Progress in Optics, Volume 26

PROGRESS IN OPTICS VOLUME XXVI EDITORIAL ADVISORY BOARD L. ALLEN, London, England M. FRANCON, Paris, France F. GO...

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PROGRESS IN OPTICS VOLUME XXVI

EDITORIAL ADVISORY BOARD L. ALLEN,

London, England

M. FRANCON,

Paris, France

F. GORI,

Rome, Italy

E. INGELSTAM,

Stockholm, Sweden

A. KUJAWSKI,

Warsaw, Poland

A. LOHMANN,

Erlangen, F.R.G.

M. MOVSESSIAN,

Armenia, U.S.S.R.

G . SCHULZ,

Berlin, G.D.R.

J . TSUJIUCHI,

Tokyo, Japan

W. T. WELFORD,

London, England

PROGRESS IN OPTICS VOLUME XXVI

EDITED BY

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

Contributors M.C. TEICH, B.E.A. SALEH, I.C. KHOO G.P. AGRAWAL, Yu.A. KRAVTSOV, K. CREATH

1988

NORTH-HOLLAND AMSTERDAM * OXFORD * NEW YORK .TOKYO

@ ELSEVIER SCIENCE PUBLISHERS B.V., 1988

All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted,in any form or by any means, electronic, mechanical,photocopying,recording or otherwise, without the prior permission of the publisher. Elsevier Science Publishers B. V. (North-HollandPhysics Publishing Division), P.O. Box 103, 1000 A C Amsterdam, The Netherlands. Special regulations for readers in the U.S.A. :This publication has been registered 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. All other copyright questions, including photocopying outside of the U.S.A.. should be referred to the publisher.

LIBRARY OF CONGRESS CATALOG CARD NUMBER: 61-19297 ISBN: 0444870962

PUBLISHED BY:

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PRINTED IN THE NETHERLANDS

CONTENTS O F VOLUME 1(1961) THEMODERNDEVELOPMENT OF HAMILTONIAN OPTICS,R. J. PEGIS . . 1-29 WAVE OPTICS AND GEOMETRICAL OPTICS IN OPTICAL DESIGN, K. MIYAMOTO . .. . . . . . . . . .. . ... .. . . . . . 31-66 111. THEINTENSITY DISTRIBUTION AND TOTALILLUMINATION OF ABERRATIONFREEDIFFRACTION IMAGES,R. BARAKAT. . . . . . . . . . . . . . 67- 108 Iv. LIGHTAND INFORMATION, D. GABOR . . . . . . . . . . . . . 109-153 v. O N BASICANALOGIES AND PRINCIPAL DIFFERENCES BETWEEN OPTICAL AND ELECTRONIC INFORMATION, H.WOLTER. . , . . , . . . . 155-210 VI. INTERFERENCE COLOR,H. KUBOTA. . . . . . . . . . . . . . . . . 211-251 VII. DYNAMIC CHARACTERISTICS OF VISUALPROCESSES, A. FIORENTINI . . 253-288 VIII. MODERNALIGNMENT DEVICES,A. C. S.VAN HEEL . . . . . . . . , 289-329 I. 11.

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

RULING, TESTINGAND USE OF OPTICAL GRATINGS FOR HIGH-RESOLUTION SPECTROSCOPY, G. W. STROKE . . . . . . . . . .. . .. 1-72 11. THE METROLOGICAL APPLICATIONS OF DIFFRACTION GRATINGS,J. M. BURCH . . . . .. . .... . . . . .. . . .. .. . . 73-108 109-129 111. DIFFUSIONTHROUGH NON-UNIFORM MEDIA,R. G. GIOVANELLI. . . Iv. CORRECTION OF OPTICAL IMAGES BY COMPENSATION OF ABERRATIONS AND BY SPATIAL FREQUENCY FILTERING, J. TSUJIUCHI . , . . . . 131-180 v. FLUCTUATIONS OF LIGHTBEAMS,L. MANDEL . . . . . . . . . . . . 181-248 VI. METHODSFOR DETERMINING OPTICAL PARAMETERS OF THIN FILMS,F. ABELES . . . . . . . . . . . . . .. .. . . . . . . 249-288

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. . . ..

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...

I. 11. 111.

I. 11. III.

Iv. V. VI.

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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 ) THE ELEMENTS OF RADIATIVE TRANSFER,F. K O ~ L E R. . . . . . . . APODISATION, P. JACQUINOT, B. ROIZEN-DOSSIER. . . . . . . . . . MATRIXTREATMENTOFPARTIALCOHERENCE,H. GAMO . . . . . . .

1-28 29- 186 187-332

C O N T E N T S O F V O L U M E IV ( 1 9 6 5 ) HIGHERORDERABERRATION THEORY, J. FOCKE. . . . . . . . . . APPLICATIONS OF SHEARING INTERFEROMETRY, 0.BRYNGDAHL, . , SURFACEDETERIORATION OF OFTICALGLASSES, K. KINOSITA. . . . OmICAL CONSTANTS OF THINFILMS, P. ROUARD,P. BOUSQUET. . . THEMIYAMOTO-WOLF DIFFRACTION WAVE,A. RUBINOWICZ .,...

1-36 37-83 85-143 145-197 199-240

. . . . .

ABERRATION THEORYOF GRATINGSAND GRATINGMOUNTINGS,W.T. WELFORD . . . . , . . . . . . . . , . . . . . . . , . .. . VII. DIFFRACTION AT A BLACKSCREEN, PART I: KIRCHHOFF’S THEORY,F. KOTTLER . . . . . . . .. . . . . . . .. . .. . . .

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I. 11. 111.

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CONTENTS O F VOLUME V OITICALPUMPING, C. COHEN-TANNOUDJI, A. KASTLER . NON-LINEAR OPTICS,P. S. PERSHAN . . . . . . . . . TWO-BEAM INTERFEROMETRY, W. H. STEEL . . . . . . V

(1966) . . . . . .. .. . . . . . .......

241-280 281-314

1-81 83-144 145-197

v1

IV . V.

VI .

VII .

.

INSTRUMENTS FOR THE MEASURING OF OPTICAL TRANSFER FUNCTIONS. K

199-245 MURATA. . . . . . . . . . . . . . . . . . . . . . . . . . . . . LIGHTREFLECTIONFROM FILMSO F CONTINUOUSLY VARYING REFRACTIVE INDEX.R . JACOBSSON. . . . . . . . . . . . . . . . . . . . . . . 247-286 X-RAYCRYSTAL-STRUCTURE DETERMINATION AS A BRANCHOF PHYSICAL OPTICS.H . LIPSON.C.A . TAYLOR . . . . . . . . . . . . . . . . . . 287-350 THEWAVEOF A MOVINGCLASSICAL ELECTRON.J . PICHT . . . . . . . 351-370

C O N T E N T S O F V O L U M E VI ( 1 9 6 7 ) RECENTADVANCESIN HOLOGRAPHY. E.N . LEITH. J . UPATNIEKS. . . . SCATTERING OF LIGHTBY ROUGH SURFACES. P. BECKMANN. . . . . . OF THE SECOND ORDER DEGREEO F COHERENCE. M . 111. MEASUREMENT FRANCON. S . MALLICK . . . . . . . . . . . . . . . . . . . . . . IV . DESIGNOF ZOOM LENSES.K . YAMAJI. . . . . . . . . . . . . . . . OF LASERSTO INTERFEROMETRY. D . R . HERRIOTT. V . SOMEAPPLICATIONS STUDIES OF INTENSITY FLUCTUATIONS IN LASERS.J . A . VI . EXPERIMENTAL ARMSTRONG.A . W. SMITH. . . . . . . . . . . . . . . . . . . . . SPECTROSCOPY. G. A . VANASSE. H. SAKAI. . . . . . . . . . VII . FOURIER AT A BLACKSCREEN.PART11: ELECTROMAGNETIC THEORY. VIII . DIFFRACTION F. KOTTLER . . . . . . . . . . . . . . . . . . . . . . I. I1.

1-52 53-69 71-104 105-170 17 1-209 211-257 259-330 331-377

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

MULTIPLE-BEAMINTERFERENCE AND NATURAL MODES I N OPEN RESONATORS.G . KOPPELMAN . . . . . . . . . . . . . . . . . . . MULTILAYERFILTERS. E. I1. METHODS OF SYNTHESIS FOR DIELECTRIC DELANO.R .J . PEGIS . . . . . . . . . . . . . . . . . . . . . . . 1. D . ABELLA. . . . . . . . . . I11. ECHOESAND OPTICAL FREQUENCIES. IV . IMAGEFORMATION WITH PARTIALLY COHERENT LIGHT.B.J . THOMPSON THEORY OF LASERRADIATION. A . L. MIKAELIAN. M . L. V . QUASI-CLASSICAL TER-MIKAELI AN . . . . . . . . . . . . . . . . . . . . . . . . . VI . THEPHOTOGRAPHIC IMAGE. s. O O U E . . . . . . . . . . . . . . . . VII . INTERACTIONOF VERY INTENSE LIGHT WITH FREEELECTRONS.J.H. EBERLY . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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

1-66 67- 137 139-168 169-230 23 1-297 299-358 359-415

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

CONTENTS OF VOLUME IX (1971) 1.

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

PRECISE

LENGTHMEASURE-

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

1-30

VII

PICOSECOND LASERPULSES,A. J. DEMARIA. . . . . . . . . . . . . OPTICALPROPAGATION THROUGH THE TURBULENT ATMOSPHERE, J. W. STROHBEHN. . . . . . . . . . . . . . . . . . . . . . . . . . 1V. SYNTHESIS OF OPTICAL BIREFRINGENT NETWORKS, E. 0.AMMANN. . . V. MODELOCKINGIN GAS LASERS,L. ALLEN,D. G. c. JONES . . . . . . VI. CRYSTAL OPTICS WITH SPATIAL DISPERSION, 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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31-71 73- 122 123-177 179-234 235-280 281-310 3 11-407

CONTENTS OF VOLUME X (1972) BANDWIDTH COMPRESSION OF OPTICALIMAGES,T. s. HUANG. . . . . . 11. THE USE OF IMAGE TUBESAS SHUlTERS, R.w. SMITH . . . . . QUANTUM OPTICS,M. 0. SCULLY, K. G. WHITNEY 111. TOOLSOF THEORETICAL CORRECTORS FOR ASTRONOMICALTELESCOPES, c. G. WY"E . . IV. FIELD OPTICAL ABSORPTIONSTRENGTH OF DEFECTSIN INSULATORS, D. Y. V. , . . . . . . . . . . . SMITH,D. L. DEXTER . . . . . . . . . . LIGHTMODULATION AND DEFLECTION, E. K. SIITIG . . . VI. ELASTOOPTIC DETECTION THEORY, C. W. HELSTROM . . . . . . . . . . VII. QUANTUM I.

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

CONTENTS OF VOLUME XI (1973) 1-76 MASTEREQUATION METHODSIN QUANTUM OPTICS,G. S. AGARWAL. . RECENTDEVELOPMENTS IN FARINFRARED SPECTROSCOPIC TECHNIQUES, 77- 122 H. YOSHINAGA . . . . . . . . . . . . . . . . . . . . . . . . . OF LIGHTAND ACOUSTIC SURFACE WAVES,E. G. LEAN. . 123-166 111. INTERACTION WAVESIN OPTICALIMAGING, 0. BRYNGDAHL, . . . . 167-221 IV. EVANESCENT PRODUCTION OF ELECTRON PROBESUSINGA FIELDEMISSIONSOURCE, V. A.V. CREWE. . . . . . . . . . . . . . . . . . . . . . . . . . . 223-246 THEORY OF BEAMMODEPROPAGATION, J. A. ARNAUD . 247-304 VI. HAMILTONIAN INDEXLENSES,E. W. MARCHAND.. . . . . . . . . . . . 305-337 VII. GRADIENT I. 11.

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.

C O N T E N T S O F V O L U M E XI1 (1974) I. 11. 111.

IV. V.

VI.

SELF-FOCUSING, SELF-TRAPPING, AND SELF-PHASEMODULATION 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 . . . . . . . . . . . . . . . . . . . . . . . . . . THEPHASETRANSITION CONCEPT AND COHERENCE IN ATOMIC EMISSION, R. GRAHAM. . . . . . . . . , . . .. . . . . . . . BEAM-FOIL SPE~ROSCOP S.YBASHKIN , . . .. . . . .. .. . . .

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1-51 53-100 101-162 163-232

-

2 33 2 86 287-344

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

ONTHE VALIDITYOF KIRCHHOFF'S LAWOF HEATRADIATIONFOR I N A NONEQUILIBRIUM ENVIRONMENT, H. P. BALTES . . . . .

A

.

BODY

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1-25

VIII

THE CASE FOR AND AGAINSTSEMICLASSICAL RADIATIONTHEORY, L. 27-68 MANDEL. . . . . . . . . . . . . . . . . . . . . . . . . . . MEASUREMENTS OF 111. OBJECTIVE AND SUBJECTIVE SPHERICAL ABERRATION THE HUMANEYE,w. M. ROSENBLUM, J. L. CHRISTENSEN . . . . . . . 69-91 TESTINGOF SMOOTH SURFACES,G. SCHULZ, J. IV. INTERFEROMETRIC SCHWIDER. . . . . . . , . . . . . . . . . . . . . . . . . . . . 93-167 SELF FOCUSING OF LASERBEAMSIN PLASMAS AND SEMICONDUCTORS, V. M. S. SODHA,A. K. GHATAK,V. K. TRIPATHI . . . . . . . . . . . . 169-265 A N D ISOPLANATISM, w. T. WELFORD . . . . . . . . . . VI. APLANATISM 267-292 11.

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C O N T E N T S O F VOLUME XIV (1977)

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THESTATISTICS OF SPECKLE PATTERNS,J. c. DAINTY . . . . . . . . I. 11. HIGH-RESOLUTION TECHNIQUES IN OPTICAL ASTRONOMY, A. LABEYRIE. PHENOMENA IN RARE-EARTH LUMINESCENCE, L.A. RISE111. RELAXATION BERG, M. J. WEBER . . . . . . . . . . . . . . . . . . .. .. IV. THE ULTRAFAST OPTICALKERRSHUITER, M. A. DUGUAY. . . . . . . V. HOLOGRAPHIC DIFFRACTION GRATINGS, G. SCHMAHL, D. RUDOLPH . . VI. PHOTOEMISSION, P. J. VERNIER . . . . . . . . . . . . . , . . . . . VII. OPTICALFIBRE WAVEGUIDES-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 ) THEORYOF OPTICAL PARAMETRIC AMPLIFICATION AND OSCILLATION, w. BRUNNER, H. PAUL . . . . . . . . . . . . . . . . .. .. . .. 11. OPTICALPROPERTIES OF THINMETALFILMS, P. ROUARD,A. MEESSEN. 111. PROJECTION-TYPE HOLOGRAPHY, T. OKOSHI. . . . . . . . , . . IV. QUASI-OPTICAL TECHNIQUES OF RADIO ASTRONOMY, T. W.COLE , . . OF THE MACROSCOPIC ELECTROMAGNETIC THEORYOF V. FOUNDATIONS DIELECTRIC MEDIA,J. VAN KRANENDONK, J. E. SIPE . . . . . . . . I.

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1-75 71-137 139-185 187-244

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245-350

C O N T E N T S O F V O L U M E XVI (1978) LASERSELECTIVE PHOTOPHYSICS AND PHOTOCHEMISTRY, V. S. LETOKHOV 1-69 J. J. CLAIR,C. I. RECENTADVANCESIN PHASEPROFILESGENERATION, ABITBOL. . . . . . . . . . . . . . . . . . . . . . . . 71-117 111. COMPUTER-GENERATED HOLOGRAMS: TECHNIQUES AND APPLICATIONS, W.-H. LEE . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119-232 A. E. ENNOS . . . . . . . . . . . . . . 233-288 IV. SPECKLEINTERFEROMETRY, DEFORMATION INVARIANT, SPACE-VARIANT OPTICALRECOGNITION,D. V. CASASENT, D. PSALTlS . . . . . . . . . . . . . . . . . . . . . . 289-356 VI. LIGHT EMISSIONFROMHIGH-CURRENT SURFACE-SPARK DISCHARGES, R. E. BEVERLY I11 . . . . . . . . . . . . . . . . . . . . . . . . . 357-41 1 VII. SEMICLASSICAL RADIATION THEORYWITHINA QUANTUM-MECHANICAL FRAMEWORK, I. R. SENITZKY. . . . . . . . . . . . . . . . . . . 413-448 I. 11.

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IX

C O N T E N T S O F VOLUME XVII (1980) I. HETERODYNE HOLOGRAPHIC INTERFEROMETRY, R. DXNDLIKER . . . . 11. DOPPLER-FREE MULTIPHOTON SPECTROSCOPY, E. GIACOBINO, B. CAGNAC 111. THEMUTUALDEPENDENCE BETWEEN COHERENCE PROPERTIES OF LIGHT AND NONLINEAR OPTICAL PROCESSES, M. SCHUBERT, B. WILHELMI . . IV. MICHIELSONSTELLAR INTERFEROMETRY, W. J. TANGO,R. Q. TWISS . . V. SELF-FOCUSING MEDIA WITH VARIABLE INDEX OF REFRACTION,A. L. MIKAELIAN . . . . . . . . . . . . . . . . . . . . . . . . . . .

1-84 85-162 163-238 239-278 219-345

C O N T E N T S O F VOLUME XVIII (1980) I.

GRADEDINDEXOPTICALWAVEGUIDES:A REVIEW, A. GHATAK,K. THYAGARAJAN . . , . . , . . . . . . . . . . . . . . . . . . . . 1-126 11. PHOTOCOUNT STATISTICS OF RADIATION PROPAGATING THROUGH RANDOM AND 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 PAITERNS, M. v. BERRY, c. UPSTILL . . . . . . . . . . . . 257-346

CONTENTS O F VOLUME XIX (1981) THEORY OF INTENSITY DEPENDENT RESONANCELIGHTSCATTERING AND RESONANCE FLUORESCENCE, B. R. MOLLOW . . . . . . . . . . . . . 1-43 11. SURFACE AND SIZE EFFECTSON THE LIGHT SCATTERING SPECTRA OF SOLIDS,D. L. MILLS,K. R. SUBBASWAMY . . . . . . . . . . . . , . 45- 137 111. LIGHT SCATTERING SPECTROSCOPY OF SURFACE ELECTROMAGNETIC WAVES IN SOLIDS, s. USHIODA . . . . . . . . . . . . . . . . . . . 139-210 IV. PRINCIPLES OF OPTICAL DATA-PROCESSING, H. J. BUITERWECK . . . . 211-280 V. THEEFFEC~SOF ATMOSPHERIC TURBULENCE IN OPTICAL ASTRONOMY, F. RODDIER . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281-376 I.

C O N T E N T S O F VOLUME XX (1983) I.

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

DETAILLE, M. SAYSSE .

11.

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

.. ..... . c. FROEHLY, B. COLOMBEAU, M. VAMPOUILLE . . . . . . . . . . . . . . . . . . . MULTI-PHOTON SCATTERING MOLECULARSPECTROSCOPY, S. KIELICH . COLOUR HOLOGRAPHY, P. HARIHARAN. . . . . . . . . . . . , . . SHAPING AND ANALYSIS OF PICOSECOND

,

1-62

LIGHTPULSES,

111. IV. V. GENERATION OF TUNABLE COHERENT VACUUM-ULTRAVIOLET W. JAMROZ, B.P. STOICHEFF. . . . . . . . . . . . , .

63- 154 155-262 263-324

RADIATION, ,

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325-380

X

CONTENTS OF VOLUME XXI (1984) RIGOROUS I. VECTOR THEORIES OF DIFFRACTION GRATINGS,D. MAYSTRE . OF OPTICALBISTABILITY, L. A. LUGIATO. . . . . . . . . . . 11. THEORY AND ITS APPLICATIONS, H. H. BARRETT . . 111. THE RADONTRANSFORM N. M. CEGLIO, IV. ZONE PLATE CODED IMAGING: THEORYAND APPLICATIONS, . . . . . . . . . . . . . . . . . . . .. . . . . . D. W. SWEENEY FLUCTUATIONS, INSTABILITIES AND CHAOS IN THE LASER-DRIVEN NONV. LINEAR RINGCAVITY,J. C. ENGLUND, R. R. SNAPP,W. C. SCHIEVE. . .

.

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

CONTENTS OF VOLUME XXII (1985) OPTICAL AND ELECTRONICPROCESSINGOF MEDICAL IMAGES, D. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-76 MALACARA 11. QUANTUM FLUCTUATIONS IN VISION, M. A. BOUMAN, W. A. VAN DE GRIND, 17-144 P. ZUIDEMA. . . . . . . . . . . . . . . . . . . . . . . . . . . OF BROAD-BANDLASER 111. SPECTRALAND TEMPORALFLUCTUATIONS RADIATION, A. V. MASALOV. . . . . . . . . . . . . . . . . . . . 145-196 G. V. ~STROVSKAYA, IV. HOLOGRAPHIC METHODSOF PLASMADIAGNOSTICS, Yu. I. OSTROVSKY . . . . . . . . . . . . . . . . . . . . . . . 197-270 FRINGE FORMATIONS IN DEFORMATION AND VIBRATIONMEASUREMENTS V. USING LASERLIGHT,I. YAMAGUCHI . . . . . . . . . . . . . . . 271-340 IN RANDOMMEDIA:A SYSTEMSAPPROACH, R. L. VI. WAVEPROPAGATION FANTE. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 341-398

I.

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CONTENTS O F VOLUME XXIII (1986) ANALYTICAL TECHNIQUESFOR MULTIPLESCATTERING FROM ROUGH SURFACES, J. A. DESANTO,G. S. BROWN. . . . . . . . . . . . . , . 1-62 IN OPTICAL DESIGNIN TERMS OF GAUSSIAN BRACKETS, 11. PARAXIAL THEORY K. TANAKA . . . . . . . . . . . . . . . . . . . . . . . . . . . 63-1 12 PRODUCED BY ION-BASED TECHNIQUES, P. J. MARTIN, R. P. 111. OPTICALFILMS NEITERFIELD. . . . . . . . . . . . . . . . . . . . . . . . . . 113-182 IV ELECTRON HOLOGRAPHY, A. TONOMURA . . . . . . . . . . . . . . 183-220 V. PRINCIPLES OF OPTICAL PROCESSING WITH PARTIALLY COHERENT LIGHT, F.T. S. YU . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221-276 1.

.

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 COHERENTPHENOMENA, L. ROTHBERG . . . . . INTERFEROMETRY WITH LASERS,P. HARIHARAN . . . . . . . . . . . UNSTABLE RESONATOR MODES,K.E. OUGHSTUN. . . . . . . . . . . INFORMATIONPROCESSINGWITH SPATIALLYINCOHERENTLIGHT, 1. GLASER. . . . . . . . . . . . . . . . . . . . . . . . . . . .

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

C O N T E N T S O F VOLUME XXV ( 1 9 8 8 ) I.

DYNAMICAL INSTABILITIES AND PULSATIONS IN LASERS,N.B. ABRAHAM, P. MANDEL.L.M. NARDUCCI. . . . . . . . . . . . . . . . . . . . 1-190 I N SEMICONDUCTOR LASERS,M. OHTSU,T. TAKO . . . . . 191-278 11. COHERENCE 111. PRINCIPLES AND DESIGNOF OPTICAL ARRAYS, WANGSHAOMIN, L. RONCHI 279-348 G. SCHULZ. . . . . . . . . . . . . . . . . . 349-416 IV. ASPHERICSURFACES,

PREFACE Volumes in this series have been appearing at the rate of approximately one per year. Because of statistical fluctuations in the arrival of manuscripts, the present volume is the second one that is being published this year and it follows its predecessor by only a few weeks. It contains five articles on rather varied subjects, covering several areas of traditional optics as well as some of the newer areas of quantum optics. The opening article deals with certain quantum states of light which have been produced in the laboratory only within the last few years. They include so-called antibunched light and light whose photon fluctuations obey subPoisson statistics. The article provides a timely review of the characteristic features of such non-classical states of the optical field and discusses their generation and some possible uses. The second article provides a review of fundamentals of optics of liquid crystals, with special reference to some of their unusual features in the nonlinear regime. The third article describes some recent developments in the field of semiconductor lasers. Conventional lasers usually oscillate simultaneously in several longitudinal modes. In the last few years techniques have been found for making semiconductor lasers oscillate essentially in only one such mode. The article describes ways of achieving this and discusses performance of lasers of this kind, especially their spectral characteristics. Some applications of these devices are also mentioned. In the next article, entitled “Rays and caustics as physical objects”, the concept and the role of rays is discussed within the framework of physical optics. The old notion of a light ray has undergone many re-examinations and generalizations since the time when Fermat, Hamilton, Kirchhoff, Sommerfeld, Runge, Luneburg and others made important contributions to the foundations of ray optics, as the subject is customarily understood. The article describes some newer developments relating to the interrelation between ray optics and physical optics. The concluding article deals with phase-measurement interferometry. It describes recent advances that have made it possible to make phase measureXI

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PREFACE

ments with a precision that is better by one or two orders of magnitude than can be achieved by the older fringe-digitizingtechniques. The article outlines the basic principles of phase-measurement interferometry, its implementation and also describes some applications.

EMILWOLF Department of Physics and Astronomy University of Rochester Rochester, NY 14627. USA January 1988

I. PHOTON BUNCHING AND ANTIBUNCHING by M.C. TEICH(NEWYORK.NY. USA) and B.E.A.SALEH(MADISON,WI. USA)

$ 1. INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . $ 2. BUNCHEDLIGHT FROM INDEPENDENT RADIATORS . . . . . . . . . . . . . 2.1 Semiclassical theory of optical coherence . . . . . . . . . . . . . . . . 2.1.1 Coherent light . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.2 Poisson. super-Poisson. and sub-Poisson photocounts . . . . . . . . 2.1.3 Unbunched. bunched. and antibunched light . . . . . . . . . . . . 2.1.4 Chaotic light . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.5 Chaotic. superchaotic. and subchaotic light . . . . . . . . . . . . 2.1.6 Inter-event time statistics . . . . . . . . . . . . . . . . . . . . 2.2 Superposition of independent emissions . . . . . . . . . . . . . . . . 2.2.1 Chaotic emissions . . . . . . . . . . . . . . . . . . . . . . . 2.2.2 Statistically identical emissions . . . . . . . . . . . . . . . . . . 2.3 Number fluctuations . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.1 Poisson number fluctuations . . . . . . . . . . . . . . . . . . . 2.4 Emissions initiated at Poisson times . . . . . . . . . . . . . . . . . . 2.4.1 Quasi-coherent emissions . . . . . . . . . . . . . . . . . . . . $ 3. ANTIBUNCHED LIGHT FROM INDEPENDENT RADIATORS . . . . . . . . . . . 3.1 Quantum theory of optical coherence: A brief review . . . . . . . . . . . 3.2 Superposition of independent emissions . . . . . . . . . . . . . . . . 3.2.1 Quasi-coherent single-mode emissions . . . . . . . . . . . . . . . 3.3 Emissions initiated at Poisson times . . . . . . . . . . . . . . . . . . 3.3.1 Quasi-coherent single-mode emissions . . . . . . . . . . . . . . . 3.3.2 Radiation from an atomic beam . . . . . . . . . . . . . . . . . 3.4 Emissions initiated at sub-Poisson times . . . . . . . . . . . . . . . . 3.4.1 Characterization of the excitation point process . . . . . . . . . . 3.4.2 Photon statistics for emissions at antibunched times . . . . . . . . 3.4.3 Bunchmg/antibunching properties of emissions initiated at antibunched times . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 Summary: Generation of antibunched and sub-Poisson light . . . . . . . $ 4. RANDOMIZATION OF SUB-POISSON PHOTON STREAMS . . . . . . . . . . . 4.1 Bernoulli random deletion . . . . . . . . . . . . . . . . . . . . . . 4.2 Additive independent Poisson photons . . . . . . . . . . . . . . . . . 4.3 Analog relations . . . . . . . . . . . . . . . . . . . . . . . . . . $ 5 OBSERVATIONS OF ANTIBUNCHED AND CONDITIONALLY SUB-POISSON PHOTON EMISSIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Conditionally sub-Poisson photon clusters from resonance fluorescence . . . 5.2 Conditionally sub-Poisson single photons from parametric downconversion . 5.3 Destruction of sub-Poisson behavior by excitation statistics . . . . . . . .

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17 19 19 22 22 25 26 27 30 33 34 36 38 40 40 42 43 43 45 41 49 51 52 53 54 55

56 58 59

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CONTENTS

$ 6. GENERATION OF ANTIBUNCHED AND SUB-POISSON LIGHTBY PHOTON FEEDBACK 6.1 Methods using feedback intrinsic to a physical process . . . . . . . . . . 6.2 Methods using external feedback . . . . . . . . . . . . . . . . . . . 6.2.1 Correlated photon pairs from cascaded atomic emissions . . . . . . 6.2.2 Correlated photon pairs from parametric downconversion . . . . . . 6.2.3 All-optical systems using correlated photon pairs . . . . . . . . . . 6.2.4 Quantum nondemolition measurements . . . . . . . . . . . . . . 6.3 Limitations of photon-feedback methods . . . . . . . . . . . . . . . . § 7. GENERATION OF ANTIBUNCHED AND SUB-POISSON LIGHT BY EXCITATION FEEDBACK . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1 Methods using feedback intrinsic to a physical process . . . . . . . . . . 7.1.1 Space-charge-limited Franck-Hertz experiment . . . . . . . . . . 7.1.2 Space-charge-limited excitation of recombination radiation . . . . . . 7.1.3 Sub-Poisson excitations and stimulated emissions . . . . . . . . . . 7.2 Methods using external feedback . . . . . . . . . . . . . . . . . . . 7.2.1 Opto-electronic generation of sub-Poisson electrons . . . . . . . . . 7.2.2 Extraction of in-loop photons by a beamsplitter . . . . . . . . . . 7.2.3 Use of an in-loop auxiliary optical source . . . . . . . . . . . . . 7.2.4 Use of a current source with external compensation . . . . . . . . . 7.3 Limitations of excitation-feedback methods . . . . . . . . . . . . . . . $ 8. INFORMATION TRANSMISSION USING SUB-POISSON LIGHT . . . . . . . . . 8.1 Communicating with modified Poisson photons . . . . . . . . . . . . . 8.2 Communicating with sub-Poisson photons described by a self-exciting point process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3 Communicating with sub-Poisson photon counts . . . . . . . . . . . . . 8.4 Performance of a sub-Poisson photon-counting receiver . . . . . . . . . 8.4.1 Dead-time-modified-Poisson photon counts . . . . . . . . . . . . 8.4.2 Decimated-Poisson photon counts . . . . . . . . . . . . . . . . 8.4.3 Binomial photon counts . . . . . . . . . . . . . . . . . . . . . 8.5 Limitations on communicating with sub-Poisson light . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

62 63

64 65 68 70 71 72 73 77 78 80 82 83 83 85 86 89 90 91 92 93 94 95 96 96 97 98 100

I1. NONLINEAR OPTICS O F LIQUID CRYSTALS b.), I.C. KHOO( U N I V t R S I T Y PARK.PA. USA) $ 1 . INTRODUCTION

. . . . . . . . . . . . . . . . . . . 8 2. FUNDAMENTALS OF LIQUIDCRYSTALS . . . . . . . . 2.1 General . . . . . . . . . . . . . . . . . . . . . 2.2 Free energies and distortions by applied fields . . . 2.3 Optically induced director axis reorientation . . . . 2.3.1 Plane wave . . . . . . . . . . . . . . . . . 2.3.2 Finite beam size . . . . . . . . . . . . . . .

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2.4 Temperature-dependent refractive index . . . . . . . . . . . . . . . . 2.5 Dynamics of optically induced reorientations and thermal effects . . . . . 2.5.1 Thermal effect . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.2 Orientational dynamics . . . . . . . . . . . . . . . . . . . . . $ 3. NONLINEAROPTICALPROCESSES . . . . . . . . . . . . . . . . . . . . . 3.1 Summary of observed nonlinear optical effects . . . . . . . . . . . . . 3.2 General remarks on nonlinear optical processes . . . . . . . . . . . . .

107 109 109 110 115

116 120 123 127 127 130 133 133 135

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3.3 Self-focusing and self-phase modulation . . . . . . . 3.4 Optical wave mixing . . . . . . . . . . . . . . . . 3.4.1 Self-diffraction and degenerate four-wave mixing . 3.4.2 Optical wavefront conjugation . . . . . . . . . 3.4.3 Beam amplification and dynamic wave mixing . . 3.5 Optical bistability and switching . . . . . . . . . . . $ 4. FURTHER REMARKS AND CONCLUSIONS . . . . . . . . . ACKNOWLEDGEMENT . . . . . . . . . . . . . . . . . .. REFERENCES . . . . . . . . . . . . . . . . . . . . . . .

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136 145 145 146 149 153 155 157 157

I11 . SINGLE-LONGITUDINAL-MODE SEMICONDUCTOR LASERS

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by G.P. A G R A W A L . ( M ~ I RHILL K A YNJ. USA)

$ 1. INTRODUCTION. . . . . . . . . . . . . . . . . . . . . . . . . . . . $ 2. FABRY-PEROT SEMICONDUCTOR LASERS . . . . . . . . . . . . . . . . . 2.1 Laser material and structure . . . . . . . . . . . . . . . . . . . . . 2.2 Laser modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Rate equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Longitudinal-mode spectra . . . . . . . . . . . . . . . . . . . . . . $ 3. REQUIREMENTS FOR SINGLE-LONGITUDINAL-MODE OPERATION . . . . . . . $ 4. DISTRIBUTED-FEEDBACK SEMICONDUCTOR LASERS. . . . . . . . . . . . . 4.1 Longitudinal modes and gain margin . . . . . . . . . . . . . . . . . . 4.2 Fabrication and performance . . . . . . . . . . . . . . . . . . . . . 4.3 Distributed-Bragg-reflectorlasers . . . . . . . . . . . . . . . . . . . $ 5 . COUPLED-CAVITY SEMICONDUCTOR LASERS. . . . . . . . . . . . . . . . 5.1 Coupled-cavity schemes . . . . . . . . . . . . . . . . . . . . . . . 5.2Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Single-longitudinal-mode performance . . . . . . . . . . . . . . . . . $ 6. MODULATION PERFORMANCE. . . . . . . . . . . . . . . . . . . . . . 6.1 Small-signal modulation . . . . . . . . . . . . . . . . . . . . . . . 6.2 Large-signal modulation . . . . . . . . . . . . . . . . . . . . . . . LINEWIDTH . . . . . . . . . . . . . . . . . . . . . . . . . $ 7. SPECTRAL $ 8. CONCLUSION. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . REFERENCES

165 166 167 169 173 175 178 180 181 190 193 197 198 201 206 208 210 213 216 221 222

IV . RAYS AND CAUSTICS AS PHYSICAL OBJECTS by Yu.A. KRAVTSOV (Moscow. USSR)

$ 1. INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . $ 2. ELEMENTS OF GEOMETRICAL OPTICS . . . . . . . . . . . . . . . . . . 2.1 Basic equations of the method . . . . . . . . . . . . . . . . . . . 2.1.1 Scalar wave problem . . . . . . . . . . . . . . . . . . . . . 2.1.2 Equations of geometrical optics for an electromagnetic field in an isotropic medium . . . . . . . . . . . . . . . . . . . . . . 2.1.3 Independent normal waves in an anisotropic medium . . . . . . . 2.1.4 Electromagnetic fields in weakly anisotropic media . . . . . . . . 2.2 Rays as energy and phase trajectories . . . . . . . . . . . . . . . .

229 230 230 230 233 235 236 236

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8 3. FRESNELVOLUMES OF RAYS . . . . . . . . . . . . . . . . . . . . . .

3.1 Integral form of the Huygens principle for inhomogeneous media . . . . 3.2 Fresnel zones of stationary points and Fresnel volumes of rays . . . . . 3.2.1 Fresnel volume of a ray for elliptic stationary points . . . . . . . 3.2.2 Physical content of the concept of a ray . . . . . . . . . . . . . 3.2.3 Fresnel zone for saddle stationary points . . . . . . . . . . . . 3.3 Methods of actual localization of rays . . . . . . . . . . . . . . . . 3.3.1 Characterizing the degree of localization . . . . . . . . . . . . 3.3.2 Propagation of a wave through Fresnel holes . . . . . . . . . . . 3.3.3 Localization of rays by simple apertures . . . . . . . . . . . . . 3.3.4 Localization of rays by Gaussian windows . . . . . . . . . . . . 3.3.5 Unified definition of Fresnel volume for arbitrary stationary points . 3.4 Fresnel volumes of rays in anisotropic media . . . . . . . . . . . . . 3.5 Fresnel volume of a quasi-classical particle trajectory . . . . . . . . . 3.6 Fresnel volumes of space-time rays . . . . . . . . . . . . . . . . . CRITERIA FOR APPLICABILITY OF GEOMETRICAL OPTICS . . . . . 8 4. HEURISTIC 4.1 Necessary conditions of applicability . . . . . . . . . . . . . . . . . 4.2 Universal sufficient applicability criteria . . . . . . . . . . . . . . . 4.3 Stability of the geometrical optical solution with respect to small perturbations 4.4 Discrimination of rays . . . . . . . . . . . . . . . . . . . . . . . 4.5 Conditions of applicability of space-time geometrical optics . . . . . . . ESTIMATESOF THE FIELDIN DOMAINSOF INAPPLICABILITY OF 8 5 . RAY-OPTICAL GEOMETRICAL OPTICS . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Domains of inapplicability of the geometrical optical approximation . . . 5.2 Estimates of the wavefield in focal and caustic domains of inapplicability . 5.2.1 Focal field estimates from the values of the ray-optical field on the focal zone boundary . . . . . . . . . . . . . . . . . . . . . . . . 5.2.2 Energy estimates based on the conservation of the energy flux in a ray tube of finite width . . . . . . . . . . . . . . . . . . . . . . 5.2.3 Energy estimates with the initial Fresnel section . . . . . . . . . 5.3 Estimating the field inside inapplicability domains by interpolating boundary values of ray fields . . . . . . . . . . . . . . . . . . . . . . . . 5.4 Geometrical optical analysis of the wave pattern as a whole . . . . . . . 8 6. UTILITYAND APPLICATIONS OF HEURISTIC CRITERIA . . . . . . . . . . . 6.1 Diffraction of waves in a homogeneous medium . . . . . . . . . . . . 6.1.1 Form of the penumbra region for a spherical wave obstructed by a diffracting half-plane . . . . . . . . . . . . . . . . . . . . . 6.1.2 Formation of near and far fields of a laser beam . . . . . . . . . 6.1.3 Field in the vicinity of a lens focus . . . . . . . . . . . . . . . 6.1.4 Fresnel volume for the observation point lying behind the lens focus 6.2 Reflection and refraction of waves at curvilinear interfaces between two media 6.2.1 Conditions of applicability of the reflection formulas . . . . . . . 6.2.2 Estimating the radius of the Fresnel zone at a curvilinear interface . 6.2.3 Region of breakdown of reflection formulas near a ray touching a convex body . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.4 Region of inapplicability of geometrical optics near a ray which has undergone total internal reflection . . . . . . . . . . . . . . . 6.2.5 Reflected ray field of a wave incident at the Brewster angle . . . . 6.2.6 Multiple reflection of rays from a concave interface . . . . . . . . 6.3 Diffraction of waves in inhomogeneous media . . . . . . . . . . . . . 6.3.1 Shape of the Fresnel volume in a planar layered medium . . . . .

238 238 242 242 245 245 248 248 249 252 254 255 256 251 258 259 259 261 263 264 265 265 265 266 261 261 268 269 269 210 210 210 212 213 216 211 211 219 280 282 283 284 285 285

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6.3.2 Scattering by weak localized inhomogeneities

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A N D CAUSTIC ZONES. . . . . . . . . . . . . 8 7 . CAUSTICS 7.1 Caustic zones and field focusing on caustics . . . . 7.1 .1 Equation for the boundary of a caustic zone . . 7.2 Pericaustic zone in the case of a simple caustic . . .

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7.2.1 Estimate of the width of the pericaustic zone . . . . . . . . . . 7.2.2 Estimating the field on a nonsingular caustic . . . . . . . . . . . 7.3 Caustic zones in the presence of spherical aberration . . . . . . . . . 7.4 Concomitant discussion . . . . . . . . . . . . . . . . . . . . . . 7.4.1 Estimating the width of the pericaustic zone and the field at a caustic by caustic indices . . . . . . . . . . . . . . . . . . . . . . . . 7.4.2 Physical indistinguishability of rays within pericaustic zones . . . . 7.4.3 "Reality" of caustics . . . . . . . . . . . . . . . . . . . . . 7.4.4 Micromultipath rays and random caustics . . . . . . . . . . . . VOLUMES OF DIFFRACTED RAYSAND APPLICABILITY LIMITSFOR THE $ 8 . FRESNEL GEOMETRICAL THEORYOF DIFFRACTION . . . . . . . . . . . . . . . . . 8.1 Edgewaves . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1.1 Geometrical theory of diffraction . . . . . . . . . . . . . . . . 8.1.2 Fresnel volumes of edge rays . . . . . . . . . . . . . . . . . 8.1.3 GTD applicability conditions for diffraction at edges . . . . . . . 8.1.4 Admissible approach of edge rays in the presence of caustics . . . . 8.1.5 Independence of edge waves and the geometrical optical field . . . 8.1.6 Independence of edge and vertex waves . . . . . . . . . . . . . 8.2 Fresnel volumes of grazing rays . . . . . . . . . . . . . . . . . . . 8.2.1 GTD approximation for grazing waves . . . . . . . . . . . . . 8.2.2 Estimates of the Fresnel scales . . . . . . . . . . . . . . . . . 8.2.3 Conditions for GTD applicability in the problem of diffraction at a cylinder of variable curvature . . . . . . . . . . . . . . . . . 8.3 Related problems . . . . . . . . . . . . . . . . . . . . . . . . . 8.3.1 Estimates of the field in the domain of GTD inapplicability . . . . 8.3.2 Admissible approach ofobjects and choice of idealized diffraction model OF LOCALIZATION OF COMPLEX RAYS . . . . . . . . . . . . . . $ 9. DOMAIN 9.1 Complex rays . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2 Diffractional nature of fields described by complex geometrical optics . . 9.3 Some examples . . . . . . . . . . . . . . . . . . . . . . . . . . 9.4 Distinguishability of complex rays and the applicability limits of complex geometrical optics . . . . . . . . . . . . . . . . . . . . . . . . . 8 IO.RAYSKELETON OF UNIFORM CAUSTIC ASYMPTOTIC SOLUTIONS . . . . . . . 10.1Heuristic principles of spanning a wavefield on its geometrical optical skeleton 10.2Basic equations of the method of standard integrals . . . . . . . . . . . 10.3Function of rays in the method of standard integrals . . . . . . . . . . . 8 1 ICONCLUDINGREMARKS. . . . . . . . . . . . . . . . . . . . . . . . . ACKNOWLEDGEMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

287 289 289 289 290 290 292 293 297 297 299 300 301 302 302 302 304 308 311 313 313 314 314 316 319 321 321 323 324 324 326 330 335 336 336 338 342 343 344 345

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V . PHASE-MEASUREMENT INTERFEROMETRY TECHNIQUES by K . CREATH(TUCSON.AZ. USA)

$ I . INTRODUCTION. . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2. MEANSOF SHIFTING AND DETERMINING PHASE . . . . . . . . . . . . . 2.1 Means of phase modulation . . . . . . . . . . . . . . . . . . . . . 2.2 Means of determining phase . . . . . . . . . . . . . . . . . . . . . $ 3. PHASE-MEASUREMENT ALGORITHMS. . . . . . . . . . . . . . . . . . 3.1 Sampling requirements . . . . . . . . . . . . . . . . . . . . . . . . 3.2 General phase-measurement technique . . . . . . . . . . . . . . . . 3.3 Synchronous detection . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Four.bucket, or four.step, technique . . . . . . . . . . . . . . . . . 3.5 Three-bucket, or three-step. technique . . . . . . . . . . . . . . . . 3.6 Carre technique . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.7 Removal of phase ambiguities . . . . . . . . . . . . . . . . . . . . . 3.8 From wavefront to surface . . . . . . . . . . . . . . . . . . . . . . $ 4. MEASUREMENT EXAMPLE. . . . . . . . . . . . . . . . . . . . . . . . $ 5 . ERRORANALYSIS . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Phase-shifter errors . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Phase-shifter calibration . . . . . . . . . . . . . . . . . . . . . . . 5.3 Averaging-three-and-three technique . . . . . . . . . . . . . . . . . . 5.4 Detector nonlinearities . . . . . . . . . . . . . . . . . . . . . . . . $ 6. SIMULATION RESULTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . $ 7. REMOVINGSYSTEMABERRATIONS $ 8. APPLICATIONSOFPHASE-MEASUREMENTINTERFEROMETRY . . . . . . . . 8.1 Surface shape measurement . . . . . . . . . . . . . . . . . . . . . 8.2 Surface roughness measurement . . . . . . . . . . . . . . . . . . . .

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351 352 353 354 357 357 358 362 363 363 365 366 367 368 373 374 374 376 378 379 385 388 388 390 390 391 391

AUTHORINDEX . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . SUBJECT INDEX . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . CUMULATIVE INDEX.VOLUMESI-XXVI . . . . . . . . . . . . . . . . . . .

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E. WOLF, PROGRESS IN OPTICS XXVI 0 ELSEVIER SCIENCE PUBLISHERS B.V., 1988

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PHOTON BUNCHING AND ANTIBUNCHINGS BY

MALVINC, TEICH Columbia Radiation Laboratory and Centerfor Telecommunications Research Depariment of Elecirical Engineering, Columbia University New York, NY 10027, USA

BAHAA E. A. SALEH Department of Elecmkal and Computer Engineering University of Wirconsin Madiron, WI 53706, USA

* This work was supported by the Joint Services ElectronicsProgram at Columbia University and by the National Science Foundation.

CONTENTS PAGE

INTRODUCTION

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

BUNCHED LIGHT FROM INDEPENDENT RADIATORS . . . . . . . . . . . . . . .

. . . . . .

ANTIBUNCHED LIGHT FROM INDEPENDENT RADIATORS . . . . . . . . . . . . . . . . . . RANDOMIZATION STREAMS . . . . .

OF

SUB-POISSON

. . .

3

8 33

PHOTON

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

51

OBSERVATION O F ANTIBUNCHED AND CONDITIONALLY SUB-POISSON PHOTON EMISSIONS . . . . . . 55 GENERATION OF ANTIBUNCHED AND SUB-POISSON LIGHT BY PHOTON FEEDBACK . . . . . . . . . . . .

62

GENERATION OF ANTIBUNCHED AND SUB-POISSON LIGHT BY EXCITATION FEEDBACK . . . . . . . . . . 73 INFORMATION TRANSMISSION USING SUB-POISSON LIGHT . . . . . . . . . . . . . . . . . . . . . . . . REFERENCES

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

91

99

8 1. Introduction The photon correlation experiments carried out by HANBURY-BROWN and TWISS[ 1956a,b] and the invention of the laser by SCHAWLOW and TOWNES [ 19581 fostered a strong interest in photon statistics in the 1960s. As it turned out, the semiclassical theory of light provided an adequate theoretical framework for understanding the observed photon correlations from conventional sources of light as well as from lasers (MANDEL[1959a, 19631). In 1963 GLAUBER [ 1963a,b] developed a fully quantum-mechanical theory of light that encompassed the semiclassical theory. One intriguing aspect of the new quantum optics was that it admitted the possibility of photon anticorrelations as well as photon correlations. Within this expanded framework it became possible to conceive of new forms of light that had never before been realized: antibunched light, sub-Poisson light, and quadrature-squeezed light. These three characteristics of nonclassical light have recently received a great deal of attention because they have now been observed in the laboratory (KIMBLE,DAGENAISand MANDEL[ 19771, SHORTand MANDEL[ 19831, TEICHand SALEH[1985], SLUSHER,HOLLBERG,YURKE, MERTZ and VALLEY[ 19851). In any given light source these characteristics may, but need not, accompany each other (TEICH, SALEHand STOLER[ 19831, PEWINA, PERINOVAand K O ~ O U S E[ K 19841, SHORTand MANDEL[ 19841, SCHLEICH and WHEELER[ 19871, YAMAMOTO,MACHIDA,IMOTO, KITAGAWAand BJORK [ 19871). There are other manifestations of nonclassical light as well; examples include a photon interevent-time probability density function that is underdispersed relative to the exponential and a violation of Bell's inequalities in a photon correlation experiment (REID and WALLS[ 19841). Nonclassical light should be useful in providing new insights into various physical and biological processes, such as spectroscopy and the behavior of the human visual system at low levels of light (TEICH,PRUCNAL, VANNUCCI, BRETONand MCGILL[ 19821). It is also expected to find use in applications such as optical signal processing, interferometry, gravitational-wave detection, and lightwave communications. This review pertains to the generation and characterization of antibunched and sub-Poisson light (sub-Poisson light is alternatively called photon-number-

4

[I, 8 1

PHOTON BUNCHING AND ANTIBUNCHING

squeezed light). These characteristics are manifested in direct-detection experiments, in which the phase of the light wave is ignored. This is in contrast to the observation of quadrature-squeezed light, which requires the use of an auxiliary light source (a local oscillator) and a heterodyne (coherent) detection process. (Useful sources of recent literature pertinent to the generation and detection of quadrature-squeezed light include articles by WALLS[ 19831, SHAPIRO [ 19851, CAVES[ 19861, LOUDONand KNIGHT[ 19871, and TEICH and SALEH[ 19881). In the course of this presentation, we also review the generation of bunched and super-Poisson (classical) light from collections of independent radiators. The discussion is readily initiated by the analogy with a gun illustrated in fig. 1.1. Photon guns naturally generate random (Poisson) streams of photons, as shown in fig. 1,la. The production of antibunched or sub-Poisson light can be achieved in three ways: by regulating the times at which the trigger is pulled, by introducing constraints into the firing mechanism, and/or by selectively deleting some of the Poisson bullets after they are fired. Each of these techniques involves the introduction of anticorrelations, which results in a more predictable number of events. These anticorrelations are introduced by means of a feedback (or feedforward) process of one kind or another.

5T=

-a-a-a-a-*-*-**-*-a-

Trigger

Firing mechanism

( a ) Poisson

a+

Poisson bullets

*-*-*-*.*-*-**-*-*-*

(b) Dead-time deletion

(c) Coincidence decimation (d) Decimation ( e l Overflow count deletion

*-*--..a.a

I ( f ) Rate compensation

!*a aa---..aa--.-a--.-I+

p A - 4 p A 2 - j Fig. 1.1. Schematic representation of three components of a simple photon-generation system. A trigger process excites a photon emitter (firing mechanism), which in turn emits photons (Poisson bullets). Anticorrelations can be induced in any of the three elements. Mechanisms for generating antibunched and sub-Poisson light from (a) Poisson light can make use of (b) deadtime deletion, (c) coincidence decimation, (d) decimation, (e) overflow count deletion, or (f) rate compensation.

1, I11

INTRODUCrION

5

Several specific schemes for introducing such photon anticorrelations are presented in fig. 1.1. Dead-time deletion, illustrated in fig. l.lb, prohibits photons from being arbitrarily close to each other. This effect can result from a requirement that the trigger or firing mechanism reset between consecutive shots. We will show in J 5.1 that this is, in fact, the way in which isolated atoms behave in the course of emitting resonance fluorescence photons. Under appropriate conditions, dead time can instead be imposed on the bullets after they are fired, as discussed in J 6.2. The dead-time deletion process regularizes the events, as is apparent from fig. 1.1, thereby reducing the randomness of the number of events registered in the fixed counting time T. Photon anticorrelations can also be introduced by coincidence decimation, which is a process in which closely spaced pairs of photons are removed from the stream, as shown in fig. 1.lc. Optical second-harmonic generation (SHG), for example, is a nonlinear process in which two photons are exchanged for a third photon at twice the frequency (see J 6.1). Both photons must be present within the intermediate-state lifetime of the SHG process for the nonlinear photon interaction to occur. Again, the removal of closely spaced pairs of events regularizes the photon stream. The process of decimation is defined as every N t h photon (N= 2,3, ...) of an initially Poisson photon stream being passed while deleting all intermediate photons. The passage of every other photon (N= 2) is explicitly illustrated in fig. 1.ld. The regularization effect on the photon stream is similar to that imposed by dead-time deletion. This mechanism can be used when sequences of correlated photon pairs are emitted; one member of the pair can be detected and used to operate a gate that selectively passes every N t h companion photon (see J 6.2.1). Overflow count deletion is another feedback mechanism that can introduce photon anticorrelations. As shown in fig. Lle, the number of photons is counted in a set of preselected time intervals [0, TI, [ T, 2T], . .. ;the fist no photons in each interval are retained and the remainder deleted. If the average number of photons in [ 0, TI of the initial process is % no, then the transformed process will almost always contain no counts per time interval (MANDEL [ 1976a1). Finally, rate compensation is illustrated in fig. 1.If. In this case the (random) number of photons is counted in a short time T I;this information is fed back to control the future rate at which the trigger is pulled. If the random number measured in T , happens to be below average, the trigger is subsequently pulled at a greater rate and vice versa. More generally, each photon registration at time ti of a hypothetical Poisson photon process causes the rate I of the transformed

6

PHOTON BUNCHING A N D ANTIBUNCHING

[I*§ 1

point process to be modulated by the factor h(t - ti) (which vanishes for t < ti). In linear negative feedback the rate of the transformed process becomes pt = 2, - x i h ( t - ti), where A, is the rate of the Poisson process. A variety of techniques can be used to implement rate compensation, such as quantum nondemolition measurements (see $ 6.2.4) or correlated photon pairs (see $ 6.2.2). Dead-time deletion can be viewed as a special case of rate compensation in which the occurrence of each event sets the rate of the process equal to zero for a specified period of time after the registration (SHAPIRO, Ho, KUMAR,SALEHand TEICH[ 19871). SAPLAKOGLU, In the course of this review we will find that an attractive way of producing antibunched and sub-Poisson photons is to regulate the times at which the trigger is pulled, using a control mechanism located right at the trigger (see $ 7). This is most readily achieved by using an electron stream as the trigger and a collection of atoms in a solid as the firing mechanism. An electron stream exhibits natural anticorrelations in the presence of space charge. Because of their mutual Coulomb repulsion, the electrons repel each other so that the trigger can be pulled with a great deal of regularity. Indeed, the behavior of such a stream of electrons can be understood from a mathematical point of view (SRINIVASAN [1965]) in terms of rate compensation with linear negative feedback. In $ 2 we review the theory pertaining to the generation of bunched and super-Poisson light within the semiclassical theory. We elucidate the distinctions among Poisson, super-Poisson, and sub-Poisson photocounts; unbunched, bunched, and antibunched light; and chaotic, superchaotic light. The roles of wavelike and particle-like fluctuations are discussed. Particular attention is devoted to the statistical characteristics of light arising from the superposition of independent emissions where there is randomness either in the number of emissions or in the times at which the emissions are initiated. Light of this nature often arises in luminescence, scattering, and speckle experiments. The quantum theory of light generation from superpositions of independent emissions is developed in 5 3. A stationary stream of antibunched and/or sub-Poisson light can be generated if sub-Poisson statistics are obeyed by both the random times at which the emissions are initiated and the individual photon emissions themselves. For sufficiently large counting times and large detection areas, interferenceeffectsdo not appear, and the photons can be shown to behave purely as classical particles. If the emissions are (deterministic) single photons, the overall photon statistics then directly mimic the statistics of the excitations. Uniquely quantum effects can therefore be observed within the domain of linear (single-photon) optics. The implementation of physical mechanisms that lead

1.8 11

INTRODUCTION

1

to this kind of light is discussed in Q 7. Although the theory presented in 3 3 is geared principally to independent emissions, it can also be applied to certain physical processes that operate on the basis of nonindependent emissions. Several important examples considered in § 6 and Q 7 fall in this category. In Q 4 we demonstrate that the loss of photons generally randomizes the statistical properties of an anticorrelated stream of photons, ultimately converting it into random (Poisson) form. Effects such as attenuation, scattering, and the presence of background photons are deleterious to sub-Poisson light. In Q 5 we discuss two nonlinear optics mechanisms (atomic resonance fluorescence and parametric downconversion) that generate small clusters of conditionally sub-Poisson photons. A cluster may comprise 2 or 3 photons for resonance fluorescence from an isolated atom in a typical experiment, or a single photon for parametric downconversion. This effect can only be observed by gating the detector for a prespecified time window to ensure that only a single cluster is detected. Sub-Poisson behavior is destroyed by the presence of a Poisson stream of clusters, as discussed in Q 3.3.2. Thus unconditionally (cw or stationary) sub-Poisson light cannot be generated without controlling the excitations. Photon feedback is discussed as a means of producing cw unconditionally sub-Poisson light in Q 6. Photons generated by a particular process are fed back to control it, which may be accomplished by the use of a variety of nonlinear optics techniques. Methods using feedback intrinsic to a physical process, simply stated, remove selected clusters of photons from the incident pump beam, leaving behind an antibunched residue. External feedback can also be used. A simple example is a process in which photon pairs are produced (e.g., parametric downconversion),with one member of the pair being used to control its twin. Control mechanisms such as decimation, dead-time deletion and rate compensation, illustrated in fig. 1.1, can be used to achieve these ends. The use of excitation feedback for generating useful antibunched and cw unconditionally sub-Poisson light is discussed in § 7. In this case the radiators are controlled before photon emission. Many of the limitations associated with photon-feedback methods are avoided. The technique operates by initiating single-photon emissions at antibunched and sub-Poisson times. For sufficiently long detection times and areas the emitted photons behave like classical particles; their statistics mimic those of the excitations. Perhaps the simplest example of an excitation feedback scheme is provided by the space-chargelimited Franck-Hertz experiment. The anticlustering properties of an electron stream, resulting from Coulomb repulsion, are transferred to the photons. The direction of transfer is the inverse of that encountered in the usual photo-

8


[I, $ 2

detection process, in which the statistical character of the photons is imparted to the photoelectrons. Any number of solid-state implementations of this concept can be envisaged. Even a simple emitter such as a LED,driven by a constant current source from a battery, w ill emit sub-Poisson photons. Configurations in which the feedback signal is externally carried have also been suggested. Excitation feedback appears to be the most useful scheme for producing useful antibunched and sub-Poisson light. In 8 we consider the use of sub-Poisson light for carrying information, such as in a direct-detection lightwave communication system. As a matter of principle, the channel capacity of such a system, when based on the observation of the photoevent pointprocess, cannot be increased by the use of sub-Poisson light. On the other hand, the channel capacity of aphoton-counting system can be increased by the use of such light. In this latter case, error performance will either be degraded or enhanced by using sub-Poisson light, depending on where the power constraint is placed. Sources of light considered for use in directdetection lightwave communications should be strongly sub-Poisson, exhibit a large photon flux, be small in size, be fast, and produce a collimated output.

6 2. Bunched Light from Independent Radiators The fluctuation properties of light have traditionally been derived from a thermodynamic study of the interacting radiators, atoms and molecules, and the radiated field, treating the source as a continuum. It has been argued that when large numbers of individual radiators are treated discretely, the central limit theorem leads to chaotic behavior. When the number of radiators is small and random, however, it becomes possible to observe deviations from the results predicted by the central limit theorem, and to discern the dependence on the source-number fluctuations. The problem of studying the interaction of light with matter is often complicated by the fact that the radiated light itself affects the radiating matter. This is the case, for example, with black-body radiation, which results from thermodynamic equilibrium between the radiation and matter in a cavity. It is also the case with laser light in which the emitted photons are fed back to interact with the atoms by the continuing processes of absorption and stimulated emission. There are a number of problems in which the source and the ensuing radiation may be separated, however. In these cases it is possible to find an explicit relationship for the radiated field in terms of the characteristics of the source. Consider the following examples: (1) a stream of electrons or photons

1, I 21

BUNCHED LIGHT FROM INDEPENDENT RADIATORS

9

impinges on a phosphorescent screen in which each electron (or photon) produces a packet of radiation in the form of a cluster of photons. The emitted radiation escapes and does not interact with the stream of incoming electrons. (2) A laser beam illuminates a dilute solution of moving particles, resulting in the emission of scattered light. In the first Born approximation an explicit linear relation between the scattered field and the fluctuating density of the medium can be established. The radiated light does not influence the density itself. In this section we discuss the bunching properties of light generated by a collection of statistically independent radiators excited by an external source. It is assumed that the emitted photons affect neither the excitation process nor the emission process (Le., there is no feedback from the emitted photons to the source). Our approach makes use of the semiclassical theory of light; we begin with a brief review of semiclassical coherence theory. The quantum treatment is presented in 3 3.

2.1. SEMICLASSICAL THEORY OF OPTICAL COHERENCE

The semiclassical theory of light treats the radiation field classically while using the quantum theory to describe the interaction of the light with the atoms of the detector. This method has proved to be adequate for a great many purposes (see, for example, SARGENT,SCULLYand LAMB [1974] and MANDEL[ 1976b1). In the confines of this theory, light is represented by means of a random complex analytic signal V(x) [where x is the space-time point (r, t ) ] , whose squared absolute value I ( x ) = I V(x)I2 is the optical intensity (GABOR[ 19461, BORNand WOLF [ 19801). Light fluctuations are completely characterized by the statistics of the stochastic process V(x). A hierarchy of statistical moments and probability distribution functions of V(x)and I(x) for different x are defined (see, for example, MANDELand WOLF [ 19651, SALEH [ 19781 and P E ~ I N A [ 19851). At a space-time point x the most important descriptor is the probability density P ( I ) of the intensity I = I(x),its mean (I ) , and its variance Var(1). The bracket ( * ) denotes ensemble averaging. Fluctuations at two space-time points x1 and x2 are characterized by the amplitude correlation function G(l)(%

x2) =

( ‘v*(x,) V ( x 2 ) )

(2.1)

10

PHOTON BUNCHING AND ANTlBUNCHING

[I, § 2

and the intensity correlation function (also called the second-order correlation function) G‘”(x,,

~ 2 =)

(I(xi)Z(~2))

(2.2)

1

as well as their normalized versions G(Y%, x2)/[

P ) ( I l 9 x2)

=

g(2)(x,,x2)

= G‘”(x1, x2)/[ (I(X1))

(I(x1))

(I(x2)) (I(x2))

9

1

*

(2.3) (2.4)

These quantities are also known as the degrees of first- and second-order coherence, respectively. The degrees of coherence satisfy the following inequalities

0 < Ig‘”(x,,x,)I

[g'2'(x,x) - 11

9

(2.17)

indicating that the bunching/antibunching and super/sub-Poisson attributes of the light are in one-to-one correspondence (TEICH, SALEH and STOLER [ 1983)). 2.1.4. Chaotic light Another important model of light is chaotic (or thermal) light (MANDEL [ 1959a, 19631). It is characterized by an analytic signal V ( x )that is a complex Gaussian stochastic process of circular symmetry in the complex plane. At every point x, its amplitude I VI has a Rayleigh distribution, whereas its intensity I has an exponential distribution (2.18) for which

Var(Z)

=

(Zy.

(2.19)

At pairs of points x1and x2,the normalized intensity and amplitude correlation functions are related by the Siegert relation (SIEGERT[ 19541) g'2'(x1, x2) =

1 + I g y q , x2)12 .

(2.20)

The second term on the right-hand side of eq. (2.20) represents bunching. It is responsible for the effect first observed by HANBURY-BROWN and Twrss [ 1956a,b]; see HANBURY-BROWN [ 19741 for a review.

16


[I, 8 2

Photoelectrons counted in an area A and time interval T have a distribution that can be determined from eq. (2.8). If the time T and area A are saciently small, the integrated intensity W also has an exponential distribution and its Poisson transform yields the Bose-Einstein distribution, (2.21) which is characterized by the variance var(n) = ( n )

+ (n)2.

(2.22)

The first term on the right-hand side of this equation was associated by EINSTEIN[ 19091 with the photon (particle) nature of light, since it has a Poisson particle-like variance (see eq. 2.14). At the same time he associated the second term with the wavelike nature of light because of its exponential wavelike variance (see eq. 2.19). Equation (2.22) is known as Einstein's fluctuation formula for black-body radiation; it provided the first clear indication of wave-particle duality (WOLF[ 19791). In general, for arbitrary A and T, the photocount variance is given by a generalized version of eq. (2.22) that is associated with the negative-binomial rather than the Bose-Einstein distribution (MANDEL[ 1959a1). Using eq. (2.1l), and assuming that the average intensity ( I ( x ) ) is constant within D, we obtain Var(n) = ( n )

+ -.(n>' M

(2.23)

The parameter M, which is known as the degrees-of-freedom parameter [ 19631, P E ~ ~ I [N19851, A SALEH[ 19781, GOODMAN [ 1985]), is given (MANDEL by ,

r

r

(2.24) with D = AT. The corresponding Fano factor is (2.25) For small A and T it is apparent that M = 1 and that the photocounting distribution reverts to the Bose-Einstein distribution. Chaotic light is rendered

1,s 21

17


super-Poisson by the second term comprising the wavelike noise. For large A (relative to the characteristiccoherence area A,) and/or large T(re1ative to the characteristic coherence time T,), M is large so that eq. (2.23) reduces to Var(n) = ( n ) , as for the Poisson distribution. In this limit the wavelike fluctuations are averaged out, leaving only the Poisson particle-like fluctuations behind. Light from a black-body radiator is well described by the thermal model (see, and WOLF[ 19651). Black-body radiation is a result of for example, MANDEL the mutual thermal equilibrium between atoms and photons, in which emitted photons interact with atoms by absorption and stimulated emission processes. Feedback from the radiation to the atoms is an important element of the equilibrium process. It will become apparent in Q 2.2, however, that independent emissions without feedback from radiation to source can also result in radiation with the same properties as thermal light. 2.1.5. Chaotic, superchaotic, and subchaotic light

In view of the importance of chaotic light, we regard it as another benchmark with which light sources can be compared. We coin the term superchaotic to describe light for which g@)(x,x) is greater than 2. Light for which g(2)(x,x) NORMALIZED COINCIDENCE RATE g(2)(xsx) 0

ANTIBUNCHED

I

2

I

I

I

z

(a)

BUNCHED

COUNT FAN0 FACTOR F,(D)

SUPER-POISSON

Fig. 2.2. (a) Regions and boundaries of the normalized coincidence rate g(’)(x, x) that define bunched/antibunched and superchaotic/subchaoticbehavior for photoevents at the space-time point x. (b) Regions and boundaries of the Fano factor F,(D) that define Poisson, super-Poisson, and sub-Poisson behavior for photoevents counted by a detector of space-time volume D (area A and time interval T).

18

[I, § 2


is smaller than 2 is referred to as subchaotic. For photoevents counted by a small detector ( D -+ 0), these regions correspond to FJO) being greater or smaller than 1 + (n), respectively. Superchaotic light will then obviously be superPoisson, but subchaotic light may be either sub- or super-Poisson. Figure 2.2 portrays a schematic illustration of the aforedefined regions and boundaries of the normalized coincidence rate g(*)(x, x) and the count Fano factor F,(D). Representative examples for three different measures of the statistical properties of a light source are illustrated in fig. 2.3: the normalized intensity correlation function (normalized photoevent coincidence rate) g(*)(T) versus the time difference T, the inter-event-time probability density P(T)versus the inter-event time T, and the photoevent counting probability distribution p ( n ) PHOTON STATISTICS

INTER-EVENT TIME

*

.--‘E

COUNTING

-~

Coherent Thermal

Antibunched _-__-_-__-_Sub-Poisson

Fig. 2.3. Representative examples for three measures ofthe statistical properties of a light source: the normalized intensity correlation function (normalized photoevent coincidence rate) g(*)(T) versus the time difference 5, the inter-event time probability density function P ( r ) versus the inter-event time T, and the photon-counting probability distributionp(n)versus the count number n. The thin solid curve, the thick solid curve, and the dotted curve correspond to coherent, thermal (chaotic), and nonclassical light, respectively.

1. § 21


19

versus the count number n. The thin solid curve, thick solid curve, and dotted curve correspond to coherent, thermal, and nonclassical light, respectively.

2.1.6. Inter-event time

statistics

As indicated in the Introduction, there are many measures that can be used to characterize nonclassical light. One measure that has received relatively little attention in the literature is the inter-event time probability density P ( z ) versus z (SALEH[ 1978]), illustrated in fig. 2.3. A measure of this distribution that is analogous to the Fano factor is the coefficient of variation c. which is defined as C =

[var (z)] "2 3

(2.26)

( 7)

where Var(z) and ( z) are the variance and mean of the inter-event time probability density, respectively. Coherent light, associated with the Poisson distribution, displays an exponential inter-event time probability density function, as is well known (Cox [ 19621, PARZEN[ 1962]), so that c = 1. For chaotic light c > 1 (SALEH[ 1978]), as is g(')(x, x) and F,(D). T h e distribution P(z)in this case is said to be overdispersed relative to the exponential (for which c = 1). Within the framework of the semiclassical theory, light cannot be underdispersed relative to the exponential (i.e., it cannot exhibit c < 1). It will become apparent in Q 3, however, that underdispersed light is possible within the quantum theory. 2.2. SUPERPOSITION OF INDEPENDENT EMISSIONS

Many common sources of radiation comprise a number of radiators that radiate independently. Consider N such radiators (as illustrated schematically

Fig. 2.4. Schematic representation of the amplitudes of a number of statistically independent stationary emissions.

20


(135 2

in fig. 2.4). This model of light is suitable for a great variety of situations, including radiation from independent luminescent or incandescent points of a large source, as well as light from independent scatterers such as encountered in speckle (JAKEMAN [ 1980al). The complex analytic signal of the total radiation can then be expressed as the sum of N statistically independent contributions, viz., (2.27) It will be useful to relate the statistics of the total radiation V(x) to the statistics of the individual emissions { V,(x)}. We first consider the case in which N is deterministic, we then extend the results to permit N to be random in the following section. Confining our attention to one space-time point x = (r, t), we write eq. (2.27) in the form N

V=

c V,,

(2.28)

k= 1

where V = V ( x ) and V, = V,(x). For simplicity we assume that the complex random variables V, have circular symmetryin the complex plane. (This applies when the real and imaginary parts of each of the random variables V, are independent and identically distributed, or when the phase is uniformly distributed.) It follows that the mean values, as well as all odd-order moments, of V, are zero. It also follows that the sum V(n) is circularly symmetrical. The determination of the statistics of the sum V of a set of statistically independent phasors V,, whose phases are uniformly distributed, is the same as the well-known random walk problem. Its solution dates back to Lord Rayleigh. More recent work (in the context of light scattering) includes contributions by TROUP[ 19651, MITCHELL[ 19681, CHENand TARTAGLIA [1972], SCHAEFERand PUSEY[1972], PUSEY,SCHAEFERand KOPPEL [ 19741, BARAKAT[ 1974, 19761, BARAKATand BLAKE [ 19761, SCHAEFER [ 19751, JAKEMAN and PUSEY[ 19761, and ODONNELL [ 19821. The moments of Vmay be related to those of V, by the straightforward (but rather lengthy) use of eq. (2.28) in the definition of the moments, exploiting the property of statistical independence and the fact that the odd moments vanish. The simplest moments of I = I V I are

'

(2.29)

I, 8 21


21

(2.30)

where

(2.31) (2.32)

(2.33)

The parameters yk and yrepresent deviations from chaotic behavior for the kth emission and for the total field, respectively (see eq. 2.19). The symbol y (with different modifiers) will be used to denote deviations from chaotic properties throughout this chapter. We now move on to two space-time points xl, x2. The correlation functions of the total field may be related to the correlation functions of the individual emissions by the following relations, which can be obtained by systematic algebraic manipulations using the assumptions of circular symmetry and statistical independence of the different emissions: (2.34) (2.35)

where (2.36)

and (2.38)

Here gy)(xl,x2) and g‘,)(x,, x2) are the degrees of first-order coherence and degrees of second-order coherence for the kth emission, respectively. The functions y(x,, x2) and yk(xI, x2) represent deviations from chaotic behavior of the superposed light and of the kth emission, respectively (see eq. 2.20). These functions may attain positive or negative values.

22


[I, 5 2

Although it is difficult to obtain expressions for the photocount probability distribution for a detector of arbitrary area and arbitrary counting time, an expression for the photocount variance can be easily obtained by use of eq. (2.11). The corresponding Fano factor turns out to be

where 1/Mis given by eq. (2.24), and 7is determined from a similar expression: 1

(2.40)

The term ( n ) r represents the excess nonchaotic contribution to the Fano factor. For small T and A (i.e., D -+0), 7 = y = y(x, x). Since M = 1 we then have F,(O)= 1

+ (n) + (n)y.

(2.41)

2.2.1. Chaotic emissions If the individual emissions are chaotic, it is evident from eqs. (2.3 l), (2.33), (2.36) and (2.37) that yk(xl, x2), y(x,, x2), and y vanish; furthermore Var(I)

= (I)2,

g'2'(x1,x2) = 1

+ 1g(')(x1,x2)12,

indicating that the resultant total field itself becomes chaotic. This is not surprising. When the individual emissions are Gaussian (i.e., V, are Gaussian), the sum V is also Gaussian (because the sum of Gaussian random variables is Gaussian).

2.2.2. Statistically identical emissions Assume now that the emissions are statistically identical and stationary. An individual emission is described by the mean and variance of its intensity, (I, ) and Var (I,),and its correlation functions g&I)(x,, x2) and gb2)(x,,x2), where the subscript 0 denotes an individual emission. The deviation from chaotic

1 3 8


21

23

behavior is then described by the parameters (2.42)

(2.44)

If a denotes the number of photocounts associated with an individual emission in a region D, its mean is

whereas its Fano factor is (a) Fa@)= 1 +-+

M

(a)To,

(2.46)

where (2.47)

The corresponding parameters for the superposed radiation are obtained from eqs. (2.29)-(2.33): ( 1 ) =N(lo>

9

;I

Var(I)= ( I ) , 1 + -

[

,

(2.48) (2.49)

g(2)(x,, x,) = 1

+

(2.50) t--.( n > 70

N

(2.51)

24


[I. I 2

A comparison of these properties with those of chaotic light leads us to observe that the deviation from chaotic behavior is inversely proportional to the number N of superposed emissions. As N 4 00, the superposed light tends toward chaotic behavior whatever the statistics of the original emissions. This too, in fact, is expected as a result of the central limit theorem (MIDDLETON [ 1967a,b]). A theory of chaotic radiation has been constructed in terms of superpositions of discrete independent radiators (BERANand PARRENT [ 19641, KARP,GAGLIARDI and REED[ 19681, LOUDON[ 19831). Light emitted by a collision-broadened source has been modeled as a wave broken into discrete sections, each with fixed phase. The phase is assumed to change randomly when a collision occurs, and the intercollision times are taken to be random (exponentially distributed). In the limit of a large number of collisions, the result has been shown to approach chaotic behavior (LOUDON[ 19831). The terms (l/N)y,(x,, x2) and (l / N ) T o( n ) represent an excess bunching and an excess Fano factor above and beyond the chaotic values. These terms may also be negative, resulting in a reduction of bunching below the chaotic level, as will become apparent in the examples below. Deviations from chaotic behavior can be observed when N is not large (FORRESTER [ 19721, KARP [ 19751, TEICHand SALEH[ 1981a1, SALEHand TEICH[ 19821, SALEH,STOLER and TEICH[ 19831). Equation (2.51) can also be written in terms of the Fano factor of the photoevents for an independent emission,

(2.52) indicating that the excess Fano factor (above chaotic behavior) for the total radiation is equal to that for a single emission. When the degrees-of-freedom parameter Mis sufficiently large, as is the case for a large detector areaA and/or a large counting time T, so that the wave interference terms ( n ) /A4and ( a ) / M are negligible, we obtain

that is, the Fanofactor for the totalphotocount is approximately equal to the Fano factor for the single emission count. This result is obtained by simply treating photoevents as particles, arguing that the total number of photoevents is simply the sum of N independent numbers a,, a*, ...,aN of photons associated with the N emissions, each of mean ( a ) and Fano factor F,(D).

1, B 21


25

The way in which the probability distributions P(Z) and p(n) approach the distributions associated with chaotic light, as N increases, has been studied by a number of authors (for a review see BARAKATand BLAKE[ 19801). Example: Identical emissions of deterministic intensity. When the individual emissions have deterministic intensities, we have =

1

Yo = 7 0

=

lgt)(x,, x2)l = gS"(x1, YO(X,Y x2)

M

=

1,

=

-1

Y

F,(D)

x2)

Y

-1

9

1.

=

The total radiation exhibits the following properties :

;),

= (Z>2

(1 -

g(2'(x,, x2) = 2

1 - -,

Var(Z)

N

(2.54)

(2.55)

(2.56) It is evident that the superposed light is subchaotic. For N > 1, g(')(x, ,x2) > 1 and F,(D)> 1, indicating that the superposed light remains bunched and super-Poisson (as expected for semiclassically described light). Again, eqs. (2.19) and (2.20) are reproduced in the limit of large N, and chaotic behavior results.

2.3. NUMBER FLUCTUATIONS

When the number of emissions is itself random, the statistical averages considered in 5 2.2 should be regarded as conditioned on a fixed value of N . A subsequent average over the fluctuations of N is then required to yield the overall averages. This problem has been studied by a number of authors using different distributions of N . The model is applicable to radiation emitted from an ensemble of atoms that are excited at random (FORRESTER [ 19721, KARP, GAGLIARDI and REED [ 19681, KARP[ 19751) and to light scattering from a random number of scatterers (JAKEMAN [ 1980al).

26


2.3.1. Poisson numberfluctuations We assume that the number of statistically identical emissions is Poisson distributed with mean ( N ) ,

This problem has been studied by BARAKATand BLAKE[1976], PUSEY, SCHAEFERand KOPPEL [1974], SCHAEFERand PUSEY [1972], and [ 19751. Averaging the statistical moments derived in 3 2.2.2 over SCHAEFER the fluctuations of N gives rise to

(2.58)

(2.59) where

(2.60) Equation (2.59) can also be written in the form

F,(D) = 1

+ ( n ) + [F,(D) - 1 + ( a ) ] . ~

M

(2.61)

The excess coincidence rate above that for chaotic light [the third term of eq. (2.58)] is proportional to the normalized coincidence rate for a single emission, which is always nonnegative. The light is therefore superchaotic. This is to be compared with the deterministic-N case, for which the excess coincidence rate is proportional to the excess coincidence rate of an individual emission (eq. 2.50), a term that may be positive or negative. When the number of emitters N is Poisson distributed, the light is superchaotic; when N is deterministic, the light is merely bunched. We now compare the Poisson-N photocount Fano factor, given in eqs. (2.59)-(2.61), with the deterministic-N Fano factor, given in eqs. (2.51) and (2.52). The positive sign in the right-most term in eq. (2.61) reflects an increased

1, § 21


21

Fano factor resulting from randomness in the number of emissions. This distinction is important inasmuch as N is random for most natural sources of light. Example 1 Identical emissions of deterministic intensity. When the individual emissions have deterministic intensities, as in the example considered in 5 2.2.2., we have g‘*’(x,, x*) = 2

F,,(D)

=

+

1+ (n)

1

(2.62)

~

(N)’

( + (3 1

(2.63)

~

These results should be compared with those presented in eqs. (2.55) and (2.56). The light is obviously superchaotic and the photocounts are superPoisson. As the mean number of emissions ( N ) co, eqs. (2.62) and (2.63) approach their chaotic limits. It is of interest to note that this asymptotic limit, which applies when N is Poisson, does not necessarily apply for numbers of emissions governed by other distributions. The case where N is distributed in accordance with the negative-binomial (rather than the Poisson) distribution, for example, has been considered in some detail (JAKEMANand PUSEY [ 19781). If the intensity of the individual emissions is assumed to fall off in proportion to 1/( N ) , then in the limit ( N ) -+ oc) the amplitude of the total field is governed by a distribution proportional to a modified Bessel function of the second kind (the so-called K-distribution), rather than by the Rayleigh distribution as for chaotic light, regardless of the distribution of the individual emissions (JAKEMAN and PUSEY[ 1976, 19781, JAKEMAN[ 1980a,b]). This model has been applied to the study of non-Rayleigh scattering from diffusers by many authors (e.g., HOENDERS,JAKEMAN, BALTESand STEINLE[ 19791, EBELING [ 19801, JAKEMAN [ 1982, 19831, JAKEMAN and HOENDERS [ 19821, O’DONNELL [ 19821, OLIVER[ 19841). The K-distribution has also been applied to the study of laser light propagating through the atmosphere (PARRYand [ 19861). PUSEY[ 19791, ANDREWSand PHILLIPS -+

2.4. EMISSIONS INITIATED AT POISSON TIMES

The independent-radiator model considered in 9 2.2 and 5 2.3 was formulated under the assumption that the number of radiators is a random variable

28


SECONDEMISSION t2

V

)

z

A

P (0 2

A---

5s LL-

t..

OZ

B

z z

I

I I

I I

>

Fig. 2.5. Schematic representation of the amplitudes { V,(r)} of a sequence of short-duration emissions initiated at times { t , ,t,, ...,t,, ... } described by a Poisson point process. The function N ( t ) is a stochastic counting process representingthe number of radiators that have begun to radiate prior to time 1.

N (N deterministic is a special case). This model is not applicable when the number of active radiators is a random process, that is, a stochastic function of time N(t). We now proceed to this situation. Consider radiation formed by a sequence of short-duration emissions initiated at the times { t , , t,, . ..,tkr . . . }. Let N ( f ) represent the number of radiators that begin to radiate prior to time t, as illustrated in fig. 2.5. The function N ( t ) represents a stochastic counting process with jumps at the times { t , , t,, . ..,t k , . . . }, where tk is the time at which the kth emission is initiated. We limit ourselves to the case for which N ( t ) is a homogeneous Poisson counting process of rate p (events per second). We further assume that the emissions are random, statistically independent, and identically distributed (except for the fact that they begin at different times). The complex analytic signal of the total field can be written as the sum

(2.64) k = --cu

1, § 21


29

where Vk(r, t - t k ) represents the kth emission. Each individual emission is assumed to have known mean intensity (Io(r, t ) ) , amplitude correlation function Gg)(r,, t , , r,, t,), and intensity correlation function Gg)(r,, t , , r,, t,). We wish to determine the statistics of the overall radiation, particularly the correlation functions G(')(r,,r,, z) and G(,)(r1, r,, 2). It is important to note that although the individual emissions are nonstationary, the overall field is stationary because of the stationarity of the underlying emission-time point process. This is why the correlation functions of the total field are written as functions of z = t, - t,, rather than functions oft, and t, separately. The function V(r, t ) can be regarded as a filtered marked stationary Poisson process (SNYDER[ 19751). Using the properties of shot-noise processes (RICE [ 19441, GILBERTand POLLAK[ 19601, PAPOULIS[ 1984]), the total radiation can be shown to have the following statistical properties (SALEH,STOLERand TEICH[ 19831): (2.65)

(2.66)

d2)(r1,r,,

z) = 1 + I P ( r , , r,,

211,

+ y(r,, r,,

21,

(2.67)

where the excess normalized coincidence rate

(2.68) is now given by an average over the normalized coincidence rate of an individual emission. The photoevents have a normalized coincidencerate given by eq. (2.67) and a photocount Fano factor, determined by the use of eqs. (2.1 1) and (2.15), which is F , ( D ) = l + - +((n(>n ) T ,

M

(2.69)

where 7 is the average of y(r,, r,, z) as given by eq. (2.40). The result is similar to that obtained for the addition of a Poisson number of identical emissions (see $ 2.3.1).

30


2.4.1. Quasi-coherent emissions As a particular example, we assume that each emission is represented as a pulse of deterministic field with random phase. For simplicity the spatial dependence is ignored by limiting our interest to a single point r. The individual emissions may then be expressed in the form Vk(t - t k ) = Vo(t)exp(tt$)

.

(2.70)

The phases 0, are drawn from a uniform distribution. This is the model used by SALEH,STOLERand TEICH[ 1983, Sec. IIA]. The results are

(2.71)

G")(z)

=

p

Iom + V,*(t)Vo(t

z) dt ,

(2.72) (2.73)

where

(2.74)

The quantity zp is the characteristic decay time of the intensity of an individual emission, Io(t) = I Vo(t)I2.The quantity

(2.75) depends on the normalized time-averaged autocorrelation function of the intensity of an individual emission

I,(?) =

+ . Iom Zo(t) lo(t

z) d t

(2.76)

1, § 21


31

Comparing eqs. (2.73) and (2.58) leads us to see that the parameter pzp can be associated with the average number of independent emissions overlapping at a given time, that is, ( N ) = pz,. This interpretation is sensible in view of the strong underlying similarity between the two models, although in the case considered here the contributions are not strictly identical because the emissions are initiated at different times. The radiation emanating from such a process is superchaotic (as is the radiation modeled in J 2.3). In addition to the usual chaotic fluctuations manifested by the first two terms on the right-hand side of eq. (2.73), the third and term represents excess bunching. It is directly proportional to inversely proportional to pclzp, the average number of emissions per emission lifetime. The third term therefore becomes significant when pzp 4 1, that is, when the emissions are sparse and seldom overlap. This result is similar to that obtained by LOUDON[ 19801. On the other hand, when pzp % 1, that is, when the emissions overlap strongly, V(t) approaches the complex Gaussian process and that is characteristic of a chaotic field (PICINBONO,BENDJABALLAH POUGET[ 19701); in this case the third term becomes neghgible and the results for chaotic light apply (CARMICHAEL and WALLS[ 1976a,b]). We now examine the photocounting properties of such light, assuming a point detector with counting time T, quantum efficiency q, and incremental area AA (which is here taken to be unity for convenience). The mean photon count is

a2)(z)

( n > = P(U)TY

(2.77)

where (2.78)

can be interpreted as the average number of photocounts per emission, and pT is the average number of emissions initiated within T. The Fano factor, which depends on the counting time T, can be written in the form (2.79)

where (2.80)

32


T/

T~

Fig. 2.6. Dependence of the degrees-of-freedom parameters M and A on the ratio T / r , for exponentially decaying coherent emissions. M is the usual degrees-of-freedom parameter for chaotic light, whereas A is the special degrees-of-freedom parameter for shot-noise light. The dependences of M and d o n the ratio T / r , are almost symmetrically opposite. The dashed lines represent unity slope. (After SALEH,STOLERand TEICH[ 19831.)

It is also convenient to write it in the form

F,(T)= 1 +-++,

M

where A - '

=

A

(2.8 1)

ToT/zp,from which (2.82)

It is now apparent that the third term on the right-hand side of the expression for the Fano factor in eq. (2.81) is independent of the mean count ( n ) . It is therefore of the same nature as the first term; both represent particle-like noise as opposed to the wavelike noise represent by the second term. Further evidence of this is centered on the degrees-of-freedom parameters. The parameter M in eq. (2.81) is the usual degrees-of-freedomparameter for chaotic light with amplitude correlation function g(')( z), whereas the parameter A in this equation is a special degrees-of-freedom parameter that has been recently introduced to describe light whose intensity fluctuates in accordance with a shot-noise random process (TEICHand SALEH[1981a], SALEHand TEICH [ 19821, SALEH,STOLER and TEICH[ 19831). These parameters are displayed in fig. 2.6 for exponentially decaying coherent emissions. The dependences of M and d o n the ratio T/zp are almost symmetrically opposite. When T 4 zp, M 2: 1 while d z zp/T % 1. As T/zpincreases, Mincreases while A decreases. In the limit T % zp, M 2: T/zp 9 1 and A = 1.

ANTIBUNCHED LIGHT FROM INDEPENDENT RADIATORS

1.8 31

33

The value of the ratio T/zp affects the Fano factor dramatically. For T Q zp, F,(T)= 1 + ( n )

(n> +-= p=P

1+ (n)

+ ( a ) - T.

(2.83)

TP

As T/zp-,0, FJT) approaches 1 + ( n ) , which characterizes the Bose-Einstein distribution. On the other hand, for T $- zp, F,(T)

=

1+ (a> ,

(2.84)

which characterizesthe Neyman Type-A (NTA) distribution. This distribution is obtained when each of a Poisson-distributed number of primary events contributes a Poisson-distributed number of secondary events and the total number of events are counted (TEICH[ 19811, SALEHand TEICH[ 1982,19831, SALEH,STOLER and TEICH[ 19831). In our case the primary events correspond to the emissions and the secondary events correspond to the photoevents associated with each emission. The processes associated with light of this nature (shot-noise light), as well as the ensuing photocounting distributions and pulse-interval distributions, have been studied in great detail in a number of and TEICH[ 19811, papers (TEICHand SALEH[ 1981a,b], SALEH,TAVOLACCI SALEHand TEICH[ 1982,19831, MATSUO,TEICHand SALEH[ 19831, SALEH, STOLERand TEICH[ 19831, TEICHand SALEH[ 19871). Spatial effects have not been considered in this presentation. A more general treatment has been provided by TEICH,SALEHand P E ~ I N[A 19841, in which each individual emission is assumed to originate at a random position within a source of finite volume. This model leads to generalized versions of eqs. (2.65)-(2.69). Expressions have been derived for the photoevent coincidence rates at arbitrary points located in a detection plane that is at a specified distance from the source, as well as the photocount Fano factor for a detector of arbitrary area. The calculations lead naturally to the introduction of a spatial degrees-of-freedom parameter M,, which plays a role analogous to the temporal degrees-of-freedomparameter M in determiningboth the coincidence rate and the Fano factor.

0 3. Antibunched Light from Independent Radiators In 0 2 we demonstrated that light composed of many independent emissions tends toward chaotic behavior; the photoevents are bunched and the photocounts are super-Poisson. The individual emissions themselves were constrained to be either unbunched or bunched, since the semiclassical theory of light does not admit antibunched emissions.

34


[I.

53

In this section we address the same situations - a superposition of identical independent emissions, and a superposition of identical independent emissions initiated at random times. But we now use a quantum formulation of the problem, which allows the individual emissions to be antibunched and subPoisson. When the emission times are Poisson, we will see that the superposed radiation is again bunched and super-Poisson, approaching chaotic behavior when the emissions overlap strongly. We will conclude that sub-Poisson light cannot be generated by a collection of independent emissions initiated at Poisson times, even ifthe emissions are deterministic. The statistics of the times at which the emissions are initiated must be rendered sub-Poisson in order to generate antibunched and/or sub-Poisson light.

3.1. QUANTUM THEORY OF OPTICAL COHERENCE: A BRIEF REVIEW

In the quantum theory of coherence (GLAUBER [ 1963a,b], LOUDON[ 19831, P E ~ I N[A1985]), amplitude and intensity correlation functions are defined in terms of the positive- and negative-frequency parts of the optical electric field operator, k (x) and k - (x), respectively. The first- and second-order correlation functions (or coherence functions) correspond to the quantummechanical expectation values +

G(')(x,,x2) = T r { ~ k - ( x , ) k + ( x , > } ,

G(2)(x,,x2) = Tr{ ak - (x,)

k - (x2) k

a

+

(3.1) (x2) k (x,)} , +

(3.2)

respectively. Here is the density operator of the field. Normalized versions of these functions, g ( ' ) ( x , ,x2) and g(')(x,, x2), are defined in analogy with the classical theory (see eqs. 2.3 and 2.4) and these go by the same names. The probability distribution of the number of photon counts collected by a detector of area A , in the time interval T, is (KELLEYand KLEINER[ 19641, LAX andZWANZIGER [ 19731, SHAPIRO,YUEN andMACHAD0 MATA[ 19791,YUEN and SHAPIRO[ 19801, SHAPIRO[ 19851)

where

&'=

q

!D

k - (x) k

+

(x) d x .


35

Here : : denotes normal ordering and time-ordering (LOUDON[ 19831). D again denotes the region r E A and t E [0, TI, and q denotes the detector quantum efficiency. The photon-count mean and Fano factor (ratio of count variance to count mean) can be obtained from the coherence functions by use of relations that are identical to those of the semiclassical theory, eqs. (2.9) and (2.16), which we repeat here for convenience: (n)

= tj

lD

G")(x, x) d x ,

(3.4)

where D = AT. We have assumed that ( I ( x ) ) is constant within D. When the detector area and counting time are sufficiently small (D + 0), this reduces to

[F,(O)

-

11

=

( n > [g'2'(x, x) - 11

9

(3.6)

which is identical with eq. (2.17). Again, we assume that there are no feedback paths from detector to source. A quantum theory of detection that is valid in the presence of such paths has recently been developed (SHAPIRO, SAPLAKOGLU, Ho, KUMAR,SALEHand TEICH[ 19871). The difference between the semiclassical and quantum results lies in the procedures used to calculate G(')(x,, x2) and G(')(x,, x2) and in their physical interpretation. As in the semiclassical theory, qG(')(x,x ) ATAA represents the probability that a photoevent is detected within an incremental area AA and incremental time AT surrounding a point x. Likewise, as discussed in Q 2.1, g(2)(x,,x2) represents the normalized coincidence rate for two photoevents at x, and x2. However, in the quantum theory, g(2)(x,, x2) is no longer defined or interpreted as a normalized statistical correlation function of the optical intensity. It is therefore no longer bounded by the inequalities satisfied by classical correlation functions, viz. eqs. (2.6) and (2.7). The function g(')(x, x) can dip below unity, thus violating eq. (2.6). Antibunched photon registrations are consequently possible (see the reviews of LOUDON[ 19801 and PAUL [ 19821). The correlation function is required to be nonnegative, however (GLAUBER [ 1963a1). If g(2)(x,,x2) falls below unity over some subregion of D, the integral in eq. (3.5) can become negative and result in a photon-count Fano factor below unity, that is, sub-Poisson photocounts. Photon antibunching and sub-Poisson photon counting statistics are among a handful of optical phenomena that require the full quantum theory of light for an explanation.

36

tL§3


In the semiclassical theory, when there is no feedback from the detector to the source, photoevents occur in accordance with a doubly stochastic Poisson point process. In the quantum theory this is no longer the case. Photoevents follow a more general self-excitingpoint process (SHAPIRO[ 1985]), which may be characterized by its multicoincidence rates (see, for example, P E ~ ~ I N A [ 19851).

3.2. SUPERPOSITION OF INDEPENDENT EMISSIONS

An optical field generated by N independent emissions may be described by the superposition

2

+

(x)

=

c k; (x) ,

(3.7)

k

where 2; (x) is the positive-frequency part of the field operator for the kth emission. This, in turn, may be written in terms of the annihilation operators 8, of the different radiation modes as

The functions { V,,(x)} are assumed to be random classical functions with Vkl(x) and Vk,,(x)(corresponding to the kth and k'th emissions) taken to be statisticallyindependent. This defines what we mean by independent emissions. Therefore, in addition to the usual quantum-mechanical randomness embedded in the properties of the operators there is a distinct classical randomness that is associated with the stochastic functions { V,,(x)}. When determining the optical coherence functions, we must average over both kinds of randomness; therefore,

{a,},

G(')(x,, x,)

=

(Tr { &J? - (xl)

k

+

(x,)} )

,

G(,)(x, ,x, ) = ( Tr { & k - (xl ) I?- (xz) k (x, +

(3.9)

)k

+

(xl ) } ) ,

(3.10)

where ( * ) designates an average over the fluctuations of the classical functions V,,(x). In the following we assume that the { V&)l have zero mean, that is, we restrict ourselves to states for which (Tr { aE; (x)} ) = 0, so that the mean values of optical fields vanish. This restriction is convenient in simplifying the algebraic form of the results. We are now in a position to determine the coincidence rate and Fano factor for photoevents associated with the total field. It can be shown that the

1,s 31

ANTIBUNCHED LIGHT FROM INDEPENDENT RADlATORS

31

coherence functions of the total field G(’)(x,, x2) and Gc2)(xl,x2) are related to those of the individual emissions GL1)(x,,x2) and GL2)(xl,x2) by the same formulas that we reported in 2.2 using the classical formulation. This is not surprisingbecause those relations were derived with the benefit of only classical arguments. For identical emissions these relations are as follows (see eqs. 2.43, 2.44, 2.49-2.52): g(l)(x,Y

$1

= g61’(x,3

x2)

(3.12)

M

F,(D)- 1 - -( a ) ] , M

(3.13)

(3.15)

The quantity M is the degrees-of-freedom parameter in eq. (2.47) and ( a ) is the mean number of photoevents associated with an individual emission. Equation (3.11) gives g‘2’(x, x) = 2

+ -1 [gf’(x, x) - 21 , N

which was also obtained by LOUDON [ 19801 and applied to radiation from a few atoms in resonance fluorescence. When using the preceding relations, the coherence functions of the individual emissions, g&’)(x,,x2) and g&2)(x,,x2), must be calculated by using quantum rules.

38


[I, § 3

3.2.1. Quasi-coherent single-mode emissions Consider a simple example in which the individual identical emissions are single-mode emissions (TITULAER and GLAUBER [ 19651) characterized by

;?!,

(x)

=

v0(x) exp (i 0,) ii

(3.16)

where Vo(x) is an arbitrary deterministic function, the 0, are statistically independent and uniformly distributed random phases, and d is the annihilation operator for that mode. The individual emissions then have the following statistical averages: ldbl’(x1Y

x2)I

=

g t v x , , x2) = 2

1

Y

+ Yo

Y

(3.17)

(3.18) where ( a ) = ( n ) /N is the mean number of photoevents per emission and Yo =

(iitiitiiii) -2 (iitii)2

(3.19)

is a parameter that depends on the quantum state of the individual emissions (6 is the annihilation operator). Note that all averages are independent of x1 and x2 because of the single-mode assumption. It follows that the total radiation has the following properties (3.20)

=l+(n>+[F,(D)-l-(a)]=F,(D)+(N-l)(a). (3.21) Again, the deviation from chaotic behavior is inversely proportional to the number of emissions Nand is directly proportional to the deviation of the Fano factor for the individual emissions from the chaotic value. If the individual emissions are themselves chaotic [ y o = 0 or Fa@) = 1 + ( a ) 1, the overall light is chaotic. If the individual emissions are subchaotic but super-Poisson [ - 1 < yo < 0 or 1 < Fa@)], the overall light is also subchaotic and super-

1,

s 31

39

ANTIBUNCHED LIGHT FROM INDEPENDENT RADIATORS 9(2) F.

1

2

3

4

5

N

Fig. 3.1. N-dependence of the statistical properties of light that is composed of a superposition of N statistically independent and identical emissions: (a) normalized coincidence rate g(2)(x,x); (b) Fano factor F,(D). ( a ) (the slope) and F, represent the mean number of photons and the Fano factor for an individual emission, respectively. The statistics change from antibunched and sub-Poisson to bunched and super-Poisson as N increases above 1.

Poisson. If the individual emissions are Poisson [ y o 1

g(2)(x,, x2) = 2 - - ,

N

F,(D)= 1 + ( n )

( - -it-) , 1

=

- 1 or Fa@)

=

11, then (3.22)

(3.23)

reproducing the classical results in eqs. (2.55) and (2.56). Finally, if the individual emissions are sub-Poisson [Fa@) < 11, the overall light is subchaotic but super-Poisson if N > 1 + [ 1 - Fa@)]/( a ) . The dependence of g(’)(x, x) and F,(D) on N for this case is illustrated in fig. 3.1. As an example, consider individual emissions described by the one-photon number state ( a = l), for which F,(D) = 0 and yo = - 2. Then, g(2)(x,,x2) = 2

F,(D)

=

L

- -,

N - 1.

N

(3.24) (3.25)

For N = 1 (a single one-photon emission) the light is antibunched and subPoisson (by assumption). For N = 2 the light is unbunched, since gc2)(x,,x2) = F,(D) = 1. For N > 2 the light is bunched and super-Poisson. To conclude, when N statistically independent identical antibunched emissions are superposed, the resultant light has a normalized coincidence rate g(2)(x,x) that increases with increasing N, making the superposed radiation bunched even for modest N . It also has a photocount Fano factor that increases linearly with N, so that for sufficiently large N ( N > 2 in the single-photon emission case) the overall light becomes super-Poisson.

40


3.3. EMISSIONS INITIATED AT POISSON TIMES

We now examine the case of statistically independent identical emissions initiated at Poisson random times. We have already developed the main equations that relate the statistical properties of the superposed radiation to those of the individual emissions (see eqs. 2.65-2.68). Those equations are also applicable here, provided that the correlation functions of the individual emissions are determined using the quantum theory of light.

3.3.1. Quasi-coherent single-mode emissions Assume again that the individual emissions are in a single mode described by

(3.26) where Vo(t)is a deterministic pulse of intensity Zo(t) = I Vo(t)12,the 6, are random independent phases, and a is the annihilation operator of the radiation mode. Spatial effects are ignored for simplicity. An individual emission initiated at t = 0 has the following properties: gf'(t, t

+ 7) = 8,

(3.27)

(a) =

( B t B ) qjomZo(t)dt,

(3.28)

F,(m)

=

1 + ( a > ( B - 11,

(3.29)

where a is the total number of photoevents associated with an individual emission (over t E [ 0, co 1) and the coincidence rate of its photoevents fl is given by

(3.30) To determine the properties of the superposed radiation, we substitute in eqs. (2.72), (2.73) and (2.79) to obtain

(3.31)

1, I 31


41

(3.32)

F,(T)

=

( n > + ( n ) -, rLl 1+ M

(3.33)

PTP

where gb2)(z)= /?To( 2)/TO(O), To(z) being the time-averaged correlation function of the intensity of an emission pulse (see eq. 2.76), and (3.34) as in eq. (2.80). Equation (3.33) can be written in the alternate forms (3.35)

where A is the degrees-of-freedom parameter given by eq. (2.82). Equation (3.36) is the relation between the Fano factor of the total radiation and that of an individual emission. Note that eqs. (3.31)-(3.36) are the same as the results obtained from the semiclassical theory, except for the factor p, which depends on the quantum state of the individual emissions. For a quasi-coherent state, F,(oo) = 1 and p = 1, so that the classical results are reproduced exactly. For a chaotic state, F,(oo) = 1 + ( a ) and /?= 2, corresponding to superchaotic light. The state that is likely to generate the least bunching is the one-photon number state (i.e., a single-photon emission), for which ( a ) = 1, Fa(co) = 0, and /3 = 0. As evidenced by eqs. (3.32)-(3.34), even in this case the stream of emitted photons remains as bunched as chaotic light. If the observation time is long (T %- TJ, then M = T/zp%- 1 and " M m 1. In that case Fn(T)

(a>

+ Fa(a).

(3.37)

This is the result that would be obtained by ignoring photon interference, treating photons purely as particles, and arguing that n is the sum of a Poisson-distributed random number m of emissions, each containing a random

42


[I, s 3

number a of photons. For example, if each emission contains a single photon, then ( a ) = 1, F,(co) = 0, and F,(T) = 1, indicating that the resulting photon counts are Poisson. This is not surprising. We have a stream of single photons emitted about Poisson times; a Poisson number of photons is observed. 3.3.2. Radiation from an atomic beam Consider radiation emanating from a beam of atoms moving with a constant velocity u, monitored through a window of length d. The total radiation may be described by

3

+

(t)=

13 :

(t - tk)RT(t - tk) ,

k

where Ek(t) represents the kth emission, tk is the time at which the kth atom enters the window region, RT(t)is a rectangular function of value 1for t E [0, TI and zero elsewhere, and T = d/u is the atomic transit time across the window. The length d may represent a scattering region (radiation is then produced by scattering from the incoming atoms) or a region within which atoms are excited by an energy source to undergo luminescence or fluorescence. If it is assumed that the times of atomic entries into the window region are describable as a homogeneous Poisson point process (of rate p), then the theory presented in $ 3.2 (eqs. 3.11-3.15) can be used to provide

=

elsewhere,

0,

where ghz)(z) is the normalized coincidence rate for radiation from a single atom (assumed to be stationary). In particular, g‘Z’(0) = 2

1 + -gg62’ (0) PT

This result is consistent with the results of f 2.3, if pT = ( N ) is taken to be the average number of radiators. It is immediately obvious that the total radiation is not only bunched but it is superchaotic, regardless of the nature of the individual atomic emissions.


43

This reveals the root of the difficulty in obtaining sub-Poisson light from sub-Poisson atomic emissions, for example, atoms undergoing resonance fluorescence (KIMBLEand MANDEL [ 19761, CARMICHAEL and WALLS [ 1976a,b]). The antibunching and sub-Poisson properties of the resonancefluorescencephoton clusters cannot be expressed unless the randomness in the number of radiators is removed (JAKEMAN, PIKE, PUSEYand VAUGHAN [ 19771). This problem may be alleviated to some extent by gating the detector, as implemented by SHORTand MANDEL[ 19831, but the best that can then be achieved is the generation of conditionally sub-Poisson photon clusters. This issue will be addressed in 0 5 . In summary, it is apparent that independent emissions initiated at Poisson times stand no chance of producing unconditionally antibunched or subPoisson photons. The best they can achieve is unbunched, Poisson behavior. One effective way to alter the situation is to initiate the sub-Poisson emissions at sub-Poisson times (TEICH,SALEHand STOLER[ 19831, TEICH,SALEHand PERINA[ 19841). This is discussed from a mathematical point of view in the next subsection and from a physical point of view (in terms of excitation feedback) in 0 7. More delicate implementations may be provided by the methods outlined in 5 6.

3.4. EMISSIONS INITIATED AT SUB-POISSON TIMES

Consider the superposition of a sequence of statistically independent and identical emissions initiated at times { t , , t2, .. ., tk, . ..} described by an arbitrary stationary point process (no longer restricted to Poisson). This is referred to as the excitation process because these times will be determined by an excitation mechanism, as will become apparent in 0 7. Even though the individual emissions are nonstationary (typically taking the form of pulses lasting a short time), the overall radiation is stationary because of the assumed stationarity of the excitation process. 3.4.1. Characterization of the excitation point process Two of the most important descriptors of a stationary point process are the rate p (events per second) and the rate of coincidence p2gL2)(z)of pairs of events at times separated by z. These descriptors are not sufficient to characterize an arbitrary point process completely (Cox [1962], SNYDER[1975], SALEH[ 19781); in general, knowledge of the probability of multicoincidences

44


[I, 0 3

of events at k points, for k = 1,2,. .., 00, is required. If m is the number of events that occur in a time interval [0, TI, then its mean is <m>

=PT

(3.38)

and its Fano factor (ratio of variance to mean) is (3.39) where

Mi

' = 1T

jOm ): (1

-

[gk2)(z) - I] d z

(3.40)

The simplest example is the Poisson point process, for which gi2)(T ) = 1 and F,(T) = 1. If gL2)(0)< 1, the excitation process is said to be antibunched or anticorrelated,whereas ifgi2)(0)> 1, it is said to be bunched or correlated. The characteristic time associated with the function [gi2)(z)- 11 is denoted ze. Similarly, if F,(T) < 1, the excitation counts are said to be sub-Poisson (for this counting time T), whereas if F,(T)> 1, the counts are said to be super-Poisson.These are, of course, the same terms used earlier to characterize the photoevent point process. The Poisson point process has neither memory nor aftereff'ects. For the self-excitingpoint process (SEPP), on the other hand, the probability of occurrence of an event at a particular time depends on the times and numbers of previous occurrences (SNYDER[ 19751). Renewal point processes (RPPs) form an important subclass of SEPPs for which the rate p and the normalized coincidence rate gi2)(z)do characterize the process completely (COX[ 19621). These are processes for which the interevent time intervals are statistically independent and identically distributed. The following are important examples of renewal point processes that exhibit antibunched events and sub-Poisson counts: (1). The gamma-Nprocess. This process is obtained from a Poisson process by decimation, that is, by selecting every N t h event and discarding all others as illustrated in fig. l.ld (Cox [ 19621, PARZEN[ 19621). The process is so named because the inter-event-time interval distribution P ( T )(see fig. 2.3) is a gamma distribution of order X For the particular case when N = 2 (shown in fig. l.ld), it turns out that (TEICH,SALEHand P E ~ I N[A19841) g',*)(r) = 1 - exp( - 4 p T I ) ,

(3.41)

F,(T) z $ .

(3.42)

1, I 31

ANTIBUNCHED LIGHT

FROM INDEPENDENT RADIATORS

45

(2). The nonparalyzable dead-time-modified Poisson process. This process is obtained from a Poisson process by deleting events that fall within a specified dead time 7, following the registration of an event, as illustrated in fig. 1. lb (Cox [ 19621, PARZEN[ 19621, RICCIARDIand ESPOSITO[ 19661, MULLER [ 19741, CANTORand TEICH [ 19751, TEICHand VANNUCCI [ 19781). It is characterized by (TEICH,SALEHand PERINA[ 19841)

with

A=

/J

(l - pzd) ’

(3.45)

where U ( t ) is the unit step function, A is the initial rate of the process, and p is the rate after dead-time modification. Its inter-event-time density function is a decaying exponential function displaced from t = 0 to the minimum permissible inter-event time, 7,. Another example is a pulse train with random time of initiation (LOUDON [ 19801, TEICH,SALEHand P E ~ I N [A19841). 3.4.2. Photon statistics for emissions at antibunched times When the underlying excitation process has known rate p and normalized coincidence rate g a ) ( 7), but is otherwise arbitrary, what can be said about the statistics of the radiation? Because of their finite lifetime, emissions overlap and interfere (as we have seen in earlier sections). To determine their bunching properties, it is necessary to know not only the rate of coincidence of the excitation process at pairs of time instants, but it is also necessary to know coincidence rates at triple points, and so on. If such information is not available, the bunching properties of the superposed radiation cannot be determined. However, in the limit in which the counting time T is much longer than the lifetime zp of the individual emissions, interference has a negligible effect on the total number of collected photocounts. The total number of photons n is then

46

PHOTON BUNCHING AND ANTIBUNCHlNG

[I, t 3

simply the sum of the number of photons emitted independently by the individual emissions. If m is the number of emissions and ak is the number of photoevents associated with the kth emission, then n = ak. Using the fact that the { a k } are statistically independent and identical, it is not difficult to show that the mean and variance of n are

1;-

var(n) = ( a ) 2 V a r ( m )+ ( m ) Var(a),

(3.47)

from which it follows that the corresponding Fano factors are related by F,

=

(a)F,

F,

=

1

+ Fa

(3.48)

or

+ [ F a - 1 + ( a ) ] + ( a ) (F,

- 1).

(3.49)

Equation (3.47) is known as the cascade variance formula (SHOCKLEY and PIERCE[ 19381, MANDEL[ 1959b], BURGESS[ 19611). Equation (3.49) shows that the Fano factor comprises three contributions. The first term is that of a Poisson process. The second term represents excess noise due to randomness in the number of photons per emission (if a = 1, then Fa = 0 and it vanishes). The third term admits the possibility of noise reduction due to anticorrelations in the excitation process (this term vanishes if the excitation process is Poisson). As an example, we can apply this formula to the case considered in 3 3.3 in which the excitation process was a Poisson point process. When m is Poisson, F, = 1 and eq. (3.48) becomes F,

=

( a ) + Fa,

(3.50)

which reproduces eq. (3.37). We now consider another example in which each of the individual emissions is described by a one-photon number state (i.e., single-photon emissions and a = 1). This corresponds to ( a ) = 1 and Fa = 0, from which eq. (3.48) yields F,

=

F,

.

This is to be expected. For single-photon emissions the number of photons counted over a long time interval is approximately equal to the number of excitations (assuming there are no losses). If the excitation point process is sub-Poisson,the photons will also be sub-Poisson. It is of interest to note that

I , § 31


41

we need not go outside the domain of linear (single-photon) optics to see such uniquely quantum-mechanical effects. Equations (3.48) and (3.49) reveal the key to obtaining sub-Poisson photons from sub-Poisson excitations. In order to have F, < 1, F, must be < 1, as is apparent from eq. (3.48). Furthermore, a necessary condition for F, < 1 is that F, < 1 (because the second term in eq. (3.49) is nonnegative). It follows that for F, to be less than one, both Fa and F, must be less than one. Therefore, the generation of a stationary stream of sub-Poisson photons from a superposition of independent emissions requires both the excitation process and the photons of the individual emissions to be sub-Poisson. The implementation of physical mechanisms that lead to this kind of model are presented in $ 7, where various kinds of excitation feedback are used to generate sub-Poisson excitations. Other ways of producing sub-Poisson light are to generate correlated photon pairs or to use a quantum nondemolition measurement, as discussed in $ 6. One member of each correlated photon pair is used to provide a photon feedback signal that controls the twin photon beam. It might appear that the theoretical framework presented here is not appropriate for this paradigm because the photon emissions are not independent (correlated pairs are produced). However, because the control photons are annihilated they can be viewed as simply modifying the excitation statistics for the surviving photon beam, which then may be considered to comprise independent emissions. A fully quantum-mechanical analysis of photon-feedback mechanisms (SHAPIRO, SAPLAKOGLU, Ho, KUMAR,SALEHand TEICH[ 1987]), such as those considered in $ 6 , confirms that the approach presented here is suitable for describing physical processes involving photon feedback as well as excitation feedback.

3.4.3. Bunchinglantibunchingproperties of emissions initiated at antibunched times The determination of the short-time behavior of the photoevents requires knowledge of the normalized photocoincidence rate g(')( 2). This is not possible unless the excitation point process is completely specified (higher-order multicoincidence rates specified). TEICH,SALEHand P E ~ I N[ A 19841 examined this problem under the assumption that the excitation point process was a renewal point process. Under the additional assumption of single-mode individual emissions, as in eq. (3.26), they showed that

(3.51)

48


[I, 5 3

The first three terms on the right-hand side of eq. (3.51) apply when the excitation process is Poisson (see eq. 3.32). The fourth term, which is given by

(3.52)

with $(z, t ) =

jo1

[ l o ( t ’ ) I&’

+ z - t)

+ V,f(t’) Vo(t’ + z) V$(t + t’) Vo(t + t‘ + r)] dt‘ , (3.53) represents the effects of deviation of the excitation process from Poisson. When the excitation point process is antibunched, this term is negative, thereby introducing anticorrelation into the photon process. If it is sufficiently strong, it can counterbalance the bunching effects due to wave interference (second term) and to the randomness of the individual emissions (third term). With the availability of eq. (3.51), the Fano factor for the photon counts in a time interval of arbitrary duration can be determined. The result can be put in the form (TEICH,SALEHand P E ~ I N[A 19841)

(3.54) where M and A are the degrees-of-freedom parameters discussed in 3.3 and A’ is a new degrees-of-freedom parameter associated with the term r ( z ) . A‘ depends, in a complex way, on the relation between the counting time T, the emission lifetime zp, and the excitation point-process memory time z, (width of the function [gL2)(z)- 11). For counting times that are long ( T 9 zp, z,) however, it turns out that A4 = 00 and wavelike (interference) noise is washed out; A = 1 so that the role of noise in the individual emissions is enhanced; and A‘ is given by the degrees-of-freedomparameter for the excitation process Me given in eq. (3.40). It then follows that eq. (3.54) reduces to eq. (3.49), which was directly obtained by use of the cascade variance formula. This problem has been examined in considerable detail by TEICH,SALEH 19841. They also addressed the effects of different locations for and P E ~ I N[A the different emissions, and the rates of photon coincidence at pairs of positions

1. I 31


49

in the detection plane. MANDEL[1983] examined photon interference and spatial correlation effects of light produced by two independent sources, each containing either a random or a deterministic number of radiators.

3.5. SUMMARY: GENERATION OF ANTIBUNCHED AND SUB-POISSON LIGHT

It has been shown that two key effects regulate the antibunching and sub-Poisson possibilities for light generated by a two-step process of excitation and emission: (1) the statistical properties of the excitations themselves and (2) the statistical properties of the individual emissions. The role of these two factors is readily illustrated, from a heuristic point of view, in terms of the schematic presentation in fig. 3.2. In fig. 3.2a we show an excitation process that is Poisson. Consider each excitation as generating photons independently. Now if each excitation instantaneously produces a single photon, and if we ignore the effects of interference, the outcome is a Poisson stream of photons, which is neither antibunched nor, obviously, sub-Poisson. This is the least random situation that we could hope to produce, given the Poisson excitation statistics. If interference is present, it will redistributethe photon occurrences,leading to the results for chaotic light (SALEH,STOLERand TEICH[ 19831). On the other hand, the individual nonstationary emissions may consist of multiple photons or random numbers of photons. In this case, we encounter two sources of randomness, one associated with the excitations and another associated with the emissions, so that the outcome will be both bunched and super-Poisson. In particular, if the emissions are also described by Poisson statistics, and the counting time is sufficiently long, we recover the Neyman Type-A counting distribution, as has been discussed in detail elsewhere (TEICH[ 19811, TEICH and SALEH[ 1981a1, SALEHand TEICH[ 1982, 19831). Even if the individual emissions comprise deterministic numbers of photons, the end result is the fixed-multiplicative Poisson distribution, which is super-Poisson (TEICH [ 19811). Related results have been obtained when interference is permitted (SALEH,STOLER and TEICH[ 19831). It is clear, therefore, that if the excitations themselves are Poisson (or super-Poisson), there is little hope of generating antibunched or sub-Poisson light by such a two-step process. In fig. 3.2b we consider a situation in which the excitations are more regular than those for the Poisson. For illustration and concreteness we choose the excitation process to be produced by deleting every other event of a Poisson process. The outcome is the gamma-2 (or Erlang-2) renewal process, whose

50


[I, 8 3

POISSOn EXC1TRTIOnS

u u u

POISSON

SINGLE PHOTON EMISSIONS

a)

POISSON

POISSON PHOTON EMISSONS

BUNCHED SUPER-POISSON NEVMAN TYPE-A

SUB-POISSOIL EXCITATlOnS

ANTIBUNCHED SUB-POISSON (GAMMA-21

SINGLE-PHOTON EMISSIONS

b)

ANTIBUNCHED SUB-POISSON (GAIIMA-21

POISSON PHOTON E n l S S l O N S

ANTIBUNCHED SUPER-POISSON (GAMtlA- POISSON1

PULSE-TRRIn EXClTRTIOnS

TTTlTTT

ANTIBUNCHED SUB-POISSON (DELTA FUNCTION1

SINGLE-PHOTON EMISSIONS

TTTiiTT

ANTIBUNCHED SUB-POISSON (DELTA FUNCTION)

POISSON PHOTON EWlSSlONS

POISSON

Fig. 3.2. Schematic representation of a two-step process for the generation of light, illustrating stochastic excitations (first line) with either single-photon emissions (second line) or Poisson multiple-photon emissions (third line). Interference effects are ignored in this simple representation. (a) Poisson excitations; (b) antibunched, sub-Poisson excitations (gamma-2); (c) pulsetrain excitations (random phase). (After TEICH, SALEHand PERINA[1984].)

analytical properties are well understood. Single-photon emissions, in the absence of interference, result in antibunched, sub-Poisson photon statistics. Poisson emissions, on the other hand, result in super-Poisson light statistics. Of course, the presence of interference can introduce additional bunching. Clearly, a broad variety of excitation processes can be invoked for generating many different kinds of light. A process similar to the gamma-2, and for which many analytical results are available, is the nonparalyzable dead-time-modified Poisson process. Resonance fluorescence radiation from a single atom will be described by a process of this type, since after emitting a single photon the atom decays to the ground state where it cannot radiate. The superposition of light from a number of such atoms will wash out the sub-Poisson behavior, however.

RANDOMIZATION OF SUB-POISSON PHOTON STREAMS

51

Finally, in fig. 3 . 2 ~we consider the case of pulse-train excitations (with random initial time). This is the limiting result both for the gamma family of processes and for the dead-time-modified Poisson process. In the absence of interference, single-photon emissions in this case yield antibunched, ideally sub-Poisson photon statistics. Interference causes the antibunching to disappear, but the sub-Poisson nature remains in the long-counting-time limit. Poisson emissions give rise to Poisson photon statistics. The illustration presented in fig. 3.2 is intended to emphasize the importance of the excitation and emission statistics as determinants of the character of the generated light. To produce antibunched and/or sub-Poisson photons, both sub-Poisson excitations and sub-Poisson emissions are required. The statistical properties of light generated by sub-Poisson excitations, with each excitation leading to a single-photon emission, have been developed earlier in this section. The sub-Poisson excitations are characterized by a time constant ze, which represents the time over which excitation events are anticorrelated (antibunched). The single-photon emissions, on the other hand, are characterized by a photon excitation/emission lifetime zp. The detected light will be sub-Poisson provided T b z, zp; A b A,, where T is the detector counting time, A is the detector counting area, and A , is the coherence area. Different methods of sub-Poisson excitation result in different values of ,z, whereas different mechanisms of photon generation result in different values of zp and A,. Invoking these limits assures that all memory of the field from individual emissions lies within the detector counting time and area. The randomization of photon occurrences associated with interference therefore does not extend beyond these limits. Consequently the photon-count statistics are determined by the only remaining source of variability, which is the randomness in the excitation occurrences. In this limit the photons behave as classical particles, and the photon statistics are governed by the simple rules specified in eqs. (3.48)-(3.50). This kind of picture also provides the basis for understanding the generation of sub-Poisson light by a semiconductor laser (YAMAMOTO, MACHIDA,IMOTO, KITAGAWA and BJURK [ 19871).

6 4. Randomization of Sub-Poisson Photon Streams Photon streams often undergo random deletion (thinning). Obvious examples of importance in quantum optics include optical absorption and the

52


[I. 0 4

photodetection process itself, for which the quantum efficiency of the detector is invariably less than unity. It has long been known that the Poisson process, which is probably the most ubiquitous of all point processes, remains Poisson under the action of such Bernoulli random selection (PARZEN[ 19621). More recently it has been established (TEICHand SALEH[ 19821, PERINA, SALEHand TEICH[ 19831) that the DSPP photon distribution retains its form under the effects of such deletion, but with a reduced integrated rate. A specific example is the negative binomial photon distribution, which, on deletion, remains negative binomial with reduced mean and an unchanged degrees-offreedom parameter M. Another is the shot-noise-driven Poisson (SNDP) photon-counting distribution, which re-emerges on random deletion with the same degrees-of-freedom parameter A,but in this case with reduced mean and reduced multiplication parameter. There is no general result of this nature for sub-Poisson photon counting distributions. On the sub-Poisson side of the Poisson barrier, however, the binomial distribution retains its form under Bernoulli deletion. Nevertheless, on either side of the barrier, the Fano factor obeys a particularly simple relation under the effect of Bernoulli random deletion, provided that the counting window is sufficiently large. This characteristic is discussed in 5 4.1. Another source of photon-stream randomization is associated with the presence of additive independent Poisson photons, such as those arising from background radiation. The modified Fano factor reduction formula that accounts for this effect is presented in 5 4.2. Finally, we mention parenthetically that the process of photon detection often involves the use of devices that operate by means of electron multiplication. Examples are the photomultiplier tube and the avalanche photodiode. Although the electron multiplication process also adds randomness to the detected signal, it is possible to minimize the deleterious effects of multiplication noise by a judicious choice of detection device and mode of operation (TEICH, MATSUOand SALEH[ 19861).

4.1. BERNOULLI RANDOM DELETION

A photon stream with Fano factor F,(D) emerges, after random deletion, with Fano factor F,(D) in accordance with the relation

1, o 41

RANDOMIZATION OF SUB-POISSON PHOTON STREAMS

53

where q is the probability of photon survival (quantum efficiency) and ( m ) is the initial mean photon number. This relation may be obtained directly from eq. (3.48) by using the substitutions ( a ) = q and F, = 1 - q, which are appropriate for events governed by the Bernoulli (random-deletion) law. If q contains the quantum efficiency of the detector, as well as all other losses, F,(D) then represents the expected photoelectron (post-detection) Fano factor. The validity of eq. (4.1) requires that the counting time T be greater than both the characteristic electron correlation time z, and the photon correlation time zp, and that the detection area A be greater than the coherence area A , . These conditions ensure that interference has a negligible effect on the total count number (see § 3.4 and § 3.5). Furthermore, eq. (4.1) is only applicable for open-loop photodetection (see SHAPIRO, SAPLAKOGLU, Ho, KUMAR,SALEH and TEICH[ 19871). Equation (4.1) has been derived both quantum-mechanically (PAUL[ 1966, 19821, MILLER and MISHKIN[1967], GHIELMETTI [1976], YUEN and SHAPIRO [ 19781, P E ~ I N A SALEH , and TEICH[ 19831, LOUDONand SHEPHERD [ 19841) and semiclassically (TEICHand SALEH[ 19821). It is applicable for bilinear interactions of boson quantum systems in the rotating-wave approximation, which lead to Heisenberg-Langevin equations involving only annihilation operators (PERINA,SALEHand TEICH [ 19831). Such interactions leave the initial statistics of the system unchanged (in particular, a coherent initial state remains coherent). ‘Equation (4.1) is also applicable for interactions in which photons interact with electrons and atoms whose fermion properties play no role. The semiclassical derivation gives the correct result because of the correspondence between semiclassical and normally ordered correlation functions, for which vacuum fluctuations play no role. It is useful, perhaps, to point out that, in contradistinction to the Fano factor, the magnitude of the second-order correlation function g(2)(x,x) is independent of q. This is because g(’)(x, x) reflects the joint detection of pairs of photons (coincidences) at the space-time point x. Random deletion has the effect of providing fewer such pairs at each value of x, thereby reducing the accuracy of the estimated correlation function.

4.2. ADDITIVE INDEPENDENT POISSON PHOTONS

In the presence of additive independent Poisson photons, as well as Bernoulli deletion, the expression analogous to eq. (4.1) is

54


(1, § 4

Here, n represents the signal-plus-additive-background photon-count (or postdetection photoelectron) random variable and F,(D) is the overall photoncount (or photoelectron) Fano factor. The quantity fl (0 < fl < 1) is governed by the presence of Poisson additive counts (e.g., background light). In the absence of such counts, fl = 1. If the additive Poisson noise count mean is ( p ) ,then Var ( p ) = ( p ) . In the case where the Bernoulli selection occurs before the addition, p turns out to be (TEICHand SALEH[ 19821) (4.3) whereas when the Bernoulli selection occurs after the addition,

8 = ( l t x () P )

- l

.

(4.4)

Equation (4.2) clearly shows that all photon distributions move toward the Poisson barrier under the action of Bernoulli deletion and/or additive independent Poisson background events. However, no amount of deletion or additive noise will permit this barrier to be crgssed from either direction.

4.3. ANALOG RELATIONS

The analog version of eq. (4.2) is useful when the detected photocurrent (or the excitation current discussed in 0 7.1.2) is continuous rather than a sequence of discrete events. It can be used to relate the power spectral densities of an excitation current Sm(w) and a detected photocurrent S,(w). The ratios S,(w)/2e ($) may be regarded as Fano factors F,(q), wherej = m,n, and the (6) are the mean values of the respective currents. Here the counting times T, play the role of inverse bandwidths of the filters involved. In the limits T, p ,z, zp and for w 4 l/ze, l/zp, we obtain

The quantity fl accounts for the admixture of independent Poisson background events as discussed earlier.

1, $51

ANTIBUNCHED AND CONDITIONALLY SUB-POISSON PHOTON EMISSIONS

55

8 5. Observation of Antibunched and Conditionally Sub-Poisson Photon Emissions Antibunching was the first characteristic of nonclassical light to be observed in the laboratory. It is an effect that generally becomes less pronounced as the number of radiators increases (this is a result of the increase of accidental coincidences between uncorrelated photons). In 1977 KIMBLE,DAGENAIS and MANDEL[ 19771 carried out a series of experiments in which they excited sodium vapor with laser light. This led to the production of antibunched resonance fluorescence radiation (KIMBLE,DAGENAIS and MANDEL[ 1977, 19781, DAGENAISand MANDEL[1978]). Similar results were achieved by RATEIKE,LEUCHSand WALTHER,as cited in CRESSER,HAGER, LEUCHS, RATEIKE and WALTHER [ 19821, using a longer interaction time. Antibunching has also been observed in parametric downconversion by using an eventtriggered optical shutter (WALKERand JAKEMAN [ 1985b1) and in correlated atomic photon emissions (GRANGIER,ROGERand ASPECT [ 19861). More recently GRANGIER, ROGER, ASPECT, HEIDMANNand REYNAUD[ 19861 observed that a multi-atom source in a four-wave-mixing configuration gives rise to antibunched pairs of fluorescence photons traveling in opposite directions. Antibunching and sub-Poisson behavior need not necessarily accompany each other as discussed in 3 (TEICH, SALEH and STOLER [ 19831); nevertheless, they sometimes do. From an experimental point of view it is generally easier to observe antibunching than sub-Poisson statistics (SHORT and MANDEL[ 19841, WALKERand JAKEMAN[ 1985b1). The observation of antibunching and sub-Poisson behavior in resonance fluorescence reflects the fact that the atom makes a quantum jump to its ground state at the time a photon is emitted. The inability of the atom to radiate in the ground state may be viewed as an enforced dead time (TEICHand VANNUCCI [ 19781) following a photon emission, during which further emissions are prohibited (KIMBLEand MANDEL[ 19761). This regularizes the resonance-fluorescence photon emissions from a single atom, so that it produces an antibunched and sub-Poisson cluster of photons while it traverses the experimental apparatus. In 5 5.1 we discuss the generation of conditionally sub-Poisson resonancefluorescence photons from single atoms. The use of parametric downconversion for producing conditionally sub-Poisson single photons is considered in § 5.2. Finally, the destruction of sub-Poisson behavior resulting from the incorporation of excitation statistics (removal of the conditioning) is discussed in 5.3. Techniques for the generation of unconditionally subPoisson light are presented in § 6 and § 7.

56


SOURCE

DETECTOR

SOURCE EXCITATION PROCESS

CONDITIONALLY SUB-POISSON PHOTONS

Fig. 5.1. Schematic diagram illustrating the generation of conditionally sub-Poisson photons. (a) Configurationfor resonancefluorescencewhere the entry of a single atom into the field ofview of the apparatus gates the detector open for a brief time. (b) Configuration for correlated photon pairs (e.g., spontaneousparametric downconversionor 40Cacorrelatedphoton emissions), where one partner of a photon pair gates the detector open for a brief time.

5.1. CONDITIONALLY SUB-POISSON PHOTON CLUSTERS FROM RESONANCE

FLUORESCENCE

SHORTand MANDEL[1983, 19841 observed individual clusters of subPoisson emissions from isolated sodium atoms. The effect is observable only if there is a single atom in the field of view of the apparatus at any given time and if it remains there during the photon counting time. In the Short-Mandel experiment this was achieved by enforcing two conditions: (1) the beam of sodium atoms was made sufficiently weak so that the average interatomic separation was 10 ps in time or 1 cm in distance; (2) the detector was gated on for a counting time T m 200 ns by means of an auxiliary detector that registered the arrival of the atom in the apparatus. A schematic representation is provided in fig. 5.la; the excitation process was such that the source consisted of only a single atom. A block diagram of their experimental apparatus is presented in fig. 5.2. A highly collimated sodium atomic beam is intersected perpendicularly by two circularly polarized dye-laser beams, tuned to the 32S1,2, F = 2 to 32P3/2, F = 3 Na transition, and stabilized in frequency to 1-2 MHz. The two intersection regions are in a weak magnetic field parallel to the dye laser beams. Optical pumping in the first region prepares the sodium atoms to be in the 3’Slj2, F = 2, mF = 2 magnetic sublevel. The only allowed dipole transition is between this and the higher 32P3/2,F = 3, mF = 3 magnetic sublevel, so that the atoms behave essentially as a two-level quantum system. The exciting laser beam causes the atom to shuttle back and forth between the two levels, emitting resonance fluorescence photons. N

-

1,

51

ANTIBUNCHED AND CONDITIONALLY SUB-POISSON PHOTON EMISSIONS

51

Part of the fluorescence radiation in a region in the center of the second intersection is collected by a microscope objective and imaged onto a rectangular aperture where the field is split into two parts (see fig. 5.2). The light from a 50 pm-long region of the atomic beam, which the atom enters first, is directed to photomultiplier tube A (PMT A), whereas the light from an adjacent 425 pm-long region is sent to photomultiplier tube B, where the principal photon counting process takes place. When PMT A detects the arrival of an atom in the apparatus by registering a photon count, a 90 ns delay is invoked (to allow the atom to move from region A into region B), after which PMT B is gated on to allow the resonance fluorescence photons to be counted during the counting time T E 200 ns. The atom remains in region B during the measurement about 98 % of the time. A histogram is constructed of the number of photon counts registered by PMT B during this time. This is normalized to

C i rcu I a r Po I a r i z e r +

Atomic Beam

Fig. 5.2. Experimental apparatus for the observation of conditionally sub-Poisson photon [ 19841.) clusters from resonance fluorescence. (After SHORT and MANDEL

58


[I. § 5

provide the experimental photon-counting distribution p(n, T ) . Special efforts were made to minimize background and dark counts. In each of the 24 x lo6 measurements the detector was gated on when it was ascertained that an atom was in the field of view of the apparatus. The collection of a single photon-counting distribution took many hours. After collection of the data, corrections were made for additive background light, occasional pairs of atoms in the apparatus, dead time in the counting electronics, and PMT afterpulsing. The experiment provided a corrected count and a Fano factor F,(T) x 0.9978 & 0.0002, indimean ( n ) x 6.5 x cating that the resonance-fluorescence photon clusters were slightly subPoisson. The results are in good accord with theoretical calculations for resonance fluorescence (SHORTand MANDEL[ 19841, MANDEL[ 19791, COOK [ 1980, 19811, LENSTRA[ 19821).

5.2. CONDITIONALLY SUB-POISSON SINGLE PHOTONS FROM PARAMETRIC DOWNCONVERSION

It is easier to observe conditionally sub-Poisson photon emissions by means of spontaneous parametric downconversion. In this process photons from a coherent beam of light are split into lower-frequency signal and idler photons in a crystal that lacks inversion symmetry (LOUISELL,YARIVand SIEGMAN [ 19611, BURNHAMand WEINBERG[ 19701). If a signal photon is detected at some position within a short time interval T, there is an idler photon in a one-photon state at a corresponding position at the same time (HONGand MANDEL[ 19861). HONG and MANDEL[ 19861 conducted a parametric downconversion experiment that made use of an argon-ion laser pump beam at 351.1 nm and a potassium dihydrogen phosphate (KDP) crystal. The downconverted signal and idler photons were collected by lenses and sent to two counting photomultiplier tubes. The idler counter was gated on by the signal counter, as represented schematically in fig. 5.1b. The idler photon-counting probability distribution, conditioned on the occurrence of a signal photon, was therefore measured. Under the paradigm of the experiment, if the collection efficiencies were unity and there were no dark current or background photons, the conditional idler counting distribution would be very nearly ideal, that is, p ( n ) = bn,, corresponding to F x 0. The experimental results lay far from this, however, exhibiting a post-detection Fano factor F,(T) x 0.998. The dilution of sub-Poisson behavior resulted largely from the low detector quantum

I , 5 51

ANTIBUNCHED A N D CONDITIONALLY SUB-POISSON PHOTON EMISSIONS

59

efficiencies. Correction of the experimental result for background light and random deletion resulted in a photon-counting distribution near the theoretical ideal. Hong and Mandel point out that this scheme achieves a close approximation to a localized one-photon state. In principle, the same configuration could be applied to other correlated photon-emission processes, for example, cascaded two-photon atomic emisand ROGER sions. Consider the experiment performed by ASPECT,GRANGIER [ 1981, 19821, in which two photons emitted from a single 40Ca atom (the 4pz IS, + 4s4p 'PI + 4s' IS, cascade) were used in a polarization correlation experiment to demonstrate a strong violation of the generalized Bell inequalities. (It is interesting to note that REID and WALLS[ 19841 demonstrated that the violation of these inequalities is associated with nonclassical light.) Instead of the coincidence experiment they carried out, however, imagine a photon-counting experiment in which the registration of the upper (green) photon triggers a photon counter maximally sensitive to the lower (violet) photon. The violet photon will clearly be detected in Bernoulli fashion, since individual spontaneous atomic emissions represent single photons that may or may not be detected. As such, the emissions are conditionally sub-Poisson. Unlike the parametric downconversion experiment, however, the spatial atomic-emission pattern gives rise to photons that are not well localized. Thus even a perfect detection system triggered by the first photon will not result in a probability bn,l for the second photon.

5.3. DESTRUCTION OF SUB-POISSON BEHAVIOR BY EXCITATION

STATISTICS

In the experiments just discussed the detector was gated on for a brief time in such a manner as to allow only photons from a single excitation event to be detected. Most emissions in nature (e.g., atomic photon emissions) are intrinsically sub-Poisson, and this character could be readily observed if we were able to gate a detector to respond only during the appropriate short time interval. Constructing a true sub-Poisson light source from a collection of such emissions is difficult, however, because we are faced with randomness in the excitations rather than a single deterministic excitation event. In the resonancefluorescence experiment the excitation statistics are determined by the random number of atoms in the field of view; indeed this number is not usually regulated (FORRESTER [ 19721, CARMICHAEL and WALLS[ 1976a,b], CARMICHAEL, DRUMMOND, MEYSTREand WALLS[ 19781, JAKEMAN,PIKE, PUSEYand

60

[I, 8 5


VAUCHAN [ 19771). In the correlated photon-emission experiments these statistics are determined by the occurrence times of the signal photons. When these times follow a Poisson process, as is usual, the resulting stationary source of light will, in fact, be super-Poisson, as is understood from $ 3.3.2 and illustrated schematically in fig. 5.3. The sample functions in the top row represent conditionally sub-Poisson emissions from resonance fluorescence (top left) and from 4oCa violet photons (top right). The character of the results changes drastically when Poisson excitation statistics are taken into account and the gating interval is not preferentially chosen. The resulting photon counts then become super-Poisson for resonance fluorescence radiation (bottom left) and Poisson for correlated photon emissions (bottom right). The equivalent of a single excitation event can be achieved if a single atom (or atomic ion) undergoing resonance fluorescence can be trapped in the field of view. There will then be no fluctuation in the atomic number. This has recently been accomplished by DIEDRICH and WALTHER[ 19871,who showed CONDITIONAL EMISSIONS

Entrance ( e x c i t a t i o n )

Green s t a r t ( e x c i t a t i o n )

Emission

Emission

t

WITH EXCITATION S T A T I S T I C S

Excitation (Poisson)

Excitation (Poisson1

t Emission

t Emission

t Super-Poisson

Poisson

Fig. 5.3. Generation of conditionally sub-Poisson and unconditionally super-Poisson or Poisson light. Top left: sample function for a conditionally sub-Poisson resonance-fluorescence photon cluster. Bottom left: Poisson entries of atoms into the apparatus and unsynchronized gating result in unconditionally super-Poisson resonance-fluorescence radiation. Top right: sample function for a conditionally sub-Poisson 40Caviolet photon emission. Bottom right: Poisson entries of4'Ca atoms into the apparatus and unsynchronized gating lead to unconditionally Poisson green and violet photons.

1.5 51

ANTIBUNCHED A N D CONDITIONALLY SUB-POISSON PHOTON EMISSIONS

61

that the emitted radiation, though obviously very weak, was both antibunched and unconditionally sub-Poisson. It is evident that making a source of unconditionally sub-Poisson photons requires that the excitations be rendered sub-Poisson (see 3.5 and TEICH, SALEHand P E ~ I N[A19841). This is shown schematically in fig. 5.4. In the top row, sample functions are presented for conditionally sub-Poisson emissions from 40Ca pairs (top left) and for a single-photon atomic emission resulting from an electron impact excitation (top right). By the use of a suitable feedback circuit, the green photons can trigger an optical gate to provide selectivepassage of their violet partners. This can be accomplished in such a way that both the excitations and the violet emissions are unconditionally sub-Poisson (bottom left). A closely related technique was used by RARITY, TAPSTER and JAKEMAN [ 19871 to produce unconditionally sub-Poisson light in a parametric downconversion experiment. The generation of unconditionally sub-Poisson light by the use of photon feedback methods such as these is the topic of 8 6 (see CONDITIONAL EMISSIONS Green s t a r t ( e x c i t a t i o n )

Electron (excitatirm)

Emission

Emission

t

WITH EXCITATION STATISTICS E x c i t a t i o n sub-Poisson)

E x c i t a t i o n (sub-Poisson)

t Emission

t Emission

t Sub-Poisson

t sub-~oisson

Fig. 5.4. Generation of conditionally and unconditionally sub-Poisson light. Top left: sample function for a conditionally sub-Poisson 40Ca violet photon emission. Bottom left: selective gating of violet photons by green photons leads to production of unconditionally sub-Poisson violet photons (see 5 6). Top right: sample function for a conditionally sub-Poisson single-photon emission resulting from an electron impact excitation. Bottom right: sub-Poisson electron excitations (resulting, for example, from space charge) lead to unconditionally sub-Poisson photon emissions (see $ 7).

62


[I, § 6

especially Q 6.2.1 and Q 6.2.2). Electron excitations can also be made subPoisson by the use of feedback; one example entails the use of space-charge effects that operate by Coulomb repulsion. The generation of unconditionally sub-Poisson light by the use of excitation feedback methods is the topic of Q 7 (see especially Q 7.1.1).

8 6. Generation of Antibunched and Sub-Poisson Light by Photon Feedback The previous section was concerned with methods useful for observing antibunched and sub-Poisson individual emissions, which was achieved by gating the detector for a specific brief time interval. However, the generation of a stationary source of sub-Poisson light cannot be arranged so easily. The principal mechanisms for generating cw sub-Poisson light rely on photon feedback or excitation feedback. In this section we consider photon feedback, that is, configurations in which the photons generated by the process at hand also provide the feedback signal. The simplest example is a process in which photon pairs are produced with one member of the pair being used to control its twin. This condition of dual purpose means that nonlinear optics must be invoked to achieve the effect. In addition, the particular feedback process that is used may be intrinsic to the physical light-generation mechanism or it may take the form of an external feedback path. These two possibilities are illustrated schematically in fig. 6.1. In 8 6.1 we briefly discuss methods that rely on feedback intrinsic to a physical process; historically, these comprised the first proposals for generating non-

a.)

SOURCE

&

EXC lTRTl Ot1 PROCESS

tl ;I

SUB-POISSON PHOTONS

SOURCE EXC lTATlOt1 PROCESS

+/SUB-POISSON PHOTONS

Fig. 6.1. Schematic diagram illustrating the generation of sub-Poisson light by means of photon feedback. (a) Feedback process intrinsic to the physical light-generation mechanism. (b) Feedback process carried by way of an external path. The feedback may take the form of an electrical signal or an optical signal.

1, § 61

SUB-POISSON LIGHT GENERATION BY PHOTON FEEDBACK

63

classical light. This discussion is followed, in 5 6.2, by a discussion of several configurations in which the feedback signal from the photons is carried externally. In J 6.3 we discuss the limitations of such methods. A quantummechanical theory applicable to these methods shows that it is possible, at least in principle, to synthesize a quantum light beam with arbitrary prescribed photon-counting statistics (SHAPIRO,SAPLAKOGLU, Ho, KUMAR,SALEHand TEICH[ 19871).

6.1. METHODS USING FEEDBACK INTRINSIC TO A PHYSICAL PROCESS

The earliest proposals for generating nonclassical light originated some 20 years ago with TAKAHASI [ 19651, MOLLOWand GLAUBER [ 1967a,b], and STOLER[ 1970, 1971, 19741; these authors suggested the use of degenerate parametric amplification for generating antibunched and quadrature-squeezed light (see also YUEN [1976]). Since that time there have been numerous suggestions for the use of other nonlinear processes, including two-photon and multiphoton absorption, four-wave mixing, Raman and hyper-Raman scattering, interference in parametric processes, resonance fluorescence, and optical bistability and multistability. All of these interactions, in one way or another, involve higher-order (nonlinear) optical effects. Viewed in an elementary way, nonlinear optical processes can introduce antibunching by removing selected clusters of photons from the incident (usually Poisson) pump beam, leaving behind an antibunched residue. The example of a two-quantum nonlinear absorber, operating by coincidence decimation, was illustrated in fig. 1.lc. Photon pairs arriving closer in time than the intermediate-state lifetime of the absorber are successful in effecting a two-photon transition and are removed from the beam (CHANDRAand PRAKASH [ 19701, TORNAU and BACH [ 19741, SIMAANand LOUDON[ 19751, EVERY[ 19751, LOUDON[ 19761). A number of review articles and books have considered the various schemes that rely on intrinsically nonlinear optical effects (LOUDON [ 19801, PAUL[ 19821, WALLS[ 19831, P E ~ ~ I N [ 1984, A 19851, SCHUBERT and WILHELMI[ 1980, 19861); the reader is referred to these for details. Although parametric amplification was the first process suggested for generating nonclassical light, it has been so used only recently. Wu, KIMBLE, HALL and Wu [ 19861 generated strongly quadrature-squeezed light at 1.06 pm using parametric downconversion in a MgO : LiNbO, crystal. Indeed, a variety of nonlinear optical interactions have recently been used to generate

64


[I. § 6

quadrature-squeezed light (SLUSHER,HOLLBERG,YURKE, MERTZ and VALLEY [ 19851, SHELBY,LEVENSON,PERLMUTTER, DEVOE and WALLS [ 19861, MAEDA,KUMAR and SHAPIRO [ 19871).However, it appears that none of these nonlinear effects with feedback intrinsic to a physical process has yet been used to directly generate antibunched or sub-Poisson light.

6.2. METHODS USING EXTERNAL FEEDBACK

The photon feedback signal may be externally carried to the excitation process or the source, as illustrated schematically in fig. 6. lb. If the feedback is carried on the external path as an electrical signal, the photon timing information must be imparted to it in a special way. This is because the conventional process of detection involves the annihilation of photons as part of the process of creating an electrical signal. In usual circumstances the choice is therefore to have either the photons or the electronic residue of their detection. However, several special schemes have recently been suggested for imparting photon timing information onto a electrical signal while leaving the photons intact. The first such suggestion appears to have been provided by SALEHand TEICH[ 19851, who proposed a scheme making use of correlated photon pairs from cascaded atomic emissions. In this case the photons from one of the atomic transitions are detected in the conventional manner to provide an external feedback signal. This signal is used to selectively permit certain photons from the second atomic transition to be passed through a gate (by means of the decimation process illustrated in fig. 1.ld). Since the photons are always emitted in correlated pairs, the selected twins survive and contribute to the light at the output. This configuration is illustrated schematically in fig. 6.2a and discussed in 5 6.2.1. There are many variations on this theme. The external feedback signal could be used to control the twin photon beam optically by means of dead-time deletion or rate compensation, instead of by decimation as discussed above (see fig. 1.1). Or the feedback signal could be used to control the source of the photons (the atoms) or the excitation process (the furnace), as illustrated in figs. 6.2b and 6.2c, rather than the twin beam. As an example of source control, the feedback signal from the control photon beam could be used to increase the rate of 4"Ca atoms entering the system when the detection rate is low and to decrease it when the detection rate is high. (A feedback signal of this kind was used by ASPECT [ 19861 and his co-workers for atomic beam stabilization;

1, J 61


EXClTRTlON

SUB-POISSON PHOTONS

PROCESS

I

PROCESS

65

I CONTROL^ PHOTONS

Fig. 6.2. Schematicdiagram illustrating the generationof sub-Poisson light by correlated photon pairs and external feedback.One of the twin photon beams is annihilated to generate the control signal. (a) Optical control of one beam by its twin; (b) photon-source control; (c) excitation control.

however, the generation of nonclassical light requires that the characteristic feedback time z, be short in comparison with the counting time T.) Nonlinear optics schemes other than two-photon emissions can also be used to generate photon pairs; parametric downconversion is considered in $ 6.2.2. Furthermore, the feedback signal need not be carried electrically; it could be carried optically (e.g., on an optical fiber) with the attendant advantage of higher speed, as discussed in $ 6.2.3. Finally, in $ 6.2.4 we discuss the use of a quantum nondemolition (QND) scheme for generating sub-Poisson light. This technique was suggested by YAMAMOTO, IMOTO and MACHIDA[ 1986a,b]. It allows the measurement of photon number at the output of a semiconductor laser, by means of the optical Kerr effect, without photon destruction, and the subsequent rate compensation of the laser excitation. 6.2.1. Correlated photon pairs from cascaded atomic emissions The correlated atomic photon-pair emitters comprise a collection of excited atoms undergoing spontaneous cascaded emissions. For illustrative purposes we focus on the 4p2 ' S o + 4s4p 'PI + 4s' IS, green/violet cascade in 40Ca. This system was used by ASPECT,GRANGIER and ROGER11981, 19821 and

66


[I, 5 6

ASPECT,DALIBART and ROGER [ 19821 as the basis of a polarization correlation experiment that demonstrated a strong violation of the generalized Bell inequalities. A block diagram illustrating the use of 40Ca photon pairs for generating nonclassical light is presented in fig. 6.3. A beam of 40Ca atoms is selectively excited to the 4p2 ‘So state by means of optical pumping. This is the source. The atoms decay by the spontaneous emission of a green photon (at a wavelength of 551.3 nm) and a violet photon (at 422.7 nm). The two photons are correlated in emission times and in polarization. The green fluorescence light is collected by a lens and passed through a polarizer and greentransmitting interference filter to a photomultiplier tube (PMT)/discriminator, which produces the control signal. The green photoelectron events, in turn, feed a digital electronic trigger circuit that produces brief pulses of duration zg ( w 10 ns) in accordance with a rule for selection, to be discussed subsequently. The violet fluorescence light is collected from the other side of the source. It is fed through a polarizer and violet-transmitting interference filter, into an optical delay path, and then through an optical gate that is opened for a period of zg s each time a pulse arrives from the selected trigger circuit. The optical delay path is adjusted so that the electrical trigger pulse (arising from the registration of a green photon) and its companion violet photon arrive at the optical gate simultaneously.

f

POLARIZER AND VIOLET-TRANSMITTING FILTER

“Ca ATOMS E X C I T E D TO dp2’So STATE

/

P O L A R I Z E R AND GREEN-TRANSMITTING FILTER

/\ry

PMT/ SELECTED DISCRIM. 4 T R I G G E R CIRCUIT (GREEN)

J

[SUB-POISSON V I O L E T PHOTONS

Fig. 6.3. Block diagram for sub-Poisson light generation using correlated photon pairs, in this case cascaded photon emissions from 40Ca atoms. (After SALEHand TEICH[1985].)

1, § 61


nn

n

n

61

TRIGGER ’ULSES FOR IPTICAL GATE

Fig. 6.4. Sample functionsof the green and violet photon events for the scheme shown in fig. 6.3. The thinned violet process displayed in (e) represents a low-fluxsource of weakly sub-Poisson light. (After SALEHand TEICH [1985].)

Several representative sample functions of the photon events are presented in fig. 6.4. A simple picture of this kind assumes that the photon occurrences are sufficiently sparse so that their wavepackets do not overlap. This is equivalent to assuming that the degeneracy parameter 6 = pzp (where p represents the photon rate and rp is the emission lifetime) is low, which is the condition under which such an experiment must be operated. The emission of the green photons from suitably excited calcium atoms may be represented as a Poisson point process (designated “green process” in row a). Some fraction of the green photons produces unitary events at the output of the green PMT (designated “thinned green process” in row b). As discussed in Q 4, a Poisson photon process remains Poisson under Bernoulli random deletion, but with a rate that is reduced (TEICHand SALEH[ 19821, PERINA,SALEHand TEICH [ 19831). The thinned green events are then used to produce trigger pulses (of duration TJ in accordance with a prespecified rule designed to render the trigger process sub-Poisson. As an example, a trigger pulse may be produced upon the registration of every N t h green photon. (This selective deletion rule is illustrated for N = 2 in row c, designated “trigger pulses for optical gate”.) This is the same rule denoted as decimation in fig. 1.ld. It is well known from renewal point-process theory that the selective deletion of every N t h event from a Poisson process (denoted a gamma-Nprocess) leads to a counting process that becomes increasingly sub-Poisson as Nincreases, provided that the counting time Tis sufficiently long (see Q 3.4.1 and TEICH,SALEHand PERINA[ 19841, eq. A24). Alternatively, another mechanism such as dead time can be used to

68


I19

86

make the trigger pulses sub-Poisson (see 3.4.1 and TEICHand VANNUCCI [ 19781). The trick is that for each green photon, and therefore for each trigger pulse, there is a large probability that a violet companion photon is following closely behind (roughly within the intermediate-state lifetime zp = 5 ns). The optical gate permits the violet photons to pass only during the times when it is open, and those times form a sub-Poisson counting process. Assuming for the moment that no violet photons are lost, row d illustrates the “violet process” which, in our example, is clearly also described by the gamma-Ncounting process. Of course, not all of the violet photons survive, so that what actually passes through the optical gate (designated “thinned violet process” in row e) is a randomly deleted version of the gamma-Nphoton-counting process. In accordance with the results presented in § 4,a randomly deleted sub-Poisson photon point process remains sub-Poisson but moves toward the Poisson barrier. Using eq. (4. l), the Fano factor for a Bernoulli-deleted gamma-Nphotoncounting process is easily shown to be (SALEHand TEICH[ 19851)

where qv is an effective quantum efficiency for the violet photons. For an experimental configuration similar to that used by ASPECT,GRANGIER and ROGER[ 19811, the photon Fano factor is estimated to be FJT) x 0.9990. Folding in the violet-PMT quantum efficiency provides an estimated photoelectron Fano factor (SALEHand TEICH[ 19851) F,(T) = 0.9999, which is close to unity but should be measurable. 6.2.2. Correlated photon pairs from parametric downconversion

WALKERand JAKEMAN [1985a] recognized that a similar result can be achieved by using photon pairs generated by the process of spontaneous parametric downconversion (see also JAKEMAN and WALKER[1985] and JAKEMAN and JEFFERSON [1986]). This effect may be described as the splitting of a single photon into two (correlated) photons of lower frequency. The effect was first observed experimentally by BURNHAMand WEINBERG [ 19701. The time uncertainty in the emission of the photon pairs can be short; it is determined by the inverse bandwidth of the detected light (HONG and MANDEL[ 19851, FRIBERG,HONG and MANDEL[ 19851). The first parametric downconversion experiment conducted by WALKER and JAKEMAN [ 1985b] used a configuration similar to that shown in fig. 6 . 2 ~ .

1, § 61


69

One member of each correlated photon pair was detected. This event provided an electrical feedback signal, which controlled an optical shutter in the excitation beam. The optical shutter was closed for a fixed dead-time period z, following the detection of an event. Poisson photons obtained from a He-Cd laser operated at 325 nm served as the excitation process. This UV light was passed through an acousto-optic shutter (the control gate) and impinged on an ammonium dihydrogen phosphate (ADP) nonlinear crystal (the source), which produced red photon pairs by parametric downconversion. The signal light generated had an experimental second-order correlation function g(’)( z) that increased with T, as T increased from 0. This indicates photon antibunching, in accordance with the positive-derivative definition provided in Q 2.1.3. However, because g(’)(z) was always greater than unity, the light generated in this experiment was not sub-Poisson; this was attributed to long-term laser power fluctuations. More recently, RARITY, TAPSTER and JAKEMAN [ 19871 succeeded in using this technique to produce sub-Poisson light. A block diagram of their apparatus is presented in fig. 6.5. The experimental arrangement is similar to that used in the antibunching experiment just described; fiber-optic light guiding and a single-photon-counting avalanche photodiode were added. The source of downconverted photons in this case was a potassium dihydrogen phosphate (KD*P) crystal, and the control gate acted on the signal channel rather than on the excitation channel so that the configuration is similar to that shown in fig. 6.2a rather than 6 . 2 ~ .The observed effect was small but statistically significant; the postdetection Fano factor turned out to be F = 0.9998 with a photoelectron counting rate x 30 s - and a switching time x 19 ps. TRIGGER CHANNEL

Fig. 6.5. Block diagram ofthe parametric downconversion experiment used by RARITY, TAPSTER and JAKEMAN [ 19871 to generate sub-Poisson light. (After RARITY, TAPSTER and JAKEMAN [1987].)

70

[I. § 6


TAPSTER,RARITYand SATCHELL [ 19881 modified this experiment by using analog rate compensation of the pump power (see fig. l.lf) provided by the control beam, using an electro-optic modulator. They generated 60 pW of sub-Poisson light with a remarkably low Fano factor F = 0.78. This appears to be the lowest Fano factor yet reported for the direct generation of subPoisson light. However, the bandwidth over which the light was sub-Poisson was quite limited ( 60 Hz) and the overall quantum efficiency of the process was very low (-6 x lo-"). Also recently, HEIDMANN,HOROWICZ, [ 19871 used a two-mode optical paraREYNAUD,GIACOBINO and FABRE metric oscillator operating above threshold to generate high-intensity twin beams exhibiting strong quantum correlations. A number of techniques have been suggested to enhance the nonclassical degree of the light (see, for example, WALKER[ 1986, 19871). Suggestions have also been made for the use of other related schemes (YUEN[ 19861, STOLER and YURKE[ 19861, SRINIVASAN [ 1986b]), some of which are closely connected with overflow count deletion (MANDEL[ 1976a]), which is illustrated in fig. 1.le.

-

-

6.2.3. All-optical systems using correlated photon pairs In the examples discussed in the previous two subsections the feedback signal, although initiated by the photons in one of the twin beams, was canied electrically. This electrical signal was then used to control an optical beam. This external signal can also be carried optically (e.g., on an optical fiber) and used for direct optical control of the excitation or signal beams. The potential advantage in short-circuiting the electrical link is the high speed inherent in all-optical systems. A general block diagram of a system of this type is shown in fig. 6.6. The correlated photon-pair generator might be a parametric downconverter. The control signal is carried on an optical fiber to a nonlinear optical mechanism, which modifies the excitation in opposition to the control signal, that is, it provides negative feedback. Mechanisms that achieve this are second-harmonic generation and the photochromic effect (which is usually OPTICAL FIBER

LASER

PHOTON-P A IR PHOTONS

Fig. 6.6. Block diagram of a general all-optical system using correlated photon pairs, optical feedback, and a nonlinear optical mechanism that rate-compensates the excitation beam.

1, I61


71

quite slow, however). Rate compensation can be used so that operation is not restricted to a single-file photon stream. 6.2.4. Quantum-nondemolition measurements Quantum nondemolition (QND) measurements, in which an observable may be measured without perturbing its free motion, have been studied in the context of gravitational-wave detection and quantum optics (BRAGINSKY and VORONTSOV [ 19741,CAVES, THORNE, DREVER,SANDBERG and ZIMMERMAN [ 19801, BRAGINSKYand VYATCHANIN[ 1981, 19821, MILBURNand WALLS [1983]). IMOTO, HAUS and YAMAMOTO[1985] recently presented a theoretical treatment illustrating that a QND measurement of the photon number in a signal beam can be made, without photon destruction, by use of the lossless optical Kerr effect. In the proposed measurement scheme the phase of a probe wave passed through the Kerr medium provides information about the refractive-index change, which, in turn, is dependent on the signal photon number in the medium. Precision in the photon number measurement is provided at the expense of increased uncertainty in the canonically conjugate signal phase variable (CARRUTHERS and NIETO [1968]), subject to the minimum value provided by the Heisenberg uncertainty principle. YAMAMOTO, IMOTO and MACHIDA[1986a] further proposed that the results of such a QND photon-flux measurement at the output of a semiconductor diode injection laser could be negatively fed back to control the rate of excitation of the laser, thereby producing sub-Poisson photons by rate compensation, in the manner shown in fig. 1.If. They calculate that the phase noise of the signal beam is increased in an amount such that the number-phase minimum uncertainty product is preserved. Their scheme is illustrated in the block diagram of fig. 6.7. The (single) photon beam at the output of the laser gives rise, without loss of photons, to an electrical feedback signal that controls the laser excitation rate. The feedback signal is actually obtained from a probe laser in conjunction with a Kerr nonlinear interferometer, as is evident from fig. 6.7. More recently, YAMAMOTO and HAUS [ 19861 showed that under proper conditions a quasi-QND measurement of photon number, followed by a phase measurement, leads to a doubling of the noise associated with photon number and phase as required for the simultaneous measurement of two noncommuting variables. A QND signal can also be obtained by means of other nonlinear optical processes, such as four-wave mixing (MILBURNand WALLS [ 19831). The principle of QND detection has recently been verified by LEVENSON, SHELBY, REIDand WALLS[ 19861.

72


laser

laser

J

t ~

I

\

- - _I - - -- - - 1-- -1 ,

I

, sub-Poissonion

I

stole

negative feedback

Fig. 6.7. Block diagram of the scheme proposed by YAMAMOTO, IMOTO and MACHIDA[ 1986al for generating sub-Poisson (photon-number-squeezed) light. A quantum nondemolition (QND) measurement ofthe laser-outputphoton number is used to control the laser excitation rate. (After YAMAMOTOand HAUS[1986].)

6.3. LIMITATIONS OF PHOTON-FEEDBACK METHODS

As indicated in $ 1, a useful source of sub-Poisson light will exhibit a photon Fano factor F,(T) that is substantially below unity and will produce a large photon flux di (corresponding to a large average photon number (n)) with a reasonably large overall quantum efficiency 7. Ideally, the device should also be small in size and produce a directed output so that the light can be coupled to an optical fiber. The structure should be designed in such a way that light loss is minimized (see $ 4). Nonlinear optics methods (see $ 6.1) generally employ a laser pump that emits Poisson-distributed photons. In such cases FJT) and 7 will be the principal limiting factors in producing a useful source; they are determined by the efficiency with which pairs or clusters of photons can be separated from the pump beam. The photon flux will generally be unlimited and z can be small, since it is determined by the intermediate state lifetime of the nonlinear process. Methods employing correlated photon pairs (see 5 6.2.1 to 3 6.2.3) can, in principle, exhibit small values of FJT), but di, 7 and Twill be limited. The limitation on the photon flux arises from the use of dead-time and selectivedeletion gating, which require single-file events for the technique to operate. Rate compensation is a preferable feedback mechanism from this point of view, but 7 will still be limited by the photon-pair generation mechanism. The photon flux may also be limited by the necessity of avoiding photon interference effects (thereby requiring that the degeneracy parameter not exceed unity). T is limited

1, I 71

SUB-POISSON LIGHT GENERATION BY EXCITATION FEEDBACK

73

by the particular configuration: the intermediate-state lifetime for two-photon atomic emissions (z, x 5 ns in 40Ca),the dead-time (gating time) in parametric downconversion with optical dead-time gating (z,,= ns-ps); and the pair event-time difference in correlated photon-pair generation with optical-fiber feedback (z < 0.1 ns for parametric downconversion). If the light is to be used in an application such as lightwave communications, the switching time (or symbol duration) T should be able to be made small so that the device can be modulated at a high rate (SALEHand TEICH[ 19871). However, T must be sufficiently large in comparison with the characteristic response time of the system z to ensure that the sub-Poisson character of the photons is captured in the counting time. Although sub-Poisson light generated by photon feedback may not be useful for the transmission of information (see 5 8), it should be pointed out that there are specialized applications in which the use of correlated photon pairs and post-detection processing (e.g., subtraction, correlation) are potentially useful (JAKEMAN and RARITY[ 19861). The limitations of QND techniques are not yet well understood. Outstanding questions include (1) the assumption of losslessness of the Kerr medium, (2) the role of signal/probe interference (self-phase modulation), (3) the characteristic time constant of the process, and (4)the achievable value of F,(T). Some of these questions have begun to be answered, as described by LEVENSON, SHELBY,REIDand WALLS[ 19861. It will become evident in 0 7 that excitation-feedback methods are governed by a different set of constraints which are often less restrictive. tj 7. Generation of Antibunched and Sub-Poisson Light by Excitation

Feedback We now consider excitation feedback, which is an alternative technique for generating sub-Poisson light. In this case the excitation process itself‘ is rendered sub-Poisson by means of feedback, as illustrated schematically in fig. 7. l a (compare with fig. 6.1). Excitation-feedback methods provide the greatest promise for producing sources with small photon Fano factor, large photon flux, high overall efficiency, small size, and the capability of being modulated at high speeds (small T). Excitation feedback methods are also called “direct generation methods”. Excitation feedback methods effectively operate by permitting a sub-Poisson number of electrons to generate a sub-Poisson number of photons (one photon per electron); the photons may be viewed as representing a nondestructive

14


a)

11

SOURCE EXC ITATl O H PROCESS

PHOTONS

EXClTRTl ON PROCESS

SUB-PO ISSON PHOTONS

1

Fig. 7.1. Schematic diagram illustrating the generation of sub-Poisson light by means of excitation feedback: (a) feedback process intrinsic to physical excitation mechanism; (b) feedback process intrinsic to source. Excitation feedback can also be carried externally, as considered subsequently.

measurement of the electron number. This is to be distinguished from the QND configurations considered in 5 6. There, a sub-Poisson number of photons generates an electrical current, which signals the photon number without destroying the photons (by means of a phase measurement). It is far easier to achieve a measurement of the electron number than the photon number because of the robustness (non-zero rest mass) of the electrons. Unlike photons, they are not destroyed by conventional measurement techniques. A number of the limitations inherent in photon-feedback mechanisms, as discussed in the previous section, are avoided: (1) Photons naturally gravitate toward Poisson-counting statistics and shotnoise fluctuations (SALEH,STOLERand TEICH[ 19831). It is difficult for the nonlinear-optics methods to undo this natural Poisson photon noise. Electrons, on the other hand, are often governed by quieter thermal-noise fluctuations (MOULLIN [ 19381, WHINNERY [ 19591, LAMPERTand ROSE[ 1961]), thereby permitting F,(T) to be made smaller. (2) Nonlinear-optics schemes in which Poisson photons are first generated (subject to a source power constraint) and subsequently converted into subPoisson photons cannot provide an enhancement of the channel capacity for applications such as lightwave communications as discussed in 5 8 (SALEH and TEICH[ 19871). (3) Sub-Poisson electron-excitation configurations produce light by means of efficient single-photon transitions ; high overall quantum efficiencies and large values for the photon flux are therefore readily achieved. Nonlinear-optics methods, on the other hand, rely on (relatively) inefficient multiple-photon transitions. Furthermore, they are subject to photon interference effects, which can limit the degeneracy parameter (and therefore the photon flux) to small values (SALEHand TEICH[ 19851).


1,s 71

75

INVERSE PHOTOE M I S S I 0 N

PHOTO EM I SSlON hu L

(a)

e

&eta1 or otom

ENERGY RELATIONS e9

FLUX RELATIONS

E X C I TAT I0N STATISTICS

Photon p.p.

Electron p.p.

tttttttttt

ta t t t t t t t ,

I

Itn

PARTITION NOISE

E l e c l r o n p.p.

+

t

T

FINAL STATISTICS

I

I

Photon p.p.

4

Fig. 7.2. Schematic representation of photoemission and inverse photoemission (illustrated by the Franck-Hertz effect). Energy relations and flux relatons are shown, as are sample functions of the excitation and final statistics; i, represents the average electron current and I is the average light intensity (or photon flux). The generated photons behave like classical particles in a photon-counting paradigm, provided that the detector counting time and area are sufficiently large ( T % T ~ zP; , A % A c ) . (After TEICH and SALEH[1985].)

(4) Electron excitations, especially those mediated by physical processes (e.g., space charge), can attain a small characteristic response time 2, so that fast switching can be achieved. It is useful to view the conversion of electron excitations into single-photon emissions in terms of the process of inverse photoemission. A comparison between photoemission and inverse photoemission is schematically illustrated in fig. 7.2. For photoemission (fig. 7.2a) a photon impinges on a metal or atom and liberates an electron. The maximum kinetic energy of the electron (KE,,,) is equal to the photon energy (h v ) minus the work function of the material (e$)

16


[I. § 7

in accordance with Einstein's photoelectric equation, as shown in fig. 7.2b. The flux relation displayed in fig. 7 . 2 ~demonstrates that the average photocurrent i, is proportional to the average light intensity or photon flux ( I ) . Finally, sample functions of the photon point process (excitation statistics) and the resulting electron point process (final statistics) are represented in fig. 7.2d. They are related by a Bernoulli transform, which results from the non-unity quantum efficiency of the photodetector (partition noise), as discussed in $4. The electrons liberated from the photocathode accurately sample the photon number, provided that the detector counting time T and sampling area A are properly adjusted (TEICH,SALEHand P E ~ I N [A19841). 'I'he processes are essentially reversed for inverse photoemission. For clarity we expressly consider the Franck-Hertz effect as an example. In fig. 7.2a an electron strikes an atom, loses its kinetic energy, and excites the atom. The atom then decays to a lower energy state and in the process emits a single photon by means of spontaneous emission. The energy of the emitted photon is equal to the energy supplied to the electron by an external field (E) less the cathode-emitter contact potential e 0, as shown in fig. 7.2b. Only electrons with kinetic energy corresponding to the discrete energy levels of the atom (indicated by crosses) are effective in producing photons. In fig. 7 . 2 ~we show that the average photon flux is proportional to the average electron current. Because this mechanism involves ordinary spontaneous (or stimulated) emission, it is a first-order optical process and can be expected to produce a high photon flux. Finally, in fig. 7.2d we illustrate the electron point process (excitation statistics). It is portrayed as quite regular because of the space-charge regularization. The photon point process (final statistics) is a Bernoulli-deleted version of the electron point process as a result of optical loss. As in the case of sub-Poisson light generation by photon feedback, the feedback control signal may be intrinsic to the physical mechanism providing the excitation or it may be carried externally. Methods that make use of feedback intrinsic to a physical process, such as space-charge-limited excitations (in which the physical process is Coulomb repulsion), are considered in 0 7.1. In 0 7.1.1 we discuss the space-charge-limited Franck-Hertz experiment (TEICHand SALEH[ 1985]), which provided the first source of unconditionally sub-Poisson light. In 0 7.1.2 we discuss a solid-state version of the Franck-Hertz experiment that should lead to sub-Poisson recombination radiation. This is followed, in $ 7.1.3, by a discussion of the potential improvements to be realized by the cascading of sub-Poisson electron excitations and stimulated emissions.

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77

Methods that make use of an external feedback signal are considered in $ 7.2. The method reported in $ 7.2.3 makes use of an in-loop auxiliary optical source, which emits light that effectively mimics the sub-Poisson electron current. External current stabilization schemes are discussed in $ 7.2.4. Finally, a discussion of the limitations applicable to excitation-feedback methods is provided in $ 7.3. Although our attention is directed principally to excitation feedback, it is of interest to point out that there are other related schemes which may be useful. A schematic representation in which source feedback is used is shown in fig. 7. lb. For example, this mechanism could be used, at least in principle, to convert the sub-Poisson individual emissions observed by SHORTand MANDEL[ 19831 into a cw sub-Poisson source. As illustrated in fig. 5.3, the Poisson nature of the atomic entries into their apparatus (random source statistics) precludes the production of cw sub-Poisson light. However, if the source were a cold ion beam rather than an atomic beam, it could be rendered sub-Poisson by virtue of the ionic Coulomb repulsion. The source feedback could then be used to convert the sub-Poisson individual emissions into unconditionally sub-Poisson light provided, of course, that the emissions themselves were sufficiently sub-Poisson. The experiment carried out by DIEDRICH and WALTHER[ 19871, in which resonance fluoresence was observed from a single trapped ion, can be viewed as a degenerate example of source feedback.

7.1. METHODS USING FEEDBACK INTRINSIC TO A PHYSICAL PROCESS

We now consider several methods that make use of the sub-Poisson excitations inherent in an electric current. Current supplied from a dc source, such as a battery for example, is naturally sub-Poisson as a result of the intrinsic Coulomb repulsion of the electrons (the principal source of noise is Johnson noise). In such cases it suffices to drive an emitter operating by means of single-photon transitions with such a current. Thus, a simple LED driven by a constant current source should emit sub-Poisson photons. Coulomb repulsion, which is the underlying physical feedback process for space-charge-limited current flow, is ubiquitous when excitations are achieved by means of charged particle beams. The single-photon emissions may be obtained in any number of ways. In $7.1.1 they arise from spontaneous fluorescence emissions in mercury vapor, in 3 7.1.2 they represent spontaneous recombination photons in a semiconductor, and in $ 7.1.3 they are stimulated

78

[I,


87

recombination photons. These methods all operate by transferring the anticlustering properties of the electrons, ultimately arising from Coulomb repulsion, directly to the photons. 7.1.1. Space-charge-limited Franck-Hertz experiment Unconditionally sub-Poisson ultraviolet photons have been generated by the use of a space-charge-limited Franck-Hertz experiment (TEICHand SALEH [1983, 19851, TEICH, SALEH and STOLER [1983], TEICH, SALEH and LARCHUK [ 19841).The essential element of the experiment was a collection of mercury atoms excited by inelastic collisions with a low-energy space-chargelimited (quiet) electron beam. The space-charge reduction of the usual shot noise associated with thermionically emitted electrons can be substantial (MOULLIN[ 19381, THOMPSON,NORTH and HARRIS [ 1940, 19411, WHINNERY [ 19591, SRINIVASAN [ 1965, 1986a1). Fano factors for the electron stream with values F, < 0.1 are typical, and values as low as 0.01 are possible. After excitation each atom emits a (sub-Poisson) single photon by means of the Franck-Hertz (FH) effect (FRANCKand HERTZ [1914], FRANCK and JORDAN[ 19261). This scheme is of the form represented in fig. 7.la. A block diagram of apparatus used in the experiment is shown in fig. 7.3. The light was generated in a specially constructed 25 mm-diameter UV-transmitting Franck-Hertz tube, filled with 0.75 g Hg. The radiation impinged on a

O V E N AND SHIELD

DIGITAL THERMOMETER

PHOTON COUNTER

253.7-nm F RANCK- HERTZ LIGHT FILTER

'I

F,zT 1 [p.T*ysuppu.J1-1

A N D CATHODE

ADJUSTABLE NEUTRAL-DENSITY

HI-NI

FILTER

L A S E R 1632 B nml

Fig. 7.3. Block diagram of the space-charge-limited Franck-Hertz experiment that produced unconditionally sub-Poisson light at 253.7 nrn. (After TEICH,SALEHand LARCHUK[I9841 and TEICHand SALEH[1985].)

1,

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79

UV-photon-counting photomultiplier tube (PMT) in a special base that provided preamplification, discrimination, and pulse shaping. The output of this circuitry was fed to electronic photon-counting equipment, which measured the probability distribution p(n, T ) for the detection of n photoelectrons in the time T. The mean count ( n ) and the Fano factor F,(T) were calculated from p ( n , T ) . The details of the experiment have been described by TEICH,SALEH and LARCHUK[ 19841 and by TEICHand SALEH[ 19851. A representative set of raw data for the post-detection Fano factor F,(T) (the average for a set of experiments) versus the detected photon count rate p (kilocounts/s) is shown in fig. 7.4. Data are presented for Poisson filament light 1

1000

9

-t

8

G

7

0

I4

6

z

;0995 a I

y

4

W U

z 4

K

3

4

$

2

3

8 2

0

1

0

0990

9

0 988

10

I

I

I

t

I

I

I

20

30

40

50

60

70

80

DETECTED PHOTON COUNT RATE p

(kcnt/rec)

Fig. 7.4. Average post-detection photon-count variance-to-mean ratio (Fano factor) F,( T) versus detected photon count rate p (kilocounts/s), for T = 1.0 )IS. The error bracket ( f 0.0004) is the same for all data points. The Fano factor for Franck-Hertz light lies below that for Poisson light for several (sufficiently small) values of the count rate p. The overall negative slope of the data is a result of dead time in the photon-counting apparatus. (After TEICH,SALEHand LARCHUK ~ 1 ~ 8 and 4 1 IEICH and SALEH ~ I Y W ] . )

80


[I, 8 7

(open circle), Poisson filament-plus-laser light (solid circles and solid-line segments), and sub-Poisson Franck-Hertz-plus-filament light (triangles and dashed-line segments). Because of afterpulsing and dead-time effects in the measuring apparatus, the experimental results for the FH light must be compared with those for Poisson light (rather than with a theoretical Poisson distribution) at each value of p. The filament-plus-laser light provided an excellent Poisson photoelectron distribution because of the short counting time ( T = 1 ps) and the extremely low value of the PMT quantum efficiency for light at these wavelengths. The standard deviation (SD) for a measurement of F,(T) that consists of L = lo7 samples turned out to be x(2/L)’/’x 0.0004. This calculated value for the SD was experimentally verified by carrying out many series of runs and is the same for all data points. In the range p < 30 kilocounts/s (( n ) < 0.03), values of the Fano factor for the Franck-Hertz light were below those of the Poisson light by between 2 and 3 standard deviations (depending on the details of how the estimates are made). A number of corrections (PMT afterpulsing, PMT cosmic-ray events, dead time in the photon-counting system, and Poisson filament background counts) were applied to the raw data to obtain an absolute experimental estimate of the post-detection Fano factor F,(T) for the Franck-Hertz light, which turned out to be xO.998 at T = 1 ps. At higher count rates the Franck-Hertz light was consistently noisier than the Poisson light. There are several possible explanations for this observation; these include the diminished role that dead time may play for sub-Poisson processes, the increase in the degeneracy parameter of the light, and the possibility of stimulated photoluminescence. The theoretical Fano factor was calculated from eq. (4.2). Using appropriate estimates for the experimental arrangement ( q x 0.0025, /3 x 0.3, and F, x 0.1) provides an expected Fano factor F,(T) = 0.999, in good accord with the observed value. The small degree of sub-Poisson behavior is principally the result of optical losses in the experimental apparatus.

I.1.2. Space-charge-limited excitation of recombination radiation As indicated earlier a useful source of sub-Poisson light should exhibit a photon Fano factor that is substantially below unity while producing a large photon flux with high efficiency, preferably in a directed beam. It should also be small in size and rapidly switchable. This has led to a proposal for a semiconductor device structure in which sub-Poisson electron excitations are attained through space-charge-limited current flow, and single-photon emissions are achieved by means of recombi-

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81

nation radiation (TEICH,CAPASSO and SALEH[ 19871). Again, this scheme is of the form represented in fig. 7. la. A device of this nature will emit sub-Poisson recombination radiation. The energy-band diagram for such a space-chargelimited light-emitting device (SCL-LED) is illustrated in fig. 7.5. Sub-Poisson electrons are directly converted into sub-Poisson photons, as in the spacecharge-limited Franck-Hertz experiment, but these are now recombination photons in a semiconductor. In designing such a device, carrier and photon confinement should be optimized and optical losses should be minimized. The basic structure of the device is that of a p+-i-n’ diode. Near-infrared recombination radiation is emitted from the LED-like region. The current noise in such a space-charge-limited diode (SZE [ 19691) can be low. It has a thermal (rather than shot-noise) character (LAMPERT and ROSE [ 19611, NICOLET,BILGERand ZIJLSTRA [ 1975a,b]).The current noise spectral density S,(o) for a device in which only electrons participate in the conduction process is given by (TEICH,CAPASSOand SALEH[ 19871) (7.1) where ( i , ) is the average forward current in the device. ( V, ) is the applied forward-bias voltage, k is Boltzmann’s constant, B is the device temperature in K, o is the circular frequency, and e is the electronic charge. Using eqs. (4.1) and (43,the degree of sub-Poisson behavior of the detected photons is then expected to be (TEICH,CAPASSOand SALEH[ 19871)

a)

-

- -

0

-I i W- - - - - .

Fig. 7.5. Energy-band diagram of a specially constructed, solid-state space-charge-limited lightemitting device under (a) equilibrium conditions and (b) strong forward-bias conditions. The curvature of the intrinsic region under forward-bias conditions indicates the space-chargepotential. (Atter TEICH,CAPASSOand SALEH[1987].)

82


F,(T)

1+ q

[I. 8 1

( 8ke

- l), (7.2) e(ve> provided that background light is absent ( f l = 1). For a space-charge-limited diode, such as that shown in fig. 7.5, it is estimated that 8kO/e ( V, ) x 0.1 when 0 = 300 K and ( V, ) = 2 V (corresponding to ( i , ) x 33 mA). This ratio can be further reduced by cooling the device. If a dome-shaped surface-emitting GaAs/GaAlAs configuration and a Si p-i-n photodetector are used, the overall quantum efficiency is estimated to be q z 0.1125, yielding an overall estimated post-detection Fano factor F,( T ) x 0.899. A commercially available standard LED should provide F,(T) x 0.973. In both cases Tcan be as short as z 1 ns. The space-charge-limited light-emitting device therefore promises subPoisson light with properties that are significantly superior to those of the mercury-vapor space-charge-limited Franck-Hertz source discussed in the previous subsection (F,(T) x 0.998 with T x 1 ps). Indeed, the degree of sub-Poisson behavior of the recombination radiation from the SCL-LED is limited essentially only by the geometrical collection efficiency. =

~

1.1.3. Sub-Poisson excitations and stimulated emissions The properties of the light generated by the SCL-LED might be subject to improvement if stimulated emissions are permitted. These include improved beam directionality, switching speed, spectral properties, and coupling to an optical fiber. This could be achieved by the use of an edge-emitting (rather than surface-emitting)LED configuration, with its waveguidinggeometry and superfluorescence properties (single-pass stimulated emission). Although eqs. (4.1) and (4.5) were explicitly derived for independent photon emissions, they will apply even if the photon emissions are not independent, as is the case when stimulated emission plays a role, provided that T $ z, 7,; A % A , , where z, and A , are now the coherence time and coherence area of the superfluorescent emission, respectively.The effect of the stimulated emissions is to extend zp into z, and to reduce the coherence area A,. From a physical point of view the photons still behave as classical particles in this regime, since each electron gives rise to a single photon and there is no memory beyond the counting interval T. There will likely be further advantage in combining space-charge-limited current injection with a semiconductor laser structure rather than with a LED structure. This method would provide increased emission efficiency as well as additional improvement in beam directionality, switching speed, spectral

1 . 8 71


83

properties, and coupling. This will be beneficial when the laser can be drawn into a realm of operation in which it produces a state more akin to a number state than a coherent state (the coherent state has Poisson photon-number fluctuations and minimal phase fluctuations) (FILIPOWICZ, JAVANAINEN and MEYSTRE [ 19861, YAMAMOTO, MACHIDAand NILSSON[ 19861, YAMAMOTO, IMOTO and MACHIDA[ 1986b3, YAMAMOTOand MACHIDA [ 19871). MACHIDA,YAMAMOTO and ITAYA[ 19871 have shown that this mode of operation can be attained in a semiconductor laser oscillator, within the cavity bandwidth and at high photon-flux levels, if the pump fluctuations are suppressed below the shot-noise level, using external feedback to achieve the pump quieting (see 0 7.2.4). Related suggestions have been made by SMIRNOV [ 19861. and TROSHIN[ 19851 and by CARROLL 7.2. METHODS USING EXTERNAL FEEDBACK

A number of external-feedback mechanisms can be used to ensure that the current flowing in a circuit is sub-Poisson. These include both opto-electronic and current-stabilization schemes. In 0 7.2.1 we discuss the use of two negative-feedback schemes that rely on the use of a light source and detector in a feedback loop. However, under ordinary conditions the use of a beamsplitter to extract a portion of these in-loop photons is not useful for producing nonclassical light, as discussed in 0 7.2.2. In 0 7.2.3 we discuss the possibility of generating sub-Poisson photons from sub-Poisson electrons by making use of external excitation feedback and an in-loop auxiliary optical source. Sub-Poisson electrons flow through the auxiliary source and produce sub-Poisson photons en route. The photon number represents a nondestructive measurement of the electron number. The robustness of the electrons permits them to emit recombination photons without being destroyed. In this sense this configuration is like the Franck-Hertz experiment in which we begin with atoms and electrons and end with atoms, electrons and photons. The QND measurement discussed in 0 6.2.4,on the other hand, begins with atoms and photons and ends with atoms, photons and electrons, which is a more difficult process to achieve. Finally, in 0 7.2.4 we discuss the generation of sub-Poisson electrons by means of an electronic scheme, namely, external current stabilization.

7.2.1. Opto-electronic generation of sub-Poisson electrons Sub-Poisson excitations can be generated by the use of external feedback. Two opto-electronic experiments incorporating external feedback have been

84


TRIGGER

I-ASER

OPTICAL

LIGHT

GATE

-

DETECTOR

COUNTER

Fig. 7.6. Generation of antibunched and sub-Poisson electrons by external feedback, as studied by WALKERand JAKEMAN [1985a].

used to generate sub-Poisson electrons. One of these experiments was carried out by WALKERand JAKEMAN [ 1985al (see also BROWN,JAKEMAN, PIKE, RARITYand TAPSTER[ 19861) and the other by MACHIDAand YAMAMOTO [ 19861 (see also YAMAMOTO, IMOTOand MACHIDA[ 1986a1). The simplest form of the experiment carried out by WALKERand JAKEMAN [1985a] is illustrated in fig. 7.6. The registration of a photoevent at the detector operates a trigger circuit, which causes an optical gate to be closed for a fixed period of time z, following the time of registration. During this period, the power P, of the (He-He) laser illuminating the detector is set precisely equal to zero so that no photoevents are registered. This is the dead-time optical gating scheme shown schematically in fig. l.lb and discussed in 0 6.2.2. Sub-Poisson photoelectrons were observed. MACHIDAand YAMAMOTO’S [ 19861 experiment (fig. 7.7) has a similar thrust, although it is based on rate compensation (see fig. 1.lf). They used a single-longitudinal-mode GaAs/AlGaAs semiconductor injection laser diode (LD) to generate light and a Si p-i-n photodiode (PD) to detect it, as shown in fig. 7.7a. Negative electrical feedback from the detector was provided to the current driving the laser diode. A sub-shot-noise spectrum and sub-Poisson photoelectron counts were observed. The similarity in the experimental results reported by WALKERand JAKEMAN [ 1985aI and by MACHIDAand YAMAMOTO [ 19861 can be understood from a physical point of view. In the configuration used by the latter authors, the injection-laser current (and therefore the injection-laser light output) is reduced in response to peaks of the in-loop photodetector current i,. This rate compensation is essentially the same effect as that produced in the Walker-Jakeman experiment where the He-Ne laser light output is reduced (in their case to zero) in response to photoevent registrations at the in-loop photodetector. The feedback acts like a dead time, suppressing the emission of light in a manner that is correlated with photoevent occurrences at the in-loop detector.

1, I 71


(')

85

bias

Meas.urement Circuit

Fig. 7.7. (a) Generationof antibunched and sub-Poissonelectrons by external feedback using rate compensation, as investigated by MACHIDAand YAMAMOTO[1986]. (b) The removal of in-loop as photons by a beamsplitter leads to super-Poisson light at the out-of-loop detector (DB), understood from the arguments OfWALKER and JAKEMAN [ 1985al and SHAPIRO, TEICH, SALEH, KUMARand SAPLAKOGLU [1986]. (After MACHIDAand YAMAMOTO[1986].)

1.2.2. Extraction of in-loop photons by a beamsplitter Unfortunately, these simple configurations cannot generate usable subPoisson photons, since the feedback current controlling the source is generated from the annihilation of the in-loop photons. Indeed, any ordinary attempt to remove in-loop photons by means of a beamsplitter, such as that made by MACHIDAand YAMAMOTO [ 19861, as illustrated in fig. 7.7b, will lead to super-Poisson light. This result can be understood in terms of the arguments of WALKERand JAKEMAN [ 1985a1, SHAPIRO, TEICH, SALEH,KUMARand [ 19861, and SHAPIRO, SAPLAKOGLU, Ho, KUMAR,SALEHand SAPLAKOGLU TEICH[ 19871. A heuristic explanation for this phenomenon is as follows. The point process registered at the in-loop detector (DA)is a self-exciting point process (SEPP), providing sub-Poisson counts. Because there is no feedback from the out-ofloop detector (DB),however, it registers a doubly stochastic Poisson point process (DSPP). The laser-diode current fluctuations, regulated by the events at the in-loop detector, provide a form of asynchronous modulation of the light power seen by the out-of-loop detector, thereby leading to a photocount variance that is greater than the photocount mean. The result is confirmed by the experiments of WALKERand JAKEMAN [ 1985al. From a quantum-mechanical point of view, the culprit is the open port of the

86


[I, § 7

beamsplitter used for the extraction of light. It is possible, at least in principle, to use a beamsplitter to extract sub-Poisson photons if the open port of the beamsplitter is filled with squeezed-vacuum radiation (CAVES[ 19871). When components other than beamsplitters are used, the electrical feedback technique can be useful in generating sub-Poisson light. Two examples involving photon feedback have already been cited: the use of correlated photon pairs (as discussed in 6.2.1 to 5 6.2.3) and when a QND measurement may be made (as discussed in 5 6.2.4). 7.2.3. Use of an in-loop auxiliary optical source One of the more direct ways of producing antibunched and sub-Poisson light from a system making use of external feedback is to insert an auxiliary optical source in the path of the sub-Poisson electron stream, as suggested by CAPASSO and TEICH[ 19861. Two alternative configurations are shown in fig. 7.8. The character of the photon emitter is immaterial; it has been chosen to be a light-emitting diode (LED) for simplicity, but it could be a laser. In fig. 7.8a the photocurrent derived from the detection of light is negatively fed back to the LED input. It has been established both experimentally (MACHIDAand YAMAMOTO [ 19861) and theoretically (SHAPIRO, TEICH,SALEH,KUMARand SAPLAKOGLU [1986]) that, in the absence of the block labeled “source”, sub-Poisson electrons (i.e., a sub-shot-noise photocurrent) will flow in a circuit such as this. This conclusion is also valid in the presence of this block, which in this case acts simply as an added impedance to the electron flow. Incorporating this element into the system offers access to the loop and permits the sub-Poisson electrons flowing in the circuit to be converted into sub-Poisson photons by means of dipole electronic transitions. This process is achieved by replacing the detector used in the feedback configurations of MACHIDA and YAMAMOTO [ 19861 and WALKERand JAKEMAN [ 1985al with a structure that acts simultaneously as a detector and a source. The sub-Poisson electrons emit sub-Poisson photons and continue on their way. The configuration presented in fig. 7.8b is similar, except that the (negative) feedback current gates the light intensity at the output of the LED in the manner of Walker and Jakeman, rather than the current at its input in the manner of Machida and Yamamoto. Any similar scheme, such as selective deletion (SALEHand TEICH[ 19851) could be used as well. Two possible solid-state detector/source configurations have been suggested (CAPASSO and TEICH[ 19861). The scheme shown in fig. 7.9a makes use of sequential resonant tunneling (CAPASSO, MOHAMMED and CHO[ 19861) and

1,

s 71

87


SUB- POISSON PHOTONS

PHOTON EMITTER (LED)

+

PHOTON INTENSITY EMITTER W MODULATOR (LED) A

-

DETECTOR SOURCE

-

tb) Fig. 7.8. Genpration of antibunched and sub-Poisson photons by insertion of an auxiliary source into the path ofa sub-Poisson electron stream, as proposed by CAPASSO and TEICH[ 19861. Wavy lines represent photons; solid lines represent the electron current; I+ signifies the feedback time constant. The schemes represented in (a) and (b) make use ofthe sub-Poisson electron production and TEICH[1986].) methods illustrated in figs. 7.7 and 7.6, respectively. (After CAPASSO

single-photon electronic dipole transitions between the energy levels of a quantum-well heterostructure. The device consists of a reverse-biased p+-i-n' diode, where the p+ and n+ heavily doped regions have a wider bandgap than the high-field, light-absorbing/emitting i region. This arrangement ensures both high quantum efficiency at the incident photon wavelength (to which the p window layer is transparent) and high collection efficiency (due to the waveguide geometry) for the light generated by the electrons drifting in the i layer. An edge-emitting geometry is therefore appropriate. To maximize the collection efficiency, some of the facets of the device could be reflectively coated. The scheme shown in fig. 7.9b is similar, except that it uses the impact excitation of electroluminescent centers in the i region by drifting electrons. Of course, the ability of configurations such as these to generate sub-Poisson light requires a number of interrelations among various characteristic times associated with the system, much as those represented in eq. (4.1). +

88


1 h

Fig. 7.9.(a) Representative energy-band diagram of a quantum-well detector/source device (see fig. 7.8). The energy of the incident photon emitted by the LED is denoted f r q . Detection and source regions are shown. Photons of energy fro32are emitted by means of electronic quantumwell transitions. (b) Representative energy-band diagram of a detector/source device with electroluminescent centers impact-excited by energetic photoelectrons, emitting photons with energy h o , . (After CAPASSO and TEICH[1986].)

An estimate of the degree to which this mechanism will give rise to subPoisson light is, of course, provided by the Fano factor. The relevant relations are similar to those for the Franck-Hertz source, since the emissions are independent. However, in this situation a single electron may give rise to multiple photons, since there is a number of stages in the device. We consider a sub-Poisson electron counting process e, each event of which independently generates a random number of photons o! in the source. The overall photonnumber Fano factor F,(T) can then be represented in terms of the Fano factor for the electron number FJT) and the Fano factor for the source random variable F,(T). From eq. (3.48), the relationship is F,

=

(a>F, + Fa,

(7.3)


89

where ( a ) is the average number of photons generated in the source by each electron. For the case at hand it is reasonable to assume that the source random variable is Bernoulli distributed in each stage of the device, with the probability that an electron gives rise to a photon denoted u,. No generality is lost by considering the multilayer superlattice case, which consists of u independent stages. The source statistics will then be described by a binomial random variable with ( a ) = uqr and Var(a) = uq,(l - q,). In the presence of random deletion arising from other factors (e.g., finite geometrical photon-collection efficiency, absorption, external detection) and background or dark photons, these results remain valid if qr is replaced by the quantity qfi, where q is the overall quantum efficiency from electrons to detected photons and fi is the factor representing the admixture of independent dark and/or background events (see 0 4.2). Equation (7.3) then gives rise to

which differs from eq. (4.2) in that it depends on u. It is evident that sub-Poisson behavior is achieved when F, < l/u. The lowest Fano factor at the output is achieved when u = 1. In this case the photon counting process is simply a randomly deleted version of the electron counting process so that eq. (7.4) reduces to eq. (4.2). Assuming that fi x 1, numerical estimates for the Fano factor turn out to be similar for both structures illustrated in fig. 7.9, viz. F, x 0.968 (under the assumption that the photodetector has an external quantum efficiency of 0.8). This provides a substantial potential improvement over the value observed in the space-charge-limited Franck-Hertz experiment. However, the Fano factor is not as low as that attainable by the SCL-LED, principally because of low radiative efficiency in the tunneling scheme. Furthermore the external feedback mechanism is likely to be slower than the internal feedback scheme of the SCL-LED.

7.2.4. Use of a current source with external compensation Probably the simplest way of achieving sub-Poisson electron counting statistics and single-photon emissions is by discharging a capacitor C through a circuit containing a photon emitter such as a light-emitting diode (LED). The current waveform then will be a nonstationary pulse with time constant zRc = RC (where R is the resistance of the circuit). Steady-state current stabilization can be achieved by the use of a constant voltage source in series

90


[L § 7

with a sufficiently large external resistor R (YAMAMOTO, MACHIDAand NILSSON [ 1986],Y~MAMoToand MACHIDA [ 1987]), or in series with some other optoelectronic component with suitable I- V characteristic. Strong sub-Poisson light has recently been generated in two experiments that make use of external compensation. TAPSTER, RARITYand SATCHELL [ 19871 carried out an elegantly simple experiment, using a high-efficiency commercial GaAs LED fed by a Johnson-noise-limited high-impedance current source. They achieved a Fano factor F, x 0.96 over a bandwidth of about 100 kHz, with a current transfer efficiency 4 in excess of 1 1% . MACHIDA, YAMAMOTO and ITAYA [ 19871 fed a InGaAsP/InP single-longitudinal-mode distributedfeedback laser oscillator, operating at a wavelength of 1.56 pm, with a current source whose fluctuations were suppressed by the use of an external highimpedance element. These authors obtained an average Fano factor F, x 0.96 over a bandwidth of about 100 MHz, with a minimum Fano factor F, x 0.93. They calculate that the radiation produced by their device is in a near number-phase minimum-uncertainty state (JACKIW [ 1968]), in the frequency range below the cavity bandwidth (which is in excess of 100 GHz for a typical semiconductor laser). These results are impressive. It should be kept in mind, however, that the characteristic electron anticorrelation time zf in external feedback circuits such as these is likely to be larger than z, for space-chargelimited electron excitations, as pointed out earlier. 7.3. LIMITATIONS OF EXCITATION-FEEDBACK METHODS

We have shown that the generation of sub-Poisson light is best achieved by the use of sub-Poisson electron excitations, mediated by a physical mechanism such as space charge, and a single photon emission for each excitation. This method is in general superior to nonlinear-optics methods. A space-chargelimited light-emittingstructure that operates in this manner has been discussed. In all cases using external feedback, the characteristic anticorrelation time of the excitations T , is determined by the feedback time constant of the loop z., A lower limit on the feedback time constant is imposed by the response time and transit time of carriers through the device and by the RC characteristics of the feedback circuitry. In general, an internal feedback process such as space charge will provide a more effective means of providing sub-Poisson excitations than external feedback. This is because an internal physical process is likely to result in a smaller value of T, than will external electronic circuitry. Configurations making use of space-charge-limited excitations will therefore have the capacity of being switched faster than those making use of external

1, § 81

INFORMATION TRANSMISSION U S I N G SUB-POISSON L I G H T

91

feedback, although this distinction is not likely to be important if external switching can be used.

4 8. Information Transmission using Sub-Poisson Light Sub-Poisson light may find use in the study of optical interactions in various disciplines, ranging from the behavior of the human visual system at the threshold of seeing (TEICH, PRUCNAL,VANNUCCI, BRETONand MCGILL [ 19821) to optical precision measurement (JAKEMAN and RARITY[ 19861). In this section we consider the potential use of sub-Poisson light in direct-detection lightwave communication systems and other information carrying applications. Systems of this kind that have been developed to date make use of Poisson or super-Poisson light (GAGL~ARDI and KARP[ 19761, HELSTROM[ 19761, SALEH [ 19781, KOGELNIK[ 19851, HENRY[ 19851, SENIOR[ 19851). There are essentially two classes of mechanisms by means of which unconditionally sub-Poisson photons may be generated. Sub-Poisson photons can be produced from a beam of initially Poisson (or super-Poisson) photons, represented by the photon-feedback examples of 6 (see fig. 1.1). Alternatively, unconditionally sub-Poisson photons may be directly generated from subPoisson excitations, as represented by the examples of § 7. In $5 8.1 and 8.2 we discuss the channel capacity of a lightwave communication system based on the observation of the photoevent point process, demonstrating that it cannot in principle be increased by the use of sub-Poisson light. In 0 8.3, on the other hand, we show that the channel capacity of a photon-counting system can be increased by the use of sub-Poisson light (SALEH and TEICH[ 19871). The channel capacity is the maximum rate of information that can be transmitted through a channel without error. The capacity of the photon channel has been the subject of a number of studies over the years (STERN[ 19601, GORDON[ 19621, PIERCE,POSNERand RODEMICH[ 19811, YAMAMOTO and HAUS[ 19861). In 8.4,we provide an example in which the use of sub-Poisson light produced from Poisson light either degrades or enhances the error performance of a simple binary ON-OFF keying photoncounting system, depending on where the average power constraint is placed. Finally, in 8.5, we conclude with a discussion pertaining to some limitations on direct-detection communications using sub-Poisson light.

92


Poisson point process

Sub-Poisson point process

Nt(X)

Mt(X)

MODULATOR MOD IF IER

Signal X t

t

A.

Signal e s t i m a t e Xt

Fig. 8.1. Idealized lightwave communication system employing a Poisson photon source and a photon-statistics modifier.

8.1. COMMUNICATING WITH MODIFIED POISSON PHOTONS

Consider the transformation of a Poisson beam of photons (represented by a Poisson point process N, of rate p,) into a sub-Poisson beam of photons represented by a point process M, of rate I t , as illustrated in fig. 8.1. The events of the initial process N, are assumed to be observable (e.g., by the use of correlated photon beams or a QND measurement) and their registrations used to operate a mechanism which, in accordance with a specified rule, leads to the events of the transformed photon process M,. The rate I, of the process M,is thereby rendered a function of the realizations of the initial point process N, at prior times, i.e., A, = ,I,(Nrr; t’ < 2 ) . Several examples of transformations of this kind that have been suggested for use in quantum optics have been discussed earlier and were illustrated in fig. 1.1. They include dead-time deletion, coincidence decimation, decimation, and overflow count deletion. We proceed to illustrate that none of these modifications can increase the channel capacity C of a communication system based on photoevent point-process observations. If a constraint is placed on the rate of the initial Poisson process pz < pmax, then it is obvious that C cannot be increased by the modification N, -,M,. This is simply a consequence of the definition of channel capacity: it is the rate of information carried by the system without error, maximized over all coding, modulation, and modijication schemes. Can the modification N, + M, increase the channel capacity if the constraint is instead placed on the rate of the We address this question for an arbitrary modified process I , (i.e., A, < A,,)? self-exciting point process in the next section.

INFORMATION TRANSMISSION USING SUB-POISSON LIGHT

1 7 § 81

93

8.2. COMMUNICATING WITH SUB-POISSON PHOTONS DESCRIBED BY A

SELF-EXCITING POINT PROCESS

Consider a self-exciting point process M, of rate I,(M,,; t‘ < t). This is a process that contains an inherent feedback mechanism in which present event occurrences are affected by the previous event occurrences of the same point process. Of course, the modifled Poisson processes N, -+ M, introduced above are special cases of self-exciting point processes. An example of a system that generates a self-exciting point process is that of rate compensation (by linear feedback) of a source which, without feedback, would produce a Poisson process. Let each photon registration at time ti cause the rate of the process to be modulated by a factor h(t - t,) (which vanishes for t < ti). In linear negative feedback the rate is I , = I , - x , h ( t - ti), where I , is a constant. If the instantaneous photon registration rate happens to be above the average then it is reduced, and vice versa. This process is schematically illustrated in fig. l.lf for two adjacent sub-intervals T , and T,. Now consider a communication system that uses a point process M , ( X ) whose rate I , @ ) is modulated by a signal X,. The process M,(X) can be an arbitrary self-excitingpoint process (e.g., it can be sub-Poisson) which includes processes obtained by the feedforward- or feedback-modification of an otherwise Poisson process. Neither feedforward nor feedback transformations can increase the capacity of this channel, as provided by Kabanov’s theorem (KABANOV [ 1978]), and its extensions (DAVIS[ 19801, LAZAR[ 19801): Kabanov’s theorem. The capacity of the point-process channel cannot be increased byfeedback. Under the constraint I , 4 I , 4 I,,,, the channel capacity C is

where s to

=

Amax

-

I,. When I ,

=

0 (no dark counts), this expressions reduces

C = -L a x e

When the capacity is achieved, the output of the zero-dark-count point-process channel is a Poisson process with rate I , = Imax/e(the base e has been used for simplicity). The channel capacity has also been determined under added constraints on the mean rate. A coding theorem has also been proved. Kabanov’s theorem is analogous to the well-known result that the capacity of

94

[I.


58

the white Gaussian channel cannot be increased by feedback (KADOTA, ZAKAIand ZIV [ 19711). In summary, no increase in the channel capacity of a point-process lightwave communication system may be achieved by using photons that are first generated with Poisson statistics and subsequently converted into sub-Poisson statistics regardless of whether the power constraint is placed at the Poisson photon source or at the output of the conversion process. Nor may an increase in channel capacity be achieved by using feedback to generate a self-exciting point process.

8.3. COMMUNICATING WITH SUB-POISSON PHOTON COUNTS

The conclusions of $0 8.1 and 8.2 are valid only when there are no restrictions on the receiver structure. The conclusion is different if the receiver is operated in the photon-counting regime, in which information is carried by a random variable n representing the number of photoevents registered in time intervals of prescribed duration T (rather than by the photon occurrence times). The capacity of the photon-counting channel is given by (GORDON [ 19621) (8.3) where ( n ) is the mean number of counts in T and B expressions emerge:

C=Bln((n)),

(n) 4 1 .

=

1/T. Two limiting

(8.4)

If an added constraint is applied to the photon counts, such that they must obey the Poisson counting distribution, the capacity is further reduced. In that case, the limiting results analogous to eq. (8.4) are

c = :B I n ( ( n ) ) ,

(n) 4 1 .

(8.5)

In the case of photon counting, therefore, an increase in the channel capacity can in principle be realized by using sub-Poisson light. However, in the small

1, § 81

INFORMATION TRANSMISSION USING SUB-POISSON LlGHT

95

mean-count limit ( n ) 4 1 (very short T ) ,the capacity of the Poisson counting channel approaches that of the unrestricted counting channel, and the advantage of sub-Poisson light disappears. This is not unexpected in view of the result obtained from Kabanov’s theorem for the point-process channel. 8.4. PERFORMANCE OF A SUB-POISSON PHOTON-COUNTING RECEIVER

The channel capacity provides a limit on the maximum rate of error-free information transmission for all codes, modulation formats, and receiver structures. As such, it does not specify the performance (error probability) achievable by a communication system with prescribed coding, modulation, and receiver structure. It is therefore of interest to examine the performance of a system with specified structure. We consider a binary ON-OFF keying (OOK) photoncounting system (GAGLIARDI and KARP[ 19761, HELSTROM[ 19761, SALEH [ 19781, KOGELNIK[ 19851, HENRY[ 19851, SENIOR[ 19851). The information is transmitted by selecting one of two values for the photon rate A,, in time slots of (pulse) duration T. The receiver operates by counting the number of photons received during the time interval T and then deciding which rate was transmitted in accordance with a likelihood-ratio decision rule (threshold test). For simplicity, it is assumed that background light, dark noise, and thermal noise are absent so that photon registrations are not permitted when the keying is OFF (i.e., false alarms are not possible). Furthermore, the detector quantum efficiency is initially taken to be unity so that system performance is limited only by the quantum fluctuations of the light. A measure of performance for a digital system such as this is the error probability P,. In the simplified system described above, errors are possible only when the keying is ON and 0 photons are received (a miss). For a Poisson transmitter, with equal a priori probabilities for ON and OFF, P, is (HENRY [ 19851) PJPoisson) = $ exp( - (n)), (8.6) where ( n ) denotes the mean number of photons in the time T (that is, the number of photons/pulse). To minimize P,. ( n ) is made equal to its maximum allowed value This result is now compared with those obtained for sub-Poisson light derived from an initially Poisson source. The outcome will depend on where the mean photon-number constraint is placed. Two transformations are explicity considered: dead-time deletion and decimation.

96


[I, $ 8

It will become evident from these examples that system performance can be enhanced by using sub-Poisson light, provided that the power constraint is applied to the sub-Poisson light. No enhancement of system performance emerges in converting Poisson photons into sub-Poisson photons when the average power constraint is at the Poisson source. 8.4.1. Dead-time-modifid-Poisson photon counts

For a nonparalyzable dead-time modifier that is always blocked for a dead time period 7, at the beginning of the counting interval T, the passage of 0 photons arises from the emission of 0 photons in the time T - zd, independent of the number of emissions during T ~ The . error probability for this system is therefore PJdead-time)

=

$ exp[ - ( n ) (1

-

91. T

To minimize error under the conslraint ( n ) < (n),,,, we take ( n ) = ( n ) m,x. The error probability is obviously larger than that for the Poisson channel (eq. 8.6) so no performance enhancement can be achieved by use of this modifier with this constraint. If, instead, the dead-time modifier is always unblocked at the beginning of each bit then the passage of 0 photons can arise only from the emission of 0 photons in the time T, and the dead-time has no effect on the error rate in this simple system. Calculations for the unblocked counter in the presence of false alarms, however, demonstrate that the presence of dead time always does, in fact, degrade system performance with such a constraint (TEICHand CANTOR [ 19781). Although these detailed calculations were carried out for electrical dead time, the results are also applicable for optical dead time when the photon detection efficiency q = 1. On the other hand, if the constraint is placed on the mean photon count ( m ) after dead-time modification ( ( m ) < ( m ) m , x ) , it can be shown that there exists a value of ( m ) max below which performance is degraded, and above which performance is improved, relative to the Poisson channel. 8.4.2. Decimated-Poisson photon counts

We assume that the decimation parameter N = 2 (i.e., every other photon of a Poisson sequence of events is selected) and that the decimation process is reset at the beginning of each bit (i.e., the first photon in each bit is not selected).

1, § 81

INFORMATION TRANSMISSION USING SUB-POISSON LIGHT

91

The error probability is then P,(decimation)=;(l

+ (n))exp(-

(n)),

(8.8)

which again represents a degradation of performance in comparison with the Poisson channel (under a constraint ( n ) < ( n ) ).,, In this case, the error rate is increased because there are two ways for the passage of 0 photons to arise in the time T : from the emission of 0 photons or from the emission of 1 photon. However, if the constraint is placed on the modified process then, once again, there exists a value of ( m ) max below which performance is degraded and above which it is improved, relative to the Poisson channel. 8.4.3. Binomial photon counts

We conclude by considering the effects of photon deletion. We do this in the context of an ideal sub-Poisson source that generates a deterministic photon number. This is an important consideration because random photon deletion is inevitable; it results from absorption, scattering, and the finite quantum efficiency of the detector, as discussed in $4.It is well known that such deletions will transform a deterministic photon number into a binomial photon number (MANDEL[ 1976a]), which always remains sub-Poisson but approaches the Poisson boundary as the photon-survival probability q decreases (TEICHand SALEH[ 19821). MANDEL [ 1976al has shown that the information rate per symbol carried by such a counting channel will be greater than that for the Poisson channel, but will approach the latter as q approaches 0. A source that emits a binomial photon number at the outset (STOLER, SALEH and TEICH[1985], DATTOLI, GALLARDO and TORRE[1987]) retains its binomial form, but exhibits a reduced mean, in the presence of random deletion (TEICHand SALEH[ 19821). The performance of such a binary OOK photon-counting receiver, in the absence of background, is limited by the binomial fluctuations of the detected photons. In this case, it is easily shown from the binomial distribution that P,(binomial)

=

$F:2

Fig. 8.2. Error probability ( P e ) versus mean number of photons per bit ( n ' ) for the binomial channel, with the Fano factor F, as a parameter. System performance clearly improves as F, decreases below unity.

Solving eq. (8.9) for the mean number of photons per bit (n') provides (8.10) which leads to a direct-detection quantum limit that is < 10 photons/bit ( < 20 photons/pulse) for OOK, if F, < 1 and P, = 10 - '. The mean number of photons per bit (n' ) is plotted as a function of F, in fig. 8.3. The usual quantum limit ((n') = 10 photons/bit) emerges in the limit F, = 1 where the binomial distribution goes over to the Poisson. 8.5. LIMITATIONS ON COMMUNICATING WITH SUB-POISSON LIGHT

Sub-Poisson light sources can, in principle, be useful in lightwave communications systems. However, their use will only be practical if they can be made to exhibit high photon flux, low Fano factor, and a short feedback time constant, and if losses in the system as a whole are minimized. Fortunately, photomultiplier tubes and even avalanche photodiodes can (at least in principle)

99

REFERENCES

0

0.2

0.4

6.6

0.8

FAN0 FACTOR (Fn)

Fig. 8.3. Mean number of photons per bit ( h ’ ) as a fufictkm of the Fano factor F. for the binomial channel. The well-known “quantum limit” (10 photons/Bit) emerges as the binomial distribution goes over to the Poisson distribution (F, --t I).

detect sub-Poisson light in an essentially noise-ffee manner (TEICH,MAXs\K, and SALEH[ 19861). The short feedback time constunt permits the srgntdmg rate to be high. In conventional systems (i.e., those usmg Poisson Ughtb this rate is determined by the time character of the source and receiver, subject to there being a sufficiently large number of photons per bit (HENRY Il!J85])1 However, for systems using sub-Poisson photons, the symbol duratidm Tmusf exceed the anticorrelation time of the photons (teor tr)so that the sub-Poism nature of the signal is captured (see Q 7). Solid-state Implementattons of single-photon emission devices driven by sub-Poisson currents should therefore be constructed in such a way that z, is made as small as possible (TEICR, CAPASSO and SALEH[ 19871). References ANDREWS, L.C.,and R.L. PHILLIPS,1986, J. Opt. SOC.Am. A 3, 1912. ARECCHI.F.T.. E. GATTIand A. SONA, 1966, Phys. Lett. u),27. ASPECT,A., 1986, private communication. ASPECT,A., P. GRANGIERand G. ROGER, 1981, Phys. Rev. Lett. 47,460. ASPECT, A., J. DALIBART and G . ROGER,1982, Phys. Rev. b t t . 49, 1804. ASPECT,A., P. GRANGIER and G. ROGER, 1982, Phys. Rev. Lett. 49,91. BARAKAT, R., 1974, Opt. Acta 21, 903.

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BARAKAT, R., 1976, J. Opt. SOC.Am. 66, 211. BARAKAT, R., and J. BLAKE,1976, Phys. Rev. A 13, 1122. BARAKAT, R., and J. BLAKE,1980, Phys. Rep. 60,255. BERAN,M.J., and G.B. PARRENT JR, 1964, Theory of Partial Coherence (Prentice Hall, Englewood Cliffs, NJ). BORN,M., and E. WOLF, 1980, Principles of Optics, 6th Ed. (Pergamon, Oxford). BRAGINSKY, V.B., and Yu.1. VORONTSOV, 1974, Usp. Fiz. Nauk. 114,41 [1975, Sov. Phys.-Usp. 17, 6441. BRAGINSKY, V.B., and S.P. VYATCHANIN,1981, Dokl. Akad. Nauk SSSR 259, 570 [Sov. Phys.-Dokl. 26, 6861. BRAGINSKY, V.B., and S.P. VYATCHANIN, 1982, Dokl. Akad. Nauk SSSR 264, 1136 [sov. Phys.-Dokl. 27, 4781. E.R. PIKE,J.G. RARITYand P.R. TAPSTER, 1986, Europhys. Lett. BROWN, R.G.W., E. JAKEMAN, 2, 279. BURGESS, R.E., 1961, J. Phys. & Chem. Solids 22, 371. BURNHAM, D.C., and D.L. WEINBERG, 1970, Phys. Rev. Lett. 25, 84. CANTOR,B.I., and M.C. TEICH,1975, J. Opt. SOC.Am. 65, 786. F., and M.C. TEICH, 1986, Phys. Rev. Lett. 57, 1417. CAPASSO, and A.Y. CHO, 1986, Appl. Phys. Lett. 48,478. CAPASSO, F., K. MOHAMMED CARMICHAEL, H.J., and D.F. WALLS,1976a. J. Phys. B 9, L43. CARMICHAEL, H.J., and D.F. WALLS,1976b, J. Phys. B 9, 1199. CARMICHAEL, H.J., P. DRUMMOND, P. MEYSTRE and D.F. WALLS,1978, J. Phys. A 11, L121. CARROLL, J.E., 1986, Opt. Acta 33, 909. CARRUTHERS, P., and M.M. NIETO,1968, Rev. Mod. Phys. 40,411. CAVES,C.M., 1986, Amplitude and phase in quantum optics, in: Coherence, Cooperation, and Fluctuations, eds F. Haake, L.M. Narducci and D.F. Walls (Cambridge Univ. Press, Cambridge). CAVES,C.M., 1987, Opt. Lett. 12, 971. R.W.P. DREVER, V.D. SANDBERG and M. ZIMMERMAN, 1980, Rev. CAVES,C.M., K.S. THORNE, Mod. Phys. 52, 341. CHANDRA, N., and H. PRAKASH, 1970, Phys. Rev. A 1, 1696. CHEN, S.H., and P. TARTAGLIA, 1972, Opt. Commun. 6, 119. COOK,R.J., 1980, Opt. Commun. 35, 347. COOK,R.J., 1981, Phys. Rev. A 23, 1243. Cox, D.R., 1955, J. R. Stat. SOC.B 17, 129. Cox, D.R., 1962, Renewal Theory (Methuen, London). CRESSER, J.D., J. HAGER,G. LEUCHS,F.-M. RATEIKEand H. WALTHER,1982, Resonance fluorescence of atoms in strong monochromatic laser fields, in: Dissipative Systems in Quantum Optics, Topics in Current Physics, Vol. 27, eds R. Bonifacio and L.A. Lugiato (Springer, Berlin) p. 21. DAGENAIS, M., and L. MANDEL,1978, Phys. Rev. A 18, 2217. and A. TORRE,1987, J. Opt. SOC.Am. B 4, 185. DATTOLI,G.,J. GALLARDO DAVIS,M.H.A., 1980, IEEE Trans. Inf. Theory IT-26, 710. DIEDRICH, F., and H. WALTHER,1987, Phys. Rev. Lett. 58, 203. K.J., 1980, Opt. Commun. 35, 323. EBELING, EINSTEIN, A,, 1909, Phys. Z. 10, 185. EVERY, I.M., 1975, J. Phys. A 8, L69. FANO, U., 1947, Phys. Rev. 72, 26. FILIPOWICZ, P., J. JAVANAINEN and P. MEYSTRE,1986, Phys. Rev. A 34, 3077. FORRESTER, A.T., 1972, J. Opt. SOC.Am. 62, 654. FRANCK, J., and G. HERTZ,1914, Verhandl. Deutsch. Phys. Ges. (Berlin) 16, 512. FRANCK, J., and P. JORDAN, 1926, Anregung von Quantenspriingen durch StbBe, in: Struktur der Materie in Einzeldarstellungen, I11 (Springer, Berlin).

I1

REFERENCES

101

FRIBERG, S., C.K. HONG and L. MANDEL,1985, Phys. Rev. Lett. 54, 2011. GABOR,D., 1946, J. Inst. Electron. Eng. 93, 429. GAGLIARDI, R.M., and S. KARP,1976, Optical Communications (Wiley, New York). CHIELMETTI, F., 1976, Nuovo Cimento B 35, 243. GILBERT, E.N., and H.O. POLLAK,1960, Bell Syst. Tech. J. 39, 333. GLAUBER, R.J., 1963a. Phys. Rev. 130, 2529. GLAUBER, R.J., 1963b, Phys. Rev. 131, 2766. GOODMAN, J., 1985, Statistical Optics (Wiley, New York). GORDON,J.P., 1962, Proc. IRE 50, 1898. GRANGIER, P., G. ROGERand A. ASPECT, 1986, Europhys. Lett. 1, 173. GRANGIER, P., G. ROGER, A. ASPECT,A. HEIDMANN and S. REYNAUD, 1986, Phys. Rev. Lett. 57, 687. HANBURY-BROWN, R., 1974, The Intensity Interferometer (Taylor and Francis, London). HANBURY-BROWN, R., and R.Q. TWISS,1956a, Nature 177, 27. HANBURY-BROWN, R., and R.Q. Twiss, 1956b, Nature 178, 1046. E. GIACOBINO and C. FABRE,1987, Phys. Rev. HEIDMANN, A., R.J. HoRowIcz, S. REYNAUD, Lett. 59, 2555. HELSTROM, C.W., 1976, Quantum Detection and Estimation Theory (Academic, New York). HENRY,P.S., 1985, IEEE J. Quantum Electron. QE-21, 1862. HOENDERS, B.J., E. JAKEMAN, H.P. BALTESand B. STEINLE,1979, Opt. Acta 26, 1307. HONG,C.K., and L. MANDEL,1985, Phys. Rev. A 31, 2409. HONG,C.K., and L. MANDEL,1986, Phys. Rev. Lett. 56, 58. IMOTO, N., H.A. HAUSand Y.YAMAMOTO,1985, Phys. Rev. A 32, 2287. JACKIW,R., 1968, J. Math. Phys. 9, 339. JAKEMAN, E., 1980a, Proc. SOC.Photo-Opt. Instrum. Eng. 243, 9. JAKEMAN, E., 1980b. J. Phys. A 13, 31. JAKEMAN, E., 1982, J. Opt. SOC.Am. 72, 1034. JAKEMAN, E., 1983, Opt. Acta 30, 1207. JAKEMAN, E., 1986, The use of photo-event triggered optical shutters to generate sub-Poissonian photo-electron statistics, in: Frontiers in Quantum Optics (Adam Hilger, London) p. 342. JAKEMAN, E., and B.J. HOENDERS,1982, Opt. Acta 29, 1598. E., and J.H. JEFFERSON,1986, Opt. Acta 33,557. JAKEMAN, JAKEMAN, E., and P.N. PuSEY, 1976, IEEE Trans. Antennas & Propag. 24, 806. JAKEMAN, E., and P.N. PUSEY,1978, Phys. Rev. Lett. 40,546. JAKEMAN, E., and J.G. RARITY,1986, Opt. Commun. 59, 219. JAKEMAN, E., and J.G. WALKER,1985, Opt. Commun. 55, 219. JAKEMAN, E., E.R. PIKE, P.N. PUSEYand J.M. VAUGHAN,1977, J. Phys. A 10, L257. KABANOV, Yu.M., 1978, Theory Probab. & Appl. 23, 143. KADOTA,T.T., M. ZAKAIand I. ZIV, 1971, IEEE Trans. Inf. Theory IT-17, 368. KARP,S., 1975, J. Opt. SOC.Am. 65, 421. KARP,S., R.M. GAGLIARDI and I.S. REED, 1968, Proc. IEEE 56, 1704. KELLEY,P.L.,and W.H. KLEINER,1964, Phys. Rev. A 136, 316. KIMBLE,H.J., and L. MANDEL,1976, Phys. Rev. A 13, 2123. KIMBLE,H.J., M. DAGENAIS and L. MANDEL,1977, Phys. Rev. Lett. 39, 691. KIMBLE,H.J., M. DAGENAIS and L. MANDEL,1978, Phys. Rev. A 18, 201. KOGELNIK, H., 1985, Science 228, 1043. LAMPERT,M.A., and A. ROSE, 1961, Phys. Rev. 121, 26. LAX,M., and M. ZWANZIGER, 1973, Phys. Rev. A 7, 750. LAZAR,A.A., 1980, On the capacity of the Poisson-type channel, in: Proc. 14th Ann. Conf. on Information Sciences and Systems, March 2 6 2 8 , 1980 (Princeton, NJ). LENSTRA,D., 1982, Phys. Rev. A 26, 3369. LEVENSON, M.D., R.M. SHELBY, M. REID and D.F. WALLS,1986, Phys. Rev. Lett. 57, 2473. LOUDON,R., 1976, Phys. Bull. 27, 21.

102


LOUDON,R., 1980, Rep. Prog. Phys. 43, 913. LOUDON,R., 1983, The Quantum Theory of Light, 2nd Ed. (Clarendon, Oxford). LOUDON,R., and P.L. KNIGHT,1987, J. Mod. Opt. (Opt. Acta) 34, 709. LOUDON,R., and T.J. SHEPHERD,1984, Opt. Acta 31, 1243. LOUISELL, W.H., A. YARIVand A.E. SIEGMAN,1961, Phys. Rev. 124, 1646. MACHIDA,S., and Y. YAMAMOTO, 1986, Opt. Commun. 57, 290. and Y. ITAYA,1987, Phys. Rev. Lett. 58, 1000. MACHIDA,S., Y. YAMAMOTO MAEDA,M.W., P. KUMARand J.H. SHAPIRO,1987, Opt. Lett. 12, 161. MANDEL,L., 1958, Proc. Phys. SOC.(London) 72, 1037. MANDEL,L., 1959a, Proc. Phys. SOC.(London) 74, 233. MANDEL,L., 1959b, Br. J. Appl. Phys. 10, 233. MANDEL,L., 1963, Fluctuations of light beams, in: Progress in Optics, Vol. 2, ed. E. Wolf (North-Holland, Amsterdam) p. 181. MANDEL,L., 1976a, J. Opt. SOC.Am. 66, 968. MANDEI.,L., 1976b, The case for and against semiclassical radiation theory, in: Progress in Optics, Vol. 13, ed. E. Wolf (North-Holland, Amsterdam) p. 27. MANDEL,L., 1979, Opt. Lett. 4, 205. MANDEI.,L., 1983, Phys. Rev. A 28, 929. MANDEL,L., and E. WOLF, 1965, Rev. Mod. Phys. 37, 231. MATSUO,K., M.C. TEICHand B.E.A. SALEH,1983, Appl. Opt. 22, 1898. MIDDLETON, D., 1967a, IEEE Trans. Inf. Theory 13, 372. MIDDLETON, D., 1967b, IEEE Trans. Inf. Theory 13, 393. MILBURN, G.J., and D.F. WALLS, 1983, Phys. Rev. A 28, 2065. MILLER,M.M., and E.A. MISHKIN,1967, Phys. Lett. A 24, 188. MITCHELL, R.L., 1968, J. Opt. SOC.Am. 58, 1267. MOLLOW,B.R., and R.J. GLAUBER,1967a, Phys. Rev. 160, 1076. MOLLOW,B.R., and R.J. GLAUBER, 1967b. Phys. Rev. 160, 1097. MORGAN,B.L., and L. MANDEL,1966, Phys. Rev. Lett. 16, 1012. MOULLIN,E.B., 1938, Spontaneous Fluctuations of Voltage due to Brownian Motions of Electricity, Shot Effect, and Kindred Phenomena (Oxford Univ. Press, Oxford) p. 241. MOLLER,J.W., 1974, Nucl. Instrum. & Methods 117, 401. ZIJLSTRA,1975a, Phys. Status Solidi b 70, 9. NICOLET,M.A., H.R. BILGERand R NICOLET,M.A., H.R. BILGERand R.J.J. ZIJLSTRA,1975b. Phys. Status Solidi b 70, 415. ODONNELL,K.A., 1982, J. Opt. SOC.Am. 72, 1459. OLIVER, C.J., 1984, Opt. Acta 31, 701. PAPOULIS,A., 1984, Probability, Random Variables and Stochastic Processes, 2nd Ed. (McGraw-Hill, New York). PARRY,G., and P.N. PUSEY,1979, J. Opt. SOC.Am. 69, 796. PARZEN,E., 1962, Stochastic Processes (Holden-Day, San Francisco, CA). PAUL,H., 1966, Fortschr. Phys. 145, 141. PAUL,H., 1982, Rev. Mod. Phys. 54, 1061. PERINA.J., 1984, Quantum Statistics of Linear and Nonlinear Optical Phenomena (Reidel, Dordrecht). PERINA,J., 1985, Coherence of Light, 2nd Ed. (Reidel, Dordrecht). PERINA,J., B.E.A. SALEHand M.C. TEICH,1983, Opt. Commun. 48, 212. PERINA. J., V. PERINOVA and J. KODOUSEK,1984, Opt. Commun. 49, 210. PICINBONO, B., C. BENDJABALLAH and J. POUGET,1970, J. Math. Phys. 11, 2166. PIERCE,J.R., E.C. POSNERand E.R. RODEMICH,1981, IEEE Trans. Inf. Theory IT-27, 61. and D.E. KOPPEL,1974, J. Phys. A 7, 530. PUSEY.P.N., D.W. SCHAEFER RARITY,J.G., P.R. TAPSTER and E. JAKEMAN,1987, Opt. Commun. 62, 201. REID,M.D., and D.F. WALLS,1984, Phys. Rev. Lett. 53, 955. RICCIARDI, L.M., and F. ESPOSITO,1966, Kybernetik (Biol. Cybern.) 3, 148.

I1

REFERENCES

I03

RICE,S.O., 1944, Mathematical analysis ofrandom noise, Bell. Syst. Tech. J. 23, I; reprinted 1954, in: Selected Papers on Noise and Stochastic Processes, ed. N. Wax (Dover, New York) p. 133. SALEH,B.E.A., 1978, Photoelectron Statistics (Springer, Berlin). SALEH,B.E.A., and M.C. TEICH,1982, Proc. IEEE 70, 229. SALEH,B.E.A., and M.C. TEICH,1983, IEEE Trans. Inf. Theory 29, 939. SALEH,B.E.A., and M.C. TEICH,1985, Opt. Commun. 52, 429. SAI.EH, B.E.A., and M.C. TEICH,1987, Phys. Rev. Lett. 58, 2656. SALEH,B.E.A., J. TAVoLACCl and M.C. TEICH,1981, IEEE J. Quantum Electron. 17, 2341. SALEH,B.E.A., D. STOLERand M.C. TEICH,1983, Phys. Rev. A 27, 360. 111, M., M.O. SCULLYand W.E. LAMBJR, 1974, Laser Physics (Addison-Wesley, SARGENT Reading, MA). SCHAEFER, D.W., 1975, in: Laser Applications to Optics and Spectroscopy: Physics of Quantum Electronics, Vol. 2, eds S.F. Jacobs, M. Sargent, J.F. Scott and M.O. Scully (Addison-Wesley, Reading, MA) p. 245. SCHAEFER, D.W., and P.N. PUSEY,1972, Phys. Rev. Lett. 29, 843. SCHAWLOW, A.L., and C.H. TOWNES,1958, Phys. Rev. 112, 1940. SCHLEICH, W., and J.A. WHEELER, 1987, J. Opt. SOC.Am. B 4, 1715. SCHUBERT, M., and B. WILHELMI, 1980, The mutual dependence between coherence properties of light and nonlinear optical processes, in: Progress in Optics, Vol. 17, ed. E. Wolf (North-Holland, Amsterdam) p. 163. SCHUBERT, M., and B. WILHELMI, 1986, Nonlinear Optics and Quantum Electronics (Wiley, New York). SENIOR, J., 1985, Optical Fiber Communications (Prentice-Hall, Englewood Cliffs, NJ). SHAPIRO, J.H., 1985, IEEE J. Quantum Electron. QE-21, 237. SHAPIRO, J.H., H.P. YUENand J.A. MACHADO MATA,1979, IEEE Trans. Inf. Theory IT-25,179. SHAPIRO, J.H., M.C. TEICH, B.E.A. SALEH,P. KUMARand G. SAPLAKOGLU, 1986, Phys. Rev. Lett. 56, 1136. SHAPIRO, J.H., G. SAPLAKOGLU, S.-T. HO, P. KUMAR,B.E.A. SALEHand M.C. TEICH,1987, J. Opt. SOC.Am. B 4, 1604. SHELBY, R.M., M.D. LEVENSON, S.H. PERLMUTTER, R.G. DEVOEand D.F. WALLS,1986, Phys. Rev. Lett. 57, 691. SHOCKLEY, W., and J.R. PIERCE, 1938, Proc. IRE 26, 321. SHORT,R., and L. MANDEL,1983, Phys. Rev. Lett. 51, 384. SHORT,R., and L. MANDEL,1984, Sub-Poissonian photon statistics in resonance fluorescence, in: Coherence and Quantum Optics V, eds L. Mandel and E. Wolf(Plenum, New York) p. 671. SIEGERT, A.J.F., 1954, IRE Prof. Group Inf. Theory 3, 4. SIMAAN, H.D., and R. LOUDON,1975, J. Phys. A 8, 539. SLUSHER,R.E., L.W. HOLLBERG, B. YuRKE,J.C.MERTZand J.F. VALLEY,1985, Phys. Rev. Lett. 55, 2409. SMIRNOV, D.F., and A.S. TROSHIN, 1985, Opt. & Spektrosk. 59, 3 [Opt. & Spectrosc. 59, I]. SNYDER, D.L., 1975, Random Point Processes (Wiley, New York). SRINIVASAN, S.K., 1965, NUOVO Cimento 38, 979. SRINIVASAN, S.K., 1986a, Opt. Acta 33, 207. SRINIVASAN, S.K., 1986b, Opt. Acta 33, 835. STERN, T.E., 1960, IRE Trans. Inf. Theory IT-6, 435. STOLER,D., 1970, Phys. Rev. D 1, 3217. STOLER,D., 1971, Phys. Rev. D 4, 1925. STOLER, D., 1974, Phys. Rev. Lett. 33, 1397. STOLER,D., and B. YURKE,1986, Phys. Rev. A 34,3143. STOLER,D., B.E.A. SALEHand M.C. TEICH, 1985, Opt. Acta 32, 345. SZE, S.. 1969, Physics of Semiconductor Devices, 1st Ed. (Wiley, New York) p. 421, Eq. (95). TAKAHASI, H., 1965, Information theory of quantum-mechanical channels, in: Advances in Communication Systems, ed. A.V. Balakrishnan (Academic Press, New York) p. 227.

104


TAPSTER, P.R., J.G. RARITY and J.S. SATCHELL, 1987, Europhys. Lett. 4,293. TAPSTER,P.R., J.G. RARITYand J.S. SATCHELL,1988, Phys. Rev. A, to be published. TEICH,M.C., 1981, Appl. Opt. 20, 2457. TEICH,M.C., and B.I. CANTOR,1978, IEEE J. Quantum Electron. QE14, 993. TEICH,M.C., and B.E.A. SALEH,1981a, Phys. Rev. A 24, 1651. TEICH,M.C., and B.E.A. SALEH,1981b, J. Opt. SOC.Am. 71, 771. TEICH,M.C., and B.E.A. SALEH,1982, Opt. Lett. 7, 365. TEICH,M.C., and B.E.A. SALEH,1983, J. Opt. SOC.Am. 73, 1876. TEICH,M.C., and B.E.A. SALEH,1985, J. Opt. SOC.Am. B 2, 275. TEICH, M.C., and B.E.A. SALEH,1987, J. Mod. Opt. (Opt. Acta) 34, 1169. TEICH,M.C., and B.E.A. SALEH,1988, Squeezed states of light, in: Tutorials in Optics 1, 1987 OSA Annual Meeting, ed. D.T. Moore (Optical Society of America, Washington, DC). TEICH,M.C., and G. VANNUCCI, 1978, J. Opt. Soc. Am. 68, 1338. TEICH,M.C., P.R. PRUCNAL, G. VANNUCCI, M.E. BRETONand W.J. MCGILL,1982, Biol. Cybern. 44, 157. TEICH,M.C., B.E.A. SALEHand D. STOLER,1983, Opt. Commun. 46,244. TEICH,M.C., B.E.A. SALEHand J. PERINA,1984, J. Opt. SOC.Am. B 1, 366. TEICH,M.C., B.E.A. SALEHand T. LARCHUK, 1984, Observation of sub-Poisson Franck-Hertz light at 253.7 nm, in: Digest XI11 Int. Quantum Electron. Conf., 1984, Anaheim, CA (Optical Society of America, Washington, DC) paper PD-A6, p. PD-A6-I. TEICH,M.C., K. MATSUOand B.E.A. SALEH,1986, IEEE J. Quantum Electron. 22, 1184. TEICH,M.C., F. CAPASSOand B.E.A. SALEH,1987, J. Opt. SOC.Am. B 4, 1663. THOMPSON, B.J., D.O. NORTHand W.A. HARRIS,1940, RCA Rev. 4, 269,441; 5, 106, 244. THOMPSON, B.J., D.O. NORTHand W.A. HARRIS,1941, RCA Rev. 5, 371. 505; 6. 114. 1965, Phys. Rev. 140, B676. TITULAER, U.M., and R.J. GLAUBER, TORNAU, N., and A. BACH,1974, Opt. Commun. 11,46. TROUP,G.J., 1965, Phys. Lett. 17, 264. TWISS,R.Q., and A.G. LITTLE,1959, Aust. J. Phys. 12, 77. WALKER, J.G., 1986, Opt. Acta 33,213. WALKER, J.G., 1987, J. Mod. Opt. (Opt. Acta) 34, 813. WALKER, J.G., and E. JAKEMAN, 1985a, Proc. SOC.Photo-Opt. Instrum. Eng. 492, 274. WALKER, J.G., and E. JAKEMAN, 1985b. Opt. Acta 32, 1303. WALLS,D.F., 1979, Nature 280,451. WALLS,D.F., 1983, Nature 306, 141. WHINNERY, J.R., 1959, Scientia Electrica 5, 219. WOLF,E., 1979, Opt. News 5, 24. Wu, L.-A., H.J. KIMBLE, J.L. HALLand H. Wu, 1986, Phys. Rev. Lett. 57, 2520. Y., and H.A. HAUS,1986, Rev. Mod. Phys. 59, 1001. YAMAMOTO, YAMAMOTO, Y., and S. MACHIDA,1987, Phys. Rev. A 35, 5114. YAMAMOTO, Y., S. MACHIDAand 0. NILSSON,1986, Phys. Rev. A 34,4025. Y., N. IMOTOand S. MACHIDA,1986a, Phys. Rev. A 33, 3243. YAMAMOTO, YAMAMOTO, Y., N. IMOTOand S. MACHIDA,1986b. Quantum nondemolition measurement of photon number and number-phase minimum uncertainty state, in: 2nd Int. Symp. on Foundations of Quantum Mechanics, Tokyo, Japan, 1986, p. 265. N. IMOTO,M. KITAGAWA and G. B J ~ R K1987, , J. Opt. SOC.Am. YAMAMOTO, Y., S. MACHIDA, B 4, 1645. YUEN,H.P., 1976, Phys. Rev. A 13, 2226. YUEN,H.P., 1986, Phys. Rev. Lett. 56, 2176. YUEN,H.P., and J.H. SHAPIRO, 1978, IEEE Trans. Inf. Theory IT-24,657. YUEN,H.P., and J.H. SHAPIRO, 1980, IEEE Trans. Inf. Theory IT-26, 78.


I1

NONLINEAR OPTICS OF LIQUID CRYSTALS BY

IAM CHOON KHOO Department of Electrical Engineering Pennsylvania State University University Park, PA 16802, USA

CONTENTS PAGE

0 1 . INTRODUCTION . . . . . . . . . . . . . . . . . . . 107 § 2 . FUNDAMENTALS OF LIQUID CRYSTALS

. . . . . . . 109

$ 3 . NONLINEAR OPTICAL PROCESSES . . . . . . . . . . 133 § 4 . FURTHER REMARKS AND CONCLUSIONS

. . . . . . 155

ACKNOWLEDGEMENT . . . . . . . . . . . . . . . . . . . 157 REFERENCES . . . . . . . . . . . . . . . . . . . . . . .

158

8 1. Introduction Liquid crystals have been a subject of great interest for both fundamental and practical reasons (DEGENNES[ 19741, CHANDRASEKHAR [ 19771, BLINOV [ 19831). Fundamentally, the many mesophases in which liquid crystals exist and their corresponding phase transition characteristics offer a good testing ground for statistical mechanics. Practically, interests range from simple electro-optical displays to multiplex optical data trunking and large-scale television displays with unmatched colors. In these pursuits the fundamental and most important process is their linear light scattering property. In the past fifteen years the nonlinear optics of liquid crystalline material have also been actively studied. Earlier studies have concentrated mostly on the isotropic phase (PROSTand LALANNE[ 19731, WONG and SHEN [ 19731, RAO and JAYARAMAN [ 19741, LALANNE, MARTIN,POULIGNYand KIELICH[ 1977]), in which liquid crystals behave very much like an anisotropic liquid (e.g., CS,) but possess interesting pretransitional (isotropic + nematic) behavior. As the temperature of these isotropic liquid crystals is brought closer to the transition temperature T,, many of the nonlinear effects under study are enhanced while the response times begin to lengthen, indicating greater molecular correlations and macroscopic collective phenomena. This increase in the optical response (e.g., Kerr effect) near T, is one of several clues to the presence of an extraordinarily large nonlinearity in the nematic phase. This is because the transition from the isotropic phase to the nematic phase is quite unlike the usual liquid -+ solid transition, where molecular reorientations are “frozen” out completely. In the nematic phase, as is well known through years of research, a unique characteristic is that the liquid crystal will reorient with a very low externally applied field. Other well-known liquid crystal light scattering processes [ e.g., the light scattering cross-section of the nematic phase being about lo6 times that of the isotropic phase (DEGENNES[ 19741); the magnitude of the electric field, and the static nature of the field-induced nematic axis reorientation discussed in more detail in a later section) are also “glaring” hints, in retrospect of course. The discovery of the extraordinarily large nonlinearity associated with the field-induced reorientation in nematic liquid crystals is, in many ways, anaI07

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[II, § 1

logous to the development in other nonlinear materials, for example photorefractive materials. Photorefractive effects were fist observed in the early 1960s (ASHKIN,BOYD, DZIEDZIC, SMITH,BALLMAN,LEVENSTEINand NASSAU[ 19661, CHEN,LAMACCHIAand FRASER[ 1968]), but their exceptional characteristics for nonlinear optical processes (e.g., phase conjugation, beam amplification, etc.) were not fully exploited until the late 1970s and early 1980s (see, for example, the review by FEINBERG [ 19831). Liquid crystals are equally fascinating because they also possess exceptionally large thermal index gradients near the phase transition temperature. A large class of nonlinear optical processes can also be realized with laser-induced thermal effects, as reviewed in a recent article by HOFFMANN[ 1986b]. With the discovery of these highly nonlinear materials, several nonlinear optical processes (which are unimaginable in materials with small nonlinearities), such as wavefront conjugation with gain, self-oscillation, and optical bistability have been demonstrated using low power (milliwatt or microwatt) lasers. This chapter is a personal view of the interrelationships between the unusual characteristics of liquid crystals and these novel nonlinear optical processes. I will concentrate mainly on the nematic phase, but where appropriate, other mesophases of liquid crystals will also be discussed. This chapter is not intended as a treatise on liquid crystal physics. My intention is to describe, in simple but accurate terms, the essence of liquid crystal fundamentals for the benefit of the so-called “outsiders’ in the realm of liquid crystals. In the same token, space does not allow me to elaborate on all the interesting details of current nonlinear optics research. Several existing reviews and texts discuss many of these fascinating processes and materials, and they will be referred to in the appropriate sections. The field of nonlinear optics now is quite different from the time when the first review of the subject appeared in this series (PERSHAN [ 1966]), as a result of the discovery of several highly nonlinear materials. Whereas multiphoton processes (optical wave mixing) were formerly associated with signals that needed photon counting detection, most of the wave mixing effects under study now involve strong and clearly visible signal beams. Self-focusingof a milliwatt laser is almost a commonplace phenomenon in contrast with early studies involving megawatt lasers. I hope this article will convey some of the excitement that many of my colleagues and I have experienced in the “discovery” and observation of these highly nonlinear effects associated with liquid crystals.

1 1 9 8 21

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109

8 2. Fundamentals of Liquid Crystals 2.1. GENERAL

Liquid crystals are composed of fairly large organic molecules with a typical chemical structure as depicted in fig. 1. As a result of intermolecular forces, the molecules tend to align themselves in some fixed direction.From a physicist’s or engineer’s point of view, they can be thought of as cylindrically symmetrical cigar-shaped molecules. Macroscopic samples exist in three typical distinct phases (cf. figs. 2a-c) as a function of the temperature. In the nematic phase (fig. 2a) the molecules are directionally correlated but are positionally (as defined by their centers of gravity, for example) random. The direction of the macroscopic sample is described by the so-called director A, which coincides with the optical axis of the molecule. In the smectic phase there is an extra ordering because of the positional correlation; molecules exhibit layered structures (cf fig. 2b). There are actually eight known distinct smectic phases. Cholesteric liquid crystals may be viewed upon as some forms of twisted nematics, and exhibit helical structures in their alignments. Most liquid crystal molecules are uniaxial, centrosymmetrical and nonpolar, although there is a class of liquid crystals which is ferroelectric and possesses large permanent dipole moments (GRAYand GOODBY[ 19841). The present discussion will be limited to the former class, where, because of the centrosymmetry, the physical properties of the crystals are the same in the t A and the - A directions. As a result of this uniaxiality, the bulk properties of liquid and crystals can be expressed in terms of two components, one parallel to (1) the other perpendicular to (I) the director. The main bulk physical properties that play an important role in the steady-state and transient optical responses of liquid crystals are the refractive indices (nl and n , ) and the optical dielectric - cl ), the diffusion coefficients (Dland D, )the magnetic anisotropy A E = 7 X: ), and various flow coefficients and elastic constants susceptibilities( ~and that will be defined presently. C d 9 - 0 - N

-

qp

N - O - C ~ C J

-

Fig. I . Chemical structure of a typical liquid crystal molecule, terephthal-bis (-pbutylaniline) (TBBA), which exibits smectic and nematic phases.

I10

w,§ 2


a

b

C

Fig. 24a) Nematic liquid crystals are directionally correlated but positionally uncorrelated. (b) Smectic liquid crystals are directionally correlated and also positionally correlated, forming layered structures. (c) Cholesteric liquid crystals exhibit helical structures in their direction of alignments.

Most devices require chemically and photochemically stable liquid crystals with a wide operating temperature range. Most individual mesogens (e.g., 5CB, a frequently used nematic for nonlinear optics studies) have a limited liquid crystalline range. For PCB the range is 24-35.3"C. On the other hand, mixture of nematogens can be synthesized to yield a nematic range of more than 100" C. Mixtures containing up to ten nematogen components are not uncommon in many display devices. For all intents and purposes the physics of nematic liquid crystals can be closely approximated by visualizing these liquid crystal molecules as anisotropic rigid rods. The molecules can be initially aligned in several standard forms. Figures 3a and 3b are two common examples, termed homeotropic and planar (or homogeneous) alignments, respectively. Homeotropic alignment is achieved by coating the glass surfaces enclosing the liquid crystals with surfactants such as DMOAP and HTAB. Planar alignments are achieved by treating these surfaces with polyvinyl alcohol followed by unidirectional rubbing with lens tissues. A detailed description of this fabrication technique may be found in the review article by BAHADUR [ 19841.

2.2. FREE ENERGIES AND DISTORTIONS BY APPLIED FIELDS

The most successful and widely used theory for describing the response of nematics to an external field is the continuum theory (DEGENNES [1974], BLINOV[ 19831). The essence of the theory may be summarized as follows: In an ideal nematic single crystal the molecules are aligned along one common direction t A, which is the direction of preferred orientation


111

NEMATIC LIQUID /CRYSTAL

\

Fig. ).(a) A homeotropically aligned nematic liquid crystal film. Also shown are the propagation vector k and the electric field vector E of an incident linearly polarized laser beam. pis the angle made by k with the director axis A. (b) A planar nematic liquid crystal film.

(imposed by boundary plates). The directions A and - A are equivalent. At a given temperature in the nematic phase, the direction of the liquid crystal molecules fluctuates around the mean direction given by A. These orientational fluctuations and the correlation of the molecules are described by an order parameter

where 8 is the angle between the long axis of the molecules and the director of the liquid crystal (taken to be along the z-direction, for example). P , is the Legendre polynomial. In the nematic phase, S is 5 1 (typically S = 0.3-0.4 near T, and S z 0.8 at much lower temperatures). In the isotropic phase S = 0. It was shown that S is a universal function of the reduced temperature z = TV2/(TcVf),where V and V, are the volumes at T and T,, respectively. More specifically, S for most nematics can be approximated by the expression S

=

(1

-

0.98T

.

) ~ . ~ ~

The distortion of the molecular alignment in the nematic phase at a fixed temperature is described in terms of the vector field n(r). As depicted in figs.

112

NONLINEAR OPTICS OF LIQUlD CRYSTALS

a

Fig. 4.(a) Splay deformation of a nematic liquid crystal. (b) Twist deformation of a nematic liquid crystal. (c) Bend deformation of a nematic liquid crystal.

4a-c, there are three possible forms of distortion: splay, twist, and bend, with the corresponding free energy densities given by, respectively, F,

=

$ K ,(V . A)',

A),, x (V x A)]',

F2 = 4K,(n V x

,

F - IJ , [ n

splay, twist, bend,

(2.2)

where K , , K,, and K , are the corresponding elastic constants. In general, K , # K2 # K , , but they are all on the order of dyne. For some special geometries and/or for small deformations it is possible to reduce the problem to one involving only one elastic constant and one type of free energy. Also, for simplicity in estimation or calculation, one can adopt a one constant ( K , # K , # K,) approximation. In that case the total free energy F = F , + F2 + F3 reduces to F = 4Ka,n,a,n,. From these free energy terms, and using Lagrange's equation, one can derive the corresponding molecular fields for splay, twist, and bend distortions fl

=

(2.3)

K,V(V*n)

f, = - K 2 [ A V x n

+ V X (An)]

(2.4)

11, I21

FUNDAMENTALS O F LIQUID CRYSTALS

1I3

and

f3= K,(B x (V x n) t

v x (n x B ) ) ,

whereA = n . ( V * n )and B = n x (V x n). An important consideration must be remembered in the study of these bulk distortions, namely, the effect of these distortions on the surfaces enclosing the nematics. The physics of the molecular forces acting at these surfaces is interesting and important. It is also very complicated and may be found in the work of several researchers (BARBERO,SIMON]and AIELLO[ 1984, 19861). A simple view of the problem is as follows: whereas most of the distortions follow the torque balance equation obtainable from the free energies discussed earlier, in a region of molecular length near the surface the deformation depends on the detailed molecular properties. Depending on how the samples are prepared, and the geometry of the applied torques and deformations, these surface conditions will fall in either the strong or weak anchoring conditions. In the so-called strong anchoring, or “hard” boundary, condition, molecules are rigidly attached to the surface. The distortion is assumed to be vanishing on the surface. On the other hand, in the weak anchoring limit any bulk distortion (created by an external field, for example) will also deform the surface alignment. The interaction of nematics with external fields is governed by essentially the detailed physical structures of the nematics. Most nematics are diamagnetic and nonpolar. In this section I shall consider static fields; I will return to the problem of oscillatoryfields (e.g., from a laser) in Q 2.3. Furthermore, discussion will be limited to only dielectric (and diamagnetic) interactions. For a general applied electric field E the displacement D is given by D=EIEt(El - ~ ~ ) ( n * E ) n .

(2.6)

This, therefore, gives an interaction energy density of

V, -

471

joE

--E E 2 A& D . d E = A - -( ~ s E ) ~ . 871 871

(2.7)

The orientation-dependent term in eq. (2.7) can thus be identified as an additional term to the free energies, which we shall denote as FE A& FE= --(n*~?)~. 8n

114


The electric field thus creates a torque on the molecule given by fE

=

DxE

=

AE

-((n.E)(n x E ) . 471

(2.9)

Similar consideration gives a magnetic contribution FH - -LA 2

x(n’H)2

(2.10)

and fH

=

A x ( n .H ) H .

(2.11)

The new equilibrium configuration of a nematic film under an applied field can be solved by balancing the molecular torques (f l , fi, and f3)with the torque from the applied field (fE or fH) in cases where the applied fields are spatially constant. For a spatially varying field it is more appropriate to start from the free energy terms and then apply a minimization procedure (e.g., the Euler-Lagrange method). Two important points ( for nonlinear optical field effects) are embedded in the preceding well-known dc field effects, which somehow were not recognized until about 1980(ZELDOVICH and TABIRYAN [ 19801,KHOO[ 1981a1, DURBIN, ARAKELIAN and SHEN[ 1981b]), namely (1) the magnitude of the electric field needed to induce sizable nematic axis reorientation and (2) the fact that even if E contains very fast oscillations (e.g., those characterizing optical fields), the dielectric interaction (cf. eq. [ 2.91) creates a nonoscillatory torque that will effectively act as a stationary applied field to reorient the nematics. For the second point, of course, one must account for the dispersion (i.e., the frequency dependence) of the dielectric anisotropies A&(or Ax), noting in particular that A&at optical fields is different in magnitude (and sign sometimes) from A E at dc fields. Some numerical estimates will be helpful here. Consider, for example, the so-called Frederiks transition (DEGENNES[ 19741, FREDERIKS and ZOLINA [ 19331) induced by a static field E applied perpendicularly to the director axis h in a homeotropically aligned film. The original alignment is stable until the applied electric field strength exceeds a critical field (2.12) The magnitude of E , may be estimated by considering a typical nematics liquid crystal MBBA (p-methoxy-benzylidene-p-n-butylaniline), which has the fol-

11, § 21

1 I5


-

lowing molecular parameters: k x 7 x lo-', A & (static field) x 1. For a 100 pm-thick sample (i.e., d = 100 pm) E, is found to be 3 x lo2 V/cm. Consider what is needed in terms of optical intensity (in watts/cm2) if the applied field is a laser. There are, of course, some detailed differences between optical field interaction and dc field interaction with nematics (e.g., A&[optical] is about 0.6). For the purpose of making numerical estimates, we can neglect these details and consider the optical intensity associated with an optical electric C field of 340 V/cm = statV/cm. From the Poynting vector S = -E x H , 411 we get C 3 x 10'' 340 I s 1 = - E2 = erg/cmZ s 4n

4n (=)

3x --

lo3 340 ~

1271 (300) J's cm2

(2.13)

e 300 W/cm2 . This kind of intensity level can be easily achieved by a lightly focused argon laser, or a tightly focused milliwatt He-Ne laser. Because of the large birefringence of nematics (n, - n , x 0.2), it is obvious that a moderate optical intensity will cause an enormous change in the refractive index of the medium by means of the reorientation process. There is, however, another possible mechanism for a large refractive index change, namely, the extremelylarge thermal index gradients of nematics near the nematic + isotropic transition temperature T,. A temperature rise can be achieved by means of the nematogen's natural absorption or by dissolved dye modules. In the next two sections, fundamentals of these effects and how they enter into the realm of nonlinear optics will be reviewed.

2.3. OPTICALLY INDUCED DIRECTOR AXIS REORIENTATION

Speculations, theories, and some experimental observations of optically induced nonlinear effects associated with director axis reorientation appeared in the literature around 1979 and 1980 (HERMANand SERINKO[1979], LEMBRIKOV [ 19791, TABIRYAN and ZELDOVICH[ 1981a,b,c], ZOLOTKO, KITAEVA,KROO, SOBOLEVand CHILLAG [1980], KHOO and ZHUANG [ 19801). The prediction by HERMANand SERINKO[ 19791 that a low-power

116


[II, § 2

optical field can induce appreciable reorientation of the nematic axis just above a dc field-induced Frederiks transition was verified experimentally by KHOO and ZHUANG[1980] and KHOO [1982a]. Concurrently, ZELDOVICHand TABIRYAN [ 19801 and ZOLOTKO,KITAEVA,KROO,SOBOLEVand CHILLAG [ 19801proposed and experimentallyobserved the purely optically field-induced reorientation and self-focusing effects. Working on an extension of Herman and Serinko’s studies, KHOO[ 1981al independently theorized and demonstrated the possibility of optically induced Frederiks transition and the associated third-order nonlinear wave mixing effects. Later in 1981 DURBIN, ARAKELIAN and SHEN [ 1981bl published a quantitative theory and experimental measurement of the optically induced reorientation and the birefringence change. TABIRYANand ZELDOVICH[ 1981a,b,c] also presented detailed theories of optically induced molecular reorientation and the associated nonlinear effects in nematic, smectic, and cholesteric liquid crystals. In the few years following these early publications, numerous papers dealing with various detailed aspects of optical field-induced (director) reorientation in nematic, smectic, and cholesteric films have appeared. Unlike dc field-induced reorientation in nematic liquid film, where the spatial extent of the applied (uniform) field is very large compared with the liquid crystal sample thickness, an optically induced effect invariably involves a narrow laser beam incident on a small portion of nematic film. Because molecules outside the laser beam, as well as those strongly anchored at the boundary, can exert torques on the molecules within the beam, the reorientation process is actually a three-dimensional problem; the process involves two transverse dimensions of the laser beam, or the transverse intensity variation characteristic lengths (e.g., grating spacing in two-beam interference) and the thickness of the film. The mechanisms of optically induced reorientation therefore can be arbitrarity separated into two limits, namely, one in which the laser beam is large compared with the thickness and one in which the beam size is comparable with or smaller than the film thickness. 2.3.1. Plane wave For an arbitrarily large beam size compared with the thickness of the film, the equation governing the steady-state orientation of the director axis is obtained by minimizing the free energies associated with the elastic distortion and the optical fields. Detailed derivations of these equations may be found in the work of TABIRYAN and ZELDOVICH [ 1981al and DURBIN, ARAKELIAN and SHEN [1981b], and a comparative discussion was presented by ONG

11, § 21

I17


[ 19831. The theory may be simplified by considering only a small reorientation angle in geometries involving only one elastic constant. Consider, for example, the geometry as depicted in fig. 3. In steady state the torque balance between the molecular elastic and the boundary forces yields an equation of the form (KHOO[ 1982b1)

25’-

d’ 0 t (2 cos2p)O t sin28 = 0 , dz’

(2.14)

where 5’ = 4nK(Ac)- ‘ E ; , and K is the elastic constant for splay (assuming that only one elastic constant is involved and the reorientational angle 0 is small). A more detailed derivation accounting for all the elastic constants resulted in the equation (ONG [ 19831) d’ 0 (1 -Esin’O)--EsinOcosO dz’

(:y

-

+=I

n ck,,

( p - a)sinOcos0 (1 - 8 sin2 0),/*

=

0,

(2.15)

where

Notice that for 0 4 1 and under the one-elastic-constant approximation, eq. (2.15) reduces to eq. (2.14), and the solution of (2.15) reduces to that obtained by Khoo from (2.14). Since almost all nonlinear optical effects involve very small reorientational angles (because of the large birefringence of liquid crystals), discussion will continue based on the small4 limit. For large 0 the solution may be expressed in terms of some elliptic integrals (DURBIN, ARAKELIAN and SHEN[ 198la,b]). Assuming hard boundary conditions (0 = 0 at z = 0 and at z = d ) , eq. (2.14) can be solved to yield 1 0 = -sin2P(dz - z’) 4

r2

.

(2.16)

As a result of this reorientation, the incident beam (an extraordinary ray) experiences a z-dependent refractive index change given by

An

=

n(/? t 0) - n(P),

(2.17)

where, for a nematic film (a uniaxial birefringent crystal) the refractive index

1 I8


n(a) is given by n(a) =

n//n, J n ; cos'a + n: sin2a

(2.18)

For small 0 the change in the refractive index An is found to be proportional to the square modulus of the optical electric field, that is, An

=

(2.19)

n,(z) E & ,

that is, An

= a2(z) I

(2.20)

,

with a,(z) given by ( A E ) , sin'(2P) a2(z) =

4 Kc

(dz - 2')

(2.21)

The dependence of the refractive index change on E&, or the intensity of the optical field, is the central point from which many nonlinear optical processes emerge. What is truly special to liquid crystals in their nematic phase is the enormous magnitude of n, (compared with most nonlinear materials). For a film thickness d = 100 pm, AE 0.6, K = P = 45", we have

-

-

a, z 5 x

cm2/W,

where Z2 is the average of the value of a,. This nonlinear coefficient is about 8 orders of magnitude larger than that of CS,, a highly nonlinear organic liquid which has become a standard for comparison. The study by WONC and SHEN[ 19731 has shown that the nonlinearity of the isotropic phase of nematics can be two orders of magnitude larger than that of CS,. Hence, the nonlinearity of the nematic phase is about lo6 times the nonlinearity of the isotropic phase. As was mentioned earlier in Q 1, the light scattering cross-section of the nematic phase is also lo6 times that of the isotropic phase. This reflects the fundamental link between the nonlinear and the linear light scattering properties of nematic liquid crystals, namely, the ease with which orientation fluctuations can be induced with small applied perturbation fields. Figure 5 shows the experimental phase shift b$ experienced by a probe beam as a result of the refractive index change bn induced by a pump laser (DURBIN, ARAKELIAN and SHEN[ 1981a,b]). The angle between the propagation vector K and h is there labelled as a, which is denoted as P in this article. For = 0

11, § 21

1 I9


50

40

30

/2lr

+ I

2c 10

0

I00

200

I

3 00

(W/crn2 ) XBL 8 1 3 - 5 4 3 4

Fig. 5. Experimentally observed laser-induced phase shift in a nematic liquid crystal (PCB, pentyl-cyano-biphenyl) film.

there is a sharp “threshold”; that is, the orientation is vanishingly small until the applied optical electric field is greater than a threshold value, the so-called optical Frederiks transition field E , (KHOO [ 1981a1, DURBIN, ARAKELIAN and SHEN[ 1981b1, ZELDOVICH and TABIRYAN [ 19801, ZOLOTKO,KITAEVA, KROO,SOBOLEVand CHILLAG[ 19801). This is the optical analog of the dc Frederiks transition, with Atop playing the role of LIE,,. In either planar or homeotropic nematic samples, considerations of the optical field-induced reorientation process do not have to account for the surface anchoring energy (cf. 0 2.2). This is not the case for hybrid alignment, SIMONIand AIELLO[ 19841 and BARBEROand where studies by BARBERO, SIMONI[1982] have shown that such anchoring energies play a highly important role in the process. Experimentally, SIMONI and BARTOLINO [ 19851 observed that in hybrid aligned cells there was no threshold for the onset of the reorientation. ZELDOVICH and TABIRYAN [ 19821 and AKOPYANand ZELDOVICH [ 19821 have also presented a detailed discussion of the variational principles involved in treating all the energy terms used in the nematic axis reorientation. In the case where the laser beam size is large when compared with the thickness of the film, but the optical intensity distribution contains an oscillatory function (e.g., dark and bright intensity fringes resulting from interference of two overlapping laser beams), the correlated nature of the molecular torques

120

[II, § 2


in the nematics is manifested in the dependence of the magnitude of the reorientation on both the film thickness and the characteristic length of the spatial oscillation (e.g., the grating constant). This problem has been considered by KHOO[ 19831 for the case of purely optically induced reorientation and by DURBIN, ARAKELIAN and SHEN[ 19821 for magnetic-plus-optical field-induced reorientation. The essence of the calculation for the purely optical field-induced effect may be simply stated as follows: The applied optical field is of the form

E,,

=

E: t E:

+ 2E,E2 COSqX

(2.22)

where q = 1 k , - k, j is the magnitude of the grating wave vector. The reorientation is given by

O=

3 A E sin 2p E: 16nK

+ E ; + 20E,E, cosqx q2d2 t 10

(dz - z’) .

(2.23)

Notice that 0 contains a factor (10 t q2d2)which accordingly, shows the expected result of the two characteristic lengths d and q - of the system on the reorientational grating term, which is the term involvingE l E2 in (2.23). This dependence has also been experimentally verified explicitly in the studies by these authors. The diffraction efficiencies in optical wave mixing experiments were found to be critically dependent on the grating constant.

’

2.3.2. Finite beam size

In situations involving a focused laser beam incident on the nematic film, the transverse characteristic length is the beam size w,, which, if smaller than the film thickness, will obviously bring about the transverse correlation effect mentioned in the preceding paragraph; that is, molecules situated “outside” the laser beam will exert a torque on molecules “inside” the beam to inhibit their reorientation. Or, conversely, molecules “inside” the beam could exert torques on those “outside” to reorient them. The result is that the transverse dependence of the reorientation profile is not the same function as the laser beam’s transverse profile. This problem was recognized in the early work by CSILLAG, JANOSSY, KITAEVA,KROO and SOBOLEV[1982] (who treat the case of normally, = 0, incident beam) and by ZELDOVICH and TABIRYAN[ 19821. The case of an obliquely incident ( p # 0) beam was later studied by KHOO,LIU and NORMANDIN [ 19851, who presented a more exact calculation, where the exact profile of the laser beam, assuming a Gaussian function, is used. An exact numerical calculation was also employed by SANTAMATO and SHEN[ 19851 in

11, § 21


121

their paper on the spatial rings associated with transverse self-phase modulation effects, but the details were not discussed. There are several important differences between the oblique-incidence case and the normal-incidence case. A detailed recent study by KHOO,LIUand YAN [ 19871, in which both theory and experimental verifications are represented for both b = 0 and 8 # 0 cases, showed that the equations governing the transverse dependence of the reorientation R (r) (assuming cylindrical symmetry, one-elastic-constant approximation, and small O), are given by: (1) p = 0 case: d2R + -dR + dr2 R dr

~

[

-

- b-e-Qr1R3 = 0 ,

(2.24)

2

f l # 0 case:

(2)

(:>'I

b

R

+

ecar2sin28 = 0 , (2.25)

where

A&

b=E:p 4nK

1

a=-

2

w; '

and we have defined the reorientation angle O(r, z ) by

The solutions of these two equations may be summarized as follows: (1) For b = 0 there is a threshold intensity for finite reorientation to occur. The threshold intensity depends on both the thickness of the film and the beam size w,. For w, 4 d the threshold intensity increases dramatically (compared with the value for a plane wave) and the measured values are in agreement with the work by CSILLAG, JANOSSY, KITAEVA, KROOand SOBOLEV [ 19821 and the detailed theory of KHOO,YAN and LIU [ 19871. There is no threshold intensity for field-induced reorientation in the # #I0 case. The existence of the threshold is simply the natural consequence of eq. (2.24). (2) For a Gaussian laser beam input the reorientation profile is not a Gaussian. In general, the reorientation profile is a smooth bell-shaped function with a halfwidth w,different from 0,. Figures 6a and b show plots of the ratio of we to o, as a function of w,/d for the j? = 0 and /3# 0 cases, respectively.

122


,,,

,2 :o ’

I,

~

, ,

I,

, , ,

I,,,

I , ,,,

I,,,,

0.0 0

I

2

3

4

5

,

(b)

6

W./d

Fig. 6.(a) Plot of w,/w, versus wold for oblique incidence (/I# 0). (b) Plot of o,/w, versus wold for normal incidence (/I = 0).

In general, for the fl = 0 case the halfwidth oois always larger than w, for all values of o,,approaching w, for large values of mold. On the other hand, for the fl = 0 case wo can be larger than o, or smaller than w,,,depending on whether oois smaller or larger than d. These exact, or at least more detailed, treatments of the transverse profile of the reorientation are very important in the study of transverse self-phase

11, § 21

I23


modulation effect involving a cw laser beam and should be exercised not only in a nematic liquid crystal film but also in other nonlinear materials. The point is that the nonlinear response of the materials (e.g., semiconductor electronic processes, thermal effects) invariably involves some form of diffusion mechanism. The diffusion equation naturally imposes constraints on the system such that the transverse profile is different from the laser input profile. These considerations have led to a critical examination of, for example, optical bistability (HAGAN,MACKENZIE, REID,WALKERand TOOLEY [ 19851, FIRTH, GALBRAITH and WRIGHT[ 19851). These researches have observed that, for example, the existence of a hysteresis loop depends critically on the diffusion length compared with beam spot size, and the laser-switching intensities also depend critically on the spot size. A more detailed discussion of some of these effects appears in 5 3.

2.4. TEMPERATURE-DEPENDENT REFRACTIVE INDEX

The two principal refractive indices n, and nl/of the uniaxial liquid crystal and the anisotropy (nl - n,) have been the subject of intensive studies for their fundamental importance in the understanding of phase transitions and for their vital role in applied electro-optic devices. A good discussion of the historical development may be found in a standard text (DEJEU[ 19801). Since it is the dielectric constants E, (&/ = n$ and E , = :n ) that enter directly in the constitutive equation relating the polarization P,dielectric displacement D, and electric field E, through the relationship

D

=

E ~ +EP ,

(2.26)

it is theoretically more convenient to discuss the fundamentals of these temperature dependences of the refractive indices by considering the dielectric constants E . The dielectric anisotropy arises because of the anisotropy in the electric susceptibility tensor x, whose elements in the three spatial coordinates are given by zap,a, f l = x, y, z. The induced polarization P is related to the applied field E by (2.27) that is, if we write D &=I+X,

=

eE, the dielectric constant tensor

E

is related to

x

by

(2.28)

where 1 is the unit tensor. Usually z is taken as the optical axis (or director axis)

124


111, 5 2

of the liquid crystal. In that case we have E,

= [+(n;

+ n$)]''2 = n ,

(2.29)

and E/ =

nl = n:/ ,

(2.30)

The microscopic derivations of the dielectric constants E / and E~ are rather complex (DEJEU [ 19801). If one denotes Ei as the internal field, seen by the liquid crystal molecule, and E as the applied macroscopic field, their relationship may be represented by

(2.3 1)

Ei = k - E ,

where k is an ordinary second-rank internal field tensor. In this manner the polarization P may be written in the form

P

=

(2.32)

N ( a .k ) E , 8

where a is the polarizability tensor of the molecule, Nis the number of molecules per unit volume, and the brackets denote averaging over the orientations of all molecules. From eqs. (2.28) and (2.32) one thus obtains E = 1t

N -

(2.33)

(aek)

&

and

At

= E / - E,

N

=-((ask)/

-

(2.34)

(a.k),).

EO

From these considerations, the observation (DEJEU and BORDEWYK[ 19781) that

(2.35)

A E K ~ S , and

where S is the order parameter,

E~

and

E,

can be written as

(2.37)

11, I 21


125

and = :n

EL

=

1

t

N -

[ a,k,( 1 - S ) + gk,(2 t S ) ] ,

(2.38)

3%

where k,and k, are the values of k along the principal axes. Equations (2.37) and (2.38) can be rewritten as $A&

(2.39)

- $AE,

(2.40)

= E, t

and E,

= E,

where N AE = - [qk, - g k , ] S %

=

*

-

(cr,K, - gK,) pS 3%M

(2.41)

and E, =

1 t ~NAP ( c t , k t, 2gkJ

3&,M

= 1 t c0nst.p.

Notice that N has been replaced by (NAp)/M,where NA is Avogadro’s number, p is the density, and M is the mass number. The final explicit form of the E’S depends on the choice or determination of the internal field tensor k. However, it is important to note in terms of the temperature dependence of the E’S that E,N

1 t const.p= I t C,p

(2.42)

and B E - pS

=

C,pS.

(2.43)

In other words and E , (and the corresponding n, and n,) depend on the temperature ( T ) through the dependences of p and S on T. The temperature dependence of p has been measured for many nematics. On the other hand, the temperature dependence of the order parameter can be well approximated by expression (2.1). Perhaps the most important point on the temperature dependence of the refractive indices is that the thermal index gradients (dn, /d Tand dn, /d T )are extraordinarily large and rank among the largest of all known materials. From

126

“I, $ 2


(2.42), (2.43), and (2.37) or (2.38) it is possible to obtain dn//dTand dn, /dT. We have (2.44)

(2.45) Figure 7 shows the plot dn,/dT and dn,/dT for PCB as a function of temperature, with experimental data deduced from some recent detailed measurements (HORN[ 19781). From these equations it can be noted that both (n;) and tl ( n t ) are functions of the density p and the order parameters. Given an impulse type of temperature change, therefore, a density change (which could be manifested in some acoustic wave generation) and a nonpropagating order parameter change 6s will occur. If the temperature impulse change is a spatially periodic function, some intensity interference effect occur (KHOO and NORMANDIN [ 1985a1) similar to laser-generated, high-frequency acoustic waves in other materials (FAYER[ 19861). 60

100

30

-

I

I

Y

40 d

50

I

-0

d I

-0

IU \

-4

c

20

U

20

3

-15 0

-12.5

-10.0

-7

5

-5.0

-2.5

0.0

T,-T Fig. 7. Experimentally observed thermal index gradients, dn,/dT and dn, /dT for PCB: ( x ) dn,/dT; ( 0 ) dn,/dT. Some theoretical points are also shown.(KHOOand NORMANDIN [ I985al.)

148 21

127


2.5. DYNAMICS OF OPTICALLY INDUCED REORIENTATIONS AND THERMAL

EFFECTS

2.5.1. Thermal efect The dynamics of heat generation by means of the absorption of laser radiation and subsequent dissipation and the resulting refractive index change have been studied in many contexts (HSIUNG, SHI and SHEN [1984], KHOO and Normandin [ 1984a,b]). Using stimulated thermal Rayleigh scattering (URBACH, HERVETand RONDELEZ[ 19781, HERVET, URBACH and RONDELEZ [ 1978]),for example, thermal diffusivity in both the nematic and smectic phases has been measured. It is obvious, even at a casual glance, that the dynamics of the thermally induced refractive index change is a complex issue involving not only various temperature-, and therefore phase transition-, sensitive parameters such as heat capacity and densities but also various time scales (phonons, diffusions, input laser pulse widths) in addition to molecular electronic absorption constants and the geometry and boundary conditions. An example of some of the basic considerations that need to be applied may be found in studies (BATRA,ENNS and POHL[ 19711, HOFFMANN[ 1986b1) of short pulse-induced, nonlinear optical effects in simple liquids. In liquid crystals, we also need to focus on the anisotropies of the various physical parameters as well as the role played by the order parameter (i.e., the molecular correlation) in the thermalization process that follows photo absorption. Before proceeding any further we recall in table 1 a number of physical TABLE1 Some physical constants defined for the principal axes, and typical values for some nematics. Constant

Value (units)

Thermal diffusivity

1.25 x

Velocity of sound

1.5 x 105cm/s

(1.5 - A) x lo5 cm2/s A % 1.5 x 10-3

MBBA

Refractive index

1.72

1.52

PCB at 21 " C

Acoustic altenuation

1.67 x IO-'s (at T, - T = 9.5 K)

4.97 x l o - 8 s (at T, - T = 1.0 K)

MBBA

cm2/s

Nematic

7.9 x

cm*/s

PCB

128


[It, 8 2

constants defined for the two principal axes, that is, parallel (1) and perpendicular (I) to the director axis, along with typical values for some well-known and nematic materials (MULLEN, LUTHI and STEPHEN [ 19721, EDEN, GARLANDand WILLIAMSON [ 19731). For any meaningful nonlinear optical process of application to be realized, a judicious choice of the geometry to simplify this complicated threedimensional problem is a prerequisite. Indeed, most reported nonlinear optical studies (HSIUNG,S H I and SHEN [ 19841, KHOO and NORMANDIN [ 1984a1) have employed geometries of lower dimensionality, whereby some effective relaxation time constants may be deduced. In the case of heating induced by a single laser beam, HSIUNG,S H I and SHEN[ 19841 have observed that the heat diffusion process is characterized by two exponential decay time constants (to and t , ) that differ by almost an order of magnitude. The experimentally deduced values of the thermal diffusivity (assumed to be independent of director orientation) are, however, in poor agreement with values from the literature. The authors attribute the discrepancy to the one-dimensional approximation, namely, the beam size is large compared with the thickness so that the heat diffusion process is predominantly along the laser propagation direction, that is, from one cell wall to the other. The other most probable reason for the discrepancy is that the sample is first oriented by a magnetic field (which would induce different initial reorientations for different thicknesses) and the heat diffusion involves both D , and D , . With a laser beam size on the order of 250 pm, and film thickness of about 130 pm, Hsiung and colleagues observed a slow relaxation time (to)of about 100 ms and a faster relaxation time ( t , ) of 10 ms. In the same sample they also observed optical field-induced reorientation effects. KHOOand NORMANDIN [1985a] and ARMITAGEand DELWART [1985] employed transient thermal grating diffraction to study the thermal diffusion process. In their studies a thermal index grating is generated on a liquid crystal film by two overlapping short (nanosecond) laser pulses, and the light of a cw probe laser is diffracted from the grating (c.f., fig. 8). Both the turn-on time and decay time can be measured. Moreover, the grating constant can be varied by simply varying the angle of crossing of the two pump beams. By changing the polarization and/or using planar film rather than homeotropic film, the anisotropy in the diffusion constant can also be measured. In this case the thermal diffusion time constants (obtained under conditions where a onedimensional diffusion model is applicable) are in good agreement with results obtained by other techniques. The study by KHOOand NORMANDIN [ 1985al also showed that if the crossing angle is such that the grating spacing matches


II,§ 21

129

I

f =914mrn

Fig. 8. Experimental setup for observing nanosecond laser-created thermal grating diffraction in a nematic film. The inserts show the temporal evolution of the laser pulse (from a frequencydoubled Nd:Yag laser) operating in either single mode or two mode.

the acoustic wavelength, and the incident laser contains two frequency components (whose separation matches the acoustic frequencies), high-frequency (100 MHz) acoustic waves are generated. These acoustic waves interfere with the “static” thermal index grating and produce modulations on the diffracted probe beam. They were found to have a decay time constant of about 100 ns. On the other hand, ARMITACEand DELWART [1985] employed singlefrequency laser excitation on samples maintained at various temperatures and showed that the rise time of the thermal grating diffractions buildup lengthens considerably as the nematic c1 isotropic phase transition temperature (T,) is approached (from about 20 ns at Tfar from T, to more than 400 ns at T x T,). This “slowing down” of the thermal index change in response to the heat deposited by the short (10 ns) laser pulse is attributed to the temporal response of the order parameter S near T,. For some geometries employed in these studies, for example, where the beam sizes are large compared with the film thickness, whereas the grating period is much smaller than the film thickness, the thermal diffusion process can be closely approximated by a one-dimensional model (i.e., the heat diffusion occurs only along the grating wave vector, denoted by x). For f parallel to

130


[II, § 2

A, the decay time constant is given by 1 TI=-.

(2.46)

D I 4;

For

P perpendicular to A the decay time constant is (2.47)

For typical values d = 40 pm,and a grating constant (q - ') of about 17 pm (KHOOand NORMANDIN [ 1985a]), 7/ x 50 ps and 7* z 100 ps, which agree well with experimentally observed values. These rise and fall times are important parameters in optical wave mixing (e.g., phase conjugation) involving thermal index grating for the optimization of the diffraction or phase conjugation reflectivity with the appropriate laser pulse lengths (HOFFMANN[ 1986b1). 2.5.2. Orientational dynamics The time constant of the director axis relaxation in the nematic phase, a collective phenomenon, is orders of magnitude slower than the individual molecular motion. Typically, director axis response times range from milliseconds to seconds (DEGENNES[ 1974]), whereas individual molecular response times lie in the picoseconds to nanoseconds regime (PROSTand LALANNE [ 19731; LALANNE,MARTIN,POULIGNY and KIELICH[ 19771). The details of these dynamical processes are complex. In fact, under intense optical fields the rise time for director reorientation can approach the nanosecond regime (KOVALEV,NEKRASOV,PILIPOVICH, RAZVINand SERAK[ 19791, ARAKELIAN, KARAIAN and CHILINGARIAN [ 19821, HSIUNG,SHIand SHEN[ 19841, KHOO,MICHAELand YAN [1987]). The rise time of optical field-induced reorientation is also dependent on the direction and magnitude of any applied dc fields (Wu [ 19851, KARN,ARAKELIAN, SHENand ONG [ 19861). Because of their complex fluid-crystal nature, liquid crystals demand a good deal more parameters to characterize when it comes to their orientational dynamics. In general (DEGENNES[ 19741) a complete solution of the problem requires solving equations for the flow (or velocity field) v(r, t), the director n(r, t), and the pressurep(r, t ) with five independent viscosity coefficients and various other liquid crystalline parameters such as elastic constants, density, bulk modulus, electrostrictive coefficients, moments of inertia, etc. Although flow and other associated effects have been alluded to briefly in some recent

11, § 21

131


studies (HSIUNG, SHI and SHEN [1984]), so far most studies of nonlinear optical processes have avoided this complication. Furthermore, under appropriate geometrical-optical configuration the dynamics of the process can be closely approximated with one effective viscosity coefficient and one elastic constant. Consider, for example, an obliquely incident linearly polarized laser beam on a homeotropically aligned film as shown in fig. 3. Assuming that the laser beam size is very large compared with the film thickness, the equation of motion for the director axis reorientation angle is given by (KHOO,MICHAELand YAN [ 19871) (2.48) Assuming hard boundary conditions and a small reorientation angle 8, the temporal and spatial dependence of e(t, z ) can be separated by assuming that e(t, z ) is of the form (2.49) which, when substituted into eq. (2.48), gives dT

-=

dt

(~Y

2bk

cos 2p -

5)T d2Y

(y

4bK sin 28) T2 + -sin 28. (2.50) KY

Solution of eq. (2.50) gives

0.5 A

T(A = B \

+D

("D + ~

- 0'5 e 2 A D r - 1)

0.5

I

(2.51)

1 + - - O a 5 e2ADt D + 0.5

where (2.52)

(2.53)

132


and

4Kb C=-sin2P.

(2.54)

=Y These solutions allow one to identify the so-called rise time. For E$ S E t we get

Y kse =

A E E ~ cos 28

1

(2.55)

411

and for E&, -g E; we have

(2.56) For general field strength, qiseis given by

(2.57) where

f=

AEE& cos 28 4 KK

(2.58)

g=

4c2E& sin228 3n4K2

(2.59)

and

In general, for optical intensities on the order of lo2 MW/cm2, a rise time of a few tens of nanoseconds is possible. Details may be found in the work of HSIUNG,SHI and SHEN[ 19841 and KHOO,MICHAELand YAN [ 19871. The interesting feature in eq. (2.55), and the solution for T(t) involving a high-intensity optical field, is the disappearance of the film thickness d and the elastic constant K; that is, the macroscopic correlated molecular torques play an insignificant role compared with the viscous forces. Although one may be tempted to conclude that under these conditions nematic liquid crystals lose their liquid crystal mesophase characteristics, the appearance of the sin 28 and

11,s31

NONLINEAR OPTICAL PROCESSES

133

cos 2p factors, nevertheless, reminds us of the directional correlation characteristics of nematics. Despite the extraordinarily large orientational nonlinearity, the slow decay time of the reorientation remains as a major limitation in practical device applications. The decay time is simply given by

for reorientation induced by a single laser beam (see also, FUH, CODEand Xu [ 19831) for the case involving a grating). For a nematic film thickness of 10 pm, and y z 0.1 poise, K = 10 - dyne, one has z~~~~~z 100 ms. The response time scales up or down depending on the square of the thickness (and/or grating spacing). Several techniques and means of accelerating the relaxation processes in nematics have been proposed and attempted. Obviously it is helpful to choose materials with lower viscosity and higher elastic constants. In this respect synthesis of new liquid crystals to remove this slow response limitation is clearly necessary. There also have been attempts at using two-frequency fields to promote the relaxation process (DEJEU, GERRITSMA, VAN ZANTENand GOOSENS [ 19721, SCHADT[ 1972]), and recently, dual field effects have been employed (Wu [1985], HUIGNARD[1986]). In the study by Wu [1985] a magnetic field is used in conjunction with a dc electric field to demonstrate that submillisecond relaxation times can be achieved with the application of field strengths much higher than the Frederiks field. HUIGNARD [ 19861 has reported a submillisecond decay time of optical field-induced reorientation, when a dc field (from specially designed electrodes) is applied to stabilize the nematic back to its original orientation.

8 3.

Nonlinear Optical Processes

3.1. SUMMARY OF OBSERVED NONLINEAR OPTICAL EFFECTS

Since the first experimental verification in the 1980s of the extraordinarily large optical nonlinearity associated with nematic axis reorientation and the recognition of the extremely large thermal index gradients of nematics for nonlinear optical applications, there has been tremendous progress. The accelerated pace in this field closely parallels similar advances in other areas

I34


[II, 5 3

(e.g., photorefractive crystals, semiconductors, and multiple quantum wells). The following is a summary of some of the nonlinear optical processes that have been observed: (1). Optical wave mixings, including beam amplification (HERMANand SERINKO [ 19791, KHOOand ZHUANG[ 1980]), optical wavefront conjugation (GARIBYAN, KOMPANETS, PARFYONOV, PILIPETSKII, SHKUNOV, SUDARKIN, SUKHOV, TABIRYAN, VASILIEV and ZELDOVICH [ 19811, KHOOand ZHUANG [ 19821, LEITH,CHEN,CHENG,SWANSON and KHOO[ 1983]), self-diffraction (KHOO [ 1981a,b, 1982b, 19831, DURBIN, ARAKELIAN and SHEN [ 19821, ARAKELIAN and CHILINGARIAN [ 1986]), phase conjugation with gain (KHOO [1985], RICHARD,MAURINand HUIGNARD[1986]), and thermal grating diffraction (KHOO and NORMANDIN[ 1984a,b, 1985a,b], ARMITAGEand DELWART[ 19851). (2). Self-focusing and self-phase modulation, including external self-focusing of a low-power cw laser (ZELDOVICH and TABIRYAN [ 19801, ZOLOTKO, KITAEVA, SOBOLEVand SUKHORUKOV [ 19811, KHOO, ZHUANGand SHEPARD [ 1981]), self-phase modulation ring formation (VOLTERRA and WIENERAVNEAR [ 19741, ZOLOTKO,KITAEVA, KROO,SOBOLEV and CHILLAG[ 19801, DURBIN,ARAKELIAN and SHEN[ 1981a1, SANTAMATO and SHEN[ 1985]), self-limiting and bending (KHOO, FINN,MICHAELand LIU [1986]), optooptical modulation (MARTIN-PEREDA and LOPEZ[ 1982]), and suppression of self-focusing (KAWACHI,KAWASAKI and HILL [ 19821). (3). Optical bistability and switching, including Fabry-Perot type (CHEUNG,DURBINand SHEN[1983], KHOO, H o u , NORMANDIN and So [ 1983]), transverse bistability (KHOO,LIU,YAN,SHEPARDand H o u [ 1984]), intrinsic bistability (NERSISYAN and TABIRYAN [ 19841, ONG [ 19851, KARN, and SHEN[ 1986]), cavityARAKELIAN, SHENand ONG [ 19861, SANTAMATO less switching and other forms of bistability (WINFUL[ 19821, ZELDOVICH and TABIRYAN [ 19841, KHOOand H o u [ 1985]), and hybrid bistability (KOMPANETS, PARFYONOV and POPOV [ 19811, SONG,SHIN and KWON[ 19841). (4). Nonlinear interface, including frustrated total internal reflection (KHOO [ 1982c]), planar waveguide (VACH,SEATON,STEGEMAN and KHOO[ 1984]), and fiber-optic coupling (GOLDBURT and RUSSELL[ 19851). (5). Stimulated scattering (ZELDOVICH and TABIRYAN [ 19801,ZELDOVICH, MERSLIKIN, PJLIPETSKY and SUKHOV[ 19851). These studies involve various forms of lasers (pulsed, cw, low power, high power, visible, infrared, etc.) in conjunction with special characteristics of the liquid crystal films, and nonlinear optical processes and configurations. Since it would perhaps require a treatise to examing all these processes in a

11, § 31


135

quantitatively meaningful manner, interested readers should consult the literature for a detailed discussion. This section will concentrate on a few selected nonlinear processes in which the extraordinarily large nonlinearity of liquid crystals has shed new light on our fundamental understanding of them, on processes that appear to be on the threshold of being applicable to practical devices, and on special processes that can be obtained only with highly nonlinear materials. For further insight into the variety of nonlinear optical processes, the reviews by some research groups (FEINBERG [ 19831, YARIVand FISHER[ 19831, HELLWARTH [ 19831, JAIN and KLEIN [ 19831, CHEMLA, MILLERand SMITH[ 19851, HUIGNARD,RAJBENBACH, REFREGIERand SOLYMAR [ 19851, ZELDOVICH, PILIPETSKY and SHKUNOV[ 19851) may be consulted. The main motivation in using nematic films for these studies are their well-standardized fabrication techniques (production of high-quality liquid crystal films of planar, homeotropic, or hybrid alignments with special electrodes); the low-cost, low-power requirements and tremendous flexibility in the optical-polarization-dc-electric-field-nematic-axisgeometrical configurations and their juxtaposition; and transparency of the liquid crystals throughout the visible, near-, and far-infrared regime. In these respects, and since there are few room-temperature highly nonlinear optical materials in the infrared regime, nematic liquid crystals will undoubtedly play an important role in infrared nonlinear optics in the near future. Their linear scattering properties, for example, have been recognized in some infrared optical modulator studies (PASKO,TRACYand ELSER [1979], Wu, EFRON,GRINBERG, HESS and WELKOWSKY [ 19851). The extremely sensitive dependence of the refractive index on the temperature traditionally serves either as a good testing ground for fundamental phase transition studies or as a source of nuisance for some liquid crystal devices. In the light of nonlinear optics based on thermal effects, it now becomes a new source for interesting nonlinear processes and possible applications.

3.2. GENERAL REMARKS ON NONLINEAR OPTICAL PROCESSES

The basic mechanism of all these nonlinear processes is actually very simple. In a linear medium, where the induced polarization (response) of the medium is linearly proportional to the optical electric field (applied field), one light wave cannot interact with another. In a nonlinear medium the medium will mix all the input optical electric fields together, thus providing new temporal or spatial frequency components. New temporal frequency components can be generated

136


[II, I 3

if the medium’s response is fast enough, resulting in phenomena such as second-, third-, and higher-harmonic generations. On the other hand, if the nonlinearity is slow, such as the orientational and thermal nonlinearities in liquid crystals, only temporally slowly varying components in these mixed fields will be manifested. Thus, if two fields with frequencies o1and o, are mixed, forming an intensity interference pattern with a frequency of oscillation I o,- o,1, the medium will produce a sizable response (e.g., reorientation) only if I w , - w, I < z, where z is the relaxation time constant of the reorientation process. By purposely offsetting the frequency of one beam relative to another by an amount equal to z - ’ , new “responses” and nonlinear coupling and energy transfer in the two-wave mixing process can occur in situations where such couplings are not allowed (YEH [ 19861). In the following sections we will be concerned mainly with processes that can be described by representing the effects of the applied field with an intensity-dependent refractive index that is stationary in time. For optically induced reorientation involving a single laser beam, the change in refractive index An is given by eq. (2.20), that is An cc Z, the optical intensity. If the intensity involves some sinusoidal interference pattern, then we have an index grating, whose modulation ratio will depend on several geometrical, optical, and material parameters. The magnitude of this index grating is again proportional to the optical intensity (DURBIN, ARAKELIAN and SHEN [ 19821, KHOO[ 19831). In the case of thermally induced refractive index change, the dependence of An on the optical intensity is a complicated function of the geometry, as was discussed earlier, but in many situations it may be reduced also to a form analogous to eq. (2.20), that is, proportional to Z, the optical intensity. As a result of these refractive index changes, any beam traversing this nonlinear medium could experience gain (from another beam), loss (to another beam), diffraction, focusing, defocusing and reflection (with amplification!), bistable and multistable transmission, frequency broadening, and many other interesting effects. Some of these effects that occur in other nonlinear media, as well as fundamental theories about them, have been discussed in recent texts (REINTJES [ 19831, SHEN [ 19841). In the following section discussion will be limited, therefore, to relatively new theoretical formalisms and insights and to experimental observations. 3.3. SELF-FOCUSING AND SELF-PHASE MODULATION

The passage of a laser beam through a nonlinear optical material is inevitably accompanied by intensity-dependent phase shift on the wavefront of the laser,

I I , § 31


137

as a result of the intensity-dependent refractive index and the finite beam size. This and other possible mechanisms (e.g., generation of new frequency components such as Brillouin or Raman) lead to severe distortions on the laser in the form of self-focusing, defocusing, trapping, beam breakups, filamentations, spatial ring formations, and others (AKHMANOV, KHOKHLOV and SUKHORUKOV [ 19721, MARBURGER [ 19751, SHEN[ 19751). For thick media the problem of calculating the beam profile within the medium is extremely complicated. Numerical solutions are almost always the rule. A good discussion of these so-called self-action effects may be found in the bibliography quoted. Basically, the problem of self-action in thick, but not really highly nonlinear, media becomes important only in conjunction with high-power lasers. For example, the critical size o,below which a laser beam becomes unstable in propagating through a medium (i.e., begins to form filaments) with a refractive index of the form (2.19) is given by (BESPALOVand TALANOV [ 19661) (3.1) where Pc is the critical power at which a beam will experience a self-focusing effect (for n2 > 0) that just balances the diffraction (i.e., self-trapping),

Pc N (1.221)2c/128n2.

(3.2)

Combining (3.1) and (3.2) gives 0,x

1.221& (32 n n , 1 ) ~ / 2*

(3.3)

For a typical nonlinear medium, n2 x 10- esu. Assuming a wavelength 1 = 1.06 pm (Nd:Yag laser) and an intensity I x 200 MW/cm2, one has o c x 10pm. The situation in liquid crystalline materials and other highly nonlinear materials (e.g., multiple quantum wells) is obviously very different, with n2 on the order of esu or larger. On the other hand, these nonlinear media are usually very thin (microns), and the problem of optical propagation through these materials takes on a simplicity that allows for analytical solutions. The simplicity comes from the assumption that in the passage through such a nonlinear film, the laser beam is only phase modulated but otherwise suffers negligible amplitude change. (This assumption is not always valid, of course. For example, in optical wave mixing involving several beams, some beams could undergo > 100% amplitude change.)

138


[II, § 3

Consider now a steady-state self-phase modulation effect (e.g., that induced by a cw laser). If the response of the medium to the laser field is local, the spatial profile of the phase shift follows that of the laser; that is, given that the laser transverse profile is I ( r ) (assuming cylindrical symmetry), the nonlinear part of the phase shift is simply $(r) = dAn(r) 2 ./A, where dis the thickness of the film. In nematic films two important points should be made. First, the nonlinearity is dependent on the distance z into the film. Thus n2 is a function of z as well. Second, since the reorientation is a nonlocal response, the transverse profile of the phase shift is not necessarily coincident with the laser’s beam profile. The first problem can be simply solved by writing $(r) as an integral over 2n z of An@, z), that is, ~

1

The second problem, that is, finding the r dependence of An, can only be solved by solving eq. (2.24) or (2.25) in accordance with the laser-nematic interaction geometry. In early studies (ZOLOTKO, KITAEVA,KROO,S o B o L E v and CHILLAG [ 19801, DURBIN, ARAKELIAN and SHEN[ 1981a]), these transverse nonlocal effects were ignored and An(r, z ) was simply assumed to have the transverse profile of the laser intensity, that is, An(r, z) = n2(z)I(r). For a Gaussian laser input, that is, I ( r ) = I, exp( - 2 r 2 / o ; ) ,where o,is the beam waist, An(r, z ) and therefore $(r) is a Gaussian function in r. In that case, because there exist two radial positions where the radial wave vectors k,, i.e., d$/dr, are equal, light with different intensity (i.e., phase shift) will interfere in the far field, and interference effects in the form of bright and dark rings will occur (cf. fig. 9). It is interesting (and surprising) to note that by using such a qualitative picture,

Fig. 9. Typical rings observed in the far field due to transverse self-phase modulation effect in the passage of a Gaussian laser beam through a nematic film.

11, § 31

I39


these authors were able to obtain good agreement between the estimated number of rings and the actual observed values. To describe the process quantitatively and accurately, it is necessary to know not only the nonlocal exact radial profile of the phase shift, which is not always Gaussian, but also the effects of all the diffraction processes not properly accounted for in these early studies. A quantitative theory for various input laser profiles may be seen in the work of HERMANN[ 19841. The present discussion will be limited to a Gaussian beam. For the case of obliquely incident (Gaussian) laser beams and for beam sizes much larger than the film thickness, the nonlocal factor may be ignored. The diffraction, however, can be accounted for by employing Kirchhoff s diffraction integral (BORNand WOLF[ 19701) to calculate the far field transmission (YAN [ 19831, FERNANDEZ[ 19831, KHOO, Hou, LIU, YAN, MICHAELand FINN [ 19871). In fig. 10 the intensity distribution at a distance lfrom the film is given by the integral

{

x exp ik[Ti,dZ,

exp

(:3(L)’> (q) +

t

21

2R

r;]}

dr, ,

J,

(3.5) where Z, is the on-axis (rl = 0) intensity of the laser beam on the sample, r, is the radial dimension on the film, r is the radial coordinate on the observation plane, w is the beam waist of the laser at the plane of the nonlinear film, and Ti, is the average value of n2(z). The problem of calculating the intensity distribution Z(r, I ) is thus reduced to a simple integral. It has recently been shown (KHOO,LIU, YAN and Hou [ 19861, KHOO,Hou, LIU, YAN, MICHAEL and FINN[ 19871) that Z(r, I ) can P

___---

Fig. 10. Schematics of a laser beam incident on a nonlinear thin film. The partially reflecting mirror is needed for the transverse bistability experiment.

140

[II, § 3


be grouped into three distinct forms in accordance with the sign of the diffraction prameter (I - I + R - ') :

The intensity distribution evolves from a Gaussian shape to one with a bright central spot and concentric bright and dark rings similar to those in fig. 9. Figure 11 is a computed I(r, 1) for increasing intensity I,.

The intensity distribution evolves from a Gaussian beam to one where the central area tends to be dark (low intensity) surrounded by bright rings. Figure 12a is the experimentally observed pattern and fig. 12b is the numerical plot for increasing values of I,.

(3)

(I + i) x 0 1 R

case:

RADIUS Fig. 11. Numerical solution of the intensity distribution in the case where ( I

-

'+R-

I)

> 0.

141


11.8 31

4

I 3 N

T E N s 2

I

T Y 1

0 0

I

2

3

4

5

6

RADIUS

Fig., 12.(a) Photograph of I ( r , I ) for the case ( I - + R - I ) c 0. (b) Numerical results of I(r, I ) for ( I - + R - I ) < 0 case, for increasing values of I,.

’

This is the so-called intermediate case, where the intensity distribution at the central area assumes forms that are intermediate between those of cases ( 1 ) and (2) above (cf., fig. 13). For the case of n, < 0 the intensity distribution is almost a “mirror” image of the n, > 0 case; that is, the distribution in cases (1) and (2) is interchanged.

142

[II, § 3


RADIUS

Fig. 13. Numerical solution of I(r, I ) for the intermediate case ( I -

’ + R - ’) s 0.

It is obvious from these preceding discussions that merely counting the “number of rings”, without taking into account both I and R , can at best give only a qualitative estimate of the magnitude of nonlinearity or even its sign and the number of observed rings. In this regard, several studies of other nonlinear materials, where this rough approach is employed to estimate n2, probably deserve a second look. The case of a normally incident laser beam is complicated by two factors. One factor is the nature of the reorientation process, which, as was discussed in an earlier section, will also give a response that has a “width” considerably different from the laser beam waist. In most of the studies of optically induced Frederiks transition, the laser beam size is a few times the thickness of the film. This means that the reorientation profile will be at least two to three times narrower than the laser beam waist. The second factor is that the response is not Gaussian. This problem has been partially addressed by SANTAMATO and SHEN[ 19851, who numerically evaluated the reorientation and accounted for the radius of curvature R of the wavefront. As long as the factor (I - + R - ’) has the same positive sign, the conclusions regarding the intensity distribution obtained with this approach (ignoring 1/f) will be accurate.

’

143


’

’

The effect of the diffraction parameter (1 - + R - ) on the “brightening” or “darkening” of the intensity at a distance 1 is of particular importance when the intensity at I is imaged back onto the film by a partially reflecting mirror. From the preceding discussion one notes that for the n, > 0 case a positive feedback (i.e., intensification of the central part of the illuminated film) is achieved if (I - + R - ’) > 0. For then, < 0 case (2), (1 - + R - < 0) will give a positive feedback. For a given I (which is always positive), therefore, positive feedback is obtained for R > 0 (n, > 0) or R < 0 (n, < 0). Such positive feedback is crucial in obtaining optical bistability in the transmission (KHOO,LIU,YAN, SHEPARD and H o u [ 19841, KHOO,LIU, YAN and H o u [ 19861). In 0 2.5 it was pointed out that at higher optical intensity the director axis will reorient faster. At 10, MW/cm2 the initial portion of a nanosecond laser pulse is sufficient to induce a phase shift so that the later part of the pulse will be phase modulated to produce far-field diffraction rings. Figure 14 shows a photograph of the transmitted laser beam profile obtained in a recent study (KHOO, FINN, MICHAELand LIU [1986]), using a (single-shot) doubled Nd:Yag laser (20 ns, about 100 MW/cm2) on a nematic film (homeotropically aligned, flz 22”). A more detailed study by KHOO,MICHAELand YAN[ 19871 shows that, indeed, the diffraction rings are formed in a time-dependent

’

’

’

Fig. 14. Photograph of a far field transmitted nanosecond laser pulse showing diffraction rings.

144


[II, 8 3

manner. Figure 15 is the intensity observed at an off-axis position. As the transverse phase modulation effect intensifies, outgoing rings are detected as oscillations in time. This dependence on time is analogous to the numerical results (fig. 12b) for the dependence on intensity, if we note that the temporal dependence reflects the integrated effects from the laser pulse. In the same study a coincident cw He-Ne laser was used to probe the phase shift, especially the relaxation portion. It was noted that the diffraction rings of the He-Ne persist for a few seconds after the Nd:Yag laser pulse, which is in agreement with the expected reorientation time for a 100 pm sample. Based on this transmitted intensity modulation effect (in thick or thin samples) several other schemes for optical switching have been developed (BJORKHOLM, SMITH,TOMLINSON and KAPLAN[ 19811, SOILEAU, WILLIAMS and VAN STRYLAND [ 19831, TAI, GIBBS,PEYGHAMBARIAN and MYSYROWICZ [ 19851) in conjunction with other nonlinear materials. Optical self-limiting of a cw laser and a nanosecond-pulsed laser have been observed with nematic film (KHOO,FINN,MICHAELand LIU [ 19861) involving the orientational and thermal nonlinearities, and in the isotropic phase of liquid crystals (SOILEAU, GUHA,WILLIAMS,VAN STRYLANDand VANHERZEELE[ 19851) involving two-photon absorptions.

Fig. 15, Oscilloscopetrace of the time evolutionofthe transmittedoff-axislaser intensity showing “switching”to oscillations after about 10 ns.

11, I 31


145

3.4. OPTICAL WAVE MIXING

3.4.1. Selfsdflraction and degenerate four-wave mixing The formation of rings and self-focusing of a laser beam with a finite beam size and radius of curvature of the wavefront (i.e., a finite spread of its spatial frequency k) may be viewed as a form of mixing of light. In this type of mixing the temporal frequency of all the spatial frequencies involved is fixed, and the effect is the creation of a new spatial frequency component (the beam selffocuses or defocuses externally). The fundamental mechanism is that the nematic film imparts a spatially varying (e.g., a Gaussian or some bell-shaped function) pase shift on the wavefront of the incident laser. An equally important form of optical wave mixing process, called the degenerate four-wave mixing process, dates back to the early days of holography (GABOR [ 19491, LEITHand UPATNIEKS [ 19621). In holography a reference light beam and an object beam of the same frequency form an even more complex but interesting time-independent phase (or amplitude) grating. The end result is that, when illuminated with a reconstruction beam, a diffracted or image beam is generated. If, for simplicity, we illustrate all the beams involved as plane wave with wave vectors k,, k2, k, and k, (for reference, object, reconstruction, and image beam, respectively), the process may be represented by the “phase-matching” diagram in fig. 16. Notice that if the wavelengths (A’s) of all the light involved are the same, the vector diagram involving these k’s (of magnitude 2 x / L ) can only be approximately “closed”. There exists, therefore, a small phase mismatch Ak. The magnitude of A k depends on the geometry and can be made vanishingly small by choosing the directions of these k vectors appropriately. Its effect can also be minimized by using very thin film. In actual experiments k, and k3 may be derived from one single laser beam (k,11 k,). In this case one may say that it is a self-diffraction effect; that is, beam 1 forms a grating with beam 2 and diffracts from the grating to generate

Fig. 16. Schematics of degenerate four-wave mixing involving four beams: k, (reference), k, (object), k, (reconstruction) and k4 (image).

146


“I, § 3

beam4. Beam2 can also diffract from the same grating to give its own self-diffracted beam (not shown in fig. 16). In fact, if beams 1 and 2 are of the same intensity, a great number of self-diffracted beams at 2 8, 2 28, 2 3 4 etc., may be generated (cf. fig. 17). These have indeed been observed by KHOO [ 1982a,b] and DURBIN, ARAKELIAN and SHEN[ 19821. Because the diffractions are generated in directions different from the incident laser beam, and because the effects are extremely pronounced and polarization dependent, they have been studied in various contexts for understanding the mechanisms of reorientation with applied dc fields and for determining liquid crystalline parameters (see, for example, FUH,CODE and Xu [ 19831). For thermal nonlinearity, studies of these self-diffraction effects have provided useful information on the dynamics of thermal grating buildup and decay and on the interference effects from acoustic waves (KHOOand NORMANDIN [ 1984a,b]). Such studies also allow one to study ultra-high-frequency acoustic velocities and acoustic attenuation. A variant of this process, in which the thermal grating is formed with light of one wavelength and probed by light of another wavelength, allows one to convert images from one wavelength to an other, as demonstrated by MARTIN and HELLWARTH [ 19791with ordinary liquids and by KHOOand NORMANDIN [ 1985bl with liquid crystals. 3.4.2. Optical wa vefront conjugation

A special case of optical wave mixing arises when the reconstruction beam is counterpropagating to the reference beam. In that case the object beam is

Fig. 17. Photograph of the multiorder diffraction from a nematic film illuminated with two equal-intensity (about a few watts/cm2) laser beams.


147

preferentially generated in the direction counterpropagating to the input image beam (fig. 18). In fact, it can be shown from coupled Maxwell wave equation [ 19771, YARIVand PEPPER[ 19771)that the generatand theory (HELLWARTH ed image beam is a phase conjugate of the input image. It follows, therefore, that if the input beam suffers a phase distortion (e.g., by some bad optics), the phase-conjugated image beam will, in traversing back through the distortion, be reconstructed with the aberration removed. This process is analogous to the early holographic phase conjugation work of LEITHand UPATNIEKS [ 1962, 19651 and KOGELNIK [ 19651, involving static holograms. In the present case, because of the dynamical dependence of the process on the various timecoincident interacting beams and the medium, there are many more possible holographic gratings that can be generated. Therefore, many new aspects not possible in static holography can be achieved with optical phase conjugation (FISHER[ 19831, ZELDOVICH, PILIPETSKY and SHKUNOV [ 19851). In particular, under favorable conditions (YARIV[ 19791, FEINBERC and HELLWARTH [ 19801, FISHER, CRONIN-GOLOMB, WHITEand YARIV[ 19811, CAROand GOWER[1982]) the phase-conjugated signal beam can be more intense than the input beam (i.e., the image is amplified). In that case self-oscillations between a mirror and the nonlinear film can also occur. Amplified reflections and self-oscillations using thermal effects of nematic film have been observed by KHOO[ 19851 and RICHARD, MAURINand HUIGNARD [ 19861. The first demonstration of optical phase conjugation using nematic film was reported by KHOOand ZHUANG[ 19821,in which the aberration correction capability of the process was also verified with a low-power (z1 W) cw laser. Optical phase conjugation and image correlation studies were also demonstrated by FEKETE, Au YEUNGand YARIV[ 19801,using an isotropic-phase nematic liquid crystal, for which much greater laser power (of the order of kilowatts) was needed because of the much smaller orientational nonlinearity. An interesting variant of phase conjugation, namely, one using a spatially partially coherent laser, was employed by LEITH,CHEN,CHENC,SWANSON and KHOO[ 19831. Figure 19 illustrates the experimental setup. A spatially Nonlinear Medium

Fig. 18. Geometry for optical phase conjugation.

148


“I, 8 3

, MI

BS

Fig. 19. Experimental setup for phase conjugation with spatially partially coherent laser. Key: L, lenses; M, mirrors; BS, beam splitter; 0, object; Im, image; P, phase conjugator; D, diffuser (rotating ground glass).

coherent laser (Ar ) is rendered spatially partially coherent after focusing and recollimating it through a rotating ground glass. The beam is then split into two, one part acting as the reference beam and the other part acting as the object beam and going through an aberrator. The two beams are then brought to coincidence on the nematic liquid crystal film. As a result of the partially incoherent nature of the beams, speckly noise from the system is reduced. On the other hand, the phase aberration correction capability of the imaging process is preserved. Figure 20a is a picture of the reconstructed beam with a great deal of background noise, when a coherent laser is used. In great contrast to this fig 20b shows the phase-conjugated signal with very little background noise. This experiment demonstrates again that many holographic imaging (LEITH and SWANSON [ 1982]), image correlation, and processing techniques can be applied to phase conjugation with the added advantage of obtaining a real-time and possibly amplified signal. More recently, PUANG-NGERN and ALMEIDA [ 19861reported using nematic liquid crystal film for phase-conjugated image correlation with the input image. There are obviously many other image-processingpossibilities that are currently being investigated, for which nematic film could be used. In general, phase conjugation processes involving orientational nonlinearity are quite slow +

I49

NONLINEAR OFTICAL PROCESSES

11, § 31

Fig. 20. Phase-conjugated signal with coherent laser (noise background a bit exaggerated). (b) Phase-conjugated signal with spatially partially coherent laser.

(z1 s), whereas those employing thermal nonlinearity are a bit faster (milliseconds). A common limitation is the maximum thickness of the aligned nematic film; nematic films cannot be made thicker than about 300 pm before instabilities and flows and large scattering loss set in to degrade the nonlinear process.

3.4.3. Beam amplifiation and dynamic wave mixing The first experimental verification (KHOO and ZHUANG[ 19801) of the degenerate four-wave mixing process in a nematic film theoretically predicted by HERMANand SERINKO [ 19791, is the amplification of a weak probe beam by a strong pump beam (cf. fig. 21). The explanation by Herman and Serinko of the probe beam amplification process is that the pump and the probe beam create a phase grating by means of optical-plus-dc-field-inducedmolecular reorientation. The pump beam, in traversing the medium, therefore experiences a phase shift and scatters light in the probe beam direction, thereby increasing its output intensity. This qualitative explanation is not valid for very large beam

,-LIQUID CRYSTAL IW

/

L Fig. 21. Schematic of geometry for probe beam amplification in a Kerr medium.

150


[It, 8 3

amplification, and it cannot account for other dynamical coupling between the pump, probe, and diffracted beam. These optical wave-mixing processes can be accurately described by the coupled Maxwell wave equations. Denoting the amplitude of the pump, the probe, and the diffracted beams by A,, A ,respectively, the usual coupled wave approach (REINTJES[ 19831) yields

,

aA, - i&,k

aZ

2n2

(IA

+ 4AdA , A 2 cos @) ,

I A,e'+

(3.7)

aA2

aZ

-

iE1k(lAo~2A2 +AEA?)e'@, 2n2

where E , comes from the expression of the dielectric constant E of the medium by E = ~0 + E , IE 12, ~0 is the unperturbed dielectric constant, E is the optical electric field,

EJ. = '2(A J (z)

ei(br-

+ c.c) ,

01)

k is the magnitude of the wave vector, w is the frequency, and @ is the phase shift between the intensity grating and the refractive index grating. For simplicity and illustrative purposes (while preserving the physics), we have neglected the phase mismatch, absorption, and/or scattering loss in the preceding equations. A highly interesting effect can be readily obtained from these equations. If we neglect pump depletion and consider a Kerr-like medium (e.g., liquid crystal) where @ = 0, then we have

(3.9)

and (3.10)

where

CI

=

Elk ~.

2n2

151

NONLINEAR OPlICAL PROCESSES

11,s31

These, upon differentiation, give a2A 1 --o=-* az2

a2A2

(3.11)

az2

From the boundary conditions z = 0, A , = A,(O) and A , = A,(O) and A , that is, using two input beams (pump and probe), we therefore have

=

0,

(3.12)

or A , ( z ) = ia(A,(0)12Al(0)z + A,(O) = A,(0)(1

+ ia(A,12z).

(3.13)

The intensity of the probe beam is thus given by I,(z) = I,@) (1

+ a21A,14 2,).

(3.14)

This means that the probe beam experiences a gain of (3.15) that is, the gain is proportional to the square of the pump beam intensity - an effect first observed by KHOO and ZHUANG[ 19801. This theoretical exercise is very important, since it shows that by including the effect of the coupling between the pump beam A , and the probe beam A,, by means of the term AzAz in eq. (3.9), the probe beam can be amplified; that is, beam amplification in a Kerr-like medium is possible without resorting to a moving grating technique proposed recently by YEH [ 19861. In order for beam amplification in a Kerr medium to work, of course, the nonlinear medium must be very thin so that phase mismatch is negligible. The nonlinearity must also be very large in such a thin medium in order for the amplification effect to be of observable magnitude. Nematic liquid crystal films, therefore, are naturally among the best candidates. The effect was recently studied more quantitatively by KHOO and LIU [1987], where up to 600% increase in the probe beam was observed. Figure 22 shows the observed probe beam amplification as a function of the angle of crossing between the pump and the probe beam. The crossing angles used are very small so that the grating spacings are comparable to or larger than the nematic film thickness. It is interesting to note that a pump beam (A,? laser) intensity of about 1 W/cm2 can produce such a large amplification of the probe beam in a thin film

152

[II,


000

.002

,004

.006

.OOB

J3

.010

CROSSING A N G L E ( d e g r e e )

Fig. 22. Observed probe beam gain as a function of the crossing angle between the pump and the probe beam.

( z 100pm). A theory of this multiwave-mixing effect, taking into account all conceivably relevant parameters (e.g., loss, absorption, phase mismatch, presence of other higher-order diffracted beams, intensity, beam ratio, medium’s orientational response, frequency offset between the input beams, etc.) has been worked out (Lru [ 19871). Based on this beam amplification effect, ring oscillators may be constructed, in which the nonlinear material is placed within a ring cavity and externally pumped by another laser (FEINBERG and HELLWARTH [ 19801, RAJBENBACH and HUIGNARD[ 19851). Dynamic interferometry (HERRIAU,HUIGNARD, APOSTOLIDIS and MALLICK[ 19851) and laser beamsteering (RAK, LEDOUX and HUIGNARD[ 19841) have also been successfully demonstrated. A more extensive list of the literature on these and other related effects may also be found in the text by REINTJES [ 19831. Research efforts are currently underway that involve using nematic crystals, because of their inherently large nonlinear effect and application feasibility in the infrared regime (3-5 pm and 8-1 1 pm areas) by KHOOand by HUICNARD[ 19861. As mentioned earlier, the application of nematic film to the infrared is highly desirable in view of the scarcity of highly nonlinear infrared materials.

11, § 31


153

3.5. OPTICAL BISTABILITY AND SWITCHING

One class of nonlinear optical processes that has received considerable interest recently is optical bistability - the phenomenon in which the outputs (transmission, reflections, polarizations, etc.) are multiple valued for a given input (GIBBS[ 19851). From early studies ofmainly intellectual curiosity, optical bistability has evolved to the present state, where fundamental issues (e.g., chaos and instability) are mixed in with a great deal of device-oriented (e.g., optical computing) pursuits (GIBBS,MANDEL,PEYGHAMBARIAN and SMITH [ 1986]), and the forms and classes of optical bistability have greatly increased. In liquid crystals the early work by BISCHOFFBERGER and SHEN[ 1976, 19791 with an isotropic liquid crystal in a Fabry-Perot cavity provides good insight into the interplay between the various time scales (cavity lifetime, medium response time, and laser pulse length) and steady-state or quasitransient bistability effects. The study also demonstrates how the various modes of operations (differential gain, bistability, etc.) depend on the initial phase of the Fabry-Perot cavity in which the liquid crystal is placed. Because of the small nonlinearity of the liquid crystal in the isotropic phase, megawatt laser power is required. When it comes to the nematic phase, it is no suprise that the same kind of Fabry-Perot type of bistability can be observed with laser power of approximately a watt or so. Differential gain and bistability (KHOO,Hou, NORMANDIN and So [ 19831) were observed in a Fabry-Perot cavity containing a nematic film. CHEUNG, DURBINand SHEN[ 19831 have also shown that, as a result of the competition between thermal and reorientation effects on the nonlinear phase, periodic oscillations are produced. A new class ofoptical bistability was proposed by KAPLAN[ 1976,19771 and experimentally demonstrated by KHOO[ 1982~1and KHOO,LIU,YAN, SHEPARD and H o u [ 19841 that involves a transverse intensity-dependent phase shift. As discussed in 0 3.3, the transverse phase shift modifies the intensity at the observation plane. If this intensity is partially reflected back on the film (cf., fig. lo), the resulting intensity distribution on the observation plane will exhibit bistability. The theory of Kaplan based on the lens approximation was extended by KHOO,LIU,YAN,SHEPARD and H o u [ 19841 to include also the diffraction and higher-order terms (in the series expansion of a Gaussian function). More recently, it has been shown by KHOO,LIU, YAN and Hou [1986] that the existence of bistability is related to the conditions for on-axis intensification, discussed in detail in 0 3.2. The first experimental verification of this type of external self-focusing bistability was performed using a nematic film (KHOO

154


PI, Q 3

[1982c]). Subsequently, similar effects have also been observed in other materials (TAI, GIBBS, PEYGHAMBARIAN and MYSYROWICZ [ 19851). Figure 23 is a typical observed on-axis intensity as a function of the input intensity, and fig. 24 shows the typical off-axis switching behavior. Bistabilities in optical waveguides have also been observed where the role of the liquid crystal is to provide anonlinear cladding (VACH,SEATON, STEGEMAN and KHOO[ 19841, GOLDBURT and RUSSELL[ 19851). The “evanescent” wave of the light propagating within the waveguide senses the intensity-depending refractive index of the cladding and modifies the transmission modes accordingly. GOLDBURT and RUSSELL[ 19851 have observed intensity-dependent switching of light between fibers separated by a thin film of liquid crystals. In general, the nonlinearity of liquid crystals is too slow for this type of switching to be practically useful, but the feasibility study serves a helpful guide for practical and faster nonlinear cladding materials. More recently, cavityless optical bistability involving nonlinear electrodynamics at an interface (KAPLAN[ 1976, 19771, KHOOand Hou [ 19851) and intrinsic bistability (SANTAMATO and SHEN [ 19861, KARN, ARAKELIAN, SHENand ONG[ 19861) involving the unique physical characteristics of nematic liquid crystals in conjunction with an applied dc field have also been demons-

INCIDENT L A S E R POWER ( W A T T S )

Fig. 23. Experimentally observed on-axis intensity as a function of input intensity showing a bistability loop.

11.8 41

155

FURTHER REMARKS AND CONCLUSIONS

> a 4

a

=

I-

a

2.0

5

0.0 0.0

0.5

I .o

I .5

INCIDENT LASER POWER ( W A T T S )

Fig. 24. Off-axis intensity bistability characteristics.

trated. By biasing the nematic film with an optical field close to the Frederiks transition, it is possible to switch the intensity of this beam with a weak beam to obtain the optical equivalence of transistor action (SANTAMATO and SHEN [ 19861). In most of these studies the switching time is a few seconds to hundreds of seconds (for Frederiks transition type), and thus praeical devices are out of the question. Nevertheless, one gains good insights into the basic mechanisms of all these various forms of bistabilities and switchings, including the geometrical and optical configurations for switching, the required power levels, and also the optimum or desirable material characteristics.

8 4. Further Remarks and Conclusions I have discussed in detail the mechanisms for optically induced refractive index change in principally the nematic phase of liquid crystals, and related nonlinear optical processes such as optical wave mixings, self-focusing and bistabilities. There are, needless to say, several other possible mechanisms in other phases or types of liquid crystals that deserve equal attention in our search for new nonlinear materials or insights into the working of nonlinear optical devices and processes. Some of the nonlinear optical processes that were

156


PI, 5 4

carried out in the isotropic phase of liquid crystals have been pointed out in the text (e.g., self-focusing, wavefront conjugation, and bistability). Such studies have provided quantitative insights into both the nonlinear effects as well as confirmations of the near-T, phase transition theories. A detailed review may be seen in the article by ARAKELIAN, LYAKHOV and CHILINGARIAN [ 19801. Recently, MADDEN,SAUNDERSand SCOTT [1986] presented a detailed analysis and experimental study of the relationships between molecular characteristics (shape, polarizability, viscosity, etc.) and degenerate four-wave mixing efficiencies based on the optical field-induced alignment of the molecules. The studies are conducted with laser pulses on the order of a few nanoseconds. On the other hand, FAYER[ 19861 has used a transient grating technique (with picosecond laser pulses) to study the interference between the (individual) molecular reorientational effects and the laser-induced acoustic waves in several phases (nematic, isotropic, and smectic) of liquid crystals. Optical harmonic generation that was studied earlier by SAHAand WONG [ 19791 (using dc field-assisted second-harmonic generation) was recently employed as a useful technique for probing a monolayer or thin film of organic molecules (e.g., liquid crystal) (SHEN[ 19861). In particular it was shown that even though an individual liquid crystal molecule may possess very high electronic nonlinearity, the macroscopic (interlayer and intralayer) arrangements of the molecules are such that individual dipolar anisotropies add “destructively” to yield smaller than expected macroscopic response. Clearly, nonlinear optics can also be successfully employed as techniques to study or probe molecular parameters and complement other well-known techniques such as Raman scattering and nuclear magnetic resonance (JEN, CLARK, PERSHAN and PRIESTLEY [ 19771). There have been relatively few nonlinear optics studies in the smectic (LIPPEL and YOUNG [1983]) or the cholesteric phase. The bistability or optical wavefront conjugation processes predicted in the smectic and cholesteric phase have yet to be quantitatively and experimentally verified. Part of the problem may be because smectic-C films (in which orientation by an optical field of finite optical intensity is possible) are difficult to fabricate, and also the polarization of light going through such a film is invariably scrambled. In any case the magnitude of the nonlinearity in these mesophases is not expected to be any greater than the well-documented nematic phase, although the response time of the reorientation process in smectic-C film is expected to be faster. From a wider perspective, liquid crystals fall under the general class of organic materials or polymers that have received considerable attention lately (WILLIAMS [ 19831, GARITO,TENG,WONGand ZAMANI-KHAMIRI [ 19841).

111

REFERENCES

157

Interest is focused on the electronic nonlinearity of some of these conjugated organic polymeric systems because of their fast response times (< 10- s) and the relatively large nonlinearity (nonlinearities a few times larger than that of typical second-harmonic generation crystals). Furthermore, a great variety of chemical modifications or “molecular engineering” approaches may be employed to synthesize a vast assortment of crystalline structures with improved response and or stability (ZYSS,CHEMLA and NICOUD[ 19811). In particular, liquid crystalline polymers have been found to possess one of the largest electronic nonlinearities. These new materials present fundamentally challenging problems and are potentially excellent candidates for applications in optical signal processing and optical switching and logic operations. At the time of this writing, progress in the synthesis of new liquid crystals and mixtures, ferroelectric liquid crystals, and organic polymers, and in the use of these materials in nonlinear optical processes and devices is extremely rapid and involves many dedicated individuals. Hopefully, this review will be succeeded by even more fascinating and important developments in these fields in the near future.

Acknowledgement

I would like to express my appreciation for several helpful conversations, discussions and collaborations over the last few years with E. N. Leith, A. E. Kaplan, R. Normandin, J. P. Huignard, B. Ya. Zeldovich, and all my co-authors on various aspects of liquid crystals and nonlinear optics. My research program has been supported by the National Science Foundation and the Air Force Office of Scientific Research.

References AKHMANOV, S.A., R.V. KHOKHLOV and A.P. SUKHORUKOV, 1972, in: Laser Handbook 11, eds F.T. Arecchi and E.O. Schulz-Dubois (North-Holland, Amsterdam) p. 1151. AKOPYAN, R.S.,and B.YA. ZELDOVICH, 1982, Sov. Phys.-JETP 56, 1239. S.M., and Y.S. CHILINCARIAN, 1986, IEEE J. Quantum Electron. QE-22,1276. ARAKELIAN, ARAKELIAN, S.M., G.A. LYAKHOV and Y u S . CHILINGARIAN, 1980, Sov. Phys.-Usp. 23, 245. ARAKELIAN, S.M., A S . KARAIAN and Yu.S. CHILINGARIAN, 1982, Sov. J. Quantum Electron. 12, 1619. ARMITAGE, D., and S.M. DELWART,1985, Mol. Cryst. & Liq. Cryst. 122, 59. H.J. LEVENSTEIN and ASHKIN,A., G.D. BOYD,J.M. DZIEDZIC,R.G. SMITH,H.A. BALLMAN, K. NASSAU,1966, Appl. Phys. Lett. 9, 72.

158

REFERENCES

BAHADUR,B., 1984, Mol. Cryst. & Liq. Cryst. 301, 1. BARBERO, G., and F. SIMONI,1982, Appl. Phys. Lett. 41, 504. G., F. SIMONI and P. AIELLO,1984, J. Appl. Phys. 55, 304. BARBERO, G., E. MIRALDI, C. OLDANO,M.L. RASTELLO and P. TAVERNA VALABREGA, 1986, BARBERO, J. Phys. (France) 47, 1411. BATRA,I.P., R.H. ENNSand D. POHL,1971, Phys. Status Solidi (b) 48, 11. BESPALOV,V.I., and V.I. TALANOV, 1966, JETP Lett. 3, 307. BISCHOFFBERGER,T., and Y.R. SHEN,1976, Appl. Phys. Lett. 32, 156. T., and Y.R. SHEN,1979, Phys. Rev. A 19, 1169. BISCHOFFBERGER, BJORKHOLM, J.E., P.W. ShmH, W.J. TOMLINSON and A.E. KAPLAN,1981, Opt. Lett. 6, 345. BLINOV, L.M., 1983, Electro-Optical and Magneto-Optical Properties of Liquid Crystals (Wiley, New York). BORN,M., and E. WOLF, 1970, Principles of Optics, 4th Ed. (Pergamon Press, Oxford). CARO,R.G., and M.C. COWER,1982, IEEE J. Quantum .Electron. QLl8, 1375. CHANDRASEKHAR, S., 1977, Liquid Crystals (Cambridge Univ. Press, London). CHEMLA, D.S., D.A.B. MILLERand P.W. SMITH,1985, Opt. Eng. 24, 556. 1968, Appl. Phys. Lett. 13, 223. CHEN,F.S., J.T. LAMACCHIA and D.B. FRASER, CHEUNG,M.M., S.D. DURBIN and Y.R. SHEN,1983, Opt. Lett. 8, 39. V.F. KITAEVA, N. KROOand N.N. SOBOLEV, 1982, Mol. Cryst. & Liq. CSILLAG, L., 1. JANOSSY, Cryst. 84, 125. DEGENNES, P.G., 1974, The Physics of Liquid Crystals (Oxford Univ. Press, London). DEJEU,W.H., 1980, Physical Properties of Liquid Crystalline Materials (Gordon and Breach, New York). DEJEU,W.H., C.J. GERRITSMA, P. VAN ZANTENand W.J.A. GOOSENS,1972, Phys. Lett. A 39, 355. DURBIN, S.D., S.M. ARAKELIAN and Y.R. SHEN,1981a, Opt. Lett. 6, 411. DURBIN, S.D., S.M. ARAKELIAN and Y.R. SHEN,1981b, Phys. Rev. Lett. 47, 1411. DURBIN, S.D., S.M. ARAKELIANand Y.R. SHEN,1982, Opt. Lett. 1, 145. EDEN,D., C.W. GARLAND and R.C. WILLIAMSON, 1973, J. Chem. Phys. 58, 1861. FAYER, M.D., 1986, IEEE J. Quantum Electron. QE-22, 1437. FEINBERG, J., 1983, in: Optical Phase Conjugation, ed. R.A. Fisher (Academic Press, New York) ch. 11. FEINBERG, J., and R.W. HELLWARTH, 1980, Opt. Lett. 5, 519. FEKETE,D., J. Au YEUNG and A. YARIV,1980, Opt. Lett. 5, 51. FERNANDEZ, F.J.L., 1983, Ph.D. Thesis (Universidad Politechnica de Madrid, Spain). and E.M. WRIGHT,1985, J. Opt. SOC.Am. 2, 1005. FIRTH,W.J., I. GALBRAITH FISHER,B., M. CRONIN-GOLOMB, J.P. WHITEand A. YARIV,1981, Opt. Lett. 6, 519. V., and V. ZOLINA,1933, Trans. Farad. SOC.29, 919. FREDERIKS, FUH,Y.G., R.F. CODEand G.X. Xu, 1983, J. Appl. Phys. 54, 6388. GABOR,D., 1949, Proc. R. SOC.London Ser. A 197,454. G A R I B Y A N , I.N. ~ . ~ KOMPANETS,A.V. ., PARFYONOV, N.F. PILIPETSKII, V.V. SHKUNOV,A.N. 1981, Opt. SUDARKIN, A.V. SUKHOV, N.V. TABIRYAN, A.A. VASILIEVand B.YA.ZELDOVICH, Commun. 38.64. GARITO,A.F.,C.C.TENG,K.Y. W o ~ ~ a 0. n dZAMANI-KHAMIRI, 1984, Mol. Cryst. & Liq. Cryst. B 106, 219. GIBBS,H.M., 1985, Optical Bistability: Controlling Light with Light (Academic Press, New York). GIBBS,H.M., P. MANDEL, N. PEYGHAMBARIAN and S.D. SMITH,1986, Optical Bistability Ill (Springer, New York). GOLDBURT, E.S., and P.ST.J. RUSSELL,1985, Appl. Phys. Lett. 46,338.

111

REFERENCES

159

GRAY,G.W., and J.W. GOODBY,1984, Smectic Liquid Crystals (Leonard Hill, Glasgow). HAGAN,D.J., H.A. MACKENZIE, J.J.E. REID,A.C. WALKERand F.A.P. TOOLEY,1985, Appl. Phys. Lett. 47, 203. HELLWARTH, R.W., 1977, J. Opt. SOC.Am. 67, 1. HELLWARTH, R.W., 1983, in: Optical Phase Conjugation, ed. R.A. Fisher (Academic Press, New York). HERMAN, R.M., and R.J. SERINKO,1979, Phys. Rev. A 19, 1757. HERMANN, J.A., 1984, J. Opt. SOC.Am. B 1, 729. HERRIAU, J.P., J.P. HUIGNARD, A.G. APOSTOLIDIS and S.MALLICK, 1985, Opt. Commun. 56, 141. HERVET,H., W. URBACHand F. RONDELEZ,1978, J. Chem. Phys. 68, 2725. HOFFMANN, H.J., 1986a, J. Opt. SOC.Am. B 3, 253. HOFFMANN, H.J., 1986b, IEEE J. Quantum Electron. QE-22, 552. HORN,R.G., 1978, J. Phys. (France) 39, 105. HSIUNG,H., L.P. SHI and Y.R. SHEN,1984, Phys. Rev. A 30, 1453. HUIGNARD, J.P., 1986, private communication. H U I G N A RJ.P., D , H. RAJBENBACH, PH. REFRECIER and L. SOLYMAR, 1985, Opt. Eng. 24, 586. J A I N R.K., , and M.B. KLEIN,1983, in: Optical Phase Conjugation, ed. R.A. Fisher (Academic Press, New York). JEN, S., N.A. CLARK,P.S. PERSHAN and E.B. PRIESTLEY, 1977, J. Chem. Phys. 66, 4635. KAPLAN,A.E., 1976, Sov. Phys.-JETP 45, 896. KAPLAN,A.E., 1977, JETP Lett. 24, 114. KARN,A.J., S.M. ARAKELIAN, Y.R. SHENand H.L. ONG, 1986, Phys. Rev. Lett. 57, 448. KAWACHI, M., B.S. KAWASAKI and K.O. HILL, 1982, Electron. Lett. 18, 609. KHOO,I.C., 1981a, Phys. Rev. A 23, 2077. KHOO,I.C., 1981b, App!. Phys. Lett. 38, 123. KHOO,I.C., 1982a, Phys. Rev. A 25, 1040. Ktioo, I.C., 1982b, Phys. Rev. A 25, 1636. KHOO,I.C., 1982c, Appl. Phys. Lett. 40, 645. KHOO, I.C., 1983, Phys. Rev. A 27, 2747. KHOO,I.C., 1985, Appl. Phys. Lett. 47, 908. KHOO,I.C., and J.Y. H o u , 1985,J. Opt. SOC.Am. B 2, 761. Kiioo, I.C., and T.H. LIU, 1987, IEEE J. Quantum Electron. QE-23, 171. KHOO,I.C., and R. NORMANDIN, 1984a, Opt. Lett. 9, 285. KHOO,I.C., and R. NORMANDIN, 1984b, J. Appl. Phys. 55, 1416. KHOO,I.C., and R. NORMANDIN, 1985a, IEEE J. Quantum Electron. QE-21, 329. KHOO,I.C., and R. NORMANDIN, 1985b, Appl. Phys. Lett. 47, 350. KHOO,I.C., and S.L. ZHUANG,1980, Appl. Phys. Lett. 37, 3. KHOO,I.C., and S.L. ZHUANG,1982, IEEE J. Quantum Electron. QE-18, 246. KHOO,I.C., S.L. ZHUANGand S. SHEPARD,1981, Appl. Phys. Lett. 39, 937. and V.C.Y. So, 1983, Phys. Rev. A 27, 3251. KHOO,I.C., J.Y. Hou, R. NORMANDIN Ktioo, I.C., T.H. LIU, P.Y. YAN,S. SHEPARDand J.Y. Hou, 1984, Phys. Rev. A 29, 2756. KHOO,I.C., T.H. LIU and R. NORMANDIN, 1985, Mol. Cryst. & Liq. Cryst. 131, 315. KHOO,I.C., T.H. LIU,P.Y. YAN and J.Y. H o u , 1986, SPIE Proc. 667, 220. Ktioo, I.C., G.M. F I N N ,R.R. MICHAELand T.H. LIU, 1986, Opt. Lett. 11, 227. Knoo, I.C., J.Y. Hou, T.H. LIU,P.Y. YAN,R.R. MICHAELand G.M. FINN,1987, J. Opt. SOC. Am. B 4, 886. KHOO,I.C., P.Y. YAN and T.H. LIU, 1987, J. Opt. SOC.Am. B 4, 115. KHOO,I.C., R.R. MICHAEL and P.Y. YAN, 1987, IEEE J. Quantum Electron. QE-23, 267. KOGELNIK, H., 1965, Bell Syst. Tech. J. 44,2451.

160

REFERENCES

[I1

KOMPANETS, I.N., A.V. PARFYONOV and J.M. POPOV, 1981, Opt. Commun. 36,415. KovALEv,A.A.,G.L. NEKRASOV, V.A. PILIPOVICH,YU.V. R A Z V I NS.V. ~ ~SERAK, ~ 1979, Pis'ma v Zh. Tekh. Fiz. 5, 159. LALANNE, J.R., B. MARTIN,B. POULIGNY and S. KIELICH,1977, Mol. Cryst. & Liq. Cryst. 42, 153. LEITH, E.N., and G.J. SWANSON,1982, Opt. Lett. 7, 596. LEITH,E.N., and J. UPATNIEKS,1962, J. Opt. SOC.Am. 52, 1123. LEITH,E.N., and J. UPATNIEKS,1965, SPIE J. 4, 3. LEITH,E.N., H. C H E N , ~CHENG, . G. SWANSONand I.C. KHOO,1983, Proc. 5th Rochester Conf. on Coherence and Quantum Optics, Rochester, NY,eds E. Wolfand L. Mandel (Plenum Press, London, 1984) p. 1155. LEMBRIKOV, B.I., 1979, Zh. Tekh. Fiz. 49, 667. LIPPEL,P.H., and C.Y. YOUNG,1983, Appl. Phys. Lett. 43, 909. LIU,T.H., 1987, Ph.D. Thesis (Pennsylvania State University). MADDEN,P.A., F.C. SAUNDERS and-A.M. Scorn, 1986, IEEE J. Quantum Electron. QE-22, 1287. MARBURGER, J.H., 1975, Prog. Quantum Electron. 4, 35. MARTIN,G., and R.W. HELLWARTH,1979, Appl. Phys. Lett. 34, 371. MARTIN-PEREDA, J.A., and F.J. LOPEZ, 1982, Opt. Lett. 7, 590. MULLEN,M.E., B. LUTHIand M.J. STEPHEN,1972, Phys. Rev. Lett. 28, 799. NERSISYAN, S.R., and N.V. TABIRYAN, 1984, Mol. Cryst. & Liq. Cryst. 116, 111. ONG,H.L., 1983, Phys. Rev. A 28, 2393. ONG, H.L., 1985, Appl. Phys. Lett. 46, 822. PASKO,J.G., J. TRACY and W. ELSER,1979, SPIE Proc. 202, 82. PERSHAN, P.S., 1966, Nonlinear optics, in: Progress in Optics, Vol. V, ed. E. Wolf(North-Holland, Amsterdam) p. 83. PROST.J., and J.R. LALANNE,1973, Phys. Rev. A 8, 2090. PUANG-NGERN, S., and S.P. ALMEIDA,1986, Technical Digest, 1986 Ann. OSA Meeting, p. 34. RAJBENBACH, H., and J.P. HUIGNARD, 1985, Opt. Lett. 10, 137. RAK,D., I. LEDOUXand J.P. HUIGNARD,1984, Opt. Commun. 49, 302. RAO,D.V.G.L.N., and S. JAYARAMAN, 1973, Appl. Phys. Lett. 23, 539. RAO, D.V.G.L.N., and S. JAYARAMAN, 1974, Phys. Rev. A 10, 2457. REINTJES, J.F., 1983, Nonlinear Optical Parametric Processes in Liquids and Gases (Academic Press, New York). 1986, Opt. Commun. 57, 365. RICHARD, L., J. MAURINand J.P. HUIGNARD, RUBIN,R.L., 1985, Sov. J. Quantum Electron. 12, 2344. SAHA,S.K., and G.K.L. WONG, 1979, Appl. Phys. Lett. 34, 423. SANTAMATO, E., and Y.R. SHEN,1985, Opt. Lett. 9, 564. SANTAMATO, E., and Y.R. SHEN,1986, Opt. Lett. 11, 452. SCHADT,M., 1972, J. Chem. Phys. 56, 1494. SHEN,Y.R., 1975, Prog. Quantum Electron. 4, 1. SHEN,Y.R., 1984, The Principles of Nonlinear Optics (Wiley, New York). SHEN,Y.R., 1987, Proc. Topical Meeting on Optics of Liquid Crystals, Mol. Cryst. & Liq. Cryst. 143, 1. SIMONI, F., and R. BARTOLINO,1985, Opt. Commun. 53, 210. SOILEAU, M.J., E.W. WILLIAMS and E.W. VAN STRYLAND, 1983, IEEE J. Quantum Electron. QE-19, 731. M.J., S. GUHA,W.E. WILLIAMS, E.W. VAN STRYLAND and H. VANHERZEELE, 1985, SOILEAU, Mol. Cryst. & Liq. Cryst. 127, 321. SONG,J.W., S.Y. S H I Nand Y.S. KWON, 1984, Appl. Opt. 23, 1521.

111

REFERENCES

161

TABIRYAN, N.V., and B.YA. ZELDOVICH, 1981a, Mol. Cryst. & Liq. Cryst. 62, 637. TABIRYAN, N.V., and B.YA.ZELDOVICH, 1981b, Mol. Cryst. & Liq. Cryst. 69, 19. N.V., and B.YA.ZELDOVICH, 1981c, Mol. Cryst. & Liq. Cryst. 69, 31. TABIRYAN, TAI,K., H.M. GIBBS,N. PEYGHAMBARIAN and A. MYSYROWICZ, 1985, Opt. Lett. 10, 220. 1978, Mol. Cryst. & Liq. Cryst. 46,209. URBACH, W., H. HERVETand F. RONDELEZ, and I.C. KHOO, 1984, Opt. Lett. 9, 238. VACH,H., C.T. SEATON,G.I. STEGEMAN VOLTERRA, V., and E. WIENER-AVNEAR, 1974, Opt. Commun. 12, 194. WILLIAMS, D., ed., 1983, Nonlinear Optical Properties of Organic and Polymeric Materials (Plenum Press, New York). WINFUL,H.G., 1982, Phys. Rev. Lett. 49, 1179. WONG,G.K.L., and Y.R. SHEN,1973, Phys. Rev. Lett. 30,895. WONG,G.K.L., and Y.R. SHEN,1974, Phys. Rev. Lett. 32, 527. Wu, S.T., 1985, J. Appl. Phys. 58, 1419. L.D. HESS and M.S. WELKOWSKY, 1985, SPIE Proc. 572, Wu, S.T., U. EFRON,J. GRINBERG, 94. YAN, P.Y., 1983, M.S. Thesis (Wayne State University) unpublished. YARIV,A., 1979, IEEE J. Quantum Electron. QE-15, 256. YARIV, A., and B. FISHER,1983,in: Optical Phase Conjugation, ed. R.A. Fisher (Academic Press, New York). YARIV, A,, and D.M. PEPPER,1977, Opt. Lett. 1, 16. YEH, P., 1986, J. Opt. SOC.Am. B 3, 747. ZELDOVICH, B.YA., and N.V. TABIRYAN, 1980, Sov. J. Quantum Electron. 10, 440. ZELDOVICH, B.YA., and N.V. TABIRYAN, 1982, Sov. Phys.-JETP 55, 656. ZELDOVICH, B.YA., and N.V. TABIRYAN, 1984, Sov. J. Quantum Electron. 14, 1599. ZELDOVICH, B.YA.,S.K. MERSLIKIN, N.F. PILIPETSKY and A.V. SUKHOV, 1985, JETP Lett. 41, 418. ZELDOVICH, B.YA.,N.F. PILIPETSKY and V.V. SHKUNOV, 1985, Principles of Phase Conjugation (Springer, Berlin). ZOI.OTKO,A.S., V.F. KITAEVA, N. KROO,N.N. SOBOLEV and L. CHILLAG, 1980. JETP Lett. 32, 158. ZOLOTKO. A S . , V.F. KITAEVA, N.N. SOBOLEV and A.P. SUKHORUKOV, 1981, Sov. Phys.-JETP 81, 933. ZYSS,J., D.S. CHEMLA and J.F. NICOUD, 1981, J. Chem. Phys. 74, 4800.

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I11

SINGLE-LONGITUDINAL-MODE SEMICONDUCTOR LASERS BY

GOVIND P. AGRAWAL A T&T Bell Laboratories

Murray Hill>NJO7974, USA

CONTENTS PAGE

$ 1 . INTRODUCTION

. . . . . . . . . . . . . . . . . . . 165 $ 2. FABRY-PEROTSEMICONDUCTORLASERS . . . . . . 166 $ 3 . REQUIREMENTS FOR SINGLE-LONGITUDINAL-MODE OPERATION . . . . . . . . . . . . . . . . . . . . . 178 $ 4 . DISTRIBUTED-FEEDBACK SEMICONDUCTOR LASERS

. . . . 197 . . . . . . . . . 208 . . . . . . . . . 216 . . . . . . . . 221 ........ 222

$ 5 . COUPLED-CAVITY SEMICONDUCTOR LASERS

$ 6. MODULATION PERFORMANCE

. . $ 7. SPECTRAL LINEWIDTH . . . . . . 5 8. CONCLUSION . . . . . . . . . . . . REFERENCES . . . . . . . . . . . . . .

. . . .

180

8 1. Introduction Semiconductor lasers have experienced a phenomenal growth in recent years, motivated mainly by their applications in optical fiber communications (KRESSELand BUTLER[1977], CASEYand PANISH [1978], THOMPSON [ 19801, AGRAWALand DUTTA[ 19861).The first generation of optical communication systems utilized multimode fibers and multimode GaAs lasers operating at a wavelength of about 0.85 pm. Because of modal dispersion occurring in multimode fibers, the data transmission rate or bit rate was limited to below 100 Mb/s. The use of single-mode fibers eliminated the problem of modal noise; the system performance was then limited by chromatic dispersion in silica fibers. This realization led to the development of InGaAsP semiconductor lasers oscillating near 1.3 pm, the wavelength at which silica fibers have minimum dispersion. However, the relatively large spectral width (0.5-1 THz or 3-6 nm) of multimode InGaAsP lasers coupled with the residual fiber dispersion still limited the bit rate to 1 Gb/s. Furthermore, the transmission distance is limited to about 50 km because of a relatively high fiber loss (about 0.5 dB/km) near 1.3 pm. The recent trend in optical fiber communications is to operate the lightwave system near 1.55 pm, where silica fibers have the lowest loss (about 0.2 dB/km) and therefore allow a transmission distance of up to 100 km or more. Two routes are being pursued to overcome the resulting dispersion problem. In one approach the fiber design is suitably modified to move the minimum-dispersion wavelength region from 1.3 to 1.55 pm (AINSLIEand DAY[ 19861). In the other approach the effect of fiber dispersion is minimized by reducing the spectral width of the 1.55 pm InGaAsP laser source. This reduction is achieved by designing semiconductor lasers so that they are forced to oscillate predominantly in a single longitudinal mode (SLM). Such lasers are sometimes referred to as single-frequency lasers (BELL[ 1983]), which is a misnomer since an SLM semiconductor laser typically has a spectral width of 10-100 MHz. The name SLM laser is also inaccurate, since the dominant mode in such lasers is always accompanied by several less intense side modes. Its use generally implies that these side modes are suppressed by a factor of 100 or more. The objective of this article is to review the recent progress in the field of

-

165

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SINGLE-LONGITUDINAL-MODE SEMICONDUCTOR LASERS

[III, 5 2

SLM semiconductor lasers. In 5 2 we briefly discuss the physics of semiconductor lasers and introduce the material system and the structure commonly employed to make them. We then consider the modes supported by a conventional Fabry-Perot (FP) type of semiconductor laser with particular attention paid to the longitudinal modes and their respective threshold gains. To see why such lasers generally -oscillate in several longitudinal modes, we use the rate equations to obtain the longitudinal-mode spectrum. The fundamental reason behind the multimode operation is that an FP cavity has the same loss for all longitudinal modes ; it therefore provides no inherent mode selectivity. Since the gain spectrum is generally much wider than the FP-mode spacing, the resulting mode discrimination from the gain roll-off is poor. As discussed in 5 3, the key to SLM operation of semiconductor lasers is to make the cavity loss wavelength dependent. Two mechanisms have been used for this purpose. Section 4 considers the distributed-feedback (DFB) mechanism and 5 5 examines the coupled-cavity mechanism. The degree of side-mode suppression is governed by the gain margin between the main mode and the most intense side mode. To obtain the gain margin, we consider the longitudinal modes and their respective threshold gains for both the DFB and coupled-cavity lasers. The performance achieved by the state-of-the-art SLM lasers is also discussed. Since SLM lasers are generally modulated at high frequencies (-a few gigahertz) during their application in optical communication systems, 6 considers the modulation performance through an analysis of the single-mode rate equations. Particular attention is paid to the phenomenon of frequency chirping, since it is often the limiting factor for the performance of SLM lasers at high bit rates. The spectral linewidth of SLM lasers is considered in 5 7. The analysis is again based on the single-mode rate equations modified to include fluctuations through Langevin noise sources. Particular attention is paid to narrowlinewidth SLM lasers in view of their applications in coherent communication systems. Finally, 5 8 concludes the article through a description of the current research in the field of SLM semiconductor lasers.

# 2. Fabry-Perot Semiconductor Lasers This section is intended to provide a basic understanding of the physics behind semiconductor lasers. Two things are required to operate a laser: (1) a gain medium that can amplify the electromagnetic radiation propagating inside it and (2) a feedback mechanism that can confine the optical field in the vicinity

111, § 21

FABRY-PEROT

I67

SEMICONDUCTOR LASERS

of the gain medium. The gain medium for a semiconductor laser consists of a thin layer ( 5 0 . 2 pm) of a semiconductor material having a direct band gap, which is pumped electrically using a p-n junction. Figure 1 shows schematically the commonly used heterostructure laser and its typical dimensions. Radiative recombination of the injected electrons and holes inside the active region generate photons and can lead to optical gain when the injected carrier density exceeds a critical value required for “population inversion”. The optical feedback is obtained by forming a cavity. In contrast to other kinds of lasers, however, no external mirrors are required. The cleaved facets of a semiconductor slab in effect form an F P cavity by virtue of their natural reflectivity of about 30%.For this reason and to distinguish them from the more elaborate feedback schemes considered later in this article, such lasers are often referred to as FP semiconductor lasers.

2.1. LASER MATERIAL AND STRUCTURE

A state-of-the-art semiconductor laser generally employs a buried-heterostructure design (e.g. fig. 2), wherein a thin slab (typical dimensions 0.15 pm x 1.5 pm x 250 pm) of one semiconductor material is buried inside another semiconductor material. The two semiconductors are chosen such that they have nearly identical lattice constants while the band gap of the buried region is smaller than that of the surrounding layers. Furthermore, the surrounding layers are suitably doped with p-type or n-type impurities to make

/ Ii

/ P -TYPE

I/

?

ACTIVE LAYER

Al

’ ’ ’ ’ ’ ’ ’ ‘ 1 8 , 1 1 1 1

n -TYPE

FACET

Fig. 1. Schematic illustration of a heterostructure semiconductor laser and its typical dimensions. The active layer (hatched region) is sandwiched between the two cladding layers of a higher band-gap material suitably doped to form a p-n junction.

168

[III, 5 2


I n - inp

n - inp

InGOASP (ACTIVE 1

I

n - inp

I

n - InP (SUBSTRATE 1

Fig. 2. Schematic illustration of a buried heterostructure InGaAsP laser. The active region (shown hatched) is buried on all sides by various InP layers doped judiciously to force the current to flow through the active region.

a p-n junction. Under forward bias, charge carriers (electrons and holes) are injected into the central region and remain confined because of the band-gap difference; their subsequent recombination inside the active region provides the optical gain. Although simple in concept, the fabrication of a semiconductor laser requires the use of sophisticated epitaxial growth techniques (NAKAJIMA[ 19821, OLSEN[ 19821). The choice of the semiconductor material is dictated by the laser wavelength 1 determined by the band gap E, of the active region (E, N hell). In the wavelength range of 0.8-0.9 pm, GaAs is the material of choice. Its band gap can be tailored by forming the ternary compound AI,Ga, -..As, while the lattice constant remains almost unchanged. The bandgap difference between the active and surrounding layers can be selected by an appropriate choice of the mole fraction x. For the purpose of long-haul optical fiber communications, the wavelength range of interest is 1.3-1.6 pm. The quaternary alloy In, -,Ga,As,P, is used to make such lasers. The appropriate choice of mole fractions x and y can provide the desired wavelength while at the same time the lattice constant remains matched to InP, which is used as a substrate. As an example, for 1.3 pm InGaAsP lasers, x = 0.28 and y = 0.6. In this article we do not intend to discuss the fabrication details of semiconductor lasers and only refer to the existing literature (CASEYand PANISH [ 19781, SUEMATSU, IGAand KISHINO[ 19821, AGRAWAL and DUTTA [ 19861).

-,

11198 21

FABRY-PEROT SEMICONDUCTOR LASERS

I69

2.2. LASER MODES

For the successful operation of a laser it is necessary to confine the emitted radiation in the vicinity of the gain medium, which in semiconductor lasers is accomplished through dielectric waveguiding. Since the refractive index of the active region is slightly larger than that of the surrounding material (because of the band-gap difference), the active region forms a rectangular dielectric waveguide, which supports only a few modes in both the lateral and transverse directions. In almost all semiconductor lasers the active layer is chosen to be thin enough (50.2 pm) to support only a single transverse mode. In the buried heterostructure design (see fig. 2) the width of the active region is also controlled ( 52 pm) so that either the waveguide supports only a single lateral mode or the higher order lateral modes remain unexcited during laser operation. Since the mode confinement occurs through the refractive-index difference, such lasers are sometimes referred to as index guided. This is in contrast to gainguided semiconductor lasers where the lateral confinement occurs through an inhomogeneous gain distribution. Even when a semiconductor laser is designed to support a single mode in the lateral and transverse directions, it can oscillate simultaneously in several longitudinal modes, since an FP cavity can support a large number of modes whose wavelength separation is governed by the optical path length inside the cavity (BORN and WOLF[ 19801). The number of FP modes that actually lase depends on the gain and the loss experienced by each longitudinal mode. A particular mode is said to reach the threshold when the gain is large enough to balance the mode losses. To obtain the longitudinal modes and their respective threshold gains, one should consider wave propagation with appropriate boundary conditions at the laser facets. Figure 3 shows schematically a semiconductor laser and its associated FP cavity. The intracavity field E satisfies Maxwell’s wave equation

where c is the velocity of light in vacuum and E ( X , y ) is the dielectric constant whose spatial variation depends on the specific laser structure and is responsible for the mode confinement. If we consider a single longitudinal mode oscillating at the frequency w, the optical field can be written as E(x, y, z, t ) = 2 Re [ U(x,y ) E(z) exp( - iwt)] ,

(2.2)

I70


[III, § 2

CURRENT INJECTION

CLEAVED

FACETS

Z=O

2.L FABRY - PEROT C A V I T Y

Fig. 3. Schematic illustration of a semiconductor laser and its associated Fabry-Perot cavity. The cleaved facets act as partially reflecting mirrors.

where P is the polarization unit vector of the single mode supported by the waveguide with the spatial distribution U(x,y) and E(z)is the axial distribution of the longitudinal mode. From eqs. (2.1) and (2.2), U(x,y ) and E ( z ) are obtained by solving

d2E + b2E = 0 dz2

-

where k,

= o/c

fi = jik,

and the mode-propagation constant fi is given by

- $5

.

(2.5)

The mode index ji and the mode gain Z for the fundamental waveguide mode are obtained by solving eq. (2.3) with the appropriate boundary conditions. One finds that (AGRAWAL and DUITA [ 19861) -

Cl

=

rg -

Clint

,

(2.6)

where g is the active-region gain generated by carrier recombination. The confinement factor rrepresents the fraction of the mode energy confined to the active region. The internal loss aintaccounts for the total mode losses arising

111, t 21

FABRY-PEROT SEMICONDUCI'OR LASERS

171

from the processes such as free-carrier absorption and interfacial scattering. The axial distribution is readily obtained from eq. (2.4), whose general solution is E(z)

=

+ Eb(z) = A exp(ipz) + B exp( - ipz) ,

E,(z)

where the constants A and B are determined from the boundary conditions at the facets located at z = 0 and z = L (see fig. 3):

Eb(L)

=

r2Ef(L)

(2.8)

9

where r, and r, are the amplitude-reflection coefficients. Using eqs. (2.7) and (2.8), we obtain A

=

r,B,

r,A exp(2ipL) = B .

(2.9)

These relations are satisfied nontrivially only when rlr2 exp(2ipL)

=

1.

(2.10)

Using eq. (2.5) in eq. (2. lo), we obtain the relations pk,L

=

mn,

(2.11) (2.12)

-

a =

%ir,

where m is an integer and the mirror loss (2.13)

k,

Eqauation (2.11) determines the frequencies of the longitudinal modes. Using = 2nv/c, the mode frequencies are given by v=v,=m-,

c

2pL

(2.14)

where v, are the resonance frequencies of an FP cavity of the optical length pL. The spacing between the successive longitudinal modes can be obtained from eq. (2.14). However, it is important to account for chromatic dispersion or the frequency dependence of p. The longitudinal-mode spacing in an FP semiconductor laser is thus given by (2.15)

172


[III, § 2

where (2.16) is the group index of the active region. Equation (2.12) determines the threshold condition for the F P modes. Using eq. (2.6), it can be written as (2.17) Thus a particular mode reaches threshold when its gain just balances the total cavity loss (about 80-100 cm- '). Since the cavity loss is the same for all modes, an FP cavity does not provide any mode discrimination. However, all modes do not have the same gain g , at a given pumping level. Figure4 shows schematically the gain and loss profiles. The F P mode closest to the gain peak reaches threshold first and becomes the dominant mode; other F P modes are discriminated by the gain roll-off. A characteristic feature of semiconductor lasers is that the spectral width of the gain profile is much larger ( - 5 THz) compared with the FP-mode spacing (typically 0.1 THz). As a result, mode discrimination provided by the gain roll-off is poor. For a typical F P semiconductor laser the gain margin Aa between the dominant FP mode and the neighboring FP mode is 0.1 cm - . Under ideal conditions even a small amount of gain margin should completely suppress the neighboring F P side modes as long as the gain profile is homogeneously broadened. In practice, however, several phenomena such as spontaneous emission, spatial hole-burning and spectral hole-burning forbid a complete suppression of side modes. Figure 5 shows the observed longitudinalN

-

'

t

FREQUENCY

Fig. 4. Schematic illustration of the gain and loss profiles and the longitudinal modes in a semiconductor laser. For the dominant lasing mode the threshold is reached when the peak gain nearly equals loss.

FABRY-PEROT

SEMICONDUCTOR LASERS

173

Fig. 5. Longitudinal-modespectra of a Fabry-Perot semiconductor laser at three different power levels shown by solid dots on the light-current curve. (After NELSON,WILSON,WRIGHT, BARNESand DUWA[1981]; 0 IEEE.)

mode spectra of a buried heterostructure laser at several power levels (NELSON, WILSON,WRIGHT,BARNESand DUTTA [ 19811).Generally speaking, the main mode is accompanied by several FP side modes, whose amplitude decreases with an increase in the main-mode power. This behavior can be understood if one thinks of a laser as a regenerative noise amplifier (GORDON[ 19641). All modes for whom the round-trip gain is positive are amplified; once the threshold is reached, the gain is approximately clamped and the power in the side modes saturates. To understand this behavior more quantitatively, it is necessary to consider the multimode rate equations that describe the laser dynamics.

2.3. RATE EQUATIONS

The dynamic behavior of a laser is generally modeled through a coupled set ofthe Maxwell-Bloch equations (SARGENT,SCULLY and LAMB[ 19741). In the case of semiconductor lasers the relaxation time of the induced polarization

174


[III, fi 2

(the parameter T, in the terminology of the Bloch equations) is short ( 5 0.1 ps) compared with the photon and carrier lifetimes. It is thus possible to make the so-called rate equation approximation and to consider only the dynamic interaction between the photon and carrier populations. If Pmrepresents the number of intracavity photons in the mth longitudinal mode and N represents the number of carriers inside the active region, the multimode rate equations take the form (THOMPSON [ 19801, ACRAWAL and DUTTA[ 19861): (2.18) I "=--yeN-CGmPm, 9 m

(2.19)

with I the current injected into the active region, q the charge of the electron, and ye the spontaneous recombination rate of the injected carriers.

is the net rate of stimulated emission, (2.21) is the decay rate of intracavity photons, and R,, is the rate of spontaneous emission into the mth mode. Since the width of the spontaneous-emission spectrum is generally much larger than the mode spacing, R,, is nearly the same for all modes. In eqs. (2.20) and (2.21) ug is the group velocity, that is ug = c/pg, where pg is the group index defined by eq. (2.16). In general, ye in eq. (2.19) includes the contributions of both radiative and nonradiative (e.g., Auger) recombination mechanisms and is therefore a function of N . The form of the rate equations is self-evident if one considers all possible mechanisms through which photons and electrons are generated and lost. The quantity of practical interest is the output power Poutemitted from each facet. Noting that u , ~ represents ~ , the escape rate of photons through both facets, (2.22) To get a feeling of the magnitude of various quantities, we note that the photon lifetime zp = y - N 2 ps, the carrier lifetime ze = y; N 2 ns, N N los, and P, N lo5 at a power level of 2 to 3 mW for a typical semiconductor laser. Furthermore R,, N nspy, where nsp is the population-inversion factor (n,, 1: 2).

'

'

111, § 21


175

2.4. LONGITUDINAL-MODE SPECTRA

To solve the multimode rate equations (2.18) and (2.19), we need to know the gain spectrum. The gain spectrum is either obtained experimentally or calculated numerically; its functional form is not generally available. A simple approximation is that the gain decreases quadratically from its peak value, that is (2.23) where wo is the frequency at which the gain takes its maximum value Go and Awg is the frequency spread over which gain is nonzero on either side of the gain peak (see fig. 6). We assume that the dominant mode is located at the gain-peak frequency 0,. The other longitudinal modes have frequencies w,,, = w,

+ mAo,,

(2.24)

where Am, is the longitudinal-mode spacing (see eq. 2.15) and the integer m varies from - M to + M . The value of M is determined by the largest integer contained in the ratio

Aw

L -

Y

za W

m= I 2 3

-3-2-1 Wg

- Awg

w0

Wo+AWg

FREQUENCY-

Fig. 6. Schematic illustration of the longitudinal-mode spectrum for a parabolic gain profile. The separation between the peak gain Go and the cavity-loss level is exaggerated for clarity.

I76


[III, 0 2

and 2M + 1 represents the number of longitudinal modes for which the gain is positive. Using eqs. (2.23)-(2.25), the mode gain G , is approximated by (2.26) The photon number P, can be obtained using eq. (2.18) and is given by (2.27)

P,=--. RSP Y-Grn

The substitution of eqs. (2.26) and (2.27) into eq. (2.19) leads to an implicit relation that can be solved to obtain the steady-state carrier number N . This, in turn, determines the mode gains G, and the steady-state photon population P, for each value of the current I. The results of such a numerical calculation show that below or near threshold the power increases in all of the longitudinal modes. However, the side-mode power saturates in the above-threshold regime. Numerical simulations can explain the qualitative features of fig. 5, particularly at low power levels. At high powers several other physical processes such as spatial or spectral hole-burning may become important (LEE, BURRUS,COPELAND, DENTAIand MARCUSE[ 19821). To gain physical insight in the multimode behavior of semiconductor lasers, we note from eq. (2.27) that the main-mode gain G, asymptotically approaches y and their difference is inversely proportional to the main-mode power. If we substitute m = 0 and use (2.28)

Go = y(1 - 6)

in eq. (2.27), we find that (2.29) YPO

Using eqs. (2.26) and (2.28) in (2.27), the photon number in the mth side mode is given by

P , 4R( Y

1

).

6 + (m/W2

(2.30)

This equation shows that the power distribution among various side modes is approximately Lorentzian, and the side-mode power is one half of the mainmode power when m = f i M . The width (full width at half maximum) of the

111, § 21


177

spectral envelope IS thus given by Am,

=

2m(A04)

=

$(2Amg),

(2.31)

where we have used eq. (2.25). Since 6 decreases with an increase in the main-mode power, the spectral width Am, decreases continuously in the abovethreshold regime. Figure 6 shows schematically the longitudinal-mode spectrum and its dependence on the gain profile. A measure of the spectral purity of a semiconductor laser is the modesuppression ratio (MSR) defined as the ratio of the main-mode power and the power of the most intense side-mode: (2.32) where we used eq. (2.30). Clearly MSR increases with an increase in the main-mode power Po, which reduces 6. Using eqs. (2.25) and (2.29) in eq. (2.32), the explicit dependence of MSR on the laser parameters is given by (2.33) where zp = y - ’ is the photon lifetime. This equation shows that the MSR increases linearly with the main-mode power. The previous analysis shows that the multimode characteristics of a semiconductor laser can be described in terms of two dimensionless parameters M and 6. The number 2 M + 1 corresponds to the total number of longitudinal modes that fit within the gain spectrum and experience gain. The parameter 6, defined by eq. (2.28), is a measure of how closely the peak gain approaches the total cavity loss (see fig. 6 ) and decreases with an increase in the laser power. The numerical value of 6 depends on the spontaneous-emission rate Rsp. For index-guided lasers R,, 2: 2y, and at a power level of few mW 6 N 5 x 10- ’. Using a typical value M = 30 for a 250 pm-long InGaAsP laser, eq. (2.32) predicts that a MSR of 20 can be achieved under continuous-wave (cw) operation. This value is not large enough to qualify the F P laser as a SLM laser. Furthermore when the laser is modulated, the transient mode spectrum becomes wider than the cw spectrum, and the MSR drops typically below 5 (MARCUSEand LEE [ 19831). To avoid the effect of chromatic dispersion, such multimode F P semiconductor lasers are generally used only for optical communication systems operating near the zero-dispersion wavelength of the single-mode silica fibers ( N 1.3 pm).

178

SINGLE-LONGITUDINAL-MODESEMICONDUCTOR LASERS

[III, § 3

With a careful matching of the laser wavelength to the zero-dispersion wavelength (within 10-15 nm), F P semiconductor lasers have been used up to a bit rate of about 2 Gb/s. The dominant factor limiting the performance of multimode lasers is the phenomenon of mode-partition noise (OGAWA[1985]), wherein the power in individual modes fluctuates widely even though the total power remains relatively constant.

8 3.

Requirements for Single-Longitudinal-Mode Operation

The basic requirement for a SLM laser is that its side modes should be suppressed sufficiently so that their presence does not degrade the system performance through mode-partition noise. Several models (HENRY,HENRY and LAX[ 19841, LINKE,KASPER,BURRUS,KAMINOW, KO and LEE [ 19851, SHENand AGRAWAL [ 19851) have shown that this can occur when the SLM laser exhibits a MSR of 100 (20 dB) or more. The question then becomes how one can design a semiconductor laser with such high values of MSR. An inspection of eq. (2.27) shows that the power in each mode is strongly influenced by how closely the mode gain approaches the cavity-loss level. One way to control the MSR is to make the cavity loss wavelength dependent so that different longitudinal modes have different cavity-decay rates y,, . With this generalization the MSR is given by

As in eq. (2.28), we introduce the dimensionless parameter 6 using Go = yo(l - 6). If we use eqs. (2.20) and (2.21) and yo 1: uggo, y1 - yo = Q a ,

(3.2)

Go - GI = V,Ag,

(3.3)

the MSR becomes MSRz-.

Aa

+ Ag

go 6

(3.4)

Here go is the threshold gain of the main mode (go = 80-100 cm-I), Ag accounts for the gain roll-off (Ag 1: 0.1 cm- '/nm), and A a is the cavity-loss difference between the main and side modes. The parameter 6 N 5 x 10- at a power level of 2-3 mW. For an FP laser, A a = 0, and MSR N 20 for typical

REQUIREMENTS FOR SLM OPERATION

111,s31

179

parameter values. However, an MSR of 200 can be achieved by a relative loss difference Aa/go of 1 % or A a N 1 cm- ' . Clearly, the loss discrimination among the cavity modes is very effective in suppressing the side modes. The foregoing discussion applies only for cw operation. To be useful in optical communication systems an SLM laser should maintain its MSR even under high-bit-rate modulation. To estimate the required loss difference A a under dynamic conditions, one can solve the rate equations (2.18) and (2.19) after considering two modes with different cavity-decay rates 'yo and yl. Figure 7 shows the results of such a calculation (MATSUOKA,YOSHIKUNIand MOTOSUGI [ 19851) for 2 Gb/s modulation. The MSR, defined as the ratio of the pulse energies for the main and side modes, is plotted as a function of Aa/go for several bias levels. An MSR of 25 dB or better can be maintained when the relative loss difference exceeds 5 % This corresponds to Act = 4 cm- ifwe take a typical value of 80 cm- for the threshold gain go.

I

- 50

Ib

I

I

I

Fig. 7. Dependence ofthe mode-suppressionratio on the relative loss difference between the main and side modes and on the bias level I,. The inset shows the isolated current pulse corresponding YOSHIKUNIand MOTOSUGI[1985].) to a bit rate of 2 Gb/s. (After MATSUOKA,

180


[III, § 4

Two distinct mechanisms have been used to make the cavity loss wavelength dependent. In the DFB mechanism an internal grating provides distributed feedback for modes whose wavelength satisfies the Bragg condition. In the coupled-cavity mechanism the FP laser is coupled to an external cavity. The feedback from the external cavity results in a wavelength-dependent effective reflectivity for the facet facing the external cavity. Both of these schemes are discussed in the next two sections.

4. Distributed-Feedback Semiconductor Lasers The feedback in a DFB laser is not localized at the cavity facets but is distributed throughout the cavity length. This is achieved through an internal grating, which leads to a periodic perturbation of the refractive index; feedback occurs by means of backward Bragg scattering, which couples the waves propagating in the forward and backward directions. Mode selectivity of the DFB mechanism results from the Bragg condition; coupling occurs only for wavelengths ,IB satisfying A = m -AB, 2Tz

where A is the grating period, ii is the effective-mode index, and the integer m is the order o i B r a g diffraction. By choosing A appropriately, such a device can be made to provide distributed feedback only at selected wavelengths. KOGELNIK and SHANK[ 19711 were the first to observe the lasing action in a periodic structure that utilized the DFB mechanism. Although most of the early work related to GaAs lasers (YARIVand NAKAMURA [1977], WANG [ 19771, STREIFER,SCIFRESand BURNHAM[ 1977]), the recent work has focused on the development of InGaAsP DFB lasers operating at 1.55 pm (ITAYA,MATSUOKA, KUROIWA and IKEGAMI [ 19841, UTAKA,AKIBA,SAKAI [ 1984a1, KITAMURA, YAMAGUCHI, MURATA,MITO and and MATSUSHIMA KOBAYASHI [ 1984a1, SUEMATSU, KISHINO,ARAI and KOYAMA[ 19851, NAGAI,MATSUOKA,NOGUCHI,SUZUKIand YOSHIKUNI[ 19861, AKIBA, USAMIand UTAKA[ 19871). From the point of view of device operation, semiconductor lasers employing the DFB mechanism can be classified into two broad categories: DFB lasers and distributed B r a g reflector (DBR) lasers. These are shown schematically in fig. 8. Whereas the feedback occurs throughout the cavity length in DFB lasers, it does not take place in the central pumped region in DBR lasers. In effect the end regions of a DBR laser act as

DFB SEMICONDUCTOR LASERS

181

DFB LASER

n -TYPE

ACTIVE

m + DBR LASER

LPUMPED R E G I O N 4

Fig. 8. Schematic illustration of distributed-feedback (DFB) and distributed-Brag-reflector (DBR) lasers. Shaded area shows the active region of the device.

effective mirrors whose reflectivity is wavelength dependent because of the DFB mechanism. In this section we first discuss the coupled-wave theory of DFB lasers and obtain the longitudinal modes and their respective threshold gains. We then briefly discuss the fabrication details and the performance of the state-of-the-art DFB lasers. The last subsection considers the DBR lasers and the theory behind their operation.

4.1. LONGITUDINAL MODES AND GAIN MARGIN

Similar to the case of an FP laser, the longitudinal modes of a DFB laser are obtained by solving the wave equation (2.1), with the difference that the dielectric constant E ( x , ~z), is a periodic function of z. It is useful to write E(X,

Y , 4 = E ( x , Y ) + A&, y, z ) ,

(4.2)

where E ( x , y ) is the average value. We use eqs. (2.2) and (4.2) in eq. (2.1) and assume that the perturbation A Eis sufficiently weak so that it does not perturb the waveguide mode U ( x ,y ) determined by E ( x , y ) . The axial variation of the optical field is then governed by d2E + P2E = - kiAEE, dz2

~

where k,

=

w/c and

(4.3)

fl is given by eq. (2.5). We introduce the Bragg wave

182


[IIL § 4

number fl0 by defining

and assume a general solution of eq. (4.3) of the form E(z)

= A(z) exp(ib0z)

+ B(z) exp( - ip0z) .

(4.5)

We substitute eq. (4.5) into (4.3) and equate the coefficients of exp( & ip0z) on both sides. Since AE is a periodic function of z, it can be expanded in'a Fourier series.

By keeping only the nearly phase-matched terms [corresponding to m = n in eq. (4.6)], we obtain the coupled-wave equations (KOGELNIKand SHANK [ 19721)

dA = iApA + iKB, dz

(4.7)

dB - = -iApB - ircA , dz where

A p = J/'

-

b0 = (pk0 - Po) - $.

The coupling coefficient

K

(4.9)

is given by

(4.10)

It shows that the Fourier component A&,,, for which the B r a g condition is approximately satisfied couples the forward and backward propagating waves. The contribution to the integral in eq. (4.10) comes only from the grating region where Acrn # 0. In general, K can be complex, even when AE is real, if V(x, y) i s complex. For index-guided lasers, however, K is approximately real. Its

111, I 41

183

D F B S E M I C O N D U C T O R LASERS

magnitude depends on various device parameters such as the thickness and the refractive indices of various layers and the shape and the depth of grating corrugations (STREIFER,SCIFRESand BURNHAM[ 19751). Typical values of K are in the range of 40-80 cm- I . The general solution of the coupled-wave equations take the form A(z) = A , exp(iqz) + A , exp( -iqz), B(z) = B, exp(iqz) + B, exp( - i4z) ,

(4.11)

where 4 is the complex wave number to be determined from the boundary conditions. The constants A , , A,, B,, and B, are interdependent. If we substitute the general solution in eqs. (4.7) and (4.8) and equate the coefficients of exp( f iqz), we obtain ( 4 - A/?)A, = KB,

( 4 + AB)B, = - K A ~

( 4 - AB)B2 = K A , ,

(4

+ A/3)A2 =

(4.12)

- KB, .

(4.13)

,

These relations are satisfied with nonzero values of A ,A,, B , , and B, if the possible values of 4 obey the dispersion relation (obtained by setting the determinant of the coefficient matrix to be zero) 4

=

[(A/?)’ -

~’3’’~.

(4.14)

The plus and minus signs correspond to the forward and backward propagating waves, respectively. Furthermore, we can define the DFB reflection coefficient as (4.15)

and eliminate A , and B, in eqs. (4.11) and (4.12) in favor of 4 4 ) . The general solution of the coupled-mode equation then becomes

+ r ( 4 ) B, exp( -iqz), B(z) = B, exp( -i4z) + r ( q ) A , exp(iqz).

A(z) = A , exp(iqz)

(4.16) (4.17)

Since r = 0 if K = 0, it is evident that r ( 4 ) represents the fraction of the forward-wave amplitude that is reflected back toward the backward wave and vice versa. The sign ambiguity in the expression (4.15) for r ( 4 ) can be resolved by choosing the sign of 4 such that I r ( 4 )I I 1. Mode selectivity of DFB lasers stems, as will become clear, from the dependence of r on 4. The eigenvalue 4 and the ratio B,/A, are determined by applying the appropriate boundary conditions at the laser facets.

184


[III, 8 4

In applying the boundary conditions an additional feature of DFB lasers should be taken into account. Consider a laser of length L. Since L is not necessarily an exact multiple of the grating period A, the last period of the grating close to the facet is generally not complete. Even though no significant feedback occurs over these incomplete grating periods, the phase shift in this region plays an important role in determining DFB-laser characteristics and should be properly accounted for. A simple way to do this is to assume that the effective facet reflection coefficient is complex, rj=JR,.exp(i&),

j = 1,2,

(4.18)

where R, is the facet reflectivity and I$j is the round-trip phase shift encountered by the field in traveling the extra distance (a fraction of A ) corresponding to the last incomplete grating period. A characteristic feature of DFB semiconductor lasers is that the phase shifts and I$2 are expected to vary from device to device, since the relative distance between the cleaved facet and the last complete grating period (a small fraction of 1 pm) is uncontrollable at present except through the use of controlled etching or coating of individual facets. The boundary conditions at the two facets are

+,

A(0) = r,B(O),

B(L) = r,A(L).

(4.19)

If we use them in the general solution given by eqs. (4.16) and (4.17), we obtain two homogeneous equations for the unknown constants A and B,:

,

( r , - r)B, - (1 - rrl)A1 = 0 ,

(4.20)

(r2 - r ) exp (%iqL)A,- (1 - rr2)B2= 0 .

(4.21)

These equations have a nontrivial solution only for values of q satisfying the eigenvalue equation (STREIFER, BURNHAM and SCIFRES[ 19751) (4.22) This is the threshold condition for DFB lasers. It is similar to the threshold condition obtained for an FP laser (eq. 2.10) and reduces to it if the DFB contribution is neglected by setting r = 0. On the other hand, if facet reflectivities are neglected by setting r , = r, = 0, we obtain

r2 exp(2iqL)

=

1,

(4.23)

implying that r is the effective reflection coefficient for a DFB laser. The DFB eigenvalues q are obtained from eq. (4.22) after noting that r given

111, § 41

DFB S E M I C O N D U C T O R LASERS

185

by eq. (4.15) itself depends on q. Each eigenvalue q is complex and can be used to calculate Afi = + ( q 2

+ K~)”*

(4.24)

after using eq. (4.14). From eq. (4.9) the real and imaginary parts of

Afi = 6 - $ 2

(4.25)

yield the mode detuning

6 = jTk0

- fio = ( - ~ I T ~ , / I ~ ) A I

(4.26)

and the mode gain a.In eq. (4.26), A I is the deviation of the mode wavelength from the Bragg wavelength I,, and po is the group index defined by eq. (2.16). The threshold gain is determined by using eq. (2.6) and is given by g,, = rg = ti

+ qnt.

(4.27)

The values of S corresponding to different eigenvalues indicate how far that longitudinal mode is displaced from the B r a g wavelength I , . The physical interpretation of eq. (4.27) is clear. The mode gain Cr can be thought of as the gain required to overcome the cavity loss and plays the same role as the mirror loss aminfor an FP laser (see eq. 2.13). In contrast to FP lasers, however, Cr can be different for different modes because of the DFB mechanism. For given values of the parameters I , , r, and K L , the eigenvalue equation (4.22) yields the mode gains Cr and the mode detunings 6 corresponding to various longitudinal modes of the DFB laser. The mode with the lowest value of Cr (denoted by a,)becomes the main mode, which carries most of the laser power in the above-threshold regime. The mode with the second-lowest value of 72 (denoted by Cr, ) becomes the dominant side mode. The extent of side-mode suppression is governed by the difference Aa = a,- tlo. As discussed in 3, Aa should typically exceed 4 cm- to maintain a MSR of 25 dB or more under high-speed modulation of the laser. The longitudinalmodes of a DFB laser with nonreflectingfacets ( r , = r2 = 0) were first discussed by KOGELNIKand SHANK[1972]. Such a DFB laser supports a large number of equispaced longitudinal modes with a spacing nearly identical to that of an FP laser with the same length. The spectrum is symmetrical with respect to the Bragg wavelength I,, and the lowest-loss mode does not occur at I,. The two modes closest to I , (one on each side) have the lowest threshold gain and can lase simultaneously. The separation between these two degenerate modes is referred to as the stop band. The stop band is

’

186

[I14 0 4


determined by the coupling coefficient IC and can be used for its estimation. For K L>> 1 the width of the stop band is approximately 2 ic. Although a DFB laser with no facet feedback is predicted to oscillate in two modes of equal intensity, this seldom occurs in practice because even a small facet reflectivity can break the gain degeneracy. It is thus essential to include the effect of facet reflectivities on the longitudinal modes of a DFB laser (STREIFER,BURNHAM and SCIFRES[ 19751, HENRY[ 19851). Note that the amplitude reflection coefficients rl and r, in the eigenvalue equation (4.22) are complex to account for the corrugation phases and +z at the facets. Furthermore the phases $+ and GZin eq. (4.18) can vary from device to device, making it necessary to consider their entire range from 0 to 2 A. To illustrate the main qualitative features, we consider the case of a DFB laser with cleaved facets and take R , = R, = 0.32. Figure 9 shows the mode gain ZL and the detuning 6L for six DFB modes as is varied over its 2 a range for a fixed value of $, and two values of K L .The open circles show the location of DFB modes for = 0. As increases, the modes shift to the left on the curve. Their position for $, = n is marked by closed circles, whereas arrows denote the modes for $ -1 I

- 277.

-10

-5

0

5

1 0

DETUNING, SL

Fig. 9. Threshold gain and detuning from the Brag wavelength of six longitudinal modes of a DFB laser with two cleaved facets. Continuous curves are obtained by varying 9, over its entire range from 0 to 2n; DFB modes for three specific values of 9, are marked on the solid curve for KL = 2. The horizontal dashed line shows the threshold gain of a FP laser ( K = 0). (AAer AGRAWAL and DUTTA[1986].)

187


Several conclusions can be drawn from fig. 9. It is evident that facet reflections break the gain degeneracy that occurs in the absence of reflections for the two modes separated by the stop band. Which of the two modes has a lower threshold gain depends on the relative values of the phases $1 and $,. For the case of $, = f n shown in fig. 9, the mode on the low-frequency or long-wavelength side with respect to the B r a g wavelength (negative 6) has a lower gain for - f n I +1 I in, whereas the reverse occurs (positive 6) for ~n 1 I+1 I $n.The mode spectrum is degenerate for = in and $1= In. The arrows in fig. 9 mark the location of degenerate modes for $1 = f n . Another feature arising from facet reflections is the existence of a high-gain mode inside the stop band. It is of interest to compare the threshold of the DFB laser with that of an FP laser, shown by a horizontal dashed line (no mode selectivity) in fig. 9. The DFB threshold is always lower than the FP threshold because of the additional distributed feedback. It should be stressed, however, that the answer to the question about whether the DFB or FP mode would lase in a given device also depends on the relative location of the Bragg wavelength within the gain profile. The DFB mode would lase if the gain peak occurs in the vicinity of the Bragg wavelength. When the two are significantly far apart an FP mode close to the gain peak might have an overall lower threshold gain and reach threshold first. This behavior can be understood by noting that a mismatch between the gain peak and the Bragg wavelength leads to a negative Ag in eq. (3.4). As a result, the net gain margin A a + Ag decreases. In general, DFB lasers with cleaved facets require a close matching of the Bragg wavelength with the gain peak in order to suppress the FP modes. The performance of such DFB lasers can be improved by the use of an antireflection coating on one of the facets. A residual reflectivity of 1-5% is enough to suppress the FP modes to the extent that a mismatch of up to 10-15 nm can be tolerated between the Bragg wavelength and the gain peak. Figure 10 shows the longitudinal modes of a DFB laser with R, = 0.32 and R , = 0 and should be compared with fig. 9. The effect of reducing the reflectivity of one facet is to improve the mode selectivity at the expense of a threshold increase. It is clear from figs. 9 and 10 that both the threshold gain clo and the gain margin Act vary with the phases $1 and $*. Furthermore, the gain margin A a vanishes for some specific phase combinations, indicating that such a device will not oscillate in a single longitudinal mode. Since $1 and $2 can vary randomly from device to device, the performance of DFB lasers can be characterized only in a statistical sense (Buus [ 19851, MATSUOKA, YOSHIKUNI and

+,

188


- 10’

I.0tl

I

I

I

’

-5

I

’

I

’

I

I

I

’

’

0 5 DETUNING ,SL

I

I

I

I

I

[III, 8 4

I

10

Fig. 10. Similar to fig. 9, now for a DFB laser with one cleaved facet and one nonreflecting facet A instead of two cleaved facets. (After AGRAWALand D U ~ [1986].)

MOTOSUGI [ 19851, HENRY[ 19851, AGRAWAL, DUTTAand ANTHONY[ 19861, GLINSKI and MAKINO[ 19871). The “yield” of acceptable single-mode devices can be estimated numerically by finding the fraction of phase combinations for which A a exceeds a certain value. As discussed in 3 3, Aa should typically exceed 4 cm - in order to suppress the side modes by 25 dB or more under high-speed modulation. For a 250 pm-long laser the corresponding singlemode condition is AaL 0.1. Figure 11 shows the numerically calculated yield as a function of AaL for several combinations of the facet reflectivities R , and R , (given in parentheses) after choosing KL = 2. The yield of single-mode devices approaches 90% by a proper combination of the facet reflectivities. Other laser characteristics such as the threshold current density Jth and the differential quantum efficiency q also depend on the facet reflectivities and the corrugation phases $J1 and $J,. Yield curves similar to those shown in fig. 11 can be generated by considering the phase combinations that satisfy not only the single-mode condition but also the preset conditions on Jth and q. The and MAKINO[ 19871) that it is possible to achieve an results show (GLINSKI overall yield of about 90% by a proper choice of values for the parameters KL, R , and R,. In particular, a combination of low-reflection (- 1%) and highreflection (-80%) coatings on the laser facets has resulted in DFB lasers exhibiting high output power (up to 45mW) and an MSR of >30dB (YOSHIDA, ITAYA,NOGUCHI, MATSUOKAand NAKANO[ 19861).

’

=-

189


100

80

--a9

60

n

-J

w

*

40

20

n -0

I

0.1

I

\ \ ,\

0.3 G A I N MARGIN, AaL

0.2

I

0.4

0.5

Fig. 1 1 . Theoretical yield as a function of the gain margin AaL and its dependence on the facet reflectivities (given in parentheses). The yield for a given AaL corresponds to the fraction of phase combinations for which the gain margin exceeds that specific value.

An alternative approach to high-performance DFB lasers is provided by phase-shifted DFB lasers, where a constant phase shift is introduced in the middle of the laser cavity. Although the concept of phase-shifted DFB lasers was known from the early work of HAUSand SHANK[ 19761, it is only recently such lasers have drawn considerable attention. AKIBA,USAMIand UTAKA [ 19871 have reviewed the performance of phase-shifted DFB lasers. The coupled-wave theory presented here can be used to obtain the longitudinal modes of a phase-shifted DFB laser (NISHIHARAand MURAKAMI [ 19841, EDA, FURUYA, KOYAMAand SUEMATSU [ 19851 MCCALLand PLATZMAN [ 19851, UTAKA,AKIBA,SAKAIand MATSUSHIMA [ 19861). For a phase shift of $ n the lowest-threshold mode occurs exactly at the Bragg wavelength. Since such a phase shift requires a shift of the grating corrugation by a quarter wave ($A), such lasers are also referred to as :A-shifted DFB lasers. Several schemes have been proposed and demonstrated to achieve the required quarter-wave shift (SKARTEDJO, EDA, FURUYA,SUEMATSU, KOYAMAand TANBUN-EK [ 19841, TADA,NAKANOand USHIROKAWA [ 19841, SODA,WAKAO,SUDO, TANAHASHI and IMAI [ 19841, KOENTJORO, BROBERG, KOYAMA,FURUYA and SUEMATSU [ 19841, UTAKA,AKIBA,SAKAIand MATSUSHIMA [ 1984b1). The main attraction of phase-shifted DFB lasers is that the yield of single-mode

190


[I14 $ 4

devices approaches 100% when the coupling coefficient and the facet reflectivities are appropriately chosen (Buus [ 19861, GLINSKIand MAKINO [ 19871).

4.2. FABRICATION AND PERFORMANCE

The fabrication details of a DFB laser are similar to those of FP lasers with the exception that a grating is etched onto one of the cladding layers surrounding the active layer. The choice of whether to use the upper or the lower cladding layer for the grating depends on practical considerations. The placement of the grating on the upper cladding layer has the advantage that the grating period A, given by eq. (4.1), can be adjusted for each wafer after estimating the mode index ji and the location of the gain peak. At the same time, however, this choice has the disadvantage that it requires an additional epitaxial growth. In practice the grating is often etched directly onto the substrate to reduce the epitaxial cycles. The cladding and the active layers are then deposited over the grating. Figure 12 shows a photomicrograph of a 1.55 pm InGaAsP laser wafer fabricated in this manner. The wafer is processed in a manner similar to that used for F P lasers to obtain a buried-heterostructure design (see fig. 2). The most critical step in the fabrication of DFB lasers is the formation and the preservation of the grating during regrowth. For a 1.55 pm InGaAsP laser the grating period A N 0.23 pm for a first-order grating if we use m = 1 and p = 3.4 in eq. (4.1). This value doubles for a second-order grating. Two techniques have been used for the formation of a grating with submicrometer periodicity. In the holographic technique, optical interference is used to form a fringe pattern on the photoresist deposited on the wafer surface. In the electron-beam lithographic technique an electron beam scans the wafer surface to produce the desired pattern on the electron-beam resist. Both methods generally use chemical etching for the grating corrugations, with the patterned resist acting as a mask. Once the grating has been etched on the substrate or on an epitaxial layer, the wafer can be processed in the usual way to obtain a specific laser structure. The effectiveness of the grating is governed by the coupling coefficient K given by eq. (4.10). The optimum values of IC lie in the range 40-80 cm- ' for a 250 pm-long laser so that K L = 1-2. A critical control of the layer thicknesses is needed to obtain such high values of K . In particular, the placement of the grating with respect to the active layer is critical, since only the evanescent field

co?

P

Y

5 E

v1

m

Fig. 12. Photomicrographof a 1.55 pn InGaAsP laser wafer showing a second-ordergrating etched directly onto the substrate and other layers grown over the grating using liquid-phase epitaxy. (Courtesy N. K. Dutta.)

192


[I14 I 4

associated with the fundamental waveguide mode interacts with the grating; typically the separation between the grating and the active layer should not exceed 0.1 pm. The corrugation depth h is another critical parameter. Whereas values of h as small as 0.03 pm are acceptable for first-order gratings, h should exceed 0.1 pm for a second-order grating. The coupling coefficient also depends on the shape of the grating corrugations and is particularly sensitive to such details for second-order gratings (STREIFER,SCIFRESand BURNHAM[ 19751, SAKAI,UTAKA,AKIBAand MATSUSHIMA [ 19821). In spite of the technological complexities involved, high-performance DFB lasers have been fabricated using both first-order and second-order gratings. Figure 13 shows, as an example, the light-current characteristics and the longitudinal-mode spectra (at Z = 1.5 It,,) at several temperatures for a 1.55 pm InGaAsP laser. This device has a first-order grating with an estimated corrugation depth of 0.03 pm (KITAMURA, YAMAGUCHI, MURATA,MITOand KOBAYASHI [ 1984b1). At room temperature the threshold current was 21 mA with a differential quantum efficiency of 15% per facet. The performance of these DFB lasers is comparable to similar F P lasers as far as light-current characteristics are concerned. However, as expected, DFB lasers outperform conventional FP lasers with respect to their spectral purity. In fig. 13 the DFB laser maintains the same longitudinal mode in the entire temperature range of 20-108 C. By contrast an F P laser would exhibit several mode jumps over this temperature range because of the temperature-induced shift of the gain peak. The stability of the DFB mode results from the built-in grating whose period determines the laser wavelength. The wavelength changes slightly with O

30 10

1.4.5 I + h

-3 -E

.

1

$ 5

2 0 0

400

200

CURRENT ( m A )

1.56

4.57 WAVELENGTH ( p n )

Fig. 13. Light-current characteristics and longitudinal-mode spectra at several temperatures for a 1.55pm DFB laser. (After K I T A M U R A , YAMAGUCHI,MURATA,MITO and K o B A Y A s H I [1984b].)

111, § 41


193

temperature at a rate of about 0.1 nm/" C because the mode index Ji in eq. (4.1) varies with temperature. The spectral purity of DFB lasers is measured by the degree of side-mode suppression; the device of fig. 13 exhibited an MSR of 35 dB at a power level of 5 mW. In general, the MSR degrades at high power levels. This is particularly true for DFB lasers with uncoated facets whose FP side modes do not generally have a large gain margin (see fig. 9). The gain margin is further degraded because of a mismatch between the Bragg wavelength and the gain peak. This problem can be overcome by suitably coating the laser facets. A combination of 1% and 80% facet reflectivities has led to DFB lasers with high spectral stability and purity (> 30 dB) up to a power level of 45 mW (YOSHIDA, ITAYA, NOGUCHI,MATSUOKA and NAKANO[ 19861). Recently phase-shifted DFB lasers have attracted considerable attention to improve the spectral stability. Such lasers have maintained single-longitudinal-mode operation up to a power level of 50 mW under pulsed conditions (AKIBA,USAMIand UTAKA[ 19871).

4.3. DISTRIBUTED-BRAGG-REFLECTOR LASERS

The DBR laser provides an alternative scheme wherein the frequency dependence of the DFB mechanism is utilized to select a single longitudinal mode of an FP cavity. In contrast to DFB lasers the grating is etched outside the active region (see fig. 8). In effect, a DBR laser is an F P laser whose mirror reflectivity varies with wavelength; the lasing occurs at the wavelength for which the reflectivity is maximum. In this section we briefly describe the operating principle and characteristics of DBR lasers. For a detailed discussion we refer to the work of SUEMATSU, KISHINO,ARAIand KOYAMA[ 19851). A problem specific to the DBR laser is that when the grating is etched onto the unpumped active material, the optical losses inside the DBR region are high and the resulting DBR reflectivity is poor. The problem of material loss can be overcome by using a different material that is relatively transparent at the laser wavelength. For InGaAsP lasers the InP substrate or a quaternary cladding layer (with a band gap higher than that of the active layer) can be used for this purpose. However, in this case the DBR and active regions form two separate waveguides, and transfer of the optical mode between them leads invariably to coupling losses that reduce the effective reflectivity of the DBR. The design of a DBR laser involves minimization of the coupling losses, and many coupling schemes have been used for this purpose. If C, is the power coupling efficiency between the DBR and active waveguides, the effective

194


[III, I 4

amplitude reflection coefficient of the DBR is given by reff =

tor,

(4.28)

9

where rg is the amplitude-reflection coefficient of the DBR at the junction of the two waveguides. The power coupling efficiency appears in eq. (4.28), since coupling losses occur twice during each reflection. An expression for rg can be obtained using the analysis of 5 4.1. Consider a distributed Bragg reflector of length A forward-propagating wave is incident from the left at its surface (at z = 0) and excites the counterpropagating waves A ( z ) and B ( z ) inside the DBR. The coupled-wave equations (4.7) and (4.8) govern their propagation with the general solution given by eqs. (4.16) and (4.17). The amplitude-reflection coefficient rg is obtained from

z.

(4.29) where 4 and r ( 4 ) are given by eq. (4.14) and (4.15). If we use the boundary condition B(z)= 0 at the other end of the DBR, we obtain from eq. (4.17) the relation B,

=

r ( 4 ) A I exp (2iqz) .

(4.30)

Using eq. (4.30) in eq. (4.29), we obtain (4.31) This expression can be written in terms of K and AD using eq. (4.15) and becomes i K sin ($5) rg = lrgl exp(iq5) = (4.32) 4 cos (4L)- AD sin ( 4 t ) ’ where, similar to eq. (4.25),

AD = 6 + $ag.

(4.33)

The imaginary part of AD is positive to account for the material losses inside the DBR medium, and ag is the corresponding power-absorption coefficient. The parameter 6 given by eq. (4.26) accounts for the detuning of the laser wavelength from the Bragg wavelength. Figure 14 shows the wavelength dependence of the power reflectivity I rg 1’ and the phase 4 for a& = 0.1. The reflectivity is maximum at the Bragg wavelength (6 = 0), and a 50% reflectivity

195


I )

l

l

I

l

l

l

l

l

l

-8 -6 -4 -2

0

2

4

6

8

10

DETUNING,~~

Fig. 14. Reflectivity and phase of a distributedBrag reflector as a function of detuning from the Brag wavelength for two values of KL.(AAer SUEMATSU,KISHINO, ARAI and KOYAMA [ 19851.)

can be obtained even for a relatively small value of K L = 1. Of course, the coupling loss C, in eq. (4.28) would reduce the effective DBR reflectivity by Ci. The threshold gain and the longitudinal modes of a DBR laser can be obtained in a manner similar to that discussed for FP lasers in 5 2.2. We assume for simplicity that the two Bragg reflectors are identical and can be described through the same effective reflection coefficient re*. The threshold condition becomes (see eq. 2.10) (reff)2exp(2ipL)

=

1.

(4.34)

where

p = pk 0 - 1.2'a

(4.35)

196

SINGLE-LONGITUDINAL-MODE S E M I C O N D U n O R LASERS

[Ill, 8 4

is the mode-propagation constant. Equating the modulus and the phase on the two sides of eq. (4.34), we obtain Cglr,l2 exp(aL)

pk,L

=

+ @ = mn,

1,

(4.36) (4.37)

where we have used eq. (4.28) and @ is the phase of rg. As in the case of an FP laser, it is convenient to define the DBR loss using (4.38) Equation (4.36) then simply becomes CX = aDBR.The phase equation (4.37) determines the longitudinal modes of a DBR laser. In contrast to FP lasers, however, the modes are not equispaced, since @dependson the detuning of the mode wavelength from the Bragg wavelength. Similarly, the DBR loss aDBR is different for different longitudinal modes, and the lowest-threshold mode occurs at a wavelength for which aDBRis smallest. The lowest-threshold mode occurs close to the Bragg wavelength, since the DBR reflectivity peaks at that wavelength. In contrast to DFB lasers, DBR lasers do not exhibit a stop band in their longitudinal-mode spectra. Conceptually, a DBR laser is just an FP laser whose end mirrors exhibit frequency-dependent reflectivity. The emission characteristics of DBR lasers are similar to those of DFB lasers. Generally, the threshold current of a DBR laser is relatively high because of the coupling losses, which increase aDBR given by eq. (4.38). For the same reason the differential quantum efficiency is also relatively low. Several different [ 19851) have been used to designs (SUEMATSU, KISHINO,ARAIand KOYAMA reduce the coupling losses in DBR lasers. In one specific design known as the bundle-integrated-guide DBR laser, a threshold current of 39 mA and a differential quantum efficiency of 18% per facet were achieved (TOHMORI, KOMORI,ARAIand SUEMATSU [ 19851). This laser could be operated in a single longitudinal mode up to 6.5 mW with an MSR of 30 dB or more. As an alternative to the DBR laser, several hybrid schemes have recently been proposed where a Bragg reflector is coupled to a conventional multimode FP laser (HAMMER,NEIL, CARLSON,DUFFY and SHAW[1985], BRINKMEYER,BRENNECKE, ZURN and ULRICH[ 19861, WHALEN,TENNANT, ALFERNESS, KORENand BOSWORTH[ 19861, OLSSON,HENRY,KAZARINOV, LEEand ORLOWSKY [ 19871). To distinguish them from the DBR lasers where the Bragg reflector is monolithically integrated, such lasers are referred to as external Bragg reflector (EBR) lasers. In spite of their hybrid nature, EBR

111, § 51

COUPLED-CAVITY SEMICONDUCTOR LASERS

197

lasers are promising, since both the linewidth and the frequency chirp (discussed later) can be significantly reduced compared with those obtained from DFB and DBR lasers (MEISSNER and PATZAK[ 19861, KAZARINOV and HENRY[ 19871).

8 5. Coupled-Cavity Semiconductor Lasers The coupled-cavity concept for longitudinal-mode selection is known from the early work on gas lasers (KOGELNIKand PATEL[ 19621, KLEINMAN and KISLINK[ 19621). CROWEand CRAIG[ 1964lwere the first to couple the semiconductor laser to an external cavity. At about the same time LASHER[ 19641 proposed the concept of monolithically integrated coupled-cavity devices, which attracted attention in the context of optical bistability (NATHAN, MARINANCE, RUTZ, MICHELand LASHER[ 1965]), optical amplification (KOSONOCKY and CONNELY[ 1968]), and longitudinal-mode selectivity (ALLEN,KOENIG and RICE [ 19781). SALATHB[ 19791 has reviewed the research in this field prior to 1979. The interest in the coupled-cavity scheme has revived recently in an attempt to obtain semiconductor lasers offering the potential of mode selectivity together with wavelength tunability (TSANG [1985], AGRAWAL and DUTTA [1986]). Both monolithic and hybrid techniques have been employed. In the hybrid technique a multimode FP laser is coupled to an external cavity. To distinguish them from monolithic devices, such lasers are often referred to as external-cavity semiconductor lasers. The mechanism of mode selectivity in coupled-cavity lasers can be understood by referring to fig. 15. As a consequence of the feedback from the external cavity, the effective reflectivity of the laser facet facing the external cavity becomes wavelength dependent. In effect, the mirror loss given by eq. (2.13) becomes different for different longitudinal modes of the laser cavity. The mode selected by the device is the F P mode that has the lowest cavity loss and is closest to the peak of the gain profile. Because of the periodic nature of the loss profile, other FP modes with relatively low cavity losses may exist. Such modes are discriminated against by the gain roll-off because of their large separation from each other. Thus, a combination of cavity loss modulation and gain roll-off can lead to a net gain margin Aa. + Ag in eq. (3.4) sufficiently large (24 cm I ) so that all side modes are suppressed by 30 dB or more. ~

198


[III, $ 5

MIRROR-’

LASER CAVITY

EFFECTIVE MIRROR REFLECTIVITY R ( A )

MAIN MODE

D EI WAVELENGTH X

Fig. 15. Schematic illustration of longitudinal-mode selectivity in a coupled-cavity laser. The effect of the external cavity is to make the effective reflectivity of the facet wavelength dependent. The resulting loss profile discriminates the side modes, since they experience higher losses. Compare with fig. 3 where the loss profile is flat.

5.1. COUPLED-CAVITY SCHEMES

Most coupled-cavity semiconductor lasers can be classified into two broad categories. We shall refer to them as active-passive and active-active schemes, depending on whether the second cavity remains unpumped or can be pumped to provide gain. The devices in the latter category are sometimes also called three-terminal devices, since three electrical contacts are used to pump the two optically coupled but electrically isolated cavity sections. Figures 16 and 17 show specific examples of the two kinds of devices. In the active-passive scheme the semiconductor laser is coupled to an external cavity that is umpumped and plays a passive role. In the simplest design a plane or spherical mirror is placed at a short distance from the laser facet, which may be antireflection coated to increase the coupling between the two cavity sections. The spherical-mirror geometry for InGaAsP lasers has attracted considerable attention (PRESTON,WOOLLARDand CAMERON

111, § 51


DIODE L A S E R

199

A N T I - REFLECTION COATING C A P I L L A R Y TUBE

, GRIN-ROD

LENS

GOLD F I L M REFLECTOR

Fig. 16. Schematic illustration of a coupled-cavity laser employing the active scheme. In this scheme a graded-index fiber-rod (GRIN-ROD) lens acts as an external cavity. (After LIOU, BURRUS, LINKE, KAMINOW,GRANLUND, SWANand BESOMI [ 19841.)

19811, VAN DER ZIEL and MIKULYAK[ 19841). Several variations of the plane-mirror geometry have been adopted for InGaAsP lasers. In the shortcoupled-cavity scheme (LIN, BURRUS, LINKE, KAMINOW,KO, DENTAI, LOGANand MILLER[ 1983]), a short laser cavity (- 50 pm) is coupled to a

Fig. 17. Schematic illustration of a coupled-cavity laser employing the active-active scheme. In this C3-typedevice, a single longitudinal mode can be selected and tuned by adjusting the currents I, and I, in the two cavity sections. (ARer TSANG,OLSON and LOGAN[ 19831.)

200


“11,

55

short external cavity (- 50 pm). The gold-coated facet of a semiconductor chip acts as a plane mirror. In another scheme (Lrou [ 19831) the external cavity consists of a graded-index fiber lens to provide coupling and to avoid diffraction losses. A short rod (- 100 pm) of graded-index multimode fiber with one end gold coated is used for this purpose (see fig. 16). Etching techniques have also been used to make monolithic coupled-cavity devices with integrated active and passive sections (CHOI and WANG [ 19831, MATSUDA,FUJITA, OHYA,ISHINO,SATO,SERIZAWA and SHIBATA[ 19851). In the active-active scheme both sections can be independently pumped, giving an additional degree of freedom that can be used to control the device behavior. A natural choice is to use identical active materials for both cavities. Furthermore the alignment between the active regions is automatically achieved if the two cavity sections are created by forming a gap in a conventional semiconductor laser. Etching and cleaving techniques have been used for this purpose (COLDREN, MILLER,IGAand RENTSCHLER [ 19811,TSANG,OLSSON and LOGAN[ 19831). In another scheme the interferometric property of a coupled-cavity device is obtained by bending the active region (ANTREASYAN and WANG[ 19831).Whatever the technique employed, the qualitative behavior of such three-terminal coupled-cavity lasers is similar with respect to mode selectivity and wavelength tunability. The advantage of the cleaving technique lies in providing a device whose four facets are parallel to each other. The cleaved-facet reflectivity (about 32%) allows reasonable coupling between the two cavity sections as long as the gap is not too wide. Coupled-cavity lasers made by the cleaving technique are sometimes referred to as cleaved-coupled-cavity, or C3, lasers. Their operating characteristics have been extensively studied and reviewed recently (TSANG [ 19851). While considering three-terminal devices we shall refer to C3 lasers only, but keep in mind that similar behavior would occur for other kinds of active-active devices. The mechanism of mode selectivity is the same for both active-active and active-passive devices. The main difference between the two kinds of coupledcavity lasers lies in the external means used to shift the FP modes of the two cavities. In the active-passive case the modes of the external cavity can be shifted by changing its length or by changing its temperature. By contrast, the active-active case offers the possibility of electronic shifting, since the current in the two cavities can be independently controlled. If one of the cavities is operated below threshold, a change in its drive current significantly changes the carrier density inside the active region. Since the refractive index of a semiconductor laser changes with the carrier density, the longitudinal modes shift with

111, 8 51

20 1


a change in the drive current and different FP modes of the laser cavity can be selected. In the design of a coupled-cavity laser, the cavity lengths L, and L, are adjustable to some extent. The performance of such lasers depends on the relative optical lengths p l L I and p2L2of the two cavities, where p , and p2 are the mode indices. Two cases should be distinguished, depending on whether the optical lengths are similar ( p I L lz p 2 L 2 ) or differ significantly ( p I L I b p2L,), and are referred to as the long-long and long-short geometries, respectively. These geometries, although both capable of mode selection and wavelength tuning, differ in one important aspect of wavelength stability. For the long-long geometry even a relatively small shift of the F P mode can lead to mode hopping. By contrast, a shift of about one mode spacing is required to achieve mode hopping in the long-short geometry. Clearly, unintentional mode hopping because of temperature and current fluctuations is less likely to occur in the long-short case. By the same token, wavelength tuning would require larger current or temperature changes for a long-short device than for a long-long device.

5.2. THEORY

The first step in the analysis of coupled-cavity lasers is to determine the extent of coupling between the two cavities. Figure 18 shows the geometry and notation. The coupling between the cavities is governed by an air gap of width L,. The air gap itself forms a third FP cavity, and the intercavity coupling is affected by the loss and phase shift experienced by the optical field while traversing the gap. In the scattering-matrix approach the fields in the two cavities

r;

CAVITY 1

1

r2

CAVITY

2

r;

GAP

Fig. 18. Geometry and notation used in the theoretical description of a coupled-cavity laser. In the most general case all four facet reflectivities may differ from each other.

202


[I14 I 5

are related by (COLDREN and KOCH [ 19841)

The scattering-matrix elements can be obtained by considering multiple reflections inside the gap. More explicitly, the gap is treated as an F P cavity on which the field E l is incident from the left, and the reflected field E ; and the transmitted field E ; are obtained in terms of E l after considering multiple round trips inside the cavity. The procedure yields S , = E / E l and S,, = E ; / E , . The same method is used to obtain S,, and S , , after assuming that only the field E , is incident on the FP cavity from the right. The result is

,

sl,

=

s,, = [t,(l

r:) (1 - r:)]I/, 1 - rlr2tg

-

1

where t,

=

exp(2i&Lg) = exp(2ikOl,) exp( -

with & = k , + iia, accounts for the phase shift and the loss inside the gap. Furthermore, rl and r, are the amplitude reflection coefficients at the two facets forming the gap, that is, r,, = (p,,- 1)/(pn + l), where pl and p, are the modes indices in the two cavities. Physically, the gap has been replaced by an interface whose effective reflection coefficients are S , and S,,, whereas S , , and S,, are the effective transmission coefficients from cavity 1 to cavity 2 and vice versa. It is useful to define a complex coupling parameter

,

C = Cexp(i8) =

-

where C governs the strength of mutual coupling and 8 is the coupling phase, which will be seen to play an important role. The magnitudes of C and 8 depend on a large number of device parameters. The simplest case occurs for a semiconductor laser coupled to an external mirror. In this case the coupling element is just the laser-air interface. Since

111, § 51

203


L, = 0, t, = 1. Furthermore, p2 = 1 and therefore r2 = 0. Using these values in eqs. (5.2)-(5.6), we find that C = (1 - rf)1'2/rl and 6' = ;IT.For a C3laser the two cavities have nearly equal index of refraction, and therefore rI = r, = r. Using eqs. (5.2) to (5.6),we now obtain

The coupling depends on t,, and the evaluation of C and Orequires a knowledge of the gap loss a,. These losses arise mainly from diffraction spreading of the beam inside the gap. Both C and Ovary considerably with small changes in L,. The in-phase coupling (0 = 0) occurs whenever L, = im1 (m is an integer), and C also goes through a maximum for that value of L,. In practice, however, the gap width may vary from device to device. The corresponding large variations occurring in C and O imply that the performance of C3 lasers would also be device dependent. Typical values of C are in the range of 0.5-1 for a few-micrometer-wide gap (HENRYand KAZARINOV[ 19841, KOCH and COLDREN [ 19851). We now obtain an eigenvalue equation whose solutions yield the wavelength and the threshold gain associated with the longitudinal modes of the coupled system. A simple way to do this is to consider the relationship between El and E ; , using fig. 18. The field E ; results from reflection of E, and transmission of E, and is given by (see eq. 5.1) E;

=

S l l E l+ S,,E,.

(5.8)

On the other hand, the round trip through cavity 1 results in the relation El

=

r ; exp(2iPlLl)E; = r;t,E; ,

(5.9)

where the complex progagation constant

P,, = p,,k,

- $3,

,

n

=

1,2,

(5.10)

governs wave propagation in the nth cavity, ko = 2x11,1 is the device wavelength, and 3, is the mode gain. From eq. (2.6)the mode gain is related to the material gain g,, by a,, =

rg,,- a:,

(5.11)

where r is the confinement factor and a:' is the internal loss. For an active-active device g , and g , can be individually varied by changing the current passing through each cavity section. For an active-passive device g, = 0 and Z, accounts for the absorption loss in the passive cavity.

204


[III, 8 5

Equations (5.8) and (5.9) can be combined to obtain the relation (1 - r ; t , S , , ) E ,= r ; t , S , , E , .

(5.12)

Similar considerations for cavity 2 lead to (1 - r;r2S2,)E2 = r ; t , S , , E , .

(5.13)

These two homogeneous equations have nontrivial solutions only if the secular condition (1 - r ; r , S , , ) ( l - ~ ; z ~= Sr ~~r ; r~l t )2 S 1 2 S 2 ,

(5.14)

is satisfied. Equation (5.14) is the desired eigenvalue equation for the coupled system and has been studied extensively (COLDREN and KOCH[ 19841, HENRY and KAZARINOV [ 19841, MARCUSE and LEE[ 19841, CHOI,CHENand WANG [ 19841, STREIFER, YEVICK,PAOLIand BURNHAM[ 19841). In the absence of coupling S , , = 0, S,, = r,, and we recover the threshold condition for uncoupled cavities. Note that t,,

=

e~p(2iP,,L,~) = exp(2ip,k0L) exp( - Z,L,)

(5.15)

incorporates the phase shift and the gain (or loss) experienced by the optical field during a round trip in each cavity. The loss and the phase shift inside the gap are included through tg and S,, as given by eqs. (5.2)-(5.5). The eigenvalue equation (5.14) is applicable for all kinds of coupled-cavity devices with arbitrary reflectivities at four interfaces. It is useful to introduce the concept of effective mirror reflectivity. In many cases the role of one cavity (say cavity 2) is to provide a control through which a single FP mode of the laser cavity 1 is selected. The effect of cavity 2 on mode selectivity can be treated by an effective reflectivity for the laser facet facing the cavity 2. Equation (5.14) can be written in the equivalent form (1 - ' ; R e n t , )= 0 ,

(5.16)

where the effective reflection coefficient is given by (5.17) and f2 = r;t,S,, is the fraction of the amplitude coupled back into the laser cavity after a round trip in the cavity2. Equation (5.16) suggests that the coupled-cavity laser is equivalent to a single-cavity laser with facet reflection coefficients r ; and R e , as far as mode selectivity is concerned. Although eq. (5.16) is formally exact, its practical utility is limited to the case of weak

111, 8 51


205

coupling so that a change in t , (through operating conditions of cavity 1) does not affect R,, through a change in the feedback fraction f. This is often the case for active-passive devices. In the case of active-active devices the effective reflectivity concept is reasonably valid when cavity 2 is biased below threshold. To illustrate the extent of mode selectivity offered by the coupled-cavity mechanism, we consider solutions of the eigenvalue equation (5.14) for a specific C3-type device for which r, = r ; = r2 = r ; N 0.56. We assume that cavity 2 is biased below threshold in such a way that a2 = 0. Equation (5.14) is used to obtain if, and the wavelength ;1 = 2 n / k , corresponding to various longitudinal modes. Figure 19 shows the longitudinal modes and their respective gains for the gap width L, = 1.55 pm and for the cavity lengths L , and L, corresponding to long-long and long-short geometries (COLDREN and KOCH [ 19841). Since the concept of effective reflectivity is approximately valid, 1 R,,I versus I is also known. The effect of the second cavity is to modulate the effective reflectivity, and the minimum threshold gain is required for modes for which Re, is maximum. The gain difference A a between the lowest-gain mode and the neighboring mode provides mode discrimination and leads to sidemode suppression. Figure 19 is drawn for the optimum case (Lg = A) using L , = 240pm 0.6 [L

0.4

WAVELENGTH ( p m )

WAVELENGTH ( p m )

Fig. 19. Threshold gain and effective facet reflectivity of the longitudinal modes (solid dots) for coupled-cavity lasers employing long-short (left column) and long-long (right column) geometries. The gain margin Aa determines the extent of side-mode suppression. (After COLDREN and KOCH [1984]; 0 IEEE.)

206


WI,§ 5

,

S , = S,, = 0.409 and S , , = S,, = 0.371, which from eq. (5.6) give C = 0.907 and O = O . A comparison of the long-long and long-short device in fig. 19 reveals an interesting feature. Even though the wavelength variation I R,,I is much faster in the long-long case, the two devices behave similarly as far as the threshold gains are concerned. In particular, the mode pattern repeats after eight modes (M = 8) in fig. 19. For the long-short geometry the repeat mode M N p,L1/p2L2is determined by the ratio of the optical lengths in each cavity. For the long-long geometry, by contrast, M 2: p,L,/(plLl - p2L,). For a given coupled-cavity device the repeat mode is thus determined by p2L, or p , L , - p,L,, depending on which one is smaller. As mentioned earlier, the repeat mode is discriminated by the gain roll-off (see fig. 15). A detailed modeling of the C3-type devices requires that the mode thresholds for various longitudinal modes should be known as a function of the currents I , and 1, passing through the two cavity sections. HENRYand KAZARINOV [ 19841 have considered this behavior using a simple model for the gain and index variations with the current. Single-mode operation above threshold is described by discrete zones in the 1,-Z2 plane, and mode hopping occurs at the zone boundaries. These zones are, however, extremely sensitive to the coupling phase 0, and the optimum performance for C3 lasers is expected to occur for in-phase coupling ( 0 = 0). For other values of O the steady-state operation can become unstable or display bistable behavior.

5.3. SINGLE-LONGITUDINAL-MODE PERFORMANCE

Both active-active and active-passive schemes have been successfully employed to fabricate SLM semiconductor lasers with an MSR of 30 dB or more. In a simple active-passive scheme the external cavity was formed by placing a concave spherical mirror (radius of curvature about 200 pm) at a distance of about 200 pm from the laser facet (PRESTON, WOOLLARDand CAMERON [ 19811). The laser operated in a single longitudinal mode whose wavelength could be tuned over a 4 nm range by changing the drive current. The mode wavelength can also be tuned by changing the external cavity length by ;A. A tuning range of about 10 nm was observed by VAN DER ZIELand MIKULYAK [ 19841 for an external cavity length of 95 pm. It should be stressed that the mode wavelength changes in a discrete manner; as the length of the external cavity is varied over a distance of :A using a piezoelectric transducer, the mode hops from one FP mode to the neighboring one in succession.

111, § 51


207

In another active-passive scheme the external cavity consists of a short section (- 100-200 pn long) of graded-index optical fiber, which also acts as a lens (LIOU[ 19833, LIOU,BURRUS,LINKE,KAMINOW, GRANLUND, SWAN and BESOMI[ 19841). The front end of the lens is antireflection coated, whereas the other end has a high-reflectivity gold coating. Such coupled-cavity lasers have maintained SLM operation with an MSR of 30 dB or more even when modulated at 2 Gb/s. An active-passive scheme that has attracted considerable attention recently in the context of coherent communication systems uses a grating in place of the mirror for forming the external cavity (FLEMINGand MOORADIAN [ 1981a1, WYATTand DEVLIN[ 19831). This scheme leads not only to a narrow linewidth ( - 10 kHz) but also to a wide tuning range (> 50 nm) achieved by simply rotating the grating. In effect the grating acts as a narrow-band reflecting filter, and any suitable designed filter can be used in place of the grating (DUTTA, GORDON,SHEN,ANTHONYand ZYDZIK[ 19851). Recently several schemes have employed a Bragg reflector for this purpose (HAMMER, NEIL,CARLSON, DUFFYand SHAW[ 19851, BRINKMEYER,BRENNECKE, ZURN and ULRICH [ 19861, WHALEN,TENNANT,ALFERNESS,KORENand BOSWORTH[ 19861, OLSSON,HENRY,KAZARINOV, LEE and ORLOWSKY [ 19871). Such external Bragg reflector (EBR) lasers are expected to be useful for optical communication systems. Monolithic integration of the passive cavity with the semiconductor laser is desirable from the viewpoint of system applications. This approach, however, requires sophisticated processing steps with multiple epitaxial growths and often leads to a high threshold current because of weak coupling between the active and passive sections. In one scheme, monolithic devices had threshold currents of 60-70 mA and oscillated in a single longitudinal mode with an MSR of about 30 dB (MATSUDA,FUJITA,OHYA,ISHINO,SATO,SERIZAWA and SHIBATA [ 19851). In the active-active scheme C3 lasers have been studied extensively (TSANG [ 19851, COLDREN,EBELING,SWARTZand BURRUS [ 19841) and are capable of showing a wide tuning range while maintaining a single longitudinal mode. Figure 20 shows the longitudinal mode spectra of a C3 laser that showed a tuning range of 26 nm through 13 discrete mode hops of about 2 nm each. This is one of the largest tuning ranges achieved with the use of the coupled-cavity scheme. The current change AZL needed for successive mode hops is also shown as a function of the laser wavelength. Small values of AZL are needed when the controller section is biased below threshold. However, as I , increases, AZL needed for the next mode jump increases rapidly, which occurs because the carrier density begins to saturate as the controller

208

[I14 8 6


2.8I

1

24 2.0 I

a

-

1.6 -

E

22

aI

..

.

18.2

-25.4

--

27.7

17.0 16.5 ~

-

16.1

i-

A 1.49pm

20.8

j6.7

,6.3

I

I

I

I

I

0 2 4 6 8 1012 (A,+ NAX)

Fig. 20. Longitudinal-mode spectra of a C3 laser obtained by changing the current I, in the left section while the current I, in the right section is kept fixed. The current change AIL as a function of the mode number N is also shown. (After TSANG,OLSSON,LINKEand LOGAN[1983].)

approaches its threshold. The current step AIL required for a mode hop also depends on the relative lengths L , and L,. The long-long geometry was used in obtaining fig. 20. For a long-short device much larger values of AZ, would be required, since the FP mode of the short cavity has to shift by one mode spacing of the long cavity for a mode hop to occur. When the controller is operating above threshold, the carrier density is approximately pinned and its FP modes stop shifting with an increase in I,. Under ideal conditions no mode hopping is expected. In practice, however, occasional mode hopping occurs because of a partial clamping of the carrier density or because of temperature-induced index variations. Nonetheless, when both sections of a C' laser are biased above threshold, the device can maintain the same longitudinal mode over a considerable current range. This mode of operation is generally preferred from the viewpoint of mode stability.

0 6. Modulation Performance The preceding two sections described the performance of SLM semiconductor lasers under cw operation. However, when used in optical communi-

111, § 61

MODULATION PERFORMANCE

209

cation systems, such lasers are often modulated at frequencies as high as the laser is able to respond ( a few gigahertz). This section considers the modulation performance of SLM lasers. Since the side modes remain suppressed during direct modulation of a well-designed SLM laser (see 3), the theoretical analysis can be based on the single-moderate equations. These can be obtained from eqs. (2.18) and (2.19) by considering only a single mode. However, an important modification should be made for a realistic description of the modulation behavior. A characteristic feature of semiconductor lasers is that intensity or amplitude modulation (AM) is always accompanied by phase or frequency modulation (FM). The physical mechanism behind AM and FM occurring simultaneously in current modulation is related to the index change that invariably occurs when the optical gain changes in response to variations in the carrier population. The index change leads to a transient shift of the mode frequency, a phenomenon referred to as frequency chirping. The amount of frequency chirp is governed by a dimensionless parameter fl, defined as N

where fl = p' + ip" is the propagation constant given by eq. (2.5). This parameter plays a central role in determining the spectral linewidth (see § 7) and is often referred to as the linewidth enhancement (or broadening) factor (HENRY [ 19821, OSINSKIand Buus [ 19871). Typical values of fl, are in the range 4-6. Frequency chirp is incorporated into the rate-equation formalism by considering the dynamic variation of the phase @ associated with the intracavity optical field together with the photon and electron populations P and N. With this modification and by using eqs. (2.18) and (2.19), the single-mode rate equations become (AGRAWAL and DUTTA [ 19861, HENRY[ 19861) P = [G(1 - ENLP)- YIP+ R,, ,

(6.2)

. I

N = - - y e N - GP, 4

d = $,(G

- y)

.

(6.4)

In eq. (6.2) the gain G has been multiplied by a gain-reduction factor of 1 - cNLP.The dominant physical mechanism behind this nonlinear (powerdependent) gain reduction is spectral hole burning (TUCKER[ 19851, MANNING, FYEand POWAZINIK [ 19851, ASADAand SUEMATSU [ 19851, OLSHANSKY,

210


W1,§ 6

AGRAWAL [1987]). Even though the reduction is typically less than a few percent at power levels of 2-3 mW, it affects significantly the dynamic response of semiconductor lasers and should be accounted for. The nonlinear-gain parameter eNL governs this reduction and is 10 - ’. Equations (6.2)-(6.4) are suitable for modeling the dynamic response of a single-cavity laser such as a DFB laser. In the case of coupled-cavity lasers it is sometimes necessary to consider the rate equations for each cavity separately with additional feedback terms resulting from the intercavity coupling (AGRAWAL [ 1984a, 1985a], MARCUSE[ 19851). Making use of the modes of the composite cavity, it is also possible to consider a single rate equation for the photon population (MARCUSE and LEE[ 19841, LANGand YARN[ 19851) together with two rate equations for the carrier population in each cavity. For an active-passive type of coupled-cavity device the single-cavity rate equations of the form (6.2)-(6.4) can be employed after including the effect of external feedback through the concept of effective facet reflectivity (VAHALA and YARIV [ 19841, AGRAWAL and HENRY[ 19881). Under direct modulation the current I ( t ) in eq. (6.3) consists of two parts:

-

where 1, is the bias level and I,@) is the time-dependent modulation current. Depending on the relative magnitudes of I,, and Zm(t), two cases should be distinguished,which are referred to as the small-signal and large-signalmodula 0 and ,$ , < 0, the equation for the first Fresnel zone takes on the form

$=by

(3.17) Whereas for an elliptic stationary point the Fresnel zone is an ellipse (fig. 3.2a), for a saddle point it is confined by hyperbolas extended along the separatrix-asymptotes y ' / x ' = I $=I '121 $(, - ' I 2 ; that is, the Fresnel zone has endless tails of separatrices (fig. 3.2b). The hyperbolas approach the origin as far as a , and a,, which are given by the same expressions (3.14) as the semiaxes of the Fresnel ellipse (3.13). At first sight the infinite tails prevent the rays from being localized near saddle stationary points. A physical analysis of the problem indicates, however, that the region of actual formation of the field is, in fact, finite (YARYGIN

246

RAYS A N D CAUSTICS AS PHYSICAL OBJECTS

[IV, 8 3

[ 19701, ASATRYAN [ 19851) and that for a majority of applications the Fresnel volume of a saddle point may be described by the same equation (3.13) as that for an elliptic point. Essentially all the subsequent considerations in 5 3.2 will be devoted to substantiation of this point. In ordinary optical instruments and devices, saddle fronts are a seldom observed phenomenon. Nonetheless, situations allowing one to observe saddle singularities experimentally are not hard to pinpoint. First, such a front occurs when a divergent spherical wave passes through a collecting cylindrical lens: The wave remains divergent in the meridional plane and experiences focusing in the plane perpendicular to the lens axis. Second, a saddle phase function also emerges between the near and far foci of a strongly astigmatic lens. In both cases the saddle form of a phase function can be observed within a limited range: Once the wave has passed the region of focusing, the phase function of negative Gaussian curvature acquires a positive curvature and the saddle singularity becomes elliptic. Saddle points occur without fail in multipath propagation in inhomogeneous media. General considerations lead us to assert that almost one half of all stationary points must be saddle points because between any two extrema there is always a saddle region. As a final point we note that saddle fronts are created for fun with curvy mirrors. Thus saddle points are not less typical than elliptic points in inhomogeneous media and in reflection from nonplane surfaces. However, many important details of field forming near saddle points have failed to attract the attention of researchers. Let us look at equation (3.17) for the boundary of the Fresnel zone. In the variables 5 = x ' and q = y ' this equation rewrites as

,/m

d

m

*

IS(-n=I &). Corresponding to this width is the maximum length of tails L,,, x a;xa;y/Smln,where a,, and a&, are defined in (3.24). Both L,,, and S,,, are shown in fig. 3.2b. The boundary of Kirchhoff’s approximation corresponds to the value S,,, = Ao, for which L,, z af,a&,/l,.Letting a& x aiy z af z where R is the typical curvature radius of the phase front, yields L,,, x AoR/S,,,. For Lo = 0.5 pm, ,,,6 = 1001, = 50 pm, and R = 100 cm (a typical radius of curvature when experimenting on a laboratory bench), we obtain L,,, x 1 cm. This quantity is about uf/Snllntimes the distance between the hyperbola branches at the saddle point. With the preceding values of the parameters, Lmax/afx 15. The area of the truncated Fresnel zone (fig. 3.2b) may be estimated as

m,

or, with

= afX= a;,

(3.3 1) These estimates indicate that the actual area of the Fresnel zone is considerably under LL,, and is defined by the area of the comparatively small central portion. For the effective radius of the truncated Fresnel zone, defined by S , = na:fi, we obtain, with a& af, a;,

-

N

The preceding parameter values yield an estimated a,, = 2.1 la,, which indicates that long tails extending 15times the least transversedistancea; almost fail to participate in field formation. Accordingly, the contribution of the truncated Fresnel zones turns out to be rather moderate. The calculation of the integral (3.26) for the area qT containing a’ open saddle Fresnel zones

252

RAYS AND CAUSTICS AS PHYSICAL OBJECTS

(truncation is effected at the level bmi, for each zone) yields

or, with af,

= a;, = a;,

The parameter yT of (3.27) is plotted in fig. 3.3b as a function of the number of open Fresnel zones a’ = 2 I I / a at the aforementioned values of the parameters. For odd numbers of open Fresnel zones (a’ = 1,3,...; I I = +a,$n,* . *), yT reaches values around 22. This level is higher than yT = 4 for elliptic zones, but the difference is less striking than it is for the Fresnel zones with infinite tails.

s

s

3.3.3. Localization of rays by simple apertures For a wave having survived a simple-shape aperture (rectangle, rhombus, ellipse), the difference between elliptic and saddle points is reduced further: Interference maxima of yT(a) diminish and minima become “diffusive.” Consider the degree of localization of a ray corresponding to a saddle point in a rectangular area qT of transmission factor defined as follows: T ( x ’ , y ‘ ) = 1 for

Ix’I < A x ,

for I x ’ I > A x ,

=O

ly’I < A y , Iy’(> A , ,

where the sides of the rectangle, A, and A,, are proportional to a:, and aiY: A x =AA = a’ a;,

(3.33)

9

a;y

ah and a;y are given in (3.34), and the rectangular aperture proper is depicted

above the plot in fig. 3.4a. The general equations (3.26) and (3.27) yield yT

=

(3.34)

4[c2(cr’) t ~ ’ ( a ’ ) ] ,

where c o s ( ; ~ [ ~d) r ,

S(a‘)=

IV,5 31

FRESNEL VOLUMES OF RAYS

253

5 2

Fig. 3.4. Degree of localization of a ray in the vicinity of a saddle point as a function of relative aperture size a'. (a) For a rectangular aperture, (b) for a rhombus aperture, (c) for an elliptic aperture, and (d) for the Gaussian window.

Figure 3.4a plots the dependence of yT on the dimensionless parameter a' derived by YARYGIN[ 19701 and ASATRYAN[1985]. The quantity yT is seen to be greatest exactly at a' x 1.2, that is, when the rectangle sides A, and A, almost coincide with the Fresnel scales u& and a:,, respectively. The parameter values a;,2 = 0.6,that is, A, x 0.6aiX,corresponds to a value of yT = 4. It is important to note that we arrive at exactly the same result (3.34) for an a for a ' , suggestingthat both elliptic stationary point by merely substituting types of stationary points are almost indiscernible with rectangular holes. For a rhombus (fig. 3.4b) whose semidiagonalsA, and A, are collinear with the x and y axes, respectively, and

fi

A,.a&

= A,la;, = a'

,

254


[IV, 5 3

the degree of localization is derived as

(3.35)

The corresponding plot of yT versus a ’ , depicted in fig. 3.4b, is seen to be whereas yr = i corresponds to a;,z = 0.87. maximum at a’ = If an elliptic aperture with semiaxes ratio (3.33) is placed near a saddle stationary point, the dependence y T ( a ’ ) assumes the form plotted in fig. 3 . 4 ~ . It reaches a maximum at a’ = 1.26, and yT = 21 at a’ x 0.67. In all plots of fig. 3.4, yr is a maximum at a’ x 1, which corresponds to the in (rather than n) neighborhood of the stationary point.

fi,

3.3.4. Localization of rays by Gaussian windows The examples just discussed indicate that the localization of rays by passing the wave through simple-shape apertures considerably diminishes interference maxima and “erodes” zeros of yr(a’),but the interference effects are still clear cut. To damp out even weak manifestations of interference due to sharp edges of the apertures, it would pay to employ a Gaussian window with gradually falling transparency

(;Y

T ( x ’ , y ’ )= exp - -

-

ky

(3.36)

This solution has been proposed by BERTONI,FELSENand HESSEL[ 19711, who studied domains of influence in an absorbing medium. In optics a Gaussian window is easy to manufacture by vacuum deposition. In the millimetric and centimetric radio wave bands and in acoustics, Gaussian windows can be simulated by means of starshaped (iris) apertures. Assume that the localizing Gaussian window of an elliptic section is matched with the Fresnel scales in the sense that A,/ah = A,/ai,, = a’. Then as ameasure of localization, my, we obtain - 14n2 (a’)4 (1

+ + ~ ~ ( a ’ ) ~. ) - ’

(3.37)

The corresponding plot of yT as a function of a is depicted in fig. 3.4d. The ratio a’ = A,/aix = 0.79 corresponds to the condition of localization yr = i, that is a rather small neighborhood of the saddle stationary point. For the elliptic front the degree of localization is also defined by eq. (3.37)

IV, I 31


255

but with A J a , = A,,/a,,, = a substituted for a ‘ . The value yT = $ corresponds to a = 0.56. Thus the window dimensions that ensure the same value of yT for identical magnitudes of $xx and $,,y differ for saddle and elliptic points only by a factor of $.Of course, there can be no complete coincidence between these situations. Specifically, if we turn the Gaussian window (i.e., change the angle the major window axis makes with the x axis), then for the elliptic point yTis a maximum when the axes of the window and ellipse coincide and a minimum when they coincide with the axes of the hyperbola.

3.3.5. Unified dejinition of Fresnel volume for arbitrary stationary points The aforementioned numerical evidence demonstrates that for both elliptic and saddle points the domain of ray localization attempted by simple apertures and by Gaussian windows is confined to the Fresnel scales a , and a,, of the problem at hand. On the basis of this evidence, ASATRYAN[1985] has suggested that eq. (3.13) should be used not only for elliptic but also for saddle points. It will be recalled that the equation for the elliptic Fresnel zone (3.13) differs from the equation for the saddle Fresnel zone (3.17) in that eq. (3.13) involves the absolute values of the derivatives I $xxl andl $,J. We adopt Asatryan’s proposition and shall apply eq. (3.13) to both types of stationary points. Before considering arbitrary types of stationary points, we should observe that the zone of influence may be increased to L,,, z af2/6mi,and even up to Kirchhoff s limit L,,, x a:/&, in principle. However, this possibility is realized only in specific cases when the field at the point of observation is formed by some equiphase subarea from the “tailed” Fresnel zone (3.17). Such a situation occurs, for instance, in equiphase exitation of a narrow long slit extended along the separatrix. Another example results from the emission of cophase edge waves on an edge oriented along the separatrix. If we depart slightly from the specific situations just described, however, we are faced with a sharply contracted area of influence. In general, situations where the area of influence is defined by the tails occur rarely and should be treated as nontypical cases. In other words, one has to provide artificially a collection of factors necessary to observe the equiphase effect; these include a clear-cut saddle phase function and an edge or a slit oriented along the tails. In experimental conditions such exclusive combinations of factors can seldom be realized, and in most situations we can ignore the effect of tails altogether. Thus far, discussion has been confined to stationary points of the simplest

256


[IV, 5 3

type, in the vicinity of which the phase has a square (Morse) form (3.12). Once the observation point occurs on a caustic, the second derivatives at the stationary point vanish, and higher order terms, that is, cubic, quartic, and so on, need to be considered. Keeping in line with the general procedure, we specify the Fresnel zone by eqs. (3.5) and (3.6) on the condition that the long tails extended along the separatrices 5 = k $ = 0 will be truncated. The truncation procedure may be formalized if the phase 5 is represented as a sum of normal forms, for example, 3 = i1+ 5, (see, for [ 1980]), and then these normal forms are added example, BERRYand UPSTILL in magnitude. In this case in place of (3.6) we obtain

IS,l+ 1i21- a = O .

(3.38)

The transition from the saddle singularity (3.17) to eq. (3.13) is exactly this type of operation. By way of a simple example, consider a situation with ,$, = 0, but # 0. Equation (3.13) then gives way to

I&,

41

+ $1 Ij;yyyyf31 = in,.

$xx~xf2

The scale a, is defined, as before, by eq. (3.14), whereas for a,,, we obtain the estimate (3.39)

-

It can be readily verified that an aperture of width 2a,, or a Gaussian window made to a scale A!, a,, does localize the area of field formation. This entitles us to consider the area a, x a,, as the domain of localization of the ray.

3.4. FRESNEL VOLUMES OF RAYS IN ANISOTROPIC MEDIA

It is well known that in anisotropic media rays lose their functions as phase trajectories, that is, the lines normal to the phase fronts, but they retain their functions of energy transfer paths because the flux of energy is directed along the ray vector s = dr/dz [see $2.1.2, eq. (2.19)]. The energy functions of a ray are intimately connected with the interference nature ofwavefield forming at the observation point (BORNand WOLF[ 19751). Let the initial phase front 1(1' = const. coincide with the plane z = 0, as shown in fig. 3.5. The Huygens sources on this front emit waves uniformly in

IV, $31


251

Fig. 3.5. Fresnel volume of a ray in an anisotropic medium.

all directions. However, the field at the point of observation is formed by only a limited region of the plane z = 0, namely, by a domain around the origin of the ray vector s into the point r. Although the ray vector s does not coincide with the normal to the phase front p, the field-forming region is defined, as before, by the Fresnel reasoning outlined in 3 3.2 and 3.3. Figure 3.5 portrays the contour of the Fresnel volume of the ray (KRAVTSOV and ORLOV [ 1980al). Despite the relative simplicity of the aforementioned constructions, no detailed analysis of Fresnel zones and volumes in anisotropic media has been actually attempted.

3.5. FRESNEL VOLUME OF A QUASI-CLASSICAL PARTICLE TRAJECTORY

Although numerous reports have been devoted to the transition to the quasi-classical approximation, one aspect of this problem seems to have received inadequate treatment, namely, the spatial localization of the trajectory of a quasi-classical particle. This topic may be readily handled on the basis of Huygens-Fresnel representations on the nature of wavefunction forming. In observing the analogy between the geometrical optical ray and the classical trajectory, it would be natural to evaluate the localization of a quasi-classical particle path by analogy with the treatment of the Fresnel volume of a ray. Let S,,,be the action calculated along the classical path and Svirt the action

258


[IV, I 3

along a virtual trajectory connecting a point on the plane q with the observation point r. For the quasi-classical path the equation for the boundary of the Fresnel volume then may be represented in a form analogous to eq. (3.6) (KRAVTSOVand ORLOV[ 1981b]), namely,

1 s I - nh = I svirt - Srefl- nh = 0 , I

(3.40)

where h is Planck’s constant. All discussion associated with the need for saddle singularity tail truncation (see J 3.2 and 3.3) relates to this equation as well. The Fresnel volume defined in this manner represents the domain essential for the formation of the quasi-classical wavefunction. It can be interpreted as a measure of “thickness” of the quasi-classical trajectory. This thickness can, in principle, be measured by inserting a screen with an aperture or Gaussian window in the way of the particle beam. It is advisable to note the difference between the virtual trajectories (and, respectively, virtual rays) considered here and the virtual paths which occur in Feynman’s quantum mechanics. Essentially we discuss here only two-leg trajectories (rays), which transfer a wavefunction from the initial surface Q to the intermediate surface q and further to the observation point r (fig. 2.1). On the sections Q -+ 4 and q r the rays obey the classical equations of motion (2.5);they suffer a singlebreak on the surfaceq. Ifwe apply the Huygens-Fresnel representation twice, we would obtain the three-leg rays Q -+q , -+ q2 -+ r, showing kinks at the surfaces q , and q2. The Feynman virtual paths occur in the limit of infinitely large numbers of intermediate integrations. Although the two-leg trajectories bear only a remote resemblance to the general Feynman paths, they possess a rather important property; namely, the Fresnel volume bounded by the condition (3.40) [or eq. (3.5) for rays] includes all Fresnel volumes of multileg trajectories defined by analogy with eqs. (3.40) and (3.6). Accordingly, the Fresnel volume of the most primitive two-leg trajectories envelops the region of constructive interference of all Feynman paths. This fact gives additional evidence supporting the Fresnel volume as a region essential for forming a wavefunction at the observation point. -+

3.6. FRESNEL VOLUME O F SPACE-TIME RAYS

In a dispersive medium, pulses undergo dispersive alterations of shape, which may be described as resulting from space-time diffraction. The fieldforming region at a given space-time point (z, t ) may be determined from the familiar Fresnel considerations. To illustrate, for a constant frequency pulse

Iv, 8 41

259

HEURISTIC CRITERIA FOR APPLICABILITY O F GEOMETRICAL OFTICS

that propagates in a uniform medium with a dispersion law k Fresnel time interval is defined as

=

k ( o ) , the

(3.41) where g = (dk/dw)-’ is the group velocity of the wave, and z - z’ is the distance traversed by the pulse. A generalization of eq. (3.41)for the case of frequency-modulated pulses may be found in the works by KRAVTSOV and ORLOV[ 1980a,b].

8 4. Heuristic Criteria for Applicability of Geometrical Optics 4.1. NECESSARY CONDITIONS OF APPLICABILITY

The method of geometrical optics is an effective tool for the evaluation of wavefields in smoothly inhomogeneous and slowly nonstationary media. In spite of the extremely broad application of the method, which sometimes appears as “geometrical acoustics” or “geometrical seismics” for elastic media and which has a quantum-mechanical “twin brother” (the quasi-classical approximation), thus far the limits of applicability of this method in threedimensional problems have been established for some specific cases only. In the following discussion we formulate universal sufficient conditions for applying the method on a heuristic basis, which rests on the concept of the Fresnel volume of a ray. The Fresnel volume formalism is helpful in solving a number of related problems, such as the estimation of the field in regions of inapplicability of geometrical optics (see 8 5 ) and localization of caustics (see 9 7), to name just a few. It is common to confine the consideration to the zero approximation (2.13) and think of the limits of applicability of geometrical optics as coinciding with the validity bounds of the zero approximation. The necessary conditions of applicability of geometrical optics require that there should be no sharp variations in the zero-approximation amplitude A,, that the phase fronts be sufficiently smooth, which is equivalent to the momentum components pi

260


[IV, § 4

varying smoothly, and that the refractive index n change only slowly over the wavelength; namely, kIVAoI 4 A o ,

k/Vpjl e p ,

klVnl e n ,

(4.1)

where k = AO/2nn= ko/n. In view of eq. (2.12)the first of these conditions bounds the rate of divergence of the rays, klV,$i 4g whereas for the main radii of curvature of the phase front, R , , , , the first and second conditions yield

14 IR*,,l

(4.2)

’

The zero approximation of geometrical optics (2.13) obeys Helmholtz’s equation only approximately and is accurate to the terms of order j2 = l/(k,&)’. Letting u = uo + ii = Aoeiko*+ ii ,

where ii is a correction for the geometrical optical field uo, it is an easy matter to verify by (2.3) and (2.4) th& ii satisfies Aii + kgn2ii = - eiko*AAo,

(4.3)

where AAo is of the order of p z with respect to Auo or kin2uo. Although AAo exp(ik, $) is small, at large distances along the primary rays this may cause accumulation of errors associated with diffraction effects. (Strictly speaking, all phenomena leading to deviations from the laws of geometrical optics are conventionally referred to as diffraction phenomena.) The accumulating nature of the errors may be gleaned from the formal solution of eq. (4.3), ii(r)

=

-

s

exp[ik,$(r’)] AAo(r‘)g(r, r’)dV’ ,

(4.4)

where g(r, r’) is the exact Green function for an inhomogeneous medium, and d V‘ is the elementary volume. From the ray optical representation of the Green function (3.1), we may conclude that for adjacent ray paths the sum of the eikonals $(r‘) + &,rJ r‘) is practically the same. This enables us to factor exp {ikJ $(r’) + $Jr, r’)]} outside the integral, leaving in the integrand the nonoscillating factors AAo(r’) G(r, r’ ), which are responsible for accumulating errors. The accumulation of errors ensues also from the formal solutions of the transfer equation (2.4) for higher approximation amplitudes written as integrals

IV, $41

HEURISTIC CRITERIA FOR APPLICABILITY OF GEOMETRICAL OPTICS

26 1

over the ray path

where dz’ = da’/n and the initial amplitudes A: are tacitly assumed to equal zero. Equation (4.5) suggests that at sufficiently far distances the amplitude, say, of first order A I , can exceed the zero approximation amplitude A,, which is inadmissible. In any event the conditions (4.1) and (4.2) are necessary but not sufficient for the applicability of geometrical optics. Sufficient conditions must limit accumulating errors in some way or another. 4.2. UNIVERSAL SUFFICIENT APPLICABILITY CRITERIA

According to the Huygens-Fresnel principle, the field at the point of observation forms as a result of interference of the secondary wavelets that emanate from each point of the primary phase front. The crucial role in this process belongs to the first Fresnel zone because the secondary wavelets from the first Fresnel zone differ in phase at most by II and cannot mutually cancel, whereas the cumulative contribution of higher Fresnel zones is fairly small since it involves many oscillating pairs opposite in phase. This feature highlights a special role of the Fresnel volume, which realizes the physical concept of a ray. It was already noted in $ 3 that the subject of applicability limits of geometrical optics is closely tied with the validity conditions of the method of stationary phases. Essentially, in order to transfer from the integral representation (3.1) to the geometrical optical approximation (3. lo), it is necessary for the quantity M defined by eq. (3.7) and all its constituting factors (Aq, G and a+JaN,,) to be almost constant within the first Fresnel zone. This constraint must be satisfied in all the first Fresnel zones associated with the ray, that is, within the entire Fresnel volume of the ray, resulting in the following formulation of the principal criterion of applicability of geometrical optics : Criterion (i) The parameters of the medium and of the wave (amplitude and phase gradient) must not vary appreciably over the cross-section of the Fresnel volume. This condition assumes that afl%Aol 4 A 0 9

afl%nl < n ,

afIQ,I Q P , ,

(4.6)

where 5 is the differential operator transverse to the ray and related to the q plane, p N = a +/aNq, and a, is the transverse dimension of the Fresnel volume

262


[IV, 0 4

defined by eqs. (3.14) for both elliptic and saddle points. The choice of the direction of differentiation in (4.6) (a/ax’ or a / a y ’ )must be matched with the selected semiaxes a , or af,. Substituting the amplitude A,, expressed in terms of the divergence 2 into (4.6), yields also the constraints on the radii of curvature of the phase front,

Similar constraints are imposed on the polarization of the electromagnetic field. If fk is a component of the polarization vector (see $ 2. l), it must satisfy the inequality

Most problems are concerned with distances exceeding the wavelength A,, and the Fresnel radius in this case also exceeds A,. Therefore, the inequalities (4.6) and (4.7) are much more severe than their counterparts in the necessary conditions (4.1) and (4.2). If several rather than one ray arrive at the same point of observation from the same primary phase front, that is, the rays belong to one and the same ray congruence, the resultant field is described by the sum of the fields associated with individual rays. Near a caustic, where the rays converge strongly, the first of inequalities (4.6) breaks down due to both the unbounded growth of the amplitude and the growth of the Fresnel radius a,. Consider two rays belonging to the same caustic congruence, that is, touching upon one and the same caustic. If both rays pass through the same observation point, one of them can be shown to lie inside the Fresnel volume of the adjacent ray once the first inequality of (4.6) does not hold. It would be convenient, therefore, to formulate one more, auxiliary condition of applicability, which is a corollary of criterion (i), but which facilitates the analysis for a number of applications.

Criterion (ii) The Fresnel volumes of rays belonging to the same caustic congruence and arriving at one and the same point, must not appreciably overlap. One ray belonging to the Fresnel volume of another ray implies that the phase difference of these rays is less than II and the respective difference of the eikonals is less than +A,. When criterion (ii) is violated, the Fresnel zones of the two rays overlap appreciably. In writing the resultant field as the sum of two ray fields u = A,, exp(ik,+,) + A,, exp(ik,+d

9

(4.9)

Iv, $ 4 1

HEURISTIC CRITERIA FOR APPLICABILITY OF GEOMETRICAL OPTICS

263

the overlap area of the Fresnel zones is, in fact, accounted for twice. Therefore, criterion (ii) may be formalized as

The criteria discussed here have been employed in some form (most often implicitly) in many studies, starting with the fundamental studies of Fresnel. In a number of cases they allow for a rigorous substantiation, for example, where there is a possibility to compare the geometrical optics approximation with the exact solution of the wave problem. In such situations the suggested criteria arise as an ascertainment of the known facts. The new point to be stressed here is the universality and sujiciency of these heuristic criteria. The sufficiency and universality of criteria (i) and (ii) are confirmed by the fact that in all of the numerous specific cases known to us these criteria agree with other methods of determining the limits of applicability of the geometrical optics approximation, namely by comparison with the exact, asymptotic, or numerical solutions, by estimating the leading term discarded in the ray series (2.2), and so on. Relevant examples and comparisons are presented in the ensuing sections. Another new point is the purely ray-optical recipe proposed in f 3 for constructing the Fresnel volume in an inhomogeneous medium. With this recipe the aforementioned criteria are expressed in ray language. Therefore it would not be a great exaggeration to say that geometrical optics has acquired internal criteria of applicability.

4.3. STABILITY OF THE GEOMETRICAL OPTICAL SOLUTION WITH RESPECT

TO SMALL PERTURBATIONS

The criteria formulated in the previous section enable us not only to answer the key question posed, that is, the sufficient conditions of applicability of the ray method, but also to elucidate a number of related problems that offer independent interest. One of these problem is concerned with the stability of geometrical optical solutions under small perturbations of the parameters of the medium, interfaces, and/or the initial conditions of the problem. The essence of this evaluation can be explained with the following simple example. Consider a plane wave that corresponds to a bundle of parallel rays (fig. 4.la). If we subject the initial phase front to a weak, small-scale, periodic perturbation of amplitude 6 and period X , the ray structure is cardinally

264


jb= const.

[IV, § 4

4 = const.

~

Fig. 4.1. Although the parallel ray bundle (a) is considerably distorted under a small-scale sinusoidal disturbance (b), the wavefield remains almost intact for S g 1.

distorted (fig. 4.1b). The field as calculated by geometrical optics also emerges as substantially distorted. It is obvious, however, that for 6 < I , the actual field of the wave remains almost unchanged (only perturbations of order 6/Io will be evident). This example indicates that the geometrical optical approximation is unstable (or, more exactly, is very sensitive) with respect to small perturbations of the initial conditions. The same type of intability arises under weak, small-scale perturbations of the parameters of the medium and of the interfaces (phase boundaries). The resolution of the problem of the stability of geometrical optical solutions consists simply of the fact that weak, small-scale perturbations should not be described in ray language, since for these perturbations the conditions of applicability of geometrical optics are no longer valid. The point is that the "new" rays that appear in fig. 4.1 b as the result of the perturbation of the plane phase front are fictitious and nonphysical because already at comparatively small distances L from the screen the Fresnel volumes of the new rays grow to dimensions of order which exceed the period of the perturbation X . In this case geometrical optics is inapplicable.

m,

4.4. DISCRIMINATION OF RAYS

Another problem solvable on the lines of the Fresnel volume formalism is concerned with distinguishing rays. Assume that the point of observation is hit

RAY ESTIMATES O F FIELDS IN REGIONS OF INAPPLICABILITY

265

by two rays whose Fresnel volumes appreciably overlap in space. If these rays have no common caustic, the intersection of their Fresnel volumes does not inflict invalidation of the applicability conditions of geometrical optics but prevents these rays from being actually separated. For the general case the condition of actual indiscernibility of rays may be stated as IS,

-&I 5 n.

(4.11)

4.5. CONDITIONS OF APPLICABILITY OF SPACE-TIME GEOMETRICAL

OPTICS

By analogy with criterion (i) the conditions for applicability of space-time geometrical optics may be stated as (4.12) In agreement with these inequalities, the parameters of a one-dimensional wave A = A(z, 1) and o = o(z, t ) must be constant within the Fresnel interval q. In an analysis of plane or spatial nonstationary wave problems, the inequalities (4.12) should be employed with the corresponding conditions of 5 4.2. The conditions (4.12) pave the way for estimating the domain of inapplicability for a number of applications, for example, near the leading front of a pulse or in the vicinity of space-time caustics and foci. Computations of this type have been performed by KRAVTSOV and ORLOV[ 1980a,b].

8 5.

Ray-Optical Estimates of the Field in Domains of Inapplicability of Geometrical Optics

5.1. DOMAINS OF INAPPLICABILITY OF THE GEOMETRICAL OPTICAL

APPROXIMATION

The inequalities (4.6) can be rewritten in another form by incorporating the parameter A,

PN

n

266


UV, J 5

Whereas the inequality 6 4 1 corresponds to the domain of applicability of geometrical optics, the opposite inequality 6 >, 1 obviously corresponds to the domain where geometrical optics is inapplicable. It is desirable to adopt the intermediate situation of

6% 1

(5.2)

as the conditional boundary of inapplicability of ray theory, denoted by the symbol r. Under certain conditions the square of this parameter may be used as an estimate of the relative error of the first approximation of geometrical optics. To illustrate the point, the ratio of the first term in the expansion (2.2) to A , is A , / k , A o . The quantity A , can be estimated by means of (4.5); namely, let L be the distance traversed by the wave and L A be the characteristic transverse scale of amplitude variation, then by order of magnitudeA , x LAA, x LA,L; with A, L -x-. (5.3) kOA0 k o L i At the same time, for the first component in (5.1) we have the estimate Comparing this estimate with (5.2) and observing afx x leads to the conclusion that

6 % af/L,.

a

JLIk,

Under the conditions where the inapplicability of geometrical optics results from a caustic focusing of waves, it would be convenient to establish the boundary of inapplicability with the help of criterion(ii) by replacing the inequality symbol in (4.9) with the approximate equality sign, viz., 144 -

$21

=9 0 .

(5.5)

In the estimates that follow both the condition (5.2) and the simpler condition (5.5) shall be exploited.

5.2. ESTIMATES OF THE WAVEFIELD IN FOCAL AND CAUSTIC DOMAINS OF INAPPLICABILITY

At first glance geometrical optical calculations seem to be appropriate only in the domain of applicability of the ray method, where its error is small, that

IV, $51

267

RAY ESTIMATES OF FIELDS IN REGIONS OF INAPPLICABILITY

is, 6'4 1. In a number of cases, however, the ray approach can yield an estimate of the field that is correct in order of magnitude, although crude, in a region of inapplicability of the ray-optical method, specifically,in the vicinity of foci and caustics. The boundaries of caustic and focal zones can be determined with the condition (5.2), but one may also invoke the corollary (5.5) of criterion (ii). In what follows we assume that the boundaries of caustic and focal zones are known and we estimate the field inside these zones. 5.2.1. Focal field estimates from the values of the ray-optical field on the focal zone boundary In this case the field ufoc is to be estimated by the ray optical field (2.13) computed directly on the boundary r of the focal zone of inapplicability:

From general considerations only one may conclude that the accuracy of this estimation technique for the focal field should not be better than the order of magnitude, particularly since the ray optical field itself is in error of about 100% on the boundary of the focal zone of inapplicabilitywhere 6 = 1. However, the comparison of the heuristic estimates derived with (5.6) or with the related formulas (5.8) and (5.9) against the computations of the field in the domain of inapplicability by the diffraction formulae indicates that they seldom differ by more than a factor of 2. 5.2.2. Energy estimates based on the conservation of the energyjiux in a ray tube of jinite width Another method of estimating ufoc that differs in form, but is equivalent in essence, is based on the conservation of the energy flux in a ray under the additional assumption that the initial energy flux A n o = n o J A o ( 2 A sspreads o more or less uniformly over the focal (caustic) zone, that is, A n o = nO1uO1'AsO= nlufo,12Asfo,

(5.7)

Here, As,,, is the width of the ray tube, correspondingto the focal (caustic)zone (fig. 5.1), and Aso is the initial cross section of the ray tube corresponding to

268


[IV, $ 5

Fig. 5.1. Illustrating estimation of the focal field from energy considerations.

AsFoc.From ( 5 . 7 ) we obtain the estimate

which is very close to the estimate (5.6) as on the boundary of inapplicability

4 1 z nAsfi,,/nOAsO. 5.2.3. Energy estimates with the initial Fresnel section One more type of energy estimate relies on the concept of the Fresnel volume of a ray. The area of the first Fresnel zone AsF is taken to serve as the initial section of the ray tube. With this choice of the initial section we confine ourselves to exactly the portion of the initial field u" that actually forms the field at the focal point. Given that Asfinis the final section of the ray tube corresponding to the initial value AsF, then

This estimate is equivalent to (5.8), since from the very manner in which the ray constructions have been effected it follows that the ratios of the initial sections (As" and A s F ) and the final section (AsFocand Asfin)are identical: Aso - b o c AsF Asfin

The only difference is that it is assumed that in eq. (5.8) Asfocis given and Aso is computed, whereas in eq. (5.9) AsF is given and Asfinis computed.

IV, I 51

RAY ESTIMATES OF FIE1.DS IN REGIONS OF INAPPLICABILITY

269

A need for these estimates may arise, for example, in assessing the feasibility of nonlinear processes, such as harmonic generation, self-interaction, and breakdown in the regions of field concentration at foci and caustics. The efficiency of such estimates will be demonstrated in the next subsection.

5.3. ESTIMATING THE FIELD INSIDE INAPPLICABILITY DOMAINS BY

INTERPOLATING BOUNDARY VALUES OF RAY FIELDS

Let the domain of inapplicability N of the ray optical method be confined between two curves r, and r2embracing the regions of applicability M I and M2 where the ray fields differ, for example, in the number of rays. The crudest way to estimate the field in N is based on linear interpolation by the boundary values url and un. Let 5, be the coordinate corresponding to the boundary r,, then for any cross-section in N with a coordinate 5,

An example of such an interpolation is given in Q 8. The linear interpolation is by no means optimal, and we believe it to be the simplest means of estimating some intermediate values of the field. More efficient interpolations have been derived in solving similar types of problems. Still more efficient are uniformly asymptotic methods, which yield ray fields in regions M I and M, and smoothly match the boundary values url and U r 2 in the intermediate domain N.

5.4. GEOMETRICAL OPTICAL ANALYSIS OF THE WAVE PATTERN AS A

WHOLE

The geometrical optical analysis is of great interest where there is a need for a general, nondetailed assessment of the structure of a short-wave field to be used either in application estimates or as guidelines in analytical or numerical computations. Thus ordinary geometrical optics augmented by the Fresnel volume formalism enables one to carry out a multifaceted analysis of high-frequency fields, which not only evaluates their qualitative structure but also derives quantitative

270

[IV, 8 6


estimates of the fields in regions where geometrical optics is inapplicable. Such an analysis assumes that the following procedures will be carried out. 1. The geometrical optical field will be determined, including the evaluation of rays, phase fronts, caustics, and amplitudes. 2. The Fresnel volumes of the rays will be determined. 3. The domains of applicability and inapplicability of geometrical optics will be evaluated on the basis of the heuristic criteria. 4. The heuristic estimation of ray-field errors in the region of applicability of geometrical optics will be performed and the field amplitude will be estimated in domains of inapplicability. The aforementioned procedure, although overstated, is equivalent to solving the wave problem without recourse to solving the wave equation. All the steps involved can, of course, be relegated to a computer. The coresponding software to perform an “express analysis” of the wavefield, to our mind, would facilitate the solution of many application problems in optics, where usually the order of magnitude rather than the exact amplitude of the wavefield is essential.

6 6. Utility and Applications of Heuristic Criteria 6.1. DIFFRACTION OF WAVES IN A HOMOGENEOUS MEDIUM

6.1.1. Form of the penumbra region for a spherical wave obstructed by a

difracting half-plane

As a first application illustrating the utility of the heuristic criteria, we take the simplest problem on the width of the penumbra region in the diffraction of a spherical or cylindrical wave by a semi-indefinite plane x 0, z = 0, shown in fig. 6.1. If the source S is at a point with coordinates x = 0, z = - zo, the geometrical boundary between light and shadow will coincide with the positive branch of the z axis. Above and below the positive semiaxis there is a region of penumbra indicated by parallel hatching in the figure. The shadow zone is indicated by crosshatching. Let the conditional boundary of the penumbra region be described by a function x = f(z) such that for any point P on this boundary the Fresnel volume of ray SP touches the edge of the opaque screen, point 0 in fig. 6.1. The radius a, of the Fresnel volume section at the half-plane z = 0 can be determined with

-=

IV, § 61

UTILITY A N D APPLICATIONS OF HEURISTIC CRITERIA

27 1

Fig. 6.1. Illustrating the estimation of the shadow and penumbra regions upon the diffraction of a spherical light wave at a half-plane.

(3.16) by letting L , = z, L ,

=z

to obtain

From fig. 6.1 it will be readily seen that the distance x p of a point P from the z axis is given as

The lower branch of the boundary is similarly derived and is described by the equation x = - f ( z ) . The penumbra boundary (6.1) has the shape of a hyperbola with the rectilinear asymptotes x, = f (z + fzo)&. The half-width of the angular sector occupied by the domain of inapplicability is as follows:

In the light region, for all rays outside this sector, geometrical optics is valid at infinitely large distances. For these rays the error of the method of geometri-

212


[IV, 8 6

cal optics does not accumulate, in contrast to the rays that find themselves in the penumbra region. For a plane incident wave (z, + a),the boundary of the penumbra region In this case all rays propagating (6.1) takes the form of a parabola, x = from left to right sooner or later appear in the penumbra region. The validity of these heuristic estimates may be substantiated by comparing them with the evidence derived from diffractional computations, as outlined, for instance, in Chapters 8 and 11 of BORN and WOLF[ 19751. It is worth noting that eq. (6.1) is applicable not only to a spherical wave but, rather, to a wave having an arbitrary phase front ifz, is taken to be the local radius of curvature of the geometrical optical phase front.

6.

6.1.2. Formation of near and farfilds of a laser beam The followingexample is to emphasize the salient features of Fresnel volumes in the near field and far field of a laser beam. Consider a collimated beam of diameter D = 2b. Near a laser the geometrical optical approximation describes the bundle of parallel rays (projector beam) with the same amplitude distribution as that at the output laser aperture. In the near zone this approximation is applicable as long as, in agreement with criterion (i), the radius of the first Fresnel zone anear= ,/& is small compared with b, that is, as long as

b2

Z5-.

(6.3)

LO

In the far zone of the laser aperture the beam field is described by a directional spherical wave having a width of its radiation pattern of the order 8- Ao/b. Thus the characteristic scale of the field amplitude variation at a distance rfrom the output aperture amounts to 1 , r e - &r/b. According to eq. (3.16), when the observation point lies at infinity, the dimension of the Fresnel zone on a sphere of radius r is afar= f i r . Hence, the condition for applicability of geometrical optics assumes the form

-

Naturally, this condition coincides with the ordinary condition for the far field. Thus the geometrical optical description is applicable in both the far and near zones of the beam. We should stress, however, the difference in the initial conditions. In describing the near field, we impose the initial conditions at the

IV, § 61

213


laser output aperture, whereas in the far zone we take as the initial values the field of a directional spherical wave, that is, the far field already formed. In the intermediate zone where r b2/1,, neither of the described approaches is applicable. However, the average value of the amplitude in the intermediate zone can be estimated by using the value of the field near this zone or by way of energy estimations. In this case the value of the amplitude in the intermediate zone, averaged over the beam cross-section, will approximate the field amplitude in the primary beam.

-

6.1.3. Field in the vicinity of a lens focus Let us examine an ideal lens with an aperture of diameter 2b and a focal length F. This type of lens converts an incident plane wave into a convergent spherical wave with radius of curvature F. If the point of observation is at a distance z from the lens, the Fresnel radius in the plane of the lens z = 0 is

This expression can be most readily derived with eq. (3.16) for the spherical wave if we substitute z, the distance from the lens to the observation point, for L , , and substitute - F, the distance from the virtual source to the lens (the minus sign accounts for the convergent wave), for L,. A regular, although lengthy, way is to expand the phase = +virt - Jlref in powers of x ’ , as in $ 3. In so doing we not only obtain the radius of the first Fresnel zone on the lens but also the complete equation of the Fresnel volume, viz .,

4

Equation (6.5) indicates that as we approach the focus, that is z + F, the Fresnel radius grows without bounds. Once uf becomes equal to the aperture radius b, the geometrical optical approximation breaks down because the field formation region now involves the aperture edges, where the initial field amplitude suffers a step variation. Upon requiring a, ,5 b, we obtain the following condition from (6.5) for the distance from the observation point to a focus:

214


Fig. 6.2. Illustrating estimation of the longitudinal I,, and transverse I* dimensions of the focal region of inapplicability of geometrical optics.

The quantity I,, = &F2/b2is readily recognizable as the focal spot size. The ray A F arriving to the focus F i n fig. 6.2 from the edge hits the plane z = F - I,, at a point B elevated by

above the lens axis. This quantity is known to characterize the transverse dimension of the focal spot (BORN and WOLF [ 19751). We now estimate the field I ufocI at the lens focus on the basis of the energy considerations outlined in 3 5 . The energy flux no= (A0)2nb2through the lens is equated to the energy flux I ufoc12nl: through the focal spot (we are assuming that the energy flux at the lens focus is uniform over a circle of radius I , ) to obtain for the three-dimensional problem (6.9a) and for the two-dimensional problem 7

(6.9b)

IV, § 61


215

These estimates agree well with the values of ufOcfrom diffraction theory. To demonstrate, we make use of the single-term Green formula 1

u(p, 2) =

-271

{

a

(-) aZ

u"(p') -

eikoR

R

d'p'

the second line of which is valid at R = J ( p - p')' +.z' 9 A,: Here, p = (x, y ) is a two-dimensional vector in the plane z = constant, and the integral is taken over the plane z = 0. In the Fresnel approximation we have

(6.10)

In the case of a thin lens the initial field u0 in the plane z for p'

u " ( p ' ) = A"exp(*)

=o

=

0 has the form

F. The Fresnel volume is defined by the condition I $(x ‘ z ’ )I = ;Ao. Expanding the eikonal into a series in powers of X ’ and retaining, as usual, the terms up to quadratic ones yields for the boundary of Fresnel up to quadratic ones yields for the boundary of the Fresnel volume x ’ = f ( z ’ ) the equation X’ =

,/& F - z’ I ( z - z ’ ) / ( z - F ) ;

0 < z’ < z , F < z .

(6.13)

Unlike eq. (6.6), the difference F - z’ enters this expression as a modulus so that eq. (6.13) remains valid not only for z‘ < F but also for z‘ > F. The shape of the Fresnel volume for this case is illustrated in fig. 6.3. To the right of the focus the Fresnel volume is seen to be the same as if the point source were in the focus. According to eq. (6.13), as it approaches the focal spot, that is, for z’ + F, the Fresnel volume contracts to zero. A more detailed analysis indicates,

Fig. 6.3. Shape of the Fresnel volume when the observation point lies behind the focal spot.

IV, 8 61

UTILITY AND APPLICATIONS OF HEURISTIC CRITERIA

211

however, that for z’ x F the Fresnel volume has a thin waist of about 21, in diameter and -2l,, long, where geometrical optics is inapplicable. The existence of such a waist follows as a matter of common sense; namely, if we insert a screen with an aperture into the focal plane, the field at the observation point will “sense” the presence of the screen when the diameter of the aperture is less than the diameter of the focal spot 21,.

6.2. REFLECTION AND REFRACTION OF WAVES AT CURVILINEAR

INTERFACES BETWEEN TWO MEDIA

6.2,l. Conditions of applicability of the reflection formulas When a wave is incident on the curvilinear interface between two media, it gives rise to both reflected (u,) and refracted (u,) waves. In the geometrical optical approximation the amplitudes of these waves A, and A, at the interface S are connected with the amplitude of the incident wave Ai by the local relationships (6.14) where 9 and 5 are the coefficients of reflection (reflectance) and of transmission (transmittance), respectively. Because of the geometrical optical principle of locality, these quantities are defined by the formulas for a plane wave incident on the plane tangent to the interface between the two media at the point of reflection. The values of refractive indices involved are taken in the vicinity of the point of reflection. The quantities 9 and Yfor two types of polarization may be found, specifically in the work by BORNand WOLF [ 19751. The respective fields far from the interface may be derived by treating the amplitudes (6.14) as the initial values in the formulas of geometrical optics (see 5 2). The resultant expressions are referred to as the reflection formulas (FOCK [ 19651, DESCHAMPS [ 19721, FELSEN and MARCUVITZ [ 19731). In the following discussion we formulate conditions for the applicability of the reflection formulas and elucidate a few examples where these conditions break down. When the incident ray is not normal to the interface, the waveforming region of the reflected and refracted waves depends on the slant section of the ray’s Fresnel volume. Let us examine the waveforming region in the plane of incidence of the ray, as depicted in fig. 6.4. We denote by 2afthe cross-section size of the Fresnel volume of the reflected ray and by 6, = Oi the angle of reflection equal to the angle of incidence. In this notation the Fresnel radius at the

278


P

I

/

/

Fig. 6.4. Fresnel volume of the reflected ray.

interface is (6.15)

For the transmitted (refracted) ray, b, = a,/cos 0,. The conditions for the applicability of the reflection formulas boil down to the requirement of slowly varying parameters of the wave and interface within the region of length b,. If t denotes the arc length of the interface in the plane of incidence of the ray, then similarly with (4.6) the constraints on the incident wave parameters and on the geometrical parameters of the interface are

(6.16a) As specific consequences, these constraints require that

IV, § 61

219


(1) the reflectance and transmittance vary slowly: (6.16b) (2) the Fresnel radius b, be small compared with the radius of curvature a of the interface in the plane of incidence: b, /b the branches cease to be discernible, and the caustic loop becomes unobservable, that is, nonreal (fig. 7.4~).Thus the obvious condition for caustics to be distinguishable is

1, c lb .

(7.29)

7.4.4. Micromultipath rays and random caustics The concept of micromultipath propagation, which has become rather commonplace today (see, for example, FLA'ITI~,DASHEN, MUNK, WATSONand ZACHARIASEN [ 1979]), is conditional in nature and is employed to describe the corrugated character of the phase of each of the macro-rays incoming to the point of observation in a randomly inhomogeneous medium. Unlike macrorays micro-rays disappear once the random inhomogeneities have been removed. One may speak of the reality of micro-rays, that is, of their physical distinguishability, on the condition that the Fresnel volumes of these rays do not overlap. In a random medium with large-scale disturbances, this condition is met only in a region of "unsaturated" fluctuations where relative fluctuations of intensity are small, that is, 6Z/1< 1. KRAVTSOV [ 1968~1pointed out that unsaturated fluctuations occur where caustics are yet to form, whereas clear-cut caustics appear in the region of focusing where strong fluctuations of intensity, hZ/ZZ 1, are observed. The major statement of this paper is that there is a high probability, say 30% to

302


--1

l7

m

Fig. 7.5. Caustic pattern in a randomly inhomogeneousmedium. (I) Region of weak fluctuations, (11) region of strong fluctuations, (111) region of saturated fluctuations.

SO%, for caustics to appear exactly on approach to the region of strong fluctuations; the state of the art in the field of strong fluctuations has been elucidated by TATARSKII and ZAVOROTNYI [ 19801 and YAKUSHKIN [ 19851. In the region of saturated fluctuations caustics fill the space so densely (fig. 7.5) that the conditions of distinguishability (4.12) for individual rays and (7.29) for branches of caustics may break down and the concept of multiple micro-rays may be treated as a conditional reflection of the involved nature of the field of fluctuations. It should be noticed also that random caustics arise not only in randomly inhomogeneous media but also upon the reflection of rays by irregular surfaces. The key aspects of this problem, such as the density of caustics, critical exponents, and the like, have been analyzed by BERRY and UPSTILL [ 19801.

8 8.

Fresnel Volumes of Diffracted Rays and Applicability Limits for the Geometrical Theory of Diffraction

8.1. EDGE WAVES

8.1.1. Geometrical theory of difraction The geometrical theory of diffraction (GTD) deals with ray-type diffraction fields, that is, fields which allow ray treatment. If we do not intend to return

IV,5 81

FRESNEL VOLUMES OF DIFFRACTED RAYS

303

to quasi-ray generalizations of GTD, that is, to the uniform asymptotic descriptions of the field based on special diffraction functions, a typical expression for a diffraction field used in GTD (we again take up the scalar case for simplicity) has the form

D

Here, uo = A exp (iko$') is the incident field at the point of diffraction, that is. at the origin of the diffracted ray, $D is the eikonal on the diffracted ray counted from the point of diffraction, I) = $" + $D is the complete eikonal, yD is the divergence of diffracted rays, and D is the coefficient of diffraction determined by solving the respective canonical diffraction problem, for example, for the diffraction by a wedge or for a circular cylinder. The term O( l/k,,) represents the higher terms of the ray series that are generally ignored because they are difficult to derive. The geometrical theory of diffraction has been devised by KELLER[ 1956, 1957, 19581 (see also KELLER,LEWISand SECKLER[ 19571) and has rapidly advanced into the key positions in modem wave theory. The optical aspects of the theory have been elucidated by KELLER[ 19621. The up-to-date status [ 19731, of GTD has been examined in papers by FELSENand MARCUVITZ KOUYOUMJIANand PATHAK[ 19741,JAMES [ 19761, and in a superb textbook by VAGANOVand KATSENELENBAUM [ 19821. The diffraction of short wavelengths is considered by GTD as being constituted by individual local dispersions at edges, vertices, wedges, or convex formations, for example. It is assumed that at a distance from the dispersive point the field has a ray optical structure and is described by equations such as (8.1). In this approach we may invoke for each subsequent diffraction process one of the set of canonical (model or standard) solutions. The major advantage of GTD is that it provides a physical insight into the suggested recipes of field computation. Another intrinsic feature of the theory is the high accuracy of the field values it yields. It would not be an exaggeration to claim that one cannot compute the diffraction of short waves by nontrivial shapes without having recourse to the GTD techniques and related methods, such as the edge wave technique of UFIMTSEV[1962]. Recently the G T D techniques are being used increasingly for optical instrument design, specifically in integrated optics. Despite extensive application of GTD, no universal conditions of its applicability have been formulated until recently. Some external estimates of GTD

304


[IV, 8 8

applicability may be inferred from certain specific cases, allowing for rigorous or more accurate solutions than GTD can provide. Such estimates may also be obtained by comparison with the results of the numerical analysis or experimental evidence. However, this approach to GTD applicability estimation is not always feasible and does not have uniformly valid application. Internal conditions of GTD applicability require that the higher terms of the field expansion be small compared with the leading term in (8.1). These conditions can be derived for those problems that allow a GTD solution as a power series in inverse wavenumber k, (more accurately in powers of the inverse large dimensionless parameter of the problem, k,,L 9 1). We refer here to internal a posteriori estimates of applicability because initially we need to construct the solution (8.1) along with all correcting terms. The following discussion examines a heuristic approach to the evaluation of GTD applicability limits that relies on the representation of the Fresnel volumes of diffraction rays introduced by KRAVTSOV and ORLOV[ 1982al. Once the Fresnel volumes of diffracted rays have been established, we are in a position to formulate heuristic conditions for GTD applicability. These conditions require that the quantities which were used as constant parameters in the model diffraction problem be almost constant inside the Fresnel volumes. As a result, the criteria suggested in the next section are merely internal a priori conditions. In other words, applicability conditions can be formulated prior to solving the problem along GTD lines. We shall be concerned mainly with the applicability conditions for GTD as it has been devised by Keller, that is, in the form of nonuniform (ray) asymptotic descriptions for edge waves and creeping waves. However, many statements made in the following section are true also for the uniform asymptotic representation, which extend the boundaries of applicability of the ordinary GTD to, for example, penumbra regions, pericaustics, and the like. Because in the GTD approximation the field, in the general case, is defined as the sum of ordinary (geometrical optical) rays and d8ractional rays, in addition to the criteria discussed in f 8.1.2, we should be careful also to control the conditions for applicability of geometrical optics outlined in the preceding sections. 8.1.2. Fresnel volumes of edge rays We define the Fresnel volume for edge rays with the same equation (3.5) as that used for ordinary rays, recognizing here that the eikonals I+9virt and relate to edge rays.

305


Fig. 8.1. Fresnel scale b,corresponding to the edge ray.

Let us determine the width b,of the Fresnel volume along the curvilinear edge of a wedge in homogeneous medium (fig. 8.1). Let $O be the eikonal of the incident wave at the edge defined by the equation r = rO(t), 5 being the arc length along the edge, and let r(tSt)+ r be the reference edge ray, and r O ( < + ) r the virtual edge ray (tstand 5 are the values corresponding to the reference and virtual rays at the edge). In this case $ref=

$O[rO(t,t)I+ Ir - r0(tSt)I

$virt(t)

=

$O[r"(t)I + Ir - rO(t)I

so that eq. (3.5) defining the boundary of the Fresnel volume takes the form

F(t) = 1 $O(rO(t)l- $O[rO(t,t)l+ Ir - r"t)I

-

Ir - r0(5,t)l - 3

0 =

0*

(8.2) An approximate equation for the Fresnel volume boundary can be derived by expanding (8.2) in powers of the difference 5 - tst.The resultant approximate expression for the Fresnel volume dimension b, = I r O ( < )- r(tst)l at the edge is

306


[IV, 8 8

where R = r - r"(tst) is the radius vector of the reference ray connecting a stationary point at the edge with the point of observation. The diffraction angle p is determined from the condition that a diffracted ray cone is formed, cos p = I, T = I, * T, where I, = V 1(1' is the unit vector of the incident ray at the edge, 1, = R / R is the unit vector of the reference edge ray, and T = d r O / d tis the unit vector of the tangent to the edge. Finally, p1 denotes the principal radius of curvature of the phase front of the edge wave,

-

where pp = sin2p(Tdl,/d() is the radius of curvature of the incident front in the plane { I,, T}, p ; = 1 d2ro/dt2I is the radius of curvature of the edge, and N = p; 'd2r/dt2 is the unit vector of the principal normal to the edge. It is assumed that for a divergent front p , > 0 and pp > 0, and that they are negative otherwise. It is worth noting the values of b,that follow from eq. (8.3) for two limiting situations: for R 9 p1 , 1

=-@

sin p

for R

< p, .

According to (8.3), the Fresnel volume width b, becomes infinite at the caustic of edge rays ( R = - p1 > 0); therefore, in expanding (8.2) into a power series in t' - tSt, it is necessary to retain the next, following the square, term, that is, the cubic. Then

(8.5) In the general case when m( > 3) first derivatives of F(5) with respect to ( fall off to zero, we obtain

where am-3 a5m-3


307

P A

Fig. 8.2. Fresnel scale of on the shadow face of the wedge.

Equations (8.5) and (8.6) define the size of the Fresnel volume along the edge in the situation where the observation point occurs at the caustic of edge rays. In an inhomogeneous medium eq. (8.2) incorporates the eikonals Jl(ro((r)- r ) and Jl(rO( 0. Hence, for a < +A, the edge wave representation of the field becomes invalid in the entire space. 8.1.5. Independence of edge waves and the geometrical opticalfild The nonuniform geometrical theory of diffraction separates the overall field uR and edge uD fields, so that uGTD = uR + uD. This separation is feasible provided that the domains of formation for these fields are essentially different or, similarly, if the condition (8.15) is satisfied where $l,z are assumed to be the eikonals of the geometrical optical and edge fields arriving at the point of observation P. In accord with (8.15) the assumption that these fields are independent breaks down if k,, - $Dl x x. This approximate equality is assumed to be the conditional boundary of the penumbra region. Compare this estimate with that derived from the penumbral uniform asymptotic description, which expresses the field in the penumbra region through the Fresnel integral u into the sum of the geometrical optical

u x F ( ( ) = (in)- ’/’

d1 -,

S,u

exp(izZ)d z ,

(8.22)

the upper sign corresponding to the lit zone (the where ( = T lower sign to the shadow). Figure 8.4 illustrates the field structure in the penumbra region as defined by eq. (8.22). The dashed line plots the GTD approximation corresponding to the asymptotic behavior of the function (8.22), and the arrows indicate the limits of GTD applicability evolved from criterion (8.15): c,, = f,,h = f 1.77. The difference in the field at the penumbra boundary ,c = T 1.77 amounts to less than 5 % . These estimates, of course, agree with those for the size of the penumbra region derived in $ 6 by a Fresnel volume analysis for ordinary (geometrical optical) rays. 8.1.6. Independence of edge and vertex waves The representation of the field as an edge wave (8.8) is valid provided that within the Fresnel zone bf there are no bends of the edge contour, which can become sources of vertex waves. Accordingly, the reference ray must not approach on the edge to a bend closer than 6 , that is, A1 2 bp In particular, the condition for edge and vertex waves to be independent on diffraction of a

314


-3

-2

-I

0

Fig. 8.4. Comparison of the approximate GTD solution uOTD = I uR + uD I (dashed line) with the exact solution (8.22) 1 U J = IF({)\ (continuous line), to the problem on diffraction of a plane wave at a half-plane. The dotted line shows the interpolation produced by connectin the field values at the boundary of the GTD inapplicability domain {, = f

&.

- m.

plane wave incident normally onto the edge with a bend is A1 2 b, If an edge of length I,, is confined between two bends, the GTD formulas for edge waves are applicable only in the near zone of the edge, where b, ;5 lo. Specifically,for a cophase rectangular aperture this constraint implies that the GTD applicability for edge waves is limited by the distance R < li/Ao.

8.2. FRESNEL VOLUMES O F GRAZING RAYS

8.2.1. GTD approximation for grazing waves According to the geometrical theory of diffraction (KELLER[ 19561, LEVY and KELLER[ 1959]), the incident ray grazing the surface of a smooth body


IV, § 81

315

Fig. 8.5. Incident ray QQo, creeping ray QoPo, and diffraction grazing ray POPon diffraction of a wave at a contex body.

(fig. 8.5) at a point Qo gives rise to a creeping ray propagating along the geodesic. When this creeping ray breaks away from the surface, it becomes a diffraction grazing ray tangent to the body at a point Po. The field of the scalar diffracted wave associated with the grazing ray is defined as

where u" f

=

= A"

exp(ik,$')

is the field of the incident wave at point Qo;

yCreepygraze is the overall divergence of the diffracted rays including the

divergence of the creeping rays over the surface Ycreep = dC(P,)/dC(Q,), d C being the spacing between creeping rays of all degrees of closeness, and the = R ( 1 + R / p , ) , R = 1 POPI being the spatial divergence of grazing rays ygraze length of the ray from the breakaway point Po to the observation point P, and p1 being the principal radius of curvature of the diffracted-wave phase front; s is the length of the creeping ray (geodesic) measured from point Qo; D is the coefficient of diffraction; and a is the attenuation constant on the creeping ray. The quantities D and a depend on the local properties of the surface of the body and are determined from the solution of canonical diffraction problems, specifically for the diffraction of a plane wave at a circular cylinder, as

(8.24)

316


[IV, 8 8

where ats) is the radius of curvature of the creeping ray (geodesic), and 7; and 7 ; are the first zeros of the Airy function and its derivative: Ai( - 7;) = 0 and Ail( - z;) = 0. The upper rows in (8.24) relate to the Dirichlet boundary condition on the surface, and the lower rows relate to the Neumann condition. For a diffracted electromagnetic wave, by analogy with (6. l), we have

-

4-112 exp [ik,(s + R ) ] {e,(e:Eo)K(N)+ e , ( e y E o ) K ( D ) },

(8.25)

1

H = -eR x E . W

Here

'ii is the dyadic coefficient of diffraction defined as ct

D

=

K(N)e, 60 e:

+ K(D)e, 60 e y ,

where

D"qD) and a(N*D) are defined in (8.24), en and e: are the unit normals to the surface at points Po and Q,, respectively, e , = eR x en and eO, = lo x e: are the unit binormals to the creeping ray at the same points, lo = Vt,bo is the unit vector of the incident ray at Q,, and t?R = R / R is the unit vector of the diffracted ray at its breakaway point Po (fig. 8.5). 8.2.2. Estimates of the Fresnel scales We deduce first the longitudinal Fresnel radius b,. With reference to fig. 8.6 we examine the virtual grazing ray QNQ,P,P, which differs from the reference ray QQoP,P by having one bend. The boundary of the Fresnel volume will be determined, in agreement with the general rule, by an equation such as (3.5), which for this case takes the form F=IR' +Al-R,j -ido=O,

(8.26)

where R , = QQ,, R ' = QN, and A1 = NQ, is the arc measured along the surface of the body. Let /3 be the angle subtended by the arc Al, so that A1 = pa, with a being the radius of curvature at the point of tangency Q,. For a source rather distant from

317

FRESNEL VOLUMES OF D I F F R A f l E D RAYS

Fig. 8.6. Computition of the longitudinal b, and transverse a,Fresnel scales on wave diffraction at a convex body; the Fresnel volume (dotted) of a diffracted grazing ray.

the point of grazing Q, ( R $- a), we have R , - R ' z a sinp so that R' + A1 - R , x a(/? - sinp) z &p3. Equation (8.26) then yields p x ( ~ A , / u ) ' / ~ . It would be natural to identify the arc length A1 = pa, satisfying the condition (8.26), with the longitudinal Fresnel radius b,: bf --

=

( 3 2 , ~ ~ ) =' /(3~ n)"3a M ~

=

2.1 la ~, M

(8.27)

where M = ( $ k ~ ) '$-/ ~1 is the standard large parameter of the problem introduced by FOCK[ 19651. The calculation for the virtual ray QQ,P,LP, which has a bend on the right from the breakaway point P, (fig. 8.26), also leads to formula (8.27), but the radius of curvature a of the surface should now be referred to Po. Two transverse Fresnel scales can be assigned to the edge rays, namely, a, in the normal to the surface of the body (fig. 8.6) and c, in the plane tangent to the surface (fig. 8.5). The Fresnel scale a, transverse to the ray and the surface may be estimated as the largest distance between the reference ray QQo and the virtual ray QN (fig. 8.6), namely, af z ~ ( -l C O S ~- );up2

=

M

M

0 . 5 ( 9 , ? , 8 ~ ) '=/ ~( 3 ~ ) ~ =/ 4.45 ~ - . (8.28) k, ko

This scale defines the extension of the field-forming region in the normal to the surface. The general contour of the Fresnel volume near the surface of a smooth body is depicted in fig. 8.6. It should be noted that the transverse scale afcan be determined as the width of the caustic zone A surrounding the surface because the body itself is a caustic

318


[IV, 8 8

of grazing rays. This estimation derived from the width of the pericaustic, borrowed from 5 7.2, yields an expression fairly close to (8.27), namely,

-

The caustic origin of the scale a, gives its dependence on I, as a two-thirds power law, a, A i l 3 , whereas the ordinary geometrical consideration yields a,

-Iy2. The scales

a, and b, are physically significant because these characteristic distances depict the behavior of the Fock function V ( Z , Y, q), which describes

the diffraction on smooth bodies (FOCK[ 19651). As a demonstration, if 1 is the arc length measured along the surface of the body from the light-shadow boundary, and q is the distance from the surface, arguments of the Fock function are

M

5

r

2 = - x 2.11 - , a bf

rl Y = k0 - q % 4.45 - . M a,

(8.29)

In other words, a, and b, are natural scales of the wave problem under examination. Finally, the Fresnel scale cf in the plane tangent to the surface of the body may be derived from the general equation by Taylor-series expansion near the diffracted reference ray, namely, c,

=

+

JI"('

R

y,

(8.30)

PI

where R is the distance between the observation point and the breakaway point of the diffracted ray, and p, is the principal radius of curvature of the phase of the diffracted wave. It can be noted that p, enters the GTD formula and can be readily obtained from geometrical considerations. If the observation point is on the caustic of diffracted rays so that R = - p1, then c,, defined to a square approximation by the formula (8.30), becomes infinite. Therefore, in the vicinity of caustics it is necessary to retain higher terms in the Taylor expansion. An estimate of cf on a caustic may be obtained with eq. (8.30) on the condition that near the caustic IR + p1I z (apl/aalAa, where CJ is the distance along the front of the creeping wave. Then, assuming that eq. (8.30) is valid up to ACT- c,, we obtain

319

FRESNEL VOLUMES OF DIFFRACI'ED RAYS

IV, $81

whence

(8.31) Similarly, for the general case with the Taylor expansion of the eikonal difference $ = tjvirt- tjref,beginning with a term of order(Aop 2 , m > 1,we obtain +

(8.32) This expression allows for vanishing derivatives of p1 up to order m - 1. The necessary conditions of applicability for the GTD formulas (8.23) and (8.25) are equivalent to

(8.33)

&4L,

where L is the characteristic scale over which the features of the surface of the body and the parameters of the diffraction field are assumed to vary appreciably. The heuristic conditions of GTD applicabilityto diffraction at smooth bodies require slow variation of all the quantities characterizing the grazing waves within the Fresnel scales a,, b,, or c,, that is,

La,

b, 4 L, ,

c f 0: u(x, z ) = D ( 0 , ) exp(ik,xn, sin 0, - kz,,/n: sin'tl, - nf) .

(9.15)

To evaluate in the plane z = 0 the area that forms the field at the observation point ( x , z ) , we establish in this plane a Gaussian window of halfwidth w

332


centered on

[IV, § 9

5, as follows: (9.16)

The field u,surviving the window can be determined with the Kirchhoff integral (9.9) upon the substitution of the boundary field (9.10) multiplied by the window transmittance (9.16). Assuming that the window is sufficiently wide, we calculate the resultant integral by the saddle path technique to obtain U,(X,

(

z ) = u(x, 2) 1 + 2 ;:>"eXp(

-

(5st

w2

)

- { 1 + Y cos t $ ( x , Y)l)

$.I>

9

9

Y ) = l o ( & Y ) (1 + Y cos [ $ ( x , Y ) + $ 749

(3.23) (3.24) (3.25)

where y = 0 . 8 3 for ~ ~ integration over a f n phase shift. For these intensity measurements the phase is

(3.26) and the detected intensity modulation is

(3.27) For a phase shift other than in or fn, the phase can be calculated using

where phase shifts of - a , 0, and tl are assumed. There are many more permutations of these equations, but the foregoing are the most commonly used.

v, I 31

PHASE-MEASUREMENT ALGORITHMS

365

3.6. C A R R ~TECHNIQUE

In the previous equations the phase shift is known either by calibrating the phase shifter or by measuring the amount of phase shift each time it is moved. C A R R[ 19661 ~ presented a technique of phase measurement that is independent of the amount of phase shift. It assumes that the phase is shifted by a between consecutive intensity measurements to yield four equations

where the phase shift is assumed to be linear. From these equations the phase shift can be calculated using

and the phase at each point is

To calculate the phase modulo to yield

II, the

(3.34) preceding two equations are combined

For this technique the intensity modulation is

where this equation assumes that a is near in. If the phase shift is off by lo", the estimation of y will be off by 5 10%. An obvious advantage of the Carre technique is that this phase shift does not need to be calibrated. It also has the advantage of working when a linear phase shift is introduced in a converging or diverging beam where the amount of phase shift varies across the beam.

366

[V, 8 3

PMI TECHNIQUES

Equation (3.35) will calculate the phase modulo 2 n at each point in the interferogram without worrying about errors resulting from phase calibration difference across the beam.

3.7. REMOVAL OF PHASE AMBIGUITIES

Because of the nature of arctangent calculations, the equations presented for phase calculation are sufficient for only a modulo A calculation. To determine the phase modulo 2n, the signs of quantities proportional to sin $J and cos $J must be examined. For eq. (3.9) and for all techniques but Carrc's, the numerator and denominator give the desired quantities. Table 1 shows how the phase is determined by examining the signs of these quantities after the phase is calculated modulo in using absolute values in the numerator and denominator to yield a modulo 2n calculation. For the Carrt technique, simply looking at numerators and denominators is not sufficient to determine phase modulo 211 (CREATH[ 19851). In this technique the signs of quantities proportional to sin+ and cosq must be examined when using Table 1. One such set of quantities is (12

-

=

13)

[ 21, y sin a ] sin $J .

(3.37)

(I, + z3)- (I,+ 14)= [21,y cos a sin' a ] cos $J

(3.38)

Figure 5 shows the result of a modulo 2 n phase calculation with phase ambiguities that must be removed. Once the phase has been determined to be modulo 2n, the measured wavefront can now be reconstructed using an integration technique that sums up the phases to remove jumps between TABLE 1 Determination of the phase modulo 2n. Range of phase values

Numerator [sin 41

Denominator [cos 41

Adjusted phase

positive positive negative negative 0 positive negative

positive negative negative positive anything 0 0

4

0-4

n-4

i7r-n n-f n

n+$

2n-4 R

R

R-2 X

n

in

4.

;n

:n

v, 8 31

PHASE-MEASUREMENT ALGORITHMS

361

Fig. 5. (A)The results of a modulo 2 n calculation and (B) the same data after 2n phase ambiguities have been removed.

adjacent pixels of greater than n. The phase ambiguities due to the modulo 2 n calculation can be removed by comparing the phase difference between adjacent pixels. When the phase difference between adjacent pixels is greater than n, a multiple of 2 n is added or subtracted to make the difference less than n. For the reliable removal of discontinuities the phase must not change by more than n (;A in optical path) between adjacent pixels. As long as the data are sampled as described in the sampling requirements, the wavefront can be reconstructed.

3.8. FROM WAVEFRONT TO SURFACE

Now that the phase of the wavefront is known, the surface shape can be determined from the phase. The surface height H at the location (x, y ) is (3.39)

368

PMI TECHNIQUES

[V,5 4

TEST SURFACE

Fig. 6. Definition of the illumination and viewing angles of a surface.

where 1 is the wavelength of illumination, and 8 and 8' are the angles of illumination and viewing with respect to the surface normal (fig. 6). For a Twyman-Green interferometer this equation is simply

(3.40) This technique yields a direct measurement of the test surface relative to the reference surface. A more accurate measurement of the test surface can be made by measuring the errors due to the interferometer and subtracting them from the results (as shown in 5 7). The subtraction eliminates errors caused by aberations in the interferometer or from irregularities in the reference surface.

0 4. Measurement Example There are several different equations for calculating the phase of a wavefront from interference fringe intensity measurements. Even though all equations should yield the same result, some algorithms are more sensitive to certain system errors than others (CREATH [ 1986a1). An example illustrating different results from four different algorithms is shown in this section. A more detailed comparison of these algorithms is given in the section on simulation results. The data for this example were taken using an optical profiler with a flat mirror as the test sample. The mirror was tilted to have two fringes across the diagonal of the field of view. Results are shown for both a calibrated and an uncalibrated system to illustrate the inherent variations between different algorithms. A detector with noticeable nonlinearity was chosen. Figure 7 shows the intensity data taken using a Reticon 256 x 256 detector array with relative phase shifts of 90" between consecutive data frames. Five frames are shown where the first and the last frames should have a 360" phase shift between them. This shows

MEASUREMENT EXAMPLE

369

370 PMI TECHNIQUES

[V,§ 4

Fig. 8. Results of calculating phase using four different algorithms with the same fringe intensity data containing two fringes across a flat mirror. The interferometer is calibrated for 90" ofphase shift between data frames. All plots in figs. 8 . 9 , and I2 are on the same height scale.

Fig. 9. Results of calculating phase using four different algorithms with the same fringe intensity data containing two fringes across a flat mirror. The interferometer is miscalibrated for 82" of phase shift between data frames.

v, I 51

ERROR ANALYSIS

313

that the phase shifter is indeed calibrated, since frame A and frame E overlap very well. The spikes in the intensity data are due to either bad pixels in the array or dust and defects on the sample. The results of calculating the phase using four different equations with the same four sets of intensity data are shown in fig. 8. Similar calculations for a miscalibration of the phase shifter are shown in fig. 9 after new data were taken at the same location on the surface with a relative phase shift of 82 O . Some of the results for the miscalibrated phase shifter show an error with a sinusoidal dependence, whereas other do not. When the phase shifter is calibrated (fig. 8), much of the sinusoidal error goes away, but there is still some waviness noticeable in the calculated phases. The behavior of these errors will be examined in the next section.

8 5. Error Analysis The precision of a phase-measuring interferometer system can be determined by taking two measurements, subtracting them, and looking at the root-meansquare of the difference wavefront. For a well-calibrated system this result should be less than &A. However, this method does not tell us much about the actual accuracy of the measurement. Accuracy is normally determined relative to some standard. The measurement accuracy will be degraded by system errors such as miscalibration of the phase shifter, nonlinearities due to the detector, quantization of the detector signal, the reference surface, aberrations in the optics of the interferometer, air turbulence, and vibrations. Air turbulence and vibrations are dynamic variables that contribute to both the system measurement precision and the accuracy. By placing the interferometer on a vibration-isolated table, enclosing the beam paths, and taking data fast, the effects of vibration and air turbulence can be minimized. In order to achieve a &A measurement, the detector signal should be digitized to at least 8 bits, and the interferometer intensity should be adjusted to cover the full range of the detector. Errors caused by miscalibration of the phase shifter can be eliminated by careful calibration of the system. Errors caused by an inaccurate reference surface or aberrations in the interferometer optics can be subtracted out by the methods outlined in 3 7. However, some errors such as a nonlinear phase shifter or a nonlinear detector will limit the ultimate accuracy of the measurement. These errors are discussed in more detail in the following section, The choice of phase-measurement algorithm can reduce one error at the expense of others. A simulation comparing the behavior of different algorithms for phase-shifter and detector errors is shown in Q 6.

314

PMI TECHNIQUES

5.1. PHASE-SHIFTER ERRORS

Phase errors caused by inaccurate phase-shifter calibration can be minimized by adjusting the interferometer for a single fringe. However, with large amounts of aberration present, it may not be possible to obtain a single fringe. If a constant calibration error is present, the phase shift may be written as a’

=

a(l

+E),

(5.1)

where a is the desired phase shift, a‘ is the actual phase shift, and E is the normalized error. For phase stepping it has been shown that the errors in phase resulting from a calibration error or nonlinearity in the phase shifter will decrease as the number of measurements increases (SCHWIDER, BUROW, ELSSNER, GRZANNA, SPOLACZYKand MERKEL [ 19831). The same should be true for integrating-bucket techniques. For a consistent phase-shift error, such as a miscalibration, a periodic error is seen in the calculated phase, which has a spatial frequency of twice the fringe spacing (see fig. 9). Nonlinear phase-shift errors are not as easy to deal with or detect. A quadratic nonlinear phase-shift error can be written as a’ = a(l

+ &a).

In normal operation a nonlinear phase shifter will be partially compensated in the calibration of the interferometer by adding a linear bias to its movement. The most straightforward approach to calibration is to make sure that the phase shifter actually moves 2 n over a 2 K desired change in phase. This error term can be realized by adding a normalized linear compensation term of an equal and opposite amplitude to eq. (5.2). The phase shift is then replaced with a’ = a(1

+ &a- 8 ) .

(5.3)

This function minimizes the error caused by nonlinear phase-shifter motion. Nonlinear phase-shifter errors can be reduced by applying certain algorithms such as the Carre technique and the averaging-three-and-three technique described later; however, they cannot be eliminated.

5.2. PHASE-SHIFTER CALIBRATION

The value of the phase shift a can be determined in a number of ways. A good indication of a can be obtained by taking four frames of intensity data and using eq. (3.33) to calculate the phase shift at each detector point. The phase-shift

v, B 51

315

ERROR ANALYSIS

20,000

u)

.-c0

-

n m

m

u

10,000

0

L

0

a

E,

z

-4

0

80"

900

100"

Phase shift (degrees)

Fig. 10. Histogram showing the distribution of phase shlns tor a well-calibrated phase shifter. The distribution should be a narrow Gaussian centered around the desired phase shift.

controller should be adjusted so that the average phase shift is at the desired value and the spread in calculated phase shifts is small (see fig. 10). A simpler equation can be used when five intensity measurements are taken with the same phase shift a between them (SCHWEIDER, BUROW, ELSSNER,GRZANNA, and MERKEL [ 19831, CHENGand WYANT[ 1985a1). The phase SPOLACZYK shift is then calculated from

Another technique to determine phase shift involves extracting linear slices across a set of fringes for each of the phase shifts. These slices are then differentiated and the fringe peaks and valleys are found by interpolation to determine the amount of phase shift. A more visual means of adjusting the phase shift is to plot these linear sections of fringes for N + 1 intensity measurements with shifts in increments of 2 a / N . The (N + 1)th measurement should overlap the first measurement (see fig. 11). Alternatively,the actual phase shift can be determined over the shifter's range by observing a reference signal that indicates the phase difference between the two interferometer beams. The reference signal can be generated by splitting off some of the light from the reference arm of the interferometer and directing it into a small interferometer and separate detector (HAYES[ 19841). When a

316

PMI TECHNIOUES

Fig. 11. Calibration of the phase shifter by overlapping linear traces of the interference fringes for the fist and (N t 1)th data frames.

computer controls the phase shifter, the driving voltage for the device is usually stored as an array, which is read out each time the shifter is employed. A sinusoidal intensity modulation can be obtained by linearly changing the phase of the reference beam. However, the signal that provides a linear phase shift will not necessarily be linear because of shifter nonlinearities. The signal that provides a linear phase shift can be determined by using a reference detector (HAYES[ 19841) and measuring the actual phase shift obtained, and then calculating the proper shifter motion, or by doing a least-squares fit (Ax and WYANT[ 19871). If a phase-stepping technique is being used, the actual phase shift can be determined by the computer and reference detector each time the phase shift is changed. With this information each phase step can be corrected to use algorithms requiring equal phase shifts, or the actual phase shift for each data frame can be plugged into the least-squares calculation of eq. (3.6).

5.3. AVERAGING-THREE-AND-THREE TECHNIQUE

SCHWIDER,BUROW,ELSSNER,GRZANNA, SPOLACZYKand MERKEL [ 19831 proposed a technique of reducing errors that averages two phase measurements taken with a relative phase shift of in between the two measurements. One realization of this technique involves taking four measurements as in the four-bucket technique, calculating the phase using eq. (3.21) for the first three buckets, and averaging this with the phase calculated using eq. (3.21) with the last three of the four buckets. This procedure can be expressed mathemati-

v, § 51

ERROR ANALYSIS

317

Fig. 12. Example showing the improvement realized by averaging the phase calculated using the first three of four buckets with the phase calculated using the last three of four buckets.

cally as

(5.5)

This tevchnique has the advantage of being simple to calculate and yet has the ability to average out errors. Figure 12 shows sample data where the phase from the first three buckets is shown to have sinusoidal errors caused by a phase-shift miscalibration that is directly out of phase with the phase calculated from the last three buckets. When the phases from the first three and last three buckets are averaged, the sinusoidal error term is significantly reduced.

378

[V. § 5

PMI TECHNIQUES

5.4. DETECTOR NONLINEARITIES

A nonlinear response from a detector can introduce phase errors, which are especially noticeable if they are not consistent from detector to detector in an array. Many CCD-type detector arrays read out the odd and even rows through different shift registers. If the gains in the two sets of registers are not equal and nonlinear, bothersome errors arise that must be removed. When a detector has a second-order nonlinear response, the measured optical irradiance I ’ can be written in terms of the incident optical irradiance I as

I‘

=

I + &I2,

(5.6)

where E is the nonlinear coefficient. Expanding the detected irradiance of the fringe pattern, the interference equation (3.1) becomes I ’ = l 0 [ l + ycos($+ a ) + d , Z [ 1 + 2 y c o s ( $ + a ) + y2cos2($+ a ) ] , (5.7)

I’

=

I,[ 1 + &IO]+ I,[ 1 + 2~l,]ycos($ + a ) + d ; y 2 cos2($

+ a)] , (5.8)

I‘

=

I;, + I;; ycos($

+ a ) + + E [ Z o y ] 2 { 1 + COS[2($ + a ) ] } ,

(5.9)

where a indicates the phase shift for a particular exposure. The nonlinearity in eq. (5.9) will cause phase errors. When eq. (5.9) is substituted into the four-bucket calculation of eq. (3.16), the 2r$ dependent terms will cancel in the numerator and denominator. Once the third terms cancel, the coefficients of the other terms in eq. (5.9) only reduce the measured fringe modulation and will not affect the measurement. If, however, eq. (5.9) were substituted into the three-bucket calculation of eq. (3.21), the nonlinearities add and cause large

TABLE 2 Harmonics due to detection nonlinearities. Number of buckets

Harmonic order 2

3

4

5

6

I

8

9

10

11

v, § 61

SIMULATION RESULTS

319

phase errors. Thus, when a second-order nonlinearity is present in the detected irradiance, a minimum of four measurements is necessary to obtain an accurate phase calculation. For higher-order nonlinearities, STETSONand BROHINSKY [1985] have determined which orders of detection errors will affect the measurement for small numbers of phase steps. The dashes in Table 2 indicate which detection nonlinearity orders do not contribute to phase errors in the various algorithms. In most cases the largest distortions affecting the measurement are probably due to third-order harmonics, 11

=

I

+ E13,

(5.10)

so that five steps should be enough to reduce most effects of detector nonlinearities.

8 6. Simulation Results The experimental results shown earlier indicate that the results of PMI calculations depend on the algorithm used. The most desirable algorithm depends on the particular PMI system. In general, the more intensity measurements, the less error there will be in the calculated phase values. This section endeavors to find the best algorithm using the fewest number of measurements for systems that are susceptible to phase-shifter and detector errors. The techniques compared in these simulations are: (1) three-bucket technique of eq. (3.21); (2) four-bucket technique of eq. (3.16); (3) Carre technique of eq. (3.35); (4) averaging-three-and-three technique of eq. (5.5); and ( 5 ) a fivebucket technique using the synchronous detection outlined in eq. (3.11). The simulated error terms are plotted in fig. 13 for particular values of E in eqs. (5.1), (5.3), (5.6) and (5.10). Because the errors in the phase calculations are not symmetrical, both positive and negative errors are shown. The actual phase shifts with errors are plotted versus the desired degrees of phase shift. Note that the linear compensation error for the nonlinear error shifts the actual phase by 360" when 360" is desired, but intermediate values are wrong. In the integrating bucket techniques the phase shift is continuously moved along these values, but for phase-stepping techniques there are discrete values at the desired phase shifts. The detection nonlinearities show plots of the detector output signal versus the incident intensity, assuming a perfect detector behaves linearly. The simulations are performed by starting with a known phase function, which varies from 0" to 360" over 1000 data points. The fringe intensity

W m 0

(B)

NONLINEAR PHASE SHIFT SIMULATION

LINEAR PHASE SHIFT SIMULATION

Actual Degrees Shift

-0-

Calibrated

0-

+lO%Error

-9-

-10% Error

Actual Degrees Shift

Desired Degrees Shift

Desired Degrees Shift

(D) PND-ORDER DETECTION NONLINEARITY

3RD-ORDER DETECTION NONLINEARITY 1.oo

0.75

-*- Linear

-.-

3etected Signal

.o- +lo% Error

-0-

Detected Signal

o,50

+lo% Error -1. -10% Error

-10% Error

0.25 0.00 0.00

Incident Intensity

Linear

0.25

0.50

0.75

Incident Intensity

Fig. 13. Plots of error functions used in the computer-calculated error simulations.

1.00

V, § 61

38 1

SIMULATION RESULTS

measurements for the different techniques are then calculated using the appropriate equations by both the integrating-bucket technique and the phasestepping technique with an equivalent fringe modulation. If a phase-shift error is present, it is applied when the fringe intensities are calculated. Detection errors are added after the intensities have been calculated. Once the intensity data are fabricated, the phase is calculated from these data. The error in the calculation is the difference between the calculated phases and the original phase function. These simulations are performed for the five phase-measurement algorithms at 21 different error values ranging from - 20% to + 20%. Phase-shift errors in waves (number of wavelengths) are shown in fig. 14 for -18%

-

PZT CRLIBRATION ERROR

0.024

v1

? 0.012

3-Buckets

t

4-Buckets C a r k Exs

0.000

Rvg

L

w

3a3

-_

W

2-0.012

5-Buc k e t s

R L

-0.024 0.0

(B)-lO%

0.2 0.4 0.6 0.8 Distance i n F r i n g e s

1.0

2ND-ORDER PZT NONLIN. W/lB% LINEAR

3-Buckets 4-Bucke t s

5 --B- uc k- -et s-

0.0

6.2 0.4 0.6 0.8 Distance i n F r i n g e s

1.0

Fig. 14. Results for simulationofphase-shiftererrors with (A) - 10% linear error and ( B ) - 10% second-order error, compensated by + 10% linear error.

382

[V,§ 6

PMI TECHNIQUES

a simulated wavefront containing one fringe of tilt in optical path difference across the wavefront. All five techniques show a quasi-sinusoidal dependence in the error terms for both miscalibration and a nonlinear phase shifter. The frequency of the sinusoid is twice that of the input fringe intensity data. Figure 15 shows the calculated peak-to-valley (P-V) phase-shifter errors in waves plotted versus percent simulated error. The Came technique shows no errors for the linear phase-shift error because it is tailored for this situation. A miscalibration error can be reduced substantially by the averaging-three-andthree technique (see fig. 12). With nonlinear phase-shifter errors present, the

(A) LINEAR PZT CALIBRATION ERROR 0.10

i

0.08

*. 3-buckets

Calc‘ 0.06 Phase Error 0.04 (Waves)

0 . 4-buckeIs

CarrdEqs

0. Avg 363 buckel

+ SbUCkelS

0.02

0.00 m-m-m-m-a-m-20

-10

0

10

20

Simulated Error (“A)

NONLINEAR PZT CALIBRATION ERROR (WITH LINEAR COMPENSATION)

3-buckBIS

Calc. Phase Error (Waves)

0.4.buckels

m. CardEqs 0. Avg 363 buckel

.A- Sbuckets

Simulated Error (%)

Fig. 15. Simulation results of phase-shifter errors for all five algorithms with error amounts ranging from -20% to +20%.

v, § 61

383

SIMULATION RESULTS

CarrC equations reduce the error to give a slightly better result than averagingthree-and-three; however, they do not completely eliminate it. For all phaseshifter errors the three-bucket and four-bucket techniques show the most error. Even though these techniques yield the largest error, the magnitude of that error is less than $ of a wave for a 10% error in the phase shift. The errors from the detection nonlinearities are shown versus one fringe in optical path in fig. 16. These results show a more complicated frequency structure than the phase-shift errors. The frequency of the errors is four times the fringe frequency. Figure 17 shows plots of the second- and third-order detection nonlinear errors

v)

5 8.812-.-._ 4-Buckets

B

C a r g Esf

8.880~ ,-n L

RVg 3a3

t s--S-Buc - -ke -- -

0)

2 -8. 8 12-

r a

(B)

-10% 3 R D - O R D E R

8

.

8

DETECTION NONLINEARITY

2

4

7

1

v)

P 8.812.. E

x

a

-8.824 8.8

8.2 6.4 6.6 8.8 Distance in Fringes

1.6

Fig. 16. Results for simulation of detection errors with (A) - 10% second-order error and (B) - 10% third-order error.

PMI TECHNIQUES

384

(4 2ND-ORDER NONLINEAR DETECTION ERROR

0. 3-buckets


4-buckets

.a. Carf6Eqo 0. Avg 363 bucket 4-

Bbuckets

Simulated Error ("A)

3RD-ORDER NONLINEAR DETECTION ERROR

e. 3-buckels


0.

4-buckets Carm'Eqo

0. Avp 3h3 bucket

4-%buckets

Simulated Error (%) Fig. 17. Simulation results ofnonlinear detection errors for all five algorithms with error amounts ranging from - 20% to + 20%.

versus the percentage of simulated error. Note that these have been plotted on the same scale as the phase-shift errors. As shown in the last section, the four-bucket and five-bucket techniques are insensitive to second-order detection nonlinearities, and the five-bucket technique is insensitive to third-order nonlinearities, whereas the four-bucket technique shows very little sensitivity to the third-order nonlinearity. The three-bucket technique again shows the greatest sensitivity to the simulated errors. For both second and third orders the averaging-three-and-three technique and the Carre technique have similar

REMOVING SYSTEM ABERRATIONS

385

amounts of error, which are a factor of two better than the three-bucket technique, but not nearly as good as the four- and five-bucket techniques. Looking back at the experimental results in Q 4 (figs. 8 and 9), which had two fringes of tilt in the field view, it is obvious that the Carrt technique behaves the best for a miscalibration and that the three- and four-bucket techniques are the worst. When the system phase shifler is calibrated, there still is a waviness in the results most likely caused by a detection nonlinearity. Because the four-bucket results are best, most of the error is probably due to a detection nonlinearity. This simulation study has shown that certain phase-measurement algorithms yield better results in the presence of some system errors than others. It also shows that the P-V magnitude of these errors is well within & of a wave even with a 20% error. In general, the integrating-bucket methods give the same results as the phase-stepping methods except in the case of nonlinear phaseshift errors, where the integrating-bucket method is superior. The Carrt algorithm is the best to use for phase-shifting errors, and the four- and fivebucket techniques are best for eliminatingeffects due to second- and third-order detection nonlinearities. If speed of calculation is a factor, the averaging-threeand-three technique, which averages errors, can give passable results in all cases. This study also found that the more buckets used, the less error due to the system is seen in the result.

0 7. Removing System Aberrations Errors that reduce the measurement accuracy can be caused by reference surface errors or aberrations present in the interferometer. The elimination of these errors depends on the type of measurement being performed. In all cases a measurement of system errors can be made using a very good mirror as the test object. When the test mirror is of better quality than the optics contained in the interferometer, the wavefront measured from this surface will represent the aberrations in the interferometer. This aberrated wavefront can then be subtracted from subsequent measurements using the test objects. The reference wavefront must be measured again whenever the focus, tilt, or zoom of an interferometer is changed, because these factors change the amount of aberration. If random high-frequency errors are present, and a very good surface is not available, a more involved technique is needed, for which many phase measurements of the flat must be averaged (WYANT[ 19851). In between measurements

386

PMI TECHNIQUES

[V.§ 7

the test surface is moved by a distance greater than the correlation length of irregularities on the surface. This ensures statistically independent measurements. With the averaging of statistically independent portions of the test surface, the test surface errors are reduced by the square root of the number of measurements. The errors in the interferometer are then the major contributors to the averaged wavefront. Once the reference wavefront is obtained, it is subtracted from subsequent tests to improve accuracy. If only the root-meansquare (rms) of the test surface is desired, two measurements are needed (WYANT[ 19851). First, the surface is measured, and then the test surface is moved a distance greater than the correlation length and measured again. The rms of the difference in the phases for the two measurements yields J2 times the rms of the test surface independent of any errors in the reference surface. The preceding technique works well for surface roughness measurement, but when surface figure is measured, a different approach is necessary. Absolute calibration of any curved surface can be performed using three measurements (JENSEN[ 19731, BRUNING[ 19781). The test surface is first measured to yield the wavefront phase W o o ,then rotated 180” and measured again to obtain Wj800. A third measurement Wfocusis taken by translating the test surface until the apex of the test surface is at the focal point of the diverger (or converger) lens. The three measurements can be summed up as follows: WOO

= Wsurf + Wref

+

Wdiv

9

(7.1)

wdiv

5

(7.2)

w180°

= Wsurf + Wref +

wfocus

= Wref + %wdiv +

wdiv)

9

(7.3)

where “surf” indicates the surface we are trying to measure, “ref’ refers to errors due to system aberrations in the reference arm, “div” refers to errors in the test arm and the diverger lens, and the overscores indicate a 180” rotation of that wavefront contribution. When all three wavefronts have been calculated, the wavefront resulting from the surface under test is given by

Likewise, the aberrations in the interferometer including the reference surface and the diverger lens can be calculated using

Once the system aberrations are measured, this wavefront (Waber) can be subtracted from subsequent measurements to provide an accurate measure of the test surface as long as the reference surface, diverging optics, and imaging

v, I 71

REMOVING SYSTEM ABERRATIONS

387

optics are not moved. Flats can also be measured absolutely by comparing three surfaces (SCHULZand SCHWIDER [ 19761, FRITZ[ 19841). An example using eq. (7.5) to create a reference to subtract from test measurements is shown in fig. 18. Figure 18A shows a measurement taken

Fig. 18. PMI results of measuring (A) a spherical mirror using a 20 x microscope objective as the diverger lens, (B) the reference wavefront created using the technique of eq. (7.5), and (C) the measurement of the mirror minus the reference wavefront, which yields an absolute measurement of the spherical mirror.

388

PMI TECHNIQUES

[V,8 8

using a Twyman-Green interferometer with a PZT pushing the reference mirror as the phase shifker. Tilt and focus have been subtracted from all the wavefronts in the figure. A standard 20 x microscope objective was used to generate a spherical wave to test an F/2 mirror, which was good to $1 over the F/8 measurement area. Because the microscope objective was used at incorrect conjugates, a lot of spherical aberration in the measurement limited the accuracy of the measurement. A reference wavefront was generated using the technique of eq. (7.5) to produce a wavefront containing the aberrations in the interferometer and the microscope objective (fig. 18B). When this reference is subtracted from the measurement of fig. 18A, the result (fig. 18C) is independent of the interferometer accuracy. The results show a mirror with an rms surface quality of $1. Under normal circumstances better diverging optics would be used to test a spherical mirror; however, the best results will be obtained when a reference wavefront containing interferometer errors is subtracted from the test wavefront.

6 8. Applications of Phase-Measurement Interferometry Phase-measurement interferometry (PMI) can be applied to any two-beam interferometer, including holographic interferometers. Applications can be divided into three major types of measurement: surface figure, surface roughness, and metrology. The measurement of surface figure finds the shape of a test surface (usually an optical surface) relative to a reference mirror. Surface roughness measurements are interested in the surface microstructure and its statistics rather than the shape of the piece. Metrology is used to find out properties of a sample such as the homogeneity of an optical material or the deformation of a surface.

8.1. SURFACE SHAPE MEASUREMENT

The traditional measurements in interferometry have been to measure the shape of an optical component such as a lens or mirror. Surface figure measurements can also include desensitized tests such as using computergenerated holograms, two-wavelength holography, or two-wavelength phaseshifting interferometry to measure surfaces with large departures from the reference surface. The desensitized tests either use a reference surface close to the shape of the test surface created by a null lens or a computer-generated

v, $81

APPLICATIONS

389

hologram, or they synthesize a longer wavelength using interferograms from different wavelength measurements (two-wavelength holography and twowavelength phase-shifting interferometry) or by projecting fringes onto the object surface (digital moire). Surface figure measurements are used to test both smooth surfaces such as optical flats, spheres, and aspheres, and rough surfaces of machined parts with arbitrary shapes. Recent techniques for measuring surface figures are the use of a radial shear or lateral shear interferometer to measure aspheric surfaces (HARIHARAN, OREBand ZHOUWANZHI[ 19841, YATAGAIand KANOU[ 1983,19841). Since the measurement is proportional to the slope of the surface under test, the sensitivity can be varied by changing the amount of shear. Another shearing interferometer using PMI techniques utilizes a Ronchi grating to produce the shear (YATAGAI[ 19841). For aspheric surfaces a useful technique based on [ 19841) for two-wavelength holography (WYANT,OREB and HARIHARAN measuring surface shape is two-wavelength phase-shifting interferometry (CHENGand WYANT[ 1984,1985b1, FERCHER, H u and VRY[ 19851, CREATH, CHENGand WYANT[ 19851, CREATHand WYANT[ 19861, CREATH [ 1986bl). In this technique the phases measured at two different wavelengths are subtracted and then 2 II ambiguities are removed. This test synthesizes a wavefront as if it were measured at an equivalent wavelength I,, = A , I J Al - A21. The test sensitivity is varied by changing wavelengths. For measurements where the interferometer system is subject to vibration or air turbulence, many techniques have been developed to obtain all the information in a very short period of time. One method uses a grating to code the information on a single detector array (MCLAUGHLIN[ 19861). Another method uses a grating in a Smartt point-diffraction interferometer to produce four phase-shifted interferograms on four different detectors simultaneously (KWON [ 19841). Still other techniques produce the phase-shifted interferograms in different ways (SMYTHEand MOORE[ 19841). For measurements of very irregular surfaces PMI has been applied to both holographic interferometry techniques (WYANT, OREB and HARIHARAN [ 19841, HARIHARAN and OREB[ 19841, THALMANN and DANDLIKER [ 1984, 1985a]), and moire techniques (YATAGAI,IDESAWA, YAMASHIand SUZUKI [ 19821, WOMACK[ 1984a,b], BELLand KOLIOPOULOS [ 19841). In holographic techniques a hologram of the object is illuminated using either a different wavelength of illumination or by changing the angle of the reference beam. In moire techniques, fringes are projected on the object and viewed (with or without a reference grating) as the projected fringes are phase shifted using a detector array.

390

PMI TECHNIQUES

8.2. SURFACE ROUGHNESS MEASUREMENT

To measure the microstructure of a surface, an interference microscope is used to resolve 1-pm surface areas laterally with height resolutions in the Angstrom range. These instruments use the same phase-measurement techniques as the surface figure measurements. But rather than measuring a smooth, continuous surface figure, these instruments measure profiles of random-looking surfaces. Optical profilers have been used to test super-smooth optical surfaces as well as magnetic tape, floppy disks, and magnetic read/write heads. Optical profilers using PMI techniques have been based on different types of interference microscopes. Some have been based on the Nomarski type of microscope, which splits the illumination into two polarizations and compares one to the other, giving a measure of the surface slope (SOMMARGREN [ 19811, EASTMANand ZAVISLAN[1983], ZAVISLAN and EASTMAN[1985], JABR [ 19851). Others utilize objectives that compare the surface to a reference surface with a Mirau, Michelson, or Linnik interferometric objective (WYANT, KOLIOPOULOS, BHUSHANand GEORGE[ 19841, BHUSHAN,WYANTand KOLIOPOULOS [ 19851, WYANT, KOLIOPOULUS,BHUSHANand BASILA [ 19851). Still other optical profilers look more like standard interferometers than [ 19861, SASAKIand microscopes (MATTHEWS,HAMILTONand SHEPPARD OKAZAKI [ 19861, PANTZER, POLITCHand EK [ 19861).

8.3. METROLOGY

The use of PMI techniques in optical metrology is relatively new. In addition to measuring beam profiles (HAYESand LANCE[ 19831) and the homogeneity or index profile of an optical material (MOOREand RYAN[ 1978,1982]), these techniques enable the measurement of sample properties by determining the deformation of an object caused by temperature changes, pressure changes, stress, or strain, as well as studying the vibration properties of mechanical components. Holographic interferometry has traditionally been used to measure object deformations and vibrations, but only qualitative information was available. PMI has been applied to all types of holographic interferometric measurement, from looking at deformations (SOMMARGREN [ 19771,HARIHARAN, OREBand BROWN[ 1982, 1983a,b], DANDLIKER and THALMANN [ 19851, HARIHARAN [1985], KREIS[1986]) to measuring the amplitude and phase of an object

REFERENCES

VI

39 1

vibration (OSHIDA,IWATA and NAGATA [ 19831, NAKADATE[ 1986b], HARIHARAN and OREB[ 19861). Recently, PMI measurements have also been applied to speckle interferometry techniques, which are similar to holographic interferometry techniques but do not require the making of an intermediate hologram (WILLEMINand DANDLIKER [ 19831, NAKADATEand SAITO [ 19851, STETSONand BROHINSKY [ 19851, CREATH [ 1986~1,ROBINSONand WILLIAMS[ 19861, NAKADATE [ 1986a1).

8.4. FUTURE POSSIBILITIES

Future developments in phase-measurement interferometry will most likely be continuations of these applications to more irregular surfaces. Larger detector arrays will enable the measurement of steeper surfaces and allow holographic applications without the need to produce an intermediate hologram. Likewise, faster computers and parallel processing will allow us to view wavefront measurements in real time.

References AI, C., and J.C. WYANT,1987, Appl. Opt. 26, 1112. BELL,B.W., and C.L. KOLIOPOULOS, 1984, Opt. Lett. 9, 171. BHUSHAN, B., J.C. WYANTand C.L. KOLIOPOULOS, 1985, Appl. Opt. 24, 1489. BRUNING, J.H., 1978, Fringe scanning interferometers, in: Optical Shop Testing, ed. D. Malacara (Wiley, New York). BRUNING, J.H., D.R. HERRIOTT,J.E. GALLAGHER, D.P. ROSENFELD, A.D. WHITEand D.J. BRANGACCIO, 1974, Appl. Opt. 13, 2693. CARRB,P., 1966, Metrologia 2, 13. CHENG,Y.-Y., and J.C. WYANT,1984, Appl. Opt. 23, 4539. CHENG,Y.-Y., and J.C. WYANT,1985a. Appl. Opt. 24,3049. CHENG,Y.-Y., and J.C. WYANT,1985b, Appl. Opt. 24, 804. CREATH, K., 1985, Appl. Opt. 24, 3053. CREATH, K., 1986a, Proc. SPIE 680,19. K., 1986b, Proc. SPIE 661, 296. CREATH, CREATH,K., 1986c, Direct measurements of deformations using digital speckle-pattern interferometry, in: Proc. SEM Spring Conf. on Experimental Mechanics, New Orleans (Society for Experimental Mechanics, Bethel, CT)p. 370. CREATH,K., and J.C. WYANT,1986, Proc. SPIE 645, 101. CREATH, K., Y.-Y. CHENGand J.C. WYANT,1985, Opt. Acta 32, 1455. DANDLIKER, R., and R. THALMANN, 1985, Opt. Eng. 24,824. EASTMAN, J.M., and J.M. ZAVISLAN, 1983, Proc. SPIE 429, 56. FERCHER, A.F., H.Z. Hu and U. VRY, 1985, Appl. Opt. 24, 2181.

392

PMI TECHNIQUES

FRITZ,B.S., 1984, Opt. Eng. 23, 379. GREIVENKAMP, J.E., 1984, Opt. Eng. 23, 350. HARIHARAN, P., 1985, Opt. Eng. 24,632. HARIHARAN, P., and B.F. OREB,1984, Opt. Commun. 51, 142. HARIHARAN, P., and B.F. OREB,1986, Opt. Commun. 59, 83. HARIHARAN, P., B.F. OREBand N. BROWN,1982, Opt. Commun. 41, 393. HARIHARAN, P., B.F. OREBand N. BROWN,1983a, Appl. Opt. 22, 876. HARIHARAN, P., B.F. OREBand N. BROWN,1983b, Proc. SPIE 370, 189. HARIHARAN, P., B.F. OREBand ZHOU WANZHI,1984, Opt. Acta 31, 989. HAYES,J., and S. LANCE,1983, Proc. SPIE 429, 22. HAYES,J.B., 1984, Linear Methods of Computer Controlled Optical Figuring, Ph.D. Dissertation (Optical Sciences Center, University of Arizona, Tucson, AZ). Hu, H.Z., 1983, Appl. Opt. 22, 2052. JABR,S.N., 1985, Opt. Lett. 10, 526. JENSEN,A.E., 1973, J. Opt. SOC.Am. 63, 1313. JOHNSON, G.W., D.C. LEINERand D.T. MOORE,1979, Opt. Eng. 18,46. KOTHIYAL, M.P., and C. DELISLE,1985, Appl. Opt. 24, 2288. KREIS,T., 1986, J. Opt. SOC.Am. A 3, 847. KWON,O.Y., 1984, Opt. Lett. 9, 59. MASSIE,N.A., 1980, Appl. Opt. 19, 154. MATTHEWS, H.J., D.K. HAMILTON and C.J.R. SHEPPARD, 1986, Appl. Opt. 25, 2372. MCLAUGHLIN, J., 1986, Proc. SPIE 680, 35. MOORE,D.T., and D.P. RYAN,1978, J. Opt. SOC.Am. 68, 1157. MOORE,D.T., and D.P. RYAN,1982, Appl. Opt. 21, 1042. MOORE,D.T., and B.E. TRUAX,1979, Appl. Opt. 18, 91. MOORE,D.T., R. MURRAY and F.B. NEVES,1978, Appl. Opt. 17, 3959. MOORE,R.C., and F.H. SLAYMAKER, 1980, Appl. Opt. 19, 2196. MORGAN, C.J., 1982, Opt. Lett. 7, 368. NAKADATE, s., 1986a, Appl. Opt. 25, 4155. NAKADATE, S., 1986b, Appl. Opt. 25, 4162. NAKADATE, S., and H. SAITO,1985, Appl. Opt. 24, 2172. OSHIDA,Y., K. IWATAand R. NAGATA,1983, Opt. Lasers Eng. 4, 67. PANTZER, D., J. POLITCHand L. EK, 1986, Appl. Opt. 25, 4168. REID,G.T., 1986, Opt. Lasers Eng. 7, 37. ROBINSON, D.W., and D.C. WILLIAMS, 1986, Opt. Commun. 57, 26. SASAKI, O., and H. OKAZAKI, 1986, Appl. Opt. 25, 3137. SCHULZ, G., and J. SCHWIDER, 1976, Interferometric testing of smooth surfaces, in: Progress in Optics, Vol. XIII, ed. E. Wolf (North-Holland, Amsterdam) pp. 93. SCHWIDER,J.R., R. BUROW,K.-E. ELSSNER,J. GRZANNA, R. SPOLACZYK and K. MERKEL, 1983, Appl. Opt. 22, 3421. SHAGAM, R.N., and J.C. WYANT,1978, Appl. Opt. 17, 3034. SMYTHE, R., and R. MOORE,1984, Opt. Eng. 23, 361. SOMMARGREN, G.E., 1975, J. Opt. SOC.Am. 65,960. SOMMARGREN, G.E., 1977, Appl. Opt. 16, 1736. SOMMARGREN, G.E., 1981, Appl. Opt. 20, 610. STETSON,K.A., and W.R. BROHINSKY, 1985, Appl. Opt. 24,3631. TAKEDA, M., H. INAand S. KOBAYASHI, 1982, J. Opt. SOC.Am. 72, 156. THALMANN, R., and R. DANDLIKER, 1984, Proc. SPIE 492, 299. THALMANN, R., and R. DANDLIKER, 1985a. Opt. Eng. 24, 930. THALMANN, R., and R. DANDLIKER, 1985b, Proc. SPIE 599, 141. WILLEMIN, J.-F., and R. DANDLIKER, 1983, Opt. Lett. 8, 102.

VI

REFERENCES

393

WOMACK, K.H., 1984a, Opt. Eng. 23, 391. WOMACK,K.H., 1984b, Opt. Eng. 23, 396. WYANT,J.C., 1975, Appl. Opt. 14, 2622. WYANT,J.C., 1982, Laser Focus (May), p. 65. WYANT,J.C., 1985, Acta Polytech. Scand. Phys. 150, 241. WYANT,J.C., and K. CREATH,1985, Laser Focus (November), p. 118. WYANT,J.C., and R.N. SHAGAM,1978, Use of electronic phase measurement techniques in optical testing, in: Optica Hoy y Mailana, Proc. 1 Ith Congr. of the International Commission for Optics, Madrid, 10-17 September 1978, eds J. Bescos, A. Hidalgo, L. Plaza and J. Santamaria (Sociedad Espafiola de Optica, Madrid) p. 659. WYANT,J.C., B.F. OREBand P. HARIHARAN, 1984, Appl. Opt. 23, 4020. WYANT,J.C., C.L. KOLIOPOULOS, B. BHUSHAN and O.E. GEORGE,1984, ASLE Trans. 27,101. WYANT,J.C., C.L. KOLIOPOULOS, B. BHUSHANand D. BASILA,1985, J. Tribology, Trans. ASME 108,l. YATAGAI, T., 1984, Appl. Opt. 23, 3676. YATAGAI, T., and T. KANOU,1983, Proc. SPIE 429, 136. YATAGAI, T., and T. KANOU,1984, Opt. Eng. 23, 357. YATAGAI, T., M. IDESAWA, Y. YAMASHI and M. SUZUKI,1982, Opt. Eng. 21, 901. ZAVISLAN, J.M., and J.M. EASTMAN, 1985, Proc. SPIE 525, 169.


AUTHOR INDEX ADAMS,M.J., 219, 222 AGRAWAL,G.P., 165,168,170,174,178,186, 188,197,209-21 1,214,215,220-222,224, 225 AI, C., 376, 391 AIELLO,P., 113, 119, 158 AINSLIE,B.J., 165, 222 AKHMANOV, S.A., 137, 157 AKIBA,S., 180, 189, 192, 193, 222,224,225 AKOPYAN, R.S., 119, 157 ALFERNESS, R.C., 196, 207, 216, 223, 225 ALLEN,L.B., 197,222 S.P., 148, 160 ALMEIDA, ANDERSON, J.K., 215,224 ANDREWS, L.C., 27,99 ANTHONY, P.J., 188, 207, 220, 222 ANTREASYAN, A., 200,222 APOSTOLIDIS, A.G., 152, 159 ARAI,S., 180, 193, 195, 196, 225 S.M., 114, 116-120, 130. 134. ARAKELIAN, 136, 138, 146, 154, 156-159 ARECCHI,F.T., 13,99 ARMITAGE, D., 128, 129, 134, 157 ARNOLD,V.I., 297,298, 337, 345 ASADA,M., 209, 222 ASATRYAN, A.A., 246,253,255,345 ASHKIN,A., 108, 157 ASPECT, A., 55, 59, 64-66, 68, 99, 101 Au YEUNG,J., 147, 158

BASILA,D., 390, 393 BATRA,I.P., 127, 158 BELL,B.W., 389, 391 BELL,T.E., 165, 222 BENDJABALLAH, C., 31, 102 BERAN,M.J.. 24, 100 BERRY,M.V., 256,297,298, 302, 338, 345 BERTONI,H.L., 243, 254, 345 BESOMI,P., 199,207,224 BESPALOV, V.I., 137, 158 BHUSHAN, B., 358, 363, 390, 391, 393 BICKERS,L., 215, 222 BILGER,H.R., 81, 102 BISCHOFFBERGER, T., 153, 158 BJORK,G., 3, 51, 104 BJORKHOLM, J.E., 144, 158 BLAKE,J., 20, 25, 26, 100 BLINOV,L.M., 107, 110, 158 BORN,M., 9, 100, 139, 158, 169, 222, 231, 241,256,272,274, 277, 345 BOSWORTH,R., 196,207, 225 BOWERS,J.E., 213, 215, 222, 223 BoYD,G.D., 108, 157, 215, 222 BRAGINSKY, V.B., 71, 100 BRANGACCIO, D.J., 351, 358, 362, 391 BREKHOVSKIKH, L.M., 282, 345 BRENNECKE, W., 196,207, 222 BRETON,M.E., 3, 91, 104 BRIDGES,T.J., 216, 223 BRINKMEYER, E., 196, 207, 222 BROBERG,B., 189, 223 BROHINSKY, W.R., 379, 391, 392 BROWN,N., 390,392 BROWN,R.G.W., 84, 100 BRUNING,J.H., 351, 358, 362, 386, 391 BUHL,L.L., 216, 223 BULDYREV, V.S., 284, 345 BURGESS,R.E., 46, 100 E.G., 216,223 BURKHARDT, BURNHAM, D.C., 58,68, 100

B BABICH,V.M., 284, 345 BACH,A,, 63, 104 BAHADUR, B., 110, 158 BALLMAN, H.A., 108, 157 BALTES,H.P., 27, 101 BARAKAT, R., 20, 25, 26, 99, 100 G., 113, 119, 158 BARBERO, BARNES,P.A., 173,224 BARTOLINO, R., 119, 160

395

396

AUTHOR INDEX

BURNHAM, R.D., 180, 183, 184, 186, 192, 204, 225 BUROW,R.,374-376, 392 BURRUS, C.A., 176, 178, 199,207, 215, 216, 220, 222-224 BUTLER,J.K., 165,223 BUTORIN, D.I., 307, 345 B u m . J., 187, 190, 209, 222, 224

CRAIGJR, R.M., 197, 222 CREATH,K., 351, 354-357, 359, 366, 368, 389, 391, 393 CRESSER,J.D., 13, 55, 100 CRONIN-GOLOMB, M., 147, 158 CROWE,J.W., 197, 222 CSILLAG,L., 120, 121, 158 CURTIS,P.R.,342, 345

C

D

CAMERON, K.H., 198, 206,224 CAMPBELL JR, J.C., 216,223 CANTOR,B.I., 45, 96, 100, 104 CAPASSO,F., 81, 8 6 8 8 , 9 9 , 100, 104 CARLSON, N.W., 196, 207,223 CARMICHAEL, H.J., 13, 31.43, 59, 100 CARO,R.G., 147, 158 CARRE,P., 351, 358, 365, 391 CARROLL, J.E., 83, 100 CARRUTHERS, P., 71, 100 CASEYJR, H.C., 165, 168, 222 CAVES,C.M., 4, 71, 86, 100 CHANDRA, N., 63, 100 CHANDRASEKHAR, S., 107, 158 CHEMLA, D.S., 135, 157, 158, 161 CHEN,F.S., 108, 158 CHEN,H., 134, 147, 160 CHEN,K.L., 204, 222 CHEN,S.H., 20, 100 CHENG,Y., 134, 147, 160 CHENG,Y.-Y., 375, 389, 391 CHERNYI, F.B., 243, 282, 345 CHERRY, T.M., 337, 345 CHEUNG,M.M., 134, 153, 158 Y.S., 134, 157 CHILINGARIAN, CHILINGARIAN, YuS., 130, 156, 157 CHILLAG, L., 115, 116, 119, 134, 138, 161 CHO, A.Y., 86, 100 CHOI,H.K., 200,204, 222 S., 329, 345 CHOUDHARY, CLARK, N.A., 156, 159 CODE,R.F., 133, 146, 158 COHEN,L.G., 216,223 L.A.,200,202-205,207,215,222, COLDREN, 223 CONNELY, R.H., 197, 223 CONNOR, J.N.L., 342, 345 COOK,R.J., 58, 100 COPELAND, J.A., 176,223 Cox, D.R., 10, 19, 43-45. 100

DAGENAIS, M., 3, 13, 55, 100, 101 J., 66, 99 DALIBART, DANDLIKER, R., 389-392 DASHEN,R., 301, 346 DAITOLI, G., 97, 100 DAVIS,M.H.A., 93, 100 DAWSON,R.W., 216,223 DAY,C.R., 165, 222 DEGENNES, P.G., 107, 110, 114, 130, 158 DEJEU, W.H., 123, 124, 133, 158 DELISLE,C., 353, 392 DELWART,S.M., 128, 129, 134, 157 DENTAI,A.G., 176, 199, 223, 224 DESCHAMPS, G.A., 277, 327, 345 DEVLIN,W.J., 207, 221, 225 DEVOE,R.G., 64, 103 F., 60, 77, 100 DIEDRICH, DORODNITSYN, A.A., 337, 345 DREVER,R.W.P., 71, 100 DRONOV,I.F., 342, 345 P., 59, 100 DRUMMOND, DUFFY,M.T., 196, 207, 223 DUISTERMAAT, J.J., 297, 338, 345 DURBIN,S.D., 114, 116120, 134, 136, 138, 146, 153, 158 DUTTA, N.K., 165, 168, 170, 173, 174, 186, 188, 197, 207, 209, 215, 220, 222, 224 DZIEDZIC,J.M., 108, 157

E EASTMAN,J.M., 390, 391, 393 EBELING,K.J.,27, 100, 207, 222 EDA, N., 189, 222,225 EDEN,D., 128, 158 EFRON,U., 135, 161 EINSTEIN,A,, 16, 100 EISENSTEIN, G., 216, 223 EK, L., 390, 392 ELSER,W., 135, 160 ELSSNER,K.-E., 374-376, 392

AUTHOR INDEX

ENNS,R.H., 127, 158 ESPOSITO,F., 45, 102 EVERY, I.M., 63, 100

F FABRE,C., 70, 101 FANO, U., 12, 100 FARRELLY, D., 342, 345 FAYER,M.D., 126, 156, 158 FEDORYUK, M.V., 337, 347 FEINBERG, E.L., 108, 135, 158,243,282,346 FEINBERG, J., 147, 152, 158 Z.I., 240, 346 FEIZULIN, FEKETE, D., 147, 158 FELSEN,L.B., 243, 254, 277, 303, 324, 329, 336, 345, 346 FERCHER, A.F., 389, 391 FERNANDEZ, F.J.L., 139, 158 P., 83, 100 FILIPOWICZ, FINN,G.M., 134, 139, 143, 144, 159 FIRTH,W.J., 123, 158 B., 135, 147, 158, 161 FISHER, FISHER, R.A., 147, 158 FLATT~,S.M., 301, 346 FLEMING, M.W., 207, 219, 222 FOCK,V.A., 277, 282, 317, 318, 320, 346 FOKRESTER, A.T., 24, 25, 59, 100 FRANCK, J., 78, 100 FRASER,D.B., 108, 158 FREDERIKS, V., 114, 158 S., 68, 101 FRIBERG, FRITZ,B.S., 387, 392 FUH,Y.G., 133, 146, 158 FUJITA,T., 200, 207, 215, 222, 224 FUJITO,K., 215, 222 FURUYA, K., 189, 222, 223, 225 FYE,D., 215, 224 FYE,D.M., 209, 224 G GABOR,D., 9, 101, 145, 158 GAGLIARDI, R.M., 10, 24, 25, 91, 95, 101 I., 123, 158 GALBRAITH, GALLAGHER, J.E., 351, 358, 362, 391 GALLARDO, J., 97, 100 GARIBYAN, O.V., 134, 158 GARITO, A.F., 156, 158 GARLAND, C.W., 128, 158 GATTI,E., 13, 99 GAZARYAN, Yu.L., 290, 346

397

GEORGE,O.E., 358, 363, 390, 393 GERRITSMA, C.J., 133, 158 CHIELMETTI, F., 53, 101 GHIONE,G., 324, 336, 346 GIACOBINO, E., 70, 101 GIBBS,H.M., 144, 153, 154, 158, 161 GILBERT, E.N., 29, 101 GLAUBER, R.J.,3,13,34,35,38,63,101,102, 104 GLINSKI,J., 188, 190, 222 GNAUCK, A.H., 216,223 E.S., 134, 154, 158 GOLDBURT, GOODBY,J.W., 109, 159 GOODMAN, J., 16, 101 GOOSENS,W.J.A., 133, 158 E.I., 91,94,101,173,207,222,223 GORDON, COWER,M.C., 147, 158 P., 55, 59, 65, 68, 99, 101 GRANGIER, GRANLUND, S.W., 199, 207, 224 GRAY,G.W., 109, 159 GREIVENKAMP, J.E., 359, 360, 362, 392 GRINBERG, J., 135, 161 GROSHEV, V.YA., 290, 346 GRZANNA, J., 374-376, 392 GUHA,S., 144, 160 GUSEIN-ZADE, S.M., 337, 345

H HAGAN,D.J., 123, 159 HKGER,J., 13, 55, 100 HALL,J.L., 63, 104 HAMILTON, D.K., 390, 392 HAMMER, J.M., 196, 207, 223 HANBURY-BROWN, R., 3, 15, 101 HARDER, C., 217, 218, 225 HARIHARAN, P., 389-393 HARRIS, W.A., 78, 104 HAUS,H.A., 71,72,91, 101, 104, 189, 223 HAYES,J., 390, 392 HAYES,J.B., 375, 376, 392 HEIDMANN, A., 55, 70, 101 HELLWARTH, R.W., 135, 146, 147, 152, 158-160 HELSTROM, C.W., 91.95, 101 HENNING, I.D., 219, 222 HENRY,C.H., 91,95,99, 101, 178, 186, 188, 196, 197, 203, 204, 206, 207, 209, 210, 215-219, 221-224 HENRY,P.S., 178, 221, 223, 224 HERMAN, R.M., 115, 134, 149, 159

398

AUTHOR INDEX

HERMANN, J.A., 139, 159 HERRIAU, J.P., 152, 159 HERRIOTT, D.R., 351, 358, 362, 391 HERTZ,G., 78, 100 HERVET,H., 127, 159, 161 HESS,L.D., 135, 161 HESSEL,A,, 243, 254, 345 HILL,K.O., 134, 159 Ho, S.-T., 6, 35, 47, 53, 63, 85, 103 HOENDERS,B.J., 27, 101 HOFFMANN, H.J., 108, 127, 130, 159 HOLLBERG, L.W., 3, 64, 103 HONG,C.K., 58,68, 101 HORN,R.G., 126, 159 HOROWICZ, R.J., 70, 101 Hou, J.Y., 134, 139, 143, 153, 154, 159 HSIUNG,H., 127, 128, 130-132, 159 Hu, H.Z., 353, 389, 391, 392 HUICNARD, J.P., 133-135,147,152,159,160 Huo, T.C.D., 216,223

1 IDESAWA, M., 389, 393 IGA,K., 168, 200, 216, 222, 223, 225 IKEGAMI,T., 180, 223 IMAI, H., 189, 212, 223, 225 IMOTO,N., 3,51,65,71,72,83,84, 101, 104 INA,H., 356, 392 IPATOV,E.B., 342, 345 IPATOV,E.V., 342, 346 ISHIKAWA, H., 212,223 ISHINO,M., 200, 207, 224 S., 215, 222 ISHIZUKA, ISOZUMI, s.,212, 223 I T A Y A , Y . , ~ ~102,180,188,193,223,225 ,~O, IWATA,K., 391, 392 J JABR,S.N., 390, 392 JACKIW,R., 90, 101 JAIN,R.K., 135, 159 JAKEMAN, E., 14,20,25,27,43,55,59,61,68, 69, 73, 84-86, 91, 100-102, 104 JAMES,G.L., 303, 346 JANOSSY, I., 120, 121, 158 JAVANAINEN, J., 83, 100 JAYARAMAN, S., 107, 160 JEFFERSON, J.H., 68, 101 JEN,S., 156, 159 JENSEN,A.E., 386, 392

JOHNSON,G.W., 355,392 JORDAN,P., 78, 100 K KABANOV, Yu.M., 93, 101 KADOTA,T.T., 94, 101 KAMINOW, I.P., 178, 199, 207, 224 KAMITE,K., 212,223 KANOU,T., 389, 393 KAPLAN,A.E., 144, 153, 154, 158, 159 KARAIAN, AS., 130, 157 KARAL,F.C., 324, 346 KARN,A.J., 130, 134, 154, 159 KARP,S., 10, 24, 25, 91, 95, 101 KASPER,B.L., 178, 216, 223, 224 KATSENELENBAUM, B.Z., 303, 348 M., 134, 159 KAWACHI, KAWASAKI, B.S., 134, 159 KAZARINOV, R.F., 196, 197, 203, 204, 206, 207,215,221,223, 224 KELLER,J.B., 229, 303, 314, 324, 327, 346, 347 KELLEY,P.L., 34, 101 KHOKHLOV, R.V., 137, 157 KHOO, I.C., 114-117, 119-121, 126-128, 130-132,134,136,139,143,144, 146, 147, 149, 151, 153, 154, 159-161 KIELICH,S., 107, 130, 160 KIHARA,K., 212, 223 KIKUCHI,K., 220, 223 KIMBLE,H.J., 3, 13, 43, 5 5 , 63, 101, 104 KISHINO,K., 168, 180, 193, 195, 196, 225 KISLINK,P.P., 197, 223 KITAEVA,V.F.,115, 116, 119-121, 134, 138, 158, 161 KITAGAWA, M., 3, 51, 104 KITAMURA, M., 180, 192,223 KLEIN,M.B., 135, 159 KLEINER,W.H., 34, 101 KLEINMAN, D.A., 197, 223 KNIGHT,P.L., 4, 102 KO, J.S., 178, 199, 224 KOBAYASHI, K., 180, 192, 223 KOBAYASHI, S., 356, 392 KOCH,T.L., 202-205,213-216, 222-224 KOCHAROVSKY, V.V., 236, 348 KOCHAROVSKY, V L . ~ .236, , 348 KO~OUSEK, J., 3, 102 KOENIG,H.G., 197,222 S., 189, 223 KOENTJORO,

AUTHOR INDEX

KOGELNIK, H., 91, 95, 101, 147, 159, 180, 182, 185, 197,223 KOJIMA,K., 219, 223 KOLIOPOULOS, C.L., 358,363,389-391,393 KOMORI,K., 196, 225 KOMPANETS, I.N., 134, 158, 160 KOPPEL,D.E., 20, 26, 102 KOREN,V., 196,207, 225 KOROTKY, S.K., 216, 223 KOSONOCKY, W.F., 197, 223 S., 221, 224 KOTAJIMA, KOTAKI,Y., 212, 223 KOTHIYAL, M.P., 353, 392 KOUYOUMJIAN, R.G., 303, 308, 346 KOVALEV, A.A., 130, 160 KOYAMA,F., 180, 189, 193, 195, 196, 214, 216, 222, 223,225 Yu.A., 229-231, 234, 236, 240, KRAVTSOV, 257-259,265,288,290,291,297,301,304, 324,327,337,338,340,341,343,346,347 KREIS,T., 356, 390, 392 KRESSEL,H., 165, 223 KROO,N.,115, 116, 119-121, 134, 138, 158, 161 KRYUKOVSKII, A S . , 342, 347 KUMAR,P., 6, 10, 35, 47, 53, 63, 64, 85, 86, 102, 103 KUROIWA, K., 180, 223 KWON,O.Y., 389,392 KWON,Y.S., 134, 160 KYUMA, K., 219, 223

L LALANNE, J.R., 107, 130, 160 LAMACCHIA, J.T., 108, 158 LAMBJR, W.E., 9, 12, 103, 173, 216, 224 LAMPERT,M.A., 74, 81, 101 LANC,R.J., 210, 223 LANCE,S., 390, 392 LANCER,RE., 337, 347 LARCHUK, T., 78, 79, 104 LASHER,G.J., 197, 223, 224 LAX,M., 34, 101, 178, 216, 218, 219, 223 LAZAR,A.A., 93, 101 LEDOUX,I., 152, 160 LEE, H.J., 196, 207, 215, 221, 224 LEE, T.P., 176-178, 204, 210, 220, 223, 224 LEINER,D.C.,355, 392 LEITH,E.N., 134, 145, 147, 148, 160 LEMBRIKOV, B.I., 115, 160

399

LENSTRA,D., 58, 101 LEUCHS,G., 13, 55, 100 LEVENSON, M.D., 64,71,73,101, 103 LEVENSTEIN, H.J., 108, 157 LEVY,B.P., 314, 347 LEWIS,R.M.,303, 346 LIN, C., 199, 215, 224 LINKE,R.A., 178, 199, 207, 208, 213, 214, 216,221,223-225 LIOU,K.Y., 199,200, 207, 224 LIPPEL,P.H., 156, 160 LITTLE,A.G., 13, 104 LIU, T.H., 120, 121, 134, 139, 143, 144, 151-153, 159, 160 LOGAN,R.A., 199, 200, 208, 224, 225 LOPEZ,F.J., 134, 160 LOUDON,R., 4, 1 3 , 2 4 , 3 1 , 3 4 , 3 5 , 3 7 , 4 5 , 5 3 , 63, 101-103 LOUISELL,W.H., 58, 102 LUDWIG,D., 229, 291, 337, 340, 347 LUKIN,D.S., 342, 345-347 LUTHI,B., 128, 160 LYAKHOV, G.A., 156, 157

M MACHADOMATA,J.A., 34, 103 MACHIDA,S., 3, 51, 65, 71, 72, 83-86, 90, 102, 104 MACKENZIE, H.A., 123, 159 MADDEN,P.A., 156, 160 MAEDA,M.W., 64, 102 MAKINO,T., 188, 190, 222 MALLICK,S., 152, 159 MANDEL,L., 3, 5 , 9-13, 15-17, 43, 46, 49, 55-58, 68, 70, 77, 97, 100-103 MANDEL,P., 153, 158 MANNING, J., 209, 224 MARBURGER, J.H., 137, 160 MARCUSE,D., 176, 177, 204,210,223,224 MARCUVITZ, N., 277, 303, 346 MARINANCE, J.C., 197, 224 MARTIN,B., 107, 130, 160 MARTIN,G.. 146, 160 MARTIN-PEREDA, J.A., 134, 160 MASLOV,V.P., 229, 324, 337, 347 MASSIE,N.A., 351, 392 MATSUDA,K., 200,207, 224 MATSUDA,M., 212,223 MATSUO,K., 33, 52, 99, 102, 104 MATSUOKA,T., 179, 180, 187, 188, 193, 223-225

400

AUTHOR INDEX

MATSUSHIMA, Y.,180, 189, 192, 224, 225 MATTHEWS, H.J., 390, 392 MAURIN,J., 134, 147, 160 MCCALL,S.L., 189, 224 MCCORMICK,A.R., 216, 223 MCGILL,W.J., 3, 91, 104 MCLAUGHLIN, J., 389, 392 MEISSNER, P., 197, 221, 224 MENGEL,F., 215,224 MERKEL,K., 374376,392 MERSLIKIN, S.K., 134, 161 MERTZ,J.C., 3, 64, 103 MEYSTRE,P., 59, 83, 100 MICHAEL,R.R.,130-132,134, 139,143,144, 159 MICHEL,A.E., 197, 224 MIDDLETON, D., 24, 102 MIKULYAK, R.M., 199,206,225 G.J., 71, 102 MILBURN, MILLER,B.I., 199, 200, 222,224 MILLER,D.A.B., 135, 158 MILLER,M.M., 53, 102 MIRALDI,E., 113, 158 MISHKIN, E.A., 53, 102 MITCHELL, R.L., 20, 102 MITO,I., 180, 192, 223 MOHAMMED, K., 86, 100 MOLLOW,B.R., 13,63, 102 MONTROSSET, I., 324, 336, 346 MOORADIAN, A., 207, 219, 222 MOORE,D.T., 355, 390, 392 MOORE,R., 389, 392 MOORE,R.C., 351, 392 MORGAN,B.L., 13, 102 MORGAN,C.J., 358, 360, 362, 392 MOTOSUGI,G., 179, 187, 224 MOULLIN, E.B., 74, 78, 102 MUKAI,T., 220, 221, 224 MULLEN,M.E., 128, 160 MOLLER,J.W., 45, 102 MUNK,W.H., 301, 346 MURAKAMI, M., 189,224 MURATA,S., 180, 192,223 MURRAY, R., 355, 392 MYSYROWICZ, A., 144, 154, 161 N NAGAI,H., 180, 224 NAGATA,R., 391, 392 S., 391,392 NAKADATE,

NAKAJIMA,K., 168, 224 NAKAMURA, M., 180,225 NAKANO,Y.,188, 189, 193, 225 T., 219,223 NAKAYAMA, NASSAU,K., 108, 157 NATHAN,M.I., 197,224 NEIL, C.C., 196, 207, 223 NEKRASOV, G.L., 130, 160 NELSON,K.C., 216,223 NELSON,R.J., 173, 224 NERSISYAN, S.R., 134, 160 NEVES,F.B., 355, 392 NICOLET,M.A., 81, 102 NICOUD,J.F., 157, 161 NIETO, M.M., 71, 100 NILSSON,O., 83,90, 104,221,224 NISHIHARA, A., 189,224 NOGUCHI,Y.,180. 188, 193, 224, 225 NORMANDIN, R., 120, 126128, 130, 134, 146, 153, 159 NORTH,D.O., 78, 104 0 ODONNELL,K.A., 20.27, 102 OGAWA,K., 178,224 OHTSU,M., 221, 224 OHYA,J., 200,207, 2 15, 222, 224 OKAZAKI,H., 390,392 OKOSHI,T., 220, 221, 223, 224 OLDANO,C., 113, 158 OLESEN,H., 220,221, 224 OLIVER,C.J., 27, 102 OLSEN,G.H., 168,224 OLSHANSKY, R., 209, 215, 224 OLSSON,N.A., 196, 199,200,207,208,215, 22 1,222,224,225 ONG,H.L., 116, 117, 130, 134, 154, 159, 160 OREB,B.F., 389-393 ORLOV,Yu.I., 229-231, 234, 236, 257-259, 265,288,293,297,304,337,338,340,343, 346, 347 ORLOWSKY, K.J., 196,207, 215, 221, 224 OSHIDA,Y.,391, 392 OSINSKI,M., 209, 224

P PALKIN,E.A., 342, 345-347 PANISH,M.B., 165, 168,222 PANTZER,D., 390, 392

AUTHOR INDEX

PAOLI,T.L., 204, 225 PAPOULIS,A., 29, 102 PARFYONOV, A.V., 134, 158, 160 PARRENTJR, G.B., 24, 100 PARRY,G., 27, 102 PARZEN,E., 19,44,45, 52, 102 PASKO,J.G., 135, 160 PATEL,C.K.N., 197,223 PATHAK,P.G., 303, 308, 346 PATZAK,E., 197, 220, 221, 224 PAUL,H., 13, 35, 53, 63, 102 PEPPER,D.M., 147, 161 PERINA,J., 3,9, 13, 16,33,34,36,4345,47, 48, 50, 52, 53, 61, 63, 67, 76, 102, 104 PERINOVA,V., 3, 102 PERLMUTTER, S.H., 64, 103 PERSHAN, P.S., 108, 156, 159, 160 PEYGHAMBARIAN, N., 144, 153, 154, 158, 161 PHILLIPS, R.L., 27, 99 PICINBONO, B., 31, 102 PIERCE,J.R., 46, 91, 102, 103 PIKE,E.R., 43, 59, 84, 100, 101 PILIPETSKII, N.F., 134, 158 PILIPETSKY, N.F., 134, 135, 147, 161 PILIPOVICH, V.A., 130, 160 PLATZMAN, P.M., 189, 224 POHL,D., 127, 158 POLITCH,J., 390, 392 POLLAK,H.O., 29, 101 POPOV,J.M., 134, 160 POSNER,E.C., 91, 102 POTASEK,M.J., 214, 222 POUGET,J., 31, 102 POULIGNY, B., 107, 130, 160 W., 209,224 POWAZINIK, PRAKASH,H., 63, 100 PRESTON,K.R., 198,206,224 E.B., 156, 159 PRIESTLEY, PROST,J., 107, 130, 160 PRUCNAL, P.R., 3, 91, 104 S., 148, 160 PUANG-NGERN, PUSEY, P.N., 20, 26, 27, 43, 59, 101-103

R RAJBENBACH, H., 135, 152, 159, 160 RAK, D., 152, 160 RAO, D.V.G.L.N., 107, 160 R A R I ~J.G., , 61, 69. 70, 73, 84, 90, 91, 100-102, 104

40 1

RASTELLO,M.L., 113, 158 RATEIKE,F.-M., 13, 5 5 , 100 RAZVIN,Yu.V., 130, 160 REED,I.S., 24, 25, 101 REFREGIER,PH., 135, 159 REID,G.T., 359, 392 REID,J.J.E., 123, 159 REID, M., 71,73, 101 REID, M.D., 3, 59, 102 REINTJES,J.F., 136, 150, 152, 160 RENTSCHLER, J., 200,222 REYNAUD,S., 55, 70, 101 RICCIARDI,L.M., 45, 102 RICE, R.R., 197,222 RICE, S.O., 29, 103 RICHARD,L., 134, 147, 160 ROBINSON,D.W., 391, 392 RODEMICH,E.R., 91, 102 ROGER,G., 5 5 , 59, 65, 66, 68, 99, 101 RONDELEZ,F., 127, 159, 161 ROSE, A., 74, 81, 101 ROSENFELD,D.P., 351, 358, 362, 391 RUBIN,R.L., 160 RUSSELL,P.ST.J., 134, 154, 158 RUTZ,R.F., 197, 224 RYAN,D.P., 390, 392 RYTOV, S.M., 231, 234, 347

S SAHA,S.K., 156, 160 SAITO,H., 391, 392 SAITO,S., 220,221,224 SAKAI,K., 180, 189, 192, 224, 225 SALATHB,R.P., 197, 224 SALEH,B.E.A., 3,4, 6.9-11, 14-16, 19, 24, 29,30,32,33,35,4345,47-50,52,53,55, 61,63,64,66-68,73-76,78,79,81,85,86, 91.95, 97, 99, 102-104 SANDBERG, V.D., 71, 100 SANTAMATO,E., 120,134,142,154,155,160 SAPLAKOGLU, G., 6,10,35,47,53,63,85,86, 103 SARGENT111, M., 9, 12, 103. 173, 216, 224 SASAKI,O., 390,392 SATCHELL, J.S., 70, 90, 104 SATO, H., 200,207, 215, 222, 224 SAUNDERS,F.C., 156, 160 SCHADT,M., 133, 160 SCHAEFER,D.W., 20.26, 102, 103 SCHAWLOW, A.L.. 3, 103

402

AUTHOR INDEX

SCHLEICH, w., 3, 103 SCHUBERT, M., 63, 103 SCHULZ,G., 387, 392 SCHWIDER, J., 387, 392 SCHWIDER, J.R., 374-376, 392 SCIFRES,D.R., 180, 183, 184, 186, 192, 225 SCOTT, A.M., 156, 160 SCULLY, M.O., 9, 12, 103, 173, 216, 224 SEATON, C.T., 134, 154, 161 SECKLER,B.D., 303, 324, 346, 347 SENIOR, J., 91,95, 103 SERAK, S.V., 130, 160 SERINKO, R.J., 115, 134, 149, 159 SERIZAWA, H., 200, 207, 224 SHAGAM, R.N., 353, 355, 392, 393 SHANK,C.V., 180, 182, 185, 189, 223 J.H.,4,6,10,34-36,47,53,63,64, SHAPIRO, 85, 86, 102-104 SHAW,J.M., 196, 207, 223 SHELBY, R.M., 64, 71, 73, 101, 103 SHEN,T.M., 178, 207, 211, 222, 224,225 SHEN,Y.R., 107, 114, 116-120, 127, 128, 130-132,134, 136138,142,146, 153-156, 158-1 6 1 SHEPARD, S., 134, 143, 153, 159 SHEPHERD, T.J., 53, 102 SHEPPARD, C.J.R., 390, 392 S H I , L.P., 127, 128, 130-132, 159 SHIBATA, J., 200, 207, 224 SHIN, S.Y., 134, 160 SHKUNOV, V.V., 134, 135, 147, 158, 161 W., 46, 103 SHOCKLEY, SHORT,R., 3,43, 55-58, 77, 103 SIEGERT,A.J.F., 15, 103 SIEGMAN, A.E., 58, 102 SIMAAN, H.D., 63, 103 SIMONI, F., 113, 119, 158, 160 SKARTEDJO, K., 189, 225 F.H., 351, 392 SLAYMAKER, SLUSHER, R.E., 3,64, 103 SMIRNOV, D.F., 83, 103 SMITH,P.W., 135, 144, 158 SMITH,R.G., 108, 157 SMITH,S.D., 153, 158 SMYTHE, R., 389, 392 SNYDER, D.L., 10, 29,43, 44, 103 So, V.C.Y., 134, 153, 159 SOBOLEV,N.N., 115,116,119-121, 134, 138, 158, 161 SODA,H., 189, 212, 223, 225

SOILEAU,M.J., 144, 160 SOLYMAR, L., 135, 159 SOMMARGREN, G.E., 35 1, 358, 390, 392 SONA,A., 13, 99 SONG,J.W., 134, 160 SPOLACZYK, R., 376376,392 SRINIVASAN, S.K., 6, 70, 78, 103 STAVRODIS, O.N., 300, 347 STEGEMAN, G.I., 134, 154, 161 STEINLE,B., 27, 101 STEPHEN,M.J., 128, 160 STERN,T.E., 91, 103 STETSON,K.A., 379, 391, 392 STOLER,D., 3, 13-15, 24, 29, 30, 32, 33, 43, 49, 55, 63, 70, 74, 78, 97, 103, 104 STREIFER, w., 180, 183, 184, 186, 192, 204, 225, 327, 346 STULZ,L.W., 216,223 SUDARKIN, A.N., 134, 158 SUDO, H., 189, 212,223, 225 SUEMATSU,Y.,168, 180,189, 193, 195, 196, 209, 214, 222,223, 225 SUGIMURA, A., 220, 221, 224 A.P., 134, 137, 157, 161 SUKHORUKOV, SUKHOV,A.V., 134, 158, 161 SUZUKI, M., 389, 393 SUZUKI, Y., 180, 224 SWAN,C.B., 199,207, 224 SWANSON, G., 134, 147, 160 SWANSON,G.J., 148, 160 SWARTZ,R.G., 207, 222 SZE, s.,81, 103

T TABIRYAN, N.V., 114-116, 119, 120, 134, 158, 160, 161 TADA,K., 189, 225 TAI, K., 144, 154, 161 TAKAHASI, H., 63, 103 TAKEDA,M., 356, 392 TALANOV, V.I., 137, 158 TALMAN,J.R., 216, 223 TANAHASHI, T., 189, 225 TANBUN-EK, T., 189, 225 TAPSTER,P.R., 61, 69, 70, 84, 90, 100, 102, 104 TARTAGLIA, P., 20, 100 TATARSKII, V.I., 302, 347 TAVERNA VALABREGA, P., 113, 158 TAVOLACCI, J., 33, 103

AUTHOR INDEX

TEICH,M.C.,3,4,6, 10, 14, 15,24,29,30,32, 33,35,43-45,47-50,52,53,55,61,63,64, 6668,73-76,78,79,81,85-88,91,96,97, 99, 100, 102-104

TENG,C.C., 156, 158 TENNANT, D.M., 196,207, 225 THALMANN, R., 389-392 THOMPSON, B.J., 78, 104 G.H.B., 165, 174, 225 THOMPSON, THORNE,K.S., 71, 100 TITULAER, U.M., 38, 104 TOIiMORI, Y., 196, 225 TOMLINSON, W.J., 144, 158 TOOLEY,F.A.P., 123, 159 TORNAU,N., 63, 104 TORRE,A., 97, 100 TOWNES,C.H., 3, 103 TRACY,J., 135, 160 TROSHIN,AS., 83, 103 TROUP,G.J., 20, 104 TRUAX,B.E., 355, 392 TSANG,W.T., 197, 199, 200, 207, 208, 225 TUCKER,R.S.,209, 212, 225 Twiss, R.Q., 3, 13, 15, 101, 104 U

UFIMTSEV, P.YA., 303, 307, 345, 347 ULRICH,R., 196, 207, 222 UPATNIEKS, J., 145, 147, 160 UPSTILL,C., 256, 297, 298, 302, 338, 345 URBACH,W., 127, 159, 161 USAMI,M., 180, 189, 193, 222 USHIROKAWA, A., 189, 225 UTAKA,K., 180, 189, 192, 193,222,224,225

V VACH,H.,134, 154, 161 R.B.,303, 348 VAGANOV, VAHALA,K., 210,216-218, 221, 225 VALLEY,J.F., 3, 64, 103 VAN DER ZIEL, J.P., 199, 206, 225 VAN STRYLAND, E.W., 144, 160 VAN ZANTEN,P., 133, 158 H.,144, 160 VANHERZEELE, G., 3, 45, 55, 68, 91, 104 VANNUCCI, VARCHENKO, A.N., 337, 345 VASILIEV, A.A., 134, 158 VAUGHAN,J.M., 43, 59, 101 VESELKA, J.J., 216, 223 VLASOV, S.A., 337, 347

403

VOLTERRA. V., 134, 161 VORONTSOV, Yu.I., 71, 100 VRY, u., 389, 391 VYATCHANIN, S.P., 71, 100

W WAKAO,K., 189, 212, 223, 225 WALKER,A.C., 123, 159 WALKER,J.G., 55,68,70, 84-86, 101, 104 WALLS,D.F., 3.4, 13, 31,43,59,63,64,71, 73, 100-104 WALTHER,H., 13, 5 5 , 60, 77, 100 WANG,S., 180, 200, 204, 222, 225 WATSON,K.M., 301, 346 WEINBERG,D.L., 58, 68, 100 WELKOWSKY, M.S., 135, 161 WESTBROOK, L.D., 215, 222 WHALEN,M.S., 196, 207, 225 WHEELER,J.A., 3, 103 WHINNERY, J.R., 74, 78, 104 WHITE,A.D., 351, 358, 362, 391 WHITE,J.P., 147, 158 WIENER-AVNEAR, E., 134, 161 B., 63, 103 WILHELMI, WILLEMIN, J.-F., 391, 392 WILLIAMS, D., 156, 161 WILLIAMS, D.C., 391, 392 WILLIAMS, E.W., 144, 160 WILLIAMS,W.E., 144, 160 WILLIAMSON, R.C., 128, 158 WILSON,R.B.,173, 224 WILT, D.P., 216, 220, 223 WINFUL,H.G., 134, 161 WOLF,E., 9, 16, 17, 100, 102, 104, 139, 158, 169,222,23 I , 24 1,256,272,274,277,345 356, 389, 393 WOMACK.K.H., WONG,G.K.L., 107, 118, 156, 160, 161 WONG,K.Y., 156, 158 WOOLLARD, K.C., 198,206,224 WRIGHT,E.M., 123, 158 WRIGHT,P.D., 173, 224 Wu, H., 63, 104 WU, L.-A,, 63, 104 Wu, S.T., 130, 133, 135, 161 WYANT,J.C., 351, 353-356, 358, 359, 363, 375, 376, 385, 386, 389-393 WYAIT, R., 207, 221, 225 X

Xu, G.X., 133, 146, 158

404

AUTHOR INDEX

Y YAKUSHKIN, I.G., 302, 348 YAMAGUCHI, M., 180, 192,223 YAMAKOSHI, S., 212,223 YAMAMOTO, Y., 3,51,65,71,72,83-86,90, 91, 101, 102, 104, 216, 221, 224, 225 YAMASHI, Y., 389,393 YAN,P.Y.,121, 130-132, 134, 139, 143, 153, 159, 161 YARIV,A,, 58, 102, 135, 147, 158, 161, 180, 210,216-218, 221,223,225 YARYGIN, A.P., 245,253, 348 YATAGAI,T., 389, 393 YEH,P., 136, 151, 161 YEN, R.T., 216, 223 YEVICK, D., 204,221, 224, 225 YOSHIDA, J., 188, 193, 225 YOSHIKUNI, Y., 179, 180, 187, 224 YOUNG,C.Y., 156, 160 YUEN,H.P., 34, 53, 63, 70, 103, 104 YURKE,B., 3, 64, 70, 103

2 ZACHARIASEN, F., 301, 346 ZAKAI,M., 94, 101 ZAMANI-KHAMIRI, O., 156, 158 ZAVISLAN, J.M., 390, 391, 393 ZAVOROTNYI, V.D., 302,347 ZELDOVICH, B.YA., 114-116, 119, 120, 134, 135, 147, 157, 158, 161 ZHELEZNYAKOV, V.V., 236, 348 ZHOU, WANZHI, 389,392 ZHUANG,S.L., 115, 116, 134, 147, 149, 151, 159 ZIILSTRA, R.J.J., 81, 102 ZIMMERMAN, M., 71, 100 ZIV, I., 94, 101 ZOLINA,V., 114, 158 ZOLOTKO,A.S., 115, 116,119, 134, 138, 161 ZORN,M., 196,207,222 ZWANZIGER, M., 34, 101 ZYDZIK,G., 207,222 ZYSS,J., 157, 161

SUBJECT INDEX A

D

acousto-optic modulator, 353 ADP crystal, 69 Airy function, 292, 337 amplitude modulation, 209 antibunched light, 3, 6, 13, 33, 49, 62, 64, 73 antibunching, 15, 35,47, 55 anticorrelation, photon, 3

dead-time deletion, 7 diffraction, geometrical theory of, 229, 302-304,309-311, 313,314,318-321,323, 324 - grating, 353 Dirichlet boundary condition, 308, 3 16

E B eikonal, 239,242,286, 288,290,291, 304 Einstein's fluctuation formula, 16 electro-optic device, 123 --- modulator, 353 Euler-Lagrange method, 114 evanescent wave, 154

Bell's inequalities, 3 Bernoulli deletion, 52-54, 67 transform, 76 black-body radiation, 8, 16 Bose-Einstein distribution, 16 B r a g cell, 353 - condition, 182 Brewster angle, 283 bunched light, 4, 6, 8, 13 bunching, 15, 47

-

F Fabry-Perot cavity, 153, 166, 167, 171, 172 Fano factor, 12, 16, 19,22-24,26,29, 33, 36, 41,48,52,54,58,68, 70,78-80,82, 88,97, 98 Fermat principle, 325 Fock function, 318, 321 four-wave mixing, 71, 145 Franck-Hertz effect, 76, 78 -_experiment, 7, 76, 78, 83, 89 -_light, 80 Frederiks transition, 116 FrenBt reference frame, 235 frequency modulation, 209 Fresnel integral, 322 radius, 262, 330 scale, 316-319, 321, 322 volume. 229, 230, 238, 245, 246, 255-259, 261,262,264, 268-270, 272, 276, 280, 281.

C

Carri: equation, 383

- technique, 365, 382, 384, 385 catastrophe theory, 338, 343 caustic, 289, 290, 297, 302, 31 1, 336, 337 field, 229 zone, 289,290,293,299-301 central limit theorem, 24 chaotic light, 6, 15-17, 27 coherence, degree of, 10, 21 function, 37 coherent light, 12 correlation function, 22, 53 --, amplitude, 9, 32, 34 --,intensity, 10, 34 -, photon, 3

-

-

405

406

SUBJECT INDEX

284-286, 289, 301, 304-307, 310, 314, 317, 322, 324, 344 -zone, 229,237,239,243-245,247-252,255, 256, 261, 262, 268, 272, 279, 293, 294, 3 11

G gamma& process, 44, 68 Gaussian stochastic process, 15 geometrical optics, 230, 233, 259, 260, 263-265,269, 271, 281, 291, 326, 340 -- approximation, 234, 236, 242 Green formula, 275 - function, 260

H Heisenberg-Langevin equation, 53

- uncertainty principle, 71

Helmholtz equation, 230, 231, 260, 339 heterodyne detection, 4 hole-burning, spectral, 176 holography, 145 Huygens-Fresnel principle, 238, 261 --- representation, 257, 258

I interference microscope, 352 interferometer, Mach-Zehnder, 352 -, Smartt Point-Diffraction, 352 -, Twyman-Green, 352 interferometry, holographic, 389, 390 -, phase-measurement, 351, 352, 388, 390

-noise, 216 laser, argon-ion, 58 cleaved-coupled-cavity, 200, 20 I , 206-208 -, coupled-cavity, 166, 197 distributed B r a g reflector, 180, 181, 193-197 -, distributed feedback, 166, 180, 181, 183-190, 192, 193, 196, 197, 210-212, 219, 220 -, external B r a g reflector, 196 -, He-Ne, 115, 144 -, InGaAsP, 165, 168, 177, 190, 198, 199 -, Nd: Yag, 137, 144 --,semiconductor, 165-167,169,171,174,177, 178, 202, 208,216 -,single longitudinal mode, 165,166,178,206, 208, 209, 216, 218, 221, 222 light-emitting diode, 86, 89 liquid crystal, 107-109, 118, 127, 154-157 --, cholesteric, 116 --, isotropic, 153 --,nematic,111,112,114,116,132,148,151, I52 --, smectic, 116 local oscillator, 4

-. -.

M Markoffian approximation, 217 Maxwell equations, 236 -wave equation, 150, 169, 173 mode-suppression ratio, 177 multiphoton absorption, 63 multistability, 63

K

N Kabanov’s theorem, 93 K-distribution, 27 KDP crystal, 58 Kerr effect, 65, 71, 107 -medium, 71, 73, 151 Kirchhoff approximation, 250, 25 1 - integral, 295 radius, 330

Neumann condition, 308,316 Neyman Type-A distribution, 33, 49 nonclassical light, 19, 55, 56, 65 nonlinear optical process, 133, 135

-

0

L

- coherence, quantum theory of, 34

optical bistability, 63, 134, 153, 154

Lagrange’s equation, 112 Langevin force, 217, 219

--, semiclassical theory of, 9 - wave mixing, 145, 146

opto-optical modulation, 134

SUBJECT INDEX

407

P

R

parametric amplification, 63 - downconversion, 7, 58, 59, 63, 65, 68 path integral, 342 phase conjugation, 108, 134, 147, 148 photochromic effect, 70 photon statistics, 45 --, sub-Poisson, 50 --, super-Poisson, 50 Photorefractive effect, 108 - material, 108 piezo-electric transducer, 354 p-n junction, 168 point process, Poisson (see Poisson point process) --, renewal, 44 --, self-exciting,44 Poisson distribution, 12, 17, 26 - photo counts, 6, 12 - point process, 10, 36, 44, 46, 67 - transform, 1 1 population inversion, 167, 174 power spectrum, 216,219 Poynting vector, 115, 237

Raman scattering, 63 rate equations, 173, 175 Rayleigh distribution, 15 resonance fluorescence, 5, 7, 37, 55, 58, 60 Rytov's law, 235

S

second-harmonic generation, 5, 70, 156 self-focusing, 134, 136, 156 shot noise, 81 Siegert relation, 15 smectic-C film, 156 squeezed light, 3, 63, 64 subchaotic light, 17, 18 sub-Poisson light, 3,4,7, 8, 13,34,43,49, 62, 64, 70, 72-74, 80, 91, 94, 96, 98, 99 --- photocounts, 6, 12, 94 superchaotic light, 6, 17 super-Poisson light, 4, 91 --- photocounts, 6, 12, 33 synchronous detection, 362

Q

Z

quantum nondemolition, 65,71,73,74,86,92

Zeeman laser. 353


CUMULATIVE INDEX - VOLUMES I-XXVI ABELBS,F., Methods for Determining Optical Parameters of Thin Films ABELLA,I. D., Echoes at Optical Frequencies ABITBOL, C. I., see J. J. Clair Dynamical Instabilities and ABRAHAM, N. B., P. MANDEL,L. M. NARDUCCI, Pulsations in Lasers AGARWAL, G. S., Master Equation Methods in Quantum Optics AGRAWAL, G. P., Single-longitudinal-mode Semiconductor Lasers Crystal Optics with Spatial Dispersion AGRANOVICH, V. M., V. L. GINZBURG, ALLEN,L., D. G. C. JONES,Mode Locking in Gas Lasers AMMANN, E. O., Synthesis of Optical Birefringent Networks J. A,, A. W. SMITH,Experimental Studies of Intensity Fluctuations ARMSTRONG, in Lasers ARNAUD, J. A., Hamiltonian Theory of Beam Mode Propagation 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 BARRETT,H. H., The Radon Transform and its Applications BASHKIN, S., Beam-Foil Spectroscopy BECKMANN, P., Scattering of Light by Rough Surfaces BERRY,M. V., C. UPSTILL,Catastrophe Optics: Morphologies of Caustics and their Diffraction Patterns 111, R. E., Light Emission from High-Current Surface-Spark Discharges BEVERLY BLOOM,A. L., Gas Lasers and their Application to Precise Length Measurements Quantum Fluctuations in BOUMAN, M. A,, W. A. VAN DE GRIND,P. ZUIDEMA, Vision BOUSQUET, P., see P. Rouard BROWN,G. S., see J. A. DeSanto W., H. PAUL,Theory of Optical Parametric Amplification and OscillaBRUNNER, tion BRYNGDAHL, O., Applications of Shearing Interferometry O., Evanescent Waves in Optical Imaging BRYNGDAHL, BURCH,J. M., The Meteorological Applications of Diffraction Gratings BU~TERWECK, H. J., Principles of Optical Data-Processing CAGNAC,B., see E. Giacobino CASASENT, D., D. PSALTIS,Deformation Invariant, Space-Variant Optical Pattern Recognition 409

11, 249 VII, 139 XVI, 71

xxv, XI, XXVI, IX, IX, IX,

I 1

163 235 179 123

VI, 211 XI, 241 XII,

1

I, 67 XXI, 217 XII, 287 VI, 53 XVIII, 259 XVI, 357 IX, 1 XXII, 77 IV, 145 XXIII, 1 xv, 1 IV, 37 XI, 167 11, 73 XIX, 21 1 XVII. 85 XVI, 289

410

CUMULATIVE INDEX

CEGLIO, N. M., D. W. SWEENEY, Zone Plate Coded Imaging: Theory and ApplicaXXI, 287 tions XIII, 69 CHRISTENSEN, J. L., see W. M. Rosenblum XVI, 71 CLAIR,J. J., C. I. ABITBOL,Recent Advances in Phase Profiles Generation XIV, 321 CLARRICOATS, P. J. B., Optical Fibre Waveguides - A Review COHEN-TANNOUDJI, C., A. KASTLER,Optical Pumping v, 1 XV, 187 COLE,T. W., Quasi-Optical Techniques of Radio Astronomy XX. 63 COLOMBEAU, B., see C. Froehly G., P. CRUVELLIER, M. DETAILLE,M. SA~SSE, Some New Optical COURT~S, xx, 1 Designs for Ultra-Violet Bidimensional Detection of Astronomical Objects XXVI, 349 CREATH, K., Phase-Measurement Interferometry Techniques XI, 223 CREWE,A. V., Production of Electron Probes Using a Field Emission Source xx, 1 CRUVELLIER, P., see C. G. Courtes Light Beating Spectroscopy VIII, 133 CUMMINS, H. Z., H. L., SWINNEY, XIV, 1 DAINTY, J. C., The Statistics of Speckle Patterns XVII, 1 DXNDLIKER, R., Heterodyne Holographic Interferometry DECKERJr., 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. Courtes 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 Measurement of the Second Order Degree of CoheFRANCON, M., S. MALLICK, VI. 71 rence 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, 109 111, 187 GAMO,H., Matrix Treatment of Partial Coherence XIII, 169 GHATAK, A. K., see M. S. Sodha XVIII, 1 GHATAK, A., K. THYAGARAJAN, Graded Index Optical Waveguides: A Review XVII, 85 GIACOBINO, E., B. CAGNAC,Doppler-Free Multiphoton Spectroscopy

41 I

CUMULATIVE INDEX

GINZBURG, V. L., see V. M. Agranovich GIOVANELLI, R. G., Diffusion Through Non-Uniform Media GLASER, I., Information Processing with Spatially Incoherent Light GNIADEK, K., J. PETYKIEWICZ, Applications of Optical Methods in the Diffraction Theory of Elastic Waves GOODMAN, J. W., Synthetic-Aperture Optics GRAHAM, R., The Phase Transition Concept and Coherence in Atomic Emission HARIHARAN, P., Colour Holography HARIHARAN, P., Interferometry with Lasers HARWIT, M., J. A. DECKERJr., Modulation Techniques in Spectrometry HELSTROM, C. W., Quantum Detection Theory HERRIOTT, D. R., Some Applications of Lasers to Interferometry HUANG,T. S., Bandwidth Compression of Optical Images JACOBSSON, R., Light Reflection from Films of Continuously Varying Refractive Index Apodisation JACQUINOT, P., B. ROIZEN-DOSSIER, JAMROZ,W., B. P. STOICHEFF,Generation of Tunable Coherent Vacuum-Ultra. violet Radiation JONES,D. G. C., see L. Allen KASTLER,A,, see C. Cohen-Tannoudji KHOO,I. C., Nonlinear Optics of Liquid Crystals KIELICH,S., Multi-Photon Scattering Molecular Spectroscopy KINOSITA, K., Surface Deterioration of Optical Glasses K O P P E L M ~ N G., N , Multiple-Beam Interference and Natural Modes in Open Resonators KOTTLER,F., The Elements of Radiative Transfer KOTTLER,F., Diffraction at a Black Screen, Part I: Kirchhoffs Theory KOTTLER,F., Diffraction at a Black Screen, Part 11: Electromagnetic Theory KRAVTSOV, Yu. A,, Rays and Caustics as Physical Objects KUBorA, H., Interference Color A,, High-Resolution Techniques in Optical Astronomy LABEYRIE, LEAN,E. G., Interaction of Light and Acoustic Surface Waves LEE, W.-H., Computer-Generated Holograms: Techniques and Applications LEITH,E N., J. UPATNIEKS, Recent Advances in Holography V. S., Laser Selective Photophysics and Photochemistry LETOKHOV, LEVI,L., Vision in Communication X-Ray Crystal-Structure Determination as a Branch LIPSON,H., C. A. TAYLOR, of Physical Optics LUGIATO,L. A,, Theory of Optical Bistability D., Optical and Electronic Processing of Medical Images MALACARA, MALLICK,L., see M. Francon MANDEL,L., Fluctuations of Light Beams MANDEL,L., The Case for and against Semiclassical Radiation Theory MANDEI.,P., see N. B. Abraham

IX. 235 11, 109 XXIV, 389 IX, VIII, XII, XX, XXIV, XII, X, VI, x.

281 1

233 263 103 101 289 171 I

V, 241 111. 29

X X , 325 IX, 179

v,

1

XXVI, 105 X X , 155 IV, 85 VII, 1 111, 1 IV, 281 V1, 331 XXVI, 227 I, 211 XIV, 47 XI, 123 XVI, 119 VI, 1 XVI, 1 VIII, 343 V, 287 XXI. 69 XXII, 1 VI. 71 11, 181 XIII, 27

xxv,

1

412

CUMULATIVE INDEX

XI, 305 MARCHAND, E. W., Gradient Index Lenses Optical Films Produced by Ion-Based TechMARTIN,P. J., R. P. NETTERFIELD, XXIII, 113 niques MASALOV, A. V., Spectral and Temporal Fluctuations of Broad-Band Laser XXII, 145 Radiation XXI, I MAYSTRE, D., Rigorous Vector Theories of Diffraction Gratings xv, 77 MEESSEN, A,, see P. Rouard VIII, 373 MEHTA,C. L., Theory of Photoelectron Counting 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 ScatteMILLS,D. L., K. R. SUBBASWAMY, XIX, 43 ring 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 K., Instruments for the Measuring of Optical Transfer Functions V, 199 MURATA, MUSSET,A., A. THELEN,Multilayer Antireflection Coatings VIII, 201 NARDUCCI, L. M., see N. B. Abraham xxv, 1 XXIII, 113 NETTERFIELD, R. P., see P. J. Martin NISHIHARA, H., T. SUHARA, Micro Fresnel Lenses XXIV, 1 XXV, 191 OHTSU,M., T. TAKO,Coherence in Semiconductor Lasers XV, 139 OKOSHI,T., Projection-Type Holography VII, 299 OOUE,S., The Photographic Image G. V., Yu. I. OSTROVSKY, Holographic Methods in Plasma OSTROVSKAYA, XXII, 197 Diagnostics OSTROVSKY, Yu. I., see G. V. Ostrovskaya XXII, 197 K. E., Unstable Resonator Modes XXIV, 165 OUGHSTUN, xv, 1 PAUL,H., see W. Brunner PEGIS,R. J., The Modern Development of Hamiltonian Optics 1, 1 PEGIS,R. J., see E. Delano VII, 67 J., Photocount Statistics of Radiation Propagating through Random and PERINA, XVIII, 129 Nonlinear Media V, 83 PERSHAN, P. S., Non-Linear Optics IX, 281 PETYKIEWICZ, J., see K. Gniadek PICHT,J., The Wave of a Moving Classical Electron V, 351 D., see D. Casasent PSALTIS, XVI. 289 RISEBERG,L.A., M. J. WEBER,Relaxation Phenomena in Rare-Earth LumiXIV, 89 nescence VIII, 239 RISKEN,H., Statistical Properties of Laser Light XIX, 281 RODDIER, F., The Effects of Atmospheric Turbulence in Optical Astronomy 111, 29 B., see P. Jacquinot ROIZEN-DOSSIER, XXV. 279 RONCHI,L., see Wang Shaomin ROSENBLUM, W. M., J. L. CHRISTENSEN, Objective and Subjective Spherical XIII, 69 Aberration Measurements of the Human Eye

CUMULATIVE INDEX

413

ROTHBERG, L., Dephasing-Induced Coherent Phenomena XXIV, 39 ROUARD,P., P. BOUSQUET, Optical Constants of Thin Films IV, 145 ROUARD, P., A. MEESSEN,Optical Properties of Thin Metal Films XV, 17 RUBINOWICZ, A,, The Miyamoto-Wolf Diffraction Wave IV, 199 RUDOLPH, D., see G. Schmahl XIV, 195 SA'ISSE,M., see G. Courtts xx, 1 SAKAI,H., see G. A. Vanasse VI, 259 SALEH,B. E. A., see M. C. Teich XXVI, 1 SCHIEVE, W. C., see J. C. Englund XXI, 355 SCHMAHL, G., D. RUDOLPH, Holographic Diffraction Gratings XIV, 195 The Mutual Dependence between Coherence ProSCHUBERT, M., B. WILHELMI, perties of Light and Nonlinear Optical Processes XVII, 163 SCHULZ,G., J. SCHWIDER, Interferometric Testing of Smooth Surfaces XIII, 93 SCHULZ,G., Aspheric Surfaces xxv, 349 J., see G. Schulz SCHWIDER, XIII, 93 M. O., K. G. WHITNEY,Tools of Theoretical Quantum Optics SCULLY, x. 89 I. R., Semiclassical Radiation Theory within a Quantum-Mechanical SENITZKY, Framework XVI, 413 SIPE,J. E., see J. Van Kranendonk XV, 245 SITTIC,E. K., Elastooptic Light Modulation and Deflection X, 229 SLUSHER, R. E., Self-Induced Transparency XII, 53 SMITH,A. W., see J. A. Armstrong VI, 21 1 SMITH,D. Y., D. L. DEXTER,Optical Absorption Strength of Defects in Insulators X, 165 SMITH,R. W., The Use of Image Tubes as Shutters x, 45 SNAPP,R. R., see J. C. Englund XXI, 355 V. K. TRIPATHI, Self Focusing of Laser Beams in SODHA,M. S., A. K. GHATAK, Plasmas and Semiconductors XIII, 169 STEEL,W. H., Two-Beam Interferometry V, 145 STOICHEFF,B. P., see W. Jamroz XX, 325 J. W., Optical Propagation Through the Turbulent Atmosphere STROHBEHN, IX, 73 STROKE, G. W., Ruling, Testing and Use of Optical Gratings for High-Resolution 11, 1 Spectroscopy SUBBASWAMY, K. R., see D. L. Mills XIX, 43 XXIV, 1 SUHARA, T., see H. Nishihara SVELTO,O., Self-Focusing, Self-Trapping, and Self-phase Modulation of Laser XII, 1 Beams SWEENEY, D. W., see N. M. Ceglio XXI, 287 SWINNEY, H. H., see H. Z. Cummins VIII, 133 TAKO,T., see M. Ohtsu XXV, 191 TANAKA, K., Paraxial Theory in Optical Design in Terms of Gaussian Brackets XXIII, 63 TANGO,W. J., R. Q. TWISS,Michelson Stellar Interferometry XVII, 239 V. I., V. U. ZAVOROTNYI, Strong Fluctuation in Light Propagation TATARSKII, XVIII, 207 in a Randomly Inhomogeneous Medium V, 287 C. A., see H. Lipson TAYLOR,

414

CUMULATIVE INDEX

XXVI, 1 TEICH,M. C., B. E. A. SALEH,Photon Bunching and Antibunching VII, 231 TER-MIKAELIAN, M. L., see A. L. Mikaelian VIII, 201 THELEN, A., see A. Musset VII, 169 THOMPSON, B. J., Image Formation with Partially Coherent Light THYAGARAJAN, K., see A. Ghatak XVIII, 1 TONOMURA, A., Electron Holography XXIII, 183 XIII, 169 V. K., see M. S. Sodha TRIPATHI, J., Correction of Optical Images by Compensation of Aberrations and TSUJIUCHI, 11, 131 by Spatial Frequency Filtering XVII, 239 TWISS, R. Q., see W. J. Tango VI, 1 UPATNIEKS, J., see E. N. Leith XVIII, 259 UPSTILL,C., see M. V. Berry USHIODA,S.,Light Scattering Spectroscopy of Surface Electromagnetic Waves in XIX, 139 Solids XX, 63 VAMPOUILLE,M., see C. Froehly VI, 259 VANASSE, G. A., H. SAKAI,Fourier Spectroscopy XXII, 77 VAN DE GRIND,W. A., see M. A. Bouman I, 289 VAN HEEL,A. C. S.,Modern Alignment Devices VAN KRANENDONK, J., J. E. SIPE,Foundations of the Macroscopic ElectromagneXV, 245 tic Theory of Dielectric Media XIV, 245 VERNIER,P., Photoemission XXV, 279 WANG,SHAOMIN, L. RONCHI,Principles and Design of Optical Arrays XIV, 89 WEBER,M. J., see L. A. Riseberg IV, 241 WELFORD,W. T., Aberration Theory of Gratings and Grating Mountings XIII, 267 WELFORD,W. T., Aplanatism and Isoplanatism XVII, 163 WILHELMI, B., see M. Schubert X, 89 WITNEY,K. G., see M. 0. Scully WOLTER,H., On Basic Analogies and Principal Differences between Optical and I, 155 Electronic Information X, 137 WYNNE,C. G., Field Correctors for Astronomical Telescopes YAMAGUCHI,I., Fringe Formations in Deformation and Vibration Measurements XXII, 271 Using Laser Light VI. 105 YAMAJI,K., Design of Zoom Lenses YAMAMOTO, T., Coherence Theory of Source-Sue Compensation in Interference VIII, 295 Microscopy XI, 77 YOSHINAGA,H., Recent Developments in Far Infrared Spectroscopic Techniques XXIII, 227 Yu, F. T. S.,Principles of Optical Processing with Partially Coherent Light XVIII, 207 ZAVOROTNYI, V. U., see V. I. Tatarskii XXII, 77 ZUIDEMA, P., see M. A. Bouman

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