart can be attributed either to the phase to be measured @orto p itself if it is appropriate to do so. This freedom of choice has been used to help with the classification of the different methods.

216

ADVANCED EVALUATION TECHNIQUES IN INTERFEROMETRY

[IV, 0 2

zation of the intensity values also occurs if computer evaluations are to be carried out. Before considering the phase evaluation techniques, we shall discuss a rough survey of the physical effects leading to phase deviations. The phase deviations in an optical device evolve through different physical effects such as refraction, reflection, diffraction, and scattering. Typical optical elements or objects in interferometric testing devices can be attached to the aforementioned effects: 1. Reflection: mirrors, technical flats. 2. Refraction: lenses, prisms, double refracting crystals or elements, plasma discharges. 3. Diffraction: gratings, holograms, and synthetic holograms. 4. Scattering: most of the objects in holographic interferometry.

4 2. Classification and Characteristics of Real-Time Interferometric Methods General features of real-time interferometers include the following: 1. One- or two-dimensional photoelectric detection of interference patterns. 2. On-line calculation of the phase and extraction of relevant merit functions or figures of merit. 3. Solution of the modulo 2n and the sign problem. The classification can be carried out with the help of the chosen reference phase cp in every special case. In the following, 4 groups shall be distinguished: 1. Fringe evaluations cp = p,, + p , x . 2. Phase-lock interferometry cp = a sinwt + cp". 3. Heterodyne interferometry cp = of. 4. Phase sampling or phase shifting interferometry cp = (I - l)cp,,.

2.1. CONVENTIONAL EVALUATION TECHNIQUES

In the past interference patterns were evaluated visually or were photographed. After development the loci of the extrema were measured either by graphical means (SCHWIDER[ 19661, SCHWIDER,SCHULZ,RIEKHERand MINKWITZ[ 1966]), or by using commercial densitometers. In most cases interference fringes were used and the fringe deviations of the wedge-type fringes from an equidistant and parallel raster were measured. To enhance the accuracy, either multiple-beam interference fringes after TOLANSKY [ 19481 or equidensities (LAUand KRUG[ 19681) or densitometers (JONESand KADAKIA [ 19681) were applied.

IV, § 21

211

REAL-TIME INTERFEROMETRIC METHODS

CROSS SECTION THROUGH OBJECT BRANCH

INTERFERENCE PATTERN

INCOMING WAVE

PHASESHIFT:

PHASESHIFT:

60

-

2h

x In

*

6@

-

2n

'

d D

Fig. 2. Measurement of phase steps.

The well-known proofglass method is the most widespread shop testing procedure and relies on estimates by the test person. The sign of the phase deviations can be derived either from the succession of interference color or by mechanical pressure and observation of the accompanying fringe shift. The result of such estimations is the number of interference rings (in the case of spherical surfaces giving also the radius of the test piece provided that the radius or curvature of the proofglass is known), and the ring asymmetries show the deviation of the surface from a true sphere. The accuracy of such estimations is of the order of All0 or worse. Another typical example is the determination of the step height of an optical layer, as illustrated in fig. 2. Here, the estimation can be carried out directly by the measurement of the fringe shift at the edge of the step. If the shift overshoots one fringe distance, some uncertainty may occur, which can be overcome by the use of white light fringes as marker fringes to find the correct attachment of the fringes on either side of the edge. Unfortunately, these techniques are only of limited value if high accuracy and speed are required in the evaluation process, which becomes especially evident in the case of two-dimensional evaluations as in optical testing (SCHULZ and SCHWIDER[ 19761).

278

ADVANCED EVALUATION TECHNIQUES IN INTERFEROMETRY

[IV, 8 3

2.2. SOME FORERUNNERS OF AUTOMATIC EVALUATION TECHNIQUES

In the case of parallel adjustment indicated by a fluffed-out fringe pattern, the phase deviation can be obtained by means of direct measurements of the intensity variations across the field of view. For this purpose ROESLER[ 19621 adjusted the average phase difference in the interferometer in such a way that the intensity values vary only in the region of steepest slope, that is, in the neighborhood of the zeros of the cosine function. In this region the relation between intensity and phase is nearly linear. In an earlier study by KOPPELMANN and KREBS[ 19611 this method was invented for the testing of Fabry-Perot plates with the help of multiple-beam fringes. In this technique the sensitivity against small surface deviations is very high but the measuring range is rather restricted. CRANE[ 19691 used the first modulation interferometer to detect small phase differences of the order of x/4.

0 3. Fringe Evaluation (cp = po + p , x ) This section will discuss those methods which rely on the adjustment of a fringe pattern. It is known that for the fringe extrema the phase is 2xm, where m is the order number of the fringes. This advance knowledge can be used to establish a data set of known phase values in the interference pattern. The phase deviations are then stored as deviations of the fringes from equidistance and straightness. This section is divided into two parts: The first part examines the historically older interpolation methods where the phase information is extracted from the positions of the extrema. The second part discusses more general methods that deliver the phase in a more continuous form.

3.1. PHASE MEASUREMENT BY INTERPOLATION BETWEEN FRINGE POSITIONS

The initial discussion of fringe evaluation techniques will focus on the description of automated methods for fringe interpolation. In conventional interferometric terms the evaluation begins with the generation of a fringe pattern that has a sufficient spatial frequency. Since only one intensity pattern shall be evaluated, some prior knowledge is required. The fringe adjustment is designated by the reference phase cp being a linear function, for example, in the

FRINGE EVALUATION

219

x coordinate cp = Po + P I X

(7)

9

where po and p 1 are a constant phase term and the slope of the linear function, respectively. The phase to be measured, @(x,y), modulates the carrier interference pattern

I

=

1 + cos(po + P I X ) .

(8)

This modulation is indicated by changes in the position and spatial frequency of the fringes. To avoid problems connected with closed fringes, the carrier frequency should be chosen accordingly, that is,

over the whole interference pattern. The sign of p 1 must also be known beforehand and entered into the computer. Generally, the modulation of the carrier by @isonly slight, that is, Apl < 0 . 2 or, ~ ~equivalently,the fringe density is rather high for this type of evaluation process. Under this assumption it is sufficient to detect only the positions of the extrema of the intensity pattern. At the extremal points the phase difference (@ - cp) = d at the maxima (minima) of I ( x , y ) takes on the values

;:={'

Max 1 ) n , Min,

9

( N = 1,2,.*.).

The phase between the extremal positions can be calculated by linear or higher order interpolation as long as inequality (9) is fulfilled. The sign of the slope p 1 must be known to obtain the correct interference order sequence. ZEEVI, GAVRIELY and SHAMAI[ 19871 have extensively discussed the reconstruction of a function from the sine-wave zero crossings and showed that the reconstruction of a bandlimited function is satisfactory if the condition p , > 3 x (bandwidth) holds for the carrier frequency. Thus the previously assumed carrier modulation keeps well within these limits. The interpolating procedure can be understood with the help of fig. 3. The positions of the minima will be considered as the measured values. The phase $(xi) for a point lying between the positions X , and x N + of the Nth and the (N + lYh minima will now be derived. Since x N < x ~ < x N + ~ ,

280

ADVANCED EVALUATION TECHNIQUES IN INTERFEROMETRY

'I

INCREASING

6)

IIV, B 3

, I

I

ri -

ORDER FRINGE

IISTANT OINATE GRID b

X Fig. 3. Linear interpolationscheme for the determinationof the phase from the positions of the extrema.

and linear interpolation will be assumed, one obtains for d(x,)

The extension to two dimensions is straightforward, since the same procedure can be repeated for all sections y = const. In this way a set of phase data in an equidistant ( x , y ) raster can be determined, which is the starting point for the calculation of relevant merit functions or test results. As a result of the method chosen, d = (Q - (p) is monotonously rising or falling. Although the carrier interference fringes are the backbone of this procedure, the carrier should be removed later to obtain the relevant data. Therefore, after the phase determination procedure a fitting process follows, where from the phase du a functional F(i, j; pk) representing adjustment aberrations is removed. The best fitting procedure relies on least squares (lsq) approximations. The functional F(i, j;pk)depends on a set of adjustment parameters. Here,

IV,0 31

FRINGE EVALUATION

28 1

only the most common functionals for planeness and sphericity testing in polynomial form will be given: Planeness testing: F = p o + p , i + p 2j , Sphericity testing: F = po + p , i + p 2 j + p 3 ( i z + j 2 ) .

(12)

The Isq fitting delivers the set of pk values that are later used for data reduction @re,

=

- F(Pk)

(124

+

3.2. ASSESSMENT OF FRINGE POSITIONS

So far only the general philosophy of interpolation between a set of (x, y) locations for which the phase is known has been outlined. Now we will discuss the determination of positions x and y from the intensity pattern having known phase values modulo 2n. In real interferometry the intensity distribution given in eq. (6) is disturbed by noise. One essential source for noise contributions is dust diffraction in the interferometer. To the intensity distribution (eq. (6)) a noise term n(x, y) should

COMPLEX PLANE

re

CONSTRUCTIVE INTERFERENCE Fig.4. Influence of speckle noise on the definition of the maxima and the minima of the interference pattern.

282

ADVANCED EVALUATION TECHNIQUES IN INTERFEROMETRY

[IV, B 3

be added:

w ,Y )

=

1,(1 + ~ c o s ( @ ( xY,) -

0)+ n(x, Y )

*

(13)

It can be shown that the maxima of the interference pattern are hampered especially by coherent noise modulations. This can be understood with the help of fig.(4), where the relative position of the complex amplitudes of the reference u, and the object u, wave are given for the case of constructive (maxima) and destructive (minima) interference. The statistical amplitude part is u,, and uosr respectively. The statistical character of these small amplitude contributions is hinted at by the circles at the tip of the arrows of u, and uo, respectively. In the case of constructive interference the intensity IM becomes

ZM

= uM&

=

Ic, + do12

z41ur12+ Iurs+uos12+41urI

Iuos+

ursIcos(qrs-qos),

(14)

since it is assumed that Ju,I = Iu,I and Iu,,[ = ~ u , , holds. ~ Neglecting quadratic terms in u,, and uos,the modulation due to the statistical phase q,, - qosis of the order 2J1;1,cos(c~,s

- qos) *

(15)

In the case of a minimum the intensity I , is

Since I u,, I is small compared with the mean intensity, the fluctuation is also small. The amplitude of the fluctuations is for constructive interference: 2 destructive interference : N 2 Z,, , if Z, is put to 1. Since I,, 1, the square root law for the maxima provokes a considerable speckle fluctuation of the interference maxima. Therefore several methods have been developed to cope with the problem of noise reduction and the separation of the influence of the slowly varying mean intensity and the visibility of the fringes. JONES and KADAKIA[ 19681 evaluated photographic negatives in a photoelectric densitometer. The transparency values were digitized and the positions of the interference minima were determined by means of reference levels and and ALTE [ 19781 a special discriminating software. Similarly, ROSENZWEIG described a flying-spot scanner to get the data into an on-line computer. The determination of the minimum positions is similar to the procedure described earlier.

- A,

+

IV. 0 31

FRINGE EVALUATION

283

CRESCENTINI and FIOCCO [ 19881 recently performed fringe analysis with a similar method. The extrema were determined by level slicing and taking the median values along straight lines perpendicular to the fringes. From these locations the phase is determined by fitting a second order polynomial to the data. To fill the gap between adjacent extrema, the mean phase can be shifted by a PZT in one arm of the Michelson interferometer. Comparisons with the FFT algorithm (5 3.4) were made on the basis of intensity data from a 6-bit frame grabber. They reported accuracies of 1/30 and A/lOO, respectively, for the two methods. In recent years the application of digitized vidicon cameras and CCD cameras has become increasingly common, and thus real-time evaluations became possible. For obvious reasons the interference pattern is adjusted perpendicularly to the scan direction. The intensity values are typically digitized as 8-bit words and fed to an on-line computer. A large number of methods for the determination of fringe locations have been developed and shall be discussed separately. 3.2.1. Low passfiltering and dc subtraction From ac-length measuring interferometry it is known that the zero crossings of the cosine function provide sufficient accuracy. NAKADATE,MAGOME, HONDAand TSUJIUCHI[ 19811 applied low pass filtering to the video signal of a vidicon camera to obtain zero crossings that were free from disturbances due to speckling and illumination inconstancies. While the electron beam scans the fringe pattern, the spatial distribution is transformed into the time domain. Figure 5 shows the image plane with the filtering process underlying the fringe detection technique. The intensity according to eq. (13) is low pass filtered with two windows with different widths. One window passes the dc term and the other, the cosine function. The speckle noise is greatly reduced by the limited width of the two windows. The detected signal is

-

U F- [ F(Z)rectZ - F(Z)rect 11 ,

(17)

where F and F- I denote the Fourier and the inverse Fourier transform, and rect represents the window function. The resulting signal transformation is schematically shown in the lower part of fig. 5 . The noisy intensity picture is transformed into a purely ac-cosine signal. The positions of the extrema are obtained by taking the average of two neighboring zeros. This procedure also guarantees a highly effective suppression of nonuniform illumination effects because of the opposite sign of the slopes in the neighboring zero regions.

284

ADVANCED EVALUATION TECHNIQUES IN INTERFEROMETRY

X

I;--I---

FOURIER SPACE

1

I I

I I

I I I

I I

p=o

"[

rect I

V

PI

DETECTED SIGNAL

Fig. 5. Low pass filtering combined with dc subtraction. Top, intensity degraded by speckle noise; middle, filtering windows in the Fourier plane; bottom, resulting electronic signal.

IV, 8 31

285

FRINGE EVALUATION

3.2.2. Parabolic approximation of minimum positions

AUGUSTYN,ROSENFELDand ZANONI [ 19781, MARCUSEand PRESBY [ 19801, and BIRCH,JACKSON, PUGH and WEST [ 19821 used similar methods to define the position of fringe minima. Since the minima are better defined because of the reduced speckle influence, the accuracy for finding the minimum positions is greater than that for the maxima. For an explanation of the possible fitting let us consider a one-dimensional scan through the intensity pattern in the neighborhood of one of the minima (fig. 6). If the intensity is sampled in the x direction, as assumed in fig. 6, the neighborhood of the minimum with the index i can be determined by 5 (or 7) values 1,- through I, 2. The variance +

i+2

C

u=

( I , - a - bl

- cl‘)’ ,

I=i-2

is minimized, resulting in the 3 values for a, b, and c. The position of the minimum follows from

ax

-=

31

b -+ 2 c l = 0,

and finally

lmin= - b / 2 c . This procedure is repeated for each scanline y = const. and each minimum position. A rough estimate of the minimum position can be found by using a discriminator level or similar techniques.

i-2

i-l

i

it1

it2

(PIXEL NUMBER)

Fig, 6. Parabolic fitting procedure to derive minimum positions.

1

286

ADVANCED EVALUATION TECHNIQUES IN INTERFEROMETRY

[IV,B 3

3.2.3. Low pass filtering combined with dgerentiation

SNYDER [ 19801 introduced a convolution algorithm that enables the determination of the extrema by means of the zeros of the first derivative of the intensity distribution. As is well known, the first derivative of a function can also be determined by convolution of this function with a suitable kernel. Such a kernel is the first derivative of the Dirac S function I’(x)= -

J-,

+a

a

- S(X - x ’ ) I ( xdx’ ’ ). ax

In modem applications of interferometry the intensity values are sampled with a scanning electronic detector array, digitized, and fed into an on-line computer. Since fast algorithms implemented on smaller computers allow only for simple convolution kernels, here a square wave approximation (or hook function) for the first derivative of a Sfunction can be used. As a result of sampling, the convolution degenerates to a simple summation that is easily carried out on a microcomputer i+h

i

where i and b are whole numbers. The square wave period 2b is chosen so that the noise terms are sufficiently suppressed. It should be noted in this connection that the methods given by DEW[ 19671 and DYSON[ 19631 working on an analog basis are essentially similar. To understand the filtering and differentiation process more clearly, a discussion in the spatial frequency domain might be useful (fig. 7). In Fourier FOURIERPLANE

1 I

cos-FRINGES

Fig. 7. Low pass filtering combined with differentiation. In the illustration the Fourier plane is given together with the filter fhction.

IV,B 31

FRINGE EVALUATION

287

space a convolution corresponds with a multiplication of the Fourier transforms. The transform H(v) of the convolving square wave kernel is H(v)= 2jbsinc(vn:>sin(vn~), where S is the fringe spacing and vis the spatial frequency. The optimum choice of b is determined by the position of the first order extremum of H(v) in relation to the center of gravity of the ac term of the Fourier transform of the intensity pattern. This occurs for values b/S w 0.37. Since H(v) changes the sign at v = 0, the cosine term is transformed into a sine pattern that conforms with the differentiation feature of the filtering process. The zeros of the filtered sine function coincide with the extrema of I ( x ) . Since sinc(vnb/S) suppresses high frequencies, and since sin(vnb/S) is zero for v = 0, the dc term is eliminated. Nevertheless, one should keep in mind that the filtering process together with the necessary sampling operations will generate spurious information, especially in the rim region of an interferogram. 3.2.4. Image subtraction and level slicing

LANZLand SCHLUETER[1978] and SCHLUETER[1980] developed a method that makes use of two interference patterns in phase opposition. The first pattern is stored, for example, on videotape or in the memory of a computer, and is subtracted from a second pattern that has undergone a phase shift of II.The resulting intensity pattern is freed from the dc term and also from the low-frequency variations of the latter, so that level slice operations can be carried out even if the signal is strongly distorted by nonuniform illumination effects. The positions of the extrema follow in the usual way by averaging the slice positions. 3.2.5. Fringe skeletonizing or thinning operations With the help of morphological image-processingmethods the fringe maxima or minima can be found. For an overview on morphological image processing methods the reader is referred to MARAGOS[ 19871. FUNELL [1981] developed a fringe evaluation technique based on image processing with interactive software. The operator starts the fringe evaluation process by choosing a starting point either on a fringe maximum (ridge) or a fringe minimum (valley). The program evaluates in a field of 3 x 3 pixels the

288

ADVANCED EVALUATION TECHNIQUES IN INTERFEROMETRY

[IV,8 3

direction along the max (min) of the stored intensity data. The starting direction is chosen interactively. The program searches in a +90 degree forward direction. If an obstacle is met or if the fringe position is wrongly determined, for example, due to high noise levels, the software allows the control and correction of the data. The operator can link data points belonging together or correct wrong decisions. The fringe following process can be observed on the graphics display. The rim of the interference pattern is also determined by the operator. HUNTLEY[ 19861 applied two-dimensional Walsh spectral analysis to estimate the frequency components of Youngs ftinges. The latter fringes are used as a measure for lateral movements (ERF [ 19741) in speckle photography. It has been shown that Walsh transformation in combination with cross-correlation techniques with square wave functions give satisfactory results after shorter processing compared with the normal Fourier techniques. YATAGAIand KANOU[1983] applied image processing to fringe evaluations. The intensities are read in a frame memory from an 8-bit 512 x 512 pixel vidicon camera. The on-line computer is equipped with local pattern processing hardware. The main steps in fringe evaluation comprise shading correction, binarization, and skeletonizing of the fringes. The thinning operation uses logical filtering operations by means of hardware logic. After smoothing, masking, and level slice operations the intensity data are scanned by a 3 x 3 pixel matrix. The content of the matrix is a binary number that gives the address for a lookup table, enabling the thinning operation by appropriate coding of the table. Depending on the amount of noise, several intermediate smoothing operations can be carried out. [ 19851 also made thinning operations by excluding MASTINand GHIGLIA a set of logical neighborhood transformations. The eroding operation for the fringe pixels was done symmetrically, avoiding losses in the continuity of the fringe skeletons. WOMACK,JONAS,KOLIOPOULOS, UNDERWOOD, WYANT,LOOMISand HAYSLETT[ 19791 experimented with different fringe defining operations and implemented a method in a video processor that obtains the fringe positions by comparing a slice level voltage with the video signal from the scanning vidicon and then determines the fringe positions by software operations. One possibility would be to take the mean position between the positions of the slices.

IV, 8 31

289

FRINGE EVALUATION

3.3. ANALOG PROCESSING OF SCANNED FRINGE INTENSITY

ICHIOKA and INUIJA [ 19721 developed an electronic real-time network (fig. 8) that delivers the phase Q of an interference pattern in an analogous manner. By electronic scanning the intensity becomes a time-dependent function Z(t) = Z ,( 1

+ Y cos [ Q ( t ) - 211vt]) .

(23)

By using an oscillator tuned to 2nv and a suitable phase shifter, signals of cos 2n vt and of sin 2n vt-type can be obtained also. The electronic signal from the vidicon tube is multiplied by cos2nvt and sin2nvt, respectively, and followed by a low pass filtering operation

C= S=

T

T

Iff+

Z ( t ) cos 2n vt dt ,

j z ' + T I ( t )sin2nvtdt.

The two signals C and S are fed into a special network providing that

The scan speed here was 40 kHz and the frame rate was 5 frames/sec. The attained accuracy is of the order 24100 rms with the restriction that in the neighborhood of the values Q = in, $ n some systematic errors occur because of deficiencies of the electronic circuitry.

INTERFERO

VIDICON sin w t ,I

OSCILLATOR

-

PHASE SHIFT

:

ot

MULTIPLlER MULTI-

-PLlER

- LOW

PASS

- LOW PASS

~

-1

s

tan C

7

~

Fig. 8. Analog processingoffringe patternsby electronicscanningof the spatial light distribution and processing in the time domain.

290

[IV,i3 3

ADVANCED EVALUATION TECHNIQUES IN INTERFEROMETRY

3.4. FRINGE ANALYSIS BY MEANS OF FOURIER TRANSFORM OPERATIONS

If the intensity values of a fringe pattern are stored in a digital computer, the application of digital image processing to extract the phase is only a consequent step. In recent years the introduction of high-quality CCD-photoelectric sensors with nearly constant sensitivity across the device has considerably increased the possibilities for the application of image processing. TAKEDA, INA and KOBAYASHI [ 19821 gave a procedure that relies on the FFT algorithm and RAo [ 19821). (COOLEYand TUCKEY[ 19651; see also ELLIOTISince the spatial frequency vo of the interference pattern obeys the condition of eq. (9), one-dimensional Fourier transforms can be used to obtain the phase @. Let the cosine function be represented as a sum of complex exponentials, then eq. (6) can be rewritten as follows: I ( x , y ) = a(x, y ) + c(x, y) ejZnvox+ c*(x, y) e-jznvox,

(26)

c ( x , y ) = +IoI/ej@(x*J”.

(27)

with

After a one-dimensional FFT and taking the Fourier shift theorem (see, for [ 19681) into account, the following holds: example, GOODMAN F ( I ) = A ( v , v) + C(V -

yo,

y ) + C*(V +

Yo, y ) ,

(28)

where A, C and C* are complex Fourier amplitudes. By means of digital filtering, one sideband of the intensity distribution is singled out (fig. 9) and undergoes an inverse FFT. In this way the quantity c(x, y) can be separated.

A FILTERING WINDOW

/

-v

0

vo

V

Fig. 9. Phase determination by single-sideband filtering. The spatial carrier signal is eliminated by shifting the sideband in the Fourier domain to zero spatial frequency.

IV,8 31

29 1

FRINGE EVALUATION

The phase can be calculated as usual from the complex quantity c ( x , y): @ ( x , y ) = tan-

Im c 'Re c

modn ,

or by using the complex logarithm

log[c(x,y)]

=

IO~[;Z,V]+ i@(x,y).

Phase ambiguities in the form of phase jumps can be eliminated if the sampling theorem is fulfilled, and consequently more than 2 detector elements per period of the cosine function were used for scanning the intensity pattern. RODDIERand RODDIER[ 19871 applied the Fourier technique to interferograms with strong variations of the mean intensity and with a high degree of noise. The varying mean intensity was then determined by measuring a fringeless intensity pattern or by transforming an interferogram with the fringes parallel to the scan direction. In the spectrum of the normal fringe interferogram the parts of the spectrum centered around vo are exchanged for the Fourier amplitudes contained in the second interferogram. After an inverse Fourier transform the mean intensity is used to normalize the intensity pattern. A further problem with the Fourier technique occurs in the rim region of the interferogram. Because of the finiteness of the interferogram and the band pass filtering process, the phase accuracy in the rim region is low. Therefore an extrapolation algorithm for the fringe pattern is applied. It works iteratively in the following way: After a first FFT the spectral region outside of a circular region around the carrier frequency is put to zero. After an inverse FFT the fringe pattern in the region of the initial interferogram is replaced by the original one. Then the iteration is repeated in the same way until a sufficient continuous extrapolation for the fringe pattern is obtained. From this data field the phase values within the initial interferogram are then calculated, giving much better results. KREIS[ 1986a,b] expanded the FFT technique to a two-dimensional approach, where the filter function covers nearly a halfplane of the Fourier space. Equation (28) becomes, in the two-dimensional case, F ( I ) = A(p, v ) + C(p, v) + C*(p, v).

(31)

The advantage of the two-dimensional FFT is the possibility for the evaluation of more complex interference patterns, especially where the condition eq. (9) is violated. Closed fringes are an obvious violation of the preceding condition. In this case the information about the sign of the phase is lost in some parts of the interference pattern. Therefore, KREIS[ 1986a,b] proposed the use

292

ADVANCED EVALUATION TECHNIQUES IN INTERFEROMETRY

[IV,8 3

of two interference patterns of the same object, where the second pattern is shifted by about a = i n in comparison with the first pattern. Let the two interference patterns have the intensities I, and I , :

a(x, y ) = tan-

Rec, Imc, Rec, Rec,

Imc, Rec, + Imc, Imc, -

(34)

From the sign a the decision on the sign @ at each point of the evaluated field can be made, so that the sign-corrected phase unwrapping procedure under the limitations given by the sampling theorem can be carried out unambiguously. Since a is nearly constant apart from the sign indication, the accuracy of the phase retrieval can be improved by averaging the results of the two interference patterns. Only recently ICHIKAWA, LOHMANN and TAKEDA[ 19881 showed that the FFT algorithm can be applied to phase retrieval problems of a wave field from intensity measurements. The first partial derivatives of the phase function can be calculated from the intensity transport equation. To enable a solution of the transport equation by means of the FFT algorithm, a periodic amplitude mask is put into the ray path. A certain similarity with the Ronchi test is obvious. Ru, HONDA,TSUJIUCHIand OHYAMA[ 19881 extended the FFT formalism to spatially quadratic carrier functions by using the Fresnel transform in order to obtain the phase. Because of the limited separability of the carrier from the dc term and the phase conjugate, there is some phase error on axis, similar to the disturbances through twin images known from in-line holography.

3.5. SPATIALLY SYNCHRONOUS FRINGE ANALYSIS

WOMACK [ 1984al introduced this technique into fringe analysis. This technique is strongly analogous to the phase sampling technique (PSI, see § 7) introduced by BRUNING,HERRIOIT,GALLAGHER, ROSENFELD, WHITEand BRANGACCIO [ 19741 in the time domain. Whereas the Fourier transform

IV, § 31

FRINGE EVALUATION

293

method of TAKEDA,INA and KOBAYASHI[1982] carries out the filtering proposed 3 solutions in the process in the spatial frequency domain, WOMACK direct space domain. The first solution is called the “quadrature multiplicative moire“ algorithm, which works as follows. The intensity distribution of the fringe pattern is multiplied with cos/sin functions that have a linear phase dependence with the same mean spatial carrier frequency as the interference pattern. These cos/sin functions are generated in the computer. An optical equivalent to this process would be a multiplicative moirC of two transparencies, with transparencies proportional to the interference pattern, on the one hand, and to a cosine grating, on the other. Since the interference pattern is usually scanned, a one-dimensional representation can be used. By multiplicative processing, sum and difference frequencies are generated: M I = jzo Vcos[Q(x, y ) - 2A(T - vo)x],

(35)

~ ~ = $ ~ ~ ~ s i n [ Q i 2( 1x1,( y~ -)vo)x], -

(36)

where T is the mean spatial frequency of the interferogram and vo is a rough estimate of this frequency derived from a line scan in the interferogram. The phase to be measured, Q, is retrieved from the expressions of eqs. (35) and (36) by a low pass filtering process, that is, a convolution with a window function H ( x , y ) several fringe distances wide, which delivers

where

a, =

N

c c

Z(Xi, y ) cos 211 voxiH(x - x,, y ) ,

i = -N

N

K12 = i=

z(xi, y) s i n 2 x v o x i ~ ( x- xi, y) ,

-N

holds true. The low pass window should be of the hanning window type (ELLIOTT and RAO [ 19821)to suppress periodic phase disturbances. The convolution process is carried out with the sampled data set which is why the integration is replaced by a summation. In this case the finite extent of the interferogram has been neglected. The second method is called the “sinusoidal window” algorithm, which is

294

ADVANCED EVALUATION TECHNIQUES IN INTERFEROMETRY

WINDOW

FUNCTIOh t

H'

= COs(Znv,x) I * H(x,y)

H' = sin(Zrrv,x) t

FOURIER TRANSFOF t

n

*

H(x,Y)

of WINDOW ?

t

Fig. 10. Summary of methods for spatially synchronous fringe analysis. On the left side the case of "quadrature multiplicative moirt" and on the right the case of "sinusoidal window" is shown.

a high pass filteringwith a convolution kernel containing cos/sin of 27t vox; that is,

FRINGE EVALUATION

@=tan-'

Fl

-

.

295

(39)

F2

Figure 10 shows the respective filter functions. The third algorithm is a complex single-sideband algorithm strongly [ 19821 but in the analogous to the method of TAKEDA,INA and KOBAYASHI direct space. The phase @ is also given by an expression identical to eq. (39). The underlying algorithm is an alternative interpretation of the second method. From the standpoint of the numerical computation there is no difference.

3.6. SINUSOIDAL FITTING

A procedure proposed by MERTZ [1983] and applied to interferogram analysis by MACY [ 19831 is very similar to the previous method. For this technique the spatial carrier frequency is chosen in such a manner that 3 pixels span one period of the interference pattern. The intensity distribution can then be written

~ ( xy,) = a(x, y ) + d(x, y ) cosfnx

+ e(x, y ) sinfnx ,

(40)

where d = b cos @(x,y ) and e

=

b sin @(x,y ) .

Under the usual condition of slowly varying phase fluctuations, one can define @(i,j ) by its neighboring values; that is,

and from this follows

296

ADVANCED EVALUATION TECHNIQUES IN INTERFEROMETRY

[IV,8 4

The sign of @ should be determined by other means, and the phase unwrapping can be carried out in the usual way by the removal of the phase jumps in the phase values according to eq. (42). The sinusoidal fitting method is susceptible to some periodic noise, which can be smoothed out by convolution of the phase data with a rectangular window function. The periodic phase error is caused by a sampling mismatch, either by the phase variations across the aperture or by a mismatch of the fringe period during the adjustment operation of the interferometric setup. RANSOMand KOKAL[ 19861 therefore discussed a modified version of the sinusoidal fitting method. They could show that the mismatch can be accounted for by a multiplicative factor in the argument of the arctangent of eq. (42). For this correction factor an estimation procedure using the neighboring f 5 intensity values was given using the sinc functions due to the requirements of sampling theory for interpolation. The improvement of the accuracy was demonstrated with the help of computer simulations only. HOTand DUROU [ 19791made a sinusoidalfitting by aleast squares algorithm for one scan line through the interference pattern perpendicular to the fringes. They claim a All00 accuracy with a 4-bit interface to the vidicon camera.

8 4. Phase Modulators So far the evaluation of static interference patterns has been discussed. In this section methods will be described where the reference phase is a timevarying function considering the phase retrieval procedure as a whole. This necessitates a discussion of the principles of the major phase modulation techniques. A survey of the most common possibilities is given in fig. 11. A very simple phase modulator is a translating mirror (fig. 1la). The phase shift resulting from a piston movement through the distance s is rp= ~ ~ S C O S U ,

(43)

where a is the incidence angle and k = 27c/A. The maximum elongation of the mirror is necessarily limited, especially if piezoelectric transducers (PZT) are used. The shifter of fig. 1lb uses a glass wedge translation perpendicular to the light path. The phase shift belonging to a,lateral shift of the wedge s is r p = k(n - l ) s t a 9 ,

(44)

where 9 is the wedge angle and n is the refractive index of the glass wedge.

IV,§ 41

PHASE MODULATORS

297

Fig. 11. Overview of the physical principles suitable for phase modulators.

In addition, the inclination by an angle a of a plane-parallel glass plate of thickness t generates a phase shift cp = kt[J-a

- cosa] .

(45)

DOERBAND, HERTELand STOCKMANN [ 19781 applied this type of a phase shifter in a phase sampling interferometer (see 3 7) for the measurement of wave aberrations of optical systems (fig. 1lc). In interferometry grating phase shifters are very common (fig. lld). It is

298

ADVANCED EVALUATION TECHNIQUES IN INTERFEROMETRY

[IV, 8 4

known that light diffracted at a translating grating is shifted in its phase. For an explanation let us assume that a sine grating is translating with a velocity L?/2nf. The transparency of such a grating can then be described by z(x) = 1 + sin[2nfx

+ at] .

(46)

The far-field amplitude of such a grating is

J -m

Substituting (46) into (47) and applying the Fourier theorems result in u(() =

a(() + f j ej"6(5 -f)- i j e-jn'6(( +f),

(48)

where 6 is the Dirac delta function. The sidebands at +f have a timedependent phase shift cp = a t , which enables the use of the first diffraction orders as reference waves in interferometric experiments. The gratings can be of different types, either amplitude or phase gratings. STUMPF[1979] and KOLIOPOULOS [ 19801 used rotating radial gratings. INEICHEN,DAENDLIKER and MASTNER[1977] and MOTTIER[1978] proposed the application of acousto-optic modulators (AOM), where running density waves in optical media are used as Bragg-diffraction cells (DRISCOLLand VAUGHAN[ 19781). The phase increases linearly with time:

*

cp= a t .

In polarizing interferometers, where the two separated wave fields are perpendicularly polarized to each other, the phase shift can be brought about by rotating half-wave plates (in double transmission also $ 2 plates) or by the rotation of polarizers (SOMMARGREN [ 19751, CRANE[ 19691) (fig. 1le). A 2n rotation of the polarizing element corresponds with a 8 n phase shift. The function of the polarizing phase shifter can be understood with the help of the JONES-calculus[ 19411. Only a short description is given here and for a detailed analysis the reader is referred to SOMMARGREN [ 19751. Figure 12a shows a scheme of a polarizing phase shifter. A linearly polarized plane wave impinges upon a half-wave plate (HWP). The direction of the electrical field strength here coincides with the x direction, and the frequency is denoted by w. The HWP is rotating with an angular velocity w' about the optical axis (z axis). The wave leaving the HWP is again linearly polarized, but the plane of polarization is rotating with the angular velocity 20'. This wave can be interpreted as a superposition of a right and a left circularly polarized wave with the temporal frequencies w + 2w' and 0 - 20', respectively. These waves hit

299

PHASE MODULATORS

SAMPLEM &

POLARIZER

t &

POLARIZER

Fig. 12. Principle of polarizing phase shifter. (a) Frequency shift introduced by the rotating half-wave plate (HWP). Emerging circular polarized wave vectors rotate with the frequencies w t 2 0 ' and w - 212'. (b) InterferometricTwyman-Green and Mach-Zehnder setups supplied with an external phase modulator. Polarizing beam splitters in combination with quarter-wave plates separate the waves having different frequencies,so that each branch of the interferometer is passed by only one of the waves. Polarizer in the output plane brings the two waves to interference. (c) Similar to (b) but with double refracting components as beam splitter and recombiner.

300

ADVANCED EVALUATION TECHNIQUES IN INTERFEROMETRY

[IV, 8 4

a quarter-wave plate (QWP), whose fast axis is oriented under 45 degrees to the x axis. Two plane waves leave the QWP polarized perpendicularly to each other with the frequencies w - 20' and w + 2w'. These waves enter a polarizing interferometer and are separated by a polarizing beam splitter (PBS) (cf. fig. 12b) or Wollaston prisms (cf. fig. 12c). The two arms of the interferometer are perpendicularly polarized to each other. The situation is especially easy to understand in the case of a Mach-Zehnder interferometer (fig. 12b). Two perpendicularly linear polarized waves leave the second polarizing beam splitter (PBS2) and so exit the interferometer. Both pass a polarizer oriented under 45 degrees to the ( x , y ) coordinate system, giving rise to an interference pattern I

-

:[ 1 + A2 + 24 cos(9 + 4 d t ) l ,

(49)

where A indicates the modulus of the complex amplitude of the test wave; the modulus of the complex amplitude of the reference wave has been put to 1. Equation (49) indicates a phase modulation with the frequency 4w', as mentioned earlier. Another realization of a phase shifter is a pressure chamber (fig. 1If) situated in one arm of the interferometer. This technique is well known from Fabry-Perot interferometers or from length-measuring devices. In addition, optical single mode fibers can be used as a phase shifter. For this purpose the fiber is fixed on a piezoelectric ceramic, and the mechanical stress caused by the piezoelectric effect when a high voltage is applied to the ceramic causes a phase shift (STOLENand DE PAULA [ 19871). In principle all physical effects causing phase shifts in the fiber could be used. In a similar manner integrated optics devices can be used as phase shifters. Here the linear electro-optic effect provides for the shift of the phase while the light passes through a wave guide on a LiNbO, substrate (KISTand KERSTEN [ 19841). If the interferometer is not balanced in its optical path difference (OPD), which means that the OPD # 0, wavelength changes also bring about phase variations (PRIMAK[ 19811, KIKUTA,IWATAand NAGATA[ 19861, KIKUTA [ 19871). It should be noted that this technique is of a rather restricted value, since the wavelength change and the OPD must be controlled very carefully. The wavelength tuning of diode laser radiation has been applied to distance measurements by KIKUTA,IWATAand NAGATA[ 19871, where a phase change A @ = 2nLA1/A2 with L as the distance (or the optical path difference) and A 1 the shift of the wavelength is attained by tuning the diode current. I

301

PHASE-LOCK INTERFEROMETRY

0 5. Phase-lock Interferometry (PLI) (cp

=

a sin wt)

The possibilities of phase modulation open up new branches of ac interferometry delivering highly accurate results under quasi-real-time conditions. The first example for such a type of interferometry shall be PLI, where the reference phase is modulated, on the one hand, sinusoidally and, on the other hand, in a monotonous manner; that is, rp(x, y, t)

=

a sin wt

+ rp' (x, y, t ) .

(50)

The periodic part can be realized by oscillating a plane mirror in the reference arm, and the nonperiodic part rp' can be realized by a piston movement of the same mirror (fig. 13). If eq. (50) is substituted into eq. (6),the resulting intensity detected by a photoelectric sensor is Z(x,y,t)=1,[1

+ Ycos(Q(x,y)-

(51)

cp'(t)-asinwt)].

Equation (51) contains terms of the type cos (a sin 02) and sin (a sin wt). An expansion of these functions into a Neumann series has Bessel functions as coefficients (WATSON[ 19581). The photoelectric signal U can be written as

u = u,= uw+uzw+u 3 w + ' . '.

(52)

The time-varying voltages at the frequencies w and 2w can be separated by

DISPLAY lofyr

1

i OSClL LATOR

DETECTOR

SENSITIVE Fig. 13. Scheme of a phase-lock interferometer.

p7-l

CONTROL

302

ADVANCED EVALUATION TECHNIQUES IN INTERFEROMETRY

[IV,f 5

electronic filters resulting in U,-IOVusinwtsin(@-

u,,

-

cp’),

+IOVu2 cos2ot cos(@- cp‘) ,

(53)

where the modulation amplitude u has been assumed to be very small. For values (54)

(@-cp’)=Nrt,

the following holds:

U,

=

0,

U,,

=

Max.

The signal component U, of U is obtained by a suitable electronic filter. With a phase-sensitive detection (e.g., a phase-sensitive rectifier controlled by the electronic oscillator with the frequency a),the phase cp’ can be steered in such a way that the state U, = 0 is locked. In this way the change of 8 with varying positions ( x , y ) of the scanning detector is monitored by a synchronous variation of cp‘, which means that the value of cp‘ gives @ more or less directly. A scheme of the phase-locked interferometer is given in fig. 13. The reference phase shift cp‘ is taken over by a plane mirror driven by a piezoelectric transducer in a pistonlike manner (MOORE,MURRAY and NEVES[ 19781). But, of course, other phase shifters have also been used (FREITAG, GROSSMANN and TANDLER [ 19791). A simple solution for the phase-sensitive detector is a phase-sensitive and rectifier (PSR) under the restriction that I cp’ I keeps well below (TIETZE SCHENK [ 19801). Figure 14 gives the general scheme for a PSR,which delivers a dc signal with a sign due to the sign of sin(@- cp’). The signal flow is as

Ref

F,, I

PHASE-SENSITIVE DETECTOR

I

Fig. 14. Scheme of a phase-sensitive rectifier.

IV, B 61

303

HETERODYNE INTERFEROMETRY

follows: The incoming voltage U is multiplied with a square wave reference signal derived from the driving oscillator for the PZT in the reference arm of the interferometer. The product voltage is low pass filtered and results in a dc voltage

whose sign is dependent on the sign of (@ - q’). This signal becomes zero if eq. (54) is fulfilled ; otherwise, the PSR provides a signal proportional to the amplitude aZoY sin (@ - rp’) of U,. Because of the orthogonality of the trigonometric functions, the following holds:

+

r~

U,, Uref dt = 0 , k > 1, whole number. Jo

If the phase excursion of @ exceeds several periods 2 x and the PZT cannot follow over the whole domain during the scanning of @(x, y), special solutions for the phase-sensitive detector are necessary. One example has been given by MOORE,MURRAYand NEVES[ 19781. In addition, the driving voltage of the PZT is chosen in a saw-tooth form, where the number of the phase jumps 2 x is counted with the correct sign to provide for the whole span of possible @variations. There are, however, two major problems connected with this type of interferometry: 1. The phase detection is very sensitive to phase drifts of the whole apparatus during the scanning procedure of the interferometric aperture and is practically limited to rms accuracies of A/30. 2. The scanning region must be determined beforehand with some type of interactive software, since otherwise the phase tracking during the scan could be interrupted if the rim of the aperture is crossed, jeopardizing the measurement as a whole.

4 6. Heterodyne Interferometry (cp

= wt)

The principle of heterodyne interferometry is simple. The interference fringes are modulated by a continuous phase modulator and photoelectrically detected. The phase difference between two photoelectrical detectors can be measured by electronic phasemeters. The role of the local oscillator is assumed

304

[IV, B 6

ADVANCED EVALUATION TECHNIQUES IN INTERFEROMETRY

by one of the photodetectors, for example, the detector at rest. Let the intensity at the position (x, y) be I ( x , y) and at the position (xI,y l ) be I ( x l , yl), then the following relations hold: I = lo[1 + vcos { @ ( x , y) + ot}], I,

=

1,[1

(55)

+ vcos{@(x,,y,)+ or}].

If the position (xl, yl) is used as reference and (x, y) as the position of the scanning detector, the phase difference 9' (x, y ) = @(x, y ) - 9(x,, y l ) can be measured. In Q 4 it has been pointed out that polarizing, grating, and acoustooptic modulators are especially suitable for a continuous phase modulation. In some applications a two-frequency laser has been used, where the emitted frequencies are in orthogonal states of polarization. Two alternatives have been reported: one is a Zeeman-type laser with a frequency difference of about 1 MHz and the other alternative laser type is a two axial mode laser with a frequency difference of about 100 MHz. Especially in the latter case the phase measurement is difficult to perform. A possible philosophy for an electronic phasemeter used in heterodyne interferometry will be discussed with reference to fig. 15. In a first step the periodic ac signal is separated from the dc offset by means of high pass filters. The phasemeter should provide for a voltage proportional to the phase difY o

-

Q

: = - c1

'

1-10

d

-R

LOW PASS

:

I1

-R

--

DIFFERENCE AMPLIFIER

6

-

(W mod2w)

*

AND

1-ID

Ul

~

C1

P

COUNTER 1

-

__ COUNTER

TRACT

whole part of 0'

Fig. 15. Electronic phasemeter allowing for the detection of phase differences greater than 2x. The branch above evaluates the phase mod 2r, whereas the two counter-results give the whole part of the phase difference on subtraction.

IV,8 61

HETERODYNE INTERFEROMETRY

305

ference between the two signals and also provide for the solution of the modulo 271 problem if the phase difference exceeds 27t during the scanning operation. A first step of signal processing is the transformation of the sinusoidal signals into square wave signals U,and U,by means of Schmitt triggers. The description will be divided into two parts: One will deal with the detection of the phase mod 27t and the other with the sign-correct counting of the difference number of phase jumps of one signal in relation to the other. (1) Phase detection mod 2x: As phase detector two parallel D flip-flops with a feedback by means of an AND gate can be used. The two voltages U,and U, are fed to the upper/lower set input of the phase detector. If U, leads or lags behind in time against U,,the phase difference will be defined as positive or negative, respectively. The AND gate delivers a reset pulse for the two D flip-flops if both flip-flops are set by the positive slope of the square-wave signal at the input. The setting of the second flip-flop determines the length of the rectangular output and, therefore, the voltage being proportional to the phase difference of the two signals. The output voltage is a positive rectangular pulse at the lower/upper output of the phase detector. The two outputs of the flip-flops are fed into a difference amplifier, as indicated in fig. 15, whose output goes through a low pass filter to obtain a voltage proportional to 9'.This detector gives 9' in the range & 27t with the correct sign. For I 9' I = 27t the phase jumps to zero, but the phase 9' mod2x remembers the correct sign and goes up (down) for 9' > 2n (< - 2x). Although this phase detector works in an analogous manner, the pulse length could also be detected with the help of a gated high-frequency pulse train (see KIKUTA[ 19871). In this case the pulse length being proportional to 9' serves as gate pulse for a counter. The sign can be derived from the fact that the positive/negative phase pulse stems from the lower/upper output of the two D flip-flops. (2) Detecting the number of phase jumps: The two voltages U,and U, drive a forward counter each in parallel to the aforementioned detector. Both counters are driven by the low/high slopes of the input signals. The counting results from these counters are subtracted, giving a whole number that represents the difference of phase jumps at the carrier frequency between the two photoelectric detectors. If both detectors are at rest, the difference will be zero. If one is moving, the relative phase is changing and each time the phase outruns 211 a difference count will occur. In this way the whole part of 9' can be obtained. This result and the fractional part of 9' are combined with the correct sign derived from the fraction to obtain 9' as wanted. The range of the counters should be sufficiently large to avoid overflow. Before the start of a measurement

306

ADVANCED EVALUATION TECHNIQUES IN INTERFEROMETRY

[IV, 5 6

all counters should be reset to zero. MASTNERand MASEK[ 19801 developed a phasemeter with a 211. x accuracy, which was used in heterodyne and INEICHEN[ 19771. The claimed accuracy experiments by DAENDLIKER presupposes some preprocessing of the signals entering the phasemeter. This preprocessing embraces automatic gain control and band pass filtering with a bandwidth of about 1/10 of the carrier frequency. For the application of heterodyne interferometry to holographic inter[ 19801 in ferometry the reader is referred to a review article by DAENDLIKER this series. A related electrical detection scheme for phase changes has been described [ 19871. Here, the sinusoids by, for example, KIKUTA[ 19871and WEISSMANN of the signal and the reference wave signals enter after rect-wave-shaping an AND gate, resulting in a square wave signal whose duty cycle is proportional to the phase shift. The phase measurement is performed with the help of a counter and a clock oscillator, with a frequency at the upper end of the possible counting rates (e.g., lo4 times the heterodyne frequency). The clock pulses are gated by the AND signal, and the counter is reset by the high/low transition of this signal. The phase resolution is of the order 2x x 10- 4. (2~)-Ambiguities can be resolved by using frequency-divided signals that furnish a selectable extension of the measuring range. STETSONand BROHINSKY [ 19861 discussed a similar phasemeter, that relies on the use of 3 counters, enabling also an extension of the measuring range to 2xm, where m is a whole number. The cyclic ambiguity is avoided because the reference is not used as a time marker. The phase change given in radians is measured in N cycles of the reference period known from heterodyning. The fractional parts of 2x are saved by using a very high-frequency clock and a counter gated by the heterodyne reference. In the following an interferometric device for optical testing shall be described. MASSIE[ 19781 and MOTTIER[ 19781 worked with external modulators in connection with a polarizing interferometer of the Twyman-Green type. The external modulator uses two acousto-optic modulators (AOM) tuned to frequencies differing by some hundred kHz. As an example, a heterodyne interferometer for optical testing will be described (fig. 16). The two AOMs are placed into the arms of a polarizing Mach-Zehnder interferometer.Two linearly polarized waves with the frequencies w and w + B emerge from this external interferometer, and are fed into a Twyman-Green interferometer (TWGI). This interferometer is also a polarizing type instrument with a polarizing beam splitter (PBS) at the entrance plane. The PBS directs the two linearly and perpendicularly polarized waves to different arms of the

IV,0 61

307

HETERODYNE INTERFEROMETRY

7 COMPUTER

jdDIVERGER SURFACE UNDER TEST

Fig. 16. Heterodyne interferometer used in optical testing. The test interferometer and the modulating Mach-Zehnder interferometer are arranged in series.

TWGI. In both arms quarter-wave plates (QWPs) are situated rotating the plane of polarization by 90 degrees in double transmission, enabling the light to leave the PBS at the other exit and thus isolating the interferometer from the laser. A further advantage is the higher light economy. Outside the interferometer on the entrance side a half-wave plate (HWP) is put in the ray path, which enables a balancing of the intensity ratio in the two arms of the interferometer. The two waves emerging from the TWGI are forced to interfere by passing a suitably adjusted polarizer. One possibility, given here, is the use of a further PBS in combination with a HWP to obtain two outputs, one for the reference detector and the other for an image dissector camera (IDC) to scan the image. The phase difference between the signals from the single detector and the IDC is measured by a phasemeter, which actually delivers the phase to be measured. The IDC scans the aperture with a rather high frequency. By means of an on-line computer the scan path in the aperture plane should be predetermined to avoid an interruption of the heterodyne signal. By choosing a closed scan path, control of the phase measurement is possible. Pistonlike phase drifts and vibrations are automatically eliminated, since the reference signal is changing its phase in the same manner. MASSIE[ 19801 proposed the use of 3 reference

308

ADVANCED EVALUATION TECHNIQUES IN INTERFEROMETRY

[IV, I 7

detectors to allow also for the elimination of tilt drifts or vibrations. For adaptive optics experiments the photoelectric signal is amplified with a gaincontrolled amplifier. With such methods rather fast and accurate measurements have been carried out. Similarly, this technique has been applied for surface profile measurements (MAKOSCHand SOLF[ 19811, HUANG[ 19831, PANTZER, POLITCHand EK [ 19861, PANTZER[ 19871, BIEDERMANN [ 19871, YOSHIZUMI,MURAO,MASUI,IMANAKA and OKINO[1987], LAERIand STRAND [ 1987]), where very high depth resolution has been reported. Further interferometers with internal modulators (contained in one or in both arms of the interferometer) were proposed for plasma diagnostics with high temporal and DEYOUNG[ 19771, resolution (LAVAN,VAN DAMME,CADWALLENDER HUGENHOLTZ and MEDDENS[ 19791).

6 7. Phase Sampling Interferometry (PSI) (rp = (r - l)rp,,) HERRIOTT, Phase sampling interferometry (PSI) was invented by BRUNING, [ 19741. Other names GALLAGHER, ROSENFELD, WHITEand DRANCACCIO for this technique are electronic phase shifting interferometry or, simply, phase shifting interferometry (WYANT[ 1982]),digital phase measuring interferometry (BRUNING, HERRIOTT, GALLAGHER, ROSENFELD, WHITEand DRANCACCIO [ 1974]), fringe scanning interferometry (BRUNING[ 19781) and often also simple phase measuring interferometry (MAHANYand BUZAWA[ 1979]), [ 19861 uses even the although the last name is tautological. FREISCHLAD terminus heterodyne interferometry, where the local oscillator assumes the form of a filter function. The mathematical treatment is based on correlation with a correlation length zero. In this way the numerous proposals for PSI can be summarized as a filtering process with two filtering functions being in phase quadrature. Recently a review article in this series on this subject by CREATH [ 19881 appeared. We have chosen the terminus technicus “phase sampling interferometry”, since for the performance of the technique a set of reference phase values, and thus a set ofintensity values are necessary (SCHWIDER, ELSSNER, SPOLACZYK and MERKEL [ 19851, SCHWIDER, BUROW, ELSSNER, FOELLMER, GRAZANNA, SPOLACZYK, WALLBURGand MERKEL[ 19851). In the same sense the term phase shifting interferometry seems satisfactory and also the abbreviation PSI, which can denote phase sampling as well as phase shifting or even phase stepping interferometry. The principle of PSI rests on the fitting of a cosine function to the set of r

IV, 8 71

309

PHASE SAMPLING INTERFEROMETRY

0

1

2

3

4

5

6

1

8

9

I

Fig. 17. Phase sampling understood as curve fitting.

intensity values I, at every single detector point (x, y). In fig. 17 the situation for one point in the aperture is given. The r intensity values are fitted to a cosine-type distribution underlying the two-beam interference phenomenon (eq. (6)). From a general point of view only 3 intensities are necessary to separate the 3 position-dependent functions Z,(x, y), V ( x ,y), and @ ( x , y) of eq. (6). In the case R > 3 a lsq-fitting process delivers higher accuracy. If the reference phase values are equidistantly distributed over one, or a number of periods, the orthogonality relations of the trigonometric functions provide a great simplification for the algorithm. To explain the technique, eq. (6) can be rewritten as I ( x , y ) = L +Mcoscp+Nsincp,

(56)

with L = I,, M = I, Y cos @, N = I, V sin @. If eq. (56) is multiplied with coscp,/sincp, and summed up over one period, where rp, = (r - l)q, and r = 1,2, . . .,R holds, then due to the orthogonality relations (MORSEand FESHBACH [ 19531, BAULE[ 19561) follows:

r= 1

r= 1

where f,.symbolizes the measured intensity values belonging to the reference phase values cp,. From the eqs. (56) and (57) follows R

310

ADVANCED EVALUATION TECHNIQUES IN INTERFEROMETRY

[IV,8 7

Equation (58) is the basic equation for PSI. If only 4 (R = 4) reference phase values are taken, that is, 9,. = 0, fx, x, and Zx, eq. ( 5 8 ) degenerates to a very simple equation

This formula is especially suitable for microcomputer usage, since the mathematical operations are easy to carry out. The tan- ' values can be stored in a look-up table to attain a high-speed algorithm. Proposals for 3 reference phase steps have been made by GALLAGHER and HERRIOTT[ 19711 and DOERBAND[ 19821. There are at least two ways of implementing a three-step method: (1) cpl = 0, cp2 = in, cp3 = x resulting in: @=tan-' (2) cpl

=

-1, + 2 f 2 - f3

f, - f3

4 x resulting , in: 0, q2 = ~2 x cp3, = ~

@=tan-'

J3i2- J3f3 21' - f2 + f3

Since the lsq-fitting process results in eq. (58), the whole procedure can also be understood as a Fourier analysis for the first coefficients of sin q, cos cp in a Fourier series approach. In a similar sense the technique may also be interpreted as a synchronous detection technique, since electronic noise and other fluctuations have a reduced influence on the measuring result because of a restricted electronic bandwidth (being inversely proportional to the measuring time). A standard scheme for a PSI is given in fig. 18. The light from a constant light source (in most applications a stabilized laser) enters a beam-splitting cube, dividing the light into a reference and test branch. The reference arm is equipped with a piezoelectricallydriven plane reference mirror, allowing for the adjustment of the reference phase values. The PZT is actuated by a high-voltage amplifier, which is controlled by an on-line computer. The design of the test arm depends on the measuring task and the test object; examples are given in 8 10. Generally speaking, the phase @ can be attributed to the test sample, although interferometer components contribute also to the deviations. Therefore calibration operations are a necessary step for high-quality measurement

IV, § 71

311

PHASE SAMPLING INTERFEROMETRY

ON -LINE COMPUTER

LI

. & REFERENCE

HIGH VOLTAGE AMPLIFIER

7 I

ARM

Fig. 18. Standard scheme of a phase sampling interferometer.

(see 0 9). A scanning detector array is situated in the exit plane of the PSI, which detects the intensities f r belonging to the reference phase values rp, and feeds the data by means of an ADC interface to the computer. The measuring process involves the following steps : adjustment of qryreading of the intensities fryand calculation of @ and other related quantities. WYANT[ 19751, STUMPF[ 19791, and KOLIOPOULOS [ 19811 used a related PSI technique, where the reference phase was not stepped but varied in a saw-toothlike manner. The intensity is integrated while the reference phase is tuned. Strict synchronization between the integration time of the detector and the movement of the piezoelectrically driven mirror should be maintained. The integrated intensity values are called "buckets". Especially simple are the 4- and 8-bucket methods. WYANT [1975] gave the following equations for 4 and 8 buckets: (4 buckets)

(62)

and

For an extensive discussion of the possibilities with this technique the reader is referred to KOLIOPOULOS [ 19811.

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ADVANCED EVALUATION TECHNIQUES IN INTERFEROMETRY

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Even periodic modulations of the reference phase (see 5) can be used in PSI, as has been shown by SASAKIand OKAZAKI[1986a], SASAKIand OKAZAKI[ 1986b], SASAKI,OKAZAKIand SAKAI[ 19871. The sinusoidal modulation is combined with an integrating detection mode. The synchronization phase 8 and the modulation amplitude z influence the accuracy of the measurement, since both quantities are contained in the formula for the phase to be measured. The integration time of the intensity covers one fourth of a period of the modulation function. As a result of the sinusoidal modulation, the integrated intensities depend on a series of Bessel functions ofthe modulation amplitude and on trigonometric functions of the phase difference 8between the modulating voltage of the PZT driver and the starting point of integration. An optimum choice for the modulation amplitude z and the phase 8 was derived by the aforementioned authors to be z = 2.45 and 8 = 56 degrees. The authors claim an accuracy of about 1 nm in a reflected light interferometer. Most of the PSI publications rest on the Twyman-Green interferometer (TWGI), but the Fizeau interferometer has also been applied in PSI. For [ 19801 described a spherical spherical surfaces, MOOREand SLAYMAKER Fizeau interferometer, which is also used in the commercial version of the ZYGO M IV instrument. An aperture-dependent phase shift occurs in connection with the axial translation of a spherical mirror. Therefore, either a mathematical compensation (MOOREand SLAYMAKER [ 19801) or a comBUROW,ELSSNER,GRZANNA,SPOLACZYK pensated form of PSI (SCHWIDER, and MERKEL[ 19831)is necessary. The PSI technique has been applied in many commercial interferometers because of its simple operation with high accuracy and its automatic detection of the interferometer aperture. SCHWIDER,BUROW, ELSSNER, GRZANNA and SPOLACZYK[ 19861 applied the PSI technique to the evaluation of frozen fringe interferograms. For this purpose a double diffraction apparatus and a grating phase shifter are suitable. The evaluation technique as a whole is equivalent to PSI with the restriction that the sign of the phase should be known beforehand.

7.1. PHASE UNWRAPPING

As aresult of the periodic character of the two-beam interference pattern and the even character of the cos-type intensity distribution, it is necessary to discuss the phase ambiguities that each phase measuring method poses for phase unwrapping. ITOH [ 19821 summarized the unwrapping algorithm as follows: Phase

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313

unwrapping can be understood as an integration of the wrapped phase differences. The general assumption is a fulfilment of the sampling theorem; that is, at least two samples should be taken from one period of the interference pattern (Nyquist limit), or equivalently, the phase difference of adjacent pixels should lie in the region

- n < Gn - G n - , < + n , where a,, Gn- are the phase values of the adjacent pixels. GHIGLIA, MASTIN and ROMERO[ 19871 emphasized that in the case of a continuous function the phase would jump by 2n if the principal values are calculated from the arctangent. Of course, this is not true for the sampled function. The phase difference approaches n if the sampling distance is chosen equal to the Nyquist limit. Such neighborhood operations enable the removal of phase discontinuities (i.e., phase unwrapping) in two dimensions. GHIGLIA, MASTIN and ROMERO[ 19871proposed the use of cellular automata to solve the unwrapping problem fast and unambiguously. They devised iterative procedures to unwrap patterns disturbed by “salt-and-pepper” noise effects. Evaluations based on image processing methods of a single interference pattern have the most serious loss of information on the sign of the phase in the interference pattern. If, in addition, the carrier frequency in the interference pattern cannot be chosen adequately (i.e., the spatial frequency is greater than the maximum of the wavelength slope), closed fringes may appear. This situation is rather common, especially in holographic interferometry (OSTROWSKY [ 19871). The common procedure to resolve phase ambiguities is the use of a priori knowledge. To enable manipulations of the phase pattern, interactive software is necessary, as was discussed in $ 3.2.5. This software allows for the numbering of the fringes. To accomplish this, either touch screens in combination with a graphics display (YATAGAI, NAKADATE,IDESAWAand SAITO[ 1982a1) or graphics hardware, with the possibility of overlaying the fringe pattern from the vidicon or CCD matrix with fringe skeletons or the like (LIU and BIRCH[ 1984]), can be used. By drawing intersecting vector lines between adjacent zones with closed fringes (locations where valleys or mountains exist) and using the a priori knowledge, a sufficient fringe numbering can be carried out. Furthermore, if additional measurements are made, for example, one additional measurement with a phase offset (KREIS[ 1986a]), the sign retrieval in two dimensions can be accomplished without a priori knowledge. The latter applies more generally to all methods where the reference phase is varied in

314

ADVANCED EVALUATION TECHNIQUES IN INTERFEROMETRY

[IV, 8 7

time, that is, where several intensity values at one point of the aperture are taken to retrieve the phase. To attain this, different provisions need to be made, depending on the phase measuring method. In the case of PLI it is necessary to have a phase shifter that covers the whole excursion of the phase to be measured (MOORE,MURRAYand NEVES [ 19781). Similarly, for heterodyne interferometry the phasemeter should be equipped with a bidirectional counter to count the 2n phase jumps in one or the other direction. PSI requires the fulfilment of the sampling theorem; that is, the detector should sample the fringe pattern at more than 2 points per period throughout the whole interference pattern. If this condition is met and the phase difference of adjacent pixels exceeds n, a phase jump will occur and a phase of 2n should be added with the sign of the phase of the previous pixel. Thus it is possible to correct the phase data and obtain a continuous phase function. This procedure should be carried out in two dimensions, first for one diameter and after this in the perpendicular direction from this diameter proceeding over the rest of the aperture. In fig. 19 this procedure is indicated for the case where the sampling theorem is fulfilled and where it is not. Since PSI delivers the phase solely by means of the arctangent function, the raw phase values should undergo an extension from the range of ( - n, n) to ( - n, n)by taking the sign of the numerator and denominator of eq. (58) into account. GREIVENKAMP [ 19871 pointed out that the usual limitation posed by the sampling theorem can be overcome by using a priori knowledge of the wavefront to be measured, a technique named sub-Nyquist PSI. This case often occurs in optical testing of smooth surfaces, especially with aspheric testing. If a sparse-array detector is used, that is, a detector with small light-sensitive areas compared with the distance of adjacent detecting elements, the undersampling of the fringe pattern leads to aliasing effects. This can be traced by the moire-effect between the detector geometry and the fringe pattern. In the simple case of an equidistant and parallel set of fringes (wedge fringes), there are zero fringe patterns seen by the detector if the number of fringes equals a multiple of the number of detecting elements. The contrast of the fringes for the higher order aliases is, of course, diminished. This puts a principal limit on the workable aliasing degree. For arbitrary wavefronts the aliasing effects are strongly localized. A typical example may be a rotationally symmetrical aspheric. In the center the normal phase unwrapping procedure can be used. An ambiguous reconstruction of the phase @ from a set of values mod2n assumes a sufficient region where the normal PSI phase unwrapping procedure works. The continuation algorithm uses the continuity of the first and second

4 4

PHASE SAMPLING INTERFEROMETRY

Y

315

Fig. 19. Phase unwrapping failure resulting from undersampling. Left, number of fringes below Nyquist limit; right, number of fringes just above the Nyquist limit in the lower part of the picture.

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ADVANCED EVALUATION TECHNIQUES IN INTERFEROMETRY

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derivative. The starting region has the extent of (n + 1) x (n t 1) pixel of the nth derivative is used in the procedure. GREIVENKAMP [1987] claims an extension of the measuring range by two orders of magnitude. Another possibility to extend the measuring range is the use of two or multiple wavelength interferometry. This method has many similarities to the one just described. Here, agreement between the phase values for the two wavelengths delivers the criterion for the phase unwrapping algorithm (CHENGand WYANT [ 1985a,b]). Details are discussed in 9.6. In addition, the contrast of interference fringes in polychromatic light can be used to indicate the fringe order number (STRANDand KATZIR [ 1987]), which enables the extension of the phase unwrapping algorithm to the sub-Nyquist range. To obtain unambiguous results, nonwavelength-dependent contrast variations should be excluded. This restricts the range in the presence of additive noise accordingly. 7.2. CALIBRATION METHODS FOR REFERENCE PHASE SHIFTERS

This subsection describes different methods for the calibration of phase shifters. Calibration is a necessary step in PLI and PSI measurements, since the accuracy depends very much on the linearity and to some extent also on the reproducibility of the phase shifter. The typical phase shifter in PSI, one of the most widespread techniques, is the piezoelectric transducer (PZT). Unfortunately, these transducers are susceptible to hysteresis and other nonlinear responses. The preference of PZT-driven phase shifters is based on their simplicity and their freedom from rotatory movements of optical elements. In this way the modulation of the intensity by co-rotating dust diffraction patterns is avoided, guaranteeing higher stability of the method against systematic errors. SCHWIDER,BUROW, ELSSNER,GRZANNA,SPOLACZYKand MERKEL [ 19831 proposed the use of 4 values: 0, qo, 3q0, and 4q0, to obtain the calibration constant of the high-voltage generator driving the PZT. If the intensities are numbered I, through Z4, the value for qo is

I -Io

qo = arccosi 4 , 13 - I1

provided qois known to some degree, so that the sign ambiguities of the cosine function can be resolved. This method has been modified and used in the intergrating mode of PSI by CHENGand WYANT [ 19851. By measuring with

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317

different values qo, one can determine the necessary information about the characteristic curve of the PZT. Of course, such methods have value only if the phase shifter works more or less linearly. This can be realized, for example, by using of a capacitive-controlled PZT (SPOLACZYK, ADAMSBERGER and [ 19791). SCHWIDER Assuming linearity of the PZT or other phase shifter, the method of CARRE [ 19661 can be applied to the PSI, as has been done extensively by the group at the Optical Science Center in Tucson, Arizona. The phase is stepped from - 3q0, - qo,qo,and 3q0, with the corresponding intensity values I, through I,. The phase to be measured follows then from

Since the sensitivity of the results against reference phase errors is highest for a small number of steps, several methods have been discussed to attain high accuracy for the reference phase values 0, in, x , and $ x using the cos-type intensity distribution itself. KOLIOPOULOS [ 19811 discussed the use of the phase-lock technique to get the preceding values. For this purpose the PZT has to be wobbled sinusoidally with a small amplitude. The signal from a photodetector is detected by a lock-in amplifier at the frequencies o and 20, where w is the wobble frequency. With some a priori knowledge the zeros of the w and the 2w signal deliver the values for 0, x and in, and In, respectively (see also eq. (53)). KOLIOPOULOS [ 19811 also gave a second method that delivers the preceding values by convolving the intensities I, of q,. with a square-wave function (method proposed by SNYDER [ 19801) once and twice, thus obtaining the zeros of the first and second derivative (see 5 3.2.3) of the function Z,(q,). The zeros of the first and second derivative deliver the reference phase values 0, x , fx, and $ x . KINNSTAETTER, LOHMANN, SCHWIDER and STREIBL [ 19881 proposed using the signal of two photodetectors to obtain a set of point pairs being in phase quadrature. If the two signals I , (q,) and I,(q,) are displayed on the x and y axis, respectively, of a graphics display, a Lissajous figure can be observed, which degenerates to a circle if the signals are in exact phase quadrature and the mean photovoltage at the two points is the same. The Lissajous figure enables the detection of the linearity, the periodicity, and the uniformity of the phase shifter. Also, nonlinearities of the detector can be detected from the deviations of the Lissajous figure from a circle or, more generally, from an ellipse.

318

ADVANCED EVALUATION TECHNIQUES IN INTERFEROMETRY

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The set of intensity values for 32 phase steps, for example, can be used to implement an iterative correction method. Corrected phase values (or a corrected table of control bytes for the high-voltage amplifier) can be obtained iteratively by calculating $r from

l -

z01.2 =

The error q

R

cR I],>,

r=l

$: - qr can be used to correct

=

+ 1) = u,(n) - u(er)/4,

u,(n

$r

and therewith u,, that is,

with 4 > 1 .

By changing the voltages u, for each r in a sign-correct fashion, because of the characteristic curve one obtains an improvement of the phase stepper perform25OfU2

200

oo

0 0

0

2oo/

0

0

0

0

0

50!

0

loot

0

0

0

0

0

0

0

0

5 0 i

0

0

0

0

0

50

0

0 0

0

0

O 0 0 0 0

0 0

50

LOO

0 0 0

0

0

0

0

1

0

0

150

200

250

Fig. 20. A sequence of Lissajous figures resulting from the self-correctingalgorithm underlying eq. (66). Left,Lissajous figure for an uncalibrated phase shitler; right, Lissajous figure after one iteration cycle. Note the closing of the gap in the Lissajous figure.

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PHASE SAMPLING INTERFEROMETRY

319

ance. Figure 20 shows a sequence of two such pictures. This technique can be further improved by averaging over many such point pairs in phase quadrature, since the reference phase error is periodic with the phase to be measured (see eq. (71)). W=l 1371 RI1s=0 3001

-8

1841

Rrs=e.e411

12

27.1.86 F\IU-EZ I

Fig. 21. Deviations of a spherical surface. Top, relevant data screened by tilt and defocus terms; middle, tilt and defocus terms eliminated; bottom, sections through the surface along the x and y axis.

320

ADVANCED EVALUATION TECHNIQUES IN INTERFEROMETRY

[IV,0 8

CHENGand WYANT[ 19851monitored the features of the PZT by displaying the intensity curves for Z,(cp, + i n n ) with n = 0, 1,2,3,4. From the agreement of 1, with I4 in this way one C M detect very small phase step errors. Nevertheless, one should emphasize that these methods presuppose a very good mechanical stability during the phase stepping procedure, since all changes of the optical path difference (OPD) show up as reference phase or intensity variations. A technique that greatly helps against all disturbances is to average different measuring results, which can be accomplished easily due to the high speed in the evaluation process.

6 8. Relevant Data and Merit Functions The phase measured with the techniques discussed has different meanings, depending on the measuring task at stake. In optical testing the relevant information is, in general, concealed by adjustment aberrations of the interferometric setup. Therefore the phase data have to undergo a lsq-fitting operation, where according to the type of surface or component under test, linear, quadratic, or even more involved phase terms are fitted and their contribution to the aberration is subtracted. The remaining data can be considered as relevant, provided the interferometer has been calibrated beforehand (calibration procedures are discussed in Q 9). The possible functionals for plane and spherical surfaces have already been given (eq. (12)). Figure 21 shows the situation for a spherical surface, where the continuous data are freed from linear and quadratic phase terms. The wave aberrations or surface deviations are usually displayed in pseudo-3D plots, as contour line plots, or as sections in the x or y direction of the aperture.

Fig. 22. Level slice plot (right) of a deviation pattern (left). S, slicing level; P,percentage of area within two slicing levels; + , - ,part of the surface lying above or below slicing limits.

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321

For the optician in the workshop a picture on a color display or as a level slice plot (fig. 22) is also suitable. If the surface shall be characterized by a few numbers the peak to valley deviation or the mean square deviation is given by

where M is the number of data points in the aperture. For a display of the data it is essential that data in the rim region of the test sample be excluded from the calculation, since due to fabrication strong deviations in this region are inevitable. Optical systems are not only characterized by the wave aberration data, but also by their characteristic system functions, such as the point spread function and the optical transfer function. For the latter the modulus (MTF) primarily 12

27.1.86

aw-Ezi

12

27.1.86 anu-Ez i

Fig. 23. Point spread function (top, left) and two-dimensional modulation transfer function (MTF) (top, right). Bottom, one-dimensional sections through the MTF; ideal curve shows the MTF for a diffraction-limited system of the same aperture. The objective under test was a micro-objective with the numerical aperture N.A. = 0.40; V, Strehl definition.

322

[IV,S 8

ADVANCED EVALUATION TECHNIQUES IN INTERFEROMETRY

is of importance. For the two limiting cases, that is, incoherent and coherent illumination, these functions can be derived from the wave aberrations (BORN and WOLF[ 19641). Commonly, the incoherent case is calculated by means of Fourier transformation from the pupil function. In a first step the PSF is calculated as

and from these data the MTF follows as the modulus of the inverse transform MTF

=

IF-’(PSF)I

.

(69)

In addition, the Strehl definition is calculated and given in the display of the PSF (fig. 23). The MTF is point-symmetrical, per definition, and therefore one-dimensional sections in the spatial frequency plane are displayed. For a comparison the ideal curve is given with the objective MTF. For very small deviations it has become common practice to add a linear function to the phase data and displaying the sum (fig. 24). The contour line plot then shows strong similarity to a normal interferogram but with a higher depth resolution from “fringe to fringe” (fig. 24). In the analysis of the microstructure of surfaces, other parameters are relevant. The surface profile (one-dimensional) or the surface geometry (twodimensional) are the quantities measured with PSI, PLI, or heterodyne techniques. BHUSHAN,WYANT and KOLIOPOULOS [ 19851 developed an interference microscope with PSI evaluation. From the profile data the autocovariance and the power spectrum can be evaluated (fig. 25). The power spectrum and the autocovariance are especially valuable if the surface shows periodic deviations, as for example, diamond-turned optical surfaces.

, 8 , I

-

. \

\

I

Fig. 24. Addition of a linear function for a clear display of small deviations.

3 1 I

RELEVANT DATA AND MERIT FUNCTIONS

323

Fig. 25. A three-dimensional plot of a diamond-turned mirror measured at 100 x . The middle figure is a plot of the averaged power spectrum in the x and y directions. The bottom figure is the autocovariance showing the statistical correlation of the surface. (Courtesy of J. Wyant, Wyko Corp., Tucson, AZ.)

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ADVANCED EVALUATION TECHNIQUES IN INTERFEROMETRY

[IV,§ 9

8 9. Calibration Methods for Interferometers Since the interferometers used in the assessment of test data are rather complex, there is some necessity for calibration. The interferometer aberrations, even with ideal test samples, are caused mainly by deviations of optical elements and to some extent also by the state of adjustment. In optical testing the accuracy requirements are rather stringent, and calibration seems to be of great importance. Typical reference waves in interferometry are either plane or spherical. Therefore, the establishment of flatness and sphericity standards is a necessary step. If such standard surfaces have been established, the interferometer can be calibrated with the help of the standard surface, whose known deviation data are used to correct the content of the interferometer data in the memory of the on-line computer. Recently, FRITZ[ 19841 worked out a planeless test using the scheme outlined by SCHULZ[ 19671 (see also SCHULZand SCHWIDER [ 19761). He combined an algorithm given by PARKS[ 19781for the evaluation of rotational shear interference patterns using Zernike polynomials for interpolation and the 4-combination method of SCHULZ [1967]. The basic idea of absolute planeness tests is to measure the deviation sums of 3 combinations in pairs of 2 plates. This leads to a linear system of equations, which can be solved only for one diameter in a first approach. To obtain deviations across the whole flat rotations (SCHULZ[ 19671) or translations (DEW[ 19671, SCHWIDER [ 19671) of one of the surfaces relative to its partner should be added providing deviations on other diameters or sections. In this way absolute planeness HILLERand KICKER deviations have been measured (SCHULZ,SCHWIDER, [ 19711, FRITZ[ 19841). GRZANNA, BUROW,SCHULZand VOGEL[ 19881 used 5 combinations of three normals, that is, 3 basic positions, one shifted and one rotated position (by 90 degrees), to establish absolute flatness deviations over a Cartesian net of data points given by the geometry of the detecting matrix. In this way interpolations over a polar net become superfluous. In the case of spherical surfaces a similar procedure (HARRIS[1971], HARRISand HOPKINS[ 19791) can be followed. Much more efficient is a method in which the even and odd part of the absolute surface deviations is [ 19761). The measuredetermined (JENSEN[ 19731, SCHULZand SCHWIDER ment of the odd part of a surface deviation is rather straightforward, since only two interference patterns have to be subtracted, where the second pattern is made with a 180"-rotated position of the surface to be tested. BRUNING, HERRIOTT, GALLAGHER, ROSENFELD, WHITEand BRANGACCIO [ 19741 and SCHWIDER, ELSSNER,GRZANNAand SPOLACZYK [ 19851 gave test results based on 3 interference patterns.

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325

The accuracy of the test greatly depends on the quality of the adjustment stage and the special auxiliary software to guarantee the coincidence of the coordinate systems through all combinations. In fig. 26 a typical example of an absolute sphericity test is shown. When making absolute tests, one quickly realizes that the coherent noise is the limiting factor for accuracy. Since the last test mentioned also assumes a position with an imminent wavefront inversion, spatial coherence is a necessity in the experiments. Therefore the influence of coherent noise becomes apparent if the deviations are rather small, as can be expected. To overcome these difficulties, two different approaches exist, that is, the global fit of an orthogonal set of polynomials (e.g., Zernike polynomials) or digital low pass filtering operations. SCHWIDER, ELSSNER,BUROW,GRZANNA and SPOLACZYK [ 19851 (see also SCHWIDER, ELSSNERand GRZANZA [ 19861, and SPOLACZYK[ 19891) developed a very ELSSNER,BUROW,GRZANNA efficient filtering technique on the basis of iterative convolutions (BURGERand VAN CIITERT [ 19321). The computing time for this type of filter is about 2 s on a slow 280 computer. KUECHEL[ 19861 applied PSI to calibrate big lightweight plane mirrors with the help of a Ritchey-Common test combined with an absolute test for the auxiliary sphere used in the Ritchey-Common test. Other surfaces cannot be tested in such a simple way for absolute deviations apart from parabolic and cylindrical surfaces (SCHULZ and SCHWIDER [ 19761). Nevertheless, the application of difference measurements will be helpful in many cases, especially if some sort of normal (e.g., a computer-generated hologram [CGH]) exists already. In general, calibration enables the use of medium-quality optics in interferometers for high-accuracy measurements. In roughness measurements using an interference microscope

Fig. 26. Absolute deviations from an ideal sphere.

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ADVANCED EVALUATION TECHNIQUES IN INTERFEROMETRY

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(KOLIOPOULOS[1981]), the reference surface may influence the r m s or peak/valley (P/V)figures. Here, either the difference of two surface profiles is taken from the same sample but at different locations, thus freeing the results from the reference surface contribution, or the reference surface data are determined by taking the average of a big ensemble of different but high-quality surfaces. If the surface deviations of the test samples are statistically distributed, the average will represent the deviations of the reference surface.

4 10. Applications Real-timeinterferometry has been applied to nearly all test situationsprevailing in optics. The most common interferometer types are the Fizeau, Twyman-Green (TWGI),and Mach-Zehnder interferometers. Fizeau interferometers are especially suited for testing plane surfaces. A typical phase sampling Fizeau interferometer for planeness testing is shown in fig. 27, and this type has been used for absolute testing (FRITZ[ 19841) and for the test of microgeometrical features of optical surfaces (see 10.2). The big advantage of a Fizeau design is its stability against shock and vibration and its immunity against dust diffraction if parallel adjustment is maintained during the measurement. 10.1. OPTICAL TESTING AND FLATNESS TESTS

Optical systems, prisms, and gratings are mostly tested in a TWGI configuration because of its variability and adaptability to different reflection

SURFACE SURFACE Fig. 27. Phase sampling Fizeau interferometer for planeness testing.

APPLICATIONS

321

Fig. 28. Phase sampling Twyman-Green interferometerfor testing spherical surfaces and objectives.

coefficients. BRUNING,HERRIOTT, GALLAGHER, ROSENFELD, WHITEand BRANGACCIO [ 19741, MAHANY and BUZAWA[ 19791, BALASUMBRAMANIAN and DEBELL[ 19801, and SCHAHAM [ 19811 gave measuring and application examples for spherical testing and optical system characterization. Figure 28 shows a typical TWGI taken from SCHWIDER, BUROW,ELSSNER, GRZANNA, SPOLACZYK and MERKEL [ 19831, SCHWIDER, BUROW,ELSSNER, FOELLMER, GRZANNA, SPOLACZYK, WALLBURGand MERKEL[1985]. This is a polarizing TWGI described, for example, by DYSON[ 19801. The need for calibrationoperations becomes evident, especiallyin this case, since a multitude of optical elements contribute to the interferometer aberrations. MERKEL, GIGGEL,ELSSNER and SPOLACZYK [ 19881 demonstrated the use of PSI with a calibrated TWGI in the fabrication of high-quality optics for systems used in microlithography. HARIHARAN and OREB[ 19841 applied the TWGI to the measurement of a plane Fabry-Perot etalon. Because of the high accuracy and the possibility of making difference measurements, very high parallelism could be reached. Testing aspheric surfaces can be carried out with the help of synthetic holograms (SH). SCHWIDER and BUROW[1975] and SCHWIDER [1977] used an SH with rotational symmetry to test steep aspheric surfaces that generate wavefronts with a tendency to wavefront warping. To overcome such problems, the SH should be placed in one arm of the interferometer. Here, the computer-orientedPSI could help to solve the problem of wavefront distortions stemming from the SH glass support. Figure 29 shows a test interferometer in transmitted light, where the SH compensates the wavefront generated by the

328

ADVANCED EVALUATION TECHNIQUES IN INTERFEROMETRY

ASPHERIC

[IV, B 10

CG H

COMPENSATING

t

HIGH AMPLIFIER

ON -LINE COMPUTER

Fig. 29. Phase sampling Mach-Zehnder interferometer for testing of aspherics in transmitted light using computer-generated holograms.

aspheric to a plane wave. The error stemming from the thickness variations of the glass support can be measured if the aspheric plus compensating lens is removed from the ray path and the interference pattern of the undiffracted wave from the SH is superimposed with the plane reference wave. If these data are stored in the computer, the interferometer and SH deviations can be subtracted in the final measuring step from the aspheric/SH deviation picture. In the measuring process the minus first diffraction order is used. The wavefront outside the aspheric/SH combination is plane with good approximation, as during the calibration operation. This calibration also eliminates the interferometer errors of the empty instrument. DOHERTY[ 19791 and DOERBAND and TIZIANI[ 19851 used a TWGI with a plane-parallel plate phase shifter to measure aspherics with PSI. The computer-generated hologram (CGH) was placed outside the interferometer. The latter also gave test results using the FFT method (5 3.4) and compared the two methods with a big set of data points. MACGOVERN and WYANT[ 19711 and HARIHARAN, OREBand BROWN [1982] tested aspherics with the SH outside the interferometer. The interferometer is again of the PSI type. Shearing interferometry has also been applied to the testing of aspherics by YATAGAI[ 19841. In addition, technical flats can be tested by interferometric methods, for example, semiconductor wafers (TROPEL [ 19841, SCHWIDER,BUROW, ELSSNER,GRZANNA and SPOLACZYK [ 19861). The necessary depth resolu-

IV, B 101

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329

tion is determined by the depth of focus of the optical repeater and is of the order of 0.1 pm. On the other hand, the range of surface deviations can be of the order of 10 pm, and thus it is the usual procedure to enlarge the effective wavelength. One very common solution is grazing incidence interferometry, for [ 19691).The example, by means of a prism-Fizeau interferometer (ABRAMSON effective wavelength can be varied by changing the angle of incidence a. For normal incidence one fringe corresponds to a surface deviation of L/2, but for grazing incidence the surface deviation per fringe amounts to L/(2 cos a). The sensitivity is greatly reduced in the neighborhood of 90degrees. The PSI technique has been applied with different phase modulators. TROPEL[ 19841 used polarizing means, and SCHWIDER, BUROW,ELSSNER,GRZANNAand SPOLACZYK [ 19861 used a grating interferometer in series to scan the interference pattern. Alternatively, Fizeau interferometry has also been applied by YATAGAI,INABA,NAKANOand SUZUKI[1984]. By using rather dense sampling (512 x 512 pixels) and hardware logic, a wafer tester has been built. 10.2. MICROSTRUCTURE MEASUREMENTS

The microstructure of optical and other surfaces can be measured with the help of profilometers working with different principles. LEINERand MOORE [ 19781 described an interference microscope using PLI. SOMMARGREN [ 19811, HUANG[ 19831, and recently, PANTZER,POLITCHand EK [ 19861 and LAERIand STRAND[ 19871 built instruments for roughness testing on the basis of heterodyne interferometry. HUANG[ 19831 claimed a depth resolution of about 0.1 A by comparing a wave reflected from the tiny measuring spot with a wave reflected from an extended region in the neighborhood of the measuring point. The surface should be scanned mechanically, which may cause some noise and endanger the claimed accuracy values somewhat. KOLIOPOULOS [ 19811 and WYANT,KOLIOPOULOS, BHUSHANand BASILA [ 19811 applied PSI techniques to a profiling interference microscope of the Mirau type. The measurement is carried out with a CCD-line camera with 1024 pixels. The Mirau interferoscope can be understood as a Mach-Zehnder interferometer under oblique illumination. The depth resolution is of the order of several Angstrom units. In the measurement of microstructure the coherent noise (see 8 11.3.5) is a serious problem. Therefore KOLIOPOULOS [ 19811 worked with an incoherent light source and investigated the influence of the degree of coherence (spatial and temporal) on the attainable accuracy. Another problem with the measurement of very small surface deviations is the contribution of the sharply imaged reference flat itself. Here, an averaging of different surface profiles can be used in a type of calibration operation.

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p-t: t

t

ARRAY

Fig.30. Micro-Fizeau interferometer.

CHENand MAIMON[ 19811 used a Fizeau interferometer (fig. 30), in which the reference surface is the front surface of a A14 plate, which diminishes the influenceof coherent noise and in combination with the polarizing beam splitter (PBS) achieves a decoupling of the detector array from the illuminating laser. The quarter-waveplate (QWP) is phase stepped by a PZT. As detector, CCD arrays were used with 128 x 128 pixels and 488 x 380 pixels. The macrogeometrical wavefront aberrations were removed by fitting a Zernike polynomial of degree 8 and subtractingthe polynomialfrom the measured data. The difference is the microstructure data. Such measurements are of importance in connection with the scattering in optical systems, scattering in X-ray optics, especially with X-ray telescopes, and for the transmission factor of dielectric interference filters.

10.3.

SHAPE MEASUREMENTS OF GROUND SURFACES

The measurement of the shape of ground surfaces is the counterpart to microstructure studies. There are two competing methods. That is, infrared interferometryand two-wavelengthinterferometry. The COz laser, with a wavelength of 10.6 pm, has been used by WYANTand CREATH[ 19851. The IR

APPLICATIONS

.-a

s 8

z

e

c)

I..

0

Fig. 31. False color picture of the deviations of a telescope mirror in an intermediatefabrication stage; all deviations are given in nanometers. (Courtesy M. Kuechel, CZ Oberkochen, FRG.) 33 1

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ADVANCED EVALUATION TECHNIQUES IN INTERFEROMETRY

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radiation is detected with the help of a chopped pyroelectric vidicon. FERCHER and VRY [ 19861 used a two-wavelength method to obtain the surface profile of an optical surface. The effective wavelength

resulting from the superposition of two neighboring Ar-laser wavelengths, can be adjusted to the measuring task. DAENDLIKER, THALMANNand PRONGUE[ 19881 proposed a superheterodyne technique for range-finding applications that could also be used in surface measurements. A polarizing Michelson interferometeris illuminated by two lasers with different wavelengths. Each laser beam is split and modulated with the help of an acoust-optic modulator (AOM) in a similar manner to that given in fig. 16. The two AOMs provide for frequencies v, + fl and v, + f,. The resulting interference pattern at the exit of the measuring Michelson interferometer is, in fact, an additive moirk. This moire is transformed by nonlinear detection into a multiplicative moire, and after amplitude demodulation of this signal and a signal derived at the entrance of the interferometer the phase difference at the effective wavelength (eq. (70)) is measured. All drifts are eliminated by the superheterodyne technique. Because of the nonlinear detection, the phase unwrapping is done as usual with heterodyne methods. KUECHEL and HEYNACHER [ 19881 used a PS interferometer measuring with a CO, laser during the grinding process and a He-Ne laser in the polishing stage of telescopic mirror fabrication. The aspherical mirror deviations were compensated by null lenses. The measurement requires a special stabilization loop to get rid of vibrations of the long focus optics. A typical test result of an intermediate fabrication stage is given in fig. 3 1 in a false color representation.

10.4. MOIRe TOPOGRAPHY

Moire topography is one of the noninterferometric fields for advanced phase measurement methods, since as in the case of moire fringes, the intensity distribution is periodic and with good approximation also of the cos type, as with two-beam interference fringes. No wonder that nearly all conceivable evaluation methods have been tried in this field. Applications of the moire topographic techniques are known from mechanics, hydrodynamics, and last but not least, biostereometrics.

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Reviews on moirC techniques have been published recently by POST [ 19821 and REID [1984]. The evaluation methods comprise the whole range of possibilities, range from image-processing methods (YATAGAI,NAKADATE, IDESAWA and SAITO[ 1982a1, YATAGAI,IDESAWA,YAMAASHI and SUZUKI [1982]) to phase-lock moire (MOOREand TRUAX[1979]) to heterodyne detection (SRINIVASAN, LIU and HALIOUA[ 1984]), and to PSI techniques (YATAGAIand IDESAWA[ 19821, REID,RIXONand MESSER[ 19841). Since all modern methods use on-line computers, data sets of different states of objects can be subtracted or otherwise superimposed, enabling the detection of variations in the state of objects very easily. 10.5. MEASUREMENT OF REFRACTIVE INDEX DISTRIBUTIONS

The measurement of refractive index distributions is a further field of application, either in the form of measuring inhomogeneities or the spatial distribution of the refractive index. Beginning with the latter, PRESBYand ASTLE[ 19781 determined the refractive index profile, either perpendicularly to the preform or axially in an interference microscope. The fringes are evaluated for the fringe extrema by image processing methods. The application of PSI to measure small inhomogeneitiesin high-quality glass plates has been carried out by SCHWIDER, BUROW,ELSSNER,GRZANNA and SPOLACZYK [ 19851. To separate the influence of the glass body inhomogeneities from the surface deviations, a combination of 4 interference patterns needs to be evaluated. 10.6. TWO-WAVELENGTH INTERFEROMETRY

Two-wavelength interferometry can also be used in cases where a reduction of the number of detected fringes is advisable, for example, in testing aspherics. This method was originally proposed by WYANT[ 19721and has been upgraded by the use of the phase shifting technique (CHENGand WYANT [1984], CREATH, CHENGand WYANT[ 19851). Two-wavelength interferometry is, in fact, an additive moirC technique. Therefore the photoelectric detector needs resolve the fringe pattern. To cope with the problem of different reference phase values for the two wavelengths, the phase evaluation process of CARRE[ 19661 has been used in this connection, since the ratio of the reference phase values for the phase evaluation formula (see 5 7.2) is essential and not the actual reference phase value. In this way good results for aspheric surfaces could be

334

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ADVANCED EVALUATION TECHNIQUES IN INTERFEROMETRY

obtained, although there is a loss in accuracy because of the long effectivc wavelength. Therefore CHENGand WYANT[ 1985al combined the capabilities of the two wavelengthswith that of the single-wavelengthPSI. The underlying philosophy is the following: Two-wavelength interferometry allows for the unambiguous surface reconstruction, and also, in the case of steep surfaces, single-wavelength interferometry delivers the accuracy. The background information of TWPSI enables the elimination of the 2x phase jumps in the SWPSI, which follows from the undersampling of the interference pattern. The application of this undersampling technique from the point of view of single-wavelength PSI assumes a &like sampling detector or a higher pixel capacity of the detector so that every second pixel or so can be omitted. The usual array detectors would average the intensity modulation and in this way make this type of PSI impossible.

10.7. HOLOGRAPHIC INTERFEROMETRY

Holographic interferometry is a main field of application for nearly all fringe [ 19801 gave a evaluation techniques discussed in this survey. DAENDLIKER survey on heterodyne techniques in holographic interferometry, but static processing methods have also been applied. HARIHARAN, OREBand BROWN[ 19821 and HARIHARAN [ 19851 demonstrated PSI-holographic interferometry with the 3 reference-phase method.

LASER

TESTPIECE

Fig. 32. Heterodyne holographic interferometer using two reference waves.

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335

With a two-reference-wave holographic interferometer “frozen fringes” can also be evaluated, either with heterodyne (DAENDLIKER [1980]) or with PSI (DAENDLIKERand THALMANN[ 19851) techniques. THALMANNand DAENDLIKER [ 19871 compared the heterodyne and PSI for applications involving the two-reference-wave holography. They found out that heterodyne techniques are only applicable in the laboratory but guarantee very high accuracies, whereas PSI is fast and in most applications sufficiently accurate, especially in industrial environments. The principle of two-reference-wave holography is sketched in fig. 32. The laser light is diffracted by a rotating radial grating furnishing 3 waves with the frequencies w, w f Aw (where Aw represents the frequency shift due to the rotation). During the exposure of the holographic interferogram, the reference wave R, is taken for the object state 1 and R, for the state 2. During exposure, the grating is at rest. Upon reconstruction the radial grating rotates, generating a moving interference pattern that is detected by the two detectors D, and D,. D, is taken as reference while D, is scanning the hologram in the case of heterodyne interferometry. For PSI the radial grating (a linear grating could be used just as well) is stepped through the R reference-phase values. The detector is in this case a CCD array. For the measurement of the three-dimensional displacement field several holograms with differing exposure geometries have to be made. Therefore the system of equations for the solution may be ill-conditioned. For this reason high accuracies in the phase measurement seem and necessary, which can be done by heterodyne or PSI (BREUCKMANN THIEME[ 19851). KUJAWINSKA and ROBINSON[1988] applied a phase shifting method invented by KWONand SHOUGH[ 19851 to holographic interferometry. This method works with 3 intensity pictures with a relative phase shift of 120 degrees. The 3 pictures are generated simultaneously by a grating placed into the object beam in front of the hologram. This grating fulfils two functions. On the one hand, it generates 3 object images whose lateral distances can be controlled by the choice of the grating-object distance and, on the other hand, it provides for the phase shifts necessary in PSI. The phase shift is generated by a small shift of the grating between the exposures. Since the - lth,Oth, and + lth diffraction orders are used, the phase shift is accordingly - rp,, 0, and + (po. With a CCD array 3 times 256 x 256 pixel images can be sampled in one scan. The technique is especially useful in pulsed holography. KWON,SHOUGH [ 19871 used this technique for stroboscopic PSI. The grating and WILLIAMS is adjusted in such a way that a 90degree phase shift between the 3 single interferograms occurs. Since each frame provides enough information for the PSI technique, time-resolved experiments can be carried out.

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HARIHARAN, OREBand FREUND [ 19871 extended the PSI technique to stroboscopic holographic interferometry to study vibrating objects.

10.8. WAVEFRONT SENSORS FOR ADAPTIVE OPTICS

Wavefront sensors for adaptive optics is another field of application (WYANT [ 19811) of real-time interferometry. Two main approaches and KOLIOPOULOS have been discussed and partially realized. 10.8.1. Shearing sensors One possibility is the use of shearing interferometers as wavefront sensors. Lateral shear interferometers deliver lirst partial derivatives of a wavefront (STUMPF[1979]). As has been shown by HUNT [1979], WYANT and KOLIOPOULOS [ 19811, and others, the wavefront aberrations can be reconstructed mathematically from the two first derivatives using relaxation algorithms (HARDY[ 19781). High-speed operation is especially necessary with atmospheric scintillation compensation, since the time constant of these scintils. FREISCHLAD [ 19861 developed a lateral lations is of the order of shearing interferometer in which the two first partial derivatives in x and y are obtained by using a grating shear interferometer and a special Koesters prism beam splitter, in which the beam-splitting mirror consists of a checkerboard of reflecting and transparent squares. By an appropriate orientation of the beam splitter, grating, and detector array, a simultaneous measurement of the derivatives in x and y is carried out by using the PSI technique and a translation of the grating for attaining the phase shifts. The wave aberrations are calculated with the help of a special Fourier transform algorithm. The measured phase data representing the first derivatives are integrated by fitting a series expansion of modal complex exponentials to the phase values. So far, laboratory tests have been carried out. The application in an adaptive telescope seems feasible. 10.8.2. Point reference sensors Point reference interferometers allow for direct wavefront measurements. SMARTTand STRONG[1972] gave a solution for a point diffraction interferometer, which has been developed further by WYANTand KOLIOPOULOS [ 19811 to obey the needs of wavefront sensing with the help of PSI (fig. 33). The wavefront to be tested is focussed onto the point diffraction screen (here

331

APPLICATIONS

DIFFRACTED SPHERICAL (REFERENCE) WAVEFRONT

HALF- WAVEPLATE

Fig. 33. Real-time wavefront sensor using the Smartt interferometer.

a HWP) with a very small hole at the center. In this way a diffracted but unrotated plane polarized wave and an undiffracted but %-degree rotated wave are generated. The diffracted wave serves as a purely spherical reference wave. With a series arrangement of QWP, rotating HWP, and polarizer, a phase variation (see 5 4) can be obtained. In addition, point reference interferometers with a separated ray path have been automated for PSI. This interferometer is usable for testing semiconductor laser wavefronts (HAYESand LANGE [ 19831). A nearly ideal plane reference wavefront can also be generated by extreme radial shear. The strongly expanded beam serves as a reference for the demagnified object beam. Because of the extreme expansion of the reference, the wave aberrations are considerable flattened, so that their influence on the measuring result is reduced. The limited accuracy of such methods is not disturbing in adaptive optics applications, since in most cases a rough estimate of the wavefront deviations is sufficient in view of the limited accuracy of wavefront compensators.

10.9. SPECKLE INTERFEROMETRY

Speckle interferometry is a very valuable tool (JONESand WYKES[ 19831) for testing in real time with television techniques, called electronic speckle interferometry (ESPI). Phase evaluation techniques have also been applied to this field for obvious reasons. ROBINSON[1983], NAKADATEand SAITO [ 19851, and CREATH[ 1985al used PSI. There are some problems with the modulation of the intensity pattern, since the speckle pattern contains the intensity value zero with the highest probability. This leads to locations with zero intensity variation in the PSI process, giving rise to a loss of information.

338

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10

These locations must be filled with neighboring information, limiting the possible phase excursions in ESPI. Digital phase shifting interferometry uses the interferometric equipment typical for ESPI but with a phase shifter included in the reference arm.The phase values of the speckles are measured by PSI and stored in the memory of a computer. To attain a sufficient resolution, the imaging sensor should have more than 256 x 256 pixels, although CREATH [ 1985al has shown that even 100 x 100 pixels are sufficient to demonstrate the principle. NAKADATEand SAITO [1985] showed the capability of this approach with a digitized vidicon camera with 512 x 512 pixels. To suppress the so-called salt-and-pepper noise, a median filter is used to smooth the phase data. The speckle size, that is, the regions with sufficiently defined phase, should be greater than the pixel distance to obtain a resolved speckle picture. After deformation or other changes of the macroscopic features of the speckle field, the PSI measurement is repeated and the data frames of the two runs are subtracted. For modest phase changes the typical raw phase picture of PSI becomes visible and can be used to obtain the deviation picture similar as in holographic interferometry. To smooth the phase pattern, some a posteriori processing of the raw data is necessary. Another possibility for smoothing the phase data is by averaging the speckle pattern distributions resulting from changes in the illumination of the object under test. This applies only to the difference pictures made by means of the PSI process, since the statistical carrier is necessary to save the low-frequency information due to object movements. One problem with the PSI-speckle interferometry is speed, since the unavoidable big data sets seriously strain the computer capacity. Here, two possibilities exist: Either special processors could be used or the very fast PSI method devised by SMYTHEand MOORE[ 19831. A further problem is the extreme range of intensity values in a fully developed speckle pattern. The intensity in the “hot spots” can be handled by using a strong reference wave, for example, with 4 times the average object intensity. SLETTEMOEN and WYANT[ 19861 discussed the optimum intensity of the reference and object wave in connection with the minimum detected modulation and the dynamic range of a speckle pattern in a theoretical estimation. CREATHand SLETTEMOEN [1985] and CREATH[1985b] discussed the application of digital speckle methods to vibration measurements. They gave several techniques, relying on reference phase offsets to suppress the influence of cross-interference terms in the speckle of the object., achieving improved contrast in time averaged patterns. Other evaluation techniques, especially those relying on image processing, have also been applied successfully. NAKADATE, YATAGAIand SAITO[ 19831

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ERROR SOURCES AND MEASURING LIMITATIONS

339

extracted the information about the speckle fringes from the difference of two speckle states made in the computer by taking either the absolute value of the difference or by applying an exclusive OR operation to the binarized data set obtained from level slicing and nonlinear processing steps. This fringe evaluation process includes median filtering with different filter windows. In order to obtain skeletons of the fringe maxima without ambiguities, an interactive software package allows the operator to make the skeletons continuous.

10.10. MISCELLANEOUS

BONKHOFER,KUEHLKEand VON DER LINDE[ 19881 measured the phase dispersion of laser mirrors by means of heterodyne interferometry. The wavelength dependence was assessed by a tunable dye laser as a light source. and Accuracies in the range of 1/300 of a period were obtained. GEORGE STONE[ 19881 gave several schemes for achromatic grating phase shifters that can be used in PSI and other ac interferometric methods.

8 11. Error Sources and Measuring Limitations A comparison of the different methods for phase evaluations necessitates a and CRANE[ 19791 consideration of the error sources and limitations. GROSSO discussed to some extent these error sources for the case of PSI. They classified the error causes into statistical ones, that is, measuring errors which can be diminished in their influence by averaging and systematic errors which, in general, cannot be quenched. Here, we identify 3 types of error sources, namely, environmental errors, errors that are dependent on the method, and errors typically encountered in interferometry.

11.1. ENVIRONMENTAL ERROR SOURCES

11.1.1. Air turbulence and stratification

Since, in most cases, the test arm of the interferometer is rather complex, big air volumes are encountered in the ray path. In this way the measuring result can be greatly disturbed by turbulence (see fig. 34). In the same figure it is

340 -. 134A R)(s=B.BBsA

ADVANCED EVALUATION TECHNIQUES IN INTERFEROMETRY

[IV,8

11

2.1.&. 1-

Fig. 34. Iduence of air turbulence on accuracy.Top, data without averaging; bottom, averaging of 16 runs, resulting in a considerable improvement of accuracy; for a clear display a linear function was added.

shown that averaging independent measurements brings an improvement, since, to some extent, the character of the disturbance is quasi-random. This is not true for stratification and laminar air streams in the interferometer. Here, only working in a perpendicular optical arrangement or in a vacuum can help. 11.1.2. Thermal drips and mechanical relaxation

Thermal drifts can influence either the components to be tested or the whole setup. In the latter case the drifts concern primarily the average phase of the measuring interferometer and can be attributed to instabilities of the reference phase, which will be discussed later. Thermal nonequilibrium of the component Fig. 35. Demonstration of heat transfer and thermal relaxation. Sequence of pictures taken with different time lag to the heating of a plane plate at one side. Top to bottom, immediately after heating the left side, 10 sec, 1 min, and 12 min after the heat source was removed.

ERROR SOURCES AND MEASURING LIMITATIONS

'y

34 1

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ADVANCED EVALUATION TECHNIQUES IN INTERFEROMETRY

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under test gives rise to mechanical strain and deformation. An example is shown in fig. 35, where an optical surface has been heated at one side. The heat transfer and the return of the sample to tne state of thermal equilibrium can be followed in this sequence of deviation pictures. To avoid such effects, thermal insulation of the component and a sufficient settling time after installation of the sample are necessary. Since the decay of thermal disturbances follows an exponential law, much depends on the material constants of the material at hand, that is, its thermal expansion coefficient and the heat conductivity. Materials that have low expansion coefficients and high heat conductivity reach the state of equilibrium fast (one example is fused silica). Normal surfaces that are used as calibration standards should be made from low-expansion glass ceramics like Zerodur to make frequent recalibrations of the normals unnecessary. High-precision work with big samples requires good climate control of the measuring laboratory. 11.1.3. Mechanical strain Mechanical forces exerted on the test specimen should be avoided by suitable support systems or suspensions. In the past, investigations on the bending of plane mirrors under gravitational forces have been carried out. SCHULZand [ 19761 summarized some of these efforts. Much depends on the SCHWIDER dimensions of the sample, its orientation to the gravitational field, and the type of support system. Small samples, for example, about 5 cm in diameter and 1 cm thickness, can be mounted with the common techniques. The design of the mechanical mounts influences the stability of an optical system rather strongly. For bigger samples the rigidity (see TIMOSHENKO and WOINOWSKYKRIEGER[ 19591) influences the design of the support system predominantly. The influence of the gravitational forces can be greatly reduced in the testing stage by supporting the sample in a ribbon attached to the rim (SCHWESINGER [ 19541). But this support is very special and can only be used during the testing stage. Therefore the bigger samples (e.g., telescope mirrors) rest on three pads or, more probably, on a suspension system of pads or a similar arrangement. Starting from the three-point suspension, it is possible to proceed to 6, 9, 18, etc., supporting points (MEHTA[ 19831). The optimum choice of the point distribution follows from a finite-element analysis (BUDIANSKI and NELSON [ 19831). In this way it becomes possible to build big lightweight mirrors. In general, mirror deformations cause about 4 times bigger wavefront aberrations than lens surface deformations caused by mechanical strain.

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11.1.4. Vibrations

Vibrations of the interferometer parts relative to each other will lead to different effects, depending on the frequency spectrum of the disturbance, their amplitude, the method of phase evaluation, and the measuring time of the phase. By suitable design and skill of the experimenter the amplitudes of the vibrations can be kept well under one wavelength of light. Low-frequency vibrations, that is, the vibration period is smaller or comparable with the measuring time, can be considered as drifts and they give rise to reference phase errors. Their influence can be reduced by averaging independent measurements. High-frequency vibrations give rise to visibility losses of the interference pattern ifthe integration time of the detector is greater than the LOHMANN, SCHWIDER period of the vibration (e.g., with PSI,KINNSTAETTER, and STREIBL [ 19881). Otherwise, they cause phase jitter, especially in the case of PLI. Heterodyne interferometers are rather immune against vibrations and drifts due to the compensated measurement scheme (the reference detector that delivers the signal of the local oscillator is situated in the same interference field that yields the compensation). Fringe evaluation methods rely on a single frame and are therefore insensitive to vibration errors, provided the fringe contrast is not wiped out totally. Most common are pistonlike movements of the optical elements, but in some cases torsional vibrations may occur. Such vibrations may cause systematic errors, since the contrast loss is not uniform across the interference pattern. A redesign of the mechanical adjustment stages seems advisable in this case.

11.2. ERRORS DEPENDING ON THE EVALUATION METHOD

The investigation of method-dependent errors is most developed in the case of PSI.GROSSOand CRANE[ 19791 investigated the contribution of different influences for PSI but gave only measuring results. SCHWIDER,BUROW, ELSSNER,GRZANNA,SPOLACZYK and MERKEL[ 19831 derived formulas for reference phase errors and compensation schemes. 11.2.1. Reference phase errors

For small reference phase errors due to PZT malfunction, drifts, and vibrations, the error A@ for the phase to be measured can be derived (SCHWIDER,

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BUROW, ELSSNER, GRZANNA,

l

-

A @ = tan-'

R

[IV, B 11

SPOLACZYK and MERKEL[ 19831) by

CR E? - Ccos2@- Ssin2@ ?=I

1 - CSin2@+ SCOS2@

7

(71)

with

and the deviation E, = cp, - $,., where q,.is the ideal reference phase and $?the real one; that is, the reference phase prevailing during the integration time of the photoelectric detector. The most obvious feature of eq. (71) is the 2 4 dependence of A@. This 24dependence can be deduced from fig. 36. The coefficients C and S are becoming smaller with an increasing number of phase steps R, since the sums containing cos and sin functions will increase at a much slower rate because positive and negative contributions are equally probable. It has also been shown that the use of a second in-offset measurement and the calculation of the phase according to the formula

enables a cancellation of linear reference phase deviations or drifts. N and D are the numerator and denominator according to eq. (58). By applying a 5-phase step method derived from eq. (72) with reference

Fig. 36. Reference phase error. Interferogram showed 3 fringes.

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phase values cp, = 0, in, n, $n, and 2n, the following equation is valid: @=tan-'

2 4 - 2i4

i, - 2i3 + i, ,

(73)

which guarantees a drift-stable regime of phase acquisition in case of linear drifts and miscalibrations of the reference phase shifter. AI and WYANT[ 19871 discussed the influence of quadratic nonlinearities of the reference phase shifter on the measuring error. For the integrating-bucket method they could show that the Fresnel integrals known from diffraction problems can be used for a description of the errors. Different sets of intensity measurements were compared. As was pointed out earlier, averaging of phase values of independent measurements, where the second is out of phase by 90 degrees, is the most effective countermeasure against this type of problem. The results are a special case of eq. (72). Since reference phase errors may occur through drifts and low-frequency vibrations during the data-gathering time of a single run, it makes sense to reduce such errors a posteriori if possible. For small errors, eq. (71) can be expanded into a simpler form: @=a

+ c c o s 2 8 + s sin2@.

Now, the task at hand is the determination of the parameters a, c, and s and afterward the correction of the unwrapped phase data by subtracting such a functional. With the help of computer simulations, SCHWIDER[ 19891 showed that this method furnishes improvements of the data accuracy by about one order of magnitude. The method works fairly well as long as the variation of the phase across the interferometer aperture is greater than 2n. In addition, polarization phase shifters give rise to reference phase errors, as has been discussed by Hu [ 19831. These errors can have two main origins, one being imperfections of the 1/4plates used in the interferometric setup, and the other being an azimuth angle error of the fast axis of these plates. Hu gives for the different errors estimates for their influence on the accuracy. Since all modulations of the intensity are interpreted by PSI as phase shifts, some care is necessary to prevent dust or other defects on the rotating elements. Otherwise the intensity is changed locally because of the disturbances rotating with the phase shifting element causing errors that intensify other errors. In the case of sinusoidal modulation PSI, SASAKIand OKAZAKI[ 1986a,b], and SASAKI,OKAZAKIand SAKAI[ 19871 showed that the modulation amplitude accuracy influences the ultimate accuracy of the measurement. Accurate

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knowledge of this value is necessary to obtain a sufficientaccuracy. The reader interested in details concerning the special errors of this technique is referred to the preceding publications. 1 1.3. ERRORS TYPICALLY ENCOUNTERED IN INTERFEROMETRY

11.3.1. Errors due to spurious fringes

Any fringe system other than the desired one should be considered as a disturbance. In many cases lasers are used as light sources, giving rise to extraneous fringes, which will cause phase errors in almost all cases and with all methods of phase acquisition. BRUNING, HERRIOTT,GALLAGHER, ROSENFELD, WHITE and BRANGACCIO [ 19741 gave an expression for the influence of three-beam interferences on the phase accuracy for the special case 9 = 0. The complete formula showing a @dependence of the error has been derived by SCHWIDER, BUROW, ELSSNER,GRZANNA, SPOLACZYK and MERKEL[ 19831:

A@ = tan-'

q sin(?

- @)

1 +qcos(u]- @) '

(74)

where q exp ( j u]) has been assumed for the disturbing complex amplitude generating the spurious fringes. As eq. (74) indicates, the error depends on the phase difference ( q - @) and the amplitude q. An extensive discussion of this error type in connection with PSI has been given in the aforementioned publication, together with a compensation method presupposing an additional measurement with a phase offset of about n for the object amplitude. In this way a total cancellation of phase errors due to spurious fringes can be attained. AI and WYANT[ 19881 confirmed the result obtained with the error removal algorithm given by SCHWIDER, BUROW,ELSSNER,GRZANNA, SPOLACZYK and MERKEL[ 19831 and modified it in the following way: Two measurements are made, one while the reflected light from the object surface is not obstructed and the other while the test surface is screened. The measured intensity values of the two runs are pairwise subtracted, and these differences are later used in the phase calculation. The result is then free from phase errors due to eq. (74). In the case of heterodyne and phase lock interferometry periodic phase errors could occur, since for the indication of the phase the zero crossings of the sinusoidal signals are used. In addition, methods relying on image processing for the evaluation of wedge fringe patterns will show some sensitivity to this type of error, either by means

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341

of intensity modulations (i.e., very low-frequency modulations of the intensity pattern) or by local phase variations of the position of the extrema. Although MORRIS,MCILRATH and SNYDER [ 19841 have shown for a Fizeau wavemeter that strong modulations do not spoil the accuracy of averages for fringe distances and a common phase, this is not true for the local phase 9 ( x , y) derived from the positions of the fringe extrema. The influence of the spurious fringes on the phase accuracy will greatly depend on their spatial frequency. If they are eliminated in the filtering processes connected with the phase extraction algorithm, the accuracy will not be impaired; otherwise the discussed periodic disturbances may occur. 11.3.2. Detector noise The intensity is detected in most modern applications photoelectrically and converted to a digital signal with the help of an ADC, which is interfaced to an on-line computer. The photon noise will, therefore, be one of the limiting factors, since the signal-to-noiseratio depends on the square root of the number [ 19811, as well of detected photons. WYANT[ 19751 and later KOLIOPOULOS as BRUNING [ 19781, obtained similar results for error estimates concerning the phase uncertainty due to detector noise. This type of noise is commonly handled as an additive noise contribution to the signal. From the rules of error propagation of a, (intensity variance due to photon noise) on the phase one obtains an error estimate for the case of PSI

where R is the number of steps in the phase evaluation process and S/N is the signal-to-noise ratio. Similar results can be obtained for the integrating mode. The deviations from eq. (75) are only slight, and the interested reader is referred to KOLIOPOULOS [ 198 11. Since dark current noise is also additive, the preceding formula can be applied for the phase error dependence on the S/N. 11.3.3. Quantization noise Estimations of the influence of the least significant bit (lsb) uncertainty [ 19821 gave a table for the on the phase error also exist for PSI. DOERBAND [ 19811 wavefront aberration for the 3 reference phase PSI. KOLIOPOULOS derived a more general estimate l + v A 9= 2 V d Q ’

348

ADVANCED EVALUATION TECHNIQUES IN INTERFEROMETRY

[IV, 3 11

where Q = 2" is the number of grey levels of the ADC and V is the visibility of the fringes. This estimate is only valid if the whole dynamic range of the detector is exhausted. Low-intensity patterns will give rise to larger errors. 11.3.4. Nonlinearities of the photodetector Depending on the phase evaluation method, the effect of nonlinearities on the phase accuracy is very different. The common feature is the generation of periodic errors, depending on the type of the nonlinearity. If interferograms are recorded photographically and the evaluation is carried out with the FFT method, much depends on the choice of the carrier frequency and the filtering window. Such errors have been reported by NUGENT[ 19851, who dealt with the nonlinearities of an interferogram for plasma diagnostics using the knowledge that the interference pattern is purely sinusoidal. STETSONand BROHINSKY[ 19851 showed that second-order nonlinearities do not affect the accuracy of the PSI process if a symmetrical formula such as eq. (59) is being used. They maintained without proof that the degree of the nonlinearity affecting the accuracy is determined by the number of samples per fringe period. KINNSTAETTER, LOHMANN, SCHWIDERand STREIBL[ 19881 considered the case of 4 samples per period and also found a contribution of the fourth-order coefficientto the error, in contrast to the preceding statements. CREATH[ 19861 made computer simulations for some special cases. 11.3.5. Coherent noise Smooth wavefronts are an unfounded idealization if laser light is used in interference experiments. A wave propagating through an optical channel encounters scattering obstacles along its path in the form of dust, inhomogeneities, surface irregularities, and others. The scattered light interferes with the background wave and gives rise to a modulated intensity pattern. This means that the amplitude as well as the phase of the complex light amplitude are modulated. Since the scatterers are distributed at random in space, the resulting wavefront also shows statistical phase fluctuations. This is the situation in optical testing, where only the slowly varying wavefront aberrations are of interest. With strong scatterers the typical speckle pattern occurs and low-frequency interference patterns can only be observed if two correlated statistical wavefronts interfere. This typical situation occurs in holographic and speckle interferometry. In the case of holographic interferometry the imaging aperture should be

IV, § 111

ERROR SOURCES AND MEASURING LIMITATIONS

349

large enough to guarantee a sufficient definition of the fringes or, equivalently, of the mean phase. TANNER[ 19681 has shown that more than 10 speckles are necessary to define a fringe.

11.4.

OPTICAL LIMITATIONS

In this section limitations of the attainable accuracy in phase measurements caused by aberrations of optical components in the interferometer and by the special measuring geometry will be examined. Depending on the aim of the measurement and on the specific interferometric setup, different limitations occur. The discussion will therefore be restricted to some typical cases and will only deal with optical testing in the macro-region and micro-region and with holographic interferometry. 11.4.1. Testing of smooth sufaces

The testing of smooth optical surfaces poses the highest accuracy requirements for phase measurements. Since several phase evaluations are necessary, especially in the case of absolute measurements, the accuracy requirements for a single measurement are more stringent. In addition, the measured phase data have to correspond with surface deviations and should not represent other systematic deviations due to aberrations of the collimating and imaging optics in the setup. The situation is similar in planeness and sphericity testing. Each surface testing interferometer contains some beam-shaping optics. This is a collimating, or focusing, optical system in cases of testing for planeness and sphericity, respectively. The optical system should be corrected to a maximum permissible tolerance for spherical aberration and sine condition (BIDDLES [ 19691, HOPKINS[ 19701, HARRIS[ 19711, SCHULZ[ 19771). The influence of spherical aberration can be estimated as follows: At a Fizeau interferometer, for example, the optical path difference (OPD) between the interfering waves, depending on the thickness of the air gap t between the two surfaces under test and the inclination angle cc of the incoming ray, is

OPD = 2t C

,

O S ~

(77)

where the refractive index of air has been taken to be unity. In a Twyman-Green interferometer eq. (77) holds with t being virtually the difference between the two surfaces. In our estimation the light will be collimated

350

ADVANCED EVALUATION TECHNIQUES IN INTERFEROMETRY

[IV, 0 11

sufficiently so that the variation S(0PD) across the aperture can be approximated by

6(OPD) = t a 2 .

(78)

If one makes assumptions concerning the maximum value of S(OPD), the admissible collimation error or the angular aberration a of the beam-shaping optics becomes

a < , / m t .

(79)

With increasing surface distance t the requirements on the collimating optics increase, which in planeness testing can be coped with by testing the combination of the two surfaces in close contact (i.e., t < 1 mm). In spherical surface testing the situation is similar, but here the distance between the surfaces is equal to the difference of the two radii of the normal surface and the test surface. If we assume that the interferometer will be calibrated with the help of a normal surface having the radius of curvature, RN,then t is t = RT - RN

,

(80)

where RT is the radius of curvature of the test sample. If, for example, a TWG interferometer has been calibrated with the help of a normal with R N ,one may ask which parameter range for RT is admissible. A rough estimate can be derived from eqs. (78) and (80):

RT

=

RN

6(OPD)/a2.

For a given amount of lateral ray aberrations q of the beam-shaping optical system, it follows that

< RT < R, + { S(OPD)/q2}R h , (81) with the additional constraint R, < RN because of geometrical reasons. This RN - { b(OPD)/q2}R&

relation has been plotted for S(0PD) = L/400,q = 5 pm, and L = 0.5 pm. If it is assumed that the interferometer has been calibrated with a normal surface of RN = + 15 mm, then with this interferometer radii between RT = 10 . . . 15 mm and RT < - 30 mm could be measured with the preceding accuracy. From fig. 37 it follows that the graduation for the normals becomes very close for small radii of curvature. The influence of an offence against sine condition is connected with carrier frequency fringe patterns. So far a parallel fringe pattern has been assumed in our estimates. But many evaluation methods are relying on a carrier fringe pattern; that is, the OPD is composed of a linear wedge function and the

IV, 8 111

ERROR SOURCES AND MEASURING LIMITATIONS

351

Fig. 37. Tolerance curves limiting the region of radii of curvature of test spheres measured with a Twyman-Green interferometer that has been calibrated with a sphere of radius RN.For the case of a calibrating sphere with RN = 15 mm the allowed test radii are indicated.

aberration function to be measured. Wedge adjustment means different ray paths for the two interfering waves travelling through the imaging optics between the intermediate image of the surface under test in the interferometer and the detector plane. Since interference takes place in the detector plane, the differences of all aberrations accumulated along the two ray paths are taken in this way and show up as a phase difference. The relevant phase data are the deviations of the fringes from straightness and parallelism. Thus the beamshaping optics should obey a sine condition to avoid fringe distortion. In the case of nontelecentric imaging of a spherical surface onto a plane detector surface the sine condition should be replaced by a sine-tangent condition (GATES[ 19581). For the telecentric case the sine condition can be formulated (HOPKINS [ 19461)

6(OPD)

= Ab 1 y/f'

- sins( ,

(82)

where Ab is the lateral distance of the center of curvature test/normal surface (Ab determines the number of fringes/diameter); y is the entering height of a ray onto the beam-shaping optics; sin CY is the surface diameter/2 x radius of curvature; and f' is the focus length of the beam-shaping optics.

352

ADVANCED EVALUATION TECHNIQUES IN INTERFEROMETRY

[IV,8 11

To some extent this condition should be obeyed by every interference picture, since at least a small wedge will remain after the necessary readjustments have been carried out. SLOMBA and FIGOSKI [ 19781 discussed the situation from a different point of view. As is known, straight fringes can be produced on plane surfaces by plane waves inclined relative to each only. Therefore the imaging system should be telescopic to guarantee plane waves in both the input and output plane. If this condition is violated, hyperbolical fringes will be generated, causing systematic measuring errors. Telecentric imaging is also advisable because small amounts of defocusing could result in changes of the scale factor causing measuring errors if wave aberrations of different runs are subtracted. Interference pictures should be imaged sharply and free from distortion onto the detector plane (HARRIS[ 19711, MALACARAand MENCHACA[ 19851). Then the measured phase distribution does not represent surface deviations, for example, but the phase distribution in the particular plane imaged sharply onto the detector plane. In optical testing such effects may go undetected, since the surface is smooth and focusing is difficult. Unsharp imaging of the object onto the detector can be detected by edge ringing and fringe bending in the rim region of the interferogram. 11A.2. Microstructure testing

Micro-interferometric measurements are confronted with the problem that the lateral resolution is limited by the aperture of the imaging system. Several authors have discussed an uncertainty relation for the resolution in the lateral and the depth dimensions (for a review of this subject see SCHULZand SCHWIDER [ 19761). 1 1.4.3. Limitations in holographic interfrometry

Holographic interferometry poses other optical problems. A typical task in holographic interferometry is the determination of surface movements of deformed solid bodies from holographic interferometry. The connection between the measured phase and the surface movement components depends on the illumination and observation directions. Furthermore, several holograms are commonly necessary to find all movement components. The movement quantities follow from solving an inversion problem. The condition of the corresponding matrix equation strongly depends on the chosen geometry of the and ERLER[ 19801, ERLER, holographic interferometer (WENKE,SCHREIBER

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WENKEand SCHREIBER [ 19801). A detailed discussion of this subject is out of the scope of this article. The interested reader should consult the special literature on this subject (see, for example, VEST [ 19791, OSTROVSKY, BUTUSOVand OSTROVSKAYA [ 19801).

Acknowledgement Most of the work on this article was done while I was working with the Central Institute for Optics and Spectroscopy in East Berlin. The majority of the pictures that represent measuring results also derive from the experiments which were carried out with my former colleagues in East Berlin. While working with the optical computing group Erlangen-University, Professor Lohmann and the editor of this series encouraged me to complete the article despite the organizational problems involved in my leaving the German Democratic Republic. The h a l draft was completed during my work with the Heinrich Hertz Institute in Berlin. My former colleague, Dr. K.-E. Elssner, helped with the pictures and the editor played the role of a never-tiring mediator.

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E. WOLF, PROGRESS IN OPTICS XXVIII 0 ELSEVIER SCIENCE PUBLISHERS B.V., 1990

V

QUANTUM JUMPS BY

RICHARD J. COOK Frank J. Seiler Research Laboratory, US Air Force Academy, Colorado Springs, CO, USA

CONTENTS PAGE

$ 1 . INTRODUCTION

. . . . . . . . . . . . . . . . . . . 363 $ 2. ION TRAPPING AND COOLING . . . . . . . . . . . . 368 $ 3 . THEORY OF TELEGRAPHIC FLUORESCENCE . . . . . 377 $ 4. THE NATURE OF QUANTUM JUMPS . . . . . . . . . 397 5 5 . OBSERVATION OF QUANTUM JUMPS . . . . . . . . . 407 ACKNOWLEDGEMENT . REFERENCES

. . . . . . . . . . . . . . . . . . 413

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

414

6 1. Introduction The concept of quantum jumps began with Bohr’s 1913 theory of atomic structure (BOHR[ 19131). According to this theory, an atom at rest can exist in any of a discrete set of energy levels, and the fundamental radiative processes (absorption, induced emission, and spontaneous emission) proceed by abrupt jumps between these levels. The first of these ideas survived the transition to modern quantum mechanics. It is still believed that the bound states of an atom have a discrete spectrum. But Bohr’s concept of quantum jumps did not fare as well. After the development of wave mechanics the question of whether or not the picture of quantumjumps was consistent with the new quantum theory was addressed by Schradinger. He concluded, on the basis of his early interpretation of the wave function, that the notion of quantum jumps was incom[ 1927, 19521). patible with the principles of wave mechanics (SCHR~DINGER There were discussions between Bohr and Schradinger on this point, apparently without agreement. The heat of these discussions is captured in the now famous statement by Schrbdinger that “If all this damned quantum jumping were really to stay, I should be sorry I ever got involved with quantum theory”, after which Bohr reportedly replied, “But we others are very gratefulto you that you did, since your work did so much to promote the theory” (HEISENBERG [ 19551). An essential difference between Bohr’s theory and wave mechanics is that in the former the atom is at each instant in one, and only one, energy state, whereas in the latter the atom can exist in a coherent superposition of states. In the Bohr theory a dynamic history of the atom is described, as illustrated in fig. 1, by a discontinuous, piecewise constant function of time. At each instant the state of the atom is definite. The notion that the atom can exist in two or more states at the same instant is entirely foreign to the Bohr picture of atomic dynamics. By contrast, the modern quantum theory describes the dynamic history of the atom in terms of probability amplitudes Ci(t) for each of the energy levels Ei, and states exist for which two or more of the amplitudes are nonzero. For such coherent superposition states, the atom is, in some sense, simultaneously in more than one energy state at a given instant, or at least the atom is not in a definite energy state at every instant. This is clearly incompatible with the Bohr picture of atomic dynamics. 363

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TIME Fig. 1. A typical dynamic history of an atom in the Bohr theory. Discontinuities of the energy versus time curve are called quantum jumps.

Coherent superposition states are commonly produced when atoms are coherently excited from the ground state by laser light. For instance, a n/Zpulse applied to a two-level atom creates a coherent superposition with the two energy states equally represented. The important point is that such states can have properties which are different and distinguishablefrom those of any of the definite energy states. We can see this by means of a simple example. As is usual in atomic physics, let us take the atomic energy states to be states of definite angular momentum (eigenstates of .fz and .f2). Then it is easy to construct coherent superposition states for which the expectation value of the x component of the angular momentum is nonzero ((.f,) # 0). For example, if an atom initially in a spin-zero ground state absorbs a left circularly polarized photon traveling in the x direction, the atom is promoted to a superposition of the j, states for which the x component of angular momentum has the definite value h, the angular momentum of the absorbed photon. Hence, after the absorption process (S,) = h. But if, as the quantum jumps picture demands, the atom were at each instant in a definite one of the .f, angular momentum states, the expectation value of the x component of angular momentum would always be zero, since the expectation value of .fxin any eigenstate of 3’ and j’ is zero. This means that a quantum-jumps picture on the .f, states is not capable of correctly describing the absorption of a photon with one unit of angular momentum in the x direction. The idea that the atom is always in some state with a definite value of .f2 is simply and obviously incompatible with the existence of coherent superpositions of these states. Generalizing from this idea, it is easy to see that a quantum-jumps picture on a given set of states is

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consistent with modern quantum mechanics only if the temporal evolution of the system is such that a coherent superposition of the states does not develop. This restriction precludes a quantum-jump interpretationof the coherent evolution of any quantum system, such as the Rabi oscillations of a two-level atom, and seems so restrictive as to eliminate the quantum-jump picture as a useful interpretation of quantum behavior. Despite these general arguments against quantumjumps, a number of experiments have been reported in recent years whose interpretation in terms of quantum jumps seems natural (NAGOURNEY, SANDBERGand DEHMELT [ 19861, BERGQUIST,HULET, ITANOand WINELAND[ 19861, SAUTER, BLATT and TOSCHEK[ 19861). The reported observations of NEUHAUSER, quantum jumps were made possible by the ion-trapping technology pioneered by Hans Dehmelt and collaborators. NEUHAUSER, HOHENSTATT, TOSCHEK and DEHMELT[ 19801 succeeded in trapping a single atomic ion in a highvacuum radiofrequency Paul trap and observed the laser-induced fluorescence of the isolated single atom. This event marks the beginning of single-atom spectroscopy, a field that promises the highest resolution atomic spectra yet achieved. Ultra-high resolution spectra are obtained by cooling the ion with laser light. By illuminating the ion with laser radiation tuned slightly below the atomic resonant frequency, the ion experiences a radiation force that damps its translationalmotion. In this way the ion is brought to rest in the trap, and both the first- and second-order Doppler shifts are eliminated. The result is a nearly ideal spectroscopic sample, a single atom at rest whose spectrum is distorted by neither the Doppler shift nor collisional processes. Perhaps the most interesting effect that has been observed by means of single-atom spectroscopy is telegraphic atomic fluorescence,the random starting and stopping of the resonance fluorescence of certain atoms when illuminated with cw laser beams. Telegraphic fluorescence occurs when the optical electron intermittentlyexits the atom’s resonance fluorescencecycle by jumping to a metastable level and thereby turns off the fluorescencefor a period of time. The idea that the resonance fluorescence of a single atom would turn off when the atomic electron is “shelved” in a metastable level was first suggested by HANSDEHMELT[ 1975, 1982, 19831 as a technique for detecting very weak, and hence very narrow, atomic transitions. It was pointed out by COOKand KIMBLE[ 19851 that the switching off and on of the single-atom resonance fluorescence was in one-to-one correspondencewith the quantumjumps of the atomic electron to and from the metastable level. The proposal that individual quantum jumps of a single electron in a single atom could be monitored in this way revived the old debate about the existence of quantum jumps (ERBERand

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\

WEAK

STRONG

I

0 Fig. 2. Energy level scheme for observing telegraphic atomic fluorescence, the V-configuration.

PUTTERMAN[ 19851, JAVANAINEN [ 19861, PEGG, LOUDONand KNIGHT [ 19861, KNIGHT,LOUDONand PEGG[ 19861, ARECCHI,SCHENZLE, DEVOE, JUNGMAN and BREWER[ 19861, SCHENZLE,DEVOE and BREWER[ 19861, SCHENZLE and BREWER[ 19861). To see how single-atom fluorescence can be intermittent and how the fluorescence serves as a monitor of quantum jumps, consider the three-level atom depicted in fig. 2, the so-called V-configuration. The atom has two transitions, 0 t)1 and 0t)2, with a common lower level (the ground state 0). Let the transition 00 1 be a strong dipole-allowed transition with an upper state lifetime z, of, say, lo-* s. Let the other transition (002) be a weak dipoleforbidden transition. Level 2 is then a metastable state with a lifetime of, say, 1 s. With laser radiation applied to the strong transition, the atomic electron is quickly promoted from the ground state 0 to level 1. From there it spontaneously returns to the ground state in about lO-'s with the emission of a fluorescence photon. The excitation and spontaneous emission cycle 0 -+ 1 + 0

t

TIME

Fig. 3. Single-atom fluorescence intensity versus time. Interruptions in the fluorescence signal quantum jumps to the metastable level.

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361

then repeats, and for a strong applied field, fluorescencephotons are given out at the saturated rate R = 1/22, 108s-’. Incidentally, a source of this strength in the visible can be seen with the unaided eye. (With a 10 x magnifying lens a point source of this strength would appear about as bright as one of the stars in the Big Dipper [constellation Uris Major].) With the first laser continuing to drive the strong transition, let a second laser be applied to the weak transition. Then occasionally, when the electron occupies level 0, it is promoted to the metastable level 2, say once every few seconds. As long as the electron remains “shelved” in level 2, it is unavailable for the fluorescence cycle 0 + 1 + 0 + 1 + .. .. Therefore the atomic fluorescence switches off and remains off until the electron returns to level0 by either spontaneous or stimulated emission. The times of transitions to and from the metastable level are, to a large extent, random. Hence, with two steady laser beam driving the atom, the atomic fluorescence I(?) has the appearance of a random telegraph signal, as depicted in fig. 3. The important point is that the switching off and switching on of the atomic fluorescence are in one-to-one correspondence with the upward and downward quantum jumps on the weak transition. The fluorescent signal is a direct monitor of the quantumjumps, and the individual jumps may be detected in this way with near 100% efficiency. The preceding discussion seems to show a discrepancy between the fundamental principles of wave mechanics, which appear to rule out a quantum-jump interpretation of atomic dynamics, and the simple shelving argument together with the corroborativeexperiments on telegraphic fluorescence,which are most simply explained in terms of quantum jumps. Before the experiments demonstrating telegraphic fluorescence, this apparent discrepancy led several authors to question the shelving argument (JAVANAINEN [ 19861, PEGG, LOUDON and KNIGHT[ 19861, ARECCHI,SCHENZLE, DEVOE,JUNGMAN and BREWER[ 19861, SCHENZLE, DEVOEand BREWER[ 19861, SCHENZLE and BREWER[ 19861). The central question was whether there actually would be dark periods in the single-atom fluorescence. The alternative view was that the resonance fluorescencemight decrease in intensity to a lower steady value when the weak transition is driven, as in many-atom double resonance experiments (BITTERand BROSSEL[ 19501). An unequivocal answer to this question was given by COHEN-TANNOUDJI and DALIBARD [ 19861 by calculating the distribution function for the delays between successive fluorescence photons, a subject discussed here in detail in 3 3. The question of how the quantum-jump interpretation of telegraphic fluorescence can be consistent with the postulates of quantum mechanics is taken up in 8 4 and answered in terms of quantum measurementtheory. The theory of ion trapping and cooling is briefly addressed

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in 2. The reader who is not interested in this aspect of the subject may skip to 3 without loss of continuity. In 5 5 selected experiments on quantumjumps are discussed. A short review of quantum jumps in atomic systems has been given by BLATTand ZOLLER[ 19881 and a more comprehensive one by ERBER, HAMMERLING, HOCKNEY, PORRATIand PUTTERMAN [ 19891.

0 2. Ion Trapping and Cooling The observation of quantum jumps requires a spectroscopic sample consisting of a single atom or, at most, a very few atoms. To date only ions have been trapped as single isolated particles. The trapping of a single atomic ion and the observation of its resonance fluorescence were first achieved by NEUHAUSER, HOHENSTATT, TOSCHEKand DEHMELT[ 19801. In this section we discuss the principal method that has been used to trap and cool single ions. In view of the ease with which force may be applied to a charged particle when compared with a neutral one, it is natural that ions would have been trapped before neutral atoms. A variety of ion traps exists. Among these are DRULLINGER and WALLS the Penning trap (PENNING[ 19361, WINELAND, [ 19781, WINELANDand ITANO [ 19811, DRULLINGER,WINELANDand BERGQUIST [ 1980]), the Paul trap (PAULand RAETHER[ 19551, WUERKER, GOLDENBERG and LANGMUIR [ 19591, FISHER[ 19591, WINELAND, ITANO [ 1967,1969]], NEUHAUSER, HOHENSTATI-, andVAN DYCK[ 19831, DEHMELT TOSCHEK and DEHMELT[ 1978]), the Kingdon trap (KINGDON[ 1923]), and magnetic bottles of the type used for plasma confinement. Perhaps the best of

Fig. 4. Radio frequency Paul trap (adapted from NEUHAUSER,HOHENSTATT, TOSCHEK and DEHMELT [ 19801).

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these for the purpose of high-resolution spectroscopy is the radio frequency Paul trap because it does not employ a magnetic field, which greatly distorts the atomic spectrum under study. The first single-ion trapping experiment and most subsequent experiments have made use of the Paul trap. Here we shall discuss in detail only this type of trap. The Paul trap employs two cap electrodes and a ring electrode, as illustrated in fig. 4, to produce a quadrupole electrostatical potential AV @(r)= - ( 2 3a2

+ y2 - 2 z 2 ) ,

where a is half the distance between the cap electrodes (or the inner radius of the ring electrode) and AV is the potential difference between the ring electrode and the cap electrodes. Strictly speaking, the electrodes must be infinite hyperboloids of revolution in order for the potential (1) to be exact everywhere outside of the electrodes. In practice only the potential near the origin is significant. There is a variety of finite electrode geometries that closely approximate the quadrupole potential near the origin. According to Earnshaw’s theorem, there is no point of stable equilibrium for a charge in an electrostatic field (STRATTON[ 19411). But if the field oscillates in time, there can be points of dynamic equilibrium that are stable. Let the potential difference AV oscillate rapidly in time at the angular frequency v (AV = V,, cos vt). When the period of oscillation is long compared with the time required for light to propagate between the electrodes (a/c), the field at each instant is well approximated by the electrostatic formula (l), and the magnetic field is negligible. Typically a = 0.1 to 1 cm. Thus radio frequency fields are accurately electrostatic. The potential energy of a charge e in the field (1) is V(r, t) = e@(r,t) = evci - (x2 + y 2 3a2

- 2 z 2 ) cos vt .

Under the action of this potential an ion executes a rapid oscillatory motion with no net displacement and a superimposed slower guiding-center motion. Classical treatments of this problem leading to the so-called pondermotive force have been given by KAPITSA[ 19511, GAPONOV and MILLER[ 19581, LANDAUand LIFSHITZ[1960], and WEIBELand CLARK[1960]. Here we follow the quantum-mechanical treatment of COOK,SHANKLAND and WELLS [ 19851 because quantum-mechanical features of the motion turn out to be important in the treatment of laser cooling. An analogous classical treatment of the ion trapping and cooling problem has recently been given by COOK [ 19891.

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Potential (2) is of the general form V(r, t ) = o(r) cos vt .

In this potential the motion of the particle is governed by the Schrodinger equation

When v is a high frequency, the potential in eq. (3) has only a small effect on the particle motion because the force averages to zero over a short time interval. There is, however, a small secular effect, calculated as follows. Suppose that only the potential-energy term were present on the right-hand side of eq. (3). Then the solution of the equation would be $(r, t ) = $(r, 0) exp[ - io(r) sin(vt)/hv] .

Thus the dominant effect of the potential is to add an oscillating phase factor to the wave function. This suggest that we look for a solution to the full equation of the form $(r, t)

=

+(r, t ) exp[ - io(r) sin(vt)/hv] ,

(4)

where +(r, t) may be regarded as a slowly varying function of time, since the dominant effect of the oscillating potential is already contained in the phase factor. Observe that, because $(r, t) and +(r, t) differ only by a phase factor, the position probability density is the absolute square of either of these functions P(r,O=

I$(r,012=

I+(r,012,

that is, +(r, t) may be regarded as the wave function in position space. On substituting eq. (4) into eq. (3), we obtain the exact equation of motion for +:

In this equation the coefficientsof sin v t and sin2 v t are slowly varying functions of time. Therefore, when the frequency v is large, we may replace the trigonometric functions by their average values over the very short time interval 2 n/v. The time averages of the trigonometric functions are 0 and 1/2, respectively. This is a type of “rotating wave approximation” in which rapidly oscillating

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terms in the Hamiltonian are discarded and slowly varying terms are kept. The Schrodinger equation for $(r, t ) becomes

ih

a$

-= at

fi2

-- V2$ 2m

+ Veff(r)$,

with a time-independent effective potential energy

This potential energy is the same as that obtained from the classical treatments of particle motion in a rapidly oscillatingfield (KAPITSA[ 19511, GAPONOV and MILLER[ 19581, LANDAU and LIFSHITZ[ 19601, WEIBELand CLARK [ 19601). Here we see that the same effective potential energy may validly be used in the Schrddinger equation for the quantum-mechanical motion of the particle, provided the frequency of the field is sufficiently high. The correction to the time evolution of the wave function $(r, t ) due to the oscillating terms, which have been neglected in the present approximation, has been studied by COOK,SHANKLAND and WELLS[ 19851 and by COMBESCURE [ 19861. For the oscillating quadrupole potential of the Paul trap (2), the effective potential energy ( 5 ) reads

The effective potential is that of an anisotropic three-dimensional harmonic oscillator. The potential has a point of stable equilibrium at the center of the trap (r = 0). An important feature of the Paul trap is that the oscillating trap field, the negative gradient of @, is zero at the point of stable equilibrium. Thus there is neither a magnetic field nor an electric field to disturb the atomic spectrum when the atom is trapped at the equilibrium point of the effective potential. In current practice the trap is loaded by crossing a neutral atomic beam and a weak electron beam near the center of the trap. Ionization by electron impact creates atomic ions, which because they are charged, are confined by the effective potential (6). This method of loading the trap produces ions with rather large kinetic energies (of the same order as the thermal energy of the atomic beam) and, hence, large amplitudes of oscillation in the harmonic trap. For high-resolution spectroscopy this motion must be damped to eliminate the

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associated Doppler spread of atomic lines. It was first pointed out by WINELAND and DEHMELT[ 19751 that each line of the ion’s spectrum gains side bands due to its harmonic motion in the trap and that a laser beam tuned to a side band below the ion’s resonance line would damp the motion of the trapped ion and bring it to rest at the center of the trap. By bringing the ion to rest, both first- and second-order Doppler shifts are eliminated. It is worth noting here that most other methods of “Doppler free” spectroscopy do not eliminate the second-order or time-dilation part of the Doppler shift. A method of laser cooling similar to that of Wineland and Dehmelt, but intended for a neutral atomic vapor rather than trapped ions, was proposed at about the same [ 19751. Here we discuss only the cooling time by HANSCHand SCHAWLOW of trapped ions. We have seen that the trapped ion is a three-dimensional harmonic oscillator. Consider a laser beam propagating in the x direction. Momentum transfer from the beam affects the motion of the ion in the x direction only. Therefore we may ignore the motion in the y and z directions and treat the ion as a onedimensional harmonic oscillator. For simplicity we treat the internal motion of the ion as a two-level system with energy levels E , and E,, and Bohr transition frequency wo = (E, - E , ) / h . In the following we shall discuss laser cooling of a trapped ion from a fully quantum-mechanical point of view. But before doing so it is instructive to look at a simple classical model of side band formation. The classical model clarifies many features of the quantum-mechanical treatment. Consider an ion that oscillates along the x axis with amplitude a and frequency [ x ( r ) = a sinat]. Let an electromagnetic wave of frequency w propagate in the x direction [E(x,r) = 8 cos (kx - w t ) ] .At the position of the moving ion the electric field is E(t) = Q cos(ka sinat - or).

(7)

It is this field that drives the internal motion of the ion. Note that the field at the ion is not monochromatic. On expanding the cosine function in (7) in terms of exponentials and using the identity 00

exp(iak sinat) =

C

Jm(ak)exp(imat),

m= -m

we obtain 00

~ ( t=) d

C m=

-m

~ , ( a k )cos [(a- ma)r] ,

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ION TRAPPING AND COOLING

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whereJ,(x) is the Bessel function of order m.Thus the field at the ion contains components at frequencies w - m P (m = 0, k 1, f 2, . . .) with amplitudes gm = 8Jm(ak),respectively. The ion will be in resonance with one of these components when one of the field frequencies equals the transition frequency of the ion (w - m a = wo) or, equivalently, when the applied field frequency w has one of the values om= coo + m a .

Thus an oscillatingion has side bands on its absorption line displaced from line center by multiples of the oscillation frequency a. If the absorption crosssection for the stationary ion is ao(w), the cross-section for the oscillating ion is m

a(w) =

C

J:(ak) ao(o - mP),

m=--4)

as depicted in fig. 5. To resolve the side bands, the width of the absorption line and, in particular, the natural width must be less than the side band spacing 62. Note that the sum of the squares of the Bessel functions is unity, 00

C

J:(ak)

=

1

m=--00

Consequently the total area under the absorption profile, including the side bands, is not altered by oscillation of the ion. The magnitudes of the Bessel functions drop off rapidly with m for m > ak. Therefore, for all practical purposes, there is a finite number of side bands on each side of wo, the number being of order ak. For ak 4 1, that is, when the amplitude of oscillation is small

Fig. 5. Absorption cross-section for an oscillating two-level atom. Side bands are displaced from the atomic resonant frequency w, by multiples of the oscillation frequency Q.

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compared with the wavelength, all side bands are weak compared with the central resonance. In the cooling processes the amplitude a is driven to zero. Therefore the side bands decrease in number and then disappear altogether as the ion is cooled. The amplitude of the ion's motion and the degree of cooling can be determined spectroscopically by observing the side bands. Laser cooling or damping of the ions motion occurs when the laser is tuned to a side band below the primary resonance at frequency coo. In this case a photon of energy E = h(w0 - ma)is absorbed by the ion, whereas the average 0 0

0

Fig. 6. Energy level diagram for a two-level ion in a one-dimensional harmonic trap. Induced transitions are denoted by solid arrows,spontaneous transitions by dashed arrows.

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photon energy given out in spontaneous emission is E’ = h a , . It turns out that the energy deficit E‘ - E = m h 9 is subtracted from the translational energy of the ion. To see how this occurs, we look next at the fully quantum-mechanical picture of the ion’s motion. The total energy of the trapped ion consists of its internal energy (E, or E,) and its translational energy ET = h 9 n (n = 0, 1, 2, 3, .. .), the energy of a quantum-mechanical harmonic oscillator of frequency 9.An energy level diagram for the trapped ion is given in fig. 6. Each of the internal energy levels becomes the bottom rung of a ladder of levels describing the state of translational motion of the ion. The state with energy E , = Ei+ ET is the direct product of the ion’s internal state I i ) and its translational oscillator state In)=: Ii,n> =

10, I n h .

(8)

Interaction with the electromagnetic field causes transitions between these states. In the electric dipole approximation the interaction Hamiltonian reads

A,= - p * B ( @ y ,

(9)

where jl is the ion’s electric dipole operator and &i) is the electric field operator evaluated at the position tji of the ion (4 is the operator representing the displacement of the ion in the x direction). From the matrix elements of the interaction (9) between states (8), we fmd that the electric field couples a given “ground state” I 1, n) to several “excited states” 12, n’ ) with n’ in the neighborhood of n. These transitions have frequencies oo+ m 9 (m = n’ - n) and correspond to the various side bands of the classical picture. As expected from the classical model, the effective number of side bands increases with n, the excitation number being a crude measure of the amplitude a of the oscillation. A simple energy argument shows that a m (hn/m9)’/2,here m is the mass of the ion. There are many strong side bands in absorption from a state 11, n) with n 9 m 9 / h k 2 .(This corresponds to the classical condition ak % 1.) Conversely, all side bands are weak for n m 9 / h k 2 ,or in classical terms when ak << 1. The latter condition is known as the Lamb-Dicke limit. It obtains when the oscillator amplitude is much less than the wavelength. In the Lamb-Dicke limit the central resonance at o = coo is strong, the first side bands at o = wo 9 are much weaker, and the higher-order side bands are weaker still. Therefore in the Lamb-Dicke limit, side band cooling is most efficient when the laser is tuned to the first side band below the principal resonance (o= wo - 9).The transitions induced by a field of this frequency are indicated by solid arrows in fig. 6. Note that, for each photon absorbed at frequency wo - 9,the translational excitation number n decreases by unity

+

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(n’ = n - l), that is, the translational energy of the ion is decreased by the amount ha for each photon absorbed. After the absorption of a photon the ion is internally excited and can spontaneously emit. From the interaction (9) with the quantized field we 6nd that, from a given “excited state” 12, n) , the ion can spontaneously decay to several “ground states” 11, n’ ) with the emission of frequencies o, - m a (rn = n - n‘) in the neighborhood of a,. That is to say, the side bands appear in fluorescence as well as in absorption. The important point for side band cooling is that the branching ratios for spontaneous emission into the various side bands are symmetrically distributed about the center frequency 0,. Consequently, the mean frequency of a fluorescence photon is ooand the mean energy given out in spontaneous emission is h a , . Therefore in an absorption process [energy absorbed is h ( o , - a)]followed by spontaneous emission (energy emitted is hw,), the ion loses energy ha. Also, as noted earlier, the decrease occurs in the translational energy of the ion at the time of absorption. In the Lamb-Dicke limit the side bands are weak and spontaneous emission occurs primarily at the center frequency coo (n -+ n‘ = n). The spontaneous transitions for this case are indicated by dashed arrows in fig. 6. Repeated absorption and spontaneous emission cause the ion to walk down the ladder of oscillator levels, and eventually it ends up in the ground oscillator state I 1,O). This is the side band cooling process. It is essentially a type of optical pumping cycle that pumps the ion to states of lower translational energy. Recently the laser cooling of a Hg’ ion to the ground state of the translational motion was demonstrated by DIEDRICH,BERGQUIST, ITANOand WINELAND [ 19891. It is easy to see that if the applied field is tuned to the 6rst side band above w, rather than below o,,then energy h(o,+ 0) is absorbed, energy h a , is emitted, and the ion gains translational energy h G?for each spontaneous event. In this case the ion is optically pumped up the oscillator ladder until its amplitude is so large that, in practice, it is lost from the trapping region. Thus the ion is cooled by radiation tuned below wo and is heated by radiation tuned above 0,. We have considered side band cooling in the idealized case, where the natural width of the ion’s resonance line is small compared with the side band spacing. The picture becomes more complicated when, as is often the case in practice, the natural width exceeds the side band spacing. In this case the side bands are not resolved in absorption or emission. In spontaneous emission the mean energy of an emitted photon is still h a , , but in absorption the broad overlapping absorption lines of the side bands allow simultaneous pumping of

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several side bands by a monochromatic field. We shall not discuss this case here but refer the interested reader to the extensive literature on ion cooling and [ 19861. trapping that has recently been reviewed by STENHOLM

83. Theory of Telegraphic Fluorescence The theory of telegraphic fluorescence began with Dehmelt’s intuitive concept of electron shelving (DEHMELT [ 19751). Initially the theory developed with approaches based on atomic rate equations (COOKand KIMBLE[ 19851) and on the atomic density matrix (PEGG,LOUDONand KNIGHT[ 19861, ARECCHI, SCHENZLE, DEVOE,JUNGMAN and BREWER[ 19861, SCHENZLE and BREWER [ 19861, SCHENZLE, DEVOEand BREWER[ 19861, KIMBLE,COOKand WELLS [ 19861). With certain additional assumptions concerning the nature of the fluorescence, in particular it being assumed that the fluorescence is intermittent, these approaches seem to provide a complete description of the fluorescent signal. But the density matrix appoach tends to be algebraically cumbersome, and it is doubtful whether the atomic density matrix alone contains sufficient information to prove the existence of dark periods in the fluorescence. For example, in the steady state the probabilities to be in the various levels of the three-state atom are time independent, and there is no indication from these components of the density matrix that the atomic fluorescence is flashing on and off. Perhaps the simplest and, at the same time, the most complete and rigorous and theory of telegraphic fluorescence is that of COHEN-TANNOUDJI DALIBARD[ 19861 and WYNAUD,DALIBARDand COHEN-TANNOUDJI [ 19881 (for similar treatments see also ZOLLER,MARTEand WALLS[ 19871, and ERBER,HAMMERLING, HOCKNEY,PORRATIand PUTTERMAN [ 19891). In this theory the atomic fluorescence is treated as a sequence of discrete photon emission processes, and the photon statistics are characterized by the delay function D(z), which is the probability density for the delay z between consecutive photons. That is to say, D(z) dz is the probability that, if a photon is emitted at time t, the next photon is emitted in the interval d z at time t + z. To understand how the delay function is calculated, we look fist at the simple case of a driven two-level atom. We then proceed to the analysis of certain three-level systems that fluoresce intermittently and show how the delay function provides a complete statistical description of the telegraphic fluorescence and unequivocal proof of the existence of dark periods in the fluorescence. We do not address in this review the question of quantum jumps

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Fig. 7. Energy levels of atom plus single-mode field,

in systems consisting of two or more atoms or ions (SAUTER,BLATT and TOSCHEK[ 19871, SAUTER,BLATT,NEUHAUSERand TOSCHEK[ 19871, LEWENSTEIN and JAVANAINEN [ 1987, 19881, AGARWAL,LAWANDEand D’SOUZA[ 1988a,b]).

3.1. DELAY FUNCTION FOR TWO-LEVEL ATOM

Consider a two-level atom with states 11) and 12) at energies E l and E,, respectively, that is driven by a near resonant single-mode quantized field. When the field is strong, the atom and field are strongly coupled, and it is appropriate to treat the atom together with the field as a single system. The energy of the quantized field mode is h on (n = 0, 1,2, . ..), where o is the mode frequency. Accordingly the energy levels of the atom + mode system are E,

=

E , + hon.

The state I i, n) belonging to this energy is the direct product of an atomic state l i ) (i = I, 2) and the field state In), ti, n) = li) In). The energy levels and states of the atom + mode system are illustrated in fig. 7. In the electric dipole approximation the atom and field interact through the electric dipole Hamiltonian

A,= - # . &

(10)

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where 1is the atomic dipole moment operator and E is the mode electric field operator at the position of the atom

E = i(E*d - Edt)/2, where d and d are the photon creation and annihilation operators for the mode and E is the “one-photon amplitude” of the field. The electric dipole operator of the atom takes the form

1= p a + p * b + , where a+ = 12) ( 1I and b = 11) (21

are the atomic excitation and deexcitation operators, respectively, and p = (1 I 112) is the dipole transition moment. In the rotating wave approximation the energy nonconserving terms, those involving bd and cr + at, are discarded and the interaction Hamiltonian (10) becomes fiI=i(p.Edtb-p**E*b+d)/2. This Hamiltonian induces the transitions indicated by the solid curved arrows in fig. 7. These transitions dominate the energy nonconserving process when the mode frequency is near resonance (o= oo). In the absence of any spontaneous emission, a pair of levels connected by these induced transitions behaves as an isolated two-level system. Let Cp)be the amplitude to be in state 11, n ) and C!.) the amplitude to be in state 12, n - 1). With the state vector on this manifold of states written as

I Y(y(t))= [Cp) 11, n ) + C!.) 12, n - l ) ] exp{ - i[E, + hwn]t/h} , the Schrddinger equations for this two-level system read

cp) = n, c p / 2 , =

iA C$’” - n:Cp’/2

(1W

,

(1 1b)

where A = o - oois the detuning frequency and

61,

=p

*

En ‘I2/h

is the n-photon Rabi frequency. Equations (lla,b) describe the process in which the energy of one photon is transferred back and forth between the atom and the single-mode field. In addition to the induced processes the atom can spontaneously emit a photon into any of the other modes of the quantized field. In such a process the atom proceeds from its upper state to its lower state, and the number of

380

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photons in the driving mode does not change. The spontaneous transitions are indicated by dashed arrows in fig. 7. Resonance fluorescence consists of the sequence of spontaneous emission processes that occur as the atom + driving mode system cascades down the harmonic ladder of fig. 7. It would appear from the figure that, eventually, the fluorescence would stop when the cascade reaches the bottom rung of the ladder. This would be the process in which the driving mode repeatedly excites the atom and the excitation energy is carried away by spontaneous emission until all of the energy of the driving mode is depleted. However, in practice this does not occur because, for a laser-driven atom, the driving mode (the laser field) is continually excited by the laser medium. Such excitation of the driving mode would be described by upward transitions between the rungs of the double ladder in fig. 7. Fortunately such complications are not necessary for an understanding of resonance fluorescence. If the driving mode is a mode of infinite free space, the one-photon amplitude has the form

where V is the quantization volume and 8 is the polarization vector, and as V -,co,E tends to zero. For a given interaction strength between the atom and the driving field, as measured by some fixed Rabi frequency

the number of photons in the mode n must become infinite as Y + co . Therefore, in free space the atom is infinitely high on the excitation ladder of fig. 7, and the resonance fluorescence will not stop in any finite time. Moreover, for any finite change in n, the Rabi frequency (12) is unchanged, and eqs. (lla,b) become simply Qn)

$Cl"), = i A c g ) - $J*Cp), =

(134 ( 13b)

with 0 independent of n for all of the states of interest. Consider now the emission of a fluorescence photon at time t. The photon results from one of the spontaneous transitions in fig. 7. Immediately after the emission process the system is in one of the states on the left ladder in the figure, say the state 11, n ) . From here the atom undergoes induced transitions to and

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from the state 12, n - 1) ,and while it is in this state, may emit a fluorescence photon by spontaneous emission to state I 1, n - 1) . The delay function D ( z ) is the probability density for the time z between the spontaneous event, which brings the system to state I 1, n), and the next emission event, which carries the system to state I 1, n - 1) . To describe the latter process, we must modify eqs. (13a and b) to take account of the spontaneous emission from state 12, n - 1) . Following the usual Weisskopf-Wigner procedure, we calculate the rate of spontaneous emission from state 12, n - 1) to state I 1, n - 1) (WEISSKOPF and WIGNER[ 19301). The emitted photon goes into any singlephoton state I kA) with frequency cok = c I k 1 near the transition frequency coo (here I ( = 1,2) is the polarization index of the photon). We find that the spontaneous emission rate is simply 28 I Cp)I2, the Einstein A coefficient for the two-level atom (A = 28) times the probability to be in state 12, n - 1). Formally, eq. (13b) gains the relaxation term - fiCp), which decreases the probability to be in state 12, n - 1) at the rate of spontaneous emission:

=

-1a*C(n) 2 1 + iACp) - BCp),

(14b)

We are assuming here that the system starts out in the subspace spanned by the states I 1, n) and I 2, n - 1 ) . Therefore, a term in eqs. (14a,b) representing the flow of probability from the higher lying states in fig. 7 is unnecessary. The Weisskopf-Wigner approximation also leads to a shift of the atomic transition frequency coo, which turns out to be infinite. However, after mass renormalization and an appropriate cutoff procedure the observable shift (Lamb shift) is found to be small and of no particular interest in the present context (MILONNI[ 19761). Here we assume that the radiative shift is already included in the measured atomic transition frequency coo. We may now calculate the delay function D(z) for the two-level atom. Immediately after the photon emission at time t, the system is in state I 1, n) : CP)(t)= 1 ,

(15a)

C?)(t) = 0 .

( 15b)

From this initial condition the amplitudes Cp)(f + z) and Cp)(t+ z) for any later time are determined by eqs. (14a,b). These are the amplitudes for the atom to be in its ground state or its excited state, respectively, without having emitted the next photon. As noted earlier, the probability for next-photon emission in the interval d z at time t + z is D(z) d 2. However, from the definition of the

382

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transition rate we know that 2BI C y ) ( t+ z)l dz is also the probability for next-photon emission in the interval dz at time t + z. Hence, the delay function at delay zis simply the spontaneous emission rate from state 12, n - 1) at time t + z, that is, D(z) = 2BICy)(t +

(16)

2)".

The solution to eqs. (144b) with initial conditions (15a,b) is readily obtained by means of Laplace transforms. The solution for C$"(t + z) reads C,(t

+ z) =

-

61*[exp(S+z) - exp(S2(S+ - S - )

41 9

where S, =

- $ [ B - iA f ,/b2 - (Az + lRIZ+ 2iflA)I.

Hence, the delay function (16) is

The delay function (17) tends to zero as z+O. Thus, after the atom has emitted a photon, some time is required before the next photon can be emitted. This is the phenomenon of photon antibunching, appearing here in the nextphoton delay function (KIMBLEand MANDEL [1976], DAGENAISand MANDEL[ 19781, KIMBLE,DAGENAIS and MANDEL[ 1977, 19781). The delay function simplifies somewhat for the case of exact resonance (A = 0). For 161) < B2 it consists of three decaying exponentials D(z)=

filolz[e-Y+'+ e-Y-7- 2e-871 2(PZ -

I alZ)

9

where y* = B (1' - 1 6112)'/2. For a weak driving field (I 611' G B') the delay function grows from zero to approximately I 611 2/2flin time fl- and then slowly decays to zero as (1 611' / 2 f i ) exp( - I 611 'z/2B). The rise time B- is the time required for the atom to reach its steady-state excitation probability P2 = (1 611 /2/3)'. The broad delay function for this case reflects the fact that a

',

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weak field is slow to excite the atom, and hence, the next photon is slow to be emitted. In general, the mean time required for excitation and emission of the next photon is -C =

7D(7)d7 =

2($

+ A2) + I 621' BlQ12

For a strong field (1 Ql' > B') the delay function oscillates at frequency (I Ql ' - B2)'l2 and decays exponentially at the rate 8:

It is interesting that the next photon cannot be emitted with delays 7 = 2arn/( 1 01 - /I2)'/', m = 1,2, 3, .. .,because the delay function is zero at these times. It should be emphasized that the delay function for a two-level atom is rather difficult to measure directly. To measure it, one would have to detect every emitted photon with near unit efficiency in order to ensure that the next photon, rather than some later photon, is registered. A quantity that is more easily measured, and the one that has been measured in photon antibunching experiand MANDEL ments (KIMBLEand MANDEL[1976], KIMBLE,DAGENAIS [ 1977,1978]), is the probability that a photon, and not necessarily the next one, is registered in the interval d 7 at delay 7. This quantity is more easily measured because it is not necessary to ensure that no photons are emitted in the interval (t, t + 7 ) preceding the interval d7 of interest.

3.2. INTERRUPTION OF FLUORESCENCE DUE TO SHELVING

Perhaps the simplest case of intermittent atomic fluorescence is that which results from the energy level configuration of fig. 8. Energy levels 1 and 2 form what is essentially the two-level atom considered earlier. The transition 1 c,2 is driven by a field of frequency o near the transition frequency wo = ~ 0The ~ strength of the driving field is described by the Rabi frequency 0,and level 2 spontaneously decays to level 1 at the rate pzl. Transition 1 c,2 is a strong, lo8 s - I ) , whose resonance fluorescence is dipole-allowed transition (f12' readily observed. The new feature is a metastable level 3 between levels 1 and 2. Let the

-

~

.

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W

FAST

/ /

Fig.8. Energy level scheme for observing quantum jumps using a single driving field, the A-configuration.

-

spontaneous decay rate from level 2 to level 3 (823) and from level 3 to level 1 (831)be very small, say 8 2 3 , 8 3 1 1 s - ' . During the resonance fluorescence cycle 1 -+ 2 -+ 1 -+ 2 -+ ..., the atomic electron repeatedly decays from level 2 to level 1. This decay channel is overwhelmingly probable compared with (- 10's- I ) is very much greater than the decay to level 3 because flI3 (- 1 s-'). (The branching ratio for decay from level 2 to level 1 is PI = + 8 2 3 ) % 1, whereas that for decay to level 3,

Fig. 9. Fluorescence cascade for the three-level atom of fig. 8. Transitions labeled F are fast spontaneous decays, and those labeled S are slow.

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-

P3 = 823/(821 + 823) lo-*, is very much less than 1.) Although the branching ratio to level 3 is very small, eventually the electron is optically pumped into level 3, and while “shelved” in this level the atomic fluorescence is switched off. The fluorescence remains off until the electron spontaneously decays to the ground state (level l), where it is again available for the fluorescence cycle. A practical advantage of this particular scheme is that only a single driving laser is needed for the observation of quantum jumps. The essential transitions involved in the atomic fluorescence and in the turning off of the fluorescence, 1t)2 and 24+ 3, form a figure resembling the Greek letter A. Therefore, we shall refer to this configuration of energy levels and transitions as a A-configuration. To analyze the statistics of the intermittent fluorescence signal, we again idealize the driving field as a single quantized mode with energy levels En = h wn. The energy levels of the atom + field system are the energies E ,E,, and E3 of the atom together with replicas of these levels shifted up in energy by all multiples of h o.These levels correspond to the atom being in one of its three energy states and the field being occupied by any integer number of photons. The full energy level diagram, with the levels labeled by the states I i, n) belonging to the energies E , = El + hon,is depicted in fig. 9 for the case of exact resonance (w = wo). If a fluorescence photon, one of frequency wo = (E2 - E , ) / h , has just been emitted, the system is in one of the atomic ground states I 1, n) . From here there are two routes to the next lowest atomic ground state on the excitation ladder (state 11, n - 1)). The transitions of interest are illustrated in fig. 10. Beginning in the ground state I 1, n), the atom is promoted to the excited state 12, n - 1) by the applied field (lower solid curved arrow in fig. 10)which can also return the atom to the initial state (upper curved arrow in fig. 10). From the excited state 12, n - 1) the electron can decay directly to the ground state I 1,n - 1) with high probability (branching + or it can decay to the metastable state 13, n - 1 ) ratio P, = 821/(j321

,

-, ’ FAST ,

‘

/

,

/

/

11,

n-1)

SLGW I

13. n-1)

SLOW /

A ’

FINISH Fig. 10. Routes from state I 1, n ) to state I 1, n

- 1 ) in the fluorescence cascade.

386

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[V,4 3

with very low probability (branching ratio P3 = 8 2 3 / ( 8 2 1 + 8 2 3 ) ) . With very high probability the decay from state 12, n - 1) is to state I 1, n - 1) exactly as in the case of the two-level atom. The next decay is also very likely to be of this type, and the next, and the next. Consequently, there are long sequences of decays down the excitation ladder that do not go through level 3. Such a decay sequence describes a period during which the resonance fluorescence is on. The expected duration of a fluorescence-on period is calculated as follows. Except for a negligible correction due to the possibility of decay to level 3, the mean time Z required for the electron to go from state I 1, n) through state 12, n - 1 ) to state I 1, n - 1) is the same as the mean delay between fluorescence photons in the two-level case. If there are n such decays in the sequence, the duration of the fluorescence-on period is t = nZ. As noted earlier, the probability for a single fast decay, as opposed to a decay through level 3, is P , = 821/(821 + 8 2 3 ) . Thus the probability of n consecutive fast decays is ( P I)". This is the a priori probability for n consecutive fast decays, given that slow decays through level 3 may also occur. Normalizing the distribution to unity, we obtain the probability P,, for the length n of a sequence of fast decays: P,

=

On converting this discrete distribution over n to a continuous probability density for t = n5, we find, for n large, that the durations of fluorescence-on periods are exponentially distributed with mean value ton = 8 2 1 ' 2 / 8 2 3 . The probability density for t reads

A fluorescence-on period is terminated by a transition to level 3. Consider again the initial condition where, just after the emission of a fluorescence photon, the system is in state 11, n). It is clear from a comparison with the two-level cascade, fig. 7, that the amplitudes CP)and Cp)to be in states I 1, n) and I 2, n - 1) ,respectively, obey the same equations as for the two-level atom (eqs. (14a,b)), with 8 equal to the total decay rate from state 12, n - l ) ,

8 = 821 + 8 2 3

*

For a strong ( 1611 > 8) resonant (o= coo)driving field our approximate solution indicates that the amplitude Cp)is near zero after a time z of order 8s. Thus the decay to level 3, when it occurs, takes place in a very N

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short period of time, a period that is negligible on the scale of the durations of fluorescence-on and fluorescence-off periods. The possibility of detecting the photon given out on the transition 2 + 3 projects the system into state 13, n - 1 ) . This state simply decays by spontaneous emission to state I 1, n - 1 ) at the slow rate 8 3 1 (the probability decays at rate 2P3,).It follows that the time spent in level 3, and hence the durations of the fluorescence-off periods, are exponentially distributed,

Notice that, when the time spent in the shelving level is unaffected by the driving field, as in the present case, the mean duration of the fluorescence-off periods (“dark periods”) is simply the lifetime of the shelving level, to* = l/2831.Thus the observation of telegraphic atomic fluorescence provides a method for the measurement of certain long atomic lifetimes that might be difficult to measure by other means.

3.3. DELAY FUNCTION FOR THE V-CONFIGURATION

The V-configuration of energy levels depicted in fig. 2 has been considered by many authors for the observation of quantum jumps (COOKand KIMBLE [ 19851, PEGG,LOUDONand KNIGHT[ 19861, ARECCHI,SCHENZLE, DEVOE, JUNGMAN and BREWER[ 19861, SCHENZLE,DEVOE and BREWER[ 19861, KIMBLE,COOKand WELLS[ 19861, and others). This energy level scheme is realized in the Hg+ ion, which was used by BERGQUIST,HULET,ITANOand WINELAND[ 19861 in one of the early observations of quantum jumps. The present analysis of telegraphic fluorescence in the V-system follows the delayfunction approach of COHEN-TANNOUDJI and DALIBARD [ 19861. In the V-system the strong and weak transitions are driven by Rabi frequencies 62, and f12, respectively, and levels 1 and 2 decay to the ground state at rates and P2, with 3 P2 (the Einstein coefficients are A = 28, and A, = 28,). From the qualitative argument given in the introduction, telegraphic fluorescenceis expected when the induced and spontaneous transition rates on the strong transition are much larger than the induced and spontaneous rates on the weak transition, 621,

81 % 622, P2

With two laser fields acting on the atom, each described by a quantized field mode, the full energy level diagram of the atom + field system is a little com-

388

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[V,8 3

plicated. So, instead of working with the full energy level diagram, we consider only the essential states involved in the emission of the next photon. The states I i, n ,n2 ) of the atom t field system are labeled by the index i of the atomic energy Ei (i = 0, 1,2) and by the numbers of photons n, and n2 in the modes that drive transitions 00 1 and 002, respectively. The energy of state li, n,, n 2 > is = Ei

+ h u l n , + hw2n2

9

where w , and 0,are the frequencies of the driving modes. We assume that the driving frequencies w , and 0,are near to the transition frequencies w,, and w,,,, respectively. Immediately after the emission of a photon from either excited state the atom is in the ground level 0 with, say, n, and n2 photons in the driving modes (state 10, n,, n,)). From this state the applied fields can promote the atom to level 1 (state I 1, n - 1, n, )), by absorbing a photon from driving mode 1 ,or to level 2 (state 12, n ,n, - 1 )), by absorbing a photon from driving mode 2. The applied fields can also return the atom to the ground state I 0, n ,n, ) by stimulated emission. In the absence of spontaneous emission and when the driving fields are close to resonance with their respective transitions, an atom initially in the state 10, n,, n , ) remains in the manifold I O , n , , n , ) , I l , n l - 1,n2), and 12,n,,n, - 1 ) to an excellent approximation. Let CoyC , , and C , be the amplitudes to be in the essential states IO,n,,n,), I l , n , - l , n 2 ) , and 12,nl,n,- l),respectively.Intheelectricdipole and rotating-wave approximations the radiative processes induced by the driving modes are described by the equations

, ,

C0 -- r,a, c, + $,c,

Y

-$fC,

+ iA,C, ,

C, = -$;C,

+ iA2C2,

Cl

=

(18)

where A , = w , - wl0 and A, = w2 - w2, are the detunings of the mode frequencies with respect to the transition frequencies. For later convenience we will take the Rabi frequency 0,to be real and positive. This involves no loss of generality because a, can always be made real by properly choosing the phases of C, and C , . The essential states are illustrated in fig. 11 with induced transitions denoted by curved solid arrows. Equations (18) describe the coherent excitation of the atom in the absence of spontaneous emission. Spontaneous emission from either excited state creates a fluorescence photon and removes the system from the manifold of essential states. Specifically, spontaneous emission from state I 1, n, - 1, n,)

v. B 31

389

THEORY OF TELEGRAPHIC FLUORESCENCE

I

I

;,/

LAST FLUORESCENCE PHOTON

I a1

C1 11,n1-1, nz)

'

I

n \_t

I

0 2

4 co 10, n,, n2)

U I 12, n,, n z - l )

I -4

I+--I

c2

NEXT FLUORESCENCE PHOTON

I P2

P1

I

I

c

L 10,

10, nl-l, n2)

nl. nz-1)

Fig. 11. Essential states of the V-configuration. Induced processes are denoted by solid arrows, and spontaneous emission by dashed arrows.

carries the system to state 10,n, - 1, n,), and spontaneous emission from state 12, n n, - 1) takes the system to state 10, n,, n2 - 1) .These processes cause decay of the amplitudes C , and C, at the rates /3, and /I2, respectively. When spontaneous emission is included, eqs. (18) become c0 --_:a, c, + $,c, ( 194

,,

Y

+ (iAl - /3,)C,, c2 -- - ' 2a 2*co+ (iA, - /32)C,

Cl

=

-$,C0

9

(19b) (19c)

With initial conditions CO(t) = 1 9 Cdt) = 0 9

C,(t) = 0

9

at the time t of the last fluorescence photon, eqs. (19a-c) determine the amplitudes Co(tt z), C,(t t z), and C,(t t z) for the atom to be in levels 0, 1,

390

QUANTUM JUMPS

[V,8 3

and 2, respectively, at time t + z without having emitted the next fluorescence photon. The total probability for not emitting the next fluorescence photon in time z is P ( z ) = IC,(t+

z)12+

IC,(t+ 2)12+ IC,(t+ z)1?

(20)

Therefore the probability for emitting the next photon during the interval (t, t t z) is F(z) = 1 - P(z), and the derivative of this distribution function is the probability density for emission at time z, that is, it is the delay function,

It follows from eqs. (20) and (21) and the equation of motion (19) that

D(?)= 28,1C,(t

+ z)IZ + 2/?21CZ(t+ z)l’.

The delay function is simply the total spontaneous emission rate from the two excited states. An explicit expression for the delay function requires solution of eqs. (19a-c). To simplify the solution of eqs. (19a-c), we specialize to the case in which the field driving the strong transition is on resonance (A, = 0) and strong (I 62, I >> &). We also take the time t of emission of the last photon as the zero of time (t = 0). With the atom initially in level 0 it is highly probable that the strong applied field (a,)promotes the atom to level 1, from which it rapidly decays out of the manifold of essential states (see fig. 11). Therefore, to a first approximation, transitions to and from level 2 may be ignored. The second term on the right in eq. (19a) represents transitions from level 2 to level 0. Dropping this term, eqs. (19a,b) become

c0 -1. - ,a,c,

9

c, = -:62,co - /?,c, .

(234 (23b)

These equations are formally identical to the two-level eqs. (14a and b) studied earlier. That the amplitudes for levels 0 and 1 should behave approximately as those of a two-level system is expected, because transition 0- 1 is strongly driven and the weak transition acts as a small perturbation to the two-level behavior. Equations (23a and b) are readily solved by means of Laplace transforms. For 162, I >> PI, the case under consideration, the solution reads

C0(4= exp ( - PI 2/21 cos (a,2/21

(244

exp( -&2/2) sin(a,2/2).

(24b)

C,(z)

=

THEORY OF TELEGRAPHIC FLUORESCENCE

39 1

Here we see a rapid Rabi flopping of the probability between levels 0 and 1, and a decay of both amplitudes at the rate &/2, which is due to spontaneous emission from level 1. To the next order of approximation, there is a small chance for a transition to level 2. Equation (19c) indicates that C, acts as a source term for the amplitude C,. The general solution of eq. (19c) for C2(z) in terms of C,(t) reads C,(z)

=

- $2 exp [(iA2 - f i 2 ) 2 ]

s,'

C,(z) exp [ - (iA, - f i 2 ) t ] dt .

Into this exact expression for C2(4 we substitute the approximate expression (24a) for C,(t). Note that C,(t) becomes exponentially small for t % 8; So, for z % 8; l , the upper limit on the integral can be taken to infinity, and the factor exp ( - P2t) in the integrand may be set to unity because, for pzQ fll, this factor differssignificantly from unity only when t % 8; l . These approximations yield

C,(z)

=

--

1

i(2A2 - a , )

+ fll

+

i(2A2 + 0,) + fil

The amplitude for a transition to level 2 has a resonance structure with two peaks. There is a resonance of width at A2 = 01/2 and another of the same width at A, = - Q1/2. This is the Autler-Townes splitting of the weak absorption line. The splitting results from the strong driving of transition 0 t)1. The probability of transition to level 2 is quite small, since 10,I 4 /I1. To maximize the transition probability, we tune the radiation acting on the weak transition to one component of the Autler-Townes doubles (A2 = n1/2). Then

C,(z)

02 exp[(inl/2 - /3,)z]

= --

281 This expression is not valid for arbitrarily large z because, in dropping the second term on the right in eq. (19a), we have ignored the possibility of a transition from level 2 back to level 0 (stimulated emission by the field 0,)This . term becomes important when it is not overwhelmed by the fist term on the right in this equation. From eq. (25) the second term is found to be of the order 10,12/f11, whereas from (24b) the first term is seen to be of the order l0,l exp( -fi1z/2). The second term starts to become significant when

392

QUANTUM JUMPS

I Q2 I '/& z I a, I exp ( - fll 2/2), or when z exceeds 2

z * = -- l n [ s ] , 81

/I1 = loss-', la,l = lo's-,, and lazl = 1 s - l we have z* = 7 x s, which is considerably longer than the decay time 2/& ( = 0.2 x s) of the amplitudes in eqs. (24a and b), but much shorter than the decay time 1/& ( = 1 s) of level 2. We conclude that (25) is an accurate expression for C,(z) when 2//3, d z < z*, in which case &zd 1 and con-

With

sequently (25) reduces to

',

During the buildup of this amplitude, for z < 8; there is a small reaction back on the amplitude C, that is required for conservation of probability. Although the correction to C,(t) and Cl(t) is easy to evaluate, its effect on those amplitudes is so slight that we prefer to ignore it with the understanding that in so doing the unitarity of the induced processes is slightly compromised. After the initial surge of probability into level 2, the amplitudes for levels 0 and 1 have decayed essentially to zero, and the probability in level 2 slowly decays by two routes (see fig. 11). There is the direct decay to state 10,n,, n - 1) at the rate p2 with the emission of a fluorescence photon of frequency 02, (spontaneous emission), and there is the possibility that the field acting on the weak transition will drive the atom back to the ground state 10, n,, n2) (induced emission), from which it is rapidly transferred to level 1 with decay to state 10, n, - 1, n2) and the emission of a fluorescence photon of frequency olO. The latter route is slow because of the time required for induced emission from level 2 to level 1 by the weak field a,. To analyze this portion of the decay, we use the fact that the probability for level 2 is a slowly varying function of time. Specifically, the amplitude to be in level 2 has the form C,(z)

= a2(z)exp( + i n , 2/2),

(27)

where a&) is slowly varying. From eq. (26) we read the initial condition for a,:

at .

a,(O) = --

28,

v, B 31

THEORY OF TELEGRAPHlC FLUORESCENCE

393

Here we are ignoring the short time 1/& required for the buildup of this amplitude. This is appropriate when considering the long time scale of the decay from level 2. The amplitude C, may be viewed as a source term in the equations for Co and C,,eqs. (19a and b). In order to work with slowly varying amplitudes, we set C,(t) = ao(t)exp( + i n , 2/2),

(294

+ i62, 2/2) .

(29b)

C,(t) = a&) exp(

Then eqs. (19a,b) become

Because the amplitudes a, and a, are strongly coupled by the field a , , they relax together at a rate -8, determined by the last term in eq. (30b). (This behavior is evident in the solution (24). Thus these amplitudes quickly decay to the “steady state” values determined by the instantaneous value of the driving term Q2a,(2)/2. That is to say, the amplitudes a, and a, adiabatically follow the slowly varying amplitude a,, the values of a , and a, being determined by the solution of eqs. (30a,b) with h , and h , set to zero. The solution reads

i 62,

= - a 2( 2 )

28,

Hence the decay rate from level 1 is proportional to the level 2 probability,

R , = 28, la,I2 =

lQ212

la2I2/281 *

(32)

This is the rate at which probability flows back from level 2 through level 0 to level 1 and is lost by spontaneous emission. This decay must have a back reaction on the amplitude a, in order to conserve probability. On substituting eqs. (27) and (29a) into eq. (19c), the equation for a,, with A, = 62,/2, is seen to be a, = -$62z*a,

- p,a,.

394

[V, B 3

QUANTUM JUMPS

With the amplitude a, adiabatically following a, according to eq. (31a), this becomes a 2 = -+[2fi2+---

1M2

iln,12]a2,

2/31

0 1

z) ( y

and using the initial condition (28), we obtain C,(z)

=

--62; exp (-i(2/3, 2/31

+

z+ii

+ ol) z}. (33)

This result indicates that the probability to be in level 2 decays at the rate R, = 2/3, + I 62, I '/2&, which is the rate 21, for direct spontaneous emission from level 2 plus the rate I 62, I ,/2/3, for induced emission with a transition to level 0 followed by excitation to level 1 and spontaneous decay from that level. The latter rate is the same as the rate (32) of spontaneous emission from level 1 during this phase of the decay. The delay function (22) now follows from the amplitudes (24b) and (33):

D(z) = 28, exp( -jlz) sin2(62,z/2)+-'21

'2'

' exp { - (28. + %) lW2 z}

28; (34) The delay function is drawn in fig. 12. The first term on the right in eq. (34)

Fig. 12. Delay function for the V-configuration (not to scale).

.

v, 8 31

395

THEORY OF TELEGRAPHIC FLUORESCENCE

gives rise to the large peaks at small values of z (z 6 8; I ) , whereas the second term gives the function a long exponential tail at large values of z (z 8; I ) . The total probability for fast decay PF (the area under the peaks at z 8; I ) is very near unity, and the probability for slow decay Ps (the area under the exponential tail) is very small. The integral of the second term in eq. (34) is

-

N

and P, = 1 - Ps z 1. The high probability for short delay times implies that long strings of fluorescence photons are emitted with short delays between them, delays of the order 8; Such strings represent bright periods in the atomic fluorescence. In the present case the time required for a fast decay is the decay time of the probability I C, 1 + I C,1 2, which according to eqs. (24a,b) is 8; Arguing as in the previous subsection, we obtain from the probability for n consecutive fast decays, P,,= Ps( 1 - Ps)", an exponential probability density for the duration of fluorescence-on periods:

with mean fluorescence-on time

The long exponential tail on the delay function, with very small total probability Ps, implies that occasionally there is a long delay between fluorescence photons. Such a delay is a dark period in the atomic fluorescence. The probability density for the duration of fluorescence-off periods is simply the probability density for a long delay, that is, it is the tail of the delay function normalized to unit probability,

with mean fluorescence-off time

In summary, the high probability for short delays leads to long sequences of

396

[V, 8 3

QUANTUM JUMPS

fluorescence photons with short delays between them. These are the “bright periods” in the fluorescence. The long exponential tail on the delay function, with very small total probability, represents infrequent but long “dark periods” in the fluorescence. In this way the delay function formalism of COHEN-TANNOUDJI and DALIBARD[ 19861 provided the first unequivocal proof of the existence of dark periods in the atomic fluorescence prior to the experimental demonstrations of the effect.

3.4. DELAY FUNCTION FOR THE A-CONFIGURATION

We conclude this section with a calculation of the distribution of delays between fluorescence photons for the A-configuration. For the V-configuration we calculated the distribution of delays between fluorescence photons without regard for which transition radiated the next photon. Strictly speaking, what we are interested in is the distribution of delays between the photons radiated by the resonance transition. It turns out that this distinction is not important for the V-configuration. For the A-configuration it is more convenient to concentrate on the delays between resonance-transition photons. Recall that, immediately after the emission of a fluorescence photon on the transition 2+ 1, the atom is in its ground state with, say, n photons in the driving mode (the state 11, n ) in fig. 10). According to eqs. (14a,b), which approximate the time development of the amplitudes Cy) and Cy),the probability I C p ) ) 2+ I C$”)I2to remain in state 11, n ) or state 12, n - 1) decays to zero very rapidly. (For 1 0 1 B 821the solution is given by eqs. (24a,b) and the probability decays in a time of the order /I&’.) The two decay channels (to state 1 and to state 3) have distinguishable final states. Consequently, the probabilities for these alternatives simply add without interference. If the atom decays directly to the ground state, the distribution of decay times for this emission is clearly the same as for the two-level atom (eq. (17)). On the other hand, if the atom decays to the metastable level 3, then after the short time required for this transition (a time of the order &;l), the distribution of delay times is the distribution of dwell times in level 3. Thus the total delay function is the two-level-atom delay function weighted by the probability for decay to state 1 (which is approximately unity) plus the exponential distribution 831exp( - 831z) of level 3 lifetimes weighted by the probability for decay to level 3 (823/(823 + BZl) * 8231821)- For a strong driving field (I GI B 821), ~ ( z= ) 2821exp( - 821z) sin2(1 0 1 z/2) + 831823 e x p ( - ~ ~ ~ z ) . B2 1

v,

41

THE NATURE OF QUANTUM JUMPS

397

The delay function is of the same form as that for the V-configuration and has the same interpretation. The high probability for short delays (the first term) implies long sequences of fluorescence photons with short delays between them, and the low probability for long delays (the second term) describes infrequent but long dark periods in the fluorescence.

6 4. The Nature of Quantum Jumps Telegraphic fluorescence is only one effect that permits the observation of quantum jumps. Before the observation of telegraphic fluorescence, “quantum jumps” of a single isolated electron between its two spin states were monitored by VAN DYCK,SCHWINBERG and DEHMELT [ 1976, 19771. The change of the axial oscillation frequency of a single electron in an anharmonic Penning trap is shown in fig. 13. The arrival and departure of a single electron in a miniature SKOCPOL, metal-oxide-semiconductor junction has been monitored by RALLS, JACKEL, HOWARD,FETTER, EPWORTH and TENNANT[ 19841. These rather different observations of single-electron transitions call for a generalization of h

N

I

Y

t

TIME (minutes)

- -2

I

5?

4 Fig. 13. Record of the frequency shift of the axial oscillation of a single electron in a Penning trap. The bottom edge of the “grass” indicates the axial frequency, which changes with changes in the spin state of the electron. The frequent upward excursions of the trace are due to thermal SCHWINBERG and fluctuations in the cyclotron motion of the electron. (From VAN DYCK, DEHMELT [1977].)

398

QUANTUM JUMPS

[V,B 4

our notion of what it means to observe quantum jumps - one that goes beyond the specific technique of monitoring telegraphic fluorescence. As noted in the introduction, the Bohr picture of quantum jumps assumes that a quantum system is at each instant in one, and only one, of its energy eigenstates, and this is diametrically opposed to the notion of coherent superposition states. Therefore, if the observations of telegraphic fluorescence, electron spin flips, and so on, are to be interpreted as true observations of quantum jumps, we must understand how these observations circumvent the objection that a quantum-jump picture cannot correctly describe the coherent evolution of a quantum system. Answers to this question have been given by several authors with various degrees of completeness (PORRATIand PUTTERMAN [ 1987,19891, PEGGand KNIGHT [ 19881). Here we expand on the answer given by COOK[ 19881.

4.1. ATOMIC DYNAMICS DURING FREQUENT MEASUREMENTS

In all of the methods used to date for the observation of quantum jumps, the system under consideration is, more or less, continuosly monitored. That is to say, measurements of the state of the system are made frequently and the results of the measurements are recorded in some fashion. But energy measurements cannot be made with arbitrary rapidity. According to the time-energy uncertainty relation, a minimum time At of order R/AE is required to distinguish states separated in energy by BE. In practice the time required for a state determination is usually much longer than this theoretical minimum. In any case the state measurements are not infinitely frequent but are necessarily separated by some time interval z. It is the purpose of this section to show that, if the time z between measurements is sufficiently small, then, to a good approximation a coherent superposition of energy states does not develop and the Bohr picture of quantum jumps accurately depicts the system dynamics. To make the argument general, we base it on the measurement postulates (VON NEUMANN[1955], JAMMER [1974]) rather than on any particular monitoring technique such as telegraphic fluorescence. We shall consider only jumps between discrete energy levels. According to the measurement postulates, a measurement of energy projects the system into the energy state indicated by the result of the measurement. If the measurement is immediately repeated, the same result is obtained and the system remains in the same state. But if there is a short time interval z before the next measurement, there is a small probability that a driven system will make a transition to a different

v, v, 88 41 41

399 399

THE NATURE NATURE OF OF QUANTUM QUANTUM JUMPS JUMPS THE

energy state. The probability of transition necessarily tends to zero as z+ 0 because an immediate repetition of the measurement must find the system in the same state. Hence, the probability of transition in time T can be made as small as desired by choosing z to be small enough. (We are assuming here that the minimum measurement time h/AE is so small that it need not be considered in this argument.) When zis sufficiently small, the probability is very near unity that the system will be found in the same state at the end of the interval z as it was at the beginning. For a sequence of measurements separated by sufficiently short intervals T, the probability is high that long strings of measurements will find the system in the same state. Only rarely will a transition take place between the measurements. When a transition does take place between measurements and the system is found in a new energy state, the probability is again high that a long string of measurements will find the system in the new state. Thus the measurement postulates indicate that the results of frequent energy measurements on a system with a discrete spectrum are qualitatively similar to those depicted in fig. 14. The record of the results of frequent energy measurements is in accord with Bohr’s picture of quantum jumps. During a string of measurements with the same result E,, we say that the system is in the state belonging to this energy. When the measured energy switches from one value to another, we say that a quantum jump has occurred. It is tempting to go beyond the picture presented by the measured results and to ask questions such as, “When did the quantum jump actually occur between the two measurements that gave different results?’ But such questions are undoubtedly inappropriate because it is the measurement itself that projects the system into the new state. The state of the system just before a measurement that yields a new result has only a very small probability of being in the new state. Hence, quantum jumps seem to be a property of the measurement process and, as defined here, exist only in the context of frequent measurements.

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

E3

-

El

t

Fig. 14. Results of frequent energy measurements.

400

QUANTUM JUMPS

[V.B 4

This conclusion is supported by the result that the rate of quantum jumps depends on the frequency of the measurements. To see this and other interesting properties of frequent measurements, we consider the simple case of a two-level atom driven on resonance by a field of Rabi frequency 0.If a state measurement has just found the atom in the ground state, the probability to be in the excited state at time z after this measurement is P2(z) = sin2(62z/2).We say that a sequence of measurements is frequent when the probability of transition between measurements is very small. In the present case the measurements are frequent when separated by intervals z 4 1/62 The probability of transition in time z is then p = sin2(flz/2) = O2z2/4, and the same transition probability applies for an atom initially in the excited state. Two important conclusions follow from this simple result. First, when a sequence of frequent measurements find the atom in the same state, say E 1, the probability of being in the other state between any two measurements (p = Q2t2/4for t < z) is very small, and the probability of being in the state indicated by the last measurement result (1 - p = 1 - n2t2/4for t < z) is very near unity. This justifies our statement that a string of energy measurements yielding the result E, indicates, with high probability, that the system is in the state belonging to this energy. More importantly, during such a string of measurements, the small probability of being in any state other than that indicated by the measured results implies that frequent measurements prevent the development of a coherent superposition state in which more than one of the energy states is strongly represented. In this way the process of frequent measurement circumvents the superposition state objection to the quantumjump picture. Frequent measurements prevent the development of superposition states and thereby allow an interpretation of the results of frequent measurements in terms of quantum jumps. The second conclusion to be drawn from the two-level example is that the rate of quantum jumps is slower the more frequent the measurements are. From either state the probability of transition in time z is p = 62’ z2/4. Hence the rate of transition (probability per second) R = p / z = 0 2 z / 4 . This result, together with the fact that the system is, with high probability, in one or the other of its states at each instant, allows us to write the following rate equations for the probabilities P, and P2 to be in level 1 and level 2:

P,

=

R(P2 - P1),

P2 = R(Pl - P 2 ) .

(354 (35b)

During frequent measurements, the dynamics of the two-level system is

THE NATURE OF QUANTUM JUMPS

40 1

governed by these rate equations rather than by the Schr6dinger equation. The probabilities P , and P2 represent averages over the ensemble of possible quantum jump histories. The all important point is that the rate of quantum jumps R = Q2z/4 = Q2/4f decreases as the frequency f of the measurements increases. In the limit of continuous measurement (f+ a),which strictly speaking is not possible, the rate of quantum jumps tends to zero. This is the so-called “Zeno-effect”,in which the observation of a quantum system inhibits transitions in that system. (MISRA and SUDARSHAN[1976], FONDA, GHIRARDI and RIMINI [ 19781, LEVITAN, HOROWITZ and REPHAELI[ 19871). The Zen0 effect clearly illustrates that quantum jumps are as much a property of the measurement process as they are a property of the system being measured. Very recently the Zen0 effect was demonstrated experimentally at the National Institute of Standards and Technology (ITANO,HEINZEN, BOLLINGERand WINELAND [ 19901). Analogous experiments may be possible for the quantum jumps between the spin states of a single electron (DEHMELT [ 1986a,b]). It appears that the Zen0 effect does not occur for spontaneous emission. Frequent measurements slow the transition rate when the atom is coherently driven, because in this case the probability for transition after a measurement grows quadratically with time (p = Q2z2/4). Thus the transition rate ( R = p / z = Q2z/4) tends to zero with the time z between measurements. Spontaneous emission, on the other hand, is a rate process. The transition probability from an occupied level grows linearly with time. Consequently spontaneous emission is not slowed by frequent measurements. To take account of spontaneous emission from level 2 to level 1 during frequent measurements, we simply add the usual spontaneous relaxation terms to eqs. (35a and b), PI = R(P, - P I ) + AP,

,

P2 = R(P, - P,) - AP, , where A is the Einstein spontaneous emission rate. A closer examination of the spontaneous emission process shows that when zis very small (z 4 1/a2,)the transition probability is proportional to z2. The quadratic time dependenceof the initial transition probability occurs during the short interval before the Fermi golden rule is applicable. However, the timeenergy uncertainty principle does not permit the system energy to be resolved in such a short time span. Thus it appears that state measurements cannot be made sufficiently rapidly to slow the spontaneous rate and a Zen0 effect for spontaneous emission cannot be inferred from the initial quadratic transition probability.

402

[V,9 4

QUANTUM JUMPS

The foregoing discussion defines quantum jumps in terms of the results of repeated abstract measurements made on the system of interest. The connection with telegraphic fluorescence as a monitor of quantum jumps is made by noting that the illuminated strong transition in, say, the V-configuration of energy levels (fig. 2) is, in effect, a measurement device which repeatedly measures the state of the two-level system comprising levels 0 and 2. This is most clearly seen by considering the case where a short pulse of radiation is applied to the strong transition. If the two-level system is in level 0, the atomic electron is available to undergo the fluorescence cycle on the strong transition and a fluorescence pulse is given out by the atom. If, on the other hand, the two-level system is in state 2, the electron is unavailable for the fluorescence cycle and the pulse applied to the strong transition does not give rise to a pulse offluorescence. Thus a strong optical pulse applied to transition 0- 1measures the state of the two-level system; a pulse of fluorescence points to state lo), and no fluorescence indicates state 12). Moreover, the state of the two-level system after the measurement is in accord with the result of the measurement. If no fluorescence is observed, the system is left in level 2 at the end of the measurement, and, when fluorescence is observed, either the electron is left in level 0 or it returns to this level very quickly (in the lifetime T~ 10- * s). The pulse measurement is a nearly ideal quantum-mechanical measurement. The time required to perform a state measurement by means of an optical pulse is the time required to generate some fluorescence. The measurement is unequivocal if one fluorescence photon is emitted when the electron is in level 0. The measurement will be “good” if the probability for at least one fluorescence photon is high, given that the system is initially in level 0. This requires that the optical pulse be long enough. For an intense saturating pulse the photon statistics are Poissonian with mean rate (COOK[ 19811). This being the case, the probability of emitting no photons in time t is Po = exp( - &t), and the probability of at least one fluorescence photon is P = 1 - Po - exp ( - 8, t). For pulse length t = 102, = 10/2/3, approximately 7 out of 1000 measurements fail to give fluorescence when the system is initially in level 0. For t = 1002, only about 2 out of loz2 measurements yield incorrect results. When the strong transition is continuously illuminated, as in recent observations of quantum jumps, fluorescence photons are emitted at the saturated rate 8,. The effective rate of state measurements made on the two-level system (states 0 and 2) is of this order. According to the preceding discussion, the rapid measurement sequence should have two effects on the dynamics of the twolevel system: It should transform the coherent dynamics of the system into a rate process described by eqs. (36a,b), and the rate of transitions between

-

/?,

v, B 41

THE NATURE OF QUANTUM JUMPS

403

-

levels 0 and 2 should be decreased from the coherent Rabi value 61, to the Zeno-slowed incoherent rate R = 61:/4 f 61:/4&. However, the continuous illumination of the strong transition is not purely a measurement process. A strong field on this transition shifts and splits the energy levels of the weak transition (Autler-Townes effect), and this disturbance of the two-level system must be taken into account in a correct calculation of the transition rate. The transition rate has been calculated by KIMBLE,COOKand WELLS[ 19861.

4.2. KNOWLEDGE-INDUCED TRANSITIONS

According to the preceding discussion, a lack of fluorescence from the driven strong transition indicates that the atom is shelved in the metastable upper state of the weak transition. If the atom is initially in a state that is a coherent superposition of the three energy states, an observation of no fluorescence must somehow project the system into the metastable level. The details of this projection process have been discussed by PORATTIand PUTTERMAN [ 19871 and by PEGGand KNIGHT[ 19881 in the context of quantum jump theory. This work provides a detailed theoretical explanation of Dehmelt’s intuitive electron[ 19751). The shelving projection is an example of shelving concept (DEHMELT a type of process discussed by DICKE[ 1981,19861,wherein null measurements alter the future behavior of the system. Probably the simplest example of this kind of projection occurs for pure spontaneous emission, where two levels decay to a single lower level. In this simple context the nature of the projection process is more clearly displayed than in the quantum jump monitoring schemes and some, perhaps, counterintuitivefeatures of observed spontaneous emission come to light. Consider a single atom with two excited states 1 1) and 12) that can decay only to the ground state 10) by spontaneous emission. At t = 0 the atom is prepared in a coherent superposition of the excited states, and thereafter the atom is monitored for photon emission by a perfect photon detector, one that registers every emitted photon. We ask the question: “If no photons are emitted in the time interval (0, t), how do the upper-state probabilities P , and P, change during this interval?” This question is interesting because, on first thought, one’s intuition might lead to the conclusion that, in the absence of spontaneous emission, the upper-state probabilities do not change, whereas the quantum theory makes a different prediction. The intuition is rooted in the notion that atomic level probabilities change as a result of photon absorption and emission processes. In the present example the only possible radiative process is spon-

404

[V.I 4

QUANTUM JUMPS

taneous emission, and this is ruled out by direct observation. It might appear, therefore, that the probabilities P I and P2should not change during an interval of no photon emission. On the other hand, as shown later, quantum electrodynamics predicts that our knowledge of no photon emission induces transitions between the excited atomic states. Specifically, there is a flow of probability to the excited state with the smaller spontaneous emission rate. The flow continues until there is near unit probability in a single exicted state. To see how this result comes about, we shall analyze the process in detail. The initial state of the atom + field system is a coherent superposition of the excited atomic states with the field in the vacuum state, where lo), is the photon vacuum state and @ denotes a tensor product of atom and field states. From this state the atom can spontaneously emit a photon of frequency 0,= ( E l - E,)/h or one of frequency 0,= ( E , - E,)/h, where E,, E l , and E, are the ground and excited state energies. Upon detection of the emitted photon the atom is projected into its ground state lo), and the atom remains in this state thereafter. Here we are interested in the change of the atomic state over a time interval (0, t), during which the detector does not register a photon. With the atom initially in state (37), transitions are possible to states 10) @ I kI) with the atom in its ground state 10) and a single photon of wave vector k and polarization I ( = 1,2) in the field. At an arbitrary time t there are amplitudes C,(t) and C,(t) to be in atomic states 11) and 12), respectively, with no photons present, and amplitudes C,,(t) to be in the atomic ground state with a photon in the field mode labeled by krl. The system state vector at time t reads

,

I y(f)>= c l ( t ) I l >@ lo>,

+ cZ(t) 12)

@

lo>,

+

ckA(t)

kl

lo>

@

lk')F*

(38) The system dynamics is given, to sufficient accuracy, by the Weisskopf-Wiper approximation (WEISSKOPF and WIGNER[ 19301). We find that the amplitudes C, and C, decay at their respective spontaneous emission rates, A , and A,, C,(t) = C,(O) exp( - iE,t/h - A ,t/2) ,

(394

C,(t) = C,(O) exp( - iE,t/h - A , t / 2 ) ,

(39b)

and the amplitudes C,,(t) grow in such a way that the total probability I C, I + I C2I + C,, I C,, I is conserved. In addition, the Weisskopf-Wigner approximation predicts shifts of the atomic energy levels, which turn out to be infinite. However, after mass renormalization and an appropriate cutoff proce-

v, B 41

THE NATURE OF QUANTUM JUMPS

405

dure the physical shift is found to be small and of no particular interest here. As before, it is assumed that the radiative level shifts are already included in the measured atomic energies E l and E,. The amplitudes C,(t), C,(t), and C,,(t), being solutions of the SchrOdinger equation, describe the time development of the system in the absence of measurement. With the perfect photon detector present and operating, we have additional information that partially collapses the state vector. If in time t the detector registers a photon, then the state vector is in the subspace of onephoton states 10) @ I kA) F. Conversely, if no photon is registered in time t, the state vector is in the zero-photon subspace, the subspace spanned by states I 1) @ 10) and 12) @ I O),. According to the rules of measurement theory (COHEN-TANNOUDJI, DIUand LALOE[ 1977]), we must project the full state vector (38) onto the subspace indicated by the result of the measurement and then renormalize the state vector so that the coefficients in the new expansion may again be interpreted as probability amplitudes. The state vector at time t, when no photon is detected, reads

This result, together with eqs. (39a,b), gives the following probabilities for excited atomic states

For A, > A, and any initial conditions other than P,(O) = 0, P,(O) = 1, the probability for state 12) approaches unity as t + co,and P l ( t ) tends to zero. We offer the following interpretation of this result. Suppose the rate A , is very much larger than A,. Then an atom in state 11) will decay in a time of order z, = l/A,, whereas an atom in state 12) will remain in that state much longer, for a time of the order z2 = 1/A2. If the atom is initially in the coherent superposition (37), there is some probability of being in each of these states. But if the atom does not emit a photon in time t % z,, then we have learned that the atom was probably not in state I l),since if it were in that state, it would have emitted a photon in time t with high probability. In other words, our knowledge that the atom did not emit in a time t % z, tends to project the

406

[V,t 4

QUANTUM JUMPS

system into state 12). For short times (of the order z,) the projection is not complete because it might be that the atom is in state I 1) and simply has not yet emitted a photon. The probabilities (40) describe the gradual collapse of the atomic state vector to state 12) as our knowledge of no photon emission increases with time. The preceding calculation might be viewed as artificial in that, in effect, we first calculated the a priori level probabilities p , = I C , I p 2 = I C , I 2, and p o = C,, IC,,12, as if the system were not being observed. From these we calculated the a posteriori or conditional, probabilities P, = p , / ( p , + p , ) , P, = p , / ( p , + p , ) , and Po = 0, which include our knowledge of no photon emission. It seems more natural to work from the start with amplitudes a and a, or probabilities P, = 1 a I and P2 = I a, I that include our knowledge of no photon emission. To this end, we note that the a priori probabilities p , ,p , , and p o satisfy the rate equations

’,

,

,

P1=

-Alp,

9

P2 =

-A,P,

9

Po = A l p , + A 2 P Z

*

It follows that the conditional probabilities P, = p , / ( p , + p , ) and P, = p 2 / ( p 1+ p 2 ) obey the nonlinear rate equations

P, =

- ( A , - A,)P,P, ,

P, = ( A , - A , ) P , P , .

(4 1 4 (41b)

The solution (40) is obtained directly from these equations without introducing the a priori probabilities. These equations describe the flow of probability from level 1 to level 2 in the absence of spontaneous emission. Similarly the equations of motion for the amplitudes a , = C,/(I C , 1’ + I C21z)and a2 = C2/(I C , I 2 + 1 C,Iz)follow from eq. (39), ihu, = E,U, - iih(A, - A , ) P , ~ ,, ihu,

=

E2a2 + $h(A,

-A,)P,a2,

(424 (42b)

When single-atom spontaneous emission is observed, the upper-state amplitudes obey these equations until the photon is detected. After photon detection the upper state amplitudes are zero and the ground state probability is unity. When the state I 1) can spontaneously relax to state 12) at the rate A 12, and this emission is not observed, the terms - A 12 P, and + A 12 P, are added to eqs. (41a,b), respectively. (Here and in the following we assume E l > E,.)

v, 8 51

OBSERVATION OF QUANTUM JUMPS

407

The upper-state transitions described by eqs. (41a,b) or eqs. (42a,b) are not the result of any physical process. They are due to the increase in our knowledge of the state of the atom. In probability theory the change in probabilities resulting from new information is described by Bayed theorem. Therefore, for lack of a better term, we shall refer to such transitions as Bayesian transitions. This nomenclature distinguishes these transitions from physical ones that involve the absorption and emission of photons. The changes in level probabilities resulting from Bayesian transitions are as "real" as those caused by physical transitions. The Bayesian changes in probability are measurable and influence the future behavior of the system. In the example with A % A, the Bayesian transition to state 12) causes a substantial increase in the expected atomic lifetime. With an initial high probability to be in state I 1),the expected lifetime of the atom is near T,. However, after a time t % T~ without photon emission the Bayesian transition to state ( 2 ) gives the atom a new expected lifetime of the order z., Bayesian transitions also occur when the atom is driven by applied fields. An atom with the V-configuration of levels initially in any superposition of the excited states is projected into the metastable excited state by the Bayesian flow of probability when no fluorescencefrom the strong transition is observed for a time that is long compared with the lifetime of the upper level of the strong transition. This is the quantum-mechanical explanation of Dehmelt's intuitive shelving concept. It is interesting that the quantum formalism attributes electron shelving to the lack of fluorescence, whereas the intuitive picture of the process attributed the lack of fluorescence to electron shelving.

8 5. Observation of Quantum Jumps Although the observation of photon antibunching in single-atom resonance fluorescence (KIMBLE,DAGENAISand MANDEL[ 19771, DAGENAISand [ 19781)and other experiments(FINN,GREENLEES and LEWIS[ 19861) MANDEL provide evidence for the discrete nature of quantum transitions in individual HAGER,LEUCHS,RATEIKE and quantum systems (WALLS[ 19791, CRESSER, WALTHER [ 1982]), the first direct observation of quantum jumps in a single atom was that of NAGOURNEY, SANDBERG and DEHMELT[ 19861. These authors observed the telegraphic fluorescence of a single Ba' ion in a radio frequency Paul trap. The energy level diagram for Ba' is given in fig. 15. An level by laser ion initially in the 2S,,2 ground state is promoted to the radiation. From the 'P,,, level the ion can return to the ground state with the

408

QUANTUM JUMPS

\

f

,614.2nm

4931.4nm

455.4nm

\

649.9

Fig. 15. Energy level diagram for the Ba+ ion.

emission of a 493.4 nm photon or it can decay to the metastable ’D3/*level. The branching ratio for decay to the 2D3/2level is substantial. Therefore, with a single laser applied to the 493.4 nm transition, the fluorescence on this transition does not persist. The ion is optically pumped into the 2D3/2level, and the resonance fluorescence is switched off after emission of, at most, a few photons. To prevent this from occurring, the 2P1/2+ 2D3/2transition is also illuminated with laser radiation. With both transitions driven the ion repeatedly cycles between the 2S,,2, 2Pl/z, and 2D3/2levels and fluoresces continuously in the “green” (493.4 nm) and in the red (649.9 nm), the green being the stronger of the two signals. The fluorescence becomes telegraphic when weak radiation is applied to the 2S 1,2 + 2P3/2transition. In the experiment this transition was illuminated with v, al “ L’”’’’”’’’’’

time (sec) Fig. 16. Measured fluorescenceintensityfor the Ba + ion. The dark periods average 30 s duration. (From NAGOURNEY, SANDBERG and DEHMELT [ 19861.)

v, 8 51

OBSERVATION OF QUANTUM JUMPS

409

light from a barium hollow cathode lamp. When the 455.4nm transition is weakly driven, the ion is occasionally removed from the 2 S 1/2w2P1/2w2D3/2 fluorescence cycle by a transition from the ’ S 1,2 level to the 2P3/2level. From there it can quickly return to the 2S1/2 ground state, with no perceptible interruption in the fluorescence, or it can drop into the metastable 2D5/2level, in which case the fluorescence remains off for a period of time. When the ion is “shelved” in the 2D5/2 level, it is unaffected by any of the applied fields. Accordingly, the dwell time in this state is simply the natural lifetime of the 2D5/2level, which is about 30 s. The fluorescence switches on again when the ion decays from the ’D5/2 level to the ’ S 1/2 ground state. This returns the ion to the fluorescence cycle. The observed fluorescence intensity at 493.4 nm versus time is shown in fig. 16. The turning on of the hollow cathode lamp (weak illumination of transition ’ S 1/2~*~P3/2) gives rise to distinct dark periods in the atomic fluorescence, the duration of the dark periods being about 30 s. As noted earlier, the visible resonance fluorescence of a single atom can be seen with the unaided eye. This is the case in the present experiment. It seems remarkable that by simply looking at the Ba’ ion with the dark adapted eye, one can observe the quantum jumps of a single electron in a single atomic ion and that the individual quantum jumps can be detected in this way with near 100% efficiency. The durations of the dark periods are the dwell times in the metastable ’DSl2 level. Because the shelving time in this level is just the time required for spontaneous emission from this state, it is expected that the durations of the dark periods are exponentiallydistributed with mean value equal to the lifetime

Dwell Time in 5’ level (sec) Fig. 17. Histogram distributibn of the durations of fluorescence-offtimes. (From NAGOURNEY, SANDBERG and DEHMELT [1986].)

410

QUANTUM JUMPS

zP3,2

-

5d" 6p

Fig. 18. Energy level diagram for the Hg+ ion.

of the 2D5/2 level. Figure 17 shows a histogram of the fluorescence-off times. By fitting an exponential to the histogram, NAGOURNEY, SANDBERG and DEHMELT [ 19861 obtained the value 32 & 5 s for the lifetime of the 'DsI2level. This was the first lifetime determination by means of telegraphic fluorescence. The second observation of quantum jumps occurred at nearly the same time as the Ba' experiment when BERGQUIST,HULET,ITANOand WINELAND [ 19861observed quantum jumps in a single Hg ion. The relevant energy levels of the Hg+ ion are shown in fig. 18. The Hg+ ion is an example of the V-configuration discussed in the introduction and in $ 3. The strong transition in Hg' is the dipole-allowed 2S1/2-zP1/2 transition at 194nm. The upper state of this transition has a spontaneous lifetime of 2.3 ns and, accordingly, a spontaneous emission coefficient fl w 2 x lo8 s - (ITANO,BERGQUIST, [ 1987a,b]). This is the saturated rate of photon emisHULETand WINELAND sion for the strong transition. In the experiment the strong transition was driven near saturation by frequency-doubled laser radiation. For the purpose of laser cooling the 194nm radiation was tuned slightly below the center of the resonance line. The weak transition in Hg+ is the quadrupole transition at 281.5 nm between the 2S1/2ground state and the metastable 'D5/2 level. The upper state of this transition has a lifetime of about 0.1 s (GARSTANG [ 19641, [ 1987b1). The weak transition ITANO,BERGQUIST, HULETand WINELAND was also driven by frequency-doubled dye laser radiation. The measured fluorescence intensity of a single Hg+ ion with both transitions driven is shown in fig. 19b. Again the dark periods in the fluorescence, here corresponding to electron shelving in the 'D5/z level, are clearly evident. +

'

41 1

OBSERVATION OF QUANTUM JUMPS

0

1

2

-

3

4

Time (s)

Fig. 19. Measured fluorescence intensity of the Hg+ ion. (a) A single ion with only the 194 nm transition driven; (b) a single ion with the 194 nm and 281.5 nm transitions driven; and (c) two ions with both transitions driven. (From BERGQUIST, HULET, ITANO and WINELAND [1986].)

With the 281.5 nm radiation turned off, fig. 19a, the interruptions in the fluorescence become much less frequent but do not disappear entirely. The dark periods that occur when the weak transition is not illuminated are due to infrequent spontaneous transitions from the 2P,/2level to the metastable 2D3/2 level, which slowly decays to the ground state, and to the 2D5/2 level, which subsequently decays to ground. Shelving in the 'D3/2or the 'D5/2level causes dark periods in the fluorescence when only the 194 nm resonance transition is driven. This is essentially an example of quantum jumps in a A-system, the 2D3/2-2D5manifold /2 of levels replacing the single shelving level of the idealized model discussed earlier. When more than one ion is in the trap, the fluorescence signal is more complicated. With two ions present three levels of fluorescence intensity are expected: (1) both ions off, (2) one ion on, and (3) both ions on. The measured fluorescence intensity for two Hg+ ions is shown in fig. 19c. The three levels of 194 nm fluorescence are easily identilied.

412

QUANTUM JUMPS

[V,0 5

The third of the early observations of quantum jumps was made by SAUTER, NEUHAUSER,BLATT and TOSCHEK[1986]. As in the experiment of NAGOURNEY, SANDBERGand DEHMELT[ 19861, these authors observed quantum jumps in a single Ba' ion with both the 493.4 nm (green) and the 649.9nm (red) transition (see fig. 15) driven by dye laser radiation. In this experiment, however, the dark periods in the 493.4 nm fluorescence occur without any radiation at the frequency of the 2 S ,/2*2P3/2 transition. Most of the dark periods are attributed to Raman-Stokes scattering, which leaves the ion in the metastable 2D5/2level. While shelved in this level, the fluorescence remains off. The fluorescence turns on again when the 'D5/2 level decays to ground. The experiments of 1986 mark the end of the period in which the basic concepts of quantum jump theory were questioned and tested. There is no longer any doubt about the essential correctness of Dehmelt's shelving concept, and the quantum jump interpretation of telegraphic fluorescence seems to be generally accepted. In the years following 1986 the theory of quantum jumps has matured. The formalism describing the observation of quantum jumps and the interpretation of this formalism seem to be fairly complete. It now appears that most, if not all, of the questions one can ask about single-atom spectroscopy can be answered by straightforward application of existing formalisms. On the experimental side there have been a number of significant developments since 1986. Generally speaking, most of the recent experiments involve applications of the jump monitoring technique to the investigation of other phenomena in single-atom or few-atom spectroscopy. We list here a few of these applications. Photon antibunching and sub-Poissonian photon statistics have been studied in single-ion experiments by DIEDRICHand WALTHER [ 19871 and by ITANO,BERGQUIST and WINELAND[ 19881. SAUTER,BLATT, NEUHAUSER and TOSCHEK[ 19861 have reported that the rate of simultaneous quantum jumps for two or more Ba+ ions exceeds that which would be expected for independent random jumps of the individual ions. This suggests that a collective or cooperative interaction of the atoms with the radiation field may be operative. The possibility of a collective interaction was investigated by LEWENSTEIN and JAVANAINEN [ 19871, with the conclusion that such effects are small when the ions are separated by more than the radiation wavelength. The radiative decay rates of several levels in a single Hg+ ion were recovered from the statistics of dark periods in the fluorescence under single-laser illumination by ITANO,BERGQUIST,HULETand WINELAND[ 1987bl. The Zen0 effect, which was discussed in $4.1, has recently been demonstrated experi-

VI

ACKNOWLEDGMENTS

413

mentally using the 24Mg+ion (ITANO,HEINZEN,BOLLINGER and WINELAND [ 1990]), and a precise test of quantum jump theory was made using the same ion (HULET,WINELAND,BERGQUIST and ITANO[ 19881). The detection of quantum jumps by means of pulsed radiation has been discussed by BERGQUIST, ITANOand WINELAND[ 19871, a topic of considerable importance in view of the possibility of improved frequency standards based on this technique. In the V-configuration, for example, the light shift and broadening of the weak-transition resonance due to the field acting on the strong transition can be eliminated ifthe strong and weak transitions are driven alternately rather than simultaneously. In this way the weak-transition resonance assumes its natural width and can be used as a reference for a frequency standard. A single laser cooled ion is an ideal medium for an atomic clock because, being at rest, it does not suffer from transit time broadening, the principle source of inaccuracy in current atomic clocks. The first- and second-order Doppler shifts are also negligible for a trapped ion that is well cooled. In this way the accuracy of atomic clocks might be improved by several orders of magnitude (DEHMELT [ 19751, WINELAND [ 19841). It is always dangerous to speculate about the direction of future research developments, but the following are reasonable expectations. At present only a few species have been trapped as single ions, namely Ba+, Hg+, and Mg' . One can expect that, as the technology develops, more ions will become available for single-atom spectroscopy and some of these will have energy level configurations suitable for quantum jump experiments. It seems likely that before long a single trapped ion together with some type of quantum jump monitoring technique will be used as the basis of a more accurate time or frequency standard. This goal is presently being pursued in several laboratories. Perhaps the most important advance in this field will come with the stable trapping and cooling of neutral atoms by means of radiation pressure. Although clouds of neutral atoms have been cooled and trapped for limited periods of time (CHU, HOLLBERG,BJORKHOLM, CABLEand ASHKIN[1985], CHU, BJORKHOLM, ASHKINand CABLE[ 19861, see also the review by STENHOLM [ 1986]), the stable trapping and cooling of isolated uncharged species will allow application of quantum-jump spectroscopy to neutral atoms.

Acknowledgments

I wish to thank Peter W. Milonni, Hans G. Dehmelt, H.J. Kimble, D.J. Wineland, W.M. Itano and Edward Teller for helpful discussions.

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AUTHOR INDEX

A AARTS,E. H. L., 58,86 ABRAMOWITZ, M., 212,267 N., 329,353 ABRAMSON, ADAMSBERGER, K., 3 17,358 G. S., 378,414 AGARWAL, Y.,166, 176 AHARANOV, AI, CH., 345,346, 353 H.,45,71,84 AKAHORI, AKHMANOV, S. A., 192,267 ALLEBACH, J. P., 17, 53, 58, 59,84-86 ALLEN,L., 255,269 ALTE,B., 282,357 ANGUS,J. C., 72,85 F. T., 366, 367, 377, 387,414 ARECCHI, J., 188,267 ARMSTRONG, E., 101, 146, 176 ARTHURS, ASHBY,V. J., 98, 177 A., 413,414 ASHKIN, ASTLE,H. W., 333,357 W. H.,285,353 AUGUSTYN,

B BACHOR, H.-A., 353 N., 327,353 BALASUMBRAMANIAN, BAR-SHALOM, A., 193,245,252,256,268,270 R., 16, 17,84 BARAKAT, E., 57,84 BARNARD, R. J., 273,356 BARTLEIT, BASILA,D., 329,358 J. M., 52,84 BASTIAANS, BATES, R. H. T., 16,84,85 BAULE,B., 309,353 BECKER,K., 273,353 G.,236,267 B ~DARD, J. D., 168, 176 BEKENSTEIN, BENIOFF,P., 167,176 BENNETT, C. H., 167, 176 BENNEIT,V. P., 7,86 BENTON,S. A., 24,84 417

BERGQUIST, J. C., 365,368,376,387,410-413, 414,415 BERNHARDT, M., 79,84 D. H.,9,22,85 BERRY, C. H., 95,177 BETHEA, BHUSHAN, B., 322, 329,353,358 B. J., 349,353 BIDDLES, BIEDERMANN, K., 308,353 BILLING, H., 89, I76 BIRCH,K. G., 285,313,353,356 BISCHEL,W. K., 246,267 BI-ITER,F., 367, 414 BJBRK,G., 108, 121, 122, 125, 146, 153, 176 BJORKHOLM, J. E., 413,414 R., 365, 367, 378,412,414,415 BLA-IT, N., 184, 188-190, 196, 211, BLOEMBERGEN, 213,267,268,270 BOBBS,B., 257,267 BOHM, D., 127, 166,176 BOHR, N., 363,414 BOLLE,A,, 71,85 J. J., 401,413,415 BOLLINGER, BONDURANT,R. S., 173, 174, 176 BONE, D. J., 353 T., 339,353 BONKHOFER, BORN,M., 241,268, 274, 275,322,353 BOWDEN,C. M., 194,225, 260,268 BOYD, T. L., 110, I78 V. B., 89, 135, 141, 165, 176 BRAGINSKY, BRANGACCIO,D. J., 292, 308, 324, 327, 346, 353 BRECHA, R. J., 110, I78 BRENNER, K.-H., 17, 65.85 BRET,G., 189,267 BREUCKMANN, B., 335,353 BREWER,R. G., 366,367,377,387,414,416 L., 159, 165, 176 BRILLOUIN, BROHINSKY,W. R., 306,348,358 BROIA,M., 61,84 BROSSEL,J., 367,414

418

AUTHOR INDEX

BROWN, B. R., 5,6, 8, 22,40,42,84 BROWN,N., 328,334,355 BRUCK,Yu.M., 16,84 BRUNING, J. H., 292, 308, 324, 327, 346, 347, 353 BRYNGDAHL, O., 17, 18,24,26,36,45,47,57, 58,61, 70, 75, 79, 84-86 J. A., 39,53,84 BUCKLEW, BUDIANSKI, M. P.,342,353 BURCH,J. J., 7, 35, 53,84 BURCKHARDT, C. B., 24, 39,84 BURGER, H. C., 325,354 BURNHAM, D. C., 122, 176 R.,308,312,316,324,325,327-329, BUROW, 333, 343, 345, 346,354,355,357,358 BUTUSOV, M.M., 353,356 M., 308,327,356 BUZAWA, BYER,P.W., 110, 140, 178 BYER,R.L., 107, 176 C

CABLE,A., 413,414 W. K., 308,356 CADWALLENDER, K., 39,84 CAMPBELL, G., 123, 177 CAMY, J. L., 194,210,230,239,257-260, CARLSTEN, 268,269 R. L., 189,211, 213,268 CARMAN, CARRE, P., 317, 333, 354 P.,93, 112, 176 CARRUTHERS, C A V E S ,M., ~ . 89,90,96,97,102,127,128,147, 165, 173-175, 176 CHANG, J. C., 236,267 CHAVEL, P.,79, 86 CHEN, CH., 330, 354 316,320, 333, 334,354 CHENG,Y.-Y., CHERNOBROD, B. M., 194,195,260,262-264, 268-270 A. S., 192,267 CHIRKIN, CHOW, W.W., 89,98, 175, 176 CHU, D. C., 29,48-51, 64,69, 70,84 CHU, s.,413,414 CLARK, G. L., 369,371,416 CLINE,T. W., 354 F. E., 72,85 COFFIELD, COHEN-TANNOUDJI, C., 367, 377, 387, 396, 405,414,415 COLE,J. A., 89, 99, 175, 177 C O L L E M. ~ , J., 108, I76 R.J., 24,84 COLLIER,

M., 371,414 COMBESCURE, COOK, R. J., 365,369,371,377,387,398,402, 403,414,415 J. W., 5,84,290,354 COOLEY, CRANE,R.,278,298, 339, 343,354,355 K., 308,330,333,337,338,348,354, CREATH, 358 CRESCENTINI, L., 283,354 J. D., 407,414 CRESSER, D DAENDLIKER, R.,298,306,332,334,335,354, 355,358 M., 382, 383,407,414,415 DAGENAIS, J., 367,377, 387, 396,414,415 DALIBARD, W.J., 9,23,24,37,40,45,73,79,84 DALLAS, DAMMANN, H., 68, 72,74, 75,84 DAVIES,E. B., 129, 176 DAVIS,J. L., 99, 176 DE PAULA, R., 300,358 DEBELL,G. W., 327,353 H., 365, 368, 372, 377, 397, 401, DEHMELT, 403,407-410,412,413,414-416 DEIGHTON, H. V., 16,84 DER WEDUWE, J. J., 264,270 DEVOE,R.G.,110, 116, 137, 139-141, 178, 366, 367, 377, 387,414,416 DEW,G. D., 286,324,354 T. F., 308,356 DEYOUNG, DICKE,R.H., 403,414 F., 376,412,414 DIEDRICH, DIRAC,P. A. M., 187,268 DIU,B., 405,414 DOERBAND, B., 297,310,328, 347,354 DOHERTY, V.J., 328,354 DORSCHNER, T. A., 89,98, 99, 175, 176 K. N., 192,267 DRABOVICH, R.W. P., 89,97, 127, 165, 176 DREVER, W. G., 298,354 DRISCOLL, DRUHL,K., 260,268,269 DRULLINGER, R. E., 368,414,416 D'SOUZA, R., 378,414 J., 188,267 DUCUING, DULING,I. N., 194,253,270 DUROU, C., 296,355 DYER,M. J., 246,267 J., 286, 327,354 DYSON,

E ECKHARDT, G., 188,268 EDELSTEIN, W. A,, 96, 176

AUTHOR INDEX

EDWARDS, R. V.,72,85 EK, L., 308, 329,357 ELLIOTT, D. E., 290,293,354 ELSSNER,K.-E., 308, 312, 316, 324, 325, 327-329, 333, 343, 345,346,354,356-358 EMEL'YANOV, V. I., 193, 195,210, 219,268 ENGLUND, J. C., 194,225, 260,268 EPWORTH, R. W., 397,415 T., 365, 366, 368, 387,414 ERBER, ERF,R. K., 288,354 ERLER,K., 352,353,354,358 ESCHBACH, R., 53,84 EZEKIEL, S., 89, 99, 175, 176-178

F FABRE, C., 123, 177 N., 193,246, 248-250,268,269 FABRICIUS, FAIN,V. M., 190,268 Y.,53,84 FAINMAN, FARHOOSH, H., 53,84 FELDMAN, M. R., 53,63,84 A. F., 332,354 FERCHER, FERWERDA, H. A., 16,84 H.,309,356 FESHBACH, FETTER,L. A., 397,415 FEYNMAN, R. P.,167,177 FIDDY,M.A., 16,84 FIENUP, J. R., 17,29,48,50,51,64,65,69,84 FIGOSKI,J. W., 352,358 FINN,M. A., 407,414 FIOCCO,G., 283,354 FISHER,E., 368,415 FLUSBERG, A. M., 194,260,268 FOCK,V., 166, 177 K., 308,327,357 FOELLMER, FOLIN,K. G., 195,262-264,269 FONDA,L., 401,415 FREDKIN, E., 165, 167, 177 K. R., 308,336,354 FREISCHLAD, FREITAG, W., 302,354 FRBRE,C., 25-27,84,85 FREUND, C. H., 336,355 S., 122, 177 FRIBERG, FRIGHT,W. R., 16,84 FRITZ,B. S., 324, 326,354 FUNELL, W. R. J., 287,354

G GABEL,R.A., 43, 53,84 D., 165, 177 GABOR,

419

GALLAGHER, J. E., 292, 308, 310, 324, 327, 346,353,354 GALLAGHER JR, N. C., 17, 39, 45,46, 53,65, 71,72,84-86 GANCI,S., 25,85 A. V., 369,371,415 GAPONOV, K. L., 16,85 GARDEN, GARMIRE, E., 188,268 GARSTANG, R. H.,410,415 GATES,J. W., 351,354 GAVRIELY, A., 279,359 GEA-BANACLOCHE, J., 89,95,98,175,176,178 N., 339,354 GEORGE, GERCHBERG, R. W., 17,85 D. C., 288,313,354,356 GHIGLIA, GHIRARDI, G. C., 401,415 GIACOBINO, E., 123, 177 T. G., 99,177 GIALLORENZI, GIGGEL,V., 327,356 GIORDMAINE, J. A,, 188,210,268 J.,91,92,177,184,191,192,196, GLAUBER,R. 213,215, 216,218,219,231,260,268-270 GLOGWER, J., 93,179 GOLDENBERG, H. M., 368,416 J. W., 29,48-50,64,69,70,13,74, GOODMAN, 84,85,215,268,290,354 GBPPERT-MAYER, M., 187,268 GORDON, J.P., 89, 101, 146, 159,177 K., 68,75,84 GBRTLER, GOY,P., 110,177 A. S., 194, 254,268 GRABCHIKOV, A. Z., 184,268 GRASYUK, GREENLEES, G. W., 407,414 J. E., 314, 316,355 GREIVENKAMP, D., 189,209,268 GRISCHKOWSKY, GROB,K., 190,268 GROSS,M., 110, 177 W., 302,354 GROSSMANN, GROSSO,R. P.,339, 343,355 GRZANNA,J., 308, 312, 316, 324, 325, 327-329, 333, 343,345, 346,354,355,357, 358 L. G., 17,85 GUBIN, GUEST,C. C., 53,63,84

H HAAKE, F., 193,216, 231, 235,260,268 HAGENLOCKER, E. E., 189,268 HAGER,J., 407,414 HAKEN,H., 115,221,268

420

AUTHOR INDEX

HALIOUA, M., 333,358 HALL,J., 109, 179 P., 368,387,414 HAMMERLING, S.W., 98, I77 HAMMONS, HAN,D., 165,177 HKNSCH,T. W., 372,415 HARD,S., 71,85 HARDY, J. W., 336,355 P.,273, 327, 328, 334, 336,355 HARIHARAN, S., 110,177 HAROCHE, HARRIS, J. H., 324,355 HARRIS,J. S., 324, 349, 352,355 HARRISON, J., 89,99, 175, 177 HARTMANN, S. R., 189,268 HAUCK,R., 18, 36,45, 57,85,86 HAUS,H.A., 89, 98, 99, 101, 102, 123, 131, 141, 145, 146, 159, 163-165, 175, 176, 177, 179 HAUS,J., 260,268 HAYES,J., 337,355 HAYSLETT, C. R., 288,358 HAZAK,G., 193,245, 252,256,268 HEFFNER, H., 101, 146, 178 A., 110, 123, 177 HEIDMANN, HEINZEN, D. J., 401,413,415 HEISENBERG, W., 127,177,186,268,363,415 HELLWARTH, R., 188,268 HELSTROM, C. W., 89, 129, 159, 177 HENESIAN, M. A., 194,255,268 R., 190,270 HERENDEEN, H E R R I OD. ~ ,R., 292,308,310,324,327,346, 353,354 HERTEL,J., 297,354 HEYNACHER, E.,273,332,353,355,356 HILLER,C., 324,357 P.M., 6, 8, 15, 17, 22,45, 70, 71,85 HIRSCH, Ho, S. T., 124, 178 HOCKNEY, G., 368, 387,414 HOHENSTATT, M., 365,368,415 L.W., 109,178,413,414 HOLLBERG, HOLLY,S.,356 HOLZ,M., 89,98,99, 175, 176 HONDA, T., 283,292,356.357 HONG,C. K., 122, 177 HONG,K., 120,177 HONOLD, A., 141,179 HOPKINS,H. H., 324,349, 351,355 HOROWICZ, R., 123,177 HOROWITZ, L. P., 401,415 HOT,J. P., 296,355

HOUGH,J., 89,96,176, 177 R. E., 397,415 HOWARD, HSUEH,C. K., 50,51,64,85 Hu, C. K., 195,268 Hu. H. Z.,345,355 HUANG, C.-Y., 195,268 HUANG,C. C., 308, 329,355 T. S., 7, 9, 35,85 HUANG, HUGENHOLTZ, C. A. J., 308,355 HULET,R. G., 365,387,410-413,414,415 HUNT,B. R., 336,355 HUNTLEY, J. L., 288,355 1

ICHIKAWA, K.,292,355 ICHIOKA, Y., 8, 23,85.289,355 IDESAWA, M., 313,333,358,359 R.,308,359 IMANAKA, IMOTO,N., 92, 123, 131, 134, 135, 138, 139, 145, 177, I79 INA,H., 290,293,295,358 S., 329,359 INABA, B., 298,306,354,355 INEICHEN, INUIYA, M., 289,355 IODO, N.M., 194,254,268 ITANO,W.M., 365, 368, 376, 387, 401, 410-413,414-416 ITAYA,Y., 118, 178 ITOH,K., 312,355 K., 300,355 IWATA, IWAZAWA, H., 193,268 IZUMI,M., 8, 23, 85

J JACKEL, L. D., 397,415 R., 111,177 JACKIW, JACKSON, K., 285,353 JACOBSSON, S., 71,85 E., 124, 178, 179 JAKEMAN, JAMMER, M., 398,415 R.B., 354 JANDER, JAVANAINEN, J., 366,367,378,412,415 A., 187,268 JAYARAMAN, B.K., 59,85 JENNISON, JENSEN, A. E.,324,355 JOHNSON, G. W., 355 JONAS,J. A., 288,358 JONES,A. L., 70, 85 JONES,R., 337,355 JONES,R. A., 276,282,355

AUTHOR INDEX

JONES,R. C., 298,355 JORDAN JR,J. A., 6,8,15,17,22,45,70,71,85 JUNGMAN, K., 366,367,377, 387,414 JUST,D., 57,85

K KADAKIA, P. L., 276,282,355 KAFKA,J. D., 194,253,270 KAISER, W., 184, 188, 190, 198,206,210,211, 268,269 KALUZNY, Y., 110,177 KANIA,D. R., 273,356 KANOU,T., 288,359 KAPITSA,P. L., 369, 371,415 KATO,M., 45,85 KATYL,R. H., 45,85 KATZIR,Y.,316,358 KELLEY,P. L., 189,214, 215,268,269 KELLY,D. L., 7.85 KELLYJR, J. L., 101, 146, 176 KERMISCH, D., 71,85 KERN,S.,190,270 KERSTEN,R. TH., 300,355 KHALILI, B.Y., 89, I76 R. V., 188,269 KHOKHLOV, KICKER,B., 324,357 KIELICH,S.,196,270 KIKUTA,H., 300,305,306,355 KILIN,S.YA., 194,254,268 KIM,Y.S., 165,177 KIMBLE,H.J., 109, 110, 173, 178, 179, 365, 377, 382, 383, 387,403,407,414,415 KING,H., 260,268 KING,M. C., 9,22,85 KINGDON, K. H., 368,415 KINNSTAETTER, K., 317,343,348,355 KIRIN,Yu. M., 195,262,268 KIRK,J. P., 70,85 KIST, R., 300,355 M., 113, 134, 135, 177 KITAGAWA, KLAUDER, J., 91, 177 KLEIN,N., 110, I78 KLEIN,O.,186,268 W. H., 214,215,268 KLEINER, KLOTZ,E., 68, 75,84 KNIGHT,P. L., 366, 367, 377, 387, 398, 403, 415 S.,290,293, 295,358 KOBAYASHI, KOKAL,J. V., 296,357

42 1

C. L., 288,298,311,317,322, KOLIOPOULOS, 326,329, 336, 347,353,355,358 G., 278,355 KOPPELMANN, KOZICH,V. P., 194,254,268 KOZMA,A., 7,85 KRACKHARDT, U., 68,85 H. A., 186,268 KRAMERS, KRAVIS,S. P., 255,269 KREBS,K., 278,355 KREIS,TH., 291, 313, 355 KRISHNAN, K. S., 187,269 KROLL,N. M., 189,211,269 KRUG,W., 276,356 KRYLOV, N., 166,177 KUBO,R., 159, 177 F. M., 325,332,356 KUECHEL, KUEHLKE, D., 339,353 KUJAWINSKA, M., 335,356 KUMAR,P., 110, 116, 124, 178 KUO, S.J., 194,255, 256,269 KWON,0. Y., 335,356

L LAERI,F., 308, 329,356 P., 189,267 LALLEMAND, LALOE,F., 405,414 LAMBJR, W. E., 126, 179, 185,269 LANDAU, L.D., 159,177,369, 371, 415 LANDAUER, R., 167, 177 G., 187,269 LANDSBERG, LANGE,S., 337,355 P., 273,356 LANGENBECK, R. V., 368,416 LANGMUIR, LANZL,F., 287,356 A., 143,178 LAPORTE, LAU,E.,276,356 LAUBEREAU,A., 184,190, 198,206,210,211, 269 M. J., 308,356 LAVAN, S.V.,378,414 LAWANDE, LEE, P., 273,356 LEE, S.H., 53, 84 LEE,W.-H., 4.6, 7,9, 39, 53, 57, 81-83,85 LEINER,D. C., 329,356 LEITH,E.N., 45,85 D., 24-27,84,85 LESEBERG, LESEM,L. B., 6, 8, 15, 17,22,45, 70, 71,85 LEUCHS,G.,401,414 D. S., 89, 159, 167,177 LEVEDEV,

422

AUTHOR INDEX

LEVENSON,M.D., 110, 116, 137, 139-141, 143,178 LEVI,A., 17,85 LEVINE,D. F., 95, I77 LEVITAN, J., 401,415 LEVITAN,L. B., 89, 159, 167, 177 LEWENSTEIN, M., 194,225,232,253,254,269, 378,412,415 LEWIS,D. A., 407,414 LI, 2.W., 234,235,237,251,269 LIAO,P. F., 189,209,268 LIFSHITZ,E. M., 159, 177, 369, 371,415 LIN, L. H., 24,84 LIN, S., 99, I77 LINKE,R. A., 95, 177 LIU,B., 17,43,45,46,65, 71,84,85 LIU, H. C., 333,358 LIU, R., 313,356 LOHMANN, A. W., 5,6,8,22,40,42,43,79,84, 85,292, 317, 343, 348,355 LOOMIS,J. S., 288,356, 358 LOUDON,R., 96, 97, 177, 188, 192, 198, 199, 269,270, 366,367, 377, 387,415 LOUISELL, W. H.,101,102,146,177,178,185, 221,231,269 Lou, M. M. T., 189,209,268 LYNCH,R., 90, 128, 178

M MACGOVERN, A. J., 7,85,328,356 S.,92,102,116-118,123,131,135, MACHIDA, 139,178, 179 MACK,M. E., 189,268 MACPHERSON, D. C., 194,230,239,257-260, 268,269 MACY,W., 295,356 MAEDA,M. W., 110, 116, 178 MAGOME,N., 283,356 MAHANY, R.,308,327,356 MAIER,M., 184,210,268 MAIMON,P., 330,354 K., 89, I76 MAISCHBERGER, MAIT,J. N., 17,65, 75,85 MAKER,P. D., 188,269 MAKOSCH, G., 308,356 D., 352,356 MALACARA, MANDEL,L., 120,122,177,215,236,267,269, 382,383,407,414,415 MANDELSTAMM, L. I., 166,178 L. I., 187,269 MANDEL’STAMM,

MANNJR, J. A., 72,85 MANSFIELD,C. R., 39,84 MARAGOS,P., 287,356 MARCUSE,D., 285,356 MARTE,M., 377,416 MARTIN,W., 96,176 MASEK,V., 306,356 MASSIE,N. A., 306, 307,356 MASTIN,G. A., 288, 313,354,356 MASTNER,J., 298, 306,355,356 MASUI,J., 308,359 MAXWELL, J. C., 165, 178 MCCLUNG,F., 188,268 MCILRATH,T. J., 347,356 MCNEIL,K. J., 191, 192,269 MEDDENS,B. J. H., 308,355 MEHTA,C. L., 236,269 MEHTA,P. K., 342,356 C., 352,356 MENCHACA, MERKEL,K., 308,312,316,327,343,345,346, 356,357 MERTZ,J. C., 109, I78 MERTZ,L., 295,356 MERZLYAKOV, N. S.,9,86 MESCHEDE, D., 110, 178 MESSER,H. I., 333,357 MESSIAH,A., 91, 103, 178 MILBURN, J., 135, 178 MILLER,M. A., 369, 371,415 MILONNI,P. W., 381,415 MINCK,R. W., 189,268 MINKWITZ, G., 276,357 MISHKIN,E. A., 191, 192, 196, 213,269 MISRA,B., 401,415 MOLLOW,B. R., 191,213,269 MOORE,D. T., 302, 303, 314, 329, 333, 355, 356 MOORE,R., 312, 338,356,358 MORRIS,M. B., 347,356 MORSE,P., 309,356 MOSTOWSKI, J., 193,204,207,210, 212, 219, 226,229,232,239-241,243,244,252,269 MOTTIER,F.M., 298, 306,356 MUKAI,T., 103, 178 MULLEN,J. A., 102, 177 MULLER,G., 110,178 MURAO,T., 308,359 MURRAY, J. R., 194,255,268 R., 302,303,314,356 MURRAY, MUTOH,K., 358

AUTHOR INDEX

N NAGATA,R., 300,355 NAGOURNEY, W., 365,407-410,412,415 NAKADATE, S., 283, 313, 333, 337, 338, 356, 359

NAKANO, H., 141,179, 329,359 NAKAYAMA, Y., 45,85 NAMIKI,M., 131, 178 NATTERMANN, K., 193,246,248-250,268,269 NEITO,M. M., 93, 112, 176 NELSON,J. E., 342,353 NELSON,R. D., 356 NEUGEBAUER, G., 36,86 NEUHAUSER, W., 365,368,378,412,415 NEVES,F. B., 302,303, 314,356 NEWSAM,G., 16, 17,84 NEWSTEIN, M., 245,269 NG, W. K.,187,270 NI, W.T., 90, 128, 174, 175, 178 NILSSON,O., 102, 116, 179 NOLL,A. M., 9,22,85 Noz, M. E., 165, 177 NUGENT,K. A., 348,356 NYQUIST,H.,159, 178 0

OHYAMA, N., 292,357 OKAZAKI, H., 312,345,357 OKINO,Y., 308,359 ONEILL,E. L., 16,86 OREB,B. F., 327, 328, 334, 336,355 OREG,J., 193,245,252, 256,270 OROZCO, L.A., 110, 178 OSTEN,W., 356,358 OSTROVSKAYA, G. V., 353,356 OSTROVSKY, Yu. I., 313, 353,356 OZAWA, M., 129, 174, 175, 178

P PANDARESE, F., 188,268 PANTZER, D., 308,329,357 PARIS,D. P., 6, 43, 85 PARKS,R. E., 324,357 PARTOVI, M. H., 166, 178 PATORSKI, K., 25,86 PAUL,W.,368,415 PAULI,W., 130, 178 PEDERSEN, H. M., 357 PEDROTTI,L. M., 89,98, 175, 176 PEGG,D. T., 366,367,377,387,398,403,415

423

PENDRY,J. B., 167,178 PENNING, F. M., 368,415 PENZKOFER, A., 184, 190, 198,206,210,211, 269

PERINA,J., 196,270 PERINOVA, V., 196,270 PERLMU'ITER,S.H., 110, 116, 137, 139-141, 178

PERRY,B. N., 245,269 PERSHAN, P. S., 188,267 PIERCE,J. R., 89, 162, 178 PINE,A., 189,267 PIVTSOV, V. S., 195,262-264,269 PLACZEK, G., 187,204,269 PLATONENKO, V.T., 188,269 POLDER,D., 260,261,269,270 POLITCH,J., 308, 329,357 POLYAK, B. T., 17,85 POPOV,Yu. N., 195,262,268 PORRATI,M., 368,387, 398,403,414,415 POSNER,E. D., 89, 162, 178 POST,D.,333,357 PRASADA, B., 7, 35,85 PRENTISS, M. G., 178 PRESBY,H.,285,356 PRESBY,H.M., 333,357 PRETTYJOHNS, K. N., 357 PRIMAK, W., 300,357 PRONGUE, D., 332,354 PUGH,D. J., 285,353 PUGH,J. R., 96, 176 PUITERMAN, S., 365,366,368,387,398,403, 414,415

R RABINOWITZ, P., 245,269 RADO,W. G., 189,268 RADZEWICZ, C., 194,255, 256,269 RAETHER, M., 368,415 RAIK,E., 17,85 RAIMOND, J. M., 110,177 RAIZEN,M. G., 110, 178 RALLS,K. S., 397,415 RAMAN, C.V., 187,269 RAMDAS, A. K., 187,268 RANSOM,P. L., 296,357 RAO,K. R., 290,293,354 RARITY,J. G., 124, 178 RATEIKE, M., 407,414

424

AUTHOR INDEX

RAUTIAN,S.G., 194, 195, 260, 262-264, 268-270 RAYMER, M. G., 193, 194,204,207,210,212, 219,226,229,232,234,235, 237,239, 241, 243,244,246-248,250,251, 253,255-257, 269,270 REID,G. T., 333,357 REMPE,G., 110,178 Y.,401,415 REPHAELI, S., 110, 123, 177, 377,415 REYNAUD, RIEKHER, R., 276,357 RIFKIN,J., 257,268 A., 401,415 RIMINI, RINES,K. D., 65,86 RIXON,R. C., 333,357 D. W., 335,337,356,357 ROBINSON, RODDIER, C., 291,357 RODDIER, F., 291,357 RODEMICH, E. R., 89, 162, 178 F. L., 278,357 ROESLER, L. A., 313,354 ROMERO, ROOT,W. L., 16,86 A. H., 285,353 ROSENFELD, D. P.,292,308,324,327,346,353 ROSENFELD, ROSENZWEIG, D., 282,357 H., 356 ROTTENKOLBER, Ru, Q.-S.,292,357 A., 89, 176 RUDIGER, RZ~ZEWSKI, K., 193, 194, 232, 255,269,270

S SAEDLER, J., 356 V. P., 195,262-264,268-270 SAFONOV, SAGNAC, G., 98,178 SAITO,H., 313, 333, 337, 338,356,359 SAITO,S., 138, 177, 178, 359 SAITOH, T., 103,178 SAKAI, M., 312, 345,357 J. J., 198,269 SAKURAI, SALEH,B., 218,235,236,269 B. E. A., 124, 178 SALEH, J., 365,407-410,412,415 SANDBERG, V. D., 89,97, 127, 165,176 SANDBERG, SANDEMANN, R. J., 353 G. A., 89,99, 175,177,178 SANDERS, V. E., 89,98, 175, 176 SANDERS, G., 124,178 SAPLAKOGLU, M., 126, 179, 185,269 SARGENT, SASAKI, O., 312,345,357 Y.,139, 177 SASAKI,

SAUTER, TH., 365,378,412,415 A. A., 50, 51, 64,85 SAWCHUK, W.O., 17.85 SAXTON, SCHAHAM, M., 327,357 A. L., 372,415 SCHAWLOW, SCHENK, CH., 302,358 A., 366,367,377,387,414,416 SCHENZLE, SCHILLING, R., 89, 176 W., 89, 98, 106, 175, 176, 178 SCHLEICH, SCHLUETER, M., 287,356,357 SCHNUPP, L., 89, I76 W., 352, 353,354,358 SCHREIBER, D., 9.86 SCHREIER, G., 260,268 SC HR ~ D ER, E., 363,416 SCHR~DINGER, SCHUBERT, M., 206,269 S C H U U , G., 276,277,324,325,342,349,352, 355,357 B. L., 89, 176 SCHUMAKER, SCHUURMANS, M. F. H., 260,261,269,270 S.,188,268 SCHWARZ, SCHWESINGER, G., 342,357 SCHWIDER, J., 276, 277, 308, 312, 316, 317, 324,325,327-329,333,342,343,345, 346, 348, 352,355,357,358 P. B., 397,416 SCHWINBERG, SCIVIER, M. S., 16,84 SCULLY,M.O.,89,95,98, 126, 173-175,176, 178,179, 185,269 SELDOWITZ, M. A., 58,86 V. N., 193, 195, 210, 219,268 SEMINOGOV, I. R., 100, 178 SENITZKY, S.,279,359 SHAMAI, D. G., 369, 371,414 SHANKLAND, SHANNON, C. E., 159, 178 J. H.,89, 105, 110, 116, 124, 143, SHAPIRO, 174, 176, 178, 179 SHE,C. Y.,89, 101, 146, 178 SHELBY, R. M., 110, 116, 137, 139-141, 143, 178 SHEN,Y.R., 188, 190-192,196,267,270 SHIMIZU, F., 189,211,213,268 SHIMODA, K., 194,260,270 D. M., 335,356 SHOUGH, A. E., 102, 178 SIEGMAN, A. M., 73, 74,85 SILVESTRI, H. D., 192, 213,270 SIMAAN, P., 189,267 SIMOVA, SIZER,T., 194,253,270 B. S., 91,177 SKAGERSTAM,

AUTHOR INDEX

W. J., 397,415 SKOCPOL, F., 312,356 SLAYMAKER, G. A., 338,354,358 SLETTEMOEN, SLOMBA, A. F., 352,358 R. E., 109, 143, 178 SLUSHER, SMARTT, R. N., 336,358 A., 186,270 SMEKAL, SMITH,H. M., 82,86 SMITH,I. W., 89,98, 99, 175, 176 SMITH,R. G., 178 SMYTHE,R., 338,358 SNYDER, J. J., 286,317, 347, 356,358 SOBOLEWSKA, B., 193,232,239-241,243,244, 252,269 SODIN,L. G., 16,84 A. P., 192,267 SOKHORUKOV, SOLF,B., 308,356 G. E., 298,329,358 SOMMARGREN, P. P., 198,270 SOROKIN, R., 308, 312,316,317,324, 325, SPOLACZYK, 327-329,333, 343,345, 346,354,356-358 SRINIVASAN, V., 333,358 H., 17,85 STARK, STATZ,H., 89,98,99, 175, 176 I. A., 212,267 STEGUN, S., 377,413,416 STENHOLM, STERN,T. E., 89, 159, 179 K. A,, 306,348,358 STETSON, M., 297,354 STOCKMANN, STOLEN,R. H., 300,358 STOLER,D., 89, 104, 106,179 STONE,T., 339,354 STRAND, T. C., 308,316,329,356,358 STRATTON, J. A., 369,416 M., 193,245,252,256,270 STRAUSS, N., 68,85, 317, 343, 348,355 STREIBL, J., 336,358 STRONG, STUMPF,K. D., 298, 311, 336,358 SUDARSHAN, E. c. G., 91,179, 196,218,270, 401,415 SUSSKIND, L., 93,179 M., 329, 333,358,359 SUZUKI, SUZUKI, T., 8,23,85 R. C., 194,230,239,258,259,269 SWANSON, SWEENEY, D. W., 58, 59,85,86 SWIFT,C. D., 194,255,268 SZE, s. M.,117, 179 SZILARD, L., 165, 179 SZLACHETKA, P., 196,270

425

T TAKAHASHI, H., 89, 104, 159, 179 M., 189,209,270 TAKATSUJI, TAKEDA, M., 290,292,293,295,355,358 TAMM,I., 166, 178 H., 302,354 TANDLER, TANG,C. L., 189, 196,270 P. E., 190,270 TANNENWALD, H. L., 349,358 TANNER, P. R., 124, 178 TAPSTER, TEICH,M. C., 124,178 TENNANT, D. M., 397,415 R. W., 188,269 TERHUNE, R., 332, 335,354,358 THALMANN, W., 335,353 THIEME, THORNE, K. S., 89,97, 127, 141, 165, 176 TIETZE,U., 302,358 TIMOSHENKO, S., 342,358 H. J., 328,354,358 TIZIANI, TOFFOLI,T., 165, 167, 177 TOLANSKY, S., 276,358 TORII,Y.,45,86 TOSCHEK, P. E.,365, 368, 378,412,415 C. H., 188,268 TOWNES, TRICOLES, G., 9,86 TRIPPENBACH, M., 194,255,270 TROPEL,328,329,358 B. E., 333,356 TRUAX, J., 283, 292,356,357 TSUJIUCHI, TUCKEY, J. W., 5,84, 290,354 TURUNEN, J., 68, 72,86

U UNDERWOOD, K. L., 288,358 W. G., 89, 179 UNRUH, UPATNIEKS, J., 45, 85

V VALLEY, G., 190,270 VALLEY, J. F., 109, 178 VAN CIITERT, P.H.,325,354 VAN DAMME, G. E., 308,356 VAN DYCKJR, R. S., 368, 397,416 VAN LAARHOVEN, P. J. M., 58,86 A., 68,72,86 VARASA, W., 298,354 VAUGHAN, VEST,C. M., 353,358 VOGEL,A., 324,355 VON DER LINDE,D., 193, 246, 248-250,268, 269, 339,353

426

AUTHOR INDEX

VON FOERSTER, T.,192,219,270 VON NEUMANN, J., 129, 130, 160, 179, 398, 416 VORONTSOV,Y. I., 89, 141, 165, 176 VREHEN,Q.H. F., 260,261,264,269,270 VRY, u., 332,354 VYATCHANIN,S. P., 135,176

WODKIEWICZ, K., 90, 128,179 WOINOWSKY-KRIEGER, S.,342,358 WOLF, E., 215, 236, 241,268,269, 214, 275, 322,353 WOMACK,K. H.,288,292,293,358 WOODBURY, E. J., 187, 188,268,270 Wu, H., 109, I79 Wu, L.A.,109, 173,179 Wu, Y. H.,79,86 W WUERKER,R. F., 368,416 WALKER,J. G., 124, 179 WYANT,J. C.,7,85,86,288,308,311,316,320, WALLBURG, S., 308,321,354,357 WALLS,D.F., 89,105,106,108,110,116,135, 322, 323, 328-330, 333, 334, 336, 338, 345-347,353,354,356,358 131, 139-141, 176, 178, 179, 191, 192, 194, WYKES,C., 337,355 196,213,260,269,270,311,401,416 WYKO, 323,358 WALLS,F. L., 368,416 WALMSLEY,~. A., 193,194,232,234,235,237, WYNNE,J. J., 198,270 F., 17, 18, 44,45,47,56-59, 61, 239,241,243,244,246-248,250,251,253. WYROWSKI, 68,10-12,15,19,84,86 251,269,270 WALTHER, A., 16,86 X WALTHER, H., 110,178,401,412,414 W A N G , ~S., . 184,188,189,204,210,211,213, XIAO,M., 110, 173, 178, 179 268,270 WARNER, C., 251,267 Y WATANABE, K.,141,177,179 YAMAASHI, Y., 333,358 WATERS,J. P., 8, 22,86 YAMAMOTO, Y., 89,92,95,101,102,108,113, WATKINS, S.,139, 177 116-118, 121-123, 125, 131, 134, 135, 139, WATSON,G.N.,301,358 141,145,146,153, 159,163-165,176-179 WECKSUNG, G.W.,39,84 YARIV, A,, 102,178 WEIBEL,E.S., 369, 371,416 YAROSLAVSKII, L.P., 9,86 WEINBERG, D.L., 122,176 YASCHIN, E. G.,190,268 WEINER,D.,188,268 YATAGAI,T.,22,86,288,313,328,329, 333, WEISSBACH, S., 57, 58, 10,15,86 338,356,358,359 WEISSKOPF,V. F.,381,404,416 YOSHIZUMI, K.,308,359 WEISSMANN, H., 306,358 YOULA,D.G.,17,86 WELLS,A. L.,369,371,371,387,403,414,415 YUEN,H.P., 89, 90, 104-106, 128, 133, 144, WENKE,L.,352,353,354,358 146, 115,179 WENZEL,R.G.,260,268 YURKE,B., 109, 143,178,179 WERNICKE, G.,358 WEST, P. D., 285,353 WESTERHOLM, J., 68,12,86 Z WHEELER, J. A.,106, 129,178, 179 ZABOLOTSKII, A. A., 195,263,270 ZANONI,C.A., 285,353 WHITE,A. D.,292,308,324,321,346,353 ZEEVI,Y.Y., 219,359 WIGNER,E.P., 161,179,381, 404,416 ZEIGER,H. J., 190,270 WILHELMI,B., 206,269 ZIMMERMANN, M.,89,91,121, 165, 176 WILLIAMS,R.A.,335,356 WINELAND, D.J., 365,368,312,376,387,401, ZOLLER,P., 367,311,414,416 410-413,414-416 ZUREK,W. H.,129, 131,179 ZYGO, 359 WINKLER, W., 89,176

SUBJECT INDEX

F

A acousto-optic modulator, 298,306, 324 adaptive optics, 336 amplified spontaneous emission, 185 Autler-Townes doublet, 39 1

Fabry-Pkrot etalon, 327 interferometer, 145, 300 Fano factor, 143 Fermi golden rule, 199, 401 Fizeau interferometer, 312, 329, 330 fluctuation-dissipation theorem, 99, 100 Fourier transform, discrete, 13, 38 - fast, 5,6, 8, 14, 15, 19,24, 291, 292 Four-wave mixing, 109, 135 Fresnel number, 203,240,245,249-251,255,

-

B back action evading, 90, 132 Bayes’ theorem, 406 Bloch equations, 209 Bohr theory, 363 Born-Oppenheimer approximation, 188, 205 Bose-Einstein distribution, 216 Brillouin amplifier, 102 - scattering, 110, 140, 189 C

Casimir effect, 167 cat paradox of SchrOdinger, 184 channel capacity, quantum mechanical, 158,

256

- transform, 18, 19, 25, 26 G gamma distribution, 236 Gaussian moment theorem, 217 Gibb’s phenomenon, 13 Glauber-Sudarshan distribution, 190, 195 gravitational wave detection, 89,95, 165, 172

159, 161-163

coherent state, 90,91, 93, 94, 102, 104, 143 superposition of states, 363, 364 contractive state measurement, 128, 129, 174,

-

175

Cooley-Tukey algorithm, 5 correlation function, 240 D densitometer, 276, 282 diffraction efficiency, 3 1,43 Dirac delta function, 12 Doppler shift, 365, 372

H half-wave plate, 298, 307 Heisenberg picture, 185, 193, 220 Hilbert space, 91 hologram - amplitude, 34, 51 - binary, 67, 81 - bleached, 7, 66 - computer generated, 3-8,10,11,14,

-

E

-

Earnshaw’s theorem, 369 Einstein A coefficient, 381 electron shelving, 377 electro-optic effect, 300 entropy, 160

-

24, 28, 29, 38,42, 52, 81, 82, 325 delayed sampled, 6 detour phase, 5 digital, 20, 21 Fresnel, 20, 21 phase, 65, 72 referenceless on-axiscomplex, 29 synthetic, 276, 327

holography

- analog, 4 427

15,23,

428

- digital, 3-9, 16, 17, 28,29, 36,40, - Fourier, 11, 20, 30, 33, 45, 77-79 - Fresnel, 76-79

SUBJECX INDEX

66, 68

homodyne detection, 105, 165

I intensity fluctuations - spatial, 255 - temporal, 255 interferogram, computer generated, 7 interferometry - heterodyne, 303, 306 - holographic, 273, 306, 334, 335, 352 - phase -- lock, 301 -- sampling, 308 -- shifting, 308 - speckle, 337, 338 - two-wavelength, 333 ion trapping, 361-369

M Mach-Zehnder interferometer, 145, 153,299, 306, 326,330

magnetic bottle, 368 Manley-Rowe relation, 120 Maxwell-Bloch equations, 189, 193, 194,223, 245 theory, 188 Maxwell demon, 165 equations, 95, 185, 188, 189 Michelson interferometry, 283, 334 MoirC topography, 334

-

-

N nonclassical light, 90, 103 Nyquist function, 161 - modes, 161, 165

-

0

optical

J Jones calculus, 298

- communication, 89, 94, 105 - homodyne detector, 93, 148

K

P

Karhunen-Loeve expansion, 232, 235,241 Kerr - effect, 113 medium, 113, 115, 131, 132, 137, 153, 154,

Paul trap, 365, 368, 369, 371 parametric amplifier - degenerate, 107, 109, 147, 151, 152 nondegenerate, 121, 147 parametric downconversion, 123 oscillator, nondegenerate, 123, 125, 152,

-

156, 173

Kingdom trap, 368 kinoform, 6,7,70, 72 Kramers-Heisenberg dispersion formula, 187, 188, 192, 193,200, 204

L Lamb - -Dicke limit, 375, 376 shift, 381 Laplace transform, 382, 390 laser cooling, 372,376 correlated spontaneous emission, 173, 175 gyroscope, 89, 98,99, 175 - pump-noise-suppressed, 116 linear amplifier, 101 local oscillator, 93, 137 Lorentzian line-shape function, 201

-

-

-

169-172

Penning trap, 368, 397 phase-locked - loop, 148, 151, 171 oscillator, 149, 152 phase matching, 188 - modulator, 296 operator, 93 - retrieval problem, 16 Photon antibunching, 382, 383,407, 412 - number fluctuation, 120 statistics, 93 Placzek's model, 206 polarizability ansatz, 188

-

-

SUBJECT INDEX

Poissonian distribution, 94 pulse position modulation, 162, 163

Q @representation, 104 quantum - jump, 363,365,366,368,387,397-400,407, 409,411

- limit, standard, 89, 90, 93-95,

98, 99, 111, 112, 116-118, 122, 129, 137, 146, 169, 171-173 nondemolition measurement, 89, 125, 129, 131, 133, 135, 137-144, 153, 168, 173 - reservoir theory, 221 quarter-wave plate, 300, 307

-

103, 147, 127, 169,

429

Schmitt trigger, 305 SchrMinger - equation, 370, 371 picture, 207, 214 shot noise limit, 89 soliton, 259 spatial filter, 3 speckle - noise, 283 - pattern, 194, 243, 257 squeezed state, 89, 105, 107, 110, 144, 174 number-phase, 90, 103, 111-113, 116, 118,

-

121, 123, 133, 143 - quadrature amplitude, Stokes

- field, 183-185,

- frequency, 183

R Rabi

- frequency, 111,167,209,379,380,383,387, 388,391,400 - oscillation, 365 Raman - active medium, 183 - amplifier, 102, 184, 189, 202 linewidth, spontaneous, 189 - scattering -- cooperative, 194, 195, 260-264 spontaneous, 183, 187 -- stimulated, 183, 184, 187, 245 rate-equation approach, 188, 189, 197 resonance fluorescence, 367,380,383,384,407 Ritchey-Common test, 325 Ronchi grating, 55 rotating-wave-approximation, 370 Rydberg atom, 110

-

--

90, 103-106, 109,

115, 116, 121, 123 191

sub-Poisson

- distribution,

- light, 123

105, 106, 412

superfluorescence, 185 super-Poisson distribution, 105, 106

T telegraphic fluorescence, 377, 397 two-photon process, stimulated, 187 Twyman-Green interferometer, 299,306,312, 326, 349

V vidicon camera, 288

W Weisskopf-Wigner approximation, 381,404

S Sagnac

- constant, 169 - effect, 98 sampling theorem, 12, 159 Schawlaw-Townes linewidth, 92,97,98, 169, 175

Y Youngs fringes, 288 Z

Zen0 effect, 401, 412 Zernike polynomial, 324, 325, 330

This Page Intentionally Left Blank

CUMULATIVE INDEX - VOLUMES I-XXVIII ABELBS,F., Methods for Determining Optical Parameters of Thin Films ABELLA, I. D., Echoes at Optical Frequencies ABITBOL,C. I., see J. J. Clair ABRAHAM, N. B., P. MANDEL,L. M. NARDUCCI,Dynamical Instabilities and Pulsations in Lasers AGARWAL, G. S., Master Equation Methods in Quantum Optics AGRAWAL, G. P., Single-longitudinal-mode Semiconductor Lasers AGRANOVICH, V. M., V. L. GINZBURG, Crystal Optics with Spatial Dispersion ALLEN,L., D. G. C. JONES,Mode Locking in Gas Lasers AMMANN, E. O., Synthesis of Optical Birefringent Networks ARMSTRONG, J. A., A. W. SMITH,Experimental Studies of Intensity Fluctuations in Lasers J. A., Hamiltonian Theory of Beam Mode Propagation ARNAUD, BALTES,H. P., On the Validity of Kirchhoffs Law of Heat Radiation for a Body in a Nonequilibrium Environment BARAKAT, R., The Intensity Distribution and Total Illumination of AberrationFree Diffraction Images H. H., The Radon Transform and its Applications BARRETT, BASHKIN, S., Beam-Foil Spectroscopy BASSEIT,I. M., W. T. WELFORD,R. WINSTON,Nonimaging Optics for Flux Concentration BECKMANN, P., Scattering of Light by Rough Surfaces BERRY,M. V., C. UPSTILL,Catastrophe Optics: Morphologies of Caustics and their Diffraction Patterns BERTOLOTII, M., see D. Mihalache BEVERLY 111, R. E., Light Emission from High-Current Surface-Spark Discharges B J ~ R KG., , see Y. Yamamoto BLOOM,A. L., Gas Lasers and their Application to Precise Length Measurements BOUMAN, M. A., W. A. VAN DE GRIND,P. ZUIDEMA, Quantum Fluctuations in Vision BOUSQUET, P., see P. Rouard BROWN,G. S., see J. A. DeSanto BRUNNER, W., H. PAUL,Theory of Optical Parametric Amplification and Oscillation 43 1

11, 249 VII, 139 XVI, 71

xxv,

1

XI, 1 XXVI, 163 IX, 235 IX, 179 IX, 123 VI, 211 XI, 247 XIII,

1

I, 67 XXI, 217 XII, 287 XXVII, 161 VI, 53 XVIII, 259 XXVII, 227 XVI, 357 XXVIII, 87 IX, 1 XXII, 77 IV, 145 xxIII, 1 xv,

1

432

CUMULATIVE INDEX

BRYNGDAHL, O., Applications of Shearing Interferometry IV, 37 BRYNGDAHL, O., Evanescent Waves in Optical Imaging XI, 167 O., F. WYROWSKI, Digital holography Computer-generated BRYNGDAHL, holograms XXVIII, 1 BURCH,J. M., The Meteorological Applications of Diffraction Gratings 11, 73 B U ~ E R W E CH.KJ., , Principles of Optical Data-Processing XIX, 21 1 CAGNAC, B., see E. Giacobino XVII, 85 CASASENT, D., D. PSALTIS,Deformation Invariant, Space-Variant Optical Pattern Recognition XVI, 289 CEGLIO, N. M., D. W. SWEENEY, Zone Plate Coded Imaging: Theory and ApplicaXXI, 287 tions CHRISTENSEN, J. L., see W.M. Rosenblum XIII, 69 CLAIR,J. J., C. I. ABITBOL, Recent Advances in Phase Profiles Generation XVI, 71 CLARRICOATS, P. J. B., Optical Fibre Waveguides A Review XIV, 327 v, 1 COHEN-TANNOUDJI, C., A. KASTLER,Optical Pumping COLE,T. W., Quasi-Optical Techniques of Radio Astronomy XV, 187 COLOMBEAU, B., see C. Froehly XX, 63 COOK,R. J., Quantum Jumps XXVIII, 361 COURTBS,G., P. CRUVELLIER, M. DETAILLE,M. SAYSSE,Some New Optical Designs for Ultra-Violet Bidimensional Detection of Astronomical Objects xx, 1 CREATH,K., Phase-Measurement Interferometry Techniques XXVI, 349 CREWE, A. V., Production of Electron Probes Using a Field Emission Source XI, 223 CRUVELLIER, P., see C. G. Courtds xx, 1 CUMMINS, H. Z., H. L., SWINNEY, Light Beating Spectroscopy VIII, 133 DAINTY, J. C., The Statistics of Speckle Patterns XIV, 1 DXNDLIKER, R., Heterodyne Holographic Interferometry XVII, 1 DECKER Jr., J. A,, see M. Harwit XII, 101 DELANO,E., R. J. PEGIS,Methods of Synthesis for Dielectric Multilayer Filters VII, 67 DEMARIA, A. J., Picosecond Laser Pulses IX, 31 DESANTO,J. A., G. S. BROWN,Analytical Techniques for Multiple Scattering from Rough Surfaces XXIII, 1 DETAILLE, M., see G. Courtds XX, 1 DEXTER,D. L., see D. Y. Smith X, 165 DREXHAGE, K. H., Interaction of Light with Monomolecular Dye Layers XII, 163 DUGUAY, M. A., The Ultrafast Optical Kerr Shutter XIV, 161 EBERLY, J. H., Interaction of Very Intense Light with Free Electrons VII, 359 ENGLUND, J. C., R. R. SNAPP,W. C. SCHIEVE,Fluctuations, Instabilities and Chaos in the Laser-Driven Nonlinear Ring Cavity XXI, 355 ENNOS,A. E., Speckle Interferometry XVI, 233 FANTE, R. L., Wave Propagation in Random Media: A Systems Approach XXII, 341 FIORENTINI, A., Dynamic Characteristics of Visual Processes I, 253 FOCKE,J., Higher Order Aberration Theory IV, 1

-

-

CUMULATIVE INDEX

433

FRANCON, M., S.MALLICK,Measurement of the Second Order Degree of Coherence VI, 71 FRIEDEN, B. R., Evaluation, Design and Extrapolation Methods for Optical Signals, Based on Use of the Prolate Functions IX, 311 FROEHLY, C., B. COLOMBEAU, M. VAMPOUILLE, Shaping and Analysis of Picosecond Light Pulses XX, 63 FRY,G. A., The Optical Performance of the Human Eye VIII, 51 GABOR,D., Light and Information I, 109 GAMO,H., Matrix Treatment of Partial Coherence 111, 187 GHATAK, A. K., see M. S. Sodha XIII, 169 GHATAK, A., K. THYAGARAJAN, Graded Index Optical Waveguides: A Review XVIII, 1 GIACOBINO, E., B. CAGNAC, Doppler-Free Multiphoton Spectroscopy XVII, 85 V. L., see V. M. Agranovich GINZBURG, IX, 235 GIOVANELLI, R. G., Diffusion Through Non-Uniform Media 11, 109 GLASER, I., Information Processing with Spatially Incoherent Light XXIV, 389 GNIADEK, K., J. PETYKIEWICZ, Applications of Optical Methods in the Diffraction Theory of Elastic Waves IX, 281 GOODMAN, J. W., Synthetic-Aperture Optics VIII, 1 GRAHAM, R., The Phase Transition Concept and Coherence in Atomic Emission XII, 233 HARIHARAN, P., Colour Holography XX, 263 HARIHARAN, P., Interferometry with Lasers XXIV, 103 HARWIT,M., J. A. DECKERJr., Modulation Techniques in Spectrometry XII, 101 HELSTROM, C. W., Quantum Detection Theory X, 289 HERRIO'IT,D. R., Some Applications of Lasers to Interferometry VI, 171 HUANG,T. S.,Bandwidth Compression of Optical Images x, 1 IMOTO,N., see Y. Yamamoto XXVIII, 87 JACOBSSON, R., Light Reflection from Films of Continuously Varying Refractive Index V, 247 JACQUINOT, P., B. ROIZEN-DOSSIER, Apodisation 111, 29 JAMROZ, W., B. P. STOICHEFF, Generation of Tunable Coherent Vacuum-Ultraviolet Radiation XX, 325 JONES,D. G. C., see L. Allen IX, 179 v, 1 KASTLER, A., see C. Cohen-Tannoudji KHOO,I. C., Nonlinear Optics of Liquid Crystals XXVI, 105 KIELICH,S., Multi-Photon Scattering Molecular Spectroscopy XX, 155 KINOSITA, K., Surface Deterioration of Optical Glasses IV, 85 KITAGAWA, M., see Y. Yamamoto XXVIII, 87 KOPPELMANN, G., Multiple-Beam Interference and Natural Modes in Open Resonators VII, 1 KOTTLER,F., The Elements of Radiative Transfer 111, 1 KOTTLER,F., Diffraction at a Black Screen, Part I: Kirchhoffs Theory IV, 281 KO-ITLER,F., Diffraction at a Black Screen, Part 11: Electromagnetic Theory VI, 331

434

CUMULATIVE INDEX

XXVI, 227 KRAVTSOV, Yu. A., Rays and Caustics as Physical Objects I, 211 KUBOTA,H., Interference Color XIV, 47 LABEYRIE, A., High-Resolution Techniques in Optical Astronomy XI, 123 LEAN,E. G., Interaction of Light and Acoustic Surface Waves XVI, 119 LEE, W.-H., Computer-Generated Holograms: Techniques and Applications VI, 1 LEITH,E. N., J. UPATNIEKS, Recent Advances in Holography XVI, 1 LETOKHOV, V. S.,Laser Selective Photophysics and Photochemistry VIII, 343 LEVI,L., Vision in Communication LIPSON,H., C. A. TAYLOR, X-Ray Crystal-Structure Determination as a Branch V, 287 of Physical Optics XXI, 69 LUGIATO,L.A., Theory of Optical Bistability MACHIDA, M., see Y. Yamamoto XXVIII, 87 XXII, 1 MALACARA, D., Optical and Electronic Processing of Medical Images VI, 71 MALLICK,L., see M. Franqon 11, 181 MANDEL,L., Fluctuations of Light Beams XIII, 27 MANDEL,L., The Case for and against Semiclassical Radiation Theory xxv, 1 MANDEL,P., see N. B. Abraham XI, 305 MARCHAND, E. W., Gradient Index Lenses MARTIN,P. J., R. P. NETTERFIELD, Optical Films Produced by Ion-Based TechXXIII, 113 niques MASALOV, A. V., Spectral and Temporal Fluctuations of Broad-Band Laser XXII, 145 Radiation XXI, 1 MAYSTRE, D., Rigorous Vector Theories of Diffraction Gratings xv, 77 MEESSEN,A., see P. Rouard VIII, 373 MEHTA,C. L., Theory of Photoelectron Counting MIHALACHE, D., M. Bertolotti, C. Sibilia, Nonlinear wave propagation in planar XXVII, 227 structures MIKAELIAN, A. L., M. I. TER-MIKAELIAN, Quasi-Classical Theory of Laser RadiaVII, 231 tion XVII, 279 MIKAELIAN, A. L., Self-Focusing Media with Variable Index of Refraction Surface and Size Effects on the Light MILLS,D. L., K. R. SUBBASWAMY, XIX, 43 Scattering Spectra of Solids I, 31 MIYAMOTO, K., Wave Optics and Geometrical Optics in Optical Design MOLLOW,B. R., Theory of Intensity Dependent Resonance Light Scattering and XIX, 1 Resonance Fluorescence v, 199 MURATA,K., Instruments for the Measuring of Optical Transfer Functions VIII, 201 MUSSET,A., A. THELEN,Multilayer Antireflection Coatings xxv, 1 NARDUCCI, L. M., see N. B. Abraham XXIII, 113 NETTERFIELD, R. P., see P. J. Martin XXIV, 1 NISHIHARA, H., T. SUHARA,Micro Fresnel Lenses XXV, 191 OHTSU,M., T. TAKO,Coherence in Semiconductor Lasers XV, 139 OKOSHI,T., Projection-Type Holography

435

CUMULATIVE INDEX

OOUE,S.,The Photographic Image OSTROVSKAYA, G. V., YIJ. I. OSTROVSKY, Holographic Methods in Plasma Diagnostics OSTROVSKY, Yu. I., see G. V. Ostrovskaya OUGHSTUN, K. E., Unstable Resonator Modes PATORSKI, K. P., The Self-Imaging Phenomenon and its Applications PAUL,H., see W. Brunner PEGIS,R. J., The Modem Development of Hamiltonian Optics PEGIS,R. J., see E. Delano PERINA, J., Photocount Statistics of Radiation Propagating through Random and Nonlinear Media PERSHAN, P. S., Non-Linear Optics PETYKIEWICZ, J., see K. Gniadek PICHT,J., The Wave of a Moving Classical Electron PORTER, R. P., Generalized Holographywith Application to Inverse Scattering and Inverse Source Problems PSALTIS,D., see D. Casasent RAYMER, M. G., I. A. WALMSLEY, The Quantum Coherence Properties of Stimulated Raman Scattering RISEBERG, L. A., M. J. WEBER,Relaxation Phenomena in Rare-Earth Luminescence RISKEN,H., Statistical Properties of Laser Light RODDIER,F., The Effects of Atmospheric Turbulence in Optical Astronomy ROIZEN-DOSSIER, B., see P. Jacquinot RONCHI,L., see Wang Shaomin ROSENBLUM, W. M., J. L. CHRISTENSEN, Objective and Subjective Spherical Aberration Measurements of the Human Eye ROTHBERG, L., Dephasing-Induced Coherent Phenomena ROUARD,P., P. BOUSQUET, Optical Constants of Thin Films ROUARD, P., A. MEESSEN,Optical Properties of Thin Metal Films RUBINOWICZ, A., The Miyamoto-Wolf Diffraction Wave RUDOLPH, D., see G. Schmahl SA'IssE, M., see G. Court& SAITO,S., see Y. Yamamoto SAKAI,H., see G. A. Vanasse SALEH,B. E. A., see M. C. Teich SCHIEVE, W. C., see J. C. Englund SCHMAHL, G., D. RUDOLPH, Holographic Diffraction Gratings SCHUBERT, M., B. WILHELMI, The Mutual Dependence between Coherence Properties of Light and Nonlinear Optical Processes SCHULZ, G., J. SCHWIDER, Interferometric Testing of Smooth Surfaces SCHULZ, G., Aspheric Surfaces

VII, 299 XXII, 197 XXII, 197 XXIV, 165 XXVII, 1

xv,

1

I, 1 VII, 67 XVIII, 129 V, 83 IX, 281 V, 351 XXVII, 315 XVI, 289 XXVIII, 179 XIV, 89 VIII, 239 XIX, 281 111, 29 XXV, 279 XIII, 69 XXIV, 39 IV, 145

xv,

77

IV, 199 XIV, 195 xx, 1 XXVIII, 87 VI, 259 XXVI, 1 XXI, 355 XIV, 195 XVII, 163 XIII, 93

xxv, 349

436

CUMULATIVE INDEX

SCHWIDER, J., see G. Schulz J., Advanced Evaluation Techniques in Interferometry SCHWIDER, SCULLY,M. O., K. G. WHITNEY,Tools of Theoretical Quantum Optics SENITZKY, I. R., Semiclassical Radiation Theory within a Quantum-Mechanical Framework SIBILIA,C., see D. Mihalache SIPE,J. E., see J. Van Kranendonk SITTIG,E. K., Elastooptic Light Modulation and Deflection SLUSHER, R. E., Self-Induced Transparency SMITH,A. W., see J. A. Armstrong SMITH,D. Y.,D. L. DEXTER,Optical Absorption Strength of Defects in Insulators SMITH,R. W., The Use of Image Tubes as Shutters SNAPP,R. R., see J. C. Englund V. K. TRIPATHI, Self Focusing of Laser Beams in SODHA,M. S.,A. K. GHATAK, Plasmas and Semiconductors SOROKO,L. M., Axicons and Meso-Optical Imaging Devices STEEL,W. H., Two-Beam Interferometry STOICHEFF, B. P., see W. Jamroz STROHBEHN, J. W., Optical Propagation Through the Turbulent Atmosphere STROKE,G. W., Ruling, Testing and Use of Optical Gratings for High-Resolution Spectroscopy SUBBASWAMY, K. R., see D. L. Mills SUHARA, T., see H. Nishihara SVELTO,O., Self-Focusing, Self-Trapping, and Self-phase Modulation of Laser Beams SWEENEY, D. W., see N. M. Ceglio SWINNEY, H. H., see H. Z. Cummins TAKO,T., see M. Ohtsu TANAKA, K., Paraxial Theory in Optical Design in Terms of Gaussian Brackets TANGO,W. J., R. Q. TWISS,Michelson Stellar Interferometry TATARSKII, V. I., V. U. ZAVOROTNYI, Strong Fluctuation in Light Propagation in a Randomly Inhomogeneous Medium TAYLOR, C. A., see H. Lipson TEICH,M.C., B. E. A. SALEH,Photon Bunching and Antibunching TER-MIKAELIAN, M. L., see A. L. Mikaelian THELEN,A., see A. Musset THOMPSON, B. J., Image Formation with Partially Coherent Light K., see A. Ghatak THYAGARAJAN, TONOMURA, A., Electron Holography TRIPATHI, V. K., see M. S. Sodha TSUJIUCHI, J., Correction of Optical Images by Compensation of Aberrations and by Spatial Frequency Filtering

XIII, 93 XXVIII, 271 X, 89 XVI, 413 XXVII, 227 XV, 245 X, 229 XII, 53 VI, 21 1 X, 165 x, 45 XXI, 355

WII, XXVII, V, XX, IX,

169 109 145 325 13

11, 1 XIX, 43 XXIV, 1 XII, 1 XXI, 287 VIII, 133 XXV, 191 XXIII, 63 XVII, 239 XVIII, 207 V, 287 XXVI, 1 VII, 231 VIII, 201 VII, 169 XVIII, 1 XXIII, 183 XIII, 169 11, 131

CUMULATIVE INDEX

437

TWISS, R. Q., see W. J. Tango XVII, 239 UPATNIEKS, J., see E. N. Leith VI, 1 UPSTILL,C., see M. V. Berry XVIII, 259 USHIODA, S., Light Scattering Spectroscopy of Surface Electromagnetic Waves in Solids XIX, 139 VAMPOUILLE, M., see C. Froehly XX, 63 VANASSE,G. A., H. SAKAI,Fourier Spectroscopy VI, 259 VAN DE GRIND,W. A., see M. A. Bouman XXII, 77 VAN HEEL,A. C. S., Modem Alignment Devices I, 289 VAN KRANENDONK, J., J. E. SIPE,Foundations of the Macroscopic Electromagnetic Theory of Dielectric Media XV, 245 VERNIER,P., Photoemission XIV, 245 WALMSLEY, I. A,, see M. G. Raymer XXVIII, 179 WANG,SHAOMIN, L. RONCHI,Principles and Design of Optical Arrays XXV, 279 WEBER,M. J., see L. A. Riseberg XIV, 89 WELFORD,W. T., Aberration Theory of Gratings and Grating Mountings IV, 241 WELFORD,W. T., Aplanatism and Isoplanatism XIII, 267 WELFORD,W. T., see I. M. Bassett XXVII, 161 WILHELMI, B., see M. Schubert XVII, 163 WINSTON,R.,see I. M. Bassett XXVII, 161 WITNEY,K. G., see M. 0. Scully X, 89 WOLTER,H., On Basic Analogies and Principal Differences between Optical and Electronic Information I, 155 WYNNE,C. G., Field Correctors for Astronomical Telescopes X, 137 F., see 0. Bryngdahl XXVIII, 1 WYROWSKI, YAMAGUCHI, I., Fringe Formations in Deformation and Vibration Measurements Using Laser Light XXII, 271 M. KITAYAMAMOTO, Y., S. MACHIDA,S. SAITO,N. IMOTO,T. YAMAGAWA, GAWA, G. BJORK,Quantum Mechanical Limit in Optical Precision Measurement and Communication XXVIII, 87 YAMAJI, K., Design of Zoom Lenses VI, 105 YAMAMOTO, T., Coherence Theory of Source-Size Compensation in Interference Microscopy VIII, 295 YANAGAWA, T., see Y. Yamamoto XXVIII, 87 YOSHINAGA, H., Recent Developments in Far Infrared Spectroscopic Techniques XI, 77 Yu, F. T. S., Principles of Optical Processing with Partially Coherent Light XXIII, 227 ZAVOROTNYI, V. U., see V. I. Tatarskii XVIII, 207 ZUIDEMA, P., see M. A. Bouman XXII, 77

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