Progress in Optics, Vol. 29

PROGRESS IN OPTICS VOLUME X X I X EDITORIAL ADVISORY BOARD G . S. AGARWAL, Hyderabad, India C. COHEN-TANNOUDJI, Pa...

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PROGRESS IN OPTICS VOLUME X X I X

EDITORIAL ADVISORY BOARD

G . S. AGARWAL,

Hyderabad, India

C. COHEN-TANNOUDJI, Paris, France

F. GORI,

Rome, Italy

A. KUJAWSKI,

Warsaw, Poland

A. LOHMANN,

Erlangen, Germany

M. SCHUBERT,

Jena, Germany

J . TSUJIUCHI,

Chiba, Japan

H. WALTHER,

Garching, Germany

W. T. WELFORD~,

London, England

B. ZEL’DOVICH,

Chelyabinsk, U.S.S.R.

PROGRESS I N OPTICS VOLUME XXIX

EDITED BY

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

Contributors YLLN.BARABANENKOV, 1.P. CHRISTOV, C. FLYTZANIS, F. HACHE, D.G. HALL, M.C. KLEIN, Yu.A. KRAVTSOV, V.D. OZRIN, D . RICARD, PH. ROUSSIGNOL, A.I. SAICHEV. G . WEIGELT

1991

NORTH-HOLLAND AMSTERDAM. OXFORD, NEW YORK . TOKYO

0 ELSEVIER SCIENCE PUBLISHERS B.v., 1991

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, photoropying, recording or otherwise. without the writtenpermission ofthe Publisher, Elsevier Science Publishers B. V..P. 0.Box 211, I000 AE Amsterdam. The Netherlands. Special regulationsfor readers in the U.S.A.: This publication has been regirtered 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, unless otherwise specified. No responsibility is assumed by the Publisher for any injury andlor damage to persons or property as a matter of products liability. negrigence or otherwise, or from any use or operation of any methods. products. instructions or ideas contained in the material herein. LIBRARY OF CONGRESS CATALOG CARD NUMBER: 61-19297 ISBN: 044488951 5

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NORTH-HOLLAND ELSEVIER SCIENCE PUBLISHERS B.V.

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PRINTED ON ACID-FREE PAPER PRINTED IN THE NETHERLANDS

CONTENTS OF PREVIOUS VOLUMES

VOLUME 1(1961) 1-29 THEMODERNDEVELOPMENT OF HAMILTONIAN OPTICS,R. J. PEGIS . . WAVE OPTICS A N D GEOMETRICAL OPTICS IN OPTICAL DESIGN, K. MIYAMOTO . . . . . . . . . . . . . . . . . . . . . . . . . . . , 31-66 111. THEINTENSITY DISTRIBUTION AND TOTAL ILLUMINATION OF ABERRATIONFREEDIFFRACTION IMAGES, R.BARAKAT. . . . . . . , . . . . , . 67-108 D. GABOR. . . . . . . . . . . . . . . . 109- 153 IV. LIGHTA N D INFORMATION, ON BASICANALOGIES V. AND PRINCIPAL DIFFERENCES BETWEEN OPTICAL A N D ELECTRONIC INFORMATION, H. WOLTER. . . . . . . . . , . . . 155-210 VI. INTERFERENCE COLOR,H. KUBOTA . . . . . . . . . . . . . . . . . 211-251 VII. DYNAMIC CHARACTERISTICS OF VISUAL PROCESSES, A. FIORENTINI . . . 253-288 VIII. MODERNALIGNMENT DEVICES, A. C. S. VAN HEEL . . . . . . . . . . 289-329

I.

11.

V O L U M E I1 ( 1 9 6 3 ) I. 11.

111. Iv.

v. VI.

RULING, TESTING AND USEOF OPTICAL GRATINGS FOR HIGH-RESOLUTION SPECTROSCOPY, G. W. STROKE . . . . . . . . . . . . . . . . . . . 1-72 THE METROLOGICAL APPLICATIONS OF DIFFRACTION GRATINGS,J. M. BURCH . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73-108 DIFFUSION THROUGH NON-UNIFORM MEDIA,R. G. GIOVANELLI . . . . 109-129 CORRECTION OF OPTICAL IMAGES BY COMPENSATION OF ABERRATIONS A N D BY SPATIAL FREQUENCY FILTERING, J. TSUJIUCHI . . . . . . . . 131-180 FLUCTUATIONS OF LIGHTBEAMS,L. MANDEL . . . . . . . . . . . . 181-248 OPTICAL PARAMETERS OF THINFILMS, F. METHODSFOR DETERMINING ABELBS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249-288

V O L U M E I11 ( 1 9 6 4 ) I. 11.

III.

THEELEMENTS OF RADIATIVE TRANSFER, F. KOTTLER . . . . . . . . APODISATION, P. JACQUINOT, B. ROIZEN-DOSSIER. , . . . . . . . . MATRIXTREATMENTOFPARTIALCOHERENCE, H. GAMO . . . . . . . V

1-28 29- 186 187-332

CONTENTS O F PREVIOUS VOLUMES

VI

V O L U M E IV ( 1 9 6 5 ) I. I I.

HIGHER ORDER

ABERRATION THEORY. J . FOCKE. . . . . . . . . . .

.

APPLICATIONS O F S H E A R I N G INTERFEROMETRY. 0 BRYNCDAHL . SURFACE DETERIORATION O F O P T I C A L GLASSES. K . KINOSITA. .

. . . . . . 111. IV . O P T I C A L C O N S T A N T S O F T H I N FILMS.P . ROUARD.P . BOUSQUET. . . . V . THEMIYAMOTO-WOLFD I F F R A C T I O N WAVE. A . R U B I N O W I C Z . . . . . THEORY O F GRATINGS A N D GRATING MOUNTINGS. W . T. VI . ABERRATION WELFORD . . . . . . . . . . . . . . . . . . . . . . . . . . . . F. VII . DIFFRACTION AT A BLACK SCREEN. P A R T 1: KIRCHHOFF'S THEORY. KOTTLER . . . . . . . . . . . . . . . . . . . . . . . . . . . .

VOLUME V (1966) O P T I C A L P U M P I N G . c. COHEN.TANNOUDJI. A . K A S ~ L E R. . . . . . . . I. I1. NON-LINEAR OPTICS. P. s. P E R S H A N . . . . . . . . . . . . . . . . INTERFEROMETRY. W . H . STEEL 111. TWO-BEAM

1v. V.

v1. VII .

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

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

1-81 83- 144 145-197

INSTRUMENTS FOR T H E M E A S U R I N G O F O P T I C A L TRANSFER FUNCTIONS. K.

MURATA. . . . . . . . . . . . . . . . . . . . . . . . . . . . . L I G H T REFLECTION FROM FILMS O F CONTINUOUSLY VARYING REFRACTIVE INDEX.R . JACOBSSON. . . . . . . . . . . . . . . . . . . . . . . X-RAYCRYSTAL-STRUCTURE DETERMINATION AS A BRANCHO F PHYSICAL OPTICS.H . LIPSON.C. A . TAYLOR . . . . . . . . . . . . . . . . . . THEW A V E O F A M O V I N G CLASSICAL ELECTRON. J. PlCHT . . . . . . .

V O L U M E VI ( 1 9 6 7 ) RECENTADVANCESI N HOLOGRAPHY. E. N . LEITH. J . U P A T N I E K S . . . . SCA'ITERING O F L I G H T BY ROUGHSURFACES. P. BECKMANN. . . . . .

I. I I. OF T H E S E C O N D O R D E R DEGREEOF COHERENCE. M . I11 . MEASUREMENT FRANCON. S . MALLICK . . . . . . . . . . . . . . . . . . . . . . O F Z O O M LENSES.K . Y A M A J l . . . . . . . . . . . . . . . . IV . DESIGN S O M E APPLICATIONS O F LASERST O INTERFEROMETRY. D . R . H E R R I O T T . V. VI . EXPERIMENTAL S T U D I E S O F INTENSITY FLUCTUATIONS IN LASERS.J . A . ARMSTRONG.A . w . S M I T H . . . . . . . . . . . . . . . . . . . . . SPECTROSCOPY. G . A . VANASSE. H . SAKAI. . . . . . . . . . VII . FOURIER THEORY. VIIl DIFFRACTION AT A BLACKSCREEN. P A R T 11: ELECTROMAGNETIC F. KOTTLER . . . . . . . . . . . . . . . . . . . . . . . . . . .

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

1-52 53-69 71-104 105- 170 171-209 211-257 259-330 331-377

V O L U M E VII (1969) I.

MULTIPLE-BEAMINTERFERENCEA N D NATURAL MODES I N OPEN 1-66 RESONATORS. G . KOPPELMAN . . . . . . . . . . . . . . . . . . . FILTERS. E . I1 . METHODSO F SYNTHESIS FOR DIELECTRIC MULTILAYER 67-137 DELANO. R.J. P E G I S . . . . . . . . . . . . . . . . . . . . . . . 111. ECHOESA N D O P T I C A L FREQUENCIES. I . D . ABELLA . . . . . . . . . . 139- 168 WITH PARTIALLY COHERENT LIGHT. B . J . THOMPSON 169-230 IV . IMAGEFORMATION THEORY OF LASERRADIATION. A . L. MIKAELIAN. M . L. V . QUASI-CLASSICAL 231-297 TER-MIKAELIAN . . . . . . . . . . . . . . . . . . . . . . . . . VI . THEP H O T O G R A P H I C IMAGE. s. O O U E . . . . . . . . . . . . . . . . 299-358 VII . INTERACTION O F VERY I N T E N S E L I G H T WITH FREEELECTRONS.J . H . EBERLY . . . . . . . . . . . . . . . . . . . . . . . . . 359-415


I. 11.

111.

VII

VOLUME VIII (1970) SYNTHETIC-APERTURE OPTICS,J. W. GOODMAN. . . . . . . . . . . THEOPTICAL PERFORMANCE O F THE HUMANEYE,G. A. FRY . . . . L I G H T BEATING SPECTROSCOPY, H. z. C U M M I N S , H. L. SWINNEY . . , ,

,

1-50 51-131 133-200 20 1-237 239-294

A. MUSSET,A. THELEN. . . V. STATISTICAL PROPERTIES O F LASERLIGHT, H. RISKEN . . . . . . . . v1. COHERENCE THEORY OF SOURCE-SIZE COMPENSATION IN INTERFERENCE MICROSCOPY, T. YAMAMOTO . . . . . . . . . . . . . . . . . . . . 295-341 VII. VISION IN COMMUNICATION, H. LEVI . . . . . . . . . . . . . . . . 343-372 VIII. THEORY OF PHOTOELECTRON COUNTING, C. L. MEHTA . . . . . . . . 373-440 IV.

MULTILAYER ANTIREFLECTION COATINGS,

V O L U M E IX (1971) I. 11. 111.

GAS LASERSAND THEIR APPLICATION T O PRECISE LENGTHMEASUREMENTS, A. L. BLOOM . . . . . . . . . . . . . . . . . . . . . . . PICOSECOND LASERPULSES,A. J. DEMARIA, . . . . . . . . . . . . OPTICAL PROPAGATION THROUGH T H E TURBULENT ATMOSPHERE,J. w. STROHBEHN

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

.

. .

. . . . . . . .

.

NETWORKS, E. 0.AMMANN. . . MODELOCKINGI N GASLASERS,L. ALLEN,D. G. C. JONES . . . . . . CRYSTAL OPTICS WITH SPATIAL DISPERSION, v. M. AGRANOVICH,v. L. GINZBURG . . . . . . , . . . . . . . . . . . . . . . . . . . . . VII. APPLICATIONS OF OPTICAL METHODSIN THE DIFFRACTION THEORY OF IV. V. VI.

SYNTHESIS O F OPTICAL BIREFRINGENT

235-280

.

281-310 3 11-407

K.G N I A D E K , J. PETYKIEWICZ .

. . .

. .

73- 122 123-177 179-234

VIII. EVALUATION, D E S I G N AND EXTRAPOLATION METHODS FOR OPTICAL SIGNALS, BASEDO N U S E O F T H E PROLATE FUNCTIONS, B. R. F R I E D E N .

ELASTIC WAVES,

. . .

1-30 31-71

. .

V O L U M E X (1972) BANDWIDTH COMPRESSION O F OPTICAL IMAGES, T. s. HUANG. . . . . 1-44 THEUSE OF IMAGE Tunes AS SHUTTERS, R. w. S M I T H . . . . . . . . 45-87 111. TOOLS OF THEORETICAL QUANTUM OPTICS, M. 0. SCULLY, K. G. WHITNEY 89-135 IV. FIELD CORRECTORS FOR ASTRONOMICAL TELESCOPES, C. G. W Y N N E . . 137-164 v. OPTICAL ABSORPTION STRENGTH O F DEFECTSI N INSULATORS, D. Y. SMITH, D. L. DEXTER . . . . . . . . . . . . . . . . . . . . . . . 165-228 VI. ELASTOOPTIC LIGHTMODULATION AND DEFLECTION, E. K. SITTIC . . . 229-288 VII. Q U A N T U M D E T E C r l O N THEORY, c.w. HELSTROM . . . . . . . . . . 289-369

I. 11.

VOLUME XI (1973) MASTEREQUATION METHODSIN QUANTUM OPTICS,G. S. AGARWAL. . 1-76 RECENTDEVELOPMENTS IN FARINFRARED SPECTROSCOPIC TECHNIQUES, H. YOSHINAGA . . . . . . . . . . . . . . . . . . . . . . . . . . 77-122 111. INTERACTION O F LIGHTAND ACOUSTICSURPACE WAVES, E.G . LEAN . . 123-166 Iv. EVANESCENT WAVES IN OPTICAL IMAGING, 0. BRYNGDAHL . . . . . . 167-221 v. PRODUCTION O F ELECTRONPROBES U S I N G A F I E L D EMISSIONSOURCE, A. V. CREWE. . . . . . . . . . . . . . . . . . . . . . . . . . . 223-246 VI. HAMILTONIAN THEORY OF BEAMMODE PROPAGATION, J. A. ARNAUD . 247-304 VII. GRADIENT INDEXLENSES,E. W. M A R C H A N D.. . . . . . . . . . . . 305-337 I.

11.

VIII


V O L U M E XI1 ( 1 9 7 4 ) I 11.

Ill. IV. V.

v1.

SELF-TRAPPING, AND SELF-PHASE MODULATION OF LASERBEAMS,0. SVELTO. . . . . . . . . . . . . . . . . . . . . 1-51 SELF-INDUCED TRANSPARENCY, R. E. SLUSHER , . . . . . . . . . . . 53-100 MODULATION TECHNIQUES IN SPECTROMETRY, M. HARWIT, J. A. DECKER JR. . . . . . . . . . . , . , . . . . , . . . . . . . . . . . . . 101-162 INTERACTION O F LIGHT WITH MONOMOLECULAR DYE LAYERS,K. H. DREXHAGE. . . . . . . . . . . . . . . . . . , . . . . . . . . . 163-232 THEPHASE TRANSITION CONCEPT AND COHERENCE IN ATOMICEMISSION, R. GRAHAM. . . . . . . . . . . . . . , . . . . . . . . . . . . 233-286 287-344 BEAM-FOIL SPECTROSCOPY. s. BASHKIN. , . , . , , . . . . . . . . SELF-FOCUSING,

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

LAWOF HEATRADIATION FOR A BODY NONEQUILIBRIUM ENVIRONMENT, H. P. BALTES . . . . . . . . . 1-25 THEC A S E FOR AND AGAINSTSEMICLASSICAL RADIATION THEORY, L. M A N D E L .. . . . , . . . . . . . . . . , . , . . . . . . . . . , 27-68 OBJECTIVE AND SUBJECTIVE SPHERICAL ABERRATION MEASUREMENTS OF EYE,w. M. ROSENBLUM, J. L. CHRISTENSEN . . . . . . . 69-91 THE H U M A N INTERFEROMETRIC TESTINGOF SMOOTHSURFACES, G . SCHULZ, J. SCHWIDER. . . . , . . . . . . . . . , . , . , . . . . . . . . . 93-167 SELF FOCUSING OF LASERBEAMSI N PLASMAS AND SEMICONDUCTORS, M. S. SODHA,A.K. GHATAK, V.K. TRIPATHI. . . . . . . . . . . . 169-265 APLANATISMAND ISOPLANATISM, w. T. W E L F O R D . . . . . . . . . . 267-292 O N THE VALIDITY O F KIRCHHOFF’S IN A

11.

Ill. IV. V. VI.

V O L U M E XIV (1977) THESTATISTICS O F SPECKLE PATTERNS, J. c. DAINTY. . . . . . . . . I. TECHNIQUES IN OPTICAL ASTRONOMY, A. LABEYRIE . 11. HIGH-RESOLUTION PHENOMENA IN RARE-EARTH LUMINESCENCE, L. A. RISE111. RELAXATION BERG, M. J. WEBER . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . IV. THEULTRAFAST OPTICAL KERRSHUTTER. M.A. DUGUAY HOLOGRAPHIC DIFFRACTION GRATINGS, G . SCHMAHL, D. RUDOLPH . . V. P. J. VERNIER. . . . . . . . . . . . . . . . . . . VI. PHOTOEMISSION, - A REVIEW,P. J. B. CLARRICOATS . . . , VII. OPTICAL FIBRE WAVEGUIDES

1-46 47-87

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

VOLUME XV (1977) I. 11.

Ill. IV. V.

THEORY O F OPTICAL PARAMETRIC AMPLIFICATION AND OSCILLATION, w. BRUNNER,H. PAUL . . . . . . . . . . . , . . . . . . . . . . . . 1-75 OPTICAL PROPERTIES O F T H I N METALFILMS, P. ROUARD,A. MEESSEN. 77-137 PROJECTION-TYPE HOLOGRAPHY, T. O K O S H l . , . . . . . . . . . . . 139- 185 QUASI-OPTICAL TECHNIQUES O F RADIOASTRONOMY, T. w. C O L E . . . 187-244 FOUNDATIONS O F THE MACROSCOPIC ELECTROMAGNETIC THEORYO F DIELECTRIC MEDIA,J. VAN KRANENDONK, J. E. S l P E . . . . . . . , . 245-350


IX

V O L U M E XVI (1978) LASERSELECTIVE PHOTOPHYSICS A N D PHOTOCHEMISTRY, V. S. LETOKHOV 1-69 RECENTADVANCESIN PHASEPROFILESGENERATION, J. J. CLAIR,C. I. ABITBOL. . . . . . . . . , . . . . . . . . . . . . , . . . . . . 71-117 HOLOGRAMS: TECHNIQUES A N D APPLICATIONS, 111. COMPUTER-GENERATED W.-H. LEE . . . . . . . . . . . . . , . . . . . . . . . . . . . . 119-232 INTERFEROMETRY, A. E. ENNOS . . . . . . . . . . . . . . 233-288 IV. SPECKLE DEFORMATION INVARIANT, SPACE-VARIANT OPTICALRECOGNITION, D. V. CASASENT, D. PSALTIS . . . . . . . . . . . . . . . . . . . . . . 289-356 VI LIGHT EMISSIONFROMHIGH-CURRENT SURFACE-SPARK DISCHARGES, R. E. BEVERLY111 . . . . . . . . . . . . . . . . . . . . . . . . . 357-411 RADIATION THEORYWITHINA QUANTUM-MECHANICAL VII. SEMICLASSICAL FRAMEWORK, I. R. SENITZKY, . . . . . . . . . . . . . . . . . . . 413-448 I. 11.

VOLUME XVII (1980) HETERODYNE HOLOGRAPHIC INTERFEROMETRY, R. DANDLIKER . . . . 1-84 I. E. GIACOBINO, B. CAGNAC 85-162 MULTIPHOTON SPECTROSCOPY, 11. DOPPLER-FREE BETWEENCOHERENCE PROPERTIES OF LIGHT 111. THEMUTUALDEPENDENCE A N D NONLINEAR OPTICALPROCESSES, M. SCHUBERT, B. WILHELMI . . 163-238 INTERFEROMETRY, W. J. TANGO,R. Q. TWISS . . . 239-278 IV. MICHELSON STELLAR SELF-FOCUSING MEDIAWITH VARIABLE INDEX OF REFRACTION,A. L. V. MIKAELIAN, . . . . . . , , . . . . . . . . . . . . . , . . . . 279-345

V O L U M E XVIII (1980) I.

GRADED INDEX OPTICALWAVEGUIDES:A REVIEW,A. GHATAK,K. THYAGARAJAN . . . . . . , . . . . , . . . . . . . , , , , . . . 1- 126 11. PHOTOCOLJNT STATISTICS OF RADIATIONPROPAGATING THROUGH MEDIA,J. PERINA . . . . . . . . . . . . 127-203 RANDOMA N D NONLINEAR IN LIGHTPROPAGATION I N A RANDOMLY INHOMO111. STRONG FLUCTUATIONS GENEOUS MEDIUM,V. I. TATARSKII, V. U. ZAVOROTNYI . . . . . . . . 204-256 OF CAUSTICSAND THEIR DIFOPTICS: MORPHOLOGIES IV. CATASTROPHE FRACTION PATTERNS, M. V. BERRY,C. UPSTILL. . . . . . . . . . . . 257-346

V O L U M E X I X (1981) THEORY OF INTENSITY DEPENDENT RESONANCE LIGHTSCATTERINGA N D RESONANCE FLUORESCENCE, B. R. MOLLOW . . . . . . . . . . . . . 1-43 11. SURFACE A N D SIZE EFFECTS ON THE LIGHT SCATTERING SPECTRA OF SOLIDS,D. L. MILLS,K. R. SUBBASWAMY . . . . . . . . . . . . . . 45-137 111. LIGHT SCATTERINGSPECTROSCOPY OF SURFACEELECTROMAGNETIC WAVESI N SOLIDS,S. USHIODA. . . . . . . . . . . . . . , . . . . 139-210 IV OF OPTICAL DATA-PROCESSING, H. J. BUTTERWECK. . . . 21 1-280 PRINCIPLES V. TURBULENCE I N OPTICAL ASTRONOMY, F. THEEFFECTS OF ATMOSPHERIC RODDIER . . . . . . . . , , . . . . . . . . . . . . . . . . . . 281-376 I

X


VOLUME X X (1983) I.

SOME NEWOPTICAL DESIGNSFOR ULTRA-VIOLET BIDIMENSIONAL DETECASTRONOMICAL OBJECTS, G. COURTBS, P. CRUVELLIER, M. DETAILLE, M. S A ~ S S E. . . . . . . . . . . . . . . . . . . . . . . 11 SHAPING AND ANALYSIS OF PICOSECOND LIGHTPULSES,C. FROEHLY, B. COLOMBEAU, M. VAMPOUILLE. . . . . . . . . . . . . . . . . . . 111. MULTI-PHOTON SCATTERING MOLECULAR SPECTROSCOPY,S. KIELICH . HOLOGRAPHY, P. HARIHARAN. . . . . . . . . . . . . . . IV. COLOUR V. GENERATION OF TUNABLE COHERENT VACUUM-ULTRAVIOLET RADIATION, W. JAMROZ,B. P. STOICHEFF . . . . . . . . . . . . . . . . . . . . TION OF

1-62 63-154 155-262 263-324 325-380

VOLUME XXI (1984) RIGOROUS VECTOR THEORIES OF DIFFRACTION GRATINGS, D. MAYSTRE . L. A. LUGlATO . . . . . . . . . . . 11. THEORYOF OPTICAL BISTABILITY, A N D ITS APPLICATIONS, H.H. BAR RE^ . . . 111. THE RADONTRANSFORM THEORY A N D APPLICATIONS, N. M. CEGLIO, IV. ZONE PLATE CODED IMAGING: D. W. SWEENEY . . . . . . . . . . . . . . . . . . . . . . . . . . FLUCTUATIONS, INSTABILITIES A N D CHAOSI N THE LASER-DRIVEN NONV. LINEAR RINGCAVITY. J . c. ENGLUND, R. R. SNAPP,w. c. SCHIEVE . . . 1.

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

VOLUME XXII (1985) OPTICALA N D ELECTRONICPROCESSINGOF MEDICALIMAGES, D. . . . . . . . . . . . . . . . . . . . . . . . . . . . . MALACARA 1-76 FLUCTUATIONS IN VISION, M.A. BOUMAN, W. A. VAN DE GRIND, 11. QUANTUM 77-144 P. ZUIDEMA. . . . . . . . . . . . . . . . . . . . . . . . . . . OF BROAD-BANDLASER 111. SPECTRAL A N D TEMPORALFLUCTUATIONS 145-196 RADIATION, A. V. MASALOV. . . . . . . . . . . . . . . . . . . . METHODSOF PLASMA DIAGNOSTICS, G. V. OSTROVSKAYA, IV. HOLOGRAPHIC Yu. I. OSTROVSKY . . . . . . . . . . . . . . . . . . . . . . . . . 197-270 FRINGE FORMATIONS I N DEFORMATION A N D VIBRATION MEASUREMENTS V. USING LASERLIGHT,I. YAMAGUCHI . . . . . . . . . . . . . . . . 271-340 VI. WAVEPROPAGATION I N RANDOMMEDIA:A SYSTEMSAPPROACH, R. L. FANTE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 341-398 1.

VOLUME XXIII (1986) ANALYTICAL TECHNIQUES FOR MULTIPLESCATTERING FROM ROUGH 1-62 SURFACES, J. A. DESANTO,G . S. BROWN. . . . . . . . . . . . . . . PARAXIAL THEORYIN OPTICAL DESIGNIN TERMSOF GAUSSIAN BRACKETS, 11. K. TANAKA . . . . . . . . . . . . . . . . . . . . . . . . . . . 63-1 12 FILMS PRODUCED BY ION-BASED TECHNIQUES, P. J. MARTIN, R. P. 111. OPTICAL NETTERFIELD . . . . . . . . . . . . . . . . . . . . . . . . . . . 113-182 A. TONOMURA . . . . . . . . . . . . . . 183-220 IV. ELECTRONHOLOGRAPHY, PRINCIPLES OF OPTICALPROCESSING WITH PARTIALLY COHERENT LIGHT, V. F.T. S. YU . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221-276 I.


XI

V O L U M E XXIV (1987) I. 11. 111. IV. V.

FRESNEL LENSES,H. NISHIHARA, T. SUHARA, . . . . . . . . PHENOMENA, L. ROTHBERG . . . . . DEPHASING-INDUCED COHERENT INTERFEROMETRY WITH LASERS,P. HARIHARAN . . . . . . . . . . . UNSTABLERESONATOR MODES,K. E. OUGHSTUN. . . . . . . . . . INFORMATION PROCESSING WITH SPATIALLY INCOHERENT LIGHT, I. GLASER. . . . . . . . , . . . . . . . . . . . . . . . . . . .

MICRO

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

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DYNAMICAL INSTABILITIES AND P. MANDEL,L. M . NARDUCCI

PULSATIONS I N

.

LASERS,N. B. ABRAHAM,

. , , . . . . . . 1-190 191-278 11. COHERENCE IN SEMICONDUCTOR LASERS,M. OHTSU, T. TAKO. . . . . A N D DESIGN OF OPTICAL ARRAYS,WANGSHAOMIN, L. RONCHI 279-348 111. PRINCIPLES IV. ASPHERICSURFACES, G. SCHULZ. . . . . . . . . . . . . . . . . . 349-416 . . . . . . . . .

V O L U M E XXVI (1988) I. 11.

111. IV. V.

PHOTON BUNCHING A N D ANTIBUNCHING, M. c . TEICH, B. E. A. SALEH . NONLINEAR OPTICS OF LIQUID CRYSTALS, I. c. KHOO . . . . . . . . . SINGLE-LONGITUDINAL-MODE SEMICONDUCTOR LASERS, G. P. AGRAWAL RAYSA N D CAUSTICS AS PHYSICAL OBJECTS, YU. A. KRAVTSOV. . . . . PHASE-MEASUREMENT INTERFEROMETRY TECHNIQUES. K. CREATH. . .

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

V O L U M E XXVII (1989) I. THESELF-IMAGING PHENOMENON A N D ITS APPLICATIONS, K. PATORSKI 11. AXICONS A N D MESO-OPTICAL IMAGING DEVICES, L. M. SOROKO . . . . 111. NONIMAGING OPTICS FOR FLUXCONCENTRATION, I. M. BASSETT, w. T. WELFORD,R. WINSTON . . . . . . . . . . . . . . . . . . . . . . IV. NONLINEAR W A V E PROPAGATION IN P L A N A R STRUCTURES, D. MIHALACHE, M. BERTOLOTTI,c. S I B I L I A . . . . . . . . . . . . . . . . . . . . . V. GENERALIZED HOLOGRAPHY WITH APPLICATION TO INVERSE SCATTERING A N D INVERSE SOURCE PROBLEMS, R. P. PORTER . . . . . . . . . . .

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

V O L U M E X X V I I I (1990) 1. 11.

DIGITAL HOLOGRAPHY- COMPUTER-GENERATED HOLOGRAMS,0. BRYNGDAHL,F. WYROWSKI QUANTUM M E C H A N I C A L LIMITIN OPTICAL PRECISION MEASUREMENT AND COMMUNICATION,Y. YAMAMOTO, S. MACHIDA, s. SAITO,N. IMOTO, T. YANAGAWA, M. KITAGAWA, G. BJORK . . . . . . . . . . . . . . . THE QUANTUMCOHERENCE PROPERTIES OF STIMULATED RAMAN SCATTERING, M.G. RAYMER,I.A. WALMSLEY. . . . . . . . . . . . ADVANCED EVALUATION TECHNIQUES I N INTERFEROMETRY, J. SCHWIDER QUANTUM JUMPS,R . J . COOK . . . . . . . . . . . . . . . . . . . I

III IV. V.

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PREFACE This volume follows the tradition of most of its predecessors in presenting five authoritative review articles on optics and related subjects. The first article deals with important components of many opto-electronic systems, namely waveguide diffraction gratings. Such components are used as input/output couplers, filters, lenses and reflectors, for example. The article presents an account of the use of waveguide gratings as well as a quantitative review of the properties of optical waveguides. The second article discusses the phenomenon of enhanced backscattering. which has attracted a good deal of attention in recent years. Enhanced backscattering is a subtle manifestation of coherence effects in multiple scattering in random media, and it is somewhat analogous to effects associated with electron localization in solids, which were discovered some years earlier. The article presents accounts of research carried out in this area chiefly, but not entirely, in the Soviet Union. Scientists in other countries will undoubtedly welcome the opportunity to learn about these investigations from a review article written in the English language. In the next article the generation and propagation of ultrashort optical pulses is discussed, as well as some linear and non-linear effects which arise when such pulses propagate in free space or in material media. The article also includes accounts of the use of ultrashort pulses in the fields of optical communications and data processing. The fourth article presents a brief review of several interferometric methods for overcoming the degradation of image quality caused by atmospheric fluctuations. These include the so-called speckle masking method, speckle spectroscopy methods and optical long baseline interferometry with arrays of large telescopes. The concluding article deals with non-linear optical properties of semiconductors and metal crystallites in dielectric matrices. Good understanding of these properties is required when choosing the most appropriate materials for manufacturing devices which utilize several non-linear optical effects. Such devices would be particularly useful in connection with processing and transmission of information and their performance might eventually surpass those of present-day electronics.

XIV

It is with sadness that I record here the death last September of Walter T. Welford, a valuable member of the Editorial Advisory Board of Progress in Optics. Welford was a member of the Board since the inception of this series and was, in fact, the only member who still served on it thirty years later. He not only provided the Editor with much helpful advice, but was himself the author or co-author of three articles published in these volumes. Welford has, of course, been well-known for his numerous contributions to optics and for some fine textbooks and monographs. Those of us who were fortunate to have known him will remember him with affection. EMILWOLF Department of Physics and Astronomy University of Rochester Rochester, NY 14627, USA March 1991

CONTENTS 1. OPTICAL WAVEGUIDE DIFFRACTION GRATINGS: COUPLING BETWEEN

G U I D E D MODES by DENNISG . HALL(ROCHESTER.NY. USA)

$ 1. INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . $ 2. USES FOR WAVEGUIDE GRATINGS. . . . . . . . . . . . . . . . . . . . 2.1. General discussion . . . . . . . . . . . . . . . . . . . . . . . . . 2.2. Interactions between guided waves . . . . . . . . . . . . . . . . . . 2.3. Interactions between guided waves and the radiation field . . . . . . . . Q 3 . MODESSUPPORTED BY PLANAROPTICALWAVEGUIDES . . . . . . . . . . . 3.1. Bound modes of the step-index optical waveguide . . . . . . . . . . . . 3.2. Bound modes of the graded-index optical waveguide . . . . . . . . . . . 3.3. Bound modes of the nonlinear optical waveguide . . . . . . . . . . . . 3.4. Radiation modes of the step-index waveguide . . . . . . . . . . . . . . $ 4 . NONPLANAR OPTICALWAVEGUIDES. . . . . . . . . . . . . . . . . . . $ 5 . COUPLING BETWEENGUIDED WAVES. . . . . . . . . . . . . . . . . . . 5.1. Ideal-mode expansion and coupled-mode equations . . . . . . . . . . . 5.2. Ideal-mode expansion - An alternative approach (TE) . . . . . . . . . . 5.3. Solution of the coupled-mode equations . . . . . . . . . . . . . . . . 5.4. Coupling between TM-guided waves . . . . . . . . . . . . . . . . . 5.5. Local normal mode expansion and coupled-mode equations (TM) . . . . . 5.6. Summary of coupled-mode treatments . . . . . . . . . . . . . . . . . 5.7. Perturbative treatment . . . . . . . . . . . . . . . . . . . . . . . 5.8. TE-TM mode conversion . . . . . . . . . . . . . . . . . . . . . . $ 6. SUMMARY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . LISTOF SYMBO~.S. . . . . . . . . . . . . . . . . . . . . . . . . . . . . REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3 5 5 8 13 14 14 22 23 26 29 30 32 39 42 46 49 53 54 58 59 60 62

I1. ENHANCED BACKSCATTERING IN OPTICS by Yu.N. BARABANENKOV (Moscow, USSR). Yu.A. KRAVTSOV (Moscow. USSR).

V.D. OZRIN(Moscow. USSR) and A.I. SAICHEV(NIZHNINOVGOROD.USSR) $ 1 . INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . $ 2. ENHANCED BACKSCATTER FROM SOLIDSIMMERSED I N A TURBULENT MEDIUM 2.1. Absolute effect of enhanced backscatter: A point transmitter and a point scatterer in a turbulent medium . . . . . . . . . . . . . . . . . . . 2.1.1. Pure effect of enhanced backscatter . . . . . . . . . . . . . . . 2.1.2. A phase screen . . . . . . . . . . . . . . . . . . . . . . . 2.1.3. Spatial redistribution of the scattered intensity . . . . . . . . . .

67 69 69 69 72 12

XVI

CONTENTS

Backscatter enhancement under weak fluctuations of intensity . . Saturated fluctuations of intensity . . . . . . . . . . . . . . A lens interpretation of backscatter enhancement . . . . . . . . Backscatter-enhancement interpretation relying on multipath coherent effects . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.8. Experimental evidence . . . . . . . . . . . . . . . . . . . . 2.1.9. Enhancement of backscattered intensity fluctuations: Residual correlation of the intensity . . . . . . . . . . . . . . . . . . . . . . 2.1.10. Scattering from small inhomogeneities in a turbulent medium: A hybrid approach . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.1 1. Polarization effects . . . . . . . . . . . . . . . . . . . . . . 2.2. Extended transmitters, scatterers and receivers . . . . . . . . . . . . . 2.2.1. Wave description within the parabolic equation framework . . . . . 2.2.2. Statistical description of backscattered waves in the region of saturated fluctuations of intensity . . . . . . . . . . . . . . . . . . . . 2.2.3. Effect of extended size of a reflector . . . . . . . . . . . . . . 2.2.4. Effect of long-distance correlations and partial reversal of the wavefront 2.2.5. Enhanced backscattering in the focal plane of a lens . . . . . . . 2.2.6. Enhancement of radiant intensity . . . . . . . . . . . . . . . . 2.2.7. Giant backscatter enhancement in laser sounding of the ocean . . . 2.2.8. Backscattering of pulse signals . . . . . . . . . . . . . . . . . 2.2.9. Moving random inhomogeneities of the medium . . . . . . . . . 2.3. Reflection from wavefront-reversing mirrors embedded in a random inhomogeneous medium . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.1. Compensation of the effect of random inhomogeneities upon the reflectedwave . . . . . . . . . . . . . . . . . . . . . . . . 2.3.2. Average intensity of a wave reflected for a WFR mirror: Effect of superfocusing . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.3. Effect of a drift of random inhomogeneities on the efficiency of WFR mirrors . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.4. Magic-cap effects: Compensation of backscattering from small-scale inhomogeneities by a WFR mirror . . . . . . . . . . . . . . . $ 3. ENHANCED BACKSCATTERING BY A RANDOM MEDIUM . . . . . . . . . . . 3.1. Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2. General theory of multiple scattering: Ladder and maximally crossed diagrams 3.3. Transfer equation and enhanced backscattering . . . . . . . . . . . . . 3.4. Angular distribution of backscattered intensity . . . . . . . 3.5. Diffusion approximation . . . . . . . . . . . . . . . . . . . . . . . 3.6. Polarization effects . . . . . . . . . . . . . . . . . . . . . . . . . 3.7. Coherent backscattering in the presence of time-reversal noninvariant media 3.7.1. A weakly gyrotropic medium in a magnetic field . . . . . . . . . 3.7.2. Brownian motion of scatterers . . . . . . . . . . . . . . . . . 3.8. Coherent effects in the average field: Influence on backscatter intensity envelope $ 4. MULTIPATH COHERENT EFFECTS IN SCATTERING FROM A LIMITEDCLUSTER OF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . SCATTERERS 4.1. Enhanced backscattering from a particle . . . . . . . . . . . . . . . . 4.1.1. Single particle near an interface . . . . . . . . . . . . . . . . 4.1.2. Combined action of a rough surface, turbulence and multipath coherent effects . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.3. Existence of backscatter enhancement under time-varying conditions 2.1.4. 2.1.5. 2.1.6. 2.1.7.

14 74 75 16 79 82 84 87 87 87 91 92 95 98 101

105 110 110 111

111

113 117

119 123 123 125 135 139 142 148 162 163 166 167 168 168 168 170 171

CONTENTS

XVII

4.1.4. Kettler effect . . . . . . . . . . . . . . . . . . . . . . . . 4.1.5. Particle in a waveguide . . . . . . . . . . . . . . . . . . . . 4.2. Enhanced backscattering by a system of two scatterers . . . . . . . . . 4.2.1. Watson equations (scalar problem) . . . . . . . . . . . . . . . 4.2.2. Polarization effects . . . . . . . . . . . . . . . . . . . . . . 4.3. More involved scatterer system and geometries . . . . . . . . . . . . . 4.3.1. Cluster of N scatterers: Paired and single scattering channels . . . 4.3.2. Scattering by bodies of intricate geometries . . . . . . . . . . . 4.3.3. Coherent effects in diffraction by large bodies . . . . . . . . . . $ 5. ENHANCED BACKSCATTERING FROM ROUGHSURFACES . . . . . . . . . . . 5.1. Trend to intensity peaking in the antispecular direction . . . . . . . . . 5.2. Backscatter enhancement involving surface waves . . . . . . . . . . . . EFFECTS I N ALLIEDFIELDSOF PHYSICS . . . . . . . . . . . . . $ 6. RELATED 6.1. Enhanced backscattering in acoustics . . . . . . . . . . . . . . . . . 6.2. Effects in the radio wave band . . . . . . . . . . . . . . . . . . . . 6.3. Other effects of double passage through random media . . . . . . . . . 6.4. Coherent backscattering of particles from disordered media . . . . . . . $ 7. CONCLUSION. . . . . . . . . . . . . . . . . . . . . . . . . . . . . ACKNOWLEDGMENT . . . . . . . . . . . . . . . . . . . . . . . . . . . . REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

171 172 173 173 175 176 176 180 181 183 183 186 186 186 187 188 188 189 190 190

111. GENERATION AND PROPAGATION OF ULTRASHORT OPTICAL PULSES by I.P. CHRISTOV (SOFIA.BULGARIA)

3 1. INTRODUCTION. . . . . . . . . . . . . . . . . . . . . . . . . . . . $ 2. THEORETICAL BACKGROUND. . . . . . . . . . . . . . . . . . . . . . 2.1. Propagation of optical pulses through a resonant medium . . . . . . . 2.2. Propagation in a transparent linear medium . . . . . . . . . . . . . 2.2.1. Regular pulses . . . . . . . . . . . . . . . . . . . . . . . . 2.2.2. Partially coherent pulses . . . . . . . . . . . . . . . . . . . . 2.3. Nonlinear propagation of optical pulses . . . . . . . . . . . . . . . 2.3.1. Regular pulses . . . . . . . . . . . . . . . . . . . . . . . 2.3.2. Partially coherent pulses . . . . . . . . . . . . . . . . . . . 9 3. GENERATION OF FEMTOSECOND OPTICALPULSES . . . . . . . . . . . . 3.1. Broadband media . . . . . . . . . . . . . . . . . . . . . . . . . 3.2. Mode-locking techniques . . . . . . . . . . . . . . . . . . . . . 3.2.1. Passive mode-locking . . . . . . . . . . . . . . . . . . . . 3.2.2. Synchronously pumped mode-locked (SPML) lasers . . . . . . . 3.2.3. Miscellaneous techniques . . . . . . . . . . . . . . . . . . . 3.3. Amplification of femtosecond pulses . . . . . . . . . . . . . . . . . 3.4. Pulse compression . . . . . . . . . . . . . . . . . . . . . . . . $ 4 . PROPAGATION EFFECTS . . . . . . . . . . . . . . . . . . . . . . . . 4.1. Free-space propagation . . . . . . . . . . . . . . . . . . . . . . 4.1.1. Regular pulses . . . . . . . . . . . . . . . . . . . . . . . 4.1.2. Partially coherent pulses . . . . . . . . . . . . . . . . . . . 4.2. Transmission through optical components . . . . . . . . . . . . . . 4.2.I . Ray-optics approach . . . . . . . . . . . . . . . . . . . . . 4.2.2. Wave-optics approach . . . . . . . . . . . . . . . . . . . .

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201 202 202 209 209 212 213 213 218 220 220 221 222 227 231 235 239 244 245 245 247 249 250 253

CONTENTS

XVllI

4.3. Propagation through dispersive systems . . . . . . . . . . . . . . . . 4.3.1. Temporal modes representation of a propagating pulse . . . . . . . 4.3.2. Propagation of a short pulse in a dispersive medium . . . . . . . . 4.3.3. Pulse shaping . . . . . . . . . . . . . . . . . . . . . . . . . 4.4. Propagation in a nonlinear medium . . . . . . . . . . . . . . . . . . 4.4.1. Formation of bright solitons . . . . . . . . . . . . . . . . . . 4.4.2. Formation of dark solitons . . . . . . . . . . . . . . . . . . . 4.4.3. The soliton self-frequency shift . . . . . . . . . . . . . . . . . 4.4.4. Nonlinear propagation of chirped and noise pulses . . . . . . . . . 4.5. Femtosecond pulses in information systems . . . . . . . . . . . . . . 4.5.1. Soliton-based communication systems . . . . . . . . . . . . . . 4.5.2. Image processing by optical pulses . . . . . . . . . . . . . . . . $ 5. CONCLUSION. . . . . . . . . . . . . . . . . . . . . . . . . . . . . ACKNOWLEDGMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

256 256 257 261 269 269 271 274 275 276 276 279 284 284 284

IV . TRIPLE-CORRELATION IMAGING IN OPTICAL ASTRONOMY by G. WEIGELT (BONN. FED. REP. GERMANY) $ 1. INTRODUCTION

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

$ 2. SPECKLE MASKING: BISPECTRUMOR TRIPLECORRELATION PROCESSING . . . $ 3. OBJECTIVE PRISM SPECKLE SPECTROSCOPY . . . . . . . . . . . . . . . . $ 4 . WIDEBAND PROJECTION SPECKLE SPECTROSCOPY . . . . . . . . . . . . . $ 5. OPTICAL LONG-BASELINE INTERFEROMETRY AND APERTURE SYNTHESIS . . . $ 6. CONCLUDING REMARKS. . . . . . . . . . . . . . . . . . . . . . . . ACKNOWLEDGMENT . . . . . . . . . . . . . . . . . . . . . . . . . . . . APPENDIX A. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

295 296 309 309 31 1 315 316 316 317

V . NONLINEAR OPTICS IN COMPOSITE MATERIALS

.

1 Semiconductor and Metal Crystallites in Dielectrics by C. FLYTZANIS, F . HACHE.M.C. KLEIN.D . RICARDand PH. ROUSSIGNOL (Palaiseau.

France) $ 1. INTRODUCTION

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

$ 2. FABRICATION AND CHARACTERIZATION TECHNIQUES . . 2.1. Fabrication techniques . . . . . . . . . . . . . . 2.1.1. Metal crystallites . . . . . . . . . . . . . . 2.1.2. Semiconductor crystallites . . . . . . . . . 2.2. Characterization techniques . . . . . . . . . . . . 2.2.1. Structure and size determination . . . . . . 2.2.2. Optical techniques . . . . . . . . . . . . . $ 3. CONFINEMENT EFFECTS. . . . . . . . . . . . . . . . 3.1. Basic model . . . . . . . . . . . . . . . . . . . 3.2. Dielectric confinement . . . . . . . . . . . . . . 3.2. I . Linear regime: Effective-medium approach . . 3.2.2. Nonlinear regime . . . . . . . . . . . . . .

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

323 325 325 325 327 331 331 334 338 338 339 339 341

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xtx

3.3. Quantum confinement . . . . . . . . . . . . . . . . . . . . . . . . 345 3.3.1. Basic model . . . . . . . . . . . . . . . . . . . . . . . . . 345 3.3.2. Quantum-confined states and wave functions . . . . . . . . . . . 350 3.3.2.1. Metal crystallites . . . . . . . . . . . . . . . . . . . . 351 3.3.2.2. Semiconductor crystallites . . . . . . . . . . . . . . . . 353 3.3.3. Level broadening . . . . . . . . . . . . . . . . . . . . . . . 356 3.3.3.1. Metal crystallites . . . . . . . . . . . . . . . . . . . . 356 3.3.3.2. Semiconductor crystallites . . . . . . . . . . . . . . . . 359 $ 4. NONLINEAR OPTICAL PROPERTIES OF METALCOMPOSITES. . . . . . . . . 368 $ 5. NONLINEAR OPTICAL PROPERTIES OF SEMICONDUCTOR COMPOSITES: THEORY 315 $ 6. N O N L I N E A R ~ P TPROPERTIES ~CAL OF SEMICONDUCTOR COMPOSITES: EXPERIMENTAL STUDIES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 383 6.1. Large semiconductor crystallites . . . . . . . . . . . . . . . . . . . 384 6.1.1. Frequency and intensity dependence of optical nonlinearities . . . . 384 6.1.2. Temporal evolution of optical processes: Photodarkening . . . . . . 390 6.2. Quantum-confined crystallites . . . . . . . . . . . . . . . . . . . . 399 6.2. I . Enhancement of the optical Kerr effect . . . . . . . . . . . . . . 399 6.2.2. Electroabsorption: Static Stark shift and Franz-Keldysh effect . . . 40 1 $ 7. CONCLUSIONS AND EXTENSIONS . . . . . . . . . . . . . . . . . . . . . 404 REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 406 AUTHOR INDEX . . . . . . . . . . . SUBJECT INDEX . . . . . . . . . . . CUMULATIVE INDEX. VOLUMES I-XXIX

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E. WOLF, PROGRESS IN OPTICS XXIX 0 ELSEVIER SCIENCE PUBLISHERS B.V., 1991

I

OPTICAL WAVEGUIDE DIFFRACTION GRATINGS: COUPLING BETWEEN GUIDED MODES BY

DENNISG. HALL The Institute of Optics University of Rochester Rochester, New York 14627. USA

CONTENTS PAGE

§ 1. INTRODUCTION

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

3

. . . . . . . . . .

5

§ 2 . USES FOR WAVEGUIDE GRATINGS

§ 3. MODES SUPPORTED BY PLANAR OPTICAL WAVE-

GUIDES

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

14

§ 4 . NONPLANAR OPTICAL WAVEGUIDES

. . . . . . . . 29

§ 5 . COUPLING BETWEEN GUIDED WAVES

. . . . . . . . 30

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

59

LIST OF SYMBOLS . . . . . . . . . . . . . . . . . . . . .

60

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

62

§ 6 . SUMMARY

REFERENCES

0

1. Introduction

STEWART MILLERintroduced the term “integrated optics” in 1969 to refer to the miniaturized optical systems he envisioned as important for the future of optical communications. Two subsequent decades of research and development in this area, along with major breakthroughs in the optical fiber and semiconductor laser arenas, have led to the demonstration of many integrated optical components, devices, and systems, and to the introduction of commercial products that make use of this technology. Furthermore, interest in integrated optics as a basic technology has broadened to include not only telecommunications, but also other applications such as optical sensors, information storage and processing, medical instrumentation, navigation, and computing, to name just a few. In addition, there is a renewed emphasis on the importance of making the technology of integrated optics compatible with that of integrated electronics. The currently widespread use of the term “integrated optoelectronics” is a reflection of the attitude that optics and electronics are complementary technologies. The central idea behind the concept of an integrated optical system is the ability to process and manipulate light that is trapped within the confines of an optical waveguide. Here, the term “light” is used in a loose sense. The wavelengths (A) of interest in both integrated and fiber optics are, for the most part, in the near-infrared region of the spectrum, with wavelengths 0.8 < A < 2 pm, rather than in the visible region. Most, but not all, optical waveguide structures confine light by the mechanism of total internal reflection (TIR). Although there are many specific types of optical waveguides, the most important distinction to be drawn is based on dimensionality. A planar, or slab, optical waveguide consists of a layer of elevated refractive index bounded above and below by regions of lower refractive index. Such a structure provides confinement along only one transverse coordinate axis, as illustrated in fig. l a for a step-index, planar optical waveguide. A geometrical optics construct that illustrates a ray trapped by TIR between two surfaces also appears in fig. la. Another type of optical waveguide provides confinement along two transverse coordinate axes (fig. lb). The refractive index boundaries in fig. 1 are depicted as sharp, but this is not an essential feature of an optical waveguide. Both graded-index and step-index structures are in common use. 3

4

WAVEGUIDE DIFFRACTION GRATINGS

F 4 ) ........

(b)

Fig. 1. (a) Planar, or slab, optical waveguide. The refractive index n,ofthe film layer of thickness h must exceed that for each of the substrate (n,) and cover (n,) media. Refractive index barriers appear only along the x-direction.(b) Three-dimensional optical waveguide. The refractive index n , within the guiding structure exceeds that outside the structure along both transverse directions.

This chapter focuses on one important structure for integrated optical/optoelectronic systems: the waveguide diffraction grating. Since the diffraction grating is a familiar component for conventional optical systems, it is logical to assume that it will be for integrated optical systems as well. This has been demonstrated by the use of waveguide gratings in integrated optics for input/output couplers, filters, lenses, Bragg reflectors, distributed reflectors in lasers, and as phase-matching elements for nonlinear interactions. The fact that electromagnetic waves propagating within an optical waveguide exhibit spatial profiles that depend on the transverse coordinates complicates theoretical treatments of the interaction with waveguide diffraction gratings. Despite numerous theoretical investigations, one case has proved particularly troublesome: the Bragg reflection of a guided wave within a corrugated planar optical waveguide. The planar waveguide supports modes with either of two polarizations - transverse electric (TE) or transverse magnetic (TM). These are defined later in this chapter. A guided wave of either polarization incident on a waveguide grating generates a strong back-reflected guided wave if the Bragg

I , § 21

USES FOR WAVEGUIDE GRATINGS

5

condition is satisfied at least approximately. Almost all theoretical treatments of this problem are in agreement when both the incident and Bragg-reflected waves are TE waves. This is not the case, however, for TM waves, for which theoretical treatments are in serious disagreement. Recent theoretical and experimental efforts appear to have resolved this issue satisfactorily. This chapter describes the essential features of the guided-wave Bragg reflection problem that are crucial for a proper treatment of the problem. Sufficient preliminary material on the properties of optical waveguide modes in several structures is included to introduce the reader unfamiliar with the subject to the more important features common to all optical waveguides. Since a full discussion of both the theoretical controversy and its resolution has not yet appeared, sufficient theoretical detail has been included, particularly in the later sections, to allow others to carry out the various calculations. Hence, the introductory material is essential to make this chapter self-contained. A qualitative review of the uses of the waveguide gratings mentioned earlier is followed by a more quantitative review of the properties of optical waveguides, with emphasis on the step-index planar waveguide. The step-index planar waveguide lends itself to relatively straightforward analysis while revealing the essential qualitative features that are common to all optical waveguides. Finally, the interactions between guided waves and waveguide gratings are considered from several theoretical points of view.

g 2. Uses for Waveguide Gratings 2.1. GENERAL DISCUSSION

Waveguide diffraction gratings can be fabricated as a periodic or near-periodic modulation of either the refractive index or one, or more, of the boundaries of an optical waveguide as illustrated in fig. 2. The surface corrugation grating is the more common, since it can be implemented in almost any solid material. Such surface gratings are usually prepared by recording the interference pattern, formed when the two halves of a laser beam recombine at a selected angle, in a layer of photoresist deposited onto the substrate of interest. After the photoresist has been developed, it serves as a mask for substrate etching by techniques such as ion-milling or reactive ion etching. The photoresist mask protects certain areas of the substrate while the etchant attacks the exposed areas. In this way the mask pattern is transferred into the substrate material.

6


PLANAR OPTICAL WAVEGUIDE


(b)

Fig. 2. Two types ofwaveguide diffraction gratings with period A. (a) A periodic variation of the refractive index near the surface. (b) A periodic surface corrugation.

A similar procedure can be used based on electron-beam lithography rather than photolithography. There are two main uses for waveguide gratings in integrated optics. The first use, illustrated in fig. 3a, involves coupling between the radiation field and a bound mode of the optical waveguide. As the bound modes use total internal reflection, there is no exterior angle of incidence for which an external beam of light can be made to excite a bound mode of a waveguide with flat surfaces by refraction. Similarly, it is not possible for a guided mode to radiate in the absence of some coupling mechanism. The grating provides the necessary coupling when the following condition is fulfilled :

B=

2 zm n, (y) sine + -, C A

where /?is the propagation constant (along z ) of the guided wave, A is the grating period, m is an integer, o is the (angular) frequency of the optical wave, c is the speed of light in vacuum, and the angle B and the refractive index n, are identified in the figure. This type of interaction is clearly useful for coupling light into or out of an optical waveguide. The second use, illustrated in fig. 3b involves coupling between two waves that are both bound modes of the optical waveguide. The grating can be used to deflect an incident guided mode into a different direction, or to convert a guided mode of one order into a guided wave of another order, or both. This


SIDE VIEW

INCIDENT LIGHT

nC

A

GUIDED WAVE

P

b

Z

“s

(4 TOP VIEW

GUIDED WAVE

CORRUGATED

GUIDE WAVE PLANAR OPTICAL WAVEGUIDE

Fig. 3. (a) Light incident on a corrugated section of an optical waveguide can excite a guided mode of the structure. The grating acts as a phase-matching element to permit coupling between a guided mode and the radiation field. (b) A corrugated section of an optical waveguide can also provide coupling between two guided waves. In this example, a guided wave is Bragg reflected into a different direction within the waveguide.

type of interaction can be used for “in-plane” functions, examples of which appear in the following sections. It is the period of the grating that determines which type of interaction takes place. A specific example will make this clearer. Consider the waveguide configuration in fig. 3a, but from the point of view ofthe guided wave interacting with the grating to produce another optical wave. If we define the effective index of refraction N according to

where , Iis the optical wavelength (in vacuum), then it is not difficult to show that for guided wave propagation along z, the following first-order (rn = 1)

8


[I, 5 2

phenomena occur for the indicated ranges of the ratio of the grating period to the wavelength, A/A:

( N + n,)- < A/1 < ( N - n,)Radiation into the cover medium (x > h ) : Radiation into the substrate medium (x < 0): ( N + n,)- < A/1 < ( N - n,)Back reflection (first-order) : A/A = ( 2 N ) - I.

I. I.

First-order back reflection (or Bragg reflection) occurs when a guided wave propagating along t z interacts with the grating to produce a guided wave of the same type propagating in the - z direction. Note that since n, < N < n,for n, 2 n,, a point that will be discussed later in this chapter, the smallest period in the preceding list is required for backreflection; radiation into either the substrate or cover media requires a period A/A > ( 2 N ) - I. There is some degree of overlap of the range of periods that produce radiation into the two media. For the usual case of n, 3 n,, this means that radiation into the cover medium is always accompanied by radiation into the substrate, but that a range of 1 exists that produces radiation into only the substrate (refractive index n = n,).

2.2. INTERACTIONS BETWEEN GUIDED WAVES

An extensive literature exists that describes various demonstrations of the use of waveguide gratings. In one of the first such demonstrations, PENNINGTON and KUHN [ 197 11 used gratings formed in a layer of photoresist deposited onto a planar, glass, optical waveguide to fabricate a multistage beam-splitter. After the photoresist was developed, lines of photoresist remained to serve as perturbations of the effective index of refraction of the glass waveguide. This is illustrated in fig. 4, which shows a guided wave, incident from the lower left, split into two beams, both still contained within the waveguide, by means of diffraction. This process is repeated for the other two gratings to produce a total of eight beams emerging from the grating on the right. A similar system was reported by HANDA, SUHARA, NISHIHARA and KOYAMA[ 19801that used refractive-index gratings (fig. 2a), instead of surface gratings, made by direct electron-beam writing in arsenic trisulfide (As$,) waveguides. FLANDERS, KOGELNIK, SCHMIDTand SHANK[ 19741 demonstrated the spectral filtering property of a waveguide grating in the back-reflection geometry that appears in fig. 5. A surface corrugation grating was formed in the upper surface of a glass waveguide by first recording an interference pattern in a layer of photoresist deposited onto the glass layer. The pattern that remained

1 9 5

21


9

TOP VIEW PLANAR OPTICAL WAVEGUIDE

Fig. 4. Top view of a multistage beam-splitter fabricated in a planar optical waveguide.

TOP VIEW PLANAR OPTICAL WAVEGUIDE

INCIDENT

f---

REFLECTED CORRUGATED SECTION

Fig. 5. Top view of the arrangement for a Bragg-reflection experiment using a planar optical waveguide.

after developing the photoresist was then transferred into the glass layer by means of ion-beam etching, resulting in an approximately 50 nm modulation in the thickness of the waveguide ( 0.85 pm). A tunable dye laser was used to excite a guided wave propagating to the right (in fig. 5 ) , which was subsequently back-reflected when the incident wavelength satisfied the Bragg condition. They reported reflectivitiesgreater than 75% and reflection bandwidths less than 0.2 nm, thereby demonstrating that the grating can function as a N

10

[I, $ 2


narrow-band reflector for use in integrated optics. The emphasis in their work was on narrow-band filters, although broad-band filters are also of interest (SHELLAN, HONGand YARN [ 19771). Aperiodic gratings can also be useful for the coupling of two guided waves. LIVANOS, KATZIR,YARIVand HONG[ 19771 made use of a so-called “chirped” grating as a wavelength demultiplexer in the scheme illustrated in fig. 6. Here, the term “chirp” refers to the nearly linear variation in the grating period along the grating axis (z), which causes the wavelength that satisfies the Bragg condition to vary along z. When collinear guided waves excited by two independent sources with wavelengths A I and Az interact with the grating, the different wavelength components are diffracted at different locations along the grating. A glass waveguide was used in the experiment of Livanos and co-workers, along with a surface corrugation grating made by holographic exposure of photoresist followed by ion-beam etching, as discussed in the previous paragraph. The grating period varied between 0.293 < A < 0.321 pm over a distance of 6.5 mm. This produced a separation of 4 mm between diffracted waves for I , = 0.607 pm and A z = 0.627 pm. It is important, however, to note that waveguide gratings used at non-normal incidence (as in fig. 6) usually depolarize the incident wave. As will be discussed later in this chapter, a planar optical waveguide supports waves of two polarizations: transverse electric (TE) and transverse magnetic (TM). FUKUZAWA and NAKAMURA [ 1979) demonstrated this effect by showing that an incident guided wave of the TE polarization produced both TE- and TM-diffracted waves. The TE- and TM-components are spatially separated, since the Bragg condition is slightly different for the two polarizations due to waveguide dispersion (the effective index of refraction N depends on the polarization, even TOP VIEW

Z AXIS ___+

‘‘CHIRPED GRATING


Fig. 6. Spatial separation ofguided waves of two wavelengths using a “chirped”grating, for which the grating period varied along the length of the grating.

1, § 21

11


I


A,

CROSSED GRATINGS

Fig. 7. Two crossed gratings fabricated in a planar optical waveguide. Incident guided waves with two different wavelengths are diffracted into opposite directions.

for a fixed wavelength). Therefore, although fig. 6 shows only two diffracted components, there will, in general, be four diffracted components, because of this polarization effect, a fact that could be important if a high degree of wavelength discrimination is required. It is possible to use multiple exposure techniques to create a grating that diffracts guided waves of two wavelengths in opposite directions. The scheme used by YI-YAN, WILKINSON and LAYBOURN[ 19801, illustrated in fig. 7, makes use of crossed gratings, shown here as solid and dotted lines, on the surface of a glass optical waveguide to achieve the greatest possible spatial separation between the two wavelength components. HATAKOSHI and TANAKA[ 19781 pointed out that a waveguide grating can function as a lens. They reported the use of a glass waveguide and a grating fabricated by electron-beam writing to focus a collimated input of wavelength A = 488 nm. Here, as shown in fig. 8, the orientation of the grating rulings is changed along the grating to make certain that each segment of the incident light is diffracted toward a common point. One of the most important uses of waveguide gratings for the coupling of two guided waves occurs in the distributed feedback (DFB) and distributed Braggreflector (DBR) semiconductor lasers. The DFB laser was first discussed by KOGELNIK and SHANK[ 1971, 19721, and was first implemented in a semiconductor (waveguide) laser by NAKAMURA, YARIV,YEN, SOMEKHand GARVIN [ 19731. The ability of a waveguide grating to couple forward- and backwardgoing guided waves was discussed in connection with fig. 5 . A strong reflection

12


1


GRATING LENS FOCAL POINT

Fig. 8. A waveguide grating lens. The grating period and orientation can be adjusted continuously to deflect different portions of the incident wave toward a common point.

can be achieved within a narrow spectral bandwidth. The gain region of the laser is corrugated in a DFB laser so that the coupling between forward- and backward-going guided waves takes place throughout the laser cavity; hence, the term “distributed feedback”. The DBR laser is somewhat similar to the DFB laser, except that only the unpumped end regions of the laser are corrugated in the DBR case; the gratings are used as passive reflectors. Both approaches take advantage of the narrow Bragg bandwidth of the corrugated waveguide to reduce the spectral width of the laser emission. Waveguide gratings are also useful as phase-matching elements in nonlinear optics. This seems to have first suggested by SOMEKHand YARIV[ 19721. In the case of second-harmonic generation, for example, it is necessary that the propagation constant of the wave at frequency 2 o (nearly) equal twice that of the wave at o.This cannot be easily achieved in all materials, but in a corrugated waveguide the grating constant provides an extra contribution to the phase-matching argument so that the matching condition becomes p(2o) = 2p(w) + 2 x / A , in first order. The ability to vary both the grating period and the waveguide thickness within reasonable limits allows greater control over the phase-matching condition in a periodic waveguide than in a nonperiodic medium. The many uses that have been found for grating-induced coupling between guided waves makes it clear that a quantitative description of the strength of the coupling interaction is essential. This subject constitutes the main emphasis of this chapter.

1, § 21

13


2.3. INTERACTIONS BETWEEN GUIDED WAVES A N D THE RADIATION FIELD

Waveguide gratings can be used for the excitation of a bound mode by an incident optical beam or to allow a bound waveguide mode to radiate. This point was discussed earlier in this section. DAKSS,K U H N , HEIDRICH and SCOTT[ 19701 appear to have been the first to use a grating to excite a guided wave. They used photolithographic techniques to form a photoresist grating with a period A = 0.665 pm on the surface of a planar glass optical waveguide. Light from a helium-neon laser (A = 0.6328 pm), incident as shown in fig. 3a, was used to excite either the T E or TM modes of the waveguide for the proper choice of source polarization. They reported an input coupling efficiency of 40%. Input coupling efficiencies that exceed 40% are also possible. DALGOUTTE [ 19731 achieved an efficiency of 70% using a photoresist grating and a glass optical waveguide. One interesting feature of this experiment was the use of “reverse coupling”, shown in fig. 9. In the actual experiment, light was incident on the lower surface of the waveguide through a prism (not shown) placed in contact with the substrate. Efficient coupling occurs when there is only one incident beam that can couple to the guided mode of interest. As pointed out earlier, there is a range of the grating period A for which a guided mode can radiate into the substrate, but not into the cover medium. The guided wave can be excited most efficiently when light is incident at this same unique angle of radiation. In Dalgoutte’s experiment a grating period of 0.222 pm was used to achieve this. Many similar experimental results have been reported using different materials, different fabrication techniques, or different types of gratings. The use of blazed gratings has been explored by GRUSS,TAMand TAMIR [ 19801.

nC

-SIDE VIEW

rllllllL

A

GUIDED WAVE

P

T

* z

“s

INCIDENT LIGHT

Fig. 9. Scheme for exciting a guided wave using the reverse-coupling technique.

14

WAVEGUIDE DIFFRACI‘ION GRATINGS

[I, 8 3

The use of ion-implanted gratings has been demonstrated by KURMERand TANG[ 19831. The importance of absorption losses on grating performance was considered by STONEand AUSTIN[ 19761. Most recently, gratings have been used as output-couplers to make surface-emitting semiconductor lasers (EVANS,HAMMER, CARLSON,ELIA,JAMESand KIRK[ 19861, MACOMBER, Mom, NOLL,GALLATIN, GRATRIX, O’DWYERand LAMBERT [ 19871) and as focusing couplers for integrated read/write heads for optical data storage systems (SUHARAand NISHIHARA[ 19861).

8 3. Modes Supported by Planar Optical Waveguides 3.1. BOUND MODES OF THE STEP-INDEX OPTICAL WAVEGUIDE

The planar, step-index, optical waveguide (fig. la) supports electromagnetic modes of two polarizations : transverse electric (TE) modes, and transverse magnetic (TM) modes. The term “mode”, as it is used here, refers to a solution to the wave equation that satisfies the appropriate boundary conditions. Each such mode is an electromagneticwave with a unique transverse field profile and propagation constant fl (MARCUSE[ 19741, KOGELNIK[ 19751, ADAMS [ 19811, HALL[ 19871). Optical waveguides are open structures that support both bound modes and radiation modes. For bound modes only certain discrete values of /? are allowed. For radiation modes is continuous within a certain prescribed range of values. This section considers the bound modes. TE modes are characterized by a single electric field component that is oriented perpendicular to the direction of propagation. TE modes are thus specifled by an electric field E of the form

E

=

jiE,(x) $02-

(1)

where the hat ( A ) designates a unit vector, in this case along the y-direction, E,(x) is the TE mode function, m is an integer, /?is the propagation constant with propagation assumed in the z-direction, and w is the (angular) frequency. TM modes are, in like manner, specified by a single transverse component of the magnetic field H according

H

= y ~ , ( x ) ei(BZ-wr)

(2)

where H,(x) is the TM mode function. Since fi is discrete, it would be reasonable to attach the mode-integer subscript m,as in &, but we will suppress this subscript to Pfor the present to preserve simplicity of notation. When the above

1, B 31

MODES SUPPORTED BY PLANAR OPTICAL WAVEGUIDES

15

fields are inserted into the usual wave equations for each medium, and

V 2 E + n 2 ( ~ / ~ )=20E,

(3)

V 2 H + n2(o/c)2H = 0 ,

(4)

with n ( x ) defined in piecewise fashion, n2(x)= n,‘

x >h ,

=nf‘ O c x c h , =n,2

xtO,

we find that the TE mode function is given by E,(x)= E, exp[ - y,(x - h)] x > h , =

Ef cos (kfx - $Js)

0 <x h,

=n,Z+A+n,IEI2 =

n,2

O<xh

9

=

E:

cos {qfx - d e ) } sin

O<x z and one for z' < z, f(z)

=A

+

(z) eipZ + A - (z) e - iflz

,

(62)

where A + and A - are the z-dependent amplitudes of forward-going and backward-going waves, respectively, given by

and

It is now easy to show that eqs. (63) and (64) correspond to a set of coupled, lirst-order differential equations that determine the amplitudes A and A - . First, differentiate eqs. (63) and (64) to obtain +

1 dA+ e-'pz Q(z), dz 2iB and

Next, substitute eqs. (61) and (62) into eqs. (65) and (66). The results have the simple form

and dA dz

~

= - iK(z) A

+

(z) ei2pz- iK(z) A - (z)

These are the coupled-amplitude, or coupled-mode, equations.

32


11, J 5

Equation (58) contains no dependence on the transverse coordinates, something that is essential for a proper description of interactions in an optical waveguide. The wave equation, however, often reduces to that in eq. (58) within some convenient approximation that allows integration over the transverse coordinate(s). As soon as such an integration becomes possible, coupled equations of the form given in eqs. (67) and (68) can be expected to emerge from the analysis. Several treatments of the problem of grating coupling between guided waves will be presented and discussed here. Not all of these start with the wave equation, but coupled amplitude equations of the form that appears in eqs. (67) and (68) nevertheless emerge from all these analyses.

5.1. IDEAL-MODE EXPANSION AND COUPLED-MODE EQUATIONS

Most of the published coupled-mode formulations of the problem of the interaction of a guided wave with a waveguide grating have been based on the so-called ideal-mode expansion. Slightly different versions of this approach to the grating problem have been used by YARN [1973], MARCUSE[1974], KOCELNIK[ 19751, STREIFER,SCIFRES and BURNHAM[ 19751, WAGATSUMA, SAKAKIand SAITO[ 19791, and others (YAMAMOTO, KAMIYAand YANAI [1978], LIN, ZHOU, CHANG, FOUOUHAR and DELAVAUX [1981]). All versions of the theory provide a good description for TE-polarized guided waves, but there is evidence that the approach fails for the TM polarization. KOCELNIK'S [ 19751 treatment is particularly instructive, and is included here to illustrate the ideal-mode technique. Only planar waveguides are considered. The transverse (to z) components of the waveguide mode functions form a complete set of orthonormal functions that can serve as the basis set for an expansion of the fields of interest. This is true strictly for real refractive indices and real values of the propagation constant /?. The expansion includes both bound and radiation modes. If E, and H,represent the transverse components of the fields of interest for a forward-going wave (propagating in the + z direction), the mode expansion can be written as

E;'

a: ( z ) E:i(x)

=

+

n

s,1

a ( z ;4 ) E;')(x;4 ) dq , +

(69)

and H;'

=

C a,: I,

+

( z ) H:?(X)

a (z;4) H ~ " ( x 4 ); d4 . +

(70)

1, I 51

COUPLING BETWEEN GUIDED WAVES

33

The superscript + designates a forward-going wave, the superscript (i) designates a mode of the ideal waveguide. In each of the above equations there is a discrete sum over the bound modes and a continuous one, expressed as an integral, over the radiation modes. The quantity q in the latter represents the spatial frequency associated with radiation in a given direction. The expansion “coefficients”, a,(z) and a(z), depend on z. The expansion is based on the idea that the fields of interest can be expanded in the modes of a particular unperturbed waveguide, the ideal waveguide, for which the modes are known and given by eqs. (6), (lo), (50), and (51). The superscript (i) is assigned here to make the identification of an ideal mode as clear as possible. The form written in eqs. (69) and (70) assumes that each term in the expansion can be factored into a product that separates the x- and z-dependence. In what follows, a somewhat simpler notation will be used to represent the mode expansions in eqs. (69) and (70). Namely, a single summation symbol will be used to represent both the discrete and the continuous sums in the mode expansions. The emphasis here is on the bound modes, but Kogelnik’s formalism applies equally well to the radiation modes. An alternative expansion, the local normal mode (LNM) expansion, to be discussed later, is based on a different idea. At each z the fields are expanded in terms of the modes of the unperturbed waveguide with the local thickness. This means that in the LNM expansion, the modal fields depend on z, since the perturbed waveguide has a thickness that varies with z. The results of the two types of expansions do not always agree. KOGELNIK[ 19751 examines the problem of a guided-wave propagating along the z-direction, perpendicular to the “rulings” of a surface grating that is very nearly of the correct period for Bragg reflection (see, e.g. fig. 5). The grating is presumed to be very wide so that the fields exhibit no y-dependence. The development of the basic equations of the ideal-mode approach proceeds as follows. Consider an unperturbed waveguide with permittivity E ( X ) = n 2 ( x ) t o [see eq. ( 5 ) ] . This waveguide structure is then perturbed, by corrugating one interface, e.g., so that the permittivity becomes E ( X ) + Ae(x, z); the specific details about the corrugation are contained in A E ( x ,z). Maxwell’s two curl equations for the fields of the perturbed structure are [assume exp ( - iwt)]

V x E = ipwH,

(71)

and

VxH

= - iW(E

+ AE)E .

(72)

34

[I. 5 5


Let the subscripts 1 and 2 refer to two waves, each of which is described by fields that satisfy eqs. (71) and (72) for either A& = 0 (the ideal waveguide) or A&# 0. If wave 2 propagates in the ideal waveguide and wave 1 propagates in the perturbed waveguide, it is straightforward to show that

V (El x H,* + E,* x N,)= io(A&)El E,* ,

(73)

where the complex conjugates of eqs. (71) and (72) have been used. Next, integrate eq. (73) over x and separate the z-derivative from the x-derivative on the left-hand side of the resulting equation, m

*(E,xH,*+E,*xH,)dx=iw

(74) The integration over the a/ax term vanishes if either or both of waves 1 or 2 is a bound mode, since the fields for a bound mode vanish at x = & co.Wave 2, by hypothesis, has fields of the form given in eq. (1) and (2); assume for the moment that this is a forward-going wave:

E,

=

f E z ( x ) exp[i(flz - at)] and H ,

=

fH$,)(x) exp[i(flz - ot)].

(75)

The fields for wave 1 can be expanded according to eqs. (69) and (70), the ideal-mode expansion, after one small change. Since E, and H , will both contain forward- and backward-going waves due to the Bragg interaction, terms must be added to eqs. (69) and (70) to represent the latter. The replacements a,+ (z) + a,+ (z)

+ a;

(2)

q) + a ( z ; q) + a- (z; q)

(76)

and a + ( z ; q ) + a + ( z ; q ) - a-(z;q)

(77)

and a

+

(2;

+

in eq. (69), along with - u,(z) u,+(z)+~,+(z)

in eq. (70), make it explicit that both forward-going ( + ) and backward-going ( - ) waves are included, and make sure that the direction of Poynting’s vector E x H is correct in both cases (H, changes sign for a backward-going wave; E, does not). The use of the orthogonality relation (recall that the bound modes are orthogonal to the radiation modes), eq. (35), after these substitutions in eq. (74) gives the result

I-

m

~da,+ ( 4

dz

ifl,a;

(z) = aio

m

-

(AE)E, { E z + } *d x .

(78)


1,s 51

35

Had wave 2 been chosen to be a backward-going wave, the result would have been da' dz

+ ifima;

s-

00

(z) = - biw

(A&),??, { E c - } * d x ,

(79)

a,

where the superscripts + and - designate quantities associated with forwardand backward-going waves. These can be further reduced by introducing the amplitudes A (z) and A -(z), +

a; (z) = A; (z) exp(iflz)

and a, (z) = A , (z) exp( - iflz) ,

(80)

with the results

and

The transverse component of the field El that appears on the right-hand side of these equation-scan be expanded in the same way as described above, using eqs. (69) and (76), but the z-component is handled differently in Kogelnik's treatment. It is easy to show that H , , and E l , are related according to

V, x H , , =

-iW(&

+ A&)E,,.

(83)

H , , can be expanded using eqs. (70) and (77), which means that eq. (83) can be used to determine an expansion for E l , . The result is &

4 , = __

1 {u;

& + A &m

(z) - a, (z)}@,(x)

.

(84)

It is convenient to define two quantities that describe the interaction in the waveguide,

and

76


[I, 8 5

The superscripts + and - have been dropped from the modal fields in eqs. (85) and (86), since the signs used with a,(z) [see eqs. (76) and (77)] in themodeexpansion ensure the proper choice of signs for forward- and backwardgoing waves. The right-hand sides of the coupled-mode equations, eqs. (8 1) and (82), can now be expanded according to eqs. (69), (76), and (84) to obtain

and

(88) These are the coupled-mode equations as derived by KOGELNIK[ 19751 for the two-dimensional case (no y-dependence). Once the perturbation A Ehas been defined, the quantities in eqs. (85) and (86) can be determined, since the ideal modes are known, and the system of coupled differential equations in eqs. (87) and (88) can be solved, at least in principle. Figure 14 shows a typical perturbed waveguide structure, a segment of a waveguide with a cosine corrugation on the upper surface. The unperturbed,

I

Ac4ual Surface

ns Fig. 14. A planar optical waveguide with a corrugated upper surface. The surface grating is made up oftwo perturbation regions, labeled a and b. The grating depth is 2 Ah; the ratio Ah/h is taken to be small. The grating has a period A.


31

or ideal, waveguide is taken to be the mean waveguide of thickness h shown in the figure. The location of the upper surface x = d of the perturbed, or actual, waveguide is given by d

=

+ Ah cos(K0z),

h

(90)

where Ah gives the strength of the grating and KO = 2 n/A is the grating constant, with A the grating period. The permittivity E ( X ) = ~ ' ( X ) Efor~ the unperturbed waveguide appears in eq. (5). The perturbation AE(x,z ) is the difference between the permittivities of the actual and ideal waveguides,

A E = Eo(nf - n f ) =

h < x < d, as for region a,

~ , ( n f - n f ) d < x < h, as for region b.

(91)

The expressions in eqs. (85) and (86) can be evaluated very simply for the case of a small corrugation depth and TE polarization for both the forward-going and backward-going waves, for which KLn(z)= 0. We find, e.g., form = n = 0, Kho(Z) = 2 K

COS (&Z)

,

(92)

where

n Ah nf' - N 2 ___ (TE-TE) . 1 he, N

K = - -

(93)

In the above equations the normalization condition in eq. (24) has been used, along with eq. (29); K is referred to as the coupling coefficient. The corresponding expression for the coupling coefficient K for TM-polarized waves that emerges from Kogelnik's treatment is

(94) where qc was defined in eq. (32), and the values of N and he, appropriate for TM modes must be used [see eq. (34)]. There is strong evidence that eq. (93) is correct and eq. (94) is incorrect, as will be discussed later in this chapter. It appears that the correct TM-TM result can be obtained by setting the quantity in curly brackets { } to unity in eq. (94). The number of terms that must be retained in eqs. (87) and (88) to provide an acceptable quantitative description for a given situation is a matter of great importance. If we consider a waveguide that is sufficiently thin so that it

38

WAVEGUIDE D I F F R A a I O N GRATINGS

[I.§ 5

supports only the lowest order TE mode, we can assume that all the amplitudes are zero except n = 0: A,+ (z) = An-

(2) =

0 for n # 0 .

(95)

The coupled-mode equations then reduce to

and

equations that are clearly of the same form as eqs. (67) and (68), which were obtained in a different way. Coherent coupling between forward- and backward-going waves can only occur (in the first Bragg order) when the propagation constant and the grating constant very nearly satisfy the Bragg condition 2f10 = KO = 2 z / A Only those terms on the right-hand sides of eqs. (96) and (97) that are properly phase-matched will be significant; the rest can be neglected, an approximation often termed the synchronous approximation,which leads to

and

where 6 is a small detuning parameter; 26 = 28, - KO (6 = 0 when the Bragg condition is satisfied exactly). These coupled fist-order equations have a relatively simple solution for many problems of interest. It is important to remember, however, that the simple form of eqs. (98) and (99) is based on the approximation in eq. (95). In many cases of practical importance this approximation works quite well. Before turning to the solutions of eqs. (98) and (99), an alternative derivation that is not limited to a two-dimensional geometry will be considered for the TE polarization.

1, § 51


39

5.2. IDEAL-MODE EXPANSION - AN ALTERNATIVE APPROACH (TE)

The coupled-mode equations were developed in the previous section by starting with the full mode expansions, eqs. (69) and (70), and then manipulating them in various ways using two of Maxwell’s equations. This is in contrast to the method illustrated in eqs. (58) - (68), which showed, for a one-dimensional case, that coupled-mode equations emerge directly from the wave equation. Since only one spatial mode of the waveguide is important for most applications, the full mode expansion is an unnecessary complication. In what follows, the problem of a TE-guided wave propagating in the corrugated structure of fig. 14 will be treated, but the restriction to propagation along z, perpendicular to the grating “rulings” will be lifted. The theoretical development parallels that of eqs. (58)-(68). Figure 15 illustrates the first-order Bragg interaction considered here. A TE-guided wave propagating in a single-mode planar waveguide at angle 0 with respect to the z-axis interacts with the periodic structure (having a period A ) to produce a backward-going wave. A view of the x-z plane for the corrugated waveguide appears in fig. 14. Once again we assume that the Bragg condition is very nearly satisfied, so that 6, defined below eq. (99) with p, replacing Po, is small. We seek a solution of the wave equation for the electric field E, E(X)

+ AE(x,z) EO

1

ki E(x, y, z, 0 = 0,

TOP VIEW Z=O

Z=L

Fig. 15. Top view of a corrugated section of length L of a planar optical waveguide. The grating width along y is taken to be large. A forward-going guided wave with propagation vector 8, oriented at angle @withrespect to the z-axis,generates a backward-going guided wave by means of the Bragg interaction with the periodic perturbation.

40


[I, § 5

where E, is the permittivity of free space, k, = W / C = 2 4 A , E ( X ) = n2(x)eo and A& are as defined in eqs. ( 5 ) and (91), and the usual time dependence, exp ( - iot), is assumed. We adopt the central view of the ideal-mode expansion by considering AE to be a perturbation on the structure of the mean, or ideal, waveguide of refractive index n(x), as shown in fig. 14. The field E is oriented parallel to the y-z plane, and is written in the product form E(x, Y , 4 = f(z) exp (ifl,y) ~ $ ) ( x.)

(101)

The superscript (i) labels the lowest-order ideal mode of the unperturbed, single-mode waveguide [see eq. ( 6 ) ] . This is equivalent to neglecting the radiation modes in the full mode expansion, acknowledging that the period of the corrugation is such that it provides no coupling between the bound mode and the radiation field. The ideal mode satisfies the equation a2

-

[a,.

1

+ n2(x)k,z

E$’(x) = fl’E$’(x).

Now, insert eq. (101) into eq. (loo), make use of eq. (102), and note that

fl = (fly, fl,) to obtain

[$+

]:?j

f(z) E $ ) ( x ) = - pw2 A E ~ ( zE) $ ) ( x ) ,

where p = po = the permeability of free space. The x-dependence can be eliminated from eq. (103) by first multiplying by the complex conjugate of E,(x)* (i.e. E,(x)) integrating over all x, and using the normalization condition in eq. (24), with the result

where m

AE E $ ’ ( x )E(d“(x)*d x ,

analogous to eq. (85). Equation (104) has the same form as eq. (58), the only difference being that f ( z ) is now a vector amplitude. This offers no significant complication, however, due to the simple form of the right-hand side of eq. (104). The same steps that led from eq. (58) to eq. (62), when applied to eq. (104)

4 5 51


41

give the result f(z) = A+(z) eipzZ+ A-(z) ecipZz,

(106)

where A ( 2 ) and A - (z) are the vector amplitudes of forward- and backwardgoing (along z ) waves. They are given by +

A (z)= +

e - i & Z ' K ( ')f(z') dz' ,

~

(107)

and A -(z)=

~

cos I3

eiflrz'K(z') f(z') dz' .

These can be reduced to a pair of coupled, first-order equations by writing the vector amplitudes in terms of unit vectors e , according to A + ( z ) = A + ( z ) e + and

A-(z)

=

A-(z)e- ,

(109)

where e , - e + = 1 and

e- .e-

=

1.

(1 10)

The unit vectors specify the directions of the electric field vectors for the forward- and backward-going waves. We first form the dot products of eq. (107) and (108) with e + and e - , respectively, noting that

e,

me- =

cos(28),

to obtain the scalar equations

and dA - - - iK(z) [ A cos(28) ei2pZz+ A - ] . dz cos0 +

K ( z ) was evaluated in the previous section [see eq. (92)] for a cosine grating of the type specified in eq. (90): K(z) = 2 IC cos(K,,z), where K is given by eq. (93). As with eqs. (96) and (97), we retain only the phase-matched terms (synchronous approximation) with the results

42


[I, I 5

and

where 26

=

2/3, - KO, and

K(e) =

K

cos (28) (TE-TE) , cos 6

with K as given in eq. (93) for TE polarization. Equations (1 13) and (1 14) are in complete agreement with the coupled-mode equations derived earlier using Kogelnik's formalism for B = 0. They are more general, however, in that they apply for arbitrary angle B (see fig. 15). ) the TE-TE, firstEquation (1 15) identifies the coupling coefficient ~ ( 0 for order B r a g reflection of guided waves. All theoretical treatments of this problem obtain this same result for TE-guided waves for the case of a small surface perturbation Ah.

5.3. SOLUTION OF THE COUPLED-MODE EQUATIONS

The coupled-mode equations in eqs. (113) and (114) can be solved in a straightforward fashion after specifying the appropriate boundary conditions (KOGELNIK[ 19751). Here, we consider a surface-corrugation grating of finite length L along the z-axis, but of infinite extent along the y-axis. The upper surface of the perturbed waveguide is, then, given by

d

=

h

+ Ah cos(K0z)

=h

0

=

FI(R)G(P).

(2.44)

Taking a additional averaging over the ensemble of realizations of the rough screen in eqs. (2.43) and (2.44), we obtain

T 1 ( R , pL , )=

s

F , ( R ' ) ( I ( R ' , L ) ) Tsph(R- R ' , p , L)d2R' .

(2.45)

2.2.3. Effect of extended size of a reflector

To be clear, we shall consider a specular reflector with the Gaussian reflection factor f(P)

=

exp( - p2/a2)'

(2.46)

In view of eq. (2.18), the average intensity of a reflected wave that has been emitted by a point source is

(4(P)>

=

0 and falling off as ( P I as lpI - 0 0 , eqs. (3.73) transform to a system of Wiener-Hopf equations, namely,

Asp; a ” v (4,P) Fa,,p; a’p’ (4,PI (3.76)

(3.77)

and L is the Laplace transform of F“(O). We shall assume that the values 1, 2, and 3, through which c1 and fl run, correspond to the x, y , and z projections of the vectors. As in the preceding section, we are willing to elicit as many analytical corollaries as possible from the derived system (3.76) of Wiener-Hopf equations. Specifically,we wish to analyze the angular dependence of enhanced backscattering predicted by the solution of this system. We intend also to investigate the results of a diffusion approximation constructed by STEPHEN and CWILICH [ 19861. At 4 = 0 the matrix { A ( q , p ) }and the matrix ( F ( q , p ) }become sparse and exhibit a block structure: an entry is nonzero provided that its index pair belongs to one of the following four groups +

(3.78) (iv) . As a consequence, system (3.76) can be partitioned into four systems. In view of the symmetry Amp;a,P = APE; which is true for functions Fap;a,p(0, p ) as well, the solutions to systems (ii)-(iv) can be obtained in an explicit form. As an example, for index group (ii) we have F,2;12(O,P) =

i[H+(i/P) + H-(i/P)l

F12,2,(0,~) = :[H-(i/p)

-

-

H-(~/P)I

17

(3.79)

11, J 31

RANDOM MEDIA

153

where H , ( w ) and H - ( w ) are Chandrasekhar's functions which are constructed according to the principle used for H ( 0 , w ) of eq. (3.42) and, which for Re{w} > 0, have the form (3.80) It is essential that the functions n,(O,P)

=

1-

[Ll2;12(O,P)

~L12;21(01P)I

(3.81)

are analytic on thep-plane except the imaginary axis Re { p } = 0, I Im { p } I > 1, and have simple zeros within the intervals < IIm{p} < 1. Therefore, solutions (3.79) have the respective poles and cuts in the lower half-plane Im { p } < 0, and their inverse Laplace transforms Rap; a,P' (0; z, 0) decay exponentially as z -+ a. Instead of the functions FaB;a.p' with group (i) indices of eq. (3.78) it is convenient to introduce linear combinations, @I

=

Form; 1 1

1

y1 - 12 ( F l l ; l l

XI

-k F22;11) - F33;11

(3.82)

3

= Fll;ll - F 2 2 ; I l .

The counterparts a2,Y2, X,, and a,, Y,, X , are constructed by replacing on the right-hand sides of eq. (3.82) the second pair of indices (1, 1) with (2,2) and (3, 3), respectively. Recognizing that these functions are symmetrical with respect to transposing the index pairs (1, 1) and (2,2), i.e., x and y at q = 0, we obtain

(3.83)

The system of equations for functions (3.82) derived from eq. (3.76) separates into a pair of equations for @a and !Paand an independent equation for X,. Solving the latter yields X , ( O ? P ) = F12;12(09P) + Fl2;2l(O,P)

7

(3.84)

which can be expressed by means of H , and H - with the aid of eq. (3.79).

154

ENHANCED BACKSCATTERING IN OPTICS

[II, 8 3

Unfortunately, an explicit solution for the remaining three pairs of equations for functions @a and Ya, a = 1,2,3, defies evaluation. Nevertheless, a straightforward analysis indicates that the solutions of these equations may be represented in the form @o(o,P) = H(O9 ilp) [1 + X A P ) I - 1 ~ y , ( O , P=) 9-fl(i/P) [ 1 + X ; ( P ) I -

1

;

Y3(O,P) = 1 - Hl(i/P) [ 1 + X$(P)l

3

(3.85)

¶

where a = 1,2, 3; H ( 0 , w ) is the Chandrasekhar function (3.42) with w = 1; and the expression for H,(w) may be obtained from eq. (3.80), where A.(O,p) should be replaced with 4(0,P)

=

1 - f[Lll;ll(O,P) + Lll;22(0,P) + 2L33;33(07P)- 4 ~ , , ; 3 3 ( 0 l P ) 1 ~

(3.86)

It can be established that ~ , ( p and ) xL(p) are analytic in thep plane except the imaginary axis in the interval Im { p } < - and as I pI -+ co fall off as I p I I . HI(i/p) exhibits similar properties. Consequently, the inverse Laplace transform for Yo(O,p ) , a = 1,2,3, decays exponentially as z -, 00. At the same time the Chandrasekhar function H ( 0 , i/p) has a simple pole at p = 0, and the inverse Laplace transform of aa(0,p ) demonstrates a “diffusive” behavior. Therefore, we should expect that the behavior of the intensity near the direction of backscatter will be defined by the components FaB;a,P’diagonal in the indices a, /I, and a’, p’ . In the general case of an arbitrary q # 0 the system of eqs. (3.76) has a rather complicated structure. It defies an explicit solution but lends itself to an analysis of the behavior of the solution at large and small values of q. First, we look at the range of q 4 1 to evaluate the peak line shape in backscattering. If we differentiate the right- and left-hand sides of eq. (3.76) with respect to q at q = 0, it is not hard to verify that for the derivatives ~

with the indices from eq. (3.78) we obtain four closed systems of equations, (0, p ) in their homogeneous which differ from their counterparts for Fas; form only. a,B’ (0,p ) have a trivial solution only, The systems (ii)-(iv) for bus; Fms;a,P’( 0 , p ) = 0. For system (i) we again introduce the linear combinations

11, § 31

RANDOM MEDIA

I55

da, Y a ,and X a , composed of Fap;a, according to the scheme of eq. (3.83). The solution of the equation for Xa also yields ka(O,p)= 0. However, the system for &a and Ya has a nontrivial solution, since A ( 0 , p ) = [H(O,i/p) H(0, - i/p)] - I has at p = 0 a (diffusion) zero of second order, namely

(3.87) where t,ba and I& possess properties similar to those of xn and x;. Now we return to the formulas (3.69)-(3.71) for the albedo, Consider the case of a normal incidence where so, = 1 and the vector eo of polarization of the incident wave lies in the xy-plane; for clarity we let eo,, = 1 (fig. 3.9). For small angles of backscattering 0 -% 1, for which s, x - 1, p x i, and q x kolO, we may neglect in (3.70) and (3.71) the contribution of terms that contain the z-projection of the polarization vector e, which is proportional to sin 19,and assume that e,’ + e,2 z 1. Then a @ ,so,e, e,)

= a ( @cp)

3 3 cos’ cp + - [ cos’ ~ I FI ,I I (0, i, i) + sin’ ~ I FI I (0, ~ i,~i)]; 16n 8n 3 (3.88) + - [cos2cpFII~ ll(q;i, i) + sin’qIF,,;,,(q, i, i)] , 8n

--

Fig. 3.9. Geometry of the normal-incidencescatteringproblem: so and s represent the directions of the incident and detected fields, e, and e represent the correspondingpolarizationvectors, and q z represents the plane of scanning with q = k,,l(so + sL ).

156


[II, B 3

where cos cp = e * e,. The first term in this expression corresponds to single scattering, the second to the contribution of the ladder diagrams, and the third to the contribution of the maximally crossed diagrams. It suggests that for the parallel configuration when the polarizations of the incident and detected radiation are identical and cp = 0, the contributions due to the ladder and maximally crossed diagrams in the backscattered intensity coincide, and the enhancement factor K ( 0 , cp) = u(0, $)/ucl(O,cp) deviates from two, resulting from the contribution due to the single scattering only. For the orthogonal configuration of cp = n, no doubling of the intensity is observed any longer. Making use of the results of an analysis of the exact solution, we may establish that

dL’(O, f 71) > u‘C’(O,$ n) , 2dL’(0, 0) > U‘L’(0,

$ n) + u y o , ;n)

(3.89) *

These inequalities indicate that at cp = $ n, the peak of enhancement factor (8 = 0) is below 2 and lower than that of the parallel configuration, 1 < K ( 0 , f n) < K(0,O) 6 2

.

The behavior of albedo as a function of (3 in the range of q 4 1 is appreciably dependent on the mutual orientation of the polarization vectors. Observing the structure ofthe solutions for functions tiaa;a,B’ (0, p ) and tiaa; (0, p ) and using eq. (3.75), one can easily verify that differentiation of the functions Fap;a,B’ (4, p , p ’ ) with respect to q at q = 0 leads to a nonzero result for the functions diagonal in a, /3, and a ’ , /3’. Therefore, from eq. (3.88) it follows

(3.90) where h is derived from eqs. (3.75), (3.82), and (3.83) as h

i) + ~ ~ (i) 0+ ,$1 $ [dI(o,i) + YI(o, i)] - $ [ d3(0,i) + Y3(0, i)] [ ~ ~ (i) 0+ , ~ ~ (i)]0 . ,

= -

=-

(3.91)

Resorting to eq. (3.89), we can demonstrate that h 0. Thus, for the parallel configuration, the angular profile of the albedo u (8, cp) is a symmetrical triangular peak centered at 8 = 0 (fig. 3.10). When cpincreases, the included angle at the vertex of the peak widens, and the peak albedo decreases in magnitude to be a minimum at the perpendicular configuration, rp = f n, when the peak rounds off and the curve becomes smooth. The curve of the enhancement factor K(8, cp) parallels these variations.

11, § 31

157

RANDOM MEDIA

-2

I

0

-1

2

q=k0,lfl Fig. 3.10. Enhancement factor K ( 0 , cp) for normal incidence of linearly polarizedlight and various angles between the polarizationvectors of incident e,, and detected e light (cos cp = e * e,). Curves (a), (b), and (c) correspond to cp = 0, K, and 4 n, respectively, with e, lying in the plane of scanning;i.e., q e, = q. q = k&, I + sL ). Curve (d) corresponds to the case of parallel configuration (9= 0) and scanning in a plane orthogonal to e,, i.e., with q - e,, = 0.

-

The origin of these results can be evaluated with the aid of the diffusion approximation, which in the case of a vector field is applicable, of course, for small q. As in the scalar problem, now the behavior of a(0, cp) is governed by the behavior of the Green function of the transfer equation, Pap; (r, r ’ ) , at far transverse distances I p - p’ I. To be more specific, the diffusion law I p - p’ I - leads to a linear dependence of 0, to a triangular peak line shape, and at a faster, say power-exponential decay we find ourselves with a rounded peak. Consider the integrals of Green functions Yap; a,8’ (r, r ’ ) of the transfer equation with a concentrated isotropic source. These integrals describe the averages of field component products ( E , ( r ) E J ( r ’ ) ) . In an unconfined scattering medium, at distances r from the source greater than the mean free path or extinction length 1, the averages (ExE,*) are factorized and fall off as exp ( - p r / l ) with a constant j?. In view of the isotropic property the averages oftype (lEx12- IE,I’) or ( $ ( l E x 1 2+ lEy12) - (EZl2),relatedrespectively with the functions X , and YGof eq. (3.82), exhibit the same behavior. An exception is the diagonal combination ( I Ex I + 1 Ey I + I E, I ’) ,which can be expressed through the function @, of eq. (3.82), which is proportional to the energy density and falls off as r In the case of a half-space, as p+oo the averages (ExE,*) and ( I E,I - I E.,( 2 , decay. as before, exponentially fast at a fixed z, and the energy density law of r - ’ gives way to P - ~ However, . now that the isotropic property is broken by the medium boundary, at finite distances z from this

’

’.

’

158


PI, 8 3

boundary the average (+(I E xI + I E,, I 2, - I E, I 2 , decays as p -+ co,according to the same law as (E,E,*) . This is the reason why the coefficient of the linear term in eq. (3.91) is expressed by means of the derivatives Yu and &a. It is worth noting that the relation of these two “modes” manifests itself at finite distances from the interface, in that the pairs of Wiener-Hopf equations for Qa and !Padefy separation into independent equations. If we neglect this relationship, i.e., let xu = x: = $u = IG.:, = 0 in eqs. (3.87) and (3.85), the coefficient h in eq. (3.91) assumes the form it had in the scalar problem, namely, h wf H 2 ( 0 , 1). Although the evaluation of estimates of a(0, rp), K ( 0 , q), and h in the exactsolution approach needs a rather cumbersome computational procedure assothe ciated with the solution of the system of integral equations for Qa and !Pa, diffusion approximation of STEPHENand CWILICH [ 19861 yields these estimates in a rather straightforward manner. If we take for each function FaS;a,B’ (q, z, z’) the boundary condition (3.57b) of the absorbing plane type, then in the diffusion approximation I

r r m

FED,.‘P’ ( 4 ,PI P‘ 1

=J J

dz dz’ exp(ipz t ip’z’) 0

-

exp [ - ip, (z t z’ t 2z,)]

I

G$‘L.s.

(q,p , ) .

(3.92)

Here, G;$L.,. stands for the diffusion asymptotic expansion obtained for the Fourier transform Gab; (q, p) of the Green function of the transfer equation derived for an infinite scattering medium. Unlike eq. (3.73), in the equation for this function the integral term is a convolution over the entire z-axis. The equation is solvable with the aid of Fourier transformation resulting in (3.93) where A ( k ) is a matrix with elements A,p;,rs.(q,p) given in eq. (3.77), and Mep;,.p.(q, p) is the cofactor of A,,B’;&,p). By virtue of the invariance of the determinant, the function det A(k) depends only on the magnitude of k with components k, = p , k = q. A simple algebraic calculation yields det{A(k)}

=

[n+(O,k)n:(O,k)nl(O,k)]*A_(O,k) x “0,

kMl(0, k) - B2(k)l

7

R A N D O M MEDIA

I59

where the functions A’+ - differ from the functions A * defined in eq. (3.8 1) by having L13; l 3 k L 1 3 ; 3 1 in place of L12;1 2 $ L , , ; , , . The function B ( k ) can also be expressed in terms of LaB:a,B’ (0, k ) which for k tending to zero, falls off as k2. Note also that the Wiener-Hopf equations for QU and Yuare combined into a system resulting from B # 0. In order to construct the diffusion asymptotics we solve the problem and obtain the eigenvalues A,(k) and eigenvectors j$!)(q,p ) of the matrix {Aap; a,p (q, p ) } . Seven in the nine eigenvalues coincide with the functions A - (0, k ) and A’+ - (0, k), whereas the respective eigenvectors are independent of k at q = 0. The remaining two eigenvalues are the solution of the equation +

(A -

A) ( A , - A) - B2 = 0 ,

and for k tending to zero they coincide, accurate to k2, with A(0, k ) and AI(0, k ) . The function Gab;a,B’ ( q , p )is written as an expansion in eigenmodes; i.e., the ratio Malt ..,./detA in eq. (3.93) is replaced with the sum Xif:$f:!bz/Ai. If we keep only the leading terms of the expansion for k -+ 0 in the numerator and denominator of these fractions, we obtain precisely GLT$p (q, p). It should be noted that only one of these eigenvalues exhibits a purely diffusive behavior, namely, A(0, k ) z fk2 as k -+ 0. The expansion of the other eigenvalues has the form Ai(O, k ) z Ci(1 + at k2). For example, for the diagonal components of the Green function we have (3.94)

5,

where k2 = p 2 + q2, a: = g, and a , = and the minus sign of the last term relates to GJ;;?. Substituting this expression into eq. (3.92) yields

(3.95) where, for the sake of simplicity, we put zo = 0. It will be useful to emphasize that in deriving this expression we diagonalized the exact matrix {Aafi; therefore, eq. (3.95) refines the results that STEPHEN and CWILICH [ 19861 have obtained with perturbation theory. The first term on the right-hand side of (3.95) is related to the pole of Giy:f/l at k = 0. For the parallel configuration this term gives a contribution to the albedo, which coincides with the solution of the scalar problem. The second

160


[II, I 3

and third terms are associated with the poles of non-diffusion modes. Expressions of this type enter the function F2,;,2(q; i, i). Therefore, the angular profile of a(&, cp) and K(B, cp) is an approximately Lorentzian shape at cp = n. STEPHENand CWILICH [ 19861 have performed an albedo calculation, also taking into account the contribution of single scattering events. The experimental estimates are K(0,O) = 1.9 and K(0, in) = 1.2. The estimates obtained with the diffusion approximations agree well with the experimental evidence for the enhancement factor K (0, 0), for the parallel configuration, and q 5 1 (see fig. 3.7). For the perpendicular configuration ROSENBLUH, EDREI,KAVEHand FREUND[1987] noted that the diffusion approximation yields no satisfactory agreement with experiment although it predicts a correct qualitative behavior of K(B, n). Comparison of an exact solution with the experimental evidence is yet to be done. Characteristic experimental plots for cp = $71 are given in fig. 3.11. For large values of q where the diffusion approximation is no longer applicable, a (0, q) falls off as q - and levels off to a plateau. In this range the albedo is formed basically by double-scattering events, the contribution of which in eq. (3.71) corresponds to the first term in eq. (3.73). Straightforward calculation gives for q $- 1

4

'

a(e, 9)- a,,(cp)

9 =[cos(tp + q,)cos~coscp, + $sin2(rp + 9,)sin2cpo], 649

(3.96) where cos p, = e, 4/4. This expression indicates that at sufficiently large q, in

-

t -70

I

I

I

-5

0

5

70 S/6*

Fig. 3.1 1. Enhancement factor K(0, rp) for perpendicular configuration of rp = i n measured for the scattering of light in a 10% water suspension of polystyrene spheres 0.109.0.305,0.46, and 0.797 pm in diameter. Correspondingcurves are (a) through (d); 8* is defined from K ( P ,0) = 1.4 and is dependent on the size of scatterers.

11, 5 31

RANDOM MEDIA

161

contrast to the range of q 4 1, backscattering exhibits the anisotropy noted by VAN ALBADAand LAGENDIJK [ 19871. To be more precise, the backscattered intensity begins to depend not only on the angle between the polarization vectors e and e,, but also on the orientation of the plane of scanning with respect to the polarization of the incident radiation, on e, * q. By way of an example for the parallel configuration rp = 0, and given q, the backscattering intensity is a maximum when the scanning plane is parallel to e, and 4po = 0. This effect was observed by VAN ALBADA,VAN DER MARKand LAGENDIJK [ 19871. Results obtained for linearly polarized light show how the angular dependence of the albedo looks for various polarizations of the incident and detected light. We focus on the case of circular polarization (MACKINTOSHand JOHN [ 19881). In this case the products e,eaand eom,eOp in eqs. (3.69)-(3.71) should be replaced with the tensors PasP.(s) and P2p (so) having components (3.97) where easy is an absolutely antisymmetrical unit tensor, and a is plus or minus unity, depending on the direction of rotation of the polarization vector. Given soz = 1 and 8 4 1, the components of P:&) with a = 3 or = 3 may be deemed equal to zero. Let a = a,, i.e., the directions of the circular polarization of the incident (with respect to so) and detected (with respect to s) rotation are identical. It is not hard to verify that the contributions of the ladder and maximally crossed diagrams for s = -so coincide, and at small q we have for the albedo a(e, a,,

0) ,

a (8, a,, a(O,o,,a,)

0) ,

= ( - 2 K)hq ,

-

a (0, a,,

=

(i~)[~,~;~,@,i,i) ~~~;~A~,i,i)l,

0) ,

(3.98)

where h is the same as in eqs. (3.90) and (3.91). The profile of K ( 0 ; o,, a,) differs only insignificantly from the case of a parallel configuration of linear polarizations. It is essential, however, that for circular polarizations with a = a,, in contrast to the case of linear polarizations with eee, = f 1, single-scattering events do not affect the intensity at small backscattering angles 8. Therefore, such a configuration is convenient to test experimentally if the enhancement is 2 at the maximum (ETEMAD,THOMPSON, ANDREJCO, JOHNand MACKINTOSH[ 19871). If the incident and detected waves are circularly polarized in opposite directions a = - a,, the contribution of the maximally crossed diagrams to the

162


111, § 3

albedo is smaller than that resulting from the ladder diagrams. Thus, K ( 0 ; - a,, a,,) < 2 and its dependence on I3 are almost the same as in the case of perpendicular linear polarizations e . e, = 0, the difference being that singlescattering events contribute to circular polarizations and do not affect the intensity of linear polarizations. A detailed investigation of backscattering for and JOHN [ 19881 circular polarizations has been conducted by MACKINTOSH on the basis of the diffusion approximation. The model of a medium constituted by point-like isotropic scatterers occupying a half-space describes the main features of the polarization effects pertinent to backscattering. STEPHEN and CWILICH [ 19861 have demonstrated that the anisotropy and polarizability of the particles do not qualitatively affect the results. These authors and CWILICHand STEPHEN[ 19871, ETEMAD, THOMSON, ANDREJCO, JOHN and MACKINTOSH[ 19871, MACKINTOSH and JOHN [ 19881, and AKKERMANS, WOLF,MAYNARDand MARET[ 19881 have analyzed the effect of absorption and finite thickness of the scattering layer on the angular distribution of the intensity of polarized light. These factors manifest themselves significantly for the parallel linear and identical circular (helicity-preserving channel) polarizations of the incident and detected radiation. In these situations they cause a rounding off of the coherent backscattering peak, and so qualitatively the pattern does not differ from the case of scalar waves. The situation appears the same for a medium of large-scale scatterers, where polarization effects have been poorly documented thus far. 3.7. COHERENT BACKSCATTERING IN THE PRESENCE OF TIME-REVERSAL

NONINVARIANT MEDIA

In Q § 3.4 and 3.5 we have demonstrated that absorption and confined geometry of the scattering medium round off the backscatter intensity peak and reduce its magnitude at B = 0. Nevertheless, the property of reversibility of the scattering operator and the Green function remain invariant under these conditions and the coherence is preserved. Therefore, in the case of a scalar field or linear parallel or circular identical polarizations, the enhancement factor at I3 = 0 (maximum) is, as before, equal (or almost equal) to 2 due to the coincidence of the contributions of the ladder and maximally crossed diagrams. In the following subsection we intend to sketch the factors that d o not affect the classical (ladder) part of the backscatter peak practically but suppress the interference processes described by maximally crossed (cyclical) diagrams. This suppression is effected through the mechanisms destroying the timereversal invariance.

11, § 31

163

RANDOM MEDIA

3.1.1. A weakly gyrotropic medium in a magnetic field Consider an electromagnetic wave scattering in a nonabsorbing, weakly gyrotropic medium constituted by point-like scatterers (MACKINTOSHand JOHN [ 19881). If we put such a medium into a magnetic field B, its permittivity becomes a tensor (3.99) &,p(r) = [ 1 + Z(r)l 6,p + ie,flygy I

and the refractive index depends on the direction of propagation s and helicity of the polarization vector as

n,

N

n - ag.s / 2 n ,

where CT = f 1, g = fB is the gyration vector such that g 4 1, and f is the Faraday constant. To understand what changes in the pattern of coherent backscattering when the medium is brought in a magnetic field, we resort to the qualitative argument of 5 3.2 with one essential addition. Now, to each step Rj+ - Rj of path y we put a corresponding a wave vector kj and parameter $, = k 1, indicating the helicity of the polarization vector. For B # 0 the propagation velocities of radiation with right-hand and left-hand helicities differ from one another. Therefore, the product of u y and u *_ corresponding to the contributions of the path y and the time-reversed path - y in the field has the form up*

=

1 uyI exp(iAcp) .

The phase difference Arp is the sum of the phase increments in individual steps R,, - R,, and it does not vanish even at s = so. Assume that in the path - y described by the set k,, CT,; - k,- 1 , oh- ; . .. ; - k , , a;; k,, a, we have for the helicity (ri' = C T , - ~ . Then, Arp may be represented in the form

,

,

(3.100) Let c 3 N 1 and g .k, N gk, cos q-,and assume that 5 and cos f$ are uncorrelated random variables so that (Acp)rms N k o l g , / w l , where S , = Nl. Since for constructive interference (AV)~,,,~ < 1, then a helicity-preservingmagnetic field will not destroy coherence if the path length S,,, < S,,, l/l(k,g)2. At the same time the contribution of events with higher multiplicity of scattering having S , > S,,,, which determine the backscattering intensity for angles 6' < Om,, l / k , E , will be suppressed. Hence, the magnetic field rounds off the backscattering peak for angles 8 < g .

-

-

I64


[IL 5 3

The pattern for the albedo (not for the enhancement factor) appears to be almost the same as in the absorption case if we choose the parameter ( = l/ko1Omaxin the form 5 = l/kolg (see fig. 3.8). We note that when on the time-reversed path - y the helicity meets the condition q! = - o,,,-~,the phase difference Acpvanishes. Hence, Faraday rotation does not affect, or affects only insignificantly, the backscattered intensity in opposite helicity channels. and JOHN[ 19881 have considered In addition to gyrotropy, MACKINTOSH the effect of the natural optical activity on coherent backscattering. In optically active materials the dielectric constant assumes different values for right-hand and left-hand helicity states of light and is independent of the direction of light propagation, the refractive index being nu = n - af/2n.In this case the parity is not conserved, but the invariance to time-reversed paths remains. Therefore, natural activity does not manifest itself in helicity-preserving channels (in estimates like eq. (3. loo), Acp N 0) and does not affect the backscatter intensity peak line shape. A quantitative theory describing the effect of Faraday rotation on coherent backscattering of circularly polarized light is developed in a scheme which differs in some details from that outlined in 9 3.6. The equation for E , is derived by adding the term ikoecla,gr, connected with the off-diagonal part of the dielectric tensor (3.99), to the expression in the brackets in eq. (3.62). Therefore, in the far zone the averaged Green function of the Maxwell equation in the medium, calculated for E 6 1 in the effective-wavenumber approximation, has the form (G ,,,

(r, r ' ) ) 1: - (47rR)-

' 1

u= * l

"I

P:,, (s) exp ik,R( 1 + ig s) - - , 21

(3.101) where r - rf = R = sR, and PlPis the polarization matrix defined by eq. (3.97). We note that in the presence of a constant magnetic field B the Green function, like the approximate expression, eq. (3.101) for its average, obeys the reversibility condition in the form G,,.(r, r', B ) = Car.a(r', r, - B ) ,

(3.102)

which follows from the time-reversal symmetry in the system "medium + light + magnetic field". Hence, for the subsystem of a medium and light we may speak of the breakdown of this symmetry (GOLUBENTSEV [ 1984b]), which manifests itself in that at B # 0 the equality (3.67) for the contributions of ladder and maximally crossed diagrams in the vertex function is no longer valid.

i 1 . 8 31

165

RANDOM MEDIA

At B = 0 and circular polarizations of the incident and detected light, the formulas for the albedo a(s, so;a, a,) can be derived from eqs. (3.69)-(3.71) with the substitution e,ep for P&(s) and e,,.eOB for P 2 F . Now, if B # 0, the expressions for the ladder part dL) and cyclic part dc)of the albedo involve different functions Fhk,,.a.(q,p,p') and F$!asB(q,p,p'). With the aid of eq. (3.75) the calculation of either of these functions is reduced to solving the system of the Wiener-Hopf equation of the type (3.76) and (3.77), where the coefficients Lapia,8' for F ( L )and the coefficients Lkp;a,P' for F(c) are expressed through the Fourier transforms of the products ( G,,, (r, 0)) ( G& (r, 0)) and (G,,.(r, 0)) (C$,(O, r ' ) ) . Using eq. (3.101), we find Lap;a'P. =

1

0, 0' =

*1

L"." ap;a'/?'

7

(3.103)

where the components of k are k, = p and k , These coefficients satisfy the conditions

=

q.

L2p:a,p( k ,g) = L2p:,x.p( k + k,lg, 0) , L&Up

(3.104)

( k ,g ) = L:p;-aUB (k, 0) ,

L2b;,.,y ( k , 0) = LzB; &(A, 0) . The solution of the system for F2S!,.a. and for Fii,,,8' has a more complicated structure than in the case of B = 0, although the technique of its construction for g 4 1 remains almost the same. We only consider those salient features of the solution that control the behavior of the albedo near the zero angle, 8 = 0. This is the case of normal incidence with N 1, 0 1, q FS: k,l$, and p z i. As indicated in the preceding section, at 0 = a, the linear dependence of a ( $ , a, a,), i.e., a triangular line shape, is associated with a diffusion pole at p = 0 exhibited by functions Fas;a,B' ( 0 , p ) diagonal in c1, p, and a', 8'. In turn, this pole occurs because one of the eigenvalues of matrix { A(0,p ) } behaves as A,( p ) z f p 2 as p + 0, whereas for the others A,(O) # 0. At B = 0 (or g = 0) the matrix { A ' ( O , p ,g ) } coincides with ( A ( 0 , p ) ) .Corrections to its eigenvalues due to a magnetic field can be computed with the use of perturbation theory. A simple calculation based on symmetry conditions

+

I66

ENHANCED BACKSCAlTERING IN OPTICS

(3.104) yields

where both p and kolg Q 1. Thus, the solution has no diffusion pole, and the dependence of the albedo on 4 = ko18 is devoid of a linear term. A more detailed analysis indicates that for B # 0 and q, k,lg Q 1, the quantity 4 on the right-hand side of (3.98) should be replaced with.-/, This bears out the preceding qualitative estimates of peak rounding in the helicitypreserving channel. A thorough analysis of the albedo based on the diffusion approximation has been performed by MACKINTOSHand JOHN [ 19881.

3.1.2. Brownian motion of scatterers Time-reversal noninvariance also takes place for light scattered by a system of moving particles. Examples of such media may be water suspensions of spherical particles of latex or polystyrene frequently employed in coherent backscattering experiments. Because of collisions with water molecules, these particles of 0.1-1.0 pm typical diameter are in constant Brownian movement. Clearly, the system consisting of a radiation and water suspension is invariant with respect to time inversion, since it also assumes the inversion of velocity of all particles. However, in a given medium, for a subsystem of light and scatterers, such a symmetry is no longer present and the Green function G ( r , t ; r ' , t ' ) # G ( r ' ,t ; r, t ' ) while the coherence of the forward and reverse paths is destroyed. The effect of Brownian motion has been analyzed by GOLUBENTSEV [ 1984a1, and similar reasoning has been explored by MARETand WOLF [ 19871 and AKKERMANS, WOLF, MAYNARDand MARET[ 19881. , 1: Nl. Light travels this distance in time Consider a path y of length S t Nl/c, and the distance between every pair of scatterers alters in this time on average by (AR,)rms a t , where DB is the diffusion coefficient of Brownian motion. Since the increments ARj are uncorrelated, the length of the entire path m t . For interference between paths y and - y to can change by AS, occur it is required that AS, < 1.Therefore, for paths S , ,/=, the Brownian motion destroys coherence, suppresses the contribution of scattering the backevents of higher multiplicity, and for angles B < (D,/CZ~~:)'/~ scattering intensity peak is rounded off. In experiments on coherent backscattering in solid disordered media, KAVEH,ROSENBLUH,EDREI and FREUND [ 19861 observed considerable jumps of intensity as a function of angle 19(speckle noise) associated with the

-

N

-

=-

11,

J 31

167

RANDOM MEDIA

fixed disorder in the placement of scatterers. A common backscattering intensity peak at B = 0 is obtained by rotating the specimen to attain averaging over positions of scatterers. In experiments with suspensions the role of the averaging factor is played by Brownian motion. Therefore, the observation time in such systems is chosen to be sufficiently large (or the scanning speed over angle 8 sufficiently small, see, e.g., WOLF, MARET, AKKERMANSand MAYNARD[ 19881). This time exceeds the characteristic time to destroy the time-invariance t , Under ordinary experimental circumstances t , is in the order of 10-8s, and the characteristic angle is 0, 10-*(k0l)- '.

-d

m .

-

3.8. COHERENT EFFECTS IN THE AVERAGE FIELD: INFLUENCE ON

BACKSCATTER INTENSITY ENVELOPE

When scattering particles are embedded in a medium of effective dielectric constant I > 1 (which is usually the case in backscattering experiments), the effects of coherent interaction of waves with the medium-vacuum interface may become significant. These effects affect the refraction of the incident and backscattered waves and the process of multiple scattering of this intensity in the medium. If I E - 1 I 4 1, then for grazing propagation of incident and scattered waves the coherent effects in the average field will be significant only at shallow depths (GORODNICHEV, DUDAREV, ROGOZKINand RYAZANOV[ 19871). Therefore, we may conclude that the effects of refraction and coherent scattering affect only the transmission of waves through the interface and do not affect the scattering in the medium. Corrections for the energy density due to the interaction with the interface are small, of the order of the angular size of the region where there is internal scattering from the interface related to the entire span of the scattering angles. The results of G ~ R ~ D N I ~ H EDUDAREV v, and ROGOZKIN[ 19891 enable us to analyze how coherent effects in the average field ( u ( r ) ) affect enhancement. Specifically, the dependence ofthe enhancement factor K ( - so, so)on the angle of incidence $, f o r = cos 0, is monotonic. In an optically dense medium of 5 > 1, the grazing angle of an incident wave is larger than in vacuum. This leads to a higher effective multiplicity of scattering in the medium and accordingly to a higher enhancement factor. If the medium permittivity is significantly different from unity ( I - 1 2 l), the total internal reflection from the interface becomes significant and may alter the character of the multiple scattering and interference of waves in the medium.

168

ENHANCED BACKSCATTERING IN OFTfCS

[II, I 4

A phenomenological treatment by LAGENDIJK,VREEKER and DE VRIES [ 19891 on the basis of the diffusion approximation with the boundary conditions involving almost total internal reflection resulted in the following conclusions. As the reflection from the medium-vacuum interface increases, the effective multiplicity of scattering in the medium also increases, producing a sharper peak of coherent backscatter intensity. This effect may be described with the aid of a renormalized diffusion coefficient, i.e., by substituting for D the quantity D * = D(1 + 8 ’ ) - 413, where E’ = R / ( 1 - R), and R is the coefficient of coherent reflection from the interface.

8 4. Multipath Coherent Effects in Scattering From a Limited Cluster of Scatterers 4.1.

ENHANCED BACKSCATTERING FROM A PARTICLE

4.1.1 Singfe particle near an interface In his early model WATSON[I9691 interpreted scatterers as centers of elementary volumes of the scattering medium. However, situations exist where scatterers are centers of actual small bodies randomly distributed in space. In the preceding section we discussed the scattering from a very large number of scatterers that paved the way for an approximation of a continuous scattering medium. In this section we consider the opposite case of a small number of scatterers in which summation cannot be replaced with integration. The possibility of enhanced backscattering due to multi-path coherent effects was recognized by KRAVTSOV and NAMAZOV [ 1979, 19801 who studied the single scattering of radio waves reflected from the ionosphere. However, the pure effect of enhanced backscattering from a single scatterer was evaluated by AKHUNOV and KRAVTSOV [ 1983bI somewhat later for acoustic waves. The reasoning of this paper relates to all types of waves and may be readily extended to optical phenomena. Consider a point-like scatterer placed near the interface between two media. A wave from a source 0 travels to the scatterer S by way of two paths, as shown in fig. 4.la; the direct path is labelled 1, and the path involving a reflection from the interface is labelled 2 . Likewise there are two paths, 1’ and 2 ’ , that propagate the scattered field to the point of observation 0’. Hence, there are four channels of single scattering, namely, 11’ , 12‘, 2 1’, and 22‘. Correspondingly the total scattered field at point 0’ is represented by the sum of four

169

LIMITED CLUSTER OF SCATTERERS

a

d

C

Fig. 4.1. Ray geometry of (a) scattered transmitter 0 and receiver 0‘ for a scatterer S near the interface. When the locations of the transmitter and receiver coincide, the cross channels (b) 1-2 and (c) 2-1 becomes coherent.

contributions us =

u11.

+ u12. + u21. + u 2 2 . ,

(4.1)

and the total intensity is I,

= Iu,I2 = I U I 1 ’

+ u12. + U21’ + u22.12.

(4.2)

Let us assume that the scatterer S is placed at random in a volume V, embracing many interference fringes of the prime field. Averaging the intensity I , over the possible positions of the scatterer r,, i.e., integration of eq. (4.2) with the weight function w(rJ being the probability density of r,, eliminates all the interference terms in eq. (4.2) except the contributions characterizing the interference between the channels 12’ and 21‘. The point is that for 0’ = 0, i.e., for the location of the receiver to coincide with the transmitter’s location, paths 12’ and 21’ become identical and the respective fields become completely coherent, as illustrated in fig. 4. l b and 4. lc, u12 = u21

(4.3)

.

Thus, in the particular case of backscattering with r’ (Ibsc)

= (Ill)

+

(122)

=

r,, (4.4)

+ 4(112)

whereas in the general case (1,) =

(111)

+

(122)

+ ( Iu12, + u 2 1 4 2 >

?

(4.5)

where the angular brackets imply averaging over the ensemble of positions of scatterer r,. Thus

and by virtue of eq. (4.3), ( lu12 + u 2 J 2 ) = 4 ( I 1 2 ) . When the transmitter and receiver are separated by a sufficiently large

170

ENHANCED BACKSCATI'ERING IN OPTICS

[II, § 4

distance for the interference between channels 12' and 2 1' to vanish, we obtain instead of eq. (4.5)

(Lp)=

(Ill)

+ ( I , * ) +2(112).

(4.6)

This intensity corresponds to an incoherent addition of the fields u12,and u,, ,. If we introduce the backscatter factor as the ratio of ( Ibsc) to ( I , , , ) , then (4.7)

For a perfectly reflecting interface and about equal path lengths traversed by the wave in channels 11, 22, 12, and 21, eq. (4.7) yields the estimate K x 1.5. This figure suggests that the effective cross section of scattering of a small body placed near the interface is about 1.5 times as large as in bistatic observation and about 6 times as large as in free space. This simple and somewhat unexpected effect is directly related to the existence of coherent channels of the Watson-Ruffine type. It is useful to note that one may average over a finite band of frequencies (al,w, t A w ) rather than over the realizations of the body in (4.4-4.7). It is required only that a sufficiently large number of interference fringes AN should pass through the scatterer as the frequency sweeps the band. Where the condition A N + 1 is satisfied, one can observe enhancement in a single measurement employing a wideband signal. Essentially, under the circumstances a self-averaging over the frequency band is realized.

4.1.2. Combined action of a rough suface, turbulence, and multipath coherent efects

If the interface is rough, strong focusing, as for a random phase screen, is possible in path 22, and in eq. (4.4) I,, acquires a factor Ksurfto describe the backscatter enhancement in double reflection from the surface (ZAVOROTNYI and TATARSKII [ 19821). If the incoming and scattered waves pass through a turbulent medium, all terms in eq. (4.4) should be multiplied by a factor Kturb. If we take, for the purpose of estimation, Kturbx 2, as for saturated fluctuations, and Ksurfx 2 (moderate focusing), then for (Ibsc ) we obtain (2 t 2 x 2 t 2 x 4)I, = 141,, . This implies that, given the preceding circumstances, the effective backscatter cross section may be 14 times as strong as the scattering of a body in free space (AKHUNOV and KRAVTSOV [ 1983a1).

11, s 41

171


4.1.3. Existence of backscatter enhancement under time-varying conditions In situations where the parameters of the medium or interface vary in time, the coherence of paths 12 and 2 1 breaks down, and we cross over from eq. (4.4) for coherent addition of fields u , and ~ u21 to formula (4.6) for incoherent addition. The transition from eq. (4.4) to eq. (4.6) actually occurs once the phase difference of paths 12 and 21 exceeds n. From this condition we may derive a requirement imposed on the velocity u, of vertical motion of the surface that would not destroy the coherence of fields u12and u21.If t' is the time for u 1 2to travel from source to surface and t " is the similar time for u z , , then in (AKHUNOVand time t' - t" the surface should not go further than KRAVTSOV[ 19821); i.e., u,(t' - t " ) 5 $1.

(4.8)

4.1.4. Kettler effect The class of phenomena under consideration includes the Kettler effect, which was already known to Newton. It consists of observing iridescent rings on a dusty mirror viewed from a point close to a source of light. This effect can be explained as follows. If the distance p between the source r, and the observer r' is comparatively small (fig. 4.2), at a frequency o,waves 1' and 2' add up at a certain angle 0, which depends on the frequency and glass thickness h. In this case, averaging occurs due to the wide band of common sources of light and to the summation over numerous dust particles occupying the outer surface of the glass.

dust

Fig. 4.2. In Kettlefs experiment, coherent scattering channels occur when the point of observation r' approaches the point of transmission r,.

172


PI, § 4

4.1.5. Particle in a waveguide

For a waveguide we may expect higher values of the enhancement factor than for a particle near the interface, because the waveguide sharply increases the number of coherent channels. If m rays are incident on a scatterer, the total number of backscatter paths is m2, of which m(m - 1) paths make i m ( m - 1) = Mcoh coherent pairs (ray j induces a scattered ray p , and vice versa), and m paths have no coherent counterparts (ray j reproduces itself, i.e., also ray j ) . Therefore, after averaging over all the realizations of the scatterer (the domain of averaging should embrace sufficientlymany interference maxima of the prime field), the detected intensity in monostatic reception is estimated as Ibsc z ml,,

+ Mcoh4II1= m(2m - 1)1,1,

and in separated (bistatic) reception as I s e p z m I l , +Mcoh2II1= m 2 1 , , . Hence, an estimate for the backscatter enhancement factor is (AKHUNOV, KRAVTSOVand KUZKIN[ 19841)

One may arrive at this estimate from the mode consideration, where the transformation of rays in scattering is treated as the transformation of the modes and m is treated as the number of propagating modes. The mode analysis suggests that the maximum number of distinct rays in a waveguide m equals the number of propagating normal waves. Hence, eq. (4.9) allows dual interpretation. According to eq. (4.9), in a single-mode waveguide ( m = 1) no enhancement of the backscattered intensity is evident ( K = 1). In a two-mode waveguide ( m = 2), K = 1.5, as for the case of a scatterer near an interface. This coincidence is not by chance: both situations involve four scattering channels, of which two are single (1 1 and 22) and the other two form a coherent pair. Finally, for many propagating modes (m D 1) we have K -,2. The effect of doubling the effective scattering cross section should be taken into account when interpreting the backscattering data gathered in fiber multimode light guides. If the waveguide possesses a clearcut property of focusing the field of a point source, which is the case with a parabolic index waveguide, then for a scatterer placed in a focal spot the scattered field increases by a factor off where

’,

11, I 41

173


j” is the focusing factor indicating how many times the field at the scatterer

exceeds that produced by the source in free space. For a waveguide the backscatter enhancement factor is

K

= Ibscllsep =

(IW

t ?

r J

“>I ( I

W

t

9

r,) G(r9 r,) I 2 ,

9

where, as before, the angular brackets indicate that the ensemble average has been performed over positions of the scatterer. If the domain of averaging is limited by a focal spot, then K is rather high, K f’ % 1. When the averaging domain exceeds the distance between adjacent focal spots, then K + 2, since this follows from eq. (4.9) for m $ 1. For a single-mode propagation the Green function is devoid of interference structure, hence K = 1.

-

4.2. ENHANCED BACKSCATTERING BY A SYSTEM OF TWO SCATTERERS

4.2.1. Watson equations (scalar problem)

The system of two small scatterers is interesting since it enables an exact solution of the wave problem to any desired order of multiple scattering. First, we consider the scattering problem in the scalar formulation. Let u1 and u2 be the field of an external source at the locations of the first and the second scatterer, and let a, and a2 be the “polarizabilities” of the scatterers. The moments induced on the scatterers, pI and p 2 , combine from those due to the external field a i , 2 u 1 , 2and those due to adjacent particles a l g 1 2 p 2and a 2 g Z 1 p i , where g , = g , I = - exp (ik1)/4n1 are the Green functions corresponding to the distance I between the particles. This simple argument leads to the following system of equations (4.10)

which is an example of the equations derived by WATSON[1969]. Having determined the “moments” p1 and p 2 from eq. (4.10), the scattered field is u,(r) = Plg(ri7 r) + pzg(r2, r ) SO

7

(4.11)

that p I and p 2 have the meaning of the scattering amplitudes. For identical particles ( a , = a2 = a) the solution to eq. (4.10) has the form

(4.12)

174


[II, § 4

If we expand the denominator in a series in the powers of the parameter (org,2)2,we obtain an expansion of moments pl,, into orders of multiple scattering. When the parameter ag,, is small, we can only retain in eq. (4.12) the numerator that corresponds to the double-scattering approximation. We formulate the main results without going into great detail. Let us assume that the direction of the axis connecting the centers of the particles is uniformly distributed over a unit sphere and the distance between the particles, I assumes random values with probability density w,(l). If the source of the prime field is at a considerable distance from the system of particles (r B 1, where 1 is the mean distance between the scatterers), eqs. (4.11) and (4.12) may be used to calculate the averaged (over I and axis orientations) cross section of scattering a(0), which is a function of the angle 0 between the directions to the source and the detector. If 0, = (a/4n), is the cross section for a single particle, the plot of the angular dependence for the normalized cross section of scattering 0(0)/20, may be viewed as the profile of the enhancement factor (fig. 4.3) (4.13) In the forward direction (0 = n) there is always a maximum of K ( 8 ) = 2 corresponding to the in-phase addition of single scattered fields cr(n)x 40,. Another maximum of considerably lower height Kbsc - 1 = K ( 0 ) - 1

N

(~lgl,)* N CTO/I~

(4.14)

is evident in the backscatter direction.

Fig. 4.3. Enhancement factor K ( 0 ) for two identical, randomly located scatterers.

11, I 41

175


Thus, the averaged cross section of backscattering o,,, = o(0) always exceeds the sum of single cross sections 20,. This small enhancement is observed in a comparatively narrow cone of halfwidth AO l/kj. Despite its small magnitude the effect is of major significance because a maximum in K ( 8 ) suggests that the scatterer should have an internal structure which is often hard to reveal by other methods. N

4.2.2. Polarization efsects For an electromagnetic field the system of Watson equations takes the form (4.15) , the ~ “true” polarizabilities, pl,zhave the meaning of where this time M ~ are induced dipole moments, and the tensor operators g,2 and gzl yield the field due to the dipole moments p1 and p 2 at the adjacent particles. Because of the random orientation of vector I = r, - r1 connecting the particle centers, the polarization of the scattered field E, = g(r, r, ) p , + g(r, r2)p2differs from the polarization of the prime wave. Let the center of the system of two particles lie at the origin, and the source at a distance L % 7 from this center along the z-axis radiates an intensity polarized along the x-axis. For a detector receiving the co-polarized component of the scattered field, E,,, we introduce the angular dependences of the enhancement factor K,, on angles ,O and 0,. lying in the mutually orthogonal planes yz and xz (fig. 4.4a). Figure 4.4b illustrates an analysis of such dependences for the case where

K-7

a

b

Fig. 4.4. (a) System of coordinates and (b) angular profiles of K ( 0 ) for the different measurement schemes: (1) K.x(Oyz). (2) K x x ( L ) ,and (3) KY(Q

176


[II, § 4

the interparticle distance I is distributed uniformly in the interval (A, lOA). The enhancement factor in the backscatter direction, K,,, differs from unity by a value of about ao/f2, i.e., of the same magnitude as in the scalar problem. In the xz plane the peak is 1.5 times as wide as in the yz plane (which may be attributed to the different interference pattern of secondary electromagnetic waves), but for both cases he- l/k7 (curves 1 and 2). For the orthogonal y-polarization difference K,,,(B) - 1, curve 3, is one tenth as high as K,, - 1. It is hoped that the polarization features of backscattering from a system of two particles will take place in the case of many particles if double scattering is the dominant mechanism. A proof of this hypothesis can be obtained by comparing the experimental data of VAN ALBADA and LAGENDIJK [ 19871, who established that the intensity of a depolarized scattered field is about one tenth as strong as the intensity of the polarized component. It should be noted that the considered model of two scatterers yields very small enhancement as compared with the many-particle experiment, namely, K - 1 I ag12 ao/f23 1. For N scatterers, there will be about i N pairs, and K - 1 will increase many times.

- -

4.3. MORE INVOLVED SCATTERER SYSTEMS A N D GEOMETRIES

4.3.1. Cluster of N scatterers: Paired and single scattering channels For a system consisting of more than two scatterers, it would be reasonable to evaluate the classes of paired and single scatterings from the entire family of multiple scatterings (BUTKOVSKII, KRAVTSOVand RYABYKIN[ 19871). Consider N scatterers that are more or less uniformly distributed within a volume V. The scattered field us can be represented as a series into the orders of multiple scattering (4.16)

where in turn, every term may be written as a sum of the fields that have experienced scattering by certain scatterers. Let us consider a specific path Osisj...spO’ of scattering of order n along with the corresponding field uoii., Clearly the number of partners in such a path can be less than n, due to repeated scattering, but all adjacent indices in the series i, j , . . . , p must be different in order to prevent self-scattering from

11,s 41

1I1

LIMITED CLUSTER OF SCAlTERERS

entering into consideration. In other words, a field scattered by one particle will have another scattering event at a different particle. The single, double, and triple scattered fields are represented, respectively, by the sums

c c N

u p=

UOi0’,

i= I

N

u\?=

uoijo.,

i,J= 1

UP’ =

5

UOijkO’

,

i,J, k = 1

where the primes correspond to the requirement that two adjacent indices should not coincide. All in all there are N terms for a single scattered field, N ( N - 1) terms for double scattering, N ( N - 1)2 for triple scattering, etc. When the locations of the transmitter and receiver coincide (0 = O’), expansion (4.16) acquires coherent Watson-Ruf?ine pairs; specifically, the field uoi,.. equals the field corresponding to the reverse sequence of scatterers %ij

-

(4.17)

. . . p 0 - u ~ ... pj i o *

Some sequences, however, remain without a coherent partner. These are primarily single scattered fields uol0 and the fields of multiplicity 2m i- 1that have been scattered m times in the forward direction, say, via an index series j , , . . . ,j,, and rn + 1times in the reverse direction via a seriesj , ,j,, . . . ,j, . For such fields a reversed row of indices p , . . .,j , i coincides with the forward row i, j, . . ., p , so that the fields uoij,.. and uop,,.j i are identical, as is, for example, uo 123210 or uo765670.A typical scattering pattern corresponding to such unpaired, or single, channels is shown in fig. 4.5. Single channels of an even order of scattering (n = 2 m ) are absent. Let us use the sum of coherent pair fields (denoted by 28) and the sum of +

.

,

Fig. 4.5. Example of a simple scattering channel Ojlj2...j , + I .. j a l O , for which the forward and reversed sequence of symbols coincide.

178


[II, I 4

single fields li, including the single-scattered fields u=ii+2ii m

m

(4.18) All cross terms in these sums vanish because of the averaging over the positions of the scatterers (or over the frequencies), so that the intensity of the backscattered field may be written as

-

=

Ib,,=I+41,

(4.19)

where

I=

c

1 f i ( 2 m + 1 )I 2

9

m=O

c 03

f'=

(16(2m)12+ If(Zm+l)

I2

1

9

m=O

and the average is implied but not indicated. When the point of observation 0' moves away from the source 0, the fields uoij,. and uop,,,,io, are no longer in phase, although the intensities of these fields remain almost unchanged. As a result, the coherent effects manifest themselves only within a certain coherence zone surrounding the source. Let a cloud of scatterers of diameter L be seen from a source at a distance R at an angle 0 L/R. If the source is in the near zone with respect to the cloud ( R < L2/A),the longitudinal dimension of the coherence zone I , , (along the line from the source to the center of the cloud) is estimated as Ale2 and the transverse dimension as I , Ale (fig. 4.6a). (These estimates are similar to and TATARSKII [ 1989al.) those given in the monograph of RYTov, KRAVTSOV If the source is in the far (Fraunhofer) zone of R > Lz/A,the transverse dimension of the coherence zone is given by the previous formula N

N

a

R

3!

Fig. 4.6. Region of enhanced backscatter intensity in the vicinity of the transmitter rt for manyscatterer cases with the source (a) in the near zone (R < LZ/A)and (b) in the far zone (R> LZ/L).


I,

- 110- 1R/L,

179

but in the longitudinal direction the coherence zone extends from the Fresnel length R L2/lnto infinity, as shown in fig. 4.6b. Outside the coherence zone the coherent addition of paired channels gives way to an incoherent addition, so that instead of eq. (4.19) we have

-

Ki

Isep= r”+ 2 1 .

(4.20)

The ratio of Ibsc to Isepyields the enhancement factor

7+ 4 7

Kbsc

= _ _*=

T + 21

27 1 + _ _,=1+-,

i + 21

where M = 2$characterizes to single channels, M

=

215/7= ( K -

2M 1+2M

(4.21)

the contribution of paired channels with respect

(4.22)

1)/(2 - K ) .

Values of K close to unity imply that single scattering predominates and M < 1. Converseiy, when K + 2, multiple scattering prevails. In this case the contribution of unpaired channels, the principle of which is single scattering, tends to zero, hence M -+ 00. Thus the magnitude of an enhancement factor conveys information about the ratio of the contribution of paired channels to that of single scattering channels. Let us estimate the contribution of paired channels on the assumption that in expansion (4.16)we may limit ourselves to single and double scattering only. Let a, be the scattering cross section of a single scatterer and I, be the characteristic distance for most events of double scattering. If w(1) is the probability density of interparticle spacing 1, then

This distance I compares in the order of magnitude, with the diameter of the scatterer cluster, L. If a prime field of intensity Z, is incident upon a scatterer, an individual single scattering event produces a field of intensity I ‘ I , a,/R2, in the neighborhood ofthe source, where R is the distance from the source to the center ofthe cluster. All N scatterers of the cluster give the intensity I ( ’ ) NI’. Likewise, a single event of double scattering produces the intensity I” I , o,Z/R21: near the source, and the total number of such events is N(N - 1) fi: N 2 . As a result, the total intensity of double scattering is I ( 2 ) N 2 1 ” I , + N2a,Z/R21:. The total intensity of a scattered field outside the coherence zone may be

-

N

--

N

180


PI, § 4

written as

-

where the correction p N oo/l: is considered to be small. Continuing this argument, we may think of the triple scattered field as being of the order of p 2 P 1 ) ,etc. As long as p is small compared with unity p 6 1, we can neglect the contribution of triple scattering; then, M p and

-

(4.24) Thus, by the backscatter enhancement data, we can judge the magnitude of the ratio NCn/l:. It should be stressed that the evaluation of N by the preceding method does not require that the magnitudes of intensities be measured and, consequently, eliminates the need for calibration of the transmitter and receiver. Therefore, the method suggested to estimate p = NgJZ: can be an addition to the traditional techniques of scattering media analysis. It can be used either by measuring the intensity ofthe scattered field, which is proportional to No0 when single scattering predominates, or by measuring the extinction coefficient, which is proportional to Noo/V. 4.3.2. Scattering by bodies of intricate geometry We say that a scattering body has an intricate geometry if the intensity scattered from this body exhibits a number of spatially separated light spots due to specular reflections and scattering from edges, vertices, and such. The Fresnel criterion for physical independence of these light spots has been outlined by KRAVTSOV [ 19881.The multipath coherent effects leading to enhanced backscattering in this case stem from the fact that the incident wave suffers sequential scattering (diffraction) on a complex envelope as is the case with multiple scattering by a cluster of N individual scatterers. There exists, however, an important distinction between such a body and a system of independent scatterers; namely, some light spots are tightly associated with the characteristic elements of diffraction on the surface of the body (bosses, vertices, and sharp peaks). Accordingly, the averaging to reveal backscatter enhancement in this case is performed over the orientations of the body, rather than over the locations of the scatterer. Despite this difference, many features of backscattering for a body of intricate

11, I 41


181

geometry are essentially the same as those for a cluster of scatterers. These include, e.g., the envelope of the coherence zone and relationships (4.21) and (4.22) between the enhancement factor K and factor of multiple scattering M . The importance of eqs. (4.21) and (4.22) is that they qualitatively characterize the intricacy of the shape of a body, e.g., in laser detection and ranging. Specifically, the value of M = ( K - 1)/(2 - K ) can be viewed as a criterion in target identification. 4.3.3. Coherent eflects in difraction by large bodies

In the systematic analysis of scattering by bodies of regular shape (e.g., discs, spheres, cylinders, bodies of revolution) the multipath coherent effects are automatically incorporated into consideration. However, their contribution to the total scattering cross section has not been treated separately, perhaps because it has not occurred to anyone to break down symmetrical bodies into individual elements that alone are capable of inducing the multipath transverse effects. The “elementary” approach to scattering may be of methodological and practical significance in much the same way as the approximate methods of diffraction theory, which were tried out initially for elementary solids, have been extended to bodies of more intricate geometry. In fact, the method of edge [ 19711 and the geometrical theory of diffraction due waves due to UFIMTSEV to J. B. KELLER[ 19581, along with their generalizations, have been developed precisely in this manner. As an example, consider the scattering by a conducting sphere and focus attention on the Keller dfiraction rays returning toward the source (fig. 4.7a). All such rays represent mutually coherent fields, with a forward and reverse channel corresponding to each ray. Therefore, the axis connecting the source to the sphere is the focus where focusing of the Keller diffraction rays will occur. Depending on the phase difference between the Keller rays and a ray secularly reflected from the sphere, the corresponding fields will be added or subtracted. This explains the noteworthy oscillating behavior of the cross section of the

Fig. 4.7. Coherent paths formed by rays diffracted on (a) a large conducting sphere and (b) on a large conducting ellipsoid. (c) In dielectric bodies, coherent paths can form due to total internal reflection.

182


PI, § 4

sphere as a function of frequency when the sphere perimeter 2xa is several wavelengths long. Although the amplitude of the Keller rays is markedly attenuated on traversing around the sphere, this attenuation is compensated to a large degree by the “number” of rays taking part in the constructive interference. If we supply each ray with the Fresnel width A1 2,,& in a fairly natural manner, the sphere perimeter will accommodate about N = m / 2 @ rays. Accordingly, the focused field will be about N 2 n2a/41 = i x k a times stronger than the field of one ray; e.g., for a = 41, N 2 10. For a deformed sphere the number of the Keller rays whose fields add coherently in backscattering drops sharply. For example, only two pairs of coherent rays survive in the scattering by an ellipsoid, as shown in fig. 4.7b. For a dielectric sphere the coherent effects can be associated not only with the Keller grazing diffraction rays, but also with the rays that suffered internal reflection (fig. 4 . 7 ~ )Such . rays occurring in small water droplets help to explain the phenomenon of a halo when sunlight incident from the observer’s back to a cloud or a mist gives rise to a light nimbus around the head of the shadow. A dark ring around the nimbus corresponds to the subtraction of the diffracted waves. This phenomenon can be observed high in the mountains, above the clouds, or in an airplane for a certain position with respect to the sun. In the latter situation a halo is observed around the airplane shadow. A diffraction theory for this phenomenon (without evaluation of coherent channels) has been proposed by NUSSENZVEIG [ 19771. Similar effects take place in an optical phenomenon observed when automobile headlights illuminate modern road signs. An enhanced backscattering is achieved here with the aid of tiny glass spheres added in the coating of the road sign. These balls scatter the light in a backward direction as in the case of a water droplet. A similar effect occurs with the reflection of light from retroreflectors, specifically those mounted on the moon for laser ranging. It is useful to note the difference in the action of cat’s eyes and the effect of backscatter enhancement. Cat’s eye devices are usually arranged as sets of retroreflecting studs that concentrate the reflected rays toward the radiant source. The action of these devices is underlaid by the incoherent addition of the fields from all elements of the device. Accurate measurements of reflected fields near the source may reveal the coherent addition of the fields corresponding to coherent pairs of rays. As far as we know, no coherent experiments with cat’s eye devices have been reported. The coherent effects may manifest themselves as a very narrow peak of angular width in the order of AID, where D is the cat’s eye diameter, with the intensity in the close neighborhood of the source being twice that of the background.

-

N

ROUGH SURFACES

183

cs Fig. 4.8. Typical ray pattern in laser sounding ofgrain crops. The laser return to the source gives rise to the hot-spot effect.

The double magnitude of the backscattered intensity peak will be observed on the average over the various positions (orientations) of the device. Certain realizations may exhibit both enhancement by a factor N of the number of reflecting studs in the device, and attenuation of the intensity down to zero, which corresponds to an equal number of elements in phase and out of phase; but on the average the quantity K = ( Ibsc) / ( Isep) will be around two. The analogy with the cat’s eye is useful in considering another interesting effect, referred to as the hot-spot (GERSTL,SIMMER and POWERS[ 19861 and Ross and MARSHAK[ 19881). This effect is observed in the laser scanning of grain crops when a considerable proportion of the beam energy is reflected from the plant stem and blade almost in the backscatter direction (fig. 4.8), giving rise to the name of this phenomenon. In general, in the circumstances one may also expect an enhanced backscattering due to coherent scattering channels, but actually this is hardly feasible for in-flight laser scanning of grain crops from an airplane or helicopter.

4 5. Enhanced Backscattering from Rough Surfaces 5.1. TREND TO INTENSITY PEAKING IN THE ANTISPECULAR DIRECTION

An early indication of enhanced backscattering from randomly rough surfaces seems to have been given by KRAVTSOVand SAICHEV [ 1982bl for very rough, steep surfaces that reflect the rays back to the source with a high probability (fig. 5. la), and by ZAVOROTNYI and OSTASHEV [ 19821 for rough

184


Q

6

C

Fig. 5.1. Coherent channels arising in scattering from statisticallyrough surfaces,specifically due to double scattering by small inhornogeneities.

surface areas illuminating one another. ZAVOROTNYI [ 19841 extended these considerations on a two-scale surface (fig. 5.lb). In the treatment of ZAVoRoTNYI and OSTASHEV [.19821 and ZAVOROTNYI [ 19841, one of the reflections in fig. 5.lb, say, at point A , is specular (the field is reflected from the large-scale component of the surface roughness profile), and the other reflection at B is diffusive. The latter is due to the small-scale component and does not obey the laws of geometrical optics. Hence, the relevant coherence scattering channels occur because of single Bragg scattering and single specular reflection. For large and steep roughness heights as in fig. 5. la, coherence channels occur due to multiple (at least double) scattering of the rays. One more mechanism is capable of producing coherent channels, namely, that due to double scattering from small surface inhomogeneities (fig. 5. lc). It is weaker than its counterparts, but it does not involve specular channels and, in this respect, is a more universal mechanism; weak effects of double scattering always co-exist with the stronger mechanisms. Of the theories developed thus far to describe backscatter enhancement, the [ 1987,19891 is worth mentionfull-wave approach of BAHARand FITZWATER ing. According to the authors’ terminology, it deals with single scattering, but actually represents a second-order iterative solution. In fact, the enhancement effect is “hidden” in the ordinary theory of double scattering, but it has avoided an explicit elucidation as far as we know. On the other hand, computer simulations performed with great ingenuity by NIETO-VESPERINAS and SOTO-CRESPO [ 19871, MACASKILL and KACHOYAN[ 19881, and SOTO[ 19891 have revealed a backscatter CRESPOand NIETO-VESPERINAS intensity peak and certain polarization effects. Neither analytical nor numerical methods, however, have been able to produce an effect that compares with the experimental data of MENDEZand O’DONNELL[ 19871, O’DONNELL and MENDEZ[ 19871, SANT,DAINTYand KIM[ 19891, KIM,DAINTY, FRIBERG and SANT[ 19901. These workers studied

11,s 51

ROUGH SURFACES

185

scattering from a specially prepared, very rough surface, i.e., an aluminium coated rough surface of a photoresist resulted after speckle-field irradiation. These experiments revealed a sizeable maximum in the antispecular direction and a very strong depolarization - the intensity of the depolarized backscattered component was almost 50%. The qualitative interpretation of the backscatter intensity peak given by these authors bears on the ray optics representations and essentially parallels the arguments of KRAVTSOVand SAICHEV[ 1982bI. The ray optics interpretation provides an explanation for certain features of the polarization, specifically for the absence of axial symmetry of the scattered field. A reasonable explanation of the polarization characteristics has been given in the full-wave theory of BAHARand FITZWATER [ 1987, 19891. As long as a well-developed theory of scattering from large, steep, and rough heights is unavailable, it is logical to resort to a model description of antispecular scattering. A simple model of a unipolar, very rough surface has been devised by KRAVTSOVand RYABYKIN[ 19881. This model does not pretend to explain polarization phenomena and has been constructed as a collection of upright waveguides of random depth and width, as illustrated in fig. 5.2a. A beam launched at an angle Oo with the axis of the waveguides excites in them eigenwaves of different types. If the waveguides are sufficiently wide and deep compared to the wavelength, the reflection of the incident wave from the side walls and bottom of the waveguides may be described in the framework of geometrical optics. In this approximation the incident beam is split into two parts - one portion of the energy is reflected in the specular direction, as shown in fig. 5.2b, and the other portion is reflected backwards, i.e., in the antispecular direction, as illustrated in fig. 5 . 2 ~ . Averaged over all the waveguides, one half of the energy is reflected in the mirror direction, and the other half in the antispecular direction, so that the angular distribution of intensity will exhibit two sharp maxima of equal magnitudes. If we observe the diffraction nature of the reflection, these peaks acquire

Fig. 5.2. (a) Model of a very rough surface made of open waveguide sections of random depth

and width; (b) and (c) specular and antispecular ray paths.

186


5.2. BACKSCATTER ENHANCEMENT INVOLVING SURFACE WAVES

A well-known method of exciting electromagnetic surface waves by light involves diffraction gratings that launch one of the diffraction spectra along the metallic surface. These surface waves can suffer multiple scattering in view of the imperfections of the grating and roughness of the metal surface. Among other directions the scattered waves will emerge from the grating in the specular or antispecular direction. In the presence of paired coherent channels for surface waves one can expect enhanced backscattering for spatial light waves. These types of effect have been the focus of theoretical and numerical considerations of CELL], MARADUDIN,MARVINand MCGURN [ 19851, MCGURN,MARADUDINand CELLI[ 19851, ARYA,S u and BIRMAN [ 19851, MCGURN and MARADUDIN[1987], TRAN and CELL] [1988], and MARADUDIN, MENDEZand MICHEL[1989]. An important event was the experimental observation of enhanced backscattering for spatial light waves by Gu, DUMMER,MARADUDIN and MCGURN[ 19891. The effects involving surface waves (polaritons) are interesting because they are accompanied by wave-type transformation : light -+ polariton + light, the enhancement occurring in the transformed wave. It is likely that this is not the only example of scattering in the transformed wave process. Specifically, the scattering after a nonlinear transformation of a wave type or frequency seems feasible as a result of a parametric interaction.

6. Related Effects in Allied Fields of Physics 6.1. ENHANCED BACKSCATTERING IN ACOUSTICS

In acoustics, enhanced backscatter effects are almost as diverse as in optics. At the same time there are some specific acoustic manifestations caused by the small value of the velocity of sound. We note the possibility of the multipath coherent phenomena in a confined volume. A beam launched in a confined space by a transmitter t (fig. 6.1) gives rise to paired channels like tabcdt and tdcbat, as well as single channeis like tAt. In measuring pulse signals the different reflections from the walls may be resolved in time to discover that the amplitudes in the paired channels have been doubled and the intensities quadrupled. When the transmitter and receiver locations are separated, the transverse effect vanishes. This explains why we hear our own voices differently from our

11, 8 61

RELATED EFFECTS IN ALLIED FIELDS O F PHYSICS

187

d

C

Fig. 6.1. Single (tAt) and paired (tabcdt) scattering channels in a confined space of rectangular cross section.

roommates, but alas fails to explain the origin of misunderstanding. A theory of coherent effects in confined geometries is outlined by BUTKOVSKII, KRAVTSOVand RYABYKIN [ 19861 and the relevant experimental evidence by GINDLER,KRAVTSOVand RYABYKIN[ 19861. In view of the small velocity of sound the acoustic coherent effects find themselves destroyed faster than their optical counterparts. This circumstance can be utilized to monitor the stationary status of a medium by recording the front where the enhancement effect vanishes (AKHUNOVand KRAVTSOV [ 19841). The variety of acoustical manifestations of this effect has been examined by KRAVTSOV and RYABYKIN[ 19881.

6.2. EFFECTS IN THE RADIO WAVE BAND

An early indication of the important role of backscatter enhancement in radio sounding of the ionosphere can be found in the work of VINOGRADOV and KRAVTSOV[1973]. It is devoted to the evaluation of the concentration of electrons in the upper ionosphere by the method of incoherent scattering that has been incapable of determining the electronic concentration by the power of the scattered field. The backscatter enhancement increases this power (in monostatic observations), thus leading to concentration estimates K times higher than the true concentration values. Similar problems have been addressed by YEH [ 19831 and YANG and YEH [ 19851 for scatterers of other physical origin. Multi-channel coherent effects can also be observed in the scattering of radio waves from the ionosphere. These effects occur when the inhomogeneities are irradiated simultaneously by a direct wave from the transmitter and a wave reflected from the ionosphere (KRAVTSOV and NAMAZOV [ 1979, 19801).

188


[II, § 6

In microwave scattering from vegetation, coherent channels occur due to the reflection of the wave from the earth’s surface (LANG[ 19811 and LANGand SIDHU[ 19831). If the coefficient of reflection of microwaves from the earth’s surface is close to unity, then, on average, one may expect a growth of the effective cross section of scattering from leaves, branches, blades, and stems by a factor of 1.5 compared with the value in free space. The estimate of vegetable biomass will increase accordingly.

6.3. OTHER EFFECTS OF DOUBLE PASSAGE THROUGH RANDOM MEDIA

In addition to the intensity the backscattered wave has its other parameter altered, specifically, the phase. Let be the variance of phase for a single passage of distance L in a random medium. As has been shown, in backscattering the variance of phase increases four times over ) @ : a rather than twice, as might be expected from a common-sense consideration. For a rather large separation of the transmitter and receiver when the forward and reverse paths propagate through different inhomogeneities of the medium, the relevant variance exceeds only twice. These and other features of phase fluctuations have been investigated in the review paper of KRAVTSOV and SAICHEV[ 1982bl. The growth of the variance of the phase leads to an additional widening of the partial spectrum because the fluctuations of frequency occur as the derivative of the fluctuation of phase. Such a broadening of the spectrum has been observed experimentally in radio communications with the Venera space probe VYSHILOV, NABATOV,RUBTSOVand SHEVERDYAEV (EFIMOV,YAKOVLEV, [ 1989]), when the fluctuations of phase were caused by the motion of inhomogeneities in space plasma (solar wind).

6.4. COHERENT BACKSCATTERING OF PARTICLES FROM DISORDERED MEDIA

The coherent backscattering of particles other than photons has been approached only from the theoretical standpoint. IGARASHI[ 19871 has considered the effect of backscattering for isotopic and spin incoherent scattering of neutrons in the framework of the double-collision model. When the predominant channel of incoherent scattering is by spin-spin (magnetic) interactions, the enhancement may give way to the attenuation of backscattering.

11, § 71

CONCLUSION

189

The backscattering of electrons of middle energy (in the order of several hundred eV) and an unusual behavior of the enhancement factor as a function of the cross section of spin-orbital interaction have been discussed qualitatively by BERKOVITSand KAVEH[1988]. They have noted that the spin-orbital interaction can bring about a coherent antienhancement of backscattering and a sharp minimum in the angular distribution of the backscattered intensity. GORODNICHEV, DUDAREV and ROGOZKIN[ 1990al have obtained an exact solution to the problem of scattering of spin-$ particles, which participate in magnetic and spin-orbital interactions with a disordered medium and with a medium with an Anderson's type of disorder. These authors have also developed a theory of coherent enhancement of the backscattering process. The effect of enhancement for neutrons scattered backwards and in certain other directions has been taken up by DUDAREV [ 19881for neutrons diffracted in imperfect crystals, i.e., in crystals with isotopic and spin disorder, which corresponds to an Anderson model of disorder. The sharp resonance peaks revealed in the angular spectrum of reflected particles is associated with the diffraction of the particles at a regular part of the crystal potential. Moreover, resonance peaks occur when the periods of oscillations of the coherent field density in the nodes of the crystal lattice coincide as the particle is approaching a scatterer and returns to the surface of the crystal. DUDAREV[ 19881 and GORODNICHEV, DUDAREV,ROGOZKINand RYAZANOV[ 19891 noted in their studies of scattering in ordered periodic structures with fluctuating potentials that the effect of an additional enhancement of incoherent intensity owes its existence to the fact that the system of scattering centers possesses translational symmetry. The coherent enhancement of backscattering is caused by the effect of weak localization of particles in multiple scattering in the medium. We have avoided discussing the problem of weak localization of electrons in metals and semiconductors in this review, since it is too large a topic to be addressed in the space allotted to this article. The interested reader is referred to the review paper by BERGMANN[ 19841.

8 7. Conclusion It is not uncommon in the history of physics that a chance remark, trivial at first glance, has been a seedling for an entire branch of new physical phenomena, giving birth (not immediately but in 15 or 20 years) to a developed system of theoretical representations and experimental evidence. This is exactly

190


[I1

what happened with the private communication between Ruffine and Watson about coherent channels in 1969. It is high time to reconsider the evolution of this beautiful physical idea. Has it been completely exhausted, or will new and interesting facets be revealed to researchers? Whatever happens, we admit that we have been completely satisfied by our participation in the solution of problems associated with enhanced backscattering and weak localization phenomena. Acknowledgement We are indebted to Prof. E. Wolf for the interest shown in our work. We want to thank M. S. Belenkii, A. S. Gurvich, V. L. Mironov and D. V. Vlasov for communicating their results to us and A. I. Fedulov and F. I. Ismagilov for their assistance in computations. References* ABRAHAMS, E.,P. W. ANDERSON, D. c. LICCARDELLoandT. v. RAMAKRISHNAN, 1979, Scaling theory of localization: absence of quantum diffusion in two dimensions, Phys. Rev. Lett. 42( lo), 613-616. AGROVSKII, B. s., A. N. BOGATOV, A. S. GURVICH, s. v . KIREEV and V. A. MYAKININ,1991, Enhanced backscattering from a plane mirror viewed through turbulent phase screen, J. Opt. SOC.Am. A 8 (in press). AKHUNOV, KH.G., and Yu. A. KRAVTSOV, 1982, Coherent effects accompanying backscattering of sound from bodies near rough sea surface, Akust. Zh. 28(4), 438-440. AKHUNOV, KH.G., and Yu. A. KRAVTSOV, 1983a, Development of the enhanced backscatter effect in reflection from a phase conjugated mirror, Izv. VUZ Radiofiz. 26(5), 635-638. AKHUNOV, KH. G., and Yu. A. KRAVTSOV, 1983b, Effective cross section of a small body placed near the interface between two random media, Kratk. Soobshch. Fiz. FIAN 8, 8-11. AKHUNOV, KH.G., and Yu. A. KRAVTSOV, 1984, Conditions for coherent addition of backscattered sound waves under multipath propagation, Akust. Zh. 30(2), 145-148. 1982, Efficiency of AKHUNOV, KH.G., F. V. BUNKIN, D. V. VLASOV and Yu. A. KRAVTSOV, wavefront inversion in media with time-varying fluctuations, Kvant. Elektron. 9(6), 1287-1289. AKHUNOV, KH. G., Yu. A. KRAVTSOV and V. M. KUZKIN, 1984, Effect of enhanced backscattering from a body in a regular multimode waveguide, Izv. VUZ Radiofiz. 27(3), 319-323. D. V. VLASOVand Yu. A. KRAVTSOV, 1984, On the efficiency AKHUNOV, KH. G., F. V. BUNKIN, of phase conjugated focusing of waves in turbulent media, Radiotekh. Elektron. 29(1), 1-4.

* The titles of Russian papers have been translated for convenience. We note also that some Soviet journals are translated into English on a cover-to-cover basis, e.g., Akust. Zh. [Sov. Phys.-Acoust.]. Dokl. Akad. Nauk SSSR [Sov. Phys.-Dokl.], Izv. VUZ Radiofiz. [Radiophys. & Quantum Electron.], Radiotekh. & Elektron. [Radio Eng. & Electron. Phys.], Kvant Elektron. [Sov. J. Quant. Electronics], Zh. Eksp. & Teor. Fiz. [Sov. Phys.-JETP], Kratk. Soobsh. Fiz. [Sov. Phys. Lebedev Inst. Reports].

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ROSENBLUH, M., I. EDREI,M. KAVEH and I. FREUND, 1987, Precision determination of the line shape for coherently backscattered light from disordered solids: comparison of vector and scalar theories, Phys. Rev. B 35(10), 4458-4460. Ross, J. K., and A. L. MARSHAK, 1988, Calculation ofcanopy bidirectional reflectance using the Monte-Carlo method, Remote Sens. Environ. 24(2), 2 13-225. RUFFINE, R. S., and D. A. DE WOLF,1965, Cross-polarized electromagnetic backscatter from turbulent plasmas, J. Geophys. Res. 70(17), 4313-4321. RYBICKI, G. B., 1971, Searchlight problem with isotropic scattering, J. Quantum Spectros. & Rad. Trans. 11, 827-844. RYTOV,S. M., Yu. A. KRAVTSOV and V. I. TATARSKII, 1989a, Principles of Statistical Radiophysics, Vol. 3, Random Fields (Springer, Berlin, Heidelberg). RYTOV,S. M., Yu. A. KRAVTSOV and V. I. TATARSKII, 1989b, Principles of Statistical Radiophysics, Vol. 4, Wave Propagation through Random Media (Springer, Berlin). SAICHEV, A. I., 1978, Relation of statistical characteristics of transmitted and reflected waves in a medium with large-scale random inhomogeneities, Izv. VUZ Radiotiz, 21(9), 1290-1293. SAICHEV, A. I., 1980, Analysis of backscattering in turbulent medium under multiple scattering in the direction of propagation: the diffuse approximation, Izv. VUZ Radiofiz. 23(1 I), 1305- 13 13. SAICHEV, A. I.. 1981, Reflection from a wavefront inverting mirror, Izv. VUZ Radiofiz. 24(9), 1165-1167. SAICHEV, A. I., 1982, Compensation of wave distortions by a phase-conjugated mirror in an inhomogeneous medium, Radiotekh. Elektron. 27(9), 1961- 1968. SAICHEV, A. I., 1983, Ray description of waves reflected with phase conjugation, Radiotekh. Elektron. 28(10), 1889-1894. SALPETER, E. E., 1967, Interplanetary scintillations. I. Theory, Astrophys. J. 147(2), 433-XXX. SANT,A. J., J. C. DAINTYand M.-J. KIM, 1989, Comparison of surface scattering between identical randomly rough metal and dielectric diffusors, Opt. Lett. 14(1), 1183-1185. SCHMELTZER, D., and M. KAVEH,1987, Back-scattering of electromagnetic waves in random dielectric media, J. Phys. C 20, L175-L179. SHISHOV, V. I., 1974, Dependence of oscillation spectrum on spectrum of refractive index inhomogeneities. I. Phase screen, Izv. VUZ Radiofiz. 17( 1I), 1684-1692. SOBOLEV, V. V., 1963, Transfer of Radiation Energy in the Atmospheres of Stars and Planets (Van Nostrand, New York). J. M., and M. NIETO-VESPERINAS, 1989, Electromagnetic scattering from very SOTO-CRESPO, rough random surfaces and deep reflection gratings, J. Opt. SOC.Am. A 6(3), 367-384. STEPHEN, M. J., and G. CWILICH,1986, Rayleigh scattering and weak localization: effects of polarization, Phys. Rev. B 34(1 I), 7564-7572. TATARSKII, V. I., 1967, Estimation of light depolarization by turbulent inhomogeneities of the armosphere, Izv. VUZ Radiofiz. 10(12), 1762-1765. TATARSKII, V. I., 1971, Effects of turbulent atmosphere on wave propagation, Nat. Technol. Inform. Service USA, TT-68-50464. TRAN,P., and V. CELLI,1988, Monte-Carlo calculation of backscattering enhancement for a randomly rough grating, J. Opt. SOC.Am. A 5(10), 1635-1637. TSANG, L., and A. ISHIMARU, 1984, Backscattering enhancement of random discrete scatterers, J. Opt. Am. 1(6), 836-839. TSANG, L., and A. ISHIMARU, 1985, Theory of backscattering enhancement of random discrete isotropic scatterers based on the summation of all ladder and cyclical terms, J. Opt. SOC.Am. A 2(8), 1331-1338. UFIMTSEV, P. YA., 1971, The Method of Edge Waves in the Physical Theory of Diffraction (US Air Force, Wright-Patterson AFB, OH).

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VAN ALBADA,M. P., and A. LAGENDIJK, 1985, Observation of weak localization of light in a random medium, Phys. Rev. Lett. 55(24), 2692-2695. VAN ALBADA,M. P., and A. LAGENDIJK,1987, Vector character of light in weak localization: spatial anisotropy in coherent backscattering from a random medium, Phys. Rev. B 36(4), 2353-2356. VAN ALBADA,M. P., M. B. V A N DER MARKand A. LAGENDIJK,1987, Observation of weak localization of light in a finite slab: anisotropy effects and light-path classification, Phys. Rev. Lett. 58(4), 361-364. V A N DE R MARK,M. B., M. P. VAN ALBADA and A. LAGENDIJK, 1988, Light scattering in strongly scattering media: multiple scattering and weak localization, Phys. Rev. B 37(7), 3575-3592. VINOGRADOV, A. G., 1974, Enhanced backscattering in reflection of waves from rough surfaces placed in a randomly inhomogeneous medium, Izv. VUZ Radiofiz. 17(10), 1584-1586. VINOGRADOV,A. G . ,and Yu. A. KRAVTSOV, 1973, A hybrid approach t o the calculation offield fluctuations in a medium with large- and small-scale random inhomogeneities, Izv. VUZ Radiofiz. 16(7), 1055-1063. VINOGRADOV, A. G., Yu. A. KRAVTSOV and V. I. TATARSKII, 1973, Enhanced backscattering from bodies immersed in a random inhomogeneous medium, Izv. VUZ Radiofiz. 16(7), 1064-1070. VINOGRADOV, A. G., A. G. KOSTERIN,A. S. MEDOVIKOV and A. I. SAICHEV,1985, Effect of refraction on the propagation of a beam in a turbulent medium, Izv. VUZ Radiofiz. 25(10), 1227-1235. VLASOV, D. V., 1985, Laser sounding of an upper layer of the ocean, Izv. Akad. Nauk SSSR Ser. Fiz. 49(3), 463-472. WATSON,K. M., 1969, Multiple scattering of electromagnetic waves in an underdense plasma, J. Math. Phys. 10(4), 688-702. WOLF,P. E., and G . MARET,1985, Weak localization and coherent backscattering of photons in disordered media, Phys. Rev. Lett. 55(24), 2696-2699. and R. MAYNARD,1988, Optical coherent backWOLF, P. E., G . MARET,E. AKKERMANS scattering by random media: an experimental study, J. Phys. (France) 49(1), 63-75. I. G., 1978, Moment of field intensity propagated in a randomly inhomogeneous YAKUSHKIN, medium in the region of saturated fluctuations, Izv. VUZ Radiofiz. 21(8), 1194-1201. I. G., 1985, Intensity fluctuations due to small-angle scattering of wave fields, Izv. YAKUSHKIN, VUZ Radiofiz. 28(5), 535-565. YANG,C. C., and K. C. YEH, 1985, The behavior of the backscattered power from an intensely turbulent ionosphere, Rad. Sci. 20(3), 319-324. YEH,K. C., 1983, Mutual coherence functions: intensities of backscattered signals in a turbulent medium, Rad. Sci. 18(2), 159-165. ZAVOROTNYI, V. U., 1984, Backscattering of waves from two-scale rough surface: account of reillumination, Izv. VUZ Radiofiz. 27(2), 196-202. ZAVOROTNYI, V. U., and V. E. OSTASHEV, 1982, On the enhanced backscattering ofwaves from rough surfaces, Izv. VUZ Radiofiz. 25(1 I), 1291-1295. V. U., and V. I. TATARSKII, 1982, Backscattering enhancement of waves by a body ZAVOROTNYI, near a random interface, Dokl. Akad. Nauk SSSR 265(3), 608-612. ZAVOROTNYI, V. U., V. I. KLYATSKIN and V. I. TATARSKII, 1977, Strong fluctuations of electromagnetic intensity in randomly inhomogeneous media, Zh. Eksp. & Teor. Fiz. 73(2), 481-497. ZUEV,V. E., V. A. BANAKHand V. V. POKASOV,1988, Optika turbulentnoi atmosfery (Optics of a turbulent atmosphere) (Gidromet, Leningrad).


E. WOLF, PROGRESS IN OPTICS XXIX 0 ELSEVIER SCIENCE PUBLISHERS B.V., 1991

GENERATION AND PROPAGATION OF ULTRASHORT OPTICAL PULSES BY

IVANP. CHRISTOV* Faculty of Physics, Sofia University 1126 Sofa, Bulgaria

* Present address: Max-Planck-Institiit f i r Quantenoptik, 8046 Garching, Germany.

CONTENTS PAGE

$ 1 . INTRODUCTION

. . . . . . . . . . . . . . . . . . . 201

$ 2. THEORETICAL BACKGROUND

. . . . . . . . . . . . 202

$ 3 . GENERATION OF FEMTOSECOND OPTICAL PULSES . 220 $ 4 . PROPAGATION EFFECTS

. . . . . . . . . . . . . . . 244

$ 5. CONCLUSION . . . . . . . . . . . . . . . . . . . . . ACKNOWLEDGMENTS REFERENCES

284

. . . . . . . . . . . . . . . . . . . 284

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

284

1. Introduction Since the development of the glass lens, it has become clear that light waves are very suitable for research into a spatial scale in which the objects are almost invisible for the unaided eye. This concept embodies the idea of locality in space. The development of our knowledge about these optical phenomena has included the interesting feature of “space-time analogy”. A general example of this feature is the analogy between the diffraction and dispersion of light waves. In fact, many of the problems arising from the analysis of objects with a size of the order of the wavelength of the probing light also appear in some research in the temporal domain. It is well known that for the successful study of the dynamic behavior of some processes it is necessary to apply a shock to excite the medium. Thus the idea of locality in time requires the development of proper sources that deliver short optical pulses. However, the production of short light pulses is more difficult than the corresponding spatial problem. The primary reason derives from the difference between the spatial and the temporal spectrum of the radiation. Focusing in space involves the production of new spatial frequencies, which is not a serious problem, whereas the derivation of new time frequencies requires a change of the radiation spectrum (energy). The interest in short optical pulses started between 1962 and 1963, when the gained lasers with Q-switching allowed the production of giant pulses with a duration of about 10-8-10-9 s and powers up to 10’ W, which stimulated much research in nonlinear optics. With the discovery of active and passive mode-locking techniques around the late 1970s, the picosecond boundary was overcome. The new laser schemes that gave pulses as short as 1 ps and powers up to 10” W enabled the observation of large variety of interesting nonlinear effects, such as self-phase modulation and tunable parametric generation. At the beginning of the 1980s the advanced technique for colliding pulse mode locking led to a shortening of the generated pulses down to several tens of femtoseconds. In fact, such pulses only consist of several cycles of the carrier wave and are very promising for the investigation of ultrafast processes in atomic and molecular systems, such as gases, liquids, and semiconductors. In general, the generation of ultrashort pulses can be recognized as a problem for “making in phase” a great number of frequencies, whereas the interaction 20 1

202

GENERATION AND PROPAGATION OF ULTRASHORT OPTICAL PULSES

[III, $ 2

processes lead to dephasing of these frequencies. Therefore, in this review we shall consider these two aspects of the light-matter interaction from a unified point of view. Section 2 presents the theoretical basis necessary to study this interaction. In 0 3 we describe some of the most promising methods for generating femtosecond optical pulses that have been developed so far. Section 4 deals with linear and nonlinear effects that appear when a femtosecond pulse propagates in free space or in substances, and with some applications in the field of optical communications and data processing.

5 2. Theoretical Background 2.1. PROPAGATION OF OPTICAL PULSES THROUGH A RESONANT MEDIUM

The most widely used approach in laser physics is the semiclassical approximation. As is well known, it treats the electromagnetic field as a classical wave described by the Maxwell equations, and the medium as a set of atoms or molecules with discrete energy levels, the dynamics of which are studied by quantum theory. The importance of this approach is demonstrated by the fact that in most cases of light-matter interaction, the effects caused by the quantum-noise fluctuations are negligible or they can be described phenomenologically. On the other hand, some interesting problems such as laser line width, buildup from vacuum, and photofi statistics require a quantum treatment of both the atoms and the field. The crucial point in the semiclassical model is the so-called self-consistency condition (see, eg., SARGENT,SCULLYand LAMB[ 19741,which requires that the eiectric field E(r, t ) induces in the medium dipole moments pi according to the laws of quantum theory. These moments are then summed to result in a macroscopic polarization P(r, t), participating as a source in the right-hand side of the wave equation. The self-consistency condition then requires that the reaction field E’(r, t ) generated by the polarization must be equal to the initial field E(r, t). It should be noted that the most convenient description of the field-induced dipole moments is based on the density matrix formalism rather than on the Schrtrdinger equation, because it facilitates a statistical averaging over the individual dipole moments in order to obtain the macroscopic polarization. Thus, we can write the following basic equations governing the light-matter interaction in a semiclassical approximation (see also PANTELLand PUTHOFF[ 19691):

111, 8 21

203

THEORETICAL BACKGROUND

(i) Quantum-mechanical equation for the density operator p ( r , t ) ih -aP=

[H,p],

at

where H is the Hamiltonian, which is the sum of the unperturbed atomic Hamiltonian H , and the interaction Hamiltonian H' ,

H

=

H,

+ H'

.

(2.2)

The unperturbed Hamiltonian obeys the equation H,, I uk) = Ek 1 u k ) ,where the Ek are the stationary energy levels corresponding to the atomic states 1 u k ) . In the dipole approximation the interaction Hamiltonian is H' = - p E, where p is the dipole momentum operator ( p = - er for an electron system). After some mathematics, from eqs. (2.1) and (2.2) one obtains two equations for the diagonal and off-diagonal elements of the density matrix,

-

(2.3a)

(2.3b) where wij is the transition frequency, and T , and T2 are the decay times corresponding to the excited atomic state and to polarization, respectively. In eq. (2.3b), p i is the equilibrium value of the diagonal element. (ii) The macroscopic polarization of the medium is given as the sum of all averaged individual dipole moments,

In eq. (2.4) the overbar corresponds to averaging all particles with density N,. Then, using the formula for the expectation value of an operator A , ( A ) = Sp(pA), together with eqs. (2.3a) and (2.4) it follows that the equation for the macroscopic polarization of a two-level isotropic medium can be written as

-d+2 -P dt2

2 dP -++:,P=T2 d t

2021 lp1212(N,-N2)E, 3h

where N , - N2 = N,(p, - pZZ)= - N is the population difference per unit

204


[Ill, 2

volume, which obeys [from eq. (2.3b)l the equation aN

N-N"

at

T,

- +---

2

-

ap -*E.

hw,, at

In fact, this is an energy balance equation in which the right-hand side contains the energy lost by the field in polarizing the medium. (iii) The one-dimensional wave equation for the electric field, induced by the polarization P(z, t) is

However, this equation is unnecessarily compiicated for most practical cases. For instance, even for pulses with a duration of several tens of femtoseconds the pulse envelope is a slowly varying function of z and t compared with the carrier. Therefore, when the optical wave is linearly polarized, we can write both the field and the polarization in a scalar form: E

=

:{Eexp[i(w,t

P

=

${Fexp[i(w,i - k,z)]

- k,z)]

+ c.c.}, + c.c.},

(2.8a) (2.8b)

where k, = wo/c is the wave number. Then, substituting eqs. (2.8a,b) into eq. (2.7), we find

aE I a.??

-+ aZ

-=

at

I

-2inkOP,

where we have neglected the terms containing a2E/i3z2, a2E/at2,and In the same approximation eqs. (2.5) and (2.6) become

a2P/at2. (2.10a)

(2. lob) where we have assumed that w,, = w,. Equations (2.9) and (2.10a,b) compound the reduced set of equations describing the interaction between the optical field and the resonance two-level medium. Provided that, in addition, a more stringent requirement is met,

111, § 21

205


varies little over time of the order of T2, we obtain a rate namely that equations approximation (REA) which is valid when the signal bandwidth is narrow compared with T;'. In particular, for pulsed laser systems this means that the phase-decay time T, is negligible compared with the pulse duration. Thus, neglecting aP/at in eq.(2. lOa), we find that

-

P = iN1PI2l2T2 E',

(2.11)

3h Then eqs. (2.9) and (2. lob) yield

aE 1 aE -+--= aZ c at

2

~ 4 3 hc

p

~ ~ NE,

1

~

~

~

(2.12a)

(2.12b) Multiplying eq. (2.12a) by obtain

E*

a@ 1 a@ -+--=ON@,

aZ

at

aN

N-N'

-t-=

aZ

and adding to the conjugated equation, we

(2.13a)

NO@,

(2.13b)

T,

= 4~w,,1p,~1~T~/3hc where@ = cn/8nhoolE"12isthephotonfluxdensityando is the transition cross section. Equations (2.13a,b) comprise a rate equations approximation (REA) for a two-level system. We should note that in the more general case when oo# 021, the time constant T2in the expression for o should be replaced by the line shape function g ( o , , , coo) (see PANTELL and PUTHOFF [ 19691). The system of rate equations given by eq. (2.13a,b) enables us to describe the behavior of both the generators and the amplifiers of optical radiation (see $ 3). Although eqs. (2.13a,b) describe the dynamics of a two-level system, their simple structure permits a generalization in cases where additional levels should be taken into account. An interesting example, closely related to the amplification of ultrashort pulses, gives the set of levels corresponding to the dye molecule (see, e.g., SCHAFER[ 19731). Because of the fast nonradiative decay processes within each manifold of levels (the lowest singlet SF), the first

206


[III, $ 2

12>

II>

I o>

a

Fig. 1. Energy level scheme ofthe active dye (a) and the absorber (b). Straight arrows correspond to radiative transitions, wavy arrows correspond to nonradiative transitions.

excited singlet S(la),and the first triplet TI),the system is effectively a three-level system (see fig. la). Thus, if @* denotes the photon fluxes per unit wavelength propagating in k z directions, we can write the following set of coupled rate equations (see also GANIEL, HARDY,NEUMANN and TREVES [ 19751):

ae(A)(@+ at

+ N o s a,(A)(@++ @-)dA, -aNT - ksTN, - NT -, at

+ @-)dl (2.14b) (2.14~)

TT

No+Nl+NT=N.

(2.14d)

In these equations the quantities are defined as follows: Ni(i = 0, 1, T) is the population density of the levels SF), S(la),and T,, respectively, and W(t)is the pump rate, given by

in which a,(A) is the absorption cross section from S$" to S'f), and f(A) is the normalized spectral distribution of the pump power P(t). Moreover, in eqs. (2.14) TI and TT are the lifetimes of S(la)- and T,-states (in the absence of

111, § 21


207

stimulated emission), aJA) is the stimulated emission cross section, k,, is the S y )-+ TI crossing rate, +(A) is the absorption cross section from T , to higher triplet states, and the g' are geometrical factors accounting for the threedimensional nature of the real amplifier, whose significance we will discuss in 8 3.3. Another interesting example is the theoretical treatment of passively modelocked systems (see § 3.2. l). It is based on the assumption that a steady-state regime of the laser is reached when only a single pulse circulates within the cavity with a group velocity ug, so that the laser emits a continuous train of pulses (HERRMANN and WEIDNER[ 19821). The interaction between the pulse and the dye molecules can be successfully studied using a set of rate equations in which the active dye is considered as a four-level system (drawn in fig. la) with a very fast relaxation time from level I 3) to the upper laser level I2), and from the lower laser level I 1) to the ground state 10). On the other hand, the absorber may be regarded as a three-level system (fig. lb) with a negligible population of level I 2). Then, neglecting the population of levels I 3 ) and 1 1) of the active dye, the following set of equations can be written: (2.15a)

(2.15b)

(2.15~) where the indexes a and b correspond to the active medium and the absorber, respectively. The model based on eqs. (2.15) enables us to study the stable single-pulse region as well as the dependence of some important pulse parameters (e.g., energy, duration, and asymmetry) as a function of the laser parameters. The set of rate equations describing the performance of a synchronously mode-locked CW dye laser (see $ 3.2.2) is similar, but an equation describing the changes in the photon flux density of the pump wave should be included (STAMM[ 19881). Despite the successful application of REA For describing the dynamics of lasers and amplifiers, there are some circumstances limiting their use. One limitation origmates From the experimental observation of a significant phase modulation (chirp) exhibited by the pulses delivered from passively modelocked dye lasers. This effect may be adequately introduced in theory only by

208


[III, 5 2

considering both the saturation and the phase memory, connected with the off-resonance interaction between the pulse and the dye (RUDOLPH[ 19841, RUDOLPHand WILHELMI [ 1984a1). In such a case the time derivative of the polarization in eq. (2.10a) cannot be neglected, and a set of equations in terms of the density matrix elements is utilized directly. For example, to analyze the performance of a CW passively mode-locked dye laser RUDOLPH[ 19841 and PETROV,RUDOLPHand WILHELMI[ 19871 used a set of partial differential equations for two-level systems representing the dye molecules (2.16a)

(2.16b)

(2.16~) where eqs. (2.16a) and (2.16b) follow directly from eqs. (2.3a) and (2.3b), and eq. (2.16~)follows from eqs. (2.4) and (2.7). We should also note that a negligible energy relaxation during the pulse passage takes place. The set (2.16) is valid for both the gain medium and the absorber. By using eqs. (2.16) PETROV,RUDOLPHand WILHELMI[ 19871 found an approximate expression for the modification of the complex pulse envelope after passing through the gain and absorber media. This is useful for estimating the total round-trip contribution of the intracavity components, which is necessary to obtain the steady-state solution of the self-consistency equation (see $ 3.2). Another approach, based on the density matrix equations, was developed by CASPERSON [ 19831. The procedure is the same as in eqs. (2.16a,b), but in addition, an isotropic orientation distribution of the dye molecules is taken into account. MACFARLANE and CASPERSON [ 19891 performed further studies based on the same approach. Until now we have based our considerations on the assumption that the pulse duration considerably exceeds the phase decay times of the medium, which condition is fulfilled for dye systems and other optically active substances. The opposite case, when the time of interaction is much smaller than the characteristic times of all the relaxation processes, is also of great interest and has been extensively studied (see, e.g., MCCALLand HAHN[ 19691,ALLEN and EBERLY[ 19751).

1 1 1 9 8 21

209


2.2. PROPAGATION IN A TRANSPARENT LINEAR MEDIUM

2.2.1. Regular pulses When the radiation wavelength is far from the absorption bands of the particles, there are no resonance transitions and the picture differs considerably from that described in Q 2.1. This is valid also in the resonant case when the inversion induced by the pulse is negligible (small-area pulse approximation, CRISP[1970]). The nonresonant propagation of a short optical pulse in a transparent medium is accompanied by some inherent linear and nonlinear effects, such as dispersive spreading, self-phase modulation, and generation of higher-order harmonics. Here, we will start with some approaches currently being used in Fourier optics. At a small inversion of the atoms the basic set of equations (2.5), (2.6), and (2.7) reduces to -a2p t - - + w2O ,ap zP= at2 T, at

a2E az2

2waIA2Nv E , 3fi

(2.17a)

1 a 2 E - 411 a2P

c2

at2

(2.17b)

c 2 at2 '

where ma = w21is the atomic resonance frequency. This set includes both the nonresonant and the small-area case. It was shown by CHRISTOV [ 1988a,b] that for optical pulses with a duration much smaller than the relaxation time T2, the set (2.17) possesses a solution given by E(z, t ) = where tl =

s

E(0, w) exp[ - iazf(w)

+ iwq] dw ,

411NvI p ) 2 w a f ( w ) / 3 h cf(o) , = w/(w,' -

0 2 )and

(2.18)

q

=

t - z/c .

In eq. (2.18), E(0, w) is the Fourier transform of the input pulse E(0, t ) (the boundary condition at z = 0). The solution given by eq. (2.18) is valid either in the case of slowly varying envelopes [see eq. (2.8)] or in the case of a sufficiently rare medium. We should note also that when the orientation of all dipoles is parallel to the field polarization, it is necessary to replace 4 lpl by Ip I in eq. (2.17a). The solution of eq. (2.17a,b) corresponding to the resonant propagation of small-area pulses was given by CRISP[ 19701, E(0, w ) exp [ + iaz/2(R - i/T2) + iwq] d o ,

(2.19)

210


[III, 8 2

where 62 = w - wo is the frequency shift. For an infinite relaxation time (T2-,a),eq. (2.19) tends to the resonance solution given by CHRISTOV [ 1988al. Let us denote the spectral transmittance function of the medium as =

(2.20a)

exp [ - i q w l ,

where $(62)

= - iazf(l2)

(2.20b)

is the dispersive phase. Then eqs. (2.18) and (2.19) can be written in a more general form,

EOut(t)=

s s

&(a) H(62) eintd62,

(2.2 1a)

or, equivalently, as

Eout(t)= where

H(t - t ' ) =

Ein(l)H ( t -

s

t ' )dt ,

H ( 0 ) ein('-'') d62

(2.2 Ib)

(2.22)

is the shock response function (Green function) corresponding to this case. Equations (2.21a,b) and (2.22) could be considered as the essence of the classical Fourier optics (see GOODMAN [ 19681) applied to optical pulses. Despite the convenient integral representations (2.21) and (2.22), there is an alternative approach describing the propagation of optical pulses in a dispersive medium, which is based on partial differential equations. Let us expand the dispersive phase $(a) in a power series,

(2.23) Then the familiar second-order approximation of dispersion theory corresponds to truncation of the series given by eq. (2.23) up to the second power of 62, and the pulse evolution is governed by a parabolic differential equation VYSLOUKH and CHIRKIN[ 19881) (see, e.g., AKHMANOV, (2.24)

111, § 21


21 1

where q = t - z/v,, with ug = (ak/ao);,’ is the group velocity, and k, = (a’k/ao’),, is a parameter specifying the value of the group velocity dispersion (GVD). Equation (2.24) is similar to the parabolic equation describing the diffraction of an optical beam (YARIV[ 19751). This is an example of the space-time analogy mentioned in the introduction (see also AKHMANOV, CHIRKIN, DRABOVICH, KOVRIGIN, KHOKHLOV and SUKHORUKOV [ 19681). For a Gaussian input pulse E“( q, 0) = E , exp ( - t2/2 T 2 ) , eq. (2.24) has the solution (2.25a) where

(2.25b) It can be seen from eq. (2.25b) that at a distance z = L , = T2/k2 the pulse duration becomes twice as long. Moreover, the pulse spreading is accompanied by the appearance of a positive frequency sweep (up-chirp) in the region of positive (or normal) dispersion (k2 > 0), and a negative sweep (down-chirp) when k, < 0 (negative or anomalous dispersion). The value of the chirp parameter is a’ = 0.5 a2p/aq2 = z/[k2(z2 + L L ) ] . When the mutual dependence between the temporal and spatial properties of the propagating radiation is of primary interest, the four-dimensional wave equation must be solved. However, because this is difficult in many practical cases, here we will focus our attention on propagation in free space, where the right-hand side of the wave equation vanishes. One useful approach to studying this problem is the angular spectrum approximation (see, e.g., BOUWKAMP [ 19541, FRIBERG and WOLF[ 19831). In fact, the solution of the wave equation can be written in the form

where r = (x, y ) is the radius vector in the plane transversal to the direction of propagation (which we choose as z-axis), k , = (kx, ky), and the integration over k, has been performed by means of the vacuum dispersive relation k, = (w2/c2- k:)’” (see CHRISTOV [ 1985b], COOPER and MARX[ 19851). In

212


[III, 3 2

eq. (2.26), E ( k , , w ) is the Fourier transform of the field at the source plane E(r, z = 0, t). The angular spectrum approach can be simplified provided the ratiation angular spectrum is sufficiently narrow, i.e.; when k , < w/c, where w varies in limits for which the spectral components of the radiation have considerable amplitude. This is valid especially in the case of laser radiation. The calculations can also be simplified by the requirement for cross-spectral purity of the source field (MANDEL[ 19611). This means, in particular, that the spatial and spectral properties of the source are independent; i.e., a factorization of the field in the source plane can be done. Thus, in the temporal domain we can write E(r, z

=

0, t ) = f ( r ) g ( t ) .

(2.27)

2.2.2. Partially coherent pulses Until now we have examined radiation with a regular temporal modulation. However, this is an idealization, since the real sources deliver optical pulses that possess only partial temporal and spatial coherence. The coherence properties of the propagation of a partially coherent field are described by introducing the second-order coherence function (BORN and WOLF [ 19681):

where the angle brackets denote an ensemble averaging. The afore-mentioned angular spectrum approach is also convenient for studying the propagation of partially coherent fields (JAISWALand MEHTA [ 19721). Moreover, the problem for spectral purity of partially coherent fields has been discussed by some authors (see, e.g., WOLF and CARTER[1975, 19761). As MANDELand WOLF [ 19761 showed, when light is cross-spectrally pure, its mutual coherence function can be expressed as the product of two correlation functions, one of which characterizes the spatial coherence and the other the temporal coherence. Thus we can write a formula for the coherence function of a spatially pure source, similar to the corresponding relation, eq. (2.27), for regular fields,

Then, from eqs. (2.26)-(2.29) it follows that an integral representation of the second-order coherence function for a radiation modulated both spatially and

111, § 21

213


temporally, and, in addition, with partial spatial and temporal coherence, m , , ZIT 1 1 1 ;

r21 z21 112)

= ( W I , ZI. I

111)E*(r29 z2, 1 1 2 ) )

r r

where Tl(w,, w 2 ) and T2(k,,k 2 ) are Fourier transforms of T , ( t , ,t 2 ) and T2(r,,rZ),respectively. It should be noted that typical lasers generate so-called “globally shape-invariant’’ fields, which are generally not spectrally pure (GORI and GRELLA [ 19841). However, with a suitable experimental set-up it is possible to transform the initial field into a field that is spectrally pure (MANDEL [ 19613).

2.3. NONLINEAR PROPAGATION OF OPTICAL PULSES

2.3.1. Regular pulses The propagation of a short optical pulse in a nonlinear medium can be described by using a phenomenological approach where the polarization is represented as a power-series expansion over the optical field (see, e.g., SHEN [19841),

P

=

Pi-+ PNL= P ,

t

:E E

x ( ~ )

+ x ( ~ :)E E E + x ( ~ :) EEEE + .

* *

(2.3 1) We should note that eq. (2.31) is valid when the nonlinear reaction of the medium is almost instantaneous (quasistatic approximation). Moreover, if the medium is isotropic, only the odd powers of the field participate in eq. (2.31). In the microscopic theory based on the density matrix formalism, analytical expressions for the permeability x(”) in terms of the atomic parameters can be obtained (SHEN[ 19841). Here we will consider some effects caused by the cubic nonlinearity only, because they are of primary importance for the optics of femtosecond pulses. Thus, neglecting the terms in eq. (2.31) that correspond

214


[III, $ 2

to harmonics generation, we obtain an equation describing the propagation of an optical pulse in a dispersive medium whose index of refraction is modified by the pulse intensity, the so-called self-phase modulation (SPM). In the second-order approximation of the material dispersion [see eq. (2.24)], together with eqs. (2.7) and (2.8alb), we find (see HASEGAWA and TAPPERT [ 1973a,b], NAKATSUKA, GRISCHKOWSKY and BALANT[ 19811) (2.32) , coefficient determining where q = t - z/u,, k = kon2/2no;11, = 3 n ~ ( ~ ) /isn the the nonlinear correction of the refractive index: n = no + in2 lE12 = no + n i l . Here I = cn0/8n IEl' is the pulse intensity. Equation (2.32) represents the famous nonlinear SchrBdinger equation (NSE) and also describes the selffocusing of a light beam due to propagation in a medium with cubic nonlinearity (SHEN[ 19841). By setting E(z, q) = A(z, q) exp[icp(z, q)], eq. (2.32) can be written in the form of two equations for the real amplitude A (z, q) and the phase d z , rl), (2.33a) acp

k2

aZ

2

0=-74a',Ak+] ' )% !(

['

A a42

(2.33b)

An understanding of the effects described by eq. (2.32) will provide the solution for the case without a group velocity dispersion (k, = 0), E(z, q) = E(0, q) exp [ - ik I E"(0,q) 1 'z] .

(2.34)

This formula shows that the instantaneous frequency of the pulse changes during the propagation according to (see SHIMIZU[ 19671, STOLENand LIN [ 19781) (2.35) where leflisthe effective intensity in the fiber core, and a Gaussian input pulse with half-duration T is assumed. Equation (2.35) shows that the self-phase modulation induces a positive chirp over the central region of the pulse and a negative chirp over the pulse wings. We shall consider the effects caused by the cubic nonlinearity in more detail in 0 2.3. If the input pulse is not too strong,

111.8 21

215


the dispersive term in eq. (2.32) cannot be neglected and a more precise analysis is necessary. An inspection of eq. (2.33b) shows that it is similar to the Hamilton-Jacobi equation used in mechanics, where the phase cp corresponds to the mechanical action, and the potential V is given by

Using this analogy, we can consider the trajectory of the e - “point” on the front of a pulse with initial Gaussian profile (2.36a) Then, from energy conservation, it follows that the equation for a ( z ) is (2.36b) whose solution is given by a ( z ) = 1 + (L,2

* L,Z)z2,

(2.36~)

where L,,

=

~/(lk21kon~~o/~o)1/2

is a characteristic nonlinear length. The positive and negative signs in eq. (2.36b,c) correspond to the positive and negative signs of the dispersive parameter k,, respectively. Thus, for positive k, (normal dispersion) both the GVD and SPM lead to the appearance of up-chirp, and the pulse spreads faster than in a linear medium. In the opposite case, when k, < 0 (anomalous dispersion), the SPM acts contrary to the GVD, and from eq. (2.36~)it follows that when L , = L,, the dispersive spreading of the pulse is compensated for by the nonlinearity and the pulse propagates without any change of its shape and duration [ a ( z )= 1 in eqs. (2.36)]. This is an example of the famous solitonlike solution of the nonlinear Schrodinger equation. The equality of L , and L,, requires a critical power P, of the input radiation, P,

=

no I k2 I s,, T2kon; ’

(2.37)

when S,, is the effective cross area of the fiber. To analyze the solutions of the

216


[III,§ 2

NSE in more detail, we shall write it in a more convenient, normalized form (2.38) where the abbreviations used are as follows: = - z/L,, T = q/T, and q(z, t) = T(k0n,/2n, I k, I )‘/,E(z, The positive sign in eq. (2.38)corresponds to the anomalous dispersion regime (k2 < 0). In fact, solution (2.36a) of the NSE was obtained by prescribing its time dependence preliminary. However, ZAKHAROV and SHABAT[ 19711 showed that the NSE possesses a class of exact solutions that are shape-preserving during the propagation (i.e. solitons). To find the simplest soliton solution of NSE, we assume the following factorization

r).

d t , z)

=

~ ( z exp($it>. )

(2.39)

Then, substituting eq. (2.39) into eq. (2.38) and setting the soliton condition a A p z = 0, one finds q ( t , 7) = A , sech(r) exp($i 0) q ( t , z)

=

A , tanh(z) exp(i5).

(2.40b)

As can be seem from eq. (2.40b), this solution does not have a vanishing amplitude at infinity, and it is, therefore, called the “dark soliton”. Moreover, this is an odd pulse with an abrupt phase shift at z = 0 (fig. 2, dashed line). It is known that for silica-based fibers the GVD parameter k , passes through a zero value at about 1 = 1.3 pm (MARCUSE[ 1980]), which makes it possible to produce both bright and dark solitons experimentally (see 0 4.4). The discovery of the inverse scattering method has resulted in the finding of a more general solution of the NSE. For example, let the input pulse be given by qo sech(z),

(2.41a)

4 = qo tanh(r),

(2.41b)

q(0, 7) q(0,

=

where qo is the initial field amplitude. Then, in the case of a bright input pulse given by eq. (2.41a), and for q, = N + tl ( N 2 1 is an integer and I a / < $)

217

THEORETICAL BACKGROUND ” 7

TIME

Fig. 2. Pulse profiles of a bright soliton (solid) and dark soliton (dashed). The time variable is in arbitrary units.

the pulse evolves at infinity into a nonlinear superposition of N solitons. These solitons move together and exhibit an oscillatory behavior. For instance, for N = 2 and a = 0, the input pulse has twice the amplitude of the fundamental soliton, and it represents the first periodic solution, called a “breather” (SATSUMAand YAJIMA[ 19741) q2(5r

4=4

cosh(3z) + 3 cosh(z) exp( - 4i5) exp( - $ i t ) . cosh(4z) + 4 cosh(z) + 3 cos(45)

(2.42)

After travelling a distance equal to 5 = (or z = nT2/2 lk21), called the “soliton period”, the pulse repeats its shape (fig. 3). In the case of a dark input pulse [eq. (2.41b)], and qo = N - a (0 < a < 1) the pulse always evolves into one fundamental soliton, accompanied by a generation of 2(N - 1) secondary dark solitons under the same background, plus some nonsoliton parts (ZHAO and BOURKOFF [ 1989a1). The behavior of the higher-order solitons is more complex, but in all cases the pulse exhibits a sequence of narrowings and splittings. n

e

n.( 1-2 0

Fig. 3. Pulse profiles ofan N

=

2

4

n 1-2

0

2

4

4

2 soliton (the so-calledbreather). (a) ( = 0; (b) ( = n;(c) ( = i n ; (d) ( = in; (e) ( = in.

218


[III, $ 2

2.3.2. Partially coherent pulses We will now consider some aspects of the nonlinear propagation of partially coherent pulses. By using the Feynman path integral approach (see, e.g., [ 1980]), the NSE [eq. (2.38)] can be represented in the form MARINOV (2.43) where q(5 = 0, 0) is a boundary condition, and G(8, z, () is given by the path integral (FATTAKHOV and CHIRKIN[ 19831) G(8, z,

5) =

1

{ jot

exp -

I

U[z(x), dzldx] d x Dz(x) ,

(2.44)

where the external integral is over the space of paths connecting the points (0, 8) and (z,(), where 8 = z(0) and z = ~ ( 5 )In . eq. (2.44), U is the Lagrangian Y[z(x),dz/dx]

=

(-)

1 dz 2 dx

-

+ 1qI2.

(2.45)

Generally, it is difficult to find an exact analytical solution of eqs. (2.43) and (2.44). However, FATTAKHOV and CHIRKIN[1984, 19851 used an iteration procedure to obtain an approximate solution for these equations in the case of a propagating ultrashort pulse with random phase modulation in an optical fiber. Obviously, a predominant contribution in the integral (2.44) gives these paths, which are optimal; i.e., they satisfy the classical Euler equation,

a ax

a9 a(az/ax)

a 3-0.

(2.46)

az

Then, from eqs. (2.45) and (2.46), it follows that (2.47) As a zero-order approximation, the solution of the NSE without dispersion

can be employed [z d o ) ( x , 7)

=

-= L,, + L,,

see also eq. (2.34)],

d o , z) expb I d O , z)l’xI .

(2.48)

It can be seen from eq. (2.48) that Iq(O)(x,z)l’ = Iq(0, z)12, which physically means that the pulse propagates in a medium whose parameters are determined


III,§ 21

219

by the input pulse. FATTAKHOV and CHIRKIN [ 19831 have called this approach the “prescribed channel approach”. Thus solving eq. (2.47) with q(z, x) given by eq. (2.48) and then calculating G(0, 2, 5) from eq. (2.44), the desired approximate solution of NSE can be obtained in the form (2.43). In a simple demonstration for a regular Gaussian input pulse this method gives for the propagating pulse duration (2.49a)

7x0 = 4 < ) T , where a’( 0) (see $ 4.2). The single-stack mirrors possess k, > 0 for the red-shifted and k, < 0 for the blue-shifted pulse spectrum (DE SILVESTRI, LAPORTAand SVELTO[ 19841). Greater dispersion is exhibited by the double-stack mirrors (WEINER,FUJIMOTOand IPPEN [ 19851, LAPORTAand MAGNI [ 19851). In $ 4.2.1 it is shown that the angular dispersion of the prisms leads to negative GVD irrespective of the sign of the material dispersion. Some typical values of GVD for different intracavity components are presented in table 1. We will now examine the effects that cause SPM of the intracavity radiation. The main factors are the transient saturation of both the absorption and the gain near the dye’s resonance, and the nonlinear refraction index of the solvent (usually ethylene glycol). The SPM effect due to the propagation in the resonant medium can be described successfully using the considerations of SILVESTRI, LAPORTAand SVELTO[ 19841 and MIRANDA,JACOBOVITZ, BRITOCRUZand SCARPARO[ 19861). The phase modulation-induced chirp arises due to the temporal change of both the resonance dispersion and saturation during the pulse passage. Hence, the refractive index experiences a change equal to IZ, (w, - w)g(w) a(t), where w, is the resonance frequency, g(w) is the line shape, and a(t) exp [ - j‘I(t’) dt’ 1, where I ( t )is the pulse shape. With typical experimental conditions the generated wavelengths fall on the long-wavelength side of both the gain and the absorption spectral lines. Therefore, the absorption saturation causes a down-chirp of the pulse, whereas the gain saturation leads to an up-chirp (KUHLKE,RUDOLPHand WILHELMI[ 19831, DIETEL, DOPEL,RUDOLPHand WILHELMI[ 19861). The evidence of a phase memory of the resonance medium for pulses as short as the phase relaxation time T, [see eq. (2.17a)l leads also to a chirp, even for a weak signal. The combined action of the gain saturation and the phase memory has been considered by RUDOLPHand WILHELMI [ 1984a,b]. It has been shown that the inclusion of T2 into the model lowers the total chirp. Another phenomenon introducing a frequency chirp is the fast Kerr effect into the solvent ($ 2.3.1). Thus, the proper N

-

111,

0 31

GENERATION OF FEMTOSECOND OPTICAL PULSES

225

TABLEI Typical values of GVD for different intracavity components. Component

Wavelength

d2+/dw2 (fs2)

Chirp sign

References

(nm) Jet (100 Fm)

610

- 8.4

(+)

DE SILVESTRI, LAPORTA and SVELTO [I9841

Quartz (1 mm)

610

- 54

(+)

DE SILVESTRI, LAPORTA and SVELTO[I9841

Anomalous dispersion of DODCI

610

1.5

(-1

DE SILVESTRI, LAPORTA and SVELTO[1984]

DODCIphotoisomer

610

- 32

(+)

DE SILVESTRI, LAPORTA and SVELTO [ 19841

Single-stack dielectric mirror

610

240

(-1

DE SILVESTRI, LAPORTA and SVELTO[1984]

Broadband double-stack mirror

-

f (10-6300)

(T)

LAPORTA and MAGNI [1985]

Four quartz prisms, 1 = 25 cm, (minimum path into glass)

620

350

(-1

FORK,MARTINEZ and GORDON[ 19841

~

choice of the intracavity components could lead to mutual compensation of GVD and SPM and hence to the shortest output pulse. For example, MIRANDA, JACOBOVITZ, BRITOCRUZand SCARPARO [ 19861have shown that for pulses with an energy of 10 nJ and a duration of 30 fs, the chirp parameter near the peak is of about 4 x 10 - fs - ’. To compensate for this chirp, it is necessary to use the negative GVD introduced by a set of four prisms that have a base equal to 16 cm (see fig. 4 and FORK,MARTINEZ and GORDON [ 19841). In Q 2.1 we discussed some approaches of the laser performance. However, a unified theoretical model of the passive mode-locking in dye lasers does not exist. In a first analysis NEW [1974] has shown that when the saturable absorber has a long relaxation time compared with the pulse duration, the pulse shortening occurs due to the positive gain at the pulse peak and the negative gain at the pulse fronts. However, this model does not include various limitations of the generated spectral bandwidth. Therefore, it cannot give any

226


ABSORBER JET

[III, $ 3

GAIN JET

t Fig. 4. Ring-cavity configuration with intracavity prisms. The mirrors (M,, M,) and (M3, M4) focus the laser beam into the jets. Mirror M, also focuses the pump beam into the active dye DELIGEORGIEV, PETROV and TOMOV [ 19891). jet (MICHAILOV,

information about features such as the pulse shape or duration. By considering frequency limitations as an effective filter into the cavity, HAUS[ 19751 has obtained a steady-state solution for the output pulse shape l ( t ) sech2(t). The essence of the applied approach is in the estimation of the transmission functions of all intracavity components. By setting a condition for selfreproduction of the pulse shape after the cavity round-trip, we obtain a selfconsistency equation. Its solution enables us to determine the range of laser parameters that ensure a stable single-pulse regime. Furthermore, by the use and WEIDNER[ 19821 found a more realistic approximate of REA, HERRMANN solution yielding the energy of the output pulse as well as its duration and asymmetry. Steady-state pulses with intracavity compensated phase modulaand tion have been obtained in a semiclassical approximation by RUDOLPH WILHELMI [ 1984bl and DIELS,DIETEL, FONTAIN, RUDOLPHand WILHELMI [ 19851. These results partially explain the dependence of the output pulse duration on the amount of intracavity glass. By developing the Haus model, MARTINEZ,FORKand GORDON [ 1984, 19851 found that the pulse formation can be explained by a soliton-like mechanism taking place in the cavity (as in optical fibers). Accordingly, the round-trip time can be considered as a soliton period. Employing this idea, VALDMANISand FORK[1986] achieved considerable shortening of the generated pulses (down to 27 fs). In order to control the value of GVD, a four-prism configuration is used with a variable amount of glass on the beam path into the cavity (see fig. 4). The shortest pulses delivered so far by the CPML technique with an intracavity control are 19 fs (FINCH, CHEN,SLEATand SIBBETT [ 19881). An effect supporting the soliton-like mechanism in CPML lasers is the evidence of output pulses with periodic temporal modulation similar to the behavior of high-order solitons (see 5 2.3.1 and 5 4.4.1). SALIN,GRANGIER,

-

-


221

Roger and BRUN[ 19861 observed N = 3 soliton-like pulses at the output of a CPML laser. A simultaneous generation of two pulse trains at different wavelengths, one of which is considered as N = 3 solitons, was reported by WISE,WALMSLEY and TANG[ 19881. A good agreement between a theoretical model, not including a condition for pulse self-reproduction after a round-trip, and the experimental results was reported by AVRAMOPOULOS, FRENCH, [ 19881, AVRAMOPOULOS and NEW[ 19891, and WILLIAMS, NEWand TAYLOR AVRAMOPOULOS, FRENCH,NEW, OPALINSKA,TAYLORand WILLIAMS [ 1989). It has been shown that the observed periodic pulse evolution is a result of the interplay between SPM and GVD, but it is not attributed to the solitonlike mechanism. Another interesting regime of CPML, delivering two trains of pulses at different wavelengths, is the so-called “double mode-locking”. In this regime the saturable absorber generates a train of pulses when being intracavity pumped by the fundamental pulses. By using a new saturable absorber in a Rh6G-based ring laser, MICHAILOV, CHRISTOV and TOMOV[ 19901 reported femtosecond double mode locking with output pulses at 630 and 655 nm, as short as 200 and 270fs, respectively. Some results obtained by the PML technique are presented in table 2. 3.2.2. Synchronously pumped mode-locked (SPML) lasers A schematic diagram of a synchronously pumped laser is shown in fig. 5. To achieve a synchronous regime, the repetition rate of the pump pulses must be equal to, or a multiple of, the round-trip frequency of the “slave’s” cavity. In this way the active medium possesses again only when the pulse passes through it. Some advantages of the SPML compared with the CPML regime are the higher pulse energy and the possibility for spectral tuning. On the other hand, the pulses delivered by SPML are longer than those in the CPML regime. As a pump source, a frequency-doubled Nd: YAG or Nd: glass laser (e.g., SOFFERand LINN[ 19681) and a mode-locked Ar+-ion laser (e.g., ADAMS, BRADLEY,SIBBETTand TAYLOR[ 19801) are used. In the earlier configurations, as the active medium Rh6G in solution was used, giving output pulses in the 1-10 ps range (JAINand HERITAGE[ 19781). Pulses with femtosecond duration (600 fs) were first obtained by a rhodamine B laser, synchronously pumped by a Rh6G source (HERITAGEand JAIN [ 19781). One of the best results in “pure” SPML was reported by JOHNSONand SIMPSON[1985], where pulses as short as 210 fs were delivered in the 40 nm tuning range by the use of subpicosecond pump pulses (see also KAFKAand BAER [ 19851).

228

GENERATION A N D PROPAGATION OF ULTRASHORT OPTICAL PULSES

[III, $ 3

TABLE2 Different parameters of passive mode-locked dye lasers. Active dye

Absorber

Operation wavelength (nm)

Minimum pulse duration

Pumping laser

References

Coumarin 102 DOC1

460-512

80 (479 nm)

UV Ar

FRENCH and TAYLOR [1987,1988]

Coumarin 6

DI

518-554

96 (523 nm)

Ar

Rh 110

HICI, DASBTI

553-585

70 (583 nm)

Ar

Rh 6G

DASBTI

570-600

520

Ar+

FRENCH, DAWSON and TAYLOR [ 19861

Rh 6G

DODCI

630

19

Ar

FINCH,CHEN,SLEAT and SIBBETT [I9881

Rh 6G

TCETI

640

43

Ar+

MICHAILOV, DELIGEORGIEV, CHRISTOV and TOMOV[I9901

Rh B

DQTCI

616-658

220

Ar+

FRENCH and TAYLOR

+

+

+

+

FRENCH, OPALINSKA and TAYLOR [ 19891 FRENCH and TAYLOR [1986a]

[1986b] Rh 6G + Sulforhodamine 101

DQTCI

652-68 I

120

Ar

Rh 700

DDI

775

36

Kr

Piridine 1 + Rh 800

Neocyanine

783-815

260

Ar+

+

+

FRENCH and TAYLOR [1986c]

GEORGES, SALIN and BRUN[I9891

FRENCH WILLIAMS, TAYLOR and [ 19881 GOLDSMITH

SYNCHRONOUSLY PUMPED DYE LASER

MODE-LOCKED ARGON ION [OR KRYPTON

ION) LASER

a

Fig. 5. Design of a synchronously pumped CW dye laser (ADAMS,BRADLEY,SIBBETTand TAYLOR[ 19801).

111, § 31

229


However, considerable progress in the femtosecond pulse generation was gained when a saturable absorber into the cavity of a SPML laser was added (see, e.g., SIZER11, KAFKA,DULING, GABELand MOUROU[ 19831). The role of the absorber is similar to that in the passive mode-locking regime. It shortens the pulse fronts due to nonlinear saturation. This case corresponds with so-called hybrid mode-locking (HML). Output pulses with a duration of 70 fs were reported by MOUROUand SIZER[ 19821, where the gain medium (Rh6G) and the absorber (DQOCI) were mixed in a common jet. Similarly to the passive mode-locking case, shorter pulses can be obtained by compensating for the group velocity dispersion and the self-phase modulation experienced by the pulse due to the intracavity components. By using a linear resonator with two intracavity prisms DAWSON, BOGGESS, GARVEYand SMIRL[ 19861 have reported pulses with a duration of 69 fs. By means of a CPML synchronously pumped ring cavity laser JOHNSON and SIMPSON119831 and DOBLER, SCHULZand ZINTH[ 19861 produced a unidirectional generation of pulses as short as 150 fs and 65 fs, respectively. In order to ensure an easier performance of the laser in both SPML and CPML regimes, the so-called antiresonant ring is used, replacing one of the end mirrors of a linear cavity (see fig. 6). The two paths from the beam-splitter to the absorber are equal, which ensures the collision of the pulses into it. Similarly with the case of CPML, the generation of pulses shorter than 100 fs is attributed to a soliton-like mechanism. This is supported by research of HML with an antiresonant ring with four prisms in the cavity (CHESNOY and FINI[ 198611, where output pulses as short as 64 fs are produced. The shortest pulses ( 29 fs) produced so far by the HML technique are generated by a linear-cavity synchronously pumped dye laser without using the CPML regime (KUBOTA,KUROKAWA and NAKAZAWA [ 19881). A non-CW HML system using a pulsed actively/passively modelocked Nd : glass laser as the pump was reported by ANGEL,GAGELand LAUBEREAU [ 19891, who obtained pulses as short as 25 fs with an energy of 10 nJ. The first theoretical studies of SPML dye lasers were based on the set of rate equations for a two-level model of the active medium (see, e.g., YASA and TESCHKE [ 19751, and $ 2.1). For zero mismatch between the cavities of the pumping and the dye laser, the set of rate equations enables us to estimate the intensity and duration of the generated pulses: rout l / ~ *and , zOut where 6w is the effective bandwidth and zp is the pump duration (NEKHAENKO, and PODSHIVALOV [ 19861). However, research has shown that REA PERSHIN leads to some contradictions (NEW and CATHERALL [ 19841). The criticism made by CATHERALL, NEWand RADMORE [ 19821 demonstrated that there are N

-

N

ds,

(a)

230

[III, 5 3

GENERATION A D PROPAGATION OF ULTRASHORT OPTICAL PULSES

1

7-----------

!

1 n

A ? \ T ~ ~=, 97 , , fS rp=63fs

I

- 500

I

I

0

500

Delay, fs

I

- 200

I

I

0

I

I

200

Delay, fs

Fig. 6. (a) Schematic diagram of an antiresonant-ring dye laser: Ml-M6, mirrors; BS, SO% beam-splitter; OC, output coupler. (b) Typical zero-background intensity autocorrelationof laser output (FWHM I 97 fs). (c) Interferometric autocorrelation (LOTSHAW, MCMORROW, DICKSON and KENNEY-WALLACE [ 19891).

no analytical solutions at zero, positive, and negative mismatches between the cavities. Further progress in this field has required an introduction of multilevel models of the dyes, as well as a nonstationary polarization of the laser transition [see CASPERSON [ 19831, and eq. (2.17)]. KOVRIGIN,NEKHAENKO and PERSHIN [ 19851 developed a theory of SPML based on a four-level model of the gain medium. The propagation effects into the cavity and pump depletion [ 19811 found that were also included. By analytical estimations, NEKHAENKO the minimum output pulse duration is proportional to where T, is the polarization relaxation time (see 5 2.1). This dependence was experimentally verified by JOHNSON and SIMPSON[ 19851. Some other models describing SPML, based on a ring-cavity configuration, have been used by SCHUBERT, STAMMand WILHELMI [ 19851 and by CATHERALL and NEW[ 19861. These models require no preliminary assumptions about the shape of the steady-state solution and allow study of the transient pulse evolution and the influence of

a,

111, !i 31


23 1

both the spontaneous emission and the pump fluctuations on the SPML regime. A unified analysis of the limitations concerning CPML, SPML, and HML was made by PETROV,RUDOLPH, STAMMand WILHELMI [ 19891. The basic approach developed by STAMMand WEIDNER[1987] and STAMM [ 19881 was applied. The process of HML is regarded as an extension of SPML by a saturable absorber. It has been shown that the spontaneous emission which acts as a stochastic background disturbing the pulse parameters is a determinant factor for the SPML regime. The role of the absorber is to suppress this factor. The steady-state regime is shown to be limited by the combined action of GVD, SPM, and the spontaneous emission. These results can also be related to the aforementioned soliton-like mechanism for pulse formation. Some results obtained by the HML technique are presented in table 3. 3.2.3. Miscellaneous techniques We shall now consider some sources of femtosecond pulses based on other active media (not dyes). In recent years some authors have used stimulated Raman scattering in a fiber-ring amplification system in order to generate tunable femtosecond pulses in the near-infrared. In a regime of solitonlike shaping, where the pulse broadening due to SPM is balanced by negative GVD (see 0 2.3.1 and 0 4.4.1), the broad Raman gain bandwidth of silicabased fibers allows the production of soliton Stokes pulses with sub-100 fs and SERKIN[ 19831). DIANOV,KARASIK, MAMISHEV, duration (VYSLOUKH PROKHOROV, SERKIN,STELMAKH and FOMICHEV [ 19851 first demonstrated a generation of femtosecond pulses by means of stimulated Raman scattering. A scheme of a soliton Raman laser is shown in fig. 7. The pump radiation (pulse train delivered by CW mode-locked Nd: YAG laser) is directed towards the fiber, using a dichroic beam-splitter BS and microscope objective L , . The radiation leaving the fiber is partially reflected again to the fiber input by the mirrors M. Thus, the generated Stokes pulse at every round-trip into the fiber “sees” the synchronized pump pulse and experiences a soliton-like shaping. In such a synchronously pumped scheme, the cavity round-trip time for the Stokes pulse must be equal to, or an integral multiple of, the pumping period. By using a color-center pump laser, ISLAM, MOLLENAUER and STOLEN[ 19861obtained pulses as short as 250 fs. In a single-pass scheme GOUVEIA-NETO, GOMESand TAYLOR [ 19881reported a generation of soliton pulses with a duration of 80 fs, tunable up to 1.5 pm. DA SILVA,GOMESand TAYLOR[ 19881 demonstrated a similar technique using a high-order Stokes generation, which gives sub200 fs pulses centered arount 1.5 pm.

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TABLE 3 Different parameters of hybrid mode-locked dye lasers. Active dye

Absorber

Operation wavelength (nm)

Dusodium fluorescein

RhB

535-575

450

Ar

Rh 110

Rh B

545-585

250

Ar

Rh 110

DODCl

561

580

Ar

Rh 110

DASBTI

560

283

Nd:YAG

DAWSON, BOGGESS and SMIRL [I9871

Rh 6G

DODCl t DQOCl

583

69

Nd: YAG

DAWSON, BOGGESS, GARVEY and SMIRL [1986]

Rh 6G

DODCI

622

< 150

Nd: YAG

DAWSON, BOGGESS, GARVEY and SMIRL [ 19861

Rh 6G

DODCI

595-620

85

Nd: YAG, A.R.’b)

NORRIS,SIZERI1 and MOUROU[I9851

Rh 6G

DODCl

619

64

Nd: YAG, A.R.

CHESNOY and FINI[ 19861

Kiton red S

DODCl t DQOCI

615

29

Nd: YAG

KUBOTA, KUROKAWA and NAKAZAWA [I9881

Rh B

DTDCI

628

320

Nd : YAG

DAWSON, BOGGESS, GARVEY and SMIRL [19a7]

Rh 6G

DODCl

640

55

Nd: YAG

LOTSHAW, MCMORROW DICKSONand KENNEY-WALLACE [I9891

Piridine I

DDI

695

103

Nd: YAG

DAWSON,BOGGESS and SMIRL [1987]

Styryl9

IR 140

840-880

65

Ar’ -m.l

DOBLER, SCHULZ and ZINTH[ 19861

- mode-locked source ‘ b ’ ~ .-~antiresonant . ring arrangement (O’m.1

Minimum pulse duration

Pumping laser

+

+

+

References

- m.1.’”)

ISHIDA,NAGANUMA and YAJIMA[I9821

-

m.1.

ISHIDA,NAGANUMA and YAJIMA (19821

-

m.1.

ISHIDA, NAGANUMA and YAJIMA(1982)

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Fibre

Fig. 7. Schematic diagram of a synchronously pumped soliton Raman fiber laser (GOUVEIA-NETO, GOMESand TAYLOR [1988]).

The soliton formation mechanism was used by MOLLENAUER and STOLEN I19841 in order to design a compound cavity configuration (soliton laser), which delivers pulses as short as 200 fs (A 1.4-1.6 pm). It consists of a synchronously pumped, mode-locked color-center laser, tunable in the 1.5 pm region, coupled to a second cavity containing a single-mode polarizationpreserving optical fiber. When the generation starts, the initial pulse narrows considerably due to the passage through the fiber. The feedback from the fiber enables the laser to produce shorter and shorter pulse until the pulses into the fiber become solitons. Since the tunability in this case is limited only by power requirements for soliton formation, it is greater compared with the ordinary mode-locked lasers. It is interesting to note that, in practice, the soliton laser tends to favor production of the N = 2 soliton (see Q 2.3.1) rather than the fundamental N = 1 soliton ( ~ s e c h ~The ) . theory of the soliton laser is still under investigation (see, e.g., HAUS and ISLAM [1985], BLOWand WOOD [ 19861). In recent years the generation of femtosecond pulses in the ultraviolet (UV) has also become an object of extensive research. Unfortunately, despite their broadband gain, the excimer lasers cannot be successfully mode-locked because of their short storage time. GLOWNIA, ARJAVALINGAM, SOROKIN and ROTHENBERG [ 19861reported a configuration in which a single pulse delivered by a synchronously pumped dye laser is amplified in dye amplifiers and then passed through a nonlinear frequency-doubling crystal. After this process, the doubled pulse is amplified by an excimer (XeCl) UV amplifier from whose

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output it appears with a duration of 350 fs (A = 308 nm). Pulses as short as 365 fs at 248 nm (KrF-amplifier) have been reported by HUTCHINSON, MCINTYRE, GIBSONand RHODES[ 19871. A powerful system delivering pulses with terawatt powers, as short as 200 fs at A = 248 nm, has been reported by SZATMARI,RACZ and SCHAFER [ 19871 and SZATMARI,SCHAFER, MULLER-HORSCHEand MUCKENHEIM[ 19871. By using extra-cavity (GLOWNIA, MISEWICHand SOROKIN [ 1987]), and intra-cavity (FOCHTand DOWNER[ 19881) frequency-doubling schemes in a CPML laser, UV femtosecond pulses at 310 nm have been produced. We should note that the new nonlinear crystal BBO (beta-barium borate) possesses considerably greater nonlinearity compared with the conventionally used crystal KDP (potassium dihydrogen phosphate) (see, e.g., EIMERL,DAVIS,VELSKO,GRAHAM and ZALKIN[1987]). By using BBO as a doubling medium, EDELSTEIN, WACHMAN, CHENG,BOSENBERGand TANG[ 19881have reported UV pulses of about 43 fs generated in an intracavity doubling scheme (in a CPML ring dye laser) with a high conversion of the output into UV. An alternative approach for producing broad-tunable femtosecond pulses is based on the process of a parametric generation. EDELSTEIN, WACHMAN and TANG[ 19891 demonstrated the first femtosecond optical parametric oscillator. Their scheme uses a thin crystal KTP (KTiOPO,) synchronously pumped by intracavity femtosecond pulses at 620 nm in a CPML dye laser. Continuous tuning of pulses -200 fs from 720 to 4500 nm has been observed. There are several techniques giving femtosecond pulses in the far-infrared. One is based on the passage of a C0,-laser pulse through a regenerative amplifier, where the pulse shortening occurs due to the formation of an electron density wave (CORCUM[ 1983, 19851). Output pulses with a duration of 600 fs and intensity 10l2W/cm2 have been obtained (CORCUM[ 19831). Pulses as short as 130 fs in the mid-infrared (A = 9.5 pm) were generated by the use of semiconductor ultrafast switching (ROLLANDand CORCUM[ 19861). These pulses consist of only about four optical cycles, and they are the shortest reported so far with respect to the ratio of pulse duration to carrier period. The generation of a difference frequency is also a promising method for delivering femtosecond pulses. By focusing amplified femtosecond pulses into a cell containing ethanol, subpicosecond continuum generation occurs. After following the focusing of both the continuum and the remainder of the input pulse into a nonlinear crystal (LiNbO,), pulses of 200 fs at a difference frequency, tunable in the 1.7-4 pm range, have been obtained (MOOREand SCHMIDT[ 19871).

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3.3. AMPLIFICATION OF FEMTOSECOND PULSES

There are many experimental situations where femtosecond pulses with megawatt to terawatt powers are necessary. These include, e.g. the continuum generation (FORK,SHANK,HIRLIMANand YEN [ 19831) or the relativistic nonlinear optical effects (see 8 3.4). Since the femtosecond lasers deliver pulses with a peak power of about 1 kW, additional amplification of these pulses is necessary. The short-pulse amplification needs to satisfy some general requirements. First, it is clear that the amplifier gain bandwidth should be larger than the bandwidth of the expected output pulse. Second, for an efficient energy extraction the input intensity flux should be near the amplifier saturation level. Whereas the amplifiers of nanosecond pulses have an efficiency of about 20%, the femtosecond pulses cannot be amplified with an efficiency exceeding 0.5%. On the one hand, the time necessary for energy exchange into the amplifying medium prevents appreciable energy storage into a pulse of about 100 fs. On the other hand, the amplified spontaneous emission (ASE) is a serious limitation for femtosecond amplifiers because it reduces the gain. Another general limitation is the difference between the generator wavelength and the peak of the amplifier gain (see, e.g., GANIEL,HARDY,NEUMANN and TREVES[ 19751). There are two main directions for developing femtosecond amplifiers. The first deals with amplification of less-than-100 fs pulses with a high peak power and high repetition rate, mainly for spectroscopic purposes. The second is aimed at storing a high energy (up to Joule level) into a femtosecond pulse. The media used are substantially the same as in the generators (see 3 3.1). The dyes and excimers possess broad gain bandwidth ( 102-103 cm- I ) , but their low saturation fluency ( - 1 mJ/cm2) and short storage time (- 10 ns) prevent application for the amplification of femtosecond pulses to high energy levels. This dficulty can be overcome by using some solid-state media such as Nd-glass, alexandrite, and titanium sapphire, which have long storage times (few hundred microseconds) and high saturation level (- 1 J/cmZ). First, we will consider some of the most widely used laser-pumped dye amplifiers. A three-stage scheme pumped by a Q-switched Nd : YAG system, amplifying 500 fs pulses, has been developed by IPPENand SHANK[ 19781. An improved four-stage design of FORK,SHANKand YEN [ 19821 allows the amplification of pulses of sub-100 fs duration up to 1 mJ. The first three stages are transversely pumped, whereas the fourth stage is longitudinally pumped. The total gain after the consecutive stages is 750, 20, 10, and 40, respectively. A grating pair is placed at the output in order to compensate for the frequency chirp induced into the pulse due to its passage through the resonance media

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and optical components (see 5 3.2). The next step has been the design of high-repetition rate amplifiers, pumped by an actively mode-locked Nd : YAG laser (see, e.g., HALBOUT and GRISCHKOWSKY [ 19841). One advantage of this configuration is the possibility of synchronizing the oscillator and amplifier. Another promising pump source is the copper-vapor laser, which works at a repetition rate of several kHz. A singlejet amplifier in a multipass geometry was reported by KNOX,DOWNER,FORKand SHANK[ 19841. In order to produce the shortest optical pulses so far (- 6 fs), FORK,BRITOCRUZ,BECKERand SHANK[ 19871 amplified a pulse train delivered by a CPML laser by means of a copper-vapor laser pumped amplifier, yielding pulses as short as 50 fs with an energy of 1 mJ at an 8 kHz repetition rate. An improved multipass femtosecond amplifier at a high repetition rate was proposed by NICKEL,KISHLKE and VON DER LINDE[ 19891. It uses a fused-silica dye cell instead of a jet stream, and delivers pulses 60 fs with an energy of 50 pJ. BOYER,FRANCO, CHAMBARET, MIGUS, ANTONETTI,GEORGES, SALINand BRUN [ 19881 designed a scheme combining different stages of amplification and compression, giving pulses in the microjoule range with a duration 16 fs ( A = 620 nm) at a repetition rate of 1 1 kHz. The production of high-power pulses in the UV regime is of great importance because they can be used in laser photochemistry, plasma physics, etc. The high gain of some rare-gas halide excimers enables the design of powerful amplifiers in the UV. For instance, an XeCl excimer exhibits gain around 308 nm, which is near the second-harmonic wavelength corresponding to a typical CPML dye laser. By means of two cascaded XeCl amplifiers, GLOWNIA, MISEWICHand SOROKIN[ 19871 obtained pulses as short as 160 fs with an energy of 12 mJ. By using a KrF-based amplifier in a two-pass geometry, SZATMARI, SCHAFER, MOLLER-HORSCHE and MUCKENHEIM [ 1987 J have amplified 80 fs pulses at 248 nm with an energy of 15 mJ. It is generally difficult to extract much energy from the amplifying medium by means of a short pulse, but even if it is possible, the amplified pulse destroys the amplifying medium due to nonlinear effects (e.g., self-focusing). Here we will consider one of the most interesting techniques that is capable of yielding light pulses with enormous intensity (- 1020 Wjcm’). This technique uses pulses initially stretched by a factor of 100 to 1000 times (and correspondingly chirped) in a dispersive system (see 0 2.2.1). The stretched pulse, whose intensity is considerably lower than the initial one, can be amplified with great efficiency (chirped pulse amplification (CPA); see, e.g., STRICKLAND and MOUROU[ 19851). After the amplification, the pulse passes through a dispersive delay line (grating pair, see Q 3.4), where compression occurs. The

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GENERATION O F FEMTOSECOND OPTICAL PULSES

231

compressor scheme proposed by MARTINEZ[1987] was used by PESSOT, MAINEand MOUROU[ 19871 who have achieved an expansion/compression factor of 1000 times. The first demonstration of CPA was made in the picosecond domain, where 50 ps pulses are coupled in a 1 km-long optical fiber in order to stretch them to about 300 ps due to the fiber dispersion, and to increase their bandwidth due to SPM. After the Nd:glass stages, which amplify the pulse over nine orders of magnitude, the pulse is compressed to 1 ps with a stored energy of 0.5 J. Because of the low divergence of the output beam, this value corresponds to a brightness greater than 10l8W/cm2 (see EBERLY,MAINE,STRICKLAND and MOUROU[ 19871). PESSOT, SQUIER, BADO,MOUROUand HARTER[ 19891 have applied the CPA technique to an amplifier based on alexandrite. This material possesses 4 times higher saturation level ( 20 J/cm2) than Nd : glass and very good energy storage capabilities. The amplified pulse appears with a duration of 305 fs and an energy of 1.5 mJ, which corresponds to gigawatt power level. By using intracavity prisms in an alexandrite regenerative amplifier, PESSOT,SQUIER,BADO and MOUROU[ 19891 have achieved a CPA of 106 fs pulses with an energy of 2 mJ. As is known, if a standard dye amplifier for amplification of broadband pulses (e.g., with a duration of tens of femtoseconds) is used, a spectral narrowing due to the competition between the different spectral components occurs (see, e.g., MIGUS,SHANK,IPPEN and FORK[ 19821). However, provided these components are amplified in slightly shifted regions into the gain medium, the suppression of the low-gain frequencies would be greatly reduced. [ 1989al reported an oscillator design based on this DANAILOV and CHRISTOV idea, in which the effect of “lateral walk-off’’ (transverse displacement of the frequency components after passing a grating pair) is used (see Q 4.2.2). An ultrabroadband (up to 30 nm) generation of a Rh6G nanosecond dye laser is obtained. The same idea can be realized by using a prism pair instead of a grating pair (DANAILOV and CHRISTOV[ 1990a,b]). For amplification purposes a configuration such as that in fig. 20 could be used. It consists of a confocal lens pair placed between a pair of conjugated gratings. This scheme is nondispersive (see 5 4.3.3); i.e., the transmission through it does not change the pulse shape and duration. The amplifying medium is placed in the mutual focal plane F, of the lenses, where the frequencies are well selected. That is why the radiation in the jet plane is frequency swept in the transverse direction, and thus the amplification can be considered as a spatial variant of CPA. Such a system would be especially useful for the amplification of extremely short pulses (down to 10 fs), where the amplitude and phase properties of the radiation need to be preserved after the amplifier.

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Next, we shall briefly discuss some theoretical aspects of femtosecond amplifiers.As we have already noted, in a high-gain system the ASE is the main limiting factor because it lowers the stored energy before the signal injection, considerably depleting the gain. However, the inclusion of ASE in the theory is connected with some difficulties caused by the three-dimensional character of the spontaneous emission. GANIEL,HARDY,NEUMANNand TREVES [ 19751 analyzed a transversely pumped dye amplifier based on some simple geometrical considerations in order to estimate correctly the influence of ASE. In a case of a pencil-like (cylindrical) medium, whose radius is much smaller than its length, a simple geometrical factor g(z) is introduced into the basic set of rate equations, eqs. (2.14). In fact, g(z) is the fraction of spontaneous emission emitted into the solid angle over which the fluorescence is amplified. The numerical results have shown that the amplifier saturates rapidly due to ASE. A discussion about the validity of the geometrical factor was made by HNILO,MARTINEZ and QUEL[ 19861. The problem for amplification of femtosecond pulses was considered in detail by MIGUS, SHANK,IPPEN and FORK [ 19821. It was shown that when the amplified pulse is short compared with the pump pulse, the gain depends on the steady-state excited population, determined by the balance between the generation rate and the gain depletion due to ASE. The computer solution of the rate equations (2.14) gives the distortion of the temporal pulse shape as a function of both the input and the stored energy density. The theoretical results are compared with the experimental results obtained from a three-stage dye amplifier, which amplifies pulses as short as 500 fs with an energy of 2 nJ up to the millijoule level. As a rule, however, the numerical solutions do not allow a flexible estimate of the real experimental situation. On the other hand, most of the theoretical formulas for the smallsignal gain contain adjustable parameters. An attempt to overcome some of these difficulties was made by HNILOand MARTINEZ[1986, 19871, who deduced a formula allowing calculation of the small-signal gain G,, by only measuring ASE emitted by the amplifier

where a, = i(z) = a, T , I; I is the ASE intensity measured by a T , Ipump is the detector placed at a distance z from the amplifier, w = aa(Apump) is the pump intensity. The geometrical factors g(z) and pump rate, and Ipump g( 1) indicate the solid angles subtended by the detector measuring ASE and by the exit aperture of the amplifier, respectively.

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3.4. PULSE COMPRESSION

Pulse compression techniques take an intermediate place between the methods of generation of femtosecond pulses and the effects accompanying pulse propagation in linear and nonlinear media. Following the space-time analogy, wa may say that the pulse compression represents a focusing in time, similar to the focusing of a light beam by a lens. The original idea for optical pulse compression has arisen in microwave radar systems, where a preliminary frequency chirped pulse transmits through a dispersive delay line (KLAUDER, PRICE,DARLINGTON and ALBERSHEIM [ 19601). Insofar as the different temporal parts of the input pulse (having different instantaneous frequency) propagate with different group velocities, at the output of a well-adjusted delay line the leading edge of the pulse overlaps the trailing edge, yielding a transformlimited output pulse. A suitable delay line for optical frequencies is a pair of conjugated gratings, which possess a great negative GVD (see TREACY [ 19691 and fig. 8). As we noted in $ 2.2.1, the transmission of a wave packet through a linear dispersive system can be described as a Fourier transform with a transmittance function H(o)= exp [i$(m)], where $(w) is the phase shift introduced by the system. TREACY [ 19691 showed that the GVD parameter of the grating pair is determined by

where b is the distance between the gratings, d is the grating groove spacing,

Fig. 8. Geometrical arrangement of a grating-pair compressor (CHRISTOV and TOMOV [1986]).

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an y is the angle of incidence (see fig. 8). If the input pulse is Gaussian with half-duration T and its up-chirp parameter is cc, = - a’, then the minimum pulse duration achievable by the compressor is TE; = 4/a,T. However, a simple calculation shows that the ratio of the third to the fourth terms in the series given by eq. (2.23) is 6 = (cubic/quadratic) = Aw/wo, where Aw is half of the spectral width of the input pulse. For a typical case when w o = 3 . 1 4 x 10’5s-‘ (1=600nm), d = 1 . 7 ~10-4cm-* (600l/mm), and y = 10-50”, we find that the cubic term should be taken into account for input pulses for which Am 0.1 a,. The latter corresponds to a transform-limited duration of about 10 fs. The parameter a” representing the value of the cubic term is given by (CHRISTOV and TOMOV[ 19861)

-

1 + (2ac/w0d)sin y - sin2 y 1 - (2ac/w0d - sin 7)’

1

.

(3.6)

CHRISTOVand TOMOV[1986] also showed that the cubic term causes broadening and oscillations of the compressed pulse shape. Since pulses of a duration less than 10 fs have already been produced (KNOX,FORK,DOWNER, STOLEN,SHANKand VALDMANIS[1985]), the cubic term has obviously obscured the compression process (see also BRORSONand HAUS[ 19881. To reduce this effect, FORK,BRITOCRUZ,BECKERand SHANK[ 19871and BRITO CRUZ,BECKERand SHANK[ 19881designed a suitable combination of gratings and prisms in which the cubic term is compensated for by the opposite-sign cubic term of the prism material (glass). As a result, the shortest pulse 6 fs (FWHM) until now has been obtained. A configuration with a negative dispersion consisting only of prisms was proposed by FORK, MARTINEZand GORDON[ 19841. Its GVD is given by eq. (4.9), and it is adjustable through zero value by varying the separation of the prisms. The combination of four prisms has an additional advantage in that there is no “lateral walk-off” effect. This effect is also present in the grating-pair compressor, especially for small beam size, but it may be avoided by means of a double-pass scheme (MARTINEZ [ 1986b1). By placing a telescope between the gratings (see fig. 20), the phase shift can be modified in such manner that the designed compressor exhibits a large positive GVD (MARTINEZ[ 19871).The value of GVD depends on the position of the gratings with respect to the lenses, and its largest value is a‘ = - 1/[2kop2M(f, + f2)], wherep = 2nc/(w,2dcos yo), a n d M = f,/f2is the magnification of the telescope if the focal distances of the lenses are not equal. This system may provide a compression ratio as high as 3000 times and may be successfully used for chirped pulse amplification (see $ 3.3). An alter-

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native design, working simultaneously in both positive and negative GVD regimes with a chirp parameter given by a' = 1/(28, p 2 k o f ) ,was proposed by CHRISTOV[ 19891. In this case all gratings are situated in the focal planes of the lenses, which enables an additional Fourier-filtering of the transmitted radiation. Recently, MARTINEZ [ 19881 has introduced a matrix formalism for grating and prism compressors that is an extension of the standard ABCD formalism for Gaussian beams. We should mention that the long (up to tens of kilometers) monomode optical fibers (JANNSON[ 1983]), as well as some [ 19641, dielectric multilayer interferometers (see, e.g., GIRESand TOURNOIS KUHL and HEPPNER[ 19861, also exhibit dispersive properties capable of compressing light pulses. Although the idea for optical pulse compression was introduced in the 1960s, its practical application was hindered by the absence of a device producing ultrabroadband radiation with a linear frequency chirp. Progress in the technology of high-quality monomode optical fibers supplied the desired equipment. The main advantage ofthe fiber is the possibility of retaining a small cross section of the propagating beam over a long distance. Thus, because of the nonlinear effect of self-phase modulation, a considerable spectral broadening accompanied by a large positive chirp takes place (see, e.g., SHIMIZU[ 19671). From eq. (2.37) it follows that the spectral width of the pulse after the fiber is equal to A m Aw,k,n;Ioz, where Awo is the initial spectral width (see also STOLENand LIN [ 19781). Moreover, in the case in which the influence of GVD is neglected, the pulse chirp is positive near the peak and negative on the wings (see Q 2.3.1). This hinders the following compression, causing oscillations in the compressed pulse shape. By using NSE [see eq. (2.32)] GRISCHKOWSKY and BALANT[ 19821 demonstrated that although the effect of GVD generates a chirp with a sign opposite to that due to SPM, the simultaneous action of both GVD and SPM leads to a strong positive linear chirp entirely covering the pulse profile. This results in a high-quality compression (TOMLINSON,STOLENand SHANK[ 19841). Let A = (P/PI)'/2, where P is the peak power of the input pulse, and P, = n,cl,S,, 10-7/16nLDn;; where S,, is the effective core area of the fiber. Then, if T' is the pulse duration after the compressor, the optimum compression ratio in the case of a long fiber is TIT' x 0.63A, which corresponds to optimum fiber length zOpt= 1.6LD/A, and the optimum distance between the gratings is bopt = 6.4 nc2d2cos2(y') T2/,12A,where y' is the diffraction angle. For a long input pulse if the optimum fiber length becomes inconveniently large, the regime without GVD (short fiber) is then more suitable. In this case the preceding equations are modified as follows: T/T' % 1 + 0.9(A2z/LD), and bopt x m 2 d 2 c o s 2 ( y ' ) T 2 L D / , l ~ A 2Inz . fact, N

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some circumstances limit the performance of the fiber-gratings compressor. KNOX,FORK,DOWNER,STOLEN,SHANKand VALDMANIS [ 19851 observed significant distortions of the spectrum at the output of a 8 mm-long fiber for input intensities above lo'* W/cm2. This is attributed to some complicated nonlinear processes in addition to SPM. Despite the obtained 150 nm broad spectrum, corresponding to a transform-limited pulse of about 1 fs, the shortest pulse after the compressor has been 8 fs. Another important limiting factor is the stimulated Raman scattering arising from the fiber core. The critical power P, at which the fundamental and Raman intensities become comparable is P, = 16S,,/gz, wheregis the Raman gain (SMITH[ 19721). For example, for g 10- '' cm/W (A, = 1.064 pm), with a core diameter of 5 pm and fiber length z = 100 m, the critical power is P, = 12.6 W. An interesting alternative method of obtaining spectral broadening with a linear chirp for compression purposes is by using electro-optic phase modulation. Although this method was proposed before the fiber-based methods (see, e.g., GIORDMAINE, DUGUAYand HANSEN[1968]), its application in the picosecond and femtosecond domains was difficult because of the low-speed performance. Progress in the waveguide microwave modulators has allowed the generation of broadband spectra with a linear frequency chirp directly from CW radiation. By passing CW Ar+-ion laser radiation (A, = 514.5 nm) through a LiTa0,-based waveguide modulator followed by a grating pair, KOBAYASHI, YAO, AMANO,FUKUSHIMA, MORIMOTOand SUETA [ 19881 synthesized pulses as short as 2.1 ps. This corresponds to agenerated spectrum of 640 THz. It seems that, in the near future, spectral broadening of tens of nanometers may be expected, which will be a revolution in optics because of the high stability and controllability of the electro-optic modulators. Finally, we will consider one nonstandard method for pulse compression that could be used when the initial pulse possesses an enormous power. As we showed in 3.3, the development of the chirped-pulse amplification technique has made it possible to attain an intensity as high as 10l8 W/cmZ,and intensities of about lo2, W/cm2 could be expected. An interesting situation exists when such a powerful light pulse interacts with free charges (electrons), since a strong nonlinearity caused by relativistic effects arises. The electron motion and radiation was considered in detail by EBERLY[I9691 and SARACHIK and SHAPPERT[ 19701. However, it is difficult to do an analytical treatment of the problem for interaction between electrons and a strong ultrashort pulse. CHRISTOV and DANAILOV [ 1988al simulated this interaction by computer to investigate the possibility of using it for pulse compression. The model is based on the relativistic equation of motion of the electron and on the far-zone

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24 3

-0.5

-1.00

-0.50

w.cos

0.00

0.50

1.00

( 8 ) NORMALIZED

Fig.9. Angular dependence of the energy W(B) scattered by the electron: ( 1 I I.n = I019W/cmZ; (--------I 1.I" = 2 x 1o*l W/cmz (CHRISTOV and DANAILOV [1988a]).

I,, = l0'"W/cmZ; (-------)

solution of the wave equation. The incident pulse is supposed to be Gaussian with a duration of 6 fs (A = 600 nm). Figure 9 shows the calculated angular dependence of the scattered energy by the electron. When the incident intensity is about 10" W/cm2, the well-known case of Thomson scattering is still valid. By increasing the intensity, the scattered energy in the forward and backward directions decreases, and scattering in the perpendicular direction develops. At intensities as high as 1021W/cm2, almost all of the electron radiation is scattered in narrow angular peaks. The temporal dependence of the radiation emitted at the maximum efficiency angle (about 8 FS 54" in fig. 9) for an input intensity of 2 x 10" W/cm2 is presented in fig. 10. It can be seen that the scattered pulse consists of two main peaks with a duration of about 0.5 fs. Figure 11 shows a considerable spectral broadening due to the relativistic nonlinearity of the electron motion. It is clear, therefore, that at certain angles an efficient pulse compression may be obtained. One advantage of this method is that the electron scattering can be used in different spectral regions, in contrast with the other methods mentioned here. It should be noted also that the spectral enriching in this case leads to the disappearance of the well-defined carrier frequency (see fig. 10). This demonstrates that there is no principal limitation for the synthesis of a pulse with an arbitrary small duration, contrary to focusing in space, where the spot size cannot be smaller than the corresponding wavelength.

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!2

rl 4

r Ly

0 Z

1.00

X

w

0.50 0.00 I

0.0

I

I

I

4.0

8.0

12.0

TIME (fs) Fig. 10. Time dependence of the scattered radiation by the electron (bottom) for a Gaussian and DANAILOV incident pulse with a duration of6 fs (top), with I,, = 2 x lo2' W/cm2 (CHRISTOV [ 1988a1).

1.00 -

0.50-

0.00I

0.0

2.0

I

4.0

I

,

I

6.0

8.0

10.0

FREQUENCY~ 1 0 ' 5HZ)

Fig. 11. Spectral intensities ofthe incident pulse (dashed) and scattered pulse (solid), calculated for the case shown in fig. 10 (CHRISTOV and DANAILOV [1988a]).

Q 4. Propagation Effects In this section we shall consider some effects accompanying the free-space diffraction of an ultrashort transform-limited pulse, which propagates as a beam with a narrow angular spectrum. Obviously, the designed laser sources of femtosecond pulses obey this requirement. The influence of a more complicated spatial modulation on the time evolution of the pulse is also discussed.

111, I 41

245

PROPAGATION EFFECTS

4.1. FREE-SPACE PROPAGATION

4.1.1. Regular pulses Our analysis is based on the integral representation given by eq. (2.26). Expanding k, in a power series (CHRISTOV [1985b], COOPERand MARX [ 1985]), we obtain

In the case of narrow angular spectrum ( k x ,k, 4 w/c), the rest of the terms in eq. (4.1) may be omitted if their phase contribution in eq. (2.26) is small. This is fulfilled when the following inequality is valid: z -g 20’ na4/c3,

(4.2)

where a is the beam radius at the source plane (z = 0), and w varies in limits for which the spectral components have considerable amplitude (e.g., at least 1 % from the maximum). Meanwhile, the truncation of the series (4.1) to the second term is similar to the well-known paraxial approximation widely used in optics (see, e.g., KOGELNIKand Lr [ 19661, AKHMANOV, SUKHORUKOV and CHIRKIN [ 19681). Some evaluations concerning the error made in using this approximation were presented by COOPERand MARX [ 19851. Thus, for an input radiation with both Gaussian temporal and spatial modulation, from eqs. (2.26) and (4.1) we obtain in the far zone (z % wa2/2c) (CHRISTOV [ 1985133) E(r, q)

=

2T Eo (2 (q:) + T4)lI2 c z T: x exp

[--((>’

q: aw,rT Tf 2zcT,

+i

(5

q1 + cp + in)],

(4.3)

where r 2 = x2 + y2, T: = T 2 + (arlzc)’, q1 = q - r2/2zc, and cp = arctan(2ql/ w, T 2 ) .Moreover, in eq. (4.3), Tand a are the halves of the input pulse duration and the beam radius at e level, respectively, and w, is the carrier frequency. ~

There are two primary features following from eq. (4.3). First, at a given z the pulse duration T, increases with increasing the shift r from the beam axis, with no change on the axis. Second, the oscillating part in eq. (4.3) shows that with increasing r, the carrier frequency w,!, = wo(T/Tl )’ decreases (a shift toward the

246


[III, 8 4

long-wavelength spectral region takes place). These effect can be explained bearing in mind that the longer-wavelength spectral components spread faster than the shorter-wavelength ones, as diffraction theory predicts (BORN and WOLF[ 19681). It is clear, therefore, that with increasing r the longer-wavelength components predominate, which causes the pulse expansion in time. Moreover, from eq. (4.3) we find for the beam radius

where a m ( z ) = 2zc/woa is the radius of a CW beam in the far zone (YARIV [ 19751). It can be seen from eq. (4.4) that a decrease in the initial pulse duration leads to an increase in the beam divergency. Note also that the utilized wave approach assumes that the source field possesses a well-defined carrier frequency, i.e., T > 2/w0 in eq. (4.4). A step toward a more detailed description of the time evolution of an optical pulse with a wave front of any profile, propagating in both the near and the far zone, was achieved by the transRAJI position of Huygens principle in the time domain (GOEDGEBUER, and FERRIERE [1987]). The procedure is based on evaluating the Helmholtz-Kirchhoff integral for every spatial component of the initial pulse. Accordingly, the pulse shape E(P, t ) at a given point P can be carried out by calculating the contributions e(M, t ) arising from every element M of the wave front W, E(P,t) = 1 2nc ~

S

1 1 + cos[/?(M)] d [e@,r-:)],

-

r

2

dt

(4.5)

where r is the distance between points P and M , and p ( M ) is the angle between the local normal to the wave front at point M and the direction of observation MP. It can be seen from eq. (4.5) that the optical signal arriving at P is an amplitude superposition of “secondary” elementary pulses emitted from the wave front W, which is assumed to be locally flat. In general, it seems to be difficult to obtain a closed-form analytical expression for E ( P , t ) from eq. (4.5). By using a numerical computer technique, GOEDGEBUER, RAJI and FERRIERE [ 19871 showed that the spatial modulation of the wave front leads to considerably distortions of the output pulse profile in both the near and the far zone.

111, $41

PROPAGATION EFFECTS

241

4.1.2. Partially coherent pulses Using eq. (2.30),CHRISTOV [ 19861 considered the behavior of the coherence function for a source field with regular Gaussian spatial modulation (beam with radius a ) and regular Gaussian time modulation (pulse with half-duration T ) , under the assumption that the source field possesses an additional statistical spatial modulation with a Gaussian correlation function whose half-width is r, (r, is the radius of coherence of the source). Thus, the coherence properties of the source in space are determined by

where T,,,is a dimensional factor and r,, is the effective radius of coherence at z = 0, defined by r i = +a + r i A source with such spatial coherence is known as a Gaussian-Schell-model source (see, e.g., FRIBERGand SUDOL [ 19831). In addition, let the source spectrum be broadened by statistical temporal modulation with a Gaussian correlation functions whose half-width is T, (Tk is the time of coherence of the source). Then, similar to eq. (4.6), we have

’.

where To is the effective time of coherence at z = 0, defined by T; = T - + T; ’. In order to perform the integrations in eq. (2.30), we can use the approximation of the narrow angular spectrum [see eqs. (4.1)and (4.2)] for both k , , and kzzrand the far-zone approximation zi B o i p 2 / c ; here i = 1,2 and p2 = f(r; + 2a2). Then, substituting eqs. (4.6) and (4.7) in eq. (2.30) and performing the integrations, one finds a closed-form solution for the coherence function in the far zone (CHRISTOV [1986]). If we wish to consider some interesting features of the time behavior of the propagating pulse, a degree of temporal coherence can be introduced, whose modulus is given by

’

‘

(4.8) where I(r, z, q) = T(r, = r, = r, zI = z2 = z, q I = q2 = q) is the average field intensity. In eq. (4.8) the coherence time is defined by 9;’

=

Til‘

+ 0.5 T ; ’,

(4.9)

248


[III, 8 4

where To, = [T: + ( r , r / z ~ ) ~ ] ' ~It~ can . be seen from eq. (4.9) that 6, = Tk at r = 0, which means that for all points on the z-axis the coherence time is not changed by the diffraction. To further clarify the influence of the spatial structure on the temporal coherence, we will study the dependence 6k(r = a) on z for two cases. (i) Tk 9 T and r, -4 a, i.e., the pulse is nearly transform-limited in the source plane, but its spatial coherence is low. Then, from eq. (4.9) it follows that 6, = T 2 c z / a 2 ,so that the coherence time 6, increases linearly with z and it '/~ remains much greater than the pulse duration T, = [ T Z + ( a r / z ~ ) ~ ] [compare with the regular case, eq. (4.3)]. (ii) Tk4 T and r, -4 a ; i.e., in this case both the temporal and the spatial coherence of the source are low. Then eq. (4.9) yields 6, = T,; Le., the poor spatial coherence does not influence the time coherence. Figure 12 shows the radial behavior of 6, for z = 104cm, w, = 3.8 x 1015 Hz, 2T = 4 fs, and a = 0.1 cm. For comparison the pulse duration is presented as a solid line. It can be seen that when T, 4 Tand r, 9 a, the coherence time increases faster than T , , leading to a regularization of the field. On the other hand, when Tk9 T, 6 k ( r ) has a minimum, which tends to T when r,-+O. Thus, the poor spatial coherence of the source results in irregularization of the field. In the case in which Tk < T and r, < a, 6, increases with z but the increase is slow compared with that of T,. Let us now consider the spatial behavior of the field. The beam radius in the

0

2

L 6 rlcml

8

Fig. 12. The dependence of the coherence time 0, and pulse duration T , on the distance r from the z-axis:)-( T,(r); (---) Ok(r), when Tk< T and rk % a ; (- . . -. U r ) . when Tk9 T; (- . - . -) Ok(r), when T, Q T and r, C a. In all cases 2T = 4 fs and a = 0.1 cm (CHRISTOV [1986]).

111.8 41

249

PROPAGATION EFFECTS

far zone is given by (CHRISTOV [ 19861) (4.10)

which, in the regular case, corresponds to eq. (4.4). Introducing a degree of spatial coherence similar to that in the temporal case [see eq. (4.8)], it can be seen that, particularly when r, -+ 0 and T, Tk + co, the radius of coherence increases linearly with z (4.11) which is in accordance with the well-known van Cittert-Zernike theorem (see DYAKOV and CHIRKIN[ 19811). We MANDELand WOLF[ 19651, AKHMANOV, shall consider the following two cases. (i) r, 4 a and Tk 4 T. Then, we find for the coherence radius pk(z) = + , which means that the poor temporal coherence slows the increase in spatial coherence (see also KURASHOV, KISYLand KHOROSHKOV [ 19761). (ii) rk 4 a and Tk & T. Then, we obtain (4.12)

which corresponds to eq. (4.4) for the CW monochromatic case (T, Tk a). The coherence radius pk(z) is plotted in fig. 13. It can be seen that the increase of &(z) in case (i) is slower than that for a monochromatic, spatidly incoherent source field. However, for a propagating short pulse the coherence radius may increase more rapidly than for a stationary field. Note also that a common feature characterizing the propagation of partially coherent pulses is that the spectral purity of the source is disturbed due to the diffraction. ---f

4.2. TRANSMISSION THROUGH OPTICAL COMPONENTS

In § 3.4 we considered a transformation of an optical pulse by means of a pair of conjugated gratings, and we showed that such a device possesses a

250


2

L

6

8

1

[III, $ 4

0

z x l o L (cmi

Fig. 13. The dependence of the coherence radius pk on the distance z in the far zone: ( ) and T k < T ; (----) r k < a and T k % T ;(---.) r k < a and T k 9 T % I/w, (CHRISTOV [1986]).

rk 5. In this model one distinguishes two stages. In the first stage, nucleation centers are formed and grow out of the supersaturated solution; in the second stage, the grains coalesce, the larger ones absorbing the smaller ones. The preceding expressions are related to the latter stage. The same considerations may also apply to the case of metal particles in glasses (HUGHESand JAIN [ 19791, PERENBOOM, WYDERand MEIER[ 19811, HALPERIN [ 19861). As the different processes are still neither well understood nor controlled and the concentrations and type of defects are not yet identified to assess the impact of the crystallite size on their physical properties, it is important to have samples where all other factors remain unaltered from sample to sample. To achieve this, one performs the striking process on a well-controlled nucleated glass rod kept in a uniform temperature gradient (REMITZ,NEUROTH and SPEIT [ 1989]), the temperature increasing regularly from 500 to 750°C.It should

330

NONLINEAR OPTICS IN COMPOSITE MATERIALS

[V, § 2

stressed that the nucleation and striking stages must be well separated in time and not be permitted to occur concomitantly under any circumstances; this clearly imposes the taking of severe precautions during the nucleation stage. Semiconductor crystallites can also be obtained (WANG and HERRON [ 1987a,b], PARISE,MACDOUGALL,HERRON,FARLEE, SLEIGHT,WANG, BEIN, MOLLER and MORONEY[1988], WANG, SUNA, MAHLER and [ 19871) in gels, polymers, zeolites, and other porous materials by KASOWSKII processes that substantially differ from the one just described. Most often, the host material is introduced successively in solutions containing the chemicals; during each immersion, ions of a particular constituent element penetrate the host material and bind with the ions of a different type introduced in the preceding immersion to form clusters and crystallites of the desired type and average size. The size distribution can be narrower than that formed in the glass matrix by the thermal diffusion process; the problem here, however, is the optical quality of the host matrix, which is optically inhomogeneous, leading to substantial scattering of light. The procedure to prepare the colloidal solutions of semiconductor crystallites resembles that of the metal particles; these solutions are obtained first by preparing the clusters or crystallites by arrested precipitation in reverse micelles, with subsequent derivatization of the surface atoms with different groups in order to isolate the clusters from the micellar medium and make them stable against dissolution or aggregation ;furthermore, their solubility in hydrophobic solvents or polymers is increased by the surface derivatization (HENGLEIN [ 19821, ROSSETTI, NAKAHARAand BRUS [ 19831, NOZIK, WILLIAMS, NENADOVIC, RAJHand MICIC[ 19851, WELLER,SCHMIDT,KOCH, FOJTIK, BARAL,HENGLEIN,KUNATH,WEISS and DIEMAN[ 19861, SANDROFF, HWONGand CHUNG[ 19861). The preparation techniques of semiconductor colloids lead to narrower size distributions of the crystallites than in solid matrices, but the surface of the semiconductor crystallites is also very different and affects the photocarrier dynamics and optical nonlinearities differently. In all these cases involving “soft” matrices, polymers, liquids, or porous media, the chemistry plays a very important role, which presumably is also true in the case of the glasses where the introduction of different elements other than the ones forming the semiconductor crystallite plays a crucial role and affects its interface with the surrounding dielectric. Recently there has also been an effort to grow crystallites in “hard” solid matrices (ITOHand KURHAVA [ 19841, ITOH, IWABUCHI and KATOKA[ 1988]), e.g., in ionic crystals, CuCl or CuBr in NaCl or KCl. Here, considerations other than just chemistry play an important role because of the regular close packing of atoms in the host crystal.

v 7

I21

FABRICATION AND CHARACTERIZATION TECHNIQUES

33 1

2.2. CHARACTERIZATION TECHNIQUES

2.2.1. Structure and size determination The metal and semiconductor crystallites have been characterized by different physical, physicochemical, or chemical techniques. In the past, most attention was directed toward the metal crystallites, and HUGHES and JAIN [ 19791, PERENBOOM, WYDERand MEIER[ 19811, and HALPERIN [ 19861 give a fairly detailed account of the state of the art. The situation is being reversed, however, and the semiconductor crystallites are in the center of a growing number of characterization studies relative to their structure (DUVAL,BOUKENTERand CHAMPAGNON [ 19861, CHAMPAGNON, ANDRIANASOLO and DUVAL[ 19911, POTTERand SIMMONS [ 19881, YANAGAWA,SASAKIand NAKANO[ 19891, ROUSSIGNOL[ 19891, ALLAISand GANDAIS[ 19901, PETIAU[ 19891, DE GIORGIO, BANFI,RIGHINIand RENNIE[ 19901. X-ray diffraction and transmission electron microscopy have been used to some extent to study the structure, average size, and size distribution of the metal and semiconductor crystallites in the different matrices. Application of EXAFS for the study of the stoichiometry of the semiconductor crystallites of the type AB, C, - ~,in particular CdS,Se, -, in a glass matrix, has also been reported (PETIAU [ 19891) and work is in progress (DE GIORGIO, BANFI,RIGHINI and RENNIE [1990]) to use neutron scattering for the study of these composites. Their average size can also be determined by the light scattering technique (DUVAL, BOUKENTERand CHAMPAGNON [ 19861; for an application of this technique to semiconductor microcrystallites see CHAMPAGNON, ANDRIANASOLO and DUVAL[ 19911). Each of these techniques is sensitive to a different aspect of the structure and size distribution, and they actually complement each other, although most information until now has been obtained by X-ray scattering, in particular small-angle scattering and TEM. With respect to the latter method, the introduction of filtering techniques to sweep off the background signal originating from the glass (ALLAISand GANDAIS[1990]) has produced particularly clear pictures of the crystallite (fig. 1). Much work with these techniques is still needed to explain fully even the simplest cases ; however, some gross features have already emerged from these preliminary studies. Thus, it appears that after the metal and semiconductor clusters have grown beyond the nucleation stage, which roughly corresponds to the cluster size where volume and surface energies are equal, they acquire the crystalline structure and the stoichiometry of the bulk material. In particular, the lattice constant is the same as for the bulk. For certain semiconductor compounds that can be easily obtained in the wurtzite or zinc-blende structures by slight

332


Fig. 1. CdS crystallite (Hoya 450) view obtained by high-resolution transmission electron microscopy and filtering through Fourier transformation (From ALLAISand GANDAIS [1990].)

uniaxial pressure in the bulk along the c- respectively the 111-axis, there are indications that their nanocrystallites initially can adopt either structure, but they revert to the final one once they grow to a sufficient size inside the glass matrix. The crystallites show crystalline facets like large bulk crystals and are not WYDER perfectly spherical in shape (HUGHESand JAIN [ 19791, PERENBOOM, and MEIER[ 19811, HALPERIN [ 19861, ALLAISand G A N D A I S [ 19901). In fact, for larger semiconductor crystallites with a wurtzite structure there are strong presumptions that their average size in any three different directions generally are not equal (ALLAISand GANDAIS[1990]). However, because of their random orientation, the optical properties of the material appear to be globally isotropic, and as a good first approximation one may effectively assume spherical crystallites. The average size of these supposedly spherical crystallites, metal (HUGHES and JAIN [ 19791, PERENBOOM, WYDER and MEIER [ 19811, HALPERIN

v, $21

FABRICATION A N D CHARACXERIZATION TECHNIQUES

333

[ 19861) or semiconductor (EKIMOV,ONUSHCHENKO and TSEKHOMSKII [1980], BORRELLI,HALL, HOLLAND and SMITH [1987], POTTER and SIMMONS[ 1988]), is in rough agreement with that predicted from eq. (2.1) derived from the LIFSHITZ-SLEZOV [ 19591 theory, but the size distribution around this value is still a question of debate both in respect to its width and asymmetry. Both these features seem to depend strongly on the fabrication technique; crystallites in liquid suspensions (HENGLEIN[ 19821, ROSSETTI, NAKAHARA and BRUS [ 19831, NOZIK,WILLIAMS,NENADOVIC,RAJHand MICIC[ 19851, SANDROFF,HWONGand CHUNG[ 19861, WELLER,SCHMIDT, KOCH, FOJTIK,BARAL,HENGLEIN,KUNATH,WEISSand DIEMAN[ 19861) appear to have a narrower size distribution, which can reach ~ 7 of % the average value in total width, than that in solid matrices, where the total width can be at best (BORRELLI, HALL,HOLLANDand SMITH[ 19871, POTTERand SIMMONS[ 19881) of the order of 10-12%. An asymmetrical size distribution, but not exactly the one predicted by eq. (2.2) has been demonstrated (BORRELLI, HALL, HOLLANDand SMITH[1987], POTTER and SIMMONS [ 19881) in some binary semiconductor compounds, e.g., in CdS in glass matrix, indicating that there crystallites grow by a strict diffusion-limited process where only the size of the particles changes with time. On the other hand, for the mixed ternary compound CdS,Se, - x , there is no evidence of asymmetrical size distribution, and presumably the stoichiometry, or equivalently the coefficient x, changes in the course of the heat treatment because of the different diffusion coefficients for S and Se. Certainly these and other factors such as surface charges, crystalline anisotropy, and the constituency of the matrix, also affect the shape of the size distribution in a way not taken into account by the Lifshitz-Slezov statistical approach. Here we wish to point out that the average size, and to a lesser degree the size distribution, can also be determined by optical spectroscopy to the extent that these correlate with the spectral features of the quantum confinement, as will be discussed later. A recent study of BANFI, neutron scattering from semiconductor-doped glasses (DE GIORGIO, and RENNIE[ 19901) also revealed that the semiconductor crystallites RIGHINI have an apparent volume that is larger than the real one, presumably because of excluded (self-avoiding) volume effects. Information about the shape and surface states of the crystallite, and in particular its interface with the surrounding dielectric, is scarce or nonexistent. Yet, this information is badly needed to explain several essential physicochemical and spectroscopic features. Our extremely limited information here indirectly stems from optical studies or in certain cases from chemical treatment of the surface (HUGHESand JAIN [ 19791, PERENBOOM,WYDERand

334


[V, I 2

MEIER[ 19811, HENGLEIN[ 19821, ROSSETTI,NAKAHARA and BRUS[ 19831, HALPERIN [ 19861, NOZIK, WILLIAMS, NENADOVIC, RAJHand MICIC[ 19851, SANDROFF, HWONGand CHUNG[ 19861,WELLER,SCHMIDT,KOCH,FOJTIK, BARAL,HENGLEIN,KUNATH, WEISS and DIEMAN[ 19861, WANG and HERRON[ 1987a,b], WANG,SUNA,MAHLERand KASOWSKII [ 19871,PARISE, MACDOUGALL,HERRON,FARLEE,SLEIGHT,WANG, BEIN, MOLLERand MORONEY[ 1988]), as in the case of colloidal suspensions, but this does not necessarily apply to the case of crystallites in solid matrices. Since the latter case is more relevant for optoelectronic devices, the situation will certainly change drastically. 2.2.2. Optical techniques The previous techniques only yield information about the static structural features of these crystallites. Their dynamical properties that arise from the electronic and nuclear motion in the confined crystallite and their interaction with the crystallite walls can only be obtained by optical techniques. For this purpose several techniques have been used to study the so-called quantumconfinement effects, which are particularly conspicuous in semiconductor crystallites, as will be discussed later, and their quantitative relation to the crystallite size. In addition to the conventional absorption and transmission spectroscopy, which are routinely used to find the position and other spectral characteristics of the transitions between quantum-confined electron states, photoluminescence and resonant Raman spectroscopy give important information about the electron-phonon coupling, which as we will see, may play an important role in the broadening of the optical transitions in semiconductor crystallites along with the size distribution and other mechanisms. Important progress has been made recently by the introduction and exploitation of the nonlinear optical techniques to study the relaxation processes of the optical transitions and the dynamics of the nonlinear optical response for both semiconductor (BRET and GIRES [1964], JAIN and LIND [1983], RUSTAGIand FLYTZANIS [ 19841, ROUSSIGNOL [ 19891) and metal (RICARD, ROUSSIGNOL and FLYTZANIS [ 19851; see also RICARD[ 19861) crystallites in different matrices. By the same token, these techniques also allow the study of their nonlinear optical properties themselves, which are the topic of the present review. In these materials, even-order coherent nonlinear processes cannot take place because of the random distribution in space and direction of the crystallites in an inherently centrosymmetrical matrix. Therefore, the nonlinear optical techniques we have in mind here are those related to the odd-order

v, § 21

FABRICATION A N D CHARACTERIZATION TECHNIQUES

335

nonlinear effects (see, e.g., SHEN [1984]) and, in particular, the intensitydependent changes of the absorption and of the refractive index. Among the nonlinear optical techniques the optical phase conjugation (see, e.g., FISHER [ 19831, ZELDOVICH, PILIPETSKY and SHKUNOV [ 19851) through degenerate four-wave interaction has proved to be powerful for this purpose and is the most widely used, since it gives, in particular, direct information about the magnitude and dynamics of the optical Kerr effect, which has many potential applications in nonlinear optical devices. Its principle is depicted in fig. 2. A weak probe beam E,, together with two equally intense counterpropagating pump beams E, and E,, all three of the same frequency w, induce in the at the same fremedium under investigation a third-order polarization P& quency, which generates a phase-conjugated beam counterpropagating to the probe beam because of the phase-matching condition. As can be seen in fig. 2, in practice the probe and two pump beams are issued from the same laser. The intensity of the counterpropagating phase-conjugated beam is a direct measure of the third-order susceptibility x ( 3 ) (o,- o,a),which is related to the optical Kerr coefficient n, through the relation

where no is the linear refractive index. Furthermore, one can measure the anisotropy of n2 by an appropriate choice of the field polarizations and, more importantly, determine its temporal evolution and dynamics by using pulsed beams and introducing time delays between the pulses. Indeed, the coefficient n, as defined in (2.3) pertains to the stationary regime (monochromatic beams). These materials, when implemented in nonlinear optical devices, will operate in a nonstationary regime (pulsed beams where the pulse repetition rate may exceed several GHz) and the relevant coefficient is a time dependent n,,(t),

Fig. 2. Interaction scheme for optical phase conjugation through degenerate four-wave mixing.

336


[V, !i 2

whose temporal evolution is usually described by a Debye-type equation

or, in integral form,

where T is the decay time of the optical Kerr effect and, together with the magnitude of n2,plays a crucial role in assessing the potential use of an optical Kerr material. We also wish to point out that, in general, n, is a complex quantity and, therefore, in addition to its magnitude and time constant, its phase is also important. Only in the extreme cases of purely dispersive and purely absorptive nonlinearity is n2 real or purely imaginary. The first case occurs when cu is very far from any resonance, and one then expects z x 0, whereas the second case occurs close to a resonance, and z then is related to the relaxation processes of the resonance. One defines figures of merit for each of these two extreme cases, fd =

wX'3'/n, ,

(2.6)

which serve as measures of the potential usefulness of the material in a nonlinear device. These different capabilities of the optical phase conjugation technique can be easily appreciated by introducing the optical gratings description (FISHER [ 19831, ZELDOVICH, PILIPETSKY and SHKUNOV[ 19851) of the underlying nonlinear process (fig. 3). For an isotropic centrosymmetrical medium, which is the case with the composite materials, the nonlinear polarization source for the phase-conjugated signal beam at frequency cu can be written

PNLS= a(@ (EraE,*)&

+ b(n - 8) (Eb E,*)Ef+ c(Ef'&)E,*

, (2.8)

where 8 is the angle between the forward pump and probe wave vectors; apart from some trivial geometrical factors, the coefficients a, b, and c are actually directly related to tensorial components of x ( ~ ) .The first two terms in (2.8) correspond to spatial gratings of large ( A , )and small (A,) spacings, respectively, and the third term is the so-called self-diffraction term, which has no spatial analogue. By cross polarizing one of the three input beams with respect to the

FABRICATION A N D CHARACTERIZATION TECHNIQUES

small spacing

331

grating ( A s )

self diffraction

Fig. 3. Holographic interpretation of the nonlinear contributions leading to optical phase conjugation. The third term has no holographic interpretation (see text).

other two, each of the three terms in (2.8) can be isolated and measured. If the beams are pulsed with appropriate time delays between them, in addition, one can also study the temporal evolution of these terms and concomitantly of the nonlinear optical polarization. It should be pointed out that the preceding considerations are not only limited to third-order processes but also apply to higher odd-order processes, which in the conventional degenerate four-wave interaction when processes of all order are summed up again, lead to a nonlinear polarization of the form of eq. (2.8), but with the coefficients a, b, and c now being pump intensity dependent. Along with the degenerate four-wave interaction, one can use other nonlinear optical techniques to study the optical nonlinearities and, in particular, their time decay and spectral features. In this respect the saturation and hole burning spectroscopy (see, e.g., DEMTR~DER [ 1982]), the photon echo, and the different excite and probe time-resolved techniques give important information, as will be discussed in 5 3. In addition to these optical techniques, modulation spectroscopic techniques, like electroabsorption (CARDONA [ 19661) which is related to the static field-induced change of the absorption, is a powerful technique for studying the impact of quantum confinement in these crystallites. We conclude this list of optical techniques with the study of magneto-optical effects in these compounds. Although it has not yet been attempted, it contains much potential

338


[V, 8 3

for research and applications. It should be noted that much of the impetus for studying electron quantum confinement in metallic particles originated (KUBO [1962], GORKOVand ELIASHBERG [1965], HUGHES and JAIN [1979], PERENBOOM, WYDERand MEIER[ 19811, HALPERIN[ 19861)from theoretical work on the magnetic properties of these compounds.

6 3. Confinement Effects 3.1. BASIC MODEL

Despite their disparity, all the materials that are formed by uniformly dispersed metal and semiconductor crystallites in a liquid or solid transparent dielectric share two important features that have an essential impact on their properties in the optical frequency range. First, in the metal or semiconductor nanocrystals or microcrystals, the otherwise delocalized valence electrons in the bulk can find themselves confined in regions much smaller than their delocalization length, which is infinite in the ideal perfect metal and of the order of several tens to a hundred hgstrbms in a perfect semiconductor; this drastically modifies their quantum motion as probed by optical beams but also their interaction with other degrees of freedom. Second, because the size of the crystallites is much smaller than the wavelength and their dielectric constant is very different from that of the surrounding transparent dielectric, the electric field that acts on and polarizes the charges of these crystallites can be vastly different from the macroscopic Maxwell field. These two effects, the first quantum-mechanical and the second classical, go under the names of quantum and dielectric confinements, respectively, and are particularly conspicuous in the optical frequency range. The first requires the solution of the SchrOdinger equation in a spatially confined region whose boundary conditions impose a significantlydifferent eigenfunction and eigenenergy spectrum from those of the bulk, and the second requires introduction of the effective dielectric medium approach. In order to follow these effects and extract some qualitative and quantitative features, one is compelled to introduce some drastic simplifications regarding the composites and set up an idealized model for a composite; then one can progressively introduce complications that occur in real composites. For the surrounding dielectric, liquid or solid, we will assume that it is an ideal isotropic dielectric of dielectric constant E,, a scalar that shows no resonances and hence no absorption or dispersion in the frequency range of interest. The metal or

V, 8 31

CONFINEMENT EFFECTS

339

semiconductor particles are uniformly and randomly dispersed in small volume concentration in this dielectric; they will be assumed to be spherical in shape, with a diameter d = 2a that is much smaller than the optical wavelength 1.In the linear regime the relevant optical coefficient of such a crystallite of volume V is the polarizability a,, which has real and imaginary parts, or a, = a: + ia; , and we may formally define its dielectric constant E by the relation E, =

1 + 4 n z I " = 1 + 41E-a" , V

(3.11

which, in general, is expected to be a function of crystallite size and form but its limit for large crystallites must be the bulk value E, which also has real and imaginary parts, E = E' + i E " ;in the following we shall use e to denote both the dielectric constant of the bulk and the crystallite. We shall use this ideal composite to discuss the two main confinement effects.

3.2. DIELECTRIC CONFINEMENT

3.2.1. Linear regime: Efective-medium approach Let us assume that the volume fraction of the crystallites in the transparent dielectric is p 6 1, so that each crystallite is entirely surrounded by the dielectric and the interparticle distance is large with respect to the size of the crystallites, which is taken to be much smaller than the probing optical wavelength 2, i.e., d/A 4 1. One can then introduce an effective dielectric constant I for this composite medium, whose relation to t o , E, and p is given by the [ 1904, 19061 expression MAXWELL-GARNETT

This relation is a straightforward consequence of the familiar ClausiusMossotti approximation for the local field corrections for spherical polarizable particles and can also be generalized to ellipsoidal particles with a randomly oriented distribution. It is also intimately related to the MIE [ 19081 theory of light scattering from a diluted gas of spherical particles by imposing the vanishing of the forward scattering amplitude and neglecting all terms of higher order than the dipolar one. For the subsequent discussion it is useful to give a simple derivation of eq. (3.2) by referring to fig. 4.

340


Fig. 4. A small sphere of dielectricconstant E embedded in a matrix of dielectricconstant e, and submitted to a uniform electric field E. This leads to dielectric confinement.

The dipole induced by an applied field E in a spherical particle surrounded by a dielectric is (see, e.g., B~TTCHER[ 19731 or JACKSON [ 19801)

and the field inside the particle is

where E , is the local field in the vicinity of the particle. The presence of such polarizable particle results in an additional polarization P,

4 n P = 3P&,

- E,

& ~

&

+ 2Eo

EL= ( Z

-

E~)E,

(3.5)

which also defines the effective-medium dielectric constant 5, and E , is given by

4 nP E,=E+38,

(3.6)

Inserting eq. (3.6) in eq. (3.5), one recovers eq. (3.2) and, assumingp 4 1, one obtains the simpler expression

E

= &,

+ 3P&,

- E,

E ~

&

+ 2.50

.

(3.7)

To the extent that E is complex and frequency dependent, we see that all expressions show an enhancement close to the frequency w,, such that

+ 2Eo = 0 ,

&’(Us)

(3.8)

v, I 31

CONFINEMENT EFFECTS

34 1

which is the condition for the surface excitation or surface plasmon frequency. The width of this resonance is determined by E “ , and one can also obtain the extinction coefficient

which also determines the color of the composite. The preceding description implies a marked asymmetry in the treatment of the two materials, the crystallite inclusions and the surrounding transparent dielectric, and is valid only for p 4 1. When this is not the case, the two components must be treated on an equal footing, as in BRUGGEMAN’S [ 19351 effective-medium theory; and one obtains

P-

E - I Eg - I + ( 1 - p ) ~-0, E+2E Eo t 2;

which for p G 1 reduces to eq. (3.7). It is not clear (HUGHESand JAIN [ 19791, PERENBOOM, WYDERand MEIER[ 19811, HALPERIN [ 191361) whether Bruggeman’s theory accounts for the experimental results whenever the simpler Maxwell-Garnett theory fails to do so, and in the following we shall only use the latter.

3.2.2. Nonlinear regime The previous description only concerned the linear optical properties. In the presence of an intense electric field the induced polarization may be written (see, e.g., FLYTZANIS[ 19751, SHEN[ 19841) A p = p(1)+ p(2)+ p(3)+

... ,

(3.10)

where Pen) with n > 1 is the nonlinear polarization term of order n. In our case of an isotropic composite with random distribution of inclusions P(2n)= 0 and in particular F 2 )= 0, while

where X(’) and j ( 3 ) are, respectively, the linear and third-order effective susceptibilities of the composite. This latter quantity is a fourth-rank tensor

342


P.I 3

and, in general, has 81 components, but for the present case the number of independent components reduces to three, i.e., x$?,, , x:;:,, and xlr3.)yx. Here we focus mainly on the optical Kerr effect which, for monochromatic beams, is related to the third-order polarization at frequency w induced by an intense field E of frequency o by the relation

P"'(0)

=

3x'3'(0, - 0,0)IE(o)I2E(w),

(3.13)

and may also be described as an optically induced change of the optical dielectric constant, i.e., 61= 1 2 ~ p 1 ~ ( 4 2 .

(3.14)

This change of I contains contributions from both the embedding dielectric and the inclusions, denoted by 8co and 8e, respectively. However, close to the surface plasmon resonance w, for metal crystallites and generally for semiconductor ones, the contribution from the former can be neglected with respect to the latter, even for p Q 1. From eq. (3.7), then, a change 6~ of the dielectric and FLYTZANIS [ 19841, RICARD, constant ofthe inclusions willlead (RUSTAGI ROUSSIGNOLand FLYTZANIS [ 19851) to a change of I equal to (3.15) If we designate by x(3)the third-order susceptibilityrelevant to the internal field of the inclusion, eq. (3.4), then, in analogy with eq. (3.14) we can write

where Ei is the field (3.4); hence (3.17) and, formally, (3.18) This expression can also be derived directly by writing P 3 )as the density of third-order dipoles induced in the medium and taking into account the local field corrections. The important points to notice in eq. (3.18) are that i ( 3can ) be enhanced either by

v, s 31

CONFINEMENT EFFECTS

343

(i) the fourth power local field enhancement close to the surface plasmon resonance, or by (ii) quantum confinement mediated enhancement of x ( ~ ) . The first is the dielectric confinement effect, whereas the second is the quantum confinement effect, as will be discussed in the next section. For the discussion of the dielectric confinement we have purposely adopted an apparently phenomenological approach. The derivation can be made more rigorous and also extended to nonspherical particles (AGARWAL and DUTTA GUPTA[ 19881, HACHE[ 19881, STROUDand HUI [ 19881, HAUS,INGUVA and BOWDEN [ 19891, HAUS,KALYANIVALLA, INGUVA, BLOEMERand BOWDEN [ 19891, NEEVESand BIRNBOIM [ 1988,19891, STROUDand WOOD[ 19891) by introducing appropriate statistical averages or using the T-matrix approach (AGARWALand DUTTAGUPTA [ 19881) and microscopic considerations (HACHE[ 19881) for the electric fields and induced dipoles. The final conclusions and results pertinent to the experimental investigations, however, remain the same as described earlier. There are also predictions (LEUNG [ 19861, CHEMLA and MILLER[ 19861, SCHMITT-RINK, MILLER and CHEMLA [ 19871) that in the composite materials one may have local-field-mediated intrinsic bistability; such an effect may be difficult to observe, however, because of the large absorption that is always present whenever the local-field enhancement condition, eq. (3.8), is satisfied. The induced change 6 6 of the dielectric constant (or equivalently 6 E ) as defined above pertains to the stationary regime. Since these materials will operate in a pulsed nonstationary regime, when implemented in nonlinear devices, the temporal evolution ofn, is of central importance. The characterization of the composite materials is beset with many uncertainties that strongly affect the precise determination of the magnitude of n, as a function of the frequency, its decay time T, anisotropy, and phase. The uncertainties stem from different causes, some of them still unidentified, but are certainly related to the fabrication techniques, which are rather primitive (HUGHESand JAIN [ 19791, PERENBOOM, WYDERand MEIER[ 19811, HALPERIN [ 19861) when compared with the ones used to make quantum wells and other artificial microstructures, (see, e.g., KELLYand WEISBUCH [ 19861). The linear and third-order susceptibilities of a particle can be calculated (see, e.g., FLYTZANIS [ 19751, SHEN[ 19841) using their quantum-mechanical expressions in the dipole approximation. Since we are considering particles of a size much smaller than the optical wavelength, we may also introduce the linear and third-order polarizabilities a(w) and y(w, ,w 2 ,w3), respectively, whose expressions can be easily derived with perturbation techniques (see, e.g., FLYTZANIS

344


[V, § 3

[ 19751, SHEN[ 19841) once the spectrum of the unperturbed Hamiltonian H is known; one has

(3.19)

and

t

47 similar terms

(3.20)

Here r, s, t, and u represent the quantum states with energies E,, E,, E,, and E,, respectively; !?ma,, = E,-E,; and are the damping processes: fa,, = 1/T2 if a # b and r,, = l/Tl, where T, and TI are dephasing and energy relaxation times. Sometimes it is more convenient to transform these expressions by introducing

r,,

II=-

e

m

p,

where p is the momentum operator of the electron that satisfies the identity

h i [ H , x ]= - p m We shall use these transformed expressions in several parts of our discussion. The linear and third-order susceptibilities are simply related to the polarizabilities by

v. 8 31

CONFINEMENT EFFECTS

345

In the following we shall only consider the case 0 ,=

-w* =

w3 =

0,

which is relevant to the optical Kerr effect. We point out that in addition to the electronic contribution just considered, there are additional contributions from the nuclear motion and, in particular, a thermal one. These additional contributions are much slower than the electronic one and will be disregarded.

3.3. QUANTUM CONFINEMENT

3.3.1. Basic model The metal or semiconductor nanocrystals occupy a position intermediate between a molecule and the bulk crystal. Therefore, the choice of a model that accounts for the co-existence of features from both extremes is delicate and to a certain extent is dictated by the prominence of one feature over the other but also by the complexity of the underlying calculations. For a system consisting of very few atoms in an arbitrary configuration, the most usual approach is the one using molecular orbitals, but this quickly becomes intractable when the number of atoms is large, of the order of lo3, like it is in the nanocrystals in which we are interested. On the other hand, the description is greatly simplified for an infinitely extended periodic system where the Floquet theorem allows one to set up the space of the electron states in terms of Bloch band states (see, e.g., HARRISON [ 19801 or ASHCROFTand MERMIN[ 19811) qnPnk(r) = e i k . runk(r)

and

on

finding the real-space periodic function is any lattice vector; the corresponding band energy E,,(k) is a reciprocal space periodic function, i.e., E,,(k + K ) = E,(k), where K is any reciprocal lattice vector and k is the wave vector that labels the electron state in the band n within the first Brillouin zone that has dimensions of the order of the inverse of the lattice constant. The essential point is that for an infinite perfect crystal, eq. (3.23) has the form of a free wave and k is a good quantum number; if the periodicity is broken, e.g., by a defect or by reducing the extension of the crystal in one or more directions as in the nanocrystals, k ceases to be a good number, but to a reasonable approximation one may represent the electron motion as a wave unk(r

concentrate

(3.23)

+ R ) = unPnk(r), where R

346


[V, § 3

packet of Bloch states. If the defect encompasses several unit cells, uflkremains essentially unaffected, and close to k z 0 one may then write $ = F,(r) u,W

(3.24)

for the wave packet and the problem reduces to that of finding the envelope Ffl(r).This constitutes the basis of the effective-mass approximation (see, e.g., HARRISON [ 19801 or ASHCROFTand MERMIN[ 19811) initially introduced to treat the shallow defects and the electron-hole interaction in semiconductors. It will be assumed valid in the nanocrystals, metal (KUBO[ 19621, GORKOV and ELIASHBERG [ 1965]), or semiconductor (EFROSand EFROS[ 1982]), but we wish to point out that its justification still relies on qualitative arguments and on its a posteriori success in accounting for the essential quantum-confinement features in the optical spectrum of these nanocrystals. As we will see later, the envelope in eq. (3.24) satisfies the same equation and boundary conditions both for metal and semiconductor nanocrystals, although the underlying physical assumptions are different. In metals one has a single half-filled band up to the Fermi level E , with electron and hole states on either side of it which behave as free particles and have infinite delocalization. Accordingly one may set u(r) x 1, in the bulk E + ( k )=

h2 2m

- k2,

(3.25)

where the t is appropriate for electrons and the - for holes, and m,which is the same for electrons and holes, is very close to the free-electron mass. This half-filled band, which is usually formed with s- and p-orbitals, can, for most purposes, be replaced by an equivalent pair of parabolic bands, mirror images to each other, the upper one for the electrons and the lower one for the holes, that touch at k = 0 and are situated on either side of the Fermi level E , (see fig. 5). The wave-vector-dependent dielectric constant ~ ( kthen ) being infinite for k = 0, the electron-hole potential is completely screened to within a distance rF x l/k,, the inverse of the Fermi wave vector, which is of the order of a few angstroms or roughly equal to the lattice constant; thus, the electrons and holes can behave and move as free noninteracting particles over any distance in the perfect crystal. In the metal crystallite their motion will be hindered by the interface with the surrounding dielectric which, based on the simplification adopted in 8 2.3.1, will be visualized as a spherical potential well of infinite height. The electron and hole wave functions then will be of the form

v, I 31

CONFINEMENT EFFECTS

(a)

341

(b)

Fig. 5. Conduction (electron) band and valence (hole) band for a metal (a) and for a semiconductor (b).

(3.24) with u,(r)

(--h 2 :V 2m

=

1, and F satisfies the equation

+ W(r)

(3.26)

where the wall potential W = 0 for r < a and W = co at r = a ; with such a sharp infinite boundary, F(a) = 0. One may relax these conditions and let the wave packet leak out of the crystallite, but this will not lead to strikingly different behavior as far as the quantum-confinement features are concerned. In semiconductors the situation at the outset is extremely different. Here, too, for most purposes it is sufficient to use the two-band model, a filled valence band and an empty conduction band on either side of the Fermi energy, also designated hole and electron bands, respectively. In contrast to the metal case, at k = 0 the two bands are separated by a finite energy gap Eg. Furthermore, the two bands are not symmetrical with respect to the Fermi level, since each originates from a different basis of atomic states (fig. 5). We note that at zero temperature the Fermi level for an intrinsic semiconductor is situated halfway between the top of the valence band and the bottom of the conduction band. For states close to the bottom of the conduction band or to the top of the valence band, one may again assume parabolic bands, i.e., (3.27)

(3.28)

where m: and m z are the effective electron and hole masses, respectively, and, in general, rn: < mz.Because of the finite gap E, that now separates the hole

348


w

9

I3

from the electron spectrum, the wave vector dielectric constant e(k) is finite for k = 0, i.e., ~ ( 0 =) E, and accordingly the screening of the electron-hole Coulomb potential is only partial and bound states may exist. Hence, the electron and hole pair states have a finite extension, which can be characterized by the exciton Bohr radius aexc= h 2 E / p e 2 ,

with l / p = l/m,* + l/m,*, which is the Bohr radius for the SchrBdinger equation

The electron and hole states, when bound to an impurity, also have finite extensions characterized by the electron and hole Bohr radii a, = h2e/m,*e 2 ,

(3.30)

ah = h2e/m,*e 2 .

(3.31)

They indicate at what distance from the impurity the kinetic and potential energies for the electron and the hole, respectively, compensate each other and a, > a,,. The validity of this effective-mass approximation has been extensively discussed in the literature in connection with the states of shallow defects and the excitons; the basic condition imposed there is that a, % c, where c is the lattice constant. In a semiconductor crystallite the presence of the interface with the surrounding dielectric will introduce an additional potential term W in eq. (3.29), so that the equation for F will be

(3.32) where for simplicity we will assume that W is the same as for the metallic crystallites; namely, W = 0 for r < a and W = co at r = a. This approach initially proposed by EFROSand EFROS[ 19821 was subsequently extended by BRUS[ 1984, 19861 to allow for more realistic interface potentials where, in addition to the electron-hole Coulomb term, a potential term due to the dielectric discontinuity at the crystallite surface, the so-called surface polarization term, is included. At this stage we shall maintain the simple spherical wail potential with sharp infinite height to illustrate the main features of the

v. I 31

CONFINEMENT EFFECTS

349

quantum confinement. In contrast to the metallic particles, the characteristic respectively, lengths (3.30) and (3.31), the electron and hole radii a, and introduce three distinct quantum-confinement regimes for the semiconductor particles (EFROSand EFROS[ 19821); this is because

(3.33) where i = e, h and L is the smallest of the lengths a and aexc.We summarize here the main aspects of these three confinement regimes without going into the details (see EFROSand EFROS[ 19821).

Strong confiement a < ah < a,. Here, because of eq. (3.33), in a first approximation the electron-hole potential term can be neglected with respect to the wall potential and the kinetic energy; hence, the SchrOdinger equations for the electron and hole are decoupled, and each reduces to that of a free particle of effective mass mi*in an infinite spherical well potential (3.26) with the reminder that the electron and hole masses are now different, in contrast to the metallic crystallite. This only introduces a trivial length scale difference, however. For spherical crystallites and neglecting all anisotropy effects, the electron and hole states are labelled with three quantum numbers: a radial n and two angular I and m. Intermediate conjinement ah < a < a,. Here the electrons can still be treated as before, and their states are the same as in the strong confinement regime. On the other hand, for holes the situation is radically different, since the electron-hole interaction cannot be neglected with respect to the hole kinetic energy. Since the electrons are lighter than the holes, one can assume that the adiabatic approximation applies and proceed as in the Born-Oppenheimer treatment of the nuclear motion in molecular systems. If we label the electron state with (nlm), the holes move in an average potential (3.34) whose effect is only felt for small quantum numbers n and 1; then one can replace eq. (3.34) by the lowest terms of its Taylor development in powers of r, close to the center of the spherical crystallite, and for the s-states (I = 0) one obtains

(3.35)

350


[V.8 3

which is the isotropic three-dimensional harmonic oscillator potential, and the constants j?, and oncan be calculated from the derivatives of eq. (3.34) with respect to r,, at rh = 0. Weak confinement ah < a, < a. In this range of crystallite dimensions the bulk properties are established. In particular, the electron-hole potential now can allow bound electron-hole states or exciton states, which are only slightly distorted with respect to those prevailing in the bulk because of the presence of the infinite spherical wall potential. The essential difference with respect to the bulk is that here the exciton translational motion is confined. This can be taken into account by treating the exciton as a free particle of mass M = m,* + m l in a spherical potential well, which apart from a trivial length scale is the same as solving eq. (3.26). Thus, although the quantum confinement in semiconductor crystallites is at the outset far more complicated than that in metallic crystallites, in the final analysis the problem reduces to the solution of eq. (3.26), the SchrBdinger equation for a free particle in a spherical potential well with sharp infinite height whose solutions will be given later (see $ 3.3.2), together with the necessary modifications that allow one to incorporate more realistic aspects of the interactions inside the crystallites. Referring back to the band picture of the electron and hole states in a crystal, one may qualitativelyvisualize the quantum confinement in metal and semiconductor crystallites as resulting from the exclusion of the band states with wave vector k < l / a around the center of the Brillouin zone and their replacement by wave packets of the form (3.24); the band states with k > l / a , on the other hand, remain essentially unaffected.

3.3.2. Quantum-confined states and wave functions The solution of the SchrBdinger equation for a spherical quantum-confined crystallite in general can only be tackled numerically, and this only for the lowest states. The problem is somewhat simplified in the extremely idealized cases. Thus, within the approximation framework just discussed, the Schrbdinger equation for the wave envelope, eq. (3.26), for semiconductor crystallites in the strong confinement regime (a < a,, < a,) and for the metal crystallites irrespective of their size reduces to that of the free particle in a spherical potential well with sharp infinite boundary (3.36)

CONFINEMENT EFFECTS

35 1

where W = 0 for r < a and W = cc for r = a. The solutions of this equation with the boundary condition q ( r = a ) = 0 in spherical coordinates take the form (3.37) where the Y;l are the spherical harmonics ( - I < m < I), j,(r) is the spherical Bessel function of order I, and a,, is its nth zero, or j,(an,) = 0. With E , = h2/2ma' and k,,, = a,,,/a, the energies of these eigenstates are E,,

=

( C ( , , , ) ~=E h2k,:/2m ~ ,

(3.38)

where k,, plays the role of a quasimomentum and is independent of the quantum number m ; in addition to this (21 + 1)-fold degeneracy, the energy distribution is fairly complicated. The simplicity of this analytical treatment is quickly lost when more realistic potentials are introduced in eq. (3.36), as discussed by BRUS[ 1984, 19861. In such cases one has to resort to variational (KAYANUMA[ 19861, NAIR, SINHA and RUSTAGI [ 19871, BAWENDI, and BRUS[ 19901) or other numerical techniques (BRUS[ 1984, STEIGERWALD 19861) to derive approximate forms of the wave function of the ground state, the 1 s state, and estimates of its energy. Later, we shall refrain from going into the technicalities of these methods, summarized for the semiconductor crystaland BRUS [1990], and shall use the prelites by BAWENDI,STEIGERWALD viously outlined analytical model to discuss the salient points of the optical transitions and linear and nonlinear optical susceptibilities of quantumconfined particles. This will be done separately for metal and semiconductor crystallites ;they correspond to the intraband and interband confinement cases, respectively (FLYTZANISand HUTTER [ 19911). 3.3.2.1. Mela1 crystallites Since the cell periodic part in eq. (3.23) is u(r) x 1 and m,* = m$ x m for metals, the electron and hole wave functions (3.24) are identical and coincide with the envelope wave function (3.37).The energy spectrum given by eq. (3.38) for n 9 I can be simplified (HACHE,RICARDand FLYTZANIS[ 19861, HACHE [ 19881) by replacing a,, with its asymptotic form a,, x

;(2n + I) 7c ,

(3.39)

and, hence, for a given value of 2n + I, the energy levels plotted versus I would fall on a horizontal line; in reality, this line bends down for large I, which has implications on the density of states v(E). When such a density can be defined,

352


it is given by the bulk value

v

v(E)= 2x2

(?) 2m

3f2

2 Elf2 Ell2 = - - , including spin degeneracy, 371 E:l2 (3.40)

where V is the volume of the spherical crystallite. The energy difference between an I state of energy E and the nearest I f: 1 state (which is the nearest state attainable through a dipolar transition) is nearly AE

x(EE,)’12.

=

For simplicity we may assign the same value to the energy of all states between E and E + AE whose number is N ( E ) = v ( E )A E

2E

=-.

(3.41)

3Eo Finally, the Fermi level EF is independent of the crystallite size, i.e., (3.42) With these wave functions one can calculate all transition dipole moment matrix elements between any two states with quantum numbers nlm and n’1‘ m’ , respectively, and the corresponding oscillator strengths (HACHE, RICARDand FLYTZANIS [ 19861, HACHE[ 19881). Since the cell periodic part in eq. (3.24) for metals is U ( Y ) = 1, the dipole transition selection rules, in particular, can be easily derived, 1=1’*

1 and m = m ’ , m ’ * 1,

and the transition dipole moment between states where 11 - I’ 1 = 1 is x,, =

4a e E , (E,E,)’/2 Amm, ih E, - E, ~

7

(3.43) I =

nlm and s = n ’ l ‘ m ’ ,

(3.44)

where Ammris an angular factor. With these transition dipole moment elements, one can then proceed to compute the linear polarizability a(w) and the thirdorder polarizability y(w, - w, w), using their quantum-mechanical expressions described in !j 3.2. Since we shall always be in the resonance regime in order to take advantage of the quantum confinement, only a few transition elements

v, I 31

CONFINEMENT EFFECTS

353

are needed, those relevant to the resonance, but the broadening characteristics of the resonance must also be known. These are subtle and will be discussed shortly; we only anticipate here that, formally, one can attribute a uniform dephasing time T2 to all transitions and similarly for the energy lifetime T,. 3.3.2.2. Semiconductor crystallites In semiconductor crystallites one must distinguish between the three quantum-confinement regimes we discussed in the preceding section; this problem was examined by EFROSand EFROS[ 19821 and further extended in the case ofstrongconfinement by BAWENDI, STEIGERWALDandBRUS[ 19901usingmore realistic assumptions concerning the wall potential. Following EFROSand EFROS [ 19821, in the strong-confinement regime (BRUS [ 1984, 19861, BAWENDI,STEIGERWALD and BRUS [ 1990]), the wave functions that are of the form (3.24) for the electron (e) and hole (h) can be written $vn/rn(rv) = F v n ( m ( r v )

uv(rv)7

(3.45)

where the envelope function for v = e, h is given by eq. (3.37) with - 14 m 4 1, I = 0, 1,2, . . . , n = 1, 2,3, . . . ; the corresponding energies are (3.46) and k , , are defined by the boundary condition j l ( k n ~=~ 0)

(3.47)

and were also defined in eq. (3.38). Here m: and m$ are the electron and hole effective masses, respectively, with rn: < m$ in general, and u, and uh are the cell periodic parts of the bulk Bloch function at the band edge that we expect to be unaffected by the confinement. The transition dipole moment matrix elements and the corresponding oscillator strengths and selection rules can also be easily derived using these wave functions. Noting that the envelope function F i n the wave packet (3.45) varies slowly over several cells, one can safely set

where p,, is the usual momentum matrix element between valence (hole) and conduction (electron) bands ; this implies that for a perfectly spherical isotropic semiconductor crystallite, optical transitions can only occur between hole and electron states with identical envelope functions (complete overlap). Accord-

3 54

[V,§ 3


ingly, in contrast to eq. (3.43), the selection rules for dipole transitions in strongly confined semiconductor particles are

n=n', l = l '

and m = m ' ,

(3.49)

and the lowest energy allowed transition is the 1s-1s transition. The most conspicuous features of the strong confinement are (1) the replacement of the continuous band-to-band transition spectrum in the bulk by a discrete transition spectrum (fig. 6) between electron and hole states given by eq. (3.46) and the selection rules of eq. (3.49), and (2) the shift of the onset of absorption from Eg in the bulk to (3.50) in the crystallite, where 1/p = l/m,* + l/m,*. Actually, the spin-orbit splitting of the valence band into two sub-bands corresponding to the total angular momentaJ = a n d J = $with different gaps Eg and EL with respect to the conduction band and different hole effective masses m,*and mi* introduces a slight complexity in the previous energy level scheme and the corresponding allowed transitions spectrum (BAWENDI, STEIGERWALD and BRUS [ 19901). Instead, one now has a spectrum resulting from the simple superposition of two energy ladders, each satisfying the previ-

6 5 0 8 0 0

550

500

450

400

350

Fig. 6 . Absorption spectra for three semiconductor-doped glass samples grown from an RG 610 melt. The mean particle radii are: (1) 6 nm, (2) 2.5 nm, and (3) 1.5 nm.

CONFINEMENT EFFECTS

355

ous rules with the same electron effective mass but with different gaps and spacings because of the different hole masses. In addition to this minor complication the actual positions of the levels may deviate slightly from the ones calculated by eq. (3.46), and this may be attributed to different physical origins. One cause is the neglect of the Coulomb interactions and surface polarization terms discussed by BRUS [1984, 19861 and BAWENDI,STEIGERWALD and BRUS [ 19901. In addition, the actual interface potential is neither sharp nor infinite, so that the envelope of the wave function (3.45) does not vanish at the boundary but may leak out of the confined crystallite (see, e.g., HENGLEIN [ 19881). Finally, the surface impurities may substantially perturb the energy spectrum. The Coulomb interaction may actually influence the spectrum if an electron-hole pair has been previously created, which is the situation when intense optical fields are used to induce resonant nonlinear effects. This residual pair-pair interaction cannot be treated analytically and its influence cannot be properly assessed; one expects that it can be neglected in the strong-confinement regime but not in the intermediate and weak regimes. In the latter regime this interaction is actually responsible for the biexciton formation. Returning to the simple description of the strong confinement regime, expressions (3.19)-(3.22) of the linear and nonlinear polarizabilities can be easily calculated in the resonant regime, using the expressions for the energies and dipole moment matrix elements when the broadening of the allowed resonance is known; this will be discussed in the following section. The situation becomes more complicated in the intermediate confinement regime, because electron-hole coupling cannot be disregarded. As stated in 0 3.3.1,the electron energy spectrum remains the same as in the strong confinement regime and is given by eq. (3.46) and the corresponding wave functions aregiven by (3.45); the hole spectrum, on the other hand, for each such electron state must be calculated by applying the adiabatic approximation. In the case of 1 = 0, the problem reduces to that of a three-dimensional isotropic harmonic oscillator with the potential given by eq. (3.35), and the energy spectrum for the hole is given by E,'soo=

e2 --

&a

Qn+ttwn(2r+s+;),

where r = 0, 1, 2, and s is the equivalent of 1. Accordingly, one finds that each electronic transition is converted into a series of closely spaced lines with an asymmetrical envelope. We shall not dwell further on this case since the broadening of each of these closely spaced levels introduces a strong overlap

356

[V,§ 3


between them and leads to an overall asymmetrical broadening of the electronic transition. Finally, in the weak-confinement regime the electron and hole states resume their bulk features, and the electron-hole interaction may lead to bound states or excitons as in the bulk but with their translational motion confined within the crystallite volume; one finds that the lowest exciton transition is shifted with respect to its position in the bulk E,,, by an amount

A = - h2 n2 2Ma2 ’

(3.51)

where M = m,* + m z . As previously stated, in the intermediate and weak confinement regimes, additional interactions such as the Coulomb interactions play an important role, and this will be discussed later (see $ 5).

3.3.3. Level broadening The previous discussion of the quantum confinement in a spherical metal or semiconductor crystallite leads to an optical absorption spectrum consisting of infinitely narrow discrete transitions or Im x

=

A

c

nl, n ’ l ’

gn,I

< rill P I n’1’ > I

fnl(

1 - f n , I ’ 1w

n , I’

- En, -

9

(3.52)

where fnr is the occupation probability of the (21 + 1)-fold degenerate level nl and the double summation is extended over all electric dipole allowed transitions. Experimentally, one always observes broad spectral features instead. This broadening is caused by the various ever-present perturbations, intrinsic or extrinsic to the crystallite, which couple to the electron and hole motion and introduce temporal or spatial disorder. Because of the crucial role both these random perturbations play in the resonant nonlinear optical processes, we will analyze these mechanisms in some detail and give a short account of their origin and impact. Although in the last analysis the microscopic origin may be the same, at a more phenomenological level it is preferable to discuss the broadening in metals and semiconductor crystallites separately. 3.3.3.1. Metal crystallites In the bulk metal the free electrons in the conduction band suffer collisions with phonons and other electrons with a rate l/q,, where zb is the mean time

v, I 31

CONFINEMENT EFFECTS

357

lapse between successive scattering events and will be termed scattering time. In a crystallite with d < I,, where I, is the electron mean free path, electrons also undergo collisions with the spherical wall at an average rate uF/a,where uF is the speed of the electrons close to the Fermi level E,, from which the essential contribution comes. To the extent that the two processes are uncorrelated, one may introduce an effective collision time z,~, 1 zeff

-

1

+ -U F

‘b

a

(3.53)

One also introduces a dephasing time T,, $T, = q,, the same for all dipoleallowed transitions, which leads to a homogeneous broadening of the transitions, independent of the crystallite radius; accordingly, the delta functions in eq. (3.52) are replaced by Lorentzian functions. This classical argument is also corroborated by a detailed quantum-mechanical calculation (KAWABATA and KUBO[ 19661, GENZEL,MARTINand KREIBIG[ 19751, RUPPINand YATOM [ 19761, KREIBIG and GENZEL [ 19851, HEILWEIL and HOCHSTRASSER [ 19851, [ 19&6]),where eq. (3.53) is only the limiting HACHE,RICARDand FLYTZANIS expression for w -+0 of

- 1= - + -1

UF

zeff

a

Tb

gs(v)=-+-,

vF

zb

where

x3l2(x

+ v)I/,

dv,

(3.54)

with v = hw/EF. For a statistical assembly of metal crystallites in a dielectric, as was the case in all samples studied, one must perform an average of eq. (3.52) over the size distribution P(a/Z), where P(a/Z) da/Z is the probability for the radius a being in the interval da. This introduces in principle an inhomogeneous broadening that, however, in the optical frequency range around the surface plasmon resonance where the level spectrum and density become essentially identical to those of the bulk metal, has an inconspicuous impact on the overall broadening and can be disregarded there. This is no longer true in the far-infrared, where the quantum confinement has a stronger impact but the density of states is also substantially reduced. In addition to the dephasing mechanism of broadening, there is also an

358

[V,8 3


energy relaxation mechanism (HEILWEIL and HOCHSTRASSER [ 1985]), which will be accounted for with a time T,, which is the same for all transitions. With the introduction of these two relaxation times, T, and T, ,which determine the dephasing and energy decay rates, respectively, one can proceed to calculate the linear and nonlinear polarizabilities using the corresponding quantummechanical expressions. The size-dependent broadening of the surface plasmon resonance as predicted by eq. (3.53) has been experimentally confirmed for gold particles both in [ 1964, 19651, colloids and solid matrices (glass) (DOYLE[ 19581, DOREMUS KREIBIGand GENZEL [ 19851). In particular, using the experimental values (JOHNSON and CHRISTY [ 19721) for the dielectric constant for the bulk and expression (3.53) for the dephasing time, the variation of the absorption coefficient as a function of the average crystallite radius could be accounted for (fig. 7). On the other hand, there is little (HEILWEIL and HOCHSTRASSER [ 19851) or no information concerning the energy relaxation time T , .

.513

,382 ,191

0 370

470

570

670

A (nm)

," ,126 (2.6)

= pe(r) + ~ h ( r ).

(3.55)

For each eigenmode of quasimomentum k,,, the total Hamiltonian can then be exactly diagonalized, and the problem is fully equivalent to a shift of a harmonic oscillator potential and energy levels by a relative amount A&, a dimensionless quantity. Neglecting the dispersion of the LO phonon branch, the total coupling is then characterized by(HUANG and RHYS[ 19501, DUKEand MAHAN[ 19651, MERLIN,G~NTHERODT, HUMPHREYS, CARDONA,SURYANARAYANAN and HOLTZBERG [ 19781, NEUMARK and KOSAI [ 19831)

A2

=

C A:,

(3.56)

k

which is exactly equal to the Huang-Rhys parameter S. This parameter, which is temperature and polarity dependent, can also be defined as AElhw,,, where A E is the total lattice-forced shift in energy of the harmonic oscillator potential

CONFINEMENT EFFECTS

36 1

of the upper electronic level with respect to that of the lower hole level in the 1s-1s transition, and wLo is the level spacing in these potentials (the LO phonon frequency). In order to treat the problem properly, the quantum confinement of the phonons must be taken into account (MORI and ANDO [1989], KLEIN, HACHE,RICARDand FLYTZANIS[1990]); this does not affect the phonon frequencies because of the large ionic masses, but it does strongly modify the eigenfunctions, which has profound consequences on the magnitude of the coupling (NEUMARK and KOSAI[ 19831).Without going into the technicalities, we state here that the main consequence is that, if the size of the electronic charge distribution scales as the radius a, the electron-phonon coupling S is size independent and this results from the exact compensation of two sizedependent effects (MORI and ANDO [ 19891, KLEIN,HACHE,RICARDand FLYTZANIS [ 19901). On the one hand, reducing the size of the sphere leads to an increasing overlap of the electron and hole wavefunctions, implying a decrease in the coupling; on the other hand, the same reduction of size should lead to an increasing coupling to short-wavelength phonons. The absorption profile of the 1s-1 s transition then is simply the overlap of the shifted harmonic oscillator states of the hole and electron levels (HUANG and RHYS [1950], DUKEand MAHAN [1965], MERLIN,GUNTHERODT, HUMPHREYS,CARDONA, SURYANARAYANAN and HOLTZBERG[ 19781, NEUMARK and KOSAI[ 19831, SCHMITT-RINK, MILLERand CHEMLA[ 19871, KLEIN,HACHE,RICARDand FLYTZANIS [1990]), and this leads to a series of satellite lines spaced by oLo,i.e., at 0 K

(3.57) where B,,(SZ) is a Lorentzian centered at Q with width rofor the zero-phonon line ( p = 0), rl for the one-phonon line ( p = l), and p1I2rl for the p-phonon line. Because of the increasing width with increasing p and the weighting factor e-SS”/p!, which becomes maximum at p NN S , for sufficiently large T,/w,,, one usually obtains a broad line centered at SZ, + So,, with the zero-phonon line of width rosuperimposed; S $- 1 refers to strong coupling and S 4 1 to weak coupling. There is no precise relation between r, and r,,although both result from coupling to acoustic phonons and other dephasing degrees of freedom.

362


[V, § 3

Inhomogeneous broadening. The profile (3.57) applies to a single spherical crystallite, and for an assembly of such crystallites with a size distribution P(a/ii)eq. (3.57) must be convoluted with the latter to obtain the absorption coefficient of the sample, i.e., .

1

a(w) =

n

1 a3 a,(w) P ( a / Z )dZ/a.

(3.58)

a’ J

This leads to an additional broadening of the transition (EFROSand EFROS [ 1982]), which is inhomogeneous in character. In contrast to the temperaturedependent homogeneous broadening due to electron-phonon coupling, the inhomogeneous broadening due to the size distribution is temperature independent and is present even for the ideal case of zero electron-phonon coupling (rigid lattice). It is instructive to derive the expression of the absorption coefficient taking into account this size distribution with the LIFSHITZ-SLEZOV [ 19591 expression for P(u) given by eq. (2.2) and assuming zero electron-phonon coupling. Averaging (3.52) over the distribution (2.2) and inserting eq. (3.46) for the dipole-allowed transitions, one obtains (EFROSand EFROS[ 19821)

(3.59) nl

i.e., a series of broad lines with the profiles and positions given by the functions Pb). The convolution procedure leads to a compound broadening, which can be related to the experimentally measured crude broadening of the quantum-confined line spectrum. The relative importance of the homogeneous and inhomogeneous mechanisms can be inferred by nonlinear optical techniques, like spectral hole burning, photon echo, or indirectly by the temperature dependence of the broadening. An estimation of S and the intrinsic phonon-broadening can also be obtained by Raman and luminescence spectroscopy (KLEIN,HACHE, RICARDand FLYTZANIS[ 19901). In contrast to the case of metal crystallites, where the dephasing mechanisms of the resonances and their lifetime are to a certain extent understood (DOYLE [ 19581, DOREMUS [ 1964,19651, KAWABATA and KUBO[ 19661, KREIBIGand FRAGSTEIN[ 19691, KREIBIG[ 1970, 1974, 19771, GENZEL,MARTINand KREIBIG[ 19751, RUPPINand YATOM[ 19761, KREIBIGand GENZEL [ 19851, HEILWEIL and HOCHSTRASSER [ 19851) and at least qualitatively account for the experimental observations, in the case of semiconductor crystallites the situation is far from clear, both at the experimental and theoretical levels. The

CONFINEMENT EFFECTS

363

relative impact of the two mechanisms we singled out as being the most important in perfectly spherical isotropic semiconductor crystallites, namely, the electron-optic phonon coupling (SCHMITT-RINK, MILLERand CHEMLA [ 19871) and the size distribution, in principle can be assessed by nonlinear optical spectroscopic techniques and particularly by the hole-burning technique (DEMTR~DE [ 19821, R HAYES,GILLIE,TANGand SMALL[ 19881). The complications arise because the semiconductor crystallites are never perfectly spherical nor isotropic and certainly have a large concentration of unidentified defects and impurities, especially on their surface; in addition, lack of inversion symmetry together with the Coulomb effects eventually introduce deviations from the idealized level spectrum and the selection rules we previously derived. As a consequence, in crystallites there are numerous other broadening and lifetime-limiting mechanisms, comparable in strength to the two explicitly considered, which overlap or interfere with each other. The discussion and summary that follows should, therefore, be taken with caution and in the expectation that the situation will soon clarify, since this point is essential for understanding the nonlinear mechanisms and decay of the optical Kerr effect in the quantum-confined crystallites. The existence of homogeneous and inhomogeneous broadening in the quantum-confined resonances of semiconductor crystallites has been demonstrated with time-resolved hole-burning studies, using pulsed lasers in the nanosecond (ALIVISATOS, HARRIS,LEVINOS,STEIGERWALD and BRUS [ 1988]), picosecond (HILINSKI, LUCAS and WANG [ 19881, ROUSSIGNOL, RICARD, FLYTZANIS and NEUROTH [ 1989]), and femtosecond (PEYGHAMBARIAN, FLUEGEL,HULIN,MIGUS,JOFFRE, ANTONETTI, KOCH and LINDBERG [ 19891, ROTHBERG,JEDJU, WILSON, BAWENDI, STEIGERWALD and BRUS [1990]) time domains; all these studies actually concern CdS, Se, -,crystallites in glasses (ROUSSIGNOL,RICARD, FLYTZANIS and NEUROTH[ 19891, PEYGHAMBARIAN, FLUEGEL,HULIN, MIGUS,JOFFRE, ANTONETTI,KOCH and LINDBERG[ 19891) or colloidal suspensions (ALIVISATOS, HARRIS,LEVINOS,STEIGERWALD and BRUS [ 19881, ROTHBERG, JEDJU, WILSON,BAWENDI,STEIGERWALD and BRUS [ 19901). The hole burning was only observed at low temperature (liquid-helium temperature), whereas at room temperature only uniform saturation of the absorption was observed similar to the one commonly observed in a homogeneously broadened atomic two-level system (fig. 8) (DEMTR~DER [ 19821). That a temperature-dependent mechanism is operative can also be inferred from linear absorption spectroscopy (ROUSSIGNOL, RICARD,FLYTZANISand

3 64


lYL

/

3

1

.c '$ C

i

1

550

500

Wavelength (nm)

Fig. 8. Transmissionspectra at room temperature before (0) and immediately after (1) excitation by a picosecond pulse at t = 532 nm. The mean radius of the CdSSe particles is about 2.5 tun.

NEUROTH[ 1989]), and, as shown in fig. 9, the linewidth is reduced by more than 30% as we go from room down to liquid-helium temperatures. This may be attributed to the electron-phonon coupling, whose effect is reduced as the temperature is decreased, whereas a large part of the residual linewidth at liquid-helium temperature is certainly due to the size distribution and leads to hole burning as observed. In the studies with nanosecond (ALIVISATOS, and BRUS [ 19881) and picosecond HARRIS,LEVINOS, STEIGERWALD (ROUSSIGNOL, RICARD,FLYTZANISand NEUROTH [ 19891) time resolution, the width of the burnt note is large (figs. 10 and 1l), which if totally attributed to electron-optic-phonon coupling, implies a rather large value for S ; preliminary estimations indicated S = 2.5-3. An estimate of S can also be independently extracted from the relative intensities of the lirst- and higher-order resonant Raman spectra (BARANOV, BOBOVICHand PETROV [ 19881, ALIVISATOS, HARRIS,CARROLL,STEIGERWALD and BRUS [ 19891, KLEIN,

v, B 31

365

CONFINEMENT EFFECTS

550

Bso

450

600

550

500

Fig. 9. Absorptionspectra at room temperature (solid line) and at 12 K (dashed line) for CdSSedoped glasses. The mean radii are: (a) 1.5 MI and (b) 2.5 nm.

Hache, Ricard and FLYTZANIS [ 19901) and photoluminescence spectra (HACHE,KLEIN, RICARDand FLYTZANIS [ 19911); thus, for CdS,Se, --x crystallites it was estimated that S % 0.5-1, which seems to be in line with other rough estimates but cannot account for the whole width of the burnt hole. In the picosecond hole-burning experiments the position of the burnt hole depended on the position of the pump frequency with respect to the absorption peak, which is size dependent, whereas the theoretical discussion of the electron-optic-phonon or Frahlich coupling as well as the experimental Raman spectra indicate that S is size independent. Incidentally, for some samples and linear absorption spectroscopy (ROUSSIGNOL,RICARD, FLYTZANIS NEUROTH[ 19891) clearly shows that at room temperature a size-dependent broadening mechanism is operative, whose strength increases as the crystallite size is decreased, which also implies that the 1s-1 s transition is narrowest at an intermediate crystallite size, as experimentally observed. These observations

366


[V,§ 3

Energy ( e v )

-A(o.D.) 0.3

b

x' -

0.0 -

450

550

6 50

Wavelength ( n m )

Fig. 10. Absorption spectrum (a) and negative differential absorption spectrum (b) for small CdSe particles at a low temperature showing hole-burning on the nanosecond scale. (From ALIVISATOS, HARRIS,LEVINOS,STEIGERWALD and BRUS [ 19881.)

indicate that in addition to their coupling with polar optic phonons, which is size insensitive, the electronic transitions also strongly couple to other dephasing degrees of freedom sensitive to the confinement, like phonons from the acoustic branch. With femtosecond laser pulses (PEYGHAMBARIAN, FLUEGEL,HULIN, MIGUS, JOFFRE, ANTONETTI,KOCH and LINDBERG [ 19891, ROTHBERG,JEDJU, WILSON,BAWENDI,STEIGERWALD and BRUS [ 19901) (fig. 12) the burnt hole is not as evident as with the longer pulses, but another feature was observed, namely, the occurrence of an induced absorption, which can be attributed (BANYAI,Hu, LINDBERGand KOCH [ 19881) to the Coulomb interaction either between two photocreated electron-hole pairs in a single crystallite or between an electron-hole pair and an impurity-trapped electron-hole pair (HENGLEIN,KUMAR, JANATA and WELLER [ 19861, ROTHBERG, JEDJU, WILSON,BAWENDI,STEIGERWALD and BRUS [ 19901). Both mechanisms can be qualitatively represented with an equivalent pumpbeam photoinduced electric field that redistributes the spectrum and oscillator strengths of the probe-beam photocreated electron-hole pair. As will be discussed (see 0 5), this would also imply the operation of two distinct nonlinear

v. 5 31

367

CONFINEMENT EFFECTS

1

550

500 W a w h g t h (nm)

Fig. 1 I. Transmission spectra of a quantum-confined CdSSe-doped sample at various delays of the probe pulse (T = 12 K) showing hole-burning on the picosecond scale.

50

I

Wavelength ( nm 1

Fig. 12. Absorption spectrum (dashed line) and negative differential absorption spectra for small CdSe particles at 10 K and for two pump wavelengths in the subpicosecond time scale. (From PEYGHAMBARIAN, FLUEGEL,HULIN,MIGUS,JOFFRE,ANTONETTI,KOCH and LINDBERG [ 19891.)

368

[V,§ 4


mechanisms for the optical Kerr effect, along with the commonly assumed two-level saturation mechanism (SCHMITT-RINK,MILLER and CHEMLA [ 19871). In view of the present uncertainties concerning the size, shape, and surface of the crystallites and the complicated considerations and assumptions one must introduce when Coulomb interactions are invoked, we shall concentrate mostly on the two-level saturation model and also assume that the homogeneous broadening can be represented with a roughly size-independent dephasing time T,, whose value is in the range of 10 to 50 fs.

8 4.

Nonlinear Optical Properties of Metal Composites

The main optical properties of a composite consisting of metal particles in a transparent dielectric will be discussed within the framework outlined in the previous section. The relevant frequency domain is that close to the surface plasmon resonance w, defined by (3.8) and we shall only consider those nonlinear optical properties, related to light-induced changes of the dielectric constant, namely, the optical Kerr effect (see 5 3.2.2). As we saw in 3.2.2, the expressions of the optical coefficients of a composite containing a volume concentration p of spherical metal crystallites of average radius a can be easily related within the effective-medium approach to those of a single crystallite, using the relations (3.7), (3.9), and (3.18) when p 4 1; for our purpose the relevant optical coefficients of such a metal crystallite are the linear and third-order polarizabilities, a(o) and y ( o , - w, a),respectively, whose quantum-mechanical expressions near a resonance were given earlier, i.e., expressions (3.19) and (3.20), respectively. Referring to the notations of the previous section and to eq. (3. l), one obtains for the linear susceptibility

(4.1) where is the plasma frequency, = 4 II Ne2/mV, nrs is given by eq. (3.44), and A, is the angular part which for I & 1 is close to $. The first term is the Drude term, the same as for the bulk if we disregard the small term Tg in the denominator; the second term is the intraband term, which results from transitions between the quantum-confined electron states of the s-p conduction band or, equivalently, between the electron and hole band states (3.25);

'

PROPERTIES OF METAL COMPOSITES

369

the third term is the interband term and results from transitions between states in the d-valence band and quantum-confined states of the s-p conduction band. Taking into account expression (3.44) for nrs, we expect that the main contributions to the intraband term come from transitions with a,,x 0 or w,, z w. The first one amounts to a small correction in the real part of the Drude term, which actually renormalizes the plasma frequency, whereas the second one, after reverting to an integration using the density of states (3.40) and the identity l/(x + iT) --t P ( l/x) - in6(x) when T-+0 +,reads

where g,(v) is given by eq. (3.54). This term, lumped together with the first term in eq. (4. l), again gives a Drude term with mean collision time zeffthat considers the encounters of the electrons with the surface as well (KAWABATA and KUBO [ 19661, GENZEL,MARTINand KREIBIG[ 19751, RUPPINand YATOM[ 19761, HACHE[ 19881); in the large sphere limit, zeff reduces to T~ = T,. The imaginary part of the interband term, on the other hand, can be written as

where P and J ( w ) are, respectively, an average matrix element of the momentum operator and the joint density of states between the d and s-p bands. The third-order susceptibility ~ ( ~ ’-(o, 0w) , of a metal crystallite has a far more complex structure and diverse origins that have been discussed in detail by HACHE,RICARDand FLYTZANIS[1986], HACHE [1988], and HACHE, RICARD,FLYTZANISand KREIBIG[ 19881. The main results pertinent to the discussion are given here. They have shown that ~ ( ~ ’-( o, 0a) , can be separated in three independent contributions and can be written

The first and second contributions on the right-hand side result from the same coherent transitions as those in the linear susceptibility, the intraband and interband ones, respectively. The third term is an incoherent contribution that results from the modification of the populations of the electron states (HACHE, RICARD,FLYTZANISand KREIBIG [ 19881) caused by the elevation of their temperature subsequent to the absorption of photons in the resonant process but before the heat is released to the lattice of the crystallite. This latter process

370


[V,§ 4

takes a few picoseconds to occur (SCHOENLEIN, LIN,FUJIMOTO and EESLEY [ 1987]), and the conduction electron system, because of its weak heat capacity, attains very high temperatures during this time lapse, whose duration has the same order of magnitude as the short light pulses used in the experiments. We disregard the normal thermal contribution due to the subsequent lattice heating for reasons indicated in 0 2. Keeping (HACHE,RICARD and FLYTZANIS[1986], HACHE[1988]) the dominant resonant terms in the expressions of each of the three contributions in eq. (4.4), the first contribution x:2)ra in eq. (4.4), which results from electric dipole transitions between the quantum-confined states of the s-p conduction band, is approximately given by (HACHE,RICARD,FLYTZANIS and KREIBIC ~9881)

where T , and T2 are the energy lifetime and the dephasing time, respectively, and a, is given by a, = T2(2E,/m)”2g,(v)/[g2(v)+ g3(v)1 9

(4.6)

-

where g2(v ) and g3( v), like g,( v ) in eq. (4.5), are numbers of order 1; eq. (4.5) is negative imaginary and size dependent, i.e., ~{2),~ l / u 3for a < a,. Actually, this term vanishes rigorously for the bulk metal, since it results from electric dipole transitions. The second contribution &!er in eq. (4.4) results from electric dipole transitions between states of the d-valence band and quantum-confined electron states of the s-p conduction band and, using the same assumptions as before in deriving eq. (4.3), is approximately given by

where A , is an angular form factor ( % $) and Ti and T ; are the energy lifetime and the dephasing time, respectively, for the interband transitions, all unknown and not related to the T I and T2 relevant to the intraband transitions. x{:)~, is also negative imaginary but size independent, since the d-electrons are unafTected by the quantum confinement. Finally, the third term xi:) results from the modification of the Fermi-Dirac distribution (HACHE[ 19881, HACHE, RICARD,FLYTZANISand KREIBIG [ 1988]), since the electron temperature is elevated subsequent to the supply of

v. I 41

PROPERTIES OF METAL COMPOSITES

371

heat through the absorption process. This leads to a modification of E, which can be identified as the hot-electron contribution and is approximately given by (HACHE,RICARD,FLYTZANISand KREIBIG[ 19881)

where zo is the electron cooling time, yT is the specific heat of the conduction electrons, and E& and .!$ are the imaginary parts of the Drude and the interband contributions to E of the free electrons. The important point to notice here is that eq. (4.8) is positive imaginary and size independent. The linear optical properties of noble-metal composites, in particular gold and silver composites, have been extensively studied (HUGHES and JAIN [ 19791, PERENBOOM, WYDERand MEIER[ 19811, HALPERIN[ 1986]), both theoretically and experimentally confirming the main trends of the previous discussion. As an example, fig. 7 shows the linear absorption spectrum taken in gold-doped silicate glasses for different average sizes of the gold particles. We observe that this spectrum is dominated by the broad surface plasmon resonance as predicted by eqs. (3.8) and (3.9) and its width depends on the particle size in accordance with relation (3.53). We shall not dwell further on the linear optical properties, which have been amply covered in the literature (HUGHES and JAIN [1979], PERENBOOM,WYDER and MEIER [1981], HALPERIN[ 19861) but proceed to the nonlinear ones, in particular the optical Kerr effect related to the coefficient n2, which was measured for the first time by RICARD,ROUSSIGNOLand FLYTZANIS[ 19851. Subsequent studies confirmed the role played by the local field and clarified the actual mechanism of the nonlinearity. The optical Kerr coefficient was measured for gold and silver colloids (RICARD,ROUSSIGNOLand FLYTZANIS[ 19851, BLOEMER, HAUS and ASHLEY[ 19901) and for gold-doped glasses (HACHE,RICARDand FLYTZANIS [ 19861, HACHE,RICARD,FLYTZANIS and KREIBIG[ 1988]), using the optical PILIPETSKYand phase-conjugation technique (FISHER[ 19831, ZELDOVICH, SHKUNOV[ 19851) in the degenerate four-wave mixing configuration. The temporal behavior of the nonlinear response was obtained by measuring the normalized conjugated signal as a function of the backward pump pulse delay and allowed to assign the non-linearity to the electrons of the gold spheres. Since the conjugate beam intensity is proportional to lPNLI 2, from eq. (3.18) we expect an enhancement factor Ifi(w) I 8, which implies an eightfold resonance at w,; this too was confirmed by measuring the conjugate beam intensity

372


[V,8 4

as a function of frequency. Finally, by studying the anisotropy, phase, and size dependence of j ( 3 ) ( w ,- w, w), it was inferred that the hot-electron contribution xi:) is the dominant one (HACHE,RICARD,FLYTZANISand KREIBIG [ 19881). The temporal behavior of the normalized I ~ ( ~ ' ( w w, , w ) I for a gold colloid of mean crystallite radius x 50 A and optical density 0.5 at the absorption peak is shown in fig. 13 (HACHE,RICARDand FLYTZANIS [ 19861, HACHE[ 19881). The source was a Q-switched mode-locked Nd: phosphate glass laser, which together with a pulse switch and amplifier delivered a single pulse of wavelength 1.054 pm, pulse duration 5 ps, and energy 1 mJ with a repetition rate of 1 Hz; this pulse was frequency doubled. As we see in fig. 13, after correction for the pulse transit time in the sample, the temporal response is the same as that of the pulse, which implies a fast nonlinear mechanism of electronic origin. From the expression of the absorption coefficient eq. (3.9), close to the surface plasmon resonance w, where E" is roughly constant, one obtains a(w) If,(w)12,so that the phase-conjugated beam intensity which is x If,(w)l should scale as the absorption coefficient to the fourth power when w is tuned across the surface plasmon resonance w,. The experimental confir[ 19861) of this relation was obtained mation (HACHE,RICARDand FLYTZANIS on a gold colloid and on gold ruby glass, whose absorption spectra peak at 520 and 530 nm, respectively, and the results are shown in fig. 14. In these measurements a Q-switched and mode-locked Nd: YAG laser, a pulse-switch, an amplifier, and a frequency doubler were used to deliver 28 ps pulse duration beams at four wavelengths: I = 532 nm and its Raman-shifted I = 5616,5730, and 6302 A in benzene, nitrobenzene, and ethanol, respectively. The resonant enhancement is clearly seen in fig. 14 and is in good agreement with the enhancement factor calculated from the local-field correction presented in $ 3.2.2. Although the magnitude of the three contributions in eq. (4.4) can, in principle, be calculated from eqs. (4.4), (4.7) and (4.8), in practice the uncertainties in the values of the different physical parameters do not provide sufficient accuracy to do so. However, their phase, anisotropy, and dependence on the crystallite radius are distinctly different, which can be experimentally investigated (HACHE,RICARD,FLYTZANIS and KREIBIG[ 19881). The size dependence of x ( ' ) for gold particle suspensions in silicate glass was measured for 11 different samples with CS, as reference, using the same laser source as before at sufficiently low intensities to avoid saturation and at wavelengths I = 532 and 527 nm, which are close to the surface plasmon resonance at 530 nm. The nonlinear susceptibility f 3 ) of the gold particles turned out to

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